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+{"text":"\\section{Historical Introduction}\n\nThe late 1500's and early 1600's were a remarkable period in the evolution of human thought. It might be reasonably argued that during this period Galileo Galilee put into practice the modern scientific method for describing and understanding natural processes. An equally important advancement in our way of thinking about the world was an emerging conviction of the universality of causes. This extraordinary new way of understanding the world around us is often associated with a slightly later period and with Isaac Newton. The notion that the laws of nature applied equally everywhere was indeed imagined in this earlier period by Johannes Kepler. In particular Kepler proposed that the principles that governed the movement of the planets was the same as on Earth. Kepler's thinking of a universal nature of physical properties both celestial and terrestrial is evident in his own words \\cite{Holton88} : ``I am occupied with the investigation of the physical causes. My aim in this is to show that the celestial machine is to be likened not to a divine organism but rather to a clockwork ..., insofar as nearly all the manifold movements are carried out by means of a single, quite simple magnetic force, as in the case of a clockwork all motion are caused by a simple weight. Moreover, I show how this physical conception is to be presented through calculation and geometry.\" Kepler's way of thinking about the motions of the planets and the universality of the laws of physics would be completely recognizable to every modern physicist.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=75mm]{Faraday.jpeg}\n\\caption{This figure is from Faraday's laboratory notebook describing the apparatus he constructed to conduct experiments on the relationship between gravity and electromagnetic induction \\cite{Faraday1885}. \\label{FaradayFig}}\n\\end{figure}\n\nWhile Kepler's conviction of the relationship between the motion of the planets and processes on Earth helped inspire our modern way of thinking, he was of course mistaken in making the association between gravity and a ``simple magnetic force''. However, even this mistake was an inspired effort to describe the world around us in terms of physical causes. The universality of physical principles quickly became a central theme in the development of physics and an inspiration for Newton and those that followed. This expectation of the universality of celestial and terrestrial processes and Kepler's expectation of  universality in a connection between magnetism and the motion of the planets is evident in Faraday's experimental investigations. Some time around the 1850's Faraday conducted experiments to demonstrate the possible connection between the gravitational field and the electromagnetic field. Faraday constructed an experimental apparatus in an effort to measure the magnitude of electromagnetic induction associated with a gravitational field as shown in Fig. \\ref{FaradayFig}. Faraday's results failed to demonstrate any relation between gravity and electricity but his commitment to this idea of universality was unwavering, \\cite{Faraday1885}, ``Here end my trials for the present. The results are negative. They do not shake my strong feeling of the existence of a relation between gravity and electricity, though they give no proof that such a relation exists.\"\n\nWhile Faraday's experiments were not successful, later theoretical research by Skobelev \\cite{Skobelev75} in 1975 supported Faraday's ``strong feeling\" by demonstrating a non-zero amplitude for the interaction of gravitons and photons in both scattering and annihilation. This association between gravity and electromagnetism was also described around the same time by Gibbons \\cite{gibbons} in noting that, ``Indeed since a `graviton' presumably in some sense carries light-like momentum the creation of one or more particles with time-like or light-like momentum would violate the conservation of momentum unless the created particles were massless and precisely aligned with the momentum of the graviton\". These kinematic restrictions for conversion of massless particles have also been studied more recently and in greater detail by Fiore and Modanese \\cite{Fiore96,Modanese95}. The processes of graviton and photon interaction described by Skobelev and Gibbons are exceedingly small \\cite{Skobelev75} but are non-zero. Our more recent research has expanded on this interaction of gravity\/gravitons and electromagnetism\/photons through annihilation and scattering processes by recognizing the contribution of the external gravitational field associated with a gravitational wave \\cite{Jones15,Jones16,Jones17,Jones18,Gretarsson18}. Our study of the vacuum production of light by a gravitational wave differs from Skobelev in that the amplitudes of the ``tree level diagrams\" would be dependent on the strength of the external gravitational field or equally the strain amplitude of the gravitational wave. This type of semi-classical conversion process between gravitational and electromagnetic fields was described more broadly by Davies \\cite{Davies01} ``One result is that rapidly changing gravitational fields can create particles from the vacuum, and in turn the back-reaction on the gravitational dynamics operates like a damping force.\" The back-reaction on the gravitational wave was shown to be small compared to the gravitational wave luminosity but sufficient to be detectable under the right circumstances \\cite{Jones17}. \n\nIn this brief review we will specifically outline the relationship between gravity and electricity for the special case of gravitational and electromagnetic radiation. While we take a historical perspective leading to current research no effort will be made to present the historical formalism. Instead we will present the ideas relating the association between gravitational and electromagnetic radiation and in particular the vacuum production of electromagnetic radiation by a gravitational wave using modern notation and mathematical formalism.\n\n\n\\section{Electromagnetism and light}\n\nA general review of the research on the relationship between gravity and electricity would completely preclude any possibility of being brief. We will instead focus our attention on the radiation regimes. The current understanding of the radiation regime for electricity began with James Clerk Maxwell's modification of Ampere's law \\cite{Maxwell2} to include the displacement current. This modification led to a wave equation solution to the equations of electromagnetism. Maxwell immediately recognized this wave equation as a description of the phenomena of light. In keeping with our intent to discuss the historical development of gravitational wave production of electromagnetic radiation using modern notation, the equations describing electromagnetic radiation will be presented in a form that is completely covariant. The Maxwell equations are written in terms of tensor relations and will have the same form in Minkowski space and curved space-time. The physical properties of electromagnetic radiation, such as luminosity, will be developed in terms of Newman-Penrose scalars  \\cite{Newman61,Teukolsky73}, in a form that is well suited for the comparison of electromagnetic and gravitational radiation \\cite{Jones17}.\n\nIn order to write the field equations for electrodynamics in a suitable \nform for curved space-time two tensors are defined in terms of the \nelectric and magnetic fields. The field strength tensor \n\\cite{Ellis73,Senego98,Hogan09,Palenzuela10,Lehner09,Lehner12_85,\nLehner12_86,Lehner16} is defined as  \\footnotemark \\footnotetext{Great \ncare is required with sign conventions in any covariant representation. \nThis is particularly true in the case of Maxwell's equations and here we \nare following Palenzuela {\\it et al.} \\cite{Palenzuela10}, which is consistent\nwith our metric. It is prudent to check the signs by confirming that the \ncovariant relations reduce correctly to the Maxwell equations in a \nLorentz inertial frame.}\n\n\\begin{equation}\nF^{\\mu \\nu }  = u^{\\mu} E^{\\nu}  - u^{\\nu} E^{\\mu} + e^{\\mu \\nu \\alpha \n\\beta} B_{\\alpha} u_{ \\beta},\n\\label{Faraday}\n\\end{equation}\n\n\\noindent and the dual to the field strength tensor as,\n\n\\begin{equation}\n^*F^{\\mu \\nu }  = u^{\\mu} B^{\\nu}  - u^{\\nu} B^{\\mu}  - e^{\\mu \\nu \n\\alpha \\beta} E_{\\alpha} u_{ \\beta},\n\\label{dFaraday}\n\\end{equation}\n\n\\noindent where $e^{\\mu \\nu \\alpha \\beta}$ is the ``Levi-Civita \npseudotensor of the space-time\" and $u_{\\nu}$ is the field frame 4-\nvelocity. The covariant expression for the field strength tensor \n\\eqref{Faraday} and the dual \\eqref{dFaraday} was originally developed \nby Ellis \\cite{Ellis73}. While these expressions are perhaps not widely \nknown, expanding \\eqref{Faraday} in a Lorentz inertial frame produces \nthe expected components for the field strength tensor. This covariant \nform of the field strength tensor has proven to be very useful in \nstudies of the relation between gravity and electromagnetism \n\\cite{Hogan09,Palenzuela10,Lehner09,Lehner12_85,Lehner12_86,Lehner16}. \nConversely, the electric and magnetic fields are found from the \ncontractions of the tensors with the 4-velocity,\n\n\\begin{equation}\nE^{\\mu} = F^{\\mu \\nu } u_{ \\nu},~ ~ B^{\\mu} = {^*}F^{\\mu \\nu } u_{ \\nu}.\n\\label{EBfieldsF}\n\\end{equation}\n\n\\noindent Using the field strength tensor and its dual the Maxwell \nequations can be written in a  covariant form that is the same in both Minkowski space and curved \nspace-time. The inhomogeneous Maxwell equations ({\\it i.e.} Gauss's law and Ampere's) \nlaw are,\n\n\\begin{equation}\n\\frac{1}{\\sqrt{-\\left| g_{\\mu \\nu} \\right|}} \\partial_{\\nu} \\left( \\sqrt{-\\left| g_{\\mu \\nu} \\right|} F^{\\mu \\nu } \n\\right)=  4 \\pi J^{\\mu}.\n\\label{Maxwell1}\n\\end{equation}\n\n\\noindent where $ \\left| g_{\\mu \\nu} \\right|= det [g_{\\mu \\nu}]$ is the determinant of the metric. The homogeneous \nGauss's law for magnetism and Faraday's law are \\cite{Greiner96, \nPalenzuela10},\n\n\\begin{equation}\n\\partial_{\\nu} \\left( \\sqrt{-\\left| g_{\\mu \\nu} \\right|} ~{^*}F^{\\mu \\nu } \\right)  = 0.\n\\label{Maxwell2}\n\\end{equation}\n\n\\noindent The covariant form of the conservation law is,\n\n\\begin{equation}\n\\partial_{\\mu} \\left( \\sqrt{-\\left| g_{\\mu \\nu} \\right|} J^\\mu \\right)= 0.\n\\label{Conservation}\n\\end{equation}\n\n\\noindent The field strength tensor can also be expressed in terms of \nthe electromagnetic 4-vector potential,\n\n\\begin{equation}\nF_{\\mu \\nu } = \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu} ~.\n\\label{FourPotential}\n\\end{equation}\n\n\\noindent Maxwell came to the wave equation from the bottom up by recognizing that the displacement current term was missing from the traditional form of Ampere's law. In the modern notation the wave equation is a mathematical identity in the absence of source terms in the Maxwell equations \\cite{Tsaga05}. \n\nThe covariant form of the equations for the electromagnetic field appears naturally in the radiative expression for electrodynamics in the Newman-Penrose formalism \\cite{Newman61,Teukolsky73} through the introduction of the Newman-Penrose electromagnetic scalar, to be discussed shortly. In order to provide a means of comparison between electromagnetic and gravitational radiation using the Newman-Penrose formalism we will require the Lagrangian density for the electromagnetic field in curved space-time. Including the electric source terms the Lagrangian density is,\n\n\\begin{equation}\n \\mathcal{L}_\n{em}  =  - \\frac{1}{4}\\left( {\\partial _\\nu  A_\\mu   - \\partial _\\mu  A_\\nu  } \\right)\\left( {\\partial ^\\nu  A^\\mu   - \n\\partial ^\\mu  A^\\nu  } \\right) +J_\\mu A^\\mu.\n\\label{emLagrangian}\n\\end{equation}\n\n\\noindent The Lagrangian density can be simplified using the Lorenz gauge \\cite{Greiner96}, $\\partial_\\mu A^\\mu = 0$, and by restricting our attention to be source free ({\\it i.e.} $J_\\mu =0$) so that \\eqref{emLagrangian} becomes,\n\n\\begin{equation}\n \\mathcal{L}_{em}  =  - \\frac{1}{2}\\partial _\\mu  A_\\nu  \\partial ^\\mu\n A^\\nu .\n\\label{emLagrangianLG}\n\\end{equation}\n\n\\noindent Since we are only considering the radiation regime for the \nfield equations, we assume a plane wave solution for the \nelectromagnetic field and a massless vector field can then be \nexpressed in terms of a mode expansion \\cite{Greiner96} as follows,\n\n\\begin{equation}\nA_\\mu  \\left( {k,\\lambda ,x} \\right) = \\epsilon_{\\mu} \n^{(\\lambda)} \\phi ^{(\\lambda)} \\left( {k ,x} \\right).\n\\label{ModeExpN}\n\\end{equation}\n\n\\noindent with $k$ being the momentum, $x$ being the \nspace-time coordinates and $\\epsilon ^{(\\lambda)}$ being the \npolarization vectors with $\\lambda = 0, 1, 2, 3$. For example, the two \ntransverse modes are often labeled $\\lambda =1 , 2$ with $\\epsilon \n_\\mu ^{(1)} = (0,1,0,0)$ and $\\epsilon _\\mu ^{(2)} =  (0,0,1,0)$. The \n$\\lambda = 0, 3$ components are the time-like and longitudinal polarizations. The\nfour polarization vectors satisfy the orthogonality relationship\n$\\epsilon _\\mu ^{(\\lambda)} \\epsilon ^{\\mu ~(\\lambda')} = \n\\eta ^{\\lambda \\lambda'}$, with $\\eta ^{\\lambda \\lambda'}$ being the\nMinkowski metric.\n\nOne can define a complex scalar field using the $\\lambda = 1, 2$ \ncomponents of the real scalar fields {\\it i.e.} \n$\\phi ^{(1,2)} (k, x)$ as $\\varphi =\\frac{1}{\\sqrt{2}}\n(\\phi ^{(1)} + i \\phi ^{(2)})$. In terms of this complex scalar field \nthe Lagrange density of \\eqref{emLagrangianLG} can be written as\n\\begin{equation}\n\\label{emLagrangian2}\n{\\cal L}_{em} = - \\partial_\\mu \\varphi \\partial ^\\mu \\varphi ^* ~,\n\\end{equation}\nwhich is the Lagrange density for a massless complex scalar field. Below we will use a massless,\ncomplex scalar field as a stand-in for the massless photon. This substitution is justified \nby equation \\eqref{emLagrangian2}\n\nThe field equations following from the Lagrange density in \\eqref{emLagrangian2} are,   \n\n\\begin{equation}\n\\frac{1}{{\\sqrt{-\\left| g_{\\mu \\nu} \\right|}}} \\partial_{\\mu} \\sqrt{-\\left| g_{\\mu \\nu} \\right|} g^{\\mu \\nu} \\partial_{\\nu}\\varphi = 0.\n\\label{eomvarphi}\n\\end{equation}\n\n\\noindent This expression for the field equations is the same for both curved space-time and for Minkowski space-time. If we restricted our attention to plane waves propagating in Minkowski space-time the solution to \\eqref{eomvarphi} takes the form, \n\n\\begin{equation}\n\\varphi = B e^{i  \\text{k} u} + C,\n\\label{PlaneWave}\n\\end{equation}\n\n\\noindent where the $z$ axis is assumed to be along the direction of propagation, $B$ and $C$ are constants, and $k$ is the wave number. The solution \\eqref{PlaneWave} is written in the standard light cone coordinate $u = z - t$ and $\\partial_{z} - \\partial_{t} = 2 \\partial_{u}$ \\cite{Jones16}.\n\nThe previous discussion provides an outline of Maxwell's electromagnetic radiation in modern covariant form. The principle goal of this review is to realize Faraday's expectation of a relationship between gravity and electricity. We will demonstrate this relationship for gravitational and electromagnetic radiation by comparing the gravitational radiation luminosity to the luminosity of the corresponding electromagnetic radiation produced from the vacuum by the gravitational radiation. We have found that the Newman-Penrose formalism is a good method for calculating the luminosity of both gravitational radiation and the corresponding electromagnetic radiation. The electromagnetic luminosity is presented here in terms of the Newman-Penrose formalism and the gravitational radiation luminosity will be presented in the following section.\n\nThe radiated electromagnetic power per unit solid angle is found from the projection of the electromagnetic field strength onto the elements of a null tetrad ($l_\\mu, m^\\mu, n^\\mu , \\bar m^\\mu$) which gives us the electromagnetic Newman-Penrose scalar $\\Phi _2$. In the Newman-Penrose formalism the power per unit solid angle of emission for electromagnetic radiation is written as \\cite{Teukolsky73,Lehner09},\n\n\\begin{equation}\n\\frac{d E_{em} }{dt d\\Omega}  = \\mathop {lim}\\limits_{r \\to \\infty } \\frac{{r^2 }}{{4\\pi }}\\left| {\\Phi _2 } \\right|^2.\n\\label{emFlux0}\n\\end{equation}\n\n\\noindent The Newman-Penrose electromagnetic scalar in \\eqref{emFlux0}  \\cite{Newman61,Teukolsky73,Lehner09,Lehner12_86} is,\n\n\\begin{equation}\n\\Phi _2  = F_{\\mu \\nu } \\bar m^\\mu  n^\\nu.\n\\label{Phi2}\n\\end{equation}\n\n\\noindent The null tetrad of the Newman-Penrose formalism in \\eqref{Phi2} can be defined as \\cite{Lehner12_85},\n\n\\begin{equation}\n\\begin{array}{*{20}c}\n   {l^\\mu   = \\frac{1}{{\\sqrt 2 }}\\left( {1,0,0,1} \\right),} & {n^\\mu   = \\frac{1}{{\\sqrt 2 }}\\left( {1,0,0, - 1} \\right),}  \\\\\n   {m^\\mu   = \\frac{1}{{\\sqrt 2 }}\\left( {0,1,i,0} \\right),} & {\\bar m^\\mu   = \\frac{1}{{\\sqrt 2 }}\\left( {0,1, - i,0} \\right),}  \\\\\n\\end{array}\n\\label{null1}\n\\end{equation}\n\n\\noindent and\n\n\\begin{equation}\n l \\cdot n =  - 1,~~m \\cdot \\bar m = 1 ,~~\nl \\cdot l = n \\cdot n = m \\cdot m = \\bar m \\cdot \\bar m = 0.\n\\label{null2}\n\\end{equation}\n\n\\noindent The electromagnetic field strength tensor in \\eqref{Phi2} is $F_{\\mu \\nu }  = \\partial _\\mu  A_\\nu  {\\kern 1pt}  - \\partial _\\nu  A_\\mu$, where $A_\\mu   = \\epsilon ^{(\\lambda)}  _\\mu \\phi ^{(\\lambda )} \\left( {t,z} \\right)$, and the plane polarization vectors are $\\epsilon ^{(1)} _\\mu   = \\left(0, 1, 0, 0 \\right), \\;\\epsilon ^{(2)} _\\mu   = \\left(\n0, 0, 1,0 \\right)$ \\cite{Jones17}. The electric and magnetic fields are determined by taking the derivatives of the scalar field \\eqref{PlaneWave}: $\\,\\partial _t \\varphi  =  - i  k B e^{i k \\left( {z - t} \\right)}$ and $\\partial_z \\varphi = i k B e^{i  k \\left( {z - t} \\right)}$. Collecting terms for the Newman-Penrose scalar of the ``out\" state of \\eqref{PlaneWave} \\cite{Lehner12_85,Jones17},\n\n\\begin{equation}\n\\Phi _2  = F_{\\mu \\nu } \\bar m^\\mu  n^\\nu =\n\\frac{1}{{\\sqrt 2 }}e^{ - i\\frac{\\pi }{4}} \\left( { \\partial _z \\varphi - \\partial _t \\varphi} \\right) =\ni e^{ - i\\frac{\\pi }{4}} {\\sqrt 2 }  k B e^{i k \\left( {z - t} \\right)}.\n\\label{emNPscalar0}\n\\end{equation}\n\n\\noindent The square of the electromagnetic scalar is then,\n\n\\begin{equation}\n\\left| {\\Phi _2 } \\right|^2  = 2  k^2 B^2.\n\\label{emNPscalar}\n\\end{equation}\n\nThe square of the Newman-Penrose scalar in  \\eqref{emNPscalar} is proportional to the electromagnetic flux, $F_{em} \\sim  \\left| {\\Phi _2 } \\right|^2 $. In Section \\ref{gRad} on gravitational radiation and in Section \\ref{production} on scalar field production we will show that by using the Newmam-Penrose scalars for the gravitational and the electromagnetic fields respectively, one can compare the fluxes of gravitational and electromagnetic radiation \\cite{Jones17}.\n\n\\section{Gravitational radiation}  \\label{gRad}\n\nA wave like solution to the vacuum equations for general relativity exist similar to that of electromagnetism \\cite{Schutz00}. This was recognized by Einstein soon after the development of general relativity and proposed even earlier by Poincar{\\' e} \\cite{Smoot16}. Initially there was doubt as to whether or not gravitational waves were physically real. Unlike the production of electromagnetic radiation there is no dipole source for gravitational radiation. This is because mass dipole production of radiation would violate conservation of 4-momentum \\cite{Schutz00,Smoot16}. However, there are also quadrupole source terms which lead to a wave solution and does not violate any conservation principles \\cite{Schutz00}.\n\n\nSince our interest here is in the relation between gravity and electromagnetism, in the radiation regime, we will restrict our attention to the plane wave solution of general relativity. The metric of a gravitational plane wave traveling in the $+z$ direction and with the $+$ polarization\ncan be written as \\cite{Schutz},\n\n\\begin{equation}\nds^2 = g_{\\mu \\nu} dx^{\\mu} dx^{\\nu} = -dt^2 + dz^2 + f^2 dx^2 + g^2 dy^2, \n\\label{GWmetric}\n\\end{equation}\n\n\\noindent were we set $c=1$. This metric is oscillatory with $f = 1 + \\varepsilon (u) $, $g = 1 - \\varepsilon(u)$ with $\\varepsilon(u) =h_+ e^{iku}$.  The coefficient $h_+$ is the gravitational wave strain amplitude and $k$ is the wave number. The coordinate variable in the metric is the standard light cone coordinate $u=z-t$. This metric only includes the ``plus\" polarization. Similar to electromagnetic radiation there are two degrees of freedom corresponding to two polarization states for gravitational radiation. The two polarization states for gravitational radiation are ``plus\" and ``cross\" polarization. They differ by an angle of $\\frac{\\pi}{4}$ in contrast to a phase angle difference of $\\frac{\\pi}{2}$ for electromagnetism \\cite{Schutz00, Schutz09}. Including the ``cross\" polarization would not change our discussion.\n\n\\begin{figure}[H]\n\\centering\n{\\caption{The ``discovery paper\" spectrogram of gravitational waves produced by a binary black hole in-spiral \\cite{LIGO}.} \\label{spectrogram}}\n{\\includegraphics[width=12cm]{SpectrogramBW.jpeg}}\n\\end{figure}\n\nThe luminosity of the gravitational radiation can be calculated using the Newman-Penrose formalism \\cite{Teukolsky73,Lehner09}. The relevant scalar for gravitational radiation is a projection of the Riemann tensor onto elements of a null tetrad. This projection is identified as the gravitational Newman-Penrose scalar $\\Psi _4$. The power per unit of solid angle for the gravitational radiation is written in terms of the gravitational scalar as,\n\n\\begin{equation}\n\\frac{d E_{gw} }{dt d\\Omega}  = \\mathop {lim}\\limits_{r \\to \\infty } \\frac{{r^2 }}{{16\\pi k^2 }}\\left| {\\Psi _4 } \\right|^2 .\n\\label{GWluminsity}\n\\end{equation}\n\n\\noindent Substituting the metric for the gravitational wave \\eqref{GWmetric} into the Riemann tensor for an outgoing gravitational plane wave in vacuum the gravitational Newman-Penrose scalar \\cite{Teukolsky73,Jones17} is,\n\n\\begin{equation}\n\\Psi _4  =  - R_{\\alpha \\beta \\gamma \\delta } n^\\alpha  \\bar m^\\beta  n^\\gamma  \\bar m^\\delta   = f\\partial_{u}^{2} f- g\\partial_{u}^{2} g .\n\\label{Psi}\n\\end{equation}\n\n\\noindent The partial derivatives, $\\partial_{u}$, are with respect to the light cone coordinate, $u$. For the plane wave metric, where $\\varepsilon = h_+ e^{iku}$, the gravitational Newman-Penrose scalar and the square are given by,\n\n\\begin{equation}\n \\Psi _4  =  - 2 h_+ k^2 e^{ik\\left( {z - t} \\right) } \\rightarrow |\\Psi _4 | ^2 = 4 h_+ ^2 k^4.\n \\label{Psi4}\n \\end{equation}\n\n\\noindent The Newman-Penrose scalars in \\eqref{emNPscalar} and \\eqref{Psi4} provide a convenient method for the comparison of the power of the gravitational radiation and the counterpart vacuum production of electromagnetic radiation which will follow. The flux of the gravitational wave can be calculated from \\eqref{GWluminsity} and  \\eqref{Psi} as \\cite{Schutz,Schutz09,Gretarsson18},\n\n\\begin{equation}\nF_{gw}= \\frac{c^3}{16 \\pi G} \\left| \\dot \\varepsilon \\right| =  \\frac{c^3 h_+ ^2 \\omega^2}{16 \\pi G},\n \\label{GWfluxh}\n \\end{equation}\n \n \\noindent which is a function of the strain amplitude $h$ and gravitational wave frequency $\\omega = 2 \\pi f$.\n \nThis review of the relationship between gravity and electromagnetism in the radiation regime might be considered of academic interest only except for the recent and remarkable discovery by the LIGO scientific collaboration of gravitational waves \\cite{LIGO}. The spectrogram from the discovery papers is provided in Fig. \\ref{spectrogram} for the binary in-spiral of two black holes. The detection of gravitational waves makes the discussion of the potential production of electromagnetic radiation by gravitational waves immediately relevant to current research in both the fundamental relation between gravity and electromagnetism as well as potential applications in astrophysics. If there were any lingering doubt about the certainty of the detection of gravitational waves the more recent detection of gravitational waves from a kilonova event \\cite{ligo2} has laid these doubts to rest. The kilonova event was first identified through the detection of gravitational waves by the LIGO scientific collaboration.  The kilonova was immediately verified across the electromagnetic spectra through the coordination of an international collaboration of observatories based around the World and in space. The kilonova observations have not only eliminated any reasonable doubt of the existence of gravitational waves but also ushered in a new era of ``multi-messenger\" astronomy and astrophysics.\n\n\n\\section{Electromagnetic fields in gravitational wave background}\n\n\\subsection{Gravitational waves and uniform magnetic field}\n\nThe first modern attempt to connect gravitational radiation and \nelectromagnetic fields was work by Gertsenshtein \\cite{Gertsenshtein60}\nwhich considered the linearized Einstein field equations coupled to an \nelectromagnetic plane wave.\n\n\\begin{equation}\n\\label{gert-1}\n\\Box {\\tilde h}^{\\mu \\nu} = -\\kappa t^{\\mu \\nu} ~,\n\\end{equation}\n\n\\noindent where $t^{\\mu \\nu} = \\frac{1}{4 \\pi} (F^{\\mu \\tau} F^\\nu _\\tau -\ng^{\\mu \\nu} F^{\\alpha \\beta} F_{\\beta \\alpha})$ is the energy-momentum\ntensor for the electromagnetic field, ${\\tilde h}^{\\mu \\nu} = h^{\\mu \\nu} \n- \\frac{1}{2} g^{\\mu \\nu} h$ is the trace reduced metric deviation \nof the metric tensor ({\\it i.e.} $g^{\\mu \\nu} = \\eta ^{\\mu \\nu} + h^{\\mu \n\\nu}$), and $\\kappa = 16 \\pi G$. Now the proposal in \n\\cite{Gertsenshtein60} was to generate gravitational waves by sending \nelectromagnetic waves through a constant magnetic field. This potential \neffect was compared to radio physics phenomenon of wave resonance. The \nidea being that despite the weak coupling of gravity one could \nnevertheless generate some significant amount of gravitational radiation \nby this method. \n\nNow if one takes the electromagnetic field to have a constant magnetic \nfield part (whose field strength tensor we denote by $F^{(0) \\mu \\nu}$) \nand a plane wave part (whose field strength we denote by $F ^{\\mu \\nu}$),\nand if we feed this into \\eqref{gert-1}, dropping the squared terms in\n$F^{(0) \\mu \\nu}$ and $F^{\\mu \\nu}$ and keeping only the cross terms \nwe arrive at\n\n\\begin{equation}\n\\label{gert-2}\n\\Box {\\tilde h}^{\\mu \\nu} = -\\frac{\\kappa}{2} \\left( F^{(0) \\mu \\tau} \nF^{\\nu} _\\tau - \\frac{1}{4} g^{\\mu \\nu} F^{(0) \\alpha \\beta} F_{\\alpha \\beta}    \\right) ~.\n\\end{equation}\n\n\\noindent One now assumes that the electromagnetic plane wave field and\ngravitational field propagate along the $z$ direction with wave number \n$k$ and have the form\n\n\\begin{equation}\n\\label{gert-3}\nF^{\\mu \\nu} = b(z) \\epsilon^{\\mu \\nu} e^{i(kz-\\omega t)}  ~~~;~~~\n{\\tilde h}^{\\mu \\nu} = a(z) \\zeta ^{\\mu \\nu} \\sqrt{\\frac{\\kappa}{k^2}}\ne^{i(kz - \\omega t)} ~,\n\\end{equation}\n\n\\noindent where $\\epsilon ^{\\mu \\nu} , \\zeta ^{\\mu \\nu}$ are the \nelectromagnetic and gravitational polarization tensors respectively. \nUsing \\eqref{gert-3} in \\eqref{gert-2} and assuming slowly varying\namplitudes $a(z) , b(z)$ one obtains the following relationship \nbetween the amplitudes\n\\begin{equation}\n\\label{gert-4}\ni \\frac{da (z)}{dz} = \\sqrt{\\frac{\\kappa}{16}} F^{(0) \\mu \\nu}\n\\epsilon_{\\beta \\nu} \\zeta^{\\beta} _\\mu b(z) ~.\n\\end{equation}\nUnder the assumption that $b(x) \\approx const.$ \\eqref{gert-4} can be \nintegrated to obtain $a(x)$ as\n\\begin{equation}\n\\label{gert-5}\n\\left| \\frac{a (z)}{b(0)} \\right|^2 = \\frac{\\kappa}{16 \\pi^2} \nB_0 ^2 T^2 ~,\n\\end{equation}\nwhere $B_0 \\simeq | F^{(0) \\mu \\nu} |$ is the constant magnetic field \nstrength, $T$ is the time that the electromagnetic wave traverses the\nuniform magnetic field, and $b(0)$ is the initial amplitude of the\nelectromagnetic wave. If one takes the cosmological sized magnetic fields\n($B_0 \\simeq 10 ^{-5}$ G) and assumes cosmological times for the \nelectromagnetic wave to travel through this constant magnetic field\n($T \\simeq 10^7$ years) one finds that the ratio of gravitational to \nelectromagnetic amplitude is of the order $|a\/b|^2 \\simeq 10^{-17}$. One \ncould increase this by having stronger magnetic fields and\/or longer \nperiods of travel. \n\nOne of the most interesting features of the above mechanism is that the\ngravitational wave frequency is the same as that of the electromagnetic \nwave. This gives the possibility of generating very high frequency \ngravitational waves compared to the ``natural'' sources of gravitational \nwaves from the first series of direct detections-- merging black hole, \nmerging neutron stars. These natural or astrophysical sources have \nfrequencies in the 100s to 1000s of Hertz, whereas electromagnetic \nradiation has a much broader range of frequencies which have been\nobserved -- from 1000s of Hertz to Gigahertz and beyond.\n\nIn the original work by Gertsenshtein \\cite{Gertsenshtein60} the focus \nwas on generating gravitational waves from electromagnetic waves. In this\nreview our focus is the exact opposite -- we are interested in \nelectromagnetic radiation generated from gravitational waves. This \nreversed possibility was pessimistically noted by Gertsenshtein with the\nconcluding comment ``From general relativity follows also the possibility\nof the inverse conversion of gravitational waves into light waves, \nbut this problem is hardly of interest.'' Nevertheless, several years \nafter Gertsenshtein's paper, Lupanov \\cite{lupanov67} did examine the \ninverse process of generating electromagnetic waves from gravitational\nwaves.  \n\nWe will follow the work in \\cite{lupanov67} by examining the reverse \nprocess of electromagnetic radiation generated from gravitational waves. \nThere are two  reasons for our focus on the reverse process: (i) \nelectromagnetic radiation, even weak radiation, is easier\nto detect, and (ii) we argue, beginning in the next subsection, that the\nconversion of gravitational waves to electromagnetic radiation occurs \neven in vacuum.\n\nBefore concluding this subsection we mention that there is more recent work\nin the spirit of Gertsenshtein's work \\cite{Gertsenshtein60}\nwhere the magnetic field is replaced by a Bose-Einstein condensate\n\\cite{fuentes}. In this case the interaction of the gravitational\nwave with the Bose-Einstein condensate is conjectured to lead to \nthe creation of phonons, just as in Gertsenshtein's work the \ninteraction of the gravitational wave with the magnetic field\nlead to the creations of photons. This creation of phonons\nwith a Bose-Einstein condensate has been put forward as a \npotential alternative mechanism to interferometers like LIGO\nto detect gravitational waves.  \n\n\n\\subsection{Feynman diagram approach to gravitational and electromagnetic\nradiation}\n\nTo check the assertion that electromagnetic radiation can be created in \nvacuum from gravitational radiation we turn to tree-level Feynman \ndiagrams for graviton-photon scattering. The four basic diagrams for this \nprocess are given in Fig. \\eqref{Skobelev} with curly lines \nrepresenting gravitons and wavy lines represent photons. The original \ncalculation was carried out by Skobelev \\cite{Skobelev75} with more \nrecent and more extensive calculations being found in references \n\\cite{Bohr14}. The diagrams in Fig. \\eqref{Skobelev} represent\n$graviton + photon \\to graviton + photon$ scattering. By rotating the \ndiagrams one can get $graviton + graviton \\to photon + photon$ or\n$photon + photon \\to graviton + graviton$ which can be viewed as \ncreation of photons (gravitons) from gravitons (photons). \n\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=110mm]{Skobelev.jpeg}\n\\caption{Tree level Feynman diagrams of graviton-photon transitions \n\\cite{Skobelev75}. Wavy lines represent photons and curly lines \nrepresent gravitons \\label{Skobelev}}\n\\end{figure}\n\nIn \\cite{Skobelev75} the process $graviton + photon \\to graviton + \nphoton$ is calculated first. After a long calculation the \ndifferential scattering cross section is found to be\n\\begin{equation}\n\\label{sko-1}\n\\frac{d \\sigma}{d \\cos \\Theta} =\\frac{\\kappa ^2 \\omega ^2}{64 \\pi}\n\\left( \\frac{1+ \\cos ^8 (\\Theta \/2) }{\\sin ^4 (\\Theta \/ 2)} \\right) ~,\n\\end{equation}\nwhere $\\kappa = 16 \\pi G$ as previously, $\\omega$ is the energy of \nthe system, and $\\Theta$ is the scattering angle.\n\nNow the process of interest in this review is where gravitational waves \ncreate electromagnetic waves or photons. In the Feynman diagram language \nthis means $graviton + graviton \\to photon + photon$. This process \ncan be obtained by rotating the diagrams in Fig. \\ref{Skobelev}  by $90^0$ degrees. Upon \ndoing this the differential cross section for $graviton + graviton \\to \nphoton + photon$ is found to be \\cite{Skobelev75}\n\\begin{equation}\n\\label{sko-3}\n\\frac{d \\sigma}{d \\cos \\Theta} =\\frac{\\kappa ^2 \\omega ^2}{64 \\pi}\n(\\cos ^8 (\\Theta \/2) + \\sin ^8 (\\Theta \/ 2) ) ~.\n\\end{equation}\nIntegrating \\eqref{sko-3} to obtain the total cross section \n\\begin{equation}\n\\label{sko-4}\n\\sigma =\\frac{\\kappa ^2 \\omega ^2}{160 \\pi} ~.\n\\end{equation}\nIf one takes the energy of the system to be the rest mass energy of \nthe electron, $\\omega = m_e c^2$, \\footnote{This energy is much larger\nthan the energy implied by the frequencies of the observed gravitational \nwave signals \\cite{LIGO}. For the energy associated with the frequencies\nimplies by the LIGO observations the cross section would be even smaller\n} one finds \\cite{Skobelev75} that  $\\sigma \\simeq 10^{-110} cm ^2$. \nThis is a very small number and indicates that at the level of individual\nphotons and gravitons this is not a large effect. However, given the \nenormous energy of the observed gravitational wave signals, which implies a\nlarge number of gravitons, we will argue that there are cases where the \nsmall cross section of \\eqref{sko-4} may be compensated for by the \nlarge number of gravitons\/strength of the gravitational wave. \n\n\\subsection{Massive Scalar field in Gravitational Plane Wave Background}\n\nThe idea of particle creation from a time dependent space-time was \nfirst considered in a series of papers by Parker \\cite{parker-1,parker-2,parker-3} which investigated particle production \nfrom the time dependent FRW cosmological space-time. The next major \ntime-dependent space-time to be studied in terms of particle production \nwas the gravitational plane wave space-time. The initial studies were \ncarried out by Gibbons \\cite{gibbons} and Deser \\cite{deser}, who \nconsidered the  production of a massive scalar field in a pulsed \ngravitational wave space-time. Reference \\cite{garriga} has a \nmore extensive discussion of particle creation from a gravitational \nwave background, again in the context of a massive scalar field. \nThe conclusion of all of these works was that massive scalar \nfields would not be created from such a plane wave  gravitational \nbackground. \n\nThis conclusion, of no particle creation from a plane gravitational \nwave background, appears at odds with recent work \\cite{Jones15,Jones16,Jones17,Jones18}. However these recent\nworks focus on the case of massless particles whereas references\n\\cite{gibbons,deser,garriga} focus on massive particles.\nAs noted by Gibbons one can already guess that the production of a \nmassive field from a gravitational plane wave would be forbidden\nby energy momentum conservation. A massless graviton can not \ndecay\/transform into massive particles since this would violate \nenergy-momentum conservation. It is the same reason that forbids \na photon from decaying into an electron-positron pair in free space.\n(A photon can decay\/transform into electron-positron pair in the\npresence of a heavy nuclei which acts to conserve energy and momentum). \n\nGeneral studies of when one type of massless field quanta can \ndecay\/transform into other massless field quanta can be found\nin \\cite{Modanese95} and \\cite{Fiore96}. Using energy-momentum \nconservation these two works show that some number of massless\nquanta can transform into some number of other massless field quanta\nso long as the incoming and outgoing particles lie along the same \ndirection. This is consistent with the Feynman diagram calculations\nof reference \\cite{Skobelev75} where the decay of gravitons to photons\n({\\it i.e.} $graviton + graviton \\to photon + photon$) is possible so long as the\nmomenta of all particles lie along the same direction. This is also the\ncondition under which the creation of massless fields\/field quanta \noccurs in references  \\cite{Jones15,Jones16,Jones17,Jones18}.\n\nIn the rest of this section we review the calculations of \\cite{gibbons,deser,garriga} which demonstrate the absence of\nparticle creation when the \nparticles are massive, since this will provide a nice segue to\nthe case of massless particles. We will follow the notation of\nreference \\cite{garriga}.\n\nGarriga and Verdaguer \\cite{garriga} begin by considering the plane\nwave metric of the form\n\\begin{equation}\n\\label{garr-1}\nds^2 = -dt^2 + dz^2 + g_{ab} (z, t) dx^a dx^b = \n-du dv + g_{ab} (u) dx^a dx^b ~,\n\\end{equation}\nwhere in the last expression the metric has been transformed to light\nfront coordinates defined as $u=z-t$ and $v=z+t$ with $c=1$. The \nindices $a, b =1, 2$ and run over the $x, y$ directions, which are \nperpendicular to the $+z$ direction of travel of the gravitational wave.\nFor a gravitational wave traveling in the $-z$ direction one would take\nthe metric components to be functions of the light front coordinate\n$v$ {\\it i.e.} $g_{ab} (v)$.\n\nNext a massive scalar field, $\\varphi$, is placed in the metric given by \n\\eqref{garr-1}. The equation for a massive scalar field in a curved\nbackground is given by\n\\begin{equation}\n\\label{garr-2}\n\\left [ \\frac{1}{{\\sqrt { -\\left| {g_{\\mu \\nu } } \\right|} }}\\left( \n{\\partial _\\mu  g^{\\mu \\nu } \\sqrt { - \\left| {g_{\\mu \\nu } } \\right|} \n\\partial _\\nu  } \\right) - m^2 \\right] \\varphi  = 0.\n\\end{equation}\nUsing  the metric \\eqref{garr-1} in equation \\eqref{garr-2}\nand applying separation of variables, one finds solutions for the \nscalar field of the form\n\\begin{equation}\n\\label{garr-3}\n\\varphi (u, v, x^a)  = \n\\frac{1}{(p_{-})^{1\/2} (det g_{ab} (u))^{1\/4} (2 \\pi)^{3\/2} }\n\\exp \\left[i p_a x^a - i  p_{-} v -\\frac{i}{4 p_{-}} \n\\int _0 ^u (g_{ab} p^a p^b +m^2) du\\right]~,\n\\end{equation}\nwith $p_{-}$ and $p_a$ being separation constants which physically\ncorrespond to momenta connected with the coordinates,\n$v$ and $x^a$ respectively. \n\nThe next step is to calculate the Bogoliubov  coefficients \\cite{davies} \nfor this scalar field for a sandwich space-time {\\it i.e.} one has a \nplane wave space-time for $u_1 < u < u_2 <0$, sandwiched between two \nMinkowski space-times for $u<u_1$ and $u >u_2$. The \n``in'' and ``out'' states for this sandwich space-time are \\cite{garriga}\n\\begin{eqnarray}\n\\label{garr-4}\n\\varphi ^{in} (u, v, x^a)  &=& \n\\frac{1}{(p_{-})^{1\/2} (2 \\pi)^{3\/2} }\n\\exp \\left[i p_a x^a - i  p_{-} v -\\frac{i}{4 p_{-}} \n(p_a p^a +m^2) u + i \\Delta \\right]~,\\\\\n\\varphi ^{out} (u,v, x^a) &=&\n\\frac{1}{(k_{-})^{1\/2} (2 \\pi)^{3\/2} }\n\\exp \\left[i k_a x^a - i  k_{-} v -\\frac{i}{4 k_{-}} \n(k_a k^a +m^2) u \\right]~,\n\\label{garr-5}\n\\end{eqnarray}\nwhere $\\Delta$ is a constant phase. For the exact expression for this\nphase as well as for the full details of the calculation and some \nsubtleties in the definition of the coordinates we refer the reader to\n\\cite{garriga}. The light front momentum $p_-$ and $k_-$ are given by \n\\begin{equation}\n\\label{garr-5a}\np_- = \\frac{\\omega_p - p_z}{2} ~~~;~~~ k_-=\\frac{\\omega_k - k_z}{2} ~,\n\\end{equation}\nwith $p_z , k_z$ being the three-momentum in the $z$ direction and \n$\\omega_p = \\sqrt{{\\bf p}^2 +m^2}$ and $\\omega_k = \\sqrt{{\\bf k}^2\n+m^2}$ are the energies associated with the wave solutions. \n\nFrom \\eqref{garr-4} and \\eqref{garr-5} one can calculate the\nBogoliubov beta coefficients to be\n\\begin{equation}\n\\label{garr-6}\n\\beta = - \\langle \\varphi ^{in} | \\varphi ^{out ~*}\\rangle \\propto \n\\delta (p_- + k_-) ~,\n\\end{equation}\nwith the Dirac delta being a function of $p_- , k_-$ coming from \nintegration over $dv$. Now if the scalar field is massive it is easy \nto see, using the expressions for $\\omega _p \\omega_k$ in equation \n\\eqref{garr-5a} that\n$p_- + k_- \\ne 0$ so that $\\beta =0$. However, if $m=0$ and the 3-\nmomentum are in the same direction ${\\bf p} = p_z = {\\bf k} = k_z$\nthen $p_- + k_- =0$ and $\\beta \\ne 0$ meaning that production of \nthe scalar field from the gravitational wave occurs. This is \nconsistent with the Feynman\ndiagram calculations of \\cite{Skobelev75,Bohr14} as well as the\ndiscussion in term of energy-momentum conservation of \nparticle decay\/production\/scattering of massless fields \n\\cite{Modanese95,Fiore96}. In the next section we investigate \nin more detail the possibility of producing massless fields\/particles \nfrom a gravitational wave background.   \n\n\\section{Particle Production from a Gravitational Wave Background}\n\nIn this section we review some recent work \\cite{Jones15,Jones16,Jones17,Jones18} on the production of massless fields from\ngravitational wave backgrounds. We use a massless scalar field\nto carry out the analysis, but our results also apply to the more\nrealistic case of a massless vector field from the results and\ndiscussion around equations \\eqref{emLagrangian},  \\eqref{emLagrangianLG}, \n\\eqref{ModeExpN}, and  \\eqref{emLagrangian2}.\nWe also look at the response of an Unruh-Dewitt \ndetector in a gravitational plane wave background which supports the\npicture of gravitons decaying\/transforming into photons. \n            \n\\subsection{Scalar field production} \\label{production}\n\nWe now repeat some of the calculations of the previous section but for a \nmassless scalar field. We will follow the work of \\cite{Jones16}. For\nthe gravitational plane wave background we take the metric of \n\\eqref{garr-1} to have the more specific form\n\n\\begin{equation}\n\\label{jones-1}\nds^2 = -dt^2 + dz^2 + f^2 (z,t) dx^2 + g^2 (z,t) dy^2 = \n-du dv +  f^2 (u) dx^2 + g^2 (u) dy^2  ~,\n\\end{equation}\nwhere we have again transformed to light front coordinates, $u, v$ and \ntaken $c=1$. The form of the metric in \\eqref{jones-1} assumes the\nplus-polarization for the gravitational plane wave, which we take \nwithout loss of generality. We further assume that the ansatz functions\nhave a oscillatory behavior of the form $f(u) = 1 + h_+ e^{iku}$ and\n$g(u) =1 - h_+ e^{iku}$, where $h_+$ is the dimensionless amplitude\nof the plus polarization and $k$ is the gravitational wave \nnumber. With $m=0$ the field equation for $\\varphi$ from \n\\eqref{garr-2} becomes\n\n\\begin{equation}\n\\label{jones-2}\n\\frac{1}{{\\sqrt { -\\left| {g_{\\mu \\nu } } \\right|} }}\\left( \n{\\partial _\\mu  g^{\\mu \\nu } \\sqrt { - \\left| {g_{\\mu \\nu } } \\right|} \n\\partial _\\nu  } \\right)  \\varphi  = 0.\n\\end{equation}\n\nUsing the plane wave metric from \\eqref{jones-1} along with the oscillatory form of the ansatz functions $f(u), g(u)$ equation \\eqref{jones-2}  becomes\n\n\\begin{equation}\n \\left( {4F (ku) \\partial _u \\partial _v  - 4ikG (ku) \\,\\partial _v  + \n H(ku) \\left( {\\partial _x^2  + \\partial _y^2 } \\right)} \\right)\\varphi  \n = 0,\n\\label{jones-3}\n\\end{equation}\n\n\\noindent where the functions $F(ku), G(ku), H(ku)$, are given by\n\n\\begin{equation}\n\\begin{array}{l}\n \\quad F\\left( {ku} \\right)  \\equiv  \\left( 1 - h_ + ^2 e^{2iku}  \\right) \n ^2, \\\\ \n \\quad G\\left( {ku} \\right)  \\equiv  h_ + ^2 e^{2iku} \\left( 1  -\n h_ + ^2 e^{2iku}  \\right), \\\\ \n \\quad H\\left( {ku} \\right)  \\equiv  \\left( {1 + h_ + ^2 e^{2iku} } \n \\right). \\\\ \n \\end{array}\n\\label{jones-4}\n\\end{equation}\n\n \\noindent In arriving at \\eqref{jones-3} we have assumed that the\n behavior of $\\varphi$ in the perpendicular $x, y$ directions are the \n same so that\n $\\partial _x \\varphi = \\partial _y \\varphi$ and \n $\\partial _x ^2 \\varphi = \\partial _y ^2 \\varphi$.\n\nTo solve \\eqref{jones-3} we employ separation of variables as\n$\\varphi (u, v, x, y) = U(u) V(v) X(x) Y(y)$. The ansatz functions\nin the $v, x, y$ directions are plane waves of the form\n\n\\begin{equation}\n X (x) = e^{ik_{xy} x} ~~~;~~~ Y(y) = e^{ik_{xy} y} ~~~;~~~~ \n V(v) =  e^{ik_v v} ~,\n\\label{jones-5}\n\\end{equation}\n\n\\noindent where we have enforced the equality of the $x$ and $y$ \ndirections by taking a common wave number, $k_{xy}$. With this \nset up and the solutions from \\eqref{jones-5} the solution for $U(u)$\nis \\cite{Jones16}\n\\begin{equation}\nU = B e^{\\frac{\\lambda }{k}} e^{  \\frac{- \\lambda }{{k\\left( {1 - h_ + \n^2 e^{2iku} } \\right)}}} \\left( {1 - h_ + ^2 e^{2iku} } \n\\right)^{\\frac{1}{2}\\left( {\\frac{\\lambda }{k} - 1} \\right)} e^{ - \ni\\lambda u} + C ~,\n \\label{jones-6}\n\\end{equation} \n\\noindent with $B, C$ being integration constants and \n$\\lambda = \\frac{k_{xy}^2}{2 k_v}$. Putting equations\n\\eqref{jones-5} \\eqref{jones-6} together, and taking $C=-B$ the \nscalar field in the plane wave background becomes\n\\begin{equation}\n\\varphi (u, v, x, y)   = B e^{\\frac{\\lambda }{k}} e^{ - \\frac{\\lambda }\n{{k\\left( {1 - \nh_ + ^2 e^{2iku} } \\right)}}} \\left( {1 - h_ + ^2 e^{2iku} } \n\\right)^{\\frac{1}{2}\\left( {\\frac{\\lambda }{k} - 1} \\right)} e^{ - \ni\\lambda u} e^{ik_v v} e^{ik_{xy} x} e^{ik_{xy} y} - B.\n\\label{jones-7}\n\\end{equation}\nIn the limit $h_+ \\to 0$ ({\\it i.e.} the gravitational background is  \nturned off) the scalar field in \\eqref{jones-7} becomes\n\\begin{equation}\n\\varphi _0  (t,x,y,z) = \nB e^{ - i\\lambda u} e^{ik_v v} e^{ik_{xy} x} e^{ik_{xy} y} \n-B \\rightarrow\n B e^{i\\left( {k_v  + \\lambda } \\right)t} e^{i\\left( {k_v  - \\lambda } \n \\right)z} e^{ik_{xy} x} e^{ik_{xy} y} -B ~,\n \\label{jones-8}\n\\end{equation}\n\\noindent where in the last step we have converted back to the original\n$t,x,y,z$ coordinates. One can see that $k_v + \\lambda$ plays the role\nof the wave energy and $k_v - \\lambda$ momentum in the $z$ direction. \nThe result in \\eqref{jones-8} is expected, since if the gravitational\nwave background is turned off one should recover a plane wave traveling\nin free space, which is what the solution in \\eqref{jones-8} represents. \n\nTaking the limit where all the wave numbers\/momenta go to zero \n({\\it i.e.} $k_v, \\lambda, k_{xy} \\to 0$) in equation \\eqref{jones-7}\none would expect the scalar field to vanish. However on taking this\nlimit one finds instead that \n\\begin{equation}\n\\varphi (u, v, x, y)   = B \\left[\\left( {1 - h_ + ^2 e^{2iku} } \n\\right)^{-\\frac{1}{2}} - 1 \\right] \\approx \\frac{B}{2} h_ + ^2 e^{2iku}\n+ \\frac{3B}{8} h_ + ^4 e^{4iku} ~.\n\\label{jones-9}\n\\end{equation}\nThe result in \\eqref{jones-9} shows that even when one tries to take\nthe wave to its vacuum state, namely $k_v , \\lambda, k_{xy} \\to 0$,\nthere is a non-vanishing and non-trivial scalar field. This \nnon-vanishing scalar field is the field\/field quanta created by the \ngravitational wave background. Note that if one takes $h_+ \\to 0$\nin \\eqref{jones-9} that one does get the expected value for the scalar\nfield $\\varphi \\to 0$ \\footnote{Setting the constant $C=-B$ in\n\\eqref{jones-6} is done to get $\\varphi \\to 0$ in this limit. If one takes \n$C=0$ the $h_+ \\to 0$ limit of \\eqref{jones-9} would be\n$\\varphi \\to B$ which is also a vacuum solution to the wave equation\nfor the massless $\\varphi$, but having $\\varphi \\to 0$ is more ``natural\".}\nThe four-current associated with $\\varphi$ is given by the standard\nexpression $j_\\mu = -i (\\varphi \\partial _\\mu \\varphi ^* - \\varphi^* \n\\partial_\\mu \\varphi)$. Inserting this solution from \\eqref{jones-7} in the expression for the four-current and time averaging gives \\cite{Jones16}\n\\begin{equation}\n\\langle j_\\mu \\rangle = -2 B^2 \\lambda - B^2 h_+ ^4 \\left(\n\\frac{9}{2} \\frac{\\lambda ^3}{k^2} -\\frac{12 \\lambda ^2}{k}\n+ \\frac{13}{2} \\lambda - k \\right) ~.\n\\label{jones-10}\n\\end{equation}\nThe constant $B$ is determined by choosing a normalization condition or\nconvention. Following references \\cite{stahl} and \\cite{Jones16} \nwe pick the normalization condition $B = \\frac{1}{\\sqrt{2 k V}}$. \nOther possible normalization conditions for $B$ are \ndiscussed in \\cite{halzen}. With this normalization the vacuum \nscalar field from \\eqref{jones-9} reads\n\\begin{equation}\n\\varphi (u, v, x, y)   =    \\frac{1}{\\sqrt{2 k V}}\n\\left[\\left( {1 - h_ + ^2 e^{2iku} } \n\\right)^{-\\frac{1}{2}} - 1 \\right] \\approx \n\\frac{1}{2 \\sqrt{2 k V}} h_ + ^2 e^{2iku}\n\\left( 1 + \\frac{3}{4} h_ + ^2 e^{2iku} \\right) ~.\n\\label{jones-11}\n\\end{equation}\nTime averaging this vacuum current from \\eqref{jones-10} gives\n\\begin{equation}\n\\langle j_\\mu \\rangle =   \\frac{\\rm{sign} (k) h_+ ^4 }{2 V}~.\n\\label{jones-12}\n\\end{equation}\n\n\\noindent In \\eqref{jones-11} we are using a normalization that assumes the \nscalar field is in a box of volume $V$. In the previous section we took  \n$B=\\frac{1}{(2 \\pi) ^{3\/2}}$ -- see \\eqref{garr-4} \\eqref{garr-5}.\n\nEquation \\eqref{jones-10} gives the effect, in terms of the \nfour-current, of passing a massless scalar field through a \ngravitational wave. On setting all the energy-momentum of the\nscalar field to zero one finds, from equations \\eqref{jones-11} and \n\\eqref{jones-12}, that the scalar field and scalar field current\ndo not vanish. This represents the production \nof scalar field\/scalar field quanta from the gravitational wave\nbackground. \n\nFollowing \\cite{Jones17} the ratio of the \nproduced electromagnetic radiation \\eqref{emFlux0} \nto the gravitational \\eqref{GWluminsity} radiation can be written \ndown in terms of electromagnetic and gravitational Newman-Penrose\nscalars from  \\eqref{Phi2} and \\eqref{Psi4},  \n\n\\begin{equation}\n\\frac{{dE_{em} }}{{dE_{gm} }} = \\frac{{\\left( {\\frac{1}{{4\\pi }}\\left| {\\Phi _2 } \\right|^2 } \\right)}}{{\\left( {\\frac{1}{{16\\pi k^2 }}\\left| {\\Psi_4 } \\right|^2 } \\right)}} = \\frac{F_{em}}{F_{gw}}.\n\\label{emgwRatioA}\n\\end{equation}\n\n\\noindent Switching to a normalization where $B=1$ in \\eqref{jones-9} \nthe amplitude of the leading term of the scalar field is $\\frac{h_+^2}{2}$.\nUsing this in the expression for $|\\Phi_2|^2$  calculated \nin \\eqref{emNPscalar}  we get $|\\Phi_2|^2 =  2 k^2 h_+^4$. \nNext from \\eqref{Psi4} we recall that \n$|\\Psi _4 |^2 = 4 h_+^2 k^4$ for a gravitational plane we. Using all this\nin \\eqref{emgwRatioA} yields a relationship between the electromagnetic wave \nflux and gravitational wave flux\n\n\\begin{equation}\nF_{em}  = 2 h_+^2 F_{gw}.\n\\label{emgwRatioB}\n\\end{equation}\n\n\\noindent Since the gravitational radiation is proportional to $h_+^2$ the electromagnetic production will be proportional to $h_+^4$. The result in equation \\eqref{emgwRatioB} is consistent with the result in equation\n\\eqref{jones-12}.\n\n\\subsection{Unruh-Dewitt detector approach}\n\nAnother approach to study the connection between gravitational and \nelectromagnetic radiation is through the use of an Unruh-DeWitt detector\n\\cite{unruh-det,dewitt-det}. An Unruh-DeWitt detector is a two-state, \nquantum system which is placed in a given space-time background. If the\nUnruh-DeWitt detector is excited from the low energy state to the \nhigh energy state, this is taken to indicate that the given space-time\nhas produced field quanta in order to excite this transition. \nTwo common examples of the use of an Unruh-DeWitt detector are placing \nit in the Schwarzschild space-time \\cite{davies,hawking} of a black hole or \nplacing it the Rindler space-time of an accelerating observer \\cite{unruh}. \nIn the first case the Unruh-DeWitt detector will detect the photons from \nHawking Radiation and in the second case the Unruh-DeWitt detector will \ndetect the photons from Unruh Radiation.\n\nIn this subsection we will summarize the work of reference \\cite{Jones15}\nwhich calculates the response of an Unruh-DeWitt detector interacting with \na plane gravitational wave. The expression for the spectrum, $S(E)$, \nof an Unruh-DeWitt detector is given by \n\\begin{equation}\n\\label{UD-1}\nS(E) = n_{general} - n_{inertial} = 2 \\pi \\rho (E) F(E) ~.\n\\end{equation}\nIn equation \\eqref{UD-1} $n_{general}$ and $n_{inertial}$ are the\nphoton density of a general space-time and inertial space-time\nrespectively. The difference between these two is a measure of the\nphotons created due to the general space-time. The terms $\\rho (E)$\nand $F(E)$ are, respectively, the density of states and response \nfunction both as a function of energy \\cite{davies,letaw,akhmedov,wilburn,rad}. \n\nThe detector response function is given by\n\\begin{equation}\n\\label{UD-2}\nF(E) = \\int _{-\\infty} ^{+\\infty} e^{-i \\Delta \\tau \\Delta E}\n(G^+ _{general} (\\Delta \\tau) - G^+ _{inertial} (\\Delta \\tau))\nd (\\Delta \\tau ) ~,\n\\end{equation}\nwhere we recall that $\\hbar =1$ and $c=1$ in the above\nformulas. $\\Delta E = E_{up} - E_{down}$ is the energy difference \nbetween the two states of the Unruh-DeWitt detector. For simplicity \nwe assume $E_{down} = 0$ so that $\\Delta E \\to E_{up} \\to E$ and \nthus the response function is written as just a function of $E$. \nThe terms $G^+ _{general} (\\Delta \\tau)$ and  $G^+ _{inertial} \n(\\Delta \\tau))$ are the  Wightman functions \\cite{davies,Jones15} for the detector path in a general space-time and \nthe detector path in the inertial space-time. The\nWightman function depends on the proper time difference \n$\\Delta \\tau$ \nfor the path through the given space-time. The space-time path for \nthe inertial  detector is $x^\\mu (\\Delta \\tau ) = (\\Delta \\tau , 0, \n0, 0)$. The Wightman function for this inertial detector is\n\\begin{equation}\n\\label{UD-3}\nG_{inertial} ^+ = \\frac{1}{4 \\pi ^2 x^\\mu x_\\mu} =\\frac{1}{4 \\pi ^2 \n\\Delta \\tau ^2} ~.\n\\end{equation}\nFor a gravitational wave traveling in the $+z$ direction and having\n$+$ polarization the space-time path is given by $x^\\mu (\\Delta \n\\tau ) = ( \\gamma \\Delta \\tau , \\Delta x, 0, 0)$, where $\\Delta x$ \nis the spatial displacement of the detector due to the gravitational \nwave and $\\gamma ^{-2} = 1 - \\Delta {\\dot x} ^2$. Without loss of \ngenerality the detector is taken to be aligned along the $x$ direction. \nThe Wightman function for the gravitational wave is \n\\begin{equation}\n\\label{UD-4}\nG_{wave} ^+ = \\frac{1}{4 \\pi ^2 x^\\mu x_\\mu} =\\frac{1}{4 \\pi ^2 \n(\\gamma ^2 \\Delta \\tau ^2 - \\Delta x^2)} ~.\n\\end{equation}\nUsing these two Wightman functions from \\eqref{UD-3} \\eqref{UD-4} in \n\\eqref{UD-2} we find\n\\begin{equation}\n\\label{UD-5}\nF(E) = \\frac{1}{2\\pi^2 }\\int _0 ^{+\\infty} \\cos (E \\Delta \\tau)\n\\left( \\frac{1}{ (\\gamma ^2 \\Delta \\tau ^2 - \\Delta x^2)}- \n\\frac{1}{\\Delta \\tau ^2} \\right)\nd (\\Delta \\tau ) ~.\n\\end{equation}\nTo evaluate \\eqref{UD-5} we need to give $\\Delta x$ as a function of \n$\\Delta \\tau$. This is done using the ${\\dot x} = (1 + \\frac{1}{2}h)$ \n\\cite{EFTaylor} which is the expression for the trajectory along a null \ngeodesic, to first order, for a gravitational wave background \ncharacterized by the amplitude $h (\\Delta \\tau, \\theta, \\psi) = h_0 \n\\sin^2 (\\theta) \\sin (2 \\psi ) \\sin (\\omega \\Delta \\tau )$. The angles\n$\\theta$ and $\\psi$ give the orientation of the axis of the detector \nwith respect to the incoming gravitational wave \\cite{hendry}. The \nseparation $\\Delta x$ between the particle undergoing geodesic \nmotion in the gravitational wave background characterized by \n$h (\\Delta \\tau, \\theta, \\psi)$ and an inertial observer is then\ngiven by $\\Delta x = (1 + \\frac{1}{2}h) \\Delta \\tau - \\Delta \\tau =\n\\frac{1}{2}h (\\Delta \\tau, \\theta, \\psi) \\Delta \\tau$. Using these\nresults in \\eqref{UD-5} and integrating over $\\Delta \\tau$ as well as \nintegrating over the orientation direction $\\theta$ and $\\psi$ gives\nthe detector response function as \\cite{Jones15}\n\\begin{equation}\n\\label{UD-6}\nF(E) = \\frac{3 \\pi}{256} h_0 ^2 (2 \\omega  - E)   \\Theta (2 \\omega - E),\n\\end{equation}\nwhere $\\Theta$ is the Heaviside step function. Thus $F(E) = 0 $ when\n$E > 2 \\omega$ which is a similar type of cut-off to that in \nmuon decay \\cite{halzen,griffiths}. This suggests a picture of\ngravitons ``decaying\" into photons -- $graviton + graviton \\to photon + \nphoton$ or $graviton \\to graviton + photon + photon$.\n\nUsing the response function from \\eqref{UD-6} and a density of\nstates $\\rho (E) = \\frac{E^2}{2 \\pi^2}$ \\cite{letaw} the spectrum can \nbe found from \\eqref{UD-1} as\n\\begin{equation}\n\\label{UD-7}\nS(E) = \\frac{3 }{256 \\pi \\hbar ^3 c^3} E^2 h_0 ^2 \n(2 \\hbar \\omega  - E)  \\Theta (2 \\hbar \\omega - E) ~.\n\\end{equation}\nWe have restored factors of $\\hbar$ and $c$ temporarily. The\nfunctional form of the spectrum from \\eqref{UD-7} is that of a\n$Beta (3,2)$ distribution which is reminiscent of particle decays. This\nagain supports the picture of gravitons decaying into photons. \n\nThe analysis of the present subsection is different from the proceeding\nsubsection in that here we place an Unruh-DeWitt detector\nin the presence of a gravitational plane wave background, whereas in\nthe previous subsection we focused on the response of the vacuum to \na gravitational wave. The Unruh-DeWitt calculation is closer in spirit to \nthe work of Gertsenshtein \\cite{Gertsenshtein60} where the gravitational \nwave interacts with a magnetic field. In both these cases there is some \nphysical object --  the Unruh-DeWitt detector or a magnetic field -- which \ninteracts with the gravitational wave. In the previous subsection the \ngravitational wave interacts with the quantum vacuum. Nevertheless all of \nthese calculations indicate that\na gravitational wave can create electromagnetic radiation, or in\nparticle language that gravitons can transform\/decay into photons. The work in \\cite{Calmet16}\nalso looks into this possibility of gravitons decaying\/transforming in to  photons and thus\nweakening the gravitational wave.\n\n\\section{Possible Observational Consequences\/Signatures}\n\nWhile gravitational waves have only been directly detected very recently, electromagnetic \nradiation has been observed for all of human existence. If gravitational waves produce \ncounterpart electromagnetic radiation as is outlined above, it is natural to ask what the \npossible observable consequence of this would be. In this section we address two possible observational\nconsequence: (i) the attenuation\/decay of the gravitational wave due to production electromagnetic radiation;\n(ii) the direction detection of the electromagnetic radiation produced by the gravitational wave. \n\n\\subsection{Decay\/attenuation of the gravitational wave}\n\nIf electromagnetic waves are produced from a gravitational wave, as suggested above, this\nshould weaken and attenuate the gravitational wave since this electromagnetic radiation must be\ncreated at the expense of the gravitational wave \\cite{Jones15,Jones16}. This is similar \nto how a black hole is conjectured to lose mass as a result of Hawking radiation -- the \nHawking radiation comes at the expense of the mass of the black hole. \n\nOne can connect particle\/field production rate, $\\Gamma$, with a current, $j_\\mu$, as in equation\n\\eqref{jones-12} via the relationship \\cite{stahl,nikolic,frob}\n\\begin{equation}\n    \\label{decay-1}\n    \\frac{\\Gamma}{V} \\Delta T \\approx | j_\\mu | ~,\n\\end{equation}\nwhere $\\Delta T$ is some characteristic time for the system and $V$ is the volume. Using $|j_\\mu| = \\frac{h_+ ^4}{2 V}$\nfrom \\eqref{jones-12} and taking $\\Delta T \\approx \\frac{1}{\\omega}$ (where $\\omega$ is the frequency of the gravitational\nwave) as the characteristic time we arrive at\n\\begin{equation}\n    \\label{decay-2}\n    \\Gamma \\approx \\frac{\\omega h_+ ^4}{2}~.\n\\end{equation}\nIf we denote the number of gravitons in the volume $V$ as, $N_g$ one can write out a rate of change of $N_g$ as\n\\begin{equation}\n    \\label{decay-3}\n    \\frac{d N_g}{dt} = - \\Gamma N_g \\to   c \\frac{d N_g}{dz} = - \\Gamma N_g ~.\n\\end{equation}\nIn the last step we have replaced $dt$ by $dz\/c$ since we want the decay as a function of distance rather than \ntime. Taking the number of gravitons to be proportional to the amplitude squared \\footnote{This is similar to QED where\nthe number of photons is proportional to the square of the vector potential -- $N_\\gamma \\propto A_\\mu A^\\mu$}\n({\\it i.e.} $N_g \\propto h_+ ^2$) and using the expression of $\\Gamma$ from \\eqref{decay-2} we arrive at an\nequation for how the amplitude, $h_+$, varies with distance, $z$,\n\\begin{equation}\n    \\label{decay-4}\nc \\frac{d {h_+ ^2}}{dz} = - \\frac{\\omega h_+ ^4}{2} (h_+ ^2) \\to \\frac{dh_+}{dz} = - \\frac{k h_+ ^5}{4}~.\n\\end{equation}\nOne can solve \\eqref{decay-4} for $h_+ (z)$ and find\n\\begin{equation}\n    \\label{decay-5}\nh_+ (z) = (k z + K_0) ^{-1\/4}~,\n\\end{equation}\nwhere $K_0 = (h_+ ^{(0)} ) ^{-4}$ and $h_+ ^{(0)}$ is the reference amplitude at $z=0$. From equation \n\\eqref{decay-5} one sees that the fall off of $h_+$ as a function of distance, $z$, is very slow. This is expected,\nsince this slow fall off tells us that the transformation of gravitational radiation (gravitons) into electromagnetic\nradiation (photons) is a very weak process. If a gravitational wave background did not produce electromagnetic radiation\nthen $h_+ (z)$ should remain constant (recall that in this plane wave approximation we do not take into account\nthe $\\frac{1}{r}$ fall of a real three dimensional wave).  \n\nTo get an idea of how weak the effect is we can calculate the ``half-distance\", $\\Lambda$,  which we define \nas the distance for the amplitude of the plane wave to fall to half of its initial value, $h_+ ^{(0)}$.  \nTaking $\\omega \\approx 300$ Hz, the approximate frequency of the signal from the first LIGO detection\n\\cite{LIGO}, gives $k = \\frac{\\omega}{c} = 10 ^{-6}$ m. Setting $h_+ (\\Lambda) = \\frac{1}{2} h_+ ^{(0)}$ \ngives the ``half-distance\" as\n\\begin{equation}\n    \\label{decay-6}\n\\Lambda = \\frac{15}{k (h_+ ^{(0)} )^4} = \\frac{1.5 \\times 10^7}{ (h_+ ^{(0)} )^4} \\rm{m}~,\n\\end{equation}\nIf one sets the ``half-distance\", $\\Lambda$,  equal to the size of \nthe observable Universe -- $\\Lambda = 10^{27}$ m -- then \nequation \\eqref{decay-6} gives an amplitude of $h_+ ^{(0)} \n\\approx 10^{-5}$, which is a very large amplitude. Equation\n\\eqref{decay-6} implies that as $h_+ ^{(0)}$ gets larger the \nhalf-distance, $\\Lambda$, gets smaller. Taking  $h_+ ^{(0)} \n\\approx 10^{-3}$ would give $\\Lambda \\approx 10 ^{19}$ m, which \nis 100 times smaller than the size of the Milky Way. We also want to stress again that\nthe above estimates based on equation \\eqref{decay-6} are for planes waves and do not take into account the\n$\\frac{1}{r}$ fall off of a more realistic three dimensional wave, but regardless they\nindicate that the decay\/attenuation of the gravitational wave due to vacuum production of\nelectromagnetic radiation is a small effect, except perhaps close to the source where \none might have amplitudes like $h_+ ^{(0)} \\approx 10^{-3}$ or larger.  \n\n\\subsection{Detection of electromagnetic radiation produced by gravitational waves}\n\nNext we look at the possibility of directly detecting the electromagnetic radiation that is\nproduced from the vacuum by the gravitational wave. Looking at \\eqref{jones-9} one can \nsee that the counterpart electromagnetic radiation production would have twice the\nfrequency of the gravitational wave. From \\eqref{jones-9} one can see there are also components that\nare at four times the frequency of the gravitational wave, but they are down by an extra factor of\n$h_+ ^2$ compared to the component at twice the frequency. \n\nThe first problem that occurs in potentially detecting the counterpart electromagnetic radiation is that\nit will have a very low frequency (VLF) and thus a very large wavelength. For example the discovery paper \n\\cite{LIGO} reported frequencies for the gravitational wave on the order of $100~\\rm{Hz}$. Even doubling this,\nthe electromagnetic radiation would have a frequency and wavelength of  $200~\\rm{Hz}$ and $1.5 \\times 10 ^6 \\rm{m}$\nrespectively. \n\nA second problem with detecting the VLF counterpart electromagnetic radiation is that there are various cutoff\nfrequencies due to the plasma in space. In the illustration and table below we give the plasma cutoffs for a detector\nlocated in one of three locations: on the Earth, in space but near Earth's orbit, and finally in interstellar space\nas shown in Fig. \\eqref{CutoffsFig}. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=130mm]{Interstellar.jpeg}\n\\begin{tabular}{|c|c|}\n\\hline  ~~ Region ~~  & \\  ~~ Observable frequency range ~~       \n\\\\   \n\\hline  On Earth & \\ $> 10 $ MHz          \n\\\\  \n\\hline  Interplanetary space  (near Earth's orbit) & \\ $>$ $20~\\rm{kHz} - 30~\\rm{kHz} $  \n\\\\\n\\hline  Interstellar space (outside the heliosphere) & $ > 2 $ kHz \n\\\\   \n\\hline\n\\end{tabular}\n\\caption{Log scale illustration of the regions of space within our solar system and galaxy \\cite{UCSB} \\label{MediumFig} and the associated plasma frequency cutoff in each region \\cite{Jones17,Lacki10}.}\n\\label{CutoffsFig}\n\\end{figure}\n\nFor a detector on Earth one can detect electromagnetic radiation with a frequency of $10~\\rm{MHz}$ or larger. \nAssuming this electromagnetic wave came from production by a gravitational wave, this would require a\ngravitational wave frequency of $5~\\rm{MHz}$. Since non-exotic gravitational waves sources are expected to have \nfrequencies that are orders of magnitude lower than this, this rules out an Earth based detector for such VLF electromagnetic \nradiation. \n\nFor a detector at the outer edge of the Solar System, near interstellar space, one has a plasma cutoff of \n$2~\\rm{kHz}$ which would require a gravitational wave frequency of $1~\\rm{kHz}$. The fundamental ({\\it i.e.} \nf-modes) of neutron star quakes have frequencies in the range $1 - 3~\\rm{kHz}$ \\cite{Kokkotas97} and thus upon \ndoubling this frequency could produce counterpart VLF electromagnetic radiation above the $2~\\rm{kHz}$ cutoff. \nIn fact \nthe Voyager probes did detect such VLF electromagnetic radiation \\cite{Kurth84} in the range of $2-3 ~\\rm{kHz}$\nshowing that detection of such VLF electromagnetic radiation is possible \\footnote{The source of the Voyager \ndetection of this VLF electromagnetic radiation was a mystery for some time, but the source of this\nVLF radiation is now thought to be due to the interaction of the solar wind with ions in the outer heliosphere \nduring times of intense solar activity \\cite{Kurth03, Webber09}.} Thus\none could detect counterpart VLF electromagnetic radiation from\nthe $f$-modes of neutron star quakes if the neutron star were\nclose enough. However, it requires sending a probes to the edge of\nthe Solar System or beyond. \n\nGiven that sending probes to the edge of the Solar System is costly both in terms of time and money \none could ask if there are gravitational wave sources which would give rise to VLF electromagnetic \nradiation, which could be detected near Earth's orbit. From Fig. \\eqref{CutoffsFig}  one can see that for\ndetection one needs the electromagnetic radiation to have a frequency greater than $20 - 30$ kHz. This implies that\nthe gravitational wave vacuum producing the electromagnetic radiation would need to have a frequency in the range\n$10-15$ kHz. Theoretical models show that of gravitational wave of this kHz frequency range  could be \nproduced from neutron star oscillations \\cite{Kokkotas97,Kokkotas01}. There are different types of \nneutron star oscillation modes. Three\nof the most common are: (i) {\\it p}-modes or ``pressure modes\" \\cite{Kokkotas97} with a frequency \nrange $5 - 9~ \\rm{kHz}$; (ii) {\\it f}-modes or  ``fundamental modes\" \\cite{Kokkotas97} with a\nfrequency range of $1 - 3~\\rm{kHz}$; (iii) {\\it w}-modes or ``space-time modes\" \\cite{Anderson96} with \na frequency range of $8 - 16~\\rm{kHz}$ or greater. From this list of oscillation modes \nthe {\\it w}-modes have the most promising frequency range in terms of detection of the \ncounterpart VLF electromagnetic radiation. \n\nWe now want to give a rough estimate of the strength of the electromagnetic flux produced by \ngravitational waves coming from a  {\\it w}-mode oscillation of a neutron star quake. First from\n\\cite{abadie} the gravitational wave amplitude at Earth for {\\it f}-mode generated gravitational waves \nfrom a neutron star that is $1$ kpc  or $3\\times 10^{19}$ m distant from Earth would be of order\n$h_+ \\sim 10^{-23}$. The associated {\\it w}-modes gravitational wave amplitude is expected to be\ndown from this by at least one order of magnitude $h_+ \\sim 10^{-24}$. Using this {\\it w}-mode \namplitude and the $1\/r$ fall off relation \n\\begin{equation}\n\\label{detect-1}\n    h_+ = 10^{-24} \\frac{1 ~ {\\rm kpc}}{r}\n\\end{equation}\nwe can determine the amplitude at some point close to the source. For this we take \n$r^{(0)}=3 \\times 10^4$ m \\cite{Jones17} -- this is far enough from the neutron star that the plane wave \napproximation we have used throughout should apply, at least roughly. Using \\eqref{detect-1} and \n$r^{(0)}=3 \\times 10^4$ m we find that the {\\it w}-mode amplitude near the source would be of \norder $h_+^{(0)} \\sim 10^{-9}$. Using this amplitude, a frequency of 10 kHz in the {\\it w}-mode\nrange $8-16$ kHz, we can determine the flux of the gravitational wave near the source \n({\\it i.e.} at $r^{(0)}=3 \\times 10^4$ m) using the formula \\cite{Schutz96}\n\\begin{equation}\n    \\label{detect-2}\n    F^{(0)} _{gw} = \\frac{c^3}{16 \\pi G} |{\\dot \\epsilon}|^2 = \n    \\left( 3 \\times 10 ^{35} \\frac{W s^2}{m^2} \\right) h_+^2 f^2 =\n    3 \\times 10^{25} \\frac{W}{m^2} ~,\n\\end{equation}\nwhere $\\epsilon = h_+ e^{iku}$ as defined below equation \\eqref{GWmetric}. We can now calculate\nthe flux of the counterpart VLF electromagnetic radiation using \\eqref{detect-2} in\n\\eqref{emgwRatioB} to give \n\\begin{equation}\n    \\label{detect-3}\n    F^{(0)} _{em} = 2 \\times (10^{-9})^2 \\times (3 \\times 10^{25} \\frac{W}{m^2})\n    = 6.0 \\times 10^7 \\frac{W}{m^2}~.\n\\end{equation}\nFrom \\eqref{detect-2} and \\eqref{detect-3} we find that $F^{(0)} _{gw} \\gg F^{(0)} _{em}$\nwhich is expected -- the electromagnetic radiation produced is much smaller than the\ngravitational wave which produced it. However, $F^{(0)} _{em}$ is nevertheless large enough\nthat even taking into account the $1\/r$ fall off one could potentially detect this\nelectromagnetic radiation at the location of the Earth's orbit. Taking\nthe flux $F^{(0)} _{em}$ from \\eqref{detect-3} one can determine the flux at the \nlocation of the Earth assuming that the neutron star is 1 kpc away.\n\\begin{equation}\n    \\label{detect-4}\n    F_{em}= F^{(0)} _{em} \\left( \\frac{r^{(0)}}{1 kpc}\\right) \\sim \n    6.0 \\times 10 ^{-23} \\frac{W}{m^2} ~,\n\\end{equation}\nwhere $r^{(0)} = 3 \\times 10 ^4$ m from before. A flux of the magnitude in\n\\eqref{detect-4} could be detected \\cite{Jones17} and given the frequency range\nof the {\\it w}-modes the associated VLF electromagnetic radiation would \nhave a\nfrequency that is above the plasma cutoff at the location of Earth's\norbit as given in Fig. \\eqref{CutoffsFig}. Thus the proposal to detect such\nthe hypothesized co-produced VLF electromagnetic radiation, coming {\\it w}-modes of\nneutron star quakes, would be to place a satellite capable of detecting such\nradiation near earth's orbit \\cite{Jones17, Gretarsson18}. The old Explorer 49\nsatellite was capable of detecting such VLF electromagnetic radiation. The Explorer\n49 satellite was a Lunar orbiting satellite which was periodically occulted\nby the Moon in order to block out interference from Solar emissions. The\noccultation allows one to detect weak signals like \\eqref{detect-4} above\nthe interference from the Sun. \n            \n\\section{Conclusion and Future prospects}\n\nFor over 100 years there was nothing to support Faraday's expectation of a relationship between \ngravity and electromagnetism. However, in the past 50 years we have seen the development of considerable \ntheoretical support for this relationship and in particular the relation between gravitational and \nelectromagnetic radiation. The earliest work was by Gertsenshtein \\cite{Gertsenshtein60} \ndemonstrating that electromagnetic radiation can produce gravitational radiation. This \nwas followed in 1975 by Skobelev \\cite{Skobelev75} who calculated the small but non-zero \namplitudes for the $graviton + graviton \\to photon + photon$ processes. Beginning around the same time as reference \\cite{Skobelev75}, there was work that \nexamined the production of electromagnetic fields\/multiple photons from a gravitational background \\cite{unruh-det,dewitt-det,hawking,unruh,Jones15}. Rrcently we have worked on calculations of the production of electromagnetic radiation \nby gravitational waves propagating in vacuum \\cite{Jones16,Jones17,Jones18}. Perhaps \nin the next 50 years we will see empirical evidence  of the relation between \ngravitational and electromagnetic radiation by either direct or indirect observation.\n\nThe most promising possibility for direct observation is the detection of VLF counterpart \nproduction by gravitational waves from neutron star quakes  \\cite{Jones17}. This would \nonly be possible using space based detectors such as the Voyager missions \n\\cite{Kurth84,Kurth03,Webber09} or with a lunar occulted detector similar to the \nExplorer 49 mission  \\cite{Gretarsson18}. However, detection of counterpart production \nlocally would be limited to the highest frequencies of the counterpart production \nfrom neutron star gravitational waves. Detectors in the outer heliosphere would be \nmuch more effective and rather remarkably the Voyager space craft are still making \nobservations in the $2-4~\\rm{kHz}$ range of expected counterpart production \\cite{Gurnett15}.\n\nEven without direct observation it is possible that the counterpart production of electromagnetic radiation would have important applications in astrophysical processes. One intriguing possibility is in the energetics of core collapse supernovae. The prompt production of gravitational waves from the core collapse would produce gravitational waves with quadruple amplitudes on the order of $1~\\rm{m}$ and strain amplitude of something like $10^{-5} - 10^{-4}$ in the star layers just outside the core. These strain amplitudes have the potential of producing counterpart radiation of sufficient energy to contribute to the energetics of the supernovae. Previous work on fully general relativistic magnetohydrodynamics \\cite{Font07} (MHD) have assumed the ``ideal\" MHD condition. This assumption suppresses any potential production of electromagnetic radiation from the strong gravitational wave background. More recent work \\cite{Obergaulinger14,Just18,Obergaulinger18} on the effects of magnetic fields and rotation on the energetics of core collapse supernovae have not been fully general relativistic and again could not include the energy from production of electromagnetic radiation by the outgoing gravitational wave.  Fully general relativistic MHD simulations have been implemented \\cite{Lehner12_86} for collapsing hyper-massive neutron stars but not for the study of core collapse supernovae. It is possible that the energy associated with electromagnetic  production by gravitational waves outside the iron core could contribute importantly to the supernova, but only fully general relativistic MHD simulations would account for this phenomena in the processes of core collapse and explosion.\n\nSince the production of electromagnetic radiation by gravitational waves is so fundamental it is likely that further study of this phenomena could illuminate our understanding of nature. One recent example of the potential importance of production of photons in a gravitational wave background is the investigation of graviton-photon oscillations in alternatives to general relativity \\cite{Cembranos18, ejlli}. This investigation did not directly study counterpart production and potential general relativity violations but does describe the significance of this phenomena in investigating theories of gravity. Counterpart production by gravitational waves \\cite{Ricciardone17} could also be important in studies of cosmology. Following the Planck epoch the Standard Model fields were still massless\nfor some time. It would be interesting to consider the production of the massless Standard Model particles by primordial gravitational waves during the grand unification epoch and prior to the Standard Model particles acquiring mass via the Higgs mechanism.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{To Do list}\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nIn the study of CP violation by CKM unitary triangle analysis,\nhadron matrix elements of four-fermion operators,\nsuch as $B_{\\rm K}$, play a vital role.\nAccurate calculations of this quantity from first principles\nare an important task for the lattice QCD community.\nIn such calculations, having chiral symmetry\nis crucial to avoid an operator mixing problem\nwhich causes uncontrollable systematic errors.\nAlthough lattice chiral fermions\n \\cite{Kaplan:1992bt,Shamir:1993zy,Neuberger:1997fp}\nare a clean formulation,\nthey require enormous computing power to perform dynamical simulations.\nIn comparison, ordinary fermion formulations, like Wilson type fermions\nand starggared fermions are relatively cheap.\nNowadays, however, thanks to the development of computer architecture\nand algorithms, dynamical simulations with\nlattice chiral fermions have become feasible\neven for three flavors \\cite{Noaki:2008gx}.\nIn particular,\nthe RBC\/UKQCD collaboration \\cite{Allton:2008pn} is currently using\ndomain-wall fermions (DWFs) to compute $B_{\\rm K}$.\nIn the course of their computation,\nthere are many sources of systematic errors which one has to control.\nAmong them, the non-perturbative renormalization (NPR)\ncould be serious.\nAt the moment, the collaboration has been using conventional schemes, such as,\nthe RI\/MOM scheme and its variants \\cite{Aoki:2007xm,Sturm:2009kb}.\nHowever, these schemes potentially contain ``large scale\nproblem\" which requires a quite large lattice volume.\nTo avoid such difficulties,\na new scheme was invented, known as the Schr\\\"odinger functional\n(SF) scheme \\cite{Luscher:1992an}.\nThis scheme provides a reliable way of estimating errors in the NPR.\nIf one wants to use this scheme for the renormalization\nof $B_K$ given by the RBC collaboration,\nfirst of all, one has to formulate DWF in the SF setup.\nThis is the purpose of this paper.\n\n\n\nWhile chiral fermions are useful\nfor computing the bare $B_{\\rm K}$\nto avoid the mixing problem, a formulation for such fermions\nin the SF setup was a non-trivial task\nbecause SF boundary conditions break chiral symmetry explicitly.\nWe will address this issue in the next section.\nHowever, Taniguchi \\cite{Taniguchi:2004gf} made the first attempt to formulate overlap fermions\nby using an orbifolding technique.\nSubsequently he provided a formulation for domain-wall fermions\nand then he and his collaborators \\cite{Nakamura:2008xz} calculated\na renormalized $B_{\\rm K}$ in quenched QCD.\nSint \\cite{Sint:2010eh} developed such\ntechniques by combining with a flavor twisting trick.\nHowever, these orbifolding formulations are constrained by\nthe requirement that the number of flavors be even.\nThus, apparently such formulations\nare incompatible with current trends toward dynamical three flavor simulations.\nTo overcome this difficulty, L\\\"uscher \\cite{Luscher:2006df} gave\na completely different approach\nrelying on a universality argument,\ndimensional power counting and symmetry considerations.\nSome perturbative calculations were performed in Ref.~\\cite{Takeda:2007ga}.\nA crucial property of this formulation is that\nthere is no restriction on the number of flavors.\nSince only overlap fermions were considered in Ref.~\\cite{Luscher:2006df},\nour main purpose here is to formulate\nthe other chiral fermions, namely, domain-wall fermions.\n\n\n\nThe rest of the paper is organized as follows.\nSection \\ref{sec:formulation} gives the formulation of domain-wall\nfermions in the SF setup, after a brief\nreview of the universality argument.\nWe present several pieces of numerical evidence\nin Section \\ref{sec:spectrum} and \\ref{sec:oneloop}\nto show that our formulation is working properly.\nWe also discuss the lattice artifacts\nfor the step scaling function in Section \\ref{sec:SSF}.\nIn the last section, we conclude by giving some remarks and outlook.\n\n\n\n\n\n\\section{Formulation}\n\\label{sec:formulation}\nIn the following, we assume that the reader\nis familiar with the SF in QCD \\cite{Luscher:1992an,Sint:1993un}.\nAfter giving a brief reminder of the universality argument,\nwe give a formulation for DWF\nand finally check the chiral symmetry breaking structure numerically.\n\n\\subsection{Universality argument}\n\nIn the massless continuum theory,\nthe Dirac operator $D$ satisfies the anti-commutation relation with\n$\\gamma_5$\n\\begin{equation}\n\\gamma_5D+D\\gamma_5=0.\n\\label{eqn:Dg5}\n\\end{equation}\nThe above is true even in the SF setup, although the boundary conditions,\n\\begin{eqnarray}\nP_+ \\psi(x)=0 &\\mbox{ at }& x_0=0,\n\\label{eqn:SFBC0}\n\\\\\nP_- \\psi(x)=0 &\\mbox{ at }& x_0=T,\n\\label{eqn:SFBCT}\n\\end{eqnarray}\nwith $P_\\pm=(1\\pm\\gamma_0)\/2$,\nbreak chiral symmetry explicitly.\nEq.(\\ref{eqn:Dg5}) means that the operator itself\ndoes not know about boundary conditions.\nIn the continuum theory, information such as boundary conditions\nis embedded in the Hilbert space.\nIn fact, the corresponding propagator,\nwhich is a solution of the inhomogeneous equation,\n\\begin{equation}\nDS(x,y)=\\delta(x-y),\n\\end{equation}\nfails to satisfy the anti-commutation relation.\nInstead, it follows\n\\begin{eqnarray}\n\\lefteqn{\\gamma_5S(x,y)+S(x,y)\\gamma_5=}\\nonumber\\\\\n&&\n\\int_{z_0=0}d^3{\\bf z} S(x,z)\\gamma_5 P_- S(z,y)\n+\n\\int_{z_0=T}d^3{\\bf z} S(x,z)\\gamma_5 P_+ S(z,y).\n\\label{eqn:Sg5}\n\\end{eqnarray}\nThis can be derived by using partial integration\non the SF manifold which has two boundaries\nat time slice $x_0=0$ and $T$.\nThe non-vanishing right-hand side in eq.(\\ref{eqn:Sg5})\nshows an explicit chiral symmetry breaking.\nSince such a breaking term is supported\nonly on the time boundaries,\nthe chiral symmetry is preserved in a bulk.\n\n\nIf someone naively tries to formulate\nchiral fermions on the lattice,\none may define an overlap operator, for example,\nwith the Wilson kernel in the SF setup \\cite{Sint:1993un}.\nHowever such an operator immediately satisfies\nthe Ginsberg-Wilson relation and thus cannot reproduce\neq.(\\ref{eqn:Sg5}) in the continuum limit.\nThis indicates that such naive formulation\ndoes not work and furthermore may\nbelong to another boundary universality class\nwhich is not what we want.\nIn this way, it is a non-trivial task to\nformulate chiral fermions in the SF setup.\n\n\nSome years ago, L\\\"uscher \\cite{Luscher:2006df} proposed a clever way\nto overcome this situation.\nFirst, consider the relation for the propagator\nin eq.(\\ref{eqn:Sg5}).\nThis indicates that the GW relation has to be modified\nby boundary effects.\nThus one has to find a modified overlap operator\nwhich breaks the GW relation near the time boundaries and correctly\nreproduces eq.(\\ref{eqn:Sg5}) in the continuum limit.\nActually, finding such a modified operator is not so hard.\nHowever, a new question naturally arising\nis how the SF boundary conditions emerge.\nFor the Wilson fermion case \\cite{Sint:1993un},\nbecause there is a transfer matrix,\nit is natural for fermion fields\nto follow the SF boundary conditions.\nHowever for chiral fermions,\nthere is no such transfer matrix\nwhich can be defined from nearest neighbor interaction\nin the time direction.\nTherefore it is not an easy task.\n\n\nL\\\"uscher \\cite{Luscher:2006df} gave another point of view\nto see how fields respect the boundary condition.\nIn the quantum field theory,\nthe correlation function can tell you\nwhat kinds of boundary conditions are imposed.\nAs an example, let us see\nhow the boundary conditions emerge for Wilson fermions\nwhose action is given by\n\\begin{eqnarray}\nS_{\\rm w}\n&=&\n\\sum_x\\bar\\psi(x)\nD_{\\rm w}(m)\n\\psi(x),\n\\\\\nD_{\\rm w}(m)\n&=&\n\\frac{1}{2}\n\\left[\n\\sum_\\mu (\\nabla_\\mu+\\nabla^{\\ast}_\\mu)\\gamma_\\mu\n-a\\sum_\\mu \\nabla_\\mu^{\\ast}\\nabla_\\mu\n\\right]+m,\n\\end{eqnarray}\nwhere $\\nabla_\\mu$ and $\\nabla_\\mu^{\\ast}$ are\nforward and backward covariant difference operators respectively,\n\\begin{eqnarray}\n\\nabla_\\mu\\psi(x)\n&=&\n\\frac{1}{a}\n\\left[\nU(x,\\mu)\\psi(x+a\\hat\\mu)-\\psi(x)\n\\right],\n\\\\\n\\nabla_\\mu^{\\ast}\\psi(x)\n&=&\n\\frac{1}{a}\n\\left[\n\\psi(x)-U(x-a\\hat\\mu,\\mu)^{-1}\\psi(x-a\\hat\\mu)\n\\right].\n\\end{eqnarray}\nIn the SF setup, the sum over $x$ in the action is a little bit subtle.\nWe assume that the dynamical fields are \n$\\psi(x)$ with $a\\le x_0 \\le T-a$ and\nthe fields $\\psi(x)$ with $x_0 \\le 0$ and $T\\le x_0$ are set to zero.\nFor this setup,\nthe propagator may be defined by\n\\begin{eqnarray}\n\\langle\\eta(x)\\bar\\psi(y)\\rangle&=&a^{-4}\\delta_{x,y},\n\\label{eqn:etapsi}\n\\\\\n\\eta(x)&=&\\frac{\\delta S_{\\rm w}}{\\delta\\bar\\psi(x)}.\n\\label{eqn:etaS}\n\\end{eqnarray}\nFor $2a \\le x_0 \\le T-2a$, eq.(\\ref{eqn:etaS}) turns out to be\n\\begin{equation}\n\\eta(x)=D_{\\rm w}(m)\\psi(x).\n\\end{equation}\nOn the other hand, at $x_0=a$, we obtain\n\\begin{eqnarray}\n\\eta(x)&=&\n\\frac{1}{a}P_+\\psi(x)-\\nabla_0P_-\\psi(x)\n\\nonumber\\\\\n&+&\n\\frac{1}{2}\n\\left[\n \\sum_k (\\nabla_k+\\nabla^{\\ast}_k)\\gamma_k\n-a\\sum_k \\nabla_k^{\\ast}\\nabla_k\n\\right]\n\\psi(x)\n+m\\psi(x).\n\\label{eqn:eta}\n\\end{eqnarray}\nBy substituting eq.(\\ref{eqn:eta}) into eq.(\\ref{eqn:etapsi}) with $x\\neq y$\nwe obtain\n\\begin{equation}\n\\frac{1}{a}P_+\\langle\\psi(x)\\bar\\psi(y)\\rangle|_{x_0=a}\n-\n\\nabla_0P_-\\langle\\psi(x)\\bar\\psi(y)\\rangle|_{x_0=a}\n+...\n=0.\n\\end{equation}\nIn the continuum limit, the first term is dominant\n\\begin{equation}\n\\frac{1}{a}P_+\\langle\\psi(x)\\bar\\psi(y)\\rangle|_{x_0=0}=0.\n\\end{equation}\nThis shows that in the naive continuum limit,\nthe Dirichlet type boundary condition ($P_+\\psi|_{x_0=0}=0$)\nis stable against the Neumann one ($\\nabla_0P_-\\psi|_{x_0=0}=0$), and in the end\nthe SF boundary conditions in eq.(\\ref{eqn:SFBC0}) emerge.\nIt is plausible that similar things happen also for the chiral fermions case,\nas long as the locality and symmetry\nare kept in a proper way,\nalthough we expect that the coefficient of the lowest dimensional operators\n($\\frac{1}{a}P_+\\psi$)\nmay be different from the above case,\nand more higher dimensional terms may appear in eq.(\\ref{eqn:eta}).\nThe important point here is that\ncontinuum SF boundary conditions emerge dynamically\nin the continuum limit of the correlation function.\nThis boundary condition is natural\nand automatically guaranteed to emerge\nfrom the dimensional order counting argument.\nTherefore, when we construct chiral fermions in the SF,\nwe only have to prepare a modified operator\nby introducing an additional term\nwhich breaks the chiral symmetry near the time boundaries.\nOnce this is fulfilled, then\nsuch an operator automatically\nturns out to be the desired one in the continuum limit without fine tuning.\nA final important note is that the form of the boundary term is irrelevant\nas long as it will go into a preferred boundary universality class.\nTherefore, there is a large amount of freedom when choosing boundary terms\nand one can use this freedom for practical purposes.\n\n\n\nFollowing these guiding principles,\nL\\\"uscher \\cite{Luscher:2006df} proposed the operator:\n\\begin{eqnarray}\n\\bar{a}D_{\\rm N}&=&1-\\frac{1}{2}(U+\\tilde U),\n\\\\\nU&=&A(A^{\\dag}A+caP)^{-1\/2},\\hspace{7mm}\\tilde U = \\gamma_5 U^{\\dag} \\gamma_5,\n\\\\\nA&=&1+s-aD_{\\rm w}(0),\\hspace{7mm}\\bar{a}=a\/(1+s),\n\\label{eqn:overlap}\n\\end{eqnarray}\nwith the parameter in the range $|s|<1$.\n$D_{\\rm w}(0)$ is the massless Wilson operator in the SF.\nThe key point here is the presence of the $P$\nterm in the inverse square root which is given by\n\\begin{equation}\naP(x,y)\n=\n\\delta_{{\\bf x},{\\bf y}}\\delta_{x_0,y_0}(\\delta_{x_0,a}P_-+\\delta_{x_0,T-a}P_+).\n\\end{equation}\nNote that this term is supported near the time boundaries\nand thus called a boundary operator.\nThe presence of this term breaks the GW relation explicitly\nand the breaking is given by\n\\begin{eqnarray}\n\\Delta_{\\rm B}=\\gamma_5D_{\\rm N}+D_{\\rm N}\\gamma_5-\\bar{a}D_{\\rm N}\\gamma_5D_{\\rm N}.\n\\end{eqnarray}\nIt was shown in ref.~\\cite{Luscher:2006df} that this term is local and\nsupported in the vicinity of the boundaries up to the\nexponentially small tails.\n\n\nAlthough this operator breaks chiral symmetry explicitly,\nother symmetries (the discrete rotational symmetries, $C$, $P$ and $T$,\nflavor symmetry and so on)\nhave to be maintained\nsince the boundary conditions in eq.(\\ref{eqn:SFBC0},\\ref{eqn:SFBCT}) are\ninvariant under these symmetries.\nIn addition, this operator has $\\gamma_5$-Hermiticity.\nIn this way, the universality formulation\ncan avoid breaking important symmetries, such as the flavor symmetry.\nThis is a distinctive feature of this formulation\ncompared with the orbifolding technique,\nwhere flavor symmetries cannot be maintained\nor, there is a constraint on the number of flavors.\n\n\n\nBefore leaving this subsection,\nlet us summarized the guiding principles\nof formulating chiral fermions in the SF setup.\nWhat we learned from this construction\nis that, for an original chiral fermion operator,\none has to introduce an additional term\nto break the chiral symmetry and\nthen demand that\nsuch breaking only appears near the time boundaries.\nFurthermore, one must\nmaintain important symmetries as well as $\\gamma_5$-Hermiticity.\nOnce these conditions are fulfilled,\nit is automatically guaranteed that the such a lattice operator\nwill correctly reproduce the continuum results\naccording to the universality argument.\n\n\n\\subsection{Formulation of domain-wall fermions }\nLet us apply the guiding principles given in the previous subsection\nto domain-wall fermions.\nWe propose a massless\\footnote{The mass term can be introduced\nin the usual way, namely\n$a^4m_{\\rm f}\\sum_{{\\bf x}}\\sum_{x_0=a}^{T-a}[\\bar\\psi(x,1)P_R\\psi(x,L_s)+\\bar\\psi(x,L_s)P_L\\psi(x,1)]$.}\ndomain-wall fermion action\n\\begin{equation}\nS=\na^4\\sum_{x,x^\\prime}\\sum^{L_s}_{s,s^\\prime=1}\n\\bar\\psi(x,s)(D_{\\rm DWF})_{xs,x^\\prime s^\\prime}\\psi(x^\\prime,s^\\prime),\n\\end{equation}\nwhere a massless operator with $L_s=6$\nfor example\\footnote{We restrict ourselves to an even number of $L_s$,\nwhich is the case usually implemented.}\nin four dimensional block form is given by\n\\begin{equation}\naD_{\\rm DWF}=\n\\left[\n\\begin{array}{cccccc}\na\\tilde{D}_{\\rm w}&-P_L&0&0&0&cB\\\\\n-P_R&a\\tilde{D}_{\\rm w}&-P_L&0&cB&0\\\\\n0&-P_R&a\\tilde{D}_{\\rm w}&-P_L+cB&0&0\\\\\n0&0&-P_R-cB&a\\tilde{D}_{\\rm w}&-P_L&0\\\\\n0&-cB&0&-P_R&a\\tilde{D}_{\\rm w}&-P_L\\\\\n-cB&0&0&0&-P_R&a\\tilde{D}_{\\rm w}\\\\\n\\end{array}\n\\right],\n\\label{eqn:DWF}\n\\end{equation}\nwith the chiral projections,\n\\begin{equation}\nP_{R\/L}=(1\\pm\\gamma_5)\/2.\n\\end{equation}\nWe also assume that the dynamical fields\nare $\\psi(x,s)$ with $a\\le x_0 \\le T-a$.\nThe block elements in eq.(\\ref{eqn:DWF}) are four dimensional operators\nand $a\\tildeD_{\\rm w}$ is given by\n\\begin{equation}\na\\tilde{D}_{\\rm w}=aD_{\\rm w}(-m_5)+1.\n\\end{equation}\nThe domain-wall height parameter usually takes a value in a range $0<am_5<2$.\n\n\nAn important ingredient here is the presence\nof $B$ in eq.(\\ref{eqn:DWF}) terms in the cross diagonal elements.\nThe reason for this $s$-dependence\nis to break the chiral symmetry\nin a similar way to the usual mass term \\cite{Christ:2004gc}.\nAs mentioned before, such chiral symmetry breaking should be\npresent only near time boundaries, therefore, we chose\nthe $B$ term as\n\\begin{equation}\nB(x,y)=\\delta_{{\\bf x},{\\bf y}}\\delta_{x_0,y_0}\\gamma_5\n(\\delta_{x_0,a}P_-+\\delta_{x_0,T-a}P_+),\n\\label{eqn:Bpm}\n\\end{equation}\nwhich is supported near the boundaries.\nIn this way, the time dependence is fixed.\nThe spinor structure ($\\gamma_5 P_\\pm$) is determined by imposing\nthe discrete symmetries C, P and T and $\\Gamma_5$-Hermiticity.\nThese requirements are not so strong to determine\nthe spinor structure completely and therefore there is some freedom.\nThe structure proposed here\nis only one of many solutions.\nActually, we examined several choices\nof the spinor structure in the boundary term\nand confirmed numerically the universal results\nin the continuum limit for the lowest eigenvalue.\nIn the following,\nwe take this boundary term in eq.(\\ref{eqn:Bpm}).\n\n\nThe boundary coefficient $c$ is supposed to be non-zero\nto correctly reproduce the continuum theory as we will see in\nSection \\ref{sec:spectrum}.\nIt also plays an important role to cancel\nboundary O($a$) cutoff effects and has\na perturbative expansion\n\\begin{equation}\nc=c^{(0)}+c^{(1)}g_0^2 + O(g_0^4).\n\\end{equation}\nIn the same way as in \\cite{Takeda:2007ga},\nwe tune the first coefficient $c^{(0)}$\nas a function of the domain-wall hight $am_5$,\n\\begin{eqnarray}\nc^{(0)}&=&0.5089-0.0067(am_5-1)+0.0488(am_5-1)^2\n\\nonumber\\\\\n&&\n-0.0216(am_5-1)^3+0.0673(am_5-1)^4.\n\\label{eqn:ctreeL}\n\\end{eqnarray}\nThis is valid in the region where\nbulk O($a$) can be neglected\n\\footnote{\nActually, we observe that\nO($a$) improvement program does not work for small $L_s$\nand values of $am_5$ which are far from $1$.\nFor example O($a$) terms in $f_A$ ad $f_P$\n(defined in appendix \\ref{subsec:operators})\nat tree level\ndo not vanish simultaneously with the same value of $c^{(0)}$.\n},that is, for sufficient large $L_s$.\nIn the process of this determination, one needs to define the operators\nof the axial vector current and pseudo scalar density.\nWe give their definition together\nwith that of the conserved axial current in appendix \\ref{subsec:operators}.\n\n\n\n\n\n\n\\subsection{Structure of chiral symmetry breaking at tree level}\n\\begin{figure}[!t]\n\\begin{center}\n  \\begin{tabular}{ccc}\n  \\hspace{-30mm}\n  \\scalebox{1.2}{\\includegraphics{Ls4_L20.eps}}&\n  \\hspace{-40mm}\n  \\scalebox{1.2}{\\includegraphics{Ls12_L20.eps}}&\n  \\hspace{-40mm}\n  \\scalebox{1.2}{\\includegraphics{Ls24_L20.eps}}\\\\\n  \\end{tabular}\n\\caption{\nThe color ranging $[-14,0]$ corresponds to the value of\n$\\ln||\\Delta^{(L_s)}(x_0,y_0)||_{\\rm spin}$\nfor the zero spatial momentum configuration with the parameters\n$T\/a=20$,\n$m_{\\rm f}=0$,\n$am_5=1$ and $c=1$.\n\\label{fig:GWbreaking}}\n\\end{center}\n\\end{figure}\n\nIn this subsection, \nlet us check an important properties of the operator defined\nin the previous section.\nA reader may worry that even though the additional\nboundary term is localized to time boundary,\nafter integrating over the fifth dimensional\ndegree of freedom such breaking effects may leak into the 4-dim bulk and\nruin the bulk chiral symmetry.\nTo settle this question, we numerically investigate the\nstructure of the chiral symmetry breaking\nby looking at the breaking of the GW relation\n\\begin{equation}\n\\Delta^{(L_s)}\n=\n\\gamma_5D_{\\rm eff}^{(L_s)}+D_{\\rm eff}^{(L_s)}\\gamma_5\n-2aD_{\\rm eff}^{(L_s)}\\gamma_5D_{\\rm eff}^{(L_s)},\n\\label{eqn:DeltaLs}\n\\end{equation}\nwith the effective four dimensional operator\n\\cite{Kikukawa:1999sy,Kikukawa:1999dk}\n\\begin{equation}\n\\detD_{\\rm eff}^{(L_s)}= \\det[D_{\\rm DWF}\/D_{\\rm PV}].\n\\end{equation}\nThe Pauli-Villars (PV) operator is defined as\nthe massive DWF operator with $am_{\\rm f}=-1$.\nTo obtain the effective operator,\nfirst of all, we have to define physical quark fields\n\\begin{eqnarray}\nq(x)&=&P_L\\psi(x,1)+P_R\\psi(x,L_s),\n\\label{eqn:q}\n\\\\\n\\bar{q}(x)&=&\\bar{\\psi}(x,1)P_R+\\bar{\\psi}(x,L_s)P_L.\n\\label{eqn:barq}\n\\end{eqnarray}\nIn terms of the propagator of domain-wall fermions defined from\n\\begin{equation}\nD_{\\rm DWF} S_{\\rm DWF}(x,y;s,t)=a^{-4}\\delta_{x,y}\\delta_{s,t},\n\\end{equation}\nthat of the physical field is expressed\n\\begin{eqnarray}\n\\left[q(x)\\bar{q}(y)\\right]_{\\rm F} \n&=&\nS_{\\rm q}(x,y) \n\\label{eqn:Sq}\n\\nonumber\\\\\n&=&\n P_LS_{\\rm DWF}(x,y;1,L_s)P_L\n+P_LS_{\\rm DWF}(x,y;1,  1)P_R\n\\nonumber\\\\\n&+&\n P_RS_{\\rm DWF}(x,y;L_s,L_s)P_L\n+P_RS_{\\rm DWF}(x,y;L_s,  1)P_R.\n\\end{eqnarray}\nIn terms of $S_{\\rm q}$\nthe effective operator is given by\n\\begin{equation}\naD_{\\rm eff}^{(L_s)}=(1+a^3S_{\\rm q})^{-1}.\n\\end{equation}\n\n\n\nIn the SF setup, $\\Delta^{(L_s)}$ in eq.(\\ref{eqn:DeltaLs})\ncontains not only the bulk chiral symmetry breaking\nbut also the boundary breaking.\nThe former is supposed to be removed by taking $L_s$ to infinity.\nIn such limit, boundary breaking effects remain\nand they are expected to be localized near time boundaries.\nTo see this situation, we numerically compute $\\Delta^{(L_s)}$\nfor a free operator.\nIn the free case, \nwe can perform the Fourier transformation for spatial directions.\nWe study the momentum configuration ${\\bf p}=(0,0,0)$ in the following.\nThe remaining dimensions are only the time direction and spinor space,\ntherefore for a given $L_s$ and the fixed spatial momentum configuration,\n$\\Delta^{(L_s)}$ is a matrix with dimension $4(T\/a-1)$.\nFigure \\ref{fig:GWbreaking} shows the magnitude of\n$\\ln(||\\Delta^{(L_s)}(x_0,y_0)||_{\\rm spin})$,\nwhere the norm is taken for the spinor space only.\nBy increasing $L_s$, the bulk symmetry breaking is reduced.\nFinally at $L_s=24$, only boundary breaking effects remain\nand they are localized exponentially near the time boundaries.\nThis is the expected behavior for overlap fermions \\cite{Luscher:2006df}.\nWe conclude that the presence of the boundary term\ncauses chiral symmetry breaking, which decays exponentially away from\nthe time boundaries\nfor the effective four dimensional operator.\n\n\n\n\n\n\\section{Spectrum of free operator}\n\\label{sec:spectrum}\nIn this section, we investigate\nthe free spectrum of the DWF operator\nto confirm universality at the tree level.\nWe set $T=L$ in this section.\n\n\\subsection{Spectrum of $D_{\\rm DWF}^{\\dag}D_{\\rm DWF}$}\n\\begin{figure}[!t]\n\\begin{center}\n  \\scalebox{1.1}{\\includegraphics{Lowest.eps}}\n\\caption{\nThe lowest eigenvalue of $L^2D_{\\rm DWF}^{\\dag}D_{\\rm DWF}$ with $a m_5=1$.\nSome combinations of parameters $c$ and $L_s$ are shown.\nThe continuum value is $\\pi^2\/4=2.467..$.\nFor $c=1$ case, since the $L_s$-dependence is so weak on on this scale,\nwe show only $L_s=4$ results as representative.\n\\label{fig:DWFEV}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\begin{center}\n  \\begin{tabular}{cc}\n  \\scalebox{1.1}{\\includegraphics{Lowest_c10.eps}}&\n  \\scalebox{1.1}{\\includegraphics{Lowest_c00.eps}}\\\\\n  \\end{tabular}\n\\caption{$||\\psi(x_0,s)||_{\\rm spin}$\nwith zero spatial momentum and the parameters\n$T\/a=L_s=40$, $\\theta=0$ and $am_5=1$.\nThe norm for the eigenvector is taken in the spinor space.\nThe left (right) panel is for $c=1$ ($c=0$).\n\\label{fig:WF}}\n\\end{center}\n\\end{figure}\n\nTo achieve better chiral symmetry,\nthe (physical) eigenmodes of the domain-wall operator should\nbe localized near the boundaries of the fifth direction\nand propagate in the space-time directions.\nThis should also be true in the SF setup,\nsince the chiral symmetry is supposed to be maintained in the bulk.\nTo observe such phenomena, we numerically compute the lowest eigenmode\nof the operator $D_{\\rm DWF}^{\\dag}D_{\\rm DWF}$ with the trivial gauge configuration $U(x,\\mu)=1$.\nIn the free case, \nwe can perform the Fourier transformation for spatial directions,\nand project out the momentum configuration ${\\bf p}=(0,0,0)$.\nThus, remaining indexes of the vector space\nare now the spinor, the time $x_0$ and the extra dimension $s$,\n\\begin{equation}\nD_{\\rm DWF}^{\\dag}D_{\\rm DWF}\\psi(x_0,s)=\\lambda\\psi(x_0,s),\n\\end{equation}\nwhere the spinor indexes are suppressed.\nWe set input parameters $am_5=1$ and $\\theta=0$.\n$\\theta$ is the parameter which\ncontrols the spatial boundary condition for fermion fields.\n(For more details, we refer to \\cite{Sint:1995ch}.)\n\nWe numerically compute the lowest eigenvalue and the corresponding\neigenfunction $||\\psi(x_0,s)||_{\\rm spin}$.\nWe examine not only for $c=1$ but also for $c=0$\nto investigate the importance of the presence of the boundary operator.\nThe scaling behavior of the eigenvalue is shown in Figure \\ref{fig:DWFEV}.\nFor $c=1$, $L_s$ dependence is too small to see on this scale.\nThe lowest eigenvalues converge to their continuum values properly,\ntherefore universality is confirmed.\nFurthermore, the associated eigenfunction shows\nnice localization behavior, namely\nbeing localizing for the fifth direction and propagating for the time direction,\nas shown in the left panel of Figure \\ref{fig:WF}.\nThis shows that this mode is a physical one.\n\nFor $c=0$ in Figure \\ref{fig:DWFEV},\nalthough all $L_s=4,16,32$ results tend to converge to the continuum limit,\nlarge $L_s$ results have a bending phenomenon\nin small $a\/L$ region and show no power decay in terms of $a\/L$. \nThis indicates that if one takes $L_s$ to infinite before taking $a\/L=0$ limit,\nthe eigenvalue will likely converge to zero.\nIf this is so, the theory with $L_s=\\infty$ does not\nbelong to a correct universality class.\nFurthermore, the eigenfunction in the right panel in Figure \\ref{fig:WF}\nis localized on edges in the time-$s$ plane.\nThis is a typical unphysical mode.\nOn the other hand, interestingly for small $L_s$,\nthe scaling behavior is rather mild.\nIn the small $L_s$ case, the chiral symmetry breaking\nof domain-wall fermions are rather similar\nto that of the ordinary Wilson fermions.\nAs in the Wilson fermions case,\nthe bulk chiral symmetry breaking for DWFs due to finite $L_s$\nplays some role in producing the correct continuum limit.\nThis is the reason why DWFs with smaller $L_s$ and no\nboundary term $B$ can produce the continuum results.\n\n\n\n\nThe results shown in this subsection show\nthat the boundary term with $c\\neq 0$ plays an important role\nfor the theory to be in the correct universality class.\n\n\n\n\n\\subsection{Spectrum of $D_{\\rm q}^{\\dag}D_{\\rm q}$}\n\n\\begin{figure}[!t]\n\\begin{center}\n  \\begin{tabular}{cc}\n  \\scalebox{1.1}{\\includegraphics{MIN10_theta00_BG1.eps}}&\n  \\scalebox{1.1}{\\includegraphics{MIN10_theta05_BG1.eps}}\\\\\n  \\end{tabular}\n\\caption{The $a\/L$ dependence of the lowest ten eigenvalues\nof $L^2D_{\\rm q}^{\\dag}D_{\\rm q}$ in the presence of the background gauge field.\nThe left (right) panel is for $\\theta=0$ ($\\theta=\\pi\/5$).\nThe parameters are set to $am_5=1$ and $c=1$.\nThe red points at $a\/L=0$ are continuum values \\cite{Sint:1995ch}.\n\\label{fig:lowest10}}\n\\end{center}\n\\end{figure}\n\n\nNot all eigenmodes of $D_{\\rm DWF}$\nare physical ones and elimination of unphysical mode is not clear.\nTo extract physical modes only, let us study the eigenmodes\nof $D_{\\rm q}$, where unphysical modes are excluded.\nThe operator $D_{\\rm q}$ is defined from $S_{\\rm q}$ in eq.(\\ref{eqn:Sq})\n\\begin{equation}\nD_{\\rm q} S_{\\rm q} (x,y)=a^{-4}\\delta_{x,y}.\n\\end{equation}\n\nWe numerically compute the lowest ten eigenvalues of\n$D_{\\rm q}^{\\dag}D_{\\rm q}$\nwith the parameter set $am_5=1$, $\\theta=0,\\pi\/5$ and $c=1$\nin the presence of the the background gauge field (choice A in\nRef.~\\cite{Luscher:1993gh}).\nThe values obtained for $L\/a=T\/a=6,12,24$ are summarized in Table \\ref{tab:MIN10}.\nAll tables are given in appendix \\ref{sec:tables}\nThe scaling behavior of the eigenvalues\nare shown in Figure \\ref{fig:lowest10}.\nAlthough we show two cases of $L_s$, namely $L_s=4$ and $L_s=32$,\nit is hard to see the difference on this scale.\nWe observe that they converge to the continuum values given\nin Ref~\\cite{Sint:1995ch}.\nThis behavior persists for a variety of values of $c$,\n$0.5 \\le c\\le 1.5$.\nThis confirms universality at the tree level.\n\n\n\n\n\\section{One-loop analysis of SF coupling}\n\\label{sec:oneloop}\nTo check further universality at the quantum level\nand renormalizability, we perform the one-loop order calculation\nof the SF coupling.\n\n\\subsection{Definition and results}\n\nWe compute the fermion contribution to\nthe SF coupling \\cite{Sint:1995ch}\n$p_{1,1}(L\/a,L_s)$\n(we set $L=T$ as usual)\nat one-loop order\nfor massless domain-wall fermions.\nThe one-loop coefficient is given as\n\\begin{equation}\np_{1,1}(L\/a,L_s)\n=\n\\left.\n\\frac{1}{k}\n\\frac{\\partial}{\\partial\\eta}\n\\ln \\det(D_{\\rm DWF}\/D_{\\rm PV})\n\\right|_{\\eta=\\nu=0},\n\\label{eqn:p11}\n\\end{equation}\nwith a normalization (See \\cite{Sint:1995ch} for details.)\n\\begin{equation}\nk=12(L\/a)^2[\\sin(\\gamma)+\\sin(2\\gamma)],\n\\hspace{5mm}\n\\gamma=\\frac{1}{3}\\pi(a\/L)^2.\n\\end{equation}\nThe parameters $\\eta$ and $\\nu$ parameterize\nthe background gauge field \\cite{Luscher:1993gh}.\nIn the actual calculation,\nwe expand the $\\eta$ derivative\nand use the fact that the determinant\nis factorized for individual spatial momentum ${\\bf p}$ and color sector $b$,\n\\begin{eqnarray}\np_{1,1}(L\/a,L_s)\n&=&\n\\frac{1}{k}\n{\\rm Tr}\n\\left[\nD_{\\rm DWF}^{-1}\\frac{\\partialD_{\\rm DWF}}{\\partial\\eta}\n-D_{\\rm PV}^{-1}\\frac{\\partialD_{\\rm PV}}{\\partial\\eta}\n\\right]\n\\nonumber\\\\\n&=&\n\\frac{1}{k}\n\\sum_{{\\bf p}}\n\\sum_{b=1}^{3}\n{\\rm tr}\n\\left[\n(D_{\\rm DWF}^b)^{-1}({\\bf p})\\frac{\\partialD_{\\rm DWF}^b({\\bf p})}{\\partial\\eta}\n\\right.\n\\nonumber\\\\&&\n\\left.\n-(D_{\\rm PV}^b)^{-1}({\\bf p})\\frac{\\partialD_{\\rm PV}^b({\\bf p})}{\\partial\\eta}\n\\right].\n\\label{eqn:p11expand}\n\\end{eqnarray}\nThe trace\n${\\rm tr}$\nconcerns the spinor, the time indices and fifth coordinate only.\nIt is maybe worthwhile to note that for our definition of DWF,\n\\begin{equation}\n\\frac{\\partialD_{\\rm DWF}}{\\partial\\eta}\n=\n\\frac{\\partialD_{\\rm PV}}{\\partial\\eta}\n\\end{equation}\nholds since the mass term does not involve the gauge field.\n\n\nWe compute $p_{1,1}$ on the lattices of size $L\/a=4,6,...,48$ and\n$L_s=6,8,10,12,16$ with parameters $0.7 \\le am_5 \\le 1.3$\nand $\\theta=\\pi\/5$.\nSubsets of the results are summarized in Table \\ref{tab:p11}\nfor $am_5=1$, $L_s=6$ and $L\/a=4,6,...,48$.\nSeparate contributions from DWF and PV are\nalso shown there.\n\n\n\n\\subsection{Coefficients of Symanzik's expansion}\n\\begin{figure}[!t]\n\\begin{center}\n  \\scalebox{1.3}{\\includegraphics{A0Ls.eps}}\n\\caption{\n$L_s$-dependence of $A_0$ for $0.7 \\le am_5 \\le 1.3$.\nThe horizontal lines show \nthe values of $A_0$ in the infinity $L_s$ limit,\nwhich are obtained by combining the results\nof previous literature\n\\cite{Sint:1995ch,Aoki:2003uf}.\n\\label{fig:A0Ls}}\n\\end{center}\n\\end{figure}\n\nFrom the Symanzik's analysis of the cutoff dependence\nof Feynman diagrams on the lattice,\none expects that the one-loop coefficient has\nan asymptotic expansion in terms of $a\/L$\n\\begin{equation}\np_{1,1}(L\/a,L_s)\n=\n\\sum^{\\infty}_{n=0}\n(a\/L)^n[A_n(L_s) + B_n(L_s) \\ln (L\/a)].\n\\label{eqn:p11symanzik}\n\\end{equation}\nNote that the coefficients $A_n$ and $B_n$ ($n=0,1,2,...$)\ndepend on $L_s$.\nWe can reliably extract the first few coefficients\nby making use of the method described in Ref.~\\cite{Bode:1999sm}.\n\n\nFor the usual renormalization of the coupling constant,\n$B_0$ at $L_s=\\infty$ should be $2b_{0,1}$ where $b_{0,1}$ is\nthe fermion part of the one-loop coefficient\nof the $\\beta$-function for $N_{\\rm f}$ flavors QCD,\n\\begin{eqnarray}\nb_0\n&=&\nb_{0,0}+N_{\\rm f}b_{0,1},\n\\\\\nb_{0,0}\n&=&\n\\frac{11}{(4\\pi)^2},\n\\\\\nb_{0,1}\n&=&\n-\\frac{2}{3}\n\\frac{1}{(4\\pi)^2}.\n\\end{eqnarray}\nWe confirmed that\n$B_0(L_s)$ for large $L_s$ (say $L_s=16$) converges to $2b_{0,1}=-0.008443...$\nup to three significant digits\nfor the values of $am_5$ which we investigated.\nWhen the tree-level O($a$) improvement is realized,\nwe expect that $B_1 = 0$ holds.\nWe check this to $10^{-3}$ for the same parameter region as before.\nThis shows that the formula\nfor the boundary coefficient in eq.(\\ref{eqn:ctreeL})\nworks well to achieve the tree-level O($a$) improvement\nto the precision considered here.\nIn the following analysis, we set exact values $B_0=2b_{0,1}=-1\/(12 \\pi^2)$\nand $B_1=0$.\n\n\n\n\n$A_0$ gives information about a ratio of $\\Lambda$-parameters.\nThe obtained values of $A_0(L_s)$\nas a function of $L_s$ are shown\nin Figure \\ref{fig:A0Ls}.\nBy combining the previous results\nfrom Ref.~\\cite{Sint:1995ch,Aoki:2003uf},\nthe values of $A_0$ at infinity $L_s$ can be obtained, and\nare shown in Figure \\ref{fig:A0Ls} as the horizontal lines.\nWe observe that our results at finite $L_s$\nproperly converge to the known results at infinity $L_s$.\n\n\nTo achieve one-loop O($a$) improvement,\nwe need to determine the coefficient\nof the fermion part of the boundary\ncounter-term, $c_{\\rm t}^{(1,1)}$ \\cite{Sint:1995ch}\nat one-loop order.\nIf one imposes an improvement condition\n\\cite{Sint:1995ch}, one finds that\n\\begin{equation}\nc_{\\rm t}^{(1,1)}=A_1\/2,\n\\label{eqn:ct11A1}\n\\end{equation}\ntherefore we need the value of $A_1$.\nThe obtained values of $A_1$ are given in Table \\ref{tab:A1}.\nFor future reference, we provide an interpolation formula for\n$c_{\\rm t}^{(1,1)}$ as a polynomial of $am_5$\nfor larger $L_s$, where value of $A_1$ is saturated,\n\\begin{equation}\nc_{\\rm t}^{(1,1)}\n=\n0.00434+0.01102(am_5-1)-0.00858(am_5-1)^2,\n\\label{eqn:ct11}\n\\end{equation}\nfor $0.7 \\le am_5 \\le 1.3$.\n\n\n\\section{Lattice artifacts of the step scaling function to one-loop order}\n\\label{sec:SSF}\n\\begin{figure}[!t]\n \\begin{center}\n  \\begin{tabular}{cc}\n  \\resizebox{70mm}{!}{\\includegraphics{deviation0.eps}}\n  &\n  \\resizebox{70mm}{!}{\\includegraphics{deviation1.eps}}\n  \\\\\n  \\end{tabular}\n  \\caption{We show the relative deviation with the various actions for tree level\n$O(a)$ improvement, $\\delta_{1,1}^{(0)}$ (Left),\nand one-loop $O(a)$ improvement, $\\delta_{1,1}^{(1)}$ (Right),\nas a function of $a\/L$ and $(a\/L)^2$ respectively.\nUpper part is for $\\theta=0$, and lower is for $\\theta=\\pi\/5$.\nFor comparison, those of the Wilson type fermion\nwith $c_{\\rm t}^{(1,1)}=0$ and the clover fermion with\n$c_{\\rm t}^{(1,1)}=0.019141$ \\cite{Sint:1995ch}\nare included in the plot of $\\delta_{1,1}^{(0)}$ and\n$\\delta_{1,1}^{(1)}$ respectively.\n  \\label{fig:deviation}}\n \\end{center}\n\\end{figure}\n\n\nIn this section, we investigate lattice artifacts of\nthe step scaling function (SSF) \\cite{Luscher:1991wu}\n$\\sigma(2,u)$, which describes the evolution\nof the running coupling $\\bar{g}^2(L)=u$\nunder changes of scale $L$ by a factor $2$,\n\\begin{equation}\n\\sigma(2,u)=\\bar{g}^2(2L),\n\\hspace{5mm}\nu=\\bar{g}^2(L).\n\\end{equation}\nThe lattice version of the step scaling function is\ndenoted by $\\Sigma(2,u,a\/L)$ which\ncontains lattice artifacts.\nSuch lattice artifacts\nare described by the relative deviation\n\\begin{equation}\n\\delta (u,a\/L) \n\\equiv\n\\frac{\\Sigma(2,u,a\/L) \n - \\sigma(2,u)}{\\sigma(2,u)}.\n\\end{equation}\nBy expanding the relative deviation\nin terms of the coupling constant $u$, one obtains\n\\begin{equation}\n\\delta (u,a\/L) \n= \n\\delta_1(a\/L) u  \n+ O(u^2),\n\\end{equation}\nwhere the one-loop deviation, $\\delta_1(s,a\/L)$,\nmay be decomposed into pure gauge and fermion part \\cite{Sint:1995ch},\n\\begin{equation}\n\\delta_1(a\/L)\n=\n\\delta_{1,0}(a\/L)\n+N_{\\rm f}\n\\delta_{1,1}(a\/L).\n\\end{equation}\nWe are currently only interested in the fermion part.\nWe consider domain-wall fermions,\nthus the fermion part of the one-loop deviation\n$\\delta_{1,1}(a\/L,L_s)$\ncontains $L_s$ dependence.\nIn terms of the\none-loop coefficient of the SF coupling $p_{1,1}$,\nthe one-loop deviation is given by\n\\begin{equation}\n\\delta_{1,1}(a\/L,L_s)\n= \n p_{1,1}(2L\/a,L_s)\n-p_{1,1}(L\/a,L_s)\n-2 b_{0,1} \\ln(2).\n \\label{eqn:deviation1}\n\\end{equation}\nDepending on the value of the\nboundary counter term $c_{\\rm t}^{(1,1)}$,\nwe denote with $\\delta^{(0)}_{1,1}$\nthe tree level O($a$) improved version\nwith $c_{\\rm t}^{(1,1)}=0$,\nand $\\delta^{(1)}_{1,1}$\nthe  one-loop O($a$) improved one\nfor $c_{\\rm t}^{(1,1)}$ in eq.(\\ref{eqn:ct11A1}).\n\n\nWe show numerical results for the one-loop deviation in\nTables \\ref{tab:deviation0} and \\ref{tab:deviation1}\nand the plots in Figure \\ref{fig:deviation},\nwhere we include those of the Wilson type fermions \\cite{Sint:1995ch}\nand overlap fermion \\cite{Takeda:2007ga}\nfor comparison.\n$L_s$-dependence of DWFs is small.\nIn the case of the clover action, $c_{\\rm t}^{(1,1)}$ is set\nto be the proper value to achieve one-loop $O(a)$ improvement,\nand for Wilson fermions\nit is set to $c_{\\rm t}^{(1,1)}=0$.\nWe observe that the lattice artifacts\nfor domain-wall fermions are small for\ntree level boundary O($a$) improvement case\ncompared with other fermions, while\nthey are large for one-loop boundary O($a$) improvement case.\n\n\n\n\n\n\\section{Conclusion and outlook}\n\\label{sec:conclusion}\n\nIn this paper, we provide a new formulation of domain-wall fermions\nin the SF setup by following the universality argument of L\\\"uscher.\nIn contrast to the previous formulation by Taniguchi,\nours can deal with the boundary O($a$) improvement properly,\nand there is no constraint on the number of flavors.\nTo check that our formulation works properly,\nwe investigate the spectrum and eigenmodes of the free operator,\nand perform a one-loop analysis of the SF coupling constant.\nThen we confirm universality at tree and the one-loop level\nand observe that all results investigated\nshow the desired behaviors.\n\n\nBefore starting simulations,\nthe boundary improvement coefficient $c$\nshould be determined to one-loop order.\nThis involves calculations of the SF\ncorrelators, $f_{\\rm A}$, $f_{\\rm P}$ etc. given in\nappendix \\ref{sec:operator}.\nThis could be done in a similar way to the case of the Wilson fermion.\n\n\nAs mentioned before, one of the most important\nproperties of the universality formulation\nis that there are no restriction of the number of flavors.\nBy taking advantage of this property,\nwe may compute the renormalization factor\nof $B_{\\rm K}$ for $N_{\\rm f}=3$ QCD.\n\n\n\n\\section*{Acknowledgments}\nWe would like to thank Sinya Aoki, Yasumichi Aoki,\nNorman Christ, Michael Endres,\nTaku Izubuchi, Changhoan Kim, Robert Mawhinney\nand the members of the RBC Collaboration\nfor helpful discussions.\nWe are grateful to Stefan Sint for his hospitality\nduring our stay at Trinity College in Dublin,\nwhere this work was initiated.\nThis work is supported by the U.S. Department of Energy\nunder Grant No. DE-FG02-92ER40699.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\thesectionumber. #1}}\n\n\\newcounter{subsectionumber}\n\\newcommand{\\ssubsection}[1]{\\refstepcounter{subsectionumber}\\subsection{\\thesectionumber.\\thesubsectionumber. #1}}\n\n\n\\renewcommand{\\figurename}{\\normalfont\\bfseries Extended Data Figure}\n\\renewcommand{\\thefigure}{\\arabic{figure}}\n\n\n\n\\newcommand{\\Schrader}[1]{\\ensuremath{_{\\text{Sch[#1]}}}}\n\\newcommand{*}{*}\n\\newcommand{\\inner}[2]{\\left(#1,\\,#2\\right)}\n\\newcommand{\\kket}[1]{| #1 \\!\\!\\succ}\n\n\\newcommand{\\refgordon}{28\\xspace}\n\\newcommand{\\refzakharov}{24\\xspace}\n\\newcommand{\\refKivsharAndMalomed}{23\\xspace}\n\n\n\\newcommand{1\\xspace}{1\\xspace}\n\\newcommand{5\\xspace}{5\\xspace}\n\n\\renewcommand\\theequation{S\\arabic{equation}}\n\n\\ssection{Energetically allowed fractional loss of particles for a single soliton}\n\\label{allowed_loss}\n\nConsider the stationary, single-soliton solution (normalized to 1) of Eq.~(1) with $V=0$ in the main text, $\\Psi_{\\text{1}}(z)=\\sqrt{c\/2}\\sech (c z)$, where $c=|g_{\\textrm{1D}}| N_{\\textrm{a}} m \/(2\\hbar^2)$.\n\nThe energy-per-particle of the soliton is given by\n\\begin{align*}\nE\/N_{\\textrm{a}}\n&=\\int_{-\\infty}^{\\infty}\\,dz\\,\\left(\\frac{\\hbar^{2}}{2m}|\\partial_{z}\\Psi_{\\text{1}}(z)|^2+\\frac{g_{\\textrm{1D}}N_{\\textrm{a}}}{2}|\\Psi_{\\text{1}}(z)|^{4}\\right)\n\\\\\n&=-d N_{\\textrm{a}}^{2},\\text{ with } d=\\frac{mg_{\\textrm{1D}}^{2}}{24\\hbar^2},\n\\end{align*}\nwhere $g_{\\textrm{1D}}<0$. A fracturing of a single soliton into two smaller ones, $N_{\\textrm{a}} \\to N_{1} + N_{2}$, will raise the energy-per-particle by $2d N_{1}N_{2}$. If there are multiple fragments, their energies are even less negative, so the increase in the energy-per-particle will be even greater; same if some of the particles are radiated away, as their energy will actually be positive. So the assumption of a fracture into exactly two smaller solitons will give the maximal possible particle loss. If the fracturing is to occur due to the soliton (with center-of-mass speed $v$) colliding with a barrier, this energy can come only from the kinetic energy per particle of the center of mass, $E_{\\text{K},1}=\\frac{1}{2}mv^{2}$. Let us write $N_{1}=N_{\\textrm{a}}-\\Delta N$ and $N_{2}=\\Delta N$, where, without loss of generality, we impose the condition that $0\\leqslant \\Delta N<N_{\\textrm{a}}\/2$. We therefore have $0\\leqslant 2d(N_{\\textrm{a}}-\\Delta N)\\Delta N\\leqslant E_{\\text{K},1}$; dividing through by $2d N_{\\textrm{a}}^{2}$, we get\n\\begin{equation}\n0\\leqslant \\frac{\\Delta N}{N_{\\textrm{a}}}-\\left(\\frac{\\Delta N}{N_{\\textrm{a}}}\\right)^{2}\\leqslant\\frac{E_{\\text{K},1}}{2d N_{\\textrm{a}}^{2}}\\,.\n\\label{ineqality}\n\\end{equation}\nNow we invoke the assumption that $E_{\\text{K},1} \\ll |E\/N_{\\textrm{a}}|$. Thus the right-hand side of the above relation is much less than one. The equation $x-x^{2}=\\epsilon$ has the solutions $x_{1,2}=\\frac{1}{2}\\pm\\sqrt{\\frac{1}{2}-\\epsilon}$, i.e. one close to zero, the other close to one. Because we imposed that $\\Delta N<N_{\\textrm{a}}\/2$, it follows that $\\Delta N\/N_{\\textrm{a}}$ must be much less than one; so we may neglect the quadratic term in Eq.~(\\ref{ineqality}), obtaining\n\\[0\\leqslant\\frac{\\Delta N}{N_{\\textrm{a}}}\\leqslant\\frac{E_{\\text{K},1}}{2d N_{\\textrm{a}}^{2}}\\,.\\]\n\nSee also Ref.~\\onlinecite{Kivshar1987_968} (the paragraph before Sec.~3).\n\n\\ssection{The natural units for the 1D nonlinear Schr{\\\"o}dinger equation}\n\\label{unit_conversions}\n\nConsider the 1D nonlinear Schr{\\\"o}dinger equation\n\\begin{equation}\ni\\hbar\\partial_{t}\\Psi=-\\frac{\\hbar^{2}}{2m}\\partial^{2}_{z}\\Psi+g_{\\textrm{1D}}N_{\\textrm{a}}|\\Psi|^{2}\\psi+\\tilde{V}(z)\\Psi\\,,\n\\label{SI_NLSE}\n\\end{equation}\nwhere $\\int_{-\\infty}^{\\infty}\\,|\\Psi(z,\\,t)|^{2}\\,dz=1$. By choosing to work in ``natural units,'' it may be written as\n\\[\ni\\partial_{t}\\psi=-\\frac{1}{2}\\partial^{2}_{z}\\psi\\pm\\,|\\psi|^{2}\\psi+V(z)\\psi\\,,\n\\]\nwhere the sign of the $|\\psi|^{2}\\psi$ term is the sign of $g_{\\textrm{1D}}$. In the present work, this is negative (i.e. attractive, or ``focusing''), so that it can support bright solitons. $\\int_{-\\infty}^{\\infty}\\,|\\psi(z,\\,t)|^{2}\\,dz=\\mathcal{N}$ is the new\nnormalization, which may be chosen at will. In the resulting system of units, the unit of length is $u_{L}=\\mathcal{N} \\hbar^{2} \/(m g_{\\textrm{1D}} N_{\\textrm{a}})$, of time, $u_{T}=\\mathcal{N}^{2} \\hbar^{3}\/(m g_{\\textrm{1D}}^{2} N_{\\textrm{a}}^{2})$, and the field is transformed as\n$\\Psi(z,\\,t)=\\psi(z\/u_{L},\\,t\/u_{T})\/\\sqrt{\\mathcal{N}u_{L}}$.\n\n\n\n\\ssection{The $N$-soliton solution of the nonlinear Schr{\\\"o}dinger equation}\n\\label{N-soliton_solution}\n\nThe multisoliton solutions of the equation\n\\begin{equation*}\ni\\partial_{t}\\psi(z,\\,t)=-\\frac{1}{2}\\partial^{2}_{z}\\psi(z,\\,t)-|\\psi(z,\\,t)|^{2}\\psi(z,\\,t)\n\\end{equation*}\nmay be constructed as follows \\cite{gordon1983_596}: given the $4N$ real\nparameters $A_{j}>0$, $v_{j}$, $z_{j,0}$, and $\\phi_{j,0}$, $j=1,\\,\\ldots,\\,N$,\nwe have that\n\\[\n\\psi(z,\\,t)=\\sum_{j=1}^{N}u_{j}(z,\\,t)\\,,\n\\]\nwhere the $u_{j}$'s are the solutions of the system of $N$ equations\n\\[\n\\sum_{k=1}^{N}\\,\\frac{1\/\\gamma_{j}+\\gamma_{k}^{*}}{\\lambda_{j}+\\lambda_{k}^{*}}u_{k}=1\\,,\\quad j=1,\\,\\ldots,\\,N,\n\\]\nwhere\n\\[\n\\lambda_{j}=A_{j}\/2+i v_{j}\n\\] and\n\\[\n\\gamma_{j}=\\exp\\left[\\lambda_{j}(z-z_{j,0})+i\\lambda_{j}^{2}t\/2+i\\phi_{j,0}\\right]\\,.\n\\]\nHere the parameters $z_{j,0}$, and $\\phi_{j,0}$ are \\emph{almost but not quite}\nthe position and phase, at $t=0$, of the $j$th soliton when it is spatially\nseparated from the others (see below), but the first two are precisely its norm\nand velocity, as we now explain. The norm of $\\psi$ is $\\sum_{j=1}^{N} A_{j}$;\nas $t\\to \\pm \\infty$, if the velocities $v_{j}$ are all distinct, $\\psi$\nbecomes a sum of $N$ 1-soliton solution, of norms $A_{j}$, traveling at\nvelocities $v_{j}$. More precisely, in the limit as the $j$th soliton becomes\nmore and more spatially separated from the others, its form converges to\n\\[\n\\frac{A_{j}}{2}\\sech\n\\left[\\frac{A_{j}}{2}(z-z_{j})+q_{j}\\right]\\exp[i(\\phi_{j}+\\Psi_{j})]\\,,\n\\]\nwhere\n\\begin{align*}\nz_{j}&=z_{j,0}+v_{j}t\\,,\n\\\\\n\\phi_{j}&=v_{j}(z-z_{j})+\\frac{1}{2}\\left(A_{j}^{2}\/4+v_{j}^{2}\\right)t+\\phi_{j,0}\\,,\n\\end{align*}\nand the real numbers $q_{j}$ and $\\Psi_{j}$ capture the interaction with the\nother solitons. They are given as\n\\[\nq_{j}+i\\Psi_{j}=\\sum_{\\substack{k=1 \\\\\nk\\neq j}}^{N}\\sign (z_{k}-z_{j})\\ln\n\\frac{A_{j}+A_{k}+2i(v_{j}-v_{k})}{A_{j}-A_{k}+2i(v_{j}-v_{k})}\\,,\n\\]\nwhere $\\sign{z}$ is -1, 0, or +1 if  $z<0$, $z=0$, or $z>0$, respectively.\n\nNow we see that the parameters $z_{j,0}$, and $\\phi_{j,0}$ \\emph{differ},\nthrough $q_{j}$ and $\\Psi_{j}$, respectively, from the actual position and\nphase at $t=0$ of the $j$th soliton when it is spatially separated from the\nothers. But the actual position and phase at $t=0$ is easily characterized:\nsince the displacements $q_{j}$ depend only on the ordering of the spatial\npositions of the solitons (and not on the magnitudes of the relative\ndistances), it follows that if one wants isolated solitons to sit at\n$\\bar{z}_{0,j}$ at $t=0$, one may proceed as follows: first set each $z_{j,0}$\nto $\\bar{z}_{0,j}$, and compute the $q_{j}$'s. Then set\n$z_{j,0}=\\bar{z}_{0,j}+2q_{j}\/A_{j}$, and proceed to compute the $N$-soliton\nsolution; a similar correction may be applied for the initial phases.\n\nThe case of degenerate $\\lambda_{j}$, when two or more constituent solitons\nhave both the same norm and the same velocity, may be treated by taking the\nappropriate limit of the system of $N$ equations above. In this case one\nobtains solutions that are qualitatively different from those discussed thus\nfar. For example, in the two-soliton case, as $t\\to \\pm \\infty$, one finds \\cite{zakharov1972_118} that\nthe distance between the solitons increases proportionally to $\\ln (A^{2} t)$\n(in natural units). Since here the solitons separate on\ntheir own on the time scale of $1\/A^{2}$, the collision experiment should last\nshorter than that. On the other hand, the breather needs to start sufficiently\nfar from the barrier so that it begins in an approximately integrable regime,\nand it needs to be sufficiently slow so that the kinetic energy per particle is\nmuch less than the chemical potential. It turns out that these constraints are\nimpossible to satisfy simultaneously, and thus the degenerate case is not of\ninterest for us.\n\n\n\n\\label{conjecture_argument}\n\n\n\n\n\\ssection{Derivation of the exact expression for $d\\lambda\/dt$, Eq.~(5\\xspace) in the main text}\n\\label{derivation_of_DlambdaDt}\n\nWe are dealing with the 1D nonlinear Schr{\\\"o}dinger equation in Eq.~(1\\xspace) in the main text,\n\\begin{multline}\ni\\hbar\\frac{\\partial}{\\partial t}\\Psi(z,\\,t)=-\\frac{\\hbar^{2}}{2m}\\frac{\\partial^{2}}{\\partial z^{2}}\\Psi(z,\\,t)\n\\\\\n+g_{\\textrm{1D}}N_{\\textrm{a}}|\\Psi(z,\\,t)|^{2}\\Psi(z,\\,t)+\\tilde{V}(z,\\,t)\\Psi(z,\\,t)\\,,\n\\label{td_NLSE}\n\\end{multline}\nwhere $g_{\\textrm{1D}}<0$, \\emph{in the presence of} the external integrability-breaking potential $\\tilde{V}(z,\\,t)$ (note that here we will allow this external potential to explicitly depend on time). In order to facilitate comparison with Ref.~\\onlinecite{Kivshar1989_763}, which treats the same kind of problem, we will work in the units in which $\\hbar=1$, $m=1\/2$, and $|g_{\\textrm{1D}}|N_{\\textrm{a}}=2$. Thus the NLSE becomes\n\\begin{multline}\ni\\frac{\\partial}{\\partial t} \\psi(z,\\,t)\n=\n\\left[- \\frac{\\partial^2}{\\partial z^2}  - 2 |\\psi(z,\\,t)|^2 \\right] \\psi(z,\\,t)\n\\\\\n + \\epsilon v_{ext}(z,t) \\psi(z,\\,t)\n\\label{td_NLSE_perturbed}\n\\,,\n\\end{multline}\nwhere the external potential in (\\ref{td_NLSE}) is factorized as\n\\begin{eqnarray*}\n&&\n\\tilde{V}(z,\\,t) \\equiv V_{0} v_{ext}(z,t)\n\\\\\n&&\n\\max_{z}[v_{ext}(z,\\,0)]=1\n\\,,\n\\end{eqnarray*}\nand the small parameter $\\epsilon$ is\n\\begin{eqnarray*}\n\\epsilon \\equiv \\frac{2 \\hbar^2 V_{0}}{(g_{\\textrm{1D}}N_{\\textrm{a}})^{2} m}\n\\,.\n\\end{eqnarray*}\n\nThe first Lax operator reads\n\\begin{eqnarray}\n\\hat{\\cal L}\n=\n\\left(\n\\begin{array}{cc}\n\\hat{L}            &   \\hat{M}\n\\\\\n-\\hat{M}^{\\dagger} & -\\hat{L}\n\\end{array}\n\\right)\n\\label{Lax}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n&&\n\\hat{L} = -i \\frac{\\partial}{\\partial z}\n\\label{Lax_L}\n\\\\\n&&\n\\hat{M} = \\psi^{*}(z,\\,t)\n\\label{Lax_M}\n\\,\\,.\n\\end{eqnarray}\nFor each instance of time $t$, one can set up an eigenstate-eigenvalue problem, which is the central object of interest of this derivation:\n\\begin{eqnarray*}\n\\hat{\\cal L} | w \\!\\!\\succ = \\lambda | w \\!\\!\\succ\n\\,\\,,\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*}\n| w \\!\\!\\succ = \\left(\\begin{array}{c} u(z,\\,t) \\\\ v(z,\\,t) \\end{array}\\right)\n\\,.\n\\end{eqnarray*}\n\nTo convert expressions in this text to the ones appearing in Ref.~\\onlinecite{Kivshar1989_763}, one\nshould use the following replacement table:\n\\begin{center}\n\\begin{minipage}{.3\\textwidth}\n\\begin{eqnarray*}\n&&\nz \\to x\n\\\\\n&&\n\\psi(z,\\,t) \\to u(x,\\,t)\n\\\\\n&&\nu(z,\\,t) \\to \\psi^{(1)}(x,\\,t)\n\\\\\n&&\nv(z,\\,t) \\to \\psi^{(2)}(x,\\,t)\n\\\\\n&&\nv_{ext}(z,\\,t) \\psi(x,\\,t) \\to i (P[u])(x,\\,t)\\,.\n\\end{eqnarray*}\n$\\left.\\right.$ \\vspace{-.5\\baselineskip}\n\\end{minipage}\n\\end{center}\\vspace{\\baselineskip}\n\n\\ssubsection{Relevant functional analysis}\nFrom the fact that $\\hat{L}$ is Hermitian, $\\hat{L}^{\\dagger} = \\hat{L}$, it follows that the Lax operator (\\ref{Lax}) possesses the following property:\n\\begin{eqnarray*}\n\\hat{\\cal L}^{\\dagger} = \\hat{\\sigma}_{3} \\hat{\\cal L} \\hat{\\sigma}_{3}\n\\,.\n\\end{eqnarray*}\nThis property induces a particular Hermitian form $\\inner{\\cdot}{\\cdot}$, a pseudo-inner product:\n\\begin{align}\n\\inner{\\kket{w_{1}}}{\\kket{w_{2}}}&=\\prec \\!\\!\n  w_{1} | w_{2}\n\\!\\!\\succ\n\\equiv\n\\langle u_{1} | u_{2} \\rangle - \\langle v_{1} | v_{2} \\rangle\n\\notag\n\\\\\n&=\n\\int \\! dz \\,\n\\{\n  u_{1}^{*}(z) u_{2}(z) - v_{1}^{*}(z) v_{2}(z)\n\\}\n\\label{scalar_product}\n\\,\\,.\n\\end{align}\nThis Hermitian form lacks the property of being positive definite (i.e. lacks the property that, for all $| w \\rangle$, $\\langle w | w \\rangle \\ge 0$ and $\\langle w | w \\rangle = 0$ if and only if $| w \\rangle = | 0 \\rangle$). The rest of the inner product axioms, on the other hand, remain intact:\n\\begin{eqnarray*}\n&&\\prec \\!\\! w_{2} | w_{1} \\!\\!\\succ = \\prec \\!\\! w_{2} | w_{1} \\!\\!\\succ^{*}\n\\\\\n\\text{and}&&\n\\\\\n&&\\prec \\!\\! w_{1} | a w_{2} + b w_{3}\\!\\!\\succ = a\\! \\prec \\!\\! w_{1} | w_{2} \\!\\!\\succ + b\\! \\prec \\!\\! w_{1} | w_{3}\\!\\!\\succ\\,.\n\\end{eqnarray*}\nThe Lax operator $\\hat{\\cal L}$ from Eq.~(\\ref{Lax}) is symmetric with respect to this form:\n\\begin{eqnarray}\n\\inner{\\kket{w_{1}}}{ \\hat{\\cal L}\\kket{w_{2}}}=\\inner{ \\hat{\\cal L}\\kket{w_{1}}}{\\kket{w_{2}}}\n\\label{pseudo-Hermiticity}\n\\,\\,.\n\\end{eqnarray}\nThe property above justifies a standard notation\n$\n\\inner{\\kket{w_{1}}}{ \\hat{\\cal L}\\kket{w_{2}}}\n\\equiv\n\\prec \\!\\!\n  w_{1} | \\hat{\\cal L} | w_{2}\n\\!\\!\\succ\n$\nthat we are going to employ below.\n\nThe pseudo-Hermiticity property (\\ref{pseudo-Hermiticity}) implies the following properties of the eigenstates of $\\hat{\\cal L}$: Let\n$\\hat{\\cal L} | w_{1} \\!\\!\\succ = \\lambda_{1} | w_{1} \\!\\!\\succ$,\n$\\hat{\\cal L} | w_{2} \\!\\!\\succ = \\lambda_{2} | w_{2} \\!\\!\\succ$, and\n$\\hat{\\cal L} | w \\!\\!\\succ = \\lambda | w \\!\\!\\succ$.\nThen\n\\begin{itemize}\n\\item[1.]\neigenvectors whose eigenvalues are not complex conjugates of each other are mutually orthogonal,\n\\begin{eqnarray*}\n&&\n\\lambda_{1}^{*} \\not= \\lambda_{2}^{} \\,\\, \\Rightarrow \\,\\,\\, \\prec \\!\\! w_{1} | w_{2} \\!\\!\\succ = 0\\,;\n\\end{eqnarray*}\n\\item[2.]\nnon-zero norm eigenstates of $\\hat{\\cal L}$ correspond to real eigenvalues\n(a corollary of the above):\n\\begin{eqnarray*}\n&&\n\\lambda^{*} \\not= \\lambda^{} \\,\\, \\Rightarrow \\,\\,\\, \\prec \\!\\! w | w \\!\\!\\succ = 0\\,.\n\\end{eqnarray*}\n\\end{itemize}\nNote that the eigenspectrum of $\\hat{\\cal L}$ is not necessarily complete. In cases when the Lax operator (\\ref{Lax})\nrepresents a linear stability analysis equation of a nonlinear PDE, the missing states are associated with the continuous\nsymmetries of the PDE that is broken by the solution in question \\cite{castin2009_317}. (For a ``flat'' condensate, $\\psi(z) = \\mbox{const}$,\nwe found one missing state; there could be more. There seem to be none for a single soliton.)\n\nThe operator (\\ref{Lax}) also possesses properties specific to a particular form of the matrix elements, Eqs.~(\\ref{Lax_L}) and (\\ref{Lax_M}):\n\\begin{eqnarray*}\n&&\n\\hat{L}^{*} = - \\hat{L}\n\\,\\,.\n\\end{eqnarray*}\nThis property implies that:\n\\begin{itemize}\n\\item[1.]\nreal eigenvalues $\\lambda$ are doubly degenerate. The corresponding eigenstates,\n\\begin{eqnarray*}\n&&\n| w \\!\\!\\succ \\stackrel{\\cdot}{=} \\left(\\begin{array}{c}u(z) \\\\ v(z)\\end{array}\\right)\n\\\\\n&&\n| \\tilde{w} \\!\\!\\succ \\stackrel{\\cdot}{=} \\left(\\begin{array}{c} \\tilde{u}(z) \\\\ \\tilde{v}(z)\\end{array}\\right)\n\\,\\,,\n\\end{eqnarray*}\nare related\nby\n\\begin{eqnarray*}\n&&\n\\tilde{u}(z) = - v^{*}(z)\n\\\\\n&&\n\\tilde{v}(z) = + u^{*}(z)\n\\,\\,.\n\\end{eqnarray*}\nHere,\n\\begin{eqnarray*}\n&&\n\\hat{\\cal L}| w \\!\\!\\succ \\stackrel{\\cdot}{=} \\lambda | w \\!\\!\\succ\n\\\\\n&&\n\\hat{\\cal L}| \\tilde{w} \\!\\!\\succ \\stackrel{\\cdot}{=} \\lambda | \\tilde{w} \\!\\!\\succ\n\\,.\n\\end{eqnarray*}\n\n\\item[2.]\nComplex eigenvalues $\\lambda$ come in complex conjugate pairs, $\\lambda_{+}$, $\\lambda_{-}$ such that $\\lambda_{-} = (\\lambda_{+})^{*}$. The corresponding eigenstates,\n\\begin{eqnarray*}\n&&\n| w_{+} \\!\\!\\succ \\stackrel{\\cdot}{=} \\left(\\begin{array}{c} u_{+}(z) \\\\ v_{+}(z)\\end{array}\\right)\n\\\\\n&&\n| w_{-} \\!\\!\\succ \\stackrel{\\cdot}{=} \\left(\\begin{array}{c} u_{-}(z) \\\\ v_{-}(z)\\end{array}\\right)\n\\,\\,,\n\\end{eqnarray*}\nare related\nby\n\\begin{eqnarray}\n&&\nu_{-}(z) = - (v_{+})^{*}(z)\n\\nonumber\n\\\\\n&&\nv_{-}(z) = + (u_{+})^{*}(z)\n\\label{plus-minus_relation}\n\\,\\,.\n\\end{eqnarray}\nHere,\n\\begin{eqnarray*}\n&&\n\\hat{\\cal L}| w_{+} \\!\\!\\succ \\stackrel{\\cdot}{=} \\lambda_{+} | w_{+} \\!\\!\\succ\n\\\\\n&&\n\\hat{\\cal L}| w_{-} \\!\\!\\succ \\stackrel{\\cdot}{=} \\lambda_{-} | w_{-} \\!\\!\\succ\n\\\\\n&&\n\\lambda_{-} = (\\lambda_{+})^{*}\n\\,.\n\\end{eqnarray*}\n\\end{itemize}\n\nWithin the context of the Inverse Scattering Transform, the wavefuction $\\psi(x,\\,t)$ in the parent NLSE, Eq.~(\\ref{td_NLSE}), is assumed to be localized in space, while the\neigenstates of the Lax operator $\\hat{\\cal L}$ of Eq.~(\\ref{Lax}) are required to be finite at $x=\\pm\\infty$.\nIn this case, the real eigenvalues of $\\hat{\\cal L}$ form a continuum spectrum, while the complex eigenvalues are discrete.\nFinally, in the parent NLSE, the complex eigenvalues correspond to\nthe solitonic part of the scattering data, while the real eigenvalues correspond to the thermal noise.\n\nFrom now on, we will assume that the eigenstates of $\\hat{\\cal L}$ with substantially complex eigenvalues (i.e. the ``discrete spectrum'',  or ``bound states'') are normalized\nas\n\\begin{eqnarray}\n\\prec \\!\\! w_{-} | w_{+} \\!\\!\\succ = 1\\,.\n\\label{normalization}\n\\end{eqnarray}\n\nFor the case of a single soliton,\n\\begin{eqnarray*}\n&&\n\\psi(z) = -i \\, \\mbox{sech}(x)\n\\,,\n\\end{eqnarray*}\nthe corresponding eigenvalues and eigenstates are\n\\begin{eqnarray*}\n&&\n\\lambda_{+} = - \\frac{i}{2}\n\\\\\n&&\n| w_{+} \\!\\!\\succ \\stackrel{\\cdot}{=} \\frac{+i}{\n2\n} \\exp(x\/2) \\left(\\begin{array}{c} -1 + \\tanh(x) \\\\ \\mbox{sech}(x) \\end{array}\\right)\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n&&\n\\lambda_{-} = + \\frac{i}{2}\n\\\\\n&&\n| w_{-} \\!\\!\\succ \\stackrel{\\cdot}{=}  \\frac{-i}{\n2\n} \\exp(x\/2) \\left(\\begin{array}{c} -\\mbox{sech}(x)  \\\\ -1 + \\tanh(x) \\end{array}\\right)\n\\,.\n\\end{eqnarray*}\n\n\\ssubsection{The Hellmann-Feynman theorem}\nLet $\\hat{\\cal L}$ depend on a parameter $\\xi$: $\\hat{\\cal L} = \\hat{\\cal L}(\\xi)$. Its discrete eigenvalues $\\lambda_{\\pm}$ and the corresponding eigenstates, $| w_{\\pm} \\!\\!\\succ$ then also depend on $\\xi$:\n$\\lambda_{\\pm} = \\lambda_{\\pm}(\\xi)$, and\n$| w_{\\pm} \\!\\!\\succ = | w_{\\pm}(\\xi) \\!\\!\\succ$.\n\nLet us express the eigenvalue $\\lambda_{+}$ as\n\\begin{eqnarray*}\n\\lambda_{+}(\\xi) = \\prec \\!\\! w_{-}(\\xi) | \\hat{\\cal L}(\\xi) | w_{+}(\\xi) \\!\\!\\succ\\,.\n\\end{eqnarray*}\nThen, the derivative of $\\lambda_{+}$ with respect to $\\xi$ becomes\n\\begin{multline*}\n\\frac{d}{d\\xi}  \\lambda_{+}\n=\n\\inner{\\frac{d}{d\\xi}\\kket{w_{-}}}{\\hat{\\cal L}\\kket{w_{+}}}\n\\\\\n+\n\\inner{\\kket{w_{-}}}{\\left(\\frac{d}{d\\xi}\\hat{\\cal L}\\right)\\kket{w_{+}}}+\n\\inner{\\kket{w_{-}}}{\\hat{\\cal L}\\frac{d}{d\\xi}\\kket{w_{+}}}\n\\end{multline*}\nAs in the proof of the usual Hellmann-Feynman theorem, the sum of the first and the last term will turn out to be proportional to the derivative of the norm; and since the norm is one, the derivative is zero. Indeed, the first term gives\n\\begin{align*}\n\\inner{\\frac{d}{d\\xi}\\kket{w_{-}}}{\\hat{\\cal L}\\kket{w_{+}}}\n&\n=\n\\inner{\\frac{d}{d\\xi}\\kket{w_{-}}}{\\lambda_{+}\\kket{w_{+}}}\n\\\\\n&\n=\n\\lambda_{+}\\inner{\\frac{d}{d\\xi}\\kket{w_{-}}}{\\kket{w_{+}}}\\,.\n\\end{align*}\nSimilarly, the last term gives\n\\begin{align*}\n\\inner{\\kket{w_{-}}}{\\hat{\\cal L}\\frac{d}{d\\xi}\\kket{w_{+}}}\n&\n=\n\\inner{\\hat{\\cal L}\\kket{w_{-}}}{\\frac{d}{d\\xi}\\kket{w_{+}}}\n\\\\\n&\n=\n\\inner{\\lambda_{-}\\kket{w_{-}}}{\\frac{d}{d\\xi}\\kket{w_{+}}}\n\\\\\n&\n=\n(\\lambda_{-})^{*}\\inner{\\kket{w_{-}}}{\\frac{d}{d\\xi}\\kket{w_{+}}}\\,.\n\\end{align*}\nBut $(\\lambda_{-})^{*}= \\lambda_{+}$; thus, the sum of the two terms gives\n\\begin{multline*}\n\\lambda_{+}\\,\\left[\\inner{\\frac{d}{d\\xi}\\kket{w_{-}}}{\\kket{w_{+}}}+\\inner{\\kket{w_{-}}}{\\frac{d}{d\\xi}\\kket{w_{+}}}\\right]\n\\\\\n=\\lambda_{+}\\,\\frac{d}{d\\xi}\\inner{\\kket{w_{-}}}{\\kket{w_{+}}}=\\lambda_{+}\\,\\frac{d}{d\\xi}\\,1=0\\,.\n\\end{multline*}\nThus, we get the following generalization of the Hellmann-Feynman theorem:\n\\begin{equation}\n\\frac{d}{d\\xi}  \\lambda_{+}  =  \\prec \\!\\! w_{-} | \\left(\\frac{d}{d\\xi} \\hat{\\cal L}\\right) | w_{+} \\!\\!\\succ\\,.\n\\label{HF}\n\\end{equation}\n\n\n\\ssubsection{The exact expression for $d\\lambda\/dt$ from the Hellmann-Feynman theorem}\nLet us set\n\\begin{eqnarray*}\n&&\n\\xi = t\n\\\\\n&&\n\\hat{\\cal L}(t)\n=\n\\left(\n\\begin{array}{cc}\n-i \\frac{\\partial}{\\partial z}           &   \\psi^{*}(z,\\,t)\n\\\\\n-\\psi^{}(z,\\,t) & +i \\frac{\\partial}{\\partial z}\n\\end{array}\n\\right)\n\\\\\n&&\n\\hat{\\cal L}(t)  | w(t)\\!\\!\\succ =  \\lambda(t) | w(t)\\!\\!\\succ\n\\\\\n&&\n\\frac{d}{dt}\\hat{\\cal L}(t) =\n\\\\\n&&\n\\left(\n\\begin{array}{cc}\n0           &   +(F[\\psi] + \\epsilon P[\\psi])^{*}(z,\\,t)\n\\\\\n-(F[\\psi] + \\epsilon P[\\psi])^{}(z,\\,t) & 0\n\\end{array}\n\\right)\n\\\\\n&&\n| w(t) \\!\\!\\succ =\n\\left(\n\\begin{array}{cc}\nu(z,\\,t)\n\\\\\nv(z,\\,t)\n\\end{array}\n\\right)\n\\\\\n&&\n2 \\int_{-\\infty}^{+\\infty} u(z,\\,t) v(z,\\,t) = 1\n\\,,\n\\end{eqnarray*}\nwhere $F[\\psi](z,\\,t) = -i \\left[- \\frac{\\partial^2}{\\partial z^2}  - 2 |\\psi(z,\\,t)|^2 \\right] \\psi(z,\\,t)$, and $P[\\psi](z,\\,t) = -i v_{ext}(z,\\,t)\\psi(z,\\,t)$.\nAccording to the Hellmann-Feynman theorem, Eq.~(\\ref{HF}), the time derivative of the Lax eigenvalue is\n\\begin{eqnarray*}\n\\frac{\\partial}{\\partial t}&& \\lambda\n=\n\\\\\n&&\n- \\int_{-\\infty}^{+\\infty} \\! dz \\,\n\\left\\{\n\tu_{+}^2(z,\\,t) \\epsilon P[\\psi](z,\\,t)\n\\right.\n\\\\\n&&\\hspace{10em}\\left.\n- v_{+}^2(z,\\,t) \\epsilon^{*} P^{*}[\\psi](z,\\,t)\n\\right\\}\n\\\\\n&&\n=\n(+i)\n\\int_{-\\infty}^{+\\infty} \\! dz \\,\n\\left\\{\n\t\\epsilon u^2(z,\\,t)\\psi(z,\\,t)\n\\right.\n\\\\\n&&\\hspace{10em}\\left.\n+ \\epsilon^{*} v^2(z,\\,t) \\psi^{*}(z,\\,t)\n\\right\\}v_{ext}(z,\\,t)\n\\,.\n\\end{eqnarray*}\nNotice that the contribution to $d\\lambda\/dt$ from $F[\\psi]$ disappears. Indeed this contribution\ndescribes the time derivative of the Lax eigenvalue in the time evolution according to the {\\it unperturbed} NLS; this derivative\nindeed vanishes as a consequence of integrability of the NLSE.\n\n\nA translation to the Kivshar-Malomed\nnotation system of Ref.~\\onlinecite{Kivshar1989_763} gives\n\\begin{multline*}\n\\frac{\\partial}{\\partial t} \\lambda_{n}\n=\n- \\frac{1}{2}\n\\frac{1}\n{\\int_{-\\infty}^{+\\infty} \\! dz \\, \\psi^{(1)}(x,\\,t) \\psi^{(2)}(x,\\,t) }\n\\\\\n\\times \\int_{-\\infty}^{+\\infty} \\! dz \\,\n\\left\\{\n\t(\\psi^{(1)})^2(x,\\,t,\\,\\lambda_{n}) \\epsilon P[\\psi](x,\\,t)\n\\right.\n \\\\\n\\left.\n - (\\psi^{(2)})^2(x,\\,t) \\epsilon^{*} P^{*}[\\psi](x,\\,t)\n\\right\\}\n\\,.\n\\end{multline*}\n\\mbox{}\\\\\n\\mbox{}\\\\\n\\mbox{}\\\\\n\\mbox{}\\\\\n\\mbox{}\\\\\n\\mbox{}\\\\\n\n\n\n\n\n\\newpage\n\n\\begin{widetext}\n\n\\begin{center}\n\\textbf{\\textsc{\\relsize{1}Extended Data}}\n\\end{center}\n\n\\vfill\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth,keepaspectratio=true,draft=false]{Fig_4_01a_font}\n\\end{center}\n\\caption { \\label{Fig_4_01a}\n\\textbf{Integrable vs. nonintegrable case.} The initial state for the nonintegrable\ncase was prepared by time-propagating the breather at rest while the\nnonlinearity was slowly ramped from $|\\psi|^{2}$ to $|\\psi|^{2p}$ with $p=3\/2$.\nThe result was Galilei-boosted and scattered off of a Gaussian barrier whose\nwidth (for numerical reasons) was twice the reference value. All other\nparameters were at their reference values, including the number of particles. Also plotted is the integrable case, all of whose parameters are at their\nreference values. In particular, the number of particles is four times smaller than that for Fig.~3 in the main text, resulting in a degraded, but\nstill noticeable plateau at around 25\\% transmission. In the nonintegrable\ncase, no plateau can be discerned.\n          }\n\\end{figure}\n\n\\vfill\n\\mbox{}\n\\newpage\n\\mbox{}\n\\vfill\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth,keepaspectratio=true,draft=false]{Fig_4_01b_font}\n\\end{center}\n\\caption { \\label{Fig_4_01b}\n\\textbf{Dependence on the phase of the\nbreathing cycle.} The $y$-axis: the value of $E_{\\text{kin}}\/V_{0}$ for which\ntransmission jumps from 0 to 1\/4; the $x$-axis: the time offset in the\nbreathing cycle for the initial state, relative to that in\nFig.~2 in the main text; all other parameters are as in that Figure.\n          }\n\\end{figure}\n\n\n\\vfill\n\\mbox{}\n\\newpage\n\\mbox{}\n\\vfill\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth,keepaspectratio=true,draft=false]{Fig_4_01c_font}\n\\end{center}\n\\caption { \\label{Fig_4_01c}\n\\textbf{Transmission plot for a three-soliton breather solution.} The  constituent solitons have norms 1\/6,\\, 1\/3,\\, and 1\/2 (1:2:3 norm ratio, which does \\emph{not} belong to the sequence of odd number ratios). The dash-dotted horizontal lines are at transmissions values of 1\/6, 1\/2, and 1, corresponding, respectively, to only the smallest, norm-1\/6 soliton being transmitted, to the norm-1\/6 and norm-1\/3 solitons being transmitted, and to all three constituent solitons being transmitted.\n          }\n\\end{figure}\n\n\\vfill\n\\mbox{}\n\\newpage\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth,keepaspectratio=true,draft=false]{solitons_and_breather_workv}\n\\end{center}\n\\caption { \\label{Fig_4_01d}\n\\textbf{Uncoupled solitons vs. breather.} How the ``staircase'' plot of Fig.~3 in the main text would look if the constituent solitons of the breather were completely uncoupled, as compared to now it is in reality. \\textbf{a,} The transmission plots for the scattering of single solitons, of norms $1\/4$ and $3\/4$, off a barrier. All parameters are as in Fig.~3 in the main text, with barrier width $w=w_{0}$. \\textbf{b,} dashed line: the weighted sum of the single-soliton transmission curves from panel a, with the norms used as weights. Solid line: the transmission curve for the breather for the same set of parameters. This is the same curve as the $w=w_{0}$ curve in Fig.~3 in the main text.\n          }\n\\end{figure}\n\n\\newpage\n\\mbox{}\n\\vfill\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth,keepaspectratio=true,draft=false]{phase_imprinting_split_05_workv}\n\\end{center}\n\\caption { \\label{Fig_4_01e}\n\\textbf{Superheated integrability in the context of phase imprinting.} At $t=0$, the breather wavefunction is multiplied by the space-dependent pure phase $e^{i\\epsilon \\varphi(x)}$. Here \\mbox{$\\epsilon=0.01$} and $\\varphi(x) = \\sqrt{\\frac{2}{L}}\\sqrt{\\frac{3}{2M}}\\sum_{m=1}^{M}\\left[c_{m}\\cos (2\\pi m x\/L)+s_{m}\\sin (2\\pi m x\/L)\\right]$, with \\mbox{$L=16$} and \\mbox{$M=5$}; $c_{m}$ and $s_{m}$ were drawn from the uniform distribution on $[-1,\\,1]$, and in the realization shown here had the values $(c_{1},\\,\\ldots,\\,c_{5})=(0.307,\\, 0.622,\\, 0.648,\\, -0.738,\\, 0.304)$ and $(s_{1},\\,\\ldots,\\,s_{5})=(0.353,\\,  -0.0422,\\, -0.794 ,\\,  -0.746,\\,  0.721)$. The function $\\varphi(x)$ is plotted in the inset. The main plot shows the time evolution of the density $|\\psi|^{2}$, during which the constituent solitons separate. Once they are well-separated, one can verify that their norms are 0.25 and 0.75. In the context of Eq.~(5) in the main text: if one uses the approximation $e^{i\\epsilon \\varphi(x)}\\approx 1 + i\\epsilon \\varphi(x)$, then the process depicted in this Figure corresponds to $v_{\\text{ext.}}(x,\\,t)\\,\\psi(x,t)=i\\delta(t)\\,\\varphi(x)\\,\\lim_{\\tau\\to t^{-}}\\psi(x,\\tau)$. Just as in the case of collision with a barrier, the separation of scales between the real and imaginary parts of the Lax-operator eigenvalues $\\lambda$, which correspond to the constituent solitons, again imply that the soliton velocities ($\\sim \\text{Re}~\\lambda$) change but norms ($\\sim \\text{Im}~\\lambda$) do not.\n          }\n\\end{figure}\n\n\\vfill\n\\mbox{}\n\\newpage\n\n\\end{widetext}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Intro}\n The heavy ion collisions produce matter at extreme temperatures and densities where \nit is expected to be in the form of Quark Gluon Plasma \n(QGP), a phase in which the quarks and gluons can move far beyond  the size of a nucleon \nmaking color degrees of freedom dominant in the medium. \n  The experimental effort to produce such matter started with low energy CERN accelerator \nSPS and evolved through voluminous results \nfrom heavy ion collision at Relativistic Heavy Ion Collider (RHIC) \\cite{INTRO_Arsene, INTRO_Back, INTRO_Adams, INTRO_Adcox}.\nThe recent results from Large Hadron Collider (LHC) experiments \\cite{QGP_Tc} are \npointing towards formation of high temperature system in many ways similar to the matter\nproduced at RHIC. \n  One of the most important signal of QGP is the suppression of \nquarkonium states \\cite{SATZ}, both of the charmonium ($J\/\\psi$, $\\psi(2S)$, $\\chi_{c}$, etc) \nand the bottomonium ($\\Upsilon(1S)$ , $\\Upsilon(2S)$, $\\chi_{b}$, etc) families. This is thought to be a \ndirect effect of deconfinement, when the binding potential between the constituents of a quarkonium state, \na heavy quark and its antiquark, is screened by the colour charges of the surrounding light quarks and gluons. \n The ATLAS and CMS experiments have carried out detailed quarkonia measurements in PbPb collisions \nwith the higher energy and luminosity available at the LHC.\n The ATLAS measurements \\cite{ATLAS} show suppression of inclusive $J\/\\psi$ with high transverse momenta $p_T$  \nin central PbPb collisions compared to peripheral collisions at $\\sqrt s_{NN} = 2.76$ TeV. \n  Similarly, CMS measured a steady and smooth decrease of suppression \nof prompt $J\/\\psi$ as a function of centrality with nuclear modification factor $R_{\\rm AA}$ remaining $<$ 1 even \nin the peripheral bin \\cite{JCMS}. \n\n The melting temperature of the quarkonia states depends on their binding energy. The ground states, \n$J\/\\psi$ and $\\Upsilon(1S)$ are expected to dissolve at significantly higher temperatures than the \nmore loosely bound excited states. The difference in binding energies among different quarkonia indicate that\nthey melt in a hot QGP at different temperatures and the quarkonium spectrum can\nserve as plasma thermometer \\cite{SATZ2,Mocsy_Strik}.\n  The $\\Upsilon(2S)$ and $\\Upsilon(3S)$ have smaller binding energies as compared to ground\nstate $\\Upsilon(1S)$ and hence are expected to dissolve at a lower temperature. \n With the 2011 PbPb run the CMS published results on sequential suppression of \n$\\Upsilon(nS)$ states as a function of centrality \\cite{CMSU2} with enlarged statistics\nover their first measurement \\cite{UCMS}\nwhere a suppression of the excited $\\Upsilon$ states with respect to the ground state have been observed  \nin PbPb collisions compared to pp collisions at $\\sqrt s_{NN} = 2.76$ TeV.\n\n  The quarkonia yields in heavy ion collisions are also modified due to non-QGP effects such as\nshadowing, an effect due to change of the parton distribution functions inside the nucleus,\nand dissociation due to nuclear or comover interactions \\cite{Vogt}. Due to higher mass, the \nnuclear suppression is expected to be less for bottomonia over charmonia.\n  If large number of heavy quarks are produced in initial heavy ion collisions at LHC energy \nthis could even lead to enhancement of quarkonia via statistical recombination \\cite{Rapp1,Rapp2}. \n The effect of regeneration is expected to be less significant for bottomonia as compared \nto charmonia since bottom quarks are much smaller in number as compared to charm quarks. \n In addition, due to higher bottom mass the bound state properties obtained from \npotential models are more reliable. Thus recent years witness a shift in the \ninterest to bottomonia. \n The ratios of the yields of excited states to the ground states is considered \neven more robust QGP probe as the cold nuclear matter effects if any cancel out and can be \nneglected in the ratios. The calculation of ratios of $\\Upsilon$ states was also made in \nfew works e.g. \\cite{UPsi_Blaiz,UPsi_Guni} in past which showed that the $p_T$ dependence \nof such ratio would show large variations and this would be a direct probe of the QGP. \n \n In this paper, we calculate the bottomonia suppression due to color screening in an expanding\nQGP using the model by Chu and Matsui \\cite{CHU},\nwhich takes into account the finite QGP lifetime and spatial extent. \n We start by describing the properties of quarkonia obtained from potential models and then \ngive a brief description of the model which is extended to get the survival \nprobabilities of $\\Upsilon$ states as a function of centrality of the collisions. \n Finally we compare the model calculations with the experimental data recently \nmeasured by the CMS experiment.\n\n\\section{Properties of the $\\Upsilon$ states from potential models}\nInteraction between the heavy quark and its antiquark inside the quarkonium at zero temperature  \ncan be described by Cornell potential \\cite{QPOT1, QPOT2,UPsi_KarschMehr}.\nThe solution of the Schrodinger equation for such potential gives mass, bound state radius and\nthe formation time $\\tau_{F}$, the time needed to form a bound state after the production of heavy quark pairs.\n All parameters obtained with zero temperature \npotential using the parameter values given in \\cite{UPsi_KarschMehr,QPROP} are summarized in \nfirst three rows of Table I, which describe well the experimentally \nobserved quarkonia spectroscopy.          \n\nThe potential model can be extended to finite temperature with the main assumption that medium effects can be \naccounted for as a temperature-dependent potential. Instead of just looking at the individual\nbound states (at $T$ = 0 where quarkonium is well defined), one could rather obtain a unified treatment of bound states, \nthreshold and continuum by determining the spectral function. \n  Using a class of screened potentials based on lattice calculations of the static quark-antiquark free energy, \nspectral functions at finite temperature are calculated in a work \\cite{UPsi_Mocsy2,UPsi_Mocsy3} and it was found that all \nquarkonium states, except the 1S bottomonium, dissolve in the deconfined phase at temperatures smaller than 1.5$T_C$.\nAn upper limit on binding energy and the thermal width of different quarkonia states are then estimated using  \nspectral functions in the quark-gluon plasma. \n Corresponding upper bounds on their dissociation temperatures $T_{D}$ \\cite{UPsi_Mocsy3} are\ngiven in second last row of Table I. We used slightly lower values of $T_{D}$ given in the last row \nto obtain a good match with measured $R_{\\rm AA}$. \n\\renewcommand{\\arraystretch}{1.4}\n\\begin{table}[ph]\n\\tbl{Quarkonia properties from non-relativistic potential theory \\cite{UPsi_KarschMehr,UPsi_Mocsy3}.}\n{\\begin{tabular}{@{}cccccc@{}} \\toprule \n\\hline\\noalign{\\smallskip}\n  {\\rm Bottomonium properties}        &    $\\Upsilon(1S)$   & $\\chi_b(1P)$  &  $\\Upsilon(2S)$      & $\\Upsilon(3S)$  &  $\\chi_b(2P)$ \\\\\n\\hline\\noalign{\\smallskip}\n {\\rm Mass~[GeV\/$c^{2}$]}     &    9.46             &   9.99        &    10.02             & 10.36           &   10.26 \\\\\n\\hline\n{Radius \\rm [fm]}          &    0.28             &   0.44        &     0.56             & 0.78            &0.68    \\\\\n\\hline \n$\\tau_{F}$ \\rm [fm] \\cite{UPsi_KarschMehr}  &    0.76             &   2.60        &     1.9              & 2.4             &     \\\\\n\\hline \n$T_D$ \\rm [GeV] upper limit \\cite{UPsi_Mocsy3}    &    2~$T_C$          &   1.3~$T_C$   &     1.2~$T_C$        & 1~$T_C$       &     \\\\\n\\hline\n$T_D$ \\rm [GeV] used in the present work  &    1.8~$T_C$          &   1.15~$T_C$   &     1.1~$T_C$        & 1.0~$T_C$       &     \\\\\n\\hline\n\\end{tabular} }\n\\label{prop}\n\\end{table}\n\n\\section{Quarkonia suppression in finite size QGP}\n The bottomonia survival probabilities due to color screening in an expanding QGP\nare estimated using a dynamical model which takes into account \nthe finite lifetime and spatial extent of the system \\cite{CHU}. The competition between the resonance formation \ntime $\\tau_{F}$ and the plasma characteristics such as temperature, lifetime and spatial extent decide the \n$p_{T}$ dependence of the survival probabilities of $\\Upsilon$ sates.  We describe the essential \nsteps used to develop the model which is then extended to get \nthe survival probabilities as a function of centrality of the collision.\n\n  The model assumes that quark gluon plasma is formed at some initial entropy density \n$s_0$ corresponding to initial temperature $T_0$ at time $\\tau_{0}$ which undergoes an \nisentropic expansion by Bjorken's hydrodynamics~\\cite{UPsi_Bjork}. The plasma cools to an entropy density $s_D$ \ncorresponding to the dissociation temperature $T_D$ in time $\\tau_{D}$ which is given by \n\\begin{equation}\\label{bjork}\n \\tau_{D} = \\tau_0 \\left( { s_0 \\over s_D} \\right) = \\tau_0 \\left( \\frac{T_{0}}{T_{D}} \\right)^{3},\n\\end{equation}\n  As long as $\\tau_{D}$\/$\\tau_{F}$ $>$ 1, quarkonium formation will be suppressed. \n\n In the finite system produced in heavy ion collision, the suppression and entropy depend on the size \nof the system. The initial entropy density is assumed to be dependent on radius $R$ (decided by the\ncentrality of the collision) of the QGP \\cite{CHU} as \n\\begin{equation}\\label{eprofile}\n   s_0(r) = s_0 ~\\left(1 - \\left(\\frac{r}{R}\\right)^2\\right)^{1\/4},\n\\end{equation}\n  Using Eq.~(\\ref{bjork}) and Eq.~(\\ref{eprofile}) one can obtain the $r$ dependence of $\\tau_D$ as\n\\begin{eqnarray}\\label{tDr}\n   \\tau_{D}(r) & = &  \\tau_{D}(0)\\left(1 - \\left(\\frac{r}{R}\\right)^2\\right)^{1\/4}.\n\\end{eqnarray}\nwhere $\\tau_{D}(0)$ is the value of $\\tau_{D}$ for resonances produced in the center of the system.\n\n  Let a $Q\\overline{Q}$ pair is created at the position ${\\rm \\bf r}$ in the transverse plane with a \ntransverse momentum ${\\rm \\bf p_T}$ and transverse energy $E_{T}$ = $\\sqrt{M^2 + p_{T}^2}$.\n The $\\Upsilon$ formation time is $\\tau_{F}\\gamma$ which on equating with the screening duration $\\tau_{D}(r)$  given \nin Eq~(\\ref{tDr}) one obtains the critical radius $r_D$, which is the boundary of the suppression region as \n\\begin{equation}\n  r_D  = R\\left(1 - \\left(\\frac{\\gamma \\tau_{F}}{\\tau_{D}(0)}\\right)^{4}\\right)^{1\/2}.\n\\end{equation}\nwhere $\\gamma$ = $E_T\/M$ is the Lorentz factor associated with the transverse motion of the pair. \n A bottom-quark pair can escape the screening region $r_D$ and form $\\Upsilon$ if the position at \nwhich it is created satisfies\n\\begin{equation}\\label{taumax}\n| {\\rm \\bf r} + {\\tau_{F} {\\rm \\bf p_{T}} \\over M} | > r_D,\n\\end{equation}\nwhere the screening region $r$ $<$ $r_D$ is shrinking because of the cooling of the system.\n Defining $\\phi$ to be the angle between ${\\rm \\bf p_{T}}$  and ${\\rm \\bf r}$, the Eq.~(\\ref{taumax}) leads to a range \nof $\\phi$ for which the bottom-quark pair can escape:\n\\begin{equation}\\label{cosmax}\n   {\\rm cos} \\, \\phi \\ge z   ~~~~{\\rm where } ~~~~ \\nonumber \\\\ \n    z = \\frac{ r_D^2 - r^2 - (\\tau_{F}p_{T}\/M)^2}{2r \\,(\\tau_{F}p_{T}\/M)},\n\\end{equation}\n  With this we can then calculate probability for the pair created at ${\\rm \\bf r}$ with transverse momentum ${\\rm \\bf p_T}$\nto survive as\n\\begin{eqnarray}\n\\phi(r,p_{T})  &  =     1                 & \\,\\,\\, z\\le -1  \\nonumber \\\\\n               &  =  \\left( { {\\rm cos}^{-1}z \\over \\pi} \\right)    & \\,\\,\\, |z| < 1  \\nonumber \\\\\n               &  =    0                 & \\,\\,\\, z\\ge 1,   \\nonumber  \n\\end{eqnarray}\n  If the probability $\\rho(r)$ of a quark pair to be created at $r$ which is symmetric in transverse plane \nis parameterized as\n\\begin{equation}\n\\rho(r) = \\left(1 - \\left( {r \\over R} \\right)^2\\right)^{1\/2},\n\\end{equation}\nthe survival probability of quarkonia becomes \n\\begin{equation}\nS(p_{T}, R) = \\frac{\\int_0^Rdr~r~\\rho(r)~\\phi(r,p_{T})}{\\int_0^Rdr~r~\\rho(r)}.\n\\end{equation}\n\nThe survival probability as a function of centrality can be obtained by integrating over \n$p_T$ as follows \n\\begin{equation} \n  S(N_{\\rm part})  = \\int S(p_{T}, R(N_{\\rm part}) ) \\, Y(p_T) \\,dp_T.\n\\end{equation}\nHere Y($p_T$) is $p_T$ distribution (normalized to one) obtained from Pythia. \nThe size $R = R(N_{\\rm part})$ as a function of centrality is obtained in terms of the radius of the Pb \nnucleus given by $R_0 = r_0\\, A^{1\/3}$($r_0 = 1.2 \\,$ fm) and the total number of participants $N_{\\rm part0}=2A$ in head-on collisions as \n\\begin{equation}\\label{rnpart}\nR(N_{\\rm part}) = R_0 \\, \\sqrt{N_{\\rm part} \\over N_{\\rm part0} }.\n\\end{equation}\nWe assumed initial temperature $T_0$ is the temperature in  0-5\\% central collisions and calculated it\nfor a given initial time $\\tau_0$ by\n\\begin{equation}\\label{Int1}\nT_{0}^{3}\\tau_{0} = \\frac{3.6}{4a_{q}\\pi R_{0-5\\%}^{2}}\\left(\\frac{dN}{d\\eta}\\right)_{0-5\\%},\n\\end{equation}\nHere $(dN\/d\\eta)_{0-5\\%}$ = 1.5$\\times$1600 obtained from the charge particle multiplicity measured in \nPbPb collisions at 2.76 TeV \\cite{MULT} and $a_{q}$ = 37$\\pi^{2}$\/90 is the degrees of freedom we take in \nquark gluon phase. Using Eq.~(\\ref{rnpart}) we can obtain the transverse size of the system\nfor 0-5\\% centrality as $R_{0-5\\%}$ = 0.92$R_0$. \nFor $\\tau_{0}$ = 0.1 fm\/$c$, we obtain $T_{0}$ as 0.65 GeV using Eq.~(\\ref{Int1}). \nThe critical temperature is taken as $T_{C}$ = 0.170 GeV \\cite{QGP_Tc}. \nThe initial temperature as a function of centrality is calculated by \n\\begin{equation}\\label{Int2}\nT(N_{\\rm part})^3 = T_0^3 \\, \\left({dN\/d\\eta \\over N_{\\rm part}\/2}\\right) \/ \\left({dN\/d\\eta \\over N_{\\rm part}\/2}\\right)_{0-5\\%}.\n\\end{equation}\nwhere $(dN\/d\\eta)$ is the multiplicity as a function of number of participants measured by ALICE experiment \\cite{MULT}. \nBoth ALICE and CMS \\cite{CMSmult} measurements on multiplicity agree well with each other.\n  Equation~(\\ref{Int2}) giving the variation of initial temperature as a function of centrality \ndiffers from the approach taken in the work of Ref.~\\cite{STR1} where it is taken to vary\nas a third root of number of participants.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70\\textwidth]{Fig1_dNdEta.eps}\n\\caption{\na) Measured $(dN\/d\\eta)\/(N_{\\rm part}\/2)$ ~\\cite{MULT} as a function \nof $N_{\\rm part}$ along with the function $(dN\/d\\eta)\/(\\pi R^2)$. \n(b) The initial temperature obtained from measured multiplicity using Eq.~(\\ref{Int2})\n}\n\\label{fig:upsiRatio1}\n\\end{center}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n  \\includegraphics[width=0.48\\textwidth]{Fig2a_rD1s.eps}\n  \\includegraphics[width=0.48\\textwidth]{Fig2b_rD2s.eps} \\\\\n\\caption{ The screening radius $r_D$ (in fm) as a function of $p_T$ for $R=6.8$ fm \n(corresponding to head-on collisions) and $R=3.7$ fm (corresponding to minimum bias collisions)\nfor (a) $\\Upsilon(1S)$ and (b) $\\Upsilon(2S)$.\n The straight lines $|$ ${\\rm \\bf r} + {\\tau_{F} {\\rm \\bf p_{T}} \\over M}$ $|$\nmark the distance a bottom quark pair (created at $r=0$) will travel before forming a bound state.\n The mesh region in both the figures marks the escape region for \nbottom quark pair in case of head-on collisions and total shaded (mesh+lines) region marks\nthe escape region in case of minimum bias collisions. }\n\\label{fig:upsiRatio2}\n\\end{center}\n\\end{figure}\n\n  The nuclear modification factor, $R_{\\rm AA}$ is obtained from survival probability taking into account \nthe feed-down corrections as follows,\n\n\\begin{eqnarray}\n R_{\\rm AA}(3S) &=& S(3S)  \\nonumber \\\\\n\n R_{\\rm AA}(2S) &=& f_1~S(2S) + f_2~S(3S)  \\nonumber \\\\\n R_{\\rm AA}(1S) &=& g_1 ~S(1S) + g_2~S(\\chi_b(1P)) + g_3~S(2S) + g_4~S(3S)\n\\end{eqnarray}\nThe factors $f$'s and $g$'s are obtained from CDF measurement \\cite{CDF}. The values \nof $g_{1}$, $g_{2}$, $g_{3}$ and $g_{4}$ are 0.509, 0.27, 0.107 and 0.113 respectively.\nHere it is assumed that the survival probabilities of $\\Upsilon(3S)$ and $\\chi_{b}$(2P) \nare same and $g_4$ is their combined fraction. \nThe values of $f_{1}$ and $f_{2}$ are taken as 0.50 guided by the work from Ref.~\\cite{STR2}. \n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{Fig3a_Surv_MB.eps}\n\\includegraphics[width=0.48\\textwidth]{Fig3b_SurvFd_MB_3s.eps}\\\\\n\\caption{ (a) The survival probability as a function of \n$p_{T}$ for $\\Upsilon(1S)$, $\\Upsilon(2S)$, $\\Upsilon(3S)$ and $\\chi_{b}(1P)$ for $R=3.7$ fm \n(corresponding to average $N_{\\rm part}$ = 114 for minimum bias collisions). \n (b) The nuclear modification factor \nfor $\\Upsilon(1S)$, $\\Upsilon(2S)$ and $\\Upsilon(3S)$ which is obtained from survival probabilities including \nfeed down corrections.  The solid squares are $\\Upsilon(1S)$ $R_{\\rm AA}$ measured in the minimum \nbias PbPb collisions at $\\sqrt{s_{NN}} = 2.76$ TeV by CMS experiment \\cite{JCMS}. \n}\n\\label{fig:upsiRatio3}\n\\end{center}\n\\end{figure*}\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.65\\textwidth]{Fig4_RaaALL.eps} \n\\caption{The nuclear modification factor, $R_{\\rm AA}$\nas a function of $N_{\\rm part}$ for $\\Upsilon(1S)$, $\\Upsilon(2S)$ and $\\Upsilon(3S)$. The solid squares and circles are \nmeasured $R_{\\rm AA}$ by CMS experiment in PbPb collisions at $\\sqrt{s_{\\rm NN}}$ = 2.76 TeV \\cite{CMSU2} for  \n$\\Upsilon(1S)$ and $\\Upsilon(2S)$ respectively and solid triangles are the minimum bias data points. \nThe boxes at unity are the common systematic uncertainties in pp luminosity \nmeasurement and the pp yield. The lines(solid for $\\Upsilon(1S)$ , dashed for $\\Upsilon(2S)$ and dotted for $\\Upsilon(3S)$) \nrepresent the present model calculations. }\n\\label{fig:upsiRatio4}\n\\end{center}\n\\end{figure*}\n\n\\section{Results and discussions}\n  Figure~\\ref{fig:upsiRatio1} (a) shows measured $(dN\/d\\eta)\/(N_{\\rm part}\/2)$ ~\\cite{MULT} as a function \nof $N_{\\rm part}$. The function $(dN\/d\\eta)\/(\\pi R^2)$ gives the multiplicity divided by transverse \narea obtained using Eq.(\\ref{rnpart}). Figure~\\ref{fig:upsiRatio1} (b) gives the initial temperature \nobtained from measured multiplicity using Eq.~(\\ref{Int2}). Except in peripheral collisions, the initial \ntemperature has weak dependence on centrality of collisions. \nFigure~\\ref{fig:upsiRatio2} demonstrates working of the model. It shows\nthe screening radius $r_D$ (in fm) as a function of $p_T$ for $R=6.8$ fm \n(corresponding to head-on collisions) and $R=3.7$ fm (corresponding to minimum bias collisions)\nfor (a) $\\Upsilon(1S)$ and (b) $\\Upsilon(2S)$.\n The straight lines $|$ ${\\rm \\bf r} + {\\tau_{F} {\\rm \\bf p_{T}} \\over M}$ $|$\nmark the distance a bottom quark pair (created at $r=0$) will travel before forming a bound state.\n The mesh region in both the figures marks the escape region for \nbottom quark pair in case of head-on collisions and total shaded (mesh+lines) region marks\nthe escape region in case of minimum bias collisions. \n If $r$ is non-zero, the region where a bottomonium can escape screening, enlarges.\n\n Figure~\\ref{fig:upsiRatio3} (a) shows the survival probability as a function of \n$p_{T}$ for $\\Upsilon(1S)$, $\\Upsilon(2S)$, $\\Upsilon(3S)$ and $\\chi_{b}(1P)$ for $R=3.7$ fm \n(corresponding to average $N_{\\rm part}$ = 114 for minimum bias collisions). \n  The survival probability $S(p_{T})$ has a unique $p_T$ dependence decided by the \n$T_D$ and $\\tau_{F}$ of each $\\Upsilon$ state. \n  In general, the survival probabilities of resonance states increase with increasing $p_T$ \nand become unity at different $p_T$ for different states corresponding to complete survival. \n Since $\\Upsilon(1S)$ is expected to dissolve at a higher temperature it has more probability to survive\nthe plasma region even at lower p$_{T}$ as compared to the cases of other bottomonia states.\n The model gives very similar survival probabilities for $\\Upsilon(2S)$ and $\\Upsilon(3S)$.\nThis is due to the fact that $\\Upsilon(3S)$ has large formation time even though its \ndissociation temperature is smaller in comparison to $\\Upsilon(2S)$.\n Figure~\\ref{fig:upsiRatio3} (b) shows the nuclear modification factor \nfor $\\Upsilon(1S)$, $\\Upsilon(2S)$ and $\\Upsilon(3S)$ which is obtained from survival probabilities \nincluding feed down corrections. \n The solid squares are $\\Upsilon(1S)$ $R_{\\rm AA}$ measured in the minimum bias PbPb collisions at \n$\\sqrt{s_{NN}} = 2.76$ TeV by CMS experiment \\cite{JCMS}. \n The model reproduces the trend of the $p_T$ dependence of low statistics \nmeasurements of $R_{\\rm AA}$ from 2010 PbPb collisions by CMS.  \n\n  Figure~\\ref{fig:upsiRatio4} shows the nuclear modification factor, $R_{\\rm AA}$\nas a function of $N_{\\rm part}$ for $\\Upsilon(1S)$, $\\Upsilon(2S)$ and $\\Upsilon(3S)$. The solid squares and circles are \nmeasured $R_{\\rm AA}$ by CMS experiment in PbPb collisions at $\\sqrt{s_{\\rm NN}}$ = 2.76 TeV \\cite{CMSU2} for  \n$\\Upsilon(1S)$ and $\\Upsilon(2S)$ respectively and solid triangles are the minimum bias data points. \nThe lines(solid for $\\Upsilon(1S)$, dashed for $\\Upsilon(2S)$ and dotted for $\\Upsilon(3S)$) represent \nthe present model calculations. \nThe common systematic uncertainties in pp luminosity measurement and the pp yield are \nrepresented by the boxes at unity. The model correctly reproduces the measured nuclear modification \nfactors of both $\\Upsilon(1S)$ and $\\Upsilon(2S)$ for \nall centralities using the parameters given in the Table I. The \nsurvival probabilities for $\\Upsilon(2S)$ and $\\Upsilon(3S)$ are very similar.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.65\\textwidth]{Fig5_DRatio.eps}\n\\caption{Double ratio, $[\\Upsilon(2S)\/\\Upsilon(1S)]_{PbPb}$\/$[\\Upsilon(2S)\/\\Upsilon(1S)]_{pp}$ as a \nfunction of $N_{\\rm part}$ measured by CMS experiment \\cite{CMSU2} \nalong with the present calculation (solid line). The box at unity is\nthe common systematic uncertainty in the pp yield. \n}\n\\label{fig:upsiRatio5}\n\\end{center}\n\\end{figure}\n\n We also calculated the ratio of $R_{\\rm AA}$ of $\\Upsilon(2S)$ to that of $\\Upsilon(1S)$ \nwhich is equivalent to the so called double ratio $[\\Upsilon(2S)\/\\Upsilon(1S)]_{PbPb}$\/$[\\Upsilon(2S)\/\\Upsilon(1S)]_{pp}$.\nThe double ratio has the advantage that the effects such as initial-state nuclear effects and regeneration\nwhich we ignore in our calculations are supposedly canceled out. \n Figure~\\ref{fig:upsiRatio5} shows the double ratio measured by CMS experiment \\cite{CMSU2} \nalong with the present calculation. The calculations reproduce the measured double ratio \neven for the most peripheral data point.\n\n The most important parameters in above study are formation time \nand dissociation temperatures of bottomonia states. There are reliable calculations of formation time \nobtained from zero temperature potential models which reproduce the bottomonia spectroscopy very well. \nUpper limits are available for dissociation temperatures which are obtained from potential models \nat finite temperature. We used slightly lower values of the dissociation temperature to get a good\ndescription of the measured nuclear modification factors of $\\Upsilon(1S)$ and $\\Upsilon(2S)$. \n The dynamics of the system is affected by the initial conditions which in the present calculations are \nfixed using measured charged particle multiplicity at LHC.\nThere can be suppression due to initial nuclear effects which we assume to be \nmuch smaller than that due to colour screening and hence are ignored in the present work. \n  The calculations of shadowing in PbPb show that it will effect the bottomoina yields by \napproximately 20 \\% for most central collisions \\cite{Shadow}. Thus, the dissociation temperatures \nobtained by us are still considered to be the upper limits. Conversly there are other views which say that  \n$\\Upsilon$ ground state is not much affected by the color screening \nup to the temperatures of $\\sim 3-4T_C$ and regeneration of the states are not negligible at the LHC \\cite{Rapp}.\n  The bottom quark mass is 10 times higher than the temperature we \nare considering for the system and hence the regeneration effect can be safely ignored in calculating\nnuclear modification for bottomonia. The uncertainties in the measurements of feed-down fractions \nwould introduce uncertainties in the calculated nuclear modification factor. \n Finally we mention that the uncertainties arising from the effects other than colour screening are small and supposedly \nwill have little or no effect on the double ratio.\n\n\\section{Conclusions}\n  In summary, we calculate the survival probabilities of $\\Upsilon$ states and obtain the nuclear modification\nfactors due to colour screening in an expanding quark gluon plasma of finite lifetime and size produced \nduring PbPb collisions $\\sqrt{s_{NN}}=$ 2.76 TeV.\n  The formation time and dissociation temperatures of bottomonia states \nobtained from potential models are used as input parameters in the model. \nWe used slightly lower values of the dissociation temperatures to get a good\ndescription of the measured nuclear modification factors of $\\Upsilon(1S)$ and $\\Upsilon(2S)$. \n  The model reproduces the centrality dependence of measured nuclear modification \nfactors of $\\Upsilon(1S)$ and $\\Upsilon(2S)$ and the double ratio very well at $\\sqrt{s_{\\rm NN}}$ = 2.76 TeV.\n  The trend of $p_T$ dependence of low statistics measurements of nuclear modification factor \nfrom 2010 PbPb collisions of CMS is reproduced as well.  \n The uncertainties arising from effects other than colour screening are assumed to be \nsmall and supposedly will have little or no effect on the double ratio calculations.\n\\section{Acknowledgement}\nWe thank Vineet Kumar, Ramona Vogt and CMS heavy ion colleagues for many fruitful discussion.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the last fifteen years, large dimensional stationary factor models have achieved great success in the economic profession, especially in forecasting macroeconomic variables \\citep[see, e.g.,][]{Nowcasting}, and are now a common tool in several policy institutions. However, macroeconomic time series are typically non-stationary due to the presence of common and idiosyncratic stochastic trends, and the practice of differencing the data to achieve stationarity is a problem that not always has a clear-cut solution. Take for example the case of the unemployment rate, which is a highly-persistent time series, but at the same time economic theory forbids it to have a unit root; or, take as another example the case of inflation, which shows  periods of high-persistence in the late 70s early 80s, while more recently displays clear mean reversion. To avoid the risk of over- or under-differencing data, a Non-Stationary Dynamic Factor Model (NS-DFM) is then desirable, and it is studied in this paper. \n\nThe NS-DFM proposed in this paper captures several features of macroeconomic data as it takes into account the presence of common trends generating permanent fluctuations in the economy, as well as common transitory forces generating cyclical fluctuations. More technically, in our model, the common factors are a cointegrated vector process, thus containing both $I(1)$ trends and stationary components. Moreover, the NS-DFM addresses the possible presence of idiosyncratic trends, as well as the presence of secular (linear) trends, which can have either a constant slope (deterministic linear trends) or a time-varying slope (local linear trends). \n\nIn this paper, we study estimation of the NS-DFM by Quasi Maximum Likelihood (QML) implemented through the Expectation Maximization (EM) algorithm and the Kalman smoother (KS). Specifically, we extend the results in \\citet{BLqml} for the stationary case, to prove that when the common factors are the only source of non-stationarity, the common component estimated at a given point in time and for a given unit is $\\min(\\sqrt n,\\sqrt T)$-consistent. We also discuss extensions to the cases of (i) unit roots in the idiosyncratic components, and (ii) local levels and local linear trends. \n \nEstimation is implemented in two steps. First, given the observed data, by means of the KS we estimate the conditional mean of the latent factors,  which, together with its associated conditional covariance matrix, we use to compute the expected log-likelihood of the model (E-step).\\footnote{In a non-stationary setting the existence of the conditional mean of the factor as a minimizer of the mean-squared prediction error has been proved by \\citet[Theorem 1]{hannan67} and \\citet[Theorem 1]{sobel67}.} Second, we maximize the expected log-likelihood with respect to the loadings and the other parameters of the model (M-step). The use of an iterative procedure to extract unobserved components in the case of non-stationary data was proposed since the original work by \\citet{kalman60}. Although this is not the first paper using these techniques for non stationary data, this is the first paper to address consistency of factors. Moreover, QML estimation of autoregressive processes with unit roots is a classical problem studied at length by the literature \\citep{simsstockwatson,johansen91}.\\footnote{Solutions based on spectral analysis are also in \\citet{bell84} and \\citet{CT76}.}\n\nEstimation of NS-DFMs has also been studied by \\citet{baing04} and \\citet{BLL2} by PC analysis on differenced data. Both approaches are designed to account for non-stationary idiosyncratic components; however, only the latter is designed to deal with linear deterministic trends. \\citet{bai04} has used a factor model to estimate common trends via PC on data in levels. However, because of its nature, that approach is valid only if all idiosyncratic components are stationary, i.e., only if data are cointegrated.\n\nCompared to those PC based estimators, our approach has a number of practical advantages. First, it allows estimating the model even in the presence of missing values, which is crucial when using the model in real-time because macroeconomic data are published with delays and at non-synchronized dates. Second, it allows estimating jointly stochastic trends as well as (deterministic or local) linear trends, whereas \\citet{baing04} and \\citet{BLL2} are forced to remove the deterministic trends before running PC analysis. Third, it allows having time-varying parameters, such as, for example, the slope of linear trends. Fourth, it allows putting restrictions on the parameters, such as national accounts identities, or restrictions coming from economic theory.\n\nFrom a theoretical point of view, our estimator converges at a faster rate than those of \\citet{baing04} and \\citet{BLL2}. However, this faster convergence does not come for free. Indeed, our estimator is based on stronger assumptions than those of PC analysis: namely, it is derived under the assumption that we know which idiosyncratic components are $I(1)$ and which ones are stationary, and which series have a linear trend component. Under this assumption, we can model the $I(1)$ idiosyncratic components, and the time-varying slopes or means, as additional latent states in the KS, thus allowing to simultaneously estimate the entire model. This strategy is shown to work well in practice, provided the number of additional latent states is not too large.\n\nThe use of the EM in time series dates back to \\citet{SS77}, \\citet{shumwaystoffer82}, \\citet{watsonengle83}, \\citet{quahsargent93}, and \\citet{SAZ13}, among others. However, with the exception of the last two, none of the above works has considered the case of non-stationary data. Moreover, to the best of our knowledge, no asymptotic theory exists for the setting considered in this paper.\n\nThe rest of the paper proceeds as follows: in Section \\ref{sec:modelNS}, we present the NS-DFM and its assumptions. Estimation is outlined in Section \\ref{sec:estNS} where we also prove consistency. The extension to non-stationary idiosyncratic states is discussed in Section \\ref{sec:idioI1}. Numerical results are in Section \\ref{sec:mc2}. Section \\ref{sec:conclusion} concludes.\n\n\\section{Model and assumptions}\\label{sec:modelNS}\nWe define a NS-DFM driven by $q$ factors as \n\\begin{align}\nx_{it}&=\\alpha_{it}+\\beta_{it} t+\\bm b_i^\\prime(L) \\bm f_t +\\xi_{it},\\label{eq:NSDFM1}\\\\\n\\bm f_t&= \\bm{\\mathcal A}(L)\\bm f_{t-1}+\\bm u_t,\\label{eq:NSDFM2}\\\\\n\\xi_{it}&=\\rho_i\\xi_{it-1}+e_{it},\\label{eq:NSDFM3}\\\\\n\\alpha_{it}&=\\alpha_{it-1}+\\omega_{it},\\label{eq:NSDFM5}\\\\\n\\beta_{it}&=\\beta_{it-1}+\\eta_{it},\\label{eq:NSDFM4}\n\\end{align}\nfor  $i=1,\\ldots, n$, and $t=1,\\ldots, T$. We let $\\chi_{it}=\\bm b_i^\\prime(L) \\bm f_t$. Then, $\\bm\\chi_{nt}=(\\chi_{1t}\\cdots\\chi_{nt})^\\prime$ is the common component, $\\bm\\xi_{nt}=(\\xi_{1t}\\cdots\\xi_{nt})^\\prime$ the idiosyncratic component, $\\bm {\\mathcal B}_n(L)=(\\bm b_1(L)\\cdots\\bm b_n(L))^\\prime$ the $n\\times q$ polynomial matrix of factor loadings, $\\bm f_t=(f_{1t}\\cdots f_{qt})^\\prime$ the $q$ factors, $\\bm u_t=(u_{1t}\\cdots u_{qt})^\\prime$ the $q$ common shocks, $\\mathbf e_{nt}=(e_{1t}\\cdots e_{nt})^\\prime$ the idiosyncratic shocks, and we also define $\\bm\\omega_{nt}=(\\omega_{1t}\\cdots\\omega_{nt})^\\prime$ and $\\bm\\eta_{nt}=(\\eta_{1t}\\cdots\\eta_{nt})^\\prime$.\n\n\nWe make the following assumptions.\n\n\\begin{ass}\\label{ass:dynamic}\n\\begin{inparaenum}[(a)]\n\\item for all $i\\in\\mathbb N$ and $z\\in\\mathbb C$, $\\bm b_i(z)=\\sum_{k=0}^s \\bm b_{ik}z^k$, such that $\\bm b_{ik}$ are $q\\times 1$ and $s$ is a finite integer with $s\\ge 0$;\n\\item for all $n\\in\\mathbb N$, let $\\bm{\\mathcal B}_{kn}=(\\bm b_{1k}\\cdots \\bm b_{nk})^\\prime$, then $\\lim_{n\\to\\infty}\\Vert n^{-1}\\bm{\\mathcal B}_{kn}^\\prime\\bm{\\mathcal B}_{kn}-\\bm\\Sigma_{k}\\Vert=0$, with $\\bm\\Sigma_{k}$ being $q\\times q$, and $\\bm \\Sigma_0$ positive definite, while $\\mbox{rk}(\\bm\\Sigma_k)\\le q$ for $k=1,\\ldots,s$, \nmoreover, for all $i\\in\\mathbb N$ and $k=0,\\ldots, s$, $\\Vert\\bm b_{ik}\\Vert \\le M_B$ for some finite positive real $M_B$ independent of $i$ and $k$; \\item $\\bm\\Gamma^{\\Delta f}=\\E_{\\varphi_n}[\\Delta\\bm f_t\\Delta\\bm f_t^\\prime]$ is $q\\times q$ positive definite and there exists a finite positive real $M_f$, such that $\\Vert\\bm\\Gamma^{\\Delta f}\\Vert\\le M_f$;\n\\item $q$ is a finite positive integer, such that $q<n$ and is independent of $n$; \\item $\\bm{\\mathcal A}(z)=\\sum_{k=1}^{p} \\bm{\\mathcal A}_{k}z^{k-1}$, such that $\\bm{\\mathcal A}_{k}$ are $q\\times q$ and $p$ is a finite positive integer, and $\\det(\\mathbf I_q-\\bm{\\mathcal A}(z))\\ne 0$ for all $z\\in\\mathbb C$ such that $|z|< 1$; \\item $\\mbox{\\upshape rk}(\\bm{\\mathcal A}(1))=d$ with $0<d\\le q$; \\item $|\\rho_i|\\le 1$ for all $i\\in\\mathbb N$; \n\\item $\\alpha_{i0}$ and $\\beta_{i0}$ are finite reals.\n\\end{inparaenum}\n\\end{ass}\n\n\\begin{ass}\n\\begin{inparaenum}[(a)]\\label{ass:modelNS}\n \\item for all $t\\in\\mathbb Z$, $\\bm u_t\\sim \\mathcal N(\\mathbf 0_q,\\bm\\Gamma^u)$, such that $\\bm\\Gamma^u$ is $q\\times q$ and positive definite, and  $\\E_{\\varphi_n}[\\bm u_{t}\\bm u_{t-k}^\\prime]=\\mathbf 0_{q\\times q}$ for all $k\\ne 0$; \\item for all $t\\in\\mathbb Z$ and all $n\\in\\mathbb N$, $\\mathbf e_{nt}\\sim\\mathcal N(\\mathbf 0_n,\\bm\\Gamma_n^e)$, such that $\\bm\\Gamma_n^e$ is $n\\times n$ and positive definite, and $\\E_{\\varphi_n}[\\mathbf e_{nt}\\mathbf e_{nt-k}]=\\mathbf 0_{n\\times n}$ for all $k\\ne 0$; \n \\item for all $n\\in\\mathbb N$, $\\Vert\\bm\\Gamma_n^e \\Vert \\le M_e$, for some positive real $M_e$ independent of $n$; \\item $\\E_{\\varphi_n}[\\mathbf e_{nt}\\bm u_{s}^\\prime]=\\mathbf 0_{n\\times q}$ for all $n\\in\\mathbb N$ and $t,s\\in\\mathbb Z$; \\item for all $t\\in\\mathbb Z$ and all $n\\in\\mathbb N$, $\\bm \\omega_{nt}\\sim\\mathcal N(\\mathbf 0_n,\\bm\\Gamma_n^\\omega)$ and $\\bm \\eta_{nt}\\sim\\mathcal N(\\mathbf 0_n,\\bm\\Gamma_n^\\eta)$, such that $\\bm\\Gamma_n^\\omega$ and $\\bm\\Gamma_n^\\eta$ are diagonal, respectively with entries $0\\le \\sigma_{i\\omega}^{2} < M_\\omega$ and $0\\le \\sigma_{i\\eta}^{2} < M_\\eta$, for some positive reals $M_\\omega$ and $M_\\eta$ independent of $i$, and $\\E_{\\varphi_n}[\\bm \\omega_{nt}\\bm \\omega_{nt-k}]=\\mathbf 0_{n\\times n}$ and $\\E_{\\varphi_n}[\\bm \\eta_{nt}\\bm \\eta_{nt-k}]=\\mathbf 0_{n\\times n}$ for all $k\\ne 0$; \\item \n $\\E_{\\varphi_n}[\\bm\\omega_{nt}\\bm u_{s}^\\prime]=\\mathbf 0_{n\\times q}$,\n $\\E_{\\varphi_n}[\\bm\\eta_{nt}\\bm u_{s}^\\prime]=\\mathbf 0_{n\\times q}$, for all $n\\in\\mathbb N$ and $t,s\\in\\mathbb Z$;\n \\item  $\\E_{\\varphi_n}[\\bm\\omega_{nt}\\mathbf e_{ns}^\\prime]=\\mathbf 0_{n\\times n}$,\n $\\E_{\\varphi_n}[\\bm\\eta_{nt}\\mathbf e_{ns}^\\prime]=\\mathbf 0_{n\\times n}$, and\n $\\E_{\\varphi_n}[\\bm\\omega_{nt}\\bm \\eta_{ns}^\\prime]=\\mathbf 0_{n\\times n}$,\n for all $n\\in\\mathbb N$ and $t,s\\in\\mathbb Z$.\n\\end{inparaenum}\n\\end{ass}\n\n\\begin{ass}\\label{ass:I1}\nFor any given $n\\in\\mathbb N$, there exists sets $\\mathcal I_1\\in\\{1,\\ldots,n\\}$, $\\mathcal I_a\\in\\{1,\\ldots,n\\}$,  and $\\mathcal I_b\\in\\{1,\\ldots,n\\}$, such that:\n\\begin{inparaenum}[(a)]\n\\item if $i\\in\\mathcal I_1$ then $\\rho_i=1$, while $\\rho_i=0$ otherwise, moreover $\\# \\mathcal I_1=n_1$ such that $n_1n^{-1}\\to 0$, as $n\\to\\infty$;\n\\item if $i\\in\\mathcal I_a$ then $\\sigma_i^{2\\omega}\\ge C_\\omega$ for some positive real $C_\\omega$, while $\\sigma_i^{2\\omega}=0$ otherwise, moreover $\\# \\mathcal I_a=n_a$ such that $n_an^{-1}\\to 0$, as $n\\to\\infty$;\n\\item if $i\\in\\mathcal I_b$ then $\\sigma_i^{2\\eta}\\ge C_\\eta$ for some positive real $C_\\eta$, while $\\sigma_i^{2\\eta}=0$ otherwise, moreover $\\# \\mathcal I_b=n_b$ such that $n_bn^{-1}\\to 0$, as $n\\to\\infty$.\n\\end{inparaenum}\n\\end{ass} \n\n\nBy Assumption \\ref{ass:dynamic}(a) we are considering the case in which factors are loaded dynamically with a finite number of lags. We do not consider here the case of autoregressive filters, which has been studied in \\citet{FHLZ17} in the stationary case. By Assumption \\ref{ass:dynamic}(b) we are assuming for simplicity that all $q$ factors are pervasive at lag-zero, while at higher lags they might or might not have a pervasive effect depending on the rank of $\\bm\\Sigma_k$. In other words, \\eqref{eq:NSDFM1} can be seen as a factor model with $q(s+1)$ factors of which $q$ are strong factors, i.e., having an effect on all series, and the remaining $qs$ are weak factors, i.e., having an effect only on a subset of series. \n\nBy Assumptions \\ref{ass:dynamic}(d) and \\ref{ass:dynamic}(e), when $d<q$ we allow the dynamics of the factors to be driven by $(q-d)<q$ unit roots implying the presence of $(q-d)$ common trends \\citep{stockwatson88JASA}. Clearly, in this setting, the factors are cointegrated with cointegration rank $d$, thus representing the permanent and transitory aspects of macroeconomic dynamics. When $d=0$---i.e., the dynamics of the factors are driven by $q$ unit roots---the VAR for the common factors in levels in \\eqref{eq:NSDFM2} does not exist; instead, it exists a VAR for $\\Delta \\bm f_t$, or the factors can be modeled as $q$ independent random walks as in \\citet{bai04}. That said, the case $d>0$ is the relevant one, as there is full agreement in the economic profession that while some fluctuations in the economy are permanent (common trends), some others are only temporary.\n\nAssumption \\ref{ass:modelNS} characterizes the innovations of the model. In particular, by part (c)  the idiosyncratic innovations $e_{it}$ are allowed to be mildly cross-correlated, thus implying that $\\Delta x_{it}$ follows an approximate factor model. Moreover, by part (e) we allow some series to be driven by a time-varying intercept and\/or a trend with time-varying slope, modeled as in a local level and local linear trend model, respectively \\citep[Section 2.3.6, page 45]{harvey90}. Notice that, if we set $\\sigma_{i\\eta}^2=0$, then the trend becomes deterministic with slope $\\beta_{i0}$, which is fixed to a constant by Assumption \\ref{ass:modelNS}(h), and similarly if we set $\\sigma_{i\\omega}^2=0$, we have a deterministic, hence constant, intercept term equal to $\\alpha_{i0}$. Finally, by parts (d) and (f) all innovations are independent. Notice that gaussianity is not strictly needed, but it is a reasonable assumption in macroeconomics.\n\nUnder these assumptions, it can be shown that the covariance matrix of the differenced common component $\\Delta\\bm\\chi_{nt}$ has at least $q$ and at most $q(s+1)$ eigenvalues that diverge linearly as $n\\to\\infty$. In particular, letting the covariance matrix of $\\Delta \\bm \\chi_n$ be $\\bm\\Gamma_n^{\\Delta\\chi}$, and denoting as $\\mu_{jn}^{\\Delta\\chi}$ the $j$-th largest eigenvalue of $\\bm\\Gamma_n^{\\Delta\\chi}$, Assumptions \\ref{ass:dynamic}(b) and \\ref{ass:dynamic}(c) imply that, for $j=1,\\ldots, q$,\n\\begin{equation}\\label{eq:divevaldiff}\n\\underline K_j\\le \\lim\\inf_{n\\to\\infty} n^{-1} \\mu_{jn}^{\\Delta\\chi}\\le\\lim\\sup_{n\\to\\infty} n^{-1} \\mu_{jn}^{\\Delta\\chi}\\le \\overline K_j, \n\\end{equation}\nfor some positive reals $\\underline K_j$ and $\\overline K_j$. Moreover,  letting the covariance matrix of $\\Delta \\bm \\xi_n$ be $\\bm\\Gamma_n^{\\Delta\\xi}$, and denoting as $\\mu_{jn}^{\\Delta\\xi}$ the $j$-th largest eigenvalue of $\\bm\\Gamma_n^{\\Delta\\xi}$, Assumption \\ref{ass:modelNS}(c), implies that\n\\begin{equation}\n\\sup_{n\\in\\mathbb N} \\mu_{1n}^{\\Delta\\xi}\\le M_\\xi, \\label{eq:divevaldiff2}\n\\end{equation}\nfor some positive real $M_\\xi$. From \\eqref{eq:divevaldiff} and \\eqref{eq:divevaldiff2}, and Assumption \\ref{ass:modelNS}(e), by Weyl's inequality,  the $q$ largest eigenvalues of the covariance matrix of $\\Delta \\mathbf x_{nt}$ diverge linearly in $n$, while all other $(n-q)$ eigenvalues stay bounded for all $n\\in\\mathbb N$. \n\nMoreover, it can be shown that the $q$ largest eigenvalues of the spectral density of $\\Delta \\mathbf x_{nt}$ diverge with $n$ at all frequencies, but at zero-frequency, where, due to the presence of common trends, only $(q - d)$ eigenvalues diverge, all the others eigenvalues being bounded for all $n$ and all frequencies. Hence, by looking at the eigenvalues of the spectral density matrix of $\\Delta \\mathbf x_{nt}$ we can determine $q$ and $d$ (see \\citealp{hallinliska07}, and \\citealp{BLL2}, respectively). Moreover, notice that when all factors are pervasive at all lags, i.e., in Assumption \\ref{ass:dynamic}(b) we let $\\mbox{rk}(\\bm\\Sigma_k)=q$ for all $k=0,\\ldots,s$,  then \\eqref{eq:divevaldiff} holds for all $j=1,\\ldots,q(s+1)$.  Therefore, by looking at the eigenvalues of the covariance matrix of $\\Delta \\mathbf x_{nt}$, we can also determine $s$ \\citep{dagostinogiannone12}.\n\nThe model defined in \\eqref{eq:NSDFM1}-\\eqref{eq:NSDFM4} has $q$ latent states, given by the common factors $\\bm f_t$, and additional latent states given by those idiosyncratic components that are autocorrelated as in \\eqref{eq:NSDFM3}, and by the time-varying intercepts and trend slopes as in \\eqref{eq:NSDFM5} and \\eqref{eq:NSDFM4}. These additional latent states are such that they satisfy the following assumption. \n\nIn other words, we are assuming that some, but not all, idiosyncratic components are $I(1)$, and that some, but not all, series have a time-varying intercept and\/or a linear trend with time-varying slope. For simplicity, we are also assuming that stationary idiosyncratic components are serially uncorrelated. \n\nWe then  make the following identifying assumptions.\n\n\\begin{ass} \\label{ass:identNS}\nLet $\\mathbf M_n^{\\Delta \\chi}$ be the $q\\times q$ diagonal matrix with elements $\\mu_{1n}^{\\Delta \\chi},\\ldots,\\mu_{qn}^{\\Delta \\chi}$, and let \n$\\mathbf V_n^{\\Delta\\chi}$ be the $n\\times q$ matrix having as columns the corresponding normalized eigenvectors. Then: \\begin{inparaenum}[(a)]\n \\item $\\Delta \\bm f_t=(\\mathbf M_n^{\\Delta\\chi})^{-1\/2}\\mathbf V_n^{\\Delta\\chi\\prime} \\Delta\\bm\\chi_{nt}$;\n \\item the entries of $\\mathbf M_n^{\\Delta\\chi}$ are such that they satisfy \\eqref{eq:divevaldiff} and $\\overline K_{j+1}<\\underline K_j$ for $j=1,\\ldots, q-1$;\n \\item the entries of $\\mathbf V_n^{\\Delta\\chi}$ are such that $[\\mathbf V_n^{\\Delta\\chi}]_{1j}>0$ for all $j=1,\\ldots, q$.\n\n\\end{inparaenum}\n\\end{ass}\n \nParts (a) and (b) are standard in factor model literature for stationary processes and allow to identify the differenced factors up to a multiplication by a sign \\citep[see, e.g.,][]{FGLR09,FLM13}. We identify the first difference of the factors with the $q$ normalized principal components of $\\Delta\\bm\\chi_{nt}$ and this implies in Assumption \\ref{ass:dynamic}(b) that $\\bm\\Gamma^{\\Delta f}=\\mathbf I_q$. It can then be seen that the following must hold for the loadings\n\\begin{align}\n\\mathbf V_n^{\\Delta\\chi\\prime} \\bm{\\mathcal B}_{0n}= (\\mathbf M_n^{\\Delta\\chi})^{1\/2},\n\\end{align}\ntherefore we can choose $\\bm{\\mathcal B}_{0n}=\\mathbf V_n^{\\Delta\\chi}(\\mathbf M_n^{\\Delta\\chi})^{1\/2}$, and in Assumption \\ref{ass:dynamic}(a) we have that $\\bm\\Sigma_0$ is diagonal with entries given by $\\lim_{n\\to\\infty} (n^{-1}\\mu_{jn}^{\\Delta \\chi})$, which as requested are finite and positive because of \\eqref{eq:divevaldiff}. Part (c) is a way to fix the sign indeterminacy in the identification of the factors. Once $\\Delta\\bm f_t$ and $\\bm{\\mathcal B}_{0n}$ are identified, then the remaining loadings are obtained by projecting $\\Delta\\mathbf x_{nt}$ onto the lagged factors.\n\n\nThe identifying restrictions in Assumption \\ref{ass:identNS} are particularly useful for initializing the EM algorithm with the PC estimator (see the next section). However, it has to be stressed that this identification does not provide any economic meaning to the factors. In other words we are not interested here in giving any interpretation of the factors, but we are just interested in the common component, which is always identified.\n\n\n\\section{Estimation and asymptotic properties}\\label{sec:estNS}\nThroughout the rest of the section we assume to observe the $nT$-dimensional vector $\\bm X_{nT}=(\\mathbf x_{n1}^\\prime\\cdots\\mathbf x_{nT}^\\prime)^\\prime$ satisfying \\eqref{eq:NSDFM1}-\\eqref{eq:NSDFM4}. In order to derive an estimator of the common component, we need to estimate the factors vector $\\bm f_T=(\\bm f_{1}^\\prime\\cdots \\bm f_T^\\prime)^\\prime$ and the vector containing the true values of all parameters is \n \\[\n \\bm\\varphi_n=\\l(\\text{vec}(\\bm{\\mathcal B}_{0n}\\cdots \\bm{\\mathcal B}_{sn})^\\prime, \\text{vech}(\\bm\\Gamma_n^e)^\\prime, \\rho_1,\\ldots, \\rho_{n_1} ,\\text{vec}(\\bm{\\mathcal A}_1\\cdots \\bm{\\mathcal A}_p)^\\prime, \\text{vech}(\\bm\\Gamma^u)^\\prime,\\sigma_{1\\omega}^{2}\\cdots\\sigma^{2}_{n_a\\omega},\\sigma_{1\\eta}^{2}\\cdots\\sigma^{2}_{n_b\\eta}\\r)^\\prime, \n \\]\n where, without loss of generality, we assumed that $\\mathcal I_1=\\{1,\\ldots, n_1\\}$,  $\\mathcal I_a=\\{1,\\ldots, n_a\\}$, and  $\\mathcal I_b=\\{1,\\ldots, n_b\\}$.\n\nIn this Section, we provide asymptotic results when $n_1=0$, $n_a=0$, and $n_b=0$, thus assuming that all idiosyncratic component are stationary and that no time-varying term is present. At first sight this might seem as a strong requirement, but notice that in our framework introducing non-stationary idiosyncratic components and\/or local levels and\/or local linear trends implies just adding latent states. We discuss this extension in Section \\ref{sec:idioI1}. Moreover, in \\ref{app:prestNS}, we give all details of the EM algorithm together with explicit expressions for all estimators in the general case. \n\nWithout loss of generality, we fix $s=1,$ and we fix the VAR order in \\eqref{eq:NSDFM2} to $p=2$, thus $\\bm{\\mathcal A}(L)\\equiv(\\bm{\\mathcal A}_1 L+\\bm{\\mathcal A}_2L^2)$, and, in this way the stationary component of $\\bm f_t$ follows a non-trivial dynamics. For simplicity, we also assume that $\\alpha_{i0}=0$ and $\\beta_{i0}=0$. \n\nThe EM algorithm is an iterative procedure, which starts with an initial value of the parameters $\\widehat{\\bm\\varphi}_n^{(0)}$, and at each iteration $k\\ge 0$ produces estimates of the factors $\\bm f_{t|T}^{(k)}$ (KS and E-step) and of the parameters $\\widehat{\\bm\\varphi}_n^{(k+1)}$ (M-step).  When the EM algorithm converges, say at iteration $k^*$, it gives the estimated common component $\\widehat{\\chi}_{it}=\\widehat{\\bm b}_{i0}^{(k^*+1)\\prime}\\bm f_{t|T}^{(k^*+1)}+\\widehat{\\bm b}_{i1}^{(k^*+1)\\prime}\\bm f_{t-1|T}^{(k^*+1)}$. \n\nMore in detail, the NS-DFM in \\eqref{eq:NSDFM1}-\\eqref{eq:NSDFM2} can be written as\n\\begin{align}\n\\mathbf x_{nt}&=\\l(\\bm{\\mathcal B}_{0n}\\ \\bm{\\mathcal B}_{1n}\\r)\n\\l(\\begin{array}{c}\n\\bm f_t\\\\\n\\bm f_{t-1}\n\\end{array}\n\\r)+\\mathbf e_{nt},\\label{eq:SSNS001}\\\\\n\\l(\\begin{array}{c}\n\\bm f_t\\\\\n\\bm f_{t-1}\n\\end{array}\n\\r)&=\\l(\\begin{array}{cc}\n\\bm{\\mathcal A}_1&\\bm{\\mathcal A}_2\\\\\n\\mathbf I_q&\\mathbf 0_{q\\times q}\n\\end{array}\n\\r)\n\\l(\\begin{array}{c}\n\\bm f_{t-1}\\\\\n\\bm f_{t-2}\n\\end{array}\n\\r)\n+\\l(\\begin{array}{c}\n\\bm u_t\\\\\n\\mathbf 0_q\n\\end{array}\n\\r),\\label{eq:SSNS002}\n\\end{align} \nBy defining  $\\mathbf F_t=(\\bm f_t^\\prime\\; \\bm f_{t-1}^\\prime)^\\prime$ and $\\bm\\lambda_i=(\\bm{b}_{0i}^\\prime\\;\\bm{b}_{1i}^\\prime)^\\prime$, we see that, for given values of the parameters $\\widehat{\\bm\\varphi}_n^{(k)}$, we can easily estimate the factors via the KS applied to the state-space form in \\eqref{eq:SSNS001}-\\eqref{eq:SSNS002}. The estimated states are then $\\mathbf F_{t|T}^{(k)}=\\E_{\\widehat{\\varphi}_n^{(k)}}[\\mathbf F_t|\\bm X_{nT}]$, the first $q$-components of which give $\\bm f_{t|T}^{(k)}=\\E_{\\widehat{\\varphi}_n^{(k)}}[\\bm f_t|\\bm X_{nT}]$. Then, using the output of the KS, we can compute the expected log-likelihood, which is maximized by the loadings estimator $\\widehat{\\bm \\lambda}_{i}^{(k+1)}\\equiv(\\widehat{\\bm b}_{i0}^{(k+1)\\prime} \\; \\widehat{\\bm b}_{i1}^{(k+1)\\prime})^\\prime$, such that\n\\begin{align}\n\\widehat{\\bm \\lambda}_{i}^{(k+1)}\\!=\n\\l\\{\\sum_{t=1}^T\\E_{\\widehat{\\varphi}_n^{(k)}}\\!\\!\\l[\\l(\\begin{array}{ll}\n\\bm f_t\\bm f_t^\\prime&\\bm f_t\\bm f_{t-1}^\\prime\\\\\n\\bm f_{t-1}\\bm f_t^\\prime&\\bm f_{t-1}\\bm f_{t-1}^\\prime\n\\end{array}\\r)\\!\\!\\bigg\\vert \\bm X_{nT}\n\\r]\\r\\}^{\\!\\!-1}\\!\\!\\!\n\\l\\{\\sum_{t=1}^T\\E_{\\widehat{\\varphi}_n^{(k)}}\\!\\!\\l[\\l(\\begin{array}{l}\n\\bm f_tx_{it}\\\\\n\\bm f_{t-1}x_{it}\n\\end{array}\\r)\\!\\!\\bigg\\vert \\bm X_{nT}\n\\r]\\r\\}.\\label{eq:param1NS}\n\\end{align}\n\nThe initial value of the parameters $\\widehat{\\bm\\varphi}_n^{(0)}$ is determined as follows. For the loadings and the factors we use the approach proposed in \\citet{BLL2}, which makes use of the $q$ leading PCs of the model in first differences. Two comments are worth making. \nFirst, it important to stress that initializing the model in first differences (including when determining $q$ and $s$) is crucial, since it allows us to use PCs without incurring in spurious effects due to the presence of idiosyncratic unit roots (\\citealp{OW19}), or  linear trends (\\citealp{ngCG}). Second, in light of the previous comment, this approach provides consistent estimates of the loadings, even in the case in which Assumption \\ref{ass:I1} is satisfied with $n_1>0$ and $n_b>0$, but for constant intercepts and trend slopes \\citep[see also][in the case of no linear trends]{baing04}. In particular, our initialization delivers estimates of $\\alpha_{i0}$ and $\\beta_{i0}$, which, together with a given small initial value of the variances $\\widehat{\\sigma}_{i\\omega}^{2(0)}$ and $\\widehat{\\sigma}_{i\\eta}^{2(0)}$,  can be used to update the slope state in \\eqref{eq:NSDFM4}. Notice that the pre-estimators of those initial conditions do not need to be consistent for our results to hold.\nThe initialization is completed by estimating the parameters of  \\eqref{eq:NSDFM2} from an unrestricted VAR fitted on the estimated factors. This is a valid procedure when estimating an autoregressive model for cointegrated data (see \\citealp{simsstockwatson}). Consistency of the pre-estimators of the loadings and VAR coefficients is proved in \\citet[Lemma 3 and Proposition 2]{BLL2} (see also \\ref{app:consN}). \n\nFinally, notice also that we initialize the KF by setting the initial value of the covariance of the factors, ${\\mathbf P}_{0|0}$, to a very large value, as suggested by \\citet[Section 3.3.4, page 121]{harvey90}. \n\nConsistency of the estimated common component follows.\n\\newpage\n\\begin{prop}\\label{th:chiNS}\nUnder Assumptions \\ref{ass:dynamic}, \\ref{ass:modelNS}, \\ref{ass:identNS}, and if $\\mbox{rk}(\\bm\\Sigma_k)=q$ for all $k=0,\\ldots,s$, and $n_1=0$, $n_a=0$, and $n_b=0$,  as $n,T\\to\\infty$, for any given $i=1,\\ldots, n$ and $k=0,1$, $\\min(\\sqrt n,\\sqrt T)\\Vert \\widehat{\\bm b}_{ki}-\\bm b_{ki}\\Vert = O_p(1)$, \nand, for any given $t=\\bar t,\\ldots, T$,\n$\\min(\\sqrt n,\\sqrt T)\\Vert \\widehat{\\bm f}_{t|T}-\\bm f_{t}\\Vert= O_p(1)$. Moreover, $\\min(\\sqrt n,\\sqrt {T})\\,\\Vert \\widehat{\\chi}_{it}-{\\chi}_{it} \\Vert = O_p(1)$,\nfor any given $i=1,\\ldots,n$ and $t=\\bar t,\\ldots, T$, with $\\bar t\\ge 2$.\n\\end{prop}\n\n\n\nThe convergence rate depends on different ingredients. First, we show that the KS reaches a steady state within $\\bar t$ periods, where $\\bar t$ depends on the initial value $\\mathbf P_{0|0}$ and, as shown in Section \\ref{sec:mc2}, $\\bar t$ is typically very small.  Then, for the KS we show that,  given the true parameters, the factors are $\\sqrt N$-consistent. Third, given the true factors, the loadings estimator are consistent, with convergence rate $T$ for the loadings of the $I(1)$ components of the factors and convergence rate $\\sqrt T$ for the loadings of the stationary \ncomponent of the factors. As a result, for any given $i$, the whole loadings vector is $\\sqrt T$-consistent, unless $d=q$, in which case each all $q$ factors are random walks and then the loadings vector would be $T$-consistent. \n\nUnder the assumption $n_1=0$, $n_a=0$, and $n_b=0$, our model is equivalent to the model studied in \\citet{bai04}, who considers estimation by means of PCs in levels. In this respect, we notice that the rates in Proposition \\ref{th:chiNS} are very similar to those in  \\citet[Theorem 6 for the case $d<q$ and Theorem 4 for the case $d=q$]{bai04}. In other words, the QML estimator converges at the same rate than the estimator based on PC on the levels, which resembles the result in \\citet{BLqml} for the stationary case. However, in the simulation study in Section \\ref{sec:mc2} show that our estimator behave much better in finite samples.\n \n \n\n\n\n\n\\section{$I(1)$ idiosyncratic components, local levels, local linear trends}\\label{sec:idioI1}\n\nIf some idiosyncratic components are non-stationary, we can no longer use the EM algorithm described in the previous section. Indeed, when the residuals of \\eqref{eq:NSDFM1} are non-stationary, the M-step estimator of the loadings cannot be obtained by regressing $x_{it}$ is $I(1)$ onto  $\\bm f_t$ and $\\bm f_{t-1}$. However, the case in which some idiosyncratic components are $I(1)$ is the relevant one for large macroeconomic datasets, since otherwise all data would be cointegrated. This is, for example, shown by the empirical results in  \\citet{OGAP}, where the methodology proposed by \\citet{baing04} for testing for idiosyncratic unit roots is applied on a standard US macroeconomic dataset.\n\nIn this Section, we adapt the EM algorithm to model non-stationary idiosyncratic components as well as local levels and local linear trends. In particular, we borrow from the literature on nowcasting with stationary factor models which models autocorrelated idiosyncratic components by treating them as additional latent states (see, e.g., \\citealp{banburamodugno14}; and \\citealp{BGMR13}). \n\nLet us define $m=(n_1+n_a+n_b)$, as the number of additional latent states and recall that by Assumption \\ref{ass:I1}, $n^{-1}m\\to 0$, as $n\\to\\infty$. Define also the set $\\mathcal I_{m}=\\mathcal I_1\\cup\\mathcal I_a\\cup\\mathcal I_b$, and notice that $\\#\\mathcal I_m\\le m$, since it is possible that a variable has both a non-stationary idiosyncratic component as well as, for example, a linear trend. Then for all $i\\in\\mathcal I_{m}$,  we replace the measurement equation \\eqref{eq:NSDFM1} with\n\\begin{align}\nx_{it}&=\\alpha_{it}+\\beta_{it}t + \\bm b_i^\\prime(L) \\bm f_t + \\xi_{it} + \\nu_{it},\\label{eq:NSDFM1bis}\n\\end{align}\nsuch that Assumption \\ref{ass:dynamic} still hold, and, moreover, letting $\\bm\\nu_{mt}=(\\nu_{1t}\\cdots\\nu_{mt})^\\prime$, for all $t\\in\\mathbb Z$, we assume $\\bm\\nu_{mt}\\sim\\mathcal N(\\mathbf 0_m,\\phi\\mathbf I_m)$, with $\\phi>0$, and $\\E_{\\varphi_n}[\\bm\\nu_{mt}\\bm\\nu_{mt-k}] = 0_{m\\times m}$ for all $k\\ne 0$. If $i\\notin\\mathcal I_m$, then \\eqref{eq:NSDFM1} stays the same. Moreover, we leave the dynamics of the factors in \\eqref{eq:NSDFM2} unchanged, while we change \\eqref{eq:NSDFM3} to\n\\begin{align}\n\\xi_{it}&=\\xi_{it-1} + e_{it},\\;\\text{ if }\\; i\\in \\mathcal I_1,\\;\\text{ and }\\; \\xi_{it}= e_{it},\\;\\text{ if }\\; i\\notin \\mathcal I_1,\\label{eq:NSDFM3bis}\n\\end{align}\nwhere Assumptions \\ref{ass:modelNS}(b) and  \\ref{ass:modelNS}(c) still hold, and $\\E_{\\varphi_n}[\\nu_{it} e_{js}]=0$, for all $t,s\\in\\mathbb Z$, all $i\\in\\mathcal I_m$ and all $j=1,\\ldots, n$.\nFinally, according to \\eqref{eq:NSDFM5} and \\eqref{eq:NSDFM4}, we have the state equations\n\\begin{align}\n&\\alpha_{it}=\\alpha_{it-1} + \\omega_{it},\\;\\text{ if }\\; i\\in \\mathcal I_a,\\label{eq:NSDFM5bis}\\\\\n&\\beta_{it}=\\beta_{it-1} + \\eta_{it},\\;\\text{ if }\\; i\\in \\mathcal I_b,\\label{eq:NSDFM4bis}\n\\end{align}\nsuch that $\\E_{\\varphi_n}[\\nu_{it} \\omega_{js}]=0$, and $\\E_{\\varphi_n}[\\nu_{it} \\eta_{js}]=0$, for all $t,s\\in\\mathbb Z$, all $i\\in\\mathcal I_m$, and  all $j\\in\\mathcal I_a$ or $j\\in\\mathcal I_b$.  \n\nThe model, which has as measurement equation either \\eqref{eq:NSDFM1} or \\eqref{eq:NSDFM1bis} if $i\\in\\mathcal I_m$, and which has as state equations  \\eqref{eq:NSDFM2}, and, if needed, also equations \\eqref{eq:NSDFM3bis}, \\eqref{eq:NSDFM5bis} and \\eqref{eq:NSDFM4bis}, has a compact state space form which is given in  \\ref{app:prestNS}, together with the details on its estimation via the EM algorithm. In particular, letting $w_{it}=\\alpha_{it}+\\beta_{it}t+\\xi_{it}$, for all $i\\in\\mathcal I_m$ we show that, at a given iteration $k\\ge 0$ of the EM algorithm, the M-step gives the loadings estimators:\n\\begin{align}\n\\widehat{\\bm \\lambda}_{i}^{(k+1)}\\!=\n\\l\\{\\sum_{t=1}^T\\E_{\\widehat{\\varphi}_n^{(k)}}\\!\\!\\l[\\l(\\begin{array}{ll}\n\\bm f_t\\bm f_t^\\prime&\\bm f_t\\bm f_{t-1}^\\prime\\\\\n\\bm f_{t-1}\\bm f_t^\\prime&\\bm f_{t-1}\\bm f_{t-1}^\\prime\n\\end{array}\\r)\\!\\!\\bigg\\vert \\bm X_{nT}\n\\r]\\r\\}^{\\!\\!-1}\\!\\!\\!\n\\l\\{\\sum_{t=1}^T\\E_{\\widehat{\\varphi}_n^{(k)}}\\!\\!\\l[\\l(\\begin{array}{l}\n\\bm f_t(x_{it}-w_{it})\\\\\n\\bm f_{t-1}(x_{it}-w_{it})\n\\end{array}\\r)\\!\\!\\bigg\\vert \\bm X_{nT},\n\\r]\\r\\}.\\nonumber\n\\end{align}\nwhere $\\bm\\lambda_i=(\\bm{b}_{0i}^\\prime\\;\\bm{b}_{1i}^\\prime)^\\prime$, while for $i\\notin\\mathcal I_m$ the loadings estimator is the same as in \\eqref{eq:param1NS}. Formulas for all other estimators are given in \\ref{app:prestNS}. In order to be able to compute $\\widehat{\\bm \\lambda}_{i}^{(k+1)}$, we have to estimate the $m$ additional latent states $w_{it}$ and therefore we also need modify the KS accordingly (see \\ref{app:prestNS} for details). \n\nIn \\ref{app:misidioNS}, we provide an overview of the challenges involved by this task and we provide an informal derivation of the conditions necessary for consistent estimation, together with the related convergence rates.  Three main results emerge. First, the new latent states can be recovered only if they display also some degree of cross-sectional correlation, as if they were driven by some common factor which is weakly pervasive for the whole panel. The intuition is that, if the additional latent states are completely uncorrelated across the components of $\\mathbf x_{nt}$, then pooling many series does not help in recovering them, since their effect is always dominated by the factors. \n\nSecond, when the previous condition is verified, then we can still achieve $\\sqrt n$-consistency for the estimated factors (as in the proof of Proposition \\ref{th:chiNS}), regardless of $m$, but provided that the variance of the measurement error $\\nu_{it}$ in \\eqref{eq:NSDFM1bis} is fixed in such a way that $\\phi=o(n^{-1})$, that is, it is asymptotically negligible. Indeed, the presence of $\\nu_{it}$ represents a mis-specification of the original model in \\eqref{eq:NSDFM1}, which needs to be introduced only as a numerical device, since the KF is not be defined if $\\phi=0$. The smaller is $\\phi$, the smaller the effect of the mis-specification is, and, therefore, the estimation of the factors is unaffected by the additional states. \n\nAs a consequence of this result, our estimator converges at a faster rate than those proposed by \\citet{baing04} and \\citet{BLL2}, which are based on PC analysis on the differenced data. This faster convergence rate comes from the fact that we distinguish \\textit{a priori} between $I(1)$ and stationary idiosyncratic components. By contrast, due to differencing the estimator of \\citet{baing04} and \\citet{BLL2} essentially treat all idiosyncratic components as if they were $I(1)$. Of course, for the implementation of our estimator, it is crucial to be able to determine consistently which idiosyncratic component is $I(1)$---for example, using the test for idiosyncratic unit roots proposed by \\citet{baing04}.\n\nThird, to achieve consistency of the additional latent states a necessary condition is $mn^{-1}\\to 0$. This reflects the obvious intuition that the more latent states we need to estimate, the worse the performance of our estimator is going to be. Moreover, $\\sqrt n$-consistency for the new states can be obtained for any $m$, but only if we choose an even smaller value of $\\phi$, namely $\\phi=o((m\\sqrt n)^{-1})$. \n \nWe conclude with three remarks. First, the requirement that the new latent states display some degree of cross-sectional correlation is perfectly in line with Assumption \\ref{ass:modelNS}(c) according to which the idiosyncratic components can be cross-correlated. Moreover, we can relax Assumptions \\ref{ass:modelNS}(e) and \\ref{ass:modelNS}(g) to allow for some correlation across the innovations $e_{it}$, $\\omega_{it}$, and $\\eta_{it}$ in \\eqref{eq:NSDFM3bis}, \\eqref{eq:NSDFM5bis} and \\eqref{eq:NSDFM4bis}. Indeed, it is reasonable to assume that local linear trends are shared by real variables (e.g., GDP and GDI), or that local levels are more apt to capture time-varying mean of groups of variables belonging, for example, to the labor market. Nevertheless, as shown in the proof of Proposition \\ref{th:chiNS}, the fact that we estimate the $I(1)$ idiosyncratic components without modeling the cross-correlation between their innovations, will add miss-specification to our model, but will not affect the consistency of our estimates. \n\nSecond, as a far as estimation of the parameters given estimates of the states is concerned, we conjecture that nothing changes with respect to the results used in the proof of Proposition \\ref{th:chiNS}, provided the states estimators are $\\sqrt n$-consistent. Third, since the above are just  asymptotic arguments, the choice of $\\phi$ is not straightforward. A common way to proceed consists in initializing $\\phi$ to be very small for all $m$ additional states and then update its estimate at each iteration of the EM algorithm, thus adding $m$ additional parameters. This is the way we implement the EM algorithm in the next section (see also \\ref{app:prestNS}).\n\n\n\n\n\n\n\n\n\\section{MonteCarlo results}\\label{sec:mc2}\nThroughout, we let $n\\in\\{75,100,200,300\\}$, $T\\in\\{75,100,200,300\\}$, $q\\in\\{2,4\\}$, and $s\\in\\{0,1\\}$, and we simulate data according to \\eqref{eq:NSDFM1}, \\eqref{eq:NSDFM2}, \\eqref{eq:NSDFM3}, and \\eqref{eq:NSDFM4} as follows.\n\nFirst, the factor loadings are such that $[\\bm{\\mathcal B}_{kn}]_{ij} \\sim N(1,1)$ for $k=0,\\ldots, s$, and then if $s=1$, for all $j=1,\\ldots q$, we take $n\/2$ randomly selected elements of $[\\bm{\\mathcal B}_{1n}]_{\\cdot j}$ and we set them to zero. Second, for the common factors we set the VAR order $p=2$, and to generate $\\bm{\\mathcal A}(L)$ we use the Smith-McMillan factorization according to which $\\bm{\\mathcal A}(L)=\\mathbfcal{U}(L) \\mathbfcal{M}(L) \\mathbfcal{V}(L)$, where $\\mathbfcal{M}(L)= \\mbox{diag} \\left( (1-L)\\mathbf I_{q-d}, \\mathbf I_d\\right)$, $\\mathbfcal{V}(L)=\\mathbf I_q$, and $\\mathbfcal{U}(L)=(\\mathbf I_q-\\mathbfcal{U}_1 L)$, where $\\mathbfcal{U}_1=\\mu\\,\\widetilde{\\mathbfcal{U}}_1(\\nu^{(1)}(\\widetilde{\\mathbfcal{U}}_1))^{-1}$, where the diagonal elements of $\\widetilde{\\mathbfcal{U}}_1$ are drawn from a uniform distribution on $[0.5,0.8]$, while the off-diagonal elements from a uniform distribution on $[0,0.3]$, and $\\mu=0.5$. In this way, $\\bm f_t$ follows a VAR(2) with $q-d$ unit roots, or, equivalently, a VECM(1), where the number cointegration relations is set to $d=1$. The common innovations are such that $\\bm u_t\\stackrel{iid}{\\sim} \\mathcal{N}(\\mathbf 0_q,\\mathbf I_q)$, or $\\bm u_t\\stackrel{iid}{\\sim} t_4(\\mathbf 0_q,\\mathbf I_q)$.\n\nThird, each idiosyncratic component follows an AR(2) with roots $\\rho_{i1}$ and $\\rho_{i2}$, such that $\\rho_{i1}=1$ if $\\xi_{it}\\sim I(1)$, while $\\rho_{i1}=0$ otherwise, and $\\rho_{i2}$ is drawn from a uniform distribution on $[0.2,0.6]$. We randomly select $n_1$ idiosyncratic components to have a unit root, with $n_1\\in\\{0,25,50,75,100\\}$, provided $n_1<n$. The innovations are such that $\\mathbf e_t\\stackrel{iid}{\\sim} \\mathcal{N}(\\mathbf 0_n,\\bm\\Gamma^e_n)$, or $\\mathbf e_t\\stackrel{iid}{\\sim} t_4(\\mathbf 0_n,\\bm\\Gamma^e_n)$\n with $[\\bm\\Gamma^e_n]_{ij}=\\tau^{|i-j|}$ if $\\tau>0$, while, if $\\tau=0$, $\\bm\\Gamma^e_n$ is diagonal with entries drawn from a uniform distribution on $[0.5,1.5]$. We set $\\tau\\in\\{0,0.5\\}$.\n\nFourth, we randomly select $n_b$ variables to have a non-zero linear trend, with $n_b\\in\\{0,25,50,75,100\\}$, provided $n_b<n$. For those variables we draw $\\beta_{i0}$ from a uniform distribution on ${[0.3,0.5]}$, but we set $\\sigma_i^2=0$, thus considering only linear trends with constant slopes.\n\n\nLast, we rescale the first differences of each common and idiosyncratic component in such a way that the share of variance of the $i$-th variable explained by the common component is $\\theta(1+\\theta)^{-1}$. We set $\\theta=0.5$.\n\n\nWe consider $B=1000$ replications and we run the EM algorithm to estimate the NS-DFM by running the KS with $(q+n_1)$ latent states and estimating in the M-step only the diagonal terms of the idiosyncratic covariance matrix even when $\\tau>0$. Similarly we do not add idiosyncratic states even when $\\delta>0$. In other words, we always estimate a mis-specified model and, in this way, we are able to assess how robust our-estimators are with respect to mis-specifications. \n\nIn Table \\ref{tab:PttNS}, we report for different values of $n$ and for $t=1,\\ldots, 10$, the trace of the one-step-ahead, KF, and KS MSEs when $q=2$, $s=1$, $T=100$, $\\tau=0.5$, and $\\delta=0.2$ (serially and cross-correlated idiosyncratic components). The MSEs are computed using the true simulated value of the parameters in order to verify numerically convergence to the steady-state. First, as $n$ grows, the one-step-ahead MSE reaches a steady state within maximum five time periods and $\\mbox{tr}(\\mathbf P_{t|t-1})\/q\\simeq 1$. This is consistent with the fact that due to the presence of unit roots we inizialize the filter with a vary large value of $\\mathbf P_{0|0}$. Second, the KF and KS MSEs are very similar and both decrease to zero as $n$ grows and $\\mbox{tr}(\\mathbf P_{t|t})n\/q$ and $\\mbox{tr}(\\mathbf P_{t|T})n\/q$, computed when $t=10$, stabilize as $n$ grows thus showing that the rate of decrease is $n$.\n\n\\begin{table}[t!]\n\\setlength{\\tabcolsep}{0\\textwidth}\n\\caption{Simulation results NS-DFM}\\label{tab:PttNS} \\centering \\smallskip\n\n\\textsc{Kalman filter and Kalman smoother MSEs} \\smallskip\n\n\\scriptsize\n\\vskip .2cm\n\\begin{tabular}{L{.04\\textwidth} L{.08\\textwidth} | C{.11\\textwidth}C{.11\\textwidth}C{.11\\textwidth}C{.1\\textwidth}C{.11\\textwidth}C{.11\\textwidth}C{.11\\textwidth}C{.11\\textwidth}}\n\\multicolumn{10}{c}{Serially and cross-correlated idiosyncratic ($\\tau=0.5$, $\\delta=0.2$),  $n_1=0$, $n_b=0$, $q=2$, $s=1$}\\\\\n\\multicolumn{10}{c}{Gaussian innovations}\\\\[-4pt]\n\\\\\n\\hline\n\\hline\n&&\\\\[-4pt]\n \t&$n$&\t$5$\t&\t$10$\t&\t$25$\t&\t$50$\t&\t$75$\t&\t$100$\t&\t$200$\t&\t$300$\t\\\\[4pt]\n\\hline\n&&\\\\[-4pt]\n\\multicolumn{2}{l}{$\\mbox{tr}(\\mathbf P_{0|0})\/q$}\t\\vline \n\t&\t147.6161\t&\t154.1094\t&\t133.2495\t&\t152.7333\t&\t138.4666\t&\t109.5626\t&\t120.8897\t&\t134.7097\t\\\\[4pt]\n\\hline\n&&\\\\[-4pt]\n&$t=1$\t&\t2.874610\t&\t1.601115\t&\t1.132925\t&\t1.076572\t&\t0.981511\t&\t1.020192\t&\t0.947337\t&\t0.980840\t\\\\\n&$t=2$\t&\t2.768656\t&\t1.543904\t&\t1.110419\t&\t1.064744\t&\t0.971928\t&\t1.014110\t&\t0.945185\t&\t0.979096\t\\\\\n\\multirow{4}{*}{\\rotatebox{90}{$\\mbox{tr}(\\mathbf P_{t|t-1})\/q$}}\n&\t$t=3$\t&\t2.748894\t&\t1.537389\t&\t1.109305\t&\t1.064269\t&\t0.970874\t&\t1.013479\t&\t0.945034\t&\t0.978928\t\\\\\n&\t$t=4$\t&\t2.746597\t&\t1.536414\t&\t1.109239\t&\t1.064245\t&\t0.970735\t&\t1.013398\t&\t0.945022\t&\t0.978908\t\\\\\n&\t$t=5$\t&\t2.746357\t&\t1.536248\t&\t1.109235\t&\t1.064244\t&\t0.970716\t&\t1.013387\t&\t0.945021\t&\t0.978906\t\\\\\n&\t$t=6$\t&\t2.746329\t&\t1.536220\t&\t1.109235\t&\t1.064244\t&\t0.970714\t&\t1.013386\t&\t0.945020\t&\t0.978906\t\\\\\n&\t$t=7$\t&\t2.746325\t&\t1.536215\t&\t1.109235\t&\t1.064244\t&\t0.970713\t&\t1.013385\t&\t0.945020\t&\t0.978906\t\\\\\n&\t$t=8$\t&\t2.746324\t&\t1.536214\t&\t1.109235\t&\t1.064244\t&\t0.970713\t&\t1.013385\t&\t0.945020\t&\t0.978906\t\\\\\n&\t$t=9$\t&\t2.746324\t&\t1.536214\t&\t1.109235\t&\t1.064244\t&\t0.970713\t&\t1.013385\t&\t0.945020\t&\t0.978906\t\\\\\n&\t$t=10$\t&\t2.746324\t&\t1.536214\t&\t1.109235\t&\t1.064244\t&\t0.970713\t&\t1.013385\t&\t0.945020\t&\t0.978906\t\\\\\n[4pt]\n\\hline\n&&\\\\[-4pt]\n&\t$t=1$\t&\t1.665780\t&\t0.548360\t&\t0.161240\t&\t0.107121\t&\t0.049408\t&\t0.036570\t&\t0.015982\t&\t0.011897\t\\\\\n&\t$t=2$\t&\t1.386910\t&\t0.400762\t&\t0.116701\t&\t0.079362\t&\t0.037964\t&\t0.030533\t&\t0.011515\t&\t0.007687\t\\\\\n\\multirow{4}{*}{\\rotatebox{90}{$\\mbox{tr}(\\mathbf P_{t|t})\/q$}}\n&\t$t=3$\t&\t1.369813\t&\t0.393790\t&\t0.114352\t&\t0.078298\t&\t0.037113\t&\t0.030087\t&\t0.011295\t&\t0.007466\t\\\\\n&\t$t=4$\t&\t1.366557\t&\t0.392627\t&\t0.114174\t&\t0.078221\t&\t0.037001\t&\t0.030029\t&\t0.011276\t&\t0.007446\t\\\\\n&\t$t=5$\t&\t1.365967\t&\t0.392442\t&\t0.114163\t&\t0.078216\t&\t0.036985\t&\t0.030021\t&\t0.011274\t&\t0.007443\t\\\\\n&\t$t=6$\t&\t1.365898\t&\t0.392411\t&\t0.114163\t&\t0.078216\t&\t0.036983\t&\t0.030020\t&\t0.011274\t&\t0.007443\t\\\\\n&\t$t=7$\t&\t1.365890\t&\t0.392406\t&\t0.114163\t&\t0.078216\t&\t0.036983\t&\t0.030020\t&\t0.011274\t&\t0.007443\t\\\\\n&\t$t=8$\t&\t1.365889\t&\t0.392405\t&\t0.114163\t&\t0.078216\t&\t0.036983\t&\t0.030020\t&\t0.011274\t&\t0.007443\t\\\\\n&\t$t=9$\t&\t1.365889\t&\t0.392404\t&\t0.114163\t&\t0.078216\t&\t0.036983\t&\t0.030020\t&\t0.011274\t&\t0.007443\t\\\\\n&\t$t=10$\t&\t1.365889\t&\t0.392404\t&\t0.114163\t&\t0.078216\t&\t0.036983\t&\t0.030020\t&\t0.011274\t&\t0.007443\t\\\\\n[4pt]\n\\hline\n&&\\\\[-4pt]\n\\multicolumn{2}{l}{$\\mbox{tr}(\\mathbf P_{10|10})n\/q$}\t\\vline &\t3.414722\t&\t1.962022\t&\t1.427032\t&\t1.955403\t&\t1.386863\t&\t1.501011\t&\t1.127376\t&\t1.116435\t\\\\[4pt]\n\\hline\n&&\\\\[-4pt]\n&\t$t=1$\t&\t0.938577\t&\t0.368873\t&\t0.110011\t&\t0.074161\t&\t0.028644\t&\t0.017577\t&\t0.010943\t&\t0.008560\t\\\\\n&\t$t=2$\t&\t0.708279\t&\t0.253087\t&\t0.073988\t&\t0.051268\t&\t0.021393\t&\t0.014262\t&\t0.007372\t&\t0.005221\t\\\\\n\\multirow{4}{*}{\\rotatebox{90}{$\\mbox{tr}(\\mathbf P_{t|T})\/q$}}\n&\t$t=3$\t&\t0.697724\t&\t0.250786\t&\t0.072065\t&\t0.050450\t&\t0.020933\t&\t0.014096\t&\t0.007206\t&\t0.005061\t\\\\\n&\t$t=4$\t&\t0.696501\t&\t0.250387\t&\t0.071913\t&\t0.050388\t&\t0.020872\t&\t0.014075\t&\t0.007192\t&\t0.005046\t\\\\\n&\t$t=5$\t&\t0.696180\t&\t0.250328\t&\t0.071903\t&\t0.050384\t&\t0.020864\t&\t0.014072\t&\t0.007190\t&\t0.005045\t\\\\\n&\t$t=6$\t&\t0.696142\t&\t0.250318\t&\t0.071903\t&\t0.050384\t&\t0.020863\t&\t0.014072\t&\t0.007190\t&\t0.005045\t\\\\\n&\t$t=7$\t&\t0.696138\t&\t0.250317\t&\t0.071903\t&\t0.050384\t&\t0.020863\t&\t0.014072\t&\t0.007190\t&\t0.005045\t\\\\\n&\t$t=8$\t&\t0.696137\t&\t0.250316\t&\t0.071903\t&\t0.050384\t&\t0.020863\t&\t0.014072\t&\t0.007190\t&\t0.005045\t\\\\\n&\t$t=9$\t&\t0.696137\t&\t0.250316\t&\t0.071903\t&\t0.050384\t&\t0.020863\t&\t0.014072\t&\t0.007190\t&\t0.005045\t\\\\\n&\t$t=10$\t&\t0.696137\t&\t0.250316\t&\t0.071903\t&\t0.050384\t&\t0.020863\t&\t0.014072\t&\t0.007190\t&\t0.005045\t\\\\\n[4pt]\n\\hline\n&&\\\\[-4pt]\t\t\t\t\t\t\t\t\t\t\t\n\\multicolumn{2}{l}{$\\mbox{tr}(\\mathbf P_{10|T})n\/q$}\t\\vline &\t1.740343\t&\t1.251582\t&\t0.898783\t&\t1.259595\t&\t0.782350\t&\t0.703592\t&\t0.719033\t&\t0.756698\t\\\\[4pt]\n\\hline\n\\hline\n\n\\end{tabular}\n\\end{table}\n\nIn Table \\ref{tab:mc1NS} and in Table \\ref{tab:mc1NSt}, we report the relative MSE of our estimator over the MSE of the common component estimators obtained by PC as in \\citet{bai04}, and by PC in first differences as in \\citet{baing04} and \\citet{BLL2}. Overall our estimator outperforms the others with the exception of the latter, which is show to perform better when $n_1$ becomes very large and about the same order of magnitude as $n$. This reflects the additional computational burden of our estimator which requires increasing the number of latent states when the idiosyncratic components are non-stationary and therefore we must include their dynamics in the model.\n\n\\begin{table}[ht!]\n\\setlength{\\tabcolsep}{0\\textwidth}\n\\caption{Simulation results - Common components}\\label{tab:mc1NS}\n\\centering\n\\small \\textsc{Relative Mean Squared Errors} \\smallskip\n\n\\footnotesize\n\\begin{tabular}{C{.085\\textwidth}C{.085\\textwidth}C{.085\\textwidth}C{.085\\textwidth} | C{.11\\textwidth} C{.11\\textwidth} C{.11\\textwidth} | C{.11\\textwidth} C{.11\\textwidth} C{.11\\textwidth} }\n\\multicolumn{10}{c}{Serially and cross correlated idiosyncratic components ($\\tau=0.5$, $\\delta =0.2$). Gaussian innovations}\\\\\n\\hline\n\\hline\n&&&&\\multicolumn{3}{c|}{$q=2$, $s=0$}&\\multicolumn{3}{c}{$q=2$, $s=1$}\\\\\\hline\n$n$ & $T$ & $n_1$& $n_b$& \\tiny{B}& \\tiny{BN}& \\tiny{BLL}& \\tiny{B}& \\tiny{BN}& \\tiny{BLL}\\\\\n\\hline\n75\t&\t75\t&\t0\t&\t0\t&\t0.58\t&\t0.00\t&\t0.28\t&\t0.54\t&\t0.01\t&\t0.53\t\\\\\n100\t&\t100\t&\t0\t&\t0\t&\t0.54\t&\t0.00\t&\t0.22\t&\t0.54\t&\t0.00\t&\t0.49\t\\\\\n200\t&\t200\t&\t0\t&\t0\t&\t0.45\t&\t0.00\t&\t0.12\t&\t0.59\t&\t0.00\t&\t0.39\t\\\\\n300\t&\t300\t&\t0\t&\t0\t&\t0.40\t&\t0.00\t&\t0.08\t&\t0.63\t&\t0.00\t&\t0.32\t\\\\\n\\hline\n75\t&\t75\t&\t25\t&\t25\t&\t0.01\t&\t0.02\t&\t0.44\t&\t0.04\t&\t0.04\t&\t0.83\t\\\\\n100\t&\t100\t&\t25\t&\t25\t&\t0.01\t&\t0.01\t&\t0.47\t&\t0.02\t&\t0.02\t&\t0.68\t\\\\\n200\t&\t200\t&\t25\t&\t25\t&\t0.00\t&\t0.00\t&\t0.53\t&\t0.01\t&\t0.01\t&\t0.66\t\\\\\n300\t&\t300\t&\t25\t&\t25\t&\t0.00\t&\t0.00\t&\t0.77\t&\t0.00\t&\t0.00\t&\t0.73\t\\\\\n\\hline\n75\t&\t75\t&\t50\t&\t50\t&\t0.05\t&\t0.21\t&\t1.55\t&\t0.11\t&\t0.23\t&\t1.47\t\\\\\n100\t&\t100\t&\t50\t&\t50\t&\t0.02\t&\t0.12\t&\t1.45\t&\t0.06\t&\t0.12\t&\t1.53\t\\\\\n200\t&\t200\t&\t50\t&\t50\t&\t0.00\t&\t0.02\t&\t0.82\t&\t0.01\t&\t0.02\t&\t0.75\t\\\\\n300\t&\t300\t&\t50\t&\t50\t&\t0.00\t&\t0.01\t&\t0.93\t&\t0.00\t&\t0.01\t&\t0.80\t\\\\\n\\hline\n100\t&\t100\t&\t75\t&\t75\t&\t0.03\t&\t0.23\t&\t1.44\t&\t0.08\t&\t0.23\t&\t1.48\t\\\\\n200\t&\t200\t&\t75\t&\t75\t&\t0.00\t&\t0.03\t&\t0.80\t&\t0.02\t&\t0.06\t&\t1.52\t\\\\\n300\t&\t300\t&\t75\t&\t75\t&\t0.00\t&\t0.02\t&\t1.11\t&\t0.01\t&\t0.02\t&\t0.92\t\\\\\n\\hline\n200\t&\t200\t&\t100\t&\t100\t&\t0.01\t&\t0.12\t&\t1.87\t&\t0.04\t&\t0.15\t&\t2.08\t\\\\\n300\t&\t300\t&\t100\t&\t100\t&\t0.00\t&\t0.03\t&\t1.08\t&\t0.01\t&\t0.04\t&\t1.45\t\\\\\n\\hline\n\\hline\n\\end{tabular}\n\n\\begin{tabular}{p{\\textwidth}}\\tiny\nThis table reports relative MSEs of the QML estimator proposed in this paper over the MSE of the common component estimators obtained by PC as in \\citet{bai04} (B), and by PC in first differences as in \\citet{baing04} (BN) and \\citet{BLL2} (BLL).\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}[t!]\n\\setlength{\\tabcolsep}{0\\textwidth}\n\\caption{Simulation results NS-DFM - Common components}\\label{tab:mc1NSt}\n\\centering\n\\small \\textsc{Relative Mean Squared Errors} \\smallskip\n\n\\footnotesize\n\\begin{tabular}{C{.085\\textwidth}C{.085\\textwidth}C{.085\\textwidth}C{.085\\textwidth} | C{.11\\textwidth} C{.11\\textwidth} C{.11\\textwidth} | C{.11\\textwidth} C{.11\\textwidth} C{.11\\textwidth} }\n\\multicolumn{10}{c}{Serially and cross correlated idiosyncratic components ($\\tau=0.5$, $\\delta =0.2$). Student $t_4$ innovations}\\\\\n\\hline\n\\hline\n&&&&\\multicolumn{3}{c|}{$q=2$, $s=0$}&\\multicolumn{3}{c}{$q=2$, $s=1$}\\\\\n$n$ & $T$ & $n_1$& $n_b$&Rel-MSE&Rel-MSE&Rel-MSE&Rel-MSE&Rel-MSE&Rel-MSE\\\\\n&&&&\\tiny B&\\tiny BN&\\tiny BLL&\\tiny B&\\tiny BN&\\tiny BLL\\\\\n\\hline\n75\t&\t75\t&\t0\t&\t0\t&\t0.58\t&\t0.00\t&\t0.30\t&\t0.58\t&\t0.01\t&\t0.57\t\\\\\n100\t&\t100\t&\t0\t&\t0\t&\t0.55\t&\t0.00\t&\t0.25\t&\t0.56\t&\t0.00\t&\t0.53\t\\\\\n200\t&\t200\t&\t0\t&\t0\t&\t0.45\t&\t0.00\t&\t0.12\t&\t0.60\t&\t0.00\t&\t0.40\t\\\\\n300\t&\t300\t&\t0\t&\t0\t&\t0.41\t&\t0.00\t&\t0.09\t&\t0.67\t&\t0.00\t&\t0.35\t\\\\\n\\hline\n75\t&\t75\t&\t25\t&\t25\t&\t0.01\t&\t0.02\t&\t0.43\t&\t0.06\t&\t0.04\t&\t0.83\t\\\\\n100\t&\t100\t&\t25\t&\t25\t&\t0.01\t&\t0.01\t&\t0.39\t&\t0.03\t&\t0.02\t&\t0.70\t\\\\\n200\t&\t200\t&\t25\t&\t25\t&\t0.00\t&\t0.00\t&\t0.40\t&\t0.01\t&\t0.00\t&\t0.64\t\\\\\n300\t&\t300\t&\t25\t&\t25\t&\t0.00\t&\t0.00\t&\t0.56\t&\t0.01\t&\t0.00\t&\t0.68\t\\\\\n\\hline\n75\t&\t75\t&\t50\t&\t50\t&\t0.06\t&\t0.17\t&\t1.51\t&\t0.14\t&\t0.17\t&\t1.45\t\\\\\n100\t&\t100\t&\t50\t&\t50\t&\t0.04\t&\t0.11\t&\t1.57\t&\t0.09\t&\t0.10\t&\t1.46\t\\\\\n200\t&\t200\t&\t50\t&\t50\t&\t0.00\t&\t0.01\t&\t0.57\t&\t0.01\t&\t0.01\t&\t0.69\t\\\\\n300\t&\t300\t&\t50\t&\t50\t&\t0.00\t&\t0.01\t&\t0.63\t&\t0.01\t&\t0.01\t&\t0.71\t\\\\\n\\hline\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n100\t&\t100\t&\t75\t&\t75\t&\t0.04\t&\t0.18\t&\t1.36\t&\t0.12\t&\t0.21\t&\t1.48\t\\\\\n200\t&\t200\t&\t75\t&\t75\t&\t0.01\t&\t0.02\t&\t0.65\t&\t0.03\t&\t0.05\t&\t1.41\t\\\\\n300\t&\t300\t&\t75\t&\t75\t&\t0.00\t&\t0.01\t&\t0.71\t&\t0.01\t&\t0.01\t&\t0.81\t\\\\\n\\hline\n200\t&\t200\t&\t100\t&\t100\t&\t0.02\t&\t0.10\t&\t1.68\t&\t0.06\t&\t0.13\t&\t1.99\t\\\\\n300\t&\t300\t&\t100\t&\t100\t&\t0.00\t&\t0.02\t&\t0.72\t&\t0.02\t&\t0.04\t&\t1.39\t\\\\\n\\hline\n\\hline\n\\end{tabular}\n\n\\begin{tabular}{p{\\textwidth}}\\tiny\nThis table reports relative MSEs of the QML estimator proposed in this paper over the MSE of the common component estimators obtained by PC as in \\citet{bai04} (B), and by PC in first differences as in \\citet{baing04} (BN) and \\citet{BLL2} (BLL).\n\\end{tabular}\n\\end{table}\n\nSome clarifications on the competing methods considered are necessary in order to interpret the results in Table \\ref{tab:mc1NS} and in Table \\ref{tab:mc1NSt} (we refer to the original papers for details). First, notice that all alternative approaches considered here do not allow for dynamic loadings, so here they are implemented by computing the first $q(s+1)$ PCs. \n\nSecond, despite the common practice in the literature, \\citet{bai04} did not propose its approach for factor model estimation, but rather to estimate common trends, and it is based on the crucial assumption of all idiosyncratic components being stationary. Indeed, we see from Table \\ref{tab:mc1NS} that when $n_1>0$ this approach fails completely. \n\nThird, the \\citet{baing04} approach delivers estimates of the common component which are obtained \n\\begin{inparaenum}[($i$)]\n\t\\item by  detrending the data by estimating the slope of the trend with the mean of the data in first difference; then\n\t\\item by estimating the factors in first differences; and, finally,\n\t\\item by cumulating the differenced estimator to obtain an estimate of the levels. \n\\end{inparaenum} \nAs such, this estimator is always subject to a location shift---it can be shown to converge to a Brownian bridge. Notice that this approach was introduced to test for the presence of unit roots rather than for factor model estimation, and, while the test is unaffected by location shifts, the use of the cumulated estimator for other scopes is not justified in general. As we see from Table \\ref{tab:mc1NS}, this approach fails to consistently reconstruct the common component in all cases considered. \n\nFourth, the approach in \\citet{BLL2} is based on the same ideas of \\citet{baing04}, but it takes care of the above mentioned issues related to detrending and cumulation, and, therefore, it is a valid alternative.\n\n\n\\section{Concluding remarks}\\label{sec:conclusion}\n\nThis paper considers estimation of large non-stationary approximate dynamic factor models by means of the Expectation Maximization algorithm, implemented jointly with the Kalman smoother. In our model the factors are a cointegrated vector process, thus containing both common $I(1)$ trends and stationary (cyclical) components.  We show that, as the cross-sectional dimension $n$ and the sample size $T$ diverge to infinity, the common factors, the factor loadings, and the common component estimated are $\\min(\\sqrt n,\\sqrt T)$-consistent at each $i$ and $t$.\n\nFurthermore, we show that the model can be extended to account for the possible presence of idiosyncratic trends, as well as the presence of secular (linear) trends, which can have either a constant slope (deterministic linear trends) or a time-varying slope (local linear trends). Consistent estimation of this case is also considered. \n\nFinally, the results in this paper provides the theoretical background for the application considered in \\citet{OGAP}, where the NS-DFM  is used to estimate the output gap in the US.\n\n\\singlespacing\n{\\small{\n\\setlength{\\bibsep}{.2cm}\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}