Effective Coupling Constant of Plasmons

PHYSICAL REVIEW D 100, 056020 (2019)

Effective coupling constant of plasmons

Margaret E. Carrington Department of Physics, Brandon University, Brandon, Manitoba R7A 6A9, Canada and Winnipeg Institute for Theoretical Physics, Winnipeg, Manitoba, Canada

Stanisław Mrówczyński Institute of Physics, Jan Kochanowski University, Kielce PL-25-406, Poland and National Centre for Nuclear Research, Warsaw PL-02-093, Poland

(Received 13 July 2019; published 25 September 2019)

We study an ultrarelativistic QED plasma in thermal equilibrium. Plasmons—photon collective excitations—are postulated to correspond not to poles of the retarded photon propagator but to poles of the propagator multiplied by the fine-structure constant. This product is an invariant of the renormalization group that is independent of an arbitrarily chosen renormalization scale. In addition, our proposal is physically motivated since one needs to scatter a charged particle off a plasma system to probe its spectrum of collective excitations. We present a detailed calculation of the QED running coupling constant at finite temperature using the Keldysh-Schwinger representation of the real-time formalism. We discuss the issue of how to choose the renormalization scale and show that the temperature is a natural choice which prevents the breakdown of perturbation theory through the generation of potentially large logarithmic terms. Our method could be applied to anisotropic systems where the choice of the renormalization scale is less clear, and could have important consequences for the study of collective modes.

DOI: 10.1103/PhysRevD.100.056020

I. INTRODUCTION see e.g., Ref. [1]. It is therefore of interest to develop a method to study this issue for nonthermalized systems. The spectrum of collective excitations is a fundamental This subject is extremely important in the context of characteristic of any plasma system as it controls thermo- quark-gluon plasmas which are studied experimentally dynamic and transport properties of the system. In weakly using relativistic heavy-ion collisions. The dynamics of coupled QED or QCD plasmas, nontrivial plasmon dispersion relations can be obtained perturbatively at the the plasma is governed by QCD which is asymptotically one-loop level. Quantities like the plasma frequency and free. This means that the plasma becomes weakly interact- the screening mass depend linearly on α ¼ e2=4π for QED ing at sufficiently high momentum scales, and perturbative 2 methods are applicable. or α ¼ g =4π for QCD. However, in both QED and QCD We are particularly interested in nonequilibrium aniso- the coupling constant depends on a characteristic energy or momentum scale which emerges from the renormalization tropic QCD plasmas, which exist during some early phase of procedure—one says that the coupling “runs.” We need to the evolution of the system that is produced in a heavy-ion know the relevant energy scale at which to define the collision. Anisotropic plasmas produce unstable modes, as coupling constant when studying collective excitations. discussed at length in the review [2]. These unstable modes If the system is in equilibrium, there are two possible scales are damped through interparton collisions, but the damping to choose from: the momentum of the mode under con- effect is higher order in the coupling constant. In the sideration, or the temperature of the plasma. The situation is perturbative regime, the damping effect is small and unstable even less clear when we deal with nonequilibrium plasmas, modes can play an important role. In a strongly coupled where spectra of collective modes are often very rich; plasma, unstable modes would not have a significant effect on the dynamics of the system. The physics of the early-stage plasma therefore crucially depends on whether or not the regime of asymptotic freedom is reached. Published by the American Physical Society under the terms of We intend to study this problem systematically in the the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to context of quark-gluon plasma, both in and out of equilib- the author(s) and the published article’s title, journal citation, rium. In this paper we will start with the simpler case of a and DOI. Funded by SCOAP3. thermal QED plasma using the real-time Keldysh-Schwinger

2470-0010=2019=100(5)=056020(9) 056020-1 Published by the American Physical Society CARRINGTON and MRÓWCZYŃSKI PHYS. REV. D 100, 056020 (2019) formalism [3,4], which is applicable to both equilibrium and structure as the pressure in terms of its dependence on the nonequilibrium systems. Since quark masses are usually renormalization scale: it is given perturbatively by a series neglected in QCD plasma, we consider here an ultrarelativ- of terms depending on eðμÞ and lnðμ=TÞ that can be istic QED plasma where electrons and positrons are treated as rewritten by absorbing all log terms into a coupling eðTÞ. massless. We focus on photon collective modes, which are Throughout the paper we use natural units where called plasmons. To obtain their dispersion relations, we ℏ ¼ c ¼ kB ¼ 1. The indices i, j, k ¼ 1, 2, 3 and μ, ν ¼ 0, study the resumed retarded propagator at the one-loop level. 1, 2, 3 label, respectively, the Cartesian spatial coordinates In equilibrium systems, collective modes are usually and those of Minkowski space. The signature of the metric studied at the leading order of the hard-thermal-loop (HTL) tensor is ðþ; −; −; −Þ. approximation [5], which assumes that the system’s temperature is much bigger than the frequency and wave vector II. RESUMMED PHOTON PROPAGATOR of the collective mode. In these calculations, it is conventional to not consider the effect of the polarization of the A. Vacuum propagator vacuum, since this contribution is subleading within the The resumed time-ordered propagator is defined through HTL approximation. The effective coupling emerges beyond the Dyson-Schwinger equation as leading order, as a result of vacuum polarization effects. −1 μν −1 μν μν In this paper, we will study this issue in a more general ðD Þ ðkÞ¼ðD0 Þ ðkÞ − Π ðkÞ; ð1Þ context, with particular emphasis on the question of how the renormalization scale should be chosen in an anisotropic where ΠμνðkÞ is the time-ordered self-energy or polarization system. μν tensor and D0 ðkÞ is the free propagator. We consider two To explain the problem, let us discuss briefly the gauges: the general covariant gauge (GCG) and temporal example of the thermodynamic pressure. A perturbative axial gauge (TAG). In these two gauges the noninteracting calculation produces a series of terms that depend on the propagators are renormalized coupling eðμÞ and logarithms of the form lnðμ=TÞ, where μ is the renormalization scale and T is the 1 μ ν ∶ μν μν − 1 − ξ k k temperature. If we choose μ ¼ T, we obtain a simpler GCG D0 ðkÞ¼ 2 g ð Þ 2 ; ð2Þ k þ i0þ k expression for the pressure with the logarithms set to zero and the surviving factors of eðμÞ replaced by eðTÞ.Itis μ ν μν 1 μν k k commonly said that we resum to all orders the logarithm TAG∶ D0 ðkÞ¼ g þð1 þ ξÞ k2 þ i0þ ðk · nÞ2 terms into a running coupling constant which is defined at μ ν μ ν the appropriate scale. This has been verified to order e5ðμÞ k n þ n k − ; ð3Þ by explicit calculation [6–8]. We also note that if the ðk · nÞ renormalization scale μ were chosen to be some number much bigger or smaller than the temperature, the perturba- where ξ is an arbitrary gauge parameter that physical results tive expansion would break down, due to the appearance of should be independent of and the four-vector nμ ¼ large log terms. For this reason one says that the choice ð1; 0; 0; 0Þ defines the reference frame where the gauge μ ¼ T prevents the appearance of large logs. The method to condition is imposed. absorb the logarithm terms into the running coupling In vacuum, gauge and Lorentz invariance dictate that constant is discussed in the context of vacuum field theory the self-energy depends only on k2 (not k) and that it can be in Chapter 18 of Ref. [9]. We emphasize that in the written as the product of a four-dimensionally transverse calculation of a thermodynamic quantity, the only scale in tensor and a scalar function Pðk2Þ in the form the problem is the temperature and therefore it is natural to set μν μν 2 μ ν 2 the renormalization scale equal to the temperature, as ΠvacðkÞ¼ðg k − k k ÞPðk Þ: ð4Þ discussed above. The study of plasmons is more complicated because more than one momentum scale comes into play. In GCG inverting the Dyson-Schwinger equation (1) gives Physically meaningful quantities should be scale independent, but the renormalized photon propagator is not 1 kμkν kμkν DμνðkÞ¼ gμν − þ ξ ; ð5Þ renormalization group invariant. However, the product of k2ð1 − Pðk2ÞÞ k2 k2 e2ðμÞ and the propagator is. We therefore postulate that plasmons correspond not to poles of the retarded photon and in strict (ξ ¼ 0) TAG we obtain propagator but to poles of this product. We stress that this proposal is also motivated physically, since one needs to 1 1 − ij ij k ij k scatter a charged particle off the plasma system to probe its D ðkÞ¼ 2 2 T ð Þþ 2 2 L ð Þ; k ð1 − Pðk ÞÞ k0ð1 − Pðk ÞÞ spectrum of collective photon excitations. The product of the squared coupling and the propagator has the same ð6Þ

056020-2 EFFECTIVE COUPLING CONSTANT OF PLASMONS … PHYS. REV. D 100, 056020 (2019) where we have defined TABLE I. Multiplication table for the tensors defined in Eq. (10). kikj kikj ij k ≡ ij k ≡ δij − L ð Þ k2 ;Tð Þ k2 : ð7Þ AEB C AA 00 0 From now on when we refer to TAG we always mean with E 0 E 0 C the choice of the gauge parameter ξ 0. One advantage of B 00BC ¼ 2 working in TAG is that the components of the propagator C 0 CC−k ðB þ EÞ that have time-like indices are identically zero, and the components with only spatial indices are decomposed in ν μ terms of two projection operators: one three-dimensionally which is the component of n transverse to k , and define k transverse, Tð Þ, and the other three-dimensionally longi- μ ν μ ν μ ν tudinal, LðkÞ. μν ≡ μν − k k − nTnT μν ≡ nTnT A ðkÞ g 2 2 ;BðkÞ 2 ; The one-loop vacuum contribution to the self-energy can k nT nT μ ν be calculated in Euclidean space using dimensional regu- μ k k CμνðkÞ ≡ kμnν þ n kν;EμνðkÞ ≡ : ð10Þ larization. The procedure is standard and the result can be T T k2 found in many textbooks; see e.g., Chapter 11.2 of Ref. [10]. Rotating back to Minkowski space gives The sum of A and B is the four-dimensionally transverse Z tensor e2 1 1 1 P k2 − − γ dxx 1 − x ð Þ¼ 2π2 6 δ þ ð Þ μ ν 0 μν μν μν − k k A ðkÞþB ðkÞ¼g 2 : ð11Þ 4πM2 k ; ×ln 2 − 1 − 2 ð8Þ me xð xÞk The tensors A and B are individually four-dimensionally transverse, and A is three-dimensionally transverse. We M where is a mass parameter introduced by the regularization note for future reference the relations procedure. The parameter δ is related to the dimension d of the momentum space through d ¼ 4 − 2δ,andγ ≈ 0.5772 TijðkÞ¼−AijðkÞ; is the Euler-Mascheroni constant. In the formula (8) we keep k2 k2 k2 a finite electron mass me, as is usually done in vacuum QED. ij k − ij − ij ij 2 0 L ð Þ¼ 2 B ðkÞ¼ 2 C ðkÞ¼ 2 E ðkÞ: ð12Þ We note that when k < the argument of the logarithm is k0 2k0k k positive definite and Pðk2Þ is real and analytic. When k2 > 0 the logarithm has a branch cut when its argument becomes The full multiplication table for the tensors (10) is shown in negative. The maximum value of the factor xð1 − xÞ for x ∈ Table I. 2 2 ½0; 1 is 1=4 and therefore the branch cut begins at k ¼ 4me, The polarization tensor is four-dimensionally transverse which is the threshold for particle-antiparticle production. (due to the Ward identity) and therefore ΠμνðkÞ can be The sign of the imaginary part is determined by including the decomposed using only the tensors A and B as appropriate infinitesimal imaginary regulator. ΠμνðkÞ¼ΠTðkÞAμνðkÞþΠLðkÞBμνðkÞ; ð13Þ B. Medium propagator where the indices T and L indicate the scalar coefficients of Thermal field theory can be formulated in a Lorentz the tensors that are three-dimensionally transverse and covariant way [11]; nevertheless there is a preferred reference longitudinal, respectively. frame in the problem which is the rest frame of the heat ν We will calculate the retarded polarization tensor. We note bath, which we define with the four-vector n ¼ð1; 0; 0; 0Þ, that the Dyson-Schwinger equation (1) gives the resummed as in Eq. (3). The reason TAG is particularly useful at finite retarded propagator in terms of the noninteracting retarded temperature is that the gauge condition is imposed in the rest propagator and retarded polarization tensor, without cou- frame of the heat bath. pling to other causal structures (which is not true if one works An arbitrary symmetric tensor can no longer be decom- with the time-ordered propagator). Decomposing the free posed in terms of the coefficients of two tensors, and we propagator in terms of the tensors (10) and inverting the must extend the basis to include four independent tensors. Dyson-Schwinger equation (1) gives the propagator in GCG Using the notation of Ref. [5] (see Sec. 5.2.2), we introduce the four-vector DμνðkÞ¼DTðkÞAμνðkÞþDLðkÞBμνðkÞ μ ν μ μν k k ξ μν n ≡ g − nν; T 2 ð9Þ þ 2 þ E ðkÞ; ð14Þ k k þ ik00

056020-3 CARRINGTON and MRÓWCZYŃSKI PHYS. REV. D 100, 056020 (2019) and in TAG transverse tensor has a pole when k2 ¼ 0, but the coefficient of the three-dimensionally longitudinal tensor has a pole at k2 2 0 − ij T ij k L ij k k0 ¼ . Physically this tells us that in vacuum there is no D ðkÞ¼D ðkÞT ð Þþ 2 D ðkÞL ð Þ; ð15Þ k0 propagating three-dimensionally longitudinal mode, and the appearance of these modes is a medium effect. where 1 III. RETARDED POLARIZATION TENSOR DT;LðkÞ ≡ : ð16Þ k2 − ΠT;LðkÞ Weldon computed [11] the real part of the time-ordered polarization tensor of an ultrarelativistic QED plasma, which The positions of poles of the photon propagator should be equals the real part of the corresponding retarded polarization independent of the chosen gauge and the formulas (14), (15), tensor. He obtained the leading HTL contribution and the and (16) are clearly consistent with gauge independence. In next-to-leading-order contribution from the one-loop dia- either gauge, a transverse plasmon is found as a solution of gram. We have calculated the real part of the one-loop the dispersion equation k2 − ΠTðkÞ¼0 and a longitudinal retarded polarization tensor, to next-to-leading order. We plasmon is the solution of k2 − ΠLðkÞ¼0. apply the Keldysh representation of the real-time formulation In the next section we will show that the retarded of statistical field theory. Our method does not explicitly polarization tensor can be divided into a vacuum piece, require the use of thermal distribution functions, and we and a medium contribution that vanishes in the limit that the expect it to be generalizable to anisotropic systems that can distribution function goes to zero. We write be described by a distribution function which has the same asymptotic form as a thermal distribution. We have shown ΠT;L ΠT;L ΠT;L ðkÞ¼ vac ðkÞþ medðkÞ; ð17Þ that in equilibrium our method reproduces the result of Ref. [11]. Our calculation is described in Appendix A, and with further details can be found in Refs. [4,12]. A related calculation was done recently using an on-shell effective ΠT ΠL 2 2 vacðkÞ¼ vacðkÞ¼k Pðk Þ: ð18Þ field theory approach [13,14]. The retarded polarization tensor is We comment briefly on the fact that we have discussed the time-ordered vacuum polarization tensor in Sec. II A,while Z 3 X d p 1 − 2n ðjpjÞ in this section we work with the retarded polarization tensor ΠμνðkÞ¼2e2 f ð2πÞ3 jpj which is directly relevant to the study of collective modes. n¼1 μ ν μ ν μ ν μν The time-ordered vacuum polarization tensor can be Wick 2p p þ p k þ k p − g p · k rotated to Euclidean space where we can perform the × 2 þ ; ð20Þ ðp þ kÞ þ iðp0 þ k0Þ0 p dimensional regularization. On the other hand, the Dyson- p0¼nj j Schwinger equation defined on the Keldysh-Schwinger p ≡ jpj=T 1 −1 contour can be most easily solved for the retarded polariza- where nfðj jÞ ðe þ Þ is the distribution function tion tensor, as explained above Eq. (14).InSec.IV we will of massless fermions. need to combine thevacuum and medium contributions to the The vacuum and medium contributions can be easily self-energy, but this is straightforward because the real parts separated in the expression (20) as required by Eq. (17).The of the time-ordered and retarded self-energies are equal to real part of the vacuum piece is ultraviolet divergent and must each other [5], and the imaginary parts are finite and play be renormalized. In Sec. II Awe have renormalized the time- no role in our calculation. In Eqs. (17) and (18), and all ordered vacuum self-energy, by performing a Wick rotation T;L following equations, it should be understood that Πvac ðkÞ to Euclidean space and using dimensional regularization. will be assigned retarded boundary conditions. (As already mentioned, the real parts of the retarded and time- Using Eqs. (17) and (18), we can rewrite the transverse ordered self-energies are equal.) The medium part is ultra- and longitudinal contributions to the propagator (16) as violet finite due to the distribution function which goes exponentially to zero as jpj approaches infinity. 1 The polarization tensor (20) is symmetric [ΠμνðkÞ¼ DT;LðkÞ ≡ : ð19Þ νμ μν 2 1 − 2 − ΠT;L Π ðkÞ] and four-dimensionally transverse [kμΠ ðkÞ¼0]. k ð Pðk ÞÞ medðkÞ According to Eq. (13), such a tensor depends on only two ΠT;L → 0 ΠT In the vacuum limit medðkÞ and using Eqs. (11) and independent scalar functions, which we have called ðkÞ (13) we recover the usual form for the vacuum polarization and ΠLðkÞ and defined in Eq. (13). The easiest way to tensor (4). We see also that Eqs. (14) and (15) reduce to obtain ΠTðkÞ and ΠLðkÞ is to calculate the zero-zero Eqs. (5) and (6). In addition, we have from Eqs. (15) and (19) component and the trace of the polarization tensor, and that in vacuum the coefficient of the three-dimensionally to use the relations

056020-4 EFFECTIVE COUPLING CONSTANT OF PLASMONS … PHYS. REV. D 100, 056020 (2019) 2 1 μ k 00 adding hats to the former ones and explicitly showing their ΠT k Πμ k Π k ; ð Þ¼2 ð Þþk2 ð Þ dependence on μ. Our renormalization condition is that in 2 the vacuum limit, where the medium part of the polarization k 2 2 ΠL k − Π00 k : tensor vanishes, the renormalized propagator with k → −μ ð Þ¼ k2 ð Þ ð21Þ coincides with the free propagator. This condition can be We have calculated the medium part of the one-loop enforced by shifting retarded polarization tensor to next-to-leading order in the Pðk2Þ → Pˆ ðk2; μÞ¼Pðk2Þ − Pð−μ2Þ; ð25Þ k0=T; k =T expansion in ð j j Þ. We give some details of our μν Πˆ 2 2 0 method in Appendix B. To leading order one obtains the which gives vacðkÞjk ¼−μ ¼ and provides the renormal- familiar HTL results ization constant as 2 2 2 e T k0 jkjþk0 Z3ðμÞ¼1 þ Pð−μ Þ: ð26Þ Π00 k 1 − − iπΘ −k2 ; ½ medð ÞLO ¼ 3 2 k ln k − ð Þ j j j j k0 We note that to obtain the results in Sec. III for the 2 2 μ e T medium contribution to the self-energy we set the electron Π k : ½ medμð ÞLO ¼ 3 ð22Þ mass to zero, because we are interested in an ultrarelativistic plasma where the electron mass is assumed negligible The next-to-leading-order (NLO) contributions are compared to the temperature. In the vacuum calculation ffiffiffi 2 2 p 2 that we are discussing here, we must therefore also set e k 8k ½Π00 ðkÞ ¼ ln ; m ¼ 0 for consistency. The m → 0 limit is conventional med NLO 12π2 T2 e e in the vacuum calculation anyway, because the integrals 2 2 4 2 Πμ − e k k that are calculated are dominated by their ultraviolet ½ medμðkÞNLO ¼ 2 ln 2 : ð23Þ 0 4π T contributions. With me ¼ Eqs. (8) and (25) give 2 − 2 We note that Eqs. (22) and (23) show that there is a nonzero ˆ 2 μ e k Pðk ; Þ¼ 2 ln 2 : ð27Þ imaginary part when k2 < 0. This is exactly the opposite 12π μ behavior from what was found in vacuum [see Eq. (8)] We mention again that the argument of the logarithm in where we saw that the imaginary part is nonzero only for Eq. (27) indicates that the self-energy has an imaginary part time-like momenta, in which case the virtual photon can for k2 > 0, which means physically that a virtual time-like decay into physical final states. For the medium contribu- photon can decay into physical final states. Using retarded tion, the nonzero imaginary part of the self-energy appears 2 2 þ boundary conditions we take k → k þ ik00 and rewrite for space-like momenta and corresponds physically to the the formula (27) as scattering of electrons and positrons with momenta of order T on the low-momentum photon. In plasma physics, this 2 2 ˆ 2 e jk j 2 P k ; μ − iπΘ k k0 : phenomenon is known as Landau damping. We also note ð Þ¼12π2 ln μ2 ð Þsgnð Þ ð28Þ that the results (22) and (23) agree to next-to-leading order with those given in Ref. [11]. Substituting the expression (28) into Eq. (24) provides

IV. RENORMALIZATION Dˆ T;Lðk; μÞ Renormalization is usually done by including counter- 1 ¼ 2 2 ; terms in the Lagrangian and reexpressing bare quantities in 2 1 − eˆ ðμÞ jk j − πΘ 2 − ΠT;L k ½ 12π2 ðlnð μ2 Þ i ðk Þsgnðk0ÞÞ ðkÞ terms renormalized ones. The divergent part of the counter- med terms are chosen to cancel the divergences in the n-point ð29Þ functions, and the finite parts are determined by enforcing a chosen renormalization condition. We write the renormal- where the medium contributions are given in Eqs. (22) and ized propagator (23). The coupling constants in Eq. (28) are renormalized coupling constants (which are defined in the next section) 1 1 and should be properly written as a function of the scale μ. ˆ T;L μ ≡ T;L D ðk; Þ D ðkÞ¼ 2 ˆ 2 T;L ; This is made explicit in Eq. (29), and the coupling constants Z3ðμÞ k ð1 − Pðk ; μÞÞ − Π ðkÞ med in the medium contribution in this equation should also be ð24Þ taken to be renormalized coupling constants. Using Eqs. (22) and (23) and replacing eˆ2ðμÞ by 4παˆðμÞ we obtain where the first part of the equation defines the renormalization constant Z3, and μ is a new scale that enters through 1 Dˆ T;Lðk; μÞ¼ ; ð30Þ the renormalization condition. We distinguish renormalized 2 αˆðμÞ T2 T;L k 1 − ln 2 − αˆ μ π k quantities from their nonrenormalized counterparts by ½ 3π ðμ Þ ð Þ ð Þ

056020-5 CARRINGTON and MRÓWCZYŃSKI PHYS. REV. D 100, 056020 (2019) where πT;LðkÞ indicates contributions without potentially αˆðμÞ 1 αˆðμÞDˆ T;Lðk; μÞ¼ large logarithmic factors, which are therefore of no interest to 1 − αˆðμÞ T2 k2 − αˆ μ πT;L k 3π lnðμ2 Þ ð Þ ð Þ us. We stress that πT;LðkÞ are defined so that they do not 1 include the overall factor αˆðμÞ, and should not be confused αˆ ¼ ðTÞ 2 ; ð36Þ with ΠT;LðkÞ. k − αˆðμÞπT;LðkÞ where the first line is obtained using Eq. (30) and the V. RENORMALIZATION GROUP INVARIANT 2 equality holds up to Oðαˆ Þ. The factor αˆðTÞ in the second AND COLLECTIVE MODES line comes from Eq. (35), which also makes it clear that We postulate that collective modes correspond not to the remaining factor αˆðμÞ can be written as αˆðTÞ since the poles of the photon propagator but to poles of the photon difference between these two quantities is of order αˆ 2, and propagator multiplied by the fine-structure constant α. we have already dropped terms of this order. Equation (36) One motivation is that physical quantities should be inde- therefore becomes pendent of the renormalization scale μ, and the product αˆðTÞ αˆðμÞDˆ L;T ðk; μÞ is a renormalization group invariant, as αˆ μ ˆ T;L μ αˆ ˆ T;L ð ÞD ðk; Þ¼ 2 T;L ¼ ðTÞD ðk; TÞ; discussed at length in Chapter 9 of the classical text [15]. k − αˆðTÞπ ðkÞ Our idea is also motivated physically: to probe a photon ð37Þ collective mode one needs to scatter a charged particle off the plasma system. The charge should therefore be consid- which shows that the natural renormalization scale of ered together with the propagator from which the collective collective modes in the thermal plasma is the temperature. mode will be determined. In vacuum it is natural to choose thep renormalizationffiffiffiffiffiffiffi VI. CONCLUSIONS AND OUTLOOK scale to be equal to the momentum scale jk2j, which is the only physical scale available. We will show below that We claim that dispersion relations of plasmons should at finite temperature one should choose μ ¼ T. not be calculated by finding the poles of the retarded We start by deriving the equation for the running coupling photon propagator, but rather by finding poles of the constant, which describes how the coupling constant evolves propagator multiplied by the fine-structure constant. This with the scale at which it is defined. To satisfy the Ward product is a renormalization group invariant, as the ampli- identity the charge must be renormalized using the previ- tude of a physical process should be. We have given a ously introduced renormalization constant Z3ðμÞ given by physical argument to use this product to define plasmons. ˆ T;L Eq. (26) We have also shown that the statement that αˆðμÞD ðk; μÞ is a renormalization group invariant is equivalent to the αˆ μ Z3 μ α: 31 ð Þ¼ ð Þ ð Þ statement that the polarization tensor is given perturbatively αˆ μ μ Since the bare coupling constant on the right side of Eq. (31) by a series of terms depending on ð Þ and lnðTÞ that can be is independent of μ, the evolution of αˆðμÞ is determined by rewritten by absorbing all log terms into a running coupling the equation αˆðTÞ using Eq. (35). dαˆðμÞ We are going to extend the analysis presented in this μ ¼ βðμÞ; ð32Þ dμ paper to the case of fermionic collective modes in QED plasma, which are conventionally determined by the poles where the beta function is defined as of the retarded electron propagator. In this case however, dZ3ðμÞ αˆðμÞ the propagator multiplied by the coupling constant is not a βðμÞ ≡ μ : ð33Þ dμ Z3ðμÞ renormalization group invariant, but instead, the quantity that is independent of the renormalization scale is the Using Eq. (26) together with the formula (8), the well-known product of the electron propagator and the vertex function. beta function at the one-loop level is found as Our ultimate goal is to study collective modes of QCD 2 plasma, with a particular emphasis on anisotropic systems. βðμÞ¼ αˆ 2ðμÞ; ð34Þ 3π The structure of anisotropic plasmas is much more com- and the solution of the evolution equation (32) equals plicated and the choice of the renormalization scale is not clear, but our method is rather general and hopefully can be αˆðμ0Þ applied to such a system. αˆðμÞ¼ 2 ; ð35Þ αðμ0Þ μ 1 − 3π lnð 2Þ μ0 ACKNOWLEDGMENTS with the famous Landau pole structure. Equations (24) and (31) clearly show that the product We are very grateful to Alina Czajka and Erik Kofoed for αðμÞDT;Lðk; μÞ is a renormalization group invariant. We see numerous fruitful discussions. This work was partially this explicitly by writing supported by the National Science Centre, Poland under

056020-6 EFFECTIVE COUPLING CONSTANT OF PLASMONS … PHYS. REV. D 100, 056020 (2019) Z 2 X 4 Grant No. 2018/29/B/ST2/00646, and by the Natural μν μν ð−ieÞ d p iΠ k iΠ k γμΓ Sciences and Engineering Research Council of Canada. retð Þ¼ arð Þ¼ 2 2π 4 aij ii0jj0 ð Þ ν × G 0 ðp þ kÞG 0 ðpÞγ Γ 0 0 : ðA6Þ APPENDIX A: RETARDED ii jj ri j POLARIZATION TENSOR The sum over Keldysh indices fi; i0;j;j0g ∈ fr; ag is easily In this Appendix we give some details of our calculation done because Gaa ¼ 0 and a vertex function with an odd of the one-loop retarded polarization tensor (20). We work number of a indices vanishes. The result is in the real-time formulation of finite-temperature field Z theory, using the Keldysh representation. The one-loop 2 4 μν e d p μ ν contribution to the retarded polarization tensor can be Π ðkÞ¼i Tr½γ ðp= þ m Þγ ð=k þ p= þ m Þ ret 2 ð2πÞ4 e e found rather easily starting with what is often called the 1-2 basis; see e.g., Sec. IVA of Ref. [16], but we sketch × ½fðpÞrðp þ kÞþaðpÞfðp þ kÞ: ðA7Þ here a more general method reviewed in Ref. [4] which is applicable to multiloop diagrams. From this point on we assume that the system is in thermal 2 2 The electron propagator is a × matrix of the form equilibrium, we set the electron mass to zero, and we take jpj=T −1 nfðpÞ¼n¯ fð−pÞ¼ðe þ 1Þ . Performing the trace Grr Gra Gsym Gret G ¼ ¼ ; ðA1Þ over gamma matrices, and changing variables to combine 0 Gar Gaa Gadv the two terms in the square brackets in Eq. (A7), one obtains where the retarded, advanced and symmetric propagators are given by Z 4 Πμν 4 2 d p 2 μ ν μ ν μ ν retðkÞ¼ ie 4 ½ p p þ p k þ k p GretðpÞ¼ðp þ meÞrðpÞ; ð2πÞ μν GadvðpÞ¼ðp þ meÞaðpÞ; − g p · ðp þ kÞfðpÞrðp þ kÞ: ðA8Þ

GsymðpÞ¼ðp þ meÞfðpÞ; ðA2Þ It is straightforward to perform the integral over p0 using with the delta function in the symmetric propagator [see Eq. (A3)]. The resulting expression for the photon self- 1 ≡ energy is given in Eq. (20). rðpÞ 2 2 þ ; p − me þ i0 sgnðp0Þ 1 ≡ aðpÞ 2 2 þ ; APPENDIX B: MEDIUM CONTRIBUTION p − me − i0 sgnðp0Þ We present here some details of our calculation of the fðpÞ ≡ −2πi½ð1 − 2nfðpÞÞΘðp0Þ medium contribution to the retarded polarization tensor in 1 − 2¯ −p Θ − δ 2 − 2 þð nfð ÞÞ ð p0Þ ðp meÞ; ðA3Þ Eq. (20). The integrals of interest are where n ðpÞ and n¯ ðpÞ are the fermion and antifermion Z Z f f 2 2 ∞ X 1 1 distribution functions. The self-energy has the form Π00 − e p p p 00 medðkÞ¼ 2 dj jj jnfðj jÞ dxI ; π 0 2 −1 n¼1 0 Π Π Π adv Π rr ra A4 ðB1Þ ¼ ¼ Π Π ð Þ Πar Πaa ret sym Z Z 2 2 2 2 2 ∞ X 1 1 and the vertex function is a × × tensor which can be Πμ − e p p p μ medμðkÞ¼ 2 dj jj jnfðj jÞ dxI μ; written as π 0 2 −1 n¼1 fΓrrr; ΓrragfΓrar; Γraag ðB2Þ Γμ ¼ −ieγμ fΓarr; ΓaragfΓaar; Γaaag with f0; 1gf1; 0g p 2p2 p k ¼ −ieγμ : ðA5Þ 00 ≡ nj jk0 þ þ · 1 0 0 1 I 2 þ ; f ; gf; g 2ðnjpjk0 − p · kÞþk þ in0 2 p − p k The contribution to the retarded self-energy from the μ ≡ − ðnj jk0 · Þ I μ 2 þ ; one-loop diagram is 2ðnjpjk0 − p · kÞþk þ in0

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p·k 00 μ ω ω where x ≡ . To calculate Π k and Π k to − þ þ jpjjkj medð Þ medμð Þ trχ1 ¼ ½lnðjpj − ω− þ i0 Þ 00 μ p ω − ω leading order we expand I and I μ in ðk0=jpj; jkj=jpjÞ j jð − þÞ − p ω 0þ − p − ω 0þ which gives lnðj jþ − þ i Þ lnðj j þ þ i Þ 2p2k2 00 − þ lnðjpjþω þ i0þÞ; ðB6Þ ILO ¼ þ 2 þ ð2njpjk0 − 2p · k þ in0 Þ 2 ω ≡ k 2 2p þ njpjk0 þ p · k where we have defined ðk0 j jÞ= . We rewrite the þ þ ; 2njpjk0 − 2p · k þ in0 arguments of the logs using the relations

μ 2jkjx − 2nk0 þ þ jpj − ω þ i0 ¼jpj − ω∓ þ i0 ∓ jkj; I μLO ¼ þ : 2nk0 − 2jkjx þ in0 p ω 0þ p ω 0þ k j jþ þ i ¼j jþ ∓ þ i j j; Performing the x integral gives the familiar HTL results (22). k p ω 0þ As a check of our notation we observe that Eq. (21) and expanding the logarithms in j j=ðj j þ i Þ, shows that Eqs. (B5) and (B6) become 2 1 k ω ω − ω ω ω − ω 00 −ð þ −Þ þð þ −Þ lim 2 L ¼ lim 2 2 00 ðB3Þ χ −1 − k0→0 − Π k0→0 k Π 1 ¼ 2 2 þ 2 2 k medðkÞ k ð þ medðkÞÞ 4 p − ω 0þ 4 p − ω 0þ ð − þ i Þ ð þ þ i Þ k ðω− − ω Þð2ω− þ ω Þ ðω− − ω Þðω− þ 2ω Þ and from Eq. (22) we find the pole at imaginary j j¼imD − þ þ þ þ þ ; 2 2 2 3 12 p2 − 2ω2 0þ 12 p2 − 2ωþ 0þ that corresponds to the screening mass mD ¼ e T = . ð − þ i Þ ð þi Þ We find the NLO contribution from Z ðB7Þ 2e2 ∞ Π − p p p ω ω ω ω NLOðkÞ¼ 2 dj jj jnfðj jÞ − þ − þ π 0 trχ1 ¼ − − : ðB8Þ Z p2 − ω2 0þ p2 − ω2 0þ X 1 1 − þ i þ þ i − × dxðI ILOÞ : ðB4Þ 2 −1 −1 n¼1 The term on the right side of the first line in Eq. (B7) 00 cancels the contribution from χ2 . The remaining terms in 00 The square brackets in Eq. (B4) will be denoted χ or Eqs. (B7) and (B8) are substituted into Eq. (B4). trχ. There are two kinds of terms in the integrand: those The last step is to perform the remaining integral over − 0þ −1 of the form ðA Bx þ i Þ , and those with the form jpj. We use the identity nfðjpjÞ ¼ nbðjpjÞ − 2nbð2jpjÞ, − 0þ −2 ðA Bx þ i Þ . We separate the contributions from where nfðjpjÞ and nbðjpjÞ are fermionic and bosonic χ00 χ00 χ00 these two types of terms by writing ¼ 1 þ 2 and equilibrium distribution functions, and rescale variables χ χ χ χ00 1 tr ¼ tr 1 þ tr 2. It is straightforward to show that 2 ¼ so that all terms have a factor nbðjpjÞ. Finally we take the and trχ2 ¼ 0. Performing the x integral for the type 1 terms, real part of each self-energy component and use we obtain Z 1 ∞ 1 1 T2 00 p p p P − χ1 ¼ ½ðjpj − ω−Þðjpj − ω Þ dj jj jnBðj jÞ 2 2 ¼ ln 2 þ ðB9Þ 2 p ω − ω þ 0 p − 4 j jð − þÞ M M p − ω 0þ − p − ω 0þ × ðlnðj j − þ i Þ lnðj j þ þ i ÞÞ where M is assumed positive and real, and the dots indicate − p ω p ω p ω 0þ ðj jþ −Þðj jþ þÞðlnðj jþ − þ i Þ terms higher order in M=T. The final next-to-leading-order result is given in Eq. (23). − p ω 0þ lnðj jþ þ þ i ÞÞ; ðB5Þ

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