# Infrared Subtleties and Chiral Vertices at NLO: An Implicit Regularization
Analysis

Ricardo J. C. Rosado  CFisUC, Department of Physics, University of Coimbra,
P-3004-516 Coimbra, Portugal  Adriano Cherchiglia  Instituto de Física Gleb
Wataghin, Universidade Estadual de Campinas,
Rua Sérgio Buarque de Holanda, 777, Campinas, SP, Brasil  Departamento de
Física Teórica y del Cosmos, Universidad de Granada, Campus de Fuentenueva,
E–18071 Granada, Spain  Marcos Sampaio  Universidade Federal do ABC, 09210-580
, Santo André, Brasil  Brigitte Hiller  CFisUC, Department of Physics,
University of Coimbra, P-3004-516 Coimbra, Portugal

###### Abstract

We employ implicit regularization (IReg) in quark-antiquark decays of the Z,
or of a scalar (CP-even or odd) boson at NLO, and compare with dimensional
schemes to reveal subtleties involving infrared divergence cancellation and
$\gamma_{5}$-matrix issues. Besides the absence of evanescent fields in IReg,
such as $\epsilon$-scalars required in certain schemes that operate partially
in the physical dimension, we verify that our procedure preserves gauge
invariance in the presence of the $\gamma_{5}$ matrix without requiring
symmetry preserving counterterms while the amplitude is infrared finite as
stated by the KLN theorem.

##  1 Introduction

Different regularization frameworks have been used in quantum field theory.
Each of these frameworks has its advantages and disadvantages, and the choice
of the appropriate framework depends on the specific problem being considered
[1, 2]. On the other hand, the evaluation of precision observables is a
challenging task, primarily involving the numerical evaluation of higher-order
perturbative cross sections and decay processes where ultraviolet and infrared
divergences appear in intermediate steps.

A fully mathematical consistent regularization scheme that prevents the
occurrence of symmetry breaking terms or spurious anomalies for the Standard
Model and its extensions, and that is valid to all orders in perturbation
theory is not available yet. In perturbative calculations at next-to-leading
order (NLO) and beyond, infrared and ultraviolet divergences commonly arise
due to the presence of loop diagrams and radiation from external legs in
Feynman diagrams. Finitude theorems guarantee the finiteness of perturbative
calculations order by order under the hypothesis that a unitarity-preserving
regularization method was employed to regulate these divergences. However,
some regularization methods may not be completely consistent, especially when
applied to extensions of the Standard Model or chiral theories.

For example, the cross section of single photon emission exhibits an infrared
divergence in the limit of vanishing photon energy. This problem was initially
solved by Bloch and Nordsieck (BN) [3], who showed that the infrared
divergence can be canceled out by considering inclusive processes where the
bremsstrahlung contribution is combined with radiative corrections order by
order in perturbation theory. Early in 1960, Kinoshita-Lee-Nauenberg (KLN) [4,
5] stated that S-matrix elements squared are IR finite when a sum is performed
over final states and initial states within an energy window (degenerate
states). In other words cancellation of IR divergences follows directly from
unitarity provided the measurement is inclusive enough: a hard parton can not
be distinguished from a hard particle plus a soft gluon or from two collinear
partons with the same energy. Schematically,

$\text{KLN-theorem:}\sum_{i,f\in[E-\Delta,E+\Delta]}|\langle
f|S|i\rangle|^{2}=\text{finite}.$ (1)

In this way, the BN theorem is a special case of the KLN theorem.The
computational challenge is to find the minimal set of diagrams needed for IR
finiteness.

At a certain order in perturbation theory, a subtle cancellation happens
between IR divergences coming from coherently summed amplitudes (at the level
of phase space integrals) and virtual IR (and possibly UV) divergent
amplitudes.Therefore, a well-chosen regularization scheme is needed to handle
the UV divergences, and also to ensure that the IR divergences cancel
properly. Ideally, the regularization scheme must also respect the symmetries
of the theory, and should not introduce spurious anomalies or breaking of
symmetries.

Standard (conventional) dimensional regularization (CDR) [6] and the closely
related dimensional scheme of ’t Hooft and Veltman (HV) [7] are the natural
choice for Feynman amplitude calculations in gauge theories. Momenta and loop
internal gauge fields are treated as $d$-dimensional objects, with
$d=4-2\epsilon$ and $\epsilon\rightarrow 0$, while external gauge fields are
$d$-dimensional in CDR and strictly four dimensional in HV. However
dimensional extensions meet some challenges in the case of theories involving
dimension specific quantities, such as the $\gamma_{5}$ matrix [8] and the
Levi-Civita tensor in chiral and topological theories [9], as well as in
supersymmetric gauge theories [10]. In the latter, a mismatch between the
number of degrees of freedom of gauge fields (d) and gauginos (4) breaks
supersymmetry in CDR.

Regarding $\gamma_{5}$ matrix issues in dimensional schemes, in the HV scheme,
the loss of the anti-commuting property of $\gamma_{5}$ in d dimensions breaks
BRST symmetry. The solution is the addition of symmetry restoring counterterms
(CT) using the Breitenlohner-Maison (BM) [11] scheme order by order in
perturbation theory [12, 13]. A gauge invariant procedure has been advocated
in [14, 15] which allows to reduce the number of CT structures in the BM
scheme, often denoted as ”rightmost-position” method. Some alternatives have
also been considered to maintain the anti-commuting property of $\gamma_{5}$
in d dimensions whereas preserving gauge invariance and BRST symmetry at the
cost of giving up the cyclic properties of the trace [16, 17] 111See [18] for
recent subtleties when employing this scheme..

Apart from CDR and HV, other dimensional schemes have been developed to
operate partially in the physical dimension. In such schemes, fields are
treated differently with the help of additional metric spaces [19], as in
dimensional reduction (DRED) [20, 21] and four dimensional helicity (FDH) [22,
23]. Both consider that gauge fields in the loop live in quasi four
dimensional spaces (Q4S) while external gauge fields are strictly four
dimensional (4S) in FDH. DRED allows to consistently treat supersymmetric
theories to 2-loop order [24, 25] and FDH makes use of the efficient spinor
helicity technique for the spin algebra of observables as these are defined in
the physical dimension. This comes nevertheless at the cost of introducing
extra fields that transform as scalars under Lorentz transformations, known as
evanescent fields or $\epsilon$-scalars. They renormalize differently from the
gauge field, and it is essential to kept track of these differences in order
to have a scheme that respects unitarity [26].

To fully explore the advantages of each of the schemes it is important that
conversions between results in CDR, HV, FDH, and DRED can be made at different
steps of the calculation of a cross section. This can be done, for instance,
using the scheme dependence of beta functions and anomalous dimensions [27,
28].

On the other hand, methods that do not rely on dimensional extensions are
emerging and being explored in diverse venues, with the intent of
circumventing the above mentioned increase in complexity of dimensional
schemes, such as the four dimensional unsubtracted (FDU) method [29, 30], the
four dimensional regularization (FDR) [31], and the implicit regularization
(IReg) [32, 33, 34, 35]. For recent views on some of these methods see [36,
37]. One of the central objectives of these methods is to analytically
implement a clear separation of UV and IR divergent content of Feynman
amplitudes from the finite parts. The motivation is that this separation is
valuable for simplifications in both the renormalization and the realization
of the KLN theorem, while finite integrals are efficiently evaluated
numerically.

The FDU method is anchored on the loop-tree duality theorem and enables the
cancellation of IR divergences at integrand level. In FDR the UV divergent
content of an amplitude is isolated and discarded under certain conditions
with the purpose of yielding directly a renormalized quantity at each order.
IReg isolates the UV content in form of basic divergent integrals (BDI) and
establishes all order relations among them which can be conveniently used in
the computation of renormalization functions (see section 2 for an overview).
The prominent feature of BDIs is that they do not depend on masses and
external momenta and endow a UV renormalization scale parameter.

A direct set of conversion rules between non-dimensional and dimensional
schemes at intermediate steps of a calculation is seen only up to NLO
processes [1]. Explicit calculations at NNLO processes and higher in IReg show
that there are restrictions to a direct extension of transition rules [38].
Indeed this is also the case in a calculation at NNLO order comparing FDH and
FDR showing that these schemes lack correspondence at intermediate steps.
While the technical obstacle of transferring results between different schemes
cannot be ignored, it is important to recognize that it may also present an
opportunity for reordering calculational steps that could be advantageous in
certain cases. Further investigation is required to fully comprehend the
implications of such deviations in intermediate steps [2].

The purpose of this contribution is to use strong corrections to the vector-
axialvector (V-A) decay $Z\rightarrow q\bar{q}$ and to the charge neutral
scalar (and pseudoscalar) decay $S\rightarrow q\bar{q}$ to NLO as a playground
that encompasses different regularization subtleties: $\gamma_{5}$ matrix
vertex, UV divergences, and virtual-real IR cancellation within the IReg
framework. Such calculation lends insight in the relations among schemes
involving the $\gamma_{5}$ matrix and IR finitude within IReg guaranteed by
the KLN theorem. Moreover we make a direct comparison with CDR and DRED aiming
at generalizing our strategy to physically more challenging and interesting
setups beyond NLO.

The paper is organized as follows. In section 2 we present an overview of the
IReg method. In section 3 various processes involving scalar, pseudoscalar,
pseudovector and vector decays are calculated and compared in section 4 with
well known results of CDR/HV as well as with results of DRED/FDH and non-
dimensional methods for the same processes. Conclusions summarize our results.
We also present a set of appendixes. In the first, we perform an analysis of
the role of the $\gamma_{5}$ matrix in IReg, justifying the usage of the
rightmost approach when analyzing the decays $Z\rightarrow q\bar{q}$, and
$S\rightarrow q\bar{q}$. In the second, we discuss the delicate issue of
observables containing an odd number of $\gamma_{5}$ matrices.

##  2 Overview of Implicit Regularization

In this section we present the rules of IReg focusing on one loop order and in
the massless limit as for simplicity we consider decays into massless quarks
(to study both soft and collinear infrared divergences). A complete $n$-loop
set of rules can be found in [37, 38].

In IReg, the extraction of the UV divergent content of a Feynman amplitude is
done by using algebraic identities at the integrand level. This is done in
alignment with Bogoliubov’s recursion formula [39, 40, 41], implying that the
way the method defines an UV convergent integral respects locality, Lorentz
invariance and unitarity [35]. IReg has been shown to respect abelian gauge
invariance to $n$-loop order [42, 43], as well as non-abelian and SUSY
symmetries in specific examples up to two-loop order [38, 44, 45, 46, 47].
This is achieved in a constrained version of the method, in which surface
terms (ST’s), which are related to momentum routing of loops in Feynman
diagrams, are set to zero. In the realm of applications, processes such as
$h\rightarrow\gamma\gamma$ [48], $e^{-}e^{+}\rightarrow\gamma^{*}\rightarrow
q\bar{q}(g)$ [1], and $H\rightarrow gg(g)$[49] were studied at NLO.

Consider a general $1$-loop Feynman amplitude where we denote by $k$ the
internal (loop) momenta, and $p_{i}$ the external momenta. To this amplitude,
we apply the set of rules:

  1. 1.

Perform Dirac algebra in the physical dimension.

  2. 2.

In order to respect numerator/denominator consistency, as described in the
reference [50], it is necessary to eliminate terms involving internal momenta
squared in the numerator by dividing them out from the denominator. For
instance,

$\int_{k}\dfrac{k^{2}}{k^{2}(k-p)^{2}}\bigg{|}_{\text{IREG}}\neq
g^{\alpha\beta}\int_{k}\dfrac{k_{\alpha}k_{\beta}}{k^{2}(k-p)^{2}}\bigg{|}_{\text{IREG}}\quad\mbox{where}\quad\int_{k}\equiv\int
d^{4}k/(2\pi)^{4}.$ (2)

  3. 3.

Include a fictitious mass $\mu^{2}$ in all propagators, where the limit
$\mu\rightarrow 0$ must be taken at the end of the calculation. In the
presence of IR divergences, a logarithm with $\mu^{2}$ will remain. Assuming
that we have an implicit regulator, we apply the following identity in all
propagators dependent on the external momenta $p_{i}$

$\displaystyle\frac{1}{(k-p_{i})^{2}-\mu^{2}}=\sum_{j=0}^{n-1}\frac{(-1)^{j}(p_{i}^{2}-2p_{i}\cdot
k)^{j}}{(k^{2}-\mu^{2})^{j+1}}+\frac{(-1)^{n}(p_{i}^{2}-2p_{i}\cdot
k)^{n}}{(k^{2}-\mu^{2})^{n}\left[(k-p_{i})^{2}-\mu^{2}\right]}.$ (3)

Here $n$ is chosen such that the UV divergent part only has propagators of the
form $(k^{2}-\mu^{2})^{-j}$.

  4. 4.

Express UV divergences in terms of Basic Divergent Integrals (BDI’s) of the
form222UV divergences of quadratic nature (or higher) could also be kept in
the framework of IReg, however, they will always cancel in theories that are
multiplicative renormalizable. For some examples showing the explicit
cancellation, see [34, 51, 52, 49].

$\displaystyle I_{log}(\mu^{2})$ $\displaystyle\equiv$
$\displaystyle\int_{k}\frac{1}{(k^{2}-\mu^{2})^{2}},\quad\quad
I_{log}^{\nu_{1}\cdots\nu_{2r}}(\mu^{2})\equiv\int_{k}\frac{k^{\nu_{1}}\cdots
k^{\nu_{2r}}}{(k^{2}-\mu^{2})^{r+2}}.$ (4)

  5. 5.

Surface terms (weighted differences of loop integrals with the same degree of
divergence) should be set to zero on the grounds of momentum routing
invariance in the loop of Feynman diagrams. This constrained version
automatically preserves gauge invariance:

$\displaystyle\int_{k}\frac{\partial}{\partial
k_{\mu}}\frac{k^{\nu}}{(k^{2}-\mu^{2})^{2}}$ $\displaystyle=$ $\displaystyle
4\Bigg{[}\frac{g_{\mu\nu}}{4}I_{log}(\mu^{2})-I_{log}^{\mu\nu}(\mu^{2})\Bigg{]}=0.$
(5)

  6. 6.

A renormalization group scale can be introduced by disentangling the UV/IR
behavior of BDI’s under the limit $\mu\rightarrow 0$. This is achieved by
employing the identity

$I_{log}(\mu^{2})=I_{log}(\lambda^{2})+\frac{i}{(4\pi)^{2}}\ln\frac{\lambda^{2}}{\mu^{2}},$
(6)

It is possible to absorb the BDI’s in the renormalisation constants (without
explicit evaluation) [53], and renormalisation functions can be readily
computed using

$\lambda^{2}\frac{\partial
I_{log}(\lambda^{2})}{\partial\lambda^{2}}=-\frac{i}{(4\pi)^{2}}.$ (7)

The above rules will be applied in the virtual contributions of the processes
studied in this work. For the real contributions, we consider that the
massless particles in the final state have the same fictitious mass $\mu$
introduced in step 3. This allows to also parametrise the infrared divergences
coming from the real part in terms of logarithms of $\mu^{2}$, when performing
the integration over a massive phase-space region. Regarding the matrix-
element, it can still be computed in the massless limit, as we show in our
examples.

Finally, since the treatment of the $\gamma_{5}$ matrix presents many
subtleties, we will explain in appendix A how the $\gamma_{5}$ can be
consistently treated in connection with IReg.

##  3 Decays to quarks and antiquarks within IReg

In this section we present our main results. This work completes the study of
NLO strong corrections to the decay of bosons to massless quarks and
antiquarks within IReg. The off-shell photon decay was first considered in
[1], while here we will consider the decay of the Z-boson as well as of
neutral scalars. Even though in the SM only one physical scalar particle is
introduced (the Higgs boson), which is CP-even, for completeness we will also
provide the result for the decay of a pseudo-scalar particle, which is present
in many Beyond Standard Model extensions.

###  3.1 NLO strong corrections to $Z\rightarrow q\bar{q}$

In order to set our notation, we begin with the tree level decay rate of
$Z\rightarrow q\bar{q}$. As standard, we need to compute

$\Gamma_{t}=\frac{1}{2m_{z}}\int\frac{d^{3}q}{(2\pi)^{3}2q_{0}}\frac{d^{3}\overline{q}}{(2\pi)^{3}2\overline{q}_{0}}\sum_{spin}|M_{\rm{tree}}|^{2}(2\pi)^{4}\delta^{4}(z-q-\overline{q})\,,$
(8)

where $z^{\mu},m_{z},q^{\mu},\bar{q}^{\mu}$ are the four-momentum of the
Z-boson, its mass, quark and anti-quark momenta, respectively, and
$M_{\rm{tree}}$ is the tree-level amplitude

$M_{\rm{tree}}=\overline{u}(q)\cdot\frac{-ie\gamma^{\mu}Z_{-}}{\sin(2\omega)}\cdot
v(\overline{q})\epsilon_{\mu}(z)\,.$ (9)

We define $Z_{\pm}=(g_{V}\pm\gamma^{5}g_{A})$ with $g_{V}$ the vector
component of the interaction, given by
$g_{V}=I_{3}-2Q^{\prime}sin^{2}(\omega)$, while $g_{A}$ is the axial
component, given by $g_{A}=I_{3}$. $I_{3}$ is the third component of the
particles’ isospin, $Q^{\prime}$ is the unitary charge and $\omega$ the weak
mixing angle [54]. As can be noticed, there is a $\gamma_{5}$ matrix which may
cause ambiguities under regularization. At the present stage, no particular
treatment is required, since we are still at tree-level. After a
straightforward calculation, one obtains [55]

$\Gamma_{t}=\frac{e^{2}(g_{V}^{2}+g_{A}^{2})m_{z}}{4\pi\sin^{2}(2\omega)}.$
(10)

####  3.1.1 Virtual decay rate

Regarding the NLO correction, we begin discussing the virtual decay rate,
which stems from the diagram of fig.1.

Figure 1: Feynman diagram for the virtual contribution to decay $Z\rightarrow
q\bar{q}$.

We consider massless quarks, which renders the following amplitude

$M_{v}=\epsilon_{\mu}(z)\cdot\int_{k}\overline{u}(q)\cdot(-ig_{s}\gamma^{\alpha}t^{a})\cdot\frac{-i}{\not{q}+\not{k}}\cdot\frac{-ie}{\sin(2\omega)}\gamma^{\mu}Z_{-}\cdot\frac{i}{\not{\overline{q}}-\not{k}}\cdot(-ig_{s}\gamma^{\beta}t^{b})\cdot\frac{-ig_{\alpha\beta}\delta_{ab}}{k^{2}}\cdot
v(\overline{q}).$ (11)

Here $t^{a}$ are the color Gell-Mann matrices and $g_{s}$ the strong
interaction constant (see also definitions below eq.20). In order to deal with
the $\gamma_{5}$ matrix, we will adopt the rightmost approach [14, 15] which
implies that $Z_{\pm}$ must be moved to the rightmost position. In the
Appendix A we justify the use of this procedure in the context of IReg for the
processes calculated in the present work. This allows us to rewrite the
amplitude as follows

$\begin{split}M_{v}=4\frac{g_{s}^{2}(t^{a})^{2}e}{\sin(2\omega)}\epsilon_{\mu}(z)\overline{u}(q)[(q.\overline{q})I\gamma^{\mu}+(q^{\mu}-\overline{q}^{\mu})\not{I}-\gamma^{\mu}(q-\overline{q})_{\alpha}I^{\alpha}+\gamma_{\delta}I^{\delta\mu}-\frac{I_{2}}{2}\gamma^{\mu}]Z_{-}v(\overline{q}).\end{split}$
(12)

The integrals are regularized within IReg as below

$\displaystyle I$
$\displaystyle=\int_{k}\frac{1}{k^{2}(q+k)^{2}(\overline{q}-k)^{2}}=\frac{b}{2m_{z}^{2}}[\ln^{2}(\mu_{0})+2i\pi\ln(\mu_{0})-\pi^{2}],$
(13) $\displaystyle I^{\mu}$
$\displaystyle=\int_{k}\frac{k^{\mu}}{k^{2}(q+k)^{2}(\overline{q}-k)^{2}}=\frac{b}{m_{z}^{2}}(q^{\mu}-\overline{q}^{\mu})\left[\ln\left(\mu_{0}\right)+i\pi+2\right],$
(14) $\displaystyle I^{\mu\nu}$
$\displaystyle=\int_{k}\frac{k^{\mu}k^{\nu}}{k^{2}(q+k)^{2}(\overline{q}-k)^{2}}=\left[I_{log}(\mu^{2})+b(\ln(\mu_{0})+i\pi+3)\right]\frac{g^{\mu\nu}}{4}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\frac{b}{2m_{z}^{2}}\left[(q^{\mu}q^{\nu}+\overline{q}^{\mu}\overline{q}^{\nu})(\ln(\mu_{0})+2)+q^{\mu}\overline{q}^{\nu}+q^{\nu}\overline{q}^{\mu}\right],$
(15) $\displaystyle I_{2}$
$\displaystyle=\int_{k}\frac{k^{2}}{k^{2}(q+k)^{2}(\overline{q}-k)^{2}}=I_{log}(\mu^{2})+b[\ln(\mu_{0})+i\pi+2],$
(16)

where $\mu_{0}=\mu^{2}/m_{z}^{2}$.

As can be seen, some of the integrals contain a UV divergence,
$I_{log}(\mu^{2})$, that must still be removed by adopting a regularization
scheme. We will choose the on-shell scheme, noticing that the Z-boson behaves
as a spectator in our entire calculation. Thus, apart from factors such as
$g_{V},g_{A}$, and $\sin^{2}(2\omega)$ we will obtain a similar result to the
process $e^{+}e^{-}\rightarrow\gamma^{*}\rightarrow q\overline{q}$. To be
precise, in the on-shell scheme, the electromagnetic charge is renormalized by
requiring $\overline{u}(p)\Gamma_{\mu}^{\gamma
ee}(p,p)v(p)=ie\bar{u}(p)\gamma_{\mu}v(p)$, where $\Gamma_{\mu}^{\gamma
ee}(p,p)$ is the amputated vertex function for $Ae\bar{e}$ and $e$, $A$ are
the electron and photon fields respectively [56]. Our calculation is
completely analogous, yielding

$\displaystyle M_{v}(z=0\;;\;q=\overline{q})=$
$\displaystyle\frac{e}{\sin(2\omega)}\epsilon_{\mu}(z)\int_{k}\overline{u}(q)\cdot(g_{s}\gamma^{\alpha}t^{a})\cdot\frac{1}{\not{q}+\not{k}}\cdot\gamma^{\mu}Z_{-}\cdot\frac{1}{\not{q}-\not{k}}\cdot(g_{s}\gamma_{\alpha}t^{a})\cdot\frac{1}{k^{2}}\cdot
v(q)$ $\displaystyle=$
$\displaystyle-\frac{ie}{\sin(2\omega)}\epsilon_{\mu}(z)\overline{u}(q)\gamma^{\mu}Z_{-}v(\overline{q})\left[C_{F}\frac{\alpha_{s}}{4\pi}\frac{I_{log}(\mu^{2})}{b}\right]\;.$
(17)

Notice that the term left of the bracket is exactly the tree-level vertex
$Zq\overline{q}$, and $e$ is a bare charge. By expressing it in terms of the
renormalized charge $e=Z_{e}e_{r}=(1+\delta_{e})e_{r}$, it is possible to
obtain the counterterm $\delta_{e}$ by

$\delta_{e}=-C_{F}\left(\frac{\alpha_{s}}{\pi}\right)\frac{I_{log}(\mu^{2})}{4b}\;.$
(18)

In section 3.2.1 we will present the renormalization function for the fermion
field, $Z_{2}$, which fulfills $Z_{e}=Z_{2}$ as expected by virtue of the Ward
identity.

Once the amplitude is regularized, it is straightforward to obtain the decay
rate which, at NLO, is given by the interference term between the tree-level
and one-loop amplitudes

$\Gamma_{v}=\frac{1}{16\pi m_{z}}[2Re(M_{t}^{\dagger}M_{v})],$ (19)

with the result

$\Gamma_{v}=\Gamma_{t}C_{F}\left(\frac{\alpha_{s}}{\pi}\right)\left[-\frac{\ln^{2}(\mu_{0})}{2}-\frac{3}{2}\ln(\mu_{0})-\frac{7}{2}+\frac{\pi^{2}}{2}\right].$
(20)

As standard, we used $(t^{a})^{2}=C_{F}$, and $\alpha_{s}=g_{s}^{2}/(4\pi)$.

It is clear from the equation above that all dependence on the Z-boson vertex
is included in $\Gamma_{t}$. Thus, the result above is compatible with the one
obtained before for the process $e^{+}e^{-}\rightarrow\gamma^{*}\rightarrow
q\bar{q}$ [1], where the off-shell photon would play the same role as the
Z-boson here.

####  3.1.2 Real decay rate

Once the virtual correction was obtained, we compute on this section the real
contributions which are given by the diagrams of fig.2.

Figure 2: Feynman diagrams for the real contribution to the decay
$Z\rightarrow q\bar{q}$.

The decay rate is obtained from the amplitude

$M_{r}=\epsilon_{\mu}(z)\overline{u}(q)\left[(-ig\gamma^{\alpha}t^{a})\cdot\frac{-i}{\not{q}+\not{k}}\cdot\frac{-ie\gamma^{\mu}Z_{-}}{\sin(2\omega)}+\frac{-ie\gamma^{\mu}Z_{-}}{\sin(2\omega)}\cdot\frac{i}{\not{\overline{q}}+\not{k}}\cdot(-ig\gamma^{\alpha}t^{a})\right]v(\overline{q})\epsilon_{\alpha}(k).$
(21)

We introduce the following notation

$\displaystyle\chi$
$\displaystyle\equiv\frac{(z-q)^{2}}{m_{z}^{2}}-\frac{\mu^{2}}{m_{z}^{2}}=\frac{(\overline{q}+k)^{2}}{m_{z}^{2}}-\frac{\mu^{2}}{m_{z}^{2}},$
(22) $\displaystyle\overline{\chi}$
$\displaystyle\equiv\frac{(z-\overline{q})^{2}}{m_{z}^{2}}-\frac{\mu^{2}}{m_{z}^{2}}=\frac{(q+k)^{2}}{m_{z}^{2}}-\frac{\mu^{2}}{m_{z}^{2}}.$
(23)

In terms of $\chi$, and $\overline{\chi}$, the modulus squared of the
amplitude is given by

$\begin{split}|M_{r}|^{2}=\frac{8(t^{a})^{2}e^{2}g^{2}(g_{V}^{2}+g_{A}^{2})}{\sin^{2}(2\omega)}\left[\frac{2-2\chi-2\overline{\chi}+\overline{\chi}^{2}+\chi^{2}}{(\chi+\mu_{0})(\overline{\chi}+\mu_{0})}\right].\end{split}$
(24)

We recall that we regularize the phase space integrals by introducing a
fictitious mass $\mu$ in the propagator of the massless particles, which
explains the presence of this term in $\chi$, and $\overline{\chi}$. To
proceed we will make use of the results [57, 1].

$\displaystyle\int^{1-2\sqrt{\mu_{0}}}_{3\mu_{0}}\int^{\overline{\chi}^{+}}_{\overline{\chi}^{-}}\frac{1}{(\chi+\mu_{0})(\overline{\chi}+\mu_{0})}d\chi
d\overline{\chi}=\frac{\ln^{2}(\mu_{0})-\pi^{2}}{2}$
$\displaystyle\int^{1-2\sqrt{\mu_{0}}}_{3\mu_{0}}\int^{\overline{\chi}^{+}}_{\overline{\chi}^{-}}\frac{\chi^{a}}{(\overline{\chi}+\mu_{0})}d\chi
d\overline{\chi}=-\frac{1}{a+1}\ln(\mu_{0})-\frac{1}{a+1}\left(\frac{1}{a+1}+2\sum^{a+1}_{n=1}\frac{1}{n}\right)$
(25)

where
$\overline{\chi}^{\pm}=\frac{1-\chi}{2}\pm\sqrt{\frac{(\chi-3\mu_{0})[(1-\chi)^{2}-4\mu_{0}]}{4(\chi+\mu_{0})}}$.
The same results hold if we replace $\chi\rightarrow\overline{\chi}$ and vice-
versa in the integrand. Finally, the end result for the real contributions is

$\Gamma_{r}=\Gamma_{t}C_{F}\left(\frac{\alpha_{s}}{\pi}\right)\left[\frac{\ln^{2}(\mu_{0})}{2}-\frac{\pi^{2}}{2}+\frac{\ln(\mu_{0})}{2}+\frac{17}{4}\right],$
(26)

which is once again compatible with the result of [1].

Once the virtual and real contributions were calculated, we can obtain the NLO
decay rate for the Z-boson to a pair of quark and antiquarks in the framework
of IReg as

$\Gamma_{\rm
NLO}=\Gamma_{t}+\Gamma_{v}+\Gamma_{r}=\Gamma_{t}\left(1+\frac{3C_{F}\alpha_{s}}{4\pi}\right)\;.$
(27)

Specializing to QCD, we have $C_{F}=4/3$, which renders the well-known result
[55]

$\Gamma_{\rm NLO}=\Gamma_{t}\left(1+\frac{\alpha_{s}}{\pi}\right)\;.$ (28)

###  3.2 NLO strong corrections to $S\rightarrow q\bar{q}$

In this section we focus on the decay rate of a scalar (CP-even or odd) to a
quark-antiquark pair. As before, we begin with the tree-level analysis.

Similarly to the Z-boson decay case, eq.(8), we have the following decay rate
for the scalar particle:

$\Gamma_{t}^{s}=\frac{1}{2m_{s}}\int\frac{d^{3}q}{(2\pi)^{3}2q_{0}}\frac{d^{3}\overline{q}}{(2\pi)^{3}2\overline{q}_{0}}\sum_{spin}|M_{\rm
tree}^{s}|^{2}(2\pi)^{4}\delta^{4}(s-q-\overline{q})\,,$ (29)

where $s$ is the four-momentum of the scalar, and $m_{s}$ is its mass. By
denoting the coupling of the scalar to the quarks by $\xi_{s}T$, where
$T=\mathbb{1}(\gamma_{5})$ for the CP-even (odd) scalar, it is straightforward
to obtain the result

$\Gamma_{t}^{s}=\xi_{s}^{2}\frac{m_{s}}{8\pi}\,.$ (30)

Once again, since we are at tree-level, no ambiguities related to the
$\gamma_{5}$ matrix occur. Notice that, in the context of the SM, $\xi_{s}\sim
m_{q}$, where $m_{q}$ is the mass of the quark to which the scalar
decays.333To be precise, in the SM we only have a CP-even scalar. However, the
same reasoning applies for other BSM models, in which the coupling between the
extra CP-odd boson is proportional to the quark mass. Therefore, we cannot
naively perform the NLO calculation for massless quarks, otherwise a null
result would be obtained. We will return to this point in the next section.

####  3.2.1 Virtual decay rate

Similarly to the Z-boson decay case, the virtual correction is due to the
diagram of fig.3.

Figure 3: Feynman diagram for the virtual contribution to decay $S\rightarrow
q\bar{q}$.

For simplicity, we will treat only the case in which the mass of the quarks is
arbitrarily small (massless limit), but the coupling $\xi_{s}$ is still non-
null. In this case, we have the amplitude below:

$M_{v}^{s}=\int\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(q)\cdot(-ig_{s}\gamma^{\alpha}t^{a})\cdot\frac{-i}{\not{q}+\not{k}}\cdot(-i\xi_{s}T)\cdot\frac{i}{\not{\overline{q}}-\not{k}}\cdot(-ig_{s}\gamma^{\beta}t^{b})\cdot\frac{-ig_{\alpha\beta}\delta_{ab}}{k^{2}}\cdot
v(\overline{q})\,.$ (31)

We recall that we are adopting the rightmost approach to deal with the
$\gamma_{5}$ matrix, which allows us to write

$M_{v}^{s}=4g_{s}^{2}(t^{a})^{2}\xi_{s}[(q\cdot\overline{q})I+(\overline{q}_{\mu}-q_{\mu})\cdot
I^{\mu}-I_{2}]\overline{u}(q)Tv(\overline{q})\,.$ (32)

The integrals in IReg are defined in eqs. (13-16), with the end result

$\displaystyle
M_{v}^{s}=C_{F}\frac{\alpha_{s}}{\pi}\xi_{s}\left[-\frac{(\ln(\mu_{0})+i\pi)^{2}}{4}+\frac{1}{b}I_{log}(\mu^{2})\right]\overline{u}(q)Tv(\overline{q})\,,$
(33)

where in this case $\mu_{0}=\mu^{2}/m_{s}^{2}$.

At this point, we would like to discuss some subtleties related to the
massless limit we are adopting. As mentioned, for non-null $\xi_{s}$, we are
implicitly assuming

$(\xi_{s})_{0}\sim(m_{q})_{0}=Z_{m_{q}}(m_{q})_{r}\,,$ (34)

where $x_{0/r}$ denotes a bare/renormalized quantity, and $Z_{m_{q}}$ is the
renormalization function of the quark mass. To obtain $Z_{m_{q}}$ in the
context of the on-shell subtraction scheme, we will need to evaluate the
strong corrections to the quark self-energy diagram which is given by fig.4.

Figure 4: Feynman diagram for the quark self-energy.

The on-shell renormalized mass is defined in such a way that it corresponds to
the pole of the renormalized fermion propagator. In the context of IReg, we
obtain

$Z_{m_{q}}=1+\delta_{m_{q}}=1-C_{F}\left(\frac{\alpha_{s}}{\pi}\right)\left\\{\frac{5}{4}+\frac{3}{4}\left[\ln\left(\frac{\mu^{2}}{m_{q}^{2}}\right)+\frac{1}{b}I_{log}(\mu^{2})\right]\right\\}\,.$
(35)

We will also need the renormalization function of the fermion field which is
given by [58]

$Z_{2}=1+\delta_{2}=1-C_{F}\left(\frac{\alpha_{s}}{\pi}\right)\frac{1}{4b}I_{log}(\mu^{2})\,.$
(36)

Finally, we recall that, in order to obtain the decay rate, we need to compute

$\Gamma^{s}=\frac{1}{16\pi
m_{s}}\sum_{spin}\left|M_{t}^{s}+M_{v}^{s}\right|^{2}\,.$ (37)

For consistency, $M_{t}^{s}$ must be expressed with the renormalized coupling
$(\xi_{s})_{r}$, and we need to account for the renormalization of the
external fermion legs [59] which implies

$\Gamma^{s}=\frac{m_{s}}{8\pi}(\xi_{s})_{r}^{2}\left|1+\delta_{m_{q}}+\delta_{2}+\frac{M_{v}^{s}}{(\xi_{s})_{r}}\right|^{2}\,.$
(38)

By defining $\Gamma_{v}^{s}$ as the virtual contribution to the decay rate we
are interested in, we obtain:

$\Gamma_{v}^{s}=\Gamma_{t}^{s}C_{F}\left(\frac{\alpha_{s}}{\pi}\right)\left\\{-\frac{\ln^{2}(\mu_{0})}{2}+\frac{\pi^{2}}{2}-\frac{5}{2}-\frac{3}{2}\ln\left(\frac{\mu^{2}}{m_{q}^{2}}\right)\,.\right\\}$
(39)

Notice that the UV integrals have canceled, as they should. The above result
was obtained in the context of the on-shell renormalization scheme. In order
to compare our result to dimensional schemes in the next section, we will
translate it to the $\overline{MS}$ scheme of CDR, in which the relation below
holds [60]:

$m_{q}(\lambda)=m_{q}\left[1-C_{F}\left(\frac{\alpha_{s}}{\pi}\right)\left(\frac{3}{4}\ln\left(\frac{\lambda^{2}}{m_{q}^{2}}\right)+1\right)\right]$
(40)

where $\lambda$ is a renormalization group scale. We then need to replace

$\xi_{s}\rightarrow\xi_{s}(\lambda)\left[1+C_{F}\left(\frac{\alpha_{s}}{\pi}\right)\left(\frac{3}{4}\ln\left(\frac{\lambda^{2}}{m_{q}^{2}}\right)+1\right)\right]\,,$
(41)

in eq.38. The final result will be

$\Gamma_{v}^{s}=\Gamma_{t}^{s}C_{F}\left(\frac{\alpha_{s}}{\pi}\right)\left\\{-\frac{\ln^{2}(\mu_{0})}{2}+\frac{\pi^{2}}{2}-\frac{1}{2}-\frac{3}{2}\ln\left(\mu_{0}\right)\right\\}\,,$
(42)

where we have adopted $\lambda^{2}=m_{s}^{2}$.

####  3.2.2 Real decay rate

The real contributions for the NLO strong correction to the decay
$S\rightarrow q\bar{q}$ are given by the diagrams in fig.5.

Figure 5: Feynman diagrams for the real contribution to the decay
$S\rightarrow q\bar{q}$.

Their amplitudes are easily obtained

$M_{r}^{s}=\overline{u}(q)\left[(-ig\gamma^{\alpha}t^{a})\cdot\frac{-i}{\not{q}+\not{k}}\cdot(-i\xi_{s}T)-(-i\xi_{s}T)\cdot\frac{i}{\not{\overline{q}}+\not{k}}\cdot(-ig\gamma^{\alpha}t^{a})\right]v(\overline{q})\epsilon^{*}_{\alpha}(k)\,,$
(43)

whose modulus squared is given by

$\left|M_{r}^{s}\right|^{2}=16g^{2}\xi_{s}^{2}\;\left[\frac{2-2\chi_{s}-2\overline{\chi_{s}}+(\chi_{s}+\overline{\chi_{s}})^{2}}{(\chi_{s}+\mu_{0})(\overline{\chi_{s}}+\mu_{0})}\right]\;.$
(44)

For simplicity, we introduced the notation

$\displaystyle\chi_{s}$
$\displaystyle\equiv\frac{(s-q)^{2}}{m_{s}^{2}}-\frac{\mu^{2}}{m_{s}^{2}},$
(45) $\displaystyle\overline{\chi_{s}}$
$\displaystyle\equiv\frac{(s-\overline{q})^{2}}{m_{s}^{2}}-\frac{\mu^{2}}{m_{s}^{2}},$
(46)

where $s=q+\overline{q}+k$. Using the results of eq.(3.1.2) together with

$\int^{1-2\sqrt{\mu_{0}}}_{3\mu_{0}}\int^{\overline{\chi}^{+}}_{\overline{\chi}^{-}}d\chi
d\overline{\chi}=\frac{1}{2}\;,$ (47)

we obtain the end result for the real decay rate

$\Gamma_{r}^{s}=\Gamma_{t}^{s}C_{F}\left(\frac{\alpha_{s}}{\pi}\right)\left[\frac{\ln^{2}(\mu_{0})}{2}-\frac{\pi^{2}}{2}+\frac{3}{2}\ln(\mu_{0})+\frac{19}{4}\right]\;.$
(48)

Finally, adding the virtual, eq.(42), and the real, eq.(48), corrections we
obtain the well-known result [59, 61, 62, 63, 64]

$\Gamma^{s}=\Gamma_{t}^{s}\left[1+C_{F}\frac{17\alpha_{s}}{4\pi}\right]\,.$
(49)

##  4 Comparison with dimensional methods

Once we have obtained the NLO results for the decay of a gauge boson or scalar
to a pair of quark and antiquark in the framework of IReg, we aim to compare
them to the results for the same processes obtained by using dimensional
methods. As extensively discussed in [1, 2], the Dimensional Reduction method
(DRED) can be viewed as the most general of the dimensional schemes, allowing
to reproduce the results in Conventional Dimensional Regularization (CDR), for
instance, under certain limits. Moreover, in [2], a detailed analysis of the
decays $e^{-}e^{+}\rightarrow\gamma^{*}\rightarrow q\bar{q}$, and
$h\rightarrow q\bar{q}$ were reviewed, both at NLO and NNLO. By adopting the
$\gamma_{5}$ rightmost positioning approach, $\gamma_{5}$ matrices can be
completely isolated out of the integrals, allowing us to directly use the
results of [2]. See, for instance, eqs.12,32 in IReg which will have a
completely analogous counterpart in dimensional schemes.

###  4.1 NLO strong corrections to $S\rightarrow q\bar{q}$

In the case of the scalar decay, there are no external gauge bosons present,
implying that the treatment of the virtual corrections will require that only
the internal gluon is split according to the DRED approach. We denote the bare
amplitude at NLO as

$M_{\rm dim}=M_{\rm
dim}^{(0)}\left[1+e^{-\epsilon\gamma_{E}}(4\pi)^{\epsilon}\left(\frac{\mu_{\rm
dim}^{2}}{-m_{s}^{2}}\right)^{\epsilon}F^{(1)}_{\rm bare}\right]\,,$ (50)

where $\mu^{2}_{\rm dim}$ is the renormalization scale for dimensional
methods, $F^{(1)}$ is a form factor, and $M_{\rm
dim}^{(0)}=-i\overline{u}(q)\xi_{s}Tv(\overline{q})$ is the tree-level
amplitude using our notation.

The form factor $F^{(1)}_{\rm bare}$ is given by [2]

$F^{(1)}_{\rm
bare}=C_{F}\frac{\alpha_{s}^{0}}{\pi}\left[-\frac{1}{2\epsilon^{2}}-\frac{1}{2}+\frac{\pi^{2}}{24}\right]+C_{F}\frac{\alpha_{e}^{0}}{\pi}n_{\epsilon}\left[\frac{1}{4\epsilon}+\frac{1}{2}\right]\,,$
(51)

where $n_{\epsilon}=2\epsilon$ and $\alpha_{e}$ is related to the coupling of
the evanescent gluon to fermions. In the equation above, both couplings are to
be considered bare, although, at NLO, the distinction will not be essential.
On the other hand, since $\xi_{s}$ appear at tree-level, it is important to
consider its renormalization

$\xi^{0}_{s}=\xi_{s}\left[1+C_{F}\left(\frac{\alpha_{s}}{\pi}\right)e^{\epsilon\gamma_{E}}(4\pi)^{-\epsilon}\left(-\frac{3}{4\epsilon}-\frac{n_{\epsilon}}{8\epsilon}\right)\right]\,.$
(52)

It is straightforward to obtain the modulus squared of the amplitude, which
yields the following decay rate

$\Gamma^{s}_{v(\rm
dim)}=\Gamma^{s}_{t}C_{F}\Phi_{2}(\epsilon)c_{\Gamma}(\epsilon)m_{s}^{-\epsilon}\left[\frac{\alpha_{s}}{\pi}\left(-\frac{1}{\epsilon^{2}}-\frac{3}{2\epsilon}-1+\frac{\pi^{2}}{2}+O(\epsilon)\right)+\frac{\alpha_{\epsilon}}{\pi}\left(\frac{n_{\epsilon}}{4\epsilon}+O(\epsilon)\right)\right]\,,$
(53)

where

$c_{\Gamma}(\epsilon)=(4\pi)^{\epsilon}\frac{\Gamma(1+\epsilon)\Gamma^{2}(1-\epsilon)}{\Gamma(1-2\epsilon)}=1+O(\epsilon),\quad\Phi_{2}(\epsilon)=\left(\frac{4\pi}{m_{s}}\right)^{\epsilon}\frac{\Gamma(1-\epsilon)}{\Gamma(2-2\epsilon)}=1+O(\epsilon).$
(54)

It is valuable to make a comparison with the IReg result, as given by equation
(42). As previously noticed in [1], the matching between the IR divergences in
dimensional methods and IReg is given by
$\frac{1}{\epsilon}\rightarrow\ln(\mu_{0})$ and
$\frac{1}{\epsilon^{2}}\rightarrow\frac{1}{2}\ln(\mu_{0})^{2}$.

Moreover, by identifying $n_{\epsilon}=2\epsilon$ and setting
$\alpha_{s}=\alpha_{e}$, we also recover the finite term. We notice that the
result of CDR is obtained by setting $n_{\epsilon}=0$.

Regarding the real contribution, the decay rate is given by [2]

$\Gamma^{s}_{r(\rm
dim)}=\Gamma^{s}_{t}C_{F}\Phi_{3}(\epsilon)\left[\frac{\alpha_{s}}{\pi}\left(\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+\frac{21}{4}-\frac{\pi^{2}}{2}+O(\epsilon)\right)+\frac{\alpha_{\epsilon}}{\pi}\left(-\frac{n_{\epsilon}}{4\epsilon}+O(\epsilon^{0})\right)\right]\,,$
(55)

where

$\Phi_{3}(\epsilon)=\left(\frac{4\pi}{h}\right)^{2\epsilon}\frac{1}{\Gamma(2-2\epsilon)}=1+O(\epsilon)\,.$
(56)

Notice that we may convert into the IReg result expressed by Eq. (48) under
the same conditions imposed to virtual contributions. By adding both
corrections we reproduce the well-known result expressed by Eq. (49).

###  4.2 NLO strong corrections to $Z\rightarrow q\bar{q}$

In [1] the decay $e^{-}e^{+}\rightarrow q\bar{q}$ was computed at NLO using
DRED. In that reference, only the strong correction was considered, which
stands for modification only in the external legs containing quarks. Moreover,
the mediator between leptons and quarks was an off-shell photon. The same
process could occur with a Z-boson replaced mediator. If we are not interested
in the initial states that will eventually generate the Z-boson, one can
extract from the results of [1] the decay rate for the Z boson into a pair of
quark and antiquark. In this case the virtual and real corrections are given
by

$\displaystyle\Gamma_{v(\rm dim)}$
$\displaystyle=\Gamma_{t}C_{F}\Phi_{2}(\epsilon)c_{\Gamma}(\epsilon)m_{s}^{-\epsilon}\left[\frac{\alpha_{s}}{\pi}\left(-\frac{1}{\epsilon^{2}}-\frac{3}{2\epsilon}-4+\frac{\pi^{2}}{2}+O(\epsilon)\right)+\frac{\alpha_{\epsilon}}{\pi}\left(\frac{n_{\epsilon}}{4\epsilon}+O(\epsilon^{0})\right)\right]\,,$
(57) $\displaystyle\Gamma_{r(\rm dim)}$
$\displaystyle=\Gamma_{t}C_{f}\Phi_{3}(\epsilon)\left[\frac{\alpha_{s}}{\pi}\left(\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+\frac{19}{4}-\frac{\pi^{2}}{2}+O(\epsilon)\right)+\frac{\alpha_{\epsilon}}{\pi}\left(-\frac{n_{\epsilon}}{4\epsilon}+O(\epsilon^{0})\right)\right]\,.$
(58)

As discussed before, the correspondence
$\frac{1}{\epsilon}\rightarrow\ln(\mu_{0})$ and
$\frac{1}{\epsilon^{2}}\rightarrow{\frac{1}{2}}\ln^{2}(\mu_{0})$ is verified
and the IReg result is reproduced after setting $n_{\epsilon}=2\epsilon$ and
identifying $\alpha_{s}=\alpha_{e}$ as seen in Eqns. (20) and (26)

##  5 Concluding remarks

To improve theoretical accuracy in precision observables, there has been
considerable efforts to obtain beyond NLO corrections to processes probed at
the LHC. It is generally acknowledged that the regularization of UV and IR
poses challenges in automating higher order calculations. In response to this,
alternative approaches have been developed that avoid partially or totally the
use of dimensional continuation in the spacetime dimension such as IReg. These
approaches offer the potential to simplify calculations or may be applicable
in dimensional specific models such as chiral and topological ($\gamma_{5}$
matrix and Levi-Civita symbol issues) as well as supersymmetry models.

In this work, we studied at NLO the decay of bosons (spin zero or one) into
$q\bar{q}$ pairs in the framework of IReg, which is a fully quadridimensional
regularization scheme. In particular, we computed the NLO strong corrections
to the decay rates $\Gamma(Z\rightarrow q\bar{q})$ and $\Gamma(S\rightarrow
q\bar{q})$, where $S$ can be a CP-even or odd scalar. We have verified that
the KLN theorem is satisfied in our framework, and it is not necessary to
introduce evanescent particles, unlike in partially dimensional methods such
as FDH and DRED. We also compared IReg with these methods, showing that,
regarding IR divergences, there is a precise matching rule between IReg and
dimensional results at NLO, which was previously noticed in [1] regarding the
process $e^{-}e^{+}\rightarrow\gamma^{*}\rightarrow q\bar{q}$ and later
confirmed for the process $H\rightarrow gg$ as well [49]. Finally, since we
considered a gauge boson with axial couplings, the presence of $\gamma_{5}$
matrices can potentially lead to ambiguities in regularization and
renormalization. To tackle this problem, we adopted the $\gamma_{5}$ rightmost
position approach which is sufficient to render IReg a gauge invariant
procedure in this case while reproducing the results obtained with more
involved schemes in the literature. Therefore, IReg seems to be a feasible
alternative to dimensional schemes, whose application and automation to NLO
and beyond are active lines of research.

## Acknowledgements

We acknowledge support from Fundação para a Ciência e Tecnologia (FCT) through
the projects CERN /FIS-PAR /0040 /2019, CERN /FIS-COM /0035 /2019, UID /FIS
/04564 /2020, and the grant FCT 2020.07172.BD. A.C. acknowledges support from
National Council for Scientific and Technological Development – CNPq through
projects 166523/2020-8 and 201013/2022-3 and M. Sampaio acknowledges support
from CNPq through grant 302790/2020-9.

##  Appendix A Consistent treatment for $\gamma_{5}$

In this appendix we present a more detailed analysis of the ambiguities that
can be introduced by the treatment of the $\gamma_{5}$ matrix, and we show why
the rightmost approach is sufficient in the examples presented.

As it is well-known, the $\gamma_{5}$ matrix is a strictly four dimension
object, and it cannot be extended to $d$-dimensional. It means that, to treat
chiral theories in dimensional schemes, one must first adopt a definition for
$\gamma_{5}$, and check which of its four-dimensional properties are still
fulfilled. Soon after the development of CDR, it was proposed in [11] to still
define $\gamma_{5}$ in four-dimensions, while the remaining objects (Dirac
matrices and momenta) are promoted to $d$-dimensions. This approach (BMHV
scheme) necessarily invalidates the property
$\\{\gamma_{\mu},\gamma_{5}\\}=0$, and requires a careful treatment of the
sub-spaces in which CDR is defined. To be specific, CDR is defined in the
space $QdS=4S\oplus Q(-2\epsilon)S$, while $\gamma_{5}\in 4S$. Even though
this approach breaks gauge invariance, it is the only alternative that
delivers consistent (and unitarity-preserving) results at arbitrary loop order
in CDR.

In the case of methods defined in four-dimensions, somehow surprisingly, it
was also shown that inconsistent results can be obtained in the presence of
the $\gamma_{5}$ matrix within divergent integrals [65, 50, 66, 67]. In [50]
it was proposed a similar construction of dimensional schemes, in particular
Dimensional Reduction, where Dirac matrices, with the exception of
$\gamma_{5}$, are defined in a quasi-dimensional space $Q4S=QdS\oplus
Q(2\epsilon)S$ [24]. In contrast to DRED, where the momenta still need to be
treated in $QdS$, in these methods momenta are also defined in $Q4S$, and we
have the hierarchy $Q4S=4S\oplus XS$. Here, $X$ is an auxiliary space, which
does not need to be explicitly defined. Similar to the BHMV scheme, we obtain
a consistent method at the price of not fulfilling some properties,
$\\{\gamma_{\mu},\gamma_{5}\\}\neq 0$, and breaking gauge invariance.

In the specific case of IReg, the inconsistencies boil down to the contraction
of internal momenta in Feynman amplitudes. To illustrate this point, consider
the following results obtained in the framework of IReg

$\int_{k}\dfrac{k^{2}}{k^{2}(k-p)^{2}}=\int_{k}\dfrac{1}{(k-p)^{2}}=\lim_{\mu^{2}\rightarrow
0}\int_{k}\dfrac{1}{(k-p)^{2}-\mu^{2}}=\lim_{\mu^{2}\rightarrow
0}\int_{k}\dfrac{1}{k^{2}-\mu^{2}}=0$ (59) $\displaystyle
g^{\alpha\beta}\int_{k}\dfrac{k_{\alpha}k_{\beta}}{k^{2}(k-p)^{2}}$
$\displaystyle=g^{\alpha\beta}\left\\{\left(\frac{p_{\alpha}p_{\beta}}{3}-\frac{g_{\alpha\beta}p^{2}}{12}\right)\left[I_{\text{log}}(\lambda^{2})-b\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)+\frac{13b}{6}\right]-\frac{g_{\alpha\beta}bp^{2}}{24}\right\\}$
(60) $\displaystyle=-\frac{bp^{2}}{6}$

It is clear that if one insists to use $\\{\gamma_{\mu},\gamma_{5}\\}=0$, an
ambiguity arises

$\displaystyle\int_{k}\dfrac{\not{k}\gamma_{5}\not{k}}{k^{2}(k-p)^{2}}$
$\displaystyle\overset{?}{=}\frac{bp^{2}}{6}\gamma_{5},\quad\quad\mbox{using
eq.\ref{gab_kakb} and $\\{\gamma_{\mu},\gamma_{5}\\}=0$,}$ (61)
$\displaystyle\overset{?}{=}0,\quad\quad\quad\quad\mbox{using
$\\{\gamma_{\mu},\gamma_{5}\\}=0$ and eq.\ref{Nk2}.}$ (62)

In order to avoid these ambiguities, one defines

$\gamma_{5}=-\frac{i}{4!}\epsilon_{abcd}\bar{\gamma}^{a}\bar{\gamma}^{b}\bar{\gamma}^{c}\bar{\gamma}^{d}$
(63)

where we use a $\bar{}\;$ to denote an object pertaining to $4S$. Since the
Dirac matrices are defined in $Q4S$, we have the properties

$\displaystyle\\{\gamma_{\mu},\gamma_{\nu}\\}$
$\displaystyle=2g_{\mu\nu}\mathbb{1};\quad\\{\bar{\gamma}_{\mu},\bar{\gamma}_{\nu}\\}=\\{\gamma_{\mu},\bar{\gamma}_{\nu}\\}=2\bar{g}_{\mu\nu}\mathbb{1};\quad\gamma_{\mu}\gamma^{\mu}=\gamma_{\mu}\bar{\gamma}^{\mu}=4\;\mathbb{1}$
(64) $\displaystyle\\{\bar{\gamma}_{\mu},\hat{\gamma}_{\nu}\\}$
$\displaystyle=0;\quad\\{\gamma_{\mu},\hat{\gamma}_{\nu}\\}=\\{\hat{\gamma}_{\mu},\hat{\gamma}_{\nu}\\}=2\hat{g}_{\mu\nu}\mathbb{1};\quad\gamma_{\mu}\hat{\gamma}^{\mu}=\bar{\gamma}_{\mu}\hat{\gamma}^{\mu}=\hat{\gamma}_{\mu}\hat{\gamma}^{\mu}=0.$
(65)
$\displaystyle\\{\bar{\gamma}_{\mu},\gamma_{5}\\}=0;\quad\\{\gamma_{\mu},\gamma_{5}\\}=2\gamma_{5}\hat{\gamma_{\mu}};\quad[\hat{\gamma}_{\mu},\gamma_{5}]=0$
(66)

where we denoted by $\hat{}\;$ an object belonging to the $X$ space. In view
of the above properties, the previous integral is given by [67]

$\displaystyle\int_{k}\dfrac{\not{k}\gamma_{5}\not{k}}{k^{2}(k-p)^{2}}$
$\displaystyle=2\gamma_{5}\int_{k}\dfrac{\hat{\not{k}}\not{k}}{k^{2}(k-p)^{2}}-\int_{k}\dfrac{\gamma_{5}k^{2}}{k^{2}(k-p)^{2}}$
$\displaystyle=2\gamma_{5}\int_{k}\dfrac{k^{2}}{k^{2}(k-p)^{2}}-2\gamma_{5}\int_{k}\dfrac{\bar{k}^{2}}{k^{2}(k-p)^{2}}$
$\displaystyle=-2\bar{g}_{ab}\gamma_{5}\int_{k}\dfrac{\bar{k}^{a}\bar{k}^{b}}{k^{2}(k-p)^{2}}=\frac{bp^{2}}{3}\gamma_{5}.$
(67)

After laying down the main ideas, we tackle the specific examples studied in
this work. For ease of the reader, we repeat below the virtual contribution to
the decay $Z\rightarrow q\bar{q}$

$M_{v}=\epsilon_{\mu}(z)\cdot\int_{k}\overline{u}(q)\cdot(-ig_{s}\gamma^{\alpha}t^{a})\cdot\frac{-i}{\not{q}+\not{k}}\cdot\frac{-ie}{sin(2\omega)}\gamma^{\mu}(g_{V}-\gamma^{5}g_{A})\cdot\frac{i}{\not{q}_{b}-\not{k}}\cdot(-ig_{s}\gamma^{\beta}t^{b})\cdot\frac{-ig_{\alpha\beta}\delta_{ab}}{k^{2}}\cdot
v(\overline{q})\,.$ (68)

where we redefined the momentum of the antiquark by $q_{b}$ to avoid
confusion.

The only possibly ambiguous part is proportional to $g_{A}$. Focusing only on
this part, we can perform an analogous computation to eq. 67. One finds that
on top of the result already found using the rightmost approach (eq. 11), we
will have terms of the form

$M_{v}\supset\int_{k}\frac{(k^{2}-\overline{k}^{2})\gamma_{\mu}\gamma_{5}}{k^{2}(k+q)^{2}(k-q_{b})^{2}}\,,\int_{k}\frac{(\overline{k}^{\mu}-k^{\mu})\not{\overline{k}}\gamma_{5}}{k^{2}(k+q)^{2}(k-q_{b})^{2}}\,,\int_{k}\frac{k^{\mu}(\not{k}-\not{\overline{k}})\gamma_{5}}{k^{2}(k+q)^{2}(k-q_{b})^{2}}$
(69)

However, by close inspection of the on-shell result of these integrals in the
framework of IReg (eqs. 13-16), we find, for instance,

$\int_{k}\frac{(k^{2}-\overline{k}^{2})}{k^{2}(k+q)^{2}(k-q_{b})^{2}}=\int_{k}\frac{k^{2}}{k^{2}(k+q)^{2}(k-q_{b})^{2}}-\overline{g}_{ab}\int_{k}\frac{k^{a}k^{b}}{k^{2}(k+q)^{2}(k-q_{b})^{2}}=0\,.$
(70)

A similar result holds for the other integrals. Therefore, for the particular
example of the decay $Z\rightarrow q\bar{q}$ at NLO, we find that there is no
extra term arising from consistently treating the $\gamma_{5}$ matrix in IReg.
This justifies the usage of the rightmost approach in our calculation. A
complete analogous reasoning can be applied to the decay $S\rightarrow
q\bar{q}$ at NLO.

##  Appendix B Note on the chiral anomaly

The processes analyzed so far dealt with the $\gamma_{5}$ matrix placed in an
open fermionic line (Z and pseudoscalar amplitudes), for which we were able to
show that the rightmost method is compatible with the approach devised for
IReg mentioned in the appendix A. For closed fermionic lines, the occurrence
of the $\gamma_{5}$ matrix must be treated in connection with the trace
properties of the Dirac algebra within divergent integrals. As mentioned in
the introduction cyclic properties of the trace may or may not be kept,
depending on the method applied. To illustrate the procedure within IReg we
very shortly review the diagrammatic calculation of the chiral anomaly in the
context of the anomalous pion decay in two photons,
$\pi^{0}\rightarrow\gamma\gamma$, proceeding through one-loop quark diagrams,
where a trace must be taken over an odd number of $\gamma_{5}$ matrices
occurring in the amplitude and related Adler-Bardeen-Bell-Jackiw (ABJ) anomaly
[68, 69].

In IReg the cyclic property of the trace is maintained, therefore the
rightmost position method results in using the anti-commutator to place the
$\gamma_{5}$ matrix in any position within the trace. This turns out not to be
sufficient to secure the anomaly in the pertaining Ward identity. In other
words, although the value of the anomaly is correctly reproduced, it appears
in the vectorial instead of axial Ward identity (WI), when surface terms (ST)
are set to zero. If one insists that gauge invariance should result from
setting ST to zero, the problem is resolved within IReg by recurring to the
symmetrization of the trace, which is implemented by taking the definition
$\gamma_{5}=\frac{1}{4!}\epsilon_{\mu\nu\alpha\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}$.
These issues have been thoroughly addressed in several works of IReg, see e.g.
[65, 50, 66, 67]. With this rule the direct calculation of the pertinent quark
triangle Feynman diagrams in IReg, see Figure 6

Figure 6: Triangle diagrams which contribute to the ABJ anomaly. Following
Ref. [66], the internal lines are labeled with arbitrary momentum routing.

, yields for the axial vector vector ($AVV$) amplitude

$\displaystyle
T^{\mu\nu\alpha}=-\frac{1}{4\pi^{2}}(1+a)\epsilon^{\mu\nu\alpha\beta}(q-p)_{\beta}+T^{\mu\nu\alpha}_{fin}$

and the following vectorial and axial Ward identities

$\displaystyle
p_{\mu}T^{\mu\nu\alpha}=-\frac{1}{4\pi^{2}}(1+a)\epsilon^{\alpha\nu\beta\lambda}p_{\beta}q_{\lambda}$
$\displaystyle
q_{\nu}T^{\mu\nu\alpha}=-\frac{1}{4\pi^{2}}(1+a)\epsilon^{\alpha\mu\beta\lambda}p_{\beta}q_{\lambda}$
$\displaystyle
l_{\alpha}T^{\mu\nu\alpha}=2mT_{5}^{\mu\nu}+\frac{1}{2\pi^{2}}a\epsilon^{\mu\nu\beta\lambda}p_{\beta}q_{\lambda}.\
$ (72)

with $l=p+q$. Here $T^{\mu\nu\alpha}_{fin}$ is the part of the amplitude that
contains only finite integrals after an arbitrary valued surface term, denoted
by $1+a$, has been isolated, signalizing that the WI can not be simultaneously
satisfied. Choosing $a=-1$ gauge invariance is ensured. In operator language
the axial vector current density
$j^{5}_{\mu}(x)=\bar{\psi}(x)\gamma_{\mu}\gamma_{5}\psi(x)$ acquires then the
quantum correction

$\partial^{\mu}j^{5}_{\mu}(x)=2imj^{5}-\frac{1}{16\pi^{2}}\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}$
(73)

where $j^{5}(x)=\bar{\psi}(x)\gamma_{5}\psi(x)$ is the pseudoscalar current
density present at classical level and $F^{\mu\nu}$ denotes the
electromagnetic field strength. These results are a stringent test of IReg.
The $T_{5}^{\mu\nu}$ represents the pseudoscalar vector vector (PVV) amplitude
and only involves UV finite integrals after evaluation of the Dirac trace,
being therefore regularization independent.

The remaining discussion relating these WI to the pion radiative decay is
beautifully addressed in many text books, for instance [70, 71, 72, 73]. Here
we use it to show that once the WI have been correctly identified within IReg,
the model independent aspects of the radiative decay must follow as well. The
above WI are of special relevance to particles of Goldstone nature as the
pion. The axial current acquires dynamical significance through the partial
conservation of the axial current approximation (PCAC), which states that upon
spontaneous chiral symmetry breaking the axial symmetry current
$A_{\mu}^{5,a}$ of the hadronic model has non-vanishing matrix elements
between the vacuum and a Goldstate state and is conserved in the Goldstone
limit $m_{\pi}^{2}=0$,
$<0|\partial^{\mu}A_{\mu}^{5,a}(x)|\pi^{b},p>=if_{\pi}m_{\pi}^{2}e^{-ip.x}\delta_{ab}$,
where $f_{\pi}\sim 93$ MeV is the weak decay constant of the pion and $a,b$
denote isospin components. However in the presence of electromagnetic gauge
fields it must be modified to accommodate the anomaly contribution

$\partial^{\mu}A_{\mu}^{5,3}=f_{\pi}m_{\pi}^{2}\pi^{0}-\frac{\alpha}{8\pi}\epsilon_{\mu\nu\alpha\beta}F^{\mu\nu}F^{\alpha\beta},$
(74)

with $\alpha=\frac{e^{2}}{4\pi}$, $\pi^{0}$ the neutral pion field and the
index $3$ stands for the third component of the axial vector current The first
term contains information about the pseudo-Goldstone nature of the pion and
the second is the anomalous divergence of the axial vector current.

The modification of the PCAC relation can be sketched as follows. We start by
considering $T_{5}^{\mu\nu}$ in eq. (B). It has the same Lorentz structure as
the $\pi^{0}\rightarrow\gamma\gamma$ amplitude in chiral quark models of the
pion with $g_{\pi}(\bar{q}i\gamma_{5}\tau_{a}\pi^{a}q)$ effective Lagrangian
description, such as in the $SU(2)_{A}\times SU(2)_{V}$ linear sigma model
(LSM) with quark degrees of freedom, therefore it is convenient to adopt the
LSM for the present discussion. The hadronic coupling $g_{\pi}$ is given by
the celebrated Goldberger-Treiman (GT) relation
$g_{\pi}=\frac{mg_{A}}{f_{\pi}}$ in the limit $l^{2}\rightarrow 0$, where
$g_{A}$ is associated to the neutron beta decay (set here to $g_{A}=1$ [71])
and $m$ is the constituent quark mass. The Noether current
$A_{\mu}^{5,3}=\bar{q}\gamma_{\mu}\gamma_{5}\frac{\tau^{3}}{2}q+$ bosonic
contributions. The $T_{5}^{\mu\nu}$ amplitude can be readily converted to the
physical radiative decay amplitude of the pion, after taking into account
isospin and $N_{c}$ color factors appropriately. A factor of 2 also arises
associated with the LSM coupling $\tau_{i}$ as compared to $\tau_{3}/2$ in the
fermionic current. Direct evaluation yields in the Goldstone limit for the
$\pi^{0}\rightarrow\gamma\gamma$ amplitude

$\displaystyle
T_{\pi}^{\mu\nu}=g_{\pi}N_{c}2Tr[\frac{1}{2}\tau_{3}\\{Q,Q\\}]T_{5}^{\mu\nu}$
(75) $\displaystyle
T_{5}^{\mu\nu}=\frac{\alpha}{m}\frac{1}{\pi}\epsilon_{\mu\nu\alpha\beta}p^{\alpha}q^{\beta}\
$ (76)

where $Q=\frac{1}{2}(\frac{1}{3}+\tau_{3})$ is the quark charge matrix at the
electromagnetic vertex, the trace is over isospin, and a smooth behavior as
$l^{2}\rightarrow 0$ is implied. With
$h=N_{c}Tr[\frac{1}{2}\tau_{3}\\{Q,Q\\}]=\frac{1}{2}$ one obtains
$T_{\pi}^{\mu\nu}=\frac{\alpha}{\pi
f_{\pi}}\epsilon_{\mu\nu\alpha\beta}q^{\alpha}p^{\beta}$.

On the other hand the pion field in the evaluation of the quark loop coupling
to photons can be reduced using the Lehmann, Symanzik, Zimmermann (LSZ)
reduction formula [74] yielding after using the modified PCAC relation eq.
(74)

$l_{\alpha}{\bar{T}}^{\mu\nu\alpha}(l^{2})=\frac{f_{\pi}m_{\pi}^{2}}{(m_{\pi}^{2}-l^{2})}T_{\pi}^{\mu\nu}(l^{2})-\frac{\alpha}{\pi}\epsilon^{\mu\nu\beta\lambda}p_{\beta}q_{\lambda}.$
(77)

where ${\bar{T}}^{\mu\nu\alpha}$ carries the isospin and color factor $h$
defined above. The first term on the right hand side stems from the original
PCAC relation. In the absence of the anomalous contribution the conservation
of the axial vector current as $l^{2}\rightarrow 0$ can only be achieved if
lim $T_{\pi}^{\mu\nu}(0)=0$, as observed by Sutherland [75] and Veltman [76],
which would be in contradiction with the explicit evaluation of the amplitude.
From eq. (77) one obtains finally that the model independent leading
contribution to the radiative pion amplitude $T_{\pi}^{\mu\nu}(0)$ is given by
the anomalous term and coincides with the expression eq. (75).

To summarize, although IReg operates in the physical dimension, processes
involving $\gamma_{5}$ still require some care, as explained in the Appendix
A. We have found at one loop level that in open fermionic lines the
$\gamma_{5}$ can be treated as in the rightmost position method, where it
takes a spectator role and WI are preserved. In closed fermionic lines
involving an odd number of $\gamma_{5}$ matrices, we have discussed that this
procedure is however not sufficient in the case of IReg. We have chosen one of
the most subtle examples in the literature for illustration, the abelian
chiral anomaly. Symmetrization of the trace444In the language of Appendix A,
this is equivalent to using eq.63 before evaluating the trace over the Dirac
matrices. in this case provides the correct positioning of the anomaly in the
WI, in accordance with momentum routing invariance (or the vanishing of the
ST) in gauge invariant processes.

## References

  * [1] C Gnendiger, A Signer, D Stöckinger, A Broggio, AL Cherchiglia, F Driencourt-Mangin, AR Fazio, B Hiller, P Mastrolia, T Peraro, et al.  To d, or not to d: recent developments and comparisons of regularization schemes.  The European Physical Journal C, 77(7):1–39, 2017.
  * [2] WJ Torres Bobadilla, GFR Sborlini, P Banerjee, S Catani, AL Cherchiglia, L Cieri, PK Dhani, F Driencourt-Mangin, T Engel, G Ferrera, et al.  May the four be with you: Novel ir-subtraction methods to tackle nnlo calculations.  The European Physical Journal C, 81:1–61, 2021.
  * [3] F. Bloch and A. Nordsieck.  Note on the Radiation Field of the electron.  Phys. Rev., 52:54–59, 1937.
  * [4] Toichiro Kinoshita.  Mass singularities of feynman amplitudes.  Journal of Mathematical Physics, 3(4):650–677, 1962.
  * [5] Tsung-Dao Lee and Michael Nauenberg.  Degenerate systems and mass singularities.  Physical Review, 133(6B):B1549, 1964.
  * [6] John Collins.  Renormalization cambridge up.  Press, Cambridge, 1984.
  * [7] Martinus Veltman et al.  Regularization and renormalization of gauge fields.  Nuclear Physics B, 44(1):189–213, 1972.
  * [8] Fred Jegerlehner.  Facts of life with $\gamma_{5}$.  The European Physical Journal C-Particles and Fields, 18(4):673–679, 2001.
  * [9] M. Chaichian and W. F. Chen.  Inconsistency of naive dimensional regularizations and quantum correction to non-abelian chern-simons-matter theory reexamined.  Phys. Rev. D, 58:125004, Nov 1998.
  * [10] Dominik Stöckinger.  Regularization of supersymmetric theories.  Nuclear Physics B - Proceedings Supplements, 160:250–254, 2006\.  Proceedings of the 8th DESY Workshop on Elementary Particle Theory.
  * [11] P Breitenlohner and D Maison.  Dimensionally renormalized green’s functions for theories with massless particles. ii.  Communications in Mathematical Physics, 52(1):55–75, 1977.
  * [12] Hermès Bélusca-Maïto, Amon Ilakovac, Marija Madjor-Božinović, and Dominik Stöckinger.  Dimensional regularization and breitenlohner-maison/’t hooft-veltman scheme for $\gamma$5 applied to chiral ym theories: full one-loop counterterm and rge structure.  Journal of High Energy Physics, 2020(8):1–71, 2020.
  * [13] Hermès Bélusca-Maïto, Amon Ilakovac, Paul Kühler, Marija Mador-Božinović, and Dominik Stöckinger.  Two-loop application of the breitenlohner-maison/’t hooft-veltman scheme with non-anticommuting $\gamma$5: full renormalization and symmetry-restoring counterterms in an abelian chiral gauge theory.  Journal of High Energy Physics, 2021(11):1–32, 2021.
  * [14] Er-Cheng Tsai.  Gauge invariant treatment of $\gamma$ 5 in the scheme of’t hooft and veltman.  Physical Review D, 83(2):025020, 2011.
  * [15] Er-Cheng Tsai.  Maintaining gauge symmetry in renormalizing chiral gauge theories.  Physical Review D, 83(6):065011, 2011.
  * [16] Dirk Kreimer.  The $\gamma$5-problem and anomalies—a clifford algebra approach.  Physics Letters B, 237(1):59–62, 1990.
  * [17] Dirk Kreimer.  The role of $\gamma_{5}$ in dimensional regularization.  arXiv preprint hep-ph/9401354, 1994.
  * [18] Long Chen.  An observation on Feynman diagrams with axial anomalous subgraphs in dimensional regularization with an anticommuting $\gamma_{5}$.  hep-ph, 2304.13814.
  * [19] Christoph Gnendiger, Adrian Signer, and Dominik Stöckinger.  The infrared structure of qcd amplitudes and h→ gg in fdh and dred.  Physics Letters B, 733:296–304, 2014.
  * [20] Warren Siegel.  Supersymmetric dimensional regularization via dimensional reduction.  Physics Letters B, 84(2):193–196, 1979.
  * [21] Warren Siegel.  Inconsistency of supersymmetric dimensional regularization.  Physics Letters B, 94(1):37–40, 1980.
  * [22] Zvi Bern and David A Kosower.  The computation of loop amplitudes in gauge theories.  Nuclear Physics B, 379(3):451–561, 1992.
  * [23] Zvi Bern, A De Freitas, L Dixon, and HL Wong.  Supersymmetric regularization, two-loop qcd amplitudes, and coupling shifts.  Physical Review D, 66(8):085002, 2002.
  * [24] Dominik Stöckinger.  Regularization by dimensional reduction: consistency, quantum action principle, and supersymmetry.  Journal of High Energy Physics, 2005(03):076, 2005.
  * [25] Adrian Signer and Dominik Stöckinger.  Using dimensional reduction for hadronic collisions.  Nuclear physics B, 808(1-2):88–120, 2009.
  * [26] William B. Kilgore.  The Four Dimensional Helicity Scheme Beyond One Loop.  Phys. Rev. D, 86:014019, 2012.
  * [27] A. Broggio, Ch. Gnendiger, A. Signer, D. Stöckinger, and A. Visconti.  SCET approach to regularization-scheme dependence of QCD amplitudes.  JHEP, 01:078, 2016.
  * [28] Christoph Gnendiger, Adrian Signer, and Andrea Visconti.  Regularization-scheme dependence of QCD amplitudes in the massive case.  JHEP, 10:034, 2016.
  * [29] Roger J. Hernandez-Pinto, German F. R. Sborlini, and German Rodrigo.  Towards gauge theories in four dimensions.  JHEP, 02:044, 2016.
  * [30] German F. R. Sborlini, Felix Driencourt-Mangin, and German Rodrigo.  Four-dimensional unsubtraction with massive particles.  JHEP, 10:162, 2016.
  * [31] Roberto Pittau.  A four-dimensional approach to quantum field theories.  JHEP, 11:151, 2012.
  * [32] O. A. Battistel, A. L. Mota, and M. C. Nemes.  Consistency conditions for 4-D regularizations.  Mod. Phys. Lett. A, 13:1597–1610, 1998.
  * [33] A. P. Baeta Scarpelli, M. Sampaio, and M. C. Nemes.  Consistency relations for an implicit n-dimensional regularization scheme.  Phys. Rev. D, 63:046004, 2001.
  * [34] A. P. Baeta Scarpelli, M. Sampaio, B. Hiller, and M. C. Nemes.  Chiral anomaly and CPT invariance in an implicit momentum space regularization framework.  Phys. Rev. D, 64:046013, 2001.
  * [35] Adriano Lana Cherchiglia, Marcos Sampaio, and Maria Carolina Nemes.  Systematic implementation of implicit regularization for multiloop feynman diagrams.  International Journal of Modern Physics A, 26(15):2591–2635, 2011\.
  * [36] José de Jesús Aguilera-Verdugo, Félix Driencourt-Mangin, Roger José Hernández-Pinto, Judith Plenter, Renato Maria Prisco, Norma Selomit Ramírez-Uribe, Andrés Ernesto Rentería-Olivo, Germán Rodrigo, German Sborlini, William Javier Torres Bobadilla, et al.  A stroll through the loop-tree duality.  Symmetry, 13(6):1029, 2021.
  * [37] Dafne Carolina Arias-Perdomo, Adriano Cherchiglia, Brigitte Hiller, and Marcos Sampaio.  A brief review of implicit regularization and its connection with the bphz theorem.  Symmetry, 13(6):956, 2021.
  * [38] A Cherchiglia, DC Arias-Perdomo, AR Vieira, M Sampaio, and B Hiller.  Two-loop renormalisation of gauge theories in 4d implicit regularisation and connections to dimensional methods.  The European Physical Journal C, 81(5):1–26, 2021.
  * [39] N. N. Bogoliubov and O. S. Parasiuk.  On the Multiplication of the causal function in the quantum theory of fields.  Acta Math., 97:227–266, 1957.
  * [40] Klaus Hepp.  Proof of the Bogolyubov-Parasiuk theorem on renormalization.  Commun. Math. Phys., 2:301–326, 1966.
  * [41] W. Zimmermann.  Convergence of Bogolyubov’s method of renormalization in momentum space.  Commun. Math. Phys., 15:208–234, 1969.
  * [42] A. R. Vieira, A. L. Cherchiglia, and Marcos Sampaio.  Momentum Routing Invariance in Extended QED: Assuring Gauge Invariance Beyond Tree Level.  Phys. Rev. D, 93(2):025029, 2016.
  * [43] Luellerson C. Ferreira, A. L. Cherchiglia, Brigitte Hiller, Marcos Sampaio, and M. C. Nemes.  Momentum routing invariance in Feynman diagrams and quantum symmetry breakings.  Phys. Rev. D, 86:025016, 2012.
  * [44] Adriano Cherchiglia.  Two-loop gauge coupling $\beta$-function in a four-dimensional framework: the Standard Model case.  SciPost Phys. Proc., 7:043, 2022.
  * [45] A. L. Cherchiglia, Marcos Sampaio, B. Hiller, and A. P. Baêta Scarpelli.  Subtleties in the beta function calculation of N=1 supersymmetric gauge theories.  Eur. Phys. J. C, 76(2):47, 2016.
  * [46] H. G. Fargnoli, B. Hiller, A. P. Baeta Scarpelli, Marcos Sampaio, and M. C. Nemes.  Regularization Independent Analysis of the Origin of Two Loop Contributions to N=1 Super Yang-Mills Beta Function.  Eur. Phys. J. C, 71:1633, 2011.
  * [47] David E. Carneiro, A. P. Baeta Scarpelli, Marcos Sampaio, and M. C. Nemes.  Consistent momentum space regularization / renormalization of supersymmetric quantum field theories: The Three loop beta function for the Wess-Zumino model.  JHEP, 12:044, 2003.
  * [48] A. L. Cherchiglia, L. A. Cabral, M. C. Nemes, and Marcos Sampaio.  (Un)determined finite regularization dependent quantum corrections: the Higgs boson decay into two photons and the two photon scattering examples.  Phys. Rev. D, 87(6):065011, 2013.
  * [49] Ana Pereira, Adriano Cherchiglia, Marcos Sampaio, and Brigitte Hiller.  Higgs boson decay into gluons in a 4d regularization: Ir cancellation without evanescent fields to nlo.  The European Physical Journal C, 83(1):73, 2023.
  * [50] A. M. Bruque, A. L. Cherchiglia, and M. Pérez-Victoria.  Dimensional regularization vs methods in fixed dimension with and without $\gamma_{5}$.  JHEP, 08:109, 2018.
  * [51] Marcos D. Sampaio, A. P. Baeta Scarpelli, J. E. Ottoni, and M. C. Nemes.  Implicit regularization and renormalization of QCD.  Int. J. Theor. Phys., 45:436–457, 2006.
  * [52] A. L. Cherchiglia, A. R. Vieira, Brigitte Hiller, A. P. Baêta Scarpelli, and Marcos Sampaio.  Guises and Disguises of Quadratic Divergences.  Annals Phys., 351:751–772, 2014.
  * [53] L. C. T. Brito, H. G. Fargnoli, A. P. Baeta Scarpelli, Marcos Sampaio, and M. C. Nemes.  Systematization of Basic Divergent Integrals in Perturbation Theory and Renormalization Group Functions.  Phys. Lett. B, 673:220–226, 2009.
  * [54] Jorge C. Romao and Joao P. Silva.  A resource for signs and Feynman diagrams of the Standard Model.  Int. J. Mod. Phys. A, 27:1230025, 2012.
  * [55] VA Novikov, LB Okun, Alexandre N Rozanov, and MI Vysotsky.  Theory of z boson decays.  Reports on Progress in Physics, 62(9):1275, 1999.
  * [56] Ansgar Denner.  Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200.  Fortsch. Phys., 41:307–420, 1993.
  * [57] Roberto Pittau.  QCD corrections to H $\rightarrow$ gg in FDR.  The European Physical Journal C, 74(1):1–7, 2014.
  * [58] Marcos Sampaio, AP Baêta Scarpelli, JE Ottoni, and MC Nemes.  Implicit regularization and renormalization of qcd.  International Journal of Theoretical Physics, 45:436–457, 2006\.
  * [59] Eric Braaten and JP Leveille.  Higgs-boson decay and the running mass.  Physical Review D, 22(3):715, 1980.
  * [60] Matthias Neubert.  Renormalization theory and effective field theories.  Effective Field Theories in Particle Physics and Cosmology: Lecture Notes of the Les Houches Summer School: Volume 108, July 2017, 108:1, 2020.
  * [61] N Sakai.  Perturbative quantum-chromodynamic corrections to the hadronic decay width of the higgs boson.  Physical Review D, 22(9):2220, 1980.
  * [62] Takeo Inami and Takahiro Kubota.  Renormalization group estimate of the hadronic decay width of the higgs boson.  Nuclear Physics B, 179(1):171–188, 1981.
  * [63] S. G. Gorishnii, A. L. Kataev, and S. A. Larin.  The Width of Higgs Boson Decay Into Hadrons: Three Loop Corrections of Strong Interactions.  Sov. J. Nucl. Phys., 40:329–334, 1984.
  * [64] A Djouadi.  The anatomy of electro-weak symmetry breaking. i: The higgs boson in the standard model. phys. rept. 457, 1–216 (2008).  arXiv preprint hep-ph/0503172, 2008.
  * [65] JS Porto, AR Vieira, AL Cherchiglia, Marcos Sampaio, and Brigitte Hiller.  On the bose symmetry and the left-and right-chiral anomalies.  The European Physical Journal C, 78:1–11, 2018.
  * [66] A. C. D. Viglioni, A. L. Cherchiglia, A. R. Vieira, Brigitte Hiller, and Marcos Sampaio.  $\gamma_{5}$ algebra ambiguities in Feynman amplitudes: Momentum routing invariance and anomalies in $D=4$ and $D=2$.  Phys. Rev. D, 94(6):065023, 2016.
  * [67] Adriano Cherchiglia.  Step towards a consistent treatment of chiral theories at higher loop order: The abelian case.  Nucl. Phys. B, 987:116104, 2023.
  * [68] J. S. Bell and R. Jackiw.  A PCAC puzzle: $\pi^{0}\to\gamma\gamma$ in the $\sigma$ model.  Nuovo Cim. A, 60:47–61, 1969.
  * [69] Stephen L. Adler.  Axial vector vertex in spinor electrodynamics.  Phys. Rev., 177:2426–2438, 1969.
  * [70] Reinhold A Bertlmann.  Anomalies in quantum field theory, volume 91.  Oxford university press, 2000.
  * [71] Claude Itzykson and Jean-Bernard Zuber.  Quantum field theory.  Courier Corporation, 2012.
  * [72] Ta-Pei Cheng and Ling-Fong Li.  Gauge theory of elementary particle physics.  Oxford university press, 1994.
  * [73] Michael E Peskin and Daniel V Schroeder.  An introduction to quantum field theory (boulder, co, 1995.
  * [74] Harry Lehmann, Kurt Symanzik, and Wolfhart Zimmermann.  Zur formulierung quantisierter feldtheorien.  Il Nuovo Cimento (1955-1965), 1:205–225, 1955.
  * [75] DG Sutherland.  Current algebra and some non-strong mesonic decays.  Nuclear Physics B, 2(4):433–440, 1967.
  * [76] M Veltman.  I. theoretical aspects of high energy neutrino interactions.  Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 301(1465):107–112, 1967.