# Higgs Decays to $ZZ$ and $Z\gamma$ in the SMEFT:
an NLO analysis

S. Dawson and P.P. Giardino  Department of Physics,
Brookhaven National Laboratory, Upton, N.Y., 11973, U.S.A.

###### Abstract

We calculate the complete one-loop electroweak corrections to the inclusive
$H\rightarrow ZZ$ and $H\rightarrow Z\gamma$ decays in the dimension-$6$
extension of the Standard Model Effective Field Theory (SMEFT). The
corrections to $H\rightarrow ZZ$ are computed for on-shell $Z$ bosons and are
a precursor to the physical $H\rightarrow Zf{\overline{f}}$ calculation. We
present compact numerical formulas for our results and demonstrate that the
logarithmic contributions that result from the renormalization group evolution
of the SMEFT coefficients are larger than the finite NLO contributions to the
decay widths. As a by-product of our calculation, we obtain the first complete
result for the finite corrections to $G_{\mu}$ in the SMEFT.

##  I Introduction

The LHC experimental discovery of the Higgs boson, along with the measurement
of Higgs properties that are in rough agreement with the Standard Model (SM)
predictions, suggests that the SM is a valid effective theory at the weak
scale. The lack of new particles up to the $\rm TeV$ scale makes possible the
parameterization of possible high scale physics effects in terms of higher
dimension operators containing only SM fields Buchmuller:1985jz . In this
paper we study new physics contributions to Higgs decays in the context of the
dimension-$6$ Standard Model Effective Field Theory (SMEFT). When compared
with precise theoretical calculations, measurements of Higgs properties serve
to constrain the coefficients of the higher dimension operators and restrict
possible beyond the SM (BSM) physics at energies $\Lambda>>v$.

We study Higgs decays to $Z$ boson pairs and to $Z\gamma$ in the context of
the SMEFT, where new physics is described by a tower of operators,

${\cal L}={\cal
L}_{SM}+\Sigma_{k=5}^{\infty}\Sigma_{i=1}^{n}{\mathcal{C}_{i}^{k}\over\Lambda^{k-4}}O_{i}^{k}\,.$
(1)

The dimension-$k$ operators are constructed from SM fields and the BSM physics
effects reside in the coefficient functions, $C_{i}^{k}$. For large $\Lambda$,
it is sufficient to retain only the lowest dimensional operators. We assume
lepton number conservation, so the lowest dimension relevant operators are
dimension-$6$. Ignoring flavor, there are $59$ dimension-$6$ operators that
are $SU(3)\times SU(2)\times U(1)$ invariant combinations of the SM fields
Buchmuller:1985jz ; Grzadkowski:2010es . The operators have been classified in
several different bases, which are related by the equations of motion
Buchmuller:1985jz ; Grzadkowski:2010es ; Hagiwara:1993ck ; Giudice:2007fh . In
this paper we will use the Warsaw basis of Ref. Grzadkowski:2010es .

A detailed understanding of Higgs properties requires the inclusion of the
dimension-$6$ tree level SMEFT effects, along with radiative corrections in
the effective field theory. Measurements at the Higgs mass scale, $M_{H},$
that are sensitive to a set of SMEFT coefficients, $\mathcal{C}_{i}(M_{H}$),
at leading order will develop logarithmic sensitivity to other coefficients
when renormalization group evolved to the scale $\Lambda$, due to the
renormalization group mixing of the coefficients Elias-Miro:2013mua ; Elias-
Miro:2013gya ; Jenkins:2013zja ; Jenkins:2013wua ; Alonso:2013hga . There are
also finite contributions that may be of the same numerical size as the
logarithmic terms.

We compute the one-loop electroweak SMEFT contributions to the decays
$H\rightarrow ZZ$ and $H\rightarrow Z\gamma$. These corrections include the
one-loop SM electroweak corrections, along with the one-loop corrections due
to the SMEFT operators of Eq. (1). Our results are interesting from a purely
theoretical perspective and we present the first complete one-loop SMEFT
renormalization of $G_{\mu}$. The one-loop SMEFT corrections to $H\rightarrow
b{\overline{b}}$ Gauld:2015lmb ; Gauld:2016kuu  and $H\rightarrow\gamma\gamma$
Hartmann:2015oia ; Hartmann:2015aia  are known, as well as a general NLO SMEFT
calculation of $2$-body Higgs decays Ghezzi:2015vva . The physical process for
$M_{H}=125~{}\rm TeV$ is $H\rightarrow Zf{\overline{f}}$ and our calculation
is a precursor to the eventual one-loop $3$-body SMEFT calculation.

We review the SMEFT framework in Sec. II and our renormalization framework in
Sec. III. Sec. IV contains numerical results for $H\rightarrow ZZ$ and
$H\rightarrow Z\gamma$ and a discussion of the phenomenological impact of our
calculation. A comparison with the physical off-shell process, $H\rightarrow
Zf{\overline{f}}$, is in Appendix A. Appendix B contains numerical fits for
the one-loop SMEFT result for $H\rightarrow ZZ$ and Appendix C has analytic
formulas for the logarithmic contributions to the one-loop SMEFT result for
$H\rightarrow ZZ$. Lastly, Appendix D contains the one-loop SMEFT calculation
of $G_{\mu}$.

##  II smeft Basics

In this section we briefly introduce the SMEFT. We consider the Lagrangian in
Eq. (1) truncated at dimension$-6$,

$\mathcal{L}=\mathcal{L}_{\text{SM}}+\mathcal{L}^{(6)}_{\text{EFT}};~{}~{}~{}~{}\mathcal{L}^{(6)}_{\text{EFT}}=\sum_{i}{\mathcal{C}^{(6)}_{i}\over\Lambda^{2}}\mathcal{O}^{(6)}_{i},$
(2)

where $\mathcal{L}_{\text{SM}}$ is the Standard Model Lagrangian, and
$\mathcal{L}^{(6)}_{\text{EFT}}$ is the most general $SU(3)\times SU(2)\times
U(1)$ invariant EFT Lagrangian that can be built using only dimension-$6$
operators. In the following we drop the superscript $(6)$. Only a few
operators contribute to the $H\to ZZ$ and $H\to Z\gamma$ decays at tree-level,
while more operators contribute at one- loop. In total, 19 of the 59
independent dimension-$6$ operators of the Warsaw basis enter our calculation,

$\displaystyle\mathcal{O}_{W},\mathcal{O}_{\phi},\mathcal{O}_{\phi\square},\mathcal{O}_{\phi
D},\mathcal{O}_{u\phi},\mathcal{O}_{\phi W},\mathcal{O}_{\phi
B},\mathcal{O}_{\phi WB},\mathcal{O}_{uW},$
$\displaystyle\mathcal{O}_{uB},\mathcal{O}_{\phi l}^{(1)},\mathcal{O}_{\phi
l}^{(3)},\mathcal{O}_{\phi e},\mathcal{O}_{\phi q}^{(1)},\mathcal{O}_{\phi
q}^{(3)},\mathcal{O}_{\phi u},\mathcal{O}_{\phi
d},\mathcal{O}_{ll},\mathcal{O}_{lq}^{(3)}\,,$ (3)

where the operators are defined in Tab. 1,

$D_{\mu}=\partial_{\mu}+ig^{\prime}B_{\mu}Y+ig{\tau^{I}\over
2}W_{\mu}^{I}+ig_{s}T^{A}G_{\mu}^{A}\,,$ (4)

$\tau^{I}$ are the Pauli matrices and $l_{L}$ and $q_{L}$ are the $SU(2)_{L}$
doublet lepton and quark fields. For simplicity, we assume a diagonal flavor
structure for the coefficients $\mathcal{C}$, i.e.
$\underset{p,r}{\mathcal{C}_{i}}=\mathcal{C}_{i}\underset{p,r}{\mathbb{1}}$,
where $p,r$ are flavor indices. Furthermore, we assume
$\underset{e,\mu,\mu,e}{\mathcal{C}_{ll}}=\underset{\mu,e,e\mu}{\mathcal{C}_{ll}}\equiv\mathcal{C}_{ll}$
and
$\underset{\mu,\mu,t,t}{\mathcal{C}_{lq}^{(3)}}=\underset{e,e,t,t}{\mathcal{C}_{lq}^{(3)}}\equiv\mathcal{C}_{lq}^{(3)}$.

In general, the presence of the dimension-$6$ operators changes the structure
of the Lagrangian and the correlations between the Lagrangian quantities and
the measured observables Falkowski:2015fla ; Brivio:2017bnu . In the following
we discuss these modifications, as they are relevant to the one-loop SMEFT
calculations of $H\rightarrow ZZ$ and $H\rightarrow Z\gamma$.

s  ${\cal O}_{W}$ $\epsilon^{IJK}W_{\mu}^{I\nu}W_{\nu}^{J\rho}W_{\rho}^{K\mu}$
${\cal O}_{\phi}$ $(\phi^{\dagger}\phi)^{3}$ ${\cal O}_{\phi\Box}$
$(\phi^{\dagger}\phi)\raisebox{-1.42262pt}{$\Box$}(\phi^{\dagger}\phi)$ ${\cal
O}_{\phi D}$
$\left(\phi^{\dagger}D^{\mu}\phi\right)^{*}\left(\phi^{\dagger}D_{\mu}\phi\right)$
$\underset{p,r}{{\cal O}_{u\phi}}$
$(\phi^{\dagger}\phi)(\bar{q}^{\prime}_{p}u^{\prime}_{r}\widetilde{\phi})$
${\cal O}_{\phi W}$ $(\phi^{\dagger}\phi)\,W_{\mu\nu}W^{\mu\nu}$ ${\cal
O}_{\phi B}$ $(\phi^{\dagger}\phi)\,B_{\mu\nu}B^{\mu\nu}$ ${\cal O}_{\phi WB}$
$(\phi^{\dagger}\tau^{I}\phi)\,W^{I}_{\mu\nu}B^{\mu\nu}$ ${\cal O}_{uW}$
$(\bar{q}^{\prime}_{p}\sigma^{\mu\nu}u^{\prime}_{r})\tau^{I}\widetilde{\phi}\,W_{\mu\nu}^{I}$
$\underset{p,r}{{\cal O}_{uB}}$
$(\bar{q}^{\prime}_{p}\sigma^{\mu\nu}u^{\prime}_{r})\widetilde{\phi}\,B_{\mu\nu}$
$\underset{p,r}{{\cal O}_{\phi l}^{(1)}}$
$(\mbox{${\phi^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,\phi}$})(\bar{l}^{\prime}_{p}\gamma^{\mu}l^{\prime}_{r})$
$\underset{p,r}{{\cal O}_{\phi l}^{(3)}}$
$(\mbox{${\phi^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}^{\,I}\,\phi}$})(\bar{l}^{\prime}_{p}\tau^{I}\gamma^{\mu}l^{\prime}_{r})$
$\underset{p,r}{{\cal O}_{\phi e}}$
$(\mbox{${\phi^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,\phi}$})(\bar{e}^{\prime}_{p}\gamma^{\mu}e^{\prime}_{r})$
$\underset{p,r}{{\cal O}_{\phi q}^{(1)}}$
$(\mbox{${\phi^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,\phi}$})(\bar{q}^{\prime}_{p}\gamma^{\mu}q^{\prime}_{r})$
$\underset{p,r}{{\cal O}_{\phi q}^{(3)}}$
$(\mbox{${\phi^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}^{\,I}\,\phi}$})(\bar{q}^{\prime}_{p}\tau^{I}\gamma^{\mu}q^{\prime}_{r})$
$\underset{p,r}{{\cal O}_{\phi u}}$
$(\mbox{${\phi^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,\phi}$})(\bar{u}^{\prime}_{p}\gamma^{\mu}u^{\prime}_{r})$
$\underset{p,r}{{\cal O}_{\phi d}}$
$(\mbox{${\phi^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,\phi}$})(\bar{d}^{\prime}_{p}\gamma^{\mu}d^{\prime}_{r})$
$\underset{p,r,s,t}{{\cal O}_{ll}}$
$(\bar{l}^{\prime}_{p}\gamma_{\mu}l^{\prime}_{r})(\bar{l}^{\prime}_{s}\gamma^{\mu}l^{\prime}_{t})$
$\underset{p,r,s,t}{{\cal O}_{lq}^{(3)}}$
$(\bar{l}^{\prime}_{p}\gamma_{\mu}\tau^{I}l^{\prime}_{r})(\bar{q}^{\prime}_{s}\gamma^{\mu}\tau^{I}q^{\prime}_{t})$

Table 1: Dimension-6 operators relevant for our study (from Grzadkowski:2010es
). For brevity we suppress fermion chiral indices $L,R$. $I=1,2,3$ is an
$SU(2)$ index, $p,r$ are flavor indices, and
$\mbox{${\phi^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,\phi}$}\equiv\phi^{\dagger}D_{\mu}\phi-(D_{\mu}\phi^{\dagger})\phi$.

###  II.1 Higgs sector

In the SMEFT, the Higgs Lagrangian takes the form,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle(D^{\mu}\phi)^{\dagger}(D_{\mu}\phi)+\mu^{2}\phi^{\dagger}\phi-\lambda(\phi^{\dagger}\phi)^{2}$
(5) $\displaystyle+$
$\displaystyle\frac{1}{\Lambda^{2}}\biggl{(}\mathcal{C}_{\phi}(\phi^{\dagger}\phi)^{3}+\mathcal{C}_{\phi\square}(\phi^{\dagger}\phi)\square(\phi^{\dagger}\phi)+\mathcal{C}_{\phi
D}(\phi^{\dagger}D^{\mu}\phi)^{*}(\phi^{\dagger}D_{\mu}\phi)\biggr{)}$

where $\phi$ is the Higgs doublet:

$\displaystyle\phi=\left(\begin{array}[]{c}\phi^{+}\\\
\frac{1}{\sqrt{2}}(v+H+i\phi^{0})\end{array}\right),$ (8)

and $v$ is the vacuum expectation value (vev) defined as the minimum of the
potential,

$v\equiv\sqrt{2}\langle\phi\rangle=\sqrt{\frac{\mu^{2}}{\lambda}}+\frac{3\mu^{3}}{8\lambda^{5/2}}\frac{\mathcal{C}_{\phi}}{\Lambda^{2}}.$
(9)

Due to the presence of $\mathcal{O}_{\phi\square}$ and $\mathcal{O}_{\phi D}$
in Eq. (5), the kinetic terms in the resulting Lagrangian are not canonically
normalized. As a consequence we need to shift the fields,

$\displaystyle H$ $\displaystyle\to$ $\displaystyle
H\biggl{(}1-\frac{v^{2}}{\Lambda^{2}}(\frac{1}{4}\mathcal{C}_{\phi
D}-\mathcal{C}_{\phi\square})\biggr{)},$ $\displaystyle\phi^{0}$
$\displaystyle\to$
$\displaystyle\phi^{0}\biggl{(}1-\frac{v^{2}}{\Lambda^{2}}(\frac{1}{4}\mathcal{C}_{\phi
D})\biggr{)},$ $\displaystyle\phi^{+}$ $\displaystyle\to$
$\displaystyle\phi^{+},$ (10)

and the physical mass of the Higgs, defined as the pole of the propagator,
becomes,

$M_{H}^{2}=2\lambda
v^{2}-\frac{v^{4}}{\Lambda^{2}}(3\mathcal{C}_{\phi}-4\lambda\mathcal{C}_{\phi\square}+\lambda
C_{\phi D}).$ (11)

As anticipated, the relation between the Lagrangian parameters and the
measured observable ($M_{H}$) is altered by the presence of the dimension-$6$
operators Falkowski:2015fla ; Brivio:2017bnu .

###  II.2 Gauge sector

The introduction of the operators in Eq. (3) also alters the form of the
kinetic terms for the Lagrangian of the gauge sector. The relevant Lagrangian
terms are:

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}W{I,\mu\nu}W_{\mu\nu}^{I}-\frac{1}{4}B^{\mu\nu}B_{\mu\nu}$
(12) $\displaystyle+$
$\displaystyle\frac{1}{\Lambda^{2}}\bigg{(}\mathcal{C}_{\phi
W}(\phi^{\dagger}\phi)W^{I,\mu\nu}W_{\mu\nu}^{I}+\mathcal{C}_{\phi
B}(\phi^{\dagger}\phi)B^{\mu\nu}B_{\mu\nu}+\mathcal{C}_{\phi
WB}(\phi^{\dagger}\tau^{I}\phi)W^{I,\mu\nu}B_{\mu\nu}\bigg{)},$

It is convenient to define ”barred” fields,
${\overline{W}}_{\mu}\equiv(1-\mathcal{C}_{\phi W}v^{2}/\Lambda^{2})W_{\mu}$
and ${\overline{B}}_{\mu}\equiv(1-\mathcal{C}_{\phi
B}v^{2}/\Lambda^{2})B_{\mu}$ and ”barred” gauge couplings,
${\overline{g}}_{2}\equiv(1+\mathcal{C}_{\phi W}v^{2}/\Lambda^{2})g_{2}$ and
${\overline{g}}_{1}\equiv(1+\mathcal{C}_{\phi B}v^{2}/\Lambda^{2})g_{1}$ so
that ${\overline{W}}_{\mu}{\overline{g}}_{2}=W_{\mu}g_{2}$ and
${\overline{B}}_{\mu}{\overline{g}}_{1}=B_{\mu}g_{1}$. The ”barred” fields
defined in this way have their kinetic terms properly normalized and preserve
the form of the covariant derivative. The masses of the W and Z fields (poles
of the propagators) are then expressed in terms of the ”barred” couplings
Dedes:2017zog ; Alonso:2013hga :

$\displaystyle M_{W}^{2}$ $\displaystyle=$
$\displaystyle\frac{{\overline{g}}_{2}^{2}v^{2}}{4},$ $\displaystyle
M_{Z}^{2}$ $\displaystyle=$
$\displaystyle\frac{({\overline{g}}_{1}^{2}+{\overline{g}}_{2}^{2})v^{2}}{4}+\frac{v^{4}}{\Lambda^{2}}\left(\frac{1}{8}({\overline{g}}_{1}^{2}+{\overline{g}}_{2}^{2})\mathcal{C}_{\phi
D}+\frac{1}{2}{\overline{g}}_{1}{\overline{g}}_{2}\mathcal{C}_{\phi
WB}\right).$ (13)

It is interesting to note that the extra terms in the definition of the $Z$
mass are due to the rotation, $(W_{\mu}^{3},B_{\mu})\to(Z_{\mu},A_{\mu})$,
that is proportional to $\mathcal{C}_{\phi WB}$ and the shift of $\phi^{0}$ in
Eq. (10) that is proportional to $\mathcal{C}_{\phi D}$.

###  II.3 Fermion sector

Lastly, we study the fermion sector. We notice that the presence of the
dimension-6 operators do not alter the kinetic terms in the Lagrangian, so we
concentrate on the mass terms111We neglect flavor mixing, so
$\mathcal{C}_{u\phi}$ represents generically $\mathcal{C}_{u\phi}$,
$\mathcal{C}_{c\phi}$, $\mathcal{C}_{t\phi}$, etc.:

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-(y_{e}\bar{l}_{L}e_{R}\phi+y_{u}\bar{q}_{L}u_{R}\tilde{\phi}+y_{d}\bar{q}_{L}d_{R}\phi+h.c.)$
(14) $\displaystyle+$
$\displaystyle\frac{1}{\Lambda^{2}}\left(\mathcal{C}_{e\phi}(\phi^{\dagger}\phi)(\bar{l}_{L}e_{R}\phi)+\mathcal{C}_{u\phi}(\phi^{\dagger}\phi)(\bar{q}_{L}u_{R}\tilde{\phi})+\mathcal{C}_{d\phi}(\phi^{\dagger}\phi)(\bar{q}_{L}d_{R}\phi)+h.c.\right).$

The masses of all fermions are shifted by the interactions of Eq. (14). The
lepton and light quark masses do not enter the $1$-loop result for
$H\rightarrow ZZ$ and can be safely set to $0$ there. However, the masses of
the leptons and lighter quarks contribute logarithmically to the
$\gamma\gamma$ wave-function in the one-loop $H\to Z\gamma$ calculation and we
retain finite fermion masses there. However, since the lowest order (LO) $H\to
Z\gamma$ amplitude is ${\cal{O}}({v^{2}\over\Lambda^{2}})$, the contributions
of the terms proportional to $\mathcal{C}_{e\phi}$ and $\mathcal{C}_{d\phi}$
to light fermion masses, $m_{f}$, are
${\cal{O}}({m_{f}v^{4}\over\Lambda^{4}})$ and can be neglected. We concentrate
on the definition of the top pole mass,

$M_{t}=\frac{v}{\sqrt{2}}(y_{t}-\frac{1}{2}\mathcal{C}_{u\phi}\frac{v^{2}}{\Lambda^{2}}).$
(15)

Dimension-6 operators involving fermions also give contributions to the decay
of the $\mu$ lepton, thus changing the relation between the vev, $v$, and the
Fermi constant $G_{\mu}$ obtained from the measurement of the $\mu$ lifetime.
Considering only contributions that interfere with the SM amplitude, we obtain
the tree level result,

$\displaystyle G_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}v^{2}}-\frac{1}{2\sqrt{2}\Lambda^{2}}(\underset{e,\mu,\mu,e}{\mathcal{C}_{ll}}+\underset{\mu,e,e\mu}{\mathcal{C}_{ll}})+{\sqrt{2}\over
2\Lambda^{2}}(\underset{e,e}{\mathcal{C}_{\phi
l}}^{(3)}+\underset{\mu,\mu}{\mathcal{C}_{\phi l}}^{(3)})$ (16)
$\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}v^{2}}-\frac{1}{\sqrt{2}\Lambda^{2}}\mathcal{C}_{ll}+{\sqrt{2}\over\Lambda^{2}}\mathcal{C}_{\phi
l}^{(3)},$

where in the last equality we assumed flavor universality of the coefficients.

##  III Renormalization

The SM one-loop electroweak corrections to $H\to ZZ$ are well known and we
reproduce the results of Ref. Kniehl:1990mq . The decay $H\to Z\gamma$ first
occurs at one-loop in the SM and analytic results are in Ref. Cahn:1978nz ;
Bergstrom:1985hp .

The calculations of the radiative corrections to $H\to ZZ$ and $H\to Z\gamma$
in the SMEFT proceed in the usual way Denner:1991kt  by the choice of a
renormalization scheme, the definition of a suitable set of input parameters
and the calculation of the 1PI amplitudes involved. However, since the SMEFT
theory is only renormalizable order by order in the $(v^{2}/\Lambda^{2})$
expansion, we need to add an extra requirement and drop all terms proportional
to $(v^{2}/\Lambda^{2})^{a}$ with $a>1$. These terms would need counterterms
of dimension-$8$ that are not included in our study. Dropping them makes it
possible for us to proceed with our renormalization program. The one-loop
SMEFT calculation contains both tree level and one-loop contributions from the
dimension-6 operators, along with the full electroweak one-loop SM amplitude.
We chose a modified on shell (OS) scheme, where the SM parameters are OS
quantities, while the SMEFT coefficients are defined as ${\overline{MS}}$
quantities.

The tadpole counterterms are defined such that they cancel completely the
tadpole graphs Fleischer:1980ub . This condition forces us to identify the
renormalized vacuum to be the minimum of the renormalized scalar potential at
each order of perturbation theory. Notice that, due to this choice, the
intermediate quantities defined here are gauge dependent, while the final
result is gauge independent as expected.

We choose the $G_{\mu}$ scheme, where we take the physical input parameters to
be222The light quark masses and lepton masses enter into the $\gamma$ wave-
function renormalization for $H\rightarrow Z\gamma$. We take
$m_{\tau}=1.776~{}\rm GeV,\,m_{\mu}=0.105~{}\rm
GeV\,\text{and}\,m_{e}=0.0005~{}\rm GeV$ for the lepton masses, while the
light quark contribution has been replaced with the hadronic contribution to
the vacuum polarization $\Delta\alpha_{\rm had}^{(5)}=0.02764\pm
0.00007$Tanabashi:2018oca .,

$\displaystyle G_{\mu}$ $\displaystyle=$ $\displaystyle 1.1663787(6)\times
10^{-5}\rm GeV^{-2}$ $\displaystyle M_{Z}$ $\displaystyle=$ $\displaystyle
91.1876\pm.0021\rm GeV$ $\displaystyle M_{W}$ $\displaystyle=$ $\displaystyle
80.385\pm.015~{}\rm GeV$ $\displaystyle M_{H}$ $\displaystyle=$ $\displaystyle
125.09\pm 0.21\pm 0.11~{}\rm GeV$ $\displaystyle M_{t}$ $\displaystyle=$
$\displaystyle 173.1\pm 0.6~{}\rm GeV\,.$

Since the coefficients of the dimension-6 operators are not measured
quantities, it is convenient to treat them as $\overline{{MS}}$ parameters, so
the renormalized coefficients are
$\mathcal{C}(\mu)=\mathcal{C}_{0}-\text{poles}$, where $\mathcal{C}_{0}$ are
the bare quantities. The poles of the coefficients $\mathcal{C}_{0}$ are
obtained from the renormalization group evolution of the coefficients computed
in the unbroken phase of the theory in Refs. Jenkins:2013zja ; Jenkins:2013wua
; Alonso:2013hga . In general, one can write,

$\mathcal{C}_{i}(\mu)=\mathcal{C}_{0,i}-\frac{1}{2\hat{\epsilon}}\frac{1}{16\pi^{2}}\gamma_{ij}\mathcal{C}_{j},$
(17)

where $\mu$ is the renormalization scale, $\gamma_{ij}$ is the one-loop
anomalous dimension,

$\mu\frac{d\mathcal{C}_{i}}{d\mu}=\frac{1}{16\pi^{2}}\gamma_{ij}\mathcal{C}_{j},$
(18)

and $\hat{\epsilon}^{-1}\equiv\epsilon^{-1}-\gamma_{E}+\log(4\pi)$ is related
to the regulator $\epsilon$ for integrals evaluated in $d=4-2\epsilon$
dimensions.

At one-loop, the tree level parameters of the previous section (denoted with
the subscript 0 in this section) must be renormalized. The renormalized SM
masses are defined by,

$M_{V}^{2}=M^{2}_{0,V}-\Pi_{V}(M^{2}_{V}),$ (19)

where $\Pi_{V}(M^{2}_{V})$ is the one-loop correction to the 2-point function
of either Z or W computed on-shell. The gauge boson $2$\- point functions in
the SMEFT can be found in Refs. Chen:2013kfa ; Ghezzi:2015vva .

The one- loop relation between the vev and the Fermi constant is defined by
the equation,

$G_{\mu}+{\mathcal{C}_{ll}\over\sqrt{2}\Lambda^{2}}-\sqrt{2}{\mathcal{C}_{\phi
l}^{(3)}\over\Lambda^{2}}\equiv\frac{1}{\sqrt{2}v_{0}^{2}}(1+\Delta r),$ (20)

where $v_{0}$ is the unrenormalized minimum of the potential and $\Delta r$ is
obtained from the one-loop corrections to $\mu$ decay and is given by

$\displaystyle\Delta r$ $\displaystyle=$ $\displaystyle
2v^{2}\mathcal{B}+\mathcal{V}(1+\frac{v^{2}}{\Lambda^{2}}\mathcal{C}_{\phi
l}^{(3)})+\mathcal{E}(1+\frac{v^{2}}{\Lambda^{2}}(2\,\mathcal{C}_{\phi
l}^{(3)}-\mathcal{C}_{ll}))$ (21)
$\displaystyle-\frac{A_{WW}}{M_{W}^{2}}(1+2\frac{v^{2}}{\Lambda^{2}}\mathcal{C}_{\phi
l}^{(3)})+\frac{1}{16\pi^{2}}\frac{1}{2\hat{\epsilon}}\frac{v^{2}}{\Lambda^{2}}(2\gamma_{\phi
l,j}^{(3)}\mathcal{C}_{j}-\gamma_{ll,j}\mathcal{C}_{j}).$

In Eq. (21), $\mathcal{B}$ is the box contribution, $\mathcal{V}$ is the
vertex contribution, $A_{WW}=\Pi_{W}(0)$ is the W boson self-energy at zero
momentum and $\mathcal{E}$ is the sum of the lepton
($\mu,\,e,\,\nu_{\mu},\,\nu_{e}$) wave-function renormalizations. All the
quantities are calculated at zero external momenta. Notice that the definition
of $\Delta r$ is modified with respect to the SM result due to the presence of
dimension-6 operators in the tree-level relation between $G_{\mu}$ and $v$
given in Eq. (20). Additionally, we absorb the poles of the coefficients
$\mathcal{C}$ into the definition of $\Delta r$. The renormalization of the
vev is then,

$\displaystyle v^{2}$ $\displaystyle=$ $\displaystyle v_{0}^{2}-\delta v^{2}$
$\displaystyle\delta v^{2}$ $\displaystyle=$ $\displaystyle\frac{\Delta
r}{\sqrt{2}G_{\mu}}\biggl{(}1-\frac{1}{\sqrt{2}G_{\mu}\Lambda^{2}}\mathcal{C}_{ll}+\frac{\sqrt{2}}{G_{\mu}\Lambda^{2}}\mathcal{C}_{\phi
l}^{(3)}\biggr{)}\,.$ (22)

Analytic expressions for $\Delta r$ in both the SM and the SMEFT at
dimension-$6$ are given in Appendix C in the $R_{\xi}$.

In the following, we indicate with the symbol $\Delta\mathcal{A}^{\mu\nu}$ the
sum of the contributions from the renormalization of the vev, the masses, and
the SMEFT coefficients described above. The other contributions to the one-
loop corrections are the proper one-particle irreducible amplitudes
$\mathcal{A}_{1PI}^{\mu\nu}$, the particle reducible contributions
$\mathcal{A}_{PR}^{\mu\nu}$ due to the $Z/\gamma$ mixing which arises in the
$H\rightarrow ZZ$ process, and the external leg wave-function renormalization
$\delta Z_{i}=-\partial\Pi_{i}(k^{2})/\partial k^{2}|_{M_{i}^{2}}$. The
calculation of these contributions is relatively straightforward. We start
with the $R_{\xi}$ Feynman rules for the SMEFT in the Warsaw basis presented
in Dedes:2017zog  and convert them to a FeynArts Hahn:2000kx  model file,
using the FeynRules Alloul:2013bka  routines, from which we obtain the
amplitudes needed for our calculation. We reduce the integrals in terms of
Passarino-Veltman integrals Passarino:1978jh , using FeynCalc Mertig:1990an ;
Shtabovenko:2016sxi  and lastly we use LoopTools Hahn:2000jm  to numerically
calculate the integrals. Once we compute all the terms that contribute, the
one-loop correction can be simply written as

$\mathcal{A}^{1l,\mu\nu}=\mathcal{A}_{1PI}^{\mu\nu}+\mathcal{A}^{0l,\mu\nu}\frac{1}{2}\sum_{i}\delta
Z_{i}+\mathcal{A}_{PR}^{\mu\nu}+\Delta\mathcal{A}^{\mu\nu}.$ (23)

We verified that $\mathcal{A}^{1l,\mu\nu}$ is UV and IR finite and we
confirmed its gauge invariance by computing the amplitudes for the $H\to ZZ$
and $H\to Z\gamma$ processes in $R_{\xi}$ gauge.

We conclude this section with a few remarks on the truncation of the expansion
in loops and powers of $\frac{v^{2}}{\Lambda^{2}}$. To clarify our explanation
we consider the generic form of the $H\rightarrow ZZ$ amplitude where we re-
introduce the notation $\mathcal{C}^{(6)}$ and $\mathcal{C}^{(8)}$ for the
coefficients of dimensions 6 and 8 operators:

$\mathcal{A}\sim{\hat{a}}_{0}g_{SM}+{\hat{a}}_{1}\mathcal{C}^{(6)}\frac{v^{2}}{\Lambda^{2}}+{\hat{a}}_{2}\frac{g^{3}_{SM}}{16\pi^{2}}+{\hat{a}}_{3}\mathcal{C}^{(6)}\frac{v^{2}}{\Lambda^{2}}\frac{g^{2}_{SM}}{16\pi^{2}}+\frac{v^{4}}{\Lambda^{4}}({\hat{a}}_{4}(\mathcal{C}^{(6)})^{2}\frac{g_{SM}}{16\pi^{2}}+{\hat{a}}_{5}\mathcal{C}^{(8)})+\dots$
(24)

In Eq. (24) we assumed the following ordering
$g_{SM}>\mathcal{C}^{(6)}\frac{v^{2}}{\Lambda^{2}}>\frac{g^{3}_{SM}}{16\pi^{2}}>\mathcal{C}^{(8)}\sim(\mathcal{C}^{(6)})^{2}\frac{g_{SM}}{16\pi^{2}}$,
and we ordered the terms from larger to smaller according to it. Note that in
Eq. (24), we have only included one insertion of dimension-6 operators at tree
level, $({\hat{a}}_{1})$. If we were considering a more complicated process,
it would in general be possible to have two (or more) tree level insertions of
dimension-6 operators, which would give a contribution of
${\cal{O}}({v^{4}\over\Lambda^{4}})$ that would be of the same order as terms
we have included. The squared amplitude then is,

$\displaystyle|\mathcal{A}|^{2}$ $\displaystyle\sim$
$\displaystyle{\hat{a}}_{0}^{2}g^{2}_{SM}+2{\hat{a}}_{0}{\hat{a}}_{1}\mathcal{C}^{(6)}\frac{v^{2}}{\Lambda^{2}}g_{SM}+{\hat{a}}_{1}^{2}(\mathcal{C}^{(6)})^{2}\frac{v^{4}}{\Lambda^{4}}$
(25) $\displaystyle+$ $\displaystyle
2{\hat{a}}_{0}{\hat{a}}_{2}\frac{g^{4}_{SM}}{16\pi^{2}}+2({\hat{a}}_{1}{\hat{a}}_{2}+{\hat{a}}_{0}{\hat{a}}_{3})\mathcal{C}^{(6)}\frac{v^{2}}{\Lambda^{2}}\frac{g^{3}_{SM}}{16\pi^{2}}$
$\displaystyle+$
$\displaystyle\frac{v^{4}}{\Lambda^{4}}g_{SM}(2({\hat{a}}_{0}{\hat{a}}_{4}+{\hat{a}}_{1}{\hat{a}}_{3})(\mathcal{C}^{(6)})^{2}\frac{g_{SM}}{16\pi^{2}}+({\hat{a}}_{0}{\hat{a}}_{4})\mathcal{C}^{(8)})+\dots$

As explained at the beginning of this section, in order to obtain a result
that is finite in the UV in the dimension-$6$ SMEFT, we need to drop terms
that are of order $\frac{v^{4}}{\Lambda^{4}}$ in the amplitude $\mathcal{A}$.
However Eqs. (24) and (25) show that at the squared amplitude level the
requirement is slightly different: while dropping the terms
$\sim(\mathcal{C}^{(6)})^{2}\frac{v^{4}}{\Lambda^{4}}$ would be inconsistent
since they are in principle larger than the SM one-loop contributions, we have
to drop the terms
$\sim(\mathcal{C}^{(6)})^{2}\frac{v^{4}}{\Lambda^{4}}\frac{g^{2}_{SM}}{16\pi^{2}}$
that are of the same order of the contributions from dimension 8 operators.
From a practical point of view, however, those terms are for the most part
numerically irrelevant, and simply squaring the amplitude (24) after dropping
the terms of order $\frac{v^{4}}{\Lambda^{4}}$ is a valid option. Lastly, we
notice that the condition
$\mathcal{C}^{(6)}\frac{v^{2}}{\Lambda^{2}}>\frac{g^{3}_{SM}}{16\pi^{2}}$
ensures that the fourth term in Eq. (24) is larger than a generic SM 2-loop
contribution.

##  IV Results

###  IV.1 $H\rightarrow ZZ$

In terms of the physical input parameters, $M_{W},~{}M_{Z}$ and $G_{\mu}$, the
tree level SMEFT amplitude for the on-shell decay $H\rightarrow
Z^{\mu}(p_{1})Z^{\nu}(p_{2})$ is,

$\displaystyle{\cal{A}}^{0l,\mu\nu}$ $\displaystyle=$ $\displaystyle{\cal
A}_{0,SM}\biggl{\\{}\biggl{[}T^{0l}_{SM}+{1\over\Lambda^{2}}\Sigma_{i}\mathcal{C}_{i}T_{i}^{0l}\biggr{]}\biggl{(}g^{\mu\nu}-{p_{2}^{\mu}p_{1}^{\nu}\over
p_{1}\cdot p_{2}}\biggr{)}$ (26)
$\displaystyle+\biggl{[}T^{0l}_{SM}+{1\over\Lambda^{2}}\Sigma_{i}\mathcal{C}_{i}{\hat{T}}_{i}^{0l}\biggr{]}\biggl{(}{p_{2}^{\mu}p_{1}^{\nu}\over
p_{1}\cdot p_{2}}\biggr{)}\biggr{\\}}\,,$

where ${\cal{A}}_{0,SM}=2^{5/4}\sqrt{G_{\mu}}M_{Z}^{2}$, $T_{SM}^{0l}=1$, the
sum is over all of the contributing Warsaw basis coefficients
$\mathcal{C}_{i}$, and the tree level SMEFT contribution is,

$\displaystyle\Sigma_{i}T_{i}^{0l}\mathcal{C}_{i}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}G_{\mu}}\biggl{(}{c_{k}\over
2}\biggr{)}-{2p_{1}\cdot p_{2}\over\sqrt{2}G_{\mu}M_{Z}^{2}}c_{ZZ}\,,$
$\displaystyle\Sigma_{i}{\hat{T}}_{i}^{0l}\mathcal{C}_{i}$ $\displaystyle=$
$\displaystyle{2\over\sqrt{2}G_{\mu}M_{Z}^{2}}c_{ZZ}\,.$ (27)

Note that the tree level amplitude depends on only $2$ combinations of
coefficients,

$\displaystyle c_{k}$ $\displaystyle\equiv$ $\displaystyle{{\mathcal{C}}_{\phi
D}\over 2}+2{\mathcal{C}}_{\phi\square}+{\mathcal{C}}^{ll}_{\mu
ee\mu}-2{\mathcal{C}}_{\phi l}^{(3)}\,,$ $\displaystyle c_{ZZ}$
$\displaystyle\equiv$ $\displaystyle\biggl{[}{\mathcal{C}}_{\phi
W}{M_{W}^{2}\over M_{Z}^{2}}+(1-{M_{W}^{2}\over M_{Z}^{2}}){\mathcal{C}}_{\phi
B}+{M_{W}\over M_{Z}}\sqrt{1-{M_{W}^{2}\over M_{Z}^{2}}}{\mathcal{C}}_{\phi
WB}\biggr{]}\,.$ (28)

The combination $c_{ZZ}$ is limited from global fits to the SMEFT
contributions to Higgs decays. The $95\%$ confidence level limit is
Falkowski:2015fla 333 Note the differing normalization of $c_{ZZ}$ from Ref.
Falkowski:2015fla .

$-1.2\biggl{(}{1~{}\rm
TeV\over\Lambda^{2}}\biggr{)}<c_{ZZ}<1.6\biggl{(}{1~{}\rm
TeV\over\Lambda^{2}}\biggr{)}\,.$ (29)

The tree level decay width in the SMEFT is,

$\displaystyle\Gamma(H\rightarrow ZZ)^{0l}_{EFT}$ $\displaystyle=$
$\displaystyle{\beta G_{\mu}M_{H}^{3}\over
16\pi\sqrt{2}}\left(12\,x^{2}-4\,x+1\right)\biggl{\\{}1+{1\over\sqrt{2}G_{\mu}\Lambda^{2}}c_{k}\biggr{\\}}$
(30) $\displaystyle+{3\,\beta M_{H}\over
4\pi}{M_{Z}^{2}\over\Lambda^{2}}c_{ZZ}(2x-1)$ $\displaystyle=$
$\displaystyle\Gamma(H\rightarrow
ZZ)_{SM}\biggl{\\{}1+{1\over\sqrt{2}G_{\mu}\Lambda^{2}}\biggl{[}c_{k}+{24x(2x-1)\over
12x^{2}-4x+1}c_{ZZ}\biggr{]}\biggr{\\}}\,,$

where $\beta=\sqrt{1-{4M_{Z}^{2}\over M_{H}^{2}}}$ and $x\equiv
M_{Z}^{2}/M_{H}^{2}$. For $M_{H}$ between $2M_{Z}$ and $200~{}\rm GeV$ the
dependence on $M_{H}$ is minimal Giudice:2007fh ,

$\displaystyle R^{0l}$ $\displaystyle\equiv$
$\displaystyle{\Gamma(H\rightarrow ZZ)^{0l}_{EFT}\over\Gamma(H\rightarrow
ZZ)_{SM}^{0l}}$ (31) $\displaystyle\sim$ $\displaystyle
1+{1\over\sqrt{2}G_{\mu}\Lambda^{2}}\biggl{[}c_{k}-4c_{ZZ}\biggr{]}+{\cal{O}}\biggl{(}\mathcal{C}{{}^{2}v^{4}\over\Lambda^{4}}\biggr{)}\,.$

Using the results of Appendix A, we can compare the on-shell tree level result
of Eq. (31) with the off-shell result appropriate for the physical Higgs mass
Contino:2013kra ; Grinstein:2013vsa ; Isidori:2013cla ,

$\displaystyle R^{off}$ $\displaystyle\equiv$
$\displaystyle{\Gamma(H\rightarrow
Zf{\overline{f}})^{0l}\over\Gamma(H\rightarrow Zf{\overline{f}})_{SM}}$ (32)
$\displaystyle\sim$ $\displaystyle
1+{1\over\sqrt{2}G_{\mu}\Lambda^{2}}\biggl{[}c_{k}-.97c_{ZZ}\biggr{]}+{\cal{O}}\biggl{(}\mathcal{C}{{}^{2}v^{4}\over\Lambda^{4}}\biggr{)}\,.$

Comparing Eqs. (31) and (32), we note that the off-shell effects are large
Boselli:2017pef .

In the SMEFT, the dimension-$6$ NLO results contain both the complete SM
$1-$loop electroweak corrections to $H\rightarrow ZZ$ and the one-loop
corrections from the dimension-$6$ SMEFT operators. The complete, renormalized
NLO amplitude is,

$\displaystyle{\cal A}^{\mu\nu}_{NLO}={\cal A}^{0l,\mu\nu}+{\cal
A}^{1l,\mu\nu}\,.$ (33)

The one-loop SMEFT contribution to the amplitude can be expanded as,

$\displaystyle{\cal{A}}^{1l,\mu\nu}$ $\displaystyle=$ $\displaystyle{\cal
A}_{0,SM}\biggl{\\{}\biggl{(}T^{1l}_{SM}+\mathcal{F}_{g}\log\biggl{(}{\Lambda^{2}\over
M_{Z}^{2}}\biggr{)}+{(1~{}TeV)^{2}\over\Lambda^{2}}\sum_{i}T_{i}^{1l}\mathcal{C}_{i}\biggr{)}\biggl{(}g^{\mu\nu}-\frac{p_{1}^{\nu}p_{2}^{\mu}}{p_{1}\cdot
p_{2}}\biggr{)}$ (34) $\displaystyle+$
$\displaystyle\biggl{(}{\hat{T}}^{1l}_{SM}+\mathcal{F}_{p}\log\biggl{(}{\Lambda^{2}\over
M_{Z}^{2}}\biggr{)}+{(1~{}TeV)^{2}\over\Lambda^{2}}\sum_{i}{\hat{T}}_{i}^{1l}\mathcal{C}_{i}\biggr{)}\frac{p_{1}^{\nu}p_{2}^{\mu}}{p_{1}\cdot
p_{2}}\biggr{\\}}\,,$

where the terms $\mathcal{F}_{g}\log\biggl{(}{\Lambda^{2}\over
M_{Z}^{2}}\biggr{)}$ and $\mathcal{F}_{p}\log\biggl{(}{\Lambda^{2}\over
M_{Z}^{2}}\biggr{)}$ contain the residual dependence on the renormalization
scale, due to our choice of renormalization scheme. Retaining terms to ${\cal
O}\biggl{(}{v^{2}\over\Lambda^{2}}\biggr{)}$, we parameterize the exact one-
loop SMEFT result,444This fit is valid for $M_{H}<2M_{t}$.

$\displaystyle T_{i}^{1l}$ $\displaystyle=$ $\displaystyle
a_{0,i}+a_{1,i}{M_{H}^{2}\over M_{Z}^{2}}+a_{2,i}\biggl{(}{M_{H}^{2}\over
M_{Z}^{2}}\biggr{)}^{2}+a_{3,i}\log\biggl{(}{M_{H}^{2}\over
M_{Z}^{2}}\biggr{)}+a_{4,i}\log\biggl{(}{4M_{t}^{2}-M_{H}^{2}\over
M_{Z}^{2}}\biggr{)}$ $\displaystyle{\hat{T}}_{i}^{1l}$ $\displaystyle=$
$\displaystyle b_{0,i}+b_{1,i}{M_{H}^{2}\over
M_{Z}^{2}}+b_{2,i}\biggl{(}{M_{H}^{2}\over
M_{Z}^{2}}\biggr{)}^{2}+b_{3,i}\log\biggl{(}{M_{H}^{2}\over
M_{Z}^{2}}\biggr{)}+a_{4,i}\log\biggl{(}{4M_{t}^{2}-M_{H}^{2}\over
M_{Z}^{2}}\biggr{)}\,.$ (35)

Numerical values for the fit parameters, together with the analytical
expressions for $\mathcal{F}_{g}$ and $\mathcal{F}_{p}$ are given in Appendix
B.

The SM result is easily recovered,

${\cal A}^{1l,\mu\nu}_{SM}={\cal
A}_{0,SM}\biggl{\\{}T^{1l}_{SM}\biggl{(}g^{\mu\nu}-\frac{p_{1}^{\nu}p_{2}^{\mu}}{p_{1}\cdot
p_{2}}\biggr{)}+{\hat{T}}^{1l}_{SM}\frac{p_{1}^{\nu}p_{2}^{\mu}}{p_{1}\cdot
p_{2}}\biggr{\\}}\,,$ (36)

where $T_{SM}^{1l}=a_{0,SM}$ and ${\hat{T}}^{1l}_{SM}=b_{0,SM}$. We have
verified that our one-loop electroweak SM corrections are in agreement with
previous results Kniehl:1990mq .

Figure 1: Dependence of $H\rightarrow ZZ$ decay width on ${\mathcal{C}}_{\phi
W}$ in different expansions as explained in the text.

Figure 2: Dependence of the $H\rightarrow ZZ$ decay width on SMEFT
coefficients that contribute at tree level. Note that the curve labelled
$1-$loop EFT is the complete SMEFT result and contains both the one-loop SM
result and the one-loop contribution from the dimension-6 SMEFT coefficients.
Coefficients not specified are set to $0$.

In Fig. 1, we illustrate the dependence of the decay widths on the terms
retained in the expansion of Eq. (25). The curves labelled “$1/\Lambda^{2}$
expansion” drop the ${\hat{a}}_{1}^{2}$ and
$({\hat{a}}_{0}{\hat{a}}_{4}+{\hat{a}}_{1}{\hat{a}}_{3})$ contributions in Eq.
(25). It is apparent that for large values of the $\mathcal{C}$ (here
${\mathcal{C}}_{\phi W}$), the expansion is nonsense. For $\mathcal{C}_{\phi
W}>2$, the dimension-$6$ approximation breaks down and the dimension-$6$
approximation to the amplitude-squared becomes negative. The curves labelled
$\mid{\cal A}\mid^{2}$ contain all of the terms in the square of Eq. (24)
(with ${\hat{a}}_{4}\rightarrow 0$) and are well behaved for all
$\mathcal{C}$. For ${\mathcal{C}}_{\phi W}<2$, the full amplitude-squared is
well approximated by the terms linear in the coefficient functions and the
SMEFT d=6 approximation is valid. We have checked that this condition is
satisfied for the following plots in this section.

Figure 3: Dependence of $H\rightarrow ZZ$ decay width on SMEFT coefficients.
Coefficients are varied one at a time and coefficients not specified are set
to $0$. Figure 4: Same as Fig. 3, but normalized to $\Gamma_{1-loop}^{SM}$.

In Fig. 2, we show the contribution relative to the LO SM prediction for
representative values of the SMEFT coefficients that contribute at tree level.
Choosing all coefficients positive, for the parameters we have chosen,
inclusion of the tree level SMEFT coefficients decreases the rate by $\sim
2\%$ (red curve) . The SM one-loop corrections (black curve) increase the rate
by roughly $5\%$, leading to a partial cancellation between the 1-loop SM and
tree level SMEFT contributions. This makes it clear that global fits to SMEFT
coefficients that do not contain the electroweak corrections cannot be more
accurate than $\sim{\cal{O}}(5\%)$. The one loop corrections from the the
SMEFT operators are much smaller than the SM electroweak corrections. In this
example, the complete NLO SMEFT calculation is well approximated by including
only the logarithmic contributions from the SMEFT coefficients. On the RHS of
Fig. 2, we flip the sign of the SMEFT coefficients, which increases their
numerical significance. Note that the dependence on $M_{H}$ is rather mild.

In Figs. 3 and 5 , we show the dependence on SMEFT coefficients as a function
of the $\mathcal{C}$ for fixed $M_{H}=200~{}\rm GeV$, normalized to the LO
SMEFT result. Fig. 3 shows the dependence on coefficients that enter at tree
level and in Fig. 4, we replot the same results, normalized to the one-loop
SM. The change from the one-loop SM results in the SMEFT is of
${\cal{O}}(5\%)$ for ${\mathcal{C}}_{\phi B}\sim{\cal{O}}(-1)$ and for
${\mathcal{C}}_{\phi W}\sim{\cal{O}}(-1)$ this change is ${\cal{O}}(20\%)$.
These values are consistent with current fits to LHC Higgs decays to $ZZ$.

Fig. 5 shows the dependence on a selection of coefficient functions that do
not enter at tree level. An interesting feature of our results is that they
can be used to obtain limits on coefficients that first enter at loop level.
For example, from a global fit Berthier:2016tkq ; Falkowski:2015fla ;
Brivio:2017bnu ,

$\displaystyle\mathcal{C}_{W}$ $\displaystyle=$
$\displaystyle{\Lambda^{2}\over v^{2}}(1.14\pm.68)$ (37)
$\displaystyle\rightarrow$ $\displaystyle(18.8\pm
11.2)\biggl{(}{\Lambda^{2}\over 1~{}\rm TeV^{2}}\biggr{)}\,.$

Such a large value of $\mathcal{C}_{W}$ would increase the LO SMEFT width to
$ZZ$ by $\sim 20\%$ as observed in Fig. 5. Loop corrections to Higgs decays
therefore have the possibility of new constraints on the SMEFT coefficients.

Our corrections to the on-shell $H\rightarrow ZZ$ one-loop SMEFT result must
be considered as a first step in a full SMEFT calculation for $H\rightarrow
Zf{\overline{f}}$. Our results suggest, however, that the results of Figs. 3
and 5 can be thought of as $K$ factors that multiply the LO SMEFT off-shell
result of Eq. (32). The numerical size of our results implies that a full
SMEFT calculation is needed for $H\rightarrow Zf{\overline{f}}$ in order to
obtain reliable results. The SM electroweak corrections for $H\rightarrow
Zf{\overline{f}}$ are known and can be implemented using the PROPHECY4F
program Bredenstein:2006ha ; Bredenstein:2006nk  .

Figure 5: Dependence of the one-loop corrected SMEFT width on various
$\mathcal{C}$ for $M_{H}=200~{}GeV$.

###  IV.2 $H\rightarrow Z\gamma$

The one-loop SMEFT results for $H\rightarrow
Z^{\mu}(p_{1})\gamma^{\nu}(p_{2})$ can be obtained in a straightforward manner
from the results of the previous section. At tree level, there is an SMEFT
contribution, while the SM contribution begins at one- loop. The Ward identity
for the photon requires $p^{\nu}\cdot{\cal{A}}_{\mu\nu}=0$, so the tensor
structure is fixed,

${\cal{A}}^{\mu\nu}_{Z\gamma}=\biggl{[}{\cal{A}}_{Z\gamma,EFT}^{0l}+{\cal{A}}_{Z\gamma,SM}^{1l}+{\cal{A}}_{Z\gamma,EFT}^{1l}\biggr{]}\biggl{(}g_{\mu\nu}-{p_{1}^{\nu}p_{2}^{\mu}\over
p_{1}\cdot p_{2}}\biggr{)}\,.$ (38)

The tree level SMEFT contribution is

${\cal{A}}_{Z\gamma,EFT}^{0l}=-2^{3/4}G_{\mu}^{-1/2}\biggl{(}{M_{H}^{2}-M_{Z}^{2}\over\Lambda^{2}}\biggr{)}c_{Z\gamma}\,,$
(39)

with

$c_{Z\gamma}\equiv{M_{W}\over M_{Z}}\sqrt{1-{M_{W}^{2}\over
M_{Z}^{2}}}\biggl{(}{\mathcal{C}}_{\phi W}-{\mathcal{C}}_{\phi
B}\biggr{)}+\frac{1}{2}\biggl{(}1-2{M_{W}^{2}\over
M_{Z}^{2}}\biggr{)}{\mathcal{C}}_{\phi WB}\,.$ (40)

Numerically, for $M_{H}=125~{}\rm GeV$ and $\Lambda=1~{}\rm TeV$,

${\cal{A}}_{Z\gamma,EFT}^{0l}=1.5~{}\rm GeV\biggl{(}{\mathcal{C}}_{\phi
W}-{\mathcal{C}}_{\phi B}-0.67{\mathcal{C}}_{\phi
WB}\biggr{)}\biggl{(}{1~{}\rm TeV\over\Lambda^{2}}\biggr{)}\,.$ (41)

The SM contribution is well-known,

$\displaystyle{\cal{A}}_{Z\gamma,{SM}}^{1l}$ $\displaystyle=$
$\displaystyle\biggl{(}{M_{H}^{2}-M_{Z}^{2}\over
2}\biggr{)}{M_{W}^{2}\over\pi^{2}}G_{\mu}^{3/2}2^{-1/4}\sqrt{1-{M_{W}^{2}\over
M_{Z}^{2}}}\biggl{\\{}\Sigma_{f}N_{c}{Q_{f}M_{Z}\over
M_{W}}v_{f}A_{1/2}(\tau_{f},\lambda_{f})$ (42)
$\displaystyle+A_{1}(\tau_{W},\lambda_{W})\biggr{\\}}$ $\displaystyle=$
$\displaystyle 0.209~{}\rm GeV\,{\hbox{for}}~{}M_{H}=125~{}\rm GeV\,,$

where the sum is over all fermions, $N_{c}=1(3)$ for leptons (quarks),
$\tau_{i}=4M_{i}^{2}/M_{H}^{2}$, $\lambda_{i}=4M_{i}^{2}/M_{Z}^{2}$,
$v_{f}=2T^{3}_{f}-4Q_{f}\biggl{(}1-{M_{W}^{2}\over M_{Z}^{2}}\biggr{)}$, and
analytic expressions for the functions $A_{1}$ and $A_{1/2}$ can be found in
Refs. Cahn:1978nz ; Bergstrom:1985hp ; Gunion:1989we

We report our one-loop SMEFT corrections555In the previous version of this
paper we calculated the corrections to the vacuum polarization using the light
quark masses instead of the more accurate $\Delta\alpha_{had}^{(5)}$. The
difference between the two calculations is of order percent and noticeable
only in the corrections proportional to $C_{\phi W}$, $C_{\phi B}$ and
$C_{\phi WB}$ since they contribute already at LO. We thank the authors of
Ref. Dedes:2019bew  for pointing out the numerical effect of
$\Delta\alpha_{\rm had}^{(5)}$ and for noticing a typo in Eqs. (40) and (41).
to the $H\rightarrow Z\gamma$ amplitude, $\mathcal{A}^{1l}_{Z\gamma}$ ,
numerically for $M_{H}=125~{}\rm GeV$,

$\displaystyle{\cal{A}}_{Z\gamma,EFT}^{1l}$ $\displaystyle=$
$\displaystyle\frac{(1TeV)^{2}}{\Lambda^{2}}\biggl{\\{}-0.038\,\mathcal{C}_{\phi
l}^{(3)}+0.00185(\mathcal{C}_{\phi q}^{(1)}-\mathcal{C}_{\phi
q}^{(3)}+\mathcal{C}_{\phi u})-0.0126\,\mathcal{C}_{\phi D}$ (43)
$\displaystyle+0.0127\,\mathcal{C}_{\phi\square}+0.000753\,\mathcal{C}_{u\phi}+0.019\,\mathcal{C}_{ll}+(0.0564-0.0362\log\frac{\Lambda^{2}}{M_{Z}^{2}})\mathcal{C}_{\phi
B}$
$\displaystyle+(-0.0502+0.0154\log\frac{\Lambda^{2}}{M_{Z}^{2}})\mathcal{C}_{\phi
W}+(-0.0201-0.0269\log\frac{\Lambda^{2}}{M_{Z}^{2}})\mathcal{C}_{\phi WB}$
$\displaystyle+(-0.00999+0.0042\log\frac{\Lambda^{2}}{M_{Z}^{2}})\mathcal{C}_{uB}+(0.0669-0.0295\log\frac{\Lambda^{2}}{M_{Z}^{2}})\mathcal{C}_{uW}$
$\displaystyle+(0.00559-0.0213\log\frac{\Lambda^{2}}{M_{Z}^{2}})\mathcal{C}_{W}\biggr{\\}}~{}\rm
GeV\,.$

An interesting feature of Eq. (43) is the dependence on coefficients not
arising at tree level.

The contribution from the SMEFT operators to $H\rightarrow Z\gamma$ at $1-$
loop is much smaller than the SM $1-$loop contribution, due to the
${v^{2}\over\Lambda^{2}}$ suppression. The contribution to the amplitude, Eq.
(43) is shown in Fig. 6

Figure 6:  Contribution to $H\rightarrow Z\gamma$ in the SMEFT. The vertical
axis is the ratio of the one-loop contribution of Eq. (43) to the SM one-loop
result of Eq. (42).

##  V Conclusions

We have computed the one-loop electroweak corrections in the SMEFT to the on-
shell $H\rightarrow ZZ$ process. Numerically, the logarithmic SMEFT
contributions dominate over the finite NLO contributions for most of parameter
space. Appendix B contains a numerical fit to the finite NLO SMEFT
contributions and Appendix C has analytic results for the logarithmic
contributions to $H\rightarrow ZZ$. Our complete result can be obtained from
the ancillary files posted with this paper and at
https://quark.phy.bnl.gov/Digital_Data_Archive/dawson/zzNLO_18. The
calculation of the on-shell decay is a first step towards a full NLO
calculation of the physical $H\rightarrow Zf{\overline{f}}$ process and our
results can be used to approximate a $K$ factor for the SMEFT $H\rightarrow
Zf{\overline{f}}$ decay.

The full $H\rightarrow Z\gamma$ SMEFT NLO result is presented as a by-product
of our calculation. Finally, the complete result for the one-loop SMEFT
renormalization of $G_{\mu}$ is given for the first time.

## Acknowledgements

We thank Giuseppe Degrassi for discussions. After the publication of this
paper, Ref. Dedes:2019bew  came out. We thank its authors, Athanasios Dedes,
Kristaq Suxho and Lampros Trifyllis, for noticing a typo in Eqs. (38) and
(39), and for pointing out the impact of $\Delta\alpha_{\rm had}^{(5)}$ in the
calculation of the NLO corrections to $\mathcal{C}_{\phi W},\mathcal{C}_{\phi
B}$ and $\mathcal{C}_{\phi WB}$. S.D. and P.P.G are supported by the U.S.
Department of Energy under Grant Contract de-sc0012704.

##  Appendix A Off-shell Production

The decay width for the off-shell decay, $H\rightarrow
f_{1}(p_{1})f_{2}(p_{2})Z(p_{3})$, is

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\int_{0}^{(M_{H}-M_{Z})^{2}}dq^{2}\int dm_{23}^{2}\,{\mid
A\mid^{2}\over 256\pi^{3}M_{H}^{3}}\,,$ (44)

where $m_{ij}=(p_{i}+p_{j})^{2}$, $m_{12}^{2}\equiv q^{2}$,and
$m_{12}^{2}+m_{23}^{2}+m_{13}^{2}=M_{H}^{2}+M_{Z}^{2}$,
$\lambda(M_{H}^{2},M_{Z}^{2},q^{2})\equiv
q^{4}-2q^{2}(M_{H}^{2}+M_{Z}^{2})+(M_{H}^{2}-M_{Z}^{2})^{2}$,
$m_{23}^{2}\mid_{max,min}\equiv{1\over
2}\biggl{(}M_{H}^{2}+M_{Z}^{2}-q^{2}\pm\sqrt{\lambda}\biggr{)}$. We write the
amplitude-squared to ${\cal O}({1\over\Lambda^{2}})$ in the SMEFT as,

$\displaystyle\mid A_{EFT}\mid^{2}$ $\displaystyle=$ $\displaystyle\mid
A_{SM}\mid^{2}+\mid\delta
A_{EFT}\mid^{2}+{\cal{O}}\biggl{(}\mathcal{C}^{2}{v^{4}\over\Lambda^{4}}\biggr{)}$
(45)

where,

$\displaystyle\mid A_{SM}\mid^{2}$ $\displaystyle=$ $\displaystyle
32\,(g_{L}^{\,2}+g_{R}^{2})\,G_{F}^{2}\,M_{Z}^{4}\biggl{[}{2M_{Z}^{2}q^{2}-m_{13}^{2}q^{2}-M_{H}^{2}M_{Z}^{2}+m_{13}^{2}M_{Z}^{2}+m_{13}^{2}\,M_{H}^{2}-m_{13}^{4}\over(q^{2}-M_{Z}^{2})^{2}+\Gamma_{Z}^{2}M_{Z}^{2}}\biggr{]}$
$\displaystyle\mid\delta A_{EFT}\mid^{2}$ $\displaystyle=$ $\displaystyle\mid
A_{SM}\mid^{2}{1\over\sqrt{2}G_{F}\Lambda^{2}}c_{k}$ (46)
$\displaystyle+64(g_{L}^{2}+g_{R}^{2})\frac{\sqrt{2}G_{F}}{\Lambda^{2}}M_{Z}^{4}{q^{2}(q^{2}+M_{Z}^{2}-M_{H}^{2})\over(q^{2}-M_{Z}^{2})^{2}+\Gamma_{Z}^{2}M_{Z}^{2}}\,c_{ZZ}\,.$

Note that we have not included effects due to possible anomalous $H\rightarrow
Z\gamma$ vertices here, although they are included in the results of Sec.
IV.2.

Integrating over $dm_{23}^{2}$,

$\displaystyle{d\Gamma\over dq^{2}}\mid_{SM}$ $\displaystyle=$
$\displaystyle(g_{L}^{\,2}+g_{R}^{2})\,G_{F}^{2}\,\sqrt{\lambda(M_{H}^{2},M_{Z}^{2},q^{2})}{M_{Z}^{4}\over
48\pi^{3}M_{H}^{3}}\biggl{[}{(12M_{Z}^{2}q^{2}+\lambda(M_{H}^{2},M_{Z}^{2},q^{2}))\over(q^{2}-M_{Z}^{2})^{2}+\Gamma_{Z}^{2}M_{Z}^{2}}\biggr{]}$
$\displaystyle{d\Gamma\over dq^{2}}\mid_{EFT}$ $\displaystyle=$
$\displaystyle{d\Gamma\over
dq^{2}}\mid_{SM}\biggl{\\{}1+{1\over\sqrt{2}G_{F}\Lambda^{2}}c_{k}\biggr{\\}}$
(47)
$\displaystyle+(g_{L}^{\,2}+g_{R}^{2})\,\frac{G_{F}}{\Lambda^{2}}\,\sqrt{\lambda(M_{H}^{2},M_{Z}^{2},q^{2})}{M_{Z}^{4}\over
2\sqrt{2}\pi^{3}M_{H}^{3}}$
$\displaystyle\cdot{q^{2}(3M_{Z}^{2}+M_{H}^{2})-(M_{Z}^{2}-M_{H}^{2})^{2}+\lambda(M_{H}^{2},M_{Z}^{2},q^{2})\over(q^{2}-M_{Z}^{2})^{2}+\Gamma_{Z}^{2}M_{Z}^{2}}c_{ZZ}\,.$

Finally, integrating over $q^{2}$ and setting $M_{H}=125~{}\rm GeV$,

$\displaystyle R^{off}$ $\displaystyle\equiv$
$\displaystyle{\int~{}dq^{2}~{}d\Gamma/dq^{2}\mid_{EFT}\over\int~{}dq^{2}~{}d\Gamma/dq^{2}_{SM}}$
(48) $\displaystyle\sim$ $\displaystyle
1+{1\over\sqrt{2}G_{F}\Lambda^{2}}\biggl{\\{}c_{k}-.97c_{ZZ}\biggr{\\}}$

This is in agreement with Refs. Brivio:2017vri ; Grinstein:2013vsa ;
Buchalla:2013mpa . Note that if we require a minimum $q^{2}\equiv q^{2}_{cut}$
, the result is altered. Define,

$\displaystyle R^{off}(q^{2}_{cut})$ $\displaystyle\equiv$
$\displaystyle{\int_{q^{2}_{cut}}^{(M_{H}-M_{Z})^{2}}~{}dq^{2}~{}d\Gamma/dq^{2}\mid_{EFT}\over\int_{q^{2}_{cut}}^{(M_{H}-M_{Z})^{2}}~{}dq^{2}~{}d\Gamma/dq^{2}_{SM}}$
(49) $\displaystyle=$ $\displaystyle
1+{1\over\sqrt{2}G_{F}\Lambda^{2}}\biggl{\\{}c_{k}+f(q_{2}^{cut})c_{ZZ}\biggr{\\}}\,.$

The effects of the $q^{2}_{cut}$ are shown in Fig. 7.

Figure 7: Dependence of the off-shell decay rate on the $q_{cut}$ defined in
Eq. 49.

Refs. Contino:2013kra ; Boselli:2017pef  note the the numerically large effect
of off-shell $Z$’s, which is seen clearly in the sensitivity to the
$q^{2}_{cut}$.

##  Appendix B Numerical values for the on-shell decay $H\rightarrow ZZ$

Numerical values for the $1-$loop SMEFT on-shell decay $H\rightarrow ZZ$ in
terms of the parameterization of Eq. 35 are given in Tables 1-3.

$\displaystyle\left(\begin{array}[]{cccccc}&a_{0}&a_{1}&a_{2}&a_{3}&a_{4}\\\
\text{SM}&-3.09\times 10^{-2}&-1.41\times 10^{-2}&5.43\times
10^{-4}&4.93\times 10^{-2}&1.57\times 10^{-2}\\\ \mathcal{C}_{W}&-3.08\times
10^{-3}&-8.96\times 10^{-5}&4.76\times 10^{-5}&1.43\times 10^{-4}&8.29\times
10^{-4}\\\ \mathcal{C}_{\phi}&-2.12\times 10^{-3}&-2.11\times
10^{-4}&6.59\times 10^{-6}&1.41\times 10^{-3}&1.2\times 10^{-4}\\\
\mathcal{C}_{\phi\square}&2.99\times 10^{-3}&2.11\times 10^{-4}&2.9\times
10^{-5}&-1.14\times 10^{-4}&1.23\times 10^{-3}\\\ \mathcal{C}_{\phi
D}&1.42\times 10^{-3}&-1.63\times 10^{-5}&7.09\times 10^{-6}&4.86\times
10^{-4}&1.51\times 10^{-4}\\\ \mathcal{C}_{u\phi}&1.56\times
10^{-3}&-6.49\times 10^{-6}&1.3\times 10^{-6}&3.29\times 10^{-5}&-4.65\times
10^{-4}\\\ \mathcal{C}_{\phi W}&1.14\times 10^{-2}&8.69\times
10^{-4}&-1.75\times 10^{-4}&-1.02\times 10^{-2}&-7.56\times 10^{-4}\\\
\mathcal{C}_{\phi B}&3.52\times 10^{-3}&2.64\times 10^{-5}&-8.25\times
10^{-5}&-2.02\times 10^{-3}&-5.46\times 10^{-4}\\\ \mathcal{C}_{\phi
WB}&5.88\times 10^{-3}&-1.63\times 10^{-3}&-5.93\times 10^{-5}&1.62\times
10^{-3}&-8.97\times 10^{-4}\\\ \mathcal{C}_{uW}&-1.72\times 10^{-3}&7.28\times
10^{-4}&-4.99\times 10^{-6}&-5.86\times 10^{-5}&1.36\times 10^{-4}\\\
\mathcal{C}_{uB}&9.2\times 10^{-4}&-3.9\times 10^{-4}&2.67\times
10^{-6}&3.14\times 10^{-5}&-7.3\times 10^{-5}\\\ \mathcal{C}_{\phi
l}^{(1)}&6.82\times 10^{-7}&0&0&0&0\\\ \mathcal{C}_{\phi l}^{(3)}&3.56\times
10^{-3}&2.56\times 10^{-3}&-9.87\times 10^{-5}&-8.97\times 10^{-3}&-2.85\times
10^{-3}\\\ \mathcal{C}_{\phi e}&6.26\times 10^{-5}&0&0&0&0\\\
\mathcal{C}_{\phi q}^{(1)}&4.45\times 10^{-3}&-2.23\times 10^{-5}&-9.49\times
10^{-7}&-1.64\times 10^{-5}&5.43\times 10^{-5}\\\ \mathcal{C}_{\phi
q}^{(3)}&-6.06\times 10^{-3}&2.23\times 10^{-5}&9.49\times 10^{-7}&1.64\times
10^{-5}&-5.43\times 10^{-5}\\\ \mathcal{C}_{\phi u}&-4.71\times
10^{-3}&5.2\times 10^{-5}&2.14\times 10^{-6}&3.67\times 10^{-5}&-1.2\times
10^{-4}\\\ \mathcal{C}_{\phi d}&6.26\times 10^{-5}&0&0&0&0\\\
\mathcal{C}_{ll}&-3.17\times 10^{-3}&-1.28\times 10^{-3}&4.93\times
10^{-5}&4.49\times 10^{-3}&1.43\times 10^{-3}\\\
\mathcal{C}_{lq}^{(3)}&8.9\times 10^{-4}&0&0&0&0\\\ \end{array}\right)$ (71)

Table 1: Numerical values for the coefficients defined in Eq. 35 relevant for
the on-shell process $H\rightarrow ZZ$.

$\displaystyle\left(\begin{array}[]{cccccc}&b_{0}&b_{1}&b_{2}&b_{3}&b_{4}\\\
\text{SM}&-2.68\times 10^{-2}&-6.16\times 10^{-3}&2.83\times
10^{-4}&2.49\times 10^{-2}&1.03\times 10^{-2}\\\ \mathcal{C}_{W}&-1.6\times
10^{-3}&-4.44\times 10^{-4}&1.62\times 10^{-5}&1.76\times 10^{-3}&3.44\times
10^{-4}\\\ \mathcal{C}_{\phi}&-2.04\times 10^{-3}&-1.88\times
10^{-4}&5.84\times 10^{-6}&1.26\times 10^{-3}&1.06\times 10^{-4}\\\
\mathcal{C}_{\phi\square}&3.2\times 10^{-3}&7.73\times 10^{-4}&1.15\times
10^{-5}&-1.74\times 10^{-3}&8.63\times 10^{-4}\\\ \mathcal{C}_{\phi
D}&1.51\times 10^{-3}&1.23\times 10^{-4}&3.13\times 10^{-6}&8.87\times
10^{-5}&4.19\times 10^{-5}\\\ \mathcal{C}_{u\phi}&1.52\times
10^{-3}&-7.6\times 10^{-6}&1.08\times 10^{-6}&2.86\times 10^{-5}&-4.49\times
10^{-4}\\\ \mathcal{C}_{\phi W}&6.37\times 10^{-3}&1.07\times
10^{-3}&-3.92\times 10^{-5}&-5.3\times 10^{-3}&-8.58\times 10^{-4}\\\
\mathcal{C}_{\phi B}&7.33\times 10^{-4}&9.47\times 10^{-5}&-3.39\times
10^{-6}&-4.85\times 10^{-4}&-7.1\times 10^{-5}\\\ \mathcal{C}_{\phi
WB}&8.71\times 10^{-4}&-1.19\times 10^{-4}&6.47\times 10^{-6}&1.17\times
10^{-5}&1.92\times 10^{-4}\\\ \mathcal{C}_{uW}&1.09\times 10^{-5}&-4.3\times
10^{-8}&-1.47\times 10^{-8}&-3.18\times 10^{-7}&1.48\times 10^{-6}\\\
\mathcal{C}_{uB}&-5.85\times 10^{-6}&2.3\times 10^{-8}&7.87\times
10^{-9}&1.7\times 10^{-7}&-7.93\times 10^{-7}\\\ \mathcal{C}_{\phi
l}^{(1)}&6.82\times 10^{-7}&0&0&0&0\\\ \mathcal{C}_{\phi l}^{(3)}&2.8\times
10^{-3}&1.12\times 10^{-3}&-5.14\times 10^{-5}&-4.52\times 10^{-3}&-1.88\times
10^{-3}\\\ \mathcal{C}_{\phi e}&6.26\times 10^{-5}&0&0&0&0\\\
\mathcal{C}_{\phi q}^{(1)}&4.36\times 10^{-3}&-3.33\times 10^{-5}&-1.75\times
10^{-6}&-3.11\times 10^{-5}&1.06\times 10^{-4}\\\ \mathcal{C}_{\phi
q}^{(3)}&-5.97\times 10^{-3}&3.33\times 10^{-5}&1.75\times 10^{-6}&3.11\times
10^{-5}&-1.06\times 10^{-4}\\\ \mathcal{C}_{\phi u}&-4.71\times
10^{-3}&3.33\times 10^{-5}&1.75\times 10^{-6}&3.11\times 10^{-5}&-1.06\times
10^{-4}\\\ \mathcal{C}_{\phi d}&6.26\times 10^{-5}&0&0&0&0\\\
\mathcal{C}_{ll}&-2.79\times 10^{-3}&-5.6\times 10^{-4}&2.57\times
10^{-5}&2.26\times 10^{-3}&9.4\times 10^{-4}\\\
\mathcal{C}_{lq}^{(3)}&8.9\times 10^{-4}&0&0&0&0\\\ \end{array}\right)$ (93)

Table 2: Numerical values for the coefficients defined in Eq. 35 relevant for
the on-shell process $H\rightarrow ZZ$.

##  Appendix C Analytical expressions of $\mathcal{F}_{g}$ and
$\mathcal{F}_{p}$

Here we report the explicit results for the coefficients $\mathcal{F}_{g}$ and
$\mathcal{F}_{p}$ introduced in eq. (34) are

$\displaystyle\mathcal{F}_{g}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\frac{1}{\Lambda^{2}}\biggl{(}\frac{12\sqrt[4]{2}\sqrt{G_{\mu}}\mathcal{C}_{W}M_{W}^{3}\left(M_{H}^{2}-2M_{Z}^{2}\right)\left(M_{Z}^{2}-6M_{W}^{2}\right)}{M_{Z}^{4}}$
(94) $\displaystyle+$
$\displaystyle\mathcal{C}_{\phi\square}(-6M_{H}^{2}-12M_{t}^{2}+9M_{W}^{2}-\frac{2M_{Z}^{2}}{3})+\frac{1}{12}\mathcal{C}_{\phi
D}\left(-9M_{H}^{2}-36M_{t}^{2}+8M_{W}^{2}-35M_{Z}^{2}\right)$
$\displaystyle+$ $\displaystyle\frac{\mathcal{C}_{\phi
W}M_{W}^{2}\left(M_{H}^{2}-2M_{Z}^{2}\right)\left(9M_{H}^{2}+18M_{t}^{2}-56M_{W}^{2}+3M_{Z}^{2}\right)}{3M_{Z}^{4}}$
$\displaystyle+$ $\displaystyle\frac{\mathcal{C}_{\phi
B}\left(M_{H}^{2}-2M_{Z}^{2}\right)\left(M_{Z}^{2}-M_{W}^{2}\right)\left(9M_{H}^{2}+18M_{t}^{2}-100M_{W}^{2}+85M_{Z}^{2}\right)}{3M_{Z}^{4}}$
$\displaystyle+$ $\displaystyle\frac{\mathcal{C}_{\phi
WB}M_{W}\left(M_{H}^{2}-2M_{Z}^{2}\right)\sqrt{M_{Z}^{2}-M_{W}^{2}}\left(3M_{H}^{2}+18M_{t}^{2}-42M_{W}^{2}+56M_{Z}^{2}\right)}{3M_{Z}^{4}}$
$\displaystyle+$
$\displaystyle\frac{2\sqrt{2}\,\mathcal{C}_{uW}M_{t}M_{W}\left(2M_{Z}^{2}-M_{H}^{2}\right)\left(8M_{W}^{2}-5M_{Z}^{2}\right)}{M_{Z}^{4}}$
$\displaystyle+$
$\displaystyle\frac{2\sqrt{2}\,\mathcal{C}_{uB}M_{t}\left(M_{H}^{2}-2M_{Z}^{2}\right)\left(8M_{W}^{2}-5M_{Z}^{2}\right)\sqrt{M_{Z}^{2}-M_{W}^{2}}}{M_{Z}^{4}}$
$\displaystyle+$ $\displaystyle\mathcal{C}_{\phi
l}^{(1)}\left(8M_{Z}^{2}-8M_{W}^{2}\right)+\mathcal{C}_{\phi
l}^{(3)}\left(6M_{t}^{2}-20M_{W}^{2}\right)+\mathcal{C}_{\phi
e}\left(8M_{Z}^{2}-8M_{W}^{2}\right)$ $\displaystyle+$
$\displaystyle\mathcal{C}_{\phi
q}^{(1)}\left(-12M_{t}^{2}+8M_{W}^{2}-8M_{Z}^{2}\right)+6\,\mathcal{C}_{\phi
q}^{(3)}\left(3M_{t}^{2}-4M_{W}^{2}\right)+4\,\mathcal{C}_{\phi
u}\left(3M_{t}^{2}+4M_{W}^{2}-4M_{Z}^{2}\right)$ $\displaystyle+$
$\displaystyle\mathcal{C}_{\phi
d}\left(8M_{Z}^{2}-8M_{W}^{2}\right)-3\,\mathcal{C}_{ll}M_{Z}^{2}-6\,\mathcal{C}_{lq}^{(3)}M_{t}^{2}\biggr{)}$

$\displaystyle\mathcal{F}_{p}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\frac{1}{\Lambda^{2}}\biggl{(}\mathcal{C}_{\phi\square}(-6M_{H}^{2}-12M_{t}^{2}+9M_{W}^{2}-\frac{2M_{Z}^{2}}{3})+\frac{1}{12}\mathcal{C}_{\phi
D}\left(-9M_{H}^{2}-36M_{t}^{2}+8M_{W}^{2}-35M_{Z}^{2}\right)$ (95)
$\displaystyle+$ $\displaystyle 8\,\mathcal{C}_{\phi
l}^{(1)}\left(M_{Z}^{2}-M_{W}^{2}\right)+\mathcal{C}_{\phi
l}^{(3)}\left(6M_{t}^{2}-20M_{W}^{2}\right)+8\,\mathcal{C}_{\phi
e}\left(M_{Z}^{2}-M_{W}^{2}\right)$ $\displaystyle+$
$\displaystyle\mathcal{C}_{\phi
q}^{(1)}\left(-12M_{t}^{2}+8M_{W}^{2}-8M_{Z}^{2}\right)+6\,\mathcal{C}_{\phi
q}^{(3)}\left(3M_{t}^{2}-4M_{W}^{2}\right)+4\,\mathcal{C}_{\phi
u}\left(3M_{t}^{2}+4M_{W}^{2}-4M_{Z}^{2}\right)$ $\displaystyle+$
$\displaystyle 8\,\mathcal{C}_{\phi
d}\left(M_{Z}^{2}-M_{W}^{2}\right)-3\,\mathcal{C}_{ll}M_{Z}^{2}-6\,\mathcal{C}_{lq}^{(3)}M_{t}^{2}\biggr{)}$

##  Appendix D Analytical expression for $\Delta r$

Here we report our result for $\Delta r$ in $R_{\xi}$ gauge:

$16\pi^{2}\Delta r=\frac{\sqrt{2}G_{\mu}}{2}\Delta
r_{\text{SM}}+\frac{1}{\Lambda^{2}}\Delta r_{\text{EFT}},$ (96)

where $\Delta r_{\text{SM}}$ and $\Delta r_{\text{EFT}}$ are

$\displaystyle\Delta r_{\text{SM}}$ $\displaystyle=$
$\displaystyle\left(\frac{6\,A_{0}\left(M_{W}^{2}\right)\left(M_{H}^{2}\left(M_{W}^{2}-2M_{Z}^{2}\right)+M_{W}^{2}\left(3M_{Z}^{2}-2M_{W}^{2}\right)\right)}{\left(M_{H}^{2}-M_{W}^{2}\right)\left(M_{W}^{2}-M_{Z}^{2}\right)}+4A_{0}\left(M_{W}^{2}\xi\right)+\frac{6M_{W}^{2}\,A_{0}\left(M_{H}^{2}\right)}{M_{H}^{2}-M_{W}^{2}}\right.$
(97) $\displaystyle-$
$\displaystyle\left.12\,A_{0}\left(M_{t}^{2}\right)+\frac{6\,A_{0}\left(M_{Z}^{2}\right)\left(2M_{W}^{2}-M_{Z}^{2}\right)}{M_{W}^{2}-M_{Z}^{2}}+2A_{0}\left(M_{Z}^{2}\xi\right)-M_{H}^{2}+6M_{t}^{2}-2M_{W}^{2}-M_{Z}^{2}\right)$

and

$\displaystyle\Delta r_{\text{EFT}}$ $\displaystyle=$
$\displaystyle\frac{1}{\hat{\epsilon}}\left(4\,\mathcal{C}_{\phi
l}^{(3)}\left(3M_{t}^{2}-4M_{W}^{2}\right)+24\,\mathcal{C}_{\phi
q}^{(3)}M_{W}^{2}+\frac{2}{3}\,\mathcal{C}_{\phi\square}M_{W}^{2}-6\,\mathcal{C}_{ll}M_{Z}^{2}-12\mathcal{C}_{lq}^{(3)}M_{t}^{2}\right)$
(98) $\displaystyle+$ $\displaystyle\Delta
r_{\text{SM}}\left(\frac{1}{4}\mathcal{C}_{\phi D}-\mathcal{C}_{\phi
l}^{(3)}+\frac{1}{2}\mathcal{C}_{ll}\right)+\frac{6M_{W}^{2}\left(\text{A}_{0}\left(M_{H}^{2}\right)-\text{A}_{0}\left(M_{W}^{2}\right)\right)-M_{H}^{4}+M_{W}^{4}}{M_{H}^{2}-M_{W}^{2}}\left(\mathcal{C}_{\phi\square}-\frac{1}{2}\mathcal{C}_{\phi
D}\right)$ $\displaystyle+$ $\displaystyle
6\left(M_{t}^{2}-2\text{A}_{0}\left(M_{t}^{2}\right)\right)\left(\mathcal{C}_{\phi
l}^{(3)}+\mathcal{C}_{\phi
q}^{(3)}-\mathcal{C}_{lq}^{(3)}-\frac{1}{4}\mathcal{C}_{\phi
D}\right)+6\text{A}_{0}\left(M_{Z}^{2}\right)\left(\mathcal{C}_{ll}+c_{W}^{2}\left(4C_{\phi
l}^{(3)}-\mathcal{C}_{\phi D}\right)\right)$ $\displaystyle-$ $\displaystyle
12M_{Z}^{2}\left(c_{W}s_{W}\mathcal{C}_{\phi
WB}-\frac{5}{3}c_{W}^{2}\mathcal{C}_{\phi
l}^{(3)}-\frac{5}{12}\mathcal{C}_{ll}-\frac{1}{6}\mathcal{C}_{\phi D}\right)$
$\displaystyle+$ $\displaystyle
12\left(\text{A}_{0}\left(M_{W}^{2}\right)-c_{W}^{2}\text{A}_{0}\left(M_{Z}^{2}\right)\right)\left(\mathcal{C}_{ll}+\mathcal{C}_{\phi
l}^{(3)}+\mathcal{C}_{\phi l}^{(1)}\right)$ $\displaystyle-$
$\displaystyle\left(\text{A}_{0}\left(\xi
M_{W}^{2}\right)+\text{A}_{0}\left(\xi
M_{Z}^{2}\right)+2\left(M_{W}^{2}+M_{Z}^{2}\right)\right)\mathcal{C}_{\phi
D},$

and as usual $c_{W}=\frac{M_{W}^{2}}{M_{Z}^{2}}$, $s_{W}=\sqrt{1-c_{W}^{2}}$
and

$A_{0}(x)=-i\frac{(2\pi\Lambda)^{2\epsilon}}{\pi^{2}}\int\frac{d^{d}k_{1}}{(k_{1}^{2}-x)}=\frac{x}{\hat{\epsilon}}+x(1-\log(\frac{x}{\Lambda^{2}}))$
(99)

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