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

# The Flavourful Present and Future of 2HDMs at the Collider Energy Frontier

Oliver Atkinson b Matthew Black a Christoph Englert b Alexander Lenz b Aleksey
Rusov c James Wynne<EMAIL_ADDRESS>Matthew.Black@uni-
siegen.de<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the intersection of flavour and collider physics for Two-Higgs-
Doublet models of Type I and II. Drawing from the flavour precision-LHC
exotics search complementarity, we also provide a projection of the future
sensitivity that can be achieved in light of currently available analyses. On
the one hand, we find that the parameter space of the 2HDM can be explored
significantly further with more data from the LHC with some complementarity
with flavour physics. On the other hand, flavour physics results alongside
their projections remain powerful tools to constrain the model space in
regions where direct sensitivity to new states via exotics searches is lost.
Our results further highlight the recently observed flavour physics anomalies
as important drivers of new physics searches in the future; we also touch on
implications for a strong first order electroweak phase transition.

SI-HEP-2022-03

##  1 Introduction

Searches for physics beyond the Standard Model (BSM) are key pillars of the
current particle physics programme that stretches across many different
experimental arenas. Shortcomings of the Standard Model (SM), e.g. the
possible lack of enough CP violation or the insufficiently strong electroweak
phase transition to address the Sakharov criteria Sakharov:1967dj , provide
the motivation for extending the interaction and particle spectrum of the SM.
Especially well-motivated scenarios along these lines are Higgs sector
extensions. While the existence of a non-trivial electroweak vacuum structure
had been verified with the discovery of the $W$ and $Z$ bosons at UA1/UA2, it
was only in 2012 that the discovery of the Higgs boson ATLAS:2012yve ;
CMS:2012qbp  confirmed the mechanism of spontaneous symmetry breaking
Englert:1964et ; Higgs:1964ia ; Higgs:1964pj  through the Higgs boson’s
couplings to the massive gauge bosons. Subsequent investigations of the 125
GeV Higgs boson after its discovery have shown that this new state closely
follows the SM expectation. Nonetheless, current experimental constraints only
limit modification of the Higgs boson’s interactions to be below $\sim 10\%$
CMS:2018ipl ; ATLAS:2020qdt  so there are reasonable margins for new, weakly-
coupled modifications of the Higgs sector. Consistency with electroweak fits
Baak:2014ora  provides an additional constraint for extensions of the symmetry
breaking Higgs sector that can be competitive with current direct
measurements.

Out of the Higgs sector extensions that are typically considered for BSM
investigations, two Higgs doublet models (2HDMs) are particularly well-
motivated theories (for reviews see Refs. Branco:2011iw ; Gunion:1989we ). On
the one hand, they implement custodial isospin analogous to the SM Higgs
field, thus avoiding tensions and fine-tuning with electroweak precision
measurements that occur in higher dimensional representations of electroweak
$SU(2)_{L}\times U(1)_{Y}$ breaking Gunion:1990dt . On the other hand, 2HDMs
can be considered as harbingers of supersymmetric theories, which demand a
second Higgs doublet due to holomorphy, and (non-)perturbative anomaly
cancellation to maximise the $S$-matrix symmetry.

Leaving aside the theoretical motivation for introducing a second weak doublet
contributing to electroweak symmetry breaking and fermion mass generation, the
experimental avenues to constrain or even observe a 2HDM extension of the SM
are plentiful: links of fermion vs. gauge boson mass generation provide a
motivation to combine precision flavour physics investigations with those
performed at the high energy frontier, chiefly at the Large Hadron Collider
(LHC). In fact, a recent investigation into the flavour and Higgs signal
strength constraints of the 2HDM Atkinson:2021eox  demonstrated the power of
flavour measurements in driving BSM mass scales into a region where the direct
detection at the LHC could become challenging. Additionally, current flavour
anomalies highlight flavour physics as a particularly relevant area to inform
future investigations at the high energy frontier.

Building on Ref. Atkinson:2021eox , we investigate the intersection of flavour
and collider physics for the 2HDMs of type I and II in this work, with a view
towards the High Luminosity (HL) LHC phase. Clarifying and extrapolating the
currently available search strategies in flavour physics and for the exotic
states that are predicted in the 2HDM enables us to discuss flavour-collider
complementarity and identify regions that require targeting to enhance the BSM
potential during the upcoming LHC runs.

We organise this work as follows: in Section 2, we review the basics of the
2HDM to make this work self-consistent. Section 3 provides details on
theoretical and electroweak constraints that set the baseline of our
investigation, then in Section 4 we discuss the status of the SM Higgs signal
strengths in the 2HDM type I (2HDM-I). In Section 5 we discuss in detail the
flavour constraints on the 2HDM-I, extending the work of Ref. Atkinson:2021eox
, before we move on to contrast LHC searches and measurements in Section 6 for
the 2HDM-I and 2HDM-II. We also comment on expected improvements of both
flavour and collider constraints before discussing cosmological implications
in Section 7. We provide a summary and conclusions in Section 8.

##  2 The Two Higgs Doublet Model

We follow the notation of Refs. Branco:2011iw ; Gunion:1989we , where, instead
of using the SM Higgs doublet and its (symplectic) charge-conjugated version
to gain mass terms for both up-type and down-type quarks, one can introduce
two distinct doublets

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

with $i=1,2$. After electroweak symmetry breaking (EWSB), these doublets lead
to 5 physical Higgs bosons (the other 3 degrees of freedom construct the
longitudinal $W^{\pm}$, $Z^{0}$ boson polarisations). One finds two charged
Higgs, $H^{\pm}$, two neutral scalars, $h^{0},H^{0}$, and a neutral
pseudoscalar, $A^{0}$. We take $h^{0}$ to be the lighter of the two neutral
scalars and the currently observed Higgs particle with mass $125.25\pm
0.17\,$GeV ParticleDataGroup:2020ssz . The potential for a general 2HDM with a
softly broken $\mathbb{Z}_{2}$ symmetry is 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)

We can use this potential in its “lambda basis” to construct mass terms for
each of the physical Higgs particles, which, alongside the vacuum expectation
value (VEV) $v^{2}=v_{1}^{2}+v_{2}^{2}$, the ratio of VEVs
$\tan\beta=v_{2}/v_{1}$, the mixing parameter $\cos(\beta-\alpha)$, and the
softly $\mathbb{Z}_{2}$ breaking term $m_{12}$ form the “mass basis” which
more simply translates to physical observables, see e.g. Refs.
Atkinson:2021eox ; Basler:2016obg ; Arnan:2017lxi ; Han:2020zqg ;
Kling:2016opi  for transformations between bases.

The Yukawa couplings in the 2HDM can be generally expressed as

${\cal
L}_{Y}=-y_{ij}^{1}\bar{\Psi}^{i}_{L}\Phi_{1}\psi^{j}_{R}-y_{ij}^{2}\bar{\Psi}^{i}_{L}\Phi_{2}\psi^{j}_{R}\,.$
(3)

In general the 2HDM Yukawa couplings need not be flavour diagonal and so it is
possible to allow tree-level flavour-changing neutral currents (FCNCs) which
are absent in the SM. In order to maintain this natural flavour conservation,
one can consider fermions coupling to the Higgs doublets in specific ways. The
four types of 2HDM with natural flavour conservation are described in Table 1.

Model | I | II | X | Y
---|---|---|---|---
$u^{i}_{R}$ | $\Phi_{2}$ | $\Phi_{2}$ | $\Phi_{2}$ | $\Phi_{2}$
$d^{i}_{R}$ | $\Phi_{2}$ | $\Phi_{1}$ | $\Phi_{2}$ | $\Phi_{1}$
$e^{i}_{R}$ | $\Phi_{2}$ | $\Phi_{1}$ | $\Phi_{1}$ | $\Phi_{2}$
Table 1: Types of 2HDM with natural flavour conservation from demanding
fermions couple to specific doublets.

After spontaneous symmetry breaking, we can write the Yukawa sector of the
2HDM Lagrangian as (see e.g. Refs. Crivellin:2019dun ; Branco:2011iw )

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

Imposing the fermion interactions to specific doublets in Table 1, the $\xi$
coupling strengths are expressed in Table 2.111Note that we differ from Ref.
Branco:2011iw  by a sign in the $H^{+}$ Yukawa interaction which in turn leads
to a difference in the fermion coupling strengths. This is the same convention
as found in e.g. Ref. Crivellin:2019dun .

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

##  3 Theory and Electroweak Precision Constraints

###  3.1 Perturbativity, Unitarity, and Vacuum Stability

Perturbativity in the scalar sector can be simply expressed as Chen:2018shg ;
Ginzburg:2005dt

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

As in Ref. Atkinson:2021eox  we also consider the less conservative bounds of
$|\lambda_{i}|\leq 4$ – inspired by the results in Refs. Grinstein:2015rtl ;
Cacchio:2016qyh . Looking at the couplings in Eq. (4), one can further
consider the perturbativity constraints from the Yukawa sector:

$\displaystyle\frac{\sqrt{2}\,V_{tb}\,m_{t}\cot\beta}{2v}\leq{\sqrt{4\pi}}$
$\displaystyle\quad\Longrightarrow\quad\tan\beta>{0.14}\,,$ (6)

where we have chosen a very conservative range. For the range of
$\log[\tan\beta]$ we will consider the conservative lower bound
$\tan\beta=10^{-1.5}\approx 0.03$. For the upper bound we adopt the same
$\tan\beta=10^{+2.5}\approx 300$ as in Ref. Atkinson:2021eox , for ease of
comparison of results between 2HDM-II Atkinson:2021eox  and 2HDM-I (this
work).

We apply the conditions for a stable vacuum as set out in Ref.
Deshpande:1977rw , whilst demanding the vacuum to be the global minimum of the
potential Barroso:2013awa . We also consider the conditions from tree-level
unitarity, see Refs. Ginzburg:2005dt ; Arhrib:2000is 222 For a similar
discussion in an alternative lambda basis see Ref. 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 .

We get the same constraints on the 2HDM parameters as found in our previous
study from theory, where these are not dependent on the specific couplings
defining the different types of 2HDM. We refer to Fig. 1 of Ref.
Atkinson:2021eox  for these results, where the main implication is a close
mass degeneracy for the new Higgses enforced from $\sim 1\,$TeV and becomes
stricter as the mass scale increases.

###  3.2 Oblique corrections

The expressions for the electroweak precision observables (EWPOs): $S$, $T$,
and $U$ Peskin:1990zt ; Peskin:1991sw  (also referred to as oblique
corrections) in the 2HDM (derived from Ref. Grimus:2008nb ) are explicitly
given in Appendix C of Ref. Atkinson:2021eox . New physics generally has only
a small effect on $U$ in comparison to the $S$ and $T$ parameters as the
latter correspond to distinct dimension 6 operators in the effective field
theory expansion, in contrast to $U$ Barbieri:2004qk . It is therefore
justified to follow the approach that neglects $U$ from fits by setting $U=0$
as outlined in Ref. Baak:2014ora ; ParticleDataGroup:2020ssz . 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.

##  4 Higgs Signal Strengths

Many measurements of the properties of the observed Higgs boson have been
made. Of these, the most relevant for our discussion are the Higgs signal
strengths $\mu^{f}_{i}$, which are defined as a ratio between experimental and
SM values of the product of the cross section and branching fraction in a
given channel with production mode $i$ and decay products $f$:

$\mu^{f}_{i}=\frac{(\sigma_{i}\cdotp{\cal
B}_{f})_{\text{Exp.}}}{(\sigma_{i}\cdotp{\cal B}_{f})_{\text{SM}}}.$ (7)

Measurements of $\mu^{f}_{i}$ essentially serve as an indicator of how closely
the observed Higgs matches the SM expectations. We include results from 31
channels CMS:2018uag ; CMS:2019hve ; ATLAS:2019nkf ; ATLAS:2020qcv ;
ATLAS:2020fzp ; CMS:2020gsy ; ATLAS:2020qdt  (collected in Table 4 of Ref.
Atkinson:2021eox ).

If the observed Higgs is not that of the SM, but part of an extended Higgs
sector such as the 2HDM, its couplings to other particles will differ from the
SM, a difference that will feed through into the values of the signal
strengths. This is the case in the 2HDM-I, with the couplings being modified
by the factors shown in Table 2. It is therefore possible to use the signal
strength measurements to constrain the parameters on which these couplings
depend, namely the mixing angles. We do this by performing a fit using
analytical calculations of the signal strengths in the 2HDM-I as functions of
the input parameters $\tan\beta$ and $\cos(\beta-\alpha)$ from expressions
given in Ref. Gunion:1989we , with results presented in Fig. 1. Opposed to the
2HDM-II case (see Ref. Atkinson:2021eox ) we find that more sizeable
deviations from the alignment limit $\cos(\beta-\alpha)=0$ are allowed for the
2HDM-I. We observe that larger deviations from the alignment limit are
possible with increasing $\tan\beta$, and we find $|\cos(\beta-\alpha)|\leq
0.21\;(0.4)$ at $2\;(5)\sigma$ for sufficiently large $\tan\beta\geq 10$.

Figure 1: Combined fits of the SM Higgs signal strengths in the 2HDM-I (in the
($\tan\beta-\cos(\beta-\alpha)$) plane). Contours are shown representing
allowed parameter space at 1, 2, 3, 4, 5$\sigma$ confidence from darkest to
lightest.

##  5 Flavour Constraints in 2HDM-I

In Section 5.1, we cover the constraints on the 2HDM-I parameter space from
flavour observables, and then in Section 5.2 we combine these with the Higgs
signal strengths and the EWPOs. We follow the procedure of the earlier study
Atkinson:2021eox  in the 2HDM-II, where we now exchange the Type II couplings
for the Type I. All observables we consider here are listed with SM
predictions and experimental measurements in Tables 4-8 of Ref.
Atkinson:2021eox ; the measurements are collected from various results
throughout literature ATLAS:2020qdt ; CMS:2020gsy ; CMS:2021kom ;
ATLAS:2021upe ; ATLAS:2020qcv ; ATLAS:2020fzp ; CMS:2020xwi ; ATLAS:2020bhl ;
CMS:2018uag ; ATLAS:2020fcp ; CMS:2019hve ; ParticleDataGroup:2020ssz ;
HFLAV:2019otj ; LHCb:2020cyw ; LHCb:2021qbv ; McLean:2019qcx ;
Altmannshofer:2021qrr ; LHCb:2014cxe ; LHCb:2020gog ; LHCb:2020lmf ;
ATLAS:2018gqc ; CMS:2015bcy ; CMS:2017rzx ; BELLE:2019xld ; Belle:2019oag ;
LHCb:2015wdu ; BaBar:2013qry ; Belle:2016fev ; LHCb:2015ycz ; CMS:2018qih ;
LHCb:2015tgy ; LHCb:2018jna ; LHCb:2021trn ; LHCb:2014vgu ; LHCb:2017avl ;
BaBar:2012mrf  and the SM predictions are calculated in flavio unless
otherwise stated.

###  5.1 Individual Fits

For the tree-level (semi-)leptonic flavour-changing charged transitions, the
2HDM-I contributes through the effective operators

$\displaystyle\mathcal{O}_{S-P}=(\bar{u}P_{L}\,d)(\bar{\ell}P_{L}\nu_{\ell}),\qquad\mathcal{O}_{S+P}=(\bar{u}P_{R}\,d)(\bar{\ell}P_{L}\nu_{\ell}),$
(8)

with $P_{L(R)}=(1\mp\gamma_{5})/2$. The Wilson coefficients of these operators
in terms of the 2HDM-I parameters are

$C_{S-P}=-\frac{m_{u}\,m_{\ell}\cot^{2}\beta}{m_{H^{+}}^{2}},\qquad\quad
C_{S+P}=\frac{m_{d}\,m_{\ell}\cot^{2}\beta}{m_{H^{+}}^{2}}\,.$ (9)

For a description of all operators gaining contributions from the 2HDM for the
observables we consider and the translation of basis for these, we refer to
Atkinson:2021eox , taking expressions for the Wilson coefficients from Refs.
Crivellin:2019dun ; Ilisie:2015tra ; Borzumati:1998nx  where these are
expressed in terms of general 2HDMs and one can simply insert the appropriate
couplings for the 2HDM-I.

First 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 $\ell=e,\mu$. The latter implies a
$2.8\sigma$ tension between experimental measurements and SM predictions, and
the combined tension of the two between experiment and the SM is $3.2\sigma$
(using flavio). We show in Fig. 2 the allowed 2HDM-I parameter space for these
two observables in individual fits. Within the theoretically-motivated
parameter limits of our fits, we find that $R(D)$ allows a large part of our
parameter space within $2\sigma$, whereas $R(D^{*})$ allows most of our
parameter space at $2.8\sigma$ or above. In fact, for both observables, we
find the best fit for $m_{H^{+}}\sim 1\,$GeV which is outside the physical
domain of our model. Within the considered limits of our parameters, we find a
minimum of $3.5\sigma$ combined tension between the 2HDM-I and experiment for
$R(D^{(*)})$. In Fig. 3, we present the combined fit of tree-level flavour-
changing charged currents including leptonic and semi-leptonic decays of
mesons, hadronic decays of $\tau$ leptons, and $R(D^{(*)})$; see Table 6 of
Ref. Atkinson:2021eox  for all channels considered. The SM predictions in
flavio for the leptonic and semi-leptonic channels are based on Refs.
Bernlochner:2017jka ; Caprini:1997mu ; Sakaki:2013bfa ; Bharucha:2015bzk ;
Gubernari:2018wyi ; Detmold:2016pkz ; Bernard:2006gy ; Bernard:2009zm ;
FlaviaNetWorkingGrouponKaonDecays:2010lot , and the new 2HDM-I contributions
are described by Eq. (9).

Figure 2: Contour plots of the allowed 2HDM-I parameter space in the
($\tan\beta-m_{H^{+}}$) plane for $R(D)$ (left) and $R(D^{*})$ (right).
Contours are shown representing allowed parameter space at 1, 2, 3, 4,
5$\sigma$ confidence from darkest to lightest. Figure 3: Contour plot of the
allowed 2HDM-I parameter space in the ($\tan\beta-m_{H^{+}}$) plane for 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
$R(D^{*})$. The lighter contour indicates the allowed parameter space at
$2\sigma$ confidence level while the darker contour corresponds to $1\sigma$.

Next we consider the mixing of neutral $B_{d,s}$ mesons. The mass differences
of this meson mixing are experimentally known very precisely at the level
$\sim{\cal O}(0.1\%)$ ParticleDataGroup:2020ssz . On the theory side, however,
the uncertainties are still dominated by the non-perturbative determinations
of the matrix elements of the $\Delta B=2$ operators. We use the averages
presented in Ref. DiLuzio:2019jyq  combining HQET Sum Rules King:2019lal ;
Kirk:2017juj ; Grozin:2016uqy  and lattice calculations Dowdall:2019bea ;
Boyle:2018knm ; FermilabLattice:2016ipl , yielding a theory precision of
${\cal O}(5\%)$. Terms proportional to the $tH^{-}$ coupling dominate the 2HDM
contributions in the box diagrams with the down-type quark effects largely
mass-suppressed. The predictions in the 2HDM-I for $B$-meson mixing are
therefore very similar to those in the 2HDM-II where the dominating up-type
$H^{\pm}$ couplings remain the same. The combined fit for $\Delta m_{d,s}$ is
shown in Fig. 4, where we find a correlation between $\tan\beta$ and
$m_{H^{+}}$ constraining $\tan\beta$ from below. We take the expressions for
the $\Delta B=2$ Wilson coefficients from Ref. Crivellin:2019dun , converting
to the flavio basis as described in Ref. Atkinson:2021eox .

Figure 4: Contour plot of the allowed 2HDM-I parameter space in the
($\tan\beta-m_{H^{+}}$) plane for the combined fit to $\Delta m_{d,s}$. The
darker contour indicates allowed parameter space at $1\sigma$ confidence, and
the lighter at $2\sigma$.

A signature observable for new physics analysis is the branching ratio of the
$\bar{B}\to X_{s}\gamma$ decay. In the SM, this is known to NNLO in QCD
Misiak:2020vlo  (based on the previous Refs. Misiak:2006ab ; Misiak:2015xwa ).
We take the experimental average HFLAV:2019otj  formed from Refs. CLEO:2001gsa
; BaBar:2012fqh ; Belle:2016ufb . In the 2HDM-II, $\bar{B}\to X_{s}\gamma$
provides a distinct constraint on the lower bound for the charged Higgs mass
$m_{H^{+}}$; in the Type I however, this is much more correlated with
$\tan\beta$, and we therefore cannot find a clear constraint from this
observable alone. In Fig. 5 we show the fit to ${\cal B}(\bar{B}\to
X_{s}\gamma)|_{E_{\gamma}>1.6\,{\rm GeV}}$ in the 2HDM-I where we calculate
the contributions to the Wilson coefficents $C_{7},C_{8}$ at NLO using Ref.
Borzumati:1998nx .

Figure 5: Contour plot of the allowed 2HDM-I parameter space in the
($\tan\beta-m_{H^{+}}$) plane for ${\cal B}(\bar{B}\to X_{s}\gamma)$. The
darker contour indicates allowed parameter space at $1\sigma$ confidence, and
the lighter at $2\sigma$.

We also consider the FCNC processes $B_{d,s}\to\mu^{+}\mu^{-}$, which are
sensitive to BSM contributions to scalar operators, making them important
observables to test the 2HDM. In recent years, ATLAS, CMS, and LHCb
LHCb:2017rmj ; ATLAS:2018cur ; CMS:2019bbr ; LHCb:2020zud ; LHCb:2021awg ;
LHCb:2021vsc  have all produced measurements for $B_{s}\to\mu^{+}\mu^{-}$, and
developed further the upper limit on $B_{d}\to\mu^{+}\mu^{-}$. In our analysis
we make use of the combination of these measurements from Ref.
Altmannshofer:2021qrr . The theory prediction is formed of the perturbative
calculations Buchalla:1993bv ; Bobeth:2013uxa ; Beneke:2019slt  and the
determinations of the non-perturbative decay constants, for example
ETM:2016nbo ; Bazavov:2017lyh ; Hughes:2017spc . The 2HDM Wilson coefficients
(taken again from Ref. Crivellin:2019dun ) for the operators contributing to
these processes, ${\cal O}_{10}^{(^{\prime})},{\cal O}_{S}^{(^{\prime})},{\cal
O}_{P}^{(^{\prime})}$, also depend on further 2HDM parameters:
$\cos(\beta-\alpha),m_{H^{0}},m_{A^{0}}$. Motivated by constraints from theory
and the Higgs signal strengths (see Sections 3 and 4), we present in Fig. 6 a
2D fit in the ($\tan\beta-m_{H^{+}}$)-plane fixing $\cos(\beta-\alpha)=0$ and
$m_{H^{0}}=m_{A^{0}}=m_{H^{+}}$. In this approach, we find here similarly to
$\bar{B}\to X_{s}\gamma$ a strong correlation between $\tan\beta$ and
$m_{H^{+}}$. Later in our global fit (Section 5.2), we will discuss further
the dependence on these additional parameters.

Figure 6: Contour plot of the allowed 2HDM-I parameter space in the
($\tan\beta-m_{H^{+}}$) plane for the combined fit to
${\cal}(B_{d,s}\to\mu^{+}\mu^{-})$, taken in the limits of alignment
($\cos(\beta-\alpha)=0$) and degenerate masses
($m_{H^{0}}=m_{A^{0}}=m_{H^{+}}$). The darker contour indicates allowed
parameter space at $1\sigma$ confidence, and the lighter at $2\sigma$.

The LFU ratios $R_{K}$ and $R_{K^{*}}$ Bobeth:2007dw  have become well-studied
processes in recent years (see e.g. Ref. Bordone:2016gaq ; Hurth:2020ehu ;
Geng:2017svp ; Ciuchini:2019usw ; Datta:2019zca ; Kowalska:2019ley ;
Hurth:2020rzx ; Ciuchini:2020gvn ), driven by LHCb measurements LHCb:2014vgu ;
LHCb:2017avl  finding discrepancies with the SM, most noticeably the deviation
of $3.1\sigma$ in $R_{K^{+}}$ LHCb:2021trn . Model-independent analyses (for
example, Refs. Alguero:2019ptt ; Alok:2019ufo ; Aebischer:2019mlg ;
Altmannshofer:2017yso ) show these processes favour vector-like new physics
contributing to ${\cal O}_{9,10}$ for their deviations to be resolved. In
Table 3 we show the best fit point only including the 10 $R_{K^{(*)}}$ bins we
consider, where we find that this fit favours the limits $\tan\beta\to\infty$,
$\cos(\beta-\alpha)\to 0$ and $m_{H^{+}}\sim m_{H^{0}}\sim m_{A^{0}}\sim
50\,$GeV. This result is purely numerics-driven; we note that it lies outside
the bounds of our model given by theoretical considerations. In the large
$\tan\beta$ limit we recover the SM predictions, as the 2HDM-I induced
coupling deviations will go to zero. With the SM result equating to our best
fit point, no point in our parameter space can reduce the tensions in these
observables beyond their current status. In Fig. 7, we show our combined fit
for the 10 $R_{K^{(*)}}$ bins within a more similar parameter region to other
fits, where we see that most of our parameter space considered lies within
$1\sigma$ of the best fit (or SM) result. A discovered 2HDM-I within the
physical bounds of the model could only increase the tensions in $R_{K^{(*)}}$
as we find currently them in the SM.

Figure 7: Contour plot of the allowed 2HDM-I parameter space in the
($\tan\beta-m_{H^{+}}$) plane for the combined fit to the 10 $R_{K^{(*)}}$
bins in Table, taken in the limits of alignment ($\cos(\beta-\alpha)=0$) and
degenerate masses ($m_{H^{0}}=m_{A^{0}}=m_{H^{+}}$). The darker contour
indicates allowed parameter space at $1\sigma$ confidence, and the lighter at
$2\sigma$.

In Fig. 8, we consider more $b\to s\ell^{+}\ell^{-}$ observables (see Table 7
in Ref. Atkinson:2021eox ), where some of these also find tension with
experiment in the SM. For detailed descriptions and analyses focused
specifically on this interesting group of observables, 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 . Similarly to $R_{K}$ and $R_{K^{*}}$
above, these processes tend to favour new physics (NP) contributions from
${\cal O}^{(^{\prime})}_{9,10}$. In the 2HDM-I, we find that the contributions
to these operators (and also to ${\cal O}^{(^{\prime})}_{S,P}$) can be
sufficient to reduce the tension with experiment for some of these
observables. This effect is most significant for $m_{H^{0}}\sim m_{A^{0}}\sim
m_{H^{+}}\sim 1000\,$GeV as suggested by the best-fit point in Table 3, where
we find an improvement over the SM with a pull of $2.6\sigma$ (excluding
$R_{K^{(*)}}$); we then find a large portion of our parameter space within
$1\sigma$ confidence of this result. Varying the fit parameters around the
best fit point, we also find that this group of observables imposes a lower
bound on $\tan\beta$ as

$\tan\beta\gtrsim 0.14\,(0.16)\quad{\rm at}\;\>2\sigma\,(1\sigma),$ (10)

which is compatible with the lower bound coming from perturbativity in Section
3.

###  5.2 Global Fits

Figure 8:  Contour plot of the allowed 2HDM-I parameter space in the
($\tan\beta-m_{H^{+}}$) plane for the combined fit to the leptonic
$B_{d,s}\to\mu^{+}\mu^{-}$ decays and the semi-leptonic $b\to
s\ell^{+}\ell^{-}$ observables, excluding $R_{K^{(*)}}$ and fixing the
additional parameters to their best-fit points. The darker contour indicates
allowed parameter space at $1\sigma$ confidence, and the lighter at $2\sigma$.

We now consider our global fits to the 2HDM-I combining flavour observables,
Higgs signal strengths, and the EWPOs $S,T,U$. Within the contours of a global
fit, the LFU ratios $R_{D^{(*)}}$, $R_{K^{(*)}}$ still have a tension $\sim
3.5\sigma$ with experiment that is worsened compared to the SM.333Some other
$b\to s\ell\ell$ observables still find some tension, but these are not as
severe and are lessened compared to the SM. This motivates two scenarios for
the global fits: either including these LFU ratios, or considering a fit
excluding them. In Table 3, we summarise the results of the various fits
performed, indicating their best-fit parameter points and the corresponding
statistics. In Fig. 9, we show the fit to all observables (excluding
$R_{D^{(*)},K^{(*)}}$) where we fix the additional 2HDM parameters to their
best fit point as shown in Table 3. In Fig. 10, we show the fit to all
observables (excluding $R_{D^{(*)},K^{(*)}}$) in the alignment limit and with
degenerate masses. We find the shapes of the contours to be much the same for
excluding/including $R_{D^{(*)},K^{(*)}}$, so we do not show both sets of
plots here; the impact of including $R_{D^{(*)},K^{(*)}}$ is clearer in Table
3, where we find distinctly poorer quality of fit (low $p$-value) in this
scenario. Excluding $R_{D^{(*)},K^{(*)}}$, we find a pull of $1.9\sigma$
improvement from the SM to 2HDM-I; including $R_{D^{(*)},K^{(*)}}$, the
improvement is $1.5\sigma$.

From Table 3, we see that $\cos(\beta-\alpha)=0$ (the alignment limit) is
closely favoured from the fits, although Fig. 9 shows that larger deviations
from alignment are allowed than in the 2HDM-II. We find preference in the
global fits for $\tan\beta\sim{\cal O}(100)$ and $m_{H^{+}}\sim m_{H^{0}}\sim
m_{A^{0}}\sim 1.7\,$TeV, where a closer mass degeneracy at this scale is
enforced from theory constraints. Within the parameter region found we scan
around the best fit point to find the full constraints on our parameters. As
one can see throughout this section, much of the small $\tan\beta$ parameter
space is excluded by most observables individually and globally. We cannot
state an explicit bound on $\tan\beta$ from the global fit however as the
lower limit is clearly correlated with the charged Higgs mass. For the mass of
the charged Higgs, within the parameter region we test, we find bounds only at
$1\sigma$ confidence:

$m_{H^{+}}\leq 83.4\,{\rm TeV}\quad{\rm at}\;1\sigma.$ (11)

In our global fit of all observables, our constraints on $\cos(\beta-\alpha)$
are much improved from considering the Higgs signal strengths alone, where we
find the constraints in the full analysis to follow very closely to the
representative plot in Fig. 9:

$\displaystyle|\cos(\beta-\alpha)|$ $\displaystyle\leq 0.11$
$\displaystyle\quad{\rm at}$ $\displaystyle\;1\sigma,$ (12)
$\displaystyle|\cos(\beta-\alpha)|$ $\displaystyle\leq 0.14$
$\displaystyle\quad{\rm at}$ $\displaystyle\;2\sigma,$
$\displaystyle|\cos(\beta-\alpha)|$ $\displaystyle\leq 0.17$
$\displaystyle\quad{\rm at}$ $\displaystyle\;3\sigma,$
$\displaystyle|\cos(\beta-\alpha)|$ $\displaystyle\leq 0.19$
$\displaystyle\quad{\rm at}$ $\displaystyle\;4\sigma,$
$\displaystyle|\cos(\beta-\alpha)|$ $\displaystyle\leq 0.21$
$\displaystyle\quad{\rm at}$ $\displaystyle\;5\sigma.$

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. $R_{D^{(*)},K^{(*)}}$ | $275$ | $\\{70,1720\,{\rm GeV},1680\,{\rm GeV},1670\,{\rm GeV},-0.001\\}$ | $317.0$ | $\phantom{0}2.6\%$
All excl. $R_{D^{(*)},K^{(*)}}$ | $263$ | $\\{80,1720\,{\rm GeV},1770\,{\rm GeV},1770\,{\rm GeV},-0.003\\}$ | $281.1$ | $15.5\%$
Flavour incl. $R_{D^{(*)},K^{(*)}}$ | $241$ | $\\{60,1650\,{\rm GeV},1600\,{\rm GeV},1600\,{\rm GeV},-0.002\\}$ | $289.1$ | $\phantom{0}1.0\%$
Flavour excl. $R_{D^{(*)},K^{(*)}}$ | $229$ | $\\{30,1790\,{\rm GeV},1750\,{\rm GeV},1740\,{\rm GeV},-0.002\\}$ | $249.4$ | $11.7\%$
$b\to s\ell\ell$ incl. $R_{K^{(*)}}$ | $202$ | $\\{40,1030\,{\rm GeV},920\,{\rm GeV},1010\,{\rm GeV},-0.003\\}$ | $262.1$ | $\phantom{0}0.1\%$
$b\to s\ell\ell$ excl. $R_{K^{(*)}}$ | $192$ | $\\{35,1020\,{\rm GeV},960\,{\rm GeV},970\,{\rm GeV},-0.002\\}$ | $234.4$ | $\phantom{0}1.1\%$
Only $R_{K^{(*)}}$ | $10$ | $\\{>1000,50\,{\rm GeV},60\,{\rm GeV},50\,{\rm GeV},0.0001\\}$ | $\phantom{0}27.9$ | $0.004\%$
Higgs Signals | $31$ | $\\{>1000,--,--,--,0.0\\}$ | $\phantom{0}28.2$ | $50.7\%$
Alignment Limit, $\cos(\beta-\alpha)=0$
All incl. $R_{D^{(*)},K^{(*)}}$ | $275$ | $\\{40,1760\,{\rm GeV},1730\,{\rm GeV},1690\,{\rm GeV}\\}$ | $318.5$ | $\phantom{0}2.5\%$
All excl. $R_{D^{(*)},K^{(*)}}$ | $263$ | $\\{50,1810\,{\rm GeV},1750\,{\rm GeV},1760\,{\rm GeV}\\}$ | $283.7$ | $13.9\%$
$\cos(\beta-\alpha)=0$, $m_{H^{+}}=m_{H^{0}}=m_{A^{0}}$
All incl. $R_{D^{(*)},K^{(*)}}$ | $275$ | $\\{90,1630\,{\rm GeV}\\}$ | $324.3$ | $\phantom{0}1.8\%$
All excl. $R_{D^{(*)},K^{(*)}}$ | $263$ | $\\{80,1750\,{\rm GeV}\\}$ | $286.4$ | $13.9\%$
Table 3: Best fit points of 2HDM-I parameter fits for various groups of
observables using the constraints from theory to inform the physical parameter
values.

Figure 9: Combined fit of all flavour observables (excluding
$R_{D^{(*)},K^{(*)}}$), Higgs signal strengths, and EWPOs in the 2HDM-I (in
the $\tan\beta-m_{H^{+}}$ plane), fixing the additional parameters to their
best-fit points. Contours are shown representing allowed parameter space at
1,2,3,4,5$\sigma$ confidence from darkest to lightest. Figure 10: Combined fit
of all flavour observables (excluding $R_{D^{(*)},K^{(*)}}$), Higgs signal
strengths, and EWPOs in the 2HDM-I (in the $\tan\beta-m_{H^{+}}$ plane), taken
in the limits of alignment ($\cos(\beta-\alpha)=0$) and degenerate masses
($m_{H^{0}}=m_{A^{0}}=m_{H^{+}}$). Contours are shown representing allowed
parameter space at 1,2,3,4,5$\sigma$ confidence from darkest to lightest.

###  5.3 Comment on the Muon Anomalous Magnetic Moment

It is worthwhile to comment on the anomalous magnetic moment of the muon,
$a_{\mu}$, which we do not include in our overall fit. The recent results from
Run 1 at Fermilab Muong-2:2021ojo  confirmed the previous result from BNL
Muong-2:2006rrc , where the combined experimental value now yields a
$4.2\sigma$ deviation from the SM result predicted by the Theory Initiative
White Paper (WP) Aoyama:2020ynm  (based on Refs. Aoyama:2012wk ;
Aoyama:2019ryr ; Czarnecki:2002nt ; Gnendiger:2013pva ; Davier:2017zfy ;
Keshavarzi:2018mgv ; Colangelo:2018mtw ; Hoferichter:2019mqg ; Davier:2019can
; Keshavarzi:2019abf ; Kurz:2014wya ; Melnikov:2003xd ; Masjuan:2017tvw ;
Colangelo:2017fiz ; Hoferichter:2018kwz ; Gerardin:2019vio ; Bijnens:2019ghy ;
Colangelo:2019uex ; Blum:2019ugy ; Colangelo:2014qya ). There is also a
competing Lattice QCD prediction for the $a_{\mu}^{\rm HVP}$ contribution from
the BMW Collaboration Borsanyi:2020mff  which results in only a $1.6\sigma$
discrepancy from experiment.444There are several lattice calculations
FermilabLattice:2017wgj ; Budapest-Marseille-Wuppertal:2017okr ; RBC:2018dos ;
Giusti:2019xct ; Shintani:2019wai ; FermilabLattice:2019ugu ; Gerardin:2019rua
; Aubin:2019usy ; Giusti:2019hkz  which were included in the WP, however the
precision of these are much lower than either data-driven approach or the BMW
calculation. Here, we fit $a_{\mu}$ to the 2HDM-I parameters considering both
the WP and the BMW SM scenarios, shown in Fig. 11. We take the one- and two-
loop contributions of the 2HDM to $a_{\mu}$ from Ref. Ilisie:2015tra  where
one can insert the expressions for the 2HDM-I couplings and we then convert
the results to flavio’s WET-3 basis as described in Section 7.2 of Ref.
Atkinson:2021eox . As before, to present information in terms of two-
dimensional fits, we fix the additional 2HDM parameters $\cos(\beta-\alpha)=0$
and $m_{H^{+}}=m_{H^{0}}=m_{A^{0}}$ as motivated by theory.

We find that in the 2HDM-I, $a_{\mu}$ strongly favours low $m_{H^{+}}\sim
100\,$MeV, low $\tan\beta\sim 0.5$, where the tension between experiment and
theory using either SM prediction can be reduced to much less than $1\sigma$.
This result is however below the physical domain for $m_{H^{+}}$. Within the
physical domain, we find that $m_{H^{+}}\gtrsim 100\,$GeV can yield a tension
between the WP and experiment of less than $5\sigma$, with a minimum of
$4.3\sigma$, and between BMW and experiment less than a $2\sigma$ tension,
with a minimum of $1.6\sigma$.

Figure 11: Contour plot of allowed 2HDM-I parameter space in the ($\tan\beta-
m_{H^{+}}$) plane for $a_{\mu}$, taken in the limits of alignment
($\cos(\beta-\alpha)=0$) and degenerate masses
($m_{H^{0}}=m_{A^{0}}=m_{H^{+}}$). In the left plot, the SM prediction taken
from the theory initiative is used; in the right, the SM result from the BMW
collaboration. Contours are plotted representing allowed parameter space at
1,2,3,4,5$\sigma$ confidence from darkest to lightest: in the left plot, only
the $5\sigma$ contour is visible; in the right, the $2,3,4,5\sigma$ contours.

###  5.4 2HDM-I and II Flavour Prospects at Future Colliders

Finally, we would like to discuss the prospects of the 2HDM flavour sector at
future colliders. An approximate prediction for B-physics at future colliders
such as the HL-LHC or beyond is that the precision of the measurements will
roughly double from that of today’s Cerri:2018ypt ; FCC:2018byv . With this in
mind, we explore several scenarios for flavour observables in the future.
Denoting the present-day experimental value of some observable as $M\pm E$,
where $M$ is the central value and $E$ the uncertainty, for each observable we
now set $E\to E/2$ and consider 5 scenarios for a shift in $M$ in the range
$[M-E,M+E]$, where $M$ for all observables shifts in the same way. We also
consider an “Ideal” future scenario whereby $M$ for each observable is
individually shifted within $[M-E,M+E]$ to minimise its best-fit point
$\chi^{2}$ contribution; this scenario generally brings $M$ to the SM
prediction or as close as possible within $[M-E,M+E]$, and can be considered
qualitatively similar to an extrapolation assuming the SM result.

We perform this process on $\sim$200 of our 275 observables which are
identified as being
“experimentally-limited” in their $\chi^{2}$ contribution: that is, their
present $\chi^{2}$ contribution is dominated by the size of their experimental
uncertainty. It is assumed in these tests that the theoretical calculations
and precision do not change significantly by the time of these future
measurements.

In Table 4, we list the comparison of the $\chi^{2}$ best fits in the SM,
2HDM-I, 2HDM-II for first the present situation and then each future scenario
considered. We also list the lower bound of $m_{H^{+}}$ from $\bar{B}\to
X_{s}\gamma$ in the 2HDM-II for each scenario, as this has been an important
bound in the history of the 2HDM and can remain so in the future. It is
unlikely that any one of the exact scenarios would come to pass from future
measurements for such a large group of observables, however comparison between
all 6 scenarios sheds some light on the future suitability of the SM and the
2HDMs. In the present, we find that both the 2HDM-I and 2HDM-II perform better
than the SM by $\sim 2\sigma$. In all future scenarios, the 2HDM-II still
performs better than the SM, with a pull $>1\sigma$ in all but one of these.
The 2HDM-I’s performance over the SM is narrowed significantly however, where
the best future scenarios for this have a reduced pull of $1\sigma$. In the
“Ideal” future, the SM, 2HDM-I, and 2HDM-II all significantly improve upon
their current positions; the majority of observables in this “Ideal” scenario
simply move much closer to their SM predictions, where the small contributions
from the 2HDMs either are not significant or are sufficient to improve upon
the remaining small tensions with the SM.

Overall, it is found that increased precision of flavour observables at future
colliders can lead to much poorer fits in both the SM and the 2HDM. The
exception to this is when measurements are shifted much closer to the SM
predictions and the small 2HDM contributions can be sufficient to resolve much
of the remaining differences. While the future measurements’ central values
may not follow the scenarios considered here, the limiting factor in the
$\chi^{2}$ values in Table 4 is the increased precision of measurements.
Unless future measurements would indeed conform closely to our “Ideal” future
scenario, the performances of the SM, 2HDM-I, 2HDM-II would all become very
poor. While assuming no significant gaps in our theory predictions, this
suggests some New Physics other than the 2HDM should be explored.

Scenario | SM | 2HDM-I | 2HDM-II | Min($m_{H^{+}}$)${}^{b\to s\gamma}_{\rm 2HDM-II},2\sigma$
---|---|---|---|---
Present | 292 | 281 (+$1.9\sigma$) | 282 (+$1.8\sigma$) | 790 GeV
“Ideal” future | 174 | 169 (+$0.8\sigma$) | 166 (+$1.4\sigma$) | 680 GeV
$M\pm E/2$ | 721 | 717 (+$0.7\sigma$) | 711 (+$1.7\sigma$) | 940 GeV
$(M-E)\pm E/2$ | 811 | 808 (+$0.4\sigma$) | 808 (+$0.4\sigma$) | 1050 GeV
$(M-E/2)\pm E/2$ | 815 | 812 (+$0.4\sigma$) | 804 (+$2.0\sigma$) | 1000 GeV
$(M+E/2)\pm E/2$ | 896 | 891 (+$0.8\sigma$) | 887 (+$1.6\sigma$) | 680 GeV
$(M+E)\pm E/2$ | 899 | 893 (+$1.0\sigma$) | 887 (+$2.1\sigma$) | 770 GeV
| Best-fit point
$\\{\tan\beta,m_{H^{+}},m_{H^{0}},m_{A^{0}},\cos(\beta-\alpha)\\}$
2HDM-I | $\\{80,1720\,{\rm GeV},1770\,{\rm GeV},1770\,{\rm GeV},-0.003\\}$
2HDM-II | $\\{4.3,2340\,{\rm GeV},2380\,{\rm GeV},2390\,{\rm GeV},0.009\\}$
Table 4: Summary of future prediction scenarios and their $\chi^{2}$ best
fits. For the 2HDMs in brackets are the pulls from the SM where a positive
value indicates an improvement and a negative value a worsening. The results
are shown for excluding $R_{D^{(*)},K^{(*)}}$. Also shown is the variation of
the lower bound of the charged Higgs mass in the 2HDM-II from $\bar{B}\to
X_{s}\gamma$.

##  6 LHC Constraints, Flavour Comparison and Outlook

Having explored flavour constraints and their projections in detail in the
previous section, we now turn to constraints from collider measurements of
exotics searches. Such direct searches for Higgs bosons have a long history.
Indeed, searching for the Higgs boson of the SM was the raison d’être of the
LHC, picking up where other colliders such as the LEP left off. In the years
since the Higgs discovery, many precise measurements of the 125 GeV scalar
have been made (see above), while searches for other, more exotic, Higgs
bosons have continued alongside the precision measurements of the observed
Higgs (see also Refs. Chen:2013rba ; Chen:2013kt ). These precision
measurements closely match the phenomenology predicted of the SM Higgs. In
this Section we look to combine the wealth of experimental data on direct
searches for new Higgs bosons to further constrain the allowed parameter space
of the 2HDM, independently from the constraints in the prior sections, before
combining these results in Section 6.3. We also extrapolate the current LHC
bounds to the expected results at the HL-LHC operating at 3 ab-1 in the same
channels.

As in the rest of this work, the parameters of primary interest here are the
masses of the new Higgs bosons and the angles $\tan\beta$ and
$\cos(\beta-\alpha)$. Following the results of Section 5.2, we examine the
favoured scenario in which the new Higgs bosons have degenerate masses and the
alignment limit holds.

###  6.1 Current Collider Bounds

In order to leverage the power of the vast array of search data we use the
packages HiggsBounds Bechtle:2008jh ; Bechtle:2011sb ; Bechtle:2012lvg ;
Bechtle:2013wla ; Bechtle:2015pma ; Bechtle:2020pkv ; Bahl:2021yhk  and
2HDecay Krause:2018wmo ; Djouadi:1997yw ; Djouadi:2018xqq ; Krause:2016xku ;
Denner:2018opp ; Hahn:1998yk . We use 2HDecay to calculate the branching
ratios and decay widths of each 2HDM Higgs boson for a given point in
parameter space, and interface this data with HiggsBounds to check if the
search data excludes such a point. For the key channel of $H^{+}$ production
in association with $t\bar{b}$, we use MadGraph5_aMC@NLO Alwall:2014hca  to
generate cross sections which also form part of the input to HiggsBounds,
along with the couplings of the Higgs bosons as given in Table 2. We perform
scans using 50000 randomly generated points in the parameter space following
this method and show the allowed points for both type I and II 2HDMs in Fig.
12. The historic baseline sensitivity from LEP ALEPH:2013htx  excludes charged
Higgs masses below 72.5 GeV and 80 GeV for the Type I and Type II 2HDM models
respectively.

Figure 12: Scan of 50000 randomly generated points compared against LHC data
using 2HDecay and HiggsBounds, with allowed points shown in blue. On the left,
the scan in the 2HDM-I, on the right the 2HDM-II.

In the 2HDM-I scan on the left of Fig. 12, at low values of $\tan\beta$,
leptonic decays of the neutral Higgses (e.g. $H^{0}\to\mu^{+}\mu^{-}$ in Ref.
CMS:2019lwf ) exclude lower masses. Sensitivity to
$H^{+}\to\tau^{+}\nu_{\tau}$ peaks at $\sim 85$ GeV CMS:2019bfg  and then the
$H^{+}\to t\bar{b}$ ATLAS:2020jqj  decay provides the exclusion from $\sim
250$ GeV until falling cross sections lead to a loss of sensitivity at $\sim
2.5$ TeV, beyond which points are allowed. This exclusion limit falls with
increasing $\tan\beta$ as the $H^{+}tb$ coupling is proportional to
$\cot^{2}\beta$.555The relevant Yukawa coupling is
$2((m_{t}\xi^{u}_{H^{+}})^{2}+(m_{b}\xi^{d}_{H^{+}})^{2})/v^{2}$, where
$\xi^{u,d}_{H^{+}}=\xi^{u,d}_{A}$ in Table 2. At moderate $\tan\beta$ the low
mass exclusion limit shifts to $H^{0}/A^{0}\to 4b$ ALEPH:2006tnd . The kink at
$\sim 100$ GeV in the moderate $\tan\beta$ region is a result of
$\mathcal{B}(H^{+}\to\tau^{+}\nu_{\tau})$ falling to zero as
$\mathcal{B}(H^{+}\to t\bar{b})$ rises to unity in this mass region, with
$\mathcal{B}(H^{+}\to t\bar{b})\approx 1$ from $m_{H^{\pm}}\gtrsim m_{t}$.
Sensitivity is lost from $H^{0}\to\ell\ell$ as
$\kappa^{H^{0}}_{\ell}\propto\cot\beta$, see Table 2. The flat cut off at $80$
GeV results from $H^{+}\to qq/\tau^{+}\nu_{\tau}$ ALEPH:2013htx .

For the case of the 2HDM-II at low $\tan\beta$, $H^{0}\to\gamma\gamma$ and
$H^{+}\to t\bar{b}$ compete to give the most stringent exclusion bounds
ATLAS:2018xad ; ATLAS:2014jdv ; ATLAS:2020jqj , with $H^{+}\to t\bar{b}$ being
more sensitive at higher masses before the production cross sections fall at
high masses. At moderate values of $\tan\beta$, $H^{+}\to\tau^{+}\nu_{\tau}$
excludes masses up to $\sim$ 90 GeV, before the branching ratio for this
channel falls, which leads to the allowed region at $\tan\beta$ of order 1 and
masses of $\sim$ 100 GeV. This region is then ended by the
$H^{0}\to\tau^{+}\tau^{-}$ search in Ref. ATLAS:2020zms . This search also
gives the exclusion up to high masses in the large $\tan\beta$ region,
exceeding the limits from $H^{+}\to t\bar{b}$.

We are able to find lower mass bounds on the new Higgs bosons of 82 GeV and 86
GeV in the Type I and II 2HDM respectively, which are less stringent bounds
than those found from the combined fit to flavour observables in Section 5.2.
These bounds improve at low $\tan\beta$ in both models, and also at high
$\tan\beta$ in the 2HDM-II, owing to the dependence of the couplings of these
models to $\tan\beta$. This is in line with the patterns we see in the flavour
sector, where these couplings are also crucial.

###  6.2 Collider Outlook

Looking to the future, the LHC is due to be upgraded considerably, increasing
the integrated luminosity to $\mathcal{L}_{\text{HL-LHC}}=3$ ab-1. This will
have a sizeable impact, making it significantly harder for new particles to
remain hidden. Here, we extrapolate the LHC data currently in HiggsBounds to
this new luminosity by scaling the limits by a factor
$\sqrt{(\mathcal{L}_{0}/\mathcal{L}_{\text{HL-LHC}})}$ for a search with a
reference luminosity $\mathcal{L}_{0}$. There are important caveats to this
extrapolation. One such caveat is that a number of searches are designed to
precisely measure the SM Higgs behaviour. In these cases we match the bounds
to what would be expected of the SM Higgs, as here we have a 125 GeV scalar in
the exact alignment limit, which exactly matches the phenomenology of the SM
Higgs. We make use of the SM Higgs predictions of Ref.
LHCHiggsCrossSectionWorkingGroup:2016ypw  for this. Additionally, a number of
searches included in HiggsBounds are from LHC run 1, during which the LHC
operated at a centre of mass energy of $\sqrt{s}=7-8$ TeV, whilst the HL-LHC
will operate at $\sqrt{s}=13-14$ TeV. We therefore need to reflect increased
cross sections for these searches together with luminosity improvements (see
also Ref. Basler:2018dac ). We again make use of the results from Ref.
LHCHiggsCrossSectionWorkingGroup:2016ypw  to perform this extrapolation for SM
Higgs searches. For BSM Higgs searches we use MadGraph5_aMC@NLO Alwall:2014hca
to calculate the increase in the production cross sections at the higher
centre of mass energy as a function of the BSM Higgs mass and scale the search
data limits accordingly. We again perform a scan consisting of 50000 random
points, the results of which are shown in Fig. 13, which includes the outline
of the scan in Fig. 12 for ease of comparison.

The permitted parameter space for the 2HDM is reduced at the future collider,
based on extrapolating existing searches. In the 2HDM-I case this effect is
only apparent at low $\tan\beta$, as the sensitivity to new Higgs bosons is
minimal at high $\tan\beta$ because the relevant couplings are proportional to
$\cot^{2}\beta$. The lower limit in this region is from a LEP search
ALEPH:2013htx , which we do not extrapolate here, hence the same cut off is
present here as in Fig. 12. There is no such issue in the 2HDM-II, and we find
that the bounds improve significantly at the HL-LHC, bringing the lower mass
bound into competition with those found from flavour constraints in Section
5.4, save for a small region with masses of $\approx 95$ GeV and
$\tan\beta\approx 2$.

Figure 13: Scan of 50000 randomly generated points compared against LHC data,
extrapolated to an integrated luminosity of 3 ab-1 using 2HDecay and
HiggsBounds, with allowed points shown in blue and the current data exclusion
contour in black. On the left, the scan in the 2HDM-I, on the right the 2HDM-
II.

###  6.3 Comparison with Flavour

In order to compare the bounds found above from collider searches to those we
find from the flavour sector, we overlap results from both sections. No
statistical combination between the two is attempted here as we do not use
HiggsBounds to give a combined exclusion from all search data but only check
if a point is excluded by any one individual search. We again present the
results from the scenario in which all new Higgs bosons have degenerate masses
and the alignment limit is exactly realised, meaning the combination presented
in Fig. 14 takes the results from Fig. 12 for collider results, and from Fig.
10 and the corresponding global fit in Ref. Atkinson:2021eox  for flavour
results, where we simply extend these contours to lower charged Higgs masses
to be more compatible with the collider results.

Figure 14: Combined results from the global flavour fit and collider searches,
with allowed points from collider searches in blue and contours from the
flavour sector at 1,2,3,4,5$\sigma$ confidence, from darkest to lightest, with
2HDM-I results on the left and 2HDM-II on the right.

We find that there is some degree of complementarity between the two sectors
for the types of 2HDM that are examined here. In the 2HDM-I the LEP searches
ALEPH:2013htx  that set a lower mass bound on the new Higgses outperform the
exclusion from flavour observables, which lack sensitivity in the high
$\tan\beta$ region. Conversely, in both cases, the flavour constraints
following from precise measurements are more successful constraining
$\tan\beta$ at at high masses, reflecting the loss of sensitivity in direct
collider searches once the new particles are sufficiently heavy. Looking at
the contours in Fig. 13, we can see that the extrapolation to the HL-LHC
phase, which improves the sensitivity to new physics, enhances the
complementary nature of the results in Fig. 14. In the case of the 2HDM-II the
lower mass bound may exceed that determined from the flavour sector, depending
on how future measurements in that sector line up with the scenarios outlined
in Section 5.4.

##  7 A Note on Cosmological Context

As motivated by previous investigations Dorsch:2014qja ; Dorsch:2016tab ;
Su:2020pjw ; Dorsch:2016nrg ; Wang:2021ayg ; Basler:2016obg ; Dorsch:2017nza ;
Atkinson:2021eox ; Basler:2021kgq , we consider the possibility of generating
a strong first order electroweak phase transition (SFOEWPT) through the 2HDM-I
in order to fulfil the criterion for EW baryogenesis. Scanning large parameter
regions, these find possibilities for a SFOEWPT to be achieved in the 2HDM-I,
however this is when considering lower masses ($\lesssim 1\,$TeV) than we find
are favoured from our fits.

To evaluate the EWPT in the 2HDM-I at chosen benchmark points, we use the
BSMPT package Basler:2018cwe ; Basler:2020nrq , where we refer to the
package’s documentation666https://phbasler.github.io/BSMPT/ for further
information on the package, and we make use of it as discussed in Section 7.3
of Ref. Atkinson:2021eox . In Table 5 we present the results for the strength
of the EWPT at our selected parameter points. We first choose a benchmark of
$M=50\,$TeV as a test of extreme masses, and then consider points motivated by
our global fits in Table 3 as well as some variation in $\tan\beta$. As the
high masses of our best fits are suggested to be above the limit to achieve a
SFOEWPT, we also consider a range of points with masses below $1\,$TeV,
including the best fit point for the $b\to s\ell\ell$ observables. The
alignment limit, $\cos(\beta-\alpha)=0$, is taken for all benchmark points.

$\tan\beta$ | 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)
$11.4$ | $50000$ | $50000$ | $50000$ | $0.26$ | $0$ | $0$ | $3.1\times 10^{7}$ | $0.58$ | $164$ | $0.004$
$80$ | $1750$ | $1750$ | $1750$ | $0.26$ | $0$ | $0$ | $3.8\times 10^{4}$ | $23$ | $162$ | $0.14$
$50$ | $1810$ | $1750$ | $1760$ | $7.3$ | $-6.5$ | $-0.6$ | $6.1\times 10^{4}$ | $26$ | $174$ | $0.15$
$10$ | $1810$ | $1750$ | $1760$ | $7.3$ | $-6.5$ | $-0.6$ | $3.0\times 10^{5}$ | $26$ | $174$ | $0.15$
$150$ | $1810$ | $1750$ | $1760$ | $7.3$ | $-6.5$ | $-0.6$ | $2.0\times 10^{4}$ | $26$ | $174$ | $0.15$
$35$ | $1020$ | $960$ | $970$ | $4.2$ | $-3.6$ | $-0.3$ | $2.6\times 10^{4}$ | $24$ | $169$ | $0.14$
$80$ | $860$ | $710$ | $860$ | $8.0$ | $-3.9$ | $-3.9$ | $6.3\times 10^{3}$ | $142$ | $174$ | $0.82$
$80$ | $860$ | $690$ | $860$ | $9.0$ | $-4.3$ | $-4.3$ | $6.0\times 10^{3}$ | $177$ | $174$ | $1.02$
$80$ | $680$ | $470$ | $680$ | $8.2$ | $-4.0$ | $-4.0$ | $2.8\times 10^{3}$ | $211$ | $147$ | $1.43$
$80$ | $570$ | $320$ | $570$ | $7.6$ | $-3.7$ | $-3.7$ | $1.3\times 10^{3}$ | $226$ | $125$ | $1.81$
$80$ | $490$ | $250$ | $490$ | $6.1$ | $-2.9$ | $-2.9$ | $7.8\times 10^{2}$ | $207$ | $126$ | $1.65$
$80$ | $490$ | $490$ | $490$ | $0.26$ | $0$ | $0$ | $3.0\times 10^{3}$ | $23$ | $161$ | $0.14$
Table 5:  Table of results for the EWPT in the 2HDM-I.
$\xi_{c}=\omega_{c}/T_{c}$ is the parameterisation of the strength of the
EWPT, with $\omega_{c}$ the high-temperature VEV and $T_{c}$ the critical
temperature. A SFOEWPT is indicated by $\xi_{c}>1$. In the limits chosen in
Ref. Atkinson:2021eox  (and similarly discussed in Ref. BhupalDev:2014bir ),
$\lambda_{1}=\lambda_{2}=m_{h^{0}}^{2}/v^{2}=0.26$ for all benchmark points,
and so are not explicitly shown in the table. Additionally, $\lambda_{3,4,5}$
are not independent, but
$\lambda_{3}+\lambda_{4}+\lambda_{5}=m_{h^{0}}^{2}/v^{2}=0.26$.

The results from these benchmark points are similar to those of the 2HDM-II.
We find that the high mass scales around or above our best fit point (also
enforcing close mass degeneracy) limit the strength of the phase transition.
Sufficiently below our best fit points, where theoretical considerations now
allow sampling with mass differences similar to those favoured in Ref.
Su:2020pjw , we now find scenarios which support a SFOEWPT. In contrast to the
2HDM-II, these points with $\xi_{c}\geq 1$ are each within $1\sigma$
confidence of our best fit and are allowed by direct search data. In these
scenarios, the global fit performs only slightly worse than the best fit
points described in Table 3 with improvement over the SM at $\sim 1.5\sigma$.

##  8 Conclusions

The search for new physics beyond the Standard Model is key to improving our
understanding of physics at the smallest distances. Searches at the LHC for
new interactions and states have so far not revealed any significant deviation
from the SM expectation, while the precision of flavour and electroweak
precision measurements corner the SM from different directions with some
potential hints for LFU violation.

In this work we have performed a detailed investigation and comparison of the
flavour and LHC exotics measurement programme, alongside its future
extrapolation. We find that both flavour and current exotics searches can be
significantly improved with increasing luminosity, thus allowing the
experimental collaborations to cover a significant proportion of parameter
space that is currently unconstrained.

Extending the results of Ref. Atkinson:2021eox  for 2HDM-II constraints from
indirect searches using electroweak precision, Higgs signal strengths, and
flavour observables to the type I 2HDM, we find that the 2HDM-I statistically
outperforms both the 2HDM-II and the SM in fits to the data. The best fit
point for the 2HDM-I lies around

$\displaystyle m_{H^{+}}\approx m_{H^{0}}$ $\displaystyle\approx
m_{A^{0}}\approx 1.7\,\mbox{TeV}\,,$ (13) $\displaystyle\tan\beta\approx 70,$
$\displaystyle\,\cos(\beta-\alpha)\approx-0.001\,.$

The parameters of the 2HDM-I are found however to be much less constrained
than in the 2HDM-II. An upper bound on the charged Higgs mass is only found at
$1\sigma$ of $83\,$TeV, where for masses at or above $\sim 1\,$TeV, the
additional Higgses are expectedly to be closely degenerate. For a lower
charged Higgs mass, $\tan\beta$ is strongly constrained from below to be
${\cal O}(1)$ or higher, however this constraint is lessened with increasing
charged Higgs mass. The alignment limit, $\cos(\beta-\alpha)=0$, is favoured
by our fits, however larger deviations than in the 2HDM-II are allowed, up to
$|\cos(\beta-\alpha)|=0.4$ at $5\sigma$.

Expecting an increase in precision of flavour measurements from future
colliders, we consider our flavour fits in several future measurements’
projections, where these predictive fits encourage consideration of other
models instead of (or in addition to) a 2HDM of type I/II to improve upon the
SM.

Furthermore, we consider the implications of these flavour fits on the
electroweak phase transition of the early universe. It is found that the high
masses encouraged by our fits do not support a strong EWPT for baryogenesis,
however due to the 2HDM-I being less constrained than the 2HDM-II, sufficently
low masses are possible within $1\sigma$ of our best fit.

Using data from direct searches for new fundamental particles at colliders we
have been able to exclude large regions of the parameter space for the 2HDM of
types I and II, highlighting possible areas to search for possible new
physics. Looking forward, we have extrapolated a significant amount of data to
match the expected performance of the HL-LHC and find that this further
improves the constraints on new physics signals, to a level that can be
competitive with indirect results from the flavour sector.

While some complementarity between flavour, Higgs physics and direct exotics
searches remains, the finite energy coverage of exotics searches typically
means a loss of direct LHC sensitivity for large masses approaching the
decoupling limit.

### Acknowledgements

O.A. is funded by a STFC studentship under grant ST/V506692/1. The work of
M.B. is supported by Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) through TRR 257 “Particle Physics Phenomenology after the Higgs
Discovery”. C.E. is supported by the STFC under grant ST/T000945/1, the
Leverhulme Trust under grant RPG-2021-031, and the Institute for Particle
Physics Phenomenology Associateship Scheme. Computations carried out for this
work made use of the OMNI cluster of the University of Siegen.

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