Numerical Study of the Chiral Separation Effect in Two-Color QCD at Finite Density
aa r X i v : . [ h e p - l a t ] D ec Numerical Study of the Chiral Separation Effect in Two-Color QCD at Finite Density
P. V. Buividovich ∗ Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK
D. Smith † Institut f¨ur Theoretische Physik, Justus-Liebig-Universit¨at, 35392 Giessen, GermanyHelmholtz Research Academy Hesse for FAIR (HFHF), Campus Giessen, 35392 Giessen, Germany andFacility for Antiproton and Ion Research in Europe GmbH (FAIR GmbH), 64291 Darmstadt, Germany
L. von Smekal ‡ Institut f¨ur Theoretische Physik, Justus-Liebig-Universit¨at, 35392 Giessen, Germany andHelmholtz Research Academy Hesse for FAIR (HFHF), Campus Giessen, 35392 Giessen, Germany (Dated: December 10th, 2020)We study the Chiral Separation Effect (CSE) in finite-density SU (2) lattice gauge theory withdynamical fermions. We find that the strength of the CSE is close to that for free quarks inmost regions of the phase diagram, including the high-temperature quark-gluon plasma phase, thelow-temperature phase with spontaneously broken chiral symmetry, and the diquark condensationphase which is specific for the SU (2) gauge theory. The CSE is significantly suppressed only atlow temperatures and low densities, where the chemical potential is roughly less than half of thepion mass. This suppression can be approximately described by assuming that the CSE currentis proportional to the charge density, rather than to the chemical potential, as suggested in theliterature [Phys. Rev. D , 085020 (2018)]. We also provide an upper bound on the contributionof disconnected fermionic diagrams to the CSE, which is consistent with zero within our statisticalerrors and small compared to that of the connected diagrams. I. INTRODUCTION
Anomalous transport phenomena are transport re-sponses of quantum matter which originate in quantumanomalies, inevitable violations of classical symmetriesupon quantization [1, 2]. In particular, in strongly in-teracting matter described by Quantum Chromodynam-ics (QCD) the classical symmetry between left-handedand right-handed fermions is violated by the Adler-Bell-Jackiw axial anomaly [3]. This violation manifests itselfin the infamous Chiral Magnetic Effect [4] - the genera-tion of an electric current along a magnetic field in chi-rally imbalanced matter - as well as the closely relatedChiral Separation Effect (CSE) [5, 6] - the generation ofan axial current along a magnetic field in a dense medium(see Fig. 1).In the last decade, anomalous transport phenomena inQCD matter were systematically and intensely studiedin heavy-ion collision experiments at the RHIC [7] andLHC [8] colliders, and will also be studied at the NICAcollider [9]. These studies are not conclusive yet due tolarge background effects, which contaminate the signa-tures of anomalous transport [8, 10–14]. A dedicated runwith isobar nuclei has been recently completed at RHICin order to disentangle these background effects [7, 15],and the produced experimental data is currently beinganalyzed [16]. ∗ [email protected] † [email protected] ‡ [email protected] FIG. 1. a) Feynman diagrams which contribute to the ChiralSeparation Effect at leading order and b) one of the possiblecorrections to it in a gauge theory with dynamical fermions.
Just as the viscosity of the quark-gluon plasma isrelated to hadronic elliptic flow [17], anomalous trans-port coefficients characterizing the strengths of the Chi-ral Magnetic and Chiral Separation Effects can be re-lated to correlations of angular distributions of oppositelycharged hadrons in heavy-ion collisions [18, 19].One of the most popular ways to interpret ex-perimental data on these correlations relies on theanomalous-viscous fluid dynamics (AVFD) framework[20–22]. AVFD is based on anomalous hydrodynamics[23–25] which incorporates anomalous transport alongwith more conventional transport responses such as vis-cosity and electric conductivity. Hydrodynamic simula-tion codes which include anomalous transport phenom-ena on an event-by-event basis are currently being ac-tively developed [21, 26–28] and are becoming more andmore realistic.The anomalous hydrodynamic description of QCDmatter requires the values of anomalous transport co-efficients as an input. For a single-component chiralfluid, anomalous transport coefficients are fixed by ther-modynamic consistency [24, 29]. On the other hand,in a quark-gluon plasma (nearly) chiral quarks interactwith dynamical non-Abelian gauge fields, which them-selves behave as a viscous fluid and in fact dominate thehydrodynamic flow. When interactions with dynamicalgauge fields are present, all anomalous transport coeffi-cients might receive both perturbative [30] (see Fig. 1) aswell as non-perturbative [6, 31–35] corrections. However,at present not much is known about the magnitude ofthese corrections.In a few lattice gauge theory simulations, the ChiralMagnetic [36] and Chiral Vortical [37] Effects were stud-ied by measuring the responses of naively discretized ax-ial currents and energy momentum tensors to constantexternal magnetic or axial magnetic fields (the study [37]of the Chiral Vortical Effect used a trick to replace rota-tion by a background axial gauge field). In both works[36, 37] the CME and CVE transport coefficients werefound to be 5 to 20 times smaller than those obtainedfor free fermions, both in high- and in low-temperaturephases. If the corrections in the full gauge theory shouldindeed have the effect to make the anomalous transportresponses so small, they might well be unobservable incurrent heavy-ion collision experiments.However, the works [36, 37] used non-chiral latticefermions with non-conserved currents and an energy-momentum tensor without proper renormalization. Onthe other hand, a numerical study of the Chiral Separa-tion Effect in quenched SU (3) lattice gauge theory withexactly chiral overlap fermions and properly defined ax-ial current [38] found no noticeable corrections to the freefermion result, j Ai = µ C em N c π B i ≡ σ B i , (1)where j Ai = P f ¯ q f γ γ i q f is the axial current density withquark fields ¯ q f , q f of flavor f and N c colors, µ is thefermion chemical potential, B i is the magnetic field, and C em = P f Q f is the electromagnetic charge factor inwhich Q f denotes the electric charge of quark flavor f .The result (1) corresponds to the triangular diagram inFig. 1 a). To simplify notation, in what follows we assumethat C em , which appears in all formulae as a simple pre-factor, is equal to unity: C em = 1. The correct value of C em can be restored in all results in an obvious way.In this work we study the Chiral Separation Effectin the full gauge theory with dynamical fermions, tak-ing into account the contributions of virtual fermionloops and disconnected fermionic diagrams like the onein Fig. 1 b). These contributions are expected to mod-ify the free fermion result (1) and are thus important toknow for the detectability of the Chiral Separation Ef-fect in heavy-ion collisions. Rather than studying the theoretically clean, but rather academic, limit of exactlychiral quarks, we address the fate of the CSE in a morerealistic setup with finite quark and pion masses and atfinite temperatures in the vicinity of the chiral crossover.While giving some general insight into the magnitude ofnon-perturbative corrections to anomalous transport co-efficients, this might also help to estimate the observableconsequences of the CSE, such as the electric quadrupolemoment of the quark-gluon plasma [39] due to ChiralMagnetic Waves [40].Since the CSE is a feature of finite-density fermions,studying it in QCD would require simulations at fi-nite baryon density, which are complicated by the infa-mous fermion-sign problem [41]. With current simulationmethods one could only obtain first-principle lattice QCDresults for µ/T ≪
1. In particular, a study of the CSEin chiral effective field theory [35] suggests that for suf-ficiently low temperatures it is suppressed at µ < m π / m π is the pion mass. However, this is preciselythe region where the QCD sign problem is expected tobe small, and there are little chances to obtain any reli-able results outside of this region in near future.In this work we circumvent the fermion sign problemby using two-color QCD, i.e. the SU (2) gauge theory with N f = 2 light quark flavours instead of QCD. The path in-tegral weight is manifestly positive in this case, thus thesign problem is absent and the theory can be simulated atfinite density. The SU (2) gauge theory is expected to bequalitatively similar to QCD at small densities µ < m π / SU (2) gauge theory is also at leastqualitatively relevant for real QCD at small densities.At larger densities, for µ > m π / SU (2) gauge theoryis no longer similar to QCD, since the chiral condensate h ¯ qq i is rotated into the diquark condensate h qq i . Di-quarks are bound states of two quarks which are colorsinglets and hence “bosonic baryons” in the SU (2) gaugetheory. Instead of the first-order liquid-gas transition ofnuclear matter, one therefore observes Bose-Einstein con-densation together with a BEC-BCS crossover inside thediquark condensation phase [50, 51]. Similarity to QCD,although at a different conceptual level, can be againexpected at very large densities and low temperatures,in the conjectured quarkyonic and color-superconductingphases [52, 53]. II. LINEAR RESPONSE APPROXIMATIONFOR THE CHIRAL SEPARATION EFFECT
Within linear response theory, the Chiral SeparationEffect is characterized by the correlator of vector andaxial-vector currents h j A ( k ) j V ( − k ) i , where the onlynonzero momentum component is the spatial component k [54]. At small momenta this correlator behaves as h j A ( k ) j V ( − k ) i = σ CSE k , (2)where σ CSE is the anomalous transport coefficient in (1)characterizing the strength of the CSE. For free fermions, σ CSE = σ = µ N c π . (3)It is therefore convenient to define a momentum-depen-dent transport coefficient σ CSE ( k ) as σ CSE ( k ) ≡ h j A ( k ) j V ( − k ) i /k . (4)In the low-momentum hydrodynamic regime, the anoma-lous transport coefficient σ CSE in (1) is given by the zero-momentum limit of σ CSE ( k ).For exactly chiral fermions which may interact, but notwith other dynamical degrees of freedom (like dynamicalgauge fields), σ CSE is expected to be universal and equalto the free fermion result due to the relation with theAdler-Bell-Jackiw axial anomaly [5]. However, correc-tions are still possible for nonzero quark mass [6] and dueto fermionic disconnected diagrams like 1b) on Fig. 1. Ascalculations of [6] suggest, corrections to the CSE can berelated to the amplitude g π γγ for the π → γγ decay: σ CSE = µN c C em π (cid:0) − g π γγ + O ( µ ) (cid:1) . (5)Within the linear sigma model g π γγ = ζ (3) m π T , where m is the constituent quark mass.Another calculation of the flavor non-singlet CSE axialcurrent ~j aA generated by a finite isospin chemical potentialwas carried out within chiral effective field theory in [35].It suggests that in the low-temperature phase, where theCSE current is saturated by pions, σ CSE is proportionalto the isospin charge density ρ aV rather than the isospinchemical potential, ~j aA = N c Tr ( Q )4 π f π ρ aV ~B. (6)While this calculation is not directly applicable to theflavor-singlet axial current in (1) and (5), at least in thelarge- N c limit the flavor-singlet axial current should be-have similarly to the flavor non-singlet one [55]. Ournumerical results presented in Section IV below suggestthat a parametrization similar to (6) might also work atfinite density in the SU (2) gauge theory with dynamicalfermions. III. LATTICE SETUP
In this work we use the same lattice setup and the sameensembles of gauge field configurations as in our recentwork [44], so here we will provide only a brief summary.We use the standard Hybrid Monte-Carlo algorithm witha tree-level improved Symanzik gauge action and N f = 2 flavours of mass-degenerate rooted staggered fermionswith bare lattice quark mass am stagq = 5 · − . Thisyields a pion mass of am stagπ = 0 . ± .
002 and theratio of pion to ρ -meson mass m π /m ρ = 0 . ± . L s = 24 ( am π L s = 3 . L s = 30 ( am π L s = 4 .
7) and varying temporal extent L t = 4 , , . . . ,
22 to control temperature T = 1 /aL t . Weuse a single value of the lattice gauge coupling β = 1 . L t ≥ λqq in theaction with aλ = 5 · − in order to facilitate diquarkcondensation, which would otherwise be impossible in afinite volume. This diquark source has very little effecton current-current correlators outside of the diquark con-densation phase [44], see also Fig. 4. Estimates of phaseboundaries based on our data sets are shown in Fig. 2(see [44] for full details). y − y > „ yy > „
0, diquark condensationQuark−gluon plasma T / m p L t m / m p m=0.005, b =1.7a m diquarkchiral FIG. 2. Numerical estimate of the phase diagram of finite-density SU (2) gauge theory with N f = 2 rooted staggeredfermions. Blue and red points correspond to inflection pointsin the L t dependence of the chiral and diquark condensates,respectively. Configuration sets with lattice size L s = 24 onlyare shown as empty circles, and sets with both L s = 24 and L s = 30 are shown as double circles. We use Domain Wall (DW) and Wilson-Dirac (WD)valence fermions to measure the correlators of axial andvector currents in (2). On the one hand, for DW fermionsthe renormalization factor Z A for the flavour-singlet ax-ial current is expected to deviate from unity by at mostfew percent [56], which is below our the statistical uncer-tainty of our Monte-Carlo simulations (and also well be-low experimental uncertainties). On the other hand, DWfermions are computationally very expensive, and we usethe cheaper WD fermions to produce results with betterprecision covering more points on the phase diagram. Acomparison between the results obtained with DW andWD fermions further demonstrates the smallness of axialcurrent renormalization. We do not use staggered valencefermions in order to avoid artifacts related to unphysicaltaste symmetry. Such a mixed lattice action with stag-gered sea fermions and DW valence fermions has alreadybeen used in a number of studies of the nucleon axialcharge [56, 57].We tune the bare quark masses am DWq = 0 .
01 and am W Dq = − .
21 in the DW/WD Dirac operators tomatch the pion mass am stagπ = 0 . ± .
002 obtainedwith staggered valence quarks. To improve the chiralproperties of DW and WD fermions without using muchfiner and larger lattices, we follow [57] and use HYPsmearing [58] for gauge links in the DW and WD Diracoperators. For DW fermions the lattice size in the fifthdimension is L = 16, which is typically sufficient to sup-press additive mass renormalization [56, 57].For WD fermions, we use the conserved vector current j Vz,µ = X x,y ¯ q x ( j z,µ ) x,y q y , ( j z,µ ) x,y ≡ ∂D xy ∂θ z,µ (7)= iP + µ U z,µ δ x,z δ y,z +ˆ µ − iP − µ U † z,µ δ x,z +ˆ µ δ y,z , where D xy is the Dirac operator, x, y, z, . . . label latticesites, γ µ are the Euclidean gamma-matrices, with P ± µ =(1 ± γ µ ) / U z,µ are the SU (2)-valued link variables, ˆ µ denotes the unit lattice vector in the direction µ , and θ z,µ is an external U (1) gauge field. We also use theconventional point-split definition of the axial current forWD fermions [59], (cid:0) j Az,µ (cid:1) x,y ≡ ∂D xy ∂θ z,µ (8)= iγ µ γ U z,µ δ x,z δ y,z +ˆ µ − iγ µ γ U † z,µ δ x,z +ˆ µ δ y,z . For DW fermions, the four-dimensional vector and ax-ial currents are defined in the standard way by summingthe five-dimensional conserved current over the fifth di-mension. For the vector current a unit weight is used,for the axial current the summation weight changes from+1 to − µ takes five valuesand x , y , z live on the five-dimensional lattice with openboundary conditions along the fifth dimension.We measure the contributions of both connectedand disconnected fermionic diagrams to the axial-vectorcurrent-current correlator in (2). In coordinate spacethese contributions are: h j Ax,µ j Vy,ν i conn = h Tr (cid:0) j Ax,µ D − j Vy,ν D − (cid:1) i , (9) h j Ax,µ j Vy,ν i disc = h Tr (cid:0) j Ax,µ D − (cid:1) Tr (cid:0) j Vy,ν D − (cid:1) i , (10)where the traces are taken over the lattice site, spinor andcolor indices of the quark fields ¯ q , q . The disconnectedcontribution is measured using standard stochastic esti-mator techniques. After measuring h j Ax,µ j Vy,ν i conn and h j Ax,µ j Vy,ν i disc in coordinate space, we perform a discreteFourier transform to obtain the momentum-space corre-lators which enter the linear response relations (2). IV. NUMERICAL RESULTS
In Fig. 3 we present our lattice results for the mo-mentum dependent CSE transport coefficient σ CSE ( k )defined in (4). For comparison, we combine the resultsobtained with DW and WD fermions, and with the spa-tial lattice sizes L s = 24 and L s = 30. For WD fermionson the L s = 24 lattices we show the contributions (9)and (10) of both connected and disconnected fermionicdiagrams, for other data sets only the connected contri-butions are shown. We also compare the gauge theoryresults with the results obtained for free WD quarks onsame lattices.For the free WD quarks, we use a bare quark mass of am W Dq = 0 .
01 (as compared to am W Dq = − .
21 in the fullgauge theory), since for free quarks there is obviously nomass renormalization. Therefore, in this case we choosethe same bare quark mass as for the DW fermions, forwhich mass renormalization is expected to be weak.In our calculations we also combine results obtainedwith zero diquark source λ at high temperatures ( L t <
14) and with aλ = 5 · − at low temperatures ( L t ≥ λ the CSE transport coefficients σ CSE ( k ) are practicallyindistinguishable.We see from Fig. 3 that for most values of temperatureand chemical potential the momentum-dependent CSEtransport coefficient σ CSE ( k ) is very close to the corre-sponding free-fermion result for all momenta. An explicitcalculation of σ CSE ( k ) for free quarks at finite tempera-ture in the continuum is sketched in Appendix A. The re-sults of this continuum calculation are shown in all plotsof Fig. 3 as black solid lines. We note that σ CSE ( k ) onlybecomes noticeably smaller than the free quark result atlow temperatures and small values of chemical potential,less than half the pion mass (see e.g. the plot for L t = 20and aµ = 0 .
05 corresponding to µ = 0 . m π in Fig. 3).In this regime the SU (2) gauge theory is expected to bequalitatively similar to real QCD, thus the observed sup-pression of the CSE in the confined and chirally brokenphase is also likely to happen in low-temperature, low-density QCD.The contributions from disconnected fermionic dia-grams appear to be consistent with zero within statis-tical errors for all values of chemical potential and tem-perature. The upper bound which we are able to seton these disconnected contributions appears to be leaststrict for low temperatures and small µ - that is, exactlyin the corner of the phase diagram where also the con-nected contributions deviate most strongly from the freequark result (see Fig. 3, plot for L t = 20 and aµ = 0 . −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.5 1 1.5 2 2.5 3 s C SE ( k ) / m k (Lattice momentum)L t = 12, m = 0.05WD,L s =24,conn.WD,L s =30,conn.WD,L s =24,disc.WD,L s =24,freeWD,L s =30,freeFree continuum −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.5 1 1.5 2 2.5 3 s C SE ( k ) / m k (Lattice momentum)L t = 12, m = 0.20DW,L s =24,conn.WD,L s =24,conn.WD,L s =30,conn.WD,L s =24,disc.WD,L s =24,freeWD,L s =30,freeFree continuum−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.5 1 1.5 2 2.5 3 s C SE ( k ) / m k (Lattice momentum)L t = 16, m = 0.05WD,L s =24,conn.WD,L s =30,conn.WD,L s =24,disc.WD,L s =24,freeWD,L s =30,freeFree continuum −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.5 1 1.5 2 2.5 3 s C SE ( k ) / m k (Lattice momentum)L t = 16, m = 0.20DW,L s =24,conn.WD,L s =24,conn.WD,L s =30,conn.WD,L s =24,disc.WD,L s =24,freeWD,L s =30,freeFree continuum−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.5 1 1.5 2 2.5 3 s C SE ( k ) / m k (Lattice momentum)L t = 20, m = 0.05DW,L s =24,conn.WD,L s =24,conn.WD,L s =30,conn.WD,L s =24,disc.WD,L s =24,freeWD,L s =30,freeFree continuum −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.5 1 1.5 2 2.5 3 s C SE ( k ) / m k (Lattice momentum)L t = 16, m = 0.50DW,L s =24,conn.WD,L s =24,conn.WD,L s =24,disc.WD,L s =24,freeFree continuum FIG. 3. Momentum-dependent CSE transport coefficient σ CSE ( k ) as function of lattice momentum k at selected temperaturesand chemical potentials, corresponding roughly to: µ ≃ . m π for three temperatures across the chiral transition (leftcolumn), as well as µ ≃ . m π and µ ≃ . m π for temperatures approaching the boundary of diquark condensation fromabove (right column). this regime, since because of the small chemical potentialthe CSE signal is also small compared to the statisticalfluctuations.Moreover, we also note that the values of σ CSE ( k ) cal-culated with Wilson-Dirac (WD) and Domain Wall (DW)fermions appear to be very close to each other. Thissuggests that the effect of multiplicative renormalizationof the axial-current operator is small and plays a minorrole in comparison with our statistical errors, as well aswith any systematic and statistical uncertainties in the experimental detection of anomalous transport phenom-ena. Indeed, the renormalization factor for the axial sin-glet current typically appears to be close to unity on finelattices with sufficiently light pions, especially for Do-main Wall fermions [56, 57]. For these reasons, we havenot determined the precise value of Z A in this work.In order to systematically investigate the CSE trans-port coefficient σ CSE ( k ) in the low-momentum limit,which is most relevant for anomalous hydrodynamics [54],in Fig. 5 we illustrate the temperature dependence of −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.5 1 1.5 2 2.5 3 s C SE ( k ) / m k (Lattice momentum)L s = 30, L t = 12, m = 0.20WD, l =0WD, l =5·10 −4 WD, l =0,freeWD, l =5·10 −4 ,freeFree continuum, l =0 FIG. 4. CSE transport coefficient σ CSE ( k ) for different valuesof the diquark source λ . σ CSE ( k ) at the smallest nonzero value of the lattice mo-mentum, ka = 2 π/L s at different values of the chemi-cal potential µ , and compare it with the correspondingresults for free quarks on lattices of the same size. Inorder to check whether the formula similar to (6) mightdescribe our numerical results, we also show the appro-priately rescaled charge density (in lattice units) alongwith σ CSE ( k → σ CSE ( k →
0) becomes signif-icantly smaller than the free quark result only at lowtemperatures and small values of µ , in the chirally bro-ken and confined phase which should be dominated bythe pions. The data for L s = 24 and L s = 30 appear tobe reasonably close, thus suggesting that the suppressionof the CSE in this phase is not a finite-volume artifact.Comparing the data for σ CSE with the temperature de-pendence of the charge density, we furthermore observethat at low temperatures ( L t &
16) and aµ . .
20 thedependence of σ CSE ( k →
0) on both µ and temperaturecan be approximately described by a formula of the form(6), except with a flavour-singlet current and chemicalpotential: σ CSE ( k → , µ ) = α ρ V ( µ ) ,~j A = α ρ V ( µ ) ~B, (11)where the constant α is about α ≈ a in lattice units,i.e. for our particular value of lattice spacing (this doesnot imply that α scales as a with the lattice spacing a ).With our value of the vacuum pion mass, we may write α ≈ / (2 m π ) . According to (6), the coefficient α in(11) should be related to the pion decay constant f π via α = N c / (2 πf π ) . Using our estimate αa − ≈
10, fromthis relation we roughly estimate af π ≈ .
07, which hasa reasonable order of magnitude compared to the pionmass am π = 0 . aµ = 0 .
50 we observe asignificant decrease of the coefficient α down to α ≈ a which would be in line with a pion mass that increaseswith µ as expected in the diquark condensation phase.In order to present further evidence for the scaling of σ CSE with ρ V , in Fig. 6 we show the dependence of the ra-tios σ CSE ( k → /µ (on the left) and σ CSE ( k → /ρ V (on the right) on the chemical potential µ at differenttemperatures. Data points appear to collapse better ontoa single curve for the ratio σ CSE ( k → /ρ V .While these observations give some qualitative supportto the formula (11), because of a conceptually differ-ent status of axial-singlet and axial-non-singlet currentsin low-energy chiral effective theory, at the quantitativelevel one can expect further corrections to this formula.A general conclusion which we can make based on ourresults is that the CSE should be suppressed for valuesof the chemical potential roughly smaller than the pionmass. At higher temperatures and densities, the CSEtransport coefficient should approach its value for freequarks. V. CONCLUSIONS
To summarize, we have found that in a gauge the-ory with dynamical fermions the Chiral Separation Ef-fect is close to the free fermion results in all regionsof the phase diagram, including the high-temperaturequark-gluon plasma phase, the low-temperature phasewith spontaneously broken chiral symmetry and the di-quark condensation phase. This conclusion agrees withthe results of a previous study [38] in quenched SU (3)lattice gauge theory with exactly chiral valence quarks(thus formally at zero pion mass). The CSE only appearsto be suppressed in one corner of the phase diagram,namely at low temperatures and low densities, where chi-ral symmetry is spontaneously broken and the chemicalpotential is roughly below the half of the pion mass. Ex-actly this regime of SU (2) gauge theory is similar to thelow-temperature, low-density phase of real finite-densityQCD, thus our findings should be also relevant for realQCD at least qualitatively.The suppression of the CSE at low densities and tem-peratures can be approximately described if one assumesthat the CSE current is proportional to the charge den-sity rather than to the chemical potential, as in equation(6). While the formula (6) was derived in [35] for theaxial non-singlet current, we see that at the qualitativelevel it also applies to the axial singlet current, which hasa different status within the chiral effective theory.Contributions of disconnected fermionic diagrams tothe Chiral Separation Effect appear to be consistent withzero within our statistical errors. The latter are rela-tively large at low temperatures and densities. For thisreason we cannot rule out that the disconnected contribu-tion might become important when the connected one isstrongly suppressed. This scenario would certainly be in-teresting from a theoretical viewpoint. However, as thesum of both contributions still appears to be small ascompared to the free massless fermion result, it is prob-ably not very relevant for the experimental detection ofanomalous transport phenomena. s C SE / m , a r V / m x L t m = 0.05DW,L s =24WD,L s =24WD,L s =30WD,L s =24,freeWD,L s =30,freeCharge density 0 0.02 0.04 0.06 0.08 0.1 4 6 8 10 12 14 16 18 20 22 s C SE / m , a r V / m x L t m = 0.10DW,L s =24WD,L s =24WD,L s =24,freeCharge density 0 0.02 0.04 0.06 0.08 0.1 4 6 8 10 12 14 16 18 20 22 s C SE / m , a r V / m x L t m = 0.20DW,L s =24WD,L s =24WD,L s =30WD,L s =24,freeWD,L s =30,freeCharge density 0 0.02 0.04 0.06 0.08 0.1 4 6 8 10 12 14 16 18 20 22 s C SE / m , a r V / m x L t m = 0.50DW,L s =24WD,L s =24WD,L s =24,freeCharge density FIG. 5. Low-momentum limit of the CSE transport coefficient σ CSE ( k →
0) as a function of temporal lattice size (inversetemperature in lattice units) at different values of the chemical potential. The solid black lines correspond to σ CSE ( k →
0) = σ = µN c / π for free massless quarks in the continuum. s C SE / m m N c /(2 p )L s =24,L t =16L s =24,L t =18L s =24,L t =20L s =30,L t =16L s =30,L t =18L s =30,L t =20 0 2 4 6 8 10 12 14 16 18 0 0.1 0.2 0.3 0.4 0.5 s C SE / r ( m ) m L s =24,L t =16L s =24,L t =18L s =24,L t =20L s =30,L t =16L s =30,L t =18L s =30,L t =20 FIG. 6. Ratios of the low-momentum CSE transport coefficients σ CSE with chemical potential µ (left) and charge density ρ V (right). One potential source of systematic errors in our studyis the multiplicative renormalization of the axial current.However, comparison of the data obtained with DomainWall and Wilson-Dirac fermions, as well as previous lat-tice studies of axial current renormalization, suggest thatthe renormalization effects are small.
ACKNOWLEDGMENTS [1] K. Landsteiner, Acta Phys. Pol. B , 2617 (2016),1610.04413.[2] D. E. Kharzeev, K. Landsteiner, A. Schmitt, and H. Yee,Lect. Notes Phys. , 1 (2012), 1211.6245.[3] S. L. Adler, Phys. Rev. , 2426 (1969).[4] K. Fukushima, D. E. Kharzeev, and H. J. Warringa,Phys. Rev. D , 074033 (2008), 0808.3382.[5] M. A. Metlitski and A. R. Zhitnitsky,Phys. Rev. D , 045011 (2005), hep-ph/0505072.[6] G. M. Newman and D. T. Son,Phys. Rev. D , 045006 (2006), hep-ph/0510049.[7] V. Skokov, P. Sorensen, V. Koch, S. Schlichting,J. Thomas, S. Voloshin, G. Wang, and H. Yee,Chin. Phys. C , 072001 (2017), 1608.00982.[8] CMS Collaboration, Phys. Rev. C , 044912 (2018),1708.01602.[9] D. Blaschke, J. Aichelin, E. Bratkovskaya, V. Friese,M. Gazdzicki, J. Randrup, O. Rogachevsky, O. Teryaev,and V. Toneev, Eur. Phys. J. A , 267 (2016).[10] J. Adam et al. (STAR Collaboration), “Charge separa-tion measurements in p(d)+Au and Au+Au collisions;implications for the chiral magnetic effect,” (2020),2006.04251.[11] R. A. Lacey and N. Magdy, “Quantification of the ChiralMagnetic Effect in Au+Au collisions at √ s NN = 200gevs,” (2020), 2006.04132.[12] J. Zhao and F. Wang,Prog. Part. Nucl. Phys. , 200 (2019), 1906.11413.[13] L. Huang, M. Nie, and G. Ma,Phys. Rev. C , 024916 (2020), 1906.11631.[14] A. Bzdak, V. Koch, and J. Liao,Lect. Notes Phys. , 503 (2013), 1207.7327.[15] D. E. Kharzeev and J. Liao,Nucl. Phys. News , 26 (2019).[16] STAR Collaboration: J. Adam et al., “Methods for ablind analysis of isobar data collected by the STAR col-laboration,” (2019), 1911.00596.[17] D. A. Teaney, Quark-Gluon Plasma , 207 (2010),0905.2433.[18] S. A. Voloshin (STAR Collaboration),Indian J. Phys. , 1103 (2011), 0806.0029.[19] F. Becattini, I. Karpenko, M. Lisa, I. Upsal,and S. Voloshin, Phys. Rev. C , 054902 (2017),1610.02506.[20] Y. Jiang, S. Shi, Y. Yin, and J. Liao,Chin. Phys. C , 011001 (2018), 1611.04586.[21] S. Shi, H. Zhang, D. Hou, and J. Liao,Nucl. Phys. A , 539 (2018), 1807.05604.[22] S. Shi, Y. Jiang, E. Lilleskov, and J. Liao,Ann. Phys. , 50 (2018), 1711.02496.[23] J. Erdmenger, M. Haack, M. Kaminski, and A. Yarom, JHEP , 055 (2009), 0809.2488.[24] D. T. Son and P. Surowka,Phys. Rev. Lett. , 191601 (2009), 0906.5044.[25] N. Banerjee, J. Bhattacharya, S. Bhattacharyya,S. Dutta, R. Loganayagam, and P. Sur´owka, JHEP , 094 (2011), 0809.2596.[26] X. Guo, D. E. Kharzeev, X. Huang, W. Deng, and Y. Hi-rono, Nucl. Phys. A , 1 (2017), 1704.05375.[27] G. Inghirami, M. Mace, Y. Hirono, L. DelZanna, D. E. Kharzeev, and M. Bleicher,Eur. Phys. J. C , 293 (2020), 1908.07605.[28] G. Liang, J. Liao, S. Lin, L. Yan, and M. Li, “Chiralmagnetic effect in isobar collisions from stochastic hy-drodynamics,” (2020), 2004.04440.[29] A. V. Sadofyev and M. V. Isachenkov,Phys. Lett. B , 404 (2011), 1010.1550.[30] E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, andX. Wang, Phys. Rev. D , 025025 (2013), 1304.4606.[31] S. Golkar and D. T. Son, JHEP , 169 (2012),1207.5806.[32] K. Jensen, P. Kovtun, and A. Ritz,JHEP , 186 (2013), 1307.3234.[33] U. Gursoy and A. Jansen, JHEP , 92 (2014),1407.3282.[34] A. Jimenez-Alba and L. Melgar, JHEP , 120 (2014),1404.2434.[35] A. Avdoshkin, A. V. Sadofyev, and V. I. Zakharov,Phys. Rev. D , 085020 (2018), 1712.01256.[36] A. Yamamoto, Phys. Rev. Lett. , 031601 (2011),1105.0385.[37] V. Braguta, M. N. Chernodub, V. A. Goy, K. Land-steiner, A. V. Molochkov, and M. I. Polikarpov,Phys. Rev. D , 074510 (2014), 1401.8095.[38] M. Puhr and P. V. Buividovich, Phys. Rev. Lett. ,192003 (2017), 1611.07263.[39] Y. Burnier, D. E. Kharzeev, J. Liao, and H. Yee,Phys. Rev. Lett. , 052303 (2011), 1103.1307.[40] D. E. Kharzeev and H. Yee,Phys. Rev. D , 085007 (2011), 1012.6026.[41] C. Gattringer and K. Langfeld,Int. J. Mod. Phys. A , 1643007 (2016), 1603.09517.[42] J. B. Kogut, D. K. Sinclair, S. J. Hands, and S. E. Mor-rison, Phys. Rev. D , 094505 (2001), hep-lat/0105026.[43] J. B. Kogut, M. A. Stephanov, D. Toublan,J. J. M. Verbaarschot, and A. Zhitnitsky,Nucl. Phys. B , 477 (2000), hep-ph/0001171.[44] P. V. Buividovich, L. von Smekal, and D. Smith,Phys. Rev. D , 094510 (2020), 2007.05639.[45] T. Boz, P. Giudice, S. Hands, and J. Skullerud,Phys. Rev. D , 074506 (2020), 1912.10975.[46] J. Wilhelm, L. Holicki, D. Smith, B. Wellegehausen, and L. von Smekal, Phys. Rev. D , 114507 (2019),1910.04495.[47] L. Holicki, J. Wilhelm, D. Smith, B. Wellegehausen,and L. von Smekal, PoS LATTICE2016 , 052 (2017),1701.04664.[48] T. Boz, P. Giudice, S. Hands, J. Skullerud, andA. G. Williams, AIP Conf. Proc. , 060019 (2016),1502.01219.[49] S. Cotter, P. Giudice, S. Hands, and J. Skullerud,Phys. Rev. D , 034507 (2013), 1210.4496.[50] N. Strodthoff, B. Schaefer, and L. von Smekal,Phys. Rev. D , 074007 (2012), 1112.5401.[51] N. Strodthoff and L. von Smekal,Phys. Lett. B , 350 (2014), 1306.2897.[52] L. McLerran and R. D. Pisarski,Nucl. Phys. A , 83 (2007), 0706.2191.[53] V. V. Braguta, E. Ilgenfritz, A. Y. Ko-tov, A. V. Molochkov, and A. A. Nikolaev,Phys. Rev. D , 114510 (2016), 1605.04090.[54] I. Amado, K. Landsteiner, and F. Pena-Benitez,JHEP , 081 (2011), 1102.4577.[55] R. Kaiser and H. Leutwyler,Eur. Phys. J. C , 623 (2000), hep-ph/0007101.[56] E. Berkowitz, D. Brantley, C. Bouchard, C. Chang,M. A. Clark, N. Garron, B. Joo, T. Kurth, C. Monahan, H. Monge-Camacho, A. Nicholson, K. Orginos, E. Ri-naldi, P. Vranas, and A. Walker-Loud, “An accurate cal-culation of the nucleon axial charge with lattice QCD,”(2017), 1704.01114.[57] R. G. Edwards, G. T. Fleming, P. Hagler, J. W. Negele,K. Orginos, A. Pochinsky, D. B. Renner, D. G. Richards,and W. Schroers, Phys. Rev. Lett. , 052001 (2006),hep-lat/0510062.[58] A. Hasenfratz and F. Knechtli,Phys. Rev. D , 034504 (2001), hep-lat/0103029.[59] M. Bochicchio, L. Maiani, G. Martinelli, G. C. Rossi,and M. Testa, Nucl. Phys. B , 331 (1985).[60] V. Furman and Y. Shamir,Nucl. Phys. B , 54 (1995), hep-lat/9405004.[61] P. V. Buividovich, Nucl. Phys. A , 218 (2014),1312.1843. Appendix A: Momentum-dependent ChiralSeparation Effect for free quarks in the continuumat finite temperature
For free Dirac fermions, the axial-vector current-current correlator in (2) is given by the one-loop integralof the form [61]: h j Aµ ( k ) j Vν ( − k ) i = T X l Z d l (2 π ) Tr ( γ µ γ ( m − iγ α ( l α + k α / γ ν ( m − iγ β ( l β − k β / (cid:16) ( l + k/ + m (cid:17) (cid:16) ( l − k/ + m (cid:17) , (A1)where P l denotes summation over fermionic Matsubarafrequencies l = 2 πT ( n + 1 / − iµ (shifted into the com-plex plane in order to account for the chemical potential µ ). We explicitly substitute the values µ = 1, ν = 2, k = (0 , , , k ) and represent the integrand as a sum ofsimple fractions of the form l − iµ V ± i q ( ~l ± ~k/ ) + m (see e.g. Appendix A in [61] for the full derivation). Thetime-like momentum l can then be summed over usingthe identity T X l l − iǫ = i (cid:18) ǫ + µ T (cid:19) . (A2)After some algebraic manipulations, we obtain the follow-ing expression which is suitable for numerical integration: h j A ( k ) j V ( − k ) i = i + ∞ Z −∞ dl πl ∞ Z dl ⊥ π tanh µ + q m + l ⊥ + ( k / l ) T − tanh µ + q m + l ⊥ + ( k / − l ) T ++ tanh µ − q m + l ⊥ + ( k / l ) T − tanh µ − q m + l ⊥ + ( k / − l ) T ,,