Axial $U(1)$ anomaly in a gravitational field via the gradient flow

A regularization-independent universal formula for the energy--momentum tensor in gauge theory in the flat spacetime can be written down by employing the so-called Yang--Mills gradient flow. We examine a possible use of the formula in the calculation of the axial $U(1)$ anomaly in a gravitational field, the anomaly first obtained by Toshiei Kimura [Prog.\ Theor.\ Phys.\ {\bf 42}, 1191 (1969)]. As a general argument indicates, the formula reproduces the correct non-local structure of the (axial $U(1)$ current)--(energy--momentum tensor)--(energy--momentum tensor) triangle diagram in a way that is consistent with the axial $U(1)$ anomaly. On the other hand, the formula does not automatically reproduce the general coordinate (or translation) Ward--Takahashi relation, requiring corrections by local counterterms. This analysis thus illustrates the fact that the universal formula as it stands can be used only in on-shell correlation functions, in which the energy--momentum tensor does not coincide with other composite operators in coordinate space.


Introduction
Almost half a century ago, just nine months after the appearance of two seminal papers on the axial U (1) anomaly in an electromagnetic field [1,2], Kimura noticed in a lesser-known but remarkable paper [3] that a similar anomalous non-conservation of the axial vector current also occurs in a gravitational field. His result was D α ψ (x)γ α γ 5 ψ(x) − 2m 0 ψ (x)γ 5 ψ(x) = 1 384π 2 ǫ µνρσ R µνλτ (x)R ρσ λτ (x), (1.1) where m 0 is the bare mass of the fermion and R µνρσ (x) is the Riemann curvature. This axial U (1) anomaly, also obtained in Refs. [4,5] (see also Refs. [6,7]), was the first example of the quantum anomaly related to the gravitational interaction, a subject that was to be extensively explored somewhat later in a wider context [8,9]. Recently, by employing the so-called Yang-Mills gradient flow [10][11][12][13][14] (see Refs. [15,16] for reviews) and the small flow time expansion [13], a regularization-independent universal formula for the energy-momentum tensor in gauge theory in the flat spacetime has been constructed [17,18] (see also Ref. [19] for a review); the formula is then applied to the computation of thermodynamic quantities in lattice QCD [20][21][22][23][24][25][26]. References  represent a partial list of developments relating to the gradient flow.
In this paper, we examine a possible use of the universal formula in the calculation of the axial U (1) anomaly (1.1); we will obtain Eq. (1.1) by expansion around the flat spacetime. Precisely speaking, the anomaly is a clash between the axial U (1) Ward-Takahashi (WT) relation and the general coordinate (and the local Lorentz) WT relation. A general argument given in Ref. [3], which is analogous to the argument in Ref. [1], shows that the anomaly (1.1) is independent of the adopted regularization as long as the regularization is physically sensible and one imposes the general coordinate WT relation; the structure (1.1) is robust in this sense.
In what follows, we will observe that the universal formula does not automatically reproduce the correct WT relation associated with the general coordinate (or translation in the flat spacetime) WT relation. The resulting correlation functions, however, can be modified by adding appropriate local terms so that the translation WT relation holds. Then, as the general argument implies, we have Eq. (1.1). This shows that the universal formula reproduces the correct non-local structure of the (axial U (1) current)-(energy-momentum tensor)-(energy-momentum tensor) triangle diagram in a way that is consistent with the axial U (1) anomaly. This is expected without any calculation from the construction of the universal formula [17,18], but to check this point explicitly is certainly assuring. On the other hand, this analysis illustrates that the universal formula as it stands can be used only in on-shell correlation functions, in which the energy-momentum tensor does not coincide with other composite operators in coordinate space, because it does not automatically reproduce the translation WT relation when operators coincide. How to remedy this point in (a generalization of) the universal formula is a forthcoming challenge.
This paper is organized as follows. In Sect. 2, we summarize the naively expected form of the axial U (1) and the general coordinate (or translation) WT relations in the flat spacetime limit. The breaking of these relations is regarded as the quantum anomaly. In Sect. 3, using the universal formulas for the energy-momentum tensor of the Dirac fermion [18] and the axial U (1) current [35,42], we compute the total divergences of the triangle diagram and extract the parts potentially corresponding to the anomaly. Each of the axial U (1) current and the energy-momentum tensors can possess a different flow time, t 1 , t 2 , and t 3 ; these are eventually taken to be zero. We adopt a particular ordering of the limits, which turns out to considerably simplify the calculation. We find that the translation WT relation does not hold. In Sect. 4, we seek an appropriate local term added to the triangle diagram, which restores the translation WT relation. Although our analysis here is quite analogous to that of Ref. [35] on the triangle anomaly in gauge theory, partially due to the fact that the translation WT relation also contains some two-point functions, the analysis is much more complicated. Finally, in Sect. 5, by adding appropriate local terms, we obtain the expansion of Eq. (1.1) around the flat spacetime. Section 6 is devoted to the conclusion.

Naively expected form of Ward-Takahashi relations
We consider the Dirac fermion in the curved spacetime with a Euclidean signature. The curved space indices are denoted by Greek letters while the local Lorentz indices are denoted by Latin letters. Letting e a µ (x) be the vierbein, the raising and lowering of the former indices are done by the metric g µν (x) ≡ δ ab e a µ (x)e b ν (x) and its inverse matrix g µν (x); those of the latter indices are, on the other hand, done by the Kronecker deltas, δ ab and δ ab .
The action of the Dirac fermion in the curved spacetime is given by γ a is the Dirac matrix satisfying {γ a , γ b } = 2δ ab and σ ab ≡ 1 2 [γ a , γ b ]. ω ab µ (x) is the spin connection, which is defined by The coupling of the fermion to a gravitational field is given by the energy-momentum tensor: where we have defined T anti-sym.
where σ µν (x) ≡ e a µ (x)e b ν (x)σ ab . We note that the anti-symmetric part of the energymomentum tensor T anti-sym.
µν (x) is proportional to the equation of motion of the fermion. Now, in order to determine the precise form of the quantum anomalies, it is crucial to clearly recognize the form of naively expected WT relations. For simplicity, we consider the massless fermion m 0 = 0 in what follows.
We start from the WT relation associated with the axial U (1) symmetry. For this, we take the correlation function where dµ denotes the functional integration measure for the fermion field, and make the change of integration variables in the form of a localized axial U (1) transformation: 1 Noting that the action (2.1) changes under this change of variables as and considering the flat spacetime limit e a µ (x) → δ a µ , neglecting a possible breaking of the symmetry associated with the regularization, we have the identity where the first equality follows from the covariance under the Lorentz and parity transformations in the flat spacetime. The breaking of this naively expected relation is thus regarded as a quantum anomaly. Next, we consider the WT relation associated with the general coordinate invariance and the local Lorentz symmetry. For this, we start with and consider the following form of the change of integration variables: This is a particular combination of the general coordinate transformation and the local Lorentz transformation. Under this change of integration variables, from the fact that the 1 The chiral matrix γ 5 is defined by γ 5 ≡ 1 4! ǫ abcd γ a γ b γ c γ d by the totally anti-symmetric tensor ǫ abcd being normalized as ǫ 0123 = 1. action does not change if the vierbein is also changed by the same set of transformations, we have where the total energy-momentum tensor is given by Eq. (2.6). Considering the flat spacetime limit, we thus have the identity where again the first equality follows from the covariance under the Lorentz and parity transformations. We have introduced the combination On the other hand, by considering the change of integration variables of the form of the local Lorentz transformation, in Eq. (2.13), we have the following identity in the flat spacetime limit: where the last equality again follows from the covariance under the Lorentz and parity transformations and we have defined Thus, combining Eqs. (2.16) and (2.19), we have the relation in the flat spacetime limit: is the naively expected form of the WT relation associated with the general coordinate invariance and the local Lorentz symmetry (in the flat spacetime limit). Thus, the breaking of this relation should be regarded as a quantum anomaly. It can be confirmed that one can directly derive the WT relation (2.22) only by using the translational invariance in the flat spacetime (see Appendix A). The last three two-point functions including O 1 , O 2 , and O 3 play a crucial role in the following analysis of the anomaly. The contribution of such "two-sided diagrams" in addition to the triangle diagram, which have no analogue in the axial U (1) anomaly in gauge theory, has of course already been noted in Ref. [3] (through a somewhat different derivation from ours).

Computation of anomalies 3.1. Definition of the three-point function
Now, for the vector-like gauge theory in the flat spacetime, we know representations of the axial vector current [35,42] and the symmetric energy-momentum tensor [17,18,20] by the small flow time limit of flowed fields. In the zeroth order in the gauge coupling, the representations are rather trivial: where χ(t, x) andχ(t, x) are flowed fermion fields and eventually we have to take the small flow time limit t → 0 in the correlation functions. Then, using the tree-level propagator of the flowed fermion field [14], 2

4)
2 Throughout this paper, we use the abbreviation In this definition, we have adopted a particular ordering of the small time limit; we first set t 2 = t 3 → 0 and then t 1 ≡ t → 0. Because of the Gaussian damping factors e −(t1+t2)p 2 , e −(t2+t3)q 2 , and e −(t3+t1)k 2 in the first definition, the momentum integration is absolutely convergent as long as t 1 ≡ t > 0; we can thus trivially take the first limit t 2 = t 3 → 0 inside the momentum integration. It turns out that this particular ordering considerably simplifies the actual calculation of the anomalies below. 3 We also note that the expression (3.5) does not require further regularization; in other words, Eq. (3.5) is independent of the adopted regularization in the limit in which the regulator is sent to infinity. This shows the universality of the representations (3.1) and (3.2), although this finiteness is trivial in the present zerothorder perturbation theory in the gauge coupling.

Anomaly in the axial WT relation
We are primarily interested in the anomalous divergence of the axial vector current, the breaking of the WT relation (2.12). From our definition (3.5), after careful rearrangements, we find the identity (omitting the symbol lim t→0 ) 4 where F.T. denotes the Fourier transformation: On the left-hand side of Eq. (3.6), the two-point functions have been defined by These regularized two-point correlation functions identically vanish, as should be the case from the Lorentz and parity covariance. In deriving Eq. (3.6), we first apply ∂ x α to Eq. (3.5). In the integrand, this produces the factor / p − / k; each term of this is canceled by 1// p and 1// k. We then use the identities p + q = (p − k) + (q + k) and q + k = (k − p) + (p + q) and express the momentum (p − k) α by the derivative −i∂ x α . These manipulations give rise to the right-hand side of Eq. (3.6). The last two terms on the left-hand side of Eq. (3.6) are simply zero, as noted above; the 4 We have noted that the spinor trace with γ 5 requires at least other four Dirac matrices. 8 inclusion of those terms, however, clearly shows the correspondence to the naively expected axial WT relation (2.12).
Thus, comparing Eq. (3.6) and Eq. (2.12), we find that the anomalous breaking of the axial symmetry is given by the right-hand side of Eq. (3.6). We note that in Eq. (3.6) if Gaussian factors such as e −tp 2 and e −tk 2 are simply unity (i.e., if we could naively set t → 0 before the momentum integration), then the right-hand side identically vanishes. The fact is that there are Gaussian factors and they give rise to a non-vanishing result. After a straightforward calculation in the t → 0 limit, we find (3.10)

Anomaly in the translation WT relation
Next, we investigate the anomalous breaking of the translation WT relation (2.22). From Eq. (3.5), after careful rearrangements by using the relation 9 we have the identity 5 In this expression, the two-point functions on the left-hand side have been defined by and The two-point functions in Eqs. (3.13)-(3.15) identically vanish as should be the case from the Lorentz and parity covariance. On the right-hand side of Eq. (3.12), the last two lines change sign under the change of integration variables, p → −k and k → −p. Thus, those two lines identically vanish. The other three terms do not vanish and after a tedious calculation in the limit t → 0, we have

Axial anomaly in the two-point functions
As we will see, however, the axial anomaly in the two-point function (3.22) can be removed by adding an appropriate local term to the two-point function (3.18).

Local counterterms
Now, the anomalous breaking of the axial WT relation in Eq. (3.10) would have intrinsic meaning only when we require the validity of the translation WT relation (2.22). That is, we still have the freedom to modify the local part of the three-point correlation function (3.5) by adding a "local counterterm" C α,µν,ρσ (x, y, z). In the momentum space, it must be a cubic polynomial of external momenta. We see that the general form of the counterterm that is consistent with the symmetric structure of the three-point function (3.5) is given by + ǫ αµβγ p β q γ (e 1 p ν δ ρσ + e 2 p ρ δ νσ + e 3 q ν δ ρσ + e 4 q ρ δ νσ ) − ǫ αρβγ p β q γ (e 3 p σ δ µν + e 4 p µ δ νσ + e 1 q σ δ µν + e 2 q µ δ νσ ) where c i , d i , e i , and f i are constants. The basic idea is to choose the coefficients c i , d i , e i , and f i so that the right-hand side of Eq. (3.19) vanishes after the addition j 5α (x)T sym. µν (y)T sym. ρσ (z) + C α,µν,ρσ (x, y, z). Then, to the axial anomaly (3.10), the counterterm contributes by A complication arises, however, since we may also modify the two-point functions (3.16)-(3.18) appearing in the relation (3.19) by adding local terms. We choose the counterterms for the two-point functions such that the axial U (1) WT relations hold for the two-point functions.
For the two-point function (3.16), we thus require the validity of the axial WT relation, where S 1α,β,ν,ρσ (x, z) is a local term. Equation (3.20), however, shows that there is no axial anomaly in this two-point function and thus we should require ∂ x α S 1α,β,ν,ρσ (x, z) = 0. It turns out that the most general form of such a local term is Similarly, for the two-point function (3.17), requiring implies ∂ x α S 2α,µν,ρσ (x, z) = 0 because of Eq. (3.21) and the possible form of the counterterm is given by Finally, after some examination, we find that the most general form of the counterterm for the function (3.18) is given by We choose the coefficients c ′ 0 etc. so that the addition of S 1α,β,µν,ρσ (x, z) to the two-point function cancels the anomalous breaking (3.22). That is, we require This yields Now, we require that the translation WT relation (2.22) holds by adding the above local terms to the correlation functions. That is, our requirement is (4.10) The resulting relations among the coefficients in the counterterms are summarized in Appendix B. From those relations, we see that some coefficients are still left unfixed, but the coefficients in the expression (4.2) are completely determined as 14) This gives (4.17)

Final steps
We are now able to write down the axial U (1) anomaly in the three-point function (3.5) under the requirement of the translation WT relation (4.10); the latter requirement is accomplished by the counterterm (4.1). Then the axial U (1) anomaly is given by the sum of Eqs. This is the most non-trivial result of this paper. Going back to Eq. (2.6), the energy-momentum tensor also has a part that is antisymmetric under the exchange of indices, T anti-sym.
µν (x) in Eq. (2.8) is proportional to the equation of motion; its effect on the correlation functions must be at most local contact terms as the Schwinger-Dyson equation implies. We can in fact corroborate this argument by explicit calculations by using some regularizing prescription for T anti-sym.
µν (x). Here, however, we are content with the above argument and set T µν (x) → T sym.
µν (x) in what follows. We now re-express Eq. (5.1) as the anomalous divergence of the axial U (1) current in the curved spacetime. We expand D α j 5α (x) g , the divergence of the axial vector current in the curved spacetime as the power series of the vierbein around the flat spacetime: where δe µa (x) ≡ e µa (x) − δ µa and it is understood that the right-hand side is evaluated in the flat spacetime with appropriate local counterterms specified as above. Noting as far as the regularization preserves the Lorentz and parity covariance, we have where we have used [δ/δe νa (x)]S = e µ a (x)T sym. µν (x), δg µν (x) = δe µa (x)e ν a (x) + e µa (x)δe ν a (x) and Eq. (5.1) in the last equality. Comparing this with the expansion of the curvature, 6 we finally observe Eq. (1.1) for m 0 = 0.

Conclusion
In this paper, we have examined a possible use of the universal formula for the energymomentum tensor in gauge theory in the flat spacetime through the Yang-Mills gradient flow [17,18]. As a general argument indicates, after choosing local counterterms appropriately so as to restore the translation WT relation, we obtain the correct axial U (1) anomaly in Eq. (1.1) (in the flat space limit). From the present analysis, we can learn the following feature of the universal formula of the energy-momentum tensor. The universal formula is based on the gradient flow and its small flow time expansion of Ref. [13]. The latter asserts that any composite operator of flowed fields as t → 0 can be expressed as an asymptotic series of renormalized composite operators of unflowed fields with increasing mass dimensions. When two composite operators of flowed fields collide in coordinate space to form another composite operator, we have to consider the expansion in terms of another set of renormalized composite operators of unflowed fields. Consequently, it is not obvious what happens when the universal formula of the energy-momentum tensor collides with other composite operators, such as the axial U (1) current or the energy-momentum tensor, in coordinate space. Our present analysis illustrates that the formula in fact does not automatically fulfill the translation WT relation precisely when the formula coincides with other composite operators in coordinate space. On the other hand, our finding that local counterterms are sufficient to restore the translation WT relation ensures the expectation that the formula fulfills the translation WT relation when the energy-momentum tensor is in isolation in coordinate space; for this case, the translation WT relation is simply the conservation law of the energy-momentum tensor. 6 Our definition of the Riemann curvature is R α ρµν ≡ ∂ µ Γ α ρν − ∂ ν Γ α ρµ + Γ α λµ Γ λ ρν − Γ α λν Γ λ ρµ , where Γ λ µν ≡ 1 2 g λρ (∂ µ g νρ + ∂ ν g µρ − ∂ ρ g µν ) is the Christoffel symbol.
Thus, our analysis has revealed that the universal formula as it stands can be used only in on-shell correlation functions (i.e., correlation functions in which the energy-momentum tensor does not coincide with other composite operators in coordinate space). The incorporate of this point into (a generalization of) the universal formula is a forthcoming challenge. 7 A related issue is the possible generalization of the gradient flow to the curved spacetime. A possible generalization is ∂ t B µ (t, x) = g νρ (x)D ν G ρ,µ (t, x), B µ (t = 0, x) = A µ (x), (6.1) ∂ t χ(t, x) = g µν (x)D µ D ν χ(t, x), χ(t = 0, x) = ψ(x), (6.2) 3) It then appears interesting to see whether this setup improves the covariance under the general coordinate transformation and the restoration of the associated WT relations for the energy-momentum tensor.