Anomalous transport from equilibrium partition functions

Nuclear and Particle Physics Proceedings 00 (2022) 1–5 Nuclear and Particle Physics Proceedings Anomalous transport from equilibrium partition functions ∗ Eugenio Meg´ ıas Departamento de F´ısica At´omica, Molecular y Nuclear and Instituto Carlos I de F´ısica Te´orica y Computacional, Universidad de Granada, Avenida de Fuente Nueva s/n, 18071 Granada, Spain Abstract We summarize recent advances in the application of the equilibrium partition function formalism for the study of the transport coefficients of relativistic fluids induced by quantum anomalies, at first and second order in the hydrodynamic expansion. We provide results for theories with Abelian and non-Abelian chiral fermions, and discuss some features of the corresponding constitutive relations. Keywords: Relativistic fluids, Hydrodynamics, Quantum anomalies, Anomalous transport. 1. Introduction One of the most fruitful techniques to study out-ofequilibrium systems is the hydrodynamical approach, in which it is assumed local thermodynamical equilibrium. The hydrodynamical systems obey conservation laws of the energy-momentum tensor and charged currents, and the expectation values of these quantities are written in terms of fluid variables in the so-called constitutive relations, which are organized in a derivative expansion. In the presence of quantum anomalies the currents are no longer conserved, and this has important effects in the hydrodynamic description. In addition to the perfect fluid and dissipative contributions, new extra terms appear in the constitutive relations which turn out to be of non-dissipative nature, i.e. for the charged currents hJµi=nuµ+hJµidiss & anom. Two relevant phenomena appear at first order in the hydrodynamic derivative expansion as a consequence of chiral anomalies: the chiral magnetic [1] and chiral vortical [2] effects. They consist in the generation of electric currents driven by and parallel to a magnetic field and a vorticity vector, respectively, i.e. hJµianom =σBBµ+σVωµ. The corresponding susceptibilities are parity (P) odd and time reversal ∗Talk presented at QCD22, 25th International Conference in QCD (04-07/07/2022, Montpellier - FR). Email address: [email protected] (Eugenio Meg´ ıas) (T) even, the latter implying that they cannot contribute to entropy production, i.e. ∂tsanom =0. These coefficients have been computed in a wide variety of methods, including kinetic theory [3], Kubo formulae [4], fluid/gravity correspondence [5] and equilibrium partition function (EPF) formalism [6]. In this work we will focus on the latter and address the study of the anomaly-induced contributions to the constitutive relations in both Abelian and non-Abelian gauge theories. 2. Equilibrium partition function formalism We begin by giving a brief summary of the EPF formalism introduced in [6–9]. Let us consider a relativistic invariant quantum field theory with a time independent U(1) gauge connection on the manifold ds2=Gµνdxµdxν =−e2σ(~ x)(dt +ai(~ x)dxi)2+gi j(~ x)dxidxj,(1) A=A0(~ x)dx0+Ai(~ x)dxi.(2) The partition function of the system is defined as Z= Tr exp −H−µ0Q T0, where His the Hamiltonian of the theory, and Qis the conserved charged associated to the gauge connection, while T0and µ0are the temperature and chemical potential at equilibrium. The dependence of the partition function on the fields, i.e. log Z= arXiv:2209.02169v1 [hep-th] 6 Sep 2022 Eugenio Meg´ıas /Nuclear and Particle Physics Proceedings 00 (2022) 1–5 2 W(eσ,A0,ai,Ai,gi j,T0, µ0), should be consistent with invariance under: i) 3-dim diffeomorphisms; ii) KaluzaKlein (KK) transformation [t→t+φ(~ x),ai→ai− ∂iφ(~ x)]; and iii) U(1) time-independent gauge transformations, up to gauge anomalies. In particular, KK invariance implies that the dependence in the gauge fields is only through the KK invariant combinations A0≡ A0 and Ai≡ Ai−aiA0. From the partition function of the system, one can compute the energy-momentum tensor and consistent charged currents by performing appropriate t-independent variations. In doing that, one gets [6] hJiicons =T0e−σ √g3 δW δAi ,hJ0icons =−T0eσ √g3 δW δA0 ,(3) hTi 0i=T0e−σ √g3 δW δai−A0 δW δAi!,hT00i=−T0eσ √g3 δW δσ ,(4) where g3≡det(gi j), and thus Wplays the role of a generating functional for the hydrodynamic constitutive relations. 3. Abelian anomalies and hydrodynamics We will present in this section the explicit results for the constitutive relations of a gas of massless Dirac fermions with U(1) gauge symmetry. The Lagrangian is L=−iΨγµ∇µΨ,(5) where Ψ = (ψLψR)Tis a Dirac spinor, and ∇µis the covariant derivative including the gauge field Aµ. The space-time dependent Dirac matrices are related to the Minkowski matrices by γµ(x)=eµ a(x)γa, where eµ a(x) is the vierbein. We will study the properties of the EPF of this theory at first and second order in derivatives. 3.1. Anomalous transport at first order The most general expression of the EPF at first order in the hydrodynamical expansion compatible with the symmetries mentioned in Sec. 2 is [6, 9] W(1) =Zd3x√g3i jkhα1AiFjk +α2Aifjk +α3aifjki,(6) where Fi j =∂iAj−∂jAiand fi j =∂iaj−∂jai, with coefficients αi=αi(T, ν) where ν≡µ/T, with T= e−σT0and µ=e−σA0the out-of-equilibrium temperature and chemical potential, respectively. The U(1) current and energy-momentum tensor of the ideal gas of Dirac fermions write Jµ=−ΨγµΨ,(7) Tµν =i 4Ψγµ−→ ∇ν−←− ∇νγµ+(µ↔ν)Ψ.(8) The expectation values of Jµand Tµν at equilibrium may be computed from the thermal Green’s function hTψ(−iτ, ~ x)ψ†(0,~ x0)iβ=T0Pne−iωnτG(~ x,~ x0, ωn), where ωn=2πT0(n+1/2), and Tdenotes time ordering. After performing the summation over Matsubara frequencies, one gets from a computation of hJiiand hTi 0ithe following results for the chiral magnetic and chiral vortical conductivities σB=Cµ , σV=1 2Cµ2+C2T2,(9) where the coefficients C=1/(4π2) and C2=1/24 are induced by the chiral anomaly [2, 5] and mixed gaugegravitational anomaly [10, 11], respectively. These results have been obtained in a wide variety of methods, see e.g. [1, 2, 5, 9–12]. Finally, by using the variational formulae (3)-(4) with Eq. (6), and after a comparison with the explicit expressions of the constitutive relations, one gets α1(T, ν)=−C 6ν,α2(T, ν)= −1 2C 6ν2−C2and α3(T, ν)=0 [9]. For completeness, we present below the results for the constitutive relations in the theory with symmetry group U(1)V×U(1)A, i.e. one vector and one axial current with chemical potentials (µ, µ5). These are given by [4] hJµ ai(1) =(σB)aBµ+(σV)aωµ,(a=V,A),(10) hTµνi(1) =uµqν+uνqµ,qµ=σB εBµ+σV εωµ,(11) with (σB)V=µ5 2π2,(σB)A=µ 2π2,(12) (σV)V=µµ5 2π2,(σV)A=µ2+µ2 5 4π2+T2 12 ,(13) σB ε=(σV)V, σV ε=µ5 6π2(3µ2+µ2 5)+µ5 6T2.(14) Here (σB)Vis the chiral magnetic conductivity, (σB)A describes the generation of an axial current due to a magnetic field, and (σV)V(A)is the vector(axial) vortical conductivity. σB εand σV εare chiral magnetic and vortical conductivities for energy flux, respectively. 3.2. Anomalous transport at second order Let us study the EPF at second order in the derivative expansion. The most general expression writes [6] W(2) =Zd3x√g3hM1gi j∂iT∂jT+M2gi j∂iν∂jν +M3gi j∂iν∂jT+T2 0M4fi j fi j +M5Fi jFi j +T0M6fi jFi j +M7˜ R+N1i jk∂iA0fjk +T−1 0N2i jk∂iA0Fjki,(15) where ˜ Ris the Ricci scalar in 3 dim, with Mi=Mi(T, ν) and Ni=Ni(T, ν). To get W(2) it is enough to compute hJ0i(2) and hT00i(2) including only bilinear terms Eugenio Meg´ıas /Nuclear and Particle Physics Proceedings 00 (2022) 1–5 3 ∼∂iX∂jY. The explicit expression of M7turns out to be M7=−1 144 T−1 48π2Tν2+1 48π2 1 TM2log 2, where M is the renormalization scale ( ¯ M=2−3/2eγEM). This coefficient is the relevant one for the computation of the transport coefficients presented below. The results for the rest of the coefficients in Eq. (15) are in Ref. [9]. The terms proportional to M2can be renormalized by adding an appropriate counterterm. The renormalized effective action turns out to be not invariant under a Weyl rescaling due to the existence of terms ∝log ¯ M2 T2. Collecting these terms, one can identify the anomalous contribution to the partition function, a result that leads to the trace anomaly hTµ µi=−1 24π2FµνFµν [13]. The general result of the constitutive relations contains the following terms [14] hJµi(2) ⊃υ1PµαuνRνα +υ2Pµα∇νFνα ,(16) hTµνi(2) ⊃Tκ1Rhµνi+κ2uαuβRhµανiβ+κ3∇hµ∇νiν.(17) After using the variational formulae with W(2), one gets κ1=T 72 +1 24π2 µ2 T, κ2=2κ1, κ3=−µ 12π2.(18) The results for υiare provided in Ref. [9]. The results presented here for κ1,2are in agreement with Ref. [15] after setting µ=0. On the other hand, κ3and υ2 have been computed in a holographic model in 5 dim in Refs. [5, 12, 16], leading to the same parametric dependence for µT. Finally, let us mention that the non-dissipative coefficients calculated above are P-even and T-even, while the second order coefficients that are P-odd and T-even vanish, i.e. N1,2=0. 4. Non-Abelian anomalies and hydrodynamics We will study in this section the constitutive relations within a theory with a non-Abelian chiral anomaly. 4.1. The chiral anomaly Let us consider a theory of chiral fermions with symmetry group U(Nf)×U(Nf), with Lagrangian L=iψLγµ(∂µ−itaAa Lµ)ψL+iψRγµ(∂µ−itaAa Rµ)ψR,(19) where ta=t† aare the Lie algebra generators. The chiral anomaly is signaled by the non-invariance of the effective action iΓ = Wunder axial gauge transformations. This leads to the anomaly equation Aa(x)Γ[V,A]= Ga[V,A], where Gais the consistent anomaly, and Aa(x) is the local generator of axial transformations. We have defined the vector and axial gauge fields (V,A) by AL≡ V−Aand AR≡ V+A. The anomaly also leads to the (non)-conservation law DµJµ a(x)cons = Ga[V,A]. As a consequence, the chiral anomaly has effects in the hydrodynamic constitutive relations, as it has been already discussed. The Bardeen form of the non-Abelian anomaly is [17] Ga[V,A]=iNc 16π2µνρσ × ×TrntaVµνVρσ +1 3AµνAρσ −32 3AµAνAρAσ +8 3i(AµAνVρσ +AµVρσAν+VρσAµAν)o,(20) where Ncis the number of colors, while (Vµν,Aµν) are the field strengths. Gaincludes triangle, square and pentagon one-loop diagrams, in contrast to the Abelian case in which only triangle diagrams contribute. 4.2. Constitutive relations The solution of the anomaly equation can be found by using differential geometry methods based on the Chern-Simons effective action, with the result [18, 19] Γ[V,A,G]=−Nc 32π2Zdt d3x√g3i jk × ×Tr32 3i V0AiAjAk+4 3(A0Ai+AiA0)Ajk +4(V0Ai+AiV0)Vjk +8 3A2 0+3V2 0Ai∂jak.(21) We have neglected the terms related to the mixed gaugegravitational anomaly ∼C2, as these contributions demand a careful study of the Riemann tensor effects in the anomaly polynomial, see e.g. Ref. [20]. In the (uds) flavor sector of QCD, the conserved charges are the baryon number B, electric charge Q, and strangeness S. Then, instead of working in the basis of the generators of the Cartan subalgebra for Nf=3, {t0,t3,t8}, it is more convenient to work in the {B,Q,S} basis, for which we can take the following background Vµ(~ x)=VBµ(~ x)B+VQµ(~ x)Q+VSµ(~ x)S,A0=ABBand Ai=0. Then we can distinguish between the three vector currents Jµ B= ΨγµBΨ,Jµ em =eΨγµQΨand Jµ S= ΨγµSΨ, corresponding to the baryonic, electromagnetic and strangeness currents, respectively. In addition, we can define the corresponding chemical potentials as µq=e−σVq0(q=B,Q,S) and µ5=e−σA0 0. Here, µ5 controls the chiral imbalance of the system [21]. The covariant currents are defined by adding to the consistent currents the Bardeen-Zumino (BZ) terms, i.e. Jµ cov =Jµ cons +Jµ BZ [22]. These are the physically relevant currents, as can be argued using the notion of anomaly inflow [23]. To compute the constitutive relations, let us Eugenio Meg´ıas /Nuclear and Particle Physics Proceedings 00 (2022) 1–5 4 assume that the electromagnetic field is the only propagating field. Then, we can define the physical magnetic field as Bµ=1 2µναβuνVαβ, where the physical potential is Vµand its KK invariant form is Vµ, i.e. V0=V0and Vi=Vi−aiV0. Then one has VBµ=0=VSµand VQµ=eVµ, and the constitutive relations write [24, 25] hJµ emicov =e2Nc 3√6π2µ5Bµ,(22) qµ=Nc 3√6π2µ5"eµQBµ+ µ2 Q−1 4µ2 5!ωµ#,(23) where hTµνi=uµqν+uνqµ. Notice that hJµ emicov receives contribution only from the chiral magnetic conductivity. The absence of a chiral vortical effect in the U(3)V× U(3)Acase contrasts with the situation in the Abelian U(1)V×U(1)Amodel, cf. Sec. 3.1 and Refs. [4, 26]. 5. Conclusions We have studied the anomaly-induced transport effects in relativistic fluids by using the EPF formalism. By construction, this method can only account for nondissipative effects, i.e. transport coefficients multiplying quantities that survive in equilibrium. In particular, we have characterized the effects induced by external magnetic fields and fluid vorticity. In the Abelian case, the non-dissipative contributions at first order are P-odd and T-even. However, the situation is slightly different at second order, where the P-odd coefficients vanish, and the nonzero non-dissipative coefficients turn out to be P-even and T-even. In the case of non-Abelian anomalies, we have found that there are contributions to the constitutive relations at first order from the physical magnetic field, but no contribution from the vorticity. While the present study is relevant for the chiral symmetric phase of QCD at high temperatures, the computation has been extended in Refs. [18, 19] to the case of spontaneous symmetry breaking, leading to relevant information about the hydrodynamics of the Goldstone bosons interacting with external fields, with application to QCD at low temperatures. Finally, let us remark that this formalism can be used in a wide variety of systems, including other sectors of the Standard Model [27], superfluids [28], and condensed matter systems [29, 30]. Acknowledgements This work is based on Ref. [9], co-authored with M. Valle, and Refs. [18, 19], co-authored with J.L. Ma˜ nes, M. Valle and M. ´ A. V´ azquez-Mozo. I would like to thank them for collaboration and enlightening discussions. I also thank the ICTP South American Institute for Fundamental Research (SAIFR), S˜ ao Paulo, Brazil, and its Program on New Directions in Particle Physics 05-23/09/2022, for hospitality and partial financial support during the process of writing this manuscript. 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