Fluctuation and Dissipation of Axial Charge from Massive Quarks
FFluctuation and Dissipation of Axial Charge fromMassive Quarks
De-fu Hou ∗ and Shu Lin † Institute of Particle Physics (IOPP) and Key Laboratory of Quark and LeptonPhysics (MOE), Central China Normal University, Wuhan 430079, China School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai, 519082, ChinaSeptember 11, 2018
Abstract
In quantum chromodynamics (QCD), axial charge is known to be non-conserveddue to chiral anomaly and non-vanishing quark mass. In this paper, we explore therole of quark mass in axial charge fluctuation and dissipation. We present two separatecalculations of axial charge correlator, which describe dynamics of axial charge. Thefirst is free quarks at finite temperature. We find that axial charge can be generatedthrough effective quantum fluctuations in free theory. However the fluctuation doesnot follow a random walk behavior. Due to the presence of axial symmetry breakingmass term, the axial charge also does not settle asymptotically to the thermodynamiclimit given by susceptibility. The second calculation is in weakly coupled quark gluonplasma (QGP). We find in the hard thermal loop (HTL) approximation, the quark-gluon interaction leads to random walk growth of axial charge, but dissipation is notvisible. We estimate relaxation time scale for axial charge, finding it lies beyond theHTL regime. ∗ [email protected] † [email protected] a r X i v : . [ h e p - ph ] S e p Introduction
The chiral anomaly is one of the most intriguing discovery in quantum field theory. Overthe past ten years, its manifestations in macroscopic phenomena such as the chiral magneticeffect and chiral vortical effect have triggered significant interests across different communi-ties [1, 2, 3, 4, 5, 6]. There have been continuous efforts in searching for CME and CVE inquark gluon plasma produced in heavy ion collisions [7, 8, 9], as well as in Weyl semi-metal[10, 11]. For comprehensive reviews of current status, we refer to [12, 13, 14]. Effectivedescription of CME and CVE have been developed for chiral fermions, including anomaloushydrodynamics [15, 16, 17], chiral kinetic theory [18, 19, 20, 21, 22, 23, 24] and holography[25, 26, 27, 28, 29, 30]. Both theoretical frameworks reveal beautiful structures in the chirallimit. In most phenomenological applications of the two frameworks, axial charge density isneeded as an input, usually modeled by axial charge chemical potential µ . Possible issueswith using spacetime dependent µ is pointed out by one of the authors [31]. To use µ properly, a better understanding of the dynamics of axial charge is needed.One of the well-known generation mechanism of axial charge is through topologicalfluctuation of gluon field [32, 33]. Due to fluctuation-dissipation theorem, this mechanismcan also cause damping of axial charge. Most phenomenological studies ignore the dampingeffect in the dynamics of axial charge. The interplay of generation and damping is knownto lead to interesting dynamics of axial charge [34]. In addition to topological fluctuationof gluons, fermion mass violates axial charge conservation explicitly. Questions on the roleof fermion mass also arises in different context. On the theoretical side, any fundamentalfermion is known to carry mass. Knowing how fermion mass modifies the existing frame-works is a key ingredient. On the phenomenological side, quantifying the magnitude of themass effect is in need for reliable modeling. The damping effect is first discussed in [35] forelectron in neutron star. The generation effect is proposed by one of the authors [36, 37]for supersymmetric gauge theory based on a holographic model.This paper aims at providing a unified description of the two effects in the samesetting. To be specific, we study dynamics of total axial charge in QCD in the weaklycoupled regime, where perturbative calculation is possible. The results apply equally wellto QED. For pedagogical reasons, we begin with axial charge dynamics in free quark theoryin Section II. We then move on to carry out the same study in weakly coupled QCD plasmain Section III. We find that unlike free theory, weakly coupled plasma can generate axialcharge through quark mass term similar to the topological fluctuation of gluons. Therate of generation, to be coined mass diffusion rate, is numerically much smaller than the2opological fluctuation rate at physically relevant coupling for strange quark mass. Wesummarize in Section IV. For simplicity, we first study axial charge dynamics in free theory. This allows us to set thestage and gain insights into the dynamics. We first note that classical fermions satisfyingDirac equation do not have a net axial charge, even though axial symmetry is brokenby fermion mass. In order to generate net axial charge, we need quantum fluctuation topush fermions off-shell. The quantity characterizing axial charge dynamics is the followingWightman correlator (cid:90) dtd x e iq t (cid:104) ψ + γ ψ ( t, x ) ψ + γ ψ (cid:0) (cid:105) . (1)This describes dynamics of total axial charge N = (cid:82) d x ψ + γ ψ ( x ). Contribution to (1)comes from a simple quark loop diagram. The contribution is given by G > ( Q ) = (cid:90) d K (2 π ) Tr S ( K ) S ( K − Q ) , (2)with K = ( k , (cid:126)k ) and Q = ( q , q >
0. Using deltafunctions in fermion propagators S and S , we find the only possible kinematics is k = E k , and k − q = − E k , with E k = √ k + m . It is not difficult to evaluate the integral toobtain: G > ( q ) = N f N c π ˜ f (cid:16) − q (cid:17) (cid:114)(cid:16) q (cid:17) − m m q θ ( q − m ) , q > , (3)with ˜ f ( − q /
2) = 1 e − βq / + 1 . This gives the spectral function ρ ( q ) ρ ( q ) = G > ( q )1 + f ( q ) = N f N c π ˜ f ( q / f ( q / (cid:114)(cid:16) q (cid:17) − m m q θ ( q − m ) , q > , (4)with f ( q /
2) = 1 e βq / − . For the case q <
0, we use the representation G > ( q ) = ρ ( q )(1 + f ( q )) and the property ρ ( − q ) = − ρ ( q ) to obtain G > ( q ) = − N f N c π ˜ f (cid:16) − q (cid:17) (cid:114)(cid:16) q (cid:17) − m m q θ ( − q − m ) , q < . (5)3 .1 0.2 0.3 0.4 0.5 m / T0.3100.3150.3200.3250.330 χ / N c N f T Figure 1: Normalized susceptibility versus m/T for both N and N . It reduces to knownresult χ = N c N f T / δ ( q ). It is known that sucha contribution corresponds to susceptibility for conserved charge [38]. We can include thiscontribution by assuming the following decomposition of G > G > = G ( q , T, m ) + δ ( q ) F ( T, m ) . (6)We have already obtained G ( q , T, m ) in (3) and (5). F ( T, m ) can be obtained by F ( T, m ) = lim (cid:15) → (cid:90) (cid:15) − (cid:15) dq G > ( q , T, m ) (7)Using (2) for the evaluation of (7), we obtain F ( T, m ) = (cid:90) ∞ dk π k E k ˜ f ( E k )(1 + ˜ f ( E k )) . (8)It is instructive to compare (6) with Wightman correlator for N = (cid:82) d x ψ + ψ ( x ). Fol-lowing similar procedure, we would obtain the same expression (6) but without G ( q , T, m ).It is known that for conserved charge N , G > = 2 πT δ ( q ) χ , thus F ( T, m ) is simply re-lated to susceptibility F ( T, m ) = 2 πT χ . By analogy, we define susceptibility of N fromcontribution of F by χ = F/ (2 πT ). We plot the m -dependence of χ in Figure.1.The fluctuation of N is characterized by the correlator (cid:104) ∆ N ( t ) (cid:105) ≡ (cid:104) ( N ( t ) − N (0)) (cid:105) , which can be expressed by G > as (cid:104) ∆ N ( t ) (cid:105) = V (cid:90) dq π (2 − e iq t − e − iq t ) G > ( q ) , (9)with V = (cid:82) d x being the volume factor. Let us look at the contribution from F and G separately. The evaluation of the former is subtle: a naive integration of δ ( q ) gives avanishing result. However, on general ground we expect as t → ∞ , (cid:104) ∆ N ( t ) (cid:105) = (cid:104) N ( t ) (cid:105) + (cid:104) N (0) (cid:105) − (cid:104) N ( t ) N (0) (cid:105) − (cid:104) N (0) N ( t ) (cid:105) → (cid:104) N (0) (cid:105) , (10)4here we used (cid:104) N ( t ) N (0) (cid:105) → (cid:104) N ( t ) (cid:105) = (cid:104) N (0) (cid:105) . Taking contribution to (cid:104) N (0) (cid:105) from F , we would instead obtain: (cid:104) ∆ N ( t ) (cid:105) F = 2 χT V. (11)The origin of the disagreement is that the two limits q → t → ∞ ) and k → V → ∞ ) do not commute. In (9), we take V → ∞ first while in (10), we take t → ∞ first. Physically they are not equivalent: since conserved charge can fluctuate only throughcharge exchange with heat bath, taking V → ∞ requires larger and larger heat bath, andconsequently longer and longer equilibration time. To reproduce (10), we should take thelimit t → ∞ first, which amounts to dropping the rapid oscillating terms in (9). Theresulting (cid:104) ∆ N ( t ) (cid:105) indeed obtain (10).Now we turn to evaluation of the contribution from G . This is intrinsic to breaking ofthe axial symmetry. It corresponds to fluctuation by itself, not relying on charge exchangewith heat bath. It is easy to see from (3) that the fluctuation also exist in vacuum. Plugging(3) and (5) into (9), we find the integral contains a UV divergence. We regularize bysubtracting the vacuum contribution: (cid:104) ∆ N ( t ) (cid:105) G = V N f N c (cid:90) ∞ dq π − cos( q t )) π ¯ h (cid:2) ˜ f ( q / ˜ f ( q ) − (cid:3)(cid:114)(cid:16) q (cid:17) − m m q θ ( q − m ) . (12)We have restored factor of ¯ h in (12). Note that on the left hand side (LHS), N is di-mensionless. On the right hand side (RHS), the dimension reads (energy) (length) / ¯ h ,also dimensionless. We point out two counter-intuitive features of (12). The fluctuationof N contains explicit factor of ¯ h , indicating it is a consequence of quantum fluctuation.However, we know in free quark case there is no interaction to induce quantum fluctuation.The other odd feature is that the regularized fluctuation is negative (as is clear from thenegativity of the square bracket)! It means that the fluctuation at finite temperature issmaller compared to that in vacuum.The two seemingly odd features are in fact related: Although quarks are free at La-grangian level, Fermi-Dirac statistics obeyed by quarks in equilibrium provides effective in-teraction, thus quantum fluctuation is present. Furthermore, this also gives a quantitativelyexplanation of the negative sign in the regularized fluctuations. The effect of Fermi-Diracstatistics becomes prominent as we lower the temperature. In the vacuum case, quantum In [34], the same quantity is calculated in the stochastic hydrodynamics framework. χT V is obtainedinstead. The reason is we set initial N (0) = 0. This amounts to subtracting a baseline for the fluctuation. - - - - - 〈Δ N 〉 / m Figure 2: Contribution from intrinsic fluctuation (cid:104) ∆ N ( t ) (cid:105) /m for different masses: bluesolid for m = 1 /
10, red dashed for m = 1 / m = 1 /
2. The unit is setby T = 1. The fluctuation is characterized by an initial rise followed by oscillatory decay toasymptotic value. The case with larger mass shows more rapid convergence to asymptoticvalue.fluctuation is maximal, thus the vacuum fluctuation is larger than any finite temperaturefluctuation, giving rise to negative regularized fluctuation. (12) can be evaluated numeri-cally. We include the time evolution of (12) for different m in Figure. 2. The fluctuationis characterized by an initial rise followed by oscillatory decay to asymptotic value. Fig. 2suggests the initial rise satisfies the scaling (cid:104) ∆ N ( t ) (cid:105) ∝ m . Since the mass term is thesource of fluctuation, the m dependence as lowest order expansion is expected from ana-lyticty in m . Non-analyticity can occur in the presence of external field due to Schwingereffect [5, 39]. Furthermore, if we regard the period of oscillation as relaxation time, Fig. 2also implies shorter relaxation time at larger mass, which is consistent with expectation ongeneral grounds.To summarize, we find the fluctuation of N contains two contributions (10) and (12): (cid:104) ∆ N ( t ) (cid:105) = 2 V χT + V (cid:90) dq π (2 − e iq t − e − iq t ) G ( q ) , (13)where in the second line the limit t → ∞ should be taken. The χ term arises from chargeexchange with heat bath. The term proportional to G is intrinsic to breaking of the axialsymmetry. It exists without a heat bath. We could have view the second term as correctionto susceptibility. But this interpretation is misleading. Note that the second term is notnecessarily proportional to temperature as it arises from quantum fluctuation. In the nextsection, we will focus on the modification of the intrinsic fluctuation by interaction.6 Axial Charge Dynamics in Weakly Coupled QGP
After the warm-up, we move on to the calculation in weakly coupled QGP. We expect thesame structure of the Wightman correlator as (6). The term proportional to F is related tosusceptibility of N . It has been calculated in perturbation theory [40]. The other term isentirely due to non-conservation of N . We will calculate G >G in weakly coupled QGP, withthe subscript indicating it only contains the G term. We start with retarded correlator,whose imaginary part is related to the Wightman correlator. G R ( q ) ≡ (cid:90) dtd x e iq t (cid:104) (cid:2) ψ + γ ψ ( x ) , ψ + γ ψ (cid:0) (cid:105) . (14)We will proceed in imaginary time formalism and analytically continue to real time in theend. We work in the HTL approximation at one loop order. It is known from the calculationof susceptibility that one loop result of HTL is not complete ([40] and references therein).However the main purpose of this paper is to demonstrate the diffusive behavior of axialcharge from quark mass effect, we restrict ourselves to one loop order, and leave more refinedstudies for future work.At one loop order, N correlator receives contributions from three diagrams as shownin Figure. 3. The first diagram contains a soft quark loop with pseudophoton-quark vertices(˜ γqq ). The second diagram contains a soft gluon loop with two-pseudophoton-two-gluonvertex (2˜ γ g ) and the third diagram contains a quark loop with a two-pseudophoton-two-quark vertex (2˜ γ q ) vertex. Here we used ˜ γ to denote pseudophoton leg. These resummedvertices are to be evaluated for HTL diagrams. When the quark mass m = 0, we can easilyshow by commuting γ with γ µ the following relations˜ γqq = γqq × γ , γ g = 2 γ g, γ q = 2 γ q (15)We have used schematic notations. The first line of (15) means pseudophoton-quark vertexequals photon-quark vertex times γ and similarly for the second and third equalities. When m (cid:54) = 0, in general all the vertices involving pseudophoton receive corrections from m . Tosimplify the computation, we take quark mass to be soft, i.e. m ∼ gT . We stress that thisis “current quark mass”, not to be confused with thermal quark mass, which will appearbelow as m f . The current quark mass itself should be T independent. The relation m ∼ gT is only meant for numerical values for specific m , g and T .7igure 3: Three leading one loop diagrams contributing to (14) in the HTL approximation.In the spirit of HTL, we will drop any contributions at O ( m T ). This allows (15) tohold in this approximation. We are ready to write down explicit expressions of the verticesinvolved ∗ Γ µ ( P , P ) = (cid:32) γ µ − m f (cid:90) d Ω4 π ˆ K µ ˆ /K ( P · ˆ K )( P · ˆ K ) (cid:33) γ , (16) ∗ Γ µν ( P , P , Q , Q ) = − m f (cid:90) d Ω4 π ˆ K µ ˆ K ν ˆ /K ( P + Q ) · ˆ /K ( P − Q ) · ˆ /K (cid:20) P · ˆ K + 1 P · ˆ K (cid:21) , (17)with P and P being quark momenta and Q being one of the gluon momenta. m f = C F g T is thermal quark mass, not to be confused with current quark mass m . The nullvector is defined as ˆ K = ( − i, ˆ k ). The remaining 2˜ γ g vertex is obtainable by sendingtwo generators to 1 in four-gluon vertex. This leads to a vanishing result, thus the seconddiagram drops out. Unlike vertices, the resummed quark propagator does get modificationdue to quark mass as follows: ∗ S ( P ) = 1 /P + Σ + m = (∆ + + ∆ − ) iγ + (∆ + − ∆ − ) − m ∆ + ∆ − − m ∆ + ∆ − , (18)with γ p = ˆ p · (cid:126)γ and Σ = m f p (cid:2) iγ Q (cid:18) ωp (cid:19) + γ p (cid:18) − iωp Q (cid:18) ωp (cid:19)(cid:19) (cid:3) ± ( P ) = iω ∓ p − m f p (cid:2) Q (cid:18) iωp (cid:19) ∓ Q (cid:18) iωp (cid:19) (cid:3) . (19)Our calculation heavily relies on Ward identities. We note the γqq vertex and 2 γ q vertex satisfy the following Ward identities: Q µ ∗ Γ µν ( P , P , Q ) = Γ ν ( P , P − Q ) − Γ ν ( P + Q , P ) , ( P − P ) µ ∗ Γ µ ( P , P ) = Σ( P ) − Σ( P ) . (20)For our purpose, we take quark momenta as P = P = P , P = P (cid:48) and pseudophotonmomentum Q = ( − (cid:36), ) = P − P (cid:48) . Consequently ∗ Γ and ∗ Γ can be uniquely fixed by8ard identities as − (cid:36) ∗ Γ ( P, P, Q ) = ∗ Γ ( P, P − Q ) − ∗ Γ ( P + Q, P ) , − (cid:36) ∗ Γ ( P, P (cid:48) ) = Σ( P ) − Σ( P (cid:48) ) . (21)We proceed by evaluating the tadpole diagram (cid:90) d p (2 π ) tr ∗ Γ ∗ S ( P )( − , (22)The trace can be evaluated using (21) (cid:90) d p (2 π ) tr ∗ S ( P ) (cid:2) Γ ( P, P (cid:48) ) − Γ ( P + Q, P ) (cid:3) (cid:36) = (cid:90) d p (2 π ) (cid:2) tr ∗ S ( P ) ∗ Γ ( P, P (cid:48) ) − tr ∗ S ( P (cid:48) ) ∗ Γ ( P, P (cid:48) ) (cid:3) (cid:36) , (23)where in the second line, we make a change of variable: P → P (cid:48) to the second Γ . Thisexpression will be canceled by part of terms in quark-antiquark diagram. We proceedby simplifying the quark-antiquark diagram using (21). Note that ∗ Γ = ∗ Γ γ and also ∗ Γ ( P, P (cid:48) ) = ∗ Γ ( P (cid:48) , P ), which is obvious from (16), we have (cid:90) d P (2 π ) tr ∗ S ( P ) ∗ Γ ( P, P (cid:48) ) ∗ S ( P (cid:48) ) ∗ Γ ( P (cid:48) , P )( − . = (cid:90) d P (2 π ) tr 1 /P + Σ + m ∗ Γ γ /P (cid:48) + Σ (cid:48) + m (Σ − Σ (cid:48) ) γ (cid:36) , (24)where we use short hand notation Σ = Σ( P ), Σ (cid:48) = Σ( P (cid:48) ) and suppressed the argument of ∗ Γ for notational simplicity. Commuting γ through only switch sign of mass in the secondpropagator. We can further simplify the expression by splitting Σ − Σ (cid:48) = ( /P + Σ + m ) − ( /P (cid:48) + Σ (cid:48) − m ) − ( /P − /P (cid:48) + 2 m ) and using cyclic property of trace to obtain (cid:90) d P (2 π ) tr 1 /P + Σ + m ∗ Γ /P (cid:48) + Σ (cid:48) − m (Σ − Σ (cid:48) ) 1 (cid:36) = (cid:90) d P (2 π ) (cid:2) tr 1 /P (cid:48) + Σ (cid:48) − m ∗ Γ − /P + Σ + m ∗ Γ − tr 1 /P + Σ + m ∗ Γ /P (cid:48) + Σ (cid:48) − m × ( /P − /P (cid:48) + 2 m ) (cid:3) (cid:36) . (25)The retarded correlator is given by sum of (23) and (25).It is instructive to look at the result in the massless limit first. Setting m = 0, weimmediately see the first two terms of (25) cancel (23) entirely, leaving only the third termof (25). To evaluate the third term, we use (18) and the following explicit expression of Γ . ∗ Γ ( P, P (cid:48) ) = (cid:2) (cid:32) − m f i(cid:36)p δQ ( p, p (cid:48) ) (cid:33) γ + m f (cid:36) δQ ( p, p (cid:48) ) γ p (cid:3) , (26)9ith δQ n ( P, P (cid:48) ) = Q n (cid:16) iωp (cid:17) − Q n (cid:16) iω (cid:48) p (cid:17) . We can adopt a representation of Γ in terms of∆ ± ≡ ∆ ± ( P ) and ∆ (cid:48)± ≡ ∆ ± ( P (cid:48) ) by using (19). ∗ Γ ( P, P (cid:48) ) = − / ∆ + + 1 / ∆ − − / ∆ (cid:48) + − / ∆ (cid:48)− (cid:36) iγ + 1 / ∆ + − / ∆ − − / ∆ (cid:48) + + 1 / ∆ (cid:48)− (cid:36) γ p . (27)Taking the trace, we obtain (cid:90) d P (2 π ) i(cid:36) (∆ − + ∆ + − ∆ (cid:48)− − ∆ (cid:48) + ) , (28)which vanishes identically upon change of variable. This indicates that indeed the contri-bution we are after is intrinsic to breaking of the axial symmetry.Now we move on to massive case. We note that the first two terms of (25) combinewith (23) to give: (cid:90) d P (2 π ) tr ∗ Γ (cid:18) /P (cid:48) + Σ (cid:48) − m − /P (cid:48) + Σ (cid:48) + m (cid:19) (cid:36) , (29)which still vanishes upon taking the trace. The remaining terms are (cid:90) d P (2 π ) ( − )tr 1 /P + Σ + m ∗ Γ /P (cid:48) + Σ (cid:48) − m (cid:16) /P − /P (cid:48) + 2 m (cid:17) (cid:36) . (30)We aim at calculating lowest order mass correction, which begins at order O ( m ). It arisesfrom expansion of denominator and numerator of propagators and mass term in the Γ .It is easy to see that the expansion of the denominator still gives a vanishing result dueto similar cancellation as the O ( m ) result. The remaining correction can be organized asfollows (cid:90) d P (2 π ) m (cid:36) (cid:0) − ∆ + ∆ − − ∆ (cid:48) + ∆ (cid:48)− + ∆ + ∆ (cid:48)− + ∆ (cid:48) + ∆ − (cid:1) . (31)We use the following formula to perform the frequency sum:Im T Σ n g ( iω n ) g ( i ( ω n − ω )) = π (cid:16) − e βq (cid:17) × (cid:90) + ∞−∞ dp π dp (cid:48) π ˜ f ( p ) ˜ f ( p (cid:48) ) δ ( q − p − p (cid:48) ) ρ ( p ) ρ ( − p (cid:48) ) . (32)Here g and g are two generic functions. ρ and ρ correspond to their spectral densities, ρ = − Img , ρ = − Img . Note that the analytic continuation iω → q + iη is taken afterthe frequency sum and only the imaginary part of the result is kept in (32). The frequencysum of the first two terms in (31) gives a contribution with q = 0, which vanishes identically10ue to the factor 1 − e βq . The frequency sum of the remaining terms is given by (cid:90) d p (2 π ) (cid:90) dp π dp (cid:48) π π (cid:16) − e βq (cid:17) ˜ f ( p ) ˜ f ( p (cid:48) ) δ ( q − p − p (cid:48) ) 32 m q × (cid:0) Im ∆ + ( p ) Im ∆ − ( − p (cid:48) ) + Im ∆ − ( p ) Im ∆ + ( − p (cid:48) ) (cid:1) = (cid:90) d p (2 π ) (cid:90) dp π dp (cid:48) π π (cid:16) − e βq (cid:17) ˜ f ( p ) ˜ f ( p (cid:48) ) δ ( q − p − p (cid:48) ) 32 m q × (cid:0) Im ∆ + ( p ) Im ∆ + ( p (cid:48) ) + Im ∆ − ( p ) Im ∆ − ( p (cid:48) ) (cid:1) . (33)We have used the property Im ∆ ± ( − p (cid:48) ) = Im ∆ ∓ ( p (cid:48) ) in the second line. Note that Wickrotation applies N → iN , which gives an overall minus sign between correlator of N andcorrelator of N . Using KMS relation, we readily obtain G >G ( q ) = (cid:90) d p (2 π ) (cid:90) dp π dp (cid:48) π e βq ˜ f ( p ) ˜ f ( p (cid:48) ) δ ( q − p − p (cid:48) ) 32 m q × (cid:0) Im ∆ + ( p ) Im ∆ (cid:48) + ( p (cid:48) ) + Im ∆ − ( p ) Im ∆ (cid:48)− ( p (cid:48) ) (cid:1) . (34)Note that we have identified the contribution with G >G because it arises entirely from quarkmass breaking of the axial symmetry. The spectral density appearing in (34) containscontribution from poles and a cut. The convolution of two spectral densities gives rise tocontributions from the following types: pole-pole, pole-cut and cut-cut. Similar situation isencountered in the computation of soft dilepton production, showing a remarkable structure[41, 42].For the purpose of demonstrating late time dynamics of N , we focus on the small q regime. In the limit q →
0, we obtain G >G → m q , with Γ m defined asΓ m = (cid:90) d p (2 π ) dp π dp (cid:48) π m δ ( p + p (cid:48) ) ˜ f ( p ) ˜ f ( p (cid:48) ) × (cid:2) Im ∆ + ( p ) Im ∆ + ( p (cid:48) ) + Im ∆ − ( p ) Im ∆ − ( p (cid:48) ) (cid:3) . (35)Γ m characterizes the rate of fluctuation of the axial charge. To see this, we do the Fouriertransform as follows (cid:90) d x (cid:104) ( n ( t, x ) − n (0)) (cid:105) = (cid:90) dq o π (2 − e − iq t − e iq t ) G > ( q ) (cid:39) (cid:90) dq π m q (1 − cos q t ) = 4Γ m t. (36)This is the random walk growth of axial charge, with the growth rate given by (35).To evaluate Γ m , we note that the delta function constraint only allows for cut-cutcontribution in the product Im ∆ ± Im ∆ (cid:48)± : The pole-pole contribution is excluded in the11imit q →
0. The pole-cut contribution is possible only at large p , which is exponentiallysuppressed by the Fermi-Dirac distribution. The cut-cut contribution is not suppressed.We send ˜ f ( p ) → /
2, ˜ f ( p (cid:48) ) → / gT .The spectral functions scale as Im ∆ ± ∼ gT . As a result we obtain Γ m ∼ m g T . Thisis to be compared with CS diffusion rate, which scales as Γ CS ∼ g lng − T [43, 44] orΓ CS ∼ g T from extrapolation of weak coupling result [32]. At sufficient weak coupling,the quark mass diffusion rate always dominates the CS diffusion rate. It is interesting tocompare the actual number of the two rates at relevant coupling and mass. For the former,we quote the strong coupling extrapolation by Moore and Tassler [32]Γ CS ∼ α s T . (37)For the latter, we need to obtain the precise number in Γ m ∼ m ( gT ) from (35). Weobtain from numerical integration. Γ m (cid:39) . m m f . (38)We use strange quark mass m = 100 M eV and use the lattice measured thermal mass [45],which is m f (cid:39) . T . Taking T = 400 M eV and α s = 0 . CS (cid:39) . T , Γ m (cid:39) . T . (39)We found the quark mass effect is much less efficient in axial charge generation. However, theeffect of quark mass can be significantly enhanced when temperature approaches transitiontemperature from above. In this region, the relevant mass parameter is constituent quarkmass, which is enhanced by partial chiral symmetry breaking .Note that in the weakly coupled case, only the generation of axial charge is obtained,the damping effect is not visible. The reason can be understood by making an estimate ofdamping time scale. Using fluctuation-dissipation theorem, the damping time scale due toquark mass is given by τ m = χT m , (40)where χ is the axial charge susceptibility. To the leading order in g , it is given by the freetheory result χ ∼ g T . We thus obtain τ m ∼ Tm g . Obviously the relaxation is shorterfor larger mass, consistent with expectation on general grounds. Note that we assumed m ∼ gT in the calculation. The conjugate frequency to this time scale is q ∼ g T , whichlie well beyond the HTL regime. We thank Pengfei Zhuang for pointing this out for us. See also related work [46] Summary
Let us compare the fluctuation of axial charge in free theory and weakly coupled QGP.First of all, the fluctuations in both cases contain two contributions: one is proportionalto susceptibility, originating from charge exchange with heat bath; the other contributionis intrinsic to breaking of axial symmetry. Focusing on the contribution from breaking ofaxial symmetry, we find that unlike the susceptibility term, quantum fluctuation is neededto give a non-vanishing contribution. In case of free theory, the quantum fluctuation isprovided by effective Pauli repulsion. This also explains the counter-intuitive result wefind: the fluctuation maximizes at zero temperature. It also implies that it is misleading tointerpret this contribution as correction to susceptibility. In case of weakly coupled QGP,the fluctuation is given by quark-gluon interaction, which is enhanced by the presence ofthermal medium. The frequency dependence of Wightman correlator of the two cases aregiven by the following: (cid:104) G >G ( q ) (cid:105) free G ∼ (cid:113) q − m m | q | θ ( | q | − m ) , (cid:104) G >G ( q ) (cid:105) QGP G ∼ m m f q . (41)In the free case, the Wightman correlator vanishes for | q | < m . This gives a flat asymptoticbehavior for (cid:104) N ( t ) (cid:105) as shown in Fig. 2. In the weakly coupled QGP case, the Wightmancorrelator is non-vanishing, giving rise to random walk behavior for (cid:104) N ( t ) (cid:105) in long timelimit. Note that (41) is not applicable when q ∼ g T . We expect the random walk growthof axial charge to be cut off on an even longer time scale. In order to see the damping effect,we might need kinetic theory to access this time scale [42]. We leave it for future work. We are grateful Kenji Fukushima, Lianyi He, Anping Huang, Rob Pisarski and especiallyPengfei Zhuang for insightful discussions. SL would like to thank Central China NormalUniversity, Beihang University, Tsinghua University, Peking University, Institute of HighEnergy Physics for hospitalities where part of this work is done. This work is in partsupported by the Ministry of Science and Technology of China (MSTC) under the “973”Project No. 2015CB856904(4) (D. H.), and by NSFC under Grant Nos. 11375070, 11735007,11521064 (D. H.) and One Thousand Talent Program for Young Scholars (S.L.) and NSFCunder Grant Nos 11675274 and 11735007 (S.L.).13 eferences [1] A. Vilenkin. EQUILIBRIUM PARITY VIOLATING CURRENT IN A MAGNETICFIELD.
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