Centro Brasileiro de Pesquisas Físicas - CBPF

A study of anomalies using functional integration and perturbative
calculations.

Thalis José Girardi

PhD Thesis

Advisor: Prof. Dr. Sebastião Alves Dias

Rio de Janeiro, RJ

2023

### Agradecimentos111I chose to write acknowledgements in my first language,
Portuguese.

A contribuição de diversas pessoas e organizações, de forma direta ou
indireta, permitiu que eu pudesse concluir esta tese. Uso esse texto para
expressar minha gratidão a eles.

Começo agradecendo à Capes, cujo financiamento foi essencial para o
desenvolvimento dessa tese. Também agradeço ao CBPF e todos que fazem parte
dele. À Mônica, que felizmente me convenceu a participar como orientador no
Provoc. À Edna, Cláudia, Bete, Denise, Larissa e outras personalidades que
fazem esse centro de pesquisa.

Aos educadores que contribuíram para a minha trajetória, desde a educação
básica até o doutorado. Agradeço por criarem as possibilidades para que eu
pudesse expressar minha individualidade, tornando-me quem sou. Agradeço por
estimularem minhas inquietudes, mostrando que o conhecimento não está atrelado
ao passado, mas é um processo constante de invenção e reinvenção.

Ao professor Sebastião Alves Dias por sua orientação e pela serenidade que
trouxe em momentos difíceis. Ao professor Orimar Battistel pelos primeiros
passos na Física, pelas discussões que seguem e levam ao ato constante de
construir conhecimento. Ao professor José Helayël-Neto por sua presença
encorajadora e reconfortante. Agradeço aos três pelos excelentes cursos que
enriqueceram minha formação acadêmica e por influenciarem minha paixão por
ciência e educação.

À Luciana e ao José, que estão sempre próximos compartilhando felicidades,
angústias e saberes. À Grecia, que tornou meus dias mais leves com longas
conversas que parecem durar poucos minutos. Ao João, pelas memórias
compartilhadas e pelo carinho que sempre haverá entre nós. Aos colegas Erich,
Guilherme e Fábio pelo privilégio e prazer de convivermos. A todos que fizeram
parte da minha vida no Rio de Janeiro: Neusa, Ivana, Jade, Judismar, Yuri,
Gustavo, Miguel, Pablo, Guga, Manoel, Liliana, Alexandre e tantos outros. Aos
amigos que estão por perto desde a graduação: Thiago, Giuliano, Hélio e
Carine.

À minha família, cujo apoio foi fundamental para que eu pudesse concluir essa
etapa. Agradeço, em especial, aos meus pais por estarem sempre ao meu lado.
Gostaria de finalizar com um agradecimento àqueles que tiveram profunda
influência na minha vida e se foram nesses anos. À minha avó, que amo tanto e
cujas lembranças sempre me confortam. Ao meu avô, cujas provocações sempre
farão falta. À Maria, uma pessoa tão querida que será sempre lembrada pelo
carinho e cuidado que nos trouxe.

### Abstract

We present two lines of investigation involving anomalies. First, we review
mechanisms behind the classical and quantum conservation of symmetries using
functional integration. This discussion clarifies conditions for quantum
violations, as acknowledged in chiral theories. Then, we elucidate the subject
of gauge anomaly cancellation when all fields are quantized. Such an outcome
requires gauge invariance of the bosonic measure, so our first object is
proving this invariance within Fujikawa’s approach. Second, we investigate
anomalies in fermionic perturbative amplitudes using Implicit Regularization.
The discussion of the single-axial triangle fundaments this analysis, bringing
the elements necessary to approach the single-axial box. When organizing their
mathematical structure, we highlight the role of traces involving the chiral
matrix. Choosing a specific expression for them reflects on the position of
symmetry violations, which has implications regarding the linearity of
integration. Power counting and tensor structure imply the presence of surface
terms related to momenta ambiguities. We present the results without computing
these surface terms. In this neutral perspective, we explore possibilities
achieved under different prescriptions.

Keywords: Gauge and Chiral Anomalies. Divergences. Implicit Regularization.

### Resumo

Nós apresentamos duas linhas de investigação envolvendo anomalias. Primeiro,
revisamos mecanismos por trás da conservação clássica e quântica de simetrias
usando integração funcional. Essa discussão clarifica condições para a
violação quântica, como reconhecido em teorias quirais. Em seguida, elucidamos
o assunto de cancelamento da anomalia de calibre quando todos os campos são
quantizados. Isso requer a invariância de calibre da medida bosônica, então
nosso primeiro objetivo é provar essa invariância através do método de
Fujikawa. Segundo, investigamos anomalias em amplitudes perturbativas
fermiônicas usando Regularização Implícita. A discussão do triângulo com um
vértice axial fundamenta essa análise, trazendo os elementos necessários para
abordar o box com um vértice axial. Ao organizarmos suas estruturas
matemáticas, destacamos o papel de traços envolvendo a matriz quiral. Escolher
uma expressão específica para eles reflete na posição de violações de
simetria, trazendo implicações quanto à linearidade da integração. Contagem de
potências e estrutura tensorial implicam na presença de termos de superfície
relacionados a combinações ambíguas de momenta. Apresentamos esses resultados
sem calcular termos de superfície. Nesta perspectiva neutra, exploramos
possibilidades encontradas em prescrições diferentes.

Palavras-chave: Anomalias de calibre e quiral. Divergências. Regularização
Implícita.

###### Contents

  1. 1 Introduction
  2. 2 Gauge Anomaly and Invariance of the Bosonic Measure
    1. 2.1 Classical Symmetry
    2. 2.2 Quantum Symmetry
    3. 2.3 Gauge Anomaly
    4. 2.4 Gauge Invariance of the Bosonic Measure
    5. 2.5 Final Remarks and Conclusions
  3. 3 Anomalies in Fermionic Amplitudes
    1. 3.1 Model and Definitions
      1. 3.1.1 Perturbative Amplitudes
    2. 3.2 Structure of Perturbative Amplitudes
      1. 3.2.1 Two-Point Amplitudes - Preliminary Notions
      2. 3.2.2 Two-Point Amplitudes - $AV$
      3. 3.2.3 Three-Point Amplitudes - $PVV$
      4. 3.2.4 Three-Point Amplitudes - $AVV$
      5. 3.2.5 Four-Point Amplitudes - $PVVV$
      6. 3.2.6 Four-Point Amplitudes - $AVVV$
      7. 3.2.7 Comments
    3. 3.3 Strategy to Handle Divergences
      1. 3.3.1 Standard Divergent Objects
      2. 3.3.2 Finite Structure Functions - Part I
      3. 3.3.3 Finite Structure Functions - Part II
      4. 3.3.4 Finite Structure Functions - Part III
    4. 3.4 Explicit Perturbative Amplitudes
      1. 3.4.1 Two-Point Amplitudes - Feynman Integrals and $AV$
      2. 3.4.2 Three-Point Amplitudes - Feynman Integrals
      3. 3.4.3 Three-Point Amplitudes - $PVV$
      4. 3.4.4 Three-Point Amplitudes - $AVV$
      5. 3.4.5 Four-Point Amplitudes - Feynman Integrals
      6. 3.4.6 Four-Point Amplitudes - $PVVV$
      7. 3.4.7 Four-Point Amplitudes - $AVVV$
      8. 3.4.8 Comments
    5. 3.5 Analysis of the Results
      1. 3.5.1 Relations Among Green Functions - $AVV$
      2. 3.5.2 Relations Among Green Functions - $AVVV$
      3. 3.5.3 Further Explorations on Relations Among GF
      4. 3.5.4 Symmetries and Linearity
    6. 3.6 Final Remarks and Conclusions

##  Chapter 1 Introduction

When building an interacting model through a quantum field theory, one starts
by constructing a functional of free fields whose interaction dynamics one
aims to describe. In general, the original functional exhibits invariance
under global transformations, with parameters that do not depend on the space-
time position. Then, modifying this functional promotes the symmetry to a
local one. The new functional emerges after introducing a set of fields
(called gauge fields), with transformations chosen to ensure invariance when
parameters depend on the spacetime. These local transformations of the fields
are called gauge transformations, and the corresponding symmetry is called
gauge invariance. The main consequence is the generation of interaction
between the previously free fields. This proposed interaction generates
previsions (i.e., decay rates or cross sections) capable of being compared
with experimentally measured quantities.

Quantum Electrodynamics is a well-known example of this construction,
corresponding to the quantum field theory for electromagnetic interaction. The
first step is to build the Dirac action, describing free spin-(1/2) fermions
(such as electrons and positrons). Although this functional exhibits global
$U(1)$ invariance, the presence of derivatives prevents gauge invariance. The
solution comes when substituting conventional derivatives with covariant ones,
which induces coupling with a gauge field (interpreted as the photon field).
This field arises from the only generator of the Abelian symmetry group
$U(1)$. There are interacting terms involving gauge and matter fields in the
modified action, so all mentioned contributions constitute a locally invariant
object. One adds a gauge invariant term involving only the gauge field (the
Maxwell action) to furnish dynamics for the photon.

Something analogous occurs when developing Quantum Chromodynamics to describe
the strong interaction. The outset is on the Dirac action, now built with free
quarks, symmetric under global $SU(3)$ symmetry. Promoting it to be local
generates interaction terms involving gauge and matter fields. As $SU(3)$ has
eight generators, eight gluons emerge as gauge fields. The difference from the
abelian case resides in the non-commutative character of the algebra, which
implies self-interacting gauge fields. The Yang-Mills functional is introduced
to provide dynamics for gauge fields.

Regarding Electroweak Theory, the symmetry group is $SU\left(2\right)\times
U\left(1\right)$. As this theory unifies electromagnetic and weak
interactions, it adds new gauge bosons ($W^{\pm}$, $Z$) besides that
corresponding to the photon. The main difference is that these new fields are
massive, while gauge invariance does not admit this type of contribution to
the action. The strategy to deal with this problem is to start from a massless
theory, with the Higgs mechanism generating masses. That means quarks and
leptons are seen as massless Weyl fermions (with defined chiralities) instead
of Dirac fermions. Under these circumstances, the functional displays gauge
invariance before spontaneous symmetry breaking. That is crucial for the
renormalization of the theory. The masses are generated for all the
experimentally known massive fields without spoiling the renormalizability.
This mechanism is extended to the group $SU\left(3\right)\times
SU\left(2\right)\times U\left(1\right)$, defining the Standard Model (SM),
which unifies the three mentioned interactions.

There is another issue to be faced by the SM, the so-called anomalies. They
are quantum violations of symmetries originally present at the action. They
have a vast history, initiated by Johnson’s discovery of the two-dimensional
chiral anomaly [1]. A few years later, this subject received prominence due to
the Adler-Bell-Jackiw anomaly [2, 3]. Both refer to the quantum breaking of
the global (constant parameter) chiral symmetry, present in theories with
massless fermions. It also became clear the impossibility of simultaneous
maintenance of chiral and gauge symmetries at the quantum level [4]. These
symmetries are mixed in the SM before spontaneous symmetry breaking, which
means that gauge invariance is apparently broken at the quantum level. This
phenomenon is known as gauge anomaly. In the SM, gauge invariance is only
achieved through a careful adjustment of the group representation where one
puts the three families of quarks and leptons so that anomalous contributions
from both sectors cancel each other. Meanwhile, gauge invariance is necessary
to ensure renormalizability and unitarity to the theory. Gauge anomalies
modify Slavnov-Taylor identities, preventing one from relating distinct
renormalization constants with each other and canceling infinities
systematically to all orders of perturbation theory [5, 6]. We end with an
uncomfortable situation where the SM is a superposition of apparently
inconsistent theories, which result in a consistent one by a very peculiar
arrangement.

This situation motivated investigations on gauge-anomalous theories. Jackiw
and Rajaraman [7, 8] showed that chiral Quantum Electrodynamics in two
dimensions is consistent and unitary. Furthermore, the gauge field, initially
massless in the classical action, became massive after radiative corrections
without needing a Higgs mechanism. Faddeev and Shatashvili clarified the
quantization of this type of theory [9]. They introduced new quantum degrees
of freedom that provided an equivalent gauge theory (without anomalies). In
addition, Harada and Tsutsui [10] and Babelon, Schaposnik and Viallet [11]
observed that these new degrees could be obtained naturally through the
employment of the Faddeev-Popov procedure. That allowed them to express the
effective action as a gauge scalar for any space-time dimension. These results
suggested that theories with gauge anomalies could be consistent.

By taking into account gauge invariance of the gauge field measure, in the
context of functional integrals, a recent investigation [12] showed the
vanishing of the insertion of the anomaly operator in any correlator of gauge
invariant operators. This result suggested that the anomaly vanishes in the
part of the Hilbert space associated with physical states. That motivated us
to investigate gauge invariance of the boson measure in more detail. We do
this in Chapter (2), providing explicit proof of this fact that is, up to our
knowledge, absent from the literature.

We continue to investigate symmetries in the quantum context through an
approach known as Implicit Regularization (IReg), a procedure to identify and
separate the divergent part of Feynman diagrams by manipulating the integrands
before integration. The study of an amplitude associated with the neutral pion
decay (the single axial triangle) establishes the foundations for this
analysis. Afterward, we examine the possibility of one amplitude with an
analogous mathematical structure (the single axial box) exhibiting the same
characteristics. Hence, surveying features shared by these processes
highlights new aspects of the anomalies. That corresponds to the second part
of this thesis, whose development occurs in Chapter (3).

Conclusions will be presented separately for Chapters (2) and (3) since they
use different methodologies to approach the subject of anomalies.

##  Chapter 2 Gauge Anomaly and Invariance of the Bosonic Measure

Investigating the consequences of gauge symmetry in classical and quantum
theories is the general objective of this chapter. Starting with the classical
discussion in Section (2.1), we use arguments involving action invariance to
achieve current conservation. These preliminary calculations work as a guide
to explorations at the quantum level, made in Section (2.2). After finding
requirements for quantum invariance, the source of violations in functional
integration is discussed in Section (2.3). With the mathematical structure of
the anomaly in our hands, we use a simple procedure to show that its expected
value vanishes when quantizing all theory fields.

The gauge invariance of the gauge field measure is central to this
argumentation. This property has several usages in the literature, as in
investigations involving the Faddeev-Popov method. Since there is (up to our
knowledge) an absence of explicit demonstration of this invariance, our first
contribution is to provide proof of it. To do so, we use general functional
integral arguments to show that the Jacobian associated with the measure has
to be 1 (one) when inserted in correlation functions of gauge-invariant
operators. Performing the same analysis for general operators would complete
this demonstration. Since this step brings complications, we employ a
Fujikawa-like approach to calculate this Jacobian explicitly and show that it
is 1 in general.

###  2.1 Classical Symmetry

This section aims for a preliminary understanding of gauge theories,
emphasizing current conservation at the classical level. It is also the moment
to introduce notations, which follow the material from R. Jackiw’s course in
reference [13] and G. L. S. Lima’s works [14, 15].

Throughout the Introduction, we mentioned some aspects of theories employed to
describe fundamental interactions. The starting point was the functional
associated with the dynamics of free matter fields. This object is not
invariant under local transformations since it contains derivatives. So, the
idea was to implement this symmetry by making the derivative covariant. The
price paid is inducing terms of interaction with gauge fields. In other words,
gauge symmetry generates dynamics among the fields described by a theory [6].
A contribution associated with free gauge fields is also necessary. Below, we
write the action with these two sectors separated, so it is clear that each
part is invariant by itself:

$S\left[\psi,\overline{\psi},A_{\mu}\right]=S_{G}\left[A_{\mu}\right]+S_{M}\left[\psi,\overline{\psi},A_{\mu}\right].$
(2.1)

The vector $A_{\mu}=A_{\mu}^{a}T_{a}$ represents the gauge fields with $T_{a}$
being generators of the gauge group, while ($\psi$, $\overline{\psi}$)
represent fermionic matter fields.

Saying that the action is invariant means no changes occur when fields modify
through a given set of transformations. Our concern is with gauge theories, in
which case these transformations belong to special unitary groups
$SU\left(N\right)$. Its generators satisfy commutation relations like

$\left[T^{a},T^{b}\right]=if^{abc}T_{c},$ (2.2)

along with the normalization

$\mathrm{tr}\left(T^{a}T^{b}\right)=-\frac{1}{2}\delta^{ab}.$ (2.3)

The symbol $f^{abc}$ represents the structure constants, which have the
property of total antisymmetry through index permutations. Indices denoted by
Latin letters refer to internal degrees of freedom, ranging over the group
dimension (equivalent to the number of generators). As gauge fields take
values on the Lie algebra of the symmetry group, there is one field for each
generator. Greek letters in the indices refer to Minkowski space-time in the
chosen theory.

To analyze current conservation, let us adopt an arbitrary element
$g=e^{i\theta\left(x\right)}$ to perform a transformation. The parameters
depend on the space-time position
$\theta\left(x\right)=\theta^{a}\left(x\right)T_{a}$, characterizing a local
transformation. As mentioned, the action is invariant under simultaneous
changes of boson and fermion fields

$\displaystyle A_{\mu}$ $\displaystyle\rightarrow$ $\displaystyle
A_{\mu}^{g}=gA_{\mu}g^{-1}+\frac{i}{e}\left(\partial_{\mu}g\right)g^{-1},$
(2.4) $\displaystyle\psi$ $\displaystyle\rightarrow$
$\displaystyle\psi^{g}=g\psi,$ (2.5) $\displaystyle\overline{\psi}$
$\displaystyle\rightarrow$
$\displaystyle\overline{\psi}^{g}=\overline{\psi}g^{-1}.$ (2.6)

By considering small values for the parameter, we take its first-order
contribution to obtain infinitesimal transformations

$\displaystyle A_{\mu}$ $\displaystyle\rightarrow$ $\displaystyle
A_{\mu}^{g}=A_{\mu}-\frac{1}{e}\mathcal{D}_{\mu}\theta,$ (2.7)
$\displaystyle\psi$ $\displaystyle\rightarrow$
$\displaystyle\psi^{g}=\psi+i\theta\psi,$ (2.8) $\displaystyle\overline{\psi}$
$\displaystyle\rightarrow$
$\displaystyle\overline{\psi}^{g}=\overline{\psi}-i\overline{\psi}\theta.$
(2.9)

We define the covariant derivative of Lie algebra valued quantities through
the mathematical expression

$\mathcal{D}_{\mu}\theta=\partial_{\mu}\theta+ie\left[A_{\mu},\theta\right],$
(2.10)

so using the commutation relations allows specifying its components

$\mathcal{D}_{\mu}\theta=T^{a}\left(\partial_{\mu}\delta^{ac}-ef^{abc}A_{\mu}^{b}\right)\theta^{c}\equiv
T^{a}\mathcal{D}_{\mu}^{ac}\theta^{c}.$ (2.11)

Since the action is invariant under local transformations, it is also
invariant under global transformations. As the parameter is constant in the
second case, the derivative $\partial_{\mu}\theta$ cancels out within the
vector field transformation. Then, by reversing this line of reasoning,
starting from global transformations and imposing dependence on the position
is feasible. In such a case, the absence of the inhomogeneous term implies
symmetry is lost. That means the following variation must be proportional to
derivatives of the parameter

$\delta
S_{M}=S_{M}\left[\psi^{g},\overline{\psi}^{g},gA_{\mu}g^{-1}\right]-S_{M}\left[\psi,\overline{\psi},A_{\mu}\right]=\int\mathrm{dx}\text{
}\partial_{\mu}\theta^{a}\left(x\right)J_{a}^{\mu}\left(x\right),$ (2.12)

where we introduce the vector $J_{a}^{\mu}\left(x\right)$, determined by the
fields present in the model. Recalling that both sectors of the action are
invariant when considered by themselves, we focus exclusively on the matter
action. On the other hand, an infinitesimal transformation over the action
leads to another form for the same variation:

$\delta S_{M}=\int\mathrm{dx}\text{
}\theta^{a}\left(x\right)\left[\frac{\delta
S_{M}}{\delta\psi\left(x\right)}iT^{a}\psi\left(x\right)-i\overline{\psi}\left(x\right)T^{a}\frac{\delta
S_{M}}{\delta\overline{\psi}\left(x\right)}+f^{abc}A_{\mu}^{b}\left(x\right)\frac{\delta
S_{M}}{\delta A_{\mu}^{c}\left(x\right)}\right].$ (2.13)

Equating both expressions to produce one identity is feasible. By performing
an integration by parts on the contribution from (2.12), the parameter
$\theta^{a}$ factorizes inside the integration sign:

$\int\mathrm{dx}\text{
}\theta^{a}\left(x\right)\left[\partial_{\mu}J_{a}^{\mu}\left(x\right)+\frac{\delta
S_{M}}{\delta\psi\left(x\right)}iT^{a}\psi\left(x\right)-i\overline{\psi}\left(x\right)T^{a}\frac{\delta
S_{M}}{\delta\overline{\psi}\left(x\right)}+f^{abc}A_{\mu}^{b}\left(x\right)\frac{\delta
S_{M}}{\delta A_{\mu}^{c}\left(x\right)}\right]=0.$ (2.14)

Hence, the arbitrariness of this object implies that the structure in squared
brackets vanishes regardless of the integration

$\partial_{\mu}J_{a}^{\mu}\left(x\right)+\frac{\delta
S_{M}}{\delta\psi\left(x\right)}iT^{a}\psi\left(x\right)-i\overline{\psi}\left(x\right)T^{a}\frac{\delta
S_{M}}{\delta\overline{\psi}\left(x\right)}+f^{abc}A_{\mu}^{b}\left(x\right)\frac{\delta
S_{M}}{\delta A_{\mu}^{c}\left(x\right)}=0.$ (2.15)

As we have not considered local symmetry up to this point, such a result is a
consequence of global invariance.

Next, observe that equations of motion associated with fermion fields fall
over the matter action

$\displaystyle\frac{\delta S}{\delta\psi\left(x\right)}$ $\displaystyle=$
$\displaystyle\frac{\delta S_{M}}{\delta\psi\left(x\right)}=0,$ (2.16)
$\displaystyle\frac{\delta S}{\delta\overline{\psi}\left(x\right)}$
$\displaystyle=$ $\displaystyle\frac{\delta
S_{M}}{\delta\overline{\psi}\left(x\right)}=0.$ (2.17)

Hence, replacing them in Equation (2.15) cancels out some contributions, which
leads to the simplified version111As structure constants cancel out in the
Abelian theory, the conservation of the current comes directly from this
equation. That means it is unnecessary to consider gauge transformations at
any point in the calculations.

$\partial_{\mu}J_{a}^{\mu}\left(x\right)+f^{abc}A_{\mu}^{b}\left(x\right)\frac{\delta
S_{M}}{\delta A_{\mu}^{c}\left(x\right)}=0.$ (2.18)

Now, we consider gauge transformations as the final step before achieving
conservation. Invariance of the action establishes the relation

$S_{M}\left[\psi^{g},\overline{\psi}^{g},A_{\mu}\right]=S_{M}\left[\psi,\overline{\psi},A_{\mu}^{g^{-1}}\right]=S_{M}\left[\psi,\overline{\psi},g^{-1}A_{\mu}g+\frac{i}{e}\left(\partial_{\mu}g^{-1}\right)g\right].$
(2.19)

By adopting the configuration for the gauge field
$A_{\mu}^{\prime}=gA_{\mu}g^{-1}$, the variation of $S_{M}$ is achievable
again. To that end, rewrite the relation above by considering the
infinitesimal form of the transformation

$S_{M}\left[\psi^{g},\overline{\psi}^{g},gA_{\mu}g^{-1}\right]=S_{M}\left[\psi,\overline{\psi},A_{\mu}+\frac{1}{e}\partial_{\mu}\theta\right].$
(2.20)

The mentioned variation emerges through an expansion over the parameter

$\delta S_{M}=\frac{1}{e}\int\mathrm{dx}\text{
}\partial_{\mu}\theta^{a}\left(x\right)\frac{\partial S_{M}}{\partial
A_{\mu}^{a}\left(x\right)}.$ (2.21)

Therefore, a comparison between this form and Equation (2.12) generates the
following result

$\int\mathrm{dx}\text{
}\partial_{\mu}\theta^{a}\left(x\right)\left[J_{a}^{\mu}\left(x\right)-\frac{1}{e}\frac{\partial
S_{M}}{\partial A_{\mu}^{a}\left(x\right)}\right]=0.$ (2.22)

Again, the quantity in squared brackets has to vanish by itself as
transformation parameters are arbitrary. That produces the relation

$J_{a}^{\mu}\left(x\right)=\frac{1}{e}\frac{\partial S_{M}}{\partial
A_{\mu}^{a}\left(x\right)},$ (2.23)

whose replacement within Equation (2.18) allows recognizing the covariant
derivative introduced in the gauge field transformation

$\mathcal{D}_{\mu}^{ac}J_{c}^{\mu}\left(x\right)=\left(\partial_{\mu}\delta^{ac}-ef^{abc}A_{\mu}^{b}\right)J_{c}^{\mu}\left(x\right)=0.$
(2.24)

We identify the vector $J_{a}^{\mu}\left(x\right)$ as a current, while the
last equation represents its covariant conservation. Two ingredients were
necessary to achieve this outcome: local gauge invariance of the matter action
and equations of motion for fermions.

###  2.2 Quantum Symmetry

Since we finalized exploring manifestations of gauge symmetry in classical
theories, let us extend this discussion to the quantum context. To accomplish
this goal, we start by introducing the effective action
$W\left[A_{\mu}\right]$ through the functional integral

$e^{iW\left[A_{\mu}\right]}=\int d\psi
d\overline{\psi}\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right).$
(2.25)

Since gauge fields are considered external classical fields, the integration
occurs exclusively over (quantized) fermion fields.

Following the same reasoning from the previous section, we consider global
transformation and impose that parameters depend on the position. When
applying infinitesimal transformations (2.7)-(2.9), the changed expression for
the exponential follows

$e^{iW^{\prime}}=\int d\psi
d\overline{\psi}\exp\left(iS_{M}\left[\psi+i\theta\psi,\overline{\psi}-i\overline{\psi}\theta,A_{\mu}-i\left[A_{\mu},\theta\right]\right]+iS_{G}\left[A_{\mu}\right]\right),$
(2.26)

with the gauge action invariant. Although gauge fields change through the
covariant derivative, only the contribution on the commutator concerns global
invariance. Recognizing the exponential argument as the action plus a
variation allows detaching both parts

$e^{iW^{\prime}}=\int d\psi d\overline{\psi}\exp\left(i\delta
S_{M}\right)\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right).$
(2.27)

Hence, an expansion on the infinitesimal parameter leads to the exponential
variation of the effective action

$e^{iW^{\prime}}-e^{iW}=\int d\psi d\overline{\psi}\left(i\delta
S_{M}\right)\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right).$
(2.28)

As this result depends on the action variation, let us recall the information
obtained. On the one hand, we reasoned that it is proportional to the
derivative of the parameter and the current $J_{a}^{\mu}\left(x\right)$; see
Equation (2.12). In the quantum context, that leads to the expression

$e^{iW^{\prime}}-e^{iW}=-\int d\psi
d\overline{\psi}\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right)\left[i\int\mathrm{dx}\text{
}\theta^{a}\left(x\right)\partial_{\mu}J_{a}^{\mu}\left(x\right)\right],$
(2.29)

where integration by parts changes the derivative position. On the other hand,
the infinitesimal transformation produced result (2.13), which reflects on the
form

$\displaystyle e^{iW^{\prime}}-e^{iW}$ $\displaystyle=$ $\displaystyle\int
d\psi
d\overline{\psi}\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right)\times$
$\displaystyle\times i\int\mathrm{dx}\text{
}\theta^{a}\left(x\right)\left[\frac{\delta
S_{M}}{\delta\psi\left(x\right)}iT^{a}\psi\left(x\right)-i\overline{\psi}\left(x\right)T^{a}\frac{\delta
S_{M}}{\delta\overline{\psi}\left(x\right)}+ef^{abc}A_{\mu}^{b}\left(x\right)J_{c}^{\mu}\left(x\right)\right].$

We already used the association (2.23) to recognize the current within this
equation.

Since there are two forms for the same object, let us equate them to produce
an identity. Due to the arbitrariness of the transformation parameter, the
relation applies regardless of space-time integration. We emphasize that this
does not occur if the parameter is constant, as it would factor from the
integration sign without further simplifications. By identifying the covariant
derivative, the variation produces the result

$\displaystyle\int d\psi
d\overline{\psi}\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right)\left[D_{\mu}^{ab}J_{b}^{\mu}\left(x\right)\right]$
(2.31) $\displaystyle=$ $\displaystyle\int d\psi
d\overline{\psi}\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right)\left[i\overline{\psi}\left(x\right)T^{a}\frac{\delta
S_{M}}{\delta\overline{\psi}\left(x\right)}-\frac{\delta
S_{M}}{\delta\psi\left(x\right)}iT^{a}\psi\left(x\right)\right].$

In the classical discussion, the conservation law arose posteriorly to
employing equations of motion for fermions in an analogous equation. We would
expect Dyson-Schwinger equations to perform this task here, as they embody the
equations of motion within this context. In that case, current conservation
would result from the translational invariance of the fermion measure [16].
Nonetheless, gauge invariance emerges as a condition at the quantum level. Let
us integrate an arbitrary functional and explore its transformation to
understand the consequences:

$\displaystyle\int d\psi d\overline{\psi}\text{
}F\left[\psi,\overline{\psi},A_{\mu}\right]$ $\displaystyle=$
$\displaystyle\int d\psi^{g}d\overline{\psi}^{g}\text{
}F\left[\psi^{g},\overline{\psi}^{g},A_{\mu}\right]$ $\displaystyle=$
$\displaystyle\int d\psi d\overline{\psi}\text{
}F\left[\psi,\overline{\psi},A_{\mu}\right]+\int d\psi
d\overline{\psi}\int\mathrm{dx}\left[\frac{\delta
F}{\delta\psi}\delta\psi+\frac{\delta
F}{\delta\overline{\psi}}\delta\overline{\psi}\right].$

Under the hypothesis of gauge-invariance of the fermion measure

$d\psi^{g}d\overline{\psi}^{g}=d\psi d\overline{\psi},$ (2.33)

the condition applies

$\int d\psi d\overline{\psi}\int\mathrm{dx}\left[\frac{\delta
F}{\delta\psi}\delta\psi+\delta\overline{\psi}\frac{\delta
F}{\delta\overline{\psi}}\right]=0.$ (2.34)

Disregarding space-time integration, observe that this object cancels out the
right-hand side of Equation (2.31) when we set the functional. Hence, the
referred equation turns into the quantum version of the gauge current
covariant conservation:

$\int d\psi
d\overline{\psi}\left[D_{\mu}^{ab}J_{b}^{\mu}\left(x\right)\right]\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right)=0.$
(2.35)

Such an argumentation shows that gauge invariance of the fermion measure is
enough for current conservation. Invariance of the matter action does not
guarantee symmetry maintenance within quantum theory, even if it guarantees
classical conservation.

###  2.3 Gauge Anomaly

After shedding light on conditions for quantum conservation, we aim to inquire
about situations characterized by violations. The literature on functional
integrals recognizes non-trivial Jacobians for the fermion measure as the
cause of symmetry breaking [17]. This non-invariance is typical of
investigations involving chiral fermions, as in the Standard Model before
spontaneous symmetry breaking.

We approach this subject by introducing the fermion measure Jacobian as
follows

$d\psi^{g}d\overline{\psi}^{g}=J\left[g,A_{\mu}\right]d\psi d\overline{\psi}$
(2.36)

while considering the possibility of dependence on gauge fields. Although that
is unreasonable for usual integration, this type of contribution might arise
through regularization techniques when dealing with divergent objects
associated with functional integrals [18, 19]. That means integrals and
functional derivatives do not necessarily commute, requiring extra care to
avoid inconsistent results.

Given the structure of calculations developed in the previous section,
expressing the Jacobian as the exponential of another functional is convenient

$J\left[g,A_{\mu}\right]=\exp\left(i\alpha_{1}\left[g,A_{\mu}\right]\right).$
(2.37)

Thence, writing the Jacobian associated with the inverse transformation is
straightforward

$J\left[g^{-1},A_{\mu}\right]=\exp\left(i\alpha_{1}\left[g^{-1},A_{\mu}\right]\right)=\exp\left(-i\alpha_{1}\left[g,A_{\mu}\right]\right),$
(2.38)

and so is the property attributed to the exponential argument

$\alpha_{1}\left[g^{-1},A_{\mu}\right]=-\alpha_{1}\left[g,A_{\mu}\right].$
(2.39)

Besides, we consider first-order contributions on the infinitesimal
transformation parameter to build the expansion

$\alpha_{1}\left[g,A_{\mu}\right]=\alpha_{1}\left[1,A_{\mu}\right]+\int\mathrm{dx}\text{
}\theta^{a}\left(x\right)\left.\frac{\delta\alpha_{1}\left[g,A_{\mu}\right]}{\delta\theta^{a}\left(x\right)}\right|_{\theta=0}.$
(2.40)

As the first term represents the case without transformation, the Jacobian
corresponds to the identity $J\left[1,A_{\mu}\right]=1$ and implies the
vanishing argument $\alpha_{1}\left[1,A_{\mu}\right]=0$.

Since we discussed how fermionic variables change, let us explore the
implications for the effective action introduced in Equation (2.25). By
relabeling fermion fields as $\psi\rightarrow\psi^{g^{-1}}$ and
$\overline{\psi}\rightarrow\overline{\psi}^{g^{-1}}$, we get the modified
expression

$e^{iW\left[A_{\mu}\right]}=\int
d\psi^{g^{-1}}d\overline{\psi}^{g^{-1}}\exp\left(iS\left[\psi^{g^{-1}},\overline{\psi}^{g^{-1}},A_{\mu}\right]\right).$
(2.41)

After employing action invariance and inserting the Jacobian for the inverse
(2.38), we achieve another form:

$e^{iW\left[A_{\mu}\right]}=\exp\left(-i\alpha_{1}\left[g,A_{\mu}\right]\right)\int
d\psi
d\overline{\psi}\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}^{g}\right]\right).$
(2.42)

The Jacobian factors out of the integral sign as it does not depend on
quantized fermion fields. This integral corresponds to the effective action
with modified gauge fields, so expressing the Jacobian through the effective
action is feasible

$\exp\left(i\alpha_{1}\left[g,A_{\mu}\right]\right)=\exp\left(iW\left[A_{\mu}^{g}\right]-iW\left[A_{\mu}\right]\right).$
(2.43)

Taking the logarithm on both sides emphasizes that the effective action is not
invariant under this type of transformation:

$\alpha_{1}\left[g,A_{\mu}\right]=W\left[A_{\mu}^{g}\right]-W\left[A_{\mu}\right].$
(2.44)

By recalling the gauge field transformation (2.7), we expand
$W\left[A_{\mu}^{g}\right]$ to the first order on the infinitesimal parameter.
That allows writing the variation of the effective action through the integral

$W\left[A_{\mu}^{g}\right]-W\left[A_{\mu}\right]=\int\mathrm{dx}\text{
}\theta^{c}\mathcal{D}_{\mu}^{ac}\left(\frac{1}{e}\frac{\delta
W\left[A_{\mu}\right]}{\delta A_{\mu}^{a}}\right).$ (2.45)

But Equation (2.44) links this structure to the functional $\alpha_{1}$, whose
expansion is (2.40). Given the parameter arbitrariness, comparing both
equations establishes the relation

$\left.\frac{\delta\alpha_{1}\left[g,A_{\mu}\right]}{\delta\theta\left(x\right)}\right|_{\theta=0}=\mathcal{D}_{\mu}\left(\frac{1}{e}\frac{\delta
W\left[A_{\mu}\right]}{\delta A_{\mu}}\right),$ (2.46)

where the notation involving components is omitted.

For the last step of the current discussion, we recall that both effective
action and action itself are Lorentz scalars. That means the commutation
between these objects and the covariant derivative does not bring
complications. Hence, multiplying the relation above and the exponential of
the effective action leads to the mathematical expression

$\displaystyle\left.\frac{\delta\alpha_{1}\left[g,A_{\mu}\right]}{\delta\theta\left(x\right)}\right|_{\theta=0}\int
d\psi
d\overline{\psi}\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right)$
(2.47) $\displaystyle=$
$\displaystyle\mathcal{D}_{\mu}\left\\{-\frac{i}{e}\frac{\delta}{\delta
A_{\mu}}\int d\psi
d\overline{\psi}\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right)\right\\}.$

Since the gauge action is invariant, the functional derivative acts
exclusively on the matter action

$\displaystyle\left.\frac{\delta\alpha_{1}\left[g,A_{\mu}\right]}{\delta\theta\left(x\right)}\right|_{\theta=0}\int
d\psi
d\overline{\psi}\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right)$
(2.48) $\displaystyle=$ $\displaystyle\int d\psi d\overline{\psi}\text{
}\mathcal{D}_{\mu}\left(\frac{1}{e}\frac{\delta
S_{M}\left[\psi,\overline{\psi},A_{\mu}\right]}{\delta
A_{\mu}}\right)\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right).$

As the term in parenthesis is precisely the current identified in the
classical discussion (2.23), the relation applies

$\left.\frac{\delta\alpha_{1}\left[g,A_{\mu}\right]}{\delta\theta_{a}\left(x\right)}\right|_{\theta=0}=\frac{\int
d\psi
d\overline{\psi}\left(\mathcal{D}_{\mu}^{ab}J_{b}^{\mu}\right)e^{iS\left[\psi,\overline{\psi},A_{\mu}\right]}}{\int
d\psi d\overline{\psi}\text{
}e^{iS\left[\psi,\overline{\psi},A_{\mu}\right]}}.$ (2.49)

We transposed the effective action to the right-hand side to identify this
structure as the vacuum expectation value of the covariant divergence of the
current. The non-vanishing of this expression characterizes the so-called
gauge anomaly:

$\mathcal{A}_{a}\left(A_{\mu}\right)=\left.\frac{\delta\alpha_{1}\left[g,A_{\mu}\right]}{\delta\theta_{a}\left(x\right)}\right|_{\theta=0}\neq
0.$ (2.50)

This condition is what characterizes the theory as gauge anomalous. We stress
that this happens when gauge bosons are external classical fields interacting
with quantum fermion fields.

Further explorations show that the expectation value for the gauge anomaly
cancels out for the fully-quantized theory. To verify that, let us define the
generating functional

$Z\left[\eta,\overline{\eta},j_{a}^{\mu}\right]=\int d\psi
d\overline{\psi}dA_{\mu}\text{
}\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]+i\int\mathrm{dx}\left[\overline{\eta}\psi+\overline{\psi}\eta+j_{a}^{\mu}A_{\mu}^{a}\right]\right).$
(2.51)

Since our concern relates to vacuum expectation value, contributions
associated with external sources are unnecessary. The notation simplifies
under these circumstances, being viable to express this equation in terms of
the effective action

$Z\left[0,0,0\right]=\int dA_{\mu}\text{ }e^{iW\left[A_{\mu}\right]}.$ (2.52)

Following a strategy similar to previous cases, we start by relabeling the
structure above through $A_{\mu}\rightarrow A_{\mu}^{g}$. The changed version
for the effective action corresponds to the original plus a variation. After
replacing the result from the previous section (2.45), we split the
exponential argument. Then, expanding the variation part on the infinitesimal
parameter produces the equation

$Z\left[0,0,0\right]=\int dA_{\mu}^{g}\text{
}e^{iW\left[A_{\mu}\right]}\left[1+i\int\mathrm{dx}\text{
}\theta^{c}\mathcal{D}_{\mu}^{ac}\left(\frac{1}{e}\frac{\delta
W\left[A_{\mu}\right]}{\delta A_{\mu}^{a}}\right)\right].$ (2.53)

The difference between the generating functional and the first term on the
right-hand side resides in the integration variable; thus, they coincide if
the bosonic measure is gauge-invariant $dA_{\mu}=dA_{\mu}^{g}$. The second
functional integral must be zero under this condition. Since the arbitrariness
of the transformation parameter allows dropping the space-time integral, the
relation emerges

$\int dA_{\mu}\text{
}e^{iW\left[A_{\mu}\right]}\mathcal{D}_{\mu}^{ac}\left(\frac{1}{e}\frac{\delta
W\left[A_{\mu}\right]}{\delta A_{\mu}^{a}}\right)=0.$ (2.54)

At this point, we recall Equations (2.46) and (2.50) to recognize the anomaly.
Hence, by making the dependence on the fermionic variables explicit, we showed
that its vacuum expectation value is zero for the fully quantized theory:

$\int d\psi d\overline{\psi}dA_{\mu}\text{
}\mathcal{A}_{a}\left(A_{\mu}\right)\exp\left(iS\left[\psi,\overline{\psi},A_{\mu}\right]\right)=\left\langle
0\right|\mathcal{A}_{a}\left(A_{\mu}\right)\left|0\right\rangle=0.$ (2.55)

In addition to its role in the demonstration above, we stress that the bosonic
measure invariance has other applications in investigations in this area. Even
so, we did not find proof of this property in the literature. The primary
objective of this part of the thesis is to provide one, which is our next
subject.

###  2.4 Gauge Invariance of the Bosonic Measure

This section investigates the behavior of the bosonic measure under gauge
transformations. To this end, we display a preparatory argument by considering
the generating functional for correlators of gauge-invariant operators
$O_{i}\left(A_{\mu}^{g}\right)=O_{i}\left(A_{\mu}\right)$ in the pure Yang-
Mills theory (without chiral fermions):

$Z\left[\lambda^{i}\right]=\int dA_{\mu}\exp i\int\mathrm{dx}\text{
tr}\left(\frac{1}{2}F_{\mu\nu}F^{\mu\nu}+\lambda^{i}O_{i}\left[A_{\mu}\right]\right).$
(2.56)

The quantities $\lambda_{i}$ are currents, and functional derivatives with
respect to them yield the $n$-point correlators

$\left.\frac{\delta^{n}}{\delta\lambda^{1}\left(x_{1}\right)...\delta\lambda^{n}\left(x_{n}\right)}Z\left[\lambda^{i}\right]\right|_{\lambda^{i}=0}=\left\langle
0\right|T\left(O_{1}\left(A_{\mu}\right)\left(x_{1}\right)...O_{n}\left(A_{\mu}\right)\left(x_{n}\right)\right)\left|0\right\rangle.$
(2.57)

Considering the integration over $A_{\mu}$ and also over its gauge transformed
version $A_{\mu}^{g}$, we develop the comparison

$\displaystyle Z\left[\lambda^{i}\right]$ $\displaystyle=\int dA_{\mu}\exp
i\int\mathrm{dx}\text{
tr}\left[\frac{1}{2}F_{\mu\nu}F^{\mu\nu}+\lambda^{i}O_{i}\left(A_{\mu}\right)\right]$
$\displaystyle=\int dA_{\mu}^{g}\exp i\int\mathrm{dx}\text{
tr}\left[\frac{1}{2}\left(F_{\mu\nu}F^{\mu\nu}\right)^{g}+\lambda^{i}O_{i}\left(A_{\mu}^{g}\right)\right]$
$\displaystyle=\int dA_{\mu}\text{ }J\left[A_{\mu},g\right]\exp
i\int\mathrm{dx}\text{
tr}\left[\frac{1}{2}F_{\mu\nu}F^{\mu\nu}+\lambda^{i}O_{i}\left(A_{\mu}\right)\right],$
(2.58)

where the potential presence of a Jacobian $J\left[A_{\mu},g\right]$ for the
gauge transformation of the measure is allowed. Thus, we obtain the
correlators associated with both expressions for the generating functional as
follows

$\displaystyle\left\langle
0\right|T\left(J\left[A_{\mu},g\right]O_{1}\left(A_{\mu}\right)\left(x_{1}\right)...O_{n}\left(A_{\mu}\right)\left(x_{n}\right)\right)\left|0\right\rangle$
$\displaystyle=\left\langle
0\right|T\left(O_{1}\left(A_{\mu}\right)\left(x_{1}\right)...O_{n}\left(A_{\mu}\right)\left(x_{n}\right)\right)\left|0\right\rangle.$
(2.59)

Translated into words, that means all correlators involving the Jacobian
$J\left[A_{\mu},g\right]$ with gauge invariant operators are the same as those
involving the identity. Thus, in the physical Hilbert space of the theory,
both operators are the same.

This argument does not generalize to arbitrary operators that are not gauge-
invariant, as required to recover the entire Hilbert space. However, an
explicit calculation can solve this problem. Let us use the usual prescription
of defining the bosonic measure through a complete set of orthonormal
eigenfunctions $\left\\{\phi_{n}\right\\}$ of a hermitian operator $\bar{D}$:

$\bar{D}\phi_{n}=\lambda_{n}\phi_{n},$ (2.60)

with the conditions

$\int\mathrm{dx}\text{ }\phi_{n}^{{\dagger}}\phi_{m}=\delta_{nm}\text{ and
}\sum_{n}\phi_{n}\left(x\right)\phi_{n}^{{\dagger}}\left(y\right)=\delta\left(x-y\right).$
(2.61)

Posteriorly to expanding the bosonic field, we build the connection with the
measure as follows

$A_{\mu}^{a}\left(x\right)=\sum_{n}a_{\mu,n}^{a}\phi_{n}\left(x\right)\rightarrow
dA_{\mu}=\prod_{a,\mu,n}da_{\mu,n}^{a}.$ (2.62)

Next, we put the changed field into this prescription. By introducing
coefficients $\bar{a}$ to the new expansion, let us rewrite the infinitesimal
gauge transformation (2.7):

$\displaystyle A_{\mu}^{g}$
$\displaystyle=\sum_{n}\bar{a}_{\mu,n}^{a}T_{a}\phi_{n}\left(x\right)=\sum_{n}a_{\mu,n}^{a}T_{a}\phi_{n}\left(x\right)-\frac{i}{e}\mathcal{D}_{\mu}\theta$
$\displaystyle=\left[\sum_{n}\left(a_{\mu,n}^{a}+ia_{\mu,n}^{b}f_{abc}\theta^{c}\right)\phi_{n}\left(x\right)-\frac{i}{e}\partial_{\mu}\theta^{a}\right]T_{a}.$
(2.63)

Then, after decomposing parameters $\theta^{a}$ in terms of the same
eigenfunctions of $\bar{D}$

$-\frac{i}{e}\partial_{\mu}\theta^{a}\left(x\right)=\sum_{n}\tilde{a}_{\mu,n}^{a}\phi_{n}\left(x\right),$
(2.64)

obtaining a transformation rule to coefficients is feasible

$\bar{a}_{\mu,n}^{a}=\sum_{m}\left(\delta_{ab}\delta_{nm}+\int\mathrm{dx}\text{
}\phi_{n}^{{\dagger}}\left(x\right)if_{abc}\theta^{c}\left(x\right)\phi_{m}\left(x\right)\right)a_{\mu,m}^{b}+\tilde{a}_{\mu,n}^{a}.$
(2.65)

That reflects on the transformation linked to the bosonic measure

$\prod_{a,\mu,n}d\bar{a}_{\mu,n}^{a}=\det\left[\delta_{ab}\delta_{nm}+\int\mathrm{dx}\text{
}\phi_{n}^{{\dagger}}\left(x\right)if_{abc}\theta^{c}\left(x\right)\phi_{m}\left(x\right)\right]\prod_{a,\mu,n}da_{\mu,n}^{a},$
(2.66)

where the term $\tilde{a}_{\mu,n}^{a}$ does not contribute because of the
translational invariance of each measure $da_{\mu,n}^{a}$.

Following the steps of Fujikawa [17], we get the expression for the Jacobian:

$J\left[A_{\mu},\theta\right]=\exp\left[\sum_{n}\left(\text{tr}\int\mathrm{dx}\text{
}\phi_{n}^{{\dagger}}\left(x\right)if_{abc}\theta^{c}\left(x\right)\phi_{n}\left(x\right)\right)\right].$
(2.67)

This trace acts over Lie algebra indices, which cancels out the total
antisymmetric structure constant $f_{abc}$. Meanwhile, we recognize the
product of fields taken at the same point
$\phi_{n}\left(x\right)\phi_{n}^{{\dagger}}\left(x\right)$. When putting both
pieces of information together, it is easy to see that the Jacobian expression
is indefinite:

$\displaystyle\sum_{n}\left(\text{tr}\int\mathrm{dx}\text{
}\phi_{n}^{{\dagger}}\left(x\right)if_{abc}\theta^{c}\left(x\right)\phi_{n}\left(x\right)\right)$
$\displaystyle=\text{tr}\int\mathrm{dx}\text{
}if_{abc}\theta^{c}\left(x\right)\sum_{n}\phi_{n}\left(x\right)\phi_{n}^{{\dagger}}\left(x\right)$
$\displaystyle=\int\mathrm{dx}\text{
}if_{aac}\theta^{c}\left(x\right)\delta\left(0\right)=0\times\infty.$ (2.68)

Thus, let us regularize this object by introducing eigenvalues of the operator
$\bar{D}$ as

$\displaystyle J\left[A_{\mu},\theta\right]$
$\displaystyle\equiv\exp\left[\lim_{M^{2}\rightarrow\infty}\sum_{n}\left(\text{tr}\int\mathrm{dx}\text{
}\phi_{n}^{{\dagger}}\left(x\right)if_{abc}\theta^{c}\left(x\right)\exp\left(-\frac{\lambda_{n}^{2}}{M^{\alpha}}\right)\phi_{n}\left(x\right)\right)\right]$
$\displaystyle=\exp\left[\lim_{M^{2}\rightarrow\infty}\sum_{n}\left(\text{tr}\int\mathrm{dx}\text{
}\phi_{n}^{{\dagger}}\left(x\right)if_{abc}\theta^{c}\left(x\right)\exp\left(-\frac{\bar{D}^{2}}{M^{\alpha}}\right)\phi_{n}\left(x\right)\right)\right],$
(2.69)

where $\alpha$ is chosen so the exponential argument is dimensionless.

The choice of operator $\bar{D}$ usually considers the requisites of naturally
appearing in the theory, being gauge invariant, and having real eigenvalues.
Furthermore, our choice of coefficients $a_{\mu,n}^{a}$ carrying all the
dependence on $\mu$ and $a$ implies that the $\phi_{n}$ must be eigenfunctions
of a scalar colorless operator; therefore, a good choice is

$\bar{D}=\text{tr}\left(\mathcal{D}_{\mu}\mathcal{D}^{\mu}\right),$ (2.70)

where the trace is taken only over color indices. We see that the sum is
regularized under these conditions, so proceeding with the evaluation of the
Jacobian is possible. Since no additional dependence on color indices comes
from the exponential argument $\bar{D}^{2}/M^{4}$, the trace can be
immediately taken, yielding the unity

$\displaystyle J\left[A_{\mu},\theta\right]$
$\displaystyle=\exp\left[\lim_{M^{2}\rightarrow\infty}\sum_{n}\left(if_{aac}\int\mathrm{dx}\text{
}\phi_{n}^{{\dagger}}\left(x\right)\theta^{c}\left(x\right)\exp\left(-\frac{\bar{D}^{2}}{M^{4}}\right)\phi_{n}\left(x\right)\right)\right]$
$\displaystyle=\exp\left(0\right)=1.$ (2.71)

Such a result accomplishes our objective of furnishing proof for the
invariance of the bosonic measure. Of course, one could choose other
strategies so a result different from 1 could arise. Nevertheless, the “gauge
anomaly” coming from this “non-trivial” Jacobian could be removed by an
adequate choice of counterterms. To say this more precisely, we can use what
we know from the fact that Yang-Mills theories are renormalizable. In fact, ’t
Hooft’s proof [20] shows that it is possible to preserve gauge invariance at
every order in perturbation theory, which is crucial for demonstrating that
the theory is renormalizable. Algebraic renormalization results confirm this
by noticing that the cohomology of the Slavnov-Taylor operator is trivial for
a Yang-Mills theory [21]. Then, even if we would regularize the theory with
non-gauge invariant regulators (obtaining a non-trivial Jacobian), a change in
the renormalization scheme could restore gauge invariance and set the Jacobian
as the unity.

The results in this chapter are the main part of our published work [22].

###  2.5 Final Remarks and Conclusions

In the second chapter, we checked aspects related to gauge symmetry
maintenance in gauge theories. At the classical level, current conservation
arose after implementing local invariance in the theory action. Equations of
motion for fermion fields were necessary to achieve this result. This part of
the discussion established a route to follow in the quantum theory.

With this in mind, it would be reasonable for Dyson-Schwinger equations to
play a role in the current conservation due to their analogy with classical
equations of motion. It would be a consequence of the translational invariance
of the fermion measure, which is a condition to obtain the mentioned
equations. Nevertheless, we saw that gauge invariance of the fermion measure
is the new requirement for conservation.

Once the panorama was clear, we focused on gauge-anomalous theories. For them,
considering external gauge fields, the presence of a Jacobian to the fermion
measure implies a non-zero result to the expectation value of the covariant
derivative of the current (the anomaly). We saw that, when quantizing the
gauge field, the expectation value vanishes in a simple way. This outcome is a
direct consequence of considering the boson measure invariant, and the
properties of the fermion measure were unnecessary. There is no gauge anomaly
preventing current conservation in the fully quantized theory. That does not
affect the topological interpretation of the gauge anomaly since it is present
when we do not consider the integration on the gauge field.

Although our argumentation depends on gauge measure invariance, we took this
property for granted. That is usual in the literature but not explicitly
proved. This proof was achieved by G. de Lima e Silva, T.J. Girardi, and S. A.
Dias and published in reference [22]. Such a result completes the theoretical
setup for our claim that the vacuum expectation value of the gauge anomaly
vanishes. The natural course of this investigation is to define a chiral
theory perturbatively, aiming at a detailed analysis of its renormalizability
and unitarity.

##  Chapter 3 Anomalies in Fermionic Amplitudes

This chapter refers to another line of investigation in this thesis, which
concerns the occurrence of anomalies in fermionic amplitudes. As mentioned,
the single axial triangle ($AVV$) establishes the foundations for this
analysis. Although this process is largely explored in the literature, our
perspective shows new aspects of anomalies while emphasizing patterns related
to their tensor structures. The single axial box ($AVVV$) exhibits similar
elements in a more complex scene, substantiating this investigation.

Both correlators depend on traces involving the chiral matrix, which lead to
products between the Levi-Civita symbol and metric tensors. In addition to its
manifestation in anomalous amplitudes, this type of structure is common in
chiral theories and investigations developed in odd space-time dimensions.
That is part of the motivation for this work and emphasizes the significance
of mathematical resources developed throughout our calculations.

Integrals in perturbative calculus usually exhibit diverging content, which
requires using regularization techniques in intermediate steps of calculations
[23]. These prescriptions make mathematical structures finite, so
manipulations problematic to the original expressions become valid. That
implies modifying amplitudes through the introduction of non-physical
parameters. Results independent of regularizations emerge after
renormalization [24]. Then, establishing predictions to compare with
experimental data becomes feasible.

Choosing a specific regularization scheme brings consequences to the
interpretation of results. To clarify this aspect, we get back to the
impossibility of preserving chiral and gauge symmetry simultaneously [4]. This
time, however, we emphasize the issue of the maintenance of Ward identities
for the single axial triangle. This amplitude unavoidably exhibits dependence
on a diverging surface term [28], so choosing a prescription that eliminates
this object preserves some Ward identities (but not all). Methods that allow
shifts in the integration variable accomplish this task, e.g., Dimensional
Regularization [25, 26, 27]. Other prescriptions do not lead to this outcome.

Even though there is an inclination towards preserving gauge symmetry, there
are other possibilities. The reason for such is the presence of divergent
Feynman integrals having a divergence degree higher than the logarithmic one.
For them, a shift in the integration variable requires compensation through
(non-zero) surface terms to maintain the connection with the original
expression [6, 28, 29]. That implies the existence of different versions for
perturbative contributions involving loops, which differ by these surface
terms after integration. This situation is a manifestation of internal momenta
arbitrariness, although they relate to external momenta through energy-
momentum conservation [30]. We will illustrate that choices occur when taking
Dirac traces, leading to one version with a specific behavior regarding
symmetries; i.e., choosing one form sets the position of violating terms
typical of anomalous amplitudes.

The mentioned aspects motivate the perspective adopted in this investigation
and, therefore, the employment of Implicit Regularization (IReg) [37]. Its
main feature is avoiding the evaluation of divergent structures. That means we
only integrate finite contributions without modifying ill-defined objects. Our
analysis falls on the accessible values for these divergences within final
expressions for amplitudes. We also avoid choices for the internal momenta,
adopting arbitrary routings for internal lines of the graphs. This
arbitrariness is intrinsic to the perturbative calculus and received attention
in recent works [31, 32, 33]. The study of schemes to compute traces involving
chiral matrices also received attention from the authors [34, 35, 36]. This
concept characterizes another class of possibilities for anomalous amplitudes.

The discussion is organized as follows. Section (3.1) introduces the model and
the correlators that concern this investigation. We also comment on
expectations about symmetries (through Ward identities) and their relation
with the linearity of integration. Section (3.2) looks into integrands of
amplitudes, highlighting tensor arrangements associated with structures that
compound the intended organization. This feature is part of the IReg,
approached in Section (3.3). We also introduce the elements used to describe
diverging quantities and finite functions. With our perspective clear, Section
(3.4) focus on the explicit integration of amplitudes while providing a
preliminary discussion of these results. A careful analysis occurs in Section
(3.5), where we inquire about relations involving amplitudes and the
consequences of different prescriptions to evaluate divergent objects. Section
(3.6) discusses important aspects of the investigation while presenting the
conclusions.

###  3.1 Model and Definitions

We consider a $\left(1+3\right)$-dimensional model where massive spin $1/2$
fields interact with different types of bosons. The corresponding
couplings111Although some couplings do not concern this investigation at first
glance, perturbative corrections bring all these possibilities. are listed in
the interacting Lagrangian

$\displaystyle\mathcal{L}_{I}$ $\displaystyle=$ $\displaystyle
e_{S}\left(\bar{\psi}\psi\right)\phi+e_{P}\left(\bar{\psi}\gamma_{5}\psi\right)\pi-
e_{V}\left(\bar{\psi}\gamma^{\mu}\psi\right)V_{\mu}$ (3.1) $\displaystyle-
e_{A}\left(\bar{\psi}\gamma_{5}\gamma^{\mu}\psi\right)A_{\mu}+e_{T}\left(\bar{\psi}\gamma_{5}\sigma^{\mu\nu}\psi\right)H_{\mu\nu},$

where elements belonging to the set
$\left\\{\phi,\pi,V_{\mu},A_{\mu},H_{\mu\nu}\right\\}$ are respectively
scalar, pseudoscalar, vector, axial, and pseudotensor boson fields, while
$\psi$ corresponds to Dirac fermions. As coupling constants
$\left\\{e_{S},e_{P},e_{V},e_{A},e_{T}\right\\}$ do not concern the intended
discussion, we set them as the unity.

The remaining structures emerge in the context of the four-dimensional
Clifford algebra. The objects $\gamma^{\mu}$ are Dirac matrices, whose
commutator is denoted as
$\left[\gamma^{\mu},\gamma^{\nu}\right]=2\sigma^{\mu\nu}$. Since establishing
a chiral matrix that anticommutes with all gamma matrices is feasible in even
dimensions, we introduce the definition employed within this context
$\gamma_{5}=\frac{i}{4!}\varepsilon_{\mu\nu\alpha\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}$.
Even though omitted, the identity $\boldsymbol{1}$ appears within the scalar
coupling.

Those structures in parentheses within the Lagrangian correspond to Noether
currents, which couple to boson fields. Current conservation establishes
relations involving these quantities. Although violations are expected for
anomalous amplitudes, we discuss preliminary expectations here. In a case
involving fermions with different masses, the vector current divergence would
be proportional to the scalar one with a coefficient depending on the
difference between masses. Nevertheless, the vector current is conserved as we
delimit this investigation to the equal masses context

$\partial_{\mu}\left(\bar{\psi}\gamma^{\mu}\psi\right)=0.$ (3.2)

That suggests implications at the quantum level through Ward identities for
correlators involving vector vertices. The result should vanish whenever we
contract an external momentum with an index corresponding to this vertex type.
On the other hand, the axial current divergence is classically proportional to
the pseudoscalar one

$\partial_{\mu}\left(\bar{\psi}\gamma_{5}\gamma^{\mu}\psi\right)=2m\left(\bar{\psi}\gamma_{5}\psi\right).$
(3.3)

Such relation leads to Ward identities involving similar amplitudes that
differ by the corresponding vertices. Establishing an analogous association
involving the pseudotensor current is not possible.

Our objective is on the next-to-leading order corrections for processes
involving external bosons, which produces purely fermionic loops. We introduce
them in two steps. First, we employ Feynman rules to construct graphs for a
single value of the unrestricted (loop) momentum. Hence, we inspect them and
survey expectations without worrying about ill-defined mathematical
quantities. This problem arises when implementing the last Feynman rule, which
consists of momenta integration. We only consider this operation (in the
second step) after discussing a strategy to deal with the mentioned problem.
Upper and lower case letters distinguish these two versions of amplitudes

$T^{\Gamma_{i}\Gamma_{j}\cdots\Gamma_{l}}=\int\frac{d^{4}k}{\left(2\pi\right)^{4}}t^{\Gamma_{i}\Gamma_{j}\cdots\Gamma_{l}}.$
(3.4)

Such notation is extended to other integrals that emerge throughout this work.

The general form of amplitudes for a single value of the loop momentum is

$\displaystyle
t^{\Gamma_{i}\Gamma_{j}\cdots\Gamma_{l}}\left(k_{1},k_{2},\ldots,k_{n}\right)$
(3.5) $\displaystyle=$
$\displaystyle\text{tr}\left\\{\Gamma_{i}\left[S_{F}\left(k+k_{1};m\right)\right]\Gamma_{j}\left[S_{F}\left(k+k_{2};m\right)\right]\cdots\Gamma_{l}\left[S_{F}\left(k+k_{n};m\right)\right]\right\\},$

whose argument is omitted unless it associates with configurations different
from $\left(k_{1},k_{2},\ldots,k_{n}\right)$. This structure depends on
fermion propagators $S_{F}$ and vertex operators $\Gamma_{l}$. We express the
propagator of a Dirac fermion carrying momentum $K_{n}=k+k_{n}$ and mass $m$
through the structure

$S_{F}\left(k+k_{n},m\right)=\frac{1}{\not{F}_{n}}=\frac{1}{\left(\not{k}+\not{k}_{n}\right)-m}=\frac{\left(\not{k}+\not{k}_{n}\right)+m}{D_{n}}.$
(3.6)

Although we use the form $\not{F}_{n}^{-1}$ to introduce perturbative
amplitudes and derive relations among them, employing the denominator
$D_{n}=\left(k+k_{n}\right)^{2}-m^{2}$ is useful to the integration. Due to
the adopted simplifications, vertices have the following structures

$\Gamma_{l}=\left\\{\Gamma_{S},\Gamma_{P},\Gamma_{V},\Gamma_{A},\Gamma_{\tilde{T}}\right\\}=\left\\{\boldsymbol{1},\gamma_{5},\gamma^{\mu},\gamma^{\mu}\gamma_{5},\gamma_{5}\sigma^{\mu\nu}\right\\}.$
(3.7)

Capital Latin subindices denote the nature of each object. They correspond
respectively to scalar, pseudoscalar, vector, axial, and pseudotensor
vertices. We extend this notation to perturbative amplitudes, where these
labels indicate the vertex content and the specific position of each operator.

Loop corrections to processes involving two, three, and four external bosons
arise within this discussion. The Figure 3.1 shows representations through
Feynman diagrams associated with these amplitudes. We have yet to specify the
vertex content, but the momenta configuration is set. Although routings
$k_{i}$ do not have physical meaning by themselves, conservation laws on the
vertices connect external (physical) momenta with differences between
routings. The conventions adopted allow summarizing these relations into the
object $p_{i}=k_{1}-k_{i}$, whose accessible values are the following

$\begin{array}[]{ccc}p_{2}=k_{1}-k_{2}=p,&p_{3}=k_{1}-k_{3}=q,&p_{4}=k_{1}-k_{4}=r.\end{array}$

In order to proceed with definitions, we must cast the processes that concern
this investigation (by setting the vertex configurations). This subject is
covered in the sequence.

Figure 3.1: One-loop corrections to processes described by external bosons.

####  3.1.1 Perturbative Amplitudes

Since resources required to construct any fermionic amplitude are at our
disposal, let us delimit those of interest and how they relate to each other.
We consider constraints coming from their mathematical structure and symmetry
implications in this process.

The neutral pion decay in two photons has a remarkable role in studies on
anomalies, so we take the single axial triangle amplitude as the first
laboratory. It is described by one axial and two vector vertices, assuming the
form

$t_{\mu\nu\alpha}^{AVV}=\text{tr}\left\\{\gamma_{\mu}\gamma_{5}\frac{1}{\not{F}_{1}}\gamma_{\nu}\frac{1}{\not{F}_{2}}\gamma_{\alpha}\frac{1}{\not{F}_{3}}\right\\}.$
(3.8)

From now on, we use labels to refer to a particular perturbative amplitude,
i.e., we designate this one as $AVV$. Within the IReg perspective, the
systematization of results highlights certain features concerning the
mathematical structure of the $AVV$ that relate to its anomalous character.
They motivate us to pursue one higher-order amplitude exhibiting similar
elements: the single axial box amplitude

$t_{\mu\nu\alpha\beta}^{AVVV}=\text{tr}\left\\{\gamma_{\mu}\gamma_{5}\frac{1}{\not{F}_{1}}\gamma_{\nu}\frac{1}{\not{F}_{2}}\gamma_{\alpha}\frac{1}{\not{F}_{3}}\gamma_{\beta}\frac{1}{\not{F}_{4}}\right\\},$
(3.9)

denominated $AVVV$. Even though its evaluation is complex, all operations
involved are analogous to those performed in the triangle context. Thus, we
consider the first process as a guide for analyzing the second.

These amplitudes are the central elements of this work. Nevertheless, as
acknowledged in the discussion about Noether currents (3.2)-(3.3), relations
among amplitudes could be derived through contractions with the external
momenta. Thus, we explore this operation for the integrands above to introduce
the remaining correlators while discussing potential constraints on the
results.

For such purpose, let us express contractions involving physical momenta and
Dirac matrices in terms of fermion propagators (3.6):

$\not{k}_{i}-\not{k}_{j}=\not{F}_{i}-\not{F}_{j}.$ (3.10)

Now, consider the specific contraction on the index associated with the first
vector vertex of the triangle amplitude. Posteriorly to the implementation of
this identity, trace linearity leads to the result

$p^{\nu}t_{\mu\nu\alpha}^{AVV}=\text{tr}\left\\{\gamma_{\mu}\gamma_{5}\frac{1}{\not{F}_{2}}\gamma_{\alpha}\frac{1}{\not{F}_{3}}\right\\}-\text{tr}\left\\{\gamma_{\mu}\gamma_{5}\frac{1}{\not{F}_{1}}\gamma_{\alpha}\frac{1}{\not{F}_{3}}\right\\},$

where a difference between $AV$ two-point amplitudes is identified

$p^{\nu}t_{\mu\nu\alpha}^{AVV}=t_{\mu\alpha}^{AV}\left(k_{2},k_{3}\right)-t_{\mu\alpha}^{AV}\left(k_{1},k_{3}\right).$
(3.11)

An analogous relation arises for the second vector vertex through the same
steps

$\left(q-p\right)^{\alpha}t_{\mu\nu\alpha}^{AVV}=t_{\mu\nu}^{AV}\left(k_{1},k_{3}\right)-t_{\mu\nu}^{AV}\left(k_{1},k_{2}\right).$
(3.12)

As for contractions with axial vertices, we multiply the identity (3.10) by
the chiral matrix. Permuting its position is necessary to allow
identifications

$\left(\not{k}_{i}-\not{k}_{j}\right)\gamma_{5}=\not{F}_{i}\gamma_{5}+\gamma_{5}\not{F}_{j}+2m\gamma_{5}.$
(3.13)

Besides the difference between $AV$ amplitudes, one additional term
corresponds to the $PVV$ amplitude

$q^{\mu}t_{\mu\nu\alpha}^{AVV}=t_{\nu\alpha}^{AV}\left(k_{2},k_{3}\right)-t_{\alpha\nu}^{AV}\left(k_{1},k_{2}\right)-2mt_{\nu\alpha}^{PVV}.$
(3.14)

Concerning the $AVVV$ box amplitude, contractions follow the same procedure
and yield the results:

$\displaystyle r^{\mu}t_{\mu\nu\alpha\beta}^{AVVV}$ $\displaystyle=$
$\displaystyle
t_{\nu\alpha\beta}^{AVV}\left(k_{2},k_{3},k_{4}\right)-t_{\beta\nu\alpha}^{AVV}\left(k_{1},k_{2},k_{3}\right)-2mt_{\nu\alpha\beta}^{PVVV},$
(3.15) $\displaystyle p^{\nu}t_{\mu\nu\alpha\beta}^{AVVV}$ $\displaystyle=$
$\displaystyle
t_{\mu\alpha\beta}^{AVV}\left(k_{2},k_{3},k_{4}\right)-t_{\mu\alpha\beta}^{AVV}\left(k_{1},k_{3},k_{4}\right),$
(3.16) $\displaystyle\left(q-p\right)^{\alpha}t_{\mu\nu\alpha\beta}^{AVVV}$
$\displaystyle=$ $\displaystyle
t_{\mu\nu\beta}^{AVV}\left(k_{1},k_{3},k_{4}\right)-t_{\mu\nu\beta}^{AVV}\left(k_{1},k_{2},k_{4}\right),$
(3.17) $\displaystyle\left(r-q\right)^{\beta}t_{\mu\nu\alpha\beta}^{AVVV}$
$\displaystyle=$ $\displaystyle
t_{\mu\nu\alpha}^{AVV}\left(k_{1},k_{2},k_{3}\right)-t_{\mu\nu\alpha}^{AVV}\left(k_{1},k_{2},k_{4}\right).$
(3.18)

Although all operations lead to the difference between $AVV$ triangles, the
$PVVV$ four-point amplitude appears as the extra contribution in the axial
contraction.

Obtaining these relations considers only the mathematical structure of
integrands, which consist of identities at this level. Their validity after
integration represents a manifestation of linearity. Nonetheless, we will see
that the anomalous character of involved amplitudes might affect these
prospects. Then, if their verification is successful, proper relations among
Green functions (GF) are established. Expectations for contractions with the
$AVV$ triangle are the following

$\displaystyle q^{\mu}T_{\mu\nu\alpha}^{AVV}$ $\displaystyle\rightarrow$
$\displaystyle
T_{\nu\alpha}^{AV}\left(k_{2},k_{3}\right)-T_{\alpha\nu}^{AV}\left(k_{1},k_{2}\right)-2mT_{\nu\alpha}^{PVV},$
(3.19) $\displaystyle p^{\nu}T_{\mu\nu\alpha}^{AVV}$
$\displaystyle\rightarrow$ $\displaystyle
T_{\mu\alpha}^{AV}\left(k_{2},k_{3}\right)-T_{\mu\alpha}^{AV}\left(k_{1},k_{3}\right),$
(3.20) $\displaystyle\left(q-p\right)^{\alpha}T_{\mu\nu\alpha}^{AVV}$
$\displaystyle\rightarrow$ $\displaystyle
T_{\mu\nu}^{AV}\left(k_{1},k_{3}\right)-T_{\mu\nu}^{AV}\left(k_{1},k_{2}\right),$
(3.21)

while contractions involving the $AVVV$ box yield

$\displaystyle r^{\mu}T_{\mu\nu\alpha\beta}^{AVVV}$ $\displaystyle\rightarrow$
$\displaystyle
T_{\nu\alpha\beta}^{AVV}\left(k_{2},k_{3},k_{4}\right)-T_{\beta\nu\alpha}^{AVV}\left(k_{1},k_{2},k_{3}\right)-2mT_{\nu\alpha\beta}^{PVVV},$
(3.22) $\displaystyle p^{\nu}T_{\mu\nu\alpha\beta}^{AVVV}$
$\displaystyle\rightarrow$ $\displaystyle
T_{\mu\alpha\beta}^{AVV}\left(k_{2},k_{3},k_{4}\right)-T_{\mu\alpha\beta}^{AVV}\left(k_{1},k_{3},k_{4}\right),$
(3.23) $\displaystyle\left(q-p\right)^{\alpha}T_{\mu\nu\alpha\beta}^{AVVV}$
$\displaystyle\rightarrow$ $\displaystyle
T_{\mu\nu\beta}^{AVV}\left(k_{1},k_{3},k_{4}\right)-T_{\mu\nu\beta}^{AVV}\left(k_{1},k_{2},k_{4}\right),$
(3.24) $\displaystyle\left(r-q\right)^{\beta}T_{\mu\nu\alpha\beta}^{AVVV}$
$\displaystyle\rightarrow$ $\displaystyle
T_{\mu\nu\alpha}^{AVV}\left(k_{1},k_{2},k_{3}\right)-T_{\mu\nu\alpha}^{AVV}\left(k_{1},k_{2},k_{4}\right).$
(3.25)

Previously, we stated that current conservation (3.2)-(3.3) generates
implications over quantum corrections. Ward identities (WIs) relate to momenta
contractions over perturbative amplitudes. In the hypothesis that one relation
among GF applies, the maintenance of the corresponding WI requires the
cancellation of differences between amplitudes above ($AV$s in the first set
and $AVV$s in the second). We cast these expectations in the sequence, where
the required sum of channels is implicit in the notation $\mathcal{T}$.
Nevertheless, we will see that our analysis applies channel by channel. The
identities for the $AVV$ amplitude are

$\displaystyle q^{\mu}\mathcal{T}_{\mu\nu\alpha}^{AVV}$
$\displaystyle\rightarrow$ $\displaystyle-2m\mathcal{T}_{\nu\alpha}^{PVV},$
(3.26) $\displaystyle p^{\nu}\mathcal{T}_{\mu\nu\alpha}^{AVV}$
$\displaystyle\rightarrow$ $\displaystyle 0,$ (3.27)
$\displaystyle\left(q-p\right)^{\alpha}\mathcal{T}_{\mu\nu\alpha}^{AVV}$
$\displaystyle\rightarrow$ $\displaystyle 0,$ (3.28)

while those for the $AVVV$ amplitude are

$\displaystyle r^{\mu}\mathcal{T}_{\mu\nu\alpha\beta}^{AVVV}$
$\displaystyle\rightarrow$
$\displaystyle-2m\mathcal{T}_{\nu\alpha\beta}^{PVVV},$ (3.29) $\displaystyle
p^{\nu}\mathcal{T}_{\mu\nu\alpha\beta}^{AVVV}$ $\displaystyle\rightarrow$
$\displaystyle 0,$ (3.30)
$\displaystyle\left(q-p\right)^{\alpha}\mathcal{T}_{\mu\nu\alpha\beta}^{AVVV}$
$\displaystyle\rightarrow$ $\displaystyle 0,$ (3.31)
$\displaystyle\left(r-p\right)^{\beta}\mathcal{T}_{\mu\nu\alpha\beta}^{AVVV}$
$\displaystyle\rightarrow$ $\displaystyle 0.$ (3.32)

Given the impossibility of simultaneous satisfaction of gauge and axial
symmetries, these are also preliminary prospects.

Through this argumentation, we connected concepts of integral linearity and
symmetry implications. If relations among GF are identically satisfied,
canceling those differences on their right-hand side also satisfies WIs.
Nevertheless, the fact that these amplitudes exhibit diverging power counting
is problematic when testing these expectations. That is particularly important
in the anomalies context. We will return to this discussion after exploring
the perturbative amplitudes at the integrand level.

###  3.2 Structure of Perturbative Amplitudes

This work implements Feynman rules in two parts, starting with obtaining
perturbative amplitudes for a single value of the unrestricted (loop)
momentum. Thus, organizing and examining their content without worries about
the divergences that come with integration is attainable. We begin by
introducing an example illustrating the elements required for this task.
Subsequently, we inquire about two, three, and four-point functions concerning
this investigation.

####  3.2.1 Two-Point Amplitudes - Preliminary Notions

This analysis uses a simple example to familiarize with calculations while
producing tools for more complex scenes. Soon we will come across extensive
mathematical expressions that might seem vague. Thereby, designing mechanisms
to compact them and systematizing operations is part of our task.

The next-to-leading order correction to processes involving external bosons
corresponds to pure fermionic loops. We denoted these amplitudes using
uppercase letters (3.4), while their integrands use lowercase letters (3.5).
These structures contain traces of vertex operators $\Gamma_{i}$ and fermion
propagators $\not{F}_{n}^{-1}$, as seen in the example of two-point functions:

$t^{\Gamma_{i}\Gamma_{j}}=\text{tr}\left(\Gamma_{i}\frac{1}{\not{F}_{1}}\Gamma_{j}\frac{1}{\not{F}_{2}}\right).$
(3.33)

After rewriting the propagator (3.6), using the linearity of the trace makes
its matrix content explicit

$\displaystyle t^{\Gamma_{i}\Gamma_{j}}$ $\displaystyle=$
$\displaystyle\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\gamma_{B}\right)\frac{K_{1}^{A}K_{2}^{B}}{D_{12}}+m^{2}\text{tr}\left(\Gamma_{i}\Gamma_{j}\right)\frac{1}{D_{12}}$
(3.34)
$\displaystyle+m\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\right)\frac{K_{1}^{A}}{D_{12}}+m\text{tr}\left(\Gamma_{i}\Gamma_{j}\gamma_{B}\right)\frac{K_{2}^{B}}{D_{12}}.$

As several notations appear within this context, let us explain them
subsequently. We introduced compact products as that of the denominator
$D_{ij}=D_{i}D_{j}$ for propagator-like objects
$D_{i}=\left(k+k_{i}\right)^{2}-m^{2}$. Our goal in this section is to express
integrands through combinations depending on these structures

$\frac{1}{D_{i}},\frac{\left[1,k_{\mu},k_{\mu\nu}\right]}{D_{ij}},\frac{\left[1,k_{\mu},k_{\mu\nu},k_{\mu\nu\alpha}\right]}{D_{ijk}},\frac{\left[1,k_{\mu},k_{\mu\nu},k_{\mu\nu\alpha},k_{\mu\nu\alpha\beta}\right]}{D_{ijkl}},$

which leads to identifying Feynman integrals in Section (3.3). That means the
usage of the symbol $K_{i}=k+k_{i}$ is limited to the current analysis, being
another artifice to reduce expressions. We also introduced compact notations
for products of momenta or routings:

$\begin{array}[]{ccccccc}k_{\mu\nu}=k_{\mu}k_{\nu},&&&p_{\mu\nu}=p_{\mu}p_{\nu},&&&k_{1\mu\nu}=k_{1\mu}k_{1\nu}.\end{array}$

The second type of notation consists of (the possibility of) adopting
uppercase Latin letters for summed indices and neglecting their covariant or
contravariant character. This resource facilitates the recognition of sectors
with analogous index configurations inside tensor amplitudes, making
substructures promptly noticeable. Hence, identifying other amplitudes inside
the original only requires sign comparisons among options. Furthermore, other
terms receive a suitable organization through standard tensors. We also use
this notation to emphasize symmetry properties.

Since we know these tools and ideas, we implement them in the mentioned
example. It consists of the double-vector function $VV$, which associates with
the photon self-energy in the Quantum Electrodynamics context. The replacement
of Dirac matrices as vertex operators ($\Gamma_{i}=\gamma_{\mu}$ and
$\Gamma_{j}=\gamma_{\nu}$) on the integrand above generates the expression

$\displaystyle t_{\mu\nu}^{VV}$ $\displaystyle=$
$\displaystyle\text{tr}\left(\gamma_{\mu}\gamma_{A}\gamma_{\nu}\gamma_{B}\right)\frac{K_{1}^{A}K_{2}^{B}}{D_{12}}+m^{2}\text{tr}\left(\gamma_{\mu}\gamma_{\nu}\right)\frac{1}{D_{12}}$
(3.35)
$\displaystyle+m\text{tr}\left(\gamma_{\mu}\gamma_{A}\gamma_{\nu}\right)\frac{K_{1}^{A}}{D_{12}}+m\text{tr}\left(\gamma_{\mu}\gamma_{\nu}\gamma_{B}\right)\frac{K_{2}^{B}}{D_{12}}.$

Even though Dirac traces are common ingredients, we discuss them to ground
future calculations. The property of anticommutation followed by Dirac
matrices is the outset

$\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2g_{\mu\nu}.$ (3.36)

By taking the trace on both sides, linearity and invariance under cyclic
permutations lead to the equation

$\text{tr}\left(\gamma_{\mu}\gamma_{\nu}\right)=g_{\mu\nu}\text{tr}\left(\mathbf{1}\right)=4g_{\mu\nu}.$
(3.37)

Any other trace involving an even number of Dirac matrices could be reduced to
this one. For instance, we use the anticommutation property to express the
four matrices trace as the following combination

$\displaystyle\text{tr}\left(\gamma_{\mu}\gamma_{A}\gamma_{\nu}\gamma_{B}\right)$
$\displaystyle=$ $\displaystyle\text{tr}\left(2g_{\mu
A}\gamma_{\nu}\gamma_{B}-2g_{\mu\nu}\gamma_{A}\gamma_{B}+2g_{\mu
B}\gamma_{A}\gamma_{\nu}-\gamma_{A}\gamma_{\nu}\gamma_{B}\gamma_{\mu}\right)$
(3.38) $\displaystyle=$ $\displaystyle 4g_{\mu A}g_{\nu
B}-4g_{\mu\nu}g_{AB}+4g_{\mu B}g_{A\nu}.$

As for products involving an odd number of Dirac matrices, trace operation
vanishes. To prove this statement, introduce the identity
$\mathbf{1}=\gamma_{5}^{2}$ inside the argument. Using (respectively) the fact
that the chiral matrix anticommutes with any Dirac matrix and the cyclicity,
we show that these traces are equal to their negative and, therefore, vanish.
To illustrate, take the trace of one single Dirac matrix

$\text{tr}\left(\gamma_{\mu}\right)=\text{tr}\left(\gamma_{5}\gamma_{5}\gamma_{\mu}\right)=-\text{tr}\left(\gamma_{5}\gamma_{\mu}\gamma_{5}\right)=-\text{tr}\left(\gamma_{5}\gamma_{5}\gamma_{\mu}\right)=-\text{tr}\left(\gamma_{\mu}\right).$

When replacing these results on the $VV$ amplitude and rearranging it, the
sorting of free indices shows two sectors

$t_{\mu\nu}^{VV}=4\frac{K_{1\mu}K_{2\nu}+K_{1\nu}K_{2\mu}}{D_{12}}+g_{\mu\nu}\left[-\text{tr}\left(\gamma_{A}\gamma_{B}\right)\frac{K_{1}^{A}K_{2}^{B}}{D_{12}}+m^{2}\text{tr}\left(\mathbf{1}\right)\frac{1}{D_{12}}\right].$
(3.39)

The first corresponds to the symmetric version of the following standard
tensor

$t_{2\mu\nu}^{\left(s\right)}\left(k_{i},k_{j}\right)=\frac{\left(k+k_{i}\right)_{\mu}\left(k+k_{j}\right)_{\nu}+s\left(k+k_{j}\right)_{\mu}\left(k+k_{i}\right)_{\nu}}{D_{12}}.$
(3.40)

This general definition admits a numerical subindex, characterizing the number
of propagator-like objects in the denominator (two in this case
$D_{12}=D_{1}D_{2}$), and it allows different signs $s=\pm 1$. Since this
expression is a combination of structures previously mentioned, it does not
require further analysis.

As for the sector proportional to the metric tensor $g_{\mu\nu}$, we
recognized traces involving fewer matrices. They associate with a scalar
amplitude from two possibilities: $SS$ and $PP$. Thus, replace the
corresponding vertices on Equation (3.34) to determine their integrands:

$\displaystyle t^{SS}$ $\displaystyle=$
$\displaystyle\text{tr}\left(\gamma_{A}\gamma_{B}\right)\frac{K_{1}^{A}K_{2}^{B}}{D_{12}}+m^{2}\text{tr}\left(\mathbf{1}\right)\frac{1}{D_{12}},$
(3.41) $\displaystyle t^{PP}$ $\displaystyle=$
$\displaystyle-\text{tr}\left(\gamma_{A}\gamma_{B}\right)\frac{K_{1}^{A}K_{2}^{B}}{D_{12}}+m^{2}\text{tr}\left(\mathbf{1}\right)\frac{1}{D_{12}}.$
(3.42)

Since we did not rename any index, the precise identification occurs by
comparing signs, and we achieve the organization

$t_{\mu\nu}^{VV}=4t_{2\mu\nu}^{\left(+\right)}\left(k_{1},k_{2}\right)+g_{\mu\nu}t^{PP}.$
(3.43)

Exploring the $PP$ structure is still necessary, so we draw attention to its
dependence on the objects

$2K_{ij}\rightarrow 2\left(K_{i}\cdot
K_{j}-m^{2}\right)=D_{i}+D_{j}-\left(k_{i}-k_{j}\right)^{2}.$ (3.44)

This identity brings propagator-like objects to numerators, which reflects on
reductions of denominators within the amplitude integrand

$t^{PP}=-2\left[\frac{1}{D_{1}}+\frac{1}{D_{2}}-p^{2}\frac{1}{D_{12}}\right],$
(3.45)

where we identified the external momentum $p=k_{1}-k_{2}$. The recurrent
application of this resource throughout this investigation justifies generic
indices. Notice that, with the momenta integration, this identity reduces part
of the Feynman integrals to those involving one less propagator.

We do not integrate these amplitudes in the future since they are not part of
this work. Even so, take them as a guide to calculations performed from now
on.

####  3.2.2 Two-Point Amplitudes - $AV$

Given the general expression for two-point amplitudes (3.34), we replace
vertex operators to write the integrand of the axial-vector amplitude

$\displaystyle t_{\mu\nu}^{AV}$ $\displaystyle=$
$\displaystyle\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{B}\right)\frac{K_{1}^{A}K_{2}^{B}}{D_{12}}+m^{2}\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{\nu}\right)\frac{1}{D_{12}}$
(3.46)
$\displaystyle+m\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{A}\gamma_{\nu}\right)\frac{K_{1}^{A}}{D_{12}}+m\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{\nu}\gamma_{B}\right)\frac{K_{2}^{B}}{D_{12}},$

where numerators depend on $K_{i}=k+k_{i}$ and denominators are
$D_{12}=D_{1}D_{2}$. We refer to this structure as $AV$, which specifies the
first vertex as an axial $\Gamma_{i}=\gamma_{\mu}\gamma_{5}$ and the second as
a vector $\Gamma_{j}=\gamma_{\nu}$. Although these traces contain the chiral
matrix, replacing its definition
$\gamma_{5}=\frac{i}{4!}\varepsilon_{\mu\nu\alpha\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}$
suppresses this dependence. That adds four extra Dirac matrices to the
argument while introducing a global factor through the Levi-Civita symbol.
Within this perspective, we must compute even traces following steps seen in
the previous subsection and then perform contractions.

Immediately, occurrences involving an odd number of Dirac matrices plus the
chiral one vanish. That also happens in the case involving two Dirac matrices
since it leads to contractions between symmetric and antisymmetric tensors.
Hence, the only non-zero trace involves four Dirac matrices, whose computation
leads to the Levi-Civita symbol

$\text{tr}\left(\gamma_{5}\gamma_{\mu}\gamma_{A}\gamma_{\nu}\gamma_{B}\right)=4i\varepsilon_{\mu
A\nu B}.$ (3.47)

When replacing it, symmetry properties allow identifying the antisymmetric
version of the standard tensor (3.40):

$t_{\mu\nu}^{AV}=2i\varepsilon_{\mu\nu
XY}t_{2XY}^{\left(-\right)}\left(k_{1},k_{2}\right).$ (3.48)

One would expect two ingredients to compound the integrated substructure:
metric tensor and external momentum $p=k_{1}-k_{2}$. Since they combine
exclusively into symmetric quantities ($g_{XY}$ and $p_{XY}=p_{X}p_{Y}$), the
contraction should cancel out. Nevertheless, two-point functions exhibit
quadratic power counting in the physical dimension. Therefore, these integrals
are not invariant under translations, admitting the emergence of non-physical
momenta associated with surface terms. That provides another vector to build
up the substructure: the sum of arbitrary routings $k_{1}+k_{2}$. Hence, we
expect the integrated $AV$ amplitude to have the following form

$T_{\mu\nu}^{AV}\rightarrow\varepsilon_{\mu\nu
XY}\left(k_{1}-k_{2}\right)^{X}\left(k_{1}+k_{2}\right)^{Y}G_{0},$ (3.49)

where $G_{0}$ represents a surface term that is logarithmically divergent to
adjust with mass dimension.

Such dependence characterizes an ambiguity, a quantity depending on arbitrary
choices. Momenta conservation sets differences between labels as external
momenta; however, it does not attribute a particular meaning to routings
themselves or their sum. As proposed before, this arbitrariness is preserved
throughout this investigation.

####  3.2.3 Three-Point Amplitudes - $PVV$

Previously, we used lowercase letters to denote the integrand of fermionic
amplitudes (3.5). They correspond to traces containing vertex operators
$\Gamma_{i}$ and fermion propagators $\not{F}_{n}^{-1}$, as seen for the
particular case of three-point functions:

$t^{\Gamma_{i}\Gamma_{j}\Gamma_{k}}=\text{tr}\left(\Gamma_{i}\frac{1}{\not{F}_{1}}\Gamma_{j}\frac{1}{\not{F}_{2}}\Gamma_{k}\frac{1}{\not{F}_{3}}\right).$
(3.50)

Rewriting the propagators (3.6) emphasizes the coefficients as Dirac traces

$\displaystyle t^{\Gamma_{i}\Gamma_{j}\Gamma_{k}}$ $\displaystyle=$
$\displaystyle\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\gamma_{B}\Gamma_{k}\gamma_{C}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}}{D_{123}}+m\text{tr}\left(\Gamma_{i}\Gamma_{j}\gamma_{B}\Gamma_{k}\gamma_{C}\right)\frac{K_{2}^{B}K_{3}^{C}}{D_{123}}$
(3.51)
$\displaystyle+m\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\Gamma_{k}\gamma_{C}\right)\frac{K_{1}^{A}K_{3}^{C}}{D_{123}}+m\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\gamma_{B}\Gamma_{k}\right)\frac{K_{1}^{A}K_{2}^{B}}{D_{123}}$
$\displaystyle+m^{2}\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\Gamma_{k}\right)\frac{K_{1}^{A}}{D_{123}}+m^{2}\text{tr}\left(\Gamma_{i}\Gamma_{j}\gamma_{B}\Gamma_{k}\right)\frac{K_{2}^{B}}{D_{123}}$
$\displaystyle+m^{2}\text{tr}\left(\Gamma_{i}\Gamma_{j}\Gamma_{k}\gamma_{C}\right)\frac{K_{3}^{C}}{D_{123}}+m^{3}\text{tr}\left(\Gamma_{i}\Gamma_{j}\Gamma_{k}\right)\frac{1}{D_{123}},$

where numerators depend on $K_{i}=k+k_{i}$ and denominators are
$D_{123}=D_{1}D_{2}D_{3}$.

To study the structure of a specific amplitude, we set its vertex content and
evaluate corresponding traces. For the $PVV$ case, the first vertex indicates
a pseudoscalar $\Gamma_{i}=\gamma_{5}$ while the others indicate vectors
$\Gamma_{j}=\gamma_{\nu}$ and $\Gamma_{k}=\gamma_{\alpha}$. Its non-zero
contributions are the following

$\displaystyle t_{\nu\alpha}^{PVV}$ $\displaystyle=$ $\displaystyle
m\text{tr}\left(\gamma_{5}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{C}\right)\frac{K_{2}^{B}K_{3}^{C}}{D_{123}}+m\text{tr}\left(\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{\alpha}\gamma_{C}\right)\frac{K_{1}^{A}K_{3}^{C}}{D_{123}}$
(3.52)
$\displaystyle+m\text{tr}\left(\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\right)\frac{K_{1}^{A}K_{2}^{B}}{D_{123}}.$

They are proportional to the Levi-Civita symbol (3.47), leading to the
antisymmetric version of the standard tensor

$t_{3\mu\nu}^{\left(s\right)}\left(k_{i},k_{j}\right)=\frac{\left(k+k_{i}\right)_{\mu}\left(k+k_{j}\right)_{\nu}+s\left(k+k_{j}\right)_{\mu}\left(k+k_{i}\right)_{\nu}}{D_{123}},$
(3.53)

Such an object is analogous to the previous one (3.40); however, it depends on
three propagators embodied in $D_{123}$ as indicated by the numerical
subindex. With these identifications, the integrand of the amplitude exhibits
the form

$t_{\nu\alpha}^{PVV}=-2im\varepsilon_{\nu\alpha
XY}\left[t_{3XY}^{\left(-\right)}\left(k_{2},k_{3}\right)+t_{3XY}^{\left(-\right)}\left(k_{3},k_{1}\right)+t_{3XY}^{\left(-\right)}\left(k_{1},k_{2}\right)\right].$
(3.54)

Observe the analogy between the $PVV$ structure and that of the $AV$ (3.48);
both are 2nd-order tensors contracted with the Levi-Civita symbol.
Nonetheless, expectations are different now. Even though three-point functions
exhibit linear power counting, contributions involving diverging surface terms
are prohibited since only finite contributions adjust to the correct mass
dimension. On the other hand, after integration, two external momenta
($p=k_{1}-k_{2}$ and $q=k_{1}-k_{3}$) are available to build up the tensor
structure

$T_{\nu\alpha}^{PVV}\rightarrow\varepsilon_{\nu\alpha XY}p^{X}q^{Y}F_{0}.$
(3.55)

The object $F_{0}=F_{0}\left(p_{i}\cdot p_{j}\right)$ represents a finite
scalar function depending on momenta bilinears $p_{i}\cdot
p_{j}=\left\\{p^{2},\text{ }q^{2},\text{ }p\cdot q\right\\}$.

####  3.2.4 Three-Point Amplitudes - $AVV$

The $AVV$ integrand emerges by replacing the corresponding vertex operators
within Equation (3.51); they are axial $\Gamma_{i}=\gamma_{\mu}\gamma_{5}$,
vector $\Gamma_{j}=\gamma_{\nu}$ and vector $\Gamma_{k}=\gamma_{\alpha}$.
Leaving null contributions aside, we cast its initial structure:

$\displaystyle t_{\mu\nu\alpha}^{AVV}$ $\displaystyle=$
$\displaystyle\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{C}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}}{D_{123}}+m^{2}\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{\alpha}\right)\frac{K_{1}^{A}}{D_{123}}$
(3.56)
$\displaystyle+m^{2}\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\right)\frac{K_{2}^{B}}{D_{123}}+m^{2}\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{\nu}\gamma_{\alpha}\gamma_{C}\right)\frac{K_{3}^{C}}{D_{123}},$

where numerators depend on $K_{i}=k+k_{i}$ and denominators are
$D_{123}=D_{1}D_{2}D_{3}$. Terms associated with the squared mass are already
known, being proportional to the Levi-Civita symbol (3.47).

Our next task is to take the trace involving six Dirac matrices plus the
chiral one. Nevertheless, different ways to perform this operation attribute
different expressions for it. Although all forms attributed to one trace are
linked through identities, the divergent character of perturbative
calculations affects these relations after integration. Clarifying these
aspects is essential to this investigation, so we are very detailed in this
discussion.

To introduce these ideas, we use the chiral matrix anticommutation in studying
two possibilities

$\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{C}\right)=-\text{tr}\left(\gamma_{5}\gamma_{\mu}\gamma_{A}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{C}\right).$

After replacing the definition
$\gamma_{5}=\frac{i}{4!}\varepsilon_{\mu\nu\alpha\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}$
in these specific places, we obtain a trace involving only Dirac matrices. Its
computation yields combinations of the metric tensors, which are contracted
with the Levi-Civita symbol. The expression obtained through the first path is

$\displaystyle\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{C}\right)$
(3.57) $\displaystyle=$ $\displaystyle-4i\left[\varepsilon_{\mu A\nu
B}g_{\alpha C}-\varepsilon_{\mu A\nu\alpha}g_{BC}+\varepsilon_{\mu A\nu
C}g_{B\alpha}+\varepsilon_{\mu AB\alpha}g_{\nu C}-\varepsilon_{\mu
ABC}g_{\nu\alpha}\right.$ $\displaystyle+\varepsilon_{\mu A\alpha C}g_{\nu
B}-\varepsilon_{\mu\nu B\alpha}g_{AC}+\varepsilon_{\mu\nu
BC}g_{A\alpha}-\varepsilon_{\mu\nu\alpha C}g_{AB}+\varepsilon_{\mu B\alpha
C}g_{A\nu}$ $\displaystyle\left.-\varepsilon_{A\nu B\alpha}g_{\mu
C}+\varepsilon_{A\nu BC}g_{\mu\alpha}-\varepsilon_{A\nu\alpha C}g_{\mu
B}+\varepsilon_{AB\alpha C}g_{\mu\nu}-\varepsilon_{\nu B\alpha C}g_{\mu
A}\right],$

while the other is

$\displaystyle-\text{tr}\left(\gamma_{5}\gamma_{\mu}\gamma_{A}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{C}\right)$
(3.58) $\displaystyle=$ $\displaystyle-4i\left[\varepsilon_{\mu A\nu
B}g_{\alpha C}-\varepsilon_{\mu A\nu\alpha}g_{BC}+\varepsilon_{\mu A\nu
C}g_{B\alpha}+\varepsilon_{\mu AB\alpha}g_{\nu C}-\varepsilon_{\mu
ABC}g_{\nu\alpha}\right.$ $\displaystyle+\varepsilon_{\mu A\alpha C}g_{\nu
B}-\varepsilon_{\mu\nu B\alpha}g_{AC}+\varepsilon_{\mu\nu
BC}g_{A\alpha}-\varepsilon_{\mu\nu\alpha C}g_{AB}+\varepsilon_{\mu B\alpha
C}g_{A\nu}$ $\displaystyle\left.+\varepsilon_{A\nu B\alpha}g_{\mu
C}-\varepsilon_{A\nu BC}g_{\mu\alpha}+\varepsilon_{A\nu\alpha C}g_{\mu
B}-\varepsilon_{AB\alpha C}g_{\mu\nu}+\varepsilon_{\nu B\alpha C}g_{\mu
A}\right].$

Although there are other strategies to compute them, one reason to choose this
path is that the results contain all contributions with non-equivalent tensor
configurations. This feature is convenient for the organization developed
throughout this section, which is part of IReg. Furthermore, the reason for
replacing the chiral matrix definition in these specific positions (adjacent
to $\gamma_{\mu}$) is to induce a simplification.

The layout of these (equivalent) expressions highlights that they only differ
by signs on the last row, characterizing one identity:

$g_{\mu C}\varepsilon_{A\nu B\alpha}-g_{\mu\alpha}\varepsilon_{A\nu BC}+g_{\mu
B}\varepsilon_{A\nu\alpha C}-g_{\mu\nu}\varepsilon_{AB\alpha C}+g_{\mu
A}\varepsilon_{\nu B\alpha C}=0.$ (3.59)

From another perspective, note that this tensor is antisymmetric in five
indices ($\mu$ fixed); therefore, identically zero for a four-dimensional
setting. Achieving this identity is not a coincidence but a direct consequence
of comparing positions adjacent to the $\mu$-index. Finding similar identities
where other free indices play this role is within reach. That is only the
first example seen here of the so-called Schouten identities.

With this argumentation, we developed the know-how to find the same resources
in more complex expressions from four-point amplitudes. Although that
significantly reduces our efforts in these calculations, there is no damage in
ignoring these identities. We verified that these contributions produce null
integrals when evaluating perturbative amplitudes.

As a brief comment on this subject, suppose we achieve three trace expressions
corresponding to each vertex position represented by free indices ($\mu$,
$\nu$, and $\alpha$). They are equivalent since their obtainment comes from
pure algebraic manipulations. Nevertheless, due to their divergent content,
their connection might not apply after integrating the amplitude. We attribute
a central role to the $\mu$-index for now, but Subsection (3.5.3) extends this
notion. The author, L. Ebani, and J. F. Thuorst develop a broad investigation
of the behavior of different versions of odd-tensor correlators in reference
[48].

Returning to the $AVV$ triangle, replacing traces leads to its integrand

$\displaystyle t_{\mu\nu\alpha}^{AVV}$ $\displaystyle=$ $\displaystyle
4i\left(g_{\nu A}\varepsilon_{\mu\alpha BC}-g_{\nu B}\varepsilon_{\mu\alpha
CA}-g_{\nu C}\varepsilon_{\mu\alpha
AB}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}}{D_{123}}$ (3.60)
$\displaystyle+4i\left(-g_{\alpha A}\varepsilon_{\mu\nu
BC}-g_{B\alpha}\varepsilon_{\mu\nu CA}+g_{\alpha C}\varepsilon_{\mu\nu
AB}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}}{D_{123}}$
$\displaystyle+4ig_{\nu\alpha}\varepsilon_{\mu
ABC}\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}}{D_{123}}$
$\displaystyle+\varepsilon_{\mu\nu\alpha\beta}\left[\text{tr}\left(\gamma_{\beta}\gamma_{A}\gamma_{B}\gamma_{C}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}}{D_{123}}-m^{2}\text{tr}\left(\gamma_{\beta}\gamma_{A}\right)\frac{K_{1}^{A}}{D_{123}}\right.$
$\displaystyle\left.+m^{2}\text{tr}\left(\gamma_{\beta}\gamma_{B}\right)\frac{K_{2}^{B}}{D_{123}}-m^{2}\text{tr}\left(\gamma_{\beta}\gamma_{C}\right)\frac{K_{3}^{C}}{D_{123}}\right].$

We already split sectors corresponding to different tensor configurations and
identified less complex traces. As terms with the free index $\mu$ within the
metric compound the identity (3.59), we disregarded them.

Following the reasoning established in example (3.39), trace content suggests
that the last term above consists of a vector subamplitude222The trace
structure indicates this subamplitude has one Lorentz index, which links to
one axial or vector vertex. Other vertices might be scalar or pseudoscalar
combined to produce an even trace. That leads to amplitudes corresponding to
vectors: $VPP$, $VSS$, $APS$, and their permutations.. If one maintains the
notations for summed indices, comparing signs is enough to identify the $VPP$
among all possibilities. Meanwhile, the antisymmetric character of the Levi-
Civita symbol allows rewriting the remaining terms through a new standard
tensor characterized by three momenta on the numerator

$t_{3\mu;\nu\alpha}^{\left(s\right)}\left(k_{l};k_{i},k_{j}\right)=\frac{\left(k+k_{l}\right)_{\mu}\left[\left(k+k_{i}\right)_{\nu}\left(k+k_{j}\right)_{\alpha}+s\left(k+k_{i}\right)_{\alpha}\left(k+k_{j}\right)_{\nu}\right]}{D_{123}}.$
(3.61)

Following previous notations, the superindex $s$ indicates a sign choice, and
the numerical subindex indicates the association with three propagators
through the denominator $D_{123}$. Hence, we achieve the final organization

$\displaystyle t_{\mu\nu\alpha}^{AVV}$ $\displaystyle=$ $\displaystyle
2i\varepsilon_{\mu\alpha
XY}\left[-t_{3\nu;XY}^{\left(-\right)}\left(k_{3};k_{1},k_{2}\right)-t_{3\nu;XY}^{\left(-\right)}\left(k_{2};k_{3},k_{1}\right)+t_{3\nu;XY}^{\left(-\right)}\left(k_{1};k_{2},k_{3}\right)\right]$
(3.62) $\displaystyle+2i\varepsilon_{\mu\nu
XY}\left[t_{3\alpha;XY}^{\left(-\right)}\left(k_{3};k_{1},k_{2}\right)-t_{3\alpha;XY}^{\left(-\right)}\left(k_{2};k_{3},k_{1}\right)-t_{3\alpha;XY}^{\left(-\right)}\left(k_{1};k_{2},k_{3}\right)\right]$
$\displaystyle+2g_{\nu\alpha}\varepsilon_{\mu
XYZ}t_{3}^{\left(-\right)X;YZ}\left(k_{1};k_{2},k_{3}\right)-i\varepsilon_{\mu\nu\alpha\beta}t_{\beta}^{VPP}.$

After replacing the corresponding vertices333There are three vertices: one
vector $\gamma_{\beta}$ followed by two pseudoscalars $\gamma_{5}$. in the
original integrand (3.51) and taking traces, we study the vector subamplitude

$\displaystyle t_{\beta}^{VPP}$ $\displaystyle=$
$\displaystyle-4\left(g_{\beta A}g_{BC}-g_{\beta B}g_{AC}+g_{\beta
C}g_{AB}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}}{D_{123}}$ (3.63)
$\displaystyle+4m^{2}\left[\frac{K_{1\beta}}{D_{123}}-\frac{K_{2\beta}}{D_{123}}+\frac{K_{3\beta}}{D_{123}}\right].$

Scalar products on the momenta emerge with the contraction, which leads to
reducing bilinears in analogy with scalar functions used as example (3.44).
Then, some manipulations produce the structure

$\displaystyle t_{\beta}^{VPP}$ $\displaystyle=$
$\displaystyle-2p_{\beta}\frac{1}{D_{12}}-4\frac{k_{\beta}}{D_{13}}-2\left(k_{1}+k_{3}\right)_{\beta}\frac{1}{D_{13}}+2\left(q-p\right)_{\beta}\frac{1}{D_{23}}$
(3.64)
$\displaystyle+2\left(q-p\right)^{2}\frac{\left(k+k_{1}\right)_{\beta}}{D_{123}}-2q^{2}\frac{\left(k+k_{2}\right)_{\beta}}{D_{123}}+2p^{2}\frac{\left(k+k_{3}\right)_{\beta}}{D_{123}}.$

Lastly, we recall the $AV$ discussion to infer expectations regarding
integration. The objective was to compose a 2nd-order antisymmetric tensor
with available tools, namely, external and ambiguous momenta ($k_{i}-k_{j}$
and $k_{i}+k_{j}$). The only possibility was to employ them both, which
necessarily implies the presence of diverging surface terms. For this to be
consistent with the quadratic power counting, these surface terms must be
logarithmically divergent.

We find similar circumstances for any 3rd-order tensor exhibiting the property
of total antisymmetry. At least three different vectors are necessary to
compound it, which requires the presence of ambiguous momenta. This structure
brings diverging surface terms, which prevents obtaining the correct mass
dimension. As a consequence, 3rd-order antisymmetric tensors are zero under
these circumstances.

The most immediate event of this type is the (three-index) contraction between
the Levi-Civita symbol and the standard tensor. For it to be non-zero, the
tensor must have a total-antisymmetric component. As this leads to the
argumentation above, we expect its cancellation

$\varepsilon_{\nu
XYZ}T_{3}^{\left(-\right)X;YZ}\left(k_{1};k_{2},k_{3}\right)\rightarrow 0.$
(3.65)

Furthermore, we combine all non-equivalent momenta configurations to produce
an identity involving this tensor

$T_{3\mu;\nu\alpha}^{\left(-\right)}\left(k_{1};k_{2},k_{3}\right)+T_{3\mu;\nu\alpha}^{\left(-\right)}\left(k_{2};k_{3},k_{1}\right)+T_{3\mu;\nu\alpha}^{\left(-\right)}\left(k_{3};k_{1},k_{2}\right)\rightarrow
0.$ (3.66)

If these expectations realize, simplifications apply to the integrated
amplitude, yielding the expression:

$T_{\mu\nu\alpha}^{AVV}\rightarrow 4i\varepsilon_{\mu\alpha
XY}T_{3\nu;XY}^{\left(-\right)}\left(k_{1};k_{2},k_{3}\right)+4i\varepsilon_{\mu\nu
XY}T_{3\alpha;XY}^{\left(-\right)}\left(k_{3};k_{1},k_{2}\right)-i\varepsilon_{\mu\nu\alpha\beta}T_{\beta}^{VPP}.$
(3.67)

We stress that the $\mu$-index appears exclusively within the Levi-Civita
symbol as a direct consequence of its prioritized role when taking the traces;
simplification only made this clear.

####  3.2.5 Four-Point Amplitudes - $PVVV$

We still have to look into four-point amplitudes, whose integrands assume the
form

$t^{\Gamma_{i}\Gamma_{j}\Gamma_{k}\Gamma_{l}}=\text{tr}\left(\Gamma_{i}\frac{1}{\not{F}_{1}}\Gamma_{j}\frac{1}{\not{F}_{2}}\Gamma_{k}\frac{1}{\not{F}_{3}}\Gamma_{l}\frac{1}{\not{F}_{4}}\right).$
(3.68)

After replacing fermion propagators (3.6), linearity makes the matrix content
evident within Dirac traces:

$\displaystyle t^{\Gamma_{i}\Gamma_{j}\Gamma_{k}\Gamma_{l}}$ (3.69)
$\displaystyle=$
$\displaystyle\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\gamma_{B}\Gamma_{k}\gamma_{C}\Gamma_{l}\gamma_{D}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}K_{4}^{D}}{D_{1234}}+m^{4}\text{tr}\left(\Gamma_{i}\Gamma_{j}\Gamma_{k}\Gamma_{l}\right)\frac{1}{D_{1234}}$
$\displaystyle+m^{2}\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\gamma_{B}\Gamma_{k}\Gamma_{l}\right)\frac{K_{1}^{A}K_{2}^{B}}{D_{1234}}+m^{2}\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\Gamma_{k}\gamma_{C}\Gamma_{l}\right)\frac{K_{1}^{A}K_{3}^{C}}{D_{1234}}$
$\displaystyle+m^{2}\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\Gamma_{k}\Gamma_{l}\gamma_{D}\right)\frac{K_{1}^{A}K_{4}^{D}}{D_{1234}}+m^{2}\text{tr}\left(\Gamma_{i}\Gamma_{j}\gamma_{B}\Gamma_{k}\gamma_{C}\Gamma_{l}\right)\frac{K_{2}^{B}K_{3}^{C}}{D_{1234}}$
$\displaystyle+m^{2}\text{tr}\left(\Gamma_{i}\Gamma_{j}\gamma_{B}\Gamma_{k}\Gamma_{l}\gamma_{D}\right)\frac{K_{2}^{B}K_{4}^{D}}{D_{1234}}+m^{2}\text{tr}\left(\Gamma_{i}\Gamma_{j}\Gamma_{k}\gamma_{C}\Gamma_{l}\gamma_{D}\right)\frac{K_{3}^{C}K_{4}^{D}}{D_{1234}}$
$\displaystyle+m\text{tr}\left(\Gamma_{i}\Gamma_{j}\gamma_{B}\Gamma_{k}\gamma_{C}\Gamma_{l}\gamma_{D}\right)\frac{K_{2}^{B}K_{3}^{C}K_{4}^{D}}{D_{1234}}+m^{3}\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\Gamma_{k}\Gamma_{l}\right)\frac{K_{1}^{A}}{D_{1234}}$
$\displaystyle+m\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\Gamma_{k}\gamma_{C}\Gamma_{l}\gamma_{D}\right)\frac{K_{1}^{A}K_{3}^{C}K_{4}^{D}}{D_{1234}}+m^{3}\text{tr}\left(\Gamma_{i}\Gamma_{j}\gamma_{B}\Gamma_{k}\Gamma_{l}\right)\frac{K_{2}^{B}}{D_{1234}}$
$\displaystyle+m\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\gamma_{B}\Gamma_{k}\Gamma_{l}\gamma_{D}\right)\frac{K_{1}^{A}K_{2}^{B}K_{4}^{D}}{D_{1234}}+m^{3}\text{tr}\left(\Gamma_{i}\Gamma_{j}\Gamma_{k}\gamma_{C}\Gamma_{l}\right)\frac{K_{3}^{C}}{D_{1234}}$
$\displaystyle+m\text{tr}\left(\Gamma_{i}\gamma_{A}\Gamma_{j}\gamma_{B}\Gamma_{k}\gamma_{C}\Gamma_{l}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}}{D_{1234}}+m^{3}\text{tr}\left(\Gamma_{i}\Gamma_{j}\Gamma_{k}\Gamma_{l}\gamma_{D}\right)\frac{K_{4}^{D}}{D_{1234}},$

where numerators depend on $K_{i}=k+k_{i}$ and denominators are
$D_{1234}=D_{1}D_{2}D_{3}D_{4}$.

Obtaining a specific function requires replacing the corresponding vertex
operators within this expression. For the case of $PVVV$ amplitude, we use one
pseudoscalar vertex ($\Gamma_{i}=\gamma_{5}$) followed by vector ones
($\Gamma_{j}=\gamma_{\nu}$, $\Gamma_{k}=\gamma_{\alpha}$, and
$\Gamma_{l}=\gamma_{\beta}$), achieving the non-zero contributions

$\displaystyle t_{\nu\alpha\beta}^{PVVV}$ $\displaystyle=$ $\displaystyle
m\text{tr}\left(\gamma_{5}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{C}\gamma_{\beta}\gamma_{D}\right)\frac{K_{2}^{B}K_{3}^{C}K_{4}^{D}}{D_{1234}}+m\text{tr}\left(\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{\alpha}\gamma_{C}\gamma_{\beta}\gamma_{D}\right)\frac{K_{1}^{A}K_{3}^{C}K_{4}^{D}}{D_{1234}}$
(3.70)
$\displaystyle+m\text{tr}\left(\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{\beta}\gamma_{D}\right)\frac{K_{1}^{A}K_{2}^{B}K_{4}^{D}}{D_{1234}}+m\text{tr}\left(\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{C}\gamma_{\beta}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}}{D_{1234}}$
$\displaystyle+m^{3}\text{tr}\left(\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{\alpha}\gamma_{\beta}\right)\frac{K_{1}^{A}}{D_{1234}}+m^{3}\text{tr}\left(\gamma_{5}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{\beta}\right)\frac{K_{2}^{B}}{D_{1234}}$
$\displaystyle+m^{3}\text{tr}\left(\gamma_{5}\gamma_{\nu}\gamma_{\alpha}\gamma_{C}\gamma_{\beta}\right)\frac{K_{3}^{C}}{D_{1234}}+m^{3}\text{tr}\left(\gamma_{5}\gamma_{\nu}\gamma_{\alpha}\gamma_{\beta}\gamma_{D}\right)\frac{K_{4}^{D}}{D_{1234}}.$

All traces are known and can be consulted in Equations (3.47) and (3.58).
Posteriorly to their employment, our task is to group terms that share their
index configuration to recognize subamplitudes or standard tensors. We
consider each of these sectors separately since their mathematical expressions
are more extensive now.

Finding those terms where the metric tensor has exclusively free indices, we
identify the first sector:

$\displaystyle\left[t_{\nu\alpha\beta}^{PVVV}\right]_{1}$ $\displaystyle=$
$\displaystyle-4im\left(g_{\nu\alpha}\varepsilon_{\beta
BCD}-g_{\nu\beta}\varepsilon_{\alpha BCD}+g_{\alpha\beta}\varepsilon_{\nu
BCD}\right)\frac{K_{2}^{B}K_{3}^{C}K_{4}^{D}}{D_{1234}}$ (3.71)
$\displaystyle+4im\left(g_{\nu\alpha}\varepsilon_{\beta
ACD}-g_{\nu\beta}\varepsilon_{\alpha ACD}+g_{\alpha\beta}\varepsilon_{\nu
ACD}\right)\frac{K_{1}^{A}K_{3}^{C}K_{4}^{D}}{D_{1234}}$
$\displaystyle-4im\left(g_{\nu\alpha}\varepsilon_{\beta
ABD}-g_{\nu\beta}\varepsilon_{\alpha ABD}+g_{\alpha\beta}\varepsilon_{\nu
ABD}\right)\frac{K_{1}^{A}K_{2}^{B}K_{4}^{D}}{D_{1234}}$
$\displaystyle+4im\left(g_{\nu\alpha}\varepsilon_{\beta
ABC}-g_{\nu\beta}\varepsilon_{\alpha ABC}+g_{\alpha\beta}\varepsilon_{\nu
ABC}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}}{D_{1234}}.$

By following the same procedure from previous cases, axial vector amplitudes
would be achievable. Nevertheless, since quantities in parenthesis are alike,
we rename summed indices to compact them into a single object

$\left[t_{\nu\alpha\beta}^{PVVV}\right]_{1}=-4im\left(g_{\kappa\nu}g_{\alpha\beta}-g_{\kappa\alpha}g_{\nu\beta}+g_{\kappa\beta}g_{\nu\alpha}\right)f_{4\kappa}.$
(3.72)

The introduced object has the following structure

$\displaystyle f_{4\kappa}$ $\displaystyle=$ $\displaystyle\varepsilon_{\kappa
XYZ}t_{4X;YZ}^{\left(-\right)}\left(k_{2};k_{3},k_{4}\right)-\varepsilon_{\kappa
XYZ}t_{4X;YZ}^{\left(-\right)}\left(k_{1};k_{3},k_{4}\right)$ (3.73)
$\displaystyle+\varepsilon_{\kappa
XYZ}t_{4X;YZ}^{\left(-\right)}\left(k_{1};k_{2},k_{4}\right)-\varepsilon_{\kappa
XYZ}t_{4X;YZ}^{\left(-\right)}\left(k_{1};k_{2},k_{3}\right),$

which depends on the new standard tensor

$t_{4\mu;\nu\alpha}^{\left(s\right)}\left(k_{l};k_{i},k_{j}\right)=\frac{\left(k+k_{l}\right)_{\mu}\left[\left(k+k_{i}\right)_{\nu}\left(k+k_{j}\right)_{\alpha}+s\left(k+k_{i}\right)_{\alpha}\left(k+k_{j}\right)_{\nu}\right]}{D_{1234}}.$
(3.74)

Although this object is analogous to that defined in Equation (3.61), the
numerical subindex indicates the association with four propagators through the
denominator $D_{1234}$.

For the second sector, let us group components where all free indices appear
within the Levi-Civita symbol, including traces of four Dirac matrices. We
introduce a summed index $\kappa$ to isolate a global factor and recognize
less complex traces

$\displaystyle\left[t_{\nu\alpha\beta}^{PVVV}\right]_{2}$ $\displaystyle=$
$\displaystyle
i\varepsilon_{\nu\alpha\beta\kappa}\left[-m\text{tr}\left(\gamma_{B}\gamma_{C}\gamma_{D}\gamma_{\kappa}\right)\frac{K_{2}^{B}K_{3}^{C}K_{4}^{D}}{D_{1234}}+m\text{tr}\left(\gamma_{A}\gamma_{C}\gamma_{D}\gamma_{\kappa}\right)\frac{K_{1}^{A}K_{3}^{C}K_{4}^{D}}{D_{1234}}\right.$
(3.75)
$\displaystyle-m\text{tr}\left(\gamma_{A}\gamma_{B}\gamma_{D}\gamma_{\kappa}\right)\frac{K_{1}^{A}K_{2}^{B}K_{4}^{D}}{D_{1234}}+m\text{tr}\left(\gamma_{A}\gamma_{B}\gamma_{C}\gamma_{\kappa}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}}{D_{1234}}$
$\displaystyle-m^{3}\text{tr}\left(\gamma_{A}\gamma_{\kappa}\right)\frac{K_{1}^{A}}{D_{1234}}+m^{3}\text{tr}\left(\gamma_{B}\gamma_{\kappa}\right)\frac{K_{2}^{B}}{D_{1234}}$
$\displaystyle\left.-m^{3}\text{tr}\left(\gamma_{C}\gamma_{\kappa}\right)\frac{K_{3}^{C}}{D_{1234}}+m^{3}\text{tr}\left(\gamma_{D}\gamma_{\kappa}\right)\frac{K_{4}^{D}}{D_{1234}}\right].$

This structure associates with a vector subamplitude; thus, comparing signs
among the possibilities leads to the $APPP$ function444There are four
vertices: one vector $\gamma_{\kappa}$ followed by three pseudoscalars
$\gamma_{5}$.:

$\left[t_{\nu\alpha\beta}^{PVVV}\right]_{2}=-i\varepsilon_{\kappa\nu\alpha\beta}t_{\kappa}^{APPP}.$
(3.76)

As bilinears arise from traces within this subamplitude, we reduce them
through identity (3.44). The loop momentum from numerators cancels out with
this operation. Hence, the integrand associated with this function has the
final structure

$\displaystyle t_{\kappa}^{APPP}$ $\displaystyle=$ $\displaystyle
4mp_{\kappa}\frac{1}{D_{124}}+4m\left(r-q\right)_{\kappa}\frac{1}{D_{134}}$
(3.77) $\displaystyle-4m\left[\left(q^{2}-q\cdot
r\right)p_{\kappa}-\left(p^{2}-p\cdot r\right)q_{\kappa}+\left(p^{2}-p\cdot
q\right)r_{\kappa}\right]\frac{1}{D_{1234}}.$

All external momenta arose within this expression: $p=k_{1}-k_{2}$,
$q=k_{1}-k_{3}$, and $r=k_{1}-k_{4}$.

Lastly, consider those terms that mix free and summed indices

$\displaystyle\left[t_{\nu\alpha\beta}^{PVVV}\right]_{3}$ $\displaystyle=$
$\displaystyle 4im\left(g_{\nu B}\varepsilon_{\alpha C\beta D}+g_{\nu
C}\varepsilon_{B\alpha\beta D}+g_{\nu D}\varepsilon_{B\alpha
C\beta}+g_{B\alpha}\varepsilon_{\nu C\beta D}+g_{B\beta}\varepsilon_{\nu\alpha
CD}\right.$ (3.78) $\displaystyle\left.+g_{\alpha C}\varepsilon_{\nu B\beta
D}+g_{\alpha D}\varepsilon_{\nu BC\beta}+g_{C\beta}\varepsilon_{\nu B\alpha
D}+g_{\beta D}\varepsilon_{\nu B\alpha
C}\right)\frac{K_{2}^{B}K_{3}^{C}K_{4}^{D}}{D_{1234}}$
$\displaystyle+4im\left(g_{A\nu}\varepsilon_{\alpha C\beta
D}-g_{A\alpha}\varepsilon_{\nu C\beta D}-g_{A\beta}\varepsilon_{\nu\alpha
CD}-g_{\nu C}\varepsilon_{A\alpha\beta D}-g_{\nu D}\varepsilon_{A\alpha
C\beta}\right.$ $\displaystyle\left.+g_{\alpha C}\varepsilon_{A\nu\beta
D}+g_{\alpha D}\varepsilon_{A\nu C\beta}+g_{C\beta}\varepsilon_{A\nu\alpha
D}+g_{\beta D}\varepsilon_{A\nu\alpha
C}\right)\frac{K_{1}^{A}K_{3}^{C}K_{4}^{D}}{D_{1234}}$
$\displaystyle+4im\left(g_{A\nu}\varepsilon_{B\alpha\beta
D}+g_{A\alpha}\varepsilon_{\nu B\beta D}-g_{A\beta}\varepsilon_{\nu B\alpha
D}+g_{\nu B}\varepsilon_{A\alpha\beta D}-g_{\nu
D}\varepsilon_{AB\alpha\beta}\right.$
$\displaystyle\left.+g_{B\alpha}\varepsilon_{A\nu\beta
D}-g_{B\beta}\varepsilon_{A\nu\alpha D}-g_{\alpha D}\varepsilon_{A\nu
B\beta}+g_{\beta D}\varepsilon_{A\nu
B\alpha}\right)\frac{K_{1}^{A}K_{2}^{B}K_{4}^{D}}{D_{1234}}$
$\displaystyle+4im\left(g_{A\nu}\varepsilon_{B\alpha
C\beta}+g_{A\alpha}\varepsilon_{\nu BC\beta}+g_{A\beta}\varepsilon_{\nu
B\alpha C}+g_{\nu B}\varepsilon_{A\alpha C\beta}+g_{\nu
C}\varepsilon_{AB\alpha\beta}\right.$
$\displaystyle\left.+g_{B\alpha}\varepsilon_{A\nu
C\beta}+g_{B\beta}\varepsilon_{A\nu\alpha C}+g_{\alpha C}\varepsilon_{A\nu
B\beta}+g_{C\beta}\varepsilon_{A\nu
B\alpha}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}}{D_{1234}}.$

Once again, using the antisymmetric character of the Levi-Civita symbol, we
recognize combinations of the standard tensor (3.74).Then, this sector leads
to the following tensor by factorizing $2im$:

$\displaystyle f_{4\nu\alpha\beta}$ $\displaystyle=$
$\displaystyle-\left(\varepsilon_{\alpha\beta XY}g_{\nu
Z}-\varepsilon_{\nu\beta XY}g_{\alpha Z}+\varepsilon_{\nu\alpha XY}g_{\beta
Z}\right)t_{4Z;XY}^{\left(-\right)}\left(k_{1};k_{3},k_{4}\right)$ (3.79)
$\displaystyle+\left(\varepsilon_{\alpha\beta XY}g_{\nu
Z}-\varepsilon_{\nu\beta XY}g_{\alpha Z}+\varepsilon_{\nu\alpha XY}g_{\beta
Z}\right)t_{4Z;XY}^{\left(-\right)}\left(k_{1};k_{2},k_{4}\right)$
$\displaystyle-\left(\varepsilon_{\alpha\beta XY}g_{\nu
Z}-\varepsilon_{\nu\beta XY}g_{\alpha Z}+\varepsilon_{\nu\alpha XY}g_{\beta
Z}\right)t_{4Z;XY}^{\left(-\right)}\left(k_{1};k_{2},k_{3}\right)$
$\displaystyle-\left(\varepsilon_{\alpha\beta XY}g_{\nu
Z}+\varepsilon_{\nu\beta XY}g_{\alpha Z}-\varepsilon_{\nu\alpha XY}g_{\beta
Z}\right)t_{4Z;XY}^{\left(-\right)}\left(k_{2};k_{3},k_{4}\right)$
$\displaystyle+\left(\varepsilon_{\alpha\beta XY}g_{\nu
Z}+\varepsilon_{\nu\beta XY}g_{\alpha Z}-\varepsilon_{\nu\alpha XY}g_{\beta
Z}\right)t_{4Z;XY}^{\left(-\right)}\left(k_{2};k_{1},k_{4}\right)$
$\displaystyle-\left(\varepsilon_{\alpha\beta XY}g_{\nu
Z}+\varepsilon_{\nu\beta XY}g_{\alpha Z}-\varepsilon_{\nu\alpha XY}g_{\beta
Z}\right)t_{4Z;XY}^{\left(-\right)}\left(k_{2};k_{1},k_{3}\right)$
$\displaystyle+\left(\varepsilon_{\alpha\beta XY}g_{\nu
Z}-\varepsilon_{\nu\beta XY}g_{\alpha Z}-\varepsilon_{\nu\alpha XY}g_{\beta
Z}\right)t_{4Z;XY}^{\left(-\right)}\left(k_{3};k_{2},k_{4}\right)$
$\displaystyle-\left(\varepsilon_{\alpha\beta XY}g_{\nu
Z}-\varepsilon_{\nu\beta XY}g_{\alpha Z}-\varepsilon_{\nu\alpha XY}g_{\beta
Z}\right)t_{4Z;XY}^{\left(-\right)}\left(k_{3};k_{1},k_{4}\right)$
$\displaystyle+\left(\varepsilon_{\alpha\beta XY}g_{\nu
Z}-\varepsilon_{\nu\beta XY}g_{\alpha Z}-\varepsilon_{\nu\alpha XY}g_{\beta
Z}\right)t_{4Z;XY}^{\left(-\right)}\left(k_{3};k_{1},k_{2}\right)$
$\displaystyle-\left(\varepsilon_{\alpha\beta XY}g_{\nu
Z}-\varepsilon_{\nu\beta XY}g_{\alpha Z}+\varepsilon_{\nu\alpha XY}g_{\beta
Z}\right)t_{4Z;XY}^{\left(-\right)}\left(k_{4};k_{2},k_{3}\right)$
$\displaystyle+\left(\varepsilon_{\alpha\beta XY}g_{\nu
Z}-\varepsilon_{\nu\beta XY}g_{\alpha Z}+\varepsilon_{\nu\alpha XY}g_{\beta
Z}\right)t_{4Z;XY}^{\left(-\right)}\left(k_{4};k_{1},k_{3}\right)$
$\displaystyle-\left(\varepsilon_{\alpha\beta XY}g_{\nu
Z}-\varepsilon_{\nu\beta XY}g_{\alpha Z}+\varepsilon_{\nu\alpha XY}g_{\beta
Z}\right)t_{4Z;XY}^{\left(-\right)}\left(k_{4};k_{1},k_{2}\right).$

The most significant difference between both occurrences of this tensor is in
the contraction. Whereas all indices were contracted with the Levi-Civita
symbol in the previous case, only those that show the antisymmetry property
are contracted this time.

With all sectors explored, we write the $PVVV$ final form

$t_{\nu\alpha\beta}^{PVVV}=-4im\left(g_{\kappa\nu}g_{\alpha\beta}-g_{\kappa\alpha}g_{\nu\beta}+g_{\kappa\beta}g_{\nu\alpha}\right)f_{4\kappa}+2imf_{4\nu\alpha\beta}-i\varepsilon_{\kappa\nu\alpha\beta}t_{\kappa}^{APPP}.$
(3.80)

It depends on two main structures: the vector subamplitude $APPP$ and the
standard tensor with three momenta on the numerator. Even though four-point
functions have logarithmic power counting, mass dimension analysis suggests
that integrals within this particular amplitude are finite.

####  3.2.6 Four-Point Amplitudes - $AVVV$

The last correlator concerning this investigation is the $AVVV$ box, whose
structure contains one axial vertex ($\Gamma_{i}=\gamma_{\mu}\gamma_{5}$) and
three vector vertices ($\Gamma_{j}=\gamma_{\nu}$,
$\Gamma_{k}=\gamma_{\alpha}$, and $\Gamma_{l}=\gamma_{\beta}$). We obtain its
initial structure by replacing the corresponding vertices on the general
integrand of four-point functions (3.69):

$\displaystyle t_{\mu\nu\alpha\beta}^{AVVV}$ $\displaystyle=$
$\displaystyle\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{C}\gamma_{\beta}\gamma_{D}\right)\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}K_{4}^{D}}{D_{1234}}$
(3.81)
$\displaystyle+m^{2}\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{\beta}\right)\frac{K_{1}^{A}K_{2}^{B}}{D_{1234}}+m^{2}\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{\alpha}\gamma_{C}\gamma_{\beta}\right)\frac{K_{1}^{A}K_{3}^{C}}{D_{1234}}$
$\displaystyle+m^{2}\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{A}\gamma_{\nu}\gamma_{\alpha}\gamma_{\beta}\gamma_{D}\right)\frac{K_{1}^{A}K_{4}^{D}}{D_{1234}}+m^{2}\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{C}\gamma_{\beta}\right)\frac{K_{2}^{B}K_{3}^{C}}{D_{1234}}$
$\displaystyle+m^{2}\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{\nu}\gamma_{B}\gamma_{\alpha}\gamma_{\beta}\gamma_{D}\right)\frac{K_{2}^{B}K_{4}^{D}}{D_{1234}}+m^{2}\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{\nu}\gamma_{\alpha}\gamma_{C}\gamma_{\beta}\gamma_{D}\right)\frac{K_{3}^{C}K_{4}^{D}}{D_{1234}}$
$\displaystyle+m^{4}\text{tr}\left(\gamma_{\mu}\gamma_{5}\gamma_{\nu}\gamma_{\alpha}\gamma_{\beta}\right)\frac{1}{D_{1234}}.$

This subsection deals with numerous contributions that might compromise the
visualization and understanding of mathematical expressions. For this reason,
we introduce a compact notation for products of gamma matrices, e.g.,
$\gamma_{\mu}\gamma_{5}\gamma_{\nu}\gamma_{\alpha}\gamma_{\beta}=\gamma_{\mu
5\nu\alpha\beta}$. That is a temporary resource employed exclusively in $AVVV$
calculations.

Most traces above are known and can be consulted in Equations (3.47) and
(3.58). We also identify the presence of a trace involving eight Dirac
matrices plus the chiral one, which leads to products involving the Levi-
Civita symbol and metric tensors. This type of structure admits equivalent
expressions distinguished in their tensor structure. Nevertheless, this
connection is not guaranteed for perturbative amplitudes due to their
divergent character. That is analogous to the $AVV$ case and motivated us to
choose the $AVVV$ as an extension of our discussion.

Evaluating this trace follows the same procedure adopted in previous cases:
replace the chiral matrix definition, take the new trace, and perform
contractions with the Levi-Civita symbol. This strategy leads to a result
exhibiting all non-equivalent tensor contributions, which makes the existence
of identities clear. Even so, to allow a careful analysis, we chose to
approach this subject after the complete organization of the amplitude. Thus,
let us directly introduce the trace expression prioritizing the $\mu$-index:

$\displaystyle i\text{tr}\left(\gamma_{5\mu A\nu B\alpha C\beta D}\right)$
(3.82) $\displaystyle=$ $\displaystyle-\varepsilon_{\mu A\nu
B}\text{tr}\left(\gamma_{\alpha C\beta D}\right)+\varepsilon_{\mu
A\nu\alpha}\text{tr}\left(\gamma_{BC\beta D}\right)-\varepsilon_{\mu A\nu
C}\text{tr}\left(\gamma_{B\alpha\beta D}\right)+\varepsilon_{\mu
A\nu\beta}\text{tr}\left(\gamma_{B\alpha CD}\right)$
$\displaystyle-\varepsilon_{\mu A\nu D}\text{tr}\left(\gamma_{B\alpha
C\beta}\right)-\varepsilon_{\mu AB\alpha}\text{tr}\left(\gamma_{\nu C\beta
D}\right)+\varepsilon_{\mu ABC}\text{tr}\left(\gamma_{\nu\alpha\beta
D}\right)-\varepsilon_{\mu AB\beta}\text{tr}\left(\gamma_{\nu\alpha
CD}\right)$ $\displaystyle+\varepsilon_{\mu
ABD}\text{tr}\left(\gamma_{\nu\alpha C\beta}\right)-\varepsilon_{\mu A\alpha
C}\text{tr}\left(\gamma_{\nu B\beta D}\right)+\varepsilon_{\mu
A\alpha\beta}\text{tr}\left(\gamma_{\nu BCD}\right)-\varepsilon_{\mu A\alpha
D}\text{tr}\left(\gamma_{\nu BC\beta}\right)$ $\displaystyle-\varepsilon_{\mu
AC\beta}\text{tr}\left(\gamma_{\nu B\alpha D}\right)+\varepsilon_{\mu
ACD}\text{tr}\left(\gamma_{\nu B\alpha\beta}\right)-\varepsilon_{\mu A\beta
D}\text{tr}\left(\gamma_{\nu B\alpha C}\right)+\varepsilon_{\mu\nu
B\alpha}\text{tr}\left(\gamma_{AC\beta D}\right)$
$\displaystyle-\varepsilon_{\mu\nu BC}\text{tr}\left(\gamma_{A\alpha\beta
D}\right)+\varepsilon_{\mu\nu B\beta}\text{tr}\left(\gamma_{A\alpha
CD}\right)-\varepsilon_{\mu\nu BD}\text{tr}\left(\gamma_{A\alpha
C\beta}\right)+\varepsilon_{\mu\nu\alpha C}\text{tr}\left(\gamma_{AB\beta
D}\right)$
$\displaystyle-\varepsilon_{\mu\nu\alpha\beta}\text{tr}\left(\gamma_{ABCD}\right)+\varepsilon_{\mu\nu\alpha
D}\text{tr}\left(\gamma_{ABC\beta}\right)+\varepsilon_{\mu\nu
C\beta}\text{tr}\left(\gamma_{AB\alpha D}\right)-\varepsilon_{\mu\nu
CD}\text{tr}\left(\gamma_{AB\alpha\beta}\right)$
$\displaystyle+\varepsilon_{\mu\nu\beta D}\text{tr}\left(\gamma_{AB\alpha
C}\right)-\varepsilon_{\mu B\alpha C}\text{tr}\left(\gamma_{A\nu\beta
D}\right)+\varepsilon_{\mu B\alpha\beta}\text{tr}\left(\gamma_{A\nu
CD}\right)-\varepsilon_{\mu B\alpha D}\text{tr}\left(\gamma_{A\nu
C\beta}\right)$ $\displaystyle-\varepsilon_{\mu
BC\beta}\text{tr}\left(\gamma_{A\nu\alpha D}\right)+\varepsilon_{\mu
BCD}\text{tr}\left(\gamma_{A\nu\alpha\beta}\right)-\varepsilon_{\mu B\beta
D}\text{tr}\left(\gamma_{A\nu\alpha C}\right)+\varepsilon_{\mu\alpha
C\beta}\text{tr}\left(\gamma_{A\nu BD}\right)$
$\displaystyle-\varepsilon_{\mu\alpha CD}\text{tr}\left(\gamma_{A\nu
B\beta}\right)+\varepsilon_{\mu\alpha\beta D}\text{tr}\left(\gamma_{A\nu
BC}\right)-\varepsilon_{\mu C\beta D}\text{tr}\left(\gamma_{A\nu
B\alpha}\right)-\varepsilon_{A\nu B\alpha}\text{tr}\left(\gamma_{\mu C\beta
D}\right)$ $\displaystyle+\varepsilon_{A\nu
BC}\text{tr}\left(\gamma_{\mu\alpha\beta D}\right)-\varepsilon_{A\nu
B\beta}\text{tr}\left(\gamma_{\mu\alpha CD}\right)+\varepsilon_{A\nu
BD}\text{tr}\left(\gamma_{\mu\alpha C\beta}\right)-\varepsilon_{A\nu\alpha
C}\text{tr}\left(\gamma_{\mu B\beta D}\right)$
$\displaystyle+\varepsilon_{A\nu\alpha\beta}\text{tr}\left(\gamma_{\mu
BCD}\right)-\varepsilon_{A\nu\alpha D}\text{tr}\left(\gamma_{\mu
BC\beta}\right)-\varepsilon_{A\nu C\beta}\text{tr}\left(\gamma_{\mu B\alpha
D}\right)+\varepsilon_{A\nu CD}\text{tr}\left(\gamma_{\mu
B\alpha\beta}\right)$ $\displaystyle-\varepsilon_{A\nu\beta
D}\text{tr}\left(\gamma_{\mu B\alpha C}\right)+\varepsilon_{AB\alpha
C}\text{tr}\left(\gamma_{\mu\nu\beta
D}\right)-\varepsilon_{AB\alpha\beta}\text{tr}\left(\gamma_{\mu\nu
CD}\right)+\varepsilon_{AB\alpha D}\text{tr}\left(\gamma_{\mu\nu
C\beta}\right)$
$\displaystyle+\varepsilon_{ABC\beta}\text{tr}\left(\gamma_{\mu\nu\alpha
D}\right)-\varepsilon_{ABCD}\text{tr}\left(\gamma_{\mu\nu\alpha\beta}\right)+\varepsilon_{AB\beta
D}\text{tr}\left(\gamma_{\mu\nu\alpha C}\right)-\varepsilon_{A\alpha
C\beta}\text{tr}\left(\gamma_{\mu\nu BD}\right)$
$\displaystyle+\varepsilon_{A\alpha CD}\text{tr}\left(\gamma_{\mu\nu
B\beta}\right)-\varepsilon_{A\alpha\beta D}\text{tr}\left(\gamma_{\mu\nu
BC}\right)+\varepsilon_{AC\beta D}\text{tr}\left(\gamma_{\mu\nu
B\alpha}\right)-\varepsilon_{\nu B\alpha C}\text{tr}\left(\gamma_{\mu A\beta
D}\right)$ $\displaystyle+\varepsilon_{\nu
B\alpha\beta}\text{tr}\left(\gamma_{\mu ACD}\right)-\varepsilon_{\nu B\alpha
D}\text{tr}\left(\gamma_{\mu AC\beta}\right)-\varepsilon_{\nu
BC\beta}\text{tr}\left(\gamma_{\mu A\alpha D}\right)+\varepsilon_{\nu
BCD}\text{tr}\left(\gamma_{\mu A\alpha\beta}\right)$
$\displaystyle-\varepsilon_{\nu B\beta D}\text{tr}\left(\gamma_{\mu A\alpha
C}\right)+\varepsilon_{\nu\alpha C\beta}\text{tr}\left(\gamma_{\mu
ABD}\right)-\varepsilon_{\nu\alpha CD}\text{tr}\left(\gamma_{\mu
AB\beta}\right)+\varepsilon_{\nu\alpha\beta D}\text{tr}\left(\gamma_{\mu
ABC}\right)$ $\displaystyle-\varepsilon_{\nu D\beta
D}\text{tr}\left(\gamma_{\mu AB\alpha}\right)-\varepsilon_{B\alpha
C\beta}\text{tr}\left(\gamma_{\mu A\nu D}\right)+\varepsilon_{B\alpha
CD}\text{tr}\left(\gamma_{\mu A\nu\beta}\right)-\varepsilon_{B\alpha\beta
D}\text{tr}\left(\gamma_{\mu A\nu C}\right)$
$\displaystyle+\varepsilon_{BC\beta D}\text{tr}\left(\gamma_{\mu
A\nu\alpha}\right)-\varepsilon_{\alpha C\beta D}\text{tr}\left(\gamma_{\mu
A\nu B}\right).$

Since numerous components exist, we split this analysis555Although all vertex
operators appear within this context, we only comment on cases that remain in
the final form. into sectors grouping terms where free indices play similar
roles. This line of reasoning extends to all parts of the initial integrand
(3.81). Thus, we will call upon the term proportional to $K_{1}^{A}K_{2}^{B}$
to illustrate a trace involving six Dirac matrices

$\displaystyle\text{tr}\left(\gamma_{\mu 5A\nu B\alpha\beta}\right)$
$\displaystyle=$ $\displaystyle-4i\left[g_{\mu A}\varepsilon_{\nu
B\alpha\beta}-g_{\mu\nu}\varepsilon_{AB\alpha\beta}+g_{\mu
B}\varepsilon_{A\nu\alpha\beta}-g_{\mu\alpha}\varepsilon_{A\nu
B\beta}+g_{\mu\beta}\varepsilon_{A\nu B\alpha}\right.$ (3.83)
$\displaystyle+g_{A\nu}\varepsilon_{\mu
B\alpha\beta}-g_{AB}\varepsilon_{\mu\nu\alpha\beta}+g_{A\alpha}\varepsilon_{\mu\nu
B\beta}-g_{A\beta}\varepsilon_{\mu\nu B\alpha}+g_{\nu B}\varepsilon_{\mu
A\alpha\beta}$ $\displaystyle\left.-g_{\nu\alpha}\varepsilon_{\mu
AB\beta}+g_{\nu\beta}\varepsilon_{\mu AB\alpha}+g_{B\alpha}\varepsilon_{\mu
A\nu\beta}-g_{B\beta}\varepsilon_{\mu
A\nu\alpha}+g_{\alpha\beta}\varepsilon_{\mu A\nu B}\right].$

Our first step is to find those terms depending on the metric tensor with free
indices. The artifice of using uppercase Latin letters on summed indices makes
this process a lot easier. For the equation above, the following components
interest us

$\text{tr}\left(\gamma_{\mu 5A\nu B\alpha\beta}\right)\rightarrow
g_{\mu\nu}\varepsilon_{AB\alpha\beta}+g_{\mu\alpha}\varepsilon_{A\nu
B\beta}-g_{\mu\beta}\varepsilon_{A\nu B\alpha}+g_{\nu\alpha}\varepsilon_{\mu
AB\beta}-g_{\nu\beta}\varepsilon_{\mu
AB\alpha}-g_{\alpha\beta}\varepsilon_{\mu A\nu B},$

where the Levi-Civita symbols are recognized as less complex traces. Extending
this idea to the complete amplitude, we have all contributions belonging to
this sector:

$\displaystyle\left[t_{\mu\nu\alpha\beta}^{AVVV}\right]_{1}$ $\displaystyle=$
$\displaystyle-\left[g_{\alpha\beta}\text{tr}\left(\gamma_{\mu 5A\nu
BCD}\right)+g_{\nu\beta}\text{tr}\left(\gamma_{\mu 5AB\alpha
CD}\right)+g_{\nu\alpha}\text{tr}\left(\gamma_{\mu 5ABC\beta D}\right)\right.$
(3.84) $\displaystyle+g_{\mu\beta}\text{tr}\left(\gamma_{A\nu B\alpha
C5D}\right)+g_{\mu\alpha}\text{tr}\left(\gamma_{A\nu B5C\beta
D}\right)+g_{\mu\nu}\text{tr}\left(\gamma_{A5B\alpha C\beta D}\right)$
$\displaystyle\left.+\left(g_{\mu\nu}g_{\alpha\beta}-g_{\mu\alpha}g_{\nu\beta}+g_{\mu\beta}g_{\nu\alpha}\right)\text{tr}\left(\gamma_{A5BCD}\right)\right]\frac{K_{1}^{A}K_{2}^{B}K_{3}^{C}K_{4}^{D}}{D_{1234}}$
$\displaystyle+m^{2}\left[-g_{\alpha\beta}\text{tr}\left(\gamma_{5\mu A\nu
B}\right)-g_{\nu\beta}\text{tr}\left(\gamma_{5\mu
AB\alpha}\right)+g_{\nu\alpha}\text{tr}\left(\gamma_{5\mu
AB\beta}\right)\right.$
$\displaystyle\left.-g_{\mu\beta}\text{tr}\left(\gamma_{5A\nu
B\alpha}\right)+g_{\mu\alpha}\text{tr}\left(\gamma_{5A\nu
B\beta}\right)+g_{\mu\nu}\text{tr}\left(\gamma_{5AB\alpha\beta}\right)\right]\frac{K_{1}^{A}K_{2}^{B}}{D_{1234}}$
$\displaystyle+m^{2}\left[g_{\alpha\beta}\text{tr}\left(\gamma_{5\mu A\nu
C}\right)-g_{\nu\beta}\text{tr}\left(\gamma_{5\mu A\alpha
C}\right)-g_{\nu\alpha}\text{tr}\left(\gamma_{5\mu AC\beta}\right)\right.$
$\displaystyle\left.-g_{\mu\beta}\text{tr}\left(\gamma_{5A\nu\alpha
C}\right)-g_{\mu\alpha}\text{tr}\left(\gamma_{5A\nu
C\beta}\right)+g_{\mu\nu}\text{tr}\left(\gamma_{5A\alpha
C\beta}\right)\right]\frac{K_{1}^{A}K_{3}^{C}}{D_{1234}}$
$\displaystyle+m^{2}\left[-g_{\alpha\beta}\text{tr}\left(\gamma_{5\mu A\nu
D}\right)+g_{\nu\beta}\text{tr}\left(\gamma_{5\mu A\alpha
D}\right)-g_{\nu\alpha}\text{tr}\left(\gamma_{5\mu A\beta D}\right)\right.$
$\displaystyle\left.g_{\mu\beta}\text{tr}\left(\gamma_{5A\nu\alpha
D}\right)-g_{\mu\alpha}\text{tr}\left(\gamma_{5A\nu\beta
D}\right)+g_{\mu\nu}\text{tr}\left(\gamma_{5A\alpha\beta
D}\right)\right]\frac{K_{1}^{A}K_{4}^{D}}{D_{1234}}$
$\displaystyle+m^{2}\left[g_{\alpha\beta}\text{tr}\left(\gamma_{5\mu\nu
BC}\right)+g_{\nu\beta}\text{tr}\left(\gamma_{5\mu B\alpha
C}\right)+g_{\nu\alpha}\text{tr}\left(\gamma_{5\mu BC\beta}\right)\right.$
$\displaystyle\left.-g_{\mu\beta}\text{tr}\left(\gamma_{5\nu B\alpha
C}\right)-g_{\mu\alpha}\text{tr}\left(\gamma_{5\nu
BC\beta}\right)-g_{\mu\nu}\text{tr}\left(\gamma_{5B\alpha
C\beta}\right)\right]\frac{K_{2}^{B}K_{3}^{C}}{D_{1234}}$
$\displaystyle+m^{2}\left[-g_{\alpha\beta}\text{tr}\left(\gamma_{5\mu\nu
BD}\right)-g_{\nu\beta}\text{tr}\left(\gamma_{5\mu B\alpha
D}\right)+g_{\nu\alpha}\text{tr}\left(\gamma_{5\mu B\beta D}\right)\right.$
$\displaystyle\left.+g_{\mu\beta}\text{tr}\left(\gamma_{5\nu B\alpha
D}\right)-g_{\mu\alpha}\text{tr}\left(\gamma_{5\nu B\beta
D}\right)-g_{\mu\nu}\text{tr}\left(\gamma_{5B\alpha\beta