# On the trace anomaly of a Weyl fermion in a gauge background

## Abstract

We study the trace anomaly of a Weyl fermion in an abelian gauge background. Although the presence of the chiral anomaly implies a breakdown of gauge invariance, we find that the trace anomaly can be cast in a gauge invariant form. In particular, we find that it does not contain any odd-parity contribution proportional to the Chern–Pontryagin density, which would be allowed by the consistency conditions. We perform our calculations using Pauli–Villars regularization and heat kernel methods. The issue is analogous to the one recently discussed in the literature about the trace anomaly of a Weyl fermion in curved backgrounds.

## 1 Introduction

In this paper we study the trace anomaly of a chiral fermion coupled to an abelian gauge field in four dimensions. It is well-known that the model contains an anomaly in the axial gauge symmetry, thus preventing the quantization of the gauge field in a consistent manner. Nevertheless, it is useful to study the explicit structure of the trace anomaly emerging in the axial *U*(1) background.

One reason to study the problem is that an analogous situation has recently been addressed for a Weyl fermion coupled to gravity. In particular, the presence of an odd-parity term (the Pontryagin density of the curved background) in the trace anomaly has been reported in [1], and further elaborated upon in [2, 3]. This anomaly was envisaged also in [4], and discussed more recently in [5]. However, there are many indications that such an anomaly cannot be present in the theory of a Weyl fermion. The explicit calculation carried out in [6] confirms this last point of view.

One of the reasons why one does not expect the odd-parity contribution to the trace anomaly is that by CPT in four dimensions a left handed fermion has a right handed antiparticle, expected to contribute oppositely to any chiral imbalance in the coupling to gravity. To see that, one may cast the quantum field theory of a Weyl fermion as the quantum theory of a Majorana fermion. The latter shows no sign of an odd-parity trace anomaly. Indeed, the functional determinant that arises in a path integral quantization can be regulated using Pauli–Villars Majorana fermions with Majorana mass, so to keep the determinant manifestly real, thereby excluding the appearance of a phase that might produce an anomaly (the odd-parity term would carry an imaginary coefficient) [7]. Recently, this has been verified again using Feynman diagrams [8], which confirms the results of [6]. An additional piece of evidence comes from studies of the 3-point correlation functions of conserved currents in four dimensional CFTs, which exclude odd-parity terms in the correlation function of three stress tensors at non-coinciding points [9, 10], seemingly excluding its presence also in the trace anomaly (see however [11]).

Here we analyze the analogous situation of a Weyl fermion coupled to an abelian *U*(1) gauge background. The theory exhibits a chiral anomaly that implies a breakdown of gauge invariance. It is nevertheless interesting to compute its trace anomaly. Apart from the standard gauge invariant contribution (\(\sim F^2\)) and possible gauge noninvariant terms, which as we shall show can be canceled by counterterms, one might expect a contribution from the odd-parity Chern–Pontryagin density \(F\tilde{F}\). Indeed the latter satisfies the consistency conditions for trace anomalies. In addition, the fermionic functional determinant is now complex in euclidean space, and thus carries a phase (which is responsible for the known *U*(1) axial anomaly). On the other hand, the structure of the 3-point correlation function of the stress tensor with two *U*(1) currents in generic CFTs does not allow for odd-parity terms [9, 10] that could signal a corresponding anomaly in the trace of the stress tensor in a *U*(1) background. Apart from a few differences, the case seems analogous to that of the chiral fermion in curved space, and thus it is worth addressing.

To ascertain the situation we compute explicitly the trace anomaly of a Weyl fermion coupled to a *U*(1) gauge field. Using a Pauli–Villars regularization we find that no odd-parity term emerges in the quantum trace of the stress tensor. We use a Majorana mass for computing the trace anomaly, as this mass term can be covariantized (to curved space) without the need of introducing additional fields of opposite chirality, as required by a Dirac mass. The coupling to gravity (needed only at linear order) is used to treat the metric (or vierbein) as an external source for the stress tensor, and to relate the trace of the latter to a Weyl rescaling of the metric (or vierbein). The manifest covariance of the Majorana mass guarantees that the stress tensor can be kept conserved and symmetric also at the quantum level, i.e. without general coordinate (Einstein) and local Lorentz anomalies. We repeat part of our calculations with a Dirac mass as well. In addition, we calculate also the anomalies of a massless Dirac fermion which, while well-known, serve for comparison and as a test on the scheme adopted. We verify the consistency of the different regularizations, and report the local counterterms that relate them.

We organize the paper as follows. In Sect. 2 we set up the stage and review the lagrangians of the Weyl and Dirac fermions, respectively, and identify the relevant differential operators that enter our regularization schemes. In Sect. 3 we review the method that we choose for computing the chiral and trace anomalies. In Sect. 4 we present our results. We conclude in Sect. 5, confining to the appendices notational conventions, heat kernels formulas, and sample calculations.

## 2 Actions and symmetries

We first present the classical models that we wish to consider, and review their main properties to set up the stage for our calculations. The model of main interest is a massless Weyl fermion coupled to an abelian gauge field. We first describe its symmetries, and then the mass terms to be used in a Pauli–Villars regularization. For comparison, we consider also a massless Dirac fermion coupled to vector and axial abelian gauge fields, a set-up used by Bardeen to compute systematically the anomalies in vector and axial currents [12]. Our notation is commented upon and recapitulated in “Appendix A”.

### 2.1 The Weyl fermion

*U*(1) gauge field iswhere the chirality of the spinor is defined by the constraint \(\gamma ^5 \lambda =\lambda \), or equivalently by \(\lambda =\frac{1+\gamma ^5}{2}\lambda \). It is classically gauge invariant and conformally invariant. Both symmetries become anomalous at the quantum level.

*U*(1) current is conserved on-shell (i.e. using the fermion equations of motion)

*e*is the determinant of the vierbein, and is covariantly conserved, symmetric, and traceless on-shell, as consequence of diffeomorphisms, local Lorentz invariance, and Weyl symmetry, respectively

*U*(1) gauge field \(A_\mu \) and spin connection \(\omega _{\mu ab}\)

#### 2.1.1 Mass terms

*M*is a real mass parameter. Since the charge conjugation matrix

*C*is antisymmetric this term is nonvanishing for an anticommuting spinor.

^{1}

*T*is not invertible in the full 8 dimensional space on which \(\phi \) lives. An advantage of the Majorana mass term is that it can be constructed without the need of introducing extra degrees of freedom (as required by the Dirac mass term discussed below). Moreover, it can be covariantized under Einstein (general coordinate) and local Lorentz symmetries. The covariantization is achieved by multiplying it with the determinant of the vierbein

*e*

A drawback of the Dirac mass term, as regulator of the Weyl theory, is that one cannot covariantize it while keeping the auxiliary right handed spinor \(\rho \) free in the kinetic term (it cannot be coupled to gravity, otherwise it would not regulate properly the original chiral theory). One can still use the regularization keeping \(\rho \) free in the kinetic term but, as the mass term breaks the Einstein and local Lorentz symmetries explicitly, one would get anomalies in the conservation (\(\partial _a T^{ab}\)) and antisymmetric part (\(T^{[ab]}\)) of the stress tensor. Then, one is forced to study the counterterms to reinstate conservation and symmetry of the stress tensor (it can always be done in 4 dimensions [7, 14]), and eventually check which trace anomaly one is left with. As this is rather laborious, we do not use this mass term to calculate the trace anomaly in the Weyl theory.^{2}

### 2.2 The Dirac fermion

*U*(1) gauge fields \(A_a\) and \(B_a\). The lagrangian iswhere the last form is valid up to boundary terms. A chiral projector emerges when \(A_a=\pm B_a\), and we use this model to address again the issue of the chiral fermion in flat space (the limit \(A_a= B_a\rightarrow \frac{A_a}{2}\) reproduces the massless part of (24)).

#### 2.2.1 Mass terms

## 3 Regulators and consistent anomalies

To compute the anomalies we employ a Pauli–Villars regularization [15]. Following the scheme of Refs. [16, 17] we cast the calculation in the same form as the one obtained by Fujikawa in analyzing the measure of the path integral [18, 19]. This makes it easier to use heat kernel formulas [20, 21] to evaluate the anomalies. At the same time, the method guarantees one to obtain consistent anomalies, i.e. anomalies that satisfy the consistency conditions [22, 23].

*M*is a real parameter.

^{3}The mass term identifies the operator

*T*, that in turn allows to find the operator \({\mathscr {O}}\). As we shall see, in fermionic theories with a first order differential operator \({\mathscr {O}}\) in the kinetic term, the operator \({\mathscr {O}}^2\) acts as a regulator in the final formula for the anomaly. The invariance of the original action extends to an invariance of the massless part of the PV action by defining

*Z*and the one-loop effective action \(\varGamma \) are regulated by the PV field

*T*.

In the limit \(M\rightarrow \infty \) the regulating term \( ( 1- \frac{{\mathscr {O}}^2}{M^2})^{-1}\) inside (51) can be replaced by \(e^{ \frac{{\mathscr {O}}^2}{M^2}}\). This is allowed as, for the purpose of extracting the limit, these regulators cut off the ultraviolet frequencies in an equivalent way (we assume that \({\mathscr {O}}^2\) is negative definite after a Wick rotation to euclidean space) [16]. Clearly, if one finds a symmetrical mass term, then the symmetry would remain automatically anomaly free.

*R*with an insertion of

*J*

*J*is the infinitesimal part of the fermionic jacobian arising from a change of the path integral variables under a symmetry transformation, and

*R*is the regulator. The limit extracts the mass independent term (negative powers of the mass vanish in the limit, while positive (diverging) powers are made to cancel by using additional PV fields). The PV method guarantees that the regulator

*R*together with

*J*produces consistent anomalies, which follows from the fact that one is computing directly the variation of the effective action.

*R*associated to the fields assembled into \(\phi \). These are the only coefficients that survive in the limit \({M \rightarrow \infty }\) (as said, diverging pieces are removed by the PV renomalization). Running through the various cases presented in the previous section, we extract the “jacobians”

*J*and regulators

*R*to find the structure of the anomalies. For the Weyl theory we find

*U*(1) symmetry the jacobian

*J*in (54) is found from the symmetry transformations of \(\lambda \) and \(\lambda _c\) in (4)

*K*contributes, as \(\delta T\) vanishes as well as the contribution from \(\delta {\mathscr {O}}\) (it vanishes after taking the traces in (56), as will be checked in the next section). The infinitesimal parameter \(\alpha \) is eventually factorized away from (55) to obtain the local form in (56). In computing

*J*from (54), it is enough to check that the mass matrix

*T*is invertible on the relevant chiral spaces (extracted by the projectors \(P_L\) and \(P_R\)). For the Weyl symmetry one uses instead the transformation laws in (12) to find

*e*, see Eq. (23), which brings in a contribution from \(\frac{1}{2} T^{-1} \delta T\) to

*J*. This contribution is necessary to guarantee that general coordinate invariance is kept anomaly free in the regularization (\(\delta {\mathscr {O}}\) is neglected again for the same reason as before). The infinitesimal Weyl parameter \(\sigma \) is then factorized away from (55) to obtain the second equation in (56).

## 4 Anomalies

In this section we compute systematically the chiral and trace anomalies for the Weyl and Dirac theories described earlier. We use, when applicable, two different versions of the Pauli–Villars regularization with different mass terms. We verify that the final results are consistent with each other, and coincide after taking into account the variation of local counterterms.

### 4.1 Chiral and trace anomalies of a Weyl fermion

We consider first the case of a Weyl fermion.

#### 4.1.1 PV regularization with Majorana mass

The regularization of the Weyl fermion coupled to an abelian gauge field is achieved in the most minimal way by using a PV fermion of the same chirality with the Majorana mass term in Eq. (16) added. This set-up was already used in [6] to address the case of a Weyl fermion in a gravitational background, but without the abelian gauge coupling. The mass term is Lorentz invariant and does not introduce additional chiralities, but breaks the gauge and conformal (Weyl) symmetries. Therefore, one expects chiral and trace anomalies.

Thus, we have seen that the trace anomaly of a Weyl fermion does not contain any contribution from the topological density \(F\tilde{F}\) (which on the other hand enters the chiral anomaly in (63), as well-known). It can be presented in a gauge invariant form by the variation of a local counterterm, and equals half the standard trace anomaly of a Dirac fermion. These are the main results of our paper.

#### 4.1.2 PV regularization with Dirac mass

Unfortunately, we cannot proceed to compute in a simple way the trace anomaly using this regularization, as the mass term breaks the Einstein and local Lorentz symmetries as well. The ensuing anomalies should then be computed and canceled by local counterterms, to find eventually the expected agreement of the remaining trace anomaly with the one found in the previous section.

### 4.2 Chiral and trace anomalies of a Dirac fermion

For completeness, we now consider the case of the massless Dirac spinor coupled to vector and axial gauge fields with lagrangian given in Eq. (31). The results are well-known, but we wish to present them for comparison and as a check on our method. The most natural regularization is obtained by employing a Dirac mass for the PV fields, but we employ also a Majorana mass. The latter allows to take a chiral limit in a simple way, which we use to rederive the previous results on the Weyl fermion.

#### 4.2.1 PV regularization with Dirac mass

*B*field. They are canceled by the variation of a local counterterm

#### 4.2.2 PV regularization with Majorana mass

*U*(1) currents. From Eq. (43) we find the regulatorsThen, we compute from (59)

The results of this section can be projected consistently to recover the chiral and trace anomalies of a Weyl fermion. Indeed, one can consider the limit \(A_a = B_a \rightarrow \frac{1}{2} A_a\). In this limit, a chiral projector \(P_L = \frac{1+\gamma ^5}{2} \) emerges inside the Dirac lagrangian (31) to reproduce the Weyl lagrangian (1). In addition, in the coupling to gravity, the right handed component of the Dirac field can be kept free, both in the kinetic and PV mass term, preserving at the same time covariance of the mass term of the left handed part of the PV Dirac fermion. Then, the right handed part decouples completely and can be ignored altogether. Thus, one may verify that the anomalies in Sect. 4.1.1 are reproduced by those computed here, including the counterterms, by setting \(A_a = B_a \rightarrow \frac{1}{2} A_a\) (note that the current \(J^a\) in Sect. 4.1.1 corresponds to half the sum of \(J^a\) and \(J^a_5\) of this section).

As final remark, we have checked that terms proportional to \(\delta \mathscr {O}\) in (54) never contribute to the anomalies computed thus far, as the extra terms vanish under the Dirac trace.

## 5 Conclusions

We have calculated the trace anomaly of a Weyl fermion coupled to an abelian gauge field. We have found that the anomaly does not contain any odd-parity contribution. In particular, we have shown that the Chern–Pontryagin term \(F\tilde{F}\) is absent, notwithstanding the fact that it satisfies the consistency conditions for Weyl anomalies. Of course, the chiral anomaly implies that gauge invariance is broken. Nevertheless the trace anomaly can be cast in a gauge invariant form, equal to half the standard contribution of a nonchiral Dirac fermion.

While this result seems to have no direct implications for the analogous case in curved background, it strengthens the findings of ref. [6].

Recently, a generalized axial metric background has been developed in [2, 3] to motivate and explain the appearance of the Pontryagin term in the trace anomaly of a Weyl fermion, which however is in contradiction with the explicit calculation of [6]. Perhaps it would be useful to apply the methods used here in the axial metric background to clarify the situation in that context, and spot the source of disagreement.

## Footnotes

- 1.In terms of the 2-component left handed Weyl spinor \(l_\alpha \) this mass term reads as and it does not contain any other spinor apart from \(l_\alpha \) and its complex conjugate \(l^*_{\dot{\alpha }}\). In the chiral representation of the gamma matrices the 2-component spinor \(l_\alpha \) sits inside \(\lambda \), as in Eq. (A.12).
- 2.
- 3.
To be precise, one should employ a set of PV fields with mass \(M_i\) and relative weight \(c_i\) in the loop to be able to regulate and cancel all possible one-loop divergences [15]. For simplicity, we consider only one PV field with relative weight \(c=-1\), as this is enough for our purposes. The weight \(c=-1\) means that we are subtracting a massive PV loop from the original one.

## Notes

### Acknowledgements

We thank Stefan Theisen for extensive discussions, suggestions, critical reading of the manuscript, and hospitality at AEI. In addition, FB would like to thank Loriano Bonora and Claudio Corianò for useful discussions, and MB acknowledges Lorenzo Casarin for interesting discussions and insights into anomalies and heat kernel.

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