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On Ambiguities and Divergences in
Perturbative Renormalization Group Functions

Florian<EMAIL_ADDRESS>and Anders Eller
<EMAIL_ADDRESS>
${}^{1}\,$Fermi National Accelerator Laboratory,
Batavia, IL, 60510, USA ${}^{2}\,$Albert Einstein Center for Fundamental
Physics, Institute for Theoretical Physics,
University of Bern, CH-3012 Bern, Switzerland

August 27, 2024

There is an ambiguity in choosing field-strength renormalization factors in
the $\overline{\text{MS}}$ scheme starting from the 3-loop order in
perturbation theory. More concerning, trivially choosing Hermitian factors has
been shown to produce divergent renormalization group functions, which are
commonly understood to be finite quantities. We demonstrate that the
divergences of the RG functions are such that they vanish in the RG equation
due to the Ward identity associated with the flavor symmetry. It turns out
that any such divergences can be removed using the renormalization ambiguity
and that the use of the flavor-improved $\beta$-function is preferred. We show
how our observations resolve the issue of divergences appearing in previous
calculations of the 3-loop SM Yukawa $\beta$-functions and provide the first
calculation of the flavor-improved 3-loop SM $\beta$-functions in the
gaugeless limit.

Preprint: FERMILAB-PUB-21-196-T

##  1  Introduction

The renormalization group (RG) functions are fundamental quantities in quantum
field theories (QFTs). They make it possible to relate physics at various
energy scales while avoiding large logarithms in the perturbative expansion.
The $\beta$-functions are frequently used in phenomenology, but also to study
the ultimate ultraviolet fates of models to ensure the absence of Landau poles
[1, 2] or the meta-stability of the Standard Model (SM) vacuum [3, 4, 5].
Additionally, the current drive for precision physics has prompted the
calculation of the SM $\beta$-functions to the 4-loop order for gauge and
3-loop order for Yukawa and quartic couplings [6, 7, 8, 9, 10, 11, 12].

It was in connection with the 3-loop calculation of the SM matrix Yukawa
$\beta$-functions [9, 13] that the ambiguity subject to our investigation made
its first appearance. The $\overline{\text{MS}}$ field-strength
renormalization factors are typically determined by requiring finiteness of
the 2-point functions. However, starting at the 3-loop order, this
prescription leaves room for an ambiguity in the form of a nontrivial unitary
factor. This is typically identified with the ambiguity inherent in taking the
square root of a matrix renormalization factor. It was recognized that the
trivial choice of setting the unitary factor to the identity (taking Hermitian
square roots) leads to a divergent anomalous dimension of the quark fields and
the Yukawa $\beta$-functions [9, 13]. To be clear, when we refer to a
_divergent_ RG function in this context, we mean that it has poles in the
dimensional expansion. It is not a statement about the couplings diverging
because of a Landau pole in the flow. At the time, the divergence problem was
circumvented by choosing a suitable renormalization constant, which produced
finite RG functions, while also testing that various flavor invariants were
insensitive to a leftover ambiguity in the finite part of the
$\beta$-function. With the community now at the brink of new 3- and 4-loop
$\beta$-function computations, a more thorough understanding of the ambiguity
is required to confidently perform the loop calculations and settle whether
the divergences point to some deeper limitation of the regular approach. This
discussion also directly points to an ambiguity in recent 3-loop
$\beta$-function computations in Yukawa theories [14, 15] and the 6-loop
computation in pure scalar theory [16], both of which treat theories with
nontrivial flavor structure.

To elucidate the connection between the divergent RG functions and the
renormalization ambiguity in more general terms, we turn to _the local
renormalization group_ (LRG). This framework is, as one might expect, a local
extension of the RG as most theorists are familiar with it [17, 18]. The RG
functions are identified as the response of local sources to a Weyl
transformation of the theory. This construction is ideal for probing scale
invariance and has been frequently used in the search for an $A$-function [19,
20, 18, 21, 22, 23], a (monotonically) increasing function along the RG. The
Weyl symmetry of the local theory is anomalous and the consistency of the
anomaly enforces a number of nontrivial conditions on the various RG functions
known as the Weyl consistency conditions [24, 21, 25, 26, 27]. A relatively
recent result of the LRG is that a conformal field theory is signified not by
vanishing $\beta$-functions but by vanishing flavor-improved
$\beta$-functions, $B=\beta-(\upsilon\,g)$, where $\upsilon$ is an RG function
related to a background gauge field of the flavor symmetry [28].111One of the
authors (AET) realized to his dismay that in the original literature
$\upsilon$ is, in fact, a Greek upsilon rather than a Latin _v_ [21]. With
this timely warning, the reader will not have to make the same mistake. As
such, the perturbative limit cycle found of Ref. [29] does, in fact,
correspond to a zero of $B$. For our present purposes, the LRG provides a
comprehensive framework for working with the RG functions and their interplay
with the flavor symmetry.

In this paper we argue the following points:

  1. i)

the occurrence of a certain class of $\epsilon$-poles in the RG functions is
consistent with the Callan-Symanzik equation and not a sign of the theory or
the renormalization scheme breaking down;

  2. ii)

there is an ambiguity in choosing renormalization constants due to the flavor
symmetry of a theory under a simultaneous transformation of fields and bare
couplings;

  3. iii)

using the ambiguity, it is always possible to remove all the poles of the
class discussed in i) simultaneously from $\beta$-functions and field
anomalous dimensions;

  4. iv)

the flavor-improved $\beta$-function, $B$, is unambiguous and finite and
therefore a preferred choice of $\beta$-function.

We then bridge the gap from the theoretical considerations in the LRG
framework to hands-on calculations in 4-dimensional gauge-Yukawa theories, in
general. In particular, we demonstrate our points i)–iv) in the 3-loop SM
computation and present the first nontrivial computation of $B_{I}$ in the
gaugeless limit of the SM.

The remainder of the paper is organized as follows: In the next section, we
review the LRG and introduce the majority of the notation and concepts we will
need. In Section 3 we discuss the renormalization ambiguity and present our
main thesis. Section 4 discusses what the ambiguity looks like in gauge-Yukawa
theories in general and in the SM in particular. Finally, we conclude in
Section 5. We provide the explicit derivation of several nontrivial relations
crucial to the discussion of the renormalization ambiguity and a 1-loop
example showcasing the necessary computations for obtaining $\upsilon$ in the
appendix.

##  2  The Local Renormalization Group

We begin with a rather long-winded review of the LRG, the purpose of which is
twofold: there is a lot of notation to be introduced, not all of which is
standard(ized) even among those working with the LRG; and many readers may be
unfamiliar with the framework. We also take this opportunity to spell out the
inclusion of field sources in the framework.

###  2.1  Sourcing composite operators

Ultimately, our goal is to study the $\beta$-functions and field anomalous
dimensions in a dimensionally regulated 4-dimensional QFT. In the minimal
subtraction (MS) scheme, these quantities are insensitive to the inclusion of
relevant operators to the Lagrangian, which we will therefore ignore. Working
in $d=4-2\epsilon$ dimensions, the generic bare action of any such QFT is

$S[\Phi,\,g_{0},\,\mathcal{J}_{0}]=S_{\mathrm{kin}}[\Phi]+\\!\int\mathop{}\\!\mathrm{d}^{d}x\left(g_{0,I}\mathcal{O}^{I}+\mathcal{J}_{0,\alpha}\Phi^{\alpha}\right)\,,$
(2.1)

where $S_{\mathrm{kin}}$ is the kinetic term, and the scalar and fermion
quantum fields are collectively denoted $\Phi^{\alpha}(x)\,$. These matter
fields are sourced by bare field sources $\mathcal{J}_{0,\alpha}(x)\,$. While
we also allow for quantum gauge fields $A_{\mu}$ in the theory, we leave these
implicit for the discussion in this section, as we do not wish to source them.
The last piece of the action is the interaction terms with bare couplings
$g_{0,I}$, indexed with indices $I,J,\ldots\,$, and operators
$\mathcal{O}_{I}(x)$ running over all gauge-invariant marginal operators:
gauge potential, Yukawa, and quartic.

The idea behind the LRG is to allow for the computation of correlators of
composite operators, as opposed to just the field correlators sourced by
$\mathcal{J}_{0}\,$, by introducing more background sources to the theory. The
Minkowski metric $\eta_{\mu\nu}$ is replaced with a curved background
$\gamma_{\mu\nu}(x)\,$, which sources the energy-momentum tensor $T_{\mu\nu}$
in the path integral. Accordingly, all derivatives in the action are upgraded
to be covariant under diffeomorphism transformations. The marginal couplings,
$g_{I}$, are in turn upgraded to local sources $g_{I}(x)\,$, so they can
source the corresponding marginal operators in the path integral.

It is also possible to source currents of global symmetries of the theory. We
let $G_{F}$ with Lie algebra $\mathfrak{g}_{F}$ denote the largest global
symmetry group of $S_{\mathrm{kin}}$ compatible with the quantum gauge
group.222By compatibility of the quantum gauge and flavor symmetry, we mean
that $[A_{\mu},\,\omega]\phantom{}^{\alpha}\phantom{}_{\beta}=0$ in the
representation of $\Phi^{\alpha}$ for any $\omega\in\mathfrak{g}_{F}\,$. The
action of $G_{F}$ on $\Phi^{\alpha}$ dictates the action on $\mathcal{O}_{I}$
(assumed to be covariant combinations of fields and derivatives),
$\mathcal{J}\,$, and $g_{I}$ so as to keep the action invariant: under the
infinitesimal action of $\omega\in\mathfrak{g}_{F}\,$, the sources (and
fields) transforms as

$\begin{split}\delta_{\omega}\Phi^{\alpha}&=-(\omega\,\Phi)^{\alpha}=-\omega\phantom{}^{\alpha}\phantom{}_{\beta}\Phi^{\beta}\,,\\\
\delta_{\omega}g_{I}&=-(\omega\,g)_{I}=g_{J}\omega\phantom{}^{J}\phantom{}_{I}\,,\\\
\delta_{\omega}\mathcal{J}_{0,\alpha}&=-(\omega\,\mathcal{J}_{0})_{\alpha}=\mathcal{J}_{0,\beta}\omega\phantom{}^{\beta}\phantom{}_{\alpha}\,,\end{split}$
(2.2)

employing anti-Hermitian matrix representations, $\omega^{\dagger}=-\omega\,$,
as is conventional in the LRG. We will regularly use the notation
$(\omega\,\cdot)$ for the action of elements $\omega\in\mathfrak{g}_{F}$ on
various objects.

The current $J_{F}^{\mu}$ of the flavor symmetry is sourced by promoting
$G_{F}$ to a local symmetry with the background gauge field
$a_{\mu}(x)\in\mathfrak{g}_{F}$ as the source. As with any other gauge field,
the infinitesimal transformation of $a_{\mu}$ is given by

$\delta_{\omega}a_{\mu}=\partial_{\mu}\omega+\left[a_{\mu},\omega\right]\equiv
D_{\mu}\omega\,,\qquad\omega(x)\in\mathfrak{g}_{F}\leavevmode\nobreak\ .$
(2.3)

It is included in the action by replacing all derivatives with $G_{F}$
covariant versions such that they are now simultaneously covariant under
diffeomorphism, quantum gauge symmetry, and local flavor symmetry.

The inclusion of new sources to the theory introduces the need for new
counterterms, consisting of curvature terms, covariant derivatives of the
marginal couplings, as well as the field-strength tensor of the background
gauge field. We group these into an extra action piece
$S_{\mathrm{ct}}[\gamma,\,g_{0},\,a_{0}]$, though we do not need any details
for our present purposes but rather refer to Refs. [20, 21] for an exhaustive
list of the source counterterm. In principle, one should also include
dimension $d-2$ operators, $\mathcal{O}_{M}^{a}$, as these are required for a
consistent study of the Weyl anomaly even when scalar mass terms are put to
zero by hand [21, 22]. They will not play any direct role in our discussion,
so we will ignore them along with other relevant operators, which do not play
this special role.

The action with the new sources introduced in the LRG framework is

$S[\Phi,\,\gamma,\,g_{0},\,a_{0},\,\mathcal{J}_{0}]=S_{\mathrm{kin}}[\Phi,\,\gamma,\,a_{0}]+\\!\int\mathop{}\\!\mathrm{d}^{d}x\sqrt{\gamma}\left(g_{0,I}\mathcal{O}^{I}+\mathcal{J}_{0,\alpha}\Phi^{\alpha}\right)+S_{\mathrm{ct}}[\gamma,\,g_{0},\,a_{0}]\,,$
(2.4)

with the associated vacuum functional given by

$e^{i\mathcal{W}_{0}[\gamma,\,g_{0},\,a_{0},\,\mathcal{J}_{0}]}\equiv\int\mathcal{D}\Phi\,\mathcal{D}A\,e^{i\,S[\Phi,\,\gamma,\,g_{0},\,a_{0},\,\mathcal{J}_{0}]}\,,$
(2.5)

generating all bare connected Green’s functions. Anticipating the need for
renormalization, we have put subscript zeros on all sources so far, to
indicate that they are “bare.” In contrast to the conventional perturbative
approach, we give up renormalizing $\Phi$ in favor of renormalizing
$\mathcal{J}_{0}$. In this approach no kinetic counterterm or field rescaling
is required. Rather, the vacuum functional is renormalized by writing each
bare source as a function of finite renormalized sources, that is,

$\mathcal{W}[\gamma,\,g,\,a,\,\mathcal{J}]=\mathcal{W}_{0}\big{[}\gamma,\,g_{0}(g),\,a_{0}(a,g),\,\mathcal{J}_{0}(\mathcal{J},g)\big{]}\,,$
(2.6)

where $\mathcal{W}$ is the finite, renormalized vacuum functional.

The parameterization of bare sources in terms of the renormalized couplings is
ultimately a question of choosing a suitable renormalization scheme. Here we
use the MS scheme (or, equivalently, $\overline{\text{MS}}$ upon replacing the
renormalization scale with
$\mu^{2}=(4\pi)^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\bar{\mu}^{2}e^{\gamma_{\mathrm{E}}}$),
in line with the highest order SM computations. In this scheme the
counterterms exclusively remove the $\epsilon$-poles from the Greens
functions, and we can parameterize the bare sources as

$\displaystyle g_{0,I}$ $\displaystyle=\mu^{k_{I}\epsilon}(g_{I}+\delta
g_{I})\,,$ $\displaystyle\mathcal{J}_{0,\alpha}$
$\displaystyle=\mathcal{J}_{\beta}Z\phantom{}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\beta}\phantom{}_{\alpha}\,,$
$\displaystyle a_{0,\mu}$ $\displaystyle=a_{\mu}+N^{I}D_{\mu}g_{I}\,,$
$\displaystyle\delta g_{I}$ $\displaystyle=\sum_{n=1}^{\infty}\dfrac{\delta
g_{I}^{(n)}}{\epsilon^{n}}\,,$ $\displaystyle Z$
$\displaystyle=1+\sum_{n=1}^{\infty}\dfrac{z^{(n)}}{\epsilon^{n}}\,,$
$\displaystyle N^{I}$
$\displaystyle=\sum_{n=1}^{\infty}\dfrac{N^{(n)I}}{\epsilon^{n}}\in\mathfrak{g}_{F}\,,$
(2.7)

where the covariant derivative working on the marginal couplings is given by

$D_{\mu}g_{I}=\partial_{\mu}g_{I}+(a_{\mu}\,g)_{I}\,.$ (2.8)

$g_{I}$ is made dimensionless by factoring out the renormalization scale,
$\mu\,$, to the power $k_{I}\,$, which is the dimensionality of the bare
couplings in $d$ dimensions. We use the convention _not_ to count the index on
$k_{I}$ for summation conventions, i.e. only sum the index if it appears
another two times.

We proceed to discuss the various symmetries of the theory in the presence of
all sources. Several of the important equations in the RG arise as Ward
identities associated with these symmetries.

###  2.2  Weyl transformations

In four dimensions the flat space action (2.1) is scaleless at the classical
level and therefore possesses a global scale symmetry parameterized by
$\sigma\,$, that is, a transformation of the metric

$\gamma_{\mu\nu}\longrightarrow
e^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2\sigma}\gamma_{\mu\nu}\,.$ (2.9)

Invariance of the kinetic term under such a scale transformation is ensured by
letting the fields transform according to their canonical dimension:

$\Phi^{\alpha}\longrightarrow\Phi^{\sigma,\alpha}=e^{\Delta_{\alpha}\sigma}\Phi^{\alpha}\,.$
(2.10)

With the curved-space metric of the full LRG action (2.4), the scale symmetry
can be localized, $\sigma\to\sigma(x)\,$, to what is known as the Weyl
symmetry.333A non-minimal coupling of scalars to the curvature is needed to
ensure this local invariance.

In more generality, the transformation properties of the bare sources under
the Weyl symmetry are dictated by their mass dimension. Consequently, the
marginal couplings receive a nontrivial transformation when working in the
$d\neq 4$ dimensions needed for regularization (see e.g. Ref. [22]). Symmetry
of the marginal operator terms in the action is established by letting the
transformation of the bare sources compensate the canonical transformation of
the operators $\mathcal{O}^{I}\,$, and we have

$g_{0,I}\longrightarrow
g_{0,I}^{\sigma}=e^{(d-\Delta_{\mathcal{O}^{I}})\sigma}g_{0,I}=e^{k_{I}\epsilon\sigma}g_{0,I}\,.$
(2.11)

From transformation (2.11), we may infer the transformation of the
renormalized coupling. Using renormalization-scale independence of the bare
coupling yields

$g_{0,I}^{\sigma}=e^{\sigma
k_{I}\epsilon}g_{0,I}(\mu,g(\mu))=g_{0,I}(e^{\sigma}\mu,g(\mu))=g_{0,I}(\mu,g(\mu
e^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}\sigma}))\,.$ (2.12)

The transformation law for the bare couplings, thus, implies that the
renormalized couplings transform as

$g_{I}(\mu)\longrightarrow g^{\sigma}_{I}(\mu)=g_{I}(\mu
e^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}\sigma})\quad\implies\quad\delta_{\sigma}g_{I}=-\sigma\dfrac{\mathop{}\\!\mathrm{d}g_{I}}{\mathop{}\\!\mathrm{d}t}=-\sigma\beta_{I}\,,\qquad
t=\ln\mu\,,$ (2.13)

where $\delta_{\sigma}$ is the infinitesimal response to a Weyl transformation
with parameter $\sigma\,$. This transformation rule implies that the response
of the renormalized coupling to a local rescaling of space is to run to a new
energy scale with the $d$-dimensional $\beta$-function $\beta\,$.

We, also, directly infer the formula for $\beta_{I}$ from the bare coupling
parameterization (2.1). Taking an infinitesimal Weyl transformation yields

$k_{I}\epsilon\,\sigma\,\mu^{k_{I}\epsilon}(g_{I}+\delta
g_{I})=\delta_{\sigma}g_{0,I}=\mu^{k_{I}\epsilon}(\delta\phantom{}^{J}\phantom{}_{I}+\partial^{J}\delta
g_{I})\delta_{\sigma}g_{J}\,,\qquad\partial^{I}\equiv\dfrac{\partial}{\partial
g_{I}}\,.$ (2.14)

Rearranging things a little gives

$\beta_{I}+k_{I}\epsilon\,g_{I}+(k_{I}\epsilon+\beta_{J}\partial^{J})\delta
g_{I}=0\,,$ (2.15)

which is the usual relation for the MS $\beta$-function.

We proceed to the transformation rules for the field sources,
$\mathcal{J}^{\alpha}\,$. The bare sources transform according to their
canonical dimension, and applying the infinitesimal Weyl variation gives

$(d-\Delta_{\alpha})\sigma\,\mathcal{J}_{\beta}Z\phantom{}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\beta}\phantom{}_{\alpha}=\delta_{\sigma}\mathcal{J}_{0,\alpha}=\delta_{\sigma}\mathcal{J}_{\beta}Z\phantom{}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\beta}\phantom{}_{\alpha}-\delta_{\sigma}g_{I}\mathcal{J}_{\beta}[Z^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\partial^{I}Z\,Z^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}]\phantom{}^{\beta}\phantom{}_{\alpha}\,.$
(2.16)

Plugging in the transformation of the couplings (2.13), we recover

$\delta_{\sigma}\mathcal{J}_{\alpha}=\sigma\,\mathcal{J}_{\beta}\left[(d-\Delta_{\alpha})\delta\phantom{}^{\beta}\phantom{}_{\alpha}-\gamma\phantom{}^{\beta}\phantom{}_{\alpha}\right]\,,\qquad\gamma\phantom{}^{\alpha}\phantom{}_{\beta}=\beta_{I}[Z^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\partial^{I}Z]\phantom{}^{\alpha}\phantom{}_{\beta}\,.$
(2.17)

$\gamma$ is, thus, identified with the $d$-dimensional field anomalous
dimension.

Finally, for the background gauge field, we parameterize
$\delta_{\sigma}a_{\mu}=\partial_{\mu}\sigma\,\upsilon-\sigma\rho^{I}D_{\mu}g_{I}$
for $\upsilon(g)\,,\,\rho^{I}(g)\in\mathfrak{g}_{F}\,$, consistent with the
most general form respecting dimensionality and covariance under $G_{F}\,$. As
$a_{0,\mu}$ is part of the covariant derivative, it must be invariant under
the Weyl transformation, meaning that

$0=\delta_{\sigma}a_{0,\mu}=\delta_{\sigma}a_{\mu}+N^{I}(\delta_{\sigma}a_{\mu}\,g)_{I}+N^{I}(D_{\mu}\delta_{\sigma}g)_{I}+\delta_{\sigma}g_{J}\partial^{J}N^{I}D_{\mu}g_{I}$
(2.18)

per Eq. (2.1). At this point, it is sufficient to plug in the transformation
of the couplings and the parameterization of $\delta_{\sigma}a_{\mu}$ to
obtain

$\upsilon=N^{I}B_{I}\,,\qquad B_{I}=\beta_{I}-(\upsilon\,g)_{I}$ (2.19)

and

$\rho^{I}=-\beta_{J}\partial^{J}N^{I}-N^{J}\partial^{I}\beta_{J}-N^{J}(\rho^{I}\,g)_{J}$
(2.20)

from the $\partial_{\mu}\sigma$ and $\sigma$ terms, respectively. Of the two
new RG functions associated with the flavor current, $\upsilon$ is our main
interest: it bridges the gap between $\beta_{I}$ and the flavor-improved
$\beta$-function $B_{I}\,$.444$\upsilon$ is referred to as $S$ in parts of the
literature.

Collecting all the infinitesimal transformation rules of the renormalized
sources, the generator of an infinitesimal Weyl transformation acting on the
vacuum functional is

$\Delta^{\\!W}_{\sigma}=\int\mathop{}\\!\mathrm{d}^{d}x\,\bigg{(}2\sigma\gamma^{\mu\nu}\dfrac{\delta}{\delta\gamma^{\mu\nu}}-\sigma\beta_{I}\dfrac{\delta}{\delta
g_{I}}+\sigma\mathcal{J}_{\beta}\big{[}(d-\Delta_{\alpha})\delta\phantom{}^{\beta}\phantom{}_{\alpha}-\gamma\phantom{}^{\beta}\phantom{}_{\alpha}\big{]}\dfrac{\delta}{\delta\mathcal{J}_{\alpha}}\\\
+\left[\partial_{\mu}\sigma\,\upsilon-\sigma\,D_{\mu}g_{I}\,\rho^{I}\right]\cdot\dfrac{\delta}{\delta
a_{\mu}}\bigg{)}\,,$ (2.21)

where “ $\cdot$ ” denotes the inner product on $\mathfrak{g}_{F}\,$. With the
Weyl transformation being a classical symmetry of the action, the vacuum
functional is invariant down to the Weyl anomaly:

$\Delta^{\\!W}_{\sigma}\mathcal{W}=\int\mathop{}\\!\mathrm{d}^{d}x\,\mathcal{A}^{W}_{\sigma}(\gamma,\,g,\,a)\,.$
(2.22)

The anomaly, $\mathcal{A}^{W}_{\sigma}\,$, is a local term of the sources and
has been the focus of the effort to derive Weyl consistency conditions [18]
and ultimately an $A$-function in four dimensions [20, 21, 22].

The Weyl symmetry has implications for the trace of the energy-momentum
tensor. In particular, we concern ourselves with the _flat-space-constant-
coupling_ (FSCC) limit, that is, the limit where
$\gamma_{\mu\nu}=\eta_{\mu\nu}\,$, $g_{I}(x)=g_{I}\,$, and $a_{\mu}=0\,$,
which is the phenomenologically relevant limit. Taking also the field sources
to vanish, the Weyl symmetry implies the operator equation

$[T\phantom{}^{\mu}\phantom{}_{\mu}]=\beta_{I}[\mathcal{O}^{I}]+\upsilon\cdot\partial_{\mu}[J_{F}^{\mu}]-\eta_{a}\partial^{2}[\mathcal{O}^{a}_{M}]\qquad\text{(FSCC)}$
(2.23)

for the trace of the energy-momentum tensor for Green’s functions with no
coinciding points. The square brackets are taken to indicate that the
operators are renormalized. This equation is known as the scale anomaly and
signals the loss of classical scale invariance in the quantum theory. The
trace of the energy-momentum tensor will vanish if all the RG
functions—$\beta_{I}\,$, $\upsilon\,$, and $\eta_{a}$—do, and the theory
becomes conformal. Note also the presence of the dimension-2 operators
$\mathcal{O}_{M}^{a}\,$, which is why they cannot be ignored in an analysis of
the anomaly.

###  2.3  Renormalization group transformations

Osborn [18] demonstrated that the anomaly is closely related to the RG flow of
$\mathcal{W}\,$. Since $\mathcal{W}$ is dimensionless, a simultaneous scaling
of the all lengths and masses must leave the vacuum functional invariant,
resulting in the counting identity for the physical dimension:

$\Delta^{\mu}\mathcal{W}=0\,,\qquad\Delta^{\mu}=\mu\dfrac{\partial}{\partial\mu}+\\!\int\mathop{}\\!\mathrm{d}^{d}x\left(2\gamma^{\mu\nu}\dfrac{\delta}{\delta\gamma_{\mu\nu}}+(d-\Delta_{\alpha})\mathcal{J}_{\alpha}\dfrac{\delta}{\delta\mathcal{J}_{\alpha}}\right)\,.$
(2.24)

We can now identify the RG generator as the combination of $\Delta^{\mu}$ and
$\Delta^{\\!W}_{\sigma}$ that eliminates the derivative w.r.t. the metric,
thus keeping lengths fixed:

$\Delta^{\\!{\scriptscriptstyle\mathrm{RG}}}=\Delta^{\mu}-\Delta^{\\!W}_{\sigma=1}=\dfrac{\partial}{\partial
t}+\\!\int\mathop{}\\!\mathrm{d}^{d}x\bigg{(}\beta_{I}\dfrac{\delta}{\delta
g_{I}}+\mathcal{J}_{\beta}\gamma\phantom{}^{\beta}\phantom{}_{\alpha}\dfrac{\delta}{\delta\mathcal{J}_{\alpha}}+D_{\mu}g_{I}\,\rho^{I}\cdot\dfrac{\delta}{\delta
a_{\mu}}\bigg{)}\,.$ (2.25)

In the FSCC limit, $\Delta^{\mu}$ and $\Delta^{\\!W}_{\sigma}$ are both good
symmetries of $\mathcal{W}\,$. Applying
$\Delta^{\\!{\scriptscriptstyle\mathrm{RG}}}$ to the vacuum functional in this
limit, yields the Callan-Symanzik equation

$0=\Delta^{\\!{\scriptscriptstyle\mathrm{RG}}}\mathcal{W}=\bigg{(}\dfrac{\partial}{\partial
t}+\beta_{I}\partial^{I}+\\!\int\mathop{}\\!\mathrm{d}^{d}x\,\mathcal{J}_{\beta}\gamma\phantom{}^{\beta}\phantom{}_{\alpha}\dfrac{\delta}{\delta\mathcal{J}_{\alpha}}\bigg{)}\mathcal{W}\qquad\text{(FSCC)}\,.$
(2.26)

In the FSCC limit
$\Delta^{\\!{\scriptscriptstyle\mathrm{RG}}}\mathcal{W}=\mathop{}\\!\mathrm{d}\mathcal{W}\\!/\\!\mathop{}\\!\mathrm{d}t\,$,
and the Callan-Symanzik equation could also have been derived following the
usual arguments about invariance of the bare vacuum functional w.r.t. $\mu\,$.
However, as we shall now see, the flavor symmetry leaves room for an ambiguity
in the definition of $\Delta^{\\!W}_{\sigma}$ and, thus,
$\Delta^{\\!{\scriptscriptstyle\mathrm{RG}}}$ as well.

###  2.4  Flavor transformations

We have previously established the invariance of the action w.r.t. the flavor
symmetry $G_{F}$. In operator form, the infinitesimal generator of the
symmetry is

$\Delta^{\\!F}_{\omega}=\int\mathop{}\\!\mathrm{d}^{d}x\left(D_{\mu}\omega\cdot\dfrac{\delta}{\delta
a_{\mu}}-(\omega\,g)_{I}\dfrac{\delta}{\delta
g_{I}}-(\omega\,\mathcal{J})_{\alpha}\dfrac{\delta}{\delta\mathcal{J}_{\alpha}}\right),\qquad\omega\in\mathfrak{g}_{F}\,,$
(2.27)

which leaves $\mathcal{W}$ invariant down to a possible flavor anomaly:

$\Delta^{\\!F}_{\omega}\mathcal{W}=\int\mathop{}\\!\mathrm{d}^{d}x\,\mathcal{A}^{F}_{\omega}(\gamma,\,g,\,a)\,.$
(2.28)

This form of the anomaly assumes the absence of anomalies mixing with the
dynamical gauge fields $A_{\mu}\,$. Keren-Zur [30] argued that mixed anomaly
contributions with dynamical and background gauge fields are irrelevant for
perturbative contributions in any event, as they are total derivatives.

The existence of the flavor symmetry introduces an ambiguity in the definition
of the Weyl symmetry. In general, it holds that

$\Delta^{\\!W\,\prime}_{\sigma}=\Delta^{\\!W}_{\sigma}+\Delta^{\\!F}_{\sigma\alpha}\,,\qquad\alpha(g)\in\mathfrak{g}_{F}\,,$
(2.29)

where $\alpha(g)$ is a covariant function of the couplings, is an equally
valid Weyl generator [21, 22].555Or, if the reader prefers, the generator of
an equally valid Weyl transformation. The new generator maintains

$\big{[}\Delta^{\\!F}_{\omega},\,\Delta^{\\!W\,\prime}_{\sigma}\big{]}=\big{[}\Delta^{\\!W\,\prime}_{\sigma},\,\Delta^{\\!W\,\prime}_{\sigma^{\prime}}\big{]}=0\,,\qquad\Delta^{\\!W\,\prime}_{\sigma}\mathcal{W}=\int\mathop{}\\!\mathrm{d}^{d}x\,\mathcal{A}^{W\,\prime}_{\sigma}\,,$
(2.30)

where the new anomaly has the same structure as the original Weyl anomaly
$\mathcal{A}^{W}_{\sigma}$, but with modified coefficients for the CP-odd
terms [30]. One way to view this ambiguity is that due to the flavor symmetry,
a simple local scaling is indistinguishable from a local scaling in
conjunction with a flavor rotation: physics remains unchanged.

The flavor ambiguity in the definition of the Weyl generator corresponds to a
redefinition of the RG functions, namely

$\displaystyle\beta^{\prime}_{I}$ $\displaystyle=\beta_{I}+(\alpha\,g)_{I}\,,$
$\displaystyle\upsilon^{\prime}$ $\displaystyle=\upsilon+\alpha\,,$
$\displaystyle\rho^{\prime I}$ $\displaystyle=\rho^{I}-\partial^{I}\alpha\,,$
$\displaystyle\gamma$ $\displaystyle=\gamma-\alpha\,.$ (2.31)

These functions are all equally good RG functions for any $\alpha\,$, in that
they all satisfy the Callan-Symanzik equation (2.26) but with
$(\beta_{I},\,\gamma)\to(\beta^{\prime}_{I},\,\gamma^{\prime})\,$.666One needs
simply let
$\Delta^{\\!{\scriptscriptstyle\mathrm{RG}}\,\prime}=\Delta^{\mu}-\Delta^{\\!W\,\prime}_{\sigma=1}$.
We would, however, like to remind the reader that it is only the original set
of RG function that directly corresponds to the change of
couplings/renormalization constants with the renormalization scale:

$\beta_{I}=\dfrac{\mathop{}\\!\mathrm{d}g_{I}}{\mathop{}\\!\mathrm{d}t}\quad\mathrm{and}\quad\gamma=Z^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\dfrac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}Z\,.$
(2.32)

With the ambiguity in the RG functions associated to the Weyl transformation,
it can be advantageous to work with the invariant objects [28]

$\displaystyle B_{I}$ $\displaystyle=\beta_{I}-(\upsilon\,g)_{I}\,,$
$\displaystyle P^{I}$ $\displaystyle=\rho^{I}+\partial^{I}\upsilon\,,$
$\displaystyle\Gamma\phantom{}^{\alpha}\phantom{}_{\beta}=\gamma\phantom{}^{\alpha}\phantom{}_{\beta}+\upsilon\phantom{}^{\alpha}\phantom{}_{\beta}\,.$
(2.33)

Conveniently, there is even a “gauge” where these objects appear directly in
the Weyl generator defined by

$\displaystyle\widehat{\Delta}^{W}_{\sigma}$
$\displaystyle=\Delta^{\\!W}_{\sigma}+\Delta^{\\!F}_{-\sigma\,\upsilon}=\Delta^{\\!W\,\prime}_{\sigma}+\Delta^{\\!F}_{-\sigma\,\upsilon^{\prime}}$
(2.34)
$\displaystyle=\int\mathop{}\\!\mathrm{d}^{d}x\bigg{(}2\sigma\gamma^{\mu\nu}\dfrac{\delta}{\delta\gamma_{\mu\nu}}-\sigma
B_{I}\dfrac{\delta}{\delta
g_{I}}+\sigma\mathcal{J}_{\beta}\big{[}(d-\Delta_{\alpha})\delta\phantom{}^{\beta}\phantom{}_{\alpha}-\Gamma\phantom{}^{\beta}\phantom{}_{\alpha}\big{]}\dfrac{\delta}{\delta\mathcal{J}_{\alpha}}-\sigma\,D_{\mu}g_{I}\,P^{I}\cdot\dfrac{\delta}{\delta
a_{\mu}}\bigg{)}\,.$

In the limit of vanishing field sources, the Ward identity associated with
$\widehat{\Delta}^{W}_{\sigma}$ takes the operator form

$[T\phantom{}^{\mu}\phantom{}_{\mu}]=B_{I}[\mathcal{O}^{I}]-\eta_{a}\partial^{2}[\mathcal{O}^{a}_{M}]\qquad\text{(FSCC)}\,,$
(2.35)

which can equivalently be obtained from Eq. (2.23) by using the Ward identity
associated with $G_{F}$ to eliminate the current operator.

The new trace energy-momentum identity is very important for our purposes
because _it guarantees the finiteness of_ $B^{I}\,$, which is not the case for
$\beta_{I}\,$. This observation relies on
$[T\phantom{}^{\mu}\phantom{}_{\mu}]$ being finite upon insertion into (non-
coinciding) correlation functions. Meanwhile, $\eta_{a}$ can be eliminated by
adding the curvature term $-\tfrac{1}{6}\eta_{a}R\mathcal{O}_{a}$ to the
action, corresponding to improving the energy-momentum tensor [22]. This
leaves just the $B_{I}$ term on the RHS of Eq. (2.35), and with
$[\mathcal{O}^{I}]$ finite, the same must hold for $B_{I}\,$.

##  3  Renormalization Ambiguity

In this section, we discuss the source and consistency of divergences in the
RG functions and how they can ultimately be removed.

###  3.1  Deriving RG functions from counterterms

We begin our discussion of the renormalization ambiguity with a rederivation
of the formulas relating the RG functions to the poles of the counterterms. In
general, we will be concerned with $d$-dimensional RG functions rather than
their limit in $d=4\,$, as this limit is not guaranteed to exist. With the
observation that infinities can seemingly appear in the $\beta$-functions, we
expand the $d$-dimensional $\beta$-function in powers of $\epsilon$:

$\beta_{I}=\dfrac{\mathop{}\\!\mathrm{d}g_{I}}{\mathop{}\\!\mathrm{d}t}=\sum_{n=\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}^{\infty}\dfrac{1}{\epsilon^{n}}\beta^{(n)}_{I}\,,$
(3.1)

where $\beta^{(0)}_{I}$ is the standard 4-dimensional $\beta$-function.777If
one were to assume that $\beta$ had higher order terms in $\epsilon\,$, one
would find, as can be seen from Eq. (3.2), that all higher order coefficients
disappear.

To get the $\beta$-function, one can apply a $t$-derivative to the bare
coupling parameterization (2.1) to obtain Eq. (2.15). Organizing the equation
by powers of $\epsilon$ then gives

$0=\left[\beta^{(\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1)}_{I}+k_{I}\,g_{I}\right]\epsilon+\sum_{n=0}^{\infty}\dfrac{1}{\epsilon^{n}}\\!\bigg{(}\beta^{(n)}_{I}+k_{I}\delta
g_{I}^{(n+1)}+\sum_{k=\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}^{n-1}\beta^{(k)}_{J}\partial^{J}\delta
g^{(n-k)}_{I}\bigg{)}\,.$ (3.2)

From the $\epsilon$ term, we conclude that
$\beta^{(\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1)}_{I}=-k_{I}g_{I}\,$.
Plugging this back into the equation, the term constant in $\epsilon$ yields
the key formula for the 4-dimensional $\beta$-functions:

$\beta^{(0)}_{I}=\left(\zeta-k_{I}\right)\delta g_{I}^{(1)}\,,$ (3.3)

where $\zeta=k_{I}g_{I}\partial^{I}$ is twice the loop-counting operator of a
2-point function (meaning that $\zeta-k_{I}$ is counts twice the loop-order of
$\delta g_{I}$).

From Eq. (3.2) one also obtains recursive relations for the divergent part of
the $\beta$-function, which can be determined order by order in the
$\epsilon$-poles:

$\beta^{(n)}_{I}=(\zeta-k_{I})\delta
g_{I}^{(n+1)}-\sum_{k=0}^{n-1}\beta^{(k)}_{J}\partial^{J}\delta
g_{I}^{(n-k)}\,,\qquad n\geq 1\,.$ (3.4)

The traditional view holds that all the poles, $\beta^{(n)}$ for $n\geq 1$,
must vanish and, thus, that Eq. (3.4) can be used directly as a consistency
check of the calculation. As we have suggested, this is not always the case.

The field anomalous dimension can be determined from the field-strength
renormalization factors, $Z$, and the $\beta$-function. In analogy with the
$\beta$-function, we allow for $\epsilon$-poles in the $d$-dimensional
anomalous dimension and expand

$\gamma=Z^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\dfrac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}Z=\sum_{n=0}^{\infty}\dfrac{1}{\epsilon^{n}}\gamma^{(n)}\,.$
(3.5)

Seeing as $Z$ depends on the renormalization scale only through $g$, we can
plug in the $\epsilon$ expansion of $Z\,$, $\beta\,$, and $\gamma$ to obtain

$0=\big{(}\gamma^{(0)}+\zeta
z^{(1)}\big{)}+\sum_{n=1}^{\infty}\dfrac{1}{\epsilon^{n}}\\!\bigg{(}\gamma^{(n)}+\zeta
z^{(n+1)}+\sum_{k=0}^{n-1}\left[z^{(n-k)}\gamma^{(k)}-\beta^{(k)}_{I}\partial^{I}z^{(n-k)}\right]\\!\bigg{)}\,.$
(3.6)

In particular, the standard $4$-dimensional anomalous dimension is given by

$\gamma^{(0)}=-\zeta z^{(1)}\,,$ (3.7)

whereas the poles can be determined recursively from

$\gamma^{(n)}=-\zeta
z^{(n+1)}+\sum_{k=0}^{n-1}\left[\beta^{(k)}_{I}\partial^{I}z^{(n-k)}-z^{(n-k)}\gamma^{(k)}\right],\qquad
n\geq 1\,.$ (3.8)

As with the $\beta$-function, the poles are not guaranteed to vanish.

The calculation of $\upsilon$ from the counterterms follows lines similar to
the other RG functions. Expanding the defining formula
$\eqref{eq:upsilon_def}$ in powers of $\epsilon$ yields

$\upsilon^{(0)}=-k_{I}g_{I}N^{(1)I}$ (3.9)

and

$\upsilon^{(n)}=-k_{I}g_{I}N^{(n+1)I}+\sum_{k=0}^{n-1}N^{(n-k)I}\left[\beta^{(k)}_{I}-(\upsilon^{(k)}\,g)_{I}\right],\qquad
n\geq 1\,,$ (3.10)

which allows for recursive evaluation of the $\upsilon$ poles. One can derive
a formula for $\rho_{I}$ in a similar manner, but we will not need it for our
discussions.

###  3.2  Finiteness of the renormalization group equation

Having derived relations between RG functions and counterterm poles, it is
illustrative to examine how $\epsilon$-poles can arise in the field anomalous
dimension in the first place. Herren _et al._ [13] observed that the Hermitian
part of $\gamma$ is finite up to 3-loop orders in the SM; however, this is
neither true for the anti-Hermitian part of $\gamma$ nor for the
$\beta$-functions [9, 13].

The typical approach to determining the field-strength renormalization factors
prescribes choosing $Z^{\dagger}=Z\,$, which, obviously, yields a Hermitian
$\gamma^{(0)}\,$. In this case, it follows from Eq. (3.8) that the anti-
Hermitian part of $\gamma^{(1)}$ satisfies

$\gamma^{(1)}-\gamma^{(1)\dagger}=\big{[}z^{(1)},\,\zeta z^{(1)}\big{]}\,.$
(3.11)

Starting at the 3-loop order, the commutator is non-vanishing if
$\big{[}z^{(1)}_{1},\,z^{(1)}_{2}\big{]}\neq 0\,$, where $z_{\ell}^{(n)}$ is
used to denote the $\ell$-loop contribution to $z^{(n)}$. In generic theories,
or indeed the SM, this commutator will not vanish.

Despite the presence of poles in the RG functions being rather unsettling, we
should not dismiss the result out of hand: the RG functions are unphysical,
not observables. Their impact on physical quantities is contained in the
scaling behavior of Greens functions via the RG. Thus, it is the Callan-
Symanzik equation (2.26) that must be well-behaved, allowing for a smooth
4-dimensional limit. Expanding the RG functions in powers of $\epsilon$ gives

$\bigg{(}\dfrac{\partial}{\partial
t}+\left(\epsilon\beta_{I}^{(\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1)}+\beta_{I}^{(0)}\right)\\!\partial^{I}+\int\mathop{}\\!\mathrm{d}^{d}x\,\mathcal{J}_{\beta}\gamma\phantom{}^{(0)\beta}\phantom{}_{\alpha}\dfrac{\delta}{\delta\mathcal{J}_{\alpha}}\bigg{)}\mathcal{W}=-\sum_{n=1}^{\infty}\dfrac{1}{\epsilon^{n}}\bigg{(}\beta^{(n)}_{I}\partial^{I}+\int\mathop{}\\!\mathrm{d}^{d}x\,\mathcal{J}_{\beta}\gamma\phantom{}^{(n)\beta}\phantom{}_{\alpha}\dfrac{\delta}{\delta\mathcal{J}_{\alpha}}\bigg{)}\mathcal{W}\,.$
(3.12)

Clearly, the LHS is finite, but any non-vanishing poles would seem to leave
the RHS divergent. How, then, can we reconcile the presence of divergences in
the RG functions with finite evolution of the Green’s functions? The answer
lies in the flavor symmetry.

The flavor symmetry gives rise to a Ward identity, which can be derived by
applying the generator $\Delta^{\\!F}_{\omega}$ of Eq. (2.27) to the vacuum
functional. In the FSCC limit, the anomaly vanishes and the identity reads

$0=-\Delta^{\\!F}_{\omega}\mathcal{W}=\bigg{(}\\!(\omega\,g)_{I}\partial^{I}-\\!\int\mathop{}\\!\mathrm{d}^{d}x\,\mathcal{J}_{\beta}\omega\phantom{}^{\beta}\phantom{}_{\alpha}\dfrac{\delta}{\delta\mathcal{J}_{\alpha}}\bigg{)}\mathcal{W}\,.$
(3.13)

The flavor Ward identity and $\gamma^{(n)}$ being found to be anti-Hermitian
in the SM lead us to posit the theorem of RG-finiteness:

The divergent part of _any_ set of MS/$\overline{\text{MS}}$ RG functions
$(\beta_{I},\,\gamma)$ satisfy
$\gamma^{(n)}\in\mathfrak{g}_{F}\quad\mathrm{and}\quad\beta^{(n)}_{I}=-\big{(}\gamma^{(n)}\,g\big{)}_{I}\,,\qquad
n\geq 1\,.$ This property of the RG functions is referred to as RG-finiteness.

Any _RG-finite_ pair $(\beta_{I},\,\gamma)$ causes the RHS of Eq. (3.12) to
vanish due to the flavor Ward identity. Divergences of this kind in the RG
functions are, thus, consistent with finite running of the Green’s functions.
The proof that all RG functions are RG finite relies on the ambiguity of RG
functions and is presented in Section. 3.4.

In theories where all couplings are invariant under $G_{F}$, it is impossible
to construct a non-vanishing $\gamma^{(n)}\in\mathfrak{g}_{F}\,$. In such
cases, RG-finiteness implies individual finiteness of all RG functions. This
explains why higher-order calculations of RG functions in e.g. gauge-fermion
theories through five loops [31, 32, 33] or $\phi^{4}$ theory through six
loops [34] has not encountered divergences of this kind. Though it is
consistent for RG-finite RG functions to feature divergences, we concede that
for practical (and aesthetic) reasons, it is preferable to avoid any such.
Fortunately, there is an ambiguity in choosing the counterterms that allows
for choosing finite RG functions.

###  3.3  Transforming the bare sources

Other authors have made use of an ambiguity in the renormalization as a way to
eliminate the divergent parts of the RG functions. By tweaking the field-
strength renormalization constants, it has, so far, always been possible to do
so. The renormalization ambiguity, thus, seems to be linked to the divergences
of the RG functions.

Fortin _et al._ [28] pointed out that the ambiguity in choosing $Z$ is
equivalent to performing a divergent flavor rotation; however, they only
considered the impact on the finite part of the RG functions. To formalize the
idea, we introduce the notion of having the flavor transformation work on the
bare sources of the theory, by acting with the generator

$\Delta^{\\!F_{0}}_{\omega}=\int\mathop{}\\!\mathrm{d}^{d}x\left(D_{\mu}\omega\cdot\dfrac{\delta}{\delta
a_{0,\mu}}-(\omega\,g_{0})_{I}\dfrac{\delta}{\delta
g_{0,I}}-(\omega\,\mathcal{J}_{0})_{\alpha}\dfrac{\delta}{\delta\mathcal{J}_{0,\alpha}}\right),\qquad\omega\in\mathfrak{g}_{F}\,,$
(3.14)

rather than the generator (2.27) for rotations of the renormalized sources.
$\Delta^{\\!F_{0}}_{\omega}$ is the generator of the classical flavor symmetry
of the bare action and leaves $\mathcal{W}_{0}$ invariant down to the chiral
anomaly, which is irrelevant for the perturbative computation of the field-
dependent counterterms [30].888On the other hand, the chiral anomaly from the
application of $\Delta^{\\!F_{0}}_{\omega}$ will effectively generate a change
to the field-independent counterterms, $S_{\mathrm{ct}}$, essential to the the
perturbative $A$-function. We will not explore this effect here. Whenever
$\omega$ is finite, the action of $\Delta^{\\!F_{0}}_{\omega}$ and
$\Delta^{\\!F}_{\omega}$ coincide, as the renormalization factors and
counterterms are covariant under the finite flavor transformation. It is when
$\omega$ is taken to have an $\epsilon$ dependence that
$\Delta^{\\!F_{0}}_{\omega}$ becomes truly interesting, as it can generate
changes in the counterterms while leaving the renormalized sources unchanged.

Here we consider flavor rotations with parameters $u(g)\in\mathfrak{g}_{F}\,$
that are series of poles in $\epsilon\,$. Such rotations leave the finite part
of the bare sources, i.e., the renormalized sources, unchanged while changing
the counterterms, effectively generating a new set of renormalization
constants. The same holds for finite—in the sense of non-
infinitesimal—rotations given by

$U=\exp\\!\bigg{[}-\sum_{n=1}^{\infty}\dfrac{1}{\epsilon^{n}}u^{(n)}(g)\bigg{]}\,,\qquad
u^{(n)}\in\mathfrak{g}_{F}\,.$ (3.15)

The action of $U$ on the sources follows from Eq. (2.2), and the transformed
sources are

$\tilde{g}_{0,I}=(U\,g_{0})_{I}\,,\qquad\tilde{\mathcal{J}}_{0,\alpha}=\mathcal{J}_{0,\beta}U\phantom{}^{\dagger\beta}\phantom{}_{\alpha}\,,\quad\mathrm{and}\quad\tilde{a}_{0,\mu}=Ua_{0,\mu}U^{\dagger}+U\partial_{\mu}U^{\dagger}\,,$
(3.16)

after the rotation, while physics remains unchanged:

$\mathcal{W}[\gamma,\,g,\,\mathcal{J},\,a]=\mathcal{W}_{0}[\gamma,\,g_{0},\,\mathcal{J}_{0},\,a_{0}]=\mathcal{W}_{0}[\gamma,\,\tilde{g}_{0},\,\tilde{\mathcal{J}}_{0},\,\tilde{a}_{0}]\,,$
(3.17)

down to the chiral anomaly. We will take $U$ to be a flavor-covariant function
of $g_{I}$ so as not to spoil the flavor symmetry under finite $G_{F}$
transformations. Since $\mathcal{J}_{\alpha}\,$, the finite part of the bare
source, remains unchanged, the effect of the transformation is to multiply the
field-strength renormalization by a unitary factor:

$\tilde{Z}\phantom{}^{\alpha}\phantom{}_{\beta}=U\phantom{}^{\alpha}\phantom{}_{\gamma}Z\phantom{}^{\gamma}\phantom{}_{\beta}\,.$
(3.18)

This is why the ambiguity is typically identified with a unitary ambiguity
related to taking the square root of $Z^{\dagger}Z$ [20, 9, 13].

It is by no means an accident that this ambiguity enters at the 3-loop order
in the SM. The $u^{(n)}(g)$ are anti-Hermitian 2-point tensor
structures—flavor-covariant polynomials in the couplings. 1PI tensor
structures of this type are only possible beginning at the 3-loop order in
completely general gauge-Yukawa theories [27]. Furthermore, the 1-loop 2-point
tensor-structures all commute, ensuring that the leading contribution to
$u^{(n)}(g)$ is 3-loop order. Hence, all complications from the ambiguity can
be ignored in 2-loop computations. The possibility of choosing a nontrivial
contribution to $u^{(n)}$ already at the 2-loop order in the SM, as pointed
out in Refs. [9, 13], seems to ignore flavor-covariance and was extraneous to
removing any divergences in $\gamma\,$.

The renormalization ambiguity is directly relevant to the parameterization of
general gauge-Yukawa theories. If one does not fix the ambiguity with the
Hermitian choice $Z=Z^{\dagger}$, it is generally possible to add anti-
Hermitian 1PR tensor structures to $z^{(1)}$ with the $U$ rotation. This
translates directly into 1PR tensors in the finite part of $\gamma$ and
$\beta_{I}$. Similarly, examining the structure of $N^{I}$ required to cancel
the divergences of the Green’s functions outlined in Sec. 4.4, one will find
that also the simple pole of $N^{I}$ can only contain 1PR tensor structures if
$z^{(1)}$ is not Hermitian. We conclude that the parameterization of the
finite parts of the RG functions presented in Ref. [27] are valid only when
the ambiguity is fixed such that $z^{(1)}$ is Hermitian.

There is some room for differing interpretations of the renormalization
ambiguity. Since the $U$ rotation leaves the renormalized vacuum functional
and sources unchanged, it is our point of view that it parameterizes a class
of equally valid counterterms. It is not really a renormalization scheme
change, since the renormalized couplings remain unchanged under the
transformation. The only consequence of the transformation that we are aware
of is a change in the RG functions.

###  3.4  Ambiguity in the RG functions

As we have seen, the RG functions are determined by the poles of the
counterterms. Thus, the renormalization ambiguity, in the form of the flavor
rotation $U\,$, induces an ambiguity in the RG functions. The full derivation
of the transformation of the RG functions under a $U$ rotation can be found in
App. A, whereas here we just refer the result:

$\displaystyle\Delta\gamma$
$\displaystyle\equiv\tilde{\gamma}-\gamma=-\beta_{I}\,U\partial^{I}U^{\dagger}\,,$
(3.19) $\displaystyle\Delta\beta_{I}$
$\displaystyle\equiv\tilde{\beta}_{I}-\beta_{I}=-(\Delta\gamma\,g)_{I}\,,$
(3.20) $\displaystyle\Delta\upsilon$
$\displaystyle\equiv\tilde{\upsilon}-\upsilon=-\Delta\gamma\,,$ (3.21)

where $\Delta\gamma\in\mathfrak{g}_{F}\,$. For completeness, we have also
determined that

$\Delta\rho^{I}\equiv\tilde{\rho}^{I}-\rho^{I}=\partial^{I}\Delta\gamma\,.$
(3.22)

At finite order in $\epsilon\,$, Eq. (3.19) gives

$\Delta\gamma^{(0)}=\zeta u^{(1)}\,.$ (3.23)

with the parameterization (3.15) for the rotation. Similarly, all the poles,
$\Delta\gamma^{(n)}$, contain a term $\zeta u^{(n+1)}$ from the
$\beta^{(\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1)}$ contribution, while the
remaining terms only depend on the poles $u^{(k)},k\leq n\,$. $\zeta$ is the
loop-counting operator, so by carefully choosing the terms $u^{(n+1)}$, one
can engineer any covariant $\Delta\gamma\in\mathfrak{g}_{F}\,$. Hence, as long
as the original RG functions were RG finite, it is always possible to
transform the counterterms such as to make $\tilde{\gamma}$ finite, as was
found to be the case in the SM [9, 13].

From the transformation of the $\beta$-function (3.20), it further follows
that _RG-finite RG functions will remain RG finite under the renormalization
transformation_. Conversely, RG functions that were not RG finite cannot be
made so by applying the transformation. In particular, when $\tilde{\gamma}$
is made finite starting from RG-finite RG functions, then $\tilde{\beta}_{I}$
will be too.

Turning to the $\upsilon$ transformation rule (3.21), we observe that the
change in $\upsilon$ is opposite to the change in $\gamma\,$. Accordingly, the
unambiguous, or flavor-improved, RG functions $B_{I}$ and $\Gamma$ of Eq.
(2.33) are invariant not only under a redefinition of the Weyl transformation
but also under the transformation of the renormalization constants. In fact,
from comparing the ambiguity in the definition of the Weyl transformation
(2.31) to the change of the RG functions under the counterterm transformation,
it follows that it is possible to change the counterterms in such a manner as
to make the RG functions agree with any one of the possible Weyl
transformations. The two ambiguities are closely related although not quite
the same: one is an ambiguity in defining the Weyl symmetry, while the other
is an ambiguity in choosing counterterms.

The last observation we wish to make regarding the RG transformation is that
it is always possible to choose the counterterms such that
$\tilde{\upsilon}=0\,$. This is always the case as
$\upsilon\in\mathfrak{g}_{F}\,$, and we can, therefore, find a $U$ such that
$\Delta\gamma=\upsilon\,$. With this choice, the flavor-improved RG functions
coincide with the regular ones; $\tilde{\beta}_{I}=B_{I}$ and
$\tilde{\gamma}=\Gamma\,$.

We are now a position to prove RG-finiteness of all RG functions. Using the
renormalization ambiguity, we choose RG functions $(B_{I},\,\gamma)$ (for
instance, it is always possible to choose
$(\beta_{I},\,\gamma)=(B_{I},\,\Gamma)$), ensuring finiteness of the
$\beta$-function according to the discussion around Eq. (2.35). If $\gamma$
can be made finite up to arbitrarily high loop order without changing the
associated $\beta_{I}=B_{I}\,$, then the RG functions can be made separately
finite to any loop order, establishing RG-finiteness of the RG functions.

We assume for contradiction that there is a smallest loop order $\ell>0$ such
that no $(B_{I},\,\gamma)$ is finite up to order $\ell\,$. Then there exists a
$\gamma$ such that its leading contribution to the divergent part
$\gamma_{\ell}$ is $\ell$-loop order. Finiteness of the Callan–Symanzik
equation (3.12) reduces to

$K_{\epsilon}\\!\int\mathop{}\\!\mathrm{d}^{d}x\,\mathcal{J}_{\beta}\gamma_{\scriptscriptstyle\mathrm{div}}\phantom{}^{\beta}\phantom{}{}_{\alpha}\dfrac{\delta}{\delta\mathcal{J}_{\alpha}}\mathcal{W}=0\,,$
(3.24)

where $\gamma_{\scriptscriptstyle\mathrm{div}}=K_{\epsilon}\gamma\,$.
$K_{\epsilon}$ is an operator that extracts the divergent piece of its
operand. The leading contribution to the 2-point functions in Eq. (3.24) is
given between the free-field propagators and $\gamma_{\ell}\,$, providing the
constraint

$\gamma_{\ell}=-\gamma_{\ell}^{\dagger}\implies\gamma_{\ell}\in\mathfrak{g}_{F}\,.$
(3.25)

The leading contribution to several of the $n$-point functions in Eq. (3.24)
are given by the couplings $g_{I}$ dressed with free propagators and
$\gamma_{\ell}\,$. Thus,999While we have no proof, it seems to us that it
might well be impossible to construct a flavor-covariant
$\gamma_{\ell}(g)\in\mathfrak{g}_{F}$ satisfying $(\gamma_{\ell}\,g)_{I}=0$ in
fully general gauge-Yukawa theories. In that eventuality, $\gamma=\Gamma$ must
be finite along with $B_{I}\,$.

$(\gamma_{\ell}\,g)_{I}=0\,.$ (3.26)

It follows that the renormalization ambiguity can be utilized to construct a
new set of RG functions $(B_{I},\,\tilde{\gamma})$, choosing
$\Delta\gamma=\gamma_{\ell}\,$, which leaves $\tilde{\beta}_{I}=B_{I}$ by Eqs.
(3.20) and (3.26). Clearly, $\tilde{\gamma}$ is finite up to and including the
$\ell$-loop order, for contradiction. Accordingly, all RG functions are RG
finite.

##  4  Gauge-Yukawa Theories

After the somewhat abstract discussions in the previous sections, we exemplify
the discussions with the SM as a concrete example of a gauge-Yukawa theory and
show by explicit calculation that the RG functions are indeed RG finite, as we
demonstrated that they must be.

###  4.1  Generic gauge-Yukawa theories

We consider the most general class of renormalizable QFTs in four dimensions:
gauge-Yukawa theories. We leave out all relevant couplings, as they do not
impact any of the RG functions discussed in the present paper in the
$\overline{\text{MS}}$ scheme.101010Based on the arguments in the previous
section, we expect the RG functions of the relevant couplings to suffer from
similar ambiguities. The Lagrangian of completely generic gauge-Yukawa
theories can be put into the form

$\mathscr{L}=-\tfrac{1}{4}a^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}_{0,AB}F^{A}_{\mu\nu}F^{B\mu\nu}+\tfrac{1}{2}(D_{\mu}\phi)_{a}(D^{\mu}\phi)_{a}+i\psi_{i}^{\dagger}\bar{\sigma}^{\mu}(D_{\mu}\psi)^{i}+J_{a}Z_{ab}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\phi_{b}+\big{(}\eta_{i}Z^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\phantom{}{}^{i}\phantom{}_{j}\psi^{j}\;+\;\mathrm{H.c.}\;\big{)}\\\
-\tfrac{1}{2}\phi_{a}\big{(}y_{0,aij}\psi^{i}\psi^{j}\;+\;\mathrm{H.c.}\;\big{)}-\tfrac{1}{24}\lambda_{0,abcd}\phi_{a}\phi_{b}\phi_{c}\phi_{d}+\mathscr{L}_{\mathrm{ghost}}+\mathscr{L}_{\text{gauge-
fixing}}$ (4.1)

by collecting all scalars in a scalar multiplet $\phi_{a}$ and writing the
fermions as a Weyl spinor multiplet $\psi^{i}$. The gauge fields of the gauge
group $G$ (a product group with any number of simple or Abelian factors) are
packed into a single multiplet $A^{A}_{\mu}$ with collected index $A$ belong
to the adjoint representation of $G\,$. The bare gauge couplings $a_{0,AB}$
are put on the gauge kinetic term to emphasize the gauge couplings having the
structure of a two-index matrix. For the full details of the construction, we
refer the reader to Ref. [27]. $y_{0,aij}$ and $\lambda_{0,abcd}$ are the bare
Yukawa and quartic couplings, respectively, while
$\mathcal{J}_{\alpha}=(\eta_{i},\,J_{a})$ is the collection of field sources
with field-strength renormalization factors $Z_{ab}$ and
$Z\phantom{}^{i}\phantom{}_{j}\,$.

To connect the gauge-Yukawa theory to the notation employed elsewhere in the
paper, the marginal couplings and operators are identified with

$g_{0,I}=\big{\\{}a_{0,AB},\,y_{0,aij},\,\lambda_{0,abcd}\big{\\}}\quad\mathrm{and}\quad\mathcal{O}^{I}=\big{\\{}-\tfrac{1}{4}a_{0,BC}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}2}F^{A}_{\mu\nu}F^{C\mu\nu},\,-\tfrac{1}{2}\phi_{a}\psi^{i}\psi^{j},\,-\tfrac{1}{24}\phi_{a}\phi_{b}\phi_{c}\phi_{d}\big{\\}}\,.$
(4.2)

In $d$-dimensions, the mass dimensions of the bare couplings are given by
$k_{I}=\big{\\{}2,\,1,\,2\big{\\}}\,$. For the matter fields and their
normalization factors, they are

$\Phi^{\alpha}=\big{\\{}\psi^{i},\,\phi_{a}\big{\\}}\quad\mathrm{and}\quad
Z\phantom{}^{\alpha}\phantom{}_{\beta}=\big{\\{}Z\phantom{}^{i}\phantom{}_{j},\,Z_{ab}\big{\\}}\,,$
(4.3)

where $Z\phantom{}^{\alpha}\phantom{}_{\beta}$ is understood to be a block
diagonal matrix.

The flavor symmetry $G_{F}$ is the largest symmetry group of the kinetic terms
compatible with the gauge symmetry $G\,$. In a theory with $N_{f}$ Weyl
fermions and $N_{s}$ real scalars, it satisfies $G\times
G_{F}\subseteq\mathrm{U}(N_{f})\times\mathrm{O}(N_{s})$, ensuring that
$\big{[}A_{\mu},\,a_{\nu}\big{]}=0\,$.111111Assuming, of course, that all
product gauge groups have some matter charged under them. Were this not the
case, the corresponding gauge fields would completely decouple from the rest
of the theory. The transformation properties of the fields under a group
element $h\in G_{F}\,$, $\phi_{a}\to h_{ab}\phi_{b}$ and $\psi^{i}\to
h\phantom{}^{i}\phantom{}_{j}\psi^{j}\,$, determine the transformation of the
couplings and renormalization factors. In particular, the Yukawa and quartic
couplings transform as

$y_{aij}\to
h_{ab}y_{bk\ell}(h^{\dagger})\phantom{}^{k}\phantom{}_{i}(h^{\dagger})\phantom{}^{\ell}\phantom{}_{j}\quad\mathrm{and}\quad\lambda_{abcd}\to
h_{ae}h_{bf}h_{cg}h_{dh}\lambda_{efgh}\,.$ (4.4)

Due to the scalar fields being in a real representation, the elements satisfy
$h_{ab}=h^{\dagger}_{ba}\,$. The quantum gauge fields, on the other hand, do
not transform under the flavor symmetry, leaving the coupling $a_{AB}$
invariant.

The action of an element $\omega\in\mathfrak{g}_{F}$ can be ascertained by
considering the infinitesimal transformation $h=1-\omega\,$, in line with the
anti-Hermitian representation of the Lie algebra employed in Eq. (2.2). The
action is defined by the field transformation such that

$(\omega\,\phi)_{a}=\omega_{ab}\phi_{b}\quad\mathrm{and}\quad(\omega\,\psi)^{i}=\omega\phantom{}^{i}\phantom{}_{j}\psi^{j}\,.$
(4.5)

The coupling indices, meanwhile, transform in the conjugate representations
(reality of the scalar field representation ensures that
$\omega_{ab}=-\omega_{ba}$):

$\begin{split}(\omega\,y)_{aij}&=\omega_{ab}y_{bij}-y_{akj}\omega\phantom{}^{k}\phantom{}_{i}-y_{aik}\omega\phantom{}^{k}\phantom{}_{j}\,,\\\
(\omega\,\lambda)_{abcd}&=\omega_{ae}\lambda_{ebcd}+\omega_{be}\lambda_{aecd}+\omega_{ce}\lambda_{abed}+\omega_{de}\lambda_{abce}\,.\end{split}$
(4.6)

Finally, the RG-finiteness condition reads

$\begin{split}\beta^{(n)}_{aij}&=-\gamma^{(n)}_{ab}y_{bij}+y_{akj}\gamma\phantom{}^{(n)k}\phantom{}_{i}+y_{aik}\gamma\phantom{}^{(n)k}\phantom{}_{j}\,,\\\
\beta^{(n)}_{abcd}&=-\gamma^{(n)}_{ae}\lambda_{ebcd}-\gamma^{(n)}_{be}\lambda_{aecd}-\gamma^{(n)}_{ce}\lambda_{abed}-\gamma^{(n)}_{de}\lambda_{abce}\end{split}$
(4.7)

in gauge-Yukawa theories.

###  4.2  Flavor symmetry of the SM

The SM admits an
$\mathrm{SU}(3)_{q}\times\mathrm{SU}(3)_{u}\times\mathrm{SU}(3)_{d}\times\mathrm{SU}(3)_{\ell}\times\mathrm{SU}(3)_{e}\times\mathrm{U}(1)^{5}$
flavor symmetry of the kinetic term,121212One of the six $\mathrm{U}(1)$
phases of the fields is identified with $\mathrm{U}(1)_{Y}$ under which only
the Yukawa couplings,

$\mathscr{L}\supset-
y_{u}\phantom{}^{i}\phantom{}{}_{j}\overline{Q}_{\mathrm{L},i}\tilde{H}U^{j}_{\mathrm{R}}-y_{d}\phantom{}^{i}\phantom{}{}_{j}\overline{Q}_{\mathrm{L},i}HD^{j}_{\mathrm{R}}-y_{e}\phantom{}^{i}\phantom{}{}_{j}\overline{L}_{\mathrm{L},i}HE^{j}_{\mathrm{R}}\;+\;\mathrm{H.c.}\;\,,$
(4.8)

transform non-trivially. Here, $\overline{Q}_{\mathrm{L}}$ and
$\overline{L}_{\mathrm{L}}$ are the left-handed quark and lepton doublet,
whereas $U_{\mathrm{R}}$, $D_{\mathrm{R}}$ and $E_{\mathrm{R}}$ are the right-
handed up-type quark, down-type quark and lepton singlets. $H$ and $\tilde{H}$
denote the Higgs doublet and its charge conjugate.

Keeping with the non-Hermitian representation for $\mathfrak{g}_{F}\,$, the
infinitesimal transformation of the fields and Yukawas under said symmetry is
given by

$\displaystyle\delta_{\omega}Q^{i}$
$\displaystyle=-\omega_{q}\phantom{}^{i}\phantom{}{}_{j}Q^{j}\,,$
$\displaystyle\delta_{\omega}U^{i}$
$\displaystyle=-\omega_{u}\phantom{}^{i}\phantom{}{}_{j}U^{j}\,,$
$\displaystyle\delta_{\omega}D^{i}$
$\displaystyle=-\omega_{d}\phantom{}^{i}\phantom{}{}_{j}D^{j}\,,$
$\displaystyle\delta_{\omega}L^{i}$
$\displaystyle=-\omega_{\ell}\phantom{}^{i}\phantom{}{}_{j}L^{j}\,,$
$\displaystyle\delta_{\omega}E^{i}$
$\displaystyle=-\omega_{e}\phantom{}^{i}\phantom{}{}_{j}E^{j}\,,$
$\displaystyle\delta_{\omega}H$ $\displaystyle=-\omega_{h}H$ (4.9)

($\omega_{h}$ is a purely imaginary number) and

$\displaystyle\delta_{\omega}y_{u}\phantom{}^{i}\phantom{}{}_{j}$
$\displaystyle=-(\omega\,y_{u})\phantom{}^{i}\phantom{}_{j}=-\omega_{q}\phantom{}^{i}\phantom{}{}_{k}y_{u}\phantom{}^{k}\phantom{}{}_{j}+y_{u}\phantom{}^{i}\phantom{}{}_{k}\omega_{u}\phantom{}^{k}\phantom{}{}_{j}-\omega_{h}y_{u}\phantom{}^{i}\phantom{}{}_{j}\,,$
(4.10) $\displaystyle\delta_{\omega}y_{d}\phantom{}^{i}\phantom{}{}_{j}$
$\displaystyle=-(\omega\,y_{d})\phantom{}^{i}\phantom{}_{j}=-\omega_{q}\phantom{}^{i}\phantom{}{}_{k}y_{d}\phantom{}^{k}\phantom{}{}_{j}+y_{d}\phantom{}^{i}\phantom{}{}_{k}\omega_{d}\phantom{}^{k}\phantom{}{}_{j}+\omega_{h}y_{d}\phantom{}^{i}\phantom{}{}_{j}\,,$
(4.11) $\displaystyle\delta_{\omega}y_{e}\phantom{}^{i}\phantom{}{}_{j}$
$\displaystyle=-(\omega\,y_{e})\phantom{}^{i}\phantom{}_{j}=-\omega_{\ell}\phantom{}^{i}\phantom{}{}_{k}y_{e}\phantom{}^{k}\phantom{}{}_{j}+y_{e}\phantom{}^{i}\phantom{}{}_{k}\omega_{e}\phantom{}^{k}\phantom{}{}_{j}+\omega_{h}y_{e}\phantom{}^{i}\phantom{}{}_{j}\,,$
(4.12)

for $\omega\in\mathfrak{g}_{F}\,$. The subscript on $\omega$ or any other
elements of $\mathfrak{g}_{F}\,$, e.g., the generators $t^{P}$, are used to
denote the representation (effectively fundamental representations of the
$\mathrm{SU}(3)$ product groups). All other couplings and fields are in the
trivial representation of $G_{F}\,$, i.e., they do not transform.

Introducing the background gauge field
$a_{\mu}=a^{P}_{\mu}t^{P}\in\mathfrak{g}_{F}\,$, the kinetic term of the SM is

$\mathscr{L}_{\mathrm{kin}}=i\overline{Q}_{\mathrm{L}}\gamma^{\mu}(D_{\mu}+a_{\mu})Q_{\mathrm{L}}+i\overline{U}_{\mathrm{R}}\gamma^{\mu}(D_{\mu}+a_{\mu})U_{\mathrm{R}}+i\overline{D}_{\mathrm{R}}\gamma^{\mu}(D_{\mu}+a_{\mu})D_{\mathrm{R}}\\\
+i\overline{L}_{\mathrm{L}}\gamma^{\mu}(D_{\mu}+a_{\mu})L_{\mathrm{L}}+i\overline{E}_{\mathrm{R}}\gamma^{\mu}(D_{\mu}+a_{\mu})E_{\mathrm{R}}+|(D_{\mu}+a_{\mu})H|^{2}\,,$
(4.13)

where $D_{\mu}$ is the ordinary gauge-covariant derivative. The flavor current
is determined by taking an $a_{\mu}^{P}$ derivative, giving

$J_{F}^{P\mu}=i\overline{Q}_{\mathrm{L}}\gamma^{\mu}t^{P}_{q}Q_{\mathrm{L}}+i\overline{U}_{\mathrm{L}}\gamma^{\mu}t^{P}_{u}U_{\mathrm{R}}+i\overline{D}_{\mathrm{R}}\gamma^{\mu}t^{P}_{d}D_{\mathrm{R}}+i\overline{L}_{\mathrm{L}}\gamma^{\mu}t^{P}_{\ell}L_{\mathrm{L}}+i\overline{E}_{\mathrm{R}}\gamma^{\mu}t^{P}_{e}E_{\mathrm{R}}-t_{h}^{P}H^{\dagger}\overleftrightarrow{D}^{\mu}H\,.$
(4.14)

The flavor indices have been kept implicit here to avoid unnecessary clutter.
Once again, $t^{P}_{h}$ are just charges, that is, imaginary numbers.

###  4.3  Divergences in the SM RG functions

As mentioned several times already, the poles of the SM anomalous dimensions
have previously been subject to some consideration [9, 13], though how the
poles of the anomalous dimension reflect in the $\beta$-functions has not
previously been considered in detail. Using the 3-loop counterterms
$Z^{\dagger}Z$ and $\delta g_{I}$ provided by Ref. [13], we derive the
divergent parts of the RG functions and verify that they are indeed RG finite.

Since renormalization only fixes the combination $Z^{\dagger}Z\,$, we fix the
renormalization ambiguity with the Hermitian choice $Z=Z^{\dagger}\,$. As
pointed out in Section 3.2, this choice immediately removes any anti-Hermitian
contribution to $\gamma^{(0)}$ for all the fields:131313The fields are in one-
to-one correspondence with the representation of $G_{F}$. We use the same
indices for the field anomalous dimensions as for the representations despite
the finite part, $\gamma_{f}^{(0)}$, not being the representation of an
element of $\mathfrak{g}_{F}$; however, it does make the notation more
intuitive for the divergent.

$\gamma^{(0)}_{f}-\gamma^{(0)\dagger}_{f}=0\,,\qquad
f\in\big{\\{}q,u,d,\ell,e\big{\\}}\,.$ (4.15)

With the choice of Hermitian $Z$, we find that $\gamma^{(n)}_{h}=0$ for $n\geq
1$ using the counterterms of Ref. [13]. Since $H$ does not carry a flavor
index, $\big{[}z^{(j)}_{h},\,\gamma^{(k)}_{h}\big{]}=0$ for all $j,k$ and the
anti-Hermitian part of $\gamma_{h}^{(n)}$ vanishes by (3.11) along with
generalizations to higher poles. For the RG functions to be RG finite
$\gamma_{h}^{(n)}$ must therefore vanish in agreement with explicit
computation.

Starting at the 3-loop order, we find nontrivial contributions to the quark
anomalous dimensions given by

$\displaystyle(4\pi)^{6}\gamma^{(1)}_{q}$
$\displaystyle=\dfrac{a_{1}}{192}\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}+\dfrac{1}{64}\big{[}y_{u}y_{u}^{\dagger}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}+\dfrac{1}{64}\big{[}y_{d}y_{d}^{\dagger}y_{d}y_{d}^{\dagger},\,y_{u}y_{u}^{\dagger}\big{]}\,,$
(4.16) $\displaystyle(4\pi)^{6}\gamma^{(1)}_{u}$
$\displaystyle=\dfrac{1}{32}y_{u}^{\dagger}\big{[}y_{d}y_{d}^{\dagger},\,y_{u}y_{u}^{\dagger}\big{]}y_{u}\,,$
(4.17) $\displaystyle(4\pi)^{6}\gamma^{(1)}_{d}$
$\displaystyle=\dfrac{1}{32}y_{d}^{\dagger}\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{d}\,,$
(4.18)

where $a_{1}=g_{1}^{2}$ is the (squared) hypercharge gauge coupling. As
expected, the poles are anti-Hermitian in agreement with the hypothesis
$\gamma^{(1)}\in\mathfrak{g}_{F}\,$. The first poles of the quark Yukawa
$\beta$-functions are also extracted from the counterterms giving

$\displaystyle(4\pi)^{6}\beta_{y_{u}}^{(1)}=\,$
$\displaystyle-\dfrac{a_{1}}{192}\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{u}-\dfrac{1}{64}\big{[}y_{u}y_{u}^{\dagger}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{u}$
$\displaystyle-\dfrac{1}{64}\big{[}y_{d}y_{d}^{\dagger}y_{d}y_{d}^{\dagger},\,y_{u}y_{u}^{\dagger}\big{]}y_{u}+\dfrac{1}{32}y_{u}y_{u}^{\dagger}\big{[}y_{d}y_{d}^{\dagger},\,y_{u}y_{u}^{\dagger}\big{]}y_{u}\,,$
(4.19) $\displaystyle(4\pi)^{6}\beta_{y_{d}}^{(1)}=\,$
$\displaystyle-\dfrac{a_{1}}{192}\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{d}-\dfrac{1}{64}\big{[}y_{u}y_{u}^{\dagger}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{d}$
$\displaystyle-\dfrac{1}{64}\big{[}y_{d}y_{d}^{\dagger}y_{d}y_{d}^{\dagger},\,y_{u}y_{u}^{\dagger}\big{]}y_{d}+\dfrac{1}{32}y_{d}y_{d}^{\dagger}\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{d}\,.$
(4.20)

The similarities between $\gamma^{(1)}_{f}$ and $\beta_{y}^{(1)}$ are quite
evident when organized like this, and it is a simple matter to verify that
$\beta^{(1)}_{y}=-(\gamma^{(1)}\,y)$ in agreement with RG-finiteness. The
$\epsilon^{2}$-poles of the RG functions are found to be

$\displaystyle(4\pi)^{6}\gamma^{(2)}_{q}$
$\displaystyle=-\dfrac{a_{1}}{32}\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}+\dfrac{3}{32}\big{[}y_{u}y_{u}^{\dagger}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}+\dfrac{3}{32}\big{[}y_{d}y_{d}^{\dagger}y_{d}y_{d}^{\dagger},\,y_{u}y_{u}^{\dagger}\big{]}\,,$
(4.21) $\displaystyle(4\pi)^{6}\gamma^{(2)}_{u}$
$\displaystyle=-\dfrac{3}{16}y_{u}^{\dagger}\big{[}y_{d}y_{d}^{\dagger},\,y_{u}y_{u}^{\dagger}\big{]}y_{u}\,,$
(4.22) $\displaystyle(4\pi)^{6}\gamma^{(2)}_{d}$
$\displaystyle=-\dfrac{3}{16}y_{d}^{\dagger}\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{d}\,,$
(4.23)

and

$\displaystyle(4\pi)^{6}\beta_{y_{u}}^{(2)}=\,$
$\displaystyle+\dfrac{a_{1}}{32}\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{u}-\dfrac{3}{32}\big{[}y_{u}y_{u}^{\dagger}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{u}$
$\displaystyle-\dfrac{3}{32}\big{[}y_{d}y_{d}^{\dagger}y_{d}y_{d}^{\dagger},\,y_{u}y_{u}^{\dagger}\big{]}y_{u}-\dfrac{3}{16}y_{u}y_{u}^{\dagger}\big{[}y_{d}y_{d}^{\dagger},\,y_{u}y_{u}^{\dagger}\big{]}y_{u}\,,$
(4.24) $\displaystyle(4\pi)^{6}\beta_{y_{d}}^{(2)}=\,$
$\displaystyle+\dfrac{a_{1}}{32}\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{d}-\dfrac{3}{32}\big{[}y_{u}y_{u}^{\dagger}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{d}$
$\displaystyle-\dfrac{3}{32}\big{[}y_{d}y_{d}^{\dagger}y_{d}y_{d}^{\dagger},\,y_{u}y_{u}^{\dagger}\big{]}y_{d}-\dfrac{3}{16}y_{d}y_{d}^{\dagger}\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{d}\,.$
(4.25)

They also satisfy RG-finiteness, while higher-order poles cannot appear at the
3-loop order. Meanwhile, for the lepton sector, we find that all RG poles
vanish identically:

$\gamma_{\ell,e}^{(n)}=0\,,\qquad\beta_{y_{e}}^{(n)}=0\,,\qquad n\geq 1\,.$
(4.26)

This is due to $y_{e}$ being the only coupling to carry
$\mathrm{SU}(3)_{\ell,e}$ quantum numbers, making it impossible to construct a
flavor-covariant anti-Hermitian combination of the couplings. Of course, it is
well-known that the SM allows for choosing $y_{e}$ diagonal with the right
flavor rotations. If, on the other hand, one were to include right-handed
neutrinos with a Higgs coupling analogous to that of the up-quarks, anti-
Hermitian combinations become possible, and the RG poles of the lepton sector
would presumably mirror those of the quark sector. All in all, explicit
computations demonstrate that fixing $Z=Z^{\dagger}$ in the SM leads to RG-
finite RG functions up to the 3-loop order.

###  4.4  SM calculation of the $N^{I}$ counterterm

The flavor RG functions are determined by the $N^{I}$ counterterm of the
background gauge field, cf. Eq. (2.1). Although $N^{I}$ counterterms are
needed at the 1-loop order, $\upsilon$ is only nontrivial from the 3-loop
order [27]. For a direct computation of $N^{I}$ in the SM, we need to compute
the 3-loop counterterms for renormalized 1PI Green’s functions of the flavor
current $J_{F}^{P\mu}$ for all fermionic fields in the SM.141414These
corrections contain subdivergences associated with corrections to the coupling
of the current with the scalar doublet, hence we also have to compute these
corrections up to the 2-loop order. Said Green’s functions are related to
$N^{I}$ by

$\big{\langle}[\Phi^{\alpha}](x)[J_{F}^{P\mu}](y)[\Phi_{\beta}^{\dagger}](z)\big{\rangle}_{\\!{\scriptscriptstyle\mathrm{1PI}}}=\big{(}\delta^{PQ}+N^{QI}(t^{P}\,g)_{I}\big{)}Z\phantom{}^{\dagger\alpha}\phantom{}_{\gamma}\big{\langle}\Phi^{\gamma}(x)J_{F}^{Q\mu}(y)\Phi_{\delta}^{\dagger}(z)\big{\rangle}_{\\!{\scriptscriptstyle\mathrm{1PI}}}Z\phantom{}^{\delta}\phantom{}_{\beta}\,,$
(4.27)

using square brackets to denote renormalized operators and going to the FSCC
limit. Using different fermion fields for $\Phi^{\alpha}\,$, we can isolate
the various contributions to the flavor current (4.14) and corresponding
representations of $N^{I}\,$.

For the computation of the 3-loop Green’s functions, we have to extend the
computational setup employed in Ref. [12]. To this end, we introduce the field
$a_{\mu}$ as a non-dynamical degree of freedom and treat $t_{f}^{P}$ as non-
commuting matrices similar to the Yukawa couplings. All relevant Feynman
diagrams are generated using QGRAF [35] and are subsequently transformed into
FORM [36] expressions as well as mapped onto integral families, using Q2E and
EXP [37, 38]. We use the COLOR [39] package for the computation of color
factors and evaluate the 1- to 3-loop integrals using FORCER [40] after
nullifying one of the external momenta.

Having obtained the 3-loop Green’s functions expressed with bare parameters,
we employ the coupling and field-strength counterterms computed in Ref. [13]
to express them in terms of renormalized quantities. The remaining divergences
are absorbed with the current counterterms allowing for the determination of
$N^{I}$. One remark concerning the renormalization of $t_{f}^{P}$ is in order:
for the actual computation, it is advantageous to treat them in the same way
as Yukawa matrices, i.e., to extract renormalization constants for each of
them from the various contributions to
$\big{\langle}[\Phi^{\alpha}][J_{F}^{P\mu}][\Phi_{\beta}^{\dagger}]\big{\rangle}_{\\!{\scriptscriptstyle\mathrm{1PI}}}$
in the same manner as the Yukawa matrices in Ref. [13]. We can then match the
counterterms of $t_{f}^{P}$ to an Ansatz for all possible Yukawa matrix
structures in $N_{f}^{I}(t^{P}\,g)_{I}$ and are, thus, able to extract the
counterterm $N^{I}\,$. An explicit example of the procedure at the 1-loop
order is presented in Appendix B.

While the flavor-current approach captures all contributions to $N^{I}$
present in pure Yukawa theories, it is insufficient for gauge theories. The
issue is that the gauge couplings are singlets under the flavor symmetry,
meaning that $(t^{P}\,a)_{AB}=0\,$. Clearly, any such contribution drops from
the Green’s functions (4.27), and the corresponding parts of $N^{I}$ cannot be
determined with that approach. Up to 3-loop order in there is only one such
term, in $N^{I}$ given by151515It is convenient to parameterize $N^{I}$ with a
dummy variable $\hat{g}_{I}$ rather than using indices and Kronecker deltas.

$N_{q}^{I}\hat{g}_{I}\supset\dfrac{n_{a_{1}}}{(4\pi)^{6}}\hat{a}_{1}\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}$
(4.28)

for some $n_{a_{1}}$. The combined requirement that $N^{I}$ is anti-Hermitian
and that there is a flavor-covariant underlying parameterization for the
general theory (4.1) leaves just four possible terms at the 3-loop order
similar to the four gauge contribution in the general $\upsilon$
parameterization [27]. In the SM this leaves only the one term
(4.28).161616This is not to say that there are no other gauge contributions in
$N^{I}\hat{g}_{I}$ but that these come with $\hat{y}^{(\dagger)}_{f}$ and can
be obtained with the flavor-current approach.

An alternative way of determining $N^{I}$, which should address _all_
contributions is by considering Green’s function of the renormalized marginal
operators $[\mathcal{O}^{I}]$ from Eq. (4.2):

$\begin{split}\big{\langle}[\Phi^{\alpha}]&(x)[\mathcal{O}^{I}](y)[\Phi_{\beta}^{\dagger}](z)\big{\rangle}_{\\!{\scriptscriptstyle\mathrm{1PI}}}\\\
=\,&\partial^{I}g_{0,J}\,Z\phantom{}^{\dagger\alpha}\phantom{}_{\gamma}\big{\langle}\Phi^{\gamma}(x)\mathcal{O}^{J}(y)\Phi_{\delta}^{\dagger}(z)\big{\rangle}_{\\!{\scriptscriptstyle\mathrm{1PI}}}Z\phantom{}^{\delta}\phantom{}_{\beta}-Z\phantom{}^{\dagger\alpha}\phantom{}_{\gamma}\big{\langle}\Phi^{\gamma}(x)N^{I}\cdot\partial_{\mu}J_{F}^{\mu}(y)\Phi_{\delta}^{\dagger}(z)\big{\rangle}_{\\!{\scriptscriptstyle\mathrm{1PI}}}Z\phantom{}^{\delta}\phantom{}_{\beta}\\\
&+\delta^{d}(y-x)\partial^{I}Z\phantom{}^{\dagger\alpha}\phantom{}_{\gamma}\big{\langle}\Phi^{\gamma}(x)\Phi_{\delta}^{\dagger}(z)\big{\rangle}_{\\!{\scriptscriptstyle\mathrm{1PI}}}Z\phantom{}^{\delta}\phantom{}_{\beta}+\delta^{d}(y-z)Z\phantom{}^{\dagger\alpha}\phantom{}_{\gamma}\big{\langle}\Phi^{\gamma}(x)\Phi_{\delta}^{\dagger}(z)\big{\rangle}_{\\!{\scriptscriptstyle\mathrm{1PI}}}\partial^{I}Z\phantom{}^{\delta}\phantom{}_{\beta}\,,\end{split}$
(4.29)

where, again, $\partial^{I}$ denotes coupling derivatives as per Eq. (2.14).
Taking this coupling to be the hypercharge coupling of the SM requires us to
compute Green’s functions with insertions of the $\mathrm{U}(1)$ field-
strength tensor, as well as momentum-dependent insertions of derivatives of
bare coupling constants w.r.t. the $\mathrm{U}(1)$ gauge coupling. While a
similar computation was performed in Ref. [28] for the case of a Yukawa
theory,171717Yukawa theories could be done with the flavor-current approach in
a straightforward manner. it requires careful routing of momenta through
individual diagrams and cannot be achieved by simply nullifying one of the
external momenta. Consequently, we refrain from explicitly calculating this
contribution.

###  4.5  Flavor-improved RG functions in the SM

We are now in a position to present $\upsilon$ in the gaugeless limit of the
SM after applying formulas (3.9) and (3.10) to the $N^{I}$ counterterms.
Following Eqs. (4.10) and (4.11) the action of $\upsilon$ on the Yukawa
couplings is given by

$\displaystyle(\upsilon\,y_{u})\phantom{}^{i}\phantom{}_{j}=\upsilon_{q}\phantom{}^{i}\phantom{}{}_{k}y_{u}\phantom{}^{k}\phantom{}{}_{j}-y_{u}\phantom{}^{i}\phantom{}{}_{k}\upsilon_{u}\phantom{}^{k}\phantom{}{}_{j}\quad\mathrm{and}\quad(\upsilon\,y_{d})\phantom{}^{i}\phantom{}_{j}=\upsilon_{q}\phantom{}^{i}\phantom{}{}_{k}y_{d}\phantom{}^{k}\phantom{}{}_{j}-y_{d}\phantom{}^{i}\phantom{}{}_{k}\upsilon_{d}\phantom{}^{k}\phantom{}{}_{j}$
(4.30)

in terms of the representation on the three quark flavor product
groups.181818We find that $\upsilon_{\ell}=\upsilon_{e}=0$, as one would
expect. We find $\upsilon_{f}$ are given by

$\displaystyle(4\pi)^{6}\upsilon_{q}$
$\displaystyle=\left(\frac{3}{32\epsilon^{2}}+\frac{1}{64\epsilon}-1\right)\left(\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}y_{d}y_{d}^{\dagger}\big{]}+\big{[}y_{d}y_{d}^{\dagger},\,y_{u}y_{u}^{\dagger}y_{u}y_{u}^{\dagger}\big{]}\right),$
(4.31) $\displaystyle(4\pi)^{6}\upsilon_{u}$
$\displaystyle=\left(\frac{3}{16\epsilon^{2}}-\frac{1}{32\epsilon}+\frac{1}{8}\right)y_{u}^{\dagger}\big{[}y_{d}y_{d}^{\dagger},\,y_{u}y_{u}^{\dagger}\big{]}y_{u}\leavevmode\nobreak\
,$ (4.32) $\displaystyle(4\pi)^{6}\upsilon_{d}$
$\displaystyle=\left(\frac{3}{16\epsilon^{2}}-\frac{1}{32\epsilon}+\frac{1}{8}\right)y_{d}^{\dagger}\big{[}y_{u}y_{u}^{\dagger},\,y_{d}y_{d}^{\dagger}\big{]}y_{d}\leavevmode\nobreak\
.$ (4.33)

Our results for $\upsilon$ exactly cancel the divergent contributions to the
anomalous dimensions and $\beta$-functions in Eqs. (4.16–4.25), hence
explicitly rendering the flavor-improved RG functions, $B_{I}$ and $\Gamma$,
finite. Finiteness of the flavor-improved RG functions can also be used to fix
the poles of the $a_{1}$ contribution to $\upsilon$ once gauge couplings are
reintroduced. This does, however, not constrain the more interesting finite
part.

The parameterization of $\upsilon$ in the generic gauge-Yukawa theory (4.1)
has 6 free parameters for the fermion representation and 3 for the scalar and
is provided in Ref. [27]. The finite part of $\upsilon$ in the SM uniquely
fixes two of the pure-Yukawa fermion coefficients:

$\mathfrak{f}_{4}^{(3)}=-\tfrac{3}{8}\quad\mathrm{and}\quad\mathfrak{f}_{5}^{(3)}=-\tfrac{5}{16}\,.$
(4.34)

The Weyl consistency conditions derived from Osborn’s Equation [27] with input
from a (currently) partial reconstruction of the generic 4–3–2
$\beta$-functions [41], provides the independent constraint

$\displaystyle\mathfrak{f}_{4}^{(3)}-4\mathfrak{f}_{5}^{(3)}=\tfrac{7}{8}\,,$
(4.35)

which is satisfied by our result and serves to validate our computation.
Interestingly, with this input and full knowledge of the 4–3–2
$\beta$-function coefficients, the 3-loop $\upsilon$ will be fully determined
by the Weyl consistency conditions.

##  5  Conclusion

In this paper, we exhaustively discuss the origin of, and resolution to,
divergences and ambiguities in perturbative RG functions. Our discussion
reveals that the divergences of RG functions in the dimensional regulator,
first observed in 3-loop SM computations full Yukawa matrix dependence, are
generic to all models with nontrivial flavor structure. We show that said
divergences are related to $G_{F}$, the global flavor symmetry of the kinetic
terms of the matter fields. Our investigations show that the flavor symmetry
protects the theory, and, in spite of the appearance of explicit divergences
in individual RG functions, the RG flow is finite. Finiteness of the Callan-
Symanzik equation is guaranteed by the $G_{F}$ Ward identity and RG-finiteness
implies that the divergences of field anomalous dimensions are elements of the
Lie algebra of $G_{F}$ _and_ their action on the couplings produce the
divergences of the associated $\beta$-functions. We have shown this to be the
case in _all_ four-dimensional QFTs and expect similar cancellations to hold
in any spacetime dimension.

We then examined the divergences in the anomalous dimensions of the quark
fields and Yukawa $\beta$-functions in the SM. These RG functions were
obtained using the prescription of Hermitian square roots of the field
renormalization constants. We found that, indeed, the divergences are
consistent with RG-finiteness. Through a diagrammatic 3-loop computation we
obtained the counterterm of the background gauge field of $G_{F}\,$, which was
promoted to a local symmetry. This counterterm allowed us to compute the shift
from divergent $\beta$-functions to the flavor improved $\beta$-function $B$
in the SM. The computation of $B$ in the SM, along with the Weyl consistency
conditions, will allow us to generalize this shift to _all_ gauge-Yukawa
theories.

In general, all RG functions depend on the prescription used for the square
roots of field-strength renormalization constants. To apply the results
presented here one needs to choose all square roots of renormalization
constants to be Hermitian unless dealing with the flavor-improved quantities.

While the ambiguity in RG functions does not affect physical observables, as
it is related to field redefinitions, there are several scenarios where it
might affect phenomenological applications of RG functions:

  1. i)

RG evolution of couplings using divergent $\beta$-function (on account of it
being ill-defined);

  2. ii)

matching of Yukawa or scalar quartic couplings in effective field theories if
not performed via physical observables;

  3. iii)

study of the RG evolution of Yukawa couplings with textures that are not
protected by additional symmetries;

  4. iv)

numerical RG evolution with a $\beta$-function choice leading to numerical
instabilities.

As a consequence, we recommend the use of the flavor-improved
$\beta$-function, $B_{I}\,$, which is guaranteed to be finite.

###  Acknowledgments

AET would like to thank Hugh Osborn and Colin Poole for helpful discussions
and comments on the work. The work of AET has received funding from the Swiss
National Science Foundation (SNF) through the Eccellenza Professorial
Fellowship “Flavor Physics at the High Energy Frontier” project number 186866.
This document was prepared using the resources of the Fermi National
Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of
Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance,
LLC (FRA), acting under Contract No. DE-AC02-07CH11359.

Appendices

###  A  Transformation of RG functions under the renormalization ambiguity

This appendix contains the derivation of the response of the RG functions
under the counterterm transformation discussed in Sec. 3.3.

####  A.1  Alternative counterterm parameterization

Before deriving the variation of the RG functions under the divergent flavor
rotation, it is useful to consider a different parameterization of the
coupling counterterms in place of the more direct parameterization employed in
Eq. (2.1). We employ the compact notation $Z\phantom{}^{I}\phantom{}_{J}$ to
denote the tensor product of field-strength renormalization factors with open
indices to match the couplings. For gauge-Yukawa theories, we have explicitly

$Z\phantom{}^{I}\phantom{}_{J}=\big{\\{}\delta_{AC}\delta_{BD},\,Z_{a}\phantom{}^{ij}\phantom{}{}_{bk\ell},\,Z_{abcd,efgh}\big{\\}}=\big{\\{}\delta_{AC}\delta_{BD},\,Z_{ab}Z\phantom{}^{i}\phantom{}_{k}Z\phantom{}^{j}\phantom{}_{\ell},\,Z_{ae}Z_{bf}Z_{cg}Z_{dh}\big{\\}}\,.$
(A.1)

This form assumes the use of the background-field gauge, which maintains a
gauge symmetry preventing the renormalization of the gauge field itself [42,
43]. In other gauges, many additional counterterms are needed for the various
gauge interactions and a $Z_{AB}\neq\delta_{AB}$ would appear in
$Z\phantom{}^{I}\phantom{}_{J}\,$.

We extract a factor of the field-strength renormalization from the bare
couplings such that

$\mu^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}k_{I}\epsilon}g_{0,I}=(g_{J}+\delta
h_{J})Z\phantom{}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1J}\phantom{}_{I}\,,$
(A.2)

This parameterization is similar to what one would use, having renormalized
the fields $\Phi^{\alpha}$ rather than the field sources. Thus, $\delta h_{I}$
is identified as the counterterm of the 1PI vertex contribution itself.
Interestingly, under the counterterm transformation (3.16), where
$Z\phantom{}^{I}\phantom{}_{J}\to\tilde{Z}\phantom{}^{I}\phantom{}_{J}=U\phantom{}^{I}\phantom{}_{K}Z\phantom{}^{K}\phantom{}_{J}$
and $g_{0,I}\to\tilde{g}_{0,I}=g_{0,J}U\phantom{}^{\dagger
J}\phantom{}_{I}\,$, the $\delta h_{I}$ counterterm is invariant, as follows
from

$\mu^{k_{I}\epsilon}(g_{I}+\delta
h_{I})=g_{0,J}Z\phantom{}^{J}\phantom{}_{I}\longrightarrow\tilde{g}_{0,J}\tilde{Z}\phantom{}^{J}\phantom{}_{I}=g_{0,J}Z\phantom{}^{J}\phantom{}_{I}\,.$
(A.3)

The $\beta$-function formula in terms of the new parameterization can be
derived by applying a $t$ derivative to Eq. (A.2) as per usual. With the
$t$-independent bare couplings this yields191919For the coupling dimension it
holds that
$k_{I}Z\phantom{}^{I}\phantom{}_{J}=k_{J}Z\phantom{}^{I}\phantom{}_{J}\,$.

$0=\left[k_{J}\epsilon(g_{J}+\delta
h_{J})+\beta_{J}+\beta_{K}\partial^{K}\delta h_{J}-(g_{K}+\delta
h_{K})\gamma\phantom{}^{K}\phantom{}_{J}\right]Z\phantom{}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1J}\phantom{}_{I}\,.$
(A.4)

Consequently,

$\beta_{I}=-\epsilon k_{I}(g_{I}+\delta h_{I})-\beta_{J}\partial^{J}\delta
h_{I}+(g_{J}+\delta h_{J})\gamma\phantom{}^{J}\phantom{}_{I}\,,$ (A.5)

from which $\beta$ can be determined order by order in $\epsilon\,$. This form
is particularly useful for determining the effect of the divergent flavor
rotation on $\beta\,$, since $g_{I}$ and $\delta h_{I}$ are invariants.

####  A.2  Transformation of the ordinary RG functions

Given the transformation of the renormalization constants induced by the
divergent rotation $U$ (3.15), we wish to determine the associated changes in
the RG functions, $\Delta\beta_{I}=\tilde{\beta}_{I}-\beta_{I}$ and
$\Delta\gamma=\tilde{\gamma}-\gamma\,$. We will begin with the _assumption_
that $\Delta\gamma\in\mathfrak{g}_{F}\,$, and end up showing that this is
consistent with the transformation. All quantities are flavor-covariant
functions of the renormalized couplings on account of the $G_{F}$ symmetry.
Any flavor-covariant function $f(g)\,$ satisfies

$(\omega\,f)=(\omega\,g)_{I}\partial^{I}f\,,\qquad\omega\in\mathfrak{g}_{F}\,,$
(A.6)

that is, the transformation of $f$ is what one gets from transforming all the
couplings that make up $f\,$. This holds regardless of the what open indices
$f$ might have.

Since the vertex counterterm is invariant, $\delta\tilde{h}_{I}=\delta
h_{I}\,$, the change of the $\beta$-function is most easily determined from
Eq. (A.5), giving

$\Delta\beta_{I}=-\Delta\beta_{J}\partial^{J}\delta h_{I}+(g_{J}+\delta
h_{J})\Delta\gamma\phantom{}^{J}\phantom{}_{I}\,.$ (A.7)

Under the assumption $\Delta\gamma\in\mathfrak{g}_{F}\,$, it holds that
$\delta h_{J}\Delta\gamma\phantom{}^{J}\phantom{}_{I}=-(\Delta\gamma\,\delta
h)_{I}\,$. Eq. (A.6) is, thus, recast as

$\Delta\beta_{I}=-(\Delta\gamma\,g)_{I}-\left[\Delta\beta_{J}+(\Delta\gamma\,g)_{J}\right]\partial^{J}\delta
h_{I}\,,$ (A.8)

which is clearly solved by

$\Delta\beta_{I}=-(\Delta\gamma\,g)_{I}\,.$ (A.9)

Hence, the change of the $\beta$-function is determined entirely by the change
of the field anomalous dimension.

Next, we determine the change in the field anomalous dimension in terms of
$U$. Using formulas (3.5) for the definition of the anomalous dimension and
(A.9) for the transformation of the $\beta$-function, the transformation of
the anomalous dimension is

$\begin{split}\Delta\gamma&=Z^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}U^{\dagger}\tilde{\beta}_{I}\partial^{I}UZ+Z^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\Delta\beta_{I}\partial^{I}Z\\\
&=U^{\dagger}\tilde{\beta}_{I}\partial^{I}U+Z^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\big{[}U^{\dagger}\tilde{\beta}_{I}\partial^{I}U,\,Z\big{]}-Z^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}(\Delta\gamma\,g)_{I}\partial^{I}Z\end{split}$
(A.10)

Under a regular flavor transformation, $Z$ transforms like the adjoint
representation, i.e. $(\omega\,Z)=\big{[}\omega,\,Z\big{]}\,$. Furthermore,
$U^{\dagger}\partial^{I}U$ is a Maurer–Cartan form and, so, takes values in
$\mathfrak{g}_{F}$. Hence,

$\Delta\gamma=U^{\dagger}\tilde{\beta}_{I}\partial^{I}U+Z^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\big{(}[U^{\dagger}\tilde{\beta}_{J}\partial^{J}U-\Delta\gamma]\,g\big{)}_{I}\partial^{I}Z\,,$
(A.11)

which is solved by

$\Delta\gamma=U^{\dagger}\tilde{\beta}_{I}\partial^{I}U\,.$ (A.12)

Crucially, we observe that $\Delta\gamma\in\mathfrak{g}_{F}$ in line with our
original assumption. While already sufficient to determine $\Delta\gamma$
iteratively in the $\epsilon$-poles, further simplifications of Eq. (A.12) can
be obtained using flavor covariance (A.6) to write
$(\Delta\gamma\,g)_{I}\partial^{I}U=\big{[}\Delta\gamma,\,U\big{]}\,$. The
transformation properties of $U$ is due to it being the representation of an
element of $G_{F}\,$, thereby inheriting the transformation
$(\omega\,U)=\big{[}\omega,\,U\big{]}\,$. With a rearrangement of the
equation, one then finds

$\Delta\gamma=-\beta_{I}\,U\partial^{I}U^{\dagger}\,.$ (A.13)

This establishes a direct relation between $\Delta\gamma$ and the divergent
rotation $U\,$.

####  A.3  Transformation of the flavor RG functions

We will also want to determine the change
$\Delta\upsilon=\tilde{\upsilon}-\upsilon$ and
$\Delta\rho=\tilde{\rho}-\rho\,$ of the flavor RG functions under the
divergent flavor rotation $U\,$. Both $\upsilon$ and $\rho$ are determined by
the $N^{I}$ counterterm of the flavor background field $a_{0,\mu}\,$. From Eq.
(3.16), it follows that

$\tilde{a}_{0,\mu}=a_{\mu}+U\big{[}a_{\mu},\,U^{\dagger}\big{]}+U\partial^{I}U^{\dagger}\partial_{\mu}g_{I}+UN^{I}U^{\dagger}D_{\mu}g_{I}\,.$
(A.14)

While $a_{\mu}$ is finite and, so, invariant under the transformation, it
follows from covariance of $U$ that the change of $N^{I}$ is given by

$\Delta
N^{I}=\tilde{N}^{I}-N^{I}=U\big{[}N^{I},\,U^{\dagger}\big{]}+U\partial^{I}U^{\dagger}\,.$
(A.15)

The transformation of $\upsilon$ then follows from the definition (2.19):

$\begin{split}\Delta\upsilon&=\Delta B_{I}\tilde{N}^{I}+B_{I}\Delta
N^{I}=\Delta
B_{I}\tilde{N}^{I}+U\big{[}\upsilon,\,U^{\dagger}\big{]}+\beta_{I}U\partial^{I}U^{\dagger}-(\upsilon\,g)_{I}U\partial^{I}U^{\dagger}\\\
&=-\big{(}(\Delta\gamma+\Delta\upsilon)\,g\big{)}_{I}\tilde{N}^{I}-\Delta\gamma\,,\end{split}$
(A.16)

utilizing flavor covariance of various quantities and Eq. (A.13). Evidently,
this equation is solved with

$\Delta\upsilon=-\Delta\gamma\,,$ (A.17)

which determines the transformation properties of $\upsilon\,$.

Finally, to determine the change of $\rho\,$, we begin from the definition
(2.20):

$\begin{split}\Delta\rho^{I}=&-\beta_{J}\partial^{J}\Delta N^{I}-\Delta
N^{J}\partial^{I}\beta_{J}-\Delta N^{J}(\rho^{I}\,g)_{J}\\\
&-\Delta\beta_{J}\partial^{J}\tilde{N}^{I}-\tilde{N}^{J}\partial^{I}\Delta\beta_{J}-\tilde{N}^{J}(\Delta\rho^{I}\,g)_{J}\,.\end{split}$
(A.18)

Rather a lot of algebra is needed at this stage to bring things to a
reasonable manageable form. The second line can be evaluated using, covariance
of

$(\omega\,N^{I})=\big{[}\omega,\,N^{I}\big{]}+\omega\phantom{}^{I}\phantom{}_{J}N^{J}\,,$
(A.19)

where $\omega\phantom{}^{I}\phantom{}_{J}$ is the representation of the Lie
algebra produced from
$(\omega\,g)_{I}=-g_{J}\omega\phantom{}^{J}\phantom{}_{I}\,$. The second line
of Eq. (A.18) evaluates to

$(\Delta\gamma\,\tilde{N}^{I})+\tilde{N}^{J}\partial^{I}(\Delta\gamma\,g)_{J}-\tilde{N}^{J}(\Delta\rho^{I}\,g)_{J}=\big{[}\Delta\gamma,\,\tilde{N}^{I}\big{]}+\tilde{N}^{J}\big{(}(\partial^{I}\Delta\gamma-\Delta\rho^{I})\,g\big{)}_{J}\,.$
(A.20)

The first line, meanwhile, evaluates to

$\begin{split}-\beta_{J}\partial^{J}&\Delta N^{I}-\Delta
N^{J}\partial^{I}\beta_{J}-\Delta N^{J}(\rho^{I}\,g)_{J}\\\
=&-\left(\beta_{J}\partial^{J}(U\partial^{I}U^{\dagger})+\partial^{I}\beta_{J}(U\partial^{J}U^{\dagger})\right)-\beta_{J}\left(\partial^{J}U\big{[}N^{I},\,U^{\dagger}\big{]}+U\big{[}N^{I},\,\partial^{J}U^{\dagger}\big{]}\right)\\\
&-U\big{[}\beta_{J}\partial^{J}N^{I}+N^{J}\partial^{I}\beta_{J}+N^{J}(\rho^{I}\,g)_{J}+\rho^{I},\,U^{\dagger}\big{]}\,,\end{split}$
(A.21)

the entire last term of which vanishes by the definition of $\rho^{I}\,$. The
two first terms of Eq. (A.21) evaluate as

$\begin{split}-\left(\beta_{J}\partial^{J}(U\partial^{I}U^{\dagger})+\partial^{I}\beta_{J}(U\partial^{J}U^{\dagger})\right)&=\partial^{I}\Delta\gamma+\beta_{J}\big{(}\partial^{I}U\partial^{J}U^{\dagger}-\partial^{J}U\partial^{I}U^{\dagger}\big{)}\\\
&=\partial^{I}\Delta\gamma-\big{[}\Delta\gamma,\,U\partial^{I}U^{\dagger}\big{]}\end{split}$
(A.22)

and

$\begin{split}-\beta_{J}\left(\partial^{J}U\big{[}N^{I},\,U^{\dagger}\big{]}+U\big{[}N^{I},\,\partial^{J}U^{\dagger}\big{]}\right)&=-\Delta\gamma\,U\big{[}N^{I},\,U^{\dagger}\big{]}+U\big{[}N^{I},\,U^{\dagger}\Delta\gamma\big{]}\\\
&=-\big{[}\Delta\gamma,\,N^{I}\big{]}-\big{[}\Delta\gamma,\,U\big{[}N^{I},\,U^{\dagger}\big{]}\big{]}\,,\end{split}$
(A.23)

respectively. Altogether then, the first line of Eq. (A.18) reduces to

$-\beta_{J}\partial^{J}\Delta N^{I}-\Delta N^{J}\partial^{I}\beta_{J}-\Delta
N^{J}(\rho^{I}\,g)_{J}=\partial^{I}\Delta\gamma-\big{[}\Delta\gamma,\,\tilde{N}^{I}\big{]}\,.$
(A.24)

Including result (A.20) for the second line, the change of $\rho^{I}$ is
therefore

$\Delta\rho^{I}=\partial^{I}\Delta\gamma+\tilde{N}^{J}\big{(}(\partial^{I}\Delta\gamma-\Delta\rho^{I})\,g\big{)}_{J}\,.$
(A.25)

Obviously, this equation is solved by

$\Delta\rho^{I}=\partial^{I}\Delta\gamma\,.$ (A.26)

This concludes the derivation of the transformation rules for the RG functions
under a $U$-rotation of the bare sources.

###  B  Sample calculation of the $N^{I}$ counterterm

Here we outline the SM computation of the left-handed quark contribution to
$N^{I}$ to 1-loop order in the gaugeless limit. We begin by computing the
1-loop vertex corrections between $Q$ and $a_{\mu}\,$ in the limit of
vanishing $a_{\mu}$ momenta. These are renormalized by the field-strength
renormalization of the external quark legs and the current counterterm,
$N^{I}$.

For the up-type quark loop, the amplitude is

$\begin{split}\mathcal{A}^{P}_{1,\mu}&=\mu^{2\epsilon}\\!\\!\int\dfrac{\mathop{}\\!\mathrm{d}^{d}k}{(2\pi)^{d}}(-iy_{u})\dfrac{-i\not{k}}{k^{2}}(\gamma_{\mu}t_{u}^{P})\dfrac{-i\not{k}}{k^{2}}(-iy_{u}^{\dagger})\dfrac{i}{(k+p)^{2}}\\\
&=\dfrac{1}{16\pi^{2}}y_{u}t_{u}^{P}y_{u}^{\dagger}\\!\left(-\dfrac{4\pi\mu^{2}}{p^{2}}\right)^{\\!\\!\epsilon}\dfrac{d-2}{d}\dfrac{\Gamma(\epsilon)\Gamma^{2}(1-\epsilon)}{\Gamma(2-2\epsilon)}\\\
&=\dfrac{1}{32\pi^{2}\epsilon}y_{u}t_{u}^{P}y_{u}^{\dagger}\,\gamma_{\mu}\;+\;\text{finite}.\end{split}$
(B.1)

From the similar loop with an internal down-type quark, the contribution is

$\mathcal{A}^{P}_{2,\mu}=\dfrac{1}{32\pi^{2}\epsilon}y_{d}t_{d}^{P}y_{d}^{\dagger}\,\gamma_{\mu}\;+\;\text{finite}\,.$
(B.2)

Meanwhile, the counterterms are contained in the tree-level amplitude

$\begin{split}\mathcal{A}^{P}_{\mathrm{ct},\mu}&=K_{\epsilon}\\!\left[\left(\delta^{PQ}+N^{QI}(t^{P}\,g)_{I}\right)Z_{q}^{\dagger}\gamma_{\mu}t^{Q}_{q}Z_{q}\right]\\\
&=\dfrac{1}{\epsilon}N^{(1)I}_{q}(t^{P}\,g)_{I}\gamma_{\mu}+\dfrac{1}{\epsilon}z_{q}^{(1)\dagger}t^{P}_{q}\gamma_{\mu}+\dfrac{1}{\epsilon}t^{P}_{q}z^{(1)}_{1}\gamma_{\mu}+\ldots\,,\end{split}$
(B.3)

where the $K_{\epsilon}$ operator extracts the divergent piece of its
argument. At the 1-loop order the field-strength renormalization goes as

$z^{(1)}_{q}\supset-\dfrac{1}{64\pi^{2}}(y_{u}y_{u}^{\dagger}+y_{d}y_{d}^{\dagger})\,.$
(B.4)

For successful normalization,
$K_{\epsilon}(\mathcal{A}_{1}+\mathcal{A}_{2}+\mathcal{A}_{\mathrm{ct}})=0\,$,
we find

$N_{q}^{(1)I}(t^{P}\,g)_{I}=\dfrac{1}{64\pi^{2}}\left(y_{u}y_{u}^{\dagger}t_{q}^{P}-2y_{u}t_{u}^{P}y_{u}^{\dagger}+t_{q}^{P}y_{u}y_{u}^{\dagger}\right)\;+\;(u\to
d)\,.$ (B.5)

First, we notice that this result is anti-Hermitian, which is a good sign.
Next, we need to check that it is possible to find an $N^{(1)I}_{q}$, which
satisfies the above expression. To this end, we observe that we can let

$N_{q}^{(1)I}\hat{g}_{I}=\dfrac{1}{64\pi^{2}}\left(\hat{y}_{u}y_{u}^{\dagger}-y_{u}\hat{y}_{u}^{\dagger}\right)\;+\;(u\to
d)\,.$ (B.6)

Upon substituting the action of $t^{P}$ on the Yukawas (Eqs. (4.10) and
(4.11)), we see that this is the counterterm for the left-handed quark field
at 1-loop. Similar computations can be done for the other fermions.

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