# Beyond Symmetries : Anomalies in Transverse Ward–Takahashi Identities

Yi-Da Li<EMAIL_ADDRESS>Department of Physics, Tsinghua
University, Beijing 100084, P. R. China  Qing Wang<EMAIL_ADDRESS>Department of Physics, Tsinghua University, Beijing 100084, P. R. China
Center for High Energy Physics, Tsinghua University, Beijing 100084, P. R.
China

###### Abstract

Anomalies in transverse Ward–Takahashi identities are studied, allowing
discussion of the feasibility of anomalies arising in general non-symmetry
Ward–Takahashi identities. We adopt the popular Fujikawa’s method and rigorous
dimensional renormalization to verify the existence of transverse anomalies to
one-loop order and any loop order, respectively. The arbitrariness of
coefficients of transverse anomalies is revealed, and a way out is also
proposed after relating transverse anomalies to Schwinger terms and comparing
symmetry and non-symmetry anomalies. Papers that claim the non-existence of
transverse anomalies are reviewed to find anomalies hidden in their
approaches. The role played by transverse anomalies is discussed.

##  I Introduction

There are always surprises in common quantization procedures let alone the
quantization of relativistic fields, which is highly entangled with infinite
degrees of freedom, with various divergences and anomalies revealing the power
of quantum laws. We discuss such anomalies in the present paper. In various
examples, anomalies break many symmetries and manifest as various anomalous
Ward–Takahashi identitiesWard (1950); Takahashi (1957) (WTIs), such as
chiralAdler and Bardeen (1969); Bell and Jackiw (1969); Adler (1969) and
traceFujikawa (1980); Fujikawa  _et al._ (2004) anomalies. To our knowledge,
however, no paper has discussed anomalies in WTIs that do not stand for any
symmetry. Our research on anomalies in the transverse Ward–Takahashi
identityKondo (1997); He  _et al._ (2000); He (2009); xue Qin  _et al._ (2013)
(tWTI) which is not a symmetry WTI, has opened the door to non-symmetry
anomalies. The content of non-symmetry WTI is much richer than that of
symmetry WTIs, and the anomaly may be largely extended and more exposed to us
such that we may see the nature of the anomaly more deeply. However, this
requires far more examples of non-symmetry anomalies, apart from anomalies in
tWTIs as discussed in this paper. Further support for non-symmetry anomalies
must be left to further discoveries, with the present paper focusing on
anomalies in tWTIs (referred to as transverse anomalies) only.

In dealing with transverse anomalies, we find that many methods applied to
symmetry anomalies are entirely suitable for locating and analyzing a non-
symmetry anomaly. As an example, if we use dimensional renormalization,
anomalies in symmetry and non-symmetry WTIs may be treated on an equal footing
because extra dimensional operators appear in equations of motion and any WTI
that involves the use of equations of motion may acquire an anomaly because
these extra dimensional operators can often be expanded using various four-
dimensional operators that potentially include anomaly termsBonneau (1980a);
Collins (1986). Indeed, we find no difference in analyzing transverse
anomalies when adopting dimensional renormalization in Sec.IV than when
adopting a procedure to handle chiral and trace anomalies (as described in
detail in Bonneau (1980a)).

Although dimensional renormalization already allows us to go to any order in
perturbation theory, it’s interesting and inspiring to look at some semi-
classical one-loop methods in locating anomalies in WTIs. Fujikawa’s elegant
approachFujikawa (1979); Fujikawa  _et al._ (2004) tells us that anomalies
appear as long as we get non-trivial Jacobian factors by varying fields in the
path integral when obtaining WTIs. It’s therefore convenient to check whether
a WTI has anomalies if we know how to get the WTI by varying fields, and this
is the case for the tWTI. Equivalently111The equivalence of Fujikawa’s method
and the following canonical approach is proved in Tsutsui (1989)., we may
locate anomalies in the canonical framework to avoid dealing with the explicit
but somehow abstract path integral measure because it has no classical
correspondence, and we do not even have to know how to get this WTI by varying
fields. In the canonical framework without any consideration of
renormalization, it is easy to conclude that an anomaly is a matter of
definition of operators and that anomalies simply hide in equations of motion.
We consider, for example, the case of the chiral anomalyJoglekar and Misra
(1987); Tsutsui (1989). Starting from the following Dyson–Schwinger equation,
where derivatives should be outside the time ordered product (as is the
convention used throughout the present paper), we have

$\left<Ti\not{D}^{x}_{k\ell}\psi_{\ell}(x)\bar{\psi}_{n}(y)\right>=m\left<T\psi_{k}(x)\bar{\psi}_{n}(y)\right>+i\delta_{kn}\delta^{4}(x-y).$
(1)

In assembling the chiral anomaly, we contract $\left(\gamma_{5}\right)_{nk}$
on both sides of (1) and take the $y\rightarrow x$ limit, thus obtaining the
familiar expression $\mathrm{tr}[\gamma_{5}]\delta^{4}(x-x)$, a signal for
chiral anomaly222The minus sign is due to Fermi-statistics, and we drop the
time-ordered symbol in the equal time limit. :

$\left<\bar{\psi}(x)\gamma_{5}i\not{\overrightarrow{D}}\psi(x)\right>=m\left<\bar{\psi}(x)\gamma_{5}\psi(x)\right>-i\mathrm{tr}[\gamma_{5}]\delta^{4}(x-x).$
(2)

Similarly, we have (the Dyson–Schwinger equation for $\bar{\psi}$)

$\left<\bar{\psi}(x)i\not{\overleftarrow{D}}\gamma_{5}\psi(x)\right>=-m\left<\bar{\psi}(x)\gamma_{5}\psi(x)\right>+i\mathrm{tr}[\gamma_{5}]\delta^{4}(x-x),$
(3)

The anomalous partial conservation equation for axial current
$j_{5}^{\mu}\equiv\bar{\psi}\gamma^{\mu}\gamma_{5}\psi$ is then obtained
($\overrightarrow{D}_{\mu}=\overrightarrow{\partial}_{\mu}-igA_{\mu},\overleftarrow{D}_{\mu}=\overleftarrow{\partial}_{\mu}+igA_{\mu}$)
:

$\displaystyle\partial_{\mu}\left<j^{\mu}_{5}(x)\right>$ (4) $\displaystyle=$
$\displaystyle-i\left<\bar{\psi}(x)i\not{\overleftarrow{D}}\gamma_{5}\psi(x)\right>+i\left<\bar{\psi}(x)\gamma_{5}i\not{\overrightarrow{D}}\psi(x)\right>$
$\displaystyle=$ $\displaystyle
2im\left<\bar{\psi}(x)\gamma_{5}\psi(x)\right>+2\mathrm{tr}[\gamma_{5}]\delta^{4}(x-x).$

This is exactly what FujikawaFujikawa (1979) obtained333Note that Fujikawa
worked in Wick-rotated Euclidean space, thus an extra factor $i$ should be
multiplied to our anomaly terms to restore his results. This is also true for
the trace anomaly which we will talk about at once. by calculating the
transformation Jacobian of the path integral measure (before regularization).
Additionally, it is easy to generalize to other anomalies such as the trace
anomaly444Alves and Farina (1993) worked out trace anomaly of scalar field in
curved spacetime, but in the present paper we talk about trace anomaly in
quantum electrodynamics. from conformal symmetry. Starting from the following
energy-momentum tensor (see e.g. (1.2) in Adler  _et al._ (1977)) :

$\displaystyle\theta_{\mu\nu}=$
$\displaystyle\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}-F_{\mu\rho}F_{\nu}^{\
\rho}$ (5)
$\displaystyle+\frac{i}{4}\left(\bar{\psi}\gamma_{\mu}\overrightarrow{D}_{\nu}\psi+\bar{\psi}\gamma_{\nu}\overrightarrow{D}_{\mu}\psi-\bar{\psi}\gamma_{\mu}\overleftarrow{D}_{\nu}\psi-\bar{\psi}\gamma_{\nu}\overleftarrow{D}_{\mu}\psi\right),$

and contract (1) with $\delta_{kn}$, i.e.,

$\left<\bar{\psi}(x)i\not{\overrightarrow{D}}\psi(x)\right>=m\left<\bar{\psi}(x)\psi(x)\right>-i\mathrm{tr}\left[\mathbbm{1}\right]\delta^{4}(x-x).$
(6)

Together with that of $\bar{\psi}$ :

$\left<\bar{\psi}(x)i\not{\overleftarrow{D}}\psi(x)\right>=-m\left<\bar{\psi}(x)\psi(x)\right>+i\mathrm{tr}\left[\mathbbm{1}\right]\delta^{4}(x-x),$
(7)

we have

$g^{\mu\nu}\left<\theta_{\mu\nu}(x)\right>=m\left<\bar{\psi}(x)\psi(x)\right>-i\mathrm{tr}\left[\mathbbm{1}\right]\delta^{4}(x-x).$
(8)

Again, this reproduces what Fujikawa obtained by his methodFujikawa (1981)
(before regularization).

Clearly, the important step in getting the chiral and trace anomaly explicitly
is to define $\bar{\psi}(x)\gamma_{5}i\not{\overrightarrow{D}}\psi(x)$ and
$\bar{\psi}(x)i\not{\overrightarrow{D}}\psi(x)$ to be $\lim_{y\rightarrow
x}T\bar{\psi}(y)\gamma_{5}i\not{\overrightarrow{D}}_{x}\psi(x)$ and
$\lim_{y\rightarrow x}T\bar{\psi}(y)i\not{\overrightarrow{D}}_{x}\psi(x)$. If
we use the naive definition that
$\bar{\psi}(x)\gamma_{5}i\not{\overrightarrow{D}}\psi(x)\equiv\lim_{y\rightarrow
x}\bar{\psi}(y)\gamma_{5}i\not{\overrightarrow{D}}_{x}\psi(x)$ and
$\bar{\psi}(x)i\not{\overrightarrow{D}}\psi(x)\equiv\lim_{y\rightarrow
x}\bar{\psi}(y)i\not{\overrightarrow{D}}_{x}\psi(x)$, there can not be any
anomalous terms in the chiral and trace WTI. According to the above argument,
as long as equations of motion are used in derivations of a WTI and the time-
ordered product definition of operators is taken, an anomaly in the form of
singular contact terms like $\mathrm{tr}[\gamma_{5}]\delta^{4}(x-x)$ and
$\mathrm{tr}[\mathbbm{1}]\delta^{4}(x-x)$ may appear555Of course, this is only
established on one-loop order and some specific regularization schemes such as
$\zeta$ function regularization discussed in App.A.. This helps us greatly to
anticipate possible anomalies in new WTIs — not necessarily one that stands
for some symmetry — before resorting to rigorous all-order methods, such as
dimensional renormalization.

The remainder of the paper is organized as follows. We first briefly review
the tWTI in Sec.II and then derive the tWTI in Fujikawa’s paradigm (one-loop
order) in Sec.III to obtain intuitive ideas on transverse anomalies. We next
present a rigorous any-loop order analysis of the tWTI in dimensional
renormalization in Sec.IV. In Sec.V, we discuss the connection between
transverse anomalies and Schwinger terms on the basis of Sec.IV. It has been
shown many times that the naive tWTI (i.e., without transverse anomalies) is
correct on one-loop order, and we make comments in Sec.VI and App.B and App.C
relating to picking up hidden transverse anomalies in those approaches.
Symmetry and non-symmetry anomalies are then compared in Sec.VII. We conclude
the paper in Sec.VIII. It is worth emphasizing that throughout the paper
except in Sec.III and where one-loop is indicated explicitly, we work with the
accuracy to any loop order in perturbation theory.

In this paper, the space-time metric is $g^{\mu\nu}=\mathrm{diag}(+,-,-,-)$.
$\gamma_{5}\equiv i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ and
$\epsilon^{0123}=+1$. We define
$\sigma^{\mu\nu}\equiv\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$.

##  II Review of $\mathbf{tWTIs}$

Vector and axial vector tWTIs have been proposedKondo (1997); He  _et al._
(2000); He (2009); xue Qin  _et al._ (2013) in Abelian and non-Abelian cases.
In this paper, we focus on the Abelian case only while the non-Abelian
generalization is presented in App.A(see (48)) but is to be investigated in
detail elsewhere.

The Abelian vector tWTI666In most cases, this is the meaning of the tWTI. is
often presented asHe (2009); xue Qin  _et al._ (2013)(without anomaly; the
generalization that includes more $\psi\left(y_{i}\right)$ and
$\bar{\psi}\left(z_{j}\right)$ or $A^{\mu_{k}}\left(u_{k}\right)$ is obvious)

$\displaystyle\partial^{\mu}_{x}\left<Tj^{\nu}(x)\psi(y)\bar{\psi}(z)\right>-\partial^{\nu}_{x}\left<Tj^{\mu}(x)\psi(y)\bar{\psi}(z)\right>$
(9) $\displaystyle=$ $\displaystyle
i\sigma^{\mu\nu}\left<T\psi(y)\bar{\psi}(z)\right>\delta^{4}(x-y)+i\left<T\psi(y)\bar{\psi}(z)\right>\sigma^{\mu\nu}\delta^{4}(x-z)$
$\displaystyle+i\epsilon^{\mu\nu\rho\sigma}\left(\partial^{x}_{\rho}-\partial^{x^{\prime}}_{\rho}\right)\left<T\bar{\psi}(x^{\prime})\gamma_{\sigma}\gamma_{5}e^{ig\int_{x}^{x^{\prime}}dy\cdot
A}\psi(x)\psi(y)\bar{\psi}(z)\right>_{x^{\prime}\rightarrow x}$
$\displaystyle+2m\left<T\bar{\psi}(x)\sigma^{\mu\nu}\psi(x)\psi(y)\bar{\psi}(z)\right>,$

while the axial vector tWTI is

$\displaystyle\partial^{\mu}_{x}\left<Tj^{\nu}_{5}(x)\psi(y)\bar{\psi}(z)\right>-\partial^{\nu}_{x}\left<Tj^{\mu}_{5}(x)\psi(y)\bar{\psi}(z)\right>$
(10) $\displaystyle=$ $\displaystyle
i\sigma^{\mu\nu}\gamma_{5}\left<T\psi(y)\bar{\psi}(z)\right>\delta^{4}(x-y)-i\left<T\psi(y)\bar{\psi}(z)\right>\sigma^{\mu\nu}\gamma_{5}\delta^{4}(x-z)$
$\displaystyle+i\epsilon^{\mu\nu\rho\sigma}\left(\partial^{x}_{\rho}-\partial^{x^{\prime}}_{\rho}\right)\left<T\bar{\psi}(x^{\prime})\gamma_{\sigma}e^{ig\int_{x}^{x^{\prime}}dy\cdot
A}\psi(x)\psi(y)\bar{\psi}(z)\right>_{x^{\prime}\rightarrow x}.$

Before starting, it is necessary to reduce both (9) and (10) to simpler forms.
The apparently non-local expression of $\lim\limits_{x^{\prime}\rightarrow
x}i\epsilon^{\mu\nu\rho\sigma}\left(\partial^{x}_{\rho}-\partial^{x^{\prime}}_{\rho}\right)\bar{\psi}(x^{\prime})\gamma_{\sigma}\gamma_{5}e^{ig\int_{x}^{x^{\prime}}dy\cdot
A}\psi(x)$ is suitable for Fourier transformationsHe (2009) but a little
confusing because the factor $e^{ig\int_{x}^{x^{\prime}}dy\cdot A}$ is not
used in this paper. The way out is to simply work out this limit first777In
fact, the original expression is just the _result_ of the limit, so any
question about the interchange of limits is not of concern here, as easily
seen in Sec.III and Sec.IV. :

$\lim\limits_{x^{\prime}\rightarrow
x}i\epsilon^{\mu\nu\rho\sigma}\left(\partial^{x}_{\rho}-\partial^{x^{\prime}}_{\rho}\right)\bar{\psi}(x^{\prime})\gamma_{\sigma}\gamma_{5}e^{ig\int_{x}^{x^{\prime}}dy\cdot
A}\psi(x)=2\epsilon^{\mu\nu\rho\sigma}\bar{\psi}(x)\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi(x),$
(11)

where
$D_{\rho}=\frac{1}{2}\left(\overrightarrow{\partial}_{\rho}-\overleftarrow{\partial}_{\rho}\right)-igA_{\rho}$.
We hereafter use
$2\epsilon^{\mu\nu\rho\sigma}\bar{\psi}(x)\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi(x)$
rather than the non-local limit (the same as that in (10) with
$\gamma_{\sigma}\gamma_{5}\rightarrow\gamma_{\sigma}$).

We return to the tWTI. (9) and (10) are not conservation equations for any
currents because transformations in (14) leading to tWTIs with
$\alpha(x)={Const.}$ do not leave the Lagrangian or action invariant, even
with $m\rightarrow 0$. Therefore, the tWTI is a proper example that
illustrates the richness of anomalies beyond the scope of quantum obstacles to
classical symmetries.

##  III Heuristic Derivation of Transverse Anomalies using Fujikawa’s Method

In this section, we make use of Fujikawa’s method for the path integral
measure to obtain some intuitive pictures of transverse anomalies. It is
knownFujikawa  _et al._ (2004) that Fujikawa’s original method is correct only
in the sense of the background field approximation, and we thus treat
$A_{\mu}(x)$ as a background electromagnetic potential and the following
Lagrangian should be sufficient in this section (i.e., there is no need for
renormalization at the moment). The Lagrangian is

$\mathscr{L}_{\mathrm{BG}}=\bar{\psi}\left(\frac{i}{2}\overleftrightarrow{\not{\partial}}-ig_{0}\not{A}-m_{0}\right)\psi.$
(12)

The partition function $Z\left[\eta,\bar{\eta},A\right]$ is simply :

$Z\left[\eta,\bar{\eta},A\right]=\int\left[d\psi d\bar{\psi}\right]e^{i\int
d^{4}x\left(\mathscr{L}_{\mathrm{BG}}(x)+\bar{\psi}(x)\eta(x)+\bar{\eta}(x)\psi(x)\right)}.$
(13)

In the absence of the dynamics of $A_{\mu}$, (13) is simply a one-loop
approximation of quantum electrodynamics (QED).

We apply the field variation

$\displaystyle\delta\psi(x)=$
$\displaystyle\frac{1}{4}\alpha(x)\epsilon_{\mu\nu}\sigma^{\mu\nu}\psi(x),$
(14) $\displaystyle\delta\bar{\psi}(x)=$
$\displaystyle\frac{1}{4}\alpha(x)\epsilon_{\mu\nu}\bar{\psi}(x)\sigma^{\mu\nu}.$

and include its non-trivial Jacobian888See (49) for a detailed derivation. and
thus obtain the desired tWTI :

$\displaystyle\partial^{\mu}\left<j^{\nu}(x)\right>_{A}-\partial^{\nu}\left<j^{\mu}(x)\right>_{A}$
(15) $\displaystyle=$ $\displaystyle
2\epsilon^{\mu\nu\rho\sigma}\left<\bar{\psi}(x)\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi(x)\right>_{A}+2m\left<\bar{\psi}(x)\sigma^{\mu\nu}\psi(x)\right>_{A}$
$\displaystyle-\left<\bar{\psi}(x)\right>_{A}\sigma^{\mu\nu}\eta(x)-\bar{\eta}(x)\sigma^{\mu\nu}\left<\psi(x)\right>_{A}-2i\mathrm{tr}\left[\sigma^{\mu\nu}\right]\delta^{4}(x-x).$

We thus focus on the non-trivial Jacobian of the path integral measure, i.e.,
$-2i\mathrm{tr}\left[\sigma^{\mu\nu}\right]\delta^{4}(x-x)$ in (15). In
contrast, the field variation leading to the axial tWTI,

$\displaystyle\delta\psi(x)=$
$\displaystyle+\frac{1}{4}\alpha(x)\epsilon_{\mu\nu}\sigma^{\mu\nu}\gamma_{5}\psi(x),$
(16) $\displaystyle\delta\bar{\psi}(x)=$
$\displaystyle-\frac{1}{4}\alpha(x)\epsilon_{\mu\nu}\bar{\psi}(x)\sigma^{\mu\nu}\gamma_{5},$

acquires a vanishing Jacobian factor owing to the different signs of
$\delta\psi$ and $\delta\bar{\psi}$ and thus contributes no anomalies. From
now on, we will consider the vector tWTI only.

Proceeding with Fujikawa’s original regularization method (i.e., with the
regulator $e^{-\not{D}^{2}/\Lambda^{2}}$), we get the divergent result (see
App.A for details)

$-2i\mathrm{tr}\left[\sigma^{\mu\nu}\right]\delta^{4}(x-x)\overset{\Lambda}{=}\frac{g_{0}\Lambda^{2}}{2\pi^{2}}F^{\mu\nu}(x)-\frac{g_{0}}{12\pi^{2}}\partial_{\rho}\partial^{\rho}F^{\mu\nu}(x).$
(17)

However, it is knownPeskin (2018) that quadratic divergence in QED corresponds
to photon mass and thus must be discarded. An elegant way to do this is to
employ the $\zeta$ function regularization (which effectively turns
$\Lambda^{2}$ to $-m^{2}_{0}$) and thus obtain

$-2i\mathrm{tr}\left[\sigma^{\mu\nu}\right]\delta^{4}(x-x)\overset{\zeta}{=}-\frac{g_{0}m_{0}^{2}}{2\pi^{2}}F^{\mu\nu}(x)-\frac{g_{0}}{12\pi^{2}}\partial_{\rho}\partial^{\rho}F^{\mu\nu}(x).$
(18)

The final result of the tWTI is thus

$\displaystyle\partial^{\mu}\left<j^{\nu}(x)\right>_{A}-\partial^{\nu}\left<j^{\mu}(x)\right>_{A}$
(19) $\displaystyle=$ $\displaystyle
2\epsilon^{\mu\nu\rho\sigma}\left<\bar{\psi}(x)\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi(x)\right>_{A}+2m\left<\bar{\psi}(x)\sigma^{\mu\nu}\psi(x)\right>_{A}$
$\displaystyle-\left<\bar{\psi}(x)\right>_{A}\sigma^{\mu\nu}\eta(x)-\bar{\eta}(x)\sigma^{\mu\nu}\left<\psi(x)\right>_{A}$
$\displaystyle-\frac{g_{0}m_{0}^{2}}{2\pi^{2}}F^{\mu\nu}(x)-\frac{g_{0}}{12\pi^{2}}\partial_{\rho}\partial^{\rho}F^{\mu\nu}(x).$

At this point, however, it is emphasized that$\left<j^{\mu}(x)\right>_{A}$,
$\epsilon^{\mu\nu\rho\sigma}\left<\bar{\psi}(x)\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi(x)\right>_{A}$
and $\left<\bar{\psi}(x)\sigma^{\mu\nu}\psi(x)\right>_{A}$ in (19) are not
well defined (even in the background field approximation (i.e., 1-loop order))
owing to the divergence of loop diagrams with only two vertices even after
imposing gauge invariance in the external photon leg. This is unlike the case
of the chiral WTI, where degrees of divergence of triangle diagrams are
largely decreased by both an additional internal fermion propagator and gauge
invariance in two external photon legs. See Fig.1 .

Figure 1: (Divergent) one-loop diagrams for operators in the tWTI and chiral
WTI (gray vertex for those in the tWTI and black vertex for those in the
chiral WTI)

It is the divergence in these loop diagrams that makes the anomaly terms
ambiguous because999The authors are in debt to the referee for pointing this
out. they maybe only counterterms of
$\epsilon^{\mu\nu\rho\sigma}\left<\bar{\psi}(x)\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi(x)\right>_{A}$
and $\left<\bar{\psi}(x)\sigma^{\mu\nu}\psi(x)\right>_{A}$. However, we will
see that the anomaly terms survive even after renormalization. A rigorous
analysis using dimensional renormalization is presented in the next section.
Of course, the gauge fields $A_{\mu}$ present are not treated as an external
source.

##  IV Full Analysis in Dimensional Renormalization

For simplicity and clarity, the modified minimal subtraction
($\overline{\mathrm{MS}}$) is used in this section. We first specify the
effective LagrangianBonneau (1980a) (i.e., without infinite counterterms) :

$\displaystyle\mathscr{L}_{\mathrm{EFF}}=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}\left(i\not{D}-m\right)\psi$
(20)
$\displaystyle+\frac{\lambda^{2}}{2}A_{\mu}A^{\mu}-\frac{1}{2\xi}\left(\partial_{\mu}A^{\mu}\right)^{2}.$

Here the gauge fixing term is as usual, and a photon mass term is added to
regularize infrared divergencesBonneau (1980a). Now that $A_{\mu}$ is
dynamical, our analysis can be extended to all orders by virtue of dimensional
renormalization’t Hooft and Veltman (1972); Breitenlohner and Maison (1977);
Bonneau (1980b); Collins (1986).

As figured out in a series of papersBreitenlohner and Maison (1977); Bonneau
(1980b, a), anomalies in dimensional renormalization arise from extra
dimensional objects like101010We here use the same convention as used by
CollinsCollins (1986), where objects with a bar, such as $\bar{\gamma}^{\mu}$,
are genuinely four-dimensional things (but not to be confused with the bar in
$\bar{\psi}$, which indicates pseudo-Hermitian conjugation), objects with a
hat, such as $\hat{g}^{\mu\nu}$, exist in extra dimensions, and objects
without special labels, such as $p^{\mu}=\bar{p}^{\mu}+\hat{p}^{\mu}$, are
complete $d$-dimensional entities. $\hat{g}^{\mu}_{\mu}=d-4$ and
$\left\\{\gamma^{\mu},\gamma_{5}\right\\}=2\hat{\gamma}^{\mu}\gamma_{5}$ which
correspond respectively to trace and chiral anomalies, as do transverse
anomalies. Using the normal product formalismCollins (1986); Bonneau (1980b,
a) in dimensional renormalization, we can easily derive a prototype of the
tWTI :

$\displaystyle\partial^{\mu}N\left[j^{\nu}\right]-\partial^{\nu}N\left[j^{\mu}\right]=-\frac{i}{2}\partial_{\rho}N\left[\bar{\psi}\left[\gamma^{\rho},\bar{\sigma}^{\mu\nu}\right]\psi\right]$
(21) $\displaystyle=$ $\displaystyle
2N\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right]+2mN\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\psi\right]-2N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\hat{\gamma}^{\rho}iD_{\rho}\psi\right]$
$\displaystyle+N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\left(i\overrightarrow{\not{D}}-m\right)\psi\right]+N\left[\bar{\psi}\left(-i\overleftarrow{\not{D}}-m\right)\bar{\sigma}^{\mu\nu}\psi\right].$

Formally vanished
$N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\left(i\overrightarrow{\not{D}}-m\right)\psi\right]$
and
$N\left[\bar{\psi}\left(-i\overleftarrow{\not{D}}-m\right)\bar{\sigma}^{\mu\nu}\psi\right]$
obtained using equations of motion are sources of contact terms when inserted
into Green’s functionsCollins (1986), and we thus only need consider the
possible anomaly term
$N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\hat{\gamma}^{\rho}iD_{\rho}\psi\right]$.
Although
$N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\hat{\gamma}^{\rho}iD_{\rho}\psi\right]$
has a vanishing tree diagram, its loop diagrams are not zero because
$\overleftrightarrow{\partial}_{\rho}$ _inside_ the operator drops loop
momenta off111111Extra dimensional loop momenta must not be taken to be zero
before carrying out loop integrals, in contrast with the case for external
momenta. and thus has non-zero contributions. Operators with evanescent
verticesCollins (1986) like this are simply origins of anomalies in
dimensional renormalization. Thanks to the existence of Zimmermann-like
identities in dimensional renormalizationBonneau (1980b, a), transverse
anomalies that are also of the form
$N\left[\hat{g}_{\mu\nu}\mathcal{O}^{\mu\nu}\right]$ (just as trace and chiral
anomalies which were given by BonneauBonneau (1980a)) because

$N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\hat{\gamma}^{\rho}iD_{\rho}\psi\right]\equiv
N\left[\hat{g}_{\rho\sigma}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\rho}iD^{\sigma}\psi\right)\right]$
(22)

can be reduced to usual operators in physical dimensions :

$\displaystyle
N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\hat{\gamma}^{\rho}iD_{\rho}\psi\right]=$
$\displaystyle
aN\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\hat{\gamma}^{\rho}iD_{\rho}\psi\right]+b^{\prime}\left(\partial^{\mu}N\left[j^{\nu}\right]-\partial^{\nu}N\left[j^{\mu}\right]\right)$
(23)
$\displaystyle+c^{\prime}N\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right]+f^{\prime}mN\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\psi\right]$
$\displaystyle+r^{\prime}\partial^{\rho}\partial_{\rho}F^{\mu\nu}+s^{\prime}m^{2}F^{\mu\nu}.$

For simplicity, we omit the normal product symbol for a single renormalized
field $F^{\mu\nu}$. Similar to the case of trace and chiral anomaliesBonneau
(1980a), all the above coefficients can be obtained from residues of the
simple pole (r.s.p.) at $4-d=0$ of the overall divergence of specific Green’s
functions of
$N\left[\check{g}_{\rho\sigma}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\rho}iD^{\sigma}\psi\right)\right]$,
where $\check{g}_{\rho\sigma}$ defined by BonneauBonneau (1980a) is roughly
$\hat{g}_{\rho\sigma}/(d-4)$ but $1/(d-4)$ therein is not included in the
Laurent expansion when determining countertermsBonneau (1980a). The results
are as follows121212$\tilde{\psi}(p)\equiv\int d^{4}x\ e^{ip\cdot
x}\psi(x),\tilde{\bar{\psi}}(q)\equiv\int d^{4}x\ e^{-iq\cdot
x}\bar{\psi}(x),\tilde{A}^{\rho}(k)\equiv\int d^{4}x\ e^{ik\cdot
x}A^{\rho}(x).$ (here, a line over a Green’s function indicates the overall
divergence (i.e., the counterterm obtained by contracting the whole 1PI
diagram to a single vertex) has not been subtracted). We have

$\displaystyle a=$
$\displaystyle\frac{1}{48(4-d)}\mathrm{tr}\left\\{\mathrm{r.s.p.}\frac{\partial}{\partial
p_{\rho}}\overline{\left<TN\left[\check{g}_{\alpha\beta}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\alpha}iD^{\beta}\psi\right)\right]\tilde{\psi}\left(\frac{1}{2}p\right)\tilde{\bar{\psi}}\left(\frac{1}{2}p\right)\right>^{\mathrm{prop}}}|_{\check{g},p=0}\times\bar{\sigma}^{\mu\nu}\hat{\gamma}_{\rho}\right\\},$
(24) $\displaystyle b^{\prime}=$
$\displaystyle\frac{1}{-96i}\mathrm{tr}\left\\{\mathrm{r.s.p.}\frac{\partial}{\partial
p_{\rho}}\overline{\left<TN\left[\check{g}_{\alpha\beta}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\alpha}iD^{\beta}\psi\right)\right]\tilde{\psi}\left(\frac{1}{2}p\right)\tilde{\bar{\psi}}\left(-\frac{1}{2}p\right)\right>^{\mathrm{prop}}}|_{\check{g},p=0}\times\left(\bar{g}_{\mu\rho}\bar{\gamma}_{\nu}-\bar{g}_{\nu\rho}\bar{\gamma}_{\mu}\right)\right\\},$
$\displaystyle c^{\prime}=$
$\displaystyle\frac{1}{-96}\mathrm{tr}\left\\{\mathrm{r.s.p.}\frac{\partial}{\partial
p_{\rho}}\overline{\left<TN\left[\check{g}_{\alpha\beta}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\alpha}iD^{\beta}\psi\right)\right]\tilde{\psi}\left(\frac{1}{2}p\right)\tilde{\bar{\psi}}\left(\frac{1}{2}p\right)\right>^{\mathrm{prop}}}|_{\check{g},p=0}\times\epsilon_{\mu\nu\rho\sigma}\gamma^{\sigma}\gamma_{5}\right\\},$
$\displaystyle f^{\prime}=$
$\displaystyle\frac{1}{-48m}\mathrm{tr}\left\\{\mathrm{r.s.p.}\overline{\left<TN\left[\check{g}_{\alpha\beta}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\alpha}iD^{\beta}\psi\right)\right]\tilde{\psi}(0)\tilde{\bar{\psi}}(0)\right>^{\mathrm{prop}}}|_{\check{g}=0}\times\bar{\sigma}_{\mu\nu}\right\\},$
$\displaystyle r^{\prime}=$
$\displaystyle\frac{1}{-288i}\frac{\partial}{\partial
q_{\delta}}\frac{\partial}{\partial q^{\delta}}\frac{\partial}{\partial
q_{\sigma}}\mathrm{r.s.p.}\overline{\left<TN\left[\check{g}_{\alpha\beta}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\alpha}iD^{\beta}\psi\right)\right]\tilde{A}^{\rho}(q)\right>^{\mathrm{prop}}}|_{\check{g},q=0}\times\left(\bar{g}_{\mu\rho}\bar{g}_{\nu\sigma}-\bar{g}_{\mu\sigma}\bar{g}_{\nu\rho}\right),$
$\displaystyle s^{\prime}=$
$\displaystyle\frac{1}{-24im^{2}}\frac{\partial}{\partial
q_{\sigma}}\mathrm{r.s.p.}\overline{\left<TN\left[\check{g}_{\alpha\beta}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\alpha}iD^{\beta}\psi\right)\right]\tilde{A}^{\rho}(q)\right>^{\mathrm{prop}}}_{\check{g},q=0}\times\left(\bar{g}_{\mu\sigma}\bar{g}_{\nu\rho}-\bar{g}_{\mu\rho}\bar{g}_{\nu\sigma}\right).$

Finally, the tWTI is

$\displaystyle\left(1-2b\right)\left(\partial^{\mu}N\left[j^{\nu}\right]-\partial^{\nu}N\left[j^{\mu}\right]\right)$
(25) $\displaystyle=$ $\displaystyle
2\left(1-c\right)N\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right]+2m\left(1-f\right)N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\psi\right]$
$\displaystyle-2r\partial^{\rho}\partial_{\rho}F^{\mu\nu}-2sm^{2}F^{\mu\nu}$
$\displaystyle+N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\left(i\overrightarrow{\not{D}}-m\right)\psi\right]+N\left[\bar{\psi}\left(-i\overleftarrow{\not{D}}-m\right)\bar{\sigma}^{\mu\nu}\psi\right],$

where $x=x^{\prime}/(1-a)$ for $x=b,c,f,r,s$ and their one-loop values are
given in App.D.

In QED, it is necessary to ensure that all coefficients are gauge invariant
and, in particular, we should focus on coefficients of transverse anomalies,
namely $-2r$ and $-2sm^{2}$. Fortunately, following the arguments made by
Bonneau about gauge invariance of the chiral anomalyBonneau (1980a),
mainly131313Conclusions in Bonneau (1980a) are so general that nothing
essential needs modifications. (B.10), Lemma 3, and Fig.3 in Bonneau (1980a),
it is almost trivial to see that $b,c,fm,r,sm^{2}$ are all gauge invariant,
i.e., independent of $\xi$. This gauge independence is also briefly discussed
in App.D.

It is now clear that transverse anomalies exists even after renormalization.
Of course, with the presence of the four-dimensional operator
$N\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right]$
and three-dimensional operator
$N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\psi\right]$, transverse anomalies
(whose dimensions are respectively 4 and 2) may be absorbed into these
operators as finite counterterms and thus be rendered dependent on the
renormalization schemes 141414We thank the referee for making this point.. We
may represent the most general form of the tWTI as :

$\displaystyle\left(1-2b\right)\left(\partial^{\mu}N\left[j^{\nu}\right]-\partial^{\nu}N\left[j^{\mu}\right]\right)$
(26) $\displaystyle=$ $\displaystyle
2\left(1-c\right)\tilde{N}\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right]+2m\left(1-f\right)\tilde{N}\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\psi\right]$
$\displaystyle-2\tilde{r}\partial^{\rho}\partial_{\rho}F^{\mu\nu}-2\tilde{s}m^{2}F^{\mu\nu}$
$\displaystyle+N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\left(i\overrightarrow{\not{D}}-m\right)\psi\right]+N\left[\bar{\psi}\left(-i\overleftarrow{\not{D}}-m\right)\bar{\sigma}^{\mu\nu}\psi\right].$

Here, the operators with a tilde are simply linear combinations of the
original operators and transverse anomalies :

$\displaystyle
2\left(1-c\right)\tilde{N}\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right]\equiv$
$\displaystyle
2\left(1-c\right)N\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right]-2\left(r-\tilde{r}\right)\partial_{\rho}\partial^{\rho}F^{\mu\nu}-2\tilde{\alpha}\left(s-\tilde{s}\right)m^{2}F^{\mu\nu},$
(27) $\displaystyle
2m\left(1-f\right)\tilde{N}\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\psi\right]\equiv$
$\displaystyle
2m\left(1-f\right)N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\psi\right]-2\left(1-\tilde{\alpha}\right)\left(s-\tilde{s}\right)m^{2}F^{\mu\nu},$

and $\tilde{r},\tilde{s},\tilde{\alpha}$ denote arbitrary real numbers of
order $\mathcal{O}(g^{2})$.

We will talk more generally about this arbitrariness in Sec.VII by comparing
with chiral and trace anomalies. However, we suggest a way of fixing the
coefficient $\tilde{r}$ making use of Schwinger terms in the next section.

##  V Transverse anomalies and Schwinger terms

The Green’s function version of (26), together with
$\psi(y)\bar{\psi}(z)A^{\rho}(u)$, is :

$\displaystyle\left(1-2b\right)\left(\partial^{\mu}_{x}\left<TN\left[j^{\nu}\right](x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>-\partial^{\nu}_{x}\left<TN\left[j^{\mu}\right](x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>\right)$
(28) $\displaystyle=$ $\displaystyle
2\left(1-c\right)\left<T\tilde{N}\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right](x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>$
$\displaystyle+2m\left(1-f\right)\left<T\tilde{N}\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\psi\right](x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>$
$\displaystyle-2\tilde{r}\left<T\partial_{x}^{\rho}\partial_{\rho}^{x}F^{\mu\nu}(x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>-2sm^{2}\left<TF^{\mu\nu}(x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>$
$\displaystyle+i\bar{\sigma}^{\mu\nu}\left<T\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>\delta^{4}(x-y)+i\left<T\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>\bar{\sigma}^{\mu\nu}\delta^{4}(x-z).$

The crucial observation is to note the equation of motion for photon field
$A^{\rho}(u)$ :

$\displaystyle\partial_{\mu}^{x}\left<TF^{\mu\nu}(x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>+\frac{1}{\xi}\partial^{\nu}_{x}\left<T\partial_{\mu}^{x}A^{\mu}(x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>$
(29) $\displaystyle=$
$\displaystyle-g\left<Tj^{\nu}(x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>+ig^{\nu\rho}\left<T\psi(y)\bar{\psi}(z)\right>\delta^{4}(x-u).$

Taking the equation of motion together with the Bianchi identityPeskin (2018)
:

$\partial^{\mu}F^{\nu\rho}+\partial^{\nu}F^{\rho\mu}+\partial^{\rho}F^{\mu\nu}=0,$
(30)

it is easy to get

$\displaystyle\left(1-2b-2g\tilde{r}\right)\left(\partial^{\mu}_{x}\left<TN\left[j^{\nu}\right](x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>-\partial^{\nu}_{x}\left<TN\left[j^{\mu}\right](x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>\right)$
(31) $\displaystyle=$ $\displaystyle
2\left(1-c\right)\left<T\tilde{N}\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right](x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>$
$\displaystyle+2m\left(1-f\right)\left<T\tilde{N}\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\psi\right](x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>$
$\displaystyle-2i\tilde{r}\left<T\psi(y)\bar{\psi}(z)\right>\left(\partial^{\mu}_{x}g^{\nu\rho}-\partial^{\nu}_{x}g^{\mu\rho}\right)\delta^{4}(x-u)-2\tilde{s}m^{2}\left<TF^{\mu\nu}(x)\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>$
$\displaystyle+i\bar{\sigma}^{\mu\nu}\left<T\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>\delta^{4}(x-y)+i\left<T\psi(y)\bar{\psi}(z)A^{\rho}(u)\right>\bar{\sigma}^{\mu\nu}\delta^{4}(x-z).$

The contribution of $-2r\partial_{\rho}\partial^{\rho}F^{\mu\nu}$ is thus
recast to be the modification of the coefficient of the curl of
$N\left[j^{\nu}\right]$ and a new contact term,
$-2ir\left<T\psi(y)\bar{\psi}(z)\right>\left(\partial^{\mu}_{x}g^{\nu\rho}-\partial^{\nu}_{x}g^{\mu\rho}\right)\delta^{4}(x-u)$.

We recall that contact terms come from equal time commutation relations of
operators like $j^{\mu}$ and elementary fields $\psi,\bar{\psi},A^{\rho}$ in
the canonical framework, and it is thus concluded that there is a non-
canonical contribution to
$\left[N\left[j^{i}\right]\left(\vec{x},t\right),A^{\rho}\left(\vec{y},t\right)\right]$
:

$\displaystyle\delta\left(x^{0}-y^{0}\right)\left[N\left[j^{i}\right]\left(\vec{x},x^{0}\right),A^{\rho}\left(\vec{y},y^{0}\right)\right]=$
$\displaystyle\frac{-2i\tilde{r}}{1-2b-2g\tilde{r}}\left(\partial^{0}_{x}g^{i\rho}-\partial^{i}_{x}g^{0\rho}\right)\delta^{4}(x-y).$
(32)

According to SchwingerSchwinger (1959), this non-zero commutator is required
so as to not conflict with the existence of a vacuum state. As definite
operators, the commutator of $N\left[j^{\nu}\right]$ and $A^{\rho}$ should not
have arbitrariness. Then, $\frac{-2i\tilde{r}}{1-2b-2g\tilde{r}}$ is fixed and
thus $\tilde{r}$ is fixed. However, $\tilde{s}$ remains arbitrary.

Furthermore, we can also work out Schwinger terms for
$\left[N\left[j^{i}\right],N\left[j^{\rho}\right]\right]$. We consider the
following WTI 151515The last term is easily derived using Zimmermann-like
identities in dimensional renormalization proposed by BonneauBonneau (1980b,
a). Note that there are no contact terms corresponding to
$\psi,\bar{\psi},A^{\sigma}$ in this situation.:

$\displaystyle\left(1-2b-2g\tilde{r}\right)\left(\partial^{\mu}_{x}\left<TN\left[j^{\nu}\right](x)N\left[j^{\rho}\right](y)\right>-\partial^{\nu}_{x}\left<TN\left[j^{\mu}\right](x)N\left[j^{\rho}\right](y)\right>\right)$
(33) $\displaystyle=$ $\displaystyle
2\left(1-c\right)\left<T\tilde{N}\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right](x)N\left[j^{\rho}\right](y)\right>+2m\left(1-f\right)\left<T\tilde{N}\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\psi\right](x)N\left[j^{\rho}\right](y)\right>$
$\displaystyle-2\tilde{s}m^{2}\left<TF^{\mu\nu}N\left[j^{\rho}\right](y)\right>$
$\displaystyle-2\left(u\partial^{\alpha}_{y}\partial_{\alpha}^{y}+vm^{2}\right)\left(\partial^{\mu}_{y}g^{\nu\rho}-\partial^{\nu}_{y}g^{\mu\rho}\right)\delta^{4}(x-y),$

where $u\equiv u^{\prime}/(1-a),v\equiv v^{\prime}/(1-a)$ and

$\displaystyle u^{\prime}=$
$\displaystyle\frac{1}{-288i}\frac{\partial}{\partial
q_{\delta}}\frac{\partial}{\partial q^{\delta}}\frac{\partial}{\partial
q_{\sigma}}\mathrm{r.s.p.}\overline{\left<TN\left[\check{g}_{\alpha\beta}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\alpha}iD^{\beta}\psi\right)\right]N\left[\tilde{j}^{\rho}\right](q)\right>^{\mathrm{prop}}}|_{\check{g},q=0}\times\left(\bar{g}_{\mu\rho}\bar{g}_{\nu\sigma}-\bar{g}_{\mu\sigma}\bar{g}_{\nu\rho}\right)$
(34) $\displaystyle=$ $\displaystyle r^{\prime}/g,$ $\displaystyle
v^{\prime}=$ $\displaystyle\frac{1}{-12im^{2}}\frac{\partial}{\partial
q_{\sigma}}\mathrm{r.s.p.}\overline{\left<TN\left[\check{g}_{\alpha\beta}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\alpha}iD^{\beta}\psi\right)\right]N\left[\tilde{j}^{\rho}\right](q)\right>^{\mathrm{prop}}}_{\check{g},q=0}\times\left(\bar{g}_{\mu\sigma}\bar{g}_{\nu\rho}-\bar{g}_{\mu\rho}\bar{g}_{\nu\sigma}\right)$
$\displaystyle=$ $\displaystyle s^{\prime}/g.$

Therefore, the Schwinger terms of
$\left[N\left[j^{i}\right],N\left[j^{\rho}\right]\right]$ are

$\displaystyle\delta\left(x^{0}-y^{0}\right)\left[N\left[j^{i}\right]\left(\vec{x},x^{0}\right),N\left[j^{\rho}\right]\left(\vec{y},y^{0}\right)\right]$
(35) $\displaystyle=$
$\displaystyle\left(\frac{-2ir/g}{1-2b-2g\tilde{r}}\partial^{\alpha}_{y}\partial^{y}_{\alpha}+\frac{-2is/g}{1-2b-2g\tilde{r}}m^{2}\right)\left(\partial^{0}_{y}g^{i\rho}-\partial^{i}_{y}g^{0\rho}\right)\delta^{4}(x-y).$

Taking $\rho=0$, on the one-loop level, we have

$\displaystyle\delta\left(x^{0}-y^{0}\right)\left[N\left[j^{0}\right]\left(\vec{x},x^{0}\right),N\left[j^{i}\right]\left(\vec{y},y^{0}\right)\right]$
(36) $\displaystyle=$
$\displaystyle-\left(\frac{i}{12\pi^{2}}\partial^{\alpha}_{x}\partial^{x}_{\alpha}+\frac{im^{2}}{2\pi^{2}}\right)\partial^{i}_{x}\delta^{4}(x-y).$

This is comparable to results published in earlier papers. As an example, in
Chanowitz (1970), it was obtained that
$\left<\left[j^{0}\left(\vec{x},0\right),j^{i}\left(0\right)\right]\right>_{0}$
is ((10) in Chanowitz (1970)):

$\displaystyle\left<\left[j^{0}\left(\vec{x},0\right),j^{i}\left(0\right)\right]\right>_{0}=\infty\partial^{i}\delta\left(\vec{x}\right)+\frac{i}{12\pi^{2}}\partial^{i}\Delta\delta\left(\vec{x}\right),$
(37)

using spectral representation, where $\Delta\equiv\nabla^{2}$. However, we get
a finite and covariant result, in contrast with the infinite and non-covariant
result obtained in Chanowitz (1970). In any event, the reproduction of the
term161616The Schwinger term (37) was also obtained in Boulware and Jackiw
(1969) using the BJL (Bjorken, Johnson and Low) method.
$\frac{i}{12\pi^{2}}\partial^{i}\Delta\delta\left(\vec{x}\right)$ implies that
transverse anomalies are closely related to Schwinger terms.

##  VI Comments on previous articles

There are papers Kondo (1997); xin He (2001); Sun  _et al._ (2003) on the
anomalies of the tWTI, but none found an anomaly for the vector tWTI.
Additionally, He (2007) examined the vector tWTI to one-loop order in
dimensional regularization and concluded that there was no anomaly. In fact,
He (2007) has noted that $\epsilon^{\mu\nu\rho\sigma}\gamma_{\rho}\gamma_{5}$
should be replaced by
$-\frac{1}{2}\left\\{\gamma^{\rho},\sigma^{\mu\nu}\right\\}$ to ensure tWTI is
still established; otherwise, on one-loop order, additional terms of the form
a divergent integral multiplied by $(d-4)$ appear171717However, He (2007) did
not look into this one-loop anomalous term.. Equivalently, they adopted
schemes that absorb all transverse anomalies into
$N\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right]$;
i.e., what appeared in the tWTI in their paper is not
$N\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right]$
but
$N\left[-i\bar{\psi}\left\\{\gamma^{\rho},\sigma^{\mu\nu}\right\\}D_{\rho}\psi\right]$.
However, the $(d-4)$ term they discovered is actually a spurious anomaly
corresponding to corrections of coefficients of terms existing in the tWTI,
such as
$N\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right]$,
because they only examined one-loop diagrams with two external fermion legs
and did not note the crucial diagram with one external photon legs that
generates transverse anomalies. We calculated this missing diagram and
obtained exactly the results of $\zeta$ function regularization in App.A.

In Kondo (1997), the author identified the transverse vector transformation
(14) as the “local Lorentz transformation”181818In fact, only the spinor
part., and there was thus no possibility of an anomaly in the vector tWTI due
to Lorentz invariance. However, the Lorentz transformation of the Dirac
fermion mismatches the transverse transformation (14) in _signs_. The spinor
part of the Lorentz boost of the fermion isPeskin (2018)
$\delta\psi(x)=+\frac{i}{4}\alpha(x)\epsilon_{\mu\nu}\sigma^{\mu\nu}\psi(x),\delta\bar{\psi}(x)=-\frac{i}{4}\alpha(x)\epsilon_{\mu\nu}\bar{\psi}(x)\sigma^{\mu\nu}$,
where Jacobians of $\psi$ and $\bar{\psi}$ cancel each other out, regardless
of whether $\mathrm{tr}[\sigma^{\mu\nu}]\delta^{4}(x-x)$ vanishes. According
to our derivation, $\mathrm{tr}[\sigma^{\mu\nu}]\delta^{4}(x-x)$ is not zero,
and thus transverse transformation, both signs of which are _positive_ ,
cannot be protected by Lorentz symmetry to be free of anomalies.

The point-splitting method was used in xin He (2001). A spurious transverse
axial anomaly was proposed but corrected in Sun  _et al._ (2003). Meanwhile,
xin He (2001) gave a expression for the “vanishing” transverse vector anomaly;
however, following this formulation, we get a _non-vanishing_ result. Equation
(12) of xin He (2001)
is191919$U_{P}(x^{\prime},x)=\mathrm{exp}\\{-ig\int_{x}^{x^{\prime}}dy\cdot
A\\}$ differs from ours in sign, because xin He (2001) assigned
$\overrightarrow{D}_{\mu}=\overrightarrow{\partial}_{\mu}+igA_{\mu}$.

$\displaystyle\partial^{\mu}j^{\nu}(x)-\partial^{\nu}j^{\mu}(x)=$
$\displaystyle\lim_{x^{\prime}\rightarrow
x}i\left(\partial^{x}_{\lambda}-\partial^{x^{\prime}}_{\lambda}\right)\epsilon^{\lambda\mu\nu\rho}\bar{\psi}(x^{\prime})\gamma_{\rho}\gamma_{5}U_{P}(x^{\prime},x)\psi(x)$
(38) $\displaystyle+\mathrm{Symm}\lim_{\epsilon\rightarrow
0}\left\\{\bar{\psi}(x+\epsilon/2)\left[-ig\left(\gamma^{\nu}F^{\mu\rho}(x)-\gamma^{\mu}F^{\nu\rho}(x)\right)\epsilon_{\rho}\right]\psi(x-\epsilon/2)\right\\}.$

UsingPeskin (2018)
$\left<\psi(x)\bar{\psi}(y)\right>\propto\frac{\gamma^{\sigma}(x-y)_{\sigma}}{(x-y)^{4}}$,
and xin He (2001) $\mathrm{Symm}\lim_{\epsilon\rightarrow
0}\left\\{\frac{\epsilon_{\rho}\epsilon_{\sigma}}{\epsilon^{2}}\right\\}=\frac{1}{4}g^{\mu\nu}$,
we finish the calculation of the last term :

$\displaystyle\mathrm{Symm}\lim_{\epsilon\rightarrow
0}\left\\{\bar{\psi}(x+\epsilon/2)\left[-ig\left(\gamma^{\nu}F^{\mu\rho}(x)-\gamma^{\mu}F^{\nu\rho}(x)\right)\epsilon_{\rho}\right]\psi(x-\epsilon/2)\right\\}$
(39) $\displaystyle\propto$
$\displaystyle\mathrm{Symm}\lim_{\epsilon\rightarrow
0}\mathrm{tr}\left[\left(\gamma^{\nu}F^{\mu\rho}(x)-\gamma^{\mu}F^{\nu\rho}(x)\right)\gamma^{\sigma}\right]\frac{\epsilon_{\sigma}\epsilon_{\rho}}{\epsilon^{4}}$
$\displaystyle\propto$ $\displaystyle\lim_{\epsilon\rightarrow
0}\left(g^{\nu\sigma}F^{\mu\rho}(x)-g^{\mu\sigma}F^{\nu\rho}(x)\right)g_{\sigma\rho}\frac{1}{\epsilon^{2}}$
$\displaystyle\propto$ $\displaystyle\lim_{\epsilon\rightarrow
0}\frac{1}{\epsilon^{2}}F^{\mu\nu}(x)\neq 0.$

Moreover, because the above result is quadratically divergent, we need to
expand $F^{\mu\nu}(x\pm\epsilon/2)$ in intermediate steps (see App.C) to
$\mathcal{O}(\epsilon^{4})$ to extract finite contributions, which means (39)
is incomplete202020However, even if we go to $\mathcal{O}(\epsilon^{4})$,
arbitrariness of the coefficient of $\partial^{\rho}\partial_{\rho}F^{\mu\nu}$
that originates from the arbitrariness of $a\in\mathbbm{R}$ in
$\bar{\psi}(x+(a+1)\epsilon)\gamma^{\mu}\psi(x+a\epsilon)$ prevents the point-
splitting method from working for transverse anomalies; see App.C..

In brief, xin He (2001) partially worked out transverse anomalies. It is a
pity that the non-vanishing result (39) was omitted in xin He (2001).

Pauli–Villars regularizationPauli and Villars (1949) was applied to
calculating transverse anomalies in Sun  _et al._ (2003). Unfortunately Sun
_et al._ (2003) forgot a vital procedure in Pauli–Villars regularization and
thus missed transverse anomalies. This step expresses any amplitude with its
regularized form so that anomalies may appear from the WTI with mass
termsPauli and Villars (1949); Bertlmann (2000), which is the case for the
vector tWTI (19). We consider any WTI with the form

$A=mB+C,$ (40)

where $m$ is some particle’s mass. Pauli–Villars regularization
requiresBertlmann (2000) regularized WTI to be made up of regularized
amplitudes

$f^{\mathrm{phys}}=\lim_{M\rightarrow\infty}f_{m}-f_{M},$ (41)

where $f$ denotes any amplitude while $f_{m},f_{M}$ respectively denote
amplitudes calculated with physical mass $m$ and regulated mass $M$. Then, if
we proved the bare WTI

$A_{m}-A_{M}=mB_{m}+C_{m}-MB_{M}-C_{M},$ (42)

the regularized WTI may acquire an anomaly

$\displaystyle A^{\mathrm{phys}}=A_{m}-A_{M}=mB_{m}+C_{m}-MB_{M}-C_{M}$ (43)
$\displaystyle=$ $\displaystyle
mB^{\mathrm{phys}}+C^{\mathrm{phys}}+(m-M)B_{M}.$

Indeed, on the basis of the proof of the bare tWTI in Sun  _et al._ (2003), we
worked out transverse anomalies that Sun  _et al._ (2003) ignored, see App.B.

##  VII Comparison between Symmetry Anomalies and Non-symmetry Anomalies

In fact, $\partial^{\mu}j^{\nu}-\partial^{\nu}j^{\mu}$ on the left of tWTI
(25) can be recast into the divergence of some current :

$\partial^{\mu}j^{\nu}-\partial^{\nu}j^{\mu}=-\frac{i}{2}\partial_{\rho}\left(\bar{\psi}\left[\gamma^{\rho},\sigma^{\mu\nu}\right]\psi\right).$
(44)

In addition to transverse anomalies and the mass term, factors that prevent
$\bar{\psi}\left[\gamma^{\rho},\sigma^{\mu\nu}\right]\psi$ from being a
conserved current include another four-dimensional operator; i.e.,
$N\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right]$.
Without this operator, the current will become an anomalous partial conserved
current, which is the case for the tWTI in two-dimensional QED212121In two-
dimensional QED with a massless fermion,
$\left\\{\gamma^{\rho},\sigma^{\mu\nu}\right\\}=0$ and
$j^{\mu}_{5}=-\epsilon^{\mu\nu}j_{\nu}$ owing to
$\sigma^{\mu\nu}=i\epsilon^{\mu\nu}\gamma_{5}$ and
$\gamma^{\mu}\gamma_{5}=-\epsilon^{\mu\nu}\gamma_{\nu}$, and thus
$-\frac{i}{2}\partial_{\rho}\left(\psi\left[\gamma^{\rho},\sigma^{\mu\nu}\right]\psi\right)=\partial^{\mu}j^{\nu}-\partial^{\nu}j^{\mu}=\epsilon^{\mu\nu}\partial_{\rho}j^{\rho}_{5}=-\frac{g}{2\pi}\epsilon^{\mu\nu}\epsilon_{\rho\sigma}F^{\rho\sigma}=\frac{g}{\pi}F^{\mu\nu}$Peskin
(2018). Therefore, both $j^{\mu}_{5}$ and
$\partial_{\rho}\left(\psi\left[\gamma^{\rho},\sigma^{\mu\nu}\right]\psi\right)$
are anomalous partial conserved currents, and as a consequence, the tWTI in
two-dimensional QED is _not_ a non-symmetry WTI.. Therefore, the essential
difference between a symmetry WTI and a non-symmetry WTI is the presence of
extra four-dimensional operators, besides operators on the left of the
WTI222222These operators are usually in the form of a derivative of some
three-dimensional operator as is the case of chiral
($\partial_{\mu}j^{\mu}_{5}$) and trace
($\partial_{\mu}\left(x_{\nu}\theta^{\mu\nu}\right)$) anomalies.and anomalies.

It is exactly the extra four-dimensional operators that render the
arbitrariness of transverse anomalies. Obviously, this makes sense also for
any other non-symmetry anomalies that have extra four-dimensional operators.

We may proceed further. It is also possible to change coefficients of chiral
or trace anomalies at will, as long as we absorb
$N\left[\tilde{F}^{\mu\nu}A_{\nu}\right]$ or
$N\left[\bar{g}^{\mu\nu}F^{\rho\sigma}F_{\rho\sigma}\right]$ into
$N\left[\bar{\psi}\gamma^{\mu}\gamma_{5}\psi\right]$ or
$N\left[\theta^{\mu\nu}\right]$ without considering gauge invariance or energy
conservation. However, it is just these non-trivial properties or symmetries
satisfied by $N\left[\bar{\psi}\gamma^{\mu}\gamma_{5}\psi\right]$ and
$N\left[\theta^{\mu\nu}\right]$ that prevent other operators such as anomaly
terms to be absorbed into them, thus _protected_ chiral and trace anomalies
such that their coefficients cannot be adjusted arbitrarily. Therefore, once
we find some physical meanings or symmetries for
$N\left[\epsilon^{\mu\nu\rho\sigma}\bar{\psi}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi\right]$,
coefficients of transverse anomalies may be fixed naturally. As shown in
Sec.V, coefficients of transverse anomalies may be fixed partially by
resorting to Schwinger terms, but more general results for remaining
$\tilde{s}$ and other non-symmetry anomalies require deeper research.

##  VIII Conclusion and discussion

We discuss the extension of anomalies to cover those in WTIs that are not
formed by symmetry transformations, beyond the scope of symmetry, taking the
explicit example of anomalies in tWTIs. Both background field (one-loop)
analyses in Sec.III (together with some one-loop calculation in App.B and
App.C) and renormalization to all orders in dimensional renormalization in
Sec.IV indicate the existence of transverse anomalies, and we locate where
anomalies hid when Kondo (1997); xin He (2001); Sun  _et al._ (2003); He
(2007) stated the non-existence of the transverse anomalies in the vector tWTI
on one-loop order. The scheme dependence of coefficients of transverse
anomalies is also concluded temporarily, and this is partially solved by
considering Schwinger terms as in Sec.V. This needs to be investigated
further.

So far, the anomaly in all types of the local linear transformation of fermion
fields232323The local and linear transformation of $\psi(x),\bar{\psi}(x)$
must be
$\delta\psi(x)=\alpha(x)\Omega\psi(x),\delta\bar{\psi}(x)=\alpha(x)\bar{\psi}(x)\widetilde{\Omega}$,
where $\Omega$ and $\widetilde{\Omega}$ are a linear combination of $\gamma$
matrices and hence of
$\mathbbm{1},\gamma^{\mu},[\gamma^{\mu},\gamma^{\nu}],\gamma^{\mu}\gamma_{5},\gamma_{5}$.
However, the traces of odd number $\gamma$ matrices are zero; i.e.,
$\mathrm{tr}[\gamma^{\mu}]\delta^{4}(x-x)$ and
$\mathrm{tr}[\gamma^{\mu},\gamma_{5}]\delta^{4}(x-x)$ are zero even after
regularization. (not all symmetry transformations) has been exhausted. There
are only three non-zero anomalies; see Table.1 (in Fujikawa’s style for
simplicity).

Table 1: Non-trivial anomalies in all types of local linear transformation of fermion fields. anomaly type | “bare” expression | in 4-dim $SU(N)$ QCD (1 loop results)
---|---|---
trace anomalyFujikawa (1980); Fujikawa  _et al._ (2004) | $\mathrm{tr}[\mathbbm{1}]\delta^{4}(x-x)$ | $\frac{g^{2}}{48\pi^{2}}F_{a}^{\mu\nu}F_{a\mu\nu}$
transverse anomalies | $\mathrm{tr}\left\\{t_{a}[\gamma^{\mu},\gamma^{\nu}]\right\\}\delta^{4}(x-x)$ | $-\frac{gm^{2}}{4\pi^{2}}F_{a}^{\mu\nu}-\frac{g}{24\pi^{2}}\mathcal{D}^{\rho}\mathcal{D}_{\rho}F_{a}^{\mu\nu}-\frac{g^{2}}{8\pi^{2}}C_{abc}F_{b}^{\mu\rho}F_{c\rho}^{\ \ \nu}$
chiral anomalyFujikawa (1979); Fujikawa  _et al._ (2004) | $\mathrm{tr}[t_{a}\gamma_{5}]\delta^{4}(x-x)$ | $\frac{g^{2}}{32\pi^{2}}\epsilon_{\mu\nu\rho\sigma}F_{b}^{\mu\nu}F_{c}^{\rho\sigma}\mathrm{tr}[t_{a}t_{b}t_{c}]$

Table.1 shows that the transverse anomalies have many more types of operators
than the trace anomaly and chiral anomaly. In particular, the
$C_{abc}F_{b}^{\mu\rho}F_{c\rho}^{\ \ \nu}$ term may have some effect on the
present schemexue Qin  _et al._ (2013); Qin  _et al._ (2014); Albino  _et al._
(2019) making use of tWTI. However, in this scheme, the other two terms in
non-Abelian transverse anomalies and the whole Abelian transverse anomalies
(where $C_{abc}=0$) have no places to plug in, because the general methodxue
Qin  _et al._ (2013); Qin  _et al._ (2014) is to contract
$\epsilon_{\alpha\mu\nu\beta}t_{\alpha}q_{\beta},\epsilon_{\alpha\mu\nu\beta}\gamma_{\alpha}q_{\beta}$
to the vector tWTI242424So far, xue Qin  _et al._ (2013); Qin  _et al._ (2014)
discussed only the Abelian case. And here we use the Abelian tWTI for an
explanation. in the momentum space252525Eq.(4) inxue Qin  _et al._ (2013), in
the Euclidean metric; $q\equiv k-p,t\equiv k+p$. :

$\displaystyle q_{\mu}$
$\displaystyle\Gamma_{\nu}(k,p)-q_{\nu}\Gamma_{\mu}(k,p)=S^{-1}(p)\sigma_{\mu\nu}+\sigma_{\mu\nu}S^{-1}(k)$
(45)
$\displaystyle+2im\Gamma_{\mu\nu}(k,p)+t_{\lambda}\epsilon_{\lambda\mu\nu\rho}\Gamma^{A}_{\rho}(k,p)+A^{V}_{\mu\nu}(k,p),$

such that the identically vanishing left-hand side and the contracted right-
hand side serve as constraints for axial vertex $\Gamma^{A}_{\rho}(k,p)$ to be
solved. Therefore, additional terms262626$\widetilde{A}_{\mu}(k,p)$ is
$\left<TA_{\mu}(x)\psi(y)\bar{\psi}(z)\right>$ in momentum space.
$q^{2}q_{[\mu}\widetilde{A}_{\nu]}(k,p)$ and
$q_{[\mu}\widetilde{A}_{\nu]}(k,p)$ of the Abelian transverse anomalies all
vanish after contraction with $\epsilon_{\alpha\mu\nu\beta}q_{\alpha}$ because
$\epsilon_{\alpha\mu\nu\beta}q_{\alpha}q_{\mu}=0$. The Abelian transverse
anomalies are thus neglectable in current schemesxue Qin  _et al._ (2013); Qin
_et al._ (2014) making use of tWTI. However, even if the ordinary derivative
parts272727Of course, the gauge field parts (where no ordinary derivative
appears) of $F_{a}^{\mu\nu}$ and
$\mathcal{D}^{\rho}\mathcal{D}_{\rho}F_{a}^{\mu\nu}$ are not zero in general,
but these are not gauge covariant and thus may be zero by proper choice of
gauge. However, $C_{abc}F_{b}^{\mu\rho}F_{c\rho}^{\ \ \nu}$ is gauge covariant
and its contribution cannot be neglected. of $F_{a}^{\mu\nu}$ and
$\mathcal{D}^{\rho}\mathcal{D}_{\rho}F_{a}^{\mu\nu}$ vanish owing to the same
reason as the case of the Abelian tWTI, the non-Abelian transverse anomalies
have a non-vanishing contribution from $C_{abc}F_{b}^{\mu\rho}F_{c\rho}^{\ \
\nu}$ in this scheme because $C_{abc}F_{b}^{\mu\rho}F_{c\rho}^{\ \ \nu}$ is
not of the form $q_{[\mu}f_{\nu]}(k,p)$ where $f_{\nu}(k,p)$ is some
operator’s Fourier transformed Green’s function. Unfortunately, the Abelian
approximation (i.e., $\Gamma^{\mu}_{a}(\mbox{non-Abelian})\approx
t_{a}\Gamma^{\mu}(\mbox{Abelian})$) remains the backboneQin  _et al._ (2014,
2014); Binosi  _et al._ (2017); Liu  _et al._ (2019). However, once we begin
to attack the non-Abelian quark-gluon vertex directly using the non-abelian
tWTI (48), the transverse anomaly will take some responsibility. Further more,
other possible applications to the transverse anomaly are being researched.

###### Acknowledgements.

The work of Q. Wang was supported in part by the National Key Research and
Development Program of China (Grant No.2017YFA0402200) and the National
Natural Science Foundation of China (Grant No. 11475092).

## References

  * Ward (1950) J. C. Ward, Phys. Rev. 78, 182 (1950).
  * Takahashi (1957) Y. Takahashi, Nuovo Cimento 6, 371 (1957).
  * Adler and Bardeen (1969) S. L. Adler and W. A. Bardeen, Phys. Rev. 182, 1517 (1969).
  * Bell and Jackiw (1969) J. S. Bell and R. Jackiw, Nuovo Cimento A 60, 47 (1969).
  * Adler (1969) S. L. Adler, Phys. Rev. 177, 2426 (1969).
  * Fujikawa (1980) K. Fujikawa, Phys. Rev. Lett. 44, 1733 (1980).
  * Fujikawa  _et al._ (2004) K. Fujikawa, H. Suzuki, _et al._ ,  _Path integrals and quantum anomalies_ , 122 (Oxford University Press on Demand, 2004).
  * Kondo (1997) K.-I. Kondo, Int. J. Mod. Phys. A 12, 5651 (1997).
  * He  _et al._ (2000) H. He, F. Khanna, and Y. Takahashi, Phys. Lett. B 480, 222 (2000).
  * He (2009) H.-x. He, Phys. Rev. D 80, 016004 (2009).
  * xue Qin  _et al._ (2013) S. xue Qin, L. Chang, Y. xin Liu, C. D. Roberts, and S. M. Schmidt, Phys. Lett. B 722, 384 (2013).
  * Bonneau (1980a) G. Bonneau, Nucl. Phys. B 171, 477 (1980a).
  * Collins (1986) J. C. Collins,  _Renormalization: An Introduction to Renormalization, The Renormalization Group, and the Operator Product Expansion_, Cambridge Monographs on Mathematical Physics, Vol. 26 (Cambridge University Press, Cambridge, 1986).
  * Fujikawa (1979) K. Fujikawa, Phys. Rev. Lett. 42, 1195 (1979).
  * Tsutsui (1989) I. Tsutsui, Phys. Rev. D 40, 3543 (1989).
  * Joglekar and Misra (1987) S. D. Joglekar and A. Misra, Phys. Rev. D 36, 642 (1987).
  * Alves and Farina (1993) M. Alves and C. Farina, Class. Quantum Grav. 10, 1947 (1993).
  * Adler  _et al._ (1977) S. L. Adler, J. C. Collins, and A. Duncan, Phys. Rev. D 15, 1712 (1977).
  * Fujikawa (1981) K. Fujikawa, Phys. Rev. D 23, 2262 (1981).
  * Peskin (2018) M. E. Peskin,  _An introduction to quantum field theory_ (CRC Press, 2018).
  * ’t Hooft and Veltman (1972) G. ’t Hooft and M. Veltman, Nucl. Phys. B 44, 189 (1972).
  * Breitenlohner and Maison (1977) P. Breitenlohner and D. Maison, Commun. Math. Phys. 52, 11 (1977).
  * Bonneau (1980b) G. Bonneau, Nucl. Phys. B 167, 261 (1980b).
  * Schwinger (1959) J. Schwinger, Phys. Rev. Lett. 3, 296 (1959).
  * Chanowitz (1970) M. S. Chanowitz, Phys. Rev. D 2, 3016 (1970).
  * Boulware and Jackiw (1969) D. G. Boulware and R. Jackiw, Phys. Rev. 186, 1442 (1969).
  * xin He (2001) H. xin He, Phys. Lett. B 507, 351 (2001).
  * Sun  _et al._ (2003) W.-M. Sun, H.-S. Zong, X.-S. Chen, and F. Wang, Phys. Lett. B 569, 211 (2003).
  * He (2007) H.-X. He, Int. J. Mod. Phys. A 22, 2119 (2007).
  * Pauli and Villars (1949) W. Pauli and F. Villars, Rev. Mod. Phys. 21, 434 (1949).
  * Bertlmann (2000) R. A. Bertlmann,  _Anomalies in quantum field theory_ , Vol. 91 (Oxford University Press, 2000).
  * Qin  _et al._ (2014) S.-X. Qin, C. D. Roberts, and S. M. Schmidt, Phys. Lett. B 733, 202 (2014).
  * Albino  _et al._ (2019) L. Albino, A. Bashir, L. X. G. Guerrero, B. E. Bennich, and E. Rojas, Phys. Rev. D 100, 054028 (2019).
  * Binosi  _et al._ (2017) D. Binosi, L. Chang, J. Papavassiliou, S.-X. Qin, and C. D. Roberts, Phys. Rev. D 95, 031501 (2017).
  * Liu  _et al._ (2019) L. Liu, L. Chang, and Y.-x. Liu, Phys. Rev. D 99, 074013 (2019).
  * Reuter (1985) M. Reuter, Phys. Rev. D 31, 1374 (1985).
  * Schwartz (2014) M. D. Schwartz,  _Quantum field theory and the standard model_ (Cambridge University Press, 2014).

##  Appendix A Non-Abelian Generalization and one-loop Calculation through
$\zeta$ function regularization

The generalization of transverse anomalies to the non-Abelian case (with gauge
group $SU(N)$) is straightforward. Within Fujikawa’s framework, using the
LagrangianPeskin (2018) :

$\displaystyle\mathscr{L}_{SU(N)}=$
$\displaystyle\bar{\psi}\left(i\not{D}-m_{0}\right)\psi,\
D_{\mu}\equiv\frac{1}{2}\left(\overrightarrow{\partial}_{\mu}-\overleftarrow{\partial}_{\mu}\right)-igt_{a}A_{a\mu},$
(46)

applying variations of fermion fields :

$\delta\psi(x)=\frac{1}{4}\epsilon_{\mu\nu}\alpha_{a}(x)t_{a}\sigma^{\mu\nu}\psi(x),\
\delta\bar{\psi}(x)=\frac{1}{4}\epsilon_{\mu\nu}\alpha_{a}(x)\bar{\psi}(x)t_{a}\sigma^{\mu\nu},$
(47)

and considering the transformation Jacobian (of which we present a concrete
calculation later), we get :

$\displaystyle\mathcal{D}^{\mu}\left<j^{\nu}_{a}(x)\right>_{A}-\mathcal{D}^{\nu}\left<j^{\mu}_{a}(x)\right>_{A}$
(48) $\displaystyle=$
$\displaystyle\left<\epsilon^{\mu\nu\rho\sigma}\bar{\psi}(x)\gamma_{\sigma}\gamma_{5}\left\\{iD_{\rho},t_{a}\right\\}\psi(x)\right>_{A}+2m_{0}\left<\bar{\psi}(x)\sigma^{\mu\nu}t_{a}\psi(x)\right>_{A}$
$\displaystyle-\frac{g_{0}m_{0}^{2}}{4\pi^{2}}F^{\mu\nu}_{a}(x)-\frac{g_{0}}{24\pi^{2}}\mathcal{D}^{\rho}\mathcal{D}_{\rho}F_{a}^{\mu\nu}(x)-\frac{g^{2}_{0}}{8\pi^{2}}C_{abc}F_{b}^{\mu\rho}(x)F_{c\rho}^{\
\ \nu}(x).$

Renormalization of the above non-Abelian tWTI is left as further work.

We next calculate the one-loop transverse anomalies,
$-2i\mathrm{tr}[t_{a}\sigma^{\mu\nu}]\delta^{4}(x-x)$, through $\zeta$
function regularization. Identification of the transformation Jacobian of (47)
and (14) to be $-2i\mathrm{tr}[t_{a}\sigma^{\mu\nu}]\delta^{4}(x-x)$ and
$-2i\mathrm{tr}[\sigma^{\mu\nu}]\delta^{4}(x-x)$ is straightforward after
using the following equation (recall that $\ln(1+x)\cong x$ when $x\ll 1$):

$\displaystyle\mathrm{Det}\left[\frac{\delta}{\delta\psi(y)}\left(\psi(x)+\frac{1}{4}\epsilon_{\mu\nu}\alpha_{a}(x)t_{a}\sigma^{\mu\nu}\psi(x)\right)\right]$
(49) $\displaystyle=$
$\displaystyle\mathrm{Det}\left[\delta^{4}(x-y)+\frac{1}{4}\epsilon_{\mu\nu}\alpha_{a}(x)t_{a}\sigma^{\mu\nu}\delta^{4}(x-y)\right]$
$\displaystyle=$
$\displaystyle\mathrm{exp}\left\\{\mathrm{tr}\ln\left[\delta^{4}(x-y)+\frac{1}{4}\epsilon_{\mu\nu}\alpha_{a}(x)t_{a}\sigma^{\mu\nu}\delta^{4}(x-y)\right]\right\\}$
$\displaystyle\cong$
$\displaystyle\mathrm{exp}\left\\{\frac{1}{4}\epsilon_{\mu\nu}\alpha_{a}(x)\mathrm{tr}\left[t_{a}\sigma^{\mu\nu}\right]\delta^{4}(x-x)\right\\}.$

$\sigma^{\mu\nu}\equiv\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$, and it is thus
enough to calculate
$\mathrm{tr}\left\\{t_{a}[\gamma^{\mu},\gamma^{\nu}]\right\\}\delta^{4}(x-x)$.
The combination of Fujikawa’s approachFujikawa (1979) and $\zeta$ function
regularizationBertlmann (2000); Reuter (1985) leads to :

$\displaystyle\left\\{\mathrm{tr}\left\\{t_{a}[\gamma^{\mu},\gamma^{\nu}]\right\\}\delta^{4}(x-x)\right\\}_{\zeta}$
(50) $\displaystyle=$
$\displaystyle\frac{d}{ds}\Bigg{|}_{s=0}\frac{s}{\Gamma(s)}\int_{0}^{\infty}d\tau\
\tau^{s-1}\left\\{\mathrm{tr}\left[e^{-\left(\not{D}_{x}^{2}+m^{2}\right)\tau}t_{a}[\gamma^{\mu},\gamma^{\nu}]\right]\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ik\cdot(x-y)}\right\\}_{y\rightarrow
x}$ $\displaystyle=$
$\displaystyle\frac{d}{ds}\Bigg{|}_{s=0}\frac{s}{\Gamma(s)}\int_{0}^{\infty}d\tau\
\tau^{s-1}e^{-m^{2}\tau}\frac{1}{(\sqrt{\tau})^{4}}\int\frac{d^{4}k}{(2\pi)^{4}}$
$\displaystyle\times\mathrm{tr}\left[t_{a}[\gamma^{\mu},\gamma^{\nu}]\mathrm{exp}\left(k^{2}+2ik\cdot
D\sqrt{\tau}-D^{2}\tau+\frac{i}{4}gt_{b}[\gamma^{\rho},\gamma^{\sigma}]F_{b\rho\sigma}\tau\right)\right].$

In the last step, we used
$\not{D}^{2}=D^{2}-\frac{1}{4}igt_{a}[\gamma^{\mu},\gamma^{\nu}]F_{a\mu\nu}$
and rescaled $k\rightarrow k/\sqrt{\tau}$.

We note that282828In (50), we strip
$\frac{d}{ds}|_{s=0}\frac{s}{\Gamma(s)}\int_{0}^{\infty}d\tau\
\tau^{s-1}e^{-m^{2}\tau}$ away and substitute $\sqrt{\tau}$ with $1/\Lambda$,
thus arriving at the original Fujikawa’s method. From (51), it is obvious that
$\Lambda^{2}$ in the final results, like (17), is effectively regularized to
be $-m^{2}$ through $\zeta$ function regularization.

$\displaystyle\frac{d}{ds}\Bigg{|}_{s=0}\frac{s}{\Gamma(s)}\int_{0}^{\infty}d\tau\
\tau^{s-1}e^{-m^{2}\tau}\frac{1}{(\sqrt{\tau})^{4}}(\sqrt{\tau})^{n}$ (51)
$\displaystyle=$
$\displaystyle\frac{d}{ds}\Bigg{|}_{s=0}\frac{s\Gamma(s+n/2-2)}{\Gamma(s)}(m^{2})^{2-s-\frac{n}{2}}$
$\displaystyle=$ $\displaystyle\left\\{\begin{matrix}\frac{m^{4}}{2},&n=0;\\\
-m^{2},&n=2;\\\ 1,&n=4;\\\ 0,&\mbox{otherwise.}\end{matrix}\right.$

Therefore, the only contributing terms in (50) are those proportional to
$(\sqrt{\tau})^{0},(\sqrt{\tau})^{2},(\sqrt{\tau})^{4}$ in expansion of the
exponential inside the trace.

The $(\sqrt{\tau})^{0}$ term is zero because
$\mathrm{tr}\left\\{[\gamma^{\mu},\gamma^{\nu}]\right\\}=0$. The
$(\sqrt{\tau})^{2}$ term is (after finishing $\frac{d}{ds}|_{s=0}$)

$\displaystyle(-m^{2})\mathrm{tr}[t_{a}t_{b}]\mathrm{tr}\left\\{[\gamma^{\mu},\gamma^{\nu}][\gamma^{\rho},\gamma^{\sigma}]\right\\}\int\frac{d^{4}k}{(2\pi)^{4}}e^{k^{2}}\frac{i}{4}gF_{b\rho\sigma}.$
(52)

The $(\sqrt{\tau})^{4}$ term is

$\displaystyle\mathrm{tr}[t_{a}t_{b}t_{c}]\mathrm{tr}\left\\{[\gamma^{\mu},\gamma^{\nu}][\gamma^{\rho},\gamma^{\sigma}][\gamma^{\alpha},\gamma^{\beta}]\right\\}\int\frac{d^{4}k}{(2\pi)^{4}}e^{k^{2}}\frac{1}{2!}\left(\frac{i}{4}g\right)^{2}F_{b\rho\sigma}F_{c\alpha\beta}$
(53) $\displaystyle+$
$\displaystyle\mathrm{tr}\left\\{[\gamma^{\mu},\gamma^{\nu}][\gamma^{\rho},\gamma^{\sigma}]\right\\}\int\frac{d^{4}k}{(2\pi)^{4}}e^{k^{2}}\frac{1}{2!}\frac{i}{4}g(-1)\mathrm{tr}\left[t_{a}\left(D^{2}t_{b}F_{b\rho\sigma}+t_{b}F_{b\rho\sigma}D^{2}\right)\right]$
$\displaystyle+$
$\displaystyle\mathrm{tr}\left\\{[\gamma^{\mu},\gamma^{\nu}][\gamma^{\rho},\gamma^{\sigma}]\right\\}\int\frac{d^{4}k}{(2\pi)^{4}}e^{k^{2}}\frac{1}{3!}\frac{i}{4}g(2i)^{2}$
$\displaystyle\times\mathrm{tr}\left[t_{a}\left((k\cdot
D)^{2}t_{b}F_{b\rho\sigma}+(k\cdot D)t_{b}F_{b\rho\sigma}(k\cdot
D)+t_{b}F_{b\rho\sigma}(k\cdot D)^{2}\right)\right].$

We arrive at the final result by completing the integral and working out the
trace :

$\displaystyle\left\\{\mathrm{tr}\\{t_{a}[\gamma^{\mu},\gamma^{\nu}]\\}\delta^{4}(x-x)\right\\}_{\zeta}$
(54) $\displaystyle=$
$\displaystyle-\frac{gm^{2}}{4\pi^{2}}F_{a}^{\mu\nu}-\frac{g}{24\pi^{2}}\mathcal{D}_{\rho}\mathcal{D}^{\rho}F_{a}^{\mu\nu}-\frac{g^{2}}{8\pi^{2}}C_{abc}F_{b}^{\mu\rho}F_{c\rho}^{\
\ \nu}.$

The only difference between the Abelian case and non-Abelian case is the use
of $\mathrm{tr}[t_{a}t_{b}]=\frac{1}{2}\delta_{ab}$, which is not needed in
the Abelian case. The Abelian result is therefore 292929Additionally, note
that $\mathrm{tr}[t_{b}]=0$ for $SU(N)$, such that there is no contribution in
$\mathrm{tr}\ [\gamma^{\mu},\gamma^{\nu}]\delta^{4}(x-x)$ from non-Abelian
fields through observation on (52)(53) with $t_{a}$ stripped away, using
$\mathrm{tr}[t_{b}t_{c}]\mathrm{tr}\left\\{[\gamma^{\mu},\gamma^{\nu}][\gamma^{\rho},\gamma^{\sigma}][\gamma^{\alpha},\gamma^{\beta}]\right\\}F_{b\rho\sigma}F_{c\alpha\beta}=0$.

$\displaystyle\left\\{\mathrm{tr}\
[\gamma^{\mu},\gamma^{\nu}]\delta^{4}(x-x)\right\\}_{\zeta}$ (55)
$\displaystyle=$
$\displaystyle-\frac{gm^{2}}{2\pi^{2}}F^{\mu\nu}-\frac{g}{12\pi^{2}}\partial_{\rho}\partial^{\rho}F^{\mu\nu},$

where $F^{\mu\nu}\equiv\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ is the
$U(1)$ gauge field.

##  Appendix B One-loop Calculations : Pauli–Villars Regularization

This appendix calculates missing transverse anomalies in Sun  _et al._ (2003)
adopting Pauli–Villars regularization, which is also the method used by Sun
_et al._ (2003). As in Sun  _et al._ (2003), we work with an external field
$A_{\mu}$, and the Lagrangian is then again (12).

To verify (19), we go to momentum space and define

$\displaystyle\Gamma^{\mu}(q)\equiv$ $\displaystyle\int d^{4}x\ e^{-iq\cdot
x}\left<j^{\mu}(x)\right>_{A},$ (56) $\displaystyle N^{\mu\nu}(q)\equiv$
$\displaystyle\int d^{4}x\ e^{-iq\cdot
x}2\left<\bar{\psi}(x)\epsilon^{\mu\nu\rho\sigma}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi(x)\right>_{A},$
$\displaystyle T^{\mu\nu}(q)\equiv$ $\displaystyle\int d^{4}x\ e^{-iq\cdot
x}\left<\bar{\psi}(x)\sigma^{\mu\nu}\psi(x)\right>.$

The following “bare” tWTI has been verified in Sun  _et al._ (2003) (for
simplicity, we denote $m_{0}$ by $m$):

$\displaystyle
iq^{\mu}\left(\Gamma^{\nu}_{m}+\Gamma^{\nu}_{M_{1}}-2\Gamma^{\nu}_{M_{2}}\right)-iq^{\nu}\left(\Gamma^{\mu}_{m}+\Gamma^{\mu}_{M_{1}}-2\Gamma^{\mu}_{M_{2}}\right)$
(57) $\displaystyle=$ $\displaystyle
N^{\mu\nu}_{m}+N^{\mu\nu}_{M_{1}}-2N^{\mu\nu}_{M_{2}}+2\left(mT^{\mu\nu}_{m}+M_{1}T^{\mu\nu}_{M_{1}}-2M_{2}T^{\mu\nu}_{M_{2}}\right).$

Here, $\Gamma^{\mu}_{m}$ indicates an amplitude is calculated with fermion
mass $m$; There are two subtractions because the leading divergence is
quadratic. $M_{1}=\sqrt{m^{2}+2\Lambda^{2}},M_{2}=\sqrt{m^{2}+\Lambda^{2}},$
where $\Lambda$ serves as an effective cut-off, as in Sun  _et al._ (2003).

However, in the spirit of Pauli–Villars regularizationPauli and Villars
(1949); Bertlmann (2000), the WTI should be expressed by a regularized
“physical” amplitude for which any amplitude $f$ is defined as:

$f_{\mathrm{phys}}\equiv\lim_{\Lambda\rightarrow\infty}f_{m}+rf_{M_{1}}-2sf_{M_{2}}.$
(58)

Here, $r$ and $s$ should be chosen to cancel out all divergences in $f$. For
$T^{\mu\nu}$, it is easily seen that $r=m/M_{1},s=m/M_{2}$ through direct
analysis of the diagram on the lowest order of $A_{\mu}(x)$.)

The “bare” identity then gets an extra term after assembling each amplitude
into its regularized form :

$\displaystyle
iq^{\mu}\Gamma^{\nu}_{\mathrm{phys}}-iq^{\nu}\Gamma^{\mu}_{\mathrm{phys}}$
(59) $\displaystyle=$ $\displaystyle
iq^{\mu}\left(\Gamma^{\nu}_{m}+\Gamma^{\nu}_{M_{1}}-2\Gamma^{\nu}_{M_{2}}\right)-iq^{\nu}\left(\Gamma^{\mu}_{m}+\Gamma^{\mu}_{M_{1}}-2\Gamma^{\mu}_{M_{2}}\right)$
$\displaystyle=$ $\displaystyle
N^{\mu\nu}_{m}+N^{\mu\nu}_{M_{1}}-2N^{\mu\nu}_{M_{2}}+2m\left(T^{\mu\nu}_{m}+\frac{m}{M_{1}}T^{\mu\nu}_{M_{1}}-\frac{2m}{M_{2}}T^{\mu\nu}_{M_{2}}\right)$
$\displaystyle+\left[2\left(M_{1}-\frac{m^{2}}{M_{1}}\right)T^{\mu\nu}_{M_{1}}-4\left(M_{2}-\frac{m^{2}}{M_{2}}\right)T^{\mu\nu}_{M_{2}}\right]$
$\displaystyle\equiv$ $\displaystyle
N^{\mu\nu}_{\mathrm{phys}}-2mT^{\mu\nu}_{\mathrm{phys}}+\mathscr{A}^{\mu\nu}.$

We will show that

$\mathscr{A}^{\mu\nu}\equiv
2\left(M_{1}-\frac{m^{2}}{M_{1}}\right)T^{\mu\nu}_{M_{1}}-4\left(M_{2}-\frac{m^{2}}{M_{2}}\right)T^{\mu\nu}_{M_{2}}$
(60)

is exactly the anomaly we obtained in App.A up to quadratic divergences.

$T^{\mu\nu}$ is represented by the Feynman graphs303030Recall that $C$ parity
of $\bar{\psi}\sigma^{\mu\nu}\psi$ is odd, such that there are only $A^{2n+1}$
terms. :

(61)

Gauge invariance in external photon legs and dimensional analysis tell us the
diagram in (61) with $n$ photon legs diverges at worst like
$Q^{n}A^{n}\Lambda^{3-2n}$ when $\Lambda\rightarrow\infty$, where $Q$ is some
typical scale of external momenta and $A$ an abbreviation of $A^{\mu}(x)$.
Therefore, the only term contributing to transverse anomalies is the smallest
diagram :

$\displaystyle\mathscr{A}^{\mu\nu}=2\left(M_{1}-\frac{m^{2}}{M_{1}}\right)T^{\mu\nu}_{M_{1},\mbox{the
smallest
diagram}}-4\left(M_{2}-\frac{m^{2}}{M_{2}}\right)T^{\mu\nu}_{M_{2},\mbox{the
smallest diagram}}$ (62) $\displaystyle=$ $\displaystyle
2\left(M_{1}-\frac{m^{2}}{M_{1}}\right)ig_{0}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{\mathrm{tr}\left[\sigma^{\mu\nu}(\not{p}-\not{q}+M_{1})\widetilde{\not{A}}(q)(\not{p}+M_{1})\right]}{[(p-q)^{2}-m^{2}-2\Lambda^{2}][p^{2}-m^{2}-2\Lambda^{2}]}$
$\displaystyle-4\left(M_{2}-\frac{m^{2}}{M_{2}}\right)ig_{0}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{\mathrm{tr}\left[\sigma^{\mu\nu}(\not{p}-\not{q}+M_{2})\widetilde{\not{A}}(q)(\not{p}+M_{2})\right]}{[(p-q)^{2}-m^{2}-\Lambda^{2}][p^{2}-m^{2}-\Lambda^{2}]}$
$\displaystyle=$
$\displaystyle\left(iq^{\mu}\widetilde{A}^{\nu}-iq^{\nu}\widetilde{A}^{\mu}\right)\int_{0}^{1}dx\int_{0}^{x}dy\int_{0}^{y}dz$
$\displaystyle\times
g_{0}\frac{\Lambda^{4}}{\pi^{2}}\left(\frac{4}{[m^{2}+(1+x-y+z)\Lambda^{2}-y(1-y)q^{2}]}+\frac{2m^{2}+3\Lambda^{2}-[y^{2}+(1-y)^{2}]q^{2}}{[m^{2}+(1+x-y+z)\Lambda^{2}-y(1-y)q^{2}]^{2}}\right).$

Taking the limit $\Lambda\rightarrow\infty$, we get

$\mathscr{A}^{\mu\nu}=\left(\frac{g_{0}\ln
2}{\pi^{2}}\Lambda^{2}-\frac{g_{0}}{2\pi^{2}}m^{2}+\frac{g_{0}}{12\pi^{2}}q^{2}\right)\left(iq^{\mu}\widetilde{A}^{\nu}-iq^{\nu}\widetilde{A}^{\mu}\right),\
\widetilde{A}^{\mu}(q)\equiv\int d^{4}x\ e^{-iq\cdot x}A^{\mu}(x).$ (63)

This is what we get in App.A in momentum space, up to a quadratic divergent
term. However, it is well-knownPauli and Villars (1949); Schwartz (2014) that
quadratic divergence in $\left<j^{\mu}\right>_{A}$ in the Pauli–Villars scheme
corresponds to an infinite photon mass, which must be subtracted to ensure
gauge invariance. As long as we subtract quadratic divergence on both sides
of313131(64) holds at any $\Lambda$ so that quadratic divergences on the two
sides are equal.

$iq^{\mu}\Gamma^{\nu}_{\mathrm{phys}}-iq^{\nu}\Gamma^{\mu}_{\mathrm{phys}}=N^{\mu\nu}_{\mathrm{phys}}-2mT^{\mu\nu}_{\mathrm{phys}}+\mathscr{A}^{\mu\nu},$
(64)

nothing is affected by quadratic divergence. Nevertheless, it is somehow
confusing that there is no logarithmic divergence associated with this
quadratic divergence. It is not easy (as far as we are concerned) to give a
thorough explanation, but the situation may be summarized phenomenologically
as an absence of logarithmic divergence is simply a signal of an anomaly
because an anomaly is a local operator and leads to only polynomials of
external momenta on one-loop order. (Note that the coefficient of an anomaly
term is one-loop, and the matrix elements of the anomaly term are thus tree
level.)

##  Appendix C One-loop Calculations : Point-splitting Method

Similar to the case of chiral anomaly, the point-splitting methodPeskin (2018)
gives results for transverse anomalies. However, the dependence on the
splitting ratio prevents this method from working for transverse anomalies.

The point-splitting method selects a special regularization for $j^{\mu}$
(where $a$ is a real number):

$j^{\mu}(x)\rightarrow\bar{\psi}(x+(a+1)\epsilon)\gamma^{\mu}e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}\psi(x+a\epsilon).$ (65)

UsuallyPeskin (2018) $a=-1/2$ such that $x$ is the midpoint of the two split
points. However, there is no _principle_ that demands $a$ to be $-1/2$, and if
this method is to make sense, the final results must be independent of $a$, as
is the case for the chiral anomaly Peskin (2018). (Looking into concrete
process of calculating the chiral anomalyPeskin (2018), it is easy to see that
the chiral anomaly only needs expansion to $\mathcal{O}(\epsilon)$ so that the
$a$ dependence is of the form $(a+1)-a=1$.) We will see soon that the point-
splitting method cannot be applied to calculating transverse anomalies owing
to its non-trivial dependence on $a$.

We first use
$\left[\gamma^{\rho},\frac{1}{2}\sigma^{\mu\nu}\right]=ig^{\mu\rho}\gamma^{\nu}-ig^{\nu\rho}\gamma^{\mu}$
to rewrite $\partial^{[\mu}j^{\nu]}$ as :

$\displaystyle\partial^{\mu}j^{\nu}(x)-\partial^{\nu}j^{\mu}(x)$ (66)
$\displaystyle=$
$\displaystyle-i\partial_{\rho}\left(\bar{\psi}(x+(a+1)\epsilon)\left[\gamma^{\rho},\frac{1}{2}\sigma^{\mu\nu}\right]e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}\psi(x+a\epsilon)\right).$

Rearranging terms gives:

$\displaystyle\partial^{\mu}j^{\nu}(x)-\partial^{\nu}j^{\mu}(x)$ (67)
$\displaystyle=$ $\displaystyle
i\left(\bar{\psi}(x+(a+1)\epsilon)\left(\overrightarrow{\partial}_{\rho}-\overleftarrow{\partial}_{\rho}\right)\epsilon^{\mu\nu\rho\sigma}\gamma_{\sigma}\gamma_{5}\psi(x+a\epsilon)\right)e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}$
$\displaystyle-i\left(\bar{\psi}(x+(a+1)\epsilon)\overleftarrow{\partial}_{\rho}\gamma^{\rho}\sigma^{\mu\nu}\psi(x+a\epsilon)\right)e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}$
$\displaystyle+i\left(\bar{\psi}(x+(a+1)\epsilon)\sigma^{\mu\nu}\gamma^{\rho}\overrightarrow{\partial}_{\rho}\psi(x+a\epsilon)\right)e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}$
$\displaystyle+g\bar{\psi}(x+(a+1)\epsilon)\left[\gamma^{\rho},\frac{1}{2}\sigma^{\mu\nu}\right]e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}\psi(x+a\epsilon)$
$\displaystyle\times\left(\epsilon^{\sigma}\partial_{\rho}A_{\sigma}(x)+\frac{(a+1)^{2}-a^{2}}{2}\epsilon^{\sigma}\epsilon^{\lambda}\partial_{\lambda}\partial_{\rho}A_{\sigma}(x)+\frac{(a+1)^{3}-a^{3}}{6}\epsilon^{\sigma}\epsilon^{\lambda}\epsilon^{\kappa}\partial_{\lambda}\partial_{\kappa}\partial_{\rho}A_{\sigma}(x)+\mathcal{O}(\epsilon^{4})\right)$

We then use equations of motion for the massless (for simplicity) fermion
$\not{\overrightarrow{D}}\psi(x)=0$ and
$\bar{\psi}(x)\not{\overleftarrow{D}}=0$ to get

$\displaystyle\partial^{\mu}j^{\nu}(x)-\partial^{\nu}j^{\mu}(x)$ (68)
$\displaystyle=$ $\displaystyle
i\left(\bar{\psi}(x+(a+1)\epsilon)\left(\overrightarrow{\partial}_{\rho}-\overleftarrow{\partial}_{\rho}\right)\epsilon^{\mu\nu\rho\sigma}\gamma_{\sigma}\gamma_{5}\psi(x+a\epsilon)\right)e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}$
$\displaystyle-g\bar{\psi}(x+(a+1)\epsilon)A(x+(a+1)\epsilon)\gamma^{\rho}\sigma^{\mu\nu}\psi(x+a\epsilon)e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}$
$\displaystyle-g\bar{\psi}(x+(a+1)\epsilon)\sigma^{\mu\nu}\gamma^{\rho}A(x+a\epsilon)\psi(x+a\epsilon)e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}$
$\displaystyle+g\bar{\psi}(x+(a+1)\epsilon)\left[\gamma^{\rho},\frac{1}{2}\sigma^{\mu\nu}\right]e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}\psi(x+a\epsilon)$
$\displaystyle\times\left(\epsilon^{\sigma}\partial_{\rho}A_{\sigma}(x)+\frac{(a+1)^{2}-a^{2}}{2}\epsilon^{\sigma}\epsilon^{\lambda}\partial_{\lambda}\partial_{\rho}A_{\sigma}(x)+\frac{(a+1)^{3}-a^{3}}{6}\epsilon^{\sigma}\epsilon^{\lambda}\epsilon^{\kappa}\partial_{\lambda}\partial_{\kappa}\partial_{\rho}A_{\sigma}(x)+\mathcal{O}(\epsilon^{4})\right)$

We next expand $A_{\mu}$ at $x$ to $\mathcal{O}(\epsilon^{4})$ :

$\displaystyle\partial^{\mu}j^{\nu}(x)-\partial^{\nu}j^{\mu}(x)$ (69)
$\displaystyle=$ $\displaystyle
i\left(\bar{\psi}(x+(a+1)\epsilon)\left(\overrightarrow{D}_{\rho}-\overleftarrow{D}_{\rho}\right)\epsilon^{\mu\nu\rho\sigma}\gamma_{\sigma}\gamma_{5}\psi(x+a\epsilon)\right)e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}$
$\displaystyle-g\bar{\psi}(x+(a+1)\epsilon)\gamma^{\rho}\sigma^{\mu\nu}\psi(x+a\epsilon)e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}$
$\displaystyle\times\left((a+1)\epsilon^{\sigma}\partial_{\sigma}A_{\rho}(x)+\frac{(a+1)^{2}}{2}\epsilon^{\sigma}\epsilon^{\lambda}\partial_{\lambda}\partial_{\sigma}A_{\rho}(x)+\frac{(a+1)^{3}}{6}\epsilon^{\sigma}\epsilon^{\lambda}\epsilon^{\kappa}\partial_{\lambda}\partial_{\kappa}\partial_{\sigma}A_{\rho}(x)+\mathcal{O}(\epsilon^{4})\right)$
$\displaystyle-g\bar{\psi}(x+(a+1)\epsilon)\sigma^{\mu\nu}\gamma^{\rho}\psi(x+a\epsilon)e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}$
$\displaystyle\times\left(a\epsilon^{\sigma}\partial_{\sigma}A_{\rho}(x)+\frac{a^{2}}{2}\epsilon^{\sigma}\epsilon^{\lambda}\partial_{\lambda}\partial_{\sigma}A_{\rho}(x)+\frac{a^{3}}{6}\epsilon^{\sigma}\epsilon^{\lambda}\epsilon^{\kappa}\partial_{\lambda}\partial_{\kappa}\partial_{\sigma}A_{\rho}(x)+\mathcal{O}(\epsilon^{4})\right)$
$\displaystyle+g\bar{\psi}(x+(a+1)\epsilon)\left[\gamma^{\rho},\frac{1}{2}\sigma^{\mu\nu}\right]e^{ig\int_{x+a\epsilon}^{x+(a+1)\epsilon}dy\cdot
A(y)}\psi(x+a\epsilon)$
$\displaystyle\times\left(\epsilon^{\sigma}\partial_{\rho}A_{\sigma}(x)+\frac{(a+1)^{2}-a^{2}}{2}\epsilon^{\sigma}\epsilon^{\lambda}\partial_{\lambda}\partial_{\rho}A_{\sigma}(x)+\frac{(a+1)^{3}-a^{3}}{6}\epsilon^{\sigma}\epsilon^{\lambda}\epsilon^{\kappa}\partial_{\lambda}\partial_{\kappa}\partial_{\rho}A_{\sigma}(x)+\mathcal{O}(\epsilon^{4})\right).$

We finally take the $\epsilon\rightarrow 0$ limit. FromPeskin (2018); xin He
(2001), we have 323232Here we only need $\mathcal{O}(A^{0})$ of
$\left<\psi(x+a\epsilon)\bar{\psi}(x+(a+1)\epsilon)\right>$ because $C$ parity
of $j^{\mu}$ and $A^{\mu}$ are both odd, and $\mathcal{O}(A^{2})$ of
$\left<\psi(x+a\epsilon)\bar{\psi}(x+(a+1)\epsilon)\right>$ is of
$\mathcal{O}(\epsilon)$ and thus does not make a contribute. :

$\displaystyle\left<\psi(x+a\epsilon)\bar{\psi}(x+(a+1)\epsilon)\right>=\frac{i}{2\pi^{2}}\frac{\gamma^{\alpha}\epsilon_{\alpha}}{\epsilon^{4}}+\mathcal{O}(A^{1}),\
\lim_{\epsilon\rightarrow
0}\frac{\epsilon^{\mu}\epsilon^{\nu}}{\epsilon^{2}}=\frac{1}{4}g^{\mu\nu},$
(70) $\displaystyle\lim_{\epsilon\rightarrow
0}\frac{\epsilon^{\mu}\epsilon^{\nu}\epsilon^{\rho}\epsilon^{\sigma}}{\epsilon^{4}}=\frac{1}{24}\left(g^{\mu\nu}g^{\rho\sigma}+g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho}\right).$

Therefore, the final result (where the Bianchi identity is used and care is
taken with Fermi-statistics) is

$\displaystyle\partial^{\mu}j^{\nu}(x)-\partial^{\nu}j^{\mu}(x)$ (71)
$\displaystyle=$ $\displaystyle
i\bar{\psi}(x)\left(\overrightarrow{D}_{\rho}-\overleftarrow{D}_{\rho}\right)\epsilon^{\mu\nu\rho\sigma}\gamma_{\sigma}\gamma_{5}\psi(x)$
$\displaystyle+\frac{g}{\pi^{2}}\frac{1}{\epsilon^{2}}F^{\mu\nu}(x)+\frac{g\left[(a+1)^{3}-a^{3}\right]}{72\pi^{2}}\left(2\partial^{\rho}\partial_{\rho}F^{\mu\nu}(x)+2\partial^{\nu}\partial_{\rho}F^{\mu\rho}(x)-2\partial^{\mu}\partial_{\rho}F^{\nu\rho}(x)\right)$
$\displaystyle=$ $\displaystyle
2\bar{\psi}(x)\epsilon^{\mu\nu\rho\sigma}\gamma_{\sigma}\gamma_{5}iD_{\rho}\psi(x)$
$\displaystyle+\frac{g}{\pi^{2}}\frac{1}{\epsilon^{2}}F^{\mu\nu}(x)+\frac{g\left[(a+1)^{3}-a^{3}\right]}{18\pi^{2}}\partial^{\rho}\partial_{\rho}F^{\mu\nu}(x).$

This result is not only affected by quadratic divergence333333Unlike the case
in Appendix B, it seems here that we cannot find a proper way to subtract this
divergence because the quadratic divergence of $j^{\mu}$ is not shown
explicitly., but also dependent on $a$ nontrivially. The point-splitting
method is thus not suitable for transverse anomalies.

##  Appendix D Details of Dimensional Renormalization in Section IV

This appendix presents a note for one-loop calculation in dimensional
renormalization and an analysis for gauge invariance of the coefficient in
(24) including that of transverse anomalies.

To determine coefficients in tWTI (25) on one-loop order, it is not necessary
to use all the algebra in (24), and it is more convenient and simple to
calculate
$\left<TN\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\hat{\gamma}^{\rho}iD_{\rho}\psi\right]\tilde{\psi}\left(\bar{p}_{1}\right)\tilde{\bar{\psi}}\left(\bar{p}_{2}\right)\right>^{\mathrm{prop}}$
and
$\left<TN\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\hat{\gamma}^{\alpha}iD_{\alpha}\psi\right]\tilde{A}^{\rho}(\bar{q})\right>^{\mathrm{prop}}$
(instead of the Green’s functions with
$N\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\hat{\gamma}^{\rho}iD_{\rho}\psi\right]$
replaced by
$N\left[\check{g}_{\rho\sigma}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\rho}iD^{\sigma}\psi\right)\right]$)
to read out
$(b,c,f,r,s)=(b^{\prime},c^{\prime},f^{\prime},r^{\prime},s^{\prime})/(1-a)$
directly from (23). (Of course, information on $a$ is lost, but this does not
matter because $a$ is only an intermediate variable.) Only the following four
Feynman diagrams in Fig.2 are relevant.

Figure 2: One-loop diagrams of
$\left<TN\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\rho}iD_{\rho}\psi\right]\tilde{\psi}\left(\bar{p}_{1}\right)\tilde{\bar{\psi}}\left(\bar{p}_{2}\right)\right>^{\mathrm{prop}}$
and
$\left<TN\left[\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\alpha}iD_{\alpha}\psi\right]\tilde{A}^{\rho}(\bar{q})\right>^{\mathrm{prop}}$

For the first diagram,

$\displaystyle\mathscr{E}^{\mu\nu}_{(1)}(\bar{p}_{1},\bar{p}_{2})$ (72)
$\displaystyle=$
$\displaystyle-2ig^{2}\int\frac{d^{d}p}{(2\pi)^{d}}\bar{\sigma}^{\mu\nu}\left(\hat{\gamma}^{\alpha}\frac{1}{\not{p}+\not{\bar{p}}_{2}-m}\hat{\gamma}_{\alpha}-(1-\xi)\hat{\not{p}}\frac{1}{\not{p}+\not{\bar{p}}_{2}-m}\not{p}\frac{1}{p^{2}}\right)\frac{1}{p^{2}}$
$\displaystyle=$
$\displaystyle-2ig^{2}\bar{\sigma}^{\mu\nu}\int\frac{d^{d}p}{(2\pi)^{d}}\int_{0}^{1}dx\left(\frac{(4-d)\left((1-x)\not{\bar{p}}_{2}-m_{0}\right)}{\left[-p^{2}-x(1-x)\bar{p}^{2}_{2}+xm_{0}^{2}\right]^{2}}\right.$
$\displaystyle\left.+(1-\xi)\frac{2(1-x)\hat{p}^{2}\left(m-\not{\bar{p}}_{2}\right)}{\left[-p^{2}-x(1-x)\bar{p}^{2}_{2}+xm_{0}^{2}\right]^{3}}\right)$
$\displaystyle=$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\bar{\sigma}^{\mu\nu}\left(\not{\bar{p}}_{2}-4m+\left(1-\xi\right)\left(m-\not{\bar{p}}_{2}\right)\right).$

Similar to the first diagram, we have for the second diagram that

$\displaystyle\mathscr{E}^{\mu\nu}_{(2)}(\bar{p}_{1},\bar{p}_{2})$ (73)
$\displaystyle=$
$\displaystyle-2ig^{2}\int\frac{d^{d}p}{(2\pi)^{d}}\left(\hat{\gamma}^{\alpha}\frac{1}{\not{p}+\not{\bar{p}}_{1}-m_{0}}\hat{\gamma}_{\alpha}-(1-\xi)\hat{\not{p}}\frac{1}{\not{p}+\not{\bar{p}}_{1}-m}\not{p}\frac{1}{p^{2}}\right)\bar{\sigma}^{\mu\nu}\frac{1}{p^{2}}$
$\displaystyle=$
$\displaystyle-2ig^{2}\int\frac{d^{d}p}{(2\pi)^{d}}\int_{0}^{1}dx\left(\frac{(4-d)\left((1-x)\not{\bar{p}}_{1}-m\right)}{\left[-p^{2}-x(1-x)\bar{p}^{2}_{1}+xm^{2}\right]^{2}}\right.$
$\displaystyle\left.+(1-\xi)\frac{2(1-x)\hat{p}^{2}\left(m-\not{\bar{p}}_{1}\right)}{\left[-p^{2}-x(1-x)\bar{p}^{2}_{1}+xm^{2}\right]^{3}}\right)\bar{\sigma}^{\mu\nu}$
$\displaystyle=$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\left(2\not{\bar{p}}_{1}-4m+\left(1-\xi\right)\left(m-\not{\bar{p}}_{1}\right)\right)\bar{\sigma}^{\mu\nu}.$

The third diagram is a little complicated but still straightforward to
calculate :

$\displaystyle\mathscr{E}^{\mu\nu}_{(3)}(\bar{p}_{1},\bar{p}_{2})$ (74)
$\displaystyle=$ $\displaystyle
2ig^{2}\int\frac{d^{d}p}{(2\pi)^{d}}\left(\gamma^{\alpha}\frac{1}{\not{p}+\not{\bar{p}}_{1}-m}\hat{\not{p}}\bar{\sigma}^{\mu\nu}\frac{1}{\not{p}+\not{\bar{p}}-m}\gamma_{\alpha}\right.$
$\displaystyle\left.-(1-\xi)\not{p}\frac{1}{\not{p}+\not{\bar{p}}_{1}-m}\hat{\not{p}}\bar{\sigma}^{\mu\nu}\frac{1}{\not{p}+\not{\bar{p}}-m}\not{p}\frac{1}{p^{2}}\right)\frac{1}{p^{2}}$
$\displaystyle=$ $\displaystyle-2ig^{2}\int\frac{d^{d}p}{(2\pi)^{d}}\cdot
2\int_{0}^{1}dx\int_{0}^{x}dy$
$\displaystyle\times\left(\frac{\gamma^{\alpha}\left(\not{p}+(1-y)\not{\bar{p}}_{1}-(1-x)\not{\bar{p}}_{2}+m\right)\hat{\not{p}}\bar{\sigma}^{\mu\nu}\left(\not{p}-y\not{\bar{p}}_{1}+x\not{\bar{p}}_{2}+m\right)\gamma_{\alpha}}{\left[-p^{2}-y(1-y)\bar{p}^{2}_{1}-x(1-x)\bar{p}^{2}_{2}+2y(1-x)\bar{p}_{1}\cdot\bar{p}_{2}+(1-x+y)m^{2}\right]^{3}}\right.$
$\displaystyle+3(1-\xi)(x-y)\left(\not{p}-y\not{\bar{p}}_{1}-(1-x)\not{\bar{p}}_{2}\right)$
$\displaystyle\left.\times\frac{\left(\not{p}+(1-y)\not{\bar{p}}_{1}-(1-x)\not{\bar{p}}_{2}+m\right)\hat{\not{p}}\bar{\sigma}^{\mu\nu}\left(\not{p}-y\not{\bar{p}}_{1}+x\not{\bar{p}}_{2}+m\right)\left(\not{p}-y\not{\bar{p}}_{1}-(1-x)\not{\bar{p}}_{2}\right)}{\left[-p^{2}-y(1-y)\bar{p}^{2}_{1}-x(1-x)\bar{p}^{2}_{2}+2y(1-x)\bar{p}_{1}\cdot\bar{p}_{2}+(1-x+y)m^{2}\right]^{4}}\right)$
$\displaystyle=$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\left(-\frac{2}{3}\not{\bar{p}}_{1}\bar{\sigma}^{\mu\nu}+\frac{4}{3}\not{\bar{p}}_{2}\bar{\sigma}^{\mu\nu}+\frac{4}{3}\bar{\sigma}^{\mu\nu}\not{\bar{p}}_{1}-\frac{2}{3}\bar{\sigma}^{\mu\nu}\not{\bar{p}}_{2}\right.$
$\displaystyle\left.(1-\xi)\left(-2m\bar{\sigma}^{\mu\nu}+\not{\bar{p}}_{1}\bar{\sigma}^{\mu\nu}+\bar{\sigma}^{\mu\nu}\not{\bar{p}}_{2}\right)\right).$

We now come to the last but most simple diagram :

$\displaystyle\mathscr{E}^{\mu\nu\rho}_{(4)}(\bar{q})$ (75) $\displaystyle=$
$\displaystyle-2ig\int\frac{d^{d}p}{(2\pi)^{d}}\mathrm{tr}\left[\bar{\gamma}^{\rho}\frac{1}{\not{p}+\not{\bar{q}}-m}\hat{\not{p}}\bar{\sigma}^{\mu\nu}\frac{1}{\not{p}-m}\right]$
$\displaystyle=$ $\displaystyle
8g\left(\bar{q}^{\mu}\bar{g}^{\nu\rho}-\bar{q}^{\nu}\bar{g}^{\mu\rho}\right)\int\frac{d^{d}p}{(2\pi)^{d}}\int_{0}^{1}dx\frac{\hat{p}^{2}}{\left[-p^{2}-x(1-x)\bar{q}^{2}+m^{2}\right]^{2}}$
$\displaystyle=$
$\displaystyle-\frac{g^{2}}{2\pi^{2}}\left(i\bar{q}^{\mu}\bar{g}^{\nu\rho}-i\bar{q}^{\nu}\bar{g}^{\mu\rho}\right)\left(-\frac{1}{6}\bar{q}^{2}+m^{2}\right).$

When adding these terms together, all the gauge dependent terms cancel out,
which verifies our conclusion drawn in Sec.IV that all coefficients in the
tWTI (25) are gauge independent to one-loop order. We get these coefficients
on one-loop order :

$b=0,\ c=\frac{g^{2}}{6\pi^{2}},\ f=\frac{g^{2}}{4\pi^{2}},\
r=\frac{g}{24\pi^{2}},\ s=\frac{g}{4\pi^{2}}.$ (76)

As for gauge invariance of these coefficients to all orders, some general
conclusions drawn in a similar treatment of the chiral anomaly by
BonneauBonneau (1980a) are enough. We here only quote the contents for the
reader’s convenience (but with our notations).

The starting point is provided by the action principle
($N\left[\mathcal{O}(x)\right]$ is assumed to be any formally gauge invariant
operator) :

$\frac{\partial}{\partial\xi}\left<TN\left[\mathcal{O}(x)\right]X\right>=\left<T\int
d^{4}yN\left[\frac{i}{2\xi^{2}}\left(\partial_{\mu}A^{\mu}(y)\right)\right]N\left[\mathcal{O}(x)\right]X\right>,$
(77)

where
$X\equiv\prod_{i=1}^{N}\psi\left(x_{i}\right)\prod_{j=1}^{N}\bar{\psi}\left(y_{j}\right)\prod_{k=1}^{L}A^{\mu_{k}}\left(z_{k}\right)$.

Through repeated use of the gauge WTI343434$X\backslash
A_{\mu_{k}}\left(z_{k}\right)$ means $X$ with $A_{\mu_{k}}\left(z_{k}\right)$
stripped away., we have

$\displaystyle\left<T\partial_{\mu}A^{\mu}(x)N\left[\mathcal{O}(y)\right]X\right>=$
$\displaystyle-\xi\sum_{k=1}^{L}\partial_{\mu_{k}}^{x}D\left(x-z_{k}\right)\left<TN\left[\mathcal{O}(y)\right]X\backslash
A_{\mu_{k}}\left(z_{k}\right)\right>$ (78)
$\displaystyle+ig\xi\sum_{i=1}^{N}\left(D\left(x-x_{i}\right)-D\left(x-y_{i}\right)\right)\left<TN\left[\mathcal{O}(y)\right]X\right>,$

and we can recast (77) to many useful forms. We may only focus on the gauge
variance of the proper part of Green’s functions with $X=A^{\rho}$ and
$X=\psi\left(y\right)\bar{\psi}\left(z\right)$, because expression (24) for
coefficients in the tWTI only considers these two cases.

Gauge variance of Green’s functions is not the focus of this paper, and we
thus only quote two main results of Bonneau (1980a) to illustrate the gauge
invariance of coefficients in the tWTI. The first result is (B.10) in Bonneau
(1980a), for formally gauge invariant $N\left[\mathcal{O}(x)\right]$

$\frac{\partial}{\partial\xi}\left<TN\left[\mathcal{O}(x)\right]\prod_{k=1}^{L}A^{\mu_{k}}\left(z_{k}\right)\right>^{\mathrm{prop}}=0.$
(79)

This is also established for the non-overall subtracted Green function (see
the first sentence after (B.11) in Bonneau (1980a)). Thus
$\frac{\partial}{\partial\xi}r^{\prime}=0$ and
$\frac{\partial}{\partial\xi}s^{\prime}=0$ are simply special cases where
$\mathcal{O}=\check{g}_{\rho\sigma}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\rho}iD^{\sigma}\psi\right)$
and $L=1$.

The second result deals with the gauge invariance of
$\mathrm{r.s.p.}\overline{\left<TN\left[\mathcal{O}(x)\right]\tilde{\psi}\left(p\right)\tilde{\bar{\psi}}\left(q\right)\right>^{\mathrm{prop}}}$.
Figure 3 (which provides a diagrammatical representation of the gauge variance
of
$\left<TN\left[\mathcal{O}(x)\right]\tilde{\psi}\left(p\right)\tilde{\bar{\psi}}\left(q\right)\right>^{\mathrm{prop}}$)
and Lemma 3 in Bonneau (1980a) (i.e., (B.13.a) and (B.13.b) therein) indicates
that, if
$\left<TN\left[\mathcal{O}(x)\right]\tilde{\psi}\left(p\right)\tilde{\bar{\psi}}\left(q\right)\right>$
has no trivial part (where a non-trivial diagram was defined by Bonneau
(1980a) to be a graph with at least one loop), then

$\frac{\partial}{\partial\xi}\mathrm{r.s.p.}\overline{\left<TN\left[\mathcal{O}(x)\right]\tilde{\psi}\left(p\right)\tilde{\bar{\psi}}\left(q\right)\right>^{\mathrm{prop}}}=0.$
(80)

Obviously
$\left<TN\left[\check{g}_{\rho\sigma}\left(\bar{\psi}\bar{\sigma}^{\mu\nu}\gamma^{\rho}iD^{\sigma}\psi\right)(x)\right]\tilde{\psi}\left(p\right)\tilde{\bar{\psi}}\left(q\right)\right>$
has no trivial part owing to the presence of
$\check{g}_{\rho\sigma}=\hat{g}_{\rho\sigma}/(d-4)$ (which we take to be zero
after finishing all loop integrals), and obtaining the gauge independence of
$a,b^{\prime},c^{\prime}f^{\prime}$ is thus straightforawrd.

We get the gauge independence of $b,c,f,r,s$ by combining these two results.