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+{"text":"\\section{Introduction}\nOptimal investment with intermediate consumption and a stream of labor income (or liabilities) is one of the central problems in mathematical economics. \nIf borrowing against the future income is prohibited, the main  technical difficulty    lies in the fact that there are infinitely many constraints. Even in the deterministic case of no stocks and non-random income, a classical approach is based on the {\\it convexification of the constraints} that leads to a non-trivial dual problem formulated over decreasing nonnegative functions.\n\nBorrowing constraints imposed at all times  not only affect the  notion of admissibility, leading to more difficult mathematical analysis,  but also\nchange the meaning to fundamental concepts of mathematical finance such as replicability and completeness. The latter is   formulated via the attainability of every (bounded) contingent claim by a portfolio of traded  assets.   For a  labor income\/liability streams that pays off dynamically, \nthere is no a priori guarantee that such a replicating portfolio  (if it exists at all)  is admissible, i.e., satisfies the  constraints. Thus, in the terminology of \\cite{HePages}, even a  complete market becomes dynamically incomplete under the borrowing constraints. \n The  analysis of such a problem (in otherwise complete Brownian settings with a corresponding unique risk-neutral measure), is performed in \\cite{HePages} and later in \\cite{ElKarouiJeanblanc}. \nThe  nonnegative decreasing {\\it processes}   (that parametrize the dynamic incompleteness mentioned above)  play an important role in the characterizations of   optimal investment and consumption plans. The analysis  in \\cite{HePages} and \\cite{ElKarouiJeanblanc} is connected with optimal stopping techniques from~\\cite{Kar89}. \n\nIncomplete markets with no-borrowing constraints have been analyzed only in specific Markovian models  in \\cite{DuffieZarip1993} and \\cite{DuffieFlemingSonerZarip1997} based on partial differential equations techniques. \n The goal of the present paper is to study the problem of consumption and investment with no-borrowing constraints in general  (so, non-Markovian) incomplete   models. This leads to having, simultaneously, two layers of incompleteness. One comes from the many martingale measures, the other from a similar class of non-decreasing processes  (as above) that describe the dynamic incompleteness. \n We refer to \\cite{MalTrub07} for the examples of market incompleteness in finance and macroeconomics. \n\nIn contrast to \\cite{HePages} and \\cite{ElKarouiJeanblanc},   our model not only allows for incompleteness, but also for jumps. Mathematically, this means we choose to work in a general semimartingale framework.\n\nAs in  \\cite{HePages} and \\cite{ElKarouiJeanblanc}, \nour approach is based on duality.  One of the principal difficulties is in the construction of the dual feasible set and the dual value function. It is well-known that the martingale measures   drive the dual domain in many problems of mathematical finance. On the other hand, the convexification of constraints leads to the decreasing processes as the central dual object as well. We show that the dual elements in the incomplete case can be approximated by  products of the densities of martingale measures and such nonnegative decreasing processes. This is one of the primary results of this work, see section \\ref{secStructureOfDualDomain}, that leads to the complementary slackness characterization of optimal wealth in section \\ref{secSlackness}, where it is shown that the approximating sequence for the dual minimizer leads to a nonincreasing process, which decreases at most when the constraints are attained. In turn, the dual minimizer can be written as a product of such a nondecreasing process and an optional strong supermartingale deflator. In the case of  complete Brownian markets  a similar result is proved in \\cite{HePages} and   \\cite{ElKarouiJeanblanc}.\n\nIn order to implement the approach, we {\\it increase the dimensionality} of the problem and treat as arguments of the indirect utility not only the initial wealth, but also the function that specifies the number of units of labor income (or  the  stream of liabilities)  at any later time. This parametrization has the spirit of \\cite{HK04}, however unlike \\cite{HK04}, we go into (infinite-dimensional) non-reflexive spaces, which gives both novelty and technical difficulties to our analysis.\n Also, our formulation permits to price by marginal rate of substitution the whole  labor income process. This is done \n\nthrough  the subdifferentiability results in section \\ref{secSubdifferentiability}.  Note that  the sub differential elements (prices) are time-dependent, so infinite-dimensional, unlike in  \\cite{HK04}.\n\n\nAnother contribution of the paper lies in the {\\it unified framework of admissibility} outlined in section \\ref{secModel}.  More precisely, we assume that no-borrowing constraints are imposed  starting from  some pre-specified stopping time and hold up to the terminal time horizon.  \nThis framework allows us to treat in one formulation both the problem of no-borrowing constraints at all times (described above) and the one where borrowing against the future income is permitted  with a constraint only at the end. The latter is well-studied in the literature, see \\cite{CvitanicSchachermayerWang}, \\cite{Karatzas-Zitkovic-2003}, \\cite{HK04}, \\cite{Zitkovic}, and \\cite{MostovyiRandEnd}.\n In such a formulation, the constraints reduce to a single inequality and the decreasing processes in the dual feasible set become constants.  \n \nAmong the many possibilities of constraints, \\cite{Cuoco1997}  considers the problem of investment and consumption with labor income and no-borrowing constraints in Brownian market, even allowing for incompleteness. The dual problem cannot be solved directly (in part because for these constraints the dual space considered is too small) but the primal can be solved with direct methods. An approximate dual sequence can then be recovered from the primal. We generalize \\cite{HePages} and \\cite{ElKarouiJeanblanc} (complete Brownian markets) and \\cite{Cuoco1997} (possibly incomplete Brownian markets) to the case of general semi-martingale incomplete markets. Our dual approach allows us, at the same time to obtain a dual characterization (complementary slackness) of the optimal consumption plan (not present in \\cite{Cuoco1997}), similar to the complete case in \\cite{HePages} and \\cite{ElKarouiJeanblanc} and the possibility to study the dependence on labor income streams, through the parametrization of such streams.\n \n\n\n\nEmbedding path dependent problems into the convex duality framework have been analyzed in \\cite{XiangYuHabit}, \\cite{HaoMat}, \\cite[Section 3.3]{Pham01}, whereas without duality but with random endowment it is considered in \\cite{Miklos2018}, in the abstract singular control setting the duality approach is investigated in \\cite{BankKaupila}.\nOur embedding does not require any condition on labor income {replicability}, which becomes highly technical in the presence of extra admissibility constraints. Even in the case where the only constraint is imposed at maturity, in this part our approach differs from the one in \\cite{HK04}, where non-replicability of the endowment (in the appropriate sense) is used in the proofs as it ensures that the effective domain of the dual problem has the same dimensionality as the primal domain. Note that even if \n the  labor income is spanned by the same sources of randomness as the stocks, the idea of replicating the labor income and then reducing the problem to the one without it, does not necessarily work under the borrowing constraints, see the discussion in \\cite[pp. 671-673]{HePages}. \n\n\n\n\n\n\n\n Some of the more specific   technical contributions  of this paper can be summarized as follows:\n\n\\begin{itemize}\n\\item We  analyze the {\\it boundary behavior} of the value functions.  Note that the value functions are defined over infinite-dimensional spaces.  \n\n\n\\item The finiteness of the indirect utilities {\\it without} labor income is imposed  only, as a necessary and sufficient condition that allows for the standard conclusions of the utility maximization theory, see \\cite{MostovyiNec}.\n\n\n\\item We show existence-uniqueness results for the {\\it unbounded} labor income both from above and below.\n\n\n\n\\item We observe that the ``Snell envelope proposition'' \\cite[Proposition 4.3]{K96} can be extended to \n the envelope over all stopping times that exceed a  given initial stopping time $\\theta _0$.\n\\item We represent the dual value function in terms of {\\it uniformly integrable} densities of martingale measures, i.e., the densities of martingale measures under which the maximal wealth process of a self-financing portfolio that superreplicates the labor income, is a uniformly integrable martingale, see Lemma \\ref{finitenessOverZ'} below. \n\n\\end{itemize}\n\n\n\n{\\bf Organization of the paper}. In Section \\ref{secModel}, we specify the model. We state and prove existence, uniqueness, semicontinuity and biconjugacy results in Section \\ref{secProofs}, subdifferentiability is proven in Section \\ref{secSubdifferentiability}. Structure of the dual domain is analyzed in Section \\ref{secStructureOfDualDomain} and complimentary slackness is established in Section \\ref{secSlackness}.\n\\section{Model}\\label{secModel}\n\nWe consider a {financial market model} with finite time horizon\n$[0,T]$ and a zero interest rate. The price process $S=(S^i)_{i=1}^d$\nof the stocks is assumed to be a\nsemimartingale on a complete stochastic basis\n$\\left( \\Omega, \\mathcal F, \\left(\\mathcal F_t \\right)_{t\\in[0,T]}, \\mathbb\nP\\right)$, where $\\mathcal F_0$ is trivial. \n\nLet $(e)_{t\\in[0,T]}$ be an optional process that specifies the labor income rate, which is assumed to follow a certain stochastic clock, that we specify below.\nBoth processes $S$ and $e$ are given exogenously. \n\nWe define a \\textit{stochastic clock} as a\nnondecreasing, c\\`adl\\`ag, adapted process such that\n\\begin{equation}\n \\tag{finClock}\n  \\label{finClock}\n\\kappa_0 = 0, ~~ \\mathbb P\\left[\\kappa_T>0 \\right]>0,\\text{ and } \\kappa_T\\leq A\n\\end{equation}\nfor some finite constant $A$. We note that the stochastic clock allows to include multiple standard formulations of the utility maximization problem in one formulation, see e.g., \\cite[Example 2.5 - 2.9]{MostovyiNec}.\nLet us define \n\\begin{equation}\\label{defK}\nK_t {:=} \\mathbb E\\left[\\kappa_t \\right],\\quad t\\in[0,T].\n\\end{equation}\n\\begin{Remark}\nThe function $K$ defined in \\eqref{defK} is right-continuous with left limits and takes values in $[0,A]$. \n\\end{Remark}\nWe assume the income and consumption are given in terms of the clock $\\kappa$.\nDefine a {portfolio} $\\Pi$ as a quadruple $(x, q, H, c),$ where the\nconstant $x$ is the initial value of the portfolio, the function $q:[0,T]\\to\\mathbb R$ is a bounded and Borel measurable function,\n which \nspecifies the amount of labor income rate, $H =\n(H_i)_{i=1}^d$ is a predictable $S$-integrable process that\ncorresponds to the amount of each stock in the portfolio, and\n$c=(c_t)_{t\\in[0,T]}$ is the consumption rate, which we assume to be\n optional and nonnegative.\n\nThe \\textit{wealth process} $V=(V_t)_{t\\in[0,T]}$ {generated by the}\nportfolio is \n\\begin{equation}\\nonumber\nV_t = x + \\int_0^t H_sdS_s + \\int_0^t\\left(q_se_s-c_s\\right)d\\kappa_s,\\quad t\\in[0,T].\n\\end{equation}\nA portfolio $\\Pi$ with $c\\equiv 0$ and $q \\equiv 0$ is called\n\\textit{self-financing}. The collection of nonnegative wealth\nprocesses {generated by} self-financing portfolios with initial value $x\\geq 0$ is\ndenoted by $\\mathcal X(x)$, i.e.\n\\begin{equation}\\nonumbe\n\\mathcal X(x) {:=} \\left\\{X\\geq 0:~X_t=x + \\int_0^t\nH_sdS_s,~~t\\in[0,T] \\right\\},~~x\\geq 0.\n\\end{equation}\nA probability measure $\\mathbb Q$ is an \\textit{equivalent local\nmartingale measure} if $\\mathbb Q$ is equivalent to $\\mathbb P$ and\nevery $X\\in\\mathcal X(1)$ is a local martingale under $\\mathbb Q$. We\ndenote the family of equivalent local martingale measures by\n$\\mathcal M$ and assume that\n\\begin{equation}\\tag{noArb}\\label{NFLVR}\n\\mathcal M \\neq \\emptyset.\n\\end{equation}\nThis condition is  equivalent to the absence of arbitrage\nopportunities \n{in} the marke\n, see\n \\cite{DS, DS1998} as well as \n\\cite{KarKar07} for the exact statements and further\nreferences.\n\nTo rule out doubling strategies in the presence of random endowment,\nwe need to impose additional restrictions. Following  \\cite{DS97}, we say that a\nnonnegative process in $\\mathcal X(x)$ is \\textit{maximal} if its\nterminal value cannot be dominated by that of any other process in\n$\\mathcal X(x)$.\nAs in \\cite{DS97}, we define an\n\\textit{acceptable} process to be a process of the form $X = X' -\nX'',$ where $X'$ is a nonnegative wealth process {generated by} a self-financing portfolio and $X''$ is maximal.\n\n Our unified framework of admissibility is given by a  fixed stopping time $\\theta_0$. The no-borrowing constraints will hold starting at this stopping time until the end.   Let $\\Theta$ be the set of stopping times that are greater or equal than  $\\theta_0$. \n\\begin{Lemma}\\label{3311}\nLet $q^1$ and $q^2$ be bounded, Borel measurable functions on $[0,T]$, such that $q^1= q^2$, $dK$-a.e. Then, the cumulative labor income processes $\\int_0^{\\cdot}q^i_se_sd\\kappa_s$, $i = 1,2$,  are indistinguishable. \n\\end{Lemma}  The proof of this lemma is given in section \\ref{subSecCharDualDomain}.\nFollowing\n \\cite{HK04}, we denote by\n$\\mathcal X(x,q)$ the set of acceptable processes with initial {values}\n$x$, that dominate the labor income on $\\Theta$:\n\\begin{equation}\\nonumbe\n\\begin{array}{rcl}\n\\mathcal X(x, q) &{:=}& \\left\\{ \n{\\rm acceptable}~X: X_0 = x~{\\rm and}\\right.\\\\\n&&\\left.~ X_{\\tau} + \\int_0^{\\tau}q_se_sd\\kappa_s \\geq 0, ~\\mathbb P-a.s.~{\\rm for~every~}\\tau\\in\\Theta\n\\right\\}.\\\\\n\\end{array}\n\\end{equation}\nLet us set\n\\begin{equation}\\nonumbe\n\\mathcal K {:=} \\left\\{(x,q): \\mathcal X(x,q) \\neq \\emptyset \\right\\},\n\\end{equation}\nLet $\\mathring{\\mathcal K}$ denote the interior or $\\mathcal K$ in the $\\mathbb R\\times \\mathbb L^\\infty(dK)$-norm topology.\nWe characterize $\\mathcal K$ in Lemma~\\ref{lemma1} below that in particular asserts that under Assumption \\ref{asEnd}, $\\mathring{\\mathcal K}\\neq \\emptyset$. \n  \nThe set of admissible consumptions is defined as\n\\begin{equation}\\nonumbe\n\\begin{array}{rcl}\n\\mathcal A(x,q)&{:=} &\\left\\{ \noptional ~c\\geq 0:~~there~exists~X\\in\\mathcal X(x,q), ~such~that~ \\right.\\\\\n&& \n\\left.\\int_0^{\\tau}c_sd\\kappa_s\\leq X_{\\tau} + \\int_0^{\\tau}q_se_sd\\kappa_s,\n~~for~every~\\tau\\in\\Theta\\right\\},\\quad (x,q)\\in\\mathcal K.\\\\ \n\\end{array}\n\\end{equation}\nNote that $c\\equiv 0$ belongs to $\\mathcal A(x,q)$ for every $(x,q)\\in\\mathcal K$. \n\\begin{Remark}\\label{Remark-Mihai} The no-borrowing constraints can also  be written  as\n$$\n\\mathbb{P}\\left (\\int_0^{t}c_sd\\kappa_s\\leq X_{t} + \\int_0^{t}q_se_sd\\kappa_s,\n~~for~every~ \\theta _0\\leq t\\leq T \\right)=1.$$\nWe write the constraints in terms of stopping times $\\tau \\in \\Theta$ as \nwe use the stopping times $\\tau \\in \\Theta$ (and the corresponding decreasing processes that jump from one to zero at these  times) as  the building blocks of our analysis. \n\\end{Remark}\n\\begin{Remark}\\label{3312}\nIt follows from Lemma \\ref{3311}, for every $x\\in\\mathbb R$,  we have\n$$\\mathcal X(x,q^1) = \\mathcal X(x,q^2)\\quad and \\quad \\mathcal A(x,q^1) = \\mathcal A(x, q^2),$$\nwhere some of these sets might be empty\n\\end{Remark}  \n\nHereafter, we shall impose the following conditions on the  endowment process.\n\\begin{Assumption}\\label{asEnd}\nThere exists a maximal wealth process $X'$ such that\n\\begin{equation}\\nonumbe\nX'_t \\geq |e_t|, \\quad for~every~t\\in[0,T],\\quad\\Pas.\n\\end{equation}\nMoreover, Assumption \\ref{asEnd} and \\eqref{NFLVR} imply that all the assertions of Lemma \\ref{lemma1} hold.\n\\end{Assumption}\n\n\\begin{Remark}\nIf $\\theta _0 = \\{T\\}$, then Assumption~\\ref{asEnd} is equivalent to the assumptions on endowment in~\\cite{HK04} (for the case of one-dimensional random endowment).\n\\end{Remark}\n\n\n\nThe preferences of an economic agent are modeled with a\n\\textit{utility stochastic field}\n$U=U(t, \\omega, x):[0,T]\\times\\Omega\\times[0,\\infty)\\to\\mathbb R\\cup \\{-\\infty\\}$.\nWe assume that $U$\nsatisfies the conditions below.\n\\begin{Assumption}\\label{assumptionOnU}\n \nFor every $(t, \\omega)\\in[0, T]\\times\\Omega$, the function $x\\to\n  U(t, \\omega, x)$ is strictly concave, increasing, continuously\n  differentiable on $(0,\\infty)$ and satisfies the Inada conditions:\n  \\begin{equation}\\nonumbe\n    \\lim\\limits_{x\\downarrow 0}U'(t, \\omega, x) =\n    \\infty \\quad \\text{and} \\quad \\lim\\limits_{x\\to\n      \\infty}U'(t, \\omega, x) = 0,\n  \\end{equation}\n  where $U'$ denotes the partial derivative with respect to the third argument.\n At $x=0$ we suppose, by continuity, $U(t, \\omega, 0) = \\lim\\limits_{x \\downarrow 0}U(t, \\omega, x)$, {which}\n may be $-\\infty$.\nFor every\n  $x\\geq 0$ the stochastic process $U\\left( \\cdot, \\cdot, x \\right)$ is\n  optional. Below, following the standard convention, we will not write $\\omega$ in $U$.\n\\end{Assumption}\n\n\n\nThe agent can control investment and consumption. {The goal is to maximize}\n expected utility. The value function $u$ is\ndefined as:\n\\begin{equation}\\label{primalProblem}\nu(x, q) {:=} \\sup\\limits_{c\\in\\mathcal A(x, q)}\\mathbb\nE\\left[\\int_0^T U(t, c_t)d\\kappa_t\\right], \\quad\n(x, q)\\in\\mathcal K.\n\\end{equation}\nIn \\eqref{primalProblem}, we use the convention\n\\begin{displaymath}\n  \\mathbb{E}\\left[\n    \\int_0^TU(t,c_t)d\\kappa_t \\right] {:=} -\\infty\n  \\quad \\text{if} \\quad \\mathbb{E}\\left[ \\int_0^TU^{-}(t,c_t)d\\kappa_t\n  \\right]= \\infty.\n\\end{displaymath}\nHere and below, $W^{-}$ and $W^{+}$ denote the negative and positive parts of a\nstochastic field $W$, respectively.\n\n\n We employ duality techniques to obtain the standard conclusions of the utility maximization theory.  We first  define the convex conjugate stochastic field\n\n\\begin{equation}\\label{defV}\nV(t, y) {:=} \\sup\\limits_{x>0}\\left(\nU(t, x)-xy\\right),\\quad (t, y)\\in[0,T]\\times [0,\\infty),\n\\end{equation}\n and then  observe that \n$-V$ satisfies Assumption \\ref{assumptionOnU}.\nIn order to construct the feasible set of the dual problem, we define the set\n$\\mathcal L$ as the  polar cone of $-\\mathcal K$:\n\\begin{equation}\\label{defL}\n\\mathcal L {:=} \\left\\{(y,r)\\in\\mathbb R\\times\\mathbb L^1(dK): xy + \\int_0^Tq_sr_sdK_s \\geq\n0{\\rm~for~every~}(x,q)\\in\\mathcal K\\right\\}.\n\\end{equation}\n\\begin{Remark}\nUnder the conditions \\eqref{finClock}, \\eqref{NFLVR} and Assumption \\eqref{asEnd}, the set \n$\\mathcal L$ is non-empty.  By definition,  it is closed in $\\mathbb R\\times\\mathbb L^1(d K)$-norm and $\\sigma(\\mathbb R\\times\\mathbb L^1(dK), \\mathbb R\\times\\mathbb L^{\\infty}(dK))$ topologies. Also,  as shown later, the set $\\mathcal L = (-\\mathring{\\mathcal K})^o$, i.e. the polar of $-\\mathring{\\mathcal K}$.\n\\end{Remark}\nBy ${\\mathcal Z}$, we denote the set of c\\`adl\\`ag densities of\nequivalent local martingale measures:\n\\begin{equation}\\label{defZ}\n{\\mathcal Z} {:=} \\left\\{{\\rm c\\grave adl\\grave ag}~\\left(\\frac{d\\mathbb Q_t}{d\\mathbb P_t}\\right)_{t\\in[0,T]}:\\quad \\mathbb Q\\in\\mathcal M\\right\\}.\n\\end{equation}\nLet us denote by $\\mathbb L^0 = \\mathbb L^0(d\\kappa \\times \\mathbb P)$ the linear space of (equivalence classes of) real-valued optional processes on the stochastic basis $(\\Omega, \\mathcal F, (\\mathcal F_t)_{t\\in[0,T]}, \\mathbb P)$ which we equip with the topology  of convergence in measure $(d\\kappa \\times \\mathbb P)$. \nFor each $y \\geq 0$ we define\n\\begin{equation}\\label{oldY}\n\\begin{array}{c}\n \\hspace{-15mm}\\mathcal Y(y) {:=} {\\cl}\\left\\{ Y: ~Y{\\rm~is~c\\grave{a}dl\\grave{a}g~adapted~and~}\\right. \\\\\n \\hspace{35mm}\\left.0\\leq Y\\leq yZ ~\\left(d\\kappa\\times\\mathbb\nP\\right){\\rm\n~a.e.~for~some~}Z\\in{\\mathcal Z}\\right\\},\\\\\n\\end{array}\n\\end{equation}\nwhere the closure is taken in\n$\\mathbb L^0$. \nNow we are ready to set the\ndomain of the dual problem:\n\\begin{equation}\\label{defY}\n\\begin{array}{rcl}\n\\mathcal Y(y,r)&{:=} &\\left\\{\n Y: Y \\in\\mathcal Y(y)\n~{\\rm and}\\right.\\\\\n&&\n\\quad\\mathbb E\\left[\\int_0^Tc_sY_sd\\kappa_s \\right]\\leq xy + \\int_0^Tq_sr_sdK_s,\\\\\n&&\\left.\\quad{\\rm for~every}~(x,q)\\in\\mathcal K {\\rm ~and~}c\\in\\mathcal A(x,q)\\right\\}\\\\\n\\end{array}\n\\end{equation}\nNote that the definition \\eqref{defY}  requires that every element of $\\mathcal Y(y,r)$ is in $\\mathcal Y(y)$, $y\\geq 0$. Also, for every $(y,r)\\in\\mathcal L$, $\\mathcal Y(y,r)\\neq \\emptyset$, since $0\\in\\mathcal Y(y,r)$. \n\n We can now  state the dual optimization problem:\n\\begin{equation}\\label{dualProblem}\nv(y, r) {:=} \\inf\\limits_{Y\\in\\mathcal Y(y, r)}\\mathbb\nE\\left[\\int_0^T V(t, Y_t)d\\kappa_t\\right],\\quad(y, r)\\in\\mathcal L,\n\\end{equation}\nwhere we use the convention:\n\\begin{displaymath}\n  \\mathbb{E}\\left[ \\int_0^TV(t,Y_t\n    )d\\kappa_t \\right] {:=} \\infty\n  \\quad \\text{if} \\quad \\mathbb{E}\\left[ \\int_0^TV^{+}(t, Y_t )d\\kappa_t\n  \\right] = \\infty.\n\\end{displaymath}\nAlso, we set \n\\begin{equation}\\label{3313}\nv(y,r) {:=} \\infty~for~(y,r)\\in\\mathbb R\\times \\mathbb L^1(dK)\\backslash\n \\mathcal L\\quad and \\quad u(x,q){:=} -\\infty~for~(x,q)\\in\\mathbb R\\times \\mathbb L^{\\infty}(dK)\\backslash\\mathcal K.\\\\ \n\\end{equation}\nWith this definition, it will be shown below in Theorem  \\ref {mainTheorem} that   $u<\\infty$ and $v>-\\infty$ everywhere,  so  $u$ and $v$ are proper   functions  in the language of convex analysis.  \nLet us recall that in the absence of random endowment, the dual value function is defined as\n\\begin{equation}\\nonumbe\n\\tilde w(y) {:=} \\inf\\limits_{Y\\in\\mathcal Y(y)}\\mathbb E\\left[\\int_0^TV(t, Y_s)d\\kappa_s \\right],\\quad y>0,\n\\end{equation}\nwhereas the primal value function is given by\n\\begin{equation}\\label{defw}\nw(x) {:=} u(x,0),\\quad x>0.\n\\end{equation}\n\\section{Existence, uniqueness, and biconjugacy}\\label{secProofs}\n\\begin{Theorem}\\label{mainTheorem}\nLet (\\ref{finClock}) and (\\ref{NFLVR}), Assumptions~\\ref{asEnd}\nand \\ref{assumptionOnU} hold true and\n\\begin{equation}\\tag{finValue}\\label{finValue}\nw(x) > -\\infty\\quad for~every~x>0\\quad and \\quad \n \\widetilde w(y) <\\infty\\quad for~every~y>0.\n \\end{equation}\n Then we have:\n\n$(i)$ $u$ is finite-valued on $\\mathring{\\mathcal K}$ and $u<\\infty$ on $\\mathbb R\\times \\mathbb L^\\infty(dK)$.  The dual value function $v$ satisfies  $v>-\\infty$ on $\\mathbb R\\times\\mathbb L^1(dK)$,  and  the set $\\{v<\\infty\\}$ is a nonempty convex subset of $\\mathcal L$, whose closure in $\\mathbb R\\times\\mathbb L^1(dK)$ equals to $\\mathcal L$.  \n\n$(ii)$ $u$ is concave, proper, and upper semicontinuous with respect to the  norm-topology of $\\mathbb R\\times\\mathbb L^\\infty(dK)$ and the  weak-star topology $\\sigma (\\mathbb R\\times\\mathbb L^\\infty(dK), \\mathbb R\\times\\mathbb L^1(dK))$. For every $(x,q)\\in \\{u>-\\infty\\}$, there exists a unique solution to (\\ref{primalProblem}). Likewise, $v$ is convex, proper, and lower semicontinuous with respect to the norm-topology of $\\mathbb R\\times \\mathbb L^1(dK)$ and the the weak topology  $\\sigma(\\mathbb R\\times\\mathbb L^1(dK),\\mathbb R\\times\\mathbb L^{\\infty}(dK))$. For every $(y,r)\\in \\{v<\\infty\\}$, there exists a unique solution to (\\ref{dualProblem}). \n\n$(iii)$ The functions $u$ and $v$ satisfy the biconjugacy relations\n\\begin{equation}\\nonumbe\n\\begin{array}{rcll}\nu(x,q) &=& \\inf\\limits_{(y,r)\\in \\mathcal L}\\left(v(y,r) + xy + \\int_0^Tr_sq_sdK_s\\right),& (x,q)\\in \\mathcal K,\\\\\nv(y,r) &=& \\sup\\limits_{(x,q)\\in \\mathcal K}\\left(u(x,q) - xy - \\int_0^Tr_sq_sdK_s\\right),& (y,r)\\in \\mathcal L.\\\\\n\\end{array}\n\\end{equation}\n\\end{Theorem}\n\n\n\n\\begin{Lemma}\\label{lemma1}\nLet (\\ref{NFLVR}) and Assumption~\\ref{asEnd} hold. Then we have:\n\n$(i)$ for every $x>0$, $(x,0)$ belongs to $\\mathring{\\mathcal K}$ (in particular, $\\mathring{\\mathcal K}\\neq \\emptyset$),\n\n$(ii)$ for every $q\\in\\mathbb L^\\infty(dK)$, there exists $x>0$ such that $(x,q)\\in\\mathring{\\mathcal K}$,\n\n$(iii)$ $\\sup\\limits_{\\mathbb Q\\in\\mathcal M}\\mathbb {E^Q}\\left[\\int_0^T|e_s|d\\kappa_s \\right] <\\infty$,\n\n$(iv)$ there exists a nonnegative maximal wealth process $X^{''}$, such that\n\\begin{displaymath}\nX^{''}_T \\geq \\int_0^T|e_s|d\\kappa_s,\\quad \\Pas,\n\\end{displaymath}\n\n$(v)$ there exists a nonnegative maximal wealth process $X^{''}$, such that\n\\begin{equation}\\nonumbe\nX^{''}_t \\geq \\int_0^t|e_s|d\\kappa_\ns,\\quad t\\in[0,T],~\\mathbb P-a.s.\n\\end{equation}\n\n\\end{Lemma}\n\\begin{proof}\nFirst, via \\cite[Proposition I.4.49]{JS} and \\eqref{finClock}, we get\n\\begin{equation}\\nonumber\n\\begin{array}{c}\n\\int_0^T |e_s|d\\kappa_s \\leq \\int_0^T X'_sd\\kappa_s = -\\int_0^T\\kappa_{s-}dX'_s + \\kappa_TX'_T \\\\ \\leq -\\int_0^T\\kappa_{s-}dX'_s +AX'_T =AX'_0 + \\int_0^T(A - \\kappa_{s-})dX'_s.\\\\\n\\end{array}\n\\end{equation}\nTherefore, the exists a self-financing wealth process $\\bar X$, such that\n$$\\int_0^T |e_s|d\\kappa_s \\leq \\bar X_T.$$\nConsequently, \\cite[Theorem 2.3]{DS97} asserts the existence of a nonnegative {\\it maximal} process $X^{''}$, such that\n$$\\int_0^T |e_s|d\\kappa_s \\leq  X_T^{''},$$\ni.e.,  $(iv)$ holds. Therefore $(iii)$ is valid as well, by \\cite[Theorem 5.12]{DS1998}. \n\nRelation $(iv)$ and \\eqref{NFLVR} imply $(v)$.\nTo prove $(i)$ and $(ii)$, without loss of generality, we will suppose that in $(v)$, $X^{''}_0> 0.$ Let $q\\in\\mathbb L^{\\infty}(dK)$ and $\\varepsilon >0$ be fixed.  Let us define\n\\begin{equation}\\nonumbe\nx(q){:=} \\left(||q||_{\\mathbb L^{\\infty}(dK)} + 2\\varepsilon\\right)X^{''}_0.\n\\end{equation}\nWe claim that $(x(q), q)\\in\\mathring{\\mathcal K}.$\nLet \nus consider arbitrary \n\\begin{equation}\\label{454}\n|x'|\\leq \\varepsilon\\quad {\\rm and} \\quad q'\\in\\mathbb L^\\infty(dK)\n:~||q'||_{\\mathbb L^{\\infty}(dK)}\\leq {\\varepsilon},\n\\end{equation}  \nand set\n\\begin{equation}\\label{455}\n\\tilde X_t {:=} \\left(||q||_{\\mathbb L^{\\infty}(dK)} + 2\\varepsilon+x'\\right)X^{''}_t,\\quad t\\in[0,T].\n\\end{equation} Then, by item $(v)$, we have\n\\begin{displaymath}\n\\begin{array}{c}\n\\tilde X_t\\geq \\left(||q||_{\\mathbb L^{\\infty}(dK)}+{\\varepsilon}\\right)X^{''}_t\\geq \\left(||q||_{\\mathbb L^{\\infty}(dK)}+{\\varepsilon}\\right)\\int_0^t|e_s|d\\kappa_s\\geq -\\int_0^t(q_s + q'_s)|e_s|d\\kappa_s.\n\\end{array}\n\\end{displaymath}\nWe deduce that $\\tilde X\\in\\mathcal X(\\left(||q||_{\\mathbb L^{\\infty}(dK)} + 2\\varepsilon+x'\\right)X^{''}_0, q+q')$. In particular,  $$\\mathcal X(\\left(||q||_{\\mathbb L^{\\infty}(dK)} + 2\\varepsilon+x'\\right)X^{''}_0, q+q')\\neq \\emptyset.$$ As $x'$ and $q'$ are arbitrary elements satisfying \\eqref{454}, we deduce that $(x(q), q)\\in\\mathring{\\mathcal K}$. This proves $(ii)$. \n\n\nIn order to show $(i)$, first we observe that $\\mathring{\\mathcal K}$ is a convex cone. Therefore, it suffices to prove that, for a given $\\varepsilon>0$, we have\n\\begin{equation}\\label{2121}\n\\left(2\\varepsilon X^{''}_0, 0\\right)\\in\\mathring{\\mathcal K}.\n\\end{equation} \nAgain, let us consider $x'$ and $q'$ satisfying \\eqref{454} and $\\tilde X$ satisfying \\eqref{455} for $q\\equiv 0$. \nThen for every $t\\in[0,T]$, we have\n$$\\tilde X_t\\geq {\\varepsilon}X^{''}_t\\geq {\\varepsilon}\\int_0^t|e_s|d\\kappa_s\\geq -\\int_0^tq'_s|e_s|d\\kappa_s.$$\nThus, $\\mathcal X(\\left(2\\varepsilon+x'\\right)X^{''}_0, q')\\neq \\emptyset$.  Consequently, as $x'$ and $q'$ are arbitrary elements satisfying \\eqref{454}, \\eqref{2121} holds and so is $(i)$. \n This completes the proof of the lemma. \n\\end{proof}\n\\begin{Remark}\nA close look at the proofs shows that the conclusions of Theorem \\ref{mainTheorem} also hold if instead of Assumption \\ref{asEnd}, we impose any of the equivalent assertions $(iii) - (v)$ of Lemma \\ref{lemma1}.\n\\end{Remark}\n\n\n\\subsection{Characterization of the primal and dual domains}\\label{subSecCharDualDomain}\nThe polar, $A^{o}$, of a nonempty subset $A$ of $\\mathbb R\\times\\mathbb L^\\infty(dK)$, is\n the subset of\n $\\mathbb R\\times\\mathbb L^1(dK)$, defined by\n$$A^o{:=} \\left\\{(y,r)\\in \\mathbb R\\times \\mathbb L^1(dK):~xy + \\int_0^T q_sr_sdK_s \\leq 1, \\quad for~every~(x,q)\\in A\\right\\}.$$\nThe polar of a subset of $\\mathbb R\\times\\mathbb L^1(dK)$ is defined similarly.\n\\begin{Proposition}\\label{prop1} Under Assumption \\ref{asEnd} and conditions  \\eqref{finClock} and \\eqref{NFLVR}, we have:\n\n$(i)$ Let $(x,q)\\in\\mathbb R\\times\\mathbb L^{\\infty}(dK)$. Then $c\\in \\mathcal A(x,q)$ (thus, $\\mathcal A(x,q)\\neq \\emptyset$ so $(x,q)\\in \\mathcal K$ )  if and only if\n\\begin{equation}\\nonumber\n\\mathbb E\\left[\\int_0^T c_sY_sd\\kappa_s \\right]\\leq xy + \\int_0^Tq_sr_sdK_s,\n~~ {\\rm for~every}~(y,r)\\in \\mathcal L{\\rm ~ and }~ Y\\in\\mathcal Y(y,r).\n\\end{equation}\n\n$(ii)$ Likewise, for $(y,r)\\in \\mathbb{R}\\times \\mathbb{L}^1 (dK)$ we have  $(y,r)\\in \\mathcal L$ and $Y\\in \\mathcal Y(y,r)$ if and only if\n\\begin{equation}\\nonumber\n\\mathbb E\\left[\\int_0^T c_sY_sd\\kappa_s \\right]\\leq xy + \\int_0^Tq_sr_sdK_s,\n~~ {\\rm for~every}~(x,q)\\in\\mathcal K{\\rm ~ and }~c\\in\\mathcal A(x,q).\n\\end{equation}\n\nWe also have $\\mathcal K = (-\\mathcal L)^{o} = \\cl{\\mathring{\\mathcal K}}$, where the closure is taken both in norm $\\mathbb R\\times \\mathbb L^1(dK)$ and $\\sigma(\\mathbb R\\times\\mathbb L^\\infty(dK), \\mathbb R\\times\\mathbb L^1(dK))$ topologies.\n\\end{Proposition}\n\\begin{Remark}\\label{remNonEmpty}\nIt follows from Proposition \\ref{prop1} that \nfor every $(x,q)\\in\\mathcal K, \\mathcal A(x,q)\\neq \\emptyset$ as $0\\in\\mathcal A(x,q)$. Likewise, \nfor every $(y,r)\\in \\mathcal L, \\mathcal Y(y,r)\\neq \\emptyset$ as $0\\in\\mathcal Y(y,r)$. Moreover, for every $(x,q)\\in  \\mathring{\\mathcal K}$, each of the sets $\\mathcal A(x,q)$ and $\\bigcup\\limits_{(y,r)\\in \\mathcal L: ~xy + \\int_0^Tr_sq_sdK_s \\leq 1}\\mathcal Y(y,r)$ contain a strictly positive element, see Lemma \\ref{6-12-1} below. \n\\end{Remark}\nThe proof of Proposition~\\ref{prop1} will be given via several lemmas.\nLet \n\\begin{equation}\\nonumbe\n\\begin{array}{l}\n\\mathcal M'~be~the~set~of~equivalent~local~martingale~measures,~under~which\\\\X^{''}~(from~Lemma~\\ref{lemma1},~item~(v))~is~a~uniformly~integrable~martingale. \n\\end{array}\n\\end{equation}\nNote that by \\cite[Theorem 5.2]{DS97}, $\\mathcal M'$ is a nonempty, convex subset of $\\mathcal M$, which is also dense in $\\mathcal M$ in the total variation norm. \n\n\\begin{Remark}\nEven though the results in \\cite{DS97} are obtained under the condition that $S$ is a locally bounded process, they also hold without local boundedness assumption, see the discussion in \\cite[Remark 3.4]{HKS05}.\n\\end{Remark}\n\nLet $Z'$ denote the set of the corresponding c\\`adl\\`ag densities,~i.e., \n\\begin{equation}\\label{defZ'}\n\\begin{array}{rcl}\n\\mathcal Z' &{:=}& \\left\\{{\\rm c\\grave adl\\grave ag}~Z: Z_t=\\mathbb E\\left[ \\frac{d\\mathbb Q}{d\\mathbb P}|\\mathcal F_t\\right],~t\\in[0,T],~\\mathbb Q\\in\\mathcal M'\\right\\}.\\\\\n\\end{array}\n\\end{equation}\n We also set\n\\begin{equation}\\label{defUpsilon}\n\\Upsilon {:=} \\left\\{1_{[0,\\tau]}(t),t\\in[0,T]:~~\\tau\\in\\Theta\\right\\}.\n\\end{equation}\n\n\\begin{Lemma}\\label{6-16-1}\nLet   the conditions of Proposition \\ref{prop1} hold, $\\mathbb Q\\in\\mathcal M'$, ${Z} = {Z}^{\\mathbb Q}$ be the corresponding element of $\\mathcal Z'$,  and $\\Lambda \\in \\Upsilon$.  Then  there exists $r\\in \\mathbb L^1(dK)$, uniquely defined~by\n\\begin{equation}\\label{6-25-1}\n\\int_0^tr_sdK_s = \\mathbb E\\left[\\int_0^t\\Lambda_s{Z}_se_sd\\kappa_s \\right],\\quad t\\in[0,T],\n\\end{equation}\nsuch that $(1,r)\\in \\mathcal L$  and  ${Z}\\Lambda = ({Z}_t\\Lambda_t)_{t\\in[0,T]}\\in\\mathcal Y(1,r)$.\n\\end{Lemma}\n\\begin{proof}\nLet $(x,q)\\in\\mathcal K$ and $c\\in\\mathcal A(x,q)$.\nThen there exists  \n$X\\in\\mathcal X(x,q)$, such that\n\\begin{equation}\\label{246}\n\\int_0^\\tau c_sd\\kappa_s\\leq X_\\tau + \\int_0^\\tau q_se_sd\\kappa_s,\\quad for~every~\\tau\\in\\Theta.\n\\end{equation}\nIn particular, \\eqref{246} holds for the particular $\\tau$, such that $\\Lambda_t = 1_{[0,\\tau]}(t)$, $t\\in[0,T]$.  \nLet  $G_t{:=} \\int_0^t(q_se_s)^{+}d\\kappa_s$, $t\\in[0,T]$, where $(\\cdot)^{+}$ denotes the positive part. By \\cite[Theorem III.29, p. 128]{Pr}, $\\int_0^tG_{s-}d{Z}_s$, $t\\in[0,T]$, is a local martingale, so let $(\\sigma_n)_{n\\in\\mathbb N}$ be its localizing sequence. Then by the monotone convergence theorem, integration by parts formula, and the optional sampling theorem, we get\n\\begin{equation}\\label{281}\n\\begin{array}{c}\n\\mathbb E\\left[\\int_0^T {Z}_s(q_se_s)^{+}\\Lambda_sd\\kappa_s \\right] = \\lim\\limits_{n\\to\\infty}\\mathbb E\\left[\\int_0^{\\sigma_n\\wedge \\tau} {Z}_sdG \\right] \\\\\n= \\lim\\limits_{n\\to\\infty} \\mathbb E\\left[{Z}_{\\sigma_n\\wedge \\tau}G_{\\sigma_n\\wedge \\tau}\\right] \n-\\lim\\limits_{n\\to\\infty} \\mathbb E\\left[ \\int_0^{\\sigma_n\\wedge \\tau}G_{s-}d{Z}_s\\right] \n\\\\=\n\\lim\\limits_{n\\to\\infty} \\mathbb E\\left[{Z}_{\\sigma_n\\wedge \\tau}G_{\\sigma_n\\wedge \\tau} \\right] = \\lim\\limits_{n\\to\\infty} \\mathbb E^{\\mathbb Q}\\left[G_{\\sigma_n\\wedge \\tau} \\right] \n =  \\mathbb E^{\\mathbb Q}\\left[G_\\tau \\right],\\\\\n\\end{array}\n\\end{equation}\nwhere in the last equality we used the monotone convergence theorem again.\nHere finiteness of $\\mathbb E^{\\mathbb Q}\\left[G_\\tau \\right]=\\mathbb E^{\\mathbb Q}\\left[\\int_0^\\tau(q_se_s)^{+}d\\kappa_s \\right]$ follows from Assumption \\ref{asEnd} via Lemma \\ref{lemma1}, part (iii). \nIn a similar manner, we can show that\n\\begin{equation}\\label{283}\n\\begin{array}{rcl}\n\\mathbb E\\left[\\int_0^T(q_se_s)^{-}{Z}_s\\Lambda_sd\\kappa_s\\right] &= &\\mathbb E^{\\mathbb Q}\\left[ \\int_0^\\tau (q_se_s)^{-}d\\kappa_s\\right]<\\infty,\\\\\n\\mathbb E\\left[\\int_0^Tc_s{Z}_s\\Lambda_sd\\kappa_s\\right] &=& \\mathbb E^{\\mathbb Q}\\left[ \\int_0^\\tau c_sd\\kappa_s\\right]<\\infty.\\\\\n\\end{array}\n\\end{equation}\n\\cite[Lemma 4]{HK04} applies here and asserts that $X$ in \\eqref{246} is a supermartingale under $\\mathbb Q$. \nTherefore, from \\eqref{246}, using \\eqref{281} and \\eqref{283}, and taking expectation under $\\mathbb Q$, we get\n\\begin{equation}\\label{6-22-2}\n \\mathbb E\\left[ \\int_0^Tc_s\\Lambda_s{Z}_sd\\kappa_s\\right]\\leq x + \\mathbb E\\left[\\int_0^Tq_se_s{Z}_s\\Lambda_sd\\kappa_s\\right].\n\\end{equation}\nLet us define\n\\begin{equation}\\nonumbe\nR_t {:=} \\mathbb E\\left[\\int_0^t\\Lambda_s{Z}_se_sd\\kappa_s \\right],\\quad t\\in[0,T].\n\\end{equation}\nUsing the monotone class theorem, we obtain\n\\begin{equation}\\label{6-25-2}\n\\mathbb E\\left[\\int_0^T\\tilde q_se_s{Z}_s\\Lambda_sd\\kappa_s\\right]\n=\n\\int_0^T\\tilde q_sdR_s,\\quad for~every\\quad\\tilde q\\in\\mathbb L^\\infty(dK).\n\\end{equation}\nWe claim that\n$dR$ is absolutely continuous with respect to $dK$. First, using the $\\pi-\\lambda$ theorem, one can show that for every Borel-measurable subset $A$ of $[0,T]$, we have\n$$K(A) = \\mathbb E\\left[\\int_0^T 1_A(t)d\\kappa_t \\right]\\quad and \\quad R(A) = \\mathbb E\\left[\\int_0^T1_A(t)\\Lambda_t{Z}_te_td\\kappa_t\\right].$$\nThus, if for some $A$, $K(A) = 0$, then $\\int_0^T 1_A(t)d\\kappa_t=0$ a.s. and $\\kappa^A_t {:=} \\int_0^t 1_A(s)d\\kappa_s$, $t\\in[0,T]$, satisfies $\\kappa^A_T = 0$ a.s. and $\\int_0^T \\Lambda_t{Z}_te_t d\\kappa^A_t = \\int_0^T\\Lambda_t{Z}_te_t1_A(t)d\\kappa_t = 0$ a.s. \n\n\nAs $dR$ is absolutely continuous with respect to $dK$,  \nthere exists a unique $r\\in\\mathbb L^1(dK)$, such that \\eqref{6-25-1}\nholds.\nSince the left-hand side in (\\ref{6-22-2}) is nonnegative and since $(x,q)$ is an arbitrary element of $\\mathcal K$, we deduce from the definition of $\\mathcal L$, \\eqref{defL},  that $(1,r)\\in \\mathcal L$. Finally, it follows from \\eqref{6-22-2} and \\eqref{6-25-2}\nthat ${Z}\\Lambda \\in\\mathcal Y(1,r)$. This completes the proof of the~lemma.\n\\end{proof}\n\\begin{Remark}\\label{Remark-Mihai-2}\nThe  natural convexification of  the set $\\Upsilon$  consists of non-negative  left-continuous  decreasing and adapted processes $D$ such that $D_{\\theta _0}=1$. In the context of utility-maximization constraints (and for $\\theta _0=0$), this is follows  from  \\cite{HePages} and \\cite{ElKarouiJeanblanc}. \nWe investigate convexification of the constraints  in  the later Sections \\ref{secStructureOfDualDomain} and \\ref{secSlackness}, where will extend Lemma \\ref{6-16-1} to a more general set of decreasing processes than $\\Lambda$ that drives the dual domain and that allows for the multiplicative decomposition of the dual minimizer\n\\end{Remark}\n\\begin{Corollary}\\label{cor1}\nLet   the conditions of Proposition \\ref{prop1} hold, $\\mathbb Q\\in\\mathcal M'$, ${Z}$ be the c\\'adl\\'ag modification of the density process $\\mathbb E\\left[ \\frac{d\\mathbb Q}{d\\mathbb P}|\\mathcal F_t\\right]$, $t\\in[0,T]$. Then there exists $r\\in\\mathbb L^1(dK)$, such that\n ${Z}\\in\\mathcal Y(1,r)$,\nwhere\n$$\n\\int_0^tr_sdK_s = \\mathbb E\\left[\\int_0^t{Z}_se_sd\\kappa_s \\right],\\quad t\\in[0,T],\n$$\nand $(1,r)\\in \\mathcal L$.\n\\end{Corollary}\n\\begin{proof}[Proof of Lemma \\ref{3311}] Let us fix $\\mathbb Q\\in\\mathcal M'$ and let $Z\\in\\mathcal Z'$ be the corresponding density process. As in Lemma \\ref{6-16-1}, we can show that there exists $r\\in\\mathbb L^1(dK)$, such that for every bounded and Borel measurable function $q$ on $[0,T]$ we have\n\\begin{equation}\\label{5192}\n\\int_0^tq_s r_sdK_s = \\mathbb E\\left[\\int_0^t q_sZ_s|e_s|d\\kappa_s \\right],\\quad t\\in[0,T].\n\\end{equation}\nLet $\\bar q {:=} |q^1 - q^2|$, then as $\\bar q = 0$, $dK$-a.e., we get\n\\begin{displaymath}\n\\mathbb E\\left[\\int_0^T \\bar q_s|e_s|Z_sd\\kappa_s \\right] = \\int_0^T \\bar q_sr_sdK = 0.\n\\end{displaymath}\nTherefore, using integration by parts and via \\eqref{5192}, we obtain\n$$\n0 = \\int_0^T \\bar q_sr_sdK = \\mathbb E\\left[\\int_0^T q_sZ_s|e_s|d\\kappa_s \\right] = \\mathbb E^{\\mathbb Q}\\left[\\int_0^T q_s|e_s|d\\kappa_s \\right].\n$$\nConsequently, $\\int_0^T q_s|e_s|d\\kappa_s = 0$, $\\mathbb Q$-a.s., and by the equivalence of $\\mathbb Q$ and $\\mathbb P$, also $\\mathbb P$-a.s. \nAs, by construction $\\int_0^t q_s|e_s|d\\kappa_s = 0$, $t\\in[0,T]$, is a nonnegative and non-decreasing process, whose terminal value is $0$,  $\\mathbb P$-a.s., we conclude that it is indistinguishable from the $0$-valued process. The assertions of the lemma follows.\n\n\\end{proof}\n\\begin{Lemma}\\label{9-5-1}\nLet   the conditions of Proposition \\ref{prop1} hold,  $(x,q)\\in\\mathcal K$, and $c$ is a nonnegative optional process. Then $c\\in\\mathcal A(x,q)$ if and only if\n\\begin{equation}\\label{241}\n\\mathbb E\\left[\\int_0^Tc_s\\Lambda_s{Z}_sd\\kappa_s \\right]\\leq\nx + \\mathbb E\\left[\\int_0^Tq_se_s\\Lambda_s{Z}_sd\\kappa_s \\right] = x + \\int_0^Tq_sr_sdK_s,\n\\end{equation}\nfor every ${Z}\\in\\mathcal Z'$ and $\\Lambda \\in \\Upsilon$, where $r$ is given by (\\ref{6-25-1}).\n\\end{Lemma}\n\\begin{proof}\nLet $c\\in\\mathcal A(x,q)$. Then for every ${Z}\\in\\mathcal Z'$ and $\\Lambda \\in \\Upsilon$,  the validity of \\eqref{241} follows from the definition of $\\mathcal A(x,q)$, integration by parts formula and supermartingale property of every $X\\in\\mathcal X(x,q)$ under every $\\mathbb Q\\in\\mathcal M'$, which in turn follows from \\cite[Lemma 4]{HK04}. \n\nConversely, let \\eqref{241} holds for every ${Z}\\in\\mathcal Z'$ and $\\Lambda \\in \\Upsilon$. \nThen, we have\n\\begin{equation}\\label{243}\n\\mathbb E\\left[\\int_0^T(c_s-q_se_s){Z}_s\\Lambda_sd\\kappa_s \\right]\\leq x,\n\\end{equation}\nwhich, in view of  the definition of $\\Upsilon$ in \\eqref{defUpsilon}, localization, and integration by parts, implies that\n\\begin{displaymath}\n\\sup\\limits_{\\mathbb Q\\in\\mathcal M', \\tau\\in \\Theta}\\mathbb {E^Q}\\left[\\int_0^{\\tau}(c_s-q_se_s)d\\kappa_s \\right]\\leq x,\n\\end{displaymath}\nFor $X^{''}$ given by Lemma \\ref{lemma1}, item $(v)$, let us denote\n\\begin{equation}\\label{244}\nf_t{:=} ||q||_{\\mathbb L^\\infty(dK)} X^{''}_t +\\int_0^{t}(c_s-q_se_s)d\\kappa_s,\\quad t\\in[0,T].\n\\end{equation}\nIt follows from Assumption \\ref{asEnd} and item $(v)$ of Lemma \\ref{lemma1} \n\n that \n $f$ is a nonnegative process.\nWe observe that the proof of \\cite[Proposition 4.3]{K96} goes through, if we only take stopping times in $\\Theta$ and measures in $\\mathcal M'$. This proposition allows to conclude that there exists a nonnegative c\\`adl\\`ag process $V$, such that \n\\begin{equation}\\label{defV2}\nV_t = \\esssup\\limits_{\\tau\\in \\Theta: \\tau\\geq t,\\mathbb Q \\in\\mathcal M'}\\mathbb {E^Q}\\left[f_{\\tau}|\\mathcal F_t \\right],\\quad t\\in[0,T],\n\\end{equation}\n which is a supermartingale for every $\\mathbb Q \\in\\mathcal M'$. Therefore, by the density of $\\mathcal M'$ in $\\mathcal M$ in the norm topology of $\\mathbb L^1(\\mathbb P)$ and Fatou's lemma, $V$ is a supermartingale under every $\\mathbb Q\\in\\mathcal M$. Moreover, $V_0$ satisfies  \n$$V_0\\leq x + ||q||_{\\mathbb L^\\infty(dK)} X^{''}_0,$$ by \\eqref{defV2}, \\eqref{243}, and by following the argument in the proof of Lemma \\ref{6-16-1}. \n\nWe would like to apply \nthe optional decomposition theorem of F\\\"olmer and Kramkov, \\cite[Theorem 3.1]{FK}. For this, we need to show that $V$ is a local supermartingale under every $\\mathbb Q$, such that every $X\\in\\mathcal X(1)$ is a $\\mathbb Q$-local supermartingale. However, $\\mathcal M$ is dense in the set of such measures in the norm topology of $\\mathbb L^1(\\mathbb P)$, by the results of Delbaen and Schachermayer, see \\cite[Proposition 4.7]{DS1998}. Therefore, the supermartingale property of $V$ under every such $\\mathbb Q$ follows from Fatou's lemma and supermartingale property of $V$ under every $\\mathbb Q\\in\\mathcal M$ established above. Therefore, by \\cite[Theorem 3.1]{FK}, we get\n$$V_t = V_0 + H\\cdot S_t - A_t,\\quad t\\in[0,T],$$\nwhere $A$ is a nonnegative increasing process that starts at $0$. \nSubtracting the constant $||q||_{\\mathbb L^\\infty(dK)} X^{''}_0$ from both sides of \\eqref{244}, we get\n\\begin{equation}\\nonumber\\begin{array}{c}\n\\int_0^{\\tau}(c_s-q_se_s)d\\kappa_s = f_\\tau -||q||_{\\mathbb L^\\infty(dK)} X^{''}_0 \\leq V_\\tau -||q||_{\\mathbb L^\\infty(dK)} X^{''}_0\\\\= V_0 -||q||_{\\mathbb L^\\infty(dK)} X^{''}_0 + H\\cdot S_\\tau-A_\\tau \\leq x + H\\cdot S_\\tau, \\quad \\tau\\in\\Theta,\n\\end{array}\\end{equation}\nwhere $x + H\\cdot S$ is acceptable, since $V$ is nonnegative. \nConsequently, \n$c\\in\\mathcal A(x,q)$. This completes the proof of the lemma.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop1}]\nThe assertions of item $(i)$ follow from Lemma~\\ref{9-5-1}. \nIt remains to show that the affirmations of item $(ii)$ hold.\nFix a $(y,r)\\in \\mathcal L$. If $Y\\in\\mathcal Y(y,r)$, $(ii)$ follows from the definition of $\\mathcal Y(y,r)$. Conversely,\nif $(ii)$ holds for a nonnegative process $Y$, then since $(x,0)\\in \\mathcal K$ for every $x>0$, we have \n$$\\mathbb E\\left[\\int_0^Tc_sY_sd\\kappa_s \\right]\\leq 1\\quad {\\rm for~every~}c\\in\\mathcal A\\left(\\frac{1}{y},0\\right).$$\nVia \\cite[Proposition 4.4]{MostovyiNec}, we deduce that $Y\\in\\mathcal Y(y)$ and is such that $(ii)$ holds. Therefore,\n$Y\\in\\mathcal Y(y,r)$.  \n\nWe have  $\\mathring{\\mathcal K}\\neq \\emptyset$, where the interior is taken with respect to the norm-topology. According to Proposition \\ref{prop1} (i)  we have   also $\\mathcal K = (-\\mathcal L)^{o}$.  The set $\\mathcal{K}$, as the polar of $\\mathcal L$,  is convex and  closed both in  (strong) $\\mathbb R\\times \\mathbb L^{\\infty}(dK)$ and $\\sigma(\\mathbb R\\times\\mathbb L^\\infty(dK), \\mathbb R\\times\\mathbb L^1(dK))$ topologies. Having non-empty strong interior, we obtain $\\mathcal K =  \\cl{\\mathring{\\mathcal K}}$ where the closure is in the strong-topology. \nSince $\\mathcal{K}$ is also closed in the weaker $\\sigma(\\mathbb R\\times\\mathbb L^\\infty(dK), \\mathbb R\\times\\mathbb L^1(dK))$ topology we obtain that \n$$\\mathcal K = (-\\mathcal L)^{o} = \\cl{\\mathring{\\mathcal K}},$$ where the closure is taken in both\ntopologies.\n\\end{proof}\n\n\\subsection{Preliminary properties of the value functions, in particular finiteness. }\n\nLet $\\mathbf  L^0_{+}$ be the positive orthant of $\\mathbf  L^0$. The polar of a set $A \\subseteq \\mathbf  L^0_{+}$ is defined as\n\\begin{displaymath} \nA^o{:=} \\left\\{ c\\in\\mathbf  L^0_{+}:~\\mathbb E\\left[\\int_0^T c_sY_s d\\kappa_s\\right]\\leq 1,\\quad for~every~Y \\in A\\right\\}.\n\\end{displaymath}\n\nWe recall that the sets $\\mathcal Z$ and $\\mathcal Z'$ are defined in \\eqref{defZ} and \\eqref{defZ'}, respectively. \n  \n\n\\begin{Lemma}\\label{finitenessOverZ'}\nUnder the conditions of Theorem~\\ref{mainTheorem}, we have\n\\begin{equation}\\label{bipolarZ'}\n(\\mathcal Z')^{oo} = \\mathcal Y(1)\n\\end{equation}\nand\n\\begin{equation}\\label{2145}\n\\tilde w(y) = \\inf\\limits_{Y\\in\\mathcal Z'}\\mathbb E\\left[\\int_0^TV(t, yY_s)d\\kappa_s \\right]<\\infty, \\quad y>0.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nIt follows from\n \\cite[Theorem 5.2]{DS97}\n that $\\mathcal Z'$ is dense in $\\mathcal Z$ \n in $\\mathbf  L^0$. \nIt follows from Fatou's lemma that $$(\\mathcal Z')^{o} = \\mathcal Z^{o}.$$\nTherefore, \\cite[Lemma 4.2 and Proposition 4.4]{MostovyiNec} imply \\eqref{bipolarZ'}.  \n\nOne can show that $\\mathcal Z'$ is closed under countable convex combinations, where the martingale property follows from the monotone convergence theorem \\and the c\\`adl\\`ag structure of the limit is guaranteed by \\cite[Theorem VI.18]{DelMey82}, see also \\cite[ Proposition 5.1]{KLPO14} for more details in similar settings. \nNow, \\eqref{2145} follows (up to a notational change) from \\cite[Theorem 3.3]{MostovyiNec}. This completes the proof of the lemma.\n\\end{proof}\n\\begin{Lemma}\\label{6-17-1} Under the conditions of Theorem~\\ref{mainTheorem}, \nfor every $(x,q)\\in\\mathcal K$ and $(y,r)\\in \\mathcal L$, we have\n$$u(x,q) \\leq v(y,r) + xy +\\int_0^Tr_sq_sdK_s.$$\n\\end{Lemma} \n\\begin{proof}\nFix an arbitrary $(x,q)\\in\\cl\\mathcal K$, $c\\in\\mathcal A(x,q)$ as well as $(y,r) \\in \\mathcal L$, $Y\\in\\mathcal Y(y,r)$. Using Proposition\n\\ref{prop1} and \\eqref{defV}, we get\n\\begin{displaymath}\n\\begin{array}{rcl}\n\\mathbb E\\left[\\int_0^T U (t, c_s)d\\kappa_s \\right] &\\leq &\\mathbb E\\left[\\int_0^T U(t, c_s)d\\kappa_s \\right] + xy + \\int_0^Tr_sq_sdK_s - \\mathbb E\\left[\\int_0^T c_sY_sd\\kappa_s \\right]\\\\\n&\\leq &\\mathbb E\\left[\\int_0^T V(t, Y_s)d\\kappa_s \\right] + xy + \\int_0^Tr_sq_sdK_s.  \\\\\n\\end{array}\n\\end{displaymath}\nThis implies the assertion of the lemma.\n\\end{proof}\nFor every $(x,q)$ in $\\mathcal K$, we define\n\\begin{equation}\\label{auxiliarySets}\n\\begin{array}{rcl}\n\\mathcal B(x,q)&{:=}& \\left\\{(y,r)\\in \\mathcal L: ~xy + \\int_0^Tr_sq_sdK_s \\leq 1 \\right\\},\\\\\n\\mathcal D(x,q)&{:=}& \\bigcup\\limits_{(y,r)\\in \\mathcal B(x,q)}\\mathcal Y(y,r).\\\\\n\\end{array}\n\\end{equation}\nThe subsequent lemma established boundedness of $\\mathcal B(x,q)$ for $(x,q)$ in $\\mathring{\\mathcal K}$ in $\\mathbb R\\times \\mathbb L^1(dK)$.\n\\begin{Lemma}\n\\label{7-17-1}\nUnder the conditions of Theorem~\\ref{mainTheorem}, for every $(x,q)\\in\\mathring{\\mathcal K}$, \n$\\mathcal B(x,q)$ is bounded\n in $\\mathbb R\\times \\mathbb L^1(dK)$.\n\\end{Lemma}\n\\begin{proof}\nFix an $(x,q)\\in\\mathring{\\mathcal K}$. Then there exists $\\varepsilon > 0$, such that\nfor every\n\\begin{equation}\\label{7-20-1}\n|x'|\\leq\\varepsilon\\quad and \\quad||q'||_{\\mathbb L^{\\infty}} \\leq\\varepsilon,\n\\end{equation}\n we have\n$(x + x', q+q')\\in{\\mathring{\\mathcal K}}.$\nLet us fix an arbitrary $(y,r)\\in \\mathcal B(x,q)$. Then for every $(x',q')$ satisfying (\\ref{7-20-1}), by the definitions of $\\mathcal L$ and $\\mathcal B(x,q)$, respectively,~we~get \n\\begin{displaymath}\n\\begin{array}{rcl}\nxy + \\int_0^Tq_sr_sdK_s + x'y + \\int_0^Tq'_sr_sdK_s &\\geq& 0, \\\\\nxy + \\int_0^Tq_sr_sdK_s  &\\leq& 1, \\\\\n\\end{array}\n\\end{displaymath}\nwhich implies that\n\\begin{equation}\\label{7-17-2}\n-x'y - \\int_0^Tq'_sr_sdK_s \\leq xy + \\int_0^Tq_sr_sdK_s  \\leq 1.\n\\end{equation}\nTaking \n\\begin{displaymath}\nx' = -\\varepsilon\\quad{and}\\quad q'\\equiv 0,\n\\end{displaymath}\nwe deduce from (\\ref{7-17-2}) that $y\\leq \\frac{1}{\\varepsilon}$. Also, by the definition of $\\mathcal L$ and Lemma \\ref{lemma1}, item $(i)$,  $y\\geq 0$.\nIn turn, setting\n\\begin{displaymath}\nx' = 0\\quad{and}\\quad q' = -\\varepsilon 1_{\\{r\\geq 0\\}} + \\varepsilon 1_{\\{r<0\\}},\n\\end{displaymath}\nwe obtain from  (\\ref{7-17-2}) that $||r||_{\\mathbb L^1}\\leq \\frac{1}{\\varepsilon}$. This completes the proof of the lemma.\n\\end{proof}\n\\begin{Lemma}\\label{6-12-1}\nLet the conditions of Theorem~\\ref{mainTheorem} hold and $(x,q)$ be an arbitrary element of $\\mathring{\\mathcal K}$.\nThen, we have:\n\\newline\n$(i)$ $\\mathcal A(x,q)$ contains a strictly positive process.\n\\newline\n$(ii)$ The constant $\\bar y(x,q)$ given by  \n\\begin{equation}\\label{defBary}\n\\bar y(x,q) {:=} \\frac{1}{|x| + ||q||_{\\mathbb L^\\infty(dK)}\\sup\\limits_{\\mathbb Q\\in\\mathcal M}\\mathbb {E^Q}\\left[\\int_0^T|e_s|d\\kappa_s \\right]},\n\\end{equation}\ntakes values in $(0,\\infty)$ and satisfies\n\\begin{equation}\\label{6-18-1}\n\\bar y(x,q)\\mathcal Z'\\subseteq \\mathcal D(x,q).\n\\end{equation}\nIn particular, for every $z>0$, $z\\mathcal D(x,q)$ contains a strictly positive process $Y$ such that \n\\begin{equation}\\label{6-16-2}\n\\mathbb E\\left[\\int_0^TV(s, Y_s)d\\kappa_s \\right] <\\infty.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nIn order to show $(i)$, we observe that the existence of a positive process in $\\mathcal A(x,q)$ follows from the fact that $(x-\\delta, q)\\in\\mathcal K$ for a sufficiently small $\\delta.$\nNow the constant-valued consumption $\\delta\/A>0$, where $A$ is the constant that dominates the terminal value of the stochastic clock $\\kappa$ in \\eqref{finClock}, is in $\\mathcal A(x,q)$.\n\nIn order to prove $(ii)$, \nlet us consider $\\bar y(x,q)$ given by \\eqref{defBary}.\nIt follows from Lemma \\ref{lemma1}, item $(iii)$, that $\\bar y (x,q)\\in (0,\\infty)$. \nFor this $\\bar y(x,q)$, using\nCorollary \\ref{cor1}, one can show~(\\ref{6-18-1}). This and Lemma~\\ref{finitenessOverZ'}  (note that finiteness of $\\tilde w$ follows directly from \\eqref{finValue})\nimply that for every $z>0$, there exists a positive $Y\\in z\\mathcal D(x,q)$, such that (\\ref{6-16-2}) holds.\n\n\n\\end{proof}\n\\begin{Lemma}\\label{lemFinu}\nUnder the conditions of Theorem~\\ref{mainTheorem}, for every $(x,q)\\in\\mathring{\\mathcal K}$ we have\n\\begin{displaymath}\n-\\infty < u(x,q) < \\infty.\n\\end{displaymath}\nand $u<\\infty$ on $\\mathbb R\\times \\mathbb L^\\infty(dK)$.\n\\end{Lemma}\n\\begin{proof}\nLet us fix an arbitrary $(x,q)\\in\\mathring{\\mathcal K}$.  Since $\\mathring{\\mathcal K}$ is an open convex cone, there exists $\\lambda\\in(0,1)$, $(x_1, q_1)\\in\\mathring{\\mathcal K}$, and $x_2>0$, such that\n $$(x,q) =\\lambda (x_1, q_1) + (1- \\lambda)(x_2,0).$$\nNote that $(x_2, 0)\\in\\mathring{\\mathcal K}$ by Lemma \\ref{lemma1}. By \\eqref{finValue}, there $c\\in\\mathcal A(x_2,0)$, such that \n\\begin{equation}\\label{2141}\n\\mathbb E\\left[\\int_0^TU\\left(t, (1 - \\lambda) c_t\\right)d\\kappa_t  \\right] > -\\infty.\n\\end{equation}\nAs $\\mathcal A(x_1,q_1)\\neq\\emptyset$ (see Remark \\ref{remNonEmpty}), there exists $\\tilde c\\in\\mathcal A(x_1,q_1)$. As $U(t, \\cdot)$ is nondecreasing, we get\n\\begin{displaymath}\nu(x,q) \\geq \\mathbb E\\left[\\int_0^TU(t, \\lambda \\tilde c_t + (1 - \\lambda) c_t)d\\kappa_t  \\right] \n\\geq \\mathbb E\\left[\\int_0^TU(t, (1 - \\lambda) c_t)d\\kappa_t  \\right] \n> -\\infty,\n\\end{displaymath}\nwhere the last inequality follows from \\eqref{2141}. This implies finiteness of $u$ on $\\mathring{\\mathcal K}$ from below. \n\nIn order to show finiteness from above, let us fix a process $c\\in\\mathcal A(x,q)$, such~that \n\\begin{equation}\\nonumbe\n\\mathbb E\\left[\\int_0^TU(t, c_t)d\\kappa_t  \\right] \n> -\\infty.\n\\end{equation}\nBy Lemma \\ref{finitenessOverZ'}, there exists $Y\\in\\mathcal Z'$, such that \n\\begin{equation}\\label{4103}\n\\mathbb E\\left[\\int_0^TV(t,Y_t)d\\kappa_t \\right]<\\infty.\n\\end{equation\nIt follows from Lemma \\ref{6-16-1} that $Y\\in\\mathcal Y(1,\\rho)$ for some $(1,\\rho)\\in \\mathcal L$. Therefore, by Proposition \\ref{prop1}, we get\n\\begin{equation}\\label{4101}\n\\begin{array}{rcl}\n\\mathbb E\\left[\\int_0^T U (s, c_s)d\\kappa_s \\right]& \\leq& \\mathbb E\\left[\\int_0^T U(s,c_s)d\\kappa_s \\right] + x + \\int_0^T\\rho_sq_sdK_s - \\mathbb E\\left[\\int_0^T c_sY_sd\\kappa_s \\right]\\\\\n&\\leq& \\mathbb E\\left[\\int_0^T V(s,Y_s)d\\kappa_s \\right] + x + \\int_0^T\\rho_sq_sdK_s.  \\\\\n\\end{array}\n\\end{equation}\nAs $Y$ satisfies (\\ref{4103}), we conclude  that $u(x,q) < \\infty$. \nMoreover, for $(x,q)\\in\\mathcal K$, as $\\mathcal A(x,q)\\neq\\emptyset$ by Remark \\ref{remNonEmpty}, every $c\\in\\mathcal A(x,q)$ satisfies \\eqref{4101} (with the same $Y$). \nThis implies that $u<\\infty$ on $\\mathcal K$ and therefore, by \\eqref{3313}, on $\\mathbb R\\times \\mathbb L^\\infty(dK)$. This completes the proof of the lemma.\n\\end{proof}\nWe recall that, for every $(x,q)\\in\\mathcal K$, $\\mathcal D(x,q)$ is defined in \\eqref{auxiliarySets}. Let $cl \\mathcal D(x,q)$ denote the closure of $\\mathcal D(x,q)$ in $\\mathbb L^0(d\\kappa \\times \\mathbb P).$ The following lemma proves a delicate point that, for  $(x,q)\\in\\mathring{\\mathcal K}$, by passing from $\\mathcal D(x,q)$ to $cl\\mathcal D(x,q)$, we do not change the auxiliary dual value function. \n\\begin{Lemma}\\label{auxiliaryLemma}\nLet the conditions of Theorem~\\ref{mainTheorem} hold and $(x,q)\\in\\mathring{\\mathcal K}$. Then, for every $z>0$, we have\n\\begin{equation}\\label{2143}\n-\\infty<\\inf\\limits_{Y \\in\\cl \\mathcal D(x,q)}\\mathbb E\\left[\\int_0^TV(s, zY_s)d\\kappa_s \\right] =\n\\inf\\limits_{Y \\in \\mathcal D(x,q)}\\mathbb E\\left[\\int_0^TV(s, zY_s)d\\kappa_s \\right]<\\infty.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nFiniteness from above follows from Lemma \\ref{6-12-1}. To show finiteness of both infima in \\eqref{2143} from below, by Lemma~\\ref{lemFinu} we deduce\nthe existence of $c\\in\\mathcal A(x,q)$, such that\n\\begin{equation}\\label{6-18-2}\n\\mathbb E\\left[\\int_0^TU(s, c_s)d\\kappa_s \\right] > -\\infty.\n\\end{equation}\nLet $Y\\in{\\cl \\mathcal D(x,q)}$ and let $Y^n\\in\\mathcal Y(y^n,r^n)$, $n\\geq 1$, be a sequence in $\\mathcal D(x,q)$ that converges to $Y$ in $\\mathbb L^0(d\\kappa\\times\\mathbb P)$. By Fatou's lemma, Proposition~\\ref{prop1}, and the definition of the set $\\mathcal B(x,q)$ in \\eqref{auxiliarySets}, we get\n\\begin{displaymath}\n\\mathbb E\\left[\\int_0^TY_sc_sd\\kappa_s \\right] \\leq \n\\liminf\\limits_{n\\to\\infty}\\mathbb E\\left[\\int_0^TY^n_sc_sd\\kappa_s \\right] \\leq\n\\sup\\limits_{n\\geq 1}\\left(xy^n + \\int_0^Tr^n_sq_sdK_s\\right) \\leq 1. \n\\end{displaymath}\nTherefore, we obtain\n\\begin{displaymath}\n\\begin{array}{rcl}\n\\mathbb E\\left[ \\int_0^TU(s, c_s)d\\kappa_s\\right]&\\leq& \\mathbb E\\left[ \\int_0^TU(s, c_s)d\\kappa_s\\right] + 1 - \\mathbb E\\left[\\int_0^TY_sc_sd\\kappa_s \\right] \\\\\n&\\leq& \\mathbb E\\left[ \\int_0^TV(s, Y_s)d\\kappa_s\\right] + 1,\\\\\n\\end{array}\n\\end{displaymath}\nwhich together with (\\ref{6-18-2}) implies finiteness of both infima in \\eqref{2143} from below.\n\n\nLet us show equality of two infima in \\eqref{2143}. It follows from Lemma\n\\ref{6-12-1} that for every $z> 0$ there exists a process $Y\\in z\\mathcal D(x,q)$, such that\n$$\\mathbb E\\left[\\int_0^TV(s, Y_s)d\\kappa_s \\right] <\\infty.$$\nLet us fix $z>0$ and let $\\bar Y\\in\\cl\\mathcal D(x,q)$. Also, let $(Y^n)_{n\\in\\mathbb N}$ be a sequence in $\\mathcal D(x,q)$ that converges to $\\bar Y$ $(d\\kappa\\times\\mathbb P)$-a.e. \nLet us fix $\\delta >0$, then by Lemma \\ref{finitenessOverZ'}, there exists $Z'\\in\\mathcal Z'$, such that\n$$\\mathbb E\\left[\\int_0^TV(t, \\delta\\bar y(x,q) Z'_t)d\\kappa_t \\right]<\\infty,$$\nwhere $\\bar y(x,q)$ is defined in \\eqref{defBary}. Note that $\\bar y(x,q) Z'\\in \\mathcal D(x,q)$ by Lemma \\ref{6-12-1} (see \\eqref{6-18-1}). \nTherefore, using Fatou's lemma and monotonicity of $V$ in the spatial variable, we obtain\n\\begin{displaymath}\n\\begin{array}{rcl}\n\\inf\\limits_{Y \\in \\mathcal D(x,q)}\\mathbb E\\left[\\int_0^TV(t, (z+\\delta)Y_s)d\\kappa_s \\right]&\\leq&\n\\limsup\\limits_{n\\to\\infty}\\mathbb E\\left[\\int_0^TV(t, zY^n_s+\\delta \\bar y(x,q)Z'_s)d\\kappa_s \\right]\\\\ \n&\\leq  &\\mathbb E\\left[\\int_0^TV(t, z\\bar Y_s + \\delta\\bar y(x,q) Z'_s)d\\kappa_s \\right] \\\\\n&\\leq &\\mathbb E\\left[\\int_0^TV(t, z\\bar Y_s )d\\kappa_s \\right].\\\\\n\\end{array}\n\\end{displaymath}\nTaking the infimum over $Y\\in\\cl\\mathcal D(x,y)$, we deduce that\n\\begin{equation}\\label{21410}\n \\inf\\limits_{Y \\in \\mathcal D(x,q)}\\mathbb E\\left[\\int_0^TV(t, (z+\\delta)Y_s)d\\kappa_s \\right] \\leq \\inf\\limits_{Y\\in\\cl\\mathcal D(x,y)}\\mathbb E\\left[\\int_0^TV(t, zY_s )d\\kappa_s \\right].\n\\end{equation}\nLet us consider \n$$\\phi(z){:=} \\inf\\limits_{Y \\in \\mathcal D(x,q)}\\mathbb E\\left[\\int_0^TV(t, zY_s)d\\kappa_s \\right], \\quad z>0.$$\nBy the first part of the proof (finiteness of both infima), $\\phi$ is finite-valued on $(0,\\infty)$.  Convexity of $V$ in the spatial variable implies that $\\phi$ is also convex.  Therefore, $\\phi$ is continuous. As \\eqref{21410} holds for every $\\delta>0$, by taking the limit as $\\delta \\downarrow 0$ in \\eqref{21410}, we conclude that both infima in \\eqref{2143} are equal. This completes the proof of the lemma.\n\n\\end{proof}\n\nLet us define\n\\begin{equation}\\label{defE}\n\\mathcal E {:=} \\{(y,r)\\in \\mathcal L: ~v(y,r) < \\infty \\}.\n\\end{equation}\n\\begin{Lemma}\\label{lemFinv}\nUnder the conditions of Theorem~\\ref{mainTheorem}, for every $(y,r)\\in \\mathcal L$, we have\n$$v(y,r) > -\\infty.$$\nTherefore, $v>-\\infty$ on $\\mathbb R\\times\\mathbb L^1(dK)$. The set \n $\\mathcal E$ is a nonempty convex subset of $\\mathcal L$, whose closure in $\\mathbb R\\times\\mathbb L^1(dK)$ equals to $\\mathcal L$, and such that\n\\begin{equation}\\label{6-17-2}\n\\mathcal E= \\bigcup\\limits_{\\lambda \\geq 1 }\\lambda \\mathcal E.\n\\end{equation}\n\n\\end{Lemma}\n\\begin{proof}\nLet us  fix $(y,r)\\in \\mathcal L$, then finiteness of $v(y,r)$ from below follows from  \\eqref{finValue} and Lemma~\\ref{6-17-1}.\nTo establish the properties of $\\mathcal E$, we observe that the convexity of $\\mathcal E$  and (\\ref{6-17-2}) follow from convexity and monotonicity of $V$, respectively. \n\n\nIn remains to show that the closure of $\\mathcal E$ in $\\mathbb R\\times\\mathbb L^1(dK)$ contains the origin. \nIn \\eqref{auxiliarySets}, let us consider $(x,q) = (1,0)\\in\\mathcal K$. In this case, we have \n $$\\mathcal D(1,0) = \\bigcup\\limits_{(y,r)\\in\\mathcal L: y\\leq 1}\\mathcal Y(y,r)\\subseteq \\mathcal Y(1),$$\nwhere the last inclusion follows from the  very  definition of $\\mathcal Y(y,r)$'s in \\eqref{defY}.\nAs, by \\eqref{oldY}, $\\mathcal Y(1)$ is closed in $\\mathbb L^0(d\\kappa\\times\\mathbb P)$  and $\\mathcal D(1,0)\\subseteq \\mathcal Y(1)$, we deduce that \n\\begin{equation}\\label{2146}\n\\cl \\mathcal D(1,0)\\subseteq \\mathcal Y(1).\n\\end{equation}\n By Lemma \\ref{6-12-1}, $\\mathcal Z'\\subset \\mathcal D(1,0)$, as $\\bar y(1,0) = 1$. Therefore, by the bipolar theorem of Brannath and Schachermayer, \\cite[Theorem 1.3]{BranSchach}, we get\n\\begin{equation}\\label{2147}\n(\\mathcal Z')^{oo} \\subseteq\\cl\\mathcal D(1,0).\n\\end{equation}\nOn the other hand, Lemma \\ref{finitenessOverZ'} asserts that \n\\begin{equation}\\label{2148}\n(\\mathcal Z')^{oo} = \\mathcal Y(1).\n\\end{equation}\nCombining \\eqref{2146}, \\eqref{2147}, and \\eqref{2148}, we conclude\n\\begin{equation}\\nonumbe\n\\cl \\mathcal D(1,0)= \\mathcal Y(1).\n\\end{equation}\nTherefore, the sets $\\cl\\mathcal D(1,0)= \\mathcal Y(1)$ and $\\mathcal A(1,0)$ satisfy the  precise technical  assumptions of \\cite[Theorem 3.2]{MostovyiNec}, which, for every $x>0$, grants  the existence of $\\widehat c(x)\\in\\mathcal A(x,0)$, the unique maximizer to $w(x)$, where $w$ is defined in \\eqref{defw}.\nFor every $x>0$, we set\n$\nY_{\\cdot}(x){:=} U'({\\cdot}, \\widehat c_{\\cdot}(x)), \\quad \n(d\\kappa\\times\\mathbb P)-a.e\n$$\nBy \\cite[Theorem 3.2]{MostovyiNec}, for every $x>0$,  $Y(x)$  satisfies\n$$Y(x) \\in w'(x)\\cl\\mathcal D(1,0)\\quad and\\quad \\mathbb E\\left[ \\int_0^TV(t,  Y_t(x))d\\kappa_t\\right]<\\infty,\\quad x>0,$$\nwhere by \\cite[Theorem 3.2]{MostovyiNec}, $w$ is a strictly concave, differentiable function on $(0,\\infty)$ that satisfies the Inada conditions. \nTherefore, as $w'(x)$ can be arbitrary close to $0$ (by taking $x$ large enough an by using the Inada conditions) and by Lemmas \\ref{6-12-1} and \\ref{auxiliaryLemma}, we conclude that the closure of $\\mathcal E$ in $\\mathbb R\\times \\mathbb L^1(dK)$ contains origin. \n\nIn order to prove that the closure of $\\mathcal E$ in $\\mathbb R\\times \\mathbb L^1(dK)$ equals to $\\mathcal L$, let $(y,r)\\in\\mathcal L\\backslash (0, 0)$ be fixed. Let us take $\\varepsilon>0$. We want to find $(\\tilde y, \\tilde r)$, such that \n\\begin{equation}\n\\label{4171}\n|\\tilde y - y| + || \\tilde r- r ||_{\\mathbb L^1(dK)}<\\varepsilon,\n\\end{equation}\nand\n\\begin{equation}\\label{4172}\n(\\tilde y, \\tilde r)\\in\\mathcal E.\n\\end{equation}\nAs the closure of $\\mathcal E$ in $\\mathbb R\\times \\mathbb L^1(dK)$ contains origin, we can pick $(y^0, r^0)\\in\\mathcal E$, such that \n\\begin{equation}\\label{4173}\n|y^0| + ||r^0||_{\\mathbb L^1(dK)} \\leq \\varepsilon\/3\n\\end{equation}\nand $Y\\in\\mathcal Y(y^0,r^0)$, such that \n\\begin{equation}\\label{4174}\n\\mathbb E\\left[\\int_0^T V(t, Y_t)d\\kappa_t \\right]<\\infty.\n\\end{equation}\nLet us fix $\\alpha >1$, such that \n\\begin{equation}\\label{4177}\n\\frac{|y| + ||r||_{\\mathbb L^1(dK)}}{\\alpha} \\leq \\varepsilon\/3\n\\end{equation}\n and set $\\varepsilon' {:=} \\frac{1}{\\alpha}\\in(0,1)$. By \\eqref{6-17-2}, $(\\alpha y^0, \\alpha r^0)\\in\\mathcal E.$ Let \n\\begin{equation}\\nonumbe\n\\tilde y {:=}  (1-\\varepsilon')y + \\varepsilon'\\alpha y^0,\\quad \\tilde r {:=} (1-\\varepsilon')r + \\varepsilon'\\alpha r^0.\n\\end{equation} \nThen\n\\begin{displaymath}\n\\begin{array}{rcl}\n|y - \\tilde y| + || r - \\tilde r||_{\\mathbb L^1(dK)} &= & \\varepsilon' \\alpha |\\frac{y}{\\alpha} - y^0| + \\varepsilon'\\alpha || \\frac{r}{\\alpha} - r^0||_{\\mathbb L^1(dK)}\\\\\n&\\leq & \\frac{|y| + ||r||_{\\mathbb L^1(dK)}}{\\alpha} + |y^0| + ||r^0||_{\\mathbb L^1(dK)}\n\\\\\n&\\leq & \\frac{2\\varepsilon}{3},\n\\end{array}\n\\end{displaymath}\nwhere in the last inequality we have used \\eqref{4173} and \\eqref{4177}. Thus $(\\tilde y, \\tilde r)$ satisfies \\eqref{4171}. Further, as $0\\in\\mathcal Y(y,r)$, by convexity of $\\mathcal L$ and using Proposition \\ref{prop1}, we get\n\\begin{displaymath}\nY=(1-\\varepsilon') 0 + \\varepsilon' \\alpha Y \\in\\mathcal Y(\\tilde y, \\tilde r),\n\\end{displaymath}\nwhich by \\eqref{4174} implies \\eqref{4172}. This completes the proof of the lemma.\n\\end{proof}\n\n\\subsection{Existence and uniqueness of solutions to \\eqref{primalProblem} and \\eqref{dualProblem}; semicontinuity and biconjugacy of $u$ and $v$}\n\n\\begin{Lemma}\\label{existenceUniqueness}\nUnder the conditions of Theorem~\\ref{mainTheorem},\nthe value function $v$ is convex, proper, and lower semicontinuous with respect to the  topology of $\\mathbb R\\times \\mathbb L^1(dK)$. For every $(y,r)\\in\\mathcal E$, there exists a unique solution to \n(\\ref{dualProblem}). Likewise, $u$ is concave, proper, and upper semicontinuous  with respect to  the  strong  topology of $\\mathbb R\\times \\mathbb L^\\infty(dK)$. For every $(x,q)\\in\\{u>-\\infty\\}$ there exists a unique solution to~(\\ref{primalProblem}).\n\\end{Lemma}\n\\begin{proof}\nLet $(y^n, r^n)_{n\\in\\mathbb N}$ be a sequence in $\\mathcal L$ that converges to $(y,r)$ in $\\mathbb R\\times\\mathbb L^1(dK)$. \nPassing if necessary to a subsequence, we will assume that \n\\begin{equation}\\label{2151}\n\\lim\\limits_{n\\to\\infty} v(y^n,r^n)= \\liminf\\limits_{n\\to\\infty}v(y^n, r^n).\n\\end{equation}\nLet $Y^n\\in\\mathcal Y(y^n, r^n)$, $n\\in\\mathbb N$, be such that \n\\begin{equation}\\label{2152}\n\\mathbb E\\left[ \\int_0^TV(t,Y^n_t)d\\kappa_t\\right]\\leq v(y^n, r^n) +\\frac{1}{n}, \\quad n\\in\\mathbb N.\n\\end{equation}\nBy passing to convex combinations and applying Komlos'-type lemma, see e.g. \\cite[Lemma A1.1]{DS}, we may suppose that $\\widetilde Y^n\\in\\conv\\left(Y^n, Y^{n+1},\\dots\\right)$, $n\\in\\mathbb N$, converges $(d\\kappa\\times\\mathbb P)$-a.e. to some $\\widehat Y$. \n\nFor every $(x,q)\\in\\mathcal K$ and $c\\in\\mathcal A(x,q)$, by Fatou's lemma, we have\n\\begin{displaymath}\n\\begin{array}{c}\n\\mathbb E\\left[ \\int_0^T c_t\\widehat Y_td\\kappa_t\\right]\\leq \\liminf\\limits_{n\\to\\infty}\\mathbb E\\left[ \\int_0^T c_t \\widetilde Y^n_td\\kappa_t\\right]\\leq\nxy + \\int_0^Tq_sr_sdK_s.\n\\end{array}\n\\end{displaymath} \nTherefore, by Proposition \\ref{prop1}, $\\widehat Y\\in\\mathcal Y(y,r)$. With $\\bar y {:=} \\sup\\limits_{n\\geq 1}y^n$, we have $(\\widetilde Y^n)_{n\\in\\mathbb N}\\subseteq \\mathcal Y(\\bar y).$ Therefore, by \\cite[Lemma 3.5]{MostovyiNec}, we deduce that \n$V^{-}(t,\\widetilde Y^n_t)$, $n\\in\\mathbb N$, is a uniformly integrable sequence.  \nCombining uniform integrability with the convexity of $V$ in the spatial variable, we get\n\\begin{equation}\\label{4141}\n\\begin{array}{c}\nv(y,r)\\leq \\mathbb E\\left[ \\int_0^TV(t,\\widehat Y_t)d\\kappa_t\\right]\\leq \n\\liminf\\limits_{n\\to\\infty}\n\\mathbb E\\left[ \\int_0^TV(t, \\widetilde Y^n_t)d\\kappa_t\\right]\n\\\\\n\\leq \n\\liminf\\limits_{n\\to\\infty}\n\\mathbb E\\left[ \\int_0^TV(t,  Y^n_t)d\\kappa_t\\right] = \\liminf\\limits_{n\\to\\infty}v(y^n,r^n),\n\\end{array}\n\\end{equation}\nwhere in the last equality we have used \\eqref{2151} and \\eqref{2152}. Since $(y^n, q^n)$ was an arbitrary sequence that converges to $(y,r)$, lower semicontinuity of $v$ in strong  topology of $\\mathbb R\\times \\mathbb L^1(dK)$ follows. Since $\\mathcal L$ is closed and $v = \\infty$ outside of $\\mathcal L$, we deduce that $v$ is lower semicontinuous on $\\mathbb R\\times \\mathbb L^1(dK)$. The  function $v$ is proper by Lemma \\ref{lemFinv}. Note that \\eqref{4141} also implies that $\\mathcal E$ defined in \\eqref{defE} is $\\mathbb R\\times\\mathbb L^1(dK)$-norm closed. For $(y,r)\\in\\mathcal E$, by taking $(y^n, r^n)= (y,r)$, $n\\in\\mathbb N$, we deduce the existence of a minimizer to \\eqref{dualProblem}. Strict convexity of $V$ results in the uniqueness of the minimizer to \\eqref{dualProblem}. Convexity of $v$ follows.  Upper semicontinuity of $u$  with respect to the norm-topology of   $\\mathbb R\\times \\mathbb L^{\\infty}(dK)$ can be proven similarly, first proving semi-continuity on \n $\\mathcal K$  by a Fatou-type argument, then using the closedness  of $\\mathcal{K}$ and the definition of $u$ outside it. \n\\end{proof}\n\\begin{Corollary}\nUnder the conditions of Theorem~\\ref{mainTheorem},\n$-u$ and $v$ are  also  lower semicontinuous with respect to the weak topologies $\\sigma(\\mathbb R\\times\\mathbb L^{\\infty}(dK), (\\mathbb R\\times\\mathbb L^{\\infty})^*(dK))$ and $\\sigma(\\mathbb R\\times\\mathbb L^1(dK),\\mathbb R\\times\\mathbb L^{\\infty}(dK))$, respectively.\n\\end{Corollary}\n\\begin{proof}\nThe assertions of the corollary is a consequence of \\cite[Proposition 2.2.10]{BarbuPrec}, see also \\cite[Corollary I.2.2]{EkelandTemam}.\n\\end{proof}\n\nWe recall that ${\\cl \\mathcal D(x,q)}$ denotes the closure of $\\mathcal D(x,q)$ in $\\mathbb L^0(d\\kappa\\times\\mathbb P)$. \n\\begin{Lemma}\\label{5181}\nUnder the conditions of Theorem \\ref{mainTheorem}, for every $(x,q)$ in $\\mathring{\\mathcal K}$, and a nonnegative optional process $c$, we have\n$$\nc\\in\\mathcal A(x,q)\\quad{if~and~only~if}\\quad \\mathbb E\\left[\\int_0^T c_sY_sd\\kappa_s \\right]\\leq 1\\quad {for~every~}Y\\in{\\cl \\mathcal D(x,q)}.\n$$\n\\end{Lemma}\n\\begin{proof}\nLet $(x,q)$ in $\\mathring{\\mathcal K}$, $c$ is a nonnegative optional process such that \n\\begin{equation}\\label{5191}\n\\mathbb E\\left[ \\int_0^T c_sY_sd\\kappa_s \\right]\\leq 1\\quad {for~every}\\quad Y\\in{\\cl \\mathcal D(x,q)}.\n\\end{equation}\nConsider arbitrary $Z\\in\\mathcal Z'$ and $\\Lambda\\in\\Upsilon$. Let the corresponding $r$ be given by \\eqref{6-25-1} and we set $$y'{:=} x + \\int_0^Tq_sr_sdK_s.$$\n\nIf $y' = 0$, then $y\\Lambda Z\\in\\mathcal D(x,q)$ for every $y>0$. Thus,  by \\eqref{5191}, we obtain that $\\mathbb E\\left[\\int_0^T c_sy\\Lambda_sZ_sd\\kappa_s \\right]\\leq 1$. Taking the limit as $y\\to\\infty$, we get\n\\begin{equation}\\label{5183}\\mathbb E\\left[\\int_0^T c_s\\Lambda_sZ_sd\\kappa_s \\right] = 0=  x + \\int_0^Tq_sr_sdK_s= x + \\mathbb E\\left[\\int_0^Tq_se_s\\Lambda_sZ_sd\\kappa_s \\right],\n\\end{equation}\nwhere in the last equality we have used \\eqref{6-25-1}. \n\nIf $y'>0$, \nthen $\\frac{1}{y'}\\Lambda Z\\in\\mathcal D(x,q)$ and thus by \\eqref{5191}, we obtain $$\\mathbb E\\left[ \\int_0^T c_s\\frac{1}{y'}\\Lambda_sZ_sd\\kappa_s\\right]\\leq 1 = \\frac{ x + \\int_0^Tq_sr_sdK_s}{y'} = \\frac{1}{y'}\\left( x + \\mathbb E\\left[\\int_0^T q_se_s\\Lambda_sZ_sd\\kappa_s\\right]\\right),$$\nwhere in the last equality, we have used \\eqref{6-25-1} again. Consequently, we deduce\n$$\\mathbb E\\left[ \\int_0^T c_s\\Lambda_sZ_sd\\kappa_s\\right]\\leq  x + \\mathbb E\\left[\\int_0^T q_se_s\\Lambda_sZ_sd\\kappa_s\\right],$$\nwhich together with \\eqref{5183}, by Lemma \\ref{9-5-1}, imply that $c\\in\\mathcal A(x,q)$.\n\nConversely, let $(x,q)\\in\\mathring{\\mathcal K}$, $c\\in\\mathcal A(x,q)$ and $Y\\in {\\rm cl}\\mathcal D(x,q)$. Then there exists a sequence $Y^n\\in\\mathcal Y(y^n, r^n)$ convergent to $Y$, $(d\\kappa\\times\\mathbb P)$-a.e., where $(y^n,r^n)\\in\\mathcal B(x,q)$. As, \n$$\\mathbb E\\left[\\int_0^T c_sY^n_sd\\kappa_s \\right]\\leq 1,\\quad n\\in\\mathbb N,$$\nby Fatou's lemma, we get\n$$\\mathbb E\\left[\\int_0^T c_sY_sd\\kappa_s \\right]\\leq \\liminf\\limits_{n\\to\\infty}\\mathbb E\\left[\\int_0^T c_sY^n_sd\\kappa_s \\right]\\leq 1.$$\nThis completes the proof of the lemma.\n\\end{proof}\n\\begin{Lemma}\\label{conjugacy}\nUnder the conditions of Theorem \\ref{mainTheorem}, for every $(x,q)$ in $\\mathring{\\mathcal K}$, we have\n\\begin{equation}\\label{conjugacyOneDirection}\nu(x,q) = \\inf\\limits_{(y,r)\\in \\mathcal L}\\left(v(y,r) + xy + \\int_0^Tr_sq_sdK_s\\right).\n\\end{equation}\n\\end{Lemma}\n\\begin{proof} Let us fix $(x,q)\\in\\mathring{\\mathcal K}$. \nBy Lemma \\ref{6-12-1}, $\\mathcal A(x,q)$ and ${\\cl \\mathcal D(x,q)}$ contain strictly positive elements. \nTherefore, using Lemma \\ref{5181}\nwe deduce that the sets $\\mathcal A(x,q)$ and ${\\cl \\mathcal D(x,q)}$ satisfy the assumptions \nof \\cite[Theorem 3.2]{MostovyiNec}. From this theorem, Lemma~\\ref{auxiliaryLemma}, and the definition of the set $\\mathcal B(x,q)$, we get\n\\begin{equation}\\nonumbe\n\\begin{array}{rcl}\nu(x,q) &=& \\inf\\limits_{z>0}\\left(\\inf\\limits_{Y\\in{\\cl \\mathcal D(x,q)}}\\mathbb E\\left[ \\int_0^TV(t, zY_s)d\\kappa_s\\right] + z \\right) \\\\\n &=& \\inf\\limits_{z>0}\\left(\\inf\\limits_{Y\\in\\mathcal D(x,q)}\\mathbb E\\left[ \\int_0^TV(t, zY_s)d\\kappa_s\\right] + z \\right) \\\\\n&=& \\inf\\limits_{z>0}\\left(\\inf\\limits_{(y,r)\\in z\\mathcal B(x,q)}v(y,r) + z \\right) \\\\\n&\\geq& \\inf\\limits_{(y,r)\\in \\mathcal L}\\left(v(y,r) + xy + \\int_0^Tq_sr_sdK_s \\right).\\\\\n\\end{array}\n\\end{equation}\nCombining this with the conclusion of Lemma~\\ref{6-17-1}, we deduce that \\eqref{conjugacyOneDirection} holds for every $(x,q)\\in\\mathring{\\mathcal K}$. \n\n\n\\end{proof}\n\nBefore proving the biconjugacy relations of item $(iii)$, Theorem \\ref{mainTheorem}, we need a preliminary lemma. \nEssentially following the notations in \\cite{EkelandTemam}, we define \n\\begin{equation}\\label{v*}\nv^*(x,q){:=} \\inf\\limits_{(y,r)\\in\\mathbb R\\times \\mathbb L^1(dK)}\\left(v(y,r) + xy + \\int_0^Tq_sr_sdK_s \\right),\\quad (x,q)\\in\\mathbb R\\times \\mathbb L^{\\infty}(dK).\n\\end{equation}\n\\begin{equation}\\label{v**}\nv^{**}(y,r){:=} \\sup\\limits_{(x,q)\\in\\mathbb R\\times \\mathbb L^\\infty(dK)}\\left(v^*(x,q) - xy - \\int_0^Tq_sr_sdK_s \\right),\\quad (y,r)\\in\\mathbb R\\times \\mathbb L^1(dK).\n\\end{equation}\n\\begin{Remark}In \\cite{EkelandTemam}, conjugate convex functions are considered on general spaces $V$ and $V^*$ supplied with $\\sigma(V,V^*)$ and $\\sigma(V^*, V)$ topologies, which in our case are $V= \\mathbb R\\times\\mathbb L^1(dK)$, $V^* = \\mathbb R\\times\\mathbb L^\\infty(dK)$. Thus, the starting point of our analysis is $v$, not $u$. We remind the reader we have already proved that the dual value function $v$ is convex, proper and lower-semicontinuous on the space $V= \\mathbb R\\times\\mathbb L^1(dK)$. \n\\end{Remark}\n\\begin{Lemma}\\label{lemv**}\nUnder the conditions of Theorem \\ref{mainTheorem}, we have \n\\begin{equation}\n\\label{2166}\nv^{**} = v, \n\\end{equation}\n\\begin{equation}\n\\label{4151}\nv^*(x,q) = -\\infty, \\quad for~every\\quad (x,q)\\in\\mathbb R\\times\\mathbb L^\\infty(dK)\\backslash \\mathcal K.\n\\end{equation}\n\n\\end{Lemma}\n\n\\begin{proof}\nTo show \\eqref{2166}, we observe that by Lemma \\ref{existenceUniqueness}, $v$ is lower semicontinuous in the $\\mathbb R\\times\\mathbb L^1(dK)$-norm topology (and therefore, by \\cite[Corollary I.2.2]{EkelandTemam}, also in the weak topology $\\sigma(\\mathbb R\\times\\mathbb L^1(dK),\\mathbb R\\times\\mathbb  L^\\infty(dK))$). As a result, by \\cite[Proposition I.4.1]{EkelandTemam}, we get~\\eqref{2166}.\n\nThe proof of \\eqref{4151} will be done in several steps. \n\n{\\it Step 1.} Let $(x,q)\\in\\mathbb R\\times\\mathbb L^\\infty(dK)\\backslash \\mathcal K$.  According to  Proposition \\ref{prop1}, there exists $(y,r)\\in\\mathcal L$, such that\n\\begin{equation}\\nonumbe\nC{:=} xy + \\int_0^T q_sr_sdK_s <0.\n\\end{equation}\nTherefore, as $\\mathcal L$ is a cone, for every $a>0$, $(ay, ar)\\in\\mathcal L$, and we have\n\\begin{equation}\\label{4153}\nxay + \\int_0^Tq_sar_sdK_s = aC<0.\n\\end{equation}\nNote that $0\\in\\mathcal Y(ay, ar)$, $a>0$.\n\n{\\it Step 2.} Let us consider $Z\\in\\mathcal Z'$, such that \n\\begin{equation}\\label{4154}\n\\mathbb E\\left[\\int_0^TV\\left(t,\\tfrac{1}{2}Z_t\\right)d\\kappa_t\\right]<\\infty.\n\\end{equation}\nThe existence of such a $Z$ is granted by Lemma \\ref{finitenessOverZ'}.\nFurther, by Corollary \\ref{cor1}, there exists $\\rho\\in\\mathbb L^1(dK)$, such that $(1, \\rho)\\in\\mathcal L$, $Z\\in\\mathcal Y(1,\\rho)$,  and\n\\begin{equation}\\nonumbe\n\\int_0^t\\rho_sdK_s = \\mathbb E\\left[\\int_0^t Z_se_sd\\kappa_s \\right],\\quad t\\in[0,T].\n\\end{equation}\nLet us set \n\\begin{equation}\\nonumbe\nD{:=} x + \\int_0^Tq_s\\rho_sdK_s\\in\\mathbb R.\n\\end{equation}\n\n{\\it Step 3.} In \\eqref{4153}, let us pick \n\\begin{equation}\\nonumbe\na = \\frac{|D| + 1}{-C}>0.\n\\end{equation}\nThen, we have \n\\begin{equation}\\label{4158}\naC + D = -|D| + D -1<0.\n\\end{equation}\n\n{\\it Step 4.} Let us define \n\\begin{equation}\\label{4159}\nY{:=} \\tfrac{1}{2} Z, \\quad y'{:=} \\tfrac{1}{2}ay + \\tfrac{1}{2}, \\quad and \\quad r'{:=} \\tfrac{1}{2}ar + \\tfrac{1}{2}\\rho.\n\\end{equation}\nThen by \\eqref{4154}, we obtain\n\\begin{equation}\\label{41510}\n\\mathbb E\\left[\\int_0^TV\\left(t,Y_t\\right)d\\kappa_t\\right]<\\infty.\n\\end{equation}\nAs $Z\\in\\mathcal Y(1,\\rho)$ and $0\\in\\mathcal Y(ay, ar)$, by convexity of $\\mathcal L$ and Proposition \\ref{prop1}, we have\n\\begin{equation}\\label{41511}\nY\\in\\mathcal Y(y',r'),\n\\end{equation}\nwhere $y'$ and $r'$ are defined in \\eqref{4159}. Now, it follows from  \\eqref{41510} and \\eqref{41511} that $(y', r')\\in\\mathcal E$ (where $\\mathcal E$ is defined in \\eqref{defE}). \nTherefore, we obtain\n\\begin{displaymath}\n\\begin{array}{c}\n2(xy' + \\int_0^Tq_sr'_sdK_s) = (ay + 1)x + \\int_0^T(ar_s + \\rho_s)q_sdK_s \\\\= a(xy + \\int_0^Tq_sr_sdK_s) + x + \\int_0^Tq_s\\rho_sdK_s = aC + D <0,\\\\\n\\end{array}\n\\end{displaymath}\nwhere the last inequality follows from \\eqref{4158}.\nTo  recapitulate, we have shown the existence of  $(y', r')$, such that\n\\begin{equation}\\label{41512}\n(y', r')\\in\\mathcal E\\quad and\\quad xy' + \\int_0^Tq_sr'_sdK_s<0.\n\\end{equation}\n\n{\\it Step 6.} For $y'$ and $r'$ defined in \\eqref{4159}, as $v(y', r') <\\infty$ and $xy' + \\int_0^Tq_sr'_sdK_s<0$ by \\eqref{41512}, from the monotonicity of $V$, we get\n$$\\infty>v(y',r')\\geq v(\\lambda y', \\lambda r'),\\quad \\lambda \\geq 1.$$\nAs $\\bigcup\\limits_{\\lambda\\geq 1}(\\lambda y', \\lambda r')\\subset \\mathcal L$, we conclude via \\eqref{41512} that\n$$v^*(x,q) \\leq \\lim\\limits_{\\lambda \\to\\infty}\\left( v(\\lambda y', \\lambda r') + \\lambda \\left(xy' + \\int_0^Tq_sr'_sdK_s\\right)\\right) = -\\infty.$$\nTherefore, \\eqref{4151} holds. This completes the proof of the lemma.\n\\end{proof}\n\\begin{Lemma}\\label{lemBiconjugacy}\nUnder the conditions of Theorem \\ref{mainTheorem}, we have \n\\begin{equation}\\label{2161\nv(y,r) = \\sup\\limits_{(x,q)\\in\\mathcal K}\\left(u(x,q) - xy - \\int_0^Tr_sq_sdK_s\\right),\\quad (y,r)\\in\\mathcal L,\n\\end{equation}\n\\begin{equation}\\label{21610}\nu(x,q) = \\inf\\limits_{(y,r)\\in\\mathcal L}\\left(v(y,r) + xy + \\int_0^Tq_sr_sdK_s\\right),\\quad (x,q)\\in \\mathcal K.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\n\n\nLemma \\ref{conjugacy} and \\eqref{3313} imply that on $\\mathring{\\mathcal K}$, for $v^*$ defined in \\eqref{v*}, we have\n\\begin{equation}\\label{2164}\nv^{*} = u\n\\end{equation}\nBy Lemma \\ref{lemv**}, $v^* = -\\infty$ on $\\mathbb R\\times\\mathbb L^\\infty(dK)\\backslash\\mathcal K$. \nFrom \\cite[Definition I.4.1]{EkelandTemam} and Lemma \\ref{existenceUniqueness}, respectively, we deduce that both $v^*$ and $u$ are upper semicontinuous in the topology of $\\mathbb R\\times\\mathbb L^\\infty(dK)\n. Consequently, from \\eqref{2164}, using \\cite[Corollary I.2.1]{EkelandTemam},  \nwe get\n\\begin{equation}\\label{2201}v^* = u,\\quad on\\quad\\mathbb R\\times\\mathbb L^\\infty(dK).\n\\end{equation}\nAs a result, with $v^{**}$ being defined in \\eqref{v**}, for every $(y,r)\\in\\mathbb R\\times \\mathbb L^1(dK)$, we obtain\n\\begin{equation}\\label{2165}\nu^{*}(y,r){:=} \\sup\\limits_{(x,q)\\in\\mathbb R\\times \\mathbb L^\\infty(dK)}\\left(u(x,q) - xy - \\int_0^Tq_sr_sdK_s \\right)=v^{**}(y,r)\n\\end{equation}\nTherefore, from \\eqref{2166} in Lemma \\ref{lemv**} and \\eqref{2165}, we get\n\\begin{equation}\\nonumbe\nv = u^{*},\\quad on\\quad\\mathbb R\\times\\mathbb L^1(dK).\n\\end{equation}\nAs a result, applying Lemma \\ref{lemv**} again and since $u = -\\infty$ outside of $\\mathcal K$ by \\eqref{3313}, we deduce \n\\begin{equation}\\nonumbe\n\\begin{array}{rcl}\nv(y,r)  &=& \\sup\\limits_{(x,q)\\in\\mathbb R\\times\\mathbb L^\\infty(dK)}\\left( u(x,q)- xy - \\int_0^Tr_sq_sdK_s\\right)\\\\\n&=& \\sup\\limits_{(x,q)\\in\\mathcal K}\\left( u(x,q) - xy - \\int_0^Tr_sq_sdK_s\\right),\\quad(y,r)\\in\\mathcal L,\\\\\n\\end{array}\n\\end{equation}\nThus, \\eqref{2161} holds. \n\nIn turn, from \\eqref{2201} using \\eqref{3313}, \nwe conclude that\n\n\\begin{equation}\\nonumber\n\\begin{array}{rcl}\nu(x,q) &=& \\inf\\limits_{(y,r)\\in\\mathbb R\\times \\mathbb L^1(dK)}\\left(v(y,r) + xy + \\int_0^Tq_sr_sdK_s\\right),\\\\\n&=& \\inf\\limits_{(y,r)\\in\\mathcal L}\\left(v(y,r) + xy + \\int_0^Tq_sr_sdK_s\\right),\\quad (x,q)\\in\\mathcal K,\\\\\n\\end{array}\n\\end{equation}\nwhich proves \\eqref{21610} and extends the assertion of Lemma \\ref{conjugacy} to the boundary of $\\mathring{\\mathcal K}$.\n\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{mainTheorem}]\nThe assertions of item $(i)$ follow from Lemmas \\ref{lemFinu} and \\ref{lemFinv}, item $(ii)$ results from Lemma \\ref{existenceUniqueness}, whereas the validity of item $(iii)$ come from Lemma\n\\ref{lemBiconjugacy}. This completes the proof of the theorem.\n\\end{proof}\n\n\n\n\\section{Subdifferentiability of $u$}\\label{secSubdifferentiability}\nIn order to establish subdifferentiability of $u$, we need to strengthen \nAssumption~\\ref{asEnd} and to impose the following condition. \n\\begin{Assumption}\\label{asACclock}\nThere exists an a.s. bounded away from $0$ and $\\infty$  \nprocess $\\varphi$, such that\n\\begin{equation}\\nonumbe\nd\\kappa(\\omega) =\\varphi d K, \\quad for\\quad\\mathbb P-a.e.\\quad\\omega \\in\\Omega.\n\\end{equation}\n\n\\end{Assumption}\n\n\n\nLet \n\\begin{equation}\\label{defPi}\n\\begin{array}{rcl}\n\\mathcal P &{:=} &\\left\\{ \\rho:~(1,\\rho)\\in\\mathcal L\\right\\},\\\\\n\\end{array}\n\\end{equation}\n\\begin{Remark} \n\n  $\\mathcal P$ defined in \\eqref{defPi} needs to be uniformly integrable with respect to the measure $dK$ in order  for the proof of subdifferentiability of $u$ to go through.\n Assumption \\ref{asEnd} through Lemma \\ref{lemma1} only implies  \n that $\\mathcal P$ is $\\mathbb{L}^1$ bounded. A stronger condition on the stochastic clock and income stream is, therefore, needed to obtain uniform integrability. \n \n\n\\end{Remark}\nThe following theorem characterizes subdifferentiability of $u$  over  $\\mathring{\\mathcal K}$,\nwhere we are looking for an  $\\mathbb R\\times\\mathbb L^1(dK)$-valued subgradient.   Under our assumptions,  we can find elements of the sub gradient which both  belong to the  effective domain of $v$, $\\mathcal E$, and are bounded, i.e. in $\\mathbb{R}\\times \\mathbb{L}^{\\infty} (dK)$.\n\\begin{Theorem}\\label{mainTheorem2} \nLet the conditions of Theorem~\\ref{mainTheorem} and Assumption \\ref{asACclock} hold. Then for every $(x,q)\\in\\mathring{\\mathcal K}$, the subdifferential of $u$ at $(x,q)$ is a nonempty and contains an element of $\\mathcal E$, i.e.,\n\\begin{equation}\\label{subdifNonempty}\n\\partial u(x,q)\\cap \\mathcal E\\neq \\emptyset.\n\\end{equation}\nMoreover, for $(x,q)\\in\\mathring{\\mathcal K}$ and $(y,r)\\in\\mathcal L$, $(y,r)\\in\\partial u(x,q)$ if and only if the following conditions hold:\n\\begin{equation}\\label{2191}\n |v(y,r)|<\\infty,\n \\end{equation}\n thus, $(y,r)\\in\\mathcal E$, \n \\begin{equation}\\label{2192}\n \\mathbb E\\left[\\int_0^T\\widehat Y_t(y,r) \\widehat c_t(x,q)d\\kappa_t\\right] = xy + \\int_0^Tq_sr_sdK_s,\n \\end{equation}\n \\begin{equation}\\label{2193}\n \\widehat Y_t(y,r) = U'(t,\\widehat c_t(x,q)), \\quad (d\\kappa\\times\\mathbb P)- a.e.,\n \\end{equation}\n where $\\widehat c(x,q)$ and $\\widehat Y(y,r)$ are the unique optimizers to \\eqref{primalProblem} and \\eqref{dualProblem}, respectively.\n\\end{Theorem}\n\n\n\n\n\n\\subsection{Uniform integrability of $\\mathcal P$}\n\\begin{Lemma}\\label{uiP}\nLet the conditions of Theorem~\\ref{mainTheorem2} hold. Then $\\mathcal P$ is $\\mathbb{L}^{\\infty}(dK)$-bounded, and, therefore,   a uniformly integrable family.\n\\end{Lemma}\n\\begin{proof}\n{\\it Step 1.}  For an arbitrary $q:~[0,T] \\to [0,1]$, let us define\n\\begin{displaymath}\n\\beta(q) {:=} \\sup\\limits_{\\mathbb Q\\in\\mathcal M} \\mathbb {E^Q}\\left[ \\int_0^T q(s) |e_s|d\\kappa_s\\right] = \\sup\\limits_{\\mathbb Q\\in\\mathcal M, \\tau\\in\\Theta} \\mathbb {E^Q}\\left[ \\int_0^{\\tau} q(s) |e_s|d\\kappa_s\\right].\n\\end{displaymath}\nNote that by Assumption \\eqref{asEnd}, \nwe have\n\\begin{equation}\\nonumber\n\\begin{array}{rcl}\n\\beta(q) &=&\\sup\\limits_{\\mathbb Q\\in\\mathcal M} \\mathbb {E^Q}\\left[ \\int_0^T q(s) |e_s|d\\kappa_s\\right]\\\\\n&\\leq&\\sup\\limits_{\\mathbb Q\\in\\mathcal M} \\mathbb {E^Q}\\left[ \\int_0^T q(s) X'_s\\varphi d K_s\\right]\\\\\n&\\leq&\\int_0^T q(s)\\sup\\limits_{\\mathbb Q\\in\\mathcal M} \\mathbb {E^Q}\\left[  CX'_s\\right]d K_s\\\\\n&\\leq&CX'_0\\int_0^T q(s)d K_s,\\\\\n\\end{array}\n\\end{equation}\nwhere $C\\in\\mathbb R$ is such that $|\\varphi| \\leq C$. \nIt follows from \\cite[Proposition 4.3]{K96} and \\cite[Theorem 3.1]{FK} that there exists a nonnegative c\\`adl\\`ag process $V = \\beta(q) + H\\cdot S$, such that\n\\begin{displaymath}\nV_t \\geq \\esssup\\limits_{\\mathbb Q\\in \\mathcal M, \\tau \\in\\Theta, \\tau\\geq t}\\mathbb {E^Q}\\left[\\int_0^{\\tau}q(s)|e_s|d\\kappa_s | \\mathcal F_t \\right],\\quad t\\in[0,T].\n\\end{displaymath} \nConsequently, $V$ satisfies\n$$V_\\tau \\geq \\int_0^\\tau q(s)|e_s|d\\kappa_s \\geq  \\int_0^\\tau q(s)e_sd\\kappa_s,\\quad \\Pas,\\quad\\tau \\in\\Theta,$$\nand thus, $(\\beta(q), -q)\\in\\mathcal K$.  As a result, from the definitions of $\\mathcal L$ and $\\mathcal P$, we get\n\\begin{equation}\\nonumbe\n\\beta(q)\\geq \\sup\\limits_{\\rho \\in\\mathcal P}\\int_0^T q(s)\\rho(s)dK_s.\n\\end{equation}\nOne can see that\n for every $\\rho\\in\\mathcal P$, we have\n$\\rho \\leq f{:=} {CX'_0}$, $d K$-a.e. \n\n\n{\\it Step 2.} For an arbitrary $q:~[0,T] \\to [-1,0]$, let us set \n\\begin{displaymath}\n\\tilde \\beta(q) {:=} \\sup\\limits_{\\mathbb Q\\in\\mathcal M}\\mathbb {E^Q}\\left[ \\int_0^T q(s) (-|e_s|) d\\kappa_s \\right].\n\\end{displaymath}\nAs in {\\it Step 1}, we can construct a c\\`adl\\`ag process $\\tilde V = \\tilde \\beta(q) + H\\cdot S$, s.t.\n\\begin{displaymath}\n\\tilde V_t \\geq \\esssup\\limits_{\\mathbb Q\\in\\mathcal M, \\tau \\in\\Theta, \\tau \\geq t}\\mathbb {E^Q}\\left[ \\int_0^T q(s)(-|e_s|) d\\kappa_s |\\mathcal F_t\\right] \\geq \\esssup\\limits_{\\mathbb Q\\in\\mathcal M, \\tau \\in\\Theta, \\tau \\geq t}\\mathbb {E^Q}\\left[ \\int_0^T q(s)e_s d\\kappa_s |\\mathcal F_t\\right].\n\\end{displaymath}\nThis implies that $(\\tilde \\beta(q), -q)\\in\\cl\\mathcal K$. Therefore, \n\\begin{displaymath}\n\\beta(q) \\geq \\int_0^T q(s)\\rho (s) dK_s,\\quad for~every~\\rho \\in\\mathcal P.\n\\end{displaymath}\nSimilarly to {\\it Step 1},\none can see that $\\rho \\geq -f$, $d K$-a.e.  for every $\\rho \\in\\mathcal P$.\n\n{\\it Step 3.}In view of {\\it Steps 1 and 2}, uniform integrability of $\\mathcal P$ under $dK$ follows from the integrability of $f$ under $d K$.\n\\end{proof}\n\n\\begin{Lemma}\\label{uiB}\nLet the assumption of Theorem \\ref{mainTheorem2} hold and $(x,q)\\in\\mathring{\\mathcal K}$. Then \n$$\\left\\{r:(y,r)\\in\\mathcal B(x,q)\\right\\}$$ is \n $\\mathbb{L}^{\\infty}$-bounded, so  a uniformly integrable subset of \n$\\mathbb L^1(dK)$.\n\\end{Lemma}\n\\begin{proof}\nBy Lemma \\ref{7-17-1}, we deduce the existence of a constant $M>0$,\nsuch that\n\\begin{equation}\\nonumbe\ny\\leq M,\\quad for~every~(y,r)\\in\\mathcal B(x,q).\n\\end{equation}\nWe conclude that $$\\left\\{r:(y,r)\\in\\mathcal B(x,q)\\right\\}\\subseteq \\bigcup\\limits_{0\\leq \\lambda \\leq M}{\\lambda\\mathcal P},$$ and thus by Lemma \\ref{uiP}, $\\left\\{r:(y,r)\\in\\mathcal B(x,q)\\right\\}$ is a uniformly integrable family.\n\\end{proof}\n\\subsection{Closedness of $\\mathcal D(x,q)$ for every $(x,q)\\in\\mathring{\\mathcal K}$}\n\\begin{Lemma}\\label{closednessD}\nUnder the conditions of Theorem~\\ref{mainTheorem2}, for every $(x,q)\\in \\mathring{\\mathcal K}$, the set $\\mathcal D(x,q)$ is closed in $\\mathbb L^0(d\\kappa\\times\\mathbb P)$.\n\\end{Lemma}\n\\begin{proof}\nLet $(x,q)\\in\\mathring{\\mathcal K}$\nand $Y$ be an arbitrary element of $\\cl \\mathcal D(x,q)$.  We claim that there exists $(y,r)\\in \\mathcal B(x,q)$, such that $Y \\in \\mathcal Y(y,r)$. Let $Y^n\\in\\mathcal Y(y^n,r^n)$, $n\\geq 1$, be a sequence in $\\mathcal D(x,q)$, such that $\\lim\\limits_{n\\to \\infty}Y^n = Y$, $(d\\kappa\\times\\mathbb P)$-a.e. \nSince $(y^n,r^n)_{n\\geq 1}\\subset\\mathcal B(x,q)$, which is bounded in the sense of Lemma~\\ref{7-17-1}, Komlos' lemma implies the existence of a subsequence of convex combinations\n$(\\tilde y^n,\\tilde r^n)\\in \\conv((y^n,r^n), (y^{n+1}, r^{n+1}),\\dots)$, $n\\geq 1$, such that $(\\tilde y^n)_{n\\geq 1}$ converges to $y$ and $(\\tilde r^n)_{n\\geq 1}$ converges to $r$, $dK$-a.e. \nLemma~\\ref{uiB} implies that $(\\tilde r^n)_{n\\geq 1}$ is uniformly integrable. Therefore $(\\tilde r^n)_{n\\geq 1}$ converges to $r$ in $\\mathbb L^1(dK)$. Note that the corresponding sequence of convex combinations of $(Y^n)_{n\\geq 1}$, $(\\tilde Y^n)_{n\\geq 1}$  converges to $Y$, $(d\\kappa\\times\\mathbb P)$-a.e.\nThen we have\n$$ \n1 \\geq \\lim\\limits_{n\\to\\infty}\\left( x\\tilde y^n + \\int_0^T q_s\\tilde r^n_sdK_s \\right) = xy + \\int_0^T q_s r_sdK_s.\n$$\nTherefore, $(y,r)\\in \\mathcal B(x,q)$.\n\nLet us fix an arbitrary $(x',q')\\in\\mathcal K$ and $c\\in\\mathcal A(x',q')$. \nUsing Proposition~\\ref{prop1}, Fatou's lemma, and Lemma \\ref{uiB}, we obtain\n\\begin{displaymath}\n\\begin{array}{c}\n0\\leq \\mathbb E\\left[\\int_0^TY_sc_sds \\right] \\leq \n\\liminf\\limits_{n\\to\\infty} \\mathbb E\\left[\\int_0^T \\tilde Y^n_sc_sds \\right] \\\\\\leq \n \\lim\\limits_{n\\to\\infty}\\left( x'\\tilde y^n + \\int_0^T q'_s\\tilde r^n_sdK_s \\right)\n =\n  x'y + \\int_0^T q'_s r_sdK_s,\\\\\n\\end{array}\n\\end{displaymath} \nwhere the uniform integrability of $\\mathcal B(x,q)$ in needed once again in the last equality.\nFrom Proposition~\\ref{prop1}, we conclude that that $Y\\in \\mathcal Y(y,r)$. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{mainTheorem2}.] \nLet $(x,q)\\in\\mathcal K$, $\\widehat c(x,q)$ be the minimizer to (\\ref{primalProblem}), whose existence and uniqueness are established in Lemma~\\ref{existenceUniqueness}. Let us also set\n\\begin{equation}\\label{defHatY}\n\\widehat Y_{t} {:=} U'({t},\\widehat c_{t}(x,q)),\\quad (t,\\omega)\\in[0,T]\\times\\Omega,\\quad and \\quad z {:=} \\mathbb E\\left[\\int_0^T\\widehat c_s(x,q)\\widehat Y_s d\\kappa_s \\right].\n\\end{equation} \nNote that the sets $\\mathcal A(x,q)$ and ${\\cl \\mathcal D(x,q)}$ satisfy the conditions of \\cite[Theorem~3.2]{MostovyiNec}, which implies that $\\widehat Y\\in z{\\cl \\mathcal D(x,q)}$ is the unique solution to the optimization problem\n\\begin{displaymath}\n\\inf\\limits_{Y\\in z{\\cl \\mathcal D(x,q)}}\\mathbb E\\left[\\int_0^TV(t, Y_s)d\\kappa_s\\right] = \\mathbb E\\left[\\int_0^TV(t,\\widehat Y_s)d\\kappa_s\\right]\\in\\mathbb R,\n\\end{displaymath}\nwhere finiteness follows from Lemma \\ref{auxiliaryLemma}.\nNote that by Lemma~\\ref{closednessD}, $\\widehat Y\\in\\mathcal Y(zy, zr)$ for some $(y,r)\\in\\mathcal B(x,q)$.\nIt follows from the definition of $\\widehat Y$ in \\eqref{defHatY} that \nfor $\\widehat c(x,q)$ and $\\widehat Y$ we have the following relation\n\\begin{displaymath}\nU({t}, \\widehat c_{t}(x,q)) = V({t}, \\widehat Y_{t}) + \\widehat c_{t}(x,q)\\widehat Y_{t},\n\\quad (t,\\omega)\\in[0,T]\\times\\Omega\n\\end{displaymath}\nwhich together with Lemma~\\ref{conjugacy} implies that\n\\begin{equation}\\label{7-17-3}\nu(x,q) = v(zy,zr) + z\\left(xy + \\int_0^Tq_sr_sdK_s\\right),\n\\end{equation}\n\\begin{displaymath}\n\\widehat Y(zy,zr)  = \\widehat Y\n,\\quad (d\\kappa\\times\\mathbb P)-a.e.,\n\\end{displaymath}\nwhere $\\widehat Y(zy,zr)$ is the unique minimizer to the dual problem~(\\ref{dualProblem}). \nBy (\\ref{7-17-3}), $(zy,zr)\\in\\mathcal E.$ (\\ref{7-17-3}) and \\cite[Proposition I.5.1]{EkelandTemam} assert that $(zy,zr)\\in\\partial u(x,q)$. In particular, we get\n\\begin{displaymath} \n\\partial u(x,q) \\cap \\mathcal E \\neq \\emptyset,\n\\end{displaymath}\ni.e.,  \\eqref{subdifNonempty}. \nNote that even though, e.g., \\cite[Corollary 2.2.38 and Corollary 2.2.44]{BarbuPrec} imply that $\\partial u(x,q)\\neq \\emptyset$, \n over the interior of the effective domain, \nits elements are in $\\mathbb R\\times (\\mathbb L^\\infty)^*(dK)$. Relation \\eqref{subdifNonempty} shows that $\\partial u(x,q)$ contains  at least a bounded element  of $\\mathcal E\\subseteq\\mathcal L\\subset \\mathbb R\\times\\mathbb L^1(dK).$\n\nLet $(x,q)\\in\\mathcal K$ and $(y,r)\\in\\mathcal L$. Suppose that \\eqref{2191}, \\eqref{2192}, and \\eqref{2193} hold. Then by conjugacy of $U$ and $V$, we get \n\n\\begin{equation}\\label{2194}\n0 = v(y,r) - u(x,q) + xy + \\int_0^T q_sr_sdK_s.\n\\end{equation}\nLemma \\ref{lemBiconjugacy} and \n\\cite[Proposition I.5.1]{EkelandTemam} imply that $(y,r)\\in\\partial u(x,q) \\cap\\mathcal E$.\nConversely, let $(x,q)\\in\\mathcal K$ and $(y,r)\\in\\mathcal L\\cap \\partial u(x,q).$ Then by \\cite[Proposition I.5.1]{EkelandTemam}  and Lemma \\ref{lemBiconjugacy}, we deduce that \\eqref{2194} holds. Lemma \\ref{lemFinu} implies the finiteness of $u(x,q)$, which together with \\eqref{2194} results in the finiteness of $v(y,r)$, thus \\eqref{2191} holds and $(y,r)\\in\\mathcal E$. By Lemma \\ref{existenceUniqueness}, there exists a unique optimizer $\\widehat c(x,q)$, for \\eqref{primalProblem}, and $\\widehat Y(y,r)$, for \\eqref{dualProblem}, respectively. Therefore, from conjugacy of $U$ and $V$, Proposition \\ref{prop1}, and \\eqref{2194}, we obtain\n\\begin{displaymath}\n\\begin{array}{c}\n0\\leq\n \\mathbb E\\left[\\int_0^T \\left(V(t,\\widehat Y_t(y,r)) - U(t,\\widehat c_t(x,q)) + \\widehat Y_t(y,r) \\widehat c_t(x,q)\\right)d\\kappa_t\\right] \\\\\n\\leq v(y,r) - u(x,q) + xy + \\int_0^T q_sr_sdK_s = 0.\\\\\n\\end{array}\n\\end{displaymath}\nThis implies \\eqref{2192} and \\eqref{2193}. This completes the proof of the theorem.\n\\end{proof}\n\n\\section{Structure of the dual feasible set}\\label{secStructureOfDualDomain}\nBy Assumption \\ref{asACclock}, there exists at most countable subset $(s_k)_{k\\in\\mathbb N}$ of $[0,T]$, where $\\kappa$ has jumps. \nWe define $\\mathcal{D}'$ the set of non-increasing, left-continuous and adapted processes $D$ that start at $1$ and \nwith the property that \n$D_{\\theta _0} = 1$,  $ D_T\\geq 0$\nand that, there exists some $n\\in \\mathbb N$ such that  $D$ is constant off the discrete grid\n$\\mathcal T_n{:=} \\bigcup\\limits_{j = 1}^{2^n}\\{s_j\\}\\cup\\left\\{\\frac{k}{2^n}T, k = 0, \\dots,2^n\\right\\}$.\n\n\n\n\n\n\\begin{Lemma}\\label{lemSt2}\n Let $\\mathcal G'{:=} \\{ZD = (Z_tD_t)_{t\\in[0,T]}:~Z\\in\\mathcal Z', D\\in\\mathcal D' \\}$. Assume    the conditions of Proposition \\ref{prop1} hold. Then $\\mathcal G'$ is convex.\n\n\\end{Lemma}\n\\begin{proof}\nLet $Z^1D^1$ and $Z^2D^2$ are the elements of $\\mathcal G'$ and let $\\lambda \\in(0,1)$\nWe need to show that $\\lambda Z^1D^1 + (1-\\lambda)Z^2D^2 = ZD$\nfor some $Z\\in\\mathcal Z'$ and $D\\in\\mathcal D'$. There exists $n\\in\\mathbb N$, such that $D^1$ and $D^2$ decrease at most on $\\mathcal T_n$. Let  $t_k$'s be the elements of $\\mathcal T_n$ arranged in an increasing order. \nLet us define $Z_0 = D_0 = 1$ and for every $k \\in\\{0,\\dots, 2^n-1\\}$,~with \n$$\nA_t {:=} \\lambda Z^1_{t_k}D^1_{t} +(1-\\lambda)Z^2_{t_k}D^2_{t}\n\\quad and\\quad\n\\alpha_t {:=} \\frac{\\lambda Z^1_{t_k}D^1_{t}}{A_t}1_{\\{A_t\\neq 0\\}} + \\frac{\\lambda Z^1_{t_k}}{\\lambda Z^1_{t_k} + (1-\\lambda) Z^2_{t_k}}1_{\\{A_t= 0\\}},$$\n(note that $A_t = 0$ if and only if both $D^1_t=0$ and $D^2_t=0$) we  set\n\\begin{equation}\\label{351}\n\\begin{array}{rclcl}\nZ_t &{:=}& Z_{t_k}\\left(\\alpha_{t_k+}\\frac{Z^1_t}{Z^1_{t_k}} + (1 - \\alpha_{t_k+})\\frac{Z^2_t}{Z^2_{t_k}} \\right), &for& t\\in(t_k, t_{k+1}],\\\\\nD_t &{:=}& \\frac{A_{t_k+}}{Z_{t_k}},& for& t\\in(t_k, t_{k+1}].\\\\\n\\end{array}\n\\end{equation}\nOne can see that $ZD = \\lambda Z^1D^1 + (1-\\lambda)Z^2D^2$ and that $Z\\in\\mathcal Z'$, see e.g., \\cite{FK}, \\cite{Rohlin10}, \\cite{Kardaras_emery}, and \\cite{ChristaJosef15} for discussions of the sets of processes with similar convexity-type properties to the one given in \\eqref{351}. To show that $D\\in\\mathcal D'$, for $k\\geq 1$, \nwe observe that \n\\begin{displaymath}\n\\begin{array}{rcl}\nD_{t_k+} &=& \\frac{A_{t_k+}}{Z_{t_k}} \\\\\n&=& \\frac{{\\lambda Z^1_{t_k}D^1_{t_k+} +(1-\\lambda)Z^2_{t_k}D^2_{t_k+}}}{Z_{t_{k-1}}\\left(\\alpha_{t_{k-1}+}\\frac{Z^1_{t_k}}{Z^1_{t_{k-1}}} + (1 - \\alpha_{t_{k-1}+})\\frac{Z^2_{t_k}}{Z^2_{t_{k-1}}} \\right)} \\\\\n&\\leq &\\frac{{\\lambda Z^1_{t_k}D^1_{t_k} +(1-\\lambda)Z^2_{t_k}D^2_{t_k}}}{Z_{t_{k-1}}\\left(\\alpha_{t_{k-1}+}\\frac{Z^1_{t_k}}{Z^1_{t_{k-1}}} + (1 - \\alpha_{t_{k-1}+})\\frac{Z^2_{t_k}}{Z^2_{t_{k-1}}} \\right)}\\\\\n&= &\\frac{{\\lambda Z^1_{t_k}D^1_{t_k} +(1-\\lambda)Z^2_{t_k}D^2_{t_k}}}{Z_{t_{k-1}}\\left(\\frac{\\lambda Z^1_{t_{k-1}}D^1_{t_k}}{A_{t_{k-1}+}}\\frac{Z^1_{t_k}}{Z^1_{t_{k-1}}} \n+ \\frac{(1-\\lambda) Z^2_{t_{k-1}}D^2_{t_k}}{A_{t_{k-1}+}}\\frac{Z^2_{t_k}}{Z^2_{t_{k-1}}} \\right)}1_{\\{A_{t_{k-1}+}> 0\\}} +0\\cdot1_{\\{A_{t_{k-1}+}= 0\\}}\\\\\n&= &\\frac{A_{t_{k-1}+}}{Z_{t_{k-1}}}\\frac{{\\lambda Z^1_{t_k}D^1_{t_k} +(1-\\lambda)Z^2_{t_k}D^2_{t_k}}}{\\lambda D^1_{t_k}Z^1_{t_k}\n+ (1-\\lambda) D^2_{t_k}Z^2_{t_k}}1_{\\{A_{t_{k-1}+}> 0\\}}\\\\\n&= &\\frac{A_{t_{k-1}+}}{Z_{t_{k-1}}}1_{\\{A_{t_{k-1}+}> 0\\}}\\\\\n&= &D_{t_k}.\\\\\n\\end{array}\n\\end{displaymath}\nAbove, the inequality  follows from the monotonicity of $D^1$ and $D^2$.\nTherefore, $D$ is nonincreasing. Also, clearly $D$ is nonnegative. Thus  $ZD\\in \\mathcal G'$. \n\\end{proof}\nThe following lemma is an extension of Lemma \\ref{9-5-1}  and amounts to a first layer of convexification of the  set $\\Upsilon$, i.e., of the budget constraints.\n\n\\begin{Lemma}\\label{lemSt1}Let   the conditions of Proposition \\ref{prop1} hold,  $(x,q)\\in\\mathcal K$, and $c$ is a nonnegative optional process. Then $c\\in\\mathcal A(x,q)$ if and only if\n\\begin{equation}\\label{331}\n\\mathbb E\\left[\\int_0^Tc_sD_sZ_sd\\kappa_s \\right]\\leq\nx + \\mathbb E\\left[\\int_0^Tq_se_sD_sZ_sd\\kappa_s \\right],\\quad\nfor~every~Z\\in\\mathcal Z'~and~D \\in \\mathcal D'.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nThe idea is to use the assertion of Lemma \\ref{9-5-1} and to approximate a given $D\\in\\mathcal D'$ by (finite) linear combinations of the elements of $\\Upsilon$, where $\\Upsilon$ is defined in \\eqref{defUpsilon}.\n\nFor a stopping time $\\tau$, \nlet us denote $$\\Lambda^{\\tau} {:=} 1_{[0,\\tau]}\\in\\Upsilon,$$\nand fix $D\\in\\mathcal D'$. Then there exists $l\\in\\mathbb N$, such that $D$ has has jumps at most\non $\\{t_0,t_1,\\dots,\nt_l\\}$ for some increasing $t_i$'s. \nFor every $j \\in\\{ 0, \\dots, \n\\}$, $k \\in\\{0,\\dots 2^n\\}$, and $n\\in\\mathbb N$, let us set\n\\begin{equation}\\nonumbe\n\\begin{array}{rcl}\nA_{k, n, j} &{:=}& \\left\\{ \\omega:  ~D_{t_j}(\\omega) >0~and~\\frac{D_{t_j+}(\\omega)}{D_{t_j}(\\omega)} \\in \\left(\\frac{k-1}{2^n}, \\frac{k}{2^n}\\right]\\right\\},\\\\\n\\tau^{k, n, j}&{:=}&T1_{A_{k,n,j}} + t_j1_{A^c_{k,n,j}},\\\\\n\\end{array}\n\\end{equation}\nNote that $D_0 = 1$ by definition of $\\mathcal D'$, $A_{k, n, j}\\in\\mathcal F_{t_j}$, and \n$$\\frac{k-1}{2^n}\\Lambda^{\\tau^{k,n,j}}_{t_j+} = \\left\\{\n\\begin{array}{lcl}\n\\frac{k-1}{2^n}&on&A_{k,n,j}\\\\\n0&on& A^c_{k,n,j}\\\\\n\\end{array}\n \\right..$$\n By construction, we have $$D_{t_j+} = D_{t_j}\\lim\\limits_{n\\to\\infty}\\sum\\limits_{k = 1}^{2^n}\\frac{k-1}{2^n}\\Lambda^{\\tau^{k,n,j}}_{t_j+},$$ where the sequence \n $$K^n_{t_j+}{:=} \\sum\\limits_{k = 1}^{2^n}\\frac{k-1}{2^n}\\Lambda^{k,n,j}_{t_j+},\\quad n\\in\\mathbb N,$$\n is increasing on $\\{D_{t_j}>0\\}$, i.e.,\n $$K^n_{t_j+}1_{\\{D_{t_j}>0\\}}\\uparrow \\frac{D_{t_j+}}{D_{t_j}}1_{\\{D_{t_j}>0\\}}.$$\n Thus, for an arbitrary $j \\in\\{0, \\dots, l\\}$, we have constructed a sequence of elements of $\\Upsilon$, whose finite linear combinations monotonically increase to $\\frac{D_{t_j+}}{D_{t_j}}$ on $\\{D_{t_j}>0\\}$ (i.e., if $j<l$, we have approximated $D$ on the interval $(t_{j}, t_{j+1}]$). \n \n \n In order to construct a sequence that approximates $D$ at every point of its potential jumps, we first observe that for two stopping times $\\tau$ and $\\sigma$, we have\n $$\\Lambda^\\tau\\Lambda^\\sigma = 1_{[0, \\tau]}1_{[0,\\sigma]} = 1_{[0,\\tau\\wedge\\sigma]} = \\Lambda^{\\tau\\wedge \\sigma}.$$\nTherefore, for every $n\\in\\mathbb N$,\n\\begin{equation}\\label{361}\nK^n_{t_j+}K^n_{t_{j+1}+} =\\left(\\sum\\limits_{k = 1}^{2^n}\\frac{k-1}{2^n}\\Lambda^{k,n,j}_{t_j+}\\right)\\left(\\sum\\limits_{k = 1}^{2^n}\\frac{k-1}{2^n}\\Lambda^{k,n,j+1}_{t_{j+1}+}\\right) =\\sum\\limits_{i=1}^{4^n}\\lambda^{n,i} \\Lambda^{\\sigma_{n,i}}_{t_{j+1}+},\n\\end{equation}\nfor some  finite sequences of stopping times $(\\sigma_{n,i})_{i=1}^{4^n}$ and $[0,1)$-valued constants $(\\lambda^{n,i})_{i=1}^{4^n}$. \nHere $\\Lambda^{\\sigma_{n,i}}$ are such that for {\\it both} $t = t_j$ and $t=t_{j+1}$ on $\\{D_{t_{j+1}}>0\\}$, we have\n$$\\lim\\limits_{n\\to\\infty}\\sum\\limits_{i=1}^{4^n}\\lambda^{n,i} \\Lambda^{\\sigma_{n,i}}_{t+} = \\frac{D_{t+}}{D_t}.$$\nSimilarly,\n with $r(t){:=} \\max\\{i:~t_i< t\\}$, let us define\n$$D^n_t {:=} \\prod\\limits_{j = 0}^{r(t)}K^n_{t_j+}1_{\\{D_{t_j}>0\\}},\\quad t\\in[0,T], n\\in\\mathbb N.$$ \nAs in \\eqref{361}, for every $n\\in\\mathbb N$, $D^n$ can be written as a finite linear combination of $\\Lambda$'s, such that $D^n_{t+}\\uparrow D_{t+}$ for every $t\\in\\{t_0, \\dots, t_{l}\\}.$\n\nFinally, \\eqref{331} can be obtained from Lemma \\ref{9-5-1} by the approximation of $D$ by $D^n$'s as above and via  the monotone convergence theorem (applied separately to $(e_t)^{+}$ and $(e_t)^{-}$). \n\\end{proof}\n\\begin{Corollary}\\label{corRzd}\nLet   the conditions of Proposition \\ref{prop1} hold. Then, for every pair $Z\\in\\mathcal Z'$ and $D\\in\\mathcal D'$, \nthere exists $r^{ZD}\\in\\mathbb L^1(dK)$, such that $ZD\\in\\mathcal Y(1,r^{ZD})$, where $(1,r^{ZD})\\in\\mathcal L$. \n\\end{Corollary}\n\\begin{proof}\nThe existence of $r^{ZD}$, such that $ZD\\in\\mathcal Y(1, r^{ZD})$ follows from Lemma \\ref{lemSt1} (equation \\eqref{331}) and the approximation procedure in Lemma \\ref{lemSt1} (again, applied separately to $(e_t)^{+}$ and $(e_t)^{-}$) combined with the monotone convergence theorem.\nAs, the left-hand side in \\eqref{331} is nonnegative, \n$(1,r^{ZD})\\in\\mathcal L$. \n\n\\end{proof}\n\n\nFor a given $Z\\in\\mathcal Z'$ and $D\\in\\mathcal D'$, let us recall that $r^{ZD}$ is given in Corollary \\ref{corRzd}. \nWe set $$\\mathcal {B'}(x,q) {:=} \\left\\{ (y, yr^{ZD})\\in\\mathcal B(x,q):~y> 0, Z\\in\\mathcal Z', D\\in\\mathcal D'\\right\\},$$\n\n$$\\mathcal G(x,q) {:=} \\left\\{yZ'D'\\in\\mathcal D(x,q):~(y, yr^{ZD})\\in \\mathcal {B'}(x,q)\\right\\},\\quad (x,q)\\in\\mathring{\\mathcal K},$$ \n\n\n\\begin{Lemma}\\label{lemSt3}\nLet   the conditions of Proposition \\ref{prop1} hold, then for every $(x,q) \\in\\mathring{\\mathcal K}$, the closure of the convex, solid hull of $\\mathcal G(x,q)$ in $\\mathbb L^0$ coincides with $\\cl\\mathcal D(x,q)$. \n\\end{Lemma}\n\\begin{proof}\nLet $(x,q)\\in\\mathring{\\mathcal K}$ be fixed. \nAlong the lines of the proof of Lemma \\ref{5181}, one can show that\n\\begin{equation}\\nonumbe\n(\\mathcal G(x,q))^o = \\mathcal A(x,q).\n\\end{equation}\nTherefore, \n$$(\\mathcal G(x,q))^{oo} = (\\mathcal A(x,q))^o = \\cl\\mathcal D(x,q),$$\nwhere in the last equality we have used the conclusion of Lemma \\ref{5181}.\n As $\\mathcal G(x,q)\\subset \\cl\\mathcal D(x,q)$, the assertion of the lemma follows from  the bipolar theorem of Brannath and Schchermayer, \\cite[Theorem 1.3]{BranSchach}.\n\\end{proof}\n\n\n\\begin{Corollary}\\label{lemSt4}\nLet the conditions of Proposition \\ref{prop1} hold and $(x,q)\\in\\mathring{\\mathcal K}$. Then for  \nevery \\underline{maximal} element $Y$ of $\\cl\\mathcal D(x,q)$ there exists $y^n\\geq 0$, $Z^n\\in\\mathcal Z'$, $D^n\\in\\mathcal D'$, $n\\in\\mathbb N$, such that $(y^nZ^nD^n)_{n\\in\\mathbb N}\\subset\\mathcal G(x,q)$ and \n\\begin{equation}\\nonumber\n\\begin{array}{rcll}\nY\n&=& \\lim\\limits_{n\\to\\infty} y^nZ^nD^n,& \\quad (d\\kappa\\times \\mathbb P)-a.e.~and~on~ \\bigcup\\limits_{n\\in\\mathbb N}\\mathcal T_n.\n\\end{array}\n\\end{equation}\n\\end{Corollary}\n\\begin{proof}\nThe $(d\\kappa\\times \\mathbb P)$-a.e. convergence follows from Lemma \\ref{lemSt3}.  By passing to subsequences of convex combinations, we also deduce the convergence on $\\bigcup\\limits_{n\\in\\mathbb N}\\mathcal T_n$.\n\\end{proof}\n\\begin{Remark}\nIt follows from Corollary \\ref{lemSt4} and Fatou's lemma that the maximal elements of $\\cl\\mathcal D(x,q)$ are strong supermartingales. Moreover, every maximal element of $\\cl\\mathcal D(x,q)$ is an optional strong supermartingale deflator, which is optional strong supermartingale $Y$, such that $XY$ is an optional strong supermartingale for every $X\\in\\mathcal X(1)$. We refer to \\cite[Appendix 1]{DelMey82} for a general characterization and to \\cite{CzichSchachOSSup} for results on strong optional supermartingales as limits of martingales. The following section gives a more refined characterization of the dual minimizer.\n\\end{Remark}\n\n\n\\section{Complementary slackness}\\label{secSlackness}\nFor better readability of this section, we recall some notations and results that will be used below. \nThroughout this section, $(x,q)\\in \\mathring{\\mathcal{K}}$ will be fixed, $\\widehat c = \\widehat c(x,q)$ is the optimizer to \\eqref{primalProblem}, $\\widehat V$ is the corresponding wealth process, i.e., \n \\begin{equation}\\label{5211}\n\\widehat V=x+\\int_0^{\\cdot}\\widehat H_sdS_s-\\int_0^{\\cdot}\\widehat c_sd\\kappa_s+\\int_0^{\\cdot} q_se_sd\\kappa_s,\n\\end{equation}\nwhere $\\widehat H$ is some $S$-integrable process, \n$\\widehat Y$ be such that $\\widehat Y_t = U'(t,\\widehat c_t)$, $(d\\kappa\\times\\mathbb P)$-a.e., i.e., $\\widehat Y$ is the optimizer to \\eqref{dualProblem} for some \n$(y,r)\\in\\mathcal E\\cap \\partial u(x,q)$, $\\widehat Y\\in\\mathcal Y(y, r)$ and \n$$\\mathbb E\\left[\\int_0^TV(t,\\widehat Y_t)d\\kappa_t\\right] = \\inf\\limits_{Y\\in \\mathcal Y(y,r)}\\mathbb E\\left[\\int_0^TV(t, Y_t)d\\kappa_t\\right].$$\nBy Corollary \\ref{lemSt4}, $\\widehat Y$  can be approximated by a sequence $y^nD^nZ^n$, $n\\in\\mathbb N$, where $y^n$ is a nonnegative constant, $D^n\\in\\mathcal D'$, and $Z^n\\in\\mathcal Z'$, $n\\in\\mathbb N$.\nIn what follows, for any right-continuous  increasing process $A$ satisfying $$A_t=0,\\quad 0-\\leq t\\leq \\theta _{0}-,\\quad A_T=1,$$ \ni.e., for any probability measure $dA$ on  the  closed interval $[\\theta _0, T]$ (extended to $[0,T]$) we will associate a process $D$ which is left-continuous  and decreasing\n$$D_t{:=} 1-A_{t-}=dA([t,T]), \\ \\ 0\\leq t\\leq T.$$\nOne can also think that $ D_{T+}=0,$ although  this is not necessary.  It is clear that  such $A\\leftrightarrow D$ are in  bijective  correspondence.  Below, all processes $A$'s and $D$'s (with indexes) will be in such bijective correspondence, except for the case of  the limiting  process $\\widehat A$ (which is right-continuous) and  the limiting process $\\widehat D$ (that may be not  left-continuous). The  will be in a similar but more subtle correspondence. More precisely:\n\\begin{Theorem}\\label{ExactSlackness}\nLet the conditions of Theorem \\ref{mainTheorem2}  hold. Let  $\\widehat V$ be the optimal wealth process, $\\widehat c$  the optimal consumption, and $\\widehat Y$ be the dual minimizer satisfying the assertions of Theorem \\ref{mainTheorem2}. Then, there exists a strong supermartingale $\\hat Z=\\lim\\limits_{n\\to\\infty} Z^n$ (for $Z_n\\in \\mathcal{D}'$, where the limit is in the sense of \\cite{CzichSchachOSSup}) and a right-continuous  increasing process $\\widehat A$ with\n$$ \\widehat A_t=0\\ \\  \\forall 0-\\leq t \\leq \\theta _0-,\\ \\ \\  \\widehat A_T= 1, $$\nand a decreasing process $\\widehat D$ with $\\widehat D_t=1,\\ \\ 0\\leq t\\leq {\\theta _0}$ and  satisfying the complementary slackness condition \n\\begin{equation}\\label{comp-slack}\n \\mathbb{P}\\left (\\widehat D_t\\in \\left[1-\\widehat A_t, 1-\\widehat A_{t-}\\right], \\  \\forall \\ \\theta _0 \\leq t\\leq T)\\right)=1,\\ \\ \n \\mathbb{P}\\left (\\int _ {[\\theta _0,T)}1_{\\{\\widehat V_{t-}\\not= 0, \\widehat V_t \\not =0\\}} d\\widehat A_t\\right)=0,\n \\end{equation}\nsuch that the dual minimizer can be decomposed as\n$$\\hat Y= y \\hat Z\\hat D.$$\n\\end{Theorem}\n\nThe proof of the Theorem \\ref{ExactSlackness} is split in several results. \n\\begin{Lemma}\\label{lem5211} Let the conditions of Theorem \\ref{mainTheorem2}  hold. With  \n$$z{:=} xy+\\int_0^T q_sr_sdK_s,$$ there exist $y^n>0$,  $Z^n\\in\\mathcal Z'$, and $D^n\\in\\mathcal D'$, such that $y^nZ^nD^n\\in z\\mathcal G(x,q)$, $n\\in\\mathbb N$, and\n\\begin{equation}\\label{5231}\n\\begin{array}{c}\n\\widehat Y\n= \\lim\\limits_{n\\to\\infty} y^nZ^nD^n,\\quad (d\\kappa\\times \\mathbb P)-a.e.\\\\\ny^n\\to y>0;\\\\%\\textcolor{blue}{\\quad Z^n\\to \\widehat Z,\\quad D^n\\to \\widehat D,\\quad (d\\kappa\\times \\mathbb P)-a.e.}\\\\\n\\end{array}\n\\end{equation}\nFor $A^n$, $n\\in\\mathbb N$, being  in relation to $D^n$ exactly as described before Theorem \\ref{ExactSlackness} we have\n\\begin{equation}\\label{86*}\\mathbb{E}\\left [\\int _{\\theta _0}^T \\widehat V_t Z^n_t dA^n _t \\right]=\\mathbb{E}\\left [Z^n_T\\int _{\\theta _0}^T \\widehat V_t dA^n _t \\right]\\rightarrow 0.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof} Optimality of $\\widehat Y$ and Corollary \\ref{lemSt4} imply \\eqref{5231}.  \nBy \\eqref{2192} and Fatou's lemma, we get\n\\begin{equation}\\label{eqndefz}\nz = \\mathbb E\\left[\\int_0^T \\widehat Y_s\\widehat c_sd\\kappa_s \\right] \\leq \\liminf\\limits_{n\\to\\infty}y^n\\mathbb E\\left[ \\int_0^T Z^n_sD^n_s\\widehat  c_sd\\kappa_s\\right].\n\\end{equation}\nLet us fix $n\\in\\mathbb N$ and consider $\\mathbb E\\left[ \\int_0^T Z^n_sD^n_s\\widehat  c_sd\\kappa_s\\right]$. Using localization and integration by parts (along the lines of the proof of Lemma \\ref{6-16-1}), we have\n\\begin{displaymath}\\begin{array}{rcl}\n\\mathbb E\\left[ \\int_0^T Z^n_sD^n_s\\widehat  c_sd\\kappa_s\\right] &=& \\mathbb E\\left[Z^n_T \\int_0^T D^n_{s}\\widehat  c_sd\\kappa_s\\right] \\\\\n&=& \\mathbb E\\left[Z^n_T \\int_0^T \\left(1 - A^n_{s-}\\right)\\widehat  c_sd\\kappa_s\\right] \\\\\n&=& \\mathbb E\\left[Z^n_T \\left( \\int_0^T\\widehat  c_sd\\kappa_s - \\int_0^T A^n_{s-} \\widehat c_sd\\kappa_s\\right)\\right] \\\\\n&=& \\mathbb E\\left[Z^n_T \\left( \\int_0^T\\widehat  c_sd\\kappa_s - A^n_{T} \\int_0^T \\widehat c_sd\\kappa_s + \\int_0^T(\\int_0^t\\widehat c_sd\\kappa_s)dA^n_t \\right)\\right] \\\\\n&=& \\mathbb E\\left[Z^n_T \\int_0^T(\\int_0^t\\widehat c_sd\\kappa_s)dA^n_t \\right].\\\\\n\\end{array}\n\\end{displaymath}\nThe latter expression, using \\eqref{5211} and with   $\\bar X {:=} ||q||_{\\mathbb L^\\infty(dK)}X''$ (where in turn $X''$ is given by the assertion $(v)$ of Lemma \\ref{lemma1}), we can rewrite as\n\\begin{equation}\\label{5212}\n\\mathbb E\\left[Z^n_T \\int_0^T\\left((x + \\int_0^{t}\\widehat H_sdS_s +\\bar X_t) + \\left(\\int_0^tq_se_sd\\kappa_s - \\bar X_t\\right) - \\widehat V_t\\right)dA^n_t \\right].\n\\end{equation}\nLet us denote\n\\begin{equation}\\nonumbe\n\\begin{array}\n{rcl}\nT_1 &{:=} & \\mathbb E\\left[Z^n_T \\int_0^T\\left(x + \\int_0^{t}\\widehat H_sdS_s +\\bar X_t\\right)dA^n_t \\right] ,\\\\\nT_2 &{:=} & \\mathbb E\\left[Z^n_T \\int_0^T\\left(\\int_0^tq_se_sd\\kappa_s - \\bar X_t\\right)dA^n_t \\right].\\\\\n\\end{array}\n\\end{equation}\nIt follows from nonnegativity of $\\widehat V$ in \\eqref{5211}, Lemma \\ref{lemma1}, and nonnegativity of $\\widehat c$ that \n\\begin{equation}\\label{5232}\nx + \\int_0^{t}\\widehat H_sdS_s + \\bar X_t \\geq \\int_0^t\\widehat c_sd\\kappa_s + \\bar X_t -\\int_0^t q_se_sd\\kappa_s \\geq \\bar X_t -\\int_0^t q_se_sd\\kappa_s \\geq0,\\quad t\\in[0,T],\n\\end{equation}\ni.e., the integrand in $T_1$ is nonnegative. Let ${\\mathbb Q^n}$ be the probability measure, whose density process with respect to $\\mathbb P$ is $Z^n$. As $(x + \\int_0^{\\cdot}\\widehat H_sdS_s + \\bar X)$ and $\\bar X$ are local martingales under ${\\mathbb Q^n}$, by \\cite[Theorem III.27, p. 128]{Pr}, $A^n_{-}\\cdot(x + \\int_0^{\\cdot}\\widehat H_sdS_s + \\bar X)$ and $A^n_{-}\\cdot\\bar X$ are local martingales. Let $(\\tau_k)_{k\\in\\mathbb N}$ be a localizing sequence for both $A^n_{-}\\cdot(x + \\int_0^{\\cdot}\\widehat H_sdS_s + \\bar X)$ and $A^n_{-}\\cdot\\bar X$. \nBy the monotone convergence theorem and integration by parts, we get\n\\begin{equation}\\label{5214}\n\\begin{array}{rcl}\nT_1 &=& \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[\\int_0^{\\tau_k}\\left(x + \\int_0^{t}\\widehat H_sdS_s +\\bar X_t\\right)dA^n_t \\right] \\\\\n&=& \\lim\\limits_{k\\to\\infty}\\left(\\mathbb E^{\\mathbb Q^n}\\left[ \\int_0^{\\tau_k}(-A^n_{t-})d(x + \\int_0^{t}\\widehat H_sdS_s + \\bar X_t) + A^n_{\\tau_k} \\left( x + \\int_0^{\\tau_k}\\widehat H_sdS_s +\\bar X_{\\tau_k}\\right)\\right]\\right) \\\\\n&=&\\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[A^n_{\\tau_k} \\left( x + \\int_0^{\\tau_k}\\widehat H_sdS_s +\\bar X_{\\tau_k}\\right)\\right].\\\\\n\\end{array}\n\\end{equation}\nLet us consider $T_2$. With $\\mathcal E^q {:=} \\int_0^{\\cdot} q_se_sd\\kappa_s$, Lemma \\ref{lemma1} implies positivity of $\\mathcal E^q_t - \\bar X_t$, $t\\in[0,T]$, which in turn allows to invoke the monotone convergence theorem, and we obtain \n\\begin{equation}\\nonumbe\n\\begin{array}{rcl}\nT_2 &=& \\mathbb E^{\\mathbb Q^n}\\left[ \\int_0^T\\left(\\mathcal E^q_t - \\bar X_t\\right)dA^n_t \\right] \\\\\n&=&\\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[ \\int_0^{\\tau_k}\\mathcal E^q_t dA^n_t - \\int_0^{\\tau_k}\\bar X_tdA^n_t \\right].\\\\\n\\end{array}\n\\end{equation} \nBy positivity of $\\bar X$ and the monotone convergence theorem, we have $$\\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[\\int_0^{\\tau_k}\\bar X_tdA^n_t \\right]=\\mathbb E^{\\mathbb Q^n}\\left[\\int_0^{T}\\bar X_tdA^n_t \\right].$$\nTherefore, $\\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[ \\int_0^{\\tau_k}\\mathcal E^q_t dA^n_t\\right]=\\mathbb E^{\\mathbb Q^n}\\left[ \\int_0^{T}\\mathcal E^q_t dA^n_t\\right]$. We deduce that \n\\begin{displaymath}\n\\mathbb E^{\\mathbb Q^n}\\left[\\int_0^{\\tau_k}\\bar X_tdA^n_t \\right] = \\mathbb E^{\\mathbb Q^n}\\left[-(A^n_{-}\\cdot\\bar X)_{\\tau_k} +\\bar X_{\\tau_k}A^n_{\\tau_k}\\right] = \\mathbb E^{\\mathbb Q^n}\\left[\\bar X_{\\tau_k}A^n_{\\tau_k}\\right].\n\\end{displaymath}\nWhereas, in the other term in $T_2$, we get\n\\begin{displaymath}\n\\begin{array}{rcl}\n\\mathbb E^{\\mathbb Q^n}\\left[ \\int_0^{T}\\mathcal E^q_t dA^n_t\\right] &= &\\mathbb E^{\\mathbb Q^n}\\left[\\mathcal E^q_{T}A^n_{T} - (A^n_{-}\\cdot \\mathcal E^q)_{T}\\right] \\\\\n&= &\\mathbb E^{\\mathbb Q^n}\\left[\\mathcal E^q_{T} - (A^n_{-}\\cdot \\mathcal E^q)_{T}\\right] \\\\\n&= &\\mathbb E^{\\mathbb Q^n}\\left[((1- A^n_{-})\\cdot \\mathcal E^q)_{T}\\right] \\\\\n&= &\\mathbb E^{\\mathbb Q^n}\\left[(D^n\\cdot \\mathcal E^q)_{T}\\right].\\\\\n\\end{array}\n\\end{displaymath}\nusing integration by parts and localization, as in the proof of Lemma \\ref{6-16-1}, we can rewrite the latter expression as\n$$\n\\mathbb E\\left[\\int_0^T D^n_sZ^n_sq_se_sd\\kappa_s\\right].\n$$\nWe conclude that\n$$T_2 = -\\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[\\bar X_{\\tau_k}A^n_{\\tau_k}\\right] +\\mathbb E\\left[\\int_0^T D^n_sZ^n_sq_se_sd\\kappa_s\\right].$$\nCombining this with \\eqref{5214}, we obtain \n$$T_1 + T_2  = \\mathbb E\\left[\\int_0^T D^n_sZ^n_sq_se_sd\\kappa_s\\right] + \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[A^n_{\\tau_k}\\left(x + \\int_0^{\\tau_k}\\widehat H_sdS_s +\\bar X_{\\tau_k} \\right)\\right] - \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[A^n_{\\tau_k}\\bar X_{\\tau_k} \\right].$$\nAs both limits in the right-hand side exist and by positivity of $\\left(x + \\int_0^{\\tau_k}\\widehat H_sdS_s +\\bar X_{\\tau_k} \\right)$, established in \\eqref{5232}, we can bound the difference of the limits as\n\\begin{displaymath}\\begin{array}{c}\n \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[A^n_{\\tau_k}\\left(x + \\int_0^{\\tau_k}\\widehat H_sdS_s +\\bar X_{\\tau_k} \\right)\\right] - \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[A^n_{\\tau_k}\\bar X_{\\tau_k} \\right] \\\\\n \\leq \n  \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[x + \\int_0^{\\tau_k}\\widehat H_sdS_s +\\bar X_{\\tau_k}\\right] - \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[A^n_{\\tau_k}\\bar X_{\\tau_k} \\right] \\\\\n   \\leq \n   \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[x + \\int_0^{\\tau_k}\\widehat H_sdS_s\\right] + \n \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[(1-A^n_{\\tau_k})\\bar X_{\\tau_k} \\right].\\\\\n\\end{array}\n\\end{displaymath}\nBy definition of $\\mathcal M'$, $\\bar X$ is a uniformly integrable martingale under $\\mathbb Q^n$. Therefore, as $(1-A^n_{\\tau_k})$ is bounded, we can pass the limit inside of the expectation to obtain $\\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[(1-A^n_{\\tau_k})\\bar X_{\\tau_k} \\right] = 0$. In turn $x + \\int_0^{\\cdot}\\widehat H_sdS_s$ is a supermartingale under $\\mathbb Q^n$ (see the argument in the proof of Lemma \\ref{6-16-1}). We conclude that \n$$\nT_1 + T_2 \\leq x + \\mathbb E\\left[\\int_0^T D^n_sZ^n_sq_se_sd\\kappa_s\\right] = x + \\int_0^T q_s\\rho^n_sdK_s,\n$$\nfor some $\\rho^n$, which is well-defined by Corollary \\ref{corRzd}, and such that $y^n(1,\\rho^n)\\in z\\mathcal B(x,q)$, since $y^nZ^nD^n\\in z\\mathcal G(x,q)$.   \nThus, from \\eqref{eqndefz} and \\eqref{5212}, we get\n\\begin{displaymath}\n\\begin{array}{rcl}\nz&\\leq &(x + \\int_0^T q_s\\rho^n_sdK_s)y^n  -\\lim\\limits_{n\\to\\infty}y^n\\mathbb E\\left[Z^n_T \\int_0^T\\widehat V_tdA^n_t \\right].\\\\\n\\end{array}\n\\end{displaymath}\nBy optimality of $\\widehat Y$, $z\\geq (x + \\int_0^T q_s\\rho^n_sdK_s)y^n\\geq 0$. Therefore, by nonnegativity of $\\mathbb E\\left[Z^n_T \\int_0^T\\widehat V_tdA^n_t \\right]$, and since $y^n$ converges to a strictly positive limit, we conclude that\n$$\\lim\\limits_{n\\to\\infty}\\mathbb E\\left[Z^n_T \\int_0^T\\widehat V_tdA^n_t \\right] = 0.$$\nApplying integration by parts and localization we deduce the assertion of the lemma.\n\\end{proof}\n\\begin{Remark} We emphasize again that  $dA^n$ are probability measures on $[\\theta _0,T]$, which can have mass at the endpoints, and \n $D^n_t=1-A^n_{t-}, \\theta _0\\leq t\\leq T$, $D^n_{T+}=0$.\n\\end{Remark}\n\nIn the subsequent part, we will follow the notations of Lemma \\ref{lem5211} and we will work with a further subsequence, still denoted by $n$, such that\n$y_n\\rightarrow  y>0$, \n\\begin{equation}\n\\label{cone-product}\nZ^nD^n\\rightarrow \\frac{\\widehat Y}{y}, \\quad (d\\kappa\\times\\mathbb P)-a.e, \n\\end{equation} and\n\\begin{equation}\\label{fast}\n\\sum _{k=n}^{\\infty}\\mathbb E\\left[Z^n_T \\int_0^T\\widehat V_tdA^n_t \\right]\\leq \\frac{2^{-n}}{n},\n\\end{equation}\nwhere the existence of a subsequence satisfying \\eqref{fast} follows from \\eqref{86*}.\nThe next results is a Komlos-type lemma, largely based on the results in  \\cite{CzichSchachOSSup}, applied to the \\emph{double-sequence} of processes $(Z^n, \\left (D^{n}\\right)^{-1})$.   We observe that the process $\\left ( D^n\\right )^{-1}$ takes values in $[1, \\infty]$ is increasing,  left-continuous,  and satisfies\n$$\\left ( D^n _t\\right )^{-1}=1,\\quad  0\\leq t\\leq \\theta _0.$$\n\\begin{Lemma}\\label{5219} \nLet the conditions of Theorem \\ref{mainTheorem2}  hold. In the notations of Lemma \\ref{lem5211},\nfor each $n$, there exist  a finite index $N(n)$ and convex weights\n$$\\alpha _{n,k}\\geq 0,\\quad   k=n,..., N(n),\\quad \\sum _{k=n}^{N(n)}\\alpha _{n,k}=1,$$ \nand there exists a strong optional super-martingale $\\widehat Z$ and a non-decreasing  (not necessarily left-continuous) process $\\widehat D$ \nwith $\\widehat D_t=0, \\ \\forall 0\\leq t\\leq \\theta_0$, \nsuch that, simultaneously, \n\\begin{enumerate}\n\\item $\\tilde Z^n{:=} \\sum _{k=n}^{N(n)}\\alpha _{n,k}Z^k\\rightarrow \\widehat Z$ in the sense of \\cite{CzichSchachOSSup} i.e. for any stopping time $0\\leq \\tau\\leq T$ we have \n$$\\tilde Z^n _{\\tau} {\\longrightarrow}^{\\mathbb{P}}  \\widehat Z_{\\tau},$$\nand \n\\item \n$$ \\mathbb{P}\\left (\\sum _{k=n}^{N(n)}\\alpha _{n,k}(D^k_t)^{-1}\\rightarrow (\\widehat D_t )^{-1}, \\quad  \\forall  0\\leq t \\leq T+ \\right)=1$$\n\n We have set $D_{T+}=D^k_{T+}=0$. \\end{enumerate}\nFurthermore,  with the notation\n\\begin{equation}\\label{a-hat}\n \\widehat A _t {=} 1-\\widehat D_{t+}, \\quad  0\\leq t\\leq T,\\quad \\widehat A_{0-}=0, \n\\end{equation}\nwe have the probability measure  $d\\widehat A$ on $[\\theta _0,T]$ such that\n$$\\mathbb{P}\\left (\\widehat D_t\\in \\left[1-\\widehat A_t, 1-\\widehat A_{t-}\\right], \\  \\forall \\ \\theta _0 \\leq t\\leq T)\\right)=1.$$\nDenoting by \n\\begin{equation}\\label{a-tilde-n}\n\\tilde A_t^n{:=} 1- \\left (\\sum _{k=n}^{N(n)}\\alpha _{n,k}(1-A^k_t)^{-1}\\right)^{-1}=1-\n\\left (\\sum _{k=n}^{N(n)}\\alpha _{n,k}(D^k_{t+})^{-1}\\right)^{-1}, \\quad  0\\leq t\\leq T, \\quad \\tilde A ^n_{0-}=0,\n\\end{equation}\nthe point-wise convergence in time (at all times where there is continuity)  additionally implies \n$$d\\tilde A^n\\rightharpoonup d\\widehat A,\\quad \\mathbb{P}-a.e.$$ in the sense of weak convergence of probability measures on $[\\theta _0, T]$.  \\end{Lemma}\n\\begin{proof}\nThe proof reduces to applying  the  Komlos-type results in \\cite{CzichSchachOSSup} to the sequence $Z^n$ and  \\cite[Proposition 3.4]{campi-schachermayer} to the sequence $(D^{n})^{-1}$, simultaneously. We observe that: \n\\begin{enumerate}\n\\item first, no bounds are needed    for $D^{-1}$'s  since we can apply the unbounded Komlos lemma in \\cite[Lemma A1.1]{DS} (and Remark 1 following it) to the proof from \\cite[Propositions 3.4]{campi-schachermayer}, and this works even for infinite values (according to  Remark 1 after \\cite[Lemma A1.1]{DS}). Also,  predictability  can be replaced by optionality, without any change to the proof.\n\\item the Komlos arguments can be applied to both sequences $Z^n$ and $D^n$ simultaneously, with the same convex weights. In order to do this,   we first apply Komlos  to one sequence, then replace both original sequences by their  convex combinations with the weights just  obtained, and then apply Komlos again for the other sequence, and update the convex weight to both sequences.\n\\end{enumerate}\n\n\\end{proof}\n\n\\begin{Corollary} Let the conditions of Theorem \\ref{mainTheorem2}  hold. In the notations of Lemma \\ref{5219}, we have the representation\n$$\\widehat Y= y\\widehat Z\\widehat D,\\quad (d\\kappa\\times\\mathbb P)-a.e.$$\n\\end{Corollary}\n\\begin{proof} Consider an observation that, for non-negative\nnumbers  $a_k, b_k$, $k=n,\\dots, N(n)$, we have\n$$\\min _{k=n,...,N(n)} \\left (\\frac{a_k}{b_k}\\right)\\leq \\frac{\\sum _{k=n}^{N(n)}\\alpha _{n,k}a^k}{\\sum _{k=n}^{N(n)}\\alpha _{n,k}b_k}\\leq  \\max _{k=n,...,N(n)} \\left (\\frac{a_k}{b_k}\\right).$$\\\nWe apply this to $a_k=Z^k, b_k=(D^k)^{-1},$ to obtain\n$$\\min _{k=n,...,N(n)} \\left (Z^kD^k\\right)\\leq \n \\frac{\\sum _{k=n}^{N(n)}\\alpha _{n,k}Z^k}{\\sum _{k=n}^{N(n)}\\alpha _{n,k}(D^k)^{-1}}=\\tilde Z^n \\tilde D^n \\leq\n  \\max _{k=n,...,N(n)} \\left (Z^kD^k\\right),$$\npointwise a.e.  in the product space, where $\\tilde D^n {:=} \\frac{1}{\\sum _{k=n}^{N(n)}\\alpha _{n,k}(D^k)^{-1}}$. \n  Since $Z^nD^n\\rightarrow \\widehat Y\/  y$, $(d\\kappa\\times \\mathbb P)$-a.e., we conclude that both\n  $\\left(\\min _{k=n,...,N(n)} \\left (Z^kD^k\\right)\\right)_{n\\in\\mathbb N}$ and  $\\left(\\max _{k=n,...,N(n)} \\left (Z^kD^k\\right)\\right)_{n\\in\\mathbb N}$ converge to $\\frac{\\widehat Y}{y}$, $(d\\kappa\\times \\mathbb P)$-a.e.,\n  therefore\n    \\begin{equation}\n    \\label{conv-product}\\tilde Z^n \\tilde D^n    \\rightarrow \\frac{\\widehat Y}{y},\\quad (d\\kappa\\times \\mathbb P)-a.e.\n   \\end{equation}\nUsing  \\eqref{cone-product} above, if we did not plan to identify the limit $\\widehat Z$   as a strong-supermartingale, but only as a $(d\\kappa\\times \\mathbb P)$-a.e. limit in the product space,  we could only  apply Komlos arguments to the single  sequence $D^{-1}$,  to conclude convergence of the other  convex combination $\\tilde Z^n$ to $\\frac{\\widehat Y}{y\\widehat D}$, defined up to a.e. equality in the product space.\n  \n  \n  With our  (stronger) double Komlos argument, we have that, in addition to \\eqref{cone-product} we have   \\begin{equation}\n  \\label{conv-tau}\\widehat Z_{\\tau}\\widehat D_{\\tau}=\\mathbb{P}-\\lim _n \\tilde Z^n_{\\tau}\\tilde D^n _{\\tau},\\quad \\textrm{for\\ every~[0,T]-valued~stopping \\ time \\ }\\tau,\n  \\end{equation}\n  (where $\\widehat Z$ is a strong supermartingale, and $\\widehat D$ is well defined at all times).\n\n  It remains to prove that $\\widehat Y\/y=\\widehat Z\\widehat D$, $(d\\kappa\\times\\mathbb P)$-a.e. We point out that the convergence \\eqref{conv-tau} is topological. Let us consider an arbitrary optional set $O\\subset \\Omega \\times [0,T]$ and fix an upper bound $M$.  It follows from \\eqref{conv-product} that\n  $$1_O \\min \\left \\{\\tilde Z^n \\tilde D^n, M \\right \\}\\rightarrow  1_O \\min \\left \\{\\frac{\\widehat Y}{y}, M \\right \\},\\quad (d\\kappa\\times \\mathbb P)-a.e.\n$$\nTherefore, we get\n\\begin{equation}\\label{hatY}\\mathbb{E}\\left [\\int _0^T  1_O(t, \\cdot)  \\min \\left \\{\\tilde Z^n_t \\tilde D^n _t, M \\right \\} d\\kappa_t\\right]\\rightarrow  \n\\mathbb{E}\\left [\\int _0^T  1_O(t, \\cdot)  \\min \\left \\{\\frac{\\widehat Y_t}{y}, M \\right \\} d\\kappa_t\\right].\n\\end{equation} \nRecall that the stochastic clock has a density $d\\kappa_t=\\varphi  _t dK_t$ with respect to the deterministic clock $dK _t$.  For each fixed $t$, from \\eqref{conv-tau} we have\n$$1_O(t, \\cdot)  \\min \\left \\{\\tilde Z^n_t \\tilde D^n _t, M \\right \\} \\varphi _t\\rightarrow 1_O(t, \\cdot)  \\min \\left \\{\\widehat Z_t \\widehat D _t, M \\right \\} \\varphi _t,\\quad \\textrm{in}-\\mathbb{P}.$$\nRecalling that $\\mathbb{E}[\\varphi _t]<\\infty$ we have, for fixed $t$, that \n$$ M \\times  \\mathbb{E}[\\varphi _t]\\geq \\mathbb{E}\\left [1_O(t, \\cdot)  \\min \\left \\{\\tilde Z^n_t \\tilde D^n _t, M \\right \\}  \\varphi _t \\right ]\\rightarrow \\mathbb{E}\\left [1_O(t, \\cdot)  \\min \\left \\{\\widehat Z_t \\widehat D _t, M \\right \\} \\varphi _t\\right ].\n$$\nWe integrate the above with respect to the deterministic clock $dK_t$ to obtain\n\\begin{equation}\n\\begin{split}\n\\mathbb{E}\\left [\\int _0^T  1_O(t, \\cdot)  \\min \\left \\{\\tilde Z^n_t \\tilde D^n _t, M \\right \\} d\\kappa_t\\right] =\\int _0^T \\mathbb{E}\\left [1_O(t, \\cdot)  \\min \\left \\{\\tilde Z^n_t \\tilde D^n _t, M \\right \\}  \\varphi _t \\right ]dK_t\\rightarrow \\\\\n\\rightarrow \\int _0^T \\mathbb{E}\\left [1_O(t, \\cdot)  \\min \\left \\{\\widehat Z_t \\widehat D _t, M \\right \\}  \\varphi _t \\right ]dK_t=\n\\mathbb{E}\\left [\\int _0^T  1_O(t, \\cdot)  \\min \\left \\{\\widehat Z_t \\widehat D _t, M \\right \\} d\\kappa_t\\right]\\end{split}\n\\end{equation}\nTogether with \\eqref{hatY} we have\n$$\\mathbb{E}\\left [\\int _0^T  1_O(t, \\cdot)  \\min \\left \\{\\frac{\\widehat Y_t}{\\hat y}, M \\right \\} d\\kappa_t\\right]=\\mathbb{E}\\left [\\int _0^T  1_O(t, \\cdot)  \\min \\left \\{\\widehat Z_t \\widehat D _t, M \\right \\} d\\kappa_t\\right],$$\nwhich holds for any optional set $O$ and any bound $M$, therefore $\\widehat Y= y\\widehat Z\\widehat D$, $(d\\kappa\\times \\mathbb P)$-a.e.\n  \\end{proof}\n\\begin{proof}[Proof of the Theorem \\ref{ExactSlackness}]\nSince$$\\sum _{k=n}^{N(n)}\\alpha _{n,k}(D^k_t)^{-1}\\rightarrow (\\widehat D_t )^{-1}, \\quad t\\in[0,T], $$\nrecalling that $\\widehat A$ was defined from $\\widehat D$ and \\eqref{a-tilde-n}. \nAs the processes $A^n$, therefore $\\tilde A^n$ only increase by jumps.\nTherefore\nwe get\n \\begin{equation}\\label{11291}\n \\begin{array}{rcl}\n \\Delta \\tilde A^n_s &= & \\frac{1}{\\sum\\limits_{k=n}^{N(n)}\\alpha _{n,k}\\frac{1}{D^k_{s}}} - \\frac{1}{\\sum\\limits_{k=n}^{N(n)}\\alpha _{n,k}\\frac{1}{D^k_{s+}}}\\\\\n  &= & \\sum\\limits_{k=n}^{N(n)}\\underbrace{\\frac{\\alpha _{n,k}\\frac{1}{D^k_{s+}}}{\\sum\\limits_{k=n}^{N(n)}\\alpha _{n,k}\\frac{1}{D^k_{s+}}}}_{\\leq 1}\\frac{\\Delta A^k_s}{D^k_{s}}  \n  \\underbrace{\\frac{1}{\\sum\\limits_{k=n}^{N(n)}\\alpha _{n,k}\\frac{1}{D^k_{s}}}}_{\\leq 1}.\\\\\n \\end{array}\n \\end{equation}\n As $\\frac{\\alpha _{n,k}\\frac{1}{D^k_{s+}}}{\\left(\\sum\\limits_{k=n}^{N(n)}\\alpha _{n,k}\\frac{1}{D^k_{s+}}\\right)}\\leq 1$ and $\\sum\\limits_{k=n}^{N(n)}\\alpha _{n,k}\\frac{1}{D^k_{s}} \\geq 1$, we can bound the latter term in \\eqref{11291} by $\\sum\\limits_{k=n}^{N(n)}\\frac{\\Delta A^k_s}{D^k_{s}}$ for $s\\in[0,T]$.\n We deduce that\n $$\\Delta \\tilde A^n_s \\leq \\sum\\limits_{k=n}^{N(n)}\\frac{\\Delta A^k_s}{D^k_{s}},\\quad s\\in[0,T].$$ \nTherefore, we have\n$$\\int_{0}^{t} \\widehat V_u d\\tilde A^n _u  \\leq \\sum _{k=n}^{N(n)}(D^k_{t})^{-1} \\int_{0}^{t} \\widehat V_udA^k_u,\\quad t\\in[0,T],$$\nand thus, we obtain\n\\begin{equation}\\label{main-inez}\n\\left(\\min _{k=n, \\dots, N(n)} (Z^k_tD^k_t) \\right) \\int_{0}^{t} \\widehat V_u d\\tilde A^n _u  \\leq \\sum _{k=n}^{N(n)}(Z^k_tD^k_t)(D^k_{t})^{-1} \\int_{0}^{t} \\widehat V_udA^k_u= \\sum _{k=n}^{N(n)}Z^k_t\\int_{0}^{t} \\widehat V_udA^k_u,\\quad t\\in[0,T].\n\\end{equation}\nSince the process\n$$L^n_t:=\\sum _{k=n}^{N(n)}Z^k_t\\int_{0}^{t} \\widehat V_udA^k_u, \\quad 0\\leq t\\leq T$$\nis a non-negative right-continuous  submartingale, the maximal inequality  and  \\eqref{fast} together imply\n $$\\mathbb{P}\\left  (\\sup _{0\\leq t\\leq T}L^n_t\\geq \\frac 1n \\right )\\leq n\\mathbb{E}[L^n_T]\\leq 2^{-n},\n$$\nso \n\\begin{equation}\n\\label{l}\\sup _{0\\leq t\\leq T}L^n_t\\rightarrow 0,\\quad \\mathbb{P}-a.s.\n\\end{equation}\nSince $Z^nD^n \\rightarrow \\frac{1}{y}\\widehat Y>0$, $(d\\kappa\\times \\mathbb P)$-a.e.,  consequenlty \n$$\\min _{k=n, \\dots, N(n)} (Z^kD^k)\\rightarrow \\frac{1}{y}\\widehat Y>0\\quad \\left(d\\kappa \\times \\mathbb{P}\\right)-a.e.,$$\nwe obtain from\\eqref{main-inez} and \\eqref{l} that \nthe increasing RC process\n$$\\tilde L^n_t:=\\int _0^t \\widehat V_u d\\tilde A^n_u,\\quad 0\\leq t\\leq T,$$\nconverges to zero in the product space.  Denoting by $\\mathcal{O}\\subset \\Omega\\times [0,T]$ the exceptional set where convergence to zero does not take place,  and taking into account that $\\tilde L^n$ are increasing,  we have that for \nfor $$T^{\\mathcal{O}}(\\omega)=\\inf \\{0\\leq t\\leq T: (\\omega ,t)\\in \\mathcal{O}\\},$$\nwe have \n$$(T^{\\mathcal{O}},T]\\subset \\mathcal{O}.$$\nNow \n$$(d\\kappa\\times \\mathbb P)(\\mathcal{O})=0, $$\nimplies that \n$T^{\\mathcal{O}}\\geq T,~\\mathbb{P}-a.s.$ (here used an assumption that $T$ is the minimal time horizon in the sense that the deterministic clock $K$ is such that $K_t<K_T$, for every $t\\in[0,T)$, this assumption does not restrict generality). Thus, there exists a \n a set $\\Omega ^*$  of full probability $\\mathbb{P}(\\Omega ^*)=1$  such that, for each $\\omega \\in \\Omega ^*$ and $t<T$ we have\n $$\\int _0^t \\widehat V_u (\\omega)d\\tilde A^n_t (\\omega)\\rightarrow 0.$$\n\n\nLet us fix an $\\omega\\in \\Omega ^*$ and such that, \nfor this $\\omega$, the probability measure \n$d\\tilde A^{n}(\\omega)$ converges weakly to $d\\widehat A(\\omega)$ over the interval $[\\theta _0(\\omega), T]$.\nThe set of such $\\omega$'s still has probability $1$. The Skorokhod representation theorem asserts that there exists a new probability space  $\\Omega ^{\\omega}$ and a  sequence of random times \n$(t^{n}(\\omega))_{n\\in\\mathbb N}$ as well as $\\widehat t (\\omega)$ such that the distribution of $t^{n} (\\omega)$ is $d\\tilde A^{n}$,  the distribution of $\\widehat t (\\omega)$ is $d \\widehat A (\\omega)$ and\n$$t^{n}(\\omega)\\rightarrow \\widehat t (\\omega),\\quad \\mathbb{P}^{\\omega}-a.s. $$\non the new, artificial, probability space. Fix $t<T$. We have\n$$\\mathbb{E}^{\\omega}[\\widehat V _{t^{n}(\\omega)} (\\omega) 1_{\\{t^n(\\omega)\\leq t\\}}]=\\int _{\\theta _0}^t \\widehat V_u (\\omega)  d\\tilde A^{n} _u(\\omega) \\rightarrow 0.$$\nOne can see that \n$$\\xi (\\omega):=\\liminf _n \\widehat V _{t^{n}(\\omega)}(\\omega)\\in \\{ \\widehat V _{\\widehat t(\\omega)-}(\\omega),  \\widehat V _{\\widehat t(\\omega)}(\\omega)\\}, \\quad \\mathbb{P}^{\\omega}-a.s.$$ \nand \n$$1_{\\{t^n(\\omega)\\leq t\\}}\\rightarrow 1_{\\{\\widehat t (\\omega)\\leq t\\}}\\  \\textrm{on} \\ \\ \\{\\widehat t (\\omega)<t\\},\\ \\mathbb{P}^{\\omega}-a.s.$$ \nTherefore,  \napplying Fatou's lemma on $\\Omega ^{\\omega}$, we obtain\n$$\\mathbb{E}^{\\omega}[\\xi (\\omega) 1_{\\{\\widehat t(\\omega)< t\\}}]=0.$$\nWe recall that both $\\widehat V (\\omega)$ and $\\widehat V_{-}(\\omega)$ are nonnegative, consequently we have\n$$0\\leq \\min \\{ \\widehat V _{\\widehat t(\\omega)-}(\\omega),  \\widehat V _{\\widehat t(\\omega)}(\\omega) \\}\\leq \\xi (\\omega),$$ and therefore we get \n$$\\mathbb{E}^{\\omega}[ \\min \\{ \\widehat V _{\\widehat t(\\omega)-}(\\omega),  \\widehat V _{\\widehat t(\\omega)}(\\omega) \\}1_{\\{\\widehat t(\\omega)< t\\}}]=0.$$\nThis means that the distribution  of $\\widehat t (\\omega)$ over the interval $[\\theta _0(\\omega), t)$ only charges the complement of the set of times \n$$\\left\\{u<t: \\quad \\widehat V_{u-}(\\omega)=0\\quad or\\quad \\widehat V_u (\\omega)=0\\right\\}.$$\nBy taking $t\\rightarrow T$, one completes the proof.\n\\end{proof}\n\\bibliographystyle{alpha\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Introduction}\n\nThere are two motivations for this paper. \n\nFirst, for any compact Riemannian manifold  $M$ there is a natural \nfull symbol calculus for operators acting on sections of an arbitrary \nvector bundle with connection. The symbols are $\\mathop{\\rm End}\\nolimits E$-valued \nfunctions on $T^*M$. This symbol calculus can be viewed as a \nconcrete construction of quantization for the symplectic manifold \n$T^*M$. This is more or less well-known construction (see~\\cite{Bok}, \n\\cite{Wid}, \\cite{Ge}, \\cite{Saf}, \\cite{AS}, \\cite{Pf}).\nSuppose now that the bundle $E$ is the bundle of exterior differential \nforms. It is natural to treat forms as functions themselves \n(functions of odd variables) and then to go one step farther and \n``dequantize'' the operators from $\\mathop{\\rm End}\\nolimits \\Lambda^*M$. The new symbols \nobtained in this way can be identified with  {\\sl forms\\\/} on  \nsymplectic manifold $T^*M$. This is what we do in this paper.\nThis refined symbol calculus can be useful with the relation to the Atiyah-Singer \nIndex Theorem. As an example, we give a straightforward proof of the \nGauss-Bonnet-Chern Theorem.  One of our trace formulas \n(formula~(\\ref{startrace})) can be \nutilized to prove the generalized Hirzebruch Signature Theorem.  \n\n\nThe other motivation is as follows. Suppose we have a symplectic or a \nPoisson manifold $P$. At what conditions the Poisson structure can be \nextended from functions to differential forms~? Is it possible to quantize the \nalgebra of forms~? In the present paper we provide a particular \nexample of the quantization of forms, in the case when the \nmanifold $P$ is a cotangent bundle $T^{*}M$. \n\n``Supersymmetry'' proofs of index theorems originated from the works \nof Witten  and Alvarez-Gaum\\'e ~\\cite{A}, who  used path integrals \nand ``physical'' considerations. First mathematically rigorous \ntreatment was given for the bundle-valued Dirac operator by Getzler~\\cite{Ge}, \nwho combined symbol \ncalculus on Riemannian manifold with Clifford algebra. This approach was \nsubstantially simplified \nin~\\cite{AS} by an explicit use of supergeometry and deformation quantization.\nHowever, there was some peculiarity, caused by a highly asymmetric \nway in which even and odd variables entered the \npicture (see~\\cite[p.1038-1039]{AS} and Remark~\\ref{getz} below).\n(Symbol calculus was abandoned in~\\cite{Ge2} and subsequent works in favor of the analysis \nof asymptotics of the heat kernel on the diagonal.)\nIn this paper we work with forms instead of spinors, and  this helps to achieve\nquantization which is  more ``symmetric'' though leads to index \nformulas.\n\n\\begin{Rem}[on notation]  We use the standard language of \nsupermanifolds (see~\\cite{GM}, \\cite{Manin}, \\cite{Leites}, \\cite{GIT}). \nIn particular,  \nand distinguish between cotangent vectors as elements of \n$T^{*}_{x}M$ and $1$-forms as elements of \n$\\Pi T^{*}_{x}M$, where   $\\Pi$  is the parity reversion functor. \nThe same distinction is made between tangent vectors \nand ``$1$-vectors'' (multivectors of degree $1$). So on even manifold \nvectors and covectors are even, while $1$-vectors and $1$-forms are \nodd.  (As far as I know, there is no established name for the elements of \n$\\Pi V$ for a vector space $V$. Borrowing some physical language, I \ncan suggest to call them {\\it antivectors}, and the space $\\Pi V$ \nthe {\\it antispace}.)\n\\end{Rem}\n\n\\par\\medskip\\noindent {\\bf Acknowledgments.}     The results of \nthis paper were discussed at different time with M.A.~Shubin, \nA.~Weinstein, \nM.~Pflaum, J.~Rabin, L.~Sadun, S.~Rosenberg,  \nand the participants of seminars at UC Berkeley, \nUCSD, University of Texas at Austin, and Boston University. \nN.Yu.~Reshetikhin attracted my attention to the problem of quantizing \nforms as differential algebra.\nI am very \nmuch grateful to all of them.\n\n\\section{Complete symbol calculus for Riemannian manifolds}\\label{QS}\n\n\nConsider a Riemannian manifold  $M$ and a vector bundle  $E$ over \n$M$, with a connection. There is a natural construction that allows \nto assign pseudodifferential operators (p.d.o.'s) to functions on \n$T^*M$. More precisely, we mean p.d.o.'s acting on the sections of $E$, \nand by functions we actually mean ``$\\mathop{\\rm End}\\nolimits E$-valued functions'', i.e. \nthe sections of $\\pi^*(\\mathop{\\rm End}\\nolimits E)$, where $\\pi: T^*M\\to M$ is the \nprojection. (The same can be done for \noperators $\\mathop{\\rm \\Gamma}\\nolimits(E)\\to \\mathop{\\rm \\Gamma}\\nolimits(F)$ for two bundles, $E$, $F$.) In the following, the space of smooth \nsections of {\\sl any} bundle $E$ over {\\sl any} manifold $M$ is \ndenoted by $C^{\\infty}(M,E)$\n\nLet us consider  the geodesic segment connecting two points  $x,y\\in M$ (we \nsuppose that they are sufficiently close for it to exist and be  \nunique), and denote the point with the affine parameter $s$ by \n$g_{s}(x,y)$. Here $g_{0}(x,y)=x$, $g_{1}(x,y)=y$. We shall need some more \nnotation. The points of $T_{x}^*M$ (cotangent vectors, or \n``momenta'', at point $x\\in M$) will \nbe denoted by $p$. Their coordinates will be denoted by $p_a$.  \nMomenta at the point $y$ will be denoted by the letter $q$.\nThe parallel transport $T^{*}_{y}M\\to T^{*}_{x}M$ along the geodesic, \nw.r.t. the Levi-Civita connection, will \nbe denoted by $T(x,y)$. The similar transport for the bundle $E$ will \nbe denoted by $\\tau(x,y)$. We  shall assume that there is a \nneighborhood of the diagonal $\\Delta$ in $M\\times M$ where the \ngeodesic segment exists and is unique. For example, let $M$ be compact \nor $M=\\R{n}$. We shall use a bump function $\\a(x,y)$ which is \nidentically $1$ near $\\Delta \\subset M\\times M$, nonnegative, and \nsupported inside the indicated neighborhood of $\\Delta$. The \nconstruction depends on the choice of $\\a$, though not very \nheavily.\n\n\nIn the following we shall also need  some notation for the volume elements.\nLiouville's measure on $T^*M$ is denoted by $dx\\, dp$ (resp., $dy\\, dq$). \nThe Riemannian volume element on $M$ \nat point $x$ is $\\omega(x)=\\sqrt{h(x)}\\, dx$.  Here \n$h(x)=\\det (h_{{ab}}(x))$ is the Gram determinant in a coordinate frame.\nThere are Euclidean volume \nelements on $T_x M$ and $T^*_x M$. They are denoted by \n$\\theta_x=\\sqrt{g(x)}\\,dv$ \nand $\\lambda_x=(1\/\\sqrt{g(x)})\\,dp$ respectively. Here \n$g(x)=\\det (g_{ij}(x))$ is the Gram determinant in some chosen local \nframe, not necessarily coordinate. The exponential map \n$\\exp_x: T_x M\\to M$ allows to compare volume elements on $M$ and on \nthe tangent space at a fixed point. We introduce the function \n$\\mu(x,y)$ by the following equation:\n\\begin{equation}\n  \\theta_x(v)=\\mu (x,y)\\; \\omega(y),\n\\end{equation}\nif $y=\\exp_x v$, $v\\in T_x M$. This definition is valid for $y$ \nsufficiently close to $x$.\n\n\\subsection{Quantization}\n\n\\begin{de} Fix $s \\in[0,1]$. Let $f\\in C^{\\infty}(T^*M, \\mathop{\\rm End}\\nolimits E)$. We \nassociate with $f$ the following operator $\\hat f$, acting on  \nsections of $E$. For $u\\in C^{\\infty}(M, E)$\n\\begin{multline} \\label{quant}\n(\\hat f u)(x):=\\frac{1}{(2\\pi\\hbar)^n}\\int_{T^*M}dydq \\;\ne^{\\frac{i}{\\hbar}(\\expi{y}{x})\\cdot q}\\; \\a(x,y)\\,\\mu(x,y)\\\\\n \\tau(x,{\\zs}) \\; f({\\zs},T({\\zs},y)q) \\;\\tau({\\zs},y)\\;u(y).\n\\end{multline}\n\\end{de}\n\nWe can make a change of variables: $y=\\exp_x v, q=T(y,x)p$. This \nyields\n\\begin{de}[equivalent] \n\\begin{multline} \\label{quantprime}\n(\\hat f u)(x):=\\frac{1}{(2\\pi\\hbar)^n}\\int_{T_x M\\times T_x^*M}dvdp \n\\;e^{-\\frac{i}{\\hbar}vp} \\;\\a(x,\\exp_x v)\\\\\n\\tau(x,\\exp_x sv) \\; f(\\exp_x sv,T(\\exp_x sv,x)p) \\;\n\\tau(\\exp_x sv,\\exp_x v)\\;u(\\exp_x v).\n\\end{multline}\n\\end{de}\n\n\n\n\nThese formulas might look complicated but the idea is very simple.\n\n\\begin{ex} Consider $M=\\R{n}$. Take $\\a:=1$ (no bump function is \nnecessary). \nIn affine coordinates on $\\R{n}$ and in parallel frame for the (trivial) bundle \n$E$, the formula~(\\ref{quant}) becomes\n \\begin{equation} \n(\\hat f u)(x):=\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{2n}}dydq \\;e^{\\frac{i}{\\hbar}(y-x)\\cdot \nq}\\;\n f((1-s)x+sy),q) \\;u(y),\n\\end{equation}\nand the formula ~(\\ref{quantprime}) becomes\n\\begin{equation} \n(\\hat f u)(x):=\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{n}\\times \\R{n}}dvdp \\; \ne^{-\\frac{i}{\\hbar}vp} \\; f(x +sv,p) \n\\;u(x + v).\n\\end{equation}\nFor $s=0$ this yields $xp$-quantization (= standard \npseudodifferential calculus), for $s=1$ this is $px$-quantization \n(the opposite way of ordering monomials), and the case $s=1\/2$ yields Weyl \n(symmetric) quantization (cp. ~\\cite{BS}).\n\\end{ex}\n\n\\begin{ex} Take a function $f=f(x)$, not depending on $p$. Then, by \n~(\\ref{quantprime}), \n\\begin{multline}\n\t(\\hat f u)(x) = \\int_{T_x M}\\theta_x(v) \\;\\delta(v)\\;\n\\tau(x,\\exp_x sv) \\; f(\\exp_x sv) \\;\n\\tau(\\exp_x sv,\\exp_x v)\\;u(\\exp_x v)\\\\=f(x)\\, u(x).\n\\end{multline}\n(Here $\\delta$ stands for the ``Euclidean'' delta-function on $T_xM$.)\n\\end{ex}\n\n\n \\subsection{Symbols}\n\nLet $K$ be the Schwartz kernel of $A: C^{\\infty}(M,E)\\to \nC^{\\infty}(M,E)$ with respect to the Riemannian volume element:\n\\begin{equation}\n\t(Au)(x)=\\int_M \\!\\omega(y) \\;K(x,y)\\,u(y).\n\\end{equation}\n\n\n\\begin{de} The following function on $T^*M$ is called the {\\it \nsymbol} of the operator $A$ and is denoted by $\\sigma A$:\n\n\\begin{multline} \\label{symbol}\n(\\sigma A)(x,p):=\\int_{T_x M}\\theta_x(v) \\;e^{\\frac{i}{\\hbar}vp} \n\\;\\a(\\exp_x(-sv),\\exp_x (1-s)v)\\; \\rho_s(x,v)\\\\\n\\tau(x,\\exp_x (-sv)) \\; K(\\exp_x (-sv),\\exp_x (1-s)v) \\;\n\\tau(\\exp_x (1-s)v,x).\n\\end{multline}\nHere \n\\begin{equation} \\rho_s(x,v):=\\left( \\mu(\\exp_x (-sv),\\exp_x \n(1-s)v)\\right)^{-1}.\n\\end{equation}\n\\end{de}\n\nNote that $\\sigma A\\in C^{\\infty}(T^*M,\\mathop{\\rm End}\\nolimits E)$, and that $K(x,y)\\in \n\\mathop{\\rm Hom}\\nolimits (E_y,E_x)$, for fixed $x,y$.\n\n\n\n\\begin{rem} For $s=0$  the symbol  of an operator $A$ can be calculated as follows:\n\\begin{equation}\\label{specsymb}\n\t(\\sigma A)(x,p)u_0=\\left(A_y\\left(e^{\\frac{i}{\\hbar}(\\expi{x}{y})\\cdot p} \n\\a(x,y)\\;\\tau(y,x)\\;u_0 \\right)\\right)_{|y:=x},\n\\end{equation}\nfor an arbitrary constant vector $u_0\\in E_x$.  Here $A_y$ means operator \nacting on sections which argument is denoted by $y$.\n\\end{rem}\n\n\\begin{thm}  1. The quantization map $f\\mapsto \\hat f$ and the symbol \nmap $A\\mapsto \\sigma A$ are ``almost'' mutually inverse:\n\\begin{align} \n\t\\sigma\\hat f&=f(1 + O(\\hbar^{\\infty})), \\label{sq}\\\\\n\tK_{\\widehat{\\sigma A}}(x,y)&=K_{A}(x,y)\\cdot(\\a(x,y))^2,\\label{qs}\n\\end{align}\nfor any $f$ and $A$. Here $K_A$ denotes the Schwartz kernel of an \noperator $A$.\n\n2. For polynomials in $(p_{a})$ and for differential operators the maps \n    $\\ \\hat{}$ and $\\sigma$ are mutually inverse. In this case the \nconstruction is independent of the bump function $\\a$.\n\\end{thm}\n\n\\begin{co} \\label{kerdiag}If $A$ has a continuous kernel and is of \ntrace class, then the traces of $A$ and $\\widehat{\\sigma A}$ coincide.\n\\end{co}\n\\begin{proof} By equation~(\\ref{qs}), the kernels  of  $A$  and $\\widehat{\\sigma \nA}$ coincide  on the diagonal.\n\\end{proof}\n\n\n\\begin{rem} A complete symbol calculus for \nRiemannian manifolds was for the first time considered by \nBokobza-Haggiag~\\cite{Bok} and Widom~\\cite{Wid}. Later various \nmodifications were suggested and used: see~\\cite{Ge}, \\cite{AS}, \\cite{Pf}, \\cite{Saf}, \nalso~\\cite{Gutt}, \\cite{bnw}; probably more references can be given. \nThough all approaches are \nbased on the same idea (use of connection, of the exponential map and the parallel \ntransport along geodesics), there are small subtleties leading to \ninequivalent constructions. In most approaches the standard \npseudodifferential calculus (or ``$qp$''-quantization) on $\\R{n}$ has \nbeen generalized. The beautiful idea to use a middle point on a \ngeodesic segment to generalize Weyl symbols and \nother ``$s$-symbols'', compare~\\cite{Shub}, is due to Yu.G.~Safarov. \n(I use the opportunity to thank M.A.~Shubin from whom I learned about Safarov's approach \naround 1989.)  The above definitions of \nquantization and symbols are based on  this idea.  For $s=0$ the construction is basically \nequivalent to the one used in~\\cite{AS}, up to a slightly different \nintroduction of bump function. Since we follow the philosophy \nof deformation quantization, our formulas contain Planck constant, \nwhich is of course very important.\n\\end{rem}\n\n\\subsection{Composition}\n\n\\begin{de} The operation $f\\circ g:= \\sigma (\\hat f\\hat g) $ is \ncalled the {\\it composition } of functions $f,g\\in C^{\\infty}(T^*M, \n\\mathop{\\rm End}\\nolimits E)$.\n\\end{de}\n\n\\begin{thm} For $\\hbar \\to 0$,\n\\begin{equation} f\\circ g=fg \\,(1 + O(\\hbar)).\n\\end{equation}\nIf $f,g$ are scalar functions, then\n\\begin{equation} f\\circ g-g\\circ f= -{i}{\\hbar}\\,\\{f,g\\}\\;(1 + \nO(\\hbar)),\n\\end{equation}\nwith canonical Poisson brackets on $T^*M$.\n\\end{thm}\n\\begin{proof} The first statement follows by direct computation. To \nprove the second statement, note that since in the limit $\\hbar\\to 0$ \nin scalar case we obtain \ncommutative multiplication, the commutator \nw.r.t. the composition induces {\\it some} Poisson bracket on functions. Thus it is \nsufficient to check the brackets for local coordinates $x^a,p_a$. \nThat the induced brackets for them coincide with the canonical ones, follows \nfrom the calculation below.\n\\end{proof}\n\n\n\n\n\n\\subsection{Examples: quantizing linear and quadratic Hamiltonians}\n\n\nIn the subsequent  calculations we fix some point $x\\in M  $ and work \nin normal coordinates centered at $x$. By definition, then $(\\exp_x \nv)^a=v^a$, and $x^a=0$, and $\\Gamma _{bc}^a(0)=0$. We also introduce a \nframe in $E$ parallel along the geodesic radii $y^a=tv^a$, which we  \ncall the {\\it parallel frame}. It is specified by two properties:  \n$A(0)=0$, $y^a A_a(y)=0$ for the connection $1$-form $A$ in $E$. This \nimplies that $\\d_aA_b(0)$ is antisymmetric and thus equals \n$F_{ab}(0)$, where $F$ is the curvature $2$-form. We can rewrite the \nformula~(\\ref{quantprime}) in the normal coordinates and the parallel \nframe:\n\\begin{equation}\\label{simple}\n\t(\\hat f u)(0)=\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{n}\\times\\R{n}}dvdp \n\\;e^{-\\frac{i}{\\hbar}vp} \\;\\a(x,v)\\; f(sv,T(sv,0)p) \\;u(v).\n\\end{equation}\nIt looks almost as for $\\R{n}$.\n\n\\begin{ex} Let $X=X^a\\d_a\\in \\mathop{\\rm Vect}\\nolimits M$ and consider the corresponding \nfiberwise-linear Hamiltonian $X\\cdot \\mathrm p=X^a p_a$. By~(\\ref{simple}) we have:\n\\begin{multline*} \n\t(\\widehat{X\\cdot \\mathrm p}\\,u)(0)=\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{n}\\times\\R{n}}dvdp \n\\;e^{-\\frac{i}{\\hbar}vp} \\; X^a(sv)\\,(T(sv,0)p)_a \\;u(v)\\\\\n=\n\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{n}}dp\\; p_a\\int_{\\R{n}}dv \\;e^{-\\frac{i}{\\hbar}vp} \\; \n(T(0,sv)X(sv))^a\\;u(v)\\\\\n=\n-i\\hbar \\der{}{v^a}\\left( (T(0,sv)X(sv))^au(v)\\right)_{|v=0}\\\\\n=-i\\hbar(s\\cdot \\nabla_aX^a(0)+X^a(0)\\,\\d_a u(0)),\n\\end{multline*}\nwhich we can rewrite in the invariant form:\n\\begin{equation}\n\t\\widehat{X\\cdot \\mathrm p}=-i\\hbar \\,(\\nabla_X+s\\div X).\n\\end{equation}\n\\end{ex}\n\nIf we apply this example to $X=\\d_{a}$, then we immediately obtain \ncanonical commutation relations and canonical Poisson brackets.\n\n\\begin{ex}\nConsider the Hamiltonian $\\mathrm p^2=g^{ab}p_ap_b$, corresponding to \nthe metric on $M$. In the same way,\n\\begin{multline*} \n\t(\\widehat{\\mathrm p^2}u)(0)=\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{n}\\times\\R{n}}dvdp \n\\;e^{-\\frac{i}{\\hbar}vp} \\; g^{ab}(sv)\\,(T(sv,0)p)_a\\; \n(T(sv,0)p)_b\\;u(v)\\\\\n=\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{n}\\times\\R{n}}dvdp \\;e^{-\\frac{i}{\\hbar}vp} \\; \ng^{ab}(0)p_ap_b\\;u(v)\\\\\n=-\\hbar^2 g^{ab}(0)\\;\\partial^2_{ab}u(0)=\n-\\hbar^2 g^{ab}(0)(\\nabla_a\\nabla_b u-\\Gamma_{ba}^c\\nabla_cu)(0).\n\\end{multline*}\nWe used the fact that parallel transport on $M$ is an orthogonal map. \nThus, independent of $s$,\n\\begin{equation}\\label{lapsymb}\n\t\\widehat{\\mathrm p^2}=-\\hbar^2 \\Delta,\n\\end{equation}\nwhere\n$\\Delta$ stands for the Laplace-Beltrami operator on a Riemannian \nmanifold, acting on sections of a vector bundle with a connection:\n\\begin{equation}\n\t\\Delta u:=g^{ab}(\\nabla_a\\nabla_b\\, u-\\Gamma_{ba}^c\\nabla_c\\,u),\n\\end{equation}\nwhere $\\nabla_a$ in this formula denotes partial covariant derivative \nof the sections of $E$.\n\\end{ex}\n\n\n\n\n\\subsection{The trace formula}\n\n\nIf we take as a working definition of the trace of an operator $A$ \nthe integral of its kernel over the diagonal $\\Delta\\subset M\\times \nM$, then we easily obtain the following \n\\begin{thm} \\label{trace}\nFor all $s\\in [0,1]$, the trace is calculated by the same  \nformula:\n\\begin{align}\n\t\\mathop{\\rm Tr}\\nolimits \\hat f&=\\frac{1}{(2\\pi\\hbar)^n}\\int_{T^*M} dxdp\\;\\mathop{\\rm tr}\\nolimits f(x,p),\\label{trq}\\\\\n\t\\mathop{\\rm Tr}\\nolimits A&=\\frac{1}{(2\\pi\\hbar)^n}\\int_{T^*M} dxdp\\;\\mathop{\\rm tr}\\nolimits \\,(\\sigma A)(x,p),\\label{tra}\n\\end{align}\nfor any $A$ and $f$. Here $\\mathop{\\rm tr}\\nolimits$ denotes the trace on $\\mathop{\\rm End}\\nolimits E_x$.\n\\end{thm}\n\\begin{proof} From~(\\ref{quant}) we can deduce the following formula for \nthe restriction of the kernel of $\\hat f$ to the diagonal:\n\\begin{equation}\n\tK_{\\hat f}(x,x)=\\frac{1}{(2\\pi\\hbar)^n}\\int_{T^*_xM}\\lambda_x(p)\\;f(x,p).\n\\end{equation}\nThis is true for all $s$.\nThen ~(\\ref{trq}) immediately follows. (Note that  \n$\\omega(x)\\lambda_x(p)=dxdp$.) The equality~(\\ref{tra}) then follows, \nby Corollary~\\ref{kerdiag}.\n\\end{proof}\n\n\n\n\n\n\\section{Quantization of forms on $T^{*}M$}\n\n\n\nWe are going to  apply the preceding consideration to \nthe particular case when $E=\\Lambda(T^*M)$, the bundle of exterior \ndifferential  forms. (We can also consider the case of \n$\\Lambda(T^*M)\\otimes E$, where $E$ is  some other bundle.) In this \ncase we describe endomorphisms of \n$\\Lambda=\\Lambda_x=\\Lambda(T_x^*M)$ at each $x$ also as a result of \nquantization of some symbols. The corresponding symbol calculus ``at a \npoint''  is a \nsuperanalog of the quantization on $\\R{n}$. The calculus of the \nprevious section will be combined with this pointwise calculus to \nproduce our main construction.\n\n\\subsection{Quantization at a point}\n\nConsider an arbitrary local coframe on $M$. Its \nelements are considered as $1$-forms and thus odd (see the remark in the \nintroduction). So at a \npoint $x$ we have $n=\\dim M$ odd variables  $\\xi^k=e^k(x)$. \n(Later, when we shall consider different points, we shall pick \ndifferent letters for corresponding  values of $e^k$.) Elements of $\\Lambda$ \nare functions of $\\xi^k$. All endomorphisms of $\\Lambda$ are \ndifferential operators and are generated by $\\xi^k$, \n$\\lder{}{\\xi^k}$. They also can be expressed by integral kernels:\n\\begin{equation}\n\t(Au) (\\xi)=(-1)^{n \\tilde A} \\int_{\\S{n}} \\frac{D{\\eta}}{\\sqrt{g}}\\; \nk(\\xi,{\\eta})\\;u({\\eta}).\n\\end{equation}\nHere  $g=g(x)=\\det \n(g_{kl}(x))$ is the Gram determinant of the frame $(e_k(x))$. Notice \nthat \n\\begin{equation}\\label{kernel}\nk_{A}({\\xi},{\\eta})=(-1)^{n}(A\\delta_{{\\eta}})({\\xi}),\n\\end{equation}\nwhere $\\delta_{{\\eta}}({\\xi})=\\sqrt{g}\\,\\delta({\\xi}-{\\eta})$.\n\n\\begin{lm}\\label{kerprod}\n\\begin{equation}\nk_{AB}({\\xi},{\\eta}) =(-1)^{n\\tilde A} \\int_{\\S{n}} \\frac{D{\\xi}'}{\\sqrt{g}}\\;\nk_{A}({\\xi},{\\xi}')\\;k_{B}({\\xi}',{\\eta})\n\\end{equation}\n\\end{lm}\n\n\nIn the same way as above, or, actually, in the same way as for \n$\\R{n}$, we can introduce the following symbol calculus. Consider odd \nvariables $\\theta_k$ which transform contragrediently to $\\xi^k$. \nThat means that the form $\\xi^k\\theta_k$ is invariant under change \nof (co)frame. Geometrically, \n$\\theta_k$ are the values at $x$ of the elements of the dual frame \n$e_k$ regarded as  $1$-vectors (antivectors). \nA function of $\\t_k$ is a multivector at the point $x$.\n\nIn the following we fix the point $x$ and forget about it for time \nbeing.\n\n\\begin{de} Fix $r\\in [0,1]$. \nLet $f=f({\\xi},\\theta)$. We associate with this function the \nfollowing operator on $\\Lambda$. For any $u\\in \\Lambda$ set\n\\begin{equation} \\label{squant}\n\t(\\hat f u)({\\xi}):={(-i\\hbar)^n}\\int_{\\S{2n}} \nD({\\eta},\\t)\\;e^{\\frac{i}{\\hbar}({\\xi}-{\\eta})\\t}\\; f((1-r){\\xi} + r{\\eta}, \\t)\\;u({\\eta}).\n\\end{equation}\nConversely, for an operator $A $ with the kernel $k=k(\\xi,{\\eta})$, we \ndefine its {\\it symbol} by the formula\n\\begin{equation} \\label{ssymb}\n\t(\\sigma A)({\\xi},\\t):=(-1)^{n \\tilde A}\\int_{\\S{n}}\\frac{D{\\varepsilon}}{\\sqrt{g}} \n\\;e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\;k({\\xi}-r{\\varepsilon},{\\xi}+(1-r){\\varepsilon}).\n\\end{equation}\n\\end{de}\n\n\\begin{rem} For $r=0$, we can express the symbol by the action of the \noperator on the exponential function (cp. ~(\\ref{specsymb}):\n\\begin{equation}\n\t(\\sigma A)({\\xi},\\t)=(A_{{\\eta}}(e^{\\frac{i}{\\hbar}({\\eta}-{\\xi})\\t}))_{|{\\eta}:={\\xi}},\n\\end{equation}\nwhere the subscript ${}_{{\\eta}}$ means that operator is applied to the \nfunctions of the variables ${\\eta}^a$.\n\\end{rem}\n\n\\begin{thm} The quantization map $f\\mapsto \\hat f$ and the symbol map \n$A\\mapsto \\sigma A$ are mutually inverse.\n\\end{thm}\n\\begin{proof} Since we deal with finite-dimensional spaces, it \nsuffices to check either $\\hat{ }\\circ\\!\\sigma $ or \n$\\sigma\\!\\circ\\hat{ }$\\\/. \nTake $f=f({\\xi},\\t)$. The integral kernel of $\\hat f$, \nby~(\\ref{squant}),  is\n\\begin{equation}\\label{kerhatf}\nk_{\\hat f}({\\xi},{\\eta})=(-1)^{n\\tilde f}{(-i\\hbar)^n}\\int_{\\S{n}} \\sqrt{g}\\,\nD\\t\\;e^{\\frac{i}{\\hbar}({\\xi}-{\\eta})\\t}\\; f((1-r){\\xi} + r{\\eta}, \\t).\n\\end{equation}\nSo, using  formula~(\\ref{ssymb}),\n\\begin{multline*}\n\\sigma(\\hat f)({\\xi},\\t)=(-1)^{n\\tilde \nf}\\int\\frac{D{\\varepsilon}}{\\sqrt{g}}\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\;k_{\\hat \nf}({\\xi}-r{\\varepsilon},{\\xi}+(1-r){\\varepsilon})\\\\=\n{(-i\\hbar)^n}\\int\\frac{D{\\varepsilon}}{\\sqrt{g}}\\,\\sqrt{g}\\,D\\t'\\;e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\n\\;e^{-\\frac{i}{\\hbar}{\\varepsilon}\\t'}\\;f((1-r)({\\xi}-r{\\varepsilon})+r{\\xi}+r(1-r){\\varepsilon},\\t')\\\\=\n{(-i\\hbar)^n}\\int D{\\varepsilon} \\,D\\t' \\;e^{\\frac{i}{\\hbar}{\\varepsilon}(\\t-\\t')}\\;f({\\xi},\\t')=\n\\int D\\t'\\,\\delta(\\t'-\\t)\\,f({\\xi},\\t')=f({\\xi},\\t).\n\\end{multline*}\n\\end{proof}\n\n\nThe construction of quantization is invariant under linear \ntransformations, in the following sense. Consider two spaces $V_{x}, \nV_{y}\\cong \\S{n}$ and an isomorphism $T: V_{y}\\to V_{x}$. It induces \npull-back of functions: $T^{*}:\\Lambda_{x}\\to \\Lambda_{y}$. \n(Letters $x$ and $y$ are used here as just labels.)\n\\begin{lm} \\label{spinrep}\nFor any  $f=f({\\xi}_{x},\\t_{x})$,\n\\begin{equation}\nT^{*}\\;\\hat f\\;{T^{*}}^{-1}=\\widehat{T^{*}f},\n\\end{equation}\nwhere $(T^{*}f)({\\xi}_{y},\\t_{y})=f(T{\\xi}_{y},T^{-1}\\t_{y})$. \n\\end{lm}\n\\begin{proof} Straightforward calculation, using the \nformula~(\\ref{squant}).\n\\end{proof}\n(This is a trivial case of the spinor representation.)\nWe shall use the Lemma in the next section.\n\n\n\n\n\n\nThe {\\it composition} of symbols is defined in a usual way: $f\\circ \ng:=\\sigma(\\hat f\\hat g)$.\n\n\\begin{thm} The composition can be calculated by the following \nintegral formula:\n\\begin{multline}\\label{comp}\n(f\\circ \ng)({\\xi},\\t)=\\\\(-i\\hbar)^{2n}\\!\\!\\!\\!\\!\\!\\!\\!\\int\\limits_{\\S{2n}\\times\\S{2n}}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\nD{\\varepsilon}_{1}\\,D\\tau_{1}\\,D{\\varepsilon}_{2}\\,D\\tau_{2}\\; \ne^{\\frac{i}{\\hbar}({\\varepsilon}_{1}\\tau_{2}-{\\varepsilon}_{2}\\tau_{1})}\\;f({\\xi}+r{\\varepsilon}_{1},\\t+\\tau_{1})\\;\ng({\\xi}+(1-r){\\varepsilon}_{2},\\t+\\tau_{2})\n\\end{multline}\nfor arbitrary $r\\in [0,1]$. \n\\\\ For $r=0$ this formula is simplified to\n\\begin{equation} \\label{comp0}\n(f\\circ g)({\\xi},\\t)=(-i\\hbar)^{n}\\int_{\\S{2n}}D{\\varepsilon}\\,D\\tau\\; e^{-\\frac{i}{\\hbar}\n{\\varepsilon}\\tau}\\;f({\\xi},\\t+\\tau)\\; g({\\xi}+{\\varepsilon},\\t).\n\\end{equation}\n\\\\ For $r=1$ it is simplified to\n\\begin{equation} \\label{comp1}\n(f\\circ g)({\\xi},\\t)=(i\\hbar)^{n}\\int_{\\S{2n}}D{\\varepsilon}\\,D\\tau\\;\ne^{\\frac{i}{\\hbar}{\\varepsilon}\\tau}\\;f({\\xi}+{\\varepsilon},\\t)\\;\ng({\\xi},\\t+\\tau).\n\\end{equation}\n (Notice the difference in signs in the formulas~(\\ref{comp0})) \nand~(\\ref{comp1}).)\n\\end{thm}\n\\begin{proof}\nTo calculate the composition of $f$ and $g$, one has to apply twice \nthe formula ~(\\ref{squant}) to find $\\hat f\\hat g u$ for some ``test'' \nfunction $u$, and thus obtain the integral kernel for $\\hat f\\hat g$, and then \napply~(\\ref{ssymb}). After this straightforward calculation (that we \nomit), we arrive to the following formula:\n\\begin{multline}\\label{comp00}\n(f\\circ g)({\\xi},\\t)=(-i\\hbar)^{2n}\\int D{\\varepsilon}\\,D\\t_{2}\\,D{\\eta}_{1}\\,D\\t_{1}\\;\ne^{\\frac{i}{\\hbar}({\\varepsilon}\\t + ({\\xi} - r{\\varepsilon} - {\\eta}_{1})\\t_{1} + ({\\eta}_{1} - \n{\\xi} - (1-r){\\varepsilon})\\t_{2})}\\;\\\\\nf((1-r)({\\xi} - r{\\varepsilon}) + r{\\eta}_{1}, \\t) \\; g((1-r){\\eta}_{1} + r({\\xi} + (1-r){\\varepsilon}), \n\\t_{2}).\n\\end{multline}\nNow we introduce new variables: ${\\varepsilon}_{1}={\\eta}_{1} - (1-r){\\varepsilon}  - {\\xi}$,\n ${\\varepsilon}_{2}={\\eta}_{1}  + r{\\varepsilon}  - {\\xi}$. This change of variables is \n invertible: ${\\eta}_{1}={\\xi} + r{\\varepsilon}_{1} + (1-r){\\varepsilon}_{2}$, ${\\varepsilon}={\\varepsilon}_{2}-{\\varepsilon}_{1}$, \n and its Berezinian is unity. In these variables the formula \n ~(\\ref{comp00}) becomes\n\\begin{multline}\n(f\\circ g)({\\xi},\\t)=(-i\\hbar)^{2n}\\int D{\\varepsilon}_{1}\\,D{\\varepsilon}_{2}\\,D\\t_{1}\\,D\\t_{2}\\; \ne^{\\frac{i}{\\hbar}({\\varepsilon}_{2}(\\t-\\t_{1}) + {\\varepsilon}_{1}(\\t_{2}-\\t))}\\;\\\\\nf({\\xi}+r{\\varepsilon}_{1}, \\t_{1})\\; g({\\xi}+(1-r){\\varepsilon}_{2}, \\t_{2}).\n\\end{multline}\nFinally, we substitute $\\t_{1}=\\t + \\tau_{1}$, $\\t_{2}=\\t + \\tau_{2}$ \nand obtain~(\\ref{comp}). Formulas ~(\\ref{comp0}, \\ref{comp1}) follow \nfrom ~(\\ref{comp}) by direct integration.\n\\end{proof}\n\n\\begin{co} For arbitrary $r\\in [0,1]$ \n\\begin{equation} \n(f\\circ \ng)({\\xi},\\t)=e^{-\\frac{i}{\\hbar}((1-r)\\der{}{\\t_{1}}\\der{}{{\\xi}_{2}} + \nr\\der{}{{\\xi}_{1}}\\der{}{\\t_{2}})}\\;f({\\xi}_{1},\\t_{1})\\;g({\\xi}_{2},\\t_{2})\n\\,_{\\left|\\parbox{2cm}{\\scriptsize $ \n\\begin{aligned} {\\xi}_{1}&:={\\xi},  & {\\quad } {\\xi}_{2}&:={\\xi} \\\\ \n\\t_{1}&:=\\t,  & { \\quad} \\t_{2}&:=\\t \\\\ \n\\end{aligned}$}\\right.}\n\\end{equation}\n\\end{co}\n\n\n\\begin{co} In the limit $\\hbar\\to 0$ the composition becomes \nordinary multiplication. For the commutator we have the following \nformula:\n\\begin{equation}\n\tf\\circ g-(-1)^{\\tilde f\\tilde g}g\\circ f= -i\\hbar\\,\\{f,g\\}\\,(1 + O(\\hbar)),\n\\end{equation}\nwhere the nonvanishing Poisson brackets for the  coordinates \n${\\xi}^a,\\t_a$ are:\n\\begin{equation}\n\t\\{\\t_a,{\\xi}^b\\}=\\{{\\xi}^b,\\t_a\\}=\\delta_a{}^b\n\\end{equation}\n(``canonical brackets'').\n\\end{co}\n\n\n\n\\subsection{Formulas for local traces}\nThe space $\\Lambda$ can be endowed with different ${\\mathbb Z_{2}}$-gradings. In \naddition to the natural grading by parity, \none can define the grading by the eigenvalues of an \narbitrary  operator $S$ such that $S^2=1$. \nA form $u$ is {\\it $S$-even\\\/} (resp., {\\it $S$-odd\\\/}) if $Su=u$ (resp., \n$Su=-u$\\\/).\nThe {\\it trivial grading\\\/} corresponds to $S=1$ (the identity \noperator). The operator $S$ defining the natural grading  is \n$S=(-1)^{P}$, where $P$ is the ``parity operator'', with eigenvalues \n$0, 1$. \n\n\\begin{lm}\\label{kerparity}\n\\begin{equation}\nk_{(-1)^{P}}({\\xi},{\\eta}) = \\sqrt{g}\\delta({\\xi}+{\\eta})\n\\end{equation}\n\\end{lm}\n\\begin{proof} $k_{(-1)^{P}}({\\xi},{\\eta})=(-1)^{n}((-1)^{P}\\delta_{{\\eta}})({\\xi})=\n(-1)^{n}\\sqrt{g}\\delta(-{\\xi}-{\\eta})= {\\sqrt{g}\\delta({\\xi}+{\\eta})}$ \n\\end{proof}\n \n\nAn operator $A$ that preserves the $S$-grading is called {\\it \n$S$-even\\\/}. This is equivalent to $AS=SA$. Similarly, $A$ is {\\it \n$S$-odd\\\/} if $AS=-SA$.\n\nNotable example of ${\\mathbb Z_{2}}$-grading is given by \nHodge  ``star'' operator.  It is convenient here to present a \n``super'' construction for the star.\nThe Riemannian metric induces a bilinear form on the \nspace $\\S{n}=\\Pi T^*_x M$. For ${\\xi},{\\eta}$ denote \n${\\xi}\\cdot{\\eta}:=g_{ab}{\\xi}^a{\\eta}^b$.\nWe  define the operator $*$ by the following formula:\n{\n\\renewcommand{\\C}{\\mathop{\\rm const}\\nolimits}\n\\begin{equation} \\label{star}\n\t*u({\\xi}):=\\C\\int_{\\S{n}}\\frac{D{\\eta}}{\\sqrt{g}}\\;e^{-it\\,{\\xi}\\cdot{\\eta}}\\; \nu({\\eta}).\n\\end{equation}\nHere $t$ is an arbitrary parameter, and $\\C$ stands for a normalizing factor, \nwhich can be \nchosen by convenience.\n\n\\begin{lm} \\label{starkvadrat} In components, if $u=u_{a_1,\\dots,a_k}\\,{\\xi}^{a_1}\\dots \n{\\xi}^{a_k}$, where we assume the coefficients skew-symmetric,   \n\\begin{equation}\n(*u)_{a_1,\\dots,a_{n-k}}=\n\\C\\,t^{n-k}\\,i^{(n-k)^2+n(n-1)}\\frac{{\\sqrt{g}}}{(n-k)!}\\,u^{b_1,\\dots,b_{k}}\\;\n{\\varepsilon}_{b_1,\\dots,b_k,a_1,\\dots,a_{n-k}}.\n\\end{equation}\nHere the indices are raised with the help of the metric, and \n${\\varepsilon}_{b_1,\\dots,b_k,a_1,\\dots,a_{n-k}}$ is the standard Levi-Civita \nsymbol. So the operation $*$ defined by (\\ref{star}) differs from  \nusual only by some factor, which depends on the degree of the \nform.\nFor the inverse operator the following formula:\n\\begin{equation}\n\t*^{-1}=(\\C)^{-2}\\,(it)^{-n}(-1)^{n(n-1)\/2}\\;*\n\\end{equation}\nholds for any $n$.\n\\end{lm}\n\nConsider the case $n=2m$. Then  the operator $*$ is even (in the sense of natural \ngrading).\nSet $\\C:=t^{-m}$. By Lemma~\\ref{starkvadrat}, $*^2=1$. So we can consider $*$-grading. \n$*$-Even forms are what is called ``self-dual'',  and $*$-odd are \n``anti self-dual''.\n}\n\nTo each choice of ${\\mathbb Z_{2}}$-grading corresponds the {\\it trace}, defined \nby the formula:\n\\begin{equation}\n\t\\mathop{\\rm tr}\\nolimits_S A:=\\mathop{\\rm tr}\\nolimits A_{00}-A_{11}\n\\end{equation}\nfor an $S$-even operator $A$, where the block decomposition \ncorresponds to the chosen grading. \nUp to a factor, $\\mathop{\\rm tr}\\nolimits_S$ is specified by the property that it \nannihilates ``$S$-commutators'' (commutators of operators w.r.t. the \nchosen ${\\mathbb Z_{2}}$-grading). \n\nLet us consider the traces corresponding to \n parity, trivial grading, and $*$-grading (the last one only for \neven-dimensional case). Denote them by $\\mathop{\\rm str}\\nolimits$, $\\mathop{\\rm tr}\\nolimits_1$ and $\\mathop{\\rm tr}\\nolimits_*$ \nrespectively. Obviously, $\\mathop{\\rm tr}\\nolimits_{S} A= \\mathop{\\rm tr}\\nolimits_{1} (AS)= \\mathop{\\rm tr}\\nolimits_{1} \n(SA)$ for any $S$. \n\n\\begin{thm} \\label{loctrace}\nFor any operator $A$, \n\\begin{equation}\\label{str}\n\t\\mathop{\\rm str}\\nolimits A= (-1)^{\\tilde A n}\\int\\limits_{\\S{n}}\\frac{D{\\xi}}{\\sqrt \ng}\\; k_A({\\xi},{\\xi})\n\t=(-i\\hbar)^n\\int\\limits_{\\S{2n}}\\!\\! D({\\xi},\\t)\\; \\sigma A({\\xi},\\t)\n\\end{equation}\n(independent of $r$),\n\\begin{equation}\\label{tr1}\n\t\\mathop{\\rm tr}\\nolimits_1 A =\n\t(-1)^{n\\tilde A}\\int\\limits_{\\S{n}}\\frac{D{\\xi}}{\\sqrt g}\\; k_A(-{\\xi},{\\xi})\n\t=(-i\\hbar)^n\\int\\limits_{\\S{2n}}D({\\xi},\\t) \\;\\;e^{-\\frac{2i}{\\hbar}{\\xi}\\t}\\;\\sigma \nA((2r-1){\\xi},\\t),\n\\end{equation}\nand\n\\begin{equation}\\label{trstar}\n\t\\mathop{\\rm tr}\\nolimits_* A=\n\tt^{-m}\\!\\!\\!\\int\\limits_{\\S{2m}\\times\\S{2m}}\\frac{D{\\xi} D{\\eta}}{g}\\;\n\te^{-it\\,{\\xi}\\cdot{\\eta}}\\; k_A({\\xi},{\\eta})\n\t=t^{-m}\\int\\limits_{\\S{2m}}\\frac{D{\\xi}}{\\sqrt g}\\; \\sigma \n\tA({\\xi},t\\hbar\\,{\\xi}_{\\#})\n\\end{equation}\n(independent of $r$), where ${\\xi}_{\\#a}=g_{ab} {\\xi}^{b}$.     \nThe last formula works for $n=2m$.\n\\end{thm}\n\\begin{proof}\nLet us prove the first equality in~(\\ref{str}). Consider \n$[A,B]=AB-(-1)^{\\tilde A\\tilde B} BA$. Applying Lemma~\\ref{kerprod} \nand restricting the kernel to to the diagonal, one can deduce\n$$\nk_{[A,B]}({\\xi},{\\xi})=\\int\\frac{D{\\xi}'}{\\sqrt{g}}\\,(-1)^{n\\tilde A}\\;(k_{A}({\\xi},{\\xi}')\\,\nk_{B}({\\xi}',{\\xi}) - (-1)^{n}\\,(k_{A}({\\xi}',{\\xi})\\, k_{B}({\\xi},{\\xi}')),\n$$\nwhich obviously gives zero after the integration w.r.t. ${\\xi}$. Thus the \nright hand side of our formula annihilates commutators, and hence it is \nproportional to the trace $\\mathop{\\rm str}\\nolimits$. To obtain the normalization factor, it \nsuffices to check some particular operator. Consider \n$A: u\\mapsto u(0)$. For it, $\\mathop{\\rm str}\\nolimits A =1$ and\n$k_{A}({\\xi},{\\eta})=\\sqrt{g}\\,{\\eta}^{n}\\ldots{\\eta}^{1}$, so \n$$\n\\int\\frac{D{\\xi}}{\\sqrt{g}}\\;k_{A}({\\xi},{\\xi})=\\int \nD{\\xi}\\;{\\xi}^{n}\\ldots{\\xi}^{1}=1=\\mathop{\\rm str}\\nolimits A.\n$$\nThus the normalization factor \nactually equals $1$. The expression for the trace via the symbol follows from the \nequality\n\\begin{equation}\nk_{\\hat f}({\\xi},{\\xi})=(-1)^{n\\tilde f}{(-i\\hbar)^n}\\int_{\\S{n}} \\sqrt{g}\\,\nD\\t\\; f({\\xi}, \\t)\n\\end{equation}\n(see ~(\\ref{kerhatf})). So the formula~(\\ref{str})  is completely \nproved.  To prove~(\\ref{tr1}), notice that $\\mathop{\\rm tr}\\nolimits_{1}A=\\mathop{\\rm str}\\nolimits ((-1)^{P}A)= \n\\mathop{\\rm str}\\nolimits (A(-1)^{P})$. By Lemmas~\\ref{kerprod} and \\ref{kerparity},\n$$\nk_{(-1)^{P}A}({\\xi},{\\eta})=\\int \\frac{D{\\xi}'}{\\sqrt{g}} \n\\,\\sqrt{g}\\delta({\\xi}+{\\xi}')\\, k_{A}({\\xi}',{\\eta}) = k_{A}(-{\\xi},{\\eta}). \n$$\nApplying the formula for $\\mathop{\\rm str}\\nolimits$, we obtain the first equality in \n~(\\ref{tr1}). Expressing $k_{A}(-{\\xi},{\\eta})$ via the symbol completes the \nproof of~(\\ref{tr1}). Finally, to obtain~(\\ref{trstar}), we \napply~(\\ref{tr1}) to the operator $*A$. From Lemma~\\ref{kerprod} and \nthe formula~(\\ref{star}), we obtain \n$$\nk_{*A}(-{\\xi},{\\xi}) = \n\\int\\frac{D{\\xi}'}{\\sqrt{g}}\\,t^{-m}\\;e^{it\\,{\\xi}\\cdot{\\xi}'}\\;k_{A}({\\xi}',{\\xi}). \n$$\nThus, \n\\begin{multline*}\n\\mathop{\\rm tr}\\nolimits_{1}(*A)=\\int\\frac{D{\\xi}\\,D{\\eta}}{g}\\,t^{-m}\\;e^{-it{\\xi}\\cdot{\\eta}}\\;k_{A}({\\xi},{\\eta}) \n=\\\\\n\\int\\frac{D{\\xi} \\,D{\\eta}}{g}\\,t^{-m}\\;e^{-it{\\xi}\\cdot{\\eta}}\\;\n\\int(-i\\hbar)^{2m}\\,\\sqrt{g}\\,D\\t\\;e^{\\frac{i}{\\hbar}({\\xi}-{\\eta})\\t}\\;\nf((1-r){\\xi} + r{\\eta}, \\t).\n\\end{multline*}\nSubstitute ${\\xi}={\\varepsilon}-r{\\zeta}$, ${\\eta}={\\varepsilon} + (1-r){\\zeta}$. The Berezinian of this \nchange of variables is unity. Now, $(1-r){\\xi} + r{\\eta}={\\varepsilon}$, ${\\xi}-{\\eta}=-{\\zeta}$, \nand the argument of the first exponential becomes $-it({\\varepsilon}-r{\\zeta})\\cdot \n({\\varepsilon} + (1-r){\\zeta})=-it({\\varepsilon}\\cdot(1-r){\\zeta} - r{\\zeta}\\cdot{\\varepsilon})= it{\\zeta}\\cdot{\\varepsilon}$. So,\n\\begin{multline*}\n\\mathop{\\rm tr}\\nolimits_{*}A=\\int\\frac{D{\\varepsilon}\\,D\\t\\,D{\\zeta}}{\\sqrt{g}}\\,t^{-m}(-i\\hbar)^{2m}\\;\ne^{it{\\zeta}\\cdot{\\varepsilon} -\\frac{i}{\\hbar}{\\zeta}\\t}\\;f({\\varepsilon},\\t)=\\\\\n\\int\\frac{D{\\varepsilon}\\,D\\t\\,D{\\zeta}}{\\sqrt{g}}\\,t^{-m}(-i\\hbar)^{2m}\\;\ne^{-\\frac{i}{\\hbar}{\\zeta}(\\t-t\\hbar\\,{\\varepsilon}_{\\#})}\\;f({\\varepsilon},\\t)=\\\\\n\\int\\frac{D{\\varepsilon}\\,D\\t}{\\sqrt{g}}\\,t^{-m}\\;\\delta(\\t-t\\hbar\\,{\\varepsilon}_{\\#})\\;f({\\varepsilon},\\t)=\n\\int\\frac{D{\\varepsilon}}{\\sqrt{g}}\\,t^{-m}\\;f({\\varepsilon}, t\\hbar\\,{\\varepsilon}_{\\#}),\n\\end{multline*}\nand the theorem is completely proved.\n\\end{proof}\n\n\n\\begin{rem} For $r=1\/2$ (the Weyl quantization), \nit follows from the formula~(\\ref{tr1}) that $\\mathop{\\rm tr}\\nolimits_{1} A=2^{n}\\,(\\sigma A) (0,0)$.\n\\end{rem}\n\n\n\\subsection{Global quantization}\nNow we can combine the quantization built in Section~\\ref{QS} with the \nabove ``fiberwise'' quantization.  \n\nWe describe forms on $M$ as \nfunctions on the supermanifold $\\hat M=\\Pi TM$. Locally they look as \nfunctions $u=u(x,{\\xi})$, where ${\\xi}^{a}$ are elements of a coframe at \n$x$. For the coordinate coframe, ${\\xi}^{a}=dx^{a}$. Operators on forms can \nbe described by  ``Schwartz  kernels'' of the appearance $K(x,{\\xi},y,{\\eta})$:\n\\begin{equation}\n(Au)(x,{\\xi}) = (-1)^{n\\tilde A}\\int_{\\hat M} D(y,{\\eta})\\;K(x,{\\xi},y,{\\eta})\\;u(y,{\\eta}),\n\\end{equation}\nfor ${\\xi}^{a}=dx^{a}$. In the\ncoordinate-free language, the kernel is a generalized form (de Rham's \ncurrent) on $M\\times M$, and $Au=\\pi_{2*} (K\\pi_{1}^{*}u)$, where \n$\\pi_{1},\\pi_{2}:M\\times M \\to M$ are projections. \n\n\n\n\n\nAfter certain simplifications, we \nobtain the following formulas.  Below the letter $T$ \nstands for the parallel transport along geodesics, w.r.t. \nthe Levi-Civita connection, in the bundles $T^{*}M$, $\\Pi TM$ and \n$T^{*}M\\Pi$.\n\n\n{\\sl For quantization.} Suppose we have a function $f=f(x,p,{\\xi},\\t)$.  \nGeometrically, this is a section of the pull-back \n$\\pi^*(\\Lambda(TM\\oplus T^*M))$, by $\\pi: T^{*}M\\to M$,   or a function \non the supermanifold $T^*M\\oplus\\Pi TM\\oplus T^*M\\Pi)$. Denote this \nsupermanifold by $N$ (this is a  bundle over $M$).\nTo this function we assign the following operator, acting on $\\Omega \n(M)$:\n\\begin{multline}\n\t(\\hat f u)(x,{\\xi}):=\\frac{1}{(2\\pi i)^n}\\int_N \nD(y,q,{\\eta},\\t)\\;\\a(x,y)\\mu(x,y)\\;e^{\\frac{i}{\\hbar}\\left(\\exp_y^{-1}x\\cdot \nq + (T(y,x){\\xi}-{\\eta})\\t\\right)}\\;\\\\\nf( {\\zs}, T({\\zs},y)q, \n(1-r)\\,T({\\zs},x){\\xi}  + r\\,T({\\zs},y){\\eta},\\\\  \nT({\\zs},y)\\t )\\;\\;u(y,{\\eta}).\n\\end{multline}\nEquivalent definition:\n\\begin{multline}\\label{totquant1}\n\t(\\hat f u)(x,{\\xi}):=\\frac{1}{(2\\pi i)^n}\\int D(v,p,{\\varepsilon},\\t)\\a(x,\\exp_x \nv)\\;e^{-\\frac{i}{\\hbar}(vp+{\\varepsilon}\\t)}\\;\\\\\nf(\\exp_x sv , T(\\exp_x sv,x)p,\n T(\\exp_x sv,x)({\\xi}  + r{\\varepsilon}), T(\\exp_x sv,x)\\t)\\\\ \\;u(\\exp_x \nv,T(\\exp_x v,x)({\\xi}+{\\varepsilon})).\n\\end{multline}\nHere integration is over the fiber $T_xM\\oplus T_x^*M\\oplus\\Pi \nT_xM\\oplus T_x^*M\\Pi)$.\n\n{\\sl For symbols.} Let $A$ be an operator on differential forms, with \nthe Schwartz kernel $K(x,{\\xi},y,{\\eta})$. \n(Here ${\\xi}^{a}$ are elements of a coframe at $x$, and \n${\\eta}^{a}$ are elements of a coframe at $y$.)\nThen its {\\it symbol} $\\sigma \nA=(\\sigma A)\\,(x,p,{\\xi},\\t)$ is defined by the formula\n\\begin{multline} \\label{fsymbol}\n\t(\\sigma A)(x,p,{\\xi},\\t):=\\int_{T_x M\\times \\Pi T_x \nM}D(v,{\\varepsilon})\\;\\a(\\exp_x(-sv),\\exp_x (1-s)v)\\; \\rho_s(x,v) \\\\\n\\;e^{\\frac{i}{\\hbar}(vp+{\\varepsilon}\\t)} \n\\;\\;\nK(\\exp_x (-sv),T(\\exp_x (-sv),x)({\\xi}-r{\\varepsilon}), \\exp_x (1-s)v, \n\\\\\nT(\\exp_x (1-s)v,x)({\\xi}+(1-r){\\varepsilon})).\n\\end{multline}\n\nThese formulas can be obtained from~(\\ref{quant}, \\ref{symbol}) \nand~(\\ref{squant}, \\ref{ssymb}) by more or less direct calculation. At \na certain step it is necessary to use Lemma~\\ref{spinrep} to  change \nthe points at which reference frames are taken, and to change \nvariables in the integrals.\n\n\nOur constructions depend on two real parameters $s,r\\in [0,1]$. \nNote also that without any difficulties the definitions of \nquantization and symbols can be extended from ``scalar-valued'' forms \nto forms taking values \nin an arbitrary vector bundle with a connection. Quite helpful for \npractical calculations is the appearance of the formula~(\\ref{totquant1}) \nin normal coordinates centered at $x$ (so $x^{a}=0$, \n$(\\exp_{x}v)^{a}=v^{a}$):\n\\begin{multline}\\label{totquantnorm}\n\t(\\hat f u)(0,{\\xi}):=\\frac{1}{(2\\pi i)^n}\\int D(v,p,{\\varepsilon},\\t)\\a(x,v)\\;e^{-\\frac{i}{\\hbar}(vp+{\\varepsilon}\\t)}\\;\\\\\nf(sv , T(sv,0) p,\n T(sv,0)({\\xi}  + r{\\varepsilon}), T(sv,0)\\t) \\;u(v,T(v,0)({\\xi}+{\\varepsilon})).\n\\end{multline}\n\n\n\nAs it was said, the symbols in our refined symbol calculus are \nfunctions on the supermanifold $N=T^{*}M\\oplus \\Pi TM\\oplus T^{*}M\\Pi$. \nRoughly speaking, they are tensor products of forms and multivectors on $M$, depending \nalso on a point $p$ in the cotangent space. Luckily, they can be given \na nicer description. Using the Levi-Civita connection on $M$, the supermanifold $N$ \ncan be identified with $\\widehat{T^{*}M}$. That means that our \nsymbols can be considered simply as forms on $T^{*}M$. For clarity, let us \nwrite down the change of coordinates on $N$ (assuming that we use \ncoordinate frames):\n\\begin{align}\n&\\left\\{ \n\\begin{aligned}\nx^{a}&=x^{a}(x')\\\\\np_{a}&=\\der{x^{a'}}{x^{a}}(x(x'))\\,p_{a'}\\\\\ndx^{a}&=dx^{a'}\\,\\der{x^{a}}{x^{a'}}(x')\\\\\n\\t_{a}&=\\der{x^{a'}}{x^{a}}(x(x'))\\,\\t_{a'}\n\\end{aligned}\n\\right.\\\\\n\\intertext{Compare it with  $\\widehat{T^{*}M}$:}\n&\\left\\{ \n\\begin{aligned}\nx^{a}&=x^{a}(x')\\\\\np_{a}&=\\der{x^{a'}}{x^{a}}(x(x'))\\,p_{a'}\\\\\ndx^{a}&=dx^{a'}\\,\\der{x^{a}}{x^{a'}}(x')\\\\\ndp_{a}&=d\\left(\\der{x^{a'}}{x^{a}}(x(x'))\\right)\\,p_{a'} + \\der{x^{a'}}{x^{a}}(x(x'))\\,dp_{a'}\n\\end{aligned}\n\\right.\n\\end{align}\nThe tempting idea to identify $\\t_{a}$ with $dp_{a}$ fails because of \nthe additional term in the transformation law for $dp_{a}$. However, \nusing the connection, we can identify $\\t_{a}$ with $\\nabla p_{a}$, \nwhere\n\\begin{equation}\n\\nabla p_{a}=dp_{a} - dx^{b}\\mathop{\\rm \\Gamma}\\nolimits_{ab}^{c}p_{c}.\n\\end{equation}\nNotice that the change of variables from $(x,p,dx,dp)$ to \n$(x,p,dx,\\nabla p)$ has the unit Berezinian, hence all our integrals \nover $(x,p,{\\xi},\\t)$ can be rewritten as integrals over $(x,p,dx,dp)$, \ni.e., as integrals of differential forms over the manifold $T^{*}M$.\n\n\\begin{rem} \\label{getz} In the paper~\\cite{Ge} Getzler proposed a \ncomplete symbol calculus for spinor fields, combining a symbol \ncalculus on manifolds with the ``Weyl'' isomorphism $\\L(V)\\to C(V)$ \n(where $\\L(V)$ and $C(V)$ are respectively exterior and Clifford \nalgebra of a Euclidean space $V$).  The resulting symbols were {\\sl \nhorizontal\\\/} forms on $T^{*}M$, i.e., functions  $f=f(x,p,{\\xi})$ in our notation. \nThe key element of that construction  was some nonobvious\nfiltration introduced into the algebra of symbols (by the total degree \nin $p_{a}$ and ${\\xi}^{b}$, in our notation). \nAs we showed \nin~\\cite{AS}, in the quantization language this was equivalent to a \nvery peculiar convention \n$\\hbar_{\\text{spinor}}=\\hbar_{\\text{usual}}^{2}$ (in~\\cite{Ge} \nno Planck constant was used, so this fact was not explicit there). \nThat strange convention was nevertheless essential for calculating the \nDirac index, see ~\\cite{AS}. Comparing with the complete symbol calculus \nconstructed it this section, we see that now {\\sl all\\\/} forms \non $T^{*}M$ appear as symbols, and  odd  and  even variables are on equal \nfooting (no discrimination w.r.t.  $\\hbar$).\n\\end{rem}\n\n\\subsection{Examples}\n\nThe exterior differential and the codifferential (or divergence), \nwhich is defined on Riemannian manifold, are given by the \nformulas:\n\\begin{align}\n\td&=dx^a\\der{}{x^a}={\\xi}^a\\,\\nabla_a  \\label{d}\\\\\n\t\\delta &=g^{ab}\\der{}{{\\xi}^a}\\,\\nabla_b.\\label{delta}\n\\end{align}\n(Notice that the second expression for $d$ in ~(\\ref{d}) is valid for arbitrary \nframe.) The covariant derivative of (inhomogeneous) form is\n\\begin{equation}\\label{nabla}\n\\nabla_{a}=\\der{}{x^{a}} - \\mathop{\\rm \\Gamma}\\nolimits_{ak}{}^{l}{\\xi}^{k}\\der{}{{\\xi}^{l}},\n\\end{equation}\nwhere $\\mathop{\\rm \\Gamma}\\nolimits_{ak}{}^{l}$ is the Christoffel symbol.\n\n\\begin{rem} In components: if $u=u_{a_1,\\dots,a_k}dx^{a_1}\\dots \ndx^{a_k}$, then\n\\begin{equation}\n\t(\\delta u)_{a_1,\\dots,a_{k-1}}=k\\,u_{aa_1,\\dots,a_{k-1}}{}^{;a}.\n\\end{equation}\n\\end{rem}\n\nDirect application of the formula~(\\ref{totquantnorm}) gives the \nfollowing results for the quantization of symbols ${\\xi}\\mathrm p={\\xi}^{a} \np_{a}$ and $\\t\\cdot\\mathrm p=g^{ab}\\t_{a} p_{b}$:\n\n\\begin{ex}\n\t\\begin{equation}\n\t\t\\widehat{{\\xi}\\mathrm p}=-i\\hbar\\,d\n\t\\end{equation}\n\\end{ex}\n\n\\begin{ex}\n\t\\begin{equation}\n\t\t\\widehat{\\t\\cdot\\mathrm p}=-\\hbar^2\\,\\delta\n\t\\end{equation}\n\\end{ex}\n\n\\subsection{Traces}\n\n\\begin{thm}\n For any operator $A$ acting on $\\Omega (M)$,\n\\begin{align}\n\\mathop{\\rm Str}\\nolimits A &= \\frac{1}{(2\\pi i)^n}\\int_{N}D(x,p,{\\xi},\\t)\\;\\sigma A(x,p,{\\xi},\\t) = \n\\frac{i^{n}}{(2\\pi)^{n}}\\int_{T^{*}M}\\;\\sigma A   \\label{supertrace}   \\\\\n\\intertext{(integral of differential form over $T^{*}M$), independent of \n$s$ and $r$,}\n\\mathop{\\rm Tr}\\nolimits_1 A &=\\frac{1}{(2\\pi i)^n}\\int_{N}D(x,p,{\\xi},\\t)\\;\\;e^{-\\frac{2i}{\\hbar}{\\xi}\\t}\\sigma \nA(x,p,(2r-1){\\xi},\\t),   \\label{trivtrace\n\\end{align}\nindependent of $s$.\nFor $n=2m$, and for any $A$,\n\\begin{equation}\\label{startrace}\n\\mathop{\\rm Tr}\\nolimits_*A=\\frac{1}{(2\\pi)^{2m}}\\int \\frac{D(x,p,{\\xi})}{\\sqrt{g}}\\;\\sigma \nA(x,p,{\\xi},\\hbar^{-1}\\,{\\xi}_{\\#}),\n\\end{equation}\nindependent of $s$ and $r$. Here $({\\xi}_{\\#})_a=g_{ab}{\\xi}^b$. (We set in \nthe last formula $t:=\\hbar^{-2}$.)\n\\end{thm}\n\\begin{proof} Immediately follows from Theorem~\\ref{trace} and  \nTheorem~\\ref{loctrace}.\\end{proof}\n\nNotice that these formulas contain no powers of $\\hbar$ in front of the \nintegrals, compared to~(\\ref{tra}) and ~(\\ref{str}, \\ref{tr1}, \n\\ref{trstar}). This comes naturally by cancellation of the powers of $\\hbar$  \n from the ``even''\nand the ``odd'' parts of our quantization. \n\n\\section{Natural Laplacians and their symbols}\n\n\\subsection{Weitzenb\\\"ock formula}\n\nFor any vector bundle with a connection over Riemannian $M$ there is \nan operator\n\\begin{equation}\n\t\\Delta=g^{ab}(\\nabla_a\\nabla_b-\\Gamma_{ab}{}^c\\nabla_c)\n\\end{equation}\n(here $\\nabla$ stands for the covariant derivative of the sections of \nthe bundle). This Laplacian is called  Laplace-Beltrami operator \nor Bochner Laplacian. In the case when the bundle is the exterior \nbundle $\\Lambda M =\\Lambda (T^{*}M)$, there is another natural \nconstruction of a Laplacian operator, namely the Hodge Laplacian \n$\\square=(d+\\delta)^2=d\\delta +\\delta d$. The relation between these \ntwo operators is  given by the ``Weitzenb\\\"ock formula''~\\cite[p.~397]{W} (see also \n~\\cite{Rh}). (This name \n is used, in general, for formulas relating any natural Laplacian \noperators in arbitrary bundle; by a ``Laplacian \noperator'' is meant an operator generalizing the usual Laplacian on \nfunctions in the Euclidean space, in the sense that in components  \nit's  the usual Laplacian applied  componentwise, plus \nsome lower-order terms.) We shall use it in the following form.\n\n\\begin{thm}\\label{Weitz}\n\\begin{align}\n\t\\square=(d+\\delta)^2&=\\Delta + \\mathop{\\rm Ric}\\nolimits_a{}^b\\,dx^a\\der{}{dx^b} + \nR_a{}^k{}_b{}^l\\,dx^adx^b\\der{}{dx^k}\\der{}{dx^l}  \\label{weitz1}\\\\\n&=\\Delta + \\mathop{\\rm Ric}\\nolimits_a{}^b\\,dx^a\\der{}{dx^b} + \n\\frac{1}{2}\\,R_{ab}{}^{kl}\\,dx^a dx^b\\der{}{dx^k}\\der{}{dx^l}, \\label{weitz2}\n\\end{align}\nwhere $R_{ab}{}^{kl}$ is the Riemann tensor, $\\mathop{\\rm Ric}\\nolimits_a{}^b=R_{ka}{}^{kb}$ \nis the Ricci tensor, indices are raised and lowered by the Riemannian \nmetric.\n\\end{thm}\n\\begin{proof} By the formulas~(\\ref{d}) and (\\ref{delta}),  $d+\\delta={\\hat \\gamma^{a}}\\, \\nabla_{a}$, \nwhere we define ${\\hat \\gamma^{a}}:={\\xi}^{a}+g^{ak}\\lder{}{{\\xi}^{k}}$. We need to find \n$(d+\\delta)^{2}=(1\/2)\\,[d+\\delta, d+\\delta]$. Directly:\n\\begin{multline}\n[d+\\delta, d+\\delta]=[{\\hat \\gamma^{a}}\\, \\nabla_{a},{\\hat \\gamma^{b}}\\, \\nabla_{b}]\n\\\\={\\hat \\gamma^{a}}\\,[\\nabla_{a},{\\hat \\gamma^{a}}]\\,\\nabla_{b} + [{\\hat \\gamma^{a}},{\\hat \\gamma^{b}}]\\,\\nabla_{a}\\nabla_{b} \n-{\\hat \\gamma^{b}}{\\hat \\gamma^{a}}\\,[\\nabla_{a,}\\nabla_{b}]-{\\hat \\gamma^{b}}\\,[{\\hat \\gamma^{a}},\\nabla_{b}]\\,\\nabla_{a}.\n\\end{multline}\nObserve the following facts. First, $[{\\hat \\gamma^{a}},{\\hat \\gamma^{b}}]=2g^{ab}$. Second, \n$[\\nabla_{a,}\\nabla_{b}]=-R_{abk}{}^{l}{\\xi}^{k}\\lder{}{{\\xi}^{l}}$ \n(cp.(\\ref{nabla})). Third, in \nthe frame associated with normal coordinates centered at a given \npoint $x_{0}$, the operators ${\\hat \\gamma^{a}}$ and $\\nabla_{b}$ commute at \n$x_{0}$ (since the Christoffel symbols and partial derivatives of the \nmetric vanish at $x_{0}$). Let's calculate in these coordinates. \nThus, at  $x_{0}$,\n$$\n[d+\\delta, d+\\delta]=g^{ab}\\,\\nabla_{a}\\nabla_{b}+\n{\\hat \\gamma^{b}}{\\hat \\gamma^{a}}\\,R_{abp}{}^{q}{\\xi}^{p}\\der{}{{\\xi}^{q}}\n=\n2\\Delta + {\\hat \\gamma^{a}}{\\hat \\gamma^{b}}\\,R_{abp}{}^{q}{\\xi}^{p}\\der{}{{\\xi}^{q}}.\n$$\nSince (by direct computation)\n$$\n{\\hat \\gamma^{b}}{\\hat \\gamma^{a}}={\\xi}^{b}{\\xi}^{a}+ g^{ba} +({\\xi}^{b}g^{ak}-{\\xi}^{a}g^{bk})\\der{}{{\\xi}^{k}} + \ng^{bl}g^{ak}\\der{}{{\\xi}^{l}}\\der{}{{\\xi}^{k}},\n$$\nthen, using the properties of the Riemann tensor,\n\\begin{multline*}\n[d+\\delta, d+\\delta]=2\\Delta - 2 {\\xi}^{a}g^{bk}\\,R_{abp}{}^{q}\n\\der{}{{\\xi}^{k}}{\\xi}^{p}\\der{}{{\\xi}^{q}}\n=2\\Delta -2 R_{a}{}^{k}{}_{p}{}^{q}\\,{\\xi}^{a}\\der{}{{\\xi}^{k}}{\\xi}^{p}\\der{}{{\\xi}^{q}}\n\\\\=2\\Delta + \n2 R_{a}{}^{k}{}_{p}{}^{q}\\,({\\xi}^{a}{\\xi}^{p}\\der{}{{\\xi}^{k}}\\der{}{{\\xi}^{q}}\n-\\delta_{k}{}^{p}{\\xi}^{a}\\der{}{{\\xi}^{q}})\n\\\\=2\\left(\n\\Delta + R_{a}{}^{k}{}_{b}{}^{l}\\,{\\xi}^{a}{\\xi}^{b}\\der{}{{\\xi}^{k}}\\der{}{{\\xi}^{l}}\n+ \\mathop{\\rm Ric}\\nolimits_{a}{}^{b}{\\xi}^{a}\\der{}{{\\xi}^{b}}\n\\right),\n\\end{multline*}\nand the formula~(\\ref{weitz1}) is proved. To deduce~(\\ref{weitz2}), one should \nraise all indices in $R_{a}{}^{k}{}_{b}{}^{l}$ (respectively lowering \nthem for ${\\xi}^{a}{\\xi}^{b}$), alternate in $a,b$ (the factor $1\/2$ \nappears) and apply the Ricci identity $R^{akbl}+R^{bakl}+ \n\\underbrace{R^{kbal}}_{-R^{bkal}}=0$.\n\\end{proof}\n\n\\begin{rem} Presentation of $d+\\delta$ as a ``Dirac'' operator which we \nused in the proof of Theorem~\\ref{Weitz} corresponds to the action on forms \nof the Clifford algebra $C(n)$,  induced by the Riemannian metric. \nOperators ${\\hat \\gamma^{a}}$ {\\sl do not\\\/} generate all endomorphisms of $\\L$. At the \nsame time, the quantization that we constructed in the previous subsection\nactually realizes forms as spinors for larger Clifford algebra \n$C(n,n)\\cong\\mathop{\\rm End}\\nolimits\\L$, which is independent of metric. So there are two \ndifferent Clifford module structures on the space $\\L$, for two \ndifferent Clifford algebras, which should not lead to a confusion.\n\\end{rem}\n\n\n\\begin{rem} Formulas~(\\ref{weitz1}, \\ref{weitz2}) can be  \nalso rewritten as $\\square = \\Delta + R$, with the operator \n$R=R_a{}^k{}_b{}^l{\\xi}^{a}\\der{}{{\\xi}^{k}}{\\xi}^{b}\\der{}{{\\xi}^{l}}$, \nsee~\\cite{Ros}. In such form the Weitzenb\\\"ock formula was used \nin~\\cite{A}. The ``normal'' form~(\\ref{weitz2}) is most \nsuitable for our purposes.\n\n\\end{rem}\n\\subsection{The symbol of the Hodge Laplacian}\n\n\\begin{thm} For any $s\\in [0,1]$,\n\\begin{multline} \\label{symlap}\n\t(\\sigma(\\square))\\,(x,p,{\\xi},\\t)=\\\\\n=-\\hbar^{-2}\\,\\left(\\mathrm p^2 + \n\\frac{1}{2}\\,R_{ab}{}^{kl}\\,{\\xi}^a{\\xi}^b\\t_k\\t_l\\right) + \ni\\hbar^{-1}\\,(1-2r)\\, \\mathop{\\rm Ric}\\nolimits_a{}^b\\,{\\xi}^a\\t_b + r(1-r) R\n\\end{multline}\n\\end{thm}\n\\begin{proof}\nWe apply  the\nWeitzenb\\\"ock formula~(\\ref{weitz2}).\nExample~\\ref{lapsymb} provides $\\sigma(\\Delta) = -\\hbar^{-2}\\, \\mathrm \np^2$, independent of $s$. All we need, is to calculate \nthe symbols of the remaining terms. \nDenote $A:=\\mathop{\\rm Ric}\\nolimits_a{}^b\\,{\\xi}^a\\lder{}{{\\xi}^b}$,\n$B:= R_{ab}{}^{kl}\\,{\\xi}^a {\\xi}^b\\lder{}{{\\xi}^k}\\lder{}{{\\xi}^l}$. For \nsimplicity we shall write below $R_a{}^b$ instead of $\\mathop{\\rm Ric}\\nolimits_a{}^b$. \nFirst we find the kernels:\n\\begin{align*}\nk_{A}({\\xi},{\\eta})&=R_a{}^b{\\xi}^{a}\\,\\der{}{{\\xi}^{b}}\\,\\delta({\\eta}-{\\xi})\\,\\sqrt{g},\\\\\nk_{B}({\\xi},{\\eta})&=R_{ab}{}^{kl}\\,{\\xi}^{a}{\\xi}^{b}\\der{}{{\\xi}^{k}}\\der{}{{\\xi}^{l}}\\,\\delta({\\eta}-{\\xi})\\,\\sqrt{g}.\n\\end{align*}\nNow we change variables: ${\\xi}={\\zeta}-r{\\varepsilon}, {\\eta}={\\zeta}+(1-r){\\varepsilon}$, or ${\\varepsilon}={\\eta}-{\\xi}, \n{\\zeta}=(1-r){\\xi}+r{\\eta}$. As $\\lder{}{{\\xi}}=(1-r) \\lder{}{{\\zeta}} - \\lder{}{{\\varepsilon}}$, it follows \nthat\n\\begin{align}\nk_{A}({\\zeta}-r{\\varepsilon},{\\zeta}+(1-r){\\varepsilon})\n&=-R_a{}^b({\\zeta}^{a}-r{\\varepsilon}^{a})\\der{}{{\\varepsilon}^{b}}\\delta({\\varepsilon})\\,\\sqrt{g}, \n\\label{kA}\\\\\nk_{B}({\\zeta}-r{\\varepsilon},{\\zeta}+(1-r){\\varepsilon})\n&=R_{ab}{}^{kl}({\\zeta}^{a}-r{\\varepsilon}^{a})({\\zeta}^{b}-r{\\varepsilon}^{b})\\dder{}{{\\varepsilon}^{k}}{{\\varepsilon}^{l}}\n\\delta({\\varepsilon})\\,\\sqrt{g} \\label{kB}.\n\\end{align}\nSubstituting~(\\ref{kA}) and (\\ref{kB}) into~(\\ref{ssymb}) and integrating by \nparts, we obtain\n\\begin{multline*}\n(\\sigma A)({\\xi},\\t)=\\int D{\\varepsilon}\\,\\delta({\\varepsilon})\\,\\der{}{{\\varepsilon}^{b}}\n\\left(\n-R_a{}^b\ne^{\\frac{i}{\\hbar}{\\varepsilon}\\t}({\\xi}^{a}-r{\\varepsilon}^{a})\n\\right)\n\\\\=\n\\int D{\\varepsilon}\\,\\delta({\\varepsilon})\\,\n\\left(\n-R_a{}^b\\,\\frac{i}{\\hbar}\\,\\t_{b}\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}({\\xi}^{a}-r{\\varepsilon}^{a})+\nR_a{}^b\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,r\\delta_{b}{}^{a}\n\\right) \n=\\frac{i}{\\hbar}\\,R_a{}^b\\,{\\xi}^{a}\\t_{b} + rR,\\\\\n\\shoveleft{\n(\\sigma B)({\\xi},\\t)=\n\\int D{\\varepsilon}\\,\\delta({\\varepsilon})\\,\\dder{}{{\\varepsilon}^{k}}{{\\varepsilon}^{l}}\n\\left(\nR_{ab}{}^{kl}\ne^{\\frac{i}{\\hbar}{\\varepsilon}\\t}({\\xi}^{a}-r{\\varepsilon}^{a})({\\xi}^{b}-r{\\varepsilon}^{b})\n\\right)\n}\n\\\\= \\int D{\\varepsilon}\\,\\delta({\\varepsilon})\\,\\der{}{{\\varepsilon}^{k}}\n\\left(\n\\frac{i}{\\hbar}\\,\\t_{l}\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,R_{ab}{}^{kl}\\,\n({\\xi}^{a}-r{\\varepsilon}^{a})({\\xi}^{b}-r{\\varepsilon}^{b}) + \ne^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,2 R_{lb}{}^{kl}(-r)({\\xi}^{b}-r{\\varepsilon}^{b})\n\\right)\n\\\\=\\int D{\\varepsilon}\\,\\delta({\\varepsilon})\\,\n\\Bigl(\n-{\\Bigl(\\frac{i}{\\hbar}\\Bigr)}^{2}\\,\\t_{l}\\t_{k}\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,\nR_{ab}{}^{kl}\\,({\\xi}^{a}-r{\\varepsilon}^{a})({\\xi}^{b}-r{\\varepsilon}^{b}) - \n\\frac{i}{\\hbar}\\,\\t_{l}\n\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,2 R_{kb}{}^{kl}\\cdot\n\\\\\n\\left.\n(-r)({\\xi}^{b}-r{\\varepsilon}^{b})+\n\\frac{i}{\\hbar}\\,\\t_{k}\n\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,2 R_{lb}{}^{kl}(-r)({\\xi}^{b}-r{\\varepsilon}^{b})+\ne^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,2 R_{lk}{}^{kl}(-r)(-r)\n\\right)\n\\\\=\\Bigl(\\frac{i}{\\hbar}\\Bigr)^2 R_{ab}{}^{kl}\\,{\\xi}^{a}{\\xi}^{b}\\t_{k}\\t_{l}\n+\\frac{i}{\\hbar}\\left( -r2 R_{kb}{}^{kl}\\,{\\xi}^{b}\\t_{l} + r2 \n R_{lb}{}^{kl}\\,{\\xi}^{b}\\t_{k} \\right) + r^{2}\\,2R_{lk}{}^{kl}\n \\\\=\\Bigl(\\frac{i}{\\hbar}\\Bigr)^2 R_{ab}{}^{kl}\\,{\\xi}^{a}{\\xi}^{b}\\t_{k}\\t_{l}\n +\\frac{i}{\\hbar}(-4r)\\,R_{a}{}^{k}{\\xi}^{a}\\t_{k} - 2r^{2} R.\n\\end{multline*}\n(where we restored ${\\xi}$ in the argument).\nFinally we obtain\n\\begin{multline*}\n\\sigma\\square = \\sigma\\Delta + \\sigma A + \\frac{1}{2}\\sigma B\n\\\\=\n\\sigma\\Delta +i\\hbar^{-1}R_{a}{}^{b}{\\xi}^{a}\\t_{b} + rR +\n-\\hbar^{2}\\frac{1}{2} R_{ab}{}^{kl}\\,{\\xi}^{a}{\\xi}^{b}\\t_{k}\\t_{l} +\ni\\hbar^{-1}(-2r) R_{a}{}^{b}{\\xi}^{a}\\t_{b} -r^{2}R\n\\\\=\n-\\hbar^{2}\\left(\\mathrm p^{2} +  \n\\frac{1}{2} R_{ab}{}^{kl}\\,{\\xi}^{a}{\\xi}^{b}\\t_{k}\\t_{l}\\right)\n+i\\hbar^{-1}(1-2r) R_{a}{}^{b}{\\xi}^{a}\\t_{b}+r(1-r) R,\n\\end{multline*}\nwhich completes the proof.\n\\end{proof}\n\n\n\n\n\\section{Gauss-Bonnet-Chern Theorem}\n\nAs it is very well known, the index of any elliptic operator \n$A: \\mathop{\\rm \\Gamma}\\nolimits(M, E_{0})\\to \\mathop{\\rm \\Gamma}\\nolimits(M,E_{1})$ on compact manifold $M$ can be \nexpressed as the supertrace of a suitable function of the corresponding ``Laplacian''\n$\\Delta_{A}=AA^{*}+A^{*}A : \\mathop{\\rm \\Gamma}\\nolimits(M, E_{0}\\oplus E_{1}) \\to \\mathop{\\rm \\Gamma}\\nolimits(M, E_{0}\\oplus E_{1})$:\n\\begin{equation}\n\\mathop{\\rm index}\\nolimits A = \\mathop{\\rm str}\\nolimits \\varphi(\\Delta_{A}),\n\\end{equation}\nprovided the trace is absolutely convergent, and $\\varphi(0)=1$. In \nparticular, \n\\begin{equation}\n\\mathop{\\rm index}\\nolimits A = \\mathop{\\rm str}\\nolimits e^{t\\Delta_{A}},\n\\end{equation}\nfor suitable $t$. In this section we shall apply this formula \nto obtain the expression for the Euler characteristic of the \nRiemannian manifold $M$ via curvature, i.e., to deduce the \nGauss-Bonnet-Chern Theorem. Our main tool will be the formula for the \nsupertrace~(\\ref{supertrace}).\n\nConsider $D=d+\\delta: \\O^{ev}(M)\\to \\O^{od}(M)$. We have\n\\begin{equation}\\label{euler}\n\\chi(M)=\\mathop{\\rm index}\\nolimits D = \\mathop{\\rm str}\\nolimits e^{t\\square}\n\\end{equation}\nfor suitable $t$. Since\n\\begin{equation}\n\\sigma e^{t\\square}=e^{t\\sigma(\\square)}\\,(1+ O(\\hbar)) = \ne^{t(-\\hbar^{-2}(\\mathrm p^{2} +\\ldots)(1 + O(\\hbar))}\\,(1+ \nO(\\hbar))\n\\end{equation}\n(see ~(\\ref{symlap})), $t=\\hbar^{2}$ seems a good choice. So\n\\begin{equation}\n\\sigma (\\hbar^{2}\\square)= -\\mathrm p^2 - \n\\frac{1}{2}\\,R_{ab}{}^{kl}\\,{\\xi}^a{\\xi}^b\\t_k\\t_l + O(\\hbar),\n\\end{equation}\nand\n\\begin{equation}\n\\sigma e^{\\hbar^{2}\\square}=e^{\\sigma \n(\\hbar^{2}\\square)}\\,(1+O(\\hbar)) =e^{-\\mathrm p^2 - \n\\frac{1}{2}\\,R_{ab}{}^{kl}\\,{\\xi}^a{\\xi}^b\\t_k\\t_l} + O(\\hbar).\n\\end{equation}\nPassing to the limit $\\hbar\\to 0$ (which is harmless since the \nsupertrace ~(\\ref{euler}) is independent of $\\hbar$), we obtain:\n\\begin{multline}\n\\chi(M)=\\mathop{\\rm index}\\nolimits D=\\mathop{\\rm str}\\nolimits e^{\\hbar^{2}\\square}=\n\\frac{1}{(2\\pi i)^n} \\int_{N} D(x,p,{\\xi},\\t)\\; \ne^{-\\mathrm p^2 - \\frac{1}{2}\\,R_{ab}{}^{kl}\\,{\\xi}^a{\\xi}^b\\t_k\\t_l} \n\\\\=\\begin{cases}\n0, & n=2m+1\\\\\n{\\displaystyle \\frac{(\\sqrt{\\pi})^{2m}}{(2\\pi i)^{2m}}\\,\\int\\;D(x,{\\xi})\\;\n\\mathop{\\rm Pf}\\nolimits (R_{ab}{}^{kl}\\,{\\xi}^a{\\xi}^b)}, & n=2m\n\\end{cases}\n\\\\=\n\\left\\{\n\\begin{array}{l}\n   0, \\qquad n=2m+1\\\\\n   {\\displaystyle (-1)^{m}\\int D(x,{\\xi})\\;\\mathop{\\rm Pf}\\nolimits \n   \\Bigl(%\n   \\frac{1}{2\\pi}\\underbrace{\\frac{1}{2}R_{ab}{}^{kl}\\,{\\xi}^a{\\xi}^b}%\n                  _{\\text{curvature $2$-form}\n   \\Bigr) { }}\n\\end{array}\n\\right.\n=\\begin{cases}\n0, \\qquad  n=2m+1\\\\\n{\\displaystyle\\int_{M} \\underbrace{\\mathop{\\rm Pf}\\nolimits \\left(\\frac{1}{2\\pi} \n\\frac{1}{2}R_{ab}{}^{kl}\\,dx^a\\,dx^b\\right)}_{\\text{Euler class of \n$TM$}} }.\n\\end{cases}\n\\end{multline}\nThis is the Gauss-Bonnet-Chern formula for the Euler \ncharacteristic.\n\nWe used the well-known formula for the \nPfaffian as a Gaussian integral:\n$$\n\\int_{\\S{n}}D\\t\\;e^{-\\frac{1}{2}Q^{ab}\\,\\t_{a}\\t_{b}}=\n\\begin{cases}\n0 & \\text{for $n=2m+1$}\\\\ \n\\mathop{\\rm Pf}\\nolimits Q & \\text{for $n=2m$}\n\\end{cases}\n$$\nwhere $Q^{ab}=-Q^{ba}$ (see, e.g.,~\\cite{GIT}).\n\n\\begin{rem} The fact that the formula for the trace~(\\ref{supertrace}) \ncontains no $\\hbar$  factors is crucial for our calculation. It \ncomes naturally in our symbol calculus. In the calculus for spinor \nfields~\\cite{AS} the similar cancellation of the powers of $\\hbar$ was \nachieved only by an artifial ``asymmetric'' convention for Planck constant.\n\\end{rem}\n\n\\section{Discussion}\nWith very little effort we were able to deduce the Gauss-Bonnet-Chern \nTheorem from the trace formula~(\\ref{supertrace}). Though we possess \nan analogous trace formula~(\\ref{startrace}) for the $*$-grading, more subtle \nanalysis of the symbol is needed in the case of the Hirzebruch \nsignature operator, because \nthis formula contains $\\hbar^{-1}$. \n\nWe can also discuss the results with the relation to  the  problem \nof quantization of differential algebras, in particular the algebras \nof forms. In a general setting (cp.~\\cite{wein1}), \nwe consider deformation of commutative \nmultiplication in an algebra $A$, of the form\n\\begin{equation}\nf\\mathbin{*_{\\hbar}} g =fg +(-i\\hbar)\\,\\{f,g\\} + \n(-i\\hbar)^{2}\\,\\mathop{B_{2}}\\nolimits (f,g) +\\ldots,\n\\end{equation}\nstarting from some Poisson bracket,\nand of the differential in $A$, of the form\n\\begin{equation} \\label{dh}\n{d_{\\hbar}} f= df + (-i\\hbar)\\,d_{1}f + \n(-i\\hbar)^{2}\\,d_{2}f+\\ldots,\n\\end{equation}\nall this being considered up to  ``gauge equivalences''\n$f\\mapsto f+(-i\\hbar)\\,a_{1}(f) +   \\ldots$. (Here $d_{k}$ and $a_{k}$ \nare differential operators on $A$, and $B_{k}$ are bidifferential \noperators.)\nIn the first order the derivation property for ${d_{\\hbar}}$ is equivalent to\n\\begin{equation}\\label{der}\nd\\{f,g\\}-\\{df,g\\} -(-1)^{\\tilde f}\\{f,dg\\}\n=-\\Bigl(d_{1}(fg)-d_{1}f\\,g -(-1)^{\\tilde f}f\\,d_{1}g\\Bigr)\n\\end{equation}\nThe obstruction to killing $d_{1}$ or, more generally, the first \nnonvanishing term $d_{n}$ in the expansion~(\\ref{dh}) , $n\\geq 1$,\n by a gauge transformation is the \ncohomology class $[d_{n}]$ in the complex  of operators on $A$ with \nthe differential $\\mathop{\\rm ad}\\nolimits d=[d,\\phantom{d}]$. \n\nConsider the case of forms. We hope \nto analyze the general picture elsewhere, and here we shall make just a few \nremarks.  First, there is a question whether the deformation should \nrespect the $\\mathbb Z$-grading or just ${\\mathbb Z_{2}}$-grading (parity). For the \nformer option, the induced Poisson bracket should be also $\\mathbb \nZ$-graded, for example the bracket of  $1$-forms should \nbe $2$-form (and not a function, for example). This is highly \nunlikely. Poisson structure obtained above has the canonical brackets\n\\begin{equation}\\label{brac}\n\\{p_{a},x^{b}\\}=\\delta_{a}{}^{b}, \\quad \\{\\t_{a},{\\xi}^{b}\\}=\\delta_{a}{}^{b}\n\\end{equation} \nin the coordinates $(x,p,{\\xi}=dx,\\t)$ and  is more complicated in \nthe coordinates  $(x,p,dx,dp)$. This structure  respects parity but by no means\n$\\mathbb Z$-grading. The second question concerns the relation between \nthe bracket and $d$. An easy computation produces the following formulas \nfor the action of $d$:\n\\begin{align*}\ndx^{a}&={\\xi}^{a}\\\\\ndp_{a}&=\\mathop{\\rm \\Gamma}\\nolimits_{ba}^{c}{\\xi}^{b}p_{c} + \\t_{a} \\\\\nd{\\xi}^{a}&=0\\\\\nd\\t_{a}&=\\mathop{\\rm \\Gamma}\\nolimits_{ba}^{c}{\\xi}^{b}\\t_{c} -\\frac{1}{2}R_{kla}{}^{c}{\\xi}^{k}{\\xi}^{l}p_{c}, \n\\end{align*}\nwhich resemble  Cartan's equations of structure.\nDirect check shows that $d$ does not respect the  brackets~(\\ref{brac}). (We \nomit the explicit formulas for $d\\{f,g\\}-\\{df,g\\} -(-1)^{\\tilde \nf}\\{f,dg\\}$, which contain both the Christoffel symbol and the Riemann \ntensor.) \nIt's not surprising, of course, because in our quantization we did not \ncare about $d$. The open questions are, if $d$ can be ``adjusted'' by \nadditional terms to satisfy more general condition~(\\ref{der}), or if some \nother quantization for $\\Omega(T^{*}M)$ can be found, which would better \n fit into the picture incorporating $d$.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn 2006, it was suggested by Pendry et al. \\cite{pendrycloak} that an\nobject surrounded by a coating consisting of an exotic\nmaterial becomes invisible to electromagnetic\nwaves. This device was named ``invisibility cloak'' in reference to Harry Potter, the popular character of J.K. Rowling. Beside his famous cloak, the little wizard has other spells to go unnoticed. Among the most spectacular is the ``polyjuice potion'' that is able to turn somebody into anybody else's appearance \\cite{jkrowling}. In this paper, we do not present a potion but rather an optical device able to accomplish the same task, i.e. to give an arbitrary optical response chosen in advance to any other object placed inside the device. In fact, the principle is here very similar to the design of Pendry's invisibility cloak but, instead of geometrically transforming an empty domain, it is a region containing the object to be imitated that is transformed leading thus to a generalization of cloaking.\n\n\nIn this section, we present a generalization of cloaking able to arbitrarily transform the electromagnetic appearance of an object. The basic principle is to obtain the constitutive relations of the cloak by application of a space transformation to a non-empty region.\n\nA geometric transformation is given by a map $\\varphi$ from a space $N$ to a space $M$. For all our practical purposes, $M$ and $N$ will be here the whole or parts of $\\mathbb{R}^3$. Given a Cartesian coordinate system $\\mathbf{x}$ on $M$ and an arbitrary coordinate system $\\mathbf{x}'$ on $N$, $\\varphi: N \\rightarrow M$ is described by $\\mathbf{x}(\\mathbf{x'})$, i.e. $\\mathbf{x}$ given as function of $\\mathbf{x}'$. All the useful information, i.e. necessary to transform differential forms   (indeed the formalism of differential geometry is the most natural to write the Maxwell's equation \\cite{deschamps,burke,nicolet,bossavit} specially when quite general geometric transformations are involved)   and other covariant tensors, is contained in the Jacobian matrix field $\\mathbf{J}(\\mathbf{x'})=\\partial \\mathbf{x}(\\mathbf{x}')\/\\partial \\mathbf{x}' $.\n\nThe basic principle of transformation optics is that, when you have an electromagnetic system described by the tensor fields $\\underline{\\underline{\\varepsilon}}(\\mathbf{x})$ for the dielectric permittivity and $\\underline{\\underline{\\mu}}(\\mathbf{x})$ for the magnetic permeability in the space $M$,  if you replace your initial material properties by equivalent material properties given by the following rule \\cite{pcfbook,milton,twistedEPJ,twistedJWAves}:\n      \\begin{eqnarray}\n      \\underline{\\underline{\\varepsilon'}}(\\mathbf{x'}) =\n      \\mathbf{J}^{-1}(\\mathbf{x'})\\underline{\\underline{\\varepsilon}}(\\mathbf{x}(\\mathbf{x'}))\\mathbf{J}^{-T}(\\mathbf{x'})\\det(\\mathbf{J}(\\mathbf{x'})),  \\notag \\\\\n      \\underline{\\underline{\\mu'}}(\\mathbf{x'}) =\n      \\mathbf{J}^{-1}(\\mathbf{x'})\\underline{\\underline{\\mu}}(\\mathbf{x}(\\mathbf{x'}))\\mathbf{J}^{-T}(\\mathbf{x'})\\det(\\mathbf{J}(\\mathbf{x'})),\n      \\label{equivalence_rule}\n      \\end{eqnarray}\n ($\\mathbf{J}^{-T}$ is\nthe transposed of the inverse of $\\mathbf{J}$), you get an equivalent problem on $N$. Here, an equivalent problem means that the solution of the new problem on $N$, i.e. electromagnetic quantities described as differential forms, are the pulled back of the solution \\cite{nicolet} of the original problem on $M$ and that the same Maxwell's equations (i.e. as if we were in Cartesian coordinates or, more accurately, having the same form written with the exterior derivative) are still satisfied.\n\nIn the case of the cylindrical Pendry's map \\cite{pendrycloak,opl,compel_geo}, described by the transformation of the 2D cross section, the plane $\\mathbb{R}^2$ minus a disk $D_1$ of radius $R_1$ is mapped on the whole plane $\\mathbb{R}^2$ in such a way that a disk $D_2$ of radius $R_2 > R_1$, concentric with $D_1$, is the image of the annulus $D_2\\backslash D_1$ by a radial transformation. In cylindrical coordinates, this transformation is given by:\n\n\\begin{equation}\\begin{cases}  r = (r'-R_1) R_2\/(R_2-R_1) \\; \\mathrm{for} \\; R_1 \\leq r'\\leq R_2, \\; \\\\ \\theta = \\theta', \\; z = z'. \\;\n\\label{pendrysmap}\\end{cases}\\end{equation}\n\nAs for the outside of the disk $D_2$, the map between the two copies of $\\mathbb{R}^2\\backslash D_2$ is the identity map.\n\nThe material properties given by rule (\\ref{equivalence_rule}) corresponding to this transformation provide an ideal invisibility cloak: outside $D_2$, everything behaves as if we were in free space, including the propagation of electromagnetic waves across the cloak, and is completely independent of the content of $D_1$.\n\n\nNow, rule (\\ref{equivalence_rule}) may be applied to $D_2$ containing objects with arbitrary electromagnetic properties so that a region cloaked by this device is still completely hidden but has the appearance of the objects originally in $D_2$. We may call this optical effect masking \\cite{teixeira} or ``polyjuice'' effect.\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[h\n{\\centering  \\includegraphics[width=0.49\\textwidth]{juicemesh2.pdf}\n\\caption{\\label{mesh}  This figure shows a part of the triangular mesh used for the finite element modeling of the scattering problem of  Fig. \\ref{fig2}. The singular behavior of the permittivity and of the permeability requires a very fine mesh along the inner boundary of the cloak in order to achieve a satisfactory accuracy with the numerical model.  }}\n\\end{figure}\n\n\n\\begin{figure}[h\n{\\centering \\includegraphics[width=0.49\\textwidth]{etopim2.jpg}\n\\caption{\\label{fig1}  A conducting triangular cylinder is scattering cylindrical waves.}}\n\\end{figure}\n\n\n\\begin{figure}[h\n{\\centering  \\includegraphics[width=0.49\\textwidth]{etopim1.jpg}\n\\caption{\\label{fig2}  A triangular cylinder different from the one on Fig. \\ref{fig1} is surrounded by a cloak designed to reproduce the scattering pattern of the Fig. \\ref{fig1} triangular cylinder in spite of the change of diffracting object. Of course, the diffracting object inside the cloak may be arbitrary as far as it is small enough to fit inside the cloak}}\n\\end{figure}\n\n\n\\begin{figure}[h\n{\\centering  \\includegraphics[width=0.49\\textwidth]{compare.pdf}\n\\caption{\\label{fig3}   The value of the electric field (the real part of $E_z$) on a circle of radius $4\\lambda$ concentric with the cloak is represented as a function of the position angle $\\theta$ (increasing counterclockwise and with $\\theta=0$ corresponding to the point the most on the right). The three configurations considered here are the ones of Fig. \\ref{fig2} (coated), Fig. \\ref{fig1} (original), and the triangle of Fig. \\ref{fig2} without the coating (reversed).  } }\n\\end{figure}\n\n\nFigs. \\ref{fig1} and \\ref{fig2} show the effect of masking on a scattering structure. On Fig. \\ref{fig1}, a cylindrical TM wave emitted by a circular cylindrical antenna is scattered by a conducting triangular cylinder (the longest side of the cross section is $1.62\\lambda$ and $\\varepsilon_r=1 + 40 i $). The field map  represents the longitudinal electric field $E_z(x,y)$ and the outer boundary of the cloak is shown to ease the comparison with the masked case. On Fig. \\ref{fig2}, the same cylindrical TM wave is scattered by a masked triangular cylinder (but the diffracting object inside the cloak may be arbitrary as far as it is small enough to fit inside the cloak). This triangular cylinder is the symmetric of the previous one with respect to the horizontal plane containing the central fibre of the cylindrical antenna. This bare scatterer would therefore give the Fig. \\ref{fig1} image inverted upside-down but, here, this object is surrounded by a cloak in order to give the very same scattering as before. Indeed, on both sides, the electric fields outside the cloak limit are alike.\n\n  Fig. \\ref{fig3} highlights the different scattering patterns by displaying the value of $\\Re e(E_z)$ on a circle of radius $4 \\lambda$ located around the antenna-scatterer system in the three following cases: the case of Fig. \\ref{fig1} (original) with the triangle alone, the case of Fig. \\ref{fig2} (coated), and the triangle of Fig. \\ref{fig2} without the coating (reversed). It is obvious that the coating restores the field distribution independently of the object present in the central hole.\n\n\nThe numerical computation is performed using the finite element method (via the free GetDP \\cite{getdp} and Gmsh \\cite{gmsh} software tools). The mesh is made of 148,000 second order triangles   including the Perfectly Matched Layers used to truncate the computation domain . The singularity of $\\varepsilon$ and $\\mu$ requires a very fine mesh in the vicinity of the inner boundary of the cloak (see Fig. \\ref{mesh}) and is also responsible for the small discrepancies between the numerical model and a perfect cloak (see Fig. \\ref{fig2}) --- including the non zero field in the hole of the cloak.\n\nNote that a small technical problem arises in practice when rule (\\ref{equivalence_rule}) is applied: the material properties are defined piecewise on various domains and it is very useful to know explicitly the boundaries of these domains, e.g. to build the finite element mesh (see Fig. \\ref{mesh}). These boundaries are curves in the cross section and are thus contravariant objects. Therefore, their transformation requires the inverse map $\\varphi^{-1}$ from $M$ to $N$. Fortunately, map (\\ref{pendrysmap}) is very simple to invert.\n\nMore explicitly, for a given curve $\\mathbf{x}(t)$ of parameter $t$ in the initial Cartesian coordinates, its push forward by Pendry's map is:\n\\begin{equation}\n\\mathbf{x}'(t)=\\varphi^{-1}(\\mathbf{x}(t))=(\\frac{R_2-R_1}{R_2}+ \\frac{R_1}{\\|\\mathbf{x}(t) \\|})\\mathbf{x}(t),\n\\end{equation}\nwith the same variation of the parameter $t$. Note that the most common curves used in the design of devices, i.e. line segments and arc of circles, are transformed to less usual curves except   for radial segments (with respect to the center of the cloak) and arc of circles concentric with the cloak.\n\n On Fig. \\ref{fig2}, the image by $\\varphi^{-1}$ of the triangle of Fig. \\ref{fig1} is the curvilinear triangle inside the coating region of the cloak. In practice, this anamorphosis of the triangle is described by three splines interpolating each 40 points that are images of points of the segments by $\\varphi^{-1}$.\n\n\n\n\nTransformation optics do not offer only the possibility make optically disappear objects in invisibility cloaks but also to completely tune their optical signature i.e. to give them an arbitrary appearance.\n   The possibility to place an object inside the coating of the cloak and that it will therefore appear different was already considered in \\cite{opl} where a point source was shifted creating a mirage effect. A more general case is considered here since both the shape and the position of the object placed in the coating are modified .\n Note that if the object used to create the illusion is perfectly conducting and surrounds the central point of the cloak, its anamorphosis will surround the inner boundary of the cloak and it \\emph{de facto} suppresses the singular behavior of the material properties. In this particular case,   the present approach is similar to the the hiding under the carpet idea   of Li and Pendry \\cite{carpet} but here a bounded object replaces the infinite reflecting plane. This technique can be naturally extended to cloaks of arbitrary shapes \\cite{arbitrary}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $\\I\\subset\\R$ be an interval and $f\\colon\\I\\to\\R$. The \\emph{difference operator} is given by\n\\[\n \\Delta_hf(x)=f(x+h)-f(x)\n\\]\nfor any $x\\in\\I$ and $h\\in\\R$ \\st\\ $x+h\\in\\I$. In 1954 Wright~\\cite{Wri54} considered the class of functions~$f:\\I\\to\\R$, for which the function $\\Delta_h f(\\cdot)$ is non-decreasing for any $h>0$ small enough to guarantee that all the involved arguments belong to~$\\I$. This monotonicity property is equivalent to the condition\n\\begin{equation}\\label{eq:Wright}\n f\\bigl(tx+(1-t)y\\bigr)+f\\bigl((1-t)x+ty\\bigr)\\xle f(x)+f(y)\\quad \\big(x,y\\in\\I,\\;t\\in[0,1]\\big).\n\\end{equation}\nNowadays a~function fulfilling~\\eqref{eq:Wright} is called to be \\emph{Wright-convex}. For $t=\\frac{1}{2}$ we get immediately\n\\[\n f\\Bigl(\\frac{x+y}{2}\\Bigr)\\xle\\frac{f(x)+f(y)}{2}\\quad\\bigl(x,y\\in\\I\\bigr),\n\\]\nso any Wright-convex function is necessarily Jensen-convex. It was shown by Ng~\\cite{Ng87} that any Wright-convex function is a~sum of the convex function (in the usual sense) and the additive one. Such representation leads to the simple (and now classical) proof that the inclusion in question is proper. Namely, if $a\\colon\\R\\to\\R$ is a~discontinuous additive function and $f(x)=|a(x)|$, then $f$~is discontinuous and Jensen-convex. Hence the graph of~$f$ is not dense on the whole plane. But it is well-known (\\cf~\\eg~\\cite{Kuc09}) that the graphs of discontinuous additive functions (and also -- by Ng's representation -- the graphs of discontinuous Wright-convex functions) mapping~$\\R$ into~$\\R$ are dense on the whole plane. That is why~$f$ is not Wright-convex.\n\\par\\medskip\nRecently Nikodem, Rajba and W\u0105sowicz in the paper~\\cite{NikRajWas12JMAA} considered the related comparison problem for the classes of $n$-Wright-convex functions and $n$-Jensen-convex functions (the explanation is given below in this section). If $n\\in\\N$ is odd, they have shown that the first class is the proper subclass of the latter one. The example of $n$-Jensen-convex function which is not $n$-Wright-convex was $f\\colon\\R\\to\\R$ \\st\\ $f(x)=\\bigl(\\max\\{a(x),0\\}\\bigr)^n$, where $a\\colon\\R\\to\\R$ is the certain discontinuous additive function. Next P\\'ales~\\cite{Pal13} proved that this example works for any discontinuous additive function. The authors of~\\cite{NikRajWas12JMAA} did not solve the  problem, if~$n$ is even. The objective of the present paper is to give the complete solution by showing that the inclusion is proper for all $n\\in\\N$. Moreover, the construction of our example does not depend on the parity of~$n$. Then for odd~$n$ we have a~solution different from the ones given in the papers~\\cite{NikRajWas12JMAA,Pal13}.\n\\par\\medskip\nLet us recall the concepts of higher-order Jensen-convex (coming from Popoviciu~\\cite{Pop34}) and Wright-convex functions (introduced by Gil\\'anyi and P\\'ales~\\cite{GilPal01}).\n\\par\\medskip\nThe iterates of ~difference operator are we defined as usual: for $n\\in\\N$\n\\[\n \\Delta_{h_1\\dots h_nh_{n+1}}f(x)=\\Delta_{h_1\\dots h_n}\\bigl(\\Delta_{h_{n+1}}f(x)\\bigr)\n\\]\nprovided that all the involved arguments belong to~$\\I$. If $h_1=\\dots=h_n=h$, then we write \n\\[\n \\Delta_h^nf(x)=\\Delta_{h\\dots h}f(x).\n\\]\nIn particular, $\\Delta_h^1f(x)=\\Delta_hf(x)$. \nLet $n\\in\\N$ and $f\\colon\\I\\to\\R$ be a~function. We say that $f$ is \\emph{Wright-convex of order~$n$} (or $n$-\\emph{Wright-convex} for short), if\n\\[\n \\Delta_{h_1\\dots h_{n+1}}f(x)\\xge 0\n\\]\nfor any $x\\in\\I$ and $h_1,\\ldots,h_{n+1}>0$ \\st\\ $x+h_1+\\dots+h_{n+1}\\in\\I$. If the above inequality is required only for $h_1=\\dots=h_{n+1}=h>0$, precisely, if\n\\[\n \\Delta_h^{n+1}f(x)\\xge 0\n\\]\nfor all $x\\in I$ and $h>0$ such that $x+(n+1)h\\in\\I$, then~$f$ is called \\emph{Jensen-convex of order~$n$} ($n$-\\emph{Jensen-convex} for short). It is easy to see that $1$-Wright-convexity and $1$-Jensen-convexity are equivalent to Wright-convexity and Jensen-convexity, respectively.\n\\par\\medskip\nBy the above definitions any $n$-Wright-convex function is necessarily $n$-Jensen-convex. As we announced above, in this paper we show that this inclusion is proper.\n\n\\section{Main result}\nLet us start with two preparatory notes. The following equations hold for a~power function function $f(x)=x^n$ (\\cf~\\cite[Lemma 15.9.2]{Kuc09}):\n\\begin{equation}\\label{eq:delta-xn}\n \\Delta_{h_1\\ldots h_n}\\bigl(x^n\\bigr)=n!\\cdot\\prod_{i=1}^nh_i \\qquad\\text{and}\\qquad \\Delta_h^n\\bigl(x^n\\big)=n!\\cdot h^n.\n\\end{equation}\n\\par\\medskip\nThe formula below was given in~\\cite{NikRajWas12JMAA} (as a~part of the proof of Corollary~2.2).\n\\begin{lem}\\label{lemat2}\nLet $a\\colon\\R\\to\\R$ be an additive function, $g\\colon\\R\\to\\R$ and $n\\in\\N$.\nThen\n\\[\n \\Delta_{h_1\\dots h_n}(g\\circ a) (x)=\\Delta_{a(h_1)\\dots\\,a(h_n)}g\\bigl(a(x)\\bigr)\n\\]\nfor any $x,h_1,\\dots,h_n\\in\\R$.\n\\end{lem}\nNow we are in a position to state our main result.\n\\begin{thm}\\label{th_main}\nFor any $n\\in\\N$ the class of $n$-Wright-convex functions is a~proper subclass of the class of $n$-Jensen-convex functions.\n\\end{thm}\n\n\\begin{proof}\n Because every $n$-Wright-convex function is $n$-Jensen-convex, it is enough to construct an~$n$-Jensen-convex function $f\\colon\\R\\to\\R$ which is not $n$-Wright convex. To this end consider a~Hamel basis $H$ of the linear space $\\R$ over~$\\Q$ \\st~$h_0=1\\in H$. Any $x\\in\\R$ is uniquely represented as a linear combination\n\\[\n x=\\lambda_0h_0+\\sum_{t}\\lambda_th_t,\n\\]\nwhere $\\lambda_0,\\lambda_t\\in\\Q$, $h_t\\in H$. Obviously the functions $\\alpha(x)=\\lambda_0h_0$ and $\\beta(x)=x-\\alpha(x)$ are additive. Define the function $f\\colon\\R\\to\\R$ by\n\\[\n f(x)=\\bigl(\\alpha(x)\\bigr)^{n+1}+\\bigl(\\beta(x)\\bigr)^{n+1}.\n\\]\nSince the difference operator is additive, we infer by Lemma~\\ref{lemat2} and the equation~\\eqref{eq:delta-xn} that\n\\begin{multline}\\label{eq:dj}\n\\Delta_{h_1\\dots h_{n+1}}f(x)=\\Delta_{\\alpha(h_1)\\ldots\\alpha(h_{n+1})}\\bigl(\\alpha(x)\\bigr)^{n+1}+\\Delta_{\\beta(h_1)\\ldots\\beta(h_{n+1})}\\bigl(\\beta(x)\\bigr)^{n+1}\\\\\n=(n+1)!\\biggl(\\prod_{i=1}^{n+1}\\alpha(h_i)+\\prod_{i=1}^{n+1}\\beta(h_i)\\biggr).\n\\end{multline}\nIn particular,\n\\[\n \\Delta_h^{n+1}f(x)=(n+1)!\\Bigl(\\bigl(\\alpha(h)\\bigr)^{n+1}+\\bigl(\\beta(h)\\bigr)^{n+1}\\Bigr).\n\\]\nLet $h>0$. Because of the representation $h=\\alpha(h)+\\beta(h)$, it is easy to see that $\\bigl(\\alpha(h)\\bigr)^{n+1}+\\bigl(\\beta(h)\\bigr)^{n+1}>0$. Therefore $\\Delta_h^{n+1}f(x)>0$ which shows that~$f$ is $n$-Jensen-convex.\n\nTo prove that $f$ is not $n$-Wright-convex take now $h_1=-1+\\sqrt{2}$, $h_2=1$ and (if $n\\ge 2$) $h_3=\\dots=h_{n+1}=1+\\sqrt{2}$. Then $h_i>0$, $i=1,\\dots,n+1$. Moreover $\\alpha(h_1)=-1$, $\\alpha(h_i)=1$,  $i=2,\\dots,n+1$ and $\\beta(h_2)=0$. By~\\eqref{eq:dj} we obtain \n\\[\n \\Delta_{h_1\\ldots h_{n+1}}f(x)=-(n+1)!<0.\n\\]\nIt means that~$f$ is not $n$-Wright-convex.\n\\end{proof}\n\n\\section{The classes of higher order strongly Jensen-convex functions and Wright-convex functions}\n\nLet $n\\in\\N$ and $c>0$. A~function $f\\colon\\I\\to\\R$ is called \\emph{strongly Wright-convex of order $n$ with modulus $c$} (or strongly $n$-Wright-convex with modulus $c$) if\n\\[\n\\Delta_{h_1\\ldots h_{n+1}}f(x)\\xge c (n+1)!\\,h_1\\ldots h_{n+1}\n\\]\nfor all $x\\in I$ and $h_1,\\ldots,h_{n+1}>0$ such that $x+h_1+\\dots+h_{n+1}\\in\\I$. If the above inequality is required only for $h_1=\\dots=h_{n+1}=h>0$, precisely, if\n\\[\n \\Delta_{h}^{n+1}f(x)\\xge c(n+1)!\\,h^{n+1}\n\\]\nfor all $x\\in I$ and $h>0$ such that $x+(n+1)h\\in\\I$, then $f$ is called \\emph{strongly Jensen-convex of order $n$ with modulus~$c$} (or strongly $n$-Jensen-convex with modulus $c$) (\\cf~ \\cite{GerNik11}).\nFor $c=0$ we arrive at standard $n$-Wright-convex and $n$-Jensen-convex functions, respectively.\n\\par\\medskip\nThe following characterization of strongly $n$-Wright-convex functions was recently given in~\\cite{GilMerNikPal15}.\n\\begin{thm}\\label{tw4}\nLet $n\\in\\N$ and $c>0$. A function $f\\colon\\I\\to\\R$ is strongly $n$-Wright-convex with modulus~$c$ if and only if the function $g\\colon\\I\\to\\R$ \\st\\ $g(x)=f(x)-cx^{n+1}$, is $n$-Wright-convex.\n\\end{thm}\nIt is surprising that so far nobody wrote explicitly the similar characterization of strongly $n$-Jensen-convex functions. We fill this gap now.\n\\begin{thm}\\label{tw5}\nLet $n\\in\\N$ and $c>0$. A function $f\\colon\\I\\to\\R$ is strongly $n$-Jensen-convex with modulus~$c$ if and only if the function $g\\colon\\I\\to\\R$ \\st\\ $g(x)=f(x)-cx^{n+1}$, is $n$-Jensen-convex.\n\\end{thm}\n\\begin{proof}\nSuppose first, that the function $f$ is strongly $n$-Jensen-convex with modulus $c$. Applying~\\eqref{eq:delta-xn} to $g$ we infer that\n\\[\n\\Delta_{h}^{n+1}g(x)= \\Delta_{h}^{n+1}f(x) - \\Delta_{h}^{n+1}(c x^{n+1}) \\xge c (n+1)!\\,h^{n+1}- c (n+1)!\\,h^{n+1}=0,\n\\]\nwhence $g$ is $n$-Jensen-convex.\n\nConversely, if $g$ is an $n$-Jensen-convex with modulus~$c$, use~\\eqref{eq:delta-xn} for $f(x)=g(x)+cx^{n+1}$:\n\\[\n\\Delta_{h}^{n+1}f(x)= \\Delta_{h}^{n+1}g(x) + \\Delta_{h}^{n+1}(c x^{n+1}) \\xge 0+ c (n+1)!\\,h^{n+1}=c (n+1)!\\,h^{n+1}.\n\\]\nThis proves that~$f$ is strongly $n$-Jensen-convex.\n\\end{proof}\nThe following result we derive from Theorems \\ref{th_main}, \\ref{tw4}, \\ref{tw5}.\n\\begin{thm}\nLet $n\\in\\N$ and $c>0$. The class of strongly $n$-Wright-convex functions with modulus~$c$ is a~proper subclass of the class of strongly $n$-Jensen-convex functions with modulus~$c$.\n\\end{thm}\n\\begin{proof}\nIt is enough to show, that there exists a strongly $n$-Jensen--convex function with modulus $c$, which is not strongly $n$-Wright--convex with modulus $c$. By Theorem~\\ref{th_main} there exists a function~$g$, which is $n$-Jensen-convex and $g$ is not $n$-Wright-convex. Let $f(x)=g(x)+cx^{n+1}$. Since $g$ is $n$-Jensen-convex, it follows by Theorem~\\ref{tw5} that $f$~is strongly $n$-Jensen-convex with modulus~$c$. By Theorem~\\ref{tw4}~$f$ is not strongly $n$-Wright-convex with modulus~$c$ (because the function $g(x)=f(x)-cx^{n+1}$ is not $n$-Wright-convex). This concludes the proof.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Introduction}\n\\label{se:intro}\n\nAfter the discovery of a Higgs boson at the Large Hadron Collider (LHC) \\cite{Aad:2012tfa,Chatrchyan:2012ufa} at CERN, the complete identification of this particle is ongoing. The  properties of the discovered particle, such as its couplings, are determined experimentally in order to fully identify its nature. For the endeavour of the identification of this particle, \ninput from the theory side is needed in form of precise predictions for the production and decay processes in the Standard Model (SM) as well as in its extensions that are to be tested. It is also crucial to provide reliable uncertainty estimates of the theoretical predictions. \nUnderestimating this uncertainty might lead to wrong conclusions.\nIn the SM, predictions and error estimates are well advanced, and in SM extensions they are\nconsolidating as well (see, e.g., the reviews in \nRefs.~\\cite{Dittmaier2011,Dittmaier2012,Dittmaier:2012nh,Heinemeyer:2013tqa,deFlorian:2016spz,%\nSpira:2016ztx}).\n\nOne of the simplest extensions of the SM \nis the Two-Higgs-Doublet Model (THDM) \\cite{Lee:1973iz,HHGuide:1990} where a second Higgs doublet is added \nto the SM field content. The underlying gauge group \n$\\mathrm{SU}(3)_{\\text{C}} \\times \\mathrm{SU}(2)_{\\text{W}} \\times \\mathrm{U}(1)_{\\text{Y}}$ as well as the fermion content of the SM are kept. After spontaneous symmetry breaking, there are five physical Higgs bosons where three of them are neutral and two are charged. In the CP-conserving case, which we consider, one of the neutral Higgs bosons is CP-odd and two are CP-even with one of them being SM-like. \n\nEven such a simple extension of the SM \ncan help solving some questions that are unanswered in the SM. For example, CP-violation in the Higgs sector could provide solutions to the problem of baryogenesis~\\cite{Turok:1990zg,Davies:1994id,Cline:1995dg,Fromme:2006cm}, and inert THDMs contain a dark matter candidate \\cite{PhysRevD.18.2574,Dolle:2009fn}. An even larger motivation comes from the embedding of the THDM into more complex models, such as axion \\cite{Kim:1986ax, Peccei:1977hh} or supersymmetric models \\cite{Haber:1984rc}. Some of the latter are promising candidates for a fundamental theory, and supersymmetric Higgs sectors contain a THDM (in which the doublets have opposite hypercharges). Even though the THDM is unlikely to be the fundamental theory of nature, it provides a rich phenomenology, which can be used in the search for a non-minimal Higgs sector without being limited by constraints from a \nmore fundamental theory.\n\nIn this sense, it is obvious that the THDM should be tested against data,\nand phenomenological studies have been performed recently, e.g., \n{in Refs.~\\cite{Celis:2013rcs,Celis:2013ixa,Chang:2013ona,Harlander:2013mla,%\nHespel:2014sla,Baglio:2014nea,Broggio:2014mna,%\nBernon:2015qea,Bernon:2015wef,%\nGoncalves:2016qhh,Dorsch:2016tab,%\nCacciapaglia:2016tlr,Cacchio:2016qyh,Aggleton:2016tdd,Han:2017pfo}.} \nIn order to provide precise predictions within this model, not only leading-order \n(LO), but also next-to-leading-order (NLO) contributions have to be taken into account. For the calculation of NLO contributions, a proper definition of a renormalization scheme is mandatory. There is no unique choice, and applying different renormalization schemes can help to estimate the theoretical uncertainty of the prediction that originates from the truncation of the perturbative series. \nThe renormalization of the THDM has already been tackled in several publications:\nFirst, in Ref.~\\cite{Santos:1996vt}, the fields and masses were renormalized in the on-shell scheme, \nhowever, the prescription given there does not cover all parameters.\nIn Refs.~\\cite{Kanemura:2004mg,LopezVal:2009qy}, a minimal field renormalization was applied, and the field mixing conditions were used to fix some of the mixing angles. \nIn view of an automation of NLO predictions within the THDM a tool was written by \nDegrande~\\cite{Degrande:2014vpa} \nwhere all finite rational terms and all\ndivergent terms are computed using on-shell conditions or conditions within the ``modified minimal subtraction scheme'' ($\\overline{\\mathrm{MS}}$). \nThough automation is very helpful, often specific problems occur depending on the model, the process, or the renormalization scheme considered, and, it might be necessary to solve these ``manually''. Specifically, spontaneously broken gauge theories with extended scalar sectors pose issues with the renormalization of vacuum expectation values and the related ``tadpoles'', \njeopardizing gauge independence and perturbative stability in predictions.\nRenormalization schemes employing a gauge-independent treatment of the tadpole terms were described recently in Refs.~\\cite{Krause:2016oke,Denner:2016etu,Krause:2016xku}. \n\nIn this paper, we perform the renormalization of various types of THDMs \n(Type I, Type II, ``lepton-specific'', and ``flipped''),\ndescribe four different renormalization schemes\n(for each type), \nand provide explicit results facilitating their application in NLO calculations. The comparison of results obtained in these renormalization schemes allows for checking their perturbative consistency, i.e.\\ whether the expansion point for the perturbation series is chosen well and no unphysically large corrections are introduced. Knowing in which parts of the parameter space a renormalization scheme leads to a stable perturbative behaviour  is important for the applicability of the scheme. \nIn addition, we investigate the dependence on the renormalization scale $\\mu_\\mathrm{r}$ which is introduced by defining some parameters via {$\\overline{\\mathrm{MS}}$}{} conditions. In order to investigate the $\\mu_\\mathrm{r}$ dependence consistently, we solve the renormalization group equations (RGEs) and include the running effects. We also make contact to different renormalization schemes suggested in the literature, including the recent formulations \\cite{Krause:2016oke,Denner:2016etu} \nwith gauge-independent treatments of tadpoles.%\n\\footnote{In our work we do not consider the ``tadpole-pinched'' scheme suggested\nin \\citere{Krause:2016oke}.\nFollowing the arguments of \\citeres{Denner:1994nn,Denner:1994xt}\nwe consider the ``pinch technique'' just as one of many physically\nequivalent choices to fix the gauge arbitrariness in off-shell\nquantities (related to the 't~Hooft--Feynman gauge of the\nquantum fields in the background-field gauge)\nrather than singling out ``its gauge-invariant part'' in any sense.}\nTo facilitate NLO calculations in practice, we have implemented our renormalization schemes\nfor the THDM into a \\textsc{FeynArts} \\cite{Hahn2001} model file, so that amplitudes and squared matrix elements can be generated straightforwardly. \nFinally, we apply the proposed renormalization schemes in the NLO calculation of the partial decay width of the \nlighter CP-even Higgs boson decaying into four fermions, $\\mathswitchr h \\to \\mathswitchr W\\PW\/\\mathswitchr Z\\PZ\\to 4f$,\na process class that is a cornerstone in the experimental determination of Higgs-boson\ncouplings, \n{but for which electroweak corrections in the THDM are not yet known in the literature.}\nThe impact of NLO corrections on Higgs couplings in the THDM was, for instance, investigated more\nglobally in \\citeres{Kanemura:2014dja,Kanemura:2015mxa}.\n{However, a full set of electroweak corrections to all Higgs-boson decay processes in the THDM \ndoes not yet exist in the literature, so that current predictions (see, e.g., \\citere{Harlander:2013qxa})\nfor THDM Higgs analyses globally neglect electroweak higher-order effects.}\n{Our calculation, thus, contributes to overcome this shortcoming;\nelectroweak corrections to some $1\\to2$ particle decays of heavy\nHiggs bosons were presented in \\citere{Krause:2016oke}.}%\n\\footnote{Since we consider the decays of the light Higgs boson $\\mathswitchr h$ via\n$\\mathswitchr W$- or $\\mathswitchr Z$-boson pairs, where at least one of the gauge bosons\nis off its mass shell, we have to consider the full $1\\to4$ process\nwith all off-shell and decay effects, rendering a comparison to\nresults on $\\mathswitchr H\\to\\mathswitchr W\\PW\/\\mathswitchr Z\\PZ$ not meaningful.}\n\nThe structure of the paper is as follows. We introduce the four considered types of THDMs\nand our conventions in Sect.~\\ref{sec:THDM}, and the derivation of the counterterm Lagrangian is performed in Sect.~\\ref{sec:CTLagrangian}. Afterwards we fix the renormalization constants with renormalization conditions (Sect.~\\ref{sec:renconditions}). The on-shell conditions, where the renormalized parameters correspond to measurable quantities, are  described and applied in Sect.~\\ref{sec:onshellrenconditions}. The renormalization constants of parameters that do not directly correspond to physical quantities are fixed in the $\\overline{\\mathrm{MS}}$ scheme, so that they contain only the UV divergences and no finite terms. We describe different renormalization schemes based on different definitions of the $\\overline{\\mathrm{MS}}$-renormalized parameters  in Sect.~\\ref{sec:diffrenschemes}. \nThe RGEs of the {$\\overline{\\mathrm{MS}}$}{}-renormalized parameters \nare derived and numerically solved in Sect.~\\ref{sec:running}. \nThe implementation of the results into an automated matrix element generator is described in Sect.~\\ref{sec:FAmodelfile}, and numerical results for \nthe partial decay width $\\mathswitchr h\\rightarrow 4 f$ are \npresented in Sect.~\\ref{se:numerics}. \nFinally, we conclude in Sect.~\\ref{se:conclusion}, \nand further details on the renormalization prescription as well as \nsome counterterms are given in the appendix.\n\n\\section{The Two-Higgs-Doublet Model}\n\\label{sec:THDM}\n\nThe Lagrangian of the THDM,  $\\mathcal{L}_\\mathrm{THDM}$, is composed of the following parts,\n\\begin{align}\n\\mathcal{L}_\\mathrm{THDM}=\\mathcal{L}_\\mathrm{Gauge}+\\mathcal{L}_\\mathrm{Fermion}+\\mathcal{L}_\\mathrm{Higgs}+\\mathcal{L}_\\mathrm{Yukawa}+\\mathcal{L}_\\mathrm{Fix}+\\mathcal{L}_\\mathrm{Ghost}. \n\\label{eq:LTHDM}\n\\end{align}\nThe gauge, fermionic, gauge-fixing, and ghost parts \ncan be obtained in a straightforward way from the SM ones, e.g., \ngiven in Ref.~\\cite{Denner:1991kt}. The Higgs Lagrangian and the Yukawa couplings to the fermions are discussed in the following and are mostly affected by the additional degrees of freedom \nof the THDM. A very elaborate and complete discussion of the THDM Higgs and Yukawa Lagrangians, including general and specific cases, can, e.g., \nbe found in Refs.~\\cite{Gunion:1989we,Branco:2011iw}.\n\n\\subsection{The Higgs Lagrangian}\n\\label{sec:HiggsL}\n\nThe Higgs Lagrangian, $ \\mathcal{L}_\\mathrm{Higgs}$, contains the kinetic terms and a potential $V$,\n\\begin{align}\n  \\mathcal{L}_\\mathrm{Higgs}= (D_\\mu\\Phi_1)^\\dagger (D^\\mu \\Phi_1)+(D_\\mu\\Phi_2)^\\dagger (D^\\mu \\Phi_2)- V(\\Phi_1^\\dagger \\Phi_1,\\Phi_2^\\dagger \\Phi_2,\\Phi_2^\\dagger \\Phi_1,\\Phi_1^\\dagger \\Phi_2),\n\\label{eq:Lhiggs}\n\\end{align}\nwith the complex scalar doublets $\\Phi_{1,2}$ \nof hypercharge $Y_\\mathrm{W}=1$, \n\\begin{align}\n\\Phi_1=\\begin{pmatrix} \\phi_1^+ \\\\ \\phi_1^0\\end{pmatrix}, \\qquad \\Phi_2=\\begin{pmatrix}\\phi_2^+ \\\\ \\phi^0_2\\end{pmatrix},\n\\label{eq:doublets}\n\\end{align}\nand the covariant derivative \n\\begin{align}\n  D_\\mu=\\partial_\\mu \\mp \\mathrm{i} g_2 I^a_\\mathrm{W} W^a_\\mu +\\mathrm{i} g_1 \\frac{Y_\\mathrm{W}}{2}B_\\mu,\n\\label{eq:EWcovderivative}\n\\end{align}\nwhere $I^a_\\mathrm{W}$ ($a = 1,2,3$) are the generators of the weak isospin. \nThe SU(2) and \nU(1) gauge fields are denoted \n$W^a_\\mu$ and $B_\\mu$ with the corresponding gauge couplings $g_2$ and $g_1$, respectively.\nThe sign in the \n$g_2$~term is negative in the conventions of B\\\"ohm, Hollik and Spiesberger \n(BHS)~\\cite{Denner:1991kt,Bohm:1986rj} and positive in the convention of Haber and Kane \n(HK)~\\cite{Haber:1984rc}. We implemented both sign conventions, but used the former one as default. \nIn general, the potential involves all hermitian functions of the two doublets up to dimension four and can be parameterized in the most general case as follows \\cite{Wu1994,Branco:2011iw},\n\\begin{align}\nV=&\\,m^2_{11} \\Phi^{\\dagger}_1 \\Phi_1+ m^2_{22} \\Phi^\\dagger_2 \\Phi_2\n-\\left[m^2_{12} \\Phi^\\dagger_1 \\Phi_2 + h.c.\\right]\\nonumber\\\\\n&+ \\frac{1}{2} \\lambda_1 (\\Phi^\\dagger_1 \\Phi_1)^2+\\frac{1}{2} \\lambda_2 (\\Phi^\\dagger_2 \\Phi_2)^2+ \\lambda_3 (\\Phi^\\dagger_1 \\Phi_1)(\\Phi^\\dagger_2 \\Phi_2)+ \\lambda_4 (\\Phi^\\dagger_1 \\Phi_2)(\\Phi^\\dagger_2 \\Phi_1)\\nonumber\\\\\n&+\\left[\\frac{1}{2} \\lambda_5(\\Phi^\\dagger_1 \\Phi_2)^2\n+(\\lambda_6 \\Phi^\\dagger_1\\Phi_1+\\lambda_7 \\Phi^\\dagger_2 \\Phi_2)\\Phi^\\dagger_1 \\Phi_2+h.c.\\right]\\label{eq:THDMpot}.\n\\end{align}\nThe parameters $m_{11}^2,m_{22}^2,\\lambda_1,\\lambda_2,\\lambda_3,\\lambda_4$ are real, while the parameters $m_{12}^2,\\lambda_5,\\lambda_6,\\lambda_7$ are complex, yielding a total number of 14 real degrees of freedom for the potential. However, the component fields of the two Higgs doublets $\\Phi_1$ and $\\Phi_2$ do not correspond to mass eigenstates, and these doublets can be redefined using an SU(2) transformation without changing the physics, so that only 11 physical degrees of freedom remain~\\cite{Branco:2011iw}.\nFor each Higgs doublet we demand that the fields develop a\nvacuum expectation value (vev) in the neutral component,\n\\begin{align}\n  \\langle \\Phi_1 \\rangle&=\\langle 0| \\Phi_1|0\\rangle =\\begin{pmatrix}0 \\\\ \\frac{v_1}{\\sqrt{2} }  \\end{pmatrix},&\n  \\langle \\Phi_2 \\rangle=\\langle 0| \\Phi_2|0\\rangle =\\begin{pmatrix} 0 \\\\ \\frac{v_2}{\\sqrt{2}}  \\end{pmatrix}.\n\\end{align}\nIt is non-trivial that such a stable minimum of the potential exists, \nrestricting the allowed parameter space already strongly~\\cite{Ferreira:2004yd}. \nIn general, the vevs are complex (with a significant relative phase).\nThe Higgs doublets can be decomposed as follows,\n\\begin{align}\n\\Phi_1=\\begin{pmatrix} \\phi_1^+ \\\\ \\frac{1}{\\sqrt{2}}(\\eta_1+i \\chi_1+v_1)\\end{pmatrix}, && \\Phi_2=\\begin{pmatrix}\\phi_2^+ \\\\ \\frac{1}{\\sqrt{2}}(\\eta_2+i \\chi_2+v_2)\\end{pmatrix},\n\\label{eq:decom}\n\\end{align}\nwith the charged fields $\\phi_1^+,\\phi_2^+$, the neutral CP-even fields $\\eta_1,\\eta_2$, and the neutral CP-odd fields $\\chi_1,\\chi_2$.\n\n\\paragraph{Additional constraints:}\nSince the 11-dimensional parameter space of the potential is too large for early experimental analyses, we restrict the model in our analysis by imposing two additional conditions, motivated by experimental results:\n\\begin{itemize}\n\\item absence of flavour-changing neutral currents at tree level,\n\\item CP conservation in the Higgs sector (even though this holds only approximately).\n\\end{itemize}\nThe former requirement can be ensured by adding a discrete $\\mathbb{Z}_2$ symmetry $\\Phi_1 \\rightarrow - \\Phi_1$ (see Sect.~\\ref{sec:YukawaL}). This condition implies that the parameters $\\lambda_6$ and $\\lambda_7$ vanish.  Permitting operators of dimension two that\nviolate this  $\\mathbb{Z}_2$ symmetry softly,  non-zero values of  $m_{12}$ are still allowed~\\cite{HHGuide:1990, Gunion:2002zf}.\nConcerning the second condition, the potential is CP-conserving if and only if a basis of the Higgs doublets exists in which all parameters and the vevs are real~\\cite{Gunion:2005ja}. For our description we assume that a transformation to such a basis has been done already (if the parameters or vevs were initially complex), so that we only have to deal with real parameters. This renders $m_{12}$ and $\\lambda_5$ real.\nHowever, at higher orders in perturbation theory CP-breaking terms and complex phases in the Higgs sector are generated radiatively through loop contributions\ninvolving the quark mixing matrix. For our NLO analysis, this does not present a problem as they appear only beyond NLO in the specific processes we consider.\nIn addition we assume that a basis of the doublets is chosen in which $v_1,v_2>0$ (which is always possible as a redefinition $\\Phi_i \\to - \\Phi_i$ changes the sign of the vacuum expectation value).\nThe potential~\\eqref{eq:THDMpot} has then the following form,\n\\begin{align}\nV={}&m^2_{11} \\Phi^{\\dagger}_1 \\Phi_1+ m^2_{22} \\Phi^\\dagger_2 \\Phi_2\n-m^2_{12} (\\Phi^\\dagger_1 \\Phi_2 + \\Phi^\\dagger_2 \\Phi_1)\\nonumber\\\\ \n&+ \\frac{1}{2} \\lambda_1 (\\Phi^\\dagger_1 \\Phi_1)^2+\\frac{1}{2} \\lambda_2 (\\Phi^\\dagger_2 \\Phi_2)^2+ \\lambda_3 (\\Phi^\\dagger_1 \\Phi_1)(\\Phi^\\dagger_2 \\Phi_2)+ \\lambda_4 (\\Phi^\\dagger_1 \\Phi_2)(\\Phi^\\dagger_2 \\Phi_1)\\nonumber\\\\ \n&+\\frac{1}{2} \\lambda_5\\left[(\\Phi^\\dagger_1 \\Phi_2)^2+(\\Phi^\\dagger_2 \\Phi_1)^2\\right]. \n\\label{eq:lambdapara}\n\\end{align}\nExpanding the potential using the decomposition~\\eqref{eq:decom} and ordering terms with respect to powers of the fields, leads to the form\n\\begin{align}\nV={}& -t_{\\eta_1} \\eta_1 - t_{\\eta_2} \\eta_2\\nonumber\\\\\n&+\\frac{1}{2} \\,(\\eta_1,\\eta_2) \\mathbf{M}_\\eta \\begin{pmatrix}\\eta_1 \\\\ \\eta_2 \\end{pmatrix}\n+\\frac{1}{2} (\\chi_1,\\chi_2)\\,  \\mathbf{M}_\\chi  \\begin{pmatrix}\\chi_1 \\\\ \\chi_2 \\end{pmatrix}\n+(\\phi_1^+,\\phi^+_2) \\, \\mathbf{M}_\\phi \\begin{pmatrix}\\phi^-_1 \\\\ \\phi^-_2 \\end{pmatrix}+\\ldots,\\label{eq:potinmatrix}\n\\end{align}\nwith the tadpole terms proportional to the tadpole parameters $t_{\\eta_1}$, $t_{\\eta_2}$ and linear in the fields. The mass terms contain the mass matrices \n$\\mathbf{M}_\\eta$, $\\mathbf{M}_\\chi$, and $ \\mathbf{M}_\\phi$ and are quadratic in the CP-even, CP-odd, and charged Higgs-boson fields, respectively. Terms cubic or quartic in the fields are suppressed in the notation here. Only the neutral CP-even scalar fields can develop non-vanishing tadpole terms, since they carry the quantum numbers of the vacuum. Further, in the mass terms, only particles with the same quantum numbers can mix, so that the three different \ntypes of scalars (neutral CP-even, neutral CP-odd, and charged) do not mix with one another. Of course, through the cubic and quartic terms, which are not shown here, these particles interact with each other. The tadpole parameters are\n\\begin{subequations}\n\\begin{align}\nt_{\\eta_1}=- m^2_{11} v_1- \\lambda_1v_1^3\/2 + v_2 (m^2_{12}-\\lambda_{345}v_1 v_2\/2),\\\\\nt_{\\eta_2}=- m^2_{22} v_2- \\lambda_2v_2^3\/2 + v_1 (m^2_{12}-\\lambda_{345} v_1 v_2\/2),\n\\end{align}%\n\\label{eq:tadpoleterms}%\n\\end{subequations}%\nwhere we introduced the abbreviations $\\lambda_{ij\\ldots}=\\lambda_i+\\lambda_j+\\ldots$, and the mass matrices are given by\n\\begin{subequations}\n\\begin{align}\n\\mathbf{M}_\\eta&=\\begin{pmatrix}  m^2_{11}+ 3 \\lambda_1 v_1^2\/2+ \\lambda_{345} v_2^2\/2 & -m^2_{12}+  \\lambda_{345} v_1 v_2\\\\ -m^2_{12}+  \\lambda_{345} v_1 v_2&  m^2_{22}+ 3 \\lambda_2 v_2^2\/2+ \\lambda_{345} v_1^2\/2\\end{pmatrix},\n\\\\\n\\mathbf{M}_\\chi&=\\begin{pmatrix} m^2_{11}+ \\lambda_1 v_1^2\/2+ (\\lambda_{34}-\\lambda_{5}) v_2^2\/2 & -m^2_{12}+ \\lambda_{5} v_1 v_2\\\\ \n\\hspace{-0pt}-m^2_{12}+ \\lambda_{5} v_1 v_2& \n\\hspace{-0pt}m^2_{22}+ \\lambda_2 v_2^2\/2+ (\\lambda_{34}-\\lambda_{5}) v_1^2\/2\\end{pmatrix},\n\\\\\n\\mathbf{M}_\\phi&=\n\\begin{pmatrix} m^2_{11}+ \\lambda_1 v_1^2\/2+ \\lambda_{3} v_2^2\/2 \n& -m^2_{12}+ \\lambda_{45} v_1 v_2\/2 \\\\ \n-m^2_{12}+ \\lambda_{45} v_1 v_2\/2 & m^2_{22}+ \\lambda_2 v_2^2\/2+ \\lambda_{3} v_1^2\/2\n\\end{pmatrix}.\n\\label{eq:massmatricesfirst}\n\\end{align}\n\\end{subequations}\nThe fields can be transformed into their mass eigenstate basis via\n\\begin{subequations}\n\\label{eq:rotations}\n\\begin{align}\n\\label{eq:Hhrot}\n\\begin{pmatrix}\\eta_1 \\\\ \\eta_2\\end{pmatrix} &\n= \\begin{pmatrix}\\cos{\\alpha} & -\\sin{\\alpha}\\\\ \\sin{\\alpha}& \\cos{\\alpha}\\end{pmatrix} \\begin{pmatrix}H \\\\ h\\end{pmatrix},\n\\\\\n\\begin{pmatrix}\\chi_1 \\\\ \\chi_2\\end{pmatrix}&\n= \\begin{pmatrix}\\cos{\\beta_n} & -\\sin{\\beta_n}\\\\ \\sin{\\beta_n}& \\cos{\\beta_n}\\end{pmatrix} \\begin{pmatrix}G_0 \\\\ A_0\\end{pmatrix},\n\\\\\n\\begin{pmatrix}\\phi^\\pm_1 \\\\ \\phi^\\pm_2\\end{pmatrix}&\n= \\begin{pmatrix}\\cos{\\beta_c} & -\\sin{\\beta_c}\\\\ \\sin{\\beta_c}& \\cos{\\beta_c}\\end{pmatrix} \\begin{pmatrix}G^\\pm \\\\ H^\\pm\\end{pmatrix}.\n\\end{align}\n\\end{subequations}\nwhere $h$, $H$ correspond to the CP-even, $A_0$ to the CP-odd, and $H^\\pm$ to the charged mass eigenstates\\footnote{In order to avoid a conflict in our notation, we define $\\alpha_\\mathrm{em}=e^2\/(4\\pi)$ as electromagnetic coupling constant and consistently keep the symbol $\\alpha$ for the rotation angle.}. The fields $G^\\pm$, \n$G_0$ correspond to the Goldstone bosons.\nAfter a rotation of the fields, the potential has the following form,\n\\begin{align}\n\\label{eq:NLObarepot}\nV=& -t_\\mathswitchr H H - t_\\mathswitchr h h\\nonumber\\\\*\n&+\\frac{1}{2} (H,h) \\begin{pmatrix}\\mathswitch {M_\\PH}^2 & M^2_{\\mathswitchr H\\mathswitchr h}\\\\ M^2_{\\mathswitchr H\\mathswitchr h}& \\mathswitch {M_\\Ph}^2\\end{pmatrix} \\begin{pmatrix}H \\\\ h \\end{pmatrix}\n+\\frac{1}{2} (G_0,A_0) \\begin{pmatrix}M^2_\\mathswitchr {G_0} & M^2_{\\mathswitchr {G_0}\\mathswitchr {A_0}}\\\\ M^2_{\\mathswitchr {G_0}\\mathswitchr {A_0}}& M^2_{\\mathswitchr {A_0}}\\end{pmatrix} \\begin{pmatrix}G_0 \\\\ A_0 \\end{pmatrix}\\nonumber\\\\\n&+ (G^+,H^+) \\begin{pmatrix}M_\\mathswitchr {G^+}^2 & M^2_{\\mathswitchr {G}\\mathswitchr {H^+}}\\\\ M^2_{\\mathswitchr {G}\\mathswitchr {H^+}}& \\mathswitch {M_\\PHP}^2\\end{pmatrix} \\begin{pmatrix}G^- \\\\ H^- \\end{pmatrix}+ \\textrm{interaction terms} \n\\end{align}\nwith the \ntadpole parameters\n\\begin{align}\n\\label{eq:tadpoledefs}\nt_\\mathswitchr H=\\mathswitch {c_\\alpha} t_{\\eta_1}+\\mathswitch {s_\\alpha} t_{\\eta_2}, \\qquad\nt_\\mathswitchr h=-\\mathswitch {s_\\alpha} t_{\\eta_1}+\\mathswitch {c_\\alpha} t_{\\eta_2},\n\\end{align}\nwhere general abbreviations for the trigonometric functions $s_x\\equiv\\sin x $, $c_x\\equiv\\cos x$, $t_x\\equiv\\tan x$ are introduced.\nAfter the elimination of $m_{11}$, $m_{22}$ using the above equations, the entries of the mass matrices contain also the tadpole parameters $t_h$, $t_H$. \nUsing \n\\begin{align}\nv^2=v_1^2+v_2^2, \\qquad \n\\tan{\\beta} = \\frac{v_2}{v_1},\n\\label{eq:tanbeta}\n\\end{align}\nwe obtain for the mass parameters of the CP-even Higgs bosons,\n\\begin{subequations}\n\\label{eq:neutralmassdef}\n\\begin{align}\n\\mathswitch {M_\\PH}^2={}&\\frac{2s_{\\alpha-\\beta}^2}{s_{2\\beta}}  m^2_{12}\n+ \\frac{v^2}{2} \\left( 2\\lambda_1 c_\\beta^2 c_\\alpha^2 +2\\lambda_2 s_\\alpha^2 s_\\beta^2 +s_{2 \\alpha} s_{2\\beta} \\lambda_{345}\\right)\n-2t_\\mathswitchr H \\frac{s_\\alpha^3 c_\\beta+c_\\alpha^3 s_\\beta }{ v s_{2 \\beta }}\n-t_\\mathswitchr h\\frac{s_{2 \\alpha } s_{\\alpha -\\beta }}{v  s_{2 \\beta }},\n\\\\\n\\mathswitch {M_\\Ph}^2={}&\\frac{2c_{\\alpha - \\beta}^2}{s_{2 \\beta}} m^2_{12}\n+ \\frac{v^2}{2} \\left( 2\\lambda_1 c_\\beta^2 s_\\alpha^2 +2\\lambda_2 c_\\alpha^2 s_\\beta^2-s_{2 \\alpha} s_{2\\beta} \\lambda_{345}\\right)\n-t_\\mathswitchr H\\frac{s_{2 \\alpha } c_{\\alpha -\\beta }}{v s_{2 \\beta }}\n-2t_\\mathswitchr h\\frac{c_\\alpha^3 c_\\beta-s_\\alpha^3 s_\\beta }{ v s_{2\\beta }},\n\\\\\nM_{\\mathswitchr H\\mathswitchr h}^2={}& \\frac{s_{2(\\alpha-\\beta)}}{s_{2 \\beta}}m^2_{12}\n+ \\frac{v^2}{2} \\left[ s_{2\\alpha} (-c_\\beta^2 \\lambda_1+ s_\\beta^2 \\lambda_2)+ s_{2 \\beta} c_{2\\alpha} \\lambda_{345}\\right]\n-t_\\mathswitchr H\\frac{s_{2 \\alpha } s_{\\alpha -\\beta }}{v s_{2 \\beta }}\n-t_\\mathswitchr h\\frac{s_{2 \\alpha } c_{\\alpha -\\beta }}{v s_{2 \\beta }},\n\\label{eq:Hhmassmixing}\n\\end{align}\n\\end{subequations}\nthe ones of the mass matrix of the CP-odd Higgs fields result in\n\\begin{subequations}\n\\label{eq:cpmassdef}\n\\begin{align}\n\\mathswitch {M_\\PAO}^2&=2c_{\\beta-\\beta_n}^2\\Big(\\frac{m^2_{12}}{s_{2\\beta}}- \\frac{\\lambda_5 v^2}{2}\\Big) \n- 2t_\\mathswitchr H \\frac{c_{\\beta_n}^2 c_\\beta s_\\alpha + s_{\\beta_n}^2 s_\\beta c_\\alpha}{v s_{2\\beta}}\n- 2t_\\mathswitchr h \\frac{c_{\\beta_n}^2 c_\\beta c_\\alpha - s_{\\beta_n}^2 s_\\beta s_\\alpha}{v s_{2\\beta}}, \n\\label{eq:A0massdef}\\\\\nM_{\\mathswitchr {G_0}}^2&=2s_{\\beta-\\beta_n}^2\\Big(\\frac{m^2_{12}}{s_{2\\beta}}- \\frac{\\lambda_5 v^2}{2}\\Big)\n- 2t_\\mathswitchr H \\frac{s_{\\beta_n}^2 c_\\beta s_\\alpha + c_{\\beta_n}^2 s_\\beta c_\\alpha}{v s_{2\\beta}}\n- 2t_\\mathswitchr h \\frac{s_{\\beta_n}^2 c_\\beta c_\\alpha - c_{\\beta_n}^2 s_\\beta s_\\alpha}{v s_{2\\beta}},\\\\\nM_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2&=-s_{2(\\beta-\\beta_n)}\\Big(\\frac{m^2_{12}}{s_{2\\beta}}- \\frac{\\lambda_5 v^2}{2}\\Big)- t_\\mathswitchr H \\frac{s_{2\\beta_n} s_{\\alpha-\\beta}}{v s_{2\\beta}}- t_\\mathswitchr h \\frac{s_{2\\beta_n} c_{\\alpha-\\beta}}{v s_{2\\beta}}, \\label{eq:goldstonemassmixing}\n\\end{align}\n\\end{subequations}\nand the ones of the mass matrix of the charged Higgs-boson fields are\n\\begin{subequations}\n\\label{eq:chargedmassdef}\n\\begin{align}\n\\mathswitch {M_\\PHP}^2&=c_{\\beta-\\beta_c}^2\\Big[\\frac{2m^2_{12}}{s_{2\\beta}}-\\frac{v^2}{2}(\\lambda_4+\\lambda_5) \\Big]\n- 2t_\\mathswitchr H \\frac{c_{\\beta_c}^2 c_\\beta s_\\alpha + s_{\\beta_c}^2 s_\\beta c_\\alpha}{v s_{2\\beta}}\n- 2t_\\mathswitchr h \\frac{c_{\\beta_c}^2 c_\\beta c_\\alpha - s_{\\beta_c}^2 s_\\beta s_\\alpha}{v s_{2\\beta}},\n\\label{eq:Hpmmassdef}\\\\\nM_\\mathswitchr {G^+}^2&=s_{\\beta-\\beta_c}^2\\Big[\\frac{2m^2_{12}}{s_{2\\beta}}-\\frac{v^2}{2}(\\lambda_4+\\lambda_5) \\Big]\n- 2t_\\mathswitchr H \\frac{s_{\\beta_c}^2 c_\\beta s_\\alpha + c_{\\beta_c}^2 s_\\beta c_\\alpha}{v s_{2\\beta}}\n- 2t_\\mathswitchr h \\frac{s_{\\beta_c}^2 c_\\beta c_\\alpha - c_{\\beta_c}^2 s_\\beta s_\\alpha}{v s_{2\\beta}},\n\\\\\nM_{\\mathswitchr {G}\\mathswitchr {H^+}}^2&=-\\frac{s_{2(\\beta-\\beta_c)}}{2} \\Big[\\frac{2m^2_{12}}{s_{2\\beta}}-\\frac{v^2}{2}(\\lambda_4+\\lambda_5) \\Big]\n- t_\\mathswitchr H \\frac{s_{2\\beta_c} s_{\\alpha-\\beta}}{v s_{2 \\beta}}\n- t_\\mathswitchr h \\frac{s_{2\\beta_c} c_{\\alpha-\\beta}}{v s_{2 \\beta}}.\n\\label{eq:chargedgoldstonemassmixing}\n\\end{align}\n\\end{subequations}\nAt tree level we demand vanishing tadpole terms corresponding to $t_\\mathswitchr H=t_\\mathswitchr h=0$ and diagonal propagators, so that the mixing terms proportional to $M_{\\mathrm{Hh}}^2$, $M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2$, $M_{\\mathswitchr {G}\\mathswitchr {H^+}}^2$ vanish at LO. This yields $\\beta_c=\\beta_n=\\beta$ and fixes the angle $\\alpha$ as well.\nAt NLO such a diagonalization is not possible, as the propagators receive also mixing contributions from the field renormalization and from \n(momentum-dependent) one-loop diagrams, so that there is no distinct condition to define the mixing angles. Therefore we keep the bare mass mixing parameters $M_{\\mathrm{Hh}}^2$, $M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2$, $M_{\\mathswitchr {G}\\mathswitchr {H^+}}^2$ and the tadpole terms $t_\\mathswitchr H$, $t_\\mathswitchr h$ in this section, and \nspecify defining conditions for the bare parameters $\\alpha$, $\\beta_n$, $\\beta_c$ and the tadpole terms later.\nWith the above equations, $m_{12}$, $\\lambda_1$, $\\lambda_2$, $\\lambda_3$, $\\lambda_4$ can be traded for the masses of the physical bosons $M_\\mathswitchr H$, $M_\\mathswitchr h$, $M_\\mathswitchr {A_0}$, $\\mathswitch {M_\\PHP}$, and the mixing angle $\\alpha$. The parameter $\\lambda_5$ cannot be replaced by a mass or a mixing angle as it appears only in cubic and quartic Higgs couplings and acts like an additional coupling constant. Explicit relations can be obtained by inverting Eqs.~\\eqref{eq:neutralmassdef}, \n\\eqref{eq:A0massdef}, and~\\eqref{eq:Hpmmassdef}.\nThe other parameters are related to the masses by\n\\begin{subequations}\n\\label{eq:repara}\n\\begin{align}\n\\lambda_1={}&\\frac{1}{\\mathswitch {c_\\beta}^2 v^2} \\left[\\mathswitch {c_\\alpha}^2 \\mathswitch {M_\\PH}^2+\\mathswitch {s_\\alpha}^2 \\mathswitch {M_\\Ph}^2- M_{\\mathswitchr H\\mathswitchr h}^2 s_{2\\alpha}-\\sb^2 \\biggl(\\frac{\\mathswitch {M_\\PAO}^2}{ c_{\\beta-\\beta_n}^2}+\\lambda_5 v^2\\biggr) \\right]\n\\nonumber\\\\\n&+t_\\mathswitchr H\\frac{c_{{\\beta_{n}}} \\left(2 s_{\\beta } s_{{\\beta_{n}}} c_{\\alpha }+c_{\\alpha +\\beta }c_{\\beta_n}\\right)}{v^3 c_{\\beta }^2 c_{\\beta -{\\beta_{n}}}^2}\n -t_\\mathswitchr h \\frac{c_{\\beta_{n}} \\left(2 s_{\\beta } s_{\\beta_{n}} s_{\\alpha }+s_{\\alpha +\\beta }c_{\\beta_n}\\right)}{v^3 c_{\\beta }^2 c_{\\beta -\\beta_{n}}^2},\n\\\\\n\\lambda_2={}&\\frac{1}{\\sb^2 v^2} \\left[\\mathswitch {c_\\alpha}^2 \\mathswitch {M_\\Ph}^2+\\mathswitch {s_\\alpha}^2 \\mathswitch {M_\\PH}^2+ M_{\\mathswitchr H\\mathswitchr h}^2 s_{2\\alpha}-\\mathswitch {c_\\beta}^2 \\biggl(\\frac{\\mathswitch {M_\\PAO}^2}{c_{\\beta-\\beta_n}^2}+\\lambda_5 v^2\\biggr) \\right]\\nonumber\\\\\n&+ t_\\mathswitchr H \\frac{s_{\\beta_n} (2 c_\\beta c_{\\beta_n} s_\\alpha- c_{\\alpha+\\beta} s_{\\beta_n})}{v^3 s_\\beta^2 c_{\\beta-\\beta_n}^2}+ t_\\mathswitchr h \\frac{s_{\\beta_n} (2 c_\\beta c_{\\beta_n} c_\\alpha+ s_{\\alpha+\\beta} s_{\\beta_n})}{v^3 s_\\beta^2 c_{\\beta-\\beta_n}^2},\n\\\\\n\\lambda_3={}&\\frac{1}{v^2 s_{2 \\beta}} \\left[s_{2\\alpha}(\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2) + 2 c_{2\\alpha}M_{\\mathswitchr H\\mathswitchr h}^2\\right]-\\frac{\\mathswitch {M_\\PAO}^2}{v^2c_{\\beta-\\beta_n}^2}+\\frac{2 \\mathswitch {M_\\PHP}^2}{v^2c_{\\beta-\\beta_c}^2}-\\lambda_5\\nonumber \n\\\\\n&+\\frac{2t_\\mathswitchr H}{v^3 s_{2\\beta }} \n\\left[s_{\\beta } c_{\\alpha } \\bigg(\\frac{2 s^2_{\\beta_c}}{c_{\\beta -\\beta_c}^2}-\\frac{s^2_{\\beta_n}}{c_{\\beta -\\beta_n}^2}\\bigg)+c_{\\beta } s_{\\alpha }\n   \\bigg(\\frac{2 c^2_{\\beta_c}}{c_{\\beta -\\beta_c}^2}-\\frac{c^2_{\\beta_n}}{c_{\\beta -\\beta_n}^2}\\bigg)\\right]\n\\nonumber\\\\\n&+  \\frac{2t_\\mathswitchr h}{v^3 s_{2\\beta }} \n\\left[c_{\\beta } c_{\\alpha }\\bigg(\\frac{2 c^2_{\\beta_c}}{c_{\\beta -\\beta_c}^2}-\\frac{c^2_{\\beta_n}}{c_{\\beta -\\beta_n}^2}\\bigg)+s_{\\beta } s_{\\alpha }\n   \\bigg(\\frac{s^2_{\\beta_n}}{c_{\\beta -\\beta_n}^2}-\\frac{2 s^2_{\\beta_c}}{c_{\\beta -\\beta_c}^2}\\bigg)\\right]\n\\label{eq:Lambda3NLO}\\\\\n\\lambda_4={}&\\lambda_5+\\frac{2 \\mathswitch {M_\\PAO}^2}{v^2 c_{\\beta -\\beta _n}^2}-\\frac{2 \\mathswitch {M_\\PHP}^2}{v^2 c_{\\beta -\\beta _c}^2}+\\frac{2 t_\\mathrm{H} s_{\\beta _c-\\beta _n} \\left(s_{\\alpha +\\beta -\\beta _c-\\beta\n   _n}-s_{\\beta -\\alpha } c_{\\beta _c-\\beta _n}\\right)}{v^3 c_{\\beta -\\beta _c}^2 c_{\\beta -\\beta _n}^2}\\nonumber\\\\\n&+\\frac{2 t_\\mathrm{h} s_{\\beta _c-\\beta _n} \\left(c_{\\alpha +\\beta -\\beta _c-\\beta\n   _n}+c_{\\beta -\\alpha } c_{\\beta _c-\\beta _n}\\right)}{v^3 c_{\\beta -\\beta _c}^2 c_{\\beta -\\beta _n}^2},\\\\\nm^2_{12}={}&\\frac{1}{2}\\lambda_5 v^2 s_{2\\beta }\n+\\frac{\\mathswitch {M_\\PAO}^2 s_{2\\beta }}{2c_{\\beta -\\beta _n}^2}\n+\\frac{t_\\mathrm{H} \\left(s_{\\beta } c_{\\alpha } s_{\\beta _n}^2+c_{\\beta } s_{\\alpha } c_{\\beta_n}^2\\right)}{v c_{\\beta -\\beta _n}^2}\n+\\frac{t_\\mathrm{h} \\left(c_{\\beta } c_{\\alpha } c_{\\beta _n}^2-s_{\\beta }\n   s_{\\alpha } s_{\\beta _n}^2\\right)}{v c_{\\beta -\\beta _n}^2}.\\label{eq:m12NLO}\n\\end{align}\n\\end{subequations}\nThe tree-level relations are\neasily obtained by setting $\\beta_c=\\beta_n=\\beta$ and $t_\\mathswitchr H=t_\\mathswitchr h=M_{\\mathrm{Hh}}^2=M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2=M_{\\mathswitchr {G}\\mathswitchr {H^+}}^2=0$,\n\\begin{subequations}\n\\begin{align}\n\\lambda_1&={}\\frac{1}{\\mathswitch {c_\\beta}^2 v^2} \\left[\\mathswitch {c_\\alpha}^2 \\mathswitch {M_\\PH}^2+\\mathswitch {s_\\alpha}^2\n\\mathswitch {M_\\Ph}^2-\\sb^2 (\\mathswitch {M_\\PAO}^2+\\lambda_5 v^2) \\right],\\\\\n\\lambda_2&={}\\frac{1}{\\sb^2 v^2} \\left[\\mathswitch {s_\\alpha}^2 \\mathswitch {M_\\PH}^2+\\mathswitch {c_\\alpha}^2\n\\mathswitch {M_\\Ph}^2-\\mathswitch {c_\\beta}^2 (\\mathswitch {M_\\PAO}^2+\\lambda_5 v^2) \\right],\\\\\n\\lambda_3&={}\\frac{s_{2\\alpha}}{s_{2\\beta} v^2}\n(\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2)-\\frac{1}{v^2} (\\mathswitch {M_\\PAO}^2-2\\mathswitch {M_\\PHP}^2)-\\lambda_5,\n\\label{eq:lambda3tree}\\\\\n\\lambda_4&={}\\frac{2(\\mathswitch {M_\\PAO}^2-\\mathswitch {M_\\PHP}^2)}{v^2}+\\lambda_5,\\\\\nm^2_{12}&={}\\frac{s_{2\\beta}}{2} (\\mathswitch {M_\\PAO}^2+\\lambda_5 v^2).\n\\end{align}\n\\end{subequations}\n\n\\paragraph{Parameters of the gauge sector:}\nMass terms of the gauge bosons arise through the interaction of the gauge bosons with the vevs, \nanalogous to the SM. After a rotation into fields corresponding to mass eigenstates, one obtains relations similar to the SM ones:\n\\begin{align}\n\\mathswitch {M_\\PW}=g_2 \\frac{v}{2},&& \\mathswitch {M_\\PZ}=\\frac{v}{2} \\sqrt{g_1^2+g_2^2},&& e= \\frac{g_1g_2}{\\sqrt{g_1^2+g_2^2}},\n\\label{eq:EWmassesTHDM}\n\\end{align}\nwhere the electric unit charge $e$ is identified with the coupling constant of the photon field $A_\\mu$ in the covariant derivative.\nInverting these relations and introducing the weak mixing angle $\\theta_W$ via $\\cos \\theta_W=g_2\/\\sqrt{g_1^2+g_2^2}$, one can replace $v$ and the gauge couplings $g_1$ and $g_2$ by\n\\begin{align}\nv&= \\frac{2 \\mathswitch {M_\\PW} \\mathswitch {s_{\\scrs\\PW}}}{e},& g_1&=\\frac{e}{\\mathswitch {c_{\\scrs\\PW}}},& g_2&=\\frac{e}{\\mathswitch {s_{\\scrs\\PW}}}.\n\\label{eq:EWtrafo}\n\\end{align}\n\n\\paragraph{Mass parameterization:}\nThe relations~\\eqref{eq:tadpoledefs}, \n\\refeq{eq:tanbeta},\n(\\ref{eq:repara}a,b,d,e), and \\eqref{eq:EWtrafo} between the masses, angles, and the basic parameters can be used to reparameterize the Higgs Lagrangian and change the parameters\n\\begin{align}\n\\{p_{\\mathrm{basic}}\\}=\\{\\lambda_1,\\ldots,\\lambda_5,m^2_{11},m^2_{22},m^2_{12},v_1,v_2,g_1,g_2\\},\\label{eq:basicpara}\n\\end{align}\nin favour of the bare mass parameters including one parameter from scalar self-interactions\nwhich we take as $\\lambda_3$, \n\\begin{align}\n\\{p'_{\\mathrm{mass}}\\}=\\{\\mathswitch {M_\\PH},\\mathswitch {M_\\Ph},\\mathswitch {M_\\PAO},\\mathswitch {M_\\PHP},M_\\mathrm{W},\\mathswitch {M_\\PZ},e,\\lambda_5,\\lambda_3,\\beta, t_\\mathrm{H} ,t_\\mathrm{h}\\}\n\\label{eq:physparaNLOprime}\n\\end{align}\nAdditionally, one has to keep in mind that we keep the mixing parameters $\\alpha$, $\\beta_n$, and $\\beta_c$ generic, and they have to be fixed by additional conditions (which  will be given later).\nOne can use Eq.~(\\ref{eq:repara}c) to trade $\\lambda_3$ for $\\alpha$, in which case the mixing angle becomes a free parameter of the theory. Then, one obtains the parameter set\n\\begin{align}\n\\{p_{\\mathrm{mass}}\\}=\\{\\mathswitch {M_\\PH},\\mathswitch {M_\\Ph},\\mathswitch {M_\\PAO},\\mathswitch {M_\\PHP},M_\\mathrm{W},\\mathswitch {M_\\PZ},e,\\lambda_5,\\alpha,\\beta, t_\\mathrm{H} ,t_\\mathrm{h} \\}\n\\label{eq:physparaNLO}\n\\end{align}\n\n\\subsection{Yukawa couplings}\n\\label{sec:YukawaL}\n\nThe Higgs mechanism does not only give rise to the gauge-boson mass terms (which are determined by the vevs), \nbut via Yukawa couplings, it introduces\nmasses to chiral fermions. Since both Higgs doublets can couple to fermions, the general Yukawa couplings have the form\n\\begin{align}\n\\mathcal{L}_\\mathrm{Yukawa}=&-\\sum_{k=1,2}\\sum_{i,j} \\left(\\bar{L}'^{\\mathrm{L}}_i \\zeta^{l,k}_{ij} l'^{\\mathrm{R}}_j \\Phi_k+\\bar{Q}'^{\\mathrm{L}}_i \\zeta^{u,k}_{ij} u'^{\\mathrm{R}}_j \\tilde{\\Phi}_k+\\bar{Q}'^{\\mathrm{L}}_i \\zeta^{d,k}_{ij} d'^{\\mathrm{R}}_j \\Phi_k+h.c.\\right),\n\\label{LYuk}\n\\end{align}\nwith the mixing matrices $\\zeta^{f,k}$, $k=1,2$, in generation space for the \ngauge-invariant interactions with $\\Phi_1$ and $\\Phi_2$, respectively, and the generation indices $i, j = 1,2,3$.\nThe left-handed SU(2) doublets of quarks and leptons are denoted \n${Q}'^{\\mathrm{L}}=\\left(u'^{\\mathrm{L}},d'^{\\mathrm{L}}\\right)^{\\mathrm{T}}$ and \n${L}'^{\\mathrm{L}}=\\left(\\nu'^{\\mathrm{L}},l'^{\\mathrm{L}}\\right)^{\\mathrm{T}}$, \nwhile the right-handed up-type quark, down-type quark, and lepton singlets are  $u'^{\\mathrm{R}}$, $d'^{\\mathrm{R}}$, and $l'^{\\mathrm{R}}$, respectively. \nThe primes indicate that we deal with fields \nin the interaction basis here;\nfields without primes correspond to mass eigenstates.\nThe field $\\tilde{\\Phi}_k$, $k = 1, 2$, is the charge-conjugated field of $\\Phi_k$.\nSince, in the general THDM, there are two mass mixing matrices for each type $f$ of fermions, flavour-changing neutral currents (FCNC) can occur at tree level, which, however, are experimentally known to be strongly suppressed.\nAccording to the Paschos--Glashow--Weinberg theorem~\\cite{Glashow1977,Paschos1977a}, \nFCNC are absent at tree level\nif each type of fermion couples only to one of the Higgs doublets.\n\n This can be achieved by imposing an additional discrete $\\mathbb{Z}_2$ symmetry. It should be noted that the soft-$\\mathbb{Z}_2$-breaking term  proportional to $m_{12}$ in the Higgs potential does not introduce FCNC. \nThe Yukawa Lagrangian reduces then to\n\\begin{align}\n\\mathcal{L}_\\mathrm{Yukawa}=-\\sum_{i,j} \\left(\\bar{L}'^{\\mathrm{L}}_i \\zeta^{l}_{ij} l'^{\\mathrm{R}}_j \\Phi_{n_1}+\\bar{Q}'^{\\mathrm{L}}_i \\zeta^{u}_{ij} u'^{\\mathrm{R}}_j \\tilde{\\Phi}_{n_2}+\\bar{Q}'^{\\mathrm{L}}_i \\zeta^{d}_{ij} d'^{\\mathrm{R}}_j \\Phi_{n_3}+h.c.\\right),\n\\end{align}\nwith $n_i$ being either $1$ or $2$. \nDepending on the exact form of the symmetry,\none distinguishes four types of THDMs. In Type~I models, all fermions couple to one Higgs doublet (conventionally $\\Phi_2$, but this is equivalent to $\\Phi_1$ due to possible basis changes) which can be ensured by demanding a $\\Phi_1 \\rightarrow -\\Phi_1$ symmetry. In Type~II models, down-type fermions couple to the other doublet, which can be enforced by the symmetry $\\Phi_1 \\rightarrow -\\Phi_1$, $d'^{\\mathrm{R}}_j \\to- d'^{\\mathrm{R}}_j$, $l'^{\\mathrm{R}} \\to -l'^{\\mathrm{R}}$. The other two possibilities are called ``lepton-specific'' (Type~X) and ``flipped'' (Type~Y) models. An overview over the couplings and symmetries of the different models is given in Tab.~\\ref{tab:yukcoupl}.\n\\begin{table}\n  \\centering\n\\begin{tabular}{|c|c|c|c|c|}\\hline\n&\\hspace{20pt} $u_i$\\hspace{20pt} &\\hspace{20pt} $d_i$\\hspace{20pt}  &\\hspace{20pt}  $e_i$\\hspace{20pt} & $\\mathbb{Z}_2$ symmetry \\\\\\hline\nType I & $\\Phi_2$ & $\\Phi_2$ & $\\Phi_2$ & $\\Phi_1 \\rightarrow - \\Phi_1$ \\\\\nType II& $\\Phi_2$ & $\\Phi_1$ & $\\Phi_1$ &$(\\Phi_1, d_i,e_i) \\rightarrow -(\\Phi_1, d_i,e_i)$\\\\\nLepton-specific & $\\Phi_2$ & $\\Phi_2$ & $\\Phi_1$ &$(\\Phi_1, e_i) \\rightarrow -(\\Phi_1,e_i)$ \\\\\nFlipped & $\\Phi_2$ & $\\Phi_1$ & $\\Phi_2$ & $(\\Phi_1, d_i) \\rightarrow -(\\Phi_1, d_i)$\\\\\\hline\n\\end{tabular}  \n  \\caption{Different types of the THDM having in common that only one of the Higgs doublet couples to each type of fermions. This can be achieved by imposing appropriate $\\mathbb{Z}_2$ symmetry charges to the fields. }\n\\label{tab:yukcoupl}\n\\end{table}\nFor each of the fermion types, a redefinition of the fields can be performed in order to get diagonal mass matrices, analogously to the SM case. Similar to the SM, \nthe coupling of fermions to the Z boson is flavour conserving, and a CKM matrix appears in the coupling to the charged gauge bosons. \nBy specifying the model type, the Higgs--fermion interaction is determined, and one can write them, \nwidely following the notation of \nRef.~\\cite{Aoki2009}, as\n\\begin{align}\n \\mathcal{L}_\\mathrm{Yukawa,int}=\n&- \\sum_i \\sum_{f=u,d,l} \\frac{m_{f,i}}{v}\\left( \\xi^f_\\mathswitchr h\\, \\bar{f_i}f_i h+ \\xi^f_\\mathswitchr H \\, \\bar{f}_if_iH\n-2\\mathrm{i} I^3_{\\mathswitchr W,f} \\xi^f_\\mathswitchr {A_0} \\, \\bar{f}_i\\gamma_5 f_i A_0\n- 2\\mathrm{i} I^3_{\\mathswitchr W,f} \\bar{f}_i\\gamma_5 f_i G_0\\right)\\nonumber\\\\\n&-\\sum_{i,j}\\biggl[\\frac{\\sqrt{2}V_{ij}}{v} \\bar{u}_i (-m_{u,i} \\xi^u_\\mathswitchr {A_0} \\omega_- +m_{d,j} \\xi^d_\\mathswitchr {A_0} \\omega_+)\\, d_j H^+ + h.c. \\biggr]\n\\nonumber\\\\\n&- \\sum_i \\biggl[ \\frac{\\sqrt{2}m_{l,i} \\xi^l_\\mathswitchr {A_0}}{v}\\, \\bar{\\nu}_i^{\\mathrm{L}} l_i^{\\mathrm{R}} H^+ + h.c. \\biggr]\n\\nonumber\\\\\n&-\\sum_{i,j}\\biggl[ \\frac{\\sqrt{2}V_{ij}}{v} \\bar{u}_i (-m_{u,i} \\omega_- +m_{d,j} \\omega_+)\\, d_j G^+ + h.c. \\biggr]\n- \\sum_i \\biggl[ \\frac{\\sqrt{2}m_{l,i} }{v}\\, \\bar{\\nu}_i^{\\mathrm{L}} l_i^{\\mathrm{R}} G^+ + h.c. \\biggr],\n\\label{eq:ffH}\n\\end{align}\nwhere $m_{l,i}$, $m_{u,i}$, and $m_{d,i}$ are the lepton, the up-type, and the down-type quark masses, respectively,\nand $V_{ij}$ are the coefficients of the CKM matrix.\nLeft- and right-handed fermion fields, $f^{\\mathrm{L}}$ and $f^{\\mathrm{R}}$, are obtained from the \ncorresponding Dirac spinor $f$ by applying the chirality projectors\n$\\omega_\\pm= ( 1\\pm \\gamma_5)\/2$, i.e.\\ $f=(\\omega_+ +\\omega_-)f=f^{\\mathrm{L}}+f^{\\mathrm{R}}$.\nThe coupling coefficients $\\xi^f_{\\mathswitchr H,\\mathswitchr h,\\mathswitchr {A_0}}$ are defined as the couplings relative to the canonical SM value of $m_f\/v$ and are shown in Tab.~\\ref{tab:yukint}. \n\\begin{table}\n  \\centering\n\\begin{tabular}{|c|c|c|c|c|}\\hline\n&Type I & Type II &  Lepton-specific & Flipped\\\\\\hline\n$\\xi^l_\\mathswitchr H$ & $\\sin{\\alpha}\/\\sin{\\beta}$ & $\\cos{\\alpha}\/\\cos{\\beta}$ & $\\cos{\\alpha}\/\\cos{\\beta}$  &$\\sin{\\alpha}\/\\sin{\\beta}$\\\\\n$\\xi^u_\\mathswitchr H$ & $\\sin{\\alpha}\/\\sin{\\beta}$ & $\\sin{\\alpha}\/\\sin{\\beta}$ & $\\sin{\\alpha}\/\\sin{\\beta}$ &$\\sin{\\alpha}\/\\sin{\\beta}$ \\\\\n$\\xi^d_\\mathswitchr H$& $\\sin{\\alpha}\/\\sin{\\beta}$ & $\\cos{\\alpha}\/\\cos{\\beta}$ & $\\sin{\\alpha}\/\\sin{\\beta}$ & $\\cos{\\alpha}\/\\cos{\\beta}$ \\\\\\hline\n$\\xi^l_\\mathswitchr h$ & $\\cos{\\alpha}\/\\sin{\\beta}$ & $-\\sin{\\alpha}\/\\cos{\\beta}$ & $-\\sin{\\alpha}\/\\cos{\\beta}$& $\\cos{\\alpha}\/\\sin{\\beta}$  \\\\\n$\\xi^u_\\mathswitchr h$ & $\\cos{\\alpha}\/\\sin{\\beta}$ & $\\cos{\\alpha}\/\\sin{\\beta}$ & $\\cos{\\alpha}\/\\sin{\\beta}$ &$\\cos{\\alpha}\/\\sin{\\beta}$ \\\\\n$\\xi^d_\\mathswitchr h$& $\\cos{\\alpha}\/\\sin{\\beta}$ & $-\\sin{\\alpha}\/\\cos{\\beta}$ & $\\cos{\\alpha}\/\\sin{\\beta}$ &$-\\sin{\\alpha}\/\\cos{\\beta}$  \\\\\\hline\n$\\xi^l_\\mathswitchr {A_0}$ & $\\cot{\\beta}$ & $-\\tan{\\beta}$ &$-\\tan{\\beta}$ & $\\cot{\\beta}$  \\\\\n$\\xi^u_\\mathswitchr {A_0}$ & $\\cot{\\beta}$ & $\\cot{\\beta}$ & $\\cot{\\beta}$&$\\cot{\\beta}$ \\\\\n$\\xi^d_\\mathswitchr {A_0}$& $\\cot{\\beta}$ & $-\\tan{\\beta}$ & $\\cot{\\beta}$ & $-\\tan{\\beta}$ \\\\\\hline\n\\end{tabular}  \n\\caption{The coupling strengths $\\xi^f$ of $\\mathswitchr H,\\mathswitchr h,\\mathswitchr {A_0}$ to the fermions relative to the SM value of $m_f\/v$, see Eq.~\\eqref{eq:ffH}. Note that the sign of $\\xi^f_\\mathswitchr {A_0}$ is defined relative to\nthe coupling of the Goldstone-boson field $G_0$ and the relation $\\beta_n=\\beta_c=\\beta$\nis used here.}\n\\label{tab:yukint}\n\\end{table}\nNote that we have used $\\beta_n=\\beta_c=\\beta$ in Eq.~\\refeq{eq:ffH} and \\refta{tab:yukint},\nwhich is most relevant in applications; the generalization to independent\n$\\beta_n$, $\\beta_c$, $\\beta$ is simple.\n\n\\section{The counterterm Lagrangian}\n\\label{sec:CTLagrangian}\n\nThe next step in calculating higher-order corrections is the renormalization of the theory. In this section we focus on electroweak corrections of $\\order{\\alpha_{\\mathrm{em}}}$. The QCD renormalization of the THDM is straightforward and completely analogous to the SM case, since all scalar degrees of freedom are colour singlets and do not interact strongly. \nIn the formulation of the basic Lagrangian in the previous section, we dealt with {\\it bare} \nparameters and fields. To distinguish those from {\\it renormalized} quantities, in the\nfollowing we indicate {\\it bare} quantities by subscripts ``0'' consistently.\nWe perform a multiplicative renormalization, i.e.\\ we split {\\it bare} quantities into\n{\\it renormalized} parts and corresponding counterterms,\nuse dimensional regularization,\nand allow for matrix-valued field renormalization constants in the case that there are several fields with the same quantum numbers.\nThe counterterm Lagrangian $\\delta \\mathcal{L}$ containing the full dependence on the renormalization constants can be split into several parts analogous to Eq.~\\eqref{eq:LTHDM},\n\\begin{align}\n\\label{eq:CTLagrangian}\n\\delta \\mathcal{L}_\\mathrm{THDM}\n=\\delta \\mathcal{L}_{\\mathrm{Gauge}}+\\delta \\mathcal{L}_{\\mathrm{Fermion}}+\\delta \\mathcal{L}_{\\mathrm{Higgs,kin}}-\\delta V_{\\mathrm{Higgs}}+\\delta \\mathcal{L}_{\\mathrm{Yukawa}},\n\\end{align}\nwhere the Higgs part of the Lagrangian $\\delta \\mathcal{L}_{\\mathrm{Higgs}}$ is split up into the kinetic part $\\delta \\mathcal{L}_{\\mathrm{Higgs,kin}}$ and the Higgs potential part $\\delta V_{\\mathrm{Higgs}}$.\nSince the gauge fixing is applied after renormalization, no gauge-fixing counterterms occur, and since ghost fields\noccur only in loop diagrams, a renormalization of the ghost sector is not necessary at NLO for the calculation of S-matrix elements. Though, for analyzing Slavnov--Taylor or Ward identities a complete renormalization procedure would be advisable. \n{Our renormalization procedure, thus, widely parallels the treatment described for the SM in \\cite{Denner:1991kt};\nan alternative variant that is based on the transformation of fields in the gauge eigenstate basis, as suggested\nfor the SM in \\citere{Bohm:1986rj} and for the MSSM in \\citere{Dabelstein:1994hb}, is described in\n\\citere{Altenkamp:Dis}.}\n\n\\subsection{Higgs potential}\n\\label{sec:CTpot}\nAccording to Eq.~\\eqref{eq:lambdapara}, the Higgs potential contains \n8 independent parameters which have to be renormalized. In addition, there are two vevs and two gauge couplings completing the set of input parameters of \\eqref{eq:basicpara}. \nWe have carried out different renormalization procedures and in this paper discuss the renormalization of the Lagrangian in the mass parameterization\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{(\\alph{enumi}}\n\\renewcommand{\\labelenumi}{\\theenumi)}\n\\item with renormalization of the mixing angles,\n\\item alternatively,\ntaking the mixing angles as dependent parameters and applying the field transformation after renormalization.\n\\end{enumerate}\nAdditionally we have performed a renormalization of the basic parameters and a \nsubsequent transformation to renormalization constants and parameters of the mass parameter set similar to \nDabelstein in the MSSM~\\cite{Dabelstein:1994hb}%\n\\footnote{Details about this method can be found in \\citere{Altenkamp:Dis}.}. The latter has been used, after changing to our \nconventions for field and parameter renormalization, to check the counterterm Lagrangian. \nIn the following, we give a detailed description of method (a), while method~(b) is \nbriefly described in App.~\\ref{App:renormassa}.\n\n\\subsubsection{Renormalization of the mixing angles}\n\\label{sec:renmixangles}\n\n{In this section we show that the counterterms of mixing angles \nthat are not used to replace another free parameter of the theory \nare redundant in the sense that they can be absorbed by \nfield renormalization constants.}\nWe sketch the argument for generic scalar fields $\\varphi_{1},\\varphi_{2}$, which are transformed into fields $h_1, h_2$ \ncorresponding to mass eigenstates by a rotation by the angle~$\\theta$,\n\\begin{align}\n\\begin{pmatrix}\\varphi_{1,0} \\\\ \\varphi_{2,0}\\end{pmatrix}\n& = \\mathbf{R}_\\varphi(\\theta_{0}) \\begin{pmatrix}h_{1,0} \\\\ h_{2,0}\\end{pmatrix}\n=\\begin{pmatrix}c_{\\theta,0} & -s_{\\theta,0}\\\\ s_{\\theta,0}& c_{\\theta,0}\\end{pmatrix}\n\\begin{pmatrix}h_{1,0} \\\\ h_{2,0}\\end{pmatrix},\n\\label{eq:mixanglesrot}\n\\end{align}\nwhere we added subscripts ``0'' to indicate bare quantities.\nThe general argument can be applied to the neutral CP-even, the neutral \nCP-odd, and the charged Higgs fields of the THDM by replacing\n$h_1$, $h_2$ by $H$, $h$, or $G$, $A_0$ or $G^\\pm$, $H^\\pm$, respectively, and by substituting the angle $\\theta$ by \n$\\alpha$, $\\beta_n$, or $\\beta_c$.\nThe fields corresponding to mass eigenstates are renormalized using matrix-valued renormalization constants, so that the renormalization transformation reads\n\\begin{align}\n\\begin{pmatrix}h_{1,0} \\\\ h_{2,0} \\end{pmatrix}&=\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_1 \\mathswitchr h_1} &\\frac{1}{2} \\delta Z_{\\mathswitchr h_1 \\mathswitchr h_2} \\\\\\frac{1}{2} \\delta Z_{\\mathswitchr h_2 \\mathswitchr h_1} &1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_2 \\mathswitchr h_2} \\end{pmatrix} \\begin{pmatrix}h_1 \\\\ h_2\\end{pmatrix},&&\n  \\theta_{0} = \\theta + \\delta \\theta.\\label{eq:alpherendef}\n\\end{align}\nApplying this renormalization transformation to Eq.~\\eqref{eq:mixanglesrot} leads to\n\\begin{align}\n\\begin{pmatrix} \\varphi_{1,0} \\\\ \\varphi_{2,0}\\end{pmatrix}={}\n& \\bigg[ \\begin{pmatrix}c_{\\theta}  & -s_{\\theta} \\\\ s_{\\theta}& c_{\\theta} \\end{pmatrix}\n\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_1\\mathswitchr h_1}  & \\frac{1}{2}\\delta Z_{\\mathswitchr h_1\\mathswitchr h_2} \\\\ \\frac{1}{2}\\delta Z_{\\mathswitchr h_2\\mathswitchr h_1}& 1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_2\\mathswitchr h_2} \\end{pmatrix}\n+\\begin{pmatrix}-s_{\\theta}  & -c_{\\theta} \\\\ c_{\\theta}& -s_{\\theta} \\end{pmatrix} \\delta \\theta \\bigg]\n\\begin{pmatrix}h_1 \\\\ h_2\\end{pmatrix}\n\\nonumber\\\\\n{}={}& \\begin{pmatrix}c_{\\theta}  & -s_{\\theta} \\\\ s_{\\theta}& c_{\\theta} \\end{pmatrix}\n\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_1\\mathswitchr h_1}  &\\frac{1}{2}(\\delta Z_{\\mathswitchr h_1\\mathswitchr h_2}- 2 \\delta \\theta) \n\\\\ \n\t\\frac{1}{2}(\\delta Z_{\\mathswitchr h_2\\mathswitchr h_1}+ 2 \\delta \\theta)& 1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_2\\mathswitchr h_2} \\end{pmatrix}\n\\begin{pmatrix}h_1 \\\\ h_2\\end{pmatrix}.  \n\\label{eq:mixanglesrot2}\n\\end{align}\nOne can easily remove the dependence on the mixing angle \nwith a redefinition of the mixing \nfield renormalization constants by introducing\n\\begin{align}\n\\label{eq:redeffielren}\n \\delta \\tilde{Z}_{\\mathswitchr h_2\\mathswitchr h_1}= \\delta Z_{\\mathswitchr h_2\\mathswitchr h_1}+ 2 \\delta \\theta,\\qquad\n \\delta \\tilde{Z}_{\\mathswitchr h_1\\mathswitchr h_2}= \\delta Z_{\\mathswitchr h_1\\mathswitchr h_2}- 2 \\delta \\theta.\n \\end{align}\nThen, the Eq.~\\eqref{eq:mixanglesrot2} reads\n\\begin{align}\n\\begin{pmatrix} \\varphi_{1,0} \\\\ \\varphi_{2,0}\\end{pmatrix}=\n& \\begin{pmatrix}c_{\\theta}  & -s_{\\theta} \\\\ s_{\\theta}& c_{\\theta} \\end{pmatrix}\n\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_1\\mathswitchr h_1}  &\\frac{1}{2}\\delta \\tilde{Z}_{\\mathswitchr h_1\\mathswitchr h_2} \\\\ \n\t\\frac{1}{2}\\delta \\tilde{Z}_{\\mathswitchr h_2\\mathswitchr h_1}& 1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_2\\mathswitchr h_2} \\end{pmatrix}\n\\begin{pmatrix}h_1 \\\\ h_2\\end{pmatrix}.\n\\end{align}\nObviously,\nthe dependence on $\\delta \\theta$ can always be removed from the Lagrangian \nby a redefinition of the field renormalization constants. \nAs a simple shift of the mixing field renormalization constant is performing the task, the renormalization of the mixing angle $\\theta$ can be seen as an additional field renormalization \n(as it is done, e.g., in Ref.~\\cite{Aoki2009}). \nThis argument is general and holds for any renormalization condition on $\\theta$.\nWithout loss of generality one can even assume that such a redefinition has already been performed and set $\\delta \\theta=0$ from the beginning, as done in method (b) in App.~\\ref{App:renormassa}.\nOf course, the bookkeeping of counterterms depends on the way $\\delta \\theta$ is treated.\nThis can be seen by considering the mass term of the potential. \nThe general bare mass term can be written using the rotation matrix $\\mathbf{R_\\varphi}(\\theta_{0})$ as\n\\begin{align}\nV_{\\mathswitchr h_1\\mathswitchr h_2} {}={} \n&\\frac{1}{2} \\,(h_{1,0}, h_{2,0})\\, \\mathbf{R}_\\varphi^{\\mathrm{T}}(\\theta_{0}) \\mathbf{M}_{\\varphi,0} \n\\mathbf{R}_\\varphi(\\theta_{0}) \\begin{pmatrix} h_{1,0}\\\\ h_{2,0} \\end{pmatrix}\n\\nonumber\\\\\n{}={} &\\frac{1}{2} \\, (h_{1,0}, h_{2,0})\\, \n\\mathbf{R}_\\varphi^{\\mathrm{T}}(\\theta+\\delta \\theta) \n\\left(\\mathbf{M_\\varphi}+\\delta \\mathbf{M_\\varphi}\\right) \n\\mathbf{R_\\varphi}(\\theta+\\delta \\theta) \n\\begin{pmatrix} h_{1,0}\\\\ h_{2,0} \\end{pmatrix}.\n\\end{align}\nThis expression can be expanded in terms of renormalized and counterterm contributions, yielding\n\\begin{align}\nV_{\\mathswitchr h_1\\mathswitchr h_2}=\n\\frac{1}{2} \\, (h_{1,0}, h_{2,0})\\, \\begin{pmatrix} \n\\hspace{-0pt}M_{\\mathswitchr h_1}^2+\\delta M_{\\mathswitchr h_1}^2 & \n\\hspace{-5pt}\\delta \\theta (M_{\\mathswitchr h_2}^2-M_{\\mathswitchr h_1}^2)+f_{\\theta}(\\{ \\delta p\\}) \\\\ \\delta \\theta (M_{\\mathswitchr h_2}^2-M_{\\mathswitchr h_1}^2)+f_{\\theta}(\\{ \\delta p\\}) & \n\\hspace{-0pt}M_{\\mathswitchr h_2}^2+\\delta M_{\\mathswitchr h_2}^2\\end{pmatrix} \\begin{pmatrix} h_{1,0}\\\\ h_{2,0} \\end{pmatrix},\n\\end{align}\nwhere we obtain off-diagonal terms from the counterterm $\\delta\\theta$\nof the mixing angle and from the renormalization of the mass matrix. The latter contribution depends on the independent counterterms $\\{ \\delta p\\}$ and is abbreviated by $f_{\\theta}(\\{ \\delta p\\})$.\nAt NLO, the mixing entry of the mass matrix reads\n\\begin{align}\n\\delta M_{\\mathswitchr h_1\\mathswitchr h_2}^2= \\delta \\theta (M_{\\mathswitchr h_2}^2-M_{\\mathswitchr h_1})+f_{\\theta}(\\{ \\delta p\\}), \\label{eq:NLOmassmixHh}\n\\end{align}\n{and since the renormalization of the {\\it redundant} mixing angle can be chosen freely,} \ncounterterm contributions in the Lagrangian can be shifted arbitrarily from mixing terms to mixing-angle counterterms.\n{Note that $f_{\\theta}(\\{ \\delta p\\})$ does not change\nby such redistributions, since it is fixed by the remaining renormalization \nconstants.}\n\n\\subsubsection{Renormalization with a diagonal mass matrix -- version (a)}\n\\label{sec:renormassb}\n\nIn this prescription, we use the angle $\\alpha$ as an independent parameter instead of $\\lambda_3$. \n{To define $\\alpha$ at NLO, we demand that \nthe mass matrix of the CP-even Higgs bosons (in the Lagrangian), written in terms of bare fields,\nis diagonal at all orders (i.e.\\ in terms of bare or renormalized parameters).}\nEquation~\\eqref{eq:Lambda3NLO} with\n\\begin{align}\\label{eq:MHhchoice2}\nM_{\\mathrm{Hh},0}^2 = 0 + \\delta M_{\\mathrm{Hh}}^2=0, \\qquad M_{\\mathrm{Hh}}^2 = 0\n\\end{align}\nthen defines the parameter $\\alpha$. \n{Note that (momentum-dependent)\nloop diagrams tend to destroy the diagonality \nof the matrix-valued two-point functions (inverse propagators)\nin the effective action as well.\nBelow,\nthe field renormalization will be chosen to compensate those loop effects  \nat the mass shells of the propagating particles.}\n{It is relation \\refeq{eq:MHhchoice2} that distinguishes \n$\\alpha$ from the case of a redundant mixing angle, such as $\\theta$ in the\nprevious section, which can be chosen freely or absorbed by field\nrenormalization constants.}\nThe relation between $\\alpha$ and $\\lambda_3$ of Eq.~\\eqref{eq:Lambda3NLO} is the same for bare and renormalized quantities \nand can be used to eliminate $\\lambda_3$ from the theory.\nFor each of the independent parameters of the mass parameter set \\eqref{eq:physparaNLO} we apply the renormalization transformation\n\\begin{align}\nM_{\\mathswitchr H,0}^2&=\\mathswitch {M_\\PH}^2+\\delta \\mathswitch {M_\\PH}^2,&M_{\\mathswitchr h,0}^2&= \\mathswitch {M_\\Ph}^2+\\delta \\mathswitch {M_\\Ph}^2,&M_{\\mathswitchr {A_0},0}^2 &= \\mathswitch {M_\\PAO}^2+\\delta \\mathswitch {M_\\PAO}^2,\\nonumber\\\\\nM_{\\mathswitchr {H^+},0}^2&= \\mathswitch {M_\\PHP}^2+\\delta \\mathswitch {M_\\PHP}^2,&M_{\\mathswitchr W,0}^2&= \\mathswitch {M_\\PW}^2+\\delta \\mathswitch {M_\\PW}^2,&M_{\\mathswitchr Z,0}^2&= \\mathswitch {M_\\PZ}^2+\\delta \\mathswitch {M_\\PZ}^2, \\nonumber\\\\\ne_0&= e+\\delta e,&\\lambda_{5,0} &= \\lambda_5+\\delta \\lambda_5&\\alpha_0 &= \\alpha+\\delta \\alpha,\\nonumber\\\\\n\\beta_0&= \\beta+\\delta \\beta,&t_{\\mathswitchr H,0}&= 0+\\delta t_\\mathswitchr H,&t_{\\mathswitchr h,0}&= 0+\\delta t_\\mathswitchr h, \\label{eq:physpararen2}\n\\end{align}\nso that the 12 parameter renormalization constants are\n\\begin{align}\n\\{\\delta p_\\mathrm{mass}\\}=\\{\\delta \\mathswitch {M_\\PH}^2, \\delta \\mathswitch {M_\\Ph}^2, \\delta \\mathswitch {M_\\PAO}^2, \\delta \\mathswitch {M_\\PHP}^2, \\delta \\mathswitch {M_\\PW}^2, \\delta \\mathswitch {M_\\PZ}^2, \\delta e, \\delta \\lambda_5, \\delta \\alpha,\\delta \\beta,\\delta t_\\mathswitchr H, \\delta t_\\mathswitchr h\\},\n\\label{eq:renconstsR}\n\\end{align}\ncorresponding to $\\{p_\\mathrm{mass}\\}$. \nIn this part we describe the commonly used tadpole renormalization (which is gauge dependent) and describe \na gauge-independent scheme in Sect.~\\ref{sec:FJscheme}.\n\nThe higher-order corrections of the mixing angles $\\beta_n$ and $\\beta_c$ are irrelevant according to Sect.~\\ref{sec:renmixangles} and we can choose\n\\begin{align}\n\\beta_{n,0}=\\beta_{c,0}= \\beta_0 = \\beta+\\delta \\beta,\n\\label{eq:mixanglect}\n\\end{align}\nwhich defines the mixing terms uniquely and ensures that the angles $\\beta_n,\\beta_c$, and $\\beta$ do not have to be distinguished at any order.\nFrom these conditions, we can compute the mass mixing terms from Eq.~\\eqref{eq:goldstonemassmixing} and Eq.~\\eqref{eq:chargedgoldstonemassmixing} to\n\\begin{align}\n M_{\\mathswitchr {G_0}\\mathswitchr {A_0},0}^2 &= 0+\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2\n=-e\\frac{ \n\\delta t_\\mathswitchr H s_{\\alpha -\\beta }+\n\\delta t_\\mathswitchr h c_{\\alpha -\\beta }\n}{2 \\mathswitch {M_\\PW} \\mathswitch {s_{\\scrs\\PW}}},&\\nonumber\\\\\n M_{\\mathswitchr {G}\\mathswitchr {H^+},0}^2 &= 0+\\delta M_{\\mathswitchr {G}\\mathswitchr {H^+}}^2\n=-e\\frac{ \n\\delta t_\\mathswitchr H s_{\\alpha -\\beta }+\n\\delta t_\\mathswitchr h c_{\\alpha -\\beta }\n}{2 \\mathswitch {M_\\PW} \\mathswitch {s_{\\scrs\\PW}}}. \\label{eq:Mabchoice2}\n\\end{align}\nThe field re\\-nor\\-ma\\-li\\-zation is performed \nfor each field corresponding to mass eigenstates,\n\\begin{align}\n\\begin{pmatrix}H_0 \\\\ h_0 \\end{pmatrix}&=\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_\\mathswitchr H &\\frac{1}{2} \\delta Z_{\\mathswitchr H\\mathswitchr h} \\\\\\frac{1}{2} \\delta Z_{\\mathswitchr h\\mathswitchr H} &1+\\frac{1}{2}\\delta Z_{\\mathswitchr h} \\end{pmatrix} \\begin{pmatrix}H \\\\ h\\end{pmatrix},\\nonumber\\\\\n\\begin{pmatrix}G_{0,0} \\\\ A_{0,0} \\end{pmatrix}&=\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathswitchr {G_0}} &\\frac{1}{2}\\delta  Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}} \\\\\\frac{1}{2}\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}} &1+\\frac{1}{2}\\delta Z_\\mathswitchr {A_0} \\end{pmatrix} \\begin{pmatrix}G_0 \\\\ A_0\\end{pmatrix},\\nonumber\\\\\n\\begin{pmatrix}G^\\pm_0 \\\\ H^\\pm_0 \\end{pmatrix}&=\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathrm{G}^+} &\\frac{1}{2}\\delta  Z_{\\mathrm{GH}^+} \\\\\\frac{1}{2}\\delta Z_{\\mathrm{HG}^+} &1+\\frac{1}{2}\\delta Z_{\\mathswitchr H^+} \\end{pmatrix} \\begin{pmatrix}G^\\pm \\\\ H^\\pm\\end{pmatrix},\\label{eq:physfieldren}\n\\end{align}\nwith the field renormalization constants $\\delta Z_\\mathswitchr H$, $\\delta Z_{\\mathswitchr H\\mathswitchr h}$, $\\delta Z_{\\mathswitchr h\\mathswitchr H}$, $\\delta Z_{\\mathswitchr h}$,  $\\delta Z_{\\mathswitchr {G_0}}$, \n$\\delta  Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}}$, $\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}}$, $\\delta Z_\\mathswitchr {A_0}$,  $\\delta Z_{\\mathrm{G}^+}$, $\\delta  Z_{\\mathrm{GH}^+}$, $\\delta Z_{\\mathrm{HG}^+}$, and $\\delta Z_{\\mathswitchr H^+}$ for the CP-even, the CP-odd, and the charged Higgs fields.\nWe denote the complete set of parameter and field renormalization constants with $\\{\\delta \\mathbf{R}_\\mathrm{mass}\\}$. All renormalization constants \nare of $\\order{\\alpha_{\\mathrm{em}}}$, i.e.\\ all contributions of $\\order{\\alpha_{\\mathrm{em}}^2}$ or higher are omitted. \n\n\\begin{sloppypar}\nApplying the renormalization transformations~\\refeq{eq:physpararen2}, \\refeq{eq:physfieldren} \nto the bare potential of Eq.~\\eqref{eq:NLObarepot} and linearizing in the renormalization constants results in \n$V(\\{p_\\mathrm{mass}\\})+\\delta V(\\{p_\\mathrm{mass}\\},\\{\\delta \\mathbf{R}_\\mathrm{mass}\\})$\nwith the LO potential as in Eq.~\\eqref{eq:NLObarepot}, but with renormalized quantities and the counterterm potential of $\\order{\\alpha_\\mathrm{em}}$,\n\\begin{align}\n\\delta V(\\{p_\\mathrm{mass}\\},\\{\\delta \\mathbf{R}_\\mathrm{mass}\\})=&- \\delta t_\\mathswitchr H H -\\delta t_\\mathswitchr h h \\nonumber\\\\\n&+\\frac{1}{2} \\left(\\delta \\mathswitch {M_\\PH}^2 +\\delta Z_{\\mathswitchr H} \\mathswitch {M_\\PH}^2\\right) H^2 +\\frac{1}{2} \\left(\\delta \\mathswitch {M_\\Ph}^2 +\\delta Z_{\\mathswitchr h} \\mathswitch {M_\\Ph}^2\\right) h^2 \\nonumber\\\\\n&+\\frac{1}{2} \\left(\\delta \\mathswitch {M_\\PAO}^2 +\\delta Z_{\\mathswitchr {A_0}} \\mathswitch {M_\\PAO}^2\\right) A_0^2 + \\left(\\delta \\mathswitch {M_\\PHP}^2+\\delta Z_{\\mathswitchr {H^+}} \\mathswitch {M_\\PHP}^2\\right)H^+ H^-\\nonumber\\\\\n&+ \\frac{e}{4 \\mathswitch {M_\\PW} \\mathswitch {s_{\\scrs\\PW}}} \\left(\n-\\delta t_\\mathswitchr H c_{\\alpha-\\beta}\n+\\delta t_\\mathswitchr h s_{\\alpha-\\beta}\n\\right) (G_0^2 + 2 G^+ G^-)\\nonumber\\\\\n&+ \\frac{1}{2}\\left( \\mathswitch {M_\\PH}^2\\delta Z_{\\mathswitchr H\\mathswitchr h}+\\mathswitch {M_\\Ph}^2 \\delta Z_{\\mathswitchr h\\mathswitchr H}\\right) H h\\nonumber\\\\\n&+\\frac{1}{2} \\left( \\mathswitch {M_\\PAO}^2 \\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}} \n+ 2\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2 \\right) G_0 A_0\n\\nonumber\\\\\n&+ \\frac{1}{2} \\left( \\mathswitch {M_\\PHP}^2 \\delta Z_{\\mathswitchr H\\mathswitchr {G^+}} \n+2\\delta M_{\\mathswitchr {G}\\mathswitchr H+}^2 \\right)(H^+ G^-+G^+ H^-)\n\\nonumber \\\\\n&+ \\textrm{interaction terms}.\\label{eq:dVH2p}\n\\end{align}\nThe interaction terms are derived in\nthe same way, but they are very lengthy and not shown here. \n\\end{sloppypar}\n\nThe prescription for the field renormalization~\\eqref{eq:physfieldren} is non-minimal in the sense that a renormalization of the doublets with two renormalization constants,\n\\begin{align}\n\\Phi_{1,0} = Z_{\\mathswitchr H_1}^{1\/2} \\Phi_1& \\textstyle = \\Phi_1 \\left(1+\\frac{1}{2} \\delta Z_{\\mathswitchr H_1}\\right),\\nonumber\\\\\n\\Phi_{2,0} = Z_{\\mathswitchr H_2}^{1\/2} \\Phi_2& \\textstyle = \\Phi_2 \\left(1+\\frac{1}{2} \\delta Z_{\\mathswitchr H_2}\\right),\\label{eq:inputfieldren}\n\\end{align}\nactually would be\nsufficient to cancel the UV divergences. However, the prescription with matrix-valued renormalization constants allows us to renormalize each field on-shell. The UV-divergent parts of \nthe renormalization constants in Eq.~\\refeq{eq:physfieldren}\ncannot be independent, and relations between the UV-divergent parts of the two prescriptions exist. They can be obtained by applying the renormalization prescription \\eqref{eq:inputfieldren} to the left-hand side and \\eqref{eq:physfieldren} to the right-hand side of Eqs.~\\eqref{eq:rotations}, transforming thereafter the interaction states on the left-hand side to mass eigenstates and comparing both sides.\nThis results in\n\\begin{align}\n\\delta Z_\\mathswitchr h\\big|_\\mathrm{UV}&=s_{\\alpha}^2\\;\\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+c_{\\alpha}^2\\; \\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_\\mathswitchr H\\big|_\\mathrm{UV}&=c_{\\alpha}^2\\; \\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+s_{\\alpha}^2\\; \\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_{\\mathswitchr H\\mathswitchr h}\\big|_\\mathrm{UV}&=s_{\\alpha} c_{\\alpha}\\; \\left(-\\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+\\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV}\\right)+2 \\delta \\alpha\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_{\\mathswitchr h\\mathswitchr H}\\big|_\\mathrm{UV}&=s_{\\alpha} c_{\\alpha}\\; \\left(-\\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+\\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV}\\right)-2 \\delta \\alpha\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_\\mathswitchr {A_0}\\big|_\\mathrm{UV}&=\\delta Z_{\\mathswitchr H^+}\\big|_\\mathrm{UV}=s_{\\beta}^2\\;\\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+c_{\\beta}^2\\; \\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_\\mathswitchr {G_0}\\big|_\\mathrm{UV}&=\\delta Z_{G^+}\\big|_\\mathrm{UV}=c_{\\beta}^2\\; \\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+s_{\\beta}^2\\; \\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}}\\big|_\\mathrm{UV}&=\\delta Z_{\\mathswitchr {G}\\mathswitchr {H^+}}\\big|_\\mathrm{UV}=s_{\\beta} c_{\\beta}\\; \\left(-\\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+\\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV}\\right)+2 \\delta \\beta\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}}\\big|_\\mathrm{UV}&=\\delta Z_{\\mathswitchr H\\mathswitchr {G^+}}\\big|_\\mathrm{UV}=s_{\\beta} c_{\\beta}\\; \\left(-\\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+\\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV}\\right)-2 \\delta \\beta\\big|_\\mathrm{UV}.\\label{eq:fieldrenconstrel}\n\\end{align}\nWe will use these relations to derive UV-divergent parts for specific renormalization constants in Sect.~\\ref{sec:diffrenschemes}. In App.~\\ref{App:renormassa} we discuss a different choice of the mixing angles which is suited for \nthe renormalization with $\\lambda_3$ as an independent parameter. \n\n\\subsection{The Higgs kinetic part}\n\\label{sec:Hkinren}\n\nAfter expressing the Higgs kinetic term\n\\begin{align}\n \\mathcal{L}_\\mathrm{H,kin}=(D_\\mu\\Phi_1)^\\dagger (D^\\mu \\Phi_1)+(D_\\mu\\Phi_2)^\\dagger (D^\\mu \\Phi_2)\n\\end{align}\nin terms of\nbare physical fields, mixing angles, and parameters, one can apply the renorma\\-lization transformations \\eqref{eq:physpararen2} and~\\eqref{eq:physfieldren} to obtain the counterterm part of the kinetic Lagrangian which introduces \nscalar--vector mixing terms. The explicit terms are stated in App.~\\ref{App:SV-terms}.\n\n\\subsection{Fermionic and gauge parts}\n\\label{sec:fermgauge}\nSince the THDM extension of the SM does not affect the gauge and the fermion parts of the Lagrangian,  \nthe renormalization of these parts is identical to the SM case. \n{It is described in detail in Ref.~\\cite{Denner:1991kt} in \nBHS convention, which is included in the standard implementation of the \\textsc{FeynArts} package~\\cite{Hahn2001}.\nOther renormalization prescriptions\ncan, e.g., be found in \\citeres{Denner:2004bm,Kniehl:2009kk}.}\nTherefore, we here do not repeat the renormalization procedure of the CKM matrix, which does not change in the \ntransition from the SM to the THDM, and spell out the renormalization of the fermionic parts only for the\ncase where the CKM matrix is set to the unit matrix, i.e.\\ $V_{ij}=\\delta_{ij}$.\nThe transformation of the left- and right-handed fermions and of the gauge-boson fields are\n\\begin{align}\n f^\\sigma_{i,0} & = \\left(1 + \\textstyle\\frac{1}{2}\\delta Z^{f,\\sigma}_i\\right) f^\\sigma_i,\\qquad  \n\\rlap{$f=\\nu,l,u,d, \\quad \\sigma={\\mathrm{L}},{\\mathrm{R}}, \\quad i=1,2,3,$}\\\\\n \\begin{pmatrix}Z_0 \\\\ A_\\mathrm{bare}\\end{pmatrix}&=\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathswitchr Z\\PZ} &\\frac{1}{2} Z_{\\mathswitchr Z{\\mathrm{PA}}} \\\\\\frac{1}{2} \\delta Z_{{\\mathrm{PA}}\\mathswitchr Z} &1+\\frac{1}{2}\\delta Z_{{\\mathrm{PA}}\\PA} \\end{pmatrix} \\begin{pmatrix}Z \\\\ A\\end{pmatrix}, &\nW^\\pm_0 &= \\left(1 + \\textstyle\\frac{1}{2} \\delta Z_\\mathswitchr W\\right) W^\\pm,\n\\end{align}\nwhere the bare photon field is denoted $A_\\mathrm{bare}$ to distinguish it from the \nneutral CP-odd field~$A_0$.\nMixing between left-handed up- and down-type fermions does not occur, owing to charge conservation. Inserting this into the Lagrangian directly delivers the renormalized and the counterterm Lagrangians.\n\n\\subsection{Yukawa part}\n\\label{sec:YukRen}\n\nThe renormalization of the Yukawa sector is straightforward in Type I, II, lepton-specific, and flipped models and can be done by taking the Lagrangian of Eq.~\\eqref{eq:ffH}, \nreplacing the vev $v$, and applying the renormalization transformations of \nSect.~\\ref{sec:renormassb}, as well as a renormalization of the fermion masses,\n\\begin{align}\n  m_{f,i,0} &= m_{f,i} + \\delta m_{f,i}.\n  \\end{align}\nThe corresponding  counterterm couplings are stated in App.~\\ref{App:Yuk-CT}.\n\n\\section{Renormalization conditions}\n\\label{sec:renconditions}\n\nThe renormalization constants are fixed using on-shell conditions \nfor all parameters that are accessible by experiments. \nHowever, not all parameters of the THDM correspond to measurable quantities, so that we renormalize three parameters of the Higgs sector in the $\\overline{\\mathrm{MS}}$ scheme, where the renormalization constants only contain the standard UV divergence\n\\begin{align}\n  \\Delta_\\mathrm{UV}=\\frac{2}{4-D}-\\gamma_\\mathrm{E}+\\ln{4\\pi}=\\frac{1}{\\epsilon}-\\gamma_\\mathrm{E}+\\ln{4\\pi}\n\\end{align}\nin $D=4-2 \\epsilon$ dimensions and with the Euler--Mascheroni constant $\\gamma_\\mathrm{E}$.\nIn Sect.~\\ref{sec:diffrenschemes}, four different options, resulting in four different renormalization schemes, are presented.\nAn overview over the renormalization constants introduced in the previous section is shown in Tab.~\\ref{tab:renconsts}. In the following, we adapt the notation of Ref.~\\cite{Denner:1991kt}, i.e.\\ we use the same symbols for the renormalized and the corresponding unrenormalized \nGreen function, self-energies, etc., but denoting the renormalized quantities with a caret.\n\n\\begin{table}\n  \\centering\n\\begin{tabular}{|lrl|}\\hline\nParameters:&&\\\\\\hline\n&EW (3): & $\\delta \\mathswitch {M_\\PZ}^2,\\delta \\mathswitch {M_\\PW}^2,\\delta e, (\\delta \\mathswitch {c_{\\scrs\\PW}},\\delta \\mathswitch {s_{\\scrs\\PW}})$\\\\\n&fermion masses (9): & $\\delta m_{f,i},\\qquad f=l,u,d,\\quad i=1,2,3$\\\\%\\delta m_{u,i},\\delta m_{d,i}$\\\\\n&Higgs masses (4): & $\\delta \\mathswitch {M_\\PH}^2,\\delta \\mathswitch {M_\\Ph}^2,\\delta \\mathswitch {M_\\PAO}^2,\\delta \\mathswitch {M_\\PHP}^2$\\\\\n&Higgs potential (3): & $\\delta \\lambda_3 \\text{ or } \\delta \\alpha,\\delta \\lambda_5, \\delta \\beta$\\\\\n&tadpoles (2): & $\\delta t_\\mathswitchr H,\\delta t_\\mathswitchr h$\\\\\\hline\nFields:&&\\\\\\hline\n&EW (5): & $\\delta Z_\\mathswitchr W,\\delta Z_{\\mathswitchr Z\\PZ},\\delta Z_{\\mathswitchr Z{\\mathrm{PA}}},\\delta Z_{{\\mathrm{PA}}\\mathswitchr Z},\\delta Z_{{\\mathrm{PA}}\\PA}$\\\\\n&left-handed fermions (12): & $\\delta Z^{f,{\\mathrm{L}}}_i,\\qquad f=\\nu,l,u,d,\\quad i=1,2,3$ \\\\\n&right-handed fermions (9): & $\\delta Z^{f,{\\mathrm{R}}}_i,\\qquad f=l,u,d,\\quad i=1,2,3$ \\\\\n&Higgs (12): & $\\delta Z_{\\mathswitchr H},\\delta Z_{\\mathswitchr H\\mathswitchr h},\\delta Z_{\\mathswitchr h\\mathswitchr H},\\delta Z_\\mathswitchr h$\\\\\n&& $\\delta Z_{\\mathswitchr {A_0}},\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}},\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}},\\delta Z_\\mathswitchr {G_0}$\\\\\n&&$\\delta Z_{\\mathswitchr {H^+}},\\delta Z_{\\mathswitchr H\\mathswitchr {G^+}},\\delta Z_{\\mathswitchr {G}\\mathswitchr {H^+}},\\delta Z_{\\mathswitchr {G^+}}$\\\\\\hline\n\\end{tabular}  \n  \\caption{The renormalization constants used to describe the THDM, separated into sectors of parameter and field renormalization. The renormalization constants in parentheses are not independent, but useful for a better bookkeeping. The numbers in parentheses are the numbers of independent renormalization constants. In total there are 38 field and 20 parameter renormalization constants to fix. }\n\\label{tab:renconsts}\n\\end{table}%\n\n\\subsection{On-shell renormalization conditions}\n\\label{sec:onshellrenconditions}\n\n\\subsubsection{Higgs sector}\n\n\\paragraph{Tadpoles:}\nWe start with the \n(irreducible) renormalized one-point vertex function\n\\begin{align}\n\\hat\\Gamma^{\\mathswitchr H,\\mathswitchr h}=\n{\\mathrm{i}}\\hat{T}^{\\mathswitchr H,\\mathswitchr h}=\\,\\parbox{2.5cm}{\\includegraphics[scale=1]{.\/diagrams\/tadpole1.eps}}\n\\hspace{20pt}.\n\\end{align}\nAt NLO the renormalized tadpole $\\hat{T}$ consists of a counterterm contribution $\\delta t$ and an unrenormalized one-loop irreducible one-point vertex function $T$\nresulting from the diagrams shown in Fig.~\\ref{fig:tadpoles}. \n\\begin{figure}\n\\vspace{-30pt}\n\\input{diagrams\/tadpoles}\n\\vspace{-40pt}\n\\caption{Generic tadpole diagrams. There is one diagram for each massive fermion $(f)$, \nscalar $(S)$, gauge-boson $(V)$, and ghost field $(u)$.}\n\\label{fig:tadpoles}\n\\end{figure}\nIn the conventional, but gauge-dependent tadpole treatment one demands that these two contributions cancel each other,\n\\begin{align}\n\\hat{T}_\\mathswitchr H&=\\delta t_\\mathswitchr H+T_\\mathswitchr H=0,&\\hat{T}_\\mathswitchr h&=\\delta t_\\mathswitchr h+T_\\mathswitchr h=0,\n\\label{eq:tadpoleren}\n\\end{align}\nwhich means that explicit tadpole diagrams can be omitted from the \nset of one-loop diagrams for any process. \nHowever, as a remnant of the tadpole diagrams the tadpole counter\\-terms \nappear also in various coupling \ncounterterms and need to be calculated. \nIt should be noted \nthat the condition on the tadpoles does not affect physical observables \nas long as physically equivalent renormalization conditions are imposed on\nthe input parameters. This is, in particular, the case for on-shell renormalization,\nwhere input parameters are tied to measurable quantities.\nThat means, changing the tadpole renormalization condition shifts contributions between \nGreen functions and counterterms and merely changes the bookkeeping,\nbut the dependence of predicted observables on renormalized input parameters remains the same.\nThe situation changes if an {$\\overline{\\mathrm{MS}}$}{} renormalization condition is used, where the counterterm\nis not fixed by a measurable quantity, \nbut by a divergence in a specific Green function, so that\nthe gauge-dependent tadpole terms can affect the relation between renormalized input parameters and\nobservables.\nThe gauge-independent treatment of tadpole contributions is based on a different renormalization condition and discussed in Sect.~\\ref{sec:FJscheme}. \n\\vspace{10pt}\n\n\\paragraph{Scalar self-energies:}\nFor scalars, the irreducible two-point functions  with momentum transfer $k$ are\n\\begin{align}\n\\hat{\\Gamma}^{{ab}}(k)&=\\quad \\parbox{3.0cm}{\n\\includegraphics[scale=1]{.\/diagrams\/SE1.eps}} \\quad\n=\\mathrm{i}\\delta_{{ab}} (k^2-M_{a}^2)+\\mathrm{i} \\hat{\\Sigma}^{{ab}}(k),\n\\label{eq:two-def}\n\\end{align}\nwhere both fields $a,b$ are incoming and $a,b=H,h,A_0,G_0,H^\\pm,G^\\pm$. The first term is the LO two-point vertex function, while the functions $\\hat{\\Sigma}^{{ab}}$ are the renormalized self-energies containing loop diagrams and counterterms. Generic diagrams contributing to the self-energies are shown in Fig.~\\ref{fig:selfenergies}. \n\\begin{figure}\n\\vspace{-10pt}\n\\input{diagrams\/selfenergies}\n\\vspace{-20pt}\n\\caption{Generic self-energy diagrams for the heavy, neutral CP-even Higgs self-energy, for other scalar self-energies the diagrams are analogous. Only massive particles contribute.}\n\\label{fig:selfenergies}\n\\end{figure}\nMixing occurs only between $H$ and $h$,\nbetween $A_0$ and $G_0$, and between $H^\\pm$ and $G^\\pm$.\nFor the neutral CP-even fields we obtain\n\\begin{subequations}\n\\label{eq:neutralSE}\n\\begin{align}\n\\hat{\\Sigma}^{\\mathswitchr h\\Ph}(k^2)&= \\Sigma^{\\mathswitchr h\\Ph}(k^2) + \\delta Z_\\mathswitchr h(k^2-\\mathswitch {M_\\Ph}^2)  - \\delta \\mathswitch {M_\\Ph}^2,\\\\\n\\hat{\\Sigma}^{\\mathswitchr H\\PH}(k^2)&= \\Sigma^{\\mathswitchr H\\PH}(k^2) + \\delta Z_\\mathswitchr H(k^2-\\mathswitch {M_\\PH}^2)  - \\delta \\mathswitch {M_\\PH}^2,\\\\\n\\hat{\\Sigma}^{\\mathswitchr H\\mathswitchr h}(k^2)&= \\Sigma^{\\mathswitchr H\\mathswitchr h}(k^2) + \\frac{1}{2}\\delta Z_{\\mathswitchr H\\mathswitchr h}(k^2-\\mathswitch {M_\\PH}^2) +\\frac{1}{2}\\delta Z_{\\mathswitchr h\\mathswitchr H}(k^2-\\mathswitch {M_\\Ph}^2) - \\delta M_{\\mathswitchr H\\mathswitchr h}^2, \\label{eq:twopointHh}\n\\end{align}\n\\end{subequations}\nand for the CP-odd fields\n\\begin{subequations}\n\\label{eq:CPoddSE}\n\\begin{align}\n\\hat{\\Sigma}^{\\mathswitchr {A_0}\\PAO}(k^2)&= \\Sigma^{\\mathswitchr {A_0}\\PAO}(k^2) + \\delta Z_{\\mathswitchr {A_0}}(k^2-\\mathswitch {M_\\PAO}^2)  - \\delta \\mathswitch {M_\\PAO}^2,\\\\\n\\hat{\\Sigma}^{\\mathswitchr {G_0}\\PGO}(k^2)&= \\Sigma^{\\mathswitchr {G_0}\\PGO}(k^2) + \\delta Z_\\mathswitchr {G_0} k^2  - \\delta M_\\mathswitchr {G_0}^2,\\\\\n\\hat{\\Sigma}^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(k^2)&= \\Sigma^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(k^2) + \\frac{1}{2}\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}}(k^2-\\mathswitch {M_\\PAO}^2) +\\frac{1}{2}\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}} \\;k^2 - \\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2.\n\\end{align}\n\\end{subequations}\nThe charged sector involves the following self-energies,\n\\begin{subequations}\n\\label{eq:chargedSE}\n\\begin{align}\n\\hat{\\Sigma}^{\\mathrm{H}^+ \\mathrm{H}^-}(k^2)&= \\Sigma^{\\mathrm{H}^+ \\mathrm{H}^-}(k^2) + \\delta Z_{\\mathswitchr {H^+}}(k^2-\\mathswitch {M_\\PHP}^2)  - \\delta \\mathswitch {M_\\PHP}^2,\\\\\n\\hat{\\Sigma}^{\\mathrm{G}^+\\mathrm{G}^-}(k^2)&= \\Sigma^{\\mathrm{G}^+\\mathrm{G}^-}(k^2) + \\delta Z_{\\mathrm{G}^+}\\; k^2  - \\delta M_\\mathswitchr {G^+}^2,\\\\\n\\hat{\\Sigma}^{\\mathrm{G}^\\pm\\mathrm{H}^\\mp}(k^2)&= \\Sigma^{\\mathrm{G}^\\pm\\mathrm{H}^\\mp}(k^2) + \\frac{1}{2}\\delta Z_{\\mathswitchr H\\mathswitchr {G^+}}(k^2-\\mathswitch {M_\\PHP}^2) +\\frac{1}{2}\\delta Z_{\\mathswitchr {G}\\mathswitchr {H^+}}\\; k^2 - \\delta M_{\\mathswitchr {G}\\mathswitchr {H^+}}^2.\n\\end{align}\n\\end{subequations}\nThe mass mixing constants are given in Eqs.~\\eqref{eq:MHhchoice2}  and \\eqref{eq:Mabchoice2}.\nOn these two-point functions we now impose our renormalization conditions. First, we  fix the renormalized mass parameters to the on-shell\nvalues, so that the zeros of the real parts of the one-particle-irreducible two-point functions are located at the squares of the physical masses:\n\\begin{align}\n\\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr H\\PH}(\\mathswitch {M_\\PH}^2)&=0,& \\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr h\\Ph}(\\mathswitch {M_\\Ph}^2)&=0,\\nonumber\\\\\n\\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr {A_0}\\PAO}(\\mathswitch {M_\\PAO}^2)&=0,&\\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathrm{H}^+\\mathrm{H}^-}(\\mathswitch {M_\\PHP}^2)&=0.\n\\label{eq:massrenconddef}\n\\end{align}\nUsing Eqs.~\\eqref{eq:neutralSE}, \\eqref{eq:CPoddSE}, and \\eqref{eq:chargedSE}, fixes  the mass renormalization constants to\n\\begin{align}\n\\delta \\mathswitch {M_\\PH}^2&= \\mathrm{Re}\\, \\Sigma^{\\mathswitchr H\\PH}(\\mathswitch {M_\\PH}^2),&\\delta \\mathswitch {M_\\Ph}^2&= \\mathrm{Re}\\, \\Sigma^{\\mathswitchr h\\Ph}(\\mathswitch {M_\\Ph}^2),\\nonumber\\\\\n\\delta \\mathswitch {M_\\PAO}^2&= \\mathrm{Re}\\, \\Sigma^{\\mathswitchr {A_0}\\PAO}(\\mathswitch {M_\\PAO}^2),&\\delta M_{\\mathswitchr {H^+}}^2&= \\mathrm{Re}\\, \\Sigma^{\\mathrm{H}^+\\mathrm{H}^-}(M_{\\mathswitchr H^+}^2).\n\\end{align}\nFor the propagators of the fields, we demand that the residues of the particle poles are not changed by higher-order corrections.\nThis  determines the \ndiagonal field renormalization constants by conditions on the one-particle-irreducible two-point functions,\n\\begin{align}\n\\lim_{k^2\\to \\mathswitch {M_\\PH}^2}\\mathrm{Re}\\,\\frac{\\mathrm{i}\\,\\hat{\\Gamma}^{\\mathswitchr H\\PH}(k^2)}{k^2-\\mathswitch {M_\\PH}^2}&=-1,& \n\\lim_{k^2\\to \\mathswitch {M_\\Ph}^2}\\mathrm{Re}\\,\\frac{\\mathrm{i}\\,\\hat{\\Gamma}^{\\mathswitchr h\\Ph}(k^2)}{k^2-\\mathswitch {M_\\Ph}^2}&=-1,\\nonumber\\\\\n\\lim_{k^2\\to \\mathswitch {M_\\PAO}^2}\\mathrm{Re}\\,\\frac{\\mathrm{i}\\,\\hat{\\Gamma}^{\\mathswitchr {A_0}\\PAO}(k^2)}{k^2-\\mathswitch {M_\\PAO}^2}&=-1,&\n\\lim_{k^2\\to \\mathswitch {M_\\PHP}^2}\\mathrm{Re}\\,\\frac{\\mathrm{i}\\,\\hat{\\Gamma}^{\\mathrm{H}^+\\mathrm{H}^-}(k^2)}{k^2-\\mathswitch {M_\\PHP}^2}&=-1,\n\\end{align}\nwhich implies\n\\begin{align}\n\\delta Z_\\mathswitchr H&= -\\mathrm{Re}\\, \\Sigma'^{\\mathswitchr H\\PH}(\\mathswitch {M_\\PH}^2),\n&\\delta Z_\\mathswitchr h&= -\\mathrm{Re}\\, \\Sigma'^{\\mathswitchr h\\Ph}(\\mathswitch {M_\\Ph}^2), \\label{eq:dZHdef}\\nonumber\\\\\n\\delta Z_{\\mathswitchr {A_0}}&= -\\mathrm{Re}\\, \\Sigma'^{\\mathswitchr {A_0}\\PAO}(\\mathswitch {M_\\PAO}^2),\n&\\delta Z_{\\mathrm{H}^+}&= -\\mathrm{Re}\\, \\Sigma'^{\\mathrm{H}^+\\mathrm{H}^-}(\\mathswitch {M_\\PHP}^2),\n\\end{align}\nwhere we introduced \n${\\Sigma}'(k^2)$ as the derivative w.r.t.\\  the argument $k^2$.\nTo fix the mixing renormalization constants, we enforce the condition that on-mass-shell fields do not mix, i.e.\n\\begin{align}\n\\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\PH}^2)&=0,& \\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)&=0,\n\\nonumber\\\\\n\\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(\\mathswitch {M_\\PAO}^2)&=0,&\\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr {G}^+\\mathswitchr H^-}(\\mathswitch {M_\\PHP}^2)&=0.\n\\label{eq:mixrencond}\n\\end{align}\nAfter inserting the renormalized self-energies we obtain\n\\begin{align}\n\\delta Z_{\\mathswitchr H\\mathswitchr h}&= 2 \\, \\frac{\\delta M_{\\mathswitchr H\\mathswitchr h}^2-\\mathrm{Re}\\,\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)}{\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2},\n&\\delta Z_{\\mathswitchr h\\mathswitchr H}&= 2 \\, \\frac{\\delta M_{\\mathswitchr H\\mathswitchr h}^2-\\mathrm{Re}\\,\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\PH}^2)}{\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2},\n\\nonumber\\\\\n\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}}&= 2 \\, \\frac{\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2-\\mathrm{Re}\\,\\Sigma^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(\\mathswitch {M_\\PAO}^2)}{\\mathswitch {M_\\PAO}^2},\n&\\delta Z_{\\mathswitchr {G}\\mathrm{H}^+}&= 2 \\, \\frac{\\delta M_{\\mathswitchr {G}\\mathswitchr {H^+}}^2-\\mathrm{Re}\\,\\Sigma^{\\mathrm{H}^+\\mathrm{G}^-}(\\mathswitch {M_\\PHP}^2)}{\\mathswitch {M_\\PHP}^2}.\\label{eq:mixfieldrendef}\n\\end{align}\nSince Goldstone-boson fields \ndo not correspond to physical states, we do not render Green functions with external \nGoldstone bosons finite, so that we need not fix the constants \n$\\delta Z_{\\mathswitchr {G_0}}$, $\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}}$, $\\delta Z_{\\mathswitchr {G^+}}$, $\\delta Z_{\\mathswitchr H\\mathswitchr {G^+}}$;\nwe could even set them to zero consistently.\nThe possible \n$\\mathswitchr Z\\mathswitchr {A_0}$ and $\\mathrm{W}^\\pm \\mathswitchr H^\\mp$ mixings\nvanish for physical on-shell gauge bosons due to the Lorentz structure of the two-point function \nand the fact that polarization vectors $\\varepsilon^\\mu$ are orthogonal to the corresponding momentum.\nUsing the convention\n\\begin{align}\n\\hat\\Gamma^{\\mathswitchr Z\\mathswitchr {A_0}}_\\mu(k) &= k_\\mu \\, \\hat\\Sigma^{\\mathswitchr Z\\mathswitchr {A_0}}(k^2)  = \nk_\\mu \\left[ \\Sigma^{\\mathswitchr Z\\mathswitchr {A_0}}(k^2)-\\textstyle\\frac{1}{2}\\mathswitch {M_\\PZ}\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}}\\right],\n\\nonumber\\\\\n\\hat\\Gamma^{\\mathswitchr W^\\pm\\mathswitchr H^\\mp}_\\mu(k) &= k_\\mu \\, \\hat\\Sigma^{\\mathswitchr W^\\pm\\mathswitchr H^\\mp}(k^2)  = \nk_\\mu \\left[ \\Sigma^{\\mathswitchr W^\\pm\\mathswitchr H^\\mp}(k^2)\\pm\\textstyle\\frac{\\mathrm{i}}{2}\\mathswitch {M_\\PW}\\delta Z_{\\mathswitchr {G}\\mathswitchr {H^+}}\\right],\n\\end{align}\nwhere all fields are incoming and $k$ is the incoming momentum of the gauge-boson fields,\nthe vector--scalar mixing self-energies obey\n\\begin{align}\n\\left.\\varepsilon^\\mu_\\mathswitchr Z k_\\mu \\,\\mathop{\\mathrm{Re}}\\nolimits\\,\\hat{\\Sigma}^{\\mathswitchr Z\\mathswitchr {A_0}}(k^2)\\right|_{k^2= \\mathswitch {M_\\PZ}^2}=0,\n\\qquad &\n\\left.\\varepsilon^\\mu_\\mathswitchr W k_\\mu \\,\\mathop{\\mathrm{Re}}\\nolimits\\,\\hat{\\Sigma}^{\\mathswitchr W^\\pm\\mathswitchr H^\\mp}(k^2)\\right|_{k^2= \\mathswitch {M_\\PW}^2}=0.\n\\end{align}\nThe mixing self-energies on the other on-shell points \n$k^2=\\mathswitch {M_\\PAO}^2$ and $k^2=\\mathswitch {M_\\PHP}^2$, respectively, \nare connected to the mixing of \n$\\mathswitchr {A_0}$ or $\\mathswitchr H^\\pm$ with the Goldstone-boson fields of the\n$\\mathswitchr Z$ or the $\\mathswitchr W$ boson and can be calculated from a BRST symmetry \\cite{Becchi:1975nq}. The BRST variation of the Green functions of one anti-ghost and a Higgs field\n\\begin{align}\n \\delta_\\mathrm{BRST} \\langle 0 | T \\bar{u}^\\mathswitchr Z(x) A_0(y)|0\\rangle=0, &&\n \\delta_\\mathrm{BRST} \\langle 0 | T \\bar{u}^\\pm(x) H^\\pm(y)|0\\rangle=0,\n\\end{align}\nimplies Slavnov--Taylor identities. While the variation of the anti-ghost fields yields the gauge-fixing term, the variation of the Higgs fields introduces ghost contributions which \nvanish for on-shell momentum resulting in%\n\\footnote{A particularly simple, alternative \nway to derive these identities is to exploit the gauge invariance\nof the effective action in the background-field gauge, as \ndone in \\citere{Denner:1994xt} for the SM. The respective Ward identities for the background\nfields differ from the Slavnov--Taylor identities only by off-shell terms, which vanish\non the particle poles. Generalizing the derivation of \\citere{Denner:1994xt} to the THDM\nand adapting the results to our conventions for self-energies,\nthe desired Ward identities for the unrenormalized background fields read\n\\begin{align}\n0 &= k^2 {\\Sigma}^{\\hat\\mathswitchr Z\\hat{\\mathrm{PA}}_0}(k^2)+ \\mathswitch {M_\\PZ} {\\Sigma}^{\\hat{\\mathrm{PA}}_0\\hat\\mathswitchr {G}_0}(k^2)\n+\\frac{e}{2\\mathswitch {c_{\\scrs\\PW}}\\mathswitch {s_{\\scrs\\PW}}}\\left(T^{\\hat\\mathswitchr H}s_{\\beta-\\alpha}-T^{\\hat\\mathswitchr h}c_{\\beta-\\alpha}\\right),\n\\\\\n0 &= \nk^2 {\\Sigma}^{\\hat\\mathswitchr W^\\pm\\hat\\mathswitchr H^\\mp}(k^2)\\mp \\mathrm{i} \\mathswitch {M_\\PW} {\\Sigma}^{\\hat\\mathswitchr {G}^\\pm\\hat\\mathswitchr H^\\mp}(k^2)\n\\mp\\frac{\\mathrm{i} e}{2\\mathswitch {s_{\\scrs\\PW}}}\\left(T^{\\hat\\mathswitchr H}s_{\\beta-\\alpha}-T^{\\hat\\mathswitchr h}c_{\\beta-\\alpha}\\right),\n\\end{align}\nwhere the carets on the fields indicate background fields.\nSetting $k^2$ to $\\mathswitch {M_\\PAO}^2$ or $\\mathswitch {M_\\PHP}^2$, respectively, and adding the \nrelevant renormalization constants, directly leads to the identities \\refeq{eq:ZA-WI}\nand \\refeq{eq:WH-WI}.\n}\n\\begin{align}\n\\label{eq:ZA-WI}\n \\left[k^2 \\hat{\\Sigma}^{\\mathswitchr Z\\mathswitchr {A_0}}(k^2)+ \\mathswitch {M_\\PZ} \\hat{\\Sigma}^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(k^2)\\right]_{k^2= \\mathswitch {M_\\PAO}^2}=0,\\\\\n \\left[k^2 \\hat{\\Sigma}^{\\mathrm{W}^\\pm\\mathswitchr H^\\mp}(k^2)\\mp \\mathrm{i} \\mathswitch {M_\\PW} \\hat{\\Sigma}^{\\mathrm{G}^\\pm\\mathrm{H}^\\mp}(k^2)\\right]_{k^2= \\mathswitch {M_\\PHP}^2}=0.\n\\label{eq:WH-WI}\n\\end{align}\nWe have verified these identities analytically and numerically.\nTogether with the renormalization condition of \nEq.~\\eqref{eq:mixrencond} we conclude that \n\\begin{align}\n\\hat{\\Sigma}^{\\mathswitchr Z\\mathswitchr {A_0}}(\\mathswitch {M_\\PAO}^2)=0, \\quad \\hat{\\Sigma}^{\\mathrm{W}^\\pm\\mathswitchr H^\\mp}(\\mathswitch {M_\\PHP}^2)=0.\n\\end{align}\nThis set of renormalization conditions ensures that no on-shell two-point vertex function obtains any one-loop corrections,\nand the corresponding external\nself-energy diagrams do not have to be taken into account in any calculation. \n\n\\subsubsection{Electroweak sector}\n\nThe fixing of the renormalization constants of the electroweak sector is identical to the SM case. The mass renormalization constants are fixed in such a way that the squares of the masses correspond to the \n(real parts of the) locations of the poles of the gauge-boson propagators. The field renormalization constants are fixed by the conditions that residues of on-shell gauge-boson propagators  do not obtain higher-order corrections, and that on-shell gauge bosons do not mix. For a better bookkeeping we also keep the dependent renormalization constants $\\delta \\mathswitch {c_{\\scrs\\PW}}$ and $\\delta \\mathswitch {s_{\\scrs\\PW}}$ in our calculation. This results in~\\cite{Denner:1991kt}\n\\begin{align}\n\\delta \\mathswitch {M_\\PW}^2 &=\\mathrm{Re}\\, \\Sigma^W_\\mathrm{T}(\\mathswitch {M_\\PW}^2),&\\delta Z_{\\mathswitchr W}&=-\\mathrm{Re}\\, \\Sigma'^{\\mathswitchr W}_\\mathrm{T}(\\mathswitch {M_\\PW}^2),\\nonumber\\\\\n\\delta \\mathswitch {M_\\PZ}^2&=\\mathrm{Re}\\, \\Sigma^{\\mathswitchr Z\\PZ}_\\mathrm{T}(\\mathswitch {M_\\PZ}^2),\\nonumber\\\\\n\\delta Z_{\\mathswitchr Z\\PZ}&=-\\mathrm{Re}\\, \\Sigma'^{\\mathswitchr Z\\PZ}_\\mathrm{T}(\\mathswitch {M_\\PZ}^2),&\\delta Z_{{\\mathrm{PA}}\\PA}&=-\\mathrm{Re}\\, \\Sigma'^{{\\mathrm{PA}}\\PA}_\\mathrm{T}(0),\\nonumber\\\\\n\\delta Z_{{\\mathrm{PA}}\\mathswitchr Z}&=-2\\mathrm{Re}\\, \\frac{\\Sigma^{{\\mathrm{PA}}\\mathswitchr Z}_\\mathrm{T}(\\mathswitch {M_\\PZ}^2)}{\\mathswitch {M_\\PZ}^2},&\n\\delta Z_{\\mathswitchr Z{\\mathrm{PA}}}&=2\\mathrm{Re}\\, \\frac{\\Sigma^{{\\mathrm{PA}}\\mathswitchr Z}_\\mathrm{T}(0)}{\\mathswitch {M_\\PZ}^2},\\nonumber\\\\\n\\delta \\mathswitch {c_{\\scrs\\PW}}&=\\frac{\\mathswitch {c_{\\scrs\\PW}}}{2}\\left(\\frac{\\delta \\mathswitch {M_\\PW}^2}{\\mathswitch {M_\\PW}^2}-\\frac{\\delta \\mathswitch {M_\\PZ}^2}{\\mathswitch {M_\\PZ}^2}\\right), &\\delta \\mathswitch {s_{\\scrs\\PW}}&= -\\frac{\\mathswitch {c_{\\scrs\\PW}}}{\\mathswitch {s_{\\scrs\\PW}}}\\delta \\mathswitch {c_{\\scrs\\PW}}.\n\\end{align}\nThe electric charge $e$ is defined via the $\\mathrm{ee} \\gamma $ coupling in the Thomson limit of on-shell external electrons and zero momentum transfer to the photon,\nwhich yields in BHS convention \\cite{Denner:1991kt}\n\\begin{align}\n\\delta Z_e = -\\frac{1}{2} \\left(\\delta Z_{{\\mathrm{PA}}\\PA}+ \\frac{\\mathswitch {s_{\\scrs\\PW}}}{\\mathswitch {c_{\\scrs\\PW}}} \\delta Z_{\\mathswitchr Z{\\mathrm{PA}}}\\right).\n\\label{eq:dZe}\n\\end{align}\n\n\\subsubsection{Fermions}\n\nThe renormalization conditions for the fermions are identical to the ones in the SM, described in detail in Ref.~\\cite{Denner:1991kt}. We demand that the (real parts of the locations of the)\npoles of the fermion propagators correspond to the squared fermion masses, \n and that on-shell fermion propagators do not obtain loop corrections. \nAssuming the CKM matrix \nequal to the unit matrix, the results for the renormalization constants simplify to\n\\begin{align}\n\\delta m_{f,i}&=\\frac{m_{f,i}}{2}\\mathrm{Re}\\,\\big[\\Sigma^{f,{\\mathrm{L}}}_i(m^2_{f,i})+\\Sigma^{f,{\\mathrm{R}}}_i(m^2_{f,i})+2 \\Sigma^{f,{\\mathrm{S}}}_i(m^2_{f,i})\\big],\n\\nonumber\\\\\n\\delta Z^{f,\\sigma}_i&=-\\mathrm{Re}\\, \\Sigma^{f,\\sigma}_i(m^2_{f,i})-m_{f,i}^2\\frac{\\partial}{\\partial k^2} \\mathrm{Re} \\big[\\Sigma_i^{f,{\\mathrm{L}}}(k^2)+\\Sigma^{f,{\\mathrm{R}}}_i(k^2)+2\\Sigma^{f,{\\mathrm{S}}}_i(k^2)\\big]\\Big|_{k^2=m_{f,i}^2}, \\quad \\sigma={\\mathrm{L}},{\\mathrm{R}},\n\\end{align}\nwhere we have \nused the usual \ndecomposition of the fermion self-energies into a left-handed, a right-handed, and a scalar part, $\\Sigma^{f,{\\mathrm{L}}}_i$, $\\Sigma^{f,{\\mathrm{R}}}_i$, and $ \\Sigma^{f,{\\mathrm{S}}}_i$, respectively.\nThe expressions for a non-trivial CKM matrix can be found in \\citere{Denner:1991kt}.\n\n\\subsection{\\boldmath{$\\overline{\\mathrm{MS}}$ renormalization conditions}}\n\\label{sec:diffrenschemes}\n\nIn the four renormalization schemes we \nare going to present, the imposed on-shell conditions are identical, and the differences only occur in the choice of different $\\overline{\\mathrm{MS}}$ conditions.\nThe parameters $\\alpha$ or $\\lambda_3$ governing the mixing of the CP-even Higgs bosons, and the parameters $\\beta$ and $\\lambda_5$ need to be fixed.\nA formulation of an on-shell condition for these parameters is not obvious.  \nOne could relate the parameters to \nsome physical processes, such as Higgs-boson decays, and demand that these processes do not \nreceive higher-order corrections. However, so far, no sign of further Higgs bosons has been observed, hence, there is no distinguished process, and such a prescription does not only require more calculational effort, but could introduce artificially large corrections to the corresponding parameters, which would spread to many other observables, as discussed in Refs.~\\cite{Freitas:2002um,Krause:2016oke}. Therefore, we choose to renormalize these parameters within the $\\overline{\\mathrm{MS}}$ scheme,\nthough different variables (such as $\\alpha$ or $\\lambda_3$) can be chosen to parameterize the model. Imposing an $\\overline{\\mathrm{MS}}$ condition on either of the parameters leads to differences in the calculation of observables. In addition, gauge-dependent definitions of $\\overline{\\mathrm{MS}}$-renormalized parameters spoil the gauge independence of the relations between input parameters and observables. However, gauge dependences might be even\nacceptable if the renormalization scheme yields stable results and a good convergence of the perturbation series. The price to pay is that subsequent calculations should be done in the same gauge\nor properly translated into another gauge. We will discuss different renormalization schemes based on different treatments of\n$\\alpha$ or $\\lambda_3$ parameterizing the CP-even Higgs-boson mixing, of the parameter $\\beta$, and of the Higgs coupling constant\n $\\lambda_5$ \nin the following. We begin with the so-called $\\overline{\\mathrm{MS}}(\\alpha)$ scheme.\n\n\\subsubsection{\\boldmath{$\\overline{\\mathrm{MS}}(\\alpha)$ scheme}}\n\\label{sec:alphaMSscheme}\nIn this scheme the independent parameter set is $\\{p_{\\mathrm{mass}}\\}$ of Eq.~\\eqref{eq:physparaNLO}, so that the parameters $\\beta$, $\\alpha$, and $\\lambda_5$ are renormalized in $\\overline{\\mathrm{MS}}$. The corresponding counterterm Lagrangian was derived in Sect.~\\ref{sec:renormassb}.\n\n\\paragraph{The renormalization constant \\boldmath{$\\delta \\beta$}:}\n\nThe renormalization constant \n$\\delta \\tan{\\beta}=\\delta \\beta\/{c_\\beta^2}$\nof the mixing angle $\\beta$ is\nrelated to the renormalization constants of the vevs\nby demanding the defining relation $\\tan\\beta=v_2\/v_1$ for bare and renormalized \nquantities.\nIn {$\\overline{\\mathrm{MS}}$}{}, $\\delta \\beta$ can be most easily calculated using the minimal field\nrenormalization \\refeq{eq:inputfieldren} with the following renormalization transformation\nof the vevs,\n\\begin{align}\nv_{1,0} &=  Z_{\\mathswitchr H_1}^{1\/2}(v_1+\\delta \\overline{v}_1),&v_{2,0}&=Z_{\\mathswitchr H_2}^{1\/2}(v_2+\\delta \\overline{v}_2),\n \\label{eq:vevren}\n\\end{align}\nUsing the well-known relation~\\cite{Sperling:2013eva}\n\\begin{align}\n\\delta \\overline{v}_1\/v_1-\\delta \\overline{v}_2\/v_2=\\mathrm{finite},\\label{eq:vacrencond}\n\\end{align}\nthe general form of $\\delta\\beta$ in the {$\\overline{\\mathrm{MS}}$}{} scheme\n\\begin{align}\n\\delta \\beta\n&= \\frac{s_{2\\beta}}{2}\\left(\n\\frac{\\delta \\overline{v}_2}{v_2} -\\frac{\\delta \\overline{v}_1}{v_1}- \\frac{1}{2}\\delta Z_{\\mathswitchr H_1} + \\frac{1}{2} \\delta Z_{\\mathswitchr H_2} \n\\right)\\bigg|_\\mathrm{UV}\n\\end{align}\nsimplifies to \n\\begin{align}\n\\delta \\beta\n&=\\frac{s_{2\\beta}}{4}(-\\delta Z_{\\mathswitchr H_1}+\\delta Z_{\\mathswitchr H_2})\\big|_\\mathrm{UV}\n=\\frac{s_{2\\beta}}{4c_{2\\alpha}} \\left(\\delta Z_\\mathswitchr h -\\delta Z_\\mathswitchr H\\right)\\big|_\\mathrm{UV}\n=\\frac{s_{2\\beta}}{4s_{2\\alpha}} \\left(\\delta Z_{\\mathswitchr h\\mathswitchr H} +\\delta Z_{\\mathswitchr H\\mathswitchr h}\\right)\\big|_\\mathrm{UV},\n\\label{eq:tanbetarencond}\n\\end{align}\nwhere $\\big|_\\mathrm{UV}$ indicates that we take only the UV-divergent parts, which are proportional to \nthe standard divergence $\\Delta_\\mathrm{UV}$.\nThe explicit calculation of the UV-divergent terms of $\\delta Z_\\mathswitchr h,$ $\\delta Z_\\mathswitchr H$ according to Eqs.~\\eqref{eq:dZHdef} in \n't~Hooft-Feynman gauge reveals that only diagrams with closed fermion loops contribute to the counterterm, \n\\begin{align}\\label{eq:deltabeta}\n\\delta \\beta = -\\Delta_\\mathrm{UV}\\, \\frac{e^2}{64\\pi^2\\mathswitch {M_\\PW}^2\\mathswitch {s_{\\scrs\\PW}}^2} \\sum_f c_f \\xi^f_\\mathswitchr {A_0} m_f^2,\n\\end{align}\nwith the colour factors $c_\\mathrm{quark}=3$, $c_\\mathrm{lepton}=1$ and the coupling coefficients $\\xi^f_{\\mathswitchr {A_0}}$ \nas defined in Tab.~\\ref{tab:yukint}. In the class of $R_\\xi$ gauges this result is gauge independent at one-loop order \\cite{Krause:2016oke,Denner:2016etu}.\n\n\\paragraph{Neutral Higgs mixing:}\nIn the neutral Higgs sector, relations between field renormalization constants can also be used to determine another parameter in $\\overline{\\mathrm{MS}}$.\nThe first four equations of Eqs.~\\eqref{eq:fieldrenconstrel} can be solved for $\\delta \\alpha$ in various\nways, e.g., yielding\n\\begin{align}\n\\label{eq:deltaalphaderiv}\n\\delta \\alpha\\big|_\\mathrm{UV}&= \\frac{1}{4} \\left(\\delta Z_{\\mathswitchr H\\mathswitchr h}-\\delta Z_{\\mathswitchr h\\mathswitchr H}\\right) \\big|_\\mathrm{UV}.\n\\end{align}\nThe field renormalization constants can be inserted according to Eqs.~\\eqref{eq:dZHdef}, \\eqref{eq:mixfieldrendef} using \n$\\delta M_{\\mathswitchr H\\mathswitchr h}^2 = 0$, thus \n\\begin{align}\n\\label{eq:deltaalphaderiv2}\n\\delta \\alpha&= \\left. \\mathrm{Re}\\frac{\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\PH}^2)+\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)}{2(\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2)}\\right|_\\mathrm{UV}.\n\\end{align}\nAn explicit calculation of the counterterm \nyields for the fermionic contribution, \n\\begin{align}\\label{eq:delalphaferm}\n\\delta \\alpha\\big|_\\mathrm{ferm} = \\Delta_\\mathrm{UV}\\, \\frac{e^2 s_{2\\alpha}}{64\\pi^2\\mathswitch {M_\\PW}^2\\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}(\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2)} \n\\sum_f c_f \\xi^f_\\mathswitchr {A_0} m_f^2 (\\mathswitch {M_\\PH}^2+\\mathswitch {M_\\Ph}^2-12m_f^2),\n\\end{align}\nand for the bosonic contribution\n\\begin{align}\\label{eq:delalphabos}\n\\delta \\alpha\\big|_\\mathrm{bos} = &\n-\\Delta_\\mathrm{UV} \\frac{\\lambda_5^2 \\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2 }{8\\pi^2 e^2 (\\mathswitch {M_\\Ph}^2 - \\mathswitch {M_\\PH}^2) s_{2\\beta}^2}\n   \\Big[s_{2 (\\alpha - 3 \\beta)} + 10 s_{2 (\\alpha - \\beta)} + 13 s_{2 (\\alpha + \\beta)}\\Big]\n\\nonumber\\\\\n& + \\Delta_\\mathrm{UV} \\frac{\\lambda_5 }{ 128\\pi^2 (\\mathswitch {M_\\Ph}^2 - \\mathswitch {M_\\PH}^2)  s_{2\\beta}^2} \\Big[\n   -4 \\mathswitch {M_\\PH}^2 (13 c_{2 \\alpha} + 2 c_{2 (\\alpha - 2 \\beta)} - 27 c_{2 \\beta}) s_{2 \\alpha} \n\\nonumber\\\\\n& + 4 \\mathswitch {M_\\Ph}^2 (13 c_{2 \\alpha} + 2 c_{2 (\\alpha - 2 \\beta)} + 27 c_{2 \\beta}) s_{2 \\alpha} \n  + 2 \\mathswitch {M_\\PHP}^2 (s_{2 (\\alpha - 3 \\beta)} - 6 s_{2 (\\alpha - \\beta)} + 13 s_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& - \\mathswitch {M_\\PAO}^2 (7 s_{2 (\\alpha - 3 \\beta)} + 86 s_{2 (\\alpha - \\beta)} + 91 s_{2 (\\alpha + \\beta)})\n  + 4 (2 \\mathswitch {M_\\PW}^2 + \\mathswitch {M_\\PZ}^2) s_{2 (\\alpha - \\beta)} s_{2 \\beta}^2 \n\\Big]\n\\nonumber\\\\\n& + \\Delta_\\mathrm{UV} \\frac{e^2 }{ 1024\\pi^2 (\\mathswitch {M_\\Ph}^2 - \\mathswitch {M_\\PH}^2) \\mathswitch {M_\\PW}^2  \\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}^2} \\Big[\n- 2 \\mathswitch {M_\\PH}^4 (-36 c_{2 \\alpha} + 5 c_{4 \\alpha - 2 \\beta} + 31 c_{2 \\beta}) s_{2 \\alpha} \n\\nonumber\\\\\n& + 4 \\mathswitch {M_\\Ph}^2 \\mathswitch {M_\\PH}^2 (5 c_{4 \\alpha - 2 \\beta} - 29 c_{2 \\beta}) s_{2 \\alpha} \n- 2 \\mathswitch {M_\\Ph}^4 (36 c_{2 \\alpha} + 5 c_{4 \\alpha - 2 \\beta} + 31 c_{2 \\beta}) s_{2 \\alpha} \n\\nonumber\\\\\n& + 32 \\mathswitch {M_\\PHP}^4 s_{2 (\\alpha - \\beta)} s_{2 \\beta}^2 \n+ 2 \\mathswitch {M_\\PH}^2 \\mathswitch {M_\\PHP}^2 (3 s_{4 \\alpha} + 4 s_{2 (\\alpha - \\beta)} + s_{4 (\\alpha - \\beta)} \n\t+ 9 s_{4 \\beta} - 12 s_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& - 2 \\mathswitch {M_\\Ph}^2 \\mathswitch {M_\\PHP}^2 (3 s_{4 \\alpha} -4 s_{2 (\\alpha - \\beta)} + s_{4 (\\alpha - \\beta)} \n\t+ 9 s_{4 \\beta} + 12 s_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& - 2 \\mathswitch {M_\\PAO}^4 (5 s_{2 (\\alpha - 3 \\beta)} + 42 s_{2 (\\alpha - \\beta)} + 41 s_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& + \\mathswitch {M_\\PAO}^2 \\mathswitch {M_\\PH}^2 (-49 s_{4 \\alpha} + 112 s_{2 (\\alpha - \\beta)} - 7 s_{4 (\\alpha - \\beta)} \n\t+ s_{4 \\beta} + 96 s_{2 (\\alpha + \\beta)})\n\\nonumber\\\\\n& + \\mathswitch {M_\\PAO}^2 \\mathswitch {M_\\Ph}^2 (49 s_{4 \\alpha} + 112 s_{2 (\\alpha - \\beta)} + 7 s_{4 (\\alpha - \\beta)} \n\t- s_{4 \\beta} + 96 s_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& + 4 \\mathswitch {M_\\PAO}^2 \\mathswitch {M_\\PHP}^2 (s_{2 (\\alpha - 3 \\beta)} - 6 s_{2 (\\alpha - \\beta)} + 13 s_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& + 4 (2 \\mathswitch {M_\\PW}^2 + \\mathswitch {M_\\PZ}^2) s_{2 (\\alpha - \\beta)} s_{2 \\beta} \n\t((\\mathswitch {M_\\Ph}^2 - \\mathswitch {M_\\PH}^2) s_{2 \\alpha} + 2 \\mathswitch {M_\\PAO}^2 s_{2 \\beta}) \n\\nonumber\\\\\n& + 48 (2 \\mathswitch {M_\\PW}^4 + \\mathswitch {M_\\PZ}^4) s_{2 (\\alpha - \\beta)} s_{2 \\beta}^2 \n\\Big].\n\\end{align}\nThis result, which is derived in 't~Hooft Feynman gauge, is gauge dependent \\cite{Krause:2016oke,Denner:2016etu}.\n\n\\paragraph{Higgs self-coupling:}\nThe Higgs self-coupling counterterm $\\delta \\lambda_5$ has to be fixed via a vertex correction. We define this renormalization constant  in $\\overline{\\mathrm{MS}}$ as well, as there is no distinguished process to fix it on-shell. Any 3- or 4-point vertex function with external Higgs bosons is suited to calculate the divergent terms.\nSince the $\\mathswitchr H\\mathswitchr {A_0}\\PAO$ vertex correction involves fewest diagrams, it is our preferred choice. The condition is\n\\begin{align}\n\\hat{\\Gamma}^{\\mathswitchr H\\mathswitchr {A_0}\\PAO}\\big|_\\mathrm{UV}=\n\\parbox{2.4cm}{\\includegraphics[scale=1]{.\/diagrams\/L5ren.eps}}\\left.\\rule{0cm}{1.1cm}\\right|_\\mathrm{UV}=0.\n\\end{align}\nSolving this equation for $\\delta \\lambda_5$ fixes this renormalization constant. The generic one-loop diagrams appearing in this vertex correction are shown in Fig.~\\ref{fig:dL5diags}, the contribution of the diagrams involving closed fermion loops is \n\\begin{align}\n\\label{eq:L5ferm}\n\\delta \\lambda_{5,\\mathrm{ferm}}=\\Delta_\\mathrm{UV} \n\\frac{e^2 \\lambda_5}{16\\pi^2 \\mathswitch {M_\\PW}^2\\mathswitch {s_{\\scrs\\PW}}^2}\\sum_f c_f \\left(1+\\frac{c_{2\\beta}}{s_{2\\beta}}\\xi_{\\mathswitchr {A_0}}^f\\right) m_f^2.\n\\end{align}\n\\begin{figure}\n\\vspace{-15pt}\n\\input{diagrams\/triangles}\n\\vspace{-20pt}\n\\caption{Generic diagrams contributing to the $HA_0A_0$ vertex correction used for the renormalization of $\\lambda_5$.}\n\\label{fig:dL5diags}\n\\end{figure}\nThe diagrams containing only bosons lead to \n\\begin{align}\n\\label{eq:L5bos}\n\\delta \\lambda_5\\big|_\\mathrm{bos}{}={}& {}\n\\Delta_\\mathrm{UV} \\frac{\\lambda_5}{32 \\pi^2}  \n\\left(2 \\lambda_1 + 2\\lambda_2+8\\lambda_3 +12 \\lambda_4- 9 g_2^2 - 3 g_1^2\\right)\n\\nonumber\\\\\n{}={}& - \\Delta_\\mathrm{UV}\\frac{\\lambda_5^2c^2_{2\\beta}}{4 \\pi^2 s^2_{2\\beta}}\n +\\Delta_\\mathrm{UV}\\frac{\\lambda_5 e^2}{64 \\pi^2\\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}^2} \\Big[\n  \\mathswitch {M_\\PH}^2 (2 + c_{2 (\\alpha - \\beta)} - 3 c_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& + \\mathswitch {M_\\Ph}^2 (2 - c_{2 (\\alpha - \\beta)} + 3 c_{2 (\\alpha + \\beta)})\n+ \\mathswitch {M_\\PAO}^2 (1 - 5 c_{4 \\beta})\n- 4 \\mathswitch {M_\\PHP}^2 s_{2 \\beta}^2\n- 6(2 \\mathswitch {M_\\PW}^2+\\mathswitch {M_\\PZ}^2) s_{2 \\beta}^2 \n\\Big].\n\\end{align}\nSince $\\lambda_5$ is a fundamental parameter of the Higgs potential, an $\\overline{\\mathrm{MS}}$  definition leads to a gauge-independent counterterm.\n\n\\subsubsection{\\boldmath{$\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme}}\n\\label{sec:lambda3MSscheme}\n\nIn this scheme, the independent parameter set is $\\{p'_{\\mathrm{mass}}\\}$ defined in Eq.~\\eqref{eq:physparaNLOprime}.  The renormalization of $\\beta$ and $\\lambda_5$ is identical to the previous renormalization scheme and not stated again, but now the parameter $\\lambda_3$ (instead of $\\alpha$) is an independent parameter being renormalized in $\\overline{\\mathrm{MS}}$. This has the advantage that this parameter is gauge independent, as it is a defining parameter of the basic \nparameterization of the Higgs potential\nand thus is safe against potentially gauge-dependent contributions appearing in relations between bare parameters. \nAs stated above, the {$\\overline{\\mathrm{MS}}$}{} renormalization of the \nparameter $\\beta$ generally breaks gauge \nindependence, but in $\\mathrm{R}_\\xi$ gauges the gauge dependence cancels at one loop \\cite{Krause:2016oke,Denner:2016etu}, \nso that this scheme yields gauge-independent results at NLO. \nWe take the counterterm potential of Sect.~\\ref{sec:renormassb}, but treat $\\delta \\alpha$ as a dependent counterterm. \nAs $\\alpha$ is a pure mixing angle, we choose to apply the renormalization prescription of Sect.~\\ref{sec:renormassb}, where the mixing angle diagonalizes the potential to all orders.\nThe relation between $\\delta \\alpha$ and the independent constants is given in Eq.~\\eqref{eq:NLOmassmixHh} with $ \\delta M_{\\mathswitchr H\\mathswitchr h}^2=0$,\n\\begin{align}\n\\label{eq:alphadef32}\n \\delta \\alpha=\\frac{f_\\alpha(\\{ \\delta p'_\\mathrm{mass}\\})} {\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2},\n\\end{align}\nwhere $f_\\alpha(\\{ \\delta p'_\\mathrm{mass}\\})$ can be obtained from Eq.~\\eqref{eq:Hhmassmixing} by applying the renormalization transformation of Eq.~\\eqref{eq:physpararen1} (which is identical to the renormalization transformation of Sect.~\\ref{sec:renormassb}, but renormalizing $\\lambda_3$ instead of $\\alpha$). This yields\n\\begin{align}\nf_\\alpha(\\{\\delta p'_\\mathrm{mass}\\})={}&\n\\frac{1}{2} t_{2 \\alpha } \\left(\\delta \\mathswitch {M_\\Ph}^2-\\delta \\mathswitch {M_\\PH}^2\\right)\n+\\frac{s_{2 \\beta } \\left(\\delta \\mathswitch {M_\\PAO}^2-2 \\delta \\mathswitch {M_\\PHP}^2\\right)}{2 c_{2 \\alpha }}\n\\nonumber\\\\\n&+\\frac{\\delta \\beta c_{2 \\beta } \\left(\\mathswitch {M_\\PH}^2- \\mathswitch {M_\\Ph}^2\\right) t_{2 \\alpha }}{s_{2 \\beta }}\n+\\frac{2 \\mathswitch {M_\\PW}^2 s_{2\\beta } (\\delta \\lambda_3+\\delta \\lambda_5) \\mathswitch {s_{\\scrs\\PW}}^2}{e^2 c_{2 \\alpha }}\n\\nonumber\\\\\n&+\\frac{s_{2 \\beta } \\left(\\mathswitch {M_\\PAO}^2-2 \\mathswitch {M_\\PHP}^2\\right)+(\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2)s_{2\\alpha}}{c_{2\\alpha}}\n\\left( \\delta Z_e -\\frac{\\delta \\mathswitch {s_{\\scrs\\PW}}}{\\mathswitch {s_{\\scrs\\PW}}} -\\frac{\\delta \\mathswitch {M_\\PW}^2}{2\\mathswitch {M_\\PW}^2} \\right)\n\\nonumber\\\\\n&\n-\\frac{ e  \\left[\n\\delta t_\\mathswitchr H \\left(s_{\\alpha -3 \\beta }+3 s_{\\alpha +\\beta }\\right)\n+\\delta t_\\mathswitchr h \\left(c_{\\alpha -3 \\beta }+3 c_{\\alpha +\\beta } \\right) \n\\right]}{8 \\mathswitch {M_\\PW}\n   c_{2 \\alpha } \\mathswitch {s_{\\scrs\\PW}}}.\\label{eq:MHhdependence}\n\\end{align}\nThe UV-divergent term of $\\delta \\alpha$ has been calculated in Eq.~\\eqref{eq:deltaalphaderiv2}, and by renormalizing $\\delta \\lambda_3$ in $\\overline{\\mathrm{MS}}$ scheme, it is clear that the dependent $\\delta \\alpha$ must now have a finite part in addition. \nWe choose this finite term in such a way that the finite part in $\\delta \\lambda_3$ \n(which results from $\\delta \\lambda_3$ by setting $\\Delta_{\\mathrm{UV}}$ to zero) vanishes and obtain\n\\begin{align}\n\\label{eq:alphainL3MS}\n\\delta \\alpha\\big|_{\\overline{\\mathrm{MS}}(\\lambda_3)} =\n\\left. \\mathrm{Re}\\frac{\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\PH}^2)+\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)}{2(\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2)}\\right|_\\mathrm{UV}\n+\\frac{f_\\alpha(\\{ \\delta p'_\\mathrm{mass}\\})} {\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2}\\bigg|_\\mathrm{finite},\n\\end{align}\nwhere $\\delta \\lambda_3$ drops out as it has no finite part. The divergent part of $\\delta \\lambda_3$ can be calculated by solving Eq.~\\eqref{eq:alphadef32} and using the knowledge about the divergent parts of $\\delta \\alpha$ from Eqs.~\\eqref{eq:delalphaferm} and \\eqref{eq:delalphabos}. This results in \n\\begin{align}\n\\delta \\lambda_3= &\\biggl[\n\\frac{e^2 c_{2\\alpha}}{4 \\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}} \n\\left(\\mathrm{Re}\\,\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\PH}^2)+\\mathrm{Re}\\,\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)\\right)\n-\\frac{\\delta \\beta e^2 s_{2 \\alpha} c_{2 \\beta} \\left(\\mathswitch {M_\\PH}^2- \\mathswitch {M_\\Ph}^2\\right) }{2 \\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2 s_{2 \\beta }^2}\n\\nonumber\\\\\n& -\\delta \\lambda_5\n-\\frac{e^2 s_{2 \\alpha }}{4 \\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}}  \\left(\\delta \\mathswitch {M_\\Ph}^2-\\delta \\mathswitch {M_\\PH}^2\\right)\n-\\frac{e^2 \\left(\\delta \\mathswitch {M_\\PAO}^2-2 \\delta \\mathswitch {M_\\PHP}^2\\right)}{4 \\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2}\n\\nonumber\\\\\n&-\\frac{e^2\\left(s_{2\\beta}\\left(\\mathswitch {M_\\PAO}^2-2 \\mathswitch {M_\\PHP}^2\\right)+(\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2) s_{2 \\alpha }\\right)}{2\\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}}\n\\left(\\delta Z_e- \\frac{\\delta \\mathswitch {s_{\\scrs\\PW}}}{\\mathswitch {s_{\\scrs\\PW}}} -\\frac{\\delta \\mathswitch {M_\\PW}^2}{2 \\mathswitch {M_\\PW}^2} \\right)\n\\nonumber\\\\\n& +\\frac{ e^3  \\left[\n\\delta t_\\mathswitchr H \\left(s_{\\alpha -3 \\beta }+3 s_{\\alpha +\\beta }\\right)\n+\\delta t_\\mathswitchr h \\left(c_{\\alpha -3 \\beta }+3 c_{\\alpha +\\beta } \\right) \n\\right]}{16 \\mathswitch {M_\\PW}^3 \\mathswitch {s_{\\scrs\\PW}}^3 s_{2\\beta}}\\biggr]_{\\mathrm{UV}}.\n\\label{eq:deltaL3}\n\\end{align}\nThe fermionic contribution to $\\delta \\lambda_3$ is given by\n\\begin{align}\n\\nonumber\n\\delta \\lambda_3\\big|_\\mathrm{UV,ferm}\n& = -\\delta \\lambda_5\\big|_\\mathrm{UV,ferm}\n-\\Delta_\\mathrm{UV} \\frac{3e^4}{32\\pi^2\\mathswitch {M_\\PW}^4\\mathswitch {s_{\\scrs\\PW}}^4} \\sum_i\n\\left(\\xi_\\mathswitchr {A_0}^u-\\xi_\\mathswitchr {A_0}^d\\right)^2\nm_{u,i}^2 m_{d,i}^2\n\\\\ \n&\\quad\n- \\Delta_\\mathrm{UV} \\frac{e^4}{64\\pi^2\\mathswitch {M_\\PW}^4\\mathswitch {s_{\\scrs\\PW}}^4}\n\\sum_f c_f \\left(1+\\frac{c_{2\\beta}}{s_{2\\beta}}\\xi_\\mathswitchr {A_0}^f\\right) m_f^2\n\\left[\\mathswitch {M_\\PAO}^2-2\\mathswitch {M_\\PHP}^2+\\frac{s_{2\\alpha}}{s_{2\\beta}}(\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2)\\right]\n\\end{align}\nwith the massive fermions $f = \\mathswitchr e, \\dots,\\mathswitchr t$ and the generation index $i$. For the bosonic contribution we obtain\n\\begin{align}\n\\label{eq:L3bos}\n\\delta \\lambda_3\\big|_\\mathrm{UV,bos}\n= \\,& \\Delta_\\mathrm{UV}\\frac{1}{32 \\pi^2} \\big[(\\lambda_1 + \\lambda_2) (6 \\lambda_3 + 2 \\lambda_4) + 4 \\lambda_3^2 + 2 \\lambda_4^2 + 2 \\lambda_5^2\n-3 \\lambda_3 (3 g_2^2 + g_1^2 )\n\\nonumber\\\\\n& +\\textstyle\\frac{3}{4} (3 g_2^4 + g_1^4 - 2 g_2^2 g_1^2)\n\\big]\n\\nonumber\\\\\n= &\n\\,\\Delta_\\mathrm{UV} \\frac{\\lambda_5^2}{2 \\pi^2 s_{2\\beta}^2}\n+ \\Delta_\\mathrm{UV} \\frac{e^2 \\lambda_5}{128 \\mathswitch {M_\\PW}^2 \\pi^2 \\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}^3}\\Big[\n12 (2 \\mathswitch {M_\\PW}^2 + \\mathswitch {M_\\PZ}^2) s_{2 \\beta}^3 \n+ \\mathswitch {M_\\PAO}^2 (27 s_{2 \\beta} - s_{6 \\beta}) \n\\nonumber\\\\\n&+ 2 \\mathswitch {M_\\PHP}^2 (-19 s_{2 \\beta} + s_{6 \\beta}) \n + \\mathswitch {M_\\PH}^2 (-22 s_{2 \\alpha} - 3 s_{2 (\\alpha - 2 \\beta)} - 8 s_{2 \\beta} + s_{2 (\\alpha + 2 \\beta)}) \n\\nonumber\\\\\n& + \\mathswitch {M_\\Ph}^2 (22 s_{2 \\alpha} + 3 s_{2 (\\alpha - 2 \\beta)} - 8 s_{2 \\beta} - s_{2 (\\alpha + 2 \\beta)}) \\Big]\n\\nonumber\\\\\n& + \\Delta_\\mathrm{UV} \\frac{e^4}{256 \\mathswitch {M_\\PW}^4 \\pi^2 \\mathswitch {s_{\\scrs\\PW}}^4 s_{2\\beta}^3} \\Big[\n-2 \\mathswitch {M_\\PH}^4 (-3 + c_{2 (\\alpha - \\beta)} + 2 c_{2 (\\alpha + \\beta)}) s_{2 \\alpha}  \n\\nonumber\\\\\n& + 4 \\mathswitch {M_\\Ph}^2 \\mathswitch {M_\\PH}^2 (c_{2 (\\alpha - \\beta)} + 2 c_{2 (\\alpha + \\beta)}) s_{2 \\alpha}\n- 2 \\mathswitch {M_\\Ph}^4 (3 + c_{2 (\\alpha - \\beta)} + 2 c_{2 (\\alpha + \\beta)}) s_{2 \\alpha}  \n\\nonumber\\\\\n& - \\mathswitch {M_\\PAO}^4 (-7 s_{2 \\beta} + s_{6 \\beta}) \n- \\mathswitch {M_\\PAO}^2 \\mathswitch {M_\\PH}^2 (11 s_{2 \\alpha} + s_{2 (\\alpha - 2 \\beta)} + 2 s_{2 \\beta})\n\\nonumber\\\\\n& + \\mathswitch {M_\\PAO}^2 \\mathswitch {M_\\Ph}^2 (11 s_{2 \\alpha} + s_{2 (\\alpha - 2 \\beta)} - 2 s_{2 \\beta})\n+ 12 \\mathswitch {M_\\PHP}^4 s_{2 \\beta}^3 \n+ 16 \\mathswitch {M_\\PH}^2 \\mathswitch {M_\\PHP}^2 s_{2 \\beta} s_{\\alpha + \\beta}^2\n\\nonumber\\\\\n&+ 16 \\mathswitch {M_\\Ph}^2 \\mathswitch {M_\\PHP}^2 c_{\\alpha + \\beta}^2 s_{2 \\beta}\n+ 2 \\mathswitch {M_\\PAO}^2 \\mathswitch {M_\\PHP}^2 (-11 s_{2 \\beta} + s_{6 \\beta})\n\\nonumber\\\\\n&+ 6 (2 \\mathswitch {M_\\PW}^2 + \\mathswitch {M_\\PZ}^2) s_{2 \\beta}^2 ((\\mathswitch {M_\\Ph}^2 - \\mathswitch {M_\\PH}^2) s_{2 \\alpha} + (\\mathswitch {M_\\PAO}^2 - 2 \\mathswitch {M_\\PHP}^2) s_{2 \\beta})\n\\nonumber\\\\\n&+ 6 (6 \\mathswitch {M_\\PW}^4 - 4 \\mathswitch {M_\\PW}^2 \\mathswitch {M_\\PZ}^2 + \\mathswitch {M_\\PZ}^4) s_{2 \\beta}^3\n\\Big].\n\\end{align}\n\n\\subsubsection{The FJ tadpole scheme}\n\\label{sec:FJscheme}\n\nSince tadpole loop contribution $T_S$ are gauge dependent \\cite{Degrassi:1992ff}, the connection among bare parameters potentially becomes gauge dependent \nif $\\delta t_S=-T_S$ enters the relations between bare parameters, as it is the case if renormalized tadpole parameters $t_S$ \nare forced to vanish. Note that these gauge dependences systematically cancel if on-shell renormalization conditions are employed, i.e.\\ if predictions \nfor observables are parameterized by directly measurable input parameters. If some input parameters are renormalized in the {$\\overline{\\mathrm{MS}}$}{} scheme this cancellation of gauge dependences does not take place anymore in general, and the gauge dependence is ma\\-ni\\-fest \nin relations between predicted observables and input parameters at NLO.\nIn the $\\overline{\\mathrm{MS}}(\\alpha)$ and the $\\overline{\\mathrm{MS}}(\\lambda_3)$ renormalization schemes, the bare de\\-finitions of $\\alpha$ and $\\beta$ contain tadpole terms \nleading to a gauge dependence (although the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme is gauge \nindependent at NLO in $R_\\xi$ gauges).\n \nFleischer and Jegerlehner \\cite{Fleischer:125234} proposed a renormalization scheme for the SM, referred to as the FJ scheme in the following, that preserves gauge independence for all bare parameters, including the \nmasses and mixing angles%\n\\footnote{A similar scheme, called $\\beta_h$~scheme, was suggested in \\citere{Actis:2006ra}.\nA comparison of that approach to the conventional {$\\overline{\\mathrm{MS}}$}{} and FJ schemes can be found in\n\\citere{Denner:2016etu}.}.\nIn this scheme, the parameters are defined in such a way that tadpole terms do not enter the definition of any bare parameter so that all relations among bare parameters remain gauge independent. \nThis can be achieved by demanding that {\\it bare} tadpole terms vanish,\n$t_{S,0}=0$,\nfor all fields $S$ with the quantum numbers of the vacuum. Since tadpole conditions have no effect on physical observables and change only the bookkeeping, such a procedure is possible. The disadvantage is that now tadpole diagrams have to be taken explicitly into account \nin all higher-order calculations. In particular, the one-particle reducible tadpole contributions destroy the simple relation between propagators and \ntwo-point functions. In the SM, the FJ scheme does not affect observables if all parameters are \nrenormalized using on-shell conditions---as usually done---except \nfor the strong coupling constant $\\alpha_\\mathrm{s}$, which is, however, directly related to the strong gauge coupling, a model defining parameter.\nA gauge-independent renormalization scheme for the THDM can be defined by applying the FJ prescription and imposing the {$\\overline{\\mathrm{MS}}$}{} condition on  mixing angles \\cite{Denner:2016etu, Krause:2016oke}.\nThe {\\it bare} physical parameters defined in the FJ scheme \ndiffer by NLO tadpole contributions (including divergent and finite terms)\nfrom the gauge-dependent definition of the bare parameters $\\{p_{\\mathrm{mass}}\\}$ given in Eq.~\\eqref{eq:physparaNLO}. \nExceptions are $e$ and the parameter $\\lambda_5$, which is a parameter of the basic potential and therefore gauge independent by construction. \nThe renormalization of $\\lambda_5$\nin {$\\overline{\\mathrm{MS}}$}{} is identical to the one in the previous schemes.\n\nIt should be noted that in Refs.~\\cite{Denner:2016etu, Krause:2016oke} $m_{12}^2$ is chosen as independent\nparameter in contrast to our choice of $\\lambda_5$. The latter, however, is closer to common practice used in\nthe MSSM~\\cite{Frank2007,Baro:2008bg}.\nMoreover, in  Refs.~\\cite{Krause:2016oke,Denner:2016etu}\ntadpole counterterms are reintroduced by shifting the Higgs fields according to $\\eta_i \\to \\eta_i + \\Delta v_{\\eta, i}$, $i = 1,2$, where \nthe constants $\\Delta v_{\\eta, i}$ can be chosen arbitrarily, since physical observables do not depend on this shift,\nwhich can be interpreted as an unobservable change of the integration variables in the path integral. In Refs.~\\cite{Krause:2016oke,Denner:2016etu}, this freedom of choice is exploited, \nand $\\Delta v_{\\eta, i}$ \nare chosen in such a way that the fields $\\eta_1$, $\\eta_2$ do not develop vevs at all order\n. This affects the form of the counterterm Lagrangian and the definition of the renormalization constants with the consequence that the formulae given in Eq.~\\eqref{eq:dVH2p} and Sect.\\ref{sec:onshellrenconditions} cannot be applied.\n\nWe have implemented the FJ scheme following the strategy of Ref.~\\cite{Denner:2016etu} by performing the shifts $\\eta_i \\to \\eta_i+ \\Delta v_{\\eta, i}$ and in an alternative, simpler (but physically equivalent) way. \nIn this simplified approach we keep the dependence of the Lagrangian in terms of gauge-dependent masses and couplings. In addition we keep the tadpole renormalization \ncondition~\\eqref{eq:tadpoleren}, so that the definitions of the renormalization constants of the on-shell parameters and the $Z$ factors according to Sect.~\\ref{sec:onshellrenconditions} remain valid (otherwise we   needed to take into account actual tadpole diagrams everywhere).\nIn this simplified approach the counterterms for $\\alpha$ and $\\beta$ which reproduce the\nresults in the FJ scheme result from the previously derived $\\delta\\alpha$ and $\\delta\\beta$\nby adding appropriate finite terms,\n\\begin{align}\n\\delta \\alpha\\big|_\\mathrm{FJ}&= \\delta \\alpha+\\mbox{finite terms},\\nonumber\\\\\n\\delta \\beta\\big|_\\mathrm{FJ} &= \\delta \\beta +\\mbox{finite terms},\n\\end{align}\nwhich depend on the (finite parts of the) tadpole contributions $T_\\mathswitchr H$ and $T_\\mathswitchr h$.\n\nBefore performing the full calculations, we outline the strategy of the derivation of\nthose finite terms for $\\beta$; for $\\alpha$ everything works analogously.\nWe start by exploiting the fact that the form of the tadpole renormalization cannot\nchange physical results if all counterterms for independent parameters are determined \nby the same physical conditions. This means, as mentioned above, that we can simply define\nthe bare tadpoles to vanish, but this forces us to include all explicit tadpole\ncontributions to Green functions. \nWe indicate quantities in this variant by a superscript ``$t$'' in the following. \nWe get the same physical predictions in this ``$t$-variant'' if we use the\ncounterterm\n\\begin{align}\n\\delta \\beta^t &= \\delta \\beta +\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\n\\label{eq:dbt_def}\n\\end{align}\ninstead of $\\delta\\beta$,\nwhere $\\delta\\beta^t$ is calculated in the same way as $\\delta \\beta$, but with\ntadpole counterterms omitted and \nexplicit tadpole diagrams (including divergent and finite parts)\nin the occurring Green functions taken into account.\nNote that the {$\\overline{\\mathrm{MS}}$}{} prescription to include only divergent terms, which is employed to define\n$\\delta \\beta$, is not applied to the new tadpole contribution $\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)$.\nOtherwise the new $\\Delta\\beta^t$ terms could not be fully compensated by explicit \ntadpole contributions occuring elsewhere, so that\nthere would be differences in the renormalized amplitudes.\nIn fact, applying the {$\\overline{\\mathrm{MS}}$}{} prescription to $\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)$ as well\ndefines the FJ renormalization scheme,\n\\begin{align}\n\\delta \\beta^t\\big|_\\mathrm{FJ} &= \\delta \\beta +\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{UV}.\n\\end{align}\nThe quantity $\\delta \\beta^t\\big|_\\mathrm{FJ}$ is the gauge-independent counterterm for $\\beta$\nintroduced in Ref.~\\cite{Denner:2016etu} which is to be used in the $t$-variant,\nwhere all explicit tadpole diagrams are included in Green functions \n(or equivalently are redistributed by the $\\Delta v$ shift as described Ref.~\\cite{Denner:2016etu}).\nWe can translate the FJ renormalization prescription back to our \nrenormalization scheme (with vanishing renormalized tadpoles) by the counterpart\nof Eq.~\\refeq{eq:dbt_def}, but now formulated in the FJ scheme,\n\\begin{align}\n\\delta \\beta^t\\big|_\\mathrm{FJ} &= \\delta \\beta\\big|_\\mathrm{FJ} +\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h),\n\\label{eq:dbtFJ_def}\n\\end{align}\ni.e.\\ $\\delta \\beta\\big|_\\mathrm{FJ}$ is the counterterm for $\\beta$ to be used in our\ncounterterm Lagrangian in order to calculate renormalized amplitudes in the FJ scheme.\nCombining the above formulas, we obtain the finite difference between\n$\\delta \\beta$ in the (gauge-dependent) {$\\overline{\\mathrm{MS}}$}{} scheme and \n$\\delta \\beta\\big|_\\mathrm{FJ}$ in the (gauge-independent) FJ scheme,\n\\begin{align}\n\\delta \\beta\\big|_\\mathrm{FJ} &= \\delta \\beta^t\\big|_\\mathrm{FJ} -\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\n\\nonumber\\\\\n&= \\delta \\beta +\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{UV} -\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\n\\nonumber\\\\\n&= \\delta \\beta -\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{finite}.\n\\end{align}\n\n\\paragraph{The renormalization constant \\boldmath{$\\delta  \\beta\\big|_\\mathrm{FJ}$}:}\nWe begin our calculation of $\\delta\\beta|_\\mathrm{FJ}$ with an alternative \ncomputation of $\\delta \\beta$ in the {$\\overline{\\mathrm{MS}}$}{}$(\\alpha)$ scheme, because \nEq.~\\eqref{eq:vacrencond} cannot be applied in the FJ scheme.\nTo avoid the use of Eq.~\\eqref{eq:vacrencond},\nwe calculate the counterterm in the {$\\overline{\\mathrm{MS}}$}{}$(\\alpha)$ scheme from the field \nrenormalization constant by employing the last two equations of Eq.~\\eqref{eq:fieldrenconstrel}. \nThis results in\n\\begin{align}\n \\delta \\beta=\\frac{1}{4} \\left(\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}}-\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}}\\right)\\big|_\\mathrm{UV}\n=\\left.\\frac{\n2\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2\n-\\mathrm{Re}\\,\\Sigma^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(\\mathswitch {M_\\PAO}^2)-\\mathrm{Re}\\,\\Sigma^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(0)\n}{2 \\mathswitch {M_\\PAO}^2}\\right|_\\mathrm{UV},\n \\label{eq:betaMSaltern}\n\\end{align}\nwith $\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2$ as given in Eq.~\\eqref{eq:Mabchoice2}. In the second step,  $\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}}$ from Eq.~\\eqref{eq:mixfieldrendef} and\n\\begin{align}\n\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}}=2 \\, \\frac{-\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2+\\mathrm{Re}\\,\\Sigma^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(0)}{\\mathswitch {M_\\PAO}^2}\n\\label{eq:dZAG}\n\\end{align}\nhave been used. Equation~\\refeq{eq:dZAG} results from demanding finiteness of the\n$G_0A_0$ mixing self-energy at zero-momentum transfer, $k^2=0$, but actually any other \nvalue of $k^2$ would be possible as well, since we only have to remove all UV-divergent terms\nin the mixing. \nThe non-vanishing tadpole counter\\-terms in the {$\\overline{\\mathrm{MS}}$}{}$(\\alpha)$ scheme are $\\delta t_S=-T_S$.\nIn the transition to the $t$-variant, $\\delta \\beta$ gets modified by two kind of terms:\nFirst, there are no tadpole counterterms, i.e.\\ the $\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2$ term is absent,\nand second, there are explicit tadpole contributions to $\\Sigma^{\\mathswitchr {G_0}\\mathswitchr {A_0}}$.\nThis implies\n\\begin{align}\n\\Delta\\beta^t(T_\\mathswitchr H,T_\\mathswitchr h) &= \\delta\\beta^t-\\delta\\beta\n= -\\frac{\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2}{\\mathswitch {M_\\PAO}^2} \n-\\left.\\frac{\\mathrm{Re}\\,\\Sigma^{t,\\mathswitchr {G_0}\\mathswitchr {A_0}}(\\mathswitch {M_\\PAO}^2)+\\mathrm{Re}\\,\\Sigma^{t,\\mathswitchr {G_0}\\mathswitchr {A_0}}(0)}{2 \\mathswitch {M_\\PAO}^2}\\right|_\\mathrm{T_\\mathswitchr H,T_\\mathswitchr h},\n\\end{align}\nwhere the superscript ``$t$'' indicates that one-particle-reducible tadpole diagrams are \nincluded in the self-energies.\nThe subscript ``$T_\\mathswitchr H,T_\\mathswitchr h$'' means that only the those explicit tadpole contributions\nare taken into account here.\nInserting $\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2$ from Eq.~\\refeq{eq:Mabchoice2}\nand evaluating the (momentum-independent) tadpole diagrams for the $G_0A_0$ mixing,\n$\\Delta\\beta^t$ evaluates to\n\\begin{align}\n\\lefteqn{\\Delta\\beta^t(T_\\mathswitchr H,T_\\mathswitchr h)} \n\\nonumber\\\\*\n&= -\\frac{1}{\\mathswitch {M_\\PAO}^2} \\left[\n\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2\n+\\left.\\mathrm{Re}\\,\\Sigma^{t,\\mathswitchr {A_0}\\mathswitchr {G_0}}(0)\\right|_\\mathrm{T_\\mathswitchr H,T_\\mathswitchr h} \\right]\n\\nonumber\\\\\n&=-\\frac{1}{ \\mathswitch {M_\\PAO}^2}\\left[\n\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2\n+ \\parbox[c][1cm]{2.8cm}{\\includegraphics[scale=1]{.\/diagrams\/tadpoleSEHAG1.eps}}+\\parbox[c][1cm]{2.8cm}{\\includegraphics[scale=1]{.\/diagrams\/tadpoleSEhAG2.eps}}\\right]\n\\nonumber\\\\\n&=-\\frac{e}{2 \\mathswitch {M_\\PW} \\mathswitch {s_{\\scrs\\PW}}\\mathswitch {M_\\PAO}^2} \\left[\nT_\\mathswitchr H s_{\\alpha -\\beta }\n+T_\\mathswitchr h c_{\\alpha -\\beta }\n+ T_\\mathswitchr H \\frac{\\left(\\mathswitch {M_\\PAO}^2-\\mathswitch {M_\\PH}^2\\right) s_{\\alpha -\\beta }}{\\mathswitch {M_\\PH}^2} \n+T_\\mathswitchr h \\frac{\\left(\\mathswitch {M_\\PAO}^2-\\mathswitch {M_\\Ph}^2\\right) c_{\\alpha -\\beta }}{\\mathswitch {M_\\Ph}^2}\n\\right]\n\\nonumber\\\\\n&= -\\frac{e}{2 \\mathswitch {s_{\\scrs\\PW}} \\mathswitch {M_\\PW}} \\left(\nT_\\mathswitchr H \\frac{ s_{\\alpha -\\beta }}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{ c_{\\alpha -\\beta }}{\\mathswitch {M_\\Ph}^2} \n\\right).\n\\end{align}\nThe counterterm $\\delta\\beta^t\\big|_\\mathrm{FJ}$ of the FJ scheme in the $t$-variant,\nthus, reads\n\\begin{align}\n\\delta \\beta^t\\big|_\\mathrm{FJ} &= \\delta \\beta +\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{UV}\n= \\delta \\beta \n-\\left.\\frac{e}{2 \\mathswitch {s_{\\scrs\\PW}} \\mathswitch {M_\\PW}} \\left(\nT_\\mathswitchr H \\frac{ s_{\\alpha -\\beta }}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{ c_{\\alpha -\\beta }}{\\mathswitch {M_\\Ph}^2} \n\\right)\\right|_\\mathrm{UV},\n\\label{eq:dbtFJ_expl}\n\\end{align}\nwhich is in agreement with Ref.~\\cite{Krause:2016oke,Denner:2016etu}. \nThis translates to our treatment of tadpoles as\n\\begin{align}\n\\label{eq:dbFJ_expl}\n\\delta \\beta\\big|_\\mathrm{FJ} &= \\delta \\beta \n-\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{finite}\n= \\delta \\beta\n+ \\left.\\frac{e}{2 \\mathswitch {s_{\\scrs\\PW}} \\mathswitch {M_\\PW}} \\left(\nT_\\mathswitchr H \\frac{ s_{\\alpha -\\beta }}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{ c_{\\alpha -\\beta }}{\\mathswitch {M_\\Ph}^2} \n\\right)\\right|_\\mathrm{finite},\n\\end{align}\nwhere again ``finite'' means that $\\Delta_\\mathrm{UV}$ is set to zero in the tadpole contribution.\nUsing this counterterm, it is possible to keep the form of the\ncounterterm Lagrangian derived \nin Sect.~\\ref{sec:CTLagrangian}\nto obtain results in the gauge-independent\nFJ scheme, although the above counterterm Lagrangian employs a gauge-dependent\n(but very convenient) tadpole renormalization.\n\nAlternatively, $\\delta\\beta$ could be fixed by an analogous consideration\nof the $\\mathswitchr Z\\mathswitchr {A_0}$ mixing, leading to \n\\begin{align}\n\\delta\\beta|_\\mathrm{UV}\n=\\left[ \\frac{s_{2\\beta}}{4c_{2\\alpha}}\\left(\\delta Z_{\\mathswitchr H}-\\delta Z_{\\mathswitchr h}\\right) \n+ \\frac{\\Sigma^{\\mathswitchr Z\\mathswitchr {A_0}}(k^2)}{\\mathswitch {M_\\PZ}} \\right]_\\mathrm{UV},\n\\end{align}\nwhich is independent of $k^2$ and does not use relation~\\refeq{eq:vacrencond}.\nThe transition to the FJ scheme then simply amounts to replacing the\none-particle-irreducible self-energy $\\Sigma^{\\mathswitchr Z\\mathswitchr {A_0}}$ by\n$\\Sigma^{t,\\mathswitchr Z\\mathswitchr {A_0}}$, which includes tadpole diagrams. \nThe result $\\delta \\beta^t\\big|_\\mathrm{FJ}$ of this procedure\nis again given by Eq.~\\refeq{eq:dbtFJ_expl}, as it should be.\n\n\\paragraph{The renormalization constant \\boldmath{$\\delta \\alpha\\big|_\\mathrm{FJ}$}:}\nWe apply the same method to the renormalization constant \n$\\delta \\alpha\\big|_\\mathrm{FJ}$, starting from Eq.~\\refeq{eq:deltaalphaderiv2}.\nThe difference between $\\delta \\alpha$ and $\\delta \\alpha^t$ is entirely given\nby the explicit tadpole diagrams that appear in the change from \n$\\Sigma^{\\mathswitchr H\\mathswitchr h}$ to $\\Sigma^{t,\\mathswitchr H\\mathswitchr h}$ in Eq.~\\refeq{eq:deltaalphaderiv2},\n\\begin{align}\n\\Delta\\alpha^t(T_\\mathswitchr H,T_\\mathswitchr h) &= \\delta\\alpha^t-\\delta\\alpha\n= \\left. \\mathrm{Re}\\frac{\\Sigma^{t,\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\PH}^2)+\\Sigma^{t,\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)}{2(\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2)}\n\\right|_\\mathrm{T_\\mathswitchr H,T_\\mathswitchr h},\n\\end{align}\nwhich evaluates to\n\\begin{align}\n\\Delta\\alpha^t(T_\\mathswitchr H,T_\\mathswitchr h) \n& = \\left. \\mathrm{Re}\\frac{\\Sigma^{t,\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)}{\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2} \\right|_\\mathrm{T_\\mathswitchr H,T_\\mathswitchr h}\n=\\frac{1}{\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2}\\left[\\parbox[c][1cm]{2.5cm}{\\includegraphics[scale=1]{.\/diagrams\/tadpoleSEhHh2.eps}}+\n\\parbox[c][1cm]{2.5cm}{\\includegraphics[scale=1]{.\/diagrams\/tadpoleSEHHh1.eps}} \\right]\n\\nonumber\\\\\n&=\\frac{e}{\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2} \\left( \nT_\\mathswitchr H \\frac{C_{\\mathswitchr h\\mathswitchr H\\PH}}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{C_{\\mathswitchr h\\Ph\\mathswitchr H}}{\\mathswitch {M_\\Ph}^2} \n\\right),\n\\end{align}\nwith the coupling factors of the $\\mathswitchr h\\mathswitchr H\\PH$ and $\\mathswitchr h\\Ph\\mathswitchr H$ vertices\n\\begin{subequations}\n\\begin{align}\nC_{\\mathswitchr h\\mathswitchr H\\PH} &=   \\frac{e s_{\\beta-\\alpha} }{2\\mathswitch {M_\\PW}\\mathswitch {s_{\\scrs\\PW}} s_{2\\beta}}\\biggl[ \n-(3s_{2\\alpha}+s_{2\\beta})\n\\left(\\mathswitch {M_\\PAO}^2 +4 \\lambda_5 \\frac{\\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2}{e^2}\\right)\n+ s_{2 \\alpha} \\left(\\mathswitch {M_\\Ph}^2 + 2 \\mathswitch {M_\\PH}^2\\right)  \\biggr], \n\\\\\nC_{\\mathswitchr h\\Ph\\mathswitchr H} &= \\frac{ec_{\\beta-\\alpha}}{2\\mathswitch {M_\\PW}\\mathswitch {s_{\\scrs\\PW}} s_{2\\beta}}  \\biggl[\n(3s_{2\\alpha}-s_{2\\beta})\n\\left( \\mathswitch {M_\\PAO}^2 + 4 \\lambda_5 \\frac{\\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2}{e^2}\\right)\n- s_{2\\alpha}\\left(2 \\mathswitch {M_\\Ph}^2  + \\mathswitch {M_\\PH}^2  \\right) \\biggr].\n\\end{align}\n\\end{subequations}\nThe counterterm $\\delta\\alpha^t\\big|_\\mathrm{FJ}$ of the FJ scheme in the $t$-variant,\nthus, reads\n\\begin{align}\n\\delta \\alpha^t\\big|_\\mathrm{FJ} &= \\delta \\alpha +\\Delta\\alpha^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{UV}\n= \\delta \\alpha \n+\\frac{e}{\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2} \\left( \nT_\\mathswitchr H \\frac{C_{\\mathswitchr h\\mathswitchr H\\PH}}{\\mathswitch {M_\\PH}^2}+\nT_\\mathswitchr h \\frac{C_{\\mathswitchr h\\Ph\\mathswitchr H}}{\\mathswitch {M_\\Ph}^2} \n\\right)\\bigg|_\\mathrm{UV},\n\\label{eq:datFJ_expl}\n\\end{align}\nwhich is again in agreement with Ref.~\\cite{Krause:2016oke,Denner:2016etu}. \nThis translates to our treatment of tadpoles as\n\\begin{align}\n\\delta \\alpha\\big|_\\mathrm{FJ} &= \\delta \\alpha \n-\\Delta\\alpha^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{finite}\n= \\delta \\alpha\n+\\frac{e}{\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2} \\left( \nT_\\mathswitchr H \\frac{C_{\\mathswitchr h\\mathswitchr H\\PH}}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{C_{\\mathswitchr h\\Ph\\mathswitchr H}}{\\mathswitch {M_\\Ph}^2} \n\\right)\\bigg|_\\mathrm{finite}.\n\\end{align}\nConcerning the use of $\\delta \\alpha\\big|_\\mathrm{FJ}$ in our counterterm Lagrangian\nto obtain renormalized amplitudes in the gauge-independent FJ scheme,\nthe same comments made above for $\\delta\\beta\\big|_\\mathrm{FJ}$ apply.\n\n\\subsubsection{\\boldmath{The FJ($\\lambda_3$) scheme}}\n\\label{sec:FJL3scheme}\n\nIn the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme, the parameters $\\lambda_{3}$ \nand $\\lambda_{5}$ are defining parameters of the basic para\\-meterization and gauge independent by construction. Therefore, the condition on $\\delta \\beta$ is the only renormalization condition potentially being gauge dependent. To provide a fully\ngauge-independent renormalization scheme where $\\lambda_3$ is an independent quantity, we apply the FJ scheme to the parameter $\\beta$ and keep the renormalization of \n$\\lambda_{3}$ and $\\lambda_{5}$ \nas in the $\\overline{\\mathrm{MS}}(\\lambda_3)$. We call the resulting scheme the FJ($\\lambda_3$) scheme. The renormalization of the parameters reads:\n\\begin{align}\n \\delta \\beta\\big|_\\mathrm{FJ}\\text{ as in Eq.~\\eqref{eq:dbFJ_expl}},\\nonumber\\\\\n \\delta \\lambda_3 \\text{ as in Eqs.~\\eqref{eq:deltaL3}}{-}\\refeq{eq:L3bos}.\\nonumber\n\\end{align}\n\n\\subsection{Conversion between different renormalization schemes}\n\\label{sec:inputconversion}\n\nIn the previous section, we have\npresented four different renormalization schemes, which treat the mixing parameters differently. When observables calculated in different renormalization schemes are compared, particular care has to be taken that the input parameters are consistently translated from one scheme to the other. The bare values of identical independent\nparameters are equal and independent of the renormalization scheme. \nExemplarily, for a parameter $p$, the renormalized values $p^{(1)}$ and $p^{(2)}$ in two different renormalization schemes~1 and 2 are connected via the bare parameter $p_0$, \n\\begin{align}\np_0=p^{(1)}+\\delta p^{(1)} (p^{(1)})=p^{(2)} +\\delta p^{(2)} (p^{(2)}),\n\\end{align}\nwithin the considered order. \nIf $p$ is a dependent parameter in one or both schemes, it must be calculated from the independent renormalized para\\-meters and their counterterms from the relations between bare and renormalized quantities. For converting an input value from one scheme to another, one can solve for one renormalized quantity\n\\begin{align}\n\\label{eq:convertinputparam}\n p^{(1)}=p^{(2)} +\\delta p^{(2)} (p^{(2)})-\\delta p^{(1)} (p^{(1)}).\n\\end{align}\nAt NLO, this equation can be linearized by substituting the input value of $p^{(1)}$ by $p^{(2)}$ \nin the computation of the last counterterm. The differences to an exact solution are of higher order and beyond our desired NLO\naccuracy. However, large counterterms or small tree-level values can spoil the approximation so that in this case a proper solution using numerical techniques could improve the results. \nAnother benefit of a full solution of the implicit equation is the possibility that one can \nswitch to another scheme and back in a self-consistent way, while start and end scenarios\nin scheme~(1) do not exactly coincide when switching from scheme~(1) to (2) and back to (1)\nusing the linearized approximation.\nThe comparison of both methods allows for a consistency check of the computation and for an analysis of perturbative stability.\nWe have derived the Higgs mixing angles $\\alpha$ and $\\beta$ and their counterterm in all schemes. The finite parts of the gauge-dependent counterterms $\\delta \\alpha$, $\\delta \\beta$ are given here for the different renormalization schemes indicated by the respective index: \n\\begin{subequations}\n\\label{eq:fintermsalpha}\n\\begin{align}\n \\delta \\alpha\\big|_{\\overline{\\mathrm{MS}}(\\alpha),\\mathrm{finite}} &= 0,\\\\\n \\delta \\alpha\\big|_{\\overline{\\mathrm{MS}}(\\lambda_3),\\mathrm{finite}} \n&= \\delta \\alpha\\big|_{\\mathrm{FJ}(\\lambda_3),\\mathrm{finite}} \n= \\frac{f_\\alpha\\{\\delta p'_\\mathrm{mass}\\}}{\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2}\\Big|_\\mathrm{finite},\\\\\n \\delta \\alpha\\big|_{\\mathrm{FJ}(\\alpha),\\mathrm{finite}} &= -\\Delta\\alpha^t(T_\\mathswitchr H,T_\\mathswitchr h)\\big|_\\mathrm{finite}\n=\\left.\\frac{e}{\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2} \\left( \nT_\\mathswitchr H \\frac{C_{\\mathswitchr h\\mathswitchr H\\PH}}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{C_{\\mathswitchr h\\Ph\\mathswitchr H}}{\\mathswitch {M_\\Ph}^2} \n\\right)\\right|_\\mathrm{finite}.\n\\end{align}\n\\end{subequations}\nFor the angle $\\beta$ we obtain the following finite terms \nin the $\\overline{\\mathrm{MS}}$ and the FJ schemes,\n\\begin{subequations}\n\\label{eq:fintermsbeta}\n\\begin{align}\n \\delta \\beta\\big|_{\\overline{\\mathrm{MS}(\\alpha)},\\mathrm{finite}} \n&= \\delta \\beta\\big|_{\\overline{\\mathrm{MS}}(\\lambda_3),\\mathrm{finite}} \n= 0,\\\\\n\\delta \\beta\\big|_{\\mathrm{FJ}(\\alpha),\\mathrm{finite}} \n&= \\delta \\beta\\big|_{\\mathrm{FJ(\\lambda_3)},\\mathrm{finite}} \n = -\\Delta\\beta^t(T_\\mathswitchr H,T_\\mathswitchr h)\\big|_\\mathrm{finite}=\\left.\\frac{e}{2 \\mathswitch {s_{\\scrs\\PW}} \\mathswitch {M_\\PW}} \\left(\nT_\\mathswitchr H \\frac{ s_{\\alpha -\\beta }}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{ c_{\\alpha -\\beta }}{\\mathswitch {M_\\Ph}^2} \n\\right)\\right|_\\mathrm{finite}.\n\\end{align}\n\\end{subequations}\nWith these formulae we can convert\nthe input variables for $\\alpha$ and $\\beta$\neasily into each other. \nFor instance, the conversion of the input values of $\\alpha$ and $\\beta$ \ndefined in the $\\overline{\\mathrm{MS}}(\\alpha)$ scheme into \nthe other renormalization schemes reads\n\\begin{subequations}\n\\begin{align}\n\\alpha\\big|_{\\overline{\\mathrm{MS}}(\\lambda_3)} \n& = \\alpha\\big|_{\\overline{\\mathrm{MS}}(\\alpha)}\n+\\left.\\frac{f_\\alpha\\{\\delta p'_\\mathrm{mass}\\}}{\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2}\\right|_\\mathrm{finite}, &\n\\beta\\big|_{\\overline{\\mathrm{MS}}(\\lambda_3)} &=\n\\beta\\big|_{\\overline{\\mathrm{MS}}(\\alpha)},\n\\label{eq:convertinput1}%\n\\\\[.3em]\n\\alpha\\big|_{\\mathrm{FJ}(\\alpha)}\n& = \\alpha\\big|_{\\overline{\\mathrm{MS}}(\\alpha)}\n+\\Delta\\alpha^t(T_\\mathswitchr H,T_\\mathswitchr h)\\big|_\\mathrm{finite}, &\n\\beta\\big|_{\\mathrm{FJ}(\\alpha)}\n& = \\beta\\big|_{\\overline{\\mathrm{MS}}(\\alpha)}\n+\\Delta\\beta^t(T_\\mathswitchr H,T_\\mathswitchr h)\\big|_\\mathrm{finite},\n\\label{eq:convertinput2}%\n\\\\[.3em]\n\\alpha\\big|_{\\mathrm{FJ}(\\lambda_3)}\n& = \\alpha\\big|_{\\overline{\\mathrm{MS}}(\\alpha)}\n+\\left.\\frac{f_\\alpha\\{\\delta p'_\\mathrm{mass}\\}}{\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2}\\right|_\\mathrm{finite}, & \n\\beta\\big|_{\\mathrm{FJ}(\\lambda_3)}\n& = \\beta\\big|_{\\overline{\\mathrm{MS}}(\\alpha)}\n+\\Delta\\beta^t(T_\\mathswitchr H,T_\\mathswitchr h)\\big|_\\mathrm{finite}.\n\\label{eq:convertinput3}%\n\\end{align}\n\\end{subequations}%\nWithin a given scheme, $\\lambda_3$ and $\\alpha$ can be translated into each other\nusing the tree-level relation~\\refeq{eq:lambda3tree}.\nNote that, thus, the numerical values of $\\alpha$, $\\beta$, and $\\lambda_3$ corresponding\nto a given physical scenario of the THDM are different in different renormalization schemes.\nIn turn, fixing the input values in the four renormalization schemes to the same values\ncorresponds to different physical scenarios.\nIn particular, this means that the ``alignment limit'', in which $s_{\\beta-\\alpha}\\to1$\nso that $\\mathswitchr h$ is SM like \n{(see, e.g., \\citeres{Haber:1995be,Gunion:2002zf,Bernon:2015qea}),} is a notion that depends on the renormalization scheme\n(actually even on the scale choice in a given scheme).%\n\\footnote{In this brief account of results, we do not consider the (phenomenologically\ndisfavoured, though not excluded) possibility that the heavier CP-even Higgs boson $\\mathswitchr H$ is SM-like,\nwhich is discussed in detail in \\citere{Bernon:2015wef}.}\n\nExemplarily, the \nconversions of $c_{\\beta-\\alpha}$  from the {$\\overline{\\mathrm{MS}}$}{}$(\\alpha)$ scheme into the $\\overline{\\mathrm{MS}}(\\lambda_3)$ (green), FJ$(\\alpha)$ (pink), and \nFJ$(\\lambda_3)$ (turquoise) schemes are shown in Fig.~\\ref{fig:plot_Umrechnung-von-alpha-LM}. The results of the transformations \nin the inverse directions are displayed in Fig.~\\ref{fig:plot_Umrechnung-nach-alpha-LM}, and all other conversions can be seen as a combination of the presented ones. \n\\begin{figure}\n\\centering\n\\subfigure[]{\n\\label{fig:plot_Umrechnung-von-alpha-LM}\n\\includegraphics{.\/diagrams\/plot_Umrechnung-von-alpha-LM.eps}\n}\n\\hspace{15pt}\n\\subfigure[]{\n\\label{fig:plot_Umrechnung-nach-alpha-LM}\n\\includegraphics{.\/diagrams\/plot_Umrechnung-nach-alpha-LM.eps}\n}\n\\vspace*{-.5em}\n\\caption{(a) Conversion of the value of $c_{\\beta-\\alpha}$ from $\\overline{\\mathrm{MS}}(\\alpha)$ \nto the $\\overline{\\mathrm{MS}}(\\lambda_3)$ (green), FJ$(\\alpha)$ (pink), and FJ($\\lambda_3$) \nschemes (turquoise) for scenario~A. Panel~(b) shows the conversion to the $\\overline{\\mathrm{MS}}(\\alpha)$ scheme using the same colour coding. \nThe solid lines are obtained \nby solving the implicit equations \\refeq{eq:convertinputparam} numerically, the dashed lines\ncorrespond to the linearized approximation.\nThe phenomenologically relevant region is highlighted in the centre.}\n\\label{fig:plotconversionA}%\n\\end{figure}%\nThe input values (defined before the conversion) correspond to the low-mass scenario called ``A''\nof a THDM of Type~I (based on a benchmark scenario of Ref.~\\cite{Haber2015})\nwith  \n\\begin{align}\n\\mathswitch {M_\\Ph}=125 \\text{ GeV}, \\quad\n \\mathswitch {M_\\PH}=300 \\text{ GeV}, \\quad \\mathswitch {M_\\PAO}=\\mathswitch {M_\\PHP}=460 \\text{ GeV}, \\quad \\lambda_5=-1.9,\\quad \\tan \\beta=2. \n\\label{eq:scenario}\n \\end{align}\nSpecifically, scenario~A is a scan in $c_{\\beta-\\alpha}$ in the mass parameterization, \nAa and Ab are points of the scan region used to analyze the scale dependence:\n\\begin{subequations}\n\\begin{align}\n\\mbox{A:}  \\quad \\cos{(\\beta-\\alpha)} &= -0.2\\ldots0.2, \\\\\n\\mbox{Aa:} \\quad \\cos{(\\beta-\\alpha)} &= +0.1, \\\\\n\\mbox{Ab:} \\quad \\cos{(\\beta-\\alpha)} &= -0.1. \n\\end{align}\n\\label{eq:cba_A}\n\\end{subequations}\nThe {$\\overline{\\mathrm{MS}}$}{} parameters are defined at the scale\n\\begin{align}\n\\label{eq:centralscale}\n \\mu_0=\\frac{1}{5}(\\mathswitch {M_\\Ph}+\\mathswitch {M_\\PH}+\\mathswitch {M_\\PAO}+2\\mathswitch {M_\\PHP}).\n\\end{align}\nThe motivation for this choice will become clear below.\nThe remaining input parameters for the SM part are given in \nApp.~\\ref{app:smparams}.\nIn both plots, we highlight the phenomenologically relevant region in the centre. \nThe solid lines are the result obtained \nby solving the implicit equations \\refeq{eq:convertinputparam} numerically,\nthe dashed lines correspond to a linearized conversion.\nAll curves show only minor conversion effects in the parameter values, i.e.\\ the \nsolution of the implicit equations\nagrees well with the approximate linearized conversion, affirming that the contributions of the\nhigher-order functions $\\Delta\\alpha^t$, $\\Delta\\beta^t$, and $f_\\alpha$ of \nEqs.~\\refeq{eq:convertinput1}--\\refeq{eq:convertinput3}\nare small, and perturbation theory is applicable. Since the values of the parameters change when going from one renormalization scheme to another, \nthe alignment limit does not persist in these transformations, i.e.\\\nin this scenario the alignment limit sensitively depends on the definition of the parameters at NLO.\n\n{For the schemes with $\\lambda_3$ as input parameter, some singular behaviour in the parameter\nconversion can be observed in the phenomenologically disfavoured region where $c_{\\beta-\\alpha}\\lsim-0.3$.\nThis artifact in the conversion appears when $c_{2\\alpha}\\to0$ (see, e.g., Eq.~\\refeq{eq:MHhdependence}),\nindicating the breakdown of the $\\overline{\\mathrm{MS}}(\\lambda_3)$ and FJ($\\lambda_3$) schemes in such\nparameter regions. Already this case-specific study shows that stability issues of different\nrenormalization schemes have to be carefully carried out for all interesting parameter regions\nand that the applicability of a specific scheme in general does not cover the full THDM \nparameter space, a fact that was also pointed out in \\citere{Krause:2016oke}\nfor the THDM and that is known from NLO calculations in the MSSM (see, e.g., \\citeres{Heinemeyer:2010mm,Chatterjee:2011wc}).\nSpecifically, if the $\\overline{\\mathrm{MS}}(\\lambda_3)$ and FJ($\\lambda_3$) schemes are not\napplicable in a region that might be favoured by future data analyses, it would\nbe desirable and straightforward to replace $\\lambda_3$ by $\\lambda_1$ or $\\lambda_2$\nas independent parameter, thereby defining analogous schemes like $\\overline{\\mathrm{MS}}(\\lambda_1)$, etc..\n\nTo address this issue properly, was our basic motivation to introduce and compare different renormaliztion\nschemes. We will continue this discussion in more detail in a forthcoming publication,\nwhere further THDM scenarios are considered.}\n\n\\section{\\boldmath{The running of the {$\\overline{\\mathrm{MS}}$}{} parameters}}\n\\label{sec:running}\n\nParameters renormalized in the $\\overline{\\mathrm{MS}}$ scheme depend on an unphysical renormalization scale $\\mu_\\mathrm{r}$. The one-loop $\\beta$-function of a parameter $p$  can be obtained from the UV-divergent parts of its \ncounterterm $\\delta p$, \n\\begin{align}\n\\beta_p(\\mu^2_\\mathrm{r})=\\frac{\\partial}{\\partial \\ln{\\mu_\\mathrm{r}^2}} p(\\mu_\\mathrm{r}^2)\n=\\frac{\\partial}{\\partial \\Delta_\\mathrm{UV}} \\delta p.\n\\label{eq:betafct}\n\\end{align}\nSince the renormalization constants are computed in a perturbative manner, the $\\beta$-functions have a perturbative expansion in the coupling parameters.\nNote that the last equality in Eq.~\\refeq{eq:betafct} holds in the FJ schemes\nfor $\\alpha$ and $\\beta$\nonly in the $t$-variant explained above, because the finite contributions\n$\\Delta\\alpha^t$ and $\\Delta\\beta^t$ depend on the scale $\\mu_\\mathrm{r}$.\n\nAs discussed in the previous sections, \nthe ratio of the vevs, $\\tan \\beta$, the Higgs mixing parameter $\\alpha$ or $\\lambda_3$, and the Higgs self-coupling $\\lambda_5$ are renormalized in the $\\overline{\\mathrm{MS}}$ scheme. For each renormalization scheme described in Sect.~\\ref{sec:diffrenschemes}, one obtains a set of coupled RGEs involving the $\\beta$-functions of the independent parameters. Therefore, the scale dependence varies when different schemes are applied. In the perturbative expansion of the $\\beta$-function we consider only the one-loop term, being second order in the coupling constants,\ne.g., in the {$\\overline{\\mathrm{MS}}$}$(\\alpha)$ scheme\n\\begin{align}\n\\beta_p(\\mu^2_\\mathrm{r})= A_p \\alpha_\\mathrm{em}+ B_p \\lambda_5+C_p \\lambda_5^2\/\\alpha_\\mathrm{em}.\n\\end{align}\nThe dependence on the strong coupling constant vanishes at one-loop order as the parameters renormalized in {$\\overline{\\mathrm{MS}}$}{} appear only in couplings of particles that do not interact strongly. The coefficients $A_p, B_p, C_p$ of the respective renormalized parameter can be easily read \nfrom the divergent terms which have been derived in the previous section. We have checked them against the $\\beta$-functions given for $\\lambda_{3}$ and $\\lambda_{5}$ in Ref.~\\cite{Branco:2011iw} and for $\\beta$ in Ref.~\\cite{Sperling:2013eva} (supersymmetric contributions need to be omitted).\n\nIn general, RGEs, which are a set of coupled differential equations, cannot be solved analytically. \nUsually numerical techniques, such as a Runge--Kutta method, need to be employed to solve the RGEs and to compute the values of the parameters at a desired scale.\nMoreover, we emphasize that the renormalization-group flow of a running parameter\ndepends on the renormalization scheme of the full set of independent parameters.\nThat means the fact that we use on-shell quantities, such as all the Higgs-boson masses,\nto fix most of the scalar self-couplings has a significant impact on the running\nof our {$\\overline{\\mathrm{MS}}$}{} parameters.\nThe renormalization-group flow in other schemes was, e.g., investigated in\n\\citeres{Cvetic:1997zd,Branco:2011iw,Bijnens:2011gd,Chakrabarty:2014aya,Ferreira:2015rha}.\n\nThe scale dependence of $c_{\\beta-\\alpha}$ \nfor $\\mu = 100{-}900$~GeV is plotted in Fig.~\\ref{fig:plotrunningA}, for the scenario defined in Eq.~\\eqref{eq:scenario} with $c_{\\beta-\\alpha}=0.1$  (l.h.s) and $c_{\\beta-\\alpha}=-0.1$ (r.h.s) and input values given at the central scale $\\mu_0$ stated in\nEq.~\\refeq{eq:centralscale}. \nWe observe that the choice of  the renormalization scheme has a large impact on the scale dependence. While the $\\overline{\\mathrm{MS}}(\\alpha)$ scheme introduces only a mild running, the other schemes show a much stronger scale dependence, so that excluded and unphysical values of input parameters can be reached quickly. A similar observation has also been made in supersymmetric models for the parameter~$\\tan\\beta$~\\cite{Freitas:2002um}. \nGauge-dependent $\\overline{\\mathrm{MS}}$ schemes have a small scale dependence while replacing the parameters by gauge-independent ones like in the FJ schemes introduce additional terms in the $\\beta$-functions which induce a stronger scale dependence.\nIn Fig~\\ref{fig:running_cba-0.1-LM} one can also see that the curves for the $\\overline{\\mathrm{MS}}(\\lambda_3)$ and the FJ($\\lambda_3$) schemes terminate around 250 GeV. \n\\begin{figure}\n  \\centering\n  \\subfigure[]{\n\\label{fig:running_cba0.1-LM}\n\\includegraphics{.\/diagrams\/running_cba0.1-LM.eps}\n}\n\\hspace{15pt}\n\\subfigure[]{\n\\label{fig:running_cba-0.1-LM}\n\\includegraphics{.\/diagrams\/running_cba-0.1-LM.eps}\n}\n\\vspace*{-.5em}\n  \\caption{The running of $c_{\\beta-\\alpha}$ for the low-mass scenario~A with $c_{\\beta-\\alpha}=0.1$ (a) and $c_{\\beta-\\alpha}=-0.1$ (b) in the $\\overline{\\mathrm{MS}}(\\alpha)$ (blue), $\\overline{\\mathrm{MS}}(\\lambda_3)$ (green), FJ($\\alpha)$ (pink), and FJ($\\lambda_3$) (turquoise) schemes.}\n\\label{fig:plotrunningA}\n\\end{figure}\nAt this scale, the running of $\\lambda_3$ yields unphysical values for which \nEq.~\\refeq{eq:lambda3tree}\nwith the given Higgs masses becomes \noverconstrained, and no solution with $|s_{2\\alpha}|\\le1$ exists. This is unique to the $\\lambda_3$ running as only there an \nimplicit equation needs to be solved to obtain the input parameter $\\alpha$. For the other cases we prevent the angles from running out of their domain of definition by solving the running for the tangent function of the angles.\n\n\\section{Implementation into a \\textsc{FeynArts} Model File}\n\\label{sec:FAmodelfile}\nThe \\textsc{Mathematica} package \\textsc{FeynRules} (FR)~\\cite{Christensen2009} \nis a tool to generate Feynman rules from a given \nLagrangian, providing the possibility to produce model files in various output formats\nwhich can be employed by automated amplitude generators. \nWe have inserted the Lagrangian into FR in its internal notation to obtain\nthe corresponding counter\\-term Lagrangian after the renormalization transformations. \nBefore this insertion, we have computed and simplified  \nthe Higgs potential and the corresponding counterterm potential~\\eqref{eq:dVH2p} \nwith inhouse \\textsc{Mathematica} routines.\nUsing FR,\nthe tree-level and the counterterm Feynman rules as well as the renormalization conditions \nin the {$\\overline{\\mathrm{MS}}$}{}$(\\alpha)$ and {$\\overline{\\mathrm{MS}}$}{}($\\lambda_3$) schemes have been implemented into \na model file for the amplitude generator \\textsc{FeynArts} (FA)~\\cite{Hahn2001}. \nThe renormalization conditions of the FJ($\\alpha)$ and the FJ($\\lambda_3$) have not \nbeen included in the model file, because using \nEqs.~\\eqref{eq:fintermsalpha},~\\eqref{eq:fintermsbeta} it is straightforward to \nimplement the corresponding finite terms of $\\delta \\alpha$ and $\\delta \\beta$. \nWith such a model file, NLO amplitudes for any process can be generated in an automated way. \n\nThe FA NLO model file for the THDM, obtained with FR, has the following features:\n\\begin{itemize}\n\\item Type I, II, flipped, or lepton-specific THDM;\n\\item all tree-level and counterterm Feynman rules;\n\\item renormalization conditions according to the {$\\overline{\\mathrm{MS}}$}{}$(\\alpha)$ and {$\\overline{\\mathrm{MS}}$}{}$(\\lambda_3)$  schemes;\n\\item all renormalization constants are implemented additionally in $\\overline{\\mathrm{MS}}$ \nas well, which allows for fast checks of UV-finiteness;\n\\item BHS and HK conventions;\n\\item\nCKM matrix set to the unit matrix (the generalization is straightforward).\n\\end{itemize}\nThis model file has been tested intensively, including checks of UV-finiteness for \nseveral processes, both numerically and analytically. \nThis allows for the generation of amplitudes \n(and further processing with \\textsc{FormCalc} \\cite{Hahn1999}) for any process at the one-loop level, at any parameter point of the THDM.\nThe model file can be obtained from the authors upon request.\n\n\\section{\\boldmath{Numerical results for \\boldmath$ \\mathswitchr h \\to \\mathswitchr W\\PW\/\\mathswitchr Z\\PZ \\to 4f$}}\n\\label{se:numerics}\n\nIn this section we present first results from the computation of the decay of the light, neutral \nCP-even Higgs boson of the THDM into four fermions at NLO. The computer program \n\\textsc{Prophecy4f}~\\cite{Bredenstein:2006rh,Bredenstein:2006nk,Bredenstein:2006ha}%\n\\footnote{\\tt http:\/\/prophecy4f.hepforge.org\/index.html}\nprovides a ``\\textbf{PROP}er description of the \\textbf{H}iggs d\\textbf{EC}a\\textbf{Y} into \\textbf{4 F}ermions'' and calculates \nobservables for the decay process $\\mathswitchr h {\\to} \\mathswitchr W\\PW\/\\mathswitchr Z\\PZ {\\to} 4 f$ at NLO EW+QCD in the SM. \nWe have extended this program to the calculation of the corresponding decay in the THDM in such a way that the usage of the program and its applicability as event\ngenerator basically remains the same. \n{Owing to the fact that LO and real-emission amplitudes in the THDM receive only the\nmultiplicative factor $s_{\\beta-\\alpha}$ with respect to the SM, the bremsstrahlung corrections as well\nas the treatment of infrared singularities could be taken over from the SM \ncalculation~\\cite{Bredenstein:2006rh,Bredenstein:2006ha}\nvia simple rescaling.}\nThe calculation in the THDM, the implementation in  \\textsc{Prophecy4f}, as well as results of the application will be described in detail in an upcoming publication.\nWe just mention that we employ the complex-mass scheme~\\cite{Denner:2005fg} to describe the \nW\/Z~resonances, as already done in \\citeres{Bredenstein:2006rh,Bredenstein:2006nk,Bredenstein:2006ha}\nfor the SM. Note that the W\/Z-boson masses as well as the weak mixing angle are consistently taken as complex quantities in the complex-mass scheme to guarantee gauge invariance of all amplitudes\nin resonant and non-resonant phase-space regions. \nConsequently all our renormalization constants of the THDM inherit imaginary parts from the complex\ninput values, but the impact of these spurious imaginary parts is beyond NLO and negligible\n(as in the SM). Moreover, we mention that the modified version of \\textsc{Prophecy4f}\nmakes use of the public \\textsc{Collier} library~\\cite{Denner:2016kdg} for the calculation \nof the one-loop integrals.\n{Apart from performing two independent loop calculations, we have verified\nour one-loop matrix elements by numerically comparing our results to\nthe ones obtained in \\citere{Denner:2016etu} for the related\n$\\mathswitchr W\\mathswitchr h\/\\mathswitchr Z\\mathswitchr h$ production channels (including $\\mathswitchr W\/\\mathswitchr Z$ decays)\nusing crossing symmetry.}\n\nIn this paper, we present first results in order to demonstrate the use and the \nself-consistency of our renormalization schemes, employing again the scenario\ninspired by the first benchmark scenario of Ref.~\\cite{Haber2015} where the additional \nHiggs bosons are not very heavy. \nThe input values of the THDM parameters \n{for a Type~I THDM}\nare given in Eqs.~\\refeq{eq:scenario} and \n\\refeq{eq:cba_A}.\nSince $c_{\\beta-\\alpha}$ is the only parameter of the THDM appearing at LO, \nour process is most sensitive to this parameter. We vary $c_{\\beta-\\alpha}$ \nin the range $[-0.2,+0.2]$ in scenario~A for the computation of \nthe partial decay width for $\\mathswitchr h\\to\\mathswitchr W\\PW\/\\mathswitchr Z\\PZ\\to4f$, $\\Gamma^{\\mathswitchr h\\to4f}_\\mathrm{THDM}$,\nwhich is obtained by summing the partial widths of the h~boson\nover all massless four-fermion final states $4f$.\nThe parameters of the SM part of the THDM are collected in App.~\\ref{app:smparams}.\n{Note that a non-trivial CKM matrix would not change our results,\nsince quark mass effects of the first two generations as well as mixing\nwith the third generation are completely negligible in the considered\ndecays.}\n\nTo perform scale variations we take two distinguished points named Aa and Ab with $c_{\\beta-\\alpha}=\\pm0.1$. \nFor the central renormalization scale we use the average mass \n$\\mu_0$ defined in Eq.~\\refeq{eq:centralscale}\nof all scalar degrees of freedom.\nThe scale $\\mu$ of $\\alpha_\\mathrm{s}$ is kept fixed at $\\mu=\\mathswitch {M_\\PZ}$ which is the appropriate scale for the QCD corrections (which are dominated by the hadronic W\/Z decays).\n\n\\subsection{Scale variation of the width}\n\\label{sec:LowMassscalevariation}\n\n{The running of the {$\\overline{\\mathrm{MS}}$}-renormalized parameters $\\alpha$ and $\\beta$ is\ninduced by the Higgs-boson self-energies (and some scalar vertex for $\\lambda_5$),\ni.e.\\ the relevant particles in the loops are all Higgs bosons, the W\/Z bosons,\nand the top quark. If all Higgs-boson masses are near the electroweak scale,\nsay $\\sim 100{-}200\\unskip\\,\\mathrm{GeV}$, \nwhere the W\/Z-boson and top-quark masses are located, \nthen the scale $\\mathswitch {M_\\Ph}$ turns out to be a reasonable scale, as expected.\n{However, if some heavy Higgs-boson masses increase to some generic mass\nscale $M_S$ and the mixing angle\n$\\beta-\\alpha$ stays away from the alignment limit, there is no \ndecoupling of heavy Higgs-boson effects, so that $M_S$ acts\nas generic UV cutoff scale appearing in logarithms $\\log(M_S\/\\mu_\\mathrm{r})$.\nThe renormalization scale $\\mu_\\mathrm{r}$ has to go up with $M_S$ to\navoid that the logarithm drives the correction unphysically large.\nThe optimal choice of $\\mu_\\mathrm{r}$, though, is somewhat empirical.}\nA good choice of the central scale $\\mu_0$ should come close to the\nstability point (plateau in the $\\mu_{r}$ variation) in the major part of THDM\nparameter space. Our choice \\eqref{eq:centralscale} of $\\mu_0$ \neffectively takes care of this and is eventually justified by the numerics.\n\nTo illustrate this and to estimate the theoretical uncertainties due to the\nresidual scale dependence, we compute the total width while the scale $\\mu_\\mathrm{r}$ is varied from $100{-}900\\unskip\\,\\mathrm{GeV}$.}\n{Results with central scale $\\mathswitch {M_\\Ph}$ are shown in App.~\\ref{app:mu0mh},\nproving that this would be not a good choice.}\nThe parameters $\\alpha$ and $\\beta$ are defined in the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme, and to compute results in other renormalization schemes their values are converted using\nEqs.~\\refeq{eq:convertinput1}--\\refeq{eq:convertinput3},\nwhich are solved numerically without linearization.\nThereafter the scale is varied, the RGEs solved, and the width computed using the respective renormalization scheme. \nThe results are shown \nin Fig.~\\ref{fig:plotmuscan} at LO (dashed) and NLO EW (solid) for the benchmark points Aa and Ab.\n\\begin{figure}\n  \\centering\n  \\subfigure[]{\n\\label{fig:plot_MUSCAN-Aa-L3MS}\n\\includegraphics{.\/diagrams\/plot_MUSCAN-Aa-L3MS.eps}\n}\n\\hspace{15pt}\n  \\subfigure[]{\n\\label{fig:plot_MUSCAN-Ab-alphaMS}\n\\includegraphics{.\/diagrams\/plot_MUSCAN-Ab-L3MS.eps}\n}\n\\caption{The decay width for $\\mathswitchr h \\to 4f$ at LO \n(dashed) and NLO EW (solid) \nin dependence of the renormalization scale with $\\beta$ and $\\alpha$ defined in the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme. The result is computed in all four different renormalization schemes after converting the input at NLO (also for the LO curves)\nand displayed for the benchmark points Aa~(a) and Ab~(b) using the colour code of Fig.~\\ref{fig:plotrunningA}.  }\n\\label{fig:plotmuscan}\n\\end{figure}\nThe QCD corrections are not part of the EW scale variation and therefore omitted in these results. The benchmark point Aa shows almost textbook-like behaviour with the LO computation \nexhibiting a strong scale dependence for all renormalization schemes, resulting in sizable differences between the curves. However, each of the NLO curves shows a wide extremum with a large plateau, reducing the scale dependence drastically, as it is expected for NLO calculations. \nThe central scale $\\mu_\\mathrm{r}=(\\mathswitch {M_\\Ph}+\\mathswitch {M_\\PH}+\\mathswitch {M_\\PAO}+2\\mathswitch {M_\\PHP})\/5$ lies perfectly in the middle of the plateau regions motivating this scale choice. In contrast, the naive scale choice $\\mu_0=\\mathswitch {M_\\Ph}$ is not within the plateau region, leads to large, unphysical corrections, and should not be chosen. The breakdown of the FJ($\\alpha)$ curve for small scales can be explained by the running which becomes unstable for these values (see Fig.~\\ref{fig:running_cba0.1-LM}).\nFor all renormalization schemes, the plateaus coincide and the agreement between the renormalization schemes is improved at NLO w.r.t.~the LO results. This is expected, since results obtained with different  renormalization schemes should be equal up to higher-order terms, after the input parameters are properly converted. \nThe relative renormalization scheme dependence at the central scale,\n\\begin{align}\n\\Delta_\\mathrm{RS}=2 \\,\\frac{\\Gamma^{\\mathswitchr h \\to 4f}_\\mathrm{max}(\\mu_0)-\\Gamma^{\\mathswitchr h \\to 4f}_\\mathrm{min}(\\mu_0)}{\\Gamma^{\\mathswitchr h \\to 4f}_\\mathrm{max}(\\mu_0)+\\Gamma^{\\mathswitchr h \\to 4f}_\\mathrm{min}(\\mu_0)},\n\\label{eq:renschemedep}\n\\end{align}\n expresses the dependence of the result on the renormalization scheme. It can be computed from the difference of the smallest and largest width \nin the four renormalization schemes normalized to their average. \nIn the calculation of $\\Delta_\\mathrm{RS}$, the full NLO EW+QCD corrections to the width\n$\\Gamma^{\\mathswitchr h \\to 4f}$ should be taken into account.\nIn Tab.~\\ref{tab:schemevar}, $\\Delta_\\mathrm{RS}$ is given at LO and NLO and confirms the reduction of the scheme dependence in the NLO calculation. In addition, as already perceived when the running was analyzed, the $\\overline{\\mathrm{MS}}(\\alpha)$ scheme shows the smallest dependence on the renormalization scale, which attests a good absorption of further corrections into the NLO prediction.\\\\\n  \\begin{table}\n   \\centering\n   \\renewcommand{\\arraystretch}{1.2}\n \\begin{tabular}{|c|cc|}\\hline\n  \n  & Scenario Aa & Scenario Ab\\\\\n    $\\Delta^\\mathrm{LO}_\\mathrm{RS}$[\\%]       & 0.67& 0.84  \\\\\n    $\\Delta^\\mathrm{NLO}_\\mathrm{RS}$ [\\%]     & 0.08 & 0.34  \\\\\\hline\n  \\end{tabular}  \n   \\caption{The variation $\\Delta_\\mathrm{RS}$\nof the $\\mathswitchr h {\\to} 4f$ width using different renormalization schemes \nfor input parameters defined in the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme.}\n \\label{tab:schemevar}\n \\end{table}%\nThe situation for the benchmark point Ab is more subtle. For negative values of $c_{\\beta-\\alpha}$ the truncation of the schemes involving $\\lambda_3$ at \n$\\mu_\\mathrm{r}=250{-}300$ GeV as well as the breakdown of the running of the FJ($\\alpha)$ scheme, which both were observed in the running in Fig.~\\ref{fig:running_cba-0.1-LM}, are also ma\\-ni\\-fest in the computation of the $\\mathswitchr h {\\to} 4f$ width.\nTherefore, the results vary much more, and the extrema with the plateau regions are not as distinct as for the benchmark point Aa. They are even missing for the truncated curves. Nevertheless, the situation improves at NLO. \nAs for scenario~Aa, the central scale choice of $\\mu_0$ \nis more appropriate in contrast than the choice of $\\mathswitch {M_\\Ph}$.\n \nFor both benchmark points, \nthe estimate of the theoretical uncertainties by varying the scale by a factor of two from the central value for an arbitrary renormalization scheme is generally not appropriate. \nA proper strategy would be to identify the renormalization schemes which yield reliable results, and to use only those to quantify the theoretical uncertainties from the scale variation. In addition, the renormalization scheme dependence of those schemes should be investigated. \nThis procedure should be performed for different parameter regions (and corresponding \nbenchmark points) separately, which is beyond the scope of this work. \n \n\\subsection{\\boldmath{$c_{\\beta-\\alpha}$} dependence}\n\\label{sec:cbascanA}\n\nThe decay width for $\\mathswitchr h \\to 4f$ in dependence of $c_{\\beta-\\alpha}$ in scenario~A is presented in Fig.~\\ref{fig:plotscbascanA} for all renormalization schemes with \nthe input values $\\alpha$ and $\\beta$ defined in the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme. \n\\begin{figure}\n\\centering\n\\includegraphics{.\/diagrams\/plot_cbascan-A-diffschemes-L3MS.eps}\n\\caption{The decay width for $\\mathswitchr h \\to 4f$ at LO \n(dashed) and full NLO EW+QCD (solid) for scenario A in dependence of $c_{\\beta-\\alpha}$. \nThe input values are defined in the  $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme and \nare converted to the other schemes at NLO (also for the LO curves). \nThe results computed with different renormalization schemes are displayed with the colour code of Fig.~\\ref{fig:plotrunningA}, and the SM \n(with SM Higgs-boson mass $\\mathswitch {M_\\Ph}$) is shown for comparison in red.}\n\\label{fig:plotscbascanA}\n\\end{figure}%\nThe LO (dashed) and the full NLO EW+QCD total widths (solid) are computed in the different renormalization schemes after the NLO input conversion \n(without linearization) and using the constant default scale $\\mu_0$ of Eq.~\\eqref{eq:centralscale}. The SM values are illustrated in red. \nAt tree level the widths show the suppression w.r.t. to the SM with the factor $s_{\\beta-\\alpha}^2$ originating from the \n$\\mathswitchr H\\mathswitchr W\\PW$ and $\\mathswitchr H\\mathswitchr Z\\PZ$ couplings. The differences between the renormalization schemes are due to the conversion of the input. As the conversion induces NLO differences in the LO results, a pure LO computation is identical for all renormalization schemes as the conversion vanishes at this order \nand is represented by the LO curve of the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme. The suppression w.r.t.\\ the SM computation does not change at NLO, while the shape becomes slightly asymmetric,\nand the NLO results show a significantly better agreement between the renormalization schemes. \nDeviations of the THDM results from the SM expectations can be investigated when the SM Higgs-boson mass is identified with the mass \n$\\mathswitch {M_\\Ph}$ of the light CP-even Higgs boson~h \nof the THDM. The relative deviation of the full width from the SM is then\n\\begin{align}\n \\Delta_\\mathrm{SM}= \\frac{\\Gamma_\\mathrm{THDM}-\\Gamma_\\mathrm{SM}}{\\Gamma_\\mathrm{SM}},\n\\end{align}\nwhich is shown in Fig.~\\ref{fig:plot_cbascanrelSM-A-diffschemes-L3MS} at LO (dashed) and NLO (solid) in percent for parameters defined in the the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme.\n\\begin{figure}\n  \\centering\n\\includegraphics{.\/diagrams\/plot_cbascanrelSM-A-diffschemes-L3MS.eps}\n  \\caption{The relative difference of the decay width for $\\mathswitchr h \\to 4f$ in the THDM w.r.t.\\ the SM prediction at LO (dashed) and NLO EW+QCD (solid). \nThe input values are defined in the  $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme and \nare converted to the other schemes at NLO (also for the LO curves). \nThe results computed with different renormalization schemes are displayed with the colour code of Fig.~\\ref{fig:plotrunningA}.}\n\\label{fig:plot_cbascanrelSM-A-diffschemes-L3MS}\n\\end{figure}%\nThe SM exceeds the THDM widths at LO and NLO. \nThe LO  shape which is just given by \n$c_{\\beta-\\alpha}^2$ shows minor distortions due to the parameter conversions. At NLO, the shape is slightly distorted by an asymmetry of the EW corrections, and a small offset of $-0.5$\\% is visible even in the alignment limit where the diagrams including heavy Higgs bosons still contribute. The NLO computations show larger negative deviations, and this could be used to improve current exclusion bounds or increase their significance. Nevertheless, in the whole scan region the deviation from the SM is within 6\\% and for phenomenologically most interesting region with $|c_{\\beta-\\alpha}|<0.1$ even less than 2\\%, which is challenging for experiments to measure.\n\n\\section{Conclusions}\n\\label{se:conclusion}\n\nConfronting experimental results on Higgs precision observables with theory predictions\nwithin extensions of the SM, provides an important alternative to search for\nphysics beyond the SM, in addition to the search for new particles.\nThe THDM comprises an extended scalar sector with regard to the SM Higgs sector and allows for a comprehensive study of the impact of new scalar degrees of freedom without introducing new fundamental symmetries or other new theoretical structures. \n\nIn this article, we have considered the Type I, II, lepton-specific, and flipped versions of the \nTHDM. We have introduced four different renormalization schemes which employ directly\nmeasurable parameters such as masses as far as possible and make use of fields that \ndirectly correspond to mass eigenstates.\nIn all the schemes, the masses are defined via on-shell conditions, the electric charge is fixed via the Thomson limit, and the coupling $\\lambda_5$ is defined with the $\\overline{\\mathrm{MS}}$ prescription. The fields are also defined on-shell which is most convenient in applications. The renormalization schemes differ in the treatment of the coupling $\\lambda_3$ and the mixing angles $\\alpha$ and $\\beta$: In the $\\overline{\\mathrm{MS}}(\\alpha)$ scheme, $\\alpha$ and $\\beta$ are renormalized using $\\overline{\\mathrm{MS}}$ conditions. In the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme instead $\\lambda_3$ and $\\beta$ are $\\overline{\\mathrm{MS}}$-renormalized parameters. In addition to the conventional treatment of tadpole contributions, we have\nimplemented an alternative prescription suggested by Fleischer and Jegerlehner where the mixing angles $\\alpha$ and $\\beta$ obtain extra terms of tadpole contributions, rendering these schemes gauge independent to all orders. \nIt should, however, be noted that the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme is also gauge independent at NLO in the \nclass of $R_\\xi$ gauges. \nWe have also discussed relations to renormalization procedures\nsuggested in the literature for the THDM. \n\nA comparison of these four different renormalization schemes allows for testing \nthe perturbative consistency, and, for the parameter regions and renormalization schemes fulfilling this test,  estimating the theoretical uncertainty due to the truncation of the perturbation series.\nTo further investigate the latter,\nwe have investigated the scale dependence, solving the corresponding RGEs. \nOne important observation is that it is crucial to be very careful and specific \nabout the definitions of the parameters applied, i.e.\\ the declaration of the renormalization\nscheme of the parameters is vital if one aims at precision.\nThis is already relevant in the formulation of\nbenchmark scenarios, because a conversion to a different scheme might alter the physical properties of the scenario significantly. For example, the alignment limit \nmay be reached with one specific set of parameters defined in a specific renormalization scheme, but converting these parameters consistently to parameters in a different renormalization scheme might shift the parameters away from the alignment limit.\n \nThe different renormalization schemes have been implemented into\na \\textsc{FeynArts} model file and are thus ready for applications.%\n\\footnote{The model file is restricted to a unit CKM matrix, but can be generalized\nto a non-trivial CKM matrix exactly as in the SM.}\nAs a first example, we have applied and tested the different schemes in the calculation of the decay width of a \nlight CP-even Higgs boson decaying into four massless fermions.\nWe discuss the dependence of the total $\\mathswitchr h{\\to}4f$ decay width on the renormalization scale\nand advocate a scale that is significantly higher than\nthe naive choice of $\\mu_\\mathrm{r} = \\mathswitch {M_\\Ph}$, taking care of the different mass scales in the\nTHDM Higgs sector.\nIn addition, results for various values of $\\cos (\\beta-\\alpha)$, a parameter entering the prediction already at LO, are presented. The deviations of the SM are relatively small, in the phenomenologically interesting region they are about \n$2{-}6\\%$---a challenge for future measurements. \n \nThe detailed\ndescription of the calculation of the decay width in the THDM \nand a survey of numerical results will be given in a forthcoming paper. \nThis includes a deeper investigation in the renormalization scale dependence and the comparison of different renormalization schemes \nfor more benchmark points\nas well as differential distributions.\n \n\n\\subsection*{Acknowledgements}\n\nWe would like to thank Ansgar Denner, Howard Haber and Jean-Nicolas Lang for helpful discussions\n{and especially Jean-Nicolas for an independent check of one-loop matrix elements\nagainst the crossing-related amplitudes used in \\citere{Denner:2016etu}.}\nHR's work is partially funded by the Danish National Research Foundation, grant number DNRF90. HR acknowledges also support by the National Science Foundation under Grant No. NSFPHY11-25915. \nWe thank the German Research Foundation~(DFG) and \nResearch Training Group GRK~2044 for the funding and the support and\nacknowledge support by the state of Baden--W\u00fcrttemberg through bwHPC and the \nDFG through grant no INST 39\/963-1 FUGG.\n\n\\section*{Appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}