Quantum anomalies give rise to new transport phenomena. In particular a
magnetic field can induce an anomalous current via the chiral magnetic effect
and a vortex in the relativistic fluid can also induce a current via the chiral
vortical effect. The related transport coefficients can be calculated via Kubo
formulas. We evaluate the Kubo formula for the anomalous vortical conductivity
at weak coupling and show that it receives contributions proportional to the
gravitational anomaly coefficient. The gravitational anomaly gives rise to an
anomalous vortical effect even for an uncharged fluid.Comment: 4 pages, 1 figure, v2: essentially the version published in Physcial
  Review Letter

American Physical Society

Landsteiner, Karl

Megías Fernández, Eugenio

Pena-Benítez, Francisco

English

arXiv

Quantum anomalies give rise to new transport phenomena. In particular, a magnetic field can induce an anomalous current via the chiral magnetic effect and a vortex in the relativistic fluid can also induce a current via the chiral vortical effect. The related transport coefficients can be calculated via Kubo formulas. We evaluate the Kubo formula for the anomalous vortical conductivity at weak coupling and show that it receives contributions proportional to the gravitational anomaly coefficient. The gravitational anomaly gives rise to an anomalous vortical effect even for an uncharged fluid

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Gravitational Anomaly and Transport PhenomenaKarl Landsteiner, Eugenio Megı´as, and Francisco Pena-BenitezInstituto de Fı´sica Teo´rica UAM/CSIC, C/ Nicola´s Cabrera 13-15, Universidad Auto´noma de Madrid, Cantoblanco,28049 Madrid, Spain(Received 5 April 2011; published 6 July 2011)Quantum anomalies give rise to new transport phenomena. In particular, a magnetic field can induce ananomalous current via the chiral magnetic effect and a vortex in the relativistic fluid can also induce acurrent via the chiral vortical effect. The related transport coefficients can be calculated via Kuboformulas. We evaluate the Kubo formula for the anomalous vortical conductivity at weak coupling andshow that it receives contributions proportional to the gravitational anomaly coefficient. The gravitationalanomaly gives rise to an anomalous vortical effect even for an uncharged fluid.DOI: 10.1103/PhysRevLett.107.021601 PACS numbers: 11.15.q, 11.10.Wx, 11.40.q, 67.10.JnIntroduction.—The effects quantum anomalies have onthe hydrodynamics of otherwise conserved currents haverecently received much attention. Two such effects areknown: an anomalous magnetic effect [1] and an anoma-lous vortical effect [2,3]. In the first an (external) magneticfield induces a current via the axial anomaly, whereas thesecond is the generation of a current due to the presence ofa vortex in the charged relativistic fluid. These effects havebeen argued to lead to observable event by event parityviolation and a charge separation effect in noncentralheavy ion collision at RHIC and LHC [4]. In the hydro-dynamic constitutive relations these effects lead to theexistence of a new class of transport coefficients.A first principle calculation of transport coefficients ispossible via Kubo formulas. The Kubo formula for theanomalous magnetic conductivity has been derived andapplied in [5] whereas the one for the anomalous vorticalconductivity has been established recently in [6]. They areBAB ¼ limkn!0Xij"ijni2knhJiAJjBij!¼0; (1)VA ¼ limkn!0Xij"ijni2knhJiAT0jij!¼0; (2)where JA are the (anomalous) currents and T is theenergy momentum tensor. These Kubo formulas allowthe calculation of the transport coefficients in the constit-utive relationsJiA ¼ BAB"ijk@jABk þ VA "ijk@jvk; (3)where ABk are the spatial components of a collection ofgauge fields and vk is the local fluid velocity. See [6] for adiscussion. We note that we can substitute the fluid veloc-ities by the gravitomagnetic potential. To do sowe go to therest frame of the fluid defined by u ¼ ð1; 0; 0; 0Þ butswitch on a gravitomagnetic field in the metric according tods2 ¼ dt2 þ 2 ~Agd~xdt d~x2: (4)Using this metric we can compute the local fluid velocityu ¼ ð1; ~AgÞ such that ~v ¼ ~Ag. All these expressionsare valid only up to first order in the external fieldsAAk , vk.Plugging this into the constitutive relation (3) and notingthat Aig sources T0i leads to the Kubo formula for thevortical conductivity.We will now evaluate the Kubo formulas (1) and (2) for atheory of N free right-handed fermions f transformingunder a global symmetry group G generated by matricesðTAÞfg. We denote the generators in the Cartan subalgebraby HA. Chemical potentials A can be switched on only inthe Cartan subalgebra. Furthermore, the presence of thechemical potentials breaks the group G to a subgroup G^.Only the currents that lie in the unbroken subgroup areconserved (up to anomalies) and participate in the hydro-dynamics. The chemical potential for the fermionf is givenby f ¼ PAqfAA, where we write the Cartan generatorHA ¼ qfAfg in terms of its eigenvalues, the charges qfA.The unbroken symmetry group G^ is generated by the matri-ces TfAg fulfilling TfAgg ¼ fTfAg. There is no summationover indices in the last expression. From now on we willassume that all currents ~JA lie in directions indicated in thepreceding equation.We define the chemical potential throughboundary conditions on the fermion fields around the thermalcircle [7], fðÞ ¼ effð Þ with  ¼ 1=T.Therefore the eigenvalues of@ are i ~!n þf for the fermionspecies f with ~!n ¼ Tð2nþ 1Þ the fermionic Matsubarafrequencies. A convenient way of expressing the currents is interms of Dirac fermions and writingJiA ¼XNf;g¼1TgAfgiPþf; (5)T0i ¼ i2XNf¼1fð0@i þ i@0ÞPþf; (6)where we used the chiral projector P ¼ 12 ð1 5Þ. Thefermion propagator isPRL 107, 021601 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending8 JULY 20110031-9007=11=107(2)=021601(4) 021601-1  2011 American Physical SocietySðqÞfg ¼fg2Xt¼tði ~!f; ~qÞPþq^t ; (7)tði ~!f; qÞ ¼ 1i ~!f  tEq; (8)with i ~!f ¼ i ~!n þf, q^t ¼ ð1; tq^Þ, q^ ¼ ~qEq and Eq ¼ j ~qj.We can easily include left-handed fermions as well. Theycontribute in all our calculations in the sameway as the right-handed ones up to a relative minus sign.Evaluation of Kubo formulas.—Wewill address in detailthe computation of the vortical conductivities Eq. (2) andsketch only the calculation of the magnetic conductivitiessince the latter one is a trivial extension of the calculationof the chiral magnetic conductivity in [5].Vortical conductivity.—The vortical conductivity is de-fined from the retarded correlation function of the currentJiAðxÞ (5), and the energy momentum tensor or energycurrent T0jðx0Þ (6), i.e.,GVA ðx x0Þ ¼ 12"ijni	ðt t0Þh½JiAðxÞ; T0jðx0Þi: (9)Going to Fourier space, one can evaluate this quantity asGVA ðkÞ¼14XNf¼1TfAf1X~!fZ d3qð2Þ3"ijntr½SffðqÞiSffðqþkÞð0qjþji ~!fÞ: (10)The vertex of the two quarks with the graviton is fg,and therefore we find only contributions from the diagonalpart of the group G^. Our metric is g ¼ diagð1;1;1;1Þ. We can split GVA into two contributionsGVA ðkÞ ¼ GVA;ð0jÞðkÞ þGVA;ðj0ÞðkÞ; (11)which correspond to the terms 0qj and ji ~!f in Eq. (10),respectively. We focus first on the computation of GVA;ð0jÞ.The integrand of Eq. (10) for GVA;ð0jÞ can be written asIVA;ð0jÞ ¼14qjXt;u¼"ijntr½i0Pþtði ~!f; ~qÞuði ~!fþ i!n; ~qþ ~kÞq^t dðqþ kÞu: (12)From a computation of the Dirac trace in Eq. (12) onehas two contributions"ijntr½i0ab ¼ 4"ijnðaib0 þ a0biÞ; (13)"ijntr½i05ab ¼ 4iðajbn  anbjÞ: (14)Using Eqs. (12)–(14) one can express GVA;ð0jÞðkÞ asGVA;ð0jÞðkÞ ¼18XNf¼1TfAf1X~!fZ d3qð2Þ3 qj Xt;u¼"ijntqiEqþ u ki þ qiEqþkþ i tuEqEqþkðqjkn  qnkjÞtði ~!f; ~kÞuði ~!f þ i!n; ~qþ ~kÞ:(15)Note that due to the antisymmetric tensor "ijn, the twoterms proportional to qi inside the bracket in Eq. (15)vanish. Regarding the term "ijnqjki, it leads to a contribu-tion"ijnkjki after integration in d3q, which is zero. Thenthe only term which remains is the one not involving "ijn.We can now perform the sum over fermionic Matsubarafrequencies. One has1X~!ftði ~!f; ~qÞuði ~!f þ i!n; ~qþ ~pÞ¼ tnðEq  tfÞ  unðEqþk  ufÞ þ 12 ðu tÞi!n þ tEq  uEqþk ; (16)where nðxÞ ¼ 1=ðex þ 1Þ is the Fermi-Dirac distributionfunction. In Eq. (16) we have considered that !n ¼ 2Tnis a bosonic Matsubara frequency [5]. After doing theanalytic continuation, which amounts to replacing i!n byk0 þ i" in Eq. (16), one getsGVA;ð0jÞðkÞ¼  i8XNf¼1TfAfZ d3qð2Þ3~q2kn  ð ~q  ~kÞqnEqEqþk Xt;u¼unðEq  tfÞ  tnðEqþk  ufÞ þ 12 ðt uÞk0 þ i"þ tEq  uEqþk :(17)The term proportional to  12 ðt uÞ corresponds to thevacuum contribution, and it is ultraviolet divergent. Byremoving this term the finite temperature and chemicalpotential behavior is not affected, and the result becomesultraviolet finite because the Fermi-Dirac distributionfunction exponentially suppresses high momenta. By mak-ing both the change of variable ~q!  ~q ~k and theinterchange u! t and t! u in the part of the inte-grand involving the term tnðEqþk  ufÞ, one can ex-press the vacuum subtracted contribution of Eq. (17) asG^VA;ð0jÞðkÞ ¼i8knXNf¼1TfAfZ d3qð2Þ31EqEqþk~q2  ð ~q ~kÞ2~k2 Xt;u¼unðEq fÞ þ nðEq þfÞk0 þ i"þ tEq þ uEqþk : (18)The result has to be proportional to kn, so to reach thisexpression we have replaced qn by ð ~q  ~kÞkn= ~k2 in Eq. (17).PRL 107, 021601 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending8 JULY 2011021601-2At this point one can perform the sum over u and theintegration over angles. Then one gets the final resultG^VA;ð0jÞðkÞ ¼i162knk2ðk2  k20ÞXNf¼1TfAfZ 10dqqfV ðqÞ1þ 18qkXt¼½k20  k2 þ 4qðqþ tk0Þ log2t  ðqþ kÞ22t  ðq kÞ2; (19)where t ¼ k0 þ i"þ tEq, and fV ðqÞ ¼ nðEq fÞ þnðEq þfÞ. The steps to compute GVA;ðj0Þ in Eq. (11) aresimilar. In this case the Dirac trace leads to a differenttensor structure, in which the only contribution comes fromthe trace involving 5, i.e., "ijntr½ij5ab ¼8iðanb0  a0bnÞ. The sum over fermionic Matsubarafrequencies involves an extra i ~!f. Following the sameprocedure as explained above, the vacuum subtracted con-tribution writesG^VA;ðj0ÞðkÞ ¼ i322knk3XNf¼1TfAfZ 10dqXt¼fVt ðq; k0Þ4tqkk0  ðk2  k20Þð2qþ tk0Þ log2t  ðqþ kÞ22t  ðq kÞ2; (20)where fVt ðq; k0Þ ¼ qfV ðqÞ þ tk0nðEq þ tfÞ. The resultfor G^VA ðkÞ writes as a sum of Eqs. (19) and (20), accordingto Eq. (11). From these expressions one can compute thezero frequency, zero momentum, limit. Sincelimk!0limk0!0Xt¼log2t  ðqþ kÞ22t  ðq kÞ2¼ 2kq; (21)the relevant integrals areZ 10dqqfV ðqÞ ¼Z 10dqfVt ðq; k0 ¼ 0Þ¼ ðfÞ22þ 26T2: (22)Finally it follows from Eqs. (19) and (20) that the zerofrequency, zero momentum, vortical conductivity writesVA ¼182XNf¼1TfAfðfÞ2 þ 23T2¼ 1162XB;CtrðTAfHB;HCgÞBC þ 223T2trðTAÞ:(23)Both G^VA;ð0jÞ and G^VA;ðj0Þ lead to the same contribution inVA .Equation (23) constitutes our main result. The term involv-ing the chemical potentials is induced by the chiral anom-aly. More interesting is the termT2 which is proportionalto the gravitational anomaly [8] as we will show in thediscussion section. The Matsubara frequencies ~!n ¼Tð2nþ 1Þ generate a dependence onT in the final resultas compared to the chemical potentials, and then no factorsof  show up for the term T2 in Eq. (23). Left-handedfermions contribute in the same way but with a relativeminus sign. Left and right-handed fermions do not mix.If instead of having taken the zero momentum limit atzero frequency, one took the zero frequency limit at zeromomentum, the result would be 1=3 of the result quoted inEq. (23). The same factor appears in the magnetic con-ductivity when one interchanges the two limits [5].Magnetic conductivity.—The magnetic conductivityin the case of a vector and an axial Uð1Þ symmetrywas computed at weak coupling in [5]. Following thesame method, we have computed it for the unbroken(non-Abelian) symmetry group G^. The relevant Greenfunction isGBAB ¼12Xf;gTgAfTfBg1X~!fZ d3qð2Þ3 "ijn tr½SffðqÞiSffðqþ kÞj: (24)The evaluation of this expression is exactly as in [5] so weskip the details. The result isBAB ¼142XNf;g¼1TfAgTgBff ¼ 182XCtrðTAfTB;HCgÞC:(25)No contribution proportional to the gravitational anomalycoefficient is found in this case.Discussion.—In vacuum the anomaly appears on thelevel of three point functions. In the presence of externalsources for the energy momentum tensor and the currentsthis is conveniently expressed through [8]rJA ¼ "dABC322FBFC þbA7682RR:(26)The anomaly coefficients are defined bydABC ¼ 12½trðTAfTB; TCgÞR  trðTAfTB; TCgÞL; (27)bA ¼ trðTAÞR  trðTAÞL; (28)where the subscripts R, L stand for the contributions ofright-handed and left-handed fermions.We have computed the magnetic and vortical conduc-tivity at weak coupling and we find contributions that areproportional to the anomaly coefficients (27) and (28).Nonzero values of these coefficients are a necessary andsufficient condition for the presence of anomalies [9].Therefore the nonvanishing values of the transportcoefficients (1) and (2) have to be attributed to the presenceof chiral and gravitational anomalies.PRL 107, 021601 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending8 JULY 2011021601-3Since the gravitational anomaly is fourth order in de-rivatives it is a bit surprising to find it contributing to firstorder transport coefficients. One possible intuitive expla-nation one could think of is that the gravitational field inthe presence of matter gives rise to a fluid velocity u, e.g.,through frame dragging effects, and that this might effec-tively reduce the number of derivatives that enter in thehydrodynamic expansion.The holographic calculation in AdS/CFT [6] did notshow a contribution proportional to T2. This is not surpris-ing since only a holographic gauge Chern-Simons termwas included. Holographic modeling of the gravitationalanomaly calls, however, also for inclusion of a mixedgauge-gravitational Chern-Simons term of the formA ^ R ^ R [10].We find a nonvanishing vortical conductivity propor-tional to T2 even in an uncharged fluid. In [11] similarterms in the vortical conductivities have been argued for asundetermined integration constants without any relation tothe gravitational anomaly.It is also interesting to specialize our results to the caseof one vector and one axial current with chemical poten-tials R ¼ þA, L ¼ A, charges qRV;A ¼ ð1; 1Þand qLV;A ¼ ð1;1Þ for one right-handed and oneleft-handed fermion. We find (for a vector magnetic field)BVV ¼A22; BAV ¼22; VV ¼A22;VA ¼2 þ2A42þ T212:(29)Here BVV is the chiral magnetic conductivity [5], BAVdescribes the generation of an axial current due to a vectormagnetic field [12], VV is the vector vortical conductivity,VA is the axial vortical conductivity and the only onesensitive to the gravitational anomaly.In [13] enhanced production of high spin hadrons (es-pecially  baryons) perpendicular to the reaction planein heavy ion collisions has been proposed as an observa-tional signature for the chiral separation effect. Threesources of chiral separation have been identified: the anom-aly in vacuum, the magnetic and the vortical conductivitiesof the axial current JA . Of these the contribution of thevortical effect was judged to be subleading by a relativefactor of 104. The T2 term in (29) leads, however, to asignificant enhancement. If we take  to be the baryonchemical potential  10 MeV, neglectA as in [13] andtake a typical RHIC temperature of T ¼ 350 MeV, we seethat the temperature enhances the axial chiral vorticalconductivity by a factor of the order of 104. We expectthe enhancement at the LHC to be even higher due to thehigher temperature.Beyond applications to heavy ion collisions leading tocharge and chiral separation effects [13,14] it is tempting tospeculate that the new terms in the chiral vortical conduc-tivity might play a role in the early Universe. Indeed it hasbeen suggested before that the gravitational anomaly mightgive rise to Lepton number generation, e.g., in [15]. Thelepton number separation due to the gravitational anomalycould contribute to generate regions with nonvanishinglepton number.We thank D. Kharzeev, A. Rebhan, and A. Schmitt forEmail conversation. We also would like to thank ESIVienna for hospitality during the workshop on AdSHolography and Quark Gluon Plasma. This work hasbeen supported by Plan Nacional de Altas Energı´asFPA2009-07908, CPAN (CSD2007-00042), Comunidadde Madrid HEP-HACOS S2009/ESP-1473.[1] K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys.Rev. D 78, 074033 (2008).[2] D. Kharzeev and A. Zhitnitsky, Nucl. Phys. A797, 67(2007).[3] J. Erdmenger, M. Haack, M. Kaminski, and A. Yarom, J.High Energy Phys. 01 (2009) 055; N. Banerjee, J.Bhattacharya, S. Bhattacharyya, S. Dutta, R.Loganayagam, and P. Surowka, J. High Energy Phys. 01(2011) 094; D. T. Son and P. Surowka, Phys. Rev. Lett.103, 191601 (2009).[4] B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C 81,054908 (2010); B. I. Abelev et al. 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Gravitational Anomaly and Transport

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