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Higgs decays to two leptons and a photon beyond leading order in the SMEFT

T. Corbett1*, T. Rasmussen2

1 Niels Bohr International Academy,

Niels Bohr Institute, University of Copenhagen,

Blegdamsvej 17, DK-2100, Copenhagen, Denmark

*<EMAIL_ADDRESS>
## Abstract

We present the three-body decay of the Higgs boson into two leptons and a
photon to dimension-eight in the Standard Model Effective Field Theory
(SMEFT). In order to obtain this result we interfere the full one-loop
Standard Model result with the tree-level result in the SMEFT. This is the
first calculation of the partial width of the Higgs boson into two leptons and
a photon in the SMEFT to incorporate the full one-loop dependence for the
Standard Model as well as the full tree level dimension-eight dependence in
the SMEFT. We find that this channel can aid in distinguishing strongly
interacting and weakly interacting UV completions of the SMEFT under standard
assumptions. We also find that this channel presents the opportunity to
distinguish different operator Classes within the SMEFT, potentially including
contact $H\bar{\ell}\ell\gamma$ operators which are first generated only at
dimension-eight in the SMEFT.

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Higgs decays to two leptons and a photon in the SM
    1. 3.1 Tree level decay
    2. 3.2 One loop decay
  4. 4 SMEFT contributions
    1. 4.1 SMEFT amplitudes and Dalitz plots
    2. 4.2 Parameterized partial widths
  5. 5 Conclusions
  6. A Feynman rules for the effective action
  7. B Feynman rules in the SMEFT
  8. C Matching to the effective action
  9. D Full Parameterized partial widths
    1. D.1 Full $\alpha$–scheme results
    2. D.2 $M_{W}$ scheme results
  10. E Discussion on the tree-loop interference in the SM
  11. F Dipole operators

##  1 Introduction

The coupling of the Higgs boson to two photons or a $Z$–boson and photon in
the Standard Model (SM) were developed in [1, 2, 3]. The decay
$H\to\bar{\ell}\ell\gamma$ in the SM has more recently been considered in [4,
5, 6, 7, 8, 9, 10, 11, 12]. In the case of the decay $H\to Z\gamma$ the narrow
width approximation is typically employed for the $Z$ boson. Relaxing this
assumption is expected to have a negligible effect on the overall prediction
for the Higgs boson decay to two leptons and a photon.

The Standard Model Effective Field Theory (SMEFT) is a useful tool for
studying the effects of heavy New Physics (NP) in a generally model
independent fashion. The primary assumptions of this methodology are that the
NP is too heavy to produce directly at the energies considered and that the
Higgs boson belongs to an $SU(2)_{L}$ doublet. Under these assumptions the
SMEFT is formed as a tower of operators suppressed by the mass scale of the
heavy new physics $\Lambda$:

$\displaystyle\mathcal{L}_{\rm SMEFT}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\rm
SM}+\sum_{i}^{\infty}\sum_{j}\frac{1}{\Lambda^{i}}\mathcal{Q}_{j}^{(4+i)}\,.$
(1)

Operators of dimension $(4+i)$ are suppressed by the heavy scale $\Lambda^{i}$
and generally, therefore, higher dimension operators are treated as higher
order corrections in the SMEFT approach. The sum over $j$ is over all operator
forms at a given order $i$. For LHC studies odd-dimension operators are
generally negligible as they are $B$ and/or $L$ violating and therefore
expected to be strongly suppressed. The SMEFT at dimension-six has been a
major area of investigation, particularly since the discovery of the Higgs
boson. Recently much interest has been generated around how dimension-eight
operators impact studies of the SMEFT [13, 14, 15, 16, 17, 18, 19] and how
they change our understanding of truncation error in the SMEFT [20, 21, 22,
23].

In the context of the SMEFT the present work is the first work to have studied
the effect of interfering the non-resonant SM contribution for
$H\to\bar{\ell}\ell\gamma$ with the SMEFT contributions. The SMEFT tree-level
interference with the resonant SM contributions have been used across the
field when performing global fits. One-loop studies with on-shell $\gamma$ and
$Z$ were made in [24, 25] for $HZ\gamma$ and [26, 27, 28, 29] for
$H\gamma\gamma$.

In [30], the authors calculate $H\to 4\psi$ in the SMEFT and found the many
novel couplings generated in the SMEFT could allow for larger corrections to
the typically employed narrow width calculation than anticipated. Indeed,
couplings of the Higgs which do not occur in the SM at tree level, like
$H\gamma\gamma$, $Hgg$, and $H\bar{\psi}\psi\gamma$ result in contributions to
the partial width that are entirely neglected in the narrow width
approximation. Taking from this, this work seeks to analyze the implications
of allowing for off-shell $Z$ or $\gamma$ in the decay of the Higgs boson to
two fermions and a photon.

This article is organized as follows: In Section 2 we lay out our notation and
in Sec. 3 we discuss the SM tree-level and one-loop contributions to the
process $H\to\bar{\ell}\ell\gamma$. In Sec. 4 we derive the contributions in
the SMEFT up to and including $1/\Lambda^{4}$ effects, this includes the
consistent calculation of dimension-six-squared contributions as well as
contributions from dimension-eight operators. In this section we also look at
different cuts in the invariant mass of the dilepton system and how they can
be used to potentially distinguish operators generated by tree- and loop-
processes in the UV, as well as distinguish between different classes of
operators in the SMEFT. In the conclusions, Section 5, we summarize this work
and briefly mention future perspectives.

Included are many appendices: the first, App. A lays out the Feynman rules for
the terms in the one-loop effective action in the SMEFT that contribute to the
partial width, App. B lays out the necessary Feynman rules in the SMEFT for
deriving $H\to\bar{\ell}\ell\gamma$ up to $\frac{1}{\Lambda^{4}}$, App. C
outlines and briefly discusses results of matching onto the one-loop effective
action in the SM, and App. D contains results from the main text in the
$M_{W}$ input parameter scheme. Appendix E discusses the approximately
vanishing contribution from the tree-loop interference in the SM while App. F
discusses the often neglect dipole operator contribution.

Also included in the ancillary files are Mathematica notebooks which derive
the one-loop contributions to the SM process, including demonstrating how the
one-loop effective action can be derived and employed in performing
calculations of this sort.

##  2 Notation

In this article we follow the notation laid out in [31, 32, 33, 34]. The
geoSMEFT [32] methodology is used to obtain most of the relevant vertices in
the SMEFT power counting to order $\frac{1}{\Lambda^{4}}$. Certain operators
at dimension eight have not yet been written in the geoSMEFT, here we use
those laid out in [35]. An alternative source of the dimension-eight operator
basis can be found in [36]. We only lay out the conventions relevant to this
work, further details can be found in the works cited above.

The couplings to the fermions coming from the covariant derivative are
written:

$\mathcal{L}_{\psi}=\bar{\psi}\gamma^{\mu}\left[\partial_{\mu}+i\bar{g}_{3}G_{\mu}^{A}T^{A}+i\frac{\bar{g}_{2}}{\sqrt{2}}(W^{+}_{\mu}T^{+}+W^{-}_{\mu}T^{-})+i\bar{g}_{Z}(T_{3}-s_{z}^{2}Q_{\psi})Z_{\mu}+iQ_{\psi}\bar{e}A^{\mu}\right]\psi$
(2)

All fields appearing in the above equations are mass eigenstate fields.
$T^{\pm},T_{3}$ are the generators of $SU(2)_{L}$ written in the charged basis
and $Q_{\psi}$ is the charge of the fermion. The barred quantities include
corrections from the SMEFT and are defined to all orders in the SMEFT in [32].
Barred quantities such as $\bar{g}_{2}$ can be expanded in terms of Wilson
coefficients and the unbarred gauge couplings as:

$\displaystyle\bar{g}_{2}$ $\displaystyle=$ $\displaystyle
g_{2}\left(1+2c_{HW}v^{2}+\frac{1}{2}c_{HW}^{(8)}v^{4}+\frac{3}{2}c_{HW}^{2}v^{4}\right)$
(3)

The expression for $\bar{g}_{Z}$ is too cumbersome to write to dimension eight
in text. In this article $v$ is the vacuum expectation value that minimizes
the Higgs potential in the SMEFT (frequently written as $\bar{v}_{T}$ in the
literature). The ancillary files contain an updated version of the Feynrules
model defined in [33] and can be used to derive the full expression. To
dimension six $\bar{g}_{Z}$ is given by:

$\bar{g}_{Z}=\frac{g_{1}^{2}+g_{2}^{2}}{\sqrt{g_{1}^{2}+g_{2}^{2}}}\left[1+\frac{1}{g_{1}^{2}+g_{2}^{2}}\left(g_{1}^{2}c_{HB}v^{2}+g_{2}^{2}c_{HW}v^{2}+g_{1}g_{2}c_{HWB}v^{2}\right)\right]$
(4)

The quantity $\bar{g}_{3}$ is not relevant to this work and so we do not
define it. The new mixing angle, $s_{z}$, takes into account new mixing
effects at dimension-eight. It is equivalent to the barred Weinberg angle at
dimension six:

$\bar{s}_{W}^{2}=\frac{g_{1}^{2}}{g_{1}^{2}+g_{2}^{2}}\left[1+\frac{1}{g_{1}^{2}+g_{2}^{2}}\left(2g_{2}^{2}c_{HB}v^{2}-2g_{2}^{2}c_{HW}v^{2}+\frac{g_{2}}{g_{1}}(g_{2}^{2}-g_{1}^{2})c_{HWB}v^{2}\right)\right]\,.$
(5)

The dimension-eight contributions to $s_{z}$ and $\bar{s}_{W}$ are also too
cumbersome to fit neatly in print form, but can be derived using the Feynrules
package [33] or can be looked up in the extensive Appendix of [13].

The coupling of the Higgs boson to fermions is given by:

$\mathcal{L}_{\rm Yukawa}=\left[-H[Y_{\psi}]^{\dagger}+H\left(c_{\psi
H}^{(6)}H^{\dagger}H+c_{\psi
H}^{(8)}(H^{\dagger}H)^{2}\right)\right]\Psi_{L}\psi_{R}$ (6)

$\Psi_{L}$ represents a fermionic left-handed doublet and $\psi_{R}$ a
fermionic right-handed $SU(2)_{L}$ singlet. Neglecting goldstone bosons for
simplicity we can express the doublet in terms of the Higgs mass eigenstate
$h$ as:

$\displaystyle H$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}0\\\
\frac{v+c_{H,{\rm kin}}h}{\sqrt{2}}\end{array}\right)$ (9) $\displaystyle
c_{H,{\rm kin}}$ $\displaystyle=$ $\displaystyle
1+\frac{1}{4}(c_{HD}-4c_{H\Box})v^{2}-\frac{1}{32}(4c_{HD}^{(8)}+4c_{HD,2}^{(8)}-3[c_{HD}-4c_{H\Box}]^{2})v^{4}$
(10)

Expanding Eq. 6 defines the tree-level masses of the fermions and their
relation to the SM masses:

$\hat{m}_{\psi}=\frac{v}{\sqrt{2}}\left[Y_{\psi}-\frac{v^{2}}{2}c_{\psi
H}-\frac{v^{4}}{4}c_{\psi H}^{(8)}\right]$ (11)

In the next section we perform calculations to one-loop in the SM. As such we
only include the SM dependence. Again we neglect goldstone bosons in this
discussion. The one-loop calculations below are in general $R_{\xi}$ gauge and
therefore do include goldstone boson dependence. The coupling of the Higgs
boson to gauge bosons and the relevant triple gauge couplings come from the
Higgs gauge kinetic term and the gauge boson kinetic terms of the SM:

$\displaystyle\mathcal{L}_{HVV}$ $\displaystyle=$
$\displaystyle(D^{\mu}H)^{\dagger}(D_{\mu}H)-\frac{1}{4}W^{A,\mu\nu}W^{A}_{\mu\nu}-\frac{1}{4}B^{\mu\nu}B_{\mu\nu}$
(12)

Where $W^{A,\mu\nu}$ and $B^{\mu\nu}$ are the field strength tensors of the
$SU(2)_{L}$ and hypercharge gauge fields respectively.

##  3 Higgs decays to two leptons and a photon in the SM

###  3.1 Tree level decay

In the Standard Model the Higgs boson decays to two leptons via the Yukawa
coupling, relative to the mass scale of the Higgs this is typically a very
small coupling, but can be relevant for the $\mu$ and $\tau$. Taking the
lepton mass to be zero, but keeping the Yukawa coupling nonzero, the decay
width of a Higgs boson to two leptons and a photon at tree level is given by:

$\Gamma_{h\to\bar{\ell}\ell\gamma}^{(0)}=\frac{8\pi^{2}}{64\bar{m}_{H}^{3}(2\pi)^{5}}\int
ds_{1}ds_{2}\Theta(G[s_{1},s_{2},\bar{m}_{H}^{2},\hat{m}_{\ell}^{2},0,\hat{m}_{\ell}^{2}]\leq
0)\frac{4\bar{e}^{2}\hat{m}_{\ell}^{2}}{v^{2}}\frac{\left(s_{2}^{2}+\bar{m}_{H}^{4}\right)}{s_{1}(s_{1}+s_{2}-\bar{m}_{H}^{2})}\,,$
(13)

where $\Theta$ denotes the unit step function, the superscript “(0)” indicates
this is the tree level result, and $G=0$ defines the boundary of integration
and is given by [37]:

$G[x,y,z,u,v,w]=-\frac{1}{2}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&v&x&z\\\ 1&v&0&u&y\\\ 1&x&u&0&w\\\ 1&z&y&w&0\end{array}\right|$ (14)

The kinematic invariants are defined as $s_{1}=(k_{\psi}+k_{\gamma})^{2}$ and
$s_{2}=(k_{\psi}+k_{\bar{\psi}})^{2}$. This integral contains an IR divergence
which we regulate by requiring the photon energy be above 5 GeV following the
approach of [5]. Adopting the input parameters given in Table 1 we find the
tree level results:

$\begin{array}[]{lcll}\Gamma^{(0)}_{h\to e^{+}e^{-}\gamma}&=&4.13\cdot
10^{-12}\,{\rm GeV}&(4.09\cdot 10^{-12}\,{\rm GeV})\,,\\\\[3.0pt]
\Gamma^{(0)}_{h\to\mu^{+}\mu^{-}\gamma}&=&1.23\cdot 10^{-7}\,{\rm
GeV}&(1.18\cdot 10^{-7}\,{\rm GeV})\,,\\\\[3.0pt]
\Gamma^{(0)}_{h\to\tau^{+}\tau^{-}\gamma}&=&2.10\cdot 10^{-5}\,{\rm
GeV}&(2.05\cdot 10^{-5}\,{\rm GeV})\,,\end{array}$ (15)

where the first result denotes the use of the $\hat{\alpha}$ input parameter
scheme and the result in parenthesis corresponds to the $\hat{m}_{W}$ scheme
[38, 39]. Hatted quantities correspond to input parameters while barred
quantities are the Lagrangian parameters. Therefore, for example,
$\bar{m}_{W}$ is determined from $\hat{\alpha}$ in the $\hat{\alpha}$ scheme,
while it corresponds to $\hat{m}_{W}$ in the $\hat{m}_{W}$ scheme. These
results include the full lepton mass dependence. Comparing with the leading
order results111Here in the mass expansion we retain the fermion mass
dependence in the denominators as this leads to quicker numerical convergence
of the integrals. this results in an approximately $+10\%$ correction for
taus, a $+4\%$ correction for muons, and a $-14\%$ correction for electrons.
These corrections do not scale as $\hat{m}_{\ell}/\bar{m}_{H}$, this is due to
the photon mass cut off. This behavior can be understood from $E_{\gamma}$
dependence shown in Figure 4 of [11]. The error in the phase-space
integrations is approximately 5 per mil. Phase space integrations were
performed using the Vegas routine from the Cuba library [40].

Input parameters | Value | Ref.
---|---|---
$\hat{m}_{Z}$ [GeV] | $91.1876(21)$ | [41]
$\hat{m}_{W}$ [GeV] | $80.387(16)$ | [42]
$\hat{m}_{h}$ [GeV] | $125.10(14)$ | [41]
$\hat{m}_{t}$ [GeV] | $172.4(7)$ | [41]
$\hat{m}_{e}$ [MeV] | $0.51099895000(15)$ | [41]
$\hat{m}_{\mu}$ [MeV] | $105.6583745(24)$ | [41]
$\hat{m}_{\tau}$ [GeV] | $1.77686(12)$ | [43]
$\hat{G}_{F}$ [GeV-2] | 1.1663787 $\cdot 10^{-5}$ | [43, 44]
$\hat{\alpha}_{EW}$ | 1/137.035999084(21) | [41]
$\Delta\alpha$ | $0.0590\pm 0.0005$ | [45]
$m_{W}^{\hat{\alpha}}$ [GeV] | $80.36\pm 0.01$ | –
$\Delta\alpha^{\hat{m}_{W}}$ | $0.0576\pm 0.0008$ | –
$\Gamma_{Z}^{\hat{\alpha}}$ |  $2494.4\pm 0.7$ MeV | –
$\Gamma_{Z}^{\hat{m}_{W}}$ |  $2495.7\pm 1.0$ MeV | –
Table 1: Input parameter values used in this work, the last two lines are
$m_{W}$ as predicted in the $\alpha$ scheme and $\Delta\alpha$ as predicted in
the $m_{W}$ scheme. The $Z$ widths are for the SM using the values given
above. Table taken from [20], where further details can be found. Note that
the fermion mass is always used as the input parameter.

###  3.2 One loop decay

As a result of the smallness of the lepton masses compared with the Higgs
boson mass scale we expect the chirally suppressed tree level terms to be of
the same order or smaller than the one-loop contributions. The leading term in
the interference of the tree amplitude with the loop amplitude is proportional
to $m_{\ell}^{2}/(16\pi^{2})$, because of the smallness of the lepton mass we
assume:

$\displaystyle\frac{m_{\ell}^{2}}{16\pi^{2}}$ $\displaystyle\to$
$\displaystyle 0\,.$ (16)

This is further justified in App. E.

In order to perform the relevant SM loop calculations we employ the background
field method (BFM) [46], and use the conventions and Feynman rules presented
in [33]. This choice is convenient as it reduces the number of diagrams
contributing to each process as well as preserving the naive Ward
identities[46, 47], thereby greatly simplifying many expressions. In the BFM
the bosonic fields are doubled as $\phi\to\phi+\hat{\phi}$, but only the
quantum gauge fields (corresponding to the unhatted fields) are gauge fixed.
As fermionic fields are not a part of the gauge fixing procedure their fields
are not doubled, as such the Feynman rules involving the fermions are the same
in the BFM and in traditional gauge fixing. In what follows the hatted fields,
$\hat{\phi}$, correspond to background fields. One-loop contributions to the
effective action are formed of loops with external background fields and
internal quantum fields. It can be shown that the effective action in the BFM
is equivalent to that in the traditional gauge fixing [48]. The effective
action is the generating functional of one-particle irreducible diagrams, once
derived the background fields can be gauge fixed independent of the choice of
gauge fixing for the quantum fields and the S-matrix elements can be derived.
Using unitary gauge for the background fields has the added benefit of
removing goldstone boson contributions from one-particle reducible diagrams
formed from the effective action.

In adopting the BFM we first construct the relevant contributions to the one-
loop effective action which contribute to the one-loop couplings
$\hat{h}\hat{\gamma}\hat{\gamma}$ and $\hat{h}\hat{Z}\hat{\gamma}$. We then
form the one-particle reducible diagrams contributing to the process
$\hat{h}\to\bar{\ell}\ell\hat{\gamma}$. For simplicity we do not match the
one-loop diagrams involving external fermions to the effective action, this is
possible as the Feynman rules involving fermions are the same in the
background field method as in standard $R_{\xi}$ gauge fixing. The relevant
diagrams contributing to the effective action at one loop are shown in Fig. 1.
Figure 2 shows diagrams which do not contribute in the background field method
– both diagrams vanish identically as a result of the Ward identities [46,
47]222In the case of the $\hat{Z}\hat{\gamma}$ mixing, this diagram vanishes
only for on-shell photon..

These diagrams can be matched onto the one-loop effective action given by:

$\displaystyle\mathcal{L}_{1-{\rm loop}}$ $\displaystyle=$ $\displaystyle
f_{H\gamma\gamma}^{(1)}(k_{1},k_{2})\hat{h}\hat{F}_{\mu\nu}\hat{F}^{\mu\nu}+f_{H\gamma\gamma}^{(2)}(k_{1},k_{2})\hat{h}(\partial^{\mu}\hat{F}_{\mu\nu})(\partial_{\sigma}\hat{F}^{\sigma\nu})$
$\displaystyle+f_{HZ\gamma}^{(1)}(k_{1},k_{2})\hat{h}\hat{F}_{\mu\nu}\hat{Z}^{\mu\nu}+f_{HZ\gamma}^{(2)}(k_{1},k_{2})\hat{h}(\partial^{\mu}\hat{F}_{\mu\nu})\partial_{\sigma}\partial^{\nu}\hat{Z}^{\sigma}$
$\displaystyle+f_{HZ\gamma}^{(3)}(k_{1},k_{2})\hat{h}\hat{F}_{\mu\nu}\Box\hat{Z}^{\nu}\,,$

where $\hat{F}_{\mu\nu}$ and $\hat{Z}_{\mu\nu}$ are the field strength of the
photon and $Z$-boson respectively. We note that, as we are matching below
electroweak symmetry breaking (EWSB), the appearance of the massive gauge
boson $Z$ is not restricted by gauge symmetry. As such the operator form
corresponding to $f_{H\gamma\gamma}^{(2)}$ is split between two
$\hat{h}\hat{Z}\hat{\gamma}$ operators. The $f$ are form factors coming from
matching the one-loop calculations in the background field method onto the
effective action. The $f(k_{1},k_{2})$ indicates that these form factors
depend on the momenta of the external bosons, which by conservation of
momentum can be chosen to be the momenta of the photons or photon and
$Z$-boson.

The Feynman rules for these effective operators are given in Appendix A. We
note that in both cases the operators corresponding to the form factors
$f^{(2)}$ and $f^{(3)}$ give vanishing contributions when one-particle
reducible diagrams are formed and the lepton masses are taken to be zero.

(80,60) v1 v2,v3 dashes,tension=3v1,o1 photon,tension=3o2,v2
photon,tension=3o2,v3 plain,lefto1,o2,o1

(a)

(80,60) v1 v2,v3 dashes,tension=3v1,o1 photon,tension=3o2,v2
photon,tension=3o3,v3 plaino1,o2,o3,o1

(b)

(80,60) v1 v2,v3 dashes,tension=3v1,o1 photon,tension=3o2,v2
photon,tension=3o3,v3 plaino1,o2,o3 dotso3,o1

(c)

Figure 1: Standard Model diagrams contributing to Higgs boson decays to two
photons and a $Z$ boson and photon. In (1(a)) and (1(b)) the solid line
represents the contribution of either $W$–bosons, the charged goldstone
bosons, charged ghosts, or fermions. In the case of the $HZ\gamma$ coupling
loops can be composed of one goldstone boson and two $W$–bosons or two
goldstone bosons and one $W$. This is illustrated in Figure (1(c)) where the
solid line may represent $W$–bosons while the dotted line represents goldstone
bosons, or vice versa.

(80,40) v1 v2 dashes,tension=3v1,o1 photon,tension=3o1,v2
decoration.shape=circle,decoration.filled=shaded,decoration.size=25o1
label=$\hat{\phi}^{0}$v1 label=$\hat{\gamma}$v2

(a)

(80,40) v1 v2 photon,tension=3v1,o1 photon,tension=3o1,v2
decoration.shape=circle,decoration.filled=shaded,decoration.size=25o1
label=$\hat{Z}$v1 label=$\hat{\gamma}$v2

(b)

Figure 2: Standard Model diagrams contributing to Higgs boson decays to two
photons which vanish identically. These diagrams, if nonzero, would contribute
to one-particle reducible diagrams where the $\hat{\phi}^{0}$ or $\hat{Z}$ are
contracted with a $\hat{H}\hat{\phi}^{0}\hat{Z}$ or $\hat{H}\hat{Z}\hat{Z}$
vertex.

Employing the Feynman rules of [33] and using Package-X [49], we have derived
the form factors of Eq. 3.2. We have confirmed that the $f^{(1)}$ are gauge
invariant and the $f^{(2)}$ vanish identically for on-shell external
particles. The analytic results for the form factors are reproduced in
simplifying limits in Appendix C: The $H\gamma\gamma$ operators are reproduced
for on-shell Higgs momenta and in the limit of small photon off-shell momenta
and arbitrary gauge parameter. This serves two purposes: the first is to
demonstrate that the gauge parameter dependence vanishes identically in the
limit the photon momenta vanishes as well as to show the leading off-shell
dependence of the form factors. It should be noted that for the calculation of
the process $H\to\bar{\ell}\ell\gamma$ the off-shell photon is allowed to have
a momentum-squared up to the Higgs mass, and the limit in the Appendix is not
relevant. In the case of the coupling $HZ\gamma$ we have set $\xi=1$ in order
to have more compact expressions. In this case we again find that the results
are gauge parameter invariant for on-shell external particles. In what follows
we use the full momentum dependence in order to allow intermediate particles
to go arbitrarily off shell.

In order to form the amplitude for $H\to\bar{\ell}\ell$ we must decay the off-
shell photon and $Z$ to two leptons. Gauge invariance also demands we add the
box diagrams and one-loop decays of the Higgs boson to two fermions where one
fermion radiates a photon. These diagrams can be found in Figure 3. We assume
that lepton Yukawa couplings as well as masses are zero in the loop diagrams,
this removes the contribution of goldstone bosons in the loops as well as the
loops where the lepton couples directly to the Higgs boson.

(80,60) h2,v1,h3 v2,v3,v4 dashes,tension=3v1,o1 photono2,o1 photonv2,o1
fermionv3,o2,v4 phantomv4,o1 phantom,tension=.3h2,o1
decoration.shape=square,decoration.filled=full,decoration.size=8o1

(a)

(80,60) v1 v2,v3,v4,v5 dashes,tension=2v1,o1 plain,tension=5i2,v5
plain,tension=5i1,v2 fermion,tension=2i2,o2 plaino2,o3 fermion,tension=2o3,i1
photono3,o1 photono1,o2 photon,tension=0v3,i1

(b)

(80,60) v1 v2,v3,v4 dashes,tension=2v1,o1 plain,tension=5i2,v4
plain,tension=5i1,v2 fermion,tension=2i2,o2 plaino2,i3,o3
fermion,tension=2o3,i1 photono3,o1 photono1,o2 photon,tension=0v3,i3

(c)

Figure 3: Diagrams contributing to the process $H\to\bar{\ell}\ell\gamma$. In
(3(a)) the square vertex represents an insertion of the one-loop effective
action coupling to two photons or a $Z$ and photon. In (3(b)) and (3(c)) the
Higgs boson decays to two fermions via a loop containing two $W$s or two $Z$s,
a photon is then either radiated by an external or the internal fermion line.

The one-loop amplitudes can be parameterized as:

$\displaystyle i\mathcal{M}$ $\displaystyle=$
$\displaystyle(c_{1L}\eta^{\mu\nu}+c_{2L}k_{2}^{\mu}k_{1}^{\nu}+c^{\prime}_{2L}k_{3}^{\mu}k_{1}^{\nu})\,\bar{u}_{k_{2}}\gamma_{\nu}P_{L}v_{k_{3}}\epsilon^{*}_{\mu}(k_{1})$
(18)
$\displaystyle+(c_{1R}\eta^{\mu\nu}+c_{2R}k_{2}^{\mu}k_{1}^{\nu}+c^{\prime}_{2R}k_{3}^{\mu}k_{1}^{\nu})\,\bar{u}_{k_{2}}\gamma_{\nu}P_{R}v_{k_{3}}\epsilon^{*}_{\mu}(k_{1})\,,$

where the $c_{i}$ parameterize the loop contributions, $\bar{u}_{k_{2}}$ is
the barred on-shell spinor for the lepton with momentum $k_{2}$, $v_{k_{3}}$
is the on-shell spinor for the anti-lepton with momentum $k_{3}$, and
$\epsilon^{*}_{\mu}(k_{1})$ is the on-shell polarization vector for the photon
with momentum $k_{1}$. The diagrams of Fig. 3 generate all $c_{i}$ when an
intermediate photon or $Z$ boson is involved. Diagrams (3(b)) and (3(c)) for
intermediate $W$s only generate the left handed $c_{i}$. We find that the sum
of diagrams (3(b)) and (3(c)) for intermediate $Z$ bosons are gauge invariant
on their own. As the $W$s don’t generate right-handed couplings this also
implies the right-handed components of the sum of the diagrams of (3(a)), when
including both the intermediate $Z$ and photon, are gauge invariant. The left
handed components of the triangle diagrams require the addition of diagrams
(3(b)) and (3(c)) with intermediate $W$s to be gauge invariant.

After obtaining the squared amplitudes corresponding to the one-loop results
we find the following partial widths (although the one-loop process does not
contain an IR divergence333See the discussion in [11] below Eq. 2.8 we
maintain the requirement that the photon energy be greater than 5 GeV):

$\begin{array}[]{lcll}\Gamma^{(1)}_{h\to e^{+}e^{-}\gamma}&=&5.70\cdot
10^{-7}\,{\rm GeV}&(5.40\cdot 10^{-7}\,{\rm GeV})\,,\\\\[3.0pt]
\Gamma^{(1)}_{h\to\mu^{+}\mu^{-}\gamma}&=&3.73\cdot 10^{-7}\,{\rm
GeV}&(3.88\cdot 10^{-7}\,{\rm GeV})\,,\\\\[3.0pt]
\Gamma^{(1)}_{h\to\tau^{+}\tau^{-}\gamma}&=&2.84\cdot 10^{-7}\,{\rm
GeV}&(2.92\cdot 10^{-7}\,{\rm GeV})\,,\end{array}$ (19)

where the superscript “(1)” indicates these are one-loop results. In order to
efficiently integrate the phase space for these processes we employed the
CollierLink library for Package X [49] which links the Passarino-Veltman
functions of Package X with the Collier library [50, 51, 52, 53]. Phase space
integrations were performed using gauge parameter $\xi=1$ as well as $\xi=2$
to confirm gauge invariance of the results. The differences between different
flavors of leptons is due to the reduced phase space for increasing fermion
mass: in integrating the phase space we have neglected $\hat{m}_{\ell}$ in the
matrix element, but the mass is still used in the integration region as
defined by $G$ in Eq. 14.

Dalitz plots of the tree and one-loop partial widths are shown in Figure 4 for
the muon case. This is the most interesting comparison as the two
contributions differ by less than an order of magnitude. The diagrams are
qualitatively the same for the cases of electrons and taus. Qualitatively we
can identify that the large $s_{2}$ region is favorable for isolating the
tree-level contribution, $s_{2}\sim\bar{m}_{Z}^{2}$ isolates the pole region
corresponding to $H\to\gamma Z(\to\bar{\ell}\ell)$, while the low $s_{2}$
region favors the conversion of an off-shell photon to the two leptons. In
[8], the authors studied cuts isolating various regions of the phase space for
the SM. In the next section we study how the presence of the SMEFT impacts
this decay process and seek to identify how to isolate regions of phase space
that can emphasize the SMEFT effects.

Figure 4: Dalitz plots comparing the tree (Left) and one-loop (Right)
contributions to the process $H\to\bar{\mu}\mu\gamma$ in the SM. The dotted
line shows where $s_{2}$, the invariant mass of the di-muon system, crosses
the $Z$-pole. Units are [GeV]-3.

##  4 SMEFT contributions

We have so far discussed the tree-level and one-loop contributions to the
process ${H\to\bar{\ell}\ell\gamma}$ within the SM. In this context, the
assumption of Eq. 16 removed the possibility of interference between the tree
and loop processes. As such the observable corresponding to the one-loop
amplitude corresponds to a two-loop observable, or symbolically comes with a
$1/(16\pi^{2})^{2}$ suppression. In the presence of the SMEFT, however, there
are new opportunities to generate tree-level amplitudes contributing to the
process ${H\to\bar{\ell}\ell\gamma}$ which, when interfered with the SM loops,
will not be chirally suppressed.

Following recent interest in the dimension-eight contributions in the SMEFT we
include the relevant dimension-eight operators. The inclusion of dimension-
eight terms is particularly interesting as the ${H\to\bar{\ell}\ell\gamma}$
process can be generated by a single contact interaction – at dimension-six
there is no four-vertex which couples two leptons of the same chirality, a
Higgs boson, and a photon. The relevant diagrams are shown in Fig. 5 where a
circle represents an effective vertex.

(80,60) h2,v1,h3 v2,v3,v4 dashes,tension=3v1,o1 photono2,o1 photonv2,o1
fermionv3,o2,v4 phantomv4,o1 phantom,tension=.3h2,o1
decoration.shape=circle,decoration.filled=full,decoration.size=8o1

(a)

(80,60) h2,v1,h3 v2,v3,v4 dashes,tension=3v1,o1 photono2,o1 photonv2,o1
fermionv3,o2,v4 phantomv4,o1 phantom,tension=.3h2,o1
decoration.shape=circle,decoration.filled=full,decoration.size=8o2
decoration.shape=circle,decoration.filled=full,decoration.size=8o1

(b)

(80,60) v1 v2,v3,v4 dashes,tension=3v1,o1 plain,tension=1o1,v4
plain,tension=1o1,v2 photon,tension=1v3,o1
decoration.shape=circle,decoration.filled=full,decoration.size=8o1

(c)

Figure 5: Diagrams contributing to the process $H\to\bar{\ell}\ell$ in the
SMEFT. The solid circle represents an insertion of an effective vertex. There
is also the possibility that both the $HVV$ and $V\bar{\ell}\ell$ vertex
correspond to insertions of dimension-six operators giving an overall
dimension-eight contribution. Diagram (5(a)) is generated at orders
$1/\Lambda^{2}$ and $1/\Lambda^{4}$, (5(b)) requires the insertion of two
dimension-six operators and is therefore $\mathcal{O}(1/\Lambda^{4})$, and the
four-point vertex of diagram (5(c)) occurs first at dimension eight in the
SMEFT. We do not consider diagrams with a single operator insertion at the
$V\bar{\ell}\ell$ vertex as this occurs only at one-loop.

The operators which shift the Higgs Yukawa couplings are given by the “Class
5” operators:

$\mathcal{L}_{\rm
cl5}=c_{eH}^{(6)}(H^{\dagger}H)(\bar{l}e_{R}H)+c_{eH}^{(8)}(H^{\dagger}H)^{2}(\bar{l}e_{R}H)\,.$
(20)

Those which shift gauge-lepton couplings are the Class 7 operators:

$\displaystyle\mathcal{L}_{\rm cl7}$ $\displaystyle=$ $\displaystyle
c_{Hl}^{(6),1}(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{l}\gamma^{\mu}l)+c_{Hl}^{(8),1}(H^{\dagger}H)(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{l}\gamma^{\mu}l)$
(21)
$\displaystyle+c_{Hl}^{(6),3}(H^{\dagger}i\overleftrightarrow{D}_{\mu}^{I}H)(\bar{l}\sigma^{I}\gamma^{\mu}l)+c_{Hl}^{(8),3}(H^{\dagger}H)(H^{\dagger}i\overleftrightarrow{D}_{\mu}^{I}H)(\bar{l}\sigma^{I}\gamma^{\mu}l)$
$\displaystyle+c_{He}^{(6),1}(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{e}_{R}\gamma^{\mu}e_{R})+c_{He}^{(8),1}(H^{\dagger}H)(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{e}_{R}\gamma^{\mu}e_{R})$
$\displaystyle+c_{Hl}^{(8),2}(H^{\dagger}\sigma_{a}H)(H^{\dagger}i\overleftrightarrow{D}^{\mu}H)(\bar{l}\gamma_{\mu}\sigma_{a}l)\,,$

where we have used:

$\displaystyle(H^{\dagger}\overleftrightarrow{D}_{\mu}H)$ $\displaystyle=$
$\displaystyle H^{\dagger}iD_{\mu}H-(iD_{\mu}H^{\dagger})H$ (22)
$\displaystyle(H^{\dagger}\overleftrightarrow{D}_{\mu}^{I}H)$ $\displaystyle=$
$\displaystyle H^{\dagger}i\tau^{I}D_{\mu}H-(iD_{\mu}\tau^{I}H^{\dagger})H$
(23)

While the dimension-eight contributions proportional to $c_{Hl}^{(8),1}$ and
$c_{Hl}^{(8),3}$ represent re-scalings of their dimension-six operators by
$(H^{\dagger}H)$ that corresponding to $c_{Hl}^{(8),2}$ represents a new form
of operator in Class 7 which does not exist at dimension-six. The last
remaining operator corresponding to $c_{Hl}^{(8),\epsilon}$ does not
contribute (see [32] for the full operator form). As can be seen in App. B
where the Feynman Rules for the SMEFT are collected, these operators only
result in a shift in the $\bar{\ell}\ell Z$ coupling and do not shift the
coupling of leptons to the photon. This can be understood from the operator
form as well, where the covariant derivative acts on the Higgs doublet. As the
Higgs, $h$, is chargeless there is no coupling to the photon induced by these
operators.

Class 4 operators result in anomalous $H\gamma\gamma$ and $HZ\gamma$
couplings, they are:

$\displaystyle\mathcal{L}_{\rm cl4}$ $\displaystyle=$ $\displaystyle
c_{HB}^{(6)}(H^{\dagger}H)B^{\mu\nu}B_{\mu\nu}+c_{HB}^{(8)}(H^{\dagger}H)^{2}B^{\mu\nu}B_{\mu\nu}$
$\displaystyle+c_{HW}^{(6)}(H^{\dagger}H)W^{\mu\nu}_{a}W_{a,\mu\nu}+c_{HW}^{(8)}(H^{\dagger}H)^{2}W^{\mu\nu}_{a}W_{a,\mu\nu}$
$\displaystyle+c_{HWB}^{(6)}(H^{\dagger}\sigma^{a}H)W^{\mu\nu}_{a}B_{\mu\nu}+c_{HWB}^{(8)}(H^{\dagger}H)(H^{\dagger}\sigma^{a}H)W^{\mu\nu}_{a}B_{\mu\nu}$
$\displaystyle+c_{HW,2}^{(8)}(H^{\dagger}\sigma^{a}H)(H^{\dagger}\sigma^{b}H)W_{a}^{\mu\nu}W_{b,\mu\nu}$

As in the case of the Class 7 operators there is one entirely new operator
form which occurs first at dimension-eight. These operators generate a new
momentum-dependent coupling of the Higgs boson to two vectors which is not
present at tree level in the SM.

In addition to these operators, there exist operators which shift the $H\gamma
Z$ coupling without affecting the $H\gamma\gamma$444These are referred to in
[35] and this work as “Class 8” operators. They are not to be confused with
the four-fermion operators called Class 8 in the usual dimension-six
literature, which are not relevant to this work.:

$\mathcal{L}_{\rm
cl8}=c_{WH^{4}D^{2}}^{(8),1}(H^{\dagger}H)(D^{\mu}H^{\dagger}\tau^{I}D^{\nu}H)W_{\mu\nu}^{I}+c_{BH^{4}D^{2}}^{(8),1}(H^{\dagger}H)(D^{\mu}H^{\dagger}D^{\nu}H)B_{\mu\nu}$
(25)

Finally, we can generate shifts in the coupling of the leptons to the $Z$
boson as well as a contact four-point interaction via the operators in Tab. 2.
These operators as well as the Class 7 operators are implemented in an updated
version of the Feynman Rules presented in [33], the updated package is
available in the  Feynrules model database as well as in the ancillary files
to this publication. The Class 11 operators are not included in this update as
the additional derivatives of the Higgs doublet take an excessive amount of
time to compute, they are available upon request.

Class 11: $\psi^{2}H^{2}D^{3}$
---
$Q_{l^{2}H^{2}D^{3}}^{(8),1}$ | $i(\bar{l}\gamma^{\mu}D^{\nu}l)(D_{(\mu}D_{\nu)}H^{\dagger}H)$
$Q_{l^{2}H^{2}D^{3}}^{(8),2}$ | $i(\bar{l}\gamma^{\mu}D^{\nu}l)(H^{\dagger}D_{(\mu}D_{\nu)}H)$
$Q_{l^{2}H^{2}D^{3}}^{(8),3}$ | $i(\bar{l}\gamma^{\mu}\tau^{I}D^{\nu}l)(D_{(\mu}D_{\nu)}H^{\dagger}\tau^{I}H)$
$Q_{l^{2}H^{2}D^{3}}^{(8),4}$ | $i(\bar{l}\gamma^{\mu}\tau^{I}D^{\nu}l)(H^{\dagger}\tau^{I}D_{(\mu}D_{\nu)}H)$
$Q_{e^{2}H^{2}D^{3}}^{(8),1}$ | $i(\bar{e}_{R}\gamma^{\mu}D^{\nu}e_{R})(D_{(\mu}D_{\nu)}H^{\dagger}H)$
$Q_{e^{2}H^{2}D^{3}}^{(8),2}$ | $i(\bar{e}_{R}\gamma^{\mu}D^{\nu}e_{R})(H^{\dagger}D_{(\mu}D_{\nu)}H)$

Class 15: $\psi^{2}XH^{2}D$
---
$Q^{(8),1}_{e^{2}WH^{2}D}$ | $(\bar{e}_{R}\gamma^{\nu}e_{R})D^{\mu}(H^{\dagger}\tau^{I}H)W_{\mu\nu}^{I}$
$Q^{(8),1}_{e^{2}BH^{2}D}$ | $(\bar{e}_{R}\gamma^{\nu}e_{R})D^{\mu}(H^{\dagger}H)B_{\mu\nu}$
$Q^{(8),1}_{l^{2}WH^{2}D}$ | $(\bar{l}\gamma^{\nu}l)D^{\mu}(H^{\dagger}\tau^{I}H)W_{\mu\nu}^{I}$
$Q^{(8),5}_{l^{2}WH^{2}D}$ | $(\bar{l}\gamma^{\nu}\tau^{I}l)D^{\mu}(H^{\dagger}H)W_{\mu\nu}^{I}$
$Q^{(8),9}_{l^{2}WH^{2}D}$ | $\epsilon^{IJK}(\bar{l}\gamma^{\nu}\tau^{I}l)D^{\mu}(H^{\dagger}\tau^{J}H)W_{\mu\nu}^{K}$
$Q^{(8),1}_{l^{2}BH^{2}D}$ | $(\bar{l}\gamma^{\nu}\tau^{I}l)D^{\mu}(H^{\dagger}\tau^{I}H)B_{\mu\nu}$
$Q^{(8),5}_{l^{2}BH^{2}D}$ | $(\bar{l}\gamma^{\nu}l)D^{\mu}(H^{\dagger}H)B_{\mu\nu}$

Table 2: Novel operator forms occuring first at dimension-eight which
contribute to $H\to\bar{\ell}\ell\gamma$. The parenthesis in
$D_{(\mu}D_{\nu)}$ indicate symmetrization in the Lorentz indices. $l$ is the
left handed lepton $SU(2)_{L}$ doublet, while $e_{R}$ is the right handed
leptonic singlet. Operators such as
$(\bar{l}\gamma^{\nu}l)(H^{\dagger}\overleftrightarrow{D}^{\mu}H)B_{\mu\nu}$
do not contribute as the relative sign in the derivative removes the terms
with a single Higgs. These operator forms are not yet implemented into the
geoSMEFT framework.

All SM and dimension-six contributions receive corrections from the Class 3
operators which shift couplings related to the Higgs boson. The Class 4
operators mentioned above also shift all SM and dimension-six couplings of the
vector bosons. These shifts occur due to finite field and mass
renormalizations. These operators are:

$\displaystyle\mathcal{L}_{\rm cl3}$ $\displaystyle=$ $\displaystyle
c_{H\Box}^{(6)}(H^{\dagger}H)\Box(H^{\dagger}H)+c_{HD}^{(6)}(H^{\dagger}D^{\mu}H)^{*}(H^{\dagger}D_{\mu}H)$
$\displaystyle+c_{HD}^{(8)}(H^{\dagger}H)^{2}(D_{\mu}H^{\dagger})(D^{\mu}H)+c_{HD,2}^{(8)}(H^{\dagger}H)(H^{\dagger}\sigma^{a}H)(D_{\mu}H^{\dagger})\sigma^{a}(D^{\mu}H)$

With the above, we have identified all operator forms contributing the process
$H\to\bar{\ell}\ell\gamma$, without inducing chiral flips, at tree level in
the SMEFT up to and including $1/\Lambda^{4}$ effects.

Below we discuss the potential of the decay $H\to\tau\tau\gamma$ may allow
distinguishing operators which shift the SM tree coupling from the other
operators effects. This discussion must be understood as having the
potentially strong caveat that operators inducing chiral flips, have been
neglected. As an estimate of the importance of such effects, we have included
the Yukawa-like operators and the dimension-six dipole operators in the main
text. In App F we discuss the size of the dipole contributions. We find that
the interference of the dipole operators with other contributions is small
compared with the dimension-six squared dipole contributions. We use this
result as an argument to neglect the dimension-eight operators which induce
chiral flips from our analysis. A more careful study of the viability of
measurement in the tau decay-channel should be performed before such
contributions are explored. The dipole operators we consider are given by:

$\mathcal{L}_{\rm
cl6}=c^{(6)}_{eW}(\bar{l}\sigma^{\mu\nu}e)\tau^{I}HW^{I}_{\mu\nu}+c^{(6)}_{eB}(\bar{l}\sigma^{\mu\nu}e)HB_{\mu\nu}+h.c.$
(27)

We assume real Wilson coefficients for the dipole operators as their CP-
violating parts are strongly constrained by low energy measurements [54].

In all cases of operators involving fermions we have assumed the flavor
dependence is diagonal without invoking symmetry arguments or spurions. Since
each decay channel has a separate flavor we choose to be agnostic about the
universality of such couplings.

###  4.1 SMEFT amplitudes and Dalitz plots

With the full set of SMEFT Feynman rules in Appendix B we can identify all
amplitudes with novel kinematics from those in the SM. We make the assumption
that at amplitude level:

$\frac{1}{16\pi^{2}}c_{i}^{(6)}\to 0$ (28)

This assumption “allows” us to neglect to include SMEFT
$\frac{1}{\Lambda^{2}}$ contributions in the loop. Ultimately these
contributions should be included, but are beyond the scope of this work. We
emphasize that these terms are neglected at the amplitude level as they are
still allowed to interfere with the SM loops. This neglects potential
contributions from the Class 3 and 4 operators at one loop which could
interfere with the tree level SMEFT amplitudes as in Figure 5(5(a)). Such
contributions are beyond the scope of this work and are left for future
studies. While this affects our final numerical results, the contributions
should be of the same magnitude as those included and therefore are not
expected to greatly affect the qualitative discussions of Section 4.2.

The novel SMEFT kinematic forms can be categorized into the following cases:

  1. 1.

$g_{H\gamma\gamma}$: Direct couplings of the Higgs boson to two photons as
defined in Eq. B.9 due to Class 4 operators. These include the SM and shifts
to the SM-like coupling of the photon to two leptons as defined in Eq. A.26
(implicit in the definition of $\bar{e}$). We note that $g_{H\gamma\gamma}$
mimics the one-loop coupling of $f_{H\gamma\gamma}^{(1)}$ without the full
momentum dependence.

  2. 2.

$g_{HZ\gamma}\times[(g_{L}+g_{L}^{\prime})P_{L}+(g_{R}+g_{R}^{\prime})P_{R}]$:
Direct couplings of the Higgs boson to a $Z$ and $\gamma$ as defined in Eqs.
B.9 and B.13 due to Class 4 operators. These include the SM and shifts to the
SM-like coupling of the $Z$ to two leptons as defined in Eqs. A.26 and A.30.
As above, $g_{HZ\gamma}$ mimics the one-loop coupling of $f_{HZ\gamma}^{(1)}$.

  3. 3.

Contact $(A_{11}^{\prime}P_{L}+B_{11}^{\prime}P_{R})$: the direct coupling
$H\bar{\ell}\ell\gamma$ generated by Class 11 operators as in Eq. B.53.

  4. 4.

Contact $(A_{15}P_{L}+B_{15}P_{R})$: the direct coupling
$H\bar{\ell}\ell\gamma$ generated by Class 15 operators as in Eq. B.53.

  5. 5.

Shifted Yukawa couplings: The Yukawa couplings of the SM are shifted by the
Class 5 operators.

  6. 6.

Contact $c_{DP}$: direct coupling of $H\bar{\ell}\ell\gamma$ generated by the
dipole operators

In what follows we refer to these cases as, for example, “Case 1” and “C1” in
context.

The above cases generate novel kinematic forms for the amplitudes contributing
to the decay of the Higgs boson to two leptons and a photon. Defining,

$\Pi^{\mu\nu}=\left[k_{1}^{\nu}(k_{2}+k_{3})^{\mu}-k_{1}\cdot(k_{2}+k_{3})\eta^{\mu\nu}\right]\,,$
(29)

we can write the corresponding amplitudes as follows555Here for convenience we
take $k_{\gamma}=k_{1}$, $k_{\psi}=k_{2}$, and $k_{\bar{\psi}}=k_{3}$.:

$i\mathcal{M}_{\rm
C1}=-i\frac{\bar{e}Q_{\ell}g_{HAA}v}{s_{2}}\Pi^{\mu\nu}(\bar{u}_{k_{2}}\gamma_{\nu}v_{k_{3}})\epsilon^{*}_{\mu}(k_{1})\,,$
(30) $\displaystyle i\mathcal{M}_{\rm C2}$ $\displaystyle\equiv$
$\displaystyle\mathcal{M}_{\rm C2}^{L}+\mathcal{M}_{\rm C2}^{R}\,,$ (31)
$\displaystyle i\mathcal{M}_{\rm C2}^{L}$ $\displaystyle=$
$\displaystyle-i\frac{g_{HAZ}\bar{g}_{Z}(g_{L}+g_{L}^{\prime})v}{4(s_{2}-\bar{m}_{Z}^{2}+i\bar{m}_{Z}\Gamma_{Z})}\Pi^{\mu\nu}(\bar{u}_{k_{2}}\gamma_{\nu}P_{L}v_{k_{3}})\epsilon^{*}_{\mu}(k_{1})\,,$
$\displaystyle i\mathcal{M}_{\rm C2}^{R}$ $\displaystyle=$
$\displaystyle-i\frac{g_{HAZ}\bar{g}_{Z}(g_{R}+g_{R}^{\prime})v}{4(s_{2}-\bar{m}_{Z}^{2}+i\bar{m}_{Z}\Gamma_{Z})}\Pi^{\mu\nu}(\bar{u}_{k_{2}}\gamma_{\nu}P_{R}v_{k_{3}})\epsilon^{*}_{\mu}(k_{1})\,,$
$\displaystyle i\mathcal{M}_{\rm C3}$ $\displaystyle\equiv$
$\displaystyle\mathcal{M}_{\rm C3}^{L}+\mathcal{M}_{\rm C3}^{R}\,,$ (32)
$\displaystyle i\mathcal{M}_{\rm C3}^{L}$ $\displaystyle=$
$\displaystyle\frac{iA^{\prime}_{11}\bar{e}Q_{\ell}v}{2}k_{1}^{\nu}(k_{2}+k_{3})^{\mu}(\bar{u}_{k_{2}}\gamma_{\nu}P_{L}v_{k_{3}})\epsilon^{*}_{\mu}(k_{1})\,,$
$\displaystyle i\mathcal{M}_{\rm C3}^{R}$ $\displaystyle=$
$\displaystyle\frac{iB^{\prime}_{11}\bar{e}Q_{\ell}v}{2}k_{1}^{\nu}(k_{2}+k_{3})^{\mu}(\bar{u}_{k_{2}}\gamma_{\nu}P_{R}v_{k_{3}})\epsilon^{*}_{\mu}(k_{1})\,,$
$\displaystyle i\mathcal{M}_{\rm C4}$ $\displaystyle\equiv$
$\displaystyle\mathcal{M}_{\rm C4}^{L}+\mathcal{M}_{\rm C4}^{R}\,,$ (33)
$\displaystyle i\mathcal{M}_{\rm C4}^{L}$ $\displaystyle=$ $\displaystyle
iA_{15}v\Pi^{\mu\nu}(\bar{u}_{k_{2}}\gamma_{\nu}P_{L}v_{k_{3}})\epsilon^{*}_{\mu}(k_{1})\,,$
$\displaystyle i\mathcal{M}_{\rm C4}^{R}$ $\displaystyle=$ $\displaystyle
iB_{15}v\Pi^{\mu\nu}(\bar{u}_{k_{2}}\gamma_{\nu}P_{R}v_{k_{3}})\epsilon^{*}_{\mu}(k_{1})\,.$

In addition to the above there are the contributions with operators which
induce chiral flips:

$\displaystyle i\mathcal{M}_{\rm C5}$ $\displaystyle=$
$\displaystyle\frac{\bar{e}\hat{m}_{\ell}}{v}\left(1+\Delta_{H\bar{\ell}\ell}\right)\bar{u}_{k_{2}}\gamma^{\mu}\not{k}_{1}\bar{v}_{k_{3}}\epsilon_{\mu}^{*}(k_{1})\left[\frac{1}{s_{1}}-\frac{1}{s_{3}}\right]\,,$
(34) $\displaystyle i\mathcal{M}_{\rm C6}$ $\displaystyle=$ $\displaystyle
c_{DP}\bar{u}_{k_{2}}\sigma^{\mu\nu}v_{k_{3}}k_{1}^{\nu}\epsilon^{*}_{\mu}(k_{1})\,.$
(35)

where $\Delta_{H\bar{\ell}\ell}$ is defined in Eq. B.5 of App B. We have also
defined:

$s_{3}=(k_{\gamma}+k_{\bar{\psi}})^{2}=\bar{m}_{H}^{2}-s_{1}-s_{2}\,.$ (36)

Of these cases of amplitudes, only Cases 1, 2, 5, and 6 can generate results
independent of the SM loop. For cases 1 and 2, this is because they can be
generated at dimension-six, Cases 3 and 4 are first generated at
$\mathcal{O}(1/\Lambda^{4})$. For cases 5 and 6 this is because they can
interfere with the SM tree amplitude. In Tables 6–7 we present Dalitz-like
plots for the squares of the tree-level SMEFT contributions as well as the
loop-tree interference between the SMEFT and the SM loops. These Dalitz plots
are the ratio of the SMEFT contribution to the sum of the SM tree- and loop-
contributions:

$\displaystyle R_{{\rm C}i}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{|\mathcal{M}_{{\rm C}i}|^{2}}{|\mathcal{M}_{\rm
SM}^{(0)}+\mathcal{M}_{\rm SM}^{(1)}|^{2}}$ $\displaystyle R_{{\rm C}i{\rm
C}j^{*}}^{(0)}$ $\displaystyle=$ $\displaystyle\frac{2{\rm
Re}[\mathcal{M}_{{\rm C}i}\mathcal{M}_{{\rm C}j}^{*}]}{|\mathcal{M}_{\rm
SM}^{(0)}+\mathcal{M}_{\rm SM}^{(1)}|^{2}}$ (37) $\displaystyle R_{{\rm
C}i}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{2{\rm Re}[\mathcal{M}_{\rm
SM}^{(1)}\mathcal{M}_{{\rm C}i}^{*}]}{|\mathcal{M}_{\rm
SM}^{(0)}+\mathcal{M}_{\rm SM}^{(1)}|^{2}}$

These plots are specifically for the case of the muon, but hold qualitatively
for the other charged lepton flavors. The normalization of the plots is
arbitrary as we have pulled out the dependence on the Wilson coefficients from
the expressions (as indicated by the titles). Quantities such as $\bar{g}_{Z}$
and $\bar{e}$ are set to their SM values for simplicity.

The most striking feature of these plots is that in certain regions the
overall sign of the ratio $R$ can flip. This sign flip is shown in the figures
with a solid black line across the plot, the top most region is always chosen
to have to a positive value (e.g. the plot of $R_{C1}^{(1)}$ has an explicit
minus sign in its title to ensure the top region is $+$). These sign flips
occur as these interference terms can generally have an arbitrary sign, as
$s_{2}$ changes different SM-loop contributions are emphasized and can result
in an overall sign change. While this theoretically could be used to define
cuts in $s_{2}$ which emphasize the effects of these SMEFT contributions, in
general one region’s contribution is much larger than that in another. We do,
however, explore this option in the analysis below.

These plots show separately the left and right-handed components as they can
behave differently due to their interference with the left- and right-handed
SM loops which differ (as discussed above). The most stark difference can be
seen in the plots for $R_{\rm C3}^{(1)}$ and $R_{\rm C4}^{(1)}$ where the
contribution from left handed leptons flips signs twice while for the right
handed case there is only one sign flip.

|
---|---
|
|
Figure 6: Dalitz-like plots showing the ratios defined in Eq. 37. Normalization is arbitrary as prefactors related to the Wilson coefficients have been pulled out of the ratios. In some cases a sign has been pulled out of the ratio as well. “$\pm$” is used when both the left and right hand parts are equal up to the normalizations factored out in the plot (as indicated by the title). Solid black lines show where the ratio flips signs, the top-most region is always positive. |
---|---
|
|
Figure 7: Dalitz-like plots as described in Figure 6.

###  4.2 Parameterized partial widths

Including all SMEFT corrections to the partial width of the Higgs boson decay
to $\bar{\ell}\ell\gamma$ we have the following expression for the amplitude
squared truncated at order $1/\Lambda^{4}$:

$\displaystyle|\mathcal{M}|^{2}$ $\displaystyle=$
$\displaystyle|\mathcal{M}^{(0)}_{\rm
SM}|^{2}(1+2\Delta_{H\bar{\ell}\ell}+[\Delta_{H\bar{\ell}\ell}^{(6)}]^{2})+|\mathcal{M}_{\rm
C1}+\mathcal{M}_{\rm C2}+\mathcal{M}_{\rm C6}|^{2}$
$\displaystyle+\frac{1}{16\pi^{2}}2{\rm Re}\left[\mathcal{M}^{(1)}_{\rm
SM}\left(\mathcal{M}_{\rm C1}+\mathcal{M}_{\rm C2}+\mathcal{M}_{\rm
C3}+\mathcal{M}_{\rm C4}+\mathcal{M}_{\rm C5}+\mathcal{M}_{\rm
C6}\right)^{*}\right]$
$\displaystyle+\frac{1}{(16\pi^{2})^{2}}|\mathcal{M}^{(1)}_{\rm SM}|^{2}$

where $\Delta_{H\bar{\ell}\ell}^{(6)}$ is the dimension-six part of
$\Delta_{H\bar{\ell}\ell}$ defined in B.5. Truncation at order $1/\Lambda^{4}$
of the square of the sum of amplitudes is implied. Writing the amplitude
squared in this way emphasizes the enhancement of the SMEFT terms over the SM
loops – we have explicitly written $1/(16\pi^{2})$ to demonstrate that the
SMEFT contributes to the amplitude-squared at tree- and one-loop level, where
the SM has the chirally suppressed tree level contribution and the two-loop
(one-loop squared) contribution.

In [8], the authors identify regions focused on optimizing sensitivity to the
SM contributions. For example, from Fig. 4 we can see that the large $s_{2}$
region is dominated, for muons, by the tree-level contribution, cutting
$s_{2}$ around the $Z$-pole region will isolate the resonant loop
contributions, while cutting for lower $s_{2}$ will isolate the non-resonant
loop contributions. The Dalitz-like plots of Figs. 6 and 7 show no obvious
regions for isolating the SMEFT contributions due to novel kinematics not
present in the SM. $R_{\rm C1}$ and $R_{\rm C2}$ mimic the effects of the
resonant loops in the SM, and so can be isolated by looking in the
corresponding SM kinematic regions for excesses. However, the sign changes
coming from the interference with the SM loops present an interesting
opportunity for combining different regions to emphasize the effects of the
SMEFT. In Table 3 we define possible choices of cuts on $s_{2}$ and
combinations of regions in $s_{2}$ to emphasize contributions in the SMEFT.
The last column gives the approximate number of decays $H\to\mu\mu\gamma$
expected for a SM Higgs at the HL-LHC given 3/ab of integrated luminosity. The
number of decays is determines as the production cross section multiplied by
the decay width (in the SM) multiplied by the integrated luminosity,
$\sigma_{pp\to h}\times{\rm BR}_{\rm SM}(h\to\bar{\ell}\ell\gamma)\times
3$/ab. In the first four regions the number is well over 1,000. In the case of
Regions 5 and 6 the number is lower which could be problematic statistically –
although the presence of SMEFT corrections could result in a higher number of
events and be a strong indicator of NP.

# | Region [GeV]2 | Purpose | # SM
---|---|---|---
1 | $(0\leq s_{2}\leq\bar{m}_{H}^{2})$ | Full phase space/general | $22\cdot 10^{3}$
2 | $(10^{2}\leq s_{2}\leq 40^{2})$ | SM nonresonant & $R_{\rm C1}^{(0,1)}$ | $2\cdot 10^{3}$
3 | $(70^{2}\leq s_{2}\leq 100^{2})$ | SM resonant & $R_{\rm C2}^{(0,1)}$ & $R_{\rm C3}^{(1)}$ | $10\cdot 10^{3}$
4 | $(100^{2}\leq s_{2}\leq\bar{m}_{H}^{2})$ | SM tree & $R_{\rm C5}^{(0)}$ | $3\cdot 10^{3}$
5 | $(45^{2}\leq s_{2}\leq 50^{2})-(65^{2}\leq s_{2}\leq 80^{2})$ |  $R_{\rm C3}^{(1),L}$ & $R_{\rm C4}^{(1),L}$ | “$-$”$900$
6 | $(45^{2}\leq s_{2}\leq 50^{2})+(65^{2}\leq s_{2}\leq 80^{2})$ | control for $R_{\rm C3}^{(1),L}$ & $R_{\rm C4}^{(1),L}$ | $900$
Table 3: Regions in $s_{2}$ chosen to emphasize various SMEFT effects. All
regions should be understood to be limited to the physical phase space region,
for example $\bar{m}_{H}^{2}$ may be outside the physical region and instead
$s_{2}$ is integrated to the maximum value. The combined regions 5 and 6 are
understood as the sum or difference between the integrated regions in $s_{2}$.
$s_{1}$ is always integrated from the lowest to highest physical values. The
first four regions approximately correspond to those proposed in [8]. The
column labeled “# SM” is the approximate number of SM decays
$H\to\mu\mu\gamma$ expected at HL-LHC with 3/ab integrated luminosity. The
negative sign for the number of events in region 5 simply indicates that more
decays occur in one region of the phase space than the other and we are taking
the difference.

To simplify the following expressions we adopt the following convention:

$\displaystyle\tilde{c}_{i}^{(6)}$ $\displaystyle=$
$\displaystyle\hat{v}^{2}c_{i}^{(6)}\,,$ (39)
$\displaystyle\tilde{c}_{i}^{(8)}$ $\displaystyle=$
$\displaystyle\hat{v}^{4}c_{i}^{(8)}\,,$ (40) $\displaystyle\hat{v}^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}\hat{G}_{F}}\,.$ (41)

We also define the ratio of the SMEFT partial width to that of the SM in input
parameter scheme $S$ over Region $R_{i}$ as defined in Table 3 as:

$\Delta_{S}^{R_{i}}=\left.\frac{\Gamma_{H\to\bar{\mu}\mu\gamma}^{\rm
SMEFT}}{\Gamma_{H\to\bar{\mu}\mu\gamma}^{\rm SM}}\right|_{S}^{R_{i}}-1\,.$
(42)

The relations between the various Lagrangian parameters and the two input
parameter schemes can be derived from the Appendices of [34]. We emphasize
that in determining the $\Delta$s we have used the full dependence of $\Gamma$
and not the qualitative normalization-independent quantities used to derive
the Dalitz-like plots. These cuts are very simple and purely
phenomenologically motivated. In future studies should be extended to include
kinematic limitations of the experiments (e.g. photon isolation requirements,
thresholds for various kinematic measurements). We have further neglected the
experimental efficiency for muon detection in the total number of events in
the # SM column of Table 3, the efficiencies for electrons and taus has also
been neglected in the discussion below.

Beginning with the fully integrated region (i.e. Region 1 of Tab. 3), we find
the ratio of the partial width in the presence of the SMEFT to the SM value
is:

$\displaystyle\Delta_{\alpha}^{R_{1}}$ $\displaystyle=$ $\displaystyle
0.47\tilde{c}_{H\Box}^{(6)}-0.12\tilde{c}_{HD}^{(6)}-0.059\tilde{c}_{HD}^{(8)}-0.059\tilde{c}_{HD,2}^{(8)}+0.94[\tilde{c}_{H\Box}^{(6)}]^{2}-0.47\tilde{c}_{H\Box}^{(6)}\tilde{c}_{HD}^{(6)}+0.059[\tilde{c}_{HD}^{(6)}]^{2}$
$\displaystyle-0.50\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}+0.27\frac{\hat{v}^{2}}{\hat{m}_{\mu}^{2}}[\tilde{c}_{eH}^{(6)}]^{2}-0.25\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(8)}-0.50\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{H\Box}^{(6)}+0.13\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle+0.0038\tilde{c}_{H\Box}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.00096\tilde{c}_{HD}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.0061\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}+9.4\cdot 10^{-9}\frac{\delta G_{F}}{\hat{G}_{F}}$
$\displaystyle+10^{4}\left(5.6[\tilde{c}_{HB}^{(6)}]^{2}+1.8\tilde{c}_{HB}^{(6)}\tilde{c}_{HW}^{(6)}+1.1[\tilde{c}_{HW}^{(6)}]^{2}-4.7\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}-2.4\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}+1.7[\tilde{c}_{HWB}^{(6)}]^{2}\right)$
$\displaystyle+10^{11}\left(2.4[\tilde{c}_{eB}^{(6)}]^{2}-2.7\tilde{c}_{eB}^{(6)}\tilde{c}_{eW}^{(6)}+0.73[\tilde{c}_{eW}^{(6)}]^{2}\right)$
$\displaystyle-260\left(\tilde{c}_{HB}^{(6)}+\tilde{c}_{HB}^{(8)}\right)-76\left(\tilde{c}_{HW}^{(6)}+\tilde{c}_{HW}^{(8)}+\tilde{c}_{HW,2}^{(8)}\right)+140\left(\tilde{c}_{HWB}^{(6)}+\tilde{c}_{HWB}^{(8)}\right)$
$\displaystyle-520\left[\tilde{c}_{HB}^{(6)}\right]^{2}-150\left[\tilde{c}_{HW}^{(6)}\right]^{2}+16\left[\tilde{c}_{HWB}^{(6)}\right]^{2}+560\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}+0.03\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}$
$\displaystyle-260\tilde{c}_{HB}^{(6)}\tilde{c}_{H\Box}^{(6)}+120\tilde{c}_{HB}^{(6)}\tilde{c}_{HD}^{(6)}-76\tilde{c}_{HW}^{(6)}\tilde{c}_{H\Box}^{(6)}-41\tilde{c}_{HW}^{(6)}\tilde{c}_{HD}^{(6)}+140\tilde{c}_{HWB}^{(6)}\tilde{c}_{H\Box}^{(6)}+4.0\tilde{c}_{HWB}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle-4.9\tilde{c}_{HB}^{(6)}\tilde{c}_{He}^{(6)}-7.7\tilde{c}_{HB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)+4.9\tilde{c}_{HW}^{(6)}\tilde{c}_{He}^{(6)}+7.7\tilde{c}_{HW}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)-3.2\tilde{c}_{HWB}^{(6)}\tilde{c}_{He}^{(6)}$
$\displaystyle-4.9\tilde{c}_{HWB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)-0.09\tilde{c}_{BH^{4}D^{2}}^{(8),1}+0.025\tilde{c}_{WH^{4}D^{2}}^{(8),1}$
$\displaystyle+0.000016\tilde{c}_{HB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.000016\tilde{c}_{HW}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}+0.000010\tilde{c}_{HWB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}$
$\displaystyle-1.3\tilde{c}_{e^{2}BH^{2}D}^{(8),1}-0.22\left(\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),2}\right)+0.69\tilde{c}_{e^{2}WH^{2}D}^{(8),1}-0.12\left(\tilde{c}_{L^{2}BH^{2}D}^{(8),1}+\tilde{c}_{L^{2}BH^{2}D}^{(8),5}\right)$
$\displaystyle-0.023\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),2}\right)-0.012\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),3}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),4}\right)+0.067\left(\tilde{c}_{l^{2}WH^{2}D}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),5}\right)$

The lines of this expression are organized as follows: the first through third
lines are the contribution from $\Delta_{H\bar{\ell}\ell}$, the fourth and
fifth come from the square of the tree-level SMEFT contributions, and the
subsequent lines are the interference of the SMEFT with the SM loops. All
contributions are written only to two digits accuracy. Immediately we can see
that the contribution from the square of the SMEFT tree contributions is by
far the most dominant (note the factor of $10^{4}$, and $10^{11}$ pulled out
from the expression). Interestingly, the dipole operators are dominant by
several orders of magnitude and are generally neglected in the literature,
including articles discussing NLO in the SMEFT expansion. From a purely
bottom-up EFT interpretation we might argue all other terms are negligible,
however making a flavor $U(3)^{5}$ assumption is sufficient for removing them
(see, for example, the discussion of $U(3)^{5}$ symmetry in [38]). For the
remainder of the text we will make this assumption, the dipole operator
dependence is maintained in the full expressions in Appendix D and further
dependence on the dipole operators which may be relevant to taus is discussed
in App. F.

Further, the contributions coming from the shift in the definition of the
fermi constant,

$v^{2}=\frac{1}{\sqrt{2}\hat{G}_{F}}+\frac{\delta G_{F}}{\hat{G}_{F}}\,,$ (44)

are strongly suppressed compared to other contributions. For this reason we
drop them in the remainder of the main text, they are also written in Appendix
D. $\delta G_{F}$ at leading order is given by [55]:

$\delta
G_{F}=\sqrt{2}\tilde{c}_{Hl}^{(6),3}-\frac{\tilde{c}_{ll}^{(6)}}{\sqrt{2}}\,.$
(45)

Where $c_{ll}^{(6)}$ is the four fermion operator affecting muon decay,
$(\bar{L}L)^{2}$. It can be found to dimension-eight using the system of
equations in the appendices of [34].

In the case of the contributions from $c_{eH}^{(d)}$ we could be concerned
that these explode for the lighter generations as $\frac{v}{\hat{m}}$ will be
large, however constraints on these Wilson coefficients (See e.g. [56])
constrain them for the muon and tau to be,

$\begin{array}[]{cc}-\frac{14}{[{\rm TeV}]^{2}}\leq c_{\mu
H}^{(6)}\frac{v}{\hat{m}_{\mu}}\leq\frac{12}{[{\rm
TeV}]^{2}}\,,&\qquad-\frac{2.5}{[{\rm TeV}]^{2}}\leq c_{\tau
H}^{(6)}\frac{v}{\hat{m}_{\tau}}\leq\frac{4.2}{[{\rm TeV}]^{2}}\,,\end{array}$
(46)

as such we will neglect them in the following discussion, they are reproduced
in Appendix D. Constraints on the Class 3 and 4 operators can also be found in
the literature, however since $H\to\gamma\gamma$ and $H\to Z\gamma$ are used
in inferring these constraints, they cannot be applied to this analysis. The
couplings $c_{eH}^{(6)}$ can be constrained from the Higgs decays to two
leptons independent of other channels. The coupling to the electron is not
(directly) constrained by experiment, however we will still neglect the
possibility of large changes to the electron yukawa due to $c_{eH}^{(6)}$ in
the following discussion.

If we make no assumptions about the size of these Wilson coefficients the
discussion above seemingly implies a unique opportunity to directly study the
$\frac{1}{\Lambda^{4}}$ contributions from the SMEFT while the leading
contributions at $\frac{1}{\Lambda^{2}}$ are strongly loop or chirally
suppressed. We see that the contribution of the SMEFT at one loop and
$\mathcal{O}(1/\Lambda^{2})$ interfering with a tree level SMEFT amplitude
would have a negligible effect compared to the tree-level squared SMEFT
contributions in the fourth line above. This holds throughout the discussions
below.

However, perturbativity in the SMEFT requires that, generally, higher order
terms in $1/\Lambda^{2}$ should have smaller contributions to observables than
the leading order terms. Here we understand the reason for the larger impact
of the tree-level terms of the fourth and fifth lines is the loop suppression
of the subsequent lines and the chiral suppression of the first lines.
However, we can consider some scenarios in which the perturbativity of the
expansion is more manifest:

  1. 1.

NP Scale Assumption: Restoring the NP scale and the vev, i.e.
$\tilde{c}_{i}^{(4+n)}\to\frac{\hat{v}^{n}}{\Lambda^{n}}c_{i}^{(4+n)}$, we can
assume that the $c_{i}$ are order 1 and infer a scale at which the terms of
the sixth line of Eq. 4.2 (i.e. linear in Class 4 operators) become larger
than those of the fourth (quadratic in Class 4 operators, we neglect the
dipole operators). Generally the contributions from the sixth line exceed
those of the fourth before any other line under this assumption.

  2. 2.

Loop Assumption: For weakly interacting new physics the Wilson coefficients
$c_{HB}^{(6)}$, $c_{HW}^{(6)}$, and $c_{HWB}^{(6)}$ are generally expected to
be generated at one loop in the UV [57]666It should be noted, however, that
the logic typically used to categorize these Class 4 operators as loop
generated [58] does not necessarily apply at dimension-eight [34]. These
arguments do not hold for the Class 3 and contact operators, so we can use
this assumption to infer if, for weakly interacting NP, the terms on the
fourth line of Eq. 4.2 remain dominant. In this assumption all the $c_{i}$ are
taken to be order 1 except for the class 4 operators which are taken to be
order $1/(16\pi^{2})$.

  3. 3.

Combined Assumption: We can combine the above two assumptions to infer a scale
at which the Class 3 or contact operators become dominant. Generally with
modest NP scales the Class 3 operators will dominate. Changing from muon final
states to electrons can drive down the Class 3 operator contribution resulting
in the dominance of the contact operators. The combined assumption leaves a
significant amount of freedom to manipulate the outcome that it is arguably
too model dependent for discussion. While we generally do not discuss it
below, we do occasionally invoke flavor as a way to discriminate the
contributions of different operator classes.

The above assumptions are very much model dependent. However, as noted above,
if we make no assumptions the squares and interference of the tree level
amplitudes of Eqs. 30 and 31 (Cases 1 and 2) dominate regardless of regions
considered from Table 3.

For Region 1, under the NP scale assumption, for a NP scale of $\Lambda\sim 1$
TeV, the squares of the tree level C1 and C2 amplitudes are still dominant.
This holds for $\Lambda$ up to around 3.5 TeV. Under the Loop Assumption the
$[c_{HB}^{(6)}]^{2}$ term of the fourth line become roughly order $6\cdot
10^{4}/(16\pi^{2})^{2}\sim 2$ while those linear in $c_{HB}^{(6)}$ are roughly
$270/(16\pi^{2})\sim 1.7$ so these terms are of roughly the same order. Class
4 operators still dominate the decay width. Considering the case of taus
would, however, push the contributions from the first line beyond loop
suppressed Class 4 operators. Considering both loop suppression and assuming a
scale of NP the Class 4 operators cease to be dominant below an assumed scale
of 1 TeV.

Region 2 is chosen to enhance the non-resonant SM loop contributions as well
as the Case 1 amplitude’s interference with the SM loop. We find the dominance
of the tree-level SMEFT contributions is enhanced by at most a factor of 3.
The contact operators have a larger impact as well as their contributions are
concentrated in the low $s_{2}$ area of phase space. The full expression can
be found in Appendix D, instead of giving the full expression for
$\Delta_{\alpha}^{R_{i}}$ from this point in the text we will give the
expressions under the UV assumptions presented above, keeping only the leading
and subleading contributions for clarity.

If we again assume a scale associated with new physics as described above, we
find that Class 4 operator contributions at order $\frac{1}{\Lambda^{2}}$
begin to dominate over these tree-level SMEFT contributions again at a scale
of around ${\Lambda\sim 3.5\ {\rm TeV}}$:

$\displaystyle\Delta_{\alpha}^{R_{2}}$ $\displaystyle=$ $\displaystyle
3.4[c_{HB}^{(6)}]^{2}+2.0c_{HB}^{(6)}c_{HW}^{(6)}+0.31[c_{HW}^{(6)}]^{2}-3.7c_{HB}^{(6)}c_{HWB}^{(6)}-1.1c_{HW}^{(6)}c_{HWB}^{(6)}+1.0[c_{HWB}^{(6)}]^{2}$
(47) $\displaystyle-3.6c_{HB}^{(6)}-1.1c_{HW}^{(6)}+2.0c_{HWB}^{(6)}$

If we retained the dipole operators, the scale at which the dimension-six
operators would begin to be larger than the squares of the dipole contribution
is over 100 TeV. Taking the loop assumption the contributions from the contact
operators of Classes 11 and 15 are of greater importance than in the previous
region, Class 4 operators are still dominant:

$\displaystyle\Delta_{\alpha}^{R_{2}}$ $\displaystyle=$ $\displaystyle
3.4[c_{HB}^{(6)}]^{2}+2.0c_{HB}^{(6)}c_{HW}^{(6)}+0.31[c_{HW}^{(6)}]^{2}-3.7c_{HB}^{(6)}c_{HWB}^{(6)}-1.1c_{HW}^{(6)}c_{HWB}^{(6)}+1.0[c_{HWB}^{(6)}]^{2}$
(48)
$\displaystyle-4.6\left(\tilde{c}_{HB}^{(6)}+\tilde{c}_{HB}^{(8)}\right)-1.4\left(\tilde{c}_{HW}^{(6)}+\tilde{c}_{HW}^{(8)}+\tilde{c}_{HW,2}^{(8)}\right)+2.5\left(\tilde{c}_{HWB}^{(6)}+\tilde{c}_{HWB}^{(8)}\right)$
$\displaystyle-3.1\tilde{c}_{e^{2}BH^{2}D}^{(8),1}$

The enhancement from considering final state taus changes this such that class
3 operators are dominant.

Region 3 is chosen to enhance the resonant SM contributions as well as Case 2.
However, we find this region, seemingly counterintuitively, results in a drop
in the tree-level SMEFT contributions of Cases 1 and 2. This is because in
Regions 1 and 2 the phase space integral corresponding to the square of Case 1
(and the interference of C1 and C2) are two orders of magnitude larger than
the contribution of Case 2. Region 3 successfully cuts out the dominant low
energy part of the phase space integral dropping the contribution of the
squares of the tree-level yukawa-like SMEFT contributions by an order of
magnitude. As the interference of C1 and C2 has the opposite sign of the
square of C1 some cancellation results in only a single order of magnitude
drop.

For $R_{3}$, assuming a scale of new physics, the $\frac{1}{\Lambda^{2}}$
contributions begin to dominate over the tree level $\frac{1}{\Lambda^{4}}$
class 4 operator contributions for a scale of $\Lambda\sim 8$ TeV:

$\displaystyle\Delta_{\alpha}^{R_{3}}$ $\displaystyle=$ $\displaystyle
0.013[c_{HB}^{(6)}]^{2}-0.022c_{HB}^{(6)}c_{HW}^{(6)}+0.011[c_{HW}^{(6)}]^{2}+0.013c_{HB}^{(6)}c_{HWB}^{(6)}-0.015\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}+0.0049[\tilde{c}_{HWB}^{(6)}]^{2}$
(49)
$\displaystyle-0.015c_{HB}^{(6)}+0.0085\tilde{c}_{HW}^{(6)}-0.0037\tilde{c}_{HWB}^{(6)}$

If we again consider the possibility the Class 4 operators are loop generated
then we find the contact operators become competitive with the class 4
operators:

$\displaystyle\Delta_{\alpha}^{R_{3}}$ $\displaystyle=$ $\displaystyle
0.57[\tilde{c}_{HB}^{(6)}]^{2}-0.99\tilde{c}_{HB}^{(6)}\tilde{c}_{HW}^{(6)}+0.51[\tilde{c}_{HW}^{(6)}]^{2}+0.59\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}-0.66\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}+0.22[\tilde{c}_{HWB}^{(6)}]^{2}$
(50)
$\displaystyle-1.0\tilde{c}_{e^{2}BH^{2}D}^{(8),1}-0.18\left(\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),2}\right)+0.56\tilde{c}_{e^{2}WH^{2}D}^{(8),1}+0.51\left(\tilde{c}_{L^{2}BH^{2}D}^{(8),1}+\tilde{c}_{L^{2}BH^{2}D}^{(8),5}\right)$
$\displaystyle+0.089\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),2}\right)+0.045\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),3}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),4}\right)-0.28\left(\tilde{c}_{l^{2}WH^{2}D}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),5}\right)$

Region 4 is chosen to enhance the SM tree contributions and the corresponding
shifts from $\Delta_{H\bar{\ell}\ell}$. In this region we fould that the SMEFT
C1 and C2 tree contributions are strongly suppressed, but still largely
dominant. Making assumptions about the scale of new physics as above, the
linear terms in Class 4 operators begin to dominate for $\Lambda\sim 2.5$ TeV:

$\displaystyle\Delta_{\alpha}^{R_{4}}$ $\displaystyle=$ $\displaystyle
0.046[c_{HB}^{(6)}]^{2}-0.055c_{HB}^{(6)}c_{HW}^{(6)}+0.036[c_{HW}^{(6)}]^{2}+0.024c_{HB}^{(6)}c_{HWB}^{(6)}-0.051c_{HW}^{(6)}c_{HWB}^{(6)}+0.02[c_{HWB}^{(6)}]^{2}$
(51) $\displaystyle+0.044c_{HB}^{(6)}-0.059c_{HW}^{(6)}+0.042c_{HWB}^{(6)}$

Further, should the Class 4 operators be loop suppressed this is the first
region in which the Class 3 operators of Eq. 4 become dominant for the muon:

$\displaystyle\Delta_{\alpha}^{R_{4}}$ $\displaystyle=$ $\displaystyle
1.9\tilde{c}_{H\Box}^{(6)}-0.49\tilde{c}_{HD}^{(6)}-0.24\tilde{c}_{HD}^{(8)}-0.24\tilde{c}_{HD,2}^{(8)}+3.9[\tilde{c}_{H\Box}^{(6)}]^{2}-1.9\tilde{c}_{H\Box}^{(6)}\tilde{c}_{HD}^{(6)}+0.24[\tilde{c}_{HD}^{(6)}]^{2}$
$\displaystyle+0.22\tilde{c}_{e^{2}BH^{2}D}^{(8),1}-0.12\tilde{c}_{e^{2}WH^{2}D}^{(8),1}-0.42\left(\tilde{c}_{L^{2}BH^{2}D}^{(8),1}+\tilde{c}_{L^{2}BH^{2}D}^{(8),5}\right)+0.23\left(\tilde{c}_{l^{2}WH^{2}D}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),5}\right)$

While a subset of the contact operators are second-most dominant. The
enhancement of the Class 3 operators will be improved by a factor of
$m_{\tau}^{2}/m_{\mu}^{2}$ for the tau, and heavily suppressed for the
electron in which case the contact operators become dominant.

Region 5 is chosen such that the phase space integral for the first subregion,
$45^{2}\leq s_{2}\leq 50^{2}$, approximately cancels that of the second
region, $65^{2}\leq s_{2}\leq 80^{2}$, for the tree level SMEFT contributions
from Case 1. This has the effect of reducing the contributions from the Class
4 operators while enhancing the contribution from the left-handed contact
operators (see Fig. 7).

Unfortunately the region only suppresses the tree-level Class 4 operator
contributions by a factor of about 10. This is because for Cases 1, 2, and
their interference, the regions cannot be chosen to simultaneously cancel all
contributions. A very careful analysis could outperform the regions chosen
here, but are beyond the scope of this article as they should also take into
account detector effects, such as the detector’s ability resolve different
flavors of charged leptons in these particular regions of phase space. While
this region was defined with the intention of emphasizing the left-handed
contact operators we also find enhanced contributions from the right-handed
contact operators. This is simply because the dominant area of phase space for
these operators is the low $s_{2}$ region.

The terms linear in Class 4 operators begin to dominate under an assumption of
a new physics scale of about 3 TeV:

$\displaystyle\Delta_{\alpha}^{R_{5}}$ $\displaystyle=$ $\displaystyle
1.7[c_{HB}^{(6)}]^{2}-3.9c_{HB}^{(6)}c_{HW}^{(6)}+1.6[c_{HW}^{(6)}]^{2}+2.6c_{HB}^{(6)}c_{HWB}^{(6)}-1.8c_{HW}^{(6)}c_{HWB}^{(6)}+0.48[c_{HWB}^{(6)}]^{2}$
(53) $\displaystyle+2.3c_{HB}^{(6)}-1.8c_{HW}^{(6)}+1.0c_{HWB}^{(6)}$

Under the assumption Class 4 operators are loop suppressed, the contact
operators are dominant for final state electrons and muons:

$\displaystyle\Delta_{\alpha}^{R_{5}}$ $\displaystyle=$ $\displaystyle
0.95\left(\tilde{c}_{HB}^{(6)}+\tilde{c}_{HB}^{(8)}\right)-0.76\left(\tilde{c}_{HW}^{(6)}+\tilde{c}_{HW}^{(8)}+\tilde{c}_{HW,2}^{(8)}\right)+0.43\left(\tilde{c}_{HWB}^{(6)}+\tilde{c}_{HWB}^{(8)}\right)$
$\displaystyle-9.6\tilde{c}_{e^{2}BH^{2}D}^{(8),1}-1.7\left(\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),2}\right)+5.3\tilde{c}_{e^{2}WH^{2}D}^{(8),1}+6.4\left(\tilde{c}_{L^{2}BH^{2}D}^{(8),1}+\tilde{c}_{L^{2}BH^{2}D}^{(8),5}\right)$
$\displaystyle+1.1\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),2}\right)+0.56\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),3}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),4}\right)-3.5\left(\tilde{c}_{l^{2}WH^{2}D}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),5}\right)$

Comparing the Region 5 result with Region 6, which is just the sum of the two
regions used for Region 5, we find that the tree-level contributions from
Class 4 operators as much as a factor of $5$ times larger than in the
previously considered region. The calculation of the difference between
regions in $R_{5}$ could mean the difference between resolving the impact of
contact operators or only seeing the Class 4 operator contributions.

We have found that, invoking $U(3)^{5}$ symmetry, the Class 4 operator
contributions are dominant in all defined regions unless we make certain
assumptions. Assuming a scale associated with the NP allows us to infer a
scale at which the $\frac{1}{\Lambda^{2}}$ contributions dominate over the
contributions from the squares of Class 4 operators, the lowest scale for
which this occurs is 2.5 TeV. This indicates (under the NP scale assumption)
that the $H\to\ell\ell\gamma$ decay channel presents an excellent opportunity
to directly study the $(1/\Lambda^{2})^{2}$ terms in the SMEFT. Table 4
summarizes which Class of operator is dominant for a given region under the
loop assumption for Class 4 operators. By considering the electron and muon
flavors, these different regions present the opportunity to separately study
the effects of operator Classes 3, 4, and the contact operators (Classes 11
and 15). Adding in the tau allows for clearer distinguishing in Region 3, but
may be less promising when the instability of, and ability to resolve, taus is
taken into account. In this way, Table 4 shows that this decay channel of the
Higgs boson supplemented by various cuts provides an excellent opportunity to
study the interplay between different operator classes as well as different
orders in the SMEFT power counting. It is important to note that these
statements all rely heavily on assumptions about the UV physics, and while
these assumptions may be motivated, they may be misleading. By taking a more
general approach, improving the quality of our predictions, and continuing to
perform high quality global fits we will allow the physics to speak for itself
and lead us to the correct conclusions.

| Cl4 operators | Class 3 operators | Class 11 & 15 operators | $\Lambda$
---|---|---|---|---
R1 | D | – | – | 3.5 TeV
R2 | $D_{\mu,e}$ | $D_{\tau}$ | – | 3.5 TeV
R3 | $C_{\mu,e}$ | $C_{\mu},\ D_{\tau}$ | $C_{\mu,e}$ | 8 TeV
R4 | – | Dτ | De,μ | 2.5 TeV
R5 | – | Dτ | De,μ | 3 TeV
Table 4: Summary of the dominant operator contributions under the assumption
the Class 4 operators are generated at one loop. $D_{i}$ indicates this
operator class dominates for flavor $i$, $D$ indicates for all flavors.
$C_{i}$ indicates that there is no dominant contribution and that the marked
classes are of comparable sizes.

##  5 Conclusions

After laying out our conventions we derived both the SM tree and loop
contributions to $H\to\bar{\ell}\ell\gamma$. We then derived the tree level
results in the SMEFT up to and including terms of order $1/\Lambda^{4}$.
Interfering the SMEFT results with the SM we created Dalitz-like plots
demonstrating the behavior of these SMEFT contributions over the full phase
space. By considering different regions of this phase space we were able to
deduce regions in which various SMEFT contributions would be emphasized. The
interference of SMEFT amplitudes with the SM were found to sometimes switch
signs due to the different terms in the SM loops, which presented an
interesting opportunity to sum regions of phase space in such a manner that
the dominant contributions from the tree-level SMEFT amplitudes squared would
be suppressed. In all cases, however, we found that these dominant terms
prevailed.

If we make the assumption that the UV physics which generates the SMEFT in the
IR is weakly interacting it is generally believed that the Class 4 operators
(at dimension-six) will be suppressed by a factor of $\frac{1}{16\pi^{2}}$. In
this case we have identified that the process $H\to\bar{\ell}\ell\gamma$, when
different flavors of lepton are considered, can discriminate Class 4 operators
from Class 3, and the contact operators of Classes 11 and 15. Given a measured
deviation from the SM in this channel, this provides an opportunity to attempt
to infer whether or not the UV physics is strongly or weakly interacting (i.e.
the strength of the Class 4 operator contributions777It is important to note
that if the Class 4 operators have small Wilson coefficients this does not
necessarily preclude new physics which is strongly interacting, but is
consistent with weakly interacting new physics.) as well as to
phenomenologically study the size of the $\frac{1}{\Lambda^{4}}$ terms.

The nature of the three body phase space makes three body decays an excellent
opportunity for studying the novel kinematics of the SMEFT. The only other
three body decays of the Higgs involve quarks, as such they present a
challenge to precision measurements888We neglect the narrow width
approximation of, e.g. $H\to WW$ and $H\to ZZ$ in this statement [30]..
Another key aspect of this study was that the interplay between the tree and
one-loop processes in the SM which cannot interfere. This combined with the
fact the tree level contributions vary over many orders of magnitude depending
on the flavor of the final state leptons gives a very broad phenomenology
presenting many different ways to study the contributions from the SMEFT.

In performing this analysis we neglected the dimension-six one-loop terms and
potential issues with detector resolution of the leptons and photon. As a
result our quantitative results of Appendix D have room for improvement before
and during the run of the HL-LHC. Our qualitative results are expected to
approximately hold and therefore the conclusions of the ability of this
channel to distinguish between different UV physics as well as to provide an
opportunity to look for interesting flavor phenomenology remain.

## Acknowledgements

The authors thank I. Brivio, A. Martin, H. Patel, and M. Trott for useful
discussions.

#### Funding information

TC acknowledges funding from European Union’s Horizon 2020 research and
innovation program under the Marie Sklodowska-Curie grant agreement No.
890787.

##  Appendix A Feynman rules for the effective action

The Lagrangian corresponding to the effective action is given by Eq. 3.2, it
is reproduced here:

$\displaystyle\mathcal{L}_{1-{\rm loop}}$ $\displaystyle=$ $\displaystyle
f_{H\gamma\gamma}^{(1)}(k_{1},k_{2})\hat{h}\hat{F}_{\mu\nu}\hat{F}^{\mu\nu}+f_{H\gamma\gamma}^{(2)}(k_{1},k_{2})\hat{h}(\partial^{\mu}\hat{F}_{\mu\nu})(\partial_{\sigma}\hat{F}^{\sigma\nu})$
(A.2)
$\displaystyle+f_{HZ\gamma}^{(1)}(k_{1},k_{2})\hat{h}\hat{F}_{\mu\nu}\hat{Z}^{\mu\nu}+f_{HZ\gamma}^{(2)}(k_{1},k_{2})\hat{h}(\partial^{\mu}\hat{F}_{\mu\nu})\partial_{\sigma}\partial^{\nu}\hat{Z}^{\sigma}$
$\displaystyle+f_{HZ\gamma}^{(3)}(k_{1},k_{2})\hat{h}\hat{F}_{\mu\nu}\Box\hat{Z}^{\nu}\,,$

The rules follow FeynRules formatting, e.g. a field “$\phi$” with subscript
“$1$” has corresponding four-momenta $k_{1}$ and if relevant Lorentz index
$\mu_{1}$. The rule corresponding to the form factor $f_{H\gamma\gamma}^{(1)}$
is:

$\displaystyle\left(\begin{array}[]{c}\hat{h}\\\ \hat{A}_{1}\\\
\hat{A}_{2}\end{array}\right)$ $\displaystyle=$ $\displaystyle
4if_{H\gamma\gamma}^{(1)}(k_{1},k_{2})\left[k_{1}^{\mu_{2}}k_{2}^{\mu_{1}}-k_{1}\cdot
k_{2}\,\eta^{\mu_{1},\mu_{2}}\right]$ (A.6)

The rule corresponding to the form factor $f_{H\gamma\gamma}^{(2)}$ is:

$\displaystyle\left(\begin{array}[]{c}\hat{h}\\\ \hat{A}_{1}\\\
\hat{A}_{2}\end{array}\right)$ $\displaystyle=$ $\displaystyle
2if_{H\gamma\gamma}^{(2)}(k_{1},k_{2})\left[k_{1}\cdot
k_{2}\,k_{1}^{\mu_{1}}k_{2}^{\mu_{2}}-k_{1}^{2}\,k_{2}^{\mu_{1}}k_{2}^{\mu_{2}}-k_{2}^{2}\,k_{1}^{\mu_{1}}k_{1}^{\mu_{2}}+k_{1}^{2}\,k_{2}^{2}\,\eta^{\mu_{1},\mu_{2}}\right]$
(A.10)

For the operators corresponding to the $\hat{h}\hat{Z}\hat{\gamma}$ operators
we have:

$\displaystyle\left(\begin{array}[]{c}\hat{h}\\\ \hat{A}_{1}\\\
\hat{Z}_{2}\end{array}\right)$ $\displaystyle=$ $\displaystyle
2if_{HZ\gamma}^{(1)}(k_{1},k_{2})\left[k_{1}^{\mu_{2}}k_{2}^{\mu_{1}}-k_{1}\cdot
k_{2}\,\eta^{\mu_{1}\mu_{2}}\right]$ (A.14)

The Lorentz forms corresponding to $f^{(2)}_{H\gamma\gamma}$ are split between
$f^{(2)}_{HZ\gamma}$ and $f^{(3)}_{HZ\gamma}$:

$\displaystyle\left(\begin{array}[]{c}\hat{h}\\\ \hat{A}_{1}\\\
\hat{Z}_{2}\end{array}\right)$ $\displaystyle=$ $\displaystyle
if_{HZ\gamma}^{(2)}(k_{1},k_{2})\left[k_{1}\cdot
k_{2}\,k_{1}^{\mu_{1}}k_{2}^{\mu_{2}}-k_{1}^{2}\,k_{2}^{\mu_{1}}k_{2}^{\mu_{2}}\right]$
(A.18) $\displaystyle\left(\begin{array}[]{c}\hat{h}\\\ \hat{A}_{1}\\\
\hat{Z}_{2}\end{array}\right)$ $\displaystyle=$ $\displaystyle
if_{HZ\gamma}^{(3)}(k_{1},k_{2})\left[k_{2}^{2}\,k_{1}^{\mu_{1}}k_{1}^{\mu_{2}}-k_{1}^{2}\,k_{2}^{2}\,\eta^{\mu_{1},\mu_{2}}\right]$
(A.22)

The following tree level Feynman rules are need to generate the process
$H\to\ell\ell\gamma$ from the effective action:

$\displaystyle\left(\begin{array}[]{c}\bar{\ell}\\\ \ell\\\
\hat{A}_{1}\end{array}\right)$ $\displaystyle=$
$\displaystyle-i\bar{e}Q_{\ell}\gamma^{\mu_{1}}$ (A.26)
$\displaystyle\left(\begin{array}[]{c}\bar{\ell}\\\ \ell\\\
\hat{Z}_{1}\end{array}\right)$ $\displaystyle=$ $\displaystyle
i\frac{\bar{g}_{Z}}{2}\gamma^{\mu_{1}}\left[(1+2Q_{\ell}s_{Z}^{2})P_{L}+2Q_{\ell}s_{Z}^{2}P_{R}\right]\equiv
i\frac{\bar{g}_{Z}}{2}\gamma^{\mu_{1}}(g_{L}P_{L}+g_{R}P_{R})$ (A.30)

##  Appendix B Feynman rules in the SMEFT

This appendix follows the conventions set out in App. A. Note that there is
implicit dependence on the Wilson coefficients of the SMEFT in Eqs. A.26 and
A.30 (i.e. in $\bar{e}$ and $\bar{g}_{Z}$) which needs to be considered in
addition to the rules outlined below. Shifts in the coupling of the Higgs
boson to two fermions are given by:

$\displaystyle\left(\begin{array}[]{c}H\\\ \bar{\ell}\\\
\ell\end{array}\right)$ $\displaystyle=$
$\displaystyle-i\frac{\hat{m}_{\ell}}{v}(1+\Delta_{H\bar{\ell}\ell})$ (B.4)
$\displaystyle\Delta_{H\bar{\ell}\ell}$ $\displaystyle=$ $\displaystyle
v^{2}\left[c_{H\Box}^{(6)}-\frac{1}{4}c_{HD}^{(6)}-\frac{v^{2}}{8}(c_{HD}^{(8)}+c_{HD,2}^{(8)})+\frac{3v^{2}}{32}(c_{HD}^{(6)}-4c_{H\Box}^{(6)})^{2}\right]$
(B.5)
$\displaystyle-\frac{3v^{3}}{2\sqrt{2}\hat{m}_{\ell}}\left(c_{eH}^{(6)}+\frac{v^{2}}{2}c_{eH}^{(8)}\right)$

We note in the above, and what follows $v$ is not rewritten in terms of the
input parameter $\hat{G}_{F}$ and so the shift defined in Eq. 44 is implicit
in these expressions. The direct coupling of the Higgs boson to $Z\gamma$ and
$\gamma\gamma$ is given by:

$\displaystyle\left(\begin{array}[]{c}H\\\ A_{1}\\\ A_{2}\end{array}\right)$
$\displaystyle=$ $\displaystyle
ig_{HAA}v(k_{1}^{\mu_{2}}k_{2}^{\mu_{1}}-k_{1}\cdot
k_{2}\eta^{\mu_{1}\mu_{2}})$ (B.9) $\displaystyle\left(\begin{array}[]{c}H\\\
A_{1}\\\ Z_{2}\end{array}\right)$ $\displaystyle=$
$\displaystyle-\frac{ig_{HAZ}v}{2}(k_{1}^{\mu_{2}}k_{2}^{\mu_{1}}-k_{1}\cdot
k_{2}\eta^{\mu_{1}\mu_{2}})$ (B.13)

Where we have defined:

$\displaystyle g_{HAZ}$ $\displaystyle=$
$\displaystyle\left[8(c_{HB}^{(6)}-c_{HW}^{(6)})\bar{c}_{W}\bar{s}_{W}+4c_{HWB}^{(6)}(\bar{c}_{W}^{2}-\bar{s}_{W}^{2})\right](1+\Delta_{HAZ}v^{2})$
(B.15)
$\displaystyle+v^{2}\left[8(c_{HB}^{(8)}-c_{HW}^{(8)}-c_{HW,2}^{(8)})\bar{c}_{W}\bar{s}_{W}+4c_{HWB}^{(8)}(\bar{c}_{W}^{2}-\bar{s}_{W}^{2})\right]$
$\displaystyle+\frac{\bar{e}(2c_{BH^{4}D^{2}}^{(8),1}\bar{c}_{W}-c_{WH^{4}D^{2}}^{(8),1}\bar{s}_{W})}{8\bar{c}_{W}\bar{s}_{W}}v^{2}$
$\displaystyle\Delta_{HAZ}$ $\displaystyle=$ $\displaystyle
2c_{HB}^{(6)}+2c_{HW}^{(6)}+c_{H\Box}^{(6)}-\frac{1}{4}c_{HD}^{(6)}$ (B.16)
$\displaystyle g_{HAA}$ $\displaystyle=$ $\displaystyle
4\left[c_{HB}^{(6)}\bar{c}_{W}^{2}+c_{HW}^{(6)}\bar{s}_{W}^{2}-\bar{s}_{W}\bar{c}_{W}c_{HWB}^{(6)}\right]$
$\displaystyle+v^{2}\bar{c}_{W}^{2}\left[c_{HB}^{(6)}(8c_{HB}^{(6)}+4c_{H\Box}^{(6)}-c_{HD}^{(6)})+2(\bar{c}_{HWB}^{(6)})^{2}\right]$
$\displaystyle+v^{2}\bar{s}_{W}^{2}\left[c_{HW}^{(6)}(8c_{HW}^{(6)}+4c_{H\Box}^{(6)}-c_{HD}^{(6)})+2(\bar{c}_{HWB}^{(6)})^{2}\right]$
$\displaystyle-v^{2}\bar{c}_{W}\bar{s}_{W}c_{HWB}^{(6)}\left[8c_{HB}^{(6)}+4c_{H\Box}^{(6)}-c_{HD}^{(6)}+8c_{HW}^{(6)}\right]$
$\displaystyle+4v^{2}\left[c_{HB}^{(8)}\bar{c}_{W}^{2}+(c_{HW}^{(8)}+c_{HW,2}^{(8)})\bar{s}_{W}^{2}-\bar{s}_{W}\bar{c}_{W}c_{HWB}^{(8)}\right]$

In addition to the implicit SMEFT shifts in Eq. A.30, the $Z$ coupling to
leptons is shifted by the Class 7 operators:

$\displaystyle\left(\begin{array}[]{c}\bar{\ell}\\\ \ell\\\
\hat{Z}_{1}\end{array}\right)$ $\displaystyle=$ $\displaystyle
i\frac{\bar{g}_{Z}}{2}\gamma^{\mu_{1}}\left(g_{L}^{\prime}P_{L}+g_{R}^{\prime}P_{R}\right)$
(B.21) $\displaystyle g_{L}^{\prime}$ $\displaystyle=$ $\displaystyle
v^{2}\left(c_{Hl}^{(6),1}+c_{Hl}^{(6),3}\right)\left(1+\Delta_{Z\ell\ell}\right)+\frac{v^{4}}{2}\left(c_{Hl}^{(8),1}+c_{Hl}^{(8),2}+c_{Hl}^{(8),3}\right)$
(B.22) $\displaystyle g_{R}^{\prime}$ $\displaystyle=$ $\displaystyle
v^{2}c_{He}^{(6)}\left(1+\Delta_{Z\ell\ell}\right)+\frac{v^{4}}{2}c_{He}^{(8)}$
(B.23) $\displaystyle\Delta_{Z\ell\ell}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{2}\left[2\bar{c}_{W}g_{2}c_{HW}^{(6)}+2\bar{s}_{W}g_{1}\bar{c}_{HB}^{(6)}+(g_{2}\bar{s}_{W}+g_{1}\bar{c}_{W})c_{HWB}^{(6)}\right]$
(B.24)

The $Z$ coupling is also shifted by the Class 11 operators:

$\displaystyle\left(\begin{array}[]{c}\bar{\ell}\\\ \ell_{1}\\\
\hat{Z}_{2}\end{array}\right)$ $\displaystyle=$ $\displaystyle
i\frac{\bar{g}_{Z}}{8}v^{2}\left[A_{11}(k_{1}\cdot
k_{2}\gamma^{\mu_{2}}+k_{1}^{\mu_{2}}\not{k}_{2})P_{L}+B_{11}(k_{1}\cdot
k_{2}\gamma^{\mu_{2}}+k_{1}^{\mu_{2}}\not{k}_{2})P_{R}\right]$ (B.28)
$\displaystyle A_{11}$ $\displaystyle=$ $\displaystyle
2c_{l^{2}H^{2}D^{3}}^{(8),1}-2c_{l^{2}H^{2}D^{3}}^{(8),2}+c_{l^{2}H^{2}D^{3}}^{(8),3}-c_{l^{2}H^{2}D^{3}}^{(8),4}$
(B.29) $\displaystyle B_{11}$ $\displaystyle=$ $\displaystyle
2c_{e^{2}H^{2}D^{3}}^{(8),1}-2c_{e^{2}H^{2}D^{3}}^{(8),2}$ (B.30)

However, these terms make no contribution to our a calculations as they occur
at dimension-eight and no tree level coupling $HAZ$ exists in the SM.

For dipole operators we have the following rules:

$\displaystyle\left(\begin{array}[]{c}\bar{\ell}\\\ \ell\\\ \hat{A}_{1}\\\
\hat{H}_{2}\end{array}\right)$ $\displaystyle=$ $\displaystyle-ic_{\rm
DP}\sigma^{\mu_{1}\nu}p_{3}^{\nu}$ (B.35)
$\displaystyle\left(\begin{array}[]{c}\bar{\ell}\\\ \ell\\\
\hat{A}_{1}\end{array}\right)$ $\displaystyle=$ $\displaystyle-ic_{\rm
DP}^{\prime}v\sigma^{\mu_{1}\nu}p_{1}^{\nu}$ (B.39)
$\displaystyle\left(\begin{array}[]{c}\bar{\ell}\\\ \ell\\\
\hat{Z}_{1}\end{array}\right)$ $\displaystyle=$ $\displaystyle-ic_{\rm
DP2}^{\prime}v\sigma^{\mu_{1}\nu}p_{1}^{\nu}$ (B.43) $\displaystyle c_{\rm
DP}=c_{DP}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\bar{c}_{W}c_{eB,\ell}-\bar{s}_{W}c_{eW,\ell}}{2}$ (B.44)
$\displaystyle c_{DP2}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\bar{s}_{W}c_{eB,\ell}+\bar{c}_{W}c_{eW,\ell}}{2}$ (B.45)

Finally, at dimension eight both Class 11 and Class 15 operators can generate
the four-point contact interaction:

$\displaystyle\left(\begin{array}[]{c}\bar{\ell}\\\ \ell\\\ \hat{A}_{1}\\\
\hat{H}_{2}\end{array}\right)$ $\displaystyle=$
$\displaystyle\begin{array}[]{l}\\\\[12.0pt]
i\frac{Q_{\ell}\bar{e}v}{2}k_{2}^{\mu_{1}}\not{k}_{2}(A_{11}^{\prime}P_{L}+B_{11}^{\prime}P_{R})\\\\[8.0pt]
-iv\left[A_{15}(k_{1}\cdot
k_{2}\gamma^{\mu_{1}}-k_{2}^{\mu_{1}}\not{k}_{1})P_{L}+B_{15}(k_{1}\cdot
k_{2}\gamma^{\mu_{1}}-k_{2}^{\mu_{1}}\not{k}_{1})P_{R}\right]\end{array}$
(B.53) $\displaystyle A_{11}^{\prime}$ $\displaystyle=$ $\displaystyle
2c_{l^{2}H^{2}D^{3}}^{(8),1}+2c_{l^{2}H^{2}D^{3}}^{(8),2}+c_{l^{2}H^{2}D^{3}}^{(8),3}+c_{l^{2}H^{2}D^{3}}^{(8),4}$
(B.54) $\displaystyle B_{11}^{\prime}$ $\displaystyle=$ $\displaystyle
2c_{e^{2}H^{2}D^{3}}^{(8),1}+2c_{e^{2}H^{2}D^{3}}^{(8),2}$ (B.55)
$\displaystyle A_{15}$ $\displaystyle=$
$\displaystyle\bar{c}_{W}(c_{l^{2}BH^{2}D}^{(8),1}+c_{l^{2}BH^{2}D}^{(8),5})-\bar{s}_{W}(c_{l^{2}WH^{2}D}^{(8),1}+c_{l^{2}WH^{2}D}^{(8),5})$
(B.56) $\displaystyle B_{15}$ $\displaystyle=$
$\displaystyle\bar{c}_{W}c_{e^{2}BH^{2}D}^{(8),1}-\bar{s}_{W}c_{e^{2}WH^{2}D}^{(8),1}$
(B.57)

##  Appendix C Matching to the effective action

Here we reproduce the matching expressions in the small off-shell momenta
limit. Discussions in the main text generally use the full result where
momenta are allowed to be arbitrarily off shell as appropriate to the process.
The Higgs boson is taken on shell, and the momenta of photons, $k_{i}^{2}$ are
taken to be small. We separate the bosonic (i.e. $W^{\pm}$, $\phi^{\pm}$,
inclusive of ghosts $u^{\pm}$) contribution from the top contribution.
Further, we expand in an appropriate small ratio of masses for comparison with
[39, 38, 59] and to demonstrate the agreement between our and their
expressions.

We leave the gauge parameter arbitrary, and note that the term of order
$(k_{i}^{2})^{0}$ terms are gauge invariant, as they must be, as they
correspond to physical processes. Certain short hands are used for more
complicated functions, such as the scalar Passarino-Veltman function
$C_{0}^{S}$, these can be found after the full set of expressions. For the
bosonic (including ghosts) contribution we find:

$\displaystyle f_{i}^{(j)}$ $\displaystyle=$
$\displaystyle\left(f_{i}^{(j)}\right|_{\rm B}+\left(f_{i}^{(j)}\right|_{t}$
(C.1) $\displaystyle\left.f_{H\gamma\gamma}^{(1)}\right|_{\rm B}$
$\displaystyle=$
$\displaystyle-\frac{\bar{e}^{2}\bar{m}_{W}^{4}}{16\pi^{2}\bar{m}_{H}^{4}v}\left[\bar{m}_{H}^{2}(\bar{m}_{H}^{2}+6\bar{m}_{W}^{2})-3\bar{m}_{W}^{2}(\bar{m}_{H}^{2}-2\bar{m}_{W}^{2})\log\left(1+\frac{\bar{m}_{H}(\sqrt{\bar{m}_{H}^{2}-4\bar{m}_{W}^{2}}-\bar{m}_{H})}{2\bar{m}_{W}^{2}}\right)^{2}\right]$
(C.2)
$\displaystyle-\frac{\bar{e}^{2}\bar{m}_{W}^{4}}{64\pi^{2}\bar{m}_{H}^{6}v}(k_{1}^{2}+k_{2}^{2})\left[\rule{0.0pt}{27.03003pt}2(\bar{m}_{H}^{4}+6\bar{m}_{H}^{2}\bar{m}_{W}^{2}){\rm
Disc}[\bar{m}_{H}^{2},\bar{m}_{W},\bar{m}_{W}]+(\bar{m}_{H}^{4}-6\bar{m}_{H}^{2}\bar{m}_{W}^{2}+24\bar{m}_{W}^{4})\log\left(1+\frac{\bar{m}_{H}(\sqrt{\bar{m}_{H}^{2}-4\bar{m}_{W}^{2}}-\bar{m}_{H})}{2\bar{m}_{W}^{2}}\right)^{2}\right.$
$\displaystyle\phantom{-\frac{e^{2}\bar{m}_{W}^{4}}{64\pi^{2}\bar{m}_{H}^{6}v}(k_{1}^{2}+k_{2}^{2})}\
\
-\bar{m}_{H}^{2}\left(48\bar{m}_{W}^{2}-\xi(\bar{m}_{H}^{2}-2\bar{m}_{W}^{2}\xi)\log\left[1+\frac{\bar{m}_{H}(\sqrt{\bar{m}_{H}^{2}-4\bar{m}_{W}^{2}}-\bar{m}_{H})}{2\bar{m}_{W}^{2}}\right]^{2}\right)$
$\displaystyle\phantom{-\frac{e^{2}\bar{m}_{W}^{4}}{64\pi^{2}\bar{m}_{H}^{6}v}(k_{1}^{2}+k_{2}^{2})}\
\
+2\bar{m}_{H}^{2}\left([\bar{m}_{H}^{2}+\bar{m}_{W}^{2}(\xi-1)]C_{0}^{S}[0,0,\bar{m}_{H}^{2},\sqrt{\xi}\bar{m}_{W},\bar{m}_{W},\bar{m}_{W}]\right)$
$\displaystyle\phantom{-\frac{e^{2}\bar{m}_{W}^{4}}{64\pi^{2}\bar{m}_{H}^{6}v}(k_{1}^{2}+k_{2}^{2})}\
\ +\left.(\bar{m}_{W}^{2}[\xi-1]-\bar{m}_{H}^{2})\xi
C_{0}^{S}[0,0,\bar{m}_{H}^{2},\sqrt{\xi}\bar{m}_{W},\sqrt{\xi}\bar{m}_{W},\bar{m}_{W}]\rule{0.0pt}{27.03003pt}\right]+\mathcal{O}(k_{i}^{4})$
$\displaystyle=$
$\displaystyle\frac{\bar{e}^{2}}{8\pi^{2}v}\left(-\frac{7}{4}-\frac{11}{30}a-\frac{19}{105}a^{2}-\frac{58}{525}a^{3}-\cdots\right)+\mathcal{O}(k_{i}^{2})\qquad
a\equiv\frac{\bar{m}_{H}^{2}}{4\bar{m}_{W}^{2}}$ (C.3)

In the case of the operator corresponding to $f_{H\gamma\gamma}^{(2)}$ the
gauge dependence does not vanish for the leading term, this is because this
term only contributes for off-shell photons. This can be understood from Eq.
A.10 where the first three terms are proportional to $k_{i}^{\mu_{i}}$ which
vanishes when contracted with the on-shell polarization vector, while the
$\eta^{\mu_{1},\mu_{2}}$ term is preceded by $k_{i}^{2}$ which vanishes for
on-shell photons.

$\displaystyle\left.f_{H\gamma\gamma}^{(2)}\right|_{\rm B}$ $\displaystyle=$
$\displaystyle\frac{\bar{e}^{2}}{16\pi^{2}\bar{m}_{H}^{6}v}\left[-40\bar{m}_{H}^{2}(\bar{m}_{H}^{2}+6\bar{m}_{W}^{2})+\bar{m}_{H}^{2}\left(\bar{m}_{W}^{2}(\xi-9)(\xi-1)^{2}-2\bar{m}_{H}^{2}[\xi(\xi-5)-4]\right)\log\left[\frac{1}{\xi}\right]\right.$
(C.5)
$\displaystyle\phantom{\frac{e^{2}}{16\pi^{2}\bar{m}_{H}^{6}v}}+\left(2\bar{m}_{H}^{4}-\frac{\bar{m}_{H}^{6}}{\bar{m}_{W}^{2}}-20\bar{m}_{H}^{2}\bar{m}_{W}^{2}-48\bar{m}_{W}^{4}\right)\log\left(1+\frac{\bar{m}_{H}(\sqrt{\bar{m}_{H}^{2}-4\bar{m}_{W}^{2}}-\bar{m}_{H})}{2\bar{m}_{W}^{2}}\right)$
$\displaystyle\phantom{\frac{e^{2}}{16\pi^{2}\bar{m}_{H}^{6}v}}+\frac{1}{\bar{m}_{W}^{2}}\left[\bar{m}_{H}^{6}(\xi-1)+2\bar{m}_{H}^{4}\bar{m}_{W}^{2}(\xi-9)-96\bar{m}_{H}^{2}\bar{m}_{W}^{4}\right]{\rm
Disc}[\bar{m}_{H}^{2},\bar{m}_{W},\bar{m}_{W}]$
$\displaystyle\phantom{\frac{e^{2}}{16\pi^{2}\bar{m}_{H}^{6}v}}+\frac{\bar{m}_{H}^{4}}{\bar{m}_{W}^{2}}(\xi-1)\left((\bar{m}_{H}^{2}-2\bar{m}_{W}^{2}\xi){\rm
Disc}[\bar{m}_{H}^{2},\sqrt{\xi}\bar{m}_{W},\sqrt{\xi}\bar{m}_{W}]-2(\bar{m}_{H}^{2}+(\xi-1)\bar{m}_{W}^{2}){\rm
Disc}[\bar{m}_{H}^{2},\bar{m}_{W},\sqrt{\xi}\bar{m}_{W}]\right)$
$\displaystyle\phantom{\frac{e^{2}}{16\pi^{2}\bar{m}_{H}^{6}v}}+\bar{m}_{H}^{4}\left[2\bar{m}_{W}^{2}(\xi-1)^{2}+\bar{m}_{H}^{2}(\xi+1)^{2}\right]C_{0}^{S}[0,0\bar{m}_{H}^{2},\bar{m}_{W},\sqrt{\xi}\bar{m}_{W},\bar{m}_{W}]$
$\displaystyle\phantom{\frac{e^{2}}{16\pi^{2}\bar{m}_{H}^{6}v}}+\frac{4\bar{m}_{H}^{2}}{\bar{m}_{W}^{2}}\left[\bar{m}_{H}^{6}+2\bar{m}_{W}^{6}(\xi-1)^{3}-\bar{m}_{H}^{4}\bar{m}_{W}^{2}(\xi+1)+\bar{m}_{H}^{2}\bar{m}_{W}^{4}(1+(2-3\xi)\xi)\right]C_{0}^{S}[0,0\bar{m}_{H}^{2},\sqrt{\xi}\bar{m}_{W},\bar{m}_{W},\bar{m}_{W}]$
$\displaystyle\phantom{\frac{e^{2}}{16\pi^{2}\bar{m}_{H}^{6}v}}-\frac{\bar{m}_{H}^{4}}{\bar{m}_{W}^{2}}(2\bar{m}_{H}^{2}+\bar{m}_{W}^{2}(\xi-1)^{2})(\bar{m}_{H}^{2}-2\bar{m}_{W}^{2}\xi)C_{0}^{S}[0,0,\bar{m}_{H}^{2},\sqrt{\xi}\bar{m}_{W},\bar{m}_{W},\sqrt{\xi}\bar{m}_{W}]$
$\displaystyle\left.\phantom{\frac{e^{2}}{16\pi^{2}\bar{m}_{H}^{6}v}}+4\bar{m}_{H}^{2}[\bar{m}_{H}^{2}+2\bar{m}_{W}^{2}(\xi-1)][\bar{m}_{W}^{2}(\xi-1)^{2}-\bar{m}_{H}^{2}(\xi+1)]C_{0}^{S}[0,0,\bar{m}_{H}^{2},\sqrt{\xi}\bar{m}_{W},\sqrt{\xi}\bar{m}_{W},\bar{m}_{W}]\right]+\mathcal{O}(k_{i}^{2})$

While for the top quark contributions we have:

$\displaystyle\left.f_{H\gamma\gamma}^{(1)}\right|_{\rm t}$ $\displaystyle=$
$\displaystyle\frac{Q_{u}^{2}\bar{e}^{2}N_{c}}{4\pi^{2}\bar{m}_{H}^{4}v}\left[4\bar{m}_{t}^{2}\bar{m}_{H}^{2}-(\bar{m}_{H}^{2}\bar{m}_{t}^{2}-4\bar{m}_{t}^{4})\log\left(\frac{2\bar{m}_{t}^{2}-\bar{m}_{H}^{2}+\sqrt{\bar{m}_{H}^{4}-4\bar{m}_{H}^{2}\bar{m}_{t}^{2}}}{2\bar{m}_{t}^{2}}\right)^{2}\right.$
$\displaystyle\phantom{\frac{\bar{Q}_{u}^{2}e^{2}N_{c}}{4\pi^{2}\bar{m}_{H}^{4}v}}\left.-\frac{k_{1}^{2}+k_{2}^{2}}{\bar{m}_{H}^{2}}\left((\bar{m}_{H}^{4}-16\bar{m}_{H}^{2}\bar{m}_{t}^{2}-4\bar{m}_{H}^{2}\bar{m}_{t}^{2}{\rm
Disc}[\bar{m}_{H}^{2},m_{t},m_{t}]+\bar{m}_{t}^{2}(\bar{m}_{H}^{2}-8\bar{m}_{t}^{2})\log\left[\frac{2\bar{m}_{t}^{2}-\bar{m}_{H}^{2}+\sqrt{\bar{m}_{H}^{4}-4\bar{m}_{H}^{2}\bar{m}_{t}^{2}}}{2\bar{m}_{t}^{2}}\right]^{2}\right)\right]+\mathcal{O}(k_{i}^{4})$
$\displaystyle=$
$\displaystyle\frac{Q_{u}^{2}\bar{e}^{2}N_{c}}{8\pi^{2}v}\left[\frac{1}{3}+\frac{7}{90}a+\cdots\right]+\mathcal{O}(k_{i}^{2})\qquad
a\equiv\frac{\bar{m}_{H}^{2}}{4\bar{m}_{t}^{2}}$ (C.7)
$\displaystyle\left.f_{H\gamma\gamma}^{(2)}\right|_{\rm t}$ $\displaystyle=$
$\displaystyle\frac{Q_{u}^{2}\bar{e}^{2}N_{c}}{2\pi^{2}\bar{m}_{H}^{6}v}\left[20\bar{m}_{H}^{2}\bar{m}_{t}^{2}+8\bar{m}_{t}^{2}\bar{m}_{H}^{2}{\rm
Disc}[\bar{m}_{H}^{2},\bar{m}_{t},\bar{m}_{t}]+\bar{m}_{t}^{2}(\bar{m}_{H}^{2}+4\bar{m}_{t}^{2})\log\left(\frac{2\bar{m}_{t}^{2}-\bar{m}_{H}^{2}+\sqrt{\bar{m}_{H}^{4}-4\bar{m}_{H}^{2}\bar{m}_{t}^{2}}}{2\bar{m}_{t}^{2}}\right)^{2}\right.$
$\displaystyle\phantom{\frac{Q_{u}^{2}\bar{e}^{2}N_{c}}{2\pi^{2}\bar{m}_{H}^{6}v}}\left.+\frac{k_{1}^{2}+k_{2}^{2}}{3\bar{m}_{H}^{2}}\left(\bar{m}_{H}^{4}+204\bar{m}_{H}^{2}\bar{m}_{t}^{2}+84\bar{m}_{t}^{2}\bar{m}_{H}^{2}{\rm
Disc}[\bar{m}_{H}^{2},\bar{m}_{t},\bar{m}_{t}]+12(\bar{m}_{t}^{2}\bar{m}_{H}^{2}+3\bar{m}_{t}^{4})\log\left[\frac{2\bar{m}_{t}^{2}-\bar{m}_{H}^{2}+\sqrt{\bar{m}_{h}^{4}-4\bar{m}_{H}^{2}\bar{m}_{t}^{2}}}{2\bar{m}_{t}^{2}}\right]\right)\right]$

The form factors for the $\hat{h}\hat{Z}\hat{\gamma}$ couplings are again
expanded in terms of small off-shell momenta, in this case we take the gauge
parameter $\xi\to 1$ and expand in small off-shell momenta. For the $Z$ we
take $k_{2}^{2}\to\bar{m}_{Z}^{2}+\mu^{2}$, where $\mu^{2}$ represents the
small perturbation from an on-shell $Z$:

$\displaystyle N_{1}$ $\displaystyle=$
$\displaystyle-\frac{2\bar{e}^{2}t_{W}}{16\pi^{2}(\bar{m}_{H}^{2}-\bar{m}_{Z}^{2})^{2}(\bar{m}_{Z}^{2}-\bar{m}_{W}^{2})v}$
(C.9) $\displaystyle\left.f_{HZ\gamma}^{(1)}\right|_{\rm B}$ $\displaystyle=$
$\displaystyle
N_{1}\left[\rule{0.0pt}{27.03003pt}(\bar{m}_{H}^{2}-\bar{m}_{Z}^{2})\left[2\bar{m}_{W}^{2}(\bar{m}_{H}^{2}+6\bar{m}_{W}^{2})-(\bar{m}_{H}^{2}+2\bar{m}_{W}^{2})\bar{m}_{Z}^{2}\right]+\bar{m}_{Z}^{2}\left[2\bar{m}_{W}^{2}(\bar{m}_{H}^{2}+6\bar{m}_{W}^{2})-(\bar{m}_{H}^{2}+2\bar{m}_{W}^{2})\bar{m}_{Z}^{2}\right]{\rm
Disc}[\bar{m}_{H}^{2},\bar{m}_{W},\bar{m}_{W}]\right.$
$\displaystyle\phantom{N_{1}}+\bar{m}_{Z}^{2}\left[(\bar{m}_{H}^{2}+2\bar{m}_{W}^{2})\bar{m}_{Z}^{2}-2\bar{m}_{W}^{2}(\bar{m}_{H}^{2}+6\bar{m}_{W}^{2})\right]{\rm
Disc}[\bar{m}_{Z}^{2},\bar{m}_{W},\bar{m}_{W}]$
$\displaystyle\left.\phantom{N_{1}}+\bar{m}_{W}^{2}\left[12\bar{m}_{W}^{4}+6\bar{m}_{W}^{2}\bar{m}_{Z}^{2}-2\bar{m}_{Z}^{4}+\bar{m}_{H}^{2}(\bar{m}_{Z}^{2}-6\bar{m}_{W}^{2})\right]\left(\log\left[1+\frac{\bar{m}_{H}(\sqrt{\bar{m}_{H}^{2}-4\bar{m}_{W}^{2}}-\bar{m}_{H})}{2\bar{m}_{W}^{2}}\right]+\log\left[1+\frac{\bar{m}_{Z}(\sqrt{\bar{m}_{Z}^{2}-4\bar{m}_{W}^{2}}-\bar{m}_{Z})}{2\bar{m}_{W}^{2}}\right]\right)\rule{0.0pt}{27.03003pt}\right]$
$\displaystyle+\mathcal{O}(k_{i}^{2},\mu^{2})$ $\displaystyle=$
$\displaystyle\frac{\bar{e}^{2}t_{W}}{3\cdot
16\pi^{2}v}\left[\left(11-\frac{31}{t_{W}^{2}}\right)+\frac{22}{3}\left(\frac{1}{5}-\frac{1}{t_{W}^{2}}\right)a+\frac{8}{3}\left(\frac{7}{5}-\frac{4}{t_{W}^{2}}\right)b+\mathcal{O}(a^{2},b^{2})\right]+\mathcal{O}(k_{i}^{2},\mu^{2})\,,\hskip
14.22636pta\equiv\frac{\bar{m}_{H}^{2}}{4\bar{m}_{W}^{2}}\,,\qquad
b\equiv\frac{\bar{m}_{Z}^{2}}{4\bar{m}_{W}^{2}}\,.$ (C.11)

In the above, the overall normalization was chosen to match that of [39].

$\displaystyle N_{2}$ $\displaystyle=$
$\displaystyle\frac{\bar{e}^{2}}{16\pi^{2}(\bar{m}_{H}-\bar{m}_{Z})^{3}(\bar{m}_{H}+\bar{m}_{Z})^{3}\sqrt{\bar{m}_{W}^{2}(\bar{m}_{Z}^{2}-\bar{m}_{W}^{2})}v}$
(C.12) $\displaystyle\left.f_{HZ\gamma}^{(2)}\right|_{\rm B}$ $\displaystyle=$
$\displaystyle
N_{2}\left[\rule{0.0pt}{27.03003pt}12(\bar{m}_{H}^{2}-\bar{m}_{Z}^{2})[2\bar{m}_{W}^{2}(\bar{m}_{H}^{2}+6\bar{m}_{W}^{2})-\bar{m}_{Z}^{2}(\bar{m}_{H}^{2}+2\bar{m}_{W}^{2})]\right.$
$\displaystyle\phantom{N_{2}}-4(2\bar{m}_{H}^{2}+\bar{m}_{Z}^{2})[\bar{m}_{Z}^{2}(\bar{m}_{H}^{2}+2\bar{m}_{W}^{2})-2\bar{m}_{W}^{2}(\bar{m}_{H}^{2}+6\bar{m}_{W}^{2})]{\rm
Disc}[\bar{m}_{H}^{2},\bar{m}_{W},\bar{m}_{W}]$
$\displaystyle\phantom{N_{2}}+4(\bar{m}_{H}^{2}+2\bar{m}_{Z}^{2})[\bar{m}_{Z}^{2}(\bar{m}_{H}^{2}+2\bar{m}_{W}^{2})-2\bar{m}_{W}^{2}(\bar{m}_{H}^{2}+6\bar{m}_{W}^{2})]{\rm
Disc}[\bar{m}_{Z}^{2},\bar{m}_{W},\bar{m}_{W}]$
$\displaystyle\phantom{N_{2}}-\left[6\bar{m}_{W}^{2}\bar{m}_{Z}^{4}+\bar{m}_{H}^{4}(2\bar{m}_{W}^{2}+\bar{m}_{Z}^{2})+\bar{m}_{H}^{2}(\bar{m}_{Z}^{4}-20\bar{m}_{W}^{4}-4\bar{m}_{W}^{2}\bar{m}_{Z}^{2})-48\bar{m}_{W}^{6}-4\bar{m}_{W}^{4}\bar{m}_{Z}^{2}\right]$
$\displaystyle\left.\phantom{N_{2}}\ \ \
\times\log\left(\frac{2\bar{m}_{W}^{2}+\bar{m}_{H}(\sqrt{\bar{m}_{H}^{2}-4\bar{m}_{W}^{2}}-\bar{m}_{H})}{2\bar{m}_{W}^{2}+\bar{m}_{Z}(\sqrt{\bar{m}_{Z}^{2}-4\bar{m}_{W}^{2}}-\bar{m}_{Z})}\right)\left(\log\left[1+\frac{\bar{m}_{H}(\sqrt{\bar{m}_{H}^{2}-4\bar{m}_{W}^{2}}-\bar{m}_{H})}{2\bar{m}_{W}^{2}}\right]+\log\left[1+\frac{\bar{m}_{Z}(\sqrt{\bar{m}_{Z}^{2}-4\bar{m}_{W}^{2}}-\bar{m}_{Z})}{2\bar{m}_{W}^{2}}\right]\right)\rule{0.0pt}{27.03003pt}\right]$
$\displaystyle+\mathcal{O}(k_{i}^{2},\mu^{2})$ $\displaystyle N_{3}$
$\displaystyle=$
$\displaystyle\frac{\bar{e}^{2}[2\bar{m}_{W}^{2}(\bar{m}_{H}^{2}+6\bar{m}_{W}^{2})-(\bar{m}_{H}^{2}+2\bar{m}_{W}^{2})\bar{m}_{Z}^{2}]}{16\pi^{2}\bar{m}_{W}(\bar{m}_{Z}^{2}-\bar{m}_{H}^{2})^{3}\sqrt{\bar{m}_{Z}^{2}-\bar{m}_{W}^{2}}v}$
(C.14) $\displaystyle\left.f_{HZ\gamma}^{(3)}\right|_{\rm B}$ $\displaystyle=$
$\displaystyle
N_{3}\left[\rule{0.0pt}{27.03003pt}12(\bar{m}_{H}^{2}-\bar{m}_{Z}^{2})+4(2\bar{m}_{H}^{2}+\bar{m}_{Z}^{2}){\rm
Disc}[\bar{m}_{H}^{2},\bar{m}_{W},\bar{m}_{W}]-4(\bar{m}_{H}^{2}+\bar{m}_{Z}^{2}){\rm
Disc}[\bar{m}_{Z}^{2},\bar{m}_{W},\bar{m}_{W}]\right.$
$\displaystyle\phantom{N_{3}}\left.+(\bar{m}_{H}^{2}+4\bar{m}_{W}^{2}+\bar{m}_{Z}^{2})\left(\log\left[1+\frac{\bar{m}_{H}(\sqrt{\bar{m}_{H}^{2}-4\bar{m}_{W}^{2}}-\bar{m}_{H})}{2\bar{m}_{W}^{2}}\right]-\log\left[1+\frac{\bar{m}_{Z}(\sqrt{\bar{m}_{Z}^{2}-4\bar{m}_{W}^{2}}-\bar{m}_{Z})}{2\bar{m}_{W}^{2}}\right]\right)\rule{0.0pt}{27.03003pt}\right]$
$\displaystyle+\mathcal{O}(k_{i}^{2},\mu^{2})$

For the top quark contribution we have:

$\displaystyle N_{4}$ $\displaystyle=$
$\displaystyle\frac{\bar{e}\bar{g}_{Z}(g_{L}^{t}+g_{R}^{t})\bar{m}_{t}^{2}N_{c}Q_{u}}{32\pi^{2}v(\bar{m}_{H}^{2}-\bar{m}_{Z}^{2})^{2}}\qquad
g_{L}^{t}=(1-2Q_{u}s_{z}^{2})\qquad g_{R}^{t}=-2Q_{u}s_{Z}^{2}$ (C.17)
$\displaystyle\left.f_{HZ\gamma}^{(1)}\right|_{t}$ $\displaystyle=$
$\displaystyle
N_{4}\left[4(\bar{m}_{H}^{2}-\bar{m}_{Z}^{2})+4\bar{m}_{Z}^{2}\left({\rm
Disc}[\bar{m}_{H}^{2},\bar{m}_{t},\bar{m}_{t}]-{\rm
Disc}[\bar{m}_{Z}^{2},\bar{m}_{t},\bar{m}_{t}]\right)\right.$ (C.18)
$\displaystyle\phantom{N_{4}}\left.+(\bar{m}_{H}^{2}-4\bar{m}_{t}^{2}-\bar{m}_{Z}^{2})\left(\log\left[\frac{2\bar{m}_{t}^{2}-\bar{m}_{Z}^{2}+\sqrt{\bar{m}_{Z}^{4}-4\bar{m}_{Z}^{2}\bar{m}_{t}^{2}}}{2\bar{m}_{t}^{2}}\right]^{2}-\log\left[\frac{2\bar{m}_{t}^{2}-\bar{m}_{h}^{2}+\sqrt{\bar{m}_{H}^{4}-4\bar{m}_{H}^{2}\bar{m}_{t}^{2}}}{2\bar{m}_{t}^{2}}\right]^{2}\right)\right]+\mathcal{O}(k_{i}^{2},\mu^{2})$
$\displaystyle=$
$\displaystyle\frac{\bar{e}^{2}}{4\pi^{2}}\frac{N_{c}Q_{u}(3-8\bar{s}_{W}^{2})}{12\bar{s}_{W}\bar{c}_{W}\,v}\left[\frac{1}{3}+\frac{7}{90}a+\frac{11}{90}b+\mathcal{O}(a^{2},b^{2})\right]+\mathcal{O}(k_{i}^{2},\mu^{2})\qquad
a\equiv\frac{\bar{m}_{H}^{2}}{4\bar{m}_{t}^{2}}\qquad
b\equiv\frac{\bar{m}_{Z}^{2}}{4\bar{m}_{t}^{2}}$ (C.19)

Below are the special functions used in the above expressions. Not all
$C_{0}^{S}$ functions are listed as the expressions become complicated and
depend on the relation between $b$ and $c$ which is generally a function of
the gauge parameter $\xi$. The unlisted $C_{0}^{S}$ functions can be obtained
from Package-X[49].

$\displaystyle{\rm Disc}[a^{2},b,c]$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\lambda[a^{2},b^{2},c^{2}]}}{a^{2}}\log\left(\frac{b^{2}+c^{2}-a^{2}+\sqrt{\lambda[a^{2}+b^{2}+c^{2}]}}{2bc}\right)$
(C.20) $\displaystyle\lambda[a^{2},b^{2},c^{2}]$ $\displaystyle=$
$\displaystyle a^{4}+b^{4}+c^{4}-2a^{2}b^{2}-2a^{2}c^{2}-2b^{2}c^{2}$ (C.21)
$\displaystyle C_{0}^{S}[0,0,a^{2},b,c,c]$ $\displaystyle=$
$\displaystyle\log\left(1-\frac{2a^{2}}{a^{2}+b^{2}-c^{2}+\sqrt{\lambda[a^{2},b^{2},c^{2}]}}\right)^{2}+2{\rm
Li}_{2}\left(1-\frac{b^{2}}{c^{2}}\right)$ $\displaystyle-2{\rm
Li}_{2}\left(\frac{2a^{2}}{a^{2}+b^{2}-c^{2}+\sqrt{\lambda[a^{2},b^{2},c^{2}]}}\right)+2{\rm
Li}_{2}\left(\frac{2a^{2}}{a^{2}-b^{2}+c^{2}+\sqrt{\lambda[a^{2},b^{2},c^{2}]}}\right)$
$\displaystyle=$ $\displaystyle C_{0}^{S}[0,0,a^{2},c,c,b]$ (C.23)

##  Appendix D Full Parameterized partial widths

###  D.1 Full $\alpha$–scheme results

$\displaystyle\Delta_{\alpha}^{R_{1}}$ $\displaystyle=$
$\displaystyle+0.47\tilde{c}_{H\Box}^{(6)}-0.12\tilde{c}_{HD}^{(6)}-0.059\tilde{c}_{HD}^{(8)}-0.059\tilde{c}_{HD,2}^{(8)}+0.94[\tilde{c}_{H\Box}^{(6)}]^{2}-0.47\tilde{c}_{H\Box}^{(6)}\tilde{c}_{HD}^{(6)}+0.059[\tilde{c}_{HD}^{(6)}]^{2}$
$\displaystyle-0.5\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}+0.27\frac{\hat{v}^{2}}{\hat{m}_{\mu}^{2}}[\tilde{c}_{eH}^{(6)}]^{2}-0.25\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(8)}-0.5\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{H\Box}^{(6)}+0.13\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle+0.0038\tilde{c}_{H\Box}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.00096\tilde{c}_{HD}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.0061\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}+9.4\cdot 10^{-9}\frac{\delta G_{F}}{\hat{G}_{F}}$
$\displaystyle+10^{4}\left(+5.6[\tilde{c}_{HB}^{(6)}]^{2}+1.8\tilde{c}_{HB}^{(6)}\tilde{c}_{HW}^{(6)}+1.1[\tilde{c}_{HW}^{(6)}]^{2}-4.7\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}-2.4\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}+1.7[\tilde{c}_{HWB}^{(6)}]^{2}\right)$
$\displaystyle+10^{11}\left(+2.4[\tilde{c}_{eB}^{(6)}]^{2}-2.7\tilde{c}_{eB}^{(6)}\tilde{c}_{eW}^{(6)}+0.73[\tilde{c}_{eW}^{(6)}]^{2}\right)$
$\displaystyle-260.\left(\tilde{c}_{HB}^{(6)}+\tilde{c}_{HB}^{(8)}\right)-76.\left(\tilde{c}_{HW}^{(6)}+\tilde{c}_{HW}^{(8)}+\tilde{c}_{HW,2}^{(8)}\right)+140.\left(\tilde{c}_{HWB}^{(6)}+\tilde{c}_{HWB}^{(8)}\right)$
$\displaystyle-520.\left[\tilde{c}_{HB}^{(6)}\right]^{2}-150.\left[\tilde{c}_{HW}^{(6)}\right]^{2}+16.\left[\tilde{c}_{HWB}^{(6)}\right]^{2}+560.\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}+0.03\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}$
$\displaystyle-260.\tilde{c}_{HB}^{(6)}\tilde{c}_{H\Box}^{(6)}+120.\tilde{c}_{HB}^{(6)}\tilde{c}_{HD}^{(6)}-76.\tilde{c}_{HW}^{(6)}\tilde{c}_{H\Box}^{(6)}-41.\tilde{c}_{HW}^{(6)}\tilde{c}_{HD}^{(6)}+140.\tilde{c}_{HWB}^{(6)}\tilde{c}_{H\Box}^{(6)}+4.\tilde{c}_{HWB}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle-4.9\tilde{c}_{HB}^{(6)}\tilde{c}_{He}^{(6)}-7.7\tilde{c}_{HB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)+4.9\tilde{c}_{HW}^{(6)}\tilde{c}_{He}^{(6)}+7.7\tilde{c}_{HW}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)-3.2\tilde{c}_{HWB}^{(6)}\tilde{c}_{He}^{(6)}$
$\displaystyle-4.9\tilde{c}_{HWB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)$
$\displaystyle+0.000016\tilde{c}_{HB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.000016\tilde{c}_{HW}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}+0.00001\tilde{c}_{HWB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}$
$\displaystyle-0.09\tilde{c}_{BH^{4}D^{2}}^{(8),1}+0.025\tilde{c}_{WH^{4}D^{2}}^{(8),1}$
$\displaystyle-1.3\tilde{c}_{e^{2}BH^{2}D}^{(8),1}-0.22\left(\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),2}\right)+0.69\tilde{c}_{e^{2}WH^{2}D}^{(8),1}-0.12\left(\tilde{c}_{L^{2}BH^{2}D}^{(8),1}+\tilde{c}_{L^{2}BH^{2}D}^{(8),5}\right)$
$\displaystyle-0.023\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),2}\right)-0.012\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),3}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),4}\right)+0.067\left(\tilde{c}_{l^{2}WH^{2}D}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),5}\right)$

$\displaystyle\Delta_{\alpha}^{R_{2}}$ $\displaystyle=$
$\displaystyle+0.11\tilde{c}_{H\Box}^{(6)}-0.029\tilde{c}_{HD}^{(6)}-0.014\tilde{c}_{HD}^{(8)}-0.014\tilde{c}_{HD,2}^{(8)}+0.23[\tilde{c}_{H\Box}^{(6)}]^{2}-0.11\tilde{c}_{H\Box}^{(6)}\tilde{c}_{HD}^{(6)}+0.014[\tilde{c}_{HD}^{(6)}]^{2}$
$\displaystyle-0.12\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}+0.065\frac{\hat{v}^{2}}{\hat{m}_{\mu}^{2}}[\tilde{c}_{eH}^{(6)}]^{2}-0.061\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(8)}-0.12\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{H\Box}^{(6)}+0.03\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle+0.00093\tilde{c}_{H\Box}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.00023\tilde{c}_{HD}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.0015\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}+2.3\cdot 10^{-9}\frac{\delta G_{F}}{\hat{G}_{F}}$
$\displaystyle+10^{5}\left(+1.4[\tilde{c}_{HB}^{(6)}]^{2}+0.83\tilde{c}_{HB}^{(6)}\tilde{c}_{HW}^{(6)}+0.13[\tilde{c}_{HW}^{(6)}]^{2}-1.5\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}-0.46\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}+0.42[\tilde{c}_{HWB}^{(6)}]^{2}\right)$
$\displaystyle+10^{11}\left(+9.[\tilde{c}_{eB}^{(6)}]^{2}-9.8\tilde{c}_{eB}^{(6)}\tilde{c}_{eW}^{(6)}+2.7[\tilde{c}_{eW}^{(6)}]^{2}\right)$
$\displaystyle-720.\left(\tilde{c}_{HB}^{(6)}+\tilde{c}_{HB}^{(8)}\right)-220.\left(\tilde{c}_{HW}^{(6)}+\tilde{c}_{HW}^{(8)}+\tilde{c}_{HW,2}^{(8)}\right)+400.\left(\tilde{c}_{HWB}^{(6)}+\tilde{c}_{HWB}^{(8)}\right)$
$\displaystyle-1400.\left[\tilde{c}_{HB}^{(6)}\right]^{2}-430.\left[\tilde{c}_{HW}^{(6)}\right]^{2}+30.\left[\tilde{c}_{HWB}^{(6)}\right]^{2}+1600.\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}+6.4\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}$
$\displaystyle-720.\tilde{c}_{HB}^{(6)}\tilde{c}_{H\Box}^{(6)}+340.\tilde{c}_{HB}^{(6)}\tilde{c}_{HD}^{(6)}-220.\tilde{c}_{HW}^{(6)}\tilde{c}_{H\Box}^{(6)}-110.\tilde{c}_{HW}^{(6)}\tilde{c}_{HD}^{(6)}+400.\tilde{c}_{HWB}^{(6)}\tilde{c}_{H\Box}^{(6)}+5.9\tilde{c}_{HWB}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle-18.\tilde{c}_{HB}^{(6)}\tilde{c}_{He}^{(6)}-14.\tilde{c}_{HB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)+18.\tilde{c}_{HW}^{(6)}\tilde{c}_{He}^{(6)}+14.\tilde{c}_{HW}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)-12.\tilde{c}_{HWB}^{(6)}\tilde{c}_{He}^{(6)}$
$\displaystyle-8.7\tilde{c}_{HWB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)$
$\displaystyle+0.000044\tilde{c}_{HB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.000044\tilde{c}_{HW}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}+0.000028\tilde{c}_{HWB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}$
$\displaystyle+0.05\tilde{c}_{BH^{4}D^{2}}^{(8),1}-0.014\tilde{c}_{WH^{4}D^{2}}^{(8),1}$
$\displaystyle-3.1\tilde{c}_{e^{2}BH^{2}D}^{(8),1}-0.55\left(\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),2}\right)+1.7\tilde{c}_{e^{2}WH^{2}D}^{(8),1}-2.4\left(\tilde{c}_{L^{2}BH^{2}D}^{(8),1}+\tilde{c}_{L^{2}BH^{2}D}^{(8),5}\right)$
$\displaystyle-0.42\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),2}\right)-0.21\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),3}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),4}\right)+1.3\left(\tilde{c}_{l^{2}WH^{2}D}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),5}\right)$

$\displaystyle\Delta_{\alpha}^{R_{3}}$ $\displaystyle=$
$\displaystyle+0.21\tilde{c}_{H\Box}^{(6)}-0.053\tilde{c}_{HD}^{(6)}-0.026\tilde{c}_{HD}^{(8)}-0.026\tilde{c}_{HD,2}^{(8)}+0.42[\tilde{c}_{H\Box}^{(6)}]^{2}-0.21\tilde{c}_{H\Box}^{(6)}\tilde{c}_{HD}^{(6)}+0.026[\tilde{c}_{HD}^{(6)}]^{2}$
$\displaystyle-0.22\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}+0.12\frac{\hat{v}^{2}}{\hat{m}_{\mu}^{2}}[\tilde{c}_{eH}^{(6)}]^{2}-0.11\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(8)}-0.22\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{H\Box}^{(6)}+0.056\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle+0.0017\tilde{c}_{H\Box}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.00043\tilde{c}_{HD}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.0027\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}+4.2*10^{-9}\frac{\delta G_{F}}{\hat{G}_{F}}$
$\displaystyle+10^{4}\left(+1.4[\tilde{c}_{HB}^{(6)}]^{2}-2.5\tilde{c}_{HB}^{(6)}\tilde{c}_{HW}^{(6)}+1.3[\tilde{c}_{HW}^{(6)}]^{2}+1.5\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}-1.6\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}+0.55[\tilde{c}_{HWB}^{(6)}]^{2}\right)$
$\displaystyle+10^{11}\left(+1.1[\tilde{c}_{eB}^{(6)}]^{2}-1.2\tilde{c}_{eB}^{(6)}\tilde{c}_{eW}^{(6)}+0.32[\tilde{c}_{eW}^{(6)}]^{2}\right)$
$\displaystyle-16.\left(\tilde{c}_{HB}^{(6)}+\tilde{c}_{HB}^{(8)}\right)+9.\left(\tilde{c}_{HW}^{(6)}+\tilde{c}_{HW}^{(8)}+\tilde{c}_{HW,2}^{(8)}\right)-3.9\left(\tilde{c}_{HWB}^{(6)}+\tilde{c}_{HWB}^{(8)}\right)$
$\displaystyle-31.\left[\tilde{c}_{HB}^{(6)}\right]^{2}+18.\left[\tilde{c}_{HW}^{(6)}\right]^{2}+31.\left[\tilde{c}_{HWB}^{(6)}\right]^{2}+2.7\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}-18.\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}$
$\displaystyle-16.\tilde{c}_{HB}^{(6)}\tilde{c}_{H\Box}^{(6)}+8.7\tilde{c}_{HB}^{(6)}\tilde{c}_{HD}^{(6)}+9.\tilde{c}_{HW}^{(6)}\tilde{c}_{H\Box}^{(6)}-7.1\tilde{c}_{HW}^{(6)}\tilde{c}_{HD}^{(6)}-3.9\tilde{c}_{HWB}^{(6)}\tilde{c}_{H\Box}^{(6)}+9.8\tilde{c}_{HWB}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle+4.4\tilde{c}_{HB}^{(6)}\tilde{c}_{He}^{(6)}-16.\tilde{c}_{HB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)-4.4\tilde{c}_{HW}^{(6)}\tilde{c}_{He}^{(6)}+16.\tilde{c}_{HW}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)+2.8\tilde{c}_{HWB}^{(6)}\tilde{c}_{He}^{(6)}$
$\displaystyle-10.\tilde{c}_{HWB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)$
$\displaystyle+1.3\cdot 10^{-6}\tilde{c}_{HB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-1.3\cdot 10^{-6}\tilde{c}_{HW}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}+2.4\cdot 10^{-6}\tilde{c}_{HWB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}$
$\displaystyle-0.51\tilde{c}_{BH^{4}D^{2}}^{(8),1}+0.14\tilde{c}_{WH^{4}D^{2}}^{(8),1}$
$\displaystyle-1.\tilde{c}_{e^{2}BH^{2}D}^{(8),1}-0.18\left(\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),2}\right)+0.56\tilde{c}_{e^{2}WH^{2}D}^{(8),1}+0.51\left(\tilde{c}_{L^{2}BH^{2}D}^{(8),1}+\tilde{c}_{L^{2}BH^{2}D}^{(8),5}\right)$
$\displaystyle+0.089\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),2}\right)+0.045\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),3}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),4}\right)-0.28\left(\tilde{c}_{l^{2}WH^{2}D}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),5}\right)$

$\displaystyle\Delta_{\alpha}^{R_{4}}$ $\displaystyle=$
$\displaystyle+1.9\tilde{c}_{H\Box}^{(6)}-0.49\tilde{c}_{HD}^{(6)}-0.24\tilde{c}_{HD}^{(8)}-0.24\tilde{c}_{HD,2}^{(8)}+3.9[\tilde{c}_{H\Box}^{(6)}]^{2}-1.9\tilde{c}_{H\Box}^{(6)}\tilde{c}_{HD}^{(6)}+0.24[\tilde{c}_{HD}^{(6)}]^{2}$
$\displaystyle-2.1\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}+1.1\frac{\hat{v}^{2}}{\hat{m}_{\mu}^{2}}[\tilde{c}_{eH}^{(6)}]^{2}-1.\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(8)}-2.1\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{H\Box}^{(6)}+0.52\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle+0.016\tilde{c}_{H\Box}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.004\tilde{c}_{HD}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.025\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}+3.9\cdot 10^{-8}\frac{\delta G_{F}}{\hat{G}_{F}}$
$\displaystyle+10^{2}\left(+4.9[\tilde{c}_{HB}^{(6)}]^{2}-5.8\tilde{c}_{HB}^{(6)}\tilde{c}_{HW}^{(6)}+3.8[\tilde{c}_{HW}^{(6)}]^{2}+2.6\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}-5.4\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}+2.1[\tilde{c}_{HWB}^{(6)}]^{2}\right)$
$\displaystyle+10^{10}\left(+2.4[\tilde{c}_{eB}^{(6)}]^{2}-2.7\tilde{c}_{eB}^{(6)}\tilde{c}_{eW}^{(6)}+0.73[\tilde{c}_{eW}^{(6)}]^{2}\right)$
$\displaystyle+4.6\left(\tilde{c}_{HB}^{(6)}+\tilde{c}_{HB}^{(8)}\right)-6.1\left(\tilde{c}_{HW}^{(6)}+\tilde{c}_{HW}^{(8)}+\tilde{c}_{HW,2}^{(8)}\right)+4.3\left(\tilde{c}_{HWB}^{(6)}+\tilde{c}_{HWB}^{(8)}\right)$
$\displaystyle+9.1\left[\tilde{c}_{HB}^{(6)}\right]^{2}-12.\left[\tilde{c}_{HW}^{(6)}\right]^{2}-14.\left[\tilde{c}_{HWB}^{(6)}\right]^{2}+11.\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}+5.7\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}$
$\displaystyle+4.6\tilde{c}_{HB}^{(6)}\tilde{c}_{H\Box}^{(6)}-2.\tilde{c}_{HB}^{(6)}\tilde{c}_{HD}^{(6)}-6.1\tilde{c}_{HW}^{(6)}\tilde{c}_{H\Box}^{(6)}+2.3\tilde{c}_{HW}^{(6)}\tilde{c}_{HD}^{(6)}+4.3\tilde{c}_{HWB}^{(6)}\tilde{c}_{H\Box}^{(6)}-4.8\tilde{c}_{HWB}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle-4.\tilde{c}_{HB}^{(6)}\tilde{c}_{He}^{(6)}+7.2\tilde{c}_{HB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)+4.\tilde{c}_{HW}^{(6)}\tilde{c}_{He}^{(6)}-7.2\tilde{c}_{HW}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)-2.5\tilde{c}_{HWB}^{(6)}\tilde{c}_{He}^{(6)}$
$\displaystyle+4.6\tilde{c}_{HWB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)$
$\displaystyle-2.2\cdot 10^{-7}\tilde{c}_{HB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}+2.2\cdot 10^{-7}\tilde{c}_{HW}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-9.9\cdot 10^{-7}\tilde{c}_{HWB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}$
$\displaystyle+0.28\tilde{c}_{BH^{4}D^{2}}^{(8),1}-0.076\tilde{c}_{WH^{4}D^{2}}^{(8),1}$
$\displaystyle+0.22\tilde{c}_{e^{2}BH^{2}D}^{(8),1}+0.04\left(\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),2}\right)-0.12\tilde{c}_{e^{2}WH^{2}D}^{(8),1}-0.42\left(\tilde{c}_{L^{2}BH^{2}D}^{(8),1}+\tilde{c}_{L^{2}BH^{2}D}^{(8),5}\right)$
$\displaystyle-0.074\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),2}\right)-0.037\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),3}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),4}\right)+0.23\left(\tilde{c}_{l^{2}WH^{2}D}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),5}\right)$

$\displaystyle\Delta_{\alpha}^{R_{5}}$ $\displaystyle=$
$\displaystyle+0.93\tilde{c}_{H\Box}^{(6)}-0.23\tilde{c}_{HD}^{(6)}-0.12\tilde{c}_{HD}^{(8)}-0.12\tilde{c}_{HD,2}^{(8)}+1.9[\tilde{c}_{H\Box}^{(6)}]^{2}-0.93\tilde{c}_{H\Box}^{(6)}\tilde{c}_{HD}^{(6)}+0.12[\tilde{c}_{HD}^{(6)}]^{2}$
$\displaystyle-0.99\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}+0.52\frac{\hat{v}^{2}}{\hat{m}_{\mu}^{2}}[\tilde{c}_{eH}^{(6)}]^{2}-0.49\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(8)}-0.99\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{H\Box}^{(6)}+0.25\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle+0.0076\tilde{c}_{H\Box}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.0019\tilde{c}_{HD}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.012\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}+1.9\cdot 10^{-8}\frac{\delta G_{F}}{\hat{G}_{F}}$
$\displaystyle+10^{3}\left(+7.4[\tilde{c}_{HB}^{(6)}]^{2}-17.\tilde{c}_{HB}^{(6)}\tilde{c}_{HW}^{(6)}+6.9[\tilde{c}_{HW}^{(6)}]^{2}+11.\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}-8.\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}+2.1[\tilde{c}_{HWB}^{(6)}]^{2}\right)$
$\displaystyle+10^{11}\left(+7.7[\tilde{c}_{eB}^{(6)}]^{2}-8.5\tilde{c}_{eB}^{(6)}\tilde{c}_{eW}^{(6)}+2.3[\tilde{c}_{eW}^{(6)}]^{2}\right)$
$\displaystyle+150.\left(\tilde{c}_{HB}^{(6)}+\tilde{c}_{HB}^{(8)}\right)-120.\left(\tilde{c}_{HW}^{(6)}+\tilde{c}_{HW}^{(8)}+\tilde{c}_{HW,2}^{(8)}\right)+68.\left(\tilde{c}_{HWB}^{(6)}+\tilde{c}_{HWB}^{(8)}\right)$
$\displaystyle+300.\left[\tilde{c}_{HB}^{(6)}\right]^{2}-240.\left[\tilde{c}_{HW}^{(6)}\right]^{2}-160.\left[\tilde{c}_{HWB}^{(6)}\right]^{2}+380.\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}-110.\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}$
$\displaystyle+150.\tilde{c}_{HB}^{(6)}\tilde{c}_{H\Box}^{(6)}-18.\tilde{c}_{HB}^{(6)}\tilde{c}_{HD}^{(6)}-120.\tilde{c}_{HW}^{(6)}\tilde{c}_{H\Box}^{(6)}+10.\tilde{c}_{HW}^{(6)}\tilde{c}_{HD}^{(6)}+68.\tilde{c}_{HWB}^{(6)}\tilde{c}_{H\Box}^{(6)}-74.\tilde{c}_{HWB}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle-170.\tilde{c}_{HB}^{(6)}\tilde{c}_{He}^{(6)}+92.\tilde{c}_{HB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)+170.\tilde{c}_{HW}^{(6)}\tilde{c}_{He}^{(6)}-92.\tilde{c}_{HW}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)-110.\tilde{c}_{HWB}^{(6)}\tilde{c}_{He}^{(6)}$
$\displaystyle+59.\tilde{c}_{HWB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)$
$\displaystyle+5.2\cdot 10^{-6}\tilde{c}_{HB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-5.2\cdot 10^{-6}\tilde{c}_{HW}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.000015\tilde{c}_{HWB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}$
$\displaystyle+6.1\tilde{c}_{BH^{4}D^{2}}^{(8),1}-1.7\tilde{c}_{WH^{4}D^{2}}^{(8),1}$
$\displaystyle-9.6\tilde{c}_{e^{2}BH^{2}D}^{(8),1}-1.7\left(\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),2}\right)+5.3\tilde{c}_{e^{2}WH^{2}D}^{(8),1}+6.4\left(\tilde{c}_{L^{2}BH^{2}D}^{(8),1}+\tilde{c}_{L^{2}BH^{2}D}^{(8),5}\right)$
$\displaystyle+1.1\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),2}\right)+0.56\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),3}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),4}\right)-3.5\left(\tilde{c}_{l^{2}WH^{2}D}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),5}\right)$

$\displaystyle\Delta_{\alpha}^{R_{6}}$ $\displaystyle=$
$\displaystyle+0.82\tilde{c}_{H\Box}^{(6)}-0.2\tilde{c}_{HD}^{(6)}-0.1\tilde{c}_{HD}^{(8)}-0.1\tilde{c}_{HD,2}^{(8)}+1.6[\tilde{c}_{H\Box}^{(6)}]^{2}-0.82\tilde{c}_{H\Box}^{(6)}\tilde{c}_{HD}^{(6)}+0.1[\tilde{c}_{HD}^{(6)}]^{2}$
$\displaystyle-0.87\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}+0.46\frac{\hat{v}^{2}}{\hat{m}_{\mu}^{2}}[\tilde{c}_{eH}^{(6)}]^{2}-0.43\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(8)}-0.87\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{H\Box}^{(6)}+0.22\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle+0.0066\tilde{c}_{H\Box}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.0017\tilde{c}_{HD}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.011\frac{\hat{v}}{\hat{m}_{\mu}}\tilde{c}_{eH}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-8+1.6*10\frac{\delta G_{F}}{\hat{G}_{F}}$
$\displaystyle+10^{4}\left(+3.8[\tilde{c}_{HB}^{(6)}]^{2}+0.66\tilde{c}_{HB}^{(6)}\tilde{c}_{HW}^{(6)}+0.78[\tilde{c}_{HW}^{(6)}]^{2}-2.7\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}-1.6\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}+1.1[\tilde{c}_{HWB}^{(6)}]^{2}\right)$
$\displaystyle+10^{12}\left(+1.4[\tilde{c}_{eB}^{(6)}]^{2}-1.6\tilde{c}_{eB}^{(6)}\tilde{c}_{eW}^{(6)}+0.43[\tilde{c}_{eW}^{(6)}]^{2}\right)$
$\displaystyle-50.\left(\tilde{c}_{HB}^{(6)}+\tilde{c}_{HB}^{(8)}\right)-130.\left(\tilde{c}_{HW}^{(6)}+\tilde{c}_{HW}^{(8)}+\tilde{c}_{HW,2}^{(8)}\right)+140.\left(\tilde{c}_{HWB}^{(6)}+\tilde{c}_{HWB}^{(8)}\right)$
$\displaystyle-100.\left[\tilde{c}_{HB}^{(6)}\right]^{2}-270.\left[\tilde{c}_{HW}^{(6)}\right]^{2}-82.\left[\tilde{c}_{HWB}^{(6)}\right]^{2}+660.\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}-120.\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}$
$\displaystyle-50.\tilde{c}_{HB}^{(6)}\tilde{c}_{H\Box}^{(6)}+72.\tilde{c}_{HB}^{(6)}\tilde{c}_{HD}^{(6)}-130.\tilde{c}_{HW}^{(6)}\tilde{c}_{H\Box}^{(6)}-26.\tilde{c}_{HW}^{(6)}\tilde{c}_{HD}^{(6)}+140.\tilde{c}_{HWB}^{(6)}\tilde{c}_{H\Box}^{(6)}-47.\tilde{c}_{HWB}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle-140.\tilde{c}_{HB}^{(6)}\tilde{c}_{He}^{(6)}+51.\tilde{c}_{HB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)+140.\tilde{c}_{HW}^{(6)}\tilde{c}_{He}^{(6)}-51.\tilde{c}_{HW}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)-88.\tilde{c}_{HWB}^{(6)}\tilde{c}_{He}^{(6)}$
$\displaystyle+33.\tilde{c}_{HWB}^{(6)}\left(\tilde{c}_{Hl}^{(6),1}+\tilde{c}_{Hl}^{(6),3}\right)$
$\displaystyle+0.000016\tilde{c}_{HB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.000016\tilde{c}_{HW}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-3.4\cdot 10^{-6}\tilde{c}_{HWB}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}$
$\displaystyle+4.4\tilde{c}_{BH^{4}D^{2}}^{(8),1}-1.2\tilde{c}_{WH^{4}D^{2}}^{(8),1}$
$\displaystyle-10.\tilde{c}_{e^{2}BH^{2}D}^{(8),1}-1.8\left(\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{e^{2}H^{2}D^{3}}^{(8),2}\right)+5.6\tilde{c}_{e^{2}WH^{2}D}^{(8),1}+2.8\left(\tilde{c}_{L^{2}BH^{2}D}^{(8),1}+\tilde{c}_{L^{2}BH^{2}D}^{(8),5}\right)$
$\displaystyle+0.48\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),2}\right)+0.24\left(\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),3}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),4}\right)-1.5\left(\tilde{c}_{l^{2}WH^{2}D}^{(8),1}+\tilde{c}_{l^{2}H^{2}D^{3}}^{(8),5}\right)$

###  D.2 $M_{W}$ scheme results

The parameterized partial widths in the $M_{W}$ input parameter scheme for
each region defined in the main text are given below. When solving for
$\bar{s}_{W}$ in terms of the input parameters we obtain terms proportional to
$\sqrt{[c_{HWB}^{(6)}]^{2}}=|c_{HWB}^{(6)}|$. This is not mentioned explicitly
in the Appendix of [13] where equations useful for solving barred quantities
for input parameters are found. The authors of [13] have found a work around,
but this was not apparent to the authors of this article. Note that in the
$\hat{M}_{W}$ scheme Case 5 has corrections from the Class 4 operators which
shift the definition of the $\alpha$, in the $\hat{\alpha}$ scheme this is not
the case as $\alpha$ is an input parameter.

$\displaystyle\Delta_{M_{W}}^{R_{1}}$ $\displaystyle=$
$\displaystyle+0.48\tilde{c}_{H\Box}^{(6)}-0.12\tilde{c}_{HD}^{(6)}-0.06\tilde{c}_{HD}^{(8)}-0.06\tilde{c}_{HD,2}^{(8)}+0.96[\tilde{c}_{H\Box}^{(6)}]^{2}-1.1\tilde{c}_{H\Box}^{(6)}\tilde{c}_{HD}^{(6)}+0.22[\tilde{c}_{HD}^{(6)}]^{2}$
$\displaystyle-0.51\frac{\hat{v}}{\bar{m}_{\mu}}\tilde{c}_{eH}^{(6)}+0.27\frac{\hat{v}^{2}}{\bar{m}_{\mu}^{2}}[\tilde{c}_{eH}^{(6)}]^{2}-0.25\frac{\hat{v}}{\bar{m}_{\mu}}\tilde{c}_{eH}^{(8)}-0.51\frac{\hat{v}}{\bar{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{H\Box}^{(6)}+0.8\frac{\hat{v}}{\bar{m}_{\mu}}\tilde{c}_{eH}^{(6)}\tilde{c}_{HD}^{(6)}$
$\displaystyle+0.0039\tilde{c}_{H\Box}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.00097\tilde{c}_{HD}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}-0.0062\frac{\hat{v}}{\bar{m}_{\mu}}\tilde{c}_{eH}^{(6)}\frac{\delta
G_{F}}{\hat{G}_{F}}+9.5\cdot 10^{-9}\frac{\delta G_{F}}{\hat{G}_{F}}$
$\displaystyle-0.68\tilde{c}_{H\Box}^{(6)}|\tilde{c}_{HWB}^{(6)}|+0.17\tilde{c}_{HD}^{(6)}|\tilde{c}_{HWB}^{(6)}|+0.72\frac{\hat{v}}{\bar{m}_{\mu}}\tilde{c}_{eH}^{(6)}|\tilde{c}_{HWB}^{(6)}|$
$\displaystyle+10^{4}\left(+5.8[\tilde{c}_{HB}^{(6)}]^{2}+1.7\tilde{c}_{HB}^{(6)}\tilde{c}_{HW}^{(6)}+1.[\tilde{c}_{HW}^{(6)}]^{2}-4.7\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}-2.4\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}+1.7[\tilde{c}_{HWB}^{(6)}]^{2}\right)$
$\displaystyle+10^{11}\left(+2.6[\tilde{c}_{eB}^{(6)}]^{2}-2.7\tilde{c}_{eB}^{(6)}\tilde{c}_{eW}^{(6)}+0.73[\tilde{c}_{eW}^{(6)}]^{2}\right)$
$\displaystyle-260.\left(\tilde{c}_{HB}^{(6)}+\tilde{c}_{HB}^{(8)}\right)-71.\left(\tilde{c}_{HW}^{(6)}+\tilde{c}_{HW}^{(8)}+\tilde{c}_{HW,2}^{(8)}\right)+140.\left(\tilde{c}_{HWB}^{(6)}+\tilde{c}_{HWB}^{(8)}\right)$
$\displaystyle-520.\left[\tilde{c}_{HB}^{(6)}\right]^{2}-140.\left[\tilde{c}_{HW}^{(6)}\right]^{2}-230.\left[\tilde{c}_{HWB}^{(6)}\right]^{2}+400.\tilde{c}_{HB}^{(6)}\tilde{c}_{HWB}^{(6)}+310.\tilde{c}_{HW}^{(6)}\tilde{c}_{HWB}^{(6)}$
$\displaystyle-260.\tilde{c}_{HB}^{(6)}\tilde{c}_{H\Box}^{(6)}+44.\tilde{c}_{HB}^{(6)}\tilde{c}_{HD}^{(6)}-71.\tilde{c}_{HW}^{(6)}\tilde{c}_{H\Box}^{(6)}+190.\tilde{c}_{HW}^{(6)}\tilde{c}_{HD}^{(6)}+140.\tilde{c}_{HWB}^{(6)}\tilde{c}_{H\Box}^{(6)}-190.\tilde{c}_{HWB}^{(6)}\tilde{c}_{HD}^{(6)}$