Non-Abelian anomalous constitutive relations of a chiral hadronic ﬂuid ∗

. We study the constitutive relations of a chiral hadronic ﬂuid in presence of non-Abelian ’t Hooft anomalies. Analytical expressions for the covariant currents are obtained at ﬁrst order in derivatives in the chiral symmetric phase, for both two and three quark ﬂavors in the presence of chiral imbalance. We also investigate the constitutive relations after chiral symmetry breaking at the leading order.


Introduction
Quantum anomalies have important effects in relativistic hydrodynamics, as they induce extra, parity-odd terms in the constitutive relations. These anomalous contributions together with the dissipative transport effects modify the perfect fluid behavior of the constitutive relations in the form of J µ = J µ PF + J µ diss & anom for the charge currents, and T µν = T µν PF + T µν diss & anom for the energy-momentum tensor [1]. Two well known examples of anomalous transport are the chiral magnetic effect (CME), consisting in the generation of an electric current driven by and parallel to an external magnetic field [2], and the chiral vortical effect (CVE) where the electric current is triggered by fluid vorticity [3], i.e. J µ anom = σ B B µ + σ V ω µ . The two associated susceptibilities, σ B and σ V , have been computed using a wide variety of methods, for example Kubo relations [4][5][6], fluid/gravity correspondence [7][8][9][10], as well as the equilibrium partition function formalism (EPFF) [11][12][13]. The latter technique is especially convenient in systems with spontaneous symmetry breaking (SSB), as in this case the constitutive relations can be obtained directly from the expression of the Bardeen-Zumino (BZ) current [14], thus avoiding the calculation of the, usually cumbersome, Wess-Zumino-Witten (WZW) functional describing the dynamics of Nambu-Goldstone (NG) bosons [15,16]. An example of anomalous transport phenomenon appearing only in presence of SSB is the chiral electric effect, in which the electric current is generated by an external electric field in a perpendicular direction [17,18].
Here, we apply the EPFF to study the anomalous constitutive relations in non-Abelian hadronic fluids with two [14,18] and three [19] flavors. We will study first the fluid in the unbroken chiral symmetric phase. Later on, we will extend this study to the hydrodynamics in presence of SSB, and describe the dynamics of the NG bosons at the lowest order.

Equilibrium partition function formalism
We begin by giving a brief summary of the EPFF, which allows the evaluation of nondissipative (i.e., time-reversal odd) contributions to the constitutive relations [11,13,20,21]. Let us consider the theory of a time-independent Abelian U(1) gauge connection on the stationary background The system partition function is written in terms of its Hamiltonian H and the gauge charge Q as Z = Tr exp − H−µ 0 Q T 0 , with T 0 and µ 0 the equilibrium temperature and chemical potential. This quantity should be invariant under three-dimensional diffeomorphisms, Kaluza-Klein , as well as U(1) time-independent gauge transformations, modulo possible anomalies. Moreover, KK invariance demands that Z depends on the gauge fields only through the invariant combinations A 0 ≡ A 0 and A i ≡ A i − a i A 0 . Given the general form of the partition function in terms of the background metric and field, log Z = W(e σ , A 0 , a i , A i , g i j , T 0 , µ 0 ), the expectation values of the consistent current and energy-momentum tensor can be obtained from the functional derivatives [11] where g 3 ≡ det(g i j ), and thus W plays the role of a generating functional for the constitutive relations. We study the equilibrium partition function using an expansion in derivatives, which corresponds to a hydrodynamic expansion. The general form of W at zeroth and first order in derivatives for the system of Eqs. (1)-(2) are respectively given by [11,13] with P(T, µ) the pressure, and T = e −σ T 0 and µ = e −σ A 0 the out-of-equilibrium temperature and chemical potential, respectively. In addition, The zeroth order W (0) is responsible for the perfect fluid (PF) contribution to the hydrodynamic constitutive relations, which read where ε = −P + T ∂P ∂T + µ ∂P ∂µ and n = ∂P ∂µ are the energy density and charge density, respectively, and u µ = e −σ (1, 0, . . . , 0) is the local fluid velocity. The coefficients α i (T, µ) can be determined for a particular theory by inserting Eq. (6) into Eqs. (3)-(4), and comparing the results with the constitutive relations for this theory. The anomalous contributions to the constitutive relations at first order in derivatives turn out to be where the magnetic field and vorticity are B µ = 1 2 µναβ u ν A αβ and ω µ = 1 2 µναβ u ν ∂ α u β , respectively. The transport coefficients in the laboratory rest frame 1 for an ideal gas of Dirac where C = 1/(4π 2 ) and C 2 = 1/24 are related to the axial anomaly [3,8] and the gauge-gravitational anomaly [5,9], respectively. Then, one finds T 2 − C 2 and α 3 = 0. In the following we will use the EPFF extended to non-Abelian gauge fields.

Non-Abelian anomalies
We look next at a theory of chiral fermions coupled to external gauge fields with t a = t † a the Lie algebra generators. The axial anomaly is signaled by the non-invariance of the effective action Γ 0 [A L , A R ] under axial gauge transformations, leading to the anomaly where G a is the consistent anomaly, and A a (x) is the local generator of axial gauge transformations 2 . For this theory, the Bardeen form of the chiral anomaly is [ where N c is the number of colors, are the field strengths. Notice that the non-Abelian anomaly G a includes contributions from the triangle, square and pentagon diagrams, in contrast to the Abelian case in which only triangle diagram contributes. The solution of the anomaly equation can be found using differential geometry methods based on the Chern-Simons effective action and its dimensional reduction, with the result [14,18,[24][25][26]] where V µ and A µ are KK invariant vector and axial gauge fields. From now on, we neglect the term multiplying C 2 , related to the mixed gauge-gravitational anomaly.

Constitutive relations in the unbroken chiral symmetric phase
In this section we study the constitutive relations following from the effective action in Eq. (11), corresponding to the chiral symmetric phase of a QCD fluid with two and three flavors in the static geometry (1).

Covariant currents and background
While the consistent currents are computed from the functional derivatives of the effective action, the covariant currents are the ones entering in the constitutive relations. These latter are obtained by adding the BZ polynomials to the corresponding consistent currents, i.e. J µ cov = J µ cons + J µ BZ [27]. Explicit expressions for J µ BZ are provided in e.g. Refs. [14,18,28]. From W 0 = iΓ 0 and using Eqs. (3) and (4), one obtains the following equilibrium expressions [18] and vanishing values for the components J a 0 V cov = J a 0 A cov = 0 and T 00 = T i j = 0. We particularize these results for a specific case of N f = 2, 3. Due to the non-Abelian character of the charges, the maximal number of chemical potentials that can be introduced consistently equals the dimension of the Cartan subalgebra. Therefore, we take the background where A 0 0 is constant, and the generators t a are given by with σ i and λ i the Pauli and Gell-Mann matrices, respectively. Furthermore, we define the equilibrium velocity field by u µ = −e σ (1, a i ), while the chemical potentials are µ a = e −σ V a 0 (a = 0, 3, 8) and µ 5 = e −σ A 0 0 . Here, µ 5 controls the chiral imbalance of the system [29,30], whereas µ 8 = 0 for N f = 2. External gauge fields couple through the magnetic non-Abelian vector fields B µ a = 1 2 µναβ u ν V a αβ . An explicit dependence on u µ comes through the vorticity vector ω µ .
While the constitutive relations can be expressed in the basis of the generators of the Cartan subalgebra {t 0 , t 3 , t 8 }, it would be interesting to write the results also in the alternative Cartan basis formed by the conserved charges. These quantities play a fundamental role in the hydrodynamics of relativistic fluids. In the (uds) flavor sector of QCD, the conserved charges are the baryon number B, electric charge Q, and strangeness S , while S = 0 in the (ud) flavor sector. In the {B, Q, S } basis, the background defined by Eq. (15) is given by Let Ψ be a flavor doublet (triplet) of Dirac spinors made out of quarks, i.e.
with each spinor being decomposed into its chiralities according to q = (q L q R ) T . Using that J µ a V =Ψγ µ t a Ψ, we can distinguish between two (three) vector currents for N f = 2(3), i.e. J µ em = eΨγ µ QΨ, J µ B =Ψγ µ BΨ and J µ S =Ψγ µ S Ψ, corresponding to the electromagnetic, baryonic and strangeness currents, respectively. These currents, together with the isospin current J µ I =Ψγ µ I 3 Ψ, fulfill the Gell-Mann-Nishijima relation J µ em = eJ µ I + e 2 J µ B + J µ S . By using the relation between the charges {B, Q, S } and the generators t a of the Cartan subalgebra of U(N f ), we will provide explicit expressions for the constitutive relations of J µ em and T µν .

Constitutive relations for N f = 2
The constitutive relations can be computed by particularizing Eqs. (12)- (14) to the background of Eqs. (15)- (16) with N f = 2. While these relations are in equilibrium, the corresponding out-of-equilibrium expressions are obtained by Lorentz covariantization, so that in the end they will be expressed in terms of B µ a and ω µ . The result is where T µν = u µ q ν + u ν q µ . The charges {B, Q} are related to the generators {t 0 , t 3 } by In the {B, Q} basis, the background defined by Eq. (15) is given by Eq. (17) with Besides, the baryon and electromagnetic chemical potentials are defined by µ B = e −σ V B 0 and µ Q = e −σ V Q 0 . Let us assume that the electromagnetic field is the only (physical) propagating gauge field. Then, we can define the physical magnetic field by a result that implies the following relations between chemical potentials µ 0 = 1 3 µ 3 = 1 3 µ Q with µ Q = e V 0 e −σ . After using Eq. (24), the constitutive relations of J µ em and T µν turn out to be J µ em cov gives the transport coefficient associated with the CME. The absence of a CVE in the vector current for U(2) V × U(2) A contrasts with the case U(1) V × U(1) A , cf. Refs. [6,12].

Constitutive relations for N f = 3
Extending the procedure presented in the previous section [19], one finds the following Lorentz covariant expressions of the constitutive relations for The relation between the basis of conserved charges {B, Q, S } and the basis {t 0 , t 3 , t 8 } is In the {B, Q, S } basis, the background is given by Eq. (17) with and µ Q are defined as in Sec. 4.2, and the strangeness chemical potential is µ S = e −σ V S 0 . Assuming, as above, that the electromagnetic field V µ is the only propagating gauge field, one finds a result that leads to the relations µ 0 = 0 and µ 3 = √ 3µ 8 = µ Q , with µ Q = e V 0 e −σ . Finally, the constitutive relations for the electromagnetic current and energy-momentum tensor writes As in the case of N f = 2, there is no CVE in the vector current for U(3) V × U(3) A .

Lowest order constitutive relations in presence of SSB
In this section we study the phase in which chiral symmetry is spontaneously broken. We consider the breaking pattern  [15,16,35,36], where Γ 0 is the anomalous functional in the symmetric phase, cf. Eq. (11). In this expression A g = g −1 Ag + g −1 dg is the gauge field transformed by the generic element g ≡ e iξ a t a , with ξ a the NG boson fields. In our case, we make the identifications A → (A L , A R ) and g → (U, I), where I is the identity element. The matrix U is then written in terms of the conventionally normalized NG boson fields associated with the broken SU(N f ) A symmetry as U(ξ) = e iX(ξ) , where with f π ≈ 92 MeV the pion decay constant. In the case N f = 2, X(ξ) corresponds to the 2 × 2 submatrix of pions in Eq. (34) with η 8 = 0. The NG boson field ξ 0 is absent, as the U(1) A symmetry is violated in QCD by non-perturbative effects. In the following we will focus on the action at the lowest order in derivatives, that reads where P is the pressure in absence of NG bosons, and the Lagrangian L contains all dependence on the NG bosons. The latter has the following expression in terms of KK invariant fields