Anomalous gravitomagnetic moment and non-universality of the axial vortical effect at finite temperature
aa r X i v : . [ h e p - t h ] F e b Anomalous gravitomagnetic moment and non-universality of the axial vortical effectat finite temperature
M. Buzzegoli ∗ Universit´a di Firenze and INFN Sezione di Firenze,Via G. Sansone 1, I-50019 Sesto Fiorentino (Firenze), Italy
Dmitri E. Kharzeev † Center for Nuclear Theory, Department of Physics and Astronomy,Stony Brook University, New York 11794, USA andDepartment of Physics and RIKEN-BNL Research Center,Brookhaven National Laboratory, Upton, New York 11973-5000, USA
The coupling between the spin of a massive Dirac fermion and the angular momentum of themedium, i.e. the gravitomagnetic moment, is shown here to be renormalized by QED interactionsat finite temperature. This means that the anomalous gravitomagnetic moment (AGM) does notvanish, and implies that thermal effects can break the Einstein equivalence principle in quantumfield theory, as argued previously. We also show that the AGM causes radiative corrections to theaxial current of massive fermions induced by vorticity in quantum relativistic fluids, similarly to theprevious findings for massless fermions. The radiative QCD effects on the AGM should significantlyaffect the production of polarized hadrons in heavy-ion collisions.
I. INTRODUCTION
Recent experiments [1–3] at the Relativistic Heavy Ion Collider (RHIC) have opened the possibility to study theeffects of vorticity and magnetic field on the production of polarized hadrons in relativistic heavy-ion collisions, see [4]for a recent review. The coupling between the spin and vorticity of the quark gluon plasma [5–7] induces a polarizationof the hadrons emitted from the fluid, which can be measured in experiments. The spin and the polarization of aspin 1 / h b j µ A i = n A u µ + W A ω µ T . where n A is the density of axial charge, u is the fluid velocity, ω is the vorticity of the medium, and T is thetemperature. For free massless fermions the AVE conductivity W A is found to be [8–10] W A = T µ + µ π T, where µ is the vector chemical potential and µ A is the axial chemical potential.The AVE shares many similarities with other non-dissipative macroscopic quantum effects, such as the ChiralMagnetic Effect (CME) [6, 13, 14] and the Chiral Vortical Effect (CVE) [15, 16]. However, even though both CMEand CVE originate from the chiral anomaly [17], the same has not been yet conclusively established for the AVE, ∗ matteo.buzzegoli@unifi.it † [email protected] and its anomalous origin may be questioned, see discussion in [12, 18–24]. First, both the axial current and thevorticity are axial vectors, and therefore AVE does not require parity breaking. Second, the AVE conductivity isnot protected against radiative corrections: explicit calculations in massless quantum electrodynamics (QED) andquantum chromodynamics (QCD) show that AVE conductivity is renormalized by interactions [25, 26]. FIG. 1. The Feynman diagram describing the radiative corrections to AVE conductivity. Red and blue lines represent rightand left chiralities of the fermions, and the green arrows represent the photon polarization.
This correction only affects the finite-temperature part of the AVE conductivity, while the parts proportional tothe axial chemical potential and the vector chemical potential are unaltered by interactions. Indeed, at second orderin QED coupling constant e , the AVE conductivity for a massless fermion is given by [25, 26] W A = (cid:18)
16 + e π (cid:19) T . This correction is given solely by the diagram in Fig. 1. Higher order corrections are non-vanishing only if they aregiven by diagrams in which the axial current is inserted through the anomalous triangle subdiagram [26].The radiative corrections to AVE in QED can be interpreted as driven by the interaction of a graviton with thephoton cloud surrounding the fermion. Indeed, we may understand the coupling between the chirality of fermionsand rotation of the medium induced by this diagram as follows. Suppose the lower fermion line inside the loop inFig. 1 has a right chirality. Then it couples more effectively to photons having a polarization parallel to the fermionspin. Following the loop in the figure, those photons then interact with the rotation of the medium (represented bythe insertion of stress-energy tensor in the diagram) forcing their polarization to rotate – so when they couple tothe fermion in the loop again, they can induce the chirality flip of the fermion, and transform it into a left-handedone. Note that this requires the presence of photons in the rotating fluid, which in thermal equlibrium implies afinite temperature. At zero temperature, the photon cloud is part of the coherent state of a charged fermion, andthus rotates together with it. At finite temperature, the thermal bath possesses its own photons, and mixing betweenthese photons and the photons that form the coherent cloud of the fermion can cause the rotation of the fermion’spolarization.In analogy with the magnetic moment, the quantity that describes the coupling of a fermion’s spin to rotation isa gravitomagnetic moment. The picture discussed above thus suggests that we can describe the radiative correctionto this quantity as an Anomalous Gravitomagnetic Moment (AGM), as Fig. 1 is analogous to the diagram describ-ing the anomalous magnetic moment in QED. In this work we validate the connection between the AGM and theradiative corrections to the AVE for the case of a massive fermion. We show that the gravitomagnetic moment of amassive fermion in thermal equilibrium is indeed anomalous, and that the radiative corrections to the AVE in the lowtemperature regime are due to the gravitomagnetic moment.The emergence of the AGM may seem surprising, as the Einstein equivalence principle forbids the appearanceof an anomalous spin-rotation coupling [27]. In the language of quantum field theory, the equivalence principle istranslated into the Lorentz invariance of the local coupling of gravitational fields to the stress-energy tensor [28].This is why the presence of an AGM is excluded for both elementary [28] and composite [29] particles. However,an explicit renormalization of stress-energy tensor for massive QED at finite temperature has demonstrated thatthe effects of temperature break the weak equivalence principle (i.e., gravitational and inertial mass are no longerequivalent) [30, 31]. The equivalence principle violation can be ascribed to the fact that a finite temperature breaksthe Lorentz invariance of the vacuum and that, in the presence of a thermal bath, one can genuinely discern if thesystem is under acceleration or under the effect of a gravitational field by making measurements in reference to thethermal bath. Thus, we can expect, and indeed have found, the emergence of the AGM at finite temperature.This paper is organized as follows. In Sec. II we evaluate the AVE conductivity in the low temperature regime fora non-relativistic particle and show that the radiative corrections to this conductivity can be quantitatively derivedfrom the anomalous gravitomagnetic moment. In Sec. III we introduce the gravitomagnetic moment of a Diracfermion and evaluate the radiative correction to it at one-loop level in finite temperature QED. In Sec. IV we use thescattering theory in linearized gravity to obtain a formula which connects the gravitomagnetic moment to the matrixelements of the stress-energy tensor. A summary and a discussion of the results are given in Sec. V. The details on therenormalization of the stress-energy tensor at finite temperature and the evaluation of the related Feynamn diagramsare given respectively in appendix C and D. In App. A we review how finite temperature effects lead to a differencebetween inertial and gravitational mass.
II. RADIATIVE CORRECTIONS TO THE AXIAL VORTICAL EFFECT FOR A MASSIVE FIELD
In this section we evaluate the radiative corrections to the axial vortical effect (AVE) conductivity W A for amassive Dirac field in a low temperature regime. We obtain the AVE conductivity starting from the correspondingKubo formula that involves the thermal correlator between the stress-energy tensor and the axial current operators.As any quantum operator can be written in terms of its matrix elements by expanding it in multi-particle states [32],it then follows that the AVE conductivity can be obtained by considering the radiative corrections to the followingmatrix elements: h p ′ , s ′ | b T µν (0) | p, s i , h p ′ , s ′ | b j µ A (0) | p, s i , where b T µν is the stress-energy tensor, b j A is the axial current, and | p, s i is the state representing a single fermionwith momentum p and spin s . The main strategy we adopt to evaluate the radiative corrections of W A is to includethe effects of the interactions by renormalazing the matrix elements above using finite temperature QED. Afterrenormalization, the formula for the AVE conductivity is formally identical to the one in the free field case, the onlydifference being the values of the renormalized matrix elements. In particular, the thermal correlator will act solelyon the fermionic creation and annihilation operators and can be readily written down. An equivalent approach toevaluating statistical averages is described in the textbook [33], where the density matrix is written in terms of its“in-picture” matrix elements. In this section we derive the conductivity W A , while the renormalization of the matrixelements is performed in App. C.The AVE conductivity W A for a massive Dirac field is given by the thermal correlator between the axial currentand the total angular momentum of the system and can be written as [23] W A = 2 Z β d τβ Z d x x h b T ( − i τ, x ) b j (0) i T,c , (1)where the symbol h· · ·i T denotes thermal averages performed in the rest frame of thermal bath u = (1 , ) with thefamiliar homogeneous global equilibrium density operator in the grand-canonical ensemble b ρ = 1 Z exp[ − β ( x ) · b P ]where b P µ is the total four-momentum of the system. The subscript c on the thermal average in (1) denotes theconnected part of the correlator, that is, for the simplest case of two operators: h b O b O i c ≡ h b O b O i − h b O ih b O i . As mentioned in the beginning of this section, the stress-energy tensor can be decomposed in the multi-particleHilbert space basis and can be written as a combination of creation and annihilation operators. We thus denote with b a † τ and b b † τ the creation operators of, respectively, the particle and anti-particle state of Dirac field with momentum p and spin τ , covariantly normalized as: { b a σ ( q ) , b a † τ ′ ( q ′ ) } = 2 ε q δ ττ ′ δ ( q − q ′ ) , { b b σ ( q ) , b b † τ ′ ( q ′ ) } = 2 ε q δ ττ ′ δ ( q − q ′ ) , where ε q = p q + m . The spinors for particle u τ ( q ) and for anti-particle v τ ( q ) are normalized as u † τ ( q ) u τ ′ ( q ) = δ ττ ′ , v † τ ( q ) v τ ′ ( q ) = δ ττ ′ . Furthermore, taking advantage of the fact that b T µν is an additive operator, it is uniquely determined by [32]: b T µν ( x ) = 1(2 π ) X τ,τ ′ Z d q ′ Z d q b a † τ ′ ( q ′ ) b a τ ( q )e i( q − q ′ ) · x ¯ u τ ′ ( q ′ ) A M µν ++ ( q, q ′ ) AB u τ ( q ) B ++ 1(2 π ) X τ,τ ′ Z d q ′ Z d q b a † τ ′ ( q ′ ) b b τ ( q )e − i( q + q ′ ) · x ¯ u τ ′ ( q ′ ) A M µν + − ( q, q ′ ) AB v τ ( q ) B + 1(2 π ) X τ,τ ′ Z d q ′ Z d q b b † τ ′ ( q ′ ) b a τ ( q )e i( q + q ′ ) · x ¯ v τ ′ ( q ′ ) A M µν − + ( q, q ′ ) AB u τ ( q ) B + 1(2 π ) X τ,τ ′ Z d q ′ Z d q b b † τ ′ ( q ′ ) b b τ ( q )e − i( q − q ′ ) · x ¯ v τ ′ ( q ′ ) A M µν −− ( q, q ′ ) AB v τ ( q ) B + b T µν Photons ( x ) , (2)where b T µν Photons ( x ) contains only terms in the creation and annihilation operators for the photons, the sum over thespinor indices A and B is intended, and the matrices M are inferred from the following stress-energy tensor matrixelements: ¯ u τ ′ ( q ′ ) M µν ++ ( q, q ′ ) u τ ( q ) ≡h | b a τ ′ ( q ′ ) b T µν (0) b a † τ ( q ) | i = h q ′ , τ ′ | b T µν (0) | q, τ i , ¯ u τ ′ ( q ′ ) M µν + − ( q, q ′ ) v τ ( q ) ≡h | b a τ ′ ( q ′ ) b T µν (0) b b † τ ( q ) | i = h q ′ , τ ′ | b T µν (0) | q, τ i , ¯ v τ ′ ( q ′ ) M µν − + ( q, q ′ ) u τ ( q ) ≡h | b b τ ′ ( q ′ ) b T µν (0) b a † τ ( q ) | i = h q ′ , τ ′ | b T µν (0) | q, τ i , ¯ v τ ′ ( q ′ ) M µν −− ( q, q ′ ) v τ ( q ) ≡h | b b τ ′ ( q ′ ) b T µν (0) b b † τ ( q ) | i = h q ′ , τ ′ | b T µν (0) | q, τ i . (3)In the same way the axial current b j µ A = ¯Ψ γ µ γ Ψ can be decomposed into b j µ A (0) = X σ,σ ′ Z d k (2 π ) / d k ′ (2 π ) / (cid:2) ¯ u σ ( k ) A ′ A µ ++ ( k, k ′ ) A ′ B ′ u σ ′ ( k ′ ) B ′ b a † σ ( k ) b a σ ′ ( k ′ ) ++ ¯ u σ ( k ) A ′ A µ + − ( k, k ′ ) A ′ B ′ v σ ′ ( k ′ ) B ′ b a † σ ( k ) b b σ ′ ( k ′ ) + ¯ v σ ( k ) A ′ A µ − + ( k, k ′ ) A ′ B ′ u σ ′ ( k ′ ) B ′ b b † σ ( k ) b a σ ′ ( k ′ )++ ¯ v σ ( k ) A ′ A µ −− ( k, k ′ ) A ′ B ′ v σ ′ ( k ′ ) B ′ b b † σ ( k ) b b σ ′ ( k ′ ) i , (4)where ¯ u σ ′ ( k ′ ) A µ ++ ( k, k ′ ) u σ ( k ) ≡h | b a σ ′ ( k ′ ) b j µ A (0) b a † σ ( k ) | i = h k ′ , σ ′ | b j µ A (0) | k, σ i , (5)and similarly for the others as in (3). Notice that the pure photon contribution to stress-energy tensor does not affectthe AVE conductivity. Indeed, the thermal correlator in (1) for b T µν Photons ( x ) would result in a combination of thermalaverages between one photonic and one fermionic operator which are vanishing. Furthermore, the Dirac equation andthe Poincar´e symmetry constraint the matrix elements (5) to take the form: A µs s ( k, k ′ ) = F A s s (cid:0) ( k ′ − k ) (cid:1) γ µ γ + F A s s (cid:0) ( k ′ − k ) (cid:1) ( k ′ − k ) µ m γ , (6)with s , s = ± and where F A s s (0) = 1.For the sake of clarity we illustrate the calculation only for the particle part, which we denote with W A aa . That iswe consider only the terms of (2) and (4) which contain exactly one particle creation operator b a † and one particleannihilation operator b a . The contribution from the other terms are obtained in a similar fashion and will be addedat the end. We can proceed in the calculation by plugging the expressions (2) and (4) in the formula (1). Then theconductivity W A aa involves the thermal expectation values between four creation and annihilation operators, whichcan be reduced by the thermal Wick theorem to the products of two-operator thermal expectation values as follows: h b a † b a b a † b a i c = h b a † b a b a † b a i − h b a † b a ih b a † b a i = h b a † b a ih b a b a † i . The two-operator thermal expectation values for Dirac fields with the homogeneous grand-canonical ensemble operator b ρ are given by: h b a † τ ( k ) b a σ ( q ) i β ( x ) = δ τσ δ ( k − q ) n F ( ε k ) , h b a τ ′ ( k ′ ) b a † σ ′ ( q ′ ) i β ( x ) = δ τ ′ σ ′ δ ( k ′ − q ′ )(1 − n F ( ε k ′ )) , h b b † τ ( k ) b b σ ( q ) i β ( x ) = δ τσ δ ( k − q ) n F ( ε k ) , h b b τ ′ ( k ′ ) b b † σ ′ ( q ′ ) i β ( x ) = δ τ ′ σ ′ δ ( k ′ − q ′ )(1 − n F ( ε k ′ )) , (7)where other combination are vanishing and n F is the Fermi-Dirac distribution function n F ( ε k ) = 1e ε k /T + 1 . After simple calculation, by using the (7) and the identity X σ u σ ( k )¯ u σ ( k ) = /k + m ε p , X σ v σ ( k )¯ v σ ( k ) = − /k + m ε p , the particle part of the AVE conductivity is W A aa = 2(2 π ) Z β d τβ Z d x X σ,σ ′ Z d k ε k Z d k ′ ε k ′ x e i( k − k ′ ) · x e − ( k ′ − k ) τ × M ( k, k ′ ) AB u σ ( k ) B ¯ u σ ( k ) A ′ A ( k, k ′ ) A ′ B ′ u σ ′ ( k ′ ) B ′ ¯ u σ ′ ( k ′ ) A n F ( ε k ′ )(1 − n F ( ε k ))= 2(2 π ) Z β d τβ Z d x Z d k ε k Z d k ′ ε k ′ x e i( k − k ′ ) · x e − ( k ′ − k ) τ × tr (cid:2) M ( k, k ′ ) ( /k + m ) A (cid:0) /k ′ + m (cid:1)(cid:3) n F ( ε k ′ )(1 − n F ( ε k )) . Since the matrix elements M ( k, k ′ ) can always be simplified such that they contain just one gamma matrix, whenplugging the form (6) of A we realize that only the first form factor F A can contribute to the trace. It is nowstraightforward to integrate over the coordinates by using Z d x x e i( k ′ − k ) · x = i(2 π ) ∂∂k ′ x δ ( k ′ − k ) , where k x is the first component of the spatial momentum k . Thanks to the delta function, the result W A aa = 2i(2 π ) Z d k ε k Z d k ′ ε k ′ ∂∂k ′ x δ ( k ′ − k )tr (cid:2) M ( k, k ′ ) ( /k + m ) γ γ (cid:0) /k ′ + m (cid:1)(cid:3) n F ( ε k ′ )(1 − n F ( ε k )) F A (cid:0) ( k ′ − k ) (cid:1) is easily integrated in k ′ : W A aa = i(2 π ) Z d kε k − n F ( ε k )) ∂∂k ′ x ( F A (cid:0) ( k ′ − k ) (cid:1) ε k ′ tr (cid:2) M ( k, k ′ ) ( /k + m ) γ γ (cid:0) /k ′ + m (cid:1)(cid:3) n F ( ε k ′ ) ) k ′ = k . (8)Repeating the same steps described above for all the other terms in (2) and in (4) and including a chemical potential µ we eventually obtain W A = − Z β d τβ Z d k (2 π ) (cid:20) ∂∂k ′ x S ( k, k ′ , τ ) (cid:21) k ′ = k , (9)with S ( k, k ′ , τ ) ≡ X s ,s = ± F A s s (cid:0) ( k ′ − k ) (cid:1) T s ,s ( ε k , ε k ′ , k , k ′ )4 ε k ε k ′ e − ( s ε k − s ε k ′ ) τ × ( δ s , + − n F ( ε k − s µ )) ( δ s , − − n F ( ε k ′ − s µ )) ,T s ,s ( ε k , ε k ′ , k , k ′ ) ≡ tr (cid:2) M s s ( k, k ′ ) ( /k + m ) γ γ (cid:0) /k ′ + m (cid:1)(cid:3) k = s ε k ,k ′ = s ε k ′ . Before moving on, it is important to check that the proposed method reproduces the known result for the AVEconductivity in the non-interacting case. For a free theory the matrices M µν ±± and the form factor F A ±± defined in (3)and in (6) do not depend on the signs s and s and they are all simply given by M µν Free ( q, q ′ ) = 14 [ γ µ ( q ν + q ′ ν ) + γ ν ( q µ + q ′ µ )] , F A (cid:0) ( k ′ − k ) (cid:1) = 1 . Not surprisingly, plugging these forms in (9) we reproduce the same formula found in [23] for the AVE conductivityof a free massive Dirac field. Working out the expression (9) for a free field we eventually get W A = 1(2 π ) β Z d k (cid:20) − ε k − k / ε k ( n F ( ε k − µ ) + n F ( ε k + µ )) − ε k + k / ε k ∂∂ε k ( n F ( ε k − µ ) + n F ( ε k + µ )) (cid:21) . After integrating by parts we obtain the well-known result: W A = 12 π β Z d k ε k + k ε k [ n F ( ε k − µ ) + n F ( ε k + µ )] . A. Non-relativistic limit
We now proceed to evaluate the radiative correction to the axial vortical effect (AVE) for a non-relativistic particleat low temperatures T ≪ m . In this regime the relevant contribution to the AVE conductivity is given by the particlecontribution in Eq. (8), while the other terms in (9) are sub-leading. We start evaluating the formula (8) by expandingthe derivative as: W ALT ≃ W A aa = i(2 π ) Z d kε k − n F ( ε k − µ )) (cid:26) F A (0) ε k n F ( ε k − µ ) (cid:20) ∂∂k ′ x T ++ ( k, k ′ ) (cid:21) k ′ = k ++ T ++ ( k, k ) ∂∂k ′ x " F A (cid:0) ( k ′ − k ) (cid:1) ε k ′ n F ( ε k ′ − µ ) k ′ = k ) , where the subscript LT stands for low temperature and T ++ ( k, k ′ ) = tr (cid:2) M ( k, k ′ ) ( /k + m ) γ γ (cid:0) /k ′ + m (cid:1)(cid:3) . (10)Taking advantage of Dirac equation in (3) we see that the matrix element M can always be written as M ( k, k ′ ) AB = M ( k, k ′ ) λ γ λAB . Then the trace reads T ++ ( k, k ′ ) = M ( k, k ′ ) λ tr (cid:2) γ λ ( /k + m ) γ γ (cid:0) /k ′ + m (cid:1)(cid:3) = − M ( k, k ′ ) λ ǫ λ µν k µ k ′ ν , from which we see that T ++ ( k, k ′ ) is vanishing when k ′ = k . Also, reminding that F A (0) = 1, the AVE conductivitybecomes W ALT = i(2 π ) Z d kε k − n F ( ε k − µ )) n F ( ε k − µ ) (cid:20) ∂∂k ′ x T ++ ( k, k ′ ) (cid:21) k ′ = k = − i(2 π ) β Z d kε k n ′ F ( ε k − µ ) (cid:20) ∂∂k ′ x T ++ ( k, k ′ ) (cid:21) k ′ = k , where we used the identity(1 − n F ( ε k − µ )) n F ( ε k − µ ) = − β ∂∂ǫ k n F ( ε k − µ ) = − β n ′ F ( ε k − µ ) . To evaluate the trace, we notice (see App. B) that only the following form factors of M bring a non-vanishingcontribution to T ++ ( k, k ′ ): M µν Relevant ( k, k ′ ) = I P γ ( P, q ) ( γ µ P ν + γ ν P µ ) + I uγ ( P, q ) ( γ µ u ν + γ ν u µ ) ++ I P l ( P, q ) / ˆ l (cid:16) ˆ l µ P ν + ˆ l ν P µ (cid:17) + I ul ( P, q ) / ˆ l (cid:16) ˆ l µ u ν + ˆ l ν u µ (cid:17) , (11)where u is the thermal bath velocity, P = k + k ′ , q = k ′ − k and l µ = ǫ µνρσ u ν P ρ q σ . (12)The explicit values of the form factors will be evaluated in Appendix C and the final results are reported in (16) for q = 0 and in the non-relativistic limit of P . In the rest frame of thermal bath, where the formula (8) is evaluated, thematrix elements (11) read M µν ( k, k ′ ) Rest = I P γ ( P, q ) ( γ µ P ν + γ ν P µ ) + I uγ ( P, q ) (cid:0) γ µ δ ν + γ ν δ µ (cid:1) ++ I P l ( P, q ) / ˆ l (cid:16) ˆ l µ P ν + ˆ l ν P µ (cid:17) + I ul ( P, q ) / ˆ l (cid:16) ˆ l µ δ ν + ˆ l ν δ µ (cid:17) . First, consider the term in I P γ ( P, q ). The calculation is the same as the free case except for the form factor. Thetrace reads: T P γ ( k, k ′ ) = I P γ ( P, q ) (cid:2) η λ ( k y + k ′ y ) + η λ ( ε k + ε k ′ ) (cid:3) ( − ǫ λ µν k µ k ′ ν , whose derivative is ∂∂k ′ x T P γ ( k, k ′ ) (cid:12)(cid:12)(cid:12) k ′ = k =2 I P γ ( P, q = 0) (cid:0) η λ k y + η λ ε k (cid:1) ( − ǫ λ µν k µ η ν = − I P γ ( P, q = 0) (cid:0) − ǫ k y + ǫ ε k (cid:1) = − I P γ ( P, q = 0) (cid:0) ε k + k y (cid:1) . Then this term of matrix elements contributes to the low temperature AVE conductivity as W A P γ = − π ) β Z d k I P γ ( P, q = 0) ε k + k y ε k n ′ F ( ε k − µ )= − π ) β Z d k I P γ ( P, q = 0) ε k + k / ε k n ′ F ( ε k − µ ) . Since the form factor only depend on the spatial modulus of P we can integrate by parts: W A P γ = 12 π β Z d k (cid:20) I P γ ( P, q = 0) ε k + k ε k + 4 I P γ ( P, q = 0) k ε k − k / ε k ++4 ∂I P γ ( P, q = 0) ∂ε k (cid:18) k + k ε k (cid:19)(cid:21) n F ( ε k − µ ) . At low temperature the second term is sub-leading compared to the first and we can approximate the coefficient as W A P γ = 12 π β Z d k (cid:20) I P γ ( P, q = 0) ε k + k ε k + 4 ∂I P γ ( P, q = 0) ∂ε k (cid:18) k + k ε k (cid:19)(cid:21) n F ( ε k − µ ) . (13)Explicit calculations reveal that the other terms of (11) when plugged into (8) give contribution of the same formas (13) but with a different form factor. To illustrate this we show that the other terms of (11) when traced in (10)give the same result as the first term, hence M µν can be effectively by written as proportional to ( γ µ P ν + γ ν P µ ).Firstly, since the form factors must be evaluated for q going to zero and then integrated over k , from Eq. (12) we seethat the matrix elements of (11) proportional to /l are effectively given by M µν ( k, k ′ ) Eff = I P γ ( P, q ) ( γ µ P ν + γ ν P µ ) + I uγ ( P, q ) (cid:0) γ µ δ ν + γ ν δ µ (cid:1) + − I P l ( P, q ) ( γ µ P ν + γ ν P µ ) − I ul ( P, q ) (cid:0) γ µ δ ν + γ ν δ µ (cid:1) . Secondly, in the rest frame of thermal bath where ω P = P · u = P we can write the form factors I uγ ( P, q ) and I ul ( P, q ) as M µν ( k, k ′ ) Eff = I P γ ( P, q ) ( γ µ P ν + γ ν P µ ) + I uγ ( P, q ) ω P (cid:0) γ µ δ ν + γ ν δ µ (cid:1) P + − I P l ( P, q ) ( γ µ P ν + γ ν P µ ) − I ul ( P, q ) ω P (cid:0) γ µ δ ν + γ ν δ µ (cid:1) P . Lastly, one can explicitly check that the contribution from (cid:0) γ µ δ ν + γ ν δ µ (cid:1) P is the same as ( γ µ P ν + γ ν P µ ) in thelow temperature limit and the effective matrix element is M µν ( k, k ′ ) Eff = (cid:20) I P γ ( P, q ) + I uγ ( P, q ) ω P − I P l ( P, q ) − I ul ( P, q ) ω P (cid:21) ( γ µ P ν + γ ν P µ )= g Ω ( P, q )4 ( γ µ P ν + γ ν P µ ) . As we will show in the next sections, the quantity g Ω defined from the form factors of the stress-energy tensor above isthe gravitomagnetic moment of a fermion. At finite temperature the interactions renormalize the angular momentumof the system and the spin-orbit coupling of the fermion, described by g Ω , can be different from 1. Following the stepsdescribed above, it is now clear that the AVE conductivity at low temperature is given by W ALT = 12 π β Z d k (cid:20) g Ω ( ε k ) ε k + k ε k + g ′ Ω ( ε k ) (cid:18) k + k ε k (cid:19)(cid:21) n F ( ε k − µ ) . (14)Eq. (14) connects the radiative corrections to the AVE to the anomalous gravitomagnetic moment. If the gravito-magnetic moment is not anomalous, i.e. g Ω = 1, then the AVE conductivity (14) remains the one of a free-field, andradiative corrections are not present. On the other hand, if the gravitomagnetic moment is affected by interactions,the radiative corrections to the AVE can be obtained from the formula above. Let us stress that the Eq. (14) holdsin the large mass limit, where we can factorize long- and short-distance contributions. The effects of interactionsabove some energy scale (short distances) are contained in the form factors, which we renormalize at finite temper-ature, and result in the gravitomagnetic moment. The low energy contributions (large distance) are collected intothe thermal averages of one-particle states, which correspond to an expansion in T /m . We furthermore considereda non-relativistic particle, such that the contribution coming from the anti-particle thermal distribution n F ( ε k + µ )is exponentially suppressed compared to the particle one n F ( ε k − µ ). In the following sections we show that thegravitomagnetic moment at finite temperature is anomalous and we evaluate the first QED radiative correction.To conclude this section we estimate the integral in Eq. (14). At low temperature the leading contribution isobtained by setting g Ω ( ε k ) → g Ω = lim k → g Ω ( ε k ) , g ′ Ω ( ε k ) = 0 . At 1-loop in thermal QED we found (see next Section) that for T ≪ m the anomalous gravitomagnetic moment is g Ω − − e T m , where e is the QED coupling constant. The radiative correction to the AVE then reads∆ W ALT ≃ − π β e T m Z d k ε k + k ε k n F ( ε k − µ ) = − e T m W Afree . Replacing the non-relativistic low temperature limit of the axial vortical effect conductivity [23] we finally obtain∆ W ALT ≃ − e T m ( mT ) / √ π / e − ( m − µ ) /T . III. GRAVITOMAGNETIC MOMENT
In this section we introduce the gravitomagnetic moment and present the result for the Anomalous GravitomagneticMoment (AGM) in finite temperature QED. The AGM can be easily understood in analogy to the anomalous magneticmoment. The Dirac equation in an external magnetic field and in the presence of rotation can be written using rotatingcoordinates and takes the form [i γ µ ( D µ + Γ µ ) − m ] ψ = 0with the covariant derivative D µ = ∂ µ − i eA µ , the spin connection Γ µ = − i4 ω µij σ ij and σ ij = i2 [ γ i , γ j ]. Setting themagnetic field B and the rotation Ω along the z axis we find (cid:2) i γ µ D µ + i γ Ω( y∂ x − x∂ y − i σ ) − m (cid:3) ψ = h i γ µ D µ + γ Ω( b L z + b S z ) − m i ψ = 0 . Acting on this equation with (i γ µ ( D µ + Γ µ ) + m ) we obtain the second order Dirac equation (cid:2) ∂ t − ∇ − e B ( L + 2 S ) − Ω ( L + S ) + m (cid:3) ψ = 0 . The quantity which couples the magnetic field to the spin S is called the magnetic moment g B , while the quantity g Ω that couples the rotation to the spin is the gravitomagnetic moment. Therefore, Dirac equation alone predicts thatthe spin couples to the magnetic moment and rotation with g B = 2 , g Ω = 1 . This is exactly what we expect for Dirac particles, as discussed in [34–36]; spin-rotation coupling has been reviewedin [37, 38].However, one of the most precise predictions of QED is that the magnetic moment of fermions is anomalous, i.e.the value given by the Dirac equation receives radiative corrections. The anomalous magnetic moment can be foundusing scattering theory in QED in the non-relativistic limit. Interactions with magnetic field are described by theLagrangian L = J µ A µ , therefore radiative corrections to magnetic moment are obtained by evaluating the matrixelement of the electromagnetic current, which can be written as: h p ′ , s ′ | J µ (0) | p, s i = ¯ u ( p ′ , s ′ ) (cid:26) P µ m F ( q ) + i σ µν q ν m (cid:2) F ( q ) + F ( q ) (cid:3)(cid:27) u ( p, s )where | p, s i denotes a state with a fermion of momentum p and polarization s , u ( p, s ) denotes the Dirac spinor andwe defined the momenta P = p ′ + p and q = p ′ − p ; F , F are the form factors. At first order in the transferredmomentum q the matrix element is h p ′ , s ′ | J µ (0) | p, s i = ¯ u ( p ′ , s ′ ) (cid:26) P µ m + i σ µν q ν m [1 + F (0)] (cid:27) u ( p, s ) + O ( q ) . The from factor F (0) gives directly the anomalous magnetic moment, which was first evaluated by Schwinger [39] atorder α = e / π from the vertex correction. He obtained the famous result a B ≡ g B −
22 = F (0) = α π . Following a similar argument using scattering theory (as explained in detail in Section IV), we can relate thegravitomagnetic moment and its radiative corrections to the matrix elements of the stress-energy tensor. In theweak gravitational field limit, the gravitational coupling is described to first order in the gravitational field by theLagrangian L = h µν T µν , where h are small deviations from Minkowski space-time g µν = η µν + h µν , and everythingelse is happening in flat space-time. The rotation of the medium is contained inside h and to see the effects ofinteractions we must consider the matrix element of the stress-energy tensor h p ′ , s ′ | T µν (0) | p, s i . If we perform thiscalculation at zero temperature (see Sec. IV B for the details) we realize that there are no modifications to spincoupling as dictated by the Einstein equivalence principle [28].We are now interested in finite temperature effects, therefore we consider a Dirac fermion in thermal equilibriumwith the medium. It is possible to use a manifestly Lorentz covariant form of thermal field theory to write the matrixelements of stress-energy tensor if we also take into account the thermal bath velocity u [42]. We are then usingthe S-matrix elements in the QED thermal theory for a non-relativistic fermion. Since the rotation of the mediumis taken in reference to the rest frame of the thermal bath, the S-matrix elements between incoming and outgoingfermion states of momenta p and p ′ impose the conservation equation p · u = p ′ · u or equivalently q · u = 0. To detectAGM, it is then convenient to move into the rest frame of the medium u = (1 , ), which implies q = 0 , p = p ′ = p m + ( P + q ) / , | p ′ | = | p | . In a generic frame, we take advantage of orthogonality relation q · P = 0 and introduce ω P ≡ P · u , so that we candefine a space-like four-vector ˜ P orthogonal to u , and a scalar P s :˜ P µ ≡ ( η µν − u µ u ν ) P ν = P µ − ω P u µ , P s ≡ q ω P − P . To close the tetrad, we define the space-like unit four-vector ˆ l orthogonal to u, ˜ P and q (it is also orthogonal to P ): l µ = ǫ µνρσ u ν ˜ P ρ q σ , ˆ l µ = l µ √− l . Note that ˜ P · q = 0 and q · u = 0 also hold true, meaning that the tetrad { u, ˜ P , q, ˆ l } is an orthogonal non-normalizedbasis. Using scattering theory (see Sec. IV B), the gravitomagnetic moment is obtained by g Ω = lim q → P s → (cid:18) I P γ ( P, q ) + I uγ ( P, q ) ω P − I P l ( P, q ) − I ul ( P, q ) ω P (cid:19) , (15)where the functions are the following form factors of the stress-energy tensor matrix element h p ′ , s ′ | T µν (0) | p, s i = ¯ u ( p ′ , s ′ ) n I P γ ( P, q ) ( P µ γ ν + P ν γ µ ) + I uγ ( P, q ) ( u µ γ ν + u ν γ µ )+ I P l ( P, q ) / ˆ l (cid:16) P µ ˆ l ν + P ν ˆ l µ (cid:17) + I ul ( P, q ) / ˆ l (cid:16) u µ ˆ l ν + u ν ˆ l µ (cid:17) + · · · o u ( p, s ) + O ( q ) . Apart from the higher order corrections, the finite temperature effects also give contribution to the anomalous magnetic moment, seefor instance [40, 41]. q is reported in App. C 3. In the limit q → P s →
0, the thermalcontributions to the form factors are I P γ = −
10 + 3 θ HT e T m ; I uγ ω P = − e T m ; I P l = 2 − θ HT e T m ; I ul ω P = − θ HT e T m , (16)where we introduced the function θ HT θ HT ≡ ( T ≪ m T ≫ m which turns on the high temperature contributions. It is now straightforward to compute the AGM from the rela-tion (15) which gives g Ω − − − θ HT e T m = ( − e T m T ≪ m − e T m T ≫ m, m > eT . (17)It should also be taken into account that in order to define one particle states and the form factors of the stress-energytensor, the mass of the particle must be larger than eT . Then, the high temperature limit can only be taken in theweak coupling limit. The presence of an anomalous gravitomagnetic moment is to be ascribed to the violation ofEinstein equivalence principle. At finite temperature the vacuum is not Lorentz invariant. On the contrary, we canalways distinguish a preferred reference frame, which is the one where the thermal bath is at rest. This means that wecan distinguish between acceleration and the genuine effects of gravity. Similar thermal effects in QED affect modifythe values of inertial and gravitational mass of a Dirac fermion providing an explicit breaking of the weak equivalenceprinciple [30, 31, 43–45], see App. A for details. IV. GRAVITOMAGNETIC MOMENT IN LINEARIZED GRAVITY
In this section we identify the gravitomagnetic moment of a fermion using the scattering theory in linearized gravity.The fermion interacts with an external gravitational field g µν Ext = η µν + h µν Ext through the linearized Hamiltonian b H int = 12 Z d x b T µν h µν Ext . The corresponding scattering amplitude is A = − i(2 π ) δ ( p · u − p ′ · u ) 1 p Z ( p ) Z ( p ′ ) 12 ¯ u β ( p ′ ) M µν ( p, p ′ ) u β ( p ) h µν Ext ( p ′ − p ) , where Z is the wave-function renormalization constant. To study the effect of rotation, we then just have to usethe proper metric accounting for rotation. Using the linear approximation of gravito-electromagnetism [37], themetric can be written in terms of a gravitational gauge potential A µg = (2 φ g , A g ). Let h µν be the perturbation ofmetric g µν = η µν + h µν ; we then define ¯ h µν = h µν − η µν h αα . This definition is related to the gravitation gaugepotential by ¯ h = 4 φ g and ¯ h i = 2 A ig . For the case of rotation around an axis, we have h αα = 0 and therefore h µν = ¯ h µν and φ g = 0. The only non-vanishing components of metric perturbation are ¯ h µ = 2(2 φ g , A g ). Wecan also define a gravitomagnetic field via B ig = ǫ ijk ∇ j A kg , or in terms of Fourier transform B ig = − i ǫ ijk q j A kg ( q ).Gravito-electromagnetism is particularly well suited for describing a pure rotation. For instance, consider a constantrotation around the z axis. In rotating cartesian coordinates, the deviation from flat space-time is h µν = − ( x + y )Ω y Ω − x Ω 0 y Ω 0 0 0 − x Ω 0 0 00 0 0 0 ≃ y Ω − x Ω 0 y Ω 0 0 0 − x Ω 0 0 00 0 0 0 + O (Ω ) . (18)1For the metric in Eq. (18), we simply have B g = Ω . For our system at thermal equilibrium, the rotation is taken inreference to the thermal bath velocity u and the metric deviation is therefore given by h µν = 2 u µ A ν + 2 u ν A µ . Withthis metric, the scattering amplitude is A = − i(2 π ) δ ( p · u − p · u ) 2 p Z ( p ) Z ( p ′ ) ¯ u β ( p ′ ) M µν ( p, p ′ ) u β ( p ) u µ A νg ( p ′ − p ) . (19)To identify the gravitomagnetic moment g Ω of the fermion, we then compare the previous amplitude to the oneobtained from the potential containing the spin-rotation coupling: V = − µ g · B g = − g Ω S · Ω . This potential, in the non-relativistic limit, leads to the amplitude A = − i(2 π ) δ ( p − p ′ ) h − g Ω ξ ′† σ ξ · Ω i , (20)where ξ is the two component spinor, normalized such that ξ † ξ = 1 and σ are the Pauli matrices. By matching theexplicit form of the amplitude in Eq. (19) to the one in Eq. (20) coming from the spin-rotation coupling, we can readoff the gravitomagnetic moment obtained in finite temperature field theory. A. Gravitomagnetic moment at leading order
We now obtain the leading order gravitomagnetic moment by computing the amplitude (19). First, we go to therest frame u = (1 , ) where the scattering amplitude becomes A = − i(2 π ) δ ( p − p ′ ) 2 p Z ( p ) Z ( p ′ ) ¯ u ( p ′ ) M i ( p, p ′ ) u ( p ) A ig ( p ′ − p ) . At leading order, u ( p ) is the usual Dirac spinor and the matrix elements of stress energy tensor are M i ( p, p ′ ) = 14 [ γ ( p ′ + p ) i + γ i ( p ′ + p ) ] , Z ( p ) = Z ( p ′ ) = 1 . In the limit q → P s → u ( p ′ ) γ u ( p ) = ξ ′† ξ, ¯ u ( p ′ ) γ i u ( p ) = ξ ′† (cid:20) ( p ′ + p ) i m − i ǫ ijk ( p ′ − p ) j σ k m (cid:21) ξ and findlim q → A = − i(2 π ) δ ( p − p ′ )( −
1) 12 ξ ′† (cid:20) ( p ′ + p ) i + ( p ′ + p ) m ( p ′ + p ) i − ( p ′ + p ) m i ǫ ijk ( p ′ − p ) j σ k (cid:21) ξA ig ( q ) . This expression can be simplified using the following approximation valid in the non relativistic limit δ ( p − p ′ )( p + p ′ ) = δ ( p − p ′ )2 p = δ ( p − p ′ )2 p m + ( P + q ) / δ ( p − p ′ )2 m + O (cid:0) q , P /m (cid:1) . The amplitude is then lim q → A = − i(2 π ) δ ( p − p ′ )( − ξ ′† (cid:20) ( p ′ + p ) i A ig ( q ) −
12 i ǫ ijk q j σ k A ig ( q ) (cid:21) ξ = − i(2 π ) δ ( p − p ′ )( − ξ ′† (cid:20) ( p ′ + p ) i A ig ( q ) + 12 σ · B g ( q ) (cid:21) ξ = − i(2 π ) δ ( p − p ′ ) h − p ′ + p ) i A ig ( q ) ξ ′† ξ − ξ ′† σ ξ · Ω ( q ) i . Comparing with the amplitude in Eq. (20), the gravitomagnetic moment of a fermion is g Ω = 1 , which, as expected, is the value predicted by the Dirac equation.2 B. Anomalous gravitomagnetic moment at one loop
Here we consider the radiative corrections to gravitomagnetic moment. First, we deal with the zero temperaturepart. At zero temperature, the one-loop renormalized stress-energy tensor is [46] M µν = ¯ u ( p ′ ) (cid:2) ( P µ γ ν + P ν γ µ ) I P γ ( Q ) − P µ P ν I P P ( Q ) − (cid:0) q µ q ν − η µν Q (cid:1) I qq ( Q ) (cid:3) u ( p )where in the limit of Q ≡ p − q → I P γ ( Q ) = 14 (cid:18) − απ Qm (cid:19) ; I P P ( Q ) = απ m QM ; I qq ( Q ) = − απ m mQ . Since the spin coupling can only come from a gamma matrix, only the I P γ ( Q ) formfactor can contribute to grav-itomagnetic moment. However in the limit of vanishing q this term does not affect the gravitomagnetic moment;therefore there is no AGM for QED at zero temperature. This is what we expect from the Einstein equivalenceprinciple, which is still valid in interacting quantum field theory at zero temperature.We now move to temperature modifications. The finite temperature renormalization we adopt is described inApp. C. First, we address the corrections which might come from thermal spinors, see App. C 1. We should repeat theprevious calculation of the leading order gravitomagnetic moment, except that we must now employ thermal on-shellcondition and thermal Dirac spinors:¯ u β ( p ′ ) M (0) µν u β ( p ) = ¯ u β ( p ′ ) 14 ( P µ γ ν + P ν γ µ ) u β ( p ) . In the rest frame where u = (1 , ), the thermal spinors describe the spin interaction, as can be seen by using theidentity ¯ u β ( p ′ ) γ u β ( p ) = ξ ′† (cid:20) P T m p + i σ ∧ q T m p (cid:21) ξ + O ( e ) , where P T , q T and m p contain temperature modifications according to the notation used in Sec. C 1. Here the spin-rotation coupling is divided by the thermal mass, but this thermal mass is canceled out by the term p ′ T + p T ≃ m p ;the gravitomagnetic moment is therefore unaffected.Now we include the radiative corrections coming from the stress-energy tensor matrix elements evaluated from thediagrams considered in Sec. C. First, we select the terms of the stress-energy tensor matrix element M µν ( p ′ , p ) thatactually contributes to the anomalous gravitomagnetic moment (AGM). In App. B we show that only the followingterms can bring contribution to AGM: M µν Relevant ( p, p ′ ) = I P γ ( P, q ) ( γ µ P ν + γ ν P µ ) + I uγ ( P, q ) ( γ µ u ν + γ ν u µ ) ++ I P l ( P, q ) / ˆ l (cid:16) ˆ l µ P ν + ˆ l ν P µ (cid:17) + I ul ( P, q ) / ˆ l (cid:16) ˆ l µ u ν + ˆ l ν u µ (cid:17) , which are the same one that contribute to the axial vortical effect.By comparison with the contribution to AGM from P µ γ ν + P ν γ µ evaluated in Sec. IV A, the formfactor I P γ ( P s , q ) ( P µ γ ν + P ν γ µ )leads to the gravitomagnetic moment g Ω = + lim P s → lim q → I P γ ( P s , q ) . Similarly, it is straightforward to show that the other formfactors contribute to AGM as I uγ ( P s , q ) ( u µ γ ν + u ν γ µ ) → g Ω = + lim P s → lim q → I uγ ( P s , q ) ω P ; I ul ( P s , q ) / ˆ l (cid:16) u µ ˆ l ν + u ν ˆ l µ (cid:17) → g Ω = − lim P s → lim q → I ul ( P s , q ) ω P ; I P l ( P s , q ) / ˆ l (cid:16) P µ ˆ l ν + P ν ˆ l µ (cid:17) → g Ω = − lim P s → lim q → I P l ( P s , q ) . g Ω = lim q → P s → (cid:18) I P γ ( P, q ) + I uγ ( P, q ) ω P − I P l ( P, q ) − I ul ( P, q ) ω P (cid:19) . Thoee form factors are computed in App. C and lead to the result quoted in Sec. III: g Ω − − − θ HT e T m = ( − e T m T ≪ m − e T m T ≫ m . V. SUMMARY AND DISCUSSION
In summary, we showed that in a system at thermal equilibrium the interactions with photons change the gravit-omagnetic moment of a massive fermion, i.e. the coupling between the spin and the rotation of the medium. Usingthe scattering theory, in analogy to the magnetic moment, we obtained the gravitomagnetic moment from the formfactors of the stress-energy tensor, see Eq. (15). We then renormalized the stress-energy tensor at one-loop level inthe finite temperature QED. The resulting gravitomagnetic moment, given by (17), receives radiative correctionsonly in the presence of thermal effects. We argued that this is possible because the thermal bath destroys the Lorentzinvariance of stress-energy tensor and consequently violates the Einstein equivalence principle. To the best of ourknowledge, the possibility of an anomalous gravitomagnetic moment (AGM) in these settings and the calculation ofit are new results.The effect of spin-rotation coupling has been already observed in the non-vanishing global polarization of particlesemitted by the rotating quark-gluon plasma [1]. Therefore, in principle, polarization measurements in heavy ion colli-sions could reveal the presence of an anomalous gravitomagnetic moment and the breaking of the Einstein equivalenceprinciple. To give an order of magnitude, we first extend the result (17) to QCD. By comparison with the masslessQCD radiative corrections of AVE [25, 26], we expect that it is sufficient to replace the QED coupling constant e with ( N c − g / g QCDΩ − − N c −
12 6 − θ HT g T m = ( − N c −
12 16 g T m T ≪ m − N c −
12 536 g T m T ≫ m . (21)In a simple recombination picture based on the quark model, the Λ polarization is carried predominantly by thestrange quark s ; therefore the relative importance of the AGM for Λ polarization can be inferred from the magnitudeof radiative corrections to the gravitomagnetic moment of the s quark. In the quark gluon plasma phase, due to thehigh temperature T = 175 −
300 MeV and the strong coupling regime, we estimate that the constituent mass of thestrange quark m s ≃
400 MeV is larger than gT ≃ −
450 MeV. Using (21), we find that the relative contribution ofthe AGM is quite large, about 40%. Because it depends on the temperature, the AGM contribution may be detectedin the data on Λ polarization. An anomalous gravimagnetic response in the chirally broken phase of finite densityQCD has been discussed in [47] where it was found to contribute significantly to Λ polarization. Note that the effectof the AGM on polarization has the same sign for fermions and antifermions, so we expect it to contribute equally tothe polarization of both Λ and ¯Λ hyperons.We also established the connection between the AGM and the axial vortical effect (AVE). In the limit of the large m/T ratio, we can separate the short distance interactions that renormalize the angular momentum of the system,and the long distance thermal contributions which result in the AVE. In this way we obtained the formula in Eq. (14)which relates the radiative correction to the AVE for a massive fermion to the momentum-dependent AGM.While we showed here that the thermal interpretation of the AVE together with spin-rotation coupling is able todescribe the radiative corrections to the AVE, this does not exclude a connection to the mixed gauge-gravitationalanomaly [18]. The presence of radiative corrections itself does not conflict with an interpretation based on the anomalybecause the AVE is obtained from the matrix elements of the axial current and the non-renormalization theoremsapply to the operators, and not to the matrix elements, as has been established for the case of chiral anomaly inmassless QED [48]. However the anomalous origin of the effect is not yet firmly established and can be questioned;this is particularly true for massive fermions [49].Radiative corrections to the AVE were presented previously in [25, 26] for massless fermions and can be linked to thegravitational anomaly of photons [24, 50]. Unfortunately we cannot compare these corrections with those presentedabove, since our derivation requires massive fermions, and a massless limit cannot be performed. Even our definitionof the gravitomagnetic moment can not be applied to a massless particle, as it requires going to the rest frame of4the particle. Therefore, the connection between the AVE and the AGM for a massless Dirac field is still not clear.Nevertheless, we believe that the link between the anomalous gravitomagnetic moment and the axial vortical effectfor massive fermions established above will help to understand the origin of chiral currents induced by rotation.
ACKNOWLEDGMENTS
M.B. would like to thank the Center for Nuclear Theory at Stony Brook University for hospitality and supportduring his one year visit. The work of M.B. is supported by Unifi fellowship
Polarizzazione nei fluidi relativistici and
Effetti quantistici nei fluidi relativistici . The work of D.K. was supported by the U.S. Department of Energy underawards DE-FG88ER40388 and DE-SC0012704.
Appendix A: Equivalence principle at finite temperature
It has already been demonstrated [30, 31, 43–45] that radiative corrections at finite temperature lead to a differencebetween the inertial and gravitational masses, thus providing an explicit breaking of weak equivalence principle.Indeed, one can define three distinct kinds of mass for a particle. The phase-space mass is the position of the pole inthe propagator of the field. The inertial mass is the response to acceleration caused by an external force, such as anexternal electric field. And lastly, the gravitational mass is a measure of how the fermion responds to the gravitationalforce. At finite temperature it has been found that the inertial mass and the phase-space masses coincide but thegravitational mass is different [30].
1. Phase space mass
Consider a massive Dirac fermion in a QED-like theory at finite temperature. As in [30], we refer to phase-spacemass as the location of the pole in the propagator of the fermion field. The full fermion propagator is given by: S ( p ) = 1 /p − m − Σ( p ) . The self-energy can be written in covariant form as [44, 51]Σ( p ) = a/p + b/u + c, where a, b, c are Lorentz invariant functions. These functions can depend on m, T and on the following Lorentz scalars: ω ≡ p α u α , p s ≡ p ( p α u α ) − p ;since p = ω − p s , one may interpret ω and p s as Lorentz invariant energy and three momentum. It is useful todefine a tensor and a vector orthogonal to u µ by˜ η µν = η µν − u µ u ν , ˜ p µ = ˜ η µν p ν = p µ − ωu µ . The vector ˜ p is automatically space-like: ˜ p µ ˜ p µ = − p s < . Inverting the matrices, the full propagator becomes: S ( p ) = (1 − a ) /p − b/u + m + c [(1 − a ) p − bu ] − ( m + c ) . Therefore, the location of the pole is determined by the vanishing of the denominator:(1 − a ) ( ω − p s ) + b − − a ) bω = ( m + c ) . The positive solution for the pole is ω = b − a + s p s + ( m + c ) (1 − a ) . e in the coupling constant e , it suffices to linearize in the a, b, c functions, which are already e order. The phase-space mass is then m p ≡ ω − p s = m + 2 (cid:2) a ( ω − p s ) + bω + mc (cid:3) . (A1)At a given a momentum p with scalars ω and p s , the pole of the propagator is situated at ω = E β ≡ p s + m p . The functions a, b, c are obtained with the traces over spinor indices P T = tr (cid:2) /p Σ β ( p ) (cid:3) , U T = tr (cid:2) /u Σ β ( p ) (cid:3) , T = tr (cid:2) Σ β ( p ) (cid:3) , which give a = ( p · u ) U T − P T p · u ) − p ] , b = ( p · u ) P T − p U T p · u ) − p ] , c = T . (A2)Replacing the Eq.s (A2) into Eq. (A1) we find m p = m + 12 ( P T + mT ) . In real-time formalism the self energy is given by the one-loop diagramΣ( p ) = i e Z d k (2 π ) D µν ( k ) γ µ S F ( p + k ) γ ν = Σ( T = 0) + Σ β , where S F and D µν are the fermion and the photon propagator in momentum space (see Sec. C) and we split the zeroand the finite temperature part of self-energy. With standard techniques we can evaluate the self-energy in covariantform and the explicit form of the functions a, b and c . We find that the phase-space mass can be approximatedby [30, 44, 51] m p − m = ( e T T ≪ m, e T T ≫ m , where the different behavior at high temperature arises because the fermion thermal distribution becomes comparableto the contribution from the photon thermal distribution only when the temperature is much larger than the mass ofthe particle.
2. Inertial mass
We refer, as usual, to inertial mass of the particle m I as the proportionality term between a force and the accelerationcaused by it. To test the inertial mass of a charged Dirac particle, the most natural force to consider is a constantelectric field E . In this way, it is easy to include it in the Dirac equation as a minimal coupling with an externalgauge field A µ = ( φ, ), where E = −∇ φ . The corrections to this coupling given by temperature and interactions arethe corrections to the vertex Γ µ . It is found that, when the vertex is contracted between thermal spinors u β ( p ) (seeSec. C 1), the modifications exactly compensate each other [52] e ¯ u β ( p )Γ µ u β ( p ) = e p µ E β , or in other words, the charge is not renormalized by finite temperature effects. This suggests that the inertial massis to be identified with the phase-space mass.To properly evaluate the inertial mass, we just need to consider the modified Dirac equation (cid:16) /p T − m (cid:17) ψ = e Γ µ A µ ψ. In the non-relativistic limit, one can transform the previous equation into a Schrodinger equation via a Foldy–Wou-thuysen transformation. In that form, one can easily identify the Hamiltonian H of the system, then the accelerationof the particle is identified via a = − [ H, [ H, r ]] and hence one can infer the value of inertial mass. It is indeedfound [30] that inertial mass m I and phase-space mass m p coincide.6
3. Gravitational mass
Using scattering theory in linearized gravity, we can identify the gravitational mass of a fermion. We indicate thematrix element of the stress-energy tensor with h p ′ , s ′ | T µν (0) | p, s i ≡ ¯ u ( p ′ ) M µν ( p, p ′ ) u ( p ) , where p and p ′ are the external momenta and s, s ′ the spin of the fermion. Consider an external gravitational field g µν Ext = η µν + h µν Ext ; the interaction Hamiltonian in linearized gravity is therefore given by b H int = R d x b T µν h µν Ext . Inthe leading order of perturbation theory, the S-matrix element for scattering is:i A (2 π ) δ ( p · u − p ′ · u ) = − i 12 ¯ u ( p ′ ) M µν ( p, p ′ ) u ( p ) h µν Ext ( p ′ − p ) , where h µν Ext ( p ′ − p ) is the Fourier transform of h µν Ext ( x ) and M µν is the tree-level vertex function, which is given by M µν ( p, p ′ ) = 14 [ γ µ ( p ′ + p ) ν + γ ν ( p ′ + p ) µ ] − η µν (cid:2) ( /p − m ) + ( /p ′ − m ) (cid:3) . The radiative corrections modify this expression to A = − i(2 π ) δ ( p · u − p ′ · u ) 1 p Z ( p ) Z ( p ′ ) 12 ¯ u β ( p ′ ) M µν ( p, p ′ ) u β ( p ) h µν Ext ( p ′ − p ) , where we divided by √ Z - the wave-function renormalization constant - for each fermionic leg, and u β ( p, s ) is theDirac thermal spinor which satisfies the Dirac equation including radiative and thermal corrections and all perturbativediagrams are summed in M .To identify the gravitational mass of the fermion m g , following [44], we consider the scattering of a fermion from astatic gravitational potential produced by a static mass density ρ Ext ( x ). The resulting metric is the linearized solutionof Einstein field equations with a matter stress-energy tensor given by T µν = ρ Ext u µ u µ . Taking advantage of thePoisson equation − q φ Ext = ρ Ext the Fourier transform of Einstein equation solution reads: h µν ( q ) = 2 φ Ext ( q )(2 u µ u ν − η µν ) . Therefore, inserting this in the scattering amplitude, we find A = − i(2 π ) δ ( p − p ′ ) 1 p Z ( p ) Z ( p ′ ) ¯ u β ( p ′ ) M µν ( p, p ′ ) u β ( p )(2 u µ u ν − η µν ) δ ( p ′ − p ) φ Ext ( q )= − i(2 π ) δ ( p ′ − p ) M ( p ′ , p ) φ Ext ( q ) , where we have defined M ( p ′ , p ) = 1 p Z ( p ) Z ( p ′ ) ¯ u β ( p ′ ) M µν ( p, p ′ ) u β ( p )(2 u µ u ν − η µν ) . If the gravitational field is very slowly varying over a large (macroscopic) region, φ Ext ( q ) will be concentrated around q = 0; then we can take the limit q →
0. In this way, by comparison with the scattering amplitude of a potential V ( x ) = m g φ ( x ) in the Born approximation, which is A = − i m g φ ( q ) , we can identify the gravitational mass of the fermion. From the previous expression, we see that the gravitationalmass is obtained when the spatial momenta of the fermion are vanishing: m g = lim p s → " (2 u λ u ρ − η λρ ) p Z ( p ) Z ( p ′ ) (¯ u β ( p ′ , s ) M λρ ( p, p ′ ) u β ( p, s )) On-shell p ′ → p , where E β = E β ( p ) is the on-shell energy of the particle, i.e. the position of the pole of the self-energy, and p s = p ( p · u ) − p .7At leading order, as we are now showing, gravitational mass coincides with inertial and phase space mass. Atleading order Z = 1 + O ( e ), the thermal Dirac spinor reduces to the usual free Dirac spinor and the matrix elementis simply the tree-level diagram M λρ ( p, p ′ ) = 14 [ γ λ ( p + p ′ ) ρ + γ ρ ( p + p ′ ) λ ] − η λρ (cid:0) /p − m + /p ′ − m (cid:1) . Proceeding to evaluate the gravitational mass step by step, we first find (cid:0) ¯ u ( p ′ , s ) M λρ ( p, p ′ ) u ( p, s ) (cid:1) p ′ = p = ¯ u ( p, s ) (cid:26)
12 [ γ λ p ρ + γ ρ p λ ] − η λρ (cid:0) /p − m (cid:1)(cid:27) u ( p, s ) . Then, since p is taken on-shell, we take advantage of spinor properties (see Sec. C 1 for the conventions used), whichin the limit of p s going to zero give¯ u ( p, s ) u ( p, s ) | p s =0 = 1 + O ( e ) , ¯ u ( p, s ) γ µ u ( p, s ) | p s =0 = p µ m + O ( e ) , and we obtain lim p s → (¯ u ( p ′ , s ) M λρ ( p, p ′ ) u ( p, s )) p ′ = p = lim p s → m (cid:26)
12 ( p λ p ρ + p ρ p λ ) − η λρ (cid:0) p − m (cid:1)(cid:27) = lim p s → (cid:26) m ( p λ p ρ + p ρ p λ ) (cid:27) . At last, we find m g = lim p s → (cid:20) m (2 u λ u ρ − η λρ ) ( p λ p ρ + p ρ p λ ) (cid:21) = lim p s → u · p ) − p m = m, where we used the on-shell condition p · u = E β ( p s = 0) = m + O ( e ). At leading order, gravitational and inertial massare indeed equivalent m g = m I = m p = m . Instead, for 1-loop QED it has been proved [30, 44] that gravitationalmass is different from inertial mass, in particular for small temperature T ≪ m their ratio is m I m g = 1 + e T m . This is a manifest breaking of the weak equivalence principle caused by finite temperature effects.
Appendix B: Selection of the form factors
In this appendix we show that only the following terms M µν Relevant ( p, p ′ ) = I P γ ( P, q ) ( γ µ P ν + γ ν P µ ) + I uγ ( P, q ) ( γ µ u ν + γ ν u µ ) ++ I P l ( P, q ) / ˆ l (cid:16) ˆ l µ P ν + ˆ l ν P µ (cid:17) + I ul ( P, q ) / ˆ l (cid:16) ˆ l µ u ν + ˆ l ν u µ (cid:17) , with P = p ′ + p and q = p ′ − p , can contribute to the axial vortical effect (AVE) or to the gravitomagnetic moment.First, notice that M reproduces the stress-energy tensor matrix elements when evaluated between the two Diracspinors ¯ u ( p ′ ) and u ( p ): ¯ u s ′ ( p ′ ) M µν ( p, p ′ ) u s ( p ) = h p ′ , s ′ | b T µν (0) | p, s i . Therefore, taking advantage of the equation of motion and the gamma matrices algebra, we can write each term of M in terms of the tetrad { u, P, q, ˆ l } (defined in Sec. III), the metric η and at maximum one γ matrix. Comparing thespin-rotation amplitude (20) with the thermal spinor identities¯ u β ( p ′ ) γ u β ( p ) = ξ ′† ξ + O ( e ) , ¯ u β ( p ′ ) γ u β ( p ) = ξ ′† (cid:20) P m p + i σ ∧ q m p (cid:21) ξ + O ( e ) , M which contain exactly one gamma matrix can bring contribution to thegravitomagnetic moment. We come to the same conclusion for the AVE by looking at the trace in Eq. (10). Amongthe terms of M which contains one gamma matrix, the ones that contains /P and /q give¯ u ( p ′ ) /P u ( p ) = ¯ u ( p ′ ) 2 m u ( p ) , ¯ u ( p ′ ) /qu ( p ) = 0 , and hence do not contribute to AVE or to gravitomagnetic moment.Furthermore, from Eq. (10) the AVE is evaluated with the components M in the rest frame of the thermal bath.Then, only the terms which have a non-vanishing time-space component can contribute. Similarly, the gravitomagneticmoment is evaluated with the contraction M µν ( p ′ , p ) u µ A νg , therefore a relevant term must not vanish when contractedwith u and A g (notice that A g · u = 0). Therefore between the terms P µ γ ν + P ν γ µ , u µ γ ν + u ν γ µ , q µ γ ν + q ν γ µ , ˆ l µ γ ν + ˆ l ν γ µ , since u · q = u · ˆ l = 0 only the first two are relevant. The other terms left that contain a gamma matrix and thatsatisfy the conditions stated above are the following / ˆ l P µ P ν , / ˆ l ( u µ P ν + u ν P µ ) , / ˆ l ( u µ q ν + u ν q µ ) , / ˆ l (cid:16) u µ ˆ l ν + u ν ˆ l µ (cid:17) , / ˆ l ( P µ q ν + P ν q µ ) , / ˆ l (cid:16) P µ ˆ l ν + P ν ˆ l µ (cid:17) . (B1)For the AVE we see that to obtain the thermal coefficient we must evaluate the terms in (B1) for q = 0, then theterms proportional to q µ or q ν can not bring contribution. Moreover, by plugging the term / ˆ l P µ P ν in Eq. (8) and afterevaluating the trace and the derivatives, we see that it is odd under the transformation k → − k and therefore vanishafter momentum integration. For the gravitomagnetic moment we can write each terms in (B1) as / ˆ l ( w µ v ν + w ν v µ ),where w can be either u or P and v can be P, q or ˆ l . To evaluate the gravitomagnetic moment we should consider itat thermal bath rest frame: ¯ u ( p ′ ) / ˆ l u ( p )( w µ v ν + w ν v µ ) u µ A νg = ¯ u ( p ′ )( l · γ ) u ( p ) w ( v · A g ) . Since the gravitomagnetic moment is the coupling of spin and rotation, i.e. γ · A g , we can take advantage of threevector properties and write the scalar products as( γ · l )( v · A g ) = ( γ · A g )( v · l ) − ( γ ∧ v )( l ∧ A g ) . By definition of l , ( v · l ) is non vanishing if and only if v = ˆ l . Then, only the terms with ˆ l µ can contribute togravitomagnetic moment. At the end, we found that only the following terms can bring contribution to the AVE orto the gravitomagnetic moment: u µ γ ν + u ν γ µ , P µ γ ν + P ν γ µ , / ˆ l (cid:16) u µ ˆ l ν + u ν ˆ l µ (cid:17) , / ˆ l (cid:16) P µ ˆ l ν + P ν ˆ l µ (cid:17) . Appendix C: Renormalization of stress-energy tensor
In order to evaluate the radiative corrections to gravitomagnetic moment at finite temperature we have to considerthe renormalization of the stress-energy tensor at finite temperature and at first order on momentum transfer q .The zero temperature renormalization is performed with usual techniques and we do not discuss it here. To addressthermal corrections to the matrix-element of stress-energy tensor we use the real-time formalism of thermal fieldtheory. The major modification of QFT at finite temperature is the value of the vacuum of the theory, which is notempty but contain a number of bosons and fermions given respectively by the Bose-Einstein and the Fermi-Diracdistribution functions: b a † B b a B ( p ) | i = n B ( E ) | i , b a † F b a F ( p ) | i = n F ( E ) | i , where as usual we indicate n B ( x ) = 1exp( βx ) − , n F ( x ) = 1exp( βx ) + 1 . The resulting propagators of the gauge field and of the fermionic field in real-time formalism arei S F ( p ) = (cid:0) /p + m (cid:1) (cid:20) i p − m − πδ (cid:0) p − m (cid:1) n F ( p ) (cid:21) , i D µν ( k ) = − η µν (cid:20) i k + 2 πδ (cid:0) k (cid:1) n B ( k ) (cid:21) . (a)Tree Level (b)Self Energy (c)Counter Term(d)Electromagnetic Vertex (e)Contact (f)Photon Polarization FIG. 2. The diagrams contributing to the radiative corrections to the stress-energy tensor. The curly line is the stress-energytensor, the wavy line is the photon and the solid line is the fermion.
Perturbation theory and Feynman diagrams at finite temperature are unmodified compared to usual quantum fieldtheory except for the previous propagators. We see that the propagators in real-time formalism are naturally separatedinto a temperature part and into a T = 0 part. The T = 0 part has been addressed as usual and from now on wejust consider the thermal part. The diagrams responsible for radiative corrections to the stress-energy tensor matrixelement h p ′ , s ′ | T µν (0) | p, s i = 1 p Z ( p ) Z ( p ′ ) ¯ u β ( p ′ ) M µν ( p, p ′ ) u β ( p ) , are reported in Fig. 2. Furthermore, since we are only interested in the thermal part, each integrals corresponding toa Feynman diagram is weighted with a thermal function and it is therefore ultraviolet convergent and finite. All thedivergent part are already been taken care of with the T = 0 renormalization.Note that we can distinguish between two different regimes of temperature. Indeed, finite-temperature modificationmay arise both from the boson or the photon propagators. However, for low temperatures such that T ≪ m , thefermion distribution function is suppressed by a factor exp( − m/T ), which is negligible compared to contributions oforder ( T /m ) coming from the photon distribution. In the opposite regime, when T ≫ m , the photon contributionis still the same as low temperatures, while the fermion distribution can now also contribute with terms of order( T /m ) . Therefore, as in the case of the pole of the propagator, we can expect two different values of AGM valid inthe two regimes of low and high temperatures.As first step, we need to identify the mass shift and the wave-function renormalization constant. These quantitiesare obtained starting from the self-energy of the fermion, which is discussed in Sec. A 1. We found that the self-energycan be written as Σ( p ) = a/p + b/u + c and that the full fermion propagator becomes S ( p ) = (1 − a ) /p − b/u + m + c [(1 − a ) p − bu ] − ( m + c ) , (C1)which has a pole in ω = E β = p s + m p , with m p given by Eq. (A1).0
1. Thermal Dirac spinor
The Dirac equation in momentum space is modified according to the self-energy, which also include thermal modifica-tions. The thermal Dirac spinors u β ( p ) satisfy the modified Dirac equation corresponding to the new propagator (C1): (cid:2) /p − m R − Σ β ( p ) (cid:3) u β ( p ) = (cid:2) (1 − a ) /p − b/u − m R − c (cid:3) u β ( p ) = 0 , where m R indicates the zero temperature renormalized mass. The thermal spinor u β ( p ) satisfies the previous equationwhen p is the pole of the propagator, i.e. such that p · u = E β ( p s ). The thermal Dirac spinors are actually requiredto properly account for stress-energy tensor renormalization at finite temperature [30, 51].In thermal bath rest frame, we choose the normalization u † β ( p ) u β ( p ) = 1 . For convenience, we furthermore define a four vector and a scalar p T ≡ (1 − a ) p − bu, m T ≡ m R + c, so that the modified thermal Dirac equation is written as h /p T − m T i u β ( p ) = 0 . Therefore, it follows from the modified thermal Dirac equation that the thermal spinors satisfy the following identities:¯ u β ( p, s ) u β ( p, s ) = m T (1 − a ) E β − b , ¯ u β ( p, s ) γ µ u β ( p, s ) = p T µ (1 − a ) E β − b , X s u β ( p, s )¯ u β ( p, s ) = /p T + m T − a ) E β − b . In the non relativistic limit and in the frame where u = (1 , ), we can also show the validity of the following identities:¯ u β ( p ′ ) γ u β ( p ) = ξ ′† ξ + O ( e ) , ¯ u β ( p ′ ) γ u β ( p ) = ξ ′† (cid:20) P m p + i σ ∧ q m p (cid:21) ξ + O ( e ) , where ξ is the two component spinor, normalized such that ξ † ξ = 1. We used the identities above to compute thegravitational mass in Sec. A 3 and the gravitomagnetic moment in Sec. IV.
2. Wave-function renormalization constant Z The wave-function renormalization constant Z is obtained by requesting that the fermion field is properly renor-malized ψ R ( x ) = 1 √ Z ψ ( x ) . The temperature part of the wave-function renormalization constant is [43, 44] β Z ( ω, p s ) = 2 [(1 − a ) E β − b ] ∂D∂ω (cid:12)(cid:12)(cid:12) ω = E β , where E β is the pole of the propagator and D is the denominator of the propagator (C1) D ( ω, p s ) = (1 − a ) ( ω − p s ) + b − − a ) bω − ( m + c ) . At order e we find β Z = 1 + (cid:18) a + m ω ∂a∂ω + ∂b∂ω + mω ∂c∂ω (cid:19) (cid:12)(cid:12)(cid:12) ω = √ p s + m . (C2)1For the computation of AGM we are interested to the quantity ( p Z ( p ) Z ( p ′ )) − / in the limits of q → P s = p ( P · u ) − P →
0. Therefore, we just have to evaluate Z ( p ) with p an on-shell momentum and then performthe p s → a, b, c we find:lim p s → β Z ( p ) = 1 − e T m + e π Z ∞ d k n B ( k ) k − θ HT e T m , where the θ HT is defined such that it turns on the high temperatures contribution: θ HT ≡ ( T ≪ m T ≫ m . The low-temperature part is in agreement with [30, 43, 44]. Therefore we havelim p s → p β Z ( p ) β Z ( p ′ ) = 1 + e T m − e π I A + θ HT e T m , where we denoted I A ≡ π Z ∞ d k n B ( k ) k . For future convenience, noticing that for p s = 0 we have ω P = 2 m , we can write the factor coming from wave-functionrenormalization constants using ω P ≡ u · P instead of the mass m :lim P s → lim q → p β Z ( p ) β Z ( p ′ ) = 1 + e T ω P − e π I A + θ HT e T ω P .
3. Renormalization of stress-energy tensor at finite temperature
As last step to renormalize the stress-energy tensor we have to calculate the temperature contribution of thediagrams in Fig. 2. Here we write the general procedure and we leave the details in Appendix D.First, we recap the notation used. We indicate with q and P the momenta P = p ′ + p, q = p ′ − p where p and p ′ are the momenta of external legs of the diagrams in Fig. 2. Notice that q · P = 0. Moreover, scatteringtheory imposes the conservation of the time component of p and p ′ in the thermal bath rest frame, meaning q · u = 0.We are using this constraint when evaluating all the diagrams. It is then convenient to define the following scalar andfour-vectors: ω P = P · u, P s = q ω P − P , ˜ P µ = ( η µν − u µ u ν ) P ν = P µ − ω P u µ , l µ = ǫ µνρσ u ν ˜ P ρ q σ , ˆ l µ = l µ √− l , we also denote with a the ratio a ≡ p ω P − P ω P = P s ω P , < a < . Since we are considering 1-loop corrections, a generic diagrams of Fig. 2, which we label with X , can be written as M Xµν ( p, p ′ ) = Z d k (2 π ) f X ( p, p ′ , k ) D X ( p, p ′ , k ) ¯ u ( p ′ ) N Xµν u ( p ) , (C3)where all the spinorial structure is contained inside the numerator N X . Therefore N X can be simplified using Diracequation and it can be decomposed in the following terms:¯ u ( p ′ ) N Xµν u ( p ) =¯ u ( p ′ ) h N Xkk /k k µ k ν + N XP k /k ( P µ k ν + P ν k µ )++ ( N X ( s ) P γ + N X ( k ) P γ /k )( P µ γ ν + P ν γ µ ) + ( N X ( s ) kγ + N X ( k ) kγ /k )( k µ γ ν + k ν γ µ ) + · · · i u ( p ) (C4)2where each term N Xyy can depend on the scalars { k , P , q , k · P, k · q, P · q, m } and the dots stands for terms thatto do not contribute to AGM. Indeed, we show in App. B that the only terms that are relevant for AGM are u µ γ ν + u ν γ µ , P µ γ ν + P ν γ µ , / ˆ l (cid:16) u µ ˆ l ν + u ν ˆ l µ (cid:17) , / ˆ l (cid:16) P µ ˆ l ν + P ν ˆ l µ (cid:17) . Using the orthogonal non-normalized basis { u, ˜ P , q, ˆ l } , the integration variable k can be decomposed into: k µ =( k · u ) u µ + ( k · ˜ P )˜ P ˜ P µ + ( k · q ) q q µ − ( k · ˆ l )ˆ l µ = " ( k · u ) − ω P ( k · ˜ P )˜ P u µ + ( k · ˜ P )˜ P P µ + ( k · q ) q q µ − ( k · ˆ l )ˆ l µ ≡ A u u µ + A P P µ + A q q µ + A l ˆ l µ . This decomposition is used to write the diagrams in a covariant form. Consider the term ¯ u ( p ′ ) /ku ( p ) v µ w µ , where v and w are any vector between { u, P, q, l } , the term is decomposed into¯ u ( p ′ ) /ku ( p ) v µ w ν = ¯ u ( p ′ ) h A u /u + A l / ˆ l + 2 mA P i u ( p ) v µ w ν , and for what we show in Sec. IV B only the term in /l can contribute to AGM:¯ u ( p ′ ) /ku ( p ) v µ w ν → A l ¯ u ( p ′ ) / ˆ l v µ w ν u ( p ) with ( v, w ) = ( u, l ) or ( v, w ) = ( P, l ) . With the same argument, we can select only the parts relevant to AGM of all the possible terms:¯ u ( p ′ ) /ku ( p ) ( k µ v ν + k ν v µ ) → ¯ u ( p ′ ) h A u A l / ˆ l ( u µ v ν + u ν v µ ) + A P A l / ˆ l ( P µ v ν + P ν v µ ) i u ( p ) if v = l → ¯ u ( p ′ ) h A l / ˆ l (ˆ l µ v ν + ˆ l ν v µ ) i u ( p ) if v = u or v = P ¯ u ( p ′ ) /kk µ k ν u ( p ) → ¯ u ( p ′ ) h A u A l / ˆ l ( u µ ˆ l ν + u ν ˆ l µ ) + A P A l / ˆ l ( P µ ˆ l ν + P ν ˆ l µ ) i u ( p ) . We then approximate the integrand to first order in q and we perform the loop integral in k decomposing its componentalong the tetrad { u, ˜ P , q, l } . The results at first order in q are (see Appendix D):( Z ( p ) Z ( p ′ )) − / M (0) µν = I ZP γ ( P µ γ ν + P ν γ µ ) ,M SEµν + M CTµν = I SP γ ( P µ γ ν + P ν γ µ ) ,M Vµν = (cid:0) I VP γ + θ HT I VP γ,f (cid:1) ( P µ γ ν + P ν γ µ ) + (cid:0) I Vuγ + θ HT I Vuγ,f (cid:1) ( u µ γ ν + u ν γ µ )+ (cid:0) I VP l + θ HT I VP l,f (cid:1) / ˆ l (cid:16) P µ ˆ l ν + P ν ˆ l µ (cid:17) + θ HT I Vul,f / ˆ l (cid:16) u µ ˆ l ν + u ν ˆ l µ (cid:17) ,M Cµν = (cid:0) I CP γ + θ HT I CP γ,f (cid:1) ( P µ γ ν + P ν γ µ ) + (cid:0) I Cuγ + θ HT I Cuγ,f (cid:1) ( u µ γ ν + u ν γ µ ) ,M Pµν = I PP γ ( P µ γ ν + P ν γ µ ) + I Puγ ( u µ γ ν + u ν γ µ ) + (cid:0) I PP l + θ HT I PP l,f (cid:1) / ˆ l (cid:16) P µ ˆ l ν + P ν ˆ l µ (cid:17) + (cid:0) I Pul + θ HT I Pul,f (cid:1) / ˆ l (cid:16) u µ ˆ l ν + u ν ˆ l µ (cid:17) . Here, the function θ HT ≡ ( T ≪ m T ≫ m turns on the fermionic contributions at high temperature and removes them for low temperatures; we also introducethe IR divergent integral I A ≡ π Z ∞ d k n B ( k ) k . P µ γ ν + P ν γ µ arelim P s → lim q → I ZP γ = 14 + 2 + θ HT e T ω P − e π I A ;lim P s → lim q → I SP γ = − θ HT e T ω P + 12 e π I A ; I VP γ = − e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) − e π m ω P I A − a ; I VP γ,f = − e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) ; I CP γ = 118 e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) ; I CP γ,f = 136 e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) ; I PP γ = − e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) . The form factors of u µ γ ν + u ν γ µ are I Vuγ = − e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) ; I Vuγ,f = − e T ω P (cid:20) a − − a )4 a log (cid:18) a − a (cid:19)(cid:21) ; I Cuγ = 19 e T ω P (cid:20) a − − a )4 a log (cid:18) a − a (cid:19)(cid:21) ; I Cuγ,f = 118 e T ω P (cid:20) a − − a )4 a log (cid:18) a − a (cid:19)(cid:21) ; I Puγ = − e T ω P (cid:20) a − − a )4 a log (cid:18) a − a (cid:19)(cid:21) . The form factors of / ˆ l ( P µ ˆ l ν + P ν ˆ l µ ) are I VP l = 112 e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) ; I VP l,f = 7360 e T ω P (cid:20)
127 4 m ω P (cid:18) − a )2 a (1 − a ) − a log (cid:18) a − a (cid:19)(cid:19) − (cid:18) a log (cid:18) a − a (cid:19) − a (cid:19)(cid:21) ; I PP l = I PP γ = − e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) ; I PP l,f = − e T ω P m ω P (cid:20) − a )8 a log (cid:18) a − a (cid:19) − a (cid:21) . The form factors of / ˆ l ( u µ ˆ l ν + u ν ˆ l µ ) are I Vul,f = − e T ω P (cid:20) −
23 4 m ω P (cid:18) − a )8 a log (cid:18) a − a (cid:19) − a (cid:19) + 53 (cid:18) a − − a )4 a log (cid:18) a − a (cid:19)(cid:19)(cid:21) ; I Pul = I Puγ = − e T ω P (cid:20) a − − a )4 a log (cid:18) a − a (cid:19)(cid:21) ; I Pul,f = − e T ω P m ω P (cid:20) − a )8 a log (cid:18) a − a (cid:19) − a (cid:21) . These functions are written such that the quantity inside the square brackets is 1 for a = 0, which correspond tothe non relativistic particle limit ( P s → Appendix D: Radiative correction to the stress-energy tensor
In this appendix we evaluate the temperature modification of the diagrams in Fig. 2. The general strategy and thefinal results are written in Sec. C 3, in what follows we provide a detailed calculations of all the diagrams.To perform the loop integration on the momentum k in the generic diagram (C3), since the set { u, ˜ P , q, ˆ l } is abasis, we choose k along u , k z along q µ and k x along ˜ P and k y along l ; thus, defining ǫ = p − q , a = p − ˜ P ω P = p ω P − P ω P , < a < , we have ( k · u ) = k , ( k · q ) = − ǫ k z , ( k · ˜ P ) = − aω P k x ( k · P ) = ω P ( k − ak x ) , ( k · ˆ l ) = − k y A u = k − k x a , A P = k x aω P , A q = k z ǫ , A l = k y . At last we define the following unit vectors inside the k integration:ˆ k x = k x | k | = cos φ sin θ, ˆ k y = k y | k | = sin φ sin θ, ˆ k z = k z | k | = cos θ. For the fermionic part we only consider the high temperature limit, T ≫ m, P s , ω p , q . Every fermionic form factorat first order in ǫ can be written as I = C Z d kE k k l n F ( E k ) X s = ± Z dΩ k f s (ˆ k, m, ω p , a ) , where C is a numerical constant and l an integer. Because of the Fermi-Dirac distribution function, the majorcontribution of the integral comes from k ∼ T . Then in the integrand we can consider m and ω P to be smallcompared to k . To consider the relevant part of the integral we first replace m = m r ky, ω P = ω P r ky, E k = k p m r y , where y is a dummy variable to perform a Taylor series. Then we expand the integrand in series of y around zero andwe keep only the first terms. Then we replace back y = 1 , m r = mk , ω P r = ω P k . In this way we can select the relevant contribution, which has the form I ≃ C Z d k n F ( E k ) X s = ± Z dΩ k ˜ f s (ˆ k, m, ω p , a ) = Cπ T X s = ± Z dΩ k ˜ f s (ˆ k, m, ω p , a ) . The last factor can be obtained by summing and integrating over the angles. The angular integrals have the form I Ω ( n x , n y , n z , n d ) = Z dΩ k k n x x ˆ k n y y ˆ k n z z (1 − a ˆ k x ) n d and their results are quoted in Sec. D 5.
1. Self-energy and Counter-terms
The fermion-self energy diagram, Fig. 2(b), is M SEµν = ¯ u ( p ′ ) β Σ( p ′ )i S F ( p ′ ) V µν ( p ′ , p ) u ( p ) , where V µν is the stress-energy tensor fermion coupling: V µν ( p, p ′ ) = 14 [ γ µ ( p + p ′ ) ν + γ ν ( p + p ′ ) µ ] − η µν (cid:2) /p − m + /p ′ − m (cid:3) . β Σ( p ) = (cid:0) − β Z ( p ) − (cid:1) (cid:0) /p − m (cid:1) + a/p + b/u + δm. Most of this diagram is canceled by the counter-term in Fig. 2(c). Since the temperature dependent Dirac equation is (cid:0) (1 − a ) /p − b/u − m T (cid:1) u β ( p ) = 0 , we must use the finite temperature counter-term of the momentum Lagrangian δ L = a/p + b/u + δm. Therefore, the counter term is M CTµν = − ¯ u ( p ′ ) ( a /p ′ + b/u + δm ) i S F ( p ′ ) V µν ( p ′ , p ) u ( p ) . Considering both legs we have in total M SEµν + M CTµν = (cid:0) − β Z ( p ) − − β Z ( p ′ ) − (cid:1) ¯ u ( p ′ ) V µν ( p ′ , p ) u ( p ) , whose contribution to AGM is M SEµν + M CTµν = 14 (cid:0) − β Z ( p ) − − β Z ( p ′ ) − (cid:1) ¯ u ( p ′ ) ( P µ γ ν + P ν γ µ ) u ( p )= I SP γ ¯ u ( p ′ ) ( P µ γ ν + P ν γ µ ) u ( p ) . In the limit q → P s → P s → I SP γ = − e T ω P + 12 e π I A − θ HT e T ω P , where θ HT is vanishing for T ≪ m and it goes to one for T ≫ m .
2. Photon polarization diagram
Now we want to check if gravitomagnetic moment gets finite temperature corrections from the diagram of photonpolarization Fig. 2(f): M Pµν ( p, p ′ ) = Z d k (2 π ) ¯ u ( p ′ )( − i eγ ρ )i S F ( p − k )( − i eγ σ ) u ( p ) T αβµν ( k, q + k )i D ρβ ( q + k )i D σα ( k )where the stress-energy tensor-photon coupling vertex is T αβµν ( k, p ) = − ( δ αµ δ βν + δ αν δ βµ ) k · p − g αβ ( k µ p ν + k ν p µ ) + ( k µ δ βν + k ν δ βµ ) p α ++ ( p µ δ αν + p ν δ αµ ) k β + g µν ( δ αβ k · p − k α p β ) . At low temperature T ≪ m we can neglect the part coming from the fermionic thermal distribution. The real part ofthe thermal contribution of the diagram is thenRe β M Pµν ( p, p ′ ) = e Z d k (2 π ) ¯ u ( p ′ ) γ α /p − /k + m ( k − p ) − m γ β u ( p ) T αβµν ( k, q + k ) ×× " δ (cid:0) ( q + k ) (cid:1) n B ( q + k ) k + δ (cid:0) k (cid:1) n B ( k )( q + k ) . Making the changing of variables k → k − q in the first term we haveRe β M Pµν ( p, p ′ ) = e Z d k (2 π ) ¯ u ( p ′ ) γ α " /p − /k + /q + m ( k − q − p ) − m T αβµν ( k − q, k )( k − q ) ++ /p − /k + m ( k − p ) − m T αβµν ( k, q + k )( q + k ) γ β u ( p ) δ (cid:0) k (cid:1) n B ( k ) ≡ e Z d k (2 π ) ¯ u ( p ′ ) (cid:20) N D + N D (cid:21) u ( p ) δ (cid:0) k (cid:1) n B ( k ) . (D1)6We are interested in linear order of q , therefore we are using the momenta P and q : ( P = p ′ + p, p = ( P − q ) q = p ′ − p, p ′ = ( P + q ) . We can use gamma algebra to simplify the expressions of the numerators of the diagram in Eq. (D1). Takingadvantage of the Dirac equation and setting k = 0 from the Dirac delta, we find:¯ u ( p ′ ) N u ( p ) = ¯ u ( p ′ ) n ( − /k − m ) k µ k ν + ( k · q )( P µ γ ν + P ν γ µ ) + 2( k · P )( k µ γ ν + k ν γ µ ) + (2 m + 2 /k ) ( k µ q ν + k ν q µ ) − ( k · P )( q µ γ ν + q ν γ µ ) + (cid:0) − m ( k · q ) − ( k · P ) /k − ( k · q ) /k + q /k (cid:1) η µν o u ( p );¯ u ( p ′ ) N u ( p ) = ¯ u ( p ′ ) n ( − /k − m ) k µ k ν − ( k · q )( P µ γ ν + P ν γ µ ) + 2( k · P )( k µ γ ν + k ν γ µ ) − (2 m + 2 /k ) ( k µ q ν + k ν q µ )+ ( k · P )( q µ γ ν + q ν γ µ ) + (cid:0) m ( k · q ) − ( k · P ) /k + ( k · q ) /k + q /k (cid:1) η µν o u ( p ) . Decomposing the four-vector k with the tetrad { u, ˜ P , q, ˆ l } we obtain:¯ u ( p ′ ) N u ( p ) =¯ u ( p ′ ) n (cid:16) A u /u + A l / ˆ l + m + 2 A p m (cid:17) h − A u u µ u ν − A P P µ P ν − A l ˆ l µ ˆ l ν − A u A P ( u µ P ν + u ν P µ ) − A u A l ( u µ ˆ l ν + u ν ˆ l µ ) − A P A l ( P µ ˆ l ν + P ν ˆ l µ ) + (4 A q − A q ) q µ q ν + (2 A u − A u A q )( u µ q ν + u ν q µ )+ (2 A P − A P A q )( P µ q ν + P ν q µ ) + (2 A l − A l A q )(ˆ l µ q ν + ˆ l ν q µ ) i + 2( k · P ) A u ( u µ γ ν + u ν γ µ )+ [2( k · P ) A P + ( k · q )] ( P µ γ ν + P ν γ µ ) + 2( k · P ) A l (ˆ l µ γ ν + ˆ l ν γ µ ) + [2( k · P ) A q − ( k · P )] ( q µ γ ν + q ν γ µ ) − h m ( k · q ) + (cid:0) ( k · q ) + ( k · P ) − q (cid:1) (cid:16) A u /u + A l / ˆ l + 2 mA P (cid:17)i η µν o u ( p );and¯ u ( p ′ ) N u ( p ) =¯ u ( p ′ ) n (cid:16) A u /u + A l / ˆ l + m + 2 A p m (cid:17) h − A u u µ u ν − A P P µ P ν − A l ˆ l µ ˆ l ν − A u A P ( u µ P ν + u ν P µ ) − A u A l ( u µ ˆ l ν + u ν ˆ l µ ) − A P A l ( P µ ˆ l ν + P ν ˆ l µ ) + (4 A q − A q ) q µ q ν + (2 A u − A u A q )( u µ q ν + u ν q µ )+ (2 A P − A P A q )( P µ q ν + P ν q µ ) + (2 A l − A l A q )(ˆ l µ q ν + ˆ l ν q µ ) i + 2( k · P ) A u ( u µ γ ν + u ν γ µ )+ [2( k · P ) A P − ( k · q )] ( P µ γ ν + P ν γ µ ) + 2( k · P ) A l (ˆ l µ γ ν + ˆ l ν γ µ ) + [2( k · P ) A q + ( k · P )] ( q µ γ ν + q ν γ µ )+ h m ( k · q ) + (cid:0) ( k · q ) − ( k · P ) + q (cid:1) (cid:16) A u /u + A l / ˆ l + 2 mA P (cid:17)i η µν o u ( p ) . For the denominators, using k = 0 , P · q = 0 , P = 4 m − q , we find D = ( P · k + q · k ) (cid:0) q · k − q (cid:1) D = − ( P · k − q · k ) (cid:0) q · k + q (cid:1) . Choosing the frame described at the beginning of this section to decompose the momentum k we can write thedenominators as: D = − ω P k z ǫ ( k − k x a ) + ǫ (cid:0) ω P ( k − k x a ) + 2 k z (cid:1) − k z ǫ D =2 ω P k z ǫ ( k − k x a ) + ǫ (cid:0) ω P ( k − k x a ) + 2 k z (cid:1) + k z ǫ . Since the denominators does not contains k y every term that is odd in A l = k y is vanishing. Moreover, we can nowselect only the pieces that could give contribution to AGM, they are the following¯ u ( p ′ ) N u ( p ) = ¯ u ( p ′ ) n − A u A l / ˆ l ( u µ ˆ l ν + u ν ˆ l µ ) − A P A l / ˆ l ( P µ ˆ l ν + P ν ˆ l µ ) + 2( k · P ) A u ( u µ γ ν + u ν γ µ )+ [2( k · P ) A P + ( k · q )] ( P µ γ ν + P ν γ µ ) o u ( p ) , and ¯ u ( p ′ ) N u ( p ) = ¯ u ( p ′ ) n − A u A l / ˆ l ( u µ ˆ l ν + u ν ˆ l µ ) − A P A l / ˆ l ( P µ ˆ l ν + P ν ˆ l µ ) + 2( k · P ) A u ( u µ γ ν + u ν γ µ )+ [2( k · P ) A P − ( k · q )] ( P µ γ ν + P ν γ µ ) o u ( p ) . /l ( u µ ˆ l ν + u ν ˆ l µ ) at first order in ǫ we have I Pul = − e Z d k (2 π ) (cid:20) D + 4 D (cid:21) A u A l δ (cid:0) k (cid:1) n B ( k ) ≃ lim ǫ → e ω P Z d k (2 π ) (cid:20) ω P ( k − ak x )( k z − ǫ /
4) + 2( k − ak x ) (cid:21) ( k − k x /a ) k y δ (cid:0) k (cid:1) n B ( k ) , integrating k with the delta we find that the second term in square bracket is odd on k x and so vanishing; the firstterm becomes I Pul = lim ǫ → e ω P Z d k (2 π ) k n B ( k ) Z dΩ k − ˆ k x )ˆ k y (1 − a ˆ k x )(ˆ k z − ǫ / (4 k )) . The angular integrals do not converge for ǫ = 0 but they have a finite result in the principal value sense:lim τ → Z dΩ k k y (1 − a ˆ k x )(ˆ k z − τ ) = − π a log (cid:18) a − a (cid:19) , lim τ → Z dΩ k k x ˆ k y (1 − a ˆ k x )(ˆ k z − τ ) = − π (cid:18) a log (cid:18) a − a (cid:19) − a (cid:19) and therefore I Pul ≡ I Pu = − e T ω P (cid:20) a − − a )4 a log (cid:18) a − a (cid:19)(cid:21) . For the term in / ˆ l ( P µ ˆ l ν + P ν ˆ l µ ) we have I PP l = − e Z d k (2 π ) (cid:20) D + 4 D (cid:21) A P A l δ (cid:0) k (cid:1) n B ( k ) ≃ lim ǫ → e aω P Z d k (2 π ) (cid:20) ω P ( k − ak x )( k z − ǫ /
4) + 2( k − ak x ) (cid:21) k x k y δ (cid:0) k (cid:1) n B ( k ) , integrating k with the delta we find that the second term in square bracket is odd on k x and so vanishing; the firstterm becomes I PP l = lim τ → e ω P Z d k (2 π ) k n B ( k ) Z dΩ k k x ˆ k y (1 − a ˆ k x )(ˆ k z − τ ) . After integration we obtain I PP l ≡ I PP = − e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) . Consider now the part in ( u µ γ ν + u ν γ µ ). At linear order of ǫ the scalar part in front of it is I Puγ = e Z d k (2 π ) (cid:20) D + 1 D (cid:21) k · P ) A u δ (cid:0) k (cid:1) n B ( k ) ≃ e Z d k (2 π ) (cid:20) aω P k x − ak k − ak x + k x − ak ak z (cid:21) δ (cid:0) k (cid:1) n B ( k )= e π ) X s = ± Z d k k n B ( k ) Z dΩ k " aω P ˆ k x − ass − a ˆ k x + 1 k ˆ k x − asa ˆ k z = e π ) Z d k k n B ( k ) Z dΩ k " − ω P − ˆ k x − a ˆ k x + 2 k ˆ k x a ˆ k z = − e ω P π ) Z d k k n B ( k ) Z dΩ k − ˆ k x )(1 − a ˆ k x ) . Using the angular integrals Z dΩ k − a ˆ k x = 8 π a log (cid:18) a − a (cid:19) , Z dΩ k k x − a ˆ k x = 8 π (cid:18) a log (cid:18) a − a (cid:19) − a (cid:19) I Puγ = I Pu = − e T ω P (cid:18) a − − a )4 a log (cid:18) a − a (cid:19)(cid:19) . The part proportional to ( P µ γ ν + P ν γ µ ) at first order in ǫ is the integral I PP γ = e Z d k (2 π ) (cid:16) k · ˜ P ) A P + ( k · q ) (cid:17) D + (cid:16) k · ˜ P ) A P − ( k · q ) (cid:17) D δ (cid:0) k (cid:1) n B ( k ) ≃ e Z d k (2 π ) (cid:20) ω P k − ak x − k x aω P ( k − ak x ) − k x aω P k z (cid:21) δ (cid:0) k (cid:1) n B ( k )= e ω P π ) X s = ± Z d k k n B ( k ) Z dΩ k " k s − a ˆ k x − k x aω P ( s − a ˆ k x ) − k ˆ k x a ˆ k z = e ω P π ) Z d k k n B ( k ) Z dΩ k " k a ˆ k x − a ˆ k x − k x ω P (1 − a ˆ k x ) − k ˆ k x a ˆ k z = − e ω P π ) Z d k k n B ( k ) Z dΩ k k x (1 − a ˆ k x ) , where we first integrated k with the delta and then we performed the angular integration and removed the manifestlyvanishing angular integrations. After integration we obtain: I PP γ = I PP = − e T ω P (cid:18) a log (cid:18) a − a (cid:19) − a (cid:19) . Summing all the relevant terms we found that the diagram can be written as: M Pµν = I Pu ( u µ γ ν + u ν γ µ ) + I PP ( P µ γ ν + P ν γ µ ) + I Pu / ˆ l (cid:16) u µ ˆ l ν + u ν ˆ l µ (cid:17) + I PP / ˆ l (cid:16) P µ ˆ l ν + P ν ˆ l µ (cid:17) . Using this result, the contribution to gravitomagnetic moment from the photon polarization diagram at low temper-atures is vanishing g P Ω = lim a → I PP γ + I Puγ ω P − I PP l − I Pul ω P ! = 0 . a. Fermionic part: High temperature The fermionic part is negligible at low temperature but it is comparable to the bosonic part at high temperatures.We obtain the fermionic part fromRe β M Pµν ( p, p ′ ) = − e Z d k (2 π ) ¯ u ( p ′ ) γ β (cid:0) /p − /k + m (cid:1) γ α u ( p ) T αβµν ( k, q + k ) δ (cid:0) ( p − k ) − m (cid:1) n F ( p − k )( q + k ) k . Changing variables into k → p − k we haveRe β M Pµν ( p, p ′ ) = − e Z d k (2 π ) ¯ u ( p ′ ) γ β ( /k + m ) γ α u ( p ) T αβµν ( p − k, q + p − k ) δ (cid:0) k − m (cid:1) n F ( k )( q + p − k ) ( p − k ) = − e Z d k (2 π ) ¯ u ( p ′ ) N P u ( p ) D P δ (cid:0) k − m (cid:1) n F ( k ) . The numerator can be simplified into:¯ u ( p ′ ) N P u ( p ) = ¯ u ( p ′ ) n (4 /k − m ) k µ k ν + /k q µ q ν + (cid:2) ( k · P ) − m (cid:3) ( P µ γ ν + P ν γ µ ) − /kP µ P ν + q ( k µ γ ν + k ν γ µ ) − ( k · q )( q µ γ ν + q ν γ µ ) + 2 m ( k µ P ν + k ν P µ ) + (cid:0) m − k · P ) m − mq (cid:1) η µν o u ( p ) . q is¯ u ( p ′ ) N P u ( p ) = ¯ u ( p ′ ) n /k k µ k ν + (cid:2) ( k · P ) − m (cid:3) ( P µ γ ν + P ν γ µ ) o u ( p ) . and after k decomposition the relevant part is¯ u ( p ′ ) N P u ( p ) = ¯ u ( p ′ ) n A u A l / ˆ l ( u µ ˆ l ν + u ν ˆ l µ ) + 4 A P A l / ˆ l ( P µ ˆ l ν + P ν ˆ l µ ) + (cid:0) k · P − m (cid:1) ( P µ γ ν + P ν γ µ ) o u ( p ) . The denominator is D P = ( q + p − k ) ( p − k ) = (cid:0) m − ( k · P ) (cid:1) − ( k · q ) = (2 m − ω P ( k − ak x )) − ǫ k z . For the term in / ˆ l ( u µ ˆ l ν + u ν ˆ l µ ) we have I Pul,f = − e Z d k (2 π ) A u A l D δ (cid:0) k − m (cid:1) n F ( k ) ≃ − e Z d k (2 π ) ( k − k x /a ) k y [ ω P ( k − ak x ) − m ] δ (cid:0) k − m (cid:1) n F ( k )= − e π ω P Z d k k E k n F ( E k ) X s = ± Z dΩ k ( sE k /k − ˆ k x /a )ˆ k y [( sE k /k − a ˆ k x ) − m kω P ] Using the high temperature expansion described at the beginning of this appendix we find: I Pul,f ≃ − e T m πω P (cid:0) I Ω (0 , , ,
3) + 3( a − I Ω (2 , , , − a I Ω (4 , , , (cid:1) and at the end I Pul,f ≃ − e T ω P m ω P (cid:20) − a )8 a log (cid:18) a − a (cid:19) − a (cid:21) . For the term in ( P µ ˆ l ν + P ν ˆ l µ ) we have I PP l,f = − e Z d k (2 π ) A P A l D δ (cid:0) k − m (cid:1) n F ( k ) ≃ − e aω P Z d k (2 π ) k x k y [ ω P ( k − ak x ) − m ] δ (cid:0) k − m (cid:1) n F ( k )= − e π aω P Z d k k E k n F ( E k ) X s = ± Z dΩ k ˆ k x ˆ k y h ( sE k /k − a ˆ k x ) − m kω P i . At high temperatures it becomes I PP l,f = − e m T πω P (cid:0) I Ω (2 , , ,
3) + a I Ω (4 , , , (cid:1) and hence I PP l,f = − e T ω P m ω P (cid:20) − a )8 a log (cid:18) a − a (cid:19) − a (cid:21) . The part proportional to ( P µ γ ν + P ν γ µ ) at first order in ǫ is the integral I PP γ,f = − e Z d k (2 π ) ( k · P ) − m + A P q D δ (cid:0) k − m (cid:1) n F ( k ) ≃ − e Z d k (2 π ) ω P ( k − ak x ) − m δ (cid:0) k − m (cid:1) n F ( k )= − e π ω P Z d k kE k n F ( E k ) X s = ± Z dΩ k sE k /k − a ˆ k x − m kω P . I PP γ,f = Z d k n F ( k ) X n =1 k − n f n ( a ) , which gives logarithmic and sub-leading contributions in temperature that we can neglect. Notice that there is noIR divergence, because the previous integral is just an approximation for high k , at low k when the divergence wouldoccur the mass of the particle prevent the divergence.Summing all the relevant terms we found that the diagram can be written as: M Pµν,f = I Pu,f / ˆ l (cid:16) u µ ˆ l ν + u ν ˆ l µ (cid:17) + I PP,f / ˆ l (cid:16) P µ ˆ l ν + P ν ˆ l µ (cid:17) , with I Pu,f = − e T m ω P (cid:20) − a )8 a log (cid:18) a − a (cid:19) − a (cid:21) ,I PP,f = − e m T ω P (cid:20) − a )2 a (1 − a ) − a log (cid:18) a − a (cid:19)(cid:21) . Using this result to evaluate the amplitude
A ∝ M Pµν u µ A νg we find that the gravitomagnetic moment coming fromphoton polarization diagram is g P Ω = − lim P s → lim q → I Pu,f ω P + I PP,f ! θ HT = e T m θ HT . Only at high temperatures the polarization diagram contribute to AGM.
3. Electromagnetic vertex
From electromagnetic vertex correction, Fig. 2(d), we have M Vµν = Z d k (2 π ) ¯ u ( p ′ )( − i eγ α )i S F ( p ′ − k ) V µν ( p − k, p ′ − k )i S F ( p − k )( − i eγ β )i D αβ ( k ) u ( p ) , where V µν is the stress-energy tensor fermion coupling: V µν ( p, p ′ ) = 14 [ γ µ ( p + p ′ ) ν + γ ν ( p + p ′ ) µ ] − η µν (cid:2) /p − m + /p ′ − m (cid:3) . As before, we evaluate the temperature part and the relevant part at low temperature is only given by the Bosedistribution term: β M Vµν = − e Z d k (2 π ) ¯ u ( p ′ ) γ α ( /p ′ − /k + m ) V µν ( p − k, p ′ − k )( /p − /k + m ) γ α u ( p )[( p ′ − k ) − m ][( p − k ) − m ] δ (cid:0) k (cid:1) n B ( k ) ≡ − e Z d k (2 π ) ¯ u ( p ′ ) N V u ( p ) D V . After simplification we obtain¯ u ( p ′ ) N V u ( p ) = − ¯ u ( p ′ ) n (4 /k + 4 m ) k µ k ν + 2 /kP µ P ν − ( m + 3 /k ) ( k µ P ν + k ν P µ )+ (cid:0) m − q / − ( k · P ) (cid:1) ( P µ γ ν + P ν γ µ ) + (cid:0) − m + q + 2( k · P ) (cid:1) ( k µ γ ν + k ν γ µ )+ 2 ( m + /k ) ( k · P ) η µν o u ( p ) . The denominator is D V = [( k · P ) + ( k · q )] [( k · P ) − ( k · q )] = ω P ( k − ak x ) − k z ǫ . u ( p ′ ) N V u ( p ) = ¯ u ( p ′ ) n /k k µ k ν − /k ( k µ P ν + k ν P µ ) + (cid:0) m − q / − ( k · P ) (cid:1) ( P µ γ ν + P ν γ µ )+ (cid:0) − m + q + 2( k · P ) (cid:1) ( k µ γ ν + k ν γ µ ) o u ( p ) . After decomposing k and removing odd terms in A l , we have to consider:¯ u ( p ′ ) N V u ( p ) = ¯ u ( p ′ ) n A u A l / ˆ l ( u µ ˆ l ν + u ν ˆ l µ ) + (4 A P − A l / ˆ l ( P µ ˆ l ν + P ν ˆ l µ )+ (cid:2)(cid:0) m − q / − ( k · P ) (cid:1) + (cid:0) − m + q + 2( k · P ) (cid:1) A P (cid:3) ( P µ γ ν + P ν γ µ )+ (cid:0) − m + q + 2( k · P ) (cid:1) A u ( u µ γ ν + u ν γ µ ) o u ( p ) . The part in / ˆ l ( u µ ˆ l ν + u ν ˆ l µ ) is I Vul = − e Z d k (2 π ) A u A l D V δ (cid:0) k (cid:1) n B ( k ) ≃ − e Z d k (2 π ) ( k − k x /a ) k y ω P ( k − ak x ) δ (cid:0) k (cid:1) n B ( k )= − e (2 π ) ω P Z d k k n B ( k ) X s = ± Z dΩ k ( s − ˆ k x /a )ˆ k y ( s − a ˆ k x ) = − e (2 π ) ω P Z d k k n B ( k ) Z dΩ k (cid:16) a ˆ k x − a ˆ k x + ˆ k x (cid:17) ˆ k y a (1 − a ˆ k x ) = 0 . The part in / ˆ l ( P µ ˆ l ν + P ν ˆ l µ ) is I VP l = − e Z d k (2 π ) (4 A P − A l D V δ (cid:0) k (cid:1) n B ( k ) ≃ e Z d k (2 π ) (3 − k x /aω P ) k y ω P ( k − ak x ) δ (cid:0) k (cid:1) n B ( k )= e π ) ω P Z d k k n B ( k ) X s = ± Z dΩ k (3 − k ˆ k x /aω P )ˆ k y ( s − a ˆ k x ) = 3 e π ) ω P Z d k k n B ( k ) Z dΩ k a ˆ k x )ˆ k y (1 − a ˆ k x ) = e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) . The term in P µ γ ν + P ν γ µ gives I VP γ ≃ − e Z d k (2 π ) (cid:20) m ( k − ak x ) − ω P k − ak x (cid:21) aω p − k x aω P δ (cid:0) k (cid:1) n B ( k )= − e π ) aω P Z d k k n B ( k ) X s = ± Z dΩ k " m k aω p − k ˆ k x ( s − a ˆ k x ) − ω P k aω p − k ˆ k x s − a ˆ k x = − e π aω P Z d k k n B ( k ) Z dΩ k " m k aω p − k ˆ k x )(1 + a ˆ k x )(1 − a ˆ k x ) − ω P k ( aω p − k ˆ k x )2 a ˆ k x − a ˆ k x = − e π ω P Z d k k n B ( k ) Z dΩ k " m k a ˆ k x )(1 − a ˆ k x ) + 2 2ˆ k x − a ˆ k x = − e π ω P Z dΩ k " m I A π a ˆ k x )(1 − a ˆ k x ) + 2 π T k x − a ˆ k x = − e m π ω P I A − a − e T ω P (cid:18) a log (cid:18) a − a (cid:19) − a (cid:19) . u µ γ ν + u ν γ µ gives I Vuγ ≃ e Z d k (2 π ) (cid:20) m ( k − ak x ) − ω P k − ak x (cid:21) k − k x /aω P δ (cid:0) k (cid:1) n B ( k )= e (2 π ) ω P Z d k k n B ( k ) X s = ± Z dΩ k " m k ( s − ˆ k/a )( s − a ˆ k x ) − ω p ( s − ˆ k x /a ) s − a ˆ k x = e (2 π ) ω P Z d k k n B ( k ) Z dΩ k " m k k x − a ˆ k x + a ˆ k x )) a (1 − a ˆ k x ) − ω P − ˆ k x )1 − a ˆ k x = − e (2 π ) ω P π T Z dΩ k − ˆ k x )1 − a ˆ k x = − e T ω P (cid:18) a log (cid:18) a − a (cid:19) − a (cid:19) . Summing all the relevant terms we found that the diagram can be written as: M Vµν = I Vuγ ( u µ γ ν + u ν γ µ ) + I VP γ ( P µ γ ν + P ν γ µ ) + I VP l / ˆ l (cid:16) P µ ˆ l ν + P ν ˆ l µ (cid:17) . a. Fermionic part: High temperature The fermionic part is:Re β M Vµν ( p, p ′ ) = e Z d k (2 π ) ¯ u ( p ′ ) γ α ( /p ′ − /k + m ) V µν ( p − k, p ′ − k )( /p − /k + m ) γ α u ( p ) k ×× " δ (cid:0) ( p − k ) − m (cid:1) ( p ′ − k ) − m n F ( p − k ) + δ (cid:0) ( p ′ − k ) − m (cid:1) ( p − k ) − m n F ( p ′ − k ) . We change k to p − k in the first term and k → p ′ − k in the second one:Re β M Vµν ( p, p ′ ) = e Z d k (2 π ) ¯ u ( p ′ ) γ α (cid:20) ( /p ′ − /p + /k + m ) V µν ( k, p ′ − p + k )( /k + m )[( p ′ − p + k ) − m ]( p − k ) + ( /k + m ) V µν ( p − p ′ + k, k )( /p − /p ′ + /k + m )[( p − p ′ + k ) − m ]( p ′ − k ) (cid:21) γ α u ( p ) n F ( k ) δ ( k − m ) ≡ e Z d k (2 π ) ¯ u ( p ′ ) (cid:20) N V D + N V D (cid:21) u ( p ) n F ( k ) δ ( k − m ) . After simplification the relevant parts for numerators are¯ u ( p ′ ) N V u ( p ) =¯ u ( p ′ ) n − /k k µ k ν − /k ( k µ P ν + k ν P µ ) + ( k · P + k · q ) ( k µ γ ν + k ν γ µ ) o u ( p );¯ u ( p ′ ) N V u ( p ) =¯ u ( p ′ ) n − /k k µ k ν − /k ( k µ P ν + k ν P µ ) + ( k · P − k · q ) ( k µ γ ν + k ν γ µ ) o u ( p ) . After decomposing k we only have to consider:¯ u ( p ′ ) N V u ( p ) = ¯ u ( p ′ ) n − A u A l / ˆ l ( u µ ˆ l ν + u ν ˆ l µ ) − (4 A P + 1) A l / ˆ l ( P µ ˆ l ν + P ν ˆ l µ )+ ( k · P + k · q ) A P ( P µ γ ν + P ν γ µ ) + ( k · P + k · q ) A u ( u µ γ ν + u ν γ µ ) o u ( p );¯ u ( p ′ ) N V u ( p ) = ¯ u ( p ′ ) n − A u A l / ˆ l ( u µ ˆ l ν + u ν ˆ l µ ) − (4 A P + 1) A l / ˆ l ( P µ ˆ l ν + P ν ˆ l µ )+ ( k · P − k · q ) A P ( P µ γ ν + P ν γ µ ) + ( k · P − k · q ) A u ( u µ γ ν + u ν γ µ ) o u ( p ) . D V =(4 m − k · P )( k · q ) + 2( k · q ) + q (2 m − k · P + k · q )=2 k z ǫ (cid:0) ω P k − aω P k x − m (cid:1) + ǫ (cid:0) ω P ( k − ak x ) + 2 k z − m (cid:1) + k z ǫ ; D V = − (4 m − k · P )( k · q ) + 2( k · q ) + q (2 m − k · P − k · q )= − k z ǫ (cid:0) ω P k − aω P k x − m (cid:1) + ǫ (cid:0) ω P ( k − ak x ) + 2 k z − m (cid:1) − k z ǫ . Consider the term in P µ γ ν + P ν γ µ : I VP γ = e Z d k (2 π ) (cid:20) ω P ( k − k x a ) − k z ǫD V + ω P ( k − k x a ) + k z ǫD V (cid:21) k x aω P δ (cid:0) k − m (cid:1) n F ( k ) , after expanding at first order in ǫ and in high temperature, we find I VP γ ≃ − e T πω P I Ω (2 , , , − e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) . Similarly the term in u µ γ ν + u ν γ µ is I Vuγ ≃ e Z d k (2 π ) (cid:20) ω P ( k − k x a ) − k z ǫD V + ω P ( k − k x a ) + k z ǫD V (cid:21) ( k − k x /a ) δ (cid:0) k − m (cid:1) n F ( k ) , and at high temperature it becomes I Vuγ ≃ − e T πω P ( I Ω (2 , , , − I Ω (0 , , , − e T ω P (cid:20) a − − a )4 a log (cid:18) a − a (cid:19)(cid:21) . The term in /l ( P µ ˆ l ν + P ν ˆ l µ ): I VP l ≃ − e Z d k (2 π ) (cid:20) D V + 1 D V (cid:21) (4 k x + aω P ) k y aω P δ (cid:0) k − m (cid:1) n F ( k )at first order in ǫ and at high temperature is I VP l ≃ e T πω P (cid:0) m (cid:0) I Ω (2 , , ,
3) + a I Ω (4 , , , (cid:1) + ω P (cid:0) I Ω (0 , , ,
3) + 2 I Ω (2 , , − , − a I Ω (4 , , − , − a I Ω (4 , , ,
3) + 2 a I Ω (6 , , − , (cid:1)(cid:1) = 2 e m T ω P (cid:20) − a )2 a (1 − a ) − a log (cid:18) a − a (cid:19)(cid:21) − e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) = 7360 e T ω P (cid:20)
127 4 m ω P (cid:18) − a )2 a (1 − a ) − a log (cid:18) a − a (cid:19)(cid:19) − (cid:18) a log (cid:18) a − a (cid:19) − a (cid:19)(cid:21) . Lastly, the term in /l ( u µ ˆ l ν + u ν ˆ l µ ): I Vul ≃ − e Z d k (2 π ) (cid:20) D V + 1 D V (cid:21) k − k x /a ) k y δ (cid:0) k − m (cid:1) n F ( k )gives I Vul ≃ e T πω P (cid:0) m (cid:0) I Ω (0 , , ,
3) + 3( a − I Ω (2 , , , − a I Ω (4 , , , (cid:1) + ω P ( I Ω (0 , , − , − (1 + 2 a ) I Ω (2 , , − ,
3) + 2 a I Ω (4 , , − ,
3) + a I Ω (4 , , − , − a I Ω (6 , , − , (cid:1)(cid:1) = 4 e m T ω P (cid:20) − a )8 a log (cid:18) a − a (cid:19) − a (cid:21) − e T ω P (cid:20) a − − a )4 a log (cid:18) a − a (cid:19)(cid:21) = − e T ω P (cid:20) −
23 4 m ω P (cid:18) − a )8 a log (cid:18) a − a (cid:19) − a (cid:19) + 53 (cid:18) a − − a )4 a log (cid:18) a − a (cid:19)(cid:19)(cid:21) .
4. Contact term
The contact diagram, Fig. 2(e), is M Cµν = Z d k (2 π ) e a µνρκ ¯ u ( p ′ ) [ γ κ i S F ( p − k )( − i eγ α )i D ρα ( k ) + ( p ′ ↔ p )] u ( p ) , where the contact vertex is: a βρµν = η µν η βρ − (cid:0) η βµ η ρν + η βν η ρµ (cid:1) . The temperature part given by the Bose distribution is: β M Cµν ( p ′ , p ) = − e a µνρκ Z d k (2 π ) δ ( k ) n B ( k )¯ u ( p ′ ) (cid:20) γ κ ( /p − /k + m ) γ ρ ( p − k ) − m + ( p ′ ↔ p ) (cid:21) u ( p ) ≡ − e Z d k (2 π ) ¯ u ( p ′ ) (cid:20) N C D C + N C D C (cid:21) u ( p ) n B ( k ) δ ( k ) . We refer to “1” as the first term and with “2” to the term with p ′ ↔ p : The numerator of the Bose part:¯ u ( p ′ ) N C u ( p ) =¯ u ( p ′ ) n −
12 ( P µ γ ν + P ν γ µ ) + ( k µ γ ν + k ν γ µ ) + 12 ( q µ γ ν + q ν γ µ ) + (2 m + /k ) η µν o u ( p ) , ¯ u ( p ′ ) N C u ( p ) =¯ u ( p ′ ) n −
12 ( P µ γ ν + P ν γ µ ) + ( k µ γ ν + k ν γ µ ) −
12 ( q µ γ ν + q ν γ µ ) + (2 m + /k ) η µν o u ( p ) , therefore the contribution to AGM is the same for the terms “1” and “2”:¯ u ( p ′ ) N C u ( p ) = ¯ u ( p ′ ) N u ( p ) = ¯ u ( p ′ ) n −
12 ( P µ γ ν + P ν γ µ ) + ( k µ γ ν + k ν γ µ ) o u ( p ) . Decomposing k in the numerators we have¯ u ( p ′ ) N C u ( p ) = ¯ u ( p ′ ) N u ( p ) = ¯ u ( p ′ ) n (cid:18) A P − (cid:19) ( P µ γ ν + P ν γ µ ) + A u ( u µ γ ν + u ν γ µ ) o u ( p )and the denominators are D C = − [( k · P ) − ( k · q )] = − ω P ( k − ak x ) − k z ǫ,D C = − [( k · P ) + ( k · q )] = − ω P ( k − ak x ) + k z ǫ. The part in ( P µ γ ν + P ν γ µ ) at leading order in ǫ is I CP γ = − e Z d k (2 π ) (cid:20) N C D C + N C D C (cid:21) n B ( k ) δ ( k ) ≃ − e ω P Z d k (2 π ) δ ( k ) n B ( k ) ω P − k x /ak − ak x = − e π ω P Z d k k n B ( k ) X s = ± Z dΩ k " ω P k s − a ˆ k x − a ˆ k x s − a ˆ k x = − e π ω P Z d k k n B ( k ) Z dΩ k " ω P k a ˆ k x − a ˆ k x − k x − a ˆ k x . The first term in square bracket is vanishing and the second one gives I CP γ = + e π ω P π (cid:18) a log (cid:18) a − a (cid:19) − a (cid:19) Z d k k n B ( k )= e T ω P (cid:18) a log (cid:18) a − a (cid:19) − a (cid:19) . u µ γ ν + u ν γ µ ) at leading order in ǫ is I Cuγ ≃ e ω P Z d k (2 π ) δ ( k ) n B ( k ) k − k x /ak − ak x = e π ω P Z d k k n B ( k ) X s = ± Z dΩ k s − ˆ k x /as − a ˆ k x = e π ω P Z d k k n B ( k ) Z dΩ k − ˆ k x )1 − a ˆ k x = e π ω P π T π (cid:18) a − − a )4 a log (cid:18) a − a (cid:19)(cid:19) = e T ω P (cid:18) a − − a )4 a log (cid:18) a − a (cid:19)(cid:19) . a. Fermionic part: High temperature The temperature part given by the Fermi distribution is: β M Cµν ( p ′ , p ) = e a µνρκ Z d k (2 π ) δ ( k − m ) n F ( k )¯ u ( p ′ ) (cid:20) γ κ ( /k + m ) γ ρ ( p − k ) + ( p ′ ↔ p ) (cid:21) u ( p )= e Z d k (2 π ) δ ( k − m ) n F ( k )¯ u ( p ′ ) γ κ ( /k + m ) γ ρ (cid:20) p − k ) + 1( p ′ − k ) (cid:21) u ( p ) ≡ e Z d k (2 π ) δ ( k − m ) n F ( k )¯ u ( p ′ ) (cid:20) N C D C + N C D C (cid:21) u ( p ) . The numerator is N C = − ( γ µ k ν + γ ν k µ ) → N C AGM = − A u ( u µ γ ν + u ν γ µ ) − A P ( P µ γ ν + P ν γ µ ) . The denominators are D C = (cid:2) m − ( k · P ) + ( k · q ) (cid:3) = 2 m − ω P ( k − ak x ) − k z ǫ,D C = (cid:2) m − ( k · P ) − ( k · q ) (cid:3) = 2 m − ω P ( k − ak x ) + k z ǫ. The part in ( P µ γ ν + P ν γ µ ) at leading order in ǫ is I CP γ,f ≃ e ω P a Z d k (2 π ) δ ( k − m ) n F ( k ) k x k − ak x − m /ω P = e π ω P a Z d k k E k n F ( k ) X s = ± Z dΩ k ˆ k x s E k k − a ˆ k x − m ω P k . At high temperature the leading term is I CP γ,f ≃ e T πω P I Ω (2 , , ,
1) = 136 e T ω P (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) . The part in ( u µ γ ν + u ν γ µ ) at leading order in ǫ and expanding for high temperatures is I Cuγ,f ≃ e ω P Z d k (2 π ) δ ( k − m ) n F ( k ) k − k x /ak − ak x − m /ω P = e π ω P Z d k n F ( k ) X s = ± Z dΩ k k E k sE k /k − ˆ k x /asE k /k − a ˆ k x − m ω P k ≃ e T πω P ( I Ω (0 , , , − I Ω (2 , , , e T ω P (cid:20) a − − a )4 a log (cid:18) a − a (cid:19)(cid:21) . Summing all the relevant terms we found that the diagram can be written as: M Cµν = (cid:0) I CP γ + θ HT I CP γ,f (cid:1) ( P µ γ ν + P ν γ µ ) + (cid:0) I Cuγ + θ HT I Cuγ,f (cid:1) ( u µ γ ν + u ν γ µ ) .
5. Angular integrals
Here, we report the results of the angular integrals we used to evaluate the diagrams: Z dΩ k − a ˆ k x =8 π (cid:20) a log (cid:18) a − a (cid:19)(cid:21) ; Z dΩ k k x − a ˆ k x = 8 π (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) ; Z dΩ k − a ˆ k x ) =8 π (cid:20) a log (cid:18) a − a (cid:19) + 12(1 − a ) (cid:21) ; Z dΩ k k x (1 − a ˆ k x ) = 8 π (cid:20) − a ) a − a log (cid:18) a − a (cid:19)(cid:21) ;and the integrals involving ˆ k y are Z dΩ k k y (1 − a ˆ k x ) = 8 π (cid:20) a )8 a log (cid:18) a − a (cid:19) − a (cid:21) , Z dΩ k k x ˆ k y (1 − a ˆ k x ) = 8 π (cid:20) − a )8 a log (cid:18) a − a (cid:19) − a (cid:21) , Z dΩ k k y (1 − a ˆ k x ) = 8 π (cid:20) a − a (1 − a ) + 3(1 + 3 a )32 a log (cid:18) a − a (cid:19)(cid:21) , Z dΩ k k x ˆ k y (1 − a ˆ k x ) = 8 π (cid:20) − a )16 a (1 − a ) − a )32 a log (cid:18) a − a (cid:19)(cid:21) , Z dΩ k k x ˆ k y (1 − a ˆ k x ) = 8 π (cid:20) − a )16 a (1 − a ) + 105( a − a log (cid:18) a − a (cid:19)(cid:21) , and those involving ˆ k z in the principal value sense are given by Z dΩ k k y (1 − a ˆ k x )ˆ k z = lim τ → Z dΩ k k y (1 − a ˆ k x )(ˆ k z − τ ) = − π (cid:20) a log (cid:18) a − a (cid:19)(cid:21) ; Z dΩ k k x ˆ k y (1 − a ˆ k x )ˆ k z = lim τ → Z dΩ k k x ˆ k y (1 − a ˆ k x )(ˆ k z − τ ) = − π (cid:20) a log (cid:18) a − a (cid:19) − a (cid:21) ; Z dΩ k k y (1 − a ˆ k x ) ˆ k z = − π (cid:20) a log (cid:18) a − a (cid:19) − a − − a ) (cid:21) , Z dΩ k k x ˆ k y (1 − a ˆ k x ) ˆ k z = − π (cid:20) a )8 a (1 − a ) − a log (cid:18) a − a (cid:19)(cid:21) , Z dΩ k k x ˆ k y (1 − a ˆ k x ) ˆ k z = − π (cid:20) a log (cid:18) a − a (cid:19) − a − a )8 a (1 − a ) (cid:21) , Z dΩ k k x ˆ k y (1 − a ˆ k x ) ˆ k z = − π (cid:20) a log (cid:18) a − a (cid:19) − a − a + 8 a )8 a (1 − a ) (cid:21) . [1] L. Adamczyk et al. (STAR), Nature , 62 (2017), arXiv:1701.06657 [nucl-ex].[2] J. Adam et al. (STAR), Phys. Rev. C , 014910 (2018), arXiv:1805.04400 [nucl-ex].[3] J. Adam et al. (STAR), Phys. Rev. Lett. , 132301 (2019), arXiv:1905.11917 [nucl-ex].[4] F. Becattini and M. A. Lisa, Annual Review of Nuclear and Particle Science , 395 (2020), arXiv:2003.03640 [nucl-ex].[5] Z.-T. Liang and X.-N. Wang, Phys. Rev. Lett. , 102301 (2005), [Erratum: Phys.Rev.Lett. 96, 039901 (2006)],arXiv:nucl-th/0410079.[6] D. Kharzeev, Physics Letters B , 260 (2006).[7] F. Becattini, F. Piccinini, and J. Rizzo, Phys. Rev. C , 024906 (2008), arXiv:0711.1253 [nucl-th].[8] A. Vilenkin, Phys. Rev. D20 , 1807 (1979). [9] K. Landsteiner, E. Megias, L. Melgar, and F. Pena-Benitez, JHEP , 121 (2011), arXiv:1107.0368 [hep-th].[10] J.-H. Gao, Z.-T. Liang, S. Pu, Q. Wang, and X.-N. Wang, Phys. Rev. Lett. , 232301 (2012), arXiv:1203.0725 [hep-ph].[11] D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, Prog. Part. Nucl. Phys. , 1 (2016), arXiv:1511.04050 [hep-ph].[12] M. Buzzegoli and F. Becattini, JHEP , 002 (2018), arXiv:1807.02071 [hep-th].[13] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nuclear Physics A , 227 (2008).[14] K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Physical Review D , 074033 (2008).[15] D. Kharzeev and A. Zhitnitsky, Nuclear Physics A , 67 (2007).[16] J. Erdmenger, M. Haack, M. Kaminski, and A. Yarom, Journal of High Energy Physics , 055 (2009).[17] D. E. Kharzeev, Prog. Part. Nucl. Phys. , 133 (2014), arXiv:1312.3348 [hep-ph].[18] K. Landsteiner, E. Megias, and F. Pena-Benitez, Phys. Rev. Lett. , 021601 (2011), arXiv:1103.5006 [hep-ph].[19] T. Kalaydzhyan, Phys. Rev. D , 105012 (2014), arXiv:1403.1256 [hep-th].[20] A. Avkhadiev and A. V. Sadofyev, Phys. Rev. D , 045015 (2017), arXiv:1702.07340 [hep-th].[21] P. Glorioso, H. Liu, and S. Rajagopal, JHEP , 043 (2019), arXiv:1710.03768 [hep-th].[22] A. Flachi and K. Fukushima, Phys. Rev. D98 , 096011 (2018), arXiv:1702.04753 [hep-th].[23] M. Buzzegoli, E. Grossi, and F. Becattini, JHEP , 091 (2017), [Erratum: JHEP07,119(2018)],arXiv:1704.02808 [hep-th].[24] G. Y. Prokhorov, O. Teryaev, and V. Zakharov, “Chiral vortical effect for vector fields,” (2020), arXiv:2009.11402 [hep-th].[25] D.-F. Hou, H. Liu, and H.-c. Ren, Phys. Rev. D86 , 121703 (2012), arXiv:1210.0969 [hep-th].[26] S. Golkar and D. T. Son, JHEP , 169 (2015), arXiv:1207.5806 [hep-th].[27] I. Y. Kobzarev and L. Okun, Zh. Eksperim. i Teor. Fiz. (1962).[28] C. F. Cho and N. D. Hari Dass, Phys. Rev. D14 , 2511 (1976).[29] O. V. Teryaev,
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