Second Order Transport Coefficient from Chiral Anomaly at Weak Coupling: Diagrammatic Resummation
RRBRC-1129IFT-UAM/CSIC-15-035
Second Order Transport Coefficientfrom Chiral Anomaly at Weak Coupling:Diagrammatic Resummation
Amadeo Jimenez-Alba ∗ and Ho-Ung Yee , † Instituto de Fisica Teorica IFT-UAM/CSIC, Universidad Autonoma de Madrid,28049 Cantoblanco, Spain Department of Physics, University of Illinois, Chicago, Illinois 60607 RIKEN-BNL Research Center, Brookhaven National Laboratory,Upton, New York 11973-5000
Abstract
We compute one of the second order transport coefficients arising from the chi-ral anomaly in a high temperature weakly coupled regime of quark-gluon plasma.This transport coefficient is responsible for the CP-odd current that is proportionalto the time derivative of the magnetic field, and can be considered as a first cor-rection to the chiral magnetic conductivity at finite, small frequency. We observethat this transport coefficient has a non-analytic dependence on the coupling as ∼ / ( g log(1 /g )) at weak coupling regime, which necessitates a re-summation ofinfinite ladder diagrams with leading pinch singularities to get a correct leading logresult: a feature quite similar to that one finds in the computation of electric con-ductivity. We formulate and solve the relevant CP-odd Schwinger-Dyson equationin real-time perturbation theory that reduces to a coupled set of second order dif-ferential equations at leading log order. Our result for this second order transportcoefficient indicates that chiral magnetic current has some resistance to the timechange of magnetic field, which may be called “chiral induction effect”. We alsodiscuss the case of color current induced by color magnetic field. ∗ e-mail: [email protected] † e-mail: [email protected] a r X i v : . [ h e p - ph ] A p r Introduction
The chiral anomaly is an intriguing quantum mechanical phenomenon arising from aninterplay between charge and chirality of massless particles such as chiral fermions. Ithas recently been appreciated that the chiral anomaly may induce interesting parity-oddtransport phenomena in the plasmas of such particles [1, 2, 3, 4, 5]: at the lowest order inderivative expansion of hydrodynamics (that is, at the first order) it has been shown thatthe second law of thermodynamics dictates the existence of such phenomena [6]: the chiralmagnetic effect [3] and the chiral vortical effect [7, 8]. Moreover, the magnitudes of thesetransport phenomena in the static, homogeneous limit are fixed by underlying anomalycoefficients and are not renormalized by interactions. This universality has been confirmedexplicitly in both weak [9, 10, 11, 12] and strong coupling [13, 14, 15, 16, 17] computations,and there are also evidences in favor of them in lattice simulations [18, 19, 20, 21, 22].Recent results from heavy-ion experiments at RHIC [23, 24, 25, 26, 27] and LHC [28]seem consistent with the predictions from chiral magnetic and vortical effects [29, 30](as well as chiral magnetic wave [31, 32, 33, 34, 35]), and quite interestingly, there isa successful experimental test of chiral magnetic effect in Dirac/Weyl semimetals whichfeature chiral fermionic excitations [36] (the spin degree of freedom in this case arisesfrom an internal degeneracy, in other words, it is a pseudo-spin). Therefore, the existenceand the magnitudes of these transport phenomena at lowest order (i.e. first order) inderivative expansion seem by now quite robust. One can also generalize them to all evenhigher space-time dimensions than four [37, 38, 39, 40].As we go beyond the lowest order in derivatives, the possible anomaly induced trans-port phenomena become numerous: in four dimensions there are thirteen possible secondorder anomalous transport coefficients in the current and energy-momentum tensor in aconformal plasma [37] (and more in non-conformal plasma [41]), while the second law ofthermodynamics seems to constrain only eight combinations of them [37]. Some of theseconstraints have been confirmed in a strong coupling computation [42]. The interestingfact is that the values of these anomalous second order transport coefficients, althoughthey are proportional to anomaly coefficients, do depend on the dynamics of the micro-scopic theory up to the mentioned constraints, so computing them in weak and strongcoupling regimes is a non-trivial, but worthwhile task in any theoretical model.The purpose of this work is to take a small step in computing these second orderanomalous transport coefficients in weakly coupled gauge theories, having in mind QCD1nd electro-weak theory. Our current study will be based on diagrammatic techniques, andwe hope to address a similar computation in chiral kinetic theory framework [43, 44, 45, 46,47] in a separate work. We will show that one particular second order anomalous transportcoefficient in the charge current has a non-analytic dependence on the coupling constant, ∼ /g log(1 /g ), which is similar to that one finds in the shear viscosity and electricconductivity [48, 49] (and also in the chiral electric separation conductivity [50]) ∗ . Thistransport coefficient appears in the second derivative correction to the current constitutiverelation as, ν µ (2) ∼ ξ (cid:15) µναβ u ν D α E β , (1.1)where E µ = F µν u ν is the electric field strength in a local fluid rest frame defined by u µ ,and we followed the notation introduced in Ref.[37] to denote the transport coefficient ξ . Using the Bianchi identity, one can replace (cid:15) µναβ u ν D α E β with u ν D ν B µ with B µ = (cid:15) µναβ u ν F αβ being the magnetic field strength in the local rest frame, which means that ξ can be viewed as a first correction to the static chiral magnetic effect at finite frequency.More explicitly, it appears in the anomalous part of the current density as (cid:126)J = · · · + σ χ (cid:126)B + ξ d (cid:126)Bdt + · · · , (1.2)where σ χ is the topologically protected value of chiral magnetic conductivity at zerofrequency. ξ is parity (P) and charge conjugation-parity (CP) odd, so must arise fromchiral anomaly.As was observed first by Jeon [51], in diagrammatic language, the non-analytic behav-ior in the coupling dependence is signaled by the presence of pinch singularities in multi-loop ladder diagrams of two point correlation functions, which necessitates a resummationof all ladder graphs by solving a Schwinger-Dyson type equation to get a leading log result.Previously, it was observed in Ref.[12] that the zero frequency-momentum limit of the P-odd part of the 1-loop correlation function does not have pinch singularity, reproducingthe correct static value of chiral magnetic conductivity. We first motivate our study byobserving an appearance of pinch singularity in the P-odd part of 1-loop diagram at firstorder in frequency, which enters in a Kubo formula for ξ . Following intuitions from thecomputation of electric conductivity [52, 53, 54], we then identify multi-loop ladder graphs ∗ Our definition of transport coefficients does not include a trivial e factor from the definition ofelectromagnetic current which is e times of the fermion number current. Therefore, all quantities inour work are defined with the fermion number current. For example, the electric conductivity will be ∼ / ( e log(1 /e )) and the chiral magnetic conductivity at zero frequency for a single right-handed Weylfermion is σ χ = µ π . ξ . The emerging Schwinger-Dyson equation is moredifficult to solve than that in the electric conductivity, because we need to keep finiteexternal momentum (cid:126)k (up to first order in (cid:126)k ) to extract P-odd part of the correlationfunction. In section 4, we prove an important fact that all (cid:126)k dependence in the denomi-nators of pinching propagators do not contribute to the P-odd part of our interest up tofirst order in (cid:126)k , allowing us to neglect them in the denominators of pinching propagators.The necessary (cid:126)k dependence for P-odd correlation functions arises only from the spinorprojection part of the fermion propagators. With this important simplification, we areable to reduce the leading log part of the P-odd Schwinger-Dyson equation into a coupledset of second order differential equations, which can be solved numerically. Along theway, we develop and use the sum rule for the P-odd part of Hard Thermal Loop (HTL)photon spectral density, which is summarized in the Appendix 1.For most part of our presentation, we will consider a single species of Weyl fermionin quantum electrodynamics (QED) for simplicity, and a generalization to finite numberof species of Weyl and Dirac fermions as well as to a non-abelian SU ( N c ) gauge theoryis trivial at our leading log order. We will describe this generalization in our discussionsection at the end. Our results are summarized as follows: for QED with a single right-handed Weyl fermion, we have ξ = − . e log(1 /e ) µT . (1.3)For 2-flavor massless QCD ( N c = 3) with Q u = 3 / Q d = − /
3, our result is ξ QCD5 = ( Q u + Q d ) − . g log(1 /g ) µ A T = − . g log(1 /g ) µ A T . (1.4)where µ A is an axial chemical potential. The sign of ξ compared to the zero-frequencyvalue σ χ (= µ π for QED) is a meaningful dynamical result. A relative negative signbetween the two means that the chiral magnetic current has some resistance to the changeof the magnetic field. We may call this “chiral induction effect”. In this section, let us motivate our work by observing an appearance of pinch singularityin the P-odd part of 1-loop diagram at first order in frequency ω . It will also serve tofix our notations and conventions. For simplicity, we will consider the case of single Weyl3ermion species of unit charge in QED plasma at finite equilibrium temperature T , asthe generalization to multi flavors or non-abelian gauge groups is simple (which will besummarized towards the end of the paper). Throughout our analysis, we will use the real-time Schwinger-Keldysh formalism in “ra”-basis to compute the retarded current-currentcorrelation function that contains the chiral magnetic conductivity σ χ ( k ) in its P-oddpart, (cid:104) J i ( k ) J j ( − k ) (cid:105) P − odd R = iσ χ ( k ) (cid:15) ijl k l , (2.5)where italic letters run over the three spatial dimensions, and k = ( k , k ) ≡ ( ω, k ) is anexternal four momentum. Note that in ra-basis, the retarded two-point function is equalto (cid:104) J i ( k ) J j ( − k ) (cid:105) R = ( − i ) (cid:104) J ir ( k ) J ja ( − k ) (cid:105) SK ≡ ( − i ) G ij ( ra ) ( k ) , (2.6)where the subscript SK in the second term emphasizes that it is computed in the Schwinger-Keldysh path integral with J r ≡ / J + J ) and J a ≡ J − J (1 and 2 denote the twotime contours in the Schwinger-Keldysh formalism). We follow the notations in Ref.[12]for consistency † . Explicitly, J µr = ψ † r σ µ ψ r + 14 ψ † a σ µ ψ a , J µa = ψ † r σ µ ψ a + ψ † a σ µ ψ r , (2.7)with σ µ = ( × , (cid:126)σ ) in terms of two component Weyl spinor field ψ . Therefore, the taskis to compute the P-odd part (or anti-symmetric part in i, j indices) of the (ra)-correlator G ij ( ra ) ( k ) for small frequency-momentum.The zero frequency-momentum limit of σ χ ( k ) has been shown to be universallylim k → lim ω → σ χ ( k ) = µ π , (2.8)and in particular, there appears no pinch singularity in this limit as shown in Ref.[12].Our ξ appears in first order expansion in ω = k (while still taking zero momentum limit k → k → σ χ ( k ) = µ π − iξ ω + O ( ω ) . (2.9)However, since the P-odd part of G ij ( ra ) contains a linear term in k in defining σ χ ( k ), G ij, P − odd( ra ) ( k ) = − σ χ ( k ) (cid:15) ijl k l , (2.10)we have to keep k dependence in G ij ( ra ) ( k ) up to first order in k , and then take k → σ χ ( k ). This essentially means that we need to keep finite † In literature it is often chosen to denote the retarded function by G ( ra ) , which we find confusing. in the Schwinger-Dyson equation for ladder resummation, which is in contrast to thecase of electric conductivity where one can put k = 0 from the very outset which greatlysimplifies the analysis. Despite this difficulty, we will be able to solve the Schwinger-Dysonequation for the P-odd part of G ij ( ra ) ( k ), and extract the coefficient ξ . At 1-loop thereFigure 1: Diagrams responsible for the retarded response at one loop in the ra-basis.are two Feynman diagrams for G ij ( ra ) ( k ) in real-time formalism as depicted in Fig.1. Thefermion propagators are given by S ra ( p ) = (cid:88) s = ± ip − s | p | + iζ/ P s ( p ) , (2.11) S ar ( p ) = (cid:88) s = ± ip − s | p | − iζ/ P s ( p ) , (2.12) S rr ( p ) = (cid:18) − n + ( p ) (cid:19) ρ ( p ) , (2.13)where n ± ( p ) = 1 / ( e β ( p ∓ µ ) + 1), and the spectral density ρ is ρ ( p ) = (cid:88) t = ± ζ ( p − t | p | ) + ( ζ/ P t ( p ) , (2.14)and we introduce the damping rate ζ ∼ g log(1 /g ) T in the propagators, which will beneeded to regularize possible pinch singularities, that is essential to have a non-analyticdependence on the coupling constant. It is important to observe a thermal relation S rr ( p ) = (cid:18) − n + ( p ) (cid:19) ( S ra ( p ) − S ar ( p )) , (2.15)which plays a central role in our analysis. The projection operators P ± ( p ) are defined as, P ± ( p ) ≡ (cid:18) ± σ · p | p | (cid:19) = ∓ ¯ σ · p ± | p | , (2.16)5here ¯ σ µ = ( , − σ ) and p µ ± ≡ ( ±| p | , p ). Our metric convention is ( − , + , + , +). Theoperators P ± ( p ) project onto particle and anti-particle states respectively with givenmomentum p , and the ( s , t ) summation in the above physically represents distinctivecontributions from particles and anti-particles.The 1-loop expression for G ij ( ra ) ( k ) from the two Feynman diagrams is( − (cid:90) d p (2 π ) tr (cid:2) σ i S ra ( p + k ) σ j S rr ( p ) + σ i S rr ( p + k ) σ j S ar ( p ) (cid:3) . (2.17)where ( −
1) in front comes from fermion statistics. In Ref.[12] it was shown that afterextracting (cid:15) ijl k l for the P-odd part, the limit k → ω → µ/ π , without featuring pinch singularity. However, we will see that ω → ω for ξ . Using thethermal relation (2.15) to replace S rr with ( S ra − S ar ), we have several combinations of S ra and S ar . From the well-known fact (see Ref.[51]) that the pinch singularity appearsonly from the pair of S ra and S ar sharing a same momentum ‡ , let us select only termsthat potentially contain pinch singularity, which results in G ij, Pinch( ra ) ( k ) = (cid:90) d p (2 π ) (cid:0) n + ( p + ω ) − n + ( p ) (cid:1) tr (cid:2) σ i S ra ( p + k ) σ j S ar ( p ) (cid:3) . (2.18)It is clear from this expression that ω → n + ( p + ω ) − n + ( p )) ≈ ( dn + ( p ) /dp ) ω + O ( ω ) already gives a linear factor in ω . Moreover, one can put ω → ω for ξ . In computing the above using (2.11) and (2.12) for S ra and S ar ,let us recall that the chosen s = ± representing particle or anti-particle from S ra must bethe same s chosen in S ar to have a pinch singularity in their product. Therefore, we have G ij, Pinch( ra ) ( k ) ≈ ω (cid:90) d p (2 π ) (cid:18) dn + ( p ) dp (cid:19) tr (cid:2) σ i S ra ( p + k ) σ j S ar ( p ) (cid:3) ≈ ω (cid:90) d p (2 π ) (cid:18) dn + ( p ) dp (cid:19) (cid:88) s = ± i tr [ σ i P s ( p + k ) σ j P s ( p )]( p − s | p + k | + iζ/ p − s | p | − iζ/ . (2.19)To identify the P-odd structure containing (cid:15) ijl k l in small k → k -dependence: one is from the denominator and the ‡ This is because S ra ( S ar ) has particle poles slightly below (above) the real axis by an amount ± iζ/ /ζ factor which is the (regularized) pinch singularity.This also means that s = ± in (2.11) and (2.12) must be common in S ra and S ar pair causing a pinchsingularity. k since we take k → (cid:15) ijl k l piece. If one expands thedenominator to linear order in k , we then need to put k = 0 in the projection operators.The resulting trace using (2.16) givestr (cid:2) σ i P s ( p ) σ j P s ( p ) (cid:3) = 14 | p | tr (cid:2) σ i (¯ σ · p s ) σ j (¯ σ · p s ) (cid:3) = p i p j | p | , (2.20)where we use tr (cid:2) σ µ ¯ σ ν σ α ¯ σ β (cid:3) = 2( g µν g αβ − g µα g νβ + g µβ g να ) + 2 i (cid:15) µναβ , (2.21)and p s = 0. It is clear that this contribution does not lead to a P-odd contributionwhich should be anti-symmetric in i and j . Hence, we can ignore k dependence in thedenominator, which allows us to use the ordinary techniques dealing with pinch singularityin k = 0 limit. In section 4, we will prove that this simplification generalizes to all orderladder diagrams, that is, the k -dependences in the denominators appearing in the ladderdiagrams do not contribute to a P-odd part of the correlation function up to first orderin k , and hence can be ignored.In computing P-odd k -dependence in tr [ σ i P s ( p + k ) σ j P s ( p )] using (2.16), one canreplace P s ( p + k ) = − s ¯ σ · ( p + k ) s / (2 | p + k | ) with − s ¯ σ · ( p + k ) s / (2 | p | ) by the same reasonas above, and we havetr (cid:2) σ i P s ( p + k ) σ j P s ( p ) (cid:3) ∼ | p | tr (cid:2) σ i (¯ σ · ( p + k ) s ) σ j (¯ σ · p s ) (cid:3) ∼ i | p | (cid:15) iµjν (( p + k ) s ) µ ( p s ) ν (2.22)where in the second line, we use the fact that P-odd contribution can come only from thelast P-odd term in the σ -matrix trace (2.21). Our symbol ∼ cares only P-odd part linearin k . When µ = 0, ν = l we have s i | p | (cid:15) ijl | p + k | p l ∼ s i | p | (cid:15) ijl ( p · k ) p l ∼ s i | p | (cid:15) ijl k l , (2.23)where we use | p + k | ≈ | p | + ( p · k ) / | p | + O ( k ) and replace p m p l → (1 / δ ml | p | since theangular p integration in the final expression (2.19) is isotropic. When µ = l and ν = 0we have − s i | p | (cid:15) ijl ( p l + k l ) | p | ∼ − s i | p | (cid:15) ijl k l , (2.24)7o, summing these two possibilities gives us the P-odd part of tr [ σ i P s ( p + k ) σ j P s ( p )] astr (cid:2) σ i P s ( p + k ) σ j P s ( p ) (cid:3) ∼ − s i | p | (cid:15) ijl k l , (2.25)and from (2.19) we have G ij, Pinch( ra ) ∼ iω(cid:15) ijl k l (cid:90) d p (2 π ) (cid:18) dn + ( p ) dp (cid:19) | p | (cid:88) s = ± s ( p − s | p | + iζ/ p − s | p | − iζ/ . (2.26)The remaining computation is a standard procedure dealing with pinch singularityappearing in the denominators of (2.26). The p integration can be done in the complex p plane by closing the contour in either upper or lower half plane. The leading singularityappears from the pole p = s | p | ± iζ/ / ( ± iζ ):this gives a leading order contribution at weak coupling limit since ζ ∼ g log(1 /g ) T .Once this 1 /ζ term is identified from the residue of the denominators in (2.26), one canneglect ζ in the pole location p ≈ s | p | for all other terms as it engenders only higherorder terms in g . This is because the p integration has its dominant support in the region | p | ∼ T while ζ ∼ g log(1 /g ) T (cid:28) | p | . Therefore, one can effectively replace the twodenominators in (2.26) with 1( p − s | p | + iζ/ p − s | p | − iζ/ → πiiζ δ ( p − s | p | ) , (2.27)which will be used frequently in the following sections. This gives us G ij, Pinch( ra ) ∼ iω(cid:15) ijl k l ζ (cid:90) d p (2 π ) | p | (cid:88) s = ± s (cid:18) dn + ( p ) dp (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) p = s | p | , (2.28)where we ignore the momentum dependence of ζ for now, which is not strictly valid(we will be more precise in our full ladder resummation in the next section). From dn + ( p ) /dp = − βn + ( p )(1 − n + ( p )) and n + ( −| p | ) = 1 − n − ( | p | ), the integral becomes (cid:90) d p (2 π ) | p | (cid:88) s = ± s (cid:18) dn + ( p ) dp (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) p = s | p | (2.29)= − β (cid:90) d p (2 π ) | p | ( n + ( | p | )(1 − n + ( | p | )) − ( n − ( | p | )(1 − n − ( | p | )))= − β π (cid:90) ∞ d | p | | p | ( n + ( | p | )(1 − n + ( | p | )) − ( n − ( | p | )(1 − n − ( | p | ))) = − µ π , and we finally have a 1-loop expression G ij, Pinch( ra ) ∼ − iω µ π ζ (cid:15) ijl k l ≡ iω ξ − loop5 (cid:15) ijl k l , ξ − loop5 = − µ π ζ . (2.30)8lthough the overall sign of ξ depends on the chirality, the relative negative sign com-pared to the static value of chiral magnetic conductivity σ χ (0) = µ/ π doesn’t dependon chirality and is a meaningful dynamical result. Holographic computations produce thesame negative sign between σ χ (0) and ξ .Figure 2: A generic ladder diagram that contributes to the leading log result.The above exercise shows quite a similar feature to that one finds in the electric con-ductivity, and one can follow the lessons we have learned from the computation of electricconductivity. The 1 /ζ dependence from a pair of pinching propagators S ra ( p ) S ar ( p ) sig-nals a non-analytic dependence on the coupling constant. In a multi-loop ladder diagramshown in Fig.2 for example, each pair of pinching propagators sharing the same momen-tum produces a factor of 1 /ζ ∼ / ( g log(1 /g ) T ) which compensates a g from an extragauge boson exchange, making the diagram of the same order as the 1-loop diagram in thepower counting of coupling constant. Hence, one needs to sum up all multi-loop ladderdiagrams to get a correct leading order result for ξ , which can be achieved by solving aSchwinger-Dyson type integral equation that we will describe in the next section. Moreelaborate power counting [52, 54] shows that the leading contribution comes from the softregion of gauge boson momentum Q ∼ gT , so one needs a Hard Thermal Loop re-summedgauge boson propagator [55, 56] for the internal gauge boson exchange lines. The fermionmomentum stays hard ∼ T , so fermion lines and all vertices are bare ones.From the 1-loop result of ξ − loop5 with 1 /ζ ∼ / ( g log(1 /g ) T ) dependence, the leadinglog result for ξ from solving the Schwinger-Dyson equation might be expected to be ∼ / ( g log(1 /g ) T ). However, the correct dependence turns out to be ∼ / ( g log(1 /g ) T ):this is also the same as in the electric conductivity. In both cases, a physics reason behind9his is that small angle scatterings ( θ (cid:28) g ) by transverse space-like thermal gauge bosonexcitations (whose non-zero thermal spectral density is due to Landau damping physics)cannot affect the charge transport phenomena much, since they deflect charged fermiontrajectories responsible for charge transports only slightly by small angles. On the otherhand, these small angle scatterings by ultra-soft ( p ∼ g T ) transverse gauge bosons isthe dominant source for the total decay rate ζ ∼ g log(1 /g ) T , where the log comesfrom log( m D / Λ IR ) ∼ log(1 /g ) with Λ IR ∼ g T being the non-perturbative IR cutofffor the transverse magnetic sector, and m D ∼ gT is the characteristic soft scale. Thismeans that the effective IR regulator for the pinch singularities that is meaningful for thefinal conductivities is not given by the total damping rate ζ , but is provided by largerangle scatterings ( θ (cid:29) g ) and fermion-conversion to gauge bosons, which are governedby g log(1 /g ) T rate. In the latter, the origin of the log is completely different: it isfrom log( T /m D ) ∼ log(1 /g ). In our diagrammatic approach of the Schwinger-Dysonequation, this physics manifests itself in a nice cancellation of leading log part of ζ in theequation that we will see in the following sections, and what remains is indeed somethingof g log(1 /g ) T coming from the rate of fermion-conversion to gauge boson.We end this section by recalling that the situation is quite different for color conduc-tivity where even small angle scatterings by thermal transverse gluons can change thecolor charge of charge carriers (either fermion or gauge boson) due to non-Abelian natureof color charges [57], so that the same rate responsible for the leading log damping ratealso governs the color conductivity, leading to its 1 / ( g log(1 /g )) behavior [58, 59]. In this section, we set-up the Schwinger-Dyson equation that sums up all-loop ladderdiagrams with leading order pinch singularities. The idea is essentially similar to theone in the diagrammatic computations of shear viscosity or electric conductivity [52, 53],except that we have to keep a finite external momentum k up to first order in k to extracta P-odd part (but, we can still put ω = 0 from the outset since one factor of ω comesout from kinematics, see (2.18) and (2.19)). We choose to work in real-time Schwinger-Keldysh formalism in ra-basis for our convenience, rather than the Euclidean formalismwith subsequent analytic continuation as used in some previous literature. For electricalconductivity, we check that they produce the same result as they should.10igure 3: Exemplar real time Feynman diagrams that can give leading pinch singularity.Since we are computing G ij ( ra ) ( k ), the vertex at the far right in any ladder diagramis an a-type one with one fermion leg r-type and the other a-type. Since there is noaa-propagator, the a-type leg should have r-type in the other end on its left. Becausea pinch singularity can appear only from a pair of S ra and S ar , the r-type leg from thevertex should have r-type on the other end on its left, since having a-type on the otherend gives the same type of fermion propagator to the one from the former, and does notgive a pinch singularity. See Fig.3 for an exemplar ladder diagram that can give a leadingpinch singularity. In our convention, one reads ra-types of a fermion propagator alongthe reversed direction of its momentum arrow, which can be seen in Fig.3. The reasonwhy having a rr-type propagator in the diagram can give rise to a pinch singularity is thethermal relation (2.15), S rr ( p ) = (cid:18) − n + ( p ) (cid:19) ( S ra ( p ) − S ar ( p )) , (3.31)so that one can pick either S ra or S ar piece from S rr to have a pair of S ra and S ar thatgives a pinch singularity. It is clear then that the rest of a ladder diagram on the leftother than the far right vertex should have two final fermion legs of r-type on its right, inorder to create a leading order pinch singularity: that is, it has to be an effective rr-typevertex. At 1-loop order, this was automatic since it is a bare J ir vertex. What we have todo is to sum up all loop ladder diagrams for this effective rr-vertex that appears on theleft side of the diagram.Denoting the resulting summed vertex Λ i ( p, k ) that is a 2 × p is the loop momentum and k is the (small) external momentum,11igure 4: The diagrams that need to be computed to obtain the retarded response functionto leading log order. The effective vertex on the left includes infinite number of ladderdiagrams.the final G ij, Pinch( ra ) ( k ) is obtained from two possible Feynman diagrams in Fig.4 which looksimilar to those in Fig.1 except that the vertex on the left is now Λ i ( p, k ) instead of σ i , G ij, Pinch( ra ) ( k ) = ( − (cid:90) d p (2 π ) tr (cid:2) Λ i ( p, k ) (cid:0) S ra ( p + k ) σ j S rr ( p ) + S rr ( p + k ) σ j S ar ( p ) (cid:1)(cid:3) . (3.32)Using (3.31) picking up only pairs of S ra and S ar for a pinch singularity, and expandingit in ω with the same manipulation that led to (2.19) gives, up to first order in ω , G ij, Pinch( ra ) ( k ) = − ω (cid:90) d p (2 π ) (cid:18) dn + ( p ) dp (cid:19) (cid:88) s = ± tr [Λ i ( p, k ) P s ( p + k ) σ j P s ( p )]( p − s | p + k | + iζ p + k ,s / p − s | p | − iζ p ,s / , (3.33)where we put external frequency ω ≡ k = 0 in the effective vertex Λ i ( p, k ) and otherplaces since we already have one ω factor in front. Note also that the damping rate ζ p ,s depends on the on-shell momentum as well as s = ± (that is, whether it is particle orantiparticle) as indicated in the expression § . We will be concerned with only this objectΛ i ( p, k ) after putting ω = 0 in the following.The summation of all multi-loop ladder diagrams for this effective rr-type vertex, start-ing from the bare one J ir = ψ † r σ i ψ r can be achieved by solving the associated Schwinger-Dyson type equation, which is depicted in Fig.5. The “kernel” which is made of twointernal fermion lines and one soft gauge boson (we call it photon) exchange can havethree possible Feynman diagrams that can give a leading pinch singularity as shown in § The dependence on s comes via the combination sµ in the presence of chemical potential µ we areconsidering. See our Appendix 2 for a detailed discussion. i ( p, k ) .Figure 6: The three real-time diagrams with leading pinch singularity for the kernel inthe Schwinger-Dyson equation depicted in Fig. 5. The effective vertex connects only tor-type endings of fermion propagators.Fig.6. The resulting Schwinger-Dyson equation ∗ is written as (in the following we denoteQED coupling constant by e instead of g )Λ i ( p, k ) = σ i + ( ie ) (cid:90) d Q (2 π ) σ β S ar ( p + Q )Λ i ( p + Q, k ) S ra ( p + Q + k ) σ α G ( rr ) αβ ( Q )+ ( ie ) (cid:90) d Q (2 π ) σ β S ar ( p + Q )Λ i ( p + Q, k ) S rr ( p + Q + k ) σ α G ( ar ) αβ ( Q )+ ( ie ) (cid:90) d Q (2 π ) σ β S rr ( p + Q )Λ i ( p + Q, k ) S ra ( p + Q + k ) σ α G ( ra ) αβ ( Q ) , (3.34)where G ( ab ) αβ ( a, b = r or a ) are the photon propagators in Schwinger-Keldysh contour G ( ab ) αβ ≡ (cid:104) A ( a ) α ( Q ) A ( b ) β ( − Q ) (cid:105) SK = (cid:90) d x e − iQx (cid:104) A ( a ) α ( x ) A ( b ) β (0) (cid:105) SK , (3.35) ∗ In [53] it was shown that the Ward identity requires addition of an extra term in Fig.5 involving softfermion lines. This diagram gives a sub-leading contribution to the electric conductivity and we expectthe same for our ξ . We leave the explicit computation as a future work. J J correlation functions)since the photon momentum Q is soft. We will work in the Coulomb gauge which separateslongitudinal and transverse modes in a clear way. A summary of G ( ab ) αβ in this gaugeincluding the P-odd part coming from the P-odd part of HTL photon self-energy is givenin the Appendix 1, where we also find some useful sum rules for the P-odd part of theirspectral density, which will be used importantly later. Using (3.31) and a similar thermalrelation for photons (see the Appendix 1), G ( rr ) αβ ( Q ) = (cid:18)
12 + n B ( q ) (cid:19) (cid:16) G ( ra ) αβ ( Q ) − G ( ar ) αβ ( Q ) (cid:17) = (cid:18)
12 + n B ( q ) (cid:19) ρ ph αβ ( Q ) , (3.36)where the photon spectral density is defined by ρ ph αβ ( Q ) ≡ ( G ( ra ) αβ ( Q ) − G ( ar ) αβ ( Q )) , (3.37)and n B ( q ) = 1 / ( e βq − i ( p, k ) = σ i + ( ie ) (cid:90) d Q (2 π ) [ σ β S ar ( p + Q )Λ i ( p + Q, k ) S ra ( p + Q + k ) σ α × ρ ph αβ ( Q ) ( n + ( p + q ) + n B ( q ))] . (3.38)Note that the photon spectral density ρ ph αβ ( Q ) is hermitian in ( α, β ) indices, but notnecessarily real. In fact, the P-odd self-energy leads to a purely imaginary, anti-symmetriccontribution to the spectral density. We refer the readers to the Appendix 1 for a detailedexposition.From the pair S ar ( p + Q ) S ra ( p + Q + k ) in (3.38) for small k limit, one can extractthe leading pinch singularity, S ar ( p + Q )Λ i ( p + Q, k ) S ra ( p + Q + k ) → (cid:88) t = ± i P t ( p + q )Λ i ( p + Q, k ) P t ( p + q + k )( p + q − t | p + q | − iζ p + q ,t / p + q − t | p + q + k | + iζ p + q + k ,t / . (3.39)Since the photon momentum Q is soft, Q (cid:28) | p | ∼ T , and the pinch singularity in thefinal equation for G ij, Pinch( ra ) ( k ) in (3.33) necessitates the loop-momentum p to be on-shell, p = s | p | , the only possible way to have a pinch singularity for soft q integration in (3.38)is to pick only t = s piece in the expression (3.39): it means that in a ladder diagramthe leading pinch singular contribution comes from a particle loop or an anti-particleloop without any “transition” between particle and anti-particle throughout a diagram.Physically, it is obvious that a nearly on-shell particle (anti-particle) can not change to14n anti-particle (particle) with soft photon scatterings. Therefore, for a given choice of s in (3.33), we keep only t = s piece of (3.39) in the integral equation (3.38), and thesolution of the resulting integral equation we also label by s : Λ s ( p, k ). The more correctexpression for (3.33) is then G ij, Pinch( ra ) ( k ) = − ω (cid:90) d p (2 π ) (cid:18) dn + ( p ) dp (cid:19) (cid:88) s = ± tr [Λ is ( p, k ) P s ( p + k ) σ j P s ( p )]( p − s | p + k | + iζ p + k ,s / p − s | p | − iζ p ,s / , (3.40)where Λ is ( p, k ) satisfies the integral equationΛ is ( p, k ) = σ i + e (cid:90) d Q (2 π ) (cid:20) σ β P s ( p + q )Λ is ( p + Q, k ) P s ( p + q + k ) σ α × ρ ph αβ ( Q ) ( n + ( p + q ) + n B ( q ))( p + q − s | p + q | − iζ p + q ,s / p + q − s | p + q + k | + iζ p + q + k ,s / (cid:21) . (3.41)The rest of the paper is about solving the integral equation (3.41) in leading logarithmicorder in the coupling constant e .Since our transport coefficient ξ is obtained from the P-odd part of G ij, Pinch( ra ) ( k ) by G ij, Pinch , P − odd( ra ) ( ω, k ) = iωξ (cid:15) ijl k l + O ( ω , k ) , (3.42)we would like to expand (3.40) in k up to first order, focusing only on the P-odd (cid:15) ijl k l structure at the same time. Since ξ is CP-odd which shares the same quantum numberwith the (axial) chemical potential µ , it can only contain odd powers in µ , as seen inthe 1-loop computation in the preceding section. In our work, we will only compute ξ up to linear order in µ in small µ limit, neglecting higher order terms of µ and beyond.Therefore, we will only be interested in a linear µ dependence of (3.40) and (3.41) in thefollowing. In solving (3.41) up to linear order in k , and using it to compute (3.40), there are varioussources of k dependence appearing in the equations. The problematic source is the k dependence in the denominators of the equations (3.40) and (3.41). For example in(3.40), we have p − s | p + k | + iζ p + k ,s / ≈ p − s | p | + iζ p ,s / − s ˆ p · k + i ( ∂ζ p ,s /∂ p ) · k / · · · (4.43)15iving rise to up to linear in k ,1 p − s | p + k | + iζ p + k ,s / ≈ p − s | p | + iζ p ,s / s ˆ p · k − i ( ∂ζ p ,s /∂ p ) · k / p − s | p | + iζ p ,s / + · · · (4.44)The second term is a double pole, and when used in (3.40) it engenders a ∼ /ζ de-pendence which is larger than the usual 1 /ζ pinch singularity. The same is true for the k -dependence in the denominators of the integral equation (3.41). For P-even part, thismay be what one encounters when trying to include a finite k in the current correlationfunctions, which seems to be related to the expected appearance of diffusion pole structure σ el ω − iDk ≈ σ el ω + iσ el D k ω + · · · , (4.45)since the Einstein relation gives D = σ el /χ ( χ is the charge susceptibility) and the k dependence is quadratic in electric conductivity σ el ∼ T /e log(1 /e ). However, such adiffusion pole structure is not expected in the P-odd part of our interest [12] † , and it isnatural to expect that these k -dependences from the denominators in (3.40) and (3.41)do not contribute to our P-odd structure (cid:15) ijl k l . Let us show this important simplificationin the following. As a consequence, one can ignore all k ’s in the denominators of theequations, and the only interesting k dependence comes from the projection operators inthe numerators.Let us consider a generic multi-loop ladder diagram depicted in Fig.7, and let’s choosean arbitrary internal S ra S ar pair from the “side rail” that can give rise to a pinch singu-larity in small k limit. By shifting loop momentum p , the momentum that flows in S ra can be put to p + k , then the momentum of S ar is p . The denominator of S ra ( p + k )contains a k -linear piece as in (4.43). We would like to show that this k dependence doesnot lead to any P-odd structure (cid:15) ijl k l .Once we get a term like (4.44) that is linear in k from expanding the denominatorof S ra ( p + k ), we should put k = 0 in all other parts of the diagram since it alreadysaturates the linear k dependence we are looking at. These include projection operatorsin the numerators of S ra ( p + k ) and S ar ( p ): P s ( p + k ) and P s ( p ) (recall that we need tohave a same s throughout the diagram for the leading pinch singularity), as well as the † There could arise a diffusion pole structure in the P-odd part if one considers coupling to energy-momentum sector of the theory leading to “chiral magnetic energy flow”. However, when expanded in k it would give a term of k or higher [60, 61]. Also, the coupling to energy-momentum sector is of order µ , and the resulting P-odd effect is of order µ [60, 61]. Therefore, we can ignore this possibility in ourwork. (cid:48) remaining parts of the diagram other than the chosen S ra ( p + k ) S ar ( p ) pair, which wecall effective vertices: the part on the left let us call Λ i ( p, k ) and the part on the right˜Λ j ( p, k ). See Fig.7. The value of the diagram is then proportional to (the p integral) oftr (cid:104) Λ i ( p ) P s ( p ) ˜Λ j ( p ) P s ( p ) (cid:105) ∝ tr (cid:104) Λ i ( p )(¯ σ · p s ) ˜Λ j ( p )(¯ σ · p s ) (cid:105) , (4.46)where Λ i ( p ) ≡ Λ i ( p, k = 0), ˜Λ j ( p ) ≡ ˜Λ j ( p, k = 0), etc. All ij index structure comes fromthis spinor trace.The effective vertex Λ i ( p ) is a 2 × σ µ = ( , (cid:126)σ )forms a basis for any 2 × i ( p ) = σ µ λ iµ ( p ). By invoking rotationalinvariance, we generally have λ i ( p ) = f ( p , | p | ) p i , λ il ( p ) = f ( p , | p | ) δ il + f ( p , | p | ) p l p i + f ( p , | p | ) (cid:15) iml p m , (4.47)where f i are functions only on p and | p | . Similarly, we have for ˜Λ j ( p ) = σ ν ˜ λ jν ( p ) with˜ λ j ( p ) = ˜ f ( p , | p | ) p j , ˜ λ jl ( p ) = ˜ f ( p , | p | ) δ jl + ˜ f ( p , | p | ) p l p j + ˜ f ( p , | p | ) (cid:15) jml p m . (4.48)Inserting these representations of Λ i ( p ) and ˜Λ j ( p ) into the above (4.46), and using thetrace formula we repeat heretr (cid:2) σ µ ¯ σ ν σ α ¯ σ β (cid:3) = 2( g µν g αβ − g µα g νβ + g µβ g να ) + 2 i (cid:15) µναβ , (4.49)we immediately see that the last (cid:15) -tensor term in (4.49) does not contribute since p s appears twice in (4.46), and we havetr (cid:104) Λ i ( p )(¯ σ · p s ) ˜Λ j ( p )(¯ σ · p s ) (cid:105) = 4( λ i ( p ) · p s )(˜ λ j ( p ) · p s ) , (4.50)17here we use p s = 0. Then, we have λ i ( p ) · p s = s | p | λ i ( p ) + p l λ il ( p ) = ( s | p | f ( p , | p | ) + f ( p , | p | ) + | p | f ( p , | p | )) p i , (4.51)which is proportional to p i . Note that the piece involving f drops out. The sameconclusion is true, that is, ˜ λ j ( p ) · p s ∼ p j , and hence the result for the trace in (4.46) isproportional to p i p j . Since it is symmetric with respect to ij , it is clear that the result cannot contribute to the P-odd part of (cid:15) ijl k l . In summary, we have shown that k dependencefrom the denominator of any internal fermion line in leading pinch singularity limit doesnot contribute to the P-odd structure (cid:15) ijl k l , and hence we can neglect all k ’s appearingin the denominators, especially in our equations (3.40) and (3.41).Once we remove all k ’s from the denominators, we have G ij, Pinch( ra ) ( k ) = − ω (cid:90) d p (2 π ) (cid:18) dn + ( p ) dp (cid:19) (cid:88) s = ± tr [Λ is ( p, k ) P s ( p + k ) σ j P s ( p )]( p − s | p | + iζ p ,s / p − s | p | − iζ p ,s / , (4.52)andΛ is ( p, k ) = σ i + e (cid:90) d Q (2 π ) (cid:20) σ β P s ( p + q )Λ is ( p + Q, k ) P s ( p + q + k ) σ α × ρ ph αβ ( Q ) ( n + ( p + q ) + n B ( q ))( p + q − s | p + q | − iζ p + q ,s / p + q − s | p + q | + iζ p + q ,s / (cid:21) . (4.53)The p integration in (4.52) can be computed in leading pinch singularity limit by replacing1( p − s | p | + iζ p ,s / p − s | p | − iζ p ,s / → πζ p ,s δ ( p − s | p | ) , (4.54)which enforces the on-shell condition p = s | p | on the p appearing in the integral equation(4.55). We will assume this on-shell condition throughout our computation in the followingsections. Then, the integral equation becomesΛ is ( p , k ) = σ i + e (cid:90) d Q (2 π ) (cid:20) σ β P s ( p + q )Λ is ( p + q , k ) P s ( p + q + k ) σ α × ρ ph αβ ( Q ) ( n + ( s | p | + q ) + n B ( q ))( q + s | p | − s | p + q | − iζ p + q ,s / q + s | p | − s | p + q | + iζ p + q ,s / (cid:21) , (4.55)where Λ is ( p , k ) ≡ Λ is ( p, k ) (cid:12)(cid:12) p = s | p | . The reason why we can also replace Λ is ( p + Q, k ) inthe integral kernel with its on-shell value Λ is ( p + q , k ) is that the pinch singularity in the18ernel of the integral equation 1( p + q − s | p + q | − iζ p + q ,s / p + q − s | p + q | + iζ p + q ,s / → πζ p + q ,s δ ( p + q − s | p + q | ) , (4.56)will impose the on-shell condition p + q = s | p + q | as well. With this replacement of(4.56) in (4.55), we finally haveΛ is ( p , k ) = σ i + e (cid:90) d Q (2 π ) (cid:20) σ β P s ( p + q )Λ is ( p + q , k ) P s ( p + q + k ) σ α × ρ ph αβ ( Q ) ( n + ( s | p | + q ) + n B ( q )) (2 π ) δ ( q + s | p | − s | p + q | ) /ζ p + q ,s (cid:21) , (4.57)which is our starting point in solving the integral equation in leading logarithmic orderin the next section. The integral equation (4.57) obtained in leading pinch singularity limit is a matrix equa-tion, and it is desirable to transform it into a scalar equation. In fact, we don’t need itsfull matrix structure: what we need at the end in (4.52) is the tracetr (cid:2) Λ is ( p , k ) P s ( p + k ) σ j P s ( p ) (cid:3) = tr (cid:2) P s ( p )Λ is ( p , k ) P s ( p + k ) σ j (cid:3) , (5.58)and it is obvious that we only need the component of Λ is ( p , k ) projected onto the eigenspaceof the projection operator P s ( p ) on the left and P s ( p + k ) on the right, that is, P s ( p )Λ is ( p , k ) P s ( p + k ) . (5.59)Since the spinor space is two dimensional, the above projected matrix is essentially a singlenumber. This fact manifests itself by the following statement: for any 2 × A , theprojected matrix P s ( p ) A P s ( p + k ) must be proportional to the matrix P s ( p ) P s ( p + k )composed only of the projection operators. The proportionality constant, which containsthe information on A , is easily found by comparing traces of the expected relation, P s ( p ) A P s ( p + k ) = tr [ P s ( p ) A P s ( p + k )]tr [ P s ( p ) P s ( p + k )] P s ( p ) P s ( p + k )= tr [ P s ( p ) A P s ( p + k )] (1 + (cid:98) p · (cid:92) p + k ) P s ( p ) P s ( p + k ) , (5.60)19here (cid:98) p ≡ p / | p | , and tr [ P s ( p ) P s ( p + k )] = (1 + (cid:98) p · (cid:92) p + k ). Using this, it is straightfor-ward to convert our matrix integral equation into a scalar equation, and for this purposelet us introduce three scalar functions D is ( p , k ), Σ µs ( p , k ), and F s ( p ; q , k ) by P s ( p )Λ is ( p , k ) P s ( p + k ) = D is ( p , k ) P s ( p ) P s ( p + k ) , P s ( p ) σ µ P s ( p + k ) = Σ µs ( p , k ) P s ( p ) P s ( p + k ) , P s ( p ) P s ( p + q ) P s ( p + q + k ) P s ( p + k ) = F s ( p ; q , k ) P s ( p ) P s ( p + k ) . (5.61)The expressions for Σ µs ( p , k ) and F s ( p ; q , k ) can easily be found by computing the neces-sary traces involved, for example we haveΣ s ( p , k ) = 1 , Σ is ( p , k ) = s ( (cid:98) p i + (cid:92) ( p + k ) i ) − i(cid:15) ijl (cid:98) p j (cid:92) ( p + k ) l (1 + (cid:98) p · (cid:92) p + k ) , (5.62)and the expression for F s ( p ; q , k ) can be found in the Appendix 3. Using the fact that σ µ is hermitian, we also have P s ( p + k ) σ µ P s ( p ) = (Σ µs ( p , k )) ∗ P s ( p + k ) P s ( p ) . (5.63)The scalar function D is ( p , k ) is what we would like to find by solving the integral equation,and once it is found, the final expression for the correlation function G ij, Pinch( ra ) ( k ) is givenfrom (4.52) by G ij, Pinch( ra ) ( k ) = − ω (cid:90) d p (2 π ) (cid:18) dn + ( p ) dp (cid:19) (cid:88) s = ± tr [ P s ( p )Λ is ( p , k ) P s ( p + k ) σ j ]( p − s | p | + iζ p ,s / p − s | p | − iζ p ,s / − ω (cid:90) d p (2 π ) (cid:18) dn + ( p ) dp (cid:19) (cid:88) s = ± D is ( p , k )tr [ P s ( p + k ) σ j P s ( p )]( p − s | p | + iζ p ,s / p − s | p | − iζ p ,s / − ω (cid:90) d p (2 π ) (cid:18) dn + ( p ) dp (cid:19) (cid:88) s = ± D is ( p , k )(Σ js ( p , k )) ∗ (1 + (cid:98) p · (cid:92) p + k )( p − s | p | + iζ p ,s / p − s | p | − iζ p ,s / − ω (cid:90) d p (2 π ) (cid:88) s = ± (cid:18) dn + ( p ) dp (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) p = s | p | (cid:0) Σ js ( p , k ) (cid:1) ∗ D is ( p , k ) ζ p ,s , (5.64)where we use (5.61), (5.63) in the second and third lines, and replace (1 + (cid:98) p · (cid:92) p + k ) withunity in the last line, since (cid:98) p · (cid:92) p + k = 1 up to negligible O ( k ) corrections. As the lastexpression involves the combination D is ( p , k ) /ζ p ,s , let us also define φ is ( p , k ) ≡ D is ( p , k ) ζ p ,s . (5.65)20pplying projection operators P s ( p ) on the left and P s ( p + k ) on the right to ourintegral equation (4.57), and using (5.61), (5.63) we finally get the following scalar integralequation to be solved for φ is ( p , k ), ζ p ,s φ is ( p , k ) = Σ is ( p , k ) + e (cid:90) d Q (2 π ) Σ βs ( p , q ) φ is ( p + q , k ) (Σ αs ( p + k , q )) ∗ F s ( p ; q , k ) × ρ ph αβ ( Q ) ( n + ( s | p | + q ) + n B ( q )) (2 π ) δ ( q + s | p | − s | p + q | )= Σ is ( p , k ) + e (cid:90) d Q (2 π ) K s ( p , k ; Q ) φ is ( p + q , k )(2 π ) δ ( q + s | p | − s | p + q | ) , (5.66)with an integral kernel K s ( p , k ; Q ) defined as K s ( p , k ; Q ) ≡ Σ βs ( p , q ) (Σ αs ( p + k , q )) ∗ ρ ph αβ ( Q ) F s ( p ; q , k )( n + ( s | p | + q ) + n B ( q )) . (5.67)Our task is to find a solution for φ is ( p , k ) up to first order in k which can give rise to theP-odd structure (cid:15) ijl k l in the expression (5.64) for G ij, Pinch( ra ) ( k ). We will be interested inonly the leading logarithmic order in e .The structure of φ is ( p , k ) expanded up to first order in k can be severely constrainedby rotational invariance, φ is ( p , k ) = χ is ( p ) + f ils ( p ) k l = χ s ( | p | ) ˆ p i + a s ( | p | ) k i + b s ( | p | ) ˆ p i ( ˆ p · k ) + i | p | c s ( | p | ) (cid:15) ilm ˆ p l k m , (5.68)with four scalar functions χ s , a s , b s , c s which depend only on | p | . Using the expansion of(Σ js ( p , k )) ∗ up to first order in k , (cid:0) Σ js ( p , k ) (cid:1) ∗ = s ˆ p j + s | p | (cid:0) δ jl − ˆ p j ˆ p l (cid:1) k l + i | p | (cid:15) jlm ˆ p l k m , (5.69)the expression for G ij, Pinch( ra ) ( k ) from (5.64) can be computed to first order in k to find theP-odd structure proportional to (cid:15) ijl k l as G ij, Pinch( ra ) ( k ) = − ω (cid:90) d p (2 π ) (cid:88) s = ± (cid:18) dn + ( p ) dp (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) p = s | p | (cid:0) Σ js ( p , k ) (cid:1) ∗ φ is ( p , k ) ∼ − i ω (cid:90) d p (2 π ) (cid:88) s = ± (cid:18) dn + ( p ) dp (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) p = s | p | | p | (cid:18) s c s ( | p | ) − χ s ( | p | ) (cid:19) (cid:15) ijl k l , (5.70)where ∼ above only cares about P-odd terms. Note that the two functions ( a s , b s ) donot contribute to our P-odd term, so we don’t need to compute them. Therefore, we willfocus on χ s and c s only in the following. 21 .1 Computation of χ s ( | p | ) The function χ s ( | p | ) is something that has already been known in previous computationsof electric conductivity, although we will see that there is an important correction to itlinear in µ that is relevant to our final value of ξ . It satisfies the integral equation (5.66)after putting k = 0 ζ p ,s χ s ( | p | ) ˆ p i = s ˆ p i + e (cid:90) d Q (2 π ) K s ( p , Q ) χ s ( | p + q | ) (cid:92) p + q i (2 π ) δ ( q + s | p | − s | p + q | ) . (5.71)Our treatment that follows for χ s ( | p | ) is mostly the same to that one can find in Refs.[52,53], and our computation for χ s will confirm the previous results in literature. Let ushowever present some details along which we can introduce several key elements that willbe needed in our next treatment for c s ( | p | ), which is new and more interesting to us.First it is important to observe that the integral e (cid:90) d Q (2 π ) K s ( p , Q )(2 π ) δ ( q + s | p | − s | p + q | ) , (5.72)that appears in the above equation is precisely equal to the contribution to the fermiondamping rate from soft photon scatterings at full order, which contains the leading ∼ e log(1 /e ) T part of the total damping rate. We reconfirm this fact explicitly in Appendix2 including P-odd spectral density of soft photon propagator. Denoting this by ζ sp p ,s (following notations in Ref.[53]), and writing the total damping rate as ζ p ,s = ζ sp p ,s + ζ sf p ,s where ζ sf p ,s is the other remaining contribution to the damping rate arising from softfermion scatterings (or more precisely, hard fermions making conversion to hard photonsand soft fermions) which is of order e log(1 /e ) T , the integral equation (5.71) takes a form ζ sf p ,s χ s ( | p | ) ˆ p i = s ˆ p i + e (cid:90) d Q (2 π ) K s ( p , q ) (cid:104) χ s ( | p + q | ) (cid:92) p + q i − χ s ( | p | ) ˆ p i (cid:105) × (2 π ) δ ( q + s | p | − s | p + q | ) , (5.73)where we no longer have ζ sp p ,s ∼ e log(1 /e ) T explicitly in the equation, and what remainswill be shown to be only of order ∼ e log(1 /e ) T . This cancellation of e log(1 /e ) T dependence due to the identity of (5.72) with ζ sp p ,s is the diagrammatic manifestationof the physics discussion at the end of section 2: the relevant part of damping ratethat is responsible for fermionic charge transport phenomena is not the total dampingrate governed by small-angle scatterings with ultra-soft transverse photons, but the partarising from fermion conversions to photons with soft fermion scatterings, that is ζ sf p ,s . In22ppendix 2, we present a computation of ζ sf p ,s to leading log order (see Eq.(6.204)), witha special care of its sµ -dependence we will need later.Introducing a variable z = cos ϕ where ϕ is the angle between p and q , one can showthat [53] δ ( q + s | p | − s | p + q | ) = (cid:18) | p | + sq | p || q | (cid:19) δ ( z − z )Θ (cid:0) | q | − ( q ) (cid:1) , (5.74)where Θ( x ) is the Heaviside theta function, and z | q | = ˆ p · q = sq + ( q ) − | q | | p | , (5.75)and using this, one can transform the Q integration into (cid:90) d Q (2 π ) (2 π ) δ ( q + s | p | − s | p + q | ) = (cid:90) ∞ d | q || q | (2 π ) (cid:90) | q |−| q | dq (2 π ) (cid:18) sq | p | (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ˆ p · q → sq + ( q −| q | | p | , (5.76)where one needs to replace any ˆ p · q appearing in the integrand by sq +(( q ) −| q | ) / (2 | p | ).As Q = ( q , q ) ∼ eT is soft and p ∼ T is hard for leading log contribution (which can beseen in retrospect), we expand the integrand in the integral equation (5.73) in powers of( q , q ) / | p | . For example, after some algebra we have up to O ( Q ), χ s ( | p + q | ) (cid:92) p + q i − χ s ( | p | ) ˆ p i ≈ ˆ p i (cid:18) ( q ) − | q | | p | χ s ( | p | ) + sq χ (cid:48) s ( | p | ) + 12 ( q ) χ (cid:48)(cid:48) s ( | p | ) (cid:19) , (5.77)where we use the replacement ˆ p · q → sq + (( q ) − | q | ) / (2 | p | ) in the middle of com-putation, and χ (cid:48) s ( x ) = dχ s ( x ) /dx , etc. Similarly, we need an expansion of K s ( p , q ):with n + ( s | p | + q ) + n B ( q ) = 1 βq + s (cid:18) n s ( | p | ) − (cid:19) + O ( q ) , (5.78)and F s ( p ; q ,
0) = 12 (cid:16) p · (cid:92) p + q (cid:17) = 1 + O ( q ) , (5.79)what remains in K s ( p , q ) is the polarization-contracted photon spectral density ρ ph αβ ( Q ) (Σ αs ( p , q )) ∗ Σ βs ( p , q ) , (5.80)where we need to expand the polarization part (Σ αs ( p , q )) ∗ Σ βs ( p , q ) up to first order in q for our leading log computation. In Coulomb gauge, ρ ph0 i ( Q ) = 0 ( i = 1 , ,
3) and ρ ph00 ( Q ) ≡ ρ L ( Q ) is the longitudinal part of spectral density. The transverse part is ρ ph ij ( Q ) = ρ T ( Q ) (cid:18) δ ij − q i q j | q | (cid:19) + iρ odd ( Q ) (cid:15) ijl q l , (5.81)23here the second term is the P-odd contribution proportional to µ that arises from theP-odd part of current-current correlation function (or photon self-energy) in HTL limit,whose expression can be found in our Appendix 1. Note that it is purely imaginary, butanti-symmetric in i, j , so it is a hermitian matrix in i, j . For our purpose, we would onlyneed its sum rules derived in the Appendix 1, Eqns. (6.155) and (6.156), (cid:90) | q |−| q | dq (2 π ) 1 q ρ odd ( Q ) = − e µ (2 π ) | q | + · · · , (cid:90) | q |−| q | dq (2 π ) q ρ odd ( Q ) = 0 | q | + · · · (5.82)up to less singular terms in small | q | (cid:28) eT limit. All functions ( ρ L , ρ T , ρ odd ) are oddin q → − q , so we need an extra odd power of q in the final integrand to have anon-vanishing q integral over [ −| q | , + | q | ]. The only µ dependence in the usual spectraldensities ρ L/T ( Q ) is through the Debye mass, which is m D = e T / e µ / (2 π ) fora single Weyl fermion. Since we are looking at only up to linear µ dependence, we cansafely neglect µ corrections in m D and use the µ = 0 results for ρ L/T ( Q ). After somealgebra, we have up to O ( q ) ρ ph αβ ( Q ) (Σ αs ( p , q )) ∗ Σ βs ( p , q ) = ρ L ( Q ) + (cid:18) − ( q ) | q | (cid:19) ρ T ( Q ) − s (( q ) − | q | ) | p | ρ odd ( Q ) , (5.83)where the last contribution from the P-odd spectral density, although it is quadratic in q , is presented because its power counting is something new and different from those of ρ L/T ( Q ) as can be seen in (5.82), and should be checked carefully.Let us first estimate this contribution from the P-odd spectral density in the integralequation (5.73). Collecting everything presented above, the contribution from the P-oddspectral density to the integral in (5.73) becomes − s e | p | ˆ p i (cid:90) ∞ d | q || q | (2 π ) (cid:90) | q |−| q | dq (2 π ) (cid:18) sq | p | (cid:19) (cid:18) βq + s (cid:18) n s ( | p | ) − (cid:19)(cid:19) (cid:0) ( q ) − | q | (cid:1) × (cid:18) ( q ) − | q | | p | χ s ( | p | ) + sq χ (cid:48) s ( | p | ) + 12 ( q ) χ (cid:48)(cid:48) s ( | p | ) (cid:19) ρ odd ( Q ) , (5.84)and using the sum rules derived in Appendix 1, (6.155), (6.156), it is easy to find thatthe result is at most of order ∼ e without any logarithmic enhancement. Note that the | q | integration should have an UV cutoff ∼ T since we use HTL approximation for softmomentum Q (cid:28) T . In any case, these are of higher order than e log(1 /e ) of our interest,24o can be neglected. Although we find ρ odd ( Q ) does not affect the leading log equation for χ s ( | p | ), we will find shortly that it does give an important contribution to the equationfor c s ( | p | ) at leading log which is of our more interest.The integral equation (5.73) then takes a form at leading log order as ζ sf p ,s χ s ( | p | ) = s + e (cid:90) ∞ d | q || q | (2 π ) (cid:90) | q |−| q | dq (2 π ) (cid:18) sq | p | (cid:19) (cid:18) βq + s (cid:18) n s ( | p | ) − (cid:19)(cid:19) × (cid:18) ρ L ( Q ) + (cid:18) − ( q ) | q | (cid:19) ρ T ( Q ) (cid:19) × (cid:18) ( q ) − | q | | p | χ s ( | p | ) + sq χ (cid:48) s ( | p | ) + 12 ( q ) χ (cid:48)(cid:48) s ( | p | ) (cid:19) . (5.85)The remaining computation of various integrals of spectral densities is achieved at leadingorder using well-known sum rules of ρ L/T ( Q ) [62]. The leading log contribution will comefrom the region m D (cid:28) | q | (cid:46) T , and following the notations in Ref.[53] by defining J L/Tn ≡ (cid:90) | q |−| q | dq (2 π ) ( q ) n − ρ L/T ( Q ) , (5.86)we have for | q | (cid:29) m D the sum rules ‡ J L ≈ m D | q | , J T ≈ m D | q | (cid:18) log 8 | q | m D − (cid:19) ,J L ≈ m D | q | , J T ≈ m D | q | (cid:18) log 8 | q | m D − (cid:19) ,J L ≈ m D , J T ≈ m D (cid:18) log 8 | q | m D − (cid:19) . (5.87)Using these in (5.85), one encounters a logarithmic divergence (cid:90) Tm D d | q || q | = log( T /m D ) ∼ log(1 /e ) , (5.88)where the IR cutoff is m D since the sum rule expressions used are valid only for | q | (cid:29) m D (see our footnote), and the UV cutoff is T as we assume soft Q (cid:28) T throughoutour treatment, and the modification for hard Q will dampen away the UV divergences.Picking up the logarithmically enhanced terms in the integral, we finally get the differentialequation ζ sf p ,s χ s ( | p | ) = s − e m D log(1 /e )4 π (cid:18) β | p | χ s ( | p | ) − (cid:18) β | p | + n s ( | p | ) − (cid:19) χ (cid:48) s ( | p | ) − β χ (cid:48)(cid:48) s ( | p | ) (cid:19) , (5.89) ‡ The sum rules for the case | q | (cid:28) m D take different forms, and it can be checked that the ultra softregion | q | (cid:28) m D does not give rise to logarithmic divergences. µ = 0. Note however theimportant µ dependence via n s ( | p | ) in the differential equation for χ s ( | p | ), n s ( | p | ) = 1 e β ( | p |− sµ ) + 1 ≈ e β | p | + 1 + sβµ e β | p | ( e β | p | + 1) + O ( µ ) , (5.90)as well as in ζ sf p ,s that we compute in Appendix 2 (see Eq.(6.204)), ζ sf p ,s = e π m f log(1 /e ) | p | ( n B ( | p | ) + n − s (0)) ≈ e π m f log(1 /e ) | p | (cid:18) n B ( | p | ) + 12 − s βµ (cid:19) + O ( µ ) , (5.91)which give rise to a s -independent, linear µ part in χ s ( | p | ) in addition to the usual µ -independent part proportional to s . Here m f = ( e / T + µ /π ) is the asymptoticthermal mass of fermions. The solution when expanded in µ then takes a form χ s ( | p | ) = sχ (0) ( | p | ) + µχ (1) ( | p | ) + O ( µ ) , (5.92)and both χ (0) and χ (1) give separate contributions of the same leading order to the finalexpression for ξ in (5.70). In fact, this µ dependence via n s ( | p | ) and ζ sf p ,s in (5.89) (thatis, the χ (1) in the equation (5.92)) also makes a contribution to the µ dependence ofthe electric conductivity in leading log order, which seems to have been missed in someprevious literature. Our analysis in the above (with full expressions for m D and m f )contains all necessary elements that allow us to compute full µ correction to electricconductivity, and we present the correct computation of µ correction to the electricconductivity in Appendix 4. c s ( | p | ) Let us next describe our analysis for c s ( | p | ), which appears as the P-odd component ofthe φ is ( p , k ) ∼ φ is ( p ,
0) + i ( c s ( | p | ) / | p | ) (cid:15) ilm ˆ p l k m when expanded in linear k (see (5.68)),that satisfies our original integral equation (5.66) with a finite k . Expanding the kernelfunction K s ( p , k ; Q ) defined by (5.67) up to linear in k , K s ( p , k ; Q ) = K s ( p , Q ) + K (1) s ( p , k ; Q ) + O ( k ) , (5.93)where K s ( p , Q ) is something we already use before (see Eq.(5.71) and (5.72)) to deter-mine the zeroth order solution φ is ( p ,
0) = χ s ( | p | ) ˆ p i , the part of integral equation in (5.66)26hat is linear in k gives the integral equation for c s ( | p | ), which takes the form ζ p ,s ic s ( | p | ) | p | (cid:15) ilm ˆ p l k m = − i | p | (cid:15) ilm ˆ p l k m + e (cid:90) d Q (2 π ) K s ( p , Q ) ic s ( | p + q | ) | p + q | (cid:15) ilm (cid:92) p + q l k m (2 π ) δ ( q + s | p | − s | p + q | )+ e (cid:90) d Q (2 π ) K (1) s ( p , k ; Q ) χ s ( | p + q | ) (cid:92) p + q i (2 π ) δ ( q + s | p | − s | p + q | ) , (5.94)where in the last term we understand that we extract only the P-odd term having thesame structure of (cid:15) ilm ˆ p l k m . The first term on the right arises from the fact that Σ is ( p , k )given by (5.62) contains the P-odd term when expanded linear in k Σ is ( p , k ) ∼ − i | p | (cid:15) ilm ˆ p l k m . (5.95)As before, it is important to use the fact that the integral e (cid:90) d Q (2 π ) K s ( p , Q )(2 π ) δ ( q + s | p | − s | p + q | ) , (5.96)which appears in the second term on the right side is precisely equal to the contributionto the fermion damping rate arising from soft photon scatterings, ζ sp p ,s , so that one cantransform the above integral equation into the form ζ sf p ,s ic s ( | p | ) | p | (cid:15) ilm ˆ p l k m = − i | p | (cid:15) ilm ˆ p l k m + e (cid:90) d Q (2 π ) K s ( p , Q ) (cid:18) ic s ( | p + q | ) | p + q | (cid:15) ilm (cid:92) p + q l k m − ic s ( | p | ) | p | (cid:15) ilm ˆ p l k m (cid:19) × (2 π ) δ ( q + s | p | − s | p + q | )+ e (cid:90) d Q (2 π ) K (1) s ( p , k ; Q ) χ s ( | p + q | ) (cid:92) p + q i (2 π ) δ ( q + s | p | − s | p + q | ) , (5.97)where ζ sf p ,s that appears on the left is the damping rate contribution arising from softfermion scatterings only which is of order e log(1 /e ) rather than e log(1 /e ).The computation of the first integral on the right side of (5.97) at leading log order isalmost identical to that of the previous integral in (5.73). Expanding up to quadratic in Q , with the replacement q l → ( ˆ p · q ) ˆ p l due to rotational invariance of the q integral, weget after some algebra c s ( | p + q | ) | p + q | (cid:15) ilm (cid:92) p + q l k m − c s ( | p | ) | p | (cid:15) ilm ˆ p l k m = (cid:18) ( q ) − | q | | p | ˜ c s ( | p | ) + sq ˜ c (cid:48) s ( | p | ) + 12 ( q ) ˜ c (cid:48)(cid:48) s ( | p | ) (cid:19) (cid:15) ilm ˆ p l k m + O ( Q ) , (5.98)27here ˜ c s ( x ) ≡ c s ( x ) /x . Comparing this with the previous expansion (5.77) for χ s ( | p | ),we find the identical structure appearing, so that we can simply use the previous resultof the integral in (5.73) (see Eq.(5.89)) by replacing χ s with ˜ c s ( x ) = c s ( x ) /x to get e (cid:90) d Q (2 π ) K s ( p , Q ) (cid:18) ic s ( | p + q | ) | p + q | (cid:15) ilm (cid:92) p + q l k m − ic s ( | p | ) | p | (cid:15) ilm ˆ p l k m (cid:19) × (2 π ) δ ( q + s | p | − s | p + q | )= − i e m D log(1 /e )4 π (cid:18) β | p | ˜ c s ( | p | ) − (cid:18) β | p | + n s ( | p | ) − (cid:19) ˜ c (cid:48) s ( | p | ) − β ˜ c (cid:48)(cid:48) s ( | p | ) (cid:19) (cid:15) ilm ˆ p l k m = − i e m D log(1 /e )4 π | p | (cid:18) β | p | c s ( | p | ) − (cid:18) n s ( | p | ) − (cid:19) (cid:18) c (cid:48) s ( | p | ) − c s ( | p | ) | p | (cid:19) − β c (cid:48)(cid:48) s ( | p | ) (cid:19) (cid:15) ilm ˆ p l k m , (5.99)up to leading log order.What is more complicated is the evaluation of the second integral in (5.97). Let usfirst look at the term χ s ( | p + q | ) (cid:92) p + q i in the integrand. Defining ˜ χ s ( x ) ≡ χ s ( x ) /x andusing the fact that the δ function in the integrand imposes | p + q | = | p | + sq we have χ s ( | p + q | ) (cid:92) p + q i = ˜ χ s ( | p | + sq )( p i + q i ) . (5.100)Since what we need is the P-odd structure (cid:15) ilm ˆ p l k m , it is clear that the first term pro-portional to p i can not possibly generate such structure, and therefore it is sufficient toconsider only the second piece proportional to q i , χ s ( | p + q | ) (cid:92) p + q i → ˜ χ s ( | p | + sq ) q i , (5.101)in the integral of (5.97). On the other hand, since K (1) s ( p , k ; Q ) is a rotationally scalarfunction linear in k , rotational invariance dictates that it can only have three possiblestructures K (1) s ( p , k ; Q ) = (cid:0) a (1) s ˆ p l + b (1) s q l + c (1) s (cid:15) lmn ˆ p m q n (cid:1) k l , (5.102)where ( a (1) s , b (1) s , c (1) s ) are some coefficient functions that depend only on ( p , | p | , q , | q | , p · q ). Combining these two facts, and considering rotational invariance of q integration, onecan easily find that the only way to have the resulting P-odd structure (cid:15) ilm ˆ p l k m from thesecond integral in (5.97) is via the third term in (5.102), that is, we only need to find thepart of K (1) s ( p , k ; Q ) that is proportional to (cid:15) lmn ˆ p m q n k l = (cid:15) lmn ˆ p l q m k n . This simplifiesour computation by a great amount.Since (5.101) is already linear in Q , for a leading log contribution we only need toexpand K (1) s ( p , k ; Q ) up to linear in Q which is already saturated by the wanted structure28 lmn ˆ p l q m k n . This in turn implies that one can neglect sq correction in (5.101) to have χ s ( | p + q | ) (cid:92) p + q i → ˜ χ s ( | p | ) q i = χ s ( | p | ) | p | q i , (5.103)in the integral of (5.97). Given the expressions for Σ is ( p , k ) and F s ( p ; q , k ) in (5.62) andAppendix 3, as well as the photon spectral density given in (5.81), ρ ph00 = ρ L ( Q ) , ρ ph ij ( Q ) = ρ T ( Q ) (cid:18) δ ij − q i q j | q | (cid:19) + iρ odd ( Q ) (cid:15) ijl q l , (5.104)it is straightforward to find after some amount of algebra that K (1) s ( p , k ; Q ) = (cid:0) n + ( s | p | + q ) + n B ( q ) (cid:1) × (cid:18) is | p | (cid:18) ρ L ( Q ) − (cid:18) q ) | q | (cid:19) ρ T ( Q ) (cid:19) + i | p | ρ odd ( Q ) (cid:19) (cid:15) lmn ˆ p l q m k n , (5.105)up to linear in Q , which will contribute to the leading log result of the integral in (5.97).Note that we have a non-negligible contribution from the P-odd part of the spectral density ρ odd ( Q ): from its sum rule given in (5.82) one can easily see that this term engenders aleading log contribution to the integral. When combining K (1) s ( p , k ; Q ) with (5.103) inthe integral of (5.97), one has to replace q m q i with q m q i → δ mi | q T | = 12 δ mi (cid:0) | q | − ( ˆ p · q ) (cid:1) = 12 (cid:0) | q | − ( q ) (cid:1) , (5.106)where q T is the perpendicular component of q to p , and we use (5.75) in the last equality.This comes from the rotational invariance of q integral around ˆ p axis. Collecting all theseand following the same steps as in the leading log computation of χ s ( | p | ) before, we finallyhave the second integral of (5.97) to be given by at leading log order e (cid:90) d Q (2 π ) K (1) s ( p , k ; Q ) χ s ( | p + q | ) (cid:92) p + q i (2 π ) δ ( q + s | p | − s | p + q | )= − e χ s ( | p | ) | p | (cid:90) ∞ d | q || q | (2 π ) (cid:90) | q |−| q | dq (2 π ) (cid:18) sq | p | (cid:19) (cid:18) βq + s (cid:18) n s ( | p | ) − (cid:19)(cid:19) × (cid:18) is | p | (cid:18) ρ L ( Q ) − (cid:18) q ) | q | (cid:19) ρ T ( Q ) (cid:19) + i | p | ρ odd ( Q ) (cid:19) (cid:0) | q | − ( q ) (cid:1) (cid:15) ilm ˆ p l k m = − e χ s ( | p | ) β | p | (cid:90) ∞ d | q || q | (2 π ) (cid:32) is | p | (cid:18) | q | (cid:18) J L − J T − | q | J T (cid:19) − J L + J T + 1 | q | J T (cid:19) + i | p | (cid:18) − e µ (2 π ) | q | (cid:19) (cid:33) (cid:15) ilm ˆ p l k m = i e π χ s ( | p | ) β | p | log(1 /e ) e µ π (cid:15) ilm ˆ p l k m , (5.107)29here we use the sum rules (5.87), (6.155), (6.156), and interestingly it turns out thatthe contributions from the P-even spectral densities ρ L/T cancel with each other exactly.We don’t have a good understanding whether this has to be the case by some symmetryreason or it is just by accident. Therefore, the only contribution to the second integralin (5.97) at leading log order (that is, e log(1 /e )) comes from the P-odd part of the soft(HTL) photon spectral density ρ odd ( Q ). Note that for this contribution, we have equallogarithmic contributions from both e T (cid:28) | q | (cid:28) m D and T (cid:29) | q | (cid:29) m D that add uptogether in the final result § .From the integral equation (5.97) with (5.99) and (5.107), we finally obtain the sought-for second order differential equation for c s ( | p | ) as ζ sf p ,s c s ( | p | ) = −
12 + e log(1 /e ) µ (4 π )2 π χ s ( | p | ) β | p |− e m D log(1 /e )4 π (cid:18) β | p | c s ( | p | ) − (cid:18) n s ( | p | ) − (cid:19) (cid:18) c (cid:48) s ( | p | ) − c s ( | p | ) | p | (cid:19) − β c (cid:48)(cid:48) s ( | p | ) (cid:19) , (5.108)where the first line is an inhomogeneous source, especially the second term is in termsof χ s ( | p | ) that should be obtained by solving the differential equation (5.89). We wouldneed the expansion of c s ( | p | ) up to first order in chemical potential µ , c s ( | p | ) = c (0) ( | p | ) + sµ c (1) ( | p | ) + O ( µ ) , (5.109)which can be found by solving the above differential equation order by order in µ . Wereemphasize that there are linear µ dependences coming from ζ sf p ,s and n s ( | p | ) in the abovedifferential equation, which should not be missed to get a correct leading log answer. Afterfinding χ s ( | p | ) and c s ( | p | ) from the above given differential equations, we compute ourtransport coefficient ξ by (5.70) with G ij, Pinch , P − odd( ra ) ( k ) = iωξ (cid:15) ijl k l . As a first step to compute the explicit value of ξ , we solve numerically the equationsfor ( χ s ( | p | ) , c s ( | p | )) in (5.89) and (5.108). In order to do so we define Ψ s ( | p | /T ) ≡ αm D log(1 /e ) c s ( | p | ) /T and Φ s ( | p | /T ) ≡ αm D log(1 /e ) χ s ( | p | ) /T where α = e / π . Defin- § Strictly speaking, QED does not possess ultra soft magnetic cutoff ∼ e T . Since we have in mindthe generalization to non-abelian QCD discussed in section 5, we simply assume this at this point. y ≡ | p | /T , the equation (5.108) can be rewritten asΨ (cid:48)(cid:48) s ( y ) − tanh( y/ (cid:48) s ( y ) − Ψ s ( y ) y (cid:18)
32 coth( y/ − tanh( y/
2) + 2 y (cid:19) =1 − s µπ T (cid:18) s Φ s ( y ) y + π s ( y ) y (cid:19) + s µT sech ( y/ (cid:18) Ψ s ( y ) y − Ψ (cid:48) s ( y ) (cid:19) . (5.110)In addition we have the equation for the even vertex, χ s ( | p | ) in (5.89)Φ (cid:48)(cid:48) s ( y ) + (cid:18) y − tanh( y/ (cid:19) Φ (cid:48) s ( y ) − (cid:18) y + 3 coth( y/ y (cid:19) Φ s ( y ) = − s − s µ T Φ s ( y ) y − s µ T sech ( y/ (cid:48) s ( y ) . (5.111)In the above, we expand µ dependence from ζ sf p ,s and n s ( | p | ) up to linear in µ . We thenexpand the solution to first order in µ asΨ s = Ψ A + s µT Ψ B + O ( µ ) , (5.112)Φ s = s Φ A + µT Φ B + O ( µ ) , (5.113)from which we have a coupled set of differential equationsΨ (cid:48)(cid:48) A ( y ) − tanh( y/ (cid:48) A ( y ) − Ψ A ( y ) y (cid:18)
32 coth( y/ − tanh( y/
2) + 2 y (cid:19) = 1 , Ψ (cid:48)(cid:48) B ( y ) − tanh( y/ (cid:48) B ( y ) − Ψ B ( y ) y (cid:18)
32 coth( y/ − tanh( y/
2) + 2 y (cid:19) = − π Φ A ( y ) y −
34 Ψ A ( y ) y + sech ( y/ (cid:18) Ψ A ( y ) y − Ψ (cid:48) A ( y ) (cid:19) , Φ (cid:48)(cid:48) A ( y ) + (cid:18) y − tanh( y/ (cid:19) Φ (cid:48) A ( y ) − (cid:18) y + 3 coth( y/ y (cid:19) Φ A = − , Φ (cid:48)(cid:48) B ( y ) + (cid:18) y − tanh( y/ (cid:19) Φ (cid:48) B ( y ) − (cid:18) y + 3 coth( y/ y (cid:19) Φ B ( y ) = −
34 Φ A ( y ) y −
12 sech ( y/ (cid:48) A ( y ) . (5.114)We solve the above equations by iterative method with vanishing boundary conditionsat the IR ( y = 0) and the UV ( y = ∞ ). The last step is then to obtain the expression forthe transport coefficient ξ as an integral of the above quantities. After computing thesum over s = ± and performing angular integrations in (5.70), we obtain the result for31he retarded propagator as an integral of Ψ A/B ( y ) and Φ A/B ( y ), G ij, Pinch( ra ) ( k ) = 2 iωµ(cid:15) ijl k l T πe log(1 /e ) × (cid:90) ∞ ydy (cid:32) (cid:18) Ψ B ( y ) − Φ B ( y )2 (cid:19) ( y/
2) + (cid:18) Ψ A ( y ) − Φ A ( y )2 (cid:19) tanh( y/ ( y/ (cid:33) . (5.115)From the identification G ij, Pinch , P − odd( ra ) ( k ) = iωξ (cid:15) ijl k l , we find ξ = − . e log(1 /e ) µT . (5.116) It is easy to generalize the above to the case of N F species of Dirac fermions in an SU ( N c )gauge theory (but still the current and magnetic field are with respect to the global U (1)flavor symmetry). The chemical potential appearing in the integral equation is simply theaxial chemical potential µ A . The Debye mass and asymptotic thermal mass are changedto m D = g T N c + N F ) , m f = g T N c − N c . (6.117)The soft-fermion contribution to the hard photon damping rate is given by ζ sf p ,s = g π N c − N c m f log(1 /g ) | p | ( n B ( | p | ) + n − s (0)) . (6.118)In the integral equations (5.89) and (5.108), the e log(1 /e ) in the kernel part should bereplaced by g log(1 /g ) N c − N c . (6.119)In the second source term in the first line of (5.108) which is proportional to e log(1 /e ),this e log(1 /e ) should be replaced by g log(1 /g ) N F N c − N c . (6.120)This is because one factor of e coming from the spectral density ρ odd ( Q ) is replaced by g N F , while the other e coming from fermion-gluon couplings in the kernel is replaced32y g ( N c − / (2 N c ). Finally, the expression for G ij, Pinch( ra ) ( k ) from (5.70), or equivalentlyour ξ from χ s ( | p | ) and c s ( | p | ) must be multiplied by the fermion degeneracy2 N c × (cid:88) F Q F , (6.121)where Q F are charges of F flavors in units of e (for (u,d)-quarks, it is Q u = 2 / Q d = − / N c = 3) with Q u = 2 / Q d = − /
3, theresult for ξ then becomes ξ QCD5 = − . g log(1 /g ) µT . (6.122)As mentioned at the end of section 2, the color conductivity σ c for non-Abelian gaugetheory that appears in the low energy effective theory at the scale Q (cid:46) g T , J a = σ c E a + ξ a , (6.123)where a denotes adjoint color charge and ξ a is the thermal noise via fluctuation-dissipationrelation to σ c , is governed by scatterings with ultra-soft transverse thermal gluons of mo-menta Q (cid:28) gT with the rate ∼ g log(1 /g ), leading to σ c ∼ g · / ( g log(1 /g )) ∼ / log(1 /g ). ¶ To get a correct leading log result for this, one also needs a similar dia-grammatic resummation with essentially the same technique in our computation, exceptthat charge carriers include gluons as well as quarks, and there is now no longer a pre-cise cancellation of g log(1 /g ) terms in the integral equation: this is due to the absenceof U(1) Ward identity (replaced by non-Abelian version of Slavnov-Taylor identity) thatensures the cancellation of g log(1 /g ) terms [53]. The diagrammatic resummation (donein Ref.[63]) for this is therefore somewhat simpler than that for the electric conductivity,since one does not need to go to the next order of g log(1 /g ). In fact, one gets an alge-braic equation to solve rather than a differential equation. The same resummation canalso be achieved in the language of Bodecker’s approach [58] as well as in kinetic theory[59]. In the presence of axial charge µ A breaking CP symmetry, Ref.[64] recently obtainedvia Bodecker’s approach a CP-odd contribution to the color current J a = σ c E a + ξ a + µ A N F g π B a , (6.124)which is a colored analogue of chiral magnetic effect consistent with the U (1) A SU (3) c triangle anomaly. Since triangle anomaly is topological, this contribution should be sat-urated at 1-loop diagrammatically without a need for resummation of ladder diagrams. ¶ Note that we now put an extra g in the definition of color conductivities to follow the convention inliterature. g log(1 /g ) rate that the color conduc-tivity is also sensitive to, appears when one goes to the next order in derivative J a = σ c E a + ξ a + µ A N F g π B a + ξ c d B a dt . (6.125)It is clear that ξ c , a colored analogue of our ξ , will be of order ξ c ∼ g · / ( g log(1 /g ))( µ A /T ) ∼ (1 / log(1 /g ))( µ A /T ) , (6.126)due to the absence of precise cancellation of g log(1 /g ) terms in the integral equations.The computation of ξ c at leading log order is doable, following the same steps we presentin our work keeping only g log(1 /g ) terms in the integral equations (note that it receivescontributions only from quarks, not from gluons). The leading order fermion dampingrate from soft gluon scatterings is ζ sp p ,s = N c − N c g log(1 /g ) T π , (6.127)and the solution of the integral equations which become algebraic is χ s ( | p | ) = sζ sp p ,s + N c g log(1 /g ) T π = 4 πsN c g log(1 /g ) T , c s ( | p | ) = − s χ s ( | p | ) , (6.128)which gives our result for ξ c , ξ c = − g N F (cid:90) d p (2 π ) (cid:88) s = ± dn + ( p ) dp (cid:12)(cid:12)(cid:12)(cid:12) p = s | p | | p | (cid:18) sc s ( | p | ) − χ s ( | p | ) (cid:19) = − π N F N c /g ) µ A T . (6.129)The same resummation should also be achievable in the Bodecker’s approach presentedin Ref.[64] by going to the next order in time derivatives.It is utmost important to implement the correct value of chiral magnetic current inthe presence of time-varying magnetic field in any realistic simulation of chiral magneticeffect (or any other anomaly induced transport phenomena) in heavy-ion collisions. Ourresult should be an important step at weak coupling picture toward taking into accounttime-varying nature of the magnetic field in heavy-ion collisions, and will be instrumentalin the quantitative studies of the chiral anomaly induced phenomena in the experimentsat RHIC and LHC. 34 cknowledgment
A.J. would like to thank Nuclear Theory Group at UIC for hospitality during hisvisit and Francisco Pena for useful comments. H.U.Y. thanks Dima Kharzeev, DaisukeSatow, and Misha Stephanov for discussions. A.J. has been supported by FPU fellowshipAP2010-5686, Plan Nacional de Altas Energias FPA2009-07890, Consolider Ingenio 2010CPAN CSD200-00042 and Severo Ochoa award SEV-2012-0249.
Appendix 1: Sum rules for the P-odd part of HTLphoton spectral density
Let us start from the thermal relation G rrµν ( q ) = (cid:18)
12 + n B ( q ) (cid:19) (cid:0) G raµν ( q ) − G arµν ( q ) (cid:1) , (6.130)where G abµν ( x ) ≡ (cid:104) A aµ ( x ) A bν (0) (cid:105) SK ( a, b = r, a ) are correlation functions in the Schwinger-Keldysh path integral, and G arµν ( x ) = (cid:104) A aµ ( x ) A rν (0) (cid:105) SK = G raνµ ( − x ) , (6.131)where we use translational invariance of the system. Since what we encounter in writingdown our integral equations in the main text is the combination (cid:0) G raµν ( q ) − G arµν ( q ) (cid:1) , letus naturally define the photon spectral density (including possible P-odd contributions ingeneral) ρ ph µν ( q ) ≡ G raµν ( q ) − G arµν ( q ) . (6.132)We will show that ρ ph µν ( q ) is in general a hermitian matrix in terms of µν indices. For diag-onal components that come from the usual P-even contributions, ρ ph , P − even µν ( q ) is thereforereal. For P-odd contribution which turns out to be anti-symmetric in spatial ij indices(there is no P-odd contribution to time-like component, at least up to linear order in µ ),we thus have ρ ph , P − odd ij ( q ) purely imaginary.To show that ρ ph µν ( q ) is a hermitian matrix, recall that the usual retarded propagatoris defined as G Rµν ( x ) = − iθ ( x ) (cid:104) [ A µ ( x ) , A ν (0)] (cid:105) = − iG raµν ( x ) . (6.133)It is not difficult to show, using the hermiticity of A µ , that G Rµν ( x ) is real valued, and thisis what it should be since the retarded propagator gives the response of the system in real35ime which must be real valued. Therefore, in Fourier space, one has G Rµν ( − q ) = (cid:0) G Rµν ( q ) (cid:1) ∗ , (6.134)which in turn gives G raµν ( − q ) = − (cid:0) G raµν ( q ) (cid:1) ∗ . (6.135)On the other hand, from (6.131) we have G arµν ( q ) = G raνµ ( − q ) = − (cid:0) G raνµ ( q ) (cid:1) ∗ , (6.136)where we use (6.135) in the last equality. Therefore ρ ph µν ( q ) = G raµν ( q ) − G arµν ( q ) = G raµν ( q ) + (cid:0) G raνµ ( q ) (cid:1) ∗ , (6.137)which proves that ρ ph µν ( q ) is indeed a hermitian matrix.The P-odd part of the retarded current-current correlation functions, that is the re-tarded photon self energy, in Hard Thermal Loop (HTL) limit has been recently computedin literature [65, 66]. We will work in the Coulomb gauge where G ra i = 0 ( i = 1 , , G ra ( q )) − = ( G ra (0) ( q )) − − i Σ R ( q ) , (6.138)where G ra (0) ( q ) is the bare propagator which is given by G ra (0) ij ( q ) = − iP Tij ( q ) − ( q + i(cid:15) ) + | q | . (6.139)The HTL self-energy Σ R ( q ) including P-odd contribution is given byΣ Rij ( q ) = Π T ( q ) P Tij ( q ) + i Π odd ( q ) (cid:15) ijl q l , (6.140)where Π T ( q ) = − m D (cid:18) ( q ) | q | + (cid:18) ( q ) | q | − (cid:19) q | q | log (cid:18) q − | q | + i(cid:15)q + | q | + i(cid:15) (cid:19)(cid:19) , Π odd ( q ) = − e µ π (cid:18) − ( q ) | q | − (cid:18) ( q ) | q | − (cid:19) q | q | log (cid:18) q − | q | + i(cid:15)q + | q | + i(cid:15) (cid:19)(cid:19) , (6.141)36ith m D = e ( T / µ / (2 π )). From this while keeping terms only up to linear in µ ,we have G raij ( q ) = − iP Tij ( q ) − ( q ) + | q | − Π T ( q ) + Π odd ( q )( − ( q ) + | q | − Π T ( q )) (cid:15) ijl q l , (6.142)where the first term is the usual P-even HTL photon propagator, and the second term isthe new P-odd contribution. The HTL photon spectral density ρ ph ij ( q ) is then given by ρ ph ij ( q ) = ρ T ( q ) P Tij ( q ) + iρ odd ( q ) (cid:15) ijl q l , (6.143)with ρ T ( q ) = 2 Im (cid:18) − ( q ) + | q | − Π T ( q ) (cid:19) , ρ odd ( q ) = 2 Im (cid:18) Π odd ( q )( − ( q ) + | q | − Π T ( q )) (cid:19) . (6.144)It is easy to see that ρ odd ( q ) is an odd function in q , and what we need in the main textis the value of the integral J odd n ≡ (cid:90) | q |−| q | dq (2 π ) ( q ) n − ρ odd ( q ) , n = 0 , , , · · · . (6.145)One can compute them using the well-known sum-rule techniques exploring analytic prop-erty of the function ∆ odd ( q ) defined by∆ odd ( q ) ≡ Π odd ( q )( − ( q ) + | q | − Π T ( q )) . (6.146)We briefly sketch the procedure and present the results in two different regimes | q | (cid:28) m D and | q | (cid:29) m D .The starting point is the fact that ∆ odd ( q ) in the complex q plane is analytic in theupper half plane due to the causal nature of a retarded function. Thus, the integral (cid:90) ∞−∞ dq π q − ω + i(cid:15) ∆ odd ( q ) = 0 , (6.147)vanishes for any real number ω by closing the contour with the upper hemi-circle atinfinity (and ∆ odd ( q ) → | q | → ∞ ). From 1 / ( q − ω + i(cid:15) ) = P / ( q − ω ) − iπδ ( q − ω ) where P is the principal integration, we have P (cid:90) ∞−∞ dq π q − ω ∆ odd ( q ) − i odd ( ω, | q | ) = 0 . (6.148)37onsidering the imaginary part of the above, we obtain one of the Kramers-Kronig dis-persion relations for a retarded function (the real part gives the other dispersion relation), P (cid:90) ∞−∞ dq π q − ω ρ odd ( q ) = Re [∆ odd ( ω, | q | )] . (6.149)Setting ω = 0 and using ∆ odd (0 , | q | ) = − e µ/ (4 π | q | ), one gets a sum rule (cid:90) ∞−∞ dq π q ρ odd ( q ) = − e µ π | q | . (6.150)Other sum rules are obtained from (6.149) by expanding both sides in ω → ∞ . Theleft-hand side becomes − ∞ (cid:88) n =0 ω n +1 (cid:90) ∞−∞ dq π ( q ) n ρ odd ( q ) , (6.151)while the right-hand side when expanded in large ω is − e µ/ (12 π ω ) + O (1 /ω ), whichgives other two sum rules, (cid:90) ∞−∞ dq π q ρ odd ( q ) = 0 , (cid:90) ∞−∞ dq π ( q ) ρ odd ( q ) = e µ π . (6.152)The sum rules (6.150) and (6.152) are not precisely J odd n defined in (6.145), as theintegration range for J odd n is [ −| q | , + | q | ], not [ −∞ , + ∞ ]. The imaginary part of ∆( q )(that is, ρ odd ( q )) consists of two distinct parts: one part coming from a branch cut justbelow the real line along the interval q ∈ [ −| q | , + | q | ] from the logarithms in (6.141)(originated from the Landau damping), and the other part is from the two poles ± ω satisfying − ( ω ) + | q | − Π T ( ω , | q | ) = 0 corresponding to the time-like transverse photonsin the medium. The former has a continuous support in the interval q ∈ [ −| q | , + | q | ],and hence contributes to J odd n , while the latter pole contributions sit outside the interval, ω > | q | , so do not contribute to J odd n . Therefore, the only difference between the sumrule values in (6.150), (6.152) and the J odd n is simply the latter pole contributions whichwe can compute.Near the pole location q ≈ ω − i(cid:15) , we have the expansion( q ) n − odd ( q ) ≈ A ( q − ω + i(cid:15) ) + Bq − ω + i(cid:15) + { regular } , (6.153)where B = 2( ω ) n − Π odd ( ω )(2 ω + Π (cid:48) T ( ω )) (cid:18) Π (cid:48) odd ( ω )Π odd ( ω ) + (2 n − ω − (cid:48)(cid:48) T ( ω )2 ω + Π (cid:48) T ( ω ) (cid:19) , (6.154)38ith Π (cid:48) T ( q ) ≡ d Π T ( q , | q | ) /dq , etc. The first double pole does not contribute to theimaginary part in (cid:15) → q ) n − ρ odd ( q ) as − πB δ ( q − ω ) × − πB δ ( q − ω ) (the factor 2 comes from having two poles ± ω ),leading to the difference between the sum rules values in (6.150), (6.152) and the J odd n being given by − B , that is, J odd n is obtained by adding B to the sum rule values in (6.150),(6.152).In the case | q | (cid:28) m D , the pole location is ω ≈ (cid:112) / m D (1 + (9 / | q | /m D ) + · · · and an explicit computation of B gives the values of J odd n in this regime as J odd0 ≈ − e µ π | q | + 3 e µ π m D + O ( | q | /m D ) ,J odd1 ≈ − e µ | q | π m D + O ( | q | /m D ) ,J odd2 ≈ − e µ | q | π m D + O ( | q | /m D ) . (6.155)On the other hand, in the regime | q | (cid:29) m D , the poles are located in ω ≈ | q | +(1 / m D / | q | + · · · , and we have the results in this regime | q | (cid:29) m D as J odd0 ≈ e µ π | q | (cid:18) (cid:18) m D | q | (cid:19)(cid:19) ,J odd1 ≈ e µ π | q | (cid:18) (cid:18) m D | q | (cid:19)(cid:19) ,J odd2 ≈ e µ π (cid:18)
113 + log (cid:18) m D | q | (cid:19)(cid:19) . (6.156)The above results (6.155) and (6.156) will be used in the main text in section 3. Appendix 2: Hard fermion damping rate
In this appendix, we compute the damping rate of hard fermion including possible depen-dence on the chemical potential µ . Our primary objective is two-fold: we first would liketo confirm that the integral that we have in (5.72) is indeed precisely equal to the dampingrate induced by soft photon scatterings at full order in e and µ , which was instrumentalin rewriting the integral equation to take the form (5.73) that contains only e log(1 /e )terms. Our second objective is to find a linear sµ dependence of the ζ sf p ,s , that is, in thedamping rate induced by soft fermion scatterings (or equivalently, fermion conversion-to-photon processes). This sµ dependence in ζ sf p ,s is important in finding the correct µ χ s ( | p | ), which is crucial toget the correct result for ξ as well as µ correction to usual electric conductivity.Figure 8: Real-time Feynman diagrams for the retarded fermion self energy.Let us start with the self energy resummed ra-propagator (which is equal to i timesof retarded propagator) S ra ( p ) = S ra (0) ( p ) + S ra (0) ( p )Σ ra ( p ) S ra (0) ( p ) + · · · = S ra (0) ( p ) 11 − Σ ra ( p ) S ra (0) ( p ) , (6.157)which gives ( S ra ( p )) − = ( S ra (0) ( p )) − − Σ ra ( p ) , (6.158)where S ra (0) ( p ) = (cid:88) s ip − s | p | + i(cid:15) P s ( p ) , (6.159)is the bare ra-propagator, and the self-energy Σ ra ( p ) (which is a 2 × ra ( p ) = ( ie ) σ β (cid:90) d Q (2 π ) (cid:2) G rrαβ ( Q ) S ra ( p + Q ) + G arαβ ( Q ) S rr ( p + Q ) (cid:3) σ α , (6.160)with the photon propagators G abαβ ( Q ) ( a, b = r, a ) (see the Appendix 1 for our notationalconventions). In the above expression, we haven’t specified whether the propagatorsappearing in the loop are bare or HTL resummed ones, since depending on the situationswe can consider different approximations for them to get the right leading order quantities.For example, if the external momentum p is soft and one is interested in the Hard ThermalLoop (HTL) approximation, it is enough to consider hard loop momentum Q and the bothpropagators in the loop are the bare ones. On the other hand, in the case of damping rate40ith a hard momentum p , which is proportional to the imaginary part of the self-energy,the leading contribution comes from when one of the two loop propagators carries softmomentum (that is, either Q or p + Q ), and the soft propagator must then be the HTLresummed propagator while the other hard propagator is the bare one.Rotational invariance dictates the self energy to take a formΣ ra ( p ) = A ( p , | p | ) + B ( p , | p | ) ˆ p · (cid:126)σ ≡ i (cid:88) s = ± Σ Rs ( p ) P s ( p ) , (6.161)where Σ Rs ( p ) = − i (cid:0) A ( p , | p | ) + s B ( p , | p | ) (cid:1) . (6.162)From this and (6.158) we have S ra ( p ) = (cid:88) s ip − s | p | + Σ Rs ( p ) P s ( p ) . (6.163)In deriving above, we use the following properties of the projection operators to find theinverse of S ra ( p ), P + + P − = , P ± = P ± , P + P − = P − P + = 0 . (6.164)By comparing the above expression for S ra ( p ) with the one in (2.11), we see that thedamping rate is given by the imaginary part of retarded self energy Σ Rs ( p ) at on-shellmomentum p = s | p | , ζ p ,s = 2 Im (cid:2) Σ Rs ( p ) (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) p = s | p | . (6.165)In the following, we will hence only concern about the imaginary part of Σ Rs ( p ). From(6.161) and tr[ P ± ] = 1, we haveΣ Rs ( p ) = ( − i ) tr [ P s ( p )Σ ra ( p )]= ie tr (cid:20) P s ( p ) σ β (cid:90) d Q (2 π ) (cid:2) G rrαβ ( Q ) S ra ( p + Q ) + G arαβ ( Q ) S rr ( p + Q ) (cid:3) σ α (cid:21) , (6.166)which will be the starting point of our computation.For a soft p , if one uses the HTL approximation to the retarded self energy Σ Rs ( p ), theresult is the HTL fermion propagator. For a hard p , the HTL self energy is sub-leading in e so can be negligible, and moreover its imaginary part at on-shell momentum p = s | p | ,which would give a damping rate that could regularize pinch singularities, vanishes due41o kinematic constraints. The leading contribution to the imaginary part of Σ Rs ( p ) aton-shell momentum arises when either Q or p + Q is soft, so that the correspondingpropagator in the loop is the HTL resummed one. Calling the case of soft Q the soft-photon contribution, ζ sp p ,s , and the other case of soft p + Q the soft-fermion contribution, ζ sf p ,s , the total damping rate is the sum of the two, ζ p ,s = ζ sp p ,s + ζ sf p ,s .Let us discuss ζ sp p ,s first. Since the fermion propagator is the bare one, we have afterputting on-shell momentum p = s | p | S ra ( p + Q ) = (cid:88) t = ± is | p | + q − t | p + q | + i(cid:15) P t ( p + q )= (cid:88) t (cid:20) P is | p | + q − t | p + q | + πδ (cid:0) s | p + q − t | p + q | (cid:1)(cid:21) P t ( p + q ) , (6.167)where P denotes principal value. Similarly, S rr ( p + Q ) = (cid:18) − n + ( s | p | + q ) (cid:19) (cid:88) t (2 π ) δ ( s | p | + q − t | p + q | ) P t ( p + q ) . (6.168)Looking at the structure of (6.166), we have a spinor trace appearingtr (cid:2) P s ( p ) σ β P t ( p + q ) σ α (cid:3) ≡ H βαst ( p , q ) , (6.169)which is a hermitian matrix in terms of αβ indices (this can be shown easily using her-mitian nature of σ α and P ± ). Since G rrαβ ( Q ) = (cid:18)
12 + n B ( q ) (cid:19) (cid:0) G raαβ ( Q ) − G arαβ ( Q ) (cid:1) ≡ (cid:18)
12 + n B ( q ) (cid:19) ρ ph αβ ( Q ) , (6.170)is also a hermitian matrix with (cid:0) G arαβ ( Q ) (cid:1) ∗ = − G raβα ( Q ) as shown in the Appendix 1, wesee that H βαst ( p , q ) G rrαβ ( Q ) is a real number. Therefore, one sees that the imaginary partof Σ Rs ( p ) given in (6.166) arises only from the second δ -function term in (6.167) whenused in the first term of (6.166). Similarly, from (cid:16) H βαst ( p , q ) G arαβ ( Q ) (cid:17) ∗ = − H βαst ( p , q ) G raαβ ( Q ) , (6.171)and H βαst ( p , q ) ρ ph αβ ( Q ) = (cid:16) H βαst ( p , q ) G raαβ ( Q ) − H βαst ( p , q ) G arαβ ( Q ) (cid:17) , (6.172)we see that the real part of H βαst ( p , q ) G arαβ ( Q ) is equal to − (1 / H βαst ( p , q ) ρ ph αβ ( Q ). With(6.168) the imaginary part of Σ Rs ( p ) from the second term in (6.166) only comes from the42eal part of H βαst ( p , q ) G arαβ ( Q ), and therefore we can effectively replace G arαβ ( Q ) appearingin (6.166) with − (1 / ρ ph αβ ( Q ) for the purpose of damping rate computation. One thenobserves that the pieces in (6.167) and (6.168) that contribute to the damping rate are allproportional to the δ -function δ ( s | p | + q − t | p + q | ) which has a non-zero support onlyfor t = s since Q is assumed to be soft while p is hard. After collecting all these piecescontributing to the imaginary part of Σ Rs ( p ), we finally have after some algebra, ζ sp p ,s = 2 Im (cid:2) Σ Rs ( p ) (cid:3) = e (cid:90) d Q (2 π ) (cid:0) n B ( q ) + n + ( s | p | + q ) (cid:1) H βαss ( p , q ) ρ ph αβ ( Q )(2 π ) δ ( s | p | + q − s | p + q | ) , (6.173)where H βα ( p , q ) ss = tr (cid:2) P s ( p ) σ β P s ( p + q ) σ α (cid:3) = Σ βs ( p , q ) (Σ αs ( p , q )) ∗ (cid:16) p · (cid:92) p + q (cid:17) , (6.174)using the notations introduced in (5.61). From the fact that F s ( p ; q ,
0) introduced in(5.61) is equal to (cid:16) p · (cid:92) p + q (cid:17) , and recalling our definition of kernel function in(5.67) K s ( p , Q ) ≡ Σ βs ( p , q ) (Σ αs ( p , q )) ∗ ρ ph αβ ( Q ) F s ( p ; q , n + ( s | p | + q ) + n B ( q )) , (6.175)we see that ζ sp p ,s is indeed equal to ζ sp p ,s = e (cid:90) d Q (2 π ) K s ( p , Q )(2 π ) δ ( s | p | + q − s | p + q | ) , (6.176)which is precisely what appears in (5.72) and in the integral equation, which is crucial tohave (5.73).Although we don’t need the value of ζ sp p ,s in this work, it is easy to compute it fromthe above expression. From (5.76) (cid:90) d Q (2 π ) (2 π ) δ ( s | p | + q − s | p + q | ) = (cid:90) ∞ d | q || q | (2 π ) (cid:90) | q |−| q | dq (2 π ) (cid:18) sq | p | (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ˆ p · q → sq + ( q −| q | | p | , (6.177)and the small Q expansion of K s ( p , Q ), K s ( p , Q ) ≈ βq (cid:18) ρ L ( Q ) + ρ T ( Q ) (cid:18) − ( q ) | q | (cid:19)(cid:19) , (6.178)43here ρ L/T ( Q ) are P-even longitudinal and transverse photon spectral densities definedby ρ ph00 ( Q ) = ρ L ( Q ) , ρ ph ij ( Q ) = ρ T ( Q ) (cid:18) δ ij − q i q j | q | (cid:19) + iρ odd ( Q ) (cid:15) ijl q l , (6.179)we have ζ sp p ,s ≈ e (cid:90) ∞ d | q || q | (2 π ) (cid:90) | q |−| q | dq (2 π ) 1 βq (cid:18) ρ L ( Q ) + ρ T ( Q ) (cid:18) − ( q ) | q | (cid:19)(cid:19) . (6.180)The rest is to use the sum rules for ρ L/T ( Q ) that can be derived by the same way wederive the sum rules for the P-odd part in Appendix 1 [62]. The leading log arises fromthe momentum region | q | (cid:28) m D and only from the transverse part for which we have (cid:90) | q |−| q | dq (2 π ) 1 q ρ T ( Q ) = 1 | q | + O (cid:18) m D (cid:19) , (cid:90) | q |−| q | dq (2 π ) q ρ T ( Q ) = 3 | q | m D + O (cid:18) | q | m D (cid:19) , (6.181)and this gives ζ sp p ,s ≈ e β (cid:90) m D e T d | q || q | (2 π ) 1 | q | ≈ e log(1 /e ) T π , (6.182)where we put an IR cutoff of order Λ IR ∼ e T . Strictly speaking, the e T (or g T fornon-abelian theory) magnetic confinement scale exists only for non-abelian theory, whilean abelian QED which becomes free at Q (cid:28) m f does not possess any IR cutoff. In thiscase, the damping rate ζ sp p ,s is not a useful concept [67], and the effective IR cutoff isprovided by the time-scale one is looking at, so the hard fermions decay in time t as [67] e − ζ sp p ,s t/ (cid:12)(cid:12)(cid:12)(cid:12) Λ IR ∼ /t ∼ ( m D t ) − e T π t . (6.183)Since we are ultimately interested in QCD (see our discussion in section 6), we don’tworry about this any more. Another aspect is that in realistic situations, the free natureof QED at Q (cid:28) eT means that this scale is never thermalized anyway. Since the dampingrate arises from the scattering of fermion with thermally excited soft photons in this scale(recall n B ( q ) ∼ /q term in the above), we wouldn’t have these contributions in realisticsituations in any case. This also justifies our use of Λ IR ∼ e T in the above.Let us next compute the soft-fermion contribution to the damping rate, ζ sf p ,s , with ourmain objective being to find a linear sµ -dependence. Since p + Q is soft, it is convenientto shift the loop momentum Q → Q − p to haveΣ Rs ( p ) = ie tr (cid:20) P s ( p ) σ β (cid:90) d Q (2 π ) (cid:2) G rrαβ ( Q − p ) S ra ( Q ) + G arαβ ( Q − p ) S rr ( Q ) (cid:3) σ α (cid:21) , (6.184)44here now Q is soft, and we need to use HTL resummed fermion propagators whilethe bare propagators are used for photon propagators. The HTL resummed fermionra-propagator is written as S ra ( Q ) = (cid:88) t iq − t | q | + Σ R, HTL t ( Q ) P t ( q ) , (6.185)where Σ R, HTL t ( Q ) is the HTL self-energy. An explicit computation gives (in fact, one usesthe same expression (6.184) with now both Q and p + Q being hard)Σ R, HTL t ( Q ) = − m f | q | (cid:18) t + (cid:18) − t q | q | (cid:19) log (cid:18) q + | q | + i(cid:15)q − | q | + i(cid:15) (cid:19)(cid:19) , (6.186)where m f = e (cid:18) T + µ π (cid:19) , (6.187)is the asymptotic thermal mass of fermions for a single Weyl fermion system. Since µ dependence is only quadratic for Σ R, HTL t ( Q ), we can ignore this dependence in theHTL self-energy to use µ = 0 result of Σ R, HTL t ( Q ). This means that S ra ( Q ) and S ar = − ( S ra ( Q )) † can be replaced by their µ = 0 values up to linear in µ of our interest. Onthe other hand, the rr-propagator which is given by S rr ( Q ) = (cid:18) − n + ( q ) (cid:19) ( S ra ( Q ) − S ar ( Q )) , (6.188)does contain a linear µ dependence via its statistical factor in front, n + ( q ). We willindeed find shortly that this will be the (only) source of the final sµ -dependence of ζ sf p ,s .Since the HTL resummed ra-propagator (6.185) is analytic in the upper q complexplane, one can introduce real spectral densities ρ ± ( Q ) by1 q − t | q | + Σ R, HTL t ( Q ) = (cid:90) ∞−∞ dω (2 π ) ρ t ( ω, q ) q − ω + i(cid:15) = P (cid:90) ∞−∞ dω (2 π ) ρ t ( ω, q ) q − ω − i ρ t ( q , q ) , (6.189)or equivalently ρ t ( Q ) = − (cid:20) q − t | q | + Σ R, HTL t ( Q ) (cid:21) , (6.190)in terms of which we have S rr ( Q ) = (cid:18) − n + ( q ) (cid:19) (cid:88) t ρ t ( Q ) P t ( q ) . (6.191)Introducing L βαst ( p , q ) ≡ tr (cid:2) P s ( p ) σ β P t ( q ) σ α (cid:3) , (6.192)45hich is a hermitian matrix, a similar discussion to that we have above leads us to replace S ra ( Q ) → ρ t ( Q ) P t ( q ) , (6.193)for computing the imaginary part of Σ Rs ( p ) in (6.184).On the other hand, the hard photon propagators in (6.184) are bare ones. In theCoulomb gauge we have G ra ( p ) = i | p | , G raij ( p ) = − iP Tij ( p ) − ( p + i(cid:15) ) + | p | , (6.194)where P Tij ( p ) = δ ij − ˆ p i ˆ p j is the transverse projection operator, from which we have thebare photon spectral density as ρ ph00 ( p ) = 0 , ρ ph ij ( p ) = (2 π ) P Tij ( p )sign( p ) δ (cid:0) ( p ) − | p | (cid:1) , (6.195)with G rrαβ ( p ) = (1 / n B ( p )) ρ ph αβ ( p ). As before, for the imaginary part of Σ Rs ( p ) in(6.184), we can replace G arαβ ( Q − p ) → − ρ ph αβ ( Q − p ) . (6.196)Collecting all these elements, the expression for ζ sf p ,s becomes ζ sf p ,s = e (cid:88) t = ± (cid:90) d Q (2 π ) L jist ( p , q ) P Tij ( q − p ) (cid:0) n B ( q − s | p | ) + n + ( q ) (cid:1) ρ t ( Q ) × sign( q − s | p | )(2 π ) δ (( s | p | − q ) − | p − q | ) . (6.197)Since Q is soft while p is hard, we have sign( q − s | p | ) = − s and δ (( s | p | − q ) − | p − q | ) = 12( | p | − sq ) δ ( s | p | − q − s | p − q | ) , (6.198)and using the identity n B ( q − s | p | ) + n + ( q ) = ( − s ) (cid:0) n B ( | p | − sq ) + n − s ( − sq ) (cid:1) , (6.199)we have ζ sf p ,s = e (cid:88) t = ± (cid:90) d Q (2 π ) L jist ( p , q ) P Tij ( q − p ) (cid:0) n B ( | p | − sq ) + n − s ( − sq ) (cid:1) ρ t ( Q ) × | p | − sq ) (2 π ) δ ( s | p | − q − s | p − q | ) . (6.200)46t is straightforward to compute the leading log part of the above integral by expandingthe integrand in powers of Q/T or Q/ | p | , the same procedure we use several times before.From L jist ( p , q ) P Tij ( q − p ) ≈ − st ˆ p · q | q | + O ( Q ) = 1 − t q | q | + O ( Q ) and n B ( | p | − sq ) + n − s ( − sq ) ≈ n B ( | p | ) + n − s (0) + O ( Q ), the leading log comes from the expression ζ sf p ,s = e | p | ( n B ( | p | ) + n − s (0)) (cid:88) t (cid:90) ∞ d | q || q | π (cid:90) | q |−| q | dq π (cid:18) − t q | q | (cid:19) ρ t ( Q ) , (6.201)and using the sum rules (cid:107) J ± = m f | q | (cid:32) log 4 | q | m f − (cid:33) , J ± = ± m f | q | (cid:32) log 4 | q | m f − (cid:33) , (6.202)where J ± n ≡ (cid:90) | q |−| q | dq π ( q ) n ρ ± ( Q ) , (6.203)we finally have ζ sf p ,s = e π m f log(1 /e ) | p | ( n B ( | p | ) + n − s (0)) . (6.204)There exists sµ dependence in the result (6.204) via n − s (0) = 1 − n s (0) ≈ / − (1 / sβµ , which can be easily understood as follows. The soft-fermion contribution tothe damping rate comes from the process where a hard fermion (of type s ) becomesa soft-fermion (of the same type s ) by emitting a hard photon with almost the samemomentum. The rate is proportional to ( n B ( | p | ) + 1)(1 − n s (0)) where (1 − n s (0)) isthe Pauli blocking factor of the final soft-fermion state, where we can put zero for soft-momentum at leading order in coupling. A similar process is where a hard fermion (of type s ) meets with a soft-antifermion (of the type − s ) to become a hard photon, and this rate isproportional to ( n B ( | p | )+1) n − s (0) where n − s (0) is the number density of soft-antifermion.The time reversed processes also add up to the damping rate, which is a property offermionic case. These are each proportional to n B ( | p | ) n s (0) and n B ( | p | )(1 − n − s (0)).Using n − s (0) + n s (0) = 1, the total sum can be found to be( n B ( | p | ) + 1)(1 − n s (0)) + ( n B ( | p | ) + 1) n − s (0) + n B ( | p | ) n s (0) + n B ( | p | )(1 − n − s (0))= 2 ( n B ( | p | ) + n − s (0)) , (6.205)which nicely explains our result (6.204). (cid:107) We point out that what is called m f in Ref.[53] is in fact plasmino frequency ω f which is equal to m f / m f . ppendix 3: Expression of F s ( p , q ; k ) The function F s ( p , q ; k ) is given by F s ( p , q ; k ) = A (cid:16) p · (cid:92) p + k (cid:17) | p || p + q || p + k || p + q + k | , (6.206)where A = (cid:0) | p || p + q | + | p | + p · q (cid:1) (cid:0) | p + k || p + k + q | + | p + k | + ( p + k ) · q (cid:1) + (cid:0) | p || p + k | + | p | + p · k (cid:1) (cid:0) | p + q || p + q + k | + | p + q | + ( p + q ) · k (cid:1) − ( | p || p + q + k | − p · ( p + q + k )) ( | p + q || p + k | − ( p + q ) · ( p + k ))+ is ( | p | + | p + q | + | p + k | + | p + q + k | ) (cid:15) ijl p i q j k l . (6.207)For the reasons mentioned in the main text, we are only interested in this quantity tolinear order in the external momentum k and to second order in the loop momentum q .To this order the function F s ( p , q ; k ) is given by F s ( p , q ; k ) ∼ | p | (cid:0) ( p · q ) − | q | (cid:1) + 12 | p | si(cid:15) ijl p i q j k l + 14 | p | (cid:0) p · k (cid:0) | q | − p · q ) (cid:1) + ( p · k )( p · q ) − si(cid:15) ijl p i q j k l ( p · q ) (cid:1) . (6.208) Appendix 4: µ correction to electric conductivity Our analysis in this work contains all the necessary ingredients to compute the full µ correction to the usual P-even electric conductivity at leading log order. The electricconductivity is given from χ s ( | p | ) by σ = − e (cid:90) d p (2 π ) (cid:88) s = ± (cid:18) dn + ( p ) dp (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) p = s | p | χ s ( | p | ) , (6.209)where χ s ( | p | ) satisfies the second order differential equation written in (5.89), ζ sf p ,s χ s ( | p | ) = s − e m D log(1 /e )4 π (cid:18) β | p | χ s ( | p | ) − (cid:18) β | p | + n s ( | p | ) − (cid:19) χ (cid:48) s ( | p | ) − β χ (cid:48)(cid:48) s ( | p | ) (cid:19) , (6.210)48here the soft-fermion contribution to the damping rate ζ sf p ,s is given by (6.204), ζ sf p ,s = e π m f log(1 /e ) | p | ( n B ( | p | ) + n − s (0)) . (6.211)To correctly take into account µ corrections, we need to restore full expressions for m D and m f including µ corrections, m D = e (cid:18) T + 3 µ π (cid:19) , m f = e (cid:18) T + µ π (cid:19) . 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