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Perturbative finite temperature field theory in Minkowski space Keil, Werner H. 1989

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P E R T U R B A T I V E FINITE T E M P E R A T U R E FIELD T H E O R Y IN MINKOWSKI SPACE By Werner H. Keil Diplom-Physiker Technical University of Clausthal M. Sc. (Physics) The University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1989 © Werner H. Keil, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer-ence and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Physics The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Abstract This thesis contains a perturbative analysis of decay and scattering rates in finite temperature and density environments. The discussion is based on the Niemi-Semenoff real-time formulation of quantum field theory at finite temperature. Two systems are investigated: neutron j3 decay at finite density, and Higgs boson decay with radiative QED corrections, at finite temperature. For neutron /? decay, a fully relativistic analysis at tree level is presented. An analytic formula for the free neutron decay rate is derived, and subsequently generalized to a finite-density environment. The decay rates are obtained from the imaginary part of the neutron self-energy. This method turns out to be very straightforward and elegant, since it includes all relevant decay and inverse decay modes in a nontrivial way. The decay of a Higgs boson into two fermions, with one-loop QED corrections, is used to discuss the problem of renormalization at finite temperature. It is found that the finite-temperature part of the self-energy corrections cannot be absorbed into temperature dependent mass and wave function renormalization counterterms, due to the lack of Lorentz invariance, and it is argued that finite-temperature renormalization is not an appropriate concept for decay and scattering rate calculations. A general algorithm for the calculation of thermal self-energy corrections is derived, and applied to the Higgs-fermion system. The result is explicitly shown to be free of infrared and mass singularities. Previous work on the subject is compared to this general approach, and possible applications in cosmology and astrophysics are discussed. ii Table of Contents Abstract ii List of Figures v Acknowledgment vii 1 Introduction 1 1.1 Decay and scattering rates at finite temperature and density 2 1.2 Overview of the thesis 7 2 Time path formalism and thermal Green's functions 10 2.1 Prel iminaries 11 2.2 P a t h integral formulat ion 14 2.3 Real-t ime contours and thermal Green's functions 21 3 Discontinuities of thermal Green's functions 31 3.1 T h e largest and smallest t ime equation 32 3.2 Finite-temperature Cutkosky rules for thermal Green's functions 38 4 Real-time self-energies and thermal decay rates 42 4.1 Real-t ime self-energies 42 4.2 Decay rates at finite temperature 46 5 Relativistic analysis of neutron 3 decay at finite density 50 5.1 Prel iminaries 51 5.2 Free neutron decay at zero density 53 i i i 5.3 Neutron decay in a dense electron gas 60 5.4 Neutron decay in a proton-electron plasma 62 5.5 Summary and conclusions 72 6 Radiative corrections and renormalization at finite temperature 73 6.1 Higgs boson decay at finite temperature 76 6.2 The lowest-order decay rate 77 6.3 Radiative corrections at finite temperature 82 6.3.1 The vertex-correction diagram 83 6.3.2 The self-energy correction diagram 88 6.3.3 Mass and wave function renormalization at finite temperature 96 6.4 Generalization and applications 110 6.5 Summary and conclusions 113 7 Conclusion 116 Appendix 118 A The self-energy correction to the Higgs boson decay rate 118 Bibliography 122 iv List of Figures 2.1 Integration contour C in the complex time plane 13 2.2 Contour C for real-time Green's functions 22 4.1 Dyson-Schwinger equation 43 5.1 Neutron self-energy En(</) 52 5.2 Imaginary part of the self-energy Sn 52 5.3 Neutron decay rate T in a dense electron gas with Fermi momentum PF 63 5.4 Neutron decay rate Yd in a dense electron-proton plasma with Fermi momentum pp. The dashed line is the dense-electron-gas rate from Fig. 5.3, shown for reference 67 5.5 Inverse decay rate I\- in a dense electron-proton plasma with Fermi momentum pF 69 5.6 Total decay rate Td + I\ 70 6.1 Transition matrix elements for H —• e +e~ with 0(e 2) radiative corrections: a) lowest-order vertex; b) vertex correction; c) self-energy correction; d) photon emission and absorption processes. The counterterm diagrams are omitted. . . . 78 6.2 The Higgs boson self-energy Ilii(g) with 0(e 2) radiative corrections: a) lowest-order diagram; b) "vertex correction" diagram; c) "self-energy correction" dia-gram; d) vertex counterterm diagram; e) mass counterterm diagram 79 6.3 Circled diagrams for the imaginary part of the lowest-order self-energy ITn. . . . 80 6.4 Circled diagrams for the vertex-corrected Tin a n ( i the equivalent products of transition matrix elements 84 v Circled diagrams for the self-energy corrected fin and the equivalent products of transition matrix elements 90 Self-energy correction to IIn(?) for a general decay process <p —* <f)\ ... <pn I l l vi Acknowledgment I would like to express my appreciation and gratitude to the many people who contributed to this thesis. First and foremost, my thanks to my supervisor Prof. Gordon Semenoff who introduced me to the subject of finite temperature field theory. Without his advice, patience, and financial assistance, this thesis would have never been completed. I would also like to thank my friend and collaborator Dr. Randy Kobes. His help and continuing interest in the subject were invaluable to me. I also gratefully acknowledge the many helpful discussions with my committee members Prof. Doug Beder and Prof. Nathan Weiss, and with my fellow students and friends, Tony Noble, Andre Roberge, and Sandy Rutherford. Last, but not least, my thanks to my parents, to Elena Jean, and to Libby and Randy Kay, who helped with the final draft of the thesis and who provided much appreciated encouragement and support. vii Chapter 1 Introduction Relativistic quantum field theory has been extremely successful in the description of high-energy physics. Quantum electrodynamics (QED) describes the electromagnetic interactions with almost legendary precision, and the development of nonabelian gauge theories has led to a field theoretic framework for the weak and strong interactions as well. Although far from being the ultimate theory, the so-called standard model—the Weinberg-Glashow-Salam theory for the unified electroweak interactions and quantum chromodynamics (QCD) for the strong interactions—accounts satisfactorily for today's experimental particle data. However, quantum field theory, viewed as a many-body theory with infinitely many degrees of freedom, is more than just a tool for the calculation of scattering cross sections in elementary particle physics. Field theoretic techniques were applied to statistical mechanics and nonrela-tivistic many-body theory long ago, and ideas from condensed matter physics, like spontaneous symmetry breaking, have in turn influenced the particle theory aspect of quantum field theory. From this point of view, scattering experiments at accelerators probe only the ground state, or vacuum, of a relativistic many-body system. The statistical aspect of high-energy theory becomes relevant in very hot and/or dense environments like the early universe or the interior of neutron stars. The description of relativistic quantum systems in or close to thermal equilibrium is the domain of quantum field theory at finite temperature and density, also called thermal field theory or quantum statistical field theory. Its applications range from phase transitions in the early universe to particle reactions in the interior of neutron stars to the description of the quark-gluon plasma. In this thesis we will concentrate on a problem in finite temperature field theory that has been the standard task of the traditional "vacuum" or "zero temperature" field theory: the 1 Chapter 1. Introduction 2 perturbative evaluation of elementary particle decay and scattering rates. We will now explain why this comprises an interesting and nontrivial problem, and we will illustrate its physical importance with several examples. 1.1 Decay and scattering rates at finite temperature and density First we emphasize that decay and scattering rates in a thermal environment differ in many respects from the scattering processes in a particle accelerator. In the latter, particles are prepared in a free initial state, they interact, and the reaction products evolve again into a free final state, to be identified in a detector. At finite temperature, the presence of a heatbath changes this picture drastically. Imagine, for example, a collection of Z bosons immersed in a hot plasma consisting of electrons and positrons. A Z will decay into an electron-positron pair, Z —> e +e~, but the electrons and positrons in the heatbath have a thermal probability to recombine, e+e~ —» Z. Similarly, there will be scattering reactions like e~ Z —» e~ etc. Even after an infinitely long time, the initial Z ensemble will not disappear as it would at zero temperature; instead, it will approach thermal equilibrium with its environment, driven by a combination of decay, inverse decay and scattering processes. The traditional picture of stable particles in asymptotic initial and final states, does not generalize to the heatbath. Instead the appropriate notion is that of "quasiparticle" excitation in the plasma that is "Landau damping" (thermalizing with its surroundings); the damping constant corresponds to the thermal decay rate. Quantum corrections—the emission and absorption of real and virtual particles—will enhance these effects because the system can now exchange energy with the heatbath, either directly or via intermediate states. Thus, even particles that are energetically stable at zero temperature will begin to thermalize in a heatbath. Strictly speaking, the whole basis for scattering theory in vacuum field theory—asymptotic states, the LSZ theorem—does not exist at finite temperature. However, one might intuitively expect that, if the thermal effects are "not too large," the naive particle picture (with some modifications) will give an approximate, but satisfactory description of the thermal processes. Chapter 1. Introduction 3 Another nontrivial feature is the loss of Lorentz invariance at finite temperature, because the heatbath now defines a preferred frame of reference. For a particle at rest with respect to the heatbath, the energy-momentum distribution of the surrounding plasma will be isotropic, but a moving particle will "see" a Doppler shift. Thus, the decay rates will no longer be Lorentz invariant, but depend on the frame of reference. Zero temperature field theory makes extensive use of Lorentz invariance, especially in the renormalization of ultraviolet divergencies. Therefore, some nontrivial differences between the zero and the finite temperature formalism can be expected. This brings us to our next topic: a brief and informal discussion of the different formalisms used in finite temperature field theory. Temperature can be naturally incorporated in quantum field theory. A field theory in euclidean space-time (i.e. with imaginary time ir) is formally equivalent to a statistical mechanics system at temperature T (more precisely, the trace of the density operator of the field theory corresponds to the thermodynamic partition function), with the imaginary time playing the role of an inverse temperature (3 = T~l. This equiv-alence allows a rigorous treatment of zero-temperature field theory in Minkowski space: the well-defined euclidean theory is analytically continued to real times (Minkowski space) and eventually the zero-temperature limit 8~x —* 0, or ir —• oo, is taken. It is also the basis for the oldest approach to finite temperature field theory. In the euclidean Matsubara formalism [36], the theory simply remains defined on a finite imaginary time (inverse temperature) in-terval. This formalism is well suited for the calculation of static thermodynamic quantities like free energy, pressure, etc. However, dynamical quantities—response functions, transport coefficients, damping constants—require a nontrivial analytic continuation to real time, which makes a direct formulation of the field theory in Minkowski space desirable. Such a real-time formalism was developed some time ago by Niemi and Semenoff [39]. Their time path formalism, formulated in terms of functional integrals, defines the theory on a path in the complex time plane, thereby incorporating both temperature (imaginary time) and real time. A judicious choice of the complex time contour then leads to a real-time formalism that Chapter 1. Introduction 4 allows a diagrammatic perturbation expansion with Feynman rules and diagrams very much like standard vacuum field theory, but with the degrees of freedom doubled: to every physical field corresponds an unphysical "thermal ghost field." These ghosts are necessary to maintain the correct analytic structure of the theory. An earlier version of the real-time formalism [10], based on a direct analytic continuation of the euclidean theory, neglected the doubling of the fields and turned out to be inconsistent, with ill-defined distributions appearing in higher orders in perturbation theory. Nonetheless, the Dolan-Jackiw formalism was quite popular for practical calculations with first-order quantum corrections, and it is still regarded as equivalent to the Niemi-Semenoff formalism at the one-loop level (we will comment more on this point later). The time path method is based on the functional (Feynman path) integral approach to quantum field theory. There is also a canonical (operator) approach to real-time thermal field theory, the thermo field dynamics developed by Umezawa and coworkers [47]. We will not elaborate on the details of this rather complex algebraic formalism, but merely point out that the doubling of the degrees of freedom occurs as well, and that the Feynman rules for perturbation theory are identical to the ones of the Niemi-Semenoff formalism. Finally, let us consider some concrete examples of physical systems in which thermal scat-tering rates are important. One of the current areas of interest in high energy theory is the quark-gluon plasma, a phase of QCD in which quarks and gluons are no longer bound together in hadrons but form a gas of weakly interacting free particles. Lattice gauge theory calculations indicate such a transition at temperatures of about 200 MeV, and relativistic heavy-ion collisions might actually produce the plasma in the near future. Thermal decay rate calculations might be useful, both in understanding the stability of the plasma and in identifying reactions that signal its existence. For example, two groups [4, 5] recently calculated the rates for the thermal Drell-Yan process, that is, the production of lepton pairs from quarks via an intermediate photon: Chapter 1. Introduction 5 Measuring the emission rate of these leptons might yield information about the plasma. H o w -ever, at the present stage i t is too early to make definite statements about the relevance of these calculations. Indeed, it is fair to say that the whole subject of finite temperature Q C D is st i l l poorly understood. A case i n point is the controversy over the value and sign of the gluon damping constant. Recently, several groups [21, 17, 16, 13, 28, 38] calculated the damping constant ( thermal decay rate) of a gluon and found that the value and sign depended on the gauge i n which the calculation was performed. A negative sign for a " d a m p i n g " constant w o u l d signal instabi l i ty and hence a breakdown of perturbat ion theory, but the gauge dependence of a supposedly physical quantity suggests some deeper problem. Several possible explanations and solutions to this puzzle have been discussed [41], but , at the t ime of this w r i t i n g , the sign and value of the gluon decay constant is s t i l l an open problem. In astrophysics the electroweak interactions i n a thermal environment are important for the cooling of supernovae and neutron stars. Consider the following the examples: T h e dominant mode of energy loss for a supernova during core collapse is neutrino emission generated by (i) n e u t r a l i z a t i o n reactions, l ike n —• e~ p Pe e~ p —*• n ve (ii) and thermal emission reactions l ike e+e~ —• ve ve e~ 7 -> e~ ve ve which are al l mediated by W and Z bosons at temperatures of T > 1 0 9 K (0.1 M e V ) . Neutron stars are colder (T < 10 9 K ) but extremely dense, and the ensuing degeneracy of the core has important consequences. T h e dominant energy loss (cooling) mode is s t i l l neutrino emission, but the degeneracy of the interior now suppresses the U R C A 1 reactions that would Earned after the URCA casino in Rio which provides an ideal mechanism for losing money. Chapter 1. Introduction 6 normally dominate: (3 decay is forbidden by the Pauli principle (the reason for the stability of the neutrons), and inverse /? decay is suppressed by an energy-momentum mismatch at the Fermi surfaces of electrons and protons on one side and neutrons on the other. Thus, the cooling has to proceed via the much less effective "modified URCA" reactions which require a bystander particle for momentum conservation. Of course, the interior structure of supernovae and neutron stars is still an open problem, and calculations of neutrino emissions from hot and dense environments are an important subject. In a cosmological setting, thermal neutron (3 decay with radiative QED corrections attracted attention a few years ago, because of its possible implications for the nucleosynthesis of the light elements in the early universe. Let us briefly recall the basics of primordial nucleosynthesis, according to the standard big bang model. When the temperature in the expanding universe drops to T < 1 MeV, the weak interactions become too slow to compete with the expansion rate and maintain protons and neutrons in equilibrium. The neutron to proton ratio "freezes out" at its equilibrium value, and neutron (3 decay sets in, decreasing this ratio. At T ~ 0.1 MeV the nucleons become bound in light nuclei, mostly 4 He. The standard big bang model gives definite predictions for the relative abundances of these elements, and they are in fairly good agreement with the observed values. These measurements are, of course, quite difficult, and the observational limits are far from definite. The nucleosynthesis calculations are sensititve to several factors—the initial baryon density of the universe, the number of lepton and quark families, the expansion rate of the universe—each of which can change the initial neutron to proton ratio. Because the standard big bang predictions seem to agree with the observations, the measured abundances have been used in turn to put limits on these fundamental initial parameters. Obviously a neutron decay rate enhanced by temperature effects can lead to drastic changes because it would allow a higher initial neutron to proton ratio for the same final abundances. This would then allow, for example, a higher baryon density in the universe, extra families of elementary particles, etc. Fortunately (for the standard big bang) the net effect of the Chapter 1. Introduction 7 temperature on the final abundances was shown to be negligble [8, 9]. However, some of the field theoretic aspects of these calculations are questionable, and further work may be needed to settle the issue. Indeed, it was the problem of neutron 8 decay at finite temperature that initiated the work presented in this thesis. 1.2 Overview of the thesis This thesis presents a detailed analysis of decay rates in finite temperature and density envi-ronments with one-loop quantum corrections, using techniques based on the Niemi-Semenoff time path formalism. Previous work on this subject [8, 9, 11, 12] was based on the older Dolan-Jackiw [10] for-malism and attempted to generalize the standard procedure of vacuum field theory to finite temperature: thermal ghosts are neglected, the rates are calculated by integrating squares of transition amplitudes over the available phase space, and the radiative one-loop corrections are treated as temperature dependent renormalization constants. This work follows an alternate route and considers instead the imaginary part of the two-point function which contains the information about the dissipative properties of the system, viz., the damping rate. This approach yields directly the physical decay rates and avoids an unattractive feature of the real-time formalism: the ghost fields, while necessary for the consistency of the theory, lead to a proliferation of Feynman diagrams and make practical calculations very cumbersome. However, the algorithm for the calculation of the imaginary part of thermal Green's functions—the generalized Cutkosky rules developed by Kobes and Semenoff—contains no explicit reference to the ghost fields and simplifies the computation considerably. The discussion is based on the analysis of two systems: 1. neutron 3 decay in a finite density environment, and 2. Higgs boson decay at finite temperature with one-loop radiative QED corrections. Chapter 1. Introduction 8 The results of these investigations can be summarized as follows. For the neutron system we present a fully relativistic analysis of 0 decay at finite density. For the zero density case (free neutron) decay an analytic formula is derived and generalized to finite density. To our knowledge, no such analysis exists in the literature. Furthermore, the generalized Cutkosky rules are shown to be a very elegant and straightforward technique for the calculation of thermal decay rates which also includes both decay and inverse decay modes in a nontrivial way. In the second calculation the connection between radiative corrections and renormalization at finite temperature is investigated. The decay of a Higgs boson into two electrons is used as an explicit example, but the techniques and results are quite general and can be applied to other reactions as well. It is found that the finite temperature part of the one-loop fermion self-energy cannot be absorbed into mass and wave function renormalization counterterms, due to the lack of Lorentz invariance, and it is argued that finite temperature renormalization is a problematic concept and not appropriate for decay and/or scattering rate calculations. An explicit algorithm for the direct calculation of the finite temperature self-energy correction is given and applied to the Higgs example; the result is explicitly shown to be free of infrared and mass singularities. Previous work on the subject is compared to this approach, and the implications and possible applications of the results in cosmology are discussed. These results have been published in references [22] and [23]; these two papers form the central part of the thesis. To make the discussion more accessible to nonspecialists, a basic introduction to the Niemi-Semenoff time path formalism and to the calculational techniques for decay rate calculations is included. This review is essentially self-contained, and assumes only a basic knowledge of Feynman path integrals and perturbative vacuum field theory. The thesis is organized as follows. Chapter 2 introduces the time path method for finite tem-perature field theory and the diagrammatic perturbation theory for thermal real time Green's functions. Chapter 3 derives the generalized Cutkosky rules for the calculation of the imaginary Chapter 1. Introduction 9 part of a thermal Green's function. Chapter 4 concentrates on the structure of the thermal self-energy (two-point Green's function) and discusses the relation between its imaginary part and thermal decay rates. Chapter 5 and 6 contain the results of the neutron and Higgs boson decay investigation. They are essentially identical to references [22] and [23]. Chapter 7 summarizes the main points and discusses the possible implications. Chapter 2 Time path formalism and thermal Green's functions In this chapter we w i l l present a general introduct ion to the complex-time path approach to quantum field theory at finite temperature and density which forms the basis for diagrammatic perturbat ion expansions i n M i n k o w s k i space. T i m e path methods were originally used i n the context of nonrelativist ic m a n y b o d y theory and nonequi l ibr ium quantum statistical mechanics [2, 26, 37]. A b o u t six years ago N i e m i and SemenofF applied time path methods to relativistic field theory i n thermal equi l ibr ium [39]. Us ing functional integral methods they developed a transparent diagrammatic perturbat ion theory for the direct calculation of thermal Green's functions i n M i n k o w s k i space that employs F e y n m a n diagrams and rules very much l ike the standard (zero temperature) field theory, but w i t h the degrees of freedom doubled. T h i s formalism is free of the pathologies (ill-defined distributions) that had plagued earlier attempts to formulate a real-time perturbat ion theory at finite temperature [10]. Original ly formulated for the simplest case of a scalar field theory the functional real-time formalism was subsequently extended to spin and gauge fields at finite temperature and density [31, 32, 27]. Recently Landsman and van Weert [34] gave a comprehensive review of the real and imaginary time formal ism, and this presentation follows essentially their paper. W e begin w i t h a brief introduct ion to some of the basic concepts of the t ime p a t h formal ism, using a scalar field theory as a simple example. T h e n we formulate the t ime path method i n terms of functional integrals and extend the formalism to spin and gauge fields. F i n a l l y , a judicious choice of the time contour leads to the Niemi-Semenoff perturbat ion expansion w i t h its characteristic doubling of the degrees of freedom. 10 Chapter 2. Time path formalism and thermal Green's {unctions 11 2.1 Preliminaries Consider a scalar quantum field theory at temperature T = with field operator 4>, conjugate momentum #, and Hamiltonian H(4>,TT). The time evolution of the operators in the Heisenberg picture is determined by H((f>,ft) according to and the time coordinate t is generalized to the whole complex plane: t € C. The objects of interest from which, in principle, all observable properties of the theory can be extracted, are the thermal Green's functions, that is, the Gibbs ensemble averages of time ordered products of the field operator <f>: Gp(x1,...,xn) = <Tcj>(xi)...j>(xn) > _ Tr { e - ^ T e f a ) . . . f c ) } " Yr~e~^  ( ] The complex time ordering Tc orders the operators along an oriented contour C in the complex time plane. This can be formalized by denning a contour step and ^-function for C as follows. Let C be parametrized by some real .parameter r, so that t = Z(T), then we define Sc(t-t') = and we have Oc(t-t') = 0(r-r') 6(T - T') 8T Tcj>{x)4>{x') = 6c(t - t')4>(x)j>{x') + ec(t' - t)4tx')fa) dtTcj>(x)4>(x>) = 6c(t-t')[$(x),i(x')]+Tcdt$(x)4>(x') and so on. The time derivative dt should be understood as a directional derivative along the contour. Similarly, functional differentiation can be extended to the contour by Sj(x') 6j(x) 6c(t-1?)6(x'-x*) = 8e{x-x') Chapter 2. Time path formalism and thermal Green's functions 12 for c-number functions j defined on the contour. We seek a generating functional Z[j] for the Green's functions (2.1) such that 1 8nZ\i] Gp{xu...,xn) = U 1 (2.2) Z[0]iSj(x1)...%6j(xn)lJ=o where Z[0] is an arbitrary normalization constant. A generating functional with this property is obviously z[j] = z[o) < Tc i i c > (2.3) where Z[0] is chosen to be the partition function, Z[0] = Tr , and the contour C passes through the time arguments t\,...,tn of Gp. A further restriction on C is imposed by the the requirement of analyticity for G@. Consider for example the two-point function Gp{x - x') = ec(t - t')G+(x, x') + 6(t' - t)G~(x, x') (2.4) in terms of the correlation functions G+(x,x')=<j>(x)j>(x')>=G-(x',x) (2.5) Evaluating (2.5) with a complete set of energy eigenstates \Em > one obtains the spectral representation (cf. [37]) G+(t - t') = Z- x[0] £ | < Em\$(0)\En > \ie-iEn(t-t')eiEm{t-t>+i(3) m,n Thus, G+ exists for -0 < Im(t - t') < 0, G~ for 0 < Im(t - t') < 0, and Gp is well-defined as a generalized function on the strip -0 < Im(t -t')<0 provided that 9c(t — t') = 0 for Im(t -1') > 0. Thus, the contour C must have a monotonically decreasing or constant imaginary part (downward slope) as shown in Figure 2.1. This condition is also sufficient for the existence of all higher order Green's functions (see [37]). Chapter 2. Time path formalism and thermal Green's functions 13 » flit Figure 2.1: Integration contour C in the complex time plane The generating functional and propagator can be easily calculated for the free scalar theory with commutation relation j>(x),dtj>(x')\ Sc(t - t') = i6c(x - x') and field equation -Oc<f>{x) = -(d2 - V 2 + m 2)<j>{x) = 0 Together with the definition (2.3) one obtains the Dyson-Schwinger equation for the free gen-erating functional Zo[j] _nJJM+j(x)Z0\j) = 0 'i6j(x) with solution Z0[j) = Zo[0) exp{-^ / / d 4x d 4x'j(x)D c(x - x')j(x')} (2.6) 2 JC JC where D° denotes the free contour propagator UcDc(x - x') = 6c{x - x') and from (2.2) we obtain for the two-point function Gp(x - x') = iD c(x - x') Chapter 2. Time path formalism and thermal Green's functions 14 The free thermal propagator D c can be calculated directly in Fock space. Expanding the field operators as usual in terms of creation and annihilation operators d 3k (2ir)32uk where kx = k°t — kx. For a gas of free scalar bosons in thermal equilibrium we have <a\ay> = (2ir) 32ojknB(uk)S(k -< aka\, > = (2z)32u>k(l + nB(wk))6(k - le) with k° = u>k — \Z&2 + m 2 and nB(ojk) = [e /3u>k — l ] - 1 is the Bose distribution function. Using these results for the correlation functions (2.5) one finds where the Fourier transforms are given by G+(k) = e^ k°G-(k) = p0(k)(l + nB(k 0)) (2.8) in terms of the free spectral function p0(k) = 2Tre(k°)6(k2 - m 2) (2.9) where t(k°) = 6(k°) — 0(—k°). Thus, the free thermal propagator can be written in spectral form as W c(x - x>) = J 0^Po(k) [oe(t - t') + nB(k 0)] e-ifc(*"*') (2.10) We will now extend these results in the next section, using the path integral formulation of quantum field theory. 2.2 Path integral formulation The Feynman path integral provides an elegant and powerful tool for the quantization of field theories, including spin and gauge fields. Thus it is advantageous to the reformulate the time Chapter 2. Time path formalism and thermal Green's functions 15 path method in terms of path integrals. For the most part, this is a straightforward general-ization of the zero temperature procedure. For simplicity's sake we start again with a scalar field theory. Let 4>{x, /) be the field operator in the Heisenberg picture and \<f)(x),t > an eigenstate at time t with eigenvalue <fr(x): 4>(x,t)\4>{$),t >= (f>(x)\<f>(x),t > where \<t>(x),t >= em\4>(x),Q > and t is again taken to be complex. Assuming that the \<j> > form a complete set of eigenvectors at any time t we evaluate the statistical average (2.3) and obtain for the generating functional Z[j] = Z[0] < TjJ^xiWfa) > = Z[0] Jd<p< fatile-^Tce'Sciilfati > = Z[0] Jd<t>< <f>,ti - t/3|T c e'/e^,i,- > (2.11) Now recall the standard (zero temperature) path integral representation for the matrix element of a time ordered functional F[<t>\ of field operators [1]: < <j>',t'\TT[^\\cj>",t" >=N J j V#Dv TWexp j t j f* J dtd3x[n{x)<j>(x) - JT(ar)] j (2.12) with the ^-integration subject to the boundary condition 4>(x,t') = 4>'{x) <t>{x,t") = and the time arguments in T[<p\ contained in the interval [t",t']. The path integral (2.12) can be directly generalized to complex times, that is, to matrix elements with time arguments ordered on the complex contour C discussed in the previous section. The contour can always be decomposed into pieces that allow a parametrization by a real parameter. Time ordering on the contour reduces then to ordinary time ordering in this parameter, and the path integral formula can then be applied seperately to each piece. Chapter 2. Time path formalism and thermal Green's functions 16 Thus we have for our generating functional (2.11) the path integral representation Z[j] = AT J V4>Vn exp j * J d 4x[n(x)<j>(x) - H(x) + j(x)<f>(x)]} (2.13) The (monotonically decreasing) contour C starts at some initial time U and ends at ti — if}; time derivatives are understood as directional derivatives along C. The (^ -integration in (2.13) is now over all fields satisfying the periodicity condition 4>(x,ti) = <j>(x,ti — i(3) To proceed we consider now a theory without derivative couplings, that is, with a Hamiltonian of the form H M ) = \* 2 + \<t>[-v2 + ™ 2\4> + v(<j>) for which the Gaussian 7r-integral can be performed explicitly. We obtain Z[j] = Af Jv<i> exp ^ iJj 4x[C(x) + i(aO#c)]} (2-14) in terms of the Lagrange density C(x) = ^(x)[-d 2 - m 2]<t>(x) - V{4>) = C0(x) + d{x) The ill-defined normalization factor N, into which all multiplicative constants are conveniently absorbed, is irrelevant for the calculation of Green's functions (cf. (2.2)). Finally we replace in standard fashion the field in the interaction term in (2.14) by a functional differentiation with respect to the source functions j and obtain Z [ ; ] = e x p { i / ^ £ j ( - ^ y ) } z „ b l (2.15) where ZQ[J] denotes the free generating functional Zo[j] - M J V<p exp [ i J ^ o + jtfj (2.16) Before we proceed with the evaluation of ZQ[J] let us generalize our results to spin and gauge fields. For fermions the derivation of a path integral is complicated by the anticommuting nature Chapter 2. Time path formalism and thermal Green's functions 17 of the spin 1/2 fields, and requires the introduct ion of an t i commuting Grassmann fields. F o r gauge fields the problem of gauge invariance requires a projection onto the physical states of the system i n the path integral , which leads to the introduct ion of the Fadeev-Popov ghost fields. However, these modifications are the same as at zero temperature, and the above t ime p a t h analysis goes through v ir tual ly unchanged. Thus we can proceed directly to the the relevant results. W e w i l l keep our discussion concise and treat bosons and fermions together for now. T h u s , let us consider the general case of a mult icomponent , covariant, complex field ip*a i n some representation of the Lorentz group, w i t h an arbi trary number of conserved charges Qa which generate infinitesimal symmetry transformations of the fields v i a QaJi,]=-ql Jfc where the superscripts i,j refer to the internal symmetry space and the subscript a denotes the Lorentz index. T h e free Lagrangian is wri t ten as Co = $aAap(id)fo (2.17) where A is some finite-order differential operator and $ denotes the adjoint field. B o t h fermions and bosons are included i n this generic notation; specific examples w i l l follow later. T h e free generating functional for (2.17) is given by ZoilJ] = M J 2tyZ>#exp j * + + (2.18) where ip denote c-number fields for bosons and Grassmann fields for fermions. T h e y have to satisfy the quasiperiodicity condition i}(x,ti) = rje^tl;(x,ti-id) (2.19) where rj = ± 1 for bosons and fermions, respectively, and p, = paqa. Green's functions are generated from (2.18) by functional differentiation w i t h respect to j from the right and w i t h Chapter 2. Time path formalism and thermal Green's functions 18 respect to j f rom the left: < Tcj)(xi)... $(xm)$(x'i)... i>{x'n) > Tr { e - ^ - ^ ^ T c ^ ) . . . ij>(x'n)} ~ Tr e-^-^aQ") 1 <5 & S { /7 r" 'i = Zo[0,0] iSj(Xl) • • • iSj(xm) iSJix'i) '' * WPnj °[h3]\3=i=° ( 2 - 2 0 ) A n interaction Lagrangian Ci^^ip) leads, analogous to the scalar case, to the f u l l generating functional z&:fl = «q.{/«jg.J5)}*0 . J ] (2.21) T h e Gaussian functional integral (2.18) can be calculated i n standard fashion by applying the shift M*) -*1>c-J d4x' D%p{x - x>) (2.22) where the contour propagator Dcap satisfies Aap(id)D^(x - x') = 6^6c{x - x') (2.23) Demanding invariance of the path integral under the transformation (2.22) we obtain for the free generating functional Zotfj] = Zo[0,0]exp {-iJJj4xd4x'ja(x)DcaP(x - (2.24) and hence for the thermal two-point function Gipi* ~ x') =< Tci>a(x)$p(x') >= iDcaP(x - x') (2.25) analogous to the scalar case. To find the appropriate boundary conditions we recall eqn. (2.4) and write the propagator i n terms of the correlation functions Deap(.x ~ x') = 0c(t - tf)D+0(x - x') + 0c(t> - t)D-ap{x - x') (2.26) Subst i tut ing (2.26) into the transformation (2.22) and imposing the boundary condit ion (2.19) on the shifted fields leads immediately to the relation Dt0(t - iP) = ve-toD-fi(jt) (2.27) Chapter 2. Time path formalism and thermal Green's functions 19 known as the Kubo-Martin-Schwinger (KMS) boundary condition [33, 35]. Note that the inter-acting propagators, derived from (2.21), will also satisfy the KMS condition by construction. Finally we obtain a spectral representation for the thermal propagator (2.23) by introducing a Klein-Gordon divisor d(id) with the property da0(id)Afo(id) = Aa0(id)d^(id) = 6^ U(-d 2 - m 2k) (2.28) k where the mk denote the mass spectrum of A. The propagator (2.23) can then be written as D%f}{x - x') = da(3(id)D c(x - x') (2.29) with D c the propagator for the (multi-mass) Klein-Gordon equation \\(-d 2 - m 2k)Dc(x - x') = Sc(x - x') (2.30) k For D° we have now, in obvious generalization of the scalar case, the spectral form iD c(x -*') = / -^Po(P)[0c(t -?) + « n,(pP)]e-*<*-*'> (2.31) where nv(p 0) = e ^ ^ p° ~ ^ — ?yj denotes the Bose-Einstein distribution nB and the Fermi-Dirac distribution np for rj = ±1, respectively, depending on the type of field. The spectral density po for the Klein-Gordon operator is defined by P*(p) = DiscY[^-? (2.32) where by definition Disc f(z) = hrn/(2 + ic) — f(z — ie) for a complex function f(z). These general results can now be applied to the case of a Dirac fermion. We have A(id) = (» d-m) d(id) = (-i ft - m) p(p) = 2TVC(P°)S{P 2 - m 2) and hence for the contour propagator iD^x - x>) = {-ifi- m)aP J ^2^(p°)6(p 2 - m2)[6c{t - f) - nF(jp°)]e-M"^ Chapter 2. Time path formalism and thermal Green's functions 20 Also, the canonically derived results (2.6) to (2.10) for a single scalar field are contained here as the simplest case. For gauge fields we have to modify these results only insofar as to take gauge freedom into account, that is, the path integral must include only the physical degrees of freedom of the system. This is accomplished by the standard Fadeev-Popov procedure [42] which introduces anticommuting ghost fields in the Lagrangian to subtract the spurious degrees of freedom. To be specific, let us consider the case of a SU(N) gauge theory with Lagrangian £ = -jF;uFailu (2.33) in terms of the field strength tensor F^ = d^Av-dvAll-rgfahcA hllAl with fabc the structure constants of SU(N) and a = 1,... ,N 2 — 1 the color indices. The (interacting) partition function is given by Z[0] = N J VAVuVQexp^i £d 4xCeff{x)} (2.34) with the effective Lagrangian Ceff = — F ^ F ^ - ^ ^ A l Y - i d ^ ) ^ ) -gf abc(d^u a)A by (2.35) a is the gauge fixing parameter and u,u> are the anticommuting Fadeev-Popov ghost fields. Note that the ghosts are not real physical fields in thermal equilibrium but merely represent the Fadeev-Popov determinant in the functional integral. Thus, they are subject to the same boundary conditions as the (bosonic!) gauge fields. For the gauge fields we have the usual periodicity condition Al(x,t) = Al(x,t-if3) To determine the free gauge boson propagator A^(id)D: bp(x - x')6bc = g^6 ac6c(x - x') Chapter 2. Time path formalism and thermal Green's functions 21 we proceed as before and consider the free part of the interact ing Lagrangian (2.35) -Co where the bosonic differential operator is given by + (a - l)d„dv w i t h K l e i n - G o r d o n divisor d^(id) = g^d2 + (a" 1 - l)d»dv W e now apply our general results (2.29) and (2.31) which yield for the free gauge boson prop-So far we have discussed only the general properties of generating functionals and Green's functions i n the time path formalism. To proceed from the abstract contour Green's functions to an explicit perturbat ion theory for thermal real-time Green's functions we w i l l now choose a specific contour. 2.3 Real-time contours and thermal Green's functions In order to generate Green's functions w i t h real-time arguments directly, the contour C has to include the real axis i n the complex t ime plane. A part icular ly convenient choice is the contour consisting of four pieces, C = C\ U C2 U C3 U C4, as shown i n Figure 2.2. W e set ti = —tj and eventually take the l i m i t tj —> 00. Let us now show how this contour simplifies the generating functional and leads to an elegant m a t r i x structure of the propagators. agator (2.36) where the free spectral function is given by po(k) = 27re(k°)8'(k2) Chapter 2. Time path formalism and thermal Green's functions 22 £ > * c t Figure 2.2: Contour C for real-time Green's functions Consider the free and interacting generating functionals (2.18) and (2.21). For the moment we will confine ourselves to a single scalar field to simplify the notation. Writing the contour C as C = C\2 U C34, C{j = C{ U Cj we decompose the contour integral in the free generating functional into four terms: Zo[j] = M exp { - i f / f +[ [ + [ ' [ + [ I ]d*xd*x>j(x)Dc(x-x>)j(x>)\ (2.37) We want to show that the contributions from the second and third term in the exponential vanish. Consider, for example, the case of t £ C\ = [ti,tj] and t' 6 C3, which we parametrize as t'(X) = tj — iX, AG [0,/3/2]. By definition we have 6c(t — t') = 0, hence the spectral representation (2.31) of the contour propagator D° takes the form iDc"{x - x1) = /+0° *£eV<tt-*)Dtf,Sy#) J-00 27T (2.38) Chapter 2. Time path formalism and thermal Green's functions 23 where we defined D(p°, x, Sf) = J ^ e ^ - ^ p ^ n s i p ^ (2.39) which is well-defined for 0 < A < j3, in agreement with our previous discussion. We want to show that lim DCi3 = 0 (2.40) tj —>oo and consequently lim / / d 4xd 4x'j(x)D Cl3(x-x')j(x') = 0 (2.41) </-»oo JCl Jc3 The Riemann-Lebesgue lemma states that the Fourier transform maps the function space L 1 into the space of continuous functions that vanish at infinity. Regarding D Cl3(tj — t) as the Fourier transform of D(p°) we apply the lemma and obtain (2.40). However, there are two technical complications. First we note that in (2.38) obviously t ^  tj which excludes t —> oo, and we have to impose the additional boundary condition j(x) = 0 (2.42) in order for (2.41) to hold. Furthermore, for the lemma to be applicable, D has to be integrable, that is, the spectral density po has to be an ordinary function. However, according to (2.32), po is a distribution, namely, a (^-function. Thus we require all generalized functions, viz. the (5-functions (2.32), to be properly e-regularized, with the limit e —» 0 to be taken only in the end. The same analysis can be repeated for t 6 C2, f G C4 and ti —>• —00, and we conclude that the second and third term in (2.37) do indeed vanish. Furthermore we have by definition ^ ^ = 0 forieCWeC34 Sj(x) so that the full generating functional factorizes: Z[j} = exV{i^£i(J^^ZQ[j]=Arz12[j}Z34{j} (2.43) Chapter 2. Time path formalism and thermal Green's functions 24 where Zu[j] = exp {i ( ^ ) } exp { - ' / / c i j A *Vi(.)D-<. - *');(*')} (2.44) and likewise for Z34. For the computation of real-time Green's functions Z34 is just an irrelevant multiplicative constant. Thus, the generating functional for real-time Green's functions is reduced to Z12, and the contour C to the time ordered real-time interval C\ = [—00,-foo] and the anti-time ordered C2 = [—00 — id/2, + 0 0 — i(3/2\. We can now replace j, which is defined on C 1 2 , by two independent source functions according to h(t) = j(t), j2(t) = j(t - id/2) (2.45) in terms of which the interaction exponential can be written as A'LC' (m)}=^{iCdiJd3* b (ik)-Ci (ik)}) (2-46) Similarly, we write the free generating functional in (2.44) in the form exp{-^J J dtdt'j(t)De(t-i')i(<')} = e x p { 4 / d t d t ' j r ( t ) D c r 3 ( t - t ' ) j s ( t ' ) ^ (2.47) where the indices r, s = 1,2 are summed over, that is, the propagator D° can be written as a 2 x 2-matrix with components Dn(t-t') = Dc(t-t') (2.48) D22(t-t') = Dc((t - i/3/2) - (f - id/2)) = -Dc(t - t'Y (2.49) D12{t-t') = Dc{t - ( ? - id/2)) = D-{t-t' + i/3/2) (2.50) D21(t-t') = Dc(t-i(3/2-t') = D+(t-t'-i/3/2) (2.51) where we used the time and anti-time ordering on the contours C\ and C2 and the fact that times on C2 are always "later" with respect to C\. For later applications we explicitly state here that the complex conjugation of propagators, denoted by an asterisk, acts only on free Chapter 2. Time path formalism and thermal Green's functions 25 factors of i and ie, but not on Dirac matrices, Klein-Gordon divisors etc. The full generating functional is now reduced to the form e X Pl " 2 / . o o Loo  d4xd4x'ir(xWrs(x-x')3s(x')\ (2-52) which is equivalent to the path integral Z[j] = J VfaVfa exp | i y d Ax[4>TD-}<l>s + £,•(&) - d(fa) + jrfa]} (2.53) that is, a field theory in Minkowski space with two scalar fields. The contour C has thus led to an effective doubling of the degrees of freedom and a 2 x 2 matrix structure for the propagator. Real-time thermal Green's functions are generated by functional differentiation with respect to 1 6 6 g0(XI,...,*„) = W]Js^(xT)'• • W ^ ) Z [ j u h ] ^ h = °  (2' 54) The general structure of the Feynman rules for the Green's functions follows directly from (2.53) and(2.54). The Feynman diagrams will have the same topological and combinatorial form as at zero temperature, but with the degrees of freedom doubled. There will he two types of vertices, the original "type 1" vertex corresponding to the interactions of the real-time field <f>i and a sign reversed "type 2" vertex corresponding to the shifted field. The diagonal components of the propagator matrix DTa connect vertices of the same type, the off-diagonal components connect type 1 to type 2 vertices. Equation (2.54) implies that only <f>\ fields can appear on external lines of Feynman diagrams, that is, fa plays the role of a thermal ghost field. Thus, an n-point Green's function will be represented by Feynman diagrams with n external type 1 vertices and all possible combinations of both type 1 and type 2 internal vertices, connected by lines representing the DTS propagator components. For spin and gauge fields the above analysis can be repeated virtually unchanged. The only modifications are the appropriate Lorentz indices, Klein-Gordon divisors and thermal distribution functions (Bose or Fermi) for the propagators (2.48) to (2.51). Chapter 2. Time path formalism and thermal Green's functions 26 So far we have worked in coordinate space. However, for practical applications we need the explicit form of the propagator matrix elements in momentum space. The spectral form of the free contour propagator for a general covariant field was given in eqn.'s (2.29) and (2.31), for any contour. Specifying the contour ordering to C\ and Ci we can now take the Fourier transform and obtain the explicit form of the propagator matrix elements DTS in momentum space: iDlW = [»(P0)*"AF(p) + »(V)*AJ.(p) + »?po(pK(p0)]rf„^(p) (2.55) Wlfa) = ^ / a P o ( p K ( p ° ) ^ ( p ) (2-56) Wlfrp) = Ve~^iDl}(p) (2.57) Wll(p) = -W^(p) (2.58) where Ap denotes the scalar Feynman propagator in momentum space ^-n j rd j+ fc (2-59) and hence for the spectral density po Po(p) = ie(p°)[AF(p) - AF(p)) (2.60) According to our previous discussion, the e-regulator in (2.59) has to be kept finite. Let us specify these general formulas to the fields that we will consider later on. For simplicity's sake, and for future applications, we will now simplify our notation; in particular, we will write the spectral function (2.60) as the mass shell -^function to which it would reduce in the e —• 0 limit. Of course, any ill-defined distribution ("square of <5-functions") arising from this notation is understood as properly e-regularized. We have 1. for a free scalar field of mass m with Lagrange density LZ=\<t>(d2-m2)<p Chapter 2. Time path formalism and thermal Green's functions 27 iDu(k) = -iD*22(k) — + 2nnB(k)S(k2 - m 2) k 2 — m 2 + ie iD12(k) = iD*21(k) = +2ne^2nB(k)6(k2 - m 2) 1/2 (2.61) (2.62) where nB(k) = [e^ - l] 2. For a massless gauge field with Lagrange density C = - \ F ^ - |(e?A)2 iD™(k) = [~9lUf + (1 - O f c A ^ j ) iD"(k) with the D rs the scalar propagators for m = 0. 3. For a Dirac fermion with chemical potential p iSu(k) = -iS22(k) = y_ %m + ie ~ MP + m ) [0{k°)nF(xk) - 9(-k°)nF(-xk)] 6{k2 - m 2) (2.63) iSu(k) = -ie~^S2l{k) = -2rr(fl + m)e-^l 2 [e(k°)e Xk/ 2nF(xk) - 6(-k 0)e- x»/ 2nF(-xk)] 6(k 2 - m 2) (2.64) where nF(xk) = [e37*5 + l ] - 1 / 2 , and = /3(Ar°-|-)- At zero chemical potential this reduces to iSu(k) = -iS*22(k) 2 2^  2TT(# + m)nF(k)S(k 2 - m ) (2.65) where now nF(k) = e@\ k°\ -f 1 ^ - m -f ie *512(Ar) = -iS*21(k) = -2ire(k°)(]t + m)eW 0V 2nF(k)6(k2 - m 2) 1/2 (2.66) Chapter 2. Time path formalism and thermal Green's functions 28 An alternative representation of the matrix propagator can be obtained if we define the following thermal distribution functions. Introducing the combination Nv(p°) = e(p°)n+(p°) + e ( - p ° ) n - ( - p ° ) where n^(p°) = e ^ p ° ^ ^ - rj , we define a "thermal angle" 6P by sin(h)0p = ^Nv{p Q) cos(h)0p = [e(p 0) + Tje(-p 0)]^l + vNr,(p 0) (2.67) (2.68) (2.69) where the hyperbolic functions apply to the bosonic case rj = +1, and the trigometric functions to the fermionic case rj = — 1. The momentum space propagators D rs can then be written as iDHpip) = {AF(p) + Vsm(h)2ep[AF(p)-AF(p)]}daP(p) WlW = ^/ 2sin(h)^cos(h)^[A F(p)-A>(P)]4Mp) or in matrix form DaP(p) = Mn dap(p)AF(p) 0 0 -daP(p)AF(p) j Thus, the thermal dependence is now completely contained in the matrix / cos(h)0p -neP^2 sin(h)0p \ \ 7?e-^/2 sin(h)0p cos(h)0p Let us again be more specific and give the explicit form for the three generic fields. 1. For the scalar boson case we have iDab(p) = M+(3,p) where iA(p) 0 ^ 0 -iA*(p) M+(0,p) M+((3,p) = cosh 6P sinh 6P sinh 0P cosh 6P (2.70) (2.71) (2.72) (2.73) (2.74) Chapter 2. Time path formalism and thermal Green's functions 29 cosh0 p sinh 6P — *A(p) = 1 e" -/8|p°l/2 - e-flp°l p2 — m2 + ie 2. For gauge bosons the matrix M is identical to the scalar case, the only difference being the Klein-Gordon divisor 1 - a d for the propagators. 3. For Dirac fermions we have iSab(k) = M.(k,0,p) 1 iS(k) 0 ^ where M_(fe,/3,/i) = ^ 0 -iS*(k) J COS <f>k+n M_(k,0,p) ) COS sin (pk+ti \ e{k°)e fSfl/ 2sm()>k+^ cos cpk+ti ^/exk/2 _f_ e-xk/2 6(k°)e- x^ 4 + 0(-Ar°)e**/4 yV7*/4 _|_ e-xk/4 e(fed) = e ( k ° ) - 6 ( - k ° ) iS{k) =  % ]k — m — £ + ze and for = 0 the thermal functions simplify to 1 cos <pk — sin 0fc = Vl + e~P\ k°\ e-/5|fc°l/2 Chapter 2. Time path formalism and thermal Green's functions 30 The zero-temperature limit becomes obvious in this representation. For 3~ l = 0, p, = 0 the thermal matrices M,, reduce to the unit matrix, and the propagator matrix (2.72) diagonalizes, with the standard zero-temperature propagator and its complex conjugate as the only entries. Thus, the physical and ghost fields decouple completely, in other words, the generating functional has completely factorized, Zi2[j] = zx\h\z2\n\ and we recover standard zero-temperature field theory. Chapter 3 Discontinuities of thermal Green's functions In the previous chapter we discussed the general diagrammatic structure of the pertur-bation expansion for thermal real-time Green's functions. It was found that a Green's function is represented by a sum of Feynman diagrams with physical external vertices, but with all possible type 1 and type 2 combinations for the internal vertices. Needless to say, this leads to a proliferation of diagrams beyond the one-loop level and higher order calculations will be quite cumbersome, if not impractical. However, in some situations, it suffices to evaluate only the imaginary part—the discon-tinuity across the real axis in the complex time or energy plane—of a Green's function. Examples are spectral functions (see e.g. (2.32)), dispersion relations and decay rates, which are related to the Landau damping factor in the interacting propagator. At zero temperature the Cutkosky, or cutting, rules [19] provide a straightforward algo-rithm for the calculation of the imaginary part of a Feynman graph; a particularly illu-minating discussion was given by t'Hooft and Veltman in [18]. The t'Hooft-Veltman pro-cedure was generalized to finite temperature and density by Kobes and SemenofF [29, 30] who derived a set of "finite temperature Cutkosky," or "circling," rules for the compu-tation of the imaginary part of a Feynman graph. They found that the simple notion of cutting a graph does not generalize to finite-temperature Feynman diagrams. However, in the case of thermal real-time Green's functions, for which the external vertices are all of type 1, the Cutkosky rules, while still not admitting the notion of a cut, will simplify con-siderably. In particular, for the computation of the imaginary part it suffices to consider only diagrams with physical internal vertices. 31 Chapter 3. Discontinuities of thermal Green's functions 32 In this chapter, we will give a review of the finite-temperature Cutkosky rules which will be an essential tool in our analysis of thermal decay and scattering rates. As before, we will keep our discussion rather general; the following chapters contain enough examples to illustrate these concepts. 3.1 The largest and smallest time equation To avoid a proliferation of indices and to keep the following discussion transparent we will restrict ourselves for now to the case of a scalar field theory with a generic cou-pling constant — ig, g €R. The introduction of Lorentz indices, group matrices, etc., is straightforward. We consider a typical (amputated) Feynman diagram F with n vertices of both type 1 and type 2. In coordinate space F will be represented as a function of n space-time points, F ( x i , . . . , xn). We associate the coupling constant — ig with each type 1 vertex and +ig with each type 2 vertex. The vertices are then joined by propagators Dra{x{ — Xj), depending on the type of vertex, as discussed before. Furthermore recall that the propa-gators could be decomposed as follows: Dn(x-y) = e(x°-y 0)D +(x-y) + e(y 0-x°)D-(x-y) (3.1) D22(x-y) = 6(x0-y 0)D-(x-y) + e(y 0-x°)D +(x-y) = -D*n(x-y) (3.2) D +(x-y) = -D-(x-yY (3.3) Next we define two functions F> and F< from F(xi,... ,xn) by underlining any number of points in F, F(xi,... ,x±,... ,XJ,... ,xn), and carrying out the following operations: (a) reverse the sign of the coupling constant of an underlined vertex (b) leave all D\2 and D2\ propagators unchanged (c) for the D\\ and D22 propagators Chapter 3. Discontinuities of thermal Green's functions 33 (i) leave Dn(x - y)/D22(x - y) unchanged if neither x nor y are underlined (ii) replace Dn(x - y)/D22{x — y) by 1)22(2: - y)/Dn(x - y) if both x and y are underlined (iii) for F> replace Du(x — y)/D22(x — y) by D+(x — y)/D~(x — y) if x, but not y, is underlined (iv) for F< replace D\\(x — y)/Di2(x — y) by D~(x — y)/D+(x — y) if x, but not y, is underlined (v) for F> replace Dn(x — y)/D22{x — y) by D~(x — y)/D+(x — y) if y, but not x, is underlined (vi) for -F< replace Dn(i — y)/D22(x — y) by D+(x — y)/D~(x — y) if y, but not x, is underlined In a Feynman diagram an underlined vertex is conveniently represented by circling that vertex. Note that the functions F> and F< differ only by the interchange D^ *-> D^. We would like to point out that the notion of underlining/circling a vertex, which was originally developed for vacuum field theory, has an obvious interpretation in the real-time formalism. Underlining a vertex is nothing but replacing a type 1 by a type 2 vertex and vice versa, and the above procedure becomes perhaps clearer if we recall that the functions D^ are essentially the offdiagonal 1-2 propagators. For the functions F> and F< we have now two simple but important relations. For the first one, suppose that the time component of one point Xj is larger than all other time components. The largest time equation then states that that the sum of F> with a:,- underlined and J> with a:,- not underlined vanishes, all other underlinings being the same: F>(xi,... ,xi,..., xn) + F>(xi,..., Xi,..., xn) = 0 (3.4) Likewise, if we assume instead that a:,- has the smallest time component we have the Chapter 3. Discontinuities of thermal Green's functions 34 smallest time equation: F<(a; 1 , . . . ,£i , . . . ,a; r i ) + i;i<(a;i,...,a;i,...,a;n) = 0 (3.5) These two relations are solely due to the sign reversal of the underlined vertex. They only require the existence of a largest or smallest time component, respectively, but make no assumptions about the other components. The largest and smallest time equation lead directly to the following two relations: ••. ,*„) = 0 (3.6) Xi Y^ F<(*u--->*n) = 0 (3.7) Xi where the summation runs over all combinations of underlined and nonunderlined vertices, including the one with none and all vertices underlined. For the proof, suppose that some point Xi has the largest time component. Then each term in the sum with X{ underlined has a corresponding term with Xi not underlined. These terms cancel by the largest time equation and all terms in the sum can be paired this way, so (3.6) follows. The analogous argument for F< proves (3.7). Next we observe that the term with all vertices underlined is just the complex conjugate of the one with none underlined: F(xi, ...,Xn) = F*(X!,...fXn) Thus we rewrite (3.6) and (3.7) as ^[F(x1,...,xn) + F*(xu...,xn)] = - i f> ( x i , . . . , ! „ ) (3.8) ( xi) = - ^ £ * < ( s i , . . . , * « ) (3-9) where the (x^) indicate that the summation on the right hand side runs now over all terms with both underlined and nonunderlined vertices. Denning F with an overall factor of — i Chapter 3. Discontinuities of thermal Green's functions 35 we see that the left hand side is equal to the imaginary part. Thus (3.8) and (3.9) give an explicit algorithm—the finite temperature Cutkosky rules—for calculating the imaginary part of a given Feynman diagram. Needless to say, the extension of these rules to spin and gauge fields presents no problems whatsoever. We simply have to replace the scalar propagators by the corresponding spin and gauge field two-point functions, discussed in the previous chapter. So far we have worked in coordinate space. However, practical calculations require mo-mentum space Cutkosky rules. They are obtained by Fourier transforming the functions F, F> and F<, which replaces the coordinate space propagators by their momentum space counterparts. Then (3.8) and (3.9) hold for the Fourier transforms F(k\,...,kn) and F>/ F<{k\,..., kn), which are determined by the following momentum space circling rules (a) reverse the sign of a circled vertex (b) leave all D\2 and D2\ propagators unchanged (c) for the Dn and D22 propagators between two vertices (i) leave Du(k)/D22(k) unchanged if neither of the two vertices are circled (ii) replace Du(k)/D22(k) by D22(k)/D\\{k) if both vertices are circled (iii) for F> replace Dn(k)/D22(k) by D+(k)/D~(k) if k flows away from an uncir-cled vertex towards a circled one (iv) for F< replace Dn(k)/D22(k) by D~(k)/D+(k) if k flows away from an uncir-cled vertex towards a circled one (v) for F> replace Du(k) jD22(k) by D~(k)/D+(k) if k flows away from a circled vertex towards an uncircled one (vi) for F< replace Du(k)/D22(k) by D+(k)/D~(k) if k flows away from a circled vertex towards an uncircled one. Chapter 3. Discontinuities of thermal Green's functions 36 The Fourier transforms of the coordinate space propagators for various fields were already introduced in the previous chapter. Here we list them again, together with the D±, for future reference. (a) For a free scalar field of mass m iDu(k) = -iD*22(k) = W^TTe  + 2 n n B { k ) 6 { k 2 - m 2 ) ( 3 , 1 0 ) iDl2(k) = iD2l(k) = +2Tre^2nB(k)S(k2 - m2) (3.11) iD±W = 2TT[0(±A;0) + nB(k)]6(k2 - m2) (3.12) where nB(k) = [e^k°\ - l] ~ ^ . (b) For a massless gauge field in a covariant gauge, use the scalar functions with m = 0, associating with each the prefactor (Klein-Gordon divisor) + (i - OfcA^p) (c) For a Dirac fermion with chemical potential p iSn(k) = -iS22(k) = -j - 2TT( jl + m) \e(k°)nF(xk) - 0(-ko)nF(-xk)} 6(k2 - m2)(3.13) iS\2(k) = -ie-^S21(k) = -2TT(# + m)e-^l2 \e(k°)ex^2nF(xk) - 6(-k0)e-x*'2nF(-xkj\ 6(k2 - m2) (3.14) iS^k) = -e±(3k°hSl2(k) = -2TT(# + m) {e(±k°) - [e{k°)nF(xk) + 6(-k°)nF(-xk)] } 6(k2 - m2](3.15) where nF(xk) - [eXk + l ] - 1 / 2 , and xk = 0(k° + p). For p = 0 this reduces to Chapter 3. Discontinuities of thermal Green's functions 37 (3.16) (3.17) (3.18) where nF(k) = [ e ^ ° l + l ] -1/2 In m o m e n t u m space it is easier to see how the finite-temperature circl ing rules reduce to the familiar cutt ing rules at zero temperature and density. W e observe that i n this l i m i t the D± propagators w i l l reduce to forward and backward mass shell ^-functions, that is, between the circled and uncircled vertices, and a number of terms i n the sums (3.8) and (3.9) w i l l disappear by energy conservation. Indeed, the only nonvanishing contr ibut ion for the F> comes from graphs where the circled vertices form connected regions w i t h one, or more, outgoing lines and, at the same t ime, the uncircled vertices form connected regions w i t h one, or more, incoming lines, and vice versa for the i*<. These part icular circlings are then represented by a cut, or boundary l ine, through the graph [18]. Note that at finite temperature the restrictions on the energy flow do no longer apply as the D± contain now both 6(+k 0) and 0(—k°). Consequently a l l terms i n (3.8), (3.9) w i l l contribute and the notion of a cut cannot be generalized. In the following chapters we w i l l see expl ic i t ly how noncuttable graphs contribute to the imaginary part of a finite temperature diagram. they w i l l be proport ional to either 0(+k°) or 6(—k°). T h i s i n t u r n restricts the energy flow Chapter 3. Discontinuities of thermal Green's functions 38 3.2 Finite-temperature Cutkosky rules for thermal Green's functions The Cutkosky rules described in the last section provide a systematic way to find the imaginary part of a Feynman diagram with both type 1 and type 2 vertices. Now, ac-cording to our real-time Feynman rules, a typical (coordinate space) Green's function is represented by the sum of all Feynman graphs with k external type 1 vertices yi,...,yk and p internal points z\,..., zp in all possible combinations of type 1 and type 2 vertices: G{x,y)= ]T F(y1,...,yklzi,...,zp) (3.19) *i<E{l,2} To determine the imaginary part of G we could now apply our Cutkosky rules to each individual term in (3.19) and then sum these contributions. Needless to say, this procedure leads to a considerable proliferation of terms, especially for higher order calculations and a more efficient scheme would be highly desirable. We will now show that the Cutkosky rules simplify considerably if we apply them to the whole sum in (3.19) rather than to each term individually. First we observe that a Feynman diagram with a fixed set of type 2 internal vertices is identical to the same graph with these vertices replaced by type 1 vertices underlined according to the rules for the function F<. Thus we can write (3.19) in the equivalent form G(*,y) = Y!,F<(yi>-->yi»zi>-'->zp) (3-20) where the sum is now over all possible underlinings of the -^vertices, including none and all underlined. The y-vertices remain nonunderlined. Note that all vertices in (3.20) are now of type 1, with no reference to a ghost field. To find the complex conjugate of G we note that the complex conjugate of the function F<(yi, • • •, Vki zi, • • • 12p)> with all vertices nonunderlined except for a fixed number of z-points, is just the function F>([y\,... ,yk, z\,...,zp) with all vertices underlined except Chapter 3. Discontinuities of thermal Green's functions 39 for the originally underlined z's. Thus, the complex conjugate of G is given by G*(y, z) = ^ F^yi,... ,y±,zu... ,zp) (3.21) with the sum again over all possible underlinings of the z's, but now with all y's underlined. With an overall —i factor for G we have for the imaginary part \[G(y,z) + G*(y,z)] = J2 {F<(yi,...,yk,z1,...,zp) + F>(y1,...,yk,zu...,zpj\ (3.22) Next we use the relations (3.6) and (3.7), 52F>(yi,...,yk,zi1...,zp) = 0 with the sum over all possible underlinings of both y and z vertices, to subtract zero on the right hand side of (3.22). Thus we have for the imaginary part the two equivalent representations ^[G(y,z) + G*(y,z)] = - | £ F<(^• • • >yk,*u• • • +7j]C [i?>(ll.'---'M.)2i.---.2:p) - F<(jn,...,yk,zi,...,Zp)] (3.23) = ~\  F>^'--^^\,...,Zp) (Vi),Zi + o J2 [F<(yi> •••,Vk,zu->,zP)-F>(yi,...,yk,zi,... ,zp)] (3.24) The j/-summation in the first sum in (3.23) and (3.24) includes all terms with at least one, but not all y's underlined; in the second sum all y's are underlined and nonunderlined, respectively. The z-summations are over all possible underlinings. To further simplify these relations we will now make a rather general assumption. Recall that the "circled Chapter 3. Discontinuities of thermal Green's functions 40 propagators" D^ and 5 ± are related by iD+(k) = epk°iD-{k) (3.25) iS+(k) = -e^k°-^iS-(k) (3.26) for bosons and fermions, respectively, which is nothing but the KMS boundary condition (2.27) in momentum space. Due to the hermiticity of the Lagrangian there will always be an even number of fermion species in an interaction term, and we an always orientate the fermion lines in a graph so that there are as many fermion lines entering a vertex as there are leaving that vertex. For the chemical potentials associated with these fermion lines we will now assume that the sum of the incoming chemical potentials is equal to the sum of the outgoing ones. Physically, this corresponds to chemical equilibrium, or detailed balance, for the associated particle reactions. With this rather general assumption one can now show that, in the two cases where either all or none of the y-vertices are underlined, the function F<(yi,..., yk, z\, • • • > zp) with some fixed number of z-vertices underlined is equal to F>(yi,...,yk, z\,... ,zp) with the same vertices underlined. For G and G*, as given by (3.20) and (3.21), this implies the two equivalent representations G{y,z) = $^.F<(yi,...,!te,*i,...,aTp) = ^2F>(yl,...,yk,z1,...,zp) (3.27) Zi and G*(y,z) = ^ f < ( j / i , . . . , ] / f c , 2 P ) = E F>(a»---.M^i ' --- '- 2 ; p) ( 3 - 2 8 ) Zi Using these two relations in (3.23) and (3.24) we see that the second terms cancel and we are left with our final expressions for the imaginary part of G 1 1 ImG(y,z) =-[G(y,z) + G*(y,z)} = -- £ F^yx,... ,yk,zu ... ,zp) Chapter 3. Discontinuities of thermal Green's functions 41 (3.29) = ~2 E *>(0i.---iMfe»*i>---i*j>) (3.30) with the z-sum over all possible underlinings (circlings) of the internal vertices, but the y-sum only over terms with both underlined and nonunderlined vertices. The analogous result holds, of course, for the Fourier transform in momentum space. This result is the desired simplification of the finite-temperature Cutkosky rules. To calculate the imaginary part of a Green's function with physical type 1 vertices it suffices to apply the circling rules to its Feynman diagrams with all vertices of type 1. No reference to a ghost field is necessary. Furthermore, the Cutkosky rules for the imaginary part of G are symmetric with respect to the functions F> and F<, that is, to the "circled" propagators D+ and D~: for example, in momentum space we are free to choose either D+(k) or D~{k) for k flowing towards a circled vertex, as long as we choose D~(k) and D+(k), respectively, for k flowing away from a circled vertex. We note that this simplification of the Cutkosky rules occurs only for real-time Green's functions with their external type 1 vertices. If, for some reason, we should wish to compute the imaginary part of a Green's function with some type 2 external vertices, then we would have to consider all Feynman diagrams with all combinations of type 1 and type 2 vertices. Finally, we emphasize that, although the final result (3.29), (3.30) makes no reference to type 2 vertices, one cannot dispense with the ghost field entirely. Its existence is implicit in the circling rules, and was used in the derivation of the above result. Furthermore, we will see in the following chapters that type 2 vertices are explicitly contained in certain types of noncuttable graphs. Chapter 4 Real-time self-energies and thermal decay rates One of the simplest and, at the same time, most important Green's functions is the two-point function, or more precisely, the self-energy. The real part contains the excitation spectrum of the system, while the imaginary part is related to its dissipative properties and can also be used to determine spectral functions and as input for dispersion relations. In this chapter, we are going to give a brief review of the general structure of the thermal self-energy in the real-time formalism. We will also discuss the (nontrivial) connection between the imaginary part and the thermal decay rates, on which our work is based. As usual we first consider a scalar field theory and then extend the results to spin and gauge fields. 4,1 Real-time self-energies The spectral representation for the elements of the free matrix propagator Dab has al-ready been discussed in chapter 2. For the interacting scalar propagator Vab the spectral representation will be of the form [44] where p is the fully interacting thermal spectral function. Now recall that the free prop-agator matrix could be represented as (4.1) iDab(p) = M+(0,p) tA(p) V 0 -iA*(p) ; 0 M+(0,P) (4.2) 42 Chapter 4. Real-time self-energies and thermal decay rates 43 1*1 Figure 4.1: Dyson-Schwinger equation where we defined for the scalar boson case M+(/?,p) = cosh 0P sinh 0P sinhflp cosh0p j cosh dp = sinh^ p = *A(p) = y/l - e-/3|P°l e~ -j9|p»l/2 y/l - C - / V I p2 — m2 + ie Together with (4.1) this implies that the full scalar propagator can be written as iVab{p) = M+(P,P) f iV(p) 0 ^ M+OM (4.3) \ 0 -iV\p) ) with V some complex function/distribution. Next we consider the Dyson-Schwinger equa-tion, generalized to matrix propagators and shown in terms of Feynman diagrams in Figure 4.1. This defines the self-energy matrix TIa6 as the sum of all one-particle irreducible (1PI) Feynman diagrams, with one external vertex of type a and the other of type b, a,b = 1,2. The representation (4.1) Chapter 4. Real-time self-energies and thermal decay rates 44 and (4.3) for Dab and Vab, respectively, leads to the the following form for the self-energy m a t r i x -mab(p) = (M+)-\p,P) T h i s i n turn gives for iV(p) i n (4.3) ' -iU(p) 0 N ^ o *n*o>), (M+)-l(0,p) (4.4) W ^ = p » - m » - I I ( p ) + fc ^ and for the self-energy matr ix one deduces immediately from (4.4) nn(p) = - n ; 2 ( P ) n12(p) = n2i(p) = -itanh20plmllii(p) (4.6) as well as ReE(p) = ReUu(p) Jmn(p) = e(p0)tanh(Pp°/2)ImIIu(p) (4.7) A few comments on these results are i n order. F irs t we note that iD is the propagator for a thermal quasiparticle excitat ion, but the ful l propagator m a t r i x is more complicated. For example, the propagator for a for a physical excitat ion, iT>u, is given by iVu(p) = iV(P) + nB(p){W(p) - W*(p)} We also emphasize the dist inct ion between H n and the self-energy function II. T i n is s imply the 1PI graph w i t h external type 1 vertices, whereas II turns out to be the analytic continuation of the imaginary t ime M a t s u b a r a self-energy (which is defined on a discrete set of imaginary frequencies i n the complex energy plane) [34]. T h e real parts are ident ical , but the imaginary parts differ at finite temperature according to equation (4.7). T h i s dist inction w i l l be important i n the context of thermal decay rates. F i n a l l y , we observe that T i n completely determines the whole m a t r i x II0(,: a l l other compo-nents can be obtained by relation (4.6). However, this does not i m p l y that ghost vertices Chapter 4. Real-time self-energies and thermal decay rates 45 are unnecessary. The full matrix structure is required for the iteration of the Dyson-Schwinger equation; including only the 1-1 components leads immediately to pathologies (ill-defined products of distributions) (cf. [10]). The extension to fermions presents no problems. The same derivation goes through virtually unchanged and the full fermion propagator can be written as / iSab(k) = M-(k,(3,p) iS(k) \ 0 -iS*(k) M_(*,/?,/i) where now M_(fc,/?,/0 cos<j>k+ti -e(k°)e /3^2sm(pk+ti e{ka)ePtll2 sin cos <f)k+fl 0(fc°W 4 + 6(-k°)e-x*t* COS©fc4.„ = — — sinfa+n = cos<t>k+lt(±xk ^xk) e(k°) = < iS(k) = +1 if k° > 0 -1 if A;0 < 0 i ]k — m — E + it As in the scalar boson case the self-energy matrix is given by -iXab(k) = (M_)-\k,(3,p) from which one deduces the relations 0 iE* (M_)-\k,(3,p) I as well as E22(p) = -E u (p) Si2(p) = -e-^E 2 1(p) = ie(p0)e-^/2tan2^+ M/mEn(p) ReE = ReEn ImY, = e(fc0)coth(/?(fc0 + /i)/2)/mEn (4.8) (4.9) (4.10) Chapter 4. Real-time self-energies and thermal decay rates 46 For the gauge boson case the only difference to the scalar case is, as usual, the prefactor 1 - a d for the propagators. 4.2 Decay rates at finite temperature We will now turn to the connection between thermal decay rates and the imaginary part of the self-energies. First let us recall their relation at zero temperature. The conventional method for the perturbative calculation of the decay rate T of a particle in vacuum field theory can be summarized as follows [6]. Consider a particle with mass m and four-momentum p = decaying into n different particles with four-momenta fcj = (u>j,ki), i = 1,... ,n. The decay rate T(w) is then given by The transition amplitude M is calculated to any order in perturbation theory from the relevant Feynman diagrams and rules. The above formula is valid for bosons and Dirac fermions, provided we normalize the fermion spinors to 2m. Alternatively we can use the imaginary part imll(ai) of the particle self-energy TI(u;), that is, the discontinuity of TI across the real axis in the complex energy plane Disc II(w) = limll(u; + ie) - II(a> - ic) = 2iImU(u) Im II and T are related by the optical theorem. We have ImU(u) = -<j -r(w) (4.11) and the imaginary part of the self-energy graph can be determined with the help of the Cutkosky (cutting) rules discussed in the last chapter. Chapter 4. Real-time self-energies and thermal decay rates 47 At zero temperature, these two approaches are completely equivalent. However, at finite temperature this connection is more involved. It was pointed out by Weldon [48] that (4.11) has to be modified at finite temperature. Using the euclidean (imaginary time) Matsubara formalism he calculated to lowest order the imaginary part of the self energy for a boson and a fermion undergoing two-body decay in a heat bath. The results can be summarized as follows. For a boson field <fr at temperature T with analytically continued euclidean self-energy n(u>), equation (4.11) is replaced by M ( « ) = -w • r(w) r(w) = Td(u)-Ti(u>) (4.12) Yd denotes the sum of the thermally suppressed tree level decay and scattering rates <fr —• 0102, $0i —• 02 > etc., that decrease the number of $-bosons, and i \ - is the sum of all the inverse rates like cf>\4>2 —> $ etc., that increase the number of 3>'s. This agrees with physical intuition.The 3>-boson decays, but the decay products 0i and 02 in the heatbath have a thermal probability to recombine into <&, and likewise for the scattering processes. For a fermion ip with self-energy E, however, this result has to be modified. First we contract the matrix S with the free Dirac spinors u and u (normalized to 2m) to form the scalar n(w) = Tl(w)E(w)u(u;) (4.13) and obtain Jmn(w) = -w • r(w) T = Td + Ti (4.14) This shows that Y is not simply the naive "net decay rate". Rather (4.12) and (4.14) have to be interpreted as the rate which a nonequilibrium distribution /(w,i) of unstable Chapter 4. Real-time self-energies and thermal decay rates 48 particles approaches its equilibrium value /o(w). More specifically, consider the master equation for f{w,t) -/(u;,0 = -Td(u)f(u,t) + r,-(«) • [1 + T7/(w,i)] (4.15) where n — 1 for bosons and n = — 1 for fermions as a consequence of the Pauli exclusion principle. The solution to (4.16) is given by f(u,t) = fo(u) + c(Lo)e-TMi (4.16) where r » = Td(w) - v • r.(^) f(„\ - r ' ( u ; )  / O ( W ) " T H so that f(u>,t —> oo) —• fo(oj), as stated. We emphasize that T, and not the partial rates T,- and r<£, represents the physically measurable decay rate. Even for t —> oo the initial distribution will not go to zero but approach thermal equilibrium with its environment. Strictly speaking, the whole concept of asymptotic states and S-matrix elements becomes meaningless at finite temperature. Instead the appropriate notion is that of a quasiparticle excitation in a plasma that is "Landau damping", i.e. thermalizing with its surroundings. As an aside, we note that this analysis can be generalized to other Green's functions as well. For example, the discontinuity in the four-point function can be interpreted as the thermal damping rate with two particles in the initial state. However, in this case equation (4.15) would have to be generalized; for the time being we will restrict ourselves to two-point functions. Kobes and Semenoff [29] confirmed these results, up to the one-loop level, in the real-time formalism, using the finite-temperature Cutkosky rules discussed in the previous chapter Chapter 4. Real-time self-energies and thermal decay rates 49 to calculate the imaginary part of the self-energy directly for real energies. To obtain the correct result it is important to remember the distinction between ImH and Jmllii, since it is Im II which determines the thermal decay rates. This concludes our general introduction to real-time finite-temperature field theory. In the following we will now apply this formalism to explicitly analyze the structure of thermal decay rates, up to the one-loop level. Chapter 5 Relativistic analysis of neutron (3 decay at finite density We will now apply the real-time formalism to a nontrivial physical system and analyze neutron Q decay in a zero-temperature, finite-density environment. This process is of considerable interest in astrophysics, since neutrons embedded in a cold, dense plasma are realized in nature in the core of neutron stars. Of course, this problem is not new: the first discussion of neutron stars appeared more than 50 years ago, and degenerate Fermi systems and their astrophysical applications are now standard textbook material. However, the standard approach to neutron decay uses a nonrelativistic approximation for the nucleons and finite density effects are taken into account as phase space constraints on the momenta of the decay products [45, 14]. Here we are going to present a completely relativistic analysis of neutron decay at finite density, at energy scales up to 100 MeV. In the zero density limit—the free neutron— we derive an analytic formula for the decay rate T, whose numerical value, 1/939 sec, is in good agreement with the current experimental value Texp = 1/925 ± 11 sec. To our knowledge, no such relativistic treatment exists in the literature. The free rate is then generalized to Pauli-suppressed /3-decay, that is, neutron decay in a degenerate electron gas, and to neutron decay in a dense electron-proton plasma, a process known in astrophysics as URCA reaction. No nonrelativistic reduction is made, so our results are applicable to muon and quark /?-decay as well, that is, processes with relativistic phase space. At the same time, this discussion will serve as an instructive and nontrivial example for practical calculations in the real-time formalism. The generalized Cutkosky rules, applied 50 Chapter 5. Relativistic analysis of neutron 0 decay at finite density 51 to the self energy, t u r n out to be a very straightforward and elegant technique for the calculation of the physical ly relevant decay and inverse decay rates. O u r discussion is based on the V - A theory of 0-decay, which , at our energy scales (< 100 M e V ) approximates the standard electroweak model very well . Furthermore we w i l l assume that temperature effects can be neglected and take the 0~x —• 0 l i m i t for the fermion propagators. T h i s w i l l simplify the general structure of the resulting phase space integrals, m a k i n g them easier to interpret and also allowing complete analyt ic evaluation. Physica l ly this assumption may be justified as a good approximat ion to the conditions in the interior of a neutron star. In part icular we w i l l consider these three cases: (i) s tandard free neutron decay (ii) neutron decay i n a dense electron gas (iii) neutron decay i n a dense electron-proton plasma. T h i s discussion is confined to the tree level; radiative corrections are not inc luded. T h e y are the subject of the next chapter. 5.1 Preliminaries Firs t let us establish some basic definitions and notat ion. A t the energy scale we w i l l con-sider (< 100 M e V ) , the V - A model is a good approximat ion to the standard electroweak theory. T h e V - A Lagrangian for neutron /3-decay is given by C = - A 7 5)V>n?e7"( l - 75)VV, A ~ 1.25 A s an aside, we should mention that , for nucleon /? decay i n a dense environment, Gj and A w i l l , i n general, depend on the ambient nucleon density. However, i n the fol lowing, we w i l l neglect this density dependence of the coupling parameters. T h e V - A Lagrangian leads directly to the self energy diagram E n shown i n Figure 5.1. E n is given by Chapter 5. Relativistic analysis of neutron 0 decay at finite density 52 P k+P-q Figure 5.1: Neutron self-energy £n(<z) Figure 5.2: Imaginary part of the self-energy Sn Ell(<?) = IdiPdAkGliS{p)G"h Tr^iS^GiiSik)] (5.1) where we abbreviated the 7-matrix products by G^ = 7^(1 — A7 5) for the hadron part = 7M(1 — 7 5 ) for the lepton part and omitted the 11-subscript on the propagators, S(p) for the proton, S(k) for the elec-tron, and S(r), r = k + p — q, for the neutrino. To determine the imaginary part of (5.1) we use our circling rules, discussed in chapter 3 and shown in Figure 5.2, with the result Chapter 5. Relativistic analysis of neutron j3 decay at finite density 53 -2/mSn = -^^Jd4pd4kGliS-(p)G"hTr[G^iS+(r)GliS-(k)} where the circled propagators 5 ± are given by (3.15). For JmS we use (4.11) which in our case (/3 —»• oo) reduces to ImE(g) = e( 9 0)/mS u( 9) = JmE u ( 9 ) since a physical neutron is on its mass shell: q° = u>q = y/q2 + m£. We are now going to evaluate (5.2) for three different cases. 5.2 Free neutron decay at zero density We will begin our discussion by calculating the free neutron decay rate analytically with-out the ususal nonrelativistic approximation for the nucleons. These results will also be useful for the finite density case. With T = = 0, p, = 0, the thermal circled propagator (3.15) reduces to iS±(k) = 2x(>fe + m)6(±k°)6(k2 - m2) Denoting the masses for neutron, proton and electron by respectively mn,mp and m e, (5.2) turns into -2 /mE„(?) = -^yjd4kd4pGl(Jp + mp)G'1hTrGlfi(/c+ f>- k)Glu{^+me)\ x{e(-p0)9(-k°)e(k0 +p°- q°)6(p2 - m2p)S(k2 - m2e)S((P + * - <?)2) + e(p°)e(k0)9(q0 -p°- k°)S(p2 - m2p)6(k2 - m2e)S((k + p- q) 2)} (5.3) The 6— and 6—functions contain the kinematical constraints —energy thresholds and four-momentum conservation— of the the reactions described by ZmE. They simplify to e(±p0)9(±k0)e(T(P0 + k°- q°))6(p2 - m2p)S(k2 - m2) Chapter 5. Relativistic analysis of neutron /? decay at Unite density 54 = T uP)8(k° T u>k)9(T(±t»P ± - 9 ° ) ) For a physical neutron q° = u>q > 0, hence wP - uk - t j 0 ) ) = 9(-UJp - uk — u>q) = 0 and the first term in the integral, corre-sponding to the first circled diagram, vanishes. It is convenient to transform the electron four-momentum k to / := k — q. Then (5.3) turns into xTr[G'll(/+ t>)G l„U+ A + me)} X0(-1°-LJP)6((1 + P)2) and k° = u>k, p° = up, q° = u>q is understood. The spin-averaged decay rate is obtained by contracting Jm£ = JmEn with the free Dirac neutron spinors u(q,s),u(q,s) and averaging over the spins s = With the convention ^ u(q,s)Qu(q,s)p = (^ + mn)ap this gives us «=±l/2 T(uq) = — - • i Y2 uImY,nu W « 1 s=±l/2 = -32^(L)5 /rf3/d3p Tr[Gud>+m^G^+m^ Xtr[GlV+ j>)GlU+ A + me)] x6(-l°-up)6((l + p)2) (5.4) where A;0 = uk, p° - u>p and /° = uk — uq. The evaluation of the 7-matrix traces is tedious but straightforward. Using the standard trace theorems and 7-matrix identities found e.g. in [6] we find (i) for the lepton trace Chapter 5. Relativistic analysis of neutron 6 decay at finite density 55 = 8((l + p)nkl/ + (l + p)„kti-gllv(l + p)-k + » W r(f + PfkT) (ii) for the hadron trace Tr[GU^ + mp)G^A + mn)] = 4((1 + A 2 ) ( ? V + - <T (p • 9 - \-7^rnpmn) -1iWar paqT) Contract ing over the free spinor indices yields Tr\G\ .. .}Tr[G£ ...} = 2 6 {(1 - A) 2 [(/ + p) • p][(l + q).q] + (l + A)2[(Z + p) • q][(l + q)-p] - ( 1 - A 2 ) (/ + p) • (I + q)mnmp} which gives for our decay rate r K ) = _ _ 2 _ / 4^ 0 ( _ / o _ p o M / + p ) 2 ) qj I6ir5uq J u>k(l)up X{(l + \m+p).q}[(l + q)-p) + (l-X)2[(l+p)-p][(l + q)-q] - (1 - A 2 ) (/ +p)-{l + q)mnmp} (5.5) w i t h / = (wk - q°,k - q), p = (up,p) and A = 1.25. T h e ^-function is best evaluated i n a different coordinate system. Replac ing the proton four-momentum p byz = I + p and choosing for the the neutron its rest frame w i t h ? = 0, we introduce spherical coordinates: q = K,<f) = (m„, 0 , 0 , 0 ) / = (w f c - uq,k- q)-(ui- m n , 0 , 0 , | / ] ) Chapter 5. Relativistlc analysis of neutron f3 decay at Unite density 56 z = (z°,\z\ sin # cos q>,\z\ sin 0 sin <f>,\z\ cos 6) = (ui-rWp(z)-mn,...,\z\cos9) with the obvious definitions w/ = + m \ = u>k, up{z) = \J\z — l\2 + m 2. In these coordinates the <5-function reduces to 6((l + pf) = ^ 2 ) = ^° 2 ( | 4 c o s 0 ) - | * f ) = S(f(\z\,cos9)) The zeros of f(\z\,cos9) with respect to |z|are found by solving \z\ = ±(1° + LJP(\Z\,COS6)) with the result M f-lo2 + m l l*l± 2(^/° + / cos 9) m2p-I2 2(^1° + I cos 9) The Jacobian of f(\z\,cos9) is given by / ( | 2 l ± ) = ^ P U and the -^function in the integrand reduces to 9(-(l° + cop(\z\±,cos9))) = 9(T\z\±) Thus the product of 8- and ^ -functions simplifies to 2|2|_ |/o + |Zl cos &| 1/f - l°2 + m2p with \z\ — l° + \l\cos9 Chapter 5. Relativistic analysis of neutron (3 decay at finite density 57 From the ^ -function we derive immediately an upper limit on the |/j-integration: i2p - (m - mn)2 + |/f f 2 ' 2(/° + \T\cos6) > 0 2mnue -- (mj[ + m2 - m2) 2(/° + |/fcos0) and because of the (3-decay threshold /° + |/j cos 0 < w/ - mn + \l\ < u>i + ujv - mn < 0 hence 2rnnu>i < m2 + m2 - ra2 or ^ ^ ^ /(m2 - mj + m2)2 - 4m2"ml = ^ 2mu Thus the integral (5.5) can be written as r , , _ G) r dSljdlW2 dnzd\z\\z\2 i  i [ m n ) - IQ^mJ ut u„(z) |/'(|£|_| x K\Amax - %K\A - IC) x{(l + X)2(z.q)[(l + q)-(z-l)] + {1_-X)2[z.(z-l)][(l + q).q} - (1 - X2)z • (I + q)mnmp} which, after some straightforward integrations and algebra, turns into f (1 + A ) 2 . 2 l 2 , 1 x < v 0 mn{ml - I2) 2 n v p \ p + \i\coSey + (1 + X)2 (1 - X2)mnmp - (m'l - m 2e - m lp) L 2 \ ™ , / i \ \ 2 , 2 v n 1 "\ (!° + |<lcos9)3 Chapter 5. Relativistic analysis of neutron 0 decay at finite density 58 where I2 = l°2 - | / j 2 = m2n - ml - 2m„w/, 1° = u)\ — mn. T h e ^-integration is simple and yields G2 r\i\ma* d\l\\I\2 r ( m „ ) = 16TT3 r Jo X {<-I2)2 ( 1 + A ) 2 2 ; 2 , 3 / ° 2 + |/| — mn(mp - I + (1 - \2)mnmv (1 + A )2 (ml -m 2e- ml) + [ ( l - A 2 K + ( l - A ) 2 o ; ( ] | } 21° (P)2 (5.6) T h e last integration is somewhat more involved. F irs t we change the integration variable to x = I2. Rearranging the integrand by powers of x we write the decay rate as r ( m „ ) G2 647T3? L— / dx Y, Fix'Jx2 - 2{ml + ml)x + (m* - mlf (5.7) ! = -3 where the coefficients F{ are given by F3 Fx F0 F-i F-2 F-3 -7/4 [85m 2 + 2 9 ( m 2 + m 2) - 1 8 m n m p ] / 3 2 [ - 2 m 4 + 2 7 ( m 2 - m 2 ) 2 - 8 5 m 2 ( m 2 + m 2 ) + 18(2m 2 + m 2 - m 2 ) m n m p ] / 3 2 { - 2 7 m ^ + 8 3 m 4 ( m 2 + m 2 ) - 18m3pmn[m2p + 2 ( m 2 - m 2 ) ]}/32 [ - 2 7 m ^ ( m 2 + m 2 ) - 8 1 m 4 ( m 2 - m 2 ) 2 + 1 8 m ^ m „ ( m 2 - m 2 ] / 3 2 54K(m2 - m 2 ) 2 ] / 3 2 T h e integrals i n (5.7) can be worked out analyt ical ly ( see e.g. [15] ) and we obta in the decay rate for the free neutron i n fully analytic form: Chapter 5. Relativistic analysis of neutron (3 decay at finite density 59 r(-n) = g -^r[(ro 2 + ml)F_x + F_2] ro: x log mnme + ro2 + ro2 - ro2 - yjm* - 2ro2(ro2 + m2) + (m2 - ro2)2 2 ^ ron(ro2 - ro2)2' ro2 - ro2 x log ( r o 2 - mDyJm* - 2m2(m2 + ro2) + (m2 - ro2)2 + 2ro„me(ron - roe)2 2m2ronroe Fx 1 3 "5" + 7 K + o ( m n + m e)] y'ro4 - 2ro2(ro2 + m2) + (ro2 - ro2)2 [(ro2 - (ro 2 + ro2)] (F0 + (m2 + ro2)^ + ^ [5(ro 2 + ro2)2 - (ro2 - ro2)2]) + 2 F-! + F-, mi 1 I F_3 mi ml + mj 7.? — ro,?^2 •P \ ™p (w 2 - ro2)2 J ^ Y m p - 2ro2(ro2 + ro2) + (ro 2 - ro2)2 | 2ro3 With the standard values ro„ = 939.57 MeV, rop = 938.28 MeV, roe = 0.511 MeV and Gf — 1.166 • 10 - 1 1 MeV - 2 we obtain for the numerical value which is in good agreement with the current experimental value Texp = 1/(925 ± 11) sec (for a better theoretical value we would have to include radiative corrections). By comparison, the nonrelativistic approximation essentially leads to the same result: the Sargent rule Tnr a (mn — rop)5 (see e.g. [43]) gives a value of 1/934 sec, that is, a difference of about 0.5 %. (5.8) Chapter 5. Relativistic analysis of neutron (3 decay at finite density 60 5.3 Neutron decay in a dense electron gas The previous calculation can be easily generalized to the case of a neutron embedded in a cold (T = 0), dense electron gas. This electron plasma can be characterized by an electron chemical potential p > 0. This can be seen as follows. Technically, chemical potentials are associated only with conserved charges, in this case electric charge Q, lepton number L and baryon number B, and we add a term to the Hamiltonian 311 in the functional integral (cf. Chapter 2). In terms of particle number operators NE, NU, NP, NN, for electron, neutrino, proton and neutron, this can be rewritten as PQQ-r PLL + pBB = (-PQ +PL)Ne + (VQ + MB)-^P + VLNU + HBNU = peNe+ HPNP + HvNv + VnNn and the particle chemical potentials are simply linear combinations of PQIPL,^B- Physi-cally we are considering a gas of electrons in a T = 0 Fermi-Dirac distribution with Fermi energy p, with no baryons, neutrinos or antiparticles present. Hence we set B = 0, pi, = 0 and it suffices to introduce only the electron chemical potential p = \PQ\. In this case the circled diagrams in Figure 5.2 have to be evaluated with the thermal circled propagator (3.15) for the the electron Une while the proton and neutrino lines remain unchanged. In the limit T = 3~l - 0, (3.15) simplifies to iS+(k) = 2n(/k + me)—6(k° -p)6(k° -uk) (5.9) iS-(k) = 2^{jk^me)^-{8{k° + uk)-e(p-k°)8(k 0 -W f c)) (5.10) Chapter 5. Relativistic analysis of neutron 8 decay at finite density 61 and equation (5.2) for JmS turns into xTriG'^H JP- A)GU/t + ™e)]8((k + p- qf) x {9{k° + p° - q°)6(p° + up)8(k° + uk) _ o(k° + p°- q°)9(p - k°)8(p° + up)6{k° - uk) + e{q° - k° - p°)0(k° - p)8(p° - up)8(k° - uk)} (5.11) We can now easily identify and evaluate the terms in (5.11). The first term vanishes as before. The second term represents the decay of a background electron into neutron, antiproton and neutrino. This process, however, has a threshold of almost 1900 MeV and can be discarded since other reactions become dominant at much lower energies. It should be pointed out, however, that the relative minus sign is cancelled by a negative integrand (the fermion trace) which turns out to be negative in this region of phase space (see also [48] on this point). Thus electron decay actually contributes positively to the total decay rate, as discussed in chapter 3. The remaining third term is just Pauli-suppressed beta decay, that is neutron decay with the constraint uik > p for the energy of the produced electron, by the Pauli exclusion principle. Thus (5.11) is nothing but the free decay rate from section 5.2 with an additional suppression factor 9(uk — p), which sets the lower limit on the electron momentum |/]T O l n equal to the Fermi momentum pp'• \l\min = PF = \/M2 — m2.Hence the suppressed decay rate is now given by 1 0 7 T J JPF 647r3m3 7m2 .f-^ 3 v The analytic formula for T{mn,pp) is not very illuminating and we will omit it. It is clear, Chapter 5. Relativistic analysis of neutron (3 decay at finite density 62 however, that T goes to zero as pp approaches \l\max = 1.183... MeV, and the neutron becomes stable. The exact pir-dependence of T is shown in Figure 5.3. So far the only effect of the electron plasma has been the suppression of the decay by the Pauli principle. Inverse reactions like the recombination of the decay protons with the electrons is a second order effect corresponding to the interaction with an external source which is not included at tree level. The inverse rates discussed in chapter 3 will appear when we add protons to the background. 5.4 Neutron decay in a proton-electron plasma We will now discuss the case of a neutron in a cold, dense electron-proton plasma. More precisely, we are considering a gas of electrons and protons with T = 0 Fermi-Dirac distri-butions with Fermi momenta pe, fip > 0; the plasma contains no neutrinos or antiparticles. As before the chemical potentials for the particle species can be related to the conserved charges of the theory. We had (cf. section 5.3) PQQ + PLL + HBB = (-PQ +PL)NS + OQ + PB)NP + PLNV + PE)Nn = pENE + pPNP + pUNV + pNNN Since no neutrinos are present we set pu = 0 and choose pe, Pp as the free parameters for the plasma. The neutron chemical potential does not appear since there are no neutron propagators. Again we evaluate our circled diagrams, this time with the finite density propagators (5.9) and (5.10) for both electron and proton line. We obtain, in obvious notation = -2^ /S5G^+^Gi x Tr[G 1^  ip- A)GU/k + me)\ • S((k + p- qf) x {6(k° + p°- q°)6(k° + uk)6(p° + up) _ 0(k° +p°- q°)0(pe - k°)6(k° - uk)8(p° + Up) Chapter 5. Relativistic analysis of neutron 0 decay at finite density 0.72r 0.64 h 0.56h 0.48 % 0.40 CM 0.32 h 0.24 h O.I6h 0.08 h "0 0.4 0.8 1.2 1.6 2.0 PF(MeV) Figure 5.3: Neutron decay rate T in a dense electron gas with Fermi momentum Chapter 5. Relativistic analysis of neutron 0 decay at finite density 64 _ 6(k° +p°- - p°)6(k° + uk)S(p° - up) + o(k° + p°- qQ)e{pe - k°)e(pp - P°)6(k° - uk)S(P° - up) + e(q° -k°- p°)6(k° - pe)9(p° - pp)6(k° - uk)8{p° - Wp)} (5.12) As before we can identify the individual terms in (5.12) from their kinematical constraints. The first term vanishes as usual because of the -^function constraint. The second and third term represent "thermal" electron and proton decay, that is, e~ np~ v and p —> ne+ v. Electron decay can be neglected, due to its threshold of almost 1900 MeV (see section 5.3). Proton decay, however, has a threshold of about 1.8 MeV, corresponding to a proton Fermi momentum of about 58 MeV, and has to be included at higher densities. Again we point out that the relative minus sign for these two terms is cancelled by the negative fermion trace for this region of phase space. The last two contributions, known in astrophysics as UR.CA reactions, are the most impor-tant ones. The fourth term represents inverse /3-decay e~ p -+ nv, that is, recombination of protons and electrons in the background medium, and the last term is simply neutron /?-decay with two suppression factors, 6(cjk—pe) and 6(ojp—pp), due to the the degeneracy of both the electron and proton background. These four reactions are the only ones possible in this order of perturbation theory. The plasma contains only electrons, protons and the neutron and the only possible transitions are the ones with initial states n, e, p and ep. Thus (5.12) represents precisely the situation discussed in chapter 3: both decay and inverse decay rates, Td and I\', are contained in the imaginary part of the self energy and contribute to the total decay rate T. We note that, since we are dealing with fermions, both and Ti contribute positively to the total rate, as discussed before. Of course, both rates are mutually exclusive due to their kinematical constraints: the decay rate Td vanishes by Pauli suppression as the electron Fermi momentum approaches \l\max * 1.183 Chapter 5. Relativistic analysis of neutron j3 decay at finite density 65 MeV, which is the threshold for the inverse reactions I\. We will now complete our analysis and compute the partial rates Yj, and To re-duce the number of parameters we will assume charge neutrality for the electron-proton background, which, by integration over the number densities, implies equality of the pro-ton and electron Fermi momenta: PF(P) = PF(E) = PF- Thus the chemical potentials pe and pp—and hence the decay rates— depend only on a single Fermi momentum pp: /^ e.p = \Jpp + metP- For simplicity's sake we will restrict ourselves to pp < 50 MeV, that is, we will omit proton decay in rt-. First consider the "proper" decay rate for n —> e~ pv: X 6(u>q -uk- up)0(uk - Pk)6{up ~ MP)6((k +p- q)2) Since the only difference to the previous cases is the additional phase space constraint 6(up — pp) we can repeat our previous derivations unchanged. In terms of the /- and z-variables the -^function is given by 9(up - pp) = 6(-l° - \z\_ - pp) where \A- 2(/° + |/| cos0) which reduces to the condition cos.<('° + ^ 2 + " 1 2 - ^ - -2|/l(l"+p,) Thus T^  is now given by Umn,PF) = ^-3jd\l\&(ml-l2)2 ( 1 + A ) 2 mn(m2p - I2) ( 6 v J\(i°-\T\T (l° + \l\coSemax)\ Chapter 5. Relativistic analysis of neutron j3 decay at finite density 66 + | (^1 - X2)mnmp - ( 1 + 2 A ) 2 ( m 2 - m 2 - ro2) * ( a ° - Ul)2 ~ (i° + \i\ cosemaxy) + [(1-A 2)ro p + (1-A)2u,,] l°-\l\ l° + \l\ cos emaxJm (5.13) where the limits on the |/j-integration are given by J(ml - ro2 + ro2)2 - 4m2 ro2 " £ 1 , 1 s ^ 2 ^ s i l -and the upper limit cos 6max for the -^integration is derived from the two conditions -1 < cos 61 < +1 and - -2\J\(l° + H) The actual choice of cos 6max depends on the relative values of pp and |/j and can be expressed as a -^function for |/|. The (/(-integration can be done analytically in either case, but the resulting expression is extremely cumbersome and not very instructive. Hence (5.13) is best evaluated numerically; the result is shown in Figure 5.4 and shows clearly the additional suppression of Yd due to the proton degeneracy. For the inverse decay rate Yi we proceed analogously. We have 8w 9(27r) 5 J Aukup x 6{uk + Up - uq)6(pe - uk)6(pp - up)6((k + p- q f ) Transforming to /- and z-variables we obtain for the -^functions 6(1° + up)6(ne - ui)9(pp - Up)S(z2) = 6(\z\+)e(»e - u ^ p - up(\z\+))8(\z\ - |2|+) Chapter 5. Relativistic analysis of neutron j3 decay at finite density 67 0.72 r 0 0.4 0.8 1.2 1.6 2.0 PF (MeV) Figure 5.4: Neutron decay rate TD in a dense electron-proton plasma with Fermi momentum PF- The dashed line is the dense-electron-gas rate from Fig. 5.3, shown for reference. Chapter 5. Relativistic analysis of neutron (3 decay at finite density 68 where 1*1+ = I2 2 ( - / ° - f |/j cos 6) (cf. section 5.2). Thus the previous upper integration boundaries t u r n into lower l imits and the integrand is evaluated wi th \z\+ instead of |z|_. T h e result is G2, r d\l\ \l\ Ti(mn,pF) 167T3 (1 + A ) 2 6 mn{m2 - I2) 0.7 + 2 ( C1 - A 2)m nm p (_/o + |j])3 (-/o + |/]cosdm,-B)3y (1 + A ) 2 , 2 2 2\ \ 0 « - K - m;) 1 (_/0 + |/|)2 (-/O + |/jcos0ml-n)^ 1 - [ ( 1 - A ) 2 ^ + ( 1 - A 2 ) m p ] -l° + \l\ + COB 9n (5.14) w i t h integration l imits and mm 1 < |/1 < PF < cos 0 < +1 (i° + pP)2 + \if -P2F < cos 6 2|/l(/° + /xp) A g a i n we omit the tedious analyt ic integration of (5.14) and, integrating numerical ly, obtain the result shown in Figure 5.5. In Figure 5.6 both Yd and I\- are shown. Clear ly the inverse decay rate is extremely sensitive to smal l variations i n pF. W e w i l l conclude our discussion wi th two comments. F i r s t we note that our results were obtained for neutrons but can be easily generalized to other weak decays, such as muon decay which has a highly relat ivist ic phase space. E v e n quark /3-decay is in principle inc luded, although i n this case the exact v a l u e s — i n fact, the precise def ini t ion—of the quark masses present a problem. Chapter 5. Relativistic analysis of neutron 0 decay at finite density 69 0.72 r 0.64 -0.56 -0.4 8 h PF (MeV) Figure 5.5: Inverse decay rate I\ in a dense electron-proton plasma with Fermi momentum pp. Chapter 5. Relativistic analysis of neutron 0 decay at finite density 70 0.64 0.56 0.48 0.4 0 > o> £ 0.32 o £ 0.2 4 0.16 0.08 i i J — i i i i i 0 8.0 16.0 24.0 32.0 PF (MeV) Figure 5.6: Total decay rate Td + T t. Chapter 5. Relativistic analysis of neutron 8 decay at finite density 71 Our second comment concerns the generalization of our results, which were obtained for a neutron at rest with respect to its surroundings, to a neutron with nonzero three-momentum q. For the free neutron—at zero density—our result is of course quite general since the vacuum decay rate is Lorentz invariant. At nonzero density and/or temperature, however, Lorentz invariance is lost: in the restframe of a particle travelling through a medium at temperature T and/or density p, the energies of the surrounding particles will be Doppler shifted, depending on the magnitude and direction of q, and the decay rates will in general be functions of 6q and ipq. Our results are, according to pur discussion in chapter 3 , the decay rates for the lowest energy component of some nonequilibrium distribution f(uq,t) of neutrons.However, in many astrophysical applications, such as the cooling rates of neutron stars, one is inter-ested in the high energy (\q\ > mn) components of the degenerate neutron (muon, quark, ...) ensemble. We will not go into further details but merely note that our previous analysis—namely, the introduction of the /- and z-variables and integration over the -^functions—is quite general and does not depend on any particular choice of the neutron momentum q. Thus, we can reduce our initial integral for T to a two dimensional integral over |fjand cos 6. The -^dependence of V is then contained in the integrand function and, more important, in the 0-functions which determine the integration limits on |/|and cos#. Thus the integrals are best handled numerically, especially since the angular dependence of T is usually is of no interest and has to be (numerically) averaged. That the -^dependence can have important consequences is well known: a simple energy and momentum conservation argument shows that the standard URCA rates—8- and inverse /5-decay—which would normally be a highly efficient way for a neutron star to release energy in form of neutrinos (cf. Figure 5.5), are highly suppressed in a degenerate neutron gas and have to be replaced by the much less effective modified URCA rates [45]. Chapter 5. Relativistic analysis of neutron 0 decay at Unite density 72 5.5 Summary and conclusions In this chapter we gave a fully relativistic analysis of stationary neutron 0 decay in a dense environment. For the vacuum decay rate we found an analytic expression that could be generalized to the finite density case—although in practice a numerical integration is preferable to the cumbersome analytical expressions. Since we did not make the usual nonrelativistic reduction for the nucleons we can easily generalize our results to other low-energy weak interactions like muon decay, which— unlike the neutron—may have highly relativistic phase spaces. The rates were calculated from the imaginary part of the neutron self energy, using gen-eralized Cutkosky rules for a real-time formulation of quantum field theory at finite tem-perature and density. This turns out to be a very elegant and straightforward method since it automatically includes both decay and inverse decay rates in a nontrivial manner. We expect this method to be very useful for future applications, for example cooling rates in astrophysics. Chapter 6 Radiative corrections and renormalization at finite temperature We will now turn to the one-loop (first-order) radiative corrections to the thermal and scattering rates, and the problem of renormalization at finite temperature. We observe that, in the real-time formalism, the propagators, and hence the loop corrections, seperate explicitly into zero and finite temperature parts. The zero-temperature part contains the familiar ultraviolet divergencies which are absorbed into a renormalization of the parameters in the Lagrangian. The temperature dependent part is ultraviolet finite—the thermal distribution functions regulate the high-momentum behavior—but its structure is far from trivial. Previous approaches to this problem have essentially attempted to duplicate the zero-temperature renormalization procedure. These calculations of one-loop thermal decay rates are based on the earlier Dolan-Jackiw version of the real-time formalism [10] which involves only the physical fields, This assumes, of course, that the full matrix structure of the theory is necessary only in higher orders in perturbation theory. The rates were calculated, analogous to zero temperature, by integrating squares of transition matrix elements over the available phase space. The temperature dependent part of the radiative corrections was then used to define finite-temperature renormalization constants. For example, self-energy corrections on external fermion lines are treated as at zero-temperature and absorbed into a temperature dependent mass shift and a finite-temperature wave function renormalization constant, obtained from the interacting finite-temperature 7 3 Chapter 6. Radiative corrections and renormalization at Unite temperature 74 fermion propagator (inverse Dirac operator) M(P) = -7 L, ... (6.1) where E^ is the temperature dependent part of the fermion self-energy: E = E° + £^. However, Lorentz invariance, which is a crucial ingredient in the zero-temperature renor-malization procedure, is lost at finite temperature. This obscures the identification of the renormalization constants from (6.1) and leads to such unusual and problematic fea-tures as momentum dependent counterterms, a finite-temperature Dirac equation and finite-temperature spinors [11, 12]. The problem of absorbing self-energy corrections into suitable renormalization factors has also been studied in the full matrix formalism [29, 30]. However, the matrix structure of the theory together with the lack of Lorentz invariance leads to considerable complications and makes the results unsuitable for practical applications. We will now address these problems, and present a comprehensive analysis of decay rates with radiative one-loop QED corrections at finite temperature. To keep the calculations transparent, we will set all chemical potentials to zero, that is, we work in the canonical ensemble. As before, we use the generalized Cutkosky rules to compute the relevant rates from the imaginary part of the thermal self-energy. This direct calculation can then be compared to the heuristic approaches discussed above. For definiteness and simplicity we examine the decay of a scalar (Higgs) boson into two fermions: H -* e +e~. Although currently of no physical interest, this system has the advantage of being computationally rather simple and has already been treated extensively in the literature [11,12, 3]. However, the techniques and results are quite general, and can be applied to physically more important reactions. Our results can be summarized as follows. We find that, at the one-loop level, the ghost vertices and propagators do not give finite contributions to the decay rate. They are, Chapter 6. Radiative corrections and renormalization at finite temperature 75 however, necessary to cancel the pinch singularities arising from the self-energy insertion diagrams. For the radiative corrections and their interpretation in terms of renormaliza-tion constants we find that the vertex correction and the photon emission and absorption contributions are identical to previous results. The fermion self-energy insertion, how-ever, turns out to be problematic. If we follow the generally accepted philosophy for renormalizable field theories that renormalization counterterms should be of the same form as the corresponding terms in the bare Lagrangian, then the (ultraviolet finite) tem-perature dependent part of the self-energy contribution does not admit an interpretation in terms of mass and wave function renormalization counterterms, due to the lack of Lorentz invariance. For this case, we give a general algorithm how to compute the self-energy contribution for a general decay process. For the special case of two-body decay, viz. Higgs decay, we are able to define operational analogs of on-shell renormalization counterterms, but there are still significant differences from the familiar zero-temperature procedure. We conclude that finite-temperature renormalization is not a useful concept for decay and scattering rate calculations. Previous work is discussed in our more gen-eral framework. Our results have immediate applications in cosmology, in particular for radiatively corrected neutron /3 decay which determines the abundances of light elements in primordial nucleosynthesis. We proceed as follows. First we calculate the finite-temperature decay rate for a scalar particle decaying into two fermions, both at tree level and with radiative corrections. These results are used to derive a general framework for finite-temperature decay rates at the one-loop level with special emphasis on how to absorb fermion self-energy corrections into appropriate mass and wave function renormalization counterterms. We then extend these results to general decay processes and discuss the possible consequences for reactions of cosmological and astrophysical interest, in particular for neutron /? decay at finite tempeature. The last section contains a summmary of our results and a comment on infrared divergencies. Some of the calculational details can be found in the Appendix. Chapter 6. Radiative corrections and renormalization at finite temperature 76 6.1 Higgs boson decay at finite temperature We will calculate the thermal decay rate for a scalar boson decaying into two fermions. Physically this can be considered as the decay of a Higgs boson into electron and positron. To keep the discussion transparent we will set all chemical potentials to zero, that is, we neglect density effects. We will consider Higgs decay at tree level in the Higgs-fermion coupling, and with radiative QED corrections up to second order in the electromagnetic coupling. First let us recall some basic definitions and notation. The Higgs-fermion sector in the standard electroweak model is described by the bare Lagrangian C° = ^i Pi)0 - goi}0ip0(v + h0) - e^tf^QA^ = ->Po(i ft - m0)tpo - goi>0i>oho - C O V ' O TM V ' O ^ O where the bare fermion mass mo is generated by spontaneous symmetry breaking from the Yukawa coupling go and the vacuum expectation value of the Higgs field, v. In standard notation, ipo denotes the fermion field, Ao the photon field and eo the electromagnetic coupling; ho is the dynamical part of the Higgs field. Also in standard notation, we have for the renormalized fields and parameters ipo = yfZ~2~i> mo = m - Sm Ao = VZ^A g0 = gZg/Z2 eo = eZ\lZ2^fZ~z and the Higgs field remains unrenormalized to this order in perturbation theory. To 0(e2) the bare Lagrangian is split into renormalized and counterterm part = ib(i ft — m)ib — gi>4>h — eip^^A11 + 6Z2ip(i P - m)ih + 8mij)ib - SZggtptbh - hZxcty^^A* Chapter 6. Radiative corrections and renormalization at Unite temperature 77 where 8Z = Z — 1 ~ 0(e2) and we can read off the (zero-temperature) Feynman propaga-tors and vertices for the decay m a t r i x elements, shown i n Figure 6.1 for future reference. A t nonzero temperature this represents, of course, the finite-temperature propagators and vertices. 6.2 The lowest-order decay rate W e w i l l begin our discussion by calculating the Higgs decay rate to lowest order i n the Y u k a w a coupling. T h i s w i l l also i l lustrate our the generalized C u t k o s k y rules w i t h a finite-temperature, zero density example. T h e decay rate of a Higgs boson w i t h mass mn and four-momentum q = (uq,q), uq = \Jq2 + m2H, is given by r>,) = 4/mn(«) uq = €(uq) tanh(/3wg/2)Im Hu(q) uq T h e self-energy diagram I I n , shown i n Figure 6.2 a), is given by -fflnGz) = -{-igfj j0yTr[iS^p)iS^{p - q)\ where S\\ is the free finite temperature fermion propagator (3.16). U s i n g our finite temperature Cutkosky rules, shown i n Figure 6.3, we obtain for the imaginary part - 2 Im n n ( g ) = -{-ig)(+ig) J 7^Tr[iS-(p)iS+(p - q) + iS+(p)iS-(p - «)] T h e two terms i n the integrand are related by the K M S condit ion (3.26) iS±(p) = -iS*(p)e±p° hence the decay rate simplifies to Chapter 6. Radiative corrections and renormalization at finite temperature 78 d)' Figure 6.1: Transition matrix elements for H —» e+e~ with 0(e2) radiative corrections: a) lowest-order vertex; b) vertex correction; c) self-energy correction; d) photon emission and absorption processes. The counterterm diagrams are omitted. Chapter 6. Radiative corrections and renormalization at finite temperature 79 Figure 6.2: The Higgs boson self-energy Un(q) with 0(e2) radiative corrections: a) lowest-order diagram; b) "vertex correction" diagram; c) "self-energy correction" diagram; d) vertex coun-terterm diagram; e) mass counterterm diagram. Chapter 6. Radiative corrections and renormalization at Unite temperature 80 P 2 Im I o / J P~q P P o o P-q P-9 Figure 6.3: Circled diagrams for the imaginary part of the lowest-order self-energy IIn. where we used the definition (3.18) of S±. As discussed before, this expression corresponds to the difference between thermal decay and inverse decay rates, that is, the lowest-order transition amplitudes squared and integrated over the thermal phase space. We will now proceed and evaluate T explicitly. Note, however, that Lorentz invariance is lost at finite temperature, hence the decay rate will no longer be invariant but depend on the reference frame. In the following we will choose the rest frame of the decaying particle, that is, we set q- 0. First consider the product of the mass shell -^functions. With q = (mn, 0,0,0) the compatible zeros are easily found to be X W) - nF(p)][e(-p° + q°) - nF(p - q)} xTr[U> + m){ip- jq + m)] (6.2) p° - Up - mn/2 Chapter 6. Radiative corrections and renormalization at finite temperature 81 Hence the -^functions reduce to 8{p2 - m2)8{{p - q)2 - m2) = 6(p° - u , p ) % p - mH/2) 4mfju>p and fix the momentum dependence of the integrand completely. The trace is easily eval-uated and with the momenta on shell we obtain 1 4m2 \ Tr[(jp + m)(j>- h + m)] = 4(p* - p • q + m') = -2mzH 1 - —.2 . \  M H J The thermal factors can be rewritten as hyperbolic functions which reduce on shell to tanh(/?m„/2)(l + e-^)[6»(p°) - nF(p°)}[0(qo - p°) - nF(p° - q0)] = (l-e-0m»)[l-nF(p°)}[l-nF(po-q0)] sinh(/?m#/2) 2cosh2(/?TO#/4) = tanh(/3mfl-/4) Thus we obtain for the thermal decay rate 2 / 4m2 \ 3 / 2 Ttree(m„) = tanh((3mH/4)Z-mH{l-^rj = t<mh{/3mH/4)T°(mH) (6.3) where T°(mjj) denotes the zero-temperature decay rate. Thus,to lowest order, the temperature dependence is contained in a simple multiplicative factor. Let us consider the limiting cases of (6.3). For T = 0~x = 0 we have tanh(/?ro#/4) = 1 and r t r e e = T0, as it should be. In the low temperature regime Qmji > 1, expanding the tanh yields Ttree{mH) ~ (1 - 2e-/3m*/2) T°{mH) that is, an exponentially small suppression of the T = 0 decay rate which is usually neglected [11, 3]. Chapter 6. Radiative corrections and renormalization at Unite temperature 82 At higher temperatures this suppression becomes more substantial. For &mjj ~ 1 we have T tree{mH) ~ 0.25 T°(mH) and in the high temperature limit 0mjj < 1 r t r e e (m H ) 0. This agrees with physical intuition. At higher temperatures the recombination rate of electrons in the heat bath will become more significant, hence T, the difference between decay and inverse decay rates, will decrease. However, the high-temperature limit should not be taken too seriously, because perturbation theory for bosons breaks down at high temperatures. As an aside, we note that, in order to obtain the correct result, we had to use the self-energy function Iron, and not simply Imlin. This is, of course, a consequence of the underlying matrix structure of the theory. 6.3 Radiative corrections at finite temperature We will now turn to the main topic of the chapter and analyze the radiative corrections to the decay rate, up to second order in the electromagnetic coupling e. The corrections to the transition amplitudes are shown in Figure 6.1 b) to d) for future reference; for simplicity we have omitted the counterterm diagrams. The radiative correction to the self-energy diagram Un are shown in Figure 6.2 b) and c). According to the discussion in Chapter 3, all vertices axe understood as type 1, physical vertices unless stated otherwise. We work in the Feynman gauge, that is, we choose a = 1 for the photon propagator. We will now proceed and evaluate the imaginary part of these diagrams which corre-sponds to the decay rate with 0(e 2) radiative corrections. First consider the "vertex Chapter 6. Radiative corrections and renormalization at finite temperature 83 correction diagram" Figure 6.2 b) which turns out to be structurally quite simple (the actual computation is of course quite involved). 6.3.1 The vertex-correction diagram The vertex-correction diagram Figure 6.2 b) at finite temperature is given by -<(«) = (-i9)2He)2 J ^ ^Tr[iS(p - k)ltiiS(p)iS(p - q)lu xiS(p-q-k)iD lu'(k)] where the propagators are understood as the physical 1-1 components of the finite tem-perature propagator matrices (3.10) and (3.16). For notational simplicity we will from now on omit the 11-subscript. Applying our finite temperature circling (cutting) rules, as shown in Figure 6.4, we obtain for the imaginary part -2/ronn(<z) = -(-i9)(+ig)(-ie)* J^0 xTr [iS(p - k^niS-^iS+ip - q^iSip -q- k)iD(k) - iS~(p - k^^iS'^iS^p - q^iS+ip -q- k)iD*{k) + iS+(p - k)^iS(p)iS(p - q^iS'ip -q- k)iD(k) - iS*(p - k^^iS+ip^S-ip - q^iS'ip -q- k)iD*(k) + iS~(p - k)ltiiS*(p)iS+(p - q)^iS(p -q- k)iD~(k) + iS(p- k)fniS-(p)iS*(p- q^iS+ip-q- k)iD+(k) + iS*(p - k)~/niS +(p)iS(p - q)f»iS-(p -q- k)iD~(k) 4- iS+(p- fc)7Mt5(p)i5'-(p- q^iS'ip-q- k)W +{k)] (6.4) Each of the circled (cut) diagrams is equivalent to a product of transition matrix elements Figure 6.1, integrated over thermal phase space. This correspondence has already been worked out [30] and is also shown in Figure 6.4. Chapter 6. Radiative corrections and renormalization at finite temperature 84 Figure 6.4: Circled diagrams for the vertex-corrected Tin and the equivalent products of tran-sition matrix elements. Chapter 6. Radiative corrections and renormalization at Mite temperature 85 Using this relation we can now simplify the rather unwieldy looking terms in (6.4) to a more familiar form that allows a direct interpretation. First we will concentrate on the first four terms which contain the Yukawa vertex correction diagram ( 2 ^ 4 7 M « ( P - k)iS(p - q - k)riD(k) Using the cyclicity of the trace and shifting the integration variable we combine the first two terms into = (-ig)(+i9) J •^-4Tr[iS-(p)2ReG(p,q)iS +(p-q)] ( 2 7 T ) Likewise we obtain for the other two terms ... = (-ig)(+ig) J -0-Jr[iS+(p)2ReG(p,q)iS-(p-q)] and as before we can now use the relation (3.26) to combine the first four terms into -2Jmnf 1(mH) = f / 2 ( l + e - ^ ) J ^Tr[iS+(p)2ReG(p,q)iS-(p - q)} Thus we obtain a vertex-corrected decay rate TG(mH) = —~tanh{0mH/2)ImI[^1(mH) mjj = 2m^( 1~e _ / 3 m H ) / 7 0 - 4 T r U + m ) 2 R e G ( P ^ J P - A + m)] x [9(p°) - nF(p)]{6(q° - p°) - nF(p - q)} x S(p2 - m2)6((p - q)2 - m2) (6.5) This is of course the standard zero-temperature result, generalized to finite temperature. The vertex correction diagram Figure 6.1 b) modifies the Yukawa coupling to g -v gReG(p,q) and the correction to the decay rate is found by replacing g 2 in the tree rate (6.2) by the modified vertex and expanding to 0(e 2). Chapter 6. Radiative corrections and renormalization at finite temperature 86 It remains to evaluate the vertex function G(p,q) = G°(p,q) + G^(p,q) with its external momenta on mass shell. This is of course more involved and we will simply quote the results from the literature. The zero-temperature part Re G° has been given in [7], For the finite-temperature part several approximations have been considered. The simplest one—including only the ther-mal photon distribution and neglecting the fermion contributions—leads to [11]: - £ i± f£ In i ± { | jT ¥ l | | j » B ( l * - | ) (6.6) where w = \J\ - Am 2lm 2H. We note that G^ is a scalar function with no Dirac indices. Also note the l/|fc|2 infrared divergence and the logarithmic mass singularity for ro2 —»• 0, i.e. w -* 1 in (6.6). Recently, the authors of [3] have improved this result, and taken the thermal fermion distributions into account. However, the resulting expressions and approximations are quite complicated and we refer to [3] for details. We simply note that G@ remains a scalar and that no additional infrared divergencies are introduced. In summary, the vertex correction to the decay rate at finite temperature is given by V G{mH) = 2ReG(mH) Ttree(mH) (6.7) This expression is of course both ultraviolet and infrared divergent. The ultraviolet di-vergence arises only from the zero-temperature part G° whereas the infrared divergencies are contributed both by G° and G@. The ultraviolet renormalization is straightforward. The vertex counterterm —ig6Zg leads to the two counterterm diagrams Figure 6.2 d) whose imaginary part is again determined by our circling rules. Proceeding as before we find immediately -2ImEl\(mH) = g\l + e~^) J •0-4Tr[iS +(p)2Re6Z9iS-(p - q)} Chapter 6. Radiative corrections and renormalization at finite temperature 87 Adding this counterterm contribution yields the remormalized decay rate T°m = 2Re(G-6Za)T = 2ReG where 6Zg is chosen to subtract off the ultraviolet divergence in G° plus any finite part of G. Note that, if we choose to include part of G& in 8Zg and make the coupling temperature dependent, Lorentz invariance of the Lagrangian will be lost. We will discuss this problem in more detail in the next section. The infrared divergencies (in the unrenormalized rate) and the mass singularity are can-celled by contributions from the last four terms in (6.4) which represent part of the photon emission and absorption rate, r ^ i . As before we can combine these four seemingly dif-ferent terms into a single expression, using a shift in integration variables and relation (3.26) between the circled propagators. Thus we obtain for the (partial) photon emission and absorption rate r 7 l(mH) = tanh(0mH/2)ImJl1\(mH) where Imiq\(mH) g\-ie)\l + e-13™") Tr [jS+(p)7^S(p - k)iS~(p -k- q^iS^p - ?)ilT (&)] and hence for the transition rate H ' K ) = -9^{l-e-^»)Rej?0±6{p^ x Tr[(jp + m)7M(/- ft + m)(^- jk- k + m)^(f>- jq + m)] X [e(p°) - nF(p)][6(q° + k° - p°) - nF(p - k - q)][0(-k°) + nB(k)] X Xp-k)2 l.m2 + i£-^F(P-k)S((p-kf-m2) x [{p-q? l_m2_k ~ 2™F(P ~ qWP - qf ~ m2) (6.8) Chapter 6. Radiative corrections and renormalization at finite temperature 88 This can be simplified further: splitting the complex A distributions into Cauchy principal part and tf-function and regulating the infrared divergence at k = 0 with a small photon mass A it is easy to show that the product of the five four-momentum conserving 6-functions vanishes, and we are left with r71(ro/f) = - ^ ( 1 - e-^")Re j ^ r % * - m 2)6((p - k - qf - m 2)6(k 2 - A2) xTrKft + m^^p- fi + m)(j>- <q + m)^- /q + m)} X [6{p°) - nF(p))[6(q0 + k°- p°) - nF(p - k - q)][9(-tf) + nB(k)] PP 1 (p - k)2 — m2. PP- 1 (6.9) (p — q) 2 — ro2. where PP denotes the Cauchy principal part. Of course the actual evaluation of the integrals and the demonstration of the cancellation of the infrared divergencies is quite involved and we refer to [7, 11] for the calculational details. So far our results are a straightforward generalization of the zero-temperature results: the vertex-correction matrix element renormalizes the Higgs-fermion coupling and the infrared divergencies are cancelled by contributions from photon emission and absorption processes. This is not surprising since the circled (cut) diagrams correspond directly to the relevant transition matrix elements. Things will become more interesting when we consider the fermion self-energy insertion diagram. 6.3.2 The self-energy correction diagram The fermion self-energy correction E to the boson self-energy n is shown in Figure 6.2 c) . Since the two diagrams are related by a simple shift in integration variables and a reversal of the external momentum it suffices to consider only the diagram with the electron line Chapter 6. Radiative corrections and renormalization at finite temperature 89 corrected; the other one will contribute only a factor of 2. The graph is given by and applying our circling rules as shown in Figure 6.5 we obtain for the imaginary part - iS*(pyr»iS*(P - k)ltlis+{p)is-{p - q)iD*{k) 4- »S*(p)7"*S+0> - k)lfiiS{p)iS-(p - q)iD+{k) + iS+{p)^iS~(p - k^^iS+ip^S-ip - q)iD~(k) +-iS(p)i'iiS(p - k)-t„iS-(p)iS+(p - q)iD(k) - iS*(p)^iS*(p - k)lfiiS-(p)iS+(p - q)iD*(k) - iS(p)riS-(p - k)7liiS*(p)iS+(p - q)W-(k) -»5-(p)7"t5 + (p - k^^S-ip^S+ip - q)iD+(k)] (6.10) As before each of these terms represents the product of transition matrix elements as shown in Figure 6.5. Note the circled diagrams corresponding to a photon emission and absorption diagram with a type 2 vertex. Since these diagrams cannot be represented by a cut they have to vanish at zero temperature, but at finite temperature they will contribute and, as we shall see, are indispensible for the well-definedness of the self-energy. Again not all terms in (6.10) are independent. Using the by now familiar KMS condition (3.26), it is trivial to show that the last four terms are related to first four by a factor of e-W. Thus (6.10) is reduced to i^liq) =-{-ig)\-ief j d4pd4k (2x)8 Tr [iS(p)^iS(p - k)fviS{p)iS(p - q)iD»"(k)) 2/mnpa(?) = g\-ief J x [iS+ip^iSip - k)j^S(p)iD(k) - iS'ip^iS'ip - k)lfliS+(p)iD*{k) - iS*(p)riSu(p - k)7liiS(p)iD12(k)e^°^ Chapter 6. Radiative corrections and renormalization at finite temperature 90 Figure 6.5: Circled diagrams for the self-energy corrected Tin and the equivalent products of transition matrix elements. Chapter 6. Radiative corrections and renormalization at Unite temperature 91 - iS+(p)riSu(p ~ k)1(1iS +(p)iD12(k)e-to°/ 2]} (6.11) where we used (3.12) and (3.18) to rewrite the circled photon propagators in terms of the off-diagonal propagator matrix elements. Obviously each of the four terms in (6.11) contains ill-defined distributions (pinch singularities or "squares of ^ -functions") which are guaranteed to cancel on general grounds [29]. The actual cancellation procedure, however, requires some work and is rather instructive, so we will present it here in some detail. First we use the definition of the fermion self-energy matrix elements S0j, and write ImllS(mtf) = -gl(l + e-^)J-0^Tr{iS-(p-q) x [iS +(p)iEn(p)iS(p) -iS*(p)iE22(p-k)iS +(p) + iS*(p)iX12(p - k)iS{p)e^l 2 + iS+{p)iHl2{p - k)iS +(p)e-W2}} which can be rewritten, using equation (4.10), as Iml&imn) = -9l(l + e-^)Jj^Tr{iS-(p-q) x [iS+(p)iXn(p)iS(p) + iS*(p)iX*n(p)iS +(p) - e(p°)tan20(p) (iS\p)ImHn{p)iS{p)e^ I2 +iS +(p)ImZn(p)iS +(p)e- l3p0/ 2) ]} (6.12) We note that, in terms of matrix elements, the first two terms arise from the self-energy insertion on an external fermion line, whereas the last two correspond to squares of photon emission and absorption diagrams. In order to show the cancellation of the ill-defined distributions in (6.12) we recall the Chapter 6. Radiative corrections and renormalization at finite temperature 92 matrix representation (2.74) of the propagator matrix elements, and rewrite the finite-temperature propagators as follows: iS^p) = 2ir(/p + m)[e(p°)-nF(p)}6(p 2-m 2) iS{p) = (^ + m)(cos20A(p)-sin20A*(p)) iS*{p) = (^ + m)(-cos20A'(p)-rsin20A(p)) where A(p) = i/p 2 - m 2 + ic A*(p) = -i/p 2 - m 2 - ie cos2 <j> = l / e - ^ ° l + 1 sin2 0 = l / e^° l + 1 For the A-distribution we use the well-known relation lim -= = PP * - ± irS(p 2 - m2) €-*o p* - m i ± ie p 1 - m 2 from which it is easy to derive the following useful identity 2nS(p 2 - m 2)-2 V— = in-^-zttf - m 2) ± 2ir26 2(p 2 - m 2) p 2 -m 2 ±ie dm 2 where we recall that the -^functions are understood as convenient shorthand for the sums and squares of the proper e-regularized A-distributions. Consider now the first two terms corresponding to self-energy insertions on the external fermion lines (from now on all distributions are understood as properly e-regularized). With the identities just introduced it is straightforward to show that iS +(p)iS(P) = (i> + m ) W ) - n F ( p ) ] and likewise iS +(p)iS*(P) = (/P + m)2[e(p°)-nF(p)} iir-^-^6(p2 - m2) + (cos2 <p - sin2 <p)2ir262(p2 - m 2) i7r^—r<5(p2 - m 2) - (cos2 <f> - sin2 <p)2Tt26 2(p 2 - m 2) Chapter 6. Radiative corrections and renormalization at Unite temperature 93 Thus the first two terms in (6.i2) can be combined into iS +(p)iXn(p)iS(p) + iS*(p)iXn(p)iS+(p) = -2ir{jp + m)[6(p°)-nF(p)] ReXn(p)JL6{p* - ro2) + Iro£n(p)cos2<£27r£2(p2 - m 2) (iP + m) (6.13) that is, a well-defined mass derivative of the mass shell -^function proportional to the real part of £, and a pinch singularity proportional to the imaginary part which has to be cancelled by contributions from the two remaining photon emission and absorption terms, that is, the third and fourth term in (6.12). For the third term we find iS*(p)iS(p) = (jp + m) 2 [(sin 2^cos 2^-l/2)(A + A * ) 2 + l /2(A 2 -r-A* 2 ) ] d (1/4 sin2 2<j> - 1/2)(2TT£(P2 - ro2))2 + ^ ( ^ A - iw6(p 2 - ro2)) = (j> + ™)2 and the fourth term is simply a "pure" pinch singularity (iS+(p)) 2 = (i> + m ) 2 {{0(p Q) - nF(p)]2ir6(p2 - ro2)}2 The cancellation of the 62-terms is now easy to see if we rewrite them in terms of the off-diagonal propagator matrix element iSu- Recalling that iS+(p) = -e0P O/ 2iSu(p) iS12{p) = -e(p°)(p + m)eMp°V 2nF(p)2ir6(p2 - ro2) = -e(p°)(^ + ro)sin2<M(p2-ro2) we obtain for the last two terms in (6.12) £(p°)tan2<£ [iS(p)ImXn(p)iS*(py p0/ 2 + iS +(p)Imi:n(p)iS +{p)e- f3p^ 2)] = e ( p V P ° / 2 \(JP + m)Jro£ n (p)(^ + ro) t a n 2 < £ ^ (&(p) - iwS(p2 - ro2)) +2 tan 24> sin 2(f> cos 2(f) iSi2(p)ImZii(p)iSu(p)^ (6.14) Chapter 6. Radiative corrections and renormalization at Unite temperature 94 and likewise we have for the ill-defined term in (6.13) (i> + m) 2[9(p°) - nF(p)] cos20(27T(5(p2 - m2))2 = 2e{p°)e^l 2 cot 20 (iS12(p)) 2 Combining (6.13) and (6.14) we obtain for (6.12) Jmng(«) = -l(l + e-0™»)J^Tr[iS-(p-q) x ((jp + m) \-[9(p°) - nF(p)]ReEn(p)2Tr^S(p 2 - m2) (^ + m) -e(p°)Imi:u(p)e l3p0/2 tan 20^L (»A(p) - i7r«5(p2 - m2)) + V ) J m S u ( p ) e ^ / 2 (_ c o t 2^ + t a n 2 ^ _ ( i5u(p)).j] and the ill-defined .S -^term disappears, leaving only well-defined distributions. To sim-plify the remaining part we use eqn. (4.10) for ImH\\ and obtain <p°)e^°/ 2 tan20/m£ii (p) = 2[6{p°) - nF(p)]ImE(p) and for the distribution 9 ;(iA - in8(p 2 - m 2)) = ~ P P •  1 dm 2 dm 2 p 2 — m 2 (p 2 — m2)2 by definition of the principal part. Thus the self-energy corrected decay rate is given by rS(m#) = --i-tanh(/3m///2)/mnp1(mH) ; ( l - ^ ) / ^ ( ( p - g ) 2 - m 2 ) mH mH{ }J (2TT)3 X [9(p°) - njp(p)p(fl° - jP) - nF(p - g)] x Tr |(i>- k + m)(jp + m) ReX(p)2ir—8(p 2 - m 2) 2ImZ{p) (p 2 — m 2) 2 (^ + m)} (6.15) We note that, in order to achieve complete cancellation of the ill-defined 62-terms it was necessary to include the circled but non-cuttable self-energy graph or, equivalently, the Chapter 6. Radiative corrections and renormalization at finite temperature 95 matrix element with a type 2 vertex in our calculation. Thus, strictly speaking, the ghost vertices are needed in perturbation theory even at the one-loop level to ensure the well-definedness of our results. They do, however play only a "minimal role" since they do not contribute to the finite part of the result. It is also obvious from our derivation that pinch singularities will occur (and cancel) even at zero-temperature. In this case the type 2 contribution vanishes, and the cancellation involves only the regular type 1 terms. Furthermore we emphasize that our derivation and the final result are quite general since we had to consider only products of distributions that are common to the radiative QED corrections for any decay and scattering process. Also note that our derivation involves only the properties of the distributions pa_^s±>e and hence holds for boson self-energies as well. In terms of transition matrix elements the part proportional to ReT in (6.15) arises from the self-energy insertions on the external fermion lines Figure 6.1 c). The term containing the imaginary part ImT represents the remaining part Tl* of the photon emission and absorption processes Figure 6.1 d). It can be written in a more familiar form analogous to r^ i if we use the explicit form of M S : by definition -iEnOO = He)2 / J ^ l M P + kh„iD^(k) and, applying our circling rules, it is easy to show that IroE(p) = e{p°)coth(0p°/2)ImTn(p) = e(p°)^(l + e-^) J J^^iS+ipWiD-ik) This yields for r^a r 7 2 ( ^ ) = ^ ( 1 - e - * » * ) / ^ 0KP2 ~ m a ) * ( ( p - k - of - m*)6(k>) xTr[(p- /: + m)7/l( p + m)^^- ft + m)( p- ft- k + m)] x [6(p°) - nF(p)}[e(q° + k Q - p°) - nF(p - k - q)][6(-k°) + nB(k)} Chapter 6. Radiative corrections and renormalization at Unite temperature 96 As before this expression contains both zero-temperature and temperature-dependent in-frared and mass singularities which have to cancel against the ones arising from the real part of the electron self-energy. For the problem of mass and wave function renormaliza-tion, however, r^2 is of no direct interest, hence we will concentrate in the following on the ReT part. 6.3.3 Mass and wave function renormalization at finite temperature The mass derivative of the <5-function is best evaluated as Jd*p F(p,m2, .. - ) ^ ( P 2 - ™2) = J im a ^ / * P  F(P,™\ .. .)6(p2 - ro2) and the contribution of the self-energy correction to the external fermion lines can be written as ,2 TSE(mH) = - - 9 — { l - e - P m " ) Um / mjj mi-*m2 J 9_ mn 8 dm 2 •0y[e(p°) - nF(pW(q° - p°) - nF(p - q)]S(p 2 - m2)S((p - q)2 - n X ReT{p){i> + m) + g[0(p° - nF{p))[6{q° - p°) - nF(p - q)]6(p2 - m2)S((p - q)2 - ro2) xTr(jp- /q + m)(Jp + m) X^[ReT(p)(J> + m)) (6 where the above notation is understood as integrating over the -^functions before taking the derivative. In the real-time formalism the fermion self-energy can be split into a zero-temperature and a finite-temperature part: ReT, = ReT,0 + ReT13. First consider the T=0 part. Usually ReT0 is expanded in a formal Taylor series around the mass shell point "ji = ro" (which as a matrix equation is of course nonsensical) ReT°(p) = ReT%m + -^jReT%m(p-m)+... Chapter 6. Radiative corrections and renormalization at Unite temperature 97 = Sm + SZ2(jp-m) + ... (6.18) where the first two coefficients are defined as the ultraviolet divergent mass and wave-function renormalization counterterms (the finite higher order terms can be set to zero by the mass shell renormalization condition and will be neglected). More precisely, Lorentz invariance restricts ReE° to the general form ReZ°(p) = ax(p2) jp + a2(p2) (6.19) and, expanding the coefficients around the propagator pole p2 = m2, we obtain to 0(p2 - m2) ReE°(p) = ai(m2)-f-a2(m2)(/p + m)] (^) - m) + mai(m2) + a2(m2) + • • • ~ ai(m2) + 2ma'2(m2)j (jp - m) + ma\(m2) 4- a2(m2) which justifies the formal Taylor series (6.18). To renormalize the bare mass mo in the free fermion Lagrangian to its physical on-shell value mphya £° = i>(i jd - m0)ip -* ${i - mQ - 6m)ip = ip~(i Jd - mphy3)ip we have to include the mass counterterm diagrams Figure 6.2 e) in our set of self-energy diagrams. Their imaginary part is easily determined with our circling rules and adding it to the unrenormalized rate replaces J?e£° by Ret? = i?e£° — 6m in the decay rate (6.15). With this mass-subtracted self-energy we find immediately for (6.17) lim Ret°(jp + m) = 0 A [ m , ^ R e t ° ^ + m ^ =  622 m'—tm* Om" and thus the ile£0-contribution to the decay rate reduces to r 0 s V * ) = W ^ ( l - e - ^ ) . M s / ^ - m 2 ) % - g ) 2 - m 2 ) mjj m2-*m2 J (An) X [6(p°) - nF(p)][0(q° - p°) - nF(p - q)] xTr[(d>- ^ 4-m)(^ + m)] = 26Z2Ttree(mH) (6.20) Chapter 6. Radiative corrections and renormalization at Unite temperature 98 which is precisely the result one obtains from multiplicative on-shell renormalization and the LSZ theorem for the transition matrix elements. This is usually expressed as the following Feynman rule [6]: self-energy corrections on external (fermion) lines are replaced by a factor of yfZ^ for each line, provided the mass counterterm 6m has been included. The remaining ultraviolet divergence in SZ2 can of course be eliminated by adding a suitable counterterm vertex SZ2 T(p — m) to Sm in the diagram Figure 6.2 e). The crucial property in the derivation of (6.20) is Lorentz invariance which permits the formal Taylor expansion (6.18) of J?eS°. For the finite-temperature part ReT 13, however, Lorentz invariance is lost, but we can still maintain Lorentz covariance of the theory by introducing the four-velocity u of the heat bath [49, 50]. The noncovariant approach described so far corresponds to the choice u = (1,0,0,0), that is, the heat bath is taken to be at rest with respect to the lab frame. In the following u = (1,0,0,0) is always understood, unless stated otherwise. The requirement of Lorentz covariance restricts ReYP to the general form ReT l3(p) = ai J} + a2 + a 3 /i + a 4 fi p" = ai p + a2 + a 37° + a 47° p" where the four coefficient functions a,- will now depend on the two available Lorentz scalars p 2 and p • u = p°: a,- = a,(p°,p2). Thus (6.19) is now generalized to ReT^p) = Ax(p°,p 2)(p -m) + A2(pQ,P 2) (6.21) with matrix-valued coefficient functions Ai = ai + ei470 4^2 = max + a2 + (a3 + a4)70 We could now proceed as in T = 0-case and, expanding the coefficients Ai around p 2 — m 2, introduce the matrix-valued mass counterterm Sm13 = A 2p2=mi Chapter 6. Radiative corrections and renormalization at finite temperature 99 Thus the physical mass is now defined as the pole in the finite-temperature propagator = jp-m- Re^ip) and the finite-temperature wave function renormalization factor Z2 might e.g. be taken as the residue of S. However, this straightforward and popular procedure presents several problems if we accept the general philosophy that renormalization counterterms should be of the same form as the unrenormalized quantities, that is, the renormalized Lagrangian should remain Lorentz invariant and of the same functional form as the bare Lagrangian. Lorentz invariance is obviously lost for any finite-temperature counterterm denned from ReE 13, and the renormalized Lagrangian, a dynamical quantity, will now be temperature dependent. Although one might accept this, together with the more general 7-matrix structure, as a generalization necessary for finite-temperature field theory, there remains the problem that the renormalization point p 2 = m 2 is not sufficient to eliminate the momentum dependence of the covariant coefficient functions in ReYfi. This momen-tum dependence is nonpolynomial and rather complicated (see e.g. [50] for an example); counterterms constructed from the a,'s will introduce nonlocal, and presumably nonrenor-malizable, interactions in the renormalized finite temperature Lagrangian. This leads to the folio wing problem. The original bare Lagrangian is now split into temperature dependent renormalized and counterterm parts £° = CTtn + C ct both of which have the aforementioned nonpolynomial momentum dependence. While left and right hand side are formally equivalent, this does not have to be true for the corresponding perturbation theories. Due to the problematic functional form of the finite temperature Lagrangians the higher order terms need not be small and might contribute substantially to the final result. A priori, it is not even clear that these terms are well Chapter 6. Radiative corrections and renormalization at finite temperature 100 defined and free of ambiguities. However, the predictive power of a perturbation expansion depends to a large extent on our ability to assess the behavior of the higher order terms. If the higher order behavior becomes uncontrollable, then the theory will loose its predictive power. In addition to these basic objections, we will also see in the following how momentum dependent counterterms lead to incorrect results even at the one-loop level. These problems will not arise if we take the conservative approach and introduce only the zero-temperature counterterms necessary for the removal of the ultraviolet divergencies. This is legitimate since i?eS^ is ultraviolet finite (the thermal distributions act as regula-tors). The Lagrangian remains Lorentz invariant and dynamical, and the functional form of the renormalized and counterterm parts is unchanged. The ReTP contribution will of course not reduce to a simple multiplicative Z2 factor but requires the evaluation of the integral and the masses are the physical (renormalized) zero-temperature parameters. This approach is the most general one: the decay rates are evaluated as a function of the known (and measurable) zero-temperature parameters and there is no conflict with the basic requirements of locality and renormalizability. In the case of Higgs boson decay—or any decay/scattering process with two-body phase space—the problem of momentum dependent counterterms does not arise since the mass r f ( m „ ) = ~£L(l-e-*»*) p mn m 2 -»m 2 dm 1 mH + lim 7 — ^ m2-+m2 dm* (1 _ e - 0 m » ) X [6(p°) - nF(p)][e(q° - p°) - nF(p - q)} xTr[(p- A + m)(jp + m)Retf(p)( p + ro)] ){ / d 4P6(p* - ro2)... Tr [(p- k + ™)(p + m)Al{p)] fd 4p...TrU-jq + m)(p + m)A,(p)(f> + m)] } (6.22) Chapter 6. Radiative corrections and renormalization at finite temperature 101 shell -^functions fix p° and p 2, 6(p 2 - m 2)6((p - q) 2 - m 2) -> p° = wp = mH/2 and give a natural on-shell renormalization point. Thus we can proceed and define op-erational renormalization constants. Lorentz invariance is of course still lacking, so the following prescription is heuristic and "natural" only insofar as it uses the covariant generalization of the familiar counterterms and, as we shall see, reduces the self-energy contribution to a scalar Z2 constant analogous to the T = 0 renormalization proce-dure. Moreover, the covariant finite-temperature counterterms will have features that make an interpretation as QED renormalization constants extremely problematic. Thus the following covariant on-shell renormalization scheme should be regarded only as a convenient procedure to deal with the momentum dependence of ReYP. Indeed, our discussion will demonstrate that, even without the problem of momentum dependence, finite-temperature counterterms are a physically problematic concept; thermal transition rates are hence best evaluated in terms of the (well-defined) zero-temperature parameters. From (6.21) we define the matrix valued finite-temperature mass counterterm Sm 13 = A2(p° = wp = mjy/2) (6.23) which replaces ReTP by Retf'=ReE^-6m^ as before. The matrix structure of Sm**, however, has some unusual consequences which we will discuss later on. The first term in (6.17) is eliminated by 6m@, as at zero temperature, since lim Retf(j> + m) = 0 m 2-»m J and we define the remaining term as a matrix-valued "^-function" 6 2 2 = J i ^ , r a [ ^ ) ^ + m ) ] = ^ + ^ x ^ + T O ) dm 2 d dm 2 = oi + 047° + -QTp (mai + 02) x {£> + m) + ^ r ^ ( ° 3 + ma 4)7° x (jP + m) Chapter 6. Radiative corrections and renormalization at finite temperature 102 a\ +  2P°~Q^( a^ + ma4) + 2 m ^ 2 " ( m a l + °2) at p° = uv = mjj/2, m2 —> ro2. which is of course still momentum dependent. Evaluating the trace with 22 we find ? > [ ( i > - ^ + m ) ( ^ + m)] Note that we used Tr^p1- i+ ro)(/* + ro)a47°] = (2p° - q°)ma4 = 0 at p° = Up = m#/2. Thus we have, for our present system, an effective scalar finite-temperature wave function renormalization constant Z% given by 6Z$ = Z$ - 1 = ax + 2p°-^{a3 + ma4) + 2ro^rj(roai + o 3 ) (6.24) at the renormalization point p° = up = mn/2, which is nothing but the covariant gener-alization of the zero-temperature Z2. The self-energy correction to the external fermion lines is thus reduced to an effective scalar renormalization constant T$E(m„) = 26Z% Ttree(mH) as desired for on-shell renormalization. Let us now return to the finite-temperature mass counterterm. Using the 7-matrix valued Sm13 as a counterterm in the free fermion Lagrangian C = ij)(i P - mphys)ip P - mphyS - Sm13)^ leads to a temperature dependence of the Dirac operator as can be easily seen in momen-tum space. Writing a general matrix valued mass counterterm as Sm13 = A fp® + Am*3 we have in momentum space for the Lagrangian JC = i>(i jp - mvhyS - Sm13)1^ = \j)[(p - Ap0) • 7 - {mvhya + Am?)]*!) Chapter 6. Radiative corrections and renormalization at finite temperature 103 where we used the tilde notation of [11]. Thus the mass counterterm 6m^ shifts not only the mass but also the momentum operator. Rederiving the Feynman rules for the shifted finite-temperature Dirac operator is straightforward and amounts to replacing p and m by p and m in the propagators. Thus all decay rates have to be reevaluated with tilde quantities. For the tree level decay rate we obtain r^tree . a 2mH J (27r)4 x [e(p°) - nF(PW(l0 - f ~ MP - q)] *Tr[(}- A + m)(]p + rh)} As an aside we note that the terms in the trace can be regarded as products of spinors on the external fermion legs of the transition matrix elements. Here these spinors would be solutions to the "finite temperature Dirac equation" (jp~ m — 6m@)u = 0 and are a special case of the finite temperature spinors introduced in [11]. It can be easily shown that the integral is invariant under any translation by a constant four vector: p -* p = p — ApP. Hence we have to replace only m by m = m + A m ' 3 in (6.3) and we have for the mass-renormalized tree rate „2 / A ^ 2 \ 3 / 2 V  mHj f^m# 11 _ ZZT I tanh(/3mn/4) 12m . g 1 o — ; A m p * T tree(mH) = r t r e e + r A m (6.25) myw2 to 0(e 2), where w = \Jl — Am2/m2H. The other contributions—vertex correction etc.— are already of 0(e 2) and don't have to be reevaluted in tilde variables. Thus the net effect of the mass counterterm Sm 13 is indeed a mass renormalization mphya —> mphya + Am'3. In summary the covariant renormalization prescription gives indeed the operational analog of the zero-temperature renormalization procedure. The fermion self-energy correction is absorbed into a multiplicative Z-iactor and the physical mass in the tree rate (6.3) is Chapter 6. Radiative corrections and renormalization at finite temperature 104 shifted by constant. There are, however, important differences to the zero-temperature case. The zero-temperature Q E D renormalization constants are well-defined and "universal" in the sense that they are fixed by the fermion mass shell condition alone and involve only Q E D parameters, that is, the fermion mass and the fermion-photon coupling. The finite-temperature counterterms 8m& and 8Z2 , however, depend not only on the fermion mass but also on the mass of the Higgs boson or, more generally, the specific kinematics of the reaction, due to the choice of the renormalization point (it is of course possible to choose an arbitrary point (p°,p 2) without reference to the Higgs boson, but this would not eliminate the ^m -^term in the decay rate and make the concept of thermal on-shell renormalization ambiguous). Moreover consider the Z% function from which we derived the covariant wave function renormalization constant Z2 . If we replace the simple Yukawa Higgs-fermion vertex by a general coupling (vector, axial, etc.) the trace in the decay rate will be of the general form Tr • • • (p + m)ReYP(p + m)] and hence we have to consider the trace Tr • • • + m)8Z2 . With Z2 of the general form SZ^ = ai + a 47° + ^ - j [max + a3] x (p + m) + [o3 + ma 4]7° x (p + m) = Ai+A2 +A3(jp + m)+44(p + m) and the fermion mass-shell 5-functions this is easily reduced to Tr ["(jp+m)6ZJ!] = Tr [• • • (^ + m)] [Ai +2p-A4 + 2mA3] + Tr[->.(jp+m)43] Obviously the first factor is our Z -^factor (6.24) but the second term will in general depend on the specifics of the trace (in the case of Higgs decay it vanishes). In general, the wave function renormalization factor derived from 2% will depend not only on the fermion-photon coupling but on the other fermion couplings as well. Thus thermal on-shell renormalization leads to the somewhat paradoxical situation that QED counterterms depend also on non-QED interactions. Chapter 6. Radiative corrections and renormalization at finite temperature 105 Furthermore, the mass-renormalized finite-temperature Dirac operator P=P-m-6m'3 has poles in momentum space given in by the dispersion relation p2 - m2 ~p2-m2-2p- Ap13 - 2mAm/3 = 0 which has solutions p° = Ap3 ± ^ ]u2 + 2mAm<3 by definition (6.23) of 8m?. If we define a finite-temperature mass as the solution p° at some fixed three-momentum f (see e.g. [40, 51]) it is obvious that this "dispersion relation mass" will depend on all components of 8m13 whereas the "effective decay rate mass" in (6.25) involves only a shift by Am'3. It is also obvious that the finite-temperature Dirac-operator p is not identical to the inverse finite-temperature propagator %S~X =fi — m - Re£>P(p2 = m2); the latter would correspond to a Dirac operator with momentum dependent mass counterterm which we rejected as incompatible with a local and renormalizable Lagrangian. These considerations show that the familiar zero-temperature concepts of mass and wave function renormalization cannot be extended to finite temperature in a generic way, not even for the case of two-body decay. The heuristic covariant finite-temperature coun-terterms will in general depend on the specific kinematics and couplings of all particles in a particular decay process and do not have the physical and model-indepedent inter-pretation of their zero-temperature counterparts. We conclude that the notions of finite-temperature parameters in the Lagrangian, Dirac operators and spinors are problematic concepts and of no use for the study of decay processes at finite temperature. Let us now consider an explicit example. The covariant expansion (6.21) was well-suited to discuss the general momentum dependence of jReS^ , but the actual computation of Chapter 6. .Radiative corrections and renormaiization at finite temperature 106 the coefficients a,- for the full self-energy is rather cumbersome (see e.g. [46] for a zero-temperature, finite-density example and [50] for the massless case). Instead we will con-sider the familiar low-temperature case treated in [11]. In the low-temperature regime T < m, ReT,0 is approximated on mass-shell by [11, 20] Wt - m) + 4 - d - J ^ - { v ' - fWP,f) 2u)p du>p (6.26) where IA = 4* f° §inB(|fc|) h \k\ J \k\ upk° - pk where up = ^ /p2 + m 2, A;0 = |£| and nB(|fc|) = [e^l - l]"1. The explicit form of J(w p,£) is given in the Appendix. First we observe that (6.26) is not a covariant expansion of the type (6.21) discussed so far, but an expansion of the finite-temperature photon contribution in ReT0 at the mass shell point p2 = m2. The renormalization constants given in [11, 20] are SmDH = ^ ( 6 . 2 7 ) 6ZDH = j L ( / J _ 7 ) 87TJ Up e2 (T w3T2 l + v\ , e n o S where v = \p\/up. From their explicit form it is obvious that these noncovariant coun-terterms have the same unacceptable momentum dependence that we discussed for the covariant expansion (6.21). Moreover, the momentum dependence in 6m DH is not even fixed by the mass shell -^functions, thus SmDH cannot be used as a heuristic mass coun-terterm. We will now calculate directly the decay rate correction due to ReT0 without introducing any finite-temperature counterterms (the physical value of the decay rate is Chapter 6. Radiative corrections and renormalization at finite temperature 107 of course independent of the renormalization scheme used). Thus we have to evaluate eqn. (6.22) „2 Q T$ E(mH) _ JL(i _ c - 0 m mH X [8(p°) - nF(p)][0(q° - p°) - nF(p - q)] (jp- k + m)( jp + m)ReX 0( jp + m)] x Tr 8 ^ c2 Let us split ReY,® into Bi = and hence the decay rate into A straightforward calculation, given in the Appendix, yields (jp-m) T Bl = 2<5Z. ' e2 e 2 r 2 ^ 1 + w s 4x3 3mjjW 1 — w T tree(mH) T * 2 = - r ^ f l + - ( l - ^ 2 ) l n i ± ^ ) r^ ( m H ) (6.29) (6.30) where to = - Am 2/m 2H. Thus the B\ part of i£eS^ reduces to the wave function renormalization factor 6Z2 H which is not surprising since Z2 H corresponds to the part proportional to (p' - m) in ReE* 3. Note that SZ^ H contains an infrared divergence in I A and a mass singularity in Jo for m 2 —*• 0, that is, for w —> 1. However, they both cancel against similar terms in as shown explicitly in [11]. The /^ -contribution to T SE can be regarded as a noncovariant mass correction since it corresponds to 6m DH. We note that T® 2 is infrared finite and does not contain a mass singularity, but remains finite for w —» 1. Thus, B2 does not introduce any new mass Chapter 6. Radiative corrections and renormalization at finite temperature 108 singularities in the decay rate, and there is no need to introduce a momentum dependent finite-temperature counterterm in the Lagrangian to eliminate the (potentially trouble-some) tfm^-contribution. This is important since, according to our discussion, there is no generic way to define finite-temperature renormalization constants. By comparison, at zero-temperature the Kinoshita-Lee-Nauenberg theorem guarantees the absence of mass singularities in the unrenormalized decay rate, but only after mass counterterms have been included (cf. reference [7]). In [24] we pointed out that, since the self-energy (6.26) shifts the pole in the fermion propagator by a constant e2T2 rojfcy. - *  mlhya + - g ~ = we can define mp/l3/s(/3) as the physical "finite-temperature mass" and describe the shift by the constant mass counterterm , s qi ( U 1 ) 8TTj m 12m which is independent of the reaction kinematics. If one insists on a finite-temperature mass for the fermion, then 6m0 provides a simple and physically transparent alternative to the covariant counterterms discussed before. On the other hand, this finite renor-malization of the physical mass of course neither changes the value of the decay rate nor simplifies the calculation: unlike 6mDH the counterterm 8m0 does not eliminate the "mass" contribution T02 completely, but instead replaces Bi by B'o = Bi-hm0 e 2 2TT3T2, 8VT3 \ r 3m J The corresponding decay rate is then evaluated as (see Appendix) ZmjjW - ( 2 - i ( l - ^ ) l n i ± ^ ) r * - ( m H ) (6.32) Chapter 6. Radiative corrections and renormalization at finite temperature 109 The mass shifted tree rate (6.25) for the mass shift (6.31) is easily found to be T A m = - ± ± — V t r e e (6.33) mjjw* and we have pi?2 _ -(-as it should be. We note that the shift in the propagator pole that defines 6m@, is momentum independent only for the approximation (6.26) of ReE^, which neglects the thermal fermion contribu-tions. Taking the fermion corrections into account leads again to a nontrivial momentum dependence of the propagator pole (cf. [11]), and we are faced with the same problems as before. Also, our example implies that the definition of on-shell counterterms is ambiguous in the sense that it depends on the type of expansion (covariant or noncovariant) used for ReT,^. This supports our conclusion that on-shell finite-temperature renormalization cannot be defined in a generic or unique way, which is of course a consequence of the lack of Lorentz invariance. We can now compare our result for the finite-temperature radiative corrections to the Higgs decay rate in the low-temperature limit to the one given in [11]. The virtual-photon correction is given by -nvirtual -nG i TISE Lp = Lp + lp = rf + rS i + r A r o + rB» (6.34) with rf given in (6.6) and (6.7), and with TBi, TAm, T B2 given by (6.29), (6.33), (6.32). The real-photon correction was given in general form in (6.9) and (6.16). A complete evaluation is rather difficult; however, in the low-temperature approximation it suffices to expand (6.35) in a power Chapter 6. Radiative corrections and renormalization at finite temperature 110 series in T/m# and consider only the terms up to 0(T2). This was done in [11] and the authors find the following cancellation between the thermal part of (6.35) and (6.34), to The finite-temperature correction is nonzero, contrary to the result of [11] who eliminated in addition to the consistency problems discussed before, finite-temperature counterterms do not even give the correct decay rate at the one-loop level. Finally, we note that the temperature correction in (6.36) is positive, that is, the "thermal mass" correction actually enhances the decay rate. In particular, in the massless limit w —* 1 (6.36) reduces to However, in this case the thermal fermion sector can no longer be neglected and there will be additional contributions to (6.37). 6.4 Generalization and applications It is straightforward to generalize our results to a general thermal decay process with n-body phase space <j) —> <p\... <j)n and n > 2. An important physical example is neutron /3-decay n —• pe~u with radiative corrections. The fermion self-energy correction to the self-energy for such a process is shown in Figure 15. As already mentioned, the technique for the cancellation of the pinch singularities can be applied to any such diagram, hence the EeS-contribution will be of the generic form (6.36) T B2 with the momentum dependent counterterm Sm DH. This example shows clearly that, (6.37) Chapter 6. Radiative corrections and renormalization at finite temperature 111 Figure 6.6: Self-energy correction to Tiu(q) for a general decay process <f> —• <j>\. ..<f>n. = lim -jr -^o / d "phase space"%2 - m?)Tr [... (jH- m)ReT(p)(p + m)] m2—>m2 um J (6.38) The Lorentz invariant zero-temperature part ReT,0 can of course always reduced to a SZ3-factor by including the zero-temperature mass counterterm. For the finite-temperature part ReT13 we could use either the covariant expression (6.21), expanded around p* = m 2, or a direct, noncovariant mass shell expansion of the type (6.26). Evaluating the integral (6.38) we might call the contribution arising from the coefficient function proportional to (p—m) a "wave function renormalization" correction, the remainder a generalized "finite-temperature mass" contribution. Notice, however, that for a decay process with n-body phase space, n > 3, the kinematical constraints (mass shell -^functions) are not sufficient to fix both energy and momentum for the external fermion: for example, in neutron 0 de-cay the electron is emitted with a continuous energy-momentum spectrum. Consequently the expansion coefficients of ReT0 in the integral will always be momentum dependent and do no longer admit an interpretation as mass and wave function renormalization coun-terterms Sm0 and 6Z0, not even in the heuristic sense discussed in the previous section. Chapter 6. Radiative corrections and renormalization at Unite temperature 112 Thus it is also no longer possible to eliminate the "mass contribution" in the integral with an operational constant mass counterterm, and (6.38) is indeed the generic expression for the finite-temperature self-energy correction TfE to an n-body decay process. Needless to say, this also confirms our previous conclusion that finite-temperature renormalization is not a meaningful concept for decay/scattering rate calculations. If we are willing to give up Lorentz invariance of the Lagrangian we are of course free to fix some arbitrary four-momentum and define operational finite-temperature counterterms with respect to this renormalization point (as we did for two-body decay), but these constant counterterms will obviously not eliminate the momentum dependent "mass" term in (6.38) and merely complicate the interpretation of the rate in terms of physical parameters. It is also a perfectly well-defined problem to analyze quasiparticle propagators like (6.1), and extract physical information like correlation lengths, dispersion relations, etc. However, our analysis showed that these quantities will have no direct relation to the parameters in our decay rates, which is of course a consequence of the lack of Lorentz invariance. For a direct application of these results consider neutron 0 decay with radiative correc-tions at finite temperature. This reaction is important in cosmology since it is a central ingredient for the nucleosynthesis rates of the light elements in the early universe. These rates, in turn, are measurable and hence provide a probe of the conditions in the early universe. Previous calculations [8, 9] treated the self-energy corrections to the electron in standard fashion as temperature dependent mass shift and wave function renormalization correction. The mass shift was taken into account by replacing the mass shell ^ -function for the electron by the dispersion relation det(^  — m — ReT 0) —which corresponds to a momentum dependent mass counterterm in the Lagrangian—and the wave function renor-malization function was defined by 6Z$ = ^ReT 0^m„. A rigorous treatment requires the computation of the phase space integral (6.38) for 0 decay. Since neutron 0 decay is extremely phase space sensitive it would be interesting to see how this rigorous result Ch&pter 6. .Radiative corrections and renormalization at finite temperature 113 differs numerically from the one in [8] and if there are any corrections for the nucleosyn-thesis rates [25]. The previous calculations found the corrections to the abundances to be only 0.1 - 0.2%. However, recently a debate has arisen over the reliability of the standard model of primordial nucleosynthesis and several modifications and alternatives have been proposed; a rigorous result for the temperature corrections is thus clearly important. Finally let us emphasize that our results apply also to thermal scattering rates, that is, to transition rates for initial distributions of two or more particle species. These thermal cross sections are again related to the discontinuity (imaginary part) of the relevant n-point Green's functions [48] which in turn can be determined by the finite-temperature Cutkosky rules, and the self-energy corrections to external fermions lines will again contribute a phase space integral of the form (6.38). Since the kinematics of these scattering processes does not fix both energy and momentum all conclusions for decay processes with n-body phase space apply as well. Potential applications for our results are cooling rates for neutron stars, which are in part determined by neutrino-gauge boson scattering at high densities [45]; the techniques described here for fermion self-energies can of course be extended to gauge theories. 6.5 Summary and conclusions We have analyzed the problem of radiative corrections to finite-temperature decay rates to first order in perturbation theory, using the decay of a scalar boson into two fermions as an explicit example. Our treatment was based on the Niemi-Semenoff real-time formalism of finite-temperature field theory. The following results were obtained. Ghost vertices are necessary even to first order in perturbation theory to ensure the well-definedness of the theory (cancel pinch singularities), but do not contribute to the finite part of the rate. For the radiative corrections we found the vertex-correction and the photon emission and Chapter 6. Radiative corrections and renormalization at finite temperature 114 absorption processes to be essentially identical to previous results. In particular, the vertex-diagram renormalizes the coupling constant. However, the finite-temperature part of the fermion self-energy correction does not, in general, admit an interpretation in terms of local and renormalizable mass and wave function renormalization counterterms, due to the lack of Lorentz invariance. We argued that momentum dependent finite temperature counterterms conflict with the basic requirements of locality and renormalizability, and lead to uncontrollable behavior of the higher order terms in perturbation theory; momentum independent finite temperature counterterms, on the other hand, have unphysical features and are inherently ambiguous. We concluded that finite temperature renormalization is a problematic concept and of no practical value for decay and/or scattering rate calculations. Instead we derived an explicit algorithm for the direct computation of the finite-temperature self-energy corrections to a decay process and generalized it for processes of cosmological and astrophysical interest. An important point which we treated only briefly, is the cancellation of the infrared divergencies and mass singularities. To our knowledge a finite-temperature version of the Kinoshita-Lee-Nauenberg theorem—which guarantees the absence of these singularities at zero-temperature—is still lacking. In the low-temperature limit, with only the thermal photon distribution taken into account, the cancellation of the thermal singularities was shown for the vertex correction, emission and absorption rates and the "wave function renormalization part" of the self-energy [11, 20], Here we extended these results and showed explicitly that the "finite-temperature mass" does not introduce any new infrared or mass singularities. After this work was completed we became aware of two papers [4, 5] on a similar prob-lem (lepton pair production rates in a QCD plasma). The authors employ the same technique—finite-temperature Cutkosky rules—used in this paper, but the emphasis is on the cancellation of infrared and mass singularities, and their analysis includes also the thermal fermion distributions. Both groups find complete cancellation of the thermal Chapter 6, Radiative corrections and renormalization at finite temperature 115 divergencies, a result that supports the infrared reliability of finite-temperature pertur-bation theory. We note that the authors of ref. [4] employ the finite-temperature mass counterterms of [11] whereas the authors of [5] show the cancellation of the singularities also directly without these counterterms. This supports our assertion that the thermal "mass" correction is well-defined and momentum dependent counterterms are not needed to deal with infrared singularities. Chapter 7 Conclusion In this thesis, we investigated the structure of thermal decay and scattering rates, up to the one-loop level, using the Niemi-Semenoff real-time formulation of finite-temperature field theory. The decay rates were obtained from the imaginary part of the thermal self-energy, using the generalized Cutkosky rules for thermal real-time Green's functions. This method is ideally suited to real-time finite-temperature field theory. It avoids the explicit use of the matrix structure of the theory; it is, in principle, independent of concepts like asymptotic states and the LSZ theorem that are the basis of vacuum field theory; and it yields directly the physical decay (Landau damping) rate for particles immersed in a heatbath. At tree level, this rate consists of a nontrivial combination of the lowest-order decay and inverse decay reactions of the particles in the heatbath. As a nontrivial example, we studied neutron /3 decay at finite density. We gave a fully relativistic analysis of neutron 8 decay in finite density environments and derived analytic expressions for the decay rates. At the one-loop level, the imaginary part of the self-energy is effectively equivalent to the corresponding one-loop transition rates of the physical fields; thermal ghost fields only serve to cancel pinch singularities, but do not contribute to the rates. Thus, the damping rate in the heatbath is still effectively comprised of the transition rates between the physical fields. However, the radiative corrections to these transitions cannot be absorbed into a renormalization of the physical parameters in the Lagrangian, due to the lack of Lorentz invariance. In particular, the finite-temperature part of the self-energy correction does not admit an interpretation as a temperature dependent mass and wave function 116 Chapter 7. Conclusion 117 renormalization counterterm. We argued that momentum dependent finite-temperature counterterms, which are frequently employed in the literature, are not a valid concept, and that heuristic renormalization counterterms are of no practical use for decay and scattering rate calculations. We derived a general algorithm for the direct calculation of the self-energy correction to decay rates, and applied it to Higgs boson decay. The resulting rate was free of infrared and mass singularities. These results can immediately be applied to the electroweak and QCD processes dis-cussed in the introduction. The most interesting candidate among these reactions is thermal neutron 0 decay with radiative QED corrections. The previous calculations [8, 9] were based on the naive renormalization approach, described in Chapter 6. While emis-sion and absorption rates and the vertex correction were treated correctly as phase space integrals, there remains the problem of the self-energy correction which was included as a momentum dependent mass shift (counterterm). The phase space dependence of neutron 0 decay makes it an ideal system for a quantitative comparison between the "finite-temperature renormalization" approach and our rigorous treatment. The physical importance for nucleosynthesis has already been discussed. Unfortunately, the compli-cated three-body phase space for neutron 0 decay also precludes any simple estimate, and requires a numerical analysis. Thus, the actual size of the effect, and its potential influence on nucleosynthesis, is still an open problem. Appendix A The self-energy correction to the Higgs boson decay rate We write the noncovariant low-temperature expansion of iteE^ as Re^(p) = Bx(p) + B2(up,f) where Blip) B2{vp,p) = ^3 /(wp,p) and IA = i*r^nB(\k\) Je \k\ d 3k k>> |*| upk° -pk ep\k\ _ i where k° = \k\. I th can be calculated explicitly (cf. [11]). For 1° we obtain by direct integration 7°( W p ,p) = 7T3T2, 1 + V Svojp ^ n 1 — v where v — \p\/up. For p on mass shell, that is, p° = up, we have 2w3T2 d 3k 1 1*1 e*l*l - 1 = constant 118 Appendix A. The self-energy correction to the Higgs boson decay rate 119 and hence for I r P , ro « P *3T2 / I , H » n / = ^ K/°-?./) = ^— ( - m — -2 We calculate the decay rates 8 f d 4p (2TT)2 X [8(p°) - nF(p)][9(q° - p°) - nF(p - q)} xTr[(p- A + rn)(p + m)Bi(/p + m)] Let us first consider T BL. The trace is easily reduced to Tr[(jp- jq + m)(/p + m)B1(p + m)} = ^z\TrU- A + m)(p + m)] T J. 1 9 1 I A + —P • Up 0Up (p 2 - m2) -Tr 2w, •(p2 -m 2 ) 2 Taking the derivative and limit are now trivial and T Bl reduces to T B>(mH) = -X [9(p°) - nF(p))[9(p° - q°) - nF{p - q)) *Tr[(p- A + m)(fi + m)} e" 47T3 „ 2 _ 1 5/ IA + —P-0Jp oup 4 7 T 3 r 2 1 + uA _ t p e e, , r In- P r e e (m f f ) where we used 81 ^ dUr. = -1° , and v = J l 4m2 'p V m H r s ? requires the explicit evaluation of the integral. For the mass shell ^ -functions we have = w for p° = Up = mnj1. 6(p 2 — m2)<5((p — q) 2 — m 2) = 1 4mj/Wp 6(p0-p°(m 2))6(up-up(m2)) Appendix A. The self-energy correction to the Higgs boson decay rate 120 where or - 2\ mH . rh2 - m2 2 2mu ,(m2) = 2mu and we have to evaluate X W) - nF(p)}[0(q° - p°) - nF(p - q)} xTr[(d>- jq + m)U> + m) £{jp + m)] It is easy to show that for m 2 = m2, that is, for p° = q° - p° = m#/2 j t f [Otf) - nF(pQ)][e(q° - pP) - nF(p° - g °)] ^ = m 2 = 0 and therefore TB2(mH) = -e 2 g 2 ¥ t a n h ( ^ / 4 ) J im , (\p^Tr[... /] + §^Tr[... /]) 47r3 87rm| We have for m2 = m2, that is, for u>p = mn/2 \p\ — \lu2 - m2 = ~^-w and TTT-O IPI = - r — — = — — • For the trace we find Yim Tr[(i>- A + m)(j> + m) J(jp + m)] = 4\mHI°(p2-m2) + p(m 2).I(m 2)(p 2 + Zm 2-2mHp°)\^ 2 = -4m 2Hw2(p • I) where we recall that p • I = |7r 3 T 2 = constant. For the derivative of the trace we obtain JLTr[(j>- jq + m^ + m) jt(p + m) = 4 where we used mHI° - m2Hw2-^(p{rh 2) • J(m2)) j^ ^ = 4mH(l - w 2)I° p{m l)-I{m>) 'up(m2) + ™\  m 2 ) I°(m 2) - p\m 2)I(m 2) m' — m2 t0/ - i\ = P-I+ I°(m2) Appendix A. The self-energy correction to the Higgs boson decay rate Thus the rate T Bi is given by 6 9 t a n h ^ 47r3 8xml H 2m 2Hw(l - w2)I° + 2mHwp-1 e 2 T 2 2>m 2Hw2 1 + 4 1 - w 2 , 1 + w In • w 1 — w where we used the explicit form of 1°. Note that T tree(mH) limfl - w2)ln ] W 0 To calculate the rate Vs'2 where 1 87r3 V m J it remains to compute 8 dm? \p\Tr[{Jp- k + m)(p + rn^-ip + m)}\ = Qmnwp-I Together with our result for T® 2 this yields V BKmH) = 6 2 9 2 t a n h ^ 47r3 8itm 2H 2m2Hw(\ — w2)I° — 4m#tt>p • I 2T2 e 2T 3m 2Hw 2 -2 + 4 1 - w 2 , 1 + w •In T tree(mH) w 1 - w Bibliography [1] E. S. Abers and B. W. Lee, Phys. Rep. 9, 1 (1973) [2] A. A. Abrikosov, L. P. Gorkov and I. E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Mechanics, (Prentice Hall Inc., Englewood Cliffs, 1963) [3] K. Ahmed and S. Saleem, Phys. Rev. D 35, 1861 (1987) [4] T. Altherr, P. Aurenche and T. Becherrawy, preprint no. LAPP-TH-221/88 (1988) [5] R. Baier, B. Pire and D. Schiff, Phys. Rev. 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