aainstitutetext: School of Physics & Astronomy, University of Glasgow, Glasgow
G12 8QQ, United Kingdom

# Quartic Gauge-Higgs couplings: Constraints and Future Directions

Anisha a Oliver Atkinson a Akanksha Bhardwaj a Christoph Englert a Panagiotis
Stylianou<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Constraints on quartic interactions of the Higgs boson with gauge bosons have
been obtained by the experimental LHC collaborations focussing on the so-
called $\kappa$ framework of flat rescalings of SM-like interactions in weak
boson fusion (WBF) Higgs pair production. While such approaches are admissible
to obtain a qualitative picture of consistency with the SM when the
statistical yield is low, once more statistics become available a more
theoretically consistent framework of limit setting is desirable. Reviewing
the constraints provided at the Large Hadron Collider, we first show that
these limits are robust when considered in a leading order context. Turning to
radiative corrections, we demonstrate the limitations of this approach in the
SM, and by adopting Higgs effective field theory techniques, we clarify the
sensitivity from single Higgs measurements to rescalings of quartic Higgs-
gauge couplings. We then discuss avenues for sensitivity improvements of WBF
analyses employing Graph Neural Networks to combat the large contributing
backgrounds.

##  1 Introduction

Ten years into the Higgs characterisation programme at the Large Hadron
Collider (LHC), many couplings of the Higgs boson to other SM matter have been
shown to largely follow the Standard Model (SM) expectation. Given the
insights from electroweak fits before the Higgs discovery in 2012, this was
perhaps anticipated for the gauge sector Baak:2011ze ; Falkowski:2013dza . The
observation of $H\to\gamma\gamma$ with a rate in agreement with the SM
prediction indicated alignment of the fermion-Higgs interactions as part of
the Higgs discovery for the top quark. Later analyses in other fermion decay
modes of the Higgs boson ATLAS:2021qou ; ATLAS:2022yrq ; CMS:2022kdi ;
CMS:2017odg  have further established the Higgs boson as SM-like.

Couplings that have received less attention, also because corresponding
searches are statistics-limited, are the quartic interactions of the Higgs
boson with gauge fields. Assuming a weak doublet-like character of the Higgs
boson, these interactions are fully correlated with trilinear Higgs-gauge
($HWW$, $HZZ$) interactions due to gauge invariance in the SM. Searches for
anomalous quartic interactions in the so-called $\kappa$ framework
LHCHiggsCrossSectionWorkingGroup:2012nn  are therefore theoretically
cumbersome, albeit instructive when the statistical sensitivity is low
(similar to $\kappa$-like analyses of single Higgs observables after the Higgs
boson’s discovery).

In the expansion of the Higgs boson around its vacuum expectation value (vev),
the gauge boson masses are a property of the non-linear $SU(2)_{L}\times
U(1)_{Y}$ electroweak symmetry realisation, whilst the Higgs bosons’ couplings
to $W$ and $Z$ bosons probe the alignment of quantum fluctuations around this
vev. These interactions carry important information as any departure from the
SM expectation immediately implies perturbative unitarity violation at a scale
above the vev Lee:1977eg ; Chanowitz:1978mv ; Abu-Ajamieh:2020yqi , which acts
as a strong indicator of a new scale of physics beyond the SM (BSM).

Quartic interactions $\sim H^{2}V^{2}$ ($V=W,Z$), while being less relevant
from a perturbative unitarity perspective, still carry important information
when $HVV$ couplings (dis)agree with the SM: Different BSM scenarios show
similar alignment of single Higgs interactions with the vev. In the gauge
sector this degeneracy is only lifted by considering quartic interactions. An
example for this is custodial singlet mixing Binoth:1996au ; Schabinger:2005ei
; Patt:2006fw ; Englert:2011yb  compared to minimal compositeness Higgs models
(MCHMs). In the former case Higgs interactions are modified by a
characteristic mixing angle $\sim\cos\theta$ whereas in, e.g., MCHM5
Contino:2006qr , we obtain a modification of $\sim\sqrt{1-v^{2}/f^{2}}$
Contino:2006qr ; Grober:2010yv ; Gillioz:2012se  (where $v\simeq
246~{}\text{GeV}$ is the vev and $f\gtrsim v$ is the decay constant of the
Callen-Coleman-Wess-Zumino (CCWZ) Coleman:1969sm ; Callan:1969sn
construction). On the sole basis of a precision analysis of the $HWW,HZZ$
interactions, it is therefore impossible to discriminate between Higgs mixing
and Higgs compositeness as we can always understand
$\cos^{2}\theta=1-v^{2}/f^{2}$ (note that custodial symmetry is sufficiently
conserved in both models Agashe:2006at ). Only when we consider the curvature
of the Higgs interactions is this ambiguity lifted in the gauge sector: in the
custodial singlet mixing scenario, the $H^{2}V^{2}$ vertex is modified by
$\cos^{2}\theta\neq 1-2v^{2}/f^{2}$, where the latter is the result in MCHM5.

Targeted analyses for flat rescalings

$\kappa_{2V}={g_{HHVV}\over g^{\text{SM}}_{HHVV}}\,,$ (1)

in weak boson fusion (WBF) Higgs pair production show that this channel will
be statistics-limited at the high-luminosity (HL) frontier Dolan:2013rja ;
Dolan:2015zja ; Bishara:2016kjn ; Kilian:2018bhs ; Arganda:2018ftn . To some
degree, this is similar to investigations of the Higgs boson’s self-coupling;
statistics is also a limiting factor at the HL-LHC for rescalings of the
trilinear coupling $\kappa_{3}$ Dolan:2012rv . To partially address this
issue, single Higgs probes have been suggested McCullough:2013rea ;
Maltoni:2017ims ; Degrassi:2016wml ; DiVita:2017eyz ; Voigt:2017vfz ;
Englert:2019eyl  to add sensitivity. The comparison of single-Higgs probes (or
even electroweak precision data Kribs:2017znd ; Englert:2020gcp ) to
rescalings of the Higgs self-coupling $\kappa_{3}$ should, however, also be
understood as a theoretical figure of merit related to the trustworthiness of
the obtained constraints as different orders in the perturbative (effective
field theory) series expansion are tensioned against each other.

Given that WBF Higgs pair production is significantly smaller than inclusive
Higgs pair production via gluon fusion
LHCHiggsCrossSectionWorkingGroup:2016ypw , similar issues of $\kappa_{2V}$ do
occur. It is the purpose of this work to clarify these questions in the light
of existing analyses pursued by the ATLAS and CMS experiments ATLAS:2020jgy ;
ATLAS:2022ycx ; CMS:2022hgz . We show that this is achieved by understanding
the SM as a particular parameter point of the electroweak chiral (or Higgs
Effective Theory, HEFT) Lagrangian. This enables us to consistently modify the
quartic Higgs-gauge interactions away from the SM expectation with
theoretically consistent implications for single Higgs observables. In
particular, we will focus on radiative corrections to the $H\to VV$ rates,
which we tension against direct sensitivity obtained from WBF Higgs pair
analyses. To formulate an as-good-as-possible direct search limit, we employ
machine learning techniques to the signal-background discrimination to show
that significant improvements are possible beyond the existing, more
traditional techniques discussed in the literature.

This work is organised as follows: In Sec. 2, we review the theoretical
baseline of $\kappa_{2V}$ analyses in the SM. We consider electroweak
precision constraints from oblique corrections and derive unitarity
constraints from a partial wave analysis, clarifying the validity of the
results of ATLAS:2020jgy ; ATLAS:2022ycx ; CMS:2022hgz  viewed as a leading
order signal analysis. We show that a $\kappa_{2V}$ analysis cannot be
theoretically defended when we consider quantum corrections in the SM. To
overcome this limitation, we turn to the electroweak chiral Lagrangian in Sec.
3, which provides the theoretically consistent playground to discuss
correlations between single and double Higgs modifiers at a given order of the
amplitudes’ loop expansion. Performing the next-to-leading order calculation
of $H\to ZZ^{\ast},WW^{\ast}$ alongside $H\to\gamma\gamma,\gamma Z$, we
discuss the current and expected sensitivity of single Higgs data to
$\kappa_{2V}$ coupling modifiers. This highlights the direct sensitivity
provided in weak boson fusion $HHjj$ analysis as the sensitive tool to
constrain $\kappa_{2V}$ in the future (for a recent discussion at $e^{+}e^{-}$
machines see Ref. Domenech:2022uud ). Any improvement beyond increasing
statistics of current strategies ATLAS:2020jgy ; ATLAS:2022ycx ; CMS:2022hgz
will therefore be directly correlated with an improved understanding of the
Higgs boson’s TeV scale characteristics. To this end, we show that the
application of state-of-the-art Graph Neural Network (GNN) techniques
zhou2018graph ; 9046288  to $pp\to HHjj$ final states can greatly reduce the
dominant backgrounds of such searches, thus leading to a significant
sensitivity enhancement to $\kappa_{2V}$. We summarise and conclude in Sec. 6.

##  2 Vices and Virtues of $\kappa_{2V}$

We consider first the effect of flatly rescaled quartic Higgs-gauge couplings
to electroweak precision constraints. Here, we particularly focus on oblique
corrections Peskin:1990zt ; Peskin:1991sw  and their sensitivity to coupling
modifications of _only_ the physical Higgs boson (with mass $M_{H}\simeq
125~{}\text{GeV}$). These interactions are not associated with momentum
dependencies at one-loop order so we directly obtain

$\Delta S=\Delta U=0\,,$ (2)

(we do not consider single Higgs modifiers at this point). The $T$ parameter
is sensitive to the custodial isospin violation for
$\kappa_{2W}\neq\kappa_{2Z}$

$\Delta T={\kappa_{2Z}^{2}-\kappa_{2W}^{2}\over 16\pi}{M_{H}^{2}\over
M_{W}^{2}s_{W}^{2}}\log{\Lambda^{2}\over M_{H}^{2}}\,,$ (3)

with $\Lambda$ as the UV cutoff regulator for demonstration purposes.111In the
following, we denote the sine and cosine of the Weinberg angle with
$s_{W},c_{W}$. The $W$ and $Z$ masses are labelled with $M_{W},M_{Z}$,
respectively. This means that for large cutoffs of order 10 TeV we need to
ensure $\kappa_{2W}\simeq\kappa_{2Z}$ at the 1.5% level in order to not
violate present constraints Baak:2014ora  whilst UV engineering interactions
that dominantly source custodial isospin violation through the quartic gauge-
Higgs interactions. The latter is naively possible in the $\kappa$ framework,
Eq. (3), but rather unlikely when considered in the context of more realistic
SM extensions, some of which we have alluded to above. On the other hand, by
identifying $\kappa_{2W}=\kappa_{2Z}=\kappa_{2V}$, constraints from oblique
corrections are removed at the considered one-loop level. This is the
identification we will use in the following; electroweak precision results
suggest that the Higgs boson should be considered as a custodial iso-singlet
state, which is the hypothesis on which the electroweak chiral Lagrangian is
built.

Figure 1:  Perturbative unitarity bounds on $\kappa_{2V}=1+\zeta_{2}$ at
different scales from the $VH\to VH$ channels. The expected $95$% CL
constraints from ATLAS ATLAS:2022ycx  $\kappa_{2V}\in\left[-0.05,2.12\right]$,
and from CMS CMS:2022hgz  $\kappa_{2V}\in\left[-0.6,2.8\right]$ are also
overlaid. The ATLAS results are obtained from the $HH\to b\bar{b}b\bar{b}$
final state probing energies in the shaded region $m_{HH}\leq 1200$ GeV, while
CMS probed up to $m_{HH}\sim 720$ GeV with the $b\bar{b}\tau\bar{\tau}$ final
state.

Next, we turn to unitarity constraints. The partial waves of $i_{1}i_{2}\to
f_{1}f_{2}$ scattering process for angular momentum $J$ are given by (see Ref.
Jacob:1959at )

$a_{fi}^{J}={\beta^{1/4}(s,m_{f_{1}}^{2},m_{f_{1}}^{2})\beta^{1/4}(s,m_{i_{1}}^{2},m_{i_{1}}^{2})\over
32\pi
s}\int\limits_{-1}^{1}\text{d}\cos\theta\,{\cal{D}}^{J}_{\mu_{i}\mu_{f}}\,{\cal{M}}(s,\cos\theta)\,,$
(4)

(modulo factors of $1/\sqrt{2}$ for identical particles in the initial or
final state) where we denote $\sqrt{s}$ as the centre-of-mass energy, the
scattering angle in the centre-of-mass system is $\theta$, and

$\beta(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2xz\,.$ (5)

The ${\cal{D}}^{J}_{\mu_{i}\mu_{f}}$ are the Wigner functions of the Jacob-
Wick expansion Jacob:1959at  related to the helicities of the initial and
final states $\mu_{i}={\lambda}_{i_{1}}-{\lambda}_{i_{2}}$,
$\mu_{f}={\lambda}_{f_{1}}-{\lambda}_{f_{2}}$. Unitarity of the $S$ matrix
then translates for $f=i$ to the familiar conditions

$\text{Re}|a_{ii}^{j}|\leq{1\over
2}\quad\hbox{and}\quad\text{Im}|a_{ii}^{J}|\leq 1\,.$ (6)

The process relevant for $\kappa_{2V}$ constraints is longitudinal $HV_{L}\to
HV_{L}$ scattering, so that the Wigner functions reduce to the Legendre
polynomials, with the $J=0$ partial wave providing the dominant constraint.
This is shown in Fig. 1, where we define

$\kappa_{2V}=1+\zeta_{2}\,.$ (7)

We also include the LHC results from the ATLAS and CMS experiments of Refs.
ATLAS:2020jgy ; ATLAS:2022ycx ; CMS:2022hgz . As can be seen, the unitarity
constraints are relatively loose. The measurements, taken as a leading order
result, can be considered perturbatively consistent as the energy scales
explored by these analyses do not probe the unitarity cutoff of the leading
order theory.

The discussion so far suggests that $\kappa_{2V}$ is indeed relatively
unconstrained when naively considered as a free parameter in SM-like analyses.
However, beyond tree-level, a choice of $\kappa_{2V}\neq 1$ leads to incurable
shortfalls within the SM. For instance, when considering electroweak
corrections to $H\to ZZ^{\ast}$ in general $R_{\xi}$ gauge Fujikawa:1972fe ,
we obtain a counter term amplitude (${\cal{M}}_{\text{CT}}$)-renormalised222We
employ the on-shell renormalisation scheme here; further details are given in
Sec. 3. virtual amplitude (neglecting fermion contributions and considering
$M_{H}>2M_{Z}$ for convenience)

${\cal{M}}^{\text{virt}}+{\cal{M}}_{\text{CT}}=-{\alpha\over 32\pi}{e\over
M_{W}s_{W}^{5}c_{W}^{2}}\left(M_{H}^{2}+2M_{Z}^{2}s_{W}^{2}\\{3+\xi_{Z}\\}\right)\\\
\times\Delta^{\text{UV}}(\mu^{2},M_{Z}^{2})\,\zeta_{2}\,[{\epsilon}^{\mu}(Z_{1}){\epsilon}_{\mu}(Z_{2})]^{\ast}+{\cal{O}}(\varepsilon^{0})\,.$
(8)

$\epsilon(Z_{i})$ are the transverse polarisations of the $Z$ bosons (related
to the decay current $Z\to f\bar{f}$ for massless fermions) and

$\Delta^{\text{UV}}(\mu^{2},M^{2})=\left({4\pi\mu^{2}\over
M^{2}}\right)^{\varepsilon}{\Gamma(1+\varepsilon)\over\varepsilon}={1\over\varepsilon}-\gamma_{E}+\log(4\pi)+\log{\mu^{2}\over
M^{2}}+{\cal{O}}(\varepsilon)\,,$ (9)

is the usual $\overline{\text{MS}}$-related divergence in dimensional
regularisation $D=4-2\varepsilon$ tHooft:1972tcz . This result is
qualitatively not surprising: As we have modified without much care only a
single part of the gauge-invariant $SU(2)_{L}\times U(1)_{Y}$ kinetic Higgs
field term

$D_{\mu}\Phi^{\dagger}D^{\mu}\Phi\supset-
g_{HZZ}^{\text{SM}}HZ^{\mu}Z_{\mu}-g_{HHZZ}^{\text{SM}}H^{2}Z^{\mu}Z_{\mu}\,,$
(10)

($\Phi=(\sqrt{2}G^{+},v+H+iG^{0})^{T}/\sqrt{2}$ being the Higgs doublet)
rather than the whole term, we have lost gauge invariance entirely. This is
highlighted by the non-vanishing $\xi_{Z}$ gauge fixing parameter dependence
of the $H\to ZZ$ decay. $\zeta_{2}\neq 0$ has also induced singularities that
are not mended by a renormalisation of the bare $HZZ$ SM interactions
(included in Eq. (8)).333Considering off-shell $Z$ bosons for kinematics
$M_{H}<2M_{Z}$ leads to similar issues alongside the appearance of additional
divergences $\sim\xi_{W}$. Again this is expected; radiative $\zeta_{2}$
corrections should renormalise coupling deviations

$\zeta_{1}=\kappa_{V}-1\equiv{g_{HVV}\over g_{HVV}^{\text{SM}}}-1\,,$ (11)

(the Lorentz structure of Eq. (8) is linked to the $HZZ$ interactions of Eq.
(9)). However, even if we introduce a bare $\zeta_{1}$, its renormalisation
will be gauge-dependent. Therefore, $\zeta_{1}$ cannot directly be associated
with a physical observable (in contrast to, e.g., the electromagnetic coupling
$e$ and its behaviour under renormalisation group flow).

The combination of naively weak $\kappa_{2V}$ tree-level unitarity constraints
and the breaking of gauge invariance within the SM prompts us to a practical
phenomenological issue when considering the analyses by ATLAS and CMS (or
future experiments) within the $\kappa$ framework: _What is their
phenomenological value and what is the level of $\zeta_{2}$ deviations that
can be reasonably expected?_

Naturally, this question is not a sensible one in the context of the Standard
Model, as highlighted by Eq. (8). In concrete UV extensions, some of which we
alluded to above, the answer is immediately provided by examining the
$H^{2}V^{2}$ operators and their relation to the theories’ input parameters;
they will not be free parameters and typically be predictions of BSM
electroweak input parameters that are more precisely determined from other
measurements.444In the SM these couplings are rather obviously determined by
the choice of precisely measured electroweak input parameters, e.g.
$g_{HZZ}=eM_{Z}/(s_{W}c_{W})$, $g_{HHZZ}=e^{2}/(2s_{W}^{2}c_{W}^{2})$ with
$c_{W}^{2}=1-s_{W}^{2}=M_{W}^{2}/M_{Z}^{2}$. As the analyses and measurements
of ATLAS:2020jgy ; CMS:2022hgz  are model-independent in the sense that they
modify certain parameters whilst keeping others entirely SM-like, our task is
to house these assumptions in a theoretically meaningful way. This requires an
approach with which correlations are not plagued by inconsistencies displayed
by Eq. (8) to inform the reach of $\kappa_{2V}$ deviations in the light of
constraining single Higgs measurements.

##  3 Relevant HEFT interactions at NLO

The only theoretically plausible way to treat $H^{n}VV$ interactions as
independent is by avoiding the physical Higgs boson being part of an
electroweak doublet, which also means that Standard Model EFT (SMEFT)
Grzadkowski:2010es  is not applicable. This is possible by considering the
electroweak chiral Lagrangian (or HEFT) Susskind:1978ms ; Dimopoulos:1979es ;
Dimopoulos:1981xc ; Longhitano:1980iz ; Alonso:2012px ; Brivio:2013pma , which
has received considerable attention recently, including the treatment of
radiative corrections Buchalla:2013rka ; Buchalla:2016sop ; Gavela:2016bzc ;
Buchalla:2017jlu ; Buchalla:2020kdh ; Herrero:2020dtv ; Herrero:2021iqt . HEFT
is intrinsically different from the SM, which becomes clear when radiative
corrections are considered in general gauge (chiefly analysed by Herrero and
Morales in Herrero:2020dtv ; Herrero:2021iqt ; Herrero:2022krh ), but contains
the SM as a particular parameter point choice. This enables us to achieve a
theoretically consistent correlation of different Higgs channels including
radiative corrections in the modern sense of renormalisability (i.e.
renormalisation at a given loop order).

As in any QFT, radiative corrections will source higher-dimensional operators,
yet HEFT is non-renormalisable in the classical sense. It is therefore
necessary to include all relevant operators in the renormalisation programme
from the start as their renormalisation is required for a consistent final
one-loop result. The $\kappa_{V}(\kappa_{2V})$ analysis that we would like to
achieve is then related to a particular choice of couplings for the one-loop
renormalised HEFT Lagrangian at a given renormalisation scale, as we will
discuss below.

###  3.1 HEFT Lagrangian

Following the notations of Refs. Alonso:2012px ; Brivio:2016fzo ;
Herrero:2020dtv ; Herrero:2021iqt , we consider first the leading order
(chiral dimension-two Buchalla:2013eza ; Buchalla:2014eca ) HEFT Lagrangian.
$W^{a}_{\mu}$ and $B_{\mu}$ are the SM gauge bosons associated with the
$SU(2)_{L}\times U(1)_{Y}$ gauge symmetry. Contrary to the SM, the Higgs boson
is a singlet field in HEFT and the Goldstone bosons $\pi^{a}$ are parametrised
non-linearly using the matrix $U$ (a $[SU(2)_{L}\times SU(2)_{R}]/SU(2)_{R+L}$
CCWZ model)

$U(\pi^{a})=\exp\left({i\pi^{a}\tau^{a}/v}\right)\,,$ (12)

where $\tau^{a}$ are the Pauli matrices with $a=1,2,3$. The $U$ matrix
transforms under $L\in SU(2)_{L},U(1)_{Y}\subset SU(2)_{R}\ni R$ as $U\to
LUR^{\dagger}$ and is expanded as

$U(\pi^{a})={\mathbbm{1}}_{2}+i\frac{\pi^{a}}{v}\tau^{a}-\frac{2G^{+}G^{-}+G^{0}G^{0}}{2v^{2}}{\mathbbm{1}}_{2}+\dots\,,$
(13)

where $G^{\pm}=(\pi^{2}\pm i\pi^{1})/\sqrt{2}$ and $G^{0}=-\pi^{3}$. Weak
gauging of $SU(2)_{L}\times U(1)_{Y}$ is achieved through the covariant
derivative

$D_{\mu}U=\partial_{\mu}U+ig_{W}(W^{a}_{\mu}\tau^{a}/2)\;U-ig^{\prime}UB_{\mu}\tau^{3}/2\,.$
(14)

The gauge fields in the physical basis are defined as

$W^{\pm}_{\mu}=\frac{1}{\sqrt{2}}(W^{1}_{\mu}\mp
W^{2}_{\mu})\,,\quad\begin{pmatrix}Z_{\mu}\\\
A_{\mu}\end{pmatrix}=\begin{pmatrix}c_{W}&s_{W}\\\
-s_{W}&c_{W}\end{pmatrix}\begin{pmatrix}W^{3}_{\mu}\\\
B_{\mu}\end{pmatrix}\,.$ (15)

The leading order HEFT Lagrangian relevant for this study is given as follows

$\mathcal{L}=-\frac{1}{4}W^{a}_{\mu\nu}W^{a\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+{\cal{L}}_{\text{ferm}}+\mathcal{L}_{\text{Yuk}}\\\
+\frac{v^{2}}{4}\mathcal{F}_{H}\,\text{Tr}[D_{\mu}U^{\dagger}D^{\mu}U]+\frac{1}{2}\partial_{\mu}H\partial^{\mu}H-V(H)+\mathcal{L}_{\text{GF}}+\mathcal{L}_{\text{FP}}\,,$
(16a) where the interactions of Higgs with gauge and Goldstone bosons are
parametrised using the function $\mathcal{F}_{H}$ given as
$\mathcal{F}_{H}=\Big{(}1+2(1+\zeta_{1})\frac{H}{v}+(1+\zeta_{2})\Big{(}\frac{H}{v}\Big{)}^{2}+...\Big{)}\,.$
(16b)

$\zeta_{1}$ and $\zeta_{2}$ are the new physics parameters (the choice
$\zeta_{1,2}=0$ corresponds to the gauge-Higgs interactions of the SM). The
ellipses denote the interaction terms with more than two Higgs bosons which
are not relevant for our work. ${\cal{L}}_{\text{ferm}}$ parametrises the
fermion-gauge boson interactions, which we take SM-like.
$\mathcal{L}_{\text{GF}}$ and $\mathcal{L}_{\text{FP}}$ are the gauge-fixing
and Faddeev-Popov terms Faddeev:1967fc , respectively. $V({H})$ is the Higgs
potential

$V({H})=\frac{1}{2}M_{H}^{2}H^{2}+\kappa_{3}\frac{M_{H}^{2}}{2v}H^{3}+\kappa_{4}\frac{M_{H}^{2}}{8v^{2}}H^{4}\,.$
(17)

For our analysis, we will take the potential to be SM-like, $\kappa_{3,4}=1$.
The gauge-fixing term is written in terms of gauge fixing functions as

$\mathcal{L}_{\text{GF}}=-F_{+}F_{-}-\frac{1}{2}F_{Z}^{2}-\frac{1}{2}F_{A}^{2}\,,$
(18)

where the gauge fixing functions are defined as linear

$F_{\pm}=\frac{1}{\sqrt{\xi_{W}}}(\partial^{\mu}W^{\pm}_{\mu}-\xi_{W}M_{W}G^{\pm})\,,\hskip
5.69046ptF_{Z}=\frac{1}{\sqrt{\xi_{Z}}}(\partial^{\mu}Z_{\mu}-\xi_{Z}M_{Z}G^{0})\,,\hskip
5.69046ptF_{A}=\frac{1}{\sqrt{\xi_{A}}}(\partial^{\mu}A_{\mu})\,,$ (19)

with $\xi_{W}$, $\xi_{Z}$ and $\xi_{A}$ denoting the gauge-fixing parameters
in the $R_{\xi}$ gauge. Using these above gauge-fixing functions, the Faddeev-
Popov part of the effective, quantised HEFT Lagrangian is then given as

$\mathcal{L}_{\text{FP}}=\sum_{i,j}{c^{i}}^{\dagger}\frac{\delta
F_{i}}{\delta\alpha_{j}}c^{j}~{}\quad(i,j=+,-,Z,A)\,,$ (20)

where $c^{j}$ are the ghost fields and $\alpha_{j}$ are the corresponding
gauge transformation parameters which are similar to that of SM. We
parametrise the couplings of Higgs boson with fermions via

$\mathcal{L}_{\text{Yuk}}=-\frac{v}{\sqrt{2}}\begin{pmatrix}\bar{u}^{i}_{L}&\bar{d}^{i}_{L}\end{pmatrix}U\Big{(}1+c\,\frac{h}{v}+...\Big{)}\begin{pmatrix}y^{u}_{ij}u^{j}_{R}\\\
y^{d}_{ij}d^{j}_{R}\end{pmatrix}+\text{h.c.}\,,$ (21)

where $y^{u}_{ij}$ and $y^{d}_{ij}$ are the Yukawa couplings. In this study,
we will focus on the case $c=1$, which after diagonalisation, directly links
the Yukawa couplings to the fermion mass eigenstates

$y^{u}_{33}={\sqrt{2}m_{t}\over v}\,,\quad y^{d}_{33}={}{\sqrt{2}m_{b}\over
v}\,,$ (22)

with top and bottom masses $m_{t,b}$, respectively. We will neglect the light
quark flavour and lepton masses throughout this work since these do not impact
our phenomenological findings.

When considering radiative corrections in the effective theory described by
Eq. (16), ultraviolet divergences will source new operator structures at one-
loop level. This requires the introduction of bare terms to the Lagrangian ab
initio for a consistent, one-loop renormalisation programme Herrero:2021iqt ;
Herrero:2022krh . The terms relevant at NLO one-loop level (chiral dimension
4) are

${\cal{L}}_{4}=\sum_{i}a_{i}{\cal{O}}_{i}\,,$ (23)

with operators that require UV regularisation listed in Tab. 1.

$\mathcal{O}_{0}$ | $a_{0}(M_{Z}^{2}-M_{W}^{2})\text{Tr}\Big{[}U\tau^{3}U^{\dagger}\boldsymbol{\mathcal{V}}_{\mu}\Big{]}\text{Tr}\Big{[}U\tau^{3}U^{\dagger}\boldsymbol{\mathcal{V}}_{\mu}\Big{]}$
---|---
$\mathcal{O}_{1}$ | $a_{1}\;g^{\prime}g_{W}\text{Tr}\Big{[}UB_{\mu\nu}\frac{\tau^{3}}{2}U^{\dagger}W^{a}_{\mu\nu}\frac{\tau^{a}}{2}\Big{]}$
$\mathcal{O}_{HBB}$ | $-a_{HBB}\;g^{\prime 2}\frac{H}{v}\text{Tr}\Big{[}B_{\mu\nu}B^{\mu\nu}\Big{]}$
$\mathcal{O}_{HWW}$ | $-a_{HWW}\;g_{W}^{2}\frac{H}{v}\text{Tr}\Big{[}W^{a}_{\mu\nu}W^{a\mu\nu}\Big{]}$
$\mathcal{O}_{\square\boldsymbol{\mathcal{V}}\boldsymbol{\mathcal{V}}}$ | $a_{\square\boldsymbol{\mathcal{V}}\boldsymbol{\mathcal{V}}}\frac{\square H}{v}\text{Tr}\Big{[}\boldsymbol{\mathcal{V}}_{\mu}\boldsymbol{\mathcal{V}}^{\mu}\Big{]}$
$\mathcal{O}_{H0}$ | $a_{H0}(M_{Z}^{2}-M_{W}^{2})\frac{H}{v}\text{Tr}\Big{[}U\tau^{3}U^{\dagger}\boldsymbol{\mathcal{V}}_{\mu}\Big{]}\text{Tr}\Big{[}U\tau^{3}U^{\dagger}\boldsymbol{\mathcal{V}}_{\mu}\Big{]}$
$\mathcal{O}_{H1}$ | $a_{H1}\;g^{\prime}g_{W}\frac{H}{v}\text{Tr}\Big{[}UB_{\mu\nu}\frac{\tau^{3}}{2}U^{\dagger}W^{a}_{\mu\nu}\frac{\tau^{a}}{2}\Big{]}$
$\mathcal{O}_{H11}$ | $a_{H11}\frac{H}{v}\text{Tr}\Big{[}\mathcal{D}_{\mu}\boldsymbol{\mathcal{V}}^{\mu}\mathcal{D}_{\nu}\boldsymbol{\mathcal{V}}^{\nu}\Big{]}$
$\mathcal{O}_{d1}$ | $ia_{d1}\;g^{\prime}\frac{\partial^{\nu}H}{v}\text{Tr}\Big{[}UB_{\mu\nu}\frac{\tau^{3}}{2}U^{\dagger}\boldsymbol{\mathcal{V}}^{\mu}\Big{]}$
$\mathcal{O}_{d2}$ | $ia_{d2}\;g_{W}\frac{\partial^{\nu}H}{v}\text{Tr}\Big{[}W^{a}_{\mu\nu}\frac{\tau^{a}}{2}\boldsymbol{\mathcal{V}}^{\mu}\Big{]}$
$\mathcal{O}_{d3}$ | $a_{d3}\frac{\partial^{\nu}H}{v}\text{Tr}\Big{[}\boldsymbol{\mathcal{V}}^{\mu}\mathcal{D}_{\mu}\boldsymbol{\mathcal{V}}^{\mu}\Big{]}$
$\mathcal{O}_{\square\square}$ | $a_{\square\square}\frac{\square H\square H}{v}$
Table 1: HEFT operators $\mathcal{O}_{i}$ required for the cancellations of
divergences, $a_{i}$ are the corresponding HEFT coefficients.
$\boldsymbol{\mathcal{V}}_{\mu}=(D_{\mu}U)U^{\dagger}$ and
$\mathcal{D}_{\mu}\boldsymbol{\mathcal{V}}^{\mu}=\partial_{\mu}\boldsymbol{\mathcal{V}}^{\mu}+i[g_{W}W^{a}_{\mu}\frac{\tau^{a}}{2},\boldsymbol{\mathcal{V}}^{\mu}]$.

###  3.2 Amplitudes and Renormalisation

Throughout, we will work in dimensional regularisation $D=4-2\varepsilon$
tHooft:1972tcz  and adopt the on-shell scheme for propagating degrees of
freedom Denner:1991kt . Expanding out the non-linear sigma model of Eq. (16),
we understand the electroweak vev as a derived quantity

$v={2M_{W}s_{W}\over e}\,,$ (24)

with

$s^{2}_{W}=1-c_{W}^{2}=1-{M_{W}^{2}\over M_{Z}^{2}}\,,$ (25)

following Sirlin Sirlin:1980nh ; Marciano:1980pb . These can be parametrically
fixed through choices of $e=\sqrt{4\pi\alpha},M_{W}$ and $M_{Z}$. As we
identify these quantities as the input parameters for the gauge part of the
Lagrangian (coupling constants of $SU(2)_{L}\times U(1)_{Y}$ are parametrised
as $g_{W}=e/s_{W},g^{\prime}=e/c_{W}$, respectively), we have to supply
counter terms $\delta Z_{e},\delta M_{W}^{2},\delta M_{Z}^{2}$ for the bare
Lagrangian quantities (we indicate unrenormalised quantities with a subscript
‘0’ in Eq. (16), (23) after ${\cal{L}}\to{\cal{L}}_{0}$)

$\begin{split}e_{0}&=Z_{e}e=(1+\delta Z_{e})e\,,\\\
M_{0,W}^{2}&=M_{W}^{2}+\delta M_{W}^{2}\,,\\\ M_{0,Z}^{2}&=M_{Z}^{2}+\delta
M_{Z}^{2}\,.\end{split}$ (26)

The renormalisation of the Weinberg angle

$s_{W,0}=s_{W}+\delta s_{W}\,,\quad c_{W,0}=c_{W}+\delta c_{W}\,,$ (27)

is then given by

$\delta s_{W}={c_{W}^{2}\over 2s_{W}}\left({\delta M_{Z}^{2}\over
M_{Z}^{2}}-{\delta M_{W}^{2}\over M_{W}^{2}}\right)\,,\quad\delta
c_{W}=c_{W}-{s_{W}\over c_{W}}\delta s_{W}\,.$ (28)

The field renormalisations of the bare Lagrangian quantities are

$\begin{split}W^{\pm}_{0,\mu}&=\sqrt{Z_{W}}W^{\pm}_{\mu}=(1+\delta
Z_{W}/2)W^{\pm}_{\mu}\,,\\\ H_{0}&=\sqrt{Z_{H}}H=(1+\delta
Z_{H}/2)H\,,\end{split}$ (29)

and

$\left(\begin{matrix}A_{0}\\\
Z_{0}\end{matrix}\right)_{\mu}=\left(\begin{matrix}\sqrt{Z_{AA}}&\sqrt{Z_{AZ}}\\\
\sqrt{Z_{ZA}}&\sqrt{Z_{ZZ}}\end{matrix}\right)\left(\begin{matrix}A\\\
Z\end{matrix}\right)_{\mu}=\left(\begin{matrix}1+\delta Z_{AA}&\delta
Z_{AZ}\\\ \delta Z_{ZA}&1+\delta
Z_{ZZ}\end{matrix}\right)\left(\begin{matrix}A\\\
Z\end{matrix}\right)_{\mu}\;,$ (30)

for the photon-$Z$ sector. Tadpoles are renormalised by requiring that they
fully cancel against the tadpole counter term. They are treated as part of the
parameter renormalisation Denner:1991kt ; Denner:2018opp ; Dudenas:2020ggt  so
that the tadpole counter term only enters the unphysical Goldstone sector and
we can ignore associated contributions in the following. The HEFT coefficients
are renormalised through

$\zeta_{0,i}=\zeta_{i}+\delta\zeta_{i}\,,\quad a_{0,i}=a_{i}+\delta a_{i}\,.$
(31)

We choose on-shell (OS) renormalisation conditions for the gauge and Higgs
sectors, i.e. the wave function renormalisations are chosen to yield unit
residues for the propagators at the physical masses
$M_{W}^{2},M_{Z}^{2},M_{H}^{2}$; we remove $Z-\gamma$ mixing for on-shell
kinematics, $\Sigma^{T}_{AZ}(q^{2}=0)=\Sigma^{T}_{ZA}(q^{2}=M_{Z}^{2})=0$
(where $\Sigma^{T}$ denotes the transverse part of the self-energy in case of
the vector bosons). Alongside on-shell renormalisation in the fermion-sector
$f$ (the form of which is not relevant for our work), this reduces the
renormalisation condition for the electromagnetic coupling constant to the
condition of the amputated three-point function
$\Gamma_{f\bar{f}\gamma}^{\mu}=-ie\gamma^{\mu}$ for on-shell legs and zero
momentum transfer. The renormalisation constants can then be worked out as
functions of the self-energies and their derivatives and are tabled in, e.g.,
Denner:1991kt ; Denner:2019vbn . From this follows in particular

$\delta Z_{e}=-{1\over 2}\left(\delta Z_{AA}+{s_{W}\over c_{W}}\delta
Z_{ZA}\right)\,.$ (32)

For the coefficients of Eq. (31), we choose the $\overline{\text{MS}}$ scheme.

Expanding the Lagrangian of Eq. (16) into a format compatible with FeynRules
Christensen:2008py ; Alloul:2013bka , we perform our calculations using
FeynArts, FormCalc, and LoopTools Hahn:1998yk ; Hahn:2000jm ; Hahn:2000kx ;
Hahn:2006qw  alongside PackageX Patel:2016fam  for numerical evaluations and
analytical cross-checks. Calculating in general gauges
$\sim\xi_{A},\xi_{W},\xi_{Z}$, we have verified gauge independence at the
level of divergent parts as well as finite parts (see below). Further checks
include comparisons against existing results: In particular Ref.
Herrero:2021iqt  provides a comprehensive overview of the HEFT Lagrangian at
one-loop. Given that we find agreement with the results quoted there, we limit
ourselves to a summary of the findings relevant for our analysis in Sec. 4.

###  $H\to WW^{\ast}$ amplitude renormalisation

The (off-shell) $H(q)W^{\mu}(k_{1})W^{\nu}(k_{2})$ one-loop three-point
function is pictorially represented by the Feynman diagrams of Fig. 2. We
limit ourselves to transverse polarisations. The counter term vertex (in a
convenient phase convention) follows from the procedure described in Sec. 3.2,
see also appendix A,

$\parbox{71.13188pt}{\includegraphics[width=71.13188pt]{wwct.pdf}}={eM_{W}(1+\zeta_{1})\over
s_{W}}\left(\delta Z_{e}+{\delta M_{W}^{2}\over 2M_{W}^{2}}-{\delta s_{W}\over
s_{W}}+{\delta\zeta_{1}\over 1+\zeta_{1}}+{\delta Z_{H}\over 2}+{\delta
Z_{W}}\right)g^{\mu\nu}\\\
+\delta\left[\parbox{71.13188pt}{\includegraphics[width=71.13188pt]{wwd6.pdf}}\right]\,,$
(33)

with $\delta[...]$ summarising the renormalisation contributions from Eq.
(23). Contributions proportional to the matrix element $\sim\left\langle
HW^{\mu\nu}W_{\mu\nu}\right\rangle$ are UV-finite, which imposes a
cancellation555The notation $|_{\Delta}$ refers to the part proportional to
the $\overline{\text{MS}}$ factor of Eq. (9), i.e. $\delta a|_{\Delta}=\delta
a/\Delta^{\text{UV}}$.

$\delta a_{HWW}\big{|}_{\Delta}=-{1\over 2}\delta
a_{d2}\big{|}_{\Delta}={1\over
192\pi^{2}}(1+\zeta_{1})(\zeta_{1}^{2}+2\zeta_{1}-\zeta_{2})\,.$ (34)

The remaining singularity $\sim q^{2}\left\langle
HW^{\mu}W_{\mu}\right\rangle$ then allows us to read off

$\delta
a_{\Box{\boldsymbol{\cal{V}}}{\boldsymbol{\cal{V}}}}\big{|}_{\Delta}=-{1\over
64\pi^{2}}(1+\zeta_{1})\left(2+(1+\zeta_{1})^{2}\right)\,.$ (35)

The precise form of these additional terms is not relevant for the analysis of
Sec. 4, but we quote these here to highlight the agreement with
Herrero:2021iqt . The singularity related to the bare momentum-independent
$HWW$ vertex, on the other hand, fixes

$\delta\zeta_{1}\big{|}_{\Delta}={3\alpha\over 32\pi
s_{W}^{2}M_{W}^{2}}\big{(}4(m_{b}^{2}+m_{t}^{2})\zeta_{1}+M_{Z}^{2}(3s_{w}^{2}-4)\zeta_{1}(1+\zeta_{1})(2+\zeta_{1})\\\
+M_{H}^{2}(1-\kappa_{3}+\zeta_{1})(\zeta_{1}(2+\zeta_{1})-\zeta_{2})+2M_{W}^{2}(1+\zeta_{1})\zeta_{2}\big{)}\,,$
(36)

after entering the renormalisation constants of appendix A.

Figure 2:  Representative Feynman diagram topologies contributing to $H\to
WW$. We use abbreviations ${\cal{V}}=\\{W,Z,A\\}$ for the vector bosons and
${\cal{G}}=\\{G^{0},G^{+}\\}$ for the would-be Goldstone bosons. Note that we
only consider the third family of fermions $f=\\{t,b\\}$ as massive for the
purpose of this work. Figure 3:  Representative Feynman diagram topologies
contributing to $H\to ZZ$, similar to Fig. 2. Note that due to the structure
of the HEFT Lagrangian, we obtain 5-point interactions in the $H\to ZZ$
channel (i).

###  $H\to ZZ^{\ast}$ amplitude renormalisation

Similar to the discussion above, the (off-shell) $H(q)Z(k_{1})Z(k_{2})$ one-
loop three-point function is summarised in Fig. 3. The counter term vertex is

$\parbox{71.13188pt}{\includegraphics[width=71.13188pt]{zzct.pdf}}={eM_{Z}(1+\zeta_{1})\over
s_{W}c_{W}}\left(\delta Z_{e}+{\delta M_{Z}^{2}\over 2M_{Z}^{2}}-{\delta
s_{W}\over s_{W}}\right.\\\ \left.-{\delta c_{W}\over
c_{W}}+{\delta\zeta_{1}\over 1+\zeta_{1}}+{\delta Z_{H}\over 2}+{\delta
Z_{ZZ}}\right)g^{\mu\nu}+\delta\left[\parbox{71.13188pt}{\includegraphics[width=71.13188pt]{zzd6.pdf}}\right]\,.$
(37)

and there are similar cancellations for the counter terms $\delta[\dots]$
leading to Eq. (34), which are not relevant for the analysis of Sec. 4.
Entering Eq. (36) then fixes the remaining

$\delta a_{H0}\big{|}_{\Delta}={3\over 64\pi^{2}}(1+\zeta_{1})\zeta_{2}\,,$
(38)

when including the renormalisation constants of appendix A.

###  $H\to\gamma Z$ amplitude renormalisation

The $H\to\gamma Z$ counter term amplitude is given by

$\parbox{71.13188pt}{\includegraphics[width=71.13188pt]{zzct.pdf}}={eM_{W}\over
2c_{W}^{2}s_{W}}\delta
Z_{ZA}+\delta\left[\parbox{71.13188pt}{\includegraphics[width=71.13188pt]{azd6.pdf}}\right]\,,$
(39)

which renders the off-shell amplitude UV finite (we do not show the Feynman
diagram topologies as they are similar to Fig. 3). Again the terms
$\delta[\dots]$ enforce non-trivial identities among the one-loop counter
terms of the operators of Eq. (23), which are again not relevant for our
$\kappa_{2V}-\kappa_{V}$ analysis. Entering the wave-function renormalisation
constant of appendix A, we obtain a UV-finite on-shell $H\to\gamma Z$
amplitude which is manifestly gauge-independent. Our result agrees with the
findings of Herrero:2020dtv  and the unitary gauge calculation of
deBlas:2018tjm .

###  $H\to\gamma\gamma$ amplitude renormalisation

The $H\to\gamma\gamma$ amplitude has been first computed in general $R_{\xi}$
gauge in Herrero:2020dtv  after unitary gauge results have been used in
deBlas:2018tjm . The $H\to\gamma\gamma$ is free of singularities (see also
Weinberg:1971fb ; Marciano:2011gm ) and therefore only imposes one-loop
counter term relations among the coefficients of Eq. (23). Again these are not
relevant for the analysis below, but we have cross-checked our findings
against the existing results in the literature.

###  3.3 Decay widths

The amplitudes $H\to\gamma\gamma,\gamma Z$ of the previous section can be
converted into decay widths $\Gamma(H\to\gamma\gamma),\Gamma(H\to\gamma Z)$
straightforwardly; they are only sensitive to $\zeta_{1}$. For the cases of
$H\to ZZ^{\ast}$ and $H\to WW^{\ast}$ we follow the SMEFT analysis of Refs.
Dawson:2018liq ; Dawson:2018pyl . To this end, we write the decay width
$H(p_{H})\to f(p_{1})f(p_{2})V(p_{3})$ as

$\Gamma(H\to
VV^{\ast})=\int_{0}^{(M_{H}-M_{V})^{2}}\hbox{d}m_{12}^{2}\int_{m_{23,\text{l}}^{2}}^{m_{23,\text{u}}^{2}}\hbox{d}m_{23}^{2}\,\dfrac{{{\cal{M}}}^{2}}{(2\pi)^{3}\,32\,M_{H}^{3}}\,,$
(40)

with $m_{ij}^{2}=(p_{i}+p_{j})^{2}$. The invariant masses satisfy the relation

$m_{23}^{2}+m_{23}^{2}+m_{13}^{2}=M_{H}^{2}+M_{V}^{2}\,.$ (41)

With the Källén lambda function denoted as

$\lambda(m_{12},M_{H},M_{V})=m_{12}^{4}-2m_{12}^{2}(M_{H}^{2}+M_{V}^{2})+(M_{H}^{2}-M_{V}^{2})^{2}\,,$
(42)

the limits of the $m_{23}^{2}$ integration are

$m_{23,\text{l},\text{u}}^{2}={1\over
2}\left(M_{H}^{2}+M_{V}^{2}-m_{12}^{2}\mp\sqrt{\lambda(m_{12},M_{H},M_{V})}\right)\,.$
(43)

The amplitudes are constructed by considering the Eq. (40) expanded to one-
loop level

${\cal{M}}^{2}=|{\cal{M}}_{\text{LO}}|^{2}+2\,\text{Re}\left\\{{\cal{M}}_{\text{LO}}{\cal{M}}_{\text{1-loop}}^{\ast}\right\\}\,,$
(44)

and both the leading order (LO) and one-loop parts can be written as

${\cal{M}}={\cal{M}}^{\mu}\epsilon^{\ast}_{\mu}={\cal{M}}_{HVV}^{\mu\nu}\Delta_{VV}^{\nu\rho}{{\cal{M}}_{ffV}^{\rho}}\epsilon^{\ast}_{\mu}\,,$
(45)

where the polarisation vector of the on-shell boson is denoted with
$\epsilon_{\mu}$. ${\cal{M}}_{ffV}^{\rho}$ is the $ffV$ vertex arising from
$V$-current and includes the on-shell spinors, while $\Delta_{VV}^{\nu\rho}$
is the vector boson propagator. We do not include higher order corrections for
these parts and only consider them for the $HVV$ part of the amplitude,
${\cal{M}}_{HVV}^{\mu\nu}$. In the case of $H\to ZZ^{\ast}$ we also do not
include the $H\to\gamma Z$ contribution (similar to Dawson:2018pyl ).

The renormalised $HVV$ vertices described above can be decomposed into three
different Lorentz structures using the metric tensor $g^{\mu\nu}$ and the
Levi-Civita antisymmetric tensor $\varepsilon^{\mu\nu\rho\sigma}$ as

${\cal{M}}_{HVV}^{\mu\nu}=(F_{1}^{\text{LO}}+F_{1}^{\text{NLO}})g^{\mu\nu}+F_{2}^{\text{NLO}}(p_{1}+p_{2})^{\mu}p_{3}^{\nu}+F_{3}^{\text{NLO}}\tensor{\varepsilon}{{}^{\mu}{}^{\nu}_{\rho}{}_{\sigma}}p_{H}^{\rho}(p_{1}+p_{2})^{\sigma}\;.$
(46)

The last two terms only contribute at one-loop order for the SM and HEFT for
the choices detailed in Sec. 3.2. $F_{3}^{\text{NLO}}$ is zero for the $H\to
f\bar{f}Z$ channel, while the longitudinal-like second term does not
contribute as we consider massless fermions. The non-trivial form factors are
directly related to the one-loop calculation detailed above. By considering
the unpolarised amplitude and performing the integrations of Eq. (40) we
verify that our calculations reproduce the expressions supplied by Refs.
Dawson:2018dcd ; Dawson:2018liq ; Dawson:2018pyl  for the SM case at LO (see
also Rizzo:1980gz ; Keung:1984hn ). For HEFT with NLO corrections, the
integrations are performed numerically to obtain the decay widths $\Gamma(H\to
WW^{\ast})$ and $\Gamma(H\to ZZ^{\ast})$ and subsequently the relevant
$\kappa_{V}=1+\zeta_{1}$ for a scale choice $\mu_{R}=M_{H}$. For an efficient
evaluation of the amplitudes, we furthermore choose Feynman Gauge
$\xi_{W}=\xi_{Z}=\xi_{A}=1$ and rescale our results to the values reported by
the Higgs Cross Section Working Group LHCHiggsCrossSectionWorkingGroup:2011wcg
for the SM point $\zeta_{1}=\zeta_{2}=0$ for ease of comparison.

###  3.4 Comments on $\kappa_{2V}-\kappa_{V}$ interpretations

The cancellation of gauge fixing parameters $\xi_{W,Z,A}$ in the amplitude
calculations detailed above is highly non-trivial. Indeed, individual Feynman
diagrams of Figs. 2 and 3 contain singularities
$\sim\xi_{W,Z}^{2},\xi_{W,Z,A}$, which cancel when we sum over the
contributing diagrams. For instance, the vanishing
$\xi^{2}_{W,Z}\Delta^{\text{UV}}$ dependence of $H(q)\to W(k_{1})W(k_{2})$
results from a cancellation between the gauge boson insertions of diagrams
Fig. 2 (g) and (h) like in the SM. A zero $\sim\xi_{W,Z}\Delta^{\text{UV}}$
coefficient results from cancellations between diagrams Fig. 2 (b), (c), (d),
(i) and (j), and $\sim\xi_{A}\Delta^{\text{UV}}$ is due to destructive
interference between (c), (g), (j). Similar cancellations take place for $H\to
ZZ$ (there is dependence on $\xi_{A}$).

As can be seen from Eqs. (34) and (35) (see also appendix A), terms of Eq.
(23) are sourced even for $\zeta_{1}=\zeta_{2}=0$. Although we must include
all relevant HEFT coefficient renormalisation factors in the calculation, we
can obtain a consistent $\kappa_{2V}-\kappa_{V}$ correlation at our scale
choice $\mu_{R}=M_{H}$ by projecting all amplitudes to $a_{i}=0$ after
renormalisation. Concretely this means that we will limit our attention to
$\zeta_{1,2}$ in the following and choose all other $a_{i}(M_{H})=0$ as
$\overline{\text{MS}}$ parameters. In particular, this means that operator
matrix elements $\left\langle HV^{\mu\nu}V_{\mu\nu}\right\rangle(V=A,W,Z)$ are
absent, which would otherwise impact the Higgs boson production and decay
phenomenology. These should be included in a comprehensive fit of the HEFT
Lagrangian that we do not attempt here.

Parameters | ATLAS Run 2 data ATLAS:2022vkf  | HL-LHC uncertainties | Correlation Matrix ATLAS:2022vkf
---|---|---|---
$\kappa_{Z}$ | $0.99^{+0.06}_{-0.06}$ | $\pm 0.012$ | $1$ | $0.40$ | $0.44$ | $0.09$
$\kappa_{W}$ | $1.05^{+0.06}_{-0.06}$ | $\pm 0.013$ |  | $1$ | $0.47$ | $0.08$
$\kappa_{\gamma}$ | $1.01^{+0.06}_{-0.06}$ | $\pm 0.013$ |  |  | $1$ | $0.12$
$\kappa_{Z\gamma}$ | $1.38^{+0.31}_{-0.37}$ | $\pm 0.073$ |  |  |  | 1
Table 2: Constraints used for Fig. 4, based on ATLAS:2022vkf . Columns 2 and 4
show the ATLAS Run-2 results along with the correlation matrix. In column 3,
the HL-LHC projected uncertainties are listed.

##  4 $\kappa_{V}-\kappa_{2V}$ correlations from Higgs data

Equipped with the findings of the previous sections, we are now ready to turn
to results. For the current status of the Higgs programme, we construct a
$\chi^{2}$ test from the $\kappa$ results provided in Ref. ATLAS:2022vkf . The
$\chi^{2}$ statistic defined from the $\kappa$ data at an integrated
luminosity of 139 $\text{fb}^{-1}$ is given by

$\chi^{2}(\zeta_{1},\zeta_{2})=\sum_{i,j=1}^{\text{data}}\big{(}\kappa_{i,\text{exp}}-\kappa_{i,\text{th}}(\zeta_{1},\zeta_{2})\big{)}(V_{ij})^{-1}\big{(}\kappa_{j,\text{exp}}-\kappa_{j,\text{th}}(\zeta_{1},\zeta_{2})\big{)}\,,$
(47)

where $\kappa_{i,\text{exp}}$ are the central values and $V_{ij}$ is the
covariance matrix, the elements of which are given as
$\rho_{ij}\sigma_{i}\sigma_{j}$ with $\sigma_{i}$ as uncertainties and
$\rho_{ij}$ as correlations. The details of the experimental data are provided
in Tab. 2. Using the four $\kappa$ parameters detailed above in Eq. (47), we
perform a $\chi^{2}$ fit for $\zeta_{1}$ and $\zeta_{2}$.666For 68$\%$
(95$\%$) confidence level (C.L.), $\Delta\chi^{2}=2.28$
($\Delta\chi^{2}=5.99$) for two degrees of freedom (d.o.f). The allowed
regions are shown in Fig. 4a.

(a)

(b)

Figure 4: The $\zeta_{1}$ vs. $\zeta_{2}$ parameter space obtained from the
$\chi^{2}$ analysis of the $\kappa$ parameters from ATLAS Run-2 data
ATLAS:2022vkf . The solid (dashed) contour denotes the $\Delta\chi^{2}=$ 2.28
($\Delta\chi^{2}=$ 5.99) which corresponds to confidence level of $68\%$
($95\%$). Fig. 4b shows the similar regions in blue obtained with the HL-LHC
projected data.

To gauge the improvements that can be expected at the high luminosity (HL) LHC
phase, we extrapolate these contours to an integrated luminosity of 3000
$\text{fb}^{-1}$. To achieve this, we scale the ATLAS uncertainties for the
current luminosity ($\text{Lumi}_{\text{ATLAS}}$) of 139 $\text{fb}^{-1}$ by a
naive scaling factor

$\sigma_{\text{HL}}=\frac{1}{\sqrt{\text{Lumi}_{\text{HL}}/\text{Lumi}_{\text{ATLAS}}}}\sigma_{\text{ATLAS}}\,.$
(48)

The projected HL-LHC uncertainties are tabulated in column 3 of Tab. 2 for an
unchanged correlation matrix (column 4). We stress that we use this
extrapolation purely to obtain a qualitative outlook for the HL-LHC, but note
that this rescaling provides a good approximation of the comprehensive
investigation of deBlas:2019rxi . The resulting constraints at 68$\%$ and
95$\%$ confidence, assuming consistency with the SM, are shown in Fig. 4b.

As is visible from Fig. 4, we have rescaled the $\zeta_{2}$ axes to account
for the loop suppression. Given that the correlation $\kappa_{V}(\kappa_{2V})$
is related to a weak radiative correction, the bounds on
$\kappa_{2V}=1+\zeta_{2}$ from single Higgs data are loose, even when we
consider the direct sensitivity of the early stage LHC programme shown in Fig.
1. Extrapolations to $3/$ab show that single Higgs physics will greatly
collapse the current limits along the $\zeta_{1}$ direction, however the loop
suppression implies a sensitivity $|\zeta_{2}|\sim 4$.

A coupling modifier of this size is already in tension with the current
measurement results detailed in Sec. 2. This shows that the $\kappa_{2V}$ and
$\kappa_{V}$ are largely independent parameters at the LHC as is naively
suggested by Eq. (16). This highlights the need to increase the sensitivity
coverage to $\zeta_{2}$ through direct searches further, which we will turn to
in Sec. 5.

##  5 Enhancing direct collider sensitivity with Graph Neural Networks

Figure 5:  Representative Feynman diagrams contributing to WBF
$p(f)p(f^{\prime})\to HHjj$. We consider $H\to b\bar{b}$ final states, but do
not include the decays to the diagrams.

Higgs pair production via WBF has been discussed in Refs. Dolan:2013rja ;
Dolan:2015zja ; Bishara:2016kjn ; Kilian:2018bhs ; Arganda:2018ftn  (see also
Cappati:2022skp  for a recent dimension 8 SMEFT study). The process shares the
properties of single Higgs WBF production Cahn:1983ip ; Figy:2003nv ;
Jager:2006zc ; Jager:2006cp ; Ciccolini:2007ec  especially when we consider
LHC-relevant QCD corrections Figy:2008zd ; Frederix:2014hta ; Dreyer:2018qbw .
To select the events for the WBF topology, Fig. 5, we choose two forwarded
jets in opposite detector hemispheres (opposite signs of pseudorapidity
$\eta_{j1}\cdot\eta_{j2}<1$) with the invariant mass $m_{jj}$ larger than 500
GeV. For the minimum transverse momentum $p_{T}$ of WBF jets, we choose 50
GeV. We select events with four $b$-jets in the central region
$|\eta_{b-\text{jets}}|<$ 2.5 with minimum $p_{T}\geq 20$ GeV.888We use a flat
tagging efficiency of $70\%$ and ignore mistagging. For the EdgeConv GNN
strategy we employ below, mistagging will predominantly change the background
normalisation, which can be compensated through a tuned working point of the
classifier. We comment on uncertainties from the background normalisation
further below. Events are generated in proton-proton collisions at
$\sqrt{s}=13$ TeV using MadGraph5_aMC@NLO Alwall:2014hca  at leading order
precision using the standard setting scale choices.999It is known that the NLO
QCD corrections are relatively mild (e.g. Figy:2003nv ; Frederix:2014hta ) and
can be largely captured through adapted weak-boson fusion factorisation scale
choices inspired by the “double-DIS” structure of WBF Jager:2006zc ;
Jager:2006cp . We will see below that the used approach reproduces the current
experimental outcome reasonably well to warrant a high-luminosity
extrapolation. The main SM background for the $4b+2j$ final state comes from
multijet QCD processes Dolan:2013rja ; Dolan:2015zja ; Bishara:2016kjn ;
Kilian:2018bhs ; Arganda:2018ftn . After the given VBF selection cuts, the QCD
multijet background cross-section is 4.41 $pb$, while the signal cross-section
is 0.086 $fb$, which highlights the necessity to efficiently reduce the
background.

Going beyond ‘traditional’ techniques, utilising Graph Neural Networks (GNNs)
is a motivated avenue. These strategies are tailored to accessing the graph
structure of differential cross section data under realistic conditions that
we typically parametrise through Feynman diagram calculations in theoretical
reference calculations. Hence, recent applications to particle physics data in
a range of areas stretching from QCD Dreyer:2020brq , over anomaly detection
Atkinson:2021nlt  (including IR safe formulations Konar:2021zdg ;
Atkinson:2022uzb ), to SMEFT parameter fits Atkinson:2021jnj  have highlighted
GNNs as versatile and highly sensitive tools for BSM discrimination. Given the
special phenomenological properties of WBF processes, we can therefore expect
similar performance improvements in this channel.

(a)

(b)

Figure 6:  A representative diagram of an event as a fully connected graph is
shown on the left. The network performance is shown on the right with a
receiver operating characteristic (ROC) curve along with the corresponding
AUC=0.995.

To construct a GNN for the present signal vs. background discrimination task,
we use a fully-connected bi-directional graph to represent the $4b+2j$ event
(Fig. 6a). Each jet is represented by a node, which is associated with a node
feature vector $[p_{T},\eta,\phi,E,m,PID]$ (representing transverse momentum,
pseudorapidity, azimuthal angle, energy, mass and particle identification
number, respectively). To implement the graph structure, we use the Deep Graph
Library wang2020deep  and PyTorch paszke2019pytorch  and choose an Edge
Convolution (EdgeConv) network to classify signal and background. Edge
convolution is known to be particularly suited for extracting high-level
variable information as edge features from given low-level node features
wang2019dynamic . The message passing function for the Edge convolution is
defined as

$\vec{x}_{i}^{\,(l+1)}=\frac{1}{|\mathcal{N}(i)|}\sum_{j\in\mathcal{N}(i)}{\text{\sc{ReLU}}}\left(\Theta\cdot(\vec{x}_{j}^{\,(l)}-\vec{x}_{i}^{\,(l)})+\Phi\cdot(\vec{x}_{i}^{\,(l)})\right)\,.$
(49)

Here, $\vec{x}_{i}^{\,(l)}$ represents the input node features of node $i$ in
the $l$-th message passing layer, with $l=0$ denoting the input node features
of the graph. The neighbourhood set $\mathcal{N}(i)$ consists of all nodes in
the graph connected to node $i$. The linear layers $\Theta$ and $\Phi$ take
the vectors $\vec{x}_{j}-\vec{x}_{i}$ and $\vec{x}_{i}$, respectively, and map
them to the same vector space. We apply a single message passing step
producing 40-dimensional node features.101010Further details on GNNs, EdgeConv
and architecture design can be found in Ref. Atkinson:2021jnj . Since we aim
to classify graphs, we apply a mean graph readout operation to these node
features. A classifier multilayer perceptron with single layers having 40
nodes, with ReLU activation, takes these features and outputs a vector of two
dimensions (after being properly normalised with a SoftMax activation
function). We minimise the cross-entropy loss function using the Adam
optimiser DBLP:journals/corr/KingmaB14  with a learning rate of $10^{-3}$.
Given the distinct phenomenological properties, we also find that already a
shallow network is well-suited to obtain good discrimination.

The network is trained for 100 epochs with batch sizes of 124 for a
$\kappa_{2V}$-enriched parameter point sample ($\kappa_{2V},\kappa_{V}=2,1$).
We split the total data into $80\%$, $10\%$ and $10\%$ for training,
validation and testing, respectively. We verify that the validation and the
training accuracies are comparable to avoid overtraining. The receiver
operating characteristic curve of the network is shown in Fig. 6b with an area
under curve (AUC) of 0.995, implying a very high classifier
performance.111111The AUC is slightly reduced to 0.941 when considering
parton-showering and underlying event; this reduction can be compensated by a
choice of working point, highlighting the phenomenological distinction of the
signal and QCD background. Electroweak backgrounds that are not considered
here, might require additional improvements.

(a)

(b)

Figure 7:  Exclusion limits on $\kappa_{2V}$ and $\kappa_{V}$ parameter space
obtained from the GNN output trained for the $\kappa_{2V}$ enriched sample
($\kappa_{2V},\kappa_{V}=2,1$) are shown in Fig. 7a. The cyan line represents
the limits obtained from the ATLAS analysis for the fixed value of
$\kappa_{V}=1$ ATLAS:2022ycx . The dot-dashed contour shows the effect of
increasing the background by $25\%$. In Fig. 7b, HL-LHC projected constraints
are overlaid on the 95$\%$ C.L. region shown in Fig. 4b for $\kappa_{2V}$ and
$\kappa_{V}$ parameter space.

With the classifier optimised as described above, we can obtain the limit on
$\kappa_{2V}$ by fixing $\kappa_{V}=1$ (we do not include modified trilinear
Higgs interactions as in the previous sections) to compare the performance of
the GNN with ATLAS analysis of Ref. ATLAS:2022ycx . For the optimal working
point on the ROC curve, the QCD multijet background cross-section drops to is
0.88 fb, while the signal cross-section remains 0.037 fb. The exclusion limits
are calculated using the significance $\sigma={N_{s}}/{\sqrt{N_{s}+N_{b}}}$,
where $N_{s}$ is the number of signal events and $N_{b}$ is the number of
background events at a particular luminosity. This is shown in Fig. 7a, where
we show the ATLAS constraint as a horizontal (cyan) line. Firstly, this
agreement acts as a good validation of our procedure in comparison to the
realistic experimental measurement, which gives us confidence for the HL
extrapolation that we will perform next. The relative improvement suggested in
Fig. 7a might be optimistic. Given the necessity to overcome large
backgrounds, the choice of working points and the accuracy of the background
simulation are factors that can impact the discrimination, in particular when
we deal with the large statistics of the HL phase. The competitiveness of our
idealised GNN analysis, however, demonstrates that such techniques deserve
consideration as part of realistic experimental analyses. We will also comment
on a range of potential architecture improvements that we have not included in
our analysis below.

Using the $\kappa_{2V}=2$ as a reference point also for a more general scan,
we choose a fixed working point of the classifier which maximises the
significance to obtain exclusion contours in the $\kappa_{2V},\kappa_{V}$
parameter space also shown in Fig. 7a (this includes the changes to the $H\to
b\bar{b}$ branching ratio as a function of $\kappa_{V}$). To compare the level
of sensitivity, we overlay these, projected to 3/ab HL-LHC target with the
expected sensitivity from single Higgs measurements detailed in Sec. 4 in Fig.
7b. Note that in this exploratory comparison, we have not included subdominant
background from weak multi-boson processes or top production, which will
additionally degrade the sensitivity (see e.g. Dolan:2015zja ). These
processes are, however, also characterised by different kinematic and QCD
radiation properties, and we can expect significant discrimination when
considering these additional contributions in the GNN classification. To get
an estimate of how additional backgrounds modify the direct search
sensitivity, we modify the considered dominant QCD through a naive shifting by
25% in normalisation. Including these to Fig. 7a, we see that there is minor
change in the sensitivity, and good sensitivity to $\kappa_{2V}$ can be
achieved.

The GNN could be further improved through a more dedicated inclusion of
$\kappa_{2V}-\kappa_{V}$ correlations to the classification, e.g., through a
multi-class network architecture discussed in Ref. Atkinson:2021jnj , which
could also be extended to subdominant background contributions to specifically
combat those when the QCD contribution has been removed sufficiently. A more
robust graph embedding could help in such an approach as well. While the QCD
background phenomenology is very different from the WBF signal (highlighted by
the fact that a shallow network is sufficient for very good discrimination),
different electroweak correlation structures such as top and weak boson decays
impart a more tree-like structure that could be further exploited when these
become relevant. Insensitivity to differential distributions of the node
features, as well as overall normalisations, could be achievable using
adversarial networks Louppe:2016ylz . We leave more detailed investigations
for future work.

##  6 Summary and Conclusions

In this work, we have presented a comprehensive and theoretically consistent
discussion of $\kappa_{V}-\kappa_{2V}$ correlations by employing Higgs
Effective Field Theory, which puts related analyses at the LHC ATLAS:2020jgy ;
ATLAS:2022ycx ; CMS:2022hgz  on a theoretically firm footing. We show that
single Higgs constraints can be formulated for $\kappa_{2V}$ analyses,
however, given that these effects arise as a weak radiative correction, the
single-Higgs $\kappa_{V}$ constraints are relatively loose and not competitive
when compared to the LHC’s sensitivity to $\kappa_{2V}$ already at this stage
of the programme. Indeed, the LHC’s sensitivity pattern when mapped onto the
HEFT Lagrangian Eq. (16) and its weak corrections allows us to treat different
Higgs interactions in the gauge sector as largely independent parameters.

As $\kappa_{2V}$ analyses are mainly driven by the direct investigation of
weak $HHjj$ production, an enhanced direct sensitivity of future LHC runs
provides the best motivated avenue to obtain a more fine-grained picture of
the Higgs boson’s gauge interactions along these lines. To this end, we employ
Graph Neural Network techniques to demonstrate that the direct sensitivity to
$\kappa_{2V}$ can be enhanced as GNNs formidably exploit the particular
structure of the decay, colour and kinematical correlations of the $HHjj$
signal compared to the dominant QCD backgrounds. Improvements to
$\kappa_{2V}\sim 1\pm 40\%$ could then be within the reach of the HL-LHC, in
particular as improvements through multi-class GNNs and additional
architecture improvements are likely to add further sensitivity.

## Acknowledgements

CE thanks Bill Murray for helpful discussions. The work of A. is funded by a
Leverhulme Trust Research Project Grant RPG-2021-031. O.A. is supported by the
UK Science and Technology Facilities Council (STFC) under grant ST/V506692/1.
A.B. is funded by the STFC under grant ST/T000945/1. C.E. acknowledges funding
by the Leverhulme Trust (RPG-2021-031), by the STFC under grant ST/T000945/1,
and by the Institute of Particle Physics Phenomenology Associateship Scheme.
P.S. is funded by an STFC studentship under grant ST/T506102/1.

##  Appendix A UV divergent parts of renormalisation constants

In this section, we provide the explicit form of the UV divergent parts
renormalisation constants used to renormalise the Lagrangian of Sec. 3.1
required in this work. The field strength renormalisations are obtained by
evaluating all relevant (off-shell, one-particle irreducible) two-point
function $X\to Y$ contributions derived from Eq. (16), which we represent as

$\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{sminser.pdf}}\,,$
(50)

to be contrasted with 2-point insertions from Eq. (23) represented by

$\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{d6inser.pdf}}\,.$
(51)

The counter terms are then pictorially determined from the renormalisation
conditions (RCs) through

$\left[\left(\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{sminser.pdf}}+\parbox{71.13188pt}{\vspace{0.3cm}\includegraphics[width=71.13188pt]{d6inser.pdf}}\right)+\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{ctinser.pdf}}\right]_{\text{RCs}}=0\,.$
(52)

The counter term diagrams will typically contain terms from Eq. (23) which we
highlight via

$\delta\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{d6inser.pdf}}\right]\,.$
(53)

In the conventions detailed above, the contribution to the off-shell Higgs
boson two-point function $H(q)\to H(q)$ that needs to be included is

$\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{hd6.pdf}}={e^{2}\over
4m_{W}^{2}s_{W}^{2}}q^{4}a_{\Box\Box}\,.$ (54)

The counter term, including the ${\cal{L}}_{4}$ contributions, can then be
written as

$\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{hct.pdf}}=\delta
Z_{H}(q^{2}-M_{H}^{2})+\delta
M_{H}^{2}+\delta\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{hd6.pdf}}\right]\,,$
(55)

with divergent parts given by

$\begin{split}\delta Z_{H}\big{|}_{\Delta}=&{3\alpha\over 16\pi
c_{W}^{2}s_{W}^{2}}(3-2s_{W}^{2})(1+\zeta_{1})^{2}-{3\alpha\over 8\pi
s_{W}^{2}}{m_{t}^{2}+m_{b}^{2}\over M_{W}^{2}}\,,\\\ \delta
M_{H}^{2}\big{|}_{\Delta}=&{3\alpha\over 32\pi
M_{W}^{2}s_{W}^{2}}(3\kappa_{3}^{2}+\kappa_{4})M_{H}^{4}+{3\alpha\over 16\pi
c_{W}^{2}s_{W}^{2}}M_{H}^{2}(2s_{W}^{2}-3)(1+\zeta_{1})^{2}\\\ &+{3\alpha\over
16\pi
c_{W}^{4}s_{W}^{2}}M_{W}^{2}(3-4s_{W}^{2}+2s_{W}^{4})(3+2\zeta_{1}(2+\zeta_{1})+\zeta_{2})\\\
&+{3\alpha\over 8\pi
M_{W}^{2}s_{W}^{2}}M_{H}^{2}\left(m_{b}^{2}+m_{t}^{2}\right)-{9\alpha\over
4\pi M_{W}^{2}s_{W}^{2}}(m_{t}^{4}+m_{b}^{4})\,,\\\ \delta
a_{\Box\Box}\big{|}_{\Delta}=&-{3\over
64\pi^{2}}(1+\zeta_{1})^{2}\,.\end{split}$ (56)

Note that the Higgs renormalisation is manifestly gauge-independent as the
Higgs is a singlet field in HEFT Herrero:2021iqt  (this also applies to the
tadpole). The transverse part of the $W(q)\to W(q)$ boson polarisation
function receives no contributions from Eq. (23)

$\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{wd6.pdf}}\right]_{T}=0\,,$
(57)

leading to a counter term insertion

$\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{wct.pdf}}\right]_{T}=-\delta
Z_{W}(q^{2}-M_{W}^{2})+\delta M_{W}^{2}\,,$ (58)

with

$\begin{split}\delta Z_{W}\big{|}_{\Delta}=&-{\alpha\over 48\pi
s_{W}^{2}}\left(6\xi_{W}+6s_{W}^{2}(\xi_{A}-\xi_{Z})+6\xi_{Z}+\zeta_{1}(2+\zeta_{1})-2\right)\,,\\\
\delta M_{W}^{2}\big{|}_{\Delta}=&-\frac{{\alpha}}{48\pi
c_{W}^{2}s_{W}^{2}}\big{(}3c_{W}^{2}(6(m_{b}^{2}+m_{t}^{2})+M_{H}^{2}({\zeta_{1}}({\zeta_{1}}+2)-{\zeta_{2}}))\\\
&-10{M_{W}^{2}}c_{W}^{2}{\zeta_{1}}({\zeta_{1}}+2)+{M_{W}^{2}}(11-20s_{W}^{2})\big{)}\,.\end{split}$
(59)

Similarly, we find for transverse $Z$ boson polarisation from $Z(q)\to Z(q)$

$\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{zd6.pdf}}\right]_{T}=-2{e^{2}\over
c_{W}^{2}}M_{Z}^{2}a_{0}-2{e^{2}}q^{2}a_{1}\,,$ (60)

and

$\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{zct.pdf}}\right]_{T}=-\delta
Z_{ZZ}(q^{2}-M_{Z}^{2})+\delta
M_{Z}^{2}+\delta\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{zd6.pdf}}\right]_{T}\,,$
(61)

the expressions

$\begin{split}\delta Z_{ZZ}\big{|}_{\Delta}=&-{\alpha\over 48\pi
s_{W}^{2}c_{W}^{2}}\big{(}12\xi_{W}-2+4s_{W}^{2}(1-6\xi_{W}+s_{W}^{2}(20+3\xi_{W}))\\\
&+\zeta_{1}(2+\zeta_{1})\big{)}-2e^{2}\delta a_{1}\,,\\\ \delta
M_{Z}^{2}\big{|}_{\Delta}=&\frac{\alpha}{48\pi{c_{W}^{4}s_{W}^{2}}}\left(-3c_{W}^{2}({M_{H}^{2}}({\zeta_{1}}(2+{\zeta_{1}})-{\zeta_{2}})+6({m_{b}^{2}+m_{t}^{2}}))\right.\\\
&+\left.{M_{W}^{2}}\left(44s_{W}^{4}+58s_{W}^{2}+10{\zeta_{1}}(2+{\zeta_{1}})-11\right)\right)+2{e^{2}\over
c_{W}^{2}}M_{Z}^{2}(\delta a_{0}+c_{W}^{2}\delta a_{1})\,,\\\ \delta
a_{0}\big{|}_{\Delta}=&-{9\over 128\pi^{2}}(\zeta_{1}^{2}-2\zeta_{1})\,,\\\
\delta a_{1}\big{|}_{\Delta}=&-{1\over
192\pi^{2}}(\zeta_{1}^{2}-2\zeta_{1})\,.\end{split}$ (62)

For the $\gamma-\gamma$ polarisation, we obtain

$\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{ad6.pdf}}\right]_{T}=2e^{2}q^{2}a_{1}\,,$
(63)

providing

$\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{act.pdf}}\right]_{T}=-\delta
Z_{AA}q^{2}+\delta\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{ad6.pdf}}\right]_{T}\,.$
(64)

leading to

$\begin{split}\delta Z_{AA}\big{|}_{\Delta}=&-{\alpha\over
12\pi}(20+3\xi_{W})\,.\\\ \end{split}$ (65)

Finally, the $Z(q)\to\gamma(q)$ mixing is

$\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{zad6.pdf}}\right]_{T}=-{e^{2}\over
s_{W}c_{W}}(c_{W}^{2}-s_{W}^{2})a_{1}\,,$ (66)

leading to

$\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{zact.pdf}}\right]_{T}=-(\delta
Z_{AZ}+\delta Z_{ZA})q^{2}+{\delta Z_{ZA}\over
2}M_{Z}^{2}+\delta\left[\parbox{71.13188pt}{\vspace{0.2cm}\includegraphics[width=71.13188pt]{zad6.pdf}}\right]_{T}$
(67)

with

$\begin{split}\delta Z_{AZ}\big{|}_{\Delta}=&-{\alpha\over 12\pi
c_{W}s_{W}}\left(10-3\xi_{W}+s_{W}^{2}(31+3\xi_{W})\right)-2{e^{2}\over
c_{W}s_{W}}(s_{W}^{2}-c_{W}^{2})\delta a_{1}\,,\\\ \delta
Z_{ZA}\big{|}_{\Delta}=&{\alpha\over 4\pi}{c_{W}\over
s_{W}}(3+\xi_{W})\,.\end{split}$ (68)

These results agree with Ref. Herrero:2021iqt  when mapped onto the
conventions used there. In particular, the bare quantities agree, as do the
relevant unrenormalised three-point functions of Sec. 3.2. It is worth
highlighting that the mass renormalisation constants are gauge-independent

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