aainstitutetext: CCNH, Universidade Federal do ABC,
09210-580 , Santo André - SP, Brazilbbinstitutetext: Universidade Federal do
Triângulo Mineiro,
Iturama, MG 38280-000, Brazilccinstitutetext: CFisUC, Department of Physics,
University of Coimbra,
P-3004-516 Coimbra, Portugal.

# Two-loop renormalisation of gauge theories in $4D$ Implicit Regularisation:
transition rules to dimensional methods

A. Cherchiglia,111Corresponding author. a D. C. Arias-Perdomo b A. R. Vieira a
M. Sampaio c and B. Hiller<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute the two-loop $\beta$-function of scalar and spinorial quantum
electrodynamics as well as pure Yang-Mills and quantum chromodynamics using
the background field method in a fully quadridimensional setup using Implicit
Regularization (IREG). Moreover, a thorough comparison with dimensional
approaches such as conventional dimensional regularization (CDR) and
dimensional reduction (DRED) is presented. Particularly, for our calculations
we show that the inclusion of evanescent $\epsilon$-scalar particle
contributions needed in quasi-dimensional methods such as DRED and Four
Dimensional Helicity (FDH) cancel out in the determination of the ultraviolet
(UV) structure of the models we study. Subtleties related to Lorentz algebra
contractions/symmetric integrations inside divergent integrals as well as
renormalisation schemes are carefully discussed within IREG where the
renormalisation constants are fully defined as basic divergent integrals to
arbitrary loop order. Moreover we confirm the hypothesis that momentum routing
invariance in the loops of Feynman diagramas implemented via setting well-
defined surface terms to zero deliver non-abelian gauge invariant amplitudes
within IREG just as it has been proven for abelian theories.

##  1 Motivations

Unravelling physics beyond the standard model (SM) has entreated theoretical
predictions for particle physics precision observables beyond next-to-leading-
order ($NLO$). Such predictions rely on involved Feynman diagram calculations
to evaluate scattering amplitudes both in the SM and its extensions.
Theoretical models beyond the SM (BSM) can be constructed, for instance, as an
extension in the Higgs sector by either changing the number of scalar
multiplets or considering the Higgs boson as a composite particle – the so
called Composite Higgs Models HSEC ; CHM . Supersymmetric and dark matter
extensions have also been considered in order to explain SM deviations from
experimental results REVIEW  in electroweak precision observables (EWPO) which
are known with an accuracy at the per cent level or better EWPO ; ADRISTO1 ;
EBOLI . On the other hand, precise measurements and calculations of known
particles and interactions are just as important to validate, redress or
refute new models. Also, in order to evade from unphysical scale dependence at
low order, higher order terms are needed to smooth out such a dependence in
the resulting, more accurate, predictions. For example, a full $N^{3}LO$
calculation for QCD corrections to gluon-fusion Higgs boson production was
perfomed in ANAST1  at center-of-mass energy 13 $TeV$. The considerably low
residual theoretical uncertainty ($\approx 5-6\%$) and small sensitivity to
scale variation ($\approx 2\%$ ) superseded earlier results below $N^{3}LO$.
Because experimental uncertainties are expected to drop below the accuracy of
theoretical data, as expected from future experimental measurements at the
Future Circular Collider (FCC-$e^{-}e^{+}$) HTTPS , $QCD$ theoretical
uncertainties ought to be reduced at many levels so physics BSM can be
ultimately ascertained.

Ultraviolet (UV) and infrared (IR) divergences are ubiquitous beyond leading
order in S-matrix calculations and must be judiciously removed in order to
automated computation codes for the evaluation of Feynman amplitudes. As a by-
product of such subtractions, there remain residual dependences on
renormalisation ($\lambda_{R}$) and factorisation ($\lambda_{F}$) scales in
the perturbative series that describes a physical observable. Obviously the
dependence on such scales diminish after higher terms are taken into account
and, at a given order, may be minimised to yield a result the least sensitive
to variations in the unphysical parameters STEVEN . For typical collider
observables, due to the high multiplicity of jets, the real radiation is
subjected to intricate phase-space constraints and calculations are often
performed numerically. In such cases the cancellation of soft and collinear IR
divergences becomes more involved and precise evaluations of parton
distribution functions (PDF’s), needed to calculate cross-sections with hadron
beams, become mandatory PDF . A general cross section in $QCD$ usually
includes short and long-distance behaviour and thus it is not computable
directly in perturbation theory. This remarkable problem was first addressed
by Weinberg in his pioneering work on $QED$ and quantum gravity WEINBERG . At
a given order $\alpha_{s}^{n}$ and momentum transfer $Q$ there appear large
logarithms such as renormalisation and factorisation logs
$\alpha_{s}^{2}\ln^{n}(Q^{2}/\lambda_{R}^{2})$,
$\alpha_{s}^{2}\ln^{n}(Q^{2}/\lambda_{F}^{2})$, as well as high energy logs
and Sudakov logs LOG . Renormalisation group (RG) logs are resummed via RG
evolution equations whereas Sudakov logs originate from IR and collinear
singularities and may be resummed through exponentiation of IR and collinear
poles. Such resummations at and beyond next-to-leading-log ($NLL$) assure the
validity of the perturbative series, leading to non-perturbative contributions
to high energy cross sections. For those resummations, it is crucial to rely
on factorisation theorems FACTOR1 ; GARDI ; FACTORSTOCK . Factorisation
properties separate the dynamics at different energy scales. For instance, for
$n$ external partons in the high energy limit, an amplitude factorizes as
FACTORSTOCK

$\displaystyle{\cal{M}}_{n}\Bigg{(}\frac{\mu}{\lambda_{IR}},\frac{p_{i}}{\lambda_{UV}},\alpha_{s}(\lambda_{UV})\Bigg{)}=Z\Bigg{(}\frac{\mu}{\lambda_{IR}},\frac{p_{i}}{\lambda_{IR}},\alpha_{s}(\lambda_{IR})\Bigg{)}H_{n}\Bigg{(}\frac{p_{i}}{\lambda_{UV}},\frac{\lambda_{IR}}{\lambda_{UV}},\alpha_{s}(\lambda_{UV})\Bigg{)},$
(1.1)

where $H_{n}$ is the UV renormalized piece and $Z$ contains the IR soft and
collinear divergences expressed here by $\mu\rightarrow 0$. The factor $Z$
itself obeys a RG-like equation which gives rise to anomalous dimensions. The
latter can be used to resum Sudakov logs, non-abelian exponentiation of
differential cross-sections FRENKEL , as well as to understand the mapping
between IR onto UV divergences of operators in effective field theory FACTOR1
.

In order to tackle the problems discussed above, the choice of a
regularisation scheme for UV or IR divergences in Feynman amplitudes matters
from both conceptual and practical aspects. In theory the choice of
regularisation is unphysical but in practice it is paramount to both studying
anomalies in perturbative quantum field theory and automatising loop
calculations. In recent years, novel schemes have been proposed aiming at
improving or even obliterating conventional dimensional regularisation
(CDR)REGDIM . In dimensional specific models, such as chiral GAMMA5 ,
topological CHERNSIMONS  or supersymmetric quantum field theories REGSUSY ,
CDR needs caveats or even explicit modifications such as the dimensional
reduction scheme (DRED) SIEGEL . On the other hand, such modifications on CDR
can lead to additional terms at the Lagrangian level, such as the
$\epsilon$-scalar particles EPSILONSCALAR , in order to satisfy Ward
identities and account for amplitude factorisation and other renormalisation
group properties of the model. The downsides are that besides the Feynman
rules become more involved, such fictitious particles are not protected by
gauge invariance and require coupling
$\alpha_{\epsilon},\alpha_{4\epsilon}\neq\alpha_{s}$, and thus their own
$\beta$ functions, in order to preserve unitarity.

Alternatively, some schemes that operate only on the physical dimension of the
underlying model aside from energy cut-offs have been constructed. Amidst
them, implicit regularisation (IREG) was constructed for the purpose of
shedding light on theoretical aspects of regularisation dependent quantum
corrections as well as providing novel analytical and hopefully numerical
calculational methods ZURICH . Transition rules to other schemes is
particularly perspicuous within IREG.

In this contribution we focus on the UV renormalisation of scalar/spinorial
QED and QCD to two loop order within IREG. Working entirely in four
dimensions, we calculate the $\beta$-functions using the background field
method (BFM) in a minimal subtraction scheme of the basic divergent integrals
(BDI’s) in internal momenta. BDI’s contain a natural renormalisation scale
$\lambda_{R}\equiv\lambda$ and are absorbed into renormalisation constants
whose counterparts in DRED and CDR are provided. The results show compliance
with gauge symmetry lending support to the conjecture that momentum routing
invariance (MRI) in Feynman diagrams (enforced by setting well-defined surface
terms to zero) delivers gauge invariant amplitudes. Such a conjecture was
proved to be valid in the abelian gauge theories to all orders in perturbation
theory PRD2012 ; ALEXANDRE . No modifications at Lagrangian level are
necessary as IREG operates in the physical dimension. For instance, no analog
of $\epsilon$-scalar field needed in schemes such as DRED and four-dimensional
helicity (FDH) FACTORSTOCK ; FDH  to assure unitarity and proper cancellation
of divergences is introduced in IREG as a matter of principle as it operates
in the physical dimension. We show nevertheless, that in this particular
calculation, $\epsilon$-field contributions cancel out in DRED. It is
noteworthy that $\epsilon$-scalars are crucial to understand IR structure in
DRED or FDH schemes.

In the next section we provide a brief panorama over regularisation schemes
with focus upon IREG rules that will be used throughout this contribution. We
also discuss some subtleties related to Lorentz algebra contractions/symmetric
integrations inside divergent integrals, and present a dedicated analysis of
the correspondence among IREG and dimensional methods. In section 3 we study a
series examples at two-loop order, aiming to compute the first two
coefficients of the gauge coupling $\beta$ function in scalar/spinorial QED,
pure Yang-Mills, and QCD. A careful comparison with CDR and DRED is performed.
Finally, we conclude in section 4, and present a series of appendixes
containing some technical details of the calculations.

##  2 Survey of regularisation schemes and IREG rules

In CDR REGDIM  the vector bosons are treated in $d=4-2\epsilon$ dimensions.
Despite being the most common scheme, it presents some problems in chiral,
topological and supersymmetric quantum field theories as discussed in the
introduction. Some variants of CDR such as the ’t Hooft-Veltman method (HV)
MAISON , dimensional reduction (DRED)SIEGEL  and Four Dimensional Helicity
(FDH) FDH  have been developed. Apart from the dimensional specific models,
some scattering amplitudes in $QCD$ present regularisation dependence
REGDEPQCD  mainly due to the interplay between IR and UV divergences to yield
finite results. In dimensional methods both UV and IR infinities appear as
poles $1/\epsilon^{n}$. An unclear distinction between the origin of such
poles can lead to ambiguities such as the nature of radiative contributions to
supersymmetric Yang-Mills $\beta$-functions MAS . We may schematise
dimensional methods as in table 1. Grosso modo, a quasi-4-dimensional gauge
field may me decomposed as $A^{\mu}=\hat{A}^{\mu}+\tilde{A}^{\mu},$ where
$\hat{A}^{\mu}$ is a $d$-dimensional gauge field and $\tilde{A}^{\mu}$ is a
scalar of multiplicity $N_{\epsilon}=2\epsilon$ BROGGIO . This amounts to
adding up to the original Lagrangian a term ${\cal{L}}_{\epsilon}$ that
contains evanescent Yukawa couplings between $\epsilon$-scalars and quarks of
strength $\lambda_{\epsilon}$ and quartic $\epsilon$-scalar vertices with
strength $\lambda_{4\epsilon}$ EPSILONSCALAR .

| CDR | HV | FDH | DRED
---|---|---|---|---
internal gluon | $\hat{g}^{\mu\nu}$ | $\hat{g}^{\mu\nu}$ | $g^{\mu\nu}$ | $g^{\mu\nu}$
external gluon | $\hat{g}^{\mu\nu}$ | $\bar{g}^{\mu\nu}$ | $\bar{g}^{\mu\nu}$ | $g^{\mu\nu}$
Table 1:  In CDR and HV, gluons are regularised in $d$ dimensions with metric
tensor $\hat{g}_{\mu\nu}$ (QdS). The quasi-4-dimensional space has metric
$g_{\mu\nu}$ (Q4S) and the original quadridimensional space is denoted by
$\bar{g}_{\mu\nu}$ (4S). A complementary $2\epsilon$-dimensional tensor
$\tilde{g}_{\mu\nu}$ $2\epsilon$ must be introduced such that
$g_{\mu\nu}=\hat{g}_{\mu\nu}+\tilde{g}_{\mu\nu}$. Thus $Q4S=QdS\oplus
Q2\epsilon S$ . Mathematical consistency and gauge invariance require
$Q4S\supset QdS\supset 4S$ and prohibites to identify $g_{\mu\nu}$ with
$\bar{g}_{\mu\nu}$ REGSUSY .

The resulting schemes such as DRED and FDH are crucial to study models with
supersymmetry. The latter is explicitly broken in CDR as gauge bosons are
considered as quantities with $d$ components whereas gauginos remain
$4$-dimensional which means an unbalance between fermionic and bosonic degrees
of freedom. Symmetry restauring contraterms may be added order by order in
perturbation theory in the renormalised theory KRAUSSTOCK ; KRAUS . However,
besides being a striking additional complication, we cannot discard possible
anomalous symmetry breakings in supersymmetric gauge theories in general
REGSUSY . In this sense, DRED has been shown to be an invariant supersymmetric
scheme to minimal supersymmetric gauge models up to two loop order using the
quantum action principle SUSYQCD .

For an efficient computational code to evaluate Feynman amplitudes beyond
leading order, UV and IR divergences ought to be subtracted by a scheme that
respects unitarity, causality and symmetries. The subtraction of nested and
overlapped divergences should respect the Bogolyubov recursion relations,
locality and the BPHZ algorithm BPHZ . The techniques can be divided into two
categories ZURICH :

  1. 1.

“to d”: CDR (1972), HV (1977); DRED (1979), FDH (1992), Six-dimensional
formalism SDF (2009) SDF , Four dimensional Formalism FDF (2014) FDF .

  2. 2.

“or not to d”: Differential regularisation/renormalisation DIFR (1992)
FREEDMAN , Implicit regularisation IREG (1998) PRIM2 , Loop regularisation
LORE (2003) LORE , Four-Dimensional regularisation/renormalisation FDR (2012)
FDR , Four-Dimensional Unsubtraction FDU (2016) FDU .

###  2.1 The rules of IREG

For a general $n$-loop Feynman amplitude ${\cal A}_{n}$ with $L$ external
legs, let $k_{l}$ be the internal (loop) momenta ($l=1\cdots n$) and $p_{i}$
be the external momenta. After performing the usual Diracology and spacetime
algebra in the physical dimension and internal symmetry contractions, the UV
content of ${\cal A}_{n}$ can be cast in terms of well-defined basic divergent
integrals. We assume without loss of generality that all the masses of the
underlying model are zero in order to define a massless minimal subtraction
scheme. This procedure was shown to comply with unitarity, locality and
Lorentz invariance in ADRIANO . The whole program is compatible with
overlapping divergences through the Bogoliubov’s recursion formula which means
that the divergent content of $n$-loop Feynman graph can always be cast as a
basic divergent integral.

  1. 1.

Perform the internal symmetry group and the usual Dirac algebra in the
physical dimension. Care must be exercised in avoiding symmetric integration
in divergent amplitudes, namely

$\int_{k}k^{\mu_{1}}\cdots
k^{\mu_{2m}}f(k^{2})=\frac{1}{(2m)!}\int_{k}g^{\\{\mu_{1}\mu_{2}}\cdots
g^{\mu_{2m-1}\mu_{2m}\\}}f(k^{2}),$ (2.1)

where $\int_{k}\equiv\int d^{4}k/(2\pi)^{4}$ and the curly brackets here mean
symmetrisation over Lorentz indices, as such an operation is ambiguous MPV .
Consequently Dirac algebra contractions that engender such operation must be
avoided. The anticommutation $\\{\gamma_{5},\gamma_{\mu}\\}=0$ inside
divergent amplitudes must not be used even in the physical dimension as they
lead to spurious terms as well BRUQUE ; VIGLIONI ; JOILSON .

  2. 2.

Starting at one loop, assume an implicit regulator, say a momentum cut-off, in
order to remove external momenta dependence from the divergent part of the
amplitude by judiciously applying the identity

$\displaystyle\frac{1}{(k_{l}-p_{i})^{2}-\mu^{2}}=\sum_{j=0}^{n_{i}^{(k_{l})}-1}\frac{(-1)^{j}(p_{i}^{2}-2p_{i}\cdot
k_{l})^{j}}{(k_{l}^{2}-\mu^{2})^{j+1}}+\frac{(-1)^{n_{i}^{(k_{l})}}(p_{i}^{2}-2p_{i}\cdot
k_{l})^{n_{i}^{(k_{l})}}}{(k_{l}^{2}-\mu^{2})^{n_{i}^{(k_{l})}}\left[(k_{l}-p_{i})^{2}-\mu^{2}\right]},$
(2.2)

in the propagators. Here $\mu\downarrow 0$ is a ficticious mass (infrared
regulator) and at one loop order $k_{l}$ is simply $k$. Basic divergent
integrals (BDI’s) appear as

$\displaystyle I_{log}(\mu^{2})$ $\displaystyle\equiv$
$\displaystyle\int_{k}\frac{1}{(k^{2}-\mu^{2})^{2}},\quad\quad
I_{log}^{\nu_{1}\cdots\nu_{2r}}(\mu^{2})\equiv\int_{k}\frac{k^{\nu_{1}}\cdots
k^{\nu_{2r}}}{(k^{2}-\mu^{2})^{r+2}},$ $\displaystyle I_{quad}(\mu^{2})$
$\displaystyle\equiv$
$\displaystyle\int_{k}\frac{1}{(k^{2}-\mu^{2})},\quad\quad
I_{quad}^{\nu_{1}\cdots\nu_{2r}}(\mu^{2})\equiv\int_{k}\frac{k^{\nu_{1}}\cdots
k^{\nu_{2r}}}{(k^{2}-\mu^{2})^{r+1}}.$ (2.3)

The UV finite part in the limit where $\mu\downarrow 0$ has logarithmical
dependence in the physical momenta which is the characteristic behaviour of
the finite part of massless amplitudes LOGS .

  3. 3.

BDI’s with Lorentz indices $\nu_{1}\cdots\nu_{2r}$ may be written as linear
combinations of BDI’s without Lorentz indices (with the same superficial
degree of divergence) plus well defined surface terms (ST’s), e.g.

$\displaystyle\Upsilon_{0}^{(1)\mu\nu}$ $\displaystyle=$
$\displaystyle\int_{k}\frac{\partial}{\partial
k_{\mu}}\frac{k^{\nu}}{(k^{2}-\mu^{2})^{2}}=4\Bigg{[}\frac{g_{\mu\nu}}{4}I_{log}(\mu^{2})-I_{log}^{\mu\nu}(\mu^{2})\Bigg{]},$
(2.4) $\displaystyle\Upsilon_{2}^{(1)\mu\nu}$ $\displaystyle=$
$\displaystyle\int_{k}\frac{\partial}{\partial
k_{\mu}}\frac{k^{\nu}}{(k^{2}-\mu^{2})}=2\Bigg{[}\frac{g_{\mu\nu}}{2}I_{quad}(\mu^{2})-I_{quad}^{\mu\nu}(\mu^{2})\Bigg{]}.$
(2.5)

ST’s are regularisation dependent differences between integrals with the same
superficial degree of divergence which in principle should be fixed by
symmetry requirements or phenomenology JACKIW . We have shown that ST’s vanish
if and only if momentum routing invariance (MRI) holds in the loops of Feynman
diagrams. Moreover these requirements automatically deliver gauge invariant
amplitudes PRIM2 ; PRD1 ; gaugen  as we have demonstrated for abelian gauge
theories to arbitrary loop order PRD2012 ; ALEXANDRE  and verified for non-
abelian gauge models ELOY ; IREGSUSY ; HELVECIO .

  4. 4.

An arbitrary positive (renormalisation group) mass scale $\lambda$ appears via
regularisation independent identities, for instance

$I_{log}(\mu^{2})=I_{log}(\lambda^{2})-\frac{i}{(4\pi)^{2}}\ln\frac{\lambda^{2}}{\mu^{2}},$
(2.6)

which enables us to write a BDI as a function of $\lambda^{2}$ plus
logarithmic functions of $\mu^{2}/\lambda^{2}$, $\mu$ being a fictitious mass
which is added to massless propagators. $I_{quad}(\mu^{2})$ can be chosen to
vanish as $\mu$ goes to zero as we will show through a general parametrisation
of BDI’s. The limit $\mu\rightarrow 0$ is well defined for the whole amplitude
if it is power counting infrared convergent ab initio. The BDI can be absorbed
in the renormalisation constants (without explicit evaluation) CLANT  and
renormalisation functions can be computed using the regularisation independent
identity:

$\lambda^{2}\frac{\partial
I_{log}(\lambda^{2})}{\partial\lambda^{2}}=-\frac{i}{(4\pi)^{2}}.$ (2.7)

  5. 5.

At higher loop order the divergent content can be expressed in terms of BDI in
one loop momentum after performing $n-1$ integrations. The order of such
integrations is chosen systematically to display the counterterms to be
subtracted in compliance with the Bogoliubov’s recursion formula BPHZ ;
ADRIANO . The general form of the terms of a Feynman amplitude after $l$
integrations is

$\displaystyle
I^{\nu_{1}\ldots\nu_{m}}\\!=\\!\\!\int\limits_{k_{l}}\\!\frac{A^{\nu_{1}\ldots\nu_{m}}(k_{l},q_{i})}{\prod_{i}[(k_{l}-q_{i})^{2}-\mu^{2}]}\ln^{l-1}\\!\left(\\!-\frac{k_{l}^{2}-\mu^{2}}{\lambda^{2}}\right)\\!,$
(2.8)

where $l=1,\cdots,n$ and $q_{i}$ is an element (or combination of elements) of
the set $\\{p_{1},\ldots,p_{L},k_{l+1},\ldots,k_{n}\\}$.
$A^{\nu_{1}\ldots\nu_{m}}(k_{l},q_{i})$ represents all possible combinations
of $k_{l}$ and $q_{i}$ compatible with the Lorentz structure.

  6. 6.

Apply relation (2.2) in (2.8) by choosing $n_{i}^{(k_{l})}$ such that all
divergent integrals are free of $q_{i}$. Therefore the divergent integrals are
cast as a combination of

$\displaystyle I_{log}^{(l)}(\mu^{2})$
$\displaystyle\equiv\int\limits_{k_{l}}\frac{1}{(k_{l}^{2}-\mu^{2})^{2}}\ln^{l-1}{\left(-\frac{k_{l}^{2}-\mu^{2}}{\lambda^{2}}\right)},\quad$
(2.9) $\displaystyle I_{log}^{(l)\nu_{1}\cdots\nu_{2r}}(\mu^{2})$
$\displaystyle\equiv\int\limits_{k_{l}}\frac{k_{l}^{\nu_{1}}\cdots
k_{l}^{\nu_{2r}}}{(k_{l}^{2}-\mu^{2})^{r+1}}\ln^{l-1}{\left(-\frac{k_{l}^{2}-\mu^{2}}{\lambda^{2}}\right)},$
(2.10) $\displaystyle I_{quad}^{(l)}(\mu^{2})$
$\displaystyle\equiv\int\limits_{k_{l}}\frac{1}{(k_{l}^{2}-\mu^{2})}\ln^{l-1}{\left(-\frac{k_{l}^{2}-\mu^{2}}{\lambda^{2}}\right)},\quad$
(2.11) $\displaystyle I_{quad}^{(l)\nu_{1}\cdots\nu_{r+2}}(\mu^{2})$
$\displaystyle\equiv\int\limits_{k_{l}}\frac{k_{l}^{\nu_{1}}\cdots
k_{l}^{\nu_{2r}}}{(k_{l}^{2}-\mu^{2})^{r+1}}\ln^{l-1}{\left(-\frac{k_{l}^{2}-\mu^{2}}{\lambda^{2}}\right)}.$
(2.12)

The surface terms derived from higher loop BDI’s are obtained through the
identity

$\displaystyle\Upsilon_{2i}^{(l)\nu_{1}\cdots\nu_{2j}}\equiv\int_{k}\frac{\partial}{\partial
k_{\nu_{1}}}\frac{k^{\nu_{2}}\cdots
k^{\nu_{2j}}}{(k^{2}-\mu^{2})^{1+j-i}}\ln^{l-1}\Bigg{[}-\frac{(k^{2}-\mu^{2})}{\lambda^{2}}\Bigg{]}.$
(2.13)

For instance,

$\displaystyle
I_{log}^{(l)\,\mu\nu}(\mu^{2})=\sum_{j=1}^{l}\left(\frac{1}{2}\right)^{j}\\!\frac{(l-1)!}{(l-j)!}\\!\left\\{\frac{g^{\mu\nu}}{2}I_{log}^{(l-j+1)}(\mu^{2})-\frac{1}{2}\Upsilon_{0}^{(l)\,\mu\nu}\right\\}.$
(2.14)

  7. 7.

A renormalisation group scale is encoded in BDI’s. At nth-loop order a
relation analogous to (2.6) is obtained via the regularisation independent
identity

$\displaystyle I_{log}^{(l)}(\mu^{2})$
$\displaystyle=I_{log}^{(l)}(\lambda^{2})-\frac{b}{l}\ln^{l}\left(\frac{\mu^{2}}{\lambda^{2}}\right)-b\sum_{j=1}^{l-1}\frac{(l-1)!}{(l-j)!}\ln^{l-j}\left(\frac{\mu^{2}}{\lambda^{2}}\right),$
(2.15) $\displaystyle\mbox{where}\quad\lambda^{2}$ $\displaystyle\neq
0,\;\;b\equiv\frac{i}{(4\pi)^{2}}.$ (2.16)

  8. 8.

BDI’s can be absorbed in renormalisation constants. A minimal, mass-
independent scheme amounts to absorb only $I_{log}^{(l)}(\lambda^{2})$. To
evaluate RG constants, BDI’s need not be explicitly evaluated as their
derivatives with respect to the renormalisation scale $\lambda^{2}$ are also
BDI’s. For example PRD2012 ,

$\displaystyle\lambda\frac{\partial
I_{log}(\lambda^{2})}{\partial\lambda^{2}}$
$\displaystyle=-b,\quad\lambda^{2}\frac{\partial
I_{quad}(\lambda^{2})}{\partial\lambda^{2}}=\lambda^{2}I_{log}(\lambda^{2}),$
$\displaystyle\lambda^{2}\frac{\partial
I_{log}^{(n)}(\lambda^{2})}{\partial\lambda^{2}}$
$\displaystyle=-(n-1)\,I_{log}^{(n-1)}(\lambda^{2})-b\,\,\alpha^{(n)}\,,$
$\displaystyle\lambda^{2}\frac{\partial
I_{log}^{(n)\,\mu\nu}(\lambda^{2})}{\partial\lambda^{2}}$
$\displaystyle=-(n-1)I_{log}^{(n-1)\,\mu\nu}(\lambda^{2})-\frac{g_{\mu\nu}}{2}\,b\,\beta^{(n)}.$
(2.17)

where $n\geq 2$, $\alpha^{(n)}=(n-1)!$ and $\beta^{(n)}$ may be obtained from
$\alpha^{(n)}$ via relation (2.14).

  9. 9.

Finally it is possible and sometimes useful to construct a parametrisation of
the BDI’s using (2.17) and keeping track of integration constants PRIM2 ;
PRD2012 . Thus

$\displaystyle I_{log}(\lambda^{2})$
$\displaystyle=b\ln\left(\frac{\Lambda^{2}}{\lambda^{2}}\right)+a_{1},\quad
I_{log}^{\mu\nu}(\lambda^{2})=\frac{g_{\mu\nu}}{4}\Bigg{[}b\ln\left(\frac{\Lambda^{2}}{\lambda^{2}}\right)+a_{1}^{\prime}\Bigg{]},$
(2.18) $\displaystyle I_{quad}(\lambda^{2})$
$\displaystyle=c_{1}\Lambda^{2}+b\lambda^{2}\ln\left(\frac{\Lambda^{2}}{\lambda^{2}}\right)-(a_{1}-b)\lambda^{2},$
$\displaystyle I_{quad}^{\mu\nu}(\lambda^{2})$
$\displaystyle=\frac{g_{\mu\nu}}{2}\Bigg{[}c^{\prime}_{1}\Lambda^{2}+b_{d}\lambda^{2}\ln\left(\frac{\Lambda^{2}}{\lambda^{2}}\right)-(a_{1}^{\prime}-b)\lambda^{2}\Bigg{]},$
(2.19)

where $a_{1},a_{1}^{\prime},c_{1},c_{1}^{\prime}$ are arbitrary regularisation
dependent constants. The choice for which the ST’s in (2.4) and (2.5) vanish
is evidently $a_{1}=a_{1}^{\prime}$ and $c_{1}=c_{1}^{\prime}$. Likewise, for
$n\geq 2$,

$\displaystyle I_{log}^{(n)}(\lambda^{2})$
$\displaystyle=\sum\limits_{i=1}^{n}\alpha^{(i)}\frac{(n-1)!}{(i-1)!}\\!\Bigg{[}\\!\frac{-b}{(n-i+1)!}\ln^{n-i+1}\\!\\!\left(\frac{\Lambda^{2}}{\lambda^{2}}\right)+\sum\limits_{j=0}^{n-i}\frac{a_{n-j-i+1}}{j!(n-j-i)!}\ln^{j}\\!\\!\left(\frac{\Lambda^{2}}{\lambda^{2}}\right)\\!\\!\Bigg{]},$
$\displaystyle I_{log}^{(n)\,\mu\nu}(\lambda^{2})$
$\displaystyle=\frac{g_{\mu\nu}}{2}\\!\sum\limits_{i=1}^{n}\beta^{(i)}\frac{(n-1)!}{(i-1)!}\\!\Bigg{[}\\!\frac{-b}{(n-i+1)!}\ln^{n-i+1}\\!\\!\left(\frac{\Lambda^{2}}{\lambda^{2}}\right)+\sum\limits_{j=0}^{n-i}\frac{a^{\prime}_{n-j-i+1}}{j!(n-j-i)!}\ln^{j}\\!\\!\left(\frac{\Lambda^{2}}{\lambda^{2}}\right)\\!\\!\Bigg{]},$
(2.20)

from which it can be shown that the ST in (2.14),
$\Upsilon_{0}^{(n)\mu\nu}\rightarrow 0$ only if $a_{i}=a^{\prime}_{i}$.
Finally, $I_{quad}(m^{2}\rightarrow 0)=0$ if we choose $c_{1}=0$ QUADRATIC .
This choice is compatible with the result in CDR where
$I_{quad}^{4-2\epsilon}(0)=0$. Consistently with this choice, one may write,
for $\lambda^{2}\neq 0$,

$I_{quad}(\lambda^{2})=\lambda^{2}I_{log}(\lambda^{2})+b\lambda^{2}.$ (2.21)

###  2.2 Correspondence among IREG and dimensional methods

In this section we analyse to which extent it is possible to recover results
for amplitudes evaluated by dimensional methods once the result in IREG is
known. In IREG, the UV log-divergent content of a Feynman amplitude is
expressed by the BDI’s $I_{log}^{(n)}(\lambda^{2})$, while in dimensional
methods they appear as poles in $\epsilon\rightarrow 0$. Therefore, given a
n-loop amplitude, one may wonder if extracting the residues of $\epsilon^{-n}$
by evaluating $I_{log}^{(n)}(\lambda^{2})$ in $4-2\epsilon$ dimensions is
sufficient to map the IREG result on the one obtained in CDR or DRED, for
instance. Starting at $n=1$, consider the log-divergent integral composed by
the product of two massless propagators (scalar self-energy amplitude):

$\mathcal{I}=\int_{k}\frac{1}{k^{2}(k-p)^{2}}\stackrel{{\scriptstyle\text{IREG}}}{{=}}I_{log}(\lambda^{2})-b\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]+2b.$
(2.22)

By evaluating $I_{log}(\lambda^{2})$ in $d=4-2\epsilon$ dimensions and
expanding around $\epsilon=0$ ZURICH

$I_{log}^{d}(\lambda^{2})=(\mu_{DR})^{4-d}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}-\lambda^{2})^{2}}=b\left[\frac{1}{\epsilon}-\gamma_{E}+\ln(4\pi)+\ln\left(\frac{\mu^{2}_{DR}}{\lambda^{2}}\right)\right],$
(2.23)

one obtains

$\displaystyle\mathcal{I^{\text{IREG}}}\Big{|}_{d}=b\left[\frac{1}{\epsilon}-\gamma_{E}+\ln(4\pi)-\ln\left[-\frac{p^{2}}{\mu^{2}_{DR}}\right]+2\right].$
(2.24)

Notice that the $\lambda^{2}$ dependence is automatically traded by
$\mu^{2}_{DR}$. This result coincides with CDR and DRED, since contractions of
the metric are absent, namely

$\displaystyle\mathcal{I}_{d}=b\left[\frac{1}{(4\pi)^{-\epsilon}}\left(-\frac{\mu^{2}_{DR}}{p^{2}}\right)^{\epsilon}\frac{\Gamma(\epsilon)\Gamma^{2}(1-\epsilon)}{\Gamma(2-2\epsilon)}\right]=\mathcal{I^{\text{IREG}}}\Big{|}_{d}.$
(2.25)

Therefore, for this one-loop example, evaluating BDI’s in $4-2\epsilon$
dimensions is equivalent to dimensional methods. It should be pointed out that
at one-loop order, when DRED and CDR yield different finite parts, we expect
to recover the results of the former. The reason is simply IREG does not
recourse to dimensional continuation and thus contractions such as
$g_{ab}g^{ab}$ give 4 rather than $d=4-2\epsilon$. Moreover, at least at one-
loop order, by identifying the renormalisation scales of IREG and dimensional
methods in eq. 2.23, one notices that the subtraction of BDI’s is equivalent
to the $\overline{\text{MS}}$ scheme. Interestingly, in FDR scheme ZURICH ;
FDR , $\mu^{2}$ is replaced by $\mu^{2}_{DR}$ in the final result (after
taking the limit $\mu\downarrow 0$), in order to reproduce the
$\overline{\text{MS}}$ scheme. As for quadratic divergences a similar approach
yields

$I_{quad}(\mu^{2})=(\mu_{DR})^{4-d}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}-\mu^{2})}=b\mu^{2}\left[1+\frac{1}{\epsilon}+\ln\left(\frac{\mu^{2}_{DR}}{\mu^{2}}\right)\right],$
(2.26)

which clearly vanishes in the limit $\mu\downarrow 0$. From the point of view
of IREG, one could still keep quadratic divergences as a BDI, which cancel out
in multiplicatively renormalisable theories ELOY ; PRD2013 ; QUADRATIC  and
thus they can be dismissed in massless models.

We proceed to study the correspondence between dimensional methods and IREG to
higher loop order where some subtleties appear. In analogy to the self-energy
amplitude we studied at one-loop, consider the n-loop product of massless
bubble diagrams proportional to the integrals:

$\displaystyle\mathcal{J}=\int_{k_{1}}\frac{1}{k_{1}^{2}(k_{1}-p)^{2}}\cdots\int_{k_{n}}\frac{1}{k_{n}^{2}(k_{n}-p)^{2}}.$
(2.27)

Using the rules of IREG, each integral in $k_{i}$ can be independently
performed to give

$\displaystyle\mathcal{J^{\text{IREG}}}$
$\displaystyle=\left[I_{log}(\lambda^{2})-b\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]+2b\right]^{n}.$
(2.28)

In order to establish a correspondence with dimensional methods at n-loop
order, the use of eq. 2.24 is inappropriate, since terms
$\mathcal{O}(\epsilon)$ are needed. Therefore, one should consider
$I_{log}(\lambda^{2})$ before expanding around $\epsilon=0$,

$I_{log}^{d}(\lambda^{2})=(\mu_{DR})^{4-d}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}-\lambda^{2})^{2}}=\frac{b}{(4\pi)^{-\epsilon}}\left(\frac{\mu^{2}_{DR}}{\lambda^{2}}\right)^{\epsilon}\Gamma(\epsilon),$
(2.29)

and let us rewrite $\ln\left(-p^{2}/\lambda^{2}\right)$ so that the scale
$\mu^{2}_{DR}$ emerges:

$\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)=\lim_{\epsilon\downarrow
0}\frac{\Gamma(\epsilon)}{(4\pi)^{-\epsilon}}\left[\left(\frac{\mu_{DR}^{2}}{\lambda^{2}}\right)^{\epsilon}-\left(-\frac{\mu^{2}_{DR}}{p^{2}}\right)^{\epsilon}\right],$
(2.30)

where we have used that $\ln(x)=\lim_{\epsilon\downarrow
0}\,\,[(x^{\epsilon}-1)/\epsilon+\alpha(\epsilon)]$,
$\alpha(\epsilon\downarrow 0)=0$, and to write in terms of the gamma function
we have chosen $\alpha(\epsilon)=(\ln
4\pi-\gamma_{E})[(\mu_{DR}^{2}/\lambda^{2})^{\epsilon}-(-\mu_{DR}^{2}/p^{2})^{\epsilon}]$.
Thus,

$\displaystyle\mathcal{J^{\text{IREG}}}\Big{|}_{d}=b^{n}\left[\frac{1}{(4\pi)^{-\epsilon}}\left(-\frac{\mu^{2}_{DR}}{p^{2}}\right)^{\epsilon}\Gamma(\epsilon)+2\right]^{n},$
(2.31)

which, by construction, is independent of $\lambda$. On the other hand, by
performing the computation in DRED from the start one obtains

$\displaystyle\mathcal{J}_{d}$
$\displaystyle=b^{n}\left[\frac{1}{(4\pi)^{-\epsilon}}\left(-\frac{\mu^{2}_{DR}}{p^{2}}\right)^{\epsilon}\frac{\Gamma(\epsilon)\Gamma^{2}(1-\epsilon)}{\Gamma(2-2\epsilon)}\right]^{n}.$
(2.32)

Now it can be seen that eqs. 2.31 and 2.32 only agree in the $\epsilon^{-n}$
and $\epsilon^{-n+1}$ coefficients. This implies that, in a two-loop
computation, the pole structure of an integral of the type of eq. 2.27 can be
fully recovered from the IREG result. Notice that this may not be the case for
a CDR computation if, as already pointed out, contractions of the metric are
present. For instance, consider the two-loop integral

$\displaystyle\mathcal{J}$
$\displaystyle=\int_{k_{1},k{2}}\frac{(k_{1}.k_{2})^{2}}{k_{1}^{2}(k_{1}-p)^{2}k_{2}^{2}(k_{2}-p)^{2}}=\int_{k_{1}}\frac{(k_{1})^{\alpha}(k_{1})^{\beta}}{k_{1}^{2}(k_{1}-p)^{2}}\int_{k{2}}\frac{(k_{2})_{\alpha}(k_{2})_{\beta}}{k_{2}^{2}(k_{2}-p)^{2}}$
$\displaystyle=\left[g^{\alpha\beta}p^{2}A+p^{\alpha}p^{\beta}B\right]\left[g_{\alpha\beta}p^{2}A+p_{\alpha}p_{\beta}B\right]=\left[\textbf{4}A^{2}+2p^{2}AB+p^{4}B^{2}\right]$
(2.33)

where, due to a contraction of the type $g_{ab}g^{ab}=\textbf{4}$ we can only
recover the divergent results of DRED, not CDR. Therefore, by considering a
multi-loop amplitude containing only disjoint one-loop integrals as eq. 2.27,
we can already draw some conclusions: in general, given the IREG result of a
multi-loop amplitude, it is not possible to recover the result from DRED by
just evaluating BDI’s in $4-2\epsilon$ dimensions. For specific cases, like
eq. 2.27 at two-loop order, one can recover all divergent terms (
$\mathcal{O}(\epsilon^{-2})$ and $\mathcal{O}(\epsilon^{-1})$), but not the
finite part of the amplitude. The mismatch in the finite part results from the
inclusion of $O(\epsilon)$ terms in the starting expression for the comparison
of IREG and DRED in 2.30. These terms are irrelevant at 1-loop order but
generate in the sub-leading orders of higher loops finite terms from cross
products of 1/$\epsilon$ powers and $\mathcal{O}(\epsilon)$ terms. One should
emphasize that therefore this mismatch is neither a shortcoming of dimensional
methods nor of IREG, but a simple artifact resulting from the bridging of two
approaches at one-loop order. Nevertheless, for this kind of amplitude, one
can always retrieve the $\epsilon^{-n}$ and $\epsilon^{-n+1}$ terms. This
finding may be useful to check intermediate steps in a computation done with
IREG by comparing with its counterpart in DRED, for instance. Another
conclusion to be drawn is that possibly the subtraction scheme in IREG given
by the removal of BDI’s such as $I_{ln}^{(n)}(\lambda^{2})$ may not correspond
to the $\overline{\text{MS}}$ scheme or even the $\overline{\text{DS}}$
scheme. However, since this conclusion can only be ascertained by the
computation of renormalization constants rather than amplitudes, more
investigations are necessary which we will perform elsewhere.

To conclude this section, we show that, in general, one can only expect to
reproduce the $\mathcal{O}(\epsilon^{-n})$ term of a n-loop amplitude obtained
within CDR or DRED by evaluating BDI’s in $4-2\epsilon$ dimensions. This
happens because in general one also has denominators of the form
$(k_{i}-k_{j})^{2}$. To illustrate this point, we consider a two-loop integral
as below

$\displaystyle\mathcal{T}=\int_{k,l}\frac{1}{k^{2}(k-p)^{2}}\frac{1}{l^{2}(l-k)^{2}},$
(2.34)

where, given the rules of IREG, one must first evaluate the integral in $l$

$\displaystyle\mathcal{T}^{\text{IREG}}=\int_{k}\frac{1}{k^{2}(k-p)^{2}}\left[I_{log}(\lambda^{2})-b\ln\left[-\frac{k^{2}}{\lambda^{2}}\right]+2b\right].$
(2.35)

As can be seen, due to the appearance of the denominator $(l-k)^{2}$, we have
a non-local term on the internal momenta $k$, which will generate
$I_{log}^{(2)}(\lambda^{2})$. The UV divergent part of $\mathcal{T}$ is given
by

$\displaystyle\mathcal{T}^{\text{IREG}}_{\text{div}}=I_{log}^{2}(\lambda^{2})-bI_{log}^{(2)}(\lambda^{2})-bI_{log}(\lambda^{2})\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]+4bI_{log}(\lambda^{2}).$
(2.36)

To obtain a correspondence with DRED, one could consider to replace powers of
$I_{log}(\lambda^{2})$ by eq. 2.29, as well as
$\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]$ by eq. 2.30. Regarding
$I_{log}^{(2)}(\lambda^{2})$, one obtains

$\displaystyle I_{log}^{(2)}(\lambda^{2})$
$\displaystyle=\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}-\lambda^{2})^{2}}\ln\left[-\frac{k^{2}}{\lambda^{2}}\right],$
$\displaystyle I_{log}^{(2)}(\lambda^{2})\Big{|}_{d}$
$\displaystyle=(\mu^{2}_{DR})^{\epsilon}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}-\lambda^{2})^{2}}\frac{1}{(4\pi)^{-\epsilon}}\left[\left(\frac{\mu_{DR}^{2}}{\lambda^{2}}\right)^{\epsilon}-\left(-\frac{\mu^{2}_{DR}}{k^{2}}\right)^{\epsilon}\right]\Gamma(\epsilon)$
$\displaystyle=\frac{b}{(4\pi)^{-2\epsilon}}\left(\frac{\mu_{DR}^{2}}{\lambda^{2}}\right)^{2\epsilon}\Gamma(\epsilon)\left[\Gamma(\epsilon)-\frac{\Gamma(2-4\epsilon)\Gamma(2\epsilon)}{\Gamma(2-3\epsilon)}\right],$
(2.37)

where we have used eq. 2.30 on the second line. Considering only terms up to
$\mathcal{O}(\epsilon^{-1})$ one thus obtains

$\displaystyle\mathcal{T}^{\text{IREG}}_{\text{div}}\Big{|}_{d}=\frac{b^{2}}{2\epsilon^{2}}+\frac{b^{2}}{\epsilon}\left[\frac{7}{2}-\gamma_{E}+\ln(4\pi)-\ln\left(-\frac{p^{2}}{\mu_{DR}^{2}}\right)\right].$
(2.38)

If the calculation is done in DRED from the start, one gets

$\displaystyle\mathcal{T}\Big{|}_{d}=$
$\displaystyle\frac{b^{2}(4\pi)^{2\epsilon}\Gamma(1-2\epsilon)\Gamma(1-\epsilon)^{3}\Gamma(\epsilon)\Gamma(2\epsilon)\left(-\frac{p^{2}}{\mu^{2}}\right)^{-2\epsilon}}{\Gamma(2-3\epsilon)\Gamma(2-2\epsilon)\Gamma(\epsilon+1)}$
$\displaystyle=$
$\displaystyle\frac{b^{2}}{2\epsilon^{2}}+\frac{b^{2}}{\epsilon}\left[\frac{5}{2}-\gamma_{E}+\ln(4\pi)-\ln\left(-\frac{p^{2}}{\mu_{DR}^{2}}\right)\right],$
(2.39)

which differs from eq. 2.38 in local terms of $\mathcal{O}(\epsilon^{-1})$.
Nevertheless, the terms in $\epsilon^{-2}$ as well as non-local divergent
terms can be reproduced. This observation allows us to check diagram-by-
diagram the results we obtain in further sections.

###  2.3 Practical approach to multi-loop calculations

As extensively discussed in the previous subsections, IREG can be applied in a
variety of contexts, for which the inclusion of ST’s is crucial. From a
practical point of view, however, this implies that shifts are not in general
allowed in divergent multi-loop integrals. Therefore, it proves to be useful
to develop a minimal set of rules. The rules have obviously to comply with the
major requirements of unitarity and gauge invariance. Since IREG is built in
such a way that the BPHZ theorem is respected ADRIANO , regardless of the
choice of the ST’s, unitarity is always preserved. For gauge invariance, it
has been proven that setting ST’s to zero complies with abelian gauge symmetry
VIGLIONI . However, this is not the only requirement. As discussed in BRUQUE ,
it is essential to perform tensor identification in a consistent way since,
for instance,

$\displaystyle
g_{\mu\nu}\int_{k}k^{\mu}k^{\nu}f(k)\bigg{|}_{\text{IREG}}\neq\int_{k}k^{2}f(k)\bigg{|}_{\text{IREG}}.$
(2.40)

Therefore, a normal form is required to avoid mismatches during intermediary
steps of the evaluation. We propose, regarding multi-loop calculations, to set
ST’s equal to zero allowing shifts to be performed in internal momenta of
divergent integrals and define a normal form according to the following set of
rules

  1. A

apply Feynman rules to the whole multi-loop diagram;

  2. B

perform contractions (in four-dimensions) and simplify numerator against
denominator if possible. This is particularly relevant to terms such as

$\displaystyle\int_{k,q}\frac{k^{2}}{k^{2}q^{2}(k-q)^{2}}\bigg{|}_{\text{IREG}}=\int_{k,q}\frac{1}{q^{2}(k-q)^{2}}\bigg{|}_{\text{IREG}}.$
(2.41)

However, one may not manipulate the numerator to enforce cancellations, since
this may modify the normal form. For instance,

$\displaystyle\int_{k,q}\frac{k.p}{k^{2}(k-p)^{2}q^{2}(k-q)^{2}}\bigg{|}_{\text{IREG}}\neq\int_{k,q}\frac{k^{2}+p^{2}-(k-p)^{2}}{2k^{2}(k-p)^{2}q^{2}(k-q)^{2}}\bigg{|}_{\text{IREG}}.$
(2.42)

After these steps, the rules sketched in the subsection 2.1 (with ST’s equal
to zero) can be applied. For simplicity, we will also discard terms
quadratically divergent encoded as $I_{quad}^{(l)}(\mu^{2})$ since they must
cancel in theories that are multiplicative renormalizable, which are the ones
we consider here. Before proceeding to our results, it should be emphasised
that, in this framework, one is not allowed, in general, to evaluate a sub-
diagram and join the obtained result in the full diagram. The reason can be
traced back to equations similar to (2.40). This fact does not amount in a
violation of unitarity since, according to the BPHZ theorem, only the
divergent content of the integrals in (2.40) must coincide, and they do.
However, local terms may be generated which can (possibly) violate gauge
invariance. Therefore, we argue that the normal form obtained by the steps (A)
- (B) above is the one that respects both unitarity and gauge invariance, as
our results will show.

In order to clarify the discussion we present below an example. Consider the
diagram of Fig. 1,

Figure 1: Two-loop diagram which contains a gluon loop as sub-diagram

whose amplitude (in the Feynman gauge) is schematically given by

$\mathcal{A}\propto\int_{k}\Pi_{\mu\nu\alpha\beta}(k,p)\int_{l}\frac{\mathcal{F}^{\alpha\beta}(l,k,p)}{l^{2}(l-k)^{2}},$
(2.43)

where $l$ is the internal momentum of the sub diagram (gluon loop), $k$ the
internal momentum of the complete diagram, and $p$ the external momentum. From
the many possibilities encoded in $\mathcal{F}^{\alpha\beta}$, one is simply
$l^{\alpha}l^{\beta}$ which, after contraction with
$g_{\alpha\beta}k_{\mu}k_{\nu}$, one of the many terms presented in
$\Pi_{\mu\nu\alpha\beta}(k,p)$, generates a contribution as

$\displaystyle\mathcal{B}\propto$
$\displaystyle\int_{k}\frac{k_{\mu}k_{\nu}}{k^{4}(k-p)^{2}}\int_{l}\frac{l^{2}}{l^{2}(l-k)^{2}}$
(2.44) $\displaystyle\propto$
$\displaystyle\int_{k}\frac{k_{\mu}k_{\nu}}{k^{4}(k-p)^{2}}\int_{l}\frac{1}{(l-k)^{2}}$
(2.45)

Notice that eq.(2.44) was obtained by applying the step (A) of our procedure,
while eq.(2.45) is due to step (B). Next one should evaluate the integrals in
IREG according to the algorithm presented in ADRIANO , and the rules sketched
in the last subsection. It is not difficult to perceive that a null result
will be obtained, since we are allowed to perform shifts (ST’s are null) and
the integral in $l$ will be identified with $I_{quad}(\mu^{2})$ which we drop
as already explained. On the other hand, if one opts to evaluate the sub-
diagram independently, one obtains

$\displaystyle\int_{l}\frac{l^{\alpha}l^{\beta}}{l^{2}(l-k)^{2}}\Bigg{|}_{\text{IREG}}=k^{\alpha}k^{\beta}\mathcal{T}_{1}+g^{\alpha\beta}k^{2}\mathcal{T}_{2},$
(2.46)

where $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ are scalar functions defined in
eq. B.1. By contracting the previous equation with
$g_{\alpha\beta}k_{\mu}k_{\nu}$ one now gets

$\displaystyle\mathcal{\tilde{B}}\propto$
$\displaystyle\int_{k}\left[\frac{k_{\mu}k_{\nu}k^{2}}{k^{4}(k-p)^{2}}(\mathcal{T}_{1}+4\mathcal{T}_{2})\right]=\int_{k}\left[\frac{k_{\mu}k_{\nu}k^{2}}{k^{4}(k-p)^{2}}\left(-\frac{b}{6}\right)\right]$
(2.47)

which is (clearly) different from zero.

##  3 Two-loop applications in gauge theories

In this section we apply the procedure discussed in the last section to a
variety of examples. First we discuss abelian theories such as scalar QED
(which possesses derivative vertices) and spinorial QED (which requires a
proper treatment of Dirac algebra BRUQUE ). Next we turn to non-abelian gauge
theories such as Yang-Mills, and finally QCD. We will be mainly interested in
the computation of the two-loop coefficient of the gauge couplings $\beta$
function in all these theories. As is well-known, in the QED case one may only
resort to the calculation of the two-point photon correction since
$Z_{g}=Z_{A}^{-1/2}$ due to the Ward identity. As usual $Z_{g/A}$ relates to
the coupling/photon renormalisation function. In the case of non-abelian
theories, the situation is more involved. In order to mimic the behavior of
the QED case, one can resort to the background field method Abbott , which
guarantees the relation $Z_{g}=Z_{\hat{A}}^{-1/2}$ where $\hat{A}$ is now the
gluon background field. This fact simplifies considerably the computation,
since only two-point functions will be needed.

Therefore, in this contribution we will only deal with two-point functions
with the photon (scalar/spinorial QED) or the gluon background field (Yang-
Mills and QCD) as external legs. This implies that only topologies as below
can appear

Figure 2: Two-loop topologies for two-point functions

Notice that we are already omitting tapdole-like diagrams. The reason is
twofold. Firstly, since we are only interested in the gauge coupling $\beta$
function, we can consider massless scalars/fermions. Secondly, given the
minimal set of rules presented in the last section, we drop quadratically
scaleless integrals, encoded as $I_{quad}^{(l)}(\mu^{2})$. Given the general
topologies, one has now to fill them with the field content of the theory at
hand. For instance, in spinorial QED, only topologies T3 and T4 appear. In
order to perform this task automatically, we have made use of FeynArts
FeynArts  which already has the spinorial QED model implemented. For the
background field method, only the Electroweak Theory is already implemented,
so we have adapted it to consider a background version of QCD as well. For
scalar QED, we have opted to use the Feynman rules of Schwartz , building the
amplitudes ourselves.

Once the amplitudes in all theories have been built, we adapted them to be
recognized by FormCalc FormCalc , allowing only Dirac and Lorentz algebra to
be performed with the end result for each topology given schematically by:

$\displaystyle\mathcal{A_{\text{T1}/\text{T2}}}$
$\displaystyle\propto\int_{k,l}\frac{\mathcal{F}_{T1/T2}^{\alpha\beta}(l,k,p)}{k^{2}(k-p)^{2}l^{2}(l-k)^{2}},$
$\displaystyle\mathcal{A_{\text{T3}}}$
$\displaystyle\propto\int_{k,l}\frac{\mathcal{F}_{T3}^{\alpha\beta}(l,k,p)}{k^{2}(k-p)^{2}(k-l)^{2}l^{2}(l-p)^{2}},$
$\displaystyle\mathcal{A_{\text{T4}}}$
$\displaystyle\propto\int_{k,l}\frac{\mathcal{F}_{T4}^{\alpha\beta}(l,k,p)}{k^{4}(k-p)^{2}l^{2}(l-k)^{2}},$
$\displaystyle\mathcal{A_{\text{T5}}}$
$\displaystyle\propto\int_{k,l}\frac{\mathcal{F}_{T5}^{\alpha\beta}(l,k,p)}{k^{2}(k-p)^{2}l^{2}(l-p)^{2}},$
$\displaystyle\mathcal{A_{\text{T6}}}$
$\displaystyle\propto\int_{k,l}\frac{\mathcal{F}_{T6}^{\alpha\beta}(l,k,p)}{k^{2}l^{2}(l-k+p)^{2}}.$
(3.1)

Notice that we are adopting Feynman gauge and scalars/fermions are massless as
already pointed out. The equations above are already in normal form, meaning
one can now apply the rules of IREG sketched in the last section. Another
simplification of our case is that only the divergent part of the integrals
above is needed for obtaining the gauge coupling $\beta$ function. In the next
subsections we present the results of IREG for each diagram and for each of
the theories we considered.

In order to allow a comparison with dimensional regularisation methods, we
perform Dirac and Lorentz algebra in $d$-dimensions and evaluate the integrals
also in $d$-dimensions. This will allow us to recover the results diagram-by-
diagram obtained not only in CDR, but also in naive DRED (devoid of
$\epsilon$-scalars) which we will, hereafter, denote by
$\overline{\text{DRED}}$. As discussed in EPSILONSCALAR-BETA , up to two-
loops, the gauge coupling $\beta$ function will not depend on the
$\epsilon$-scalars contributions, meaning the end result of
$\overline{\text{DRED}}$, DRED, CDR is the same. We will recover this result
for all the theories considered. For completeness we will also show, for the
Yang-Mills theory, that the $\epsilon$-scalars contributions cancel.

Before proceeding to our explicit results, it is necessary to discuss briefly
the renormalisation program in the case of the background field method. For
definiteness, we will consider the Yang-Mills theory which has all the
necessary ingredients. As discussed in Abbott , in principle, one would need
to renormalize the fields (gluon, background gluon, ghost), coupling and,
potentially, the gauge-fixing parameter. However, since only the background
gluon field occurs in external legs, the renormalisation constants related to
the gluon and ghost fields will always cancel. Therefore, one should only
consider the multiplicative renormalisation constants defined below

$\hat{A}_{o}=Z_{\hat{A}}\hat{A}_{r};\quad
g_{o}=Z_{g}g_{r};\quad\alpha_{o}=Z_{\alpha}\alpha_{r},$ (3.2)

where $\hat{A}$, $g$, $\alpha$ stand for the background gluon field, strong
coupling and gauge-fixing parameter respectively. The latter could be left
unrenormalized as well if we had considered a general gauge to perform the
calculation. However, since we have adopted Feynman gauge throughout our work,
it will be necessary to include counterterms related to gauge-fixing
renormalisation.

As standard, one can write

$\alpha_{o}=Z_{\alpha}\alpha_{r}=(1+\delta_{\alpha})\alpha_{r},$ (3.3)

which implies

$\displaystyle\mathcal{L}_{\text{GF}}$
$\displaystyle=-\frac{1}{2\alpha_{o}}\left[\partial_{\mu}A^{\mu}_{a}+g_{o}f^{abc}(\hat{A}_{o})_{\mu}^{b}A_{\mu}^{c}\right]^{2}$
$\displaystyle=-\frac{1}{2\alpha_{r}}\left[\partial_{\mu}A^{\mu}_{a}+g_{r}f^{abc}(\hat{A}_{r})_{\mu}^{b}A_{\mu}^{c}\right]^{2}+\frac{\delta_{\alpha}}{2\alpha_{r}}\left[\partial_{\mu}A^{\mu}_{a}+g_{r}f^{abc}(\hat{A}_{r})_{\mu}^{b}A_{\mu}^{c}\right]^{2}$
(3.4)

where the second term gives the counterterms related to the gauge-fixing
renormalisation. Notice that we have used the important relation
$Z_{g}=Z_{\hat{A}}^{1/2}$ as well. Therefore, to obtain an explicit formula to
$\delta_{\alpha}$ it suffices to compute the one-loop correction to the gluon
propagator

$\Pi_{\mu\nu}^{ab}|_{\text{div}}=-i5g^{2}C_{A}I_{log}(\lambda^{2})(g_{\mu\nu}p^{2}-p_{\mu}p_{\nu})\delta^{ab}\xrightarrow[\text{DRED}]{}\frac{5}{3\epsilon}\frac{g^{2}C_{A}}{(4\pi)^{2}}(g_{\mu\nu}p^{2}-p_{\mu}p_{\nu})\delta^{ab},$
(3.5)

since, given we are in Feynman gauge, we should attain the relation

$\Pi_{\mu\nu}^{ab}|_{\text{div}}+\delta_{\alpha}p_{\mu}p_{\nu}\delta^{ab}\propto
g_{\mu\nu}\delta^{ab}.$ (3.6)

The final result is

$\delta_{\alpha}=-i5g^{2}C_{A}I_{log}(\lambda^{2})\xrightarrow[\text{DRED}]{}\frac{5}{3\epsilon}\frac{g^{2}C_{A}}{(4\pi)^{2}}$
(3.7)

which agrees with Abbott .

###  3.1 Scalar QED

In scalar QED, the part of the Lagrangian containing the couplings is of the
form

$\mathcal{L}_{\text{scalar QED}}\supset-
ig_{e}A_{\mu}\left[\phi^{*}(\partial_{\mu}\phi)-(\partial_{\mu}\phi^{*})\phi\right]+g_{e}^{2}A_{\mu}A^{\mu}|\phi|^{2}.$
(3.8)

This implies that only topology T4 cannot be realised (remember the photon is
in both external legs). Regarding counterterms, we only need to consider
(possible) corrections due to renormalisation of the gauge fixing parameter.
In the present case (abelian theory), none of the couplings will depend on
this parameter, meaning we do not need to consider counterterms related to
them. Moreover, since the triple coupling contains only one photon field, it
is not possible to have a photon self-energy sub-diagram (see topology T1).
Therefore, for our calculation of the $\beta$ function of scalar QED, no
counterterm is needed. For completeness, we have indeed computed the
counterterms related to the triple and quartic coupling (obtained from
shrinking sub-diagrams in topologies T1 and T2 to a point), and the
counterterm related to the scalar self-energy (obtained by shrinking the sub-
diagram in topology T3 to a point). As expected, the counterterms cancel among
themselves.

In order to present our results, we will define the two-loop correction to the
photon field to be of the form

$\mathcal{A_{\mu\nu}}=\frac{ig_{e}^{4}}{(4\pi)^{4}}\left[Ag_{\mu\nu}p^{2}-Bp_{\mu}p_{\nu}\right].$
(3.9)

where $p$ is the momentum carried by the external photon. With this definition
at hand, the explicit results for each topology can be read from table 2 in
the case of IREG and table 3 for CDR. In the case of scalar QED, there are no
contractions of the type $g_{ab}g^{ab}=d$, which implies that results in CDR
are identical to $\overline{DRED}$, as can also be seen from table 3. We
should also comment that in the case of scalar QED, there is only one type of
diagram related to each contributing topology, although topologies T1, T3 have
multiplicity 4, 2 respectively. The results for the counterterms can be found
in tables 4 and 5, which cancel among themselves as already pointed out.

$\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup\text{Topology}}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{A}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{B}\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{2-9}\cr&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})\\\
\hline\cr
T1&-\frac{2}{b}&\frac{2}{b^{2}}&-\frac{2}{b}&\frac{87}{9b}&-\frac{2}{b}&\frac{2}{b^{2}}&-\frac{2}{b}&\frac{51}{9b}\\\
\hline\cr
T2&\frac{2}{3b}&-\frac{2}{3b^{2}}&\frac{2}{3b}&-\frac{47}{9b}&\frac{2}{3b}&-\frac{2}{3b^{2}}&\frac{2}{3b}&-\frac{35}{9b}\\\
\hline\cr
T3&\frac{4}{3b}&-\frac{4}{3b^{2}}&\frac{4}{3b}&-\frac{58}{9b}&\frac{4}{3b}&-\frac{4}{3b^{2}}&\frac{4}{3b}&-\frac{52}{9b}\\\
\hline\cr T5&0&0&0&-\frac{2}{b}&0&0&0&0\\\
\hline\cr\text{Sum}&0&0&0&-\frac{4}{b}&0&0&0&-\frac{4}{b}\\\
\hline\cr\end{array}$ Table 2: Results for Scalar QED using IREG where
$\rho_{IREG}=I_{log}(\lambda^{2})\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]$
$\begin{array}[]{|c|c|c|c|c|}\hline\cr\text{Diagram}&\text{A}_{\text{CDR}}&\text{A}_{\overline{\text{DRED}}}-\text{A}_{\text{CDR}}&\text{B}_{\text{CDR}}&\text{B}_{\overline{\text{DRED}}}-\text{B}_{\text{CDR}}\\\
\hline\cr
T1&\frac{1}{\epsilon^{2}}+\frac{13-4\rho}{2\epsilon}&0&\frac{1}{\epsilon^{2}}+\frac{9-4\rho}{2\epsilon}&0\\\
\hline\cr
T2&-\frac{1}{3\epsilon^{2}}-\frac{19-4\rho}{6\epsilon}&0&-\frac{1}{3\epsilon^{2}}-\frac{15-4\rho}{6\epsilon}&0\\\
\hline\cr
T3&-\frac{2}{3\epsilon^{2}}-\frac{13-4\rho}{3\epsilon}&0&-\frac{2}{3\epsilon^{2}}-\frac{12-4\rho}{3\epsilon}&0\\\
\hline\cr T5&-\frac{1}{\epsilon}&0&0&0\\\
\hline\cr\text{Sum}&-\frac{2}{\epsilon}&0&-\frac{2}{\epsilon}&0\\\
\hline\cr\end{array}$ Table 3: Results for Scalar QED using CDR and
$\overline{\text{DRED}}$, where $\rho=\gamma_{E}-\ln
4\pi+\ln(p^{2}/\mu^{2}_{DR})$.
$\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup\text{Counterterm}}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{A}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{B}\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{2-9}\cr&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})\\\
\hline\cr\text{Coupling}&0&\frac{4}{3b^{2}}&-\frac{4}{3b}&\frac{32}{9b}&0&\frac{4}{3b^{2}}&-\frac{4}{3b}&\frac{32}{9b}\\\
\hline\cr\text{Scalar self-
energy}&0&-\frac{4}{3b^{2}}&\frac{4}{3b}&-\frac{32}{9b}&0&-\frac{4}{3b^{2}}&\frac{4}{3b}&-\frac{32}{9b}\\\
\hline\cr\text{Sum}&0&0&0&0&0&0&0&0\\\ \hline\cr\end{array}$ Table 4:
Counterterm results for Scalar QED using IREG where
$\rho_{IREG}=I_{log}(\lambda^{2})\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]$
$\begin{array}[]{|c|c|c|c|c|}\hline\cr\text{Counterterm}&\text{A}_{\text{CDR}}&\text{A}_{\overline{\text{DRED}}}-\text{A}_{\text{CDR}}&\text{B}_{\text{CDR}}&\text{B}_{\overline{\text{DRED}}}-\text{B}_{\text{CDR}}\\\
\hline\cr\text{Coupling}&\frac{4}{3\epsilon^{2}}+\frac{32-12\rho}{9\epsilon}&0&\frac{4}{3\epsilon^{2}}+\frac{32-12\rho}{9\epsilon}&0\\\
\hline\cr\text{Scalar self-
energy}&-\frac{4}{3\epsilon^{2}}-\frac{32-12\rho}{9\epsilon}&0&-\frac{4}{3\epsilon^{2}}-\frac{32-12\rho}{9\epsilon}&0\\\
\hline\cr\text{Sum}&0&0&0&0\\\ \hline\cr\end{array}$ Table 5: Counterterm
results for Scalar QED using CDR and $\overline{\text{DRED}}$, where
$\rho=\gamma_{E}-\ln 4\pi+\ln(p^{2}/\mu^{2}_{DR})$

As one can easily notice, the end result is gauge invariant in all of the
methods considered here. Also, as is well-knwon Schwartz , although we are
considering a two-loop correction, only terms proportional to $\epsilon^{-1}$
will survive in the end result. In the framework of IREG we also have a
similar pattern, since only $I_{log}(\lambda^{2})$ terms appear in the final
result. This similarity between methods, however, will not be valid in
general, as we are going to see when studying the Yang-Mills theory. The
reason can be traced back to the appearance of diagrams of topology T4, as we
will explain latter.

###  3.2 Spinorial QED

We move to spinorial QED, where, although only one type of vertex occurs
$-e\bar{\psi}\gamma^{\mu}\psi A_{\mu}$, one needs to deal with Dirac algebra.
Given the more restrictive coupling, when compared with scalar QED, one
expects that even less diagrams will contribute. Explicitly, only topologies
T2 and T3 can be realized, each one with one diagram, and multiplicity 1,2
respectively. As before, no counterterms will be necessary. For completeness,
we compute them and check that they cancel.

Regarding the computation itself, the only point that one should be careful
about is in how the Dirac algebra is performed. Now, spurious terms
$\mathcal{O}(\epsilon)$ will appear, implying that the results in CDR and
$\overline{\text{DRED}}$ are not identical diagram by diagram, although the
final result should be. We checked that this is indeed the case. In the
framework of IREG, some care should also be exercised, in order to perform
tensor identification consistently. As we argued in section 2.3, a normal form
is obtained after performing Dirac and Lorentz algebra to the whole diagram.
After the normal is attained, manipulations in the numerator that may generate
spurious terms with $k^{2}$, for instance, cannot be performed. An example of
such forbidden manipulation is

$\displaystyle\int_{k,q}\frac{k.q}{k^{2}(k-p)^{2}q^{2}(q-p)^{2}(k-q)^{2}}\bigg{|}_{\text{IREG}}\neq\int_{k,q}\frac{k^{2}+q^{2}-(k-q)^{2}}{2k^{2}(k-p)^{2}q^{2}(q-p)^{2}(k-q)^{2}}\bigg{|}_{\text{IREG}}.$
(3.10)

The left-hand side of the equation above appears in the diagram of topology
T2, for instance. The consistent approach to treat this integral is given by
the procedure defined in ADRIANO , whose result we collect in the appendix B.

To conclude this subsection we collect our results in tables 6 to 9, adopting
the same convention of eq. 3.9. As in the case of scalar QED, the end result
is transverse in all methods and depend only in $\epsilon^{-1}$ or
$I_{log}(\lambda^{2})$ terms. Moreover, the counterterms cancel among
themselves, and we notice that the two-loop correction to the photon is the
same in both scalar and spinorial QED. This implies the same gauge coupling
$\beta$ function.

$\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup\text{Topology}}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{A}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{B}\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{2-9}\cr&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})\\\
\hline\cr
T2&\frac{8}{3b}&-\frac{8}{3b^{2}}&\frac{8}{3b}&-\frac{52}{3b}&\frac{8}{3b}&-\frac{8}{3b^{2}}&\frac{8}{3b}&-\frac{16}{b}\\\
\hline\cr
T3&-\frac{8}{3b}&\frac{8}{3b^{2}}&-\frac{8}{3b}&\frac{40}{3b}&-\frac{8}{3b}&\frac{8}{3b^{2}}&-\frac{8}{3b}&\frac{12}{b}\\\
\hline\cr\text{Sum}&0&0&0&-\frac{4}{b}&0&0&0&-\frac{4}{b}\\\
\hline\cr\end{array}$ Table 6: Results for Spinorial QED using IREG where
$\rho_{IREG}=I_{log}(\lambda^{2})\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]$
$\begin{array}[]{|c|c|c|c|c|}\hline\cr\text{Diagram}&\text{A}_{\text{CDR}}&\text{A}_{\overline{\text{DRED}}}-\text{A}_{\text{CDR}}&\text{B}_{\text{CDR}}&\text{B}_{\overline{\text{DRED}}}-\text{B}_{\text{CDR}}\\\
\hline\cr
T2&\frac{4}{3\epsilon^{2}}+\frac{8(2-\rho)}{3\epsilon}&\frac{4}{3\epsilon}&\frac{4}{3\epsilon^{2}}+\frac{2(7-4\rho)}{3\epsilon}&\frac{4}{3\epsilon}\\\
\hline\cr
T3&-\frac{4}{3\epsilon^{2}}-\frac{2(11-4\rho)}{3\epsilon}&-\frac{4}{3\epsilon}&-\frac{4}{3\epsilon^{2}}-\frac{4(5-2\rho)}{3\epsilon}&-\frac{4}{3\epsilon}\\\
\hline\cr\text{Sum}&-\frac{2}{\epsilon}&0&-\frac{2}{\epsilon}&0\\\
\hline\cr\end{array}$ Table 7: Results for Spinorial QED using CDR and
$\overline{\text{DRED}}$, where $\rho=\gamma_{E}-\ln
4\pi+\ln(p^{2}/\mu^{2}_{DR})$
$\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup\text{Counterterm}}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{A}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{B}\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{2-9}\cr&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})\\\
\hline\cr\text{Coupling}&0&\frac{8}{3b^{2}}&-\frac{8}{3b}&\frac{56}{9b}&0&\frac{8}{3b^{2}}&-\frac{8}{3b}&\frac{56}{9b}\\\
\hline\cr\text{Scalar self-
energy}&0&-\frac{8}{3b^{2}}&\frac{8}{3b}&-\frac{56}{9b}&0&-\frac{8}{3b^{2}}&\frac{8}{3b}&-\frac{56}{9b}\\\
\hline\cr\text{Sum}&0&0&0&0&0&0&0&0\\\ \hline\cr\end{array}$ Table 8:
Counterterm results for Spinorial QED using IREG where
$\rho_{IREG}=I_{log}(\lambda^{2})\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]$
$\begin{array}[]{|c|c|c|c|c|}\hline\cr\text{Counterterm}&\text{A}_{\text{CDR}}&\text{A}_{\overline{\text{DRED}}}-\text{A}_{\text{CDR}}&\text{B}_{\text{CDR}}&\text{B}_{\overline{\text{DRED}}}-\text{B}_{\text{CDR}}\\\
\hline\cr\text{Coupling}&\frac{8}{3\epsilon^{2}}+\frac{8(7-3\rho)}{9\epsilon}&0&\frac{8}{3\epsilon^{2}}+\frac{8(7-3\rho)}{9\epsilon}&0\\\
\hline\cr\text{Scalar self-
energy}&-\frac{8}{3\epsilon^{2}}-\frac{8(7-3\rho)}{9\epsilon}&0&-\frac{8}{3\epsilon^{2}}-\frac{8(7-3\rho)}{9\epsilon}&0\\\
\hline\cr\text{Sum}&0&0&0&0\\\ \hline\cr\end{array}$ Table 9: Counterterm
results for Spinorial QED using CDR and $\overline{\text{DRED}}$, where
$\rho=\gamma_{E}-\ln 4\pi+\ln(p^{2}/\mu^{2}_{DR})$

###  3.3 Pure Yang-Mills

We turn to non-abelian theories. At first, we do not include scalar/fermionic
interactions, considering pure Yang-Mills theory, see Appendix A for notations
and conventions of the Feynman rules used in the background field method to
Yang-Mills Abbott , employed in the present calculation. The calculation
proves to be involved enough since not only all topologies are realized, but
also there are diagrams with different field content for some of the
topologies, as can be seen in fig. 3.

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)

Figure 3: Two-loop correction to the two-point function of the background
field $A$.

Apart from the large number of diagrams, there are two main differences
regarding the computation in QED we would like to emphasize. First, we have
the appearance of topology T5 which, as we are going to see, implies in the
presence of not only terms with $I_{log}(\lambda^{2})$ in the end result, but
also $I_{log}^{(2)}(\lambda^{2})$, and $I_{log}^{2}(\lambda^{2})$. Second, not
only the triple coupling depends on the gauge fixing parameter, but also the
1-loop correction to the gluon self-energy appears as sub-diagram in diagrams
(h) and (i). Therefore, we will need to consider counterterms related to the
gauge fixing renormalisation.

In a similar manner to eq. 3.9, we can define

$\frac{ig_{s}^{4}C_{A}^{2}\delta^{ab}}{(4\pi)^{4}}\left[Ag_{\mu\nu}p^{2}-Bp_{\mu}p_{\nu}\right],$
(3.11)

where $p$ is the external momenta carried by the background field. Regarding
the diagrams of Fig. 3, our results, for IREG, are presented in table 10 while
for CDR/$\overline{\text{DRED}}$ they can be seen in table 11.

$\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup\text{Diagram}}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{A}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{B}\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{2-9}\cr&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})\\\
\hline\cr
a&\frac{1}{3b}&-\frac{1}{3b^{2}}&\frac{1}{3b}&-\frac{29}{18b}&\frac{1}{3b}&-\frac{1}{3b^{2}}&\frac{1}{3b}&-\frac{17}{18b}\\\
\hline\cr
b&\frac{5}{12b}&-\frac{5}{12b^{2}}&\frac{5}{12b}&-\frac{97}{72b}&\frac{5}{12b}&-\frac{5}{12b^{2}}&\frac{5}{12b}&-\frac{109}{72b}\\\
\hline\cr
c&\frac{9}{4b}&-\frac{9}{4b^{2}}&\frac{9}{4b}&-\frac{39}{8b}&\frac{9}{4b}&-\frac{9}{4b^{2}}&\frac{9}{4b}&-\frac{75}{8b}\\\
\hline\cr
d&-\frac{1}{12b}&\frac{1}{12b^{2}}&-\frac{1}{12b}&\frac{47}{72b}&-\frac{1}{12b}&\frac{1}{12b^{2}}&-\frac{1}{12b}&\frac{35}{72b}\\\
\hline\cr
e&\frac{1}{2b}&-\frac{1}{2b^{2}}&\frac{1}{2b}&-\frac{7}{4b}&\frac{1}{2b}&-\frac{1}{2b^{2}}&\frac{1}{2b}&-\frac{7}{4b}\\\
\hline\cr
f&-\frac{27}{4b}&\frac{27}{4b^{2}}&-\frac{27}{4b}&\frac{195}{8b}&-\frac{27}{4b}&\frac{27}{4b^{2}}&-\frac{27}{4b}&\frac{207}{8b}\\\
\hline\cr
g&-\frac{1}{3b}&\frac{1}{3b^{2}}&-\frac{1}{3b}&\frac{29}{18b}&-\frac{1}{3b}&\frac{1}{3b^{2}}&-\frac{1}{3b}&\frac{13}{9b}\\\
\hline\cr
h+i&-\frac{25}{3b}&\frac{25}{3b^{2}}&-\frac{25}{3b}&\frac{521}{18b}&-\frac{25}{3b}&\frac{25}{3b^{2}}&-\frac{25}{3b}&\frac{268}{9b}\\\
\hline\cr j&0&0&0&\frac{1}{4b}&0&0&0&0\\\ \hline\cr
k&0&0&0&-\frac{9}{4b}&0&0&0&0\\\ \hline\cr
l&0&-\frac{6}{b^{2}}&\frac{12}{b}&-\frac{24}{b}&0&-\frac{6}{b^{2}}&\frac{12}{b}&-\frac{24}{b}\\\
\hline\cr\text{Sum}&-\frac{12}{b}&\frac{6}{b^{2}}&0&\frac{20}{b}&-\frac{12}{b}&\frac{6}{b^{2}}&0&\frac{20}{b}\\\
\hline\cr\end{array}$ Table 10: Results for pure Yang-Mills using IREG where
$\rho_{IREG}=I_{log}(\lambda^{2})\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]$
$\begin{array}[]{|c|c|c|c|c|}\hline\cr\text{Diagram}&\text{A}_{\text{CDR}}&\text{A}_{\overline{\text{DRED}}}-\text{A}_{\text{CDR}}&\text{B}_{\text{CDR}}&\text{B}_{\overline{\text{DRED}}}-\text{B}_{\text{CDR}}\\\
\hline\cr
a&-\frac{1}{6\epsilon^{2}}+\frac{-13+4\rho}{12\epsilon}&0&-\frac{1}{6\epsilon^{2}}+\frac{-9+4\rho}{12\epsilon}&0\\\
\hline\cr
b&-\frac{5}{24\epsilon^{2}}+\frac{-41+20\rho}{12\epsilon}&0&-\frac{5}{24\epsilon^{2}}+\frac{5(-9+4\rho)}{48\epsilon}&0\\\
\hline\cr
c&-\frac{9}{8\epsilon^{2}}+\frac{-57+36\rho}{16\epsilon}&\frac{3}{4\epsilon}&-\frac{9}{8\epsilon^{2}}+\frac{-93+36\rho}{16\epsilon}&\frac{3}{4\epsilon}\\\
\hline\cr
d&\frac{1}{24\epsilon^{2}}+\frac{19-4\rho}{48\epsilon}&0&\frac{1}{24\epsilon^{2}}+\frac{15-4\rho}{48\epsilon}&0\\\
\hline\cr
e&-\frac{1}{4\epsilon^{2}}+\frac{-9+4\rho}{8\epsilon}&0&-\frac{1}{4\epsilon^{2}}+\frac{-9+4\rho}{8\epsilon}&0\\\
\hline\cr
f&\frac{27}{8\epsilon^{2}}+\frac{233-108\rho}{16\epsilon}&\frac{3}{4\epsilon}&\frac{27}{8\epsilon^{2}}+\frac{245-108\rho}{16\epsilon}&\frac{3}{4\epsilon}\\\
\hline\cr
g&\frac{1}{6\epsilon^{2}}+\frac{13-4\rho}{12\epsilon}&0&\frac{1}{6\epsilon^{2}}+\frac{3-\rho}{3\epsilon}&0\\\
\hline\cr
h+i&\frac{25}{6\epsilon^{2}}+\frac{215-100\rho}{12\epsilon}&-\frac{3}{2\epsilon}&\frac{25}{6\epsilon^{2}}+\frac{110-50\rho}{6\epsilon}&-\frac{3}{2\epsilon}\\\
\hline\cr j&\frac{1}{8\epsilon}&0&0&0\\\ \hline\cr
k&-\frac{9}{8\epsilon}&0&0&0\\\ \hline\cr
l&-\frac{6}{\epsilon^{2}}+\frac{12(-2+\rho)}{\epsilon}&0&-\frac{6}{\epsilon^{2}}+\frac{12(-2+\rho)}{\epsilon}&0\\\
\hline\cr\text{Sum}&\frac{7}{3\epsilon}&0&\frac{7}{3\epsilon}&0\\\
\hline\cr\end{array}$ Table 11: Results for pure Yang-Mills using CDR and
$\overline{DRED}$, where $\rho=\gamma_{E}-\ln 4\pi+\ln(p^{2}/\mu^{2}_{DR})$

Some comments are in order. First notice in the IREG results that, apart from
minus signs and/or global factors encoded in the parameter $b$, diagrams (a)
to (i) have the same coefficients for $I_{log}^{(2)}(\lambda^{2})$,
$I_{log}^{2}(\lambda^{2})$, and
$\rho_{IREG}=I_{log}(\lambda^{2})\ln(-p^{2}/\lambda^{2})$. This fact can be
schematically understood as described below. Consider that the amplitude of
one of those diagrams is given by

$\displaystyle\mathcal{A}=\int_{k,l}\mathcal{F}(l,k,p).$ (3.12)

Guided by the procedure presented in ADRIANO , it is possible to separate the
$F(l,k,p)$ function in different pieces, organizing the order of the
integration in $l,k$ for each of them. Suppose that one of these terms is
given as below, where one must first perform the integration in $l$, then in
$k$

$\displaystyle\mathcal{A}\supset$
$\displaystyle\int_{k}G(k,p,\mu^{2})\int_{l}F(l,k,p,\mu^{2}).$ (3.13)

We have also included the $\mu^{2}$ parameter in denominators as explained in
section 2.3. At this point, one should identify the BDI presented in the $l$
integral, encoded as $I_{log}(\mu^{2})$, and apply the scale relation to trade
$\mu^{2}$ by $\lambda^{2}$

$\displaystyle\mathcal{A}\supset$
$\displaystyle\int_{k}G(k,p,\mu^{2})\left[a_{1}I_{log}(\mu^{2})+a_{2}\right],$
$\displaystyle=$
$\displaystyle\int_{k}G(k,p,\mu^{2})\left[a_{1}I_{log}(\lambda^{2})-a_{1}b\ln\left[-\frac{k^{2}}{\lambda^{2}}\right]+\bar{a}_{2}\right],$
(3.14)

where the actual form of $a_{2}$, $\bar{a}_{2}$ are not relevant here. At this
point one has to proceed to the integral in $k$. Adopting a similar treatment
as done in the $l$ integral one obtains

$\displaystyle\mathcal{A}\supset\,$ $\displaystyle
a_{1}I_{log}(\lambda^{2})\int_{k}G(k,p,\mu^{2})-a_{1}b\int_{k}G(k,p,\mu^{2})\ln\left[-\frac{k^{2}}{\lambda^{2}}\right]$
$\displaystyle=$ $\displaystyle
a_{1}I_{log}(\lambda^{2})\left[A_{1}I_{log}(\mu^{2})+\cdots\right]-a_{1}b\left[A_{1}I_{log}^{(2)}(\mu^{2})+\cdots\right]$
$\displaystyle=$ $\displaystyle
a_{1}A_{1}\left[I_{log}^{2}(\lambda^{2})-bI_{log}(\lambda^{2})\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]-bI_{log}^{(2)}(\lambda^{2})+\cdots\right]$
(3.15)

where we made use of the relation

$\displaystyle\int_{k}G(k,p,\mu^{2})\ln^{n}\left[-\frac{k^{2}}{\lambda^{2}}\right]=A_{1}I_{log}^{(n+1)}(\mu^{2})+\cdots$
(3.16)

whose proof we perform in Appendix C.

The important lesson to be taken from eq. 3.15 is that the coefficients of
$I_{log}^{(2)}(\lambda^{2})$, $I_{log}^{2}(\lambda^{2})$, and $\rho_{IREG}$
are correlated. Explicitly, there is a minus sign and $b$ factor difference
among $I_{log}^{(2)}(\lambda^{2})$, $\rho_{IREG}$, and
$I_{log}^{2}(\lambda^{2})$, which reproduces the pattern we found for diagrams
A to I, see table 10. Notice that there is no such correlation for the
$I_{log}(\lambda^{2})$ terms.

The above reasoning cannot be applied when dealing with diagrams of topology
T5, since, in this case, the integrals in $l,k$ are independent.
Schematically,

$\displaystyle\mathcal{B}=$ $\displaystyle\int_{k}G(k,p)\int_{l}F(l,p)$
$\displaystyle=$
$\displaystyle\left[B_{1}I_{log}(\mu^{2})+\cdots\right]\left[C_{1}I_{log}(\mu^{2})+\cdots\right]$
$\displaystyle=$
$\displaystyle\left[B_{1}I_{log}(\lambda^{2})-B_{1}b\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]+\cdots\right]\left[C_{1}I_{log}(\lambda^{2})-C_{1}b\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]+\cdots\right]$
$\displaystyle=$ $\displaystyle
B_{1}C_{1}\left[I_{log}^{2}(\lambda^{2})-2b\rho_{IREG}\right]+\cdots$ (3.17)

a pattern which can once again be read from the diagram L of table 10.
Therefore, it is clear that the appearance of topology T5 will (potentially)
break the pattern found before among $I_{log}^{(2)}(\lambda^{2})$,
$I_{log}^{2}(\lambda^{2})$, and $\rho_{IREG}$ in the end result. Actually,
this can be seen also in table 10, where the sum of the results is void of
$\rho_{IREG}$, while both $I_{log}^{(2)}(\lambda^{2})$,
$I_{log}^{2}(\lambda^{2})$ are still present. This fact also explains why in
the QED case only terms proportional to $I_{log}(\lambda^{2})$ survive, since
it is not possible to realize topology T5 there.

Regarding dimensional methods, this distinction is not present. In other
words, there is a correlation among $\epsilon^{-2}$ and the $\rho$ coefficient
for all topologies. Therefore, since the end result is local (void of $\rho$
terms as in the IREG case), no term proportional to $\epsilon^{-2}$ can
survive, as can be seen in table 11. It can also be noticed that there are
some differences among CDR and $\overline{\text{DRED}}$ in diagrams (c), (f),
(h), although their sum vanishes. The reason can be traced back to
contractions of the form $g_{ab}g^{ab}=d$, which may generate terms of order
$\epsilon^{-1}$ when present in two-loop diagrams. This feature was already
observed in spinorial QED. As explained in section 2.3, a consistent treatment
of DRED requires the introduction of contributions with $\epsilon$-scalars,
which, in view of the above results, must conspire to cancel among themselves.
We will show this explicitly in the next subsection.

Finally, we notice that our end result is already gauge invariant222Actually,
the terms proportional to $\rho_{IREG}$; $\rho$ (which are correlated to
$I_{log}^{2}(\lambda^{2})$,$I_{log}^{(2)}(\lambda^{2})$; $\epsilon^{-2}$) are
gauge invariant diagram by diagram, in accordance to the findings of Abbott .
and local (non-local terms are encoded in $\rho$ or $\rho_{IREG}$, which
vanish as already pointed out), although we didn’t include any counterterms
yet. This implies that any gauge breaking or non-local terms appearing in the
counterterms must cancel among themselves. Also, with a similar reasoning
regarding eq. 3.17, it is not hard to convince oneself that, apart from
$\rho_{IREG}$, only terms with $I_{log}^{2}(\lambda^{2})$ or
$I_{log}(\lambda^{2})$ can appear in the counterterms and the end result must
be independent of $I_{log}^{2}(\lambda^{2})$. This is indeed the case as shown
in table 12 for IREG and table 13 for dimensional methods.

$\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup\text{Counterterm}}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{A}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{B}\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{2-9}\cr&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})\\\
\hline\cr AA\hat{A}\text{
Coupling}&0&-\frac{25}{9b}&\frac{25}{9b}&-\frac{140}{27b}&0&-\frac{25}{9b}&\frac{25}{9b}&-\frac{140}{27b}\\\
\hline\cr\text{Gluon self-
energy}&0&\frac{25}{9b}&-\frac{25}{9b}&\frac{230}{27b}&0&\frac{25}{9b}&-\frac{25}{9b}&\frac{230}{27b}\\\
\hline\cr\text{Sum}&0&0&0&\frac{10}{3b}&0&0&0&\frac{10}{3b}\\\
\hline\cr\end{array}$ Table 12: Counterterms results for pure Yang-Mills using
IREG where
$\rho_{IREG}=I_{log}(\lambda^{2})\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]$
$\begin{array}[]{|c|c|c|c|c|}\hline\cr\text{Counterterm}&\text{A}_{\text{CDR}}&\text{A}_{\text{DRED}}-\text{A}_{\text{CDR}}&\text{B}_{\text{CDR}}&\text{B}_{\text{DRED}}-\text{B}_{\text{CDR}}\\\
\hline\cr AA\hat{A}\text{
Coupling}&-\frac{25}{9\epsilon^{2}}+\frac{5(-28+15\rho)}{27\epsilon}&0&-\frac{25}{9\epsilon^{2}}+\frac{5(-28+15\rho)}{27\epsilon}&0\\\
\hline\cr\text{Gluon self-
energy}&\frac{25}{9\epsilon^{2}}+\frac{5(46-15\rho)}{27\epsilon}&0&\frac{25}{9\epsilon^{2}}+\frac{5(46-15\rho)}{27\epsilon}&0\\\
\hline\cr\text{Sum}&\frac{10}{3\epsilon}&0&\frac{10}{3\epsilon}&0\\\
\hline\cr\end{array}$ Table 13: Counterterms results for pure Yang-Mills using
CDR and $\overline{DRED}$, where $\rho=\gamma_{E}-\ln
4\pi+\ln(p^{2}/\mu^{2}_{DR})$

###  3.4 $\epsilon$-scalars for the YM theory

In the previous subsection we showed that some of the diagrams have different
divergent parts when comparing a naive DRED scheme (without
$\epsilon$-scalars) and CDR. However, the sum of the contributions is the
same, regardless of the scheme chosen as expected (the two-loop coefficient of
the beta function is the same in mass-independent renormalisation schemes).
This observation implicitly shows that the contributions of $\epsilon$-scalars
must conspire to render the same divergent part in the end in both schemes. In
this subsection we explicitly show that this is the case.

As is well-known, $\epsilon$-scalars must be included in DRED for consistency
EPSILONSCALAR . They occur as a split of the gauge field as

$A_{\mu}|_{4}=A_{\mu}|_{4-2\epsilon}+A_{\mu}|_{2\epsilon}$ (3.18)

where the last term is the $\epsilon$-scalar. Therefore, they will certainly
occur in diagrams that contain only gluons, namely (c), (f), (h), (k), (l)
from Fig. 3. The diagrams containing $\epsilon$-scalars are depicted in Fig.
4. The contributions related to diagram (l) vanish while the one related to
diagram (k) will be of order $\rm{O}(\epsilon^{0})$, not relevant for our
purposes. Therefore, one only needs the contribution from the other three
types of diagrams.

c1

(a) *

c2

(b) *

f1

(c) *

f2

(d) *

h1

(e) *

h2

(f) *

k

(g) *

l

(h) *

Figure 4: Two-loop corrections with $\epsilon$-scalar.

Adopting the same notation of eq. 3.11, the results are collected in table 14.
They are all gauge invariant, and cancel as they should. More interestingly,
adding the results of the $\epsilon$-scalar contributions to the
$\overline{\text{DRED}}$ correspondent diagrams, one recovers the results from
CDR diagram by diagram.

$\begin{array}[]{|c|c|c|c|c|}\hline\cr\text{Diagram}&\text{A}_{\epsilon_{\text{scalar}}}&\text{B}_{\epsilon_{\text{scalar}}}\\\
\hline\cr c_{1}+c_{2}&-\frac{3}{4\epsilon}&-\frac{3}{4\epsilon}\\\ \hline\cr
f_{1}+f_{2}&-\frac{3}{4\epsilon}&-\frac{3}{4\epsilon}\\\ \hline\cr
h_{1}+h_{2}&\frac{3}{2\epsilon}&\frac{3}{2\epsilon}\\\
\hline\cr\text{Sum}&0&0\\\ \hline\cr\end{array}$ Table 14: Results for pure
Yang-mills regarding $\epsilon$-scalar contributions

###  3.5 QCD

As our last example, we consider QCD. Since it is just a SU(3) Yang-Mills
theory appended with $n_{f}$ flavors of fermions, we can reassess the results
we obtained for the general Yang-Mills, specialize to SU(3) and include the
corrections due to fermions, which are depicted in fig. 5

(a) (b) (c) (d)

Figure 5: Two-loop correction to the two-point function of the background
field - fermionic contribution of QCD

Adopting a similar convention of eq. 3.9,

$\frac{ig_{s}^{4}n_{f}}{(4\pi)^{4}}\left[Ag_{\mu\nu}p^{2}-Bp_{\mu}p_{\nu}\right],$
(3.19)

where $n_{f}$ is the number of fermions, we can express our results in tables
15 and 16. The same patterns presented in the pure Yang-Mills theory appear
here also, namely, the correlations among the coefficients of $\rho_{IREG}$;
$\rho$ and $I_{log}^{2}(\lambda^{2})$,$I_{log}^{(2)}(\lambda^{2})$;
$\epsilon^{-2}$. Also, as there is no realisation of topology T5 in this case,
only terms proportional to $I_{log}(\lambda^{2})$ survive in the end result,
as we already noticed in the QED case.

$\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup\text{Diagram}}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{A}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{B}\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{2-9}\cr&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})\\\
\hline\cr
a&-\frac{2}{9b}&\frac{2}{9b^{2}}&-\frac{2}{9b}&\frac{35}{27b}&-\frac{2}{9b}&\frac{2}{9b^{2}}&-\frac{2}{9b}&\frac{32}{27b}\\\
\hline\cr
b&-\frac{8}{b}&\frac{8}{b^{2}}&-\frac{8}{b}&\frac{44}{3b}&-\frac{8}{b}&\frac{8}{b^{2}}&-\frac{8}{b}&\frac{50}{3b}\\\
\hline\cr
c&-\frac{16}{9b}&\frac{16}{9b^{2}}&-\frac{16}{9b}&\frac{208}{27b}&-\frac{16}{9b}&\frac{16}{9b^{2}}&-\frac{16}{9b}&\frac{184}{27b}\\\
\hline\cr
d&\frac{10}{b}&-\frac{10}{b^{2}}&\frac{10}{b}&-\frac{97}{3b}&\frac{10}{b}&-\frac{10}{b^{2}}&\frac{10}{b}&-\frac{100}{3b}\\\
\hline\cr\text{Sum}&0&0&0&\frac{70}{27b}&0&0&0&-\frac{70}{27b}\\\
\hline\cr\end{array}$ Table 15: Results for fermionic part of QCD using IREG
where
$\rho_{IREG}=I_{log}(\lambda^{2})\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]$
$\begin{array}[]{|c|c|c|c|c|}\hline\cr\text{Diagram}&\text{A}_{\text{CDR}}&\text{A}_{\text{DRED}}-\text{A}_{\text{CDR}}&\text{B}_{\text{CDR}}&\text{B}_{\text{DRED}}-\text{B}_{\text{CDR}}\\\
\hline\cr
a&\frac{1}{9\epsilon^{2}}+\frac{11-4\rho}{18\epsilon}&\frac{1}{9\epsilon}&\frac{1}{9\epsilon^{2}}+\frac{5-2\rho}{9\epsilon}&\frac{1}{9\epsilon}\\\
\hline\cr
b&\frac{4}{\epsilon^{2}}+\frac{13-8\rho}{\epsilon}&-\frac{2}{\epsilon}&\frac{4}{\epsilon^{2}}+\frac{2(7-4\rho)}{\epsilon}&-\frac{2}{\epsilon}\\\
\hline\cr
c&\frac{8}{9\epsilon^{2}}+\frac{16(2-\rho)}{9\epsilon}&\frac{8}{9\epsilon}&\frac{8}{9\epsilon^{2}}+\frac{4(7-4\rho)}{9\epsilon}&\frac{8}{9\epsilon}\\\
\hline\cr
d&-\frac{5}{\epsilon^{2}}-\frac{39-20\rho}{\epsilon}&\frac{1}{\epsilon}&-\frac{5}{\epsilon^{2}}-\frac{10(2-\rho)}{\epsilon}&\frac{1}{\epsilon}\\\
\hline\cr\text{Sum}&-\frac{7}{3\epsilon}&0&-\frac{7}{3\epsilon}&0\\\
\hline\cr\end{array}$ Table 16: Results for fermionic part of QCD using CDR
and $\overline{DRED}$

One may notice that, in all diagrams, there is a mismatch between CDR and
$\overline{DRED}$, although the sum is the same in both methods, as in all
other examples we considered here. Finally, as in the case of YM, the result
is gauge invariant and local, although we still need to add counterterms. The
results of the counterterms follows a similar pattern of the one seen in the
Yang-Mills theory, and they can be read from tables 17, and 18.

$\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\vrule\lx@intercol\hfil\hbox{\multirowsetup\text{Counterterm}}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{A}\hfil\lx@intercol&\vrule\lx@intercol\hfil\text{B}\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{2-9}\cr&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})&I_{log}^{(2)}(\lambda^{2})&I_{log}^{2}(\lambda^{2})&\rho_{IREG}&I_{log}(\lambda^{2})\\\
\hline\cr AA\hat{A}\text{
Coupling}&0&\frac{10}{3b^{2}}&-\frac{10}{3b}&\frac{56}{9b}&0&\frac{10}{3b^{2}}&-\frac{10}{3b}&\frac{56}{9b}\\\
\hline\cr\text{Gluon self-
energy}&0&-\frac{10}{3b^{2}}&\frac{10}{3b}&-\frac{92}{9b}&0&-\frac{10}{3b^{2}}&\frac{10}{3b}&-\frac{92}{9b}\\\
\hline\cr\text{Sum}&0&0&0&-\frac{4}{b}&0&0&0&-\frac{4}{b}\\\
\hline\cr\end{array}$ Table 17: Results for fermionic part of QCD using IREG
where
$\rho_{IREG}=I_{log}(\lambda^{2})\ln\left[-\frac{p^{2}}{\lambda^{2}}\right]$
$\begin{array}[]{|c|c|c|c|c|}\hline\cr\text{Counterterm}&\text{A}_{\text{CDR}}&\text{A}_{\text{DRED}}-\text{A}_{\text{CDR}}&\text{B}_{\text{CDR}}&\text{B}_{\text{DRED}}-\text{B}_{\text{CDR}}\\\
\hline\cr AA\hat{A}\text{
Coupling}&\frac{10}{3\epsilon^{2}}+\frac{2(28-15)\rho}{9\epsilon}&0&\frac{10}{3\epsilon^{2}}+\frac{2(28-15\rho)}{9\epsilon}&0\\\
\hline\cr\text{Gluon self-
energy}&-\frac{10}{3\epsilon^{2}}-\frac{2(46-15\rho)}{9\epsilon}&0&-\frac{10}{3\epsilon^{2}}-\frac{2(46-15\rho)}{9\epsilon}&0\\\
\hline\cr\text{Sum}&-\frac{4}{\epsilon}&0&-\frac{4}{\epsilon}&0\\\
\hline\cr\end{array}$ Table 18: Results for fermionic part of QCD using CDR
and $\overline{\text{DRED}}$

###  3.6 Summary of the results

In this subsection we collect the results we found, aiming to compute $Z_{A}$,
the renormalisation function of the external gauge boson (the photon for QED,
the background gluon field for QCD). Defining

$\displaystyle
Z_{A}=1+\frac{g^{2}}{(4\pi)^{2}}Z_{A}^{(1)}+\frac{g^{4}}{(4\pi)^{4}}Z_{A}^{(2)},$
(3.20)

one obtains for QED

$\displaystyle Z_{A}^{(1)}|_{\text{IREG}}$
$\displaystyle=-\frac{4}{3b}I_{log}(\lambda^{2}),\quad
Z_{A}^{(1)}|_{\text{CDR}}=-\frac{4}{3\epsilon},$ (3.21) $\displaystyle
Z_{A}^{(2)}|_{\text{IREG}}$
$\displaystyle=-\frac{4}{b}I_{log}(\lambda^{2}),\quad\quad
Z_{A}^{(2)}|_{\text{CDR}}=-\frac{2}{\epsilon},$ (3.22)

and for QCD

$\displaystyle Z_{A}^{(1)}|_{\text{IREG}}$
$\displaystyle=\left(\frac{11}{b}-\frac{2}{3b}n_{f}\right)I_{log}(\lambda^{2}),\quad
Z_{A}^{(1)}|_{\text{CDR}}=\frac{11}{\epsilon}-\frac{2}{3\epsilon}n_{f},$
(3.24) $\displaystyle Z_{A}^{(2)}|_{\text{IREG}}$
$\displaystyle=\frac{54}{b^{2}}\left[I_{log}^{2}(\lambda^{2})-2bI_{log}^{(2)}(\lambda^{2})\right]+\left(\frac{210}{b}-\frac{38}{3b}n_{f}\right)I_{log}(\lambda^{2}),$
(3.25) $\displaystyle Z_{A}^{(2)}|_{\text{CDR}}$
$\displaystyle=\frac{51}{\epsilon}-\frac{19}{3\epsilon}n_{f}.$ (3.26)

For completeness we have also computed and included the one-loop corrections,
and we are already specializing to SU(3) when writing the QCD results. Notice
that in the one-loop contribution, $I_{log}(\lambda^{2})/b$ and
$\epsilon^{-1}$ share the same coefficient, as discussed in subsection 2.2.

###  3.7 The $\beta$ function

The $\beta$ function is obtained by adopting standard procedures exemplified
in textbooks. However, in order to make the connection between the different
regularisation methods clearer, we provide some details of the calculation. As
usual, the $\beta$ function is defined by

$\beta=\lambda\frac{\partial}{\partial\lambda}g_{R},$ (3.27)

where $g_{R}$ is the (renormalized) gauge coupling of the theory considered,
and $\lambda$ is the renormalisation scale (in dimensional methods, this is
identified as $\mu_{DR}$). Until this point, no distinction between
regularisations in fixed dimension and dimensional methods was done. To
proceed further, we will adopt the framework of dimensional regularisation
techniques, in which the (renormalized) coupling is replaced by

$g_{R}=\mu^{\frac{4-d}{2}}\tilde{g}_{R}\quad\Rightarrow\quad\beta=\mu\frac{\partial}{\partial\mu}g_{R}=\frac{4-d}{2}g_{R}+\mu^{\frac{6-d}{2}}\frac{\partial}{\partial\mu}\tilde{g}_{R},$
(3.28)

where $\tilde{g}_{R}$ is an adimensional (renormalized) coupling. Notice that
the equation above reduces to eq. (3.27) when using methods in fixed
dimension. One may also introduce $Z_{g}$ which is the renormalisation
constant related to the coupling $g$ satisfying $g_{0}=Z_{g}g_{R}$, where
$g_{0}$ is the bare coupling. Therefore, the $\beta$-function can also be
given by the related equation in general

$\beta=-g_{R}\lambda\frac{\partial}{\partial\lambda}\ln Z_{g}.$ (3.29)

In the background field method, the relation $Z_{g}=Z_{A}^{-1/2}$ is valid,
which reduces the calculation of the $\beta$ function in a non-abelian theory
to the knowledge of only two-point functions. To proceed, we assume that
$Z_{A}$ can be expanded in the (renormalized) adimensional coupling constant
$\tilde{g}_{R}$

$Z_{A}=1+A_{1}\tilde{g}_{R}^{2}+A_{2}\tilde{g}_{R}^{4},$ (3.30)

where $A_{i}$ is related to the counterterm of the $i$-order that renormalize
the $i$-loop correction to the two-point function in the background field $A$.
This amounts to

$\beta=\frac{g_{R}}{2}\lambda\frac{\partial}{\partial\lambda}\left[A_{1}\tilde{g}_{R}^{2}+\left(A_{2}-\frac{A_{1}^{2}}{2}\right)\tilde{g}_{R}^{4}\right].$
(3.31)

Notice that the above formula is valid for methods in fixed dimension as well,
since for those $\tilde{g}_{R}=g_{R}$. To proceed further, one has to choose a
subtraction scheme, which will be the MS-subtraction scheme (for methods in
fixed dimension we will discuss this point later). Therefore, all $A_{i}$ will
be independent of $\mu$ which implies

$\displaystyle\beta$
$\displaystyle=\frac{g_{R}}{2}\left[A_{1}\mu\frac{\partial}{\partial\mu}(\tilde{g}_{R}^{2})+\left(A_{2}-\frac{A_{1}^{2}}{2}\right)\mu\frac{\partial}{\partial\mu}(\tilde{g}_{R}^{4})\right],$
$\displaystyle=\frac{\tilde{g}_{R}}{2}\left[2\;A_{1}\tilde{g}_{R}\left(\beta-\frac{4-d}{2}g_{R}\right)+4\tilde{g}_{R}^{3}\left(A_{2}-\frac{A_{1}^{2}}{2}\right)\left(\beta-\frac{4-d}{2}g_{R}\right)\right],$
$\displaystyle=\frac{d-4}{2}g_{R}\left[A_{1}\tilde{g}_{R}^{2}+2A_{2}\tilde{g}_{R}^{4}\right].$
(3.32)

As standard, one can also define the expansion of the $\beta$-function in the
adimensional coupling constant as

$\beta=-g_{R}\left[\beta_{0}\left(\frac{\tilde{g}_{R}}{4\pi}\right)^{2}+\beta_{1}\left(\frac{\tilde{g}_{R}}{4\pi}\right)^{4}\right];$
(3.33)

which, after careful comparison between eqs. 3.33, 3.20 and 3.30, allows the
identification

$\beta_{0}=\epsilon Z_{A}^{(1)},\quad\beta_{1}=2\epsilon Z_{A}^{(2)},$ (3.34)

since $d=4-2\epsilon$. Notice that, since the $\beta$ function is finite, even
the two-loop coefficient of $Z_{A}$ must only have terms up to
$\epsilon^{-1}$. This can be explicitly confirmed by looking at eqs. 3.21 to
3.26.

Regarding methods in fixed dimension, there are some differences among them in
the definition of the counterterms. In FDR FDR  as well as DIFR FREEDMAN , the
divergent expressions are replaced by finite ones, meaning that divergences
are automatic removed by applying the method. In IREG, divergences are kept,
being identified as basic divergent integrals such as
$I_{log}^{(2)}(\lambda^{2})$, and $I_{log}(\lambda^{2})$. As can be
immediately seen, in IREG the divergent part will depend on the
renormalisation scale $\lambda$, while in dimensional methods this does not
occur. By defining the MS-subtraction scheme in IREG as the removal of only
basic divergent integrals, the IREG version of eq.(3.32) will be

$\displaystyle\beta$
$\displaystyle=\frac{g_{R}}{2}\left[g_{R}^{2}\lambda\frac{\partial}{\partial\lambda}(A_{1})+A_{1}\lambda\frac{\partial}{\partial\lambda}(g_{R}^{2})+g_{R}^{4}\lambda\frac{\partial}{\partial\lambda}\left(A_{2}-\frac{A_{1}^{2}}{2}\right)+\left(A_{2}-\frac{A_{1}^{2}}{2}\right)\lambda\frac{\partial}{\partial\lambda}(g_{R}^{4})\right],$
$\displaystyle=\frac{g_{R}}{2}\left[g_{R}^{2}\lambda\frac{\partial}{\partial\lambda}(A_{1})+2\;A_{1}g_{R}\beta+g_{R}^{4}\lambda\frac{\partial}{\partial\lambda}\left(A_{2}-\frac{A_{1}^{2}}{2}\right)+4g_{R}^{3}\left(A_{2}-\frac{A_{1}^{2}}{2}\right)\beta\right],$
$\displaystyle=-g_{R}\left[-\frac{g_{R}^{2}}{2}\lambda\frac{\partial}{\partial\lambda}A_{1}-\frac{g_{R}^{4}}{2}\lambda\frac{\partial}{\partial\lambda}A_{2}\right],$
(3.35)

which, by comparing eqs. 3.33, 3.20 and 3.30, implies

$\beta_{0}=-\frac{1}{2}\lambda\frac{\partial}{\partial\lambda}Z_{A}^{(1)},\quad\beta_{1}=-\frac{1}{2}\lambda\frac{\partial}{\partial\lambda}Z_{A}^{(2)}$
(3.36)

As already pointed out, the $Z_{A}^{(i)}$ in IREG will depend on basic
divergent integrals, which implies that the derivatives of those with respect
to $\lambda$ will be needed. They can be obtained in a straightforward way as
shown in eqs. 2.17. For our purposes here, we only need

$\lambda\frac{\partial}{\partial\lambda}I_{log}(\lambda^{2})=-2b;\quad\lambda\frac{\partial}{\partial\lambda}I_{log}^{(2)}(\lambda^{2})=-2b-2I_{log}(\lambda^{2});\quad$
(3.37)

Notice that the second of the equations above is still divergent. However, the
combination $I_{log}^{2}(\lambda^{2})-2bI_{log}^{(2)}(\lambda^{2})$, which
appears in $Z_{A}^{(2)}|_{\text{IREG}}$, will give a finite result as it
should. Finally, by applying our results collected in eqs. 3.21 to 3.26, we
obtain the well-known one and two-loop contributions for the gauge $\beta$
coupling in QED

$\displaystyle\beta_{0}|_{\text{IREG}}=-\frac{4}{3};\quad$
$\displaystyle\beta_{0}|_{\text{CDR}}=-\frac{4}{3};$ (3.38)
$\displaystyle\beta_{1}|_{\text{IREG}}=-4;\quad$
$\displaystyle\beta_{1}|_{\text{CDR}}=-4;$ (3.39)

and QCD

$\displaystyle\beta_{0}|_{\text{IREG}}=11-\frac{2}{3}n_{f};\quad$
$\displaystyle\beta_{0}|_{\text{CDR}}=11-\frac{2}{3}n_{f};$ (3.40)
$\displaystyle\beta_{1}|_{\text{IREG}}=102-\frac{38}{3}n_{f};\quad$
$\displaystyle\beta_{1}|_{\text{CDR}}=102-\frac{38}{3}n_{f}$ (3.41)

##  4 Concluding remarks

To extract any deviation between theory and experimental data in the SM as
well as test BSM theories, precision observables demand at least $N^{2}LO$ and
$N^{3}LO$ approximations involving multi-loop Feynman diagrams. Clearly the
choice of the regularisation scheme to separate UV and IR divergencies of
multi-loop amplitudes that enter into a computer code is guided by consistency
and expediency. For the reasons we have discussed in the introduction,
practical and symmetry-preserving regularisation frameworks that work fully in
the physical dimension are desirable especially when dealing with dimensional-
specific models in which the analytical continuation in the space-time
dimension is ambiguous. This has justified to exploit quasi-dimensional
methods such as DRED and FDH. They have been successfully employed in
calculations in gauge and supersymmetric models after having their consistency
validated, order by order in perturbation theory, through verification of Ward
identities via quantum action principles. The main drawback of such schemes is
that some modifications at Lagrangian level become necessary. For instance
higher covariant derivative terms improve the ultraviolet behaviour of the
propagators at the expense of complicating the Feynman rules. In the case of
DRED or FDH, evanescent scalar $\epsilon$-particles add a $\cal{L}_{\epsilon}$
term to QCD Lagrangian as a result of decomposing the quasi-4-dimensional
gluon field. Moreover, two new coupling constants besides $g_{s}$ emerge as a
result to the coupling of $\epsilon$-scalars to (anti-)quarks, namely
$g_{\epsilon}$, and a quartic $\epsilon$-scalar coupling $g_{4\epsilon}$, with
their respective $\beta$-functions and anomalous dimensions. Whilst such
modifications are crucial for inner consistency of the method as well as
shedding light on the ultraviolet and infrared factorisation structure of the
amplitudes, they are unnecessary in fully non-dimensional methods such as
IREG.

In other to raise a non-dimensional scheme such as IREG to the level of more
conventional methods a series calculations had to be performed. Firstly, show
that a program that displays the UV (and IR) content of an amplitude as a BDI,
without recoursing to explicit evaluation, can be consistently and invariantly
extended beyond one loop respecting gauge invariance. We have explicitly
verified that this is the case by fully evaluating the contributions to the
$\beta$-function of abelian and non-abelian models. We have verified the
conjecture (proved for the abelian case) that a constrained version of IREG
that sets to zero well defined surface terms in consonance with momentum
routing invariance in the loops of Feynman diagrams automatically implements
gauge invariance. We have obtained the renormalization constants by conducting
the subtraction of subdivergences within IREG and compared with CDR and DRED.
It is well-known that CDR and DRED are not equivalent in general in the sense
that the residues of the poles in $\epsilon$ do not coincide. In this respect
it is noteworthy, as we have explicitly verified, that evaluating the BDI’s of
IREG in $4-2\epsilon$ dimensions in the end of the calculation does not yield
the same residues for the poles of arbitrary orders. Nonetheless, as we have
shown in tables 2 to 18, a systematic summation among different contributions
from Feynman graphs and counterterms renders an identical result for CDR, DRED
and $\overline{\text{DRED}}$. Matter-of-factly a tuned cancellation of
$\epsilon$-scalar contributions take place. Because IREG does not recourse to
such modifications, it would be interesting to perform a calculation where
such cancellations do not occur in $\overline{\text{DRED}}$ such as in the
$g+g\rightarrow q+\bar{q}+g$ BEENAKKER  or $H\rightarrow g+g$ BROGGIO
scatterings to $NLO$ and $N^{2}LO$. Finally, we have computed the universal
two-loop $\beta$-functions of scalar and spinorial QED as well as pure Yang-
Mills and QCD in a fully quadridimensional framework by defining the
renormalization constants as BDI’s. Derivatives of BDI’s with respect to a
renormalization scale that naturally appears through a scale relation are also
expressable as BDI’s. This enable us to perform the calculations without
explicitly evaluating the BDI’s.

In order to pursuit the IREG program to apply it to precision calculations, it
is important to show that IREG respects the factorisation properties of
infrared divergences in QCD as well as to evaluate the cusp anomalous
dimensions. This can be achieved in two ways: either by parametrising the
infrared divergences in IREG as $\ln\mu^{2}$ as $\mu\rightarrow 0$ or by using
a parametrisation of infrared divergences in the reciprocal space in terms of
infrared BDI in the coordinate space. Both approaches are under active
investigation.

##  Appendix A Feynman rules

For scalar and spinorial $QED$ we refer to Schwartz  for the Feynman rules and
conventions. As for $QCD$ with a background field $A$ and gauge fixing
parameter $\alpha$, we follow the conventions of Abbott  to yield the
following rules in figure 6.

Figure 6: Feynman rules

##  Appendix B Explicit results of integrals

Here we present explicit results for the integrals used in this work:

###  B.1 1-loop: two point functions

$\displaystyle\int_{k}\frac{1}{k^{2}(k+p)^{2}}$
$\displaystyle=I_{log}(\lambda^{2})-b\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)+2b,$
(B.1) $\displaystyle\int_{k}\frac{k_{\mu}}{k^{2}(k+p)^{2}}$
$\displaystyle=-\frac{p_{\mu}}{2}\left[I_{log}(\lambda^{2})-b\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)+2b\right],$
(B.2) $\displaystyle\int_{k}\frac{k_{\mu}k_{\nu}}{k^{2}(k+p)^{2}}$
$\displaystyle=-\frac{g_{\mu\nu}p^{2}}{12}\left[I_{log}(\lambda^{2})-b\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)+\frac{8b}{3}\right]$
$\displaystyle\quad\quad+\frac{p_{\mu}p_{\nu}}{3}\left[I_{log}(\lambda^{2})-b\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)+\frac{13b}{6}\right]$
(B.3)

###  B.2 1-loop: three point functions

$\displaystyle\int_{k}\frac{k_{\mu}k_{\nu}}{k^{4}(k+p)^{2}}$
$\displaystyle=\frac{g_{\mu\nu}}{4}\left[I_{log}(\lambda^{2})-b\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)+2b\right]+\frac{p_{\mu}p_{\nu}}{p^{2}}\frac{b}{2}$
(B.4)
$\displaystyle\int_{k}\frac{k_{\mu}k_{\nu}k_{\rho}}{k^{2}(k+p_{1})^{2}(k+p_{2})^{2}}$
$\displaystyle=\left[g_{\mu\nu}(p_{1}+p_{2})_{\rho}+g_{\nu\rho}(p_{1}+p_{2})_{\mu}+g_{\mu\rho}(p_{1}+p_{2})_{\nu}\right]\left[-\frac{I_{log}(\lambda^{2})}{12}+\mbox{finite}\right]$
(B.5)

###  B.3 2-loop: one point functions

$\displaystyle\int_{k}\frac{1}{(k+p)^{2}}\ln\left(-\frac{k^{2}}{\lambda^{2}}\right)$
$\displaystyle=\frac{p^{2}}{2}I_{log}(\lambda^{2})+\mbox{finite},$ (B.6)

###  B.4 2-loop: two point functions

$\displaystyle\int_{k}\frac{1}{k^{2}(k+p)^{2}}\ln\left(-\frac{k^{2}}{\lambda^{2}}\right)$
$\displaystyle=I_{log}^{(2)}(\lambda^{2})+\mbox{finite},$ (B.7)
$\displaystyle\int_{k}\frac{k_{\mu}}{k^{2}(k+p)^{2}}\ln\left(-\frac{k^{2}}{\lambda^{2}}\right)$
$\displaystyle=-\frac{p_{\mu}}{2}\left[I_{log}^{(2)}(\lambda^{2})+\frac{I_{log}(\lambda^{2})}{2}+\mbox{finite}\right],$
(B.8)
$\displaystyle\int_{k}\frac{k_{\mu}k_{\nu}}{k^{2}(k+p)^{2}}\ln\left(-\frac{k^{2}}{\lambda^{2}}\right)$
$\displaystyle=\frac{g_{\mu\nu}}{12}\left[-I_{log}^{(2)}(\lambda^{2})+\frac{I_{log}(\lambda^{2})}{6}+\mbox{finite}\right]$
$\displaystyle\quad\quad+\frac{p_{\mu}p_{\nu}}{3}\left[I_{log}^{(2)}(\lambda^{2})+\frac{5}{6}I_{log}(\lambda^{2})+\mbox{finite}\right],$
(B.9)

###  B.5 2-loop: three point functions

$\displaystyle\int_{k}\frac{k_{\mu}k_{\nu}}{k^{4}(k+p)^{2}}\ln\left(-\frac{k^{2}}{\lambda^{2}}\right)$
$\displaystyle=\frac{g_{\mu\nu}}{4}\left[I_{log}^{(2)}(\lambda^{2})+\frac{I_{log}(\lambda^{2})}{2}+\mbox{finite}\right]+\frac{p_{\mu}p_{\nu}}{p^{2}}\mbox{finite},$
(B.10)

###  B.6 2-loop: overlapped integrals

We will have the general structure

$I[f(k,q)]=\int_{k}\frac{f(k,q)}{k^{2}(k-p)^{2}(k-q)^{2}q^{2}(q-p)^{2}}$
(B.11) $\displaystyle I[k.q]$
$\displaystyle=bI_{log}(\lambda^{2})+\mbox{finite}$ (B.12) $\displaystyle
I[k_{\mu}q_{\nu}]$
$\displaystyle=\frac{g_{\mu\nu}}{4}\left[I_{log}(\lambda^{2})+\mbox{finite}\right]+\frac{p_{\mu}p_{\nu}}{p^{2}}\mbox{finite}$
(B.13) $\displaystyle I[k_{\mu}k_{\nu}]$
$\displaystyle=\frac{g_{\mu\nu}}{4}\left\\{I_{log}^{2}(\lambda^{2})-bI_{log}^{(2)}(\lambda^{2})+bI_{log}(\lambda^{2})\left[\frac{9}{2}-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right\\}$
$\displaystyle\quad+\frac{p_{\mu}p_{\nu}}{p^{2}}\mbox{finite}$ (B.14)
$\displaystyle I[k_{\mu}q_{\nu}q.p]$
$\displaystyle=\frac{p_{\mu}p_{\nu}}{8}\left\\{I_{log}^{2}(\lambda^{2})-bI_{log}^{(2)}(\lambda^{2})+bI_{log}(\lambda^{2})\left[\frac{11}{2}-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right\\}$
$\displaystyle\quad+\frac{g_{\mu\nu}p^{2}}{8}\left[I_{log}(\lambda^{2})+\mbox{finite}\right]$
(B.15) $\displaystyle I[k_{\mu}k_{\nu}q.p]$
$\displaystyle=\frac{g_{\mu\nu}}{8}\left\\{I_{log}^{2}(\lambda^{2})-bI_{log}^{(2)}(\lambda^{2})+bI_{log}(\lambda^{2})\left[\frac{9}{2}-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right\\}$
$\displaystyle\quad+\frac{p_{\mu}p_{\nu}}{4}\left[I_{log}(\lambda^{2})+\mbox{finite}\right]$
(B.16) $\displaystyle I[k_{\mu}k.q]$
$\displaystyle=\frac{p_{\mu}}{8}\left\\{I_{log}^{2}(\lambda^{2})-bI_{log}^{(2)}(\lambda^{2})+bI_{log}(\lambda^{2})\left[\frac{19}{2}-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right\\}$
(B.17) $\displaystyle I[k_{\mu}q_{\nu}k.q]$
$\displaystyle=\frac{g_{\mu\nu}}{24}\left\\{-I_{log}^{2}(\lambda^{2})+bI_{log}^{(2)}(\lambda^{2})-bI_{log}(\lambda^{2})\left[\frac{29}{6}-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right\\}$
$\displaystyle\quad+\frac{p_{\mu}p_{\nu}}{6}\left\\{I_{log}^{2}(\lambda^{2})-bI_{log}^{(2)}(\lambda^{2})+bI_{log}(\lambda^{2})\left[\frac{85}{12}-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right\\}$
(B.18) $\displaystyle I[k^{2}]$
$\displaystyle=I_{log}^{2}(\lambda^{2})-bI_{log}^{(2)}(\lambda^{2})+bI_{log}(\lambda^{2})\left[4-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}$
(B.19) $\displaystyle I[k^{2}k_{\mu}]$
$\displaystyle=\frac{p_{\mu}}{4}\left\\{3I_{log}^{2}(\lambda^{2})-3bI_{log}^{(2)}(\lambda^{2})+bI_{log}(\lambda^{2})\left[\frac{25}{2}-3\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right\\}$
(B.20) $\displaystyle I[k^{2}q_{\mu}]$
$\displaystyle=\frac{p_{\mu}}{2}\left\\{I_{log}^{2}(\lambda^{2})-bI_{log}^{(2)}(\lambda^{2})+bI_{log}(\lambda^{2})\left[\frac{9}{2}-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right\\}$
(B.21) $\displaystyle I[k^{2}k_{\mu}q_{\nu}]$
$\displaystyle=\frac{g_{\mu\nu}}{24}\left\\{-I_{log}^{2}(\lambda^{2})+bI_{log}^{(2)}(\lambda^{2})-bI_{log}(\lambda^{2})\left[\frac{29}{6}-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right\\}$
$\displaystyle\quad+\frac{p_{\mu}p_{\nu}}{24}\left\\{10I_{log}^{2}(\lambda^{2})-10bI_{log}^{(2)}(\lambda^{2})+bI_{log}(\lambda^{2})\left[\frac{139}{3}-10\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right\\}$
(B.22) $\displaystyle I[k^{2}k.q]$
$\displaystyle=\frac{1}{4}\left[I_{log}^{2}(\lambda^{2})-bI_{log}^{(2)}(\lambda^{2})+bI_{log}(\lambda^{2})\left[\frac{9}{2}-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right]$
(B.23) $\displaystyle I[k^{2}q^{2}]$ $\displaystyle=0$ (B.24) $\displaystyle
I[k^{2}q_{\mu}q_{\nu}]$
$\displaystyle=\frac{g_{\mu\nu}}{12}\left\\{-I_{log}^{2}(\lambda^{2})+bI_{log}^{(2)}(\lambda^{2})-bI_{log}(\lambda^{2})\left[\frac{29}{6}-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right\\}$
$\displaystyle\quad+\frac{p_{\mu}p_{\nu}}{3}\left\\{I_{log}^{2}(\lambda^{2})-bI_{log}^{(2)}(\lambda^{2})+bI_{log}(\lambda^{2})\left[\frac{29}{6}-\ln\left(-\frac{p^{2}}{\lambda^{2}}\right)\right]+\mbox{finite}\right\\}$
(B.25)

##  Appendix C Explicit algorithm to express a multiloop integral as scalar
BDI’s

In this appendix we provide an explicit algorithm to rewrite BDI’s with
Lorentz indexes in terms of scalar ones. For simplicity, we focus only on log-
divergent integrals and we set surface terms to zero. As explained in section
2.1, in the course of applying IREG rules to a general massless n-loop
amplitude, one may encounters an integral of the type

$\displaystyle\mathcal{I}=\int_{k}G(k,p_{i},\mu^{2})\ln^{n-1}\left[-\frac{(k^{2}-\mu^{2})}{\lambda^{2}}\right],$
(C.1)

where $p_{i}$ stand for external momenta, and $G$ may contain free Lorentz
indexes. For simplicity, we assume that the external momenta only appear in
denominators. Considering that the integral above is log-divergent, after
applying eq. 2.2 as many times as the number of external momenta, one obtains

$\displaystyle\mathcal{I}=\mathcal{I}_{\text{div}}+\mathcal{I}_{\text{fin}},\quad\mbox{where}\quad\mathcal{I}_{\text{div}}=\int_{k}G(k,p_{i}=0,\mu^{2})\ln^{n-1}\left[-\frac{(k^{2}-\mu^{2})}{\lambda^{2}}\right].$
(C.2)

To conclude the IREG program, one has to write $\mathcal{I}_{\text{div}}$ in
terms of BDI’s. Therefore, an explicit form for G is needed which, for the
sake of generality, we adopt to be

$\displaystyle\mathcal{I}_{\text{div}}=A\int_{k}\frac{k_{\nu_{1}}\cdots
k_{\nu_{m}}}{(k^{2}-\mu^{2})^{\frac{m+2}{2}}}\ln^{n-1}\left[-\frac{(k^{2}-\mu^{2})}{\lambda^{2}}\right]=A\left[I_{log}^{(n)}(\mu^{2})\right]_{\nu_{1}\cdots\nu_{m}},$
(C.3)

where $A$ is a constant, and $m$ is even. By setting the surface term below to
zero,

$\displaystyle\int_{k}\frac{\partial}{\partial
k_{\nu_{1}}}\frac{k^{\nu_{2}}\cdots
k^{\nu_{m}}}{(k^{2}-\mu^{2})^{\frac{m+2}{2}}}\ln^{n-1}\left[-\frac{(k^{2}-\mu^{2})}{\lambda^{2}}\right]=0,$
(C.4)

one obtains the relation

$\displaystyle\left[I_{log}^{(n)}(\mu^{2})\right]_{\nu_{1}\cdots\nu_{m}}$
$\displaystyle=\frac{1}{m+2}\left\\{g_{\nu_{1}\nu_{2}}\left[I_{log}^{(n)}(\mu^{2})\right]_{\nu_{3}\cdots\nu_{m}}+\cdots
g_{\nu_{1}\nu_{m}}\left[I_{log}^{(n)}(\mu^{2})\right]_{\nu_{2}\cdots\nu_{m-1}}\right\\}$
$\displaystyle+\frac{2(l-1)}{m+2}\left[I_{log}^{(n-1)}(\mu^{2})\right]_{\nu_{1}\cdots\nu_{m}}.$
(C.5)

Notice that by using eq. C.5, one reduces a BDI of n-loop order with $m$ free
Lorentz indexes to a BDI with two less free Lorentz indexes. Also, one relates
a BDI of n-loop order to a BDI one order below. Therefore, by successive
applications of eq. C.5, one can write $\mathcal{I}_{\text{div}}$ in terms of
scalar BDI’s only. The end result is not particularly enlightening, thus we do
not present it here. However, the coefficient of the $I_{log}^{(n)}(\mu^{2})$
term is easily obtained, which gives

$\displaystyle\mathcal{I}_{\text{div}}=\frac{A}{(m+2)!!}\;g_{\left\\{\nu_{1}\nu_{2}\cdots\nu_{m-1}\nu_{m}\right\\}}\;I_{log}^{(n)}(\mu^{2})+\cdots$
(C.6)

where $g_{\left\\{\nu_{1}\nu_{2}\cdots\nu_{m-1}\nu_{m}\right\\}}$ represents
the symmetric combination of all indexes. As can be seen, one obtains the same
coefficient for the higher order BDI, regardless of the actual value of n. A
similar reasoning can be applied for integrals with higher surface degree of
divergence, allowing us to write in general

$\displaystyle\int_{k}G(k,p_{i},\mu^{2})\ln^{n-1}\left[-\frac{(k^{2}-\mu^{2})}{\lambda^{2}}\right]=\mathcal{A}\;I_{log}^{(n)}(\mu^{2})+\cdots$
(C.7)

where $\mathcal{A}$ is independent of the actual value of $n$.

###### Acknowledgements.

M.S. acknowledges a research grant from CNPq (Conselho Nacional de
Desenvolvimento Científico e Tecnológico - 303482/2017-6). We acknowledge
support from Fundação para a Ciência e Tecnologia (FCT) through the projects
UID/FIS/04564/2016 and CERN/FIS-COM/0035/2019, and the networking support by
the COST Action CA16201. This study was financed in part by the Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code
001.

## References

  * (1) M. Gorbahn, J. M. No and V. Sanz, _Benchmarks for Higgs effective theory: extended Higgs sectors_ , JHEP 10 (2015) 036.
  * (2) O. Witzel, _Review on Composite Higgs Models_ , _PoS_ (Lattice-2018) 006.
  * (3) Halina Abramowicz, Roger Forty, and the Conveners (editors), _Physics Briefing Book – Input for the European Strategy for Particle Physics Update 2020_ , arXiv: 1910.11775v2, CERN-ESU-004.
  * (4) S. Heinemeyer, W. Hollik and G. Weiglein _Electroweak Precision Observables in the Minimal Supersymmetric Standard Model_ , Phys. Rept. 425 (2006) 265.
  * (5) Adriano Cherchiglia, Patrick Kneschke, Dominik Stöckinger, and Hyejung Stöckinger-Kim _The muon magnetic moment in the 2HDM: complete two-loop result_ , JHEP 01 (2017) 007.
  * (6) Eduardo da Silva Almeida, Alexandre Alves, N. Rosa Agostinho, Oscar J. P. Éboli and M.C. Gonzalez-Garcia, _Electroweak Sector Under Scrutiny: A Combined Analysis of LHC and Electroweak Precision Data_ , Phys. Rev. D 99 (2019) 033001.
  * (7) C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog, A. Lazopoulosa and B. Mistlbergerb, _High precision determination of the gluon fusion Higgs boson cross-section at the LHC_ , JHEP 05 (2016) 058.
  * (8) https://fcc-ee.web.cern.ch/
  * (9) P. M. Stevenson, _Optimised Perturbation Theory_ , Phys. Rev. D 23 (1981) 2916.
  * (10) Jon Butterworth et al., _PDF4LHC recommendations for LHC Run II_ , J. Phys. G 43 (2016) 023001.
  * (11) S. Weinberg, _Infrared photons and gravitons_ , Phys. Rev.140 (1965) B516.
  * (12) John C. Collins, Leonid Frankfurt, Mark Strikman, _Factorisation for hard exclusive electroproduction of mesons in QCD_ , Phys. Rev. D 56 (1997) 2982; Daniel de Florian and Massimiliano Grazzini, _The Structure of large logarithmic corrections at small transverse momentum in hadronic collisions_ , Nuc. Phys. B 616 (2001) 247.
  * (13) Thomas Becher and Matthias Neubert, _Infrared singularities of scattering amplitudes in perturbative QCD_ , Phys. Rev. Lett. 102 (2009) 162001, Phys. Rev. Lett.111 (2013) 199905 (erratum); idem, _On the structure of IR singularities of gauge theory amplitudes_ , JHEP 0906 (2009) 081.
  * (14) E. Gardi, L. Magnea, _IR singularities in QCD amplitudes_ Nuovo Cimento C 32 (2009) 137.
  * (15) C. Gnendiger, A. Signer, D. Stockinger, _The IR structure of QCD and $H\rightarrow gg$ in FDH and DRED_, Phys. Lett. B 733 (2014) 296.
  * (16) J. Frenkel and J.C. Taylor, _Nonabelian Eikonal Exponentiation_ , Nucl. Phys. B 246 (1984) 231.
  * (17) C. G. Bollini, J. J. Giambiagi, _Dimensional renormalisation: the number of dimensions as a regularizing parameter_ , Nuovo Cim. B 12 1972; G. ’t Hooft and M. Veltman, _regularisation and renormalisation of Gauge Fields_ , Nuc. Phys. B 44(1972) 189.
  * (18) P. Breitenlohner and D. Maison, _Dimensionally renormalized Green’s functions for theories with massless particles_ , Comm. Math. Phys. 52 (1977) 39.
  * (19) Er-Cheng Tsai,_Maintaining Gauge Symmetry in Renormalizing Chiral Gauge Theories_ , Phys. Rev. D 83 (2011) 065011; idem, _Gauge Invariant Treatment of $\gamma_{5}$ in the Scheme of ’t Hooft and Veltman_, Phys. Rev. D 83 (2011) 025020; C. P. Martin and D. Sanchez-Ruiz,_Action principles, restoration of BRS symmetry and the renormalisation group equation for chiral non-Abelian gauge theories in dimensional renormalisation with a non-anticommuting $\gamma_{5}$_, Nuc. Phys. B 572 (2000) 387; F. Jegerjehner, _Facts of life with $\gamma_{5}$_, Eur. Phys. J. C 18 (2001) 673; C. Gnendiger and A. Signer, _$\gamma_{5}$ in the four-dimensional helicity scheme _, Phys. Rev. D 97 (2018) 096006.
  * (20) A. M. Bruque, A. L. Cherchiglia, M. Pérez-Victoria, _Dimensional regularisation vs methods in fixed dimension with and without $\gamma_{5}$_, JHEP 1808 (2018) 109.
  * (21) C. P. Martin,_Dimensional regularisation of Chern-Simons Field Theory_ , Phys. Lett. B 241 (1990) 513; M. Asorey, F. Falceto, J. L. Lopez, G. Luzon,_renormalisation ambiguities in Chern-Simons theory_ , Phys. Rev. D 49 (1994) 5377; G. Giavarini, C. P. Martin, F. Ruiz-Ruiz, _Shift versus no shift in local regularisations of Chern-Simons theory_ , Phys. Lett. B 332 (1994) 345; M. Chaichian and W. F. Chen, _Inconsistency of naive dimensional regularisations and quantum correction to non-Abelian Chern-Simons matter theory revisited_ , Phys. Rev. D 58 (1998) 125004.
  * (22) D. Stockinger, _regularisation by dimensional reduction: consistency, quantum action principle, and supersymmetry_ , JHEP 0503 (2005) 076; W. Hollik and D. Stockinger, _regularisation and supersymmetry restoring counterterms in supersymmetric QCD_ , Eur. Phys. J. C 20 (2001) 105; I. Jack and D. T. R. Jones, _Regularisation of Supersymmetric Theories_ , Adv. Ser. Direct. High Energy Physics 18 (1998) 149.
  * (23) W. Hollik, E. Kraus, D. Stöckinger, _renormalisation and symmetry conditions in supersymmetric QED_ , Eur. Phys. J. C 11 (1999) 365.
  * (24) E. Kraus, _An Anomalous breaking of supersymmetry in supersymmetric gauge theories with local coupling_ , Nucl. Phys. B 620 (2002) 55; hep-ph/0110323.
  * (25) W. Hollik and D. Stockinger, _regularisation and supersymmetry restoring counterterms in supersymmetric QCD_ , Eur. Phys. J. C 20 (2001) 105.
  * (26) W. Siegel, _Supersymmetric Dimensional regularisation via Dimensional Reduction_ , Phys. Lett. B 84(1979) 193; idem, _Inconsistency of Supersymmetric Dimensional regularisation_ , Phys. Lett. B 94(1980) 37.
  * (27) W. B. Kilgore, _regularisation Schemes and Higher Order Corrections_ , Phys. Rev. D 83 (2011) 114005; I. Jack, D. Jones, and K. Roberts, _Equivalence of dimensional reduction and dimensional regularisation_ , Z. Phys. C 63 (1994) 151; I. Jack, D. Jones, and K. Roberts, _Dimensional reduction in nonsupersymmetric theories_ , Z. Phys. C 62 (1994) 161; A. Signer and D. Stöckinger, _Factorisation and regularisation by dimensional reduction_ , Phys. Lett. B 626 (2005) 127.
  * (28) R. Harlander, P. Kant, L. Mihaila, and M. Steinhauser, _Dimensional Reduction applied to QCD at three loops_ , JHEP 0609 (2006) 053; R. Harlander, D. Jones, P. Kant, L. Mihaila, and M. Steinhauser, _Four-loop beta function and mass anomalous dimension in dimensional reduction_ , JHEP 0612 (2006) 024.
  * (29) C. Gnendiger, et al., _To d, or not to d: Recent developments and comparisons of regularisation schemes_ Eur. Phys. J. C 77 (2017) 471.
  * (30) L. Ferreira, A. Cherchiglia, M. C. Nemes, B. Hiller, Marcos Sampaio, _Momentum routing invariance in Feynman diagrams and quantum symmetry breakings_ , Phys. Rev. D 86 (2012) 025016.
  * (31) 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 (2016) 025029.
  * (32) Zvi Bern and David A. Kosower, _The Computation of loop amplitudes in gauge theories_ , Nucl.Phys. B 379 (1992) 451; William B. Kilgore, _The Four Dimensional Helicity Scheme Beyond One Loop_ , Phys. Rev. D 86 (2012) 014019; Radja Boughezal, Kirill Melnikov, Frank Petriello, _The four-dimensional helicity scheme and dimensional reconstruction_ , Phys. Rev. D 84 (2011) 034044; Christoph Gnendiger, Adrian Signer, Marcus Sperling, Dominik Stöckinger, Alexander Voigt, _Two-loop results on the renormalisation of vacuum expectation values, infrared divergences in the FDH scheme_ , PoS LL 2014 (2014) 076; A. Broggio (PSI, Villigen), Ch. Gnendiger (Dresden, Tech. U.), A. Signer, D. Stöckinger, A. Visconti , _Computation of $H\rightarrow qq$ in DRED and FDH: renormalisation, operator mixing, and explicit two-loop results_, Eur. Phys.J. C 75 (2015) 418.
  * (33) A. Broggio, Ch. Gnendiger, A. Signer, D. Stöckinger, A. Visconti ,_SCET approach to regularisation-scheme dependence of QCD amplitudes_ JHEP 1601 (2016) 078.
  * (34) Javier Mas, M. P.-Victoria and C. Seijas, _The beta function of N=1 SYM in differential renormalisation_ , JHEP 0203 (2002) 049.
  * (35) A. Broggio,Ch. Gnendiger, A. Signer, D. Stöckinger, A. Visconti, _Computation of $H\rightarrow gg$ in DRED and FDH: renormalisation, operator mixing, and explicit two-loop results_, Eur. Phys. J. C 75 (2015) 418; C. Gnendiger, A. Signer and D. Stöckinger, _The infrared structure of QCD amplitudes and $H\rightarrow gg$ in FDH and DRED_ Phys. Lett. B 733 (2014) 296
  * (36) W. Zimmermann, _Convergence of Bogoliubov’s method of renormalisation in momentum space_ ,Comm. Math. Phys. 15 (1969); N. Bogoliubov and O. Parasiuk, _On the Multiplication of the causal function in the quantum theory of fields_ , Acta Math. 97 (1957) 227; K. Hepp, _Proof of the Bogolyubov-Parasiuk theorem on renormalisation_ , Commun. Math. Phys. 2 (1966) 301; W. Zimmermann, _Local field equation for $A^{4}$ coupling in renormalized perturbation theory._ , Commun. Math. Phys. 6 (1967) 161; O. Piguet and S. P. Sorella, _Algebraic renormalisation: Perturbative renormalisation, symmetries and anomalies_ , Lect. Notes Phys. 28 (1995) 1; H. Epstein and V. Glaser, _The role of locality in perturbation theory_ , Ann. Inst.Henri Poincaré, Section A, Vol. XIX (1973) 211.
  * (37) C. Cheung, D. O’Connell, _Amplitudes and spinor-helicity in six dimensions_ , JHEP 07 (2009) 075.
  * (38) R. A. Fazio, P. Mastrolia, E. Mirabella, W. J. Torres Bobadilla, _On the four dimensional formulation of dimensionally regulated amplitudes_ , Eur. Phys. J. C 74 (2014) 3197.
  * (39) Daniel Z. Freedman, Kenneth Johnson, Jose I. Latorre, _Differential regularisation and renormalisation: A New method of calculation in quantum field theory_ , Nucl. Phys. B 371 (1992).
  * (40) Orimar Battistel, PhD thesis (1996), Federal University of Minas Gerais, Brazil; Orimar Battistel et al., Mod. Phys. Lett. A 13 (1998) 1597.
  * (41) Yue-Liang Wu, _Symmetry Principle Preserving and Infinity Free regularisation and renormalisation of quantum field theories and the mass gap_ , Int. J. Mod. Phys. A 18 (2003) 5363.
  * (42) R. Pittau, _A four-dimensional approach to quantum field theories_ , JHEP 11 (2012) 151.
  * (43) R. J. Hernandez-Pinto, G. F. R. Sborlini, G. Rodrigo, _Towards gauge theories in four dimensions_ , JHEP 02 (2016) 044.
  * (44) M. Perez-Victória, _Physical (ir)relevance of ambiguities to Lorentz and CPT violation in QED_ , JHEP 0104 (2001) 032.
  * (45) B. Delamotte, _A hint of renormalisation_ , Am. J. Phys. 72 (2004), arXiv:hep-th/0212049.
  * (46) 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 (2016) 065023.
  * (47) J. S. Porto, A. R. Vieira, A. L. Cherchiglia, Marcos Sampaio, and Brigitte Hiller, _On the Bose symmetry and the left- and right-chiral anomalies_ , Eur. Phys. J. C 78 (2018) 160.
  * (48) A. Cherchiglia, Marcos Sampaio, M. C. Nemes, _Systematic Implementation of Implicit regularisation for Multi-Loop Feynman Diagrams_ , Int. J. Mod. Phys. A 26 (2011) 1.
  * (49) R. Jackiw, _When radiative corrections are finite but undetermined_ , Int. J. Mod. Phys. B 14 (2000) 2011.
  * (50) A. P. Baêta Scarpelli, M. Sampaio and M. C. Nemes, _Consistency relations for an implicit n-dimensional regularisation scheme_ Phys. Rev. D 63 (2001) 046004.
  * (51) E. W. Dias, A. P. B. Scarpelli, Marcos Sampaio, M. C. Nemes, _Implicit regularisation beyond one loop order: Gauge field theories_ , Eur. Phys. J. C 55 (2008) 667.
  * (52) M. Sampaio, A. P. Baêta Scarpelli, J. E. Ottoni, M. C. Nemes,_Implicit regularisation and renormalisation of QCD_ Int. J. Theor. Phys. 45 (2006) 436.
  * (53) A. Cherchiglia, B. Hiller, A. P. B. Scarpelli and Marcos Sampaio, _Subtleties in the beta function calculation of N=1 supersymmetric gauge theories_ , Eur. Phys. J. C 76 (2016) 47.
  * (54) H. Fargnoli, B. Hiller, A. Scarpelli, Marcos Sampaio, M. C. Nemes, _regularisation Independent Analysis of the Origin of Two Loop Contributions to $N=1$ Super Yang-Mills Beta Function_, Eur. Phys. J. C 71 (2011) 1633.
  * (55) L. C. Brito, Marcos Sampaio, H. Fargnoli, M. C. Nemes, _Systematisation of Basic Divergent Integrals in Perturbation Theory and renormalisation Group Functions_ , Phys. Lett. B 673 (2009) 220.
  * (56) O. A. Battistel, M. C. Nemes, _Consistency in regularisations of the gauged NJL model at one loop level_ , Phys.Rev. D 59 (1999) 055010.
  * (57) A. P. Baêta Scarpelli, M. Sampaio, B. Hiller and M. C. Nemes, _Chiral anomaly and CPT invariance in an implicit momentum space regularisation framework_ , Phys. Rev. D 64 (2001) 046013.
  * (58) M. Sampaio, A. P. Baêta Scarpelli, B. Hiller, A. Brizola, M. C. Nemes and S. Gobira, _Comparing implicit, differential, dimensional and BPHZ renormalisation_ Phys. Rev. D 65 (2002) 125023.
  * (59) D. Carneiro, A. P. Baêta Scarpelli, M. Sampaio and M. C. Nemes, _Consistent momentum space regularisation / renormalisation of supersymmetric quantum field theories: The three loop beta function for the Wess-Zumino model_ , JHEP 12 (2003) 044.
  * (60) L. A. M. Souza, Marcos Sampaio, M. C. Nemes, _Arbitrary parameters in implicit regularisation and democracy within perturbative description of 2-dimensional gravitational anomalies_ , Phys. Lett. B632 (2006) 717.
  * (61) E. W. Dias, B. Hiller, A. L. Mota, M. C. Nemes, Marcos Sampaio, A. A. Osipov, _Symmetries and ambiguities in the linear sigma model with light quarks_ , Mod. Phys. Lett. A 21 (2006) 339.
  * (62) J. E. Ottoni, A. P. B. Scarpelli, Marcos Sampaio, M. C. Nemes, _Supergravity corrections to the $(g-2)_{l}$ factor by Implicit regularisation_, Phys. Lett. B 642 (2006) 253.
  * (63) Brigitte Hiller, A. L. Mota, M. C. Nemes, A. A. Osipov, Marcos Sampaio, _The Role of hidden ambiguities in the linear sigma model with fermions_ , Nucl. Phys. A 769 (2006) 53.
  * (64) C. Pontes, A. P. B. Scarpelli, Marcos Sampaio, M. C. Nemes, _Implicit regularisation of massless theories_ , J. Phys. G 34 (2007) 2215.
  * (65) A. P. Baêta Scarpelli, Marcos Sampaio, M. C. Nemes, _Gauge invariance and the CPT and Lorentz violating induced Chern-Simons-like term in extended QED_ , Eur. Phys. J. C 56 (2008) 571.
  * (66) C. Pontes, A. P. B. Scarpelli, J. L. Acebal, Marcos Sampaio, M. C. Nemes, _On the equivalence between Implicit regularisation and Constrained Differential renormalisation_ , Eur. Phys. Journal C 53 (2008) 121.
  * (67) J. Felipe, L. Brito, Marcos Sampaio, M. C. Nemes, _Quantum gravitational contributions to the beta function of quantum electrodynamics_ , Phys. Lett. B 700 (2011) 86.
  * (68) H. Fargnoli, A. Scarpelli, L. C. Brito, B. Hiller, A. Osipov, Marcos Sampaio, M. C. Nemes, _Ultraviolet and Infrared Divergences in Implicit regularisation: A Consistent Approach_ , Mod. Phys. Lett. A 26 (2011) 289.
  * (69) G. Gazzola, H. G. Fargnoli, A. P. B. Scarpelli, Marcos Sampaio, and M. C. Nemes, _QED with minimal and nonminimal couplings: on the quantum generation of Lorentz violating terms in the pure photon sector_ , J. Phys. G: Nucl. Part. Phys. 39 (2012) 035002.
  * (70) A. Cherchiglia, L. A. Cabral, M. C. Nemes, Marcos Sampaio, _(Un)determined finite regularisation dependent quantum corrections: the Higgs boson decay into two photons and the two photon scattering examples_ , Phys. Rev. D 87 (2013) 065011.
  * (71) Gustavo Gazzola, L. A. Cabral, A. Cherchiglia, M. C. Nemes and Marcos Sampaio, _Conductivity of Coulomb interacting massless Dirac particles in graphene: regularisation-dependent parameters and symmetry constraints_ , Europhys.Lett. 104 (2013) 27002.
  * (72) J. C. C. Felipe, L. A. Cabral, L. C. T. Brito, Marcos Sampaio, and M. C. Nemes, _Ambiguities in the gravitational correction of quantum electrodynamics running coupling_ , Mod. Phys. Lett. A 28 (2013) 1350078.
  * (73) A. R. Vieira, Brigitte Hiller, M. C. Nemes, and Marcos Sampaio, _Naturalness and Theoretical Constraints on the Higgs Boson Mass_ , Int. J. Theor. Phys. 52 (2013) 3494.
  * (74) J. C. C. Felipe, A. R. Vieira, A. L. Cherchiglia, A. P. B. Scarpelli, and Marcos Sampaio, _Arbitrariness in the gravitational Chern-Simons-like term induced radiatively_ , Phys. Rev. D 89 (2014) 105034.
  * (75) A. L. Cherchiglia, A. R. Vieira, Brigitte Hiller, A. P. B. Scarpelli, Marcos Sampaio, _Guises and disguises of quadratic divergences_ Ann.of Phys. 351 (2014) 751.
  * (76) H. G. Fargnoli, L. C. T. Brito, A. P. B. Scarpelli, and Marcos Sampaio, _Dual embedding of extended models with a Lorentz-breaking mass term_ , Phys. Rev. D 90 (2014) 085016.
  * (77) A. R. Vieira, J. C. C. Felipe, G. Gazzola, Marcos Sampaio, _One-loop conformal anomaly in an implicit momentum space regularisation framework_ , Eur. Phys. J. C 75 (2015) 338.
  * (78) Y. R. Batista, Brigitte Hiller, Adriano Cherchiglia, and Marcos Sampaio, _Supercurrent anomaly and gauge invariance in the N=1 supersymmetric Yang-Mills theory_ , Phys. Rev. D 98 (2018) 025018.
  * (79) Matthew D. Schwartz, _Quantum Field Theory and the Standard Model_ , CUP (2014).
  * (80) L. F. Abbott, _The Background Field Method Beyond One Loop_ , Nucl. Phys. B 185 (1981) 189.
  * (81) T. Hahn, _Generating Feynman diagrams and amplitudes with FeynArts 3_ , Comput. Phys. Commun. 140 (2001), 418-431.
  * (82) T. Hahn and M. Perez-Victoria, _Automatized one loop calculations in four-dimensions and D-dimensions,_ Comput. Phys. Commun. 118 (1999), 153-165.
  * (83) W. Beenakker, H. Kuijf, W. L. van Neerven and J. Smith, _QCD corrections to heavy-quark production in $p\bar{p}$ collisions_, Phys. Rev. D 40 (1989), 54.