#  MUonE, muon $g-2$ and electroweak precision constraints within 2HDMs

Oliver Atkinson<EMAIL_ADDRESS>School of Physics and
Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom
Matthew Black<EMAIL_ADDRESS>Physik Department, Universität
Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany
Christoph Englert<EMAIL_ADDRESS>School of Physics and
Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom
Alexander Lenz<EMAIL_ADDRESS>Physik Department, Universität
Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany
Aleksey Rusov<EMAIL_ADDRESS>Physik Department, Universität
Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany

###### Abstract

Two Higgs doublet models are attractive scenarios for physics beyond the
Standard Model. In particular, lepton-specific manifestations remain
contenders to explain the observed discrepancy between the anomalous magnetic
moment of the muon $a_{\mu}$ predicted within the Standard Model and recent
observations at Fermilab and BNL. Dominant uncertainties that affect $a_{\mu}$
have motivated the MUonE experiment to access the hadronic vacuum polarisation
contribution that impacts $a_{\mu}$ via elastic muon-electron scattering. In
this work, we contrast the high precision that is achievable within the MUonE
context with constraints from flavour physics, precision electroweak
constraints and LHC searches as well as their extrapolations for a range of
two Higgs doublet models with a softly broken $\mathbb{Z}_{2}$ symmetry. We
find that the sensitivity of MUonE does not extend beyond the parameter
regions that are already excluded by other constraints. MUonE will therefore
provide a detailed measurement of the hadronic vacuum polarisation
contribution which then transparently informs $a_{\mu}$ interpretations in
2HDMs without modifications of correlations from beyond the Standard Model
interactions. In passing we extend earlier results of LHC and flavour
projections to lepton-specific 2HDM (Types X and Y) scenarios, and comment on
the possibility of modifying the value of the W-boson mass; we briefly discuss
the implications for a strong first-order electroweak phase transition for
these models.

###### pacs:

††preprint: SI-HEP-2022-15††preprint: P3H-22-070

##  I Introduction

Searches for new physics at the current high energy collider frontier of the
Large Hadron Collider (LHC) have not revealed any significant sign of new
interactions beyond the Standard Model (BSM). This stands in contrast to some
tensions in precision analyses in flavour physics, see e.g. Refs. Albrecht et
al. (2021); Gershon et al. (2022) as well as recent measurements of the muon
anomalous magnetic moment $a_{\mu}$ at Fermilab Abi et al. (2021). This
measurement confirmed the earlier tension observed at BNL Bennett et al.
(2006) from the standard model (SM) expectation Aoyama et al. (2020). Sources
of uncertainty that affect the SM prediction have given further motivation to
constraining the hadronic polarisation contribution to $\Delta
a_{\mu}^{\text{had}}$ as there exists significant deviation between some
lattice QCD predictions Borsanyi et al. (2021); Cè et al. (2022); Alexandrou
et al. (2022) and analytically driven techniques based on the $R$ ratio
Keshavarzi et al. (2018).

The MUonE experiment Abbiendi et al. (2017) proposed at CERN aims to achieve a
measurement of $\Delta a_{\mu}^{\text{had}}$ with a statistical uncertainty of
around 0.3% through a precision measurement of $\mu-e$ scattering of a 150 GeV
muon beam off atomic electrons, collecting $\sim 3.7\times 10^{12}$ scattering
events resulting from a (leading order) cross section of around
$250~{}\mu\text{b}$. Such scattering at relatively low centre-of-mass energy
implies a small $t$ channel momentum exchange compared to the $Z$ boson
threshold thus enabling the direct measurement of $\Delta
a_{\mu}^{\text{had}}$ (see also Keshavarzi et al. (2018); Balzani et al.
(2021)). The expected experimental precision is reflected by the continued
effort to reduce the theoretical uncertainty of the SM prediction of $\mu-e$
scattering Mastrolia et al. (2017); Gakh et al. (2018); Di Vita et al. (2018);
Carloni Calame et al. (2020); Banerjee et al. (2021); Budassi et al. (2021);
in parallel such precise measurements have implications for the BSM physics as
well, both in relation to $a_{\mu}$ Dev et al. (2020) and beyond Masiero et
al. (2020); Asai et al. (2021); Galon et al. (2022). Modifications due to new
electroweak states could be constrainable at the MUonE environment, in
particular when they appear at relatively low masses away from the decoupling
limit, where they might be difficult to observe at hadron colliders due to
overwhelming QCD backgrounds.

Particularly interesting in this context is the two Higgs doublet model, which
continues to have appealing phenomenological properties with regards to the UV
completion of the SM Dorsch et al. (2014, 2016); Basler et al. (2017, 2019,
2020); Gonçalves et al. (2022); Su et al. (2021); Wang (2021); Dorsch et al.
(2017); Atkinson et al. (2022a); Basler et al. (2021); Atkinson et al.
(2022b); Anisha et al. (2022). For $\mathbb{Z}_{2}$ symmetry assignments of
the two Higgs doublet models (2HDMs) that avoid tree-level neutral current
flavour violation Glashow and Weinberg (1977), we will show that there are
parameter regions that can explain the $a_{\mu}$ anomaly whilst avoiding Large
Electron Positron (LEP), LHC and flavour constraints (see also Athron et al.
(2021); Atkinson et al. (2022a, b)).

Any significant sensitivity to new physics beyond the hadronic vacuum
polarisation will require due caution when MUonE results are interpreted along
the latter lines. The question of the extent to which a precision experiment
like MUonE can provide additional sensitivity via modified electroweak
corrections is therefore relevant and timely. This is the focus of this work:
we include the MUonE sensitivity estimate from next-to-leading order
electroweak corrections arising in 2HDMs to a global analysis of LHC, LEP,
electroweak precision and flavour physics data. We particularly focus on the
relevance of lepton-specific $\mathbb{Z}_{2}$ charge assignments that naively
tension MUonE against flavour and collider measurements.

In addition, very recently, the CDF collaboration reported a new measurement
of the $W$-boson mass $m_{W}$ Aaltonen et al. (2022) which finds an excess of
$\sim 70\,$MeV ($7\sigma$) above the Standard Model fit Zyla et al. (2020). If
such a deviation from the Standard Model is confirmed, this can open the door
for a plethora of New Physics (NP) models which could explain this anomaly.
Previously Lopez-Val and Sola (2013) it was expected that any 2HDM would only
ever have much smaller corrections to $m_{W}$ than the deviation found in
Aaltonen et al. (2022), however revived analyses of flavour conserving 2HDMs
with a softly broken ${\mathbb{Z}}_{2}$ symmetry claim larger contributions
can arise, which could bridge this gap Bahl et al. (2022); Song et al. (2022);
Heo et al. (2022); Ahn et al. (2022); Lee et al. (2022); Benbrik et al.
(2022); Abouabid et al. (2022); Botella et al. (2022); Kim (2022); Kim et al.
(2022). It is therefore worthwhile to comment, in passing, on the corrections
to the $W$-boson mass within the 2HDM using the context of our fits from other
sectors to inform the parameter regions used, resulting in ranges of values
which $m_{W}$ could take from a particular 2HDM realisation. These ranges can
then be compared to future updates of the measurement of $m_{W}$ from any
experiment. Furthermore, we correlate these ranges with the corrections to
$a_{\mu}$ in the 2HDM, where we assume the Theory Initiative’s SM prediction
Aoyama et al. (2020), and find 2HDM parameters which can resolve either or
both of these tensions with experiment.

This work is organised as follows: Section II provides a brief overview of the
models studied in this work to make this paper self-consistent. Section III
gives a summary of these scenarios in the phenomenological context of MUonE.
Section IV is devoted to results. We summarise and conclude in Section V.

Model | I | II | X | Y
---|---|---|---|---
$u_{R}$ | $\Phi_{2}$ | $\Phi_{2}$ | $\Phi_{2}$ | $\Phi_{2}$
$d_{R}$ | $\Phi_{2}$ | $\Phi_{1}$ | $\Phi_{2}$ | $\Phi_{1}$
$e_{R}$ | $\Phi_{2}$ | $\Phi_{1}$ | $\Phi_{1}$ | $\Phi_{2}$
Table 1: The four Types of 2HDM that avoid tree level FCNC. By convention the
up-type quarks couple to $\Phi_{2}$.

##  II Models

The 2HDM invokes a pair of distinct complex $SU_{L}(2)$ doublets,
$\Phi_{1,2}$, where the SM has only a single doublet that serves to give mass
terms to both the up and down-type quarks. The vacuum expectation values
(VEVs) of these doublets are non-zero and satisfy $v_{1}^{2}+v_{2}^{2}=v^{2}$
for the SM VEV $v\approx 246$ GeV. The general potential for this 2HDM is, in
the notation of Branco et al. (2012); Gunion et al. (2000),

$V(\Phi_{1},\Phi_{2})=m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}-m_{12}^{2}(\Phi_{1}^{\dagger}\Phi_{2}+\Phi_{2}^{\dagger}\Phi_{1})\\\
+\frac{\lambda_{1}}{2}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\frac{\lambda_{2}}{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi^{\dagger}_{1}\Phi_{1})(\Phi^{\dagger}_{2}\Phi_{2})\\\
+\lambda_{4}(\Phi^{\dagger}_{1}\Phi_{2})(\Phi^{\dagger}_{2}\Phi_{1})+\frac{\lambda_{5}}{2}\left[(\Phi^{\dagger}_{1}\Phi_{2})^{2}+(\Phi^{\dagger}_{2}\Phi_{1})^{2}\right].$
(1)

This opens up the possibility of tree level flavour changing neutral currents
(FCNC), which can be removed by imposing a $\mathbb{Z}_{2}$ symmetry such that
each fermion type couples to only one of the Higgs doublets. There are four
such configurations, shown in Table 1, and thus four Types of 2HDM which we
examine here; we denote these by, for instance, 2HDM-I.

Electroweak symmetry breaking sees three of the eight degrees of freedom of
the doublets “eaten” by the weak bosons, the five remaining degrees of freedom
manifest as new particles; two neutral scalars $h^{0}$, $H^{0}$
($m_{h^{0}}<m_{H^{0}}$), two charged scalars, $H^{\pm}$ and a neutral
pseudoscalar $A^{0}$. Instead of the lambda basis of Eq. (1), in this work we
make use of the mass basis, which can be attained through the transformations
given in Atkinson et al. (2022a); Basler et al. (2017); Arnan et al. (2017);
Han et al. (2020); Kling et al. (2016), which allows us to focus on the
physical masses $m_{h^{0}}$, $m_{H^{0}}$, $m_{H^{\pm}}$, $m_{A^{0}}$, the
mixing angles $\cos(\beta-\alpha)$, $\tan\beta=v_{2}/v_{1}$ and the softly
$\mathbb{Z}_{2}$ breaking parameter $m_{12}^{2}$. We additionally take $h^{0}$
to be the scalar particle observed at the LHC, thereby fixing
$m_{h^{0}}=125.25\pm 0.17$ GeV Zyla et al. (2020).

Having transformed to the mass basis, the Yukawa sector of the Lagrangian of
the general 2HDM is given by Crivellin et al. (2019); Branco et al. (2012)

$\displaystyle{\cal L}^{\rm 2HDM}_{\rm Yukawa}$
$\displaystyle=-\\!\\!\sum\limits_{f=u,d,\ell}\\!\\!\frac{m_{f}}{v}\left(\xi_{h}^{f}\,\bar{f}fh+\xi_{H}^{f}\,\bar{f}fH+i\eta_{f}\xi_{A}^{f}\,\bar{f}\gamma_{5}fA\right)$
$\displaystyle+\left[\frac{\sqrt{2}V_{ud}}{v}\bar{u}\left(m_{d}\,\xi_{A}^{d}P_{R}-m_{u}\,\xi_{A}^{u}P_{L}\right)dH^{+}\right.$
$\displaystyle\left.+\frac{\sqrt{2}}{v}m_{\ell}\,\xi_{A}^{l}(\bar{\nu}P_{R}\ell)H^{+}+{\rm
h.c.}\right]\\!,$ (2)

where we follow the convention of Crivellin et al. (2019) for the $H^{+}$
coupling. The couplings $\xi$ are Type dependent and given in Table 2, and the
factor $\eta_{f}$ for fermion type $f=d,\ell$ is $1$ and for $f=u$ is $-1$.

In previous works Atkinson et al. (2022a, b) we have studied the 2HDM-I and
2HDM-II in depth and here bring the same machinery to bear on the 2HDM-X and
2HDM-Y (also known as lepton-specific and flipped respectively), as well as
investigating the implications from the high precision MUonE experiment for
all four models.

Model | I | II | X | Y
---|---|---|---|---
$\xi^{u}_{h}$ | $\cos\alpha/\sin\beta$ | $\cos\alpha/\sin\beta$ | $\cos\alpha/\sin\beta$ | $\cos\alpha/\sin\beta$
$\xi^{d}_{h}$ | $\cos\alpha/\sin\beta$ | $-\sin\alpha/\cos\beta$ | $\cos\alpha/\sin\beta$ | $-\sin\alpha/\cos\beta$
$\xi^{l}_{h}$ | $\cos\alpha/\sin\beta$ | $-\sin\alpha/\cos\beta$ | $-\sin\alpha/\cos\beta$ | $\cos\alpha/\sin\beta$
$\xi^{u}_{H}$ | $\sin\alpha/\sin\beta$ | $\sin\alpha/\sin\beta$ | $\sin\alpha/\sin\beta$ | $\sin\alpha/\sin\beta$
$\xi^{d}_{H}$ | $\sin\alpha/\sin\beta$ | $\cos\alpha/\cos\beta$ | $\sin\alpha/\sin\beta$ | $\cos\alpha/\cos\beta$
$\xi^{l}_{H}$ | $\sin\alpha/\sin\beta$ | $\cos\alpha/\cos\beta$ | $\cos\alpha/\cos\beta$ | $\sin\alpha/\sin\beta$
$\xi^{u}_{A}$ | $\cot\beta$ | $\phantom{-}\cot\beta$ | $\phantom{-}\cot\beta$ | $\phantom{-}\cot\beta$
$\xi^{d}_{A}$ | $\cot\beta$ | $-\tan\beta$ | $\phantom{-}\cot\beta$ | $-\tan\beta$
$\xi^{l}_{A}$ | $\cot\beta$ | $-\tan\beta$ | $-\tan\beta$ | $\phantom{-}\cot\beta$
Table 2: Coupling strengths $\xi$ in each Type of 2HDM between the Higgs
particles and fermions.

##  III 2HDMs at MUonE

Figure 1: Feynman diagram topologies contributing to $e-\mu$ scattering at one
loop level. Highlighted are renormalised vertex and propagator corrections for
the topologies (a)-(c). They include 2HDM Higgs contributions for the
propagator and $\mu$ vertex corrections. The different shading of (b)
indicates that we are neglecting electron mass contributions to the virtual
amplitude. The propagator contributions include leading order hadron
polarisation effects consistently.

The MUonE experiment aims to investigate elastic $e-\mu$ scattering. The Born-
level $e-\mu$ scattering is entirely SM-like, which highlights the precision
potential of the MUonE experiment to access virtual contributions that are
modified from the SM expectation. At (renormalised) one-loop level the virtual
${\cal{M}}_{\text{virt}}$ part of the next-to-leading order amplitude

$|{\cal{M}}|^{2}_{1}=|{\cal{M}}_{\text{Born}}|^{2}+2\text{Re}\left\\{{\cal{M}}_{\text{Born}}{\cal{M}}^{\ast}_{\text{virt}}\right\\}$
(3)

can be diagrammatically represented by the Feynman diagram toplogies of Fig.
1. As we are interested in the scenarios of Table 2 where the couplings are
lepton-flavour universal, and as $m_{e}\ll m_{\mu}$, we will neglect effects
$\sim m_{e}$ contributing to the virtual amplitude throughout this work. This
removes $t$-channel Higgs and Goldstone diagram contributions as well as
scalar contributions to the box diagrams in Fig. 1(d); we employ the on-shell
renormalisation scheme in the following. It is worthwhile highlighting that
MUonE probes a subset of oblique corrections that are included in our scan as
detailed in Section IV, however, probed at space-like $t$-channel momentum
transfers (the latter also means that width contributions to the $Z$ boson
propagators are absent).

While the $t$-channel photon contribution already results in a soft
singularity at tree-level (which is avoided by requiring a finite recoil
energy in the experimental setup), the presence of virtual massless
propagators connecting on-shell legs leads to additional soft (eikonal)
singularities of $|{\cal{M}}|^{2}_{1}$ which cancel against real photon
emission via the Kinoshita-Lee-Nauenberg theorem Kinoshita (1962); Lee and
Nauenberg (1964). For abelian gauge theories the eikonal approximation takes a
particularly compact form that is well-documented in the literature ’t Hooft
and Veltman (1979); Denner (1993) and is readily implemented in publicly
available packages like FormCalc Hahn and Perez-Victoria (1999). This real
emission part is not sensitive to the 2HDM modifications, but we consider it
for completeness, including unresolved soft photon radiation up to 10% of the
MUonE centre-of-mass energy.

Numerical expressions of the loop diagrams sketched in Fig. 1 are generated
using the FeynArts/FormCalc/LoopTools Hahn (1999); Hahn and Perez-Victoria
(1999); Hahn and Schappacher (2002) toolchain. We have checked the
cancellation of ultraviolet divergences both numerically and analytically for
the full amplitude, as well as the independence of our results of a virtual
photon mass that is introduced to regularise soft singularities at
intermediate steps. We have also validated our results against the SM by
approaching the 2HDM decoupling limits of the different scenarios considered
in this work.

As a $2\to 2$ scattering process, MUonE is entirely determined by the $t$
channel momentum transfer (we assume mono-chromatic muon beams and electron
targets), which determines the laboratory scattering angles and $\mu$ and $e$
recoil energies. For our analysis we consider 51 independent bins of a
scattering angle distribution (after validating the literature-documented
cross section of $\sim 240~{}\mu\text{b}$), which we enter in a $\chi^{2}$
hypothesis test given by

$\chi^{2}=\sum_{i}{(N_{i}-N_{i}^{\text{SM}})^{2}\over\sigma^{2}_{i,\text{syst}}+\sigma^{2}_{i,\text{stat}}}\,.$
(4)

Here, $i$ runs over the bins, $N_{i}$ are the 2HDM events in the $i$th bin and
$N_{i}^{\text{SM}}$ denotes the SM expectation (evaluated at next-to-leading
order). $\sigma_{i,\text{stat}},\sigma_{i,\text{syst}}$ denote statistical and
systematic uncertainties, respectively. In the following we will use
$\sigma_{i,\text{stat}}=\sqrt{N_{i}}$ and
$\sigma_{i,\text{syst}}=10^{-5}N_{i}$ as a common benchmark choice for the
expected sensitivity of MUonE Abbiendi et al. (2017); Dev et al. (2020). From
the $\chi^{2}$ we can infer the impact of a given parameter point on the MUonE
data and whether the experiment would be sensitive to the existence of the
2HDM with such parameters.

##  IV Constraints, Results, and MUonE relevance

Here we outline the various constraints that can be placed on the 2HDM from a
variety of sectors. We begin with an analysis of the signal strengths of the
SM-like Higgs observed at the LHC, proceeding to flavour observables where the
2HDM can have large indirect effects, and then to the impact of direct BSM
searches on the parameter space, including an extrapolation to future collider
capabilities for the latter. These analyses build on the toolchains developed
in our previous works Atkinson et al. (2022a, b); we make use of the python
package flavio Straub (2018) for the flavour fits and MadGraph5_aMC@NLO Alwall
et al. (2014), 2HDecay Krause et al. (2020); Djouadi et al. (1998, 2019);
Krause et al. (2017); Denner et al. (2018); Hahn and Perez-Victoria (1999),
and HiggsBounds Bechtle et al. (2010, 2011, 2012, 2014, 2015, 2020); Bahl et
al. (2021) for the BSM collider searches. We refer the interested reader to
Atkinson et al. (2022a, b) for full details on the fitting procedures used and
experimental data included, updated to additionally include the measurements
from Sirunyan et al. (2022); Aad et al. (2022), where we will avoid being
overly repetitive here and focus on the conclusions drawn from these analyses.

Figure 2:  Contour plots of the allowed regions at 1, 2, 3, 4, $5\sigma$ from
the $h^{0}$ signal strengths with the alignment and wrong sign limits shown as
black and red dashed lines respectively

###  IV.1 Higgs Signal Strengths

The signal strengths of the observed Higgs boson provide a measure of how
closely this boson matches the expected phenomenology of the SM Higgs, and are
defined as, for a channel with production mechanism $i$ and decay products
$f$,

$\mu^{f}_{i}=\frac{(\sigma_{i}\cdotp{\cal
B}_{f})_{\text{Exp.}}}{(\sigma_{i}\cdotp{\cal B}_{f})_{\text{SM}}}\,.$ (5)

The modifications to the couplings of $h^{0}$ given in Table 2 lead to
different phenomenology in the 2HDM compared to the SM, and we use this to
constrain the parameters on which the couplings depend; $\tan\beta$ and
$\cos(\beta-\alpha)$ in the basis we use here. As the observed Higgs matches
the SM predictions to within $\sim 10$%, we expect to find that the 2HDM is
driven towards the alignment limit, $\cos(\beta-\alpha)=0$, which precisely
recovers the SM couplings. There is another region in which SM-like couplings
can be recovered, with, in the 2HDM-X, $\xi^{u}=\xi^{d}=1$ and
$\xi^{\ell}=-1$. This is known as the wrong sign limit and is attained when

$\cos{(\beta-\alpha)}=\sin{2\beta}=\frac{2\tan\beta}{1+\tan^{2}\beta}\,.$ (6)

Similar coupling configurations in which the $\xi^{i}$ have the same magnitude
as the SM are also possible in Type II and Y models, with the down-type and
lepton and only the down-type couplings taking the negative sign respectively.
The signal strengths are calculated using the expressions in Gunion et al.
(2000), with the results of the fit shown in Fig. 2.

In the 2HDM-X we find that the wrong sign limit is allowed for all confidence
levels examined here. Outside this region, there is a maximum magnitude of
$|\cos(\beta-\alpha)|\leq 0.10$ at 2$\sigma$, which falls rapidly once
$\tan\beta$ is away from $\sim$ 1, with only very small $\cos(\beta-\alpha)$
allowed for extreme values of $\tan\beta$. The wrong sign limit being allowed
at 1$\sigma$, in contrast to the 2HDM-II in which it is excluded up to
2.7$\sigma$ Atkinson et al. (2022a), is a result of the lack of sensitivity to
the sign of $\xi^{\ell}$, due to the leptons giving minimal contributions to
the loop level processes compared to the much more important contributions
from the quarks. Indeed, the leptons do not contribute at all to the crucial
gluon fusion production mechanism and their contributions to the diphoton and
$Z\gamma$ decay channels are orders of magnitude lower than those from the
quarks and the $W^{\pm}$ boson for an SM-like $h^{0}$. The leptonic decays of
$h^{0}$ are still important here as they restrict $\cos(\beta-\alpha)$ to be
small for large values of $\tan\beta$, which differentiates these results from
the 2HDM-I.

The relative lack of sensitivity to $\xi^{\ell}$ compared to the quark
couplings can also be seen in the results for the 2HDM-Y, in which the quark
sector matches that of the 2HDM-II, with the result that the contours are very
similar between the two Types, with the wrong sign limit allowed at
2.6$\sigma$ in the 2HDM-Y, with $|\cos(\beta-\alpha)|\leq 0.049$ at 2$\sigma$.

###  IV.2 Collider Searches, Electroweak Precision and Flavour Constraints

In this section we combine the numerous collider searches for BSM Higgs states
to exclude regions of the parameter space. We take the exact alignment and
degenerate mass limits in these scans, guided by the results of Section IV.1
and the theoretical constraints Atkinson et al. (2022a). We also extrapolate
the LHC data present in HiggsBounds to a future collider with a centre of mass
energy of $\sqrt{s}=13$ TeV and $\mathcal{L}=3\,\text{ab}^{-1}$, in line with
the expected integrated luminosity of the HL-LHC. The results from a scan of
50,000 randomly generated points are shown in Fig. 3, where orange points are
allowed by current data but excluded by the extrapolated dataset while blue
points are allowed by both. In the following we outline the main channels that
give the exclusion in each region.

Figure 3:  Scans of the 2HDM parameter space with randomly generated points
shown in blue if allowed by the current and extrapolated bounds, and in orange
if currently allowed but expected to be excluded by the HL-LHC. Overlaying are
the 1, 2, 3, 4, 5$\sigma$ (darker to lighter) allowed contours from the
flavour sector to highlight regions of collider-flavour (including electroweak
precision data) complementarity.

At low $\tan\beta$ in the 2HDM-X, the exclusion is from decays of the neutral
heavy Higgses to $b$ quarks Schael et al. (2006); Sirunyan et al. (2019), up
to the top quark mass, beyond which $H^{+}\to t\bar{b}$ drives the exclusion
until the mass of the charged Higgs becomes large enough such that the
production cross section falls below the current bounds Aaboud et al. (2020).
For moderate $\tan\beta$ the LEP baseline and
$H^{+}\to\tau^{+}\bar{\nu_{\tau}}$ rule out masses up to $\sim$ 100 GeV
Sirunyan et al. (2018) before sensitivity is lost due to falling branching
ratios for this channel. The decay $H^{0}\to\tau^{+}\tau^{-}$ is responsible
for the remaining exclusion in this region Aad et al. (2020). Whilst we might
expect this channel to also exclude points with large $\tan\beta$ as the
lepton couplings become large, this is not the case as the quark couplings,
proportional to $\cot\beta$, become small and the new Higgs states essentially
decouple from the quark sector, giving very low production cross sections and
thus limited sensitivity in this region beyond the historic LEP limits Schael
et al. (2006). The extrapolation improves these bounds, leading to the
additional exclusion most notable in the moderate $\tan\beta$ region from
$H^{0}\to\tau^{+}\tau^{-}$.

In the 2HDM-Y case, below the top mass at low $\tan\beta$, leptonic decays of
$H^{0}$ exclude points Sirunyan et al. (2020, 2015), before the same
$H^{\pm}\to t\bar{b}$ channel provides the exclusion above the top mass Aaboud
et al. (2020). There is relatively little exclusion for moderate $\tan\beta$
in the current data, owing to the comparatively low branching ratios in this
region for the channels that give exclusion in other 2HDM Types, with the
lower mass bound set from the LEP data Schael et al. (2006). For high
$\tan\beta$, $H^{0}\to b\bar{b}$ gives the exclusion up to $\sim$ 250 GeV
Khachatryan et al. (2015), beyond which $H^{+}\to t\bar{b}$ is the most
sensitive channel. In the extrapolation, the improved bounds from $H^{+}\to
t\bar{b}$ and $H^{0}\to b\bar{b}$ are sufficient to rule out a swathe of
points in the moderate $\tan\beta$ region.

Furthermore, we find constraints on the 2HDM parameters by way of a global fit
to flavour and electroweak precision observables, where the $h^{0}$ signal
strengths are also included. These fits for Types I and II have been
previously covered in Atkinson et al. (2022a, b), and we also now consider
Types X and Y. Having updated several input parameters, the predictions for
many of the flavour observables have changed from the previous listings in
Atkinson et al. (2022a). Although not significant enough to have a noticeable
impact on these fits, we collect the SM predictions for the observables used
in an auxiliary file. Note that here by default we assume the global fit
including all observables apart from the LFU quantities
$R(D^{(*)}),R(K^{(*)})$. The latter (as it was shown in Atkinson et al.
(2022a, b)) cannot be accommodated within 2HDMs of Type II or Type I, and we
find the same for Types X and Y in this work. Including
$R(D^{(*)}),R(K^{(*)})$ in the fit therefore leads to very poor $p$-values. On
the other side, excluding LFU observables from the fit yields more reasonable
$p$-values, and therefore, in such scenarios, constraints on 2HDM parameter
spaces have higher credibility.

The global fit for Type X results in very similar constraints to that of Type
I, where overall there is still a lot of freedom for the 2HDM parameters. The
mass of $H^{\pm}$ can be as low as the electroweak scale, which is anyway the
lower limit of validity for the Wilson coefficient expressions used from
Crivellin et al. (2019). Above this minimum value, the correlation between the
lower bound on $\tan\beta$ and $m_{H^{\pm}}$ is again very similar to that of
the Type I fit; we are not sensitive to the upper limits on $\tan\beta$ as for
large values, we approach decoupling in the quark sector and non-perturbative
coupling for the leptons.

The charged Higgs mass is most constrained in the Type Y model, where we find
strict lower and upper limits within $2\sigma$, shown in Table 3. Beyond
$2\sigma$, we find mass constraints more similar to those of Type II where
these two Types have the same quark sector couplings, leading to agreement in
the strong constraint on $m_{H^{\pm}}$ from ${\cal B}(\bar{B}\to
X_{s}\gamma)$.

We now look to compare the results of the BSM collider searches and the global
fits to gain insight into the complementary nature of these approaches. We do
so by displaying the results on the same axes in Fig. 3, and stress that this
is not a statistical combination and simply an overlay of the contours from
the flavour sector on the scans from the collider searches in Fig. 3.

We observe a good degree of complementarity between the two sectors, as each
can probe regions of the parameter space that the other cannot. For both Types
of 2HDM examined here the flavour sector excludes regions with low $\tan\beta$
that the BSM searches lack sensitivity to, both in the current and
extrapolated datasets. For the 2HDM-X, the collider searches, particularly the
$H^{0}\to\tau^{+}\tau^{-}$ channel in the HL-LHC extrapolation, are more
sensitive than the flavour sector and can rule out a portion of the 1$\sigma$
region from the flavour observables, though both approaches lack sensitivity
above masses of 100 GeV when $\tan\beta$ is large. In the 2HDM-Y case, the
collider searches rule out the entirety of the 1$\sigma$ region in the
parameter space we examine here, and the majority of the 2$\sigma$ region. The
extrapolation of the LHC data improves on this further, and rules out very
nearly all of the 2$\sigma$ region. Outside of this high $\tan\beta$ region
the flavour constraints outperform the collider searches. These results
demonstrate the high degree of complementarity between the two datasets and
give further indications of where future searches should focus efforts to
detect or exclude a 2HDM.

###  IV.3 $a_{\mu}$ in 2HDMs

With the long-standing tension between the SM prediction Aoyama et al. (2020)
and the experimental measurement from BNL Bennett et al. (2006), now also
confirmed by FNAL Abi et al. (2021), there has been strong motivation to
consider BSM contributions to $a_{\mu}$ where one can introduce new loop
diagrams, such as from the new Higgs particles we consider here in the 2HDMs.
In the 2HDM, it is important to consider diagrams at both 1- and 2-loop, where
the ‘Barr-Zee’ diagrams yield significant contributions Barr and Zee (1990);
Ilisie (2015). We will consider the 2HDM contributions to $a_{\mu}$ in each of
the four flavour-conserving Types discussed in Section II, where we scan over
$10^{7}$ random points within the $2\sigma$ global-fit constraints for each
Type, summarised in Table 3. We also follow this procedure to analyse the 2HDM
corrections to the $W$ boson mass (see Section IV.4 below), and present
together the correlated results for $a_{\mu}$ and $m_{W}$ in 2HDMs in Fig. 4.
Note that we assume the White Paper prediction Aoyama et al. (2020) when
referring to the SM contribution to $a_{\mu}$. It is found that in order to
resolve the tension in $a_{\mu}$ from any Type of 2HDM, one must either
consider non-perturbative values of $\tan\beta$ or restrict the masses of the
new Higgses to $<1\,$TeV where a sizeable mass splitting between
$H^{\pm},\,H^{0},\,A^{0}$ is also allowed from theory constraints (see e.g.
Table 1 of Atkinson et al. (2022a)). The former scenario is one explicitly
avoided in our studies and we cannot comment on the phenomenology of a non-
perturbative 2HDM; the latter scenario can be compared to our fits of other
sectors in 2HDMs, however, masses lower than $1\,$TeV are disfavoured at
$2\sigma$ in both the Type-II and Type-Y models and thus we cannot resolve
$a_{\mu}$ within the $2\sigma$ constraints of our global fits for Types II and
Y. In Types I and X, there is much more freedom allowed for the new Higgs
masses below $1\,$TeV which leads to a much larger range of mass splittings
between $H^{\pm},\,H^{0},\,A^{0}$ allowing then for larger contributions to
$a_{\mu}$; we now find for both Types I and X many parameter points within
$2\sigma$ of our global fits which would also resolve the tension in
$a_{\mu}$.

In Fig. 4, we plot in orange data calculated from allowing all parameters to
vary within the $2\sigma$ bounds of our fits (Table 3), while in blue we
further restrict the data to lie in the alignment limit,
$\cos(\beta-\alpha)=0$. In comparing the blue and orange data, one can see
that although deviations from alignment are expected to be small (see Section
IV.1), allowing these small deviations is important in resolving the tension
in $a_{\mu}$ even in the freer Types I and X.

It is important to stress however that $a_{\mu}$ in the SM itself is still
under scrutiny, and in fact with multiple recent lattice QCD predictions with
improved precision Borsanyi et al. (2021); Cè et al. (2022); Alexandrou et al.
(2022) beginning to converge on a value closer to experiment than the data-
driven SM prediction, the 2HDM contributions may in the future be required to
be much smaller than considered here for agreement between theory and
experiment to stay.

Figure 4:  The correlation of $m_{W}$ and the muon $g-2$ in 2HDMs within their
allowed parameter space, see Table 3 (orange); restricting
$\cos(\beta-\alpha)=0$ (blue). The $1\sigma$ region for each quantity stated
in the legend are shown by the dashed lines.

###  IV.4 $W$-Boson Mass

| $m_{H^{+}}\,{\rm[TeV]}$ | $\cos(\beta-\alpha)$ | $\tan\beta$
---|---|---|---
Type I | $[0.1,10]$ | $[-0.14,0.14]$ | $[0.55^{\color[rgb]{1,0,0}*},320)$
Type II | $[0.86,10]$ | $[-0.04,0.04]$ | $[0.32^{\color[rgb]{1,0,0}*},50^{\color[rgb]{1,0,0}*})$
Type X | $[0.13,10]$ | $[-0.05,0.10]$ | $[0.38^{\color[rgb]{1,0,0}*},320)$
Type Y | $[0.74,3.61]$ | $[-0.01,0.04]$ | $[110,320)$
Table 3:  Regions of 2HDM parameters used in this work. We consider the
$2\sigma$ regions from the global fit (excluding LFU observables
$R(K^{(*)}),R(D^{(*)})$) for each Type, and also do not exceed the region of
$m_{H^{+}}\in[0.1,10]\,$TeV in order to not conflict with direct search limits
Schael et al. (2006) or approach the decoupling limit. The additional neutral
Higgs masses are constrained from the charged Higgs mass as found in Atkinson
et al. (2022a) (Table 1). * indicates quantities which are not general, but
depend on $m_{H^{+}}$; we show values for $m_{H^{+}}=10\,$TeV where the
allowed regions are at their maximum.

The relation between the $W$-boson mass $m_{W}$ and the Fermi constant
$G_{F}$, the fine-structure constant at zero momentum $\alpha_{\rm em}(0)$,
and the $Z$-boson mass $m_{Z}$ reads (e.g. López-Val and Robens (2014), Hollik
(2003)):

$m_{W}^{2}\left(1-\frac{m_{W}^{2}}{m_{Z}^{2}}\right)=\frac{\pi\alpha_{\rm
em}(0)}{\sqrt{2}G_{F}}\left(1+\Delta r\right),$ (7)

where $m_{W}$ and $m_{Z}$ are renormalised in the on-shell scheme, and $\Delta
r$ accounts for radiative corrections. Eq. (7) can be then equivalently
presented as

$m_{W}^{2}=\frac{1}{2}m_{Z}^{2}\left(1+\sqrt{1-\frac{4\pi\alpha_{\rm
em}(0)}{\sqrt{2}G_{F}m_{Z}^{2}}\left[1+\Delta r(m_{W}^{2})\right]}\,\right),$
(8)

which then is iteratively solved with respect to $m_{W}$. To first order,

$\Delta
m_{W}\simeq-\frac{1}{2}m_{W}\frac{s_{W}^{2}}{s_{W}^{2}-c_{W}^{2}}\Delta r,$
(9)

where $s_{W}^{2}\equiv\sin^{2}(\theta_{W})=1-m_{W}^{2}/m_{Z}^{2}$ is the
shorthand notation of the electroweak mixing angle in the on-shell scheme, and
$c_{W}^{2}\equiv\cos^{2}\theta_{W}=1-s_{W}^{2}$. Introducing the oblique
parameter $T$,

$T\equiv\frac{1}{\alpha_{\rm
em}(m_{Z})}\left(\frac{\Sigma_{Z}(0)}{m_{Z}^{2}}-\frac{\Sigma_{W}(0)}{m_{W}^{2}}\right),$
(10)

$\Delta r$ can then be recast as López-Val and Robens (2014)

$\Delta r=\Delta\alpha-\frac{c_{W}^{2}}{s_{W}^{2}}\alpha_{\rm
em}(m_{Z})\,T+\Delta r_{\rm rem}\,,$ (11)

where $\Delta\alpha$ includes the leading logarithmic QED corrections from the
light-fermions, and $\Delta r_{\rm rem}$ absorbs the remaining contributions.
For the oblique parameter $T$ we use the Yukawa-independent expression in a
2HDM shown in Eq. (C.1) of Atkinson et al. (2022a), calculated from Grimus et
al. (2008). As $T$ is independent of the Yukawa structure of the 2HDM, it is
the same for each Type we discuss here; the difference between the results of
the four Types will come purely through the different parameter constraints
found from the global fits (Table 3).

Some studies of $m_{W}$ in 2HDMs consider $T$ in the exact alignment limit
Bahl et al. (2022); Botella et al. (2022); Kim (2022), however here we do not
constrain ourselves to this. Instead we take the constraints on
$\cos(\beta-\alpha)$ at $2\sigma$ from the global fit to each Type and vary
within these. The impact of choosing the exact alignment limit as opposed to
our method with further freedom can be seen by comparing the blue (alignment
limit) and orange ($2\sigma$ constraints) data in Fig. 4. We do not find a
significant change in the range of values $m_{W}$ can take in the 2HDM from
the choice of treatment of $\cos(\beta-\alpha)$, instead this choice is far
more important for the consideration of $a_{\mu}$ than for $m_{W}$. In Fig. 4,
both the orange and blue data are calculated using 1-loop expressions for the
$T$ parameter in the 2HDM. There are however 2-loop calculations for the 2HDM
contributions to $m_{W}$ calculated Hessenberger and Hollik (2017);
Hessenberger (2018) and used Bahl et al. (2022) in literature, remaining in
the alignment limit. We use the package
THDM_EWPOS111https://github.com/st3v3m4n/THDM_EWPOS to compare the 1- and
1+2-loop calculations for $m_{W}$ in 2HDM. Within the region plotted in Fig.
4, it is found that the 1+2-loop calculation does not provide more information
than the 1-loop, at least within the alignment limit, and thus we do not show
additional plots for these calculations. The difference found at 2-loop is a
‘stretching out’ of $m_{W}$ values which already extend past the plotted range
at 1-loop and also that the value of $m_{W}$ becomes less correlated with
$a_{\mu}$.

Within the context of our global-fit parameter constraints (at $2\sigma$), we
find that the 2HDM corrections to the $W$-boson mass can be large, either
positively or negatively. To generate large corrections, it is favoured that
there is a large mass splitting between $H^{\pm},\,H^{0},\,A^{0}$ which
approximately implies an upper bound $m_{H^{\pm}}\lesssim 1\,$TeV, however
there is still sufficient freedom for the masses to be separate within the
range of $1-2\,$TeV such that the more constrained Types II and Y can also
find large corrections to $m_{W}$, where even the $7\sigma$ deviation of the
new CDF-II measurement is easily covered by these.

###  IV.5 Impact of MUonE

As demonstrated in the previous section, $(g-2)_{\mu}$ can be accommodated in
the 2HDM. Equipped with this discussion, we now turn to relevance of MUonE,
which seeks to measure the hadronic vacuum polarisation contribution in a
data-driven way (without relying on past measurements of the $R$ ratio). Our
aim is to clarify whether new physics contributions are relevant in this
context. To this end, we consider the parameter space of all four models of
Tables 1 and 2. Furthermore, we consider not just the degenerate mass and
alignment limits in this section, but include a wide range of possible mass
scenarios, such as varying each of the new Higgs masses independently, fixing
two masses to a range of values and varying the other, all with different
values of $\cos(\beta-\alpha)$ examined in each case, as we scan across the
mass and $\tan\beta$ parameter space.

Figure 5:  Contour plots of the allowed regions at 1, 2, 3$\sigma$ from the
MUonE analysis, where in this case we have fixed $m_{H^{0}}=m_{A^{0}}$,
$m_{H^{\pm}}=$ 5 TeV and $\cos(\beta-\alpha)=0$.

We find that the MUonE experiment is largely insensitive to the presence of a
2HDM of any Type. The total $\chi^{2}$ we calculate in almost every scenario
is far smaller than the threshold value for the 68% CL and thus we find that
almost all parameter regions cannot be probed by the MUonE experiment.
However, there are some exceptions to this and we can still glean some useful
information from this analysis. In order to observe any exclusion we must
consider extreme mass scenarios, one example of which we show for the 2HDM-X
in Fig. 5, where we have fixed $m_{H^{+}}=5$ TeV, $\cos(\beta-\alpha)=0$ and
varied $m_{H^{0}}=m_{A^{0}}$. In this scenario there is a $\tan\beta$
independent lower limit on the mass of the neutral scalars of
$m_{H^{0}}=m_{A^{0}}\gtrsim 460$ GeV at 2$\sigma$, with an upper bound at
around 8.5 TeV. Identical limits are found in this scenario for all Types of
2HDM within the parameter space we examine. Separately, there is also some
exclusion at and beyond the TeV scale for $m_{H^{+}}=m_{H^{0}}$ when
$m_{A^{0}}$ is fixed to a value below 1 TeV. Inducing a large mass splitting
between the charged and neutral Higgses yields the greatest sensitivity,
however such mass splits are conclusively ruled out by the theoretical
requirements explored in Atkinson et al. (2022a), which demand a strong degree
of mass degeneracy, which becomes tighter for higher masses.

Very low Higgs masses, below the order of 1 GeV, give large $\chi^{2}$, but
such states would already have been observed in the direct collider searches
outlined in Section IV.2 and thus are already ruled out, so MUonE adds no
extra exclusion in this case. Additionally, points with large
$\tan\beta\geq\mathcal{O}(10^{4})$, are also excluded from the MUonE analysis
in the 2HDM-X but such $\tan\beta$ is highly non-perturbative; a similar
argument applies in the 2HDM-Y for $\tan\beta\leq\mathcal{O}(10^{-4})$.

The impact of each different Type of 2HDM is very similar in the majority of
the parameter space we investigate as the top quark loops provide the dominant
contributions and each model has the same set of couplings to the top quark.
It is only in extreme $\tan\beta$ regions that we observe some differences in
the $\chi^{2}$ from each model, where we see that Types I and Y give the same
results, as do Types II and X, owing to these models having the same lepton
sectors, indicating that the lepton contributions become significant only when
the couplings become extreme; see Table 2. In all scenarios and models,
increasing $\cos(\beta-\alpha)$ does increase the sensitivity, but this effect
is relatively minor and insufficient to give any meaningful additional
exclusion regions.

This lack of sensitivity to 2HDM states in parameter regions not already
excluded by other sectors means that the MUonE experiment is unlikely to be
able to provide a distinct signal of BSM physics in the Higgs sector, which is
of benefit to the aim of the experiment of measuring the leading-order
hadronic contribution to the anomalous magnetic moment of the muon. The
measurement will be free from potential contamination in the case that a 2HDM
of any Type is realised in a fashion consistent with the theoretical and
collider bounds.

###  IV.6 Electroweak Phase Transition

Previous works Atkinson et al. (2022a, b) assessed the implications of the
global fits on Type I and II 2HDMs for a strong first-order electroweak phase
transition (SFOEWPT). The significance of the different Yukawa couplings of
different Types on the EWPT in 2HDMs is minimal, so the main difference in the
results from the four flavour-conserving 2HDMs comes from the parameter
constraints of the fits. For a EWPT to be of strong first-order, it is
favoured that there is a sizeable difference between $m_{H^{+}}$ and
$m_{H^{0}}$; it is sufficient for $m_{A^{0}}\sim m_{H^{+}}$. From the
theoretical constraints found in Atkinson et al. (2022a), one can see that
this then inhibits larger Higgs masses from producing the SFOEWPT since the
larger the mass scale, the more degenerate the new masses become. Favouring
masses $\lesssim 1\,$TeV in order to generate a sufficient mass separation, we
find that, similarly to Type I, $m_{H^{+}}$ still has enough freedom in Type X
in order to generate this SFOEWPT within $1\sigma$ of our best fit.

However in Type Y, $m_{H^{+}}$ is most constrained out of all Types and has a
similar lower bound to that of Type II. Furthermore, within $2\sigma$
$\tan\beta$ is constrained to be much larger than in other Types; typically a
SFOEWPT will require a $\tan\beta\sim{\cal O}(10)$. The constraints on both
$\tan\beta$ and $m_{H^{+}}$ together prohibit a SFOEWPT from being realised
within $2\sigma$ of our global fits for Type Y. Beyond $2\sigma$, the
constraints on $\tan\beta$ significantly relax, however the constraints on
$m_{H^{+}}$ still remain stronger than those of the other Types. While these
mass constraints reduce the allowed range of masses to test for a SFOEWPT,
they still are sufficiently low enough to generate a mass splitting which,
along with a low value of $\tan\beta$ allowed above $2\sigma$, can generate a
SFOEWPT. Considering a wide range of parameters at various $\sigma$ from our
best fit, we conclude that a SFOEWPT generated from the Type Y 2HDM would
currently lie at or above $3\sigma$ from our best fit.

##  V Conclusions

The search for new physics beyond the Standard Model is a high priority of the
particle phenomenology programme across many different experimental settings.
The sizeable tension of the anomalous magnetic moment of the muon could be a
concrete pointer towards a more comprehensive theory of particle physics. The
occurrence of this tension specifically highlights the lepton sector as a BSM
source, which, when understood in the context of two Higgs doublet extensions
motivates a combined analysis of collider and flavour physics data for 2HDM
scenarios X and Y. To our knowledge this has been presented here
comprehensively for the first time. We find that each of these analyses
exclude large amounts of the parameter space of the models, with the
combination of the two improving upon this further as the flavour and collider
constraints exhibit a good degree of complementarity.

Within the 2HDM-I and 2HDM-X we can accommodate the measurements of the muon
anomalous magnetic moment by BNL/Fermilab using as SM prediction the White
Paper result Aoyama et al. (2020). The hadronic vacuum polarisation
contribution is known to be a relevant source of uncertainty for the BSM
interpretation of the muon anomalous magnetic moment measurement. Recent
lattice results Borsanyi et al. (2021); Cè et al. (2022); Alexandrou et al.
(2022) show tension with the data-driven methods employed to formulate the SM
expectation consensus Aoyama et al. (2020). Whilst future updates can be
expected here, this discrepancy can be addressed experimentally at MUonE,
which provides a precise data-driven extraction of the vacuum polarisation.
Given the high sensitivity of MUonE to this naively subdominant contribution,
it is important to also understand the impact of lepton-philic new physics for
this experiment and the resulting impact on the interpretation of MUonE
results in terms of the hadronic vacuum polarisation contribution. This
becomes particularly relevant for the 2HDM-X and Y scenarios as they can
address the anomaly in some parameter regions that are consistent with
experimental findings Athron et al. (2021). In this work, we provide the first
comprehensive analysis of electroweak radiative corrections for these
scenarios at MUonE in conjunction with constraints from flavour collider
observables (also including the partonic vacuum polarisation contribution
consistently at leading order). We find that these electroweak corrections do
not play a constraining role for the parameter region favoured by flavour and
collider data. Consequently, MUonE data can be fully interpreted as a
measurement of the hadronic vacuum polarisation contribution in these
scenarios without the need to correct for BSM effects.

In passing, we found that all four 2HDM Types can reproduce large shifts in
$m_{W}$, as indicated by the recent CDF measurement and we identify the
possibility of having a SFOEWPT within $1\sigma$ of our best fit point in
2HDM-X and at or above $3\sigma$ of our best fit point in 2HDM-Y.

Acknowledgments — We are grateful to Christine T.H. Davies, Rusa Mandal,
Panagiotis Stylianou and Thomas Teubner for helpful discussions. We thank
Stephan Hessenberger for making available the 2-loop $m_{W}$ computation
included in Sec. IV.4. O.A. is funded by a STFC studentship under grant
ST/V506692/1. The work of M.B. is supported by Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) through TRR 257 “Particle Physics
Phenomenology after the Higgs Discovery”. C.E. is supported by the STFC under
grant ST/T000945/1, the Leverhulme Trust under grant RPG-2021-031, and the
Institute for Particle Physics Phenomenology Associateship Scheme. Parts of
the computations carried out for this work made use of the OMNI cluster of the
University of Siegen.

## References

  * Albrecht et al. (2021) J. Albrecht, D. van Dyk, and C. Langenbruch, Prog. Part. Nucl. Phys. 120, 103885 (2021), eprint 2107.04822.
  * Gershon et al. (2022) T. Gershon, A. Lenz, A. V. Rusov, and N. Skidmore, Phys. Rev. D 105, 115023 (2022), eprint 2111.04478.
  * Abi et al. (2021) B. Abi et al. (Muon g-2), Phys. Rev. Lett. 126, 141801 (2021), eprint 2104.03281.
  * Bennett et al. (2006) G. W. Bennett et al. (Muon g-2), Phys. Rev. D 73, 072003 (2006), eprint hep-ex/0602035.
  * Aoyama et al. (2020) T. Aoyama et al., Phys. Rept. 887, 1 (2020), eprint 2006.04822.
  * Borsanyi et al. (2021) S. Borsanyi et al., Nature 593, 51 (2021), eprint 2002.12347.
  * Cè et al. (2022) M. Cè et al. (2022), eprint 2206.06582.
  * Alexandrou et al. (2022) C. Alexandrou et al. (2022), eprint 2206.15084.
  * Keshavarzi et al. (2018) A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D 97, 114025 (2018), eprint 1802.02995.
  * Abbiendi et al. (2017) G. Abbiendi et al., Eur. Phys. J. C 77, 139 (2017), eprint 1609.08987.
  * Balzani et al. (2021) E. Balzani, S. Laporta, and M. Passera (2021), eprint 2112.05704.
  * Mastrolia et al. (2017) P. Mastrolia, M. Passera, A. Primo, and U. Schubert, JHEP 11, 198 (2017), eprint 1709.07435.
  * Gakh et al. (2018) G. I. Gakh, M. I. Konchatnij, N. P. Merenkov, V. N. Kharkov, and E. Tomasi-Gustafsson, Phys. Rev. C 98, 045212 (2018), eprint 1804.01399.
  * Di Vita et al. (2018) S. Di Vita, S. Laporta, P. Mastrolia, A. Primo, and U. Schubert, JHEP 09, 016 (2018), eprint 1806.08241.
  * Carloni Calame et al. (2020) C. M. Carloni Calame, M. Chiesa, S. M. Hasan, G. Montagna, O. Nicrosini, and F. Piccinini, JHEP 11, 028 (2020), eprint 2007.01586.
  * Banerjee et al. (2021) P. Banerjee, T. Engel, N. Schalch, A. Signer, and Y. Ulrich, Phys. Lett. B 820, 136547 (2021), eprint 2106.07469.
  * Budassi et al. (2021) E. Budassi, C. M. Carloni Calame, M. Chiesa, C. L. Del Pio, S. M. Hasan, G. Montagna, O. Nicrosini, and F. Piccinini, JHEP 11, 098 (2021), eprint 2109.14606.
  * Dev et al. (2020) P. S. B. Dev, W. Rodejohann, X.-J. Xu, and Y. Zhang, JHEP 05, 053 (2020), eprint 2002.04822.
  * Masiero et al. (2020) A. Masiero, P. Paradisi, and M. Passera, Phys. Rev. D 102, 075013 (2020), eprint 2002.05418.
  * Asai et al. (2021) K. Asai, K. Hamaguchi, N. Nagata, S.-Y. Tseng, and J. Wada (2021), eprint 2109.10093.
  * Galon et al. (2022) I. Galon, D. Shih, and I. R. Wang (2022), eprint 2202.08843.
  * Dorsch et al. (2014) G. C. Dorsch, S. J. Huber, K. Mimasu, and J. M. No, Phys. Rev. Lett. 113, 211802 (2014), eprint 1405.5537.
  * Dorsch et al. (2016) G. C. Dorsch, S. J. Huber, K. Mimasu, and J. M. No, Phys. Rev. D 93, 115033 (2016), eprint 1601.04545.
  * Basler et al. (2017) P. Basler, M. Krause, M. Muhlleitner, J. Wittbrodt, and A. Wlotzka, JHEP 02, 121 (2017), eprint 1612.04086.
  * Basler et al. (2019) P. Basler, S. Dawson, C. Englert, and M. Mühlleitner, Phys. Rev. D 99, 055048 (2019), eprint 1812.03542.
  * Basler et al. (2020) P. Basler, S. Dawson, C. Englert, and M. Mühlleitner, Phys. Rev. D 101, 015019 (2020), eprint 1909.09987.
  * Gonçalves et al. (2022) D. Gonçalves, A. Kaladharan, and Y. Wu, Phys. Rev. D 105, 095041 (2022), eprint 2108.05356.
  * Su et al. (2021) W. Su, A. G. Williams, and M. Zhang, JHEP 04, 219 (2021), eprint 2011.04540.
  * Wang (2021) L. Wang (2021), eprint 2105.02143.
  * Dorsch et al. (2017) G. C. Dorsch, S. J. Huber, K. Mimasu, and J. M. No, JHEP 12, 086 (2017), eprint 1705.09186.
  * Atkinson et al. (2022a) O. Atkinson, M. Black, A. Lenz, A. Rusov, and J. Wynne, JHEP 04, 172 (2022a), eprint 2107.05650.
  * Basler et al. (2021) P. Basler, M. Mühlleitner, and J. Müller (2021), eprint 2108.03580.
  * Atkinson et al. (2022b) O. Atkinson, M. Black, C. Englert, A. Lenz, A. Rusov, and J. Wynne (2022b), eprint 2202.08807.
  * Anisha et al. (2022) Anisha, L. Biermann, C. Englert, and M. Mühlleitner (2022), eprint 2204.06966.
  * Glashow and Weinberg (1977) S. L. Glashow and S. Weinberg, Phys. Rev. D 15, 1958 (1977).
  * Athron et al. (2021) P. Athron, C. Balázs, D. H. Jacob, W. Kotlarski, D. Stöckinger, and H. Stöckinger-Kim (2021), eprint 2104.03691.
  * Aaltonen et al. (2022) T. Aaltonen et al. (CDF), Science 376, 170 (2022).
  * Zyla et al. (2020) P. A. Zyla et al. (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Lopez-Val and Sola (2013) D. Lopez-Val and J. Sola, Eur. Phys. J. C 73, 2393 (2013), eprint 1211.0311.
  * Bahl et al. (2022) H. Bahl, J. Braathen, and G. Weiglein (2022), eprint 2204.05269.
  * Song et al. (2022) H. Song, W. Su, and M. Zhang (2022), eprint 2204.05085.
  * Heo et al. (2022) Y. Heo, D.-W. Jung, and J. S. Lee (2022), eprint 2204.05728.
  * Ahn et al. (2022) Y. H. Ahn, S. K. Kang, and R. Ramos (2022), eprint 2204.06485.
  * Lee et al. (2022) S. Lee, K. Cheung, J. Kim, C.-T. Lu, and J. Song (2022), eprint 2204.10338.
  * Benbrik et al. (2022) R. Benbrik, M. Boukidi, and B. Manaut (2022), eprint 2204.11755.
  * Abouabid et al. (2022) H. Abouabid, A. Arhrib, R. Benbrik, M. Krab, and M. Ouchemhou (2022), eprint 2204.12018.
  * Botella et al. (2022) F. J. Botella, F. Cornet-Gomez, C. Miró, and M. Nebot (2022), eprint 2205.01115.
  * Kim (2022) J. Kim (2022), eprint 2205.01437.
  * Kim et al. (2022) J. Kim, S. Lee, P. Sanyal, and J. Song (2022), eprint 2205.01701.
  * Branco et al. (2012) G. Branco, P. Ferreira, L. Lavoura, M. Rebelo, M. Sher, and J. P. Silva, Phys. Rept. 516, 1 (2012), eprint 1106.0034.
  * Gunion et al. (2000) J. F. Gunion, H. E. Haber, G. L. Kane, and S. Dawson, _The Higgs Hunter’s Guide_ , vol. 80 (Perseus Publishing, 2000).
  * Arnan et al. (2017) P. Arnan, D. Bečirević, F. Mescia, and O. Sumensari, Eur. Phys. J. C 77, 796 (2017), eprint 1703.03426.
  * Han et al. (2020) X.-F. Han, H.-X. Wang, and X.-F. Han, Chin. Phys. C 44, 073101 (2020), eprint 2003.06170.
  * Kling et al. (2016) F. Kling, J. M. No, and S. Su, JHEP 09, 093 (2016), eprint 1604.01406.
  * Crivellin et al. (2019) A. Crivellin, D. Müller, and C. Wiegand, JHEP 06, 119 (2019), eprint 1903.10440.
  * Kinoshita (1962) T. Kinoshita, J. Math. Phys. 3, 650 (1962).
  * Lee and Nauenberg (1964) T. D. Lee and M. Nauenberg, Phys. Rev. 133, B1549 (1964).
  * ’t Hooft and Veltman (1979) G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 153, 365 (1979).
  * Denner (1993) A. Denner, Fortsch. Phys. 41, 307 (1993), eprint 0709.1075.
  * Hahn and Perez-Victoria (1999) T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. 118, 153 (1999), eprint hep-ph/9807565.
  * Hahn (1999) T. Hahn, Acta Phys. Polon. B 30, 3469 (1999), eprint hep-ph/9910227.
  * Hahn and Schappacher (2002) T. Hahn and C. Schappacher, Comput. Phys. Commun. 143, 54 (2002), eprint hep-ph/0105349.
  * Straub (2018) D. M. Straub (2018), eprint 1810.08132.
  * Alwall et al. (2014) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, JHEP 07, 079 (2014), eprint 1405.0301.
  * Krause et al. (2020) M. Krause, M. Mühlleitner, and M. Spira, Comput. Phys. Commun. 246, 106852 (2020), eprint 1810.00768.
  * Djouadi et al. (1998) A. Djouadi, J. Kalinowski, and M. Spira, Comput. Phys. Commun. 108, 56 (1998), eprint hep-ph/9704448.
  * Djouadi et al. (2019) A. Djouadi, J. Kalinowski, M. Muehlleitner, and M. Spira, Comput. Phys. Commun. 238, 214 (2019), eprint 1801.09506.
  * Krause et al. (2017) M. Krause, M. Muhlleitner, R. Santos, and H. Ziesche, Phys. Rev. D 95, 075019 (2017), eprint 1609.04185.
  * Denner et al. (2018) A. Denner, S. Dittmaier, and J.-N. Lang, JHEP 11, 104 (2018), eprint 1808.03466.
  * Bechtle et al. (2010) P. Bechtle, O. Brein, S. Heinemeyer, G. Weiglein, and K. E. Williams, Comput. Phys. Commun. 181, 138 (2010), eprint 0811.4169.
  * Bechtle et al. (2011) P. Bechtle, O. Brein, S. Heinemeyer, G. Weiglein, and K. E. Williams, Comput. Phys. Commun. 182, 2605 (2011), eprint 1102.1898.
  * Bechtle et al. (2012) P. Bechtle, O. Brein, S. Heinemeyer, O. Stal, T. Stefaniak, G. Weiglein, and K. Williams, PoS CHARGED2012, 024 (2012), eprint 1301.2345.
  * Bechtle et al. (2014) P. Bechtle, O. Brein, S. Heinemeyer, O. Stål, T. Stefaniak, G. Weiglein, and K. E. Williams, Eur. Phys. J. C 74, 2693 (2014), eprint 1311.0055.
  * Bechtle et al. (2015) P. Bechtle, S. Heinemeyer, O. Stal, T. Stefaniak, and G. Weiglein, Eur. Phys. J. C 75, 421 (2015), eprint 1507.06706.
  * Bechtle et al. (2020) P. Bechtle, D. Dercks, S. Heinemeyer, T. Klingl, T. Stefaniak, G. Weiglein, and J. Wittbrodt, Eur. Phys. J. C 80, 1211 (2020), eprint 2006.06007.
  * Bahl et al. (2021) H. Bahl, V. M. Lozano, T. Stefaniak, and J. Wittbrodt (2021), eprint 2109.10366.
  * Sirunyan et al. (2022) A. M. Sirunyan et al. (CMS) (2022), eprint 2205.05550.
  * Aad et al. (2022) G. Aad et al. (ATLAS) (2022), eprint 2207.00348.
  * Schael et al. (2006) S. Schael et al. (ALEPH, DELPHI, L3, OPAL, LEP Working Group for Higgs Boson Searches), Eur. Phys. J. C 47, 547 (2006), eprint hep-ex/0602042.
  * Sirunyan et al. (2019) A. M. Sirunyan et al. (CMS), Phys. Rev. D 99, 012005 (2019), eprint 1810.11822.
  * Aaboud et al. (2020) M. Aaboud et al. (ATLAS) (2020), eprint ATLAS-CONF-2020-039.
  * Sirunyan et al. (2018) A. M. Sirunyan et al. (CMS) (2018), eprint CMS-PAS-HIG-18-014.
  * Aad et al. (2020) G. Aad et al. (ATLAS), Phys. Rev. Lett. 125, 051801 (2020), eprint 2002.12223.
  * Sirunyan et al. (2020) A. M. Sirunyan et al. (CMS), JHEP 03, 051 (2020), eprint 1911.04968.
  * Sirunyan et al. (2015) A. M. Sirunyan et al. (CMS) (2015), eprint CMS-PAS-HIG-14-029.
  * Khachatryan et al. (2015) V. Khachatryan et al. (CMS), JHEP 11, 071 (2015), eprint 1506.08329.
  * Barr and Zee (1990) S. M. Barr and A. Zee, Phys. Rev. Lett. 65, 21 (1990), [Erratum: Phys.Rev.Lett. 65, 2920 (1990)].
  * Ilisie (2015) V. Ilisie, JHEP 04, 077 (2015), eprint 1502.04199.
  * López-Val and Robens (2014) D. López-Val and T. Robens, Phys. Rev. D 90, 114018 (2014), eprint 1406.1043.
  * Hollik (2003) W. Hollik, J. Phys. G 29, 131 (2003).
  * Grimus et al. (2008) W. Grimus, L. Lavoura, O. M. Ogreid, and P. Osland, Nucl. Phys. B 801, 81 (2008), eprint 0802.4353.
  * Hessenberger and Hollik (2017) S. Hessenberger and W. Hollik, Eur. Phys. J. C 77, 178 (2017), eprint 1607.04610.
  * Hessenberger (2018) S. Hessenberger, Ph.D. thesis, Munich, Tech. U. (2018).