# Veltman, renormalizability, calculability

Giampiero Passarino
Dipartimento di Fisica Teorica, Università di Torino, Italy
INFN, Sezione di Torino, Italy

###### Abstract

Dedicated to the memory of Prof. Veltman, one of the founding fathers of our
discipline: his legacy lives on. Many times we have to turn back and follow
his footprints to find the right path. After reviewing general aspects of high
energy physics where he gave a seminal contribution we will introduce recent
developments in the standard model effective field theory, showing how the
whole movement from renormalization to predictions plays from Veltman to
SMEFT.

12.60.-i, 11.10.-z, 14.80.Bn.

##  1 Introduction

In his talk at Higgs Hunting $2015$ 111
https://indico.ijclab.in2p3.fr/event/2722/contributions/5996/ Martinus Veltman
said

Higgs particle $\,\dots\,$ What does it mean? What does it do? It is claimed
to give mass to all other particles. What does that mean? Can we now predict
the masses of all particles? In this talk it will be attempted to explain why
the Higgs particle is important to the theory of the Standard Model. The
importance of the Higgs construction is that it made the theory of Yang-Mills
fields renormalizable. Observable results can be calculated and compared with
experiment, and that has happened in a multitude of ways in the last 40 years,
up to and including the recent discovery of the Higgs particle.

It is clear that he meant “strictly renormalizable” but it is evident that the
main emphasis is on Observable results can be calculated and compared with
experiment. In order to understand the situation before $1971$, it is
interesting to observe [1] that most of the papers on the subject came to the
same conclusion “ $\dots\,$ it is concluded that all theories based on simple
Lie groups are unrenormalizable”. All difficulties in the Yang-Mills theories
disappeared in $1971$ and the theories became fully renormalizable, that is
all occurring infinities could be absorbed in the available free parameters.
Theories with a Yang-Mills structure were now renormalizable theories and a
precise model for the weak interactions existed already (although it had
received virtually no attention). The phase of “precise calculations” started,
extending the pioneering work of Berends, Gastmans and collaborators [2, 3, 4]
to the full Standard Model [5, 6, 7, 8, 9] and continued till the most recent
successes [10, 11, 12, 13]. For a complete list of references, see Ref. [14].

To summarize a long journey, we can say that the transition was from the Fermi
theory to the standard model (SM); for that, Veltman [15, 16] had to convince
the community that the weak interactions were some form of a Yang-Mills
theory. As he wrote: I could not have done that without the knowledge of
experimental physics that I had acquired at CERN. It is worth noting that the
journey never contemplated the extension of the Fermi theory with the
inclusion of even higher operators; therefore, the SM represented the beyond-
Fermi physics. Comparing with the present: we are now in a beyond-SM desert
looking for alternative paths, although the regulative ideal of an ultimate
theory remains a powerful aesthetic ingredient.

Almost equally important, and a landmark through the whole region of “strict
renormalizability”, is the work on quantum gravity: “In case of gravitation
interacting with scalar particles, divergencies in physical quantities remain”
[17, 1]. It is possible that at some very large energy scale, all
nonrenormalizable interactions disappear. This seems unlikely, given the
difficulty with gravity. It is possible that the rules change drastically. It
may even be possible that there is no end, simply more and more scales [18].

##  2 The standard model before LEP

The last step in the renormalization procedure is the connection between
renormalized quantities and physical observables. Since all quantities at this
stage are UV-free, we term it finite renormalization. Note that the absorption
of UV divergencies into local counterterms is, to some extent, a trivial step
(except for the problem of overlapping divergencies [19]); finite
renormalization, instead, requires more attention. For example, beyond one
loop one cannot use on-shell masses but only complex poles for all unstable
particles [20, 21, 22, 23]. The complete formulation of finite renormalization
is beyond the goal of this work. However, let us show some examples where the
concept of an on-shell mass can be employed. Suppose that we renormalize a
physical observable $\mathrm{F}$,

$\mathrm{F}=\mathrm{F}_{{\scriptscriptstyle{\mathrm{B}}}}+g^{2}\,\mathrm{F}_{1{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2})+g^{4}\,\mathrm{F}_{2{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2})\hskip
1.42262pt,$ (1)

where $M$ is some renormalized mass which appears at one and two loops in
$\mathrm{F}_{1{\scriptscriptstyle{\mathrm{L}}}}$ and
$\mathrm{F}_{2{\scriptscriptstyle{\mathrm{L}}}}$ but does not show up in the
Born term $\mathrm{F}_{{\scriptscriptstyle{\mathrm{B}}}}$. In this case we can
use the concept of an on-shell mass identifying
$M=M_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}}$ for
the two-loop term and performing a finite mass renormalization at one loop,

$\mathrm{M}^{2}=\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}}\,\Bigl{\\{}1\,+\,\frac{g^{2}}{16\,\pi^{2}}\,\Bigl{[}\,\mathrm{Re}\,\Sigma^{(1)}_{{\scriptscriptstyle{\mathrm{M}}}}\mid_{p^{2}=-\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}}}-\,\delta
Z^{(1)}_{{\scriptscriptstyle{\mathrm{M}}}}\Bigr{]}\,\Bigr{\\}}=\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}}+g^{2}\,\Delta\mathrm{M}^{2}\hskip
1.42262pt,$ (2)

where $M_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}}$
is the on-shell mass and $\Sigma$ is extracted from the required one-particle
irreducible Green function. It is worth noting that Eq.(2) is still meaningful
(no dependence on gauge parameters) and will be used inside the one-loop
result,

$\mathrm{F}=\mathrm{F}_{{\scriptscriptstyle{\mathrm{B}}}}+g^{2}\,\mathrm{F}_{1{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}})+g^{4}\,\Bigl{[}\mathrm{F}_{2{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}})+\mathrm{F}^{\prime}_{1{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}})\,\Delta\mathrm{M}^{2}\Bigr{]},$
(3)

where

$\mathrm{F}^{\prime}_{1{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}})\,=\,\frac{\partial\mathrm{F}_{1{\scriptscriptstyle{\mathrm{L}}}}(\mathrm{M}^{2})}{\partial\mathrm{M}^{2}}|_{\mathrm{M}^{2}=\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{O}}}{\scriptscriptstyle{\mathrm{S}}}}}\hskip
1.42262pt.$ (4)

If we focus on renormalization, we can safely state that all the necessary
ingredients are available. Here the crucial point is to connect a set of input
experimental data (an input-parameter set, hereafter IPS) to the free
parameters of the theory:

  * –

mass renormalization involves the calculation of self-energies;

  * –

renormalization of coupling constants requires additional elements, which
depend on the choice of the observables in the IPS.

The most-obvious selection of an IPS is based on the choice of those data
which are known with the best experimental precision, e.g. the electromagnetic
coupling constant, the Fermi coupling constant and the $\mathrm{Z}\,-$boson
mass.

Before the advent of the LEP operations we had very few options for the IPS
[24, 25]. To give an example, we consider a first approximation where the
lowest order expressions are compared with a set of low-energy data points,
electric charge, neutral currents and $\mu$-decay:

$g^{2}=4\,\pi\alpha(0)\hskip
1.42262pt,\qquad\mathrm{M}^{2}=\frac{g^{2}\sqrt{2}}{8\mathrm{G}_{\mathrm{F}}}\hskip
1.42262pt,\qquad\mathrm{s}_{{}_{\theta}}^{2}\quad{\mbox{from}}\quad
R=\frac{\sigma_{{\overline{\nu}}e}}{\sigma_{\nu e}}\hskip 1.42262pt.$ (5)

The values for $g^{2},\mathrm{M}^{2}$ and $\mathrm{s}_{{}_{\theta}}^{2}$,
correct in the lowest order, will subsequently be used in the expressions for
the radiative corrections. In the next order we replace

$g^{2}\to g^{2}\left(1+\delta g^{2}\right)\hskip
1.42262pt,\qquad\mathrm{M}^{2}\to\mathrm{M}^{2}\left(1+\delta\mathrm{M}^{2}\right)\hskip
1.42262pt,\qquad\mathrm{s}_{{}_{\theta}}^{2}\to\mathrm{s}_{{}_{\theta}}^{2}\left(1+\delta\mathrm{s}_{{}_{\theta}}^{2}\right)\hskip
1.42262pt.$ (6)

The counter-terms are chosen to compensate precisely for the radiative
corrections for $e$–$\mu$ scattering, $\mu$-decay and the ratio $R$. Having
determined these quantities we may proceed to making predictions. The ratio
$R$ is $R\left(\mathrm{s}_{{}_{\theta}}\right)$. We use the fact that it does
not depend on $g^{2}$ or $\mathrm{M}^{2}$ at lowest order and at zero energy
and momentum transfer. Thus

$R_{0}\left(\mathrm{s}_{{}_{\theta}}^{2}+\mathrm{s}_{{}_{\theta}}^{2}\delta\mathrm{s}_{{}_{\theta}}^{2}\right)=R_{0}\left(\mathrm{s}_{{}_{\theta}}^{2}\right)+R^{\prime}_{0}\,\mathrm{s}_{{}_{\theta}}^{2}\delta\mathrm{s}_{{}_{\theta}}^{2}\hskip
1.42262pt,$ (7)

with $R^{\prime}=dR/d\mathrm{s}_{{}_{\theta}}^{2}$. The one-loop radiative
corrections to $R$ will be some $R_{1}$, then
$\delta\mathrm{s}_{{}_{\theta}}^{2}$ is fixed by

$\delta\mathrm{s}_{{}_{\theta}}^{2}=-\frac{R_{1}}{R^{\prime}_{0}\mathrm{s}_{{}_{\theta}}^{2}}\hskip
1.42262pt.$ (8)

The rest is standard and gives $e^{4}\to e^{4}(\,1+\delta e^{4})$ and

$\delta g^{2}=\frac{1}{2}\delta e^{4}-\delta\mathrm{s}_{{}_{\theta}}^{2}\hskip
1.42262pt,\qquad\delta\mathrm{M}^{2}=\frac{1}{2}\left(\delta_{\mu}-\delta^{\rm
em}_{\mu}\right)+\delta g^{2}\hskip 1.42262pt.$ (9)

Let us denote by $\mathrm{s}^{2}_{\nu e}$ the low-energy weak mixing angle
defined through some $R_{{\rm exp}}$. Then we can derive masses for the vector
bosons in the low-energy convention. They are given by

$\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{W}}}}=\frac{\pi\alpha}{\sqrt{2}\mathrm{G}_{\mathrm{F}}\mathrm{s}^{2}_{\nu
e}}\hskip
1.42262pt,\qquad\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}=\frac{\pi\alpha}{\sqrt{2}\mathrm{G}_{\mathrm{F}}\mathrm{s}^{2}_{\nu
e}c^{2}_{\nu e}}\hskip 1.42262pt.$ (10)

Let us consider the amplitude for
$\overline{\nu}_{\mu}e^{-}\to\overline{\nu}_{\mu}e^{-}$. In real life many
different contributions should be considered, but to illustrate some of the
relevant points in the procedure it is enough to limit the calculation to
contributions from the heavy quark doublet, ($t$–$b$), to the
$\mathrm{Z}\mathrm{Z}$ and $\mathrm{Z}\gamma$ transitions.

In this case, we obtain

$A\left(\overline{\nu}_{\mu}e^{-}\to\overline{\nu}_{\mu}e^{-}\right)=\left(\frac{ig}{4\,\mathrm{c}_{{}_{\theta}}}\right)^{2}\gamma^{\mu}\,\gamma_{+}\otimes\gamma_{\mu}\,\bigl{(}a+b\,\gamma^{5}\bigr{)}\hskip
1.42262pt,$ (11)

where
$\gamma^{\alpha}\otimes\gamma^{\alpha}={\overline{\nu}}_{\mu}\gamma^{\alpha}\overline{\nu}_{\mu}{\overline{e}}\gamma^{\alpha}e$,
etc. and

$a=\Bigl{[}4\,\mathrm{s}_{{}_{\theta}}^{2}-1-\frac{g^{2}\mathrm{s}_{{}_{\theta}}^{2}}{4\,\pi^{2}}\,\frac{\Sigma_{{\scriptscriptstyle{\mathrm{Z}}}\gamma}(p^{2})}{p^{2}}\Bigr{]}\,\Delta_{{\scriptscriptstyle{\mathrm{Z}}}}(p^{2})\hskip
1.42262pt,\qquad b=-\Delta_{{\scriptscriptstyle{\mathrm{Z}}}}(p^{2})\hskip
1.42262pt,$ (12)

with a propagator $\Delta_{{\scriptscriptstyle{\mathrm{Z}}}}$ given by

$\Delta^{-1}_{{\scriptscriptstyle{\mathrm{Z}}}}(p^{2})=p^{2}+\mathrm{M}^{2}_{{}_{0}}-\frac{g^{2}}{16\,\pi^{2}\mathrm{c}_{{}_{\theta}}^{2}}\,\Sigma_{{\scriptscriptstyle{\mathrm{Z}}}{\scriptscriptstyle{\mathrm{Z}}}}(p^{2})\hskip
1.42262pt,$ (13)

where
$\Sigma_{{\scriptscriptstyle{\mathrm{Z}}}{\scriptscriptstyle{\mathrm{Z}}}}$ is
the $\mathrm{Z}\,$-boson self-energy and
$\Sigma_{{\scriptscriptstyle{\mathrm{Z}}}\gamma}$ is the corresponding
transition. Then the total cross-section $\sigma_{\nu e}$ can be computed and
the data point $R=\sigma_{{\overline{\nu}}e}/\sigma_{\nu e}$ used:

$R=\frac{\xi_{\nu e}^{2}-\xi_{\nu e}+1}{\xi_{\nu e}^{2}+\xi_{\nu e}+1}\hskip
1.42262pt,\qquad\xi_{\nu e}=\frac{a}{b}\hskip 1.42262pt,$ (14)

where we assume the approximation of zero momentum transfer. Subtracting the
terms involving UV poles and introducing a mass scale $\mu$ we define the
counter-term for $\mathrm{s}_{{}_{\theta}}$ and fix
${\mathrm{s}_{{}_{\theta}}}^{\mbox{$\overline{\scriptscriptstyle MS}$}}$ to
first order in $\alpha$. The whole renormalization procedure amounts to
throwing away infinities. If we subtract the terms involving UV poles then the
$\overline{MS}$ redefinition of the parameters is obtained (including
$\mathrm{s}_{{}_{\theta}}$), but we could as well assign any finite value to
$1/{\bar{\varepsilon}}$ (to be defined in Eq.(20)) and check for the
independence of the physical quantities of $1/{\bar{\varepsilon}}$. From the
$\mathrm{t}$–$\mathrm{b}$ quark doublet we obtain

$\displaystyle{\mathrm{s}_{{}_{\theta}}^{2}}^{\mbox{$\overline{\scriptscriptstyle
MS}$}}$ $\displaystyle=$ $\displaystyle\mathrm{s}^{2}_{\nu
e}+\frac{\alpha}{12\,\pi}\,\Bigl{[}2\,\left(1-\frac{8}{3}\,\mathrm{s}^{2}_{\nu
e}\right)\,\ln\frac{m^{2}_{t}}{\mu^{2}}+\left(1-\frac{4}{3}\,\mathrm{s}^{2}_{\nu
e}\right)\,\ln\frac{m^{2}_{b}}{\mu^{2}}\Bigr{]}\hskip 1.42262pt,$
$\displaystyle\mathrm{s}^{2}_{\nu e}$ $\displaystyle=$
$\displaystyle\frac{1-\xi_{\nu e}}{4}\hskip 1.42262pt.$ (15)

As expected, there are no terms quadratic in the quark masses.

At this point we are ready to make predictions. Starting from $p^{2}=0$ we can
introduce an effective $p^{2}$-dependent weak mixing angle, etc.

There is an important lesson to learn: there are Lagrangian parameters (e.g.
$\mathrm{s}_{{}_{\theta}}$), input parameters (e.g. $\xi_{\nu e}$) and
predictions (or “pseudo-observables” ). In the SM there is no one-to-one
correspondence between Lagrangian parameters and input parameters. It was
different in the older days of QED where only two Lagrangian parameters are
present, $e$ and $\mathrm{m}_{\mathrm{e}}$. In this case one uses
$\mathrm{m}_{\exp}\,(1+\Delta\mathrm{m}_{e})$ reflecting some vage intuition
about the physical meaning of the bare mass. The strategy to prescribe
precisely what a Lagrangian parameter is offers a problem when there is no
unique experimental quantity that can play the role of defining the parameter.

At LEP $\mathrm{s}^{2}_{\nu e}$ will be replaced by a
$\sin\theta^{\mathrm{f}}_{{\mbox{\scriptsize eff}}}$, related to the vector
and axial couplings of the $\mathrm{Z}\,$-boson.

##  3 LEP: the $\rho\,$-parameter and pseudo-observables

One way to explain renormalization is to say that infinities are unobservable
and can thus be absorbed into the parameters of the Lagrangian. What about
potentially large effects in the renormalized theory? When are they
observable? For instance the $m^{2}_{\mathrm{t}}\,$-terms at one-loop, which
are there and show up in physical observables. Conversely, the
$\mathrm{M}_{{\scriptscriptstyle{\mathrm{H}}}}$ dependence of one-loop
radiative corrections was another seminal contribution of Veltman and it was
described by the screening theorem [26, 27]: the one-loop
$\mathrm{M}_{{\scriptscriptstyle{\mathrm{H}}}}$-dependence in physical
observables is only logarithmic. Terms proportional to
$\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{H}}}}$ are unobservable at the
one-loop level (they start at two loops) and can be absorbed into the
parameters of the SM Lagrangian, apart from the case of one-loop diagrams with
external (on-shell) Higgs-lines. The $\rho\,$-parameter [28, 29, 30] was born.

A comment is needed for the “original” $\rho\,$-parameter, defined as the
ratio of the $\mathrm{W}$ and $\mathrm{Z}$ masses squared divided by
$\mathrm{c}_{{}_{\theta}}^{2}$. We start by introducing a bare quantity:
$\rho_{0}=\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{W}}}}/(\mathrm{c}_{{}_{\theta}}^{2}\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}})$.
The quantities appearing in this relation must be related to experimental
data. There is no ambiguity in what is meant by an experimental mass. There
remains the experimental $\mathrm{c}_{{}_{\theta}}^{2}$ which should be
extracted from data (at the time, low-energy data). In the original
formulation, where everything was extracted from low-energy data $(p^{2}=0)$,
a $\rho$ was introduced as

$\rho=1+\frac{g^{2}}{16\,\pi^{2}\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{W}}}}}\,\Bigl{[}\Sigma_{{\scriptscriptstyle{\mathrm{W}}}{\scriptscriptstyle{\mathrm{W}}}}(0)-\Sigma_{{\scriptscriptstyle{\mathrm{Z}}}{\scriptscriptstyle{\mathrm{Z}}}}(0)\Bigr{]}=1+\frac{g^{2}}{16\,\pi^{2}}\,\Delta\rho\hskip
1.42262pt,$ (16)

where the $1$ in the r.h.s. of the equation is actually $\rho_{\rm bare}$ in
the SM. The $\rho$-parameter of Eq.(16) is finite and numerically very close
to the experimental one, in any scheme where the counterterms are prescribed.
The main correction is due to the top quark mass. Usually we do not attach any
particular relevance to bare parameters, only the renormalized Lagrangian
predicts meaningful — measurable — quantities. However, the $\rho$-parameter
of the SM plays a special role. It is finite because of a residual symmetry
which is nothing but the usual isospin invariance. Individual components in
Eq.(16) are by themselves infinite, but the combination occurring in this
equation for the $\rho$-parameter is finite, as it should be.

The Nobel Prize in Physics $1999$ was awarded “for elucidating the quantum
structure of electroweak interactions in physics”’ which was crucial for LEP.
At LEP we had the SM with one missing ingredient, therefore the strategy was:
test the SM hypothesis versus $\mathrm{M}_{{\scriptscriptstyle{\mathrm{H}}}}$,
introduce Pseudo-Observables (generalizations of the $\rho$ parameter [26]),
fit them and derive limits on the Higgs boson mass [31].

Ideally, the strategy should have been to combine the results of the LEP
experiments at the level of the measured cross-sections and asymmetries - a
goal that has never been achieved because of the intrinsic complexity, given
the large number of measurements with different cuts and the complicated
structure of the experimental covariance matrices relating their errors. This
reflects computing limitations at the time.

What the experimenters did [31] was just collapsing (and/or transforming) some
“primordial quantities” (say number of observed events in some pre-defined
set-up) into some “secondary quantities” (the POs) which are closer to the
theoretical description of the phenomena. The practical attitude of the
experiments was to stay with a fit from “primordial quantities” to POs (with a
SM remnant) for each experiment, and these sets of POs were averaged. The
result of this procedure are best values for POs. The extraction of Lagrangian
parameters, was based on the LEP-averaged POs. The PO-strategy was made
possible thanks to high-precision QED calculations and tools, from the
pioneering work of Ref. [32](for an update see Ref. [33]) up to the Monte
Carlos that were instrumental at LEP in extracting the realistic observables;
for instance KKMC, the most advanced MC for $2\,$-fermion production [34]
(including second order QED with resummation, initial-final state interference
and spin polarizations, for an update see Ref. [35]).

##  4 LHC: after the discovery of the Higgs boson

After the LHC Run 1, the SM has been completed, raising its status to that of
a full theory [36]. Despite its successes, this SM has shortcomings vis-à-vis
cosmological observations. At the same time, there is presently a lack of
direct evidence for new physics phenomena at the accelerator energy frontier.
From this state of affairs arises the need for a consistent theoretical
framework in which deviations from the SM predictions can be calculated.

Theoretical physics suffers from some inherent difficulties: great successes
during the $2$0th century, increasing difficulties to do better, as the easier
problems get solved. The lesson of experiments from $1973$ to today is that it
is extremely difficult to find a flaw in the SM 222Note however the long-
standing tension between experiment and SM prediction in the anomalous
magnetic moment of the muon, recently reaffirmed by the Fermilab experiment,
T. Albahri et al. Muon $g-2$, Phys. Rev. Accel. Beams (2021).: maybe the SM
includes elements of a truly fundamental theory. The conventional vision is:
some very different physics occurs at Planck scale, the SM is just an
effective field theory (EFT). What about the next SM? A new weakly-coupled
renormalizable model? A tower of EFTs? Of course there is a different vision:
is the SM close to a fundamental theory?

We need a consistent theoretical framework in which deviations from the SM (or
next-SM) predictions can be calculated. Such a framework should be applicable
to comprehensively describe measurements in all sectors of particle physics:
LHC Higgs measurements, past electroweak precision data (EWPD), etc. Here we
outline the strategy:

  * •

Consider the SM augmented with the inclusion of higher dimensional operators
(say theory $\mathrm{T}_{1}$); not strictly renormalizable. Although workable
to all orders, $\mathrm{T}_{1}$ fails above a certain scale, $\Lambda_{1}$.

  * •

Consider any beyond-standard-model (BSM) model that is strictly renormalizable
and respects unitarity ($\mathrm{T}_{2}$); its parameters can be fixed by
comparison with data, while masses of heavy states are presently unknown.
$\mathrm{T}_{1}\not=\mathrm{T}_{2}$ in the UV but they must have the same IR
behavior.

  * •

Consider now the whole set of data below $\Lambda_{1}$. $\mathrm{T}_{1}$
should be able to explain them by fitting Wilson coefficients,
$\mathrm{T}_{2}$ adjusting the masses of heavy states (as SM did with the
Higgs mass at LEP) should be able to explain the data. Goodness of both
explanations is crucial in understanding how well they match and how
reasonable is to use $\mathrm{T}_{1}$ instead of the full $\mathrm{T}_{2}$.

  * •

Does $\mathrm{T}_{2}$ explain everything? Certainly not, but it should be able
to explain something more than $\mathrm{T}_{1}$.

  * •

We could now define $\mathrm{T}_{3}$ as $\mathrm{T}_{2}$ augmented with (its
own) higher dimensional operators; it is valid up to a scale $\Lambda_{2}$.

  * •

Continue.

This prompts the important question whether there is a last fundamental theory
in this tower of EFTs that supersede each other with rising energies
333Kuhlmann, Meinard, “Quantum Field Theory”, The Stanford Encyclopedia of
Philosophy (Winter 2018Edition), Edward N. Zalta (ed.). Some people conjecture
that this deeper theory could be a string theory, i.e. a theory which is not a
field theory any more. Or should one ultimately expect from physics theories
that they are only valid as approximations and in a limited domain? To
summarize: experiments occur at finite energy and measure
$\mathrm{S}^{\mathrm{eff}}(\Lambda)$ (an effective $\mathrm{S}\,$-matrix);
whatever QFT should give low energy
$\mathrm{S}^{\mathrm{eff}}(\Lambda)\,,\;\forall\,\Lambda<\infty$, i.e. there
is no fundamental scale above which $\mathrm{S}^{\mathrm{eff}}(\Lambda)$ is
not defined. However $\mathrm{S}^{\mathrm{eff}}(\Lambda)$ loses its predictive
power if a process at $E=\Lambda$ requires an infinite number of renormalized
parameters [37, 38, 39].

To summarize: before LHC we had the SM, a weakly coupled, strictly
renormalizable (a theory with $n$ Lagrangian parameters, requiring $n$ data
points, requiring $n$ calculations; the $(n+1)\,$th calculation is a
prediction) theory with one unknown,
$\mathrm{M}_{{\scriptscriptstyle{\mathrm{H}}}}$. The strategy was: test data
against predictions vs. $\mathrm{M}_{{\scriptscriptstyle{\mathrm{H}}}}$. At
LHC, after the discovery, with a lack of direct evidence for new physics
phenomena we have all the ingredients required to asses (in)consistency of the
SM against data.

We briefly review the SMEFT Lagrangian [40, 41, 42]: consider the standard
model, described by a Lagrangian
$\mathcal{L}^{(4)}_{\mathrm{\scriptscriptstyle{SM}}}$ with a symmetry group
$\mathrm{G}=SU(3)\,\times\,SU(2)\,\times\,U(1)$. The SMEFT extension is
described by a Lagrangian

$\mathcal{L}_{\mathrm{\scriptscriptstyle{SMEFT}}}=\mathcal{L}^{(4)}_{\mathrm{\scriptscriptstyle{SM}}}+\sum_{d>4}\,\sum_{i}\,\frac{\mathrm{a}^{d}_{i}}{\Lambda^{4-d}}\,\mathcal{Q}^{(d)}_{i}\hskip
1.42262pt,$ (17)

where $\Lambda$ is the cutoff of the effective theory, $\mathrm{a}^{d}_{i}$
are Wilson coefficients and $\mathcal{Q}^{(d)}_{i}$ are
$\mathrm{G}\,$-invariant operators of mass-dimension $d$ involving the
$\mathcal{L}^{(4)}_{\mathrm{\scriptscriptstyle{SM}}}$ fields. In this work we
will use the so-called “Warsaw basis” [43].

Unconventional approach to EFT: derivative-coupled field theories are known to
develop ghosts [44]. The EFT option [45] replaces the original $\mathcal{L}$
with some $\mathcal{L}_{{\mbox{\scriptsize eff}}}$ truncated at some order in
the $\Lambda\,$-expansion; the “dangerous” terms are substituted by using the
equations of motion where, for instance, we neglect terms of
$\mathcal{O}(\Lambda^{-2})$. We assume that $\mathcal{L}_{{\mbox{\scriptsize
eff}}}$ will be replaced by a well-behaved $\mathcal{L}^{\prime}$ at some
larger scale, therefore justifying a truncated perturbative expansion; the EFT
does not have ghosts while remaining within its regime of validity.

SMEFT and renormalization
At this point we have lost strict renormalizability but this should not come
at the price of loosing computability [46, 47]; whether the predictions of a
theory are matched by Nature is a completely different matter and can be
decided only by comparing the predictions with experiment (calculability).

A renormalizable theory is determined by a fixed number of parameters; once
these are determined (after finite renormalization) we can make definite
predictions at a fixed accuracy. An EFT theory requires at higher and higher
energies more and more counterterms; the asymptotic expansion in
$\mathrm{E}/\Lambda$ may break down completely above some scale. Given a
truncated expansion, we still have a large family of UV-complete theories with
these low order terms, which have different behavior at higher energies. Note
that the notion of UV completion adopted here [48] is the claim that a theory
is “formally” predictive up to all (possible) high energies, but we do not
include the additional criterion that the theory be a final, unified “theory
of everything”.

Therefore, it is crucial to prove that our EFT is closed under
renormalization, order-by-order in the asymptotic expansion, although the
number of counterterms will grow with the order (as mentioned above, the
predictive power is lost at scales approaching the cutoff). For any given
process the amplitude can be written as follows:

$\displaystyle\mathrm{A}$ $\displaystyle=$
$\displaystyle\sum_{n=\mathrm{N}}^{\infty}\,g^{n}\,\mathrm{A}^{(4)}_{n}+\sum_{n=\mathrm{N}_{6}}^{\infty}\,\sum_{l=1}^{n}\,\sum_{k=1}^{\infty}\,g^{n}\,g^{l}_{4+2\,k}\,\mathrm{A}^{(4+2\,k)}_{n\,l\,k}\hskip
1.42262pt,$ (18)

where $g$ is the $SU(2)$ coupling constant and
$g_{4+2\,k}=1/(\sqrt{2}\,\mathrm{G}_{\mathrm{F}}\,\Lambda^{2})^{k}=g^{k}_{6}$,
where $\mathrm{G}_{\mathrm{F}}$ is the Fermi coupling constant and $\Lambda$
is the scale around which new physics (NP) must be resolved. For each process,
$N$ defines the $\mathrm{dim}=4$ LO (e.g. $N=1$ for
$\mathrm{H}\to\mathrm{V}\mathrm{V}$ etc. but $N=3$ for
$\mathrm{H}\to\gamma\gamma$). $N_{6}=N$ for tree initiated processes and $N-2$
for loop initiated ones. Here we consider single insertions of
$\mathrm{dim}=6$ operators, which defines the so-called NLO SMEFT. To be more
precise, we define a NLO SMEFT amplitude as the one containing SMEFT vertices
inserted in tree-level SM diagrams, tree-level (SMEFT-induced) diagrams with a
non-SM topology, SMEFT vertices inserted in one-loop SM diagrams, and one-loop
(SMEFT-induced) non-SM diagrams.

The amplitude can be rewritten as

$\displaystyle\mathrm{A}$ $\displaystyle=$ $\displaystyle
g^{N}\,\mathrm{A}^{(4)}_{\mathrm{\scriptscriptstyle{LO}}}\bigl{(}\\{p\\}\bigr{)}+g^{N}\,g_{{}_{6}}\,\mathrm{A}^{(6)}_{\mathrm{\scriptscriptstyle{LO}}}\bigl{(}\\{p\\}\,,\,\\{\mathrm{a}\\}\bigr{)}+\frac{g^{N+2}}{16\,\pi^{2}}\,\mathrm{A}^{(4)}_{\mathrm{\scriptscriptstyle{NLO}}}\bigl{(}\\{p\\}\bigr{)}$
(19) $\displaystyle+$
$\displaystyle\frac{g^{N+2}\,g_{6}}{16\,\pi^{2}}\,\mathrm{A}^{(6)}_{\mathrm{\scriptscriptstyle{NLO}}}\bigl{(}\\{p\\}\,,\,\\{\mathrm{a}\\}\bigr{)}\hskip
1.42262pt,$

where $\\{p\\}$ is the set of SM parameters and $\\{\mathrm{a}\\}$ the set of
Wilson coefficients. Counterterms are introduced using

$\Delta_{\mathrm{UV}}=\frac{2}{4-\mathrm{d}}-\gamma-\ln\pi-\ln\frac{\mu^{2}_{\mathrm{R}}}{\mu^{2}}=\frac{1}{\bar{\varepsilon}}-\ln\frac{\mu^{2}_{\mathrm{R}}}{\mu^{2}}\hskip
1.42262pt,$ (20)

where $\mathrm{d}$ is the space-time dimension, $\gamma$ is the Euler-
Mascheroni constant, the loop measure is $\mu^{4-d}\,d^{d}q$ and
$\mu_{\mathrm{R}}$ is the renormalization scale. The counterterms are defined
by

$\mathrm{Z}_{i}=1+\frac{g^{2}}{16\,\pi^{2}}\,\bigl{(}d\mathrm{Z}^{(4)}_{i}+g_{6}\,d\mathrm{Z}^{(6)}_{i}\bigr{)}\,\Delta_{\mathrm{UV}}\hskip
1.42262pt.$ (21)

With field/parameter counterterms we can make UV finite (at
$\mathcal{O}(g^{2}\,g_{6})$) all self-energies and transitions and the
corresponding Dyson resummed propagators. Of course we have to prove
cancellation of UV poles for all Green’s functions. This means that we need to
make the SMEFT $\mathrm{S}\,$-matrix UV (and IR) finite, including
$\mathrm{dim}=6$ operators and, at least, $\mathrm{dim}=8$ operators
(truncation uncertainty). The verification of any claim with explicit
computations is of importance. The role of symmetry is crucial. The best way
to understand the connection between UV poles and symmetry is given by the
background-field-method [49, 17]. Let us give an example of the complexity of
proving cancellation of UV poles in any EFT. Consider a scalar theory

$\displaystyle\mathcal{L}(\phi_{\mathrm{c}}+\phi)$ $\displaystyle=$
$\displaystyle\mathcal{L}(\phi_{\mathrm{c}})+\phi_{i}\,\mathcal{L}^{\prime}_{i}(\phi_{\mathrm{c}})+\frac{1}{2}\,\partial_{\mu}\,\phi_{i}\,\partial_{\mu}\,\phi_{i}$
(22) $\displaystyle+$
$\displaystyle\phi_{i}\,\mathrm{N}^{\mu}_{ij}\,\partial_{\mu}\,\phi_{j}+\frac{1}{2}\,\phi_{i}\,\mathrm{M}_{ij}(\phi_{\mathrm{c}})\,\phi_{j}+\mathcal{O}(\phi^{3})+\mbox{tot.
der.}$

($\mathcal{L}^{\prime}_{i}(\phi_{\mathrm{c}})=0$). All one loop diagrams are
generated by $\mathcal{L}_{2}(\phi)$, the part quadratic in $\phi$.

$\displaystyle\mathcal{L}_{2}(\phi)$ $\displaystyle\to$
$\displaystyle-\frac{1}{2}\,\left(\partial_{\mu}\,\phi\right)^{2}+\phi\,\mathrm{N}^{\mu}\,\partial_{\mu}\,\phi+\frac{1}{2}\,\phi\,\mathrm{M}\,\phi\hskip
1.42262pt.$ (23)

The counter-Lagrangian is given by

$\displaystyle\Delta\,\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{8\,\pi^{2}\,(d-4)}\,\Bigl{[}a_{0}\,M^{2}+a_{1}\,\left(\partial_{\mu}\,\mathrm{N}_{\nu}\right)^{2}+a_{2}\,\left(\partial_{\mu}\,\mathrm{N}_{\mu}\right)^{2}+a_{3}\,\mathrm{M}\,\mathrm{N}^{2}$
(24) $\displaystyle+$ $\displaystyle
a_{4}\,\mathrm{N}_{\mu}\,\mathrm{N}_{\nu}\,\partial_{\mu}\,\mathrm{N}_{\nu}+a_{5}\,\left(\mathrm{N}^{2}\right)^{2}+a_{6}\,\left(\mathrm{N}_{\mu}\,\mathrm{N}_{\nu}\right)^{2}\Bigr{]}\hskip
1.42262pt.$

However, define $X=\mathrm{M}-\mathrm{N}^{\mu}\,\mathrm{N}_{\mu}$ and see that
$\mathcal{L}$ is invariant under ’t Hooft transformation ($\mathrm{H}$) ,
$\Lambda$ antisymmetric

$\displaystyle\phi^{\prime}=\phi+\Lambda\,\phi\hskip
1.42262pt,\quad\mathrm{N}^{\prime}_{\mu}=\mathrm{N}_{\mu}-\partial_{\mu}\,\Lambda+\bigl{[}\Lambda\,,\,\mathrm{N}_{\mu}\bigr{]}\hskip
1.42262pt,\quad X^{\prime}=X+\bigl{[}\Lambda\,,\,X\bigr{]}$ (25)

Therefore $\Delta\,\mathcal{L}$ also will be invariant ($\mathrm{Tr}\,X$ is
invariant)

$\displaystyle\Delta\,\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{\varepsilon}\,\mbox{Tr}\,\left(a\,X^{2}+b\,Y^{\mu\nu}\,Y_{\mu\nu}\right)\hskip
1.42262pt,$ $\displaystyle Y_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,\mathrm{N}_{\nu}-\partial_{\nu}\,\mathrm{N}_{\mu}+\bigl{[}\mathrm{N}_{\mu}\,,\,\mathrm{N}_{\nu}\bigr{]}\hskip
1.42262pt.$ (26)

and $Y$ transforms as $X$. The counter-Lagrangian is made of products of
objects transforming as $X$ and of $\mathrm{dim}=4$. As a consequence of the
$\mathrm{H}\,$-invariance the number of counterterms goes from $7$ to $2$. Any
approach to SMEFT violating invariance is doomed to failure [50]. An EFT (e.g.
SMEFT) including $\mathrm{dim}=6,8$ operators will contain a term (for
$\mathrm{dim}=6$ see Ref. [51])

$\displaystyle\frac{1}{2}\,\partial_{\mu}\,\phi_{i}\;\;\mathrm{g}^{\mu\nu}_{ij}(\phi_{\mathrm{c}})\;\;\partial_{\nu}\,\phi_{j}$
matrix-valued metric tensor (27)

We should pay attention to the fact that in the SM
$\mathrm{g}^{\mu\nu}_{ij}\propto\delta^{\mu\nu}\,\delta_{ij}$ while in quantum
gravity (QGR) it remains diagonal only in the $ij$ indices. Therefore, in
SMEFT we will have matrix-valued Riemann tensors, i.e. more invariants for the
counter-Lagrangian $\Delta\,\mathcal{L}$, i.e.
$\mathrm{Tr}(X\,\mathrm{R}),\mathrm{Tr}\,(\mathrm{R}^{2})\,\dots$; as a
consequence, EFT is computationally more complex than QGR [52]. We should
remember that the name of the game is to have the full $\Delta\,\mathcal{L}$,
not the counterterms for one or two processes. If $\mathcal{L}$ is invariant
under a group $\mathrm{G}$ then the relation between the $\mathrm{G}$
transformation and the $\mathrm{H}$ one is crucial in proving closure under
renormalization (not the same as strict renormalizability). In other words the
$\mathrm{H}\,$-invariant counterterms of Eq.(26) must also be
$\mathrm{G}\,$-invariant.

To give an example of Eq.(27) we consider a Lagrangian

$\mathcal{L}=-\partial_{\mu}\Phi^{*}\,\partial_{\mu}\Phi-\mathrm{m}^{2}\,\Phi^{*}\Phi-\frac{1}{2}\,g^{2}\,\bigl{(}\Phi^{*}\Phi\bigr{)}\hskip
1.42262pt,$ (28)

where $\Phi=\bigl{(}\phi_{1}+\mathrm{v}+i\,\phi_{2}\bigr{)}/\sqrt{2}$. We
introduce $v=\mathrm{M}/g$ and $\mathrm{m}^{2}=\beta-\mathrm{M}^{2}/2$ and add

$\mathcal{Q}^{(8)}=\Bigl{(}\frac{gg_{6}}{\mathrm{M}^{2}}\Bigr{)}^{2}\,\partial_{\mu}\Phi^{*}\,\partial_{\mu}\Phi\,\Box\,\bigl{(}\Phi^{*}\Phi\bigr{)}\hskip
1.42262pt.$ (29)

After splitting $\phi_{i}=\phi_{i\mathrm{c}}+\phi_{i}$ we obtain that the
contribution of the operator of $\mathrm{dim}=8$ to the metric tensor is
$\delta g=\delta_{1}g+\delta_{2}g+\mathcal{O}(g^{3})$, where

$\delta_{1}g^{\mu\nu}=\frac{gg^{2}_{6}}{\mathrm{M}}\,\phi_{1\mathrm{c}}\,\delta^{\mu\nu}\,\mathrm{I}\hskip
1.42262pt,$ (30)
$\delta_{2}g^{\mu\nu}=\frac{g^{2}g^{2}_{6}}{\mathrm{M}^{4}}\left(\begin{array}[]{ll}(g_{11}+g_{22})\,\delta^{\mu\nu}+4\,\partial^{\mu}\phi_{1\mathrm{c}}\,\partial^{\nu}\phi_{1\mathrm{c}}&4\,\partial^{\mu}\phi_{1\mathrm{c}}\partial^{\nu}\phi_{2\mathrm{c}}\\\
4\,\partial^{\mu}\phi_{12\mathrm{c}}\partial^{\nu}\phi_{1\mathrm{c}}&(g_{11}+g_{22})\,\delta^{\mu\nu}+4\,\partial^{\mu}\phi_{2\mathrm{c}}\,\partial^{\nu}\phi_{2\mathrm{c}}\end{array}\right)$

where
$g_{ij}=\partial^{\alpha}\phi_{i\mathrm{c}}\,\partial^{\alpha}\phi_{j\mathrm{c}}$.

Here we will summarize the main steps in the renormalization procedure for the
SMEFT. Field/parameter counterterms are not enough to make UV finite the
Green’s functions with more than two legs. A mixing matrix among Wilson
coefficients is needed:

$\mathrm{a}_{i}=\sum_{j}\,\mathrm{Z}^{{\scriptscriptstyle{\mathrm{W}}}}_{ij}\,\mathrm{a}^{\mathrm{ren}}_{j}\hskip
1.42262pt,\quad\mathrm{Z}^{{\scriptscriptstyle{\mathrm{W}}}}_{ij}=\delta_{ij}+\frac{g^{2}}{16\,\pi^{2}}\,d\mathrm{Z}^{{\scriptscriptstyle{\mathrm{W}}}}_{ij}\,\Delta_{\mathrm{UV}}\hskip
1.42262pt.$ (31)

We can start by computing the amplitude for the (on-shell) decay
$\mathrm{H}\to\gamma\gamma$ and fix a few entries in the mixing matrix; we can
continue with the $\mathrm{H}\mathrm{Z}\gamma$ and
$\mathrm{H}\mathrm{Z}\mathrm{Z}$ amplitudes. When we arrive at
$\mathrm{H}\mathrm{W}\mathrm{W}$ we find that the $\mathrm{dim}=4$ part can be
made UV finite but for the $\mathrm{dim}=6$ part there are no Wilson
coefficients left free so that the UV finiteness follows from gauge
cancellations. We then continue with the decay of the Higgs boson (and of the
$\mathrm{Z}$ boson) into fermion pairs and make all the corresponding
amplitudes UV finite.

At this point we are left with the universality of the electric charge. In QED
there is a Ward identity telling us that $e$ is renormalized in terms of
vacuum polarization, Ward-Slavnov-Taylor identities allow us to generalize the
argument to the full SM [12, 53].

We can give a quantitative meaning to the the previous statement by saying
that the contribution from vertices (at zero momentum transfer) exactly cancel
those from (fermion) wave function renormalization factors. Therefore, we need
to compute the vertex ${\overline{f}}f\gamma$ (at $q^{2}=0$) and the $f$ wave
function factor in SMEFT, proving that the WST identity can be extended to
$\mathrm{dim}=6$; this is non trivial since there are no free Wilson
coefficients in these terms (after the previous steps); (non-trivial)
finiteness of LEP processes follows.

Finite renormalization in SMEFT; let us recall that the renormalization
procedure comprises the specification of the gauge-fixing term including,
together with the corresponding FP Lagrangian, the choice of the
regularization scheme — nowadays dimensional regularization — the prescription
for the renormalization scheme and the choice of a input parameter set. For
the SM parameters we use on-shell renormalization which requires the choice of
some input parameter set (IPS); for the Wilson coefficients we use the
$\overline{MS}$ scheme. Therefore, the final answer will contain “universal”
logarithms (renormalization group) but also “non-universal” logarithms which
depend on the choice of the IPS. The choice of the IPS should be such that the
effect of “non-universal” logarithms is minimal, e.g. $\alpha(0)$ should be
avoided. Once we work with NLO SMEFT, non-local effects (e.g. normal-threshold
singularities or even anomalous thresholds [54]) will also show up [55, 56,
45] for all those observables where the light masses are small compared to the
scale at which we test the process and much smaller than the cutoff $\Lambda$.

Finally, the infrared/collinear part of the one-loop virtual and of the real
corrections shows double factorization and the total is finite [57] at
$\mathcal{O}(g^{4}\,g_{6})$.

Asymptotics of BSM models
The backround-field-method is also used in combination with the heat kernel
[58], a very convenient tool for studying various asymptotics of the effective
action; for instance in deriving the low-energy limit of some underlying BSM
theory described by $\mathcal{L}(\\{\Phi\\})$, where $\\{\Phi\\}$ includes
both heavy and light fields. We expand $\Phi=\Phi_{\mathrm{c}}+\phi$ and
derive

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{c}+\langle\,\phi\,,\,\mathrm{D}\phi\,\rangle=\mathcal{L}_{c}+\phi^{\dagger}\,\mathrm{Q}\,\phi\hskip
1.42262pt,$ $\displaystyle\mathrm{Z}\bigl{[}\Phi_{\mathrm{c}}\bigr{]}$
$\displaystyle=$
$\displaystyle\int\bigl{[}\mathcal{D}\phi\bigr{]}\,\exp\\{i\,\mathrm{S}\\}=\exp\\{i\,\mathrm{S}_{c}\\}\,\mathrm{det}^{-1/2}(\mathrm{D})\hskip
1.42262pt,$ (32)

where $\mathrm{D}$ must be self-adjoint. Using
$\mathrm{Q}=\Box-\mathrm{M}+{\hat{\mathrm{Q}}}(\Phi_{\mathrm{c}})$ the heat
kernel expansion requires computing

$\mathrm{Tr}\,\ln\mathrm{Q}(x)\,\delta^{4}(x-y)=\int
d^{4}x\,\frac{d^{4}q}{(2\,\pi)^{4}}\,\mathrm{tr}\,\ln\Bigl{[}-q^{2}-M+\Box+2\,i\,{q}\cdot{\partial}+{\hat{\mathrm{Q}}(x\,,\,\partial_{x})}\Bigr{]}\hskip
1.42262pt.$ (33)

When there is one field or $\mathrm{M}_{ij}=\mathrm{M}^{2}\,\delta_{ij}$ we
write $\ln\Bigl{[}-(q^{2}+\mathrm{M}^{2})\,(\mathrm{I}+\mathrm{K})\Bigr{]}$,
expand $\ln(\mathrm{I}+\mathrm{K})$ in powers of $\mathrm{K}$ obtaining the
large $\mathrm{M}$-expansion of $\mathrm{S}_{{\mbox{\scriptsize
eff}}}=1/2\,\ln\,\mathrm{det}(\mathrm{D})$ in terms of tadpole integrals.
Otherwise, with more heavy scales or mixed heavy-light scales, the correct
Taylor expansion [59] is

$\ln(\mathrm{A}+\mathrm{B})-\ln\mathrm{A}=\int_{0}^{\infty}\,d\mu^{2}\,\Bigl{[}\mathrm{A}^{-1}_{+}\,\mathrm{B}\,\mathrm{A}^{-1}_{+}-\mathrm{A}^{-1}_{+}\,\mathrm{B}\,\mathrm{A}^{-1}_{+}\,\mathrm{B}\,\mathrm{A}^{-1}_{+}\;+\,\dots\Bigr{]}\hskip
1.42262pt,$ (34)

where $\mathrm{A}_{+}=\mathrm{A}+\mu^{2}\,\mathrm{I}$.

##  5 Advanced SMEFT

In this Section we will use the so-called “Warsaw basis” [43]; note however
that we will rescale the Wilson coefficients: in front of an operator
$\mathcal{Q}^{(\mathrm{d})}_{i}$ of dimension $\mathrm{d}$ containing $n$
fields we will write
$g^{n-2}\,\mathrm{a}^{\mathrm{d}}_{i}/\Lambda^{\mathrm{d}-4}$, where $g$ is
the $SU(2)$ coupling constant. Our goal is to consider the SMEFT with its
anomalies [37, 60, 61, 62]; in these calculations a certain amount of
$\gamma\,$-matrix manipulation is unavoidable and we must specify the
regularization scheme to be used, in particular, we must specify how to treat
$\gamma^{5}$. We will use the scheme developed by Veltman in Ref. [63], which
is based on the work of Refs. [64, 65]. Therefore, $\gamma^{\mu},\gamma^{5}$,
and $\varepsilon^{\mu\nu\alpha\beta}$ are formal objects where

$\displaystyle\bigl{\\{}\gamma^{\mu}\,,\,\gamma^{\nu}\bigr{\\}}=2\,\delta^{\mu\nu}\,\mathrm{I}$
$\displaystyle\mathrm{Tr}\,\mathrm{I}=4\hskip 1.42262pt,$
$\displaystyle\delta^{\mu\nu}=\delta^{\bar{\mu}\bar{\nu}}+\delta^{\hat{\mu}\hat{\nu}},\;\;\delta^{\bar{\mu}\bar{\mu}}=4,$
$\displaystyle\delta^{\hat{\mu}\hat{\mu}}=\mathrm{d}-4,$
$\displaystyle\delta^{\mu\alpha}\,\delta^{\alpha\nu}$ $\displaystyle=$
$\displaystyle\delta^{\bar{\mu}\bar{\alpha}}\,\delta^{\bar{\alpha}\bar{\nu}}+\delta^{\hat{\mu}\hat{\alpha}}\,\delta^{\hat{\alpha}\hat{\nu}}\hskip
1.42262pt,$ (35)

where $\mathrm{d}$ is the space-time dimension. As described in Ref. [63] we
have the following relation

$\displaystyle\mathrm{Tr}\,\bigl{(}\cdots\,\gamma^{\alpha}\,\cdots\,\gamma^{\beta}\,\cdots\bigr{)}$
$\displaystyle=$
$\displaystyle\mathrm{Tr}\,\bigl{(}\cdots\,\gamma^{\bar{\alpha}}\,\cdots\,\gamma^{\bar{\beta}}\,\cdots\bigr{)}$
(36) $\displaystyle+$
$\displaystyle\mathrm{Tr}\,\bigl{(}\cdots\,\gamma^{\hat{\alpha}}\,\cdots\,\gamma^{\hat{\beta}}\,\cdots\bigr{)}\hskip
1.42262pt,$

where the dots indicate strings of four-dimensional gamma matrices and also
$\gamma^{5}$. The second trace in the r.h.s. of Eq.(36) is computed according
to the following rules:

  1. 1.

move all the $\gamma^{\hat{\mu}}$ matrices to the right using
$\gamma^{\hat{\mu}}\,\gamma^{\bar{\nu}}=-\gamma^{\bar{\nu}}\,\gamma^{\hat{\mu}}$,

  2. 2.

for a trace containing an odd number of $\gamma^{5}$ matrices use
$\gamma^{\hat{\mu}}\,\gamma^{5}=\gamma^{5}\,\gamma^{\hat{\mu}}$,

  3. 3.

for a trace containing an even number of $\gamma^{5}$ matrices use
$\gamma^{\hat{\mu}}\,\gamma^{5}=-\gamma^{5}\,\gamma^{\hat{\mu}}$.

As a consequence we obtain

$\mathrm{Tr}\,\bigl{(}\cdots\,\gamma^{\hat{\alpha}}\,\gamma^{\hat{\beta}}\,\cdots\bigr{)}=\mathrm{Tr}\,\bigl{(}\cdots\bigr{)}\,\mathrm{Tr}\,\bigl{(}\gamma^{\hat{\alpha}}\,\gamma^{\hat{\beta}}\bigr{)}\hskip
1.42262pt,$ (37)

where the first trace in the r.h.s. only contains four-dimensional quantities
while
$\mathrm{Tr}\,\bigl{(}\gamma^{\hat{\alpha}}\,\gamma^{\hat{\beta}}\bigr{)}=\mathrm{d}-4$
etc.

Using a $\gamma^{5}$ which does not anticommute with the other Dirac matrices
leads to “spurious anomalies” which violate gauge invariance (spoiling
renormalizability) and we must impose the relevant Ward-Takahashi (Refs. [66,
67]) and Slavnov-Taylor (Refs. [68, 69, 70]) identities (for a review see Ref.
[71]). The problems of course are related to the existence of the ABJ anomaly
(Refs. [72, 73, 74]), which cancels in the SM. The goal of this Section is to
study the ABJ anomaly in the SMEFT extension of the SM.

###  5.1 SMEFT and WTST identities

Consider the following amplitudes:

$\displaystyle\mathrm{A}^{{\scriptscriptstyle{\mathrm{Z}}}}_{\mu\,;\,\nu_{1},\,\dots\,,\nu_{n}}(p\,;\,q_{1}\,\dots\,,q_{n}\,;\,k_{1}\,,\dots\,,k_{m})\hskip
1.42262pt,$
$\displaystyle\mathrm{A}^{\phi}_{\nu_{1},\,\dots\,,\nu_{n}}(p\,;\,q_{1}\,\dots\,,q_{n}\,;\,k_{1}\,,\dots\,,k_{m})\hskip
1.42262pt,$ (38)

involving a $\mathrm{Z}\,$-boson (or a $\phi^{0}$ Higss-Kibble ghost [75]) of
momentum $p$, $n$ gauge bosons ($\mathrm{A},\mathrm{Z},\mathrm{W}$) and $m$
Higgs bosons (all momenta are flowing inwards). The corresponding WTST
identity is

$\Gamma_{\nu_{1},\,\dots\,,\nu_{n}}\;=\;i\,p^{\mu}\,\mathrm{A}^{{\scriptscriptstyle{\mathrm{Z}}}}_{\mu\,;\,\nu_{1},\,\dots\,,\nu_{n}}+\mathrm{M}_{{\scriptscriptstyle{\mathrm{Z}}}}\,\mathrm{A}^{\phi}_{\nu_{1},\,\dots\,,\nu_{n}}\;=\;0\hskip
1.42262pt.$ (39)

We will study $2$ different schemes:

  * S1)

$\mathrm{Z}$ and $\phi^{0}$ off-shell, remaining gauge bosons coupled to
physical sources, i.e. $\partial_{\mu}\,\mathrm{J}_{\mu}=0$, anti-commuting
$\gamma^{5}$;

  * S2)

$\mathrm{Z}$ and $\phi^{0}$ off-shell, remaining gauge bosons coupled to
physical sources, i.e. $\partial_{\mu}\,\mathrm{J}_{\mu}=0$, Veltman-
prescription for $\gamma^{5}$, four-dimensional, on-shell, external momenta.

It is worth noting that there is a third scheme (which will not be considered
here): $\mathrm{Z}$ and $\phi^{0}$ off-shell, remaining gauge bosons coupled
to arbitrary sources, i.e. $\partial_{\mu}\,\mathrm{J}_{\mu}\not=0$, Veltman-
prescription for $\gamma^{5}$, $\mathrm{d}$-dimensional, off-shell external
momenta, i.e. $p_{\mu}=p_{\bar{\mu}}+p_{\hat{\mu}}$.

Consider the SMEFT, a theory which is not strictly renormalizable; if there
were no anomalies, all the UV divergences could be cancelled in
$\mathrm{S}^{{\mbox{\scriptsize eff}}}(\Lambda)$, order-by-order in
$1/\Lambda$. SMEFT loses its predictive power if a process at
$\mathrm{E}=\Lambda$ requires an infinite number of renormalized parameters.
Due to the anomaly at the one-loop level, the symmetry will be broken and the
mechanism of cancelling divergences is disturbed, i.e. WTST identities break
down. For example one relevant identity concerns the amplitude for a
$\mathrm{Z}$ to decay into two photons: if this identity is violated we can
still restore it by introducing a UV-finite counterterm as it would be the
case in the SM with only an electron and a neutrino. However, this new term is
of non-renormalizable type giving rise to infinities at higher orders; see
Ref. [76] for a complete discussion. The “complete” SM is an anomaly-free
theory and we want to investigate the SMEFT from the following point of view:
are there anomalies in the SMEFT? If they show up, could we find relations
among the Wilson coefficients that cancel the anomalies? In the following
sections we will consider the relevant WTST identities for SMEFT amplitudes

$\mathrm{A}=\mathrm{A}^{(4)}+g_{{}_{6}}\,\mathrm{A}^{(6)}\hskip
1.42262pt,\qquad
g_{{}_{6}}=\frac{1}{\sqrt{2}\,\mathrm{G}_{\mathrm{F}}\,\Lambda^{2}}\hskip
1.42262pt,$ (40)

where $\mathrm{G}_{\mathrm{F}}$ is the Fermi coupling constant and the first
term in Eq.(40) is the SM contribution. In computing the WTST identity of
Eq.(39) (schemes $1$ and $2$) we assume that all the sources for the $n$ gauge
bosons and the $m$ Higgs bosons are physical and on-shell, i.e.
${q_{i}}\cdot{\mathrm{J}_{i}(q_{i})}=0$; this means that external sources
cannot absorb/emit Fadeev-Popov ghosts. Internal lines in one-loop diagrams
represent fermions. Each one-loop amplitude can be decomposed as follows:

$\displaystyle\mathrm{A}$ $\displaystyle=$
$\displaystyle\frac{\mathrm{S}}{\mathrm{d}-4}+\mathrm{R}+\sum_{\mathrm{a}}\mathrm{f}_{\mathrm{a}}\,\mathrm{A}^{{\mbox{\scriptsize
fin}}}_{0}(\mathrm{a})+\sum_{\mathrm{b}}\mathrm{f}_{\mathrm{b}}\,\mathrm{B}^{{\mbox{\scriptsize
fin}}}_{0}(\mathrm{b})$ (41) $\displaystyle+$
$\displaystyle\sum_{\mathrm{c}}\mathrm{f}_{\mathrm{c}}\,\mathrm{C}_{0}(\mathrm{c})+\sum_{\mathrm{d}}\mathrm{f}_{\mathrm{d}}\,\mathrm{D}_{0}(\mathrm{d})\hskip
1.42262pt,$

where $\mathrm{A}_{0},\,\dots\,\mathrm{D}_{0}$ are scalar one,$\,\dots\,$four
point functions [5, 6], $\mathrm{d}$ is the space-time dimension and “fin”
denotes the UV finite part. It is worth noting that anomalies can be cancelled
by adding counterterms if and only if the anomaly is of $\mathrm{R}$ type,
i.e. UV finite and local. Locality of the UV-finite counterterms is related to
the unitarity of the theory.

###  5.2 Anomalies and anomalous terms

Anomalies and anomalous terms, although correlated, are not the same thing.
Anomalies have to do with WTST identities, perhaps the best example is given
by

$\Gamma_{\alpha\beta}=ip^{\lambda}\,\mathrm{A}^{{\scriptscriptstyle{\mathrm{Z}}}}_{\lambda\,;\,\alpha\,,\,\beta}+\mathrm{M}_{{\scriptscriptstyle{\mathrm{Z}}}}\,\mathrm{A}^{\phi}_{\alpha\,,\,\beta}\hskip
1.42262pt,$ (42)

the $\mathrm{Z}\mathrm{A}\mathrm{A}$ WTST identity. If this identity is
violated, say

$\Gamma_{\alpha\beta}=\mathrm{X}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\hskip
1.42262pt,$ (43)

with $\mathrm{X}$ UV-finite and local, we can restore it by adding to the
Lagrangian a term
$\mathrm{X}\,\varepsilon_{\mu\nu\alpha\beta}\,\phi^{0}\,(\partial_{\mu}\,\mathrm{A}_{\alpha})\,(\partial_{\nu}\,\mathrm{A}_{\beta})$.
One could think of introducing two counterterms,
$\mathrm{Z}\mathrm{A}\mathrm{A}$ and $\phi^{0}\mathrm{A}\mathrm{A}$, in order
to restore the identity and to cancel, at the same time, the anomalous
$\mathrm{Z}\mathrm{A}\mathrm{A}$ and $\phi^{0}\mathrm{A}\mathrm{A}$ couplings.
However, these counterterms are not local. Having loop-induced anomalous
couplings is not a surprise, even in the SM. The important thing is the
cancellation of anomalies, not the presence of anomalous couplings. The point
was explained long ago in Ref. [21]; consider any
$\mathrm{V}\mathrm{V}\mathrm{V}$ one-loop, SM, vertex, it will contain
$\varepsilon$-terms that contribute to the triangle anomaly. The cancellation
of the anomaly requires the contribution of the massive top quark. Writing

$\mathrm{V}=\sum_{\mathrm{f}}\,\mathrm{V}_{\mathrm{f}}(m_{\mathrm{f}}=0)+\Bigl{[}\mathrm{V}_{\mathrm{t}}(m_{\mathrm{t}}\not=0)-\mathrm{V}_{\mathrm{t}}(m_{\mathrm{t}}=0)\Bigr{]}\hskip
1.42262pt,$ (44)

it follows from the anomaly-cancellation conditions that all
$\varepsilon$-terms disappear in the (massless) sum on the right-hand side.
The remainder contains $\varepsilon$-terms and is $m_{\mathrm{t}}$-dependent.
This dependence is known to produce effects of delayed unitarity cancellation,
which may become relevant at high energies. Everything becomes clear when we
consider the explicit expressions in the SM for
$\mathrm{Z}_{\mu}(P)\to\mathrm{A}_{\alpha}(p_{1})+\mathrm{A}_{\beta}(p_{2})$
and $\phi^{0}(P)\to\mathrm{A}_{\alpha}(p_{1})+\mathrm{A}_{\beta}(p_{2})$:

$\displaystyle\mathrm{A}^{\phi}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{g^{3}}{12\,\pi^{2}}\,\frac{\mathrm{s}_{{}_{\theta}}^{2}}{\mathrm{c}_{{}_{\theta}}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\Bigl{[}3\,\sum_{\mathrm{l}}\frac{m^{2}_{\mathrm{l}}}{M^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{C}_{0}(m_{\mathrm{l}})$
$\displaystyle+$
$\displaystyle\sum_{\mathrm{d}}\frac{m^{2}_{\mathrm{d}}}{M^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{C}_{0}(m_{\mathrm{d}})-4\,\sum_{\mathrm{u}}\frac{m^{2}_{\mathrm{u}}}{M^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{C}_{0}(m_{\mathrm{u}})\Bigr{]}\hskip
1.42262pt,$
$\displaystyle\mathrm{A}^{{\scriptscriptstyle{\mathrm{Z}}}}_{\mu\,;\,\alpha\beta}$
$\displaystyle=$
$\displaystyle-i\,\frac{g^{3}}{12\,\pi^{2}}\,\frac{\mathrm{s}_{{}_{\theta}}^{2}}{\mathrm{c}_{{}_{\theta}}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\Bigl{[}3\,\sum_{\mathrm{l}}\frac{m^{2}_{\mathrm{l}}}{s}\,\mathrm{C}_{0}(m_{\mathrm{l}})$
(46) $\displaystyle+$
$\displaystyle\sum_{\mathrm{d}}\frac{m^{2}_{\mathrm{d}}}{s}\,\mathrm{C}_{0}(m_{\mathrm{d}})-4\,\sum_{\mathrm{u}}\frac{m^{2}_{\mathrm{u}}}{s}\,\mathrm{C}_{0}(m_{\mathrm{u}})\Bigr{]}\,P_{\mu}\hskip
1.42262pt,$

where $P=-p_{1}-p_{2}$, $P^{2}=-s$ and $\mathrm{C}_{0}(m_{\mathrm{f}})$ is the
scalar three-point function with internal $\mathrm{f}$ lines. The sum is over
leptons, up and down quarks. The WTST identity is obviously satisfied but both
amplitudes correspond to anomalous, non-local, couplings. As far as the
$\phi^{0}\mathrm{A}\mathrm{A}$ amplitude is concerned this fact is not so
relevant since $\phi^{0}$ is not an asymptotics state. Furthermore, the
$\mathrm{Z}\mathrm{A}\mathrm{A}$ amplitude gives zero when the $\mathrm{Z}$
boson is coupled to a conserved current. To summarize, the no-anomaly scheme
does not imply the absence of anomalous couplings. Cancellation of anomalous
couplings is not the question, cancellation of anomalies is. As we will
discuss in the next paragraph, this WTST identity is violated in the SMEFT.

###  5.3 Step $0$: WTST identities for two-point functions

For two-point functions scheme $1$ and scheme $2$ give the same result, i.e.
all WTST identities are satisfied, both in the SM and in the SMEFT [40, 77].
It is worth noting that the identities are violated if we use a scheme where
$\\{\gamma^{\bar{\mu}}\,,\,\gamma^{5}\\}=0$ and
$[\gamma^{\hat{\mu}}\,,\,\gamma^{5}]=0$, i.e. without using the anomalous
trace introduced in Ref. [63]. Of course, they can always be restored by
introducing local and UV-finite counterterms even in the presence of
evanescent terms [78]. These terms are formally zero in the limit
$\mathrm{d}\to 4$ but their effect must be carefully analyzed due to the
presence of UV poles.

###  5.4 Step $1$: WTST identities for three-point functions

In our notation $g$ is the $SU(2)$ coupling constant,
$\mathrm{c}_{{}_{\theta}}=\mathrm{M}_{{\scriptscriptstyle{\mathrm{W}}}}/\mathrm{M}_{{\scriptscriptstyle{\mathrm{Z}}}}$
is the cosine of the weak-mixing angle. Furthermore, we introduce the
following combinations,

$\displaystyle\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}}}=\mathrm{s}_{{}_{\theta}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+\mathrm{c}_{{}_{\theta}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\hskip
1.42262pt,$
$\displaystyle\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}}}=-\mathrm{c}_{{}_{\theta}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+\mathrm{s}_{{}_{\theta}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\hskip
1.42262pt,$
$\displaystyle\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}}}=\mathrm{s}_{{}_{\theta}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+\mathrm{c}_{{}_{\theta}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\hskip
1.42262pt,$
$\displaystyle\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}}}=-\mathrm{c}_{{}_{\theta}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+\mathrm{s}_{{}_{\theta}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\hskip
1.42262pt,$
$\displaystyle\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}}}=\mathrm{s}_{{}_{\theta}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+\mathrm{c}_{{}_{\theta}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\hskip
1.42262pt,$
$\displaystyle\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}}}=\mathrm{c}_{{}_{\theta}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}-\mathrm{s}_{{}_{\theta}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\hskip
1.42262pt,$ (47)

where $\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}}}$, etc. are Wilson
coefficients in the Warsaw basis [43]. It is worth noting that in SMEFT we
have both triangles and bubbles due to four-point vertices like
$\mathrm{A}\phi^{0}{\overline{\mathrm{f}}}\mathrm{f}$ etc. . If they are not
included, the anomaly contains an UV-divergent term. We introduce

$\mathrm{C}^{{\scriptscriptstyle{\mathrm{A}}}}=-\partial_{\mu}\,\mathrm{A}_{\mu}\hskip
1.42262pt,\;\;\mathrm{C}^{{\scriptscriptstyle{\mathrm{Z}}}}=-\partial_{\mu}\,\mathrm{Z}_{\mu}+\mathrm{M}_{{\scriptscriptstyle{\mathrm{Z}}}}\,\phi^{0}\hskip
1.42262pt,\;\;\mathrm{C}^{\pm}=-\partial_{\mu}\,\mathrm{W}^{\pm}_{\mu}+\mathrm{M}_{{\scriptscriptstyle{\mathrm{W}}}}\,\phi^{\pm}\hskip
1.42262pt.$ (48)

In the “diagrammatic” language of Veltman the validity of the WTST identities
is equivalent to the statement that the $\mathrm{C}$ are free fields and any
Green’s function with one or more external $\mathrm{C}\,$-sources is zero.
Considering the effective action $\mathrm{S}$, we observe that diagrams
determine $\mathrm{S}$ only up to an arbitrary choice of local counterterms
and we are free to redefine $\mathrm{S}$ by adding to it
$\mathrm{S}_{\mathrm{ct}}$ with an arbitrary coefficent [37].

####  5.4.1 WTST identity for
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{Z}}}}\mathrm{A}\mathrm{A}$

In this case we have
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{Z}}}}(P)\to\mathrm{A}_{\alpha}(p_{1})+\mathrm{A}_{\beta}(p_{2})$.
Summing over fermion generations we obtain

$\Gamma_{\alpha\,\beta}=\frac{g^{3}}{8\,\pi^{2}}\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}^{3}}\,g_{{}_{6}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\,\sum_{\\{\mathrm{g}\\}}\left(\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+2\,\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}\right)\hskip
1.42262pt,$ (49)

for schemes $1$ and $2$. As expected there is no anomaly in $\mathrm{dim}=4$
but there is one in $\mathrm{dim}=6$ which is mass dependent; The standard
treatment is that the anomaly can be removed by adding to the Lagrangian a
term proportional to
$\varepsilon_{\mu\nu\alpha\beta}\,\phi^{0}\,\partial^{\mu}\mathrm{A}^{\alpha}\,\partial^{\nu}\,\mathrm{A}^{\beta}$.

####  5.4.2 WTST identity for
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{Z}}}}\mathrm{Z}\mathrm{A}$

In this case we have
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{Z}}}}(P)\to\mathrm{Z}_{\alpha}(p_{1})+\mathrm{A}_{\beta}(p_{2})$
and obtain

$\displaystyle\Gamma^{\mathrm{S}1}_{\alpha\,\beta}$ $\displaystyle=$
$\displaystyle\frac{g^{3}}{32\,\pi^{2}}\,g_{{}_{6}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\,\sum_{\\{\mathrm{g}\\}}\Bigl{[}2\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}^{3}}\,\bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+2\,\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\bigr{)}$
(50) $\displaystyle+$
$\displaystyle\frac{1}{\mathrm{c}_{{}_{\theta}}^{4}}\,\bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{v}_{\mathrm{l}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+3\,\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{v}_{\mathrm{d}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+3\,\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{v}_{\mathrm{u}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}\bigr{)}$
$\displaystyle+$
$\displaystyle\frac{4}{3}\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}^{2}}\,\bigl{(}3\,\mathrm{a}^{(3)}_{\phi\mathrm{q}}+\mathrm{a}^{(1)}_{\phi\mathrm{q}}-8\,\mathrm{a}_{\phi\mathrm{u}}-2\,\mathrm{a}_{\phi\mathrm{d}}-3\,\mathrm{a}^{(3)}_{\phi\mathrm{l}}+3\,\mathrm{a}^{(1)}_{\phi\mathrm{l}}-6\,\mathrm{a}_{\phi\mathrm{l}}\bigr{)}$
$\displaystyle-$
$\displaystyle\frac{8}{3}\,\mathrm{s}_{{}_{\theta}}\,\bigl{(}3\,\mathrm{a}^{(3)}_{\phi\mathrm{q}}+5\,\mathrm{a}^{(1)}_{\phi\mathrm{q}}-4\,\mathrm{a}_{\phi\mathrm{u}}-\mathrm{a}_{\phi\mathrm{d}}-3\,\mathrm{a}^{(3)}_{\phi\mathrm{l}}+3\,\mathrm{a}^{(1)}_{\phi\mathrm{l}}-3\,\mathrm{a}_{\phi\mathrm{l}}\bigr{)}\Bigr{]}\hskip
1.42262pt,$ $\displaystyle\Gamma^{\mathrm{S}2}_{\alpha\,\beta}$
$\displaystyle=$
$\displaystyle\frac{g^{3}}{64\,\pi^{2}}\,g_{{}_{6}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\,\sum_{\\{\mathrm{g}\\}}\Bigl{[}4\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}^{3}}\,\bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+2\,\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\bigr{)}$
(51) $\displaystyle+$
$\displaystyle\frac{1}{\mathrm{c}_{{}_{\theta}}^{4}}\,\bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{v}_{\mathrm{l}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+3\,\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{v}_{\mathrm{d}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+3\,\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{v}_{\mathrm{u}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}\bigr{)}$
$\displaystyle-$ $\displaystyle
4\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}^{2}}\,\bigl{(}\mathrm{a}^{(3)}_{\phi\mathrm{q}}+3\,\mathrm{a}^{(1)}_{\phi\mathrm{q}}+2\,\mathrm{a}_{\phi\mathrm{u}}+\mathrm{a}_{\phi\mathrm{d}}-\mathrm{a}^{(3)}_{\phi\mathrm{l}}+\mathrm{a}^{(1)}_{\phi\mathrm{l}}+\mathrm{a}_{\phi\mathrm{l}}\bigr{)}\Bigr{]}\hskip
1.42262pt,$

where
$\mathrm{v}_{\mathrm{f}}=1-8\,\mathrm{Q}_{\mathrm{f}}\,\mathrm{I}^{3}_{\mathrm{f}}\,\mathrm{s}_{{}_{\theta}}^{2}$.
There is an anomaly in $\mathrm{dim}=6$ which is mass dependent but UV finite
and local; however the anomaly is scheme dependent.

####  5.4.3 WTST identity for
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{Z}}}}\mathrm{Z}\mathrm{Z}$

In this case we have
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{Z}}}}(P)\to\mathrm{Z}_{\alpha}(p_{1})+\mathrm{Z}_{\beta}(p_{2})$
and obtain

$\displaystyle\Gamma^{\mathrm{S}1}_{\alpha\,\beta}$ $\displaystyle=$
$\displaystyle\frac{g^{3}}{16\,\pi^{2}}\,g_{{}_{6}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\,\sum_{\\{\mathrm{g}\\}}\Bigl{[}\frac{1}{\mathrm{c}_{{}_{\theta}}^{4}}\,\bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{v}_{\mathrm{l}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+3\,\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{v}_{\mathrm{d}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+3\,\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{v}_{\mathrm{u}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}\bigr{)}$
(52) $\displaystyle+$
$\displaystyle\frac{8}{3}\,\frac{1}{\mathrm{c}_{{}_{\theta}}}\,\bigl{(}3\,\mathrm{a}^{(3)}_{\phi\mathrm{q}}+\mathrm{a}^{(1)}_{\phi\mathrm{q}}-8\,\mathrm{a}_{\phi\mathrm{u}}-2\,\mathrm{a}_{\phi\mathrm{d}}-3\,\mathrm{a}^{(3)}_{\phi\mathrm{l}}+3\,\mathrm{a}^{(1)}_{\phi\mathrm{l}}-6\,\mathrm{a}_{\phi\mathrm{l}}\bigr{)}$
$\displaystyle-$
$\displaystyle\frac{8}{3}\,\mathrm{c}_{{}_{\theta}}\,\bigl{(}3\,\mathrm{a}^{(3)}_{\phi\mathrm{q}}+5\,\mathrm{a}^{(1)}_{\phi\mathrm{q}}-4\,\mathrm{a}_{\phi\mathrm{u}}-\mathrm{a}_{\phi\mathrm{d}}-3\,\mathrm{a}^{(3)}_{\phi\mathrm{l}}+3\,\mathrm{a}^{(1)}_{\phi\mathrm{l}}-3\,\mathrm{a}_{\phi\mathrm{l}}\bigr{)}$
$\displaystyle+$
$\displaystyle\frac{4}{3}\,\frac{1}{\mathrm{c}_{{}_{\theta}}^{3}}\,\bigl{(}-\mathrm{a}^{(1)}_{\phi\mathrm{q}}+8\,\mathrm{a}_{\phi\mathrm{u}}+2\,\mathrm{a}_{\phi\mathrm{d}}-3\,\mathrm{a}^{(1)}_{\phi\mathrm{l}}+6\,\mathrm{a}_{\phi\mathrm{l}}\bigr{)}\Bigr{]}\hskip
1.42262pt,$ $\displaystyle\Gamma^{\mathrm{S}2}_{\alpha\,\beta}$
$\displaystyle=$
$\displaystyle\frac{g^{3}}{32\,\pi^{2}}\,g_{{}_{6}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\,\sum_{\\{\mathrm{g}\\}}\Bigl{[}\frac{1}{\mathrm{c}_{{}_{\theta}}^{4}}\,\bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{v}_{\mathrm{l}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+3\,\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{v}_{\mathrm{d}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+3\,\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{v}_{\mathrm{u}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\bigr{)}$
(53) $\displaystyle-$
$\displaystyle\frac{4}{\mathrm{c}_{{}_{\theta}}}\,\bigl{(}\mathrm{a}^{(3)}_{\phi\mathrm{q}}+3\,\mathrm{a}^{(1)}_{\phi\mathrm{q}}+2\,\mathrm{a}_{\phi\mathrm{u}}+\mathrm{a}_{\phi\mathrm{d}}-\mathrm{a}^{(3)}_{\phi\mathrm{l}}+\mathrm{a}^{(1)}_{\phi\mathrm{l}}+\mathrm{a}_{\phi\mathrm{l}}\bigr{)}$
$\displaystyle+$
$\displaystyle\frac{2}{3}\,\frac{1}{\mathrm{c}_{{}_{\theta}}^{3}}\,\bigl{(}6\,\mathrm{a}^{(3)}_{\phi\mathrm{q}}+6\,\mathrm{a}^{(1)}_{\phi\mathrm{q}}+9\,\mathrm{a}_{\phi\mathrm{u}}+3\,\mathrm{a}_{\phi\mathrm{d}}-6\,\mathrm{a}^{(3)}_{\phi\mathrm{l}}+2\,\mathrm{a}^{(1)}_{\phi\mathrm{l}}+5\,\mathrm{a}_{\phi\mathrm{l}}\bigr{)}\Bigr{]}\hskip
1.42262pt,$

where, once again, there is an anomaly in $\mathrm{dim}=6$ which is mass
dependent, scheme dependent but UV finite and local.

####  5.4.4 WTST identity for
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{Z}}}}\mathrm{W}^{+}\mathrm{W}^{-}$

In this case we have
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{Z}}}}(P)\to\mathrm{W}^{-}_{\alpha}(p_{1})+\mathrm{W}^{+}_{\beta}(p_{2})$
and obtain

$\displaystyle\Gamma^{\mathrm{S}1}_{\alpha\,\beta}$ $\displaystyle=$
$\displaystyle-\,\frac{g^{3}}{16\,\pi^{2}}\,g_{{}_{6}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\,\sum_{\\{\mathrm{g}\\}}\Bigl{[}\bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}-\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}\bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}^{3}}{\mathrm{c}_{{}_{\theta}}^{3}}$
(54) $\displaystyle+$
$\displaystyle\bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}-\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}^{2}}{\mathrm{c}_{{}_{\theta}}^{2}}-4\,\bigl{(}\mathrm{a}^{(3)}_{\phi\mathrm{q}}-\mathrm{a}^{(3)}_{\phi\mathrm{l}}\bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}^{2}}{\mathrm{c}_{{}_{\theta}}}\Bigr{]}\hskip
1.42262pt,$ $\displaystyle\Gamma^{\mathrm{S}2}_{\alpha\,\beta}$
$\displaystyle=$
$\displaystyle-\,\frac{g^{3}}{24\,\pi^{2}}\,g_{{}_{6}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\,\sum_{\\{\mathrm{g}\\}}\Bigl{[}\bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}-3\,\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}\bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}^{3}}$
(55) $\displaystyle+$
$\displaystyle\bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}-3\,\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\bigr{)}\,\frac{1}{\mathrm{c}_{{}_{\theta}}^{2}}$
$\displaystyle+$ $\displaystyle
3\,\bigl{(}2\,\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}\bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}}+3\,\bigl{(}2\,\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\bigr{)}$
$\displaystyle-$ $\displaystyle
8\,\bigl{(}\mathrm{a}^{(3)}_{\phi\mathrm{q}}-\mathrm{a}^{(3)}_{\phi\mathrm{l}}\bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}^{2}}{\mathrm{c}_{{}_{\theta}}}\Bigr{]}\hskip
1.42262pt.$

####  5.4.5 WTST identity for
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{A}}}}\mathrm{W}^{+}\mathrm{W}^{-}$

In this case we have
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{A}}}}(P)\to\mathrm{W}^{-}_{\alpha}(p_{1})+\mathrm{W}^{+}_{\beta}(p_{2})$
and obtain

$\displaystyle\Gamma^{\mathrm{S}1}_{\alpha\,\beta}$ $\displaystyle=$
$\displaystyle\frac{g^{3}}{16\,\pi^{2}}\,g_{{}_{6}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\,\sum_{\\{\mathrm{g}\\}}\Bigl{[}\bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}-\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}\bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}^{2}}{\mathrm{c}_{{}_{\theta}}^{2}}$
(56) $\displaystyle+$
$\displaystyle\bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}-\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}}-4\,\bigl{(}\mathrm{a}^{(3)}_{\phi\mathrm{q}}-\mathrm{a}^{(3)}_{\phi\mathrm{l}}\bigr{)}\,\mathrm{s}_{{}_{\theta}}\Bigr{]}\hskip
1.42262pt,$ $\displaystyle\Gamma^{\mathrm{S}2}_{\alpha\,\beta}$
$\displaystyle=$
$\displaystyle-\,\frac{g^{3}}{8\,\pi^{2}}\,g_{{}_{6}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\,\sum_{\\{\mathrm{g}\\}}\Bigl{[}\bigl{(}2\,\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}\bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}^{2}}{\mathrm{c}_{{}_{\theta}}^{2}}$
(57) $\displaystyle+$
$\displaystyle\bigl{(}2\,\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}}+\frac{8}{3}\,\bigl{(}\mathrm{a}^{(3)}_{\phi\mathrm{q}}-\mathrm{a}^{(3)}_{\phi\mathrm{l}}\bigr{)}\,\mathrm{s}_{{}_{\theta}}\Bigr{]}\hskip
1.42262pt,$

In this case the anomaly can be removed by adding to the Lagrangian a term
proportional to
$\varepsilon^{\mu\alpha\beta\nu}\,\mathrm{A}_{\nu}\,\bigl{(}\mathrm{W}^{+}_{\beta}\partial_{\mu}\mathrm{W}^{-}_{\alpha}-\mathrm{W}^{-}_{\alpha}\partial_{\mu}\mathrm{W}^{+}_{\beta}\bigr{)}$.

####  5.4.6 WTST identity for
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{A}}}}\mathrm{Z}\mathrm{Z}$ and
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{A}}}}\mathrm{Z}\mathrm{A}$

We only present the results for scheme $1$. They are

$\displaystyle\Gamma^{\mathrm{S}1}_{\alpha\,\beta}$ $\displaystyle=$
$\displaystyle\frac{g^{3}}{12\,\pi^{2}}\,g_{{}_{6}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\,\sum_{\\{\mathrm{g}\\}}\Bigl{[}\frac{3}{2}\,\Bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+2\,\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\Bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}^{3}}$
(58) $\displaystyle+$
$\displaystyle\bigl{(}3\,\mathrm{a}^{(3)}_{\phi\mathrm{q}}-3\,\mathrm{a}^{(3)}_{\phi\mathrm{l}}-8\,\mathrm{a}_{\phi\mathrm{u}}-2\,\mathrm{a}_{\phi\mathrm{d}}+\mathrm{a}^{(1)}_{\phi\mathrm{q}}+3\,\mathrm{a}^{(1)}_{\phi\mathrm{l}}-6\,\mathrm{a}_{\phi\mathrm{l}}\bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}^{2}}$
$\displaystyle-$ $\displaystyle
2\,\bigl{(}3\,\mathrm{a}^{(3)}_{\phi\mathrm{q}}-3\,\mathrm{a}^{(3)}_{\phi\mathrm{l}}-4\,\mathrm{a}_{\phi\mathrm{u}}-\mathrm{a}_{\phi\mathrm{d}}+5\,\mathrm{a}^{(1)}_{\phi\mathrm{q}}+3\,\mathrm{a}^{(1)}_{\phi\mathrm{l}}-3\,\mathrm{a}_{\phi\mathrm{l}}\bigr{)}\,\mathrm{s}_{{}_{\theta}}\Bigr{]}\hskip
1.42262pt,$

for $\mathrm{C}^{{\scriptscriptstyle{\mathrm{A}}}}\mathrm{Z}\mathrm{Z}$ and

$\displaystyle\Gamma^{\mathrm{S}1}_{\alpha\,\beta}$ $\displaystyle=$
$\displaystyle\frac{g^{3}}{16\,\pi^{2}}\,g_{{}_{6}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\,\sum_{\\{\mathrm{g}\\}}\Bigl{[}\Bigl{(}\frac{m^{2}_{\mathrm{l}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+2\,\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\Bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}^{3}}$
(59) $\displaystyle-$
$\displaystyle\frac{4}{3}\,\bigl{(}3\,\mathrm{a}^{(3)}_{\phi\mathrm{q}}-3\,\mathrm{a}^{(3)}_{\phi\mathrm{l}}-4\,\mathrm{a}_{\phi\mathrm{u}}-\mathrm{a}_{\phi\mathrm{d}}+5\,\mathrm{a}^{(1)}_{\phi\mathrm{q}}+3\,\mathrm{a}^{(1)}_{\phi\mathrm{l}}-3\,\mathrm{a}_{\phi\mathrm{l}}\bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}^{2}}{\mathrm{c}_{{}_{\theta}}}\Bigr{]}\hskip
1.42262pt,$

for $\mathrm{C}^{{\scriptscriptstyle{\mathrm{A}}}}\mathrm{Z}\mathrm{A}$.

####  5.4.7 WTST identity for
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{Z}}}}\mathrm{G}\mathrm{G}$

In this case (where $\mathrm{G}^{a}_{\mu}$ is the gluon field) we have
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{Z}}}}(P)\to\mathrm{G}^{a}_{\alpha}(p_{1})+\mathrm{G}^{b}_{\beta}(p_{2})$
and obtain

$\Gamma^{a\,b}_{\alpha\,\beta}=\frac{gg^{2}_{\mathrm{S}}}{16\,\pi^{2}\,\mathrm{c}_{{}_{\theta}}^{2}}\,g_{{}_{6}}\,\varepsilon_{\mu\nu\alpha\beta}\,p^{\mu}_{1}\,p^{\nu}_{2}\,\delta^{a\,b}\,\sum_{\\{\mathrm{g}\\}}\,\bigl{(}\frac{m^{2}_{\mathrm{d}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{G}}}+\frac{m^{2}_{\mathrm{u}}}{\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{Z}}}}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{G}}}\bigr{)}\hskip
1.42262pt,$ (60)

with a sum over the quark generations and where $g_{\mathrm{S}}$ is the
$SU(3)$ coupling constant. There is an anomaly in $\mathrm{dim}=6$ which is
mass dependent but UV finite and local.

####  5.4.8 WTST identity for $\mathrm{C}^{-}\mathrm{W}^{+}\mathrm{Z}$ and
$\mathrm{C}^{-}\mathrm{W}^{+}\mathrm{A}$

In these two cases we have found no anomaly in scheme $1$. However, a
technical remark is needed: for amplitudes having a Born term we must take
into account the relation between bare and renormalized parameters and also
include Dyson-resummation of the propagators (when needed). The identity reads
as follows:

$i\,p^{\lambda}\,\mathrm{A}^{{\scriptscriptstyle{\mathrm{W}}}}_{\lambda\alpha\beta}+\mathrm{M}\,\mathrm{A}^{\phi}_{\alpha\beta}=0\hskip
1.42262pt,$ (61)

where $\mathrm{M}$ is the bare $\mathrm{W}$ mass and we use
$\mathrm{M}_{0}=\mathrm{M}/\mathrm{c}_{{}_{\theta}}$ for the bare $\mathrm{Z}$
mass. Inside and in front of loops we will use the on-shell masses,
$\mathrm{M}_{{\scriptscriptstyle{\mathrm{W}}}}$ and
$\mathrm{M}_{{\scriptscriptstyle{\mathrm{Z}}}}$. However, in this case there
is a lowest order where the relation between bare and on-shell masses must be
corrected at $\mathcal{O}(g^{2})$, involving the corresponding self-energies.

There is an important remark to be made here: consider the SM
($\mathrm{dim}=4$), where there is no anomaly; consider now the SM with one
electron and one neutrino, scheme $1$ gives zero anomaly which is in conflict
with the master formula for the one-loop anomalies given in Ref. [76]. The
anomalies require a counterterm proportional to

$\varepsilon_{\mu\nu\alpha\beta}\,\Bigl{(}\mathrm{F}_{\mu\nu}\,\mathrm{G}^{a}_{\alpha\beta}\,\phi^{a}+\mathrm{F}_{\mu\nu}\,\mathrm{F}_{\alpha\beta}\,\phi^{0}\,\tan\theta\Bigr{)}\hskip
1.42262pt,$ (62)

where $\mathrm{F}_{\mu\nu}$ is the field strength of the original $U(1)$
vector boson, $\mathrm{G}^{a}_{\mu\nu}$ is the field strength of the original
$SU(2)$ vector boson and $\phi^{a}$ is the Higgs-Kibble ghost (with
$\phi^{3}\equiv\phi^{0}$). The correct result is reproduced with scheme $2$;
here the
$\mathrm{C}^{-}(p_{1})\mathrm{Z}_{\mu}(P)\mathrm{W}^{+}_{\beta}(p_{2})$
identity in SMEFT is given by

$\displaystyle\Gamma^{\mathrm{S}2}_{\beta\,\mu}$ $\displaystyle=$
$\displaystyle\frac{g^{3}}{192\,\pi^{2}\,\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{W}}}}}\,g_{{}_{6}}\,\varepsilon_{\alpha\nu\beta\mu}\,p^{\alpha}_{1}\,p^{\nu}_{2}\,$
(63) $\displaystyle\times$
$\displaystyle\sum_{\\{\mathrm{g}\\}}\Bigl{[}\Bigl{(}5\,m^{2}_{\mathrm{l}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+15\,m^{2}_{\mathrm{d}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}-3\,m_{\mathrm{u}}m_{\mathrm{d}}\,(\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}})+15\,m^{2}_{\mathrm{u}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}\Bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}}{\mathrm{c}_{{}_{\theta}}}$
$\displaystyle+$ $\displaystyle
12\,\Bigl{(}m^{2}_{\mathrm{l}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+3\,m^{2}_{\mathrm{d}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+3\,m^{2}_{\mathrm{u}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\Bigr{)}$
$\displaystyle-$ $\displaystyle
2\,\Bigl{(}7\,m^{2}_{\mathrm{l}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+5\,m^{2}_{\mathrm{d}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}+m_{\mathrm{u}}m_{\mathrm{d}}\,(\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}-\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{W}\mathrm{B}}})+13\,m^{2}_{\mathrm{u}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{W}\mathrm{B}}}\Bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}^{3}}{\mathrm{c}_{{}_{\theta}}}$
$\displaystyle-$ $\displaystyle
2\,\Bigl{(}7\,m^{2}_{\mathrm{l}}\,\mathrm{a}_{\mathrm{l}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+5\,m^{2}_{\mathrm{d}}\,\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}+m_{\mathrm{u}}m_{\mathrm{d}}\,(\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}-\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{B}\mathrm{W}}})+13\,m^{2}_{\mathrm{u}}\,\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{B}\mathrm{W}}}\Bigr{)}\,\mathrm{s}_{{}_{\theta}}^{2}$
$\displaystyle+$ $\displaystyle
32\,\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{W}}}}\Bigl{(}\mathrm{a}^{(3)}_{\phi\mathrm{q}}-\mathrm{a}^{(3)}_{\phi\mathrm{l}}\Bigr{)}\,\frac{\mathrm{s}_{{}_{\theta}}^{2}}{\mathrm{c}_{{}_{\theta}}}-32\,\mathrm{M}^{2}_{{\scriptscriptstyle{\mathrm{W}}}}\,\Bigl{(}3\,\mathrm{a}^{(1)}_{\phi\mathrm{q}}+\mathrm{a}^{(1)}_{\phi\mathrm{l}}\Bigr{)}\,\frac{1}{\mathrm{c}_{{}_{\theta}}}\Bigr{]}\hskip
1.42262pt.$

###  5.5 SMEFT and anomalies: conclusions

There are SMEFT anomalies, UV-finite, local, mass-dependent and scheme-
dependent. There are different options, for instance use consistently a
scheme, e.g. the (naive) anti-commuting $\gamma^{5}$ scheme or the Veltman
scheme and introduce counterterms. This procedure is the one which we could
use working with the SM with one electron and one neutrino: due to the
anomaly, the WTST indentities break down at the one-loop level. We then
introduce counterterms and the identities are restored but the terms are of a
non-renormalizable nature and they give rise to infinities, at the earliest at
the two-loop level, as described in Ref. [76].

Alternatively, we could cancel the anomalies. For instance, within the anti-
commuting $\gamma^{5}$ scheme we obtain the following relations:

$\displaystyle\mathrm{a}_{\mathrm{f}\scriptscriptstyle{\mathrm{B}}}=\mathrm{a}_{\mathrm{f}\scriptscriptstyle{\mathrm{W}}}=0\quad\forall\mathrm{f}\hskip
1.42262pt,\quad\mathrm{a}_{\mathrm{u}\scriptscriptstyle{\mathrm{G}}}=\mathrm{a}_{\mathrm{d}\scriptscriptstyle{\mathrm{G}}}=0\hskip
1.42262pt,$
$\displaystyle\mathrm{a}^{(3)}_{\phi\mathrm{l}}=\mathrm{a}^{(3)}_{\phi\mathrm{q}}\hskip
1.42262pt,\;\mathrm{a}^{(1)}_{\phi\mathrm{l}}=-3\,\mathrm{a}^{(1)}_{\phi\mathrm{q}}\hskip
1.42262pt,\;\mathrm{a}_{\phi\mathrm{l}}=-\frac{1}{3}\,\Bigl{(}4\,\mathrm{a}^{(1)}_{\phi\mathrm{q}}+4\,\mathrm{a}_{\phi\mathrm{u}}+\mathrm{a}_{\phi\mathrm{d}}\Bigr{)}\hskip
1.42262pt.$ (64)

Similar relations, obtained within the Veltman scheme, are less “transparent”.
It is worth noting that in the SMEFT there is a breakdown of the WTST
identities, no matter which scheme is used. As long as the anomalies are UV-
finite and local (which is the “tested” case up to three-point functions) they
can be removed, within a given scheme, by counterterms.

One possible question is whether to include Eq.(64) in fits [79, 62]. We have
shown that the cancellation of SMEFT anomalies is scheme dependent (extending
to the full SMEFT the work of Ref. [62]) but there is also another point [61]:
what happens if we have a UV complete and anomaly-free underlying theory whose
low-energy behavior violates the identities? There doesn’t seem to be any
obvious solution if we insist with this strategy; note that only
$\mathrm{a}_{\mathrm{f}\scriptscriptstyle{\mathrm{B}}}$ and
$\mathrm{a}_{\mathrm{f}\scriptscriptstyle{\mathrm{W}}}$ are loop-generated
[80] $\mathrm{dim}=6$ operators that we have inserted in loops and mass-
dependent anomalies are due to loop-generated operators. Having said that, the
fact that using the relations in Eq.(64) is enough to cancel the
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{Z}}}}$ and
$\mathrm{C}^{{\scriptscriptstyle{\mathrm{A}}}}$ anomalies remains an
intriguing consideration involving the fermions.

A final comment follows from the work of Ref. [76]: one should investigate the
divergences arising in higher orders from these counterterms, something
happenning if the counterterms are part of a loop (these diagrams must be seen
as two-loop diagrams). The result is that there are several logarithmically
divergent amplitudes, e.g. an anomalous magnetic moment for the fermions.

WTST identities for four-point functions show new interesting aspect to be
presented elsewhere.

##  6 Conclusions

It is straightforward to see that when it comes to confronting theoretical
predictions with experimental results there will always be a reference to a
work of Martinus Veltman.

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