# One-loop renormalization of VBS with the electroweak chiral Lagrangian in
covariant gauges

Maria J. Herrero  and Roberto A. Morales

###### Abstract

This work presents a first full one-loop computation of vector boson
scattering (VBS) within the non-linear effective field theory given by the
bosonic sector of the usually called electroweak chiral Lagrangian (EChL). The
computation is performed in the most general case of covariant $R_{\xi}$
gauges and is compared through all this work with the Standard Model case,
whose computation in these covariant gauges is also novel and is presented
also here. The calculation of the one-loop VBS amplitude is performed using
the diagrammatic method by means of the one-particle-irreducible (1PI) Green
functions that are involved in these scattering processes. The central part of
this work is then devoted to the renormalization of all the $n$-legs one-loop
1PI Green functions involved. This renormalization is performed in the most
general off-shell case with arbitrary external legs momenta. We then describe
in full detail the renormalization program, which within this context of the
EChL, implies to derive all the counterterms for both the electroweak
parameters, like boson masses and gauge couplings, and those for the EChL
coefficients. These later are crucial for the renormalization of the new
divergences typically appearing when computing loops with the lowest chiral
dimension Lagrangian. We present here the full list of involved divergences
and counterterms in the $R_{\xi}$ gauges and derive the complete set of
renormalization group equations for the EChL coefficients. In the last part of
this work, we present the EChL numerical results for the one-loop cross
section in the $WZ$ channel and compare them with the SM results.

IFT-UAM/CSIC-21-30

##  1 Introduction

The use of Effective Field Theories (EFTs) to describe the phenomenology of
new physics beyond the Standard Model (SM) of elementary particle interactions
is nowadays a quite generalized tool when comparing theoretical predictions
with experimental data. The most appealing feature of an Effective Field
Theory (EFT) approach is that it can be used as a generic test of the new
physics without specifying the underlying ultraviolet fundamental theory that
originates such low energy theory. For a proper EFT, it is sufficient to
require it to preserve the same symmetries of the SM, in particular, the
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ gauge invariance, and to be able to
deal with quantum corrections, providing a framework for renomalization. The
physical active fields in these EFTs are as in the SM, and include fermions
(quarks and leptons), gauge bosons (both electroweak (EW) and strong) and the
Higgs field. At present, the most popular EFTs are the Standard Model
Effective Field Theory (SMEFT) and the Higgs Effective Field Theory (HEFT)
(for a review see, for instance, [1]). The main difference between these two
EFTs is the realization of the EW gauge symmetry in the scalar sector, i.e.,
in the system formed by the Higgs particle, $H$, and the three EW would be
Golstone bosons, named here $\pi^{a}$ ($a=1,2,3$). It is important to recall
that, within the SM context, this scalar system is the responsible for the
mass generation of all the SM particles, via the Higgs mechanism, once the EW
symmetry, $SU(2)_{L}\times U(1)_{Y}$, is spontaneously broken down to the
electromagnetism subgroup, $U(1)_{\rm em}$. One crucial point to notice is
that these three associated GBs to this breaking are indeed the same three GBs
that are associated to the spontaneous breaking of the EW chiral symmetry,
$SU(2)_{L}\times SU(2)_{R}\to SU(2)_{L+R}$. This EW chiral symmetry is an
exact global symmetry of the SM scalar sector in the limit of vanishing EW
gauge couplings, i.e., $g$ and $g^{\prime}$ set to zero, and is the main
responsible for protecting the mass relation $m_{W}=m_{Z}\cos\theta_{W}$ from
large radiative corrections. Thus, $SU(2)_{L+R}$ is sometimes called custodial
symmetry group.

The realization of both the EW gauge symmetry and the EW chiral symmetry in
the scalar sector is linear in the SM and the SMEFT, whereas it is non-linear
in the HEFT. The Higgs boson and the three GBs are placed together in a linear
representation in the case of SM and SMEFT, given by the usual doublet
$\Phi^{T}=(i\pi^{+}\,,\,((H+v)-i\pi^{3})/\sqrt{2})$, with
$\pi^{\pm}=(\pi^{1}\mp i\pi^{2})/\sqrt{2}$, whereas they are treated
differently in the HEFT. In this HEFT, the three GBs are put together in a
non-linear representation, usually by an exponential parametrization,
$U(\pi^{a})={\rm exp}(i\pi^{a}\tau^{a}/v)$ with $v=246\,{\rm GeV}$ and
$\tau^{a}=1,2,3$ the Pauli matrices, and the Higgs boson, in contrast, is a
singlet under all the involved symmetries. This difference between linear and
non-linear behaviour of the EW scalar sector in the two EFTs, SMEFT and HEFT,
is not just a mere choice of parametrization but it contains important
differences in their respective phenomelogical implications. The main reason
of this relevant difference is because of the different hierarchies in the
ordering and importance of the participating effective operators in both
approaches. To understand deeply this main difference one has to place the
comparison of both EFTs not only at the tree level, but also beyond the tree
level, i.e., including ${\cal O}(\hbar)$ loop corrections, and dealing with
the important renormalization issue in both theories. In contrast to the usual
ordering based on canonical dimension counting in the SMEFT, the ordering of
effective operators in the HEFT is by construction done by increasing powers
of their chiral dimension, starting in chiral dimension 2, next chiral
dimension 4, and so on. This implies that the ordering in the importance of
the HEFT effective operators is given in terms of an expansion in powers of
momenta, $(p/(4\pi v))^{n}$, where $4\pi v\sim 3\,{\rm TeV}$ sets typically
the maximum energy scale of applicability in this EFT.

In this mentioned chiral counting it is important to notice that all EW mass
scales are treated as soft mass scales, therefore, counting equally as
momentum, namely, $\partial_{\mu}\,,\,m_{\rm W}\,,\,m_{\rm Z}\,,\,m_{\rm
H}\,,\,g{\it v}\,,\,g^{\prime}{\it v}\sim\mathcal{O}(p)\,$. This also orders
the importance of the loop corrections in this EFT. This reasoning is done in
analogy to the usual Chiral Perturbation Theory (ChPT) of low energy QCD [2,
3, 4], where the most important contributions come from pion loops (the so
called chiral loops of QCD or GB loops) which naturally come with the loop
factors $(1/(4\pi f_{\pi}))^{n}$, where $4\pi f_{\pi}\sim 1.2\,{\rm GeV}$, and
the polynomial contributions in powers of the external momenta, $p^{n}$,
introduced by the derivative interactions of the GBs in this non-linear
approach. The renormalization program of ChPT is well known and very
successful and teaches us the way to proceed in the non-linear EFTs. It relays
on the key point that the chiral dimension four operators act as well as
counterterms of the one-loop corrections generated by the lowest chiral
dimension two operators. Thus, when computing an observable in ChPT to ${\cal
O}(\hbar)$, like for instance the pion-pion scattering amplitude, it is
sufficient to compute loops of pions with the lowest order chiral dimension
two Lagrangian, and to treat the chiral dimension four operators at the tree
level, acting simultaneously as counterterms to renormalize the loop
divergenges generated from those loops. The approach followed by the HEFT is
very similar to the one of ChPT. It also organizes the ordering of the
effective operators by their increasing chiral dimension as ${\cal L}_{\rm
HEFT}={\cal L}_{2}+{\cal L}_{4}+...$, and the renormalization program is
implemented in practice by computing loops with just ${\cal L}_{2}$ and
treating ${\cal L}_{4}$ to tree level and simultaneously using it as
counterterms to renormalize the loop divergences generated by those loops.
Thus, the renormalization of the EFT coefficients in ${\cal L}_{4}$ is
dictated by the loop divergences generated from ${\cal L}_{2}$. The difference
with respect to ChPT is that in the HEFT all kind of loops participate, with
gauge bosons, scalar bosons (GBs and Higgs boson), etc, not only GB loops.
This renormalization program is totally different in the case of SMEFT where
the effective operators, which are ordered by their increasing canonical
dimension, four, five, six and so on, as ${\cal L}_{\rm SMEFT}={\cal
L}_{c4}+{\cal L}_{c5}+{\cal L}_{c6}+..$, are treated all at both the tree and
the loop level, and the so-called Wilson coefficients of those effective
operators participate in the loop contributions to a given observable. This
sets important differences also on the hierarchies of the quantum corrections
on both EFT approaches.

We focus here on the HEFT and more concretely on the bosonic sector, with EW
gauge bosons, GBs and the Higgs boson, assuming that the fermion sector is as
in the SM. The corresponding effective Lagrangian of the HEFT in that case is
usually called the electroweak chiral Lagrangian (EChL). The main focus of
this work is on the one-loop renormalization program for this EFT when applied
to a practical computation of physical scattering processes. In particular, we
focus here on the most sensitive observables to the new physics in the case of
an hypothetical underlying strongly interacting UV theory, which are the
vector boson scattering (VBS) processes, with $V=W^{\pm},Z$ vector bosons in
the external legs (for a recent review on VBS see, for instance, [5]). At high
energies compared to the gauge boson masses, and by virtue of the Equivalence
Theorem (ET), the VBS amplitude for external longitudinal vector bosons, is
approximately equal to the scattering amplitude with the vector bosons
replaced by the corresponding GBs, $\pi^{\pm},\pi^{3}$, and, therefore, one
expects to find in observables like the cross section $\sigma(V_{L}V_{L}\to
V_{L}V_{L})$ the most prominent signals of the underlying UV strongly
interacting theory. This expectation is in close analogy to the case of the
pion-pion scattering predictions from ChPT being one of the best low energy
hints of the strongly interacting underlying QCD dynamics. In that case, it is
well known that a linear approach to the pions self-interactions (the so-
called sigma model) does not provide as good predictions of the pion-pion
scattering rates as it does the non-linear ChPT approach.

In this work we present the computation within the EChL of the full one-loop
scattering VBS amplitudes including all ${\cal O}(\hbar)$ radiative
corrections from the complete bosonic sector and the corresponding cross
sections. We use the standard Feynman diagrammatic approach and describe the
full renormalization program also in terms of one-loop Feynman diagrams. We
organize this computation in terms of the involved one-particle-irreducible
(1PI) Green functions, with two, three and four external legs, and perform by
the first time in the literature this full one-loop computation in the generic
$R_{\xi}$ gauges. This is a crucial point of the computation since it provides
a good check of the gauge invariance of the result. For definiteness, we will
describe the details of the computation for the particular VBS channel, $WZ\to
WZ$, and leave for future works the details of the other VBS channels. We also
present a comparison of our renormalization program for the non-linear EFT
with previous related works in the literature. The one-loop renormalization of
the EChL has been studied in [6, 7, 8]. These works are focused, in contrast
to our work, in the renormalization of the Lagrangian itself and use the path
integral formalism and the background field method to compute the one-loop
divergences that have to be renormalized by the redefinition of the EChL
coefficients. They use in addition the equations of motion to reduce the
number of independent effective operators. Our computation in contrast does
not make use of the equations of motion since our objective is to compute off-
shell renormalized 1PI Green functions. On the other hand, the one-loop VBS
amplitudes within the EChL were computed previously in [9, 10, 11] but using
the ET, namely, taking into account only the scalar sector, with external GBs
and with pure scalar loops of GBs and Higgs bosons, and considering massless
GBs. Also the one-loop renormalization program within the EChL in [12] is
performed in the pure scalar theory. In this latter reference they consider
the off-shell effects and therefore use the full set of effective operators of
the scalar sector. Our work here represents, therefore, the first full one-
loop computation of VBS with all relevant features included: off-shell
renormalized 1PI Green functions, all effective operators up to chiral
dimension four taken into account, massive gauge bosons in the VBS amplitudes
(we do not use the ET), full bosonic loops with gauge bosons and scalar bosons
and full renormalization program using the $R_{\xi}$ gauges. Our one-loop VBS
results within the EChL are, therefore, complete and include all finite
radiative corrections from the bosonic sector. The renormalization conditions
are fixed here in the on-shell scheme for the EW parameters, like masses and
couplings, and in the $\overline{MS}$ scheme for the effective operator
coefficients. One of the most interesting results in this work is our finding
of the $\overline{MS}$ renormalized EChL coefficients and their running with
the renormalization scale, that we compute from the involved renormalized 1PI
Green functions with the generic $R_{\xi}$ gauges and that we find independent
on the $\xi$ gauge-fixing parameter.

The paper is organized as follows. The main features of the EChL with
$R_{\xi}$ gauge-fixing and the relevant operators for VBS processes are
presented in Section 2. The diagrammatic computation of VBS in terms of the
1PI functions is described in Section 3. The central part of the work is in
Section 4 where the renormalization program is presented in detail, including
the prescriptions for regularization and renormalization assumed and the
summary of all the results from the renormalization process. All the divergent
counterterms and the derived renormalization group equations are also
presented in this section. A discussion on the Slavnov-Taylor identities and
the peculiarities of the Higgs tadpole in the non-linear EFT and the
comparison with the SM case is also included in this section. The numerical
results for the one-loop $WZ$ scattering process within the EChL and the SM
are presented and discussed in Section 5. Finally, we conclude in Section 6.
The appendices summarize some technical aspects of the computation. Appendix A
collects the relevant Feynman Rules (FRs) and the chosen conventions. Appendix
B contains the generic one-loop diagrams involved. Appendices C-D contain the
technical details of the comparison between the SM and the EChL computation,
including the discussion on the STI and the Higgs tadpole role.

##  2 The electroweak chiral Lagrangian using the $R_{\xi}$ gauge-fixing

In this section we shortly summarize the most relevant pieces of the EChL for
the present computation, and introduce some necessary notation. The active
fields of the EChL are the EW gauge bosons, $W^{a}_{\mu}$ ($a=1,2,3$) and
$B_{\mu}$, that are associated to $SU(2)_{L}$ and $U(1)_{Y}$, respectively,
the three GBs $\pi^{a}$ ($a=1,2,3$), and the Higgs boson $H$. The GBs are
introduced in a non-linear representation, usually via the exponential
parametrization, by means of the matrix $U$:

$U(\pi^{a})=e^{i\pi^{a}\tau^{a}/{\it v}}\,\,,$ (2.1)

where, $\tau^{a}$, $a=1,2,3$, are the Pauli matrices and $v=246$ GeV. Under a
EW chiral transformation of $SU(2)_{L}\times SU(2)_{R}$, given by $L\in
SU(2)_{L}$ and $R\in SU(2)_{R}$, the field $U$ transforms linearly as
$LUR^{\dagger}$, whereas the GBs $\pi^{a}$ transform non-linearly. This
peculiarity implies multiple GBs interactions, not just among themselves but
also with the other fields, and it is the main feature of this non-linear EFT,
which is clearly manifest in the following expansion:

$\displaystyle U(\pi^{a})$ $\displaystyle=$ $\displaystyle
I_{2}+i\frac{\pi^{a}}{{\it
v}}\tau^{a}-\frac{2\pi^{+}\pi^{-}+\pi^{3}\pi^{3}}{2{\it
v}^{2}}I_{2}-i\frac{(2\pi^{+}\pi^{-}+\pi^{3}\pi^{3})\pi^{a}}{6{\it
v}^{3}}\tau^{a}+\ldots$ (2.2)

where $I_{2}$ is the unity matrix and the dots stand for terms with four or
more GBs.

The $H$ field is, in contrast to the GBs, a singlet of the EW chiral symmetry
and the EW gauge symmetry and, consequently, there are not limitations from
symmetry arguments on the implementation of this field and its interactions
with itself and with the other fields. Usually, in the EChL, the interactions
of $H$ are introduced via generic polynomials.

The EW gauge bosons are introduced in the EChL by means of the
$SU(2)_{L}\times U(1)_{Y}$ gauge prescription, namely, via the covariant
derivative of the $U$ matrix, and by the $SU(2)_{L}$ and $U(1)_{Y}$ field
strength tensors, as follows:

$\displaystyle D_{\mu}U$ $\displaystyle=$
$\displaystyle\partial_{\mu}U+i\hat{W}_{\mu}U-iU\hat{B}_{\mu}\,,$
$\displaystyle\hat{W}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\hat{W}_{\nu}-\partial_{\nu}\hat{W}_{\mu}+i[\hat{W}_{\mu},\hat{W}_{\nu}]\,,\quad\hat{B}_{\mu\nu}=\partial_{\mu}\hat{B}_{\nu}-\partial_{\nu}\hat{B}_{\mu}\,,$
(2.3)

where $\hat{W}_{\mu}=gW^{a}_{\mu}\tau^{a}/2$, and
$\hat{B}_{\mu}=g^{\prime}B_{\mu}\tau^{3}/2$. For the construction of the EChL
and in addition to these basic building blocks, it is also customary to use
the following objects:

$\displaystyle\mathcal{V}_{\mu}$ $\displaystyle=$
$\displaystyle(D_{\mu}U)U^{\dagger}\,,\quad{\cal
D}_{\mu}O=\partial_{\mu}O+i[\hat{W}_{\mu},O]\,.$ (2.4)

The physical gauge fields are given, as usual, by:

$W_{\mu}^{\pm}=\frac{1}{\sqrt{2}}(W_{\mu}^{1}\mp iW_{\mu}^{2})\,,\quad
Z_{\mu}=c_{W}W_{\mu}^{3}-s_{W}B_{\mu}\,,\quad
A_{\mu}=s_{W}W_{\mu}^{3}+c_{W}B_{\mu}\,,$ (2.5)

where we use the short notation $s_{W}=\sin\theta_{W}$ and
$c_{W}=\cos\theta_{W}$, with $\theta_{W}$ the weak angle.

For the present computation we include in the EChL all the relevant effective
operators, organized as usual by their chiral dimension into two terms: ${\cal
L}_{2}$, with chiral dimension two and ${\cal L}_{4}$ with chiral dimension
four. As already said in the introduction, for this chiral counting, we
consider that all involved masses count equally as momentum, namely, with
chiral dimension one. Consequently, $\partial_{\mu}\,,\,m_{\rm W}\,,\,m_{\rm
Z}\,,\,m_{\rm H}\,,\,g{\it v}\,,\,g^{\prime}{\it v}\,,\lambda
v\,\sim\mathcal{O}(p)\,$. Thus, for the present work, the relevant
$SU(2)_{L}\times U(1)_{Y}$ gauge invariant EChL is given, in short, by

${\cal L}_{\rm EChL}=\mathcal{L}_{2}+\mathcal{L}_{4}\,.$ (2.6)

The relevant chiral dimension two Lagrangian is as follows,

$\displaystyle\mathcal{L}_{2}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{4}\left(1+2a\frac{H}{v}+b\left(\frac{H}{v}\right)^{2}+\ldots\right){\rm
Tr}\Big{[}D_{\mu}U^{\dagger}D^{\mu}U\Big{]}+\frac{1}{2}\partial_{\mu}H\partial^{\mu}H-V(H)$
(2.7) $\displaystyle-\frac{1}{2g^{2}}{\rm
Tr}\Big{[}\hat{W}_{\mu\nu}\hat{W}^{\mu\nu}\Big{]}-\frac{1}{2g^{{}^{\prime}\,2}}{\rm
Tr}\Big{[}\hat{B}_{\mu\nu}\hat{B}^{\mu\nu}\Big{]}+\mathcal{L}_{GF}+\mathcal{L}_{FP}\,,$

where, $\mathcal{L}_{GF}$, and $\mathcal{L}_{FP}$, are the gauge-fixing
Lagrangian and Fadeev-Popov Lagrangian, respectively, and $V(H)$ is the Higgs
potential, which we take here as,

$V(H)=\frac{1}{2}m_{\rm H}^{2}H^{2}+\kappa_{3}\lambda{\it
v}H^{3}+\kappa_{4}\frac{\lambda}{4}H^{4}\,,$ (2.8)

with $m_{H}^{2}=2\lambda v^{2}$. The new physics BSM in $\mathcal{L}_{2}$ is
encoded in the chiral coefficients $a$, $b$, $\kappa_{3}$ and $\kappa_{4}$.
These are all generically different from one, which is their reference SM
value. In Eq.(2.7) above, the dots stand for terms with more than two Higgs
bosons (allowed by the fact that $H$ is a singlet in this non-linear EFT) but
we omit them here since they do not enter in our VBS processes of interest,
neither at tree level nor at one-loop level.

The relevant chiral dimension four Lagrangian is organized here as follows:

$\displaystyle{\mathcal{L}}_{4}$ $\displaystyle=$
$\displaystyle{\mathcal{L}}_{4}^{\rm no-Higgs}+{\mathcal{L}}_{4}^{\rm one-
Higgs}+{\mathcal{L}}_{4}^{\rm two-Higgs}+\ldots,$ (2.9)

where ‘no-Higgs’ means effective operators not including the Higgs field,
‘one-Higgs’ means effective operators including one Higgs field, and so on,
and the dots again means operators which do not enter in the present
computation.

The list of operators in ${\mathcal{L}}_{4}^{\rm no-Higgs}$ were given long
ago in [13] and summarize the complete set of CP and $SU(2)_{L}\times
U(1)_{Y}$ gauge invariant effective operators, including both custodial
preserving and custodial breaking ones. We take them from this reference,
although using a different notation:

$\displaystyle{\mathcal{L}}_{4}^{\rm no-Higgs}$ $\displaystyle=$
$\displaystyle\,a_{0}\left(m_{\rm Z}^{2}-m_{\rm W}^{2}\right){\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal V}_{\mu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal V}^{\mu}\Big{]}$ (2.10)
$\displaystyle+a_{1}{\rm
Tr}\Big{[}U\hat{B}_{\mu\nu}U^{\dagger}\hat{W}^{\mu\nu}\Big{]}+ia_{2}{\rm
Tr}\Big{[}U\hat{B}_{\mu\nu}U^{\dagger}[{\cal V}^{\mu},{\cal
V}^{\nu}]\Big{]}-ia_{3}{\rm Tr}\Big{[}\hat{W}_{\mu\nu}[{\cal V}^{\mu},{\cal
V}^{\nu}]\Big{]}$ $\displaystyle+a_{4}{\rm Tr}\Big{[}{\cal V}_{\mu}{\cal
V}_{\nu}\Big{]}{\rm Tr}\Big{[}{\cal V}^{\mu}{\cal V}^{\nu}\Big{]}+a_{5}{\rm
Tr}\Big{[}{\cal V}_{\mu}{\cal V}^{\mu}\Big{]}{\rm Tr}\Big{[}{\cal
V}_{\nu}{\cal V}^{\nu}\Big{]}$ $\displaystyle+a_{6}{\rm Tr}\Big{[}{\cal
V}_{\mu}{\cal V}_{\nu}\Big{]}{\rm Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal
V}^{\mu}\Big{]}{\rm Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal
V}^{\nu}\Big{]}+a_{7}{\rm Tr}\Big{[}{\cal V}_{\mu}{\cal V}^{\mu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal V}^{\nu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal V}_{\nu}\Big{]}$
$\displaystyle-\frac{a_{8}}{4}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}\hat{W}_{\mu\nu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}\hat{W}^{\mu\nu}\Big{]}-i\frac{a_{9}}{2}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}\hat{W}_{\mu\nu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}[{\cal V}^{\mu},{\cal V}^{\nu}]\Big{]}$
$\displaystyle+a_{10}{\rm Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal
V}^{\mu}\Big{]}{\rm Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal V}_{\mu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal V}^{\nu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal V}_{\nu}\Big{]}$ $\displaystyle+a_{11}{\rm
Tr}\Big{[}{\cal D}_{\mu}{\cal V}^{\mu}{\cal D}_{\nu}{\cal
V}^{\nu}\Big{]}+a_{12}{\rm Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal D}_{\mu}{\cal
D}_{\nu}{\cal V}^{\nu}\Big{]}{\rm Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal
V}^{\mu}\Big{]}$ $\displaystyle+\frac{a_{13}}{2}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal D}_{\mu}{\cal V}_{\nu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal D}^{\mu}{\cal V}^{\nu}\Big{]}\,.$

The relation with the notation in [13] is: $a_{0}=\beta_{1}$,
$a_{1}=(g/g^{\prime})\alpha_{1}$, $a_{2}=(g/g^{\prime})\alpha_{2}$,
$a_{3}=-\alpha_{3}$, $a_{4}=\alpha_{4}$, $a_{5}=\alpha_{5}$;
$a_{6}=\alpha_{6}$, $a_{7}=\alpha_{7}$, $a_{8}=-g^{2}\alpha_{8}$,
$a_{9}=-g\alpha_{9}$, $a_{10}=\alpha_{10}/2$, $a_{11}=\alpha_{11}$,
$a_{12}=\alpha_{12}/2$ and $a_{13}=\alpha_{13}$.

The set of relevant chiral dimension four effective operators with one-Higgs
field and two-Higgs fields are taken from [14], but using a different
notation:

$\displaystyle{\mathcal{L}}_{4}^{\rm one-Higgs}$ $\displaystyle=$
$\displaystyle-a_{HBB}\frac{H}{v}\,{\rm
Tr}\Big{[}\hat{B}_{\mu\nu}\hat{B}^{\mu\nu}\Big{]}-a_{HWW}\frac{H}{v}{\rm
Tr}\Big{[}\hat{W}_{\mu\nu}\hat{W}^{\mu\nu}\Big{]}+a_{\Box\mathcal{V}\mathcal{V}}\frac{\Box
H}{{\it v}}{\rm Tr}\Big{[}{\cal V}_{\mu}{\cal V}^{\mu}\Big{]}$
$\displaystyle+a_{H0}\frac{H}{v}\left(m_{\rm Z}^{2}-m_{\rm W}^{2}\right){\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal V}_{\mu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal V}^{\mu}\Big{]}+a_{H1}\frac{H}{v}{\rm
Tr}\Big{[}U\hat{B}_{\mu\nu}U^{\dagger}\hat{W}^{\mu\nu}\Big{]}$
$\displaystyle-\frac{a_{H8}}{4}\frac{H}{{\it v}}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}\hat{W}_{\mu\nu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}\hat{W}^{\mu\nu}\Big{]}+a_{H11}\frac{H}{{\it
v}}{\rm Tr}\Big{[}{\cal D}_{\mu}{\cal V}^{\mu}{\cal D}_{\nu}{\cal
V}^{\nu}\Big{]}$ $\displaystyle+\frac{a_{H13}}{2}\frac{H}{{\it v}}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal D}_{\mu}{\cal V}_{\nu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal D}^{\mu}{\cal V}^{\nu}\Big{]}+\ldots$
$\displaystyle+a_{\Box 0}\left(\frac{m_{\rm Z}^{2}-m_{\rm W}^{2}}{{\it
v}^{2}}\right)\frac{\Box H}{{\it v}}{\rm Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal
V}_{\mu}\Big{]}{\rm Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal V}^{\mu}\Big{]}$
$\displaystyle+ia_{d1}\frac{\partial^{\nu}H}{v}{\rm
Tr}\Big{[}U\hat{B}_{\mu\nu}U^{\dagger}{\cal
V}^{\mu}\Big{]}+ia_{d2}\frac{\partial^{\nu}H}{v}{\rm
Tr}\Big{[}\hat{W}_{\mu\nu}{\cal
V}^{\mu}\Big{]}+a_{d3}\frac{\partial^{\nu}H}{{\it v}}{\rm Tr}\Big{[}{\cal
V}_{\nu}{\cal D}_{\mu}{\cal V}^{\mu}\Big{]}$
$\displaystyle+ia_{d4}\frac{\partial^{\nu}H}{v}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}\hat{W}_{\mu\nu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal
V}^{\mu}\Big{]}+a_{d5}\frac{\partial^{\nu}H}{v}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal D}_{\mu}{\cal V}^{\mu}\Big{]}{\rm
Tr}\Big{[}U\tau^{3}U^{\dagger}{\cal V}_{\nu}\Big{]}\,\,,\,$
$\displaystyle{\mathcal{L}}_{4}^{\rm two-Higgs}$ $\displaystyle=$
$\displaystyle a_{\Box\Box}\frac{\Box H\,\Box H}{{\it v}^{2}}+\ldots\,\,,$
(2.11)

where the dots again mean operators which are not relevant for the present
computation. The relation of our EChL coefficients in the above
$\mathcal{L}_{4}$, which are refer jointly as $a_{i}$ coefficients, with those
in [14] is as follows: $a_{HBB}\leftrightarrow P_{B}$, $a_{HWW}\leftrightarrow
P_{W}$, $a_{\Box{\cal V}{\cal V}}\leftrightarrow P_{7}$,
$a_{H0}\leftrightarrow P_{T}$, $a_{H1}\leftrightarrow P_{1}$,
$a_{H8}\leftrightarrow P_{12}$, $a_{H11}\leftrightarrow P_{9}$,
$a_{H13}\leftrightarrow P_{16}$, $a_{\Box 0}\leftrightarrow P_{25}$,
$a_{d1}\leftrightarrow P_{4}$, $a_{d2}\leftrightarrow P_{5}$,
$a_{d3}\leftrightarrow P_{10}$, $a_{d4}\leftrightarrow P_{17}$,
$a_{d5}\leftrightarrow P_{19}$ and $a_{\Box\Box}\leftrightarrow P_{\Box H}$.
Notice that we have used a specific notation for some of the $a_{i}$’s in
${\mathcal{L}}_{4}^{\rm one-Higgs}$ that refer explicitly to those operators
that are replicas of the structures in ${\mathcal{L}}_{4}^{\rm no-Higgs}$ but
with an extra factor given by $(H/v)$. For instance, $a_{H0}$ versus $a_{0}$,
$a_{H1}$ versus $a_{1}$, etc. Alternative sets of effective operators, with a
reduced number of operators in the list by the use of the equations of motion,
can be found in the literature (see, for instance, [8]).

Regarding the quantization of the EChL we choose here to use the same gauge-
fixing Lagrangian, $\mathcal{L}_{\rm GF}$, as in the SM for the linear
covariant $R_{\xi}$ gauges [15], in which the tree level mixing between gauge
bosons with their corresponding GB are absent. Some features of $R_{\xi}$
gauge-fixing and renormalization within the context of the EChL were already
studied long ago in [16, 17] when the Higgs particle was not included
explicitly in the Lagrangian (see also [18]). Generically, the quantization of
the EChL requires the insertion of appropriate gauge-fixing functions $F_{j}$
involving the EW gauge bosons and the GBs. This gauge-fixing Lagrangian can be
written in terms of the physical basis as follows:

$\mathcal{L}_{\rm
GF}=-F_{+}F_{-}-\frac{1}{2}F_{Z}^{2}-\frac{1}{2}F_{A}^{2}\,,$ (2.12)

where the gauge-fixing functions are:

$\displaystyle F_{\pm}=\frac{1}{\sqrt{\xi}}(\partial^{\mu}W_{\mu}^{\pm}-\xi
m_{\rm W}\pi^{\pm})\,,\quad
F_{Z}=\frac{1}{\sqrt{\xi}}(\partial^{\mu}Z_{\mu}-\xi m_{\rm Z}\pi^{3})\,,\quad
F_{A}=\frac{1}{\sqrt{\xi}}(\partial^{\mu}A_{\mu})\,,$ (2.13)

and $\xi$ is the typical gauge-fixing parameter of the $R_{\xi}$ gauges.

From the above gauge-fixing functions, $F_{\pm}$, $F_{Z}$ and $F_{A}$, one
derives the corresponding Faddeev-Popov Lagrangian [19], by:

$\mathcal{L}_{\rm FP}=\sum_{i,j=+,-,Z,A}\bar{c}^{i}\frac{\delta
F_{i}}{\delta\alpha_{j}}c^{j}\,,$ (2.14)

where $c^{j}$ are the ghost fields and $\alpha_{j}$ ($j=+,-,Z,A$) are the
corresponding gauge transformation parameters under the local transformations
$SU(2)_{L}\times U(1)_{Y}$ given by
$L=e^{ig\vec{\tau}\cdot\vec{\alpha}_{L}(x)/2}$ and
$R=e^{ig^{\prime}\tau^{3}\alpha_{Y}(x)/2}$. The corresponding gauge field
transformations are as in the SM. However, the scalar transformations in this
non-linear EFT are

$\displaystyle\delta H$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\delta\pi^{\pm}$ $\displaystyle=$ $\displaystyle\,m_{\rm
W}\alpha^{\pm}\mp\frac{ig}{2}\alpha^{\pm}\pi^{3}\pm\frac{g(c_{\rm
w}^{2}-s_{\rm w}^{2})}{2c_{\rm w}}\alpha_{Z}\pi^{\pm}\pm igs_{\rm
w}\alpha_{A}\pi^{\pm}$ $\displaystyle+\frac{m_{\rm W}}{3{\it
v}^{2}}(-\alpha^{\pm}\pi^{+}\pi^{-}+\alpha^{\mp}\pi^{\pm}\pi^{\pm}-\alpha^{\pm}\pi^{3}\pi^{3})+\frac{m_{\rm
Z}}{3{\it v}^{2}}\alpha_{Z}\pi^{3}\pi^{\pm}+\ldots$
$\displaystyle\delta\pi^{3}$ $\displaystyle=$ $\displaystyle\,m_{\rm
Z}\alpha_{Z}-\frac{ig}{2}(\alpha^{-}\pi^{+}-\alpha^{+}\pi^{-})$ (2.15)
$\displaystyle+\frac{m_{\rm W}}{3{\it
v}^{2}}(\alpha^{+}\pi^{-}\pi^{3}+\alpha^{-}\pi^{+}\pi^{3})-\frac{2m_{\rm
Z}}{3{\it v}^{2}}\alpha_{Z}\pi^{+}\pi^{-}+\ldots$

which differ from the corresponding ones in the SM. In particular, two main
differences respect to the SM arise from this ghost Lagrangian which are worth
mentioning: 1) within the EChL there are not interactions among the Higgs
boson and ghost fields, and 2) there are new interactions with multiple GBs
and two ghost fields.

As a general comment to the EChL above, notice that we do not use the
equations of motion to reduce the number of effective operators since, as
already stated in the introduction, we are interested on the computation of
VBS in terms of the off-shell renormalized 1PI Green functions, thus we keep
the track of all the involved EChL coefficients into our computation. For the
same reason, we do not apply either any Higgs field redefinition and keep
explicitly the track as well of all the off-shell Higgs effects.

The summary of all the relevant FRs for the present computation is presented
in Appendix A. We have also added the corresponding FRs within the SM, for a
clear comparison.

##  3 Diagrammatic computation: VBS from 1PI Green functions

For definiteness in the description of our computational procedure, and to fix
some convenient notation, we focus in this section on the VBS channel $WZ\to
WZ$. The detailed study of other VBS channels is postponed for future works.
The amplitude of our interest here is, therefore, $\mathcal{A}(WZ\to WZ)$.
Following the standard counting rules of the EChL, the full one-loop
scattering amplitude can be splitted into two parts, as follows:

$\mathcal{A}(WZ\to WZ)^{{\rm EChL}}=\mathcal{A}^{(0)}(WZ\to
WZ)+\mathcal{A}^{(1)}(WZ\to WZ)\equiv\mathcal{A}^{{\rm EChL}_{\rm Full}}\,,$
(3.1)

where the leading order (LO), ${\cal O}(p^{2})$, and the next to leading order
contributions (NLO), ${\cal O}(p^{4})$, are denoted as $\mathcal{A}^{(0)}$ and
$\mathcal{A}^{(1)}$ respectively, and are given by:

$\displaystyle\mathcal{A}^{(0)}(WZ\to WZ)$ $\displaystyle\equiv$
$\displaystyle\mathcal{A}^{{\rm EChL}^{(2)}_{\rm Tree}}\,,$ (3.2)
$\displaystyle\mathcal{A}^{(1)}(WZ\to WZ)$ $\displaystyle\equiv$
$\displaystyle\mathcal{A}^{{\rm EChL}^{(4)}_{\rm Tree}}+\mathcal{A}^{{\rm
EChL}^{(2)}_{\rm Loop}}\,.$ (3.3)

This means, that to compute the LO amplitude in this EChL context, one uses
the Feynman rules from ${\cal L}_{2}$ at the tree level, and to compute the
NLO correction one adds the contribution from ${\cal L}_{4}$ at the tree level
and the contribution from the loops using ${\cal L}_{2}$. This reflects the
typical double role of ${\cal L}_{4}$ in the chiral Lagrangian approach. On
the one hand, it adds an extra tree level contribution to the scattering
amplitude, and on the other hand it also acts as generator of new counterterms
that are needed to remove the extra divergences emerging from the loops
computed with ${\cal L}_{2}$ and that are not removable by a simple
redefinition (counterterms) of the parameters in ${\cal L}_{2}$. This
renormalization procedure of the extra divergences, which is well known in the
context of ChPT, works also properly here for the EChL and its application to
the computation of finite VBS amplitudes, as will be seen in detail in the
next section. Notice also that the previous splitting in Eq. (3.1) of the EChL
amplitude into $\mathcal{A}^{(0)}$ and $\mathcal{A}^{(1)}$ can be done in a
different way, attending instead to the two parts of different order in the
quantum corrections expansion, i.e., in powers of $\hbar$. Thus, one can split
the full one-loop amplitude, alternatively, as follows:

$\mathcal{A}^{{\rm EChL}_{\rm Full}}=\mathcal{A}^{{\rm EChL}^{(2+4)}_{\rm
Tree}}+\mathcal{A}^{{\rm EChL}^{(2)}_{\rm Loop}}\,,$ (3.4)

where, the tree level amplitude, ${\cal O}(\hbar^{0})$ , is:

$\mathcal{A}^{{\rm EChL}^{(2+4)}_{\rm Tree}}=\mathcal{A}^{{\rm
EChL}^{(2)}_{\rm Tree}}+\mathcal{A}^{{\rm EChL}^{(4)}_{\rm Tree}}\,,$ (3.5)

and the one-loop correction, ${\cal O}(\hbar^{1})$, is $\mathcal{A}^{{\rm
EChL}^{(2)}_{\rm Loop}}$. Since, in our posterior analysis of the full one-
loop results within the EChL we plan to compare them with the corresponding SM
full results, which we have also computed here in parallel, it is then
convenient to also fix the notation here for the SM case. The full one-loop SM
amplitude is then defined as the sum of the SM leading-order (LO)
contribution, that, in this case, is the tree level part of ${\cal
O}(\hbar^{0})$, and the SM next-to-leading order (NLO) contribution, that is
the one-loop part of ${\cal O}(\hbar^{1})$. In short, it is given by:

$\mathcal{A}^{{\rm SM}_{\rm Full}}=\mathcal{A}^{{\rm SM}_{\rm
Tree}}+\mathcal{A}^{{\rm SM}_{\rm Loop}}\,.$ (3.6)

To our knowledge, the SM computation presented here is the first full bosonic
one-loop computation in the literature of $WZ\to WZ$ scattering using the
$R_{\xi}$ gauges, and, therefore, we believe it is interesting by itself, in
addition to be the obvious reference to compare our EChL computation with.

As already said, our purpose in this paper is to organize the computation of
the full one-loop amplitude in terms of the basic building blocks in any QFT
which are the 1PI Green functions. Taking into account that we work in generic
$R_{\xi}$ gauges, this procedure by means of 1PI functions has the advantage
of requiring a more demanding renormalization program, since all the off-shell
effects of external legs in the 1PI functions must be considered. Concretely,
in the next section we will fix our renormalization program in the $R_{\xi}$
gauges that makes finite all the relevant 1PI Green functions for arbitrary
external legs (off-shell) momenta and no transversality condition for the EW
gauge bosons is applied. The $\xi$-dependence of all the divergences involved
in these 1PI functions will be followed explicitly and, in this sense, the
extraction of the new counterterms given by the EChL coefficients $a_{i}$ will
be more complicated in these $R_{\xi}$ gauges, but they will provide an
excellent check of the gauge invariance of these new counterterms. We continue
next with the presentation of the amplitude in terms of the 1PI functions and
postpone all these renormalization issues for the next section, Section 4.

Generically, the full amplitude can be organized in terms of the contributing
1PI functions as follows:

$\mathcal{A}^{{\rm EChL}_{\rm
Full}}=\mathcal{A}^{(0)}+\mathcal{A}_{1-leg}+\mathcal{A}_{2-legs}+\mathcal{A}_{3-legs}+\mathcal{A}_{4-legs}+\mathcal{A}_{res}\,,$
(3.7)

where $\mathcal{A}_{n-legs}$ means the $\mathcal{O}(\hbar)$ contribution from
the $n$-legs 1PI function to the amplitude, and we have separated explicitly
the extra contribution from the finite residues of the external gauge bosons
$\mathcal{A}_{res}$. Regarding the 1-leg 1PI function (i.e., the tadpole
$\hat{T}$ of the Higgs boson), a priori, it enters in many parts of the
different diagrams contributing to the amplitude, but as it will be seen in
the next section, one can set this renormalized Green function equals to zero
by a convenient renormalization condition, and then $\mathcal{A}_{1-leg}$
simply vanishes.

Figure 1: Full 1PI functions (black balls) contributing to the full one-loop
amplitude $\mathcal{A}(WZ\to WZ)$

We next specify the particular 1PI functions contributing to the amplitude of
our interest. For definitess, here and in the following of this work, we fix
the notation for the momenta assignments and Lorentz indexes as follows:

$W^{+}_{\lambda}(p_{+})\,Z_{\rho}(p_{0})\to
W^{+}_{\sigma}(k_{+})\,Z_{\kappa}(k_{0})\,,$ (3.8)

where $p_{+,0}$ and $k_{+,0}$ (with $p_{+}+p_{0}=k_{+}+k_{0}$) are the
incoming and outcoming momenta of the gauge bosons, with polarization vectors
$\epsilon_{\lambda}(p_{+})\,,\epsilon_{\rho}(p_{0})\,,\epsilon_{\sigma}(k_{+})$
and $\epsilon_{\kappa}(k_{0})$, respectively. Thus, the amplitude
$\mathcal{A}$ can be written as:

$\mathcal{A}=A^{\lambda\rho\sigma\kappa}\,\epsilon_{\lambda}(p_{+})\epsilon_{\rho}(p_{0})\epsilon_{\sigma}(k_{+})^{*}\epsilon_{\kappa}(k_{0})^{*}\,,$
(3.9)

where the tensor amplitude with explicit Lorentz indexes is defined by
$A^{\lambda\rho\sigma\kappa}$. The complete set of full 1PI functions and full
propagators that contribute to the full one-loop amplitude $\mathcal{A}(WZ\to
WZ)$ are displayed graphically in Fig. 1. These full one-loop functions are
represented in this figure by black balls and are denoted by quantities with a
hat, in order to be distinguished from the corresponding leading order
quantities (without the hat). These full functions include: 1) the full
propagators, $\hat{\Delta}^{HH}$, $\hat{\Delta}^{\pi\pi}$,
$\hat{\Delta}^{WW}$, $\hat{\Delta}^{W\pi}$ and $\hat{\Delta}^{\pi W}$; 2) the
full 1PI vertex functions with three-legs, $\hat{\Gamma}_{WWZ}$,
$\hat{\Gamma}_{\pi WZ}$, $\hat{\Gamma}_{HWW}$, $\hat{\Gamma}_{HZZ}$,
$\hat{\Gamma}_{ZZA}$ and $\hat{\Gamma}_{ZZZ}$; and 3) the full 1PI vertex
function with four-legs, $\hat{\Gamma}_{WZWZ}$. Notice, that some of these
full functions contain contributions of both orders, ${\cal O}(\hbar^{0})$ and
${\cal O}(\hbar^{1})$, whereas others are pure ${\cal O}(\hbar^{1})$
functions. For instance, the functions $\hat{\Delta}^{W\pi}$,
$\hat{\Delta}^{\pi W}$, $\hat{\Gamma}_{ZZA}$ and $\hat{\Gamma}_{ZZZ}$ get only
NLO contributions since they vanish at LO. Consequently, when providing the
amplitude to a given order in powers of $\hbar$ one has to keep accordingly
the same powers in the involved products of full functions. Thus, in products
like
$\hat{\Gamma}_{WWZ}^{\mu\lambda\rho}\,\hat{\Delta}_{\mu\nu}^{WW}\,\hat{\Gamma}_{WWZ}^{\nu\sigma\kappa}$
there are several contributions inside: 1) LO contributions
$\Gamma_{WWZ}^{\mu\lambda\rho}\,\Delta_{\mu\nu}^{WW}\,\Gamma_{WWZ}^{\nu\sigma\kappa}$
and 2) NLO contributions from the corresponding corrections in each of the
full functions. On the other hand, the propagators $\Delta^{AA}$,
$\Delta^{ZZ}$ and the vertex function $\Gamma_{WWA}$, enter just at leading
order in this computation. We can then simply read the various contributions
by grouping them together, for instance, by $s$, $t$, $u$ and contact $c$
channels as follows:

$A^{\lambda\rho\sigma\kappa}=A_{s}^{\lambda\rho\sigma\kappa}+A_{t}^{\lambda\rho\sigma\kappa}+A_{u}^{\lambda\rho\sigma\kappa}+A_{c}^{\lambda\rho\sigma\kappa}\,,$
(3.10)

where the amplitudes by channels in terms of the full functions that enter at
one-loop level read as follows:

$\displaystyle iA_{s}^{\lambda\rho\sigma\kappa}$ $\displaystyle=$
$\displaystyle
i\hat{\Gamma}_{WWZ}^{\mu\lambda\rho}\,(-i)\hat{\Delta}_{\mu\nu}^{WW}\,i\hat{\Gamma}_{WWZ}^{\nu\sigma\kappa}+i\Gamma_{WWZ}^{\mu\lambda\rho}\,\hat{\Delta}_{\mu}^{W\pi}\,i\Gamma_{\pi
WZ}^{\sigma\kappa}$ $\displaystyle+i\Gamma_{\pi
WZ}^{\lambda\rho}\,\hat{\Delta}_{\nu}^{\pi
W}\,i\Gamma_{WWZ}^{\nu\sigma\kappa}+i\hat{\Gamma}_{\pi
WZ}^{\lambda\rho}\,i\hat{\Delta}^{\pi\pi}\,i\hat{\Gamma}_{\pi
WZ}^{\sigma\kappa}$ $\displaystyle iA_{t}^{\lambda\rho\sigma\kappa}$
$\displaystyle=$ $\displaystyle
i\hat{\Gamma}_{HWW}^{\lambda\rho}\,i\hat{\Delta}^{HH}\,i\hat{\Gamma}_{HZZ}^{\sigma\kappa}+i\Gamma_{WWA}^{\mu\lambda\rho}\,i\Delta_{\mu\nu}^{AA}\,i\hat{\Gamma}_{ZZA}^{\nu\sigma\kappa}+i\Gamma_{WWZ}^{\mu\lambda\rho}\,i\Delta_{\mu\nu}^{ZZ}\,i\hat{\Gamma}_{ZZZ}^{\nu\sigma\kappa}$
$\displaystyle iA_{u}^{\lambda\rho\sigma\kappa}$ $\displaystyle=$
$\displaystyle
i\hat{\Gamma}_{WWZ}^{\mu\lambda\kappa}\,(-i)\hat{\Delta}_{\mu\nu}^{WW}\,i\hat{\Gamma}_{WWZ}^{\nu\sigma\rho}+i\Gamma_{WWZ}^{\mu\lambda\kappa}\,\hat{\Delta}_{\mu}^{W\pi}\,i\Gamma_{\pi
WZ}^{\sigma\rho}$ $\displaystyle+i\Gamma_{\pi
WZ}^{\lambda\kappa}\,\hat{\Delta}_{\nu}^{\pi
W}\,i\Gamma_{WWZ}^{\nu\sigma\rho}+i\hat{\Gamma}_{\pi
WZ}^{\lambda\kappa}\,i\hat{\Delta}^{\pi\pi}\,i\hat{\Gamma}_{\pi
WZ}^{\sigma\rho}$ $\displaystyle iA_{c}^{\lambda\rho\sigma\kappa}$
$\displaystyle=$ $\displaystyle
i\hat{\Gamma}_{WZWZ}^{\lambda\rho\sigma\kappa}\,.$ (3.11)

For the forthcomming section dealing with the renormalization program, it is
convenient to write the previous full propagators in terms of 1PI functions
with two-legs, therefore, in terms of the self-energies. For the case of gauge
boson propators, we introduce some convenient Lorentz tensors which are
usefull to decompose them into their transverse ($T$) and longitudinal ($L$)
parts by:

$\displaystyle-i\hat{\Delta}_{\mu\nu}^{WW}(q)$ $\displaystyle=$
$\displaystyle-i\hat{\Delta}_{T}^{WW}(q^{2})T_{\mu\nu}-i\hat{\Delta}_{L}^{WW}(q^{2})L_{\mu\nu}\,,$
(3.12)

where,

$\displaystyle T^{\mu\nu}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}}\,\,\,;\,\,\,L^{\mu\nu}=\frac{q^{\mu}q^{\nu}}{q^{2}}\,,$
(3.13)

which fulfil the following useful properties:

$\displaystyle T_{\mu\nu}+L_{\mu\nu}=g_{\mu\nu}\,,\quad
T^{\mu\nu}T_{\nu\rho}=T^{\mu}_{\rho}\,,\quad
L^{\mu\nu}L_{\nu\rho}=L^{\mu}_{\rho}\,,$ $\displaystyle\quad
T^{\mu\nu}L_{\nu\rho}=0\,,\quad T_{\mu\nu}q^{\nu}=0\quad\,,\quad
L_{\mu\nu}q^{\nu}=q_{\mu}.$ (3.14)

The full self-energies, involving gauge bosons can similarly be decomposed as:

$\hat{\Sigma}^{\mu\nu}_{WW}(q)=\hat{\Sigma}^{T}_{WW}(q^{2})T^{\mu\nu}+\hat{\Sigma}^{L}_{WW}(q^{2})L^{\mu\nu}\,.$
(3.15)

For the mixed self-energies connecting a $W$ with a $\pi$, we use the
following decomposition:

$\hat{\Sigma}^{\mu}_{W\pi}(q)=\frac{q^{\mu}}{m_{\rm
W}}\hat{\Sigma}_{W\pi}(q^{2})\,,$ (3.16)

and we have the corresponding one to the full propagator:

$\hat{\Delta}_{\mu}^{W\pi}(q)=\frac{q_{\mu}}{m_{\rm
W}}\hat{\Delta}^{W\pi}(q^{2})\,.$ (3.17)

Then, the simple expressions of the full propagators in terms of the full
self-energies, $\hat{\Sigma}$, to be used at the one-loop level, are
summarized by:

$\displaystyle i\hat{\Delta}^{HH}(q^{2})$ $\displaystyle=$ $\displaystyle
i\Delta^{HH}+i\Delta^{HH}\,(-i)\hat{\Sigma}_{HH}\,i\Delta^{HH}\,,$
$\displaystyle i\hat{\Delta}^{\pi\pi}(q^{2})$ $\displaystyle=$ $\displaystyle
i\Delta^{\pi\pi}+i\Delta^{\pi\pi}\,(-i)\hat{\Sigma}_{\pi\pi}\,i\Delta^{\pi\pi}\,,$
$\displaystyle-i\hat{\Delta}^{WW}_{T}(q^{2})$ $\displaystyle=$
$\displaystyle-i\Delta^{WW}_{T}-i\Delta^{WW}_{T}\,i\hat{\Sigma}_{WW}^{T}\,(-i)\Delta^{WW}_{T}\,,$
$\displaystyle-i\hat{\Delta}^{WW}_{L}(q^{2})$ $\displaystyle=$
$\displaystyle-i\Delta^{WW}_{L}-i\Delta^{WW}_{L}\,i\hat{\Sigma}_{WW}^{L}\,(-i)\Delta^{WW}_{L}\,,$
$\displaystyle\hat{\Delta}^{W\pi}(q^{2})$ $\displaystyle=$
$\displaystyle\Delta^{W\pi}-i\Delta^{WW}_{L}\,\hat{\Sigma}_{W\pi}\,i\Delta^{\pi\pi}\,,$
(3.18)

where all functions on the right hand side are functions of $q^{2}$ and we
have used the conventions for signs, ‘$i$’ factors, and Lorentz decompositions
as shown in Fig. 2 for the self-energies $\hat{\Sigma}$.

Figure 2: Diagrammatic Lorentz conventions corresponding to the self-energies
of two gauge bosons (left), two scalars (middle) and gauge boson with a scalar
(right). Incoming momenta are understood.

The involved $R_{\xi}$ gauges propagators at LO are summarized by:

$\displaystyle i\Delta^{HH}$ $\displaystyle=$
$\displaystyle\frac{i}{q^{2}-m_{\rm
H}^{2}}\,,\qquad-i\Delta^{WW}_{T}=\frac{-i}{q^{2}-m_{\rm W}^{2}}\,,$
$\displaystyle-i\Delta^{WW}_{L}$ $\displaystyle=$
$\displaystyle\frac{-i\xi}{q^{2}-\xi m_{\rm W}^{2}}\,,\quad
i\Delta^{\pi\pi}=\frac{i}{q^{2}-\xi m_{\rm W}^{2}}\,,\quad\Delta^{W\pi}=0\,.$
$\displaystyle-i\Delta^{AA}_{T}$ $\displaystyle=$
$\displaystyle\frac{-i}{q^{2}}\,,\quad-i\Delta^{AA}_{L}=\frac{-i\xi}{q^{2}}\,,\quad-i\Delta^{ZZ}_{T}=\frac{-i}{q^{2}-m_{\rm
Z}^{2}}\,,\quad-i\Delta^{ZZ}_{L}=\frac{-i\xi}{q^{2}-\xi m_{\rm Z}^{2}}\,.$
(3.19)

As commented previously, only $\Delta^{HH}$, $\Delta^{WW}_{T}$,
$\Delta^{WW}_{L}$ and $\Delta^{\pi\pi}$ are involved in the LO contribution to
the amplitude in Eq. (3.1). On the other hand, the relevant vertex functions
at LO are:

$\displaystyle i\Gamma_{WWZ}^{\mu\nu\rho}$ $\displaystyle=$ $\displaystyle-
igc_{\rm w}V^{\mu\nu\rho}\,,\quad i\Gamma_{\pi
WZ}^{\sigma\kappa}=g^{\prime}s_{\rm w}m_{\rm W}g^{\sigma\kappa}\,,$
$\displaystyle i\Gamma_{HWW}^{\lambda\rho}$ $\displaystyle=$ $\displaystyle
iagm_{\rm W}g^{\lambda\rho}\,,\quad i\Gamma_{HZZ}^{\sigma\rho}=iag\frac{m_{\rm
Z}}{c_{\rm w}}g^{\sigma\rho}\,,$ $\displaystyle
i\Gamma_{WZWZ}^{\lambda\rho\sigma\kappa}$ $\displaystyle=$ $\displaystyle-
ig^{2}c_{\rm w}^{2}S^{\lambda\sigma,\rho\kappa}\,,$ $\displaystyle
i\Gamma_{WWA}^{\mu\nu\rho}$ $\displaystyle=$ $\displaystyle-igs_{\rm
w}V^{\mu\nu\rho}\,,\quad
i\Gamma_{ZZA}^{\nu\sigma\kappa}=i\Gamma_{ZZZ}^{\nu\sigma\kappa}=0\,,$ (3.20)

where the Lorentz structures $V^{\mu\nu\rho}$ and
$S^{\lambda\sigma,\rho\kappa}$ are summarized in Eq. (A.1).

Finally, for completeness, we end this section presenting the computation of
the LO amplitude using the $R_{\xi}$ gauges. This can be easily done by
plugging the corresponding LO functions of Eq. (3.19) and Eq. (3.20) in Eq.
(3.11). Namely, using $\Gamma$ instead of $\hat{\Gamma}$, and $\Delta$ instead
of $\hat{\Delta}$.

Figure 3: Diagrams at LO contributing to $W^{+}Z\to W^{+}Z$ in the covariant
$R_{\xi}$ gauges.

Then, we get the result for the LO amplitude in the $R_{\xi}$ gauge, which
corresponds to the tree level diagrams in Fig. 3. It is given by:

$\displaystyle\mathcal{A}^{(0)}$ $\displaystyle=$
$\displaystyle\left(A^{(0)\,\lambda\rho\sigma\kappa}_{s}+A^{(0)\,\lambda\rho\sigma\kappa}_{t}+A^{(0)\,\lambda\rho\sigma\kappa}_{u}+A^{(0)\,\lambda\rho\sigma\kappa}_{c}\right)\,\epsilon_{\lambda}(p_{+})\epsilon_{\rho}(p_{0})\epsilon_{\sigma}(k_{+})^{*}\epsilon_{\kappa}(k_{0})^{*}\quad$
(3.21)

with:

$\displaystyle A^{(0)\,\lambda\rho\sigma\kappa}_{s}$ $\displaystyle=$
$\displaystyle g^{2}c_{\rm w}^{2}\frac{1}{s-m_{\rm
W}^{2}}V_{WWZ}^{\mu\lambda\rho}T_{\mu\nu}V_{WWZ}^{\nu\sigma\kappa}$
$\displaystyle+g^{2}c_{\rm w}^{2}\frac{\xi}{s-\xi m_{\rm
W}^{2}}V_{WWZ}^{\mu\lambda\rho}L_{\mu\nu}V_{WWZ}^{\nu\sigma\kappa}-g^{\prime
2}s_{\rm w}^{2}m_{\rm W}^{2}\frac{1}{s-\xi m_{\rm
W}^{2}}g^{\lambda\rho}g^{\sigma\kappa}\,,$ $\displaystyle
A^{(0)\,\lambda\rho\sigma\kappa}_{t}$ $\displaystyle=$
$\displaystyle-a^{2}g^{2}\frac{m_{\rm W}m_{\rm Z}}{c_{\rm w}}\frac{1}{t-m_{\rm
H}^{2}}\,g^{\lambda\sigma}g^{\rho\kappa}\,,$ $\displaystyle
A^{(0)\,\lambda\rho\sigma\kappa}_{u}$ $\displaystyle=$ $\displaystyle
g^{2}c_{\rm w}^{2}\frac{1}{u-m_{\rm
W}^{2}}V_{WWZ}^{\mu\lambda\kappa}T_{\mu\nu}V_{WWZ}^{\nu\sigma\rho}$
$\displaystyle+g^{2}c_{\rm w}^{2}\frac{\xi}{u-\xi m_{\rm
W}^{2}}V_{WWZ}^{\mu\lambda\kappa}L_{\mu\nu}V_{WWZ}^{\nu\sigma\rho}-g^{\prime
2}s_{\rm w}^{2}m_{\rm W}^{2}\frac{1}{u-\xi m_{\rm
W}^{2}}g^{\lambda\kappa}g^{\sigma\rho}\,,$ $\displaystyle
A^{(0)\,\lambda\rho\sigma\kappa}_{c}$ $\displaystyle=$
$\displaystyle-g^{2}c_{\rm w}^{2}S^{\lambda\sigma,\rho\kappa}\,.$ (3.22)

Some more comments on this LO result are in order. First, notice that the
corresponding result of the SM amplitude at LO is simply obtained from this
same formula by setting $a=1$. Second, we have checked explicitly the gauge
invariance of the LO result, namely, that the dependence on $\xi$ disappears
in the final amplitude as expected. The cancellation of the $\xi$-dependent
terms is achieved once the external gauge bosons are on-shell, i.e., by
contracting the tensorial amplitude with their corresponding polarization
vectors in $\mathcal{A}^{(0)}$ of Eq. (3.21). Concretely, the cancellation of
the $\xi$-dependent terms occurs separately in the two channels $s$ and $u$,
and it happens between the two terms in the second lines (coming from the $W$
and GB propagators) of the corresponding $s$ and $u$ amplitudes in Eq. (3.22).
More explicitly, using the transversality and on-shell conditions of the
external gauge bosons one gets in the $s$ channel:

$\displaystyle
V_{WWZ}^{\mu\lambda\rho}L_{\mu\nu}V_{WWZ}^{\nu\sigma\kappa}\epsilon_{\lambda}(p_{+})\epsilon_{\rho}(p_{0})\epsilon_{\sigma}(k_{+})^{*}\epsilon_{\kappa}(k_{0})^{*}$
$\displaystyle=$ $\displaystyle\frac{(m_{\rm Z}^{2}-m_{\rm
W}^{2})^{2}}{s}\epsilon(p_{+})\cdot\epsilon(p_{0})\,\epsilon(k_{+})^{*}\cdot\epsilon(k_{0})^{*}$
(3.23)

and the $\xi$-dependence from the two terms in the second line of the $s$
channel amplitude in Eq. (3.22) then cancels. The contribution from these two
terms to ${\cal A}_{s}^{(0)}$ then finally reduce to:

$-g^{2}\frac{s_{\rm w}^{4}}{c_{\rm w}^{2}}\frac{m_{\rm
W}^{2}}{s}\epsilon(p_{+})\cdot\epsilon(p_{0})\,\epsilon(k_{+})^{*}\cdot\epsilon(k_{0})^{*}\,,$
(3.24)

consequently, leading to a gauge invariant amplitude, as expected. The
cancellation in the $u$ channel contribution, ${\cal A}_{u}^{(0)}$ proceeds in
a similar way.

##  4 Renormalization program

###  4.1 Generalities

In this section we present our renormalization program to compute the
renormalized 1PI functions within the EChL in covariant $R_{\xi}$ gauges which
are the basic pieces in our computation of the VBS amplitudes. Within a
diagrammatic approach, these renormalized 1PI functions, denoted here
generically by $\hat{\Gamma}_{n-legs}$, can be decomposed into three pieces as
follows:

$\hat{\Gamma}_{n-legs}=\Gamma^{\rm Tree}_{n-legs}+\Gamma^{\rm
Loop}_{n-legs}+\Gamma^{\rm CT}_{n-legs}\,.$ (4.1)

where, within the context of the EChL we are working with, $\Gamma^{\rm Tree}$
means contributions from tree level Lagrangian,
$\mathcal{L}_{2}+\mathcal{L}_{4}$, $\Gamma^{\rm Loop}$ are the contributions
from the unrenormalized one-loop diagrams using the interaction vertices of
$\mathcal{L}_{2}$ only, and $\Gamma^{\rm CT}$ summarizes the contributions
from all the counterterms. For the analytical computation of these three
pieces we use the codes and tools specified next. We have implemented our
model with FeynRules [20], generated and drawn the Feynman diagrams by
FeynArts [21], performed the main calculations with FormCalc and LoopTools
[22] and added some extra checks of the involved one-loop divergences using
FeynCalc [23] and the code Package-X [24]. Also, a comparison with respect to
the SM has been perfomed in parallel and a summary of the corresponding SM
results are collected in Appendix C, for completeness.

In the following of this section we describe the details of the
renormalization program. First we set the regularization and multiplicative
renormalization prescriptions, and fix the renormalization conditions. Second,
we discuss the Slavnov-Taylor identity in these $R_{\xi}$ gauges. Third, we
present the various contributions to the 1PI functions as well as the
solutions for the counterterms involved. Finally, we derive the values of the
renormalized EChL coefficients and set the corresponding renormalization group
equations (RGEs) describing the running of these coefficients.

###  4.2 Regularization and Renormalization prescriptions

The regularization procedure of the loop contributions is performed with
dimensional regularization [25], in $D=4-\epsilon$ dimensions, as usual. This
method has the advantage that preserves all the relevant symmetries in the
bosonic sector of the theory, including chiral invariance (remember that we do
not consider the fermionic contributions and the Dirac $\gamma_{5}$ is not
involved in this work). Consequently, the scale of dimensional regularization
is set to $\mu$ and all the one-loop divergencies are expressed in terms of :

$\Delta_{\epsilon}=\frac{2}{\epsilon}-\gamma_{E}+\log(4\pi)\,.$ (4.2)

Regarding the renormalization procedure, the counterterms of all the
parameters and fields appearing in the tree level Lagrangian,
$\mathcal{L}_{2}+\mathcal{L}_{4}$, are generated by the usual multiplicative
renormalization prescription that relates the bare quantities (here denoted by
a specific sub- or super-script with a label $0$) and the renormalized ones
(here with no specific sub- or super-script labels). In our present case, the
relevant relations are summarized as follows:

$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle\sqrt{Z_{H}}H\,,\quad
B_{0\,\mu}=\sqrt{Z_{B}}B_{\mu}\,,\quad
W_{0\,\mu}^{1,2,3}=\sqrt{Z_{W}}W_{\mu}^{1,2,3}\,,\quad\pi_{0}^{1,2,3}=\sqrt{Z_{\pi}}\pi^{1,2,3}\,,$
$\displaystyle{\it v}_{0}$ $\displaystyle=$ $\displaystyle\sqrt{Z_{\pi}}({\it
v}+\delta{\it v})\,,\quad\lambda_{0}=Z_{H}^{-2}(\lambda+\delta\lambda)\,,$
$\displaystyle g^{\prime}_{0}$ $\displaystyle=$ $\displaystyle
Z_{B}^{-1/2}(g^{\prime}+\delta g^{\prime})\,,\quad g_{0}=Z_{W}^{-1/2}(g+\delta
g)\,,\quad\xi_{1,2}^{0}=\xi(1+\delta\xi_{1,2})\,,$ $\displaystyle a^{0}$
$\displaystyle=$ $\displaystyle a+\delta a\,,\quad b^{0}=b+\delta
b\,,\quad\kappa_{3,4}^{0}=\kappa_{3,4}+\delta\kappa_{3,4}\,,\quad
a_{i}^{0}=a_{i}+\delta a_{i}\,,$ (4.3)

where the $Z_{i}$ renormalization constants are set as usual to
$Z_{i}=1+\delta Z_{i}$. Some comments on this Eq. (4.3) are in order. Our
final results for both the renormalized 1PI functions and the VBS scattering
amplitudes are expressed in terms of the renormalized quantities, $m_{W}$,
$m_{Z}$, $m_{H}$, $g$, $g^{\prime}$, $v$, $a$, $a_{i}$’s, etc. It should be
noticed that for the $WZ\to WZ$ case, the renormalization of the $b$,
$\kappa_{3,4}$ and $\lambda$ parameters do not enter and, therefore, the
associated counterterms will not be present in the renormalized 1PI functions.
Besides, the ghost counterterms do not enter either, and we omit to show them
for shortness. In contrast, the renormalization of the covariant gauge
parameters do enter and are relevant for the present paper. Notice that in Eq.
(4.3) we have set a common renormalized $\xi$ parameter for all the involved
EW gauge bosons. The associated counterterms $\delta\xi_{1,2}$, introduced in
Eq. (4.26), play an important role in the renormalization of the charged
unphysical propagators. We refer here to the charged unphysical propagators as
in reference [26], namely, formed by the charged Goldstone bosons $\pi^{\pm}$
and the gauge bosons $W^{\pm}$. The discussion on the renormalization of the
involved functions for this charged unphysical sector,
$\hat{\Sigma}_{WW}^{L}$, $\hat{\Sigma}_{W\pi}$ and $\hat{\Sigma}_{\pi\pi}$,
and their relations will be presented in the subsection 4.5. Finally, one can
easily relate the corresponding counterterms for the EW parameters in the
physical basis with the previous counterterms in Eq. (4.3). In particular,

$\displaystyle\delta Z_{A}$ $\displaystyle=$ $\displaystyle c_{\rm
w}^{2}\delta Z_{B}+s_{\rm w}^{2}\delta Z_{W}\,,$ $\displaystyle\delta Z_{Z}$
$\displaystyle=$ $\displaystyle s_{\rm w}^{2}\delta Z_{B}+c_{\rm w}^{2}\delta
Z_{W}\,,$ $\displaystyle\delta Z_{ZA}$ $\displaystyle=$ $\displaystyle s_{\rm
w}c_{\rm w}(\delta Z_{W}-\delta Z_{B})=\frac{s_{\rm w}c_{\rm w}}{c_{\rm
w}^{2}-s_{\rm w}^{2}}(\delta Z_{Z}-\delta Z_{A})\,,$ $\displaystyle\delta
m_{\rm W}^{2}$ $\displaystyle=$ $\displaystyle m_{\rm W}^{2}\left(-\delta
Z_{W}+\delta Z_{\pi}+2\frac{\delta g}{g}+\frac{2\delta{\it v}}{{\it
v}}\right)\,,$ $\displaystyle\delta m_{\rm Z}^{2}$ $\displaystyle=$
$\displaystyle m_{\rm Z}^{2}\left(-\delta Z_{Z}+\delta Z_{\pi}+2c_{\rm
w}^{2}\frac{\delta g}{g}+2s_{\rm w}^{2}\frac{\delta
g^{\prime}}{g^{\prime}}+\frac{2\delta{\it v}}{{\it v}}\right)\,,$
$\displaystyle\delta m_{\rm H}^{2}$ $\displaystyle=$ $\displaystyle m_{\rm
H}^{2}\left(-2\delta Z_{H}+\frac{\delta
Z_{\pi}}{2}+\frac{\delta\lambda}{\lambda}+\frac{2\delta{\it v}}{{\it
v}}\right)\,,$ (4.4)

where, our conventions for the mass counterterms are:

$\displaystyle m_{\rm H}^{0\,2}$ $\displaystyle=$ $\displaystyle m_{\rm
H}^{2}+\delta m_{\rm H}^{2}\,,\,m_{\rm W}^{0\,2}=m_{\rm W}^{2}+\delta m_{\rm
W}^{2}\,,\,m_{\rm Z}^{0\,2}=m_{\rm Z}^{2}+\delta m_{\rm Z}^{2}\,.$ (4.5)

Finally, regarding the renormalization conditions we adopt here a hybrid
prescription in which we choose the on-shell (OS) scheme for the EW parameters
in the lowest order Lagrangian $\mathcal{L}_{2}$ and the $\overline{MS}$
scheme for all the EChL coefficients $a$ and $a_{i}$. These particular
conditions will provide the specific values of all the counterterms involved
in the present computation. Concretely, our renormalization conditions read as
follows:

  * •

Vanishing (Higgs) tadpole:

$\hat{T}=0\,.$ (4.6)

  * •

The pole of the renormalized propagator of the Higgs boson lies at $m_{\rm
H}^{2}$ and the corresponding residue is equal to 1:

${\rm Re}\left[\hat{\Sigma}_{HH}(m_{\rm H}^{2})\right]=0\,,\quad{\rm
Re}\left[\frac{d\hat{\Sigma}_{HH}}{dq^{2}}(m_{\rm H}^{2})\right]=0\,.$ (4.7)

  * •

Properties of the photon: residue equal one; no $A-Z$ mixing propagators; and
the electric charge defined like in QED, since there is a remnant $U(1)_{\rm
em}$ electromagnetic gauge symmetry:

${\rm
Re}\left[\frac{d\hat{\Sigma}^{T}_{AA}}{dq^{2}}(0)\right]=0\,,\quad\hat{\Sigma}^{T}_{ZA}(0)=0\,,\quad\hat{\Gamma}^{\mu}_{\gamma
ee}|_{\rm OS}=ie\gamma^{\mu}\,.$ (4.8)

  * •

The poles of the transverse renormalized propagators of the $W$ and $Z$ bosons
lie at $q^{2}=m_{\rm W}^{2}$ and $q^{2}=m_{\rm Z}^{2}$, respectively:

${\rm Re}\left[\hat{\Sigma}_{WW}^{T}(m_{\rm W}^{2})\right]=0\,,\quad{\rm
Re}\left[\hat{\Sigma}_{ZZ}^{T}(m_{\rm Z}^{2})\right]=0\,.$ (4.9)

  * •

The poles of the renormalized propagators in the unphysical charged sector
$\\{W^{\pm},\pi^{\pm}\\}$ lie at $q^{2}=\xi m_{\rm W}^{2}$. Therefore:

${\rm Re}\left[\hat{\Sigma}_{WW}^{L}(\xi m_{\rm W}^{2})\right]=0\,,\quad{\rm
Re}\left[\hat{\Sigma}_{\pi\pi}(\xi m_{\rm W}^{2})\right]=0\,.$ (4.10)

  * •

$\overline{MS}$ scheme for all the involved EChL coefficients.
In particular for $a$, $b$, $\kappa_{3,4}$ in Eq. (2.7) and the $a_{i}$’s in
Eqs. (2.10)-(2.11).

It is important to stress that the previous renormalization conditions on all
the EChL parameters determine both the divergent and finite parts of all the
counterterms. In particular, the physical masses $m_{W}$, $m_{Z}$ and $m_{H}$
coincide with the corresponding renormalized ones in the OS scheme. On the
other hand, notice that the residue for the Higgs and photon fields are set to
one in the previous conditions, but the residues ${\cal Z}_{W,Z}$, for the
gauge bosons $W$ and $Z$, are not set to one. These finite residues are given,
respectively, by:

${\cal Z}_{W}=1+{\rm Re}\left[\frac{d\hat{\Sigma}^{T}_{WW}}{dq^{2}}(m_{\rm
W}^{2})\right]\quad;\quad{\cal Z}_{Z}=1+{\rm
Re}\left[\frac{d\hat{\Sigma}^{T}_{ZZ}}{dq^{2}}(m_{\rm Z}^{2})\right]\,.$
(4.11)

Therefore, they contribute via $\mathcal{A}_{res}$ in Eq. (3.7) to the $WZ$
scattering as follows:

$\mathcal{A}_{res}=\left({\rm
Re}\left[\frac{d\hat{\Sigma}^{T}_{WW}}{dq^{2}}(m_{\rm W}^{2})\right]+{\rm
Re}\left[\frac{d\hat{\Sigma}^{T}_{ZZ}}{dq^{2}}(m_{\rm
Z}^{2})\right]\right)\mathcal{A}^{(0)}\,,$ (4.12)

since each external $W(Z)$ provides a factor ${\cal Z}^{1/2}_{W(Z)}$ to the
observable $S$ matrix.

###  4.3 Summary of contributions to the renormalized 1PI functions

We collect here the various contributions to the renormalized 1PI functions,
including the specific counterterms according to our previous prescriptions
especified in subsection 4.2. For the 2-legs functions and the corresponding
self-energies we use the conventions defined in Fig. 2. For the 3-legs and
4-legs functions we use the conventions defined in Fig. 4. All these functions
depend on the momenta of the external legs which are generically off-shell. By
using momentum conservation one can reduce the independent momenta to 2 and 3
for the 3-legs and 4-legs functions respectively.

Figure 4: Diagrammatic Lorentz conventions corresponding to the 3-legs
($i\hat{\Gamma}_{SVV^{\prime}}^{\mu\nu}$, $i\hat{\Gamma}_{WWV}^{\mu\nu\rho}$)
and 4-legs ($i\hat{\Gamma}_{VV^{\prime}WW}^{\mu\nu\rho\sigma}$) Green
functions, relevant for our observables of interest. The generic scalar here,
$S$, can be a Goldstone boson or a Higgs boson. The vector bosons $V$ here,
refer generically to the corresponding EW gauge boson, $W$, $Z$, $\gamma$. All
momenta are incoming.

For the generic 3-legs function with one external scalar $S$ (with momentum
$q$) and two EW gauge bosons $V_{\mu}$, $V^{\prime}_{\nu}$ (with momenta
$k_{1}$, $k_{2}$) we choose the gauge bosons momenta as independent variables
and the generic function can be decomposed into Lorentz structures as
$g^{\mu\nu}$, $k_{2}^{\mu}k_{1}^{\nu}$, $k_{1}^{\mu}k_{1}^{\nu}$,
$k_{1}^{\mu}k_{2}^{\nu}$ and $k_{2}^{\mu}k_{2}^{\nu}$. Notice that the last
three structures can be omitted if we impose the transversality conditions to
the gauge bosons, i.e. $k_{i}\cdot\epsilon(k_{i})=0$. However we keep them
here since we are interested on the full vertex functions off-shell.
Similarly, we choose the momenta of the $W$ bosons as independent variables of
the $\hat{\Gamma}_{WWV}^{\mu\nu\rho}$ (middle panel of Fig. 4) resulting in
the Lorentz structures $g^{\mu\nu}k_{1}^{\rho}$, $g^{\mu\nu}k_{2}^{\rho}$,
$g^{\nu\rho}k_{1}^{\mu}$, $g^{\nu\rho}k_{2}^{\mu}$, $g^{\rho\mu}k_{1}^{\nu}$
and $g^{\rho\mu}k_{2}^{\nu}$. For the case of $\hat{\Gamma}_{ZZA}$ and
$\hat{\Gamma}_{ZZZ}$ we choose the $Z$ bosons momenta. The 4-legs vertex
functions of the right panel have Lorentz structures
$g^{\mu\nu}g^{\rho\sigma}$, $g^{\mu\rho}g^{\nu\sigma}$ and
$g^{\mu\sigma}g^{\nu\rho}$ even at $\mathcal{O}(\hbar)$ as we found
explicitly.

We list in the following the various contributions to the renormalized
$n$-legs functions. We include not only the 1PI functions that are needed for
the present computation of $WZ\to WZ$ scattering, already mentioned in Section
3, but also some additional ones since we wish to determine separately the
renormalization of all the involved EW parameters and EChL coefficients in
Eqs. (2.10)-(2.11).

####  4.3.1 1-leg function (Higgs tadpole)

$\displaystyle i\hat{T}$ $\displaystyle=$ $\displaystyle iT^{\rm Loop}-i\delta
T\,,\quad\delta T=(-\delta m_{\rm H}^{2}/2+2\lambda{\it
v}^{2}\left(\frac{\delta{\it v}}{{\it v}}+\frac{\delta
Z_{\pi}}{2}\right)+\delta\lambda{\it v}^{2}-2\lambda{\it v}^{2}\delta
Z_{H}){\it v}$ (4.13)

####  4.3.2 2-legs functions

$\displaystyle-i\hat{\Sigma}_{HH}(q^{2})$ $\displaystyle=$
$\displaystyle-i\Sigma_{HH}^{\rm Loop}(q^{2})+i\left(\delta Z_{H}(q^{2}-m_{\rm
H}^{2})-\delta m_{\rm H}^{2}\right)+i\frac{2a_{\Box\Box}}{{\it
v}^{2}}q^{4}\,,$ (4.14) $\displaystyle i\hat{\Sigma}_{WW}^{T}(q^{2})$
$\displaystyle=$ $\displaystyle i\Sigma_{WW}^{T\,{\rm
Loop}}(q^{2})-i\left(\delta Z_{W}\left(q^{2}-m_{\rm W}^{2}\right)-\delta
m_{\rm W}^{2}\right)\,,$ $\displaystyle i\hat{\Sigma}_{AA}^{T}(q^{2})$
$\displaystyle=$ $\displaystyle i\Sigma_{AA}^{T\,{\rm Loop}}(q^{2})-i\delta
Z_{A}q^{2}-ig^{2}s_{\rm w}^{2}(a_{8}-2a_{1})q^{2}\,,$ $\displaystyle
i\hat{\Sigma}_{ZZ}^{T}(q^{2})$ $\displaystyle=$ $\displaystyle
i\Sigma_{ZZ}^{T\,{\rm Loop}}(q^{2})-i\left(\delta Z_{Z}\left(q^{2}-m_{\rm
Z}^{2}\right)-\delta m_{\rm Z}^{2}\right)$ $\displaystyle-i\left(2g^{\prime
2}m_{\rm Z}^{2}a_{0}+(2g^{2}s_{\rm w}^{2}a_{1}+g^{2}c_{\rm
w}^{2}a_{8}+(g^{2}+g^{\prime 2})a_{13})q^{2}\right)\,,$ $\displaystyle
i\hat{\Sigma}_{ZA}^{T}(q^{2})$ $\displaystyle=$ $\displaystyle
i\Sigma_{ZA}^{T\,{\rm Loop}}(q^{2})-i\left(\delta Z_{ZA}q^{2}-m_{\rm
Z}^{2}s_{\rm w}c_{\rm w}\left(\delta g/g-\delta
g^{\prime}/g^{\prime}\right)\right)$ (4.15) $\displaystyle-i\left(g^{2}s_{\rm
w}c_{\rm w}a_{8}-gg^{\prime}(c_{\rm w}^{2}-s_{\rm w}^{2})a_{1}\right)q^{2}\,.$
$\displaystyle i\hat{\Sigma}_{WW}^{L}(q^{2})$ $\displaystyle=$ $\displaystyle
i\Sigma_{WW}^{L\,{\rm Loop}}(q^{2})+i\left(-\frac{1}{\xi}\left(q^{2}-\xi
m_{\rm W}^{2}\right)\delta Z_{W}+\delta m_{\rm
W}^{2}+q^{2}\frac{\delta\xi_{1}}{\xi}\right)-iq^{2}g^{2}a_{11}\,,$
$\displaystyle\hat{\Sigma}_{W\pi}(q^{2})$ $\displaystyle=$
$\displaystyle\Sigma_{W\pi}^{\rm
Loop}(q^{2})+\frac{\delta\xi_{2}-\delta\xi_{1}}{2}m_{\rm
W}^{2}+q^{2}g^{2}a_{11}\,,$ $\displaystyle-i\hat{\Sigma}_{\pi\pi}(q^{2})$
$\displaystyle=$ $\displaystyle-i\Sigma_{\pi\pi}^{\rm
Loop}(q^{2})+i\left(\left(q^{2}-\xi m_{\rm W}^{2}\right)\delta
Z_{\pi}-\xi\delta m_{\rm W}^{2}-\xi m_{\rm
W}^{2}\delta\xi_{2}\right)-i\frac{g^{2}}{m_{\rm W}^{2}}q^{4}a_{11}\,.$ (4.16)

####  4.3.3 3-legs functions

$\displaystyle i\hat{\Gamma}_{HAA}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
i\Gamma^{\rm Loop}_{HAA}+i\frac{g^{2}s_{\rm w}^{2}}{{\it
v}}(a_{HBB}+a_{HWW}-a_{H1}+a_{H8}/2)((q^{2}-k_{1}^{2}-k_{2}^{2})g^{\mu\nu}-2k_{2}^{\mu}k_{1}^{\nu})\,,$
$\displaystyle i\hat{\Gamma}_{HAZ}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
i\Gamma^{\rm Loop}_{HAZ}+i\frac{ag^{2}s_{\rm w}{\it v}}{2c_{\rm
w}}\left(\frac{\delta g}{g}-\frac{\delta
g^{\prime}}{g^{\prime}}\right)g^{\mu\nu}$ $\displaystyle+i\frac{g^{2}s_{\rm
w}}{{\it v}c_{\rm w}}(-s_{\rm w}^{2}a_{HBB}+c_{\rm w}^{2}a_{HWW}+(1/2-c_{\rm
w}^{2})a_{H1}+c_{\rm
w}^{2}a_{H8}/2)((q^{2}-k_{1}^{2}-k_{2}^{2})g^{\mu\nu}-2k_{2}^{\mu}k_{1}^{\nu})$
$\displaystyle+i\frac{g^{2}s_{\rm w}}{4{\it v}c_{\rm
w}}(a_{d1}+a_{d2}+2a_{d4})((q^{2}+k_{1}^{2}-k_{2}^{2})g^{\mu\nu}-2(k_{1}^{\mu}+k_{2}^{\mu})k_{1}^{\nu})\,,$
$\displaystyle i\hat{\Gamma}_{HZZ}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
ia\frac{g^{2}{\it v}}{2c_{\rm w}^{2}}g^{\mu\nu}+i\Gamma^{\rm
Loop}_{HZZ}+\frac{ag^{2}{\it v}}{2c_{\rm w}^{2}}\left(\frac{\delta
a}{a}+\frac{2\delta g}{g}c_{\rm w}^{2}+\frac{2\delta
g^{\prime}}{g^{\prime}}s_{\rm w}^{2}+\frac{\delta{\it v}}{{\it
v}}+\frac{\delta Z_{H}}{2}+\frac{\delta Z_{\pi}}{2}\right)g^{\mu\nu}$
$\displaystyle-i\frac{g^{2}}{2{\it v}c_{\rm w}^{2}}\left(\left(4s_{\rm
w}^{2}m_{\rm Z}^{2}a_{H0}+\left(-2c_{\rm w}^{4}a_{HWW}-2s_{\rm
w}^{4}a_{HBB}-2s_{\rm w}^{2}c_{\rm w}^{2}a_{H1}-c_{\rm
w}^{4}a_{H8}-a_{H13}\right.\right.\right.$
$\displaystyle\left.\left.\left.\hskip 42.67912pt+s_{\rm w}^{2}a_{d1}-c_{\rm
w}^{2}a_{d2}-2c_{\rm w}^{2}a_{d4}-4s_{\rm w}^{2}\frac{m_{\rm Z}^{2}}{{\it
v}^{2}}a_{\Box 0}-2a_{\Box{\cal V}{\cal V}}\right)q^{2}\right.\right.$
$\displaystyle\left.\left.\hskip 42.67912pt+(2c_{\rm w}^{4}a_{HWW}+2s_{\rm
w}^{4}a_{HBB}+2s_{\rm w}^{2}c_{\rm w}^{2}a_{H1}+c_{\rm
w}^{4}a_{H8}+a_{H13})(k_{1}^{2}+k_{2}^{2})\right)g^{\mu\nu}\right.$
$\displaystyle\left.\hskip 28.45274pt+\left(-s_{\rm w}^{2}a_{d1}+c_{\rm
w}^{2}a_{d2}+a_{d3}+2c_{\rm
w}^{2}a_{d4}+2a_{d5}\right)\left(k_{1}^{\mu}k_{1}^{\nu}+k_{2}^{\mu}k_{2}^{\nu}\right)\right.$
$\displaystyle\left.\hskip
28.45274pt+2\left(a_{d3}+2a_{d5}-a_{H11}\right)k_{1}^{\mu}k_{2}^{\nu}\right.$
$\displaystyle\left.+2\left(2c_{\rm w}^{4}a_{HWW}+2s_{\rm
w}^{4}a_{HBB}+2s_{\rm w}^{2}c_{\rm w}^{2}a_{H1}+c_{\rm w}^{4}a_{H8}-s_{\rm
w}^{2}a_{d1}+c_{\rm w}^{2}a_{d2}+2c_{\rm
w}^{2}a_{d4}\right)k_{2}^{\mu}k_{1}^{\nu}\right)\,,$ $\displaystyle
i\hat{\Gamma}_{HW^{-}W^{+}}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
ia\frac{g^{2}{\it v}}{2}g^{\mu\nu}+i\Gamma^{\rm Loop}_{HWW}+i\frac{ag^{2}{\it
v}}{2}\left(\frac{\delta a}{a}+\frac{2\delta g}{g}+\frac{\delta{\it v}}{{\it
v}}+\frac{\delta Z_{H}}{2}+\frac{\delta Z_{\pi}}{2}\right)g^{\mu\nu}$
$\displaystyle-i\frac{g^{2}}{2{\it
v}}\left(\left(-(2a_{HWW}+a_{d2}+2a_{\Box{\cal V}{\cal
V}})q^{2}+2a_{HWW}(k_{1}^{2}+k_{2}^{2})\right)g^{\mu\nu}\right.$
$\displaystyle\left.\hskip
34.1433pt+\left(a_{d2}+a_{d3}\right)\left(k_{1}^{\mu}k_{1}^{\nu}+k_{2}^{\mu}k_{2}^{\nu}\right)+2\left(a_{d3}-a_{H11}\right)k_{1}^{\mu}k_{2}^{\nu}+2\left(2a_{HWW}+a_{d2}\right)k_{2}^{\mu}k_{1}^{\nu}\right)\,,$
$\displaystyle i\hat{\Gamma}_{\pi^{+}W^{-}A}^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{gg^{\prime}{\it v}c_{\rm w}}{2}g^{\mu\nu}+i\Gamma^{\rm
Loop}_{\pi WA}-\frac{gg^{\prime}{\it v}c_{\rm w}}{2}\left(\frac{\delta
g}{g}+\frac{\delta g^{\prime}}{g^{\prime}}+\frac{\delta{\it v}}{{\it
v}}+\delta Z_{\pi}\right)g^{\mu\nu}$ $\displaystyle+\frac{g^{2}s_{\rm w}}{{\it
v}}\left(\left(-(a_{1}-a_{2}+a_{3}-a_{8}+a_{9}-2a_{11})q^{2}\right.\right.$
$\displaystyle\left.\left.\hskip
56.9055pt+(a_{1}-a_{2}+a_{3}-a_{8}+a_{9})k_{1}^{2}+(a_{1}+a_{2}-a_{3}-a_{8}-a_{9})k_{2}^{2}\right)g^{\mu\nu}\right)$
$\displaystyle\left.+4a_{11}k_{1}^{\mu}k_{1}^{\nu}+2a_{11}k_{1}^{\mu}k_{2}^{\nu}+2\left(a_{1}-a_{2}+a_{3}-a_{8}+a_{9}\right)k_{2}^{\mu}k_{1}^{\nu}-2(a_{2}-a_{3}-a_{9})k_{2}^{\mu}k_{2}^{\nu}\right)\,,$
$\displaystyle i\hat{\Gamma}_{\pi^{+}W^{-}Z}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{gg^{\prime}{\it v}s_{\rm w}}{2}g^{\mu\nu}+i\Gamma^{\rm
Loop}_{\pi WZ}+\frac{gg^{\prime}{\it v}s_{\rm w}}{2}\left(\frac{\delta
g}{g}+\frac{\delta g^{\prime}}{g^{\prime}}+\frac{\delta{\it v}}{{\it
v}}+\delta Z_{\pi}\right)g^{\mu\nu}$ (4.17) $\displaystyle-\frac{g^{2}}{{\it
v}c_{\rm w}}\left(\left(4s_{\rm w}^{2}m_{\rm Z}^{2}a_{0}-(s_{\rm
w}^{2}a_{1}-s_{\rm w}^{2}a_{2}+s_{\rm w}^{2}a_{3}+c_{\rm w}^{2}a_{8}-c_{\rm
w}^{2}a_{9}-2s_{\rm w}^{2}a_{11}+2a_{12}+a_{13})q^{2}\right.\right.$
$\displaystyle\left.\left.\hskip 51.21495pt+(s_{\rm w}^{2}a_{1}-s_{\rm
w}^{2}a_{2}-(1+c_{\rm w}^{2})a_{3}+c_{\rm w}^{2}a_{8}-c_{\rm
w}^{2}a_{9}+a_{13})k_{1}^{2}\right.\right.$ $\displaystyle\left.\left.\hskip
51.21495pt+(s_{\rm w}^{2}a_{1}+s_{\rm w}^{2}a_{2}+(1+c_{\rm
w}^{2})a_{3}+c_{\rm w}^{2}a_{8}+c_{\rm
w}^{2}a_{9}+a_{13})k_{2}^{2}\right)g^{\mu\nu}\right.$
$\displaystyle\left.\hskip 34.1433pt+2\left(a_{3}+(s_{\rm w}^{2}-c_{\rm
w}^{2})a_{11}-a_{12}\right)k_{1}^{\mu}k_{1}^{\nu}+2(s_{\rm
w}^{2}a_{11}-a_{12})k_{1}^{\mu}k_{2}^{\nu}\right.$ $\displaystyle\left.\hskip
34.1433pt+2\left(-s_{\rm w}^{2}a_{2}-c_{\rm w}^{2}a_{3}-c_{\rm
w}^{2}a_{9}+a_{11}-2a_{12}+a_{13})\right)k_{2}^{\mu}k_{1}^{\nu}\right.$
$\displaystyle\left.\hskip 34.1433pt+2\left(s_{\rm w}^{2}a_{1}-s_{\rm
w}^{2}a_{2}+s_{\rm w}^{2}a_{3}+c_{\rm w}^{2}a_{8}-c_{\rm
w}^{2}a_{9}+a_{13})\right)k_{2}^{\mu}k_{1}^{\nu}\right)\,,$ $\displaystyle
i\hat{\Gamma}_{W^{-}W^{+}A}^{\mu\nu\rho}$ $\displaystyle=$ $\displaystyle-
igs_{\rm w}V^{\mu\nu\rho}+i\Gamma^{\rm Loop}_{WWA}-igs_{\rm w}V^{\mu\nu\rho}$
$\displaystyle+ig^{3}s_{\rm
w}\left(g^{\mu\nu}(a_{11}k_{1}^{\rho}-(a_{1}-a_{2}+a_{3}-a_{8}+a_{9}-a_{11})k_{2}^{\rho})\right.$
$\displaystyle\left.\hskip
28.45274pt+g^{\rho\nu}(a_{11}k_{1}^{\mu}+(a_{1}-a_{2}+a_{3}-a_{8}+a_{9})k_{2}^{\mu})\right)\,,$
$\displaystyle i\hat{\Gamma}_{W^{-}W^{+}Z}^{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle-igc_{\rm w}V^{\mu\nu\rho}+i\Gamma^{\rm Loop}_{WWZ}-igc_{\rm
w}V^{\mu\nu\rho}$ $\displaystyle+i\frac{g^{3}}{c_{\rm
w}}\left(g^{\mu\nu}((a_{3}-s_{\rm w}^{2}a_{11}+a_{12})k_{1}^{\rho}+(s_{\rm
w}^{2}a_{1}-s_{\rm w}^{2}a_{2}-c_{\rm w}^{2}a_{3}+c_{\rm w}^{2}a_{8}-c_{\rm
w}^{2}a_{9}-s_{\rm w}^{2}a_{11}+a_{12}+a_{13})k_{2}^{\rho})\right.$
$\displaystyle\left.+g^{\nu\rho}((a_{3}-s_{\rm
w}^{2}a_{11}+a_{12})k_{1}^{\mu}-(s_{\rm w}^{2}a_{1}-s_{\rm
w}^{2}a_{2}-(1+c_{\rm w}^{2})a_{3}+c_{\rm w}^{2}a_{8}-c_{\rm
w}^{2}a_{9}+a_{13})k_{2}^{\mu})\right.$
$\displaystyle\left.+g^{\rho\mu}(-2a_{3}k_{1}^{\nu}-a_{3}k_{2}^{\nu})\right)\,,$
$\displaystyle i\hat{\Gamma}_{ZZA}^{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle i\Gamma^{\rm Loop}_{ZZA}\,,$ $\displaystyle
i\hat{\Gamma}_{ZZZ}^{\mu\nu\rho}$ $\displaystyle=$ $\displaystyle i\Gamma^{\rm
Loop}_{ZZZ}\,.$ (4.18)

####  4.3.4 4-legs functions

$\displaystyle i\hat{\Gamma}_{AAW^{-}W^{+}}^{\mu\nu\rho\sigma}$
$\displaystyle=$ $\displaystyle-ig^{2}s_{\rm
w}^{2}S^{\mu\nu,\rho\sigma}+i\Gamma^{\rm Loop}_{AAWW}-ig^{2}s_{\rm
w}^{2}S^{\mu\nu,\rho\sigma}$ $\displaystyle-ig^{4}s_{\rm
w}^{2}a_{11}(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho})\,,$
$\displaystyle i\hat{\Gamma}_{AZW^{-}W^{+}}^{\mu\nu\rho\sigma}$
$\displaystyle=$ $\displaystyle-ig^{2}s_{\rm w}c_{\rm
w}S^{\mu\nu,\rho\sigma}+i\Gamma^{\rm Loop}_{AZWW}-ig^{2}s_{\rm w}c_{\rm
w}S^{\mu\nu,\rho\sigma}$ $\displaystyle+i\frac{g^{4}s_{\rm w}}{c_{\rm
w}}\left(2a_{3}g^{\mu\nu}g^{\rho\sigma}-(-a_{3}+s_{\rm
w}^{2}a_{11}-a_{12})(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho})\right)\,,$
$\displaystyle i\hat{\Gamma}_{ZZW^{-}W^{+}}^{\mu\nu\rho\sigma}$
$\displaystyle=$ $\displaystyle-ig^{2}c_{\rm
w}^{2}S^{\mu\nu,\rho\sigma}+i\Gamma^{\rm Loop}_{ZZWW}-ig^{2}c_{\rm
w}^{2}S^{\mu\nu,\rho\sigma}$ $\displaystyle+i\frac{g^{4}}{c_{\rm
w}^{2}}\left(2(2c_{\rm
w}^{2}a_{3}+a_{5}+a_{7})g^{\mu\nu}g^{\rho\sigma}\right.$
$\displaystyle\left.\hskip 34.1433pt-(2c_{\rm w}^{2}a_{3}-a_{4}-a_{6}+s_{\rm
w}^{4}a_{11}-2s_{\rm
w}^{2}a_{12})(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho})\right)\,,$
$\displaystyle i\hat{\Gamma}_{W^{-}W^{+}W^{-}W^{+}}^{\mu\nu\rho\sigma}$
$\displaystyle=$ $\displaystyle ig^{2}S^{\mu\rho,\nu\sigma}+i\Gamma^{\rm
Loop}_{WWWW}+ig^{2}S^{\mu\rho,\nu\sigma}$ (4.19)
$\displaystyle+i\frac{g^{4}}{c_{\rm
w}^{2}}\left(2(-2a_{3}+a_{4}+a_{8}-2a_{9}+2a_{13})g^{\mu\rho}g^{\nu\sigma}\right.$
$\displaystyle\left.\hskip
34.1433pt-(-2a_{3}-a_{4}-2a_{5}+a_{8}-2a_{9}+2a_{13})(g^{\mu\nu}g^{\rho\sigma}+g^{\mu\sigma}g^{\nu\rho})\right)\,.$

Notice that, to simplify the notation, in all the previous equations, Eq.
(4.13) through Eq. (4.19), we have dropped the superindex $0$ in all the bare
$a_{i}^{0}$ coefficients which are the ones entering in these equations.
Remember that these bare coefficients $a_{i}^{0}$ are the sum of the
renormalized coefficients plus the counterterms, according to our prescription
defined in Eq. (4.3). Therefore, the $a_{i}$ coefficients appearing in the
previous equations, Eq. (4.13) through Eq. (4.19), really mean $a_{i}+\delta
a_{i}$, where $a_{i}$ is the rermormalized EChL coefficient and $\delta a_{i}$
is the corresponding counterterm.

Regarding the contributions from the one-loop diagrams to the Green functions
(generated by $\mathcal{L}_{2}$), we classify them generically into four
categories accordingly to the particles in the loops: i) loops with just
scalar bosons, i.e., with $H$ and/or $\pi$ (called ‘chiral’ loops in short),
ii) loops with both gauge and scalar bosons (called ‘mix’ in short reference
to mixed loops), iii) loops with only gauge bosons (called ‘gauge’ loops in
short), and iv) loops with only ghosts (called ‘ghost’ loops in short). Then
we compute the loops contributions in the $R_{\xi}$ gauges, generically called
here by $\Gamma^{\rm Loop}_{n-legs}$, as the sum of all these loops:

$\Gamma^{\rm
Loop}_{n-legs}=\Gamma^{\rm{chiral}}_{n-legs}+\Gamma^{\rm{mix}}_{n-legs}+\Gamma^{\rm{gauge}}_{n-legs}+\Gamma^{\rm{ghost}}_{n-legs}\,.$
(4.20)

For this purpose, we have calculated analytically the many Feynman loop
diagrams (there are more than 500 in the computation of the $WZ\to WZ$
scattering amplitude) by using Mathematica [27]. More precisely, we have
implemented our model with FeynRules [20], generated and drawn all the Feynman
diagrams by FeynArts [21], performed the main calculations with FormCalc and
LoopTools [22] and incorporate some additional checks using FeynCalc [23] and
Package-X [24]. The generic loop diagrams entering in the specific 1PI
functions that participate in the present computation of the $WZ\to WZ$
scattering amplitude are summarized in Appendix Appendix B.

On the other hand, the loop diagrams contributing to the 1-leg funtion (EChL
Higgs tadpole), not explicitly drawn, are the loops with just scalars ($H$ and
$\pi$), and the loops with just gauge bosons ($W$ and $Z$). The analytical
result for these loop contributions to the Higgs tadpole in the $R_{\xi}$
gauges is:

$iT^{\rm Loop}=i\frac{1}{32\pi^{2}{\it v}}(3\kappa_{3}m_{\rm
H}^{2}A_{0}(m_{\rm H}^{2})+2a(6m_{\rm W}^{2}A_{0}(m_{\rm W}^{2})-4m_{\rm
W}^{4}+3m_{\rm Z}^{2}A_{0}(m_{\rm Z}^{2})-2m_{\rm Z}^{4}))\,,$ (4.21)

where
$A_{0}(m^{2})=\left(\Delta_{\epsilon}+\log(\mu^{2}/m^{2})+1\right)m^{2}$.

At this point, it is important to remark that the Higgs tadpole in the EChL is
$\xi$-independent, in contrast to the SM case. Therefore, the tadpole is a
gauge invariant quantity within the EChL. This interesting feature is a
consequence of the Higgs boson being a singlet field in this non-linear EFT
approach, in contrast to the SM (and SMEFT) case where the Higgs is a
component of a doublet field. Furthermore, the Higgs field does not couple to
the ghost fields in the EChL (in contrast to the linear realization of SM and
SMEFT where the Higgs does couple to the ghosts), as we have seen in the
presentation of the EChL Feynman rules in Section 2. Therefore, the ghosts
loops do not participate in this 1-leg function and also decouple from other
1PI functions with external $H$ fields. The comparison with the SM case is
summarized in the Appendices C-D.

###  4.4 Divergences of the one-loop contributions

Due to the lengthy full analytical results of all the involved 1PI functions
to one-loop in the $R_{\xi}$ gauge, we do not include the explicit formulas
for these $\Gamma^{\rm Loop}_{n-legs}$ here, and instead we present just their
corresponding analytical divergent (singular) contributions. The list with
these ${\cal O}(\Delta_{\epsilon})$ results is shown in the following.

The loops divergencies in 1- and 2-legs 1PI functions are:

$\displaystyle iT^{\rm Loop}|_{div}$ $\displaystyle=$ $\displaystyle
i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{3}{2{\it v}}\left(\kappa_{3}m_{\rm
H}^{4}+2a\left(2m_{\rm W}^{4}+m_{\rm Z}^{4}\right)\right)$
$\displaystyle-i\Sigma_{HH}^{\rm Loop}(q^{2})|_{div}$ $\displaystyle=$
$\displaystyle i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{3}{2{\it
v}^{2}}\left(a^{2}q^{4}-2a^{2}(2m_{\rm W}^{2}+m_{\rm
Z}^{2})q^{2}+(3\kappa_{3}^{2}+\kappa_{4})m_{\rm H}^{4}+(4a^{2}+2b)(2m_{\rm
W}^{4}+m_{\rm Z}^{4})\right)$ $\displaystyle i\Sigma_{WW}^{T\,{\rm
Loop}}(q^{2})|_{div}$ $\displaystyle=$ $\displaystyle
i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}}{12}\left((51-a^{2}-12\xi)q^{2}+3(a^{2}-b)m_{\rm
H}^{2}+3(9-3a^{2}+4\xi)m_{\rm W}^{2}-9m_{\rm Z}^{2}\right)$ $\displaystyle
i\Sigma_{AA}^{T\,{\rm Loop}}(q^{2})|_{div}$ $\displaystyle=$ $\displaystyle
i\frac{\Delta_{\epsilon}}{16\pi^{2}}e^{2}(4-\xi)q^{2}$ $\displaystyle
i\Sigma_{ZZ}^{T\,{\rm Loop}}(q^{2})|_{div}$ $\displaystyle=$ $\displaystyle
i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}}{12}\left((4-\frac{1+a^{2}}{c_{\rm
w}^{2}}+12c_{\rm w}^{2}(4-\xi))q^{2}\right.$ $\displaystyle\left.\hskip
45.5244pt+3(a^{2}-b)\frac{m_{\rm H}^{2}}{c_{\rm w}^{2}}+12m_{\rm
W}^{2}(3+\xi)-18m_{\rm Z}^{2}-9a^{2}\frac{m_{\rm Z}^{2}}{c_{\rm
w}^{2}}\right)$ $\displaystyle i\Sigma_{ZA}^{T\,{\rm Loop}}(q^{2})|_{div}$
$\displaystyle=$ $\displaystyle
i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{eg}{2c_{\rm w}}\left((1/3+2c_{\rm
w}^{2}(4-\xi))q^{2}+m_{\rm W}^{2}(3+\xi)\right)$ $\displaystyle
i\Sigma_{WW}^{L\,{\rm Loop}}(q^{2})|_{div}$ $\displaystyle=$ $\displaystyle
i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}}{4}\left(a^{2}q^{2}+(a^{2}-b)m_{\rm
H}^{2}+(9-3a^{2}+4\xi)m_{\rm W}^{2}-3m_{\rm Z}^{2}\right)$ $\displaystyle
i\Sigma_{W\pi}^{\rm Loop}(q^{2})|_{div}$ $\displaystyle=$ $\displaystyle
i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}}{4}\left(-a^{2}q^{2}-(a^{2}-b)m_{\rm
H}^{2}-(3-3a^{2}+8\xi/3)m_{\rm W}^{2}+(3-2\xi/3)m_{\rm Z}^{2}\right)$
$\displaystyle-i\Sigma_{\pi\pi}^{\rm Loop}(q^{2})|_{div}$ $\displaystyle=$
$\displaystyle i\frac{\Delta_{\epsilon}}{16\pi^{2}}\left(\frac{a^{2}}{{\it
v}^{2}}q^{4}+\frac{q^{2}}{{\it v}^{2}}((a^{2}-b)m_{\rm
H}^{2}-(3+3a^{2}-4\xi/3)m_{\rm W}^{2}-(3-4\xi/3)m_{\rm Z}^{2})\right)$ (4.22)

The loops divergencies in the $S(q)V_{\mu}(k_{1})V^{\prime}_{\nu}(k_{2})$ 1PI
functions are:

$\displaystyle\Gamma^{\rm Loop}_{HAA}|_{div}$ $\displaystyle=$ $\displaystyle
0$ $\displaystyle i\Gamma^{\rm Loop}_{HAZ}|_{div}$ $\displaystyle=$
$\displaystyle i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}s_{\rm w}}{{\it
v}}am_{\rm W}m_{\rm Z}(\xi+3)g^{\mu\nu}$ $\displaystyle i\Gamma^{\rm
Loop}_{HZZ}|_{div}$ $\displaystyle=$ $\displaystyle
i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}}{12{\it v}c_{\rm
w}^{2}}\left(\left(3a(2+a^{2})q^{2}+a(a^{2}-b)(k_{1}^{2}+k_{2}^{2})\right.\right.$
$\displaystyle\hskip 39.83385pt\left.\left.-3(a^{2}-b)(2a-3\kappa_{3})m_{\rm
H}^{2}-18abm_{\rm Z}^{2}+24am_{\rm W}^{2}c_{\rm w}^{2}(\xi+3)-36am_{\rm
W}^{2}\right)g^{\mu\nu}\right.$ $\displaystyle\left.\hskip
56.9055pt+2a(a^{2}+2b)(k_{1}^{\mu}k_{1}^{\nu}+k_{2}^{\mu}k_{2}^{\nu})+12a^{3}k_{1}^{\mu}k_{2}^{\nu}\right)$
$\displaystyle i\Gamma^{\rm Loop}_{HWW}|_{div}$ $\displaystyle=$
$\displaystyle i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}}{12{\it
v}}\left(\left(3a(2+a^{2})q^{2}+a(a^{2}-b)(k_{1}^{2}+k_{2}^{2})\right.\right.$
$\displaystyle\left.\left.\hskip 56.9055pt-3(a^{2}-b)(2a-3\kappa_{3})m_{\rm
H}^{2}-18abm_{\rm W}^{2}+6am_{\rm W}^{2}(4\xi+9)-18am_{\rm
Z}^{2}\right)g^{\mu\nu}\right.$ $\displaystyle\left.\hskip
51.21495pt+2a(a^{2}+2b)(k_{1}^{\mu}k_{1}^{\nu}+k_{2}^{\mu}k_{2}^{\nu})+12a^{3}k_{1}^{\mu}k_{2}^{\nu}\right)$
$\displaystyle i\Gamma^{\rm Loop}_{\pi WA}|_{div}$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}s_{\rm w}}{6{\it
v}}\left(g^{\mu\nu}\left(3a^{2}q^{2}+(1-a^{2})k_{2}^{2}\right.\right.$
$\displaystyle\left.\left.\hskip 79.66771pt+3(a^{2}-b)m_{\rm
H}^{2}+9(1-a^{2})m_{\rm W}^{2}-9m_{\rm Z}^{2}+2(4m_{\rm W}^{2}+m_{\rm
Z}^{2})\xi\right)\right.$ $\displaystyle\left.\hskip
56.9055pt+6a^{2}k_{1}^{\mu}k_{1}^{\nu}+3a^{2}k_{1}^{\mu}k_{2}^{\nu}-(1-a^{2})k_{2}^{\mu}k_{2}^{\nu}\right)$
$\displaystyle i\Gamma^{\rm Loop}_{\pi WZ}|_{div}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}}{12{\it v}c_{\rm
w}}\left(g^{\mu\nu}\left(6s_{\rm
w}^{2}a^{2}q^{2}+(1-a^{2})k_{1}^{2}-(1-a^{2})(c_{\rm w}^{2}-s_{\rm
w}^{2})k_{2}^{2}\right.\right.$ (4.23) $\displaystyle\left.\left.\hskip
79.66771pt+2s_{\rm w}^{2}\left(3(a^{2}-b)m_{\rm H}^{2}+9(1-a^{2})m_{\rm
W}^{2}-9a^{2}m_{\rm Z}^{2}+2(4m_{\rm W}^{2}+m_{\rm
Z}^{2})\xi\right)\right)\right.$ $\displaystyle\left.+(-1+7a^{2}-12c_{\rm
w}^{2}a^{2})k_{1}^{\mu}k_{1}^{\nu}+6s_{\rm
w}^{2}a^{2}k_{1}^{\mu}k_{2}^{\nu}+(-1+2c_{\rm w}^{2}+7a^{2}-2c_{\rm
w}^{2}a^{2})k_{2}^{\mu}k_{2}^{\nu}\right)$

The loops divergencies in the $W^{-}_{\mu}(k_{1})W^{+}_{\nu}(k_{2})V_{\rho}$
1PI functions are:

$\displaystyle i\Gamma^{\rm Loop}_{WWA}|_{div}$ $\displaystyle=$
$\displaystyle i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{3}s_{\rm
w}}{12}\left(g^{\mu\nu}\left(k_{1}^{\rho}\left(33-4a^{2}-18\xi\right)-k_{2}^{\rho}\left(33+2a^{2}-18\xi\right)\right)\right.$
$\displaystyle\left.\hskip
56.9055pt+g^{\nu\rho}(k_{1}^{\mu}(33-4a^{2}-18\xi)-2k_{2}^{\mu}(-33+a^{2}+18\xi))\right.$
$\displaystyle\left.\hskip
56.9055pt+(-33+a^{2}+18\xi)g^{\rho\mu}(2k_{1}^{\nu}+k_{2}^{\nu})\right)$
$\displaystyle i\Gamma^{\rm Loop}_{WWZ}|_{div}$ $\displaystyle=$
$\displaystyle i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{3}}{24c_{\rm
w}}\left(g^{\mu\nu}\left(k_{1}^{\rho}\left(1+5a^{2}+c_{\rm
w}^{2}(66-8a^{2}-36\xi)\right)\right.\right.$ (4.24)
$\displaystyle\left.\left.\hskip 82.51299pt-k_{2}^{\rho}\left(1-7a^{2}+c_{\rm
w}^{2}(66+4a^{2}-36\xi)\right)\right)\right.$ $\displaystyle\left.\hskip
5.69054pt+g^{\nu\rho}(k_{1}^{\mu}(1+5a^{2}+c_{\rm
w}^{2}(66-8a^{2}-36\xi))-2k_{2}^{\mu}(-1+a^{2}+c_{\rm
w}^{2}(-66+2a^{2}+36\xi)))\right.$ $\displaystyle\left.\hskip
5.69054pt+(-1+a^{2}+c_{\rm
w}^{2}(-66+2a^{2}+36\xi))g^{\rho\mu}(2k_{1}^{\nu}+k_{2}^{\nu})\right)$

The $Z_{\mu}(k_{1})Z_{\nu}(k_{2})V_{\rho}$ 1PI functions are finite.

The loops divergencies in the
$V_{\mu}V^{\prime}_{\nu}W^{-}_{\rho}W^{+}_{\sigma}$ 1PI functions are:

$\displaystyle i\Gamma^{\rm Loop}_{AAWW}|_{div}$ $\displaystyle=$
$\displaystyle-i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{4}s_{\rm
w}^{2}}{12}\left((24\xi+a^{2}-15)S^{\mu\nu,\rho\sigma}-3a^{2}(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho})\right)$
$\displaystyle i\Gamma^{\rm Loop}_{AZWW}|_{div}$ $\displaystyle=$
$\displaystyle-i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{4}s_{\rm w}c_{\rm
w}}{24}\left((48\xi-33+2c_{\rm
w}^{2}(a^{2}+1)+a^{2})S^{\mu\nu,\rho\sigma}\right.$ $\displaystyle\left.\hskip
88.20354pt+8(2-3c_{\rm
w}^{2})a^{2}(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho})\right)$
$\displaystyle i\Gamma^{\rm Loop}_{ZZWW}|_{div}$ $\displaystyle=$
$\displaystyle-i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{4}}{24c_{\rm
w}^{2}}\left((48c_{\rm w}^{4}\xi-32c_{\rm w}^{4}-a^{4}+3a^{2}b-2a^{2}s_{\rm
w}^{2}(2+c_{\rm w}^{2})\right.$ $\displaystyle\left.\hskip
73.97716pt-3b^{2}/2-1+4s_{\rm w}^{2}c_{\rm
w}^{2})S^{\mu\nu,\rho\sigma}\right.$ $\displaystyle\left.\hskip
14.22636pt-(a^{4}+3a^{2}b-3a^{2}s_{\rm w}^{2}(1+2c_{\rm
w}^{2})-3b^{2}/2+3s_{\rm w}^{2}c_{\rm
w}^{2})(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho})\right)$
$\displaystyle i\Gamma^{\rm Loop}_{WWWW}|_{div}$ $\displaystyle=$
$\displaystyle
i\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{4}}{24}\left((48\xi+2a^{4}-30)S^{\mu\rho,\nu\sigma}-3(-2a^{4}+2a^{2}b-b^{2}-2)(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho})\right)$

All these divergent contributions will set the values of the ${\cal
O}(\Delta_{\epsilon})$ counterterms, both for the EW parameters and the
$a_{i}$ coefficients, that are relevant for our computation of the VBS
amplitudes. These will be presented in Section 4.6.

###  4.5 Slavnov-Taylor identity in the unphysical charged sector

Before presenting our main results on the renormalization of the
$\mathcal{L}_{2}$ parameters and $a_{i}$ coefficients in Section 4.6, it is
illustrative to discuss first an important aspect related to the gauge
invariance of the EChL: the Slavnov-Taylor Identity (STI) corresponding to the
unphysical charged sector, in particular the equation relating the two-legs
functions of the $W$ and $\pi$ fields. The validity of this relation among the
self-energies in this unphysical sector at the renormalized level and for
arbitrary external momentum is one of the key points for the $\xi$-invariance
of the NLO EChL predictions. Setting properly this equation at the
renormalized level also guarantees the same poles structure at the NLO as in
the LO for all the unphysical propagators as reflected in Eq. (4.10), which is
very convenient from the practical computational aspects. The corresponding
Slavnov-Taylor relations among the self-energies in the unphysical neutral
sector, i.e., for $\\{A,Z,\pi^{3}\\}$, can also be set in a similar way.
However, the details in this sector are more involved and we do not present
them here since they are not needed for the present computation of $WZ$
scattering. We then focus next in the charged unphyical sector and summarize
the basic points of the STI.

After the introduction of $\mathcal{L}_{GF}$ and $\mathcal{L}_{FP}$ in Eq.
(2.7), the original $SU(2)_{L}\times U(1)_{Y}$ gauge invariance of the
classical Lagrangian is lost. However, the EChL is invariant under the BRS
transformations [28] that involve the ghost fields. The BRS symmetry induces
relations among the various Green functions involved, called the Slavnov-
Taylor identities (STIs) [29, 30]. A special case of interest here is the
relation between the propagators of the unphysical charged sector,
$\\{W^{\pm},\pi^{\pm}\\}$ which then leads to the corresponding relations
among self-energies.

In oder to get these relations and to understand how are they written in terms
of renormalized quantities let us first consider the corresponding gauge-
fixing Lagrangian of the charged sector in terms of the bare quantities. This
can be written as follows:

$\mathcal{L}_{GF}^{0}=-\left(\frac{1}{\sqrt{\xi_{1}^{0}}}\,\partial^{\mu}W_{0\,\mu}^{+}-\sqrt{\xi_{2}^{0}}\,\frac{g_{0}{\it
v}_{0}}{2}\,\pi_{0}^{+}\right)\left(\frac{1}{\sqrt{\xi_{1}^{0}}}\,\partial^{\mu}W_{0\,\mu}^{-}-\sqrt{\xi_{2}^{0}}\,\frac{g_{0}{\it
v}_{0}}{2}\,\pi_{0}^{-}\right)\,,$ (4.26)

where two independent gauge-fixing bare parameters $\xi_{1}^{0}$ and
$\xi_{2}^{0}$ have been used (for this presentation we follow closely [26] and
[18]). Then we use the prescription in Eq. (4.3), to write this Lagragian in
terms of renormalized quantities. In particular, remember that we have chosen
just one common renormalized gauge parameter $\xi$ with
$\xi_{1,2}^{0}=\xi(1+\delta\xi_{1,2})$. Then, the leading order GF Lagrangian
in terms of the renormalized quantities corresponds to the first term on the
l.h.s. of Eq. (2.12), i.e., to:

$\mathcal{L}_{GF}=-\left(\frac{1}{\sqrt{\xi}}\,\partial^{\mu}W_{\mu}^{+}-\sqrt{\xi}\,\frac{g{\it
v}}{2}\,\pi^{+}\right)\left(\frac{1}{\sqrt{\xi}}\,\partial^{\mu}W_{\mu}^{-}-\sqrt{\xi}\,\frac{g{\it
v}}{2}\,\pi^{-}\right)\,,$ (4.27)

One can use Eq. (4.26) to get the relevant STI in terms of the undressed
propagators in momentum-space. This reads as follows:

$q^{2}\Delta_{L}^{WW}(q^{2})+2\sqrt{\xi_{1}^{0}\xi_{2}^{0}}\,q^{2}\Delta^{W\pi}(q^{2})-\xi_{1}^{0}\xi_{2}^{0}m_{\rm
W}^{0\,2}\Delta^{\pi\pi}(q^{2})=\xi_{1}^{0}\,.$ (4.28)

We then write the above propagators in terms of the undressed 2-leg functions
at one-loop precision and arrive to the following relation among undressed
self-energies:

$q^{2}\Sigma_{WW}^{L\,{\rm Loop}}(q^{2})+2q^{2}\Sigma_{W\pi}^{\rm
Loop}(q^{2})-m_{\rm W}^{2}\Sigma_{\pi\pi}^{\rm Loop}(q^{2})=0\,.$ (4.29)

Finally, by adding the corresponding counterterms, given explicitly in Eq.
(4.16), one arrives to the wanted STI in terms of the renormalized quantities
at the one-loop level. This reads as follows:

$q^{2}\hat{\Sigma}_{WW}^{L}(q^{2})+2q^{2}\hat{\Sigma}_{W\pi}(q^{2})-m_{\rm
W}^{2}\hat{\Sigma}_{\pi\pi}(q^{2})=\frac{q^{2}-\xi m_{\rm
W}^{2}}{\xi}f_{ST}(q^{2})\,,$ (4.30)

where the $q^{2}$ dependent function $f_{ST}(q^{2})$ on the r.h.s of this STI,
which is finite, has been defined as:

$f_{ST}(q^{2})=-\left(\delta Z_{W}-\delta\xi_{1}\right)q^{2}+\xi\delta m_{\rm
W}^{2}+\xi m_{\rm W}^{2}\left(\delta Z_{\pi}+\delta\xi_{2}\right).$ (4.31)

Some comments about the previous Eq. (4.30) are in order. First, this STI and
the renormalization conditions of Eq. (4.10) ensure that
$\hat{\Sigma}_{W\pi}(\xi m_{\rm W}^{2})=0$, or equivalently, the poles of the
renormalized charged propagators are located at $q^{2}=\xi m_{\rm W}^{2}$ at
the one-loop level. Second, it is important to notice that the new physics
effects encoded in the $a_{11}$ coefficient, which enters separately in each
self-energy, cancel in this particular combination of renormalized self-
energies. Then, interestingly, this STI in Eq. (4.30), using the $R_{\xi}$
gauges, has the same formal expression in the EChL as in the SM [26]. However,
it is also important to notice that the role played by the tadpole in the SM
is different than in the EChL, as it is shown in Appendix D. Again, the fact
that the $H$ field is a singlet in the EChL case introduces some peculiarities
respect to the linear realizations like the SM and the SMEFT. For a
verification and discussion of the STIs in the SM and SMEFT using the
background field method see also [31].

Finally, due to the relevance of the STIs in the consistency of the theory, we
will impose the validity of Eq. (4.29) at the renormalized level. Namely, we
set in Eq. (4.30):

$f_{ST}(q^{2})=0\quad\forall q^{2}$ (4.32)

This is equivalent to set an additional renormalization condition in the
unphysical sector valid for all external momenta (i.e., not only at $q^{2}=\xi
m_{W}^{2}$), and implies that each coefficient of this linear function of
$q^{2}$ vanishes, resulting in:

$\displaystyle\delta\xi_{1}=\delta Z_{W}\quad;\quad\delta\xi_{2}=-\delta
Z_{\pi}-\frac{\delta m_{\rm W}^{2}}{m_{\rm W}^{2}}.$ (4.33)

These conditions in Eq. (4.32) (or, equivalently, in Eq. (4.33)) and Eq.
(4.10) fix then the four counterterms of the unphysical charged sector
($\delta\xi_{1}$, $\delta\xi_{2}$, $\delta Z_{\pi}$ and $\delta a_{11}$). It
is important to stress that the particular gauge-fixing and renormalization
conditions in the unphysical sector do not affect the physical $S$-matrix
elements. However, the simplifications induced by the use of the STIs and the
consequent specific cancellations among the $\xi$-dependent terms of the
various contributions in the different channel diagrams do depend on the
concrete formulation of the unphysical sector. In this sense, among the many
arbitrary prescriptions for this unphysical sector, we choose a particular one
via the renormalization conditions given by Eq. (4.10) and Eq. (4.33). This is
mainly motivated by the interplay between the unphysical self-energies
invloved in the $s$\- and $u$-channels of the $WZ$ scattering process of our
interest in Fig. 1 and the cancellations of the $\xi$-dependent terms that we
have seen occur in both channels of the LO amplitude. Our prescription then
replicates these cancellations also at the NLO.

###  4.6 Renormalization of the EFT parameters

In this section we present the results for the renormalization of the EFT
parameters. These include the EW parameters entering in $\mathcal{L}_{2}$,
like $g,$ $g^{\prime}$, etc., and the EChL coefficients, namely, the ones
entering in $\mathcal{L}_{2}$, like $a$, etc, and the ones entering in
$\mathcal{L}_{4}$, i.e., the $a_{i}$ coefficients.

With our previously described diagrammatic procedure of renormalization of all
the 1PI functions we can derive the divergent parts, i.e., of ${\cal
O}(\Delta_{\epsilon})$, for all the involved counterterms. These are called
here in short $\delta_{\epsilon}$. Alternatively, these divergent parts can
also be derived by using the renormalization conditions introduced in Section
4.2 which allow to write the parameter counterterms in terms of the undressed
1PI functions. In fact, we have used this second procedure as a check of our
results that we obtain with the first procedure. Regarding the finite
contributions to all these counterterms, they are also determined by these
renormalization conditions and, indeed, we use them in the final numerical
computation of the $WZ$ scattering in the next section. Therefore, we postpone
the introduction of the finite contributions for the next section and focus
here in the derivation of the divergent parts of the EChL counterterms.

The determination of the divergent parts of all the counterterms for the EChL
parameters follows immediately from the requirement of getting finite all the
renormalized 1PI functions at arbitrary values of the external leg momenta,
i.e., off-shell 1PI functions. This means that the cancellation of the ${\cal
O}(\Delta_{\epsilon})$ contributions must occur at all external momenta, and
it proceeds between the loop contributions in all these 1PI functions and the
involved counterterms. One then requires that these cancellations occur in
each involved Lorentz structure and in each term in the momentum powers
expansion and all these lead to a system of equations with a number of
unknowns, the $\delta_{\epsilon}$’s, than one solves globally. In practice,
one can conveniently solve this system in a sequential form, first solving the
1-leg function, then the two-legs functions, with the previous solutions then
solving the three-legs and finally the four-legs functions. This provides the
solution for all $\delta_{\epsilon}$ of both the EW parameters and the EChL
coefficients jointly. For instance, the case of the Higgs self-energy
illustrates clearly this procedure. Setting the cancellation of the ${\cal
O}(\Delta_{\epsilon})$ contributions in all the terms of
$\hat{\Sigma}_{HH}(q^{2})$ of Eq. (4.14), namely, in the terms of order
$q^{4}$, $q^{2}$ and $q^{0}$, and using our result for $\Sigma_{HH}^{\rm
Loop}(q^{2})|_{div}$ in Eq. (4.22) one derives easily
${\delta_{\epsilon}}a_{\Box\Box}$, ${\delta_{\epsilon}}Z_{H}$ and
${\delta_{\epsilon}}m_{\rm H}^{2}$. One can proceed similarly with the other
self-energies of the gauge bosons and the $\pi$ bosons in Eq. (4.15) and the
other 1PI functions. At this point, it is worth remarking that the
divergencies of the CTs for the EW parameters are determined by just the 1-
and 2-legs 1PI functions whereas the divergencies of the CTs for the EChL
coefficients are determined by all the 1PI functions. These new CTs in the
EChL coefficients cancel the extra divergencies arising in the loop diagramas
(generated by $\mathcal{L}_{2}$) of all the 1PI functions and for arbitrary
(off-shell) external momenta.

The summary of the sequential settings regarding the EW parameters can be read
as follows: 1) $\hat{\Sigma}_{HH}$ sets ${\delta_{\epsilon}}Z_{H}$ and
${\delta_{\epsilon}}m_{\rm H}^{2}$, 2) $\hat{\Sigma}_{WW}^{T}$ sets
${\delta_{\epsilon}}Z_{W}$ and ${\delta_{\epsilon}}m_{\rm W}^{2}$, 3)
$\hat{\Sigma}_{\pi\pi}$ sets ${\delta_{\epsilon}}Z_{\pi}$ (and
${\delta_{\epsilon}}\xi_{2}$), 4) $\hat{\Sigma}_{W\pi}$ and
$\hat{\Sigma}_{WW}^{L}$ set ${\delta_{\epsilon}}\xi_{1}$ and
${\delta_{\epsilon}}\xi_{2}$, 5) $U(1)_{\rm em}$ gauge invariance sets $\delta
g^{\prime}/g^{\prime}=0$, 6) $\hat{\Sigma}_{ZA}^{T}$ then sets
${\delta_{\epsilon}}g/g$, 7) with the previous ${\delta_{\epsilon}}Z_{\pi}$,
${\delta_{\epsilon}}Z_{W}$, ${\delta_{\epsilon}}m_{\rm W}^{2}$ and
${\delta_{\epsilon}}g/g$, then one derives ${\delta_{\epsilon}}v/v$, 8) the
tadpole $\hat{T}$ then sets ${\delta_{\epsilon}}\lambda/\lambda$, or it can
also be derived alternatively from ${\delta_{\epsilon}}Z_{H}$,
${\delta_{\epsilon}}m_{\rm H}^{2}$, ${\delta_{\epsilon}}g/g$ and
${\delta_{\epsilon}}v/v$, 9) finally, the CTs in the neutral gauge sector
${\delta_{\epsilon}}Z_{Z}$, ${\delta_{\epsilon}}m_{\rm Z}^{2}$ and
${\delta_{\epsilon}}Z_{A}$ are derived from $\hat{\Sigma}_{ZZ}^{T}$,
$\hat{\Sigma}_{AA}^{T}$ and $\hat{\Sigma}_{AZ}^{T}$ jointly.

Regarding the EChL coefficients, the ${\delta_{\epsilon}}$’s are fixed, in
summary, as follows: 1) $\hat{\Sigma}_{HH}$ sets
${\delta_{\epsilon}}a_{\Box\Box}$, 2) $\hat{\Sigma}_{\pi\pi}$ sets
${\delta_{\epsilon}}a_{11}$ (it is also set by $\hat{\Sigma}_{WW}^{L}$ and
$\hat{\Sigma}_{W\pi}$), 3) $\hat{\Sigma}_{ZZ}^{T}$, $\hat{\Sigma}_{AZ}^{T}$
and $\hat{\Sigma}_{AA}^{T}$ jointly set ${\delta_{\epsilon}}a_{0}$,
${\delta_{\epsilon}}(a_{1}+a_{13})$ and ${\delta_{\epsilon}}(a_{8}+a_{13})$,
4) $\hat{\Gamma}_{HWW}$ sets ${\delta_{\epsilon}}a/a$, 5)
$\hat{\Gamma}_{HWW}$, $\hat{\Gamma}_{HZZ}$, $\hat{\Gamma}_{HAZ}$ and
$\hat{\Gamma}_{HAA}$ jointly set all the ${\delta_{\epsilon}}a_{i}$’s in
${\mathcal{L}}_{4}^{\rm one-Higgs}$, 6) $\hat{\Gamma}_{WWZ}$,
$\hat{\Gamma}_{WWA}$, $\hat{\Gamma}_{\pi WZ}$ and $\hat{\Gamma}_{\pi WA}$ set
jointly ${\delta_{\epsilon}}a_{2}$, ${\delta_{\epsilon}}a_{3}$,
${\delta_{\epsilon}}a_{8}$, ${\delta_{\epsilon}}a_{9}$,
${\delta_{\epsilon}}a_{12}$ and ${\delta_{\epsilon}}a_{13}$ (and using this
latter one then solves $a_{1}$ separately from 3)) , 6) $\hat{\Gamma}_{WWWW}$
sets ${\delta_{\epsilon}}a_{4}$ and ${\delta_{\epsilon}}a_{5}$ and
$\hat{\Gamma}_{ZZWW}$ sets ${\delta_{\epsilon}}(a_{4}+a_{6})$ and
${\delta_{\epsilon}}(a_{5}+a_{7})$. Therefore, the two together also set
${\delta_{\epsilon}}a_{6}$ and ${\delta_{\epsilon}}a_{7}$. 7) Finally, one can
use extra functions and relations among them as a check of the previous
findings. For instance, one can use $\hat{\Gamma}_{AAWW}$ and
$\hat{\Gamma}_{AZWW}$ regarding the checks in the 4-legs functions, and the
relations in Eq. (4.33) regarding the two-legs functions in the unphysical
sector.

We list in the following the results for all these divergent counterterms for
the EFT parameters. We collect together the CTs for the EW parameters, and
separately those for the EChL coefficients.

####  4.6.1 Divergent CTs for the EW parameters

The results are the following:

$\displaystyle{\delta_{\epsilon}}Z_{H}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{3a^{2}}{{\it
v}^{2}}(2m_{\rm W}^{2}+m_{\rm
Z}^{2})\,,\qquad{\delta_{\epsilon}}T=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{3}{2{\it
v}}\left(\kappa_{3}m_{\rm H}^{4}+2a\left(2m_{\rm W}^{4}+m_{\rm
Z}^{4}\right)\right)\,,$ $\displaystyle{\delta_{\epsilon}}m_{\rm
H}^{2}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{3}{2{\it
v}^{2}}((3\kappa_{3}^{2}+\kappa_{4})m_{\rm H}^{4}-2a^{2}m_{\rm H}^{2}(2m_{\rm
W}^{2}+m_{\rm Z}^{2})+(4a^{2}+2b)(2m_{\rm W}^{4}+m_{\rm Z}^{4}))\,,$
$\displaystyle{\delta_{\epsilon}}Z_{B}=-\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{\prime
2}}{12}(1+a^{2})\,,\qquad{\delta_{\epsilon}}Z_{W}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}}{12}(51-a^{2}-12\xi)\,,$
$\displaystyle{\delta_{\epsilon}}m_{\rm
W}^{2}=-\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}}{12}\left(3(a^{2}-b)m_{\rm
H}^{2}+(78-10a^{2})m_{\rm W}^{2}-9m_{\rm Z}^{2}\right)\,,$
$\displaystyle{\delta_{\epsilon}}m_{\rm
Z}^{2}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}}{12c_{\rm
w}^{2}}\left(-3(a^{2}-b)m_{\rm H}^{2}+(7(1+a^{2})+2(-43+a^{2})c_{\rm
w}^{2})m_{\rm W}^{2}+(10+a^{2})m_{\rm Z}^{2}\right)\,,$
$\displaystyle{\delta_{\epsilon}}g^{\prime}/g^{\prime}=0\,,\qquad{\delta_{\epsilon}}g/g=-\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}}{2}(3+\xi)\,,$
$\displaystyle{\delta_{\epsilon}}\xi_{1}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{g^{2}}{12}(51-a^{2}-12\xi)\,,$
$\displaystyle{\delta_{\epsilon}}\xi_{2}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{1}{3{\it
v}^{2}}(6(a^{2}-b)m_{\rm H}^{2}+(69-19a^{2}+4\xi)m_{\rm W}^{2}-(18-4\xi)m_{\rm
Z}^{2})\,,$
$\displaystyle{\delta_{\epsilon}}Z_{\pi}=-\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{1}{{\it
v}^{2}}((a^{2}-b)m_{\rm H}^{2}-(3+3a^{2}-4\xi/3)m_{\rm W}^{2}-(3-4\xi/3)m_{\rm
Z}^{2})\,,$ $\displaystyle{\delta_{\epsilon}}{\it v}/{\it
v}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{2(m_{\rm W}^{2}+m_{\rm
Z}^{2})}{3{\it v}^{2}}\xi\,.$ (4.34)

Some comments about these results are in order. First of all, we notice that
the $\xi$ parameter enters in these counterterms, as it is expected in
renormalization with $R_{\xi}$ gauges. This also happens in the SM case, which
we have also analyzed for completeness and comparison and whose results are
collected in Eq. (C.5). Concretely, we find in both the EChL and the SM, that
the $\xi$ parameter enters in the divergencies of the CTs corresponding to
$g$, $Z_{W}$, $\xi_{1,2}$, $Z_{\pi}$ and ${\it v}$. As we can see by an
explicit comparison, only ${\delta_{\epsilon}}g^{\prime}/g^{\prime}$,
${\delta_{\epsilon}}g/g$ coincide in both theories.
${\delta_{\epsilon}}Z_{B}$, ${\delta_{\epsilon}}Z_{W}$ and
${\delta_{\epsilon}}\xi_{1}$ coincide for $a=1$. In general all the other EW
CTs are different in both theories. One of the new features of this non-linear
EFT is that ${\delta_{\epsilon}}T$, ${\delta_{\epsilon}}m_{\rm H}^{2}$,
${\delta_{\epsilon}}m_{\rm W}^{2}$ and ${\delta_{\epsilon}}m_{\rm Z}^{2}$ are
$\xi$-independent separately. In contrast, in the SM case, only the proper
contributions of both tadpole and mass counterterms to the pole of the
propagators ($\widetilde{\delta m}^{2}$) are $\xi$-independent, as we show in
Eq. (D.11). Finally, notice also that the divergent part of the combination
$(\delta{\it v}/{\it v}+\delta Z_{\pi}/2)$ is $\xi$-independent in the EChL.

####  4.6.2 Divergent CTs for the EChL coefficients

The results are the following

$\displaystyle{\delta_{\epsilon}}a=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{3}{2{\it
v}^{2}}\left((a^{2}-b)(a-\kappa_{3})m_{\rm H}^{2}+a\left((1-3a^{2}+2b)m_{\rm
W}^{2}+(1-a^{2})m_{\rm Z}^{2}\right)\right)\,,$
$\displaystyle{\delta_{\epsilon}}a_{0}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{3}{8}(1-a^{2})\,,\qquad{\delta_{\epsilon}}a_{1}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{1}{12}(1-a^{2})\,,\qquad{\delta_{\epsilon}}a_{2}=-{\delta_{\epsilon}}a_{3}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{1-a^{2}}{24}$
$\displaystyle{\delta_{\epsilon}}a_{4}=-\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{(1-a^{2})^{2}}{12}\,,\quad{\delta_{\epsilon}}a_{5}=-\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{1}{24}(a^{4}+4a^{2}-3a^{2}b+\frac{3}{2}b^{2}+1)\,,$
$\displaystyle{\delta_{\epsilon}}a_{6}={\delta_{\epsilon}}a_{7}={\delta_{\epsilon}}a_{8}={\delta_{\epsilon}}a_{9}={\delta_{\epsilon}}a_{10}={\delta_{\epsilon}}a_{12}={\delta_{\epsilon}}a_{13}=0\,,\qquad{\delta_{\epsilon}}a_{11}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{a^{2}}{4}\,,$
$\displaystyle{\delta_{\epsilon}}a_{HBB}={\delta_{\epsilon}}a_{HWW}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{a(a^{2}-b)}{12}\,,\qquad{\delta_{\epsilon}}a_{\Box{\cal
V}{\cal V}}=-\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{a(2+a^{2})}{4}\,,$
$\displaystyle{\delta_{\epsilon}}a_{H0}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{3a(1-b)}{4}\,,\qquad{\delta_{\epsilon}}a_{H1}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{a(a^{2}-b)}{6}\,,\qquad{\delta_{\epsilon}}a_{H11}=-\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{a(a^{2}-b)}{2}\,,$
$\displaystyle{\delta_{\epsilon}}a_{d1}=-{\delta_{\epsilon}}a_{d2}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{a(a^{2}-b)}{6}\,,\qquad{\delta_{\epsilon}}a_{d3}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{a(a^{2}+b)}{2}\,,$
$\displaystyle{\delta_{\epsilon}}a_{H8}={\delta_{\epsilon}}a_{H13}={\delta_{\epsilon}}a_{\Box
0}={\delta_{\epsilon}}a_{d4}={\delta_{\epsilon}}a_{d5}=0\,,$
$\displaystyle{\delta_{\epsilon}}a_{\Box\Box}=-\frac{\Delta_{\epsilon}}{16\pi^{2}}\frac{3a^{2}}{4}\,.$
(4.35)

Some comments about these results are in order. First of all, we wish to
remark that we have found no $\xi$-dependent piece in any of all these results
for the CTs in the EChL coefficients, in contrast to the previous results for
the CTs of the EW parameters. This $\xi$-independence in the results for the
${\delta_{\epsilon}}a_{i}$’s is not trivial at all, since its derivation
involves $\xi$-dependent terms from the loop diagrams everywhere. Obviously,
this result is welcome, and on the other hand, it is also expected since by
construction all the operators in ${\cal L}_{4}$ are separately
$SU(2)_{L}\times U(1)_{Y}$ gauge invariant. Therefore, they do not mix under
gauge transformations. Secondly, we see that just a subset of these
coefficients get divergent CTs, concretely: $a_{0}$, $a_{1}$, $a_{2}$,
$a_{3}$, $a_{4}$, $a_{5}$, $a_{11}$, $a$ (remember that the renormalization of
$b$, $\kappa_{3}$ and $\kappa_{4}$ do not enter here), $a_{HWW}$, $a_{HBB}$,
$a_{\Box{\cal V}{\cal V}}$, $a_{H0}$, $a_{H1}$, $a_{H0}$, $a_{H11}$, $a_{d1}$,
$a_{d2}$, $a_{d3}$ and $a_{\Box\Box}$. The remaining $a_{i}$’s do not get
divergent counterterms and, therefore, the corresponding operators in ${\cal
L}_{4}$ are not needed to cancel the extra divergencies generated by ${\cal
L}_{2}$. We also see in these results that some of these CTs vanish for the
choice $a=b=\kappa_{3}=\kappa_{4}=1$, and some others do not, like $a_{5}$,
$a_{11}$, $a_{\Box{\cal V}{\cal V}}$ and $a_{\Box\Box}$. The particular choice
$b=a^{2}$ also produces some simplifications and fewer divergent CTs are
found. Concretely, we find vanishing ${\delta_{\epsilon}}$’s for $a_{HWW}$,
$a_{HBB}$, $a_{H1}$, $a_{H11}$, $a_{d1}$ and $a_{d2}$ in that case.

At this point, we believe it is worth comparing our results in Eq. (4.35) with
some previous results of the EChL one-loop divergencies and counterterms in
the literature. This comparison is a partial one in any case, since our
results are the only ones that apply to the most general and complete case of
off-shell one-loop 1PI functions, including 1-, 2-, 3- and 4-legs functions,
and including all types of loop diagrams in the $R_{\xi}$ gauges.

The closest comparison of our results in the present work is with our own
previous results in [32]. In that paper we also worked with the EChL in the
$R_{\xi}$ gauges but focused exclusively on the one-loop decay amplitudes of
the Higgs boson into $\gamma\gamma$ and $\gamma Z$. There we found that some
particular combinations of EChL coefficients enter in those amplitudes and
they turned out to be finite, RGE invariant and, therefore, not needed for the
cancellation of the one-loop divergences involved in these Higgs decays. Using
the notation in the present paper, these combinations are:

$\displaystyle c_{H\gamma\gamma}$ $\displaystyle=$ $\displaystyle
a_{HBB}+a_{HWW}-a_{H1}\,,$ (4.36) $\displaystyle c_{H\gamma Z}$
$\displaystyle=$ $\displaystyle\frac{1}{c_{\rm w}^{2}}(-a_{HBB}s_{\rm
w}^{2}+a_{HWW}c_{\rm w}^{2}-\frac{1}{2}a_{H1}(c_{\rm w}^{2}-s_{\rm
w}^{2}))\,.$ (4.37)

for $H\to\gamma\gamma$ and $H\to\gamma Z$ respectively. This result is in
agreement with Eq. (4.35) since we also get here the same cancellation in
those combinations of the corresponding $\delta_{\epsilon}a_{i}$’s.

Another example of observable where there appear cancellations of divergences
in the involved combination of EChL coefficients is in $\gamma\gamma\to WW$
and $\gamma\gamma\to ZZ$ scattering. This was studied in [11] where the one-
loop amplitude for the scattering processes with final longitudinal gauge
bosons $V_{L}V_{L}$ (with $VV=WW,ZZ$) was computed using the equivalence
theorem, i.e., using ${\cal A}(\gamma\gamma\to V_{L}V_{L})\simeq{\cal
A}(\gamma\gamma\to\pi\pi)$ and within the approximation of considering only
chiral loops, i.e., loops with Higgs and GBs. The masses of GBs were set to
cero (as in the Landau gauge, i.e., for $\xi=0$). The combinations of EChL
coefficients found in [11] for ${\cal A}(\gamma\gamma\to\pi\pi)$, which are
finite and RGE invariant, using the notation of the present paper, are:

$\displaystyle c_{\gamma}=a_{HBB}+a_{HWW}-a_{H1}\,\,\,$ $\displaystyle\rm{in}$
$\displaystyle\,\,\,\pi^{+}\pi^{-}\,\,\,\rm{and}\,\,\,\pi^{0}\pi^{0}\,,$
(4.38) $\displaystyle\text{and also }\qquad(a_{1}-a_{2}+a_{3})\,\,\,$
$\displaystyle\rm{in}$ $\displaystyle\,\,\,\pi^{+}\pi^{-}\,.$ (4.39)

Again this result is in agreement with Eq. (4.35), as can be checked by the
cancellation of the corresponding combinations of $\delta_{\epsilon}a_{i}$’s.
Also in [11] the EW precision ‘oblique’ $S$ parameter was related with the
coefficient $a_{1}$. The divergent CT ${\delta_{\epsilon}}a_{1}$ in Eq. (4.35)
is also in agreement with the corresponding divergence involved in the
computation of $S$ in [11].

The VBS processes with $W_{L}$ and $Z_{L}$ in the external legs were studied
to one-loop within the EChL previously in [9, 10]. It was done by means of the
ET, i.e., replacing the external $V_{L}$’s by the corresponding $\pi$’s,
considering just chiral loops and assuming massless GBs (as in Landau gauge
again, i.e., for $\xi=0$). In those works only interactions among scalars were
involved, and only $a$, $b$, $a_{4}$ and $a_{5}$ appeared in the VBS
amplitudes, since they assumed the so-called isospin limit where no custodial
breaking operators appear. Comparing the divergences for $a$, $a_{4}$ and
$a_{5}$ (the renormalization of $b$ does not enter in VBS) found in [9, 10]
with our results in Eq. (4.35), we find agreement for $a$ and $a_{4}$. The
case of $a_{5}$ is more tricky. We have checked that when taking the isospin
limit in the VBS amplitudes with physical on-shell external gauge bosons (this
limit implies: $m_{Z}\to m_{W}$, $g^{\prime}\to 0$, $c_{W}\to 1$ and no
custodial breaking operators) and considering the full set of operators in
${\cal L}_{4}$, it turns out that $a_{5}$ allways appears combined with
others. Concretely, it appears in the following combination of EChL
coefficients:

$\displaystyle\tilde{a}_{5}$ $\displaystyle=$ $\displaystyle
a_{5}-\frac{a}{2}a_{\Box{\cal V}{\cal V}}+\frac{a^{2}}{4}a_{\Box\Box}.$ (4.40)

From our results in Eq. (4.35), our prediction for the divergence in this
$\tilde{a}_{5}$ is:

$\displaystyle{\delta_{\epsilon}}\tilde{a}_{5}$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{\epsilon}}{32\pi^{2}}\left(\frac{1}{8}(b-a^{2})^{2}+\frac{1}{12}(1-a^{2})^{2}\right)\,,$
(4.41)

which is in agreement with the result for $a_{5}$ in [9, 10], where these two
coefficients, $a_{\Box{\cal V}{\cal V}}$ and $a_{\Box\Box}$, were not
considered. At this point, it is interesting to remark that this combination
in Eq. (4.41) indeed vanishes for $a=b=1$.

The renormalization of the EChL coefficients was also studied in [12]. In this
case, they considered the pure scalar theory, i.e., only the Higgs and GBs
sector of the EChL and worked with massless GBs (as in Landau gauge, with
$\xi=0.$). No gauge or ghost fields were included and, therefore, no gauge-
fixing. The considered operators were custodial preserving. The
renormalization of one-loop 1PI functions, for 1-leg, 2-legs, 3-legs and
4-legs, was performed for off-shell external legs. We find agreement in the
divergences found for the subset of $a_{i}$’s involved in the scalar sector
(the coefficients in the notation of [12] are specified inside the
parentheses). Concretely, we agree in: $a$ ($a_{C}$), $a_{4}$ ($c_{11}$),
$a_{5}$ ($c_{6}$), $a_{11}$ ($c_{9}$), $a_{\Box{\cal V}{\cal V}}$ ($c_{7}$),
$a_{d3}$ ($c_{10}$), and $a_{\Box\Box}$ ($c_{\Box H}$).

The renormalization of the EChL was studied in the path integral formalism,
using the background field method, in [6, 7, 8]. The most complete comparison
of our results should be done with the bosonic loop results of [7, 8] since
they also included all loops of scalar and gauge particles. However, the
comparison with the path integral results is tricky since they use the
equations of motion to reduce the number of operators in the Lagrangian.
Therefore, some off-shell divergences do not appear in their results and some
others are redefined by the use of the equations of motion. They also use
redefinitions of the fields (in particular the Higgs field) to reach the
canonical kinetic term in the Lagrangian. On the other hand, the
parametrization used in [7, 8] is also very different than here and not
straightforward to compare with. In any case, solving these differences of
parametrizations and doing some algebra, we have found agreement in the
divergences of the following subset of coefficients (the coefficients in the
notation of [7, 8] are specified inside the parentheses): $a_{0}$
($\beta_{1}$), $a_{1}$ ($XU1$), $a_{2}$ ($XU7$), $a_{3}$ ($XU8$), $a_{4}$
($D_{2}$) and ${\tilde{a}}_{5}$ ($D_{1}$).

####  4.6.3 RGEs and running EChL coefficients

Finally, to close the renormalization section we include here the results for
the RGEs and running EChL coefficients. These are easily derived from the
previous results in Eq. (4.35) and taking into account the relation between
the renormalized and bare coefficients given in Eq. (4.3) by
$a^{0}_{i}=a_{i}+\delta a_{i}$. In the $\overline{MS}$ scheme (recall that
$\mu$ is the scale of dimensional regularization in $D=4-\epsilon$
dimensions), the running $a_{i}(\mu)$ can be written as follows:

$\displaystyle a_{i}(\mu)$ $\displaystyle=$ $\displaystyle a_{i}^{0}-\delta
a_{i}(\mu)\,,\,\,\,\delta
a_{i}(\mu)={\delta_{\epsilon}}a_{i}-\frac{\gamma_{a_{i}}}{16\pi^{2}}\log\mu^{2}\,,\,\,\,{\delta_{\epsilon}}a_{i}=\frac{\Delta_{\epsilon}}{16\pi^{2}}\gamma_{a_{i}}\,,$
(4.42)

where we have written the divergent ${\delta_{\epsilon}}a_{i}$ in terms of the
anomalous dimension $\gamma_{a_{i}}$ of the corresponding effective operator.
The running and renormalized $a_{i}$’s can then be related, in practice, by:

$\displaystyle a_{i}(\mu)$ $\displaystyle=$ $\displaystyle
a_{i}+\frac{\gamma_{a_{i}}}{16\pi^{2}}\log\mu^{2}$ (4.43)

The set of RGEs for all the $a_{i}$’s then immediately follow:

$\displaystyle a_{i}(\mu)$ $\displaystyle=$ $\displaystyle
a_{i}(\mu^{\prime})+\frac{1}{16\pi^{2}}\gamma_{a_{i}}\log\left(\frac{\mu^{2}}{\mu^{\prime
2}}\right)\,,$ (4.44)

where the specific value of $\gamma_{a_{i}}$ for each coefficient can be read
from Eq. (4.35). The RGEs can also be written, alternatively, as:

$\displaystyle\frac{da_{i}(\mu)}{d\log\mu}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{a_{i}}}{8\pi^{2}}$ (4.45)

Notice that the coefficients (or combination of coefficients) with vanishing
${\delta_{\epsilon}}$ do not run, therefore, they are RGE invariants. Finally,
notice also that, in the scattering amplitude of our interest in this work,
${\cal A}(WZ\to WZ)$ only the following reduced list of running coefficients
enter: $a(\mu)$, $a_{0}(\mu)$, $a_{1}(\mu)$, $a_{2}(\mu)$, $a_{3}(\mu)$,
$a_{4}(\mu)$, $a_{5}(\mu)$, $a_{11}(\mu)$, $a_{HWW}(\mu)$, $a_{HBB}(\mu)$,
$a_{H1}(\mu)$, $a_{d1}(\mu)$, $a_{d2}(\mu)$, $a_{\Box{\cal V}{\cal V}}(\mu)$
and $a_{\Box\Box}(\mu)$.

##  5 Numerical results for $WZ$ scattering

In this section we present the numerical results of the one-loop radiative
corrections to the VBS cross section for the particular channel $WZ\to WZ$ and
postpone the other VBS channels for future works. This $WZ$ channel
illustrates the main features of the complex NLO cross section computation
within the EChL and also serves as a good reference case to compare with the
SM case. For the numerical estimation of the cross section we use LoopTools
[22]. Due to the lengthy and time consuming computation with many one-loop
diagrams involved (more than 500), we have chosen in this numerical
computation the Feynman ’t Hooft gauge ($\xi=1$), and we have restricted
ourselves to the custodial preserving operators that are needed as
counterterms. Furthermore, we have assumed the relation $b=a^{2}$ and taken
the values $\kappa_{3}=\kappa_{4}=1$ in the EChL coefficients of ${\cal
L}_{2}$. With these simplifying assumptions, the EChL coefficients
participating in our computation of $\sigma(WZ\to WZ)$ are reduced to the
following ones: $a$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$, $a_{11}$,
$a_{\Box{\cal V}{\cal V}}$ and $a_{\Box\Box}$. Regarding the EW input
parameters, for the numerical estimates we have chosen $m_{\rm W}$, $m_{\rm
Z}$ and $G_{F}$.

Before presenting the numerical results, we comment briefly on three
preliminar checks that we have done on the computing procedure.

Firstly, in addition to the previously commented analytical checks of
cancellations of the divergent ${\cal O}(\Delta_{\epsilon})$ contributions, we
have also checked numerically that with this subset of EChL coefficients the
full one-loop cross section $\sigma(WZ\to WZ)$ is finite. For this check, we
use an indirect method: Concretely, we have checked numerically the
independence of the full one-loop cross section with the value of the $\mu$
parameter of dimensional regularization, which is a high precision test since
this $\mu$ parameter enters in the very many loops participating in the
radiative corrections.

Secondly, we have also checked analytically that when using the so-called
isospin limit, which implies taking $m_{\rm W}=m_{\rm Z}$, the tree level
amplitude from ${\cal L}_{4}$ depends on an even more reduced subset of
coefficients: $a$, $a_{3}$, $a_{4}$, $a_{5}$, $a_{\Box{\cal V}{\cal V}}$ and
$a_{\Box\Box}$. Then, taking the high energy limit in this amplitude,
$\sqrt{s}\gg m_{\rm W},m_{\rm H}$, the involved parameters are finally reduced
to: $a$, $a_{3}$, $a_{4}$, and ${\tilde{a}}_{5}$, with
$\tilde{a}_{5}=a_{5}-(a/2)a_{\Box{\cal V}{\cal V}}+(a^{2}/4)a_{\Box\Box}$,
already mentioned in Eq. (4.40). Here, we do not use those approximations of
taking the isosplin limit nor the high energy limit, but it is worth knowing
these analytical results for the interpretation of the final complete
numerical results presented in this section.

Thirdly, we have also checked the already known results for $\sigma(WZ\to WZ)$
at the tree level in [33] that showed that under the hypothesis of similar
size for all the $a_{i}$ coefficients, the ones that contribute the most to
the cross section are $a_{4}$ and $a_{5}$ (and $a_{3}$ to a milder level),
specially at the high energies of ${\cal O}(1\,{\rm TeV})$. In this reference
(and others, see for instance [34]) it was also shown the dominance, at large
energies, of the polarized cross section with longitudinal external gauge
bosons.

Considering all these features summarized above, we have restricted our
numerical analysis of the NLO cross-section to the longitudinal polarization
state $W_{L}Z_{L}\to W_{L}Z_{L}$ channel, studied the effects from the most
relevant EChL renormalized coefficients, $a$, $a_{4}$ and $a_{5}$, and fixed
$b=a^{2}$ and $a_{i}=0$ for the rest of renormalized coefficients in all our
plots in this section. In that case, notice that $a_{5}$ and $\tilde{a}_{5}$
coincide. Then, we analyze in the following the results for the six chosen
benchmark cases:

  * 1)

$a=1$, $a_{4}\neq 0$, $a_{5}=0$

  * 2)

$a=1$, $a_{4}=0$, $a_{5}\neq 0$

  * 3)

$a=0.9$, $a_{4}\neq 0$, $a_{5}=0$

  * 4)

$a=0.9$, $a_{4}=0$, $a_{5}\neq 0$

  * 5)

$a=1$, $a_{4}\neq 0$, $a_{5}\neq 0$

  * 6)

$a=0.9$, $a_{4}\neq 0$, $a_{5}\neq 0$

In these benchmark cases we have considered two different values of the $a$
parameter: the case of $a=1$ that connects with the SM results, and the BSM
case of $a=0.9$ (compatible with present data, see for instance [35]). The
choice of non-vanishing $a_{4}$ and $a_{5}$ have been set, to both positive
and negative values, with modulus within the interval $(10^{-3}$, $10^{-4}$)
which are compatible with present data (see, for instance [36]). However, it
should be noticed that the concrete constraints on these parameters depend, in
fact, on the unitarization method used (for a recent study see, for instance,
[33]). On the other hand, notice also that the special choice $a=1$ reduces
considerably the number of involved divergences and therefore the number of
divergent counterterms in the EChL coefficients. Concretely, for $a=1$, there
only remain non vanishing ${\delta_{\epsilon}}$’s for $a_{5}$, $a_{11}$,
$a_{\Box{\cal V}{\cal V}}$ and $a_{\Box\Box}$, whereas $a_{4}$ and the
combination defining ${\tilde{a}}_{5}$ give vanishing ${\delta_{\epsilon}}$
therefore leading to RGE invariant coefficients.

The results for the benchmark cases 1) and 2) are shown in Fig. 5, left and
right panels, respectively. Those for 3) and 4) are shown in Fig. 6, left and
right panels, respectively. Finally, the cases 5) and 6) are shown in Fig. 7,
left and right panels, respectively. In all cases we have computed: a) the
total cross section (upper panels), b) the lowest partial wave amplitude
$|a_{J}|$ with $J=0$ (lower panels) and c) the quantity $\delta_{\rm{1-loop}}$
that provides the relative size of the one-loop correction in the full one-
loop results respect to the tree level prediction. It is defined by:

$\delta_{\rm{1-loop}}=(\sigma_{\rm Full}-\sigma_{\rm Tree})/\sigma_{\rm Tree}$
(5.1)

All these evaluated quantities are displayed as functions of the center of
mass energy $\sqrt{s}$ of the $WZ\to WZ$ scattering process, in the interval
$300\,{\rm GeV}<\sqrt{s}<3000\,{\rm GeV}$. The disallowed areas by the partial
wave unitarity bound $|a_{0}|>1$ are also displayed (in grey) in the b) plots.
The corresponding predictions for the SM case are also shown in all the plots,
for comparison. Specifically, using our notation introduced in Section 3, we
include the predictions for: ${\rm SM}_{\rm Full}$, ${\rm SM}_{\rm Tree}$,
${\rm EChL}_{\rm Full}$, ${\rm EChL}_{\rm Tree}^{(2+4)}$ and the corresponding
$\delta_{\rm{1-loop}}$.

Let us first discuss the results in Fig. 5. Looking at these plots we learn
the following features. Since $a=1$ in these plots, there is a connection
between the EChL predictions and the SM ones. Indeed the matching of the EChL
and SM predictions occur for $a_{i}=0$, as expected, both at the tree and one-
loop level. Similarly, all the lines in this figure converge to the SM line at
low energies, both at the tree level and at the one-loop level.

Figure 5: Cross-section predictions for $W_{L}^{+}Z_{L}\to W_{L}^{+}Z_{L}$
within the EChL corresponding to $a=1$, $b=a^{2}$, $\kappa_{3}=\kappa_{4}=1$
and varying $a_{4}$ (left) and $a_{5}$ (right). The rest of $a_{i}$’s are set
to zero. The $\mu$ scale is set to $\mu=1$TeV. Dashed lines are the tree level
predictions. Solid lines are the full one-loop predictions. The solid red
(pink) lines correspond to the full (tree) SM prediction. The relative size of
the one-loop prediction respect to the tree level one defined by means of
$\delta_{\rm{\rm{1-loop}}}$ is also shown (lower plots in the upper panels).
The predictions for the lowest partial wave $a_{J}$ with $J=0$ are also
displayed (lower panels). The unitarity violating regions in these plots are
indicated by the shadowed areas.

The (EW) one-loop prediction within the SM is by itself interesting. To our
knowledge, the analytical EW one-loop SM result in the $R_{\xi}$ gauge has not
been provided in the literature for this particular $WZ\to WZ$ channel. The
previous full one-loop evaluations of the EW radiative corrections affecting
the VBS processes are numerical and have been done in the context of the LHC.
These numerical estimates therefore involve many more diagrams than those
considered here, since the relevant subprocesses for the production of two EW
gauge bosons at the LHC include the initial quarks in the protons and not all
of them have the VBS configuration. In addition, final state fermions from the
final EW gauge boson decays are also usually considered. In particular, the EW
one-loop corrections at the LHC for the doubly charged channel, $W^{+}W^{+}$,
were computed in [37] and for the $WZ$ channel in [38]. The EW one-loop
corrections in the polarized case for $WZ$ have been studied in [39]. Although
the comparison of our SM numerical results with those commented LHC works is
not immediate nor can be done with high precision for the reasons explained
above, it is however possible to make a rough comparison of the relative size
of the correction, $\delta_{\rm 1-loop}$. For $W^{+}W^{+}$, it was found in
[37] a large and negative one-loop radiative correction of up to $\delta_{\rm
1-loop}\sim-16\%$ in the VBS energy range of ${\cal O}(1\,{\rm TeV})$
typically reached at the LHC. For $WZ$, it was found in [38] a similar size of
the EW one-loop correction of up to $\delta_{\rm 1-loop}\sim-16\%$ at the
relevant LHC energies. Our SM numerical results for the EW one-loop radiative
corrections in $WZ\to WZ$ scattering, shown in Fig. 5, are clearly compatible
with those LHC results. First, Fig. 5 shows that the SM tree level cross-
section for the $WZ$ channel (pink line) is nearly flat with $\sqrt{s}$
(reaching about $\sim$55 pb), whereas the SM full one-loop cross-section (red
line) is slightly decreasing with energy and it lays below the tree level one
in the whole studied energy interval. Therefore the interference effect from
the one-loop contribution is destructive. The size of the one-loop correction
relative to the tree level prediction (red line in the $\delta_{\rm{1-loop}}$
plot) reaches the maximum negative value of about $\delta_{\rm
1-loop}\sim-16\%$ at the upper part of the studied interval $\sqrt{s}\sim
3000\,{\rm GeV}$ which is similar to the previous one-loop results, in [37]
and [38], commented above.

Next we comment on the EChL results in this Fig. 5. Regarding the tree level
estimates, first we see that the EChL predictions at the LO, i.e., from ${\cal
L}_{2}$, coincide with the SM predictions at LO (pink line), i.e., ${\rm
EChL}_{\rm Tree}^{(2)}|_{a=1}={\rm SM}_{\rm Tree}$, as expected. Second, we
see that the tree level EChL predictions including both ${\cal L}_{2}$ and
${\cal L}_{4}$, given by ${\rm EChL}_{\rm Tree}^{(2+4)}$ (dashed lines),
display a different behaviour with $\sqrt{s}$ than ${\rm EChL}_{\rm
Tree}^{(2)}$. This is due to the effect from the involved $a_{i}$ which can be
summarized, at the amplitude level, as $(f_{1}+f_{2}\,a_{i})$, where at high
energies $f_{1}$ and for $a=1$ tends to a constant value (like in the SM),
whereas $f_{2}$ goes as $s^{2}$, giving the polynomial ${\cal O}(p^{4})$
dependence in powers of the external momentum, typical in the chiral