aainstitutetext: SUPA, School of Physics & Astronomy, University of Glasgow,
Glasgow G12 8QQ, UKbbinstitutetext: Physik Department, Universität Siegen,
Walter-Flex-Str. 3, 57068 Siegen, Germanyccinstitutetext: IPPP, Department of
Physics, University of Durham, DH1 3LE, UK

# Cornering the Two Higgs Doublet Model Type II

Oliver Atkinson b Matthew Black b Alexander Lenz b Aleksey Rusov c James Wynne
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We perform a comprehensive study of the allowed parameter space of the Two
Higgs Doublet Model of Type II (2HDM-II). Using the theoretical framework
flavio we combine the most recent flavour, collider and electroweak precision
observables with theoretical constraints to obtain bounds on the mass spectrum
of the theory. In particular we find that the 2HDM-II fits the data slightly
better than the Standard Model (SM) with best fit values of the heavy Higgs
masses around 2 TeV and a value of $\tan\beta\approx 4$. Moreover, we conclude
that the wrong-sign limit is disfavoured by Higgs signal strengths and
excluded by the global fit by more than five standard deviations and potential
deviations from the alignment limit can only be tiny. Finally we test the
consequences of our study on electroweak baryogenesis via the program package
BSMPT and we find that the allowed parameter space strongly discourages a
strong first order phase transition within the 2HDM-II.

SI-HEP-2021-20, SFB-257-P3H-21-049

##  1 Introduction

Two Higgs Doublet Models (2HDM) Lee:1973iz  (see Ref. Gunion:1989we  for a
textbook and Ref. Branco:2011iw  for a more recent review) are very well-
studied extensions of the Standard Model (SM) and probably one of the simplest
possibilities for beyond the SM (BSM) physics. Besides the SM particles, four
new scalars arise in the 2HDM: two charged ones ($H^{+}$ and $H^{-}$), a
pseudoscalar ($A^{0}$, CP-odd) and a scalar ($H^{0}$, CP-even; when denoting
the SM Higgs with $h^{0}$). Such an extension might solve some of the
conceptual problems that arise within the SM, see e.g. Ref. Trodden:1998qg .
In particular, 2HDMs could provide the missing amount of CP violation to
explain the matter-antimatter asymmetry in the Universe and they could also
lead to a first order phase transition in the early Universe, as required by
the Sakharov criteria Sakharov:1967dj . 2HDMs further appear as subsets of
more elaborate BSM models, such as the minimal supersymmetric SM (MSSM).
Depending on the structure of the Yukawa couplings one distinguishes different
types of 2HDMs – in this work we will consider the 2HDM-II.

Since we have no direct experimental evidence so far for an extended Higgs
sector, we investigate indirect constraints on the 2HDM-II by comparing the
most recent measurements with high precision calculations. We perform a
comprehensive study of more than 250 observables, where the new Higgs
particles could appear as virtual corrections and modify the SM prediction.
Our study extends previous works like Refs. Deschamps:2009rh ;
Chowdhury:2017aav .111 There is a vast literature on 2HDM studies, which we
are not going to review here. Papers which have some minor overlap with our
study are e.g. Ref. Arco:2020ucn , which focuses on triple Higgs couplings in
the 2HDM and uses many of the same constraints as we do here, with the
addition of LHC searches for BSM Higgs searches, to set bounds on the allowed
triple coupling values. Further studies of the 2HDM have also been carried out
in e.g. Refs. Bhattacharyya:2015nca ; Das:2015qva . In 2009, Ref.
Deschamps:2009rh  studied bounds from quark flavour observables on the 2HDM-II
within the framework of CKMfitter. In particular, it was found that the 2HDM
does not perform better in the fit than the SM and that the new charged scalar
has to be heavier than 316 GeV. We will considerably extend this analysis by
using updated measurements, theory predictions and by including additional
flavour observables. In 2017, the authors of Ref. Chowdhury:2017aav  used
HEPfit to constrain the 2HDM-II with Higgs data, electroweak precision
observables and some flavour observables (the radiative decay $b\to s\gamma$
and the mass difference of neutral $B$ mesons $\Delta m_{q}$), as well as
theoretical constraints like perturbativity. The main findings are lower mass
bounds such as $m_{H^{+}}>740$ GeV and a lower limit on the mass splittings of
the heavy Higgses of the order of 100 GeV. Higgs signal strengths allow only
small deviations from the alignment limit, $|\beta-\alpha-\pi/2|<0.055$ \-
this bound is further reduced to $|\beta-\alpha-\pi/2|<0.02$ within the
overall fit (driven by theory constraints). Finally, the authors of Ref.
Chowdhury:2017aav  found that the parameter $\tan\beta$ lies in the preferred
range $[0.93;5.0]$. We will extend this analysis by updating experimental
values and considering a large number of flavour observables. Moreover we
confirm the observation made in Ref. Chowdhury:2017aav , that the wrong-sign
limit is excluded by data.

Throughout this work, we use the Python program package flavio Straub:2018kue
for flavour physics for our predictions of observables in the 2HDM-II. New
physics models are not explicitly included in flavio. Instead, they impact
observables through their contributions to the Wilson coefficients of
dimension-6 operators in the relevant effective theory: the Weak Effective
Theory (WET) below the electroweak scale, or the Standard Model Effective
Field Theory (SMEFT) above. In Section 6, the relevant 2HDM-II contributions
are included in the coefficients of the WET operators with five active
flavours (there are only three active flavours in the WET-3 basis used for the
anomalous magnetic moment of the muon $a_{\mu}$). For a full list of operators
in the flavio basis for each effective theory, see the documentation of the
WCxf package Aebischer:2017ugx .

Once 2HDM-II contributions to the relevant coefficients have been included, we
use flavio to construct fits in 2HDM parameter spaces, where we show in colour
the allowed regions of our parameter space within the confidence levels stated
in the captions. The package flavio expresses experimental measurements and
input parameters as various one- and two-dimensional probability distribution
functions. The errors of theoretical predictions are constructed by
calculating the prediction for each of a number (we use $N=10^{4}$) of
randomly-selected values of the input parameters within their probability
distributions, and then computing the standard deviation of these values. We
work with the “fast likelihood" method in flavio for constructing likelihood
functions of the form $\mathcal{L}=e^{-\chi^{2}(\vec{\xi})/2}$, where it is
assumed that the set of fit parameters $\vec{\xi}$ contributing to the
observables entering the likelihood function are taken at their central
values. This method uses the combined experimental and theoretical covariance
matrix of the observables in the fit. For further information on the “fast
likelihood" method and the package’s construction, we refer to flavio’s
documentation Straub:2018kue  or the flavio webpage222 https://flav-
io.github.io.

The program package flavio is used in Section 6 for the flavour observables,
where we work in two-dimensional parameter spaces. We make use of flavio to
construct likelihoods and combine measurements in Section 5 for the Higgs
signal strengths, however for the calculations of the signal strengths
themselves, we use analytical expressions written in Python natively. The
constraints discussed in Section 3 and 4 stemming from theoretical
considerations and the electroweak precision observables cannot be expressed
in terms of coefficients of dimension-6 operators. Therefore in Section 3, the
results are taken from a native Python code generating Monte Carlo scans of
the full parameter space. The likelihood functions for the electroweak
precision observables, taking into account the experimental correlations were
translated into native Python code to be properly combined in Section 7 with
all other observables and theory constraints using some customisations to the
flavio package.

The paper is organised as follows. Our notation for the 2HDM-II is set up in
Section 2. In Section 3 theoretical constraints like perturbativity, vacuum
stability, and unitarity are studied, while Section 4 introduces electroweak
precision observables in the form of the oblique parameters. Constraints
stemming from the SM Higgs production and decay at the LHC are studied in
Section 5. Most of our observables stem from the quark flavour sector, see
Section 6. In Section 6.1 we investigate leptonic and semi-leptonic tree-level
decays, where the charged Higgs is predominantly responsible for the new
contributions. In this section we further study a potential pollution of the
determination of CKM elements from semi-leptonic and leptonic decays by the
2HDM-II. $B$-mixing is considered in Section 6.2 and loop-induced $b$ quark
decays are studied in Section 6.3. Section 6.3.1 contains a brief description
of the $b\to s\gamma$ transition, known to give a lower bound on the mass of
the charged Higgs boson. The leptonic decays $B_{q}\to\mu^{+}\mu^{-}$ get also
sizable contributions from the new neutral Higgses, see Section 6.3.2. In
Section 6.3.3 we study semi-leptonic $b\to s\ell^{+}\ell^{-}$-transitions,
where the so-called flavour anomalies have been observed in recent years.
These anomalies can be best explained by vector-like new effects, while we
consider here the effects of new scalar couplings. The results of our global
fit, combining all constraints discussed so far are presented in Section 7. In
Section 7.2 we comment on the lepton flavour observable, $a_{\mu}$, the
anomalous magnetic moment of the muon, where recent measurements at Fermilab
Abi:2021gix  have confirmed the older BNL value Bennett:2006fi . We study two
scenarios based on using the SM prediction from the theory initiative
Aoyama:2020ynm  or a recent lattice evaluation Borsanyi:2020mff . Using the
program package BSMPT Basler:2018cwe ; Basler:2020nrq  we investigate in
Section 7.3 the question of whether our allowed parameter space can also lead
to a first order phase transition in the early Universe. Finally we conclude
in Section 8, while all our inputs are collected in Appendix A.

##  2 The Two Higgs Doublet Model of Type II

In the SM, a conjugate of the Higgs doublet is used to ensure both up-type and
down-type quarks acquire mass. This can also be achieved by using two distinct
complex scalar doublets

$\Phi_{i}=\begin{pmatrix}\phi^{+}_{i}\\\
(v_{i}+\phi^{0}_{i}+iG^{0}_{i})/\sqrt{2}\end{pmatrix},$ (1)

with $i=1,2$. These acquire different vacuum expectation values (VEVs) $v_{i}$
after electroweak symmetry breaking (EWSB), that are related to the SM Higgs
VEV $v$ by $v_{1}^{2}+v_{2}^{2}=v^{2}$ Lee:1973iz ; Gunion:1989we ;
Branco:2011iw . This construction has 8 degrees of freedom, 3 of which become
the longitudinal polarisations of the massive $W^{\pm}$ and $Z$ bosons,
leaving 5 physical Higgs bosons. Two of these are charged, $H^{\pm}$, two are
neutral scalars, $H^{0}$ and $h^{0}$, and one is a neutral pseudoscalar,
$A^{0}$. Typically $h^{0}$ is taken to be the lighter of the two neutral
scalars. The potential for a general 2HDM with a softly broken
$\mathbb{Z}_{2}$ symmetry is, in the lambda basis Branco:2011iw ,

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

The $\mathbb{Z}_{2}$ symmetry is required to forbid tree-level flavor changing
neutral currents (FCNC). Constraints on the $\lambda_{i}$ from a theoretical
perspective are considered in Section 3. The parameters $m_{11}^{2}$ and
$m_{22}^{2}$ are related to $v_{1}$ and $v_{2}$ as Basler:2016obg

$m_{11}^{2}=m_{12}^{2}\,\frac{v_{2}}{v_{1}}-\lambda_{1}\frac{v_{1}^{2}}{2}-(\lambda_{3}+\lambda_{4}+\lambda_{5})\frac{v_{2}^{2}}{2},$
(3)
$m_{22}^{2}=m_{12}^{2}\,\frac{v_{1}}{v_{2}}-\lambda_{2}\frac{v_{2}^{2}}{2}-(\lambda_{3}+\lambda_{4}+\lambda_{5})\frac{v_{1}^{2}}{2}.$
(4)

In the remainder of this work we make use of the mass basis, through a
transformation Arnan:2017lxi  333We differ with the expressions given in
Branco:2011iw  for $m_{A^{0}}$ and $m_{H^{\pm}}$.

$\displaystyle\begin{split}m_{H^{0}}^{2}&=\frac{m_{12}^{2}}{\sin\beta\cos\beta}\sin^{2}(\beta-\alpha)\\\
&\quad+v^{2}\left[\lambda_{1}\cos^{2}\alpha\cos^{2}\beta+\lambda_{2}\sin^{2}\alpha\sin^{2}\beta+\frac{\lambda_{3}+\lambda_{4}+\lambda_{5}}{2}\sin
2\alpha\sin 2\beta\right],\end{split}$ (5)
$\displaystyle\begin{split}m_{h^{0}}^{2}&=\frac{m_{12}^{2}}{\sin\beta\cos\beta}\cos^{2}(\beta-\alpha)\\\
&\quad+v^{2}\left[\lambda_{1}\sin^{2}\alpha\cos^{2}\beta+\lambda_{2}\cos^{2}\alpha\sin^{2}\beta-\frac{\lambda_{3}+\lambda_{4}+\lambda_{5}}{2}\sin
2\alpha\sin 2\beta\right],\end{split}$ (6) $\displaystyle m_{A^{0}}^{2}$
$\displaystyle=\frac{m_{12}^{2}}{\sin\beta\cos\beta}-\lambda_{5}v^{2},$ (7)
$\displaystyle m_{H^{\pm}}^{2}$
$\displaystyle=\frac{m_{12}^{2}}{\sin\beta\cos\beta}-\frac{\lambda_{4}+\lambda_{5}}{2}v^{2}.$
(8)

The mass basis allows us to concentrate on the 8 physical parameters: the
masses of the new particles, $m_{H^{\pm}}$, $m_{h^{0}}$, $m_{H^{0}}$,
$m_{A^{0}}$, the mixing angles $\alpha$ and $\beta$, the softly
$\mathbb{Z}_{2}$ breaking term $m_{12}$, and the VEV $v$. We identify $h^{0}$
as the observed SM-like boson with a mass of $125.1\pm 0.14$ GeV Zyla:2020zbs
and take $v=246$ GeV throughout this work. The angles $\alpha$ and $\beta$,
describing the mixing of the neutral and charged Higgs fields, respectively,
satisfy the following relations Arnan:2017lxi

$\displaystyle\tan\beta$ $\displaystyle=\frac{v_{2}}{v_{1}},$ (9)
$\displaystyle\tan 2\alpha$
$\displaystyle=\frac{2\left(-m_{12}^{2}+(\lambda_{3}+\lambda_{4}+\lambda_{5})v_{1}v_{2}\right)}{m_{12}^{2}\left(v_{2}/v_{1}-v_{1}/v_{2}\right)+\lambda_{1}v_{1}^{2}-\lambda_{2}v_{2}^{2}}.$
(10)

One may revert Eqs. (5) - (10) and express explicitly the $\lambda_{i}$
parameters in terms of the physical parameters, see e.g. Ref. Kling:2016opi .
We will show below that data strongly favours the small $\cos(\beta-\alpha)$
limit, where the expressions for the $\lambda_{i}$ become particularly simple
Han:2020zqg ; Kling:2016opi :

$\displaystyle v^{2}\lambda_{1}$
$\displaystyle=m_{h^{0}}^{2}-\frac{\tan\beta(m_{12}^{2}-m_{H^{0}}^{2}\sin\beta\cos\beta)}{\cos^{2}\beta},$
(11) $\displaystyle v^{2}\lambda_{2}$
$\displaystyle=m_{h^{0}}^{2}-\frac{m_{12}^{2}-m_{H^{0}}^{2}\sin\beta\cos\beta}{\tan\beta\sin^{2}\beta},$
(12) $\displaystyle v^{2}\lambda_{3}$
$\displaystyle=m_{h^{0}}^{2}+2m_{H^{+}}^{2}-m_{H^{0}}^{2}-\frac{m_{12}^{2}}{\sin\beta\cos\beta},$
(13) $\displaystyle v^{2}\lambda_{4}$
$\displaystyle=m_{A^{0}}^{2}-2m_{H^{+}}^{2}+\frac{m_{12}^{2}}{\sin\beta\cos\beta},$
(14) $\displaystyle v^{2}\lambda_{5}$
$\displaystyle=-m_{A^{0}}^{2}+\frac{m_{12}^{2}}{\sin\beta\cos\beta}.$ (15)

The Lagrangian for the Higgs and Yukawa sectors in the 2HDM is given by

$\mathcal{L}^{\rm 2HDM}_{\rm
H+Y}=\sum_{i}|D_{\mu}\Phi_{i}|^{2}-V(\Phi_{1},\Phi_{2})+\mathcal{L}_{\text{Y}},$
(16)

where $\mathcal{L}_{\text{Y}}$ depends on the details of the quark-Higgs
couplings in the 2HDM. There are four main types of 2HDM, based on which
particles each of the doublets couple to. In this work, the 2HDM of Type II
(2HDM-II) is investigated as it gives quark mass terms in the Lagrangian that
are similar to the form in the SM. This in turn yields a similar CKM matrix
through which all flavour-changing interactions can be described, with tree-
level FCNCs forbidden by the imposition of the $\mathbb{Z}_{2}$ symmetry
(which is only softly broken by $m_{12}^{2}$). The 2HDM-II model has the
doublet $\Phi_{1}$ coupling to down-type quarks and leptons, while $\Phi_{2}$
is coupling to up-type quarks, so

$\mathcal{L}_{\text{Y}}=-Y^{(u)}_{ij}\bar{Q^{i}_{L}}\tilde{\Phi}_{2}u^{j}_{R}-Y^{(d)}_{ij}\bar{Q^{i}_{L}}\Phi_{1}d^{j}_{R}-Y^{(\ell)}_{ij}\bar{L_{L}^{i}}\Phi_{1}\ell^{j}_{R}+{\rm
h.c.},$ (17)

where $\tilde{\Phi}_{i}=i\sigma_{2}\Phi^{*}_{i}$, $Y_{ij}^{(u,d,\ell)}$ are
the Yukawa couplings, $i,j=1,2,3$ run over the generations of the fermions,
and the left-handed $SU(2)$-doublets of quarks and leptons are, respectively,

$Q_{L}^{i}=\begin{pmatrix}u_{L}^{i}\\\ d_{L}^{i}\end{pmatrix},\qquad
L_{L}^{i}=\begin{pmatrix}\nu_{L}^{i}\\\ \ell_{L}^{i}\end{pmatrix}.$

After EWSB, rotating to the mass basis and focusing on the gauge and Yukawa
couplings of $h^{0}$ gives Lagrangian terms of a very similar form to the SM,
with factors $\kappa_{i}$ relating the coupling constants to those in the SM
Han:2020lta :

$\mathcal{L}_{h^{0}}=\kappa_{V}\frac{m_{Z}^{2}}{v}hZ_{\mu}Z^{\mu}+\kappa_{V}\frac{2m_{W}^{2}}{v}hW^{+}_{\mu}W^{\mu-}-\sum_{f=u,d,\ell}\kappa_{f}\frac{m_{f}}{v}h\bar{f}f,$
(18)

where, in the 2HDM-II,

$\displaystyle\kappa_{\small{V}}$ $\displaystyle=\sin{(\beta-\alpha)},$ (19)
$\displaystyle\kappa_{u}$
$\displaystyle=\sin{(\beta-\alpha)}+\cot{\beta}\cos{(\beta-\alpha)},$
$\displaystyle\kappa_{d,\ell}$
$\displaystyle=\sin{(\beta-\alpha)}-\tan{\beta}\cos{(\beta-\alpha)}.$

Naturally, in the SM all $\kappa_{i}=1$, which is consistent with LHC data
Aad:2019mbh ; ATLAS:2020qdt ; CMS:2020gsy . Setting $\cos{(\beta-\alpha)}=0$
recovers this and thus matches the phenomenology of $h^{0}$ in the 2HDM with
the SM Higgs. As such, this is known as the alignment limit. The dependence of
these, and numerous other, couplings on $\cos{(\beta-\alpha)}$ and $\tan\beta$
leads us to choose these as parameters of the model, in place of $\alpha$ and
$\beta$.

The addition of new Higgs fields introduces new interactions, where the
charged Higgs fields can replace $W^{\pm}$ fields in flavour-changing charged
interactions, as well as direct couplings of the new bosons to the weak bosons
themselves. The part of the Yukawa Lagrangian describing interactions of
$H^{\pm}$ bosons with the fermions is given by Gunion:1989we ; Branco:2011iw
444Here and hereafter, for notation simplicity, we omit indices $i,j$
indicating the generation of the up- and down-type quarks $u=u,c,t$ and
$d=d,s,b$.

$\mathcal{L}_{H^{\pm}}=\frac{\sqrt{2}\,V_{ud}}{v}H^{+}\bar{u}\left[m_{u}\cot\beta\,P_{L}+m_{d}\tan\beta\,P_{R}\right]d+\frac{\sqrt{2}}{v}m_{\ell}\tan\beta\,H^{+}\left(\bar{\nu}_{\ell}P_{R}\,\ell\right)+{\rm
h.c.},$ (20)

where $P_{L,R}=(1\mp\gamma_{5})/2$, $m_{u}$ and $m_{d}$ are the masses of the
up- and down-type quarks, $m_{\ell}$ ($\ell=e,\mu,\tau$) is the charged-lepton
mass, and $V_{ud}$ denotes the corresponding element of the quark-mixing
Cabibbo-Kobayashi-Maskawa (CKM) matrix.

The new Higgs fields, discussed above, will affect many measured particle
physics quantities including the electroweak precision parameters, Higgs
signal strengths and flavour physics observables. These effects are
investigated in Sections 4, 5, and 6 respectively and used to set bounds on
the parameter space of the model. However, we start with the discussion of
purely theoretical constraints.

##  3 Theoretical Constraints

###  3.1 Perturbativity

Within the lambda basis, perturbativity in the scalar sector can be simply
expressed as Chen:2018shg ; Ginzburg:2005dt

$|\lambda_{i}|\leq 4\pi,\qquad i=1,\ldots 5.$ (21)

Clearly this is not a strict experimental constraint like the bounds studied
below; it is more related to our ability to make meaningful predictions in the
2HDM - and the numerical value of the bound on the $\lambda_{i}$ contains a
certain amount of arbitrariness. Thus we will also consider the less
conservative bound of $|\lambda_{i}|\leq 4$ \- inspired by the results in
Refs. Grinstein:2015rtl ; Cacchio:2016qyh . None of our conclusions will
actually be changed by modifying the perturbativity bound from $4\pi$ to $4$;
to be able to study some more extreme scenarios in Section 7.3 we will,
however, use the more conservative bound $4\pi$.

Writing the expressions for $\lambda_{i}$ in terms of the additional Higgs
masses we find that for low masses, i.e. $M\ll 1000$ GeV, no bounds (below the
particle mass $M$) are set on the mass differences of the new Higgs particles,
while for high masses a form of degeneracy has to hold. In our analysis, we
will consider heavy Higgs masses ranging between $10^{2.5}\approx 300$ GeV and
$10^{5}$ GeV. Masses of the charged Higgs boson as low as 300 GeV are clearly
ruled out by flavour observables, in particular by $b\to s\gamma$, as
discussed below. The experimental lower bound on the charged Higgs mass from
direct searches is actually only $m_{H^{+}}\gtrsim 160\,$GeV Aaboud:2018gjj .
The upper bound has been chosen to be equal to the centre-of-mass energy of a
future 100 TeV collider.

Looking at the couplings in Eq. (20), one can derive two further
perturbativity constraints from the Yukawa sector

$\displaystyle\frac{\sqrt{2}\,V_{tb}\,m_{t}\cot\beta}{2v}\leq{\sqrt{4\pi}}$
$\displaystyle\quad\Rightarrow\quad\tan\beta>{0.14}\,,$ (22)
$\displaystyle\frac{\sqrt{2}\,V_{tb}\,m_{b}\tan\beta}{2v}\leq{\sqrt{4\pi}}$
$\displaystyle\quad\Rightarrow\quad\tan\beta<{300}\,,$

where we again have chosen a very conservative range. For the range of
$\log(\tan\beta)$ we will consider the conservative lower bound
$\tan\beta=10^{-1.5}=0.03$, and the upper bound $\tan\beta=10^{+2.5}=300$.

###  3.2 Vacuum Stability and Unitarity

Next we apply the conditions for a stable vacuum as set out in Ref.
Deshpande:1977rw :

$\displaystyle\lambda_{1,2}$ $\displaystyle>0,$ (23)
$\displaystyle\lambda_{3}$ $\displaystyle>-(\lambda_{1}\lambda_{2})^{1/2},$
$\displaystyle\lambda_{3}+\lambda_{4}-|\lambda_{5}|$
$\displaystyle>-(\lambda_{1}\lambda_{2})^{1/2}.$

Demanding the vacuum to be the global minimum of the potential, we further
find Barroso:2013awa

$m_{12}^{2}\left(m_{11}^{2}-m_{22}^{2}\left(\frac{\lambda_{1}}{\lambda_{2}}\right)^{\frac{1}{2}}\right)\left(\tan\beta-\left(\frac{\lambda_{1}}{\lambda_{2}}\right)^{\frac{1}{4}}\right)>0.$
(24)

We also consider the conditions from tree-level unitarity, see Refs.
Ginzburg:2005dt ; Arhrib:2000is 555 For similar discussion in an alternative
lambda basis see Horejsi:2005da . , alongside NLO unitarity and the condition
that NLO corrections to partial wave amplitudes are suppressed relative to LO
contributions, see Refs. Cacchio:2016qyh ; Grinstein:2015rtl .

To implement these constraints alongside the perturbativity conditions, we
carry out Monte Carlo scans. Starting with the allowed sets of $\lambda_{i}$
(via Eq. (21)) we then use Eqs. (6) and (10) to get a relation among the
parameters $\lambda_{1}$,…, $\lambda_{5}$, $\beta$ and the small parameter
$\delta:=\beta-\alpha-\pi/2$666Higgs data suggests $|\delta|<$0.05; see
Section 5., which is explicitly given in Appendix B. Expanding the expressions
up to second order in $\delta$, Eqs. (5)-(8) give the allowed ranges of Higgs
masses depending on the value of $\beta$ and $\delta$. We show the effect of
the theory constraints on the Higgs masses $m_{H^{+}}$, $m_{H^{0}}$,
$m_{A^{0}}$, $\tan\beta$ and $\cos(\beta-\alpha)$ as two-dimensional
projections in Fig. 1. The starting values $|\lambda_{i}|<4\pi$ are depicted
in green, while points from $|\lambda_{i}|<4$ are shown in amber – the latter
obviously leads to more constrained regions. Here we note that due to the
imposition of NLO unitarity constraints, for starting values
$|\lambda_{i}|<4\pi$, we find most successful points for
$|\lambda_{i}|\lesssim 2\pi$. Moreover, we find that for charged Higgs masses
above 1 TeV - as implied by the flavour observables - the choice of
$|\lambda_{i}|<4\pi$ or $|\lambda_{i}|<4$ has only a very minor impact.

Figure 1: Bounds on the heavy Higgs masses $m_{H^{0}}$, $m_{A^{0}}$ and
$m_{H^{+}}$, $\tan\beta$ and $\cos(\beta-\alpha)$ stemming from the theory
constraints (perturbativity, vacuum stability and unitarity conditions). For
the plots $10^{8}$ points were generated and only values of $|\delta|<0.5$
were considered. The starting values $|\lambda_{i}|<4\pi$ are given in green,
while $|\lambda_{i}|<4$ is shown in amber.

The two upper plots and the left of the middle plots of Fig. 1 show that
theory constraints require the new Higgses to be more or less degenerate in
mass. The heavier the mass, the stronger this requirement becomes, which can
also be nicely read off from Table 1. The right plot of the middle row in Fig.
1 shows that $|\delta|$ will be constrained to small values, if the new Higgs
particles are heavier than about a TeV (such a bound on the charged Higgs mass
will follow from flavour constraints discussed below). Again, the constraints
on $|\delta|$ become stronger the heavier the new Higgs masses become.
Quantitative values for the bounds on $|\delta|$ are given in Table 1. In the
lower row of Fig. 1 we see that $\tan\beta$ is not really constrained by the
theory constraints, apart from the restrictions given in Eq. (22).

In the alignment limit, the value of $\alpha$ is fixed by the value of
$\tan\beta$, and we find a similar requirement for a mass degeneracy of the
new Higgs particles as in the general case. Moreover, we do not find any bound
on $\tan\beta$ within the regions of interest.

$m_{H^{\pm}}$/TeV$\in$ |  $\Delta_{A^{0}H^{0}}$/GeV $\in$ |  $\Delta_{A^{0}H^{\pm}}$/GeV $\in$ |  $\Delta_{H^{0}H^{\pm}}$/GeV $\in$ | $\delta\in$
---|---|---|---|---
$[0.0,0.5)$ | $(-536,521)$ | $(-420,574)$ | $(-351,532)$ | $(-0.5,0.5)$
$[0.5,1.0)$ | $(-297,296)$ | $(-328,268)$ | $(-351,268)$ | $(-0.5,0.5)$
$[1.0,1.5)$ | $(-163,158)$ | $(-123,157)$ | $(-118,154)$ | $(-0.30,0.25)$
$[1.5,2.0)$ | $(-109,105)$ | $(-74,110)$ | $(-73,105)$ | $(-0.19,0.16)$
$[2.0,2.5)$ | $(-82,84)$ | $(-52,82)$ | $(-46,79)$ | $(-0.14,0.13)$
$[2.5,3.0)$ | $(-67,64)$ | $(-42,67)$ | $(-36,65)$ | $(-0.11,0.09)$
$[3.0,3.5)$ | $(-53,55)$ | $(-35,56)$ | $(-34,53)$ | $(-0.09,0.08)$
$[3.5,4.0)$ | $(-47,47)$ | $(-30,47)$ | $(-27,48)$ | $(-0.07,0.07)$
$[4.0,4.5)$ | $(-42,40)$ | $(-30,41)$ | $(-22,43)$ | $(-0.07,0.06)$
$[4.5,5.0)$ | $(-35,35)$ | $(-23,36)$ | $(-18,37)$ | $(-0.06,0.06)$
Table 1: A numerical summary of the effects of imposing the theory constraints
with starting values $|\lambda_{i}|<4\pi$. These intervals are found by
carrying out a $10^{8}$ points Monte Carlo scan with $|\delta|<0.5$,
$m_{H^{\pm}},m_{A^{0}},m_{H^{0}}<5000\,\mathrm{GeV}$ and
$-1.5<\log_{10}(\tan\beta)<2.5$. We have defined $\Delta_{ij}:=m_{i}-m_{j}$.

##  4 Electroweak Precision Observables

The oblique parameters first proposed in Ref. Peskin:1990zt  are defined as

$\displaystyle S$
$\displaystyle:=\frac{4e^{2}}{\alpha}\frac{d}{dq^{2}}\big{[}\Pi_{33}(q^{2})-\Pi_{3Q}(q^{2})\big{]}\big{|}_{q^{2}=0},$
(25) $\displaystyle T$
$\displaystyle:=\frac{e^{2}}{\alpha\sin^{2}(\theta_{W})m^{2}_{W}}\big{[}\Pi_{11}(0)-\Pi_{33}(0)\big{]},$
(26) $\displaystyle U$
$\displaystyle:=\frac{4e^{2}}{\alpha}\frac{d}{dq^{2}}\big{[}\Pi_{11}(q^{2})-\Pi_{33}(q^{2})\big{]}\big{|}_{q^{2}=0}.$
(27)

$\Pi_{XY}(q^{2})$ represents the self energies of the electroweak gauge bosons
$X$ and $Y$ with 4-momentum $q^{\mu}$, the subscript $i$ represents the
$W_{i}$ field and the subscript $Q$ represents the $B$ field. Here we stress
that in defining these oblique parameters a subtraction of the SM
contributions is made; $S=S_{\rm NP}-S_{\rm SM}$ and similarly for $T$ and
$U$, such that, by construction, in the SM $S=T=U=0$.

While most of the constraints we consider below depend on the particular type
of the 2HDM, the oblique parameters are, at one loop order, unaffected by the
Yukawa couplings and therefore universal for all 2HDMs. The expressions for
$S$, $T$, and $U$ in the 2HDM (derived from Ref. Grimus:2008nb ) have been
collected in Appendix C.

New physics generally has only a small effect on $U$; we therefore follow the
approach outlined in Ref. Zyla:2020zbs  and set $U$ to zero. This effectively
reduces the error on the experimental result for $T$, due to the correlation
between $T$ and $U$. The oblique parameters will be included in our global
fit.

##  5 Higgs Signal Strengths

Signal strengths are a key measurement in determining if the observed Higgs
boson is indeed that of the SM, or if new physics is involved. For a single
production mode $i$ with a cross section $\sigma_{i}$, and decay channel $f$
with a branching fraction ${\cal B}_{f}$, the signal strength $\mu^{f}_{i}$ is
defined as

$\mu^{f}_{i}=\frac{(\sigma_{i}\cdotp{\cal
B}_{f})_{\text{Exp.}}}{(\sigma_{i}\cdotp{\cal B}_{f})_{\text{SM}}},$ (28)

as the cross section and branching fraction cannot be separately measured
without further assumptions. Evidently if the SM is accurate, all these signal
strengths should take a value of 1.

In practice, the small branching fractions and cross sections render some
channels impossible to measure currently, but there are recent measurements
available for 31 channels from the ATLAS and CMS collaborations at
$\sqrt{s}=13\;\text{TeV}$ with an integrated luminosity of up to 139 fb-1 and
137 fb-1 respectively Sirunyan:2018koj ; Sirunyan:2019qia ; Aad:2019mbh ;
Aad:2020plj ; Aad:2020xfq ; CMS:2020gsy ; ATLAS:2020qdt . These include
various combinations of the four main $h^{0}$ production channels at the LHC:
gluon-gluon fusion (ggF), vector boson fusion (VBF), in association with a
weak vector boson (Vh) or a top-antitop pair (tth), and the decay channels
$\gamma\gamma$, $ZZ^{*}$, $WW^{*}$, $Z\gamma$, $\tau^{+}\tau^{-}$, $b\bar{b}$
and $c\bar{c}$. The recent observations of $h^{0}\to\mu^{+}\mu^{-}$
Aad:2020xfq ; Sirunyan:2020two  are also included here. In cases such as
$h^{0}\to\mu^{+}\mu^{-}$ where a combined signal strength across all channels
is given, it is interpreted as being purely from gluon-gluon fusion as this
mechanism dominates the Higgs production cross section at the LHC. In
instances where a correlation matrix between measurements is given this is
also included in the analysis. For full details on which channels have been
used and their experimental values, see Table LABEL:tab:List-of-Higgs-signal-
strengths in Appendix A. Where multiple measurements are available for a
single channel we take an average, which significantly improves our
sensitivity (see flavio’s documentation for details Straub:2018kue ).

The couplings of $h^{0}$ to the massive gauge bosons and fermions are
multiplied by factors $\kappa_{i}$ compared to the SM Higgs, as set out in
Section 2, with the relations for $\kappa_{i}$ given in Eq. (19). As the
couplings are modified, the signal strength data can be used to place
constraints on the two parameters on which the $\kappa_{i}$ depend:
$\cos{(\beta-\alpha)}$ and $\tan\beta$. A scenario in which

$\displaystyle\kappa_{\small{V}}=1,\qquad\kappa_{u}=1,\qquad\kappa_{d,\ell}=-1,$
(29)

has been introduced as one of the ways a fourth sequential chiral generation
of fermions may still exist777Simply adding a fourth sequential, chiral,
perturbative generation of fermions, without any additional new degrees of
freedom is definitely ruled out, see e.g. Refs. Eberhardt:2012gv ;
Djouadi:2012ae . while remaining hidden from current LHC data Das:2017mnu . By
enforcing $\kappa_{u}=-\kappa_{d}=1$ one finds

$\cos{(\beta-\alpha)}=\sin{2\beta}=\frac{2\tan\beta}{1+\tan^{2}\beta}\,.$ (30)

The full criteria of the wrong sign limit are satisfied exactly only when this
equation is followed at very large $\tan\beta$, giving the set of couplings in
Eq. (29). The contributions of the additional Higgs bosons to the loop-level
processes have been neglected as they are negligible for the allowed masses
found in Section 6.

Figure 2:  Contour plot of the allowed 2HDM-II parameter space in the
$(\tan\beta-\cos(\beta-\alpha))$ plane from the Higgs signal strengths, with
the dashed black line indicating the alignment limit ($\cos(\beta-\alpha)=0$)
and the dashed red line the wrong sign limit of Eq. (30). The contours
indicate the allowed space at 1, 2, 3, 4, 5$\sigma$ from dark to lighter.

The results of the analysis including all the Higgs signal strengths listed in
Table LABEL:tab:List-of-Higgs-signal-strengths are shown in Fig. 2 (see also
Table 2); we will later include the Higgs signal strengths in our global fit.
This fit is performed using the analytic expressions for the Higgs production
and decay channels in the 2HDM found in Ref. Gunion:1989we . As outlined
above, the coupling modifications, the $\kappa_{i}$, are functions of
$\tan\beta$ and $\cos(\beta-\alpha)$, allowing us to constrain this plane. For
some recent studies that include a similar analysis in the 2HDM, see, for
instance, Refs. Chowdhury:2017aav ; Dawson:2020oco ; Han:2020zqg ; Han:2020lta
; Ellis:2018gqa . We observe that the alignment limit must be closely
followed, which is unsurprising given that almost all of the experimental data
included here matches the SM within 2$\sigma$, with the majority agreeing at
1$\sigma$. As a result, the 2HDM must closely replicate the SM behaviour for
$h^{0}$. At 2$\sigma$ we find

$\big{|}\cos{(\beta-\alpha)}\big{|}\leq 0.050,$ (31)

with the maximum allowed value of $\cos{(\beta-\alpha)}$ occurring at
$\tan\beta$ $\approx$ 1\. Due to the dependence of $\kappa_{d,\ell}$ on
$\tan\beta$, this maximum value falls significantly once away from $\tan\beta$
$\approx$ 1, to the point where, at large values of $\tan\beta$, the alignment
limit must be strictly followed and it is valid to set
$\cos{(\beta-\alpha)}=0$. At 3$\sigma$ and above we find the wrong sign limit
to be allowed, which substantially increases this upper bound, provided Eq.
(30) is closely followed.

Figure 3:  Contour plot of the coupling constant modifiers, in which we have
fixed to its best fit value $\kappa_{V}=1.036$ and marked the alignment
(wrong-sign) limit in black (red). The contours indicate the allowed space at
1, 2, 3, 4, 5$\sigma$ from dark to lighter.

The wrong sign condition from enforcing $\kappa_{u}=-\kappa_{d}=1$, Eq. (30),
which is shown in Fig. 2 as a dashed red line, is here found to be excluded at
2.7$\sigma$ by the signal strength data, an improvement over previous works
such as Refs. Han:2020zqg ; Aad:2019mbh ; Bertrand:2020lyb , due in part to
our inclusion of a larger data set than in such works, which improves our
sensitivity.

To further probe the wrong-sign limit, we examine the possible values of the
coupling modification factors, independently from Eq. (19). We first perform a
fit in terms of the three $\kappa_{i}$ to find the best fit point, and then
fix one of the $\kappa_{i}$ to its best fit value, allowing a 2D fit in terms
of the remaining $\kappa_{j,k\neq i}$ to be performed. We show the results
from such a fit in the $(\kappa_{u}-\kappa_{d,\ell})$ plane in Fig. 3, where
we have fixed to the best fit value $\kappa_{V}=1.036$, and stress again that
these results do not make use of Eq. (19). This fit does not exclude
$\kappa_{d}=-1$, in line with the current lack of sensitivity to the sign of
$\kappa_{d}$ at the LHC. We then look to find solutions in terms of
$\tan\beta$ and $\cos(\beta-\alpha)$ that give values of $\kappa_{u}$ and
$\kappa_{d}$ that lie within the 2$\sigma$ allowed region close to
$\kappa_{d}=-1$ (bottom part of Fig. 3) by using Eq. (19). However, the only
solutions we find require that $|\cos(\beta-\alpha)|\gtrsim 0.96$. Such
extreme values of $\cos(\beta-\alpha)$ are not consistent, in the 2HDM-II,
with the fixed value of $\kappa_{V}=1.036$ from the best fit point that we use
in the scan. Therefore this $2\sigma$ allowed region in the
$(\kappa_{u}-\kappa_{d,\ell})$ plane cannot be attained in the 2HDM-II. Thus,
while we do not in general exclude $\kappa_{d}=-1$ we can, in the 2HDM-II that
we examine here, rule out the arrangement $\kappa_{u}=-\kappa_{d}=1$ (given by
Eq. (30)) at 2.7$\sigma$ from Fig. 2.

More generally, we observe that when $\kappa_{d,\ell}=-1$, small deviations
away from $\kappa_{u}=1$ have a significant impact, causing
$\cos(\beta-\alpha)$ to be large, which is incompatible with the signal
strength data. The debate in the literature over the possibility of the wrong-
sign limit will hopefully be resolved by future collider data, with
suggestions on fruitful search channels made in Refs. Modak:2016cdm  and
Raju:2020hpe , with the latter also including the possibility of a hidden
sequential fourth generation of fermions.

##  6 Flavour Observables

In this section, we present our fits of the 2HDM-II for selected flavour
observables. We show fits for groups of observables individually, and then in
Section 7, we combine these with all other observables in the global fit. For
each fit, we use the standard $(\tan\beta-m_{H^{+}})$ parameter space in a
log-log scale. All fits have been performed using the flavio package as
described in Section 1.

###  6.1 Leptonic and Semi-leptonic Tree-Level Decays

The Yukawa term in the 2HDM-II Lagrangian (17) leads to additional
contributions to the tree-level flavour-changing charged transitions.
Integrating out the heavy charged Higgs boson $H^{\pm}$, one obtains the
effective Hamiltonian describing the $d\to u\,\ell^{-}\bar{\nu}_{\ell}$ or
$u\to\,d\ell^{+}\nu_{\ell}$ transition in the 2HDM:

${\cal H}^{\rm eff}_{H^{+}}=-\frac{4G_{F}}{\sqrt{2}}V_{ud}(C_{S-P}\,{\cal
O}_{S-P}+C_{S+P}\,{\cal O}_{S+P})+{\rm h.c.,}$ (32)

where the new effective operators are

$\displaystyle\mathcal{O}_{S-P}=(\bar{u}P_{L}\,d)(\bar{\ell}P_{L}\nu_{\ell}),\qquad\mathcal{O}_{S+P}=(\bar{u}P_{R}\,d)(\bar{\ell}P_{L}\nu_{\ell}).$
(33)

The corresponding Wilson coefficients are related to the 2HDM-II parameters as
follows:

$\displaystyle C_{S-P}=\frac{m_{u}\,m_{\ell}}{m_{H^{+}}^{2}}\,,\qquad C_{S+P}$
$\displaystyle=\frac{m_{d}\,m_{\ell}\tan^{2}\beta}{m_{H^{+}}^{2}}\,.$ (34)

For use in flavio, we convert the latter coefficients to
$C_{SL}^{du\ell\nu_{\ell}},C_{SR}^{du\ell\nu_{\ell}}$, respectively, in the
flavio WET basis, finding

$\displaystyle C_{SL}^{du\ell\nu_{\ell}}=-C_{S-P}\,,\qquad
C_{SR}^{du\ell\nu_{\ell}}$ $\displaystyle=-C_{S+P}\,,$ (35)

where the negative sign in front of these coefficients is due to the
convention for the effective operators adopted in flavio’s basis. Feynman
diagrams describing the leptonic and semi-leptonic transitions in the 2HDM are
shown in Fig. 4.

Figure 4: Diagrams contributing to leptonic (left) and semi-leptonic (right)
decays in the 2HDM.

The full list of leptonic and semi-leptonic modes included in our fit is given
in Table LABEL:tab:List-of-flavour-observables, where the corresponding SM
predictions (produced in flavio) are based on Bernlochner:2017jka ;
Caprini:1997mu ; Sakaki:2013bfa ; Bharucha:2015bzk ; Gubernari:2018wyi ;
Detmold:2016pkz ; Bernard:2006gy ; Bernard:2009zm ; Antonelli:2010yf . In
addition to previously measured semi-leptonic decay channels, we consider for
the first time the $B_{s}\to D_{s}^{(*)}\mu\bar{\nu}_{\mu}$ modes measured
recently by the LHCb collaboration Aaij:2020hsi , and the corresponding
$B_{s}\to D_{s}^{(*)}$ form factors are determined using Lattice QCD
McLean:2019qcx . Moreover, we consider the Lepton-Flavour Universality (LFU)
observables $R(D^{(*)})\equiv{\cal B}(B\to D^{(*)}\tau\bar{\nu}_{\tau})/{\cal
B}(B\to D^{(*)}\ell\bar{\nu}_{\ell})$, where $\ell=e$ or $\mu$. The
experimental measurements of the latter were found to be in tension with the
corresponding SM predictions, giving hints of possible LFU violation. In the
2HDM, this violation is caused by different couplings of the charged Higgs
boson to the lepton pair, which are proportional to the lepton mass
$m_{\ell}$. Using the HFLAV averages for $R(D^{(*)})$ Amhis:2019ckw , we
present in Fig. 5 the allowed $(\tan\beta-m_{H^{+}})$ regions at 1 and
2$\sigma$ levels, where the left (right) plot corresponds to $R(D)$
($R(D^{*})$). Note that the semi-leptonic $b\to c\ell\bar{\nu}_{\ell}$
transitions receive negative corrections compared to the SM contributions,
therefore $R(D)$ and $R(D^{*})$ move further away from the measurements (apart
from a narrow region of the $(\tan\beta-m_{H^{+}})$ parameter space (see Fig.
5), where the 2HDM-II correction becomes more than the twice the size of the
SM contribution proportional to the scalar form factor). Since $R(D)$ is in
fact only $1.2\sigma$ away from the SM prediction (see Table LABEL:tab:List-
of-flavour-observables), one finds a wide allowed range of $\tan\beta$ and
$m_{H^{+}}$ at the $2\sigma$ level (the left plot in Fig. 5). On the other
side, $R(D^{*})$ is in greater tension ($2.8\sigma$) with the SM, therefore
one obtains just a narrow region with large $\tan\beta$ and small $m_{H^{+}}$
(the right plot in Fig. 5). The experimental combination of both $R(D)$ and
$R(D^{*})$ Amhis:2019ckw  is more than three standard deviations away from the
SM prediction, and within 2$\sigma$ one finds in the 2HDM-II only a very
narrow region at very low masses of the charged Higgs, $m_{H^{+}}\sim 1\,{\rm
GeV}$, which is far from the physical domain. We find that the 2HDM-II is not
able to accommodate the experimental data on both $R(D)$ and $R(D^{*})$ within
$3.5\,\sigma$; the corresponding tension in the SM (using flavio) is
$3.2\,\sigma$.

Combining all the leptonic and semi-leptonic tree-level decay channels
indicated in Table LABEL:tab:List-of-flavour-observables yields the allowed
range of $\tan\beta$ and $m_{H^{+}}$ at $1$ and $2\sigma$ levels as shown in
Fig. 6.

Figure 5: Contour plot of the allowed 2HDM-II parameter space in the
$(\tan\beta-m_{H^{+}})$ plane, originating from $R(D)$ (left) and $R(D^{*})$
(right). The lighter contour indicates the allowed parameter space at
$2\sigma$ confidence level while the darker contour corresponds to $1\sigma$.
Figure 6: Contour plot of the allowed 2HDM-II parameter space in the
$(\tan\beta-m_{H^{+}})$ plane, originating from the combination of tree-level
leptonic and semi-leptonic decays of $B,B_{s},D,D_{s},K$, and $\pi$ mesons and
the hadronic decays of $\tau$ leptons to $K$ and $\pi$ mesons with a tau
neutrino as well as $R(D)$ and $R(D^{*})$, see Table LABEL:tab:List-of-
flavour-observables. The lighter contour indicates the allowed parameter space
at $2\sigma$ confidence level while the darker contour corresponds to
$1\sigma$.

In addition, we would like to make an important observation regarding the
extraction of the CKM matrix elements from experimental data. Consider e.g.
the branching fraction of a leptonic meson decay in the 2HDM,

$\mathcal{B}_{\rm 2HDM}=\mathcal{B}_{\rm SM}\times(1+\delta_{\rm 2HDM})^{2},$
(36)

where $\delta_{\rm 2HDM}$ is the 2HDM correction factor. Conventionally, the
corresponding CKM element is determined from the experimental measurement,
assuming $\mathcal{B}_{\rm exp}=\mathcal{B}_{\rm SM}$. But if the 2HDM is
realistic then $\mathcal{B}_{\rm exp}=\mathcal{B}_{\rm 2HDM}$ and the CKM
element extracted from measurement is actually proportional to $(1+\delta_{\rm
2HDM})$. Therefore, the fraction $1/(1+\delta_{\rm 2HDM})$ is then interpreted
as the modification factor that the measured CKM element must receive in the
2HDM to be the true CKM element. A similar argument holds for the semi-
leptonic meson decays.

Using Eq. (36), and a similar relation for the semi-leptonic decays, the size
of the modification factor $1/(1+\delta_{\rm 2HDM})$ can be scanned across the
2HDM-II parameter space $(\tan\beta-m_{H^{+}})$ to test the significance of
the factor for each of the CKM matrix elements, where the accepted values are
commonly taken from leptonic and semi-leptonic tree-level decays. We perform
tests of this significance for $V_{us}$, $V_{ub}$ and $V_{cb}$ using the tree-
level parameterisation to consider the propagated effects to all CKM elements,
and we also take into account unitarity of the CKM matrix. We find negligible
effects to all CKM elements for most of the parameter space considered here,
and the small areas where more noticeable effects arise are excluded from
theoretical constraints, direct searches, and by the global fit in Section 7.

Were the 2HDM-II proven to be physical, it could be the case that there is
some small modification to CKM elements that, although is not enough to impact
results here, would need to be considered. This would be dependent on the
specific values found for $m_{H^{+}}$ and $\tan\beta$ from experiment.

###  6.2 Neutral $B$-Meson Mixing

Neutral $B$-meson mixing plays an important role in flavour physics. For a
recent discussion on $B$-mixing in the SM and within New Physics models see,
for example, Ref. DiLuzio:2019jyq . In the 2HDM, $B$-mixing gains additional
contributions from new box diagrams where one or both of the virtual $W^{\pm}$
bosons are replaced by charged Higgses $H^{\pm}$, see Fig. 7. Here we consider
the mass difference $\Delta m_{d,s}$ between the heavy and light eigenstates
in both $B_{d}$ and $B_{s}$ meson mixing.

Figure 7: Examples of box diagrams describing $B_{d}$\- and $B_{s}$-meson
mixing in the 2HDM.

The effective 2HDM Hamiltonian for $\Delta B=2$ processes can be expressed as
Crivellin:2019dun :

$\mathcal{H}_{\rm eff}^{\Delta
B=2}=\sum_{k=1}^{5}C_{k}\mathcal{O}_{k}+\sum_{k=1}^{3}C_{k}^{\prime}\mathcal{O}_{k}^{\prime},$
(37)

where the operators are defined (for $q=d,s$) as

$\displaystyle\mathcal{O}_{1}^{(^{\prime})}$
$\displaystyle=(\bar{q}^{\alpha}\gamma^{\mu}P_{L(R)}b^{\alpha})(\bar{q}^{\beta}\gamma_{\mu}P_{L(R)}b^{\beta}),$
(38) $\displaystyle\mathcal{O}_{2}^{(^{\prime})}$
$\displaystyle=(\bar{q}^{\alpha}P_{L(R)}b^{\alpha})(\bar{q}^{\beta}P_{L(R)}b^{\beta}),$
$\displaystyle\mathcal{O}_{4}$
$\displaystyle=(\bar{q}^{\alpha}P_{L}b^{\alpha})(\bar{q}^{\beta}P_{R}b^{\beta}),$
$\displaystyle\mathcal{O}_{3}^{(^{\prime})}$
$\displaystyle=(\bar{q}^{\alpha}P_{L(R)}b^{\beta})(\bar{q}^{\beta}P_{L(R)}b^{\alpha}),$
$\displaystyle\mathcal{O}_{5}$
$\displaystyle=(\bar{q}^{\alpha}P_{L}b^{\beta})(\bar{q}^{\beta}P_{R}b^{\alpha}),$

with $\alpha$ and $\beta$ denoting the colour indices. The contributions of
these operators to $B$-meson mixing in the 2HDM are well-known throughout
literature, see e.g. Refs. Geng:1988bq ; Urban:1997gw ; Crivellin:2019dun . In
our work, for the Wilson coefficients $C^{(\prime)}_{k}$ in Eq. (37) we use
expressions calculated in Ref. Crivellin:2019dun . We convert these to the
flavio WET basis Aebischer:2017ugx , where we find

$\displaystyle C_{VLL(RR)}^{bqbq}$ $\displaystyle=C_{1}^{(^{\prime})},$
$\displaystyle\qquad C_{SLR}^{bqbq}$ $\displaystyle=C_{4},$ (39)
$\displaystyle C_{SLL(RR)}^{bqbq}$ $\displaystyle=C_{2}^{(^{\prime})},$
$\displaystyle C_{VLR}^{bqbq}$ $\displaystyle=-\frac{1}{2}\,C_{5}.$

The operators $\mathcal{O}_{3}^{(^{\prime})}$ do not give any contributions
(at LO in QCD) to mixing in the 2HDM Crivellin:2019dun , so we do not need to
consider conversion for $C_{3}^{(\prime)}$ here. The largest theory
uncertainties by far stem from the non-perturbative values of the matrix
elements of the $\Delta B=2$ operators given in Eq. (38). In our analysis we
will use the averages presented in Ref. DiLuzio:2019jyq , which are based on
HQET Sum Rule evaluations King:2019lal ; Kirk:2017juj ; Grozin:2016uqy  and
lattice simulations Dowdall:2019bea ; Boyle:2018knm ; Bazavov:2016nty . The
perturbative SM corrections are known and implemented to NLO-QCD accuracy
Buras:1990fn . In the numerical analysis we further use the most recent
experimental averages for $\Delta m_{d}$ and $\Delta m_{s}$ Amhis:2019ckw
888Not yet including the recent, most precise value of $\Delta m_{s}$ from
LHCb Aaij:2021jky . (see Table LABEL:tab:List-of-flavour-observables). The
results of the fit are shown in Fig. 8, indicating that the mass differences
constrain the allowed $\tan\beta$ region from below.

Figure 8: Contour plot of the allowed 2HDM-II parameter space in the
$(\tan\beta-m_{H^{+}})$ plane for the mass difference in the mixing of $B_{d}$
and $B_{s}$ mesons. The lighter contour indicates the allowed parameter space
at $2\sigma$ confidence level while the darker contour corresponds to
$1\sigma$.

###  6.3 Loop-Level $b\to s,d$ Transitions

The effective Hamiltonian for flavour-changing neutral current (FCNC) $b\to
q\ell^{+}\ell^{+}$ and $b\to s\gamma$ (for $q=d,s$) processes is defined as
Crivellin:2019dun

$\mathcal{H}_{\rm eff}^{b\to
q\ell\ell}=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{tq}^{*}\left(\sum_{k=7,8}C_{k}^{(^{\prime})}\mathcal{O}_{k}^{(^{\prime})}+\sum_{k=9,10,S,P}\\!\\!\\!C_{k}^{(^{\prime})}\mathcal{O}_{k}^{(^{\prime})}\right)+{\rm
h.c.}.$ (40)

The operators in this Hamiltonian are

$\displaystyle\mathcal{O}_{7}^{(^{\prime})}$
$\displaystyle=\frac{e\,m_{b}}{16\pi^{2}}\left(\bar{q}\sigma^{\mu\nu}P_{R(L)}b\right)\\!F_{\mu\nu},$
$\displaystyle\mathcal{O}_{8}^{(^{\prime})}$
$\displaystyle=\frac{g_{s}\,m_{b}}{16\pi^{2}}\left(\bar{q}\sigma^{\mu\nu}P_{R(L)}T^{a}b\right)\\!G_{\mu\nu}^{a},$
(41) $\displaystyle\mathcal{O}_{9}^{(^{\prime})}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}}(\bar{q}\gamma_{\mu}P_{L(R)}b)(\bar{\ell}\gamma^{\mu}\ell),$
$\displaystyle\mathcal{O}_{10}^{(^{\prime})}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}}(\bar{q}\gamma_{\mu}P_{L(R)}b)(\bar{\ell}\gamma^{\mu}\gamma_{5}\ell),$
$\displaystyle\mathcal{O}_{S}^{(^{\prime})}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}}(\bar{q}P_{L(R)}b)(\bar{\ell}\ell),$
$\displaystyle\mathcal{O}_{P}^{(^{\prime})}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}}(\bar{q}P_{L(R)}b)(\bar{\ell}\gamma_{5}\ell).$

The expressions for 2HDM-II contributions to each operator’s Wilson
coefficient are given e.g. in Section 3 of Crivellin:2019dun . Since all the
operators in Eq. (40) are defined in the same way as in the flavio WET basis
(for either $\ell=e,\mu$), no Wilson coefficient conversions are required
here. Some example diagrams describing the $b\to q\ell^{+}\ell^{-}$
transitions in the 2HDM are shown in Fig. 9.

Figure 9: Example of diagrams (penguins and boxes) describing the $b\to
q\ell^{+}\ell^{-}$ transitions in the 2HDM.

####  6.3.1 Radiative Decay $\bar{B}\to X_{s}\gamma$

Figure 10: One-loop penguin contributions to $b\to s\gamma$ processes in the
2HDM.

The radiative decay $\bar{B}\to X_{s}\gamma$ is an important observable in the
2HDM-II. Historically, it has played a significant role in giving a lower
bound on the charged Higgs mass. A recent update of the SM value with NNLO
accuracy in QCD Misiak:2020vlo  (based on e.g. Refs. Misiak:2006ab ;
Misiak:2015xwa ),

$\mathcal{B}^{\rm SM}(\bar{B}\to X_{s}\gamma)\big{|}_{E_{\gamma}>1.6\,{\rm
GeV}}=(3.40\pm 0.17)\times 10^{-4},$ (42)

leads to a stronger constraint on the charged Higgs mass: $m_{H^{+}}\geq
800\,$GeV (95% CL) Misiak:2020vlo . The experimental average from
Amhis:2019ckw  is based on Chen:2001fja ; Lees:2012ym ; Belle:2016ufb . The
operators which contribute to the radiative decays $\bar{B}\to X_{s}\gamma$
are $\mathcal{O}_{7},\mathcal{O}_{8}$, given by Eq. (41). As expected, our fit
for this process provides an important lower bound for the charged Higgs mass,
predicted here as (see also Fig. 11)

$m_{H^{+}}\gtrsim 790\,(1510)\,\text{GeV}\;\text{ at }2\sigma\,(1\sigma).$
(43)

We slightly differ from Ref. Misiak:2020vlo  in that our 2HDM-II contributions
are taken at NLO Borzumati:1998nx , while they use NNLO results Hermann:2012fc
and we also use a different statistical treatment. In addition, as can be seen
in Fig. 11, the bound for $m_{H^{\pm}}$ becomes even stronger for lower values
of $\tan\beta$.

Figure 11: Contour plot of the allowed 2HDM-II parameter space in the
$(\tan\beta-m_{H^{+}})$ plane for the radiative decay $B\to X_{s}\gamma$. The
lighter contour indicates the allowed parameter space at $2\sigma$ confidence
level while the darker contour corresponds to $1\sigma$.

####  6.3.2 Leptonic $B_{d,s}\to\mu^{+}\mu^{-}$ Decays

The FCNC leptonic meson decays $B_{d,s}\to\mu^{+}\mu^{-}$ are particularly
sensitive to scalar operator contributions in NP models, and therefore can be
excellent probes of the effects of the 2HDM. These processes are particularly
worthy of consideration now, due to the recently-produced experimental
combinations which we use in our fit Altmannshofer:2021qrr , and recent ATLAS,
CMS, and LHCb results Aaij:2017vad ; Aaboud:2018mst ; Sirunyan:2019xdu ; LHCb-
CONF-2020-002 ; LHCb:2021awg ; LHCb:2021vsc . The SM prediction is based on a
perturbative element Buchalla:1993bv ; Bobeth:2013uxa ; Beneke:2019slt  and a
non-perturbative determination of the decay constants, see e.g. Ref.
Bussone:2016iua ; Bazavov:2017lyh ; Hughes:2017spc . It has been common in the
past to study these decays in the large $\tan\beta$ limit
($\tan\beta\gg\sqrt{m_{t}/m_{b}}$, see e.g. Ref. Logan:2000iv ) where the
Yukawa coupling to $b$ quarks is large and there can be simplifications to the
2HDM contributions of the operators which affect this process:
$\mathcal{O}_{10}^{(^{\prime})},\mathcal{O}_{S}^{(^{\prime})},\mathcal{O}_{P}^{(^{\prime})}$.
However, a big part of the allowed parameter space of the 2HDM-II lies not
within the large $\tan\beta$, thus we use only the general expressions given
in Ref. Crivellin:2019dun .

Figure 12:  Contour plot of the allowed 2HDM-II parameter space in the
$(\tan\beta-m_{H^{+}})$ plane for the FCNC leptonic $B_{d,s}\to\mu^{+}\mu^{-}$
decays, fixing the additional parameters as $\cos(\beta-\alpha)=0$,
$m_{H^{0}}=m_{A^{0}}=m_{H^{+}}$. The lighter contour indicates the allowed
parameter space at $2\sigma$ confidence level while the darker contour
corresponds to $1\sigma$.

These generalised expressions bring three further parameters of the 2HDM into
play: $\alpha$, $m_{H^{0}}$, and $m_{A^{0}}$. In Section 7, we will combine
all observables in a fit of the five independent parameters: $\tan\beta$,
$m_{H^{+}}$, $m_{H^{0}}$, $m_{A^{0}}$ and $\cos(\beta-\alpha)$. Here, in order
to present a 2D fit in the $(\tan\beta-m_{H^{+}})$ parameter space, we fix the
other three parameters as $\cos(\beta-\alpha)=0$, as the alignment limit is
preferred (see Section 7) by the global fit, and
$m_{H^{0}}=m_{A^{0}}=m_{H^{+}}$, which is justified from theoretical
constraints (see Section 3). The corresponding contour plot is shown in Fig.
12. From there one can see that the mass of charged Higgs is also constrained
from below, roughly:

$m_{H^{+}}\gtrsim 300\,{\rm GeV},$ (44)

however this bound is not as strong as that from the $B\to X_{s}\gamma$ decay.
Furthermore, $B_{d,s}\to\mu^{+}\mu^{-}$ yields a strong correlation between
possible values of $\tan\beta$ and $m_{H^{+}}$.

####  6.3.3 Semi-leptonic $b\to s\ell^{+}\ell^{-}$ Transitions

For the semi-leptonic $b\to s\ell^{+}\ell^{-}$ processes we are considering,
all of the operators from Eq. (41) contribute. Therefore all the 2HDM
parameters, $m_{H^{+}}$, $m_{H^{0}}$, $m_{A^{0}}$, $\tan\beta$ and
$\cos(\beta-\alpha)$, will affect these processes.

The first group of observables are the LFU ratios $R_{K}$ and $R_{K^{*}}$
Bobeth:2007dw , defined as

$R_{K^{(*)}}\equiv\frac{\mathcal{B}[B\to
K^{(*)}\mu^{+}\mu^{-}]}{\mathcal{B}[B\to K^{(*)}e^{+}e^{-}]},$ (45)

where the branching fractions are integrated over bins of squared dilepton
invariant mass $q^{2}$. These quantities are theoretically very clean, since
almost all hadronic effects drop out in the ratio. In the SM, the values of
$R_{K^{(*)}}$ are very close to 1 with tiny uncertainties. The electromagnetic
corrections for $R_{K^{(*)}}$ worked out in Ref. Bordone:2016gaq  were found
to be very small, $\approx 1-2\%$. In our analysis, we consider the ten binned
LFU ratios listed in Table LABEL:tab:RKtab. The recent measurement by LHCb
shows a deviation from the SM of $3.1\sigma$ for $R_{K^{+}}$ Aaij:2021vac .

Figure 13:  Contour plot of the allowed 2HDM-II parameter space in the
$(\tan\beta-m_{H^{+}})$ plane for the $R_{K^{(*)}}$ observables and fixing the
additional parameters at their best fit point in Table 2. The lighter contour
indicates the allowed parameter space at $2\sigma$ confidence level while the
darker contour corresponds to $1\sigma$.

In Fig. 13 we present the resulting $(\tan\beta-m_{H^{+}})$ contour plot for
$R_{K^{(*)}}$ observables, where we fixed the three other 2HDM parameters at
their best fit values for the 10 $R_{K^{(*)}}$ bins:
$\cos(\beta-\alpha)=0.001$, $m_{H^{0}}=250\,{\rm GeV},m_{A^{0}}=9.2\,$ GeV
(see Table 2). As one can see, $R_{K^{(*)}}$ could only be accommodated within
the 2HDM-II only at very low $m_{H^{+}}\lesssim 1\,{\rm GeV}$ and very high
$\tan\beta\gtrsim 300$, which are beyond the physical domain (see Section 3).
Therefore, the 2HDM-II (as well as the SM) is unable to accommodate the
current experimental values of $R_{K^{(*)}}$. The contours in Fig. 13 are not
only found in regions which are in disagreement with the allowed contours in
Figs. 6, 8, 11, 12, but also with the constraints from direct searches
Aaboud:2018gjj . Moreover, in deriving the formulae for the relevant Wilson
coefficients Crivellin:2019dun , it was assumed that $m_{H^{+}}$ is at least
of the order of the electroweak scale, meaning that the validity of contours
much lower than $\log_{10}[m_{H^{+}}/\text{GeV}]\sim 2$ must be questioned.

The combined fit of the 10 $R_{K^{(*)}}$ observables is in disagreement with
our combined fits of all observables to $4.2\,\sigma$ confidence. Due to this
disagreement between $R_{K^{(*)}}$ and other flavour observables, we work with
two approaches for our fits: excluding and including $R_{K^{(*)}}$.

Next we include further $b\to s\ell^{+}\ell^{-}$ observables (binned branching
ratios, angular distributions, asymmetries, etc.) – some of these also deviate
from the SM predictions, see Table LABEL:tab:bsll. For more detailed analyses
of these processes and their NP implications, see e.g. Refs.
Khodjamirian:2010vf ; Khodjamirian:2012rm ; Bharucha:2015bzk ;
Khodjamirian:2017fxg ; Alguero:2019ptt ; Hurth:2020ehu ; Alok:2019ufo ;
Ciuchini:2019usw ; Hurth:2020rzx ; Hurth:2021nsi ; Alguero:2021anc ;
Cornella:2021sby ; Altmannshofer:2021qrr ; Geng:2021nhg ; Ciuchini:2020gvn ;
MunirBhutta:2020ber ; Biswas:2020uaq . For all $b\to s\ell^{+}\ell^{-}$
processes listed in Table LABEL:tab:bsll and the leptonic decays
$B_{q}\to\mu^{+}\mu^{-}$ (see Table LABEL:tab:List-of-flavour-observables),
but excluding the $R_{K^{(*)}}$ observables, one gets the allowed regions for
$\tan\beta$ and $m_{H^{+}}$ shown in Fig. 14. Deviations of experiment from SM
predictions now introduce an upper bound on the charged Higgs mass at
$1\sigma$. A further inclusion of the $R_{K^{(*)}}$ observables does not
change the contours of allowed regions in Fig. 14 sizeably, but does worsen
the corresponding fit and the $p$-value, see Table 2.

Figure 14: Contour plot of the allowed 2HDM-II parameter space in the
$(\tan\beta-m_{H^{+}})$ plane for the leptonic $B_{d,s}\to\mu^{+}\mu^{-}$
decays and the semi-leptonic $b\to s\ell^{+}\ell^{-}$ observables listed in
Table LABEL:tab:bsll excluding the $R_{K^{(*)}}$ observables and fixing the
additional parameters to the best fit point from the $b\to s\ell^{+}\ell^{-}$
fit in Table 2. The lighter contour indicates the allowed parameter space at
$2\sigma$ confidence level while the darker contour corresponds to $1\sigma$.

##  7 Global Fit

###  7.1 Results

Combining all the observables collected in Table LABEL:tab:List-of-Higgs-
signal-strengths (Higgs signal strengths), Table LABEL:tab:List-of-oblique-
parameters ($S,T,U$), Table LABEL:tab:List-of-flavour-observables (flavour
observables without $b\to s\ell^{+}\ell^{-}$), Table LABEL:tab:bsll ($b\to
s\ell^{+}\ell^{-}$ observables) and Table LABEL:tab:RKtab ($R_{K^{(*)}}$) one
obtains the results shown in Table 2. We emphasize that in searching for the
best-fit point of the corresponding likelihood function we have applied the
theory constraints for the Higgs mass differences (see Table 1). As one can
see from Table 2, including all the observables yields a relatively poor fit
with a low $p\approx 1.5\%$-value. This is due to the fact that the
$R_{K^{(*)}}$ and $R(D^{(*)})$ quantities cannot be accommodated within the
2HDM-II, as was already discussed in Sections 6.1 and 6.3.3. Nevertheless the
2HDM-II still achieves a higher level of agreement than the SM and we find a
corresponding pull of $2.3\sigma$. Not considering the observables
$R_{K^{(*)}}$ yields a better fit for the 2HDM-II with a significantly higher
$p\approx 7\%$-value, again marking a better performance than the SM, with a
corresponding pull of $1.8\sigma$.

Next we find from Table 2 and Fig. 15 that the global fit prefers the
alignment limit, $\cos(\beta-\alpha)=0$, allowing only small deviations from
zero:

$\max\\{|\cos(\beta-\alpha)|\\}\leq 0.02\,(0.04),\qquad{\rm
at}\,\,1\sigma\,(2\sigma).$ (46)

This bound is considerably stronger than the one we obtained in Eq. (31) using
only data on Higgs signal strengths. Regarding the wrong sign limit solution,
we found using only the Higgs signal strengths this to be allowed at
$2.7\sigma$ or above; using the global fit of all observables (excluding
$R_{K^{(*)}}$), this is now excluded within $6\sigma$ of our best fit point
with $p$-value $\approx 7\%$. We also perform fits to all observables
requiring the wrong sign limit as defined in Eq. (30), where one can see the
quality of these fits is significantly worsened compared to the general global
fit with no input constraint on $\cos(\beta-\alpha)$, yielding a $p$-value
$\approx 0.1\%$.

The best fit further favours values of $m_{H^{+}}\approx m_{H^{0}}\approx
m_{A^{0}}\approx 2.3$ TeV and $\tan\beta\approx 4$. Performing a scan around
the best-fit point (excluding $R_{K^{(*)}}$) and keeping in mind the theory
constraints in Table 2, we find that the charged Higgs mass is constrained to
be in the region

$\displaystyle 76.3\,{\rm TeV}\geq m_{H^{+}}\geq 1.26\,{\rm TeV}$
$\displaystyle\mbox{at }1\sigma,$ $\displaystyle m_{H^{+}}\geq 0.86\,{\rm
TeV}$ $\displaystyle\mbox{at }2\sigma,$ $\displaystyle m_{H^{+}}\geq
0.68\,{\rm TeV}$ $\displaystyle\mbox{at }3\sigma,$ $\displaystyle
m_{H^{+}}\geq 0.57\,{\rm TeV}$ $\displaystyle\mbox{at }4\sigma,$
$\displaystyle m_{H^{+}}\geq 0.49\,{\rm TeV}$ $\displaystyle\mbox{at
}5\sigma.$ (47)

From the $2\sigma$-level onwards we do not see an upper bound within the area
considered ($m_{H^{+}}\leq 100\,{\rm TeV}$). The parameter $\tan\beta$ has a
strong correlation with $m_{H^{+}}$. For the best fit values of
$\cos(\beta-\alpha)$, $m_{H^{0}}$ and $m_{A^{0}}$, we find the allowed regions
in the ($\tan\beta-m_{H^{+}}$) plane given in Fig. 16 and Fig. 17 (the latter
zoomed into the best-fit point and shown in combination with the theory
constraints). Assuming both the alignment limit, $\cos(\beta-\alpha)=0$, and
the degeneracy for the Higgs masses, $m_{H^{+}}=m_{H^{0}}=m_{A^{0}}$, the
contour plot of Fig. 18 shows that $\tan\beta$ is strongly constrained for low
$m_{H^{+}}\sim 1\,{\rm TeV}$ while the bounds on $\tan\beta$ become weaker
with an increase of $m_{H^{+}}$ up to $100\,{\rm TeV}$. We stress again that
these constraints above are obtained from the fit by excluding $R_{K^{(*)}}$.
The inclusion of $R_{K^{(*)}}$ would not significantly change the allowed
parameter regions, while it would considerably worsen the quality of the fit.

Furthermore, we investigate several constrained scenarios in the fit, as
indicated in Table 2. First, we checked that the wrong-sign limit is
absolutely disfavored by the global fit. Second, in the exact alignment limit
the fit yields similar results as in the general case. Finally, assuming both
the alignment limit, $\cos(\beta-\alpha)=0$, and the degeneracy for the Higgs
masses, $m_{H^{+}}=m_{H^{0}}=m_{A^{0}}$, again leads to similar results due to
the preference of both of these assumptions by the global fit and theory
constraints.

Scenario | $\\#$ | Best-fit point | $\chi_{min}^{2}$ |  $p$-value
---|---|---|---|---
| Observables | $\\{\tan\beta,m_{H^{+}},m_{H^{0}},m_{A^{0}},\cos(\beta-\alpha)\\}$ |  |
All incl. $R_{K^{(*)}}$ | $275$ | $\\{4.2,2140\,{\rm GeV},2180\,{\rm GeV},2210\,{\rm GeV},0.0096\\}$ | $323$ | $\phantom{0}1.5\%$
All excl. $R_{K^{(*)}}$ | $265$ | $\\{4.3,2340\,{\rm GeV},2380\,{\rm GeV},2390\,{\rm GeV},0.0090\\}$ | $295$ | $\phantom{0}6.6\%$
Flavour incl. $R_{K^{(*)}}$ | $241$ | $\\{4.2,2310\,{\rm GeV},2300\,{\rm GeV},2290\,{\rm GeV},0.001\\}$ | $297$ |  $\phantom{0}0.4$%
Flavour excl. $R_{K^{(*)}}$ | $231$ | $\\{4.3,2420\,{\rm GeV},2390\,{\rm GeV},2360\,{\rm GeV},0.0001\\}$ | $269$ |  $\phantom{0}2.5$%
$b\to s\ell\ell$ incl. $R_{K^{(*)}}$ | $202$ | $\\{4.0,820\,{\rm GeV},690\,{\rm GeV},690\,{\rm GeV},0.0003\\}$ | $265$ |  $\phantom{0}0.1$%
$b\to s\ell\ell$ excl. $R_{K^{(*)}}$ | $192$ | $\\{4.0,820\,{\rm GeV},690\,{\rm GeV},690\,{\rm GeV},0.0003\\}$ | $238$ |  $\phantom{0}0.7$%
Only $R_{K^{(*)}}$ | $10$ | $\\{2370,0.14\,{\rm GeV},250\,{\rm GeV},9.2\,{\rm GeV},0.001\\}$ | $8.5$ |  $13.1$%
Higgs Signals | $31$ | $\\{\sim 100\pm 20,-,-,-,0.0003\\}$ | $24$ |  $72.9$%
Wrong Sign Limit, $\cos(\beta-\alpha)=\sin 2\beta$
All incl. $R_{K^{(*)}}$ | $275$ | $\\{12.1,2320\,{\rm GeV},2270\,{\rm GeV},2260\,{\rm GeV}\\}$ | $380$ | $0.001\%$
All excl. $R_{K^{(*)}}$ | $265$ | $\\{12.2,2320\,{\rm GeV},2270\,{\rm GeV},2260\,{\rm GeV}\\}$ | $340$ | $0.1\%$
Alignment Limit, $\cos(\beta-\alpha)=0$
All incl. $R_{K^{(*)}}$ | $275$ | $\\{4.1,2290\,{\rm GeV},2330\,{\rm GeV},2340\,{\rm GeV}\\}$ | $326$ |  $1.3$%
All excl. $R_{K^{(*)}}$ | $265$ | $\\{4.2,2260\,{\rm GeV},2300\,{\rm GeV},2310\,{\rm GeV}\\}$ | $297$ |  $6.4$%
$\cos(\beta-\alpha)=0$, $m_{H^{+}}=m_{H^{0}}=m_{A^{0}}$
All incl. $R_{K^{(*)}}$ | $275$ | $\\{4.0,2070\,{\rm GeV}\\}$ | $328$ |  $1.3$%
All excl. $R_{K^{(*)}}$ | $265$ | $\\{4.2,2320\,{\rm GeV}\\}$ | $300$ |  $5.7$%
Table 2: Best fit points of 2HDM-II parameter fits for various groups of
observables using the constraints from theory to inform the physical parameter
values. Figure 15:  Contour plot of the allowed 2HDM-II parameter space in the
$(\tan\beta-\cos(\beta-\alpha))$ plane obtained by combining all observables
excluding $R_{K^{(*)}}$, fixing the additional parameters to the best fit
point from Table 2. The contours indicate the allowed parameter space at
$1,2,3,4,5\sigma$ confidence level going from darker to lighter. Figure 16:
Contour plot of the allowed 2HDM-II parameter space in the $(\tan\beta-
m_{H^{+}})$ plane, obtained by combining all observables excluding
$R_{K^{(*)}}$, fixing the additional parameters to the best fit point from
Table 2. The contours indicate the allowed parameter space at
$1,2,3,4,5\sigma$ confidence level going from darker to lighter. Figure 17:
Contour plot of the allowed 2HDM-II parameter space in the $(\tan\beta-
m_{H^{+}})$ plane, obtained by combining all observables excluding
$R_{K^{(*)}}$, fixing the additional parameters to the best fit point from
Table 2 and including the restrictions from the theory constraints in Table 1
(shown by the red dashed lines). The contours indicate the allowed parameter
space at $1,2,3,4,5\sigma$ confidence level going from darker to lighter.
Figure 18:  Contour plot of the allowed 2HDM-II parameter space in the
$(\tan\beta-m_{H^{+}})$ plane, obtained by combining all observables excluding
$R_{K^{(*)}}$, fixing the additional parameters as $\cos(\beta-\alpha)=0$,
$m_{H^{+}}=m_{H^{0}}=m_{A^{0}}$. The contours indicate the allowed parameter
space at $1,2,3,4,5\sigma$ confidence level going from darker to lighter.

###  7.2 Comment on the Anomalous Magnetic Moment of the Muon

Figure 19: Examples of one- and two-loop 2HDM contributions to
$\mu\to\mu\gamma$.

The anomalous magnetic moment of the muon,

$a_{\mu}=\frac{g_{\mu}-2}{2},$ (48)

measures the deviation of the muon gyromagnetic moment $g_{\mu}$ from 2 due to
quantum loop effects. This can be accurately measured in experiment, with
recent measurements at Fermilab Abi:2021gix  confirming the older BNL value
Bennett:2006fi . It can also be accurately predicted in the Standard Model and
the Theory Initiative (WP Aoyama:2020ynm ) yields a $4.2\sigma$ discrepancy
with the updated combined experimental value. We will also consider a recent
Lattice QCD evaluation (BMW Collaboration Borsanyi:2020mff ) that finds the
$g_{\mu}-2$ discrepancy to be only $1.6\sigma$.

This deviation in $a_{\mu}$ can be a strong motivation for potential new
physics signals, such as supersymmetric particle loops or other BSM
contributions, see e.g. Ref. Athron:2021iuf . 2HDM contributions to $a_{\mu}$
form a subset of SUSY contributions to the anomalous magnetic moment and they
have been recently studied e.g. in Refs. Botella:2020xzf ; Jana:2020pxx ;
Ghosh:2020tfq .

Matching flavio’s WET-3 basis Aebischer:2017ugx , we write the effective
Hamiltonian for $\ell\to\ell\gamma$ processes as

$\mathcal{H}_{\rm
eff}^{\ell\to\ell\gamma}=-\frac{4G_{F}}{\sqrt{2}}\frac{e}{16\pi^{2}}m_{\mu}C_{7}(\bar{\ell}\sigma_{\rho\omega}P_{R}\ell)F^{\rho\omega}+{\rm
h.c.}$ (49)

The 2HDM contribution to $a_{\mu}$ is then defined as

$a_{\mu}^{\rm 2HDM}=\frac{G_{F}m_{\mu}^{2}}{\sqrt{2}\,\pi^{2}}C_{7},$ (50)

where we take the contributions at one- and two-loop level given in
Ilisie:2015tra . The two-loop contributions come from the Barr-Zee diagrams
(Barr:1990vd , see Fig. 19) with a fermion or boson loop which can give rise
to significant contributions to $a_{\mu}^{\rm 2HDM}$. All Higgs bosons are
present in these loops, therefore $a_{\mu}^{\rm 2HDM}$ depends not only on
$\tan\beta$ and $m_{H^{+}}$, but also on $m_{A^{0}}$ and $m_{H^{0}}$, with a
further dependence on the mixing angle $\alpha$.

Figure 20: Contour plots of the allowed 2HDM-II parameter space in the
$(\tan\beta-m_{H^{+}})$ plane for $a_{\mu}$, using the Standard Model
prediction by WP20 Aoyama:2020ynm  (BMW Borsanyi:2020mff ) (left (right)
plot), fixing the additional parameters as
$\cos(\beta-\alpha)=0,\,m_{H^{0}}=m_{A^{0}}=m_{H^{+}}$. The lighter contour
indicates the allowed parameter space at $2\sigma$ confidence level while the
darker contour corresponds to $1\sigma$.

From Fig. 20 we find that the 2HDM-II cannot explain the discrepancy between
the experimental number for the anomalous magnetic moment of the muon and the
white paper prediction as the required values of $\tan\beta$ lie in the non-
perturbative region. Obviously the BMW prediction yields a consistency of
experiment and theory within 2$\sigma$ and thus a large portion of the 2HDM-II
parameter space is still allowed.

###  7.3 Electroweak Phase Transition

The electroweak phase transition (EWPT) is a mechanism which can generate the
baryon asymmetry we observe in the Universe today through the process of EW
baryogenesis, with the requirement that the EWPT is of strong first order
(SFO). In the SM this could be achieved for $m_{h^{0}}\lesssim 70\,$ GeV
Kajantie:1996mn , however the measurement of the SM Higgs boson mass as
$m_{h^{0}}=125.1\pm 0.14\,$ GeV Zyla:2020zbs  means we now require BSM physics
to achieve a SFOEWPT; a 2HDM is in principle capable of generating this. For
recent work testing the 2HDM for a SFOEWPT across large regions in its
parameter space, see e.g. Refs. Basler:2016obg ; Su:2020pjw ; Dorsch:2016nrg ,
in which the authors find regions where the 2HDM-II could support a SFOEWPT,
however only for Higgs masses less than $1\,$TeV. In Wang:2021ayg , the
strength of the EWPT in the 2HDM-II is tested for Higgs masses $\gtrsim
600\,$GeV; a FOEWPT can be found for these masses but it is not strong enough
to support EW baryogenesis. Furthermore, a SFOEWPT appears to favour a mass
splitting ($\sim 200\,$GeV Su:2020pjw ) between $H^{+}$ and $H^{0}$, and
$m_{A^{0}}\sim m_{H^{+}}$ Su:2020pjw ; Dorsch:2016nrg . This could still be
allowed by our results in Section 3, however only for low charged Higgs
masses, in tension with the global fit of Fig. 16.

In this work, we use the BSMPT package Basler:2018cwe ; Basler:2020nrq  to
calculate the strength of the EWPT for selected parameter points across our
fit. The BSMPT package calculates the strength of the EWPT for the 2HDM in the
lambda parameter basis ($\lambda_{i}$, $m_{12}^{2}$, $\tan\beta$). We have so
far primarily worked in the mass basis
($m_{A^{0}},m_{H^{0}},m_{h^{0}},m_{H^{+}},\tan\beta,\cos(\beta-\alpha)$), and
so must convert to the lambda basis. In the alignment limit, a sufficient but
not necessary condition for vacuum stability is Kling:2016opi

$m_{12}^{2}=m_{H^{0}}^{2}\sin\beta\cos\beta\,\cap\,m_{h^{0}}^{2}+m_{H^{\pm}}^{2}-m_{H^{0}}^{2}>0\,\cap\,m_{h^{0}}^{2}+m_{A^{0}}^{2}-m_{H^{0}}^{2}>0.$
(51)

We then use this, along with Eqs. (11)-(15), to perform the conversion, while
making the assumption that $\cos(\beta-\alpha)$ is small, which is of course
in line with our constraints. This is only appropriate when the conditions of
Eq. (51) are fulfilled, i.e. primarily for $m_{H^{0}}\lesssim
m_{A^{0}},m_{H^{+}}$.

The BSMPT package parameterises the strength of the EWPT in the 2HDM by
calculating

$\xi_{c}=\frac{\omega_{c}}{T_{c}},$ (52)

where $\omega_{c}$ is the high-temperature VEV
($\omega_{c}^{2}=\omega_{1}^{2}(T_{c})+\omega_{2}^{2}(T_{c})$ and
$\omega_{i}(T=0)=v_{i}$) at the critical temperature $T_{c}$. The BSMPT
package only passes a result if $\xi_{c}>0$, with the desired SFOEWPT
indicated by $\xi_{c}>1$.

$\tan\beta$ | Mass Basis (GeV) | Lambda Basis | $m_{12}^{2}$ | $\omega_{c}$ | $T_{c}$ | $\xi_{c}$
---|---|---|---|---|---|---
$m_{H^{+}}$ | $m_{H^{0}}$ | $m_{A^{0}}$ | $\lambda_{3}$ | $\lambda_{4}$ | $\lambda_{5}$ | (GeV2) | (GeV) | (GeV)
$4.2$ | $50000$ | $50000$ | $50000$ | $0.26$ | $0$ | $0$ | $5.6\times 10^{8}$ | $0.66$ | $162$ | $0.004$
$4.2$ | $2320$ | $2320$ | $2320$ | $0.26$ | $0$ | $0$ | $1.2\times 10^{6}$ | $23$ | $162$ | $0.14$
$4.2$ | $2320$ | $2250$ | $2280$ | $10.8$ | $-8.3$ | $-2.2$ | $1.1\times 10^{6}$ | $31$ | $186$ | $0.17$
$1.0$ | $2320$ | $2320$ | $2320$ | $0.26$ | $0$ | $0$ | $2.7\times 10^{6}$ | $23$ | $162$ | $0.14$
$10$ | $2320$ | $2320$ | $2320$ | $0.26$ | $0$ | $0$ | $5.3\times 10^{5}$ | $24$ | $162$ | $0.15$
$4.2$ | $860$ | $710$ | $860$ | $8.03$ | $-3.9$ | $-3.9$ | $1.1\times 10^{5}$ | $142$ | $175$ | $0.81$
$4.2$ | $860$ | $690$ | $860$ | $8.95$ | $-4.3$ | $-4.3$ | $1.1\times 10^{5}$ | $177$ | $176$ | $1.00$
$4.2$ | $680$ | $470$ | $680$ | $8.22$ | $-4.0$ | $-4.0$ | $5.0\times 10^{4}$ | $211$ | $149$ | $1.42$
$4.2$ | $570$ | $320$ | $570$ | $7.60$ | $-3.7$ | $-3.7$ | $2.3\times 10^{4}$ | $226$ | $126$ | $1.79$
$4.2$ | $490$ | $250$ | $490$ | $6.12$ | $-2.9$ | $-2.9$ | $1.4\times 10^{4}$ | $207$ | $126$ | $1.64$
$4.2$ | $490$ | $490$ | $490$ | $0.26$ | $0$ | $0$ | $5.4\times 10^{4}$ | $24$ | $161$ | $0.15$
Table 3:  Table of results for the EWPT in the 2HDM. We consider the exact
alignment limit, $\cos(\beta-\alpha)=0$, and in addition we employ the
condition by Eq. (51). In this limit, as follows from Eqs. (11)-(15), one
gets: $\lambda_{1}=\lambda_{2}=m_{h^{0}}^{2}/v^{2}=0.26$, i.e. $\lambda_{1,2}$
are fixed and independent of the Higgs masses and $\tan\beta$ (and therefore
not again shown in the table). In addition, one obtains that $\lambda_{3,4,5}$
are independent of $\tan\beta$, and
$\lambda_{3}+\lambda_{4}+\lambda_{5}=m_{h^{0}}^{2}/v^{2}=0.26$. For similar
discussions of the lambda basis, see e.g. BhupalDev:2014bir .

We choose $M=50\,$TeV as a benchmark for more extreme masses, the best-fit
point of our degenerate mass fit, and lower limits in $m_{H^{+}}$ at $\sim
2,3,4,5\sigma$ where we select more favourable mass splittings as points to
test for a SFOEWPT. We also show some variations in $\tan\beta$ at the best-
fitting degenerate masses. In Table 3, we present our selected parameter
points in both bases, and then also the results produced by the BSMPT package.
As is generally favoured by our fits, for higher masses we mostly test
parameter points in the limit of degenerate masses, since this will satisfy
the conditions of Eq. (51) and the small mass splittings allowed at higher
mass scales have only small effects, as is shown in one benchmark point.
However for lower masses where these splittings are more significant, we
consider further benchmark points allowing for mass splitting more favourable
for a SFOEWPT Su:2020pjw , and find that demanding degeneracy for these lower
masses is a poorer choice.

We find that for most of our allowed parameter space, a SFOEWPT is not
possible due to the large mass scales involved. From the benchmark points at
higher masses, we find a maximum of $\xi_{c}=0.17$; much smaller than
required. For our benchmark points at lower masses however, the outlook
improves, where we first find a $\xi_{c}\geq 1$ for a parameter point at $\sim
2\sigma$ from the global best fit point. On the other hand, we note that all
points in Table 3 with a ratio $\xi_{c}\geq 1$ require some of the couplings
$|\lambda_{i}|>4$, which could be in conflict with perturbativity, depending
on what constraint we apply; see the beginning of Section 3.1. As we have only
chosen a few benchmark points, we cannot rule out the possibility of a
$\xi_{c}\geq 1$ at some point closer to our best fit, however this seems
unlikely given the trends shown here and in other works. Therefore to
accommodate a SFOEWPT in the 2HDM-II, we would require much lower Higgs masses
than are favoured by the global fit. But for such regions of the parameter
space the 2HDM-II will no longer perform better in the overall fit compared to
the SM.

##  8 Conclusion

In this work we have investigated theoretical bounds on the 2HDM-II stemming
from perturbativity, unitarity and vacuum stability as well as experimental
constraints stemming from electroweak precision, Higgs signal strengths,
flavour observables and the anomalous magnetic moment of the muon. All in all,
we find that the 2HDM-II can accommodate the data better than the SM. The best
fit point for the 2HDM-II lies around

$\displaystyle m_{H^{+}}\approx m_{H^{0}}$ $\displaystyle\approx
m_{A^{0}}\approx 2\,\mbox{TeV}\,,$ (53) $\displaystyle\tan\beta\approx 4,$
$\displaystyle\quad\cos(\beta-\alpha)\approx 0.01\,.$

For the charged Higgs mass we find a lower limit of 680 GeV at $3\sigma$ and
the remaining Higgses have to be largely degenerate – this requirement becomes
stronger the heavier the Higgs masses become. For low values of the charged
Higgs mass, $\tan\beta$ is severely constrained to lie around a value of 4,
while for high values of the charged Higgs mass (around 50 TeV) $\tan\beta$
should be in the region between 0.1 and 300. We further find that the wrong-
sign limit is disfavoured by Higgs signal strengths and excluded by the global
fit by more than $5\sigma$ and that only small deviations from the alignment
limit are possible.

Experiments currently observe some deviations from SM predictions, in
particular in $R(D^{(*)})$, $R_{K^{(*)}}$ and the anomalous magnetic moment of
the muon. The tensions in $R(D^{(*)})$ will either be enhanced in the 2HDM-II
or to be resolved, they would require parameter regions in the $(\tan\beta-
m_{H^{+}})$ plane which are excluded by other flavour observables. Eliminating
the tension in $R_{K^{(*)}}$ would require charged Higgs masses below 1 GeV
and non-perturbative values of $\tan\beta$. The 2HDM-II can also not resolve
the discrepancy between the measurement of the magnetic moment of the muon and
the white paper prediction.

Finally we found that the allowed parameter space of the 2HDM-II strongly
disfavours the existence of a sufficiently strong first order phase transition
for EW baryogenesis. This requires mass splittings discouraged by theoretical
constraints, and low masses of the charged Higgs – at least $2\sigma$ away
from our best fit point, which makes this model theoretically much less
attractive.

As a next step we plan to study more types of 2HDMs and also investigate
implications of our findings for future collider searches.

###### Acknowledgements.

The work of M.B. is supported by Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through TRR 257 “Particle Physics Phenomenology after the
Higgs Discovery”. O.A. is supported by a UK Science and Technology Facilities
Council (STFC) studentship under grant ST/V506692/1. We would like to thank
Otto Eberhardt, Christoph Englert, Martin Jung, Rusa Mandal, Margarete
Mühlleitner, Ulrich Nierste, Danny van Dyk for helpful discussions, Dipankar
Das for very useful comments on the arXiv version of the paper and Matthew
Kirk, Peter Stangl and David Straub for guidance with flavio. The fits were
run on the Siegen OMNI cluster.

##  Appendix A Inputs and Observables

Table 4: Table of the Higgs signal strengths used in the analysis in Section 5. Observable | Experimental value |  Source | SM prediction
---|---|---|---
Higgs signal strengths
$\mu^{\gamma\gamma}_{\text{ggF}}$ | $1.03\pm 0.11$ |  ATLAS:2020qdt  | 1
| $1.09^{+0.15}_{-0.14}$ |  CMS:2020gsy
| $1.07^{+0.12}_{-0.11}$ |  Sirunyan:2021ybb
$\mu_{\text{ggF}}^{ZZ}$ | $0.94^{+0.11}_{-0.10}$ |  ATLAS:2020qdt  | 1
| $0.98^{+0.12}_{-0.11}$ |  CMS:2020gsy
$\mu^{WW}_{\text{ggF}}$ | $1.08^{+0.19}_{-0.18}$ |  ATLAS:2020qdt  | 1
| $1.28^{+0.20}_{-0.19}$ |  CMS:2020gsy
| $1.20^{+0.16}_{-0.15}$ |  ATLAS:2021upe
$\mu^{Z\gamma}_{\text{ggF}}$ | $2.0^{+1.0}_{-0.9}$ |  Aad:2020plj  | 1
$\mu^{\tau\tau}_{\text{ggF}}$ | $1.02^{+0.60}_{-0.55}$ |  ATLAS:2020qdt  | 1
| $0.39^{+0.38}_{-0.39}$ |  CMS:2020gsy
$\mu^{\mu\mu}_{\text{ggF}}$ | $1.2\pm 0.6$ |  Aad:2020xfq  | 1
| $0.31^{+1.82}_{-1.81}$ |  CMS:2020gsy
| $1.19^{+0.42}_{-0.41}$ |  Sirunyan:2020two
$\mu^{bb}_{\text{ggF}}$ | $2.51^{+2.43}_{-2.01}$ |  ATLAS:2020qdt  | 1
| $2.45^{+2.53}_{-2.35}$ |  CMS:2020gsy
$\mu^{\gamma\gamma}_{\text{VBF}}$ | $1.31^{+0.26}_{-0.23}$ |  ATLAS:2020qdt  | 1
| $0.77^{+0.37}_{-0.29}$ |  CMS:2020gsy
| $1.04^{+0.34}_{-0.31}$ |  Sirunyan:2021ybb
$\mu^{ZZ}_{\text{VBF}}$ | $1.25^{+0.50}_{-0.41}$ |  ATLAS:2020qdt  | 1
| $0.57^{+0.46}_{-0.36}$ |  CMS:2020gsy
$\mu^{WW}_{\text{VBF}}$ | $0.60^{+0.36}_{-0.34}$ |  ATLAS:2020qdt  | 1
| $0.63^{+0.65}_{-0.61}$ |  CMS:2020gsy
| $0.99^{+0.24}_{-0.20}$ |  ATLAS:2021upe
$\mu^{\tau\tau}_{\text{VBF}}$ | $1.15^{+0.57}_{-0.53}$ |  ATLAS:2020qdt  | 1
| $1.05^{+0.30}_{-0.29}$ |  CMS:2020gsy
$\mu^{\mu\mu}_{\text{VBF}}$ | $3.18^{+8.22}_{-7.93}$ |  CMS:2020gsy  | 1
$\mu^{bb}_{\text{VBF}}$ | $3.03^{+1.67}_{-1.62}$ |  ATLAS:2020qdt  | 1
| $0.95^{+0.38}_{-0.36}$ |  ATLAS:2020bhl
$\mu^{\gamma\gamma}_{\text{Wh}}$ | $3.76^{+1.48}_{-1.35}$ |  Sirunyan:2018koj  | 1
$\mu^{WW}_{\text{Wh}}$ | $2.85^{+2.11}_{-1.87}$ |  CMS:2020gsy  | 1
$\mu^{\tau\tau}_{\text{Wh}}$ | $3.01^{+1.65}_{-1.51}$ |  CMS:2020gsy  | 1
$\mu^{bb}_{\text{Wh}}$ | $1.27^{+0.42}_{-0.40}$ |  CMS:2020gsy  | 1
| $0.95^{+0.27}_{-0.25}$ |  Aad:2020jym
$\mu^{\gamma\gamma}_{\text{Zh}}$ | $0.00\pm 1.14$ |  Sirunyan:2018koj  | 1
$\mu^{WW}_{\text{Zh}}$ | $0.90^{+1.77}_{-1.43}$ |  CMS:2020gsy  | 1
$\mu^{\tau\tau}_{\text{Zh}}$ | $1.53^{+1.60}_{-1.37}$ |  CMS:2020gsy  | 1
$\mu^{bb}_{\text{Zh}}$ | $0.93^{+0.33}_{-0.31}$ |  CMS:2020gsy  | 1
| $1.08^{+0.25}_{-0.23}$ |  Aad:2020jym
$\mu^{cc}_{\text{Zh}}$ | $37^{+20}_{-19}$ |  Sirunyan:2019qia  | 1
$\mu^{\gamma\gamma}_{\text{Vh}}$ | $1.32^{+0.33}_{-0.30}$ |  ATLAS:2020qdt  | 1
| $1.34^{+0.36}_{-0.33}$ |  Sirunyan:2021ybb
$\mu^{ZZ}_{\text{Vh}}$ | $1.53^{+1.13}_{-0.92}$ |  ATLAS:2020qdt  | 1
| $1.10^{+0.96}_{-0.74}$ |  CMS:2020gsy
$\mu^{bb}_{\text{Vh}}$ | $1.02^{+0.18}_{-0.17}$ |  ATLAS:2020qdt  | 1
| $1.03^{+0.28}_{-0.17}$ |  Aad:2020jym
$\mu^{\gamma\gamma}_{\text{tth}}$ | $0.90^{+0.41}_{-0.35}$ |  ATLAS:2020qdt  | 1
| $1.62^{+0.52}_{-0.43}$ |  CMS:2020gsy
| $1.35^{+0.34}_{-0.28}$ |  Sirunyan:2021ybb
$\mu^{ZZ}_{\text{tth}}$ | $0.25^{+1.03}_{-0.25}$ |  CMS:2020gsy  | 1
$\mu^{WW}_{\text{tth}}$ | $0.93^{+0.48}_{-0.45}$ |  CMS:2020gsy  | 1
$\mu^{VV}_{\text{tth}}$ | $1.72^{+0.56}_{-0.53}$ |  ATLAS:2020qdt  | 1
$\mu^{\tau\tau}_{\text{tth}}$ | $1.20^{+1.07}_{-0.93}$ |  ATLAS:2020qdt  | 1
| $0.81^{+0.74}_{-0.67}$ |  CMS:2020gsy
$\mu^{bb}_{\text{tth}}$ | $0.79^{+0.60}_{-0.59}$ |  ATLAS:2020qdt  | 1
| $1.13^{+0.33}_{-0.30}$ |  CMS:2020gsy
|  |  |

Table 5: List of the Oblique parameters used in the analysis in Section 4. Observable | Experimental value |  Source | SM prediction
---|---|---|---
Oblique parameters
S | $0.02\pm 0.10$ |  Zyla:2020zbs  | 0
T | $0.07\pm 0.12$ |  Zyla:2020zbs  | 0
U | $0.00\pm 0.09$ |  Zyla:2020zbs  | 0
|  |  |
Table 6: Table of flavour observables (except $b\to s\ell^{+}\ell^{-}$) used in the analysis. The stated SM predictions are our calculations through flavio, where we have modified some appropriately in order to closely replicate the most recent averages or predictions, for example in $B$-mixing and $B\to X_{s}\gamma$ (marked by ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}*}$ in the fourth column). Observable | Experimental value |  Source | SM prediction
---|---|---|---
Tree-level leptonic decays
$\Gamma(\pi^{+}\to\mu^{+}\nu_{\mu})$ |  $(2.528\pm 0.005)\times 10^{-17}\,$eV |  Zyla:2020zbs  |  $(2.521\pm 0.003)\times 10^{-17}\,$eV
${\cal B}(K^{+}\to\mu^{+}\nu_{\mu})$ | $(63.56\pm 0.11)\times 10^{-2}$ |  Zyla:2020zbs  | $(63.91\pm 0.93)\times 10^{-2}$
${\cal B}(B^{+}\to\mu^{+}\nu_{\mu})$ | $(5.3\pm 2.2)\times 10^{-7}$ |  Zyla:2020zbs  | $(4.15\pm 0.51)\times 10^{-7}$
${\cal B}(B^{+}\to\tau^{+}\nu_{\tau})$ | $(1.09\pm 0.24)\times 10^{-4}$ |  Zyla:2020zbs  | $(9.24\pm 1.13)\times 10^{-5}$
${\cal B}(D^{+}\to\mu^{+}\nu_{\mu})$ | $(3.74\pm 0.17)\times 10^{-4}$ |  Zyla:2020zbs  | $(3.97\pm 0.04)\times 10^{-4}$
${\cal B}(D^{+}\to\tau^{+}\nu_{\tau})$ | $(1.20\pm 0.27)\times 10^{-3}$ |  Zyla:2020zbs  | $(1.07\pm 0.01)\times 10^{-3}$
${\cal B}(D_{s}^{+}\to\mu^{+}\nu_{\mu})$ | $(5.49\pm 0.16)\times 10^{-3}$ |  Zyla:2020zbs  | $(5.31\pm 0.04)\times 10^{-3}$
${\cal B}(D_{s}^{+}\to\tau^{+}\nu_{\tau})$ | $(5.48\pm 0.23)\times 10^{-2}$ |  Zyla:2020zbs  | $(5.22\pm 0.04)\times 10^{-2}$
${\cal B}(\tau^{+}\to K^{+}\nu_{\tau})$ | $(6.96\pm 0.10)\times 10^{-3}$ |  Zyla:2020zbs  | $(7.07\pm 0.10)\times 10^{-3}$
${\cal B}(\tau^{+}\to\pi^{+}\nu_{\tau})$ | $(10.82\pm 0.05)\times 10^{-2}$ |  Zyla:2020zbs  | $(10.85\pm 0.13)\times 10^{-2}$
Tree-level semi-leptonic decays
${\cal B}(B^{0}\to D^{-}\ell^{+}\nu_{\ell})$ | $(2.31\pm 0.10)\times 10^{-2}$ |  Amhis:2019ckw  | $(2.26\pm 0.18)\times 10^{-2}$
${\cal B}(B^{+}\to D^{0}\ell^{+}\nu_{\ell})$ | $(2.35\pm 0.10)\times 10^{-2}$ |  Amhis:2019ckw  | $(2.45\pm 0.19)\times 10^{-2}$
${\cal B}(B^{0}\to D^{*-}\ell^{+}\nu_{\ell})$ | $(5.06\pm 0.12)\times 10^{-2}$ |  Amhis:2019ckw  | $(4.82\pm 0.44)\times 10^{-2}$
${\cal B}(B^{+}\to D^{*0}\ell^{+}\nu_{\ell})$ | $(5.66\pm 0.22)\times 10^{-2}$ |  Amhis:2019ckw  | $(5.20\pm 0.48)\times 10^{-2}$
${\cal B}(B^{+}\to\pi^{0}\ell^{+}\nu_{\ell})$ | $(7.80\pm 0.27)\times 10^{-5}$ |  Zyla:2020zbs  | $(6.37\pm 1.03)\times 10^{-5}$
${\cal B}(B^{0}\to\pi^{-}\ell^{+}\nu_{\ell})$ | $(1.50\pm 0.06)\times 10^{-4}$ |  Zyla:2020zbs  | $(1.18\pm 0.19)\times 10^{-4}$
${\cal B}(B^{+}\to\rho^{0}\ell^{+}\nu_{\ell})$ | $(1.58\pm 0.11)\times 10^{-4}$ |  Zyla:2020zbs  | $(2.19\pm 0.44)\times 10^{-4}$
${\cal B}(B^{0}\to\rho^{-}\ell^{+}\nu_{\ell})$ | $(2.94\pm 0.21)\times 10^{-4}$ |  Zyla:2020zbs  | $(4.06\pm 0.82)\times 10^{-4}$
${\cal B}(B^{0}_{s}\to D^{-}_{s}\mu^{+}\nu_{\mu})$ | $(2.40\pm 0.12\pm 0.15\pm 0.06\pm 0.10)\times 10^{-2}$ |  Aaij:2020hsi ; Aaij:2021nyr  |  $(2.36\pm 0.18)\times 10^{-2}$ * McLean:2019qcx
${\cal B}(B^{0}_{s}\to D^{*-}_{s}\mu^{+}\nu_{\mu})$ | $(5.19\pm 0.24\pm 0.47\pm 0.13\pm 0.14)\times 10^{-2}$ |  Aaij:2020hsi ; Aaij:2021nyr  |  $(4.99\pm 0.45)\times 10^{-2}$ * McLean:2019qcx
${\cal B}(D^{+}\to\bar{K}^{0}e^{+}\nu_{e})$ | $(8.73\pm 0.10)\times 10^{-2}$ |  Zyla:2020zbs  | $(9.08\pm 0.61)\times 10^{-2}$
${\cal B}(D^{+}\to\bar{K}^{0}\mu^{+}\nu_{\mu})$ | $(8.76\pm 0.19)\times 10^{-2}$ |  Zyla:2020zbs  | $(9.04\pm 0.61)\times 10^{-2}$
${\cal B}(D^{+}\to\pi^{0}e^{+}\nu_{e})$ | $(3.72\pm 0.17)\times 10^{-3}$ |  Zyla:2020zbs  | $(3.47\pm 0.20)\times 10^{-3}$
${\cal B}(D^{+}\to\pi^{0}\mu^{+}\nu_{\mu})$ | $(3.50\pm 0.15)\times 10^{-3}$ |  Zyla:2020zbs  | $(3.46\pm 0.19)\times 10^{-3}$
${\cal B}(D^{0}\to K^{-}e^{+}\nu_{e})$ | $(3.542\pm 0.035)\times 10^{-2}$ |  Zyla:2020zbs  | $(3.55\pm 0.24)\times 10^{-2}$
${\cal B}(D^{0}\to K^{-}\mu^{+}\nu_{\mu})$ | $(3.41\pm 0.04)\times 10^{-2}$ |  Zyla:2020zbs  | $(3.54\pm 0.24)\times 10^{-2}$
${\cal B}(D^{0}\to\pi^{-}e^{+}\nu_{e})$ | $(2.91\pm 0.04)\times 10^{-3}$ |  Zyla:2020zbs  | $(2.68\pm 0.15)\times 10^{-3}$
${\cal B}(D^{0}\to\pi^{-}\mu^{+}\nu_{\mu})$ | $(2.67\pm 0.12)\times 10^{-3}$ |  Zyla:2020zbs  | $(2.67\pm 0.15)\times 10^{-3}$
${\cal B}(K^{+}\to\pi^{0}e^{+}\nu_{e})$ | $(5.07\pm 0.04)\times 10^{-2}$ |  Zyla:2020zbs  | $(5.12\pm 0.06)\times 10^{-2}$
${\cal B}(K^{+}\to\pi^{0}\mu^{+}\nu_{\mu})$ | $(3.352\pm 0.033)\times 10^{-2}$ |  Zyla:2020zbs  | $(3.380\pm 0.043)\times 10^{-2}$
${\cal B}(K^{0}_{L}\to\pi^{+}e^{-}\bar{\nu}_{e})$ | $(40.55\pm 0.11)\times 10^{-2}$ |  Zyla:2020zbs  | $(40.82\pm 0.43)\times 10^{-2}$
${\cal B}(K^{0}_{L}\to\pi^{+}\mu^{-}\bar{\nu}_{\mu})$ | $(27.04\pm 0.07)\times 10^{-2}$ |  Zyla:2020zbs  | $(27.04\pm 0.30)\times 10^{-2}$
${\cal B}(K^{0}_{S}\to\pi^{+}e^{-}\bar{\nu}_{e})$ | $(7.04\pm 0.08)\times 10^{-4}$ |  Zyla:2020zbs  | $(7.15\pm 0.07)\times 10^{-4}$
${\cal B}(K^{0}_{S}\to\pi^{+}\mu^{-}\bar{\nu}_{\mu})$ | $(4.69\pm 0.05)\times 10^{-4}$ |  Zyla:2020zbs  | $(4.73\pm 0.05)\times 10^{-4}$
Tree-level LFU Ratios
$\mathcal{R}(D)$ | $(0.340\pm 0.030)$ |  Amhis:2019ckw  | $(0.303\pm 0.006)$
$\mathcal{R}(D^{*})$ | $(0.295\pm 0.014)$ |  Amhis:2019ckw  | $(0.255\pm 0.003)$
Radiative decays
${\cal B}(\bar{B}\to X_{s}\gamma)$ | $(3.32\pm 0.15)\times 10^{-4}$ |  Amhis:2019ckw  |  $(3.40\pm 0.17)\times 10^{-4}$ ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}*}$ Misiak:2020vlo
FCNC leptonic decays
${\cal B}(B_{d}\to\mu^{+}\mu^{-})$ | $(0.56\pm 0.70)\times 10^{-10}$ |  Altmannshofer:2021qrr  | $(1.08\pm 0.13)\times 10^{-10}$
${\cal B}(B_{s}\to\mu^{+}\mu^{-})$ | $(2.93\pm 0.35)\times 10^{-9}$ |  Altmannshofer:2021qrr  | $(3.48\pm 0.26)\times 10^{-9}$
$B$-Mixing
$\Delta m_{d}$ |  $(0.5065\pm 0.0019)\,$ps-1 |  Amhis:2019ckw  |  $(0.533^{+0.022}_{-0.036})\,$ps-1 * DiLuzio:2019jyq
$\Delta m_{s}$ |  $(17.757\pm 0.021)\,$ps-1 |  Amhis:2019ckw  |  $(18.4^{+0.7}_{-1.2})\,$ps-1 * DiLuzio:2019jyq
Anomalous magnetic moments
$a_{\mu}$ | $116592091(54)(33)\times 10^{-11}$ |  Zyla:2020zbs  |  $116591810(43)\times 10^{-11}$ * Aoyama:2020ynm
|  |  |  $116591954(55)\times 10^{-11}$ * Borsanyi:2020mff

Table 7:  Table of $b\to s\ell^{+}\ell^{-}$ observables (except $R_{K^{(*)}}$) used in the analysis. We include correlations between measurements when provided by the sources. The stated SM predictions are our calculations through flavio. The modes marked by * assume averaging of $B^{+}$ and $B^{0}$ decays. Mode | Observable | Source | Bin | Experimental value | SM prediction
---|---|---|---|---|---
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ | $10^{\,8}\,{\rm GeV}^{2}\times\frac{\Delta\cal B}{\Delta q^{2}}$ | LHCb Aaij:2014pli  | $[0.1,0.98]$ | $3.32\pm 0.18\pm 0.17$ | $3.34\pm 0.63$
|  |  | $[1.1,2]$ | $2.33\pm 0.15\pm 0.12$ | $3.33\pm 0.63$
|  |  | $[2,3]$ | $2.82\pm 0.16\pm 0.14$ | $3.32\pm 0.62$
|  |  | $[3,4]$ | $2.54\pm 0.15\pm 0.13$ | $3.30\pm 0.62$
|  |  | $[4,5]$ | $2.21\pm 0.14\pm 0.11$ | $3.28\pm 0.62$
|  |  | $[5,6]$ | $2.31\pm 0.14\pm 0.12$ | $3.26\pm 0.61$
|  |  | $[15,16]$ | $1.61\pm 0.10\pm 0.08$ | $2.25\pm 0.33$
|  |  | $[16,17]$ | $1.64\pm 0.10\pm 0.08$ | $2.03\pm 0.28$
|  |  | $[17,18]$ | $2.06\pm 0.11\pm 0.10$ | $1.77\pm 0.23$
|  |  | $[18,19]$ | $1.37\pm 0.10\pm 0.07$ | $1.49\pm 0.19$
|  |  | $[19,20]$ | $0.74\pm 0.08\pm 0.04$ | $1.17\pm 0.14$
|  |  | $[20,21]$ | $0.59\pm 0.07\pm 0.03$ | $0.82\pm 0.10$
|  |  | $[21,22]$ | $0.43\pm 0.07\pm 0.02$ | $0.44\pm 0.05$
$B^{0}\to K^{0}\mu^{+}\mu^{-}$ | $10^{\,8}\,{\rm GeV}^{2}\times\frac{\Delta\cal B}{\Delta q^{2}}$ | LHCb Aaij:2014pli  | $[0.1,2]$ | $1.22^{+0.59}_{-0.52}\pm 0.06$ | $3.10\pm 0.58$
|  |  | $[2,4]$ | $1.87^{+0.55}_{-0.49}\pm 0.09$ | $3.07\pm 0.58$
|  |  | $[4,6]$ | $1.73^{+0.53}_{-0.48}\pm 0.09$ | $3.03\pm 0.57$
|  |  | $[15,17]$ | $1.43^{+0.35}_{-0.32}\pm 0.07$ | $1.98\pm 0.28$
|  |  | $[17,19]$ | $0.78^{+0.17}_{-0.15}\pm 0.04$ | $1.50\pm 0.19$
$B^{+}\to K^{*+}\mu^{+}\mu^{-}$ | $10^{\,8}\,{\rm GeV}^{2}\times\frac{\Delta\cal B}{\Delta q^{2}}$ | LHCb Aaij:2014pli  | $[0.1,2]$ | $5.92^{+1.44}_{-1.30}\pm 0.40$ | $7.53\pm 1.11$
|  |  | $[2,4]$ | $5.59^{+1.59}_{-1.44}\pm 0.38$ | $4.60\pm 0.63$
|  |  | $[4,6]$ | $2.49^{+1.10}_{-0.96}\pm 0.17$ | $5.13\pm 0.74$
|  |  | $[15,17]$ | $6.44^{+1.29}_{-1.15}\pm 0.44$ | $7.19\pm 0.89$
|  |  | $[17,19]$ | $1.16^{+0.91}_{-0.76}\pm 0.08$ | $4.96\pm 0.68$
$B^{+}\to K^{*+}\mu^{+}\mu^{-}$ | $F_{L}$ | LHCb Aaij:2020ruw  | $[0.1,0.98]$ | $0.34^{+0.10}_{-0.10}\pm 0.06$ | $0.303\pm 0.052$
|  |  | $[1.1,2.5]$ | $0.54^{+0.18}_{-0.19}\pm 0.03$ | $0.767\pm 0.043$
|  |  | $[2.5,4]$ | $0.17^{+0.24}_{-0.14}\pm 0.04$ | $0.798\pm 0.035$
|  |  | $[4,6]$ | $0.67^{+0.11}_{-0.14}\pm 0.03$ | $0.712\pm 0.048$
|  |  | $[15,17]$ | $0.41^{+0.18}_{-0.14}\pm 0.02$ | $0.346\pm 0.022$
|  |  | $[17,19]$ | $0.34^{+0.11}_{-0.12}\pm 0.04$ | $0.326\pm 0.016$
$B^{+}\to K^{*+}\mu^{+}\mu^{-}$ | $P_{1}$ | LHCb Aaij:2020ruw  | $[0.1,0.98]$ | $\phantom{-}0.44^{+0.38}_{-0.40}\pm 0.11$ | $\phantom{-}0.045\pm 0.027$
|  |  | $[1.1,2.5]$ | $\phantom{-}1.60^{+4.92}_{-1.75}\pm 0.32$ | $\phantom{-}0.022\pm 0.047$
|  |  | $[2.5,4]$ | $-0.29^{+1.43}_{-1.04}\pm 0.22$ | $-0.116\pm 0.036$
|  |  | $[4,6]$ | $-1.24^{+0.99}_{-1.17}\pm 0.29$ | $-0.176\pm 0.046$
|  |  | $[15,17]$ | $-0.88^{+0.41}_{-0.67}\pm 0.07$ | $-0.531\pm 0.050$
|  |  | $[17,19]$ | $-0.40^{+0.58}_{-0.57}\pm 0.09$ | $-0.746\pm 0.036$
$B^{+}\to K^{*+}\mu^{+}\mu^{-}$ | $P_{2}$ | LHCb Aaij:2020ruw  | $[0.1,0.98]$ | $-0.05^{+0.12}_{-0.12}\pm 0.03$ | $-0.136\pm 0.008$
|  |  | $[1.1,2.5]$ | $-0.28^{+0.24}_{-0.42}\pm 0.15$ | $-0.451\pm 0.017$
|  |  | $[2.5,4]$ | $\phantom{-}0.03^{+0.26}_{-0.25}\pm 0.11$ | $-0.054\pm 0.114$
|  |  | $[4,6]$ | $-0.15^{+0.19}_{-0.20}\pm 0.06$ | $\phantom{-}0.294\pm 0.081$
|  |  | $[15,17]$ | $\phantom{-}0.45^{+0.03}_{-0.07}\pm 0.03$ | $\phantom{-}0.414\pm 0.016$
|  |  | $[17,19]$ | $\phantom{-}0.14^{+0.10}_{-0.10}\pm 0.04$ | $\phantom{-}0.320\pm 0.020$
$B^{+}\to K^{*+}\mu^{+}\mu^{-}$ | $P_{3}$ | LHCb Aaij:2020ruw  | $[0.1,0.98]$ | $-0.42^{+0.20}_{-0.21}\pm 0.05$ | $\phantom{-}0.0015\pm 0.0134$
|  |  | $[1.1,2.5]$ | $-0.09^{+0.70}_{-0.99}\pm 0.18$ | $\phantom{-}0.0040\pm 0.0242$
|  |  | $[2.5,4]$ | $-0.45^{+0.50}_{-0.62}\pm 0.20$ | $\phantom{-}0.0039\pm 0.0103$
|  |  | $[4,6]$ | $\phantom{-}0.52^{+0.82}_{-0.62}\pm 0.15$ | $\phantom{-}0.0025\pm 0.0154$
|  |  | $[15,17]$ | $-0.23^{+0.16}_{-0.20}\pm 0.02$ | $-0.0004\pm 0.0003$
|  |  | $[17,19]$ | $\phantom{-}0.12^{+0.21}_{-0.21}\pm 0.02$ | $-0.0003\pm 0.0002$
$B^{+}\to K^{*+}\mu^{+}\mu^{-}$ | $P_{4}^{\prime}$ | LHCb Aaij:2020ruw  | $[0.1,0.98]$ | $-0.09^{+0.36}_{-0.35}\pm 0.12$ | $\phantom{-}0.234\pm 0.012$
|  |  | $[1.1,2.5]$ | $\phantom{-}0.58^{+0.62}_{-0.56}\pm 0.11$ | $-0.066\pm 0.047$
|  |  | $[2.5,4]$ | $-0.81^{+1.09}_{-0.84}\pm 0.14$ | $-0.392\pm 0.048$
|  |  | $[4,6]$ | $-0.79^{+0.47}_{-0.28}\pm 0.09$ | $-0.502\pm 0.028$
|  |  | $[15,17]$ | $-0.32^{+0.23}_{-0.22}\pm 0.08$ | $-0.618\pm 0.010$
|  |  | $[17,19]$ | $-0.57^{+0.29}_{-0.36}\pm 0.13$ | $-0.660\pm 0.007$
$B^{+}\to K^{*+}\mu^{+}\mu^{-}$ | $P_{5}^{\prime}$ | LHCb Aaij:2020ruw  | $[0.1,0.98]$ | $\phantom{-}0.51^{+0.30}_{-0.28}\pm 0.12$ | $\phantom{-}0.684\pm 0.024$
|  |  | $[1.1,2.5]$ | $\phantom{-}0.88^{+0.70}_{-0.71}\pm 0.10$ | $\phantom{-}0.112\pm 0.083$
|  |  | $[2.5,4]$ | $-0.87^{+1.00}_{-1.68}\pm 0.09$ | $-0.517\pm 0.107$
|  |  | $[4,6]$ | $-0.25^{+0.32}_{-0.40}\pm 0.09$ | $-0.763\pm 0.077$
|  |  | $[15,17]$ | $-0.14^{+0.21}_{-0.20}\pm 0.06$ | $-0.673\pm 0.036$
|  |  | $[17,19]$ | $-0.66^{+0.36}_{-0.80}\pm 0.13$ | $-0.487\pm 0.035$
$B^{+}\to K^{*+}\mu^{+}\mu^{-}$ | $P_{6}^{\prime}$ | LHCb Aaij:2020ruw  | $[0.1,0.98]$ | $-0.02^{+0.40}_{-0.34}\pm 0.06$ | $-0.0476\pm 0.0376$
|  |  | $[1.1,2.5]$ | $\phantom{-}0.25^{+1.22}_{-1.32}\pm 0.08$ | $-0.0539\pm 0.0784$
|  |  | $[2.5,4]$ | $-0.37^{+1.59}_{-3.91}\pm 0.05$ | $-0.0439\pm 0.1071$
|  |  | $[4,6]$ | $-0.09^{+0.40}_{-0.41}\pm 0.05$ | $-0.0285\pm 0.1159$
|  |  | $[15,17]$ | $-0.48^{+0.21}_{-0.21}\pm 0.02$ | $-0.0029\pm 0.0015$
|  |  | $[17,19]$ | $\phantom{-}0.12^{+0.33}_{-0.33}\pm 0.04$ | $-0.0012\pm 0.0007$
$B^{+}\to K^{*+}\mu^{+}\mu^{-}$ | $P_{8}^{\prime}$ | LHCb Aaij:2020ruw  | $[0.1,0.98]$ | $\phantom{-}0.45^{+0.50}_{-0.39}\pm 0.09$ | $-0.0326\pm 0.0225$
|  |  | $[1.1,2.5]$ | $\phantom{-}0.12^{+0.75}_{-0.76}\pm 0.05$ | $-0.0271\pm 0.0354$
|  |  | $[2.5,4]$ | $\phantom{-}0.12^{+7.89}_{-4.95}\pm 0.07$ | $-0.0182\pm 0.0394$
|  |  | $[4,6]$ | $-0.15^{+0.44}_{-0.48}\pm 0.05$ | $-0.0114\pm 0.0377$
|  |  | $[15,17]$ | $-0.34^{+0.23}_{-0.22}\pm 0.04$ | $\phantom{-}0.0007\pm 0.0006$
|  |  | $[17,19]$ | $\phantom{-}0.36^{+0.37}_{-0.33}\pm 0.07$ | $\phantom{-}0.0003\pm 0.0002$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $F_{L}$ | LHCb Aaij:2020nrf  | $[0.1,0.98]$ | $0.255\pm 0.032\pm 0.007$ | $0.295\pm 0.050$
|  |  | $[1.1,2.5]$ | $0.655\pm 0.046\pm 0.017$ | $0.760\pm 0.044$
|  |  | $[2.5,4]$ | $0.756\pm 0.047\pm 0.023$ | $0.796\pm 0.036$
|  |  | $[4,6]$ | $0.684\pm 0.035\pm 0.015$ | $0.711\pm 0.048$
|  |  | $[15,17]$ | $0.352\pm 0.026\pm 0.009$ | $0.348\pm 0.022$
|  |  | $[17,19]$ | $0.344\pm 0.032\pm 0.025$ | $0.328\pm 0.016$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{1}$ | LHCb Aaij:2020nrf  | $[0.1,0.98]$ | $\phantom{-}0.090\pm 0.119\pm 0.009$ | $\phantom{-}0.0441\pm 0.0465$
|  |  | $[1.1,2.5]$ | $-0.617\pm 0.296\pm 0.023$ | $\phantom{-}0.0234\pm 0.0463$
|  |  | $[2.5,4]$ | $\phantom{-}0.168\pm 0.371\pm 0.043$ | $-0.116\pm 0.037$
|  |  | $[4,6]$ | $\phantom{-}0.088\pm 0.235\pm 0.029$ | $-0.178\pm 0.047$
|  |  | $[15,17]$ | $-0.511\pm 0.096\pm 0.020$ | $-0.534\pm 0.050$
|  |  | $[17,19]$ | $-0.763\pm 0.152\pm 0.094$ | $-0.751\pm 0.035$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{2}$ | LHCb Aaij:2020nrf  | $[0.1,0.98]$ | $-0.003\pm 0.038\pm 0.003$ | $-0.132\pm 0.008$
|  |  | $[1.1,2.5]$ | $-0.443\pm 0.100\pm 0.027$ | $-0.451\pm 0.014$
|  |  | $[2.5,4]$ | $-0.191\pm 0.116\pm 0.043$ | $-0.063\pm 0.113$
|  |  | $[4,6]$ | $\phantom{-}0.105\pm 0.068\pm 0.009$ | $\phantom{-}0.292\pm 0.081$
|  |  | $[15,17]$ | $\phantom{-}0.396\pm 0.022\pm 0.004$ | $\phantom{-}0.413\pm 0.016$
|  |  | $[17,19]$ | $\phantom{-}0.328\pm 0.032\pm 0.017$ | $\phantom{-}0.317\pm 0.020$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{3}$ | LHCb Aaij:2020nrf  | $[0.1,0.98]$ | $\phantom{-}0.073\pm 0.057\pm 0.003$ | $\phantom{-}0.0014\pm 0.0135$
|  |  | $[1.1,2.5]$ | $\phantom{-}0.324\pm 0.147\pm 0.014$ | $\phantom{-}0.0039\pm 0.0245$
|  |  | $[2.5,4]$ | $\phantom{-}0.049\pm 0.195\pm 0.014$ | $\phantom{-}0.0040\pm 0.0104$
|  |  | $[4,6]$ | $-0.090\pm 0.139\pm 0.006$ | $\phantom{-}0.0026\pm 0.0155$
|  |  | $[15,17]$ | $-0.000\pm 0.056\pm 0.003$ | $-0.0004\pm 0.0003$
|  |  | $[17,19]$ | $\phantom{-}0.085\pm 0.068\pm 0.004$ | $-0.0003\pm 0.0002$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{4}^{\prime}$ | LHCb Aaij:2020nrf  | $[0.1,0.98]$ | $\phantom{-}0.135\pm 0.118\pm 0.010$ | $\phantom{-}0.251\pm 0.017$
|  |  | $[1.1,2.5]$ | $-0.080\pm 0.142\pm 0.019$ | $-0.064\pm 0.045$
|  |  | $[2.5,4]$ | $-0.435\pm 0.169\pm 0.035$ | $-0.393\pm 0.046$
|  |  | $[4,6]$ | $-0.312\pm 0.115\pm 0.013$ | $-0.504\pm 0.028$
|  |  | $[15,17]$ | $-0.626\pm 0.069\pm 0.018$ | $-0.619\pm 0.010$
|  |  | $[17,19]$ | $-0.647\pm 0.086\pm 0.057$ | $-0.661\pm 0.007$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{5}^{\prime}$ | LHCb Aaij:2020nrf  | $[0.1,0.98]$ | $\phantom{-}0.521\pm 0.095\pm 0.024$ | $\phantom{-}0.688\pm 0.024$
|  |  | $[1.1,2.5]$ | $\phantom{-}0.365\pm 0.122\pm 0.013$ | $\phantom{-}0.140\pm 0.081$
|  |  | $[2.5,4]$ | $-0.150\pm 0.144\pm 0.032$ | $-0.500\pm 0.102$
|  |  | $[4,6]$ | $-0.439\pm 0.111\pm 0.036$ | $-0.757\pm 0.079$
|  |  | $[15,17]$ | $-0.714\pm 0.074\pm 0.021$ | $-0.670\pm 0.036$
|  |  | $[17,19]$ | $-0.590\pm 0.084\pm 0.059$ | $-0.483\pm 0.035$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{6}^{\prime}$ | LHCb Aaij:2020nrf  | $[0.1,0.98]$ | $\phantom{-}0.015\pm 0.094\pm 0.007$ | $-0.055\pm 0.036$
|  |  | $[1.1,2.5]$ | $-0.226\pm 0.128\pm 0.005$ | $-0.070\pm 0.077$
|  |  | $[2.5,4]$ | $-0.155\pm 0.148\pm 0.024$ | $-0.052\pm 0.105$
|  |  | $[4,6]$ | $-0.293\pm 0.117\pm 0.004$ | $-0.030\pm 0.117$
|  |  | $[15,17]$ | $\phantom{-}0.061\pm 0.085\pm 0.003$ | $-0.003\pm 0.002$
|  |  | $[17,19]$ | $\phantom{-}0.103\pm 0.105\pm 0.016$ | $-0.001\pm 0.001$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{8}^{\prime}$ | LHCb Aaij:2020nrf  | $[0.1,0.98]$ | $-0.007\pm 0.122\pm 0.002$ | $-0.0056\pm 0.0222$
|  |  | $[1.1,2.5]$ | $-0.366\pm 0.158\pm 0.005$ | $-0.0177\pm 0.0350$
|  |  | $[2.5,4]$ | $\phantom{-}0.037\pm 0.169\pm 0.007$ | $-0.0173\pm 0.0370$
|  |  | $[4,6]$ | $\phantom{-}0.166\pm 0.127\pm 0.004$ | $-0.0115\pm 0.0376$
|  |  | $[15,17]$ | $\phantom{-}0.007\pm 0.086\pm 0.002$ | $\phantom{-}0.0007\pm 0.0006$
|  |  | $[17,19]$ | $-0.055\pm 0.099\pm 0.006$ | $\phantom{-}0.0003\pm 0.0002$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $F_{L}$ | ATLAS Aaboud:2018krd  | $[0.04,2.0]$ | $0.44\pm 0.08\pm 0.07$ | $0.388\pm 0.057$
|  |  | $[2,4]$ | $0.64\pm 0.11\pm 0.05$ | $0.799\pm 0.035$
|  |  | $[4,6]$ | $0.42\pm 0.13\pm 0.12$ | $0.711\pm 0.048$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{1}$ | ATLAS Aaboud:2018krd  | $[0.04,2.0]$ | $-0.05\pm 0.30\pm 0.08$ | $\phantom{-}0.0435\pm 0.0280$
|  |  | $[2,4]$ | $-0.78\pm 0.51\pm 0.34$ | $-0.095\pm 0.038$
|  |  | $[4,6]$ | $\phantom{-}0.14\pm 0.43\pm 0.26$ | $-0.178\pm 0.047$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{4}^{\prime}$ | ATLAS Aaboud:2018krd  | $[0.04,2.0]$ | $\phantom{-}0.31\pm 0.40\pm 0.20$ | $\phantom{-}0.150\pm 0.018$
|  |  | $[2,4]$ | $-0.76\pm 0.31\pm 0.21$ | $-0.349\pm 0.050$
|  |  | $[4,6]$ | $\phantom{-}0.64\pm 0.33\pm 0.18$ | $-0.504\pm 0.028$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{5}^{\prime}$ | ATLAS Aaboud:2018krd  | $[0.04,2.0]$ | $\phantom{-}0.67\pm 0.26\pm 0.16$ | $\phantom{-}0.513\pm 0.033$
|  |  | $[2,4]$ | $-0.33\pm 0.31\pm 0.13$ | $-0.409\pm 0.110$
|  |  | $[4,6]$ | $\phantom{-}0.26\pm 0.35\pm 0.18$ | $-0.757\pm 0.079$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{6}^{\prime}$ | ATLAS Aaboud:2018krd  | $[0.04,2.0]$ | $-0.18\pm 0.21\pm 0.04$ | $-0.056\pm 0.042$
|  |  | $[2,4]$ | $\phantom{-}0.31\pm 0.28\pm 0.19$ | $-0.056\pm 0.042$
|  |  | $[4,6]$ | $\phantom{-}0.06\pm 0.27\pm 0.13$ | $-0.030\pm 0.117$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{8}^{\prime}$ | ATLAS Aaboud:2018krd  | $[0.04,2.0]$ | $-0.29\pm 0.48\pm 0.18$ | $-0.0084\pm 0.0233$
|  |  | $[2,4]$ | $\phantom{-}1.07\pm 0.41\pm 0.39$ | $-0.0179\pm 0.0388$
|  |  | $[4,6]$ | $-0.24\pm 0.42\pm 0.09$ | $-0.0115\pm 0.0376$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $10^{\,8}\,{\rm GeV}^{2}\times\frac{\Delta\cal B}{\Delta q^{2}}$ | CMS Khachatryan:2015isa  | $[1,2]$ | $4.6\pm 0.7\pm 0.3$ | $4.59\pm 0.61$
|  |  | $[2,4.3]$ | $3.3\pm 0.5\pm 0.2$ | $4.25\pm 0.59$
|  |  | $[4.3,6]$ | $3.4\pm 0.5\pm 0.3$ | $4.79\pm 0.71$
|  |  | $[14.18,16]$ | $6.7\pm 0.6\pm 0.5$ | $7.06\pm 0.85$
|  |  | $[16,19]$ | $4.2\pm 0.3\pm 0.3$ | $5.16\pm 0.68$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $F_{L}$ | CMS Khachatryan:2015isa  | $[1,2]$ | $0.64^{+0.10}_{-0.09}\pm 0.07$ | $0.724\pm 0.050$
|  |  | $[2,4.3]$ | $0.80^{+0.08}_{-0.08}\pm 0.06$ | $0.793\pm 0.036$
|  |  | $[4.3,6]$ | $0.62^{+0.10}_{-0.09}\pm 0.07$ | $0.703\pm 0.049$
|  |  | $[14.18,16]$ | $0.48^{+0.05}_{-0.06}\pm 0.04$ | $0.362\pm 0.027$
|  |  | $[16,19]$ | $0.38^{+0.05}_{-0.06}\pm 0.04$ | $0.333\pm 0.020$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $A_{\rm FB}$ | CMS Khachatryan:2015isa  | $[1,2]$ | $-0.27^{+0.17}_{-0.40}\pm 0.07$ | $-0.156\pm 0.031$
|  |  | $[2,4.3]$ | $-0.12^{+0.15}_{-0.17}\pm 0.05$ | $-0.026\pm 0.033$
|  |  | $[4.3,6]$ | $\phantom{-}0.01^{+0.15}_{-0.15}\pm 0.03$ | $\phantom{-}0.132\pm 0.045$
|  |  | $[14.18,16]$ | $\phantom{-}0.39^{+0.04}_{-0.06}\pm 0.01$ | $\phantom{-}0.412\pm 0.018$
|  |  | $[16,19]$ | $\phantom{-}0.35^{+0.07}_{-0.07}\pm 0.01$ | $\phantom{-}0.349\pm 0.021$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{1}$ | CMS Sirunyan:2017dhj  | $[1,2]$ | $-0.12^{+0.46}_{-0.47}\pm 0.10$ | $\phantom{-}0.044\pm 0.047$
|  |  | $[2,4.3]$ | $-0.69^{+0.58}_{-0.27}\pm 0.23$ | $-0.106\pm 0.038$
|  |  | $[4.3,6]$ | $\phantom{-}0.53^{+0.24}_{-0.33}\pm 0.19$ | $-0.180\pm 0.049$
|  |  | $[14.18,16]$ | $-0.33^{+0.24}_{-0.23}\pm 0.20$ | $-0.465\pm 0.051$
|  |  | $[16,19]$ | $-0.53^{+0.19}_{-0.19}\pm 0.16$ | $-0.680\pm 0.041$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $P_{5}^{\prime}$ | CMS Sirunyan:2017dhj  | $[1,2]$ | $\phantom{-}0.10^{+0.32}_{-0.31}\pm 0.07$ | $\phantom{-}0.289\pm 0.063$
|  |  | $[2,4.3]$ | $-0.57^{+0.34}_{-0.31}\pm 0.18$ | $-0.449\pm 0.104$
|  |  | $[4.3,6]$ | $-0.96^{+0.22}_{-0.21}\pm 0.25$ | $-0.769\pm 0.073$
|  |  | $[14.18,16]$ | $-0.66^{+0.13}_{-0.20}\pm 0.18$ | $-0.719\pm 0.035$
|  |  | $[16,19]$ | $-0.56^{+0.12}_{-0.12}\pm 0.07$ | $-0.547\pm 0.036$
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ | $10^{\,8}\,{\rm GeV}^{2}\times\frac{\Delta\cal B}{\Delta q^{2}}$ | Belle Abdesselam:2019lab  | $[0.1,4]$ | $4.51^{+1.05}_{-0.94}\pm 0.20$ | $3.33\pm 0.60$
|  |  | $[4,8.12]$ | $3.00^{+0.68}_{-0.61}\pm 0.07$ | $3.24\pm 0.61$
|  |  | $[1,6]$ | $4.60^{+0.82}_{-0.76}\pm 0.10$ | $3.30\pm 0.60$
$B^{+}\to K^{*+}\mu^{+}\mu^{-}$ | $10^{\,8}\,{\rm GeV}^{2}\times\frac{\Delta\cal B}{\Delta q^{2}}$ | Belle Abdesselam:2019wac  | $[1.1,6]$ | $2.45^{+1.84}_{-1.43}\pm 0.41$ | $4.87\pm 0.77$
|  |  | $[15,19]$ | $7.25^{+2.50}_{-2.00}\pm 0.75$ | $6.08\pm 0.73$
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ | $10^{\,8}\,{\rm GeV}^{2}\times\frac{\Delta\cal B}{\Delta q^{2}}$ | Belle Abdesselam:2019wac  | $[1.1,6]$ | $3.88^{+1.22}_{-1.02}\pm 0.61$ | $4.48\pm 0.71$
|  |  | $[15,19]$ | $5.50^{+1.25}_{-1.00}\pm 0.50$ | $5.60\pm 0.67$
$B_{s}^{0}\to\phi^{0}\mu^{+}\mu^{-}$ | $10^{\,8}\,{\rm GeV}^{2}\times\frac{\Delta\cal B}{\Delta q^{2}}$ | LHCb Aaij:2015esa ; Aaij:2021nyr  | $[0.1,2]$ | $5.54^{+0.69}_{-0.65}\pm 0.13\pm 0.27$ | $7.92\pm 0.98$
|  |  | $[2,5]$ | $2.42^{+0.40}_{-0.38}\pm 0.06\pm 0.12$ | $4.94\pm 0.69$
|  |  | $[15,17]$ | $4.28^{+0.54}_{-0.51}\pm 0.11\pm 0.21$ | $6.60\pm 0.65$
|  |  | $[17,19]$ | $3.75^{+0.54}_{-0.51}\pm 0.13\pm 0.18$ | $3.93\pm 0.41$
$B_{s}^{0}\to\phi^{0}\mu^{+}\mu^{-}$ | $F_{L}$ | LHCb Aaij:2015esa  | $[0.1,2]$ | $0.20^{+0.08}_{-0.09}\pm 0.02$ | $0.496\pm 0.037$
|  |  | $[2,5]$ | $0.68^{+0.16}_{-0.13}\pm 0.03$ | $0.811\pm 0.017$
|  |  | $[15,17]$ | $0.23^{+0.09}_{-0.08}\pm 0.02$ | $0.349\pm 0.015$
|  |  | $[17,19]$ | $0.40^{+0.13}_{-0.15}\pm 0.02$ | $0.328\pm 0.014$
$B_{s}^{0}\to\phi^{0}\mu^{+}\mu^{-}$ | $S_{3}$ | LHCb Aaij:2015esa  | $[0.1,2]$ | $-0.05^{+0.13}_{-0.13}\pm 0.01$ | $\phantom{-}0.020\pm 0.006$
|  |  | $[2,5]$ | $-0.06^{+0.19}_{-0.23}\pm 0.01$ | $-0.009\pm 0.004$
|  |  | $[15,17]$ | $-0.06^{+0.16}_{-0.19}\pm 0.01$ | $-0.180\pm 0.007$
|  |  | $[17,19]$ | $-0.07^{+0.23}_{-0.27}\pm 0.02$ | $-0.262\pm 0.006$
$B_{s}^{0}\to\phi^{0}\mu^{+}\mu^{-}$ | $S_{4}$ | LHCb Aaij:2015esa  | $[0.1,2]$ | $\phantom{-}0.27^{+0.28}_{-0.18}\pm 0.01$ | $\phantom{-}0.060\pm 0.007$
|  |  | $[2,5]$ | $-0.47^{+0.30}_{-0.44}\pm 0.01$ | $-0.148\pm 0.017$
|  |  | $[15,17]$ | $-0.03^{+0.15}_{-0.15}\pm 0.01$ | $-0.296\pm 0.004$
|  |  | $[17,19]$ | $-0.39^{+0.25}_{-0.34}\pm 0.02$ | $-0.313\pm 0.004$
$B_{s}^{0}\to\phi^{0}\mu^{+}\mu^{-}$ | $S_{7}$ | LHCb Aaij:2015esa  | $[0.1,2]$ | $\phantom{-}0.04^{+0.12}_{-0.12}\pm 0.00$ | $-0.025\pm 0.018$
|  |  | $[2,5]$ | $-0.03^{+0.18}_{-0.23}\pm 0.01$ | $-0.020\pm 0.038$
|  |  | $[15,17]$ | $\phantom{-}0.12^{+0.16}_{-0.15}\pm 0.01$ | $-0.0013\pm 0.0006$
|  |  | $[17,19]$ | $\phantom{-}0.20^{+0.29}_{-0.22}\pm 0.01$ | $-0.0005\pm 0.0003$
$B\to X_{s}\,\mu^{+}\mu^{-}$ $\\!\\!\phantom{B}{}^{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}*}$ | $10^{6}\times\Delta{\cal B}$ | BaBar Lees:2013nxa  | $[1,6]$ | $0.66^{+0.87}_{-0.80}\pm 0.07$ | $1.68\pm 0.16$
|  |  | $[14.2,25]$ | $0.60^{+0.31}_{-0.29}\pm 0.00$ | $0.35\pm 0.04$
$B\to X_{s}\,e^{+}e^{-}$ $\\!\\!\phantom{B}{}^{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}*}$ | $10^{6}\times\Delta{\cal B}$ | BaBar Lees:2013nxa  | $[0.1,2]$ | $3.05^{+0.60}_{-0.53}\pm 0.35$ | $1.24\pm 0.12$
|  |  | $[2.0,4.3]$ | $0.69^{+0.33}_{-0.29}\pm 0.07$ | $0.79\pm 0.08$
|  |  | $[4.3,6.8]$ | $0.69^{+0.34}_{-0.31}\pm 0.05$ | $0.74\pm 0.07$
|  |  | $[14.2,25]$ | $0.56^{+0.19}_{-0.18}\pm 0.00$ | $0.30\pm 0.03$
$B\to K^{*}\mu^{+}\mu^{-}$ $\\!\\!\phantom{B}{}^{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}*}$ | $P_{4}^{\prime}$ | Belle Wehle:2016yoi  | $[1,6]$ | $-0.22^{+0.35}_{-0.34}\pm 0.15$ | $-0.342\pm 0.041$
|  |  | $[14.18,19]$ | $-0.10^{+0.39}_{-0.39}\pm 0.07$ | $-0.627\pm 0.010$
$B\to K^{*}\mu^{+}\mu^{-}$ $\\!\\!\phantom{B}{}^{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}*}$ | $P_{5}^{\prime}$ | Belle Wehle:2016yoi  | $[1,6]$ | $\phantom{-}0.43^{+0.26}_{-0.28}\pm 0.10$ | $-0.431\pm 0.096$
|  |  | $[14.18,19]$ | $-0.13^{+0.39}_{-0.35}\pm 0.06$ | $-0.626\pm 0.036$
$B\to K^{*}e^{+}e^{-}$ $\\!\\!\phantom{B}{}^{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}*}$ | $P_{4}^{\prime}$ | Belle Wehle:2016yoi  | $[1,6]$ | $-0.72^{+0.40}_{-0.39}\pm 0.06$ | $-0.339\pm 0.041$
|  |  | $[14.18,19]$ | $-0.15^{+0.41}_{-0.40}\pm 0.04$ | $-0.627\pm 0.010$
$B\to K^{*}e^{+}e^{-}$ $\\!\\!\phantom{B}{}^{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}*}$ | $P_{5}^{\prime}$ | Belle Wehle:2016yoi  | $[1,6]$ | $-0.22^{+0.39}_{-0.41}\pm 0.03$ | $-0.424\pm 0.095$
|  |  | $[14.18,19]$ | $-0.91^{+0.36}_{-0.30}\pm 0.03$ | $-0.626\pm 0.036$
$B^{0}\to K^{*0}e^{+}e^{-}$ | $F_{L}$ | LHCb Aaij:2015dea  | $[0.002,1.12]$ | $0.16\pm 0.06\pm 0.03$ | $0.18\pm 0.04$
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ | $A_{\rm FB}$ | CMS Sirunyan:2018jll  | $[1,6]$ | $-0.14^{+0.07}_{-0.06}\pm 0.03$ | $0.0\pm 0.0$
|  |  | $[16,18]$ | $\phantom{-}0.04^{+0.05}_{-0.04}\pm 0.03$ | $0.0\pm 0.0$
|  |  | $[18,22]$ | $\phantom{-}0.05^{+0.05}_{-0.04}\pm 0.02$ | $0.0\pm 0.0$
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ | $F_{H}$ | CMS Sirunyan:2018jll  | $[1,6]$ | $0.38^{+0.17}_{-0.21}\pm 0.09$ | $0.0247\pm 0.0003$
|  |  | $[16,18]$ | $0.07^{+0.06}_{-0.07}\pm 0.07$ | $0.0064\pm 0.0004$
|  |  | $[18,22]$ | $0.10^{+0.06}_{-0.10}\pm 0.09$ | $0.0079\pm 0.0007$
$\Lambda_{b}^{0}\to\Lambda\mu^{+}\mu^{-}$ | $10^{\,8}\,{\rm GeV}^{2}\times\frac{\Delta\cal B}{\Delta q^{2}}$ | LHCb Aaij:2015xza  | $[1.1,6]$ | $\phantom{1}0.9^{+0.6\,+0.1}_{-0.5\,-0.1}\pm 0.2$ | $0.98\pm 0.57$
|  |  | $[15,16]$ | $11.2^{+1.9\,+0.5}_{-1.8\,-0.5}\pm 2.3$ | $6.75\pm 0.90$
|  |  | $[16,18]$ | $12.2^{+1.4\,+0.3}_{-1.4\,-0.6}\pm 2.5$ | $7.28\pm 0.88$
|  |  | $[18,20]$ | $12.4^{+1.4\,+0.6}_{-1.4\,-0.5}\pm 2.6$ | $6.12\pm 0.76$
$\Lambda_{b}^{0}\to\Lambda\mu^{+}\mu^{-}$ | $A_{\rm FB}^{\ell}$ | LHCb Aaij:2018gwm  | $[15,20]$ | $-0.39\pm 0.04\pm 0.01$ | $-0.353\pm 0.018$
| $A_{\rm FB}^{h}$ |  | $[15,20]$ | $-0.30\pm 0.05\pm 0.02$ | $-0.318\pm 0.011$
| $A_{\rm FB}^{\ell h}$ |  | $[15,20]$ | $\phantom{-}0.25\pm 0.04\pm 0.01$ | $\phantom{-}0.164\pm 0.009$
$B_{s}\to\mu^{+}\mu^{-}$ | $10^{9}\times{\cal B}$ | Comb. Altmannshofer:2021qrr  |  | $2.93\pm 0.35$ | $3.48\pm 0.26$

Table 8: Table of $R_{K^{(*)}}$ observables used in the analysis. The stated SM predictions are our calculations through flavio. Observable | Source | Bin | Experimental value | SM prediction
---|---|---|---|---
$R_{K^{+}}$ | LHCb Aaij:2021vac  | $[1.1,6]$ | $0.846^{+0.042\,+0.013}_{-0.039\,-0.012}$ | $1.0008\pm 0.0003$
$R_{K^{*0}}$ | LHCb Aaij:2017vbb  | $[0.045,1.1]$ | $0.66^{+0.11}_{-0.07}\pm 0.03$ | $0.926\pm 0.004$
|  | $[1.1,6.0]$ | $0.69^{+0.11}_{-0.07}\pm 0.05$ | $0.9964\pm 0.0006$
$R_{K}$ | BaBar Lees:2012tva  | $[0.10,8.12]$ | $0.74^{+0.40}_{-0.31}\pm 0.06$ | $1.0007\pm 0.0003$
$R_{K^{*}}$ | BaBar Lees:2012tva  | $[0.10,8.12]$ | $1.06^{+0.48}_{-0.33}\pm 0.08$ | $0.9949\pm 0.0015$
$R_{K^{+}}$ | Belle Abdesselam:2019lab  | $[1,6]$ | $1.39^{+0.36}_{-0.33}\pm 0.02$ | $1.0008\pm 0.0003$
$R_{K^{*+}}$ | Belle Abdesselam:2019wac  | $[0.045,1.1]$ | $0.62^{+0.60}_{-0.36}\pm 0.09$ | $0.928\pm 0.005$
|  | $[1.1,6]$ | $0.72^{+0.99}_{-0.44}\pm 0.15$ | $0.997\pm 0.001$
|  | $[15,19]$ | $1.40^{+1.99}_{-0.68}\pm 0.12$ | $0.99807\pm 0.00004$
$R_{K^{*0}}$ | Belle Abdesselam:2019wac  | $[0.045,1.1]$ | $0.46^{+0.55}_{-0.27}\pm 0.13$ | $0.926\pm 0.004$
|  | $[1.1,6]$ | $1.06^{+0.63}_{-0.38}\pm 0.14$ | $0.996\pm 0.001$
|  | $[15,19]$ | $1.12^{+0.61}_{-0.36}\pm 0.10$ | $0.99807\pm 0.0004$

##  Appendix B Calculation of $\delta$ from $\lambda_{i}$ and $\tan\beta$

We start from Eq. (10) and rearrange to give an expression for $m_{12}^{2}$.
Using small angle approximations on $\delta=\beta-\alpha-\pi/2$, we expand to
second order, and find that

$m_{12}^{2}\,\delta^{2}=\Gamma^{-2}_{\beta}+\Gamma^{-1}_{\beta}\delta+\Gamma^{0}_{\beta}\,\delta^{2}+\mathcal{O}(\delta^{3})$
(54)

with

$\displaystyle\Gamma^{-2}_{\beta}$
$\displaystyle=s_{\beta}\,c_{\beta}\left(m_{h^{0}}^{2}-\tilde{\lambda}_{1}c_{\beta}^{4}-\tilde{\lambda}_{2}s_{\beta}^{4}-\frac{\tilde{\lambda}_{345}\,s_{2\beta}^{2}}{2}\right),$
(55) $\displaystyle\Gamma^{-1}_{\beta}$
$\displaystyle=s_{\beta}\,c_{\beta}\left(-2\tilde{\lambda}_{1}c_{\beta}^{3}s_{\beta}+2\tilde{\lambda}_{2}s_{\beta}^{3}c_{\beta}+\tilde{\lambda}_{345}\,s_{2\beta}c_{2\beta}\right),$
(56) $\displaystyle\Gamma^{0}_{\beta}$
$\displaystyle=s_{\beta}\,c_{\beta}\left(\tilde{\lambda}_{1}(c_{\beta}^{4}-s_{\beta}^{2}c_{\beta}^{2})+\tilde{\lambda}_{2}(s_{\beta}^{4}-c_{\beta}^{2}s_{\beta}^{2})+\tilde{\lambda}_{345}\,s_{2\beta}^{2}\right),$
(57)

and where we have used as shorthand:

$s_{\theta}=\sin(\theta),\,\,c_{\theta}=\cos(\theta),\,\,\lambda_{345}=\lambda_{3}+\lambda_{4}+\lambda_{5},\,\,\text{and}\,\,\tilde{\lambda}_{i}=v^{2}\lambda_{i}.$
(58)

Then inserting Eq. (54) into Eq. (10), we find an equation for $\delta$ of the
form

$c+b\,\delta+a\,\delta^{2}+\mathcal{O}(\delta^{3})=0,$ (59)

where

$\displaystyle a$ $\displaystyle=$ $\displaystyle 2s_{2\beta}\Delta
t_{\beta}\,\Gamma^{-2}_{\beta}+2c_{2\beta}\Delta
t_{\beta}\,\Gamma^{-1}_{\beta}-2s_{2\beta}\Delta
t_{\beta}\,\Gamma^{0}_{\beta}-s_{2\beta}\tilde{\lambda}_{1}c_{\beta}^{2}+s_{2\beta}\tilde{\lambda}_{2}s_{\beta}^{2}$
(60)
$\displaystyle-2c_{2\beta}\,\Gamma^{0}_{\beta}+\tilde{\lambda}_{345}\,c_{2\beta}s_{2\beta}-4s_{2\beta}\,\Gamma^{-1}_{\beta}+4c_{2\beta}\,\Gamma^{-2}_{\beta},$
$\displaystyle b$ $\displaystyle=$ $\displaystyle 2c_{2\beta}\Delta
t_{\beta}\,\Gamma^{-2}_{\beta}-s_{2\beta}\Delta
t_{\beta}\,\Gamma^{-1}_{\beta}-2c_{2\beta}\,\Gamma^{-1}_{\beta}-4s_{2\beta}\,\Gamma^{-2}_{\beta},$
(61) $\displaystyle c$ $\displaystyle=$ $\displaystyle-s_{2\beta}\Delta
t_{\beta}\,\Gamma^{-2}_{\beta}-2c_{2\beta}\,\Gamma^{-2}_{\beta},$ (62)

having defined

$\Delta t_{\beta}=\tan(\beta)-\frac{1}{\tan(\beta)}.$ (63)

By trigonometric manipulation we find that $c=0$. We are therefore left with
the two solutions

$\delta\rightarrow 0\quad\text{and}\quad\delta=-\frac{b}{a},$ (64)

where the first solution corresponds to the alignment limit. Note that the
second solution is only valid for small $\delta$ as a second-order Taylor
expansion was used in its derivation.

##  Appendix C Formulas for the Oblique Parameters in the 2HDM

The formulae for the oblique parameters in a general 2HDM as derived from
Grimus:2008nb  are

$\displaystyle\begin{split}T^{\text{2HDM}}&=\frac{G_{F}}{8\sqrt{2}\,\alpha\pi^{2}}\Big{\\{}F(m_{H^{\pm}}^{2},m_{A^{0}}^{2})+\sin^{2}(\theta)F(m_{H^{\pm}}^{2},m_{H^{0}}^{2})+\cos^{2}(\theta)F(m_{H^{\pm}}^{2},m_{h^{0}}^{2})\\\
&\qquad\qquad-\sin^{2}(\theta)F(m_{A^{0}}^{2},m_{H^{0}}^{2})-\cos^{2}(\theta)F(m_{A^{0}}^{2},m_{h^{0}}^{2})\\\
&\qquad\qquad+3\cos^{2}(\theta)[F(m_{Z}^{2},m_{H^{0}}^{2})-F(m_{W}^{2},m_{H^{0}}^{2})]\\\
&\qquad\qquad-3\cos^{2}(\theta)[F(m_{Z}^{2},m_{h^{0}}^{2})-F(m_{W}^{2},m_{h^{0}}^{2})]\Big{\\}},\end{split}$
(65)
$\displaystyle\begin{split}S^{\text{2HDM}}&=\frac{G_{F}m_{W}^{2}\sin^{2}(\theta_{W})}{12\sqrt{2}\,\alpha\pi^{2}}\Big{\\{}(2\sin^{2}(\theta_{W})-1)^{2}\,G(m_{H^{\pm}}^{2},m_{H^{\pm}}^{2},m_{Z}^{2})\\\
&\qquad\qquad+\sin^{2}(\theta)\,G(m_{A^{0}}^{2},m_{H^{0}}^{2},m_{Z}^{2})\\\
&\qquad\qquad+\cos^{2}(\theta)\,G(m_{A^{0}}^{2},m_{h^{0}}^{2},m_{Z}^{2})+2\ln\left(\frac{m_{A^{0}}\,m_{H^{0}}}{m_{H^{\pm}}^{2}}\right)\\\
&\qquad\qquad+\cos^{2}(\theta)\left[\hat{G}(m_{H^{0}}^{2},m_{Z}^{2})-\hat{G}(m_{h^{0}}^{2},m_{Z}^{2})\right]\Big{\\}},\end{split}$
(66)
$\displaystyle\begin{split}U^{\text{2HDM}}&=\frac{G_{F}m_{W}^{2}}{48\sqrt{2}\,\alpha\pi^{2}}\Big{\\{}G(m_{H^{\pm}}^{2},m_{A^{0}}^{2},m_{W}^{2})+\sin^{2}(\theta)\,G(m_{H^{\pm}}^{2},m_{H^{0}}^{2},m_{W}^{2})\\\
&\qquad\qquad+\cos^{2}(\theta)\,G(m_{H^{\pm}}^{2},m_{h^{0}}^{2},m_{W}^{2})-[2\sin^{2}(\theta_{W})-1]^{2}G(m_{H^{\pm}}^{2},m_{H^{\pm}}^{2},m_{Z}^{2})\\\
&\qquad\qquad-\sin^{2}(\theta)\,G(m_{A^{0}}^{2},m_{H^{0}}^{2},m_{Z}^{2})-\cos^{2}(\theta)\,G(m_{A^{0}}^{2},m_{h^{0}}^{2},m_{Z}^{2})\\\
&\qquad\qquad+\cos^{2}(\theta)[\hat{G}(m_{H^{0}}^{2},m_{W}^{2})-\hat{G}(m_{H^{0}}^{2},m_{Z}^{2})]\\\
&\qquad\qquad-\cos^{2}(\theta)[\hat{G}(m_{h^{0}}^{2},m_{W}^{2})-\hat{G}(m_{h^{0}}^{2},m_{Z}^{2})]\Big{\\}},\end{split}$
(67)

where $\theta=\beta-\alpha$. The loop functions used above are defined as

$\displaystyle F(x,y)$
$\displaystyle=\begin{cases}\displaystyle\frac{x+y}{2}-\displaystyle\frac{xy}{x-y}\ln\left(\frac{x}{y}\right),&\text{for
}x\neq y,\\\ 0,&\text{for }x=y,\end{cases}$ (68) $\displaystyle G(x,y,z)$
$\displaystyle=-\frac{16}{3}+\frac{5(x+y)}{z}-\frac{2(x-y)^{2}}{z^{2}}$
$\displaystyle\qquad+\frac{3}{z}\left[\frac{x^{2}+y^{2}}{x-y}-\frac{x^{2}-y^{2}}{z}+\frac{(x-y)^{3}}{3z^{2}}\right]\ln\left(\frac{x}{y}\right)+\frac{r}{z^{3}}f(t,r),$
(69) $\displaystyle\tilde{G}(x,y,z)$
$\displaystyle=-2+\left(\frac{x-y}{z}-\frac{x+y}{x-y}\right)\ln\left(\frac{x}{y}\right)+\frac{f(t,r)}{z},$
(70) $\displaystyle\hat{G}(x,z)$
$\displaystyle=G(x,z,z)+12\,\tilde{G}(x,z,z).$ (71)

These functions depend on the sub-function,

$\displaystyle f(t,r)$
$\displaystyle=\begin{cases}r^{1/2}\displaystyle\ln\left|\frac{t-r^{1/2}}{t+r^{1/2}}\right|&\text{for
}r>0,\\\ 0&\text{for }r=0,\\\
2(-r)^{1/2}\displaystyle\tan^{-1}\left(\frac{(-r)^{1/2}}{t}\right)&\text{for
}r<0,\end{cases}$ (72)

where

$\displaystyle t$ $\displaystyle=x+y-z,$ (73) $\displaystyle r$
$\displaystyle=z^{2}-2z(x+y)+(x-y)^{2}.$

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