Equilibration of the chiral asymmetry due to finite electron mass in electron-positron plasma

We calculate the rate of collisional decay of the axial charge in an ultrarelativistic electron-positron plasma, also known as the chirality flipping rate. We find that contrary to the existing estimates, the chirality flipping rate appears already in the first order in the fine-structure constant $\alpha$ and is therefore orders of magnitude greater than previously believed. This happens because the processes of near-collinear bremsstrahlung can occur in plasma due to the modified dispersion relations and finite quasiparticles' lifetime. Moreover, even higher-order processes, e.g., the Compton scattering, also contribute to $\mathcal{O}(\alpha)$ result because of the infrared divergence in their cross section being regularized on the soft scale $\sim eT$ by thermal corrections. Our results are important for the description of the early Universe processes (such as leptogenesis or magnetogenesis) that affect differently left- and right-chiral fermions of the Standard Model, as discussed in more details in the companion Letter.

We calculate the rate of collisional decay of the axial charge in an ultrarelativistic electronpositron plasma, also known as the chirality flipping rate. We find that contrary to the existing estimates, the chirality flipping rate appears already in the first order in the fine-structure constant α and is therefore orders of magnitude greater than previously believed. This happens because the processes of near-collinear bremsstrahlung can occur in plasma due to the modified dispersion relations and finite quasiparticles' lifetime. Moreover, even higher-order processes, e.g., the Compton scattering, also contribute to O(α) result because of the infrared divergence in their cross section being regularized on the soft scale ∼ eT by thermal corrections. Our results are important for the description of the early Universe processes (such as leptogenesis or magnetogenesis) that affect differently left-and right-chiral fermions of the Standard Model, as discussed in more details in the companion Letter.
Unlike the electric charge, the axial charge is not fundamentally conserved. Typically, there exist scattering events which lead to chirality flipping and equilibration of the opposite chirality components. For this reason the applicability conditions of chiral hydrodynamics depend on the kinetic rate describing chirality flipping. In quantum electrodynamics (QED) and other weakly coupled quantum theories transport coefficients and kinetic rates are, in principle, amenable to calculation by means of perturbation theory. Computations of transport coefficients in high-temperature QED (more generally, in Standard Model) plasmas have been the subject of active research for many years (see, e.g., [33][34][35][36][37][38][39][40][41][42][43][44], see [42,43,45] for review and discussion). Notwithstanding the broad scope of this theoretical effort, rigorous methods have never been applied to the calculation of the kinetic coefficient associated with the collisional decay of the axial charge -the chirality flipping rate, Γ flip .
In our accompanying paper [46], we use the Boltzmann's kinetic theory to investigate the collisional decay of the axial charge in an electron-positron plasma at temperatures below the electroweak crossover and much greater than the electron mass. We demonstrate the existence of scattering channels which contribute to the chirality flipping rate in the first order of the fine-structure constant α, despite their nominal perturbative order being α 2 . This happens due to the power law infrared divergence in the process's matrix element, which gets regularized on the soft scale q IR ∼ eT . We also provide qualitative arguments as to why apart from the 2 ↔ 2 processes, there also might exist nearly collinear 1 ↔ 2 processes also contributing to the same order. In the present paper, we perform a comprehensive investigation of all scattering mechanisms contributing to the chirality flipping rate to the first order in α. The main result of our analysis is the leading-order (in α) asymptotic expression for the chirality flipping rate summarized in equation (47) and the rate equation (48), where we used the numerical value of the fine structure constant in all logarithmic terms.
The structure of the article is as follows. In Sec. 2 we state the problem and qualitatively discuss the scattering processes contributing to chirality flipping rate at leading order. In Sec. 3 we introduce the diagrammatic formalism which we then use for the perturbative calculation of the chirality flipping rate. We investigate the structure of infrared divergences in the leading-order terms of the perturbative expansion and then perform partial resummation of the leading singularity in all orders of perturbation theory. In Sec. 4 we summarize all our results. Appendices A-E provide the details of the computations.

STATEMENT OF THE PROBLEM AND PRELIMINARY DISCUSSION
Consider a hot QED plasma in which the typical kinetic energy of an electron is ultrarelativistic. In this regime, each electron can be characterized by an almost conserved chirality quantum number, which is defined as an eigenvalue of the operator γ 5 . 1 The approximate conservation of chirality can be expressed in the form of a continuity equation where j µ 5 =ψγ µ γ 5 ψ.
is the axial four-current. The two terms on the right-hand side of Eq. (1) represent two very different mechanisms by which the axial charge, that is the overall difference between right-handed and left-handed particles, is not conserved. The first term represents collisional decay of the axial charge due to chirality-flipping collisions. It contains the local boost parameter η µ , which in a homogeneous system has the meaning of the center-of-mass four-velocity and therefore satisfies η µ η µ = 1; and is proportional to a kinetic rate Γ flip which is the main subject of the present work. The second term on the right-hand side of (1) arises from the axial gauge anomaly, a subtle quantum phenomenon, by which the spectral flow of the single-particle Dirac operator results in a time evolution of the axial charge of electrons without any actual chirality-flipping collisions [47]. It is sometimes convenient to absorb the effects of the anomaly into an appropriate redefinition of the axial current. In particular, one can introduce a gauge invariant charge [12,13,48] which is conserved in the absence of chirality-flipping collisions and obeys the near-equilibrium decay laẇ in the presence of such processes. The chirality flipping rate has never been calculated from first principles, however the following simple-minded estimate has been used in a number of works (see, e.g., [2,21,22,49]). Consider an ultrarelativistic particle moving with a certain momentum in a given helicity state. Note, that the eigenstate of helicity coincides with the eigenstate of chirality up to (m e /E) corrections. The helicity can only change in a an act of collision, which changes the direction of the particle's momentum, therefore where Γ scat ∝ α 2 T is the (chirality-preserving) electron scattering rate in plasma with T ≫ m e and α = e 2 /(4π) is the fine structure constant. As we shall show momentarily, this crude estimate neglects some crucial aspects of the analytic structure of the scattering amplitudes and for that reason falls orders of magnitude short of the correct prediction.
Infrared singularity in 2 ↔ 2 chirality flipping processes. The problem of the decay of the axial charge from the perspective of elementary scattering processes was addressed in our companion paper [46]. Here we briefly re-iterate its argument. Consider 2 ↔ 2 chirality changing processes, starting from the massless QED and treating mass as perturbation (see Fig. 1). Such processes have a non-integrable infrared singularity at small momentum transfer [50,51]. This signals out that a partial resummation of the perturbative series is required, which should result in the answer where q IR is the infrared regulator scale. In vacuum, the only possible resummation is the m e series, cutting the infrared divergence at q IR ∼ m e . This would cancel the m 2 e factor in the numerator and lead to a strange result that the chirality flipping rate is mass-independent. However in a hot medium we also have to take into account the thermal corrections to the dispersion relations of (quasi)particles. Performing hard thermal loops (HTL) resummation (see Appendix D) we find that these corrections appear at the order eT . If the temperature is high enough, T ≫ m e /e, their effect is much more significant than that of the zero-temperature mass. With q IR ∼ eT we arrive to where the numerical coefficient C ≈ 0.24.
Other processes contributing to the flipping rate. The 2 ↔ 2 processes discussed above are not the only one that can lead to O(α) chirality flipping rate. The order O(α 0 ) candidate is the 1 ↔ 1 process shown in Fig. 2(a). Applying the mass vertex selection rules to this process one can see that the outgoing particle has the same helicity and momentum as the incoming one. This implies that the chiralities of the incoming and the outgoing particles cannot be opposite unless the momentum is exactly zero. We conclude that while the chirality flip process in this channel is possible, its contribution to the chirality relaxation rate vanishes because the momentum space for such a process in the collision integral has vanishing measure. The situation gets more subtle for 1 ↔ 2 processes, one of which is shown in Fig. 2(b). The QED interaction vertex respects the chiral symmetry therefore the minimal diagram has to include one mass vertex insertion in the incoming or outgoing electron leg. The conservation of the four-momentum imposes the constraint that the momenta of the  process with chirality flip of a free electron which is forbidden by the angular momentum conservation unless the particle's momentum vanishes; (b) one of 1 ↔ 2 collinear processes with chirality flip of the incoming electron. Although for massless particles the process (b) has a finite phase space, it is very sensitive to any modification of the particles' dispersion relations. This leads to an uncertainty in the probability of such a process which will be resolved in Sec. 3. incoming and the outgoing particles be perfectly collinear. At the same time, selection rules impose two consecutive flips: chirality flip at the mass vertex and helicity flip at the electron-photon vertex. This results in a valid on-shell out state of an electron with consistent signs of chirality and helicity, even though in the intermediate virtual state between the vertices such a consistency is absent. The phase space for such a process has a finite volume where ǫ k = |k|, and ω p = |p|. This result, however, is physically unstable. Even the slightest deformation of the dispersion relations of the particles ǫ → ǫ + δǫ, ω → ω + δω, for example due to the mean-field effects of the environment, can completely wipe out the phase space available for the process. It is worth noting that due to the infrared divergence of the propagator describing the intermediate virtual state, the probability of the process shown at Fig. 2(b) diverges at small momentum of the incoming particle. This infrared singularity is not integrable therefore its resolution requires a unitarity-restoring resummation of the perturbation theory series. Such a resummation cannot be done consistently without modifying the dispersion relations of the particles and the resulting suppression of the phase space. Thus, we arrive at an uncertainty of the 0 × ∞ type which we shall resolve in the following sections.
To conclude, the tree-level kinetic approach to the problem of chirality flipping is fraught with difficulties arising from an assortment of light-cone and phase space singularities. These singularities should presumably be resolved by a partial resummation of the perturbation theory series. We now proceed to the next section where we introduce a formalism which gives Γ flip in the form of a diagrammatic perturbation theory series. We then use the formalism to perform the required partial resummation.

CHIRALITY FLIPPING RATE FROM THE LINEAR RESPONSE FORMALISM
In this section, we use first principle quantum mechanical description to develop a perturbative expansion of the chirality flipping rate and identify in it the contributions from 1 ↔ 1, 1 ↔ 2 and 2 ↔ 2 processes described in the previous section. We demonstrate again that these perturbative contributions are IR-divergent. Then, in the subsection 3.2 we perform a partial resummation of the perturbative expansion, thus taking into account that fermion and photon propagators are dressed with thermal corrections and show that this results in resolution of IR-divergences.
We use massless electrodynamics as the zeroth-order approximation. In the massless case, the chiral charge Q 5 , Eq. (3) is conserved. This means that the expectation value of its time derivative (treated as the quantum operator) vanishes, Q 5 0 = 0. In the presence of a finite mass the time evolution of the axial charge operator is described by the Heisenberg equation: where H m = m e d 3 xψψ is the chiral symmetry breaking term. Using the canonical anticommutation relations for Thus, in the presence of mass, operatorQ 5 acquires the nonzero expectation value which can be calculated in the lowest order in perturbation by means of the linear response formalism (see, e.g., textbook [52,Chap. 3]). It should be borne in mind, however, that massless electrodynamics does not contain an intrinsic length scale, therefore the actual dimensionless parameter containing the electron mass is the ratio m e /T where T is the temperature of the system. In the high-temperature regime considered here, this parameter is small therefore its use as an asymptotic parameter of perturbative expansion is appropriate. Another small parameter of electrodynamics is the electromagnetic coupling constant α = e 2 /(4π). As is discussed above, the α → 0 limit of massive electrodynamics does not admit for chirality flipping processes, even though the axial charge is not formally conserved. For this reason, we also need to consider the effects of QED interaction beyond the zeroth-order perturbation theory. To summarize, in the analysis of the relaxation of axial charge it is natural to make use of the double asymptotic expansion in two small parameters, the fine structure constant and in the dimensionless parameter m e /T . Following through the standard steps described in detail in Appendix A, we arrive at the following expression for the rate of change of the axial charge per unit volume: where G ret (ω, q) is the momentum space retarded Green's function which is defined as follows: and Ψ R,L are the field operators of right-or left-handed fermions in the Heisenberg representation of the Hamiltonian (A.2) of massless QED in the great canonical ensemble, and the expectation value is taken over the state of massless QED. Using the near-equilibrium relation q 5 = µ 5 T 2 /3, we arrive to the rate equation for the axial charge density, similar to Eq. (4) with the microscopic expression for the kinetic coefficient Γ flip given by The retarded Green's function at a finite temperature can be calculated by analytic continuation of the corresponding Matsubara function G(iΩ n , q = 0) to the real axis iΩ n → ω + iδ, with δ → 0 + . Therefore, in the next section we will consider the perturbative expansion for the following Matsubara correlation function: Here Ω n = 2πnT , n ∈ Z is the bosonic Matsubara frequency.

Perturbative expansion for the chirality flipping rate
In this subsection we develop perturbative expansion in α for the chirality flipping rate (13). We show that in each order (except the zeroth one) the result appears to be divergent, moreover, this divergence is accumulated in higher orders (for certain types of Feynman diagrams). Similarly to the discussion in [46], these infrared divergences will be resolved by a resummation of the leading divergences in higher orders of perturbation theory (see Sec. 3.2).

Zeroth order
In the zeroth order the fermions are noninteracting. In terms of elementary processes in plasma, this corresponds to 1 ↔ 1 process discussed in Sec. 2 and depicted in Fig. 2(a). As for the Green's function (14), it is decomposed into a product of two-point correlators of left and right-handed particles separately: Feynman diagram for the chirality flipping rate in zeroth order in α. Its imaginary part is expressed as an overlap of the free electron spectral densities with the same momentum and different chiralities. For nonzero momenta they never overlap so that the diagram vanishes. Cutting this diagram we get the amplitude of 1 ↔ 1 process with the chirality flip discussed in Sec. 2.
In Fourier space, this corresponds to the Feynman diagram shown in Fig. 3, and the final result reads as where we used the expression for the free fermion propagator (B.2) and performed the summation over the Matsubara frequencies by means of the formulas listed in Appendix B. In order to calculate the rate of change of the chiral charge (11), we perform the analytic continuation iΩ n → −2µ 5 + i0 in Eq. (16) and take the imaginary part using the Sokhotski formula. We end up with because of the identity k 2 δ(k) ≡ 0. This result is in perfect accordance with the conclusion of the previous section. Once again, it means that free electrons cannot spontaneously flip their chirality because the angular momentum has to be conserved.

First order
In the first order in α one gets three diagrams depicted in Fig. 4. We will see that the first two of them give the logarithmically divergent contribution to the chirality flipping rate. In Appendix C it is shown that this divergence occurs when all three lines in diagrams (a) and (b) are on-shell. Cutting these lines, we immediately get the matrix elements of collinear 1 ↔ 2 scattering processes qualitatively discussed in Sec. 2. The subtlety with the first-order diagrams is that their divergence resulting from the collinear processes is extremely sensitive to the the dispersion relations of the involved particles. After performing resummation of leading divergences in higher order perturbation theory we will find that the dressing of the particles' dispersion relations leads to a collapse of the phase volume of the collinear processes, however this effect is counteracted by a finite lifetime of the quasiparticles resulting in a finite contribution to Γ.
Feynman diagrams for chirality flipping rate in the first order in α. Diagrams (a) and (b) contain logarithmic infrared divergence for the loop momentum k → 0. It comes from the singular behavior of the self-energy insertions in these two diagrams as well as from multiple nearly-on-shell propagators. The divergence appears when all three particles in the loop are on-shell and collinear. Cutting these diagrams, we get the matrix elements of the 1 ↔ 2 scattering processes discussed in Sec. 2, Fig. 2. In addition, all three diagrams contain the unphysical divergence associated with vanishing photon momentum; however, it is canceled in the final result due to the gauge invariance.
Coming back to the diagrams in Fig. 4, we now calculate their contributions to the chirality flipping rate. The corresponding expressions are rather cumbersome and for the sake of convenience we list them in Appendix C together with the details of further manipulations.
When we try to perform the analytic continuation of the final expression for the Matsubara Green's function to the desired point on the real axis, iΩ n → −2µ 5 + i0, the result appears to be logarithmically divergent. In order to extract the divergent part, we perform the continuation into the point shifted along the real axis, iΩ n + 2µ 5 → iδ + ǫ, and take the symmetric limit ǫ → 0. Using Eq. (13), we obtain the result It is worth noting that there are in fact two different logarithmic divergences arriving from two different regions of the phase space. The first one, which is associated with the zero photon momentum in the loop, appears in all three diagrams of Fig. 4, however it exactly cancels in their sum due to the gauge invariance of the theory. One can eliminate such a divergence by an appropriate gauge choice for the photon propagator, however, it is more convenient for us to work in the Feynman gauge. The second logarithmic divergence is caused by the fact that the particles in the loop are massless and there is a possibility for real collinear 1 ↔ 2 processes to take place. Remarkably that this divergence is present only in the first two diagrams in Fig. 4. More precisely, it comes from the phase space region where the momentum of lower electron line in diagram (a) [upper electron line in diagram (b)] as well as the momenta of both particles in the self-energy blobs are on-shell and collinear. The remaining electron propagators are off-shell because they have different chirality; however, they approach the shell when the loop momentum k → 0 and this is where this divergence appears. Let us look more precisely at the reason why does this divergence comes about. The first two diagrams correspond to the self-energy insertion into one of the fermionic lines in the diagram in Fig. 3. In fact, they can be represented as shown in Fig. 5.  Fig. 4(a) and 4(b) represented in an equivalent form by introducing the self-energy insertions (grey blobs). For convenience, the self-energy operator is discussed in Appendix D. Its singular behavior for small k as well as the presence of two identical propagators in each diagram lead to the logarithmic divergence in the result.
In terms of the general self-energy correction Σ, the imaginary part of the retarded Green's function for the first two diagrams has the form where Σ λ L,R (k 0 , k) = tr Σ L,R (k 0 , k) 1 − λσ · k/k 2 (20) and Σ L,R (k 0 + i0, k) is the retarded self-energy of the chiral fermion which is given by one-loop diagram shown in Fig. D.1.
It is important to note that k 2 in the denominator comes from the fact that we have two identical fermion propagators surrounding the self-energy insertion in Fig. 5. The momentum k is on-shell for the fermion running in the undressed line. Because of the chirality flip in mass vertices, this momentum is off-shell for the dressed line, however, for k → 0 it approaches the shell and the divergence occurs. Moreover, the imaginary part of the fermion self-energy in HTL approximation (D.3), is also singular for k → 0: Finally, substituting Eq. (19) into Eq. (13), we obtain which is indeed logarithmically divergent and after the regularization leads to Eq. (18). Thus, multiple propagators with the same momentum and singular behavior of the self-energy for k → 0 are two features which lead to the logarithmically divergent result.
Let us now discuss the diagram in Fig. 4(c) which corresponds to dressing of a vertex (not a QED interaction vertex but simply the mass insertion). This diagram contains two propagators with one momentum and two other propagators with the different momentum (any two of them cannot be simultaneously on-shell because of the different chiralities). Combining this with the fact that the singular self-energy does not appear in this diagram, we conclude that this diagram does not contain the divergence at k → 0. Nevertheless, as we mentioned earlier, it contains the fictitious infrared divergence at small photon momenta which is also contained in the diagrams (a) and (b) in Fig. 4. That is why we have to keep all three diagrams together and, moreover, regularize them in the similar way so that not to violate the cancellation of this unphysical divergence.
Before dealing with these divergences, let us check whether they appear also in higher perturbative orders.

Second order
In this subsection, we consider the diagrams of the second order in α. We have much more diagrams than in previous case which can be classified into 10 topologically different classes shown in Fig. 6. Analyzing their singularities, we will show that there are two types of divergences: the first one is of the same origin as discussed in previous subsection, however, it is more severe at this order; the second one is new and it is associated with the off-shell loop momentum.
Let us discuss the singularity of each class. Diagrams of the first line, (a), (b), and (c) represent the higher order corrections to electron's self-energy operator inserted into one of the lines. In other words, these diagrams can be combined together with the diagram in Fig. 4(a) resulting in a diagram in Fig. 5(a). It is worth noting that the two-loop self-energy correction which is present in the diagram (c) is important because it provides the finite electron damping rate which is crucial for the calculation of the chirality flipping rate.
The diagrams (d), (e), and (f) contain at most two similar propagators and one singular self-energy blob (or vertex correction). So that they cannot be more singular as the diagram in Fig. 4(a), but they are of higher order in α and thus can be neglected.
The diagrams (g) and (h) are just finite for the same reason as the diagram in Fig. 4(c) is. The are no multiple nearly-on-shell propagators and no singular self-energy blobs.
Finally, we are left with two types of diagrams, (i) and (j), which need a special attention. Obviously, they contain two singular self-energy insertions and more identical propagators in the loop. This leads to the accumulation of divergences which will be discussed for a general order in the next subsection.

Higher orders in perturbation theory
Here we concentrate our attention on the two classes of diagrams which accumulate divergences with the increase of the perturbative order. These classes were detected in previous subsections. , and (f) contain one singular self-energy or QED vertex correction and at most two identical propagators; thus, their singularity is of the same type as for diagrams in Fig. 4(a) and (b). Because of the higher order in α, they can be neglected. Diagrams (g) and (h) do not contain nontrivial singularities just as that in Fig. 4(c). Diagram (i) is more singular than the first-order graphs because it accumulates the singularity of two self-energy corrections and three identical propagators. This divergence is associated with on-shell loop momentum k and quickly grows in higher orders. Diagram (j) additionally contains the singularity coming from off-shell (spacelike) loop momenta. Such a divergence appears for the first time in diagram (j) and it is also accumulated in higher orders. The behavior of the diagrams (i) and (j) show the necessity for the resummation.
The first one is represented by a diagram in Fig. 5(a). In such a case, one of the fermion lines remains undressed and the contribution comes from the on-shell loop momentum k. Performing the power counting of the previous subsections, we would have n + 1 identical propagators giving 1/k factor each and n divergent self-energy blobs. As a result the divergence of the integral would be powerlike in the infrared, dk/k 2n−1 .
Another possibility arises when we dress both fermion lines as it is shown in Fig. 6(j), or in general case in Fig. 7(b). As we will see explicitly below, the sought ℑm G ret is expressed as the overlap of the imaginary parts of the upper and lower lines in the loop. If we have both dressed lines, the imaginary part arises not only for on-shell loop momentum [as for diagram in Fig. 7(a)], but also for off-shell (spacelike) momentum k, where the incoherent part of the self-energy operator is nontrivial. The power counting is very similar to the previous one, because for any spacelike momentum The propagators linking the self-energy blobs all behave ∼ 1/k for k → 0. As a result, for the diagram shown in Fig. 7(b) we would have the powerlike divergence of the form ∼ dk/k 2n+2m−1 . As we see, this divergence is accumulated if we go to higher perturbative orders. The two divergences described above have different physical meaning. The "on-shell" contribution whose lowest order diagrams are given by the first two graphs in Fig. 4, can be identified with the effective 1 ↔ 2 processes in plasma. Indeed, such processes appear if we cut the on-shell lines of these diagrams. The lowest order diagram containing the "off-shell" divergence is shown in Fig. 6(j). Here, the imaginary parts of the self-energy insertions for the off-shell momentum k appear when the particles inside the blobs are on-shell. Cutting these lines we arrive at  Fig. 6(j) and is quadratically infrared divergent; this divergence quickly increases in higher orders. This singularity comes from the incoherent part of the self-energy associated with on-shell particles inside the self-energy loops. The resummation of these diagrams would lead to dressing of both lines in the loop, which is shown by the diagram (c). Obviously, such a resummation absorbs also diagrams of the type (a). Since the vertex corrections do not lead to the accumulation of the divergences, the vertices in diagram (c) remain undressed. the 2 ↔ 2 Compton or annihilation diagrams shown in Fig. 1. Thus, we see the one-to-one correspondence of the different regions of the phase space with the elementary processes discussed in Sec. 2.
As we saw above, when we add more self-energy insertions, the singularities are accumulated. Thus, in order to obtain the physically meaningful result, we need to perform the resummation of the self-energy corrections to the fermion propagators in the loop. This will be done in the next subsection. We would like to note that at leading order there is no need for resummation of the vertex corrections [diagrams like in Fig. 4(c) and of higher orders]. That is because the divergences in those are much weaker and do not accumulate as quickly as in self-energy.

Resummation of leading divergences in the perturbative expansion
The divergences arising in our naive perturbative expansion signal out that the free fermions and photons are not the correct physical degrees of freedom in such a situation and we have to think of some other choice. The self-energy resummation gives us a correct choice replacing the free particles with the quasiparticles with modified dispersion relations due to thermal effects in plasma. In fact, in the kinetic approach considered in [46] we used the resummed fermion propagator and this helped to regularize the divergence. We will use the same strategy in this subsection.
At technical level, the resummation of the self-energy corrections is performed by considering the full fermion propagators instead of bold ones. Since there is no need for the vertex resummation, we obtain the diagram shown in Fig. 7(c). The full propagator of the chiral fermion is the solution of the Schwinger-Dyson equation (D.7) and in general can be decomposed into two helicity components as follows: Here ∆ ± (k 0 , k) is the denominator whose zeros give the dispersion relations for the quasiparticles with positive(negative) ratio of chirality to helicity.
Then, the expression for the diagram in Fig. 7(c) takes the form It is natural that the functions with opposite signs λ appear in our result. The sum over the Matsubara frequencies can be rewritten as the integral in a complex plane z along the contour spanning on the branchcuts at z = k 0 and z = k 0 + iΩ n , k 0 ∈ R. Further, we perform the analytic continuation iΩ n → −2µ 5 + i0 and take the imaginary part to get the following expression for the rate of change of the chiral charge per unit volume: Then, shifting the variable k 0 + µ R → k 0 in the first term and k 0 + µ L → k 0 in the second one and noting that two values of λ give equal contributions, we have the final expression where is the spectral density of the fermions with positive/negative ratio between chirality and helicity. Expanding the hyperbolic tangent in Eq. (26) and using Eq. (13), we obtain the part of the chirality flipping rate coming from the self-energy corrected diagram in Fig. 7(c) in the following form: This equation is well known in the literature. It arises, for example, in condensed matter theory in the context of particle exchange between two reservoirs across a tunneling barrier (see, e.g., Eq. (9.3.11) and the corresponding diagram 9.9 in textbook [52]). In the following subsection we will use Eq. (28) in order to calculate the chirality flipping rate in the leading order in α.

Calculation of the chirality flipping rate at leading order
In this subsection, we calculate the chirality flip at leading order in α. As we discovered in the previous subsection, there are two diagrams which must be taken into account. First of all, this is the dressed loop diagram in Fig. 7(c) which is a result of resummation of an infinite number of divergent graphs from all orders in perturbation theory containing only the self-energy corrections. The second one is the vertex correction diagram in Fig. 4(c) which is of the first order in α. Fig. 7(c) The part of the chirality flipping rate coming from the diagram in Fig. 7(c) is given by Eq. (28). In order to calculate this integral, we need to know the spectral densities of electrons with different chiralities (27). For free fermions, the spectral density is simply ρ (0) ± = 2πδ(k 0 ∓ k), and expression (26) coincides with Eq. (17) derived in the zeroth order in α. Since the spectral densities with opposite chiralities do not overlap in this case, the chirality flipping rate vanishes. This is the manifestation of the fact that free fermion cannot spontaneously flip its chirality because of the angular momentum conservation.

Diagram in
In the medium, however, the spectral function acquires a more complicated form Here, the first two terms in brackets correspond to the quasiparticle poles with the energies ǫ ± (k), residues Z ± (k), and decay width γ e . The function δ γe (k 0 ± ǫ) is the Lorentz contour The last term in Eq. (29) is the continuous (or incoherent) contribution to the spectral density. At leading order in α, the thermal corrections can be described in the HTL approximation (see, e.g., the textbook [53]). It accounts for the 1-loop self-energy correction in which the loop momentum is hard p ∼ T and much greater than the external momentum. In this approximation, the dispersion relations ǫ ± (k) and the residues Z ± (k) are wellknown and for convenience they are listed in Appendix D. The incoherent contribution is nontrivial in the region of spacelike momenta and corresponds to the Landau damping of the corresponding modes. Its explicit expression is given by Eq. (D.18).
It is important to note that the HTL approximation does not predict the finite lifetime for the quasiparticles in plasma. Indeed, in the region of timelike momenta, the imaginary part of the HTL self-energy (D.6) is absent and, therefore, the dispersion relations are real. The finite lifetime comes from the scattering processes which arise in the self-energy only in the two-loop order and, therefore, are not captured by the HTL. It was shown in Refs. [54][55][56] that the decay width of the electron can be estimated as at leading logarithmic order, i.e., the constant under the logarithm remains unknown.
Remarkably, that the energy of soft electrons with momenta k eT are of order m th = eT /2 ∼ √ αT . Therefore, the decay width γ e is parametrically smaller than the energy for all possible electron momenta. This fact allows for the quasiparticle description of the hot plasma. Now, let us apply Eq. (28) in order to calculate the part of the chirality flipping rate coming from the self-energy corrected diagram in Fig. 7(c). There are several contributions from different terms in the spectral function. We consider them separately in Appendix E in details and here we list only the main results.
3.2.1.1. Contribution from the plasmino branch. First of all, we show that the contribution of the plasmino branch in the quasiparticle spectrum is of higher order in α and can be neglected. In fact, this branch only exists in the soft momentum region, k eT , and for hard momenta the residue of the corresponding pole is exponentially suppressed, see Appendix D. In this region, however, the energy of the plasmino is displaced from the incoherent part as well as from the normal branch by energy ∆ǫ ∼ eT which is much larger than the characteristic width (31) of the pole. That is why, the integral (28) is suppressed by higher power of α if at least one of the spectral densities ρ ± corresponds to the plasmino branch, see the estimates in Appendix E.1.
3.2.1.2. Overlap of the incoherent parts. If we take only the incoherent parts in both spectral functions in Eq. (28), we will get the following result: where the expression (D.18) was used. Again, the main contribution comes from the soft momentum region, k ∼ eT , but now it is a smooth function which accumulates nonperturbatively large spectral weight, i.e., the incoherent part in this region contributes to the sum rules the amounts of order unity. Since ρ (LD) ± ∼ 1/(eT ) in the soft momentum region, the naive estimate gives for the chirality flipping rate the following result: In Appendix E.2 we explicitly calculate the integral (32) and show that the estimate (33) indeed holds with the constant C equal to The rate given by (33) and (34) exactly coincides with the result coming from the Compton diagram calculated in the kinetic approach [46]. Let us discuss what is the connection between these two approaches. At finite temperature, the imaginary part of the particle's self-energy emerges in the situations when the particle can take part in any process involving the real particles, because there is a thermal bath of such particles (in vacuum at T = 0, only in decay processes). In particular, even for soft electron with momentum k ∼ eT , the imaginary part of the its self-energy is nonzero for spacelike momenta, because this virtual electron can take part in the processes e * + γ → e, e * +ē → γ, where e (ē) and γ are real (on-shell) particles. Thus, the incoherent part of the self-energy appears when the on-shell particles run inside the loop. . The contribution of such processes was calculated in the kinetic framework [46].
The lowest order diagram in which the two incoherent parts appear simultaneously is the three-loop diagram shown in Fig. 8(a). In both self-energy loops the particles are on-shell. Cutting the corresponding lines, we reveal the matrix elements of the Compton or annihilation processes shown in Fig. 8(b). This qualitatively explains the relation between the two approaches.
Thus, we recovered the result of [46] in the linear response formalism. However, we will see below that there is also another contribution to the leading order in α which is calculated in the following subsections.
3.2.1.3. Overlap of the quasiparticle pole with the incoherent part. Finally, we have to calculate the contribution coming from the overlap of the quasiparticle pole in ρ + with the spectral density of the fermion of the negative helicity ρ − and vice versa. In the HTL effective theory such an overlap vanishes because the width of the pole is infinitely small and it is always located in the region k 0 > k where ρ − is identically zero. However, the HTL spectral density is modified by collisions in plasma which cause the finite width γ e of the pole and also wash out the region where ρ − vanishes.
For soft momenta k ∼ eT , the quasiparticle pole is displaced from the "old shell" k 0 = k by the value of order eT . However, for hard momenta, this displacement becomes smaller and asymptotically vanishes, ǫ + (k) − k ≈ m 2 th /(2k) → 0. Since the decay width parametrically behaves like γ e ∝ e 2 ln e −1 T , the above mentioned overlap will be more significant in the region of hard momenta. In fact, the contribution from the soft momentum region can be estimated in a similar way as we did for the plasmino branch and it is of higher order in α. Thus, in what follows we consider only the hard electron momenta. The corresponding contribution to the chirality flipping rate reads as where we used the fact that the hyperbolic cosine is the smooth function and neglected x as well as the thermal corrections to the energy dispersion in its argument. The only thing we need is to calculate the spectral density ρ − far from its shell.
where 1/∆ The first one comes from the region of small momenta of the photon running in the loop in the self-energy diagram. It is given by where Λ ∼ eT is the upper momentum cutoff which separates the soft photon momenta, and * ρ µν (Q 0 , Q) is the spectral density of the photon in HTL approximation. Using the approximate formula (E.18) valid for small momenta of the photon, we finally obtain the parametric dependence of this part It is worth noting that this logarithmic enhancement is the regularized version of the old (fictitious) logarithmic divergence on the zero photon momentum. We will see further that it will be canceled by the corresponding contribution from the diagram in Fig. 4(c) (the "vertex correction" diagram).
There is also another important contribution which comes from the region when the electron quasiparticle as well as photon and electron running in the loop in the self-energy diagram all are hard and nearly collinear. The corresponding expression for the chirality flipping rate is given by The combination in the argument of the Lorentz-function shows that for different choices of the signs λ ′ , λ ′′ we would get different 1 ↔ 2 processes contributing to the chirality flip. Three of them are shown in Fig. 9, the forth one, λ ′ = λ ′′ = −1 is not realized. The Lorentz function takes into account the approximate energy conservation law in the collision event. The amount for which this conservation law can be violated is determined by the decay width of the electrons. But since the decay width γ e ∼ T α log α −1 is much less than the energies of the quasiparticles, the process has to be very close to collinear so that the difference of the energies was small. In Appendix E. .
Here m γ = eT / √ 6 is the photon thermal mass. 9. Nearly collinear 1 ↔ 2 processes in plasma with the chirality flip of the incoming electron (different colors show different chirality of quasiparticles). In these processes, quasiparticles have modified dispersion relations and finite width due to the thermal corrections. The latter shift the energies from free-particle shells and simultaneously allow for a slight violation of the energy conservation. The interplay of these two factors gives rise to the finite contribution of 1 ↔ 2 processes to the chirality flipping rate. Fig. 4(c) So far we calculated only the contribution of the diagram in Fig. 7(c) which is a result of resummation of an infinite number of diagrams starting from Fig. 4(a) and 4(b). Now, we are going to compute the diagram in Fig. 4(c) since it also gives the contribution of the same order. The corresponding expression is given by Eq. (C.3) where we must replace the free fermion and photon propagators with the full renormalized ones containing the thermal corrections. Using the spectral representations for the electron and photon propagators, we obtain Eq. (E.34) and the expression for the imaginary part of the retarded Green's function is given by Eq. (E.36).

Diagram in
Like in the previous case of the self-energy corrected diagram, here we also have two interesting regions in the phase space. For the details of calculation we refer the reader to Appendix E.4 and here only report the results.
The first nontrivial result comes from the region of small photon momenta. The corresponding contribution to the chirality flipping rate equals to This expression has exactly the same form and an opposite sign compared to Eq. (37) emerging from the first two diagrams in the case of the soft photon momenta. Therefore, in the final answer they exactly cancel each other. The second important result emerges when the momenta of all three particles are hard. The corresponding expression for the retarded Green's function reads as where the functions F and B are defined in Eqs. (C.8) and (C.9), respectively. The expression in the argument of the Lorentz function can be approximately zero if the momenta of colliding particles are almost collinear. Each of three possible choices of the relative signs corresponds to one of the three collinear processes considered in the previous subsection. In fact, the case λ ′ = −λ, λ ′′ = λ corresponds to the nearly collinear bremsstrahlung e L,R (k) → e R,L (k − Q) + γ(Q), the case λ ′ = λ ′′ = −λ corresponds to the process of nearly collinear absorption of the photon e L,R (k) + γ(Q) → e R,L (k + Q), and the case λ ′ = λ ′′ = λ corresponds to the process of nearly collinear annihilation e L,R (k) +ē L,R (Q − k) → γ(Q). These processes with the chirality flip are schematically depicted in Fig. 9.
The contributions from all three processes can be combined together in the following expression:

Chirality flipping rate from nearly collinear 1 ↔ 2 processes
Picking up all the results of the previous subsections, we obtain two main contributions to the chirality flipping rate: from the incoherent part of the electron spectral function which encodes the 2 ↔ 2 processes allowed even with zero decay width [see Eq. (33)] and from the nearly collinear 1 ↔ 2 processes in plasma which are possible due to the finite lifetime of the quasiparticles. Here we summarize the second contribution.
Let us extract the leading parametric behavior of this expression. The arctangent can be replaced by (π/2) sign y if |xy| (8γ e T /m 2 th ) −1 ∼ 1/ ln e −1 . In this case, the y-integral is dominated by the lowest possible value of |y|, because it is logarithmically divergent on the lower bound. Therefore, we have Comparing this result with Eq. (18), we conclude that this is the same logarithmic contribution where the divergence is now regularized due to the thermal corrections. Indeed, Eq. (18) contains ln(k max /k min ), where k max and k min are the characteristic maximal and minimal possible values of the electron's momentum for which the collinear processes can occur. Naturally, the maximal value is cut by the temperature because the number of electrons with higher momentum is exponentially suppressed. The minimal value can be determined by the condition that the energy of the electron deviates from the "old shell" by the value not greater than the decay width, ∆ǫ = ǫ + (k) − k γ e . This immediately gives k min ∼ m 2 th /γ e ∼ T / ln e −1 . Now, we can see that this lower bound cannot be deduced by some heuristic arguments and only by consistent taking into account of thermal effects. It is important to note that this contribution hardly can be derived in the kinetic approach because of the uncertainty of the type 0 × ∞ which was discussed in Sec. 2.
Assuming that the electron damping rate is given by Eq. (31), we numerically integrate Eq. (44) and determine few first terms in the asymptotic decomposition over the inverse logarithm of the coupling constant Combining Eqs. (33) and (46)

CONCLUSION
In this work we calculated the rate of chirality flipping processes in hot QED plasma with the chiral imbalance µ 5 = 0. The final expression is given by Eq. (47). Numerically, this gives the following equation for the dissipation of the axial chargeq This rate is ≈ 1440 times larger than the naive estimate (5), c.f. [2,21,22,49]. The reasons for this striking difference are as follows. Firstly, in plasma the nearly collinear 1 ↔ 2 processes with a slight violation of the energy conservation can take place because of the finite lifetime of the quasiparticles, which counteracts the suppression of the phase volume due to the modified dispersion relations of quasiparticles. Unlike the previously considered Compton processes, the collinear process is even nominally first order in α. Secondly, the tree-level matrix elements of some 2 ↔ 2 scattering processes have severe infrared divergences which are regularized in plasma by thermal corrections on the soft scale q ∼ eT . In both cases, result in contributions to the chirality flipping rate which are of the order O(α).
The interest to the exact value of the chirality flipping rate has increased with the emergence of the chiral magnetohydrodynamics (chiral MHD) [1][2][3], see also [4,5]. Hydrodynamical description of magnetized systems of relativistic fermions in weakly coupled magnetized plasmas [1][2][3][4]48]; dynamics of quasi-particles in new materials such as graphene (see, e.g., [32]); and behavior of quark-gluon plasma formed in heavy-ion collisions (see, e.g., [57]), cannot be formulated solely in terms of the usual magnetohydrodynamic variables (flow velocity, magnetic field, density and pressure) appearing in the Navier-Stokes and the Maxwell equations. The hydrodynamics of ultrarelativistic particles necessarily contains an additional degree of freedom corresponding to the space-and time-dependent axial chemical potential [2,3,58] that can be expressed via an axion-like dynamical degree of freedom [3,12,13,58], see also [59]. The dynamics of this degree of freedom is coupled to the magnetic helicity [via Eq. (3)] and to the Maxwell equations via the chiral magnetic effect (see review [60] and refs. therein). Chiral MHD is the subject of active research [3, 5-11, 16-24, 49, 61-84]. However, its effects are operational only while the chirality flipping processes are slower than other relevant rates. In particular, chiral cascade [1, 2, 6-11] that may be responsible for the survival of the primordial magnetic fields, stops being operational once chirality flipping rate is faster than the rate Γ B of chirality transfer between fermionic (N 5 ) and magnetic (H) components of the axial charge (3). Recent microscopic studies of the axial charge dynamics [85][86][87] suggest that the rate Γ B may be much faster than one can estimate from the classical (MHD-based) description (see, e.g., [1,2,88,89]). Therefore, the range of applicability of chiral MHD in the hot plasmas, and, in particular, the evolution of primordial magnetic fields [46] remains an open research question.

A. APPLICATION OF THE LINEAR RESPONSE FORMALISM
The operatorQ 5 has vanishing expectation value in massless QED and satisfies the Heisenberg equation (10) in the case of nonzero mass. Therefore, its expectation value in the perturbed state can be calculated using the linear response formalism [52] where all field operators are considered in the Heisenberg representation with respect to massless QED Hamiltonian H 0 and the averaging is over the states of massless theory. This is marked by the index 0, which will be omitted in what follows. We consider the process of chirality flip in the hot chirally-asymmetric electron-positron plasma at high temperature m e ≪ T T EW . Therefore, the averaging should be performed at finite temperature and chemical potentials for leftand right-handed particles in the grand canonical ensemble with the Hamiltonian where the last two terms correspond to the conservation of left and right-handed particle numbers separately, µ = (µ R + µ L )/2, µ 5 = (µ R − µ L )/2. There is some subtlety in the calculation of this correlator which complicates the averaging procedure. In fact, the time evolution of the system is governed by the quantum Hamiltonian of massless QED H 0 , and the thermodynamic averaging in the grand canonical ensemble has to be performed with extended Hamiltonian (A.2). In order to overcome this inconsistency, we represent the quantum Hamiltonian in the form H 0 = K + µN + µ 5 Q 5 and, taking into account that it commutes with the number operators, we separate the corresponding exponents: Then, we substitute this into Eq. (A.1). Further simplifications could be done using the well-known formula: In each case we have: Now we can move into the Heisenberg representation with respect to the full Hamiltonian K: Introducing the notations we rewrite our linear response result in the following form: where , (A.14) The latter one contains operators which do not conserve the numbers of left-and right-handed particles. Therefore, the corresponding average equals to zero. The first term I 1 can be expressed through the following retarded Green's function: Finally, in terms of the retarded Green's function introduced here, the rate of change of the chiral charge per unit volume is given by Eq. (11).

B. FREE PROPAGATORS AND SUMMATION OVER MATSUBARA FREQUENCIES
The propagator of the free fermion in the imaginary time formalism is given by where ω m = (2m + 1)πT , m ∈ Z is the fermionic Matsubara frequency and the propagator in frequency-momentum representation reads as where σ i are the Pauli matrices, µ R,L are the chemical potentials for the right and left chiral fermions, respectively. The left and right components of the propagator are decoupled because we consider the massless case. We use the chiral representation for the Dirac matrices For further convenience, it is also useful to decompose the chiral components of the propagator into two helicity components The the free photon propagator in the Feynman gauge has the following form: where Ω p = 2πpT , p ∈ Z are the bosonic Matsubara frequencies.
The summation over the Matsubara frequencies can be performed by means of the following formulas:

C. DIAGRAMS IN THE FIRST ORDER IN α
In this Appendix we calculate the first-order contribution to the chirality flipping rate from three diagrams shown in Fig. 4. Using the Feynman rules we write down the corresponding analytical expressions: where Q = k − q is the photon momentum and we used the shorthand notations σ µ = (1, σ),σ µ = (1, −σ) for the four-vectors of the Pauli matrices. Calculating traces in Eqs. (C.1)-(C.3) and performing the summation over the Matsubara frequencies we obtain: Further, we have to perform the analytic continuation iΩ n + 2µ 5 → iδ, δ → 0 + and to take the imaginary part after that. In the majority of terms, the external frequency iΩ n enters the expressions in the combinations 1/(iΩ n + 2µ 5 + 2λk), 1/(iΩ n + 2µ 5 + 2λk) 2 and (iΩ n + 2µ 5 + 2λ ′ q). After analytic continuation and taking of the imaginary part they will contribute −πδ(2λk), πδ ′ (2λk) and −πδ(2λ ′ q), correspondingly. It should be noted that in a spherical coordinate system the measure has the form d 3 k = k 2 sin θdθdϕ. It is easy to see that in all expressions the fraction 1/(iΩ n + 2µ 5 + 2λk) is multiplied by the regular function which behaves like ∼ k, when k → 0, the fraction 1/(iΩ n + 2µ 5 + 2λk) 2 is multiplied by function ∼ k 2 when k → 0, and the fraction 1/(iΩ n + 2µ 5 + 2λ ′ q) is multiplied by function ∼ q 2 when q → 0. Therefore, by virtue of the identity k n δ(k) ≡ 0, n ≥ 1, all such terms can be discarded and not considered in what follows. Then in each of the expressions (C.4)-(C.6) only the last terms remain and they have a similar structure of the integrand and can be combined. Finally, we get the following expression for the sum of all three diagrams in Matsubara representation: 1 − λλ ′ cos θ (iΩ n + 2µ 5 + 2λk) 2 + 2 (iΩ n + 2µ 5 + 2λk)(iΩ n + 2µ 5 + 2λ ′ q) , where we introduced the following notations for the thermal factors: There are four different ways to choose signs in Eq. (C.7), λ ′ = ±λ and λ ′′ = ±λ. Below we consider each of them separately.
Feynman parametrization allows to combine two denominators into one: (C.10) The contribution of the first two diagrams could be calculated using the second formula in (C.10): (C.11) In elliptic coordinates the expression takes the form Then, in order to reduce the power in denominator we integrate by parts twice with respect to Q. Factor Q 4 provides the zero value on the lower boundary and B(λξQ, λQ) is exponentially decreasing on the upper boundary. After integration we obtain the following expression: (C.14) Now, we can perform the analytic continuation and take the imaginary part of the expression. In order to avoid the divergence, we move to the shifted point on the real axis, namely iΩ n → −2µ 5 + ǫ + iδ, δ → 0 + .
Integration over x could be done with the help of delta function. We consider the case of ǫ > 0 (the opposite sign will be discussed later), then, the nonzero contribution gives only the term with λ = −1. The delta function gives the Jacobian J = 1 Q(η+1) and the integration variable acquires the value x = ǫ/Q+1−ξ η+1 . Taking into account that 0 ≤ x ≤ 1, we obtain the integration region for the remaining variables: 1 ≤ ξ ≤ 1 + ǫ/Q and ξ + η ≥ ǫ/Q.
Lets estimate the behavior of each term as ǫ → 0. In the first integral Q < ǫ/2, therefore the functions could be replaced by their asymptotics B ∼ 1/Q, F ∼ const, and d 2 dQ 2 Q 4 F · B ∼ Q. Integral over ξ and η gives the number of order unity and integral over Q is proportional to ∼ ǫ 3 . Finally, the first term in (C.16) vanishes as ∼ ǫ.
In the second term the main contribution is made by the region ξ ∼ 1, η ∼ −1. Therefore, we put in functions B and F exactly ξ = 1, η = −1. The remaining integration over ξ and η is straightforward: After that in the leading order in ǫ the integration by parts over Q could be done: Summarizing, the leading contribution is logarithmically divergent for ǫ → 0: Now, lets consider the contribution of the third diagram. There are three different denominators in Eq. (C.7). It could be rewritten in the following form (z = iΩ n + 2µ 5 ): The first two terms after the analytic continuation and taking of the imaginary part would be proportional to δ(k) and δ(q), correspondingly. They are multiplied by the function which behaves like ∼ 1/k (or ∼ 1/q) and the integration measure gives k 2 q 2 . Therefore, these terms vanish. The third term reads as In the elliptic coordinates (C.12) it could be rewritten as The result is finite at ǫ → 0 and equals to The finite contributions from the first two diagrams also could be calculated. Then, the corresponding expression reads as ℑm G If we take ǫ < 0 the contribution would be made by the terms with λ = +1. In ideal case ǫ = 0 the delta-function would work for both λ but only at the boundary of integration. As a result, the additional factor of 1/2 is required. Therefore, we will take the half of the sum of the expressions for both λ. In summary, the final expression is the following: Using the Feynman parametrization (C.10) we rewrite the expression of the first two diagrams in the following form: (C. 26) In elliptic coordinates the expression takes the form (C.28) Then, we integrate by parts twice with respect to q and obtain the following expression: Expression for the third diagram could be rewritten in the similar form. In order to do this we use the identity (z = iΩ n + 2µ 5 ) .
First two terms would be proportional to δ(q) and δ(k), correspondingly, and finally vanish. The third term could be rewritten with the help of (C.10): (C. 31) In elliptic coordinates (C.27) we integrate by parts in order to reduce the power of the denominator. After that we obtain Then, we do the analytic continuation and take the imaginary part of (C.29) and (C.32). After integrating over x with the help of delta function the region of integration over the remaining variables is constrained by the conditions 1 ≤ ξ ≤ 1 + ǫ/q and ξ + η ≥ ǫ/q. It could be shown like in the previous subsection that the integral over the region q < ǫ/2 vanishes as ∼ ǫ for ǫ → 0. Therefore, we consider only the integral over the region q > ǫ/2.
These expressions contain the divergences of two types. The first corresponds to electron momentum k = 0 and occurs at ξ ∼ 1, η ∼ −1. The second is connected with the photon momentum Q = 0 at ξ ∼ η ∼ 1. We consider them separately.
In the first case, we put ξ = 1, η = −1 in the arguments of B and F . Then, the integrals over ξ and η could be easily taken: It is obvious that the first two diagrams give the divergent contribution while the third diagram is finite.
ℑm G In the second case, when Q → 0, we cannot put ξ = η = 1 directly in the argument of B, because it is singular: . However, we can use its asymptotic in the region ξ − η ≪ T /q. In all other nonsingular expressions we put ξ = η = 1.
And the corresponding expressions for the diagrams read: ret,a + G Integrating by parts we see that the leading divergent parts cancel each other. Therefore, the infrared divergence connected with the zero photon momentum cancels out. After calculating the finite contributions the final expression reads as In this case it could be shown that the expression is finite even without the Feynman parametrization. We expand the expression (C.7) into the sum of simple fractions (z = iΩ n + 2µ 5 ): After analytic continuation and taking the imaginary part the only terms which would give the nonzero contributions are proportional to 1/(iΩ n + 2µ 5 + λk + λ ′ (q + Q)).
In elliptic coordinates the expression has the following form: The nonzero contribution is only in the case λ ′ = −λ and it is finite: Taking all results together, we get The corresponding value of the chirality flipping rate is given by Eq. (18). The analytical expression in the momentum space is the following: where S 0 is the free electron propagator in Matsubara representation given by Eq. (B.2). Taking into account the block structure of the propagator and Dirac gamma-matrices, we conclude that the self-energy also has the block structure and does not mix the left and right components. This is a consequence of the fact that the EM interaction respects the chiral symmetry. Therefore, one can consider the left and right components of the self-energy separately. The leading contribution to the self-energy is captured by the HTL approximation when the external momentum is considered to be much less than the loop momentum and the former is neglected everywhere except the denominator. The final result has the following form [53]: where the integration is performed over all possible directions of the unit vector v and m 2 th = e 2 T 2 /4 is the electron asymptotic thermal mass (meaning that, for hard momenta, the energy dispersion of the electron quasiparticle takes the usual form ǫ = k 2 + m 2 tanh ). One of the advantages of the HTL self-energy is its gauge invariance with respect to the Lorentz covariant gauges.
Spatial isotropy leads to the simple matrix structure of the self-energy Thus, the fermion self-energy in HTL approximation is determined by the single scalar function Σ 0 L,R = Σ 0 (iω m + µ L,R , k). Let us consider it as a function of the complex variable k 0 in a complex plane with the branch cut along the real axis from k 0 = −k to k 0 = k. Then, the integration can be performed explicitly and we obtain: The full fermion propagator is determined by the Schwinger-Dyson equation whose solution reads as usual Substituting the HTL self-energy (D.3), we obtain the full propagator in the following form: It can be represented as a decomposition into the components with positive and negative helicity wherek = k/k, is the denominator whose zeros determine the quasiparticle spectrum, and the function Σ 0 is given by Eq. (D.6). The quasiparticle dispersion relations can be found from the requirement ∆ ± (k 0 , k) = 0. This gives It is possible to find the asymptotic expressions for the dispersion relations for small and large momenta [53]: 14) The spectral function ρ ± = −2 ℑm ∆ −1 ± is given by Eq. (29), where the residues Z ± (k) of the quasiparticle poles are given by For small momenta the spectral density is almost equally distributed between both poles, 16) however, for large momenta the ordinary branch survives and the plasmino branch is exponentially suppressed, Finally, the incoherent part of the spectral density has the form The plasmino branch ǫ − (k) exists only in the soft momentum region k ∼ eT . For larger momenta, the corresponding residue is exponentially suppressed. If the plasmino pole is multiplied by any other pole contribution, the following power counting takes place: where ǫ 1,2 are the energies of the poles at least one of which is of the plasmino branch, ∆ǫ = |ǫ 1 − ǫ 2 | ∼ eT for soft momenta. Similar contribution comes from the interference term between the plasmino pole and the Landau damping part Therefore, in what follows we will omit the plasmino contribution in the spectral function ρ ± .

E.2. Contribution of the incoherent part
Further, let us calculate the contribution coming from the overlap between the incoherent parts of the spectral functions. Using expression (D. 18 .

(E.4)
It is obvious that the integrand is peaked in the region of x ∼ 1 (i.e., the soft momentum region k ∼ eT ). In fact, for x ≪ 1 we have the suppression of the integrand ∼ x 5 . On the contrary, for x ≫ 1 we have ∼ ln x x 3 . Since that, we can safely replace the hyperbolic cosine with the unity. It starts to be important for x ∼ T /m th ∝ e −1 ≫ 1. That is why neglecting it, we would bring the relative error of order ∞ e −1 ln x x 3 ∼ e 2 ln e −1 , (E. 5) i.e., of higher order in α. Then, the result for the chirality flipping rate is given by Eq. (33) with the constant C given by Eq. (34).

E.3. Contribution of the quasiparticle pole
The overlap between the quasiparticle pole and the incoherent part of the spectral function is determined by Eq. (35) where the spectral function far from the quasiparticle shell is needed. According to Eq. (36), it is fully determined by the corresponding retarded fermion self-energy. As usual, we calculate the Matsubara self-energy and after that perform its analytic continuation to the real axis. The one-loop electron self-energy is shown in Fig. D.1. Its component with the certain helicity λ is given by where * D µν is the full photon propagator and Here,k = k/k. For further convenience, we list here the expressions for the components of A µν Using the spectral representation for the propagators * D µν (iΩ p , we can perform the summation over the Matsubara frequencies Then, performing the analytic continuation to the real axis, we get the spectral density in the form which leads to the chirality flipping rate This expression contains three different contributions which must be considered separately. 1. Soft electron in the loop. For soft electrons, their spectral density ρ λ ′ is distributed in comparable amounts among the quasiparticle poles and incoherent part. On the other hand, the photon is obviously hard and we can substitute its spectral density in the form * ρ µν (Q 0 , Q) = P t µν (Q) where ω t (Q) is the energy dispersion of hard transverse photon and is the transverse projector. We take only the part corresponding to the transverse photon, because it contains almost all spectral weight at hard momenta [53]. It is convenient to use the electron momentum q = k − Q as the integration variable. The nonzero contribution comes from λ ′′ = +1, because in this case ǫ + (k) − Q 0 ≈ q cos θ is small. Then, we obtain the following result: where Λ eT is the separation scale between soft and hard momenta. We can roughly estimate this expression taking into account that in the soft momentum region the electron spectral function is not singular and is always of order ∼ 1/(eT ). We get 16) meaning that this contribution can be neglected.
2. Soft photon in the loop. If we restrict the Q-integration in Eq. (E.12) to the soft region, i.e., Q 0 , Q eT , further simplifications can be done. First of all, the electron's momentum in the loop is hard and almost equal to the external one. Therefore, we can use the spectral density of the hard electron which leads to the following chirality flipping rate The Lorentz function (ensuring the approximate energy conservation) works only for λ ′ = +1 and the tensor A µν −+ (k, k − Q) ≈ P µν t (k). Finally, expanding the hyperbolic cotangent, we obtain Eq. (37). In order to estimate that expression, we can use the following approximate formula It becomes more and more exact as the momentum Q decreases. Then, we get 3. Nearly collinear processes. If both, the electron and the photon running in the loop, are hard, we can use Eq. (E.17) and the photon spectral density (E.13). This leads to Eq. (39). There are four ways of choosing the signs of λ ′ and λ ′′ which correspond to different nearly collinear processes in plasma (one of them is forbidden though).
The case λ ′ = λ ′′ = +1 corresponds to the process of nearly collinear bremsstrahlung e L,R (k) → e R,L (k − Q) + γ(Q), schematically shown by the first diagram in Fig. 9. This process is possible if Q < k. We decompose and use the asymptotic expressions for the electron's and photon's dispersion relations where m th = eT /2 and m γ = eT / √ 6 are the corresponding thermal masses. Then, we obtain Γ 7c,(pole, brem) flip . . (E.25) The case λ ′ = +1, λ ′′ = −1 corresponds to the process of nearly collinear absorption of the photon e L,R (k) + γ(Q) → e R,L (k + Q), schematically shown by the second diagram in Fig. 9. This process is possible for any Q and k. Decomposing Q = Q k + Q ⊥ , k + Q = (Q + k)k + Q ⊥ , (E. 26) we obtain Γ 7c,(pole, abs) flip . . (E.29) The case λ ′ = −1, λ ′′ = +1 corresponds to the process of nearly collinear annihilation e L,R (k) +ē L,R (Q − k) → γ(Q), schematically shown by the second diagram in Fig. 9. This process is possible if Q > k. Decomposing we obtain Γ 7c,(pole, annih) flip . Finally, all three cases can be combined and the chirality flipping rate coming from the collinear processes is given by Eq. (40).

E.4. Contribution from the vertex correction diagram
Here, we calculate the diagram from Fig. 4(c) corresponding to the vertex renormalization. Its general expression is given by Eq. (C.3). Using the spectral representation (E.9) for the propagators, we substitute them into Eq. (C.3), perform the summation over the Matsubara frequencies and get In Eq. (E.34), we took into account only the normal branch of the fermionic spectrum with the dispersion ǫ + (k). Also we neglected x 0 i in the arguments of hyperbolic tangents and cotangents because these are smooth functions and x 0 i ∼ e 2 T ≪ T . Thus, we derived the analog of Eq. (C.6) which takes into account the modified dispersion of the electrons and their finite width. Further, in order to find the retarded function, we must perform the analytic continuation iΩ n → −2µ 5 + i0. After that we will take the imaginary part of it. First of all, we should mention that the imaginary part cannot emerge from the combinations like [λ(2ǫ k + x 0 2 + x 0 3 ) + i0] −1 or [λ ′ (2ǫ q + x 0 1 + x 0 4 ) + i0] −1 , because in contrast to the case of free fermions, the real parts of these expressions never equal to zero. Indeed, the energy is always ǫ k,q eT and is of order T for hard fermions while x 0 i ∼ e 2 T ≪ eT . Thus, the first term in the square brackets will not contribute to the final answer. That is why we have the following result: λλ ′ ǫ k ǫ qÃ µν λλ ′ (k, q) × × * ρ µν (Q 0 , Q)B(λǫ k + λ ′ ǫ q , Q 0 )F (λǫ k , λ ′ ǫ q )δ λ(ǫ k + x 0 1 ) + λ ′ (ǫ q + x 0 2 ) − Q 0 , (E. 36) where the functions F and B are defined in Eqs. (C.8) and (C.9), respectively. For further convenience, we calculate the components ofÃ µν λλ ′ :Ã 00 λλ ′ (k, q) = Like in the previous subsection, we consider the contributions from the soft photon spectral density and from the nearly collinear processes separately.
1. Soft photon. In the case when the photon momentum is soft Q, Q 0 eT , ǫ q ≈ ǫ k − Q cos θ and the deltafunction works only for λ ′ = −λ. Expanding the hyperbolic cotangent, we get In order to extract the chirality flipping rate, we expand the integrand for small chemical potential Taking into account that the photon spectral density * ρ µν (Q 0 , Q) is the odd function of frequency Q 0 , we finally obtain Eq. (41). 2. Nearly collinear processes. For the hard photon momenta, we use the spectral density given by Eq. (E.13) and Eq. (E.36) takes the form of Eq. (42). This corresponds to the nearly collinear 1 ↔ 2 processes in plasma, the same as we considered in the previous subsection.
There are three independent ways to choose the signs of λ, λ ′ , λ ′′ in Eq. Eq. (42). In each of these cases, we decompose the vector Q into two components: parallel to the vector k and perpendicular to it. The latter must be soft Q ⊥ eT in order to satisfy the approximate energy conservation in the collision. Then, the integration over Q ⊥ can be performed and we obtain the following expressions: . (E. 42) In order to extract from these expressions the chirality flipping rate, we expand the integrand for small µ 5 . The only quantities containing chemical potentials are the thermal distribution functions B and F . Therefore, we have . (E.43) The chirality flipping rate can be found using Eq. (13). Choosing the integration variables in the same way as we did in Sec. E.3, we get the following results: .