Mass corrections to the hard thermal/dense loops

We compute corrections to the hard thermal (or dense) loop photon polarization tensor associated to a small mass $m$ of the fermions of an electromagnetic plasma at high temperature $T$ (or chemical potential $\mu$). To this aim we use the on-shell effective field theory, amended with mass corrections. We also carry out the computation using transport theory, reaching to the same result. Interemediate steps in the computations reveal the presence of potential infrared divergencies. We use dimensional regularization, as it is respectful with the gauge symmetry, and then show that all infrared divergencies cancel in the final result. We compare the mass corrections with both the power and two-loop corrections, and claim that they are equally important if the mass is soft, that is, of order $e T$ (or $e \mu$), where $e$ is the gauge coupling constant, but are dominat if the mass obeys $ e T


I. INTRODUCTION
Relativistic QED and QCD plasmas have attracted the interest of the physics community for their wide range of applications in both cosmological, astrophysical and also heavy-ion physics. In their weak coupling regime pertubative computations of different physical observables require the resummation of Feynman diagrams [1,2], the so called hard thermal loops (HTL), to attain a result valid at a certain order in the gauge coupling expansion (see [3] for a review and complete set of basic references). This makes the studies of relativistic plasmas particularly cumbersome.
For very large values of the temperature T (or of the chemical potential µ), a well-defined hierarchy of energy scales appears in these relativistic plasmas, that allows for effective field theory descriptions, very similar to those applied for non-thermal physics. In a series of papers [4][5][6][7][8][9], the on-shell efffective theory (OSEFT) has been developed in order to describe the physics of the hard scales, or scales of order T (or µ), which are on-shell degrees of freedom.
This effective field theory was initially developed to obtain quantum corrections to classical transport equations. Then it was realized that it could be used to improve the description of the hard scales of the plasma, and as by-product, also the soft scales of order eT (or eµ), where e is the gauge coupling constant.
The rationale and technical tools used by OSEFT are the same as that of other effective field theories, such as high density field theory (HDET) [10], or soft collinear effective field theory (SCET) [11,12], for example. After fixing the value of high energy scale, in this case the energy of the (quasi) massless fermion, which is of order ∼ T for thermal plasmas, one defines some small fluctuations around that scale. Integrating out the high energy modes, one is left with an effective theory for the lower scales or quantum fluctuations. The resulting Lagrangian is organized as an expansion of operators of increasing dimension over powers of the high energy scale.
In this manuscript we focus our attention to thermal corrections to the HTL photon polarization tensor associated to the fact that fermions on the plasma might not be strictly massless, but have indeed a small mass m much less than the temperature, m T . This is a realistic assumption, as only in the cosmological epoch before the electroweak phase transition all elementary particles were strictly massless. While the power corrections to the HTL photon polarization tensor have been computed with OSEFT in Ref. [5], here we will use the OSEFT for the computation of the leading fermion mass corrections. We also check that the same re-

II. SMALL MASS CORRECTIONS TO THE OSEFT
In this Section we derive the OSEFT Lagrangian including mass corrections to the third order in the energy expansion. Let us briefly discuss how this is achieved.
In the spirit of the OSEFT we split the momentum of the energetic fermion as where v µ is a light-like vector, p is the high scale, while k µ is the so called residual momentum, associated to the quantum fluctuation, and is such that k µ p. For the antifermion we will whereṽ µ is also a light-like vector. We will impose that where u µ is a frame vector, such that u 2 = 1, thus, it is time-like.
The OSEFT Lagrangian including small mass corrections has been derived in Ref. [9], and in an arbitrary frame it reads for the particle field χ v , where / D ⊥ = P µν ⊥ γ µ D ν , and the transverse projector is defined as . For antiparticles, the Lagrangian can be obtained after replacing p ↔ −p and v µ ↔ṽ µ . The Lagrangian has the same structure of that of SCET amended with small mass corrections [13,14]. Note that OSEFT and SCET are different theories, as the power counting is not the same (see [7] for a discussion on this point).
In writing the above Lagrangian, one assumes that the covariant derivatives, defined as iD µ = i∂ µ + eA µ , are soft, meaning that they are much less that the high energy scale, which here it is p. Equally, one assumes that the mass is such that m p. The Lagrangian can then be now expanded using that p is the hard scale of the problem.
For applications of plasma physics in thermal field theory, it is convenient to use the frame at rest with the plasma, thus u µ = (1, 0). Then one can replace / v 2 by / u = γ 0 (recall that / vχ v = 0). Expanding the Lagrangian on the high energy scale, one gets easily the first two terms in the energy expansion, which respect chirality It is convenient to introduce local field redefinitions to eliminate the temporal derivative appearing at second order, as in Ref. [5]. These simplify the computations at higher orders.
Thus, after the field redefinition the Lagrangian at second order reads Note that the term linear in the mass describes a breaking of chirality.
We will also need the Lagrangian up to third order. To eliminate temporal derivatives at that order, we perform the local field redefinition so that the final Lagrangian reads Please note that in the limit m = 0 we recover the same Lagrangians derived in Ref. [5].
The pieces which are quadratic in the mass can be recovered from those of Ref. [5] simply by The linear terms in m, originating from the expansion of the last term in Eq. (4), describe the breaking of the chiral symmetry induced by the fermion mass.
We present here how the OSEFT fermion propagators are modified in the presence of a small mass. The particle/antiparticle projectors in the frame at rest with the plasma are defined as P v = 1 2 / vγ 0 and Pṽ = 1 2 / vγ 0 , respectively. We introduce chirality projectors The propagators for a fermion of chirality χ in the Keldysh formulation of the real time formalism of thermal field theory Ref. [15] read where n F (x) = 1/(exp |x|/T + 1) is the Fermi-Dirac equilibrium distribution function. The function f (k, m) determines the dispersion law, and it is expanded also, we denote as f (n) (k, m) the n order term in the 1/p expansion. At lowest order and we have defined k = k · v, while as follows from Eqs. (6) and (8), respectively. The propagators for the antiparticle quantum fluctuations can be also be easily deduced. They read where the functionf (k, m) can be obtained from f (k, m), with the replacements v → −v and p → −p. Note the extra minus sign in the symmetric antiparticle propagator, absent in its particle counterpart.
In summary, the OSEFT fermion propagators in this case can be deduced from those of the massless case simply by replacing k 2 ⊥ → k 2 ⊥ +m 2 in the function that determines the dispersion relation at every order in the energy expansion.
Note that, for convenience, we keep the propagators above unexpanded in this Section, as done in Ref. [5], but in the explicit computation of the different diagrams they are to be expanded in a 1/p series.

RETARDED PHOTON POLARIZATION TENSOR
In this Section we compute the mass corrections to the retarded photon polarization tensor computed in OSEFT. Recall that there are two possible different topological diagrams that contribute to the computation, the bubble and the tadpole diagrams, see Fig. 1. The tadpole diagrams, absent in QED, take into account particle-photon interactions mediated by an off-shell antiparticle (and viceversa for antiparticle-photon interactions), and are needed to respect the Ward identity obeyed by the polarization tensor, as we will explicitly check in this manuscript.
In the Keldysh representation the particle contribution to the bubble diagram of the retarded polarization tensor has the structure 1 while the particle contribution to the tadpole diagram can be expressed as where the momentum dependence of the vertex functions V µ and W µν are understood. Similar expressions can be written for the antiparticle contributions to the polarization tensor.
Using the explicit expressions of the fermion propagators, one can carry out the integral in k 0 to arrive to the general expressions and for the bubble and tadpole diagrams, respectively.
The Feynman rules needed for the computation of the photon polarization tensor were given in Ref. [5] (see Tables I and II of that reference). In the presence of a mass in the OSEFT Lagrangian, new vertices appear proportional to the mass squared, which are given by There are also new vertices proportional to the mass, which imply a change in the fermion chirality. At the order we will compute the mass corrections, n = 3 in the energy expansion, these will not be needed, although they would be required at fourth order in the energy expansion. Note that at least two of these vertices would be needed in a computation of the photon polarization tensor to preserve the fermion chirality inside the loop.
We now evaluate the polarization tensor at different orders, noting that we either consider the energy expansion in the vertex functions, or in the fermion propagators, which can be used at the desired order of accuracy.
The first non-vanishing contribution to the photon polarization tensor occurs at n = 1, but it does not carry any mass dependence. This was computed in Ref. [5], and it reproduces the HTL contribution. Let us recall the main results here. Adding the bubble and the tadpole diagrams at order n = 1 gives where the retarded prescription l 0 → l 0 + i0 + is understood.
It is now important to return to the original momentum variable q µ . Using the identity [5] p,v and the relations [5] where now the symbols k ,q and k ⊥,q denote the components of k parallel and perpendicular toq ≡ q/q with q = |q|, respectively. We also define the vectors v μ q ≡ (1,q) ,ṽ μ q ≡ (1, −q) .
After adding both the particle and antiparticle contributions to the photon polarization tensor one arrives to the well-known HTL expression At second order in the energy expansion, and in the absence of chiral misbalance, the Bose-Einstein statistics and the crossing symmetry demands that the polarization tensor Π µν (l) be symmetric under the simultaneous exchange of µ ↔ ν and l ↔ −l [16]. These symmetries explain the absence of linear terms in the photon momenta in the polarization tensor, which ultimately explain why there are not n = 2 corrections in the polarization tensor in OSEFT.
This was explicitly checked in Refs. [5,17]. This reasoning applies actually to all the even orders of the energy expansion, n = 4, 6, 8, . . .. We do not expect thus mass corrections at even orders, neither, and we actually have checked that there are none at n = 2.
The first mass corrections to the photon polarization tensor occur at third order in the energy expansion, the same as the power corrections to the HTL computed in Refs. [5,17].
The mass dependent terms that arise in the bubble diagram are while in the tadpole one gets We add the two pieces, and go back to the full momentum variables. The final result, after adding also the antiparticle contributions, yields We note that the first two terms of the second line of Eq. (33) can be written as the HTL contribution, but with a coefficient proportional to e 2 m 2 rather than the Debye mass squared m 2 D = e 2 T 2 /3. Note also that in the tadpole diagram the pieces that are proportional to n F (q)/q 3 are in principle infrared divergent. These terms have to be evaluated using a regularization. We use dimensional regularization (DR), by assuming that the system is in d = 3 + 2 dimensions. In this case the momentum integrals become where θ parametrises an angle with respect to an external vector, and Γ(z) stands for the Gamma function. Furthermore, in d dimensions one has to change the coupling constant as The relevant infrared radial integral is where γ E is Euler's constant. However, when carrying out the angular integrals in d = 3 + 2 dimensions, the pole term and logarithm exactly cancel, as the angular integral turns out to be proportional to (that is, it would cancel if d = 3). If dΩ d is the solid angle element in d dimensions, and S d = 2π d /Γ(d/2) is the area of a d-dimensional unit sphere, one can check Thus, combing the two results one gets and there is no infrared divergence, but only a finite term. This finite term is ultimately needed to preserve the Ward identity obeyed by the polarization tensor, as can be checked after computing Note that if we had used a cutoff regularization of the integrals, the IR divergent terms would also vanish, but the above integral would not yield the finite contribution, the last term of Eq. (38), needed to respect the gauge invariance of the computation.
We define the longitudinal and transverse parts of the photon polarization tensor in d dimensions by We then find the following mass corrections to the longitudinal and transverse parts of the polarization tensor Let us finally stress that Eqs. (40)-(41) remain also valid in the presence of a finite chemical potential µ. In the presence of a chemical potential the particle and antiparticle contributions differ, but the final result can be recovered from Eq. (33), simply by replacing in Eq. (33) After an explicit evaluation of the corresponding integrals, one reaches to the same mass corrections to the polarization tensor which are valid at high temperature. In particular, our results still hold if we take T = 0 and keep the chemical potential µ as the high scale of the problem.

IV. COMPUTATION OF THE PHOTON POLARIZATION TENSOR FROM KINETIC THEORY
We compute in this Section the mass corrections to the photon polarization tensor as computed from kinetic theory. We use the transport approach derived from OSEFT, and focus on the vectorial component of the Wigner function. From Ref. [9], the transport equation associated to a fermion with chirality χ up to second order in the energy expansion reads where v µ χ = q µ /E q , and we take the frame vector that defines the system as u µ = (1, 0). Furthermore where n χ (X, q) is the distribution function, and the delta gives the on-shell constraint, Q χ m being a function of the momentum and the mass. The particle contribution to the electromagnetic current is expressed, at n = 2 order as We ignore in this manuscript the possible effect of the spin coherence function discussed in Ref. [9], which represent coherent quantum states of mixed chiralities. We also will ignore the terms in the transport equation, on-shell constraint, and in the vector current proportional to the spin tensor S µν χ , as they are irrelevant if the chiral chemical potential is zero, as the contribution of the two fermion chiralities makes these pieces to cancel in the macroscopic current. Those terms are relevant, though, to derive the chiral magnetic effect, which is not our goal here (see Appendix of Ref. [7] for that derivation). We thus write the on-shell constraint to the considered order of accuracy Where q = |q|. We now assume to be close to thermal equilibrium, such that where G χ (0) is the Wigner function in thermal equilibrium. Using the transport equation, we find and after computing one derives the polarization tensor as It is not difficult to find the particle contribution to the polarization tensor, which reads where for the particles. A similar expression holds for the antiparticles.
We compute all the pieces up to O(m 2 ), by noting that We thus find that the polarization tensor can be written as The HTL part, as arising from particles and antiparticles of the two possible chiralities, reads While the leading mass correction is One can check that so that the Ward identity is respected for the mass dependent pieces of the polarization tensor at this order. Note that the first integral of Eq. (58) has the same structure than the HTL contribution, but it is proportional to the fermion mass squared. This contribution was also found out in the diagrammatic computation of Sec. III. The second integral contains IR divergencies, but are of a quite different structure as those appearing in the diagrammatic computation, see Eq. (37). The apparent IR divergencies here are clearly non-local. We evaluate these integrals using DR.
Note that we only need to evaluate the integral where Li 2 stands for the Euler polylogarithmic function of order 2. All the remaining non-local integrals can de deduced from this one, after simple manipulations.
An explicit computation shows that after angular integration in d = 3 + 2 dimensions the IR divergencies exactly cancel, but there are remaining finite pieces, which in this case turn out to be non-local, and that allows one to reproduce the same value of the photon polarization tensor that we found in Sec. III.

V. DISCUSSION
We used OSEFT to assess how a small fermion mass would affect the retarded photon polarization tensor at soft scales in a ultrarelativistic electromagnetic plasma. While it could be obvious that such corrections would be of order m 2 /T 2 , the effective field theory techniques we used allowed us their proper evaluation.
Our results should be compared to both the power and two-loop corrections to the HTL tensor that have been computed in Refs. [5,17] and [18], respectively. More precisely, we will write Π I = Π htl I + Π m I + Π pow·corr Our results might be useful to obtain a better evaluation of different physical observables whenever the fermions in the plasma are not strictly massless, which is a realistic condition for most of the physical scenarios where the HTL resummation techniques have been applied so far.