diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzeqtg" "b/data_all_eng_slimpj/shuffled/split2/finalzzeqtg" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzeqtg" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\nExtracting phenomenological predictions from string theory is a subtle task.\nChief among the complications is the question of \nfinding a suitable vacuum.\nWithout solving this problem, one is limited to making generic statements that might hold across\nbroad classes of string theories.\nBut even within the context of specific string models with certain favorable characteristics,\nmost attempts at extracting the corresponding phenomenological predictions follow a common path. First, one tallies the massless states that arise in such models. Then, one constructs a field-theoretic Lagrangian which describes the dynamics of these states. Finally, one proceeds to analyze this Lagrangian using all of the regular tools of quantum field theory\nwithout further regard for the origins of these states within string theory. \n\n\nAlthough such a treatment may be sufficient for certain purposes, calculations performed in this manner have a serious shortcoming: \nby disregarding the infinite towers of string states that necessarily accompany these low-lying modes within the \nfull string theory, \nsuch calculations implicitly disregard many of the underlying string symmetries that ultimately \nendow string theory with a plethora of remarkable properties that transcend our field-theoretic expectations. \nAt first glance, it may seem that these extra towers of states cannot play an important role for \nlow-energy physics because these states typically have masses which are set by \nthe string scale (generically assumed near the Planck scale) or by the scales associated with the compactification geometry. \nFor this reason it would seem that these heavy states can legitimately be integrated out of the theory, thereby \njustifying a treatment based on a Lagrangian description of the low-lying modes alone, along with possible \nhigher-order operators suppressed by powers of these heavier scales. However, it is \ndifficult to justify integrating out {\\it infinite}\\\/ towers of states, \nmuch less towers whose state degeneracies at each mass level grow {\\it exponentially}\\\/ with mass. \nYet this is precisely the situation we face in string theory.\nIndeed, these infinite towers of states particularly affect \nthose operators (such as those associated with the Higgs mass \nand the cosmological constant) which have positive\ndimension and are therefore sensitive to all mass scales in the theory.\n\nMany of the string symmetries that rely on these infinite towers of states go beyond what can be \nincorporated within the framework of an effective field theory (EFT).~ \nFor example, strong\/weak coupling duality relations intrinsically rely on the presence of \nthe full towers of string states, both perturbative and non-perturbative. \nBut there also exist stringy symmetries that operate purely within the perturbative weak-coupling regime. \nA prime example of this is T-duality, under which the physics of closed strings compactified \non small compactification volumes is indistinguishable from the physics associated with strings \ncompactified on large compactification volumes. This sort of \nequivalence between ultraviolet (UV) and \ninfrared (IR) physics cannot be incorporated within an\nEFT-based approach in which we integrate out heavy states while treating light states as dynamical.\n\n\nBoth strong\/weak coupling duality and T-duality are spacetime symmetries.\nAs such, like all spacetime physics, they are merely the {\\it consequences}\\\/ of an underlying string theory.\nBut closed string theories have another symmetry of this sort which is even more fundamental and which \nmust be imposed for consistency \ndirectly on the worldsheet.\nThis is worldsheet modular invariance, which will be the focus of this\npaper. \nWorldsheet modular invariance is crucial\nsince it lies at the heart of many of the finiteness properties for which string theory is famous.\nMoreover, since modular invariance is an exact symmetry of all perturbative closed-string vacua, it provides \ntight constraints on the spectrum of string states at all mass scales as well as on their interactions. \nIndeed, this symmetry is the ultimate ``UV-IR mixer'',\noperating over all scales and enforcing a delicate balancing between low-scale and high-scale physics. \nThere is no sense in which its breaking can be confined to low energies, and likewise\nthere is no sense in which it can be broken by a small amount. \nAs an exact symmetry governing string dynamics,\nworldsheet modular invariance is preserved even as the theory passes through phase transitions such as the \nStandard-Model electroweak or QCD phase transitions, as might occur \nunder cosmological evolution. Indeed, any shifts in the low-energy degrees of freedom induced by such phase transitions \nare automatically accompanied by corresponding shifts in the high-scale theory such that modular invariance is maintained \nand finiteness is preserved. Yet this entire structure is missed if we integrate out the heavy states and \nconcentrate on the light states alone. \n\nWhile certain phenomenological questions are not likely to depend on such symmetries, this need not always be the case.\nFor example, these symmetries are likely to be critical for addressing fundamental questions connected with finiteness and\/or \nthe stability of (or even the coexistence of) \ndifferent scales under radiative corrections. Chief among these questions are hierarchy problems, \nwhich provide clues as to the UV theory \nand its potential connections to IR physics. Indeed, two of the most pressing mysteries in physics are the hierarchy problems \nassociated with the cosmological constant and with the masses of scalar fields such as the Higgs field.\nHowever, integrating out the heavy string states eliminates all of the stringy physics that may \nprovide alternative ways of addressing such problems. \nThe lesson, then, is clear: If we are to take string theory literally as a theory of physics, \nthen we should perform our calculations within the full framework of string theory, incorporating all of \nthe relevant symmetries and infinite towers of states that string theory provides.\n\nWith this goal in mind, \nwe begin this paper by \nestablishing a fully string-theoretic framework for calculating\none-loop Higgs masses directly from first principles in perturbative closed string theories.\nThis is the subject of Sect.~\\ref{sec2}.~\nOur framework will make no assumptions other than worldsheet modular invariance \nand will therefore be applicable to\nall closed strings, regardless of the specific string \nconstruction utilized. \nOur results will thus have a generality that extends beyond individual string models.\nAs we shall see, this framework operates independently of spacetime supersymmetry, and can \nbe employed even when spacetime supersymmetry is broken \n(or even when the string model has no spacetime supersymmetry to begin with at any scale).\nLikewise, our framework can be utilized for all scalar Higgs fields, regardless of the particular gauge symmetries they break. \nThis therefore includes the Higgs field responsible for electroweak symmetry breaking in the Standard Model. \n\nOne of the central results emerging from our framework is a relationship \nbetween the Higgs mass and the one-loop cosmological constant. \nThis connection arises as the result of a gravitational modular anomaly,\nand is thus generic for all closed string theories.\nThis then provides a string-theoretic connection between the two fundamental quantities which are \nknown to suffer from hierarchy problems in the absence of spacetime supersymmetry. \nFrom the perspective of ordinary quantum field theory, such a relation \nbetween the Higgs mass and the cosmological constant would be entirely unexpected. \nIndeed, quantum field theories are insensitive to the zero of energy.\nString theory, by contrast, unifies\ngauge theories with gravity.\nThus, it is only within a string context that such a relation could ever arise. \nAs we shall see, this relationship does not require supersymmetry in any form. \nIt holds to one-loop order, but its direct emergence as \nthe result of a fundamental string symmetry leads us to believe that \nit actually extends more generally. We stress that it is not the purpose of this paper to \nactually solve either of these hierarchy problems\n(although we shall return to this issue briefly in Sect.~\\ref{sec:Conclusions}). \nHowever, we now see that these two hierarchies are connected in a deep way within a string context.\n\nAs we shall find, the Higgs mass receives contributions from all of the states \nthroughout the string spectrum which couple to the Higgs in specified ways.\nThis includes the physical (level-matched) string states as well as the unphysical (non-level-matched) string states.\nDepending on the string model in question, \nwe shall also find that our expression for the total Higgs mass can be divergent; ultimately this will depend on the charges carried by the massless states.\nAccordingly,\nwe shall then proceed to develop a set of regulators \nwhich can tame the Higgs-mass divergences while at the same time allowing\nus to express the Higgs mass as a weighted supertrace over only the physical string states.\nDeveloping these regulators is the subject of Sect.~\\ref{sec3}.~\nTo do this, \nwe shall begin by reviewing prior results in the mathematics literature which will form the basis for\nour work.\nBuilding on these results, we will then proceed to develop\na set of regulators which are completely general, \nwhich preserve modular invariance, and which can be used in a wide variety of contexts even beyond\ntheir role in regulating the Higgs mass.\n\nIn Sect.~\\ref{sec4}, we shall then use these modular-invariant regulators in order to \nrecast our results for the Higgs mass in a form that is closer to what we might expect in field theory.\nThis will also allow us to develop\nan understanding of how the Higgs mass ``runs'' in string theory\nand to develop a physical ``renormalization'' prescription that can operate at all scales.\nTowards this end, we begin in Sect.~\\ref{UVIRequivalence}\nwith a general discussion of how (and to what extent) one can meaningfully extract an effective field theory \nfrom UV\/IR-mixed theories such as modular-invariant string theories.\nThis issue is surprisingly subtle, since modular invariance relates UV and IR divergences to each other\nwhile at the same time softening both. \nFor example, we shall demonstrate that while the Higgs mass is quadratically divergent\nin field theory, modular invariance renders the Higgs mass at most logarithmically divergent in string theory.\nWe shall then apply our regulators from Sect.~\\ref{sec3} to our Higgs-mass\nresults in Sect.~\\ref{sec2}\nand thereby demonstrate how the Higgs mass ``runs''\nas a function of an energy scale $\\mu$.\nThe results of our analysis are highlighted in Fig.~\\ref{anatomy}, which \nnot only exhibits features which might be expected in \nan ordinary effective field theory but also includes features which clearly transcend traditional quantum field-theoretic\nexpectations. The latter include the existence of a ``dual'' infrared region at high energy scales as well as an invariance under \nan intriguing ``scale duality'' transformation {\\mbox{$\\mu\\to M_s^2\/\\mu$}}, where $M_s$ denotes the string scale.\nThis scale-inversion duality symmetry in turn implies the existence of a fundamental limit on the extent to which a modular-invariant theory such as string\ntheory can exhibit UV-like behavior.\n\nAll of our results in Sects.~\\ref{sec2} through \\ref{sec4} are formulated in a fashion that assumes that our \nmodular-invariant string theories can be described through charge lattices.\nHowever, it turns out that our results can be recast in a completely general fashion\nthat does not require the existence of a charge lattice.\nThis is the subject of Sect.~\\ref{sec5}.~\nMoreover, we shall find that this reformulation has an added benefit, allowing us to extract a modular-invariant stringy effective\npotential for the Higgs from which the Higgs mass can be obtained through a modular-covariant double derivative with respect\nto fluctuations of the Higgs field. \nThis potential therefore sits at the core of our string-theoretic calculations\nand allows us to understand not only the behavior of the Higgs mass but also the overall stability of the string theory\nin a very compact form.\nIndeed, in some regions this potential exhibits explicitly string-theoretic behavior. However, in other regions,\nthis potential --- despite its string-theoretic origins --- exhibits a number of features which are\nreminiscent of the traditional Coleman-Weinberg Higgs potential.\n\nFinally, in Sect.~\\ref{sec:Conclusions},\nwe provide an overall discussion of our results and outline some possibilities for future research. \nWe also provide an additional bird's-eye perspective on the \nmanner in which modular invariance induces UV\/IR mixing \nand the reason why the passage from a full string-theoretic result to an EFT description \nnecessarily breaks the modular symmetry. \nWe will also discuss some of the possible implications of our results for addressing\nthe hierarchy problems associated with the cosmological constant and the Higgs mass.\nThis paper also has two Appendices which provide the details of calculations \nwhose results are quoted in Sects.~\\ref{chargeinsertions} and \\ref{Lambdasect} respectively.\n\nOur overarching goal in this paper is to provide a fully string-theoretic framework for the calculation of the Higgs\nmass --- a framework in which modular invariance is baked into the formalism from the very beginning.\nOur results can therefore potentially serve as the launching point for a rigorous investigation of \nthe gauge hierarchy problem in string theory.\nHowever, our methods are quite general and can easily be adapted to other quantities of phenomenological interest,\nincluding not only the masses of all particles in the theory but\nalso the gauge couplings, quartic couplings, and indeed the couplings associated with all allowed interactions.\n\nAs already noted, much of the inspiration for this work stems from our conviction that \nit is not an accident or phenomenological irrelevancy \nthat string theories contain not only low-lying modes but also infinite towers of massive states.\nTogether, all of these states conspire to enforce many of the unique symmetries for which\nstring theory is famous, and thus \ntheir effects are an intrinsic part of the predictions of string theory.\nIn this spirit, one might even view our work as a continuation of the line\n originally begun in the classic 1987 paper of Kaplunovsky~\\cite{Kaplunovsky:1987rp}\nwhich established a framework for calculating string threshold corrections in which \nthe contributions of the infinite towers of string states were included.\nIndeed, as discussed in Sect.~\\ref{higgsmin},\nsome of our results for the Higgs mass \neven resemble results obtained in Ref.~\\cite{Kaplunovsky:1987rp} for threshold corrections.\nOne chief difference in our work,\nhowever, is our insistence on maintaining modular invariance at all steps \nin the calculation, {\\it including the regulators}\\\/,\nespecially when seeking to understand the behavior of dimensionful operators.\nIt is this extra ingredient which is critical for ensuring consistency with the underlying\nstring symmetries, and which allows us to probe the unique effects \nof such symmetries (such as those induced by UV\/IR mixing) in a rigorous manner.\n\n\n\n\\section{Modular invariance and the Higgs mass: A general framework\\label{sec2}}\n\nIn this section we develop a framework for calculating the Higgs mass in any four-dimensional\nmodular-invariant string theory.\nOur framework incorporates modular invariance in a fundamental way, and ultimately leads\nto a completely general expression for one-loop Higgs mass.\nOur results can therefore easily be applied to any \nfour-dimensional \nclosed-string model.\nThroughout most of this paper, our analysis will focus on heterotic string models and will proceed under the assumption \nthat the string model in question can be described through a corresponding charge lattice.\nAs we shall see, the existence of a charge lattice provides a very direct way of performing\nour calculations and illustrating our main points.\nHowever, as we shall discuss in Sect.~\\ref{sec5}, our results \nare ultimately more general than this, and apply even for closed-string models\nthat transcend a specific charge-lattice construction.\n\n\n\\subsection{Preliminaries: String partition functions, charge lattices, and modular invariance}\n\nWe begin by reviewing basic facts about string partition\nfunctions, charge lattices, and modular invariance,\nestablishing our notation and normalizations along the way.\nThe one-loop partition function for any closed heterotic string in four spacetime dimensions is\na statistics-weighted trace over the Fock space of closed-string states, and thus takes the general form\n\\begin{equation}\n {\\cal Z} (\\tau,{\\overline{\\tau}}) ~\\equiv~ \\tau_2^{-1} \\frac{1}{\\overline{\\eta}^{12} \\eta^{24}} \n \\, \\sum_{m,n} \\, (-1)^F \\, {\\overline{q}}^m q^n~.\n\\label{Zform}\n\\end{equation}\nHere $\\tau$ is the one-loop (torus) modular parameter, \n{\\mbox{$\\tau_2\\equiv {\\rm Im}\\,\\tau$}}, \n{\\mbox{$q\\equiv \\exp(2\\pi i\\tau)$}}, \n$F$ is the spacetime fermion number, \nand the Dedekind eta-function is {\\mbox{$\\eta(\\tau)\\equiv q^{1\/24}\\prod_{n=1}^\\infty (1-q^n)$}}.\nIn this expression, the $\\overline{\\eta}$- and $\\eta$-functions represent the \ncontributions from the string oscillator states [which include\nappropriate right- and left-moving vacuum energies $(-1\/2,-1)$ respectively],\nwhile the $(m,n)$ sum tallies the contributions from\nthe Kaluza-Klein (KK) and winding excitations of the heterotic-string worldsheet fields --- \nexcitations which result from the compactification\nof the heterotic string to four dimensions from its critical spacetime dimensions ($=10$ for the \nright-movers and $26$ for the left-movers), \nwith $(m,n)$ representing the corresponding right- and left-moving worldsheet energies.\nThese KK\/winding contributions can be written in terms of the charge vectors {\\mbox{${\\bf Q}\\equiv \\lbrace {\\bf Q}_R,{\\bf Q}_L\\rbrace $}} of \na $(10,22)$-dimensional Lorentzian charge lattice ---\nor equivalently the KK\/winding momenta $\\lbrace {\\bf p}_R,{\\bf p}_L\\rbrace$ of a corresponding momentum lattice of the\nsame dimensionality ---\nvia\n\\begin{equation}\n m= {{\\bf Q}_R^2\\over 2} = {\\alpha' {\\bf p}_R^2\\over 2} ~,\n ~~~~~ n= {{\\bf Q}_L^2\\over 2} = {\\alpha' {\\bf p}_L^2\\over 2} ~,\n\\label{mnQ}\n\\end{equation}\nwhere {\\mbox{$\\alpha'\\equiv 1\/M_s^2$}} with $M_s$ denoting the string scale.\nThus the partition function in Eq.~(\\ref{Zform}) can be written as a sum over charge vectors ${\\bf Q}_L, {\\bf Q}_R$:\n\\begin{equation}\n {\\cal Z}(\\tau,{\\overline{\\tau}}) ~=~ \\tau_2^{-1} {1\\over \\overline{\\eta}^{12} \\eta^{24}} \\, \\sum_{{\\bf Q}_L,{\\bf Q}_R} \n (-1)^F {\\overline{q}}^{{\\bf Q}_R^2\/2} \n q^{{\\bf Q}_L^2\/2}~.\n\\label{preZQdef}\n\\end{equation}\n\nIn general, the spacetime mass $M$ of the resulting string state is given by \n{\\mbox{$\\alpha' M^2 = 2(m+n) + 2(\\Delta_L+\\Delta_R) + 2(a_L+a_R)$}}\nwhere $\\Delta_{R,L}$ are the contributions from the oscillator excitations and {\\mbox{$(a_R,a_L)=(-1\/2, -1)$}} are the corresponding\nvacuum energies.\nIdentifying individual left- and right-moving contributions\nto $M^2$ through the convention \n\\begin{equation}\n M^2 ~=~ {1\\over 2} (M_L^2 + M_R^2)\n\\label{masssum}\n\\end{equation}\nthen yields {\\mbox{$\\alpha' M_R^2 = 4 (m+ \\Delta_R +a_R) $}} \nand {\\mbox{$\\alpha' M_L^2 = 4 (n+ \\Delta_L +a_L) $}}.\nWriting these masses in terms of the lattice charge vectors then yields\n\\begin{eqnarray}\n {\\alpha'\\over 2} M_R^2 ~&=&~ {\\bf Q}_R^2 + 2 \\Delta_R + 2 a_R~,\\nonumber\\\\\n {\\alpha'\\over 2} M_L^2 ~&=&~ {\\bf Q}_L^2 + 2 \\Delta_L + 2 a_L~.\n\\label{eq:massRL}\n\\end{eqnarray}\nStates are level-matched (physical) if {\\mbox{$M_R^2 = M_L^2$}} and unphysical otherwise.\nIndeed, with these conventions, \ngauge bosons in the left-moving non-Cartan algebra are massless, with {\\mbox{${\\bf Q}_L^2=2$}} and {\\mbox{$\\Delta_L=0$}},\nwhile those in the left-moving Cartan algebra are massless, with {\\mbox{${\\bf Q}_L^2 =0$}} and {\\mbox{$\\Delta_L=1$}}.\n(Indeed, such results apply to all left-moving simply-laced gauge groups with level-one affine realizations; more\ncomplicated situations, such as necessarily arise for the right-moving gauge groups, are discussed in Ref.~\\cite{Dienes:1996yh}.)\nNote that the CPT conjugate \nof any state with charge vector $\\lbrace {\\bf Q}_R,{\\bf Q}_L\\rbrace$ has charge vector \n$-\\lbrace {\\bf Q}_R,{\\bf Q}_L\\rbrace$.\nThus CPT invariance requires that all states in the string spectrum come\nin $\\pm\\lbrace {\\bf Q}_R,{\\bf Q}_L\\rbrace$ pairs.\nBy contrast, \nsince the right-moving gauge group is necessarily non-chiral as a result of superconformality constraints,\nthe chiral conjugate of \nany state with charge vector $\\lbrace {\\bf Q}_R,{\\bf Q}_L\\rbrace$ has charge vector \n$\\lbrace {\\bf Q}_R,-{\\bf Q}_L\\rbrace$.\n \nOne important general property of the partition functions in Eq.~(\\ref{Zform}) ---\nand indeed the partition functions of {\\it all}\\\/ closed strings in any spacetime dimension --- \nis that they must be {\\it modular invariant}\\\/, {\\it i.e.}\\\/, invariant\nunder all transformations of the form\n{\\mbox{$\\tau \\to (a\\tau+b)\/(c\\tau+d)$}} where {\\mbox{$a,b,c,d\\in \\mathbb{Z}$}} and {\\mbox{$ad-bc=1$}}\n(with the same transformation for ${\\overline{\\tau}}$).\nModular invariance is thus an exact symmetry underpinning all heterotic strings,\nand in this paper we shall be exploring its consequences for the masses of the Higgs fields in such theories.\nFor these purposes, it will be important to understand the manner in which these\npartition functions achieve their modular invariance.\nIn general, the partition functions for heterotic strings in four dimensions\ncan be rewritten in the form\n\\begin{equation}\n {\\cal Z} (\\tau,{\\overline{\\tau}}) ~\\equiv~ \\tau_2^{-1} \\frac{1}{\\overline{\\eta}^{12} \\eta^{24}}\\, \n \\sum_{{\\overline{\\imath}},i} N_{{\\overline{\\imath}} i} ~\\overline{g_{\\overline{\\imath}} (\\tau)} f_i(\\tau)~\n\\label{partfZ}\n\\end{equation}\nwhere each $({\\overline{\\imath}},i)$ term represents the contribution from a different sector of the theory\nand where the left-moving holomorphic $f_i$ functions (and the corresponding right-moving antiholomorphic\n$g_{\\overline{\\imath}}$ functions) \ntransform covariantly under modular transformations according to\nrelations of the form\n\\begin{equation} \n f\\left( \\frac{a\\tau+b}{c\\tau+d}\\right) ~\\sim~ (c\\tau+d)^{k} \\,f(\\tau)\n\\label{fis}\n\\end{equation}\nwhere $k$ is the so-called modular weight of the $f_i$ functions\n(with an analogous weight ${\\overline{k}}$ for the $g_{\\overline{\\imath}}$ functions)\nand where the $\\sim$ notation allows for the possibility of overall \n$\\tau$-independent phases which\nwill play no future role in our arguments.\nWe likewise have \n\\begin{equation}\n \\eta\\left( \\frac{a\\tau+b}{c\\tau+d}\\right) ~\\sim~ (c\\tau+d)^{1\/2} \\, \\eta(\\tau)~.\n\\label{modfis}\n\\end{equation}\nThus, since {\\mbox{$\\tau_2\\to \\tau_2\/|c\\tau+d|^2$}} as {\\mbox{$\\tau\\to (a\\tau+b)\/(c\\tau+d)$}}, we immediately\nsee that modular invariance of the entire partition function in Eq.~(\\ref{partfZ}) requires not only that\nthe $N_{{\\overline{\\imath}} i}$ coefficients in Eq.~(\\ref{partfZ}) be chosen correctly but also that\n{\\mbox{$k=11$}} and {\\mbox{${\\overline{k}} =5$}}. \nIn general, \nfor strings realizable through free-field constructions, \nthese $f_i$ and $g_{\\overline{\\imath}}$ functions produce the lattice sum in Eq.~(\\ref{preZQdef}) because\nthey can be written in the factorized forms \n\\begin{equation}\n f_i~\\sim~ \\prod_{\\ell=1}^{22} \\vartheta\n \\begin{bmatrix} \\alpha_\\ell^{(i)} \\\\ \\beta_\\ell^{(i)} \\end{bmatrix}~,~~~~~\n g_{\\overline{\\imath}}~\\sim~ \\prod_{\\ell=1}^{10} \\vartheta\n \\begin{bmatrix} \\alpha_\\ell^{({\\overline{\\imath}})} \\\\ \\beta_\\ell^{({\\overline{\\imath}})} \\end{bmatrix}~,~~\n\\label{partft}\n\\end{equation}\nwhere each $\\vartheta$-function factor \nis the trace over the $\\ell^{\\rm th}$ direction $Q_\\ell$ of the charge lattice:\n\\begin{equation}\n \\vartheta_\\ell(\\tau) \n ~\\equiv~ \\vartheta \\begin{bmatrix} \\alpha_\\ell \\\\ \\beta_\\ell \\end{bmatrix}(\\tau)\n ~\\equiv~\n \\sum_{Q_\\ell\\in \\mathbb{Z}+\\alpha_\\ell} e^{2\\pi i \\beta_\\ell Q_\\ell} \\,q^{Q_\\ell^2\/2}~.~~\n\\label{thetadef}\n\\end{equation}\nIndeed, the $\\vartheta_\\ell$-functions transform \nunder modular transformations as in Eq.~(\\ref{fis}),\nwith modular weight $1\/2$.\nThe modular invariance of the underlying string theory\nthen ensures that there exists a special\n$(10,22)$-dimensional ``spin-statistics vector'' ${\\bf S}$ \nsuch that we may identify \nthe spacetime fermion number $F$ within Eq.~(\\ref{Zform})\nas {\\mbox{${F = 2 {\\bf Q}\\cdot {\\bf S}}$}}~(mod~2)\nfor any state with charge ${\\bf Q}$,\nwhere the dot notation `$\\cdot$' signifies the Lorentzian \n(left-moving minus right-moving) dot product.\nModular invariance \nalso implies \nthat the shifted charges ${\\bf Q} -{\\bf S}$ \nassociated with the allowed string states\ntogether form a Lorentzian lattice which is both odd and self-dual.\nIt is with this understanding that \nwe refer to the charges ${\\bf Q}$ themselves as populating a ``lattice''.\nIndeed, it is the self-duality property of the \nshifted charge lattice $\\lbrace {\\bf Q}-{\\bf S}\\rbrace$ which guarantees \nthat the $f_i$ and $g_{\\overline{\\imath}}$ functions in Eq.~(\\ref{fis})\ntransform covariantly \nunder the modular group,\nas in Eq.~(\\ref{fis}).\n\nFor later purposes, we simply observe that the\ngeneral structure given in Eq.~(\\ref{partfZ}) is \ntypical of the modular-invariant quantities that arise\nas heterotic-string Fock-space traces.\nIndeed, a general quantity of the form\n\\begin{equation}\n \\tau_2^\\kappa~\n \\frac{1}{\\overline{\\eta}^{12} \\eta^{24}}\\, \n \\sum_{{\\overline{\\imath}},i} N_{{\\overline{\\imath}} i} ~\\overline{g_{\\overline{\\imath}} (\\tau)} f_i(\\tau)~\n\\label{genexp}\n\\end{equation}\ncannot be modular invariant unless \nthe $N_{{\\overline{\\imath}} i}$ are chosen correctly\nand the corresponding $f_i$ and $g_{\\overline{\\imath}}$ functions transform\nas in Eq.~(\\ref{fis})\nwith \n\\begin{equation}\n k-12 ~=~ {\\overline{k}}-6 ~=~ \\kappa~.\n\\label{genexp2}\n\\end{equation}\nWhile {\\mbox{$\\kappa = -1$}} for the partition \nfunctions of four-dimensional heterotic strings, as\ndescribed above,\nwe shall see that other important Fock-space traces can have different\nvalues of $\\kappa$.\nFor example, the partition functions of heterotic strings\nin $D$ spacetime dimensions have {\\mbox{$\\kappa = 1-D\/2$}}, with\ncorresponding changes to the dimensionalities of \ntheir associated charge lattices.\n\n\n\\subsection{Higgsing and charge-lattice deformations} \\label{latdeform}\n\n\nIn general, different string models exhibit different spectra and thus have different charge lattices.\nHowever, Higgsing a theory changes its spectrum in certain dramatic ways, \nsuch as by giving mass to formerly massless gauge bosons and thereby breaking the associated gauge symmetries.\nThus, in string theory, Higgsing \ncan ultimately be viewed as a process of \ntransforming the charge \nlattice from one configuration to another. \n\nOf course, \nmodular invariance must \nbe maintained throughout the Higgsing process. \nIndeed, it is only in this way that\nwe can regard the Higgsing process as a fully string-theoretic \noperation that \nshifts the string vacuum state within the space of self-consistent string vacua.\nHowever, modular invariance then implies that\nthe charge-lattice transformations induced by Higgsing are not arbitrary. \nInstead, they must preserve those charge-lattice properties, \nas described above, which guarantee the modular invariance of the theory.\n\nThis in turn tells us that the process of Higgsing is likely to be far more complicated\nin string theory than it is in ordinary quantum field theory.\nIn general, \nthe charge lattice receives contributions from all sectors of the theory,\nand modular transformations mix these different contributions in highly non-trivial ways.\nThus the process of Higgsing a given gauge symmetry within a given sector of a string\nmodel generally involves not only the physics associated with that gauge symmetry \nbut also the physics of all of the {\\it other}\\\/ sectors of the theory as well, both twisted and untwisted,\nand the properties of the {\\it other}\\\/ gauge symmetries, including gravity, that might also be present \nin the string model --- even if these other gauge symmetries are apparently completely disjoint from \nthe symmetry being Higgsed. \nWe shall see explicit examples of this below.\nMoreover, in string theory the dynamics of the Higgs VEV --- and indeed the dynamics of all string moduli ---\nis generally governed by an effective potential \nwhich is nothing but the vacuum energy of the theory, expressed as a function of this VEV.~\nThus the overall dynamics associated with Higgsing can be rather subtle: \nthe Higgs VEV determines the deformations of the charge lattice, and these deformations\nalter the vacuum energy which in turn determines the VEV. \n\n\n\nIn this paper, our goal is to calculate the mass of the physical Higgs scalar field that emerges \nin the Higgsed phase ({\\it i.e.}\\\/, after the \ntheory has already been Higgsed). We shall therefore assume that our theory contains a scalar Higgs field\nwhich has already settled into the new minimum of its potential.\nThis will allow us to \nsidestep the (rather complex) model-dependent issue \nconcerning the manner in which the Higgsing itself occurs, and instead\nfocus on the {\\it perturbations}\\\/ of the field around this new minimum.\nIn this way we will be able to determine \nthe curvature of the scalar potential at this local minimum, and thereby obtain the corresponding Higgs mass. \n\nTo do this, we shall begin by \nexploring the manner in which a general charge lattice is deformed\nas we vary a scalar Higgs field away from the minimum of its potential.\nOur discussion will be completely general, and we shall defer to Sect.~\\ref{sec:EWHiggs} \nany assumptions that\nmight be specific to the particular Higgs field responsible for electroweak symmetry breaking.\nFor concreteness, we shall let\nour scalar field have\na value $\\langle \\phi \\rangle + \\phi$, where $\\langle \\phi\\rangle$ is the Higgs VEV \nat the minimum of its potential and where $\\phi$ describes the fluctuations away from this point.\nIf $\\lbrace {\\bf Q}_L,{\\bf Q}_R\\rbrace$ are the charge vectors associated with a\ngiven string state in the Higgsed phase ({\\it i.e.}\\\/, at the minimum of the potential, when {\\mbox{$\\phi=0$}}),\nthen turning on $\\phi$ corresponds to deforming these charge vectors. \nIn general, for {\\mbox{$\\phi\/\\langle \\phi\\rangle\\ll 1$}}, we shall parametrize these deformations\naccording to\n\\begin{eqnarray} \n {\\bf Q}_L &\\to& ~{\\bf Q}_L + \\sqrt{\\alpha'} \\phi {\\bf Q}_a + {1\\over 2} \\alpha' \\phi^2 {\\bf Q}_b~+~... ~~~~~ \\nonumber\\\\\n {\\bf Q}_R &\\to& ~{\\bf Q}_R + \\sqrt{\\alpha'} \\phi \\tilde {\\bf Q}_a + {1\\over 2} \\alpha' \\phi^2 \\tilde {\\bf Q}_b~+~...~,\n\\label{Qdeform}\n\\end{eqnarray}\nwhere ${\\bf Q}_a$, ${\\bf Q}_b$, $\\tilde{\\bf Q}_a$, and $\\tilde {\\bf Q}_b$ are deformation charge vectors of \ndimensionalities $22$, $22$, $10$, and $10$ respectively.\nIndeed, the forms of these vectors are closely correlated with the specific gauge symmetries\nbroken by the Higgsing process,\nand as such these vectors continue to \ngovern the fluctuations\nof the Higgs scalar around this Higgsed minimum.\n\n\nIn this paper, we shall keep our analysis as general as possible. \nAs such, we shall not make any specific assumptions regarding\nthe forms of these vectors.\nHowever, as discussed above, we know that the Higgsing process --- and even the fluctuations around the Higgsed minimum \nof the potential --- should not break modular invariance. In particular,\nthe corresponding charge-lattice deformations in Eq.~(\\ref{Qdeform}) should not \ndisturb level-matching. This means that the value of the difference ${\\bf Q}_L^2 - {\\bf Q}_R^2$ should not be \ndisturbed when $\\phi$ is taken to non-zero values, which in turn means that \nthis difference should be independent of $\\phi$.\nThis then constrains the choices for \nthe vectors ${\\bf Q}_a$, ${\\bf Q}_b$, $\\tilde{\\bf Q}_a$, and $\\tilde {\\bf Q}_b$.\n\nTo help simplify the notation, let us assemble a single 32-dimensional charge vector \n{\\mbox{${\\bf Q} \\equiv ({\\bf Q}_L,{\\bf Q}_R)^t$}} (where `$t$' signifies the transpose). \nRecalling that the dot notation `$\\cdot$' signifies \nthe Lorentzian (left-moving minus right-moving) contraction of vector indices,\nas appropriate for a Lorentzian charge lattice,\nwe therefore require that {\\mbox{${\\bf Q}^2 \\equiv {\\bf Q}^t\\cdot {\\bf Q}$}} be $\\phi$-independent for all $\\phi$.\nGiven the above shifts, we find that terms within {\\mbox{${\\bf Q}^t\\cdot {\\bf Q}$}} \nwhich are respectively linear and quadratic in $\\phi$ will cancel provided \n\\begin{eqnarray}\n ({\\bf Q}^t_a, \\tilde {\\bf Q}^t_a) \\cdot {\\bf Q} ~&=&~ 0~,~~~~~\\nonumber\\\\\n ({\\bf Q}_b^t, \\tilde {\\bf Q}^t_b) \\cdot {\\bf Q} + ({\\bf Q}_a^t, \\tilde {\\bf Q}_a^t ) \\cdot \n \\begin{pmatrix}\n {\\bf Q}_a\\\\\n \\tilde {\\bf Q}_a\n \\end{pmatrix}\n ~&=&~0~.\n\\label{levelmatching}\n\\end{eqnarray}\nThese are thus modular-invariance\nconstraints on the allowed choices for the shift vectors ${\\bf Q}_a$, ${\\bf Q}_b$, $\\tilde{\\bf Q}_a$, and $\\tilde {\\bf Q}_b$.\n\nWe can push these constraints one step further if we write these shift vectors in terms of $\\bf Q$ itself via relations of the form\n\\begin{equation}\n \\begin{pmatrix}\n {\\bf Q}_a\\\\\n \\tilde {\\bf Q}_a\n \\end{pmatrix}\n = {\\cal T} \\cdot {\\bf Q} ~,~~~~~\n \\begin{pmatrix}\n {\\bf Q}_b\\\\ \\tilde {\\bf Q}_b\n \\end{pmatrix}\n = {\\cal N} \\cdot {\\bf Q} ~\n\\label{shiftQ}\n\\end{equation}\nwhere ${\\cal T}$ and ${\\cal N}$ are {\\mbox{$(32\\times 32)$}}-dimensional matrices \nand where `$\\cdot$' retains its Lorentzian signature for the index contraction that underlies matrix multiplication. \nThe first constraint equation above then tell us that {\\mbox{${\\bf Q}^t \\cdot {\\cal T}\\cdot {\\bf Q}=0$}}, which implies that ${\\cal T}$ must be antisymmetric,\nwhile the second constraint equation tells us that {\\mbox{${\\bf Q}^t \\cdot ( {\\cal N}+ {\\cal T}^t\\cdot {\\cal T})\\cdot {\\bf Q} =0$}}, which implies that\n{\\mbox{${\\cal N}+ {\\cal T}^t \\cdot {\\cal T}$}} must also be antisymmetric. \nIt turns out that the precise value of {\\mbox{${\\cal N} + {\\cal T}^t \\cdot {\\cal T}$}} will have no bearing on the Higgs mass. We will therefore\nset it to zero (which is indeed antisymmetric), implying that {\\mbox{${\\cal N} = - {\\cal T}^t \\cdot {\\cal T}$}}.~ \nThus, while ${\\cal T}$ is antisymmetric, ${\\cal N}$ is symmetric.\nIndeed, if we write our ${\\cal T}$-matrix in terms of left- and right-moving submatrices ${\\cal T}_{ij}$ in the form\n\\begin{equation}\n {\\cal T} ~=~ \\begin{pmatrix}\n {\\cal T}_{11} & {\\cal T}_{12} \\\\\n {\\cal T}_{21} & {\\cal T}_{22} \n \\end{pmatrix}~,\n\\end{equation}\nthen we must have {\\mbox{${\\cal T}_{11}^t = -{\\cal T}_{11}$}}, {\\mbox{${\\cal T}_{22}^t = -{\\cal T}_{22}$}}, and {\\mbox{${\\cal T}_{12}^t = -{\\cal T}_{21}$}}.\nLikewise, we then find that\n\\begin{eqnarray}\n {\\cal N}_{11} ~&=&~ - {\\cal T}_{11}^t {\\cal T}_{11} + {\\cal T}_{21}^t {\\cal T}_{21}\\nonumber\\\\ \n {\\cal N}_{12} ~&=&~ - {\\cal T}_{11}^t {\\cal T}_{12} + {\\cal T}_{21}^t {\\cal T}_{22}\\nonumber\\\\ \n {\\cal N}_{21} ~&=&~ - {\\cal T}_{12}^t {\\cal T}_{11} + {\\cal T}_{22}^t {\\cal T}_{21}\\nonumber\\\\ \n {\\cal N}_{22} ~&=&~ - {\\cal T}_{12}^t {\\cal T}_{12} + {\\cal T}_{22}^t {\\cal T}_{22}~.\n\\label{NTrelations}\n\\end{eqnarray}\n \n\n\\subsection{Example: The Standard-Model Higgs\\label{sec:EWHiggs}}\n\nIn general, within any given string model, \nthe deformation vectors \n${\\bf Q}_a$, ${\\bf Q}_b$, $\\tilde{\\bf Q}_a$, and $\\tilde {\\bf Q}_b$ in Eq.~(\\ref{Qdeform}) \ndepend on the particular charge vector $({\\bf Q}_R,{\\bf Q}_L)$ being deformed.\nHowever the ${\\cal T}$- and ${\\cal N}$-matrices in Eq.~(\\ref{shiftQ}) are universal for all charge vectors within the model.\nIt is therefore these matrices which carry all of the relevant information concerning the response of the theory \nto fluctuations of the particular Higgs field under study.\nIn general, these matrices depend on how the gauge groups and corresponding Higgs field are embedded\nwithin the charge lattice.\nThus the precise forms of these matrices depend on the particular string model under study and the Higgs field\nto which it gives rise.\n\nTo illustrate this point, it may be helpful to consider the special case of the Standard-Model (SM) Higgs.\nFor concreteness, we shall work within the framework of heterotic string models \nin which the Standard Model itself is realized at affine level {\\mbox{$k=1$}} through a standard level-one $SO(10)$ embedding.\nIn the following we shall adhere to the conventions in Ref.~\\cite{Dienes:1995sq}.\nSince $SO(10)$ has rank $5$, this group can be minimally embedded within a \nfive-dimensional sublattice $\\lbrace Q_1,Q_2,Q_3,Q_4,Q_5\\rbrace$ \nwithin the full 22-dimensional left-moving lattice $\\lbrace {\\bf Q}_L\\rbrace$. \nWithin this sublattice, we shall take\nthe {\\mbox{$\\ell=1,2$}} directions as corresponding to \nthe {\\mbox{$U(2) = SU(2)\\times U(1)$}} electroweak subgroup of $SO(10)$, \nwhile the {\\mbox{$\\ell =3,4,5$}} directions will correspond to the {\\mbox{$U(3)= SU(3)\\times U(1)$}} color subgroup.\nBy convention we will take the $SU(2)_L$ representations to lie along the line perpendicular to $(1,1,0,0,0)$\nwithin the two-dimensional $U(2)$ sublattice,\nand the $SU(3)_c$ representations to lie within the two-dimensional plane perpendicular to $(0,0,1,1,1)$\nwithin the three-dimensional $U(3)$ sublattice.\nIt then follows that any state with charge vector ${\\bf Q}_L$ has $SU(2)$ quantum numbers determined by projecting ${\\bf Q}_L$ onto the \n$SU(2)$ line [thereby yielding an $SU(2)$ weight in the corresponding $SU(2)$ weight system]\nand $SU(3)$ quantum numbers determined by projecting ${\\bf Q}_L$ onto the $SU(3)$ plane [thereby yielding an $SU(3)$ weight within the \ncorresponding $SU(3)$ weight system].\nLikewise, the $SO(10)$-normalized hypercharge $Y$ of any state with left-moving charge vector\n${\\bf Q}_L$ is given by {\\mbox{$Y=\\sum_{\\ell=1}^5 a_{Y}^{(\\ell)} Q_\\ell$}} where \n\\begin{equation}\n {\\bf a}_{Y}~=~ (\\, {\\textstyle{1\\over 2}},\\, {\\textstyle{1\\over 2}}, ~ -\\mbox{\\small ${\\frac{1}{3}}$},\\, -\\mbox{\\small ${\\frac{1}{3}}$},\\, -\\mbox{\\small ${\\frac{1}{3}}$}\\, )~\n\\end{equation}\n(with all other components vanishing).\nThus, {\\mbox{$Y\\equiv {\\bf a}_{Y} \\cdot {\\bf Q}_L$}}. \nIndeed we see that {\\mbox{$k_Y\\equiv 2 {\\bf a}_{Y}\\cdot {\\bf a}_{Y} = 5\/3$}}, as appropriate for the standard $SO(10)$ embedding\n(as well as other non-standard $SO(10)$ embeddings~\\cite{Dienes:1995sq}).\nIn a similar way, the electromagnetic charge $q_{\\rm EM}$ of any state with charge vector ${\\bf Q}_L$ is given by \n{\\mbox{$q_{\\rm EM}= {\\bf a}_{\\rm EM} \\cdot {\\bf Q}_L$}}, where\n\\begin{equation}\n {\\bf a}_{\\rm EM}~=~ (\\, 0, ~ 1, ~ -\\mbox{\\small ${\\frac{1}{3}}$}, \\, -\\mbox{\\small ${\\frac{1}{3}}$},\\, -\\mbox{\\small ${\\frac{1}{3}}$} )~\n\\end{equation}\n(with all other components vanishing).\nAs a check we verify that {\\mbox{$T_3 = {\\bf a}_{T_3} \\cdot {\\bf Q}_L$}},\nwhere\n{\\mbox{${\\bf a}_{T_3} = {\\bf a}_{\\rm EM} - {\\bf a}_{Y} = (-{\\textstyle{1\\over 2}}, {\\textstyle{1\\over 2}}, 0,0,0) = {\\textstyle{1\\over 2}} {\\bf Q}_{T^+}$}}\nwhere ${\\bf Q}_{T^+}$ is the charge vector (or root vector) associated with \nthe $SU(2)$ gauge boson with positive $T_3$ charge.\n \nThus far, we have focused on the gauge structure of the theory. As we have seen, the corresponding charge vectors\nfollow our usual group-theoretic expectations, just as they would in ordinary quantum field theory.\nHowever, the charge vectors associated with the SM matter states \nin string theory are far more complex than would be expected in quantum field theory\nand actually spill beyond the $SO(10)$ sublattice. \n\n\nTo see why this is so, it is perhaps easiest to \nconsider the original $SO(10)$ theory {\\it prior}\\\/ to electroweak breaking.\nIn this phase of the theory, the SM matter content consists of massless fermion and Higgs fields transforming \nin the ${\\bf 16}$ and ${\\bf 10}$ representations of $SO(10)$, respectively.\nThe former representations has charge vectors \nwith $SO(10)$-sublattice components {\\mbox{$Q^{(f)}_\\ell =\\pm 1\/2$}} for each $\\ell$ (with an odd net number of minus signs),\nwhile the latter has\n{\\mbox{$Q^{(\\phi)}_\\ell =\\pm \\delta_{\\ell k}$}} where {\\mbox{$k=1,2,...,5$}}.\nThus, the ${\\bf 16}$ and ${\\bf 10}$ representations have \nconformal dimensions {\\mbox{$h_{\\bf 16}= 5\/8$}} and {\\mbox{$h_{\\bf 10}=1\/2$}}.\nIndeed, according to the gauge embeddings discussed above,\nthe particular Higgs states which are electrically neutral have {\\mbox{$Q_\\ell =\\pm \\delta_{\\ell 1}$}}. \nHowever, as a result of the non-trivial left-moving heterotic-string\nvacuum energy {\\mbox{$E_L= -1$}},\nany massless string state must correspond to worldsheet excitations contributing a total \nleft-moving conformal dimension {\\mbox{$h_L=1$}}.\nThus, even within the $SO(10)$ embedding specified above, string consistency constraints require \nthat the SM fermion and Higgs states carry non-trivial \ncharges not only {\\it within}\\\/ the $SO(10)$ sublattice $\\lbrace Q_1,Q_2,...,Q_5\\rbrace$ {\\it but also beyond it} --- {\\it i.e.}\\\/, \nelsewhere in the 17 remaining left-moving lattice directions {\\mbox{${\\bf Q}_{\\rm int} \\equiv \\lbrace Q_6,...,Q_{\\rm 22}\\rbrace$}}\nwhich {\\it a priori}\\\/ correspond to gauge symmetries beyond those of the SM (such as those of potential hidden sectors).\nIndeed, these additional excitations must contribute additional left-moving conformal dimensions $3\/8$ and $1\/2$ for\nthe SM matter and Higgs fields respectively, corresponding to\n{\\mbox{$[{\\bf Q}^{(f)}_{\\rm int}]^2 = 3\/4$}} for the fermions and {\\mbox{$[{\\bf Q}^{(\\phi)}_{\\rm int}]^2 = 1$}} for the Higgs.\n\n \nA similar phenomenon also occurs within the 10-dimensional {\\it right-moving}\\\/ charge lattice, with \ncomponents $\\lbrace \\tilde Q_1,..., \\tilde Q_{10}\\rbrace$.\nThe component associated with the non-zero component of the {\\bf S}-vector discussed\nbelow Eq.~(\\ref{thetadef}) --- henceforth chosen as $\\tilde Q_1$ --- \ndescribes the spacetime spin-helicity of the state. \nAs such, we must have {\\mbox{$\\tilde Q_1^{(f)}=\\pm 1\/2$}} for the SM fermions\nand {\\mbox{$\\tilde Q_1^{(\\phi)}=0$}} for the scalar Higgs. \nOf course, the right-moving side of the heterotic string has \n{\\mbox{$E_R= -1\/2$}}, requiring that all massless string states have total right-moving conformal dimensions {\\mbox{$h_R=1\/2$}}. \nWe thus find that the SM fermion and Higgs fields \nmust have additional nine-dimensional\ncharge vectors {\\mbox{$\\tilde {\\bf Q}_{\\rm int}\\equiv \\lbrace\\tilde Q_2,..., \\tilde Q_{10}\\rbrace$}}\n(presumably corresponding to additional {\\it right-moving}\\\/ gauge symmetries)\nsuch that \n{\\mbox{$[\\tilde {\\bf Q}^{(f)}_{\\rm int}]^2 = 3\/4$}}\nand \n{\\mbox{$[\\tilde {\\bf Q}^{(\\phi)}_{\\rm int}]^2 = 1$}}.\n\nWe see, then, that the electrically neutral Higgs field prior to electroweak symmetry breaking\nmust have a total $32$-dimensional charge vector of the form\n\\begin{eqnarray}\n {\\bf Q}_\\phi ~&\\equiv&~ \n ({\\bf Q}_L^{(\\phi)}\\,|\\, {\\bf Q}_R^{(\\phi)}) \\nonumber\\\\ \n ~&=&~ \n (1,0,0,0,0, \\, {\\bf Q}_{\\rm int}^{(\\phi)} \\,|\\, 0 , \\, \\tilde {\\bf Q}_{\\rm int}^{(\\phi)} ) \n\\label{Higgsform}\n\\end{eqnarray}\nwhere {\\mbox{$[{\\bf Q}_{\\rm int}^{(\\phi)}]^2 = [\\tilde {\\bf Q}_{\\rm int}^{(\\phi)}]^2 = 1$}}. \nIn general, the specific forms of\n${\\bf Q}_{\\rm int}^{(\\phi)}$\nand \n$\\tilde {\\bf Q}_{\\rm int}^{(\\phi)}$\ndepend on the specific string model and the spectrum beyond the Standard Model.\nHowever, those components which are specified within Eq.~(\\ref{Higgsform}) are guaranteed \nby the underlying $SO(10)$ structure \nand by the requirement that the Higgs be electrically neutral.\nOf course, the process of electroweak symmetry breaking can in principle alter the form of this vector.\nHowever, we know that $U(1)_{\\rm EM}$ necessarily remains unbroken.\nThus, even if the forms of the particular ``internal'' vectors ${\\bf Q}_{\\rm int}^{(\\phi)}$ and \n$\\tilde {\\bf Q}_{\\rm int}^{(\\phi)}$ \nare shifted under electroweak symmetry breaking,\nthe zeros in the charge vector in Eq.~(\\ref{Higgsform})\n ensure the electric neutrality of the Higgs field and must therefore be preserved.\nThis remains true not only for the physical Higgs field after electroweak symmetry breaking, but also\nfor its quantum fluctuations in the Higgsed phase.\n\nThis observation immediately allows us to constrain the form of the ${\\cal T}$-matrices which parametrize\nthe response of the charge lattice to small fluctuations of the Higgs field around its minimum.\nBecause the zeros in the charge vector in Eq.~(\\ref{Higgsform}) must remain vanishing --- and\nindeed because the electromagnetic charges and spin-statistics of {\\it all}\\\/ string states must remain\nunaltered \nunder such fluctuations ---\nwe see that the (necessarily anti-symmetric) ${\\cal T}$-matrix can at most have the general form\n\\begin{equation}\n{\\cal T} ~\\sim~ \\left(\n \\begin{array}{cccccc|cc}\n ~ & ~ & ~ & ~ & ~ & {\\bf t} & 0 & \\tilde{\\bf t} \\\\ \n ~ & ~ & ~ & ~ & ~ & {\\bf 0} & 0 & \\tilde{\\bf 0} \\\\ \n ~ & ~ & {\\bf 0}_{5\\times 5} & ~ & ~ & {\\bf 0} & 0 & \\tilde {\\bf 0} \\\\ \n ~ & ~ & ~ & ~ & ~ & {\\bf 0} & 0 & \\tilde {\\bf 0} \\\\ \n ~ & ~ & ~ & ~ & ~ & {\\bf 0} & 0 & \\tilde {\\bf 0} \\\\ \n -{\\bf t}^t & {\\bf 0}^t & {\\bf 0}^t & {\\bf 0}^t & {\\bf 0}^t & {\\bf t}_{11} & {\\bf 0}^t & {\\bf t}_{12} \\\\ \n \\hline \n \\rule{0pt}{2.5ex} 0 & 0 & 0 & 0 & 0 & {\\bf 0} & 0 & \\tilde{\\bf 0} \\\\ \n -\\tilde \n {\\bf t}^t & \\tilde {\\bf 0}^t & \\tilde {\\bf 0}^t & \\tilde{\\bf 0}^t & \\tilde{\\bf 0}^t & -{\\bf t}_{12}^t & \\tilde {\\bf 0}^t & \n {\\bf t}_{22} \n\\end{array}\n\\right)\n\\label{Tmatrixform}\n\\end{equation}\nwhere \n${\\bf t}$ is an arbitrary \n 17-dimensional row vector;\nwhere $\\tilde {\\bf t}$ is an arbitrary \n9-dimensional row vector;\nwhere ${\\bf t}_{11}$, ${\\bf t}_{12}$, and ${\\bf t}_{22}$\nare arbitrary \nmatrices of dimensionalities {\\mbox{$17\\times 17$}}, {\\mbox{$9\\times 17$}}, and {\\mbox{$9\\times 9$}}, respectively, with\n${\\bf t}_{11}$ and ${\\bf t}_{22}$ antisymmetric;\nand where ${\\bf 0}$ and $\\tilde {\\bf 0}$ are respectively $17$- and $9$-dimensional\nzero row vectors. \nIndeed, as we have seen,\nonly this form of the ${\\cal T}$-matrix can preserve the electromagnetic charges and spin-statistics\nof the string states \nunder small shifts in the Higgs field around its new minimum, assuming a heterotic string model\nwith a standard level-one $SO(10)$ embedding.\nThe precise forms of ${\\bf t}$, $\\tilde {\\bf t}$, \n${\\bf t}_{11}$, ${\\bf t}_{12}$, and ${\\bf t}_{22}$ then depend on \nmore model-specific details of how the Higgs is realized within the theory --- details which go beyond\nthe $SO(10)$ embedding.\n\n\nAs indicated above, this is only one particular example of the kinds of ${\\cal T}$-matrices that can occur.\nHowever, all of the results of this paper will be completely general,\nand will not rest on this particular example.\n\n\n\\subsection{Calculating the Higgs mass}\n\nWe can now use the general results in Sect.~\\ref{latdeform}\nto calculate the mass of $\\phi$.\nIn general, this mass can be defined as\n\\begin{equation}\n m_\\phi^2 ~\\equiv~ {d^2 \\Lambda(\\phi)\\over d \\phi^2 } \\biggl|_{\\phi=0}\n\\label{higgsdef}\n\\end{equation}\nwhere\n\\begin{equation}\n \\Lambda(\\phi) ~\\equiv~ -{{\\cal M}^4\\over 2} \\int_{\\cal F} \\dmu \\, {\\cal Z}(\\tau,{\\overline{\\tau}},\\phi)~.\n\\label{Lambdaphi}\n\\end{equation}\nIndeed, $\\Lambda(\\phi)$ is the vacuum energy that governs the dynamics of $\\phi$.\nHere\n$ d^2\\tau \/\\tau_2^2$ is the modular-invariant integration measure, \n${\\cal F}$ is the fundamental domain of the modular group\n\\begin{equation}\n {\\cal F} ~\\equiv~ \\lbrace \\tau :\\, -{\\textstyle{1\\over 2}} <\\tau_1\\leq {\\textstyle{1\\over 2}}, \\, \\tau_2>0,\\, |\\tau|\\geq 1 \\rbrace~,\n\\label{Fdef}\n\\end{equation}\nand {\\mbox{${\\cal M}\\equiv M_s\/(2\\pi)$}} \nis the reduced string scale.\nIn this expression, following Eq.~(\\ref{preZQdef}), the shifted partition function is given by\n\\begin{equation}\n {\\cal Z}(\\tau,{\\overline{\\tau}},\\phi) ~=~ \\tau_2^{-1} {1\\over \\overline{\\eta}^{12} \\eta^{24}} \\, \\sum_{{\\bf Q}_L,{\\bf Q}_R} \n (-1)^F {\\overline{q}}^{{\\bf Q}_R^2\/2} \n q^{{\\bf Q}_L^2\/2}~\n\\label{ZQdef}\n\\end{equation}\nwhere the left- and right-moving charge vectors ${\\bf Q}_L$ and ${\\bf Q}_R$ are now deformed as in Eq.~(\\ref{Qdeform})\nand thus depend on $\\phi$.\n\nGiven this definition, we begin by evaluating the leading contribution to the Higgs mass by taking \npartial derivatives of ${\\cal Z}$, {\\it i.e.}\\\/,\n\\begin{equation}\n {\\partial^2{\\cal Z} \\over \\partial\\phi^2} ~=~ \n \\tau_2^{-1} {1\\over \\overline{\\eta}^{12} \\eta^{24}} \\, \\sum_{{\\bf Q}_L,{\\bf Q}_R\\in L} \n (-1)^F \\,X\\, \\,{\\overline{q}}^{{\\bf Q}_R^2\/2} q^{{\\bf Q}_L^2\/2}~\n\\label{eq:Zexp}\n\\end{equation}\nwhere the summand insertion $X$ is given by\n\\begin{equation}\n X ~\\equiv~ \\pi i \n {\\partial^2\\over \\partial\\phi^2} (\\tau {\\bf Q}_L^2 - {\\overline{\\tau}} {\\bf Q}_R^2 ) \n - \\pi^2 \\left\\lbrack {\\partial\\over \\partial\\phi} (\\tau {\\bf Q}_L^2 - {\\overline{\\tau}} {\\bf Q}_R^2)\\right\\rbrack^2~.\n\\label{stuff}\n\\end{equation}\nNote that it is the {\\it partial}\\\/ derivative $\\partial^2\/\\partial\\phi^2$ in Eq.~(\\ref{eq:Zexp}) which provides the \nleading contribution to the Higgs mass;\nwe shall return to this point shortly.\nExpanding $X$ in powers of $\\tau_1$ and $\\tau_2$ and then setting {\\mbox{$\\phi= 0$}} yields\n\\begin{equation}\n X\\bigl|_{\\phi=0} ~=~ \n A \\,\\tau_1 + B \\,\\tau_2 + C \\,\\tau_1^2 + D \\,\\tau_2^2 + E \\,\\tau_1\\tau_2~,\n\\end{equation}\nwhere\n\\begin{eqnarray} \n A~&=&~0~,\\nonumber\\\\\n B~&=&~ -2\\pi\\alpha' (\n {\\bf Q}_a^2 + \\tilde {\\bf Q}_a^2 + {\\bf Q}_b^t {\\bf Q}_L + \\tilde {\\bf Q}_b^t {\\bf Q}_R )~,\\nonumber\\\\ \n C~&=&~0~,\\nonumber\\\\\n D~&=&~ 4\\pi^2 \\alpha' \\,({\\bf Q}_a^t {\\bf Q}_L + \\tilde {\\bf Q}_a^t {\\bf Q}_R)^2~ , \\nonumber\\\\\n E~&=&~0~.\n\\label{intermed}\n\\end{eqnarray} \nNote that $A$, $C$, and $E$ each vanish as the result of the constraints in \nEq.~(\\ref{levelmatching}). This is consistent, as these are the quantities which are proportional \nto powers of $\\tau_1$, which multiplies ${\\bf Q}_L^2-{\\bf Q}_R^2$ within Eq.~(\\ref{stuff}). \n\nUsing Eqs.~(\\ref{shiftQ}) and (\\ref{NTrelations}),\nwe can now express the shift vectors within Eq.~(\\ref{intermed}) directly in terms \nof ${\\bf Q}_L$ and ${\\bf Q}_R$.\nFor convenience we define\n\\begin{equation}\n {\\bf Q}_h \\equiv {\\cal T}_{21} {\\bf Q}_L~,~~~~~\n \\tilde {\\bf Q}_h \\equiv {\\cal T}_{12} {\\bf Q}_R~,\n\\label{Qhdef}\n\\end{equation}\nand likewise define \n\\begin{equation}\n {\\bf Q}_j \\equiv {\\cal T}_{11} {\\bf Q}_L~,~~~~~\n \\tilde {\\bf Q}_j \\equiv {\\cal T}_{22} {\\bf Q}_R~.\n\\label{Qjdef}\n\\end{equation}\nWe then find\n\\begin{eqnarray}\n B ~&=&~ -4\\pi \\alpha' ({\\bf Q}_h^2 + \\tilde {\\bf Q}_h^2 - \\tilde {\\bf Q}_j^t {\\bf Q}_h - {\\bf Q}_j^t \\tilde {\\bf Q}_h)~,\\nonumber\\\\\n D ~&=&~ 4\\pi^2 \\alpha' ({\\bf Q}_R^t {\\bf Q}_h - {\\bf Q}_L^t \\tilde {\\bf Q}_h)^2~.\n\\label{eq:finalb}\n\\end{eqnarray}\nNote the identity {\\mbox{${\\bf Q}_R^t {\\bf Q}_h = - {\\bf Q}_L^t \\tilde {\\bf Q}_h$}}, as a result of which\nour expression for $D$ can actually be collapsed into one term.\nHowever, we have retained this form for $D$ in order to make manifest the symmetry between left- and right-moving contributions.\nOur overall insertion into the partition function is then given by\n{\\mbox{$X|_{\\phi=0} \\equiv {{\\cal X}}\/{\\cal M}^2$}}, where\n\\begin{eqnarray}\n {\\cal X} ~&=&~ \n \\tau_2^2 \\, ({\\bf Q}_R^t {\\bf Q}_h - {\\bf Q}_L^t \\tilde {\\bf Q}_h)^2 \\nonumber\\\\\n && ~~ - {\\tau_2\\over \\pi } \\left({\\bf Q}_h^2 + \\tilde {\\bf Q}_h^2 - \\tilde {\\bf Q}_j^t {\\bf Q}_h - {\\bf Q}_j^t \\tilde {\\bf Q}_h\\right)~.~~~~~~\n\\label{Xdef}\n\\end{eqnarray}\n\n\n\\subsection{Modular completion and additional Higgs-mass contributions \\label{sec:completion}}\n\n\nThus far, we have calculated the leading contribution to the Higgs mass \nby evaluating $\\partial^2 {\\cal Z}\/\\partial \\phi^2$.\nHowever, the full contribution $d^2 {\\cal Z}\/d\\phi^2$ (with full rather than \npartial $\\phi$-derivatives) also includes\nvarious additional effects on the partition function ${\\cal Z}$ \nthat come from fluctuations of the Higgs field. For example, such fluctuations deform \nthe background moduli fields (such as the metric that contracts compactified components of ${\\bf Q}_L^2$ and ${\\bf Q}_R^2$ within the \ncharge lattice).\nSuch effects produce additional contributions to the total Higgs mass.\n\nIt turns out that \nwe can calculate all of these extra contributions in a completely general way \nthrough the requirement of modular invariance. \nIndeed, because modular invariance remains unbroken even when the theory is Higgsed,\nthe final expression for the total Higgs mass must not only be modular invariant but also arise\nthrough a modular-covariant sequence of calculational operations.\nAs we shall demonstrate, the above expression for the insertion ${\\cal X}$ \nin Eq.~(\\ref{Xdef}) does not have this property. We shall therefore determine the additional contributions\nto the Higgs mass by performing the ``modular completion'' of ${\\cal X}$ --- {\\it i.e.}\\\/, by determining the\nadditional contribution to ${\\cal X}$ which will render \nthis insertion consistent with modular invariance.\n \nIn general, prior to the insertion of ${\\cal X}$, the partition-function trace in Eq.~(\\ref{preZQdef}) \n[or equivalently the trace in Eq.~(\\ref{ZQdef}) evaluated at {\\mbox{$\\phi=0$}}]\nis presumed to already be modular invariant, as required for the consistency of the underlying string.\nIn order to determine the modular completion of the quantity ${\\cal X}$ in Eq.~(\\ref{Xdef}),\nwe therefore need to understand the modular-invariance effects \nthat arise when ${\\cal X}$ is inserted into this partition-function trace.\nBecause ${\\cal X}$ involves various combinations of components of charge vectors,\nlet us begin by investigating the effect of inserting powers of a single charge vector \ncomponent $Q_\\ell$ (associated with the $\\ell^{\\rm th}$ lattice direction) \ninto our partition-function trace.\nWithin the partition functions described in Eqs.~(\\ref{partfZ}) and (\\ref{partft}),\ninserting $Q_\\ell^n$ for any power $n$\nis tantamount to replacing\n\\begin{equation}\n \\vartheta_\\ell ~\\to~ \\sum_{Q_\\ell\\in \\mathbb{Z} +\\alpha_\\ell} e^{2\\pi i \\beta_\\ell Q_\\ell} ~Q_\\ell^n~ q^{Q_\\ell^2\/2}~.\n\\label{Qinsert}\n\\end{equation}\nHowever, one useful way to proceed is to recognize that this latter sum\ncan be rewritten as\n\\begin{equation}\n {1\\over (2\\pi i)^n} \\, \\frac{\\partial^n }{\\partial z_\\ell^n} \\vartheta_\\ell(z_\\ell|\\tau) \\biggl|_{z_{\\ell=0}}\n\\label{zderiv}\n\\end{equation}\nwhere the generalized $\\theta_\\ell (z_\\ell|\\tau)$ \nfunction is defined as\n\\begin{equation}\n \\vartheta_\\ell(z_\\ell|\\tau) \n ~\\equiv~\n \\sum_{Q_\\ell\\in \\mathbb{Z} +\\alpha_\\ell} e^{2\\pi i (\\beta_\\ell +z_\\ell) Q_\\ell} \\,q^{Q_\\ell^2\/2}~.~~\n\\label{thetadefz}\n\\end{equation}\nIndeed, we see that $\\vartheta_\\ell(\\tau)$ is nothing but $\\vartheta_\\ell (z_\\ell|\\tau)$\nevaluated at {\\mbox{$z_\\ell=0$}}.\nHowever, for arbitrary $z$, these \ngeneralized $\\vartheta(z|\\tau)$ functions have the schematic modular-transformation properties\n\\begin{equation} \n \\vartheta_\\ell\\left(z\\biggl|\\frac{a\\tau+b}{c\\tau+d}\\right) ~\\sim~ \n (c\\tau+d)^{1\/2} \\, e^{ \\pi i c(c\\tau+d) z^2}\\, \n \\vartheta_{\\ell} ((c\\tau+d) z | \\tau)~.\n\\label{modtransthetaz}\n\\end{equation}\nIt then follows that \n\\begin{equation}\n \\vartheta\\left(z\\biggl|\\frac{a\\tau+b}{c\\tau+d}\\right) \n \\biggl|_{z=0} ~\\sim~ (c\\tau+d)^{1\/2} \\,\n \\vartheta(z|\\tau)\\biggl|_{z=0}~,\n\\label{firstderiv}\n\\end{equation}\nand likewise \n\\begin{equation}\n \\frac{\\partial}{\\partial z} \n \\vartheta\\left(z\\biggl|\\frac{a\\tau+b}{c\\tau+d}\\right) \n \\biggl|_{z=0} ~\\sim~ (c\\tau+d)^{3\/2} \\,\\frac{\\partial}{\\partial z} \n \\vartheta(z|\\tau)\\biggl|_{z=0}~.\n\\label{secondderiv}\n\\end{equation}\nThis indicates that while the function {\\mbox{$\\vartheta(z|\\tau)|_{z=0}$}} transforms covariantly with modular weight $1\/2$,\nits first derivative {\\mbox{$[\\partial\\vartheta(z|\\tau)\/\\partial z]|_{z=0}$}} transforms covariantly with modular weight $3\/2$.\n\nAt first glance, one might expect this pattern to continue, with the second derivative\n{\\mbox{$[\\partial^2 \\vartheta(z|\\tau)\/dz^2]|_{z=0}$}} transforming\ncovariantly with modular weight $5\/2$. However, this is not what happens. Instead, \nfrom Eq.~(\\ref{modtransthetaz}) we find\n\\begin{eqnarray}\n \\frac{\\partial^2}{\\partial z^2} \n \\vartheta\\left(z\\biggl|\\frac{a\\tau+b}{c\\tau+d}\\right) \n\\biggl|_{z=0} &\\sim&~ (c\\tau+d)^{5\/2} \\,\\frac{\\partial}{\\partial z} \n \\vartheta(z|\\tau)\\biggl|_{z=0} \\nonumber\\\\\n && ~ + ~2\\pi i \\, c\\, (c\\tau+d)^{3\/2}\\, \\vartheta(\\tau)~.~~~~~~\\nonumber\\\\\n\\end{eqnarray}\nWhile the term on the first line is the expected result,\nthe term on the second line represents a {\\it modular anomaly} which destroys the \nmodular covariance of the second derivative.\n\nSince modular covariance must be preserved, we must perform\na {\\it modular completion}. In this simple case, this means that\nwe must replace \n${\\partial^2 \/ \\partial z^2}$ with a {\\it modular-covariant second derivative}\\\/ $D^2_z$\nsuch that $D^2_z $ not only contains\n$\\partial^2 \/\\partial z^2$ but also has the property that {\\mbox{$D^2_z \\theta(z|\\tau)|_{z=0}$}} transforms covariantly with weight $5\/2$.\nIt is straightforward to show that the only such modular-covariant derivative is\n\\begin{equation}\n D^2_z ~\\equiv~ \\frac{\\partial^2}{\\partial z^2} + \\frac{\\pi}{\\tau_2}~,\n\\label{modcovderiv}\n\\end{equation}\nand with this definition one indeed finds\n\\begin{equation}\n \\left\\lbrace \\left[ D^2_z \\vartheta( z|\\tau)\\right]_{\\tau\\to \\frac{a\\tau+b}{c\\tau+d}} \\right\\rbrace\\biggl|_{z=0} \n ~\\sim~ (c\\tau+d)^{5\/2} \\, D^2_z \\vartheta(z|\\tau)\\biggl|_{z=0}~,\n\\end{equation}\nthereby continuing the pattern set by Eqs.~(\\ref{firstderiv}) and (\\ref{secondderiv}).\nIt turns out that this modular-covariant second $z$-derivative \nis equivalent to the\nmodular-covariant $\\tau$-derivative \n\\begin{equation} D_\\tau ~\\equiv ~ {\\partial \\over \\partial \\tau} ~-~ {ik\\over 2\\tau_2}~ \n\\end{equation}\nwhich preserves the modular covariance of any modular function of weight $k$.\nIndeed, \n our $\\vartheta(z|\\tau)$ functions have {\\mbox{$k=1\/2$}} and satisfy \nthe heat equation \n$\\partial^2 \\vartheta(z|\\tau) \/\\partial z^2 = 4\\pi i \\, \\partial \\vartheta(z|\\tau) \/\\partial \\tau$.\nIn this sense, the $z$-derivative serves as a ``square root'' of the $\\tau$-derivative and gives us a precise\nmeans of extracting the individual charge insertions (rather than their squares).\n In this connection, we emphasize that there is a tight correspondence between the Higgs field and the\n $z$-parameter. Specifically, when we deform a theory through a continuous change in the value of the Higgs VEV, \n its partition function deforms through a corresponding continuous change in the $z$-parameter.\n \n\n\n \nIn principle we could continue to examine higher $z$-derivatives \n(all of which will also suffer from modular anomalies),\nbut the results we have thus far will be sufficient for our purposes.\nRecalling the equivalence between the expressions in Eqs.~(\\ref{Qinsert}) and (\\ref{zderiv}),\nwe thus see that the insertion of a single power of any given $Q_\\ell$ does not disturb the\nmodular covariance of the corresponding holomorphic (or anti-holomorphic) factor in the \npartition-function trace, but the insertion of a quadratic term $Q_\\ell^2$\nalong the $\\ell^{\\rm th}$ lattice direction\ndoes {\\it not}\\\/ lead to a modular-covariant result and must, according to Eq.~(\\ref{modcovderiv}),\nbe replaced by the modular-covariant insertion \n$Q_\\ell^2 - 1\/(4\\pi\\tau_2)$.\nThus, our rules for modular completion \nthrough second-order in charge-vector components are given by\n\\begin{equation}\n\\begin{cases} \n ~{\\bf Q}_\\ell &\\to~~ {\\bf Q}_\\ell \\\\\n ~{\\bf Q}_\\ell{\\bf Q}_{\\ell'} &\\to~~ {\\bf Q}_\\ell {\\bf Q}_{\\ell'} - \\frac{1}{4\\pi \\tau_2} \\delta_{\\ell,\\ell'}~.\n\\end{cases}\n\\label{completionrules}\n\\end{equation}\nThese general results hold for all lattice directions $(\\ell,\\ell')$ regardless of whether \nthey correspond to left- or right-moving lattice components.\nSuch modular completions have also arisen in other contexts, \nsuch as within string-theoretic threshold corrections~\\cite{Kiritsis:1994ta,Kiritsis:1996dn,Kiritsis:1998en}. \n\nWith these modular-completion rules in hand, we can now investigate the modular completion of the \nexpression for ${\\cal X}$ in Eq.~(\\ref{Xdef}).\nIt is simplest to begin by focusing on\nthe quartic terms, {\\it i.e.}\\\/, the terms in the top line of Eq.~(\\ref{Xdef}).\nGiven the identity just below Eq.~(\\ref{eq:finalb}),\nthese terms are proportional to $({\\bf Q}_L^t \\tilde {\\bf Q}_h)^2$.\nWith $Q_{L\\ell}$ denoting the $\\ell^{\\rm th}$ component of ${\\bf Q}_L$, {\\it etc.}\\\/, \nwe find \n\\begin{eqnarray}\n ({\\bf Q}_L^t \\tilde {\\bf Q}_h)^2\n &=& \\left( \\sum_{\\ell=1}^{22} \\sum_{m=1}^{10} Q_{L\\ell} ({\\cal T}_{12})_{\\ell m} Q_{Rm} \\right)^2 \\nonumber\\\\\n &=& \\sum_{\\ell,\\ell' =1}^{22} \\sum_{m,m'=1}^{10} \n ({\\cal T}_{12})_{\\ell m} ({\\cal T}_{12})_{\\ell' m'} \\nonumber\\\\ \n && ~~~~~~~~~~~\\times~ Q_{Rm} Q_{Rm'} Q_{L\\ell} Q_{L\\ell'}~.~~~~~~~~~~~~~~\n\\label{intterm}\n\\end{eqnarray}\nFollowing the rules in Eq.~(\\ref{completionrules}), we can readily \nobtain the modular completion of this expression by replacing the final line in Eq.~(\\ref{intterm})\nwith\n\\begin{equation}\n \\left( Q_{Rm} Q_{Rm'} - \\frac{1}{4\\pi \\tau_2} \\delta_{mm'} \\right) \n \\left( Q_{L\\ell} Q_{L\\ell'} - \\frac{1}{4\\pi \\tau_2} \\delta_{\\ell\\ell'} \\right)~. \n\\label{substi}\n\\end{equation}\nSubstituting Eq.~(\\ref{substi}) into Eq.~(\\ref{intterm}) and recalling \nthat {\\mbox{${\\cal T}_{12}^t = -{\\cal T}_{21}$}}, \nwe thus find that the modular completion \nof the quartic term $({\\bf Q}_L^t \\tilde {\\bf Q}_h)^2$\nwithin ${\\cal X}$\nis given by\n\\begin{equation}\n ({\\bf Q}_L^t \\tilde {\\bf Q}_h)^2 - \\frac{1}{4\\pi \\tau_2} \\left( {\\bf Q}_h^2 + \\tilde {\\bf Q}_h^2\\right) + \\frac{\\xi}{(4\\pi\\tau_2)^2}~~~ \n\\label{quadr1}\n\\end{equation}\nwhere \n\\begin{eqnarray}\n \\xi ~&\\equiv&~ \n {\\rm Tr} ({\\cal T}_{12}^t {\\cal T}_{12})\n ~=~ {\\rm Tr} ({\\cal T}_{21}^t {\\cal T}_{21})~\\nonumber\\\\\n && ~~~=~ - {\\rm Tr} ({\\cal T}_{12} {\\cal T}_{21}) ~=~ - {\\rm Tr} ({\\cal T}_{21} {\\cal T}_{12})~.~~~~~~~~\n\\end{eqnarray}\n\nRemarkably, the quadratic terms ${\\bf Q}_h^2 + \\tilde {\\bf Q}_h^2$ that are generated within Eq.~(\\ref{quadr1}) \nalready appear on the second line of Eq.~(\\ref{Xdef}).\nIn other words, even if we had not already known of these quadratic terms, we could have deduced their existence\nthrough the modular completion of our quartic terms!\nConversely, we could have generated the quartic terms through a modular completion of these quadratic terms --- {\\it i.e.}\\\/,\neach set of terms provides the modular completion of the other.\nThus, the only remaining terms within Eq.~(\\ref{Xdef}) that might require modular completion are \nthe final quadratic terms on the second line of Eq.~(\\ref{Xdef}), namely\n$\\tilde {\\bf Q}_j^t {\\bf Q}_h + {\\bf Q}_j^t \\tilde {\\bf Q}_h$.\nHowever, \n${\\bf Q}_h$ and ${\\bf Q}_j$ involve only left-moving components of the lattice while\n$\\tilde {\\bf Q}_h$ and $\\tilde {\\bf Q}_j$ involve only right-moving components.\nThus \n$\\tilde {\\bf Q}_j^t {\\bf Q}_h + {\\bf Q}_j^t \\tilde {\\bf Q}_h$\nis already modular complete.\nPutting all the pieces together, we therefore find that the total expression for ${\\cal X}$ in Eq.~(\\ref{Xdef}) has\na simple (and in fact universal) modular completion:\n\\begin{equation}\n {\\cal X} ~\\to~ \n {\\cal X} ~+~ \\frac{\\xi}{4\\pi^2} ~. \n\\label{Xmodcomplete}\n\\end{equation}\nIndeed, this sole remaining extra term generated by the modular completion stems from \nthe final term in Eq.~(\\ref{quadr1}).\nIt is noteworthy that this extra term \nis entirely independent of the charge vectors.\nThis is consistent with our expectation that such additional terms\nrepresent the contributions from the deformations of the moduli fields under Higgs fluctuations --- deformations\nwhich act in a universal (and hence $Q$-independent) manner.\n\nSome remarks are in order regarding\nthe uniqueness of the completion\nin Eq.~(\\ref{Xmodcomplete}).\nIn particular, at first glance one might wonder how the modular completion of \nthe quadratic terms $\\tilde {\\bf Q}_j^t {\\bf Q}_h + {\\bf Q}_j^t \\tilde {\\bf Q}_h$ could \nuniquely determine the quartic terms in ${\\cal X}$, given that the modular-completion rules \nwithin Eq.~(\\ref{completionrules}) only seem to generate extra terms \nwhich are of lower powers in charge-vector components.\nHowever, the important point is that the rules in Eq.~(\\ref{completionrules}) only ensure\nthe modular covariance of the individual (anti-)holomorphic components\nof the partition-function trace. \nIn particular, these rules do not, in and of themselves, ensure that \nwe continue to satisfy \nthe additional constraint in Eq.~(\\ref{genexp2}) \nthat arises when stitching these holomorphic and anti-holomorphic components together\nas in Eq.~(\\ref{genexp}). \nHowever, given that ${\\bf Q}_h^2$ increases the modular weight of the \nholomorphic component by two units without increasing the modular weight of the\nanti-holomorphic component,\nand given that \n$\\tilde {\\bf Q}_h^2$ does the opposite, the only way to properly modular-complete their sum\nis by ``completing the square'' and realizing these terms as the off-diagonal terms that are generated\nthrough a factorized modular completion as in Eq.~(\\ref{substi}).\nThis then compels the introduction of the appropriate quartic diagonal terms, as seen above.\n\nIn this connection, it is also important to note that modular completion involves\nmore than simply demanding that our final result be modular invariant.\nAfter all, we have seen in Eq.~(\\ref{Xmodcomplete}) that the modular completion of ${\\cal X}$ \ninvolves the addition of a pure number, {\\it i.e.}\\\/, the addition of \na quantity which is intrinsically modular-invariant on its own (or more precisely, \na quantity whose\ninsertion into the partition-function summand automatically preserves\nthe modular invariance of the original partition function). \nHowever, \nas we have stated above, modular completion \nensures more than the\nmere modular invariance of our final result --- it also ensures that\nthis result is obtainable\nthrough a modular-covariant sequence of calculational operations.\nAs we have seen, the extra additive constant that forms the modular completion\nof ${\\cal X}$ in Eq.~(\\ref{Xmodcomplete}) is crucial in allowing us to ``complete the square'' \nand thereby cast our results into the factorized form\nof Eq.~(\\ref{substi}) --- a form which itself emerged as a consequence of \nour underlying modular-covariant\n$z$-derivatives $D^2_z$.\nAs such, the constant appearing in Eq.~(\\ref{Xmodcomplete}) is an intrinsic part \nof our resulting expression for $m_\\phi^2$.\n\n\n\\subsection{Classical stability condition\\label{stability}}\n\nThus far, we have focused on deriving an expression for the Higgs mass, as defined in Eq.~(\\ref{higgsdef}). \nHowever, our results presuppose that we are discussing a classically stable \nparticle. In other words, while we are identifying the mass with the second $\\phi$-derivative of the\nclassical potential, we are implicitly assuming that the first $\\phi$-derivative vanishes so that we are sitting\nat a minimum of the Higgs potential.\nThus, there is an extra condition that we need to impose, namely \n\\begin{equation}\n {d \\Lambda(\\phi)\\over d \\phi } \\biggl|_{\\phi=0} ~=~ 0~.\n\\label{linearcond}\n\\end{equation}\nThis condition must be satisfied for the particular vacuum state \nwithin which our Higgs-mass calculation has been performed.\n\nIt is straightforward to determine the ramifications of this condition.\nProceeding exactly as above, we find in analogy with\nEq.~(\\ref{stuff})\nthat {\\mbox{$\\partial{\\cal Z}\/\\partial \\phi|_{\\phi=0}$}} corresponds to an insertion given by\n{\\mbox{$Y|_{\\phi=0} = {{\\cal Y}\/{\\cal M}}$}}, where\n\\begin{equation}\n {{\\cal Y}} ~\\sim~ \\tau_2 \\left( {\\bf Q}_R^t {\\bf Q}_h - {\\bf Q}_L^t \\tilde {\\bf Q}_h\\right) ~\\sim~ \\tau_2 \\,({\\bf Q}_R^t {\\cal T}_{21} {\\bf Q}_L)~. \n\\label{Ydef}\n\\end{equation}\nGiven this result, there are {\\it a priori}\\\/ three distinct ways in which \nthe condition in Eq.~(\\ref{linearcond}) can be satisfied \nwithin a given string vacuum.\nFirst, ${\\cal Y}$ might vanish for each state in the corresponding string spectrum.\nSecond, ${\\cal Y}$ might not vanish for each state in the string spectrum \nbut may vanish in the {\\it sum}\\\/ over the string states\n(most likely in a pairwise fashion between chiral and anti-chiral states with opposite charge vectors).\nHowever, there is also a third possibility:\nthe entire partition-function trace may be non-zero, even with the ${\\cal Y}$ insertion,\nbut nevertheless vanish when integrated over the fundamental domain of the modular group, as in Eq.~(\\ref{Lambdaphi}). \nIn general, very few mathematical examples are known of situations in which this latter\nphenomenon occurs~{\\mbox{\\cite{Moore:1987ue,Dienes:1990qh,Dienes:1990ij}}},\nalthough the fact that this would involve an integrand with vanishing modular weight offers\nunique possibilities.\n\n\nTwo further comments regarding this condition are in order.\nFirst, it is easy to verify that this condition respects modular invariance, as it must.\nIndeed, the quantity ${\\cal Y}$, as defined above, is already modular complete.\nAt first glance, this might seem surprising, given that the quartic terms within ${\\cal X}$ are nothing but the square of ${\\cal Y}$,\nand we have already seen that these quartic terms are not modular complete by themselves.\nHowever, it is the squaring of ${\\cal Y}$ \nthat introduces the higher powers of charge-vector components\nwhich in turn induce the modular anomaly. \nSecond, if ${\\cal Y}$ vanishes when summed over all of the string states,\nthen it might be tempting to hope that the quartic terms within ${\\cal X}$ \nalso vanish when summed over the string states.\nUnfortunately, this hope is not generally realized, since important sign information is lost when these quantities\nare squared.\nOf course, if ${\\cal Y}$ vanishes for each individual state in the string spectrum,\nthen the quartic terms within ${\\cal X}$ will also evaluate to zero in any calculation of the corresponding Higgs mass.\nThis would then simplify the explicit evaluation of ${\\cal X}$ for such a string vacuum.\n\n\n\\subsection{A relation between the Higgs mass and the cosmological constant}\n\nLet us now collect our results for the Higgs mass.\nFor notational simplicity we define \n\\begin{equation}\n \\langle A \\rangle ~\\equiv~ \\int_{\\cal F} \\dmu \n ~\\frac{\\tau_2^{-1}}{\\overline{\\eta}^{12} \\eta^{24}} \\, \\sum_{{\\bf Q}_L,{\\bf Q}_R} \n (-1)^F A~ {\\overline{q}}^{{\\bf Q}_R^2\/2} q^{{\\bf Q}_L^2\/2}~\n\\label{expvalue}\n\\end{equation}\nwhere the charge vectors $\\lbrace {\\bf Q}_L,{\\bf Q}_R\\rbrace$ in the sum over states are henceforth understood as \nunperturbed ({\\it i.e.}\\\/, with {\\mbox{$\\phi=0$}}) and thus correspond directly to the charges that arise at the minimum of the Higgs potential. \nOur results then together imply that \n\\begin{equation} \n m_\\phi^2 ~=~ -\\frac{{\\cal M}^2}{2} \\langle {\\cal X}\\rangle ~-~ \\frac{{\\cal M}^2}{2} \\frac{\\xi}{4\\pi^2} \\langle {\\bf 1} \\rangle\n\\label{preresult}\n\\end{equation}\nwhere ${\\cal X}$ is given in Eq.~(\\ref{Xdef}).\nAs indicated above, these results implicitly assume that\n{\\mbox{$\\langle {\\cal Y}\\rangle=0$}}, \nwhere ${\\cal Y}$ is defined in Eq.~(\\ref{Ydef}).\nHowever, we immediately recognize that the quantity $\\langle {\\bf 1}\\rangle$ within Eq.~(\\ref{preresult}) is nothing other than \nthe one-loop zero-point function (cosmological constant) $\\Lambda$!~\nMore precisely, we may identify $\\Lambda$ as {\\mbox{$\\Lambda(\\phi)|_{\\phi=0}$}} \n[where $\\Lambda(\\phi)$ is given in Eq.~(\\ref{Lambdaphi})], or equivalently\n\\begin{equation}\n \\Lambda ~=~ -\\frac{{\\cal M}^4}{2} \\,\\langle {\\bf 1}\\rangle~.\n\\label{lambdadeff}\n\\end{equation}\nWe thus obtain the relation\n\\begin{equation}\n m_\\phi^2 ~=~ \\frac{\\xi}{4\\pi^2} \\, \\frac{\\Lambda}{{\\cal M}^2} ~-~ \\frac{{\\cal M}^2}{2} \\, \\langle {\\cal X} \\rangle~.\n\\label{relation1}\n\\end{equation}\nIndeed, retracing our steps in arbitrary spacetime dimension $D$,\nwe obtain the analogous relation\n\\begin{equation}\n m_\\phi^2 ~=~ \\frac{\\xi}{4\\pi^2} \\, \\frac{\\Lambda}{{\\cal M}^{D-2}} ~-~ \\frac{{\\cal M}^2}{2} \\, \\langle {\\cal X} \\rangle~\n\\label{relation1b}\n\\end{equation}\nwhere the cosmological constant $\\Lambda$ now has mass dimension $D$. \n\nRemarkably, this is a general relation between the Higgs mass and the one-loop cosmological constant! \nBecause this relation rests on nothing but modular invariance,\nit holds generally for {\\it any}\\\/ perturbative \nclosed string in any arbitrary spacetime dimension $D$. \nThe cosmological-constant term in Eq.~(\\ref{relation1}) is universal, \nemerging as \nthe result of a modular anomaly that required a modular completion, or equivalently\nas the result of a universal shift in the moduli.\n By contrast, the second term depends on the particular charges that are inserted into the partition-function trace.\n\nFor weakly coupled heterotic strings,\nwe can push this relation one step further.\nIn such theories the string scale {\\mbox{$M_s\\equiv 2\\pi {\\cal M}$}} and Planck scale $M_p$ are connected through the \nrelation {\\mbox{$M_s= g_s M_P$}} where\n$g_s$ is the string coupling \nwhose value is set by the vacuum expectation value of the dilaton.\nDepending on the particular string model, $g_s$ in turn sets the values of the individual gauge couplings.\nLikewise, the canonically normalized scalar field $\\phi$ is \n{\\mbox{$\\widehat \\phi \\equiv \\phi\/g_s$}}. \nWe thus find that our relation in Eq.~(\\ref{relation1}) equivalently takes the form\n\\begin{equation}\n m_{\\widehat \\phi}^2 ~=~ \\frac{\\xi}{M_P^2} \\, \\Lambda ~-~ \\frac{g_s^2 {\\cal M}^2}{2} \\, \\langle {\\cal X} \\rangle~.\n\\label{relation2}\n\\end{equation}\n\n \nIn quantum field theory, we would not expect to find a relation between a Higgs mass\nand a cosmological constant. Indeed, quantum field theories do not involve gravity and are thus\ninsensitive to the absolute zero of energy.\nEven worse, in quantum field theory, the one-loop zero-point function is badly divergent. \nString theory, by contrast, not only unifies \ngauge theories with gravity but also yields a {\\it finite}\\\/ $\\Lambda$ (the latter\noccurring as yet another byproduct of modular invariance).\nThus, it is only within a string context that such a relation could ever arise, and indeed\nEqs.~(\\ref{relation1}) and (\\ref{relation2}) are precisely the relations \nthat arise for all weakly-coupled four-dimensional heterotic strings. \nWe expect that this is but the tip of the iceberg, and that other modular-invariant string constructions\nlead to similar results.\nIt is intriguing that such relations join together precisely the two quantities ($m_\\phi$ and $\\Lambda$) whose\nvalues lie at the heart of the two most pressing hierarchy problems in modern physics.\n\n \n\n\n\n\n\n\\section{Regulating the Higgs mass:~ From amplitudes to supertraces \\label{sec3}}\n\n\n\n\nIn Eq.~(\\ref{relation1}) we obtained a result in which the Higgs mass, via the definition in Eq.~(\\ref{expvalue}),\n is expressed in terms of certain one-loop string amplitudes\nconsisting of \nmodular integrals of various traces over the entire string spectrum.\nAs discussed below Eq.~(\\ref{eq:massRL}),\nthese traces include the contributions of not only {\\it physical}\\\/ ({\\it i.e.}\\\/, level-matched) string states with {\\mbox{$M_L^2=M_R^2$}},\nbut also {\\it unphysical}\\\/ ({\\it i.e.}\\\/, non level-matched) string states with {\\mbox{$M_L^2 \\not= M_R^2$}}.\nThis distinction between physical and unphysical string states is important because only \nthe physical string states can serve as {\\it bona-fide}\\\/ in- and out-states.\nBy contrast, the unphysical states are intrinsically stringy and have no field-theoretic analogues.\n\n\nWe now wish to push our calculation several steps further.\nIn particular, there are three aspects to our result \nin Eq.~(\\ref{relation1}) which we will need to understand \nin order to allow us to make contact with traditional quantum-field-theoretic expectations. \nThe first concerns the fact that while the one-loop vacuum energy $\\Lambda$ which appears in these\nresults is finite for all tachyon-free string models --- even without spacetime supersymmetry --- the\nremaining amplitude $\\langle {\\cal X}\\rangle$ which appears in these \nexpressions is generically divergent.\nNote that this is not in conflict with string-theoretic expectations; \nin particular, as we shall discuss in Sect.~\\ref{UVIRequivalence},\nstring theory generally softens various field-theoretic divergences \nbut need not remove them entirely. \nThus, our expression for the Higgs mass is formally divergent and requires some sort of regulator \nin order to extract finite results. \nSecond, while these results are expressed in terms of sums over the entire string spectrum,\nwe would like to be able to express\nthe Higgs mass directly in terms of supertraces over only the {\\it physical}\\\/ string states --- {\\it i.e.}\\\/, the states\nwith direct field-theoretic analogues.\nThis will ultimately allow us to express the Higgs mass in a form that might be recognizable within ordinary quantum field\ntheory, and thereby extract an effective field theory (EFT) description of the Higgs mass\nin which our Higgs mass experiences an effective renormalization-group ``running''. \nThis will also allow us\nto extract a stringy effective potential for the Higgs field. \nFinally, as a byproduct, we would also like to implicitly perform the stringy modular integrations \ninherent in Eq.~(\\ref{expvalue}). \n\nAs it turns out, these three issues are intimately related.\nHowever, appreciating these connections requires \na deeper understanding of the properties of the modular functions\non which which our Higgs-mass calculations rest.\nIn this section, we shall therefore outline the \nmathematical procedures which will enable us to address all three of our goals.\nMany of these methods originated in the classic mathematics papers of Rankin~{\\mbox{\\cite{rankin1a,rankin1b}}} and Selberg~\\cite{selberg1} \nfrom the late 1930s, and were later extended in an important way by Zagier~\\cite{zag} in the early 1980s.\nSome of the Rankin-Selberg results also later independently found their way \ninto the string literature in various forms~{\\mbox{\\cite{McClain:1986id,OBrien:1987kzw,Kutasov:1990sv}}},\nand have occasionally been studied and further developed (see, {\\it e.g.}\\\/, Refs.~\\cite{Dienes:1994np, Dienes:1995pm,\nDienes:2001se,\nAngelantonj:2010ic,\nAngelantonj:2011br,\nAngelantonj:2012gw,\nAngelantonj:2013eja,\nPioline:2014bra,\nFlorakis:2016boz}).\nOur purpose in recounting these results here is not only to pull them all together and explain their logical connections\nin relatively non-technical terms, but also to extend them in certain directions which will be important for our work in Sect.~\\ref{sec4}.~\nThis conceptual and mathematical groundwork\nwill thus form the underpinning for our further analysis of the Higgs mass in Sect.~\\ref{sec4}.\n\n\n\\subsection{The Rankin-Selberg technique\\label{sec:RStechnique}}\n\nWe are interested in \nmodular integrals such as those in Eq.~(\\ref{expvalue}) \nwhich generically take the form\n\\begin{equation}\n I~\\equiv ~\\int_{\\mathcal{F}}\\dmu \\,F(\\tau,{\\overline{\\tau}})~,\n\\label{eq:I}\n\\end{equation}\nwhere ${\\cal F}$ is the modular-group fundamental domain given in Eq.~(\\ref{Fdef}),\nwhere $d\\tau_1 d\\tau_2\/\\tau_2^2$ is the modular-covariant integration measure\n(with {\\mbox{$\\tau\\equiv \\tau_1+i\\tau_2$}}, {\\mbox{$\\tau_i\\in\\mathbb{R}$}}), \nand where the integrand $F$ is modular invariant.\nIn general the integrands $F$ take the form\n\\begin{equation}\n F~\\equiv~ \\tau_2^k\\, \\sum_{m,n} a_{mn} {\\overline{q}}^m q^n\n\\label{integrand}\n\\end{equation}\nwhere {\\mbox{$q\\equiv e^{2\\pi i\\tau}$}}\nand where $k$\nis the modular weight of the holomorphic and anti-holomorphic modular functions whose products contribute\nto $F$.\nNote that integrands of this form include those in Eq.~(\\ref{Zform}): we simply power-expand\nthe $\\eta$-function denominators and absorb these powers into \n$m$ and $n$.\nThus, with string integrands written as in Eq.~(\\ref{integrand}) we can now directly identify\n{\\mbox{$m= \\alpha' M_R^2\/4$}} and\n{\\mbox{$n= \\alpha' M_L^2\/4$}}.\nThe quantity $a_{mn}$ then tallies the number of bosonic minus fermionic string degrees of freedom\ncontributing to each $(M_R^2,M_L^2)$ term.\n\n\nInvariance under {\\mbox{$\\tau\\to \\tau+1$}} guarantees that every term within $F$ has {\\mbox{$m-n\\in\\mathbb{Z}$}}.\nThe {\\mbox{$m=n$}} terms represent the contributions from physical string states with spacetime masses {\\mbox{$\\alpha' M^2 = 2(m+n)= 4n$}}, \nwhile the {\\mbox{$m\\not=n$}} terms represent the contributions from off-shell ({\\it i.e.}\\\/, unphysical) string states.\nWithin the {\\mbox{$\\tau_2\\geq 1$}} integration subregion within ${\\cal F}$,\nthe {\\mbox{$m\\not=n$}} terms make no contribution to the integral $I$ \nbecause these contributions are eliminated when we perform the \n$\\int_{-1\/2}^{1\/2} d\\tau_1$ integral.\n[Indeed, within this subregion of ${\\cal F}$\nexpressions such as Eq.~(\\ref{eq:I}) come\nwith an implicit instruction \nthat we are to perform the $\\tau_1$ integration prior to performing the $\\tau_2$ integration.]\nHowever, the full integral $I$ does receive {\\mbox{$m\\not=n$}} contributions \nfrom the {\\mbox{$\\tau_2<1$}} subregion within ${\\cal F}$.\nThus, in general, both physical and unphysical string states contribute to amplitudes such as $I$.\n\nOur goal is to express $I$ in terms of contributions from the physical string states alone.\nClearly this could be done if we could somehow transform the region of integration within $I$\nfrom the fundamental domain ${\\cal F}$ to the positive half-strip \n\\begin{equation}\n {\\cal S} ~\\equiv~ \\lbrace \\tau :\\, -{\\textstyle{1\\over 2}} <\\tau_1\\leq {\\textstyle{1\\over 2}}, \\, \\tau_2>0 \\rbrace~,\n\\label{Sdef}\n\\end{equation}\nfor we would then have\n\\begin{equation}\n \\int_{\\cal S} \\dmu \\,F(\\tau,{\\overline{\\tau}}) ~=~ \\int_0^\\infty {d\\tau_2 \\over \\tau_2^2} \\, g(\\tau_2)\n\\end{equation}\nwhere $g(\\tau_2)$ is our desired trace over only the physical string states:\n\\begin{equation}\n g(\\tau_2) ~=~ \\int_{-1\/2 }^{1\/2} d\\tau_1 \\,F(\\tau,{\\overline{\\tau}})~=~ \\tau_2^k\\, \\sum_{n} a_{nn} \\,e^{-4\\pi\\tau_2 n}~.~~~~\n\\label{gtrace}\n\\end{equation} \n\nFortunately, there exists a well-known method for ``unfolding'' ${\\cal F}$ into ${\\cal S}$.\nWhile ${\\cal F}$ is the fundamental domain of the modular group $\\Gamma$ generated by both {\\mbox{$\\tau\\to -1\/\\tau$}} and {\\mbox{$\\tau\\to \\tau+1$}},\nthe strip ${\\cal S}$ is the fundamental domain of the modular {\\it subgroup}\\\/ {\\mbox{$\\Gamma_\\infty$}} generated solely by {\\mbox{$\\tau\\to \\tau+1$}}.\n(Indeed, this is the subgroup that preserves the cusp at {\\mbox{$\\tau=i\\infty$}}.) \nThus the strip ${\\cal S}$ can be realized as the sum of the images of ${\\cal F}$ transformed under all modular transformations $\\gamma$ (including the identity) in \nthe coset {\\mbox{$\\Gamma_\\infty \\backslash \\Gamma$}}:\n\\begin{equation}\n {\\cal S} ~=~ \\bigcup\\limits_{\\gamma\\in \\Gamma_\\infty \\backslash \\Gamma} \\gamma\\cdot {\\cal F} ~.\n\\label{stripF}\n\\end{equation}\nIt then follows \nfor any integrand $\\widetilde F(\\tau,{\\overline{\\tau}})$ \nthat\n\\begin{equation}\n \\int_{\\cal S} \\dmu \\, \\widetilde F(\\tau,{\\overline{\\tau}}) ~=~ \\int_{\\cal F} \\dmu \n \\sum_{\\gamma\\in \\Gamma_\\infty \\backslash \\Gamma} \\widetilde F_\\gamma(\\tau,{\\overline{\\tau}})~,\n\\label{unfold}\n\\end{equation}\nwhere $\\widetilde F_\\gamma(\\tau,{\\overline{\\tau}})$ is the $\\gamma$-transform of $\\widetilde F(\\tau,{\\overline{\\tau}})$.\nMoreover, if $\\widetilde F(\\tau,{\\overline{\\tau}})$ is invariant under {\\mbox{$\\tau\\to\\tau+1$}}, then the total integrand on\nthe right side of Eq.~(\\ref{unfold}) is modular invariant.\n\nAt this stage, armed with the result in Eq.~(\\ref{unfold}), \nwe see that we are halfway towards our goal.\nHowever, two fundamental problems remain.\nFirst, while choosing $\\widetilde F$ as our original integrand $F$ would allow us to express the left side of Eq.~(\\ref{unfold}) \ndirectly in terms of the desired trace in Eq.~(\\ref{gtrace}), our need to relate this to the original integral $I$ in \nEq.~(\\ref{eq:I}) would instead seem to require choosing $\\tilde F$ such that \n{\\mbox{$F= \\sum_{\\gamma\\in \\Gamma_\\infty \\backslash \\Gamma} \\widetilde F_\\gamma$}}.\nSecond, the manipulations underlying Eq.~(\\ref{unfold}), such as the exchanging of sums and regions of integration, implicitly \nassumed that the integrand on the right side of Eq.~(\\ref{unfold}) converges sufficiently rapidly as {\\mbox{$\\tau_2\\to\\infty$}} \n[or equivalently that the integrand on the left side of Eq.~(\\ref{unfold}) converges sufficiently rapidly as {\\mbox{$\\tau_2\\to 0$}}]\nso that all relevant integrals are absolutely convergent.\nHowever this is generally not the case for the physical situations that will interest us.\n\nIt turns out that these problems together motivate a unique choice for $\\widetilde F$.\nNote that $g(\\tau_2)$ generally has a form resembling that in Eq.~(\\ref{gtrace}), consisting of an infinite sum multiplied by a power of $\\tau_2$. As {\\mbox{$\\tau_2\\to 0$}}, \nthe successive terms in this sum are less and less suppressed by the exponential factor $e^{-4\\pi n\\tau_2}$. We therefore expect the infinite sum within $g(\\tau_2)$ to experience an increased tendency to diverge as {\\mbox{$\\tau_2\\to 0$}}. Let us assume for the moment that the divergence of this infinite sum grows no faster than some inverse power of $\\tau_2$ as {\\mbox{$\\tau_2\\to 0$}}. In this case, \nthe divergence of the sum within $g(\\tau_2)$ will cause \n$g(\\tau_2)$ itself to diverge as {\\mbox{$\\tau_2\\to 0$}} unless $g(\\tau_2)$ also includes a prefactor consisting of sufficiently many powers of $\\tau_2$ to hold the divergence of the sum in check. We can therefore {\\it regulate}\\\/ our calculation by introducing sufficiently many \nextra powers of $\\tau_2$ into $g(\\tau_2)$. In other words, in such cases we shall take\n\\begin{equation}\n \\widetilde F(\\tau,{\\overline{\\tau}}) ~=~ \\tau_2^s \\, F(\\tau,{\\overline{\\tau}})\n\\label{extratau2}\n\\end{equation}\nwhere $s$ is chosen sufficiently large (typically requiring {\\mbox{$s>1$}}) so as to guarantee convergence.\nIndeed, since the number of powers of $\\tau_2$ within $g(\\tau_2)$ is generally correlated in string theory\nwith the number of uncompactified spacetime dimensions,\nwe may view this insertion of extra powers of $\\tau_2$ as a stringy version of dimensional regularization,\ntaking {\\mbox{$D\\to D_{\\rm eff} \\equiv D-2s$}}.\nHowever, since our original integrand $F(\\tau,{\\overline{\\tau}})$ is presumed modular invariant,\nthe choice in Eq.~(\\ref{extratau2}) in turn implies that the integrand on the right side of Eq.~(\\ref{unfold}) must\nbe taken as\n\\begin{equation}\n \\sum_{\\gamma \\in \\Gamma_\\infty \\backslash \\Gamma} ({\\rm Im}\\, \\gamma\\cdot \\tau )^{s} \\,F_\\gamma(\\tau,{\\overline{\\tau}}) ~=~\n E(\\tau,{\\overline{\\tau}},s) \\,F(\\tau,{\\overline{\\tau}})\n\\end{equation}\nwhere $E(\\tau,{\\overline{\\tau}},s)$ is the {\\it non-holomorphic Eisenstein series}, often simply denoted $E(\\tau,s)$ \nand defined by \n\\begin{equation}\n E(\\tau,s) ~\\equiv~ \\sum_{\\gamma\\in \\Gamma_\\infty \\backslash \\Gamma} \n [{\\rm Im}\\, (\\gamma\\cdot \\tau) ]^{s} ~=~ \n {\\textstyle{1\\over 2}} \\, \\sum_{(c,d)=1} \\frac{\\tau_2^s}{|c\\tau+d|^{2s}}~\n\\label{Eisenstein}\n\\end{equation}\nwith the second sum in Eq.~(\\ref{Eisenstein}) restricted to integer, relatively prime values of $c,d$.\nThus, with these choices, we now have\n\\begin{equation}\n \\int_{\\cal F} \\dmu \\, E(\\tau,s) F(\\tau,{\\overline{\\tau}})~=~ \\int_0^\\infty d\\tau_2 \\,\\tau_2^{s-2} g(\\tau_2) ~ \n\\label{stepone}\n\\end{equation}\nwhere the expression on the right side depends on only the physical string states.\n\nThe Eisenstein series $E(\\tau,s)$ \nhas a number of important properties. \nIt is convergent for all {\\mbox{$s>1$}}, \nbut can be analytically continued to all values of $s$.\nIt is not only modular invariant (consistent with ${\\cal F}$ as the corresponding region of \nintegration), but its insertion on the left side of Eq.~(\\ref{stepone}) relative to our original starting point in \nEq.~(\\ref{eq:I}) softens the divergence as {\\mbox{$\\tau_2\\to\\infty$}}, as required. \nMost importantly for our purposes,\nhowever, this function has a simple pole at {\\mbox{$s=1$}}, with a $\\tau$-independent residue $3\/\\pi$.\nThe fact that this residue is $\\tau$-independent means that we can formally extract \nour original integral $I$ in Eq.~(\\ref{eq:I}) by taking the {\\mbox{$s=1$}} residue of both sides of Eq.~(\\ref{stepone}):\n\\begin{equation}\n I ~=~ \\frac{\\pi}{3}\\, \\oneRes\\, \\int_0^\\infty d\\tau_2 \\,\\tau_2^{s-2} \\,g(\\tau_2) ~. \n\\label{RSresult}\n\\end{equation}\nWe have therefore succeeded in expressing our original modular integral $I$ in terms of only the contributions\nfrom the physical states.\nThe result in Eq.~(\\ref{RSresult}) was originally obtained by Rankin and Selberg in 1939 \n(see, {\\it e.g.}\\\/, Refs.~{\\mbox{\\cite{rankin1a,rankin1b,selberg1}}}), \nand has proven useful for a number of applications in both physics and pure mathematics. \n\nAt this stage,\nthree important comments are in order. \nFirst, it may seem that the result in Eq.~(\\ref{RSresult}) implies that the unphysical states \nultimately make no contributions to the amplitude $I$. However, this is untrue: the result in Eq.~(\\ref{RSresult}) was derived under\nthe supposition that our original integrand $F(\\tau,{\\overline{\\tau}})$ is modular invariant, and this modular invariance \ndepends crucially on the existence of both physical and unphysical states in the full string spectrum.\nFor example, through the requirement of modular invariance,\nthe distribution of unphysical states in the string spectrum has a profound effect~{\\mbox{\\cite{Dienes:1994np,Dienes:1995pm}}} \non the values of the physical-state degeneracies $\\lbrace a_{nn}\\rbrace$ \nwhich appear in Eq.~(\\ref{gtrace}).\n\nAs our second comment, we point out that the above results can be reformulated in a manner which\neliminates the $\\tau_2$ integration completely and which depends directly on the integrand $g(\\tau_2)$.\nTo see this, we note if we define $I(s)$ as the term on the left\nside of Eq.~(\\ref{stepone}),\nthen the relation in \nEq.~(\\ref{stepone}) simply states that \n$I(s)$ is nothing but the\nMellin transform of $g(\\tau_2)\/\\tau_2$. \nOne can therefore use the inverse Mellin transform to write $g(\\tau_2)\/\\tau_2$ directly in terms of $I(s)$.\nWhile such an inverse relation is useful in many contexts, \nfor our purposes it will be sufficient to note that such an inverse relation implies a direct connection\nbetween the poles of $I(s)$ and the asymptotic behavior of $g(\\tau_2)$ as {\\mbox{$\\tau_2\\to 0$}}.\nSpecifically, one finds a correlation\n\\begin{eqnarray}\n && {\\rm poles~of}~ I(s) ~{\\rm at}~ s=s_n ~{\\rm with~residues} ~ c_n \\nonumber\\\\\n && ~~~~\\Longrightarrow~~ g(\\tau_2)\\sim \\sum_n c_n \\tau_2^{1-s_n} ~~{\\rm as}~~\\tau_2\\to 0~.~~~~~~~ \n\\label{correlations}\n\\end{eqnarray}\nAs we have seen, $I(s)$ has a single pole along the real axis at {\\mbox{$s=1$}}, with residue $3I\/\\pi$. \nHowever, $I(s)$ also has an infinite number of poles at locations {\\mbox{$s_n= \\rho_n\/2$}}, where\n$\\rho_n$ are the non-trivial zeros of the Riemann $\\zeta$-function $\\zeta(s)$. \nAccording to the Riemann hypothesis, these zeros all have the form {\\mbox{$\\rho_n = {\\textstyle{1\\over 2}} \\pm i\\gamma_n $}}\nwhere {\\mbox{$\\gamma_n\\in\\mathbb{R}$}}.\nThe fact that {\\mbox{${\\rm Re}\\\/(s_n)<1$}} for all of these additional poles of $I(s)$ \nthen implies that the amplitude $I$ dominates the leading behavior of $g(\\tau_2)$ as {\\mbox{$\\tau_2\\to 0$}},\nallowing us to write~{\\mbox{\\cite{zag,Kutasov:1990sv}}}\n\\begin{equation}\n I~=~ \\frac{\\pi}{3}\\, \\lim_{\\tau_2\\to 0} g(\\tau_2)~.\n\\label{reformulation}\n\\end{equation}\nOf course, from Eq.~(\\ref{correlations}) we see that the {\\mbox{$\\tau_2\\to 0$}} limit of $g(\\tau_2)$ also contains\nsubleading oscillatory terms~\\cite{zag} corresponding to the non-trivial zeros of the $\\zeta$-function.\nThis suggests, through Eq.~(\\ref{gtrace}), that the $a_{nn}$ coefficients tend to oscillate in sign\nas {\\mbox{$n\\to \\infty$}}. This oscillating sign is in fact a consequence of the so-called \n``misaligned supersymmetry''~{\\mbox{\\cite{Dienes:1994np,Dienes:1995pm,Dienes:2001se}}} \nwhich is a generic property of all tachyon-free non-supersymmetric string models ---\na property whose existence is a direct consequence\nof modular invariance in general situations where $I$ is finite and {\\mbox{$F(\\tau,{\\overline{\\tau}})\\not=0$}}.\n\nOur final comment, however, is perhaps the most crucial.\nAs we have seen, the results in Eqs.~(\\ref{RSresult}) and (\\ref{reformulation}) were derived under the assumption,\nas stated within the above derivation,\nthat the infinite sum within the definition of $g(\\tau_2)$ in Eq.~(\\ref{gtrace}) \ndiverges no more rapidly than some inverse power of $\\tau_2$ as {\\mbox{$\\tau_2\\to 0$}}. This requirement was needed \nso that the introduction of sufficiently many $\\tau_2$ prefactors could suppress this divergence and render a finite result.\nUndoing the modular transformations involved in Eq.~(\\ref{unfold}),\nwe see that this is equivalent to demanding that our original integrand $F(\\tau,{\\overline{\\tau}})$ either fall, remain constant,\nor grow less rapidly than $\\tau_2$ as {\\mbox{$\\tau_2\\to \\infty$}}.\nIndeed, these are the conditions under which the Rankin-Selberg analysis is valid.\nNot surprisingly, these are also the conditions under which any integrand $F$ lacking terms with {\\mbox{$m=n<0$}} will produce a finite value for $I$.\n \n\n\\subsection{Regulating divergences\\label{Regulators}}\n\nThe techniques discussed in Sect.~\\ref{sec:RStechnique} are completely adequate\nfor situations in which the original amplitude $I$ is finite, with \nan integrand $F(\\tau,{\\overline{\\tau}})$ remaining finite or diverging less rapidly than $\\tau_2$ as {\\mbox{$\\tau_2\\to \\infty$}}.\nHowever, many physical situations of interest\n(including those we shall ultimately need to consider in this paper) \nlead to integrands $F(\\tau,{\\overline{\\tau}})$ which diverge more rapidly than this as {\\mbox{$\\tau_2\\to\\infty$}}.\nAs a result, the corresponding integral $I$ formally diverges and must be regulated.\n\nIn this section we shall discuss three different methods of regulating such amplitudes.\nThese methods are appropriate for cases \n--- such as we shall ultimately face ---\nin which \nthe integrand experiences a power-law divergence $\\sim \\tau_2^p$ with {\\mbox{$p\\geq 1$}} as {\\mbox{$\\tau_2\\to\\infty$}}.\nAs we shall see, these regulators each have different strengths and weaknesses,\nand thus it will prove useful to have all three at our disposal.\nIn particular, two of these regulators will explicitly break modular invariance,\nbut are closer in spirit to those that are traditionally\nemployed in ordinary quantum field theory.\nBy contrast, the third regulator will be fully modular invariant.\nBy comparing the results we will then be able \nto discern the novel effects \nthat emerge through a fully modular-invariant regularization procedure\nand understand the reasons why such a regulator is greatly superior to the others.\n\nAll three of these regulators proceed from the same fundamental observation.\nLet us suppose that $F(\\tau,{\\overline{\\tau}})$ diverges\nat least as quickly as $\\tau_2$ as {\\mbox{$\\tau_2\\to\\infty$}}.\nClearly, this behavior \nwill cause the integral $I$ to diverge on the left side \nof Eq.~(\\ref{RSresult}).\nHowever, this behavior will also cause\n$g(\\tau_2)$ to diverge as {\\mbox{$\\tau_2\\to \\infty$}}, which means that the\nright side of Eq.~(\\ref{RSresult}) will also diverge.\nThus, in principle, a relation such as that in Eq.~(\\ref{RSresult}) will be rendered meaningless.\nHowever, if there were a consistent way of {\\it subtracting}\\\/ or {\\it regulating}\\\/ the appropriate divergence on each \nside of Eq.~(\\ref{RSresult}),\nwe can imagine that we might then obtain an analogous relation between a finite regulated \nintegral $\\widetilde I$\nand a corresponding finite regulated physical-state trace $\\widetilde g(\\tau_2)$.\nAs we shall see, all three of the regulators we shall discuss have this property and \nlead to results which are analogous to the result in Eq.~(\\ref{RSresult}) and relate regulated integrals to\nregulated physical-state supertraces.\n\n\n\\subsubsection{Minimal regulator\\label{sec:minimal}}\n\nPerhaps the simplest and most minimal regulator that can be envisioned~\\cite{zag} is one in which we\ndirectly excise the divergence from the integral $I$ without disturbing the rest of the integral. \nBecause the divergences on both sides of Eq.~(\\ref{RSresult}) arise as {\\mbox{$\\tau_2\\to \\infty$}}, \nwe can formally excise this region of integration\nfrom ${\\cal F}$ by defining a truncated region ${\\cal F}_t$ to be the same as ${\\cal F}$ but with the additional\nrestriction that {\\mbox{$\\tau_2 t$}} region (specifically as {\\mbox{$\\tau_2\\to\\infty$}}),\nwe can \n``undo'' the {\\mbox{$\\tau_2\\to \\infty$}} limit and alternatively define\n\\begin{equation}\n \\widetilde I(t) ~\\equiv~ \\int_{{\\cal F}_t} \\dmu \\, F + \\int_{{\\cal F}-{\\cal F}_t} \\dmu \\, [ F- \\Phi(\\tau_2)]~\n\\label{Itdef}\n\\end{equation} \nwhere we shall continue to assume that {\\mbox{$F(\\tau,{\\overline{\\tau}})\\sim \\Phi(\\tau_2)+...$}} as {\\mbox{$\\tau_2\\to\\infty$}}.\n Note that $\\widetilde I(t)$ is convergent for all finite $t$, as we desire. Moreover,\nbecause the second term in Eq.~(\\ref{Itdef}) has an integrand which is convergent throughout \nthe integration region ${\\cal F}-{\\cal F}_t$, taking the {\\mbox{$t\\to\\infty$}} limit eliminates the second term and\n$\\widetilde I(t)$ reproduces our original unregulated integral $I$ in Eq.~(\\ref{eq:I}).\nThus $\\widetilde I(t)$ represents an alternative, $t$-dependent method of regulating our original\nintegral $I$, one which is distinct from the minimal regularization $\\widetilde I$ \nin Eq.~(\\ref{tildeIdef}). \n\n\nThese two regularizations are deeply connected, however \n --- a fact which will also enable us to express $\\widetilde I(t)$ in terms of supertraces,\njust as we did for $\\widetilde I$.\nNote that the only $t$-dependence within $\\widetilde I(t)$ arises from the \nintegration of the subtraction term $\\Phi(\\tau_2)$\nalong the $t$-dependent lower boundary \nof the integration region ${\\cal F}-{\\cal F}_t$.\nWe thus see that the subtraction term $\\Phi(\\tau_2)$\nwhich regularizes $\\widetilde I(t)$ \nintroduces a non-trivial dependence on $t$ such that\n\\begin{equation}\n \\widetilde I(t)~=~ \\Phi_I(t) + {\\cal C}\n\\label{tdependence}\n\\end{equation}\n where we recall that $\\Phi_I(\\tau_2)$ is the anti-derivative of $\\Phi(\\tau_2)\/\\tau_2^2$\nand where ${\\cal C}$ is an as-yet unknown $t$-independent quantity.\nHowever, it is easy to solve for ${\\cal C}$.\nGiven that {\\mbox{${\\cal C}= \\widetilde I(t) - \\Phi_I(t)$}}, we immediately see \nby taking the {\\mbox{$t\\to \\infty$}} limit of both sides and comparing with Eq.~(\\ref{tildeIdef}) that\n{\\mbox{$\\lim_{t\\to \\infty}{\\cal C} = \\widetilde I$}}.\nHowever, ${\\cal C}$ is independent of $t$, which means that {\\mbox{${\\cal C}=\\widetilde I$}}\nfor {\\it any}\\\/ value of {\\mbox{$t\\geq 1$}}.\nWe thus obtain a general relation, valid for all {\\mbox{$t\\geq 1$}},\nbetween our two regulators: \n\\begin{equation}\n \\widetilde I(t) ~=~ \\widetilde I +\\Phi_I(t)~.\n \\end{equation}\nOur previous result for $\\tilde I$ in \nEq.~(\\ref{Zresult}) then yields~\\cite{zag} \n\\begin{equation}\n \\widetilde I(t) ~=~ \\frac{\\pi}{3}\\, \\oneRes \\, \\int_0^\\infty d\\tau_2 \\,\\tau_2^{s-2}\\, \\widetilde g(\\tau_2) ~+~ \n \\Phi_I(t)~+~ \\widetilde\\Phi~.\n\\end{equation}\n\nThus, just as with our minimal regulator, we find that our $t$-dependent regulator\nproduces a finite integral $\\widetilde I(t)$ which continues to be expressible\nin terms of a physical-state supertrace.\nIndeed, for the divergence structure {\\mbox{$\\Phi(\\tau_2)= c_0+c_1\\tau_2$}},\nwe find that\nour $t$-dependent regularized integral \n\\begin{equation}\n \\widetilde I(t) ~\\equiv~ \\int_{{\\cal F}_t} \\dmu \\, F + \\int_{{\\cal F}-{\\cal F}_t} \\dmu \\, ( F- c_0 - c_1\\tau_2)~\n\\label{I2}\n\\end{equation} \nis given by\n\\begin{eqnarray}\n \\widetilde I(t) \n ~&=&~ \\frac{\\pi}{3}\\, \\oneRes \\, \\int_0^\\infty d\\tau_2 \\,\\tau_2^{s-2}\\, \n \\biggl\\lbrack g(\\tau_2) -c_0-c_1 \\tau_2 \\biggr\\rbrack ~~~~\\nonumber\\\\\n &&~ ~ + \\left( \\frac{\\pi}{3} - {1\\over t}\\right) \\, c_0 + \n \\log\\left( 4\\pi \\,t\\, e^{-\\gamma}\\right) c_1~.\n\\label{Zagierresult2}\n\\end{eqnarray}\n\nOnce again, $c_0$ plays a special role in this result\nbecause the presence of the $c_0$ term within $\\Phi(\\tau_2)$ does not lead\nto a divergence.\nIndeed, given that the region ${\\cal F}-{\\cal F}_t$ has volume $1\/t$ with respect to the $\\dmu$ measure, \nwe see that the subtraction of $c_0$ within Eq.~(\\ref{I2}) simply removes\na finite quantity $c_0\/t$ from the value of $\\widetilde I(t)$.\nFor integrands having this divergence structure\nwe can therefore define a {\\it modified}\\\/ (or {\\it improved}\\\/) non-minimal regulator\n\\begin{eqnarray}\n \\widehat I(t) ~&\\equiv&~ \\widetilde I(t) + c_0\/t \\nonumber\\\\\n ~&=&~ \\int_{{\\cal F}_t} \\dmu \\, F + \\int_{{\\cal F}-{\\cal F}_t} \\dmu \\, ( F - c_1\\tau_2)~,~~~~\n\\label{I3}\n\\end{eqnarray} \nwhereupon we find from Eq.~(\\ref{Zagierresult2}) that\n\\begin{eqnarray}\n \\widehat I(t) \n ~&=&~ \\frac{\\pi}{3}\\, \\oneRes \\, \\int_0^\\infty d\\tau_2 \\,\\tau_2^{s-2}\\, \n \\biggl\\lbrack g(\\tau_2) -c_0-c_1 \\tau_2 \\biggr\\rbrack ~~~~~\\nonumber\\\\\n &&~ ~ + \\frac{\\pi}{3} \\, c_0 + \\log\\left( 4\\pi \\,t\\, e^{-\\gamma}\\right) c_1~.\n\\label{Zagierresult3}\n\\end{eqnarray}\nIn other words, the $1\/t$-dependence on the right side of Eq.~(\\ref{Zagierresult2})\nwas in some sense spurious, reflecting a corresponding $1\/t$-dependence that was\nneedlessly inserted into the regulator definition in Eq.~(\\ref{I2}) and which has\nnow been removed from Eqs.~(\\ref{I3}) and (\\ref{Zagierresult3}).\nThe right side of Eq.~(\\ref{Zagierresult3}) is then independent of $c_0$ in the manner discussed\nbelow Eq.~(\\ref{Zagierresult}) for the minimal regulator.\n\n\n\\subsubsection{Modular-invariant regulators \\label{sec:modinvregs}}\n\nAlthough our results in Eqs.~(\\ref{Zagierresult}), (\\ref{Zagierresult2}), and (\\ref{Zagierresult3}) \nwere each derived in a manner that remained true to the modular-invariant unfolding procedure,\nneither side of these relations is modular invariant \nby itself. \nIn other words, even though \nthese relations correctly\nallow us to \nexpress our regulated\nintegrals $\\widetilde I$, $\\widetilde I(t)$, and $\\widehat I(t)$ in terms of a corresponding \nregulated physical-state supertrace $\\widetilde g(\\tau_2)$,\nneither $\\widetilde I$, $\\widetilde I(t)$, nor $\\widehat I(t)$ is itself a modular-invariant quantity. \nThis is an important observation because \nthese latter quantities will ultimately correspond to physical observables \nwithin the modular-invariant string context.\nWe must therefore additionally require that these observables themselves be modular invariant.\n\nThe issue, of course, is that \nneither $\\widetilde I$, $\\widetilde I(t)$, nor $\\widehat I(t)$ \nincorporates a modular-invariant way of eliminating the associated divergences as {\\mbox{$\\tau_2\\to\\infty$}}. \nHowever, it is possible to design regulators in which such divergences are indeed eliminated in a \nfully modular-invariant way.\nIn this work we shall present a particular set of modular-invariant regulators which will have several useful properties for our purposes.\n\nIn order to define these regulators, let us first recall that the partition function of\na bosonic worldsheet field compactified on a circle of radius $R$ is given by\n\\begin{eqnarray}\n Z_{\\rm circ}(a,\\tau) &=&\n \\sqrt{ \\tau_2}\\,\n \\sum_{m,n\\in\\mathbb{Z}} \\,\n \\overline{q}^{(ma-n\/a)^2\/4} \\,q^{(ma+n\/a)^2\/4}\\nonumber\\\\\n &=& \\sqrt{ \\tau_2}\\,\n \\sum_{m,n\\in\\mathbb{Z}} \\,\n e^{-\\pi \\tau_2 (m^2 a^2 + n^2 \/a^2)} \\, e^{2\\pi i mn \\tau_1}~~~\n\\nonumber\\\\\n\\label{Zcircdef}\n\\end{eqnarray}\nwhere we have defined the dimensionless inverse radius\n{\\mbox{$a\\equiv \\sqrt{\\alpha'}\/R$}}.\nHere the sum over $m$ and $n$ represents \nthe sum over all possible KK momentum and winding modes, respectively.\nNote that {\\mbox{$Z_{\\rm circ}\\to 1\/a$}} as {\\mbox{$a\\to 0$}},\nwhile {\\mbox{$Z_{\\rm circ}\\to a$}} as {\\mbox{$a\\to \\infty$}}.\nAs expected, $Z_{\\rm circ}(a,\\tau)$ is modular invariant for any $a$.\nUsing $Z_{\\rm circ}(a,\\tau)$, we shall then \nregulate any divergent integral of the form in Eq.~(\\ref{eq:I}) \nby defining\na corresponding series of regulated integrals $\\widetilde I_\\rho (a)$:\n\\begin{equation}\n \\widetilde I_\\rho (a) ~\\equiv~ \\int_{{\\cal F}} \\dmu \\, F(\\tau) \\, {\\cal G}_\\rho(a,\\tau)\n\\label{Iadef}\n\\end{equation}\nwhere our regulator functions \n${\\cal G}_\\rho(a,\\tau)$ are defined for any {\\mbox{$\\rho\\in \\mathbb{R}^+$}}, {\\mbox{$\\rho\\not=1$}}, as\n\\begin{equation}\n {\\cal G}_\\rho(a,\\tau) ~\\equiv~ \n A_\\rho\\, a^2 \\frac{\\partial}{\\partial a} \\biggl\\lbrack Z_{\\rm circ}( \\rho a,\\tau) - Z_{\\rm circ}(a,\\tau)\\biggr\\rbrack~ \n\\label{regG}\n\\end{equation} \nwhere {\\mbox{$A_\\rho\\equiv \\rho\/(\\rho-1)$}} is an overall normalization factor.\nNote that ${\\cal G}_\\rho(a,\\tau)$ inherits its modular invariance from $Z_{\\rm circ}$, thereby rendering the regulated\nintegral $\\widetilde I_\\rho(a)$ in Eq.~(\\ref{Iadef}) fully modular invariant for any $a$ and $\\rho$. \nWe further note that ${\\cal G}_\\rho(a,\\tau)$ satisfies the identity\n\\begin{equation}\n {\\cal G}_\\rho(a,\\tau) ~=~ {\\cal G}_{1\/\\rho}(\\rho a,\\tau)~.\n\\label{rhoflipidentity}\n\\end{equation}\nWe can therefore take {\\mbox{$\\rho>1$}} without loss of generality. \n\n\n\n\nThese functions ${\\cal G}_\\rho(a,\\tau)$ have two important properties which \nmake them suitable as regulators\nwhen {\\mbox{$a\\ll 1$}}.\nFirst, as {\\mbox{$a\\to 0$}}, we find that {\\mbox{${\\cal G}_\\rho(a,\\tau)\\to 1$}} for all $\\tau$.\nThus the {\\mbox{$a\\to 0$}} limit restores our original unregulated theory.\nSecond, for any {\\mbox{$a>0$}}, we find that {\\mbox{${\\cal G}_\\rho(a,\\tau)\\to 0$}} {\\it exponentially rapidly}\\\/ as {\\mbox{$\\tau_2\\to\\infty$}}.\nThus the insertion of ${\\cal G}_\\rho (a,\\tau)$ into the integrand of Eq.~(\\ref{Iadef})\nsuccessfully eliminates whatever power-law divergence \nmight have otherwise arisen from the original integrand $F(\\tau)$. \nIndeed, we see that this now happens in a smooth, fully modular-invariant way rather than through\na sharp, discrete subtraction.\nMotivated by these two properties, we shall therefore focus on situations in which {\\mbox{$a\\ll 1$}}, \nas these are the situations in which our regulator preserves as much of the original theory as possible (as we expect\nof a good regulator) while simultaneously eliminating all power-law divergences as {\\mbox{$\\tau_2\\to\\infty$}}.\nIn fact, for the special case {\\mbox{$\\rho=2$}} and for the specific values \n{\\mbox{$a= 1\/\\sqrt{k+2}$}} \n the insertion of this regulator even has a direct physical interpretation, arising through a procedure in which \nthe various fields in the background string geometry are turned on in such \na way that the CFT associated with the flat four-dimensional spacetime\nis replaced by that associated with a $SU(2)_k$ WZW model~\\cite{Kiritsis:1994ta,Kiritsis:1996dn,Kiritsis:1998en}. \n\n\nThat said, there is one further property of these regulator functions ${\\cal G}_\\rho(a,\\tau)$ \nwhich will prove useful for our purposes.\nWhen {\\mbox{$a\\ll 1$}}, the contributions from all non-zero winding modes \nwithin $Z_{\\rm circ}$ (and ultimately within ${\\cal G}$)\nare exponentially suppressed relative to those of the KK momentum modes.\nIn other words, when {\\mbox{$a\\ll 1$}} we can effectively restrict our summation in Eq.~(\\ref{Zcircdef})\nto cases with {\\mbox{$n=0$}}.\nWe then find that ${\\cal G}_\\rho(a,\\tau)$ loses its dependence on $\\tau_1$,\nrendering ${\\cal G}_\\rho(a,\\tau)$ a function of $\\tau_2$ alone.\nIn such cases we shall simply denote our regulator function as ${\\cal G}_\\rho(a,\\tau_2)$.\n\n \nIn Fig.~\\ref{regulator_figure}, we have plotted \nthe regulator function ${\\cal G}_2(a,\\tau_2)$ within the $(a,\\tau_2)$ plane for {\\mbox{$\\tau_2\\geq 1$}} (left panel)\nand as a function of {\\mbox{$\\tau_2\\geq 1$}} for various discrete values of {\\mbox{$a\\ll 1$}} (right panel).\nWe see, as promised, that\n{\\mbox{${\\cal G}_2(a,\\tau_2) \\to 0$}} for all {\\mbox{$a>0$}} as {\\mbox{$\\tau_2\\to\\infty$}},\nwhile {\\mbox{${\\cal G}_2(a,\\tau_2)\\to 1$}} for all {\\mbox{$\\tau_2\\geq 1$}} as {\\mbox{$a\\to 0$}}.\nWe also note that this suppression for large $\\tau_2$ is quite pronounced,\neven for {\\mbox{$a\\ll 1$}}.\n\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[keepaspectratio, width=0.51\\textwidth]{KKregulator3D.pdf}\n\\hskip 0.12 truein\n\\includegraphics[keepaspectratio, width=0.45\\textwidth]{KKregulator.pdf}\n\\caption{\n{\\it Left panel}\\\/: The modular-invariant regulator function ${\\cal G}_2(a,\\tau_2)$, \nplotted within the $(a,\\tau_2)$ plane for {\\mbox{$a\\ll 1$}} and {\\mbox{$\\tau_2\\geq 1$}}.\n{\\it Right panel}\\\/: The modular-invariant regulator ${\\cal G}_2(a,\\tau_2)$, \nplotted as a function of $\\tau_2$ for\n{\\mbox{$a=0.05$}} (blue), {\\mbox{$a=0.1$}} (orange), and {\\mbox{$a=0.3$}} (green).\nIn all cases we see that {\\mbox{${\\cal G}_2(a,\\tau_2) \\to 0$}} for all {\\mbox{$a>0$}} as {\\mbox{$\\tau_2\\to\\infty$}},\nwhile {\\mbox{${\\cal G}_2(a,\\tau_2)\\to 1$}} for all {\\mbox{$\\tau_2\\geq 1$}} as {\\mbox{$a\\to 0$}}.\nIndeed, for {\\mbox{$a=0.05$}}, we see that {\\mbox{${\\cal G}_2(a,\\tau_2)\\approx 1$}} for all {\\mbox{$\\tau_2\\lsim 100$}}.\nThus for small non-zero $a$ this regulator succeeds in suppressing the divergences\nthat might otherwise arise as {\\mbox{$\\tau_2\\to\\infty$}} while nevertheless \nhaving little effect throughout the rest of\nthe fundamental domain.}\n\\label{regulator_figure}\n\\end{figure*}\n\nFor any $a$ and $\\rho$, we see \nfrom Fig.~\\ref{regulator_figure}\nthat there is a corresponding value $\\tau_2^\\ast$ \nwhich can be taken as characterizing the approximate $\\tau_2$-location of the transition between\nthe unregulated region with {\\mbox{${\\cal G}_\\rho(a,\\tau_2)\\approx 1$}} and \nthe regulated region with {\\mbox{${\\cal G}_\\rho(a,\\tau_2)\\approx 0$}}.\nFor example, we might define $\\tau_2^\\ast$ as the critical \nvalue corresponding to the top of the ``ridge'' in the left panel \nof Fig.~\\ref{regulator_figure} (or equivalently the maximum in the right panel of Fig.~\\ref{regulator_figure}).\nAlternatively, given the shapes of the functions in the right panel of Fig.~\\ref{regulator_figure},\nwe might define $\\tau_2^\\ast$ as the location \nat which ${\\cal G}_\\rho(a,\\tau_2)$ experiences an inflection from being concave-down to concave-up.\nFinally, a third possibility might be to define $\\tau_2^\\ast$ as\nthe value of $\\tau_2$ at which {\\mbox{${\\cal G}_\\rho(a,\\tau_2) =1\/2$}},\nrepresenting the ``midpoint'' between {\\mbox{${\\cal G}=1$}} and {\\mbox{${\\cal G}=0$}}.\nFor the {\\mbox{$\\rho=2$}} case shown in Fig.~\\ref{regulator_figure},\nwe then find for {\\mbox{$a\\ll 1$}} that each of these has a rather straightforward \nscaling behavior with $a^{-2}$:\n\\begin{eqnarray}\n \\hbox{ridge top:}~~~ && ~~ \\tau_2^\\ast ~\\approx~ \\frac{3}{2\\pi a^2} ~\\approx~ \\frac {0.477}{a^2}~\\nonumber\\\\\n \\hbox{inflection:}~~~ && ~~ \\tau_2^\\ast ~\\approx~ \\frac{3+\\sqrt{6}}{2\\pi a^2} ~\\approx~ \\frac {0.867}{a^2}~\n ~~~~~~\\nonumber\\\\\n \\hbox{{\\mbox{${\\cal G}=1\/2$}}:}~~~ && ~~ \\tau_2^\\ast ~\\approx~ \\frac{1.411}{a^2}~.\n\\label{tau2ast}\n\\end{eqnarray}\nIndeed, each of these results becomes increasingly accurate as {\\mbox{$a\\to 0$}}.\nMoreover, although the \nnumerical coefficient in the third case depends significantly on $\\rho$,\nthe numerical coefficients in the first two cases are actually independent of $\\rho$.\nIn all cases, however, \nwe see that ${\\cal G}_\\rho(a,\\tau_2)$ suppresses the contributions from regions of the fundamental domain\nwith {\\mbox{$\\tau_2\\gg \\tau_2^\\ast$}} while preserving the contributions from regions with {\\mbox{$1< \\tau_2\\ll \\tau_2^\\ast$}}.\nIndeed, this property holds regardless of our precise definition for $\\tau_2^\\ast$. \n\nArmed with these regulator functions ${\\cal G}_\\rho(a,\\tau_2)$, \nwe now wish to express the integral in Eq.~(\\ref{Iadef})\nin terms of an appropriately regulated supertrace over physical string states.\nHowever, given that $\\widetilde I_\\rho(a)$ is fully modular invariant and convergent as {\\mbox{$\\tau_2\\to\\infty$}},\nwe can simply use the original Rankin-Selberg result in Eq.~(\\ref{RSresult}).\nWe thus have\n\\begin{equation}\n \\widetilde I_\\rho (a) ~=~ \\frac{\\pi}{3}\\, \\oneRes \\, \\int_0^\\infty d\\tau_2 \\,\\tau_2^{s-2} \\,\\widetilde g_\\rho(a,\\tau_2) ~ \n\\label{Irhoa}\n\\end{equation}\nwhere, in analogy with Eq.~(\\ref{gtrace}), we have \n\\begin{equation}\n \\widetilde g_\\rho(a,\\tau_2) ~\\equiv~ \\int_{-1\/2}^{1\/2} d\\tau_1 \\, F(\\tau) \\, {{\\cal G}}_\\rho(a,\\tau)~.\n\\label{gFGdef}\n\\end{equation}\n\nIn general, for arbitrary $a$,\nthe regulator\n${\\cal G}_\\rho(a,\\tau)$ will have a traditional $(q,{\\overline{q}})$ power-expansion\nof the form {\\mbox{${\\cal G}\\sim \\sum_{r,s} b_{rs} {\\overline{q}}^r q^s$}}, just as we have\n{\\mbox{$F\\sim \\sum_{m,n} a_{mn}{\\overline{q}}^m q^n$}} in Eq.~(\\ref{integrand}). \nGiven this, \nwe find that the $\\tau_1$ integral in Eq.~(\\ref{gFGdef}) projects onto those states for which {\\mbox{$n-m= r-s$}}.\nHowever, ${\\cal G}_\\rho(a,\\tau)$ generally receives contributions from states with many different values of $r-s$.\nAs a result, $\\widetilde g_\\rho(a,\\tau_2)$ will generally receive contributions from not only the {\\it physical}\\\/ \n{\\mbox{$m=n$}} states within $F(\\tau)$ but also \nsome of the {\\it unphysical}\\\/ {\\mbox{$m\\not=n$}} states.\nIn other words, for general $a$, our regulator function ${\\cal G}_\\rho(a,\\tau)$ becomes entangled\nwith the physical-state trace in a way that allows unphysical states to contribute.\n\nAs we have seen, it is useful for practical purposes\nthat ${\\cal G}_\\rho(a,\\tau)$ loses its dependence on $\\tau_1$ when {\\mbox{$a\\ll 1$}}.\nIn other words, \nfor {\\mbox{$a\\ll 1$}}\nwe find that the contributions from terms with {\\mbox{$r\\not=s$}} within ${\\cal G}$ are suppressed.\nThe $\\tau_1$ integral in Eq.~(\\ref{gFGdef}) then projects onto only the {\\mbox{$m=n$}} physical states, as desired,\nand to a good approximation our expression for $\\widetilde g_\\rho(a,\\tau_2)$ in Eq.~(\\ref{gFGdef}) \nsimplifies to\n\\begin{equation}\n \\widetilde g_\\rho(a,\\tau_2) ~\\approx~ g(\\tau_2) \\, {{\\cal G}}_\\rho(a,\\tau_2)~\n\\label{gFGdef2}\n\\end{equation}\nwhere $g(\\tau_2)$ is our original unregulated physical-state trace in Eq.~(\\ref{gtrace}).\nThus, for {\\mbox{$a\\ll 1$}}, the same regulator function ${\\cal G}_\\rho(a,\\tau_2)$ which smoothly softens the \n{\\mbox{$\\tau_2\\to\\infty$}} divergence\nin the integrand $\\widetilde I_\\rho(a)$ also smoothly softens the \n{\\mbox{$\\tau_2\\to\\infty$}} divergence\nin the physical-state trace $\\widetilde g_\\rho(a,\\tau_2)$ --- all without introducing contributions from unphysical states.\nHowever, we shall later demonstrate that the integral \nin Eq.~(\\ref{Iadef}) can actually be performed exactly, yielding\nan expression in terms of purely physical states for all values of $a$.\n\n\nWhile these regulator functions ${\\cal G}_\\rho(a,\\tau_2)$ are suitable for many applications,\nit turns out that we can use these functions in order to \nconstruct additional modular-invariant regulators\nwhose symmetry properties transcend even those of ${\\cal G}_\\rho(a,\\tau_2)$.\nTo do this, we first observe from the modular invariance of ${\\cal G}_\\rho(a,\\tau_2)$ that \n\\begin{equation}\n {\\cal G}_\\rho(a, 1\/\\tau_2) ~=~ {\\cal G}_\\rho(a, \\tau_2) ~\n\\label{StransG}\n\\end{equation}\nfor any $\\rho$, $a$, and $\\tau_2$. Indeed, invariance under {\\mbox{$\\tau_2\\to 1\/\\tau_2$}} follows directly from\ninvariance under the modular transformation {\\mbox{$\\tau\\to -1\/\\tau$}} for {\\mbox{$\\tau_1=0$}}.\nSecond, the identity in Eq.~(\\ref{rhoflipidentity}) tells us that the parameters $(\\rho,a)$\nwhich define our ${\\cal G}$-functions\nhave a certain redundancy, such that the ${\\cal G}$-function with $(\\rho,a)$ is the same as\nthe ${\\cal G}$-function with $(1\/\\rho, \\rho a)$. \nIndeed, only the combination {\\mbox{$a'\\equiv \\sqrt{\\rho} a$}} is invariant under this redundancy. \n Thus, while Eq.~(\\ref{StransG}) provides a symmetry under reciprocal flips in $\\tau_2$,\nEq.~(\\ref{rhoflipidentity}) provides a symmetry under reciprocal flips in $\\rho$.\n\nGiven these two symmetries, it is natural to wonder\nwhether ${\\cal G}_\\rho(a,\\tau)$ also exhibits a reciprocal flip symmetry in\nthe one remaining variable {\\mbox{$a'\\equiv \\sqrt{\\rho} a$}}.\nThis would thus be a symmetry under {\\mbox{$a\\to 1\/\\rho a$}}, or equivalently under {\\mbox{$\\rho a^2\\to 1\/(\\rho a^2)$}}. \nIndeed, we shall find in Sect.~\\ref{sec:alignment} \nthat such an additional symmetry will be very useful and \nrender the modular symmetry manifest in certain cases where it would otherwise have been obscure. \nUnfortunately, ${\\cal G}_\\rho(a,\\tau)$ does not exhibit such a symmetry.\nOne might nevertheless wonder whether it is possible to modify this regulator\nfunction in such a way that it might exhibit this additional symmetry as well.\n\n \nIt turns out that this enhanced symmetry structure is relatively easy to arrange.\nFirst, we observe that $Z_{\\rm circ}(a,\\tau)$ is itself invariant under\n{\\mbox{$a\\to 1\/a$}} for any $\\tau$; indeed, this is the symmetry underlying T-duality for closed strings.\nGiven this, it is then straightforward to verify that \nthe functions\n\\begin{equation}\n \\widehat {\\cal G}_\\rho(a,\\tau) ~\\equiv~ \\frac{1}{1+ \\rho a^2} \\, {\\cal G}_\\rho(a,\\tau)~\n\\label{hatGdef}\n\\end{equation}\nnot only inherit all of the regulator properties and symmetries \ndiscussed above for ${\\cal G}_\\rho(a,\\tau)$ when {\\mbox{$a\\ll 1$}},\nbut are also manifestly invariant under {\\mbox{$a'\\to 1\/a'$}}, or equivalently {\\mbox{$a\\to 1\/(\\rho a)$}}, for any $\\tau$: \n\\begin{equation}\n \\widehat {\\cal G}_\\rho(a,\\tau) ~=~ \\widehat{\\cal G}_\\rho ( 1\/\\rho a, \\tau)~.\n\\label{newest}\n\\end{equation}\n We shall therefore take these $\\widehat {\\cal G}$-functions as defining our enhanced modular-invariant regulators.\nWe shall likewise define\ncorresponding enhanced regularized \nintegrals $\\widehat I_\\rho(a)$ as in Eq.~(\\ref{Iadef}), but with ${\\cal G}_\\rho(a,\\tau)$\nreplaced by $\\widehat {\\cal G}_\\rho(a,\\tau)$.\nWe then find that we can express $\\widehat I_\\rho(a)$ in terms of corresponding\nphysical-state traces $\\widehat g_\\rho(a,\\tau_2)$ as in\nEqs.~(\\ref{Irhoa}) through (\\ref{gFGdef2}),\nexcept with ${\\cal G}_\\rho(a,\\tau_2)$ replaced by $\\widehat {\\cal G}_\\rho(a,\\tau_2)$\nthroughout. \n\nThe enhanced regulators in Eq.~(\\ref{hatGdef})\ncan also be understood in a completely different way, through analogy with what we\nhave already observed for our non-minimal regulators in Sect.~\\ref{sec:nonminimal}.~\nAs discussed after Eq.~(\\ref{Zagierresult3}),\nthe quantity $\\widetilde I(t)$ defined through our non-minimal regulator\nultimately contained \na spurious $t$-dependence that could be removed without disturbing\nthe suitability of the regulator itself.\nIt is for this reason that \nwe were able to transition from our original\nnon-minimal regulator in Eq.~(\\ref{I2}) to \nour improved non-minimal regulator in Eq.~(\\ref{I3}) in which\nsuch spurious terms were eliminated.\n\nIt turns out that a similar situation arises for our original modular-invariant \nregulators ${\\cal G}_\\rho(a,\\tau_2)$.\nIndeed, as we shall find in Sect.~\\ref{sec4}, use of these regulators would have led to results with analogously spurious terms --- {\\it i.e.}\\\/,\nterms which obscure the underlying symmetries of the theory.\nHowever, just as with Eq.~(\\ref{I3}), it is possible to define\nimproved modular-invariant regulators \nin which such spurious effects are eliminated.\nIndeed, these improved modular-invariant regulators \nare nothing but the regulators $\\widehat {\\cal G}_\\rho(a,\\tau)$\nintroduced in Eq.~(\\ref{hatGdef}).\nAdditional reasons for adopting the $\\widehat {\\cal G}_\\rho(a,\\tau)$ regulators\nwill be discussed in Sect.~\\ref{sec:Conclusions}.~\nThese improved regulators will therefore be our main interest in this paper.\n\n\n\\subsubsection{Aligning the non-minimal and modular-invariant regulators\\label{sec:alignment}}\n\nNeedless to say, the most important feature of\nour modular-invariant regulators is precisely that they are modular invariant.\nUse of these regulators therefore provides a way of controlling the divergences that might\nappear in string amplitudes while simultaneously preserving the modular invariance that rests at the heart\nof all that we are doing in this paper.\n\nThis becomes especially apparent upon comparing these modular-invariant regulators with\nthe non-minimal regulators of Sect.~\\ref{sec:nonminimal}. Recall that the non-minimal regulators operate by isolating those terms\nwithin the partition function $F(\\tau)$ which would have led to a divergence as {\\mbox{$\\tau_2\\to \\infty$}}, and then performing\na brute-force subtraction of those terms over the entire region of the fundamental domain ${\\cal F}$\nwith {\\mbox{$\\tau_2\\geq t$}}.\nIn so doing, modular invariance is broken twice: first, in artificially separating those terms \nwithin the partition function which would have led to a divergence from those \nwhich do not; and second, in then selecting a particular sharp location {\\mbox{$\\tau_2=t$}} at which \nto perform the subtraction of these divergence-inducing terms, essentially multiplying these terms \nby $\\Theta(t-\\tau_2)$ where $\\Theta$ is the Heaviside function.\nBy contrast, our modular-invariant regulator keeps the entire partition function $F$ intact\nand then multiplies $F$\nby a single modular-invariant regulator function $\\widehat {\\cal G}_\\rho(a,\\tau)$.\nAs such it does not induce a sharp Heaviside-like subtraction at any particular location within the fundamental domain,\nbut rather (as illustrated in the right panel of Fig.~\\ref{regulator_figure})\ninduces a smooth damping which operates most strongly for {\\mbox{$\\tau_2\\gg \\tau_2^\\ast$}} and which can be removed (or pushed off\ntowards greater and greater values of $\\tau_2^\\ast$) as {\\mbox{$a\\to 0$}}. \nAll of these crucial differences are induced by the modular invariance of the regulator\nand render our modular-invariant regulators \nwholly different from the non-minimal regulator of Sect.~\\ref{sec:nonminimal}.\n\n\nThese two regulators do share one common feature, however: they both introduce suppressions \ninto the integrands of our string amplitudes.\nWithin the non-minimal regulator this takes the form of a \nsharp subtraction that occurs at {\\mbox{$\\tau_2=t$}},\nwhile the modular-invariant regulator\ngives rise to a smoother suppression, a transition from\n{\\mbox{$\\widehat{\\cal G}\\approx 1$}} to {\\mbox{$\\widehat{\\cal G}\\approx 0$}} \n that occurs near {\\mbox{$\\tau_2\\approx \\tau_2^\\ast$}}.\nTo the extent that these two regulators share this single common feature, it is therefore possible \nto ``align''\nthem by choosing a particular definition for $\\tau_2^\\ast$\nwithin the modular-invariant regulator and then identifying\n\\begin{equation}\n t ~=~ \\tau_2^\\ast~.\n\\label{prealignment}\n\\end{equation} \nIn general, we have seen in Eq.~(\\ref{tau2ast})\nthat $\\tau_2^\\ast$ takes the general form\n\\begin{equation}\n \\tau_2^\\ast ~=~ \\frac{\\xi}{a^2} ~=~ \\frac{\\xi \\rho}{\\rho a^2}~,\n\\label{tau2asta}\n\\end{equation}\nwhere $\\xi$ is a numerical coefficient which depends on the particular definition of $\\tau_2^\\ast$ that is chosen.\nThus, for any value of $t$,\nwe can correspondingly tune our choices for {\\mbox{$\\rho>1$}} and $\\rho a^2$ in order to enforce Eq.~(\\ref{prealignment}) \nand in this sense bring our regulators into alignment.\n\nThis alone is sufficient to align our regulators.\nHowever, in keeping with the spirit of symmetry-enhancement that motivated our transition\nfrom ${\\cal G}$ to $\\widehat {\\cal G}$,\nwe can push this one step further.\nWe have already seen that our $\\widehat{\\cal G}$ regulator has a symmetry\nunder {\\mbox{$\\tau_2\\to 1\/\\tau_2$}} (and thus under {\\mbox{$\\tau_2^\\ast \\to 1\/\\tau_2^\\ast$}})\nas well as a symmetry under {\\mbox{$a\\to 1\/\\rho a$}} [or equivalently under {\\mbox{$\\rho a^2 \\to 1\/(\\rho a^2)$}}].\nAlthough these are independent symmetries,\nthe fact that $\\tau_2^\\ast$ and $\\rho a^2$ are related through Eq.~(\\ref{tau2asta}) suggests\nthat we can align these \ntwo symmetries as well by further demanding that {\\mbox{$\\xi \\rho=1$}}.\n\nFor either of the first two $\\tau_2^\\ast$ definitions in Eq.~(\\ref{tau2ast}), this \nis a very easy condition to enforce: we simply take {\\mbox{$\\rho = 1\/\\xi$}}. \nThis is possible because the ``ridge-top'' and ``inflection'' definitions \nlead to values of $\\xi$ which are independent of $\\rho$.\nBy contrast, for the {\\mbox{${\\cal G} = 1\/2$}} condition (where {\\mbox{${\\cal G}\\approx \\widehat{\\cal G}$}} in the {\\mbox{$\\rho a^2 \\ll 1$}} limit),\nthe value of $\\xi$ is itself highly $\\rho$-dependent and it turns out that the constraint {\\mbox{$\\rho \\xi(\\rho)=1$}} has\nno solution for $\\rho$.\n\nIt is easy to understand why these different $\\tau_2^\\ast$ definitions lead to such different outcomes for $\\rho$.\nFor {\\mbox{$a\\ll 1$}} and {\\mbox{$\\rho>1$}}, the contributions to $\\widehat {\\cal G}$ from $Z_{\\rm circ}(\\rho a,\\tau)$ are hugely \nsuppressed compared with those from $Z_{\\rm circ}(a,\\tau)$.\nAs a result,\nany defining condition for $\\tau_2^\\ast$ which depends on the \nactual values of $\\widehat {\\cal G}$ will carry a sensitivity to $\\rho$ only through\nthe $\\widehat {\\cal G}$-prefactor {\\mbox{$A_\\rho = \\rho\/(\\rho-1)$}}. \nBy contrast, any defining condition for $\\tau_2^\\ast$ which depends on the \nvanishing of {\\it derivatives}\\\/ of $\\widehat {\\cal G}$ becomes insensitive to the overall\nscale factor $A_\\rho$ \nand thus independent of $\\rho$.\nIndeed, the ``ridge-top'' and ``inflection'' definitions depend on the vanishing of\nthe first and second $\\widehat {\\cal G}$-derivatives respectively. \nSuch conditions therefore lead to a vastly simpler algebraic structure for $\\tau_2^\\ast$ as\na function of $\\rho$.\n\nThus, pulling the pieces together, we see that we can align our modular-invariant regulator with\nour non-minimal regulator by adopting a particular definition for $\\tau_2^\\ast$\nand then choosing the values of the $(\\rho,a)$ parameters \nwithin our modular-invariant regulator such that\n\\begin{equation}\n t ~=~ \\frac{\\xi \\rho}{\\rho a^2}~.\n\\label{identification}\n\\end{equation}\nMoreover, we can further enhance the symmetries underlying\nthis identification by restricting our attention to $(\\rho,a)$ choices \nfor which {\\mbox{$\\xi \\rho=1$}}.\nHowever, because modular invariance essentially smoothes out the\nsharp transition at {\\mbox{$\\tau_2=t$}}\nthat otherwise existed within the non-minimal regulator, \nwe face an inevitable uncertainty in how we define $\\tau_2^\\ast$.\nIn the following, we shall therefore adopt \n\\begin{equation}\n t ~=~ \\frac{1}{\\rho a^2}~\n\\label{alignment}\n\\end{equation}\nas our alignment condition.\nDirectly enforcing this condition \nenables us to sidestep\nthe issues associated with choosing a particular value of $\\xi$ or a \nparticular definition for $\\tau_2^\\ast$.\nHowever, in enforcing this condition we should remain mindful of our regulator condition \nthat {\\mbox{$a\\ll 1$}}.\nLikewise, whenever needed, our choices for $\\rho$ should lie within\na range that is sensibly close to the approximate values of $\\xi^{-1}$ that characterize\nthe transition from {\\mbox{$\\widehat {\\cal G}\\approx 1$}} to {\\mbox{$\\widehat{\\cal G}\\approx 0$}}.\nFor example, when needed for the purposes of illustration, we shall \nchoose the fiducial value {\\mbox{$\\rho=2$}}.\nIndeed, such a value is very close to the value that would be required for the ``ridge-top''\ndefinition, yielding {\\mbox{$\\xi \\rho = 0.954$}}.\nHowever, by enforcing Eq.~(\\ref{alignment}) directly, we will be able to maintain alignment\nwithout needing to identify a particular definition \nfor $\\tau_2^\\ast$. Moreover, as already noted, the combination $\\rho a^2$ is\ninvariant under the symmetry in Eq.~(\\ref{rhoflipidentity}). \nThis combination will therefore appear naturally in many of our future calculations,\nthereby largely freeing us from the need to specify $\\rho$ and $a$ individually.\n\nOf course, we see from Eq.~(\\ref{alignment}) that choosing $\\rho$ within this range and taking {\\mbox{$a\\ll 1$}} will be possible\nonly if {\\mbox{$t\\gg 1$}}. Thus, although the choice of $t$ is completely arbitrary\nwithin the non-minimal regulator, only those non-minimal regulators with {\\mbox{$t\\gg 1$}} can\nbe aligned with our modular-invariant regulators in a meaningful way.\n\n\n\n\n \n\n\n\n\n\\section{Towards a field-theoretic interpretation: \n The Higgs mass as a supertrace over physical string states\n \\label{sec4}}\n\n\nEquipped with the mathematical machinery from Sect.~\\ref{sec3}, we now \nseek to express\nour result for the Higgs mass given in Eq.~(\\ref{relation1}) \nin terms of the supertraces over only the physical string states.\nIn so doing we will be \ndeveloping an \nunderstanding of our results from a field-theory perspective\n--- indeed, as a string-derived effective field theory (EFT) valid at low energies.\nAll of these results will be crucial for allowing us to understand how\nthe Higgs mass ``runs'' within such an EFT, and\nultimately allowing us to extract \na corresponding ``stringy'' effective Higgs potential in Sect.~\\ref{sec5}.\n\n\n\n\\subsection{Modular invariance, UV\/IR equivalence, and the passage to an EFT \\label{UVIRequivalence}}\n\n\nOur first task is to understand the manner through which\none may extract an EFT description\nof a theory with modular invariance.\nThis is a subtle issue because such theories, as we shall see, possess a certain\nUV\/IR equivalence. However, understanding this issue is ultimately crucial for the physical interpretations that we will be providing\nfor our results in the rest of this section, especially as they relate to the effects\nof the mathematical regulators\nwe have presented in Sect.~\\ref{sec3}.\n\nIn this paper, our interest has thus far focused on performing a fully string-theoretic calculation\nof the Higgs mass. Given that modular invariance is a fundamental symmetry of perturbative closed strings,\nwe have taken great care to preserve modular invariance at every step of our calculations (or to note the extent\nto which this symmetry has occasionally been violated, such as for two of the three possible regulators discussed in\nSect.~\\ref{sec3}).~ \nHowever, modular transformations\nmix the contributions of individual string states into each other in \nhighly non-trivial ways across the entire string spectrum.\nIndeed, we shall see that modular invariance even leads to a fundamental \nequivalence between ultraviolet (UV) and infrared (IR) divergences.\nThus a theory such as string theory can be modular invariant\nonly if all of its states across the {\\it entire}\\\/ string spectrum \nare carefully balanced against each other~\\cite{Dienes:1994np} and treated similarly, as a coherent whole.\nEFTs, by contrast, are predicated on an approach that treats UV physics and IR physics\nin fundamentally different ways, retaining the dynamical degrees of freedom associated with the IR physics \nwhile simultaneously ``integrating out'' the degrees of freedom associated with the UV physics.\nAs a result, any attempt to develop\na true EFT description of a modular-invariant \ntheory such as string theory inherently breaks modular invariance.\n\nIt is straightforward to see that modular invariance leads to an equivalence between UV and IR divergences.\nIn general, one-loop closed-string amplitudes are typically \nexpressed in terms of modular-invariant integrands $F(\\tau)$ which\nare then integrated over the fundamental domain ${\\cal F}$ of the modular group.\nIf such an amplitude diverges, this divergence will arise from the {\\mbox{$\\tau_2\\to \\infty$}}\nregion within ${\\cal F}$.\nGiven that the contributions from the heavy string states \nwithin the integrand are naturally suppressed as {\\mbox{$\\tau_2\\to\\infty$}},\nit would be natural to interpret this divergence\nas an IR divergence involving low-energy physics.\n\nHowever, such an interpretation would be inconsistent\nwithin a modular-invariant theory.\nIn any modular-invariant theory with a modular-invariant integrand $F(\\tau)$, \nwe can always rewrite\nour amplitude through the identity\n\\begin{equation} \n \\int_{\\cal F} \\dmu ~ F(\\tau) = \n \\int_{\\cal F} \\dmu ~ F(\\gamma \\cdot \\tau) = \n \\int_{\\gamma\\cdot {\\cal F}} \\dmu ~ F(\\tau)~~~\n\\label{anychoice}\n\\end{equation}\nwhich holds for any modular transformation $\\gamma$.\nFrom Eq.~(\\ref{anychoice}) we see that \nchoosing ${\\cal F}$ as our region of integration\nis mathematically equivalent to choosing\nany of its images {\\mbox{$\\gamma\\cdot {\\cal F}$}} under any modular transformation $\\gamma$.\nOne of these equivalent choices is {\\mbox{${\\cal F}'\\equiv \\gamma_S\\cdot {\\cal F}$}} \nwhere $\\gamma_S$ is the {\\mbox{$\\tau\\to -1\/\\tau$}} modular transformation.\nThis region is explicitly given as\n\\begin{eqnarray}\n {\\cal F}' ~&\\equiv&~ \\lbrace \\tau :\\, \\tau_2>0,\\, |\\tau|\\leq 1, \\,\n (\\tau_1 +1)^2 + \\tau_2^2 \\geq 1, ~~~~~~~~\\nonumber\\\\\n && ~~~~~~~~~~~~~~ (\\tau_1 -1)^2 + \\tau_2^2 \\geq 1\\, \\rbrace~,~~~~~\n\\end{eqnarray}\nand as such includes the {\\mbox{$\\tau_2\\to 0$}} region but no longer includes the {\\mbox{$\\tau_2\\to\\infty$}} region.\nIndeed, via the identity in Eq.~(\\ref{anychoice}) we see that the\ndivergence of our amplitude now appears as {\\mbox{$\\tau_2\\to 0$}}.\nHowever, \nthere is no suppression of the contributions from the heavy string states \nwithin the integrand\nas {\\mbox{$\\tau_2\\to 0$}}.\nInstead, any divergence as {\\mbox{$\\tau_2\\to 0$}} arises through the accumulating contributions of the heavy\nstring states and would therefore naturally be interpreted as a UV divergence. \nThus, by trading ${\\cal F}$ for ${\\cal F}'$ through Eq.~(\\ref{anychoice}), we see that we can \nalways mathematically recast what would naively appear \nto be an IR divergence as {\\mbox{$\\tau_2\\to \\infty$}} \ninto what would naively appear to be a UV divergence as {\\mbox{$\\tau_2\\to 0$}} --- all without\ndisturbing the integrand of our amplitude in any way.\nA similar conclusion holds for the many other ${\\cal F}''$ domains that could equivalently have been chosen for\nother choices of the modular transformation $\\gamma$.\n\n\n\n\n\n\nThis is a fundamental observation.\nWhen we calculate an amplitude in string theory, \nwe are equipped with an integrand which reflects\nthe spectrum of string states but we must\nchoose an appropriate fundamental domain of the modular group.\nThis choice is not something dictated within the theory itself, but instead\namounts to a {\\it convention}\\\/ which is adopted for the sake of performing a calculation.\nIt is possible, of course, that the amplitude in question diverges.\nAs we have seen, if we choose the fundamental domain ${\\cal F}$ as defined in Eq.~(\\ref{Fdef}) \nthen this divergence will manifest itself\nas an IR divergence.\nHowever, if we choose ${\\cal F}'$ as our fundamental domain, this same divergence of\nthe amplitude will manifest itself as a UV divergence.\nBoth interpretations are equally valid \nbecause the divergence \nof a one-loop modular-invariant string amplitude\nis neither intrinsically UV nor intrinsically IR.~\nIndeed, such a divergence \nis a property {\\it of the amplitude itself}\\\/ and is not intrinsically tied\nto any particular value of $\\tau$.\nSuch a divergence is then merely {\\it represented}\\\/ as a UV or IR divergence depending on our choice of\na region of integration.\n\nThis observation can also be understood through a comparison with our expectations from quantum field theory.\nAs we have seen in Eq.~(\\ref{unfold}), there is a tight relation between\nthe fundamental domain ${\\cal F}$ and the strip ${\\cal S}$ defined in Eq.~(\\ref{Sdef}): \nessentially ${\\cal F}$ is a ``folded'' version of ${\\cal S}$. Likewise, the modular-invariant integrand $F(\\tau)$ \nthat is integrated over ${\\cal F}$ is nothing but the sum of the images of the {\\it non}\\\/-invariant integrand which \nwould be integrated over ${\\cal S}$.\nThus, through the unfolding procedure in Eq.~(\\ref{unfold}), we have two equivalent representations\nfor the same physics. These are often called the ${\\cal F}$- and ${\\cal S}$-representations.\n\nIt is through the ${\\cal S}$-representation that we can most directly\nmake contact with the results that would come from a quantum field theory based on point particles. \nWithin the ${\\cal S}$-representation, we can identify $\\tau_2$ as the Schwinger proper-time\nparameter, with {\\mbox{$\\tau_2\\to\\infty$}} corresponding to the field-theoretic IR limit\nand with {\\mbox{$\\tau_2\\to 0$}} corresponding to the field-theoretic UV limit.\nIndeed, within field theory these limits are physically distinct, just as they are geometrically\ndistinct within the strip.\nHowever, \nupon folding the strip ${\\cal S}$ \ninto the fundamental domain ${\\cal F}$, we see that {\\it both}\\\/ the UV {\\it and}\\\/ IR field-theoretic regions\nwithin ${\\cal S}$ are together mapped onto the {\\mbox{$\\tau_2\\to\\infty$}} region within ${\\cal F}$.\nIndeed, the distinct UV and IR regions of the strip ${\\cal S}$ are now ``folded'' so as \nto lie directly on top of each other within ${\\cal F}$.\nThus, within the ${\\cal F}$-representation, the {\\mbox{$\\tau_2\\to\\infty$}} limit in some sense\nrepresents {\\it both}\\\/ the UV and IR field-theory limits simultaneously --- limits\nwhich would have been viewed as distinct within field theory but which are now related to each other in\nstring theory through modular invariance. \nAn identical argument also holds for the {\\mbox{$\\tau_2\\to 0$}} region within ${\\cal F}'$.\n\nWe can therefore summarize the situation as follows.\nFor a modular-invariant string-theoretic amplitude there is only one kind of divergence. \nIt can be represented as either a UV divergence or an IR divergence depending on our choice\nof fundamental domain (region of integration). However, in either case, this single \nstring-theoretic divergence can be mapped back to what can \nbe considered a modular-invariant {\\it combination}\\\/ of UV and IR field-theoretic divergences in \nfield theory ({\\it i.e.}\\\/, on the strip ${\\cal S}$). Indeed, we may schematically write\n\\begin{equation}\n \\underbrace{ \n {\\rm IR}_{{\\cal F}} \\,=\\, {\\rm UV}_{{\\cal F}'} }_{\\hbox{string theory}}\n ~\\Longleftrightarrow~\n \\underbrace{ {\\rm IR}_{{\\cal S}} \\,\\oplus\\, {\\rm UV}_{{\\cal S}} }_{\\hbox{field theory}}~~\n\\label{UVIR}\n\\end{equation}\nwhere `$\\oplus$' signifies a modular-invariant combination.\nWe shall obtain an explicit example of such a combination below.\nIt is ultimately in this way, through Eq.~(\\ref{UVIR}), that \nour modular-invariant string theory loses its ability to distinguish between UV and IR physics.\nWe will discuss these issues further in Sect.~\\ref{sec:Conclusions}.\n\n\n\nOur discussion in this paper has thus far been formulated with ${\\cal F}$ chosen\nas our fundamental domain. In this way we have been implicitly casting \nour string-theoretic divergences as infrared. In the following, we shall therefore continue along this line and attach corresponding \nphysical interpretations to our mathematical results as far as possible. \nHowever, we shall also \noccasionally\nindicate how our results might alternatively \nappear within the ${\\cal F}'$-representation, or within the \nunfolded ${\\cal S}$-representation of ordinary quantum field theory. \nThis will ultimately be important for extracting an EFT for the Higgs mass,\nfor understanding how our Higgs mass ``runs'' within such an EFT,\nand for eventually interpreting our results in terms of a stringy effective potential.\n\n\nOne further comment regarding the nature of these divergences is in order.\nThe above discussion has focused on the manner in which modular invariance mixes\nUV and IR divergences\nwhen passing from field theory to string theory.\nHowever, it is also important to remember that modular invariance likewise affects\nthe {\\it strengths}\\\/ of these divergences.\nTo understand this, we recall from Eq.~(\\ref{stripF}) that the strip ${\\cal S}$, which serves as the field-theoretic region of integration,\nis nothing but the sum of the images of ${\\cal F}$, a string-theoretic region of integration,\nunder each of the modular transformations $\\gamma$ in the coset {\\mbox{$\\Gamma_\\infty\\backslash \\Gamma$}}.\nHowever, there are an infinite number of such modular transformations within this coset.\nThis means, in essence, that our string-theoretic divergences (if any) are added together\nan infinite number of times when ${\\cal F}$ is unfolded into ${\\cal S}$,\nimplying that the resulting field-theoretic divergences are far more severe than those of the string.\nPhrased somewhat differently, we see that modular invariance softens a given\nfield-theoretic divergence by allowing us to reinterpret part of this divergence as resulting from an infinity\nof identical copies of a weaker (modular-invariant) string divergence, whereupon we are authorized to\nselect only one such copy. \n\nThis observation is completely analogous to what happens within field theory in the presence of a gauge symmetry.\nIf we were to disregard the gauge symmetry when calculating a field-theoretic amplitude, \nwe would integrate over an infinite number of gauge slices when performing our path integrals. \nThis would result in divergences which are spuriously severe.\nHowever, modular invariance is similar to gauge symmetry in the sense that both represent redundancies of \ndescription. \n(In the case of modular invariance, the redundancy arises from the fact that all values of {\\mbox{$\\gamma \\cdot \\tau$}}\nfor {\\mbox{$\\gamma\\in\\Gamma$}} correspond to the same worldsheet torus.)\nIn a modular-invariant theory, we therefore divide out by the (infinite) volume of the \nredundancy coset {\\mbox{$\\Gamma_\\infty\\backslash \\Gamma$}} and consider only one modular-invariant ``slice''. \nIndeed, this is precisely what is happening when we pass from ${\\cal S}$ to ${\\cal F}$ (or any of its images)\nas the appropriate region of integration in a modular-invariant theory, where the particular choice of image\nis nothing but the particular choice of slice.\nThis passage from ${\\cal S}$ to a particular modular-invariant slice\nthen softens our field-theoretic divergences and in some cases even eliminates them entirely.\n\nWe have already seen one example of this phenomenon: the one-loop vacuum energy (cosmological\nconstant) $\\Lambda$ is badly divergent in quantum field theory, yet finite in any tachyon-free closed string theory.\nIndeed, it is modular invariance which is alone responsible for this phenomenon.\nAs we shall see, a similar softening of divergences also occurs for the Higgs mass.\n\n\nWe conclude this discussion with one additional comment.\nIt is a common assertion that string theory lacks UV divergences.\nThe rationale usually provided for this is that string theory intrinsically has a minimum length \nscale, namely $M_s^{-1}$, and that this provides a ``cutoff'' that eliminates \nall physics from arbitrarily short length scales and thereby eliminates the associated UV divergences. \nHowever, this argument fails to acknowledge that IR divergences may still remain,\nand of course in a modular-invariant theory the UV and IR divergences are mixed.\nIndeed, as we have explained, there is no modular-invariant way of disentangling these\ntwo kinds of divergences. Thus string theory is not free of divergences.\nThese divergences are simply softer than they would have been in field theory.\n \n\n\\subsection{The divergence structure of the Higgs mass}\n\n\nWith the above comments in mind, we now consider the divergence structure of the Higgs mass.\nWe begin by recalling from Eq.~(\\ref{relation1}) that \nthe Higgs mass $m_\\phi$ \nwithin any four-dimensional heterotic string \nis given by \n\\begin{equation}\n m_\\phi^2 ~=~ -\\frac{{\\cal M}^2}{2} \\biggl(\n \\langle {\\cal X}_{1a}\\rangle \n + \\langle {\\cal X}_{1b}\\rangle \n + \\langle {\\cal X}_2\\rangle \\biggr)\n + \\frac{\\xi}{4\\pi^2} \\frac{\\Lambda}{{\\cal M}^2}~~~\n\\label{Higgsmass}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n {\\cal X}_{1a} ~&\\equiv &~\n \\frac{\\tau_2}{\\pi} \n \\left( \\tilde {\\bf Q}_j^t {\\bf Q}_h + {\\bf Q}_j^t \\tilde {\\bf Q}_h \\right) \\nonumber\\\\ \n {\\cal X}_{1b} ~&\\equiv &~\n - \\frac{\\tau_2}{\\pi} \n \\left( {\\bf Q}_h^2 + \\tilde {\\bf Q}_h^2 \\right) \\nonumber\\\\ \n {\\cal X}_2 ~&\\equiv &~ \n \\tau_2^2 \\, \\left({\\bf Q}_R^t {\\bf Q}_h - {\\bf Q}_L^t \\tilde {\\bf Q}_h \\right)^2 \\nonumber\\\\\n && ~~~ = ~ \n 4 \\tau_2^2\\, ({\\bf Q}_R^t {\\bf Q}_h)^2 ~=~ 4 \\tau_2^2 \\, ({\\bf Q}_L^t \\tilde {\\bf Q}_h )^2 ~~~~~~\n\\label{Xidef}\n\\end{eqnarray}\nand where $\\Lambda$ is the one-loop cosmological constant.\nNote that we have explicitly separated those terms ${\\cal X}_{1a}$ and ${\\cal X}_{1b}$\nwhich are quadratic in charge\ninsertions \nfrom those terms ${\\cal X}_2$ which are quartic, as these will shortly play very different roles.\nMoreover, within the quadratic terms, we have further distinguished \nthose insertions ${\\cal X}_{1a}$ within which each term consists of a paired contribution \nof a left-moving charge with a right-moving charge\nfrom those insertions ${\\cal X}_{1b}$\nin which each term consists of two charges which are \nboth either left- or right-moving.\nIndeed, we recall from Sect.~\\ref{sec2} that only \n$\\langle {\\cal X}_{1a}\\rangle$ \nand the sum \n$\\langle {\\cal X}_{1b}\\rangle + \\langle {\\cal X}_2\\rangle$ \nare modular invariant;\nin particular, \n$\\langle {\\cal X}_{1b}\\rangle$\nand \n $\\langle {\\cal X}_2\\rangle$ \nare the modular completions of each other\nand thus \nneither \nis modular invariant by itself.\nThat said, it will prove convenient in this section to \nsimply define\n\\begin{eqnarray}\n {\\cal X}_{1} ~&\\equiv &~ {\\cal X}_{1a} + {\\cal X}_{1b}\\nonumber\\\\\n &= & ~\n \\frac{\\tau_2}{\\pi} \n \\left( \\tilde {\\bf Q}_j^t {\\bf Q}_h + {\\bf Q}_j^t \\tilde {\\bf Q}_h \n - {\\bf Q}_h^2 - \\tilde {\\bf Q}_h^2 \\right) ~,\n\\label{Xidef2}\n\\end{eqnarray}\nso long as we remember that only the \nfull combination $\\langle {\\cal X}_1\\rangle +\\langle {\\cal X}_2\\rangle$ is modular invariant.\n\nAs discussed in Sect.~\\ref{sec2}, these results are completely general and apply to any\nscalar $\\phi$ whose VEV determines the vacuum structure of the theory.\nIndeed, the various charge insertions \n${\\bf Q}_h$, $ \\tilde {\\bf Q}_h$, $ {\\bf Q}_j$, and $\\tilde {\\bf Q}_j$\nin Eq.~(\\ref{Xidef})\nare defined in Eqs.~(\\ref{Qhdef}) and (\\ref{Qjdef}) in terms of the\n${\\cal T}$-matrices which encapsulate the relevant information concerning\nspecific scalar under study.\n\nUnlike the other terms in Eq.~(\\ref{Higgsmass}), the final term $\\Lambda$ \nemerges as the result of a universal shift in the background moduli.\nAs such, this quantity is wholly independent of the specific ${\\cal T}$-matrices,\nand merely provides a uniform shift to the masses of all scalars \nin the theory regardless of the specific roles these scalars might play in breaking gauge symmetries \nor otherwise affecting the vacuum state of the theory.\nIn other words, $\\Lambda$ provides what is essentially a mere ``background'' contribution\nto our scalar masses. Moreover, as the one-loop cosmological constant of the theory,\n$\\Lambda$ is an independent physical observable unto itself.\nFor this reason, we shall defer our discussion of $\\Lambda$ to Sect.~\\ref{Lambdasect} \nand focus instead on the effects coming from the ${\\cal X}_i$ insertions\nin Eq.~(\\ref{Higgsmass}).\n\nIn order to make use of the machinery in Sect.~\\ref{sec3}, we must\nfirst understand the divergence structure that can arise from each of these ${\\cal X}_i$\ninsertions as {\\mbox{$\\tau_2\\to \\infty$}}.\nFor any string model in four spacetime dimensions,\nthe original partition function prior to any ${\\cal X}_i$ insertions has\nthe form indicated in Eq.~(\\ref{Zform}),\nwith an overall factor of $\\tau_2^{-1}$. \nThus, the insertion of \n${\\cal X}_1$ leads to \nintegrands without a leading factor of $\\tau_2$,\nwhile the insertion of\n${\\cal X}_2$ leads to integrands with a \nleading factor of $\\tau_2^{+1}$. \n\nDetermining the possible divergences \nas {\\mbox{$\\tau_2\\to\\infty$}} requires that we\nalso understand the spectrum of low-lying states \nthat contribute to these integrands.\nWe shall, of course, assume that our string model\nis free of of physical (on-shell) tachyons.\nThus, expanding the partition function ${\\cal Z}$ of our string model as \nin Eq.~(\\ref{integrand}) with {\\mbox{$k= -1$}}, \nwe necessarily have {\\mbox{$a_{nn}=0$}} for all {\\mbox{$n< 0$}} in Eq.~(\\ref{integrand}).\n\nThere is, however,\nan {\\it off-shell}\\\/ tachyonic state which must always appear within the spectrum of any self-consistent heterotic string model:\nthis is the so-called {\\it proto-graviton}~\\cite{Dienes:1990ij} with {\\mbox{$(m,n)=(0,-1)$}}, and no possible GSO projection can eliminate this state \nfrom the spectrum.\nAlthough this state is necessarily a singlet under all of the gauge symmetries of the model,\nit transforms as a vector under the spacetime Lorentz group. Consequently \nthe degrees of freedom that compose this state have non-vanishing charge vectors of the form\n\\begin{equation}\n {\\bf Q}_{\\hbox{\\scriptsize proto-graviton}} ~=~ ({\\bf 0}_{22} \\, | \\pm 1, {\\bf 0}_9 )\n\\label{protocharge}\n\\end{equation}\nwhere we have written this charge vector in the same basis as used in Eq.~(\\ref{Tmatrixform}),\nwith the non-zero charge component in Eq.~(\\ref{protocharge}) lying along the spacetime-statistics direction\ndiscussed in Sect.~\\ref{sec:EWHiggs}.\n\nBecause of this non-zero charge component, the proto-graviton state has the possibility of contributing to one-loop string\namplitudes even when certain charge insertions occur.\nHowever, we have seen in Eq.~(\\ref{Tmatrixform})\nthat the ${\\cal T}$-matrices appropriate for shifts in the Higgs VEV do not disturb the spin-statistics of the states\nin the spectrum, and thus necessarily have zeros along the corresponding columns and rows. \nIndeed, these zeros are a general feature which would apply to all such ${\\cal T}$-matrices\nregardless of the specific Higgs scalar under study or its particular gauge embedding.\nAs a result, the would-be contributions from the proto-graviton state\ndo not survive either of the ${\\cal X}_i$ insertions in Eq.~(\\ref{Xidef}).\nIndeed, similar arguments also apply to potential contributions from the proto-gravitino states (such as would appear in the\nspectra of string models exhibiting spacetime supersymmetry).\n\nIn general, a heterotic string model can also contain other off-shell tachyonic $(m,n)$ states with {\\mbox{$m\\not=n$}} but {\\mbox{$m+n<0$}}. Unlike the proto-graviton state with {\\mbox{$(m,n)=(0,-1)$}}, such states would generally have {\\mbox{$(m,n)= (k+1,k)$}} where {\\mbox{$-10$}} have a built-in Boltzmann-like \nsuppression $\\sim e^{-\\pi \\alpha' M^2 \\tau_2}$ as {\\mbox{$\\tau_2\\to\\infty$}}, while massless states\ndo not. \nThus massless states are unprotected by the Boltzmann suppression factor as {\\mbox{$\\tau_2\\to\\infty$}},\nwhich is why their contributions are subtracted as part of the regularization procedure.\n\nWithin the {\\it non}\\\/-minimal regulator, however, we distinguish between\ntwo different ranges for $\\tau_2$: one range with {\\mbox{$1\\leq \\tau_2 \\leq t$}}, and a\nsecond range with {\\mbox{$t\\leq \\tau_2 < \\infty$}}.\nOnly within the second range do we subtract the contributions from the massless states;\nindeed, massless states are considered ``safe'' within the first range.\nBut for any finite $t$, it is possible that there are many light states which do not have\nappreciable Boltzmann suppression factors at {\\mbox{$\\tau_2=t$}}.\nSuch light (or ``effectively massless'') states are therefore essentially indistinguishable from \ntruly massless states as far\nas their Boltzmann suppression factors are concerned.\nIndeed, it is only as {\\mbox{$\\tau_2\\to\\infty$}} that we can distinguish the truly massless\nstates relative to all the others. \n\nThis suggests that for any finite value of $t$, we can assess whether a given state \nof mass $M$ is effectively light or heavy\naccording to the magnitude of its corresponding Boltzmann suppression factor at {\\mbox{$\\tau_2=t$}} within the \npartition function.\nRecalling that the contribution from a physical string state of \nmass $M$ to the string partition function scales as $e^{-\\pi \\alpha' M^2 \\tau_2}$,\nwe can establish an arbitrary criterion for the magnitude of the Boltzmann suppression \nof a state with mass {\\mbox{$M=\\mu$}} at the cutoff $t$:\n\\begin{equation}\n e^{-\\pi \\alpha' \\mu^2 t} ~\\sim~ e^{-\\varepsilon}\n\\label{massscale}\n\\end{equation}\nwhere {\\mbox{$\\varepsilon\\geq 0$}} \nis an arbitrarily chosen dimensionless parameter.\nAccording to this criterion, states whose Boltzmann factors at {\\mbox{$\\tau_2= t$}} \nexceed $e^{-\\varepsilon}$ have not experienced significant Boltzmann suppression and \ncan then be considered light relative to that choice of $t$, \nwhile all others can be considered heavy.\nWe thus find that our division between light and heavy states can be demarcated by\na running mass scale $\\mu (t)$ defined as\n\\begin{equation}\n \\mu^2(t) ~\\equiv~ \\frac{\\varepsilon}{\\pi \\alpha' t}~.\n\\label{mudefeps}\n\\end{equation}\nNote, as expected, that {\\mbox{$\\mu(t)\\to 0$}} as {\\mbox{$t\\to\\infty$}}. Thus, as expected, \nthe only states that can be considered light as {\\mbox{$t\\to\\infty$}} are those which are exactly massless. \n\nUltimately, the choice of $\\varepsilon$ \ndetermines an overall scale for the mapping between $t$ and $\\mu$ and is thus a matter of convention.\nFor the sake of simplicity within Eq.~(\\ref{mudefeps}) and our subsequent expressions, we shall henceforth \nchoose {\\mbox{$\\varepsilon = \\pi$}}, whereupon Eq.~(\\ref{mudefeps}) reduces to\n\\begin{equation}\n \\mu^2(t) ~=~ \\frac{1}{\\alpha' t}~.\n\\label{mut}\n\\end{equation}\n \nWith these adaptations, our result for the regulated Higgs mass $\\widehat m_\\phi^2 (t)$ in Eq.~(\\ref{nonminresult}) \ncan be rewritten as\n\\begin{eqnarray}\n \\widehat m_\\phi^2(\\mu) \\,&=&\\, \\frac{\\xi}{4\\pi^2} \\frac{\\Lambda}{{\\cal M}^2} \n +{\\textstyle{1\\over 2}} {\\cal M}^2 \\, \\biggl[\n - \\frac{\\pi}{3} \\, {\\rm Str}\\, {\\mathbb{X}}_1 \\nonumber\\\\\n && ~~~~~~ + (\\,\\newzStr \\,{\\mathbb{X}}_2) \\log \\left(\\frac{\\mu^2}{{\\cal M}_\\ast^2}\\right) \\biggr]~~~~~~~~~~\n\\label{nonminresultFT}\n\\end{eqnarray}\nwhere we have defined \n\\begin{equation}\n {\\cal M}_\\ast^2 ~\\equiv~ 4\\pi \\,e^{-\\gamma} \\,M_s^2 ~=~ 16\\pi^3\\, e^{-\\gamma}\\, {\\cal M}^2~\n\\end{equation}\nand where we have restored the additional universal $\\Lambda$-term in Eq.~(\\ref{nonminresultFT}).\nWe thus see that while the first two terms in Eq.~(\\ref{nonminresultFT}) are independent of $\\mu$ and \ntogether constitute what may be considered an overall threshold term, \nthe logarithmic $\\mu$-dependence within $\\widehat m_\\phi^2(\\mu)$ for any $\\mu$ arises from those\nphysical string states which are charged under ${\\mathbb{X}}_2$ with masses {\\mbox{$M\\leq \\mu$}}.\n\n\nAs we have seen, the enhanced non-minimal regulator we are using here \noperates by explicitly subtracting the contributions \nof the ${\\mathbb{X}}_2$-charged massless states from all regions {\\mbox{$\\tau_2\\geq t$}}.\nThis is a sharp cutoff, and it is natural to wonder \nhow such a cutoff actually maps back onto the strip under the unfolding process.\nIndeed, answering this question will give us some idea about how this sort of cutoff might be interpreted \nin field-theoretic language.\nAs expected, imposing a sharp cutoff {\\mbox{$\\tau_2\\leq t$}} within the ${\\cal F}$ representation\nproduces both an IR cutoff as well as a UV cutoff on the strip.\nThe IR cutoff is inherited directly from the string-theory cutoff and takes the same form {\\mbox{$\\tau_2\\leq t$}}, thereby excising\nall parts of the strip with $\\tau_2$ exceeding $t$, independent of $\\tau_1$.\nHowever, the corresponding UV cutoff is highly non-trivial and is actually sensitive to $\\tau_1$ as well ---\na degree of freedom that does not have a direct interpretation in the field theory.\nMathematically, this UV cutoff excises from the strip \nthat portion of the region~\\cite{zag}\n\\begin{equation}\n \\bigcup_{(a,c)=1} \\, S_{a\/c}\n\\label{circles}\n\\end{equation}\nwhich lies within the range {\\mbox{$-1\/2\\leq \\tau_1 \\leq 1\/2$}},\nwhere $S_{a\/c}$ denotes the disc\nof radius $(2c^2 t)^{-1}$ \nwhich is tangent to the {\\mbox{$\\tau_2=0$}} axis at\n{\\mbox{$\\tau_1= a\/c$}} and where the union in Eq.~(\\ref{circles}) includes all such disks\nfor all relatively prime integers $(a,c)$.\nThus, as one approaches the {\\mbox{$\\tau_2=0$}} axis of the strip from above, the excised region \nconsists of an infinite series of smaller and smaller discs which are all tangential to this axis\nin an almost fractal-like pattern.\nClearly, \nall points which actually lie along the {\\mbox{$\\tau_2=0$}} axis with {\\mbox{$\\tau_1\\in \\mathbb{Q}$}} are excised\nfor any finite $t$ \n(and strictly speaking the other points along the {\\mbox{$\\tau_2=0$}} axis with {\\mbox{$\\tau_1\\not\\in \\mathbb{Q}$}} are \nnot even part of the strip).\nThus, through this highly unusual UV regulator, all UV divergences on the strip are indeed eliminated \nfor any finite $t$.\nOf course, this excised UV region is nothing but the image of the IR-excised region {\\mbox{$\\tau_2\\geq t$}} under\nall of the modular transformations (namely those within the coset {\\mbox{$\\Gamma_\\infty\\backslash \\Gamma$}}) \nthat play a role in building the strip from ${\\cal F}$.\nHowever, in field-theoretic language this amounts to a highly unusual UV regulator indeed!\n\n\n\n\n\\subsection{Results using the modular-invariant regulator}\n\n\nFinally, we turn to the results for the Higgs mass \nthat are obtained using the fully modular-invariant regulator \n$\\widehat {\\cal G}_\\rho(a,\\tau_2)$ in Eq.~(\\ref{hatGdef}).\nAs we have stressed, only such results can be viewed as faithful to the modular symmetry\nthat underlies closed string theory, and therefore only such results can be viewed\nas truly emerging from closed string theories.\n\nWe have seen in Eq.~(\\ref{Higgsmass}) that the string-theoretic Higgs mass \n$m_\\phi^2$ has two contributions: one of these stems from the ${\\cal X}_i$\ninsertions and requires regularization, while the other --- namely the\ncosmological-constant term --- is finite within any tachyon-free modular-invariant\ntheory and hence does not.\nWhen discussing the possible regularizations of the Higgs mass using\nthe minimal and non-minimal regulators in Sects.~\\ref{higgsmin} and \\ref{higgsnonmin},\nwe simply carried the cosmological-constant \nterm along within our calculations and focused\non applying our regulators to the contributions with ${\\cal X}_i$ insertions.\nThis was adequate for the minimal and non-minimal regulators because these regulators\ninvolve the explicit subtraction of divergences \nand thus have no effect on quantities which are already finite and therefore lack\ndivergences to be subtracted.\nOur modular-invariant regulator, by contrast, operates by deforming the theory.\nIndeed, this deformation has the effect of\nmultiplying the partition function of the theory\nwith a new factor $\\widehat {\\cal G}_\\rho(a,\\tau)$.\nAs such, this regularization procedure \ncan be expected to have an effect even when acting on finite quantities such as $\\Lambda$.\nWhen regularizing the Higgs mass in this manner,\nwe must therefore consider how this regulator affects both classes of Higgs-mass contributions --- \nthose involving non-trivial ${\\cal X}_i$ insertions, and those coming from the \ncosmological constant.\nIndeed, with\n $\\widehat m_\\phi^2(\\rho,a)\\bigl|_{{\\cal X},\\Lambda}$\nrespectively denoting these two classes of contributions \nto the $\\widehat {\\cal G}$-regulated version \n $\\widehat m_\\phi^2(\\rho,a)$\nof the otherwise-divergent string-theoretic Higgs mass in Eq.~(\\ref{Higgsmass}),\nwe can write \n\\begin{eqnarray}\n \\widehat m_\\phi^2(\\rho,a) \n ~&\\equiv&~ \\widehat m_\\phi^2(\\rho,a)\\Bigl|_{\\cal X} + ~ \\widehat m_\\phi^2(\\rho,a)\\Bigl|_\\Lambda~ \\nonumber\\\\ \n ~&\\equiv&~ \\widehat m_\\phi^2(\\rho,a)\\Bigl|_{\\cal X} + ~ \\frac{\\xi}{4\\pi^2 {\\cal M}^2} \\,\\widehat \\Lambda(\\rho,a)~.~~~~~~ \n\\label{twocontributions}\n\\end{eqnarray}\nWe shall now consider each of these contributions in turn.\n\n\n\\subsubsection{Contributions from terms with charge insertions\\label{chargeinsertions}}\n\nOur first contribution in Eq.~(\\ref{twocontributions})\nis given by \n\\begin{equation}\n \\widehat m_\\phi^2(\\rho,a)\\Bigl|_{\\cal X} ~\\equiv~ -\\frac{{\\cal M}^2}{2} \n \\Bigl\\langle {\\cal X}_1+ {\\cal X}_2\\Bigr\\rangle_{\\cal G} \n \\label{Higgsmass1}\n\\end{equation}\n where\n\\begin{equation}\n \\langle A \\rangle_{\\cal G} ~\\equiv~ \\int_{\\cal F} \\dmu\n \\left\\lbrace \n \\left\\lbrack \n \\tau_2^{-1} \\sum_{m,n} (-1)^F A~ {\\overline{q}}^{m} q^{n} \n \\right\\rbrack\n \\widehat {\\cal G}_\\rho(a,\\tau)\\right\\rbrace ~\n\\label{AG}\n\\end{equation}\nwith {\\mbox{$m\\equiv \\alpha' M_R^2\/4$}}, {\\mbox{$n\\equiv \\alpha' M_L^2\/4$}}. \nIndeed, the insertion of $\\widehat {\\cal G}_\\rho(a,\\tau)$ into the integrand of \nEq.~(\\ref{AG}) is what tames the logarithmic divergence.\nFollowing the result in Eq.~(\\ref{Irhoa}) we then find that \n$\\widehat m_\\phi^2 (\\rho,a)$ can be expressed as\n\\begin{equation}\n \\widetilde m_\\phi^2(\\rho,a)\\Bigl|_{{\\cal X}} ~=~ \n \\frac{\\pi}{3}\\, \\oneRes \\, \\int_0^\\infty d\\tau_2 \\,\\tau_2^{s-2} \\,\\widehat g_\\rho(a,\\tau_2) ~ \n\\label{Irhoa2}\n\\end{equation}\nwhere \n\\begin{eqnarray}\n && \\widehat g_\\rho(a,\\tau_2) \\,\\equiv\\, \n -\\frac{{\\cal M}^2}{2}\n \\int_{-1\/2}^{1\/2} d\\tau_1 \\nonumber\\\\ \n && ~~~~~~~~~ \n \\times\\, \\left\\lbrace \n\\left\\lbrack \n \\sum_{m,n} (-1)^F \\,({\\mathbb{X}}_1 + \\tau_2 {\\mathbb{X}}_2)\\, {\\overline{q}}^{m} q^{n} \\right\\rbrack \n \\,\\widehat {\\cal G}_\\rho(a,\\tau)\\right\\rbrace ~~~\\nonumber\\\\\n&& ~~~~~ \\approx ~\n -\\frac{{\\cal M}^2}{2}\n \\left\\lbrack {\\rm Str} \\,({\\mathbb{X}}_1 + \\tau_2 {\\mathbb{X}}_2)\\, e^{-\\pi \\alpha' M^2 \\tau_2} \\right\\rbrack \n \\widehat {\\cal G}_\\rho(a,\\tau_2) ~.~~~ \\nonumber\\\\\n\\label{gFGdef3}\n\\end{eqnarray}\nNote that in passing to the approximate factorized form in the final expression of Eq.~(\\ref{gFGdef3}), we \nhave followed the result in Eq.~(\\ref{gFGdef2}) \nand explicitly restricted our attention to those cases\nwith {\\mbox{$a\\ll 1$}}, as appropriate for the \nregulator function $\\widehat {\\cal G}_\\rho(a,\\tau)$.\nIndeed, the term within square brackets in the second line of \nEq.~(\\ref{gFGdef3}) is our desired supertrace over physical string states,\nwhile the regulator function $\\widehat{\\cal G}_\\rho(a,\\tau_2)$ --- an example of which \nis plotted in the right panel of Fig.~\\ref{regulator_figure} --- generally eliminates the\ndivergence that would otherwise have arisen as {\\mbox{$\\tau_2\\to \\infty$}} for any {\\mbox{$a>0$}}.\nMoreover, we learn that\nas a consequence of the\nidentity in Eq.~(\\ref{StransG})\n--- an identity which holds\nfor $\\widehat {\\cal G}$ as well\nas for ${\\cal G}$ itself ---\nthe behavior shown in\nthe right panel of Fig.~\\ref{regulator_figure} \ncan be symmetrically ``reflected'' through {\\mbox{$\\tau_2=1$}}, resulting in the same \nsuppression behavior as {\\mbox{$\\tau_2\\to 0$}}. \n\n\nThe next step is to substitute Eq.~(\\ref{gFGdef3}) back into Eq.~(\\ref{Irhoa2})\nand evaluate the residue at {\\mbox{$s=1$}}. \nIn general, the presence of the regulator function $\\widehat{\\cal G}_\\rho(a,\\tau_2)$ within \nEq.~(\\ref{gFGdef3}) renders this calculation somewhat intricate. However, we \nknow that {\\mbox{$\\widehat {\\cal G}_\\rho(a,\\tau_2)\\to 1$}} as {\\mbox{$a\\to 0$}}.\nIndeed, having already exploited our regulator in \nallowing us to pass from Eq.~(\\ref{Higgsmass1}) to Eq.~(\\ref{Irhoa2}),\nwe see that taking {\\mbox{$a\\to 0$}} corresponds to the limit in which we subsequently remove our regulator.\nLet us first\nfocus on the contributions from massive states.\nIn the {\\mbox{$a\\to 0$}} limit, we then obtain\n\\begin{eqnarray}\n&& \\oneRes \\int_0^\\infty d\\tau_2 \\, \\tau_2^{s-2}\\, \\widehat g_\\rho(a,\\tau_2) \\nonumber\\\\\n&& ~~\\,=\\, -{\\textstyle{1\\over 2}} {\\cal M}^2 \\, \\oneRes \\, \\bigl\\lbrack\n \\Gamma(s-1) \\, \\pStr {\\mathbb{X}}_1 \\,(\\pi \\alpha' M^2)^{1-s} ~~~~\\nonumber\\\\ \n && ~~~~~~~~~~~~~~~~~~~~~~+ \\Gamma(s) \\, \\pStr {\\mathbb{X}}_2 \\,(\\pi \\alpha' M^2)^{-s}\\bigr\\rbrack \\nonumber\\\\ \n&& ~~\\,=\\, -{\\textstyle{1\\over 2}} {\\cal M}^2 \\, \\pStr {\\mathbb{X}}_1~,\n\\label{cheat}\n\\end{eqnarray}\nwhereupon we find that the contribution from massive states yields\n\\begin{equation}\n M>0:~~~\\lim_{a\\to 0} \\widehat m_\\phi^2 (\\rho,a)\\Bigl|_{\\cal X} \\,=\\, \n - \\frac{\\pi}{6} {\\cal M}^2 \\, \\pStr {\\mathbb{X}}_1~.\n\\label{prelimitingcase}\n\\end{equation}\n This result is independent of $\\rho$.\nMoreover, as expected for massive states, this contribution is finite.\nOf course, there will also be contributions from massless states.\nIn general, these contributions are more subtle to evaluate, and we know\nthat as {\\mbox{$a\\to 0$}} the effective removal of \nthe regulator will lead to divergences coming from \npotentially non-zero values of $\\zStr {\\mathbb{X}}_2$ (since\nit is the massless states which are charged under ${\\mathbb{X}}_2$ which\ncause the Higgs mass to diverge).\nHowever, massless states charged under ${\\mathbb{X}}_1$ --- like the massive\nstates --- do not lead to divergences. We might therefore imagine restricting our\nattention to cases with {\\mbox{$\\zStr {\\mathbb{X}}_2=0$}}, and deforming\nour theory slightly so that these massless ${\\mathbb{X}}_1$-charged states accrue small non-zero masses.\nIn that case, the calculation in Eq.~(\\ref{cheat}) continues to apply.\nWe can imagine removing this deformation without encountering any divergences.\nThis suggests that the full result for the regulated Higgs mass in the \n{\\mbox{$a\\to 0$}} limit should be the same as in Eq.~(\\ref{prelimitingcase}), but\nwith massless ${\\mathbb{X}}_1$-charged states also included.\nWe therefore expect\n\\begin{equation}\n \\lim_{a\\to 0} \\,\\widehat m_\\phi^2 (\\rho,a)\\Bigl|_{\\cal X} ~=~\n - \\frac{\\pi}{6} {\\cal M}^2 \\, {\\rm Str}\\, {\\mathbb{X}}_1~\n\\label{limitingcase}\n\\end{equation}\nin cases for which {\\mbox{$\\zStr {\\mathbb{X}}_2=0$}}.\nWe shall rigorously confirm this result below.\n\nAs discussed in Sect.~\\ref{sec:modinvregs},\nthe two quantities $(\\rho, a)$ that parametrize our modular-invariant regulator\nare analogous to the quantity $t$ that parametrized our non-minimal regulator.\nIndeed, these quantities effectively specify the value of the ``cutoff'' imposed by these regulators,\nand as such we can view these quantities as corresponding to a floating physical mass scale $\\mu$.\nThis scale $\\mu$ is defined in terms of $t$ for the non-minimal regulator\nin Eq.~(\\ref{mut}), \nand we have already seen that\nmaintaining alignment between this regulator and our modular-invariant regulator requires\nthat we enforce the condition in Eq.~(\\ref{alignment}).\nWe shall therefore identify a physical scale $\\mu$ for our modular-invariant regulator as \n\\begin{equation}\n \\mu^2(\\rho,a) ~\\equiv ~ \\frac{\\rho a^2}{\\alpha'}~.\n\\label{mudef}\n\\end{equation}\nSince {\\mbox{$\\rho \\sim {\\cal O}(1)$}}, \nthe {\\mbox{$a\\ll 1$}} region \nfor our regulator corresponds to the restricted region {\\mbox{$\\mu \\ll M_s$}}.\n\nThe identification in Eq.~(\\ref{mudef})\nenables us to rewrite\nour result in Eq.~(\\ref{limitingcase}) in the more suggestive form\n\\begin{equation}\n \\lim_{\\mu \\to 0} \\,\\widehat m_\\phi^2 (\\mu) \\Bigl|_{\\cal X} ~=~\n - \\frac{\\pi}{6} {\\cal M}^2 \\,{\\rm Str}\\,{\\mathbb{X}}_1~\n\\label{limitingcasemu}\n\\end{equation}\nin cases for which {\\mbox{$\\zStr{\\mathbb{X}}_2=0$}}.\nIn EFT language, we can therefore regard this result as holding in the deep infrared.\n\nThe natural question that arises, then, is to determine how our regulated Higgs \nmass $\\widehat m_\\phi^2(\\mu)$\n{\\it runs}\\\/ as a function of the scale $\\mu$.\nIn order to do this,\nwe need to evaluate \n$\\widehat m_\\phi^2 (\\rho,a)$ \nas a function of $a$ for small {\\mbox{$a\\ll 1$}} {\\it without}\\\/ taking the full {\\mbox{$a\\to 0$}} limit.\n\n\nAs indicated above, this calculation is somewhat intricate and is presented in Appendix~\\ref{higgsappendix}.~\nThe end result, given in Eq.~(\\ref{finalhiggsmassa}),\nis an expression for $\\widehat m_\\phi^2(\\rho,a)$ \nwhich is both {\\it exact}\\\/ and valid for all $a$.\nUsing the identification in Eq.~(\\ref{mudef}) and henceforth taking the benchmark value {\\mbox{$\\rho=2$}},\nthe result in Eq.~(\\ref{finalhiggsmassa}) can then be expressed in terms of the scale $\\mu$,\nyielding\n\\begin{eqnarray}\n && \\widehat m_\\phi^2(\\mu)\\Bigl|_{\\cal X} \\,=\\, \\frac{{\\cal M}^2}{1+\\mu^2\/M_s^2} \\Biggl\\lbrace \\nonumber\\\\ \n && ~~\\phantom{+} \n \\, \\zStr {\\mathbb{X}}_1 \\left\\lbrack - \\frac{\\pi}{6}\\left(1+\\mu^2\/M_s^2\\right) \\right\\rbrack \\nonumber\\\\\n && ~+ \\, \\zStr {\\mathbb{X}}_2 \\left\\lbrack \n \\log\\left( \\frac{ \\mu}{2\\sqrt{2} e M_s}\\right) \n \\right\\rbrack \\nonumber\\\\\n && ~+ \\, \\pStr {\\mathbb{X}}_1 \\, \\Biggl\\lbrace - \\frac{\\pi}{6} \n - \\frac{1}{2\\pi} \\left(\\frac{M}{{\\cal M}}\\right)^2 \\times \\nonumber\\\\\n && ~~~~~~~~~~~~ \n \\times \\left\\lbrack \n {\\cal K}_0^{(0,1)}\\!\\left( \\frac{2\\sqrt{2}\\pi M}{\\mu} \\right) + \n {\\cal K}_2^{(0,1)}\\!\\left( \\frac{2\\sqrt{2}\\pi M}{\\mu} \\right) \n \\right\\rbrack \\Biggr\\rbrace \\nonumber\\\\\n && ~+ \\, \\pStr {\\mathbb{X}}_2 \\, \n \\Biggl \\lbrack \n 2{\\cal K}_0^{(0,1)}\\!\\left( \\frac{2\\sqrt{2}\\pi M}{\\mu} \\right) \n - {\\cal K}_1^{(1,2)}\\!\\left( \\frac{2\\sqrt{2}\\pi M}{\\mu}\\right) \\Biggr\\rbrack \n \\Biggr\\rbrace\n\\nonumber\\\\\n \\label{finalhiggsmassmu}\n\\end{eqnarray}\nwhere we have defined the Bessel-function combinations\n\\begin{equation}\n {\\cal K}_\\nu^{(n,p)} (z) ~\\equiv~ \\sum_{r=1}^\\infty ~ (rz)^{n} \\Bigl\\lbrack \n K_\\nu(rz\/\\rho) - \\rho^p K_\\nu(rz) \\Bigr\\rbrack~, \n\\label{Besselcombos}\n\\end{equation}\nwith $K_\\nu(z)$ denoting\nthe modified Bessel function of the second kind.\nWe see, then, that \nthe contributions to \nthe running of \n$\\widehat m_\\phi^2(\\mu)\\bigl|_{\\cal X}$ \nfrom the different\nstates in our theory \ndepend rather non-trivially on their masses and on their various ${\\mathbb{X}}_1$ \nand ${\\mathbb{X}}_2$ charges, with\nthe contributions\nfrom each string state with non-zero mass $M$\ngoverned by various combinations of Bessel functions $K_\\nu(z)$ with\narguments {\\mbox{$z\\sim M\/\\mu$}}.\n\nThere is a plethora of physics wrapped within Eq.~(\\ref{finalhiggsmassmu}), and we shall\nunpack this result in several stages.\nFirst, it is straightforward to take the {\\mbox{$\\mu\\to 0$}} limit of Eq.~(\\ref{finalhiggsmassmu}) \nin order to verify our expectation in Eq.~(\\ref{limitingcase}).\nIndeed, in the {\\mbox{$\\mu\\to 0$}} limit, we have {\\mbox{$z\\to \\infty$}} for all {\\mbox{$M>0$}}.\nSince\n\\begin{equation}\n {\\cal K}_\\nu^{(n,p)}(z) ~\\sim~ \\sqrt{\\frac{\\pi \\rho}{2}} \\,z^{n-1\/2} \\,e^{-z\/\\rho} ~~~~{\\rm as}~ z\\to \\infty~,\n\\label{asymptoticform}\n\\end{equation}\nit then follows that\nall of the terms \ninvolving Bessel functions in Eq.~(\\ref{finalhiggsmassmu})\nvanish exponentially in the {\\mbox{$\\mu\\to 0$}} limit.\nFor cases in which {\\mbox{$\\zStr {\\mathbb{X}}_2 =0$}} [{\\it i.e.}\\\/, cases in which the original Higgs mass $m_\\phi^2$\nis finite, with no massless states charged under ${\\mathbb{X}}_2$], \nwe thus reproduce the result in Eq.~(\\ref{limitingcase}).\n\n\n \nUsing the result in Eq.~(\\ref{finalhiggsmassmu}),\nwe can also study the running of $\\widehat m_\\phi^2(\\mu)$ as a function of {\\mbox{$\\mu>0$}}.\nOf course, given that our $\\widehat {\\cal G}$-function acts as a regulator only for {\\mbox{$a\\ll 1$}},\nour analysis is restricted to the {\\mbox{$\\mu\\ll M_s$}} region.\nLet us first concentrate on the contributions from the terms within \nEq.~(\\ref{finalhiggsmassmu}) that do not involve Bessel functions.\nThese contributions are given by\n \\begin{equation}\n {{\\cal M}^2} \\,\\biggl\\lbrace \n - \\frac{\\pi}{6}\\, {\\rm Str}\\, {\\mathbb{X}}_1 \n + \\, \\zStr {\\mathbb{X}}_2 \\log\\left( \\frac{ \\mu}{2\\sqrt{2} e M_s}\\right)\\biggr\\rbrace~.\n\\label{finalhiggsmasslimit}\n\\end{equation}\nFrom this we see \nthat our deep-infrared \ncontribution to $\\widehat m_\\phi^2$ in Eq.~(\\ref{limitingcase}) \nactually persists as an essentially constant contribution for all scales {\\mbox{$\\mu\\ll M_s$}}. \nWe also see from Eq.~(\\ref{finalhiggsmasslimit}) that each massless string state \nalso contributes an additional logarithmic running \nwhich is proportional to its ${\\mathbb{X}}_2$ charge and which\npersists all the way into the deep infrared.\nGiven that massless ${\\mathbb{X}}_2$-charged states are\nprecisely the states that led to the original logarithmic \ndivergence in the {\\it unregulated}\\\/ Higgs mass $m_\\phi^2$,\nthis logarithmic running is completely expected.\nIndeed, it formally leads to a divergence in our regulated\nHiggs mass $\\widehat m_\\phi^2(\\mu)$ in the full {\\mbox{$\\mu\\to 0$}} limit\n(at which our regulator is effectively removed),\nbut otherwise produces a finite contribution for all other {\\mbox{$\\mu>0$}}.\nThe issues connected with this logarithm are actually no different from those\nthat arise in an ordinary field-theoretic calculation. \nWe shall discuss these issues in more detail in Sect.~\\ref{sec:Conclusions}\nbut in the meantime this term will not concern us further.\n\n\n\n\n\n\n\nThe remaining contributions are those arising from the terms\nwithin Eq.~(\\ref{finalhiggsmassmu}) involving supertraces over Bessel functions.\nAlthough our analysis is restricted to the {\\mbox{$\\mu\\ll M_s$}} region,\nour supertraces receive contributions from the entire string spectrum.\nThis necessarily includes states with masses {\\mbox{$M\\gsim M_s$}}, but may also include\npotentially light states with non-zero masses far below $M_s$.\nThe existence of such light states depends on our string construction and\non the specific string model in question. \nIndeed, such states are particularly relevant for the kinds of string models that motivate\nour analysis, namely (non-supersymmetric) string models \nin which the Standard Model is realized directly within the low-energy spectrum.\n\nThe Bessel functions \ncorresponding to states with masses {\\mbox{$M\\gsim M_s$}}\nhave arguments {\\mbox{$z\\sim M\/\\mu \\gg 1$}} when {\\mbox{$\\mu\\ll M_s$}}.\nAs a result,\nin accordance with Eqs.~(\\ref{finalhiggsmassmu}) and (\\ref{asymptoticform}), \nthe contributions from these states to the running of $\\widehat m_\\phi^2(\\mu)\\bigl|_{\\cal X}$ \nare exponentially suppressed.\nIt then follows that the dominant contributions to\nthe Bessel-function running of $\\widehat m_\\phi(\\mu)$ \nwithin the {\\mbox{$\\mu\\ll M_s$}} region \ncome from the correspondingly light states, {\\it i.e.}\\\/, states with masses {\\mbox{$M\\ll M_s$}}.\nHowever, for states with masses {\\mbox{$M\\ll M_s$}},\nwe see from Eq.~(\\ref{finalhiggsmassmu}) \nthat the corresponding Bessel-function contributions which are proportional to their ${\\mathbb{X}}_1$ charges\nare all suppressed by a factor $(M\/{\\cal M})^2$.\nWe thus conclude that\nthe contribution from a state of non-zero mass {\\mbox{$M\\ll M_s$}} within the string spectrum \nis sizable only when this state carries a non-zero ${\\mathbb{X}}_2$ charge.\nIndeed, we see from Eq.~(\\ref{finalhiggsmassmu}) that this contribution\nfor each bosonic degree of freedom of mass $M$ is given by\n\\begin{equation}\n 2{\\cal K}_0^{(0,1)}\\!\\left(z\\right) - {\\cal K}_1^{(1,2)}\\!\\left(z\\right) \n\\label{lightcontribution}\n\\end{equation}\nper unit of ${\\mathbb{X}}_2$ charge, \nwhere {\\mbox{$z\\equiv 2\\sqrt{2}\\pi M\/\\mu$}}.\n\n\n\n\n\nIn Fig.~\\ref{transientfigure}, we plot this contribution\nas a function of $\\mu\/M$.\nAs expected, we see that states with {\\mbox{$M\\gg \\mu$}} produce no running and can be ignored --- essentially\nthey have been ``integrated out'' of our theory at the scale $\\mu$ and leave behind only an exponential tail.\nBy contrast, states with {\\mbox{$M\\lsim \\mu$}} are still dynamical at the scale $\\mu$. \nWe see from Fig.~\\ref{transientfigure} that their effective contributions are then effectively {\\it logarithmic}\\\/.\nIndeed, as {\\mbox{$z\\to 0$}}, one can show that~\\cite{Paris}\n\\begin{eqnarray}\n {\\cal K}_0^{(0,1)}(z)~&\\sim &~ - {\\textstyle{1\\over 2}} \\log\\,z + {\\textstyle{1\\over 2}}\\left[ \\log\\,(2\\pi) - \\gamma\\right] \\nonumber\\\\ {\\cal K}_1^{(1,2)}(z)~&\\sim &~ 1~\n\\label{Kasymp}\n\\end{eqnarray}\nwhere $\\gamma$ is the Euler-Mascheroni constant.\nThis leads to an \nasymptotic logarithmic running of the form \n\\begin{equation}\n \\log\\left[ \\frac{1}{\\sqrt{2}}\\,e^{-(\\gamma+1)} \\frac{\\mu}{M}\\right]\n\\label{loglimit}\n\\end{equation}\nfor {\\mbox{$\\mu\\gg M$}} in Fig.~\\ref{transientfigure}.\nFinally, between these two behaviors, we see that the expression in Eq.~(\\ref{lightcontribution})\ninterpolates smoothly and even gives rise to a transient ``dip''.\nThis is a uniquely string-theoretic behavior resulting from the \nspecific combination of Bessel functions in Eq.~(\\ref{lightcontribution}).\nOf course, the statistics factor $(-1)^F$ within the supertrace\nflips the sign of this contribution for degrees of freedom which are fermionic. \n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[keepaspectratio, width=0.48\\textwidth]{transient.pdf}\n\\caption{\nThe expression in Eq.~(\\ref{lightcontribution}), plotted as a function of $\\mu\/M$.\nThis quantity is the Bessel-function contribution per unit ${\\mathbb{X}}_2$ charge \nto the running of the regulated Higgs mass $\\widehat m_\\phi^2(\\mu)\\bigl|_{{\\cal X}}\/{\\cal M}^2$\nfrom a bosonic state of non-zero mass $M$.\nThis contribution is universal for all $\\mu\/M$ and assumes only that {\\mbox{$\\mu \\ll M_s$}}.\nWhen {\\mbox{$\\mu\\gg M$}}, the state is fully dynamical and produces a running which is effectively logarithmic.\nBy contrast, when {\\mbox{$\\mu \\ll M$}}, the state is heavier than the scale $\\mu$ and is effectively\nintegrated out, thereby suppressing any contributions to the running. Finally, within the\nintermediate {\\mbox{$\\mu\\approx M$}} region, \nthe Bessel-function expression in Eq.~(\\ref{lightcontribution}) \nprovides a smooth connection between \nthese two asymptotic behaviors and even gives rise to a transient ``dip''\nin the overall running.\nNote that for a fixed scale $\\mu$, adjusting the mass $M$ of the relevant state\nupwards or downwards simply corresponds to shifting this curve\nrigidly to the right or left, respectively.\nIn this way one can imagine summing over all such contributions to the running\nas one takes the supertrace over the entire ${\\mathbb{X}}_2$-charged string spectrum.} \n\\label{transientfigure}\n\\end{figure}\n\n\n\nThus far we have focused on the \nHiggs-mass running, as shown in Fig.~\\ref{transientfigure},\n from a single massive string degree of freedom of mass $M$.\nHowever, the contribution from another string state with a different mass $M'$ can be\nsimply obtained by rigidly sliding this curve towards the left or right (respectively corresponding to \ncases with {\\mbox{$M'M$}}, respectively).\nThe complete supertrace contribution in Eq.~(\\ref{finalhiggsmassmu})\nis then obtained by summing over all of these curves, each with its appropriate horizontal displacement and each weighted by \nthe corresponding net (bosonic minus fermionic) number of degrees of freedom.\nThe resulting net running from the final term within \nEq.~(\\ref{finalhiggsmassmu}) is therefore highly sensitive to the properties of the \nmassive ${\\mathbb{X}}_2$-charged part of the string spectrum. \nThis will be discussed further in Sect.~\\ref{seehow}.~\nOf course, as discussed above, the contributions from states with {\\mbox{$M'\\gg \\mu$}} are\nexponentially suppressed. Thus, for any $\\mu$, the only states which contribute meaningfully to\nthis Bessel-function running of the Higgs mass are those with {\\mbox{$M\\lsim \\mu$}}. \n\n\n\nThus, combining these Bessel-function contributions with those from Eq.~(\\ref{finalhiggsmasslimit})\nand keeping only those (leading) terms which dominate when {\\mbox{$M\\ll \\mu\\ll M_s$}},\nwe see that we can approximate the exact result in Eq.~(\\ref{finalhiggsmassmu}) as\n\\begin{eqnarray}\n&& \\widehat m_\\phi^2(\\mu)\\Bigl|_{\\cal X} \\,\\approx\\,\n - \\frac{\\pi}{6}\\, {\\cal M}^2\\, {\\rm Str}\\, {\\mathbb{X}}_1 \n + {\\cal M}^2 \\, \\zStr {\\mathbb{X}}_2 \\,\\log\\left( \\frac{ \\mu}{2\\sqrt{2} e M_s}\\right)\\nonumber\\\\\n && ~~~~~~~ + {\\cal M}^2 \\,\\effStr \n {\\mathbb{X}}_2 \\,\\log\\left[ \\frac{1}{\\sqrt{2}}\\,e^{-(\\gamma+1)} \\frac{\\mu}{M}\\right]~.~~~~~~~~\n\\label{approxhiggsmassmu}\n\\end{eqnarray}\nInterestingly, we see that to leading order,\nthe ${\\mathbb{X}}_1$ charges of the string states\nonly contribute to an overall constant term in Eq.~(\\ref{approxhiggsmassmu}),\nand they do this for all states regardless of their masses.\nBy contrast, it is the ${\\mathbb{X}}_2$ charges \nof the states which induce a corresponding running,\nand this only occurs for those\nstates within the EFT at the scale $\\mu$ --- {\\it i.e.}\\\/, those light \nstates with masses {\\mbox{$M\\lsim \\mu$}}.\n\n\nThe net running produced by the final term in Eq.~(\\ref{approxhiggsmassmu})\ncan exhibit a variety of behaviors.\nTo understand this, let us consider the behavior of this term\nas we increase $\\mu$ from the deep infrared.\nOf course, this term does not produce any running at all \nuntil we reach {\\mbox{$\\mu \\sim M_{\\rm lightest}$}},\nwhere $M_{\\rm lightest}$ is the mass of the lightest massive string state\ncarrying a non-zero ${\\mathbb{X}}_2$ charge.\nThis state then contributes a logarithmic\nrunning which persists for all higher $\\mu$.\nHowever, as $\\mu$ increases still further,\nadditional ${\\mathbb{X}}_2$-charged string states \nenter the EFT and contribute their own individual logarithmic contributions. \nOf course, if these additional states \nhave masses {\\mbox{$M\\gg M_{\\rm lightest}$}},\nthe logarithmic nature of the running shown in Fig.~\\ref{transientfigure}\nfrom the state with mass $M_{\\rm lightest}$\nwill survive intact until {\\mbox{$\\mu \\sim M$}}.\nHowever, if the spectrum of states is relatively dense \nbeyond $M_{\\rm lightest}$, the logarithmic contributions from each of these states\nmust be added together, leading to a far richer behavior.\n \nOne important set of string models exhibiting the latter property\nare those involving a relatively large compactification radius $R$.\nIn such cases, we can identify {\\mbox{$M_{\\rm lightest}\\sim 1\/R$}},\nwhereupon we expect an entire tower of corresponding Kaluza-Klein (KK) states\nof masses {\\mbox{$M_k\\sim k\/R$}}, {\\mbox{$k\\in\\mathbb{Z}^+$}}, each sharing a common charge ${\\mathbb{X}}_2$ and\na common degeneracy of states $g$.\nFor any scale $\\mu$, the final term in Eq.~(\\ref{approxhiggsmassmu}) \nthen takes the form\n\\begin{eqnarray}\n && {\\cal M}^2 \\,g \\,{\\mathbb{X}}_2 \\,\\sum_{k=1}^{\\mu R}\n \\,\\log\\left[ \\frac{1}{\\sqrt{2}}\\,e^{-(\\gamma+1)} \\, \\frac{\\mu R}{k}\\right]~\\nonumber\\\\\n && ~~=~ {\\cal M}^2 \\,g \\,{\\mathbb{X}}_2 \\,\\left\\lbrace \n \\mu R \\,\\log\\left[ \\frac{1}{\\sqrt{2}}\\,e^{-(\\gamma+1)} \\, \\mu R\\right] - \\log \\,(\\mu R)!\\right\\rbrace\\nonumber\\\\ \n && ~~=~ {\\cal M}^2 \\,g \\,{\\mathbb{X}}_2 \\,\n \\, \\left\\lbrace \n \\log\\left[ \\frac{1}{\\sqrt{2}}\\,e^{-(\\gamma+1)} \\right] +1 \\right\\rbrace \\, \\mu R\n\\label{newpower}\n\\end{eqnarray}\nwhere in passing to the third line we have used Stirling's approximation\n{\\mbox{$\\log N! \\approx N \\log N - N$}}.\nWe thus see that in such cases our sum over logarithms actually produces a {\\it power-law}\\\/ \nrunning! In this case the running is linear, \nbut in general the KK states associated with $d$ large toroidally-compactified dimensions\ncollectively yield a regulated Higgs mass whose running scales as $\\mu^d$.\n\nThis phenomenon whereby a sum over KK states deforms a running from logarithmic to power-law\nis well known from phenomenological studies of theories with large extra dimensions,\nwhere it often plays a crucial role \n(see, {\\it e.g.}\\\/, Refs.~\\cite{Dienes:1998vh, Dienes:1998vg, Dienes:1998qh}).\nThis phenomenon can ultimately be understood from \nthe observation that a large compactification radius\neffectively increases the overall spacetime dimensionality of the theory,\nthereby shifting the mass dimensions of quantities such \nas gauge couplings and Higgs masses and simultaneously shifting their corresponding runnings.\nIndeed, as discussed in detail in Appendices~A and B \nof Ref.~\\cite{Dienes:1998vg}\n(and as illustrated in Fig.~11 therein),\nthe emergence of power-law running from logarithmic running is surprisingly robust. \n\nOf course, it may happen that \nthe spectrum of light states not only has a lightest mass $M_{\\rm lightest}$\nbut also a heaviest mass $M_{\\rm heaviest}$, with a significant mass gap beyond this\nbefore reaching even heavier scales. \nIf such a situation were to arise (but clearly does not within the large extra-dimension\nscenario described above),\nthen the corresponding running of $\\widehat m_\\phi^2(\\mu)\\bigl|_{\\cal X}$\nwould only be power-law within the range {\\mbox{$M_{\\rm lightest}\\lsim \\mu\\lsim M_{\\rm heaviest}$}}.\nFor {\\mbox{$\\mu > M_{\\rm heaviest}$}}, by contrast, the running would then revert back to logarithmic. \n\nIn summary, we see that while the first term within\nEq.~(\\ref{approxhiggsmassmu}) \nrepresents an overall constant contribution arising from the entire spectrum of ${\\mathbb{X}}_1$-charged states,\nthe second term represents an overall logarithmic contribution from precisely the massless ${\\mathbb{X}}_2$-charged states\nwhich were the source of the original divergence of the unregulated Higgs mass $m_\\phi^2$.\nBy contrast, the final term\nin Eq.~(\\ref{approxhiggsmassmu}) \nrepresents the non-trivial contribution to the running from\nthe massive ${\\mathbb{X}}_2$-charged states.\nAs we have seen,\nthis latter contribution can exhibit a variety of behaviors, ranging from logarithmic (in cases with\nrelatively large mass splittings between the lightest massive ${\\mathbb{X}}_2$-charged\nstates) to power-law (in cases with relatively small uniform mass splittings between\nsuch states).\nOf course, \ndepending on the details of the underlying string spectrum,\nmixtures between these different behaviors are also possible. \n \n\n\\subsubsection{Contribution from the cosmological constant \\label{Lambdasect}} \n\nLet us now turn to the \nsecond term in Eq.~(\\ref{twocontributions}).\nThis contribution lacks ${\\cal X}_i$ insertions\nand arises from the cosmological-constant term in Eq.~(\\ref{Higgsmass}).\nAlthough this contribution is the result of a universal shift in the background moduli\nand is thus independent of the specific ${\\cal T}$-matrices,\nwe shall now demonstrate that it too can be expressed as a supertrace over the physical string spectrum.\nIt also develops a scale dependence when subjected to our modular-invariant regulator.\n\nWithin the definition of $\\Lambda$ in Eq.~(\\ref{lambdadeff}),\nthe integrand function ${\\cal F}(\\tau,{\\overline{\\tau}})$ is simply $(-{\\cal M}^4\/2) {\\cal Z}(\\tau,{\\overline{\\tau}})$\nwhere ${\\cal Z}(\\tau,{\\overline{\\tau}})$ is the partition function of the string in the Higgsed phase.\nOf course, if this theory exhibits unbroken spacetime supersymmetry, \nthe contributions from the bosonic states in the spectrum cancel level-by-level against\nthose from their fermionic superpartners. In such cases we then have {\\mbox{${\\cal Z}=0$}}, implying {\\mbox{$\\Lambda=0$}}.\nOtherwise, for heterotic strings, we necessarily have {\\mbox{${\\cal Z}\\not =0$}}.\nIndeed, it is a theorem (first introduced in Ref.~\\cite{Dienes:1990ij} and discussed more recently, {\\it e.g.}\\\/, in \nRef.~\\cite{Abel:2015oxa})\nthat any non-supersymmetric heterotic string model in $D$ spacetime dimensions must contain an off-shell\ntachyonic {\\it proto-graviton}\\\/ state\nwhose contribution to the partition function remains uncancelled. This then results in a string partition function\nwhose power-series expansion has the leading behavior {\\mbox{$Z=(D-2)\/q + ...$}} \n\nIn principle this proto-graviton contribution would appear to introduce \nan exponential divergence as {\\mbox{$\\tau_2\\to\\infty$}}, thereby taking us beyond\nthe realm of validity for the mathematical techniques presented in Sect.~\\ref{sec:RStechnique}.~\nHowever, this tachyonic state is off-shell and thus does not appear in the actual physical string spectrum.\nIndeed, as long as there are no additional {\\it on-shell}\\\/ tachyons present in the theory,\nthe corresponding integral $\\Lambda$ is fully convergent\nbecause the integral over the fundamental domain ${\\cal F}$ comes\nwith an explicit instruction that we are to integrate across $\\tau_1$ in the {\\mbox{$\\tau_2>1$}} region of ${\\cal F}$\n{\\it before}\\\/ integrating over $\\tau_2$.\nThis integration therefore prevents the proto-graviton state from contributing to $\\Lambda$ \nwithin the {\\mbox{$\\tau_2>1$}} region of integration, and likewise prevents this state from contributing to $g(\\tau_2)$. \n\n\nAssuming, therefore, that we can disregard the proto-graviton contribution to ${\\cal Z}$ as {\\mbox{$\\tau_2\\to\\infty$}},\nwe find that {\\mbox{${\\cal Z}\\sim \\tau_2^{-1}$}} as {\\mbox{$\\tau_2\\to\\infty$}}.\nThus ${\\cal Z}$ is effectively of rapid decay and we can use the \noriginal Rankin-Selberg results in Eq.~(\\ref{RSresult}).\nIn this connection, we note that this assumption regarding the proto-graviton contribution\nfinds additional independent \nsupport through the arguments presented in Ref.~\\cite{Kutasov:1990sv} which \ndemonstrate that any contributions from the proto-graviton beyond those in Eq.~(\\ref{RSresult}) \nare suppressed by an infinite volume factor in all spacetime dimensions {\\mbox{$D>2$}}.\nA similar result is also true in string models\nwith exponentially suppressed cosmological constants~\\cite{Abel:2015oxa}.\n \nWith ${\\cal Z}$ taking the form in Eq.~(\\ref{integrand})\nand with the mass $M$ of each physical string state \nidentified via {\\mbox{$\\alpha' M^2=2(m+n)=4m$}},\nwe then have\n\\begin{equation}\n g(\\tau_2) ~=~ -\\frac{{\\cal M}^4}{2} \\, \\tau_2^{-1}\\, {\\rm Str}\\, e^{-\\pi \\alpha' M^2 \\tau_2}~.\n\\label{glambda}\n\\end{equation}\nInserting this result into Eq.~(\\ref{RSresult}) and performing the $\\tau_2$ integral\nthen yields~\\cite{Dienes:1995pm}\n\\begin{eqnarray}\n \\Lambda ~&=&~ - \\frac{{\\cal M}^4}{2}\\, \\frac{\\pi}{3} \\,\\oneRes\n \\biggl[\\pi^{2-s} \\,\\Gamma(s-2) \\,{\\rm Str}\\, (\\alpha' M^2)^{2-s}\\biggr\\rbrack\\nonumber\\\\\n ~&=&~ \\frac{{\\cal M}^4}{2} \\frac{\\pi^2}{3} {\\rm Str}\\, (\\alpha' M^2) \\nonumber\\\\\n ~&=&~ \\frac{1}{24} {\\cal M}^2 \\, {\\rm Str}\\, M^2.\n\\label{eq:lamlam}\n\\end{eqnarray} \nWe thus see that $\\Lambda$ is given as a universal supertrace \nover {\\it all}\\\/ physical string states,\nand not only those with specific charges relative to the Higgs field.\n\nAs evident from the form of the final supertrace in Eq.~(\\ref{eq:lamlam}), \nmassless states do not ultimately contribute within this expression for $\\Lambda$.\nStrictly speaking, our derivation in Eq.~(\\ref{eq:lamlam}) already implicitly assumed\nthis, given that the intermediate steps in Eq.~(\\ref{eq:lamlam}) are valid only for {\\mbox{$M>0$}}.\nHowever, it is easy to see that the contributions\nfrom massless states lead to a $\\tau_2$-integral whose divergence has no residue at {\\mbox{$s=1$}}.\nThus, massless states make no contribution to this expression, \nand the result in Eq.~(\\ref{eq:lamlam}) stands.\n\nThis does {\\it not}\\\/ mean that massless states do not contribute to $\\Lambda$, however.\nRather, this just means that the constraints from modular invariance \nso tightly connect \nthe contributions to $\\Lambda$ from the massless states to those from the \nmassive states \n(and also those from the unphysical string states of any mass)\nthat an expression for $\\Lambda$ as in Eq.~(\\ref{eq:lamlam}) becomes possible.\n \n\n \nFor further insight into this issue,\nit is instructive to obtain this same result through Eq.~(\\ref{reformulation}). \nWe then have\n\\begin{equation}\n \\Lambda~=~ -\\frac{\\pi}{3} \\frac{{\\cal M}^4}{2} \\lim_{\\tau_2\\to 0} \n \\biggl\\lbrack \\tau_2^{-1} \\,{\\rm Str}\\, \\exp\\left( -\\pi \\alpha' M^2 \\tau_2\\right)\\biggr\\rbrack~.\n\\end{equation}\nExpanding the exponential {\\mbox{$e^{-x}\\approx 1 -x + ...$}} and taking the {\\mbox{$\\tau_2\\to 0$}} limit of each term separately, \nwe find that the linear term leads directly to the result in Eq.~(\\ref{eq:lamlam}) while the contributions from all\nof the higher terms vanish.\nInterestingly, the constant term would {\\it a priori}\\\/ appear to lead to a divergence for $\\Lambda$.\nThe fact that $\\Lambda$ is finite in such theories then additionally tells us that~\\cite{Dienes:1995pm}\n\\begin{equation} \n {\\rm Str}\\, {\\bf 1} ~=~0~.\n\\label{eq:lamlam0}\n\\end{equation}\nAs apparent from our derivation, this constraint must hold for any \ntachyon-free modular-invariant theory \n({\\it i.e.}\\\/, any modular-invariant theory in which $\\Lambda$ is finite). \nIndeed, this is one of the additional constraints from modular invariance which\nrelates the contributions of the physical string states\nwhich are massless to the contributions of those which are massive.\nThus, we may regard the result in Eq.~(\\ref{eq:lamlam}) --- like all of the results of this paper --- \nas holding within a modular-invariant context \nin which other constraints such as that in Eq.~(\\ref{eq:lamlam0}) are also simultaneously \nsatisfied.\nWe also see from this analysis that \nour supertrace definition in Eq.~(\\ref{supertracedef})\nmay be more formally defined as~\\cite{Dienes:1995pm}\n\\begin{equation}\n {\\rm Str} \\, A ~\\equiv~ \n \\lim_{y\\to 0} \\, \\sum_{{\\rm physical}~ i} (-1)^{F_i} \\, A_i \\, e^{- y \\alpha' M_i^2}~.\n\\label{supertracedef2}\n\\end{equation}\n\nThe supertrace results in Eqs.~(\\ref{eq:lamlam}) and (\\ref{eq:lamlam0}) were first derived in \nRef.~\\cite{Dienes:1995pm}.\nAs discussed in Refs.~{\\mbox{\\cite{Dienes:1995pm,Dienes:2001se}}}, \nthese results hold for all tachyon-free heterotic strings in four dimensions,\nand in fact similar results hold in all spacetime dimensions {\\mbox{$D>2$}}.\nFor theories exhibiting spacetime supersymmetry,\nthese relations are satisfied rather trivially.\nHowever, even if the spacetime supersymmetry is broken\n--- and even if the scale of supersymmetry breaking is relatively large or at the Planck scale ---\nthese results nevertheless continue to hold.\nIn such cases, these supertrace relations do not arise as the results of pairwise cancellations\nbetween the contributions of bosonic and fermionic string states.\nRather, these relations emerge as the results of conspiracies that occur across the {\\it entire}\\\/\nstring spectrum, with the bosonic and fermionic string states \nalways carefully arranging themselves \nat all string mass levels \nso as to exhibit a so-called ``misaligned supersymmetry''~{\\mbox{\\cite{Dienes:1994np,Dienes:2001se}}}.\nNo pairing of bosonic and fermionic states \noccurs within misaligned supersymmetry,\nyet misaligned supersymmetry ensures that these supertrace relations are always satisfied.\nThese results therefore constrain the extent to which supersymmetry can be broken in tachyon-free string theories\nwhile remaining consistent with modular invariance. \n\n\nThe results that we have obtained thus far pertain to the cosmological constant $\\Lambda$.\nAs such, they would be sufficient if we were aiming to understand this quantity unto itself,\nsince $\\Lambda$ is finite in any tachyon-free modular-invariant\ntheory and hence requires no regulator.\nHowever, in this paper our interest in this quantity stems from the fact that $\\Lambda$ is\nan intrinsic contributor to the total Higgs mass in Eq.~(\\ref{relation1}),\nand we already have seen that the Higgs mass requires regularization.\nAt first glance, one might imagine regulating the terms with non-zero ${\\cal X}_i$ insertions\nwhile leaving the $\\Lambda$-term alone.\nHowever, it is ultimately inappropriate to regularize only a subset of terms that contribute\nto the Higgs mass --- for consistency we must apply the same regulator to the entire expression\nat once.\nIndeed, we recall from Sect.~\\ref{sec2} that the entire Higgs-mass expression including $\\Lambda$ forms\na modular-invariant unit, with $\\Lambda$ emerging from the modular completion\nof some of the terms with non-trivial ${\\cal X}_i$ insertions.\nFor this reason,\nwe shall now study the analogously regulated \ncosmological constant\n\\begin{equation}\n \\widehat \\Lambda (\\rho,a) ~\\equiv~ \\int_{\\cal F} \\dmu~ {\\cal Z}(\\tau) \\, \\widehat{\\cal G}_\\rho(a,\\tau)~\n\\label{Lambdahatdef}\n\\end{equation}\nand determine the extent to which this regularized cosmological constant\ncan also be expressed in terms of supertraces over the physical string states.\n\nOur discussion proceeds precisely as for the terms involving the ${\\cal X}_i$ insertions.\nFollowing the result in Eq.~(\\ref{Irhoa}) we find that \n$\\widehat \\Lambda (\\rho,a)$ can be expressed as\n\\begin{equation}\n \\widehat \\Lambda(\\rho,a) ~=~ \n\\frac{\\pi}{3}\\, \\oneRes \\, \\int_0^\\infty d\\tau_2 \\,\\tau_2^{s-3} \\,\\widehat g_\\rho(a,\\tau_2) ~ \n\\label{Irhoa3}\n\\end{equation}\nwhere \n\\begin{eqnarray}\n && \\widehat g_\\rho(a,\\tau_2) \\,\\equiv\\, \n -\\frac{{\\cal M}^4}{2}\n \\int_{-1\/2}^{1\/2} d\\tau_1 \\nonumber\\\\ \n && ~~~~~~~~~ \n \\times\\, \\left\\lbrace \n\\left\\lbrack \n \\sum_{m,n} (-1)^F \\,{\\overline{q}}^{m} q^{n} \\right\\rbrack \n \\,\\widehat {\\cal G}_\\rho(a,\\tau)\\right\\rbrace ~~~\\nonumber\\\\\n&& ~~~~~ = ~\n -\\frac{{\\cal M}^4}{2}\n \\left\\lbrack {\\rm Str} \\, e^{-\\pi \\alpha' M^2 \\tau_2} \\right\\rbrack \n \\widehat {\\cal G}_\\rho(a,\\tau_2) ~.~~~~~ \n\\label{gFGdef4}\n\\end{eqnarray}\nIn the second line the sum over $(m,n)$ indicates a sum over the entire spectrum \nof the theory, while\nin passing to the factorized form in the third line of Eq.~(\\ref{gFGdef4}) we \nhave again followed the result in Eq.~(\\ref{gFGdef2}) \nand explicitly restricted our attention to those cases\nwith {\\mbox{$a\\ll 1$}}, as appropriate for the \nregulator function $\\widehat {\\cal G}_\\rho(a,\\tau)$.\nIndeed, the term within square brackets in the second line of \nEq.~(\\ref{gFGdef3}) is our desired supertrace over physical string states,\nwhile the regulator function $\\widehat{\\cal G}_\\rho(a,\\tau_2)$ \nprovides a non-trivial $\\tau_2$-dependent weighting to the different\nterms within $g_\\rho(a,\\tau_2)$. \n\n\nOnce again, \nthe next step is to substitute Eq.~(\\ref{gFGdef4}) back into Eq.~(\\ref{Irhoa3})\nand evaluate the residue at {\\mbox{$s=1$}}. \nIn general, the presence of the regulator function $\\widehat{\\cal G}_\\rho(a,\\tau_2)$ within \nEq.~(\\ref{gFGdef3}) renders this calculation somewhat intricate. However, \njust as for the terms with non-trivial ${\\cal X}_i$ insertions,\nwe know that {\\mbox{$\\widehat {\\cal G}_\\rho(a,\\tau_2)\\to 1$}} as {\\mbox{$a\\to 0$}}.\nIn this limit, we therefore expect to obtain our original (finite) unregulated $\\Lambda$:\n\\begin{equation}\n \\lim_{a\\to 0} \\,\\widehat \\Lambda(\\rho,a) ~=~ \\Lambda ~=~ \n \\frac{1}{24} {\\cal M}^2 \\, {\\rm Str}\\, M^2~\n\\label{limitingcase2}\n\\end{equation}\nwhere in the final equality we have utilized the result in Eq.~(\\ref{eq:lamlam}).\nEquivalently, upon identifying the physical scale $\\mu$ as in Eq.~(\\ref{mut}),\nwe thus expect\n\\begin{equation}\n \\lim_{\\mu\\to 0} \\,\\widehat \\Lambda (\\mu) ~=~ \\Lambda~.\n\\label{lambdamulimit}\n\\end{equation}\n\nLet us now determine how $\\widehat\\Lambda(\\mu)$ runs\nas a function of the scale $\\mu$.\nTo do this, we need to evaluate $\\widehat \\Lambda(\\rho,a)$ explicitly as a function of $\\rho$ and $a$.\nThis question is tackled in Appendix~\\ref{lambdaappendix}, yielding the exact result\nin Eq.~(\\ref{lambdaresult}). Written in terms of the physical scale $\\mu$ in Eq.~(\\ref{mudef}) \nthis result then takes the form\n\\begin{eqnarray}\n \\widehat\\Lambda (\\mu ) ~&=&~ \\frac{1}{1+\\mu^2\/M_s^2} \\, \\Biggl\\lbrace\n \\frac{{\\cal M}^2}{24} \\, {\\rm Str}\\,M^2 \\nonumber\\\\\n && ~~ - \\frac{7}{960 \\pi^2} \\,(n_B-n_F)\\, \\mu^4 \n~\\nonumber\\\\\n && ~~ - \\frac{1}{2\\pi^2}\\,\n \\pStr M^4 \\,\\Biggl[ {\\cal K}_1^{(-1,0)}\\!\\left( \\frac{2\\sqrt{2} \\pi M}{\\mu} \\right) ~~~~~~\\nonumber\\\\\n && ~~~~~~~~~ + 4\\, {\\cal K}_2^{(-2,-1)}\\!\\left( \\frac{2\\sqrt{2} \\pi M}{\\mu}\\right)\\nonumber\\\\\n && ~~~~~~~~~ + {\\cal K}_3^{(-1,0)}\\!\\left( \\frac{2\\sqrt{2} \\pi M}{\\mu} \\right)\n \\Biggr] ~ \\Biggr\\rbrace~\n\\label{lambdamuresult}\n\\end{eqnarray}\nwhere we have again taken {\\mbox{$\\rho=2$}} as our benchmark value, \nwhere ${\\cal K}_\\nu^{(n,p)}(z)$ are the Bessel-function combinations defined in Eq.~(\\ref{Besselcombos}),\nand where $n_{B}$ and $n_F$ are the numbers of massless bosonic and fermionic degrees of freedom in the theory\nrespectively (so that {\\mbox{$\\zStr {\\bf 1} = n_B-n_F$}}).\n\n\n\nIt is straightforward to verify that \nthis result is consistent \nwith the result in Eq.~(\\ref{lambdamulimit}) in the {\\mbox{$\\mu\\to 0$}} limit.\nBecause all of the Bessel-function combinations within Eq.~(\\ref{lambdamuresult}) vanish\nexponentially rapidly as their arguments grow to infinity, \nonly the first term in Eq.~(\\ref{lambdamuresult}) survives in this limit.\nWe therefore find that the {\\mbox{$\\mu\\to 0$}} limit of Eq.~(\\ref{lambdamuresult}) \nyields the anticipated result in Eq.~(\\ref{lambdamulimit}). \n\nFrom Eq.~(\\ref{lambdamuresult}) \nwe can also understand the manner in which $\\widehat \\Lambda(\\mu)$ runs as a function of $\\mu$ for all {\\mbox{$0<\\mu\\ll M_s$}}.\nLet us first focus on the Bessel-function terms within the square brackets in Eq.~(\\ref{lambdamuresult}).\nBy themselves, these terms behave in much the same way as shown in Fig.~\\ref{transientfigure}, except without the \ntransient dip and with the asymptotic behavior for {\\mbox{$\\mu\\gsim M$}} scaling as a power (rather than logarithm) of $\\mu$.\nMore specifically, to leading order in $\\mu\/M$ and for {\\mbox{$\\mu\\gsim M$}}, we find using the techniques developed\nin Ref.~\\cite{Paris} that\n\\begin{eqnarray}\n && {\\cal K}_1^{(-1,0)}\\!\\left( z \\right) \n + 4 \\,{\\cal K}_2^{(-2,-1)}\\!\\left( z\\right) + {\\cal K}_3^{(-1,0)}\\!\\left( z \\right) ~~~~~~~\\nonumber\\\\ \n && ~~~~~~~~~~~~~~~~~~~~~~~\\sim~ \\frac{7}{480} \\left( \\frac{\\mu}{M}\\right)^4~\n\\end{eqnarray}\nwhere {\\mbox{$z\\equiv 2\\sqrt{2}\\pi M\/\\mu$}}.\nBy contrast, for {\\mbox{$\\mu \\lsim M$}}, this quantity is exponentially suppressed.\nThus, \nrecalling \nthe result in Eq.~(\\ref{eq:lamlam})\nfor our original unregulated (but nevertheless finite) cosmological constant $\\Lambda$ \nand once again keeping only those (leading) running terms which dominate for {\\mbox{$M\\ll \\mu \\ll M_s$}},\nwe find that Eq.~(\\ref{lambdamuresult}) simplifies to take the approximate form\n \\begin{eqnarray}\n \\widehat\\Lambda (\\mu ) ~&\\approx&~ \\Lambda - \\frac{7}{960 \\pi^2} \\left[ \\left(\\zStr {\\bf 1} \\right) +\n \\left( \\effStr {\\bf 1} \\right) \\right]\\! \\mu^4 ~\\nonumber\\\\\n &\\approx&~ \\Lambda - \\frac{7}{960 \\pi^2 } \\left( \\zeffStr {\\bf 1} \\right) \\mu^4 ~.\n\\label{lambdamuresult2}\n\\end{eqnarray}\nWe once again emphasize that we have retained the second term (scaling as $\\mu^4$) as this is the \nleading $\\mu$-dependent term when {\\mbox{$M\\ll \\mu \\ll M_s$}}. \nJust as for $\\widehat m_\\phi^2(\\mu)\\bigl|_{\\cal X}$, there also generally \nexist additional running terms \nwhich scale as $\\mu^2$ and $\\log\\,\\mu$, but these terms are subleading\n relative to the above $\\mu^4$ term when {\\mbox{$M\\ll \\mu \\ll M_s$}}. \nWe shall discuss these subleading terms further in Sect.~\\ref{sec5}.~\nMoreover, just as we saw for $\\widehat m_\\phi^2(\\mu)\\bigl|_{\\cal X}$, the $\\mu^4$ scaling\nbehavior can be enhanced to an even greater effective power $\\mu^n$ with {\\mbox{$n>4$}} if the spectrum of light states\nis sufficiently dense when taking the supertrace over string states.\nHowever, even this leading $\\mu^n$ scaling is generally subleading compared with the constant term $\\Lambda$.\nThus the regulated quantity $\\widehat \\Lambda(\\mu)$ --- unlike $\\widehat m_\\phi^2(\\mu)\\bigl|_{\\cal X}$ \nin Eq.~(\\ref{approxhiggsmassmu}) ---\nis dominated by a constant term and exhibits at most a highly suppressed running relative to this constant.\n\n\n\n\n\\FloatBarrier\n\\subsubsection{The Higgs mass in string theory: See how it runs!\\label{seehow}}\n\n\nWe now finally combine both contributions \n$\\widehat m_\\phi^2(\\mu)\\bigl|_{{\\cal X},\\Lambda}$\nas in Eq.~(\\ref{twocontributions})\nin order to obtain our final result for the \ntotal modular-invariant regulated Higgs mass $\\widehat m_\\phi^2(\\mu)$.\nThe exact result, of course, is given by the sum of Eqs.~(\\ref{finalhiggsmassmu})\nand (\\ref{lambdamuresult}), with the \nlatter first multiplied by $\\xi\/(4\\pi^2 {\\cal M}^2)$.\nHowever, once again taking the corresponding approximate forms in\nEqs.~(\\ref{approxhiggsmassmu}) and (\\ref{lambdamuresult2})\nwhich are valid for {\\mbox{$M\\ll \\mu\\ll M_s$}},\nwe see that the $\\mu^4$ running within \nEq.~(\\ref{lambdamuresult2})\nis no longer the dominant running for \n$\\widehat m_\\phi^2(\\mu)$ as a whole, \nas it is extremely suppressed compared with the running coming from\nEq.~(\\ref{approxhiggsmassmu}).\nWe thus find that to leading order,\nthe net effect of adding \nEqs.~(\\ref{approxhiggsmassmu}) and (\\ref{lambdamuresult2})\nis simply to add the overall constant $\\xi \\Lambda\/(4\\pi^2 {\\cal M}^2)$ to \nthe result in Eq.~(\\ref{approxhiggsmassmu}).\nWe therefore find that the total regulated Higgs mass has the leading\nrunning behavior\n\\begin{eqnarray}\n && \\widehat m_\\phi^2(\\mu) ~\\approx~\n \\frac{\\xi}{4\\pi^2} \\frac{\\Lambda}{{\\cal M}^2}\n - \\frac{\\pi}{6}\\, {\\cal M}^2\\, {\\rm Str}\\, {\\mathbb{X}}_1\\nonumber\\\\\n && ~~~~~~~~~~ \n + {\\cal M}^2 \\, \\zStr {\\mathbb{X}}_2 \\,\\log\\left( \\frac{ \\mu}{2\\sqrt{2} e M_s}\\right)\\nonumber\\\\\n && ~~~~~~~~~~ \n + {\\cal M}^2 \\,\\effStr \n {\\mathbb{X}}_2 \\,\\log\\left[ \\frac{1}{\\sqrt{2}}\\,e^{-(\\gamma+1)} \\frac{\\mu}{M}\\right]~~~~~~~~~\n\\label{totalhiggsrunning}\n\\end{eqnarray}\nwhere we have retained only the terms \nthat are leading for {\\mbox{$M\\ll \\mu\\ll M_s$}}.\nOnce again, just as for $\\widehat m_\\phi^2(\\mu)\\bigl|_{\\cal X}$,\nwe see that to this order the ${\\mathbb{X}}_2$ charges of the string states\nlead to non-trivial running while their ${\\mathbb{X}}_1$ charges only contribute\nto an overall additive constant. \nIndeed, in the {\\mbox{$\\mu\\to 0$}} limit, we find\n\\begin{equation}\n \\lim_{\\mu\\to 0} \\widehat m_\\phi^2(\\mu) ~=~\n \\frac{\\xi}{4\\pi^2} \\frac{\\Lambda}{{\\cal M}^2}\n - \\frac{\\pi}{6}\\, {\\cal M}^2\\, {\\rm Str}\\, {\\mathbb{X}}_1\n\\label{asymplimit}\n\\end{equation}\nwhen {\\mbox{$\\zStr {\\mathbb{X}}_2=0$}}.\nOf course, when {\\mbox{$\\zStr {\\mathbb{X}}_2\\not=0$}}, the {\\mbox{$\\mu\\to 0$}} limit diverges, as expected from the \nfact that the massless ${\\mathbb{X}}_2$-charged states are precisely the states that led\nto a divergence in the original unregulated Higgs mass $m_\\phi^2$. \nAs discussed in Sect.~\\ref{chargeinsertions},\nwe nevertheless continue to obtain a finite result for the regulated Higgs mass $\\widehat m_\\phi^2(\\mu)$ \nfor all {\\mbox{$\\mu>0$}} even when\n{\\mbox{$\\zStr {\\mathbb{X}}_2\\not=0$}}.\n\nIn order to understand what the running in Eq.~(\\ref{totalhiggsrunning}) looks like for {\\mbox{$0<\\mu\\ll M_s$}}, \nlet us begin by considering the contribution from \na single ${\\mathbb{X}}_2$-charged string state with a given mass {\\mbox{$0 M_{\\rm lightest}$}}, where\n$M_{\\rm lightest}$ denotes the mass \nof the lightest ${\\mathbb{X}}_2$-charged states.\nIndeed, as discussed previously, whether an effective power-law running emerges depends on the density of states in the theory\nwith masses {\\mbox{$M\\gsim M_{\\rm lightest}$}}. \nIt is for this reason that we have indicated in Fig.~\\ref{anatomy} \nthat the net running within the {\\mbox{$\\mu> M_{\\rm lightest}$}} region can be either logarithmic\nor power-law.\nHowever, as we progress to lower scales {\\mbox{$\\mu\\sim M_{\\rm lightest}$}}, \nwe enter the ``dip region''\nwhere this logarithmic\/power-law running shuts off.\nFinally, for {\\mbox{$\\mu0$}} while\nmassless states have {\\mbox{$\\beta_0=0$}}. However, we see that even massless \nstates introduce a $\\phi$-dependence prior to the truncation to {\\mbox{$\\phi=0$}}.\n\nFocusing initially on the first term within $P_\\Lambda(a)$ in Eq.~(\\ref{Pacos}), we immediately see that \n\\begin{eqnarray}\n && \\partial_\\phi^2 \\left. \\left[ \\frac{{\\cal M}^2}{24 a} \\, {\\rm Str}\\, M^2 \\right] \\right|_{\\phi=0} \n = ~ \\left. \\frac{{\\cal M}^2}{24a} \\, {\\rm Str}\\, (\\partial_\\phi^2 M^2) \\right|_{\\phi=0} \\nonumber\\\\\n && ~~~~~~~~~~~~~~~ = ~ - \\frac{{\\cal M}^2}{2} \\,\\left[ \\frac{\\pi}{3a} \\,{\\rm Str}\\, {\\mathbb{X}}_1\\right]~\n\\end{eqnarray}\nwhere we have used the result in Eq.~(\\ref{newXi}) in passing to the final expression.\nThis successfully reproduces the initial terms within $P_{\\cal X}(a)$ in Eq.~(\\ref{finalPa}).\n Next, we evaluate the second $\\phi$-derivative of\nthe Bessel-function terms within $P_\\Lambda(a)$ in Eq.~(\\ref{Pacos}).\nTo do this, we note the mathematical identity \n\\begin{eqnarray}\n && \\partial_\\phi^2 \\left[ M^2 K_2{ \\left( \\frac{r M}{a \\calM} \\right) }\\right] ~=~ \n - \\frac{r M}{2a{\\cal M}} \\left( \\partial_\\phi^2 M^2\\right) K_1{ \\left( \\frac{r M}{a \\calM} \\right) } \\nonumber\\\\\n &&~~~~~~~~~~~~~~~~~ + \\frac{r^2}{4 a^2 {\\cal M}^2} \\left(\\partial_\\phi M^2\\right)^2 K_0{ \\left( \\frac{r M}{a \\calM} \\right) }~\n\\end{eqnarray} \nwhich follows from standard results for Bessel-function derivatives along with a judicious \nrepackaging of terms.\nGiven this, and given the relations in Eq.~(\\ref{newXi}), we then find that \n\\begin{eqnarray}\n&& \\partial_\\phi^2 \\left. \\left\\lbrace \n \\frac{a}{\\pi^2 } \n \\, \\bpStr \\!\\left\\lbrack M^2 \\sum_{r=1}^\\infty\n \\frac{1}{r^2} K_2\\left(\\frac{ r M}{ a {\\cal M}}\\right)\\right\\rbrack\\right\\rbrace \\right|_{\\phi=0} \\nonumber\\\\\n&& ~~~~~~~~~ =~ \n \\frac{2}{\\pi}\\, \\pStr {\\mathbb{X}}_1 \\,\\left\\lbrack \\sum_{r=1}^\\infty \\left(\\frac{M}{r{\\cal M}}\\right)\n \\,K_1\\left( \\frac{r M}{a{\\cal M}} \\right)\\right\\rbrack \\nonumber\\\\\n&& ~~~~~~~~~ \\phantom{=}~ + \n \\frac{4}{a} \\,\\pStr {\\mathbb{X}}_2 \\left\\lbrack \\sum_{r=1}^\\infty \n K_0\\left( \\frac{ r M}{a{\\cal M}} \\right) \\right\\rbrack~.~~~~~~~~~~~~~~ \n\\label{besselderivs}\n\\end{eqnarray}\nwhere the supertrace in the first line is over all states whose mass functions $M^2(\\phi)$ have {\\mbox{$\\beta_0>0$}}. \nWe thus see that the result in Eq.~(\\ref{besselderivs}) likewise successfully reproduces \nthe Bessel-function terms within $P_{\\cal X}(a)$ in Eq.~(\\ref{finalPa}).\n\nOur final task is to evaluate $\\partial_\\phi^2$ acting on the second term in Eq.~(\\ref{Pacos}).\nAt first glance, it would appear that this term does not yield any contribution since \nit is wholly independent of the mass $M$ and would thus not lead to any $\\phi$-dependence.\nIndeed, as evident from Eq.~(\\ref{P2cosa}), this term represents a contribution to $P_\\Lambda(a)$ \nfrom purely massless states, and as such the identification {\\mbox{$M=0$}} has already been implemented\nwithin this term. This is why no factors of the mass $M$ remain within this term.\nHowever, as discussed above, for the purposes of the present calculation we are to regard the masses $M$ \nas functions of $\\phi$ before taking the $\\phi$-derivatives.\n Thus, when attempting to take $\\phi$-derivatives of the second term in Eq.~(\\ref{Pacos}),\nwe should properly go back one step to the original derivation of this term that appears in Eq.~(\\ref{P2cosa}) \nand reinsert a non-trivial mass function $M^2(\\phi)$ with {\\mbox{$\\beta_0=0$}} into the derivation.\nThe remaining derivation of this term then algebraically mirrors the derivation of the {\\it massive}\\\/ \nBessel-function term in Eq.~(\\ref{P2cosb}), only with $M^2$ now replaced by $M^2(\\phi)$ with {\\mbox{$\\beta_0=0$}}.\nIn other words, for the purposes of our current calculation, we should formally identify\n\\begin{eqnarray}\n&& \\frac {{\\cal M}^4}{2}\\, \\frac{\\pi^2}{45}\\,(n_B-n_F) \\,{a^3} \\nonumber\\\\\n&& ~~~=~ \n \\frac{{\\cal M}^2}{2}\\left. \\frac{a}{\\pi^2 } \n \\, \\bzStr \\left\\lbrack M^2 \\sum_{r=1}^\\infty\n \\frac{1}{r^2} K_2\\left(\\frac{ r M}{ a {\\cal M}}\\right) \\right\\rbrack \\right|_{\\phi=0}~~~~~~~\n\\label{substitution}\n\\end{eqnarray}\nand then evaluate the $\\phi$-derivatives before \ntruncating to {\\mbox{$\\phi=0$}}.\nAside from the overall factor of $- {\\cal M}^2\/2$, acting with $\\partial^2_\\phi$ and then truncating to {\\mbox{$\\phi=0$}} yields \nthe same result as on the right side of Eq.~(\\ref{besselderivs}),\nexcept with each supertrace over massive states \nreplaced with a supertrace over massless states.\nWe thus need to evaluate these Bessel-function expressions at zero argument.\nHowever, for small arguments {\\mbox{$z\\ll 1$}}, the Bessel functions have the leading asymptotic behaviors\n\\begin{equation}\n K_\\nu(z) ~\\sim~ \\begin{cases}\n ~-\\log (z\/2) - \\gamma +...& {\\rm for}~~\\nu=0 \\\\\n ~\\phantom{-}{\\textstyle{1\\over 2}} \\,\\Gamma(\\nu) \\,(z\/2)^\\nu + ... & {\\rm for}~~\\nu>0 \n \\end{cases}\n\\label{Bessellimits}\n\\end{equation}\nwhere $\\gamma$ is the Euler-Mascheroni constant.\nAnalyzing the ${\\rm Str} \\,{\\mathbb{X}}_1$ term,\nwe thus see that\n\\begin{eqnarray}\n && \\frac{2}{\\pi}\\, \\lim_{M\\to 0} \\,\\pStr {\\mathbb{X}}_1 \\left\\lbrack \\sum_{r=1}^\\infty \\left(\\frac{M}{r{\\cal M}}\\right)\n \\,K_1\\left( \\frac{r M}{a{\\cal M}} \\right)\\right\\rbrack \\nonumber\\\\\n && ~~~~~~~=~ \\frac{2}{\\pi}\\, \\zStr {\\mathbb{X}}_1 \\,\\lim_{M\\to 0} \\left\\lbrack \\sum_{r=1}^\\infty \\left(\\frac{M}{r{\\cal M}}\\right)\n \\left(\\frac{a {\\cal M}}{r M} \\right)\\right\\rbrack \\nonumber\\\\\n && ~~~~~~~=~ \\frac{2a}{\\pi} \\,\\sum_{r=1}^\\infty \\,\\frac{1}{r^2} ~= ~ \\frac{\\pi}{3}\\, a~,\n\\label{hidden}\n\\end{eqnarray} \nthereby successfully reproducing the corresponding term which appears in $P_{\\cal X}(a)$.\nIndeed, we see that the {\\mbox{$M\\to 0$}} limit in Eq.~(\\ref{hidden}) is convergent and continuous with the exact {\\mbox{$M=0$}} result.\n\nFor theories in which {\\mbox{$\\zStr {\\mathbb{X}}_2=0$}}, there are no further terms to consider.\nThe results of this analysis are then clear:\nwithin such theories, we have found that\n\\begin{equation}\n P_{\\cal X}(a) ~=~ \\partial_\\phi^2 \\, P_\\Lambda(a,\\phi) \\bigl|_{\\phi=0}~.\n\\end{equation}\nThrough Eq.~(\\ref{mtoP}), this then implies that \n\\begin{equation}\n \\widehat m_\\phi^2(\\rho,a)\\bigl|_{\\cal X} ~=~ \n \\partial_\\phi^2 \\, \\widehat\\Lambda( \\rho,a,\\phi) \\bigl|_{\\phi=0}~,\n\\end{equation}\nwhereupon use of Eq.~(\\ref{twocontributions}) tells us that\n\\begin{equation}\n \\widehat m_\\phi^2(\\rho,a) ~=~ \\left.\\left( \\partial_\\phi^2 + \\frac{\\xi}{4\\pi^2 {\\cal M}^2} \\right) \n \\, \\widehat\\Lambda( \\rho,a,\\phi) \\right|_{\\phi=0}~,\n\\end{equation}\nor equivalently\n\\begin{eqnarray}\n \\widehat m_\\phi^2(\\mu) ~&=&~ \n \\left.\\left( \\partial_\\phi^2 + \\frac{\\xi}{4\\pi^2 {\\cal M}^2} \\right) \n \\, \\widehat\\Lambda( \\mu,\\phi) \\right|_{\\phi=0} ~~~~~\\nonumber\\\\\n &=&~ \\left. D_\\phi^2 ~\\widehat\\Lambda( \\mu,\\phi) \\right|_{\\phi=0}~,\n\\label{finalCWresult}\n\\end{eqnarray}\nwhere we have defined the modular-covariant derivative \n\\begin{equation}\n D_\\phi^2 ~\\equiv~ \\partial_\\phi^2 + \\frac{\\xi}{4\\pi^2 {\\cal M}^2}~.\n\\label{Dphi2}\n\\end{equation}\nOf course, for theories with {\\mbox{$\\zStr {\\mathbb{X}}_2=0$}}, our original unregulated Higgs mass \nwas already finite and {\\it a priori}\\\/ there was no \nneed for a regulator. However, even within such theories, it is the use of our modular-invariant\nregulator for both $\\Lambda$ and $m_\\phi^2$ which enabled us to extract EFT descriptions \nof these quantities and to analyze their runnings as functions of an effective scale $\\mu$.\n\nThe result in Eq.~(\\ref{finalCWresult}) is both simple and profound.\nIndeed, comparing this result with our starting point in Eq.~(\\ref{higgsdef})\nand recalling the subsequent required modular completion in Eq.~(\\ref{Xmodcomplete}),\nwe see that we have in some sense come full circle.\nHowever, as stressed above, we have now demonstrated this result using only the general \nexpressions for ${\\mathbb{X}}_1$ and ${\\mathbb{X}}_2$ in Eq.~(\\ref{newXi}) and thus \n{\\it entirely without the assumption of a charge lattice}\\\/.\nThis result therefore holds for {\\it any}\\\/ modular-invariant string theory with {\\mbox{$\\zStr{\\mathbb{X}}_2=0$}}. \nIndeed, as indicated above, we can view $D_\\phi^2$ as a modular-covariant\nderivative, in complete analogy with the lattice-derived\ncovariant derivative $D_z^2$ in Eq.~(\\ref{modcovderiv}).\n\nBut more importantly, we see from Eq.~(\\ref{finalCWresult}) that \nwithin such theories\nwe can now identify $\\widehat \\Lambda(\\mu,\\phi)$ as {\\it an effective potential for the Higgs}\\\/. Strictly speaking, this is not the entire effective potential --- it does\nnot, for example, allow us to survey different minima \nas a function of $\\phi$ in order to select the global and local minima, as would be needed in order\nto determine the ground states of the theory in different possible phases (with unbroken and\/or broken symmetries).\nHowever, we see that $\\widehat \\Lambda(\\mu,\\phi)$ does provide a {\\it piece}\\\/ of the full potential, namely \nthe portion of the potential in the immediate vicinity of the assumed minimum (around which \n$\\phi$ parametrizes the fluctuations, as always).\nWith this understanding, we shall nevertheless simply refer to $\\widehat \\Lambda(\\mu,\\phi)$ as the Higgs effective potential.\nIndeed, as expected, we see from Eq.~(\\ref{finalCWresult}) that \nthe Higgs mass is related to the curvature of this potential around this minimum. \nOne can even potentially imagine repeating the calculations in this paper {\\it without}\\\/ implicitly assuming \nthe stability condition in Eq.~(\\ref{linearcond}), thereby dropping the implicit assumption that we are sitting\nat a stable vacuum of the theory. In that case, the first and second $\\phi$-derivatives of $\\widehat \\Lambda(\\mu,\\phi)$\nwould describe the slope and curvature of the potential for arbitrary values of $\\phi$, whereupon the methods in this paper\ncould provide a method of ``tracing out'' the \nshape of the full potential. However, at best this would appear to be a challenging undertaking.\n\nAs remarked above, the form of Eq.~(\\ref{finalCWresult}) makes sense from the perspective \nof Eq.~(\\ref{higgsdef}), in conjunction with the subsequent modular completion.\nAt first glance, it may seem surprising that such a result would continue to survive \neven after imposing our modular-invariant regulator \nin order to generate our regulated expressions\nfor $\\widehat m_\\phi^2(\\mu)$ and $\\widehat \\Lambda(\\mu,\\phi)$,\nand perhaps even more surprising after \nthe Rankin-Selberg techniques and their generalizations in Sect.~\\ref{sec3}\nare employed in order to express these regulated quantities in terms of supertraces over purely physical (level-matched)\nstring states.\nUltimately, however, \nthe result in Eq.~(\\ref{finalCWresult})\nconcerns the $\\phi$-structure of the theory and the response of the theory to fluctuations in the Higgs field.\nIn theories with {\\mbox{$\\zStr{\\mathbb{X}}_2=0$}},\nthese properties are essentially ``orthogonal'' to the manipulations that occurred in Sects.~\\ref{sec3} and \\ref{sec4},\nwhich ultimately concern the regulators and the resulting behavior of these quantities as functions of $\\mu$.\nIn other words, in such theories\nthe process of $\\phi$-differentiation in some sense ``commutes''\nwith all of these other manipulations.\nThus the relation in Eq.~(\\ref{finalCWresult}) holds not only for our original unregulated Higgs mass\nand cosmological constant, but also for their regulated counterparts as well as for the running which describes\ntheir dependence on the variables defining the regulator.\n\nIt is also intriguing that we are able to identify a modular-covariant \nderivative $D_\\phi^2$ within the results in Eq.~(\\ref{finalCWresult}).\nOf course, this is the {\\it second}\\\/ $\\phi$-derivative.\nBy contrast, the {\\it first}\\\/ $\\phi$-derivative does not require modular completion.\nWe have already seen this in Sect.~\\ref{stability}, where we found\nthat $\\partial_\\phi$ acting on the partition function ${\\cal Z}$ corresponds to insertion\nof the factor ${\\cal Y}$, which was already modular invariant.\nIn this sense, $\\phi$-derivatives are similar to the $z$-derivatives \ndiscussed in Sect.~\\ref{sec:completion}.~\n\nThe result in Eq.~(\\ref{finalCWresult}) holds only for theories in which {\\mbox{$\\zStr{\\mathbb{X}}_2=0$}}.\nHowever, when {\\mbox{$\\zStr{\\mathbb{X}}_2\\not=0$}}, there is an additional term \nto consider within $P_\\Lambda$.\nTaking the {\\mbox{$M\\to 0$}} limit of the $\\pStr\\,{\\mathbb{X}}_2$ result in \nEq.~(\\ref{besselderivs}) in conjunction with the limiting behavior in Eq.~(\\ref{Bessellimits}),\nwe formally obtain\n\\begin{eqnarray}\n && \\frac{4}{a} \\,\\lim_{M\\to 0}\\, \\pStr {\\mathbb{X}}_2 \\left\\lbrack \\sum_{r=1}^\\infty \n K_0\\left( \\frac{ r M}{a{\\cal M}} \\right) \\right\\rbrack\\nonumber\\\\\n && ~~~~=~ \\frac{4}{a} \\,\\zStr {\\mathbb{X}}_2 \\sum_{r=1}^\\infty \\left[ -\\log\\left( \\frac{rM}{2a{\\cal M}}\\right) -\\gamma \\right]~.~~~~~\n\\label{badstuff}\n\\end{eqnarray}\nUnfortunately, this infinite $r$-summation is not convergent. \nIt also does not correspond to what is presumably the \nexact {\\mbox{$M=0$}} result within $P_{\\cal X}$.\nWe stress that these complications arise only when {\\mbox{$\\zStr {\\mathbb{X}}_2\\not=0$}}, \nwhich is precisely the condition under which the original unregulated Higgs mass is divergent. \n\n\n\n\n\nIn order to better understand this phenomenon,\nwe can perform a more sophisticated analysis \nby analytically performing the $r$-summation \nin complete generality before taking the {\\mbox{$M\\to 0$}} limit.\nWe begin by defining the Bessel-function combinations\n\\begin{equation}\n {\\mathbb{K}}_\\nu(z) ~\\equiv~ 2\\, \\sum_{r=1}^\\infty \\, (r z)^{-\\nu} K_\\nu (r z)~.\n\\label{bbK}\n\\end{equation}\nThese Bessel-function combinations are relevant for both $P_{\\cal X}$ and $P_\\Lambda$\nin the same way that the combinations ${\\cal K}^{(n,p)}_\\nu(z)$ \nin Eq.~(\\ref{Besselcombos})\nwere relevant for $\\widehat m_\\phi^2\\bigl|_{\\cal X}$ and $\\widehat \\Lambda$, and indeed\n\\begin{equation}\n {\\cal K}^{(-\\nu,p)}_\\nu (z) ~=~ {\\textstyle{1\\over 2}} \\bigl[\n \\rho^{-\\nu} \\,{\\mathbb{K}}_\\nu(z\/\\rho) - \\rho^p \\,{\\mathbb{K}}_\\nu(z) \\bigr]~.\n\\end{equation} \nUsing the techniques in Ref.~\\cite{Paris}, it is then straightforward (but exceedingly tedious) \nto demonstrate that ${\\mathbb{K}}_\\nu(z)$ for {\\mbox{$z\\ll 1$}} has a \nMaclaurin-Laurent series representation given by\n\\begin{widetext}\n\\begin{eqnarray}\n{\\mathbb{K}}_{\\nu}(z) ~&=&~ \n \\sum_{p=1}^{\\nu}\\, \n 2^{-\\nu}\\pi^{p}\n \\frac{(-1)^{\\nu-p}}{\\left(\\nu-p\\right)!}\n \\zeta^{\\ast}(2p)\n \\left(\\frac{z}{2}\\right)^{-2p} \n ~+~ 2^{-\\nu} \\sqrt{\\pi} \\,\\Gamma\\left( {\\textstyle{1\\over 2}}-\\nu\\right)\\, \\frac{1}{z} \\nonumber\\\\\n&&~~~~+~ \n \\frac{(-2)^{-\\nu}}{\\nu!}\\left[\\gamma-\\frac{H_\\nu}{2}+\\log\\left(z\/4\\pi\\right)\\right]\n ~+~ \\sum_{p=1}^{\\infty} \\,2^{-\\nu}\\pi^{-p}\n \\frac{(-1)^{\\nu+p}}{\\left(\\nu+p\\right)!}\n \\zeta^{\\ast}(2p+1)\n \\left(\\frac{z}{2}\\right)^{2p}\n\\label{lyon}\n\\end{eqnarray}\n where {\\mbox{$H_n\\equiv \\sum_{k=1}^{n}1\/k$}} is the $n^{\\rm th}$ harmonic number \nand where {\\mbox{$\\zeta^\\ast(s) \\equiv \\pi^{-s\/2} \\Gamma(s\/2) \\zeta(s) = \\zeta^\\ast(1-s)$}} \nis the ``completed'' Riemann $\\zeta$-function. \nThe representation in Eq.~(\\ref{lyon}) \nis particularly useful for {\\mbox{$z\\ll 1$}}, allowing us to extract\nthe leading behaviors \n\\begin{eqnarray}\n{\\mathbb{K}}_{0}(z) ~&=&~\\frac{\\pi}{z}+\\left[\\gamma+\\log\\left(\\frac{z}{4\\pi}\\right)\\right]\n -\\frac{\\zeta(3)z^2}{8\\pi^{2}}+\\frac{3\\zeta(5)z^4}{128\\pi^{4}}+\\ldots\\nonumber \\\\\n{\\mathbb{K}}_{1}(z) ~&=&~ \\frac{\\pi^{2}}{3z^{2}}-\\frac{\\pi}{z}\n -\\frac{1}{2}\\left[\\gamma-\\frac{1}{2} +\\log\\left(\\frac{z}{4\\pi}\\right)\\right]\n +\\frac{\\zeta(3)z^2}{32\\pi^{2}} -\\frac{\\zeta(5)z^4}{256\\pi^{4}}+\\ldots\\nonumber \\\\\n{\\mathbb{K}}_{2}(z) ~&=&~ \\frac{2\\pi^{4}}{45z^{4}}-\\frac{\\pi^{2}}{6z^{2}}+\\frac{\\pi}{3z}\n +\\frac{1}{8}\\left[\\gamma-\\frac{3}{4}+\\log\\left(\\frac{z}{4\\pi}\\right)\\right]\n -\\frac{\\zeta(3)z^2}{192\\pi^{2}}+\\frac{\\zeta(5)z^4}{2048\\pi^{4}}+\\ldots\n\\label{Kseries}\n\\end{eqnarray}\n\\end{widetext}\nIndeed, use of the expression for ${\\mathbb{K}}_1(z)$ confirms our result in Eq.~(\\ref{hidden}).\n\nArmed with the expression for ${\\mathbb{K}}_2(z)$ in Eq.~(\\ref{Kseries}), we can now rigorously evaluate the leading terms within\n$P_\\Lambda(a)$ ---\nand by extension within $\\widehat \\Lambda(\\rho,a)$ --- in complete generality, even when massless states\nare included.\nStarting from Eq.~(\\ref{Pacos}) in conjunction with the replacement in Eq.~(\\ref{substitution}),\nwe now have\n\\begin{eqnarray}\n P_\\Lambda(a) ~&=&~ \\frac{{\\cal M}^2}{24a} \\,{\\rm Str}\\, M^2 \n - \\frac{1}{4\\pi^2 a} \\,{\\rm Str}\\, M^4 \\,{\\mathbb{K}}_2\\!\\left( \\frac{M}{a{\\cal M}}\\right)~\\nonumber\\\\\n ~&\\approx&~ \\frac{{\\cal M}^2}{24a} \\,{\\rm Str}\\, M^2 \n - \\frac{1}{4\\pi^2 a} ~\\aMeffStr\\, M^4 \\,{\\mathbb{K}}_2\\!\\left( \\frac{M}{a{\\cal M}}\\right)~\\nonumber\\\\\n\\label{Plam}\n\\end{eqnarray}\nwhere the final supertrace on the first line is over {\\it all}\\\/ states in the theory, including those that are massless,\nand where in passing to the second line we have recognized that ${\\mathbb{K}}_2(z)$ is exponentially suppressed unless {\\mbox{$z\\ll 1$}}.\nThe fact that ${\\mathbb{K}}_2(z)$ is now explicitly restricted to the {\\mbox{$z\\ll 1$}} regime implies that it is legitimate\nto insert the series expansion for ${\\mathbb{K}}_2(z)$ from Eq.~(\\ref{Kseries}) within Eq.~(\\ref{mtoP}).\nIdentifying the physical scale $\\mu$ as in Eq.~(\\ref{mudef}) \nand retaining only the leading terms for {\\mbox{$\\mu\\ll M_s$}},\nwe then obtain\n\\begin{eqnarray}\n \\widehat \\Lambda(\\mu,\\phi) \\,&=&~ \\frac{1}{1+\\mu^2\/M_s^2} \\Biggl\\lbrace \\nonumber\\\\\n && \\phantom{-}\\frac{1}{24}{\\cal M}^2 \\,{\\rm Str}\\, M^2 + \\zeffStr \\left( \\frac{M^2 \\mu^2}{96\\pi^2} \n - \\frac{7\\mu^4}{960\\pi^2}\\right) \\nonumber\\\\ \n && - \\frac{1}{32\\pi^2} \\,\\zeffStr M^4 \\log\\left( \\sqrt{2}\\, e^{\\gamma+1\/4} \\frac{M}{\\mu}\\right)+...\\Biggr\\rbrace ~\\nonumber\\\\\n ~&=&~ \\frac{1}{24}{\\cal M}^2 \\,{\\rm Str}\\, M^2 \n -{\\rm Str}\\, \\frac{M^2 \\mu^2}{96\\pi^2} \\nonumber\\\\\n && + \\zeffStr \\left( \\frac{M^2 \\mu^2}{96\\pi^2} \n - \\frac{7\\mu^4}{960\\pi^2} \\right) \\nonumber\\\\ \n && - \\frac{1}{32\\pi^2} \\,\\zeffStr M^4 \\log\\left( \\sqrt{2}\\, e^{\\gamma+1\/4} \\frac{M}{\\mu}\\right)+... \\nonumber\\\\\n\\label{intermed2}\n\\end{eqnarray}\nwhere we have continued to adopt our benchmark value {\\mbox{$\\rho=2$}}\nand where we recall that each factor of $M$ carries a $\\phi$-dependence through Eq.~(\\ref{TaylorM}).\nNote that in passing to the final expression in Eq.~(\\ref{intermed2}) we have Taylor-expanded the overall \nprefactor and kept only those terms of the same order as those already shown.\nHowever, we now see that the $\\mu^2$ term from expanding the prefactor cancels the corresponding $\\mu^2$\nterm from ${\\mathbb{K}}_2$, leaving behind a net $\\mu^2$ term which scales as the $M^2$ supertrace \nof only those states whose masses {\\it exceed}\\\/ $\\mu$.\nWe thus obtain our final result\n\\begin{eqnarray}\n \\widehat \\Lambda(\\mu,\\phi) \\,&=&\\, \n \\frac{1}{24}{\\cal M}^2 \\,{\\rm Str}\\, M^2 \n -\\antieffStr \\frac{M^2 \\mu^2}{96\\pi^2} \n - \\zeffStr \\frac{7\\mu^4}{960\\pi^2} \\nonumber\\\\ \n && - \\frac{1}{32\\pi^2} \\,\\zeffStr M^4 \\log\\left( \\sqrt{2}\\, e^{\\gamma+1\/4} \\frac{M}{\\mu}\\right) \\,+\\, ... \\nonumber\\\\\n\\label{Lambdafull}\n\\end{eqnarray}\n Indeed, this result provides the leading approximation to the exact expression\nin Eq.~(\\ref{lambdamuresult}).\n\nThe first and third terms in this result are consistent with \nthose in Eq.~(\\ref{lambdamuresult2}), and indeed the $\\mu^4$ term is the contribution\nfrom the massless states within the second supertrace in Eq.~(\\ref{Plam}). \nHowever, to this order, we now see that there \nare two additional terms.\nThe first is a term scaling as $\\mu^2$ which depends on the spectrum of states with masses {\\mbox{$M\\gsim \\mu$}}.\nThis contribution lies outside the range {\\mbox{$M\\ll \\mu$}} studied in Eq.~(\\ref{lambdamuresult2})\nbut nevertheless generally appears for {\\mbox{$M\\gsim \\mu$}}.\nThe second is a logarithmic term. \nThis term is subleading when compared to the other terms shown, and \nmassless states make no contribution to this term (divergent or otherwise) when evaluated at {\\mbox{$\\phi=0$}} because\nof its $M^4$ prefactor.\n\n \n\nThis logarithmic term is nevertheless of critical importance \nwhen we consider the corresponding Higgs mass. \nAs we have seen in Eq.~(\\ref{finalCWresult}), the Higgs mass $\\widehat m_\\phi^2(\\mu)$\nreceives a contribution which scales as $\\partial_\\phi^2 \\widehat \\Lambda(\\mu)$.\nOf course, all of the dependence on $\\phi$ is carried within the masses $M$ which\nappear in Eq.~(\\ref{Lambdafull}), and as expected $\\widehat\\Lambda(\\mu)$ depends not\non these masses directly but on their squares.\nHowever, for any function $f(M^2)$ we have the algebraic identity\n\\begin{equation}\n \\partial_\\phi^2 f(M^2) \n ~=~ (\\partial_\\phi^2 M^2) \\frac{\\partial f}{\\partial M^2} +\n (\\partial_\\phi M^2)^2 \n \\frac{\\partial^2 f}{(\\partial M^2)^2}~.~~\n\\end{equation} \nThus, identifying {\\mbox{$f\\sim \\widehat\\Lambda(\\mu)$}} \nand recalling Eq.~(\\ref{newXi}), we obtain\n\\begin{eqnarray}\n \\partial_\\phi^2 \\,\\widehat \\Lambda(\\mu) \\Bigl|_{\\phi=0} &=& \n \\left. -4\\pi \\,{\\rm Str} \\,{\\mathbb{X}}_1 \\,\\frac{\\partial \\widehat \\Lambda(\\mu)}{\\partial M^2}\\right|_{\\phi=0} \\nonumber\\\\\n && ~~~+ \\left. 16\\pi^2 {\\cal M}^2 \\,{\\rm Str} \\,{\\mathbb{X}}_2 \\,\\frac{\\partial^2 \\widehat \\Lambda(\\mu)}{(\\partial M^2)^2}\\right|_{\\phi=0}~\n ~~~~~~~~\n\\label{derivs}\n\\end{eqnarray}\nwhere we have implicitly used the fact that only non-negative powers of $\\phi$ appear within $\\widehat\\Lambda(\\mu)$,\nthereby ensuring that our truncation to {\\mbox{$\\phi=0$}} factorizes within each term.\n\nIn principle, both supertraces in Eq.~(\\ref{derivs}) include massless states.\nMoreover, we see that the ${\\rm Str} \\, {\\mathbb{X}}_1$ term is proportional to the single \n$M^2$-derivative of $\\widehat\\Lambda(\\mu)$, and when acting on the logarithm term within Eq.~(\\ref{Lambdafull})\nwe find that massless states continue to be harmless, yielding no contribution (and therefore no divergences).\nBy contrast, we see that the ${\\rm Str} \\, {\\mathbb{X}}_2$ term is proportional \nto the {\\it second}\\\/ $M^2$-derivative of $\\widehat \\Lambda(\\mu)$.\nThis derivative therefore leaves behind a logarithm with no leading $M^2$ factors remaining.\nThus, for {\\mbox{$M= 0$}}, we obtain a logarithmic divergence for the Higgs mass --- as expected ---\nso long as {\\mbox{$\\zStr \\,{\\mathbb{X}}_2\\not=0$}}.\nIndeed, all of this information is now directly encoded within the effective potential\n$\\widehat\\Lambda(\\mu)$ for this theory, as given in Eq.~(\\ref{Lambdafull}). \n\nThis situation is analogous to the behavior \nof the traditional Coleman-Weinberg potential $V(\\varphi_c)$ as originally given in Refs.~{\\mbox{\\cite{Coleman:1973jx,Weinbergpost}}}.\nIn that case, it was shown that $V(\\varphi_c)$ contains a term\nscaling as \n\\begin{equation}\n V(\\varphi_c) ~\\sim~ \\varphi_c^4 \\,\\log \\varphi_c^2\n\\end{equation}\nwhere $\\varphi_c$ are the fluctuations of the classical Higgs field around its VEV\nand where one has assumed a $U(1)$-charged scalar field subject to a $\\lambda \\phi^4$ interaction.\nThe Higgs mass (which goes as the second derivative $\\partial^2 V\/\\partial \\varphi_c^2$)\ntherefore remains finite even as {\\mbox{$\\varphi_c\\to 0$}}, whereas the {\\it fourth}\\\/ derivative $\\partial^4 V\/\\partial \\varphi_c^4$\nactually has a logarithmic singularity as {\\mbox{$\\varphi_c\\to 0$}}.\nIndeed, this fourth derivative describes the behavior of the coupling $\\lambda$.\nThe cure for this disease, as suggested in Refs.~{\\mbox{\\cite{Coleman:1973jx,Weinbergpost}}}, is to\nmove away from the {\\mbox{$\\varphi_c=0$}} origin, and instead define the coupling $\\lambda$ at this shifted point.\n\n\nOf course, in our more general string context, we see that our potential scales like $M^4 \\log M$. Moreover, within the ${\\mathbb{X}}_2$ term,\nit is not the fourth derivative with respect to $M$ which leads to difficulties --- rather, it is the {\\it second}\\\/ derivative\nwith respect to $M^2$. As a consequence, this logarithmic divergence shows up in the Higgs mass rather than in a four-point coupling.\nThat said, it is possible that the cure for this disease may be similar to that discussed in \nRefs.~{\\mbox{\\cite{Coleman:1973jx,Weinbergpost}}}.\nIn particular, this suggests that in string theories for which {\\mbox{$\\zStr {\\mathbb{X}}_2\\not=0$}}, a cure for our logarithmically divergent\nHiggs mass and the fact that\nradiative potential is not twice-differentiable there \nmay be similarly found by avoiding the sharp {\\mbox{$\\phi=0$}} truncation that originally appears in Eq.~(\\ref{higgsdef}),\nand by instead deforming our theory away from the {\\mbox{$\\phi=0$}} origin in $\\phi$-space.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\nFinally, given that we are now equipped with our effective Higgs potential $\\widehat \\Lambda(\\mu,\\phi)$, \nwe can revisit our classical stability condition, as originally discussed in Sect.~\\ref{stability}.~\nIn general, our theory will be sitting at an extremum of the potential as long as \n\\begin{equation}\n \\partial_\\phi \\widehat\\Lambda(\\mu,\\phi)\\Bigl|_{\\phi=0} ~=~0~.\n\\label{stabcond}\n\\end{equation}\nThis, then, is a supplementary condition that we have implicitly assuming to be satisfied within our analysis.\nNote that \n\\begin{eqnarray}\n \\partial_\\phi \\widehat\\Lambda(\\mu,\\phi)\\Bigl|_{\\phi=0} \n ~&=&~ \\left. (\\partial_\\phi M^2) \\frac{\\partial \\widehat \\Lambda(\\mu)}{\\partial M^2} \\right|_{\\phi=0}\\nonumber\\\\ \n ~&=&~ 4\\pi {\\cal M} \\,{\\rm Str} \\,{\\mathbb{Y}} \\left.\\frac{\\partial \\widehat \\Lambda(\\mu)}{\\partial M^2} \\right|_{\\phi=0} ~\n\\label{connecttoY}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n {\\mathbb{Y}} ~\\equiv~ \\frac{1}{ 4\\pi{\\cal M}} \\,(\\partial_\\phi M^2)\\Bigl|_{\\phi=0}~.\n\\label{Ydef2}\n\\end{equation}\nIndeed, for the case of theories with an underlying charge lattice, we have {\\mbox{${\\mathbb{Y}} = \\tau_2^{-1} {\\cal Y}$}}\nwhere ${\\cal Y}$ is given in Eq.~(\\ref{Ydef}).\nThus the stability of the theory (and the possible existence of a destabilizing $\\phi$-tadpole) is closely \nrelated to the values of ${\\mathbb{Y}}$ across the string spectrum, as already anticipated in Sect.~\\ref{stability}.~\nSubstituting the exact expression in Eq.~(\\ref{lambdamuresult})\ninto Eq.~(\\ref{connecttoY}),\nwe find\n\\begin{eqnarray}\n && \\partial_\\phi \\widehat\\Lambda(\\mu,\\phi)\\Bigl|_{\\phi=0} \n \\,=~ \\frac{{\\cal M}^3}{1+\\mu^2\/M_s^2} \n \\,{\\rm Str} \\,{\\mathbb{Y}} \\,\\Biggl\\lbrace \n \\frac{\\pi}{6}\n + \\frac{1}{2\\pi} \\left(\\frac{M}{{\\cal M}}\\right)^2 \\times \\nonumber\\\\ \n && ~~~~~~~~\\times \n \\left\\lbrack \n {\\cal K}_0^{(0,1)}\\!\\left( \\frac{2\\sqrt{2}\\pi M}{\\mu} \\right) + \n {\\cal K}_2^{(0,1)}\\!\\left( \\frac{2\\sqrt{2}\\pi M}{\\mu} \\right) \n \\right\\rbrack \\Biggr\\rbrace \\nonumber\\\\ \n && ~~~=~ \\frac{{\\cal M}^3}{1+\\mu^2\/M_s^2} ~\\Biggl\\lbrace\n \\zStr {\\mathbb{Y}} \\left\\lbrack \\frac{\\pi}{6}\\left(1+\\mu^2\/M_s^2\\right) \\right\\rbrack \\nonumber\\\\\n && ~~~~~~~~+ \\pStr {\\mathbb{Y}} \\,\\Biggl\\lbrace \n \\frac{\\pi}{6}\n + \\frac{1}{2\\pi} \\left(\\frac{M}{{\\cal M}}\\right)^2 \\times \\nonumber\\\\ \n && ~~~~~~~~\\times \n \\left\\lbrack \n {\\cal K}_0^{(0,1)}\\!\\left( \\frac{2\\sqrt{2}\\pi M}{\\mu} \\right) + \n {\\cal K}_2^{(0,1)}\\!\\left( \\frac{2\\sqrt{2}\\pi M}{\\mu} \\right) \n \\right\\rbrack \\Biggr\\rbrace \\Biggr\\rbrace~ \\nonumber\\\\ \n\\label{explicitBessel}\n\\end{eqnarray}\nwhere in passing to the second expression we have explicitly separated the\ncontributions from the massless and massive string states, \nand where \n${\\cal K}_\\nu^{(n,p)} (z)$ continue to denote\nthe combinations of Bessel functions in Eq.~(\\ref{Besselcombos}).\n\nInterestingly (but not unexpectedly), the terms multiplying ${\\rm Str}\\,{\\mathbb{Y}}$ \nin Eq.~(\\ref{explicitBessel}) are the same as the terms multiplying ${\\rm Str}\\,{\\mathbb{X}}_1$ \nin Eq.~(\\ref{finalhiggsmassmu}).\nEquivalently, we can view the quantity in Eq.~(\\ref{explicitBessel}) as the coefficient\nof the tadpole term (linear in $\\phi$) within the effective potential\n$\\Lambda(\\mu,\\phi)$ in Eq.~(\\ref{lambdamuresult}) when the masses \nare Taylor-expanded as in Eq.~(\\ref{TaylorM}).\n\nThere are several ways in which the expressions in Eq.~(\\ref{explicitBessel})\nmight vanish for all $\\mu$, as required for a stable vacuum.\nIn principle, for a given value of $\\mu$, there might exist a spectrum of states\nwith particular masses $M$ such that the contributions from the Bessel and\nnon-Bessel terms together happen to cancel when tallied across the spectrum. \nAny continuous change in the \nvalue of $\\mu$ might then induce a corresponding continuous change in the \nspectrum such that this cancellation is maintained. This is not unlike what\nhappens in the traditional field-theoretic Coleman-Weinberg potential, where changing the\nscale $\\mu$ can change the vacuum state and the spectrum of excitations built upon it.\nOf course in the present case we are \nworking within the context of string theory rather than field theory.\nAs such, we are dealing with an infinite tower of \nstring states and simultaneously maintaining modular invariance as the\nspectrum is deformed.\n\nAnother possibility is to simply demand stability in the deep infrared region,\nas {\\mbox{$\\mu\\to 0$}}. From Eq.~(\\ref{explicitBessel}) \nwe see that this would then require simply that \n\\begin{equation}\n {\\rm Str} \\,{\\mathbb{Y}} ~=~0~\n\\label{fullsutrace}\n\\end{equation}\nwhere the supertrace is over all string states, both massless and massive.\n\nA final possibility is to guarantee stability for every value of $\\mu$ by demanding\nthe somewhat stronger condition\n\\begin{equation}\n ~~~~~{\\rm Str} \\,{\\mathbb{Y}} \\,=\\,0~ ~~{\\rm for~each~mass~level~individually}.~~\n\\label{bestcondition}\n\\end{equation}\nOf course, Eq.~(\\ref{bestcondition}) implies Eq.~(\\ref{fullsutrace}), but \nthe fact that ${\\rm Str}\\,{\\mathbb{Y}}$ vanishes for each mass level {\\it individually}\\\/ ensures\nthat stability no longer rests on any $\\mu$-dependent cancellations involving the\nBessel functions.\n\nComparing Eq.~(\\ref{Ydef2}) with Eq.~(\\ref{newXi}), we see that {\\mbox{${\\mathbb{X}}_2 = {\\mathbb{Y}}^2$}}.\nHowever,\nas discussed below Eq.~(\\ref{Ydef}),\nconstraints on ${\\mathbb{Y}}$ do not necessarily become constraints on ${\\mathbb{X}}_2$, even if the values of\n${\\mathbb{Y}}$ happen to cancel pairwise amongst degenerate states across the string spectrum\n[which would guarantee Eq.~(\\ref{bestcondition})]. \nThus the requirement of stability does not necessarily lead \nto any immediate constraints on the supertraces of ${\\mathbb{X}}_2$. \n\n\n\n \n\n\nIn summary, then, we have shown that for theories with {\\mbox{$\\zStr\\,{\\mathbb{X}}_2=0$}} there exists an effective Higgs potential\n$\\widehat\\Lambda(\\mu,\\phi)$ from which the Higgs mass can be obtained through the modular-covariant double-derivative $D^2_\\phi$,\nas in Eq.~(\\ref{finalCWresult}).\nThis effective potential is given exactly in Eq.~(\\ref{lambdamuresult}), with\nthe leading terms given in Eq.~(\\ref{Lambdafull}).\nBy contrast, for theories with {\\mbox{$\\zStr \\,{\\mathbb{X}}_2\\not=0$}} we have found\nthat the effective potential $\\widehat \\Lambda(\\mu,\\phi)$ \npicks up an additional contribution whose second \nderivative is discontinuous at {\\mbox{$\\phi=0$}}.\nIn this sense, the Higgs mass is not well defined at {\\mbox{$\\phi=0$}}.\nOf course, one option is to retain the expression obtained in Eq.~(\\ref{finalhiggsmassmu});\nthis expression is not the second derivative of $\\widehat\\Lambda(\\mu,\\phi)$ when {\\mbox{$\\zStr\\,{\\mathbb{X}}_2\\not =0$}},\nbut it is indeed finite except as {\\mbox{$\\mu\\to 0$}}.\nAn alternative option is to define our Higgs mass away from the {\\mbox{$\\phi=0$}} origin.\nEither way, these features exactly mirror those seen within the traditional\nColeman-Weinberg potential. \n\n\n\n\n\n\n\n\n\\section{Pulling it all together: Discussion, top-down perspectives, and future directions \\label{sec:Conclusions}}\n\nA central question when analyzing any string theory is to understand the properties of \nits ubiquitous scalars --- its Higgs fields, its moduli fields, its axions, and so forth. \nTo a great extent the behavior of a scalar is dominated by its \nmass, and in this paper we have developed a completely general \nframework for understanding the masses of such scalars at one-loop order in\nclosed string theories.\nOur framework can be applied at all energy scales, is independent of any supersymmetry, and \nmaintains worldsheet modular invariance and hence finiteness at all times. \nMoreover, our framework is entirely string-based and does not rely on establishing any \nparticular low-energy effective field theory. Indeed the notion of an effective \nfield theory at a given energy scale ends up being an {\\it output}\\\/ of our analysis,\nand we have outlined the specific conditions and approximations under which such an EFT emerges\nfrom an otherwise completely string-theoretic calculation.\n\nBeyond the crucial role played by the scalar mass, another \nmotivation for studying this quantity is its special status as \nthe ``canary in the coal mine'' for UV completion. \nThe scalar mass term is virtually the only operator that \nis both highly UV-sensitive and also IR-divergent when coupled to massless states. \nThus, once we understand this operator, we understand much of the entire structure of the theory. \n\nWe can appreciate the special status of this operator if we think about a typical EFT.~\nWithin such an EFT, the familiar result for the one-loop contributions to the Higgs mass \ntakes the general form \n\\begin{equation}\nm_{\\phi}^{2}~=~\\frac{M_{\\text{UV}}^{2}}{32\\pi^{2}}\\,\\eftStr\\,\\partial_{\\phi}^{2}M^{2}\\,-\\eftStr\\,\\partial_{\\phi}^{2}\\left[\\frac{M^{4}}{64\\pi^{2}}\\log\\left(c\\frac{M^{2}}{M_{\\text{UV}}^{2}}\\right)\\right]~\n\\label{eq:CW}\n\\end{equation}\nwhere $M_{\\rm UV}$ is an ultraviolet cutoff, where $c$ is a constant,\nand where $\\eftStr$ denotes a supertrace over the states in the effective theory.\nThis expression has both\na quadratic UV-divergence which we would normally subtract by a counter-term as well as a logarithmic\ncutoff dependence which would normally be indicative of RG running. Thus any UV-completion such as string theory\nhas to resolve two issues within this \nexpression at once: not only must it make the quadratic term finite, but it must also\nbe able to give us specific information about the running. In particular, \nto what value does the Higgs mass actually run in the IR?~\nSuch information is critical in order to nail down the logarithmic running, anchoring \nit firmly as a function of scale. \n\nPrior to our work, such questions remained unanswered.\nIn retrospect, one clue could already be found in the earlier work of Ref.~\\cite{Dienes:1995pm}, which in turn\n rested on previous results in Ref.~\\cite{Kutasov:1990sv}.\nIn Ref.~\\cite{Dienes:1995pm}, it was shown that\nthe one-loop cosmological constant $\\Lambda$ \nfor any non-tachyonic closed string \ncan be expressed as a supertrace over\nthe entire infinite spectrum of level-matched physical string states:\n\\begin{equation}\n \\Lambda~=~ \\frac{1}{24}\\,{\\cal M}^2\\, {\\rm Str}\\, M^2~.\n\\label{eq:lam-rep}\n\\end{equation}\nThis result, which we have rederived in Eq.~(\\ref{eq:lamlam}),\nimmediately suggests two things.\nThe first is that it might be possible to derive an analogous spectral supertrace formula\nfor the one-loop Higgs mass within such strings which, like that in Eq.~(\\ref{eq:lam-rep}), depends on only the physical states in the theory.\nThe second, stemming from a comparison between the result in Eq.~(\\ref{eq:lam-rep}) and \nthe first term in Eq.~(\\ref{eq:CW}), is that there might exist a possible\nderivative-based connection between the one-loop Higgs mass and the one-loop cosmological constant.\n\n \nIn this study, we have addressed all of these issues.\nIndeed, one of the central results of our study is an\nequivalent spectral supertrace formula for the one-loop Higgs mass.\nLike the calculation of the cosmological constant,\nour calculation for the Higgs mass relies on nothing more than worldsheet modular invariance ---\nan exact symmetry which maintains string finiteness and is preserved, even today.\nAnother of our central results is a deep connection between \nthe Higgs mass and the cosmological constant.\nHowever, we also found that unlike the cosmological constant, the Higgs mass may\nactually have a leading logarithmic divergence.\nIndeed, this issue depends on the particular string model under study,\nand in particular the presence of massless states carrying specific charges.\nAs a result of this possible divergence,\nand as a result of the extreme sensitivity of the Higgs mass to physics at all scales, arriving at a fully consistent treatment of the Higgs mass \nrequired us to broach several delicate issues. \nThese encompassed varied aspects of regularization and renormalization and touched \non the very legitimacy of extracting an effective field theory from a UV\/IR-mixed theory. \nThe scope of our study was therefore quite broad, with a number of \nimportant insights and techniques developed along the way.\n\n\nOur first step was to understand how the Higgs and similar scalars\nreside within a typical modular-invariant string theory. \nIn particular, for closed string theories with charge lattices, \nwe began by examining the manner in which fluctuations\nof the Higgs field deform these charge lattices, all the while bearing in mind that these\ndeformations must preserve modular invariance. \nWe were then able to express the contributions to the Higgs mass in terms of \none-loop modular integrals with specific charge insertions ${\\cal X}_i$ incorporated into the\nstring partition-function traces.\nHowever, we found that these insertions have an immediate consequence, producing\na modular anomaly which then requires us to \nperform a ``modular completion'' of the theory.\n This inevitably introduces an additional term into the Higgs mass, one which \nis directly related to the one-loop cosmological constant.\nOur derivation of this term rested solely on considerations of modular invariance and thereby\nendows this result with a generality \nthat holds across the space of perturbative closed-string models.\nIn this way we arrived at one of the central conclusions of our work, namely the existence\nof a universal relation between scalar masses and the cosmological constant in any tachyon-free closed string theory.\nThis relation is given in Eq.~(\\ref{relation1}) for four-dimensional theories,\nand in Eq.~(\\ref{relation1b}) for theories in arbitrary spacetime dimensions $D$. \nStemming only from modular invariance, this result is exact and holds regardless of \nother dynamics that the theory may experience.\n\nHaving established the generic structure of one-loop contributions to the Higgs mass,\nwe then pushed our calculation one step further with the aim of expressing our \nresult for the Higgs mass as a supertrace over the purely physical level-matched spectrum of the theory. \nIndeed, we demonstrated that the requirements of modular covariance so deeply constrain \nthe contributions to the Higgs mass from the unphysical states that these latter\ncontributions can be expressed in terms of contributions from the physical states\nalone. However, part of this calculation required dealing with the \nlogarithmic divergences which can arise.\nThis in turn required that we somehow {\\it regularize}\\\/ the Higgs mass. \n\nFor this reason, we devoted a large portion of our study to establishing\na general formalism for regulating quantities such as the Higgs mass that\nemerge in string theory. We initially considered two forms of \nwhat could be called ``standard'' regulators.\nThe ``minimal'' regulator is essentially a subtraction of the contributions\nof the massless states. We referred to this as a minimal regulator because\nit does not introduce any additional parameters into the theory. Thus, for\nany divergent quantity, there is a single corresponding regulated quantity.\nWe also discussed what we referred to as a ``non-minimal'' regulator,\nbased on a mathematical regularization originally introduced in the mathematics\nliterature~\\cite{zag}. \nThis regulator introduces a new dimensionless parameter $t$, so that for any divergent quantity\nthere exist a set of corresponding regularized quantities parametrized by $t$, with the limit\n{\\mbox{$t\\to \\infty$}} corresponding to the removal of the regulator and the restoration of the original unregulated quantity.\nThis regulator is essentially the one used in Ref.~\\cite{Kaplunovsky:1987rp} and later in Ref.~\\cite{Dienes:1996zr}.\n\nAs we have explained in Sect.~\\ref{sec3}, both of these regulators yield finite quantities which can be expressed in terms of supertraces over only those string states which are physical ({\\it i.e.}\\\/, level-matched). Indeed, in each of these cases,\nthe relation between the regulated quantities and the corresponding supertraces respects modular invariance. Thus, the regulated quantity and the corresponding supertrace in each case transform identically under modular transformations. However, for both the minimal and non-minimal regulators, neither the regulated quantity nor the corresponding supertrace expression is modular invariant on its own. While this additional criterion \nwas not important for the purposes that led to the original development of these regulators in the mathematics literature,\nthis criterion is critical for us because we now wish these regulated quantities to correspond to physical observables (such as our regulated Higgs mass). Each of these regulated quantities must therefore be independently modular invariant on its own. \n\nWe therefore presented a third set of regulators --- those based on the functions $\\widehat{\\cal G}_\\rho(a,\\tau)$.\nThese are our modular-invariant regulators, and they depend on two free parameters $(\\rho,a)$.\nUnlike the minimal and non-minimal regulators, \nthese regulators do not operate by performing a sharp, brute-force subtraction of particular contributions \nwithin the integrals associated with one-loop string amplitudes.\nInstead, we simply multiply the integrand of any one-loop string amplitude by the regulator function $\\widehat {\\cal G}_\\rho(a,\\tau)$. These functions have two important properties which \nmake them suitable as regulators when {\\mbox{$a\\ll 1$}}.\nFirst, as {\\mbox{$a\\to 0$}}, we find that {\\mbox{$\\widehat{\\cal G}_\\rho(a,\\tau)\\to 1$}} for all $\\tau$.\nThus the {\\mbox{$a\\to 0$}} limit restores our original unregulated theory.\nSecond, {\\mbox{$\\widehat{\\cal G}_\\rho(a,\\tau)\\to 0$}} exponentially quickly as {\\mbox{$\\tau\\to i\\infty$}} for all {\\mbox{$a>0$}}. \nThese functions thereby suppress all relevant divergences\nwhich might appear in this limit.\nBut most importantly for our purposes, $\\widehat {\\cal G}_\\rho(a,\\tau)$ is completely modular invariant.\nIn particular, this function is completely smooth, with no sharp transitions in its behavior.\nAs a result, multiplying the integrand of any one-loop string amplitude by $\\widehat{\\cal G}_\\rho(a,\\tau)$ \ndoes not simply excise certain problematic contributions within the corresponding string amplitude, \nbut rather provides a smooth, modular-invariant way of deforming (and thereby regulating) the entire theory.\nThis function even has a physical interpretation in the {\\mbox{$\\rho=2$}} special case, arising as \nthe result of the geometric deformations discussed in Refs.~\\cite{Kiritsis:1994ta, Kiritsis:1996dn, Kiritsis:1998en}. \n\n\nArmed with this regulator, we then demonstrated that \nour regulated Higgs mass can be expressed as the \nsupertrace over only the physical string states.\nOur result for $\\widehat m_\\phi^2(\\rho,a)$ is given in Eq.~(\\ref{twocontributions}),\nwhere $\\widehat m_\\phi^2(\\rho,a)\\bigl|_{\\cal X}$ is given in \nEq.~(\\ref{finalhiggsmassa})\nand where $\\widehat \\Lambda(\\rho,a)$ is given in \nEq.~(\\ref{lambdaresult}).\nWe stress that this is the exact string-theoretic result for the\nregulated Higgs mass, expressed as a function of the regulator parameters $(\\rho,a)$.\nMoreover $\\widehat \\Lambda(\\rho,a)$ by itself is the corresponding regulated\ncosmological constant. As discussed in the text, \nthe one-loop cosmological constant $\\Lambda$ requires regularization in this context\neven though it is already finite in all tachyon-free closed string theories.\n\nWe originally derived these results under the assumption that our underlying\nstring theory could be formulated with an associated charge lattice.\nThis assumption gave our calculations a certain concreteness,\nallowing us to see exactly which states \nwith which kinds of charges ultimately contribute to the Higgs mass.\nHowever, we then proceeded to demonstrate that many of our results are actually more\ngeneral than this, and do not require a charge lattice at all.\nThis lattice-free reformulation also had an added benefit, allowing us to demonstrate a second\ndeep connection between the Higgs mass and the cosmological constant beyond that in Eq.~(\\ref{relation1}). \nIn particular, we were able to demonstrate that each of these quantities can be expressed\nin terms of a common underlying quantity $\\widehat \\Lambda(\\rho,a,\\phi)$ via\nrelations of the form\n\\begin{equation}\n \\begin{cases}\n ~\\widehat \\Lambda(\\rho,a) &=~ \\widehat \\Lambda(\\rho,a,\\phi)\\bigl|_{\\phi=0} \\\\\n ~\\widehat m_\\phi^2 (\\rho,a) &=~ D_\\phi^2 \\,\\widehat \\Lambda(\\rho,a,\\phi)\\bigl|_{\\phi=0}\n \\end{cases}\n\\label{effpotl}\n\\end{equation}\nwhere $D_\\phi^2$ is the modular-covariant second $\\phi$-derivative given in Eq.~(\\ref{Dphi2}). \nThese relations allow us to interpret\n$\\widehat\\Lambda(\\rho,a,\\phi)$ as a stringy effective potential for the Higgs.\nIndeed, these relations are ultimately the fulfillment of our original suspicion\nthat the Higgs mass might be related to the cosmological constant through a double $\\phi$-derivative,\nas discussed below Eq.~(\\ref{eq:lam-rep}).\nHowever, we now see from Eq.~(\\ref{Dphi2}) that this is not just an ordinary $\\phi$-derivative $\\partial_\\phi^2$, but\nrather a {\\it modular-covariant}\\\/ derivative, complete with anomaly term. \nThe second relation in Eq.~(\\ref{effpotl}) thereby {\\it subsumes}\\\/\nour original relation between the Higgs mass and the cosmological constant,\nas expressed in Eq.~(\\ref{relation1}). Moreover, we see that \nthe regulated\ncosmological constant $\\widehat\\Lambda(\\rho,a)$ is nothing but the {\\mbox{$\\phi=0$}} truncation of \nthe same effective potential $\\widehat\\Lambda(\\rho,a,\\phi)$.\nIn this way, $\\widehat \\Lambda(\\rho,a,\\phi)$ emerges as the central object \nfrom which our other relevant quantities can be obtained.\n\nAt no step in the derivation of these results was modular invariance broken.\nThus all of these results are completely consistent with modular invariance, as required.\nMoreover, expressed as functions of the worldsheet regulator parameters $(\\rho,a)$, all of our\nquantities are purely string-theoretic and there are no ambiguities in their definitions.\n\nOur next goal was to interpret these regulated quantities in terms of a physical cutoff scale $\\mu$.\nOf course, if we had been working within a field-theoretic context, \nall of our regulator parameters would have had direct spacetime interpretations in terms of a spacetime scale $\\mu$.\nAs a result, varying the values of these regulator parameters would have \nled us directly to a renormalization-group flow with an associated RGE.~\nString theories, by contrast, are formulated \nnot in spacetime but on the worldsheet --- for such strings, spacetime is nothing but a derived quantity.\nAs a result, although we were able to express our regulated quantities as functions of the two regulator parameters $(\\rho,a)$,\nthe only way to extract an EFT description from these otherwise complete string-theoretic\nexpressions was to develop\na mapping between the worldsheet parameters $(\\rho,a)$ and a physical spacetime scale $\\mu$.\n\nAs we have seen, this issue of connecting $(\\rho,a)$ to $\\mu$ is surprisingly subtle,\nand {\\it it is at this step that we must make certain choices that break modular invariance.}\\\/\nWe already discussed some the issues surrounding IR\/UV equivalence in Sect.~\\ref{UVIRequivalence} ---\nindeed, these issues already suggested that the passage to an EFT would be highly non-trivial\nand involve the breaking of modular invariance.\nBut now, with our complete results in hand, we can take a bird's-eye view and finally map out the full structure of the problem.\n\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[keepaspectratio, width=0.95\\textwidth]{mappings.pdf}\n\\caption{\nThe scale structure of physical quantities in a modular-invariant string theory.\nFor the modular-invariant regulator function \n$\\widehat {\\cal G}_\\rho(a,\\tau_2)$ discussed in this paper, \nthe mapping between the regulator parameter $\\rho a^2$ and the physical spacetime scale $\\mu$\nhas two distinct branches.\nThe traditional branch (shown in blue) identifies {\\mbox{$\\mu^2\/M_s^2 = \\rho a^2$}}, \nbut modular invariance implies \nthe existence of an invariance under the scale-inversion duality symmetry \n{\\mbox{$\\mu\\to M_s^2\/\\mu$}}. This in turn implies \nthe existence of a second branch \n(shown in green) on which we can alternatively identify\n{\\mbox{$\\mu^2\/M_s^2= 1\/(\\rho a^2)$}}.\nAlthough $\\widehat {\\cal G}_\\rho(a,\\tau_2)$ functions as a regulator for {\\mbox{$\\rho a^2<1$}},\nits symmetry under {\\mbox{$\\rho a^2 \\to 1\/(\\rho a^2)$}} \nimplies that this function also acts as a regulator when \nextended into the {\\mbox{$\\rho a^2>1$}} region.\nThis then allows us to see the full four-fold modular structure of the theory.\nThe Higgs-mass plot shown in Fig.~\\ref{anatomy} can now be understood as\nfollowing \nthe {\\mbox{$\\mu^2\/M_s^2 = \\rho a^2$}} branch from the lower-left corner of this figure \ninward toward the central location at which {\\mbox{$\\mu=M_s$}},\nafter which it then follows the \n{\\mbox{$\\mu^2\/M_s^2 = 1\/(\\rho a^2)$}} branch outward towards the upper-left corner.\nHowever, in a modular-invariant theory, all four quadrants of this figure are equivalent and describe the same physics.\nLikewise, in such theories there is no distinction between IR and UV.~\nThus one can exchange ``IR'' $\\leftrightarrow$ ``UV'' within all labels of this sketch,\nand we have simply chosen to show \nthose labels\nthat have the most natural interpretations\nwithin the lower-left quadrant. \nFinally, \nregions with beige shading indicate locations where EFT descriptions exist, \nwhile stringy effects dominate in the yellow central region.\nAs a result, focusing on any one of the four EFT regions by itself\nnecessarily breaks modular invariance\nbecause the choice of EFT region is tantamount to picking a preferred direction for the flow of the scale $\\mu$ relative to \nthe underlying string-theoretic regulator parameter $\\rho a^2$. \nHowever, even within the EFT regions, string states {\\it at all mass scales}\\\/ contribute non-trivially.\nThus even these EFT regions differ from what might be expected within quantum field theory.} \n\\label{mappings_figure}\n\\end{figure*}\n\n\nOur understanding of this issue is summarized in Fig.~\\ref{mappings_figure}. \nAlthough the specific situation sketched in Fig.~\\ref{mappings_figure} corresponds \nto our modular-invariant regulator function $\\widehat{\\cal G}_\\rho(a,\\tau)$,\nthe structure of this diagram is general. \nUltimately, the connection between worldsheet physics and spacetime physics\nfollows from the one-loop partition function, \nwhich for physical string states of spacetime masses $M_i$ takes the general form \n\\begin{equation}\n {\\cal Z}\\\/ ~\\sim~ \\tau_2^{-1} \\, \\sum_i \\,e^{- \\pi \\alpha' M_i^2 \\tau_2}~.\n\\label{pfZ}\n\\end{equation}\nThese $M_i$ are precisely the masses \nwhich ultimately appear in our physical supertrace formulas.\nHowever, our regulator function $\\widehat {\\cal G}_\\rho(a,\\tau)$ \ncannot regulate the divergences that might arise from light and\/or massless states \nas {\\mbox{$\\tau\\to i\\infty$}} unless it suppresses contributions\nto the partition function within the region {\\mbox{$\\tau_2\\gsim \\tau_2^\\ast$}} for some $\\tau_2^\\ast$. \nWe thus see that whether a given state contributes significantly to one-loop amplitudes\nin the presence of the regulator\ndepends on the magnitude of $\\alpha' M_i^2 \\tau_2^\\ast$.\nThis immediately leads us to identify a corresponding spacetime physical RG scale \n{\\mbox{$\\mu^2\\sim 1\/(\\alpha' \\tau^\\ast_2)$}}.\nIndeed, this was precisely the logic that originally led us to Eq.~(\\ref{mut}).\nMoreover, for our specific regulator function $\\widehat{\\cal G}_\\rho(a,\\tau)$, we have \n{\\mbox{$\\tau_2^\\ast \\sim 1\/(\\rho a^2)$}},\nthus leading to the natural identification {\\mbox{$\\mu^2\/M_s^2=\\rho a^2$}}.\n\nHowever, modular invariance does not permit us to \nidentify just one special\npoint {\\mbox{$\\tau_2= \\tau_2^\\ast$}} along the {\\mbox{$\\tau_1=0$}} axis within the fundamental domain. \nIndeed, for every such special point, the corresponding point with {\\mbox{$\\tau_2= 1\/\\tau_2^\\ast$}} is equally special,\nsince these two points along the {\\mbox{$\\tau_1=0$}} axis are related by the {\\mbox{$\\tau\\to -1\/\\tau$}} modular transformation.\nIn other words, although our $\\widehat{\\cal G}_\\rho(a,\\tau)$ regulator function suppresses contributions\nfrom the {\\mbox{$\\tau_2\\gsim \\tau_2^\\ast$}} region, the modular invariance of this function requires that it also\nsimultaneously suppress contributions from the region with {\\mbox{$\\tau_2 \\lsim 1\/\\tau_2^\\ast$}}.\n(In general a modular-invariant regulator function equally suppresses the contributions \nfrom the regions that approach {\\it any}\\\/ of the modular cusps, but for the purposes of mapping to a physical\nspacetime scale $\\mu$ our concern is limited to the cusps along the {\\mbox{$\\tau_1=0$}} axis.)\nWe thus see that for any value of the spacetime scale $\\mu$ that we attempt to identify as corresponding\nto $\\rho a^2$, there always exists a second scale $M_s^2\/\\mu$ which we might equally validly identify as\ncorresponding to $\\rho a^2$.\nThis is the implication of the {\\mbox{$\\mu\\to M_s^2\/\\mu$}} scale-inversion symmetry discussed in Sect.~\\ref{sec4}.\n\nThe upshot of this discussion is that the mapping from $\\rho a^2$ to $\\mu$\nin any modular-invariant theory actually\nhas {\\it two branches}\\\/, as shown in Fig.~\\ref{mappings_figure}.\nAlong the first branch we identify {\\mbox{$\\mu^2\/M_s^2=1\/\\tau_2^\\ast= \\rho a^2$}}, but along the second\nbranch we identify \n{\\mbox{$\\mu^2\/M_s^2=\\tau_2^\\ast= 1\/(\\rho a^2)$}}. \nThese branches contain the same physics, but the choice of either branch breaks modular invariance.\nIn this respect modular invariance is much like another description-redundancy symmetry, namely gauge invariance:\nall physical quantities must be gauge invariant, but the choice of \na particular gauge slice (which is tantamount to the choice of a particular branch) \nnecessarily breaks the underlying symmetry.\n\nIn most of our discussions in this paper,\nwe focused on the behavior of our \nregulator functions within the {\\mbox{$a\\ll 1$}} regime.\nHowever, as we have seen in Eq.~(\\ref{newest}),\nthese functions exhibit a symmetry under {\\mbox{$\\rho a^2 \\to 1\/(\\rho a^2)$}}.\nThe logical necessity for this extra symmetry will be discussed below,\nbut this symmetry implies that $\\widehat {\\cal G}_\\rho(a,\\tau)$ also acts as a regulator when \nextended into the {\\mbox{$a\\gg 1$}} region.\nThis then allows us to see the full four-fold modular structure of the theory,\nas shown in Fig.~\\ref{mappings_figure}.\nGiven this structure,\nwe can also revisit the Higgs-mass plot shown in Fig.~\\ref{anatomy}. We now see that\nwe can interpret this plot as following \nthe {\\mbox{$\\mu^2\/M_s^2 = \\rho a^2$}} branch from the lower-left corner of Fig.~\\ref{mappings_figure}\ntoward the central location at which {\\mbox{$\\mu=M_s$}},\nand then \nfollowing the \n{\\mbox{$\\mu^2\/M_s^2 = 1\/(\\rho a^2)$}} branch outward towards the upper-left corner.\n\nGiven the sketch in Fig.~\\ref{mappings_figure},\nwe can also understand more precisely how the passage from string theory \nto an EFT breaks modular invariance. Within this sketch,\nregions with beige shading indicate locations where EFT descriptions exist\n(and where our regulators are designed to function most effectively, with {\\mbox{$a\\ll 1$}} or {\\mbox{$a\\gg 1$}}). \nBy contrast, stringy effects dominate in the yellow central region, which is\nthe only region that locally exhibits the full modular symmetry, lying on both branches simultaneously.\nAs a result, we necessarily break modular invariance by choosing to focus on any one of the four \nEFT regions alone.\nIndeed, each EFT region intrinsically exhibits a certain direction for the flow of the scale $\\mu$\nrelative to the flow of the underlying worldsheet parameter $\\rho a^2$.\nHowever, the relative direction of this flow is not modular invariant,\nas evidenced from the fact that this flow is reversed in switching from one branch\nto the other.\n\nAt first glance, the fact that the EFT regions appear only at the extreme ends of each branch \nin Fig.~\\ref{mappings_figure} might lead \none to believe that only extremely light states contribute within the EFT\nand that the infinite towers of heavy string states can be ignored within such regions.\nHowever, as we have repeatedly stressed throughout this paper, even this seemingly mild \nassertion would be incorrect. For example, even within the {\\mbox{$\\mu\\to 0$}} limit,\nwe have seen in Eq.~(\\ref{asymplimit}) that\nthe Higgs mass receives contributions from ${\\mathbb{X}}_1$-charged states of all masses across\nthe entire string spectrum. Likewise, $\\Lambda$ receives contributions from {\\it all}\\\/ string states, \nregardless of their mass. \nWe have also seen that the Higgs mass accrues a $\\mu$-dependence which transcends\nour field-theoretic expectations, even for {\\mbox{$\\mu\\ll M_s$}}. A particularly \nvivid example of this is the unexpected ``dip'' region shown in Fig.~\\ref{anatomy} --- an\neffect which is the direct consequence of the stringy Bessel functions whose form is dictated by\nmodular invariance.\nThus modular invariance continues to govern the behavior of the Higgs mass at all scales,\neven within the EFT regions.\n\nLikewise, within such theories there is no distinction between IR and UV.~\nWe can already see this within Fig.~\\ref{mappings_figure},\nwhere the points near the upper end of the figure ({\\it i.e.}\\\/, with large $\\mu$) are \ndesignated not as ``UV'' but as ``dual IR'', since they are the images of the IR regions\nwith small $\\mu$ under the duality-inversion symmetry.\nBut even this labeling is not truly consistent with \nmodular invariance, since there is no reason to adopt the \nlanguage of the small-$\\mu$ \nregion in asserting that the bottom part of the figure corresponds to the IR.~\nThanks to the equivalence under {\\mbox{$\\mu\\to M_s^2\/\\mu$}}, we might as well have decided to label the upper\nportion of the figure as ``UV'' and the lower portion of the figure as ``dual UV''.\nIn that case, the center of the figure would represent the most IR-behavior that is possible,\nrather than the most UV.~\nThe upshot is that the mere distinction between ``IR'' and ``UV'' \nitself breaks modular invariance.\nIn a modular-invariant theory, what we would normally call a UV divergence \nis not distinct from an IR divergence --- they are one and the same. \nIndeed, we have seen that the quadratic divergences normally associated with the Higgs mass in field theory \nare softened to mere logarithmic divergences --- such is the power of modular invariance ---\nbut in string theory there is no deeper physical interpretation\nto this remaining divergence as either UV or IR in nature until we decide to introduce one. \n\n\nIn this connection, we note that it \nmight have seemed tempting to look at the EFT expression in Eq.~(\\ref{eq:CW}) and suppose that\nin a UV-complete theory one could have set about the calculation in a piecemeal\nmanner, dividing the contributions into a UV contribution\nand a much less lethal logarithmically divergent IR contribution\nand then evaluating each one separately.\nThis is certainly the kind of reasoning that is suggested by the notion\nof softly broken symmetries, for example. \nHowever, because there is no intrinsic notion of\nUV and IR in a modular-invariant theory, no such separation can exist.\nInstead, all we have in string theory \nare amplitudes which may be divergent,\nand the question as to whether such divergences are most naturally interpreted as UV or IR in nature\nultimately boils down to a {\\it convention}\\\/ as to which modular-group fundamental domain \nis selected as our region of integration.\nAlthough these arguments are expressed in terms of one-loop amplitudes, \nsimilar arguments extend to higher loops as well. \nOf course, most standard textbook recipes for evaluating one-loop modular\nintegrals in string theory adopt the \nfundamental domain which includes the cusp at {\\mbox{$\\tau\\to i\\infty$}}.\nThis choice then leads to an IR interpretation for the divergence.\nBut when we derived\nour supertrace expressions involving only the physical string states,\nour calculations required that we sum over an infinite number of such fundamental domains which \nare all related to each other under modular transformations, as in Eq.~(\\ref{stripF}).\nIt is only in this way that we were able to transition from the fundamental domain to the strip\nand thereby obtain supertraces involving only the physical string states.\nThus UV and IR physics are inextricably mixed within such supertrace expressions.\n\n\nGiven this bird's-eye view, we can now also understand in a deeper way why it was necessary \nfor us to switch from our original modular-invariant regulator functions ${\\cal G}_\\rho(a,\\tau)$ \nin Eq.~(\\ref{regG}) to our enhanced modular-invariant functions $\\widehat {\\cal G}_\\rho(a,\\tau)$ in Eq.~(\\ref{hatGdef})\nwhich exhibited the additional symmetry under {\\mbox{$a\\to 1\/(\\rho a)$}}.\nAt the level of the string worldsheet,\nour original functions ${\\cal G}_\\rho(a,\\tau)$ would have been suitable, since they already \nsatisfied the two critical criteria\nwhich made them suitable as regulators:\n\\begin{itemize}\n\\item {\\mbox{${\\cal G}_\\rho(a,\\tau)\\to 1$}} for all $\\tau$ as {\\mbox{$a\\to 0$}}, so that the {\\mbox{$a\\to 0$}} \n limit restores our original unregulated theory; and\n\\item {\\mbox{$ {\\cal G}_\\rho(a,\\tau)\\to 0$}} sufficiently rapidly for any {\\mbox{$a>0$}} as\n $\\tau_2$ approaches the appropriate cusps ({\\mbox{$\\tau\\to i\\infty$}}, or equivalently {\\mbox{$\\tau\\to 0$}}), so that\n $f$ is capable of regulating our otherwise-divergent integrands for all {\\mbox{$a>0$}}.\n\\end{itemize}\nIndeed, for any divergent string-theoretic quantity $I$, these functions would have led to\na corresponding set of finite quantities $\\widetilde I_\\rho(a)$ for each value of $(\\rho,a)$.\nWe further saw that these ${\\cal G}$-functions had a redundancy under {\\mbox{$(\\rho,a)\\to (1\/\\rho,\\rho a)$}},\nso that the only the combination $\\rho a^2$ was invariant.\n\nHowever, while such functions would have been suitable at the level of the string worldsheet, there is one additional\ncondition that must also be satisfied if we want to be able to interpret our results \nin {\\it spacetime}\\\/, with the invariant combination $\\rho a^2$ identified as a running spacetime scale $\\mu^2\/M_s^2$.\nAs we have argued below Eq.~(\\ref{pfZ}),\nmodular-invariant string theories necessarily exhibit an invariance\nunder {\\mbox{$\\mu\\to M_s^2\/\\mu$}}; indeed,\nthis scale-duality symmetry rests on very solid foundations.\nHowever, given this scale-inversion symmetry, we see that we \nwould not have been able to consistently identify $\\rho a^2$ with the spacetime scale\n$\\mu^2\/M_s^2$ unless our regulator function itself also exhibited such an inversion symmetry, with an invariance under\n{\\mbox{$\\rho a^2 \\to 1\/(\\rho a^2)$}} [or equivalently under {\\mbox{$a\\to 1\/(\\rho a)$}}].\nThis was the ultimately the reason we transitioned from the ${\\cal G}$-functions to the $\\widehat {\\cal G}$-functions,\nas in Eq.~(\\ref{hatGdef}). This not only preserved the first two properties itemized above, \nbut also ensured a third:\n\\begin{itemize}\n\\item {\\mbox{$\\widehat {\\cal G}_\\rho(a,\\tau) = \\widehat {\\cal G}_\\rho(1\/\\rho a,\\tau)$}} for all $(\\rho,a)$.\n\\end{itemize}\nIn other words, while our first two conditions ensured proper behavior for our regulator functions on the string worldsheet,\nit was the third condition which allowed us to endow our regulated string theory with an interpretation\nin terms of a renormalization flow with a spacetime mass scale $\\mu$. \nIndeed, we see from Fig.~\\ref{mappings_figure} that in some sense this extra symmetry was forced on \nus the moment we identified {\\mbox{$\\mu^2\/M_s^2 = \\rho a^2$}} and recognized the existence of the scale-duality\nsymmetry under {\\mbox{$\\mu\\to M_s^2\/\\mu$}}.\nA similar symmetry structure would also need to hold for any alternative regulator functions that might be chosen.\n \n\nGiven these insights, we then proceeded to derive expressions for\nour regulated Higgs mass $\\widehat m_\\phi^2(\\mu)$ and regulated cosmological constant\n(effective potential) $\\widehat \\Lambda(\\mu)$ \nas functions of $\\mu$.\nThe exact results for these quantities are given in Eqs.~(\\ref{finalhiggsmassmu}) \nand (\\ref{lambdamuresult}), respectively. \nOnce again, we stress that these results are fully modular invariant except for the fact\nthat we have implicitly chosen to work within the lower-left branch of Fig.~\\ref{mappings_figure}.\nFor {\\mbox{$\\mu\\ll M_s$}},\nwe were then able to derive the corresponding approximate EFT running \nfor these quantities in Eqs.~(\\ref{approxhiggsmassmu}) \nand (\\ref{lambdamuresult2}).\nIndeed, as we have seen in Eq.~(\\ref{Lambdafull}),\n our final result for the running \neffective potential $\\widehat \\Lambda(\\mu,\\phi)$ takes \nthe general form\n\\begin{eqnarray}\n \\widehat \\Lambda(\\mu,\\phi) \\,&=&\\, \n \\frac{1}{24}{\\cal M}^2 \\,{\\rm Str}\\, M^2 \n -c' \\antieffStr M^2 \\mu^2 \\nonumber\\\\ \n && - \\zeffStr \\left[ \\frac{M^4}{64\\pi^2} \\log\\left( c \\frac{M^2}{\\mu^2}\\right) \n + c''\\mu^4\\right] ~~~~~~~\n\\label{eq:lambdaconclusions} \n\\end{eqnarray}\nwhere \n{\\mbox{$c= 2e^{2\\gamma+1\/2}$}},\n{\\mbox{$c'=1\/(96\\pi^2)$}}, and\n{\\mbox{$c''= 7 c'\/10$}},\nand where of course we regard the masses $M^2$ as a functions of $\\phi$ as in Eq.~(\\ref{TaylorM}).\nThese specific values of $\\lbrace c,c',c''\\rbrace$\nwere of course calculated with our regulator function taken as $\\widehat {\\cal G}_\\rho(a,\\tau)$\nassuming the benchmark value {\\mbox{$\\rho=2$}},\nand with $\\mu$ defined along the lower-left branch in Fig.~\\ref{mappings_figure}.\nHowever, in general these constants depend on the precise profile of our regulator function.\nFinally, given our effective potential, we also discussed the general conditions under which \nour theory is indeed sitting at a stable minimum as a function of $\\phi$.\n\n\nWith the results in Eq.~(\\ref{eq:lambdaconclusions}) in conjunction with\nthe relations in Eq.~(\\ref{effpotl}),\nwe have now obtained an understanding of the Higgs mass \nas emerging from $\\phi$-derivatives \nof an infinite spectral supertrace of regulated effective potentials.\nWe can now also perceive the critical similarities and differences relative to the \nEFT expectations in Eq.~(\\ref{eq:CW}) and thereby address the questions posed\nat the beginning of this section.\nFor example, \nfrom the first term within Eq.~(\\ref{eq:lambdaconclusions})\nwe see that the Higgs mass within the full modular-invariant theory \ncontains a term of the form $\\frac{1}{24} {\\cal M}^2 {\\rm Str} \\partial_\\phi M^2$.\nComparing this term \nwith first term within Eq.~(\\ref{eq:CW}), \nwe might be tempted to identify\n{\\mbox{$M_{\\rm UV}= \\sqrt{3\/2} \\pi {\\cal M}$}}.\nHowever, despite the superficial resemblance between these terms,\nwe see that the full string-theoretic term is very different\nbecause the relevant supertrace is over the {\\it entire}\\\/ spectrum of states in the theory\nand not just the light states in the EFT.~\n\nIt is also possible to compare the logarithmic terms within\nEqs.~(\\ref{eq:CW})\nand (\\ref{eq:lambdaconclusions}).\nOf course, as in the standard treatment, the logarithmic term in Eq.~(\\ref{eq:CW}) can be regulated by subtracting \na term of the form $\\log(M_{\\rm UV}\/\\mu)$, thereby obtaining an effective running.\nWe then see that both logarithmic terms actually agree.\nWhile it is satisfying to see this agreement, it is nevertheless \nremarkable that we have obtained such a logarithmic EFT-like running \nfrom our string-theoretic result.\nAs we have seen, our full string results in \nEqs.~(\\ref{finalhiggsmassmu}) and (\\ref{lambdamuresult}) did\nnot contain logarithms --- they contained Bessel functions.\nMoreover, unlike the term discussed above, their contributions were {\\it not}\\\/ truncated\nto only the light states with {\\mbox{$M\\lsim \\mu$}} --- they involved supertraces over {\\it all}\\\/\nof the states in the string spectrum, as expected for a modular-invariant theory.\nHowever, the behavior of the Bessel functions themselves \nsmoothly and automatically suppressed the contributions from states with {\\mbox{$M\\gsim \\mu$}}.\nThus, we did not need to {\\it impose}\\\/\nthe {\\mbox{$M\\lsim \\mu$}} restriction on the supertrace of the logarithm\nterm in Eq.~(\\ref{eq:lambdaconclusions})\nbased on a prior EFT-based expectation, as in Eq.~(\\ref{eq:CW});\nthis restriction, and thus an EFT-like interpretation, \nemerged naturally from the Bessel functions themselves.\nIt is, of course, possible to verify the appearance of such a term directly within the\ncontext of a given compactification through a direct calculation of the two-point function of the Higgs field\n(and indeed we verified this explicitly for various compactification choices),\nbut of course the expression in \nEq.~(\\ref{eq:lambdaconclusions}) \nis completely general\nand thus holds regardless of the specific compactification.\n\n\nWe can also now answer the final question posed at the beginning of this section:\n to what value does the Higgs mass actually run as {\\mbox{$\\mu\\to 0$}}?\nAssuming {\\mbox{$\\zStr {\\mathbb{X}}_2=0$}}, the answer is clear from Eq.~(\\ref{asymplimit}):\n\\begin{eqnarray}\n \\lim_{\\mu\\to 0} \\widehat m_\\phi^2(\\mu) \\, &=& \\,\n \\frac{\\xi}{4\\pi^2} \\,\\frac{\\Lambda}{{\\cal M}^2}\n - \\frac{\\pi}{6} \\,{\\cal M}^2\\, {\\rm Str}\\, {\\mathbb{X}}_1 \\nonumber\\\\\n &=&\\, \n \\frac{\\xi}{96\\pi^2} \\,{\\rm Str}\\, M^2 \n + \\frac{1}{24}\\, {\\cal M}^2\\, {\\rm Str}\\, \\partial_\\phi^2 M^2 \\Bigl|_{\\phi=0} \\nonumber\\\\\n &=&\\, \\frac{{\\cal M}^2 }{24}\\,D_\\phi^2 \\,{\\rm Str}\\, M^2\\Bigl|_{\\phi=0}~. \n\\label{asymplimit2}\n\\end{eqnarray}\nFrom a field-theory perspective, this is a remarkable result: all running actually\nstops as {\\mbox{$\\mu\\to0$}}, and the Higgs mass approaches a constant whose value is set by\na supertrace over {\\it all}\\\/ of the states in the string spectrum.\nThis behavior is clearly not EFT-like.\nHowever, the underlying reason for this has to do with UV\/IR equivalence and the scale-inversion\nsymmetry under {\\mbox{$\\mu\\to M_s^2\/\\mu$}}.\nRegulating our Higgs mass ensures that our theory no longer diverges as {\\mbox{$\\mu\\to \\infty$}}; rather,\nthe Higgs mass essentially ``freezes'' to a constant in this limit.\nIt is of course natural that in this limit the relevant constant includes contributions from\nall of the string states.\nThe scale-inversion symmetry then implies that the Higgs mass must also ``freeze'' \nto exactly the same value as {\\mbox{$\\mu\\to 0$}}.\nWe thus see that although a {\\it portion}\\\/ of the running of the Higgs mass is EFT-like\nwhen {\\mbox{$\\mu\\ll M_s$}}, this EFT-like behavior does not persist all the way to {\\mbox{$\\mu=0$}} because\nthe scale-inversion symmetry forces \nthe behavior as {\\mbox{$\\mu\\to 0$}} to mirror\nthe behavior as {\\mbox{$\\mu\\to \\infty$}}.\nIndeed, the ``dip'' region is nothing but the stringy transition between these two\nregimes.\n\n\nGiven the results in Eq.~(\\ref{asymplimit2})\nwe also observe that we can now write\n\\begin{equation}\n m_\\phi^2 ~=~ \\left. \\frac{1}{24} {\\cal M}^2 \\,{\\rm Str} \\left[ D_\\phi^2 M^2(\\phi)\\right]\\,\\right|_{\\phi=0}~~~\n\\end{equation}\nThis result is thus the Higgs-mass analogue \nof the $\\Lambda$-result in Eq.~(\\ref{eq:lam-rep}).\nWe can also take the $a\\to 0$ (or equivalently $\\mu\\to 0$) limit of Eq.~(\\ref{effpotl}),\nyielding the simple relations\n\\begin{equation}\n \\begin{cases}\n ~ \\Lambda &=~ \\Lambda(\\phi)\\bigl|_{\\phi=0} \\\\\n ~ m_\\phi^2 &=~ D_\\phi^2 \\,\\Lambda(\\phi)\\bigl|_{\\phi=0}~.\n \\end{cases}\n\\label{effpotl2}\n\\end{equation}\nIndeed, for theories with {\\mbox{$\\zStr {\\mathbb{X}}_2=0$}}, \nthese are exact relations amongst finite quantities.\n\n\nThe final results of our analysis \nare encapsulated within Fig.~\\ref{anatomy}.\nIndeed, this figure graphically illustrates many of the most important conclusions of this paper.\nIn Fig.~\\ref{anatomy}, we have dissected the anatomy of the Higgs-mass running,\nillustrating how this running\npasses through different distinct stages as $\\mu$ increases. \nStarting from the ``deep IR\/UV''\nregion near {\\mbox{$\\mu\\approx 0$}}, the Higgs mass passes through the ``dip'' region and the ``EFT'' region\nbefore ultimately reaching the ``turnaround'' region. \nBeyond this, the theory enters the \n``dual EFT'' region, followed by the ``dual dip'' region\nand ultimately the ``dual deep IR\/UV'' region.\nAbove all else, this figure clearly illustrates\nhow in a modular-invariant theory our normal understanding\nof ``running'' is turned on its head. The Higgs mass does not somehow\nget ``born'' in the UV and then run to some possibly undesirable\nvalue in the IR.~ \nInstead, we may more properly consider the Higgs mass to be ``born'' at {\\mbox{$\\mu=M_s$}}.\nIt then runs symmetrically towards both lesser and greater values of $\\mu$\nuntil it eventually asymptotes to a constant as {\\mbox{$\\mu\\to 0$}} and as {\\mbox{$\\mu\\to \\infty$}}.\n\n\nWe conclude this discussion with two comments regarding technical points.\nFirst, as discussed in Sect.~\\ref{sec4}, we have freely assumed throughout this paper \nthat the residue of a supertrace sum is equivalent to the supertrace sum of the individual residues. In other words, as discussed below Eq.~(\\ref{integrateg}), we have assumed that the supertrace sum does not introduce any additional divergences beyond those already encapsulated within our assertion that the four-dimensional Higgs mass is at most logarithmically divergent, or equivalently that the level-matched integrand has a divergence structure\n{\\mbox{$g(\\tau)\\sim c_0+c_1\\tau_2$}} as {\\mbox{$\\tau_2\\to\\infty$}}. Indeed, this assumption is justified because we are working within the presence of a regulator which is sufficiently powerful to render our modular integrals finite, given this divergence structure. Moreover, the divergence structure of our original unregulated Higgs mass is completely general for theories in four spacetime dimensions, since only a change in spacetime dimension can alter the numbers of $\\tau_2$ prefactors which emerge. Of course, four-dimensional string models generically contain many moduli, and some of these moduli may correspond to the radii associated with possible geometric compactifications from our original underlying 10- and\/or 26-dimensional worldsheet theories. If those moduli are extremely large or small, one approaches a decompactification limit in which our theory becomes effectively higher-dimensional. For any finite or non-zero value of these moduli, our results still hold as before. However, in the full limit as these moduli become infinite or zero, new divergences may appear which are related to the fact that the effective dimensionality of the theory has changed. Indeed, extra spacetime dimensions generally correspond to extra factors of $\\tau_2$, thereby increasing the strengths of the potential divergences. Although all of our results in Sects.~\\ref{sec2} and \\ref{sec3} are completely general for all spacetime dimensions, our results in Sect.~\\ref{sec4} are focused on the case of four-dimensional string models \nfor which {\\mbox{$g(\\tau_2)\\sim c_0+c_1\\tau_2$}} as {\\mbox{$\\tau_2\\to\\infty$}}. \nAs a result, the supertrace-summation and residue-extraction procedures will not commute in the decompactification limit, and additional divergences can arise. However, this does not pose a problem for us --- we simply use the same regulators we have already outlined in Sect.~\\ref{sec3}, but instead work directly in a higher-dimensional framework in which $g(\\tau_2)$ as \n{\\mbox{$\\tau_2\\to\\infty$}} \ntakes a form appropriate for the new effective spacetime dimensionality. \nOnce this is done, we are once again free to exchange the orders of residue-extraction and supertrace-summation, knowing that our results must once again be finite.\n\n\nOur second technical point relates to the concern that has occasionally been expressed in the prior literature\nabout the role played by the off-shell tachyons which necessarily appear \nwithin the spectra of all heterotic strings, and the exponential one-loop divergences\nthey might seem to induce in the absence of supersymmetry\nas {\\mbox{$\\tau\\to i\\infty$}}. \nIn this paper, we discussed this issue briefly in the paragraph surrounding\nEq.~(\\ref{protocharge}). \nUltimately, however, we believe that this concern is spurious.\nFirst, as discussed below Eq.~(\\ref{protocharge}), such states typically lack the non-zero charges \nneeded in order to contribute to the relevant one-loop string amplitudes. \nSecond, \nwithin such one-loop amplitudes,\nour modular integrations \ncome with an implicit instruction \nthat within the {\\mbox{$\\tau_2>1$}} region of the fundamental domain\nwe are to perform the $\\tau_1$ integration prior to performing the $\\tau_2$ integration.\nThis then eliminates the contributions from the off-shell tachyons in the {\\mbox{$\\tau\\to i\\infty$}} limit.\nThis integration-ordering prescription \nis tantamount to replacing the divergence as {\\mbox{$\\tau\\to i\\infty$}} with its average along the line segment {\\mbox{$-1\/2\\leq \\tau_1\\leq 1\/2$}},\nwhich makes sense in the {\\mbox{$\\tau_2\\to\\infty$}} limit as this line segment moves infinitely far up the fundamental domain.\nAnother way to understand this is to realize that under a modular transformation no information can be lost, yet this entire\nline segment as {\\mbox{$\\tau_2\\to \\infty$}} is mapped to the single point with {\\mbox{$\\tau_1=\\tau_2=0$}} under the modular transformation {\\mbox{$\\tau\\to -1\/\\tau$}}.\nFinally, through the compactification\/decompactification argument in presented in Ref.~\\cite{Kutasov:1990sv}, \none can see directly that this off-shell tachyon makes no contribution in all spacetime dimensions {\\mbox{$D>2$}}. \nThus no exponential divergence arises.\nHowever, we note that even if \nan exponential divergence were to survive, it would also be automatically regulated through \nour modular-invariant regulator $\\widehat G_\\rho(a,\\tau)$ --- or sufficiently many higher powers thereof --- given\nthat $\\widehat G_\\rho(a,\\tau)$ itself exhibits an exponential suppression as {\\mbox{$\\tau\\to i\\infty$}}.\n\n\nThe results in this paper have touched on many different topics. Accordingly, there are \nseveral directions that future work may take. \n\n\nFirst, although we have focused in this paper on the mass of the\nHiggs, it is clear that this UV\/IR-mixed picture of running provides a general paradigm\nfor how one should think about the behavior of a modular-invariant theory as a whole. \n For example, one question that naturally arises from \nour discussion concerns the renormalization of the dimensionless couplings. \nThis was the subject of the seminal work in Ref.~\\cite{Kaplunovsky:1987rp}. \nEven though a regulator was chosen in Ref.~\\cite{Kaplunovsky:1987rp} which \nwas not consistent with modular invariance,\nthis was one of the first calculations in which the contributions from the full \ninfinite towers of string states were incorporated within a calculation of gauge couplings and their behavior.\nIt would therefore be interesting to revisit these issues and analyze the running and beta functions\nof the dimensionless gauge couplings that would emerge in the presence of a fully modular-invariant regulator. \nThe first steps in this direction have already been taken in Refs.~\\cite{Kiritsis:1994ta, Kiritsis:1996dn, Kiritsis:1998en}.\nHowever, using the techniques we have developed in this paper, it is now possible to\nextend these results to obtain full scale-dependent RG\nflows for the gauge couplings\n as functions of $\\mu$, and in a continuous way that simultaneously incorporates both UV and IR physics and which does not artificially separate the results into a field-theoretic running with a string-theoretic threshold correction.\nMoreover, due to the {\\mbox{$\\mu\\to M_s^2\/\\mu$}} symmetry\nwe expect that the coefficients of {\\it all}\\\/ operators in the theory \nshould experience symmetric runnings with vanishing gradients at {\\mbox{$\\mu=M_s$}}.\nFor operators with zero engineering dimension, this then translates to a vanishing\nbeta function at {\\mbox{$\\mu=M_s$}}, suggesting the existence of an unexpected (and ultimately unstable) ``UV'' \nfixed point at that location.\n\nIn the same vein, it would also be interesting to study the behavior of scattering amplitudes\nwithin a full modular-invariant context.\nWe once again expect significant deviations from our field-theoretic expectations at all scales ---\nincluding those at energies relatively far below the string scale --- but it would be interesting\nto obtain precise information about how this occurs and what shape the deviations take.\n\n \n\nGiven our results thus far,\nperhaps the most important and compelling avenue to explore concerns the gauge hierarchy problem.\nAs discussed in the Introduction,\nit remains our continuing hope that modular symmetries might provide a new perspective on this problem,\none that transcends our typical field-theoretic expectations.\nSome ideas in this direction were already sketched in Ref.~\\cite{Dienes:2001se},\nalong with suggestions \nthat the gauge hierarchy problem\nmight be connected with the cosmological-constant problem,\nand that these both might be closely connected with the question of vacuum stability.\nIt was also advocated in the Conclusions section of Ref.~\\cite{Dienes:2001se} \nthat these insights might be better understood through calculational frameworks that did not involve\ndiscarding the contributions of the infinite towers of string states, but which instead\nincorporated all of these contributions in order \nto preserve modular invariance and the string finiteness that follows.\n\n\nThe results of this paper \nenable us to begin the process of \nfulfilling these ambitions.\nIn particular, the effective potential\nin Eq.~(\\ref{eq:lambdaconclusions}) is a powerful first step because\nthis result \nprovides a ``UV-complete'' effective potential \nwhich yields the raw expressions\nfor radiative corrections written in terms of the spectrum of whatever theory\none may be interested in studying. Moreover it is an expression that\nis applicable at all energy scales, including the scales associated with the \ncosmological constant and the electroweak physics \nwhere such results are critical.\n\nGiven our results, we can develop a string-based reformulation of both \nof these hierarchy problems.\nOur expression for the cosmological constant in Eq.~(\\ref{eq:lam-rep})\n[or equivalently taking {\\mbox{$\\lim_{\\mu \\to 0} \\widehat \\Lambda(\\mu)$}}]\nimplicitly furnishes us with a constraint \nof the form {\\mbox{${\\rm Str}\\,M^2 \\sim 24 M_\\Lambda^4\/{\\cal M}^2$}}\nwhere {\\mbox{$M_\\Lambda \\sim \\Lambda^{1\/4}\\approx 2.3\\times 10^{-3}\\,$}}{\\rm eV} is the\nmass scale associated with the cosmological constant.\nLikewise, we see that\n{\\mbox{$\\Lambda\\ll 4\\pi^{2}M_{\\rm EW}^{2}\\mathcal{M}^{2}$}} where\n{\\mbox{$M_{\\rm EW}\\sim {\\cal O}(100)~{\\rm GeV}$}} denotes the electroweak scale.\nThus, \nwith $\\phi$ representing the Standard-Model Higgs \nand roughly identifying the physical Higgs mass as\n{\\mbox{$\\lim_{\\mu \\to 0} \\widehat m_\\phi^2 (\\mu) \\sim M_{\\rm EW}^2$}},\nwe see from Eq.~(\\ref{asymplimit2}) that we can obtain a second constraint\nof the form\n{\\mbox{$\\partial_\\phi^2 {\\rm Str}\\, M^2\\bigl|_{\\phi=0}\\sim 24 M_{\\rm EW}^2\/{\\cal M}^2$}}.\nWe therefore see that our two hierarchy conditions now respectively take the forms\n\\begin{equation}\n \\begin{cases}\n ~\\phantom{\\partial_\\phi^2\\,} {\\rm Str}\\, M^2 \\Big|_{\\phi=0} \\!\\! &\\sim~ 24\\, M_\\Lambda^4\/{\\cal M}^2 \\\\\n ~\\partial_\\phi^2\\, {\\rm Str}\\, M^2 \\Big|_{\\phi=0} \\!\\! &\\sim~ 24\\, M_{\\rm EW}^2\/{\\cal M}^2 \n \\end{cases}\n\\label{stringVeltman}\n\\end{equation}\nwhere we continue to regard our masses $M^2$ as functions of the \nHiggs fluctuations $\\phi$, as in Eq.~(\\ref{TaylorM}).\nTo one-loop order, these are the hierarchy conditions that must be satisfied by the\nthe spectrum of any modular-invariant string theory. \nIndeed, substituting the masses in Eq.~(\\ref{TaylorM}), these two conditions reduce to\nthe forms\n\\begin{equation}\n\\begin{cases}\n ~{\\rm Str}\\, \\beta_0 \\!\\! &\\sim~ 24\\, M_\\Lambda^4\/{\\cal M}^4 \\\\\n ~{\\rm Str}\\, \\beta_2 \\!\\! &\\sim~ 24\\, M_{\\rm EW}^2\/{\\cal M}^2 ~. \n\\end{cases}\n\\label{stringVeltman2}\n\\end{equation}\nAlthough every massive string state has a non-zero $\\beta_0$ and therefore\ncontributes to the first constraint, only those string states \nwhich couple to the Higgs field\nhave a non-zero $\\beta_2$ and thereby contribute to the second.\nOf course, given the form of Eq.~(\\ref{TaylorM}), \nthe non-zero $\\beta_i$'s for each state are still expected to \nbe $\\sim {\\cal O}(1)$, which is precisely why these constraints are so\ndifficult to satisfy. \nMoreover, as we know in the case of string models exhibiting charge lattices,\nthese $\\beta_i$-coefficients are related to the charges of the individual string states\nand therefore can be discrete in nature.\n\n\n\n Given the constraints in Eq.~(\\ref{stringVeltman2}), \n it is natural to wonder why there is no hierarchy condition \n corresponding to ${\\rm Str}\\,\\beta_1$.\n Actually, such a condition exists, although this is not normally treated as a hierarchy \n constraint. This is nothing but our stability condition \n {\\mbox{$\\partial_\\phi \\widehat \\Lambda(\\mu,\\phi)\\bigl|_{\\phi=0}=0$}} in\n Eq.~(\\ref{stabcond2}), which can be considered on the same footing as\n the other two relations in Eq.~(\\ref{effpotl}). As we have seen,\n this leads directly to the relations {\\mbox{${\\rm Str}\\,{\\mathbb{Y}}=0$}} or equivalently\n {\\mbox{$\\partial_\\phi {\\rm Str}\\,M^2\\bigl|_{\\phi=0}=0$}}, which can be considered\n alongside the relations in Eq.~(\\ref{stringVeltman}).\n This then leads to the constraint {\\mbox{${\\rm Str} \\,\\beta_1=0$}}.\n Of course, it is always possible that there exists a \n non-zero Higgs tadpole, as long as this tadpole is sufficiently small as to have\n remained unobserved ({\\it e.g.}\\\/, at colliders, or cosmologically), \n leading to string models which are not truly stable but only metastable. \n Such models would be analogous to non-supersymmetric\n string models in which the {\\it dilaton}\\\/ tadpole is non-vanishing\n but exponentially suppressed to a sufficient degree that the theory \n is essentially stable on cosmological timescales~\\cite{Abel:2015oxa}.\n In such cases involving a non-zero Higgs potential, \n we can define an associated mass scale $M_{\\rm stab}$ \n which characterizes the maximum possible Higgs instability we can tolerate\n experimentally and\/or observationally.\n Our corresponding ``hierarchy'' condition would then take the form\n\\begin{equation}\n{\\rm Str}\\, \\beta_1 ~\\lsim~ {M_{\\rm stab}}\/{{\\cal M}}~.\n\\label{stabcond2}\n\\end{equation}\n Of course, this condition differs from\n the others in that it does not describe a phenomenological constraint on a particular\n vacuum but rather helps to determine whether that vacuum even exists.\n All conditions nevertheless determine whether a given value of $\\langle \\phi\\rangle$ (in this case\n defined as {\\mbox{$\\langle \\phi\\rangle =0$}}) is viable. \n In general, such ``hierarchies'' exist for each scalar $\\phi$ in the theory.\n\nDespite their fundamentally different natures, these two types of hierarchies can actually\nbe connected to each other.\nIn the case that $\\phi$ represents the Standard-Model Higgs, \nthis connection will then allow us to relate $M_{\\rm stab}$ to $M_{\\rm EW}$.\nThe fundamental reason for this connection is that \na tadpole corresponds to a linear term in an effective potential for the Higgs.\nThis is in addition to the quadratic mass term.\nHowever, we can eliminate the linear term by completing the square, which\nof course simply shifts the corresponding Higgs VEV.~\nThe maximum size of this tadpole diagram is therefore \nalso bounded by $M_{\\rm EW}$. More precisely,\nwe find for the Standard-Model Higgs that\n\\begin{equation}\n M_{\\rm stab} ~\\sim~ 24\\, M_{\\rm EW}^3\/{\\cal M}^2~,\n\\end{equation}\nwhereupon Eq.~(\\ref{stabcond2}) takes the form\n\\begin{equation}\n{\\rm Str}\\, \\beta_1 ~\\lsim~ 24\\, M_{\\rm EW}^3\/{\\cal M}^3~.\n\\label{stabcond2alt}\n\\end{equation}\nIndeed, in this form Eq.~(\\ref{stabcond2alt}) \nmore closely resembles the relations in \nEq.~(\\ref{stringVeltman2}).\n\n\n\n \n\n\nIt is remarkable that in string theory\nthe constraints from the cosmological-constant problem\nand the gauge hierarchy problem in Eq.~(\\ref{stringVeltman2}) take such similar algebraic forms.\nIndeed in some sense $\\beta_0$ and $\\beta_2$ measure the responses of our \nindividual string states to mass (or gravity) and to \nfluctuations of the Higgs field, respectively,\nwith $\\beta_2$ related to the {\\it charges}\\\/ of these states\nwith respect to Higgs couplings.\nIt is also noteworthy that these conditions \neach resemble the so-called ``Veltman condition''~\\cite{Veltman:1980mj}\nof field theory.\nRecall that the Veltman condition for addressing the gauge hierarchy\nin an effective field theory such as the Standard Model\ncalls for cancelling the quadratic divergence of the Higgs mass\nby requiring the vanishing of the (mass)$^2$ supertrace ${\\rm Str} \\,M^2$ \nwhen summed over all light EFT states which couple to the Higgs.\nHowever, we now see that in string theory \nthe primary difference is that the supertraces ${\\rm Str}\\,M^2$ in Eq.~(\\ref{stringVeltman2}) \nare evaluated over \nthe {\\it entire}\\\/ spectrum of string states\nand not merely the light states within the EFT.~ \nThis is an important difference because the vanishing of this supertrace\nwhen restricted to the EFT generally tells us nothing\nabout its vanishing in the full theory, or vice versa.\nThese are truly independent conditions, and we see that\nstring theory requires the latter, not the former.\n \n\nOne of the virtues of modular invariance --- and indeed an indication of\nits overall power as a robust, unbroken symmetry --- is that the string naturalness\nconditions in Eqs.~(\\ref{stringVeltman}) and (\\ref{stringVeltman2}) \nnecessarily include the effects of {\\it all}\\\/ physics occurring \nat intermediate scales. This includes, for example,\nthe effects of a possible GUT phase transition.\nAs discussed earlier in this paper,\nthis is true because modular invariance is an exact symmetry governing\nnot only all of the states in the string spectrum but also their interactions.\nThus all intermediate-scale physics --- even including phase transitions ---\nmust preserve \nmodular invariance. This in turn implies that as the masses and degrees of\nfreedom within the theory evolve,\nthey all evolve together in a carefully balanced way such that modular invariance is preserved. \nThus, given that relations such as that in Eq.~(\\ref{asymplimit2}) are general and rest solely on \nmodular invariance, they too will remain intact. \nRelations such as those in Eqs.~(\\ref{stringVeltman}) and (\\ref{stringVeltman2}) \nthen remain valid.\n\n\n\n\nThus far we have reformulated the constraints associated\nwith the cosmological-constant and gauge hierarchy problems,\nproviding what may be viewed as essentially ``stringy'' versions of the traditional Veltman condition.\nHowever our results also suggest new stringy mechanisms by which such constraints might actually be satisfied ---\nmechanisms by which such hierarchies might\nactually emerge within a given theory. \nGiven the general running behavior of the Higgs mass \nin Fig.~\\ref{anatomy},\nwe observe two interesting features that may be\nrelevant for hierarchy problems.\nFirst, let us imagine that we apply our formalism for the running of the Higgs mass in the \noriginal {\\it unbroken}\\\/ phase of the theory.\nWe will then continue to obtain a result for the Higgs running \nwith the same shape as that shown in Fig.~\\ref{anatomy}, \nonly with the relevant quantities $\\Lambda$, ${\\mathbb{X}}_1$, and ${\\mathbb{X}}_2$ evaluated\nin the unbroken phase.\nConcentrating on the region with {\\mbox{$\\mu\\leq M_s$}},\nwe see that there is a relatively slow (logarithmic) running which stretches all the way from\nthe string scale $M_s$ down to the energy scales associated with the lightest massive string states,\nfollowed by a transient ``dip'' region within which the Higgs mass experiences\na sudden local minimum.\nThis therefore provides a natural scenario in which electroweak symmetry breaking\nmight be triggered at an energy scale hierarchically below the fundamental high energy scales\nin the theory.\nNote that the dip region indeed produces a {\\it minimum}\\\/ for the Higgs mass only if {\\mbox{${\\rm Str} \\,{\\mathbb{X}}_2>0$}}; \notherwise the logarithmic running changes \nsign and the Higgs mass would already be tachyonic \nat high energy scales near the string scale,\nsignifying (contrary to assumptions) that our theory was not sitting at a stable minimum in $\\phi$-space\nat high energies. (We also note that \neven though {\\mbox{${\\mathbb{X}}_2\\geq 0$}},\nthe supertrace ${\\rm Str}\\,{\\mathbb{X}}_2$ \ncan have either sign \ndepending on how these ${\\mathbb{X}}_2$-charges are distributed between\nbosonic and fermionic states.) \nHowever, with {\\mbox{${\\rm Str}\\,{\\mathbb{X}}_2>0$}},\nthis transient minimum in Fig.~\\ref{anatomy} will cause the Higgs to become tachyonic as long as\n\\begin{equation}\n \\frac{\\pi}{6}\\, {\\rm Str}\\,{\\mathbb{X}}_1 + \n \\frac{3}{10}\\, {\\rm Str}\\,{\\mathbb{X}}_2\n ~\\gsim~ \\frac{\\xi}{4\\pi^2} \\frac{\\Lambda}{{\\cal M}^4}~\n\\end{equation}\nwhere the factor of $3\/10$ represents the\napproximate value $\\approx 0.3$ parametrizing the ``dip depth'' from Fig.~\\ref{anatomy}.~\nIt is remarkable that this condition \nlinks the scale of electroweak symmetry breaking with the value of the one-loop\ncosmological constant.\nJust as with our other conditions, this condition can be also expressed as a constraint \non the values of our $\\beta_i$ coefficients:\n\\begin{equation}\n \\frac{9}{5} \\, {\\rm Str}\\,\\beta_1^2\n -4\\pi^2\\, {\\rm Str}\\,\\beta_2 \n ~\\gsim~ \\xi\\, {\\rm Str}\\,\\beta_0~.\n\\end{equation}\nThis is then our condition for triggering electroweak symmetry breaking at small scales\nhierarchically below ${\\cal M}$.\nOf course, after this breaking occurs, we would need to work in the broken phase\nwherein $\\phi$ returns to representing the Higgs fluctuations relative to the new broken-phase vacuum.\n\nThe second feature illustrated within Fig.~\\ref{anatomy} that may be relevant for the\nhierarchy problems concerns the scale-duality symmetry {\\mbox{$\\mu\\to M_s^2\/\\mu$}}.\nAs we have discussed at numerous points throughout this paper,\nthis symmetry implies an equivalence between UV physics and IR physics --- an observation\nwhich already heralds a major disruption of our understanding of the relationship between\nhigh and low energy\nscales compared with field-theoretic expectations. \nGiven that hierarchy problems not only emerge \nwithin the context of low-energy EFTs but also assume traditional field-theoretic relationships between UV and IR physics,\nit is possible to speculate that such hierarchy problems are not fundamental\nand do not survive in string theory in the manner we normally assume.\nFurthermore, we have already seen that modular invariance not only leads to this UV\/IR mixing but\nalso softens divergences so dramatically that certain otherwise-divergent amplitudes\n(such as the cosmological constant)\nare rendered finite.\nTaken together, these observations suggest that modular invariance may hold the key to an entirely new way of thinking \nabout hierarchy problems --- a point originally made in Ref.~\\cite{Dienes:2001se}\nand which we will develop further in upcoming work~\\cite{SAAKRDinprep}.\n\n\nThe results of this paper also prompt a number of additional lines of research.\nFor example, although most of our results are completely general and hold across all\nmodular-invariant string theories, much of our analysis in this paper has been restricted to one-loop order.\nIt would therefore be interesting to understand what occurs at higher loops.\nIn this connection, we note that it is often asserted in the string literature \nthat modular invariance is only a one-loop symmetry, seeming to imply that it should no longer apply at higher loops.\nHowever, this is incorrect: modular invariance is an {\\it exact}\\\/ worldsheet symmetry of (perturbative) closed strings,\nand thus holds at all orders. This symmetry is merely {\\it motivated}\\\/ \nby the need to render one-loop string amplitudes consistent with the underlying conformal invariance of the string worldsheet.\nOnce imposed, however, this symmetry affects the entire string model --- all masses and interactions, to any order.\nLikewise, one might wonder whether there are {\\it multi-loop}\\\/ versions of modular invariance \nwhich could also be imposed, similarly motivated by considerations of higher-loop amplitudes.\nHowever, it has been shown~\\cite{Kawai:1987ew}\nthat within certain closed string theories, amplitude factorization and \nphysically sensible state projections\ntogether ensure that one-loop modular invariance automatically implies multi-loop modular invariance.\nThus one-loop modular invariance is sufficient, and no additional \nsymmetries of this sort are needed.\n\nBecause modular invariance is an {\\it exact}\\\/ worldsheet symmetry,\nwe expect that certain features we have discussed \nin this paper (such as the existence of the scale-duality symmetry under {\\mbox{$\\mu\\to M_s^2\/\\mu$}})\nwill remain valid to all orders.\nWe believe that the same is true of other consequences of modular invariance,\nsuch as our supertrace relations and the ``misaligned supersymmetry''~{\\mbox{\\cite{Dienes:1994np,Dienes:1995pm,Dienes:2001se}}}\nfrom which they emerge.\n\n\nThat said, modular invariance is a symmetry of closed strings.\nFor this reason, we do not expect modular invariance to hold for Type~I strings, which contain\nboth closed-string and open-string sectors.\nHowever, within Type~I strings there are tight relations between the closed-string and open-string\nsectors, and certain remnants of modular invariance survive even into the open-string sectors.\nFor example, certain kinds of misaligned supersymmetry have been found \nto persist even within open-string sectors~\\cite{Cribiori:2020sct}.\nIt will therefore be interesting to determine the extent to which the results and techniques of this paper\nmight extend to open strings.\n\n\nThe results described in this paper have clearly covered a lot of territory, stretching \nfrom the development of new techniques for calculating Higgs masses to the development \nof modular-invariant methods \nof regulating divergences. \nWe have also tackled critical questions concerning UV\/IR mixing and the extent to which one can extract\neffective field theories from modular-invariant string theories, complete with Higgs masses and a cosmological\nconstant that run as functions of a spacetime mass scale.\nWe have demonstrated that there are unexpected relations between the Higgs mass and the one-loop cosmological\nconstant in any modular-invariant string model, and that it is possible to extract \nan entirely string-based effective potential for the Higgs. \nMoreover, as indicated in the Introduction,\nour results apply to {\\it all}\\\/ scalars in the theory --- even beyond the Standard-Model Higgs --- and apply\nwhether or not spacetime supersymmetry is present. \nAs such, we anticipate that there exist numerous areas of exploration that may be prompted by these developments.\nBut perhaps most importantly for phenomenological purposes, we believe that \nthe results of this paper can ultimately serve as the launching point for a rigorous investigation of \nthe gauge hierarchy problem in string theory.\nMuch work therefore remains to be done.\n\n\n\n\n\n\n\\begin{acknowledgments}\n\nWe are happy to thank Carlo Angelantonj, Athanasios Bouganis, and Jens Funke for insightful discussions. \nThe research activities of SAA were supported by the STFC grant \nST\/P001246\/1 and partly by a CERN Associateship and \nRoyal-Society\/CNRS International Cost Share Award IE160590.\nThe research activities of KRD were supported in part by the U.S.\\ Department of Energy\nunder Grant DE-FG02-13ER41976 \/ DE-SC0009913, and also \nby the U.S.\\ National Science Foundation through its employee IR\/D program.\nThe opinions and conclusions\nexpressed herein are those of the authors, and do not represent any funding agencies.\n\n\\end{acknowledgments}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLet $X$ be a compact K\\\"ahler manifold such that the anticanonical bundle $-K_X$ is nef, and \nlet \\holom{\\pi}{X}{T} be the Albanese map. By the work of Zhang \\cite{Zha96}, P\\v aun \\cite{Pau12}\nand the first named author \\cite{Cao13} we know that $\\pi$ is a fibration, i.e. $\\pi$ is surjective and has connected fibres.\nThe aim of this paper is to give evidence for the following:\n\n\\begin{conjecture} \\cite{DPS94} \\label{conjecturealbanese}\nLet $X$ be a compact K\\\"ahler manifold such that $-K_X$ is nef, and \nlet \\holom{\\pi}{X}{T} be the Albanese map.\nThen the fibration $\\pi$ is smooth. \nIf the general $\\pi$-fibre is simply connected, the fibration $\\pi$ is locally trivial in the analytic topology.\n\\end{conjecture}\n\nThis conjecture has been proven under the \nstronger assumption that $T_X$ is nef or $-K_X$ is hermitian \nsemipositive \\cite{CP91, DPS93, DPS94, DPS96, CDP12}, but the general case is \nvery much open: so far it is only known in the case where $q(X)=\\dim X$, i.e. the Albanese map is birational \\cite{Zha96, Fan06}. If $X$ is projective we also know that $\\pi$ is equidimensional and has reduced fibres \\cite{LTZZ10}.\nIn low dimension explicit computations based on the minimal model program (MMP) allow to say more:\n\n\\begin{theorem} \\label{theoremPS} \\cite[Thm.]{PS98} \nLet $X$ be a projective manifold such that $-K_X$ is nef, and \nlet \\holom{\\pi}{X}{T} be the Albanese map. If $\\dim X \\leq 3$, then $\\pi$ is smooth.\n\\end{theorem}\n\nWe prove Conjecture \\ref{conjecturealbanese} when the general fibre is a weak Fano manifold:\n\n\\begin{theorem} \\label{theoremmain}\nLet $X$ be a compact K\\\"ahler manifold such that $-K_X$ is nef, and \nlet \\holom{\\pi}{X}{T} be the Albanese map.\nLet $F$ be a general $\\pi$-fibre. If $-K_F$ is nef and big, then $\\pi$ is locally trivial\nin the analytic topology.\n\\end{theorem}\n\nLet us explain the strategy of proof under the stronger assumption that $-K_X$ is $\\pi$-ample:\nfor $m \\gg 0$ we have an embedding \n$$\nX \\hookrightarrow \\ensuremath{\\mathbb{P}}(\\pi_* (\\omega_X^{\\otimes -m})).\n$$\nThe main technical point is to \nshow that the direct image sheaf $\\pi_* (\\omega_X^{\\otimes -m})$ is a nef vector bundle and $(-K_X)^{\\dim X-\\dim T+1}=0$.\nIf $X$ is projective this is not very difficult, the non-algebraic case needs substantially more effort and should be\nof independent interest. \nCombining these two facts an intersection computation shows that $\\pi_* (\\omega_X^{\\otimes -m})$ is actually numerically flat, i.e.\nnef and antinef. Yet a numerically flat vector bundle is a rather special \nlocal system \\cite[Sect.3]{Sim92}, so an argument from \\cite{Cao12} \nallows to show that the equations of the fibres $X_t \\subset \\ensuremath{\\mathbb{P}}(\\pi_* (\\omega_X^{\\otimes -m})_t)$ do \nnot depend on $t \\in T$. In particular all the fibres are isomorphic, so $\\pi$ is locally trivial.\nIf $-K_X$ is only nef and $\\pi$-big, the same considerations show that the relative anticanonical fibration $X' \\rightarrow T$\nis locally trivial. We then use birational geometry to deduce that $X \\rightarrow T$ is also locally trivial.\nTheorem \\ref{theoremmain} immediately implies:\n\n\\begin{corollary} \\label{corollarymain}\nLet $X$ be a compact K\\\"ahler manifold such that $-K_X$ is nef. \nConjecture \\ref{conjecturealbanese} holds if $q(X) = \\dim X-1$. \n\\end{corollary}\n\nThis also settles the problem in low dimension.\n\n\\begin{corollary} \\label{corollarykaehler}\nLet $X$ be a compact K\\\"ahler manifold such that $-K_X$ is nef. \nConjecture \\ref{conjecturealbanese} holds if $\\dim X \\leq 3$.\n\\end{corollary}\n\nIn the second part of the paper we turn our attention to the case where the positivity of $-K_X$ is not strict, even\nalong the general $\\pi$-fibre. We use the MMP to prove Conjecture \\ref{conjecturealbanese} for fibres of low dimension.\n\n\\begin{theorem} \\label{theoremmaintwo}\nLet $X$ be a projective manifold such that $-K_X$ is nef.\nConjecture \\ref{conjecturealbanese} holds if $q(X) = \\dim X-2$. \n\\end{theorem}\n\nThe basic idea of the proof is very simple: find a Mori contraction $\\mu: X \\rightarrow Y$ onto \na projective manifold $Y \\rightarrow T$ such that $-K_{Y}$ is nef and relatively big.\nThen $-K_X-\\mu^* K_Y$ is nef and relatively big, using the birational morphism\n$$\nX \\rightarrow X' \\subset \\ensuremath{\\mathbb{P}}(\\pi_* (\\omega_X^{\\otimes -m} \\otimes \\mu^* \\omega_Y^{\\otimes -m})).\n$$\nwe can prove as in Theorem \\ref{theoremmain} that $X \\rightarrow T$ is locally trivial.\nUnfortunately it is a priori not clear that such a contraction $X \\rightarrow Y$ exists.\nIn fact the second named author constructed an example of a \n(rationally connected) projective threefold $M$ such that $-K_M$ is nef and not big\nand $M=\\mbox{Bl}_B M'$ with $B$ a smooth rational curve such that $-K_{M'} \\cdot B<0$ \\cite[Ex.4.8]{a8}. \nThis problem already appeared in the work of Peternell and Serrano and we follow\nthe same strategy to overcome this difficulty: let $X'\/T$ be a Mori fibre space birational to $X\/T$,\nthen try to prove that $-K_{X'}$ is nef. Once this property is established one can\ndescribe precisely all the steps of the MMP $X \\dashrightarrow X'$.\nThe contribution of this paper is to introduce a new method to establish this kind of statement:\nour proof is based on the idea that if we restrict the MMP to some (pluri-)anticanonical divisor $D' \\subset X'$,\nthe numerical dimension of $-K_{X'}|_{D'}$ is zero or one. This observation quickly leads to strong\nrestrictions on the MMP in a neighbourhood of $D'$, cf. Lemma \\ref{lemmanu2}.\nThe main point is thus to show the existence of global sections of $-m K_X$ for some $m \\in \\ensuremath{\\mathbb{N}}$: \nthis can be done on threefolds, but is completely open in higher dimension. \n \n\n\n{\\bf Acknowledgements.} We thank I.Biswas, S.Boucksom, T.Dinh, V. Lazi\\'c, W.Ou, T. Sano and C.Simpson for helpful communications.\nWe thank J-P. Demailly for helpful discussions and numerous suggestions.\nA. H\\\"oring was partially supported by the A.N.R. project CLASS\\footnote{ANR-10-JCJC-0111}.\n\n\n\\begin{center}\n{\\bf\nNotation and terminology\n}\n\\end{center}\n\nFor general definitions - at least in the algebraic context - we refer to Hartshorne's book \\cite{Har77}.\nWe will frequently use standard terminology and results \nof the minimal model program (MMP) as explained in \\cite{KM98} or \\cite{Deb01}.\nManifolds and varieties will always be supposed to be irreducible.\nA fibration is a proper surjective map with connected fibres \\holom{\\varphi}{X}{Y} between normal varieties.\n\nLet us recall the various positivity concepts that will be used in this paper.\n\n\\begin{definition} \\label{definitionnef} \\cite{Dem12}\nLet $(X, \\omega_{X} )$ be a compact K\\\"ahler manifold, and let $\\alpha \\in H^{1,1}(X) \\cap H^2(X, \\ensuremath{\\mathbb{R}})$ be a real cohomology class \nof type $(1,1)$. We say that $\\alpha$ is nef if for every $\\epsilon> 0$, there is a smooth $(1,1)$-form $\\alpha_{\\epsilon}$\nin the same class of $\\alpha$ such that $\\alpha_{\\epsilon}\\geq -\\epsilon\\omega_{X}$.\n\nWe say that $\\alpha$ is pseudoeffective if there exists a $(1, 1)$-current $T\\geq 0$ in the same class of $\\alpha$.\nWe say that $\\alpha$ is big if there exists a $\\epsilon> 0$ such that $\\alpha-\\epsilon \\omega_{X}$ is pseudoeffective.\n\\end{definition}\n\n\\begin{definition} \\label{definitionrelativebig}\nLet $\\alpha$ be a nef class on a compact K\\\"ahler manifold $X$, and let $\\pi: X\\rightarrow T$ be a fibration.\nWe say that $\\alpha$ is $\\pi$-big if for a general fibre $F$, the restriction $\\alpha|_F$ is big.\n\\end{definition}\n\n\\begin{definition} \\cite[Def 6.20]{Dem12} \\label{definitionnumericaldimension}\nLet $X$ be a compact K\\\"ahler manifold, and let $\\alpha \\in H^{1,1}(X) \\cap H^2(X, \\ensuremath{\\mathbb{R}})$ be a real cohomology class \nof type $(1,1)$. Suppose that $\\alpha$ is nef.\nWe define the numerical dimension of $\\alpha$ by \n$$\n\\nd (\\alpha) :=\n\\max \\{k \\in \\ensuremath{\\mathbb{N}} \\ | \\ \\alpha^{k}\\neq 0 \\mbox{ in } H^{2k}(X,\\mathbb{R})\\}.\n$$\n\\end{definition}\n\n\\begin{remark} \\label{remarknumericaldimension} In the situation above, set $m=\\nd (\\alpha)$.\nBy \\cite[Prop 6.21]{Dem12} the cohomology class $\\alpha^{m}$ can be represented \nby a non-zero closed positive $(m,m)$-current $T$.\nTherefore \n$\\int_X \\alpha^{m}\\wedge\\omega_{X}^{\\dim X - m}\\neq 0$ for any K\\\"ahler class $\\omega_{X}$.\n\\end{remark}\n\n\n\\begin{definition} \\label{definitionnefcodimone}\nLet $M$ be a projective variety, and let $L$ be a $\\ensuremath{\\mathbb{Q}}$-Cartier divisor on $M$. We say that $L$\nis nef in codimension one if $L$ is pseudoeffective and for every prime divisor $D \\subset M$,\nthe restriction $L|_D$ is pseudoeffective.\n\\end{definition}\n\n\\begin{remark} \\label{remarknefcodimone}\nIf $M$ is a normal projective variety and $L$ a $\\ensuremath{\\mathbb{Q}}$-Cartier divisor which is nef \nin codimension one, then $L^2$ is a pseudoeffective cycle, i.e. a limit of effective cycles of codimension two.\nIndeed if \n$L= \\sum \\lambda_j D_j +N$ is the divisorial Zariski decomposition \\cite{Bou04, Nak04}, we have\n$$\nL^2 = \\sum \\lambda_j L|_{D_j} + L \\cdot N.\n$$\nBy hypothesis the restriction $L|_{D_j}$ is pseudoeffective, so a limit of effective divisors on $D_j$.\nThe class $N$ is modified nef in the sense of \\cite{Bou04}, so its intersection with any pseudoeffective divisor\ngives a pseudoeffective cycle. \n\\end{remark}\n\n\n\\begin{definition} \\cite{Miy87} \\label{definitiongenericallynef}\nLet $X$ be a normal, projective variety of dimension $n$, and let \n$\\sF$ be a torsion free coherent sheaf on $X$. We say that $\\sF$ is generically nef with\nrespect to a polarisation $A$ on $X$ \nif $\\sF|_C$ is nef where\n\\[\nC := D_1 \\cap \\ldots \\cap D_{n-1}\n\\]\nwith $D_j \\in | m_j A |$ general and $m_j \\gg 0$. \n\\end{definition}\n\n\n\\section{Numerical dimension} \\label{sectionnumericaldimension}\n\nIn this section we give an upper bound for the numerical dimension of $-K_X$:\n\n\\begin{proposition} \\label{propositionnumericaldimension}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$ such that $-K_X$ is nef, and \nlet \\holom{\\pi}{X}{T} be the Albanese map. Set $r:=\\dim T$.\nIf $-K_X$ is $\\pi$-big, we have $\\nd (-K_X)=n-r$.\n\\end{proposition}\n\n\\begin{remark} \\label{remarkprojective} If the torus $T$ is projective, this statement is well-known: in this case the manifold $X$ is \nalso projective, so\nif $\\nd(-K_X)>n-r$ we can apply Kawamata-Viehweg vanishing \\cite[6.13]{Dem00} to see that \n\\begin{equation}\\label{projkawaview}\nH^{r}(X, \\sO_X) = H^{r}(X, K_X+(-K_X))=0.\n\\end{equation}\nThe pull-back of a non-zero holomorphic $r$-form from $T$ gives an immediate contradiction. \nWe will also use the following special case of \\cite[Thm.5.1]{AD11}:\n\\end{remark}\n\n\\begin{lemma} \\label{lemmanumericaldimensionprojective}\nLet $X$ be a normal projective variety and $\\Delta$ a boundary divisor on $X$ such that\nthe pair $(X, \\Delta)$ is klt. Let \\holom{\\varphi}{X}{C} be a fibration onto a smooth curve such \nthat $-(K_{X\/C}+\\Delta)$ is nef and $\\varphi$-big. \nThen we have\n$$\n(K_{X\/C}+\\Delta)^{\\dim X} = 0.\n$$\n\\end{lemma}\n\\vspace{5pt}\n\nTo prove Proposition \\ref{propositionnumericaldimension} for arbitrary K\\\"ahler manifolds,\nwe first prove that if $\\nd (-K_X)\\geq n-r+1$, then $\\pi_* ((-K_X)^{n-r+1})$ is nontrivial .\nMore precisely, we have\n\n\\begin{lemma} \\label{lemmanumericaldimension}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$, and let\n$\\pi: X\\rightarrow T$ be a surjective morphism onto a compact K\\\"ahler manifold $(T, \\omega_T)$ of dimension $r$.\nLet $L$ be a line bundle on $X$ that if nef and $\\pi$-big.\nIf $\\nd(L)\\geq n-r+1$, we have\n$$\n\\int_{X}L^{n-r+1}\\wedge(\\pi^{*}\\omega_{T})^{r-1} > 0.\n$$\n\\end{lemma}\n\n\\begin{proof}\nWe suppose that $\\nd (L)=n-r+k$ for some $k \\in \\ensuremath{\\mathbb{N}}^*$.\nSince $L$ is nef and $\\pi$-big, the class \n\\begin{equation} \\label{equationone}\n\\alpha=L+C\\cdot\\pi^{*}(\\omega_{T})\n\\end{equation}\nis a nef class for any fixed constant $C>0$, and\n$\\int_{X}\\alpha^{n} > 0$.\nThanks to \\cite[Thm. 0.5]{DP04},\nthere exists $\\epsilon> 0$, such that $\\alpha-\\epsilon\\omega_{X}$ is a pseudoeffective class.\nCombining this with the fact that $L$ is nef,\nwe have\n\\begin{equation}\\label{degrelast}\n\\int_{X} L^{n-r+k}\\wedge \\alpha^{r-k}\n\\geq \\epsilon\\int_{X} L^{n-r+k}\\wedge \\alpha^{r-k-1}\\wedge\\omega_{X}\n\\end{equation}\n$$\n\\geq \\epsilon^2\\int_{X} L^{n-r+k}\\wedge \\alpha^{r-k-2}\\wedge\\omega^{2}_{X}\\geq\n\\cdots\\geq \\epsilon^{r-k}\\int_{X}L^{n-r+k}\\wedge\\omega_{X}^{r-k}> 0,$$\nwhere the last inequality comes from Remark \\ref{remarknumericaldimension}.\nBy the definition of numerical dimension and $\\eqref{equationone}$, we have\n\\begin{equation}\\label{degrefirst}\nC^{n-k}\\cdot\\int_{X} L^{n-r+k}\\wedge (\\pi^{*}\\omega_{T})^{r-k} = \\int_{X} L^{n-r+k}\\wedge \\alpha^{r-k}.\n\\end{equation}\nNow \\eqref{degrelast} and \\eqref{degrefirst} imply that\n\\begin{equation} \\label{equationtwo}\n\\int_{X} L^{n-r+k}\\wedge (\\pi^{*}\\omega_{T})^{r-k} > 0.\n\\end{equation}\nOn the other hand, since $L$ is $\\pi$-big, we have\n\\begin{equation}\\label{equationthree}\n\\int_{X} L^{n-r}\\wedge (\\pi^{*}\\omega_{T})^{r}> 0.\n\\end{equation}\nUsing the Hovanskii-Teissier inequality in the K\\\"{a}hler case (cf. Appendix \\ref{appendixinequality}),\nthe inequalities $\\eqref{equationtwo}$ and $\\eqref{equationthree}$ imply\n$\\int_{X}L^{n-r+1}\\wedge(\\pi^{*}\\omega_{T})^{r-1}> 0$.\n\\end{proof}\n\nWe recall a vanishing theorem proved in \\cite[Prop. 2.4]{Cao12}\n\n\\begin{lemma}\\label{keyvanishing1}\nLet $L$ be a line bundle on a compact K\\\"{a}hler manifold $(X,\\omega)$ of dimension $n$, and \nlet $\\varphi$ be a metric on $L$ with analytic singularities. \nLet \n$ \\lambda_{1}(x)\\leq \\lambda_{2}(x)\\leq\\cdots\\leq\\lambda_{n}(x)$\nbe the eigenvalues of $\\frac{i}{2\\pi}\\Theta_{\\varphi}(L)$ with respect to $\\omega$.\nIf\n\\begin{equation}\\label{equation1}\n\\sum_{i=1}^{p} \\lambda_{i}(x)\\geq c\n\\end{equation}\nfor some constant $c> 0$ independent of $x \\in X$,\nthen\n$$H^{q}(X, K_{X}\\otimes L\\otimes \\mathcal{I}(\\varphi))=0 \n\\qquad \\forall \\ q\\geq p.$$\n\\end{lemma}\n\nThe following vanishing property plays an important role in the proof of Proposition \\ref{propositionnumericaldimension}, \nAlthough it was essentially proved in \\cite[Prop.5.3]{Cao12}, we give the proof since\nthe situation here is a little bit more general.\n\n\\begin{proposition}\\label{lemmavanishing}\nLet $(X, \\omega_{X})$ be a compact K\\\"ahler manifold of dimension $n$, and\n$L$ be a nef line bundle on $X$.\nSuppose that the following holds:\n\\begin{enumerate}[(i)]\n\\item $X$ admits\na two steps tower fibration \n$$\\begin{CD} X @>\\pi >> T @> \\pi_{1}>> S \\end{CD}$$ \nwhere $\\pi$ is a fibration onto a compact K\\\"ahler manifold $(T, \\omega_T)$ of dimension $r$,\nand $\\pi_{1}$ is a smooth fibration onto a smooth curve $S$.\n\\item $L$ is $\\pi$-big and satisfies\n$$\\pi_{*}(c_{1}(L)^{n-r+1})=\\pi_{1}^{*}(\\omega_{S})$$ \nfor a K\\\"ahler metric $\\omega_{S}$ on $S$.\n\\end{enumerate}\nThen we have\n$$ \nH^{q}(X,K_{X}\\otimes L)=0 \\qquad \\forall \\ q\\geq r.\n$$\n\\end{proposition}\n\n\\begin{proof}\nBy \\cite[Lemma 5.1]{Cao12}\\footnote{Note that the proof of \\cite[Lemma 5.1]{Cao12} works well in our case.} the class\n$L-d\\cdot \\pi^* \\pi_{1}^* \\omega_{S}$\nis pseudoeffective for some $d> 0$.\nTherefore there exists a singular metric $h_{1}$ on $L$\nsuch that \n$$i\\Theta_{h_{1}}(L)\\geq d\\cdot\\pi^{*}\\pi_{1}^*\\omega_{S}.$$\nSince $c_{1}(L)+ \\pi^{*} \\omega_{T}$ is nef and \n$\\int_{X}(c_{1}(L)+\\pi^{*} \\omega_{T})^{n}> 0$,\n\\cite[Thm. 0.5]{DP04}\nimplies the existence of a singular metric $h_{2}$ on $L$\nsuch that\n$$i\\Theta_{h_{2}}(L)\\geq c\\cdot \\omega_{X}- \\pi^{*} \\omega_{T}$$\nin the sense of currents for some constant $c > 0$.\nThanks to a standard regularization theorem \\cite{Dem92},\nwe can suppose moreover that $h_{1}, h_{2}$ have analytic singularities.\nSince $L$ is nef we know that for any $\\epsilon> 0$, there exists a smooth metric $h_{\\epsilon}$ on $L$\nsuch that $i\\Theta_{h_{\\epsilon}}(L)\\geq -\\epsilon\\omega_{X}$.\nNow we define a new metric $h$ on $L$:\n$$h=\\epsilon_{1} h_{1}+\\epsilon_{2} h_{2}+ (1-\\epsilon_{1}-\\epsilon_{2})h_{\\epsilon}$$\nfor some $1\\gg \\epsilon_{1}\\gg \\epsilon_{2} \\gg \\epsilon> 0$.\nBy construction, we have\n\\begin{equation}\\label{equation2}\ni\\Theta_{h}(L)=\n\\epsilon_{1}i\\Theta_{h_{1}}(L)+\\epsilon_{2}i\\Theta_{h_{2}}(L)+(1-\\epsilon_{1}-\\epsilon_{2})i\\Theta_{h_{\\epsilon}}(L)\n\\end{equation}\n$$\n\\geq d\\cdot\\epsilon_{1} \\pi^{*}(\\omega_{S})-\\epsilon_{2}\\pi^{*}(\\omega_{T})+(c \\cdot\\epsilon_{2}-\\epsilon)\\omega_{X}.\n$$\nSet\n$\\omega_{\\tau}=\\tau\\cdot\\omega_{X}+\\pi^{*}(\\omega_{T})$ for some $\\tau> 0$.\n\nWe now check that $( i\\Theta_{h}(L), \\omega_{\\tau} )$ satisfies the condition \\eqref{equation1} in Lemma \\ref{keyvanishing1} \nfor $p=r$ when $\\tau$ is small enough (i.e., we consider the eigenvalues of $i\\Theta_{h}(L)$ with respect to $\\omega_{\\tau}$,\nwhere $\\tau\\ll \\epsilon$).\nLet $x\\in X$ and let $V$ be a $r$ dimensional subspace of $(T_X)_x$.\nBy an elementary estimate, we \nhave\\footnote{In fact, since $\\pi_1$ is a submersion, $\\omega_T$ decomposes the tangent bundle of $T$ as $T_{T\/S}\\oplus \\pi_1^* (T_S)$ \nin the sense of $C^{\\infty}$. Observing that $\\pi_1$ is smooth and $\\epsilon_1\\gg \\epsilon_2$,\nwe have\n\\begin{equation}\\label{addremark}\nd\\cdot\\epsilon_{1} \\pi_1^{*}(\\omega_{S})(t,t)-\\epsilon_{2}\\omega_{T}(t, t)\\geq \\frac{d\\cdot\\epsilon_{1}}{2}\\omega_{T}(t,t)\n\\qquad\\text{for any }t\\in \\pi_1^* (T_S).\n\\end{equation}\nSince $\\dim V=r$, there exists an non zero element $v\\in V$ such that $\\pi_* (v)\\in \\pi_1^* (T_S)$.\nBy \\eqref{equation2} and \\eqref{addremark}, we obtain\n\\begin{equation}\n\\frac{ i\\Theta_h (L) (v, v )}{\\langle v, v\\rangle_{\\omega_{\\tau}}}\\geq \n\\frac{(c\\epsilon_2 -\\epsilon )\\langle v, v\\rangle_{\\omega_X}+ \\frac{d\\cdot\\epsilon_1}{2}\\langle \\pi_* (v), \\pi_* (v)\\rangle_{\\omega_T}}{\\tau\\langle v, v\\rangle_{\\omega_X}+\\langle \\pi_* (v), \\pi_* (v)\\rangle_{\\omega_T}}\n\\geq \\min\\{ \\frac{c \\epsilon_2 -\\epsilon}{\\tau}, \\frac{d\\cdot \\epsilon_1}{2}\\}.\n\\end{equation}\n}\n$$\\sup\\limits_{v\\in V}\\frac{ i\\Theta_h (L) (v, v )}{\\langle v, v\\rangle_{\\omega_{\\tau}}}\\geq \n \\min \\{ \\frac{c \\epsilon_2 -\\epsilon}{\\tau}, \\frac{d\\cdot \\epsilon_1}{2}\\}\\gg (r-1)\\cdot \\epsilon_2 $$\nby the choice of $\\tau , \\epsilon_1, \\epsilon_2$.\nMoreover, since $\\epsilon_{2}\\ll \\epsilon_{1}$, \n\\eqref{equation2} implies that $i\\Theta_{h}(L)$ has at most $(r-1)$-negative eigenvectors\nand their eigenvalues with respect to $\\omega_{\\tau}$ are larger than $ - \\epsilon_{2}$.\nBy the minimax principle, \n$( i\\Theta_{h}(L), \\omega_{\\tau} )$ satisfies the condition \\eqref{equation1} in Lemma \\ref{keyvanishing1}.\nThus we have\n$$H^{q}(X,K_{X}\\otimes L\\otimes\\mathcal{I} (h))=0\n\\qquad \\forall \\ q \\geq r.$$\nSince $\\epsilon_{1}, \\epsilon_{2}$ are small enough, \nwe have $\\mathcal{I}(h)=\\mathcal{O}_{X}$.\nTherefore we get\n$$\nH^{q}(X,K_{X}\\otimes L)=0\\qquad \\forall \\ q\\geq r.\n$$\n\\end{proof}\n\nTo prove the main theorem in this section, \nwe need another vanishing lemma. \nThe idea of the proof is essentially the same as Proposition \\ref{lemmavanishing}.\n\n\\begin{lemma}\\label{lemmavanishing3}\nLet $(X, \\omega_{X})$ be a compact K\\\"ahler manifold of dimension $n$\nwhich admits a fibration $\\pi: X\\rightarrow T$ onto a compact K\\\"ahler manifold $(T, \\omega_T)$ of dimension $r$.\nLet $L$ be a line bundle on $X$ that is nef and $\\pi$-big, and let $A$ be a line bundle on $T$\nthat is semiample. If $\\nd (A)=s$, then we have \n$$\nH^q (X, K_X \\otimes L \\otimes \\pi^* (A) )=0 \\qquad \\forall \\ q\\geq r-s+1.\n$$\n\\end{lemma}\n\n\\begin{proof}\nSince $A$ is semiample of numerical dimension $s$,\nthere exists a smooth metric $h_A$ on $A$ such that $i \\Theta_{h_A} (A)$ is semipositive\nand has $s$ strictly positive eigenvalues which admit a positive lower bound that does not depend on \nthe point $t \\in T$.\nBy the proof of Proposition \\ref{lemmavanishing}, \nthere exists a metric $h_{2}$ on $L$ with analytic singularities\nsuch that\n$$i\\Theta_{h_{2}}(L)\\geq c\\cdot \\omega_{X}- \\pi^{*} \\omega_{T}$$\nin the sense of currents for some constant $c > 0$.\nNote that $L$ is nef. Then for any $\\epsilon> 0$, there exists a smooth metric $h_{\\epsilon}$ on $L$\nsuch that $i\\Theta_{h_{\\epsilon}}(L)\\geq -\\epsilon\\omega_{X}$.\nNow we define a new metric $h$ on $L$:\n$$h=\\epsilon_{2} h_{2}+ (1-\\epsilon_{2})h_{\\epsilon}$$\nfor some $\\epsilon_{2}\\ll 1$ and $\\epsilon\\ll c\\cdot \\epsilon_2$.\nBy construction, we have\n\\begin{equation}\\label{equationadd2}\ni\\Theta_{h\\cdot h_A}(L+\\pi^* (A) )=\n\\epsilon_{2}i\\Theta_{h_{2}}(L)+(1-\\epsilon_{2})i\\Theta_{h_{\\epsilon}}(L)+\\pi^* ( i\\Theta_{h_A}(A) )\n\\end{equation}\n$$\\geq -\\epsilon_{2}\\pi^{*}(\\omega_{T})+(c \\cdot\\epsilon_{2}-\\epsilon)\\omega_{X} + \\pi^* (i\\Theta_{h_A}(A) )\n= (c \\cdot\\epsilon_{2}-\\epsilon)\\omega_{X} + \\pi^* (i\\Theta_{h_A}(A)-\\epsilon_{2}\\pi^{*}\\omega_{T}) .$$\nSince $i\\Theta_{h_A}(A)$ is fixed, we can let \n$\\epsilon_2$ small enough with respect to the smallest strictly positive eigenvalues of $i\\Theta_{h_A}(A)$.\nSet $\\omega_{\\tau}=\\tau\\omega_X+\\pi^* (\\omega_T)$ for $\\tau> 0$.\nSince the semipositive $(1,1)$-form $i\\Theta_{h_A}(A)$ contains $s$ strictly positive directions,\nby the same argument as in Proposition \\ref{lemmavanishing}, we know that the pair\n$$( i\\Theta_{h\\cdot h_A}(L \\otimes A) , \\omega_{\\tau})$$\nsatisfies the condition \\eqref{equation1} in Lemma \\ref{keyvanishing1} \nfor $p=r-s+1$ when $\\tau$ is small enough.\nUsing Lemma \\ref{keyvanishing1}, we obtain that \n$$H^q (X, K_X\\otimes L\\otimes A\\otimes \\mathcal{I}(h\\cdot h_A))=0 \\qquad\\text{for }q\\geq n-s+1 .$$\nSince $\\epsilon_2\\ll 1$, we have $\\mathcal{I}(h\\cdot h_A)=\\mathcal{O}_X$.\n\\end{proof}\n\n\n\nWe can now prove the main theorem in this section:\n\n\\begin{theorem}\\label{KVvanishing}\nLet $X$ be a compact K\\\"{a}hler manifold of dimension $n$. \nSuppose that there exists a fibration \n$\\pi: X\\rightarrow T$ onto a torus of dimension $r$.\nLet $L$ be a line bundle on $X$ that is nef and $\\pi$-big.\nIf $\\nd (L) \\geq n-r+1$, then we have\n$$\nH^{q}(X, K_{X} \\otimes L)=0 \\qquad \\forall \\ q \\geq r.\n$$\n\\end{theorem}\n\n\\begin{remark}\nBy the same argument as in Proposition \\ref{lemmavanishing}, we can easily prove that \nif $\\nd (L) =n-r$, then \n$$\nH^{q}(X, K_{X} \\otimes L)=0 \\qquad \\forall \\ q> r.\n$$\n\\end{remark}\n\n\n\\begin{proof}[Proof of Theorem \\ref{KVvanishing}]\nSince $\\nd (L)\\geq n-r+1$, \nLemma \\ref{lemmanumericaldimension} implies that \n\\begin{equation} \\label{eqna}\n\\int_{T}\\pi_{*}(c_{1}(L)^{n-r+1})\\wedge\\omega_{T}^{r-1}> 0\n\\end{equation}\nfor any K\\\"ahler class $\\omega_{T}$.\nUsing the assumption that $T$ is a torus, \nwe can represent the cohomology class $\\pi_{*}(c_{1}(L)^{n-r+1})$ by\na constant $(1,1)$-form \n$\\sum_{i=1}^{r}\\lambda_{i}d z_{i}\\wedge d\\overline{z}_{i}$\non $T$.\nSince \\eqref{eqna} is valid for any K\\\"ahler class $\\omega_{T}$, an elementary computation shows that \n$\\lambda_{i}\\geq 0$ for any $i$. Thus \n$\\pi_{*}(c_{1}(L)^{n-r+1})$ is a semipositive \n(non trivial) class in $H^{1,1} (T) \\cap H^2(T, \\mathbb{Q})$.\nUsing \\cite[Prop. 2.2]{Cao12}, we get a smooth fibration\n$\\varphi: T \\rightarrow S$ \nwhere $S$ is an abelian variety of dimension $s$, and\n$$\\pi_{*}(c_{1}(-K_X)^{n-r+1})=\\lambda \\ \\varphi^* A$$\nfor some $\\lambda>0$ and a very ample divisor $A$ on $S$.\n\nFor every $p \\in \\{ 0, \\ldots, s-1 \\}$ let $S_{p}$ be a complete intersection of $p$ general divisors in \nthe linear system $|A|$, and set $X_p:=\\fibre{(\\varphi \\circ \\pi)}{S_p}$ and $T_p:=\\fibre{\\varphi}{S_p}$.\nThen we get a tower of fibrations\n$$\\begin{CD}\n X_{p} @>\\pi|_{X_p}>> T_{p} @>\\varphi|_{T_p}>> S_{p}\n \\end{CD}\n$$\nand $X_{p}$ is smooth by Bertini's theorem.\nMoreover, we have also the equality\n$$\n(\\pi|_{X_p})_{*}(c_{1}(L)^{n-r+1})=\\lambda\\cdot (\\varphi|_{T_p})^{*} A|_{S_p}. \n$$\nNote that $(\\varphi|_{T_p})^{*} A|_{S_p}$ is semiample of numerical dimension $\\dim S_p$, so\nwe have\n\\begin{equation}\\label{intervanishing}\nH^q (X_p, K_{X_p} \\otimes (L \\otimes \\pi^* \\varphi^* A)|_{X_p})=0 \\qquad \\forall \\ q\\geq \\dim T- \\dim S +1.\n\\end{equation}\nby Lemma \\ref{lemmavanishing3}.\nUsing \\eqref{intervanishing} and the exact sequence \n$$\n0 \\rightarrow K_{X_p} \\otimes L|_{X_{p}} \\rightarrow K_{X_p} \\otimes (L \\otimes \\pi^* \\varphi^* A)|_{X_p} \\rightarrow K_{X_{p+1}} \\otimes L|_{X_{p+1}} \\rightarrow 0,\n$$\nan easy induction shows that\n\\begin{equation}\\label{surjective}\nH^{q-(s-1)}(X_{s-1}, K_{X_{s-1}} \\otimes L|_{X_{s-1}}) \\twoheadrightarrow\nH^{q}(X, K_X \\otimes L) \\qquad \\forall \\ q \\geq r. \n\\end{equation}\nApplying Proposition \\ref{lemmavanishing} to $X_{s -1}$ and the line bundle $L|_{X_{s-1}}$, \nwe get\n$$\nH^{q}(X_{s -1}, K_{X_{s -1}} \\otimes L|_{X_{s-1}})=0 \\qquad \\forall \\ q \\geq \\dim T_{s -1}.\n$$\nSince $\\dim T_{s-1} = r-(s-1)$ we conclude by \\eqref{surjective}.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition \\ref{propositionnumericaldimension}]\nIf $\\nd (-K_X)\\geq n-r+1$, by taking $L=-K_X$ in Theorem \\ref{KVvanishing}, \nwe obtain $H^{r}(X, \\sO_X)=0$. \nThe pull-back of a non-zero holomorphic $r$-form from $T$ gives an contradiction.\n\\end{proof}\n\n\n\\section{Positivity of direct image sheaves} \\label{sectionpositivity}\n\nIn this section we will prove that, under the conditions of Theorem \\ref{theoremmain}, the direct image sheaves $\\pi_* (\\omega_X^{\\otimes -m})$ are numerically flat vector bundles for $m \\gg 0$. If the torus $T$ is projective this can be done by proving\nthat $\\pi_* (\\omega_X^{\\otimes -m})$ is nef and numerically trivial on a general complete intersection curve. If $T$ is non-algebraic such a curve\ndoes not exist, so we have to refine the construction. In Subsection \\ref{subsectionsemistable} we introduce the tools necessary\nto deal with the non-algebraic case, Subsection \\ref{subsectionanticanonical} contains the core of our proof, the direct image argument.\n\n\\subsection{Semistable filtration on compact K\\\"ahler manifold} \\label{subsectionsemistable}\n\nLet us recall the following terminology:\n\n\\begin{definition} \\cite[Defn.1.5.1]{HL97} \\label{definitionjordanhoelder}\nLet $(X, \\omega_X)$ be a compact K\\\"ahler manifold, \nand let $\\sG$ be a reflexive sheaf that is semistable with respect to $\\omega_X$.\nA Jordan-H\\\"older filtration is a filtration\n$$\n0 = \\sG_0 \\subset \\sG_1 \\subset \\ldots \\subset \\sG_l = \\sG\n$$ \nsuch that the graded pieces $\\sG_{i}\/\\sG_{i-1}$ are stable for all $i \\in \\{ 1, \\ldots, l\\}$.\n\\end{definition}\n\nEvery semistable sheaf admits a Jordan-H\\\"older filtration \\cite[Prop.1.5.2]{HL97}. Moreover given a torsion-free $E$\nwith Harder-Narasimhan filtration\n$$\n0 = \\sE_0 \\subset \\sE_1 \\subset \\ldots \\subset \\sE_m = E\n$$\nwe can use the Jordan-H\\\"older filtration of every graded piece $\\sE_{j}\/\\sE_{j-1}$ to obtain a refined filtration\n\\begin{equation} \\label{stablefiltration}\n0 = \\sF_0 \\subset \\mathcal{F}_{1}\\subset \\mathcal{F}_{2} \\subset \\ldots \\subset \\mathcal{F}_{k}=E\n\\end{equation}\nsuch that the graded pieces $\\sF_{i}\/\\sF_{i-1}$ are stable for all $i \\in \\{ 1, \\ldots, k \\}$.\nWe call $\\sF_\\bullet$ the {\\em stable filtration} of $E$ with respect to $\\omega$.\n\n\\begin{lemma} \\label{lemmafiltration}\nLet $(X, \\omega_X)$ be a compact K\\\"ahler manifold of dimension $n$, \nand let $\\pi: X\\rightarrow Y$ be a smooth fibration onto a curve $Y$.\nLet $E$ be a nef vector bundle on $X$. Suppose that\n$$\nc_{1}(E)= M\\cdot\\pi^{*}\\omega_{Y}\n$$ \nfor some constant $M$ and $\\omega_Y$ a K\\\"ahler form on $Y$.\nLet $\\sF_\\bullet$ be the stable filtration \\eqref{stablefiltration} of $E$\nwith respect to $\\pi^{*}\\omega_{Y}+\\epsilon\\omega_{X}$ \nfor some $0<\\epsilon \\ll 1$. \nThen \n$$ \nc_{1}(\\mathcal{F}_{1})=a_{1}\\cdot\\pi^{*}(\\omega_{Y}) \n$$\nfor some $a_{1}\\geq 0$.\n\\end{lemma}\n\n\\begin{proof}\nIf $E$ is stable the statement is obvious, so suppose that $k \\geq 2$.\nFor $y \\in Y$ an arbitrary point, we denote by $X_y$ the fibre over $y$.\nSince $c_{1}(E)= \\lambda \\cdot\\pi^{*}(\\omega_{Y})$,\nwe have $c_{1}(E|_{X_y})=0$.\nThen $E|_{X_y}$ is numerically flat, and by the proof of \\cite[Thm 1.18]{DPS94},\nfor any reflexive subsheaf $\\mathcal{F}\\subset E$, we have\n$$c_{1}(\\mathcal{F}|_{X_y})\\wedge (\\omega_{X}|_{X_y})^{n-2}\\leq 0 .$$\nTherefore we get\n\\begin{equation} \\label{equationfourminus}\nc_{1}(\\mathcal{F})\\wedge \\pi^{*} (\\omega_{Y}) \\wedge\\omega_{X}^{n-2}\\leq 0 \\qquad \\forall \\ \\mathcal{F}\\subset E.\n\\end{equation}\nArguing as \\cite[Lemma 1.1]{Cao13}, we see that\n\\begin{equation} \\label{equationfour}\n\\sup\\{ c_{1}(\\mathcal{F})\\wedge \\pi^* (\\omega_{Y}) \\wedge(\\omega_{X})^{n-2} \n\\mid \\mathcal{F}\\subset E\\text{ and }c_{1}(\\mathcal{F})\\wedge \\pi^* (\\omega_{Y}) \\wedge(\\omega_{X})^{n-2}< 0\\} < 0.\n\\end{equation}\nWe claim that \n\\begin{equation} \\label{equationfive}\nc_{1}(\\mathcal{F}_{1})\\wedge\\pi^{*}(\\omega_{Y})\\wedge \\omega_{X}^{n-2}=0 \\qquad\\text{and}\\qquad\nc_{1}(\\mathcal{F}_{1})\\wedge(\\omega_{X})^{n-1}\\geq 0.\n\\end{equation}\nTo prove the claim, we first notice that the nefness of $E$ implies that\n\\begin{equation} \\label{equationstar}\nc_{1}(\\mathcal{F}_{1})\\wedge(\\pi^{*}\\omega_{Y}+\\epsilon\\omega_{X})^{n-1}\\geq 0. \n\\end{equation} \nThe base $Y$ being a curve we have\n$$(\\pi^{*}\\omega_{Y}+\\epsilon\\omega_{X})^{n-1}=\n\\epsilon^{n-2}\\pi^{*}(\\omega_{Y})\\wedge(\\omega_{X})^{n-2}+\\epsilon^{n-1}(\\omega_{X})^{n-1}.$$\nNote also that $c_{1}(\\mathcal{F})\\wedge(\\omega_{X})^{n-1}$ is uniformly bounded from above for \nany $\\mathcal{F}\\subset E$, cf. \\cite[Lemma 7.16]{Kob87}.\nThen \\eqref{equationstar} implies that \n$$c_{1}(\\mathcal{F}_{1})\\wedge\\pi^{*}(\\omega_{Y})\\wedge(\\omega_{X})^{n-2}\\geq -\\epsilon\\cdot M$$\nfor a constant $M$ independent of $\\epsilon$. \nSince $\\epsilon$ is sufficiently small, the uniform estimate $\\eqref{equationfour}$ and \\eqref{equationfourminus} and imply that \n$$\nc_{1}(\\mathcal{F}_{1})\\wedge\\pi^{*}(\\omega_{Y})\\wedge \\omega_{X}^{n-2}=0 .\n$$\nUsing \\eqref{equationstar} we deduce that $c_{1}(\\mathcal{F}_{1})\\wedge(\\omega_{X})^{n-1}\\geq 0$.\nThis proves the claim.\n\nCombining \\eqref{equationfive} with the assumption that $c_{1}(E)= M\\cdot\\pi^{*}\\omega_{Y}$,\nwe get\n$$c_{1}(E\/\\mathcal{F}_{1})\\wedge\\pi^{*}(\\omega_{Y})\\wedge \\omega_{X}^{n-2}=0 .$$\nCombining this with the fact that $c_{1}(E\/\\mathcal{F}_{1})$ is nef and $\\omega_{Y}^{2}=0$, \nwe obtain\n\\begin{equation} \\label{equationsix}\nc_{1}(E\/\\mathcal{F}_{1})= b \\cdot\\pi^{*}(\\omega_{Y}) \n\\end{equation}\nfor some $b \\geq 0$ by Hodge index theorem \n(cf. Remark \\ref{corHT} of Appendix \\ref{appendixinequality}).\nThus we have $c_{1}(\\sF_{1})= a_1 \\cdot\\pi^{*}(\\omega_{Y})$ for some $a_1 \\in \\ensuremath{\\mathbb{Q}}$ and\n\\eqref{equationfive} implies that $a_1 \\geq 0$. \n\\end{proof}\n\nWe come to the main result of this subsection:\n\n\\begin{proposition} \\label{propositionfiltration}\nIn the situation of Lemma \\ref{lemmafiltration}\nthe reflexive sheaves $\\sF_i$ are subbundles of $E$, in particular they and the\ngraded pieces $\\sF_{i}\/\\sF_{i-1}$ are locally free.\nMoreover each of the graded pieces $\\sF_{i}\/\\sF_{i-1}$ is projectively flat, and \nthere exist a smooth metric $h_{i}$ on $\\sF_{i}\/\\sF_{i-1}$, such that\n$$i\\Theta_{h_{i}}(\\mathcal{F}_{i+1}\/\\mathcal{F}_{i})\n=a_{i}\\pi^{*}(\\omega_{Y})\\cdot \\Id_{\\mathcal{F}_{i+1}\/\\mathcal{F}_{i}}, $$\nfor some constant $a_{i}\\geq 0$.\n\\end{proposition}\n\n\\begin{proof}\n\n{\\em Step 1. Proof of the first statement.}\nWe first prove the statement for $i=1$.\nBy \\cite[Lemma 1.20]{DPS94} it is sufficient to prove that the induced morphism\n$$\n\\det \\sF_1 \\rightarrow \\bigwedge^{\\ensuremath{rk} \\ \\sF_1} E\n$$\nis injective as a morphism of vector bundles. Note now that the set $Z \\subset X$ where\n$\\sF_1 \\subset E$ is not a subbundle has codimension at least two: it is contained \nin the union of the loci where the torsion-free sheaves $\\sF_{k+1}\/\\sF_k$ are not locally free. In particular $Z$ does not\ncontain any fibre $X_y:=\\fibre{\\pi}{y}$ with $y \\in Y$. Thus for every $y \\in Y$ the restricted morphism\n\\begin{equation} \\label{inclusiondet}\n(\\det \\sF_1)|_{X_y} \\rightarrow (\\bigwedge^{\\ensuremath{rk} \\ \\sF_1} E)_{X_y}\n\\end{equation}\nis not zero. Yet by Lemma \\ref{lemmafiltration} the line bundle $(\\det \\sF_1)|_{X_y}$ is numerically trivial\nand the vector bundle $(\\bigwedge^{\\ensuremath{rk} \\ \\sF_1} E)_{X_y}$ is numerically flat. Thus the inclusion \\eqref{inclusiondet}\nis injective as a morphism of vector bundles.\nThen $\\sF_1$ is a subbundle of $E$ \\cite[Prop.1.16]{DPS94}.\n\nNow $E\/\\sF_1$ is a nef vector bundle on $X$. Moreover, Lemma \\ref{lemmafiltration} implies that \n$c_1 (E\/\\sF_1)=M'\\cdot \\pi^* (\\omega_Y)$ for some constant $M'$.\nThen we can argue by induction on $E\/\\sF_1$, and the first statement is proved.\n\n{\\em Step 2. The graded pieces are projectively flat.}\nApplying Lemma \\ref{lemmafiltration} to $E\/\\sF_i$, we obtain that $c_1(\\sF_{i}\/\\sF_{i-1})=a_{i}\\cdot \\pi^* (\\omega_Y)$ for some constant $a_i$, in particular\n$c_1^2 (\\sF_{i}\/\\sF_{i-1} )=0$.\nSince $\\sF_{i}\/\\sF_{i-1}$ is $(\\pi^{*}\\omega_{Y}+\\epsilon\\omega_{X})$-stable,\nto prove that $\\sF_{i}\/\\sF_{i-1}$ is projectively flat, by \\cite[Thm.4.7]{Kob87} it is sufficient to prove\nthat \n$$\nc_2(\\sF_{i}\/\\sF_{i-1}) \\cdot (\\pi^{*}\\omega_{Y}+\\epsilon\\omega_{X} )^{n-2}=0 .\n$$\nSince $c_1(\\sF_{i}\/\\sF_{i-1})$ is a pull-back from the curve $Y$ for every $i \\in \\{ 1, \\ldots, k\\}$ it is easy to see that\n$$\nc_2(E) = \\sum_{i=1}^{k} c_2(\\sF_{i}\/\\sF_{i-1}).\n$$\nSince we have $c_2(\\sF_{i}\/\\sF_{i-1}) \\cdot (\\pi^{*}\\omega_{Y}+\\epsilon\\omega_{X})^{n-2} \\geq 0$ \nfor every $i \\in \\{ 1, \\ldots, k\\}$ by \\cite[Thm.4.7]{Kob87}, \nwe are left to show that $c_2(E) \\cdot (\\pi^{*}\\omega_{Y}+\\epsilon\\omega_{X})^{n-2}=0$. \nYet $E$ is nef with $c_1(E)^2=0$, \nso this follows immediately from the Chern class inequalities\nfor nef vector bundles \\cite[Cor.2.6]{DPS94}.\n\\end{proof}\n\n\\subsection{Positivity of $\\pi_*(\\omega_X^{\\otimes -m})$}\n\\label{subsectionanticanonical}\n\nLet $X$ be a normal compact K\\\"ahler space with at most terminal Gorenstein singularities, \nand let \\holom{\\pi}{X}{T} be a fibration such that $-K_X$ is $\\pi$-nef and $\\pi$-big, that\nis $-K_X$ is nef on every fibre and big on the general fibre. In this case\nthe relative base-point free theorem holds \\cite[Thm.3.3]{Anc87}, i.e. for every $m \\gg 0$ the natural map\n$$\\pi^{*}\\pi_{*}(\\omega_X^{\\otimes -m})\\rightarrow \\omega_X^{\\otimes -m}$$\nis surjective. Thus $\\omega_X^{\\otimes -m}$ is $\\pi$-globally generated and induces a bimeromorphic morphism\n\\begin{equation} \\label{eqnrelativemodel}\n\\holom{\\mu}{X}{X'}\n\\end{equation}\nonto a normal compact K\\\"ahler space $X'$. Standard arguments from the MMP show that the bimeromorphic\nmap $\\mu$ is crepant, that is $K_{X'}$ is Cartier and we have\n$$\nK_X \\simeq \\mu^* K_{X'}.\n$$\nIn particular $X'$ has at most canonical Gorenstein singularities. The fibration $\\pi$ factors through the morphism $\\mu$,\nso we obtain a fibration\n\\begin{equation} \\label{eqnrelativefibration}\n\\holom{\\pi'}{X'}{T}\n\\end{equation}\nsuch that $-K_{X'}$ is $\\pi'$-ample. Therefore we call \\holom{\\mu}{X}{X'} the relative anticanonical model of $X$ and\n\\holom{\\pi'}{X'}{T} the relative anticanonical fibration.\n\nWe start with an elementary computation:\n\n\\begin{lemma} \\label{lemmaelementary}\nLet $V$ be a nef vector bundle over a smooth curve $C$, and let $A \\subset V$ be an ample subbundle.\nLet $Z \\subset \\ensuremath{\\mathbb{P}}(V)$ be a subvariety such that $Z \\not\\subset \\ensuremath{\\mathbb{P}}(V\/A)$. Then we have\n$$\nZ \\cdot \\sO_{\\ensuremath{\\mathbb{P}}(V)}(1)^{\\dim V}>0.\n$$\n\\end{lemma}\n\n\\begin{proof} \nLet $\\holom{f}{\\ensuremath{\\mathbb{P}}(V)}{C}$ and $\\holom{g}{\\ensuremath{\\mathbb{P}}(A)}{C}$ be the canonical projections, and let \n$\\holom{\\mu}{X}{\\ensuremath{\\mathbb{P}}(V)}$ be the blow-up along the subvariety $\\ensuremath{\\mathbb{P}}(V\/A)$. \nThe restriction of $\\mu$ to any $f$-fibre \\fibre{f}{c} is the blow-up of a projective space $\\ensuremath{\\mathbb{P}}(V_c)$\nalong the linear subspace $\\ensuremath{\\mathbb{P}}(V_c\/A_c)$, so we see that we have a fibration\n$\\holom{h}{X}{\\ensuremath{\\mathbb{P}}(A)}$ which makes $X$ into a projective bundle over $\\ensuremath{\\mathbb{P}}(A)$.\n$$\n\\xymatrix{\nZ'\\ar[d]\\ar[r]& X \\ar[d]^{\\mu}\\ar[r]^{h} & \\ensuremath{\\mathbb{P}}(A)\\ar[ldd]^{g}\\\\\nZ\\ar[r] & \\ensuremath{\\mathbb{P}}(V)\\ar[d]^{f} & \\\\\n & C\n}\n$$\nSince $Z \\not\\subset \\ensuremath{\\mathbb{P}}(V\/A)$, the strict transform $Z'$ is well-defined and we have\n$$\n\\sO_{\\ensuremath{\\mathbb{P}}(V)}(1)^{\\dim Z} \\cdot Z = \\mu^* \\sO_{\\ensuremath{\\mathbb{P}}(V)}(1)^{\\dim Z} \\cdot Z'.\n$$\nWe claim that\n\\begin{equation} \\label{decompoone}\n\\mu^* \\sO_{\\ensuremath{\\mathbb{P}}(V)}(1) \\simeq h^* \\sO_{\\ensuremath{\\mathbb{P}}(A)}(1) + E,\n\\end{equation}\nwhere $E$ is the $\\mu$-exceptional divisor. Indeed we can write\n$$\n\\mu^* \\sO_{\\ensuremath{\\mathbb{P}}(V)}(1) \\simeq a h^* \\sO_{\\ensuremath{\\mathbb{P}}(A)}(1) + b E + c F,\n$$\nwhere $F$ is a $f \\circ \\mu$-fibre and $a,b,c \\in \\ensuremath{\\mathbb{Q}}$. By restricting to $F$ one easily sees \nthat we have $a=1, b=1$. Note now (for example by looking at the relative Euler sequence) that we have\n$\nN_{\\ensuremath{\\mathbb{P}}(V\/A)\/\\ensuremath{\\mathbb{P}}(V)} \\simeq f^* A^* \\otimes \\sO_{\\ensuremath{\\mathbb{P}}(V\/A)}(1)$.\nSince the exceptional divisor $E$ is the projectivisation of $N_{\\ensuremath{\\mathbb{P}}(V\/A)\/\\ensuremath{\\mathbb{P}}(V)}^*$ we deduce that\n$$\n- E|_E \\simeq \\sO_{\\ensuremath{\\mathbb{P}}(f^* A \\otimes \\sO_{\\ensuremath{\\mathbb{P}}(V\/A)}(-1))}(1) \\simeq (h^* \\sO_{\\ensuremath{\\mathbb{P}}(A)}(1))|_E + \\mu|_E^* \\sO_{\\ensuremath{\\mathbb{P}}(V\/A)}(-1).\n$$\nSince $\\mu^* \\sO_{\\ensuremath{\\mathbb{P}}(V)}(-1)|_E \\simeq \\mu|_E^* \\sO_{\\ensuremath{\\mathbb{P}}(V\/A)}(-1)|_E$ we obtain $c=0$.\n\nIn order to simplify the notation, set $\\xi_V:=\\mu^* \\sO_{\\ensuremath{\\mathbb{P}}(V)}(1)$ and $\\xi_A:=h^* \\sO_{\\ensuremath{\\mathbb{P}}(A)}(1)$.\nBy \\eqref{decompoone} we have\n$$\n\\xi_V^{\\dim Z} \\cdot Z' = \\xi_V^{\\dim Z-1} \\cdot \\xi_A \\cdot Z'\n+ \\xi_V^{\\dim Z-1} \\cdot E \\cdot Z'.\n$$\nSince $E$ does not contain $Z'$, the two terms on the right hand side are non-negative. If $\\xi_V^{\\dim Z-1} \\cdot E \\cdot Z'>0$\nwe are obviously finished, so suppose that this is not the case. Let $e$ be the dimension of $h(E \\cap Z')$. \nSince $\\sO_{\\ensuremath{\\mathbb{P}}(A)}(1)$ is ample and $\\xi_V$ is $h$-ample, we have\n$$\n\\xi_V^{\\dim Z-e-1} \\cdot \\xi_A^{e} \\cdot E \\cdot Z'>0.\n$$\nThus\n$$\nl := \\min \\{ j \\in \\ensuremath{\\mathbb{N}} \\ | \\ \\xi_V^{\\dim Z-j-1} \\cdot \\xi_A^j \\cdot E \\cdot Z'>0 \\}.\n$$\nis an integer. An easy induction now shows that\n$$\n\\xi_V^{\\dim Z} \\cdot Z' = \\xi_V^{\\dim Z-l-1} \\cdot \\xi_A^{l+1} \\cdot Z' +\n\\xi_V^{\\dim Z-l-1} \\cdot \\xi_A^l \\cdot E \\cdot Z'>0.\n$$\n\\end{proof}\n\n\n\\begin{lemma} \\label{lemmaflat}\nLet $X$ be a compact K\\\"ahler manifold of dimension $n$ such that $-K_X$ is nef.\nLet $\\pi: X \\rightarrow T$ be the Albanese fibration, and suppose that $-K_X$ is $\\pi$-big. \nLet $\\holom{\\pi'}{X'}{T}$ be the relative anticanonical fibration. \n\nThen $\\pi'$ is flat and $E_{m}:=(\\pi')_{*}(\\omega_{X'}^{\\otimes -m})$ is locally free for $m \\in \\ensuremath{\\mathbb{N}}$.\n\\end{lemma}\n\n\\begin{proof}\nThe variety $X'$ has at most canonical singularities, so it is Cohen-Macaulay. \nThe base $T$ being smooth it is sufficient to prove that $\\pi'$ is equidimensional \\cite[III,Ex.10.9]{Har77}. \nSet $r:=\\dim T$. \nBy Proposition \\ref{propositionnumericaldimension} we know that \n$$\n(-K_X)^{n-r+1}=(-K_{X'})^{n-r+1}=0.\n$$\nIf $F' \\subset X'$ is an irreducible component of a $\\pi'$-fibre, we have\n$(-K_{X'}|_{F'})^{\\dim F'} \\neq 0$\nsince $-K_{X'}|_{F'}$ is ample. By the preceding equation we see that $\\dim F' \\leq n-r$.\n\nSince $X'$ has at most canonical singularities, \nthe relative Kawamata-Viehweg theorem applies and shows that\n$R^j (\\pi')_{*}(\\omega_{X'}^{\\otimes -m})=0$ for all $j>0$.\nThe fibration $\\pi'$ being flat, the statement follows.\n\\end{proof}\n\n\n\\begin{lemma} \\label{lemmanef}\nIn the situation of Lemma \\ref{lemmaflat}, the vector bundle $E_{m}$ is nef for $m \\gg 1$.\n\\end{lemma}\n\n\\begin{remark}\nIf the fibration is smooth and the torus $T$ is abelian, \nthe nefness is proved in \\cite[Lemma 3.21]{DPS94}.\nSince $T$ is an arbitrary torus and $\\pi$ is -a priori- not necessarily smooth, \nwe use \\cite[Thm. 0.5]{DP04} and the standard regularization method (cf. \\cite[Ch.13]{Dem12}, \\cite[Sect.3]{Dem92})\nto overcome these difficulties. \n\\end{remark}\n\n\\begin{proof}[Proof of Lemma \\ref{lemmanef}]\n\nWe first notice that $-K_X =\\mu^* (-K_{X'})$ by construction.\nTherefore $E_{m}=\\pi_{*}(\\omega_X^{\\otimes -m})$.\nWe first fix a Stein cover $\\mathcal{U}=\\{U_{i}\\}$ on $T$ as constructed in \\cite[13.B]{Dem12}\n\\footnote{We keep the notations in \\cite[13.B]{Dem12}, which can also be found in \\cite[Sect. 3]{Dem92} .}, \nsuch that $U_{i}$ are simply connected balls of radius $2\\delta$ fixed.\nLet $U_i^{'}\\Subset U_i{''}\\Subset U_{i}$ be the balls constructed in \\cite[13.B]{Dem12} such that they are the balls of radius \n$\\delta$, $\\frac{3}{2}\\delta$, $2\\delta$ respectively\nand $\\{ U_i^{'} \\}$ also covers $T$.\nLet $\\theta_{j}$ be a smooth partition function with support in $U_{j}''$ as constructed in \\cite[Lemma 13.11]{Dem12}.\nLet $\\varphi_{k}: T\\rightarrow T$ be a $2^k$-degree isogeny of the torus $T$, and set $X_k :=T\\times_{\\varphi_k} X$.\nLet $L=-(m+1) K_{X_k\/T}$ and set $E_{m, k} :=\\pi_{*}(K_{X_k}+L)$.\nWe have the commutative diagram\n$$\\xymatrix{\n&X_k\\ar[r]^{\\widetilde{\\varphi}_k}\\ar[d]_{\\widehat{\\pi}}& X\\ar[d]_{\\pi}\n\\\\ \\ensuremath{\\mathbb{P}} (E_{m, k})\\ar[r]^{\\pi_{1}} &T\\ar[r]^{\\varphi_k} & T \n\\\\ & U_i\\ar@{^{(}->}[u] &}$$\nNote that the cover $\\mathcal{U}=\\{U_{i}\\}$, and the partition functions $\\theta_{i}$ are independent of $k$.\nWe first prove that there exists a smooth metric $h$ on $\\mathcal{O}_{\\ensuremath{\\mathbb{P}}(E_{m, k})} (1)$, \nsuch that \n$$i\\Theta_{h}(\\mathcal{O}_{\\ensuremath{\\mathbb{P}}(E_{m, k})} (1))\\geq -C\\cdot \\pi_1 ^{*}( \\omega_{T} )$$\nfor a constant $C$ independent of $k$\n\\footnote{ All the constants $C, C_{1},\\cdots, C_{i}$ below are independent of $k$.}.\n\nWe fix a K\\\"ahler metric $\\omega_{X_k}$ on $X_k$.\nSince $L$ is nef and $\\widehat{\\pi}$-big, \n\\cite[Thm. 0.5]{DP04} implies the existence of a singular metric $h_{\\widetilde{\\epsilon}_k }$ on $L$\nsuch that \n$$i\\Theta_{h_{\\widetilde{\\epsilon}_k}}(L)\\geq \\widetilde{\\epsilon}_k \\omega_{X_k}- C_{1}\\widehat{\\pi}^*( \\omega_{T} ),$$\nfor a constant $C_{1}$ independent of $k$, but $\\widetilde{\\epsilon}_k > 0$ is dependent of $k$.\nSince $L$ is nef, for any $\\epsilon > 0$, there exists a metric $h_{\\epsilon}$ such that\n$$i\\Theta_{h_{\\epsilon}}(L)\\geq -\\epsilon\\omega_{X_k} .$$\nBy combining these two metrics, we can easily construct a new metric $h_{\\epsilon_k}$ on $L$,\nsuch that\\footnote{We just need to take $h_{\\epsilon_k}=h_{\\widetilde{\\epsilon}_k }^{r_k}\\cdot h_{\\epsilon}^{1-r_k}$\nfor some $r_k$ small enough, and $\\epsilon\\ll r_k\\cdot \\widetilde{\\epsilon}_k$.}\n\\begin{equation}\\label{firstsingularmetric}\ni\\Theta_{h_{\\epsilon_k}}(L)\\geq \\epsilon_k \\omega_{X_k}- 2\\cdot C_{1}\\widehat{\\pi}^*(\\omega_{T})\\qquad\n\\text{and }\\qquad \\mathcal{I} (h_{\\epsilon_k})=\\mathcal{O}_{X_k}\n\\end{equation}\nfor some $\\epsilon_k > 0$.\nSince $\\mathcal{I} (h_{\\epsilon_k})=\\mathcal{O}_{X_k}$ and $U_i$ are simply connected Stein varieties, \nwe can suppose that $L^2$-bounded (with respect to $h_{\\epsilon_k}$) \nelements in $H^{0}(\\widehat{\\pi}^{-1}(U_{i}), K_{X_k}+L)$\ngenerate $E_{m,k}$ over $U_i$. \n\nLet $\\{\\widehat{e}_{i , j}\\}_{j}$ be an orthonormal base of $H^{0}(\\widehat{\\pi}^{-1}(U_{i}), K_{X_k}+L)$ \nwith respect to $h_{\\epsilon_k}$,\ni.e., $\\int_{\\widehat{\\pi}^{-1}(U_{i})} \\langle\\widehat{e}_{i , j}, \\widehat{e}_{i , j'}\\rangle_{h_{\\epsilon,k}}^2=\\delta_{j, j'}$.\nThen $\\widehat{e}_{i , j}$ induce an element $e_{i,j}\\in H^{0}(\\pi_{1}^{-1}(U_i), \\O_{\\ensuremath{\\mathbb{P}}(E_{m,k})} (1))$.\nWe now define a smooth metric $h_{i}$ on $\\O_{\\ensuremath{\\mathbb{P}}(E_{m,k})} (1) $ over $\\pi_{1}^{-1}(U_i)$ by \n$$\\|\\cdot\\|_{h_i}^2=\\frac{\\|\\cdot\\|_{h_{0}}^2}{\\sum\\limits_{j}\\|e_{i, j}\\|_{h_{0}}^2} ,$$\nwhere $h_{0}$ is a fixed metric on $\\O_{\\ensuremath{\\mathbb{P}}(E_m)} (1) $.\nThanks to the construction,\n$h_i$ is smooth and $i\\Theta_{h_i}(\\O_{\\ensuremath{\\mathbb{P}}(E_{m,k})} (1) )$ is semi-positive on $\\pi_{1}^{-1}(U_i)$.\n\nWe claim that \n\\begin{equation}\\label{equation75}\n\\frac{1}{C_{2}}\\leq \\frac{\\sum\\limits_{j} \\|e_{i, j}\\|_{h_{0}}^2 (z)}{\\sum\\limits_{j} \\|e_{i', j}\\|_{h_{0}}^2 (z)}\\leq C_{2}\n\\qquad \\text{for }z\\in \\pi_{1}^{-1}(U_{i}''\\cap U_{i'}'') ,\n\\end{equation}\nfor some $C_{2}> 0$ independent of $z, k, i, i'$.\nThe proof is almost the same as in \\cite[Lemma 13.10]{Dem12}, except that we use the metric \n$\\epsilon_k\\cdot\\omega_{X_k}+\\widehat{\\pi}^{*}\\omega_T$\nin stead of $\\omega_X$ in the estimate.\nWe postpone the proof of the claim \\eqref{equation75} in Lemma \\ref{gluing}\nand first finish the proof of Lemma \\ref{lemmanef}.\n\nWe now define a global metric $h$ on $\\mathcal{O}_{\\ensuremath{\\mathbb{P}}(E_{m})} (1)$ by\n$$\\|\\cdot\\|_{h}^{2}=\\|\\cdot\\|_{h_{0}}^{2} e^{- \\sum\\limits_{i} (\\pi_1 ^{*}(\\theta_{i}'))^{2}\\cdot\\ln (\\sum\\limits_{j} \\|e_{i, j}\\|_{h_{0}}^2)} ,\n\\qquad\\text{where }(\\theta_{i}')^{2}=\\frac{\\theta_{i}^{2}}{\\sum\\limits_{k} \\theta_{k}^{2}} .$$\nNote that\n$$i (\\theta_{j}'\\partial\\overline{\\partial} \\theta_{j}'-\\partial\\theta_{j}'\\wedge \\overline{\\partial} \\theta_{j}' )\n\\geq -C_{3}\\cdot\\omega_{T}$$\nby construction.\nCombining this with \\eqref{equation75} and \napplying the Legendre identity in the proof of \n\\cite[Lemma 13.11]{Dem12}\\footnote{Although in the proof of \\cite[Lemma 13.11]{Dem12}, $\\theta_{i}'$ is supposed to be constant on $U_i^{'}$,\nthe uniformly strictly positive of the lower boundedness of $\\theta_{i}'$ on $U_{i}'$ is sufficient for the proof.},\nwe obtain that\n$$i\\Theta_{h}(\\mathcal{O}_{\\ensuremath{\\mathbb{P}}(E_{m,k})} (1))\\geq -C\\cdot \\pi_1 ^{*}(\\omega_{T})$$\nfor a constant $C$ independent of $k$.\n\nBy \\cite[Prop. 1.8]{DPS94}, the metric $h$ on $\\mathcal{O}_{\\ensuremath{\\mathbb{P}}(E_{m,k})} (1)$ induce a smooth metric $h_k$\non $\\mathcal{O}_{\\ensuremath{\\mathbb{P}}(E_{m})} (1)$ such that\n$$i\\Theta_{h_k}(\\mathcal{O}_{\\ensuremath{\\mathbb{P}}(E_{m})} (1))\\geq -\\frac{C}{2^{k-1}}\\omega_{T}.$$\nThe lemma is proved by letting $k\\rightarrow +\\infty$.\n\\end{proof}\n\nWe now prove the claim \\eqref{equation75} in Lemma \\ref{lemmanef}, \nwhich is in some sense a relative gluing estimate.\n\n\\begin{lemma}\\label{gluing} In the situation of the proof of Lemma \\ref{lemmanef},\nwe have\n\\begin{equation}\\label{equationlemmagluing}\n\\frac{1}{C_{2}}\\leq \\frac{\\sum\\limits_{j} \\|e_{i, j}\\|_{h_{0}}^2 (z)}{\\sum\\limits_{j} \\|e_{i', j}\\|_{h_{0}}^2 (z)}\\leq C_{2}\n\\qquad \\text{for }z\\in \\pi_{1}^{-1}(U_{i}''\\cap U_{i'}'').\n\\end{equation} \n\\end{lemma}\n\n\\begin{proof}\n\nRecall that $U_i^{'}\\Subset U_i{''}\\Subset U_{i}$ are the balls of radius $\\delta$, $\\frac{3}{2}\\delta$, $2\\delta$ respectively\nas constructed in \\cite[13.B]{Dem12}.\nLet $z$ be a fixed point in $\\pi_{1}^{-1}(U_{i}''\\cap U_{i'}'')$.\nSince $e_{i,j }$ is a section of a line bundle, we have\n$$\\sum\\limits_{j} \\|e_{i, j}\\|_{h_{0}}^2 (z)= \n\\sup\\limits_{\\sum\\limits_{j} \\mid a_j \\mid^2 =1} \\|\\sum\\limits_{j} a_{j} e_{i, j}\\|_{h_{0}}^{2}(z) .$$\nTherefore, there exists a $\\widehat{e}_{i}\\in H^{0}(\\widehat{\\pi}^{-1}(U_{i}), K_{X_k}+L)$\nsuch that\n\\begin{equation}\\label{estimateonepiece}\n\\int_{\\widehat{\\pi}^{-1}(U_{i})} \\|\\widehat{e}_{i}\\|_{h_{\\epsilon_k}}^{2}=1\\qquad \\text{and}\\qquad\n\\|e_{i}\\|_{h_{0}}^{2} (z) = \\sum\\limits_{j} \\|e_{i, j}\\|_{h_{0}}^2 (z) ,\n\\end{equation}\nwhere $e_{i}\\in H^{0}(\\pi_{1}^{-1}(U_i), \\O_{\\ensuremath{\\mathbb{P}}(E_{m,k})} (1))$ is induced by $\\widehat{e}_{i}$.\nOur goal is to construct an element in $H^{0}(\\widehat{\\pi}^{-1}(U_{i'}), K_{X_k}+L)$ with controlled norm\nand equals to $\\widehat{e}_{i}$ on $\\widehat{\\pi}^{-1}(\\pi_{1} (z))$.\n\nLet $\\theta$ be a cut-off function with support in the ball of radius $ \\frac{\\delta}{4}$ \ncentered at $\\pi_{1} ( z )$ (thus is supported in $U_{i}\\cap U_{i'}$), \nand equal to $1$ on the ball of radius $\\frac{\\delta}{8}$ centered at $\\pi_{1} ( z )$.\nBy construction, $(\\widehat{\\pi}^{*}(\\theta)\\cdot\\widehat{e}_{i})$ is supported in $\\widehat{\\pi}^{-1} (U_{i}\\cap U_{i'})$, \nthus it is well defined on $\\widehat{\\pi}^{-1}(U_{i'})$.\nTherefore we can solve the $\\overline{\\partial}$-equation for $\\overline{\\partial} (\\widehat{\\pi}^{*}(\\theta)\\cdot\\widehat{e}_{i})$ on \n$\\widehat{\\pi}^{-1}(U_{i'})$\nwith respect to the metric \n\\begin{equation}\\label{newmetric}\n\\omega_{X_k,\\epsilon_k}=\\epsilon_k\\cdot\\omega_{X_k}+\\widehat{\\pi}^{*}\\omega_T \n\\end{equation}\nby choosing a good metric on $L$. \nBefore giving the good metric on $L$, we first give some estimates.\n\nSince $\\theta$ is defined on $T$, we have\n$$\\|\\overline{\\partial}\\widehat{\\pi}^{*}(\\theta)\\|_{\\omega_{X_k,\\epsilon_k}}\\leq C_4$$\nfor some constant $C_{4}$ independent of $k, \\epsilon_k$\n\\footnote{$C_4$ depends on $\\delta$. But by construction, the radius $\\delta$ is independent of $k$.}.\nTherefore we have\n\\begin{equation}\\label{equation5}\n\\int_{\\widehat{\\pi}^{-1}(U_{i'})}\\|\\overline{\\partial}(\\widehat{\\pi}^{*}(\\theta)\\cdot\\widehat{e}_{i})\\|_{h_{\\epsilon_k}, \\omega_{X_k,\\epsilon_k}}^{2}\n =\\int_{\\widehat{\\pi}^{-1}(U_{i})}\\|\\overline{\\partial}(\\widehat{\\pi}^{*}(\\theta)\\cdot\\widehat{e}_{i})\\|_{h_{\\epsilon_k}, \\omega_{X_k,\\epsilon_k}}^{2}\n\\end{equation}\n$$\n\\leq C_4 \\int_{\\widehat{\\pi}^{-1}(U_{i})}\\|\\widehat{e}_{i}\\|_{h_{\\epsilon_k}}^{2}=C_{4} ,\n$$\nwhere the first equality comes from the fact that $(\\widehat{\\pi}^{*}(\\theta)\\cdot\\widehat{e}_{i})$ is supported in \n$\\widehat{\\pi}^{-1} (U_{i}\\cap U_{i'})$.\nBy \\eqref{firstsingularmetric} and \\eqref{newmetric}, we have\n\\begin{equation}\\label{equation6}\ni\\Theta_{h_{\\epsilon_k}}(L)\\geq \\epsilon_k\\omega_{X_k}-2\\cdot C_1 \\widehat{\\pi}^{*}(\\omega_{T} )\\geq \n\\omega_{X_k,\\epsilon_k}-(2\\cdot C_1 +1) \\widehat{\\pi}^{*}(\\omega_{T} ). \n\\end{equation}\n\nWe now define a metric $\\widetilde{h}_{\\epsilon_k} = h_{\\epsilon_k}\\cdot e^{-(n+1) \\widehat{\\pi}^{*}(\\ln |t-\\pi_{1} (z)|)-\\widehat{\\pi}^{*}\\psi_{i'}(t)}$ \non $ L $ over $\\widehat{\\pi}^{-1}(U_{i'})$,\nwhere $\\psi_{i'}(t)$ is a uniformly bounded function on $U_{i'}$ satisfying \n$$dd^c \\psi_{i'}(t)\\geq (2 C_1 +1) \\omega_{T} .$$\nThen \\eqref{equation6} implies that \n$$i\\Theta_{\\widetilde{h}_{\\epsilon_k}}(L)\\geq \\omega_{X_k,\\epsilon_k} \\qquad \\text{on }\\widehat{\\pi}^{-1}(U_{i'}).$$\nBy solving the $\\overline{\\partial}$-equation for $\\overline{\\partial} (\\widehat{\\pi}^{*}(\\theta)\\cdot\\widehat{e}_{i})$ \nwith respect to $(\\widetilde{h}_{\\epsilon_k} , \\omega_{X_k,\\epsilon_k} )$ on $\\widehat{\\pi}^{-1}(U_{i'})$, \nwe can find a $g_{i'}\\in L^{2}(\\widehat{\\pi}^{-1}(U_{i'}), K_{X_k}+L)$ such that\n\\begin{equation}\\label{solutionpartial}\n\\overline{\\partial}g_{i'}=\\overline{\\partial}(\\widehat{\\pi}^{*}(\\theta)\\widehat{e}_{i})\n\\end{equation}\nand\n\\begin{equation}\\label{equation7}\n\\int_{\\widehat{\\pi}^{-1}(U_{i'})}\\|g_{i'}\\|_{\\widetilde{h}_{\\epsilon_k}}^{2}\\leq \nC_5\\int_{\\widehat{\\pi}^{-1}(U_{i'})} \\|\\overline{\\partial}(\\widehat{\\pi}^{*}(\\theta)\n\\cdot\\widehat{e}_{i})\\|_{\\widetilde{h}_{\\epsilon_k}, \\omega_{X,\\epsilon_k}}^{2}\n\\leq C_{6},\n\\end{equation}\nwhere the second inequality comes from \\eqref{equation5} \nand the fact that\n$$\n\\overline{\\partial}(\\widehat{\\pi}^{*}(\\theta)\\cdot\\widehat{e}_{i}) (z)=0 \\qquad \\text{for }z\\in \\widehat{\\pi}^{-1}(B_{\\frac{\\delta}{8}}(\\pi_{1}(z))),\n$$\nwhere $B_{\\frac{\\delta}{8}}(\\pi_{1}(z))$ is the ball of radius $\\frac{\\delta}{8}$ centered at $\\pi_{1}(z)$.\nBy \\eqref{solutionpartial}, we obtain a holomorphic section \n$$\\widehat{e}_{i'} := (\\widehat{\\pi}^{*}(\\theta)\\cdot \\widehat{e}_{i}- g_{i'} )\\in H^{0}(\\widehat{\\pi}^{-1}(U_{i'}), K_{X_k}+L) .$$\nBy the definition of the metric $\\widetilde{h}_{\\epsilon_k}$ and \\eqref{equation7}, \n$g_{i'}=0$ on $\\widehat{\\pi}^{-1}(\\pi_{1}(z))$.\nTherefore \n$\\widehat{e}_{i'} = \\widehat{e}_{i} $ on $\\widehat{\\pi}^{-1}(\\pi_{1} (z))$.\nMoreover, \\eqref{estimateonepiece} and \\eqref{equation7} imply that\n$$\\int_{\\widehat{\\pi}^{-1}(U_{i'})} \\|\\widehat{e}_{i'}\\|_{h_{\\epsilon_k}}^{2}=\\int_{\\widehat{\\pi}^{-1}(U_{i'})}\\|\\widehat{\\pi}^{*}(\\theta)\\cdot\\widehat{e}_{i}- g_{i'}\\|_{h_{\\epsilon_k}}^{2}\\leq C$$\nfor a constant $C$ independent of $k$.\nBy the extremal property of Bergman kernel, \\eqref{equationlemmagluing} is proved.\n\\end{proof}\n\n\\begin{lemma} \\label{lemmanumericallyflat}\nIn the situation of Lemma \\ref{lemmaflat}, the vector bundle $E_{m}$ is numerically flat.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lemmanef} it is sufficient to prove that $c_{1}(E_{m})=0$.\nArguing by contradiction we suppose that $c_{1}(E_{m})\\neq 0$.\nThen \\cite[Prop.2.2]{Cao12} implies that there exists a smooth fibration \n$\\pi_{1}: T\\rightarrow S$ onto an abelian variety $S$ of dimension $s$ such that \n$$\nc_{1}(E_{m})=c\\cdot\\pi^{*}_{1} c_1(A)\n$$\nfor some very ample line bundle $A$ and $c> 0$.\n\nLet $S_{1}$ be a complete intersection of $s-1$ hypersurfaces defined by $s-1$ general elements in $H^{0}(S, A)$.\nWe have thus a morphism\n$$\\begin{CD}\n X_{1} @>\\pi|_{X_1}>> T_{1} @>{\\pi_1}|_{T_1}>> S_{1}\n \\end{CD}\n$$\nwhere $X_{1} :=\\pi^{-1}\\pi_{1}^{-1}(S_{1})$ and $T_{1} :=\\pi_{1}^{-1}(S_{1})$\nare smooth by Bertini's theorem.\nFor simplicity of notation we set $E_{m}':=E_{m}|_{T_{1}}$. \nThen $E_{m}'$ is nef and $c_{1}(E_{m}')=c\\cdot (\\pi_{1}|_{T_1})^* c_1(A)$. \nApplying Proposition \\ref{propositionfiltration}, \nwe obtain a semipositive projective flat vector bundle $0\\subset F_{1}\\subset E_{m}'$ on $T_{1}$\nsuch that \nand $c_{1}(F_{1})=\\pi_{1}^{*} (\\omega_{S_{1}})$\nfor some K\\\"ahler form $\\omega_{S_{1}}$ on $S_{1}$.\n\nOur goal is to show that \n$$\n(\\sO_{\\ensuremath{\\mathbb{P}} (E_{m}')}(1))^{n-r+1} \\cdot X_{1} > 0.\n$$\nSince $\\sO_{\\ensuremath{\\mathbb{P}} (E_{m}')}(1)|_{X_1} \\equiv (-m K_X)|_{X_1}$ this gives a contradiction \nto Proposition \\ref{propositionnumericaldimension}. \nNote first that $X_1$ is not contained in any projective subbundle of $\\ensuremath{\\mathbb{P}} (E_{m}')$ since\nthe general $\\pi$-fibre $F$ is embedded by the complete linear system $|-m K_F|$, so it is linearly non-degenerate.\nThus if $\\pi_1$ is an isomorphism (which is equivalent to $\\det E_m$ being ample) \nthe statement follows from Lemma \\ref{lemmaelementary}.\nIn the general case we follow a similar construction: let \n$\\mu: Y\\rightarrow \\mathbb{P}(E_{m}')$ be the blow-up along the subvariety \n$\\mathbb{P}(E_{m}'\/F_{1})$.\nSince $X_{1}$ is not contained in $\\mathbb{P}(E_{m}'\/F_{1})$, we have a diagram\n$$\n\\xymatrix{\nX_{1}'\\ar[d]\\ar[r]^{i}& Y \\ar[d]^{\\mu}\\ar[r]^{h} & \\ensuremath{\\mathbb{P}}(F_{1})\\ar[ldd]^{g}\\\\\nX_{1}\\ar[r] \\ar[rd]_{\\pi|_{X_1}} & \\ensuremath{\\mathbb{P}}(E_{m}')\\ar[d]^{f} & \\\\\n & T_{1}\\ar[d]^{\\pi_{1}}\\\\\n & S_{1}\n}\n$$\nwhere $X_{1}'$ is the strict transformation of $X_{1}$ and $f,g$ and $h$ are the natural maps as in the proof \nof Lemma \\ref{lemmaelementary}.\nBy the same argument as in \\eqref{decompoone} of Lemma \\ref{lemmaelementary}, we have\n$$\n\\mu^* \\sO_{\\ensuremath{\\mathbb{P}} (E_{m}')}(1) \\simeq h^* \\sO_{\\ensuremath{\\mathbb{P}}(F_{1})}(1) + E,\n$$\nwhere $E$ is the $\\mu$-exceptional divisor.\nSince $\\sO_{\\ensuremath{\\mathbb{P}} (E_{m}')}(1)$ is nef,\nwe have\n\\begin{equation}\\label{equationaddd}\n(\\mu^* \\sO_{\\ensuremath{\\mathbb{P}} (E_{m}')}(1))^{n-r+1} \\cdot X_{1}'\\geq\n(\\mu^* \\sO_{\\ensuremath{\\mathbb{P}} (E_{m}')}(1))^{n-r}\\cdot h^* \\sO_{\\ensuremath{\\mathbb{P}}(F_{1})}(1)\\cdot X_{1}' .\n\\end{equation}\nBy Proposition \\ref{propositionfiltration}, there is a smooth metric $h_0$ on $F_{1}$\nsuch that \n$$i \\Theta_{h_0} (F_{1}) =\\pi_{1}^*\\omega_{S_1}\\cdot \\Id_{F_{1}} .$$\nOn $\\ensuremath{\\mathbb{P}}(F_{1})$ we have\n$$\ni \\Theta_{h_0} (g^*F_{1}) = g^*\\pi_{1}^*\\omega_{S1}\\cdot \\Id_{g^*F_{1}}.\n$$\nThe metric $h_0$ induces a natural metric $h_0 ^{'}$ on $\\sO_{\\ensuremath{\\mathbb{P}}(F_{1})}(1)$,\nand by \\cite[Prop. 1.11]{DPS94},\nwe obtain\n$$\ni \\Theta_{h_0 ^{'}}(\\sO_{\\ensuremath{\\mathbb{P}}(F_{1})}(1))\\geq g^*\\pi_{1}^*\\omega_{S_1}.\n$$ \nSince $h\\circ g=\\mu\\circ f$, we get\n$h^*i \\Theta_{h_0 ^{'}} (\\sO_{\\ensuremath{\\mathbb{P}}(F_{1})}(1))\\geq \\mu^{*} f^{*}\\omega_{S_{1}}$.\nCombining this with the fact that $f\\circ\\mu\\circ i (X_{1}')=T_1$ by construction, \nwe obtain \n$$h^* \\sO_{\\ensuremath{\\mathbb{P}}(F_{1})}(1)\\cdot X_{1}'\\geq C\\cdot X_{1, s}',$$\nwhere $X_{1,s}'$ is the general fiber of $i\\circ\\mu\\circ f\\circ \\pi_{1}$, and $C> 0$.\nCombining this with the fact that $\\sO_{\\ensuremath{\\mathbb{P}} (E_{m}')}(1)$ is $f$-ample,\nwe get\n$$(\\mu^* \\sO_{\\ensuremath{\\mathbb{P}} (E_{m}')}(1))^{n-r}\\cdot h^* \\sO_{\\ensuremath{\\mathbb{P}}(F_{1})}(1)\\cdot X_{1}' \\neq 0 .$$\nWe conclude by \\eqref{equationaddd}.\n\\end{proof}\n\n\\subsection{The projective case}\n\nIn Section \\ref{sectiondimensiontwo} we will need the following\nversions of Lemma \\ref{lemmanef} and Lemma \\ref{lemmanumericallyflat}.\n\n\\begin{lemma} \\label{lemmakltdirectimage}\nLet $X$ be a normal projective variety and $\\Delta$ a boundary divisor on $X$ such that\nthe pair $(X, \\Delta)$ is klt. Let \\holom{\\varphi}{X}{T} be a flat fibration onto an abelian\nvariety. Let $L$ be a Cartier divisor on $X$ such that $L-(K_X+\\Delta)$ \nis nef and $\\varphi$-big. Then\nthe following holds:\n\\begin{enumerate}[(i)]\n\\item Let $H$ be an ample line bundle on $T$, then $\\varphi_* (\\sO_X(L)) \\otimes H$ is ample.\n\\item The direct image sheaf $\\varphi_* (\\sO_X(L))$ is nef.\n\\end{enumerate} \n\\end{lemma}\n\n\\begin{proof}\nNote first that since $L-(K_X+\\Delta)$ is relatively nef and big, \nthe higher direct images $R^j \\varphi_* (\\sO_X(L))$\nvanish for all $j>0$ by the relative Kawamata-Viehweg theorem. By cohomology and base change the sheaf\n$\\varphi_* (\\sO_X(L))$ is locally free.\n\n{\\em Proof of (i).} Since $H$ is ample and $L-(K_X+\\Delta)$ is nef and $\\varphi$-big, \nthe divisor $L+\\varphi^*H-(K_X+\\Delta)$ is nef and big.\nSo if $P \\in \\mbox{Pic}^0(T)$ is a numerically trivial line bundle, then\nby Kawamata-Viehweg vanishing one has\n\\[\nH^j(T, \\varphi_* (\\sO_X(L)) \\otimes H \\otimes P) \n=\nH^j(X, L \\otimes \\varphi^*(H \\otimes P)) = 0 \n\\qquad\n\\forall \\ j > 0. \n\\]\nThus $ \\varphi_* (\\sO_X(L)) \\otimes H$ satisfies Mukai's property $IT_0$ \\cite{Muk81}. \nIn particular it is ample by \\cite[Cor.3.2]{Deb06}.\n\n{\\em Proof of (ii).} Let \\holom{\\mu}{T}{T} be the multiplication map $x \\mapsto 2x$.\nLet $H$ be a symmetric ample divisor, then by the theorem of the square\n\\cite[Cor.2.3.6]{BL04} we have $\\mu^* H \\simeq H^{\\otimes 4}$. \nIt is sufficient to show that for $d \\in \\ensuremath{\\mathbb{N}}$ arbitrary, the $\\ensuremath{\\mathbb{Q}}$-twisted \\cite[Sect.6.2]{Laz04b} vector bundle\n$\\varphi_* (\\sO_X(L))\\hspace{-0.8ex}<\\hspace{-0.8ex}\\frac{1}{4^d}H\\hspace{-0.8ex}>$\nis nef. Denote by $\\mu_d$ the $d$-th iteration of $\\mu$ and consider the base change diagram\n\\[\n\\xymatrix{\nX_d\n\\ar[d]_{\\tilde \\varphi} \\ar[r]^{\\tilde \\mu_d} & X \\ar[d]_\\varphi\n\\\\\nT \\ar[r]^{\\mu_d} & T\n}\n\\]\nwhere $X_d := X \\times_T T$.\nBy flat base change \\cite[Prop.9.3]{Har77} we have\n\\[\n\\mu_d^*(\\varphi_* (\\sO_X(L))\\hspace{-0.8ex}<\\hspace{-0.8ex}\\frac{1}{4^d}H\\hspace{-0.8ex}>)\n\\sim_\\ensuremath{\\mathbb{Q}}\n\\tilde \\varphi_* (\\sO_{X_d}(\\tilde \\mu_d^*L))\\hspace{-0.8ex}<\\hspace{-0.8ex}\\frac{1}{4^d}\\mu_d^* H\\hspace{-0.8ex}>\n\\]\nSince $\\frac{1}{4^d} \\mu_d^* H \\sim_\\ensuremath{\\mathbb{Q}} H$, we see that\n$\\mu_d^*(\\varphi_* (\\sO_X(L))\\hspace{-0.8ex}<\\hspace{-0.8ex}\\frac{1}{4^d}H\\hspace{-0.8ex}>)\n\\sim_\\ensuremath{\\mathbb{Q}}\n\\tilde \\varphi_* (\\sO_{X_d}(\\tilde \\mu_d^*L)) \\otimes H\n$\nwhich by the first statement is ample. \n\\end{proof}\n\n\\begin{proposition} \\label{propositionkltdirectimage}\nLet $X$ be a normal projective variety and $\\Delta$ a boundary divisor on $X$ such that\nthe pair $(X, \\Delta)$ is klt. Let \\holom{\\varphi}{X}{T} be a flat fibration onto a\nsmooth curve or an abelian\nvariety. Suppose that $-(K_{X\/T}+\\Delta)$ is nef and $\\varphi$-ample.\nThen \n$$\nE_{m}:=\\varphi_{*}(\\sO_X(-m (K_{X\/T}+\\Delta)))\n$$ \nis a numerically flat vector bundle\nfor all sufficiently large and divisible $m \\gg 0$.\n\\end{proposition}\n\n\\begin{proof} \nFor sufficiently divisible $m \\in \\ensuremath{\\mathbb{N}}$ the $\\ensuremath{\\mathbb{Q}}$-divisor $m (K_{X\/T}+\\Delta)$ is integral\nand Cartier.\n\n{\\em 1st case. $T$ is a curve.} Then $E_m$ is nef \\cite{Kol86},\nand for $m \\gg 0$ we have an inclusion\n$X \\hookrightarrow \\ensuremath{\\mathbb{P}}(E_m)$.\nWe can now argue as in Lemma \\ref{lemmanumericallyflat}: if $E_m$ is not numerically flat, there\nexists an ample subbundle $A \\subset E_m$. By \nLemma \\ref{lemmanumericaldimensionprojective} we have\n$(K_{X\/T}+\\Delta)^{\\dim X}=0$, so Lemma \\ref{lemmaelementary} implies that $X \\subset \\ensuremath{\\mathbb{P}}(E_m\/A)$.\nHowever this is not possible since the embedding of the general fibre $X_t$ in $\\ensuremath{\\mathbb{P}}((E_m)_t)$ is linearly nondegenerate.\n\n{\\em 2nd case. $T$ is an abelian variety.} By Lemma \\ref{lemmakltdirectimage}\nthe sheaf $E_m$\nis a nef vector bundle.\nIf $C$ is a general complete intersection curve in $T$,\ndenote by $X_C:=\\fibre{\\varphi}{C}$ its preimage. Then the pair\n$(X_C, \\Delta_C:=\\Delta \\cap X_C)$ is klt and the relative canonical divisor\n$-(K_{X_C\/C}+\\Delta_C)$\nis nef and relatively ample. By the first case $\\varphi_{*}(E_m \\otimes \\sO_C)$\nis numerically flat, so $\\det E_m$ is numerically \ntrivial \\cite[3.8]{Deb01}.\n\\end{proof}\n\n\n\n\\section{Proof of Theorem \\ref{theoremmain}}\n\nLet $X'$ be a normal compact K\\\"ahler space that admits a flat fibration $\\pi': X' \\rightarrow T$ \nonto a compact K\\\"ahler manifold $T$. Let $L$ be a line bundle on $X$ that is $\\pi'$-ample,\nfor all $m \\in \\ensuremath{\\mathbb{N}}$ we set $E_m := (\\pi')_* (\\sO_{X'}(mL))$. We fix a $m \\gg 0$ such that we have\nan embedding $X' \\hookrightarrow \\mathbb{P}(E_{m})$, for simplicity's sake we denote by\n$\\holom{\\pi'}{\\ensuremath{\\mathbb{P}}(E_m)}{T}$ the natural map. Let $\\sI_{X'} \\subset \\sO_{\\ensuremath{\\mathbb{P}}(E_m)}$ be the ideal\nsheaf of $X'$. Then we define for every $d \\in \\ensuremath{\\mathbb{N}}$\n$$\nS_{m, d} := (\\pi')_{*}(\\sI_{X'}\\otimes \\mathcal{O}_{\\mathbb{P}(E_{m})}(d)).\n$$\n\n\\begin{proposition} \\label{propositionloctriv}\nIn the situation above suppose that\n$E_{m}$ and $S_{m, d}$ are numerically flat vector bundles\nfor all $m \\gg 1$ and $d\\gg 1$. Then the fibration $\\pi'$ is locally trivial.\n\\end{proposition}\n\n\\begin{proof}\nWe have a natural inclusion $i: S_{m, d}\\hookrightarrow S^d E_m$.\nSince numerically flat vector bundles are local systems \n(cf. \\cite[Sect.3]{Sim92} for general case or \\cite[Lemma 6.5]{Ver04} for numerically flat vector bundles),\n$S_{m, d}$ and $S^d E_m$ are local systems on $T$.\nLet $U$ be any small Stein open set in $T$, and let $e_{1},\\cdots, e_{k}$ be a local constant coordinates of $S_{m, d}$\nover $U$.\nNote that $\\Hom (S_{m, d}, S^d E_m)$ is also a local system on $T$, and \n$i\\in H^{0}(T, \\Hom (S_{m, d}, S^d E_m))$.\nBy \\cite[Lemma 4.1]{Cao12}, $i$ is \nparallel\\footnote{It is not true that for any local system, the global sections are parallel.\nHowever, it is true for numerically flat bundles.}\nwith respect to the local system \n$\\Hom (S_{m, d}, S^d E_m)$.\nTherefore the images of $e_{1},\\cdots, e_{k}$ in $S^d E_m$ are also locally constant,\ni.e. the polynomials defining the fibers $X'_{t}$ for $t\\in U$ are locally constant.\nIn particular the fibration $\\pi'$ is locally trivial.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{theoremmain}]\n{\\em Step 1. The relative anticanonical fibration is locally trivial.}\nWe follow the argument of \\cite[Thm.3.20]{DPS94}. Denote by $X'$ the relative anticanonical model of $X$ \n(cf. Section \\ref{subsectionanticanonical}).\nFix $m \\gg 0$ such that we have an inclusion $X'\\hookrightarrow \\mathbb{P}(E_{m})$\nwhere $E_m$ is defined as in Lemma \\ref{lemmaflat} and denote by $\\pi'$ both the anticanonical fibration\n$X' \\rightarrow T$ and $\\ensuremath{\\mathbb{P}}(E_m) \\rightarrow T$. \nFor every $d \\in \n\\ensuremath{\\mathbb{N}}$ we have an exact sequence\n$$\n0 \\rightarrow \\sI_{X'}\\otimes \\mathcal{O}_{\\mathbb{P}(E_{m})}(d) \\rightarrow\n\\mathcal{O}_{\\mathbb{P}(E_{m})}(d) \\rightarrow \\omega_{X'}^{\\otimes -md} \\rightarrow 0.\n$$\nSince $\\sO_{\\ensuremath{\\mathbb{P}}(E_{m})}(1)$ is $\\pi'$-ample we get for $d \\gg 0$ an exact sequence\n$$\n0 \\rightarrow (\\pi')_{*}(\\sI_{X'}\\otimes \\mathcal{O}_{\\mathbb{P}(E_{m})}(d)) \n\\rightarrow S^d E_m \\rightarrow E_{md} \\rightarrow 0.\n$$\nThe vector bundles $S^d E_m$ and $E_{md}$ are numerically flat, so $(\\pi')_{*}(\\sI_{X'}\\otimes \\mathcal{O}_{\\mathbb{P}(E_{m})}(d))$\nis numerically flat. Now apply Proposition \\ref{propositionloctriv}.\n\n{\\em Step 2. The Albanese map $\\pi$ is locally trivial.} Let $F'$ be the general fibre of the relative anticanonical fibration, and\nlet $F_t$ be any smooth $\\pi$-fibre.\nThen $F_t$ is a weak Fano manifold, and $\\holom{\\mu|_{F_t}}{F_t}{F'}$ is a crepant birational morphism,\nso $F_t$ is a terminal model of the Fano variety $F'$.\nBy \\cite[Cor.1.1.5]{BCHM10} there are only finitely many terminal models of $F'$. \nThus there are only finitely many possible isomorphism classes for $F_t$,\nhence there exists a non-empty Zariski open subset $T_0 \\subset T$ such that $\\fibre{\\pi}{T_0} \\rightarrow T_0$ \nis locally trivial with fibre $F$. Fix now an ideal sheaf $\\sI$ on $F'$ such that $F$ is isomorphic to the blow-up of $F'$\nalong $\\sI$.\n\nLet $t \\in T$ be an arbitrary point, and let $t \\in U \\subset T$ be an analytic neighbourhood such that\n$X'_U:=\\fibre{(\\pi')}{U} \\simeq U \\times F'$. Let \\holom{\\tilde \\mu}{\\tilde X_U}{X'_U} be the blow-up\nof $X'_U$ along the ideal sheaf $\\sI \\otimes \\sO_U$, then $\\tilde X_U \\simeq U \\times F$.\nIn particular $\\tilde X_U$ is smooth and the birational morphism $\\tilde \\mu$ is crepant.\nSet $X_U :=\\fibre{\\pi}{U}$, then $X_U$ is also smooth and $\\holom{\\mu|_{X_U}}{X_U}{X'_U}$\nis crepant. Thus $X_U$ and and $\\tilde X_U$ are both minimal models over the base $X'_U$ (cf. \\cite[Defn.3.48]{KM98}),\nhence the induced birational morphism $\\tau: X_U \\dashrightarrow X'_U$ is an isomorphism in codimension one \\cite[Thm.3.52]{KM98}.\nMoreover by the universal property of the blow-up the restriction of $\\tau$\nto \n$$\n(X_U \\cap \\fibre{\\pi}{T_0}) \\dashrightarrow (X'_U \\cap \\fibre{(\\pi' \\circ \\mu)}{T_0})\n$$ \nis an isomorphism.\nLet $H$ be an effective $\\pi$-ample divisor on $X_U$, and let $H':=\\tau_* H$ be its strict transform.\nThen $H'$ is $\\pi' \\circ \\mu$-nef: indeed if $C$ is any (compact) curve in $\\tilde X_U \\simeq U \\times F$, it\ndeforms to a curve $C'$ that is contained in $e \\times F$ with $e \\in (U \\cap T_0)$. Yet the restriction \nof $H'$ to $e \\times F$ is ample, since $\\tau$ is an isomorphism in a neighbourhood of $e \\times F$.\nThus we see that $H' \\cdot C = H' \\cdot C'>0$. Since $X_U$ is smooth we satisfy the conditions\nof \\cite[Lemma 6.39]{KM98}, so $\\tau$ is an isomorphism.\nThis shows that $X_U \\rightarrow U$ is locally trivial with fibre $F$, since $t \\in T$ is arbitrary this concludes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Corollary \\ref{corollarymain}]\nSuppose first that $q(X)=\\dim X-1$. If $-K_X$ is $\\pi$-big we see by Theorem \\ref{theoremmain} that the Albanese map is a $\\ensuremath{\\mathbb{P}}^1$-bundle. If $-K_X$ is not $\\pi$-big, the general fibre is an elliptic curve. Note that the $C_{n, n-1}$-conjecture is also known\nin the K\\\"ahler case \\cite[Thm.2.2]{Uen87}, so we see that $\\kappa(X) \\geq 0$. Yet $-K_X$ is nef, so we see that $-K_X \\equiv 0$.\nWe conclude by the Beauville-Bogomolov decomposition. \n\\end{proof}\n\nThe proof of Theorem \\ref{theoremmain} works also in the following relative situation which we will use\nin Section \\ref{sectiondimensiontwo}.\n\n\\begin{corollary} \\label{corollaryMFS}\nLet $X$ be a normal $\\ensuremath{\\mathbb{Q}}$-factorial projective variety with at most terminal singularities,\nand let $\\holom{\\varphi}{X}{C}$ be a fibration onto a smooth curve \nsuch that $-K_{X\/C}$ is nef and $\\varphi$-big. If the general fibre is smooth,\nthen $\\varphi$ is locally trivial in the analytic topology.\n\\end{corollary}\n\n\n\n\\section{Two-dimensional fibres} \\label{sectiondimensiontwo}\n\nIn this section we will prove Theorem \\ref{theoremmaintwo}. While the positivity of direct image sheaves\nwill still play an important role, we need some additional geometric information to deduce the smoothness of the Albanese map.\nThis information will be obtained by describing very precisely the MMP.\n\n\\subsection{Reduction to the curve case}\n\nThe following lemma shows that the nefness condition imposes strong restrictions on the singularities\nof a fibre space.\n\n\\begin{lemma} \\label{lemmaimportant}\nLet $M$ be a normal projective variety, and let $\\holom{\\varphi}{M}{C}$ be a fibration onto a curve such\nthat $-K_{M\/C}$ is $\\ensuremath{\\mathbb{Q}}$-Cartier and nef.\nLet $\\Delta$ be a boundary divisor on $M$ such that $\\Delta \\equiv -\\alpha K_{M\/C}$ for some $\\alpha \\in [0, 1]$.\n\nIf the pair $(M, \\Delta)$ is lc (resp. klt) over the generic point of $C$, the pair \n$(M, \\Delta)$ is lc (resp. klt).\nIf the pair $(M, \\Delta)$ is lc, every lc center of $(M, \\Delta)$ surjects onto $C$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\holom{\\mu}{M'}{M}$ be a dlt blow-up \\cite[Thm.10.4]{Fuj11}, i.e. $\\mu$ is birational morphism from $M'$ a normal $\\ensuremath{\\mathbb{Q}}$-factorial\nvariety such that if we set\n$$\n\\Delta' := \\mu_*^{-1} \\Delta + \\sum_{E_i \\ \\mbox{\\tiny $\\mu$-exc.}} E_i,\n$$\nthen $(M', \\Delta')$ is dlt. Moreover one has\n$$\nK_{M'} +\\Delta' = \\mu^* (K_M+\\Delta) + \\sum_{E_i \\ \\mbox{\\tiny $\\mu$-exc.}} (a_i+1) E_i,\n$$\nand $a_i \\leq -1$ for all $i$. Let $\\Delta'=\\Delta'_{hor}+\\Delta'_{vert}$ be the decomposition\nin the horizontal and vertical part.\nSince $M'$ is $\\ensuremath{\\mathbb{Q}}$-factorial, the pair $(M', \\Delta'_{hor})$ is dlt \\cite[Cor.2.39]{KM98} and\n$$\nK_{M'} +\\Delta'_{hor}= \\mu^* (K_M+\\Delta) + \\sum_{E_i, a_i<-1} (a_i+1) E_i - \\Delta'_{vert}.\n$$\nSuppose now that $(M, \\Delta)$ is lc over the generic point of $C$.\nThen the restriction of $\\sum_{E_i \\ \\mbox{\\tiny $\\mu$-exc.}} (a_i+1) E_i$ \nto a general $(\\varphi \\circ \\mu)$-fibre $F$ is empty. Thus the anti-effective divisor\n$\n- E := \\sum_{E_i, a_i<-1} (a_i+1) E_i - \\Delta'_{vert}\n$\nis vertical. We claim that $E=0$. Assuming this for the time being, let us see how to deduce all the statements:\nsince $\\sum_{E_i, a_i<-1} (a_i+1) E_i=0$ we know that $(M, \\Delta)$ is lc. Moreover any lc centre $W$ surjects onto $C$:\notherwise there exists a vertical divisor $E_i$ with discrepancy $-1$. Thus we have $\\Delta'_{vert} \\neq 0$, a contradiction.\n\nIf $(M, \\Delta)$ is klt over the generic point of $C$, it is lc by what precedes. \nMoreover any lc centre would be contained in a $\\varphi$-fibre, so it would\ngive a non-zero component of $\\Delta'_{vert}$. However we just proved that $\\Delta'_{vert}=0$.\n\n\n{\\em Proof of the claim.}\nSince $- \\mu^* (K_{M\/C}+\\Delta) \\equiv (\\alpha-1) \\mu^* K_{M\/C}$ is nef, \nwe know that for $H$ an ample Cartier divisor on $M'$ and all $\\delta>0$, the divisor\n$-\\mu^* (K_{M\/C}+\\Delta)+\\delta H$ is ample. \nThus there exists an effective $\\ensuremath{\\mathbb{Q}}$-divisor $B \\sim_\\ensuremath{\\mathbb{Q}} -\\mu^* (K_{M\/C}+\\Delta) + \\delta H$ \nsuch that the pair $(M', \\Delta'_{hor}+B)$ is dlt. Thus we have\n$$\nK_{M'\/C} + \\Delta'_{hor} + B \\sim_\\ensuremath{\\mathbb{Q}} \\mu^* (K_{M\/C}+\\Delta) - E - \\mu^* (K_{M\/C}+\\Delta) +\\delta H \\sim_\\ensuremath{\\mathbb{Q}} - E +\\delta H. \n$$\nSince $E$ does not dominate $C$, the restriction of $K_{M'\/C} + \\Delta'_{hor} + B$ to the general \n$(\\varphi \\circ \\mu)$-fibre $F$ \nis numerically equivalent to $\\delta H|_F$. In particular for $m \\in \\ensuremath{\\mathbb{N}}$ sufficiently large and divisible the divisor\n$m (K_{M'\/C} + \\Delta'_{hor} + B)$ is Cartier and the restriction to $F$ has global sections.\nThe pair $(M', \\Delta'_{hor}+B)$ being lc we know by \\cite[Thm.4.13]{Cam04} that \n$(\\varphi \\circ \\mu)_* (m (K_{M'\/C} + \\Delta'_{hor} + B))$ is a nef vector bundle.\nThe natural morphism\n$$ \n(\\varphi \\circ \\mu)^* (\\varphi \\circ \\mu)_* (m (K_{M'\/C} + \\Delta'_{hor} + B)) \\rightarrow m (K_{M'\/C} + \\Delta'_{hor} + B)\n$$\nis not zero, so $m (K_{M'\/C} + \\Delta'_{hor} + B)$ is pseudoeffective.\nThus we see that $- E +\\delta H$ is pseudoeffective. \nTaking the limit $\\delta \\rightarrow 0$ we deduce that the anti-effective divisor $-E$ is pseudoeffective. This proves the claim.\n\\end{proof}\n\n{\\bf Attribution.}\nThe proof above is a refinement of\nthe argument in \\cite{LTZZ10}. Note that our argument can be used\nto give a simplified proof of \\cite[Thm.]{LTZZ10}.\n\n\n\n\\begin{corollary} \\label{corollaryreductioncurve}\nLet $X$ be a projective manifold such that $-K_X$ is nef. Let $\\holom{\\pi}{X}{T}$ be the Albanese map.\nFix an arbitrary point $t \\in T$ and let $C \\subset T$ be a smooth curve such that $t \\in C$\nand for $c \\in C$ general, the fibre $\\fibre{\\pi}{c}$ is smooth.\n\nThen $\\fibre{\\pi}{C}$ is a normal variety with at most canonical singularities. \n\\end{corollary}\n\n\\begin{proof}\nBy \\cite[Thm.]{LTZZ10} the fibration $\\pi$ is flat, so $\\fibre{\\pi}{C}$ is Gorenstein and has pure dimension $\\dim X-\\dim T+1$.\nThe general fibre of $\\fibre{\\pi}{C} \\rightarrow C$ is irreducible, so $\\fibre{\\pi}{C}$ is irreducible. \nSince all the $\\pi$-fibres are reduced \\cite[Thm.]{LTZZ10} we see that \n$\\fibre{\\pi}{C}$ is smooth in codimension one, hence normal. \nThe relative anticanonical divisor $-K_{\\fibre{\\pi}{C}\/C}= -K_X|_{\\fibre{\\pi}{C}}$ is Cartier and nef.\nThe general fibre of $\\fibre{\\pi}{C} \\rightarrow C$ is smooth, so $\\fibre{\\pi}{C}$ is smooth over the generic point of $C$.\nThus the pair $(\\fibre{\\pi}{C}, 0)$ is klt by Lemma \\ref{lemmaimportant}. Since $\\fibre{\\pi}{C}$ is Gorenstein, it has at most canonical singularities.\n\\end{proof}\n\n\n \nCorollary \\ref{corollaryreductioncurve} reduces Conjecture\n\\ref{conjecturealbanese} to the following problem:\n\n\\begin{conjecture} \\label{conjecturerelativealbanese}\nLet $M$ be a normal projective variety with at most canonical Gorenstein singularities.\nLet \\holom{\\varphi}{M}{C} be a fibration onto a smooth curve $C$ such that\n$-K_{M\/C}$ is nef. If the general fibre $F$ is smooth, then $\\varphi$ is smooth.\nIf $F$ is smooth and simply connected, then $\\varphi$ is locally trivial in the analytic topology.\n\\end{conjecture} \n\n\\subsection{Running the MMP for fibres of dimension two} \\label{subsectionMMP}\n\nIn the previous section we reduced Conjecture \\ref{conjecturealbanese} to the case \nof a fibration onto a curve such that the total space has canonical singularities. In this section we will\nmake the stronger assumption that the total space is $\\ensuremath{\\mathbb{Q}}$-factorial with terminal singularities.\nMoreover we will assume at some point the existence of an effective relative anticanonical divisor.\nIn Subsection \\ref{subsectionmainresult} we will see how to verify these additional conditions.\n\n\\begin{setup} \\label{setup}\n{\\rm\nLet $X_C$ be a normal $\\ensuremath{\\mathbb{Q}}$-factorial projective variety with at most terminal singularities, and \nlet \\holom{\\varphi}{X_C}{C} be a fibration onto a smooth curve $C$. Suppose that the general $\\varphi$-fibre is uniruled.\nBy \\cite{BCHM10} we know that $X\/C$ is birational to a Mori fibre space, and we denote by\n\\begin{equation} \\label{MMP}\nX_C=:X_0 \\stackrel{\\mu_0}{\\dashrightarrow} X_1 \\stackrel{\\mu_1}{\\dashrightarrow}\n\\ldots \\stackrel{\\mu_k}{\\dashrightarrow} X_k \n\\end{equation}\na MMP over $C$, that is for every $i \\in \\{0, \\ldots, k-1\\}$ the map\n$\\merom{\\mu_i}{X_i}{X_{i+1}}$ is either a divisorial Mori contraction of a\n$K_{X_i\/C}$-negative extremal ray in $\\NE{X_i\/C}$ or the flip of a small contraction \nof such an extremal ray. Note that all the varieties $X_i$ \nare normal $\\ensuremath{\\mathbb{Q}}$-factorial with at most terminal singularities and endowed with a\nfibration $X_i \\rightarrow C$.\nMoreover $X_k \\rightarrow C$ admits a Mori fibre space structure \\holom{\\psi}{X_k}{Y} onto a normal variety\n$\\holom{\\tau}{Y}{C}$. We know \nthat $Y$ is $\\ensuremath{\\mathbb{Q}}$-factorial \\cite[Prop.7.44]{Deb01}, moreover $Y$ has at most klt singularities \\cite[Thm.0.2]{Amb05}.\n}\n\\end{setup}\n\n\\begin{remark} \\label{remarkflip}\nIf $\\mu_i$ is a flip, denote by $\\Gamma_i$ the normalisation of its graph and by $\\holom{p_i}{\\Gamma_i}{X_i}$\nand $\\holom{q_i}{\\Gamma_i}{X_{i+1}}$ the natural maps. \nBy a well-known discrepancy computation \\cite[Lemma 9-1-3]{Mat02} we have\n\\begin{equation} \\label{flipdiscrepancy}\np_i^* (-K_{X_i\/C}) = q_i^*(-K_{X_{i+1}\/C}) - \\sum a_{i, j} D_{i,j}\n\\end{equation}\nwith $a_{i,j}>0$ where the sum runs over all the $q_i$-exceptional divisors. \n\\end{remark}\n\n\\begin{remark} \\label{remarkdivisorial}\nIf $\\mu_i$ is a divisorial contraction, let $D_i \\subset X_i$ be the exceptional divisor. We have \n\\begin{equation} \\label{divisorialdiscrepancy}\n-K_{X_i\/C} = \\mu_i^*(-K_{X_{i+1}\/C}) - \\lambda_i D_i\n\\end{equation}\nwith $\\lambda_i>0$.\n\\end{remark}\n\nWe explained in the introduction that the nefness of $-K_{X_C\/C}$ is usually not preserved under the MMP. However\nwe have the following:\n\n\\begin{lemma} \\label{lemmaMMPbasic}\nIn the situation of Setup \\ref{setup}, suppose that\n\\begin{enumerate}[(i)]\n\\item $-K_{X_C\/C}$ is pseudoeffective. Then $-K_{X_i\/C}$ is pseudoeffective. \n\\item $-K_{X_C\/C}$ is nef in codimension one. Then $-K_{X_i\/C}$ is nef in codimension one. \n\\item $-K_{X_C\/C}$ is nef. If $B \\subset X_i$ is a curve such that $-K_{X_i\/C} \\cdot B<0$, then $B$ is \n(a strict transform of) a curve contained in the flipping locus or the image of a divisorial contraction. \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{remark} \\label{remarkMMPdimtwo}\nA $\\ensuremath{\\mathbb{Q}}$-Cartier divisor on a surface is nef in codimension one if and only if it is nef.\nThus the lemma shows that for {\\em surfaces} the MMP preserves the property that $-K_{X_C\/C}$ is nef.\n\\end{remark}\n\n\\begin{proof} Our proof follows the arguments in \\cite[Prop.2.1, Prop.2.2]{PS98}.\nLet $\\Gamma$ be the normalisation of the graph of the birational map $X \\dashrightarrow X_i$, and\ndenote by $\\holom{p}{\\Gamma}{X}$ and $\\holom{q}{\\Gamma}{X_i}$ the natural maps. By\n\\eqref{flipdiscrepancy} and \\eqref{divisorialdiscrepancy} we have\n$$\np^* (-K_{X_C\/C}) = q^* (-K_{X_i\/C}) - D,\n$$\nwith $D$ an effective $q$-exceptional $\\ensuremath{\\mathbb{Q}}$-divisor. In particular\nif $B \\subset X'$ is a curve that is not contained in $q(D)$ and $B_\\Gamma \\subset \\Gamma$ its strict transform, \nthen we have\n\\begin{equation} \\label{increase}\np^*(-K_{X_C\/C}) \\cdot B_\\Gamma \\leq -K_{X_i\/C} \\cdot B.\n\\end{equation}\nThis proves the statements $(i)$ and $(iii)$. \n\nSuppose now that \n$-K_{X_C\/C}$ is nef in codimension one. Let $S \\subset X_i$ be a prime divisor, and\ndenote by $S_\\Gamma \\subset \\Gamma$ its strict transform. Since $X_i \\dashrightarrow X$ does not contract\na divisor, we see that $S_\\Gamma$ is not $p$-exceptional. Hence $p^*(-K_{X_C\/C})|_{S_\\Gamma}$\nis pseudoeffective. Thus we see that $q^*(-K_{X_i\/C})|_{S_\\Gamma}$ is pseudoeffective,\nhence $-K_{X_i\/C}|_S$ is pseudoeffective.\n\\end{proof}\n\nWe will now restrict ourselves to the case where $\\dim X_C-\\dim C=2$.\nThis allows to study\nthe positivity of $-K_\\bullet$ on horizontal curves. \n\n\\begin{definition} \\label{definitionnefoverC}\nLet $M$ be a projective variety admitting a fibration $\\holom{f}{M}{C}$ onto a curve $C$.\nA $\\ensuremath{\\mathbb{Q}}$-Cartier divisor $L$ on $M$ is nef over $C$ if \nfor any curve $B \\subset M$ such that $L \\cdot B<0$, the image $f(B)$ is a point.\n\\end{definition}\n\n\\begin{remark} \\label{remarkflippingloci}\nNote that if $\\dim X_C-\\dim C=2$, the exceptional locus\nof a small contraction cannot surject onto $C$, so if $\\mu_i$ is the corresponding step of the MMP,\nthe flipping loci are contained in\nfibres of the maps $X_i \\rightarrow C$ and $X_{i+1} \\rightarrow C$.\nIn particular if $-K_{X_i\/C}$ is nef over $C$, then $-K_{X_{i+1}\/C}$ is nef over $C$\nby \\eqref{increase}. The same holds if the exceptional divisor $D_i$ of a divisorial\ncontraction does not surject onto $C$.\n\\end{remark}\n\n\\begin{lemma} \\label{lemmanefoverC}\nIn the situation of Setup \\ref{setup}, suppose that $\\dim X_C-\\dim C=2$ and $-K_{X_i\/C}$\nis nef over $C$. Then $-K_{X_{i+1}\/C}$ is nef over $C$.\n\\end{lemma}\n\n\\begin{proof}\nBy what precedes it is sufficient to consider the case where \n$\\holom{\\mu_i}{X_i}{X_{i+1}}$ contracts a divisor $D_i$\nonto a curve $C_0$ such that $C_0 \\rightarrow C$ is surjective.\n\nWe will argue by contradiction, so suppose that $-K_{X_{i+1}\/C} \\cdot C_0 < 0$. \nSince $-K_{X_i\/C}$ is nef over $C$, the restriction of $-K_{X_i\/C}$ to \n$D_i \\rightarrow C_0$ is nef over $C_0$. \nSince $\\mu_i$ is a Mori contraction $-K_{X_i\/C}|_{D_i}$ is $\\mu_i|_{D_i}$-ample,\nso $-K_{X_i\/C}|_D$ is nef. By \\eqref{divisorialdiscrepancy} one has\n$$\n-K_{X_i\/C} = \\mu_i^*(-K_{X_{i+1}\/C}) - \\lambda_i D_i.\n$$\nIn particular if $B \\subset D_i$ is any curve that is not contracted by $\\mu_i$, then\n$$\n- \\lambda_i D_i \\cdot B = -K_{X_i\/C} \\cdot B + K_{X_{i+1}\/C} \\cdot (\\mu_i)_*(B) > 0.\n$$\nSince $-D_i$ is $\\mu_i|_{D_i}$-ample, this shows that $-D_i|_{D_i}$ is positive on all curves. \nMoreover we have\n$$\n(- \\lambda_i D_i|_{D_i})^2 = (-K_{X_i\/C}|_{D_i})^2 + 2 (-K_{X_i\/C}) \\cdot (\\mu_i^* K_{X_i\/C}) \\cdot D_i > 0,\n$$\nso by the Nakai-Moishezon criterion the divisor $-D_i|_{D_i}$ is ample. \nThus we see that $-(K_{X_i\/C}+D_i)|_{D_i}$ is ample. Let now \n$\\holom{\\nu}{\\hat D_i}{D_i}$ be the composition of normalisation and minimal resolution, then by the adjunction formula \\cite{Rei94}\n$$\n- K_{\\hat D_i\/C} = -\\nu^* (K_{X_i\/C}+D_i)|_{D_i} + F,\n$$\nwith $F$ an effective divisor. Since $D_i \\rightarrow C_0$ is generically a $\\ensuremath{\\mathbb{P}}^1$-bundle, the divisor $F$\ndoes not surject onto $C_0$. Thus we see that $-K_{\\hat D_i\/C}$ is nef over $C$, moreover it is big.\nLet $\\hat D_i \\rightarrow \\tilde D_i$ be a MMP over $C$ with $\\tilde D_i \\rightarrow C$ a Mori fibre space.\nThen $-K_{\\tilde D_i\/C}$ is nef and big, a contradiction to Lemma \\ref{lemmanumericaldimensionprojective}.\n\\end{proof}\n\nCombining Lemma \\ref{lemmaMMPbasic} and Lemma \\ref{lemmanefoverC} we obtain:\n\n\\begin{corollary} \\label{corollarynefoverC}\nIn the situation of Setup \\ref{setup}, suppose that $\\dim X_C-\\dim C=2$ and $-K_{X_C\/C}$ is nef.\nThen for every $i \\in \\{1, \\ldots, k\\}$ the divisor $-K_{X_i\/C}$ is nef in codimension one and nef over $C$.\n\\end{corollary}\n \nOur goal will be to prove that $-K_{X_i\/C}$ is actually nef, but this needs some serious preparation.\n\n\\begin{lemma} \\label{lemmanu1surfaces}\nLet $S$ be an integral projective surface admitting a morphism $\\holom{f}{S}{C}$ onto a curve $C$.\nLet $L$ be a $\\ensuremath{\\mathbb{Q}}$-Cartier divisor on $S$ that is pseudoeffective and nef over $C$.\n \nLet $S'$ be an integral projective surface admitting a morphism $\\holom{f'}{S'}{C}$ onto $C$. \nLet $L'$ be a nef and $f'$-ample $\\ensuremath{\\mathbb{Q}}$-Cartier divisor on $S'$ such that $(L')^2=0$.\n\nSuppose that there exists a birational morphism $\\holom{\\mu}{S}{S'}$ such that $f' \\circ \\mu = f$.\nSuppose that we have\n$$\nL \\equiv \\mu^* L' - E\n$$\nwith $E$ an effective $\\ensuremath{\\mathbb{Q}}$-Cartier divisor. Then the following holds:\n\n\\begin{enumerate}[(i)]\n\\item If $L$ is $\\mu$-nef, then $L$ is nef and not big. Moreover we have $L \\cdot \\mu^* L' = 0$.\n\\item If $L$ is $\\mu$-nef and $E$ is $f$-vertical, then $E=0$.\n\\end{enumerate}\n\\end{lemma}\n\nRecall that if $L$ is a pseudoeffective $\\ensuremath{\\mathbb{Q}}$-divisor on a smooth projective surface,\nwe can consider its Zariski decomposition\n$L = P + N$,\nwhere $P$ is a nef $\\ensuremath{\\mathbb{Q}}$-divisor and $N$ an effective $\\ensuremath{\\mathbb{Q}}$-divisor such that $P \\cdot N=0$.\n\n\\begin{proof}\nThe statement being invariant under normalisation we can suppose without loss of generality that $S$ and $S'$\nare normal. For the same reason we can suppose that $S$ is smooth.\n\nLet $L = P + N$ be the Zariski decomposition of $L$, and\nlet $N = N_{hor} + N_{vert}$ be the decomposition of the \neffective divisor $N$ in its vertical and horizontal components. \n\nSince $P \\equiv \\mu^* L' - (E+N)$ and $(L')^2=0$, we have\n$$\nP \\cdot \\mu^* L' = -(E+N) \\cdot \\mu^* L'.\n$$\nSince $\\mu^* L$ and $P$ are nef, the left hand side is non-negative. \nHowever $-(E+N) \\cdot \\mu^* L'$ is non-positive, so we get\n\\begin{equation} \\label{orthogonal}\nP \\cdot \\mu^* L' = (E+N) \\cdot \\mu^* L' = 0.\n\\end{equation}\nIn particular for every irreducible component $D \\subset N$ we have $D \\cdot \\mu^* L'=0$.\nSince $L'$ is $f'$-ample and $N_{vert} \\cdot \\mu^* L'=0$ we see that $N_{vert}$ is $\\mu$-exceptional.\n\nThe divisor $P$ being nef, we have $P^2 \\geq 0$\nand $(E+N) \\cdot P \\geq 0$.\nHowever by \\eqref{orthogonal} one has\n$$\nP^2 = [\\mu^* L' - (E+N)] \\cdot P = - (E+N) \\cdot P,\n$$\nso we get $P^2 = (E+N) \\cdot P = 0$.\nYet $(E+N) \\cdot P = 0$ and $(E+N) \\cdot \\mu^* L' = 0$ implies that \n\\begin{equation} \\label{square}\n(E+N)^2 = 0.\n\\end{equation} \n\n{\\em Proof of $(i)$.}\nSince $N_{vert}$ is $\\mu$-exceptional, we know that $L$ is nef on every irreducible\ncomponent of $N_{vert}$. Moreover $L$ is nef over $C$, so it is nef \non every irreducible\ncomponent of $N_{hor}$. Thus $L$ is nef on its negative part $N$, hence $N=0$ and $L$ is nef.\nIn particular we have $L^2=P^2=0$, so $L$ is not big.\n\n\n{\\em Proof of $(ii)$.} Since $E$ is $f$-vertical and $L'$ is $f'$-ample, \nthe equality $E \\cdot \\mu^* L'=0$ implies that $E$ is $\\mu$-exceptional.\nSince $N=0$ we know by \\eqref{square} that $E^2=0$.\nHowever the intersection matrix of an exceptional divisor is \nnegative definite, so we have $E=0$.\n\\end{proof}\n\nFor the next corollary recall that any normal surface is numerically $\\ensuremath{\\mathbb{Q}}$-factorial \\cite{Sak84}, so\nwe can define intersection numbers for any Weil divisor. \n\n\\begin{corollary} \\label{corollarysurfaces}\nLet $Y$ be a normal projective surface \nadmitting a fibration $\\holom{\\tau}{Y}{C}$ onto a smooth curve such that the general fibre is $\\ensuremath{\\mathbb{P}}^1$. \nSuppose that $-K_{Y\/C}$ is pseudoeffective and nef over $C$.\nThen $-K_{Y\/C}$ is nef and $Y \\rightarrow C$ is a $\\ensuremath{\\mathbb{P}}^1$-bundle.\n\\end{corollary}\n\n\\begin{proof} \n{\\em Step 1. Suppose that $Y$ is smooth.}\nWe argue by induction on the relative Picard number. If $\\rho(Y\/C)=1$, then $-K_{Y\/C}$ is nef\nand $\\tau$-ample. Thus $Y \\rightarrow C$ is a $\\ensuremath{\\mathbb{P}}^1$-bundle.\nIf $\\rho(Y\/C)>1$ there exists a Mori contraction $\\holom{\\mu}{Y}{Y'}$ over $C$\nand by Lemma \\ref{lemmaMMPbasic} the divisor $-K_{Y'\/C}$ is pseudoeffective and nef over $C$.\nBy induction $-K_{Y'\/C}$ \nis nef and $Y' \\rightarrow C$ is a $\\ensuremath{\\mathbb{P}}^1$-bundle. We have $(-K_{Y'\/C})^2=0$ and $-K_{Y'\/C}$ \nis ample over $C$. The $\\mu$-exceptional divisor $E$\nis $\\tau$-vertical, so $E=0$ by Lemma \\ref{lemmanu1surfaces}, a contradiction. \n\n\n{\\em Step 2. General case.} \nLet $\\holom{\\nu}{\\hat Y}{Y}$ be the minimal resolution, then we have $K_{\\hat Y} = \\nu^* K_Y - E$ with $E$ an effective divisor. \nThus the relative canonical divisor $-K_{\\hat Y\/C} = - \\nu^* K_{Y\/C} + E$\nis pseudoeffective and nef over $C$. By Step 1 we know that $\\hat Y \\rightarrow C$\nis a $\\ensuremath{\\mathbb{P}}^1$-bundle. Thus $\\nu$ is an isomorphism.\n\\end{proof}\n\n\\begin{remark} \\label{remarksurfaces}\nIn the situation of Corollary \\ref{corollarysurfaces} we can write $Y \\simeq \\ensuremath{\\mathbb{P}}(V)$ with $V$ a rank two vector bundle on $C$.\nSince $-K_{Y\/C}$ is nef, the vector bundle $V$ is semistable \\cite[Prop.2.9]{MP97}.\nMoreover the nef cone $\\mbox{Nef}(Y) \\subset N^1{Y}$ \nand the pseudoeffective cone $\\mbox{Pseff}(Y) \\subset N^1{Y}$ coincide \\cite[Sect.1.5.A]{Laz04a},\nthey are generated over $\\ensuremath{\\mathbb{Z}}$ by $\\frac{-1}{2} K_{Y\/C}$ and a fibre $F$ of the ruling $Y \\rightarrow C$.\nRecall that\non any smooth projective surface a Cartier divisor $L$ is generically nef with respect to all polarisations if and only if it is \npseudoeffective. Thus we see that $L \\rightarrow Y$ is generically nef with respect to all polarisations if and only if $L$ \nis nef if and only if\n$\nL \\equiv \\frac{-m}{2} K_{Y\/C} + n F\n$\nwith $m, n \\in \\ensuremath{\\mathbb{N}}_0$.\n\\end{remark}\n\nWe will now use Lemma \\ref{lemmanu1surfaces} to obtain strong restrictions on the MMP.\n\n\\begin{lemma} \\label{lemmanu2}\nIn the situation of Setup \\ref{setup}, suppose that $\\dim X_C-\\dim C=2$.\nLet $\\merom{\\mu_i}{X_i}{X_{i+1}}$ be a step of the MMP.\n\nLet $S' \\subset X_{i+1}$ be an irreducible surface such that\nthe map $S' \\rightarrow C$ is surjective.\nSuppose that $-K_{X_{i+1}\/C}|_{S'}$ is nef but not big. Suppose moreover that either\n\\begin{itemize}\n\\item we have $-K_{X_{i+1}\/C}|_{S'} \\equiv 0$; or\n\\item the divisor $-K_{X_{i+1}\/C}|_{S'}$ is ample on the fibres of $S' \\rightarrow C$. \n\\end{itemize}\nThen the following holds:\n\\begin{enumerate}[(i)]\n\\item Suppose that $\\mu_i$ is a flip: then $S'$ is disjoint from the flipping locus. \n\\item Suppose that $\\mu_i$ is divisorial with exceptional divisor $D_i$. \nThen either $\\mu_i(D_i)$ is disjoint from $S'$ or $\\mu_i(D_i) \\subset S'$. \nIf $\\mu_i(D_i) \\subset S'$, the map $\\mu_i(D_i)\\rightarrow C$ is surjective and \n$-K_{X_{i+1}\/C}|_{S'} \\not\\equiv 0$.\n\\item Let $S \\subset X_i$ be the strict transform of $S'$. Then $-K_{X_i\/C}|_{S}$ is nef but not big.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}{\\em Proof of (i).}\nArguing by contradiction we suppose that $S'$ is not disjoint from the flipping locus $Z$.\nSince $-K_{X_{i+1}\/C}|_{S'}$ is nef, the intersection $S' \\cap Z$ is non empty and finite.\nUsing the notation of Remark \\ref{remarkflip}, let \n$\\Gamma_S \\subset \\Gamma_i$ be the strict transform of $S'$. \nRestricting \\eqref{flipdiscrepancy} to $\\Gamma_S$ we obtain\n$$\np_i^* (-K_{X_i\/C})|_{\\Gamma_S} = q_i^*(-K_{X_{i+1}\/C})|_{\\Gamma_S} \n- (\\sum a_{i, j} D_{i, j}) \\cap \\Gamma_{S}.\n$$\nThe divisor $(\\sum a_{i, j} D_{i, j})$ being $\\ensuremath{\\mathbb{Q}}$-Cartier, the non-empty intersection\n$E:= (\\sum a_{i, j} D_{i, j}) \\cap \\Gamma_S$ is a non-zero $q_i|_{\\Gamma_S}$-exceptional effective $\\ensuremath{\\mathbb{Q}}$-divisor on $\\Gamma_S$.\nSince $\\Gamma_S$ is not $p_i$-exceptional and surjects onto $C$, it follows\nfrom Corollary \\ref{corollarynefoverC} that\n$p_i^* (-K_{X_i\/C})|_{\\Gamma_S}$ is pseudoeffective and nef over $C$,\nmoreover it is $q_i|_{\\Gamma_S}$-nef.\nIf $-K_{X_{i+1}\/C}|_{S'} \\equiv 0$ this already gives a contradiction, so suppose now\nthat $-K_{X_{i+1}\/C}|_{S'}$ is ample on the fibres of $S' \\rightarrow C$.\n\nRecall that the flipping locus $Z$ is contained in the fibres of $X_{i+1} \\rightarrow C$ \n(cf. Remark \\ref{remarkflippingloci}), so $E$ is vertical\nwith respect to the fibration $\\Gamma_S \\rightarrow C$.\nYet by Lemma \\ref{lemmanu1surfaces} applied to the birational map\n$\\holom{q_i|_{\\Gamma_S}}{\\Gamma_S}{S'}$ we see that $E=0$, a contradiction.\n\n{\\em Proof of (ii).}\nLet $S \\subset X_i$ be the strict transform of $S'$, then we have an induced fibration $S \\rightarrow C$. \nUsing the notation of Remark \\ref{remarkdivisorial} and restricting \\eqref{divisorialdiscrepancy} to $S$ we have\n\\begin{equation} \\label{relation}\n-K_{X_i\/C}|_S = \\mu_i^*(-K_{X_{i+1}\/C})|_{S'} - \\lambda_i (D_i \\cap S),\n\\end{equation}\nIf $\\mu_i(D_i)$ is not disjoint from $S'$,\nthen $D_i \\cap S$ is a non-zero effective divisor. \nNote that the restriction $-K_{X_i\/C}|_S$ is pseudoeffective and nef over $C$, moreover \nit is $\\mu_i$-ample.\nIf $-K_{X_{i+1}\/C}|_{S'} \\equiv 0$ then \\eqref{relation} shows that $-K_{X_i\/C}|_S$ is anti-effective and not zero,\na contradiction.\n\nSuppose now that $-K_{X_{i+1}\/C}|_{S'}$ is ample \non the fibres of $S' \\rightarrow C$. Then we know by Lemma \\ref{lemmanu1surfaces} that $D_i \\cap S$\nis empty unless it has a horizontal component. \nSince $D_i \\cap S$ has a horizontal component, the irreducible curve $\\mu_i(D_i)$ is contained in $S'$\nand surjects onto $C$.\n\n{\\em Proof of (iii).}\nIf the image of the exceptional (resp. the flipping locus) is disjoint\nfrom $S'$ the statement is trivial. If this is not the case, then by $(i)$ and $(ii)$ \nthe contraction $\\mu_i$ is divisorial and \n$-K_{X_{i+1}\/C}|_{S'}$ is ample on the fibres of $S' \\rightarrow C$.\nThus we can apply Lemma \\ref{lemmanu1surfaces} to see that \n$-K_{X_i\/C}|_{S}$ is nef and not big.\n\\end{proof}\n\nThe next lemma describes the Mori fibre space at the end of the MMP:\n\n\\begin{lemma} \\label{lemmapreparation}\nIn the situation of Setup \\ref{setup}, suppose that $\\dim X_C-\\dim C=2$ and that the general $\\varphi$-fibre\nis rationally connected. Suppose that $Y$ is a surface.\n\nThen $Y$ is smooth, the fibration $\\holom{\\tau}{Y}{C}$ is a $\\ensuremath{\\mathbb{P}}^1$-bundle and $-K_{Y\/C}$ is nef.\nLet $\\Delta \\subset Y$ be the $1$-dimensional part of the $\\psi$-singular locus. \n\\begin{itemize}\n\\item If $\\Delta \\neq 0$, it is a smooth irreducible curve and the map $\\Delta \\rightarrow C$ is \\'etale. We have $\\Delta \\equiv - \\lambda K_{Y\/C}$ with $\\lambda \\in \\ensuremath{\\mathbb{Q}}^+$.\nMoreover $S':=\\fibre{\\psi}{\\Delta}$ is an irreducible surface such that $-K_{X_k\/C}|_{S'}$ is nef but not big. The\nrestriction $-K_{X_k\/C}|_{S'}$ is ample\non the fibres of $S' \\rightarrow \\Delta$ and if $l \\subset S'$ is an irreducible component of a general\nfibre of $S' \\rightarrow \\Delta$, we have $-K_{X_k\/C} \\cdot l=1$.\n\\item If $\\Delta=0$, then $X_k \\rightarrow Y$ is a $\\ensuremath{\\mathbb{P}}^1$-bundle, in particular $X_k$ is smooth.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{remark*}\nIf $\\Delta \\neq 0$ it is -a priori- not clear that $X_k$ is Gorenstein and $X_k \\rightarrow Y$ is a conic bundle, \ncp. \\cite[\\S 12]{MP08} and \\cite[p.483]{PS98}.\n\\end{remark*}\n\n\\begin{proof}\nBy Corollary \\ref{corollarynefoverC} we know that $-K_{X_k\/C}$ is nef in codimension one and nef over $C$.\nBy Remark \\ref{remarknefcodimone} this implies that $K_{X_k\/C}^2$ is a pseudoeffective $1$-cycle.\nWe claim that $-K_{Y\/C}$ is pseudoeffective and nef over $C$. \n\n{\\em Proof of the claim.\\footnote{For experts it is not difficult to deduce the claim from general results on the positivity\nof direct image sheaves, cf. \\cite[Cor.0.2]{BP08} \\cite[Lemma 3.24]{a8}.}} \nThe fibration $\\psi$ does not contract a divisor, so it is equidimensional. \nSince a terminal threefold has at most isolated singularities, there are at most finitely many points \n$Z \\subset Y$ such that $(X_k \\setminus \\fibre{\\psi}{Z}) \\rightarrow (Y \\setminus Z)$\nis a conic bundle. Thus we have \\cite[4.11]{Miy83}\n\\begin{equation} \\label{miyanishi}\n\\psi_* (K_{X_k\/C}^2) = - (4 K_{Y\/C} + \\Delta).\n\\end{equation}\nThe cycle $K_{X_k\/C}^2$ is pseudoeffective, \nso its image $-(4 K_{Y\/C} + \\Delta)$ is pseudoeffective. This already proves that $-K_{Y\/C}$ is pseudoeffective.\nWe will now follow an argument from \\cite[p.482]{PS98}:\nlet $B \\subset Y$ be any irreducible curve that surjects onto $C$. Since $-K_{X_k\/C}$\nis $\\psi$-ample and nef over $C$, the restriction $-K_{X_k\/C}|_{\\fibre{\\psi}{B}}$ is nef.\nThus we see by the projection formula and \\eqref{miyanishi} that\n\\begin{equation} \\label{discrim}\n0 \\leq (-K_{X_k\/C})^2 \\cdot \\fibre{\\psi}{B} = - (4 K_{Y\/C} + \\Delta) \\cdot B.\n\\end{equation}\nIn particular we have $-K_{Y\/C} \\cdot B \\geq 0$ unless $B \\subset \\Delta$. Arguing by contradiction we suppose\nthat there exists an irreducible curve $B \\subset \\Delta$ such that $-K_{Y\/C} \\cdot B < 0$.\nSince $B \\subset \\Delta$ the inequality \\eqref{discrim} then implies\n$$\n(K_{Y\/C}+B) \\cdot B \\leq (K_{Y\/C}+\\Delta) \\cdot B = (4 K_{Y\/C} + \\Delta) \\cdot B + 3 (-K_{Y\/C} \\cdot B) < 0.\n$$\nThus if $\\tilde B$ is the normalisation of $B$, the subadjunction formula \\cite{Rei94} shows that $\\deg K_{\\tilde B\/C}<0$,\na contradiction to the ramification formula. This proves the claim.\n\nWe can now describe the surface $Y$: the general fibre of $X_k \\rightarrow C$ is rationally connected, so\nthe general fibre of $Y \\rightarrow C$ is $\\ensuremath{\\mathbb{P}}^1$. Thus we know \nby Corollary \\ref{corollarysurfaces} that $Y \\rightarrow C$ is a ruled surface and \n$-K_{Y\/C}$ is nef.\n\n{\\em 1st case. $\\Delta \\neq \\emptyset$.}\nSince $\\psi$ is a conic bundle in the complement of finitely many points, the general\nfibre over a point in $\\Delta$ is a reducible conic. It is well-known that if $l \\subset S'$ is an irreducible\ncomponent of such a reducible conic, then $S' \\cdot l=-1$ and $-K_{X_k\/C} \\cdot l=1$. Using $\\rho(X_k\/Y)=1$ standard arguments prove that $\\Delta$ and $S'$ are irreducible, cf. \\cite[Rem.2.3.3]{MP08}.\n\nIn the proof of the claim we saw that $-(4 K_{Y\/C} + \\Delta)$ is pseudoeffective.\nSince $-K_{Y\/C}$ is nef and $(-K_{Y\/C})^2=0$ we obtain\n$$\n0 \\leq -K_{Y\/C} \\cdot (-4 K_{Y\/C} - \\Delta) = K_{Y\/C} \\cdot \\Delta \\leq 0.\n$$ \nSince $Y$ is a ruled surface, the equality $-K_{Y\/C} \\cdot \\Delta=0$ implies that\nthat $\\Delta \\equiv - \\lambda K_{Y\/C}$ with $\\lambda \\in \\ensuremath{\\mathbb{Q}}^+$. In particular $\\Delta$ surjects onto $C$ and we have $\\Delta^2=0$.\nBy adjunction we see that $K_{\\Delta\/C}$ has degree $0$, so $\\Delta \\rightarrow C$ is \\'etale and $\\Delta$ is smooth.\n\nSince $\\Delta$ surjects onto $C$, the divisor\n $-K_{X_k\/C}|_{S'}$ is nef and ample on the fibres\nof $S' \\rightarrow \\Delta$. Using the projection formula and \\eqref{miyanishi} we have\n$$\n(-K_{X_k\/C})^2 \\cdot S' = - (4 K_{Y\/C} + \\Delta) \\cdot \\Delta = 0,\n$$\nso $-K_{X_k\/C}|_{S'}$ is not big.\n\n{\\em 2nd case. $\\Delta= \\emptyset$.}\nThe terminal threefold $X_k$ is Cohen-Macaulay and the fibration on the smooth base $Y$ is equidimensional,\nso $\\psi$ is flat. Moreover $\\psi$ has at most finitely many singular fibres. Thus $\\psi$ is a $\\ensuremath{\\mathbb{P}}^1$-bundle by\n\\cite[Thm.2.]{AR12}.\n\\end{proof}\n\n\\begin{proposition} \\label{propositionMMPnef}\nIn the situation of Setup \\ref{setup}, suppose that $\\dim X_C-\\dim C=2$ and that the general $\\varphi$-fibre\nis rationally connected.\nThen $-K_{X_k\/C}$ is nef.\n\\end{proposition}\n\n\\begin{proof}\nBy Corollary \\ref{corollarynefoverC} we know that $-K_{X_k\/C}$ is nef in codimension one and nef over $C$.\nBy Lemma \\ref{lemmapreparation} we know that\n$Y \\rightarrow C$ is a ruled surface such that $-K_{Y\/C}$ is nef.\nLet $\\Delta \\subset Y$ be the $1$-dimensional part of the $\\psi$-singular locus. \n\nDenote by $\\{ B_1, \\ldots, B_m\\} \\subset X_k$\nthe finite (maybe empty) set of curves such that $-K_{X_k\/C} \\cdot B_j<0$. Since $-K_{X_k\/C}$ is nef over $C$ and $\\psi$-ample,\nwe see that for all $j \\in \\{1, \\ldots, m\\}$, the curve $\\psi(B_j)$ is a fibre of the ruling $Y \\rightarrow C$.\n\n{\\em 1st case: $\\Delta \\neq \\emptyset$.} Let $S' \\subset X_k$ be the surface\nconstructed in Lemma \\ref{lemmapreparation}.\nWe will describe the MMP $X \\dashrightarrow X_k$\nin a neighbourhood of $S'$.\n\nWe set $S_k:=S'$ and for \nevery $i \\in \\{ 0, \\ldots, k-1 \\}$ we define inductively\n$S_{i} \\subset X_{i}$ as the strict transform of $S_{i+1} \\subset X_{i+1}$.\nConsider now the largest $m \\in \\{ 1, \\ldots, k\\}$ such that\nthe surface $S_{m+1}$ is not disjoint from the flipping locus of $\\mu_m$ or, if $\\mu_m$ is divisorial,\nthe image $\\mu_m(D_m)$ of the exceptional divisor. Since $\\mu_k \\circ \\ldots \\mu_{m+1}$\nis an isomorphism near $S_{m+1}$ we see that $S_{m+1} \\simeq S'$ and\nby Lemma \\ref{lemmapreparation} the divisor $-K_{X_{m+1}\/C}|_{S_{m+1}}$ is nef but not big. \nMoreover $-K_{X_{m+1}\/C}|_{S_{m+1}}$ is ample\non the fibres of $S_{m+1} \\rightarrow \\Delta$. Thus we can apply Lemma \\ref{lemmanu2} and see that\n$\\mu_m$ is divisorial, the curve $\\mu_m(D_m)$ is contained in $S_{m+1}$ and surjects onto $C$.\nSince $X_{m+1}$ has only finitely many singular points and $\\mu_m(D_m)$ is a lci curve in its general point, we see\nby \\cite[Prop.0.6]{PS98} that $\\mu_m$ is {\\em generically} the blow-up of the curve $\\mu_m(D_m)$.\nIn particular we have\n$$\n-K_{X_{m}\/C} = - \\mu_m^* K_{X_{m+1}\/C} - D_m.\n$$\nLet $l \\subset S_{m+1}$ be an irreducible component of a general fibre of $S_{m+1} \\rightarrow \\Delta$,\nand let $l' \\subset S_m$ be its strict transform. Since $\\mu_m(D_m)$ intersects $l$, we have $D_m \\cdot l' \\geq 1$.\nBy Lemma \\ref{lemmapreparation} we have \n$$\n1 = - K_{X_{m+1}\/C} \\cdot l = - \\mu_m^* K_{X_{m+1}\/C} \\cdot l', \n$$\nso $-K_{X_{m}\/C} \\cdot l'=0$. By Lemma \\ref{lemmanu2} the divisor $-K_{X_{m}\/C}|_{S_m}$ is nef.\nSince it is numerically trivial on the general fibre of $f: S_m \\rightarrow \\Delta$, we see that \n$-K_{X_{m}\/C}|_{S_m}=f^* H$ with $H$ a nef $\\ensuremath{\\mathbb{Q}}$-Cartier divisor on $\\Delta$. However by Lemma \\ref{lemmanu1surfaces}\nwe have\n$$\n(-K_{X_{m}\/C}|_{S_m}) \\cdot \\mu_m^* (-K_{X_{m+1}\/C}|_{S_{m+1}})= 0.\n$$\nSince $-K_{X_{m+1}\/C}|_{S_{m+1}}$ is ample on the fibres of $S_{m+1} \\rightarrow \\Delta$, we see that $H \\equiv 0$,\nso $-K_{X_{m}\/C}|_{S_m} \\equiv 0$. Thus we are in the first case of Lemma \\ref{lemmanu2}: \nfor every $i \\in \\{ 0, \\ldots, m-1 \\}$, the MMP\nis disjoint from $S_i$.\n\nWe will now argue by contradiction and suppose that $-K_{X_k\/C}$ is not nef. \nThen the surface $S_k$ meets the curves $B_j$ in finitely many points.\nOn the one hand we have just seen that the surfaces $S_i$ are disjoint from any flipping locus\nof the MMP and if the contraction is divisorial and $S_i$ is not disjoint from $\\mu_i(D_i)$, \nthen $\\mu_i(D_i)$ surjects onto $C$. \nOn the other hand we know by Lemma \\ref{lemmaMMPbasic}, (iii) that\n$B_j$ is contained in a flipping locus or the image of an exceptional \ndivisor. Thus we have $B_j \\cap S_k = \\emptyset$, a contradiction.\n\n{\\em 2nd case: $\\Delta = \\emptyset$.} \nBy Lemma \\ref{lemmapreparation} the variety $X_k$ is smooth and\n$X_k \\rightarrow Y$ is a $\\ensuremath{\\mathbb{P}}^1$-bundle. Using the Grothendieck-Riemann-Roch formula \\cite[App.A, Thm.5.3]{Har77}\nwe see that\n$$\nc_1(\\psi_* (\\omega^*_{X_k\/Y}))=0, \\ c_2(\\psi_* (\\omega^*_{X_k\/Y}))=\\frac{1}{2} K_{X_k\/Y}^3.\n$$\nSince $\\psi_* (\\omega^*_{X_k\/C}) \\simeq \\psi_* (\\omega^*_{X_k\/Y}) \\otimes \\omega^*_{Y\/C}$ and\n$K_{Y\/C}^2=0$ we deduce that\n$$\nc_1(\\psi_* (\\omega^*_{X_k\/C}))= - 3 K_{Y\/C}, \\ c_2(\\psi_* (\\omega^*_{X_k\/C}))=\\frac{1}{2} K_{X_k\/C}^3.\n$$\nLet $A \\subset Y$ be a general hyperplane section, then $\\fibre{\\psi}{A} \\rightarrow A$ is a $\\ensuremath{\\mathbb{P}}^1$-bundle\nand $-K_{X_k\/C}|_{\\fibre{\\psi}{A}}$ is nef, since $\\fibre{\\psi}{A} \\cap B_j$ is a finite set for every $j$.\nIn particular the direct image $\\psi_* (\\omega^*_{X_k\/C})|_A$ is nef. Thus $\\psi_* (\\omega^*_{X_k\/C})$\nis generically nef for any polarisation $A$ and by \\cite[Thm.6.1]{Miy87} one has\n$$\nc_2(\\psi_*( \\omega^*_{X_k\/C})) \\geq 0.\n$$\nWe claim that $K_{X_k\/C}^3 \\leq 0$, by what precedes this implies $c_2(\\psi_*( \\omega^*_{X_k\/C})) = 0$.\n\n{\\em Proof of the claim.} Recall that $K_{X_k\/C}^2$ is a pseudoeffective cycle and\n$\\psi_* (K_{X_k\/C}^2) = - 4 K_{Y\/C}$. Let $(K_n)_{n \\in \\ensuremath{\\mathbb{N}}}$ be a sequence of effective $1$-cycles with rational coefficients \nconverging in $N_1(X_k)$ to $K_{X_k\/C}^2$. Then we can write\n$$\nK_n = \\sum_{j=1}^m \\eta_{j, n} B_j + R_n,\n$$\nwhere $R_n$ is an effective $1$-cycle with rational coefficients such that $-K_{X_k\/C} \\cdot R_n \\geq 0$. \nIf $H$ is an ample divisor on $X_k$, the degrees $H \\cdot (\\eta_{j, n} B_j)$ and $H \\cdot R_n$ are bounded\nfor large $n \\in \\ensuremath{\\mathbb{N}}$ by $(H \\cdot K_{X_k\/C}^2)+1$. Thus, up to replacing $K_n$ by some subsequence, we can suppose\nthat the sequences $\\eta_{j, n}$ and $R_n$ converge. Thus we have\n$$\nK_{X_k\/C}^2 = \\sum_{j=1}^m \\eta_{j, \\infty} B_j + R_\\infty,\n$$\nwhere $R_\\infty$ is a pseudoeffective cycle such that \n$-K_{X_k\/C} \\cdot R_\\infty \\geq 0$. Pushing down to $Y$ we have\n$$\n- 4 K_{Y\/C} = \\sum_{i=j}^m \\eta_{j, \\infty} \\psi_*(B_j) + \\psi_*(R_\\infty).\n$$\nRecall now that $K_{Y\/C}^2=0$ and $-K_{Y\/C} \\cdot \\psi_*(B_j)>0$ for all $j$. Then the preceding equation\nshows that $\\eta_{j, \\infty}=0$ for all $j$, hence we get $K_{X_k\/C}^2 = R_\\infty$ and\n$$\n- K_{X_k\/C}^3 = -K_{X_k\/C} \\cdot R_\\infty \\geq 0.\n$$\nThis proves the claim.\n\n{\\em Conclusion.}\nWe will now prove that $\\psi_* (\\omega^*_{X_k\/C})$ is a nef vector bundle. Since the natural morphism\n$$\n\\psi^* \\psi_* (\\omega^*_{X_k\/C}) \\rightarrow \\omega^*_{X_k\/C}\n$$\nis surjective, this proves that $-K_{X_k\/C}$ is nef. If $\\psi_* (\\omega^*_{X_k\/C})$ is stable for some\npolarisation $A$ the property\n$$\nc_1^2(\\psi_* (\\omega^*_{X_k\/C}))= - 3 K_{Y\/C}^2=0, \\ c_2(\\psi_* (\\omega^*_{X_k\/C}))=0\n$$\nimplies that $\\psi_* (\\omega^*_{X_k\/C})$ is projectively flat with nef determinant \\cite[Cor.3]{BS94}, hence nef.\nWe already know that $V:=\\psi_* (\\omega^*_{X_k\/C})$ is generically nef for any polarisation $A$ on $Y$. Thus\nif $V \\rightarrow Q$ is any torsion-free quotient sheaf, then $Q$ is generically nef for any polarisation $A$ on $Y$.\nBy Remark \\ref{remarksurfaces} we then have\n$$\n\\det Q = \\frac{-m}{2} K_{Y\/C} + n F\n$$\nwith $m, n \\in \\ensuremath{\\mathbb{N}}_0$. \nSuppose now that $V$ is not stable with respect to the polarisation $\\frac{-1}{2} K_{Y\/C} + \\frac{1}{8} F$.\nThen there exists a stable reflexive subsheaf $\\sF \\subset V$ such that the quotient $Q:=V\/\\sF$ is torsion-free and\nthe slope $\\mu(Q)$ is less or equal than the slope $\\mu(V)$.\nAn elementary computation shows that\n$\\det Q=\\frac{-m}{2} K_{Y\/C}$ with $m \\leq 2 \\ensuremath{rk} \\ Q$. In particular we have $\\det \\sF= \\frac{-(6-m)}{2} K_{Y\/C}$.\nSince $c_2(Q) \\geq 0$ by \\cite[Thm.6.1]{Miy87} and $c_2(\\sF) \\geq 0$ by the Bogomolov-Miyaoka-Yau inequality\nwe see that $c_2(\\sF) = 0$ and $c_2(Q) = 0$. In particular $\\sF$ is projectively flat with nef determinant \\cite[Cor.3]{BS94},\nhence nef. The same holds for $Q$ if it is stable. If $Q$ is not stable we easily prove that it is an extension\nof two line bundles $L_1$ and $L_2$ which are non-negative multiples of $\\frac{-m}{2} K_{Y\/C}$, in particular $Q$ is nef.\nThus $V$ is an extension of nef vector bundles, hence nef.\n\\end{proof}\n\n\\begin{remark*}\nThe proof of the second case $\\Delta = 0$ is tedious and rather ad-hoc. If we could suppose the existence of\na curve $C_0 \\subset Y$ such that $-K_{Y\/C} \\cdot C_0=0$ we could argue as in the first case.\nUnfortunately the curve $C_0$ does not always exist, cf. Remark \\ref{remarkmumford}.\n\\end{remark*}\n\n\\begin{proposition} \\label{propositionMMPtotal}\nIn the situation of Setup \\ref{setup}, suppose that $\\dim X_C-\\dim C=2$ and that the general $\\varphi$-fibre\nis rationally connected. \nSuppose also that there exists an effective divisor $A_0 \\subset X_C$ such that\n$A_0 \\equiv -mK_{X_C\/C}$ for some $m \\in \\ensuremath{\\mathbb{N}}$.\nThen every $\\mu_i$ is a divisorial contraction onto some \\'etale multisection,\nand $-K_{X_i\/C}$ is nef for all $i \\in \\{ 0, \\ldots, k\\}$. \nIf $X_C$ is Gorenstein, the fibration $X_C \\rightarrow C$ is locally trivial in the analytic topology.\n\\end{proposition}\n\n\\begin{proof}\nNote first that $-K_{X_k\/C}$ is nef: if $\\dim Y=2$ this was shown in Proposition \\ref{propositionMMPnef},\nif $\\dim Y=1$ the Mori fibre space $\\psi$ and the fibration $X_k \\rightarrow C$ identify, \nso $-K_{X_k\/C}$ is relatively ample and nef over $C$, hence nef.\nLet $F$ be a general fibre of $X_k \\rightarrow C$. \n\n{\\em Step 1. Description of the MMP.}\nWe denote by $M \\in \\mbox{Pic}^0 C$ a divisor such that\n$$\nA_0 \\in H^0(X_C, -mK_{X_C\/C} + \\varphi^* M).\n$$\nFor every $i \\in \\{0, \\ldots, k\\}$ we denote by $M_i$ the pull-back of $M$ to $X_i$ via the natural map $X_i \\rightarrow C$.\nSetting inductively $A_{i+1}=(\\mu_i)_* A_i$ for every $i \\in \\{0, \\ldots, k-1\\}$\nwe have\n$$\nA_i \\in H^0(X_i, -mK_{X_i\/C} + M_i) \\qquad \\forall \\ i \\in \\{0, \\ldots, k\\}.\n$$\nThe divisor $A_i$ being effective we see that $-K_{X_i\/C}$ is nef if and only if \nits restriction to every irreducible component of $A_i$ is nef.\nNote that if the contraction $\\mu_i$ is divisorial with exceptional divisor $D_i$, we have\n$$\nH^0(X_{i+1}, (\\mu_i)_*(\\sO_{X_i}(-mK_{X_i\/C} + M_i))) = H^0(X_{i+1}, (-mK_{X_{i+1}\/C} + M_{i+1}) \\otimes {\\mathcal J}),\n$$\nwhere ${\\mathcal J}$ is an ideal sheaf whose cosupport is $\\mu_i(D_i)$. In particular $A_{i+1}$ contains $\\mu_i(D_i)$.\nSince $-K_{X_k\/C}$ is nef, the contraction $\\mu_k$ is divisorial.\n\n{\\em 1st case. Suppose that $K_{F}^2=0$.} By Lemma \\ref{lemmapreparation}\nand \\eqref{miyanishi} we have\n$\\psi_* (K_{X_k\/C}^2) \\equiv - m K_{Y\/C}$\nwith $m \\geq 0$. Restricting to the general fibre $F$ the condition $K_F^2=0$ implies that $m=0$.\nThus the pseudoeffective cycle $K_{X_k\/C}^2$\nis numerically equivalent to $\\mu l$ where $l$ is a general $\\psi$-fibre. \nThus we see that $0=(-K_{X_k\/C})^3 = 2 \\mu$.\nHence we have $K_{X_k\/C}^2 \\equiv 0$, in particular the restriction of $-K_{X_k\/C}$ to any component\nof $A_k$ is numerically trivial. If the MMP $X \\dashrightarrow X_k$ is not an isomorphism, then\n$\\mu_k(D_k)$ is not disjoint from $A_k$, a contradiction to \nLemma \\ref{lemmanu2}(ii).\n\n{\\em 2nd case. Suppose that $K_{F}^2>0$.}\nIn this case we can apply Corollary \\ref{corollaryMFS} to see \nthat $X_k \\rightarrow C$ is locally trivial in the analytic topology, in particular $X_k$ is smooth. \n\nSuppose for the moment that $-K_{X_i\/C}$ is nef and relatively big for some $i \\in \\{1, \\ldots, k\\}$. \nWe claim that the divisor $A_i$ is a union of irreducible components\n$A_i = \\sum b_{i,l} A_{i,l}$ such that for every $l$ the natural map $A_{i,l} \\rightarrow C$ is surjective\nand $-K_{X_i\/C}|_{A_{i,l}}$ is either numerically trivial or nef and relatively ample, but not big.\n\n{\\em Proof of the claim.}\nSince $-K_{X_i\/C}$ is nef but not big, the restriction $-K_{X_i\/C}|_{A_{i,l}}$ is not big.\nSince $(\\varphi_i)_*(\\omega^{\\otimes -m}_{X_i\/C}) \\otimes M$ is numerically flat we can see as in the proof\nof Proposition \\ref{propositionloctriv} that $A_i \\rightarrow C$ is locally trivial. \nIn particular all the irreducible components surject onto $C$ and\nif $-K_{X_i\/C}|_{A_{i,l}}$ is relatively big for the fibration $A_{i,l} \\rightarrow C$, it is relatively ample.\nThus we are left to show that if $-K_{X_i\/C}$ is numerically trivial on the fibres of $A_{i,l} \\rightarrow C$,\nthen its restriction to $A_{i,l}$ is numerically trivial. Note that in this case $A_{i,l}$\nis contracted by the morphism to the relative anticanonical model \n$$\n\\nu_i: X_i \\rightarrow X_i' \\subset \\ensuremath{\\mathbb{P}}((\\varphi_i)_*(\\omega^{\\otimes -d}_{X_i\/C}))=:\\ensuremath{\\mathbb{P}}(V_i)\n$$ \nwith $d \\gg 0$, and the image\nof $\\nu_i(A_{i,l})$ is an irreducible component of $(X_i')_{sing}$. Since $X_i' \\rightarrow C$ is \nlocally trivial, the curve $\\nu_i(A_{i,l})$ is an \\'etale multisection and a connected component of $(X_i')_{sing}$.\nAfter \\'etale base change we can suppose that $\\nu_i(A_{i,l})$ is a section.\nIf $\\sI_{X_i'}$ is the ideal of $X_i'$ in $\\ensuremath{\\mathbb{P}}(V_i)$ the direct image\n$(\\varphi_i)_* (\\sI_{X_i'} \\otimes \\sO_{\\ensuremath{\\mathbb{P}}(V_i)}(e))$ is numerically flat for $e \\gg 0$ (cf. the proof of Theorem \\ref{theoremmain}),\nso $(\\varphi_i)_* (\\sI_{(X_i')_{sing}} \\otimes \\sO_{\\ensuremath{\\mathbb{P}}(V_i)}(e))$ is also numerically flat. \nIn particular the section $\\nu_i(A_{i,l})$ corresponds to a numerically trivial quotient of $V_i$, hence\n$$\n0 = \\sO_{\\ensuremath{\\mathbb{P}}(V_i)}(e) \\cdot (X_i')_{sing} = - e d K_{X_i'\/C} \\cdot (X_i')_{sing}.\n$$\nSince $\\nu_i$ is crepant this proves the claim.\n\nWe will now prove by descending induction that $-K_{X_i\/C}$ is nef and relatively big for all $i \\in \\{1, \\ldots, k\\}$. \nThis is clear for $i=k$, so suppose that it holds for $i+1$.\nSince $-K_{X_{i+1}\/C}$ is nef, the contraction $\\mu_i$ is divisorial\nwith exceptional divisor $D_i$ and $\\mu_i(D_i)$ is contained in $A_{i+1}$.\nBy the claim the irreducible components $A_{i+1,l}$ satisfy the conditions of Lemma\n\\ref{lemmanu2}. Thus $\\mu_i(D_i)$ is a curve surjecting onto $C$\nand the divisor $-K_{X_i\/C}$ is nef on the strict transforms\nof all the irreducible components $A_{i+1,l}$.\nThis already proves that $-K_{X_{i}\/C}|_{A_{i}}$ is nef, unless\n$A_i$ has one irreducible component more than $A_{i+1}$, the exceptional divisor $D_i$. \nClearly $-K_{X_{i}\/C}|_{D_i}$ is relatively ample with respect to $D_i \\rightarrow \\mu_i(D_i)$.\nYet $\\mu_i(D_i)$ surjects onto $C$, so the restriction $-K_{X_{i}\/C}|_{D_i}$ is\nnef over $\\mu_i(D_i)$. This proves that $-K_{X_{i}\/C}|_{D_i}$ is nef, hence $-K_{X_{i}\/C}|_{A_{i}}$ is nef.\nSince $-K_{X_{i}\/C}$ is nef we know by \\cite[Prop.4.11]{PS98} that $\\mu_i$ \nis the blow-up along an \\'etale (multi-)section.\n\n{\\em Step 2. $\\varphi$ is locally trivial.} By what precedes the first step of MMP is a divisorial\ncontraction $\\holom{\\mu_0}{X_0}{X_1}$ contracting a divisor $D_0$ onto an \\'etale \nmultisection\\footnote{The case of a trivial MMP can be excluded as follows: \nconsider the Mori fibre space $\\psi: X_C = X_k \\rightarrow Y$.\nSince $X_C$ is Gorenstein, the fibration $\\psi$ is a conic bundle \\cite[Thm.7]{Cut88}. \nThe discriminant locus $\\Delta$ is smooth\nby Lemma \\ref{lemmapreparation}, so all the fibres over $\\Delta$ are reducible conics \\cite[Prop.1.8.3)]{Sar82}.\nThus the associated two-to-one cover $\\tilde \\Delta \\rightarrow \\Delta$ is \\'etale, hence $\\tilde \\Delta \\rightarrow C$\nis \\'etale by Lemma \\ref{lemmapreparation}. Arguing as \\cite[Prop.0.4, Rem.0.5]{PS98} we see that \n$X_C \\times_C \\tilde \\Delta \\rightarrow \\tilde \\Delta$ admits a Mori contraction that blows down exactly one (-1)-curve\nin every fibre.}.\nIn particular $-K_{X_1\/C}$ is nef and $\\varphi_1$-big where $\\holom{\\varphi_1}{X_1}{C}$ is the natural fibration.\nBy Corollary \\ref{corollaryMFS} the variety $X_1$ is smooth, so $X_0$ is smooth.\nMoreover $-K_{X_0\/C} - \\mu_0^* K_{X_1\/C}$ is nef and $\\varphi$-big,\nhence for $m \\in \\ensuremath{\\mathbb{N}}$ the direct image $\\varphi_*(\\omega_{X_0\/C}^{\\otimes -m} \\otimes \\mu_0^* \\omega_{X_1\/C}^{\\otimes -m})$\nis nef \\cite{Kol86}. The inclusion $(\\mu_0)_* (\\omega_{X_0\/C}^{\\otimes -m}) \\hookrightarrow \\omega_{X_1\/C}^{\\otimes -m}$ \nyields an inclusion \n$$\n\\varphi_*(\\omega_{X_0\/C}^{\\otimes -m} \\otimes \\mu_0^* \\omega_{X_1\/C}^{\\otimes -m})\n\\hookrightarrow\n(\\varphi_1)_* (\\omega_{X_1\/C}^{\\otimes -2m}).\n$$\nThe sheaf $(\\varphi_1)_* (\\omega_{X_1\/C}^{\\otimes -2m})$ is numerically flat for all $m \\gg 0$ by\nProposition \\ref{propositionkltdirectimage}, so \n$\\varphi_*(\\omega_{X_0\/C}^{\\otimes -m} \\otimes \\mu_0^* \\omega_{X_1\/C}^{\\otimes -m})$ is also numerically flat\nfor all $m \\gg 0$.\nBy the relative base-point free theorem the natural map \n$$\n\\varphi^* \\varphi_*(\\omega_{X_0\/C}^{\\otimes -m} \\otimes \\mu_0^* \\omega_{X_1\/C}^{\\otimes -m})\n\\rightarrow \n\\omega_{X_0\/C}^{\\otimes -m} \\otimes \\mu_0^* \\omega_{X_1\/C}^{\\otimes -m}\n$$\nis surjective for all $m \\gg 0$, so we obtain a birational morphism\n$\\holom{\\mu}{X_0}{X_0'}$ onto a normal projective variety $\\varphi': X_0' \\rightarrow C$\nembedded in $\\varphi': \\ensuremath{\\mathbb{P}}(E_m) \\rightarrow C$\nwhere $E_m := \\varphi_*(\\omega_{X_0\/C}^{\\otimes -m} \\otimes \\mu_0^* \\omega_{X_1\/C}^{\\otimes -m})$\nfor some fixed $m \\gg 0$. We can now argue as in the proof of Theorem \\ref{theoremmain}:\ndenoting by $\\sI_{X_0'} \\subset \\sO_{\\ensuremath{\\mathbb{P}}(E_m)}$ the ideal sheaf of $X_0' \\subset \\ensuremath{\\mathbb{P}}(E_m)$,\nwe have for\nevery $d \\gg 0$ an exact sequence\n$$\n0 \\rightarrow (\\varphi')_{*}(\\sI_{X_0'}\\otimes \\mathcal{O}_{\\mathbb{P}(E_{m})}(d)) \n\\rightarrow S^d E_m \\rightarrow \\varphi_*(\\omega_{X_0\/C}^{\\otimes -dm} \\otimes \\mu_0^* \\omega_{X_1\/C}^{\\otimes -dm}) \\rightarrow 0.\n$$\nThus $(\\varphi')_{*}(\\sI_{X_0'}\\otimes \\mathcal{O}_{\\mathbb{P}(E_{m})}(d)) $ \nis numerically flat, so $\\varphi': X_0' \\rightarrow C$ is locally trivial with fibre $F'$ by Proposition \\ref{propositionloctriv}.\n\nThe Cartier divisor $-K_{X_0\/C}$ is $\\mu$-trivial, \nso we have $K_{X_0\/C} = \\mu^* K_{X_0'\/C}$ \\cite[Thm.3.24]{KM98}. Thus $X_0 \\rightarrow X_0'$ is a crepant resolution of\nthe normal projective variety $X_0'$, \nand a smooth $\\varphi$-fibre $F$ is the minimal resolution of the general $\\varphi'$-fibre $F'$.\nIn particular $F$ is unique up to isomorphism, arguing exactly as in Step 2 of the proof of Theorem \\ref{theoremmain}\nwe see that $X_C \\rightarrow C$ is locally trivial with fibre $F$.\n\\end{proof}\n\n\n\\subsection{Main result} \\label{subsectionmainresult}\n\nIn this section we will prove Theorem \\ref{theoremmaintwo}. Proposition \\ref{propositionMMPtotal}\nobviously settles the main part, however we proved the statement\nunder a nonvanishing condition which is not satisfied in general (cf. Remark \\ref{remarkmumford}).\nWe will now show that these properties hold if we start with a fibration onto a torus.\n\n\\begin{lemma} \\label{lemmanonvanishing}\nLet $X$ be a projective manifold such that $-K_X$ is nef.\nLet $\\pi: X \\rightarrow A$ be the Albanese fibration, \nand suppose that $-K_F$ is nef and abundant\\footnote{Cf. \\cite{Fuj11} for the relevant definitions.} for the general $\\pi$-fibre $F$.\nThen there exists an effective divisor $A \\subset X$ such that\n$A \\equiv -mK_{X}$ for some $m \\in \\ensuremath{\\mathbb{N}}$.\n\\end{lemma}\n\n\\begin{proof}\nSince $-K_F$ is nef and abundant we know by the relative version of Kawamata's \ntheorem \\cite[Thm.1.1]{Fuj11}\nthat $-K_X$ is $\\pi$-semiample, so\nfor every sufficiently divisible $m \\gg 0$ the natural map\n$$\n\\pi^{*}\\pi_{*}(\\omega_X^{\\otimes -m})\\rightarrow \\omega_X^{\\otimes -m}\n$$\nis surjective. Thus $-mK_X$ induces a morphism\n$\\holom{\\psi}{X}{Y}$\nonto a normal projective variety $\\holom{\\tau}{Y}{A}$ \nsuch that $-K_X \\sim_\\ensuremath{\\mathbb{Q}} \\psi^* H$\nwith $H$ a nef and $\\tau$-ample Cartier divisor. \nSince $\\pi$ is equidimensional \\cite{LTZZ10}, the fibration $\\tau$ is equidimensional.\nBy \\cite[Thm.0.2]{Amb05} there exists a boundary divisor $\\Delta_Y$ on $Y$ such that the pair $(Y, \\Delta_Y)$ is klt\nand $H \\sim_\\ensuremath{\\mathbb{Q}} -(K_Y+\\Delta_Y)$. In particular the variety $Y$ is Cohen-Macaulay, so the equidimensional fibration\n$\\tau$ is flat. Thus we can apply Proposition \\ref{propositionkltdirectimage} to see that for sufficiently divisible $m \\gg 0$, \nthe direct image sheaf\n$$\n\\pi_* (\\omega_X^{- \\otimes m}) \\simeq \\tau_* (\\sO_Y(-m(K_Y+\\Delta_Y)))\n$$\nis a numerically flat vector bundle. By \\cite[Thm.1.18]{DPS94} there exists a subbundle $F \\subset \\pi_* (\\omega_X^{- \\otimes m})$\nsuch that $F$ is given by a unitary representation $\\pi_1(A) \\rightarrow U(\\ensuremath{rk} \\ F)$. The group $\\pi_1(A)$\nis abelian, so the representation splits, i.e. $F$ is a direct sum of numerically trivial line bundles. In particular\nthere exists a $M \\in \\mbox{Pic}^0 A$ such that\n$H^0(A, M \\otimes F) \\neq 0$.\n\\end{proof}\n\n\\begin{lemma} \\label{lemmaGRR}\nLet \\holom{f}{M}{C} be a fibration from a normal $\\ensuremath{\\mathbb{Q}}$-factorial threefold with\nat most Gorenstein terminal singularities onto a curve $C$ such that $-K_{M\/C}$\nis nef. Suppose that the general fibre $F$ is rationally connected and $K_F^2=0$.\n\\begin{enumerate}[(i)]\n\\item Then we have\n$c_1(f_!(\\omega^*_{M\/C}))=0$.\n\\item If $h^0(F, -K_F)=1$, there exists an effective divisor $A \\subset M$ \nsuch that $A \\equiv -K_{M\/C}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{remark} \\label{remarkmumford}\nLet $C$ be a curve of genus at least two, and let $U$ be a rank two bundle of degree $0$ on $C$\nsuch that all the symmetric powers $S^m U$ are stable (such vector bundles have been constructed by Mumford).\nSet $M:=\\ensuremath{\\mathbb{P}}(U)$, then $-K_{M\/C}$ is nef, but not numerically equivalent to any effective $\\ensuremath{\\mathbb{Q}}$-divisor.\n\\end{remark}\n\n\\begin{proof}\n{\\em Proof of (i)} By the Grothendieck-Riemann-Roch formula \n\\cite[Thm.15.2]{Ful98}\\footnote{The statement in \\cite{Ful98} is only for a smooth total space,\nbut if $\\holom{\\mu}{M'}{M}$ is a resolution of singularities one checks easily that\n$ch(-K_{M\/C}) td(T_M)=ch(-\\mu^* K_{M\/C}) td(T_{M'})$. Thus the formula holds since\nwe can apply \\cite[Thm.15.2]{Ful98} to $f \\circ \\mu$.}\nwe have\n\\[\ntd(T_C) ch(f_! (\\omega_{M\/C}^*)) = f_* (ch(-K_{M\/C}) td(T_M)).\n\\]\nWe will prove that the degree $3$ component of $ch(-K_{M\/C}) td(T_M)$ is equal to $1-g$,\nwhich by the formula above implies the statement.\n\nSince $K_{M\/C}^3=0$ and $K_F^2=0$ we have\n\\begin{equation} \\label{equationeasy}\nK_{M\/C}^2 \\cdot K_M = 0, \\qquad K_{M\/C} \\cdot K_M^2 = 0.\n\\end{equation}\nThe Chern character of $-K_{M\/C}$ is \n$ch(-K_{M\/C}) = 1 - K_{M\/C} + \\frac{1}{2} K_{M\/C}^2$, and the Todd class is\n\\[\ntd(T_M) = 1- \\frac{K_M}{2} + \\frac{K_M^2+c_2(M)}{12} + \\chi(M, \\sO_M).\n\\]\nThus the degree $3$ components of $ch(-K_{M\/C}) td(T_M)$ are given by\n\\begin{equation} \\label{equationeasy2}\n\\chi(M, \\sO_M) - K_{M\/C} \\cdot \\frac{K_M^2+c_2(M)}{12} + \\frac{1}{4} K_{M\/C}^2 \\cdot K_M.\n\\end{equation}\nSince we have $\\chi(M, \\sO_M)=-\\frac{K_M c_2(M)}{24}$ and $f^* K_C \\cdot c_2(M) =\n(2g-2) c_2(F)$ and $c_2(F)=12$ we obtain\n$\n- K_{M\/C} \\cdot \\frac{c_2(M)}{12} = 2 \\chi(M, \\sO_M) + 2g-2$. \nUsing \\eqref{equationeasy} the formula \\eqref{equationeasy2} simplifies to\n$3 \\chi(M, \\sO_M) + 2g-2$.\nThe general $f$-fibre is rationally connected, so we have \n$\\chi(M, \\sO_M)=\\chi(C, \\sO_C)=1-g$.\n\n\n{\\em Proof of (ii)} Note that for a general fibre $F$ we have \n$h^1(F, -K_F)=h^2(F, -K_F)=0$, so \n$R^j f_* (\\omega_{M\/C}^*)$ is a torsion sheaf for $j \\geq 1$. If $F_0$ is an arbitrary fibre, then by\nSerre duality $h^2(F_0, -K_{F_0})=h^0(F_0, 2 K_{F_0})=0$ since $K_{F_0}$ is by hypothesis antinef\nand not trivial.\nThus we have $R^2 f_* (\\omega_{M\/C}^*)=0$ and\n$$\nf_! (\\omega_{M\/C}^*) = f_* (\\omega_{M\/C}^*) - R^1 f_* (\\omega_{M\/C}^*).\n$$\nSince $R^1 f_* (\\omega_{M\/C}^*)$ is a torsion sheaf, statement $(i)$ implies that\n$c_1 (f_* (\\omega_{M\/C}^*)) \\geq 0$.\nThus $f_* (\\omega^*_{M\/C})$ is a line bundle of non-negative degree, so there exists a\nnumerically trivial line bundle $L$ on $C$ such that\n$H^0(C, f_* (\\omega^*_{M\/C}) \\otimes L) \\neq 0$.\n\\end{proof}\n\n\\begin{proposition} \\label{propositioneasy}\nLet $X$ be a projective manifold such that $-K_X$ is nef, and \nlet \\holom{\\pi}{X}{T} be the Albanese map. Suppose that $\\dim T=\\dim X-2$ and the general\n$\\pi$-fibre $F$ is uniruled but not rationally connected. Then there exists a finite \\'etale cover $X' \\rightarrow X$\nsuch that $q(X')=\\dim X-1$. Moreover the fibration $\\pi$ is smooth.\n\\end{proposition}\n\n\\begin{proof}\nLet $X \\dashrightarrow Y$ be a model of the MRC-fibration \\cite{Deb01} such that $Y$ is smooth. Then $Y$ is not uniruled,\nand we denote by $K_Y=P+N$ the divisorial Zariski decomposition. By \\cite[Main Thm.]{Zha05} the positive part $P$ is \nzero\\footnote{The statement in \\cite[Main Thm.]{Zha05} is only $\\kappa(Y)=0$, but the proof consists in showing that $P=0$.}.\nBy \\cite[Cor.3.4]{Dru11} the variety $Y\/T$ has a good minimal model $Y'\/T$.\nBy \\cite[Prop.8.3]{Kaw85} there exists a finite \\'etale cover $\\tilde T \\rightarrow T$ such that $\\tilde T \\times_T Y'$ is a torus.\nSince the irregularity is invariant under the MMP, we see that $q(\\tilde T \\times_T Y)=\\dim X-1$,\nthus $q(\\tilde T \\times_T X)=\\dim X-1$. \nThis proves the first statement. \n\nLet now $X' \\rightarrow X$ be an \\'etale cover such that $q(X')=\\dim X-1$, and let $X' \\rightarrow T'$ be the Albanese map.\nBy Corollary \\ref{corollarymain} we know that $X' \\rightarrow T'$ is a $\\ensuremath{\\mathbb{P}}^1$-bundle. In particular the reduction of every $\\pi$-fibre\nis a $\\ensuremath{\\mathbb{P}}^1$-bundle $F_0$ over an elliptic curve $E$. Let $\\holom{\\psi}{X}{Y}$ be a Mori contraction over $T$,\nthen $\\psi$ is a $\\ensuremath{\\mathbb{P}}^1$-bundle, and the (reductions of) fibres of $\\tau: Y \\rightarrow T$ are elliptic curves.\nThus we have $K_Y \\equiv 0$, by the Beauville-Bogomolov decomposition the fibration $\\tau$ is smooth.\nHence $\\pi= \\tau \\circ \\psi$ is smooth. \n\\end{proof}\n\n\\begin{remark} \\label{remarkeasy}\nProposition \\ref{propositioneasy} also holds if $X$ is a compact K\\\"ahler threefold: using \\cite{Pau12} we see that\nthe base $Y$ of the MRC fibration has $\\kappa(Y)=0$. Since $Y$ is a surface we can run the MMP, in the surface case Kawamata's result \\cite{Kaw85} follows from the Kodaira-Enriques classification.\n\\end{remark}\n\n\n\\begin{proof}[Proof of Theorem \\ref{theoremmaintwo}]\nLet $F$ be a general $\\pi$-fibre. Using Beauville-Bogomolov we easily exclude the case where $F$\nis not uniruled. If $F$ is uniruled but not rationally connected we conclude by Proposition \\ref{propositioneasy}. Suppose now that $F$ is rationally connected. \nIf $K_F^2>0$ we conclude by Theorem \\ref{theoremmain}, so suppose $K_F^2=0$.\n\n\nWe fix an arbitrary $t \\in T$ \nand $C \\subset T$ a general smooth curve such that $t \\in C$.\nBy Corollary \\ref{corollaryreductioncurve} the preimage\n$\\fibre{\\pi}{C}$ is a normal variety with at most canonical singularities. The divisor \n$-K_{\\fibre{\\pi}{C}\/C}$ is Cartier and nef, so if $X_C \\rightarrow \\fibre{\\pi}{C}$ is a terminal $\\ensuremath{\\mathbb{Q}}$-factorial model \n\\cite[Thm.6.23, Thm.6.25]{KM98} the divisor \n$-K_{X_C\/C}$ is Cartier and nef. By Proposition \\ref{propositionMMPtotal} the fibration $X_C \\rightarrow C$ is locally trivial if we prove\nthat there exists an effective $\\ensuremath{\\mathbb{Q}}$-divisor $A_0$ such that $A_0 \\equiv -mK_{X_C\/C}$ with $m \\in \\ensuremath{\\mathbb{N}}$.\nIf $-K_F$ is not abundant this holds by Lemma \\ref{lemmaGRR},b).\nIf $-K_F$ is nef and abundant we know by Lemma \\ref{lemmanonvanishing} that there exists an effective divisor $A$ on $X$\nsuch that $A \\equiv -mK_X$. Since $C \\subset T$ is general, the restriction $A|_{\\fibre{\\pi}{C}}$ is an effective divisor \nthat is numerically equivalent to $-mK_{\\fibre{\\pi}{C}\/C}$. The pull-back of this divisor to $X_C$ then gives $A_0$.\n\nThus we know that $X_C \\rightarrow C$ is locally trivial. In particular any curve in a fibre of $X_C \\rightarrow C$ deforms into a general fibre,\nhence $X_C \\rightarrow \\fibre{\\pi}{C}$ is an isomorphism.\nThus $X_C \\simeq \\fibre{\\pi}{C} \\rightarrow C$ is locally trivial.\n\\end{proof}\n\nThe proof of Theorem \\ref{theoremmaintwo} would be much simpler if we could classify\nmanifolds such that $-K_X$ is nef but not semiample. The following example shows that this is non-trivial\neven for threefolds, there by correcting \\cite[p.498]{PS98} and \\cite[p.600]{Pet12}.\n\n\\begin{example}\nLet $C$ be an elliptic curve, and let $L_0 \\in \\mbox{Pic}^0 C$ be a line bundle of degree $0$ that is not torsion.\nSet $L_1 := L_2 := \\sO_C$ and $L_3 := L_0^{\\otimes -2}$. Then $V:= \\oplus_{i=0}^3 L_i$ is a numerically flat\nvector bundle of rank four, and we denote by $\\holom{\\psi}{\\ensuremath{\\mathbb{P}}(V)}{C}$ the projectivisation. The vector bundle $S^3 V$ contains a subvector bundle \n$$\n(L_0^{\\otimes 2} \\otimes L_3) \\oplus L_1^{\\otimes 3} \\oplus L_2^{\\otimes 3} \\simeq \\sO_C^{\\oplus 3},\n$$ \nso $\\sO_{\\ensuremath{\\mathbb{P}}(V)}(3)$ has global sections corresponding fibrewise to the degree three monomials\n$\nX_0^2 X_3, \\ X_1^3, \\ X_2^3$.\nThe polynomial $X_0^2 X_3 + X_1^3 + X_2^3$ defines a cubic surface in $\\ensuremath{\\mathbb{P}}^3$ that is normal and has \na unique singular point in $(0:0:0:1)$, this point is of type $D_4$ \\cite[Case C]{BW79}.\nThus if we denote by $X \\subset \\ensuremath{\\mathbb{P}}(V)$ the hypersurface defined by the global section\nof $\\sO_{\\ensuremath{\\mathbb{P}}(V)}(3)$ corresponding to this polynomial, we see that $X$ is normal with at most canonical singularities.\nMoreover we have\n$$\n\\omega_X^* \\simeq (\\psi^* L_0 \\otimes \\sO_{\\ensuremath{\\mathbb{P}}(V)}(1))|_X,\n$$\nso $-K_X$ is nef. The singular locus of $X$ is the curve $C_0$ defined fibrewise by $X_0=X_1=X_2=0$, so it corresponds to the quotient $V \\rightarrow L_3.$\nSince $L_3 = L_0^{\\otimes -2}$ we see that\n$\\omega^*_X|_{C_0} \\simeq L_0^*.$ \nThe line bundle $L_0$ is not torsion, so we obtain that $C_0 \\subset {\\rm Bs} |-mK_X|$ for all $m \\in \\ensuremath{\\mathbb{N}}$. In particular $-K_X$ is not semiample.\n\nLet now $X' \\rightarrow X$ be a terminal model obtained by taking fibrewise the minimal resolution, then\n$X'$ is smooth and $-K_{X'}$ is nef and not semiample. One checks easily that $X'$ is not a product, even after finite \\'etale cover.\n\\end{example}\n\n\\begin{proof}[Proof of Corollary \\ref{corollarykaehler}]\nIf $X$ is projective we conclude by Corollary \\ref{corollarymain} and Theorem \\ref{theoremmaintwo}.\nThus we are left to deal with the case where $X$ is not projective and $q(X)=1$. Then the general fibre $F$ of\n$\\holom{\\pi}{X}{T}$ is not rationally connected, since otherwise $H^2(X, \\sO_X)=0$. If $F$ is uniruled we apply \nRemark \\ref{remarkeasy}. If $F$ is not uniruled, the canonical bundle $K_X$ is pseudoeffective \\cite{Bru06}.\nThus $K_X \\equiv 0$ and we conclude by Beauville-Bogomolov.\n\\end{proof}\n\n\\begin{appendix}\n\n\\section{A Hovanskii-Teissier inequality} \\label{appendixinequality}\n\nFor the convenience of reader, we give the proof of the Hovanskii-Teissier concavity inequality for arbitrary compact K\\\"ahler manifolds,\nwhich was first proved in \\cite{Gro}. The proof here is a direct consequence of \\cite[Thm A, C]{DN06}.\n\n\\begin{proposition} \\label{propositionHT}\nLet $(X,\\omega_{X})$ be a compact K\\\"ahler manifold of dimension $n$, \nand let $\\alpha$, $\\beta$ be two nef classes. For every $i,j, k, s \\in \\ensuremath{\\mathbb{N}}$ we\nhave\n\\begin{equation} \\label{equationseven}\n\\int_{X}(\\alpha^{i}\\wedge\\beta^{j}\\wedge\\omega_{X}^{n-i-j})\n\\end{equation}\n$$\\geq \n(\\int_{X}\\alpha^{i-k}\\wedge\\beta^{j+k}\\wedge\\omega_{X}^{n-i-j})^{\\frac{s}{k+s}}\n\\cdot(\\int_{X}\\alpha^{i+s}\\wedge\\beta^{j-s}\\wedge\\omega_{X}^{n-i-j})^{\\frac{k}{k+s}}.\n$$\n\\end{proposition}\n\n\\begin{proof}\nLet $\\omega_{1},\\cdots, \\omega_{n-2}$ be $n-2$ arbitrary K\\\"ahler classes. \nThanks to \\cite[Thm.A]{DN06}, \nthe bilinear form on $H^{1,1}(X)$\n$$Q([\\lambda], [\\mu])=\\int_{X}\\lambda\\wedge\\mu\\wedge\\omega_{1}\\wedge\\cdots\\wedge\\omega_{n-2}\n\\qquad\\lambda, \\mu\\in H^{1,1}(X)$$\nis of signature $(1, h^{1,1}-1)$.\nSince $\\alpha$, $\\beta$ are nef classes,\nthe function $f(t)=Q (\\alpha+t\\beta, \\alpha+t\\beta)$ is indefinite on $\\mathbb{R}$ \nif and only if $\\alpha$ and $\\beta$ are linearly independent.\nTherefore \n\\begin{equation}\\label{equationaddapen}\n\\int_{X}(\\alpha\\wedge\\beta\\wedge\\omega_{1}\\wedge\\cdots\\wedge\\omega_{n-2})\\geq \n(\\int_{X}\\alpha^{2}\\wedge\\omega_{1}\\wedge\\cdots\\wedge\\omega_{n-2})^{\\frac{1}{2}}\n\\cdot(\\int_{X}\\beta^{2}\\wedge\\omega_{1}\\wedge\\cdots\\wedge\\omega_{n-2})^{\\frac{1}{2}}.\n\\end{equation}\n\nIf we let $\\omega_{1}, \\cdots ,\\omega_{i-1}$ tend to $\\alpha$,\nlet $\\omega_{i},\\cdots,\\omega_{i+j-2}$ tend to $\\beta$\nand take $\\omega_{i+j-1}=\\cdots=\\omega_{n-2}=\\omega_{X}$ in \\eqref{equationaddapen},\nwe have\n$$\\int_{X}(\\alpha^{i}\\wedge\\beta^{j}\\wedge\\omega_{X}^{n-i-j})\\geq \n(\\int_{X}\\alpha^{i-1}\\wedge\\beta^{j+1}\\wedge\\omega_{X}^{n-i-j})^{\\frac{1}{2}}\n\\cdot(\\int_{X}\\alpha^{i+1}\\wedge\\beta^{j-1}\\wedge\\omega_{X}^{n-i-j})^{\\frac{1}{2}} .$$\nThen \\eqref{equationseven} is an easy consequence of the above inequality.\n\\end{proof}\n\n\\begin{remark}\\label{corHT}\nIt is easy to see that the equality holds in \\eqref{equationaddapen} if and only if $\\alpha$ and $\\beta$ are colinear. \n\\end{remark}\n\n\\end{appendix}\n\n\n\n\\def$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n{}Usually, solutions of classical string equations of motion are studied in the case of\nstrings propagating in some dynamically-inactive background fields.\nOn the other hand, one can also consider strings coupled to dynamic local fields with\ntheir kinetic terms included in the total action. In this paper\nwe consider the following system of a classical bosonic string coupled to a massless\nscalar field:\n\\begin{equation}\\label{action.string}\n S = -{1\\over 2} \\int d^2\\sigma \\left[T + \\lambda\\phi(X(\\sigma))\\right]h^{1\/2}h^{\\alpha\\beta}\\partial_\\alpha X^\\mu\\partial_\\beta X_\\mu -{1\\over 8\\pi}\\int d^Dx~\n \\partial^\\mu\\phi(x)\\partial_\\mu\\phi(x)\n\\end{equation}\nHere $T$ is the string tension; the string is described by its $D$-dimensional spacetime coordinates $X^\\mu(\\sigma)$, $\\mu = 1, \\dots, D$,\nwhich are defined on the worldsheet parametrized by the worldsheet coordinates\n$\\sigma^\\alpha = (\\tau, \\sigma)$; $h^{\\alpha\\beta}(\\sigma)$ is the worldsheet metric, and $h = \\det(-h^{\\alpha\\beta})$;\nthe scalar field $\\phi$ lives in the $D$-dimensional spacetime; $\\lambda$ measures the strength of the\ncoupling between the scalar field and the bosonic string.\n\n{}A similar system was considered by Dabholkar and Harvey \\cite{DH}. We will study the renormalizability\nof this system using the methods developed in \\cite{EM} for the classical point-particle electrodynamics.\nThese methods can also be applied to classical equations of motion of the string, which turn out to be non-renormalizable for $D>4$.\n\n{}The analysis in the subsequent sections allows to see in a rather\nsimple way that the string being an extended object, with a natural\nultraviolet cut-off at the fundamental scale $(\\alpha^\\prime)^{1\/2}$, alone is not sufficient for its finiteness.\nIt is the unique combination of its excitations that provides non-renormalization properties\nof the string. This combination of the local fields appearing in the effective action,\ncontaining both massless and massive modes, is a consequence of the conformal\ninvariance, which is crucial for the consistency of the theory.\n\n{}The paper is organized as follows. In Section 2 we review the renormalization of\nthe classical electrodynamics based on \\cite{EM}. Section 3 uses these methods to renormalize\nthe classical string in four dimensions. In Section 4 we argue that the\ntheory is not renormalizable in $D > 4$. We briefly conclude in Section 5.\n\n\\section{Renormalization of Classical Electrodynamics}\n\nConsider a relativistic point particle in $D$ dimensions coupled to the electromagnetic\nfield:\n\\begin{eqnarray}\\label{S.EM}\n {\\cal S} =&& -m \\int d\\tau \\left(U^\\mu(\\tau) U_\\mu(\\tau)\\right)^{1\/2} - e\\int d\\tau \\left[{\\cal A}^\\mu(X(\\tau)) + A^\\mu(X(\\tau))\\right] U_\\mu(\\tau) \\nonumber\\\\\n &&-{1\\over 16\\pi}\\int d^D x~ F^{\\mu\\nu}(x)F_{\\mu\\nu}(x)\n\\end{eqnarray}\nHere $m$ is the mass of the particle; $X^\\mu(\\tau)$ are its coordinates on the worldline\nparametrized by the proper time $\\tau$; $U^\\mu(\\tau) = dX^\\mu(\\tau)\/d\\tau$; $e$ is the electric charge of the\nparticle, which is coupled to the electromagnetic field; $A^\\mu(x)$ is the potential of the\nfield created by the particle itself; ${\\cal A}^\\mu(x)$ is the potential of a dynamically-inactive external electromagnetic\nfield. The electromagnetic field tensor for the self-field $A^\\mu(x)$ is given by\n\\begin{equation}\n F^{\\mu\\nu}(x) = \\partial^\\mu A^\\nu(x) - \\partial^\\nu A^\\mu(x)\n\\end{equation}\nIt enters the kinetic term in the action (\\ref{S.EM}). Note that the analogous quantity for the external field\n\\begin{equation}\n {\\cal F}^{\\mu\\nu}(x) = \\partial^\\mu {\\cal A}^\\nu(x) - \\partial^\\nu {\\cal A}^\\mu(x)\n\\end{equation}\ndoes not enter the action (\\ref{S.EM}) as the external field is dynamically inactive.\n\n{}Due to the reparametrization invariance of the action (\\ref{S.EM}), we have a constraint, which can be chosen as (here $U^2(\\tau) = U^\\mu(\\tau) U_\\mu(\\tau)$):\n\\begin{equation}\n U^2(\\tau) = 1\n\\end{equation}\nThe equations of motion then read:\n\\begin{eqnarray}\\label{self.EM}\n &&m~U^{\\prime\\mu}(\\tau) = e\\left[F^{\\mu\\nu}(X(\\tau)) + {\\cal F}^{\\mu\\nu}(X(\\tau))\\right]U_\\nu(\\tau)\\\\\n &&\\partial_\\nu F^{\\mu\\nu}(x) = -4\\pi e \\int_{-\\infty}^{+\\infty} d\\tau~\\delta\\left(x - X(\\tau)\\right) U^\\mu(\\tau)\\label{F}\n\\end{eqnarray}\nEq. (\\ref{self.EM}) describes the self-interaction of the particle and its interaction with the external field.\n\n{}In the Lorentz gauge $\\partial^\\mu A_\\mu(x) = 0$, Eq. (\\ref{F}) has the following solution\n\\begin{equation}\n A^\\mu(x) = 4\\pi e \\int_{-\\infty}^{+\\infty} d\\tau~G^{-}\\left(x - X(\\tau)\\right)U^\\mu(\\tau)\n\\end{equation}\nHere $G^-(x-y)$ is the retarded Green's function satisfying the following equation and boundary condition:\n\\begin{eqnarray}\n &&\\partial^2 G^-(x-y) = \\delta(x-y)\\\\\n &&G^-(x-y) = 0,~~~x^0 < y^0\n\\end{eqnarray}\nWe will also need another Green's function defined as:\n\\begin{equation}\\label{combo}\n {\\overline G}(x - y) = {1\\over 2}\\left[G^-(x-y) + G^-(y-x)\\right] = {\\cal G}_D\\left((x-y)^2\\right)\n\\end{equation}\nwhere we use the fact that this Green's function depends only on the quantity $(x-y)^2$ (and\nwe also explicitly indicate the $D$ dependence).\n\n{}Using (\\ref{combo}), we have\n\\begin{eqnarray}\\label{FU}\n &&F^{\\mu\\nu}(X(\\tau))U_\\nu(\\tau) = 16\\pi e\\int_{-\\infty}^\\tau d\\tau^\\prime~{\\cal G}^\\prime_D\\left((X(\\tau)-X(\\tau^\\prime))^2\\right) K^\\mu(\\tau, \\tau^\\prime)\\\\\n &&K^\\mu(\\tau, \\tau^\\prime) = \\left[\\left(X^\\mu(\\tau) - X^\\mu(\\tau^\\prime)\\right)U^\\nu(\\tau^\\prime) - (\\mu\\leftrightarrow\\nu)\\right]U_\\nu(\\tau)\n\\end{eqnarray}\n\n{}We can now see that the quantity $F^{\\mu\\nu}(X(\\tau))$, i.e., the electromagnetic field tensor on the worldline, diverges.\nThis is due to the divergent nature of the Green's\nfunction. Therefore, it should be regularized and the divergences, if possible, should be\nremoved via renormalization. From this viewpoint, renormalization\nof the classical electrodynamics is analogous to renormalization in quantum\nfield theory: it is needed because the self-interaction of the particle is divergent. But\nthe analogy stops here. Thus, the renormalized equations\nof motion of the classical point particle cannot be derived from the action principle.\n\n{}The explicit form of the function ${\\cal G}_D(z)$ reads:\n\\begin{eqnarray}\n &&{\\cal G}_D(z) = \\theta^{(k)}(z) \/ 4\\pi^k,~~~D = 2k+2,~~~k=0,1,2,\\dots\\\\\n &&{\\cal G}_D(z) = (-1)^k Q^{(k)}(z) \/ \\pi^k,~~~D = 2k+3,~~~k=0,1,2,\\dots\n\\end{eqnarray}\nwhere the superscript $(k)$ means the $k$-th derivative w.r.t. $z$, and\n\\begin{eqnarray}\n &&\\theta(z) = \\int_{-\\infty}^z d\\alpha~\\delta(\\alpha)\\\\\n &&Q(z) = (4\\pi z^{1\/2})^{-1}\n\\end{eqnarray}\n\n{}It is convenient to use different regularizations when $D$ is even and when $D$ is\nodd. When $D$ is even, we can replace the quantity $(X(\\tau)-X(\\tau^\\prime))^2$ in (\\ref{FU}) by $(X(\\tau)-X(\\tau^\\prime))^2 - \\varepsilon^2$, where $\\varepsilon$ is a positive infinitesimal parameter. Then $F^{\\mu\\nu}(X(\\tau))U_\\nu(\\tau)$ is equal to:\n\\begin{eqnarray}\n &&{\\cal O}(\\varepsilon),~~~D = 2\\\\\n &&-eU^{\\prime\\mu}(\\tau)\/2\\varepsilon + {2\\over 3}e\\left[U^{\\prime\\prime\\mu}(\\tau) + U^\\mu(\\tau)~U^{\\prime 2}(\\tau)\\right] + {\\cal O}(\\varepsilon),~~~D=4\\label{4D}\\\\\n &&-eU^{\\prime\\mu}(\\tau)\/4\\pi\\varepsilon^3 + 3e\\left[U^{\\prime\\prime\\mu}(\\tau) + 3U^\\mu(\\tau)~U^{\\prime 2}(\\tau)\/2\\right]^\\prime\/16\\pi\\varepsilon + \\nonumber\\\\\n &&~~~~~~~+\\mbox{finite terms},~~~D=6\n\\end{eqnarray}\nand so forth. When $D$ is odd, we can replace the integration limit $\\tau$ in (\\ref{FU}) by\n$\\tau-\\varepsilon$ and obtain for the same quantity:\n\\begin{eqnarray}\n &&-eU^{\\prime\\mu}(\\tau)\\ln(\\rho\/\\epsilon) + \\mbox{finite terms},~~~D=3\\\\\n &&-3eU^{\\prime\\mu}(\\tau)\/4\\pi\\varepsilon^2 + e\\left[U^{\\prime\\prime\\mu}(\\tau) + U^\\mu(\\tau)~U^{\\prime 2}(\\tau)\\right]\/\\pi\\varepsilon - 3e\\ln(\\rho\/\\epsilon)\\times \\nonumber\\\\\n &&~~~~~~~\\times\\left[U^{\\prime\\prime\\mu}(\\tau) + 3U^\\mu(\\tau)~U^{\\prime 2}(\\tau)\/2\\right]^\\prime\/8\\pi + \\mbox{finite terms},~~~D=5\n\\end{eqnarray}\nand so forth. Here $\\rho$ is an arbitrary positive infrared cut-off parameter which appears\nbecause of the logarithmic divergences.\n\n{}So, we see that in two dimensions there is no divergence and, moreover, the point\nparticle does not radiate any electromagnetic waves. The reason for this is that in 2D a\nfree electromagnetic field is a pure gauge, although the Coulomb interaction is nontrivial.\nThe electromagnetic field follows the point particle preserving its constant energy and\nspatial shape.\n\n{}In three and four dimensions the self-interaction term is divergent but nonetheless\nin both cases the divergences that appear have the form of the kinetic term in the initial\nLorentz equations of motion (\\ref{self.EM}). Therefore, we can eliminate these divergences via\nmass renormalization:\n\\begin{eqnarray}\n &&{\\widetilde m} = m + e^2\\ln(\\rho\/\\varepsilon),~~~D=3\\\\\n &&{\\widetilde m} = m + e^2\/2\\varepsilon,~~~D=4\n\\end{eqnarray}\nSince no other divergences are present in these two cases, the theory is renormalizable.\nIn particular, in four dimensions using (\\ref{4D}) we obtain the well-known Lorentz equation\nwith the radiation term \\cite{LL}:\n\\begin{equation}\n {\\widetilde m}~U^{\\prime\\mu}(\\tau) = e {\\cal F}^{\\mu\\nu}(X(\\tau)) U_\\nu(\\tau) + {2\\over 3}e^2\\left[U^{\\prime\\prime\\mu}(\\tau) + U^\\mu(\\tau)~U^{\\prime 2}(\\tau)\\right]\n\\end{equation}\nIn $D = 5,6,\\dots$ we can also eliminate one divergence by renormalizing\nthe mass of the particle, but there are other divergences which cannot be\nremoved via renormalization as the required terms are absent in the\noriginal equations of motion.\n\n\\section{Renormalization of String in Four Dimensions}\n\n{}In this section we discuss a renormalization procedure analogous to that\nin the previous section, but for the case of the four-dimensional string. The equations\nof motion for the string coordinates and the scalar field following from the action (\\ref{action.string}) read:\n\\begin{eqnarray}\\label{EOM.string}\n &&\\left[T + \\lambda\\phi(X(\\sigma))\\right]\\partial^2X^\\mu(\\sigma) = \\nonumber\\\\\n &&~~~~~~~=\\lambda\\left\\{{1\\over 2}~\\partial^\\mu\\phi(X(\\sigma))(\\partial X(\\sigma))^2 - \\partial^\\nu\\phi(X(\\sigma))\\partial_\\alpha X^\\mu(\\sigma)\\partial^\\alpha X_\\nu(\\sigma)\\right\\}\\\\\n &&\\partial^2\\phi(x) = 2\\pi\\lambda\\int d^2\\sigma~\\delta(x - X(\\sigma))(\\partial X(\\sigma))^2\n\\end{eqnarray}\nHere the gauge freedom has been used to fix the constraint as follows:\n\\begin{equation}\n \\partial_\\alpha X^\\mu(\\sigma)\\partial_\\beta X_\\mu(\\sigma) = {1\\over 2}~\\eta_{\\alpha\\beta}~(\\partial X(\\sigma))^2\n\\end{equation}\nSo, the worldsheet is flat: $h^{\\alpha\\beta} = \\eta^{\\alpha\\beta}$, where $\\eta^{\\alpha\\beta}$ is the Minkowski metric. Note that we could also include a non-dynamical external scalar field in the action (\\ref{action.string}), but this is not crucial here.\n\n{}Using the techniques discussed in the previous section, we have (note that here we are working in four dimensions):\n\\begin{equation}\\label{phi1}\n \\phi(X(\\sigma)) = \\lambda\\int d\\sigma^\\prime\/|\\sigma - \\sigma^\\prime| + \\mbox{finite terms}\n\\end{equation}\nThe one-dimensional integral over the spatial coordinate $\\sigma^\\prime$ in (\\ref{phi1}) is taken along the string. Also, the quantity\n\\begin{eqnarray}\n &&{1\\over 2}~\\partial^\\mu\\phi(X(\\sigma))(\\partial X(\\sigma))^2 - \\partial^\\nu\\phi(X(\\sigma))\\partial_\\alpha X^\\mu(\\sigma)\\partial^\\alpha X_\\nu(\\sigma)=\\nonumber\\\\\n &&~~~~~~~={\\lambda\\over 2}~\\partial^2 X^\\mu(\\sigma)\\int d\\sigma^\\prime\/|\\sigma - \\sigma^\\prime| + \\mbox{finite terms}\n\\end{eqnarray}\ncan be computed using the following formula:\n\\begin{eqnarray}\n &&\\int d\\sigma^\\prime\\int_{-\\infty}^\\tau d\\tau^\\prime~(\\sigma- \\sigma^\\prime)^\\alpha (\\sigma- \\sigma^\\prime)^\\beta ~\\delta^\\prime((X(\\sigma) - X(\\sigma^\\prime))^2) = \\nonumber\\\\\n &&~~~~~~~=-[(\\partial X(\\sigma))^2]^{-2}\\eta^{\\alpha\\beta} \\int d\\sigma^\\prime\/|\\sigma - \\sigma^\\prime| + \\mbox{finite terms}\n\\end{eqnarray}\nThus, the equation of motion (\\ref{EOM.string}) reads\n\\begin{equation}\n \\left\\{T + {\\lambda^2\\over 2}\\int d\\sigma^\\prime\/|\\sigma - \\sigma^\\prime| \\right\\}\\partial^2 X^\\mu(\\sigma) = \\mbox{finite nonlocal terms}\n\\end{equation}\n\n{}So, the situation for the four-dimensional string is analogous to that for the relativistic point particle in $D = 3$ or $D = 4$ discussed earlier. There is\nonly one divergence of the same form as the kinetic term in the original equations\nof motion. The logarithmically divergent integral can be regularized as follows:\n\\begin{equation}\n \\int d\\sigma^\\prime\/|\\sigma - \\sigma^\\prime| = \\int_{\\sigma - \\rho}^{\\sigma-\\varepsilon} d\\sigma^\\prime\/(\\sigma - \\sigma^\\prime) +\n \\int_{\\sigma+\\varepsilon}^{\\sigma + \\rho} d\\sigma^\\prime\/(\\sigma^\\prime - \\sigma) = 2\\ln(\\rho\/\\varepsilon)\n\\end{equation}\nThe infrared cut-off parameter $\\rho$ is analogous to that introduced in the previous section. The\nrenormalized string tension thus becomes:\n\\begin{equation}\n {\\widetilde T} = T + \\lambda^2\\ln(\\rho\/\\varepsilon)\n\\end{equation}\nThere is no other divergence in this case and, therefore, the string equations of motion\nare renormalizable in 4D.\n\n\\section{Strings in Higher Dimensions}\n\n{}Now we turn to the renormalizability of the string\nequations of motion in higher dimensions. First note that in the string case we\nneed not regularize the integral over the $\\tau^\\prime$ variable appearing in such quantities as\n$\\phi(X(\\sigma))$ as the extended spatial dimension of the string plays the role of the regulator for this integral. But then we\nhave to regularize the remaining integral over the $\\sigma^\\prime$ variable. So, based on the\nresults obtained in Section 2, we can see that in five dimensions there are two\ndivergences of the form $1\/\\varepsilon$ and $\\ln(\\varepsilon)$. In six dimensions we have the $1\/\\varepsilon^2$, $1\/\\varepsilon$ and $\\ln(\\varepsilon)$\ndivergences. And so forth for the higher dimensions. In 3D the string is finite.\n\n{}The preceding analysis shows that the string is not renormalizable if $D > 4$.\nIn the case of the point particle\nwe can always eliminate one of the appearing divergences via mass\nrenormalization. However, in the higher-than-four-dimensional string case no divergence can be\nremoved via renormalization.\n\n{}Thus, consider the leading divergences in $\\phi(X(\\sigma))$ and the r.h.s. of (\\ref{EOM.string}) but in $D > 4$\ndimensions. They enter the equations of motion in the following form:\n\\begin{equation}\\label{HigherD}\n \\propto \\lambda^2 \\varepsilon^{4-D} [(\\partial X(\\sigma))^2]^{(4-D)\/2} \\partial^2 X^\\mu(\\sigma)\n\\end{equation}\nTherefore, this divergence cannot be removed via string tension renormalization owing to the extra factor $[(\\partial X(\\sigma))^2]^{(4-D)\/2}$, which appears due to the following rescaling property of the Green's function in $D$ dimensions:\n\\begin{equation}\n {\\cal G}_D(\\gamma z) = \\gamma^{(2-D)\/2}{\\cal G}_D(z)\n\\end{equation}\nNote that for the point particle the quantity analogous to $(\\partial X(\\sigma))^2$ is $U^2(\\tau) = 1$, which is why for the point particle the leading divergence can always be removed via mass renormalization.\n\n\\section{Concluding Remarks}\n\n{}Let us briefly summarize the discussion in the previous sections. We have seen that\nclassical electrodynamics is a renormalizable theory only in $D < 5$ dimensional spacetimes.\nThe same holds for the bosonic string coupled to the massless scalar\nfield. It is worthwhile to compare our results with the work \\cite{DH}, in which it was argued\nthat in four dimensions the superstring tension is not renormalized at all. Indeed, there\nis a combination of the dilaton $\\phi$, metric $g_{\\mu\\nu}$ and antisymmetric tensor $B_{\\mu\\nu}$ fields,\ncoupled to the bosonic string via a non-linear sigma-model action, for which the\ntension of the string is not renormalized at all at the lowest order of the perturbation\ntheory. For the point particle there too is a certain combination of the electric charge and the mass of the\nparticle, for which in four dimensions the mass of the particle is not renormalized at\nall due to the cancelation of the contributions from the electromagnetic and gravitational self-interactions.\nHowever, in the case of the string we have shown that in $D > 4$ there are some other\ndivergences unrelated to the string tension renormalization. So, to achieve finiteness of the string, we\nmust include massless as well as an infinite tower of massive modes. For instance, the extra factor\n$[(\\partial X(\\sigma))^2]^{(4-D)\/2}$ in (\\ref{HigherD}) in $D > 5$ indicates that there are additional modes\nto be considered. This fits in the ideology of \\cite{Lepage}: non-renormalizability\nof a physical theory indicates that there is some new underlying physics to be included.\nOur analysis of the renormalizability of classical string theory shows that the extended\nnature of the string alone is insufficient for finiteness. The latter requires inclusion of\nall the excitation modes of the string. On the\nother hand, the way the string excitations combine together in the local field theory\naction functional is determined by the conformal invariance. In particular,\nthe special combination of the dilaton $\\phi$, metric $g_{\\mu\\nu}$ and antisymmetric tensor $B_{\\mu\\nu}$ fields\ndiscussed above is a consequence\nof the conformal invariance requirement.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\n\nObject manipulation tasks performed by autonomous robots require the robot to recognize and locate the object in the 3D space, typically through visual perception. Precise grasping tasks necessitate an estimation of the full 6-DoF pose of task-relevant objects from the input visual data. In the past few years, powerful CNNs have been successfully employed for this purpose enabling object pose estimation under difficult circumstances \\cite{pavlakos17object3d, xiang2018posecnn}. However, the performance of most CNN based state-of-the-art methods is dependent on the availability of large sets of real-world data --- labeled with the ground-truth pose for each object instance --- to supervise the training of the involved neural networks. This training dataset may easily consist of tens of thousands or more labeled data points per object in varying backgrounds and environments \\cite{xiang2018posecnn, zeng2017multi:zengapc, rad2017bb8:bb8}. In such cases, manual labeling of raw samples becomes unreasonable and impractical. Other pose estimation methods that claim to use artificially generated synthetic data are either still partially dependent on real data for fine-tuning or require very high quality textured 3D object models \\cite{tremblay2018corl:dope}. Nevertheless, real-world annotated data is indispensable for a meaningful and comprehensive evaluation of any pose estimator.\n\n\nTechniques for (semi-) automating the process of training data generation has gained research interest in recent years \\cite{rennie2016dataset, wong2017segicp, hodan2017t:tless, hinterstoisser2012model:linemod, suzui2019toward}. However, their reliance on additional hardware such as multi-sensor rigs, turntables with markers, or motion capture systems limits their application in arbitrary environments. Besides, hardware setup and sensor calibration may in itself be time-consuming. LabelFusion \\cite{marion2018label} is a recent but popular method that allows raw data capture through a hand-held RGB-D sensor. Yet, its dependence on the availability of pre-built object meshes restricts operation on a wider object set. EasyLabel \\cite{suchi2019easylabel} attempts to overcome the dependence on pre-built models. However, it cannot produce data on a scale that is necessary for training deep networks, as the method works on individual snapshots of the scene and there is no propagation of labels. \n\nWe address these problems by proposing a technique to produce large amounts of pose-annotated, real-world data for rigid objects that uses minimal human input and does not require any previously built 3D shape or texture model of the object. Our method also simplifies the capture of raw (unlabeled, unprocessed RGB-D) data due to independence from turntables, marker fields, motion capture setups, and manipulator arms. The key idea is to use sparse annotations provided manually and systematically by a human user over multiple scenes, and combine them under geometrical constraints to produce the labels. We define ``pose labels\" as the 2D bounding-box labels, 2D keypoint labels and the pixel-wise mask labels (although other types of labels can be deduced too). Hence, our method can effectively be used for generating large training datasets -- for 2D\/3D object detection, 2D\/3D keypoint estimation, 6-DoF object pose estimation, and semantic segmentation -- for novel objects in real scenarios outside of popular datasets such as LineMOD\\cite{hinterstoisser2012model:linemod} and T-LESS\\cite{hodan2017t:tless}.\n\nWe demonstrate the effectiveness of our method by generating keypoint labels, pixel-wise mask labels, and bounding-box labels for more than 150,000 RGB images of 11 unique objects (7 in-house + 4 YCB-Video \\cite{xiang2018posecnn} objects) in a total of 80 (single and multi-object) scenes --- in only a few minutes of manual annotation for each object. We evaluate the accuracy of the generated labels and the sparse model using ground-truth CAD models. Subsequently, we train and evaluate (1) a keypoint-based 6-DoF pose estimation pipeline and (2) an object segmentation CNN using the resulting labeled dataset. Our proposed tool with the graphical user-interface are published as a public GitHub repository \\footnote[2]{\\url{https:\/\/github.com\/rohanpsingh\/RapidPoseLabels}}.\n\n\\section{RELATED WORK}\n\nVarious works on developing open-source 6-DoF pose labeled datasets have adopted different approaches for automating the process of labeling RGB-D data \\cite{hinterstoisser2012model:linemod, hodan2017t:tless, rennie2016dataset, hua2016scenenn, dai2017scannet}. The popular Rutger's dataset \\cite{rennie2016dataset} was generated by mounting a Kinect sensor to the end joint of a Motoman robot arm. The annotation was done by a human to align the 3D model of the objects in the corresponding RGBD point cloud scene. This approach does produce high quality ground-truth data albeit at the cost of laborious manual involvement. The T-LESS dataset \\cite{hodan2017t:tless}, containing about 50K RGB-D images of 30 textureless objects, was developed with an arguably complicated setup involving a turntable with marker-field and a triplet of sensors attached to a jig. The method for dataset generation described in \\cite{zeng2017multi:zengapc} works through background subtraction, first recording data in a scene without the object and then with the object. This can produce pixel-wise segmentation labels even for model-less objects, but cannot be used for 6-DoF pose labels. SegICP \\cite{wong2017segicp} used a labeled set of 7500 images in indoor scenes, produced with the help of a motion capture (MoCap) based automated annotation system, which inevitably limits the types of environments in which data is acquired. An interesting method of collecting data through the robot in a life-long self-supervised learning fashion was presented in \\cite{deng2019self}. Here the robot learns pose estimation and simultaneously generates new data for improving pose estimation by itself, although in a very limited environment.\n\nThe complications involved in acquisition of real-world training data have inspired methods such as \\cite{tremblay2018corl:dope, tobin2017domain} which are trained only on synthetic or photo-realistic simulated data. Although demonstrating very promising results, they present an additional requirement of the availability of a very high quality textured 3D model, which may be hard to obtain in itself without dedicated special hardware.\n\nMarion et. al. presented LabelFusion \\cite{marion2018label} -- a method for generating ground-truth labels for RGB-D data minimizing the human effort involved. They perform a dense 3D reconstruction of the scene using ElasticFusion, and then ask the user to manually annotate 3 points in the 3D scene space to initiate an ICP-based registration to align a previously built 3D model to the scene point cloud. For sidestepping the need for a prior object model, Suchi et. al. proposed EasyLabel \\cite{suchi2019easylabel} wherein scenes are incrementally built and depth changes are exploited to obtain object masks. This however limits the scale at which data can be generated and so, their method is presented primarily for evaluation purposes -- as opposed to training of deep CNNs.\n\nOur method outperforms the current state-of-the-art in regard to the above-mentioned problems. It is capable of labeling large scales of datasets, suitable for supervised training of deep networks and, as stated earlier, it is not dependent on the availability of a prior 3D object model.\n\n\\begin{figure*}\n \\includegraphics[width=\\textwidth]{images\/main_approach3.png}\n \\caption{\\textbf{Proposed system for generating pose labels.} The input to the system is a set of RGB-D image sequences and a limited amount of manual labels. The output consists of (1) labels for each frame in the raw dataset, (2) a sparse, keypoint-based representation and (3) a dense model of the object. Existing techniques are used for dense reconstruction while a user-friendly GUI is implemented for providing the manual annotations. An optimization problem is solved on the manual annotations to recover a sparse model and finally, a dense model is built-up from the sparse model using a region-growing segmentation algorithm.}\n \\label{figure:main_approach}\n\\end{figure*}\n\n\\section{APPROACH}\n\nWe consider the case of generating training data for a single object-of-interest. The human user collects a set of $N_s$ RGB-D videos (alternatively referred to as \\textit{scenes}) by moving a hand-held RGB-D sensor around the relevant object --- placed in varying orientations, positions, backgrounds, and lighting conditions. This set of unlabeled videos or scenes is denoted by $\\mathcal{U} = \\{\\mathcal{U}_1, \\mathcal{U}_2, \\dots, \\mathcal{U}_{N_s}\\}$. The duration and frame rate of each video $\\mathcal{U}_s$ can be variable (2-5 minutes and 30fps in our experiments). Mathematically, a scene consisting of ${t_{s}}$ frames is defined as $\\mathcal{U}_s = \\{ (\\mathbf{I}_s^{(t)}, \\mathbf{D}_s^{(t)})_{t=1}^{t_{s}} \\}$, where $\\mathbf{I}_s^{(t)}$ and $\\mathbf{D}_s^{(t)}$ are the RGB and depth images respectively at time instance $t \\in \\{1 ,\\dots, {t_{s}}\\}$ in the scene $s \\in \\{1 ,\\dots, {N_s}\\}$. Our final objective is to generate the labeled dataset $\\mathcal{L} = \\{\\mathcal{L}_1, \\mathcal{L}_2, \\dots, \\mathcal{L}_{N_s}\\}$, i.e. associate each frame in each scene with a pose label.\n\n\\textbf{System Overview.} Figure \\ref{figure:main_approach} illustrates the overview of the process. Given $\\mathcal{U}$, our system initiates by performing dense scene reconstructions using a third-party software to obtain scene meshes and recover camera trajectories (Section: \\ref{subsec:dense_recon}). Next, we obtain the manual labels with the help of our GUI tool (Section \\ref{subsec:manual_annot}). The key idea in this step is to ask the human annotator to label an ordered set of $N_k$ arbitrarily chosen landmark points on the object (called the \\textit{keypoints set}) in each scene. The user needs to sequentially mark only a subset of the keypoints on any randomly selected RGB image in each collected scene. This produces fragments of the sparse keypoint-based object representation each defined in the respective scene's frame of reference. If there is enough overlap between the fragments, we can recover simultaneously (1) the full, 3D keypoint-based sparse model $\\mathbf{Q} \\in \\mathbb{R}^{3\\times N_k}$ and (2) the relative transformations between the scenes, explaining the annotations provided by the user. We achieve this by formulating and solving an optimization problem on the sparse, user-annotated keypoint configurations constrained on 3D space rigidity (Section \\ref{subsec:optim_step}). The resulting output of a successful optimization is (optionally, if mask labels are required) used to build a dense model of the object by segmenting out the points corresponding to the object of interest from all scene meshes and combining them together (Section \\ref{subsec:dense_model}). Finally, the produced sparse and dense models can be projected back to all the 2D image planes in $\\mathcal{U}$ to obtain the desired type of labels (Section \\ref{subsec:label_generation}). \n\n\\subsection{Dense Scene Reconstruction} \\label{subsec:dense_recon}\nEach scene $\\mathcal{U}_s$ is a sequence of image pairs where the camera trajectory through time is unknown. It is valuable to obtain this trajectory as it can be used to automatically propagate labels through all instances in the scene. We rely on an existing technique which provides camera pose tracking along with a dense 3D reconstruction, to avoid dependence on fiducial markers or robotic manipulators. With this, the dataset $\\mathcal{U}$ is now altered to become $\\mathcal{U} = \\{\\{(\\mathbf{I}_s^{(t)}, \\mathbf{D}_s^{(t)}, \\mathbf{C}_s^{(t)})_{t=1}^{t_{s}}\\}_{s=1}^{N_s}\\}$, where $\\mathbf{C}_s^{(t)} \\in \\mathbb{R}^{4\\times4}$ is the homogeneous transformation matrix giving the camera pose $\\mathcal{F}_{s}^{(t)}$ at instance $t$ in the scene $s$ relative to the initial camera pose $\\mathcal{F}_{s}^{(1)}$ of the same scene. In our experiments, we use the open-source implementation of ElasticFusion \\cite{whelan2016elasticfusion} for this step (as in \\cite{marion2018label}).\nIt is worthwhile to mention here that while recording the videos, an individual scene $\\mathcal{U}_s$ does not necessarily need to consist of a full scan of the object from all views, which may be difficult to obtain. Our method combines different object views from all scenes in $\\mathcal{U}$ to produce final output.\n\n\\subsection{Manual Annotation} \\label{subsec:manual_annot}\nKeeping in line with our desire to reduce the involved human effort and time to as low as possible, our system sources minimal annotations from the human user. The user first chooses a set of arbitrary but well-distributed, ordered, and uniquely identifiable landmark points (called \\textit{keypoints}) on the physical object. Then, with the help of our user-interface, the user sequentially labels the location of these keypoints in each scene. The labeling is done on the RGB image. As all faces of the object may not be visible in a scene, labeling is done only for the visible subset of the keypoints on the RGB image. The non-visible keypoints are skipped. Moreover, the annotations can be distributed over multiple images of the same scene.\n\nFor each user-annotated keypoint $k' \\in \\{1,\\dots,N_k\\}$ on the RGB image $\\mathbf{I}_{s'}^{(t')}$ in scene $s' \\in \\{1 ,\\dots, {N_s}\\}$ and at time instance $t' \\in \\{1,\\dots, {t_{s'}}\\}$, using camera intrinsics of the RGB sensor and the depth image $\\mathbf{D}_{s'}^{(t')}$, we obtain the 3D point position $\\prescript{}{k'}{\\mathbf{d}_{s'}^{(t')}} = [t_x, t_y, t_z]^T$ of the labeled pixel. Then, we transform this point $\\prescript{}{k'}{\\mathbf{d}_{s'}^{(t')}}$ to the coordinate frame $\\mathcal{F}_{s'}^{(1)}$ (i.e. in the camera frame at time $t=1$ in scene $s=s'$):\n\n\\begin{equation} \\label{eq:tf}\n\\begin{bmatrix}\n\\prescript{}{k'}{\\mathbf{d}_{s'}^{(1)}} \\\\[6pt]\n1\n\\end{bmatrix} = \\mathbf{C}_{s'}^{(t')} \\cdot\n\\begin{bmatrix}\n\\prescript{}{k'}{\\mathbf{d}_{s'}^{(t')}} \\\\[6pt]\n1\n\\end{bmatrix}.\n\\end{equation}\n\nFor each scene $s$, this gives us the matrix $\\mathbf{W}_{s} \\in \\mathbb{R}^{3 \\times N_k}$, where the columns hold the 3D position $\\prescript{}{k}{\\mathbf{d}_{s}^{(1)}}$ of the keypoint $k$ if it was manually annotated and $\\mathbf{0}^{3}$ otherwise. Doing so for all scenes, we obtain $\\mathcal{W} = \\{\\mathbf{W}_1, \\mathbf{W}_2, \\dots, \\mathbf{W}_{N_s}\\}$.\n \n\\textbf{Selection of Keypoints.} Although the selection of keypoints on the object is arbitrary, there are some constraints on the manual annotation which must be kept in mind by the user while choosing them. To ensure existence of a unique solution to the optimization problem, the marked keypoints should rigidly connect all the scenes together, namely by avoiding the system to suffer underdetermination in the produced constraints for the scene relative poses. For example, for two scenes, mutual sharing of 3 or more non-collinear keypoints rigidly ``ties\" them. Hence, we recommend choosing points that are more susceptible to be shared in multiple scenes, such as on the edge shared by two faces of an object.\n\n\\subsection{Optimization} \\label{subsec:optim_step}\nThus far, we have localized subsets or fragments of the object's keypoint model $\\mathbf{Q}$ in each scene of $\\mathcal{U}$. The subsets do not exist in one common frame of reference. Instead, the fragment $\\mathbf{W}_{s}$ is defined in the coordinate frame $\\mathcal{F}_{s}^{(1)}$ and the relative transformations between scenes are yet unknown. In this step we find the relative transformations and combine the fragments in a common space to recover $\\mathbf{Q}$. Let us represent the set of relative transformations by $\\mathcal{T} = \\{{\\mathbf{T}_{1}}, {\\mathbf{T}_2}, \\dots, {\\mathbf{T}_{N_s}}\\}$ where $\\mathbf{T}_{s} = \\begin{bmatrix} \\mathbf{q}_s \\quad \\mathbf{t}_s \\end{bmatrix}$ is the rigid transformation of the local frame $\\mathcal{F}_{s}^{(1)}$ with respect to the world frame $\\mathcal{F}_{w}$. $\\mathbf{q}_s \\in \\mathbb{R}^4$ represents the rotation quaternion and $\\mathbf{t}_s \\in \\mathbb{R}^3$ is the 3D position of the camera center. We set $\\mathcal{F}_{w} = \\mathcal{F}_{1}^{(1)}$ (i.e. origin of scene $1$). We formulate the following nonlinear optimization problem:\n\n\\begin{align}\n\\mathbf{Q}\\text{*},\\mathcal{T}\\text{*} & = \\underset{\\mathbf{Q},\\mathcal{T}}{\\text{argmin}}. \n\\| \\mathbf{S} \\cdot f(\\mathcal{T}, \\mathbf{Q}) - \\mathbf{W} \\| ^2, \\nonumber \\\\\n& \\textrm{s.t.} \\quad \\mathbf{q}_s^T\\cdot\\mathbf{q}_s=1, \\nonumber \\\\\n& \\forall s \\in \\{0 ,\\dots, N_s-1\\}. \\label{eq:opt_eq}\n\\end{align}\n\\noindent\nwhere $\\mathbf{W}$ is the concatenation of all non-zero points in $\\mathcal{W}$ and the function $f(\\cdot)$ successively applies each $\\mathbf{T}_{s} \\in \\mathcal{T}$ to $\\mathbf{Q}$ and returns the concatenated vector. The selection matrix $\\mathbf{S}$ selects from the vector $f(\\mathcal{T}, \\mathbf{Q})$ only those keypoints whose reference is available in $\\mathcal{W}$, that is, the keypoints which were manually annotated. The solution is given by $\\mathbf{Q}\\text{*}$ - representing the optimized keypoint representation of the object model defined in the frame $\\mathcal{F}_{w}$ - and the set $\\mathcal{T}\\text{*}$ - representing the optimized relative scene transformations.\n\nIntuitively, solving Eq. \\ref{eq:opt_eq} is equivalent to bringing together the subsets in $\\mathcal{W}$, defined in their local frames, into one common world frame of reference (as illustrated in Figure \\ref{figure:optimization}). This is possible because the user has inputted mutually overlapping annotations, and the solution to Eq. \\ref{eq:opt_eq} provides a consistent explanation of all the observations.\n\nWe used the scipy.optimize library in Python with the Sequential Least Squares Programming solver \\cite{kraft1988software} to minimize Eq. \\ref{eq:opt_eq}. We initialized the iterations by setting all elements of $f(\\mathcal{T}, \\mathbf{Q})$ to 0 except for the real part of the rotation quaternion which is set to 1.\n\n\\begin{figure}[b]\n \\includegraphics[width=\\linewidth]{images\/optimization.png}\n \\caption{\\textbf{Problem description.} An example dataset with 3 scenes `1', `2' and `3' with trajectory lengths $t_1$, $t_2$ and $t_3$ respectively are shown. The left-side shows 4 keypoints annotated on an image of Scene `2' at time $t=t'$, projected to get the 3D positions. This is done for each scene (with some amount of mutually shared keypoints). The 3D point fragments are then transformed to the origin of their respective scene trajectories, giving us $\\mathbf{W}_1$, $\\mathbf{W}_2$ and $\\mathbf{W}_3$, on the right. The intention of the optimization, essentially, is to find $\\mathbf{T}_1$, $\\mathbf{T}_2$, $\\mathbf{T}_3$ such that all fragments can be represented in world frame $\\mathcal{F}_w$. Keypoint overlapping ensures existence of a unique solution(eg. keypoints 2,3,4 are annotated in scene `1' and `2' both).}\n \\label{figure:optimization}\n\\end{figure}\n\n\\subsection{Segmentation of Dense Object Model} \\label{subsec:dense_model}\nIn this step, we use the sparse object model and the scene transformations produced above to segment out the 3D points corresponding to the object from each of the dense scenes, thereby producing a dense model of the object. A dense object model is necessary to obtain pixel-wise mask labels and is also useful for planning robotic grasp poses. We modify Point Cloud Library's (PCL) region-growing-segmentation algorithm such that the individual points of the sparse model act as seeds and the regions are grown and spread outwards from the seeds. All segmented regions from each scene are then combined using the relative transformations to create the dense model. The output may occasionally have holes in the geometry or contain points from the background scenes which require manual cropping, yet, as our experiments indicate, it remains a practical approximation for the object shape. \n\n\n\\subsection{Generation of Pose Labels} \\label{subsec:label_generation}\nHaving obtained the object sparse model, the dense model (both defined in $\\mathcal{F}_{w}$) and the set $\\mathcal{T}$ of scene transformations w.r.t. $\\mathcal{F}_{w}$, the labeled dataset $\\mathcal{L} = \\{\\mathcal{L}_1, \\mathcal{L}_2,\\dots, \\mathcal{L}_{N_s}\\}$ can be generated through back projection to the RGB image planes. For the purpose of training the 6-DoF pose estimation pipeline adopted by us \\cite{pavlakos17object3d}, we define $\\mathcal{L} = \\{\\{(\\mathbf{I}_{s}^{(t)}, \\textbf{L}_{s}^{(t)}, \\textbf{b}_{s}^{(t)})_{t=1}^{t_{s}}\\}_{s=1}^{N_s}\\}$ where $\\textbf{L}_{s}^{(t)} \\in \\mathbb{R}^{2\\times{N_k}}$ is the 2D keypoint annotation in the image $\\mathbf{I}_{s}^{(t)}$ for the $N_k$ keypoints chosen on the object and $\\textbf{b}_{s}^{(t)} \\in \\mathbb{R}^{3}$ represents the $x,y$ pixel coordinates of the center and side length $h$ of an upright bounding-box square around the object in the image.\n\nComputation of the label for the $k^{th}$ keypoint at time instance $t$ of the scene $s$, $\\prescript{}{k}{\\textbf{l}}_{s}^{(t)}$, can be done simply by transforming the point $\\mathbf{Q}_{*,k}$ to the camera frame $\\mathcal{F}_{s}^{(t)}$ to obtain the 3D point $\\prescript{}{k}{\\mathbf{d}_{s}^{(t)}} = [t_x, t_y, t_z]^T$, following Eq. \\ref{eq:tf_inverse}. And subsequently, projecting it onto the 2D image plane to get $\\prescript{}{k}{\\textbf{l}}_{s}^{(t)} = [u_x, u_y]^T$. \\par\n\n\\begin{align}\n\\begin{bmatrix}\n\\prescript{}{k}{\\mathbf{d}_{s}^{(t)}} \\\\[6pt] \n1\n\\end{bmatrix}\n & = (\\mathbf{C}_{s}^{(t)})^{-1} \\cdot\n \\begin{bmatrix}\n{\\mathbf{T}^{-1}_{s}} \\circ {\\mathbf{Q}_{*,k}} \\\\[6pt]\n1\n\\end{bmatrix}. \\label{eq:tf_inverse}\n\\end{align}\n\nThe per-pixel mask label can also be obtained similarly once the dense model has been generated - by transformation and back projection of each point of the 3D dense model onto the 2D RGB images. \n\nThe bounding-box label $\\textbf{b}_{s}^{(t)}$ can be obtained by finding the up-right bounding rectangle of the mask points or taking the bounding-box of all points in $\\textbf{L}_{s}^{(t)}$ and scaling up by a factor of $1.5$ to avoid cropping out of the object itself.\n\n\\subsection{Using the Sparse Model for New Scenes}\nApplication of deep-learning approaches for practical purposes often requires fine tuning of the trained model with labeled data collected in the real environment of operation \\cite{zeng2017multi:zengapc}. So, the user must be capable of quickly labeling a new RGB-D scene that is outside of the initial dataset $\\mathcal{U}$. This is achieved as follows:\n\nOnce a model $\\mathbf{Q}$ of an object has been generated from $\\mathcal{U}$, for each new RGB-D video, the user must uniquely mark any subset of $\\mathbf{Q}$ (at least 3 points) on a randomly selected RGB image. Then, we obtain the 3D positions of the points using depth data, and apply Procrustes Analysis (using Horn's solution \\cite{horn1987closed}) to compute the rigid transformation between the set of marked points and their correspondences in $\\mathbf{Q}$. We can generate the labels again through back projection of the dense model according to the computed transformation. As this process involves only a few seconds of manual annotation, scalability to a large number of scenes is convenient.\n\n\\begin{figure}[t]\n \\includegraphics[width=\\linewidth]{images\/objects.png}\n \\caption{\\textbf{Selected objects for experiments.} (a) In-house objects on the left and (b) objects selected from YCB-Video dataset on the right.}\n \\label{figure:objects}\n\\end{figure}\n\n\\section{EXPERIMENTS}\nWe employ our proposed system to generate the sparse models, dense models and labeled datasets for two separate sets of objects for quantitative analysis (refer to Figure \\ref{figure:objects}). The first set consists of 4 objects from the YCB-Video dataset \\cite{xiang2018posecnn} which already provides us with multi-object RGB-D video scenes and the object CAD models. The second is a set of 7 objects for which we manually collected the RGB-D scenes with an Intel RealSense D435 at $640\\times480$ resolution. The CAD models of the 7 in-house objects were either sourced from the manufacturer or drawn manually prior to the experiments. We name these objects models as GT-CAD. Nevertheless, the CAD models for all objects and experiments were used only for evaluation purposes, as a key advantage of our system is its independence from prior object models.\n\nFrom the YCB-Video dataset, we selected 25 (20 single-object + 5 multi-object) scenes, with each of the 4 chosen objects appearing in 9 unique scenes. For the in-house objects, we recorded 55 (43 single-object + 12 multi-object) RGB-D scenes using the Intel RealSense D435 sensor. Each object's dataset $\\mathcal{U}$ was carved out of these 55 + 25 scenes.\n\n\\subsection{3D Sparse Model and Label Generation} \nTo evaluate the accuracy of the predicted sparse model, we align it to the corresponding GT-CAD model using ICP (with manual initialization) and find the closest points on the CAD model from each of the points in the sparse model. The set of closest points acts as the ground-truth for the sparse model (denoted by GT-SPARSE). We report the mean Euclidean distance between the corresponding points as an estimate of the sparse-model's accuracy. Next, as the primary motivation of our system is to enable automated generation of training data, we also evaluate the accuracy of the generated labels (keypoint + pixel-wise labels). To get the ground-truth for the keypoint and mask labels, we align (manual initialization followed by ICP fine tuning) the GT-CAD to the scene reconstructions, and project back GT-SPARSE and GT-CAD respectively to the RGB images using the camera trajectories (estimated by ElasticFusion or provided in YCB-Video dataset). Note that this approach is similar to the method of label generation in \\cite{pavlakos17object3d, marion2018label}. \n\nErrors in keypoints labels are computed as the mean 2D distance between the predicted pixel coordinates and the ground-truth. For the pixel-wise mask labels, we use the popular Intersection-over-Union (IoU) metric. Table \\ref{tab:table_model_errors} lists the results of this quantitative analysis for each object along with the number of scenes and number of keypoints chosen for this set of experiments.\n\n\\begin{figure}[b]\n \\includegraphics[width=\\linewidth]{images\/pose_estimation.png}\n \\caption{\\textbf{Block diagram of adopted pose estimation pipeline.} Computed by solving a P\\textit{n}P problem on predicted 2D keypoints and their 3D correspondences.}\n \\label{figure:pose_estimation}\n\\end{figure}\n\nThe errors in the mask labels also give an indication of the accuracy of the dense model. As explained earlier, the dense model requires manual cropping (~2-3min), but our system reduces the overall time in creating labeled datasets by several orders of magnitude as compared to the completely manual approach. We note that any ground-truth (produced either by this approach or through hand-labeling of each instance) will contain impurities in itself on a large dataset, hence the reported quantities capture the accuracy of our method only to a certain extent. However, the low errors (6.56 pixels and 81.2\\% IoU on average) approximately quantify the practicality of our method. Our mean IoU metric remains at par with the 80\\% IoU reported in LabelFusion while completely eliminating the need for a prior 3D model.\n\n\\begin{figure*}\n \\includegraphics[width=\\textwidth]{images\/output_example.png}\n \\caption{\\textbf{Keypoint and bounding-box labels.} Raw real RGB-D data was captured conveniently using a handheld RealSense sensor in 55 scenes for 7 in-house objets with no other hardware requirements. We generated keypoint and bounding-box labels using our approach for all chosen objects (6 are show here) in different scenes without using a prior 3D model from minimal manual annotation. Our approach is easily scalable to a large number of scenes.}\n \\label{figure:label_examples}\n\\end{figure*}\n\n\\begin{figure}[t]\n \\includegraphics[width=\\linewidth]{images\/eval_curves1.png}\n \\caption{Evaluation of the generated sparse keypoint-based model. We measure the average errors in estimated keypoint position with respect to the total no. of keypoints in (a) and total no. of scenes in (b).}\n \\label{figure:error_curves}\n\\end{figure}\n\nNext, we perform a set of experiments on the following 3 objects: Dewalt Cut-out Tool (OBJ1), Facom Wrench (OBJ2), Hitachi Screw Gun (OBJ3) -- to measure the effect of the number of scenes in $\\mathcal{U}$ ($N_s$) and the number of chosen keypoints ($N_k$) on the optimization process. The error in keypoint 3D positions for the 3 objects as a function of the total number of defined keypoints $N_k$ are shown in Figure \\ref{figure:error_curves}(a) when $N_s=8$, while Figure \\ref{figure:error_curves}(b) shows the mean error as a function of number of scenes $N_s$ when $N_k=9$. As the curves show, the mean positional error remained fairly independent of $N_k$ while the accuracy seemed to improve with $N_s$. This is expected as manual input for the same keypoint in multiple scenes would help remove bias. For $N_k$, a lower number of keypoints makes it harder to cover all faces of the object and a higher number increases the chances of human error in the manual annotation step. In our experience, choosing 6--12 adequately spaced keypoints proved to be sufficient for all objects, while $N_s$ is entirely up to the use-case scenario.\n\n\\subsection{Application to 6-DoF Object Pose Estimation}\nAs we expect the proposed approach to be primarily used for deployment of DL based 6-DoF pose estimation, we evaluate the performance of such a pipeline trained on the labeled dataset generated through our method for OBJ1, OBJ2 and OBJ3. We adopt a keypoint-based pose estimation approach \\cite{pavlakos17object3d, rpsingh2020instance}, where a stacked-hourglass network is used for keypoint predictions on RGB images cropped by an object bounding-box detector network (such as SSD \\cite{liu2016ssd}, YOLO \\cite{redmon2016you:yolo}). Predictions from the hourglass module along with the 3D model correspondences (from $\\mathbf{Q}$) are fed through a P\\textit{n}P module to obtain the final pose. An overview is depicted in Figure \\ref{figure:pose_estimation}.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\columnwidth]{images\/mask_labels.png}\n \\caption{\\textbf{Pixel-wise mask labels.} Examples of single and multi-object scenes from the experiments. The generated mask label is shown in the middle while the intersection-over-union computation is on the right.}\n \\label{fig:my_label}\n\\end{figure}\n\n\\begin{figure*}[t]\n \\includegraphics[width=\\textwidth]{images\/sparse_and_dense.png}\n \\caption{Generated dense models of the experiment objects on the left and the sparse representations overlayed on the CAD models on the right. The dense models are produced by \\textit{growing regions} on the scene meshes, seeded by the points of the sparse-representation.}\n \\label{figure:sparse_models}\n\\end{figure*}\n\n\\begin{table*}[t]\n\\caption{Accuracy analysis of generated sparse model and labels.}\n\\label{tab:table_model_errors}\n\\begin{center}\n\\begin{tabular}{c | c | c | c | c | c | c | c}\n\\hline\n& Object & \\# KPs & \\# scenes & Mean KP Error (3D) & Mean KP Error (2D) & Mean IoU & \\# labels\\\\\n& & $(N_{k})$ & $(N_{s}=$len$(\\mathcal{U}))$ & (in \\textit{mm}) & (in \\textit{pixels}) & & (sampled@3Hz)\\\\\n\\hline\n01 & \\textit{Dewalt Cut-out Tool} & 8 & 11 & 1.00 & 4.23 & \\textbf{81.09} & 3043\\\\\n\\hline\n02 & \\textit{Facom Electronic Wrench} & 6 & 10 & 2.69 & 6.66 & 71.54 & 2188\\\\\n\\hline\n03 & \\textit{Hitachi Screw Gun} & 9 & 9 & 5.29 & 6.08 & 75.73 & 2172\\\\\n\\hline\n04 & \\textit{Cup Noodle} & 10 & 6 & 0.5 & 3.97 & \\textbf{88.95} & 860\\\\\n\\hline\n05 & \\textit{Lipton} & 11 & 6 & 2.1 & 8.02 & \\textbf{83.19} & 1029\\\\\n\\hline\n06 & \\textit{Mt. Rainier Coffee} & 11 & 8 & 1.6 & 7.40 & \\textbf{82.49} & 1036\\\\\n\\hline\n07 & \\textit{Oolong Tea} & 11 & 5 & 1.1 & 3.65 & \\textbf{83.6 }& 850\\\\\n\\hline\n08 & \\textit{Mustard Bottle} & 7 & 5 & 1.17 & 7.15 & \\textbf{83.29} & 947\\\\\n\\hline\n09 & \\textit{Potted Meat Can} & 10 & 8 & 1.02 & 8.83 &\\textbf{ 83.49} & 1391\\\\\n\\hline\n10 & \\textit{Bleach Cleanser} & 8 & 9 & 3.37 & 9.73 & \\textbf{81.5} & 1525\\\\\n\\hline\n11 & \\textit{Power Drill} & 12 & 9 & 1.52 & 6.43 & 78.25 & 1346\\\\\n\\hline\n & \\textbf{Mean} & - & - & \\textbf{1.94} & \\textbf{6.56} & \\textbf{81.2} & -\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nWe trained the YOLO and the stacked-hourglass networks for each of the 3 objects from scratch on images sampled from the RGB-D videos of all scenes at 3Hz with a 90-10 split for training and testing respectively. The YOLO network was trained on the default hyperparameters while the stacked-hourglass network was trained on mostly the same hyperparameters as \\cite{pavlakos17object3d} (though we implemented the keypoint detector in PyTorch instead of Lua originally). After the keypoint detection stage, we computed the 6-DoF object pose using OpenCV's E-P\\textit{n}P method, using the generated sparse model $\\mathbf{Q}$ in the test pipeline and the hand-modelled 3D CAD for the benchmark pipeline. For benchmarking, we trained the networks again from scratch on the same set of raw images but using the ground-truth labels and compared the errors in estimated 6-DoF poses. Rotation error was computed using the following geodesic distance formula: \\par\n\n\\begin{equation}\n\\begin{gathered}\n\\Delta(R_1, R_2)\n= \\frac{{\\lVert log(R_1^T R_2) \\rVert}_{F}}{\\sqrt{2}}.\n\\end{gathered}\n\\end{equation}\n\nThe median errors in rotation and translation for the 3 objects for the pipeline, trained on the generated dataset (OURS) and ground-truth dataset (MANUAL), are consolidated in Table \\ref{tab:table_pose_errors}. The results indicate that our proposed approach, while reducing the human time required for the entire procedure by several orders of magnitude, can be used to generate labeled datasets for training pose estimation pipelines to remarkable accuracy (comparable to that trained on manually generated dataset).\n\n\n\\subsection{Application to Object Pixel-wise Segmentation}\nWe investigate the suitability of the generated mask labels for the training of existing object segmentation methods. We selected an open-source implementation of Mask R-CNN \\cite{matterport_maskrcnn_2017} -- a popular approach for pixel-wise segmentation of objects in RGB images.\n\nFrom the labeled dataset generated in our experiments, we selected a subset of images (sampled at 3Hz) and again created a 90\/10 train-test split. Although our tool is capable of generating labels for multi-object scenes, here we limit our analysis to single-object scenes for the sake of simplicity. We trained the model for 11 classes of objects (refer to Table \\ref{tab:table_model_errors} for names), with default initialization of the weights. The ResNet101 backbone was chosen for the architecture and training images were square cropped to $512\\times512$ size. The training was performed on ``head\" layers for the first 20 epochs followed by ``all\" layers for then next 20. Other hyperparameters were set to default.\n\n\\begin{figure}[t]\n \\includegraphics[width=\\columnwidth]{images\/chart.png}\n \\caption{\\textbf{Performance of Mask R-CNN trained on our dataset.} Mean IoU scores for each of the 11 objects in the dataset.}\n \\label{figure:segmentation}\n\\end{figure}\n\n\\addtolength{\\textheight}{-2cm} \n \n \n \n \n \n \nThe purpose here was to measure the performance of the segmentation network trained on the automatically generated dataset (OURS) and compare it to the segmentation network trained on the ground-truth dataset (MANUAL). In both cases, the trained network was evaluated against the test subset of the ground-truth dataset. The Intersection-over-Union metric was measured and reported in Figure \\ref{figure:segmentation}. As the object-wise IoU metric remains comparable in both cases, we argue that training on our dataset (which takes a lot less manual effort to generate) provides similar performance for real applications.\n\\begin{table}[t]\n\\caption{Errors (Median) in Object Pose Estimation}\n\\label{tab:table_pose_errors}\n\\begin{center}\n\\begin{tabular}{c | c | c | c | c}\n\\hline\n& \\multicolumn{2}{c|}{Position} & \\multicolumn{2}{c}{Orientation}\\\\\n& \\multicolumn{2}{c|}{(in \\textit{mm})} & \\multicolumn{2}{c}{(in \\textit{degrees})}\\\\\n\\hline\n& OURS & MANUAL & OURS & MANUAL\\\\\n\\hline\n\\textit{OBJ1} & 9.93 & \\textbf{9.23} & 0.78 & \\textbf{0.69}\\\\\n\\hline\n\\textit{OBJ2} & \\textbf{3.66} & 4.75 & 1.35 & \\textbf{1.23}\\\\\n\\hline\n\\textit{OBJ3} & \\textbf{6.77} & 7.52 & \\textbf{4.46} & 4.80\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{CONCLUSION \\& FUTURE WORK}\n\nThrough this paper, we have presented a technique for rapidly generating large datasets of labeled RGB images that can be used for training of deep CNNs for various applications. Our pipeline is highly automated --- we gather input from the human user for a few keypoints in only one RGB per scene. We do not require any complicated hardware setups like robot manipulators and turntables or sophisticated calibration procedures. In fact, the only hardware requirements are -- a calibrated RGB-D sensor and the object itself -- consequently, making it easier for the user to acquire the dataset in different environments.\n\nWe validated the effectiveness of the proposed method by using it to rapidly produce more than 150,000 labeled RGB images for 11 objects and subsequently, using the dataset to train a pose estimation pipeline and a segmentation network. We evaluated the accuracy of the trained networks to establish practicality and applicability of the labeled datasets as a solution to 6-DoF pose for real-world robotic grasping and manipulation tasks.\n\nFuture work in this direction would focus on making the sparse model of an object to be used as a canonical representation of the object's class to help class-specific pose estimation. Also, improving the quality of the segmented dense models by the means of a better algorithm would help improve the quality of labels.\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Motivation}\n\\label{sec: intro}\n\nConsider a unit --- for example, a human subject in a medical or a social science study, an experimental animal in a biological experiment, a machine in an engineering or reliability setting, or a company in an economic or business situation --- in a longitudinal study monitored over a period $[0,\\tau$], where $\\tau$ is possibly random. Associated with the unit is a covariate row vector $\\mathbf{X} = (X_1, X_2, \\ldots, X_p)$. Over time, the unit will experience occurrences of $Q$ competing types of recurrent events, its recurrent competing risks (RCR) component; transitions of a longitudinal marker (LM) process $W(t)$ over a discrete state space $\\mathfrak{W}$; and transitions of a `health' status (HS) process $V(t)$ over a discrete state space $\\mathfrak{V} = \\mathfrak{V}_0 \\bigcup \\mathfrak{V}_1$, with $\\mathfrak{V}_0 \\ne \\emptyset$ being absorbing states. If the health status process transitions into an absorbing state prior to $\\tau$, then monitoring of the unit ceases, so time-to-absorption serves as the lifetime of the unit. To demonstrate pictorially, the two panels in Figure \\ref{fig: observables} depict the time-evolution for two distinct units, where $Q = 3$, $\\mathfrak{W} = \\{w_1, w_2, w_3\\}$, $\\mathfrak{V} =\\{v_0, v_1, v_2\\}$ with $v_0$ an absorbing state. In panel 1, the unit did not transition to an absorbing state prior to reaching $\\tau$; whereas in panel 2, the unit reached an absorbing state prior to $\\tau$. Two major questions arise: (a) how do we specify a dynamic stochastic model that could be a generative model for such a data, and (b) how do we make statistical inferences for the model parameters and predictions of a unit's lifetime from a sample of such data?\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=2.5in]{fi\/censor} &\n\\includegraphics*[width=2.5in]{fi\/absorb}\n\\end{tabular}\n\\end{center}\n\\caption{Realized data observables from two distinct study units. The first plot panel is for a unit which was right-censored prior to reaching an absorbing state, while the second plot panel is for a unit which reached an absorbing state prior to getting right-censored.}\n\\label{fig: observables}\n\\end{figure}\n\nTo address these questions, the major goals of this paper are (i) to propose a joint stochastic model for the random observables consisting of the RCR, LM, and HS components for such units, and (ii) to develop appropriate statistical inference methods for the proposed joint model when a sample of units are observed. Achieving these two goals will enable statistical prediction of the (remaining) lifetime of a, possibly new, unit; allow for the examination of the synergistic association among the RCR, LM, and HS components; and provide a vehicle to compare different groups of units and\/or study the effects of concomitant variables or factors. More importantly, a joint stochastic model endowed with proper statistical inference methods could potentially enable unit-based interventions which are performed after a recurrent event occurrence or a transition in either the LM or HS processes. As such it could enhance the implementation of precision or personalized decision-making; for instance, precision medicine in a medical setting.\n\nA specific situation where such a data accrual occurs is in a medical study. For example, a subject may experience different types of recurring cancer, with the longitudinal marker being the prostate-specific antigen (PSA) level categorized into a finite ordinal set, while the health status is categorized into either a healthy, diseased, or dead state, with the last state absorbing. A variety of situations in a biomedical, engineering, public health, sociology, and economics settings, where such data structure arise, are further described in Section \\ref{sec: scenarios}. Several previous works dealing with modeling have either focused in the marginal modeling of each of the three data components, or in the joint modeling of two of the three data components. In this paper we tackle the problem of {\\em simultaneously} modeling all three data components: RCR, LM, and HS, in order to account for the associations among these components, which would not be possible using either the marginal modeling approach or the joint modeling of pairwise combinations of these three components. A joint full model could also subsume these previous marginal or joint models -- in fact, our proposed class of models subsumes as special cases models that have been considered in the literature. In contrast, only by imposing restrictive assumptions, such as the independence of the three model components, could one obtain a joint full model from marginal or pairwise joint models. As such, a joint full model will be less likely to be mis-specified, thereby reducing the potential biases that could accrue from mis-specified models among estimators of model parameters or when predicting residual lifetime.\n\nA joint modeling approach has been extensively employed in previous works. For instance, joint models for an LM process and a survival or time-to-event (TE) process have been proposed in \\cite{tsiatis1995modeling}, \\cite{wulfsohn1997joint}, \\cite{song2002semiparametric}, \\cite{henderson2000joint}, \\cite{tsiatis2004joint}, and \\cite{mclain2015}. Also, the joint modeling of an LM process and a recurrent event process has also been discussed in \\cite{han2007} and \\cite{efen2013}, while the joint modeling of a recurrent event and a lifetime has also been done such as in \\cite{liu2004}. An important and critical theoretical aspect that could not be ignored in these settings is that when an event occurrence is terminal (e.g., death) or when there is a finite monitoring period, informative censoring naturally occurs in the RCR, LM, or HS components, since when a terminal event or when the end of monitoring is reached, then the next recurrent event occurrence, the next LM transition, or the next HS transition will not be observed. \n\nAnother aspect that needs to be taken into account in a dynamic modeling approach is that of performed interventions, usually upon the occurrence of a recurrent event. For instance, in engineering or reliability systems, when a component in the system fails, this component will either be minimally or partially repaired, or altogether replaced with a new component (called a perfect repair); while, with human subjects, when a cancer relapses, a hospital infection transpires, a gout flare-up, or alcoholism recurs, some form of intervention will be performed or instituted. Such interventions will impact the time to the next occurrence of the event, hence it is critical that such intervention effects be reflected in the model; see, for instance, \\cite{gonzalez2005modelling} and \\cite{han2007}. In addition, models should take into consideration the impacts of the covariates and the effects of accumulating event occurrences on the unit. Models that take into account these considerations have been studied in \\cite{pena2006dynamic} and \\cite{pena2007semiparametric}. Appropriate statistical inference procedures for these dynamic models of recurrent events and competing risks have been developed in \\cite{pena2006dynamic} and \\cite{taylor2014nonparametric}. Extensions of these joint dynamic models for both RCR and TE can be found in \\cite{liu2015dynamic}. Some other recent works in joint modeling included the modeling of the three processes: LM, RCR (mostly, a single recurrent event), and TE simultaneously in \\cite{kim2012joint}, \\cite{cai2017joint}, \\cite{krol2017tutorial}, \\cite{mauguen2013dynamic} and \\cite{blanche2015quantifying}. The joint model that will be proposed in this paper will take into consideration these important aspects.\n\nWe now outline the remainder of this paper. Prior to describing formally the joint model in Section \\ref{sec: mathform}, we first present in Section \\ref{sec: scenarios} some concrete situations in science, medicine, engineering, social, and economic disciplines where the data accrual described above could arise and where the joint model will be relevant. Section \\ref{sec: mathform} formally describes the joint model using counting processes and continuous-time Markov chains (CTMCs), and provide interpretations of model parameters. In subsection \\ref{subsec: special case} we discuss in some detail a special case of this joint model obtained using independent Poisson processes and homogeneous CTMCs. Section \\ref{sec: estimation} deals with the estimation of the parameters. In subsection \\ref{subsec: estimation - parametric} we demonstrate the estimation of the model parameters in the afore-mentioned special case to pinpoint some intricacies of joint modeling and its inferential aspects. The general joint model contains nonparametric (infinite-dimensional) parameters, so in subsection \\ref{subsec: estimation - semiparametric} we will describe a semi-parametric estimation procedure for this general model. Section \\ref{sec: Properties} will present asymptotic properties of the estimators, though we will not present in this paper the rigorous proofs of these analytical results but defer them to a more theoretically-focused paper. Section \\ref{sec-Illustration} will then demonstrate the semi-parametric estimation approach through the use of a synthetic or simulated data using an {\\tt R} \\cite{RCitation} program we developed. In Section \\ref{sec: simulation}, we perform simulation studies to investigate the finite-sample properties of the estimators arising from semi-parametric estimation procedure and compare these finite-sample results to the theoretical asymptotic results. An illustration of the semi-parametric inference procedure using a real medical data set is presented in Section \\ref{sec: realdata}. Section \\ref{sec: conclusion} contains concluding remarks describing some open research problems. \n\n\\section{Concrete Situations of Relevance}\n\\label{sec: scenarios}\n \nTo demonstrate potential applicability of the proposed joint model, we describe in this section some concrete situations arising in biomedical, reliability, engineering, and socio-economic settings where the data accrual described in Section \\ref{sec: intro} could arise.\n\n\\begin{itemize}\n\\item {\\bf A Medical Example:} Gout is a form of arthritis characterized by sudden and severe attacks of pain, swelling, redness and tenderness in one of more joints in the toes, ankles, knees, elbows, wrists, and fingers (see, for instance, Mayo Clinic publications about gout). When a gout flare occurs, it renders the person incapacitated (personally attested by the senior author) and the debilitating condition may last for several days. Since the location of the gout flare could vary, we may consider gout as competing recurrent events --- competing with respect to the location of the flare, and recurrent since it could keep coming back. Gout occurs when urate crystals accumulate in the joints, which in turn is associated with high levels of uric acid in the blood. The level of uric acid is measured by the {Serum Urate Level (SUR)}, which can be categorized as {Hyperuricemia} (if SUR $>$ 7.2 mg\/dL for males; if SUR > 6.0 mg\/dL for females), or {Normal} (if 3.5 mg\/dL $\\le$ SUR $\\le$ 7.2 mg\/dL for males; if 2.6 mg\/dL $\\le$ SUR $\\le$ 6.0 mg\/dL for females). The SUR level could be considered a longitudinal marker. Kidneys are associated with the excretion of uric acid in the body. Thus, chronic kidney disease (CKD) impacts the level of uric acid in the body, hence the occurrence of gout. The ordinal stages of CKD, based on the value of the {Glomerular Filtration Rate (GFR)}, are as follows: {Stage 1 (Normal)} if GFR $\\ge$ 90 mL\/min; {Stage 2 (Mild CKD)} if 60 mL\/min $\\le$ GFR $\\le$ 89 mL\/min; {Stage 3A (Moderate CKD)} if 45 mL\/min $\\le$ GFR $\\le$ 59 mL\/min; {Stage 3B (Moderate CKD)} if 30 mL\/min $\\le$ GFR $\\le$ 44; {Stage 4 (Severe CKD)} if 15 mL\/min $\\le$ GFR $\\le$ 29 mL\/min; and {Stage 5 (End Stage CKD)} if GRF $\\le$ 14 mL\/min. The state of Stage 5 (End Stage CKD) can be viewed as an absorbing state. The CKD status could be viewed as the ``health status'' of the person. Other covariates, such as gender, blood pressure, weight, etc., could also impact the occurrence of gout flares, uric acid level, and CKD. When a gout flare occurs, lifestyle interventions could be performed such as (i) consuming skim milk powder enriched with the two dairy products glycomacropeptide (GMP) and G600 milk fat extract; or (ii) consuming standard milk or lactose powder. The purpose of such interventions is to lessen gout flare recurrences. Of major interest is to jointly model the competing gout recurrences, the categorized SUR process, and the CKD process. A study consisting of $n$ subjects could be performed with each subject monitored over some period, with the time of gout flare recurrences of each type, SUR levels, and CKD states recorded over time, aside from relevant covariates. Based on such a data, it will be of interest to estimate the model parameters and to develop a prediction procedure for time-to-absorption to End Stage CKD for a person with gout recurrences.\n\n\n\\item {\\bf A Reliability Example}: Observe $n$ independent cars over each of their own monitoring period until the car is declared inoperable or the monitoring period ends. Cars are complex systems in that they are constituted by different components, which could be subsystems or modules, configured according to some coherent structure function \\cite{barlow1975}. For each car, the states of $Q$ components (such as its engine subsystem; transmission subsystem; brake subsystem; electrical subsystem; etc.) are monitored. Furthermore, its covariates such as weight; initial mileage; current mileage; years of operation; and other characteristics (for example, climate in which car is mostly being driven) are observed. Also, its `health status', which is either functioning, functioning with some problems, or total inoperability (an absorbing state), is tracked over the monitoring period. Meanwhile, a longitudinal marker such as its oil quality indicator (which is either excellent; good; or poor) and the occurrences of failures of any of the $Q$ components are also recorded over the monitoring period. When a component failure occurs, a repair or replacement of the component is undertaken. Given the data for these $n$ cars, an important goal is to predict the time to inoperability of another car. Note that this type of application could occur in more complex systems such as space satellites, nuclear power plants, medical equipments, etc.\n\n\\item {\\bf A Social Science Example}: Observe $n$ independent newly-married \ncouples over a period of years (say, 20 years). Over this follow-up period, the marriage could end in separation or divorce, remain intact, or end due to the death of at least one of them. Each couple will have certain characteristics: their ages when they got married; working status of each; income level of each; education level of each; number of children (this is a time-dependent covariate); net worth of couple; etc. Possibly through a regularly administered questionnaire, \n the couple provides information from which their ``marriage status'' could be ascertained (either very satisfied; satisfied; poor; separated or divorced). Competing recurrent events for the couple could be changes in job status for either; addition in the family; educational changes of children; and major disagreements. A longitudinal marker could be the financial health status of the couple reflected by their categorized FICO scores. A goal is to infer about the parameters of the joint model based on the observations from these $n$ couples, and to predict if separation or divorce will occur for a married couple, who are not in the original sample, and if so, to obtain a prediction interval of the time of such an occurrence. \n\n\\item {\\bf A Financial Example}: Track $n$ independent publicly-listed\ncompanies over their monitoring periods. At the end of its monitoring period, a company could be bankrupt, still in business, or could have been acquired by another company. Each company has its own characteristics, such as total assets, number of employees, number of branches, etc. Note that these are all time-dependent characteristics. The ``health status'' of a company is rated according to four categories (A: Exceptional; B: Less than Exceptional; C: Distressed; D: Bankrupt). The bankrupt status is the absorbing state. The company's liability relative to its asset, categorized into High, Medium, Low, Non-Existent could be an important longitudinal marker. Recurrent competing risks will be the occurrence of an increase (decrease) of at least 5\\% during a trading day in its stock share price. Based on data from a sample of these companies, it could be of interest to predict the time to bankruptcy of another company that is not in the sample. \n\n\\item {\\bf COVID-19 Example}: Consider a vaccine trial where $n$ subjects are randomized into different vaccine groups, including a no vaccine group. Group membership could be coded using dummy covariates. Each subject is monitored over an observation period, until loss to follow-up, or until death. Aside from the group membership, other covariates (e.g., age or age-group, gender, BMI, race, pre-existing conditions such whether immuno-compromised or not, political affiliation, religious affiliation, etc.) for each subject are also observed. A longitudinal marker for each subject could be the viral load, categorized into none, low, medium, or high. See \\cite{VelavanEtAl2021} for other examples of longitudinal medical markers observed in Covid-19 studies. The health status for each subject could be classified into free of Covid-19, moderately infected, severely infected, or dead. Competing recurrent events could be the occurrence of abdominal problems, severe coughing, body temperature reaching 103 degrees Fahrenheit. Possible goals of the study are to compare the different vaccine groups in terms of preventing Covid-19 infection; with respect to the mean or median time to absorption; or with respect to mean or median time to transitioning out of infection state given Covid-19 infection.\n\\end{itemize}\n\n\n\\section{Joint Model of RCR, LM, and HS Processes}\n\\label{sec: mathform}\n\n\\subsection{Data Observables for One Unit}\n\nDenote by $(\\Omega, \\mathfrak{F}, \\Pr)$ the basic filtered probability space with filtration $\\mathcal{F} = \\{\\mathcal{F}_s: s \\ge 0\\}$ where all random entities under consideration are defined. We begin by describing the joint model for the data observable components for {\\bf one unit}. \n\nLet $\\tau$, the end of monitoring period, have a distribution function $G(\\cdot)$, which may be degenerate. The covariate vector will be ${X} = (X_1, \\ldots, X_p)$, assumed to be time-independent, though the extension to time-dependent covariates are possible with additional assumptions.\nFor the RCR component, let $N^R = \\{N_q^R(s) \\equiv (N^R(s;q), q \\in \\mathfrak{I}_Q):\\ s \\ge 0\\}$, with index set $\\mathfrak{I}_Q = \\{1,\\ldots,Q\\}$, be a $Q$-dimensional multivariate counting (column) vector process such that, for $q \\in \\mathfrak{I}_Q$, $N_q^R(s)$ is the number of {\\em observed} occurrences of the recurrent event of type $q$ over $[0,s]$, with $N_q^R(0) = 0$. Thus, the sample path $s \\mapsto N_q^R(s)$ takes values in $\\mathbb{Z}_{0,+} = \\{0,1,2,\\ldots\\}$, is a non-decreasing step-function, and is right-continuous with left-hand limits. We denote by $dN_q^R(s) = N_q^R(s) - N_q^R(s-)$, the jump at time $s$ of $N_q^R$. \n\nFor the LM process, let $W = \\{W(s): s \\geq 0\\}$, where $W(s)$ takes values in a finite state space $\\mathfrak{W}$ with cardinality $|\\mathfrak{W}|$. $W(s)$ represents the state of the longitudinal marker at time $s$. The sample path $s \\mapsto W(s)$ is a step-function which is right-continuous with left-hand limits. By introducing the index set $\\mathfrak{I}_{\\mathfrak{W}} = \\{(w_1,w_2): w_1, w_2 \\in \\mathfrak{W}, w_1 \\ne w_2\\}$, we can convert $W$ into a (column) $2{\\binom{|\\mathfrak{W}|} {2}}$-dimensional multivariate counting process $\\{N^W \\equiv (N^W(s;w_1,w_2), (w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}): s \\ge 0\\}$, where $N^W(s;w_1,w_2)$ is the number of observed transitions of $W$ from state $w_1$ into $w_2$ over the period $[0,s]$, that is, for $(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}$,\n$$N^W(s;w_1,w_2) = \\sum_{t \\le s} I\\{W(t-) = w_1,W(t) = w_2\\},$$\nwith $I(\\cdot)$ denoting indicator function.\nThus, for each $(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}$, the sample path $s \\mapsto N^W(s;w_1,w_2)$ takes values in $\\mathbb{Z}_{0,+}$, is a nondecreasing step-function, right-continuous with left-hand limits, and with $N^W(s;w_1,w_2) = 0$. Furthermore, $\\sum_{(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}} dN^W(s;w_1,w_2) \\in \\{0,1\\}$ for every $s \\ge 0$.\n\nFor the HS process, let $V = \\{V(s): s \\ge 0\\}$, where $V(s)$ takes values in the finite state space $\\mathfrak{V} = \\mathfrak{V}_0 \\bigcup \\mathfrak{V}_1$, where states in $\\mathfrak{V}_0$ are absorbing states, and with $|\\mathfrak{V}_0| > 0$. $V(s)$ is the state occupied by the HS process at time $s$, so that if $V(s) \\in \\mathfrak{V}_0$ then $V(s^\\prime) = V(s)$ for all $s^\\prime > s$. Similar to the LM process, let $\\mathfrak{I}_{\\mathfrak{V}} = \\{(v_1,v): v_1 \\in \\mathfrak{V}_1, v \\in \\mathfrak{V}; v_1 \\ne v\\}$, whose cardinality is $|\\mathfrak{I}_{\\mathfrak{V}}| = |\\mathfrak{V}_1| |\\mathfrak{V}| - |\\mathfrak{V}_1|$. We convert $V$ into a (column) $|\\mathfrak{I}_{\\mathfrak{V}}|$-dimensional multivariate counting process $\\{N^V \\equiv (N^V(s;v_1,v), (v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}): s \\ge 0\\}$, where $N^V(s;v_1,v)$ is the number of observed transitions of $V$ from state $v_1$ into state $v$ over the period $[0,s]$, that is, for $(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}$,\n$$N^V(s;v_1,v) =\\sum_{t \\le s} I\\{V(t-) = v_1, V(t) = v\\}.$$\nFor each $(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}$, the sample path $s \\mapsto N^V(s;v_1,v)$ takes values in $\\mathbb{Z}_{0,+}$, and is a nondecreasing step-function, right-continuous with left-hand limits, and with $N^W(0;v_1,v) = 0$. In addition, $\\sum_{(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}} dN^V(s;v_1,v) \\in \\{0, 1\\}$ for every $s \\ge 0$.\nNext, we combine the multivariate counting processes $N^R$, $N^W$, and $N^V$ into one (column) multivariate counting process $N = \\{N(s): s \\ge 0\\}$ of dimension $Q + |\\mathfrak{I}_{\\mathfrak{W}}| + |\\mathfrak{I}_{\\mathfrak{V}}|$, where, with $^{\\tt T}$ denoting vector\/matrix transpose,\n$$N(s) = \\left[(N^R(s))^{\\tt T}, (N^W(s))^{\\tt T}, (N^V(s))^{\\tt T}\\right]^{\\tt T}.$$\n\nAn important point needs to be stated regarding the observables in the study, which will have an impact in the interpretation of the parameters of the joint model. This pertains to the ``competing risks'' nature of all the possible events at each time point $s$. The possible $Q$ recurrent event types, as well as the potential transitions in the LM and HS processes, are all competing with each other. Thus, suppose that at time $s_0$, the event that occurred is a recurrent event of type $q_0$, that is, $dN^R(s_0;q_0) = 1$. This means that this event has occurred {\\em in the presence} of the potential recurrent events from the other $Q-1$ risks, and the potential transitions from either the LM and HS processes. This will entail the use of {\\em crude} hazards, instead of {\\em net} hazards, in the joint modeling, and this observation will play a critical role in the dynamic joint model since each of the competing event occurrences at a given time point $s$ from all the possible event sources (RCR, LM, and HS) will be affected by the history of all these processes just before time $s$. This is the aspect that exemplifies the synergistic association among the three components.\n\nAnother observable process for our joint model is a vector of effective (or virtual) age processes $\\mathcal{E} = \\{(\\mathcal{E}_1(s),\\ldots,\\mathcal{E}_Q(s)): s \\ge 0\\}$, whose components are $\\mathcal{F}$-predictable processes with sample paths that are non-negative, left-continuous, piecewise nondecreasing and differentiable. These effective age processes will carry the impact of interventions performed after each recurrent event occurrence or a transition in either the LM process or the HS process. For recent articles dealing with virtual ages, see the philosophically-oriented article \\cite{FinCha21} and the very recent review article \\cite{Beu2021}.\n\nFinally, we define the time-to-absorption of the unit to be\n$\\tau_A = \\inf\\{s \\ge 0: V(s) \\in \\mathfrak{V}_0\\}$\nwith the convention that $\\inf \\emptyset = \\infty$. Using this $\\tau_A$ and $\\tau$, we define the unit's at-risk process to be $Y = \\{Y(s): s \\ge 0\\}$, with\n$Y(s) = I\\{\\min(\\tau,\\tau_A) \\ge s\\}.$\nIn addition, we define LM-specific and HS-specific at-risk processes as follows: For $w \\in \\mathfrak{W}$, define $Y^W(s;w) = I\\{W(s-) = w\\}$; and, for $v \\in \\mathfrak{V}_1$, define $Y^V(s;v) = I\\{V(s-) = v\\}$.\nFor a unit, we could then concisely summarize the random observables in terms of stochastic processes as:\n\\begin{equation}\n\\label{data: one unit}\nD = (X,N,\\mathcal{E},Y,Y^W,Y^V) \\equiv \\{X,(N(s),\\mathcal{E}(s),Y(s),Y^W(s),Y^V(s): s \\ge 0)\\}.\n\\end{equation}\nNote that the processes are undefined for $s > \\min(\\tau_A,\\tau) \\equiv \\inf\\{s \\ge 0: Y(s+) = 0\\}$ since monitoring of the unit had by then ceased.\n\n\\subsection{Joint Model Specification for One Unit}\n\\label{subsec: joint model}\n\nThe joint model specification will be through the specification of the compensator process vector and the predictable quadratic variation (PQV) process matrix of the multivariate counting process $N$. The predictable process vector $A = \\{A(s): s \\ge 0\\}$ is of dimension $Q + |\\mathfrak{I}_{\\mathfrak{W}}| + |\\mathfrak{I}_{\\mathfrak{V}}|$ and is such that the vector process $M = N - A = \\{M(s) = N(s) - A(s): s \\ge 0\\}$ is a zero-mean square-integrable martingale process with PQV matrix process $\\langle M, M \\rangle$. The vectors $A$ and $M$ are actually partitioned into three vector components reflecting the RCR, LM, and HS components, according to\n\\begin{displaymath}\nA = \\left[ (A^R)^{\\tt T}, (A^W)^{\\tt T}, (A^V)^{\\tt T} \\right]^{\\tt T} \\quad \\mbox{and} \\quad\nM = \\left[(M^R)^{\\tt T}, (M^W)^{\\tt T}, (M^V)^{\\tt T}\\right],\n\\end{displaymath}\nwhere, with $q \\in \\mathfrak{I}_Q$, $(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}$, $(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}$, and $s \\ge 0$,\n\\begin{eqnarray*}\n& A^R = \\{(A^R(s;q))\\} \\quad \\mbox{and} \\quad M^R = \\{(M^R(s;q))\\}; & \\\\\n& A^W = \\{(A^W(s;(w_1,w_2))\\} \\quad \\mbox{and} \\quad M^W = \\{(M^W(s;(w_1,w_2))\\}; & \\\\\n& A^V = \\{(A^V(s;(v_1,v))\\} \\quad \\mbox{and} \\quad M^V = \\{(M^V(s;(v_1,v))\\}, &\n\\end{eqnarray*}\nwith $A^R$ and $M^R$ of dimensions $Q$; $A^W$ and $M^W$ of dimensions $|\\mathfrak{I}_{\\mathfrak{W}}|$; and $A^V$ and $M^V$ of dimensions $|\\mathfrak{I}_{\\mathfrak{V}}|$. The matrix $\\langle M, M \\rangle$ could then be partitioned similarly to reflect these block components.\n\nWe can now proceed with the specification of the compensator process vector and the PQV process matrix. \nFor conciseness, we introduce the generic mapping $\\iota$ defined as follows: For a set $A$ with $m$ elements, $A=\\{a_1,\\ldots,a_m\\}$, let $\\iota_{A}(a)=(I(a_2=a),\\ldots,I(a_m=a))$, a row vector of $m-1$ elements. Here $\\iota_A(a)$ is the indicator vector of $a$ excluding the first element of $A$, so that $\\iota_{A}(a_1)=(\\underline{0})$. Excluding $a_1$ in the $\\iota$ mapping is for purposes of model identifiability. One may think of the mapping $\\iota$ as a converter to dummy variables. We will also need the following quantities or functions.\n\\begin{itemize}\n \\item For each $q \\in \\mathfrak{I}_Q$ there is a baseline (crude) hazard rate function $\\lambda_{0q}(\\cdot)$ with associated baseline (crude) cumulative hazard function $\\Lambda_{0q}(\\cdot)$. We also denote by $\\bar{F}_{0q}(\\cdot) = \\prodi_{v=0}^{\\cdot} [1 - \\Lambda_{0q}(dv)]$ the associated baseline (crude) survivor function, where $\\prodi$ is the product-integral symbol.\n %\n \\item For each $q \\in \\mathfrak{I}_Q$ there is a mapping $\\rho_q(\\cdot;\\cdot): \\mathbb{Z}_{0,+}^Q \\times \\Re^{d_q} \\rightarrow \\Re_{0,+}$, where the $d_q$'s are known positive integers. There is an associated vector $\\alpha_q \\in \\Re^{d_q}$.\n %\n \\item There is a collection of non-negative real numbers $\\eta = \\{\\eta(w_1,w_2): (w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}\\}$, and we define for every $w_1 \\in \\mathfrak{W}$, $\\eta(w_1,w_1) = -\\sum_{w \\in \\mathfrak{W}; w \\ne w_1} \\eta(w_1,w).$\n %\n \\item There is a collection of non-negative real numbers $\\xi = \\{\\xi(v_1,v): (v_1, v) \\in \\mathfrak{I}_{\\mathfrak{V}}\\}$, and we define for every $v_1 \\in \\mathfrak{V}$, $\\xi(v_1,v_1) = -\\sum_{v \\in \\mathfrak{V}; v \\ne v_1} \\xi(v_1,v),$ and with $\\xi(v_1,v_2) = 0$ for every $v_1 \\in \\mathfrak{V}_0$ and $v_2 \\in \\mathfrak{V}$.\n %\n\\end{itemize}\nWe then define the observables and associated finite-dimensional parameters, respectively, for each of the three model components according to\n\\begin{eqnarray*}\n& B^R(s)=[X,\\iota_{\\mathfrak{V}}(V(s)),\\iota_{\\mathfrak{W}}(W(s))] \\quad \\mbox{and} \\quad \\theta^R=[(\\beta^R)^{\\tt T},(\\gamma^R)^{\\tt T},(\\kappa^R)^{\\tt T}]^{\\tt T}; & \\\\\n& B^W(s)=[X,\\iota_{\\mathfrak{V}}(V(s)),N^R(s)] \\quad \\mbox{and} \\quad \\theta^W=[(\\beta^W)^{\\tt T},(\\gamma^W)^{\\tt T},(\\nu^W)^{\\tt T}]^{\\tt T}; & \\\\\n& B^V(s)=[X,\\iota_{\\mathfrak{W}}(W(s)),N^R(s)] \\quad \\mbox{and} \\quad \\theta^V=[(\\beta^V)^{\\tt T},(\\kappa^V)^{\\tt T},(\\nu^V)^{\\tt T}]^{\\tt T}. &\n\\end{eqnarray*}\nIn addition to the $\\lambda_{0q}$'s, $\\alpha_q$'s, $\\eta$, $\\xi$, the $\\theta^R$, $\\theta^W$, and $\\theta^V$ will constitute all of the model parameters.\nFor the proposed model, the compensator process components are given by, for $q \\in \\mathfrak{I}_Q$, $(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}$, and $(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}$:\n\\begin{eqnarray*}\n& A^R(s;q) = \\int_0^s Y(t) \\lambda_{0q}[\\mathcal{E}_q(t)] \\rho_q[N^R(t-);\\alpha_q] \\exp\\{B^R(t-) \\theta^R\\} dt; & \\\\\n& A^W(s;w_1,w_2) = \\int_0^s Y(t) Y^W(t;w_1) \\eta(w_1,w_2) \\exp\\{B^W(t-) \\theta^W\\} dt; \\\\\n& A^V(s;v_1,v) = \\int_0^s Y(t) Y^V(t;v_1) \\xi(v_1,v) \\exp\\{B^V(t-) \\theta^V\\} dt. &\n\\end{eqnarray*}\nIn the left-hand side of the equations above, we have suppressed writing the dependence on the model parameters. With $\\mbox{Dg}(a)$ denoting the diagonal matrix formed from vector $a$, the PQV process is specified to be\n\\begin{displaymath}\n\\langle M, M \\rangle (s) = \\mbox{Dg}[(A^R(s))^{\\tt T}, (A^W(s))^{\\tt T}, (A^V(s))^{\\tt T}].\n\\end{displaymath}\nObserve the dynamic nature of this model in that an event occurrence at an infinitesimal time interval $[s,s+ds)$ depends on the history of the processes before time $s$. According to the theory of counting processes, we have the the following probabilistic interpretations (statements are almost surely):\n\\begin{eqnarray*}\n& E\\{dN^R(s;q)|\\mathcal{F}_{s-}\\} = dA^R(s;q); & \\\\\n& E\\{dN^W(s;w_1,w_2)|\\mathcal{F}_{s-}\\} = dA^W(s;w_1,w_2); & \\\\\n& E\\{dN^V(s;v_1,v)|\\mathcal{F}_{s-}\\} = dA^V(s;v_1,v), &\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n& Var\\{dN^R(s;q)|\\mathcal{F}_{s-}\\} = dA^R(s;q); & \\\\\n& Var\\{dN^W(s;w_1,w_2)|\\mathcal{F}_{s-}\\} = dA^W(s;w_1,w_2); & \\\\\n& Var\\{dN^V(s;v_1,v)|\\mathcal{F}_{s-}\\} = dA^V(s;v_1,v), &\n\\end{eqnarray*}\ntogether with the conditional covariance, given $\\mathcal{F}_{s-}$, between any pair of elements of $dN(s)$ being equal to zero, e.g., $Cov\\{dN^R(s),dN^W(s;w_1,w_2)|\\mathcal{F}_{s-}\\} = 0$. However, note that we are not assuming that the components of $N^R$, $N^W$, and $N^V$ are independent, nor even conditionally independent. A way to view this model is that, given $\\mathcal{F}_{s-}$, the history just before time $s$, on the infinitesimal interval $[s,s+ds)$, $dN(s) = (dN^R(s)^{\\tt T}, dN^W(s)^{\\tt T}, dN^V(s)^{\\tt T})^{\\tt T}$ has a multinomial distribution with parameters $1$ and $dA(s) = (dA^R(s)^{\\tt T},dA^W(s)^{\\tt T},dA^V(s)^{\\tt T})^{\\tt T}$. As such, for every $s \\ge 0$, the following constraint holds:\n\\begin{displaymath}\ndN_\\bullet(s) = dN_\\bullet^R(s) + dN_\\bullet^W(s) + dN_\\bullet^V(s) \\in \\{0,1\\},\n\\end{displaymath}\nwith the notation that a subscript of $\\bullet$ means the sum over all the appropriate index set, e.g., $dN_\\bullet^R(s) = \\sum_{q \\in \\mathfrak{I}_Q} dN^R(s;q)$ and $dA_\\bullet(s) = dA_\\bullet^R(s) + dA_\\bullet^W(s) + dA_\\bullet^V(s)$. The multinomial distribution above could actually be approximated by independent Bernoulli distributions. To see this, if we have real numbers $p_k, k=1,\\ldots,K,$ with $0 < p_k \\approx 0$ for each $k = 1, \\ldots, K,$ and with $\\sum_{k=1}^K p_k \\approx 0$, then we have the approximation\n\\begin{displaymath}\n\\left(1 - \\sum_{k=1}^K p_k\\right) \\approx \\prod_{k=1}^K (1 - p_k).\n\\end{displaymath}\nConsequently, in the equation below, the multinomial probability on the left-hand side is approximately the product of (independent) Bernoulli probabilities in the right-hand side.\n\\begin{eqnarray*}\n\\lefteqn{ \\left\\{\\prod_{q \\in \\mathfrak{I}_Q} [dA_q(s)]^{dN_q^R(s)}\\right\\} \\left\\{\\prod_{(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}} [dA^W(s;w_1,w_2)]^{dN^W(s;w_1,w_2)} \\right\\} \\times } \\\\ &&\n \\left\\{\\prod_{(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}} [dA^V(s;v_1,v)]^{dN^V(s;v_1,v )} \\right\\} \\left\\{[1 - dA_\\bullet(s)]^{1 - dN_\\bullet(s)}\\right\\} \\\\ & \\approx &\n \\left\\{\\prod_{q \\in \\mathfrak{I}_Q} [dA_q(s)]^{dN_q^R(s)} [1 - dA_q^R(s)]^{1- dN_q^R(s)}\\right\\} \\\\ && \\left\\{\\prod_{(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}} [dA^W(s;w_1,w_2)]^{dN^W(s;w_1,w_2)} [1 - dA^W(s;w_1,w_2)]^{1 - dN^W(s;w_1,w_2)}\\right\\} \\\\ && \\left\\{\\prod_{(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}} [dA^V(s;v_1,v)]^{dN^V(s;v_1,v )} [1 - dA^V(s;v_1,v)]^{1 - dN^V(s;v_1,v )}\\right\\}.\n\\end{eqnarray*}\nThis approximate equivalence informs the manner in which we generate data from the model later in the sections dealing with an illustrative data set (Section \\ref{sec-Illustration}) and the simulation studies (Section \\ref{sec: simulation}) where we used this product of Bernoulli approach.\n\nConsider a unit that is still at risk just before time $s$ whose LM process is at state $w_1$ and HS process at state $v_1 \\notin \\mathfrak{V}_0$. Two questions of interest are: \n\\begin{itemize}\n\\item[(a)] What is the distribution of the next event occurrence? \n\\item[(b)] Given in addition that an event occurred at time $s+t$, what are the conditional probabilities of each of the possible events? \n\\end{itemize}\nDenote by $T$ the time to the next event occurrence starting from time $s$. Then, \n\\begin{eqnarray*}\n\\Pr\\{T > t|\\mathcal{F}_{s-}\\} & = & \\prodi_{u=s}^{s+t} \\left[1 - \\left(\\sum_{q=1}^Q \\lambda_{0q}[\\mathcal{E}_{q}(u)] \\rho_{q}[N^R(u-);\\alpha_q] \\exp\\{B^R(u-) \\theta^R\\} - \\right.\\right. \\\\ && \\left.\\left. \\eta(w_1,w_1) \\exp\\{B^W(u-)\\theta^W\\} - \\xi(v_1,v_1) \\exp\\{B^V(u-)\\theta^V\\}\\right) du \\right] \\\\\n& = & \\exp\\left\\{-\\int_{s}^{s+t} \\left(\\sum_{q=1}^Q \\lambda_{0q}[\\mathcal{E}_{q}(u)] \\rho_{q}[N^R(u-);\\alpha_q] \\exp\\{B^R(u-) \\theta^R\\} - \\right.\\right. \\\\ && \\left.\\left. \\eta(w_1,w_1) \\exp\\{B^W(u-)\\theta^W\\} - \\xi(v_1,v_1) \\exp\\{B^V(u-)\\theta^V\\}\\right) du \\right\\} \\\\\n& = & \\exp\\left\\{-\\exp\\{B^R(s-) \\theta^R\\}\\sum_{q=1}^Q \\rho_{q}[N^R(s-);\\alpha_q] \\int_{s}^{s+t} \\lambda_{0q}[\\mathcal{E}_{q}(u)] du + \\right. \\\\\n&& \\left. \\eta(w_1,w_1) \\exp\\{B^W(s-)\\theta^W\\} t + \\xi(v_1,v_1) \\exp\\{B^V(s-)\\theta^V\\} t\\right\\},\n\\end{eqnarray*}\nwith the second equality obtained by invoking the product-integral identity \n$$\\prodi_{s \\in I} [1 - dA(s)]^{1-dN(s)} = \\exp\\left\\{-\\int_{s \\in I} dA(s)\\right\\}$$ \nand since no events in $[s,s+t)$ means $dN_\\bullet^R(u) + dN_\\bullet^W(u) + dN_\\bullet^V(u) = 0$ for $u \\in [s,s+t)$, and the last equality arising since, prior to the next event, there will be no changes in the values of $N^R$, $B^R$, $B^W$, and $B^V$ from their respective values just before time $s$. In particular, if the $\\lambda_{0q}$s are constants, corresponding to the hazard rates of an exponential distribution, then the distribution of the time to the next event occurrence is exponential. Given that the event occurred at time $s+t$, then the conditional probability that it was an RCR-type $q$ event is \n$e^{\\{B^R(s-) \\theta^R\\}} \\rho_{q}[N^R(s-);\\alpha_q] \\lambda_{0q}[\\mathcal{E}_{q}(s+t)]\/C(s,t),$\nwith\n\\begin{eqnarray*}\n\\lefteqn{C(s,t) = e^{\\{B^R(s-) \\theta^R \\}} \\sum_{q^\\prime = 1}^Q \\rho_{q^\\prime}[N^R(s-);\\alpha_{q^\\prime}] \\lambda_{0q^\\prime}[\\mathcal{E}_{q^\\prime}(s+t)] - } \\\\\n&& \\eta(w_1,w_1) e^{\\{B^W(s-)\\theta^W\\}} - \\xi(v_1,v_1) e^{\\{B^V(s-)\\theta^V\\}}\n\\end{eqnarray*}\nSimilarly, the conditional probability that it was a transition to state $w_2$ for the LM process is\n$\\eta(w_1,w_2) e^{\\{B^W(s-)\\theta^W\\}}\/C(s,t),$\nand the conditional probability that is was a transition to state $v$, possibly to an absorbing state, for the HS process is\n$\\xi(v_1,v) e^{\\{B^V(s-)\\theta^V\\}}\/C(s,t).$\n %\nThese probabilities demonstrate the competing risks nature of the different possible events. They also provide a computational approach to iteratively generate data from the joint model for use in simulation studies, with the basic idea being to first generate a time to any type of event, then to mark the type of event or update each of the counting processes by using the above conditional probabilities.\n\nDenoting by $\\Theta$ the set of all parameters of the model, the likelihood function arising from observing $D$, with $p_{(W,V)}(\\cdot,\\cdot)$ the initial joint probability mass function of $(W(0),V(0))$, is given by\n\\begin{eqnarray}\n\\label{lik for one unit} \\lefteqn{ \\mathcal{L}(\\Theta|D) = p_{(W,V)}(W(0),V(0)) \\times } \\\\ \n&& \\prodi_{s=0}^\\infty \\left\\{\n\\left[ \\prod_{q \\in \\mathfrak{I}_Q} [dA^R(s;q)]^{dN^R(s;q)} \\right] \\left[ \\prod_{(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}} [dA^W(s;w_1,w_2)]^{dN^W(s;w_1,w_2)} \\right] \\right. \\times \\nonumber \\\\ \n&& \\left. \\left[ \\prod_{(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}} [dA^V(s;v_1,v)]^{dN^V(s;v_1,v)} \\right] \\left[1 - dA_\\bullet(s)\\right]^{1-dN_\\bullet(s)} \n\\right\\} \\nonumber\n\\end{eqnarray}\nThe likelihood in (\\ref{lik for one unit}) could be rewritten in the form\n\\begin{eqnarray}\n\\label{lik: one unit alt} \\lefteqn{\\mathcal{L}(\\Theta|D) = p_{(W,V)}(W(0),V(0)) \\times} \\\\\n&& \\prodi_{s=0}^\\infty \\left\\{\n\\left[ \\prod_{q \\in \\mathfrak{I}_Q} [dA^R(s;q)]^{dN^R(s;q)} \\right] \\left[ \\prod_{(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}} [dA^W(s;w_1,w_2)]^{dN^W(s;w_1,w_2)} \\right] \\right. \\nonumber \\\\\n&& \\left. \\left[ \\prod_{(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}} [dA^V(s;v_1,v)]^{dN^V(s;v_1,v)} \\right] \n\\right\\} \\exp\\{-A_\\bullet(\\infty)\\}. \\nonumber \n\\end{eqnarray}\nLet us examine the meaning of $A_\\bullet(\\infty) \\equiv A_\\bullet^R(\\infty) + A_\\bullet^W(\\infty) + A_\\bullet^V(\\infty)$. Simplifying, we see that this equals\n$A_\\bullet(\\infty) = \\int_0^\\infty Y(s) T(s) ds = \\int_0^{\\tau\\wedge\\tau_A} T(s) ds$,\nwhere\n\\begin{eqnarray*}\nT(s) & = & \\sum_{q \\in \\mathfrak{I}_Q} \\lambda_{0q}[\\mathcal{E}_q(s)] \\rho_q[N^R(s-);\\alpha_q] \\exp\\{B^R(s-) \\theta^R\\} - \\\\\n&& \\sum_{w_1 \\in \\mathfrak{W}} Y^W(s;w_1) \\eta(w_1,w_1) \\exp\\{B^W(s-) \\theta^W\\} - \\\\\n&& \\sum_{v_1 \\in \\mathfrak{V}_1} Y^V(s;v_1) \\xi(v_1,v_1) \\exp\\{B^V(s-) \\theta^V\\}.\n\\end{eqnarray*}\nRecall that $\\eta(w_1,w_1)$ and $\\xi(v_1,v_1)$ are non-positive real numbers. Thus, $T(s) ds$ could be interpreted as the total risk of an event, either a recurrent event in the RCR component or a transition in the LM or HS components, occurring from all possible sources (RCR, LM, HS) that the unit is exposed to at the infinitesimal time interval $[s, s+ds)$, given the history $\\mathcal{F}_{s-}$ just before time $s$.\n\nWe provide further explanations of the elements of the joint model. First, there is a tacit assumption that no more than one event of any type could occur at any given time $s$. Second, the event rate at any time point $s$ for any type of event is in the presence of the other possible risk events. Thus, consider a specific $q_0 \\in \\mathfrak{I}_Q$ and assume that the unit is still at-risk at time $s_0$. Then,\n\\begin{eqnarray}\n\\label{RCR probability}\n\\Pr\\{dN_{q_0}^R(s_0) = 1|\\mathcal{F}_{s_0-}\\} & \\approx & \\lambda_{0q_0}[\\mathcal{E}_{q_0}(s_0)] \n\\rho_{q_0}[N^R(s_0-);\\alpha_{q_0}] \\exp\\{B^R(s_0-)\\theta^R\\} d{s_0}\n\\end{eqnarray}\nis the conditional probability, given $\\mathcal{F}_{s_0-}$, that an event occurred at $[s_0,s_0+ds_0)$ and is of RCR type $q_0$ {\\em and} all other event types did not occur, which is the essence of what is called a crude hazard rate, instead of a net hazard rate. Third, the effective (or virtual) age functions $\\mathcal{E}_{q}(\\cdot)$s, which are assumed to be dynamically determined and not dependent on any unknown parameters, encodes the impact of performed interventions that are performed after each event occurrence. Several possible choices of these functions are:\n\\begin{itemize}\n\\item $\\mathcal{E}_q(s) = s$ for all $s \\ge 0$ and $q \\in \\mathfrak{I}_Q$. This is usually referred to as a minimal repair intervention, corresponding to the situation where an intervention simply puts back the system at the age just before the event occurrence.\n\\item $\\mathcal{E}_q(s) = s - S_{N_\\bullet(s-)}$ where $0 = S_0 < S_1 < S_2 < \\ldots$ are the successive event times. This corresponds to a perfect intervention, which has the effect of re-setting the time to zero for each of the RCRs after each event occurrence. In a reliability setting, this means that all $Q$ components (unless having exponential lifetimes) are replaced by corresponding identical, but new, components at each event occurrence.\n\\item $\\mathcal{E}_q(s) = s - S_{N_\\bullet^R(s-)}^R$ where $0 = S_0^R < S_1^R < S_2^R < \\ldots$ are the successive event times of the occurrences of the RCR events.\n\\item $\\mathcal{E}_q(s) = s - S_{qN^R(s-;q)}^R$ where $0 = S_{q0}^R < S_{q1}^R < S_{q2}^R < \\ldots$ are the successive event times of the occurrences of RCR events of type $q$.\n\\item Other general forms are possible, such as those in \\cite{dorado1997} and \\cite{gonzalez2005modelling}, the latter employing ideas of Kijima. See also the discussion on the `reality' of virtual or effective ages in the paper by \\cite{FinCha21}, as well as the recent review paper by \\cite{Beu2021}.\n\\end{itemize}\nFourth, the impact of accumulating RCR event occurrences, which could be adverse, but could also be beneficial as in software engineering applications, is incorporated in the model through the $\\rho_q(\\cdot;\\cdot)$ functions. One possible choice is an exponential function, such as $\\rho_q(N^R(s-);\\alpha_q) = \\exp\\{N^R(s-)^{\\tt T}\\alpha_q\\}$, but other choices could be made as well. Finally, the modulating exponential link function in the model is for the impact of the covariates as well as the values of the RCR, LM, and HS processes just before the time of interest, with the vector of coefficients quantifying the effects of the covariates. The use of $N(s-)$ in the model could be viewed as using them as internal covariates, or that the dynamic model is a self-exciting model.\n\nSimilar interpretations hold for the parameters $\\{\\eta(w_1,w_2): (w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}\\}$ and $\\{\\xi(v_1,v): (v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}\\}$. Thus, if just before time $s_0$, $W(s_0-) = w_1$ and $V(s_0-) = v_1$, indicated by $\\mathcal{F}_{s_0-}(w_1,v_1)$, then\n\\begin{eqnarray}\n\\lefteqn{ \\Pr\\{W(s_0+ds_0) = w_2|\\mathcal{F}_{s_0}(w_1,v_1)\\} \\approx } \\label{LM probability} \\\\ \\eta(w_1,w_2)\n&& \\exp\\{X\\beta^W + \\gamma^W_{j(v_1)} + N^R(s_0-) \\nu^W\\} ds_0; \\nonumber \\\\\n\\lefteqn{ \\Pr\\{V(s_0+ds_0) = v|\\mathcal{F}_{s_0}(w_1,v_1)\\} \\approx } \\label{HS probability} \\\\ \\xi(v_1,v)\n&& \\exp\\{ X\\beta^V + \\kappa^V_{j(w_1)} + N^R(s_0-) \\nu^V\\} ds_0, \\nonumber\n\\end{eqnarray}\nwhere $j(v_1)$ is the index associated with $v_1$ in $\\mathfrak{V}_1$ and $j(w_1)$ is the index associated with $w_1$ in $\\mathfrak{W}$.\n\n\\subsection{Special Case: Independent Poisson Processes and CTMCs for One Unit}\n\\label{subsec: special case}\n\nThere is a special case arising from this general joint model obtained when we set $\\lambda_{0q}(s) = \\lambda_{0q}$, $q \\in \\mathfrak{I}_Q$; $\\rho_q = 1$; $\\theta^R = 0$; $\\theta^W = 0$; and $\\theta^V = 0$. In this situation, we have\n\\begin{eqnarray*}\ndA^R(s;q) & = & \\lambda_{0q} ds, q \\in \\mathfrak{I}_Q; \n\\\\ dA^W(s;w_1,w_2) & = & \\eta(w_1,w_2) Y(s) Y^W(s;w_1) ds, (w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}; \n\\\\ dA^V(s;v_1,v) & = & \\xi(v_1,v) Y(s) Y^W(s,v_1) ds, (v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}.\n\\end{eqnarray*}\nIt is easy to see that this particular model coincides with the model where we have the following situations:\n\\begin{itemize}\n\\item[(i)] $N^R(\\cdot;q), q \\in \\mathfrak{I}_Q,$ are independent homogeneous Poisson processes with respective rates $\\lambda_{0q}, q \\in \\mathfrak{I}_Q$;\n\\item[(ii)] $W(\\cdot)$ is a continuous-time Markov chain (CTMC) with infinitesimal generator matrix (IGM) consisting of $\\{\\eta(w_1,w_2)\\}$;\n\\item[(iii)] $V(\\cdot)$ is a CTMC with IGM consisting of $\\{\\xi(v_1,v_2)\\}$; \n\\item[(iv)] $N^R$, $W$, and $V$ are independent; and\n\\item[(v)] Processes are observed over $[0,\\min(\\tau,\\tau_A)]$, where $\\tau$ is the end of monitoring period, while $\\tau_A$ is the absorption time of $V$ into $\\mathfrak{V}_0$.\n\\end{itemize}\nIn this special situation, the $\\lambda_{0q}$'s are both crude and net hazard rates. Also, due to the memoryless property of the exponential distribution, interventions performed after each event occurrence will have no impact in the succeeding event occurrences. This specific joint model further allows us to provide an operational interpretation of the model parameters. Thus, suppose that at time $s$, the LM process is at state $w_1$ and the HS process is at state $v_1 \\notin \\mathfrak{V}_0$. Then, the distribution of the time to the next event occurrence of any type (the holding or sojourn time at the current state configuration) has an exponential distribution with parameter $C = \\lambda_{0\\bullet} - \\eta(w_1,w_1) - \\xi(v_1,v_1)$. When an event occurs, then the (conditional) probability that it is (i) an RCR event of type $q$ is $\\lambda_{0q}\/C$; (ii) a transition to LM state $w_2 \\ne w_1$ is $\\eta(w_1,w_2)\/C$; or (iii) a transition to an HS state $v \\ne v_1$ is $\\xi(v_1,v)\/C$. This is the essence of the competing risks nature of all the possible event types: an RCR event, an LM transition, and an HS transition.\nAs such, the more general model could be viewed as an extension of this basic model with independent Poisson processes for the RCR component and CTMCs for the LM and HS components. \nFor this special case, the likelihood function in (\\ref{lik: one unit alt}) simplifies to the expression\n\\begin{eqnarray}\n\\lefteqn{\\mathcal{L}(\\Theta|D) = p_{(W,V)}(W(0),V(0))\n\\left[\\prod_{q \\in \\mathfrak{I}_Q} \\lambda_{0q}^{N^R(\\tau\\wedge\\tau_A;q)}\\right] \\times \\label{lik: special case one unit} } \\\\\n&& \n\\left[\\prod_{(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}} \\eta(w_1,w_2)^{N^W(\\tau\\wedge\\tau_A;w_1,w_2)}\\right]\n\\left[\\prod_{(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{W}}} \\xi(v_1,v)^{N^V(\\tau\\wedge\\tau_A;v_1,v)}\\right] \\times \\nonumber \\\\\n&& \\exp\\left\\{-\\int_0^{\\tau\\wedge\\tau_A} T(s) ds\\right\\}. \\nonumber\n\\end{eqnarray}\nwhere $T(s) = \\lambda_{0\\bullet} - \\sum_{w_1 \\in \\mathfrak{W}} \\eta(w_1,w_1) Y^W(s;w_1) - \\sum_{v_1 \\in \\mathfrak{V}_1} \\xi(v_1,v_1) Y^V(s;v_1).$ Note that $T(s) = T(s;\\lambda_0,\\eta,\\xi)$, that is, it is a quantity instead of a statistic.\n\n\n\\section{Estimation of Model Parameters}\n\\label{sec: estimation}\n\n\\subsection{Parametric Estimation}\n\\label{subsec: estimation - parametric}\n\nHaving introduced the joint model, we now address in this section the problem of making inferences about the model parameters. We assume that we are able to observe $n$ independent units, with the $i$th unit having data $D_i = (X_i,N_i,\\mathcal{E}_i,Y_i,Y_i^W, Y_i^V)$ as in (\\ref{data: one unit}). The full sample data will then be represented by\n\\begin{equation}\n\\label{sample data}\n\\mathbf{D} = (D_1,D_2,\\ldots,D_n),\n\\end{equation}\nwhile the model parameters will be represented by, with the convention that $q \\in \\mathfrak{I}_Q$, $(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}$, and $(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}$,\n\\begin{eqnarray*}\n\\Theta & \\equiv & \\left[\\{\\lambda_{0q}(\\cdot), \\alpha_q\\}, \\{\\eta(w_1,w_2)\\},\n\\{\\xi(v_1,v)\\}, \\theta^R, \\theta^W, \\theta^V\\right].\n\\end{eqnarray*}\nThe $\\lambda_{0q}$s could be parametrically-specified, hence will have finite-dimensional parameters, so $\\Theta$ will also then be finite-dimensional. Except for the special case mentioned above, our main focus will be the case where the $\\lambda_{0q}$s are nonparametric. The distributions, $G_i$s, of the end of monitoring periods, $\\tau_i$s, also have model parameters, but they are not of main interest. To visualize the type of sample data set that accrues, Figure \\ref{sample data picture} provides a picture of a simulated sample data with $n = 50$ units.\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics*[width=2.75in]{fi\/covariate.png} & \\includegraphics*[width=2.5in]{fi\/plotallunitsRCR.png} \\\\\n\\includegraphics*[width=2.5in]{fi\/plotallunitsLM.png} & \\includegraphics*[width=2.5in]{fi\/plotallunitsHS.png} \n\\end{tabular}\n\\caption{Simulated sample data with $n = 50$ units. {\\em Panel 1:} Covariate values of the $n=50$ units. Values of $X_{2}$ are indicated by rectangles if $X_{1}=0$, while values of $X_{1}$ are indicated by circles if $X_{1}=1$. Here $X_{1} \\sim \\mbox{BER}(0.5)$, $X_{2} \\sim N(0,1)$. $X_{1}$ and $X_{2}$ are generated independently of each other. {\\em Panel 2:} Recurrent competing risks occurrences with three types of competing risks. Each unit is either censored (``+'') or reaches the absorbing status (``$\\times$''). {\\em Panel 3:} Marker processes. {\\em Panel 4:} Health status processes, with state ``1'' absorbing.}\n\\label{sample data picture}\n\\end{figure}\n\nThe full likelihood function, given $\\mathbf{D}$, is\n$\\mathcal{L}(\\Theta|\\mathbf{D}) = \\prod_{i=1}^n \\mathcal{L}(\\Theta|\\mathbf{D}_i)$,\nwhere the $\\mathcal{L}(\\Theta|D_i)$ is of the form in (\\ref{lik: one unit alt}). If the $\\lambda_{0q}(\\cdot)$s are parametrically-specified, then estimators of the finite-dimensional model parameters could be obtained as the maximizers of this full likelihood function, and their finite and asymptotic properties will follow from the general results for maximum likelihood estimators based on counting processes; see, for instance, \\cite{Bor84} and \\cite{ander1993}.\n\nWe illustrate this situation for the special case of the model given in subsection \\ref{subsec: special case}, so that the parameter is simply $\\Theta = [\\{\\lambda_{0q}\\},\\{\\eta(w_1,w_2)\\},\\{\\xi(v_1,v)\\}]$. In this situation, from (\\ref{lik: special case one unit}), the full likelihood reduces to, with $\\tau_i^* = \\tau_i \\wedge \\tau_{iA}$,\n\\begin{eqnarray*}\n\\lefteqn{\\mathcal{L}(\\Theta|\\mathbf{D}) = \\prod_{i=1}^n p_{(W,V)}(W_i(0),V_i(0)) \\times} \\\\\n&& \\left[\\prod_{q\\in\\mathfrak{I}_Q} \\lambda_{0q}^{\\sum_{i=1}^n N_i^R(\\tau_i^*;q)} \\right] \n\\left[\\prod_{(w_1,w_2)\\in\\mathfrak{I}_{\\mathfrak{W}}} \\eta(w_1,w_2)^{\\sum_{i=1}^n N_i^W(\\tau_i^*;w_1,w_2)}\\right] \\times \\\\\n&& \\left[\\prod_{(v_1,v)\\in\\mathfrak{I}_{\\mathfrak{V}}} \\xi(v_1,v)^{\\sum_{i=1}^n N_i^V(\\tau_i^*;v_1,v)}\\right]\n\\exp\\left\\{-\\sum_{i=1}^n \\int_0^{\\tau_i^*} T_i(s;\\lambda_0,\\eta,\\xi) ds\\right\\},\n\\end{eqnarray*}\nwhere\n\\begin{displaymath}\nT_i(s;\\lambda_0,\\eta,\\xi) = \\lambda_{0\\bullet} - \\sum_{w_1\\in\\mathfrak{W}} \\eta(w_1,w_1) Y_i^W(s;w_1) - \\sum_{v_1\\in\\mathfrak{V}} \\xi(v_1,v_1) Y_i^V(s;v_1).\n\\end{displaymath}\nThe score function $U(\\Theta|\\mathbf{D}) = \\nabla_\\Theta \\log \\mathcal{L}(\\Theta|\\mathbf{D})$ has elements\n\\begin{eqnarray*}\nU^R(\\Theta;q) & = & \\frac{\\sum_{i=1}^n N_i^R(\\tau_i^*;q)}{\\lambda_{0q}} - \\sum_{i=1}^n \\tau_i^*,\\quad q \\in \\mathfrak{I}_Q; \\\\\nU^W(\\Theta;w_1,w_2) & = & \\frac{\\sum_{i=1}^n N_i^W(\\tau_i^*;w_1,w_2)}{\\eta(w_1,w_2)} - \\sum_{i=1}^n \\int_0^{\\tau_i^*} Y_i^W(s;w_1) ds,\\quad (w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}; \\\\\nU^V(\\Theta;v_1,v) & = & \\frac{\\sum_{i=1}^n N_i^V(\\tau_i^*;v_1,v)}{\\xi(v_1,v)} - \\sum_{i=1}^n \\int_0^{\\tau_i^*} Y_i^V(s;v_1) ds,\\quad (v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}.\n\\end{eqnarray*}\nEquating these equations to zeros yield the ML estimators of the parameters, which are given below and possess the interpretation of being the observed ``occurrence-exposure\" rates.\n\\begin{eqnarray*}\n\\hat{\\lambda}_{0q} & = & \\frac{\\sum_{i=1}^n N_i^R(\\tau_i^*;q)}{\\sum_{i=1}^n \\tau_i^*} = \\frac{\\sum_{i=1}^n \\int_0^\\infty dN_i^R(s;q)}{\\sum_{i=1}^n \\int_0^\\infty Y_i(s) ds}, q \\in \\mathfrak{I}_Q; \\\\\n\\hat{\\eta}(w_1,w_2) & = & \\frac{\\sum_{i=1}^n N_i^W(\\tau_i^*;w_1,w_2)}{\\sum_{i=1}^n \\int_0^{\\tau_i^*} Y_i^W(s;w_1) ds} = \\frac{\\int_0^\\infty dN_i^W(s;w_1,w_2)}{\\sum_{i=1}^n \\int_0^\\infty Y_i(s) Y_i^W(s;w_1) ds}, (w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}; \\\\\n\\hat{\\xi}(v_1,v) & = & \\frac{\\sum_{i=1}^n N_i^V(\\tau_i^*;v_1,v)}{\\sum_{i=1}^n \\int_0^{\\tau_i^*} Y_i^V(s;v_1) ds} = \\frac{\\int_0^\\infty dN_i^V(s;v_1,v)}{\\sum_{i=1}^n \\int_0^\\infty Y_i(s) Y_i^V(s;v_1) ds}, (v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}.\n\\end{eqnarray*}\nIn these estimators, the numerators are total event counts, e.g., $\\sum_{i=1}^n N_i^R(\\tau_i^*;q)$ is the total number of observed RCR type $q$ events; $\\sum_{i=1}^n N_i^W(\\tau_i^*;w_1,w_2)$ is the total number of observed transitions in the LM process from state $w_1$ into $w_2$; and $\\sum_{i=1}^n N_i^V(\\tau_i^*;v_1,v)$ is the total number of observed transitions in the HS process from state $v_1$ into $v$. On the other hand, the denominators are total observed exposure times, e.g., $\\sum_{i=1}^n \\int_0^\\infty Y_i(s) ds$ is the total time at-risk for all the units; $\\sum_{i=1}^n \\int_0^\\infty Y_i(s)Y_i^W(s;w_1) dw$ is the total observed time of all units that they were at-risk for a transition in the LM process from state $w_1$; and $\\sum_{i=1}^n \\int_0^\\infty Y_i(s) Y_i^V(s;v_1) ds$ is the total observed time of all units that they were at-risk for a transition in the HS process from state $v_1$. An important and crucial point to emphasize here is that in these exposure times, they all take into account the time after the last observed events in each component process until the end of monitoring, whether it is a censoring (reaching $\\tau_i$) or an absorption (reaching $\\tau_{iA}$). If one ignores these right-censored times, then the estimators could be severely biased. This is a critical aspect we mentioned in the introductory section and re-iterate at this point that this should not be glossed over when dealing with recurrent event models.\n\nThe elements of the observed information matrix, $I(\\Theta;\\mathbf{D}) = - \\nabla_{\\Theta^{\\tt T}} U(\\Theta|\\mathbf{D})$, which is a diagonal matrix, have diagonal elements given by:\n\\begin{eqnarray*}\nI^R(\\Theta;q) & = & \\frac{\\sum_{i=1}^n N_i^R(\\tau_i^*;q)}{\\lambda_{0q}^2}, q \\in \\mathfrak{I}_Q; \\\\\nI^W(\\Theta;w_1,w_2) & = & \\frac{\\sum_{i=1}^n N_i^W(\\tau_i^*;w_1,w_2)}{\\eta(w_1,w_2)^2}, (w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}; \\\\\nI^V(\\Theta;v_1,v) & = & \\frac{\\sum_{i=1}^n N_i^V(\\tau_i^*;v_1,v)}{\\xi(v_1,v)^2}, (v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}.\n\\end{eqnarray*}\nAbbreviating the estimators into $\\hat{\\Theta} = (\\hat{\\lambda}_0, \\hat{\\eta}, \\hat{\\xi})$, we obtain the asymptotic result, that as $n \\rightarrow \\infty$,\n\\begin{equation}\n\\label{approx asymptotic special case}\n\\hat{\\Theta} \\sim \\mbox{AsyMVN}(\\Theta,I(\\hat{\\Theta};\\mathbf{D})^{-1}),\n\\end{equation}\nwith \\mbox{AsyMVN} meaning asymptotically multivariate normal. Thus, this result seems to indicate that the RCR, LM, and HS components or the estimators of their respective parameters do not have anything to do with each other, which appears intuitive since the RCR, LM, and HS processes were assumed to be independent processes to begin with. But, let us examine this issue further. The result in (\\ref{approx asymptotic special case}) is an approximation to the theoretical result that\n\\begin{equation}\n\\label{asymptotic special case}\n\\hat{\\Theta} \\sim \\mbox{AsyMVN}\\left(\\Theta,\\frac{1}{n}\\mathfrak{I}(\\Theta)^{-1}\\right),\n\\end{equation}\nwhere $\\frac{1}{n} I(\\hat{\\Theta};\\mathbf{D}) \\stackrel{pr}{\\rightarrow} \\mathfrak{I}(\\Theta)$. Evidently, $\\mathfrak{I}$ is a diagonal matrix, so let us examine its diagonal elements. Let $q \\in \\mathfrak{I}_Q$. Then we have, with `$\\mbox{pr-lim}$' denoting in-probability limit,\n\\begin{eqnarray*}\n\\lambda_{0q}^2 \\mathfrak{I}^R(\\Theta;q) & = &\\mbox{pr-lim}_{n\\rightarrow\\infty} \\frac{1}{n} \\sum_{i=1}^n \\int_0^\\infty dN_i^R(s;q) \\\\\n& = & \\mbox{pr-lim}_{n\\rightarrow\\infty} \\frac{1}{n} \\sum_{i=1}^n \\int_0^\\infty dM_i^R(s;q) + \\lambda_{0q} \\left[\\mbox{pr-lim}_{n\\rightarrow\\infty} \\frac{1}{n} \\sum_{i=1}^n \\int_0^\\infty Y_i(s) ds \\right].\n\\end{eqnarray*}\nThe first term on the right-hand side (RHS) converges in probability to zero by the weak law of large numbers and the zero-mean martingale property. The second term in the RHS converges in probability to its expectation, hence\n\\begin{displaymath}\n\\mathfrak{I}^R(\\Theta;q) = \\frac{1}{\\lambda_{0q}} \\left[ \\int_0^\\infty \\left\\{ \\lim_{n\\rightarrow\\infty} \\frac{1}{n} \\sum_{i}^n E[Y_i(s)]\\right\\} ds \\right].\n\\end{displaymath}\nBut, now,\n\\begin{eqnarray*}\n\\lim_{n\\rightarrow\\infty} \\frac{1}{n} \\sum_{i}^n E[Y_i(s)] & = & \\lim_{n\\rightarrow\\infty} \\frac{1}{n} \\sum_{i}^n \\Pr\\{\\tau_i \\ge s\\} \\Pr\\{\\tau_i^A \\ge s\\} \\\\\n& = & \\lim_{n\\rightarrow\\infty} \\frac{1}{n} \\sum_{i}^n \\bar{G}_i(s-) \\Pr\\{V_i(u) \\notin \\mathfrak{V}_0, u \\le s\\}.\n\\end{eqnarray*}\nwith $\\bar{G}_i = 1 - G_i$. The last probability term above will depend on the generators $\\{\\xi(v_1,v): (v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}\\}$ of the CTMC $\\{V(s): s \\ge 0\\}$, so that the theoretical Fisher information or the asymptotic variance associated with the estimator $\\hat{\\lambda}_{0q}$ depends after all on the HS process, as well as on the $G_i$s, contrary to the seemingly intuitive expectation that it should not depend on the LM and HS processes. This result is a subtle one which arise because of the structure of the observation processes. If $G_i = G, i=1,\\ldots,n$, then we have that\n\\begin{displaymath}\n\\mathfrak{I}^R(\\Theta;q) = \\frac{1}{\\lambda_{0q}} \\int_0^\\infty \\bar{G}(s-) \\Pr\\{V(u) \\notin \\mathfrak{V}_0, u \\le s\\} ds,\n\\end{displaymath}\nsince $\\Pr\\{V_i(u) \\notin \\mathfrak{V}_0, u \\le s\\} = \\Pr\\{V(u) \\notin \\mathfrak{V}_0, u \\le s\\}, i=1,\\ldots,n$. If we denote by $\\Gamma$ the generator matrix of $\\{V(s): s \\ge 0\\}$ and let $\\Gamma_1$ be the sub-matrix associated with the $\\mathfrak{V}_1$ states, then if $p_0^V \\equiv (p_V(v_1), v_1 \\in \\mathfrak{V}_1)^{\\tt T}$ is the initial probability mass function of $V(0)$, we have\n\\begin{displaymath}\n\\Pr\\{V(u) \\notin \\mathfrak{V}_0, u \\le s\\} = \\Pr\\{V(s) \\in \\mathfrak{V}_1\\} = (p_0^V)^{\\tt T} \\left[e^{s\\Gamma_{11}}\\right] 1_{|\\mathfrak{V}_1|}\n\\end{displaymath}\nwhere the matrix exponential is\n$e^{s\\Gamma_{11}} \\equiv \\sum_{k=0}^\\infty \\frac{s^k \\Gamma_{11}^k}{k!}$\nand $1_K$ is a column vector of $1$s of dimension $K$. Thus, we obtain\n\\begin{displaymath}\n\\mathfrak{I}^R(\\Theta;q) = \\frac{1}{\\lambda_{0q}} (p_0^V)^{\\tt T} \\sum_{k=0}^\\infty \\left[ \\int_0^\\infty \\bar{G}(s-) \\frac{s^k}{k!} ds \\right] \\Gamma_{11}^k 1_{|\\mathfrak{V}_1|}. \n\\end{displaymath}\nFor example, if $\\bar{G}(s) = \\exp(-\\nu s)$, that is, $\\tau_i$'s are exponentially-distributed with mean $1\/\\nu$, then the above expression simplifies to\n$$\\mathfrak{I}^R(\\Theta;q) = \\frac{1}{\\lambda_{0q}} \\frac{1}{\\nu} (p_0^V)^{\\tt T} \\left[ \\sum_{k=0}^\\infty \\left(\\frac{\\Gamma_{11}}{\\nu}\\right)^k\\right] 1_{|\\mathfrak{V}_1|}.$$\nFor computational purposes, one may use an eigenvalue decomposition of $\\Gamma_{11}$: $\\Gamma_{11} = U \\mbox{Dg}(d) U^{-1}$ where $d$ consists of the eigenvalues of $\\Gamma_{11}$ and $U$ is the matrix of eigenvectors associated with the eigenvalues $d$. The main point of this example though is the demonstration that estimators of the parameters associated with the RCR, LM, or HS process will depend on features of the other processes, {\\em even} when one starts with independent processes.\n\nWe remark that the estimators $\\hat{\\lambda}_{0q}$s, $\\hat{\\eta}(w_1,w_2)$s, and $\\hat{\\xi}(v_1,v)$s could also be derived as method-of-moments estimators using the martingale structure. The inverse of the observed Fisher information matrix coincides with an estimator using the optional variation (OV) matrix process, while the inverse of the Fisher information matrix coincides with the limit-in-probability of the predictable quadratic variation matrix. To demonstrate for $\\lambda_{0q}$, we have that\n$$\\left\\{\\sum_{i=1}^n M_i^R(s;q) = \\sum_{i=1}^n \\left[N_i^R(s;q) - \\int_0^s Y_i(t) \\lambda_{0q} dt\\right]: s \\ge 0\\right\\}$$\nis a zero-mean square-integrable martingale. Letting $s \\rightarrow \\infty$, setting $\\sum_{i=1}^n M_i^R(\\infty;q) = 0$, and solving for $\\lambda_{0q}$ yields $\\hat{\\lambda}_{0q}$. Next, we have\n\\begin{eqnarray*}\n\\hat{\\lambda}_{0q} & = & \\frac{\\sum_{i=1}^n \\int_0^\\infty dN_i^R(s;q)}{\\sum_{i=1}^n \\int_0^\\infty Y_i(s) ds} = \\lambda_{0q} + \\frac{\\sum_{i=1}^n \\int_0^\\infty dM_i^R(s;q)}{\\sum_{i=1}^n \\int_0^\\infty Y_i(s) ds}\n\\end{eqnarray*}\nso that\n\\begin{eqnarray*}\n\\sqrt{n}[\\hat{\\lambda}_{0q} - \\lambda_{0q}] = \\left( \\int_0^\\infty \\frac{1}{n} \\sum_{i=1}^n Y_i(s) ds \\right)^{-1} \\frac{1}{\\sqrt{n}} \\sum_{I=1}^n \\int_0^\\infty dM_i^R(s;q).\n\\end{eqnarray*}\nWe have already seen where $\\int_0^\\infty \\frac{1}{n} \\sum_{i=1}^n Y_i(s) ds$ converges in probability, whereas by the Martingale Central Limit Theorem, we have that\n\\begin{eqnarray*}\n\\frac{1}{\\sqrt{n}} \\sum_{i=1}^n \\int_0^\\infty dM_i^R(s;q) \\stackrel{d}{\\rightarrow} N(0,\\sigma^2_R(q))\n\\end{eqnarray*}\nwith \n\\begin{eqnarray*}\n\\sigma_R^2(q) & = & \\int_0^\\infty \\left\\{\\mbox{pr-lim}_{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{i=1}^n d \\langle M_i^R(\\cdot;q), M_i^R(\\cdot;q) \\rangle (s) \\right\\} \\\\\n & = & \\int_0^\\infty \\left\\{\\mbox{pr-lim}_{n\\rightarrow\\infty} \\frac{1}{n} \\sum_{i=1}^n Y_i(s) \\lambda_{0q} \\right\\} ds.\n\\end{eqnarray*}\nTherefore, we have\n\\begin{eqnarray*}\n\\lefteqn{ \\sqrt{n} [\\hat{\\lambda}_{0q} - \\lambda_{0q}] \\stackrel{d}{\\rightarrow} } \\\\\n&& N\\left(0,\\frac{\\lambda_{0q}}{\\int_0^\\infty \\left[ \\mbox{pr-lim}_{n\\rightarrow\\infty} \\frac{1}{n} \\sum_{i=1}^n Y_i(s) \\right] ds} = \\frac{\\lambda_{0q}}{\\int_0^\\infty \\left[ \\lim_{n\\rightarrow\\infty} \\frac{1}{n} \\sum_{i=1}^n E[Y_i(s)] \\right] ds} \\right), \n\\end{eqnarray*}\nwhich is the same result stated above using ML theory. Analogous asymptotic derivations can be done for $\\hat{\\eta}(w_1,w_2)$ and $\\hat{\\xi}(v_1,v)$, though the resulting limiting variances will involve expected occupation times for their respective states of the $W_i$-processes coming from the $Y_i^W(\\cdot;w_1)$ terms and the $V_i$-processes from the $Y_i^V(\\cdot;v_1)$ terms. Note that, asymptotically, these estimators are independent, {\\em but} their limiting variances depend on the parameters from the other processes.\n\n\\subsection{Semi-Parametric Estimation}\n\\label{subsec: estimation - semiparametric}\n\nIn this section we consider the estimation of model parameters when the hazard rate functions $\\lambda_{0q}(\\cdot)$s are specified nonparametrically. We shall denote by $\\Lambda_{0q}(t) = \\int_0^t \\lambda_{0q}(u) du, q \\in \\mathfrak{I}_Q$, the associated cumulative hazard functions. To simplify notation, we let\n\\begin{eqnarray*}\n& \\psi_R(s;\\theta^R) = \\exp\\{B^R(s) \\theta^R\\}; \n \\psi^W(s;\\theta^W) = \\exp\\{B^W(s) \\theta^W\\}; \n \\psi^V(s;\\theta^V) = \\exp\\{B^V(s) \\theta^V\\}. &\n\\end{eqnarray*}\nUsing these functions, for $q \\in \\mathfrak{I}_Q$, $(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}$, and $(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}$, we then have\n\\begin{eqnarray*}\ndA^R(s;q) & = & Y(s) \\lambda_{0q}[\\mathcal{E}_q(s)] \\rho_q(N^R(s-);\\alpha_q) \\psi^R(s-;\\theta^R) ds; \\\\\ndA^W(s;w_1,w_2) & = & Y(s) Y^W(s;w_1) \\eta(w_1,w_2) \\psi^W(s-;\\theta^W) ds; \\\\\ndA^V(s;v_1,v) & = & Y(s) Y^V(s;v_1) \\xi(v_1,v) \\psi^V(s-;\\theta^V) ds.\n\\end{eqnarray*}\nWe also abbreviate the vector of model parameters into $\\Theta \\equiv (\\Lambda_0,\\alpha,\\eta,\\xi,\\theta^R,\\theta^W,\\theta^V)$. Our goal is to obtain estimators for these parameters based on the sample data $\\mathbf{D} = (D_1, D_2,\\ldots,D_n)$. In a nutshell, the basic approach to obtaining our estimators is to first assume that $(\\alpha,\\theta^R,\\theta^W,\\theta^V)$ are known, then obtain `estimators' of $(\\Lambda_0,\\eta,\\xi)$. Having obtained these `estimators', in quotes since they are not yet estimators when $(\\alpha,\\theta^R,\\theta^W,\\theta^V)$ are unknown, we plug them into the likelihood function to obtain a profile likelihood function. From the resulting profile likelihood function, which depends on $(\\alpha,\\theta^R,\\theta^W,\\theta^V)$, we obtain its maximizers with respect to these finite-dimensional parameters to obtain their estimators. These estimators are then plugged into the `estimators' of $(\\Lambda_0,\\eta,\\xi)$ to obtain their estimators.\n\nThe full likelihood function based on the sample data $\\mathbf{D} = (D_1,\\ldots,D_n)$ could be written as a product of three ``major'' likelihood functions corresponding to the three model components:\n\\begin{eqnarray}\n\\label{full lik factor} \\lefteqn{\\mathcal{L}(\\Theta|\\mathbf{D}) = \n\\left[ \\prod_{q \\in \\mathfrak{I}_Q} \\mathcal{L}^R(\\Lambda_{0q},\\alpha,\\theta^R;q|\\mathbf{D}) \\right] \\times } \\\\\n&& \\left[ \\prod_{(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}} \\mathcal{L}^W(\\eta,\\theta^V;w_1,w_2|\\mathbf{D}) \\right] \\times \n\\left[ \\prod_{(v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}} \\mathcal{L}^V(\\xi,\\theta^W;v_1,v|\\mathbf{D}) \\right], \\nonumber\n\\end{eqnarray}\nwhere, suppressing writing of the parameters in the functions,\n\\begin{eqnarray*}\n\\mathcal{L}^R(\\Lambda_{0q},\\alpha,\\theta^R;q|\\mathbf{D}) & = & \\left\\{\\prod_{i=1}^n \\prodi_{s=0}^\\infty \\left[dA_i^R(s;q)\\right]^{dN_i^R(s;q)}\\right\\} \\times \\\\ && \\exp\\left\\{-\\sum_{i=1}^n \\int_0^\\infty dA_i^R(s;q)\\right\\}; \\\\\n\\mathcal{L}^W(\\eta,\\theta^V;w_1,w_2|\\mathbf{D}) & = & \\left\\{ \\prod_{i=1}^n \\prodi_{s=0}^\\infty \\left[dA_i^W(s;w_1,w_2)\\right]^{dN_i^W(s;w_1,w_2)}\\right\\} \\times \\\\ && \\exp\\left\\{-\\sum_{i=1}^n\\int_0^\\infty dA_i^W(s;w_1,w_2)\\right\\}; \\\\\n\\mathcal{L}^V(\\eta,\\theta^V;v_1,v|\\mathbf{D}) & = & \\left\\{ \\prod_{i=1}^n \\prodi_{s=0}^\\infty \\left[dA_i^V(s;v_1,v)\\right]^{dN_i^V(s;v_1,v)}\\right\\} \\times \\\\ && \\exp\\left\\{-\\sum_{i=1}^n\\int_0^\\infty dA_i^V(s;v_1,v)\\right\\}.\n\\end{eqnarray*}\nLet $0 = S_0 < S_1 < S_2 < \\ldots < S_K < S_{K+1} =\\infty$ be the ordered distinct times of \\underline{any} type of event occurrence for all the $n$ sample units. Also, let $0 = T_0 < T_1 < T_2 < \\ldots < T_L < T_{L+1} = \\infty$ be the ordered distinct values of the set $\\{\\mathcal{E}_{iq}(S_j): i = 1, \\ldots, n; q \\in \\mathfrak{I}_Q; j=0,1,\\ldots,S_K\\}$. Recall that $\\tau_i^* = \\tau_i \\wedge \\tau_{iA}$. Observe that both $\\{S_k: k=0,1,\\ldots,K,K+1\\}$ and $\\{T_l: l=0,1,\\ldots,L,L+1\\}$ partition $[0,\\infty)$. For each $i=1,\\ldots,n$, and $q \\in \\mathfrak{I}_Q$, $\\mathcal{E}_{iq}(\\cdot)$ is observed, hence defined, only on $[0,\\tau_i^*)$. However, for notational convenience, we define $\\mathcal{E}_{iq}(s) = 0$ for $s > \\tau_i^*$. In addition, on each non-empty interval $(S_{j-1} \\wedge \\tau_i^*,S_j \\wedge \\tau_i^*]$, $\\mathcal{E}_{iq}(\\cdot)$ has an inverse which will be denoted by $\\mathcal{E}_{iqj}^{-1}(\\cdot)$. Henceforth, for brevity of notation, we adopt the mathematically imprecise convention that $0\/0 = 0$.\n\n\\begin{proposition}\n\\label{prop-1}\nFor $q \\in \\mathfrak{I}_Q$, if $(\\alpha_q,\\theta^R)$ is known, then the nonparametric maximum likelihood estimator (NPMLE) of $\\Lambda_{0q}(\\cdot)$ is given by\n\\begin{eqnarray}\n\\label{cumulative hazard}\n\\hat{\\Lambda}_{0q}(t;\\alpha_q,\\theta^R) = \\sum_{l: T_l \\le t} \\left[\\frac{\\sum_{i=1}^n \\sum_{j=1}^K I\\{\\mathcal{E}_{iq}(S_j) = T_l\\} dN_i^R(S_j;q)}{S_q^{0R}(T_l|\\alpha_q,\\theta^R)}\\right] \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n \\label{sum of at-risk} S_q^{0R}(u|\\alpha_q,\\theta^R) & = & \\sum_{i=1}^n\\sum_{j=1}^K \\left\\{ \\left[\\frac{ \\rho_q[N_i^R(\\mathcal{E}_{iqj}^{-1}(u)-);\\alpha_q] \\psi_i^R(\\mathcal{E}_{iqj}^{-1}(u)-;\\theta^R)}{\\mathcal{E}_{iq}^\\prime[\\mathcal{E}_{iqj}^{-1}(u)]}\\right] \\times \\right. \\\\ && \\left. \nI\\{\\mathcal{E}_{iq}(S_{j-1} \\wedge \\tau_i^*) < u \\le \\mathcal{E}_{iq}(S_{j} \\wedge \\tau_i^*)\\} \\right\\}. \\nonumber\n\\end{eqnarray}\n\\end{proposition}\n\n\\begin{proof}\nThe likelihood $\\mathcal{L}_q^R(\\Lambda_{0q},\\alpha_q,\\theta^R|\\mathbf{D})$ could be written as follows:\n\\begin{eqnarray*}\n\\mathcal{L}_q^R & = & \\left[\\prod_{i=1}^n \\prod_{j=1}^{K} [Y_i(S_j) \\lambda_{0q}[\\mathcal{E}_{iq}(S_j) \\rho_q[N_i^R(S_j-);\\alpha_q] \\psi_i(S_j-;\\theta^R)]^{dN_i^R(S_j;q)}\\right] \\times \\\\ && \\exp\\left\\{-\\sum_{i=1}^n\\sum_{j=1}^{K+1} \\int_{S_{j-1}}^{S_j} Y_i(s) \\lambda_{0q}[\\mathcal{E}_{iq}(s)]\n\\rho_q[N_i^R(s-);\\alpha_q] \\psi_i(s-;\\theta^R) ds\\right\\}.\n\\end{eqnarray*}\nFocusing on the nonparametric parameter $\\Lambda_{0q}(\\cdot)$, the first term of $\\mathcal{L}_q^R$ could be written as\n\\begin{eqnarray*}\n\\prod_{i=1}^n \\prod_{j=1}^K [\\lambda_{0q}[\\mathcal{E}_{iq}(S_j)]^{dN_i^R(S_j;q)} & = &\n\\prod_{l=1}^L [\\lambda_{0q}(T_l)]^{dN_\\bullet^R(\\infty,T_l;q)} = \\prod_{l=1}^L [d\\Lambda_{0q}(T_l)]^{N_\\bullet^R(\\infty,T_l;q)}\n\\end{eqnarray*}\nwhere \n\\begin{displaymath}\ndN_\\bullet^R(\\infty,T_l;q) = \\sum_{i=1}^n \\sum_{j=1}^K I\\{\\mathcal{E}_{iq}(S_j) = T_l\\} dN_i^R(S_j;q).\n\\end{displaymath}\nThe exponent in the second term of $\\mathcal{L}_q^R$ could be written as follows, the second equality obtained after an obvious change-of-variable and using the definition of $S_q^{0R}(\\cdot;\\cdot,\\cdot)$ in the proposition:\n\\begin{eqnarray*}\n\\lefteqn{\\sum_{i=1}^n\\sum_{j=1}^{K+1} \\int_{S_{j-1}}^{S_j} Y_i(s) \\lambda_{0q}[\\mathcal{E}_{iq}(s)]\n\\rho_q[N_i^R(s-);\\alpha_q] \\psi_i(s-;\\theta^R) ds} \\\\\n& = & \\sum_{i=1}^n \\sum_{j=1}^{K+1} \\int_{S_{j-1} \\wedge \\tau_i^*}^{S_j \\wedge \\tau_i^*} \\lambda_{0q}[\\mathcal{E}_{iq}(s)]\n\\rho_q[N_i^R(s-);\\alpha_q] \\psi_i(s-;\\theta^R) ds \\\\\n& = & \\int_0^\\infty S_q^{0R}(u|\\alpha_q,\\theta^R) d\\Lambda_{0q}(u) = \\sum_{l=1}^{L+1} \\int_{T_{l-1}}^{T_l} S_q^{0R}(u;\\alpha_q,\\theta^R) d\\Lambda_{0q}(u) \\\\ \n& = & \\sum_{l=1}^L S_q^{0R}(T_l|\\alpha_q,\\theta^R) d\\Lambda_{0q}(T_l) + \\sum_{l=1}^{L+1} \\int_{u \\in (T_{l-1},T_{l})} S_q^{0R}(u|\\alpha_q,\\theta^R) d\\Lambda_{0q}(u)\n\\end{eqnarray*}\nTherefore, $\\mathcal{L}_q^R$, when viewed solely in terms of the parameter $\\Lambda_{0q}$ equals\n\\begin{eqnarray*}\n\\mathcal{L}_q^R & = & \\prod_{l=}^L [d\\Lambda_{0q}(T_l)]^{dN_\\bullet^R(\\infty,T_l;q)} \\times \\\\ && \\exp\\left\\{-\\left[\\sum_{l=1}^L S_q^{0R}(T_l|\\alpha_q,\\theta^R) d\\Lambda_{0q}(T_l) + \\sum_{l=1}^{L+1} \\int_{u \\in (T_{l-1},T_{l})} S_q^{0R}(u|\\alpha_q,\\theta^R) d\\Lambda_{0q}(u)\\right]\\right\\}\n\\end{eqnarray*}\nSince $\\sum_{l=1}^{L+1} \\int_{u \\in (T_{l-1},T_{l})} S_q^{0R}(u|\\alpha_q,\\theta^R) d\\Lambda_{0q}(u) \\ge 0$, then we obtain the upper bound for $\\mathcal{L}_q^R$ by setting this term to be equal to zero:\n\\begin{displaymath}\n\\mathcal{L}_q^R \\le \\left[\\prod_{l=1}^L [d\\Lambda_{0q}(T_l)]^{dN_\\bullet^R(\\infty,T_l;q)}\\right] \\exp\\left\\{-\\sum_{l=1}^L S_q^{0R}(T_l|\\alpha_q,\\theta^R) d\\Lambda_{0q}(T_l)\\right\\}.\n\\end{displaymath}\nThe upper bound is maximized by setting \n\\begin{displaymath}\nd\\hat{\\Lambda}_{0q}(T_l|\\alpha_q,\\theta^R) = \\frac{dN_\\bullet^R(\\infty,T_l;q)}{S_q^{0R}(T_l|\\alpha_q,\\theta^R)}, l=1,2,\\ldots,L.\n\\end{displaymath}\nFor $u \\in (T_{l-1},T_l)$, we then take $\\hat{\\Lambda}_{0q}(u|\\alpha_q,\\theta^R) = \\hat{\\Lambda}_{0q}(T_{l-1}|\\alpha_q,\\theta^R)$ which will satisfy the condition $\\int_{u \\in (T_{l-1},T_{l})} S_q^{0R}(u|\\alpha_q,\\theta^R) d\\hat{\\Lambda}_{0q}(u) = 0$ for all $l = 1,2, \\ldots, L+1$. Thus,\n\\begin{displaymath}\n\\hat{\\Lambda}_{0q}(t|\\alpha_q,\\theta^R) = \\sum_{l: T_l \\le t} d\\hat{\\Lambda}_{0q}(T_l;\\alpha_q,\\theta^R) = \\sum_{l: T_l \\le t} \\frac{dN_\\bullet^R(\\infty,T_l;q)}{S_q^{0R}(T_l|\\alpha_q,\\theta^R)},\n\\end{displaymath}\nwhich is a step-function with jumps only on $T_l$s with $dN_\\bullet^R(\\infty,T_l;q) > 0$, maximizes $\\Lambda_{0q}(\\cdot) \\mapsto \\mathcal{L}_q^R(\\Lambda_{0q}(\\cdot)|\\alpha_q,\\theta^R|\\mathbf{D})$ for given $(\\alpha_q,\\theta^R)$, completing the proof of the proposition.\n\\end{proof}\n\nA more elegant representation of the Aalen-Breslow-Nelson type estimator $\\hat{\\Lambda}_{0q}(\\cdot;\\alpha_q,\\theta^R)$ in Proposition \\ref{prop-1}, which shows that the estimator is also moment-based, aside from being useful in obtaining finite and asymptotic properties, is through the use doubly-indexed processes as in \\cite{pena2007semiparametric} for a setting with only one recurrent event type and without LM and HS processes. Define the doubly-indexed processes $\\{(N_i^R(s,t;q),A_i^R(s,t; q|\\Lambda_{0q},\\alpha_1,\\theta^R): (s,t) \\in \\Re_+^2\\}$ where\n\\begin{eqnarray*}\n& N_i^R(s,t;q) = \\int_0^s I\\{\\mathcal{E}_{iq}(v) \\le t\\} dN_i^R(v;q); & \\\\\n& A_i^R(s,t;q|\\Lambda_{0q},\\alpha_q,\\theta^R) = \\int_0^s I\\{\\mathcal{E}_{iq}(v) \\le t\\} dA_i^R(v;q|\\Lambda_{0q},\\alpha_q,\\theta^R). &\n\\end{eqnarray*}\nAlso, let \n\\begin{eqnarray*}\nN_\\bullet^R(s,t;q) & = & \\sum_{i=1}^n N_i^R(s,t;q)\\ \\mbox{and}\\ A_\\bullet^R(s,t;q|\\Lambda_{0q},\\alpha_q,\\theta^R) = \\sum_{i=1}^n A_i^R(s,t;q|\\Lambda_{0q},\\alpha_q,\\theta^R).\n\\end{eqnarray*}\nThen, for fixed $t$, $\\{M_\\bullet^R(s,t;q|\\Lambda_{0q},\\alpha_q,\\theta^R) = N_\\bullet^R(s,t;q) - A_\\bullet^R(s,t;q|\\Lambda_{0q},\\alpha_q,\\theta^R): s \\ge 0\\}$ is a zero-mean square-integrable martingale. Thus, $E[N_\\bullet^R(s,t;q)] = E[A_\\bullet^R(s,t;q|\\Lambda_{0q},\\alpha_q,\\theta^R)]$.\n\n\\begin{proposition}\n\\label{prop-2}\nFor $q \\in \\mathfrak{I}_Q$, $A_\\bullet^R(s,t;q|\\Lambda_{0q},\\alpha_q,\\theta^R) = \\int_0^t S_q^{0R}(s,u|\\alpha_q,\\theta^R) d\\Lambda_{0q}(u)$, where\n\\begin{eqnarray*}\nS_q^{0R}(s,u|\\alpha_q,\\theta^R) & = & \\sum_{i=1}^n\\sum_{j=1}^K \\left\\{ \\left[\\frac{ \\rho_q[N_i^R(\\mathcal{E}_{iqj}^{-1}(u)-);\\alpha_q] \\psi_i^R(\\mathcal{E}_{iqj}^{-1}(u)-;\\theta^R)}{\\mathcal{E}_{iq}^\\prime[\\mathcal{E}_{iqj}^{-1}(u)]}\\right] \\times \\right. \\\\ && \\left. \nI\\{\\mathcal{E}_{iq}(S_{j-1} \\wedge \\tau_i^* \\wedge s) < u \\le \\mathcal{E}_{iq}(S_{j} \\wedge \\tau_i^* \\wedge s)\\} \\right\\};\n\\end{eqnarray*}\nand $dN_\\bullet^R(s,T_l;q) = \\sum_{i=1}^n \\sum_{j=1}^K I\\{S_j \\le s; \\mathcal{E}_{iq}(S_j) = T_l\\} dN_i^R(S_j;q)$.\n\\end{proposition}\n\n\\begin{proof}\nSimilar to steps in the proof of Proposition \\ref{prop-1}.\n\\end{proof}\n\nBy first assuming that $(\\alpha_q,\\theta^R)$ is known, then from the identities in Proposition \\ref{prop-2}, a method-of-moments type estimator of $\\Lambda_{0q}(t)$ is given by\n\\begin{equation}\n\\label{NAE-type 1}\n\\hat{\\Lambda}_{0q}(s,t|\\alpha_q,\\theta^R) = \\sum_{l : T_l \\le t} \\frac{dN_\\bullet^R(s,T_l;q)}{S_q^{0R}(s,T_l|\\alpha_q,\\theta^R)} =\\int_0^t \\frac{N_\\bullet^R(s,du;q)}{S_q^{0R}(s,u|\\alpha_q,\\theta^R)}.\n\\end{equation}\nWhen $s \\rightarrow \\infty$, this $\\hat{\\Lambda}_{0q}(s,t|\\alpha_q,\\theta^R)$ converges to the estimator $\\hat{\\Lambda}_{0q}(t|\\alpha_q,\\theta^R)$ in Proposition \\ref{prop-1}.\nNext, we obtain estimators of the $\\eta(w_1,w_2)$s and $\\xi(v_1,v)$s, again by first assuming first that $\\theta^W$ and $\\theta^V$ are known.\n\n\\begin{proposition}\n\\label{prop-est of nu and xi}\nIf $(\\theta^W,\\theta^V)$ are known, the ML estimators of the $\\eta(w_1,w_2)$s and $\\xi(v_1,v)$s are the ``occurrence-exposure'' rates\n\\begin{eqnarray}\n\\hat{\\eta}(w_1,w_2|\\theta^W) & = & %\n\\frac{\\sum_{i=1}^n\\sum_{j=1}^K dN_i^W(S_j;w_1,w_2)}{S^{0W}(w_1;\\theta^W)}, \\forall (w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}; \\label{eta estimator} \\\\\n\\hat{\\xi}(v_1,v|\\theta^V) & = & \n\\frac{\\sum_{i=1}^n\\sum_{j=1}^K dN_i^V(S_j;v_1,v)}{S^{0V}(v_1;\\theta^V)}, \\forall (v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}}, \\label{xi estimator}\n\\end{eqnarray}\nwhere\n$$S^{0W}(w_1;\\theta^W) = \\int_0^\\infty \\sum_{i=1}^n Y_i(s) Y_i^W(s;w_1) \\psi_i^W(s-;\\theta^W) ds;$$\n$$S^{0V}(v_1;\\theta^V) = \\int_0^\\infty \\sum_{i=1}^n Y_i(s) Y_i^V(s;v_1) \\psi_i^V(s-;\\theta^V) ds.$$\n\\end{proposition}\n\n\\begin{proof}\nFollows immediately by maximizing the likelihood functions $\\mathcal{L}^W$ and $\\mathcal{L}^V$ with respect to the $\\eta(w_1,w_2)$s and $\\xi(v_1,v)$s, respectively.\n\\end{proof}\n\nWe can now form the profile likelihoods for the parameters $(\\{\\alpha_q, q \\in \\mathfrak{I}_Q\\}, \\theta^R, \\theta^W, \\theta^V)$. These are the likelihoods that are obtained after plugging-in the `estimators' $\\hat{\\Lambda}_{0q}(\\cdot;\\alpha_q,\\theta^R)$s, $\\hat{\\eta}(w_1,w_2)$s, and $\\hat{\\xi}(v_1,v)$s in the full likelihoods. The resulting profile likelihoods are reminiscent of the partial likelihood function in Cox's proportional hazards model \\cite{cox1972,AndGil1982}. \n\n\\begin{proposition}\n\\label{profile liks}\nThe three profile likelihood functions $\\mathcal{L}_{pl}^R$, $\\mathcal{L}_{pl}^W$ and $\\mathcal{L}_{pl}^V$ are given by\n\\begin{eqnarray*}\n\\lefteqn{\\mathcal{L}^R_{pl}(\\alpha_q,q\\in\\mathfrak{I}_Q,\\theta^R|\\mathbf{D}) = } \\\\ && \\prod_{q\\in\\mathfrak{I}_Q} \\prod_{i=1}^n \\prod_{j=1}^K \\prod_{l=1}^L\n\\left[\\frac{\\rho_q[N_i^R(S_j-);\\alpha_q] \\psi_i^R(S_j-;\\theta^R)}{S_q^{0R}(T_l|\\alpha_q,\\theta^R)}\\right]^{I\\{\\mathcal{E}_{iq}(S_j)=T_l\\} dN_i^R(S_j;q)};\n\\end{eqnarray*}\n\\begin{displaymath}\n\\mathcal{L}_{pl}^W(\\theta^W|\\mathbf{D}) = \\prod_{w_1 \\in \\mathfrak{W}} \\prod_{i=1}^n \\prod_{j=1}^K \\left[\\frac{\\psi_i^W(S_j-;\\theta^W)}{S^{0W}(w_1;\\theta^W)}\\right]^{dN_i^W(S_j;w_1,\\bullet)};\n\\end{displaymath}\n\\begin{displaymath}\n\\mathcal{L}_{pl}^V(\\theta^V|\\mathbf{D}) = \\prod_{v_1 \\in \\mathfrak{V}_1} \\prod_{i=1}^n \\prod_{j=1}^K \\left[\\frac{\\psi_i^V(S_j-;\\theta^V)}{S^{0V}(v_1;\\theta^V)}\\right]^{dN_i^V(S_j;v_1,\\bullet)}.\n\\end{displaymath}\nwith $dN_i^W(S_j;w_1,\\bullet) = \\sum_{w_2 \\in \\mathfrak{W};\\ w_2 \\ne w_1} dN_i^W(S_j;w_1,w_2)$, the number of transitions from state $w_1$ at time $S_j$ for unit $i$, and $dN_i^V(S_j;v_1,\\bullet) = \\sum_{v \\in \\mathfrak{V};\\ v \\ne v_1} dN_i^V(S_j;v_1,v)$, the number of transitions from state $v_1$ at time $S_j$ for unit $i$.\n\\end{proposition}\n\n\\begin{proof}\nThese follow immediately by plugging-in the `estimators' into the three main likelihoods in (\\ref{full lik factor}) and then simplifying.\n\\end{proof}\n\nFrom these three profile likelihoods, we could obtain estimators of the parameters $\\alpha_q$s, $\\theta^R$, $\\theta^W$, and $\\theta^V$ as follows:\n\\begin{eqnarray*}\n& (\\hat{\\alpha}_q, q\\in\\mathfrak{I}_Q,\\hat{\\theta}^R) = {\\arg\\max}_{(\\alpha_q, \\theta^R)} \\mathcal{L}^R_{pl}(\\alpha_q,q\\in\\mathfrak{I}_Q,\\theta^R|\\mathbf{D}); & \\\\\n& \\hat{\\theta}^W = \\arg\\max_{\\theta^W} \\mathcal{L}_{pl}^W(\\theta^W|\\mathbf{D}) \\ \\mbox{and} \\\n\\hat{\\theta}^V = \\arg\\max_{\\theta^V} \\mathcal{L}_{pl}^V(\\theta^V|\\mathbf{D}). &\n\\end{eqnarray*}\nEquivalently, these estimators are maximizers of the logarithm of the profile likelihoods. These log-profile likelihoods are more conveniently expressed in terms of stochastic integrals as follows:\n\\begin{eqnarray*}\nl_{pl}^R & = & \\sum_{q\\in\\mathfrak{I}_Q}\\sum_{i=1}^n \\int_0^\\infty\\int_0^\\infty \\left[\\log\\rho_q[N_i^R(s-);\\alpha_q] + \\log\\psi_i^R(s-;\\theta^R) - \\right. \\\\ && \\left. \\log S_q^{0R}(t;\\alpha_q,\\theta^R)\\right] N_i^R(ds,dt;q); \\\\\nl_{pl}^W & = & \\sum_{w_1 \\in \\mathfrak{W}} \\sum_{i=1}^n \\int_0^\\infty \\left[\\log\\psi_i^W(s-;\\theta^W) - \\log S^{0W}(s;w_1|\\theta^W)\\right] N_i^W(ds;w_1,\\bullet); \\\\\nl_{pl}^V & = & \\sum_{v_1 \\in \\mathfrak{V}_1} \\sum_{i=1}^n \\int_0^\\infty \\left[\\log\\psi_i^V(s-;\\theta^V) - \\log S^{0V}(s;v_1|\\theta^V)\\right] N_i^V(ds;v_1,\\bullet).\n\\end{eqnarray*}\nAssociated with each of these log-profile likelihood functions are their profile score vector (the gradient or vector of partial derivatives) and profile observed information matrix (negative of the matrix of second partial derivatives): $U^R(\\alpha,\\theta^R)$; $U^W(\\theta^W)$ and $U^V(\\theta^V)$ for the score vectors, and $I^R(\\alpha,\\theta^R)$, $I^W(\\theta^W)$ and $I^V(\\theta^V)$ for the observed information matrices. The estimators could then be obtained as the solutions of the set of equations\n\\begin{displaymath}\nU_{pl}^R(\\hat{\\alpha},\\hat{\\theta}^R) = 0; U_{pl}^W(\\hat{\\theta}^W) = 0; U_{pl}^V(\\hat{\\theta}^V) = 0.\n\\end{displaymath}\nA possible computational approach to obtaining the estimates is via the Newton-Raphson iteration procedure:\n\\begin{eqnarray*}\n& (\\hat{\\alpha},\\hat{\\theta}^R) \\leftarrow (\\hat{\\alpha},\\hat{\\theta}^R) + I^R(\\hat{\\alpha},\\hat{\\theta}^R)^{-1} U^R(\\hat{\\alpha},\\hat{\\theta}^R); & \\\\\n& \\hat{\\theta}^W \\leftarrow \\hat{\\theta}^W + I^W(\\hat{\\theta}^W)^{-1} U^W(\\hat{\\theta}^W); \\\n\\hat{\\theta}^V \\leftarrow \\hat{\\theta}^V + I^V(\\hat{\\theta}^V)^{-1} U^V(\\hat{\\theta}^V). &\n\\end{eqnarray*}\nHaving obtained the estimates of $\\alpha$, $\\theta^W$ and $\\theta^V$, we plug them into $\\hat{\\Lambda}_{0q}(t|\\alpha_q,\\theta^R)$s, $\\eta(w_1,w_2|\\theta^W)$s and $\\xi(v_1,v|\\theta^V)$s to obtain the estimators $\\hat{\\Lambda}_{0q}(t)$s, $\\hat{\\eta}(w_1,w_2)$s and $\\hat{\\xi}(v_1,v)$s:\n\\begin{equation}\n\\label{final estimators}\n\\hat{\\Lambda}_{0q}(t) = \\hat{\\Lambda}_{0q}(t|\\hat{\\alpha}_q,\\hat{\\theta}^R);\\quad \\hat{\\eta}(w_1,w_2) = \\hat{\\eta}(w_1,w_2|\\hat{\\theta}^W);\\quad \\hat{\\xi}(v_1,v) = \\hat{\\xi}(v_1,v|\\hat{\\theta}^V).\n\\end{equation}\n\n\\section{Asymptotic Properties of Estimators}\n\\label{sec: Properties}\n\nIn this section we provide some asymptotic properties of the estimators, though we do not present the rigorous proofs of the results due to space constraints and instead defer them to a separate paper. To make our exposition more concise and compact, we introduce additional notation. Consider a real-valued function $h$ defined on $\\mathfrak{I}_{\\mathfrak{W}}$. Then, $h(w_1,w_2)$ will represent the value at $(w_1,w_2)$, but when we simply write $h$ it means the $|\\mathfrak{I}_{\\mathfrak{W}}| \\times 1$ vector consisting of $h(w_1,w_2), (w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}$. Thus, $\\eta$ is an $|\\mathfrak{I}_{\\mathfrak{W}}| \\times 1$ (column) vector with elements $\\eta(w_1,w_2)$s; $N_i^W(s)$ is an $|\\mathfrak{I}_{\\mathfrak{W}}| \\times 1$ vector consisting of $N_i^W(s;w_1,w_2)$s; $S^{0W}(s|\\theta^W)$ is an $|\\mathfrak{I}_{\\mathfrak{W}}| \\times 1$ vector with elements $S^{0W}(s;w_1|\\theta^W), (w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}$; and $\\psi_i^W(s|\\theta^W)$ is an $|\\mathfrak{I}_{\\mathfrak{W}}| \\times 1$ vector consisting of the same elements. Similarly for those associated with the HS process where functions are defined over $\\mathfrak{I}_{\\mathfrak{V}}$; e.g., $\\eta = (\\eta(v_1,v), (v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}})$, $N_i^V(s) = (N_i^V(s;v_1,v): (v_1,v) \\in \\mathfrak{I}_{\\mathfrak{V}})$, etc.\n\nLet us first consider the profile likelihood function $\\mathcal{L}_{pl}^W$ and the estimators $\\hat{\\theta}^W$ and $\\hat{\\eta}$. Using the above notation, the associated profile log-likelihood function, score function, and observed information matrix could be written as follows:\n\\begin{eqnarray*}\n& l_{pl}^W(\\theta^W) = \\sum_{i=1}^n \\int_0^\\infty \\left[\\log\\psi_i^W(s-|\\theta^W) - \\log S^{0W}(s|\\theta^W) \\right]^{\\tt T} dN_i^W(s); & \\\\\n& U_{pl}^W(\\theta^W) = \\sum_{i=1}^n \\int_0^\\infty H_{1i}^W(s|\\theta^W)^{\\tt T} dN_i^W(s) ; & \\\\ \n&I_{pl}^W(\\theta^W) = \\sum_{i=1}^n \\int_0^\\infty V_1^W(s|\\theta^W)^{\\tt T} dN_i^W(s), &\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*}\nH_{1i}^W(s|\\theta^W) & = & \\frac{\\stackrel{\\cdot}{\\psi}_i^W(s-|\\theta^W)}{\\psi_i^W(s-|\\theta^W)} - \\frac{\\stackrel{\\cdot}{S}^{0W}(s|\\theta^W)}{S^{0W}(s|\\theta^W)} \\\\\n& \\equiv & \\left(\\frac{\\stackrel{\\cdot}{\\psi}_i^W(s-|\\theta^W)}{\\psi_i^W(s-|\\theta^W)} - \\frac{\\stackrel{\\cdot}{S}^{0W}(s;w_1|\\theta^W)}{S^{0W}(s;w_1|\\theta^W)}: (w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}\\right); \n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\nV_1^W(s|\\theta^W) & = & \\frac{\\stackrel{\\cdot\\cdot}{S}^{0W}(s|\\theta^W)}{S^{0W}(s|\\theta^W)} - \\left(\\frac{\\stackrel{\\cdot}{S}^{0W}(s|\\theta^W)}{S^{0W}(s|\\theta^W)}\\right)^{\\otimes 2} \\\\\n& \\equiv & \\left(\\frac{\\stackrel{\\cdot\\cdot}{S}^{0W}(s;w_1|\\theta^W)}{S^{0W}(s;w_1|\\theta^W)} - \\left(\\frac{\\stackrel{\\cdot}{S}^{0W}(s;w_1|\\theta^W)}{S^{0W}(s;w_1|\\theta^W)}\\right)^{\\otimes 2}: (w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}} \\right).\n\\end{eqnarray*}\nA `$\\cdot$' over a function means gradient with respect to the parameter vector, while a `$\\cdot\\cdot$' over a function means the matrix of second-partial derivatives with respect to the element of the parameter vector. In obtaining $I_{pl}^W$, we also used the fact that $\\psi_i^W$ is an exponential function. We now present some results about $\\hat{\\theta}^W$. We also let \n$$\\mathfrak{I}_{pl}^W = \\mbox{pr-lim} \\left[ \\frac{1}{n} I_{pl}^W(\\theta^W)\\right].$$\nThis will be a function of {\\bf\\em all} the model parameters, since the limiting behavior of the $Y_i$s and $Y_i^W$s will depend on all the model parameters, owing to the interplay among the RCR, LM, and HS processes, as could be seen in the special case of Poisson processes and CTMCs.\n\n\\begin{proposition}\n\\label{prop-asymptotic about thetaW}\nUnder certain regularity conditions, we have, as $n \\rightarrow \\infty$,\n\\begin{itemize}\n\\item[(i)] (Consistency)\n$\\hat{\\theta}^W \\stackrel{p}{\\rightarrow} \\theta^W$;\n\\item[(ii)] (Asymptotic Representation)\n\\begin{eqnarray*} \n\\sqrt{n}[\\hat{\\theta}^W - \\theta^W] = \\left[\\frac{1}{n} I_{pl}^W(\\theta^W)\\right]^{-1} \\frac{1}{\\sqrt{n}} \\sum_{i=1}^n \\int_0^\\infty\nH_{1i}^W(s|\\theta^W)^{\\tt T} dM_i^W(s|\\eta,\\theta^W) + o_p(1);\n\\end{eqnarray*}\n\\item[(iii)] (Asymptotic Normality) \n$\\sqrt{n}[\\hat{\\theta}^W - \\theta^W] \\stackrel{d}{\\rightarrow} N\\left[0,\\left(\\mathfrak{I}_{pl}^W\\right)^{-1}\\right].$\n\\end{itemize}\n\\end{proposition}\n\n\nWe point out an important result needed for the proof of Proposition \\ref{prop-asymptotic about thetaW}(iii), which is that\n\\begin{eqnarray*}\n\\lefteqn{\\frac{1}{n} I_{pl}^W(\\theta^W) = } \\\\\n&& \\sum_{(w_1,w_2) \\in \\mathfrak{I}_{\\mathfrak{W}}} \\left[ \\frac{1}{n} \\sum_{i=1}^n \\int_0^\\infty \\left\\{H_{1i}^W(s;w_1|\\theta^W)\\right\\}^{\\otimes 2} Y_i(s) Y_i^W(s;w_1) \\eta(w_1,w_2) \\psi_i^W(s-|\\theta^W) ds \\right] \\\\ && + o_p(1) \n\\stackrel{p}{\\rightarrow} \\mathfrak{I}_{pl}^W.\n\\end{eqnarray*}\nThis asymptotic equivalence indicates where the involvement of the at-risk processes come into play in the limiting profile information matrix, hence the dependence on all the model parameters. This also shows that a natural consistent estimator of $\\mathfrak{I}_{pl}^W$ is $I_{pl}^W(\\hat{\\theta}^W)\/n$.\n\nNext, we are now in position to present asymptotic results about $\\hat{\\eta}$. First, let $s^{0W}$ and $\\stackrel{\\cdot}{s}^{0W}$ be deterministic functions satisfying\n\\begin{eqnarray*}\n& \\left|\\frac{1}{n} \\int_0^\\infty S^{0W}(z|\\theta^W) dz - \\int_0^\\infty s^{0W}(z) dz\\right| \\rightarrow 0; & \\\\\n& \\left|\\frac{1}{n} \\int_0^\\infty \\stackrel{\\cdot}{S}^{0W}(z|\\theta^W) dz - \\int_0^\\infty \\stackrel{\\cdot}{s}^{0W}(z) dz\\right| \\rightarrow 0. &\n\\end{eqnarray*}\n\n\\begin{proposition}\n\\label{prop-asymptotic about eta}\nUnder certain regularity conditions, we have as $n \\rightarrow \\infty$,\n\\begin{itemize}\n\\item[(i)] (Consistency) $\\hat{\\eta} \\stackrel{p}{\\rightarrow} \\eta$;\n\\item[(ii)] (Asymptotic Normality) $\\sqrt{n}(\\hat{\\eta} -\\eta) \\stackrel{d}{\\rightarrow} N(0,\\Sigma)$, where\n\\begin{eqnarray*}\n\\Sigma & = & \\mbox{Dg}\\left(\\int_0^\\infty s^{0W}(z) dz\\right)^{-1} \\mbox{Dg}(\\eta) + \\\\\n&& \\left[\\mbox{Dg}\\left(\\int_0^\\infty s^{0W}(z) dz\\right)^{-1} \\mbox{Dg}(\\eta) \\left(\\int_0^\\infty \\stackrel{\\cdot}{s}^{0W}(z) dz\\right)^{\\tt T} (\\mathfrak{I}_{pl}^W)^{-1\/2}\\right]^{\\otimes 2}.\n\\end{eqnarray*}\n\\item[(iii)] (Joint Asymptotic Normality) $\\sqrt{n}(\\hat{\\eta} - \\eta)$ and $\\sqrt{n}(\\hat{\\theta}^W - \\theta^W)$ are jointly asymptotically normal with means zeros and asymptotic covariance matrix\n\\begin{eqnarray*}\n\\lefteqn{\\mbox{Acov}(\\sqrt{n}(\\hat{\\theta}^W - \\theta^W),\\sqrt{n}(\\hat{\\eta} - \\eta)) = } \\\\\n&& -(\\mathfrak{I}_{pl}^W)^{-1} \\left(\\int_0^\\infty \\stackrel{\\cdot}{s}^{0W}(z) dz\\right) \\mbox{Dg}(\\eta) \\mbox{Dg}\\left(\\int_0^\\infty s^{0W}(z) dz\\right)^{-1}.\n\\end{eqnarray*}\n\\end{itemize}\n\\end{proposition}\n\n\nWe remark that in result (ii) for the asymptotic covariance matrix $\\Sigma$ in Proposition \\ref{prop-asymptotic about eta}, the additional variance term in the right-hand side is the effect of plugging-in the estimator $\\hat{\\theta}^W$ for $\\theta^W$ in the `estimators' $\\hat{\\eta}(w_1,w_2|\\theta^W)$s to obtain $\\hat{\\eta}(w_,w_2)$s.\nWithout having to write them down explicitly, similar results are obtainable for the estimators $\\hat{\\theta}^V$ and $\\hat{\\xi}$. \n\nNext, we present results concerning the asymptotic properties of the estimators of $\\alpha_q$s, $\\theta^R$, and $\\Lambda_{0q}(\\cdot)$s. Define the restricted profile likelihood for $(\\alpha_q,\\theta^R)$ based on data $\\mathbf{D}$ observed over $[0,s^*]$ via\n\\begin{eqnarray*}\n\\mathcal{L}_{pl}^R(\\alpha,\\theta^R) & \\equiv & \\mathcal{L}_{pl}^R(\\alpha,\\theta^R|s^*,t^*) \\equiv \\mathcal{L}_{pl}^R(\\alpha,\\theta^R|\\mathbf{D}(s^*,t^*)) \\\\\n& = & \n\\prod_{q=1}^Q \\prod_{i=1}^n \\prodi_{s=0}^{s^*} \n\\left[\n\\frac{\\rho_q[N_i^R(s-);\\alpha_q] \\psi_i^R(s-;\\theta^R)}{S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}\n\\right]^{I\\{\\mathcal{E}_{iq}(s) \\le t^*\\} dN_i^R(s;q)}\n\\end{eqnarray*}\nwith $S_q^{0R}(\\cdot,\\cdot|\\cdot,\\cdot)$ as defined in Proposition \\ref{prop-2}. This is restricted in the sense that we only consider events that happened when the effective age are no more than $t^*$ and happened over $[0,s^*]$. Note that as we let $t^* \\rightarrow \\infty$ and $s^* \\rightarrow \\infty$, we obtain the profile likelihood in Proposition \\ref{profile liks}, The log-profile likelihood function is\n\\begin{eqnarray*}\n\\lefteqn{ {l}_{pl}^R(\\alpha,\\theta^R|s^*,t^*) \\equiv \\log \\mathcal{L}_{pl}^R(\\alpha,\\theta^R|s^*,t^*) } \\\\ & = &\n\\sum_{q=1}^Q \\sum_{i=1}^n \\int_0^{s^*} \\left\\{\n\\log\\rho_q[N_i^R(s-);\\alpha_q] + \\log\\psi_i^R(s-|\\theta^R) - \\log S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)\n\\right\\} \\times \\\\ && I\\{\\mathcal{E}_{iq}(s) \\le t^*\\} dN_i^R(ds;q).\n\\end{eqnarray*}\nThe associated profile score function is\n\\begin{displaymath}\nU_{pl}^R(\\alpha,\\theta^R|s^*,t^*) = U_{pl}^R(\\alpha,\\theta^R) = \\sum_{i=1}^n \\sum_{q=1}^Q \\int_0^{s^*} H_{iq}^R(s|s^*,\\alpha,\\theta^R) N_i^R(ds,t^*;q)\n\\end{displaymath}\nwhere, for $q \\in \\mathfrak{I}_Q$,\n$$H_{iq}^R(s|s^*,\\alpha,\\theta^R)^{\\tt T} = \\left[ 0^{\\tt T},\\ldots, 0^{\\tt T}, H_{i1q}(s|s^*,\\alpha,\\theta^R)^{\\tt T}, 0^{\\tt T}, \\ldots, 0^{\\tt T},\nH_{i2q}(s|s^*,\\alpha,\\theta^R)^{\\tt T}\\right],$$\na $(Q + 1) \\times 1$ block matrix and with\n$$H_{i1q}^R(s|s^*,\\alpha_q,\\theta^R) = \\frac{\\nabla_{\\alpha_q} \\rho_q[N_i^R(s-);\\alpha_q)}{\\rho_q[N_i^R(s-);\\alpha_q)} -\n\\frac{\\nabla_{\\alpha_q} S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}{S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)};$$\n$$H_{i2q}^R(s|s^*,\\alpha_q,\\theta^R) = \\frac{\\nabla_{\\theta^R} \\psi_i^R(s-|\\theta^R)}{ \\psi_i^R(s-|\\theta^R)} -\n\\frac{\\nabla_{\\theta^R} S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}{S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}.$$\nNote the dimensions of these block vectors: the $q$th block is of dimension $\\dim(\\alpha_q) \\times 1$, while the $(Q+1)$th block is of dimension $\\dim(\\theta^R) \\times 1$. An important martingale representation of $U_{pl}^R(\\alpha,\\theta^R|s^*,t^*)$, which is straight-forward to establish, is\n\\begin{displaymath}\nU_{pl}^R(\\alpha,\\theta^R|s^*,t^*) = \\sum_{i=1}^n \\sum_{q=1}^Q \\int_0^{s^*} H_{iq}^R(s|s^*,\\alpha,\\theta^R) M_i^R(ds,t^*;q).\n\\end{displaymath}\nThe predictable variation associated with this score function is\n\\begin{displaymath}\n\\langle U_{pl}^R(\\alpha,\\theta^R|\\cdot,t^*) \\rangle (s^*) = \\sum_{i=1}^n \\sum_{q=1}^n \\int_0^{s^*} [H_{iq}^R(s|s^*,\\alpha,\\theta^R)]^{\\otimes 2} A_i^R(ds,t^*;q)\n\\end{displaymath}\nwith $A_i^R(ds,t;q) = I\\{\\mathcal{E}_{iq}(s) \\le t^*\\} dA_i^R(s;q)$. \nThe estimators $\\hat{\\alpha}_q(s^*,t^*), q =1,\\ldots,Q,$ and $\\hat{\\theta}^R(s^*,t^*)$ satisfy the equation\n$U_{pl}^R(\\hat{\\alpha},\\hat{\\theta}^R|s^*,t^*) = 0.$\n\nThe observed profile information matrix $I_{pl}^R(\\alpha,\\theta^R|s^*,t^*)$ is a $(Q+1) \\times (Q+1)$ symmetric block matrix with the following block elements, for $q, q^\\prime = 1, \\ldots,Q$:\n\\begin{displaymath}\nI_{pl,qq^\\prime}^R(\\alpha,\\theta^R|s^*,t^*) = \n\\left\\{\n\\begin{array}{ccc}\n\\sum_{i=1}^n \\int_0^{s^*} V_{i11qq}^R(s|s^*,\\alpha,\\theta^R) N_i^R(ds,t^*;q) & \\mbox{for} &q = q^\\prime \\\\\n0 & \\mbox{for} & q \\ne q^\\prime\n\\end{array}\n\\right.;\n\\end{displaymath}\n\\begin{displaymath}\nI_{pl,(Q+1)q}^R(\\alpha,\\theta^R|s^*,t^*) = \n\\sum_{i=1}^n \\int_0^{s^*} V_{i21q}^R(s|s^*,\\alpha,\\theta^R) N_i^R(ds,t^*;q);\n\\end{displaymath}\n\\begin{displaymath}\nI_{pl,(Q+1)(Q+1)}^R(\\alpha,\\theta^R|s^*,t^*) = \n\\sum_{i=1}^n \\sum_{q=1}^Q \\int_0^{s^*} V_{i22q}^R(s|s^*,\\alpha,\\theta^R) N_i^R(ds,t^*;q),\n\\end{displaymath}\nwhere $V_{i11qq^\\prime}^R$ is of dimension $\\dim(\\alpha_q) \\times \\dim(\\alpha_{q^\\prime})$ for $q, q^\\prime = 1, 2, \\ldots, Q$; $V_{i21q}^R$ and $(V_{i12q}^R)^{\\tt T}$ have dimensions $\\dim(\\theta^R) \\times \\dim(\\alpha_q)$; and $V_{i22q}^R$ has dimension $\\dim(\\theta^R) \\times \\dim(\\theta^R)$. These are given by the following expressions, for $q, q^\\prime = 1,\\ldots,Q$:\n\\begin{eqnarray*}\n \\lefteqn{V_{i11qq}^R(s|s^*,\\alpha,\\theta^R) = -\\left\\{\\frac{\\nabla_{\\alpha_q\\alpha_q^{\\tt T}} \\rho_q[N_i(s-);\\alpha_q]}{\\rho_q[N_i(s-);\\alpha_q]} - \\left(\\frac{\\nabla_{\\alpha_q} \\rho_q[N_i(s-);\\alpha_q]}{\\rho_q[N_i(s-);\\alpha_q]}\\right)^{\\otimes 2}\\right\\} + } \\\\\n & & \\left\\{\\frac{\\nabla_{\\alpha_q\\alpha_q^{\\tt T}} S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}{S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)} - \\left(\\frac{\\nabla_{\\alpha_q} S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}{S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}\\right)^{\\otimes 2}\\right\\}; \n \\end{eqnarray*}\n $$V_{i11qq^\\prime}^R(s|s^*,\\alpha,\\theta^R) = 0, q \\ne q^\\prime; $$\n\\begin{eqnarray*}\n\\lefteqn{V_{i21q}^R(s|s^*,\\alpha,\\theta^R) =\n-\\left\\{\\frac{\\nabla_{\\alpha_q(\\theta^R)^{\\tt T}} S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}{S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)} - \\right. } \\\\ && \\left.\n\\left(\\frac{\\nabla_{\\alpha_q} S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}{S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}\\right)\n\\left(\\frac{\\nabla_{(\\theta^R)^{\\tt T}} S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}{S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}\\right)\\right\\};\n\\end{eqnarray*}\n\\begin{eqnarray*}\nV_{i22q}^R(s|s^*,\\alpha,\\theta^R) & = & -\\left\\{\\frac{\\nabla_{\\theta^R(\\theta^R)^{\\tt T}} S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}{S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)} - \\right. \\\\ && \\left.\n\\left(\\frac{\\nabla_{\\theta^R} S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}{S_q^{0R}(s^*,\\mathcal{E}_{iq}(s)|\\alpha_q,\\theta^R)}\\right)^{\\otimes 2}\\right\\},\n\\end{eqnarray*}\nwhere for the last expression we used the fact that $\\psi_i(z) = \\exp(z)$.\n\nA condition needed for the asymptotic results is that there is an invertible matrix function $\\mathfrak{I}_{pl}^R(\\alpha,\\theta^R|s^*,t^*)$ which equals the in-probability limit of $\\frac{1}{n} I_{pl}^R(\\alpha,\\theta^R|s^*,t^*)$ as $n \\rightarrow \\infty$. Under this condition, we have the following consistency and asymptotic normality results for $(\\hat{\\alpha},\\hat{\\theta}^R)$:\n\n\\begin{proposition}\n\\label{asymptotics of alpha and thetaR}\nUnder regularity conditions and as $n \\rightarrow \\infty$,\n\\begin{itemize}\n\\item[(i)] $(\\hat{\\alpha}(s^*,t^*),\\hat{\\theta}^R(s^*,t^*)) \\stackrel{p}{\\rightarrow} (\\alpha,\\theta^R)$; \n\\item[(ii)] $\\sqrt{n}\\left[\\begin{array}{c} \\hat{\\alpha}(s^*,t^*) - \\alpha \\\\ \\hat{\\theta}^R(s^*,t^*) - \\theta^R \\end{array} \\right] \\stackrel{d}{\\rightarrow} N\\left(0,[\\mathfrak{I}_{pl}^R(\\alpha,\\theta^R|s^*,t^*)]^{-1}\\right).$\n\\end{itemize}\n\\end{proposition}\n\n\nImportant equivalences to note are, for $q, q^\\prime =1,\\ldots,Q$:\n\\begin{eqnarray*}\n\\frac{1}{n} I_{pl,qq^\\prime}^R(\\alpha,\\theta^R|s^*,t^*) & = &\n\\frac{1}{n}\\sum_{i=1}^n\\int_0^{s^*} V_{i11qq^\\prime}^R(s|s^*,\\alpha,\\theta^R) I\\{\\mathcal{E}_{iq}(s) \\le t^*\\} \\times \\\\ &&\nY_i(s) \\rho_q[N_i^R(s-);\\alpha_q] \\psi_i^R(s-;\\theta^R) \\lambda_{0q}[\\mathcal{E}_{iq}(s)] ds + o_p(1); \\\\\n\\frac{1}{n} I_{pl,(Q+1)q}^R(\\alpha,\\theta^R|s^*,t^*) & = &\n\\frac{1}{n}\\sum_{i=1}^n\\int_0^{s^*} V_{i21q}^R(s|s^*,\\alpha,\\theta^R) I\\{\\mathcal{E}_{iq}(s) \\le t^*\\} \\times \\\\ &&\nY_i(s) \\rho_q[N_i^R(s-);\\alpha_q] \\psi_i^R(s-;\\theta^R) \\lambda_{0q}[\\mathcal{E}_{iq}(s)] ds + o_p(1); \\\\\n\\frac{1}{n} I_{pl,(Q+1)(Q+1)}^R(\\alpha,\\theta^R|s^*,t^*) & = &\n\\frac{1}{n}\\sum_{i=1}^n \\sum_{q=1}^Q \\int_0^{s^*} V_{i22q}^R(s|s^*,\\alpha,\\theta^R) I\\{\\mathcal{E}_{iq}(s) \\le t^*\\} \\times \\\\ &&\nY_i(s) \\rho_q[N_i^R(s-);\\alpha_q] \\psi_i^R(s-;\\theta^R) \\lambda_{0q}[\\mathcal{E}_{iq}(s)] ds + o_p(1).\n\\end{eqnarray*}\nIn addition, the regularity conditions must imply that, we have\n\\begin{eqnarray*}\n& \\left|\\frac{1}{n} I_{pl,qq}^R(\\alpha,\\theta^R|s^*,t^*) - \\frac{1}{n}\\sum_{i=1}^n \\int_0^{s^*} [H_{i1q}^R(s|s^*,\\alpha,\\theta^R)]^{\\otimes 2} A_i^R(ds,t^*;q) \\right| \\stackrel{p}{\\rightarrow} 0; & \\\\\n& \\left|\\frac{1}{n} I_{pl,(Q+1)(Q+1)}^R(\\alpha,\\theta^R|s^*,t^*) - \\frac{1}{n}\\sum_{i=1}^n \\int_0^{s^*} [H_{i2q}^R(s|s^*,\\alpha,\\theta^R)]^{\\otimes 2} A_i^R(ds,t^*;q) \\right| \\stackrel{p}{\\rightarrow} 0; & \\\\\n& \\left|\\frac{1}{n} I_{pl,(Q+1)q}^R(\\alpha,\\theta^R|s^*,t^*) - \\frac{1}{n}\\sum_{i=1}^n \\int_0^{s^*} H_{i1q}^R(s|s^*,\\alpha,\\theta^R) [H_{i2q}^R(s|s^*,\\alpha,\\theta^R)]^{\\tt T} A_i^R(ds,t^*;q) \\right| & \\\\ & \\stackrel{p}{\\rightarrow} 0. &\n\\end{eqnarray*}\nThese conditions imply that $\\left|\\frac{1}{n} \\langle U_{pl}^R \\rangle - \\frac{1}{n} I_{pl}^R \\right| \\stackrel{p}{\\rightarrow} 0$ as $n \\rightarrow \\infty$.\nThese are the analogous results to those in the usual asymptotic theory of maximum likelihood estimators, where the Fisher information is equal to the expected value of the squared partial derivative with respect to the parameter of the log-likelihood function and also the negative of the expected value of the second partial derivative of the log-likelihood function. They are usually satisfied by imposing a set of conditions that allows for the interchange of the order of integration with respect to $s$ and the partial differentiation with respect to the parameters; see, for instance, \\cite{Bor84}, \\cite{AndGil1982}, and \\cite{pena2016} in similar but simpler settings.\n\nThe proof of Proposition \\ref{asymptotics of alpha and thetaR} then relies on the asymptotic representation\n$$\\sqrt{n} \\left[\\begin{array}{c} \\hat{\\alpha}(s^*,t^*) - \\alpha \\\\ \\hat{\\theta}^R(s^*,t^*) - \\theta^R \\end{array}\\right] = \\left[\\frac{1}{n}I_{pl}^R(\\alpha,\\theta^R|s^*,t^*)\\right]^{-1} \\left[\\frac{1}{\\sqrt{n}} U_{pl}^R(\\alpha,\\theta^R|s^*,t^*)\\right] + o_p(1).$$\nIn fact, this representation is also crucial for finding the asymptotic properties of the estimators of the $\\Lambda_{0q}(\\cdot)$s, which we will now present. By first-order Taylor expansion and under regularity conditions, we have the representations, for each $q = 1,\\ldots, Q$, given by\n\\begin{eqnarray*}\n\\hat{\\Lambda}_{0q}(s^*,t) & = & \\int_0^t \\frac{I\\{S_q^{0R}(s^*,w|\\hat{\\alpha}(s^*,t^*),\\hat{\\theta}^R(s^*,t^*)) > 0\\}}{S_q^{0R}(s^*,w|\\hat{\\alpha}(s^*,t^*),\\hat{\\theta}^R(s^*,t^*)} \\sum_{i=1}^n N_i^R(s^*,dw;q) \\\\\n& = & \\int_0^t \\frac{I\\{S_q^{0R}(s^*,w|\\alpha,\\theta^R) > 0\\}}{S_q^{0R}(s^*,w|\\alpha,\\theta^R)} \\sum_{i=1}^n N_i^R(s^*,dw;q) + \\\\ && \\left[ B_{1q}(s^*,t|\\alpha,\\theta^R),\\ B_{2q}(s^*,t|\\alpha,\\theta^R)\\right] \\left[\\begin{array}{c} \\hat{\\alpha}(s^*,t^*) - \\alpha \\\\ \\hat{\\theta}^R(s^*,t^*) - \\theta^R \\end{array}\\right] + o_p(1\/\\sqrt{n})\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*}\n\\lefteqn{B_{1q}(s^*,t|\\alpha,\\theta^R) = - \\int_0^t I\\{S_q^{0R}(s^*,w|\\alpha,\\theta^R) > 0\\} \\times } \\\\ && \\left\\{\\frac{\\nabla_\\alpha S_q^{0R}(s^*,w|\\alpha,\\theta^R)}{[S_q^{0R}(s^*,w|\\alpha,\\theta^R)]^2}\\right\\} \\sum_{i=1}^n N_i^R(s^*,dw;q);\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\lefteqn{B_{2q}(s^*,t|\\alpha,\\theta^R) = - \\int_0^t I\\{S_q^{0R}(s^*,w|\\alpha,\\theta^R) > 0\\} \\times } \\\\ && \\left\\{\\frac{\\nabla_\\theta^R S_q^{0R}(s^*,w|\\alpha,\\theta^R)}{[S_q^{0R}(s^*,w|\\alpha,\\theta^R)]^2}\\right\\} \\sum_{i=1}^n N_i^R(s^*,dw;q).\n\\end{eqnarray*}\nFor $q = 1,\\ldots,Q,$ let\n$$\\Lambda_{0q}^*(s^*,t) = \\int_0^t I\\{S_q^{0R}(s^*,w|\\alpha,\\theta^R) > 0\\} \\lambda_{0q}(w) dw.$$\nObserve that\n$$\\int_0^t \\frac{I\\{S_q^{0R}(s^*,w|\\alpha,\\theta^R) > 0\\}}{S_q^{0R}(s^*,w|\\alpha,\\theta^R)} \\sum_{i=1}^n A_i^R(s^*,dw|\\alpha,\\theta^R) = \\Lambda_{0q}^*(s^*,t),$$\nimplying that\n\\begin{eqnarray*}\n\\lefteqn{ \\int_0^t \\frac{I\\{S_q^{0R}(s^*,w|\\alpha,\\theta^R) > 0\\}}{S_q^{0R}(s^*,w|\\alpha,\\theta^R)} \\sum_{i=1}^n N_i^R(s^*,dw;q) = } \\\\\n&& \\int_0^t \\frac{I\\{S_q^{0R}(s^*,w|\\alpha,\\theta^R) > 0\\}}{S_q^{0R}(s^*,w|\\alpha,\\theta^R)} \\sum_{i=1}^n M_i^R(s^*,dw;q) + \\Lambda_{0q}^*(s^*,t).\n\\end{eqnarray*}\nLet us also define\n\\begin{eqnarray*}\n\\lefteqn{ \\hat{Z}_q^R(s^*,t) = [\\hat{\\Lambda}_{0q}(s^*,t) - \\Lambda_{0q}^*(s^*,t)] - } \\\\ && \\left[ B_{1q}(s^*,t|\\alpha,\\theta^R),\\ B_{2q}(s^*,t|\\alpha,\\theta^R)\\right] \\left[\\begin{array}{c} \\hat{\\alpha}(s^*,t^*) - \\alpha \\\\ \\hat{\\theta}^R(s^*,t^*) - \\theta^R \\end{array}\\right];\n\\end{eqnarray*}\n\\begin{displaymath}\n\\tilde{H}_{iq}^R(w|s^*) = \\sum_{j: S_j \\le s^*} H_{iq}^R[\\mathcal{E}_{iqj}^{-1}(w)|s^*] I\\{\\mathcal{E}_{iq}(S_{j-1}) < w \\le \\mathcal{E}_{iq}(S_j)\\};\n\\end{displaymath}\nand form the vectors of functions $\\hat{Z} = (\\hat{Z}_q, q=1,\\ldots,Q)$ and $\\tilde{H}_i^R = (\\tilde{H}_{iq}^R, q = 1,\\ldots,Q)$. Then, the main asymptotic representation leading to the asymptotic results for the RCR component parameters is given by, for $t \\in [0,t^*]$,\n\\begin{eqnarray}\n\\sqrt{n} \\left[\n\\begin{array}{c}\n\\hat{\\alpha}(s^*,t^*) - \\alpha \\\\\n\\hat{\\theta}^R(s^*,t^*) - \\theta^R \\\\\n\\hat{Z}(s^*,t)\n\\end{array}\n\\right] & = &\n\\left[\n\\begin{array}{cc}\n\\left(\\frac{1}{n} I_{pl}^R(s^*,t^*)\\right)^{-1} & 0 \\\\ 0 & I\n\\end{array}\n\\right] \\times \\nonumber \\\\ &&\n\\frac{1}{\\sqrt{n}} \\int_0^{t}\n\\left[\n\\begin{array}{c}\n\\tilde{H}_i^R(w|s^*) \\\\\n\\mbox{Dg}\\left(\\frac{I\\{S^{0R}(s^*,w)\/n > 0\\}}{S^{0R}(s^*,w)\/n}\\right)\n\\end{array}\n\\right] \nM_i^R(s^*,dw) + o_p(1).\n\\label{major asymptotic representation RCR}\n\\end{eqnarray}\n\nUsing this representation, we obtain the following asymptotic properties, though not stated in the most general form.\n\n\\begin{proposition}\n\\label{prop-main result RCR estimators}\nUnder certain regularity conditions, we have\n\\begin{itemize}\n\\item[(i)]\n$\\sqrt{n} \\left[ \\begin{array}{c}\n\\hat{\\alpha}(s^*,t^*) - \\alpha \\\\\n\\hat{\\theta}^R(s^*,t^*) - \\theta^R\n\\end{array}\n\\right]$ and $\\sqrt{n} \\hat{Z}(s^*,t)$ are asymptotically independent;\n\\item[(ii)]\nFor each $q = 1, \\ldots, Q$, $\\hat{\\Lambda}_{0q}(s^*,t)$ converges uniformly in probability to $\\Lambda_{0q}(t)$ for $t \\in [0,t^*]$;\n\\item[(iii)]\nFor each $q = 1, \\ldots, Q$, and for $t \\in [0,t^*]$, $\\sqrt{n}[\\hat{\\Lambda}_{0q}(s^*,t) - \\Lambda_{0q}(t)]$ converges in distribution to a normal distribution with mean zero and variance\n\\begin{displaymath}\n\\Gamma_q(s^*,t) = \\int_0^t \\frac{d\\Lambda_{0q}(w)}{s_q^{0R}(s^*,w)} +\n(b_{1q}(s^*,t),\\ b_{2q}(s^*,t)) [\\mathfrak{I}_{pl}^R(s^*,t^*)]^{-1} (b_{1q}(s^*,t),\\ b_{2q}(s^*,t))^{\\tt T},\n\\end{displaymath}\nwhere $(b_{1q}(s^*,t),\\ b_{2q}(s^*,t)) = \\mbox{pr-lim} \\left\\{ \\frac{1}{n} (B_{1q}(s^*,t),\\ B_{2q}(s^*,t))\\right\\}$.\n\\item[(iv)] More, generally, for $q=1,\\ldots,Q$, the stochastic process $\\{\\sqrt{n}[\\hat{\\Lambda}_{0q}(s^*,t) - \\Lambda_{0q}(t)]: \\ t \\in [0,t^*]\\}$ converges weakly in Skorokhod's space $(\\mathfrak{D}[0,t^*], \\mathcal{D}[0,t^*])$ to a zero-mean Gaussian process with covariance function\n\\begin{eqnarray*}\n\\Gamma_q(s^*,t_1,t_2) & = & \\int_0^{\\min(t_1,t_2)} \\frac{d\\Lambda_{0q}(w)}{s_q^{0R}(s^*,w)} + \\\\ &&\n(b_{1q}(s^*,t_1),\\ b_{2q}(s^*,t_1)) [\\mathfrak{I}_{pl}^R(s^*,t^*)]^{-1} (b_{1q}(s^*,t_2),\\ b_{2q}(s^*,t_2))^{\\tt T}.\n\\end{eqnarray*}\nThis covariance function is consistently estimated by replacing the unknown functions by their empirical counterparts and replacing the finite-dimensional parameters by their estimators.\n\\end{itemize}\n\\end{proposition}\n\n\nWe point out that the $\\hat{\\Lambda}_{0q}, q=1,\\ldots,Q,$ are asymptotically dependent, and are also not independent of $(\\hat{\\alpha}(s^*,t^*),\\hat{\\theta}^R(s*,t^*))$. The results stated above generalize those in \\cite{AndGil1982} and \\cite{pena2016} to a more complex and general situation.\n\n\\section{Illustration of Estimation Approach on Simulated Data}\n\\label{sec-Illustration}\n\nIn this section we provide a numerical illustration of the estimation procedure when given a sample data. The illustrative sample data set with $n = 50$ units is depicted in Figure \\ref{sample data picture}. This is generated from the proposed model with the following characteristics.\nFor the $i$th sample unit, the covariate values are generated according to $X_{i1}\\ {\\sim}\\ \\mbox{BER}(0.5)$, $X_{i2}\\ {\\sim}\\ N(0, 1)$ with $X_{i1}$ and $X_{i2}$ independent, where $\\mbox{BER}(p)$ is the Bernoulli distribution with success probability of $p$. The end of monitoring time is $\\tau_i\\ {\\sim}\\ \\mbox{EXP}(5)$, where $\\mbox{EXP}(\\lambda)$ is the exponential distribution with mean $\\lambda$. For the RCR component with $Q=3$, the baseline (crude) hazard rate function for risk $q \\in \\{1,2,3\\}$, is a two-parameter Weibull given by \n$$\\lambda_{0q}(t|\\kappa_q^*,\\theta_q^*)=\\frac{\\kappa_q^*}{\\theta_q^*}\\left(\\frac{t}{\\theta_q^*}\\right)^{\\kappa_q^*-1},\\quad t \\ge 0,$$ \nwith $\\kappa_q^* \\in \\{2, 2, 3\\}$ and $\\theta_q^* \\in \\{ 0.9, 1.1, 1\\}$. The associated (crude) survivor function for risk $q$ is\n$$\\bar{F}_{0q}(t|\\kappa_q^*,\\theta_q^*) = \\frac{\\kappa_q^*}{\\theta_q^*}\\left(\\frac{t}{\\theta_q^*}\\right)^{\\kappa_q^*-1} \\exp\\left\\{-\\left(\\frac{t}{\\theta_q^*}\\right)^{\\kappa_q^*}\\right\\},\\quad t \\ge 0.$$\nFor risk $q$, the effective age process function is $\\mathcal{E}_{iq}(s) = s - S_{iqN_{iq}^R(s-)}^R$, the backward recurrence time for this risk. For the effects of the accumulating event occurrences, $\\rho_q(N_i^R(s-);\\alpha_q) =(\\alpha_q)^{\\log(1+N^R_{iq}(s-))}$. \nFor the HS component, $\\mathfrak{V} = \\{1,2,3\\}$ with state `$1$' an absorbing state, so $\\gamma$ is a scalar. For the LM component, $\\mathfrak{W} = \\{1=\\mbox{High}, 2=\\mbox{Normal}, 3=\\mbox{Low}\\}$, so $\\kappa$ is a two-dimensional vector. \nThe infinitesimal generator matrices $\\eta$ for the LM process and $\\xi$ for the HS process are, respectively,\n\\begin{eqnarray*}\n\\eta=\\begin{bmatrix}\n-0.3 & 0.2 & 0.1\\\\\n0.1 & -0.2 & 0.1 \\\\\n0.1 & 0.2 & -0.3\n\\end{bmatrix}\n \\quad \\mbox{and} \\quad\n\\xi=\\begin{bmatrix}\n0 & 0 & 0\\\\\n0.2 & -0.7 & 0.5 \\\\\n0.05 & 0.5 & -0.55\n\\end{bmatrix}.\n\\end{eqnarray*}\nThe values in the first row for the $\\xi$-matrix are all zeros because state $1$ in HS is absorbing. The true values of the remaining model parameters are given in the second column of Table \\ref{oneEs}.\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{cccc}\n \\hline\nParameter & True & Estimate & Est.\\ Standard Error \\\\ \n \\hline\n $\\alpha_1$ & 1.50 & 1.58 & 0.13 \\\\\n $\\alpha_2$ & 1.20 & 1.16 & 0.15 \\\\\n $\\alpha_3$ & 2.00 & 2.10 & 0.21 \\\\\n $\\beta^R_1$ & 1.00 & 1.17 & 0.09 \\\\\n $\\beta^R_2$ & -1.00 & -0.94 & 0.10 \\\\\n $\\gamma^R_1$ & 1.00 & 1.17 & 0.09 \\\\ \n $\\kappa^R_1$ & 1.00 & 1.10 & 0.09 \\\\\n $\\kappa^R_2$ & -1.00 & -0.68 & 0.10 \\\\\n\\hline\n$\\beta^W_1$ & 1.00 & 1.02 & 0.28 \\\\ \n$\\beta^W_2$ & -1.00 & -0.78 & 0.28 \\\\ \n$\\gamma^W_1$ & 1.00 & 0.97 & 0.27 \\\\ \n$\\nu^W_1$ & 1.00 & 0.82 & 0.06 \\\\ \n$\\nu^W_2$ & 1.00 & 1.11 & 0.09 \\\\\n$\\nu^W_3$ & -2.00 & -2.01 & 0.09 \\\\ \n\\hline\n $\\beta^V_1$ & 1.00 & 0.93 & 0.17 \\\\\n $\\beta^V_2$ & -1.00 & -1.00 & 0.16 \\\\\n $\\kappa^V_1$ & 1.00 & 1.00 & 0.20 \\\\\n $\\kappa^V_2$ & -1.00 & -1.59 & 0.40 \\\\\n $\\nu^V_1$ & 1.00 & 1.24 & 0.04 \\\\\n $\\nu^V_2$ & 1.00 & 1.04 & 0.06 \\\\\n $\\nu^V_3$ & -2.00 & -2.30 & 0.06 \\\\ \n\\hline\n\\end{tabular}\n\\caption{The second column contains the true values of the parameters in the first column of the model that generated the illustrative sample data in Figure \\ref{sample data picture} and also used in the simulation study. The third column are the estimates of these parameters arising from the illustrative sample data, while the fourth column contains the information-based estimates of the standard errors of the estimators. The RCR component model parameters are the $\\alpha$s and those with superscript $R$. The LM component parameters are those with superscript of $W$. Finally, the HS component parameters are those with superscript $V$.}\n\\label{oneEs}\n\\end{table}\n\nFor each replication, the realized data for the $i$th unit among the $n$ units are generated in the following manner. At time $s=0$, we first randomly assign an initial LM state (either 1, 2, or 3) and HS state (either 2 or 3) uniformly among the allowable states and independently for the LM and HS processes. We specify a fixed length of the intervals partitioning $[0,\\infty)$, which in the simulation runs was set to $ds=0.001$, so we have intervals $I_k = [s_k,s_{k+1}) = [k(ds),(k+1)(ds)), k = 0, 1, 2, \\ldots$. A smaller value of $ds$ will make the data generation coincide more closely to the model, but at the same time will also lead to higher computational costs, especially in a simulation study with many replications. \nThe data generation proceeds sequentially over $k=0,1,2,\\ldots$. Suppose that we have reached interval $I_k = [s_k,s_{k+1})$ with $W(s_k) = w_1$ and $V(s_k) = v_1$. For $q \\in \\mathfrak{I}_Q$, generate a realization $e_q^R$ of $E_q^R$ according to a $\\mbox{BER}(p_q^R)$ with $p_q^R$ given by (\\ref{RCR probability}). Also, for $w_2 \\ne w_1$, generate a realization $e_{w_2}^W$ of $E_{w_2}^W$ according to a $\\mbox{BER}(p_{w_2}^W)$ with $p_{w_2}^W$ given by (\\ref{LM probability}). Finally, for $v \\ne v_1$, generate a realization $e_{v}^V$ of $E_{v}^V$ according to a $\\mbox{BER}(p_{v}^V)$ with $p_v^V$ given by (\\ref{HS probability}). \n\\begin{itemize}\n\\item\nIf all the realizations $e_q^R, q \\in \\mathfrak{I}_Q$, $e_{w_2}^W, w_2 \\in \\mathfrak{I}_{\\mathfrak{W}}, w_2 \\ne w_1$, and $e_v^V, v \\in \\mathfrak{I}_{\\mathfrak{V}}, v \\ne v_1$ are zeros, which means no events occurred in the interval $I_k$, then we proceed to the next interval $I_{k+1}$, provided that $\\tau \\notin I_k$, otherwise we stop.\n\\item\nIf exactly one of the realizations $e_q^R, q \\in \\mathfrak{I}_Q$, $e_{w_2}^W, w_2 \\in \\mathfrak{I}_{\\mathfrak{W}}, w_2 \\ne w_1$, and $e_v^V, v \\in \\mathfrak{I}_{\\mathfrak{V}}, v \\ne v_1$ equals one, so that an event occurred, then we update the values of $N^R, N^W, N^V$ and proceed to the next interval $I_{k+1}$, unless $e_v^V = 1$ for a $v \\in \\mathfrak{V}_0$ (i.e., there is a transition to an absorbing state) or $\\tau \\in I_k$, in which case we stop.\n\\end{itemize}\nIn our implementation, since $0 < ds \\approx 0$, the success probabilities in the Bernoulli distributions above will all be close to zeros, hence the probability of more than one of the $e_q^R, q \\in \\mathfrak{I}_Q$, $e_{w_2}^W, w_2 \\in \\mathfrak{I}_{\\mathfrak{W}}, w_2 \\ne w_1$, and $e_v^V, v \\in \\mathfrak{I}_{\\mathfrak{V}}, v \\ne v_1$ taking values of one is very small. But, in case there were at least two of them with values of one, then we randomly choose one to take the value of one and the others are set to zeros. Thus, we always have $$\\sum_{q \\in \\mathfrak{I}_Q} e_q^R + \\sum_{w_2 \\in \\mathfrak{I}_{\\mathfrak{W}};\\ w_2 \\ne w_1} e_{w_2}^W + \\sum_{v \\in \\mathfrak{I}_{\\mathfrak{V}};\\ v \\ne v_1} e_v^V \\in \\{0, 1\\},$$ which means there is at most one event in any interval. Note that whether we reach an absorbing state or not, there will always be right-censored event times or sojourn times.\n\nWe remark that the event time generation could also have been implemented by first generating a sojourn time and then deciding which event occurred according to a multinomial distribution at the realized sojourn time as indicated in subsection \\ref{subsec: joint model}. In addition, depending on the form of the baseline hazard rate functions $\\lambda_{0q}(\\cdot)$s (e.g., Weibulls) and the effective age processes $\\mathcal{E}_{iq}(\\cdot)$s (e.g., backward recurrence times), a more direct and efficient manner of generating the event times using a variation of the Probability Integral Transformation is possible, without having to do the partitioning of the time axis as done above. However, the method above is more general, though approximate, and applicable even if the $\\lambda_{0q}$'s or the $\\mathcal{E}_{iq}$'s are of more complex forms.\n\nGraphical plots associated with the generated illustrative sample data are provided in Figure \\ref{sample data picture} of Section \\ref{subsec: estimation - parametric}. \nFor this sample data set there were $36$ units that reached the absorbing state, with a mean time to absorption of about $t=1$; while $14$ units did not reach the absorbing state before hitting their respective end of their monitoring periods, with the mean monitoring time at about $t=2$. Recall that $\\tau_i$'s were distributed as $\\mbox{EXP}(5)$ so the mean of $\\tau_i$ is 5. One may be curious why the mean monitoring time for those that did not get absorbed is about 2 and not close to 5. The reason for this is because of an induced selection bias. Those units who got absorbed will tend to have longer monitoring times, hence those that were not absorbed will tend to have shorter monitoring times, explaining a reduction in the mean monitoring times for the subset of units that were not absorbed.\n\nWe have developed programs in {\\tt R} \\cite{RCitation} for implementing the semi-parametric estimation procedure for the joint model described above. We used these programs on this illustrative sample data set to obtain estimates and their estimated standard errors of the model parameters.\nThe third and fourth columns of Table \\ref{oneEs} contain the parameter estimates and estimates of their standard errors, respectively, of the finite-dimensional parameters in the first column, whose true values are given in the second column. Figure \\ref{estbase} depicts the true baseline survivor functions, which are Weibull survivor functions, together with their semi-parametric estimates for each risk of the three risks in the RCR component. The estimates of the baseline survivor functions are obtained using the product-integral representation of cumulative hazard functions. For this generated sample data, the estimates obtained and the function plots demonstrate that there is reasonable agreement between the true parameter values and true functions and their associated estimates, indicating that the semi-parametric estimation procedure described earlier appears to be viable.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth,height=7in]{fi\/plotbase} \n\\caption{The true (red, dashed) and estimated (blue, solid) baseline (crude) survivor functions for each of the three risks based on the simulated illustrative sample data set with $n=50$ units depicted in Figure \\ref{sample data picture}.}\n\\label{estbase}\n\\end{figure}\n\n\n\\section{Finite-Sample Properties via Simulation Studies}\n\\label{sec: simulation}\n\n\\subsection{Simulation Design}\n\nWe have provided asymptotic results of the estimators in Section \\ref{sec: Properties}. In this section we present the results of simulation studies to assess the finite-sample properties of the estimators of model parameters. This will provide some evidence whether the semi-parametric estimation procedure, which appears to perform satisfactorily for the single illustrative sample data set in Section \\ref{sec-Illustration}, performs satisfactorily over many sample data sets.\nThese simulation studies were implemented using {\\tt R} programs we developed, in particular, the programs utilized in estimating parameters in the preceding section. In these simulation studies, as in the preceding section, when we analyze each of the sample data, the baseline hazard rate functions are estimated semi-parametrically, even though in the generation of each of the sample data sets, two-parameter Weibull models were used in the RCR components. \nAside from the set of model parameters described in Section \\ref{sec-Illustration}, the simulation study have the additional inputs which are the sample size $n$ and the number of simulation replications \\mbox{\\tt Mreps}, the latter set to 1000. The sample sizes used in the two simulation studies are $n \\in \\{50, 100\\}$.\nFor fixed $n$, for each of the \\mbox{\\tt Mreps}\\ replications, the sample data generation is as described in Section \\ref{sec-Illustration}.\n\nTable \\ref{genob} presents some summary results pertaining to the three processes based on the \\mbox{\\tt Mreps}\\ replications. The first three rows indicate the means and standard deviations of the number of event occurrences per unit for each risk, and the mean time for the first event occurrence of each risk. For example, risk $1$ occurs about $2.6$ times per unit with a standard deviation $3.57$. The mean time for risk $1$ to occur for the first time is about $0.48$. We notice that occurrence frequencies for three risks are ordered according to $\\mbox{Risk 1} \\succ \\mbox{Risk 2} \\succ \\mbox{Risk 3}$, and consequently risk $1$ tends to have the shortest mean time to the first event. Also note that the mean number of event occurrences per unit for each risk is around $2$, which implies that there are not too few RCR events or too many RCR events (see the property of ``explosion'' as discussed in \\cite{gjess2010}) per unit. This indicates that the choice of the effective age function $\\mathcal{E}_{iq}(\\cdot)$ and the accumulating event function $\\rho_q(\\cdot)$, together with the parameter values we chose for the data generation, were reasonable. \n\nThe fourth to ninth rows show the mean and standard deviation of the number of transitions to specific states per unit, the mean and standard deviation of occupation times per unit for specific states, the mean and standard deviation of sojourn times for specific states. For example, column 4 tells us that (i) the mean number of transitions to state $2$ of the HS process (HS $2$ for short) per unit is $2.34$; (ii) a unit would stay in HS $2$ for an approximate time of $0.8$ on average; (iii) the mean sojourn time for HS $2$ is about $0.34$. We do not include information for HS $1$ since it is an absorbing state. Comparing the $V=2$ and $V=3$ columns, we find that units tend to transit to HS state $2$ more often than to HS state $3$. The mean occupation time for state $2$ per unit is longer compared to state 3. For the last three columns, there are more transitions to state $2$ than to other states. Thus, a unit tends to stay in LM state $1$ more often than in the other two LM states. \n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{c|ccc|cc|ccc}\n \\hline\n & RCR1 & RCR2 & RCR3 & V=2 & V=3 & W=1 & W=2 & W=3 \\\\ \n \\hline\nMean: Count & 2.60 & 2.06 & 1.62 & & & & & \\\\ \n SD: Count & 3.57 & 2.62 & 3.37 & & & & & \\\\ \n Mean: TimePerEvent & 0.48 & 0.61 & 0.78 & & & & & \\\\ \n \\hline\n Mean: NumTransition & & & & 2.34 & 2.10 & 1.16 & 1.36 & 1.10 \\\\ \n SD: NumTransition & & & & 2.17 & 2.02 & 1.31 & 1.24 & 1.33 \\\\ \n Mean: OccupationTime & & & & 0.80 & 0.46 & 0.59 & 0.31 & 0.36 \\\\ \n SD: OccupationTime & & & & 1.83 & 0.78 & 1.51 & 0.58 & 0.71 \\\\ \n Mean: SojournTime & & & & 0.34 & 0.22 & 0.51 & 0.23 & 0.32 \\\\ \n SD: SojournTime & & & & 1.09 & 0.54 & 1.39 & 0.49 & 0.64 \\\\ \n \\hline\n\\end{tabular}\n\\caption{Summary statistics for three processes for \\mbox{\\tt Mreps}\\ replications of data set. The first three rows are for RCR events. The first three rows indicate the mean\/standard deviation of the number of recurrent event occurrences per unit for each risk, the mean time for one recurrent event for each risk. The fourth to ninth rows are for HS and LM events. They indicate the mean\/standard deviation of the number of transitions to the specific state per unit, the mean\/standard deviation of the occupation time for the specific state per unit, the mean\/standard deviation of the sojourn time for the specific state. State $1$ of the HS process $V$ is absorbing, hence not included in the table.}\n\\label{genob}\n\\end{table}\n \nAlso, to obtain some insights into the model-induced dependencies among the components, we also obtained the correlations among RCR, LM, and HS processes over time from the simulated data. We first constructed a vector of six variables over a finite number of time points $\\mathcal{S} \\subset [0,\\tau]$ given by\n %\n \\begin{displaymath}\n \\{Z(s) \\equiv [I(V_i(s)=2), I(W_i(s)=2), I(W_i(s)=3), N_{i1}^R(s), N_{i2}^R(s), N_{i3}^R(s)]^T : s\\in \\mathcal{S}\\}. \n \\end{displaymath}\n %\nFor each $s \\in \\mathcal{S}$, we then obtained the sample correlation matrix $C(s)$ from $\\{Z_i(s), i=1,2,\\ldots,n\\}$. Each of the \\mbox{\\tt Mreps}\\ replications then yielded a $C(s)$, so we took the mean, element-wise, of these \\mbox{\\tt Mreps}\\ correlation matrices. The matrix of scatterplots in Figure \\ref{corr} provides the plots of these mean correlation coefficients over time points $s \\in \\mathcal{S}$. The point we are making here is that the joint model does induce non-trivial patterns of dependencies over time among the three model components.\n \n \\begin{figure}\n\\begin{center}\n\\includegraphics*[width=\\textwidth]{fi\/correlation2}\n\\caption{Plots of sample (Pearson) correlations over a finite set of time points in $[0,3]$. The random vector, at each time $s$ and for each sample unit, is $C(s) = [I(V_i(s)=2), I(W_i(s)=2), I(W_i(s)=3), N_{i1}^R(s), N_{i2}^R(s), N_{i3}^R(s)]$. The sample correlation matrix is computed based on the $n = 50$ sample units. The element-wise means of the \\mbox{\\tt Mreps}\\ correlation matrices were then computed. The plots depict these mean sample correlations for the pairs of variables in $C(s)$ over time $s$.}\n\\label{corr}\n\\end{center}\n\\end{figure}\n\n\\subsection{Finite-Sample Properties of Estimators}\n\nThe set of estimates obtained for one sample data set in the last section is insufficient to assess the performance of the semi-parametric estimation procedure. To get a sense of its performance we performed simulation studies with $\\mbox{\\tt Mreps} = 1000$ replications and sample sizes $n \\in \\{50, 100\\}$. For each replication, we generated a sample data set according to the same joint model, then obtained the set of estimates, via the semi-parametric procedure, for this data set. \nSummary statistics, such as the means, standard deviations, asymptotic standard errors, percentiles, and boxplots for all \\mbox{\\tt Mreps}\\ estimates were then obtained or constructed. Table \\ref{estab} shows these summary statistics of the \\mbox{\\tt Mreps}\\ estimates for each parameter. The asymptotic standard errors reported in Table \\ref{estab} are the means of the asymptotic standard errors over the \\mbox{\\tt Mreps}\\ replications. Also included are the percentile $95\\%$ confidence intervals for each of the unknown parameters based on the \\mbox{\\tt Mreps}\\ replications stratified according to $n=\\{50, 100\\}$. Since the $\\Lambda_{0q}$s are functional nonparametric parameters, we only provide the estimates at four selected time points. Due to space limitation, we also only provide the results for a small subset of the $\\eta$ and $\\xi$ parameters in Table \\ref{estab}. From these simulation results, we observe that the estimates are close to the true values of the model parameters, and the sample standard deviations are close to the asymptotic standard errors, providing some validation to the semi-parametric estimation procedure for the joint model and empirically lending support to the asymptotic results. By comparing the results for $n=50$ and $n=100$ in Table \\ref{estab}, we find that when the sample size $n$ increases, the performance of the estimators of the parameters improves with biases and standard errors decreasing. \n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{c|r|ccccc|ccccc}\n\\hline\n \\multicolumn{2}{c|}{Sample Size}\n& \\multicolumn{5}{c|}{$n=50$} \n& \\multicolumn{5}{c}{$n=100$} \\\\\n\\hline\nParameter & True & Mean & SD & ASE & PL & PU & Mean & SD & ASE & PL & PU \\\\\n\\hline\n$\\Lambda_{01}(0.3)$ & 0.11 & 0.09 & 0.03 & 0.04 & 0.05 & 0.16 & 0.1 & 0.02 & 0.03 & 0.07 & 0.15 \\\\\n$\\Lambda_{01}(0.6)$ & 0.44 & 0.43 & 0.09 & 0.07 & 0.22 & 0.66 & 0.42 & 0.06 & 0.05 & 0.3 & 0.6 \\\\\n$\\Lambda_{01}(0.9)$ & 1 & 0.97 & 0.19 & 0.18 & 0.57 & 1.61 & 0.99 & 0.12 & 0.12 & 0.71 & 1.46 \\\\\n$\\Lambda_{01}(1.2)$ & 1.78 & 1.83 & 0.43 & 0.42 & 1.02 & 2.7 & 1.79 & 0.27 & 0.27 & 1.25 & 2.65 \\\\\n$\\Lambda_{02}(0.3)$ & 0.09 & 0.07 & 0.03 & 0.04 & 0.04 & 0.13 & 0.08 & 0.02 & 0.03 & 0.05 & 0.12 \\\\\n$\\Lambda_{02}(0.6)$ & 0.36 & 0.32 & 0.08 & 0.07 & 0.18 & 0.59 & 0.34 & 0.06 & 0.05 & 0.23 & 0.48 \\\\\n$\\Lambda_{02}(0.9)$ & 0.81 & 0.78 & 0.16 & 0.15 & 0.41 & 1.36 & 0.79 & 0.12 & 0.1 & 0.54 & 1.14 \\\\\n$\\Lambda_{02}(1.2)$ & 1.44 & 1.53 & 0.33 & 0.32 & 0.81 & 2.33 & 1.46 & 0.23 & 0.23 & 1.03 & 2.24 \\\\\n$\\alpha_1$ & 1.5 & 1.52 & 0.15 & 0.13 & 1.13 & 2.06 & 1.5 & 0.1 & 0.09 & 1.25 & 1.83 \\\\\n$\\alpha_2$ & 1.2 & 1.21 & 0.15 & 0.14 & 0.8 & 1.76 & 1.2 & 0.11 & 0.09 & 0.95 & 1.58 \\\\\n$\\alpha_3$ & 2 & 2.02 & 0.23 & 0.22 & 1.43 & 2.7 & 2.01 & 0.14 & 0.14 & 1.56 & 2.6 \\\\\n$\\beta^R_1$ & 1 & 1.03 & 0.11 & 0.1 & 0.78 & 1.45 & 1.02 & 0.08 & 0.07 & 0.82 & 1.25 \\\\\n$\\beta^R_2$ & -1 & -1.03 & 0.11 & 0.11 & -1.29 & -0.85 & -1.02 & 0.07 & 0.08 & -1.17 & -0.89 \\\\\n$\\gamma^R_1$ & 1 & 1.03 & 0.1 & 0.08 & 0.8 & 1.47 & 1.02 & 0.06 & 0.06 & 0.86 & 1.25 \\\\\n$\\kappa^R_1$ & 1 & 1.03 & 0.1 & 0.09 & 0.77 & 1.52 & 1.02 & 0.08 & 0.06 & 0.83 & 1.29 \\\\\n$\\kappa^R_2$ & -1 & -1.02 & 0.15 & 0.14 & -1.51 & -0.52 & -1.01 & 0.1 & 0.1 & -1.33 & -0.74 \\\\\n\\hline\n$\\eta(2,1)$ & 0.1 & 0.1 & 0.04 & 0.03 & 0.04 & 0.18 & 0.1 & 0.02 & 0.02 & 0.05 & 0.16 \\\\\n$\\eta(3,1)$ & 0.1 & 0.1 & 0.04 & 0.03 & 0.04 & 0.19 & 0.1 & 0.02 & 0.02 & 0.06 & 0.17 \\\\\n$\\eta(1,2)$ & 0.2 & 0.2 & 0.05 & 0.06 & 0.11 & 0.29 & 0.2 & 0.04 & 0.04 & 0.12 & 0.29 \\\\\n$\\eta(3,2)$ & 0.2 & 0.21 & 0.05 & 0.05 & 0.11 & 0.29 & 0.2 & 0.03 & 0.03 & 0.13 & 0.29 \\\\\n$\\beta^W_1$ & 1 & 0.98 & 0.23 & 0.21 & 0.51 & 1.45 & 0.99 & 0.16 & 0.15 & 0.68 & 1.32 \\\\\n$\\beta^W_2$ & -1 & -0.99 & 0.18 & 0.2 & -1.24 & -0.75 & -0.99 & 0.14 & 0.14 & -1.16 & -0.81 \\\\\n$\\gamma^W_1$ & 1 & 0.99 & 0.23 & 0.23 & 0.56 & 1.48 & 0.99 & 0.15 & 0.15 & 0.71 & 1.33 \\\\\n$\\nu^W_1$ & 1 & 0.99 & 0.07 & 0.06 & 0.74 & 1.25 & 0.99 & 0.05 & 0.04 & 0.82 & 1.15 \\\\\n$\\nu^W_2$ & 1 & 0.99 & 0.09 & 0.07 & 0.71 & 1.29 & 0.99 & 0.06 & 0.06 & 0.8 & 1.17 \\\\\n$\\nu^W_3$ & -2 & -1.98 & 0.09 & 0.08 & -2.42 & -1.6 & -1.99 & 0.06 & 0.06 & -2.25 & -1.72 \\\\\n\\hline\n$\\xi(2,1)$ & 0.2 & 0.19 & 0.05 & 0.05 & 0.11 & 0.29 & 0.2 & 0.04 & 0.04 & 0.13 & 0.28 \\\\\n$\\xi(3,1)$ & 0.05 & 0.05 & 0.02 & 0.02 & 0.02 & 0.1 & 0.05 & 0.02 & 0.01 & 0.01 & 0.08 \\\\\n$\\xi(3,2)$ & 0.5 & 0.49 & 0.07 & 0.06 & 0.4 & 0.6 & 0.5 & 0.05 & 0.04 & 0.41 & 0.6 \\\\\n$\\xi(2,3)$ & 0.5 & 0.48 & 0.14 & 0.15 & 0.4 & 0.59 & 0.49 & 0.09 & 0.1 & 0.4 & 0.6 \\\\\n$\\beta^V_1$ & 1 & 0.99 & 0.16 & 0.15 & 0.65 & 1.35 & 0.99 & 0.09 & 0.1 & 0.78 & 1.21 \\\\\n$\\beta^V_2$ & -1 & -1 & 0.13 & 0.14 & -1.2 & -0.83 & -0.99 & 0.09 & 0.1 & -1.11 & -0.88 \\\\\n$\\kappa^V_1$ & 1 & 1.01 & 0.18 & 0.17 & 0.65 & 1.4 & 1 & 0.13 & 0.12 & 0.74 & 1.27 \\\\\n$\\kappa^V_2$ & -1 & -1.01 & 0.29 & 0.28 & -1.64 & -0.44 & -1.01 & 0.21 & 0.21 & -1.52 & -0.57 \\\\\n$\\nu^V_1$ & 1 & 1 & 0.05 & 0.04 & 0.83 & 1.17 & 1 & 0.04 & 0.04 & 0.89 & 1.13 \\\\\n$\\nu^V_2$ & 1 & 1.01 & 0.06 & 0.05 & 0.82 & 1.21 & 1 & 0.04 & 0.03 & 0.87 & 1.14 \\\\\n$\\nu^V_3$ & -2 & -1.99 & 0.07 & 0.05 & -2.29 & -1.73 & -2 & 0.04 & 0.04 & -2.2 & -1.81 \\\\\n\\hline\n\\end{tabular}%\n\\end{center}\n\\caption{Summary statistics of the parameter estimates for the \\mbox{\\tt Mreps}\\ replications in the simulation runs for sample sizes $n=\\{50,100\\}$. The columns are the true values of the model parameters, the sample mean of the estimates, the sample standard deviations, the asymptotic standard errors, the 2.5\\% percentile, and the 97.5\\% percentile. The sample standard deviations are estimates of the standard errors of the estimators. The asymptotic standard errors are the means of the asymptotic standard errors over the \\mbox{\\tt Mreps}\\ replications. The RCR component includes model parameters $\\Lambda$s, $\\alpha$s and those parameters with superscript $R$. The LM component includes model parameters $\\eta$s and those parameters with superscript $W$. The HS component includes model parameters $\\xi$s and those parameters with superscript $V$. \n}\n\\label{estab}\n\\end{table}\n\n\nThe graphical summary of these centered estimates for \\mbox{\\tt Mreps}\\ replications of $n=50$ units is given in Figure \\ref{estbox}. Centering for each estimator is done by subtracting the {\\em true} value of the parameter being estimated. We observe that the medians of all these {\\em centered} estimates are close to $0$. We also observe some outliers, but most of the centered \\mbox{\\tt Mreps}\\ estimates are close to $0$. In Figure \\ref{allba}, we show three types of plots for the baseline survivor function for each risk in the RCR component. The true Weibull type baseline survivor function is plotted in red color, the overlaid plots of a random selection of ten estimates of the baseline survivor functions are in green color, and the mean baseline survivor function based on the \\mbox{\\tt Mreps}\\ estimates is shown in blue color. We observe that there is close agreement between the true (red) and mean (blue) curves. Based on these simulation studies, the semi-parametric estimation procedure for the joint model appears to provide reasonable estimates of the true finite- and infinite-dimensional model parameters, at least for the choices of parameter values for these particular simulations. Further simulation and analytical studies are still needed to substantively assess the performance of the semi-parametric estimation procedure for the proposed joint model. \n\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{c}\n\\includegraphics*[width=\\textwidth,height=2.75in]{fi\/centeredest_1} \\\\\n\\includegraphics*[width=\\textwidth,height=2.75in]{fi\/centeredest_2} \\\\\n\\includegraphics*[width=\\textwidth,height=2.75in]{fi\/centeredest_3}\n\\end{tabular}\n\\caption{Boxplots of the centered parameter estimates from \\mbox{\\tt Mreps}\\ replications for simulated data sets each with $n=50$ units. Centering is done by subtracting the true parameter value from each of the \\mbox{\\tt Mreps}\\ estimates.}\n\\label{estbox}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{c}\n\\includegraphics*[width=\\textwidth,height=2.75in]{fi\/allbasesur_1} \\\\\n\\includegraphics*[width=\\textwidth,height=2.75in]{fi\/allbasesur_2} \\\\\n\\includegraphics*[width=\\textwidth,height=2.75in]{fi\/allbasesur_3}\n\\end{tabular}\n\\caption{Overlaid plots of the true baseline survivor function (in red), ten simulated estimates of the baseline survivor function (in green), and the mean baseline survivor function based on \\mbox{\\tt Mreps}\\ simulations (in blue) for each of the three risks in the RCR component.}\n\\label{allba}\n\\end{figure}\n\n\n\\section{Illustration on a Real Data Set}\n\\label{sec: realdata}\n\nTo illustrate our estimation procedure on a real data set, we apply the joint model and the semi-parametric estimation procedure to a medical data set with $n = 150$ patients diagnosed with metastatic colorectal cancer which cannot be controlled by curative surgeries. This data set was gathered in France from 2002--2007 and was used in \\cite{duc2011}. It consists of two data sets which are deposited in the \\textit{frailtypack} package in the {\\tt R} Library \\cite{RCitation}: data set \\textbf{colorectal.Longi} and data set \\textbf{colorectal}. The data set \\textbf{colorectal.Longi} includes the follow-up period, in years, of the patient's tumor size measurements. The times of first measurements of tumor size vary from patient to patient, so to have all of them start at time `zero', our artificial time origin, we consider these first measurements as their initial states. Subsequent times of measuring tumor size are then in terms of the lengths of time from their time origin. There were a total of $906$ tumor size measurements for all the patients. In order to conform to our discrete-valued LM model, we classify (arbitrarily) the tumor size into three categories (states): $1$, $2$, and $3$, if the tumor size belongs in the intervals $[3.4, 6.6]$, $[2, 3.4]$, and $(0, 2)$, respectively. Since the tumor size is only measured at discrete times, instead of continuously, we assumed that the tumor size state is constant between tumor size measurement times, and consequently the tumor size process could only transition at the times in which tumor size is measured. This assumption is most likely unrealistic, but we do so for the purpose of illustration. The data set \\textbf{colorectal} contains some information about the patient's ID number, covariates $X_1$ and $X_2$, with $X_1=1 (0)$ if patient received treatment C (S); $X_2$ consists of two dummy variables, with $X_2=(0,0),(1,0),(0,1)$ if the initial WHO performance status is $0,1,2$, respectively; the time (in years) of each occurrence of a new lesion since baseline measurement time; and the final right-censored or death time. There were $289$ occurrences of new lesions and $121$ patients died during the study. \n\nClearly, this data set is a special case of the type of data appropriate for our proposed joint model, having only one type of recurrent event, one absorbing health status state (dead), and one transient health status state (alive). We assume the effective age $\\mathcal{E}_i(s) = s - S_{iN_i^R(s-)}^R$ after each occurrence of a new lesion, and use $\\rho(k);\\alpha) =\\alpha^{\\log(1+k)}$. The unknown model parameters in the RCR (here, just a recurrent event) component of the model includes $\\alpha$ in the $\\rho(\\cdot)$ function, $\\beta^R=[\\beta^R_1,\\beta^R_2,\\beta^R_3]$ for the covariates, and $[\\kappa^R_1,\\kappa^R_2]$ for the LM state. The unknown model parameters in the LM component of the model are $\\beta^W$ for the covariates and $\\nu^W_1$ for the recurrent event process. The unknown parameters in the HS component of the model includes $\\beta^V$ for the covariates, $[\\kappa^V_1,\\kappa^V_2]$ for the LM state, and $\\nu^V_1$ for the recurrent event counting process.\n\nWe fitted the joint model to this data set. The resulting model parameter estimates along with the information-based standard error estimates are given in Table \\ref{truees}. The standard errors are obtained by taking square roots of the diagonal elements of the observed inverse of the profile likelihood information matrix. Based on these estimates, we could also perform hypothesis tests. Thus, for instance, the $p$-values associated with the two-tailed hypothesis tests are also given in the table. The null hypothesis being tested for $\\alpha$ is that $H_0: \\alpha=1$, while the null hypotheses for the other model parameters are that their true parameter values are zeros. We test $\\alpha=1$ instead of $\\alpha = 0$ because $\\alpha=1$ means that the accumulating number of recurrent event occurrences does not have an impact in subsequent recurrent event occurrences. From the values in Table \\ref{truees}, the estimate of $\\alpha$ is less than one, which may indicate that each occurrence of new lesion decreases the risk of future occurrences of new lesions, though from the result of the statistical test we cannot conclude statistically that $\\alpha < 1$. Based on the set of $p$-values, we find that the initial WHO performance state of $1$ and the tumor size state of $2$ are associated with decreased risk of new lesion occurrences, while an initial WHO performance state of $2$ is associated with an increased risk of new lesion occurrences. An initial WHO performance state of $2$ and the number of occurrences of new lesions are associated with an increased risk of death in the health status. \n\nFinally, we want to emphasize the importance of the effective age process. An inappropriate effective age may lead to misleading estimates. Parameter model estimates under a mis-specified effective age could lead to biases and potentially misleading conclusions. This is one aspect where domain specialists and statisticians need to consider when assessing the impact of interventions since the specification of the effective or virtual age have important and consequential implications. For more discussions about effective or virtual ages, see the recent papers \\cite{FinCha21,Beu2021}, the last one also touching on the situation where the virtual age function depends on unknown parameters.\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{cccc}\n \\hline\nParameter & Estimate & Est.\\ Standard Error & $p$-value \\\\ \n \\hline\n$\\alpha$ & 0.77 & 0.22 & 0.30 \\\\ \n $\\beta^R_1$ & -0.16 & 0.20 & 0.43 \\\\ \n $\\beta^R_2$ & -0.42 & 0.21 & 0.05 \\\\ \n $\\beta^R_3$ & 0.88 & 0.41 & 0.03 \\\\ \n $\\kappa^R_1$ & -0.52 & 0.27 & 0.05 \\\\ \n $\\kappa^R_2$ & -0.25 & 0.25 & 0.31 \\\\ \n \\hline\n $\\beta^W_1$ & 0.08 & 0.25 & 0.75 \\\\ \n $\\beta^W_2$ & -0.00 & 0.25 & 0.99 \\\\ \n $\\beta^W_3$ & -0.51 & 0.73 & 0.48 \\\\ \n $\\nu^W_1$ & -0.23 & 0.18 & 0.19 \\\\ \n \\hline\n $\\beta^V_1$ & 0.49 & 0.30 & 0.10 \\\\ \n $\\beta^V_2$ & -0.11 & 0.30 & 0.71 \\\\ \n $\\beta^V_3$ & 1.20 & 0.41 & 0.00 \\\\ \n $\\kappa^V_1$ & -0.43 & 0.31 & 0.16 \\\\ \n $\\kappa^V_2$ & -0.18 & 0.33 & 0.59 \\\\ \n $\\nu^V_1$ & 0.71 & 0.14 & 0.00 \\\\\n \\hline\n\\end{tabular}\n\\caption{Parameter estimates, information-based standard errors, and $p$-values for the RCR, LM, and HS processes based on the real data set. The $p$-value is based on the two-tailed hypothesis test that the model parameter is zero (except for the parameter $\\alpha$ where $H_0: \\alpha=1$). The top block includes the model parameters in the RCR component, the middle block includes the model parameters in the LM component, and the bottom block includes the model parameters in the HS component.}\n\\label{truees}\n\\end{table}\n\n\\section{Concluding Remarks}\n\\label{sec: conclusion}\n\nFor the general class of joint models for recurrent competing risks, longitudinal marker, and health status proposed in this paper, which encompasses many existing models considered previously, there are still numerous aspects that need to be addressed in future studies. Foremost among these aspects is a more refined analytical study of the finite-sample and asymptotic properties of the estimators of model parameters, together with other inferential and prediction procedures. The finite-sample and asymptotic results could be exploited to enable performing tests of hypothesis and construction of confidence regions for model parameters. There is also the interesting aspect of computationally estimating the standard errors of the estimators. How would a bootstrapping approach be implemented in this situation? Another important problem that needs to be addressed is how to perform goodness-of-fit and model validation for this joint model. Though the class of models is very general, there are still possibilities of model mis-specifications, such as, for example, in determining the effective age processes, or in the specification of the $\\rho_q(\\cdot)$-functions. What are the impacts of such model mis-specifications? Do they lead to serious biases that could potentially result in misleading conclusions? These are some of the problems whose solutions await further studies.\n\nA potential promise of this joint class of models is in precision medicine. Because all three components (RCR, LM, HS) are taken into account simultaneously, in contrast to a marginal modeling approach, the synergy that this joint model allows may improve decision-making -- for example, in determining interventions to be performed for individual units. In this context, it is of utmost importance to be able to predict in the future the trajectories of the HS process given information at a given point in time about all three processes. Thus, an important problem to be dealt with in future work is the problem of forecasting using this joint model. How should such forecasting be implemented? This further leads to other important questions. One is determining the relative importance of each of the components in this prediction problem. Could one ignore other components and still do as well relative to a joint model-based prediction approach? If there are many covariates, how should the important covariates among these numerous covariates be chosen in order to improve prediction of, say, the time-to-absorption?\n\nFinally, though our class of joint models is a natural extension of earlier models dealing with either recurrent events, competing recurrent events, longitudinal marker, and terminal events, one may impugn it as not realistic, but instead view it as more of a futuristic class of models, since existing data sets were not gathered in the manner for which these joint models apply. For instance, in the example pertaining to gout in Section \\ref{sec: scenarios}, the SUR level and CKD status are not continuously monitored. However, with the advent of smart devices, such as smart wrist watches, embedded sensors, black boxes, etc., made possible by miniaturized technology, high-speed computing, almost limitless cloud-based memory capacity, and availability of rich cloud-based databases, the era is, in our opinion, fast approaching when continuous monitoring of longitudinal markers, health status, occurrences of different types of recurrent events, be it on a human being, an experimental animal or plant, a machine such as an airplane or car, an engineering system, a business entity, etc. will become more of a standard rather than an exception. By developing the models and methods of analysis for such future complex and advanced data sets, {\\em even} before they become available and real, will hasten and prepare us for their eventual arrival.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\nSome materials in this work are based on L.\\ Tong's PhD dissertation research at the University of South Carolina.\nE.\\ Pe\\~na is currently Program Director in the Division of Mathematical Sciences (DMS) at the National Science Foundation (NSF). As a consequence of this position, he receives support for research, which included work in this manuscript, under NSF Grant 2049691 to the University of South Carolina. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF. P.\\ Liu acknowledges the 2017--2019 Summer Research Grants from Bentley University. E.\\ Pe\\~na acknowledges NSF Grant 2049691 and NIH Grant P30 GM103336-01A1.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}