###### Contents

  1. 1 Introduction
  2. 2 The 2HDM Lagrangian
    1. 2.1 The low-energy effective description
  3. 3 Model predictions and numerical Analysis
    1. 3.1 Flavour Observables
    2. 3.2 Electron electric dipole moment
    3. 3.3 Collider Observables
    4. 3.4 Triple Higgs Coupling
    5. 3.5 $m_{W}$ from CDF II
  4. 4 Conclusions
  5. A From the $Z_{2}$-original basis to the Higgs basis

##  1 Introduction

After the July 4th 2012 announcement [1, 2] of the discovery of a resonance
compatible with the Standard Model (SM) Higgs boson, no other particle has
been detected. On the other hand, experimental data from the last decade is
being used to increase the precision of our knowledge on many different
observables, resulting in very accurate indirect searches.

In the absence of a specific New Physics (NP), one approach to studying
possible deviations from the SM predictions is the Standard Model Effective
Field Theory (SMEFT) [3, 4] that is a basis of independent operators, ordered
in terms of mass dimensions, invariant under the SM gauge symmetry and
constructed out of the SM fields 111As the nature of the Higgs field is still
uncertain up to a certain degree, alternative effective descriptions have been
proposed [5, 6, 7, 8, 9, 10, 11, 12, 13, 14], providing different expectations
in certain observables.. The hope is to extract information on the ultraviolet
(UV) completion by identifying specific patterns and synergies among the SMEFT
operators.

Even restricting to the subgroup with mass dimension $d=6$, the number of such
operators is tremendous, 2499, considering all the flavour contractions [15].
Although it is possible to perform global fits on a reduced subset of
operators under certain reasonable assumptions, no clear evidence of a
specific NP has emerged (see for example Ref. [16] for a recent fit to the
electroweak Higgs data). This occurs despite the rapport on some apparent
anomalies in particle physics. To be mentioned is the new measurement from the
CDF II collaboration of the $W$-gauge boson mass [17], that exceeds the SM
prediction by approximately $7\sigma$ (but not confirmed by the last ATLAS
result [18]): assuming that the CDF II result is correct, different
explanations have been proposed to account for the deviation from the SM
prediction, from tree-level modification of the muon decay induced by exotic
leptons (see for example Refs. [19, 20, 21]), to loop quantum corrections to
the two-point functions of the SM gauge bosons (see for example Refs. [22,
23]). Another example is the charged $B$-anomalies, hinting to lepton flavour
universality violation in the $B$-meson semi-leptonic decays $B\rightarrow
D^{(\ast)}l\nu$: the experimental results, combining data from the BaBar [24,
25] and Belle [26, 27, 28] collaborations, exceeds the SM prediction by
$3.5\sigma$ [29]. Recently, the LHCb collaboration [30] announced its results
on this observable, lowering the deviation of the world average to the
$3\sigma$ level. Considering that other apparent discrepancies in the
$B$-decays have been clarified by the LHCb collaboration [31, 32],
explanations of the charged $B$-anomalies in terms of exotic charged gauge
bosons (see e.g. Ref. [33]) or a charged Higgs boson (see e.g. Refs. [34, 35,
36]) or leptoquarks (see e.g. Refs. [37, 38, 39]) appear much less plausible,
even if still possible.

This bottom-up approach has been largely adopted in the recent literature and,
given the lack of clear indications towards a specific NP, we will abandon it
here. Instead, we will assume a concrete extension of the SM and analyse its
effects in the low-energy effective Lagrangian approximation, with the aim of
exploring the associated phenomenology in the reduced NP parameter space. Our
choice is the two Higgs doublet models (2HDMs) [40, 41, 42, 43, 44, 45, 46,
47]. Although such extensions of the SM do not address the most pressing
questions like the origin of the active neutrino masses, the existence of dark
matter and the baryon asymmetry of the Universe, they are nevertheless a
useful theoretical laboratory. They permit to investigate the room left by the
present flavour and Higgs sector experimental data for NP in the scalar
potential and in the effective Yukawa couplings. They are also interesting in
the context of supersymmetric extensions of the SM.

Most of the analyses present in the literature in the framework of the EFT
approach treat data on the electroweak (EW) and Higgs sectors separately from
flavour data. Only a few examples appeared attempting to merge all the
information, although at the cost of assuming specific flavour structures.
This is the case of Refs. [48, 49, 50] and subsequent Refs. [51, 52], where
the generic conclusion is that flavour data and EW/Higgs data are close to
being comparable to each other in their sensitivity to NP effects. In this
paper, we will extend upon the study in Refs. [49, 50] based on the Minimal
Flavour Violation (MFV) ansatz [53, 54, 55, 56, 57, 58] and on the Froggatt-
Nielsen (FN) mechanism [59], two very well-known contexts in the literature,
see Refs. [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76,
77] and [78, 79, 80, 81, 82], respectively. The MFV and the FN represent two
extreme cases: the fermion-Higgs couplings are flavour conserving or largely
violating it, respectively; moreover, there is only one universal CP violating
phase for each fermion sector in the MFV case, while several phases are
present in the FN one. Our framework is the effective Lagrangian derived from
$Z_{2}$-symmetric versions of the 2HDMs, where the role of the $Z_{2}$
symmetry is to ensure protection from tree-level flavour changing neutral
current (FCNC) contributions [41], but still allowing effects in flavour
changing processes due to the presence of charged Higgses. Thus, they
represent a middle ground between the two aforementioned constructions based
on flavour symmetries. Furthermore, as we will discuss, in the effective
Yukawa couplings there is only one universal CP violating phase for all
fermions of the SM.

Our analysis is valid near the decoupling limit, which will be defined in
Sect. 2, of all the heavy scalar states. This hypothesis is motivated by the
absence of any discovery of additional scalars and we will assume them to be
heavy enough to justify an effective description. In consequence, all the
heavy scalars are almost degenerate in mass, so their low-energy effects are
to a good approximation controlled by only one mass scale. This last condition
will be relaxed when discussing the possible explanation of the CDF II
measurement of the $W$ mass.

The bulk of our analysis depends on only three new NP parameters, that is the
common mass of the heavier scalar degrees of freedom, one coupling in the
scalar potential and $\tan\beta$, characteristic of the 2HDM constructions.
The predictive power is therefore high and we make use of it providing the
expected maximum deviation in the triple Higgs coupling from its SM value for
each $Z_{2}$-symmetric 2HDM realisation.

The structure of the paper can be read in the table of Contents.

##  2 The 2HDM Lagrangian

The general 2HDM and its $Z_{2}$ symmetric versions have been extensively
discussed in the literature (see for example Refs. [83, 84, 85, 86, 87, 88,
89, 90, 91]). The main motivation for imposing a $Z_{2}$ symmetry on a 2HDM is
to forbid tree-level FCNC contributions. For the sake of easy reference to the
$Z_{2}$ symmetric versions and for fixing the notation used throughout the
paper, we collect here (and in the App. A) their key ingredients.

A discrete Abelian $Z_{2}$ symmetry quantum numbers are assigned to the two
Higgs doublets $\Phi_{1}$ and $\Phi_{2}$, even and odd under $Z_{2}$
respectively, and to the fermion fields. The assignment of the $Z_{2}$ quantum
numbers identifies the original basis for those fields. Focusing first on the
scalar potential of the model, the $Z_{2}$ symmetry restricts its form (see
App. A), and the vacuum of the model is fixed in terms of the mass parameters
and the couplings to be:

$\langle\Phi_{1}^{\dagger}\Phi_{1}\rangle=\dfrac{v_{1}^{2}}{2}\,,\qquad\langle\Phi_{2}^{\dagger}\Phi_{2}\rangle=\dfrac{v_{2}^{2}}{2}\,,\qquad\langle\Phi_{1}^{\dagger}\Phi_{2}\rangle=\dfrac{v_{12}^{2}}{2}\,,\qquad
v_{1}^{2}+v_{2}^{2}=v^{2}\,$ (2.1)

where $v_{1}$ and $v_{2}$ are real, $v_{12}$ is in general complex, and
$v=246\ \text{GeV}$. Moreover, the CP violating phase of the mixed Higgs
condensate we denote as

$\xi\equiv\arg\left(\Phi_{1}^{\dagger}\Phi_{2}\right)\,.$ (2.2)

It is convenient to rewrite the scalar potential in the Higgs basis where only
one of the two Higgses acquires a VEV. Performing $U(2)$ rotations of the
Higgs fields in a two-dimensional complex plane by the angle $\beta$ defined
as

$\tan\beta=\frac{v_{2}}{v_{1}}\,,$ (2.3)

one obtains:

$\displaystyle V(H_{1},H_{2})=\,$
$\displaystyle\widetilde{m}_{1}^{2}H_{1}^{\dagger}H_{1}+\widetilde{m}_{2}^{2}H_{2}^{\dagger}H_{2}-\left(\widetilde{m}_{12}^{2}H_{1}^{\dagger}H_{2}+\text{h.c.}\right)+\dfrac{1}{2}\widetilde{\lambda}_{1}\left(H_{1}^{\dagger}H_{1}\right)^{2}+$
$\displaystyle+\dfrac{1}{2}\widetilde{\lambda}_{2}\left(H_{2}^{\dagger}H_{2}\right)^{2}+\widetilde{\lambda}_{3}\left(H_{1}^{\dagger}H_{1}\right)\left(H_{2}^{\dagger}H_{2}\right)+\widetilde{\lambda}_{4}\left(H_{1}^{\dagger}H_{2}\right)\left(H_{2}^{\dagger}H_{1}\right)+$
(2.4)
$\displaystyle+\left[\dfrac{1}{2}\widetilde{\lambda}_{5}\left(H_{1}^{\dagger}H_{2}\right)^{2}+\widetilde{\lambda}_{6}\left(H_{1}^{\dagger}H_{1}\right)\left(H_{1}^{\dagger}H_{2}\right)+\widetilde{\lambda}_{7}\left(H_{2}^{\dagger}H_{2}\right)\left(H_{1}^{\dagger}H_{2}\right)+\text{h.c.}\right]\,.$

The expressions for the tilde parameters in terms of the parameters defined in
the original basis are given in the App. A. The parameters
$\widetilde{m}_{1}^{2}$, $\widetilde{m}_{2}^{2}$, $\widetilde{\lambda}_{1}$,
$\widetilde{\lambda}_{2}$, $\widetilde{\lambda}_{3}$ and
$\widetilde{\lambda}_{4}$ are real, while $\widetilde{m}_{12}^{2}$,
$\widetilde{\lambda}_{5}$, $\widetilde{\lambda}_{6}$ and
$\widetilde{\lambda}_{7}$ are in general complex. However, as discussed in
App. A, for the $Z_{2}$ symmetric models there are only two independent CP-
violating phases:

$\theta_{1}\equiv\dfrac{1}{2}\arg{(\widetilde{\lambda}_{6}^{2}\widetilde{\lambda}_{5}^{\ast})}\,,\qquad\qquad\theta_{2}\equiv\dfrac{1}{2}\arg{(\widetilde{\lambda}_{7}^{2}\widetilde{\lambda}_{5}^{\ast})}\,.$
(2.5)

Enforcing the condition that

$\langle H^{\dagger}_{1}H_{1}\rangle=\dfrac{v^{2}}{2}\,,\qquad\qquad\langle
H^{\dagger}_{2}H_{2}\rangle=0\,,$ (2.6)

leads to two relations between four of the parameters,

$\widetilde{m}_{1}^{2}=-\dfrac{1}{2}\widetilde{\lambda}_{1}v^{2}\,,\qquad\qquad\widetilde{m}_{12}^{2}=-\dfrac{1}{2}\widetilde{\lambda}_{6}v^{2}\,,$
(2.7)

reducing the number of independent parameters.

In the vacuum defined by the scalar potential, five degrees of freedom acquire
masses, while the remaining three remain massless, corresponding to the three
longitudinal components of the EW gauge bosons. The two charged Higgs bosons
$H^{\pm}$ receive a mass that reads

$m_{H_{\pm}}^{2}=\widetilde{m}_{2}^{2}+\dfrac{1}{2}\widetilde{\lambda}_{3}v^{2}\,,$
(2.8)

while a mixing is present in the neutral sector and the corresponding mass
eigenvalues can be identified with the SM Higgs boson $h$, a heavier CP-even
Higgs $H$, and a heavier CP-odd Higgs $A$. At the second order in the
expansion $v^{2}/\widetilde{m}_{2}^{2}$ their masses read

$\displaystyle m_{h}^{2}$
$\displaystyle=v^{2}\left\\{\widetilde{\lambda}_{1}-\left|\widetilde{\lambda}_{6}\right|^{2}\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}-\dfrac{1}{2}\left[\left(2\widetilde{\lambda}_{1}-\widetilde{\lambda}_{3}-\widetilde{\lambda}_{4}\right)\left|\widetilde{\lambda}_{6}\right|^{2}-\dfrac{1}{2}\left(\widetilde{\lambda}_{5}^{\ast}\widetilde{\lambda}_{6}^{2}+\text{h.c.}\right)\right]\left(\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\right)^{2}\right\\}\,,$
$\displaystyle m_{A,H}^{2}$
$\displaystyle=\widetilde{m}_{2}^{2}\left\\{1+\dfrac{1}{2}\left(\widetilde{\lambda}_{3}+\widetilde{\lambda}_{4}\mp\left|\widetilde{\lambda}_{5}\right|\right)\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}+\dfrac{1}{2}\left|\widetilde{\lambda}_{6}\right|^{2}\Big{(}1\mp\cos{(2\theta_{1})}\Big{)}\left(\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\right)^{2}\right\\}\,.$
(2.9)

Notice that the mass splitting between the two heavy neutral states is
determined by $\left|\widetilde{\lambda}_{5}\right|$:

$m_{H}^{2}-m_{A}^{2}=\left(\left|\widetilde{\lambda}_{5}\right|+\left|\widetilde{\lambda}_{6}\right|^{2}\cos(2\theta_{1})\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\right)v^{2}\,,$
(2.10)

neglecting higher order terms in $v^{2}/\widetilde{m}_{2}^{2}$. Notice that in
the near the decoupling limit defined as [92, 93]

$\widetilde{m}_{2}^{2}\gg\widetilde{\lambda}_{i}v^{2}\,,$ (2.11)

all the heavy states become degenerate with the mass converging to
$\widetilde{m}_{2}$. Non-degenerate effects have been studied at length in the
literature, see for example Refs. [94, 95].

Special attention has been devoted in the literature to the mixing between the
two CP-even scalars, customary defined as $\cos(\alpha-\beta)$, being $\alpha$
a rotation angle. This is refereed to as the alignment parameter [96], such
that when it vanishes, $\cos(\alpha-\beta)=0$, the $125\ \text{GeV}$ Higgs
corresponds to the direction of the VEV in the field space, that identifies
the alignment limit. Expressing this mixing parameter in an expansion in
$v^{2}/\widetilde{m}_{2}^{2}$, we find

$\cos(\beta-\alpha)=-|\widetilde{\lambda}_{6}|\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}+\mathcal{O}\left(\dfrac{v^{4}}{\widetilde{m}_{2}^{4}}\right)\,.$
(2.12)

The SM fermion couplings to the Higgs doublets $\Phi_{1}$ and $\Phi_{2}$ are
also controlled by their $Z_{2}$ charges. Any tree-level FCNC contribution
from the scalar sector is avoided if each sector of fermions, up, down, and
leptons, couples to only one of the doublets. Then the Yukawa couplings of all
three neutral Higgs mass eigenstates remain aligned to the fermion mass
matrices and are simultaneously diagonalised. There are four possible $Z_{2}$
charge assignments to fermions that ensure such a solution to the FCNC problem
in the 2HDMs. They are summarised as follows:

Model | $\Phi_{1}$ | $\Phi_{2}$ | $Q_{L}$ | $u_{R}$ | $d_{R}$ | $L_{L}$ | $e_{R}$
---|---|---|---|---|---|---|---
Type I | $+$ | $-$ | $+$ | $-$ | $-$ | $+$ | $-$
Type II | $+$ | $-$ | $+$ | $-$ | $+$ | $+$ | $+$
Type III (X) | $+$ | $-$ | $+$ | $-$ | $-$ | $+$ | $+$
Type IV (Y) | $+$ | $-$ | $+$ | $-$ | $+$ | $+$ | $-$

Although the introduction of the $Z_{2}$ symmetry guarantees that, not only
the lightest Higgs $h$, but also the heavier CP-even $H$ and CP-odd $A$
neutral scalars do not give rise to FCNC contributions, there are NP
contributions to processes with flavour changing mediated by the charged
Higgses $H^{\pm}$.

As no other scalar beyond the $125\ \text{GeV}$ one has been discovered so
far, we assume that the heavier Higgses have sufficiently large masses to
escape direct detection at present colliders. Their existence, however, may be
revealed by studying deviations from the SM predictions of various observables
at low-energies. For this reason, in the next section, we will introduce the
effective description of the $Z_{2}$-symmetric 2HD models near the decoupling
limit. In this limit, all the heavier scalars will be taken as degenerate in
mass: we will comment later on some effects associated with the departure from
this condition.

###  2.1 The low-energy effective description

Integrating out the heavy scalar degrees of freedom leads to a low-energy
effective description: near the decoupling limit, as defined in Eq. (2.11), it
has already been presented in Ref. [88]. We will report here a few results
relevant to our analysis.

The effective Lagrangian involving the lightest Higgs $h$ can be written as
the sum of several different parts, in addition to the canonically normalised
kinetic terms,

$\mathscr{L}_{h}^{\text{eff}}=\mathscr{L}^{\text{eff}}_{VV}+\mathscr{L}^{\text{eff}}_{Y}-V^{\text{eff}}\,.$
(2.13)

The mass terms and the couplings of the massive gauge bosons with $h$ can be
written as

$\displaystyle\mathscr{L}^{\text{eff}}_{VV}\supset$
$\displaystyle\,\,\dfrac{1}{2}Z_{\mu}Z^{\mu}\left(m_{Z}^{2}+g_{hZZ}\,v\,h+g_{h^{2}ZZ}\,h^{2}+\ldots\right)+$
(2.14)
$\displaystyle+W^{+}_{\mu}W^{-\mu}\left(m_{W}^{2}+g_{hWW}\,v\,h+g_{h^{2}WW}\,h^{2}\ldots\right)\,,$

where the dots refer to interactions with a higher number of $h$. The $Z$ and
$W$ masses, $m_{Z}^{2}$ and $m_{W}^{2}$, at tree level coincide with the SM
expressions,

$m_{W}=\dfrac{g\,v}{2}\,,\qquad\qquad m_{Z}=\dfrac{\sqrt{g^{2}+g^{\prime
2}}\,v}{2}\,,$ (2.15)

where $g$ and $g^{\prime}$ are the gauge coupling constants for the
$SU(2)_{L}$ and $U(1)_{Y}$ symmetries, respectively. The triple and quartic
couplings turn out to be

$\displaystyle
g_{hVV}=\dfrac{2\,m^{2}_{V}}{v^{2}}\left[1-\dfrac{1}{2}\left|\widetilde{\lambda}_{6}\right|^{2}\left(\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\right)^{2}\right]\equiv\dfrac{2\,m^{2}_{V}}{v^{2}}\kappa_{V}\,,$
(2.16) $\displaystyle
g_{h^{2}VV}=\dfrac{m^{2}_{V}}{v^{2}}\left[1-3\left|\widetilde{\lambda}_{6}\right|^{2}\left(\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\right)^{2}\right]\,,$

where, we have introduced in the first equation the definition of $\kappa_{V}$
to match the notation used in the literature for the deviations from the SM
predictions, that are proportional to $|\widetilde{\lambda}_{6}|^{2}$ and only
appear at the next-to-leading (NLO) order of the expansion in
$v^{2}/{\widetilde{m}}^{2}_{2}$.

Moving to the Higgs-fermion effective couplings, the dominant corrections
arise at the leading order (LO) and are also proportional to
$\widetilde{\lambda}_{6}$. We adopt the $\kappa$-formalism for the Yukawa
interactions [97, 98, 99],

$-\mathscr{L}^{\text{eff}}_{Y}\supset
M_{f}\overline{f}f+\dfrac{M_{f}}{v}\,h\,\left(\kappa_{f}\,\overline{f}f+\widetilde{\kappa}_{f}\,\overline{f}i\gamma_{5}f\right)+\ldots\,,$
(2.17)

where the dots refer again to interactions with a higher number of $h$, and
$M_{f}$ stand for the fermion mass matrix in the flavour basis (the flavour
contractions are left implicit). We can distinguish the CP-even and the CP-odd
contributions, $\kappa_{f}$ and $\widetilde{\kappa}_{f}$, respectively, as
follows:

$\displaystyle\kappa_{u}=$
$\displaystyle\kappa_{d}=\kappa_{e}=1-\zeta_{f}\,{\texttt{Re}}\left[\widetilde{\lambda}_{6}^{\ast}\,e^{-i\xi/2}\right]\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\equiv
1-\zeta_{f}\left|\widetilde{\lambda}_{6}\right|\cos{(\rho)}\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\,,$
(2.18) $\displaystyle\widetilde{\kappa}_{u}=$
$\displaystyle\widetilde{\kappa}_{d}=\widetilde{\kappa}_{e}=-\zeta_{f}\,{\texttt{Im}}\left[\widetilde{\lambda}_{6}^{\ast}\,e^{-i\xi/2}\right]\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\equiv-\zeta_{f}\,\left|\widetilde{\lambda}_{6}\right|\sin{(\rho)}\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\,,$

where
$\rho\equiv\arg\left[\widetilde{\lambda}_{6}^{\ast}\,e^{-i\xi/2}\right]$, and
the $\zeta_{f}$ parameters indicates the type of 2HDM under consideration,

| Type I | Type II | Type III (X) | Type IV (Y)
---|---|---|---|---
$\zeta_{u}$ | $\cot\beta$ | $\cot\beta$ | $\cot\beta$ | $\cot\beta$
$\zeta_{d}$ | $\cot\beta$ | $-\tan\beta$ | $\cot\beta$ | $-\tan\beta$
$\zeta_{e}$ | $\cot\beta$ | $-\tan\beta$ | $-\tan\beta$ | $\cot\beta$

It is worth pointing out that, within any of the 2HDM types, the NP
corrections to the Yukawa couplings are, not only flavour blind, but also
universal, up to $\tan\beta$ dependent coefficients. This is a stronger result
than that obtained in the MFV case [49, 50] (or in any other flavour-based
context), where the NP deviations are aligned with the SM Yukawas, but are
different for up-type quarks, down-type quarks and charged leptons. This has a
particular impact for CP violating observables, as we will discuss in the next
section.

Finally, the effective scalar potential can be written as

$V^{\text{eff}}=\dfrac{1}{2}m_{h}^{2}h^{2}+\dfrac{g_{h^{3}}}{3!}\,v\,h^{3}+\dfrac{g_{h^{4}}}{4!}h^{4}+\ldots\,,$
(2.19)

where the dots refer again to interactions with a higher number of $h$. The
$m_{h}^{2}$ is defined in Eq. (2.9) and, up to the second order in the
expansion $v^{2}/\widetilde{m}_{2}^{2}$, the cubic and quartic self-couplings
read

$\displaystyle g_{h^{3}}$
$\displaystyle=\dfrac{3\,m_{h}^{2}}{v^{2}}-6\,\left|\widetilde{\lambda}_{6}\right|^{2}\,\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}-\dfrac{1}{2}\left\\{\left[21\,\widetilde{\lambda}_{1}-12\left(\widetilde{\lambda}_{3}+\widetilde{\lambda}_{4}\right)\right]\left|\widetilde{\lambda}_{6}\right|^{2}-\left(6\,\widetilde{\lambda}_{5}\widetilde{\lambda}_{6}^{\ast
2}+\text{h.c.}\right)\right\\}\left(\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\right)^{2}\,,$
(2.20) $\displaystyle g_{h^{4}}$
$\displaystyle=\dfrac{3\,m_{h}^{2}}{v^{2}}-36\,\left|\widetilde{\lambda}_{6}\right|^{2}\,\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}-\left\\{\left[105\,\widetilde{\lambda}_{1}-60\left(\widetilde{\lambda}_{3}+\widetilde{\lambda}_{4}\right)\right]\left|\widetilde{\lambda}_{6}\right|^{2}-\left(30\,\widetilde{\lambda}_{5}\widetilde{\lambda}_{6}^{\ast
2}+\text{h.c.}\right)\right\\}\left(\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\right)^{2}\,.$

The dominant corrections arise at the LO and depend only on
$|\widetilde{\lambda}_{6}|^{2}$.

Besides the effective Lagrangian involving the lightest Higgs $h$, indirect
signals of the presence of the heavier Higgses manifest themselves in the four
fermion operators. Only six operators of the SMEFT Lagrangian receive
contributions in the 2HDM setup at mass dimension $d=6$: according to the
notation in Ref. [4], those are $Q^{(1)}_{qu}$, $Q^{(1)}_{qd}$, $Q^{(1)}_{\ell
e}$, $Q_{\ell edq}$, $Q^{(1)}_{quqd}$ and $Q^{(1)}_{lequ}$. We can write these
interactions after the EW symmetry breaking as the following effective
Lagrangian operators:

$\displaystyle\mathscr{L}_{4f}^{\text{eff}}=$ $\displaystyle
c_{ijrs}^{qu(0)}\overline{u_{L_{i}}}u_{R_{j}}\overline{u_{R_{r}}}u_{L_{s}}+c_{ijrs}^{qu(\pm)}\overline{d_{L_{i}}}u_{R_{j}}\overline{u_{R_{r}}}d_{L_{s}}+$
(2.21)
$\displaystyle+c_{ijrs}^{qd(0)}\overline{d_{L_{i}}}d_{R_{j}}\overline{d_{R_{r}}}d_{L_{s}}+c_{ijrs}^{qd(\pm)}\overline{u_{L_{i}}}d_{R_{j}}\overline{d_{R_{r}}}u_{L_{s}}+$
$\displaystyle+c_{ijrs}^{\ell
e(0)}\overline{e_{L_{i}}}e_{R_{j}}\overline{e_{R_{r}}}e_{L_{s}}+c_{ijrs}^{\ell
e(\pm)}\overline{\nu_{L_{i}}}e_{R_{j}}\overline{e_{R_{r}}}\nu_{L_{s}}+$
$\displaystyle+\Bigg{[}c_{ijrs}^{\ell
edq(0)}\overline{e_{L_{i}}}e_{R_{j}}\overline{d_{R_{r}}}d_{L_{s}}+c_{ijrs}^{\ell
edq(\pm)}\overline{\nu_{L_{i}}}e_{R_{j}}\overline{d_{R_{r}}}u_{L_{s}}+$
$\displaystyle\hskip
19.91692pt+c_{ijrs}^{quqd(0)}\overline{u_{L_{i}}}u_{R_{j}}\overline{d_{L_{r}}}d_{R_{s}}+c_{ijrs}^{quqd(\pm)}\overline{d_{L_{i}}}u_{R_{j}}\overline{u_{L_{r}}}d_{R_{s}}+$
$\displaystyle\hskip 19.91692pt+c_{ijrs}^{\ell
equ(0)}\overline{e_{L_{i}}}e_{R_{j}}\overline{u_{L_{r}}}u_{R_{s}}+c_{ijrs}^{\ell
equ(\pm)}\overline{\nu_{L_{i}}}e_{R_{j}}\overline{d_{L_{r}}}u_{R_{s}}+\text{h.c.}\Bigg{]}\,,$

where the associated dimensional Wilson coefficients read

$\displaystyle c_{ijrs}^{qu(0)}$
$\displaystyle=\delta_{ij}\delta_{rs}\dfrac{\widehat{M}_{u_{j}}\widehat{M}_{u_{s}}}{2v^{2}}\dfrac{\zeta_{u}^{2}}{\widetilde{m}_{2}^{2}}\,,\qquad\qquad
c_{ijrs}^{qu(\pm)}=\dfrac{V^{\ast}_{ji}\widehat{M}_{u_{j}}\widehat{M}_{u_{r}}V_{rs}}{2v^{2}}\dfrac{\zeta_{u}^{2}}{\widetilde{m}_{2}^{2}}\,,$
(2.22) $\displaystyle c_{ijrs}^{qd(0)}$
$\displaystyle=\delta_{ij}\delta_{rs}\dfrac{\widehat{M}_{d_{j}}\widehat{M}_{d_{s}}}{2v^{2}}\dfrac{\zeta_{d}^{2}}{\widetilde{m}_{2}^{2}}\,,\qquad\qquad
c_{ijrs}^{qd(\pm)}=\dfrac{V_{ij}\widehat{M}_{d_{j}}\widehat{M}_{d_{r}}V^{\ast}_{sr}}{2v^{2}}\dfrac{\zeta_{d}^{2}}{\widetilde{m}_{2}^{2}}\,,$
$\displaystyle c_{ijrs}^{\ell e(0)}$ $\displaystyle=c_{ijrs}^{\ell
e(\pm)}=\delta_{ij}\delta_{rs}\dfrac{\widehat{M}_{\ell_{j}}\widehat{M}_{\ell_{s}}}{2v^{2}}\dfrac{\zeta_{\ell}^{2}}{\widetilde{m}_{2}^{2}}\,,$
$\displaystyle c_{ijrs}^{\ell edq(0)}$
$\displaystyle=2\delta_{ij}\delta_{rs}\dfrac{\widehat{M}_{\ell_{j}}\widehat{M}_{d_{s}}}{v^{2}}\dfrac{\zeta_{\ell}\,\zeta_{d}}{\widetilde{m}_{2}^{2}}\,,\qquad\qquad
c_{ijrs}^{\ell
edq(\pm)}=2\delta_{ij}\dfrac{\widehat{M}_{\ell_{j}}\widehat{M}_{d_{r}}V^{\ast}_{sr}}{v^{2}}\dfrac{\zeta_{\ell}\,\zeta_{d}}{\widetilde{m}_{2}^{2}}\,,$
$\displaystyle c_{ijrs}^{quqd(0)}$
$\displaystyle=2\delta_{ij}\delta_{rs}\dfrac{\widehat{M}_{u_{j}}\widehat{M}_{d_{s}}}{v^{2}}\dfrac{\zeta_{u}\,\zeta_{d}}{\widetilde{m}_{2}^{2}}\,,\qquad\qquad
c_{ijrs}^{quqd(\pm)}=-2\dfrac{V^{\ast}_{ji}\widehat{M}_{u_{j}}V_{rs}\widehat{M}_{d_{s}}}{v^{2}}\dfrac{\zeta_{u}\,\zeta_{d}}{\widetilde{m}_{2}^{2}}\,,$
$\displaystyle c_{ijrs}^{\ell equ(0)}$
$\displaystyle=2\delta_{ij}\delta_{rs}\dfrac{\widehat{M}_{\ell_{j}}\widehat{M}_{u_{s}}}{v^{2}}\dfrac{\zeta_{\ell}\,\zeta_{u}}{\widetilde{m}_{2}^{2}}\,,\qquad\qquad
c_{ijrs}^{\ell
equ(\pm)}=-2\delta_{ij}\dfrac{\widehat{M}_{\ell_{j}}V^{\ast}_{sr}\widehat{M}_{u_{s}}}{v^{2}}\dfrac{\zeta_{\ell}\,\zeta_{u}}{\widetilde{m}_{2}^{2}}\,,$

in the fermion mass basis defined by

$M_{f}=V^{\dagger}_{f}\,\widehat{M}_{f}\,U_{f}\,,$ (2.23)

where $V_{i}$ and $U_{i}$ are unitary transformations over the left-handed
(right-handed) fields. The matrices $\widehat{M}_{f}$ are the diagonal mass
matrices and the CKM matrix is defined as $V=V_{d}^{\dagger}V_{u}$. These
results are obtained with massless active neutrinos, otherwise we should also
consider the presence of the PMNS matrix. As can be seen from the previous
expressions, there are no $\widetilde{\lambda}_{i}$ parameters entering those
Wilson coefficients, so $\widetilde{m}_{2}$ is the only NP parameter present
beyond $\tan\beta$.

The presented above effective Lagrangian consists of dimension $d=4$ and $d=6$
operators: the light Higgs boson self-couplings and those with the gauge
bosons are dimensionless and the NP corrections appear at LO and NLO,
respectively, in the $v^{2}/{\widetilde{m}}^{2}_{2}$ expansion; on the other
hand, the four-fermion operators contribute to both charged current processes
and to the flavour conserving neutral current processes, with their
dimensional Wilson coefficients being suppressed by
$1/{\widetilde{m}}^{2}_{2}$. As expected, in the limit of
$\widetilde{m}_{2}^{2}\to\infty$ or $\widetilde{\lambda}_{6}\to 0$, the SM
form is recovered.

One would be tempted to work in the ${\cal
O}\left(v^{2}/{\widetilde{m}}^{2}_{2}\right)$ order of precision and use the
effective Lagrangian given above at the tree level. However, as we discuss in
the next section, this is not sufficient for the analysis of the flavour
physics constraints. In the language of the effective Lagrangian, the first
corrections to the FCNC processes appear at the 1-loop level, with the loops
generated by the four-fermion operators given in Eqs. (2.21) and (2.22). Such
loops would contribute e.g. to the meson-antimeson mixings and rare radiative
meson decays, giving the corresponding 4-fermion operator with the Wilson
coefficient suppressed by $1/{\widetilde{m}}^{2}_{2}$. In the analysis of the
next section, those effects are effectively taken into account providing lower
bounds on the parameter ${\widetilde{m}}_{2}$.

##  3 Model predictions and numerical Analysis

The low-energy effective Lagrangian introduced in the previous section depends
only on few parameters and, therefore, in the leading order in the expansion
in the $1/{\widetilde{m}}^{2}_{2}$ many correlated deviations from the SM are
predicted. We summarise them briefly.

  1. a)

All corrections from 4-fermion operators to flavour-changing processes depend
on the (approximately) degenerate mass of the heavy states. In principle, this
is a powerful test of the models. Unfortunately, neither any significant
deviation from the SM predictions is observed in the precision flavour data
nor a new scalar(s) discovered to test those predictions. However, one can use
the vast number of flavour data to put lower bounds on the new scalar masses
as a function of $\tan\beta$, for each type of the 2HDM.

  2. b)

As seen in Eq. (2.18), the real and imaginary parts of the Yukawa couplings
depend, in addition to the value of $\tan\beta,$ on only one effective
parameter, that is

$C=\left|{\tilde{\lambda}}_{6}\right|\cos({\rho})\frac{v^{2}}{{\widetilde{m}}^{2}_{2}}\,,\qquad\qquad\widetilde{C}=\left|{\tilde{\lambda}}_{6}\right|\sin({\rho})\frac{v^{2}}{{\widetilde{m}}^{2}_{2}}\,,$
(3.1)

where $C$ is for the real and $\widetilde{C}$ for the imaginary parts. So, the
models predict strong same correlations between different $\kappa_{f}$’s and
$\widetilde{\kappa}_{f}$’s. Not only they are the same for all fermions in
each sector but also correlated between sectors for a given value of
$\tan\beta$. So, the Higgs decay rates to fermion pairs are correlated to each
other, and also the fermion loop contributions to the EDM are correlated. It
is also worth remembering about the searches for CP violating effects in the
Higgs decays by analysing various CP sensitive asymmetries [100]. The
interesting point is that the correlations do not depend on the values of the
parameters $C$ and $\widetilde{C}$, although the magnitude of the deviations
in the individual $\kappa$’s of course does.

Trading the parameters $C$ and $\widetilde{C}$ for e.g. $\kappa_{u}$ and
$\widetilde{\kappa}_{u}$, respectively, for the other two
$\delta\kappa_{f}=\kappa_{f}-1$ and for $\widetilde{\kappa}_{f}$ one gets:

$\delta\kappa_{f}=\frac{\zeta_{f}}{\zeta_{u}}\delta\kappa_{u}\,,\qquad\qquad\widetilde{\kappa}_{f}=\frac{\zeta_{f}}{\zeta_{u}}\widetilde{\kappa}_{u}\,.$
(3.2)

Checking the Table below Eq. (2.18), we can see which are the most favorable
Higgs boson decay modes to observe the deviations from the SM predicted by the
models. For instance, for Type II, such deviations in the down quark and
lepton Yukawa couplings, relevant for the decays into a pair of the down
quarks or leptons, are enhanced by $\tan\beta^{2}$ (for $\tan\beta>1$)
compared to the up quarks. Similarly, CP sensitive asymmetries in the
$h\rightarrow\tau\tau$ are more promising to be observed than the CP violation
effects in the $htt$ coupling.

Again, since no statistically significant deviations in the Higgs decays from
the SM predictions are observed in the data (and only upper bounds for the
EDMs exist) we perform a fit to the data to get bounds on the parameters $C$
and $\widetilde{C}$. They allow to estimate the room left by the present data
for observing the deviations from the SM predicted by the 2HDMs in the Yukawa
couplings.

  3. c)

It is interesting to resolve the bounds on the parameters $C$ and
$\widetilde{C}$ as bounds on $\widetilde{\lambda}_{6}\cos({\rho})$ and
$\widetilde{\lambda}_{6}\sin({\rho})$ as a function of the mass scale of the
heavy scalars. First, this is a check of consistency with near the decoupling
limit approximation. Second, one needs it for predicting the expectations for
the triple Higgs coupling values.

In the next subsections we present the results of our fits. They generically
depend on the type of 2HDM considered and throughout the next text we will
adopt the following colour-code for the different realisations of
$Z_{2}$-symmetry:

###  3.1 Flavour Observables

In this section, we discuss the lower bounds on the degenerate (near the
decoupling limit) masses of the heavy scalars following the data for flavour
observables. For the calculations of the SM predictions and their
uncertainties, as well as those including NP, we use the software Flavio
[101].

The relevant four-fermion operators and the corresponding Wilson coefficients
are listed in Eqs. (2.21) and (2.22), depending only on $\widetilde{m}_{2}$
and $\tan\beta$ as NP parameters. At tree level, those operators contribute
only to the charged current and flavour-conserving neutral current processes.
In this category, only the $B^{+}\to\tau^{+}\nu$ decay provides an interesting
constraint.

Turning now to the very precisely measured FCNC processes, as discussed in the
previous section they receive contributions from the loops generated by the
four fermion operators Eq. (2.21). Those loop effects cannot be neglected
[102] and $B_{d}$ and $B_{s}$ meson transitions are the most constraining
observables of the $\widetilde{m}_{2}$ vs. $\tan\beta$ parameter space. We
include them in an effective way by adopting the results presented in Refs.
[103, 104, 105, 106, 107, 108, 109] for the one-loop contributions in the
$Z_{2}$-symmetric 2HDM scenarios mediated by the five scalar bosons, and
taking the mass degenerate limit for the heavier Higgses, to be consistent
with near the decoupling formalism.

Observable | SM prediction | Exp. Measurement | Ref.
---|---|---|---
$\mathcal{B}(B^{0}\to\mu^{+}\mu^{-})$ | $1.1(1)\times 10^{-10}$ | $0.56(70)\times 10^{-10}$ | [110]
$\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$ | $3.66(15)\times 10^{-9}$ | $2.93(35)\times 10^{-9}$ | [110]
$\Delta M_{B_{d}}\leavevmode\nobreak\ [\text{GeV}]$ | $4.0(5)\times 10^{-13}$ | $3.334(13)\times 10^{-13}$ | [111]
$\Delta M_{B_{s}}\leavevmode\nobreak\ [\text{GeV}]$ | $1.25(8)\times 10^{-11}$ | $1.1688(14)\times 10^{-11}$ | [111]
$\mathcal{B}(B\to X_{s}\gamma)$ | $3.3(2)\times 10^{-4}$ | $3.27(14)\times 10^{-4}$ | [112]
$\mathcal{B}(B_{s}\to\phi\gamma)$ | $4.0(5)\times 10^{-5}$ | $(3.6\pm 0.5\pm 0.3\pm 0.6)\times 10^{-5}$ | [113]
$\mathcal{B}(B^{+}\to\tau^{+}\nu)$ | $8.8(7)\times 10^{-5}$ | $1.09(24)\times 10^{-4}$ | [114]
$\mathcal{B}(B^{0}\to e^{+}e^{-})$ | $2.7(3)\times 10^{-15}$ | $2.4(4.4)\times 10^{-9}$ | [115]
$\mathcal{B}(B_{s}\to e^{+}e^{-})$ | $8.6(4)\times 10^{-14}$ | $0.30(1.29)\times 10^{-9}$ | [115]
$\mathcal{B}(B^{0}\to K^{\ast}\gamma)$ | $4.2(8)\times 10^{-5}$ | $4.33(15)\times 10^{-5}$ | [111]
$\mathcal{B}(B^{+}\to K^{\ast}\gamma)$ | $4.2(9)\times 10^{-5}$ | $4.21(18)\times 10^{-5}$ | [111]
$A_{CP}(B\to X\gamma)$ | $2(3)\times 10^{-18}$ | $0.032(34)$ | [111]
$A_{CP}(B^{0}\to K^{\ast}\gamma)$ | $0.004(2)$ | $-0.002(15)$ | [111]
$S_{\phi\gamma}$ | $-2(3)\times 10^{-4}$ | $0.43\pm 0.30\pm 0.11$ | [116]
$C_{\phi\gamma}$ | $4(2)\times 10^{-3}$ | $0.11\pm 0.29\pm 0.11$ | [116]
$\mathcal{A}^{\Delta}_{\phi\gamma}$ | $3(2)\times 10^{-2}$ | $-0.67^{+0.37}_{-0.41}\pm 0.17$ | [116]
$\left<\dfrac{\text{d}\mathcal{B}^{\prime}(B^{+}\to K^{+}\mu^{+}\mu^{-})}{\text{d}q^{2}\leavevmode\nobreak\ \text{GeV}^{-2}}\right>_{\text{bins}}$ | - | - | [117]
$\left<\dfrac{\text{d}\mathcal{B}^{\prime}(B\to K^{0}{}^{\ast}\mu^{+}\mu^{-})}{\text{d}q^{2}\leavevmode\nobreak\ \text{GeV}^{-2}}\right>_{\text{bins}}$ | - | - | [118]
$[F_{L},P_{1,2,3},P_{4,5,6,8}^{\prime}](B\to K^{0}{}^{\ast}\mu^{+}\mu^{-})$ | - | - | [119]

Table 1: Observables used in the fit. For the last three lines, data are
collected in the corresponding references.

(a) (b) (c) (d)

Figure 1: Bounds given by flavour observables in Tab. 1 for the different
types of 2HDM. The green(yellow)[red] area corresponds to
$[1\sigma,2\sigma]$($[2\sigma,3\sigma]$)[$>3\sigma$] region.

The complete list of observables used in our fit is shown in Tab. 1. The plots
referring to Type I and Type III are very similar to each other, excluding
only the low values of $\tan\beta$, mainly due to $\mathcal{B}(B\to
X_{s}\gamma)$, $\Delta M_{B_{s}}$ and $B_{d,s}\to\mu^{+}\mu^{-}$. The plot for
Type IV shows, on top of the aforementioned limit for low $\tan\beta$ values,
a strong cut on low $\widetilde{m}_{2}$ for any values of $\tan\beta$, mainly
constrained by the rare radiative $B$ decay. Finally, the most constrained
parameter space is the one of Type II, where smallish and largish values of
$\tan\beta$ are disfavoured even for large $\widetilde{m}_{2}$. While
$\mathcal{B}(B\to X_{s}\gamma)$ is the main responsible for the lower bound on
$\widetilde{m}_{2}$, the large $\tan\beta$ region is mostly constrained by
$B_{s}\to\mu^{+}\mu^{-}$. For intermediate values of $\tan\beta$, the lower
bound on $\widetilde{m}_{2}$ is similar to the one for the Type IV scenario.
This result follows mainly from the constraints given by the combination of
the rates for the radiative $B$ decay and for the leptonic neutral $B$ decays.

The reason for those similarities and differences may be traced back to the
$\tan\beta$ and $\cot\beta$ dependence of the Wilson coefficients Eq. (2.22),
which also determine the loop effects for the different types of 2HDM. In
turn, they depend on the Yukawa couplings. As the Higgs couplings with the up-
type quarks always include $\cot\beta$, they do not play a role in
differentiating among the different constructions. On the other hand,
$\zeta_{d}=-\tan\beta$ for Type II and Type IV and therefore $\mathcal{B}(B\to
X_{s}\gamma)$ grows with larger $\tan\beta$ values: wherein the Type IV plot,
this translates in the almost vertical bound on $\widetilde{m}_{2}$ once
$\tan\beta$ is larger than $\sim 2$; in the Type II plot, instead, it
corresponds to the bounds on $\widetilde{m}_{2}$ for intermediate $\tan\beta$
values, that matches the limits aforementioned. This enhancement is absent in
Type I and III as $\mathcal{B}(B\to X_{s}\gamma)$ is suppressed with
$\tan\beta$. Finally, the differences between Type II and Type IV are mainly
due to the Higgs couplings to the charged leptons that have an effect in the
leptonic neutral $B$ decays: their branching ratios are enhanced with
$\tan\beta$ only for Type II and thus their bounds strongly constrain the
parameter space.

The results presented above are overall in agreement with the literature, see
e.g. Ref.s [120, 121, 122, 123]. One of the most updated references is Ref.
[120], which however presents two relevant differences. First, the lower bound
on $\widetilde{m}_{2}$ present in Type II and IV is about $100\ \text{GeV}$
smaller in our case. The reason is mostly to be found in the inclusion of
recent data of angular observables of $B_{0}\to K^{\ast}\mu^{+}\mu^{-}$, as
some of them show relatively large deviations from the SM predictions, even
above $2\sigma$, e.g. $P_{5}^{\prime}$. Secondly, according to Refs. [108,
120], in Type II the uppermost allowed value is $\tan\beta\sim 30$, which is a
bound almost twice as strong as ours. Such bound is mainly due to
$B_{s}\to\mu^{+}\mu^{-}$ and $B^{+}\to\tau^{+}\nu$. The cause of the
discrepancy could be one of the following reasons or a combination of them: i)
a different treatment of the theoretical errors; ii) the fact that we do not
let the masses of the different scalars float when doing the fit. Finally, our
results for Type II are in excellent agreement with those of Ref. [123].

Apart from flavour observables, one could expect significant constraints from
direct searches at collider. The most constraining ones regard the searches
for heavy neutral particles. Such bounds typically are given only for specific
types of $Z_{2}$-models, especially for Type II. They depend on other
parameters of the model besides $\widetilde{m}_{2}$ and $\tan\beta$, such as
the mixing of the CP-even scalars, $\cos(\beta-\alpha)$, Eq. (2.12). Direct
searches bounds can be relevant for $\lesssim 400\ \text{GeV}$ [124], that is
in a region of the parameter space where the near the decoupling limit starts
to be problematic. For this reason, we will not report such bounds in the
results of the following sections.

###  3.2 Electron electric dipole moment

The present experimental bound on the electric dipole moment (EDM) of the
electron provided by the ACME II collaboration [125],

$d_{e}=\left(4.3\pm 3.1\pm 2.6\right)\times 10^{-30}\text{ e cm}$ (3.3)

translated to an upper bound at $2\sigma$

$|d_{e}|<1.24\times 10^{-29}\text{ e cm}\,,$ (3.4)

represents one of the most powerful observables to constrain the CP-violating
phases in NP models. Using the effective description in terms of the
$\kappa$-formalism in Eq. (2.18), we can translate this limit into a bound on
$\widetilde{\kappa}_{f}$, the imaginary parts in the Yukawa couplings
accounting for the effects of the $d=6$ corrections.

Due to the correlations between $\widetilde{\kappa}_{f}$’s emphasized in the
previous section, the electron EDM depends to a very good approximation only
on the parameter $\widetilde{C}$ defined in Eq. (3.1) and the value of
$\tan\beta$, which fixes the relative magnitudes of the three
$\widetilde{\kappa}_{f}$’s, for the up, down and charged lepton sectors. Ref.
[126], which corrects the results given in Ref. [127], presents the explicit
expression for $d_{e}$ in the limit we are discussing here, that is in
$Z_{2}$-symmetric 2HD models near the decoupling limit:

Type I: $\displaystyle d_{e}\simeq 1.06\times
10^{-27}\,\left|\widetilde{\lambda}_{6}\right|\sin{(\rho)}\left(\dfrac{1\
\text{TeV}}{\widetilde{m}_{2}}\right)^{2}\times\cot\beta\left[1+0.07\ln\left(\dfrac{\widetilde{m}_{2}}{1\
\text{TeV}}\right)\right]\text{e cm}\,,$ (3.5) Type II: $\displaystyle
d_{e}\simeq-0.47\times
10^{-27}\,\left|\widetilde{\lambda}_{6}\right|\sin{(\rho)}\left(\dfrac{1\
\text{TeV}}{\widetilde{m}_{2}}\right)^{2}\times$ $\displaystyle\hskip
85.35826pt\times\left\\{\tan\beta\left[1+0.16\ln\left(\dfrac{\widetilde{m}_{2}}{1\
\text{TeV}}\right)\right]-1.26\cot\beta\right\\}\text{e cm}\,,$ Type III:
$\displaystyle d_{e}\simeq-0.47\times
10^{-27}\,\left|\widetilde{\lambda}_{6}\right|\sin{(\rho)}\left(\dfrac{1\
\text{TeV}}{\widetilde{m}_{2}}\right)^{2}\times$ $\displaystyle\hskip
85.35826pt\times\left\\{\tan\beta\left[1+0.16\ln\left(\dfrac{\widetilde{m}_{2}}{1\
\text{TeV}}\right)\right]-1.25\cot\beta\right\\}\text{e cm}\,,$ Type IV:
$\displaystyle d_{e}\simeq 1.06\times
10^{-27}\,\left|\widetilde{\lambda}_{6}\right|\sin{(\rho)}\left(\dfrac{1\
\text{TeV}}{\widetilde{m}_{2}}\right)^{2}\times$ $\displaystyle\hskip
85.35826pt\times\left\\{\cot\beta\left[1+0.07\ln\left(\dfrac{\widetilde{m}_{2}}{1\
\text{TeV}}\right)\right]+0.002\tan\beta\right\\}\text{e cm}\,.$

We use the above equations to find upper bounds of $\widetilde{C}$ as a
function of $\tan\beta$ for each Type of the model: these limits are
approximately independent from $\widetilde{m}_{2}$, as the
$\widetilde{m}_{2}$-dependent logarithmic terms are numerically negligible in
the considered range of $\widetilde{m}_{2}$. Then, using Eqs. (2.18) one
obtains upper bounds on the imaginary parts of the Yukawas shown in Fig. 2.
They are shown for Type II and IV. For Type I the upper bounds for all 3
$\widetilde{\kappa}_{f}$’s are the same, of order $10^{-3}$; for Type III the
bound on $\widetilde{\kappa}_{e}$ remains as for Type II and the bound on
$\widetilde{\kappa}_{d}$ is the same as the bound on $\widetilde{\kappa}_{u}$
in Type II. This behaviour can be easily understood from Eqs. (2.18).

(a) Type II (b) Type IV

Figure 2: Upper bounds on $\widetilde{\kappa}_{f}$ at 95% C.L. for Type II and
IV.

It is worth commenting on the structure of the bounds in the context of
experimental search for various CP-sensitive asymmetries in the Higgs
production and decays. At least in the framework of the $Z_{2}$ symmetric
2HDMs, the search for such effects in the $htt$ coupling is least promising.
Better chances are offered by the down quark sector in Type IV, but otherwise,
the limits are always very strong, at the level of
$\widetilde{\kappa}_{f}\approx 10^{-2}-10^{-3}$.

The individual fermion loop contributions in the Barr-Zee diagrams to the
electron EDM, and therefore the role of each loop in getting the final upper
bound for $\tilde{C}$, depend on the structure of the $\zeta_{f}$ coefficients
in Eqs. (2.18) and on the fermion masses, which enter into the loop
calculation. For the Type I case, for example, all $\widetilde{\kappa}_{f}$’s
are weighted by $\cot\beta$ and one can see that the dominant contribution to
$d_{e}$ comes from the top loop (see the discussion in Ref. [49], where the
analysis considering only one NP parameter at a time was performed).
Differently, for the Type II and Type III, the charged lepton contributions
are enhanced by $\tan\beta$, and thus the bound can be directly associated
with them. Finally, for type IV, it depends on the value of $\tan\beta$ that
only enhances the down-type quarks.

It is also worth mentioning the possible cancellation that could take place
for specific values of $\tan\beta$ between the terms in the curly brackets of
the previous expressions. Interestingly, the same pattern of relations also
holds for the neutron EDM, but with different numerical factors. As a result,
even if one may select a specific value for $\tan\beta$ that cancels the NP
contributions to the electron EDM, the bounds from the neutron EDM would still
apply, providing a smaller but still strong constraint on the
$\widetilde{\kappa}_{f}$.

(a) (b)

Figure 3: Upper bounds at $2\sigma$ from electron EDM for different types of
2HDM and values of $\tan\beta$. Each colour corresponds to a specific
realisation of 2HDM: Type I-blue, Type II-green, Type III-orange, Type IV-red.
The plot is symmetric with respect to the x-axis for negative values of
$\rho$. The area on the left-side of the vertical dot-dashed lines is excluded
at $2\sigma$ by flavour observables. In the grey shaded area the
$v/\widetilde{m}_{2}$ expansion is not precise and can be affected by
$\gtrsim\mathcal{O}(40\%)$ corrections.

It is useful to show the upper bound on the parameter $\widetilde{C}$ as the
bound on the parameter $|\widetilde{\lambda}_{6}|\sin{\rho}$ as a function of
$\widetilde{m}_{2}$ (the logarithmic terms are included). The results are
shown in Fig. 3. The four plots refers to different values of
$\tan\beta=\\{2,\,15\\}$, and the lines are the $2\sigma$ upper bounds on
$|\widetilde{\lambda}_{6}|\sin\rho$ as a function of $\widetilde{m}_{2}$. The
colours identify the 2HDM construction: Type I-blue, Type II-green, Type III-
orange, Type IV-red. The vertical lines are the $2\sigma$ lower bounds for
$\widetilde{m}_{2}$ as identified in Fig. 1 from the fit with data from the
flavour observables in Tab. 1. The values on the left of these lines are
therefore excluded.

With analogous results for $|\widetilde{\lambda}_{6}|\cos{\rho}$ as a function
of $\widetilde{m}_{2}$ obtained in the next subsection, one can infer the
acceptable values for the coupling $|\widetilde{\lambda}_{6}|$ and the phase
$\rho$, separately.

###  3.3 Collider Observables

In this section we investigate the bounds on the real parts of the Yukawa
couplings following from the Higgs boson production and decay data. We employ
the latest results from ATLAS [128] and CMS [129]. The experimental data can
be used to constrain the parameters $\kappa_{V}$ describing the deviations
from the SM predictions of the Higgs couplings to gauge bosons, defined in Eq.
(LABEL:eq:kappaVV), and the combinations

$r_{f}^{2}\equiv\kappa_{f}^{2}+\widetilde{\kappa}_{f}^{2}\,,$ (3.6)

with $\kappa_{f}$ and $\widetilde{\kappa}_{f}$ in Eq. (2.18), for the
couplings with fermions. The results reported by the experimental
collaborations are shown in Tab. 2.

| ATLAS [128] | CMS [129]
---|---|---
$r_{\mu}$ | $1.07(26)$ | $1.11(21)$
$r_{\tau}$ | $0.94(7)$ | $0.925(75)$
$r_{b}$ | $0.90(11)$ | $1.02(16)$
$r_{t}$ | $0.95(7)$ | $0.95(7)$
$\kappa_{W}$ | $1.02(5)$ | $1.03(3)$
$\kappa_{Z}$ | $0.99(6)$ | $1.02(3)$
Table 2: Values of $r_{i}$ and $k_{V}$ used for the fits.

The EDM upper bounds at the $2\sigma$ level for the $\widetilde{\kappa}_{f}$’s
are significantly stronger than the uncertainties on the $r_{f}$’s shown (at
$1\sigma$) in Tab. 2, so in our analysis of the bounds on the $\kappa_{f}$’s
we assume $\widetilde{\kappa}_{f}=0$. The accuracy of this approximation can
be judged once the results for the bounds on $\kappa$’s are obtained. Thus,
taking $r_{f}=\kappa_{f}$, we fit the parameter $C$ defined in Eq. (3.1). For
each value of $\tan\beta$, we define a $\chi^{2}$-function based on the data
on $r_{f}$ and we obtain the best-fit value of $C$ for each type of the model.
In each case, the results of the fit are most sensitive to those $r_{f}$’s in
Tab. 2 for which $\zeta_{f}=\tan\beta$. Next, using Eqs. (2.18) with fitted
values for $C$, we obtain the bounds shown at $2\sigma$ level in Fig. 4 for
Type II and IV.

(a) (b)

Figure 4: Upper and lower bounds on $\kappa_{f}$ at 95% C.L. for Type II and
IV.

As it follows from Eqs. (2.18), for Type I the bounds are $\tan\beta$
independent and the same for all $\kappa_{f}$’s in the range

$0.9<\kappa_{f}<1.02\,,$ (3.7)

For Type III, $\kappa_{d}=\kappa_{u}$ and $\kappa_{e}$ remains as in Type II.
We see that in most cases $\kappa_{f}\gg\widetilde{\kappa}_{f}$, consistently
with our approximation. Similarly as for the EDM bounds, we note that the
structure of correlations for the Yukawa couplings predicted by the models
results in very strong bounds on the $htt$ coupling. For seeing the deviations
from the SM, the Higgs decays to the down quarks and leptons (for Type II),
only to leptons (Type III) and only to the down quarks (Type IV) are much more
promising.

The best fit results obtained for the parameters $C$ translate into parabola-
like branches in the parameter space $|\widetilde{\lambda}_{6}|\cos{(\rho)}$
vs. $\widetilde{m}_{2}$. The results are shown in Fig. 5 for different values
of $\tan\beta=\\{2,\,15\\}$. The lines represent the $2\sigma$ upper limit on
the combination $|\widetilde{\lambda}_{6}|\left|\cos{(\rho)}\right|$: solid
(dot-dashed) lines correspond to positive (negative) values of $\cos{(\rho)}$.
As for Fig. 3, the colours identify the 2HDM construction, that is Type
I-blue, Type II-green, Type III-orange, Type IV-red. Moreover, the vertical
lines are again the $2\sigma$ lower bounds for $\widetilde{m}_{2}$ as
identified in Fig. 1.

(a) (b)

Figure 5: Upper bounds at $2\sigma$ from collider observables for different
types of 2HDM and values of $\tan\beta$. Each colour corresponds to a specific
realisation of 2HDM: Type I-blue, Type II-green, Type III-orange, Type IV-red.
The case with $\cos\rho>0\leavevmode\nobreak\ (<0)$ corresponds to solid
(dashed) lines. The area on the left-side of the vertical dot-dashed lines is
excluded at $2\sigma$ by flavour observables. The purple line corresponds to
the limit in Eq. (3.9) and it applies to the case $\left|\cos\rho\right|=1$.
In the grey shaded area the $v/\widetilde{m}_{2}$ expansion is not precise and
can be affected by $\gtrsim\mathcal{O}(40\%)$ corrections.

As the value of $\tan\beta$ grows, larger values for
$|\widetilde{\lambda}_{6}|\left|\cos{(\rho)}\right|$ are allowed in the Type I
2HDM case, while the contrary holds for the other cases. This is easy to
understand for the Type I 2HDM, where $\zeta_{u,d,e}=\cot\beta$, and then the
fitted values of the parameter $C$ increase with larger $\tan\beta$. On the
other hand, the analysis is more involved for the other constructions: indeed,
it is the combination of the sensitivity to the different $r_{f}$ and the
dependence on $\tan\beta$ or $\cot\beta$ that determines which is the dominant
contribution in the fit.

Comparing Figs. 3 and 5, we can conclude that generically
$|\widetilde{\lambda}_{6}|\left|\cos{(\rho)}\right|\gg\left|\widetilde{\lambda}_{6}\right|\left|\sin{(\rho)}\right|$.
This confirms once again the validity of the approximation chosen in the fit
with only the fermionic collider data and can be interpreted as
$\left|\cos\rho\right|\approx 1$. In fact, generically the bounds on
$|\widetilde{\lambda}_{6}|\left|\cos{(\rho)}\right|$ are $\mathcal{O}(10-100)$
stronger than the ones on
$|\widetilde{\lambda}_{6}|\left|\sin{(\rho)}\right|$. The only exception lies
in Type IV, as the two upper bounds become of the same order for
$\tan\beta\sim 15$. Eventually, for larger values of $\tan\beta$, the bound on
$|\widetilde{\lambda}_{6}|\left|\sin{(\rho)}\right|$ become weaker than the
one of $|\widetilde{\lambda}_{6}|\left|\cos{(\rho)}\right|$. One should thus
expect corrections of $\mathcal{O}(1)$ which, however, should not change the
magnitude of the limits.

On the other hand, some care is necessary when considering the large
$\tan\beta$ scenario, as the bounds are relatively weak and
$|\widetilde{\lambda}_{6}|$ could reach values as large as
$\widetilde{m}_{2}^{2}/v^{2}$, that are inconsistent with the decoupling
limit. This is already the case for $\tan\beta=15$ and for
$\widetilde{m}_{2}^{2}\gtrsim 850\ \text{GeV}$: in this case, the upper bound
on $|\widetilde{\lambda}_{6}|$ should be read from Eq. (2.11).

Finally, performing a fit including only $\kappa_{W}$ and $\kappa_{Z}$ from
both collaborations, where

$\displaystyle\kappa_{V}=$ $\displaystyle
1-\dfrac{1}{2}\left|\widetilde{\lambda}_{6}\right|^{2}\left(\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\right)^{2}\,,$
(3.8)

we obtain at $2\sigma$

$\left|\widetilde{\lambda}_{6}\right|\left(\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\right)\leq
0.17\,.$ (3.9)

This result is valid for all four types of 2HDM realisations and is
independent of $\tan\beta$. We show this limit in Fig. 5 as an additional
purple line. Although strictly speaking, this line refers only to the case
with $\cos\rho=1$, it helps to guide the eye in the interpretation of the
relative magnitude of the different bounds.

###  3.4 Triple Higgs Coupling

With the results of the previous sections for the bounds on the mass scale
${\tilde{m}}_{2}$ and the coupling ${\widetilde{\lambda}}_{6}$, in this
section we discuss the triple Higgs coupling, whose deviation from the SM is
reported in Eq. (2.20). The latest bounds on it are given by the ATLAS
collaboration [130] and read (at $2\sigma$)

$-1.4<\dfrac{g_{h^{3}}}{g_{h^{3}}^{\text{SM}}}<6.1\,.$ (3.10)

The leading NP contribution depends on the ratio
$|\widetilde{\lambda}_{6}|^{2}v^{2}/\widetilde{m}_{2}^{2}$. It depends on
$\tan\beta$ through the values of ${\widetilde{\lambda}}_{6}$. One then
generically expects that, even considering the largest allowed values on
$|\widetilde{\lambda}_{6}|\left|\cos{(\rho)}\right|$ and
$|\widetilde{\lambda}_{6}|\left|\sin{(\rho)}\right|$ determined in the
previous sections, the predicted deviation from the SM of $g_{h^{3}}$ is
smaller than the direct bound in Eq. (3.10) (see for example Refs. [131, 132,
133, 134, 135, 136]).

It is well known that the radiative corrections to the triple Higgs coupling
are relevant. Focussing on the $1-$loop contributions in the CP conserving
limit, that is a good approximation given the smallness of the CP violating
phases as shown in the previous sections, we can write [137],

$g_{h^{3}}=g^{TL}_{h^{3}}+\delta g_{h^{3}}\,,$ (3.11)

where $g^{TL}_{h^{3}}$ is given in Eq. (2.20) while $\delta g_{h^{3}}$ reads

$\displaystyle\delta g_{h^{3}}\approx$
$\displaystyle\dfrac{3m_{h}^{2}}{v^{2}}\left\\{\dfrac{1}{12\pi^{2}m_{h}^{2}v^{2}}\left[m_{H}^{4}\left(1-\dfrac{\widetilde{m}_{2}^{2}}{m_{H}^{2}}\right)^{3}+m_{A}^{4}\left(1-\dfrac{\widetilde{m}_{2}^{2}}{m_{A}^{2}}\right)^{3}\right]\right.$
(3.12) $\displaystyle\hskip
42.67912pt\left.+\dfrac{m_{H_{\pm}}^{4}}{6\pi^{2}m_{h}^{2}v^{2}}\left(1-\dfrac{\widetilde{m}_{2}^{2}}{m_{H_{\pm}}^{2}}\right)^{3}-\dfrac{N_{c}m_{t}^{4}}{3\pi^{2}m_{h}^{2}v^{2}}\right\\}\,,$

where the last term is due to the SM contribution mediated by the top quark.

As can be seen, the contribution can be large if the decoupling is not
realized or if the masses are not almost degenerate. In the case of almost
decoupling, substituting the explicit expressions for the masses and keeping
the leading contributions, the part containing the NP-corrections at $1$-loop
simplifies and reads

$\delta
g_{h^{3}}^{\text{NP}}\approx\dfrac{1}{16\pi^{2}}\left(\dfrac{v}{\widetilde{m}_{2}}\right)^{2}\left[(\widetilde{\lambda}_{3}+\widetilde{\lambda}_{4}+\widetilde{\lambda}_{5})^{3}+(\widetilde{\lambda}^{3}-3\widetilde{\lambda}_{5}(\widetilde{\lambda}_{3}+\widetilde{\lambda}_{4})^{2}-\widetilde{\lambda}_{5}^{3})\right]\,.$
(3.13)

Notice that they are of the same order of magnitude as the leading corrections
in Eq. (2.20) to the tree level result. Moreover, if $\widetilde{\lambda}_{i}$
are positive, they may induce destructive interference with the tree-level NP
contribution. Notably, the dependence on $\widetilde{\lambda}_{3,4,5}$ scales
with the third power, so the relative importance between tree- and loop-level
contributions highly depends on such couplings. Such dependence is illustrated
in Fig. 6 for $\widetilde{\lambda}_{6}=0.1$ and different values of
$\widetilde{\lambda}_{3,4,5}$. To maximize the effect, we take
$\widetilde{\lambda}_{3}=\widetilde{\lambda}_{4}=\widetilde{\lambda}_{5}$.

Figure 6: Relative importance of the loop- vs. tree-level NP contributions. In
the vertical axis we plot the ratio between the triple Higgs coupling
prediction including NP contributions and the purely SM quantity at $1-$loop
level. The solid-black line only contains the tree-level NP contributions,
while the purple lines also contain the $1-$loop level NP contributions for
different values of
$\widetilde{\lambda}_{3}=\widetilde{\lambda}_{4}=\widetilde{\lambda}_{5}$.

One may expect that the parameters in the potential are of the same order of
magnitude: for this reason and in order to simplify the discussion, without
any loss of generality, in what follows we shall consider that
$\widetilde{\lambda}_{3,4,5}$ are close enough to $\widetilde{\lambda}_{6}$ in
order to neglect all NP contributions at $1$-loop.

We combine the information from the electron EDM and collider Higgs data of
the previous sections to determine an upper bound on
$|\widetilde{\lambda}_{6}|^{2}v^{2}/\widetilde{m}_{2}^{2}$, summing
quadratically the upper values for the real ($\propto\cos\rho$) and imaginary
($\propto\sin\rho$) parts, for any 2HDM realisations. With this procedure, we
always obtain the strongest bound on the triple coupling as a function of
${\tilde{m}}_{2}$, except for the Type I 2HDM. In the latter case, the
strongest bound coincides with Eq. (3.9) (for the values of $\tan\beta>1$). In
Fig. 7 we show $2\sigma$ the lower bounds on
$g_{h^{3}}/g_{h^{3}}^{\text{SM}}$, for $\tan\beta=\\{2,\,15\\}$. We use the
same colour code as the previous plots. The vertical lines are again the
$2\sigma$ lower bounds for $\widetilde{m}_{2}$ as identified in Fig. 1.
Finally, the grey shaded area corresponds to the direct bound in Eq. (3.10).
When the purple and blue lines do not appear they are outside the scale of the
plots (they give much weaker bounds).

(a) (b)

Figure 7: Lower bounds on triple Higgs-coupling normalised to the SM (tree-
and $1-$loop contribution) from collider observables for different types of
2HDM and values of $\tan\beta$. The solid black line corresponds to the SM
prediction. Each colour corresponds to a specific realisation of 2HDM: Type
I-blue, Type II-green, Type III-orange, Type IV-red. The purple line shows the
universal bound from Higgs decay into gauge bosons in Eq. (3.9). Only the area
enclosed within the coloured and black solid lines is allowed. The shaded grey
area is excluded at $2\sigma$ for all models from the direct measurement of
the triple Higgs coupling. Finally, the area on the left-side of the vertical
lines is excluded at $2\sigma$ by flavour observables. In the grey shaded area
the $v/\widetilde{m}_{2}$ expansion is not precise and can be affected by
$\gtrsim\mathcal{O}(40\%)$ corrections. The horizontal black dashed line
corresponds to the lower value that HL-LHC will be able to probe at
$95\%$C.L..

For Type II and Type III, the models predict very small deviations from the SM
for the triple Higgs couplings, for any values of $\tan\beta$. This is also
true for Type IV for larger values of $\tan\beta$. On the other hand, the Type
IV 2HDM for smaller $\tan\beta$ and the Type I for any $\tan\beta$ values
sizeable deviations are allowed and can be probed in the future colliders.

Finally, HL-LHC prospects to measure much more precisely such quantity. The
current projections estimate to test
$g_{h^{3}}/g_{h^{3}}^{\text{SM}}\in[0.1,2.3]$ at $95\%$ C.L. [138]. This
remarkable improvement will be very useful to constrain Type-I, but it will
not restrict the parameter space for the other types with respect to the
current bounds from Higgs-fermion interactions.

###  3.5 $m_{W}$ from CDF II

The determination of the W-boson mass, $m_{W}$, represents the last relevant
anomaly in particle physics. The CDF II collaboration [17] measured it with
the best-achieved sensitivity,

$m_{W}^{\text{CDF II}}=80.4335\pm 0.0094\ \text{GeV}\,,$ (3.14)

showing a $7\sigma$ discrepancy relative to the SM prediction [139]

$m_{W}^{\text{SM}}=80.354\pm 0.007\ \text{GeV}\,.$ (3.15)

Interestingly, this measure is not only in tension with the SM but also with
all the other experimental determinations: the last one, from the ATLAS
collaboration [18], shows no deviation from the SM prediction, although the
uncertainties are not as good as those of the CDF II measurement.

We refrain here from taking a definite point of view and instead discuss under
which conditions the 2HDM setup we discussed so far can agree either with the
SM prediction or with the experimental CDF II result222As reported in Ref.
[140], neglecting the possible correlations, one could estimate the average
among the different experimental determinations obtaining a value for $m_{w}$
only $1\sigma$ far from the CDF II result, as expected. For this reason, we
will perform our analysis considering the value in Eq. (1.1), understanding
that the conclusions would remain invariant using instead the average
quantity..

The most precise theoretical prediction of $m_{W}$ is obtained using
$\alpha_{\text{em}}$, $G_{F}$, and $m_{Z}^{2}$ as experimental input. As the
Fermi constant is extracted experimentally from the dominant muon decay, if NP
contributes to the branching ratio of this decay, then $G_{F}$ gets redefined
and a tree-level modification of $m_{W}$ follows. However, the impact of 2HDM
the dominant muon decay induced by the operators of Eq. (2.21) is extremely
small, due to the suppression of the electron mass entering in the
corresponding Wilson coefficients. It follows that no sizeable tree-level
correction to the SM prediction is present.

Another possibility concerns the role of oblique corrections, and more
specifically of the Peskin-Takeuchi parameters, $S,T,U$ [141]. Denoted the
loop-correction to the two-points function of the gauge fields
$i\Pi^{\mu\nu}_{VV}\equiv
i\Pi_{VV}\eta^{\mu\nu}+i\Pi^{pp}_{VV}p^{\mu}p^{\nu}$, they are defined as

$\displaystyle
T\equiv\dfrac{1}{\alpha_{\text{em}}}\left(\dfrac{\Pi^{\text{NP}}_{WW}(0)}{m_{W}^{2}}-\dfrac{\Pi^{\text{NP}}_{ZZ}(0)}{m_{Z}^{2}}\right)\,,$
(3.16) $\displaystyle
S\equiv\dfrac{4c_{W}^{2}s_{W}^{2}}{\alpha_{\text{em}}}\left[\dfrac{\Pi^{\text{NP}}_{ZZ}(m_{Z}^{2})-\Pi^{\text{NP}}_{ZZ}(0)}{m_{Z}^{2}}-\dfrac{c_{W}^{2}-s_{W}^{2}}{c_{W}s_{W}}\dfrac{\Pi^{\text{NP}}_{Z\gamma}(m_{Z}^{2})}{m_{Z}^{2}}-\dfrac{\Pi^{\text{NP}}_{\gamma\gamma}(m_{Z}^{2})}{m_{Z}^{2}}\right]\,,$
(3.17) $\displaystyle
U\equiv\dfrac{4s_{W}^{2}}{\alpha_{\text{em}}}\left[\dfrac{\Pi^{\text{NP}}_{WW}(m_{W}^{2})-\Pi^{\text{NP}}_{WW}(0)}{m_{W}^{2}}-\dfrac{c_{W}}{s_{W}}\dfrac{\Pi^{\text{NP}}_{Z\gamma}(m_{Z}^{2})}{m_{Z}^{2}}-\dfrac{\Pi^{\text{NP}}_{\gamma\gamma}(m_{Z}^{2})}{m_{Z}^{2}}\right]-S\,,$
(3.18)

where $\alpha_{\text{em}}\equiv\alpha_{\text{em}}(m_{Z})$ and the superscript
“NP” indicates that the SM contribution has been subtracted and only NP
contributes. Their impact on $m_{W}$ reads [142]

$m_{W}^{2}=(m_{W}^{\text{SM}})^{2}\left(1+\dfrac{s_{W}^{2}}{c_{W}^{2}-s_{W}^{2}}\Delta
r^{\prime}\right)\,,$ (3.19)

where

$\Delta
r^{\prime}\equiv\dfrac{\alpha_{\text{em}}}{s_{W}^{2}}\left(-\dfrac{1}{2}S+c_{W}^{2}T+\dfrac{c_{W}^{2}-s_{W}^{2}}{4s_{W}^{2}}U\right)\,.$
(3.20)

The expressions for the oblique parameters for multi-higgs extensions have
already been computed in Refs. [143, 144, 145, 146, 147]. In the alignment
limit, these expressions can be expanded in terms of
$v^{2}/\widetilde{m}_{2}^{2}$ and, in the CP-conserving case, they read

$\displaystyle
S\approx\dfrac{1}{24\pi}\widetilde{\lambda}_{4}\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\,,$
(3.21) $\displaystyle
T\approx\dfrac{1}{192\pi^{2}\alpha_{\text{em}}}(\widetilde{\lambda}_{4}^{2}-|\widetilde{\lambda}_{5}|^{2})\dfrac{v^{2}}{\widetilde{m}_{2}^{2}}\,.$
(3.22)

The parameter $U$ has been neglected as it receives corrections at the next
order in the expansion $v^{2}/\widetilde{m}_{2}^{2}$. From the above
expressions, we can estimate that $|T|\sim 8\leavevmode\nobreak\ |S|$, unless
a cancellation occurs between the Lagrangian parameters
$\widetilde{\lambda}_{4}$ and $\widetilde{\lambda}_{5}$.

If one takes for true the CDF II anomaly and pretends to solve it within this
context, the relative signs of the second expression are fixed, as
$\tilde{\lambda}_{4}>|\tilde{\lambda}_{5}|$ guarantees a larger prediction
than the SM one. The $T$ parameter is related to the violation of the
custodial symmetry of the scalar potential and vanishes if all the heavy
scalars are degenerate in mass. In fact, we can see that

$\widetilde{\lambda}_{4}^{2}-|\widetilde{\lambda}_{5}|^{2}\approx\dfrac{4}{v^{4}}(m_{H_{\pm}}^{2}-m_{H}^{2})(m_{H_{\pm}}^{2}-m_{A}^{2})\,.$
(3.23)

This implies that a minimal mass-splitting is necessary in order to explain
the $m_{W}$ anomaly through a sizeable $T$ oblique parameter. Such result can
be seen in Fig. 8.

(a) (b)

Figure 8: Parameter space where it is possible to explain the CDF II
measurement of $m_{W}$ via oblique $T$ parameter. In green(yellow) the
experimental value at $1\sigma$($2\sigma$). In the grey shaded area the
$v/\widetilde{m}_{2}$ expansion is not precise and can be affected by
$\gtrsim\mathcal{O}(40\%)$ corrections.

It is interesting to notice that the explanation of $m_{W}$ fixes also the
ordering of the mass hierarchy among the heavy scalars, leaving just two
possibilities:

$m_{H_{\pm}}^{2}>m_{H}^{2},\leavevmode\nobreak\ m_{A}^{2}\,,\qquad\qquad\qquad
m_{H_{\pm}}^{2}<m_{H}^{2},\leavevmode\nobreak\ m_{A}^{2}\,.$ (3.24)

On the other side, if the CDF II anomaly will lose evidence, this may be
interpreted in the context of the $Z_{2}$-symmetric 2HDM as an indication of
the (almost) degeneracy of the heavy scalar masses.

##  4 Conclusions

We have studied the predictions of the $Z_{2}$ symmetric 2HD models near the
decoupling limit, motivated by no discovery so far of any new scalar. After
integrating out the heavy particles, the models can be formulated in terms of
effective Lagrangians that depend on only three parameters: $\tan\beta$, the
(approximately) common mass of the heavy scalars $\widetilde{m}_{2}$, and a
complex coupling $\widetilde{\lambda}_{6}$ in the scalar potential. Due to the
small number of free parameters, the effective models predict many
correlations between observables, which are relevant both for analysing the
present experimental constraints and the future discovery prospects.

The effective 2HD models are interesting theoretical laboratories for
investigating the interplay of the flavour and collider constraints on the
extensions of the SM, along the lines proposed in Refs. [49, 50]. The flavour
$Z_{2}$ symmetry eliminates the tree-level contributions to the FCNC and,
therefore, new scalars with a relatively low mass scale may couple to the SM
gauge bosons and fermions consistently with the constraints from the very high
precision flavour data. In the effective approach, the predicted contributions
to flavour-changing observables depend only on the value of $\tan\beta$
(entering through Yukawa couplings) and on the degenerate heavy scalar masses
$\widetilde{m}_{2}$. A fit to almost 60 observables gives the exclusions
limits on $\widetilde{m}_{2}$, depending on type I-IV of the 2HDM and the
value of $\tan\beta$: in particular, for the Type IV a lower bound on
$\widetilde{m}_{2}$ in the range $300-500\ \text{GeV}$ is found; a similar
limit applies in the Type-II scenario, but only for values of $\tan\beta$ in
the range $20-40$, otherwise the bounds are weaker.

With these relatively weak bounds from flavour data, a complementary source of
information on the effective Yukawa couplings and on the Lagrangian parameter
comes from collider data on the Higgs production and decays. Strong
correlations between different Yukawa couplings are predicted by the models.
They have an impact on the results of the fit to the present data and
determine the room left for deviations from the SM, important for future
discovery prospects. The striking prediction of all four types of 2HD models
is that the top quark Yukawa coupling is the least promising to show
deviations from the SM value. The correlations among the effective Yukawa
couplings are equally important for the bounds on the imaginary parts of the
Yukawas inferred from the EDM bounds.

Another prediction following from the fits to flavour and Higgs data is for
the expected range of the triple Higgs coupling: the deviations from the SM
prediction, in the absence of hierarchies between $\widetilde{\lambda}_{6}$
and $\widetilde{\lambda}_{3,4,5}$, are tiny in the Type II and III models for
any value of $\tan\beta$ and in the Type IV for large $\tan\beta$; in the
other cases the deviations may be sizeable and therefore testable in the
future colliders.

## Acknowledgements

The authors thanks Francisco Arco García and José Miguel No for useful
discussions. A.d.G. and L.M. thanks the Institute of Theoretical Physics and
the Faculty of Physics of the University of Warsaw for hospitality during the
development of this project. S.P. thanks the Institute of Theoretical Physics
of the Universidad Autónoma de Madrid for hospitality during the development
of this project. A.d.G. and L.M. acknowledge partial financial support by the
Spanish Research Agency (Agencia Estatal de Investigación) through the grant
IFT Centro de Excelencia Severo Ochoa No CEX2020-001007-S and by the grant
PID2019-108892RB-I00 funded by MCIN/AEI/ 10.13039/501100011033, by the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreements No 860881-HIDDeN and
101086085-ASYMMETRY. The work of A.d.G. is supported by the European Union’s
Horizon 2020 Marie Skłodowska-Curie grant agreement No 860881-HIDDeN. The
research of S.P. has received partial financial support by the National
Science Centre, Poland, grant DEC-2018/31/B/ST2/02283.

##  Appendix A From the $Z_{2}$-original basis to the Higgs basis

In this appendix, we define the scalar sector of the Lagrangian and its
connection to the Higgs basis used in Sec. 2 following the notation of Ref.
[88].

The scalar sector is composed of two identical complex scalar fields
$\Phi_{a}$, with $a=1,2$, which are charged under $SU_{L}(2)\times U_{Y}(1)$
and with hypercharge, $Y_{\Phi_{a}}=1/2$. The Lagrangian reads

$\mathscr{L}=(D_{\mu}\Phi_{a})^{\dagger}(D^{\mu}\Phi_{a})+V(\Phi_{1},\Phi_{2})\,,$
(A.1)

where the most general renormalizable potential for the two doublets reads

$\displaystyle V(\Phi_{1},\Phi_{2})=$ $\displaystyle-
m_{1}^{2}(\Phi^{\dagger}_{1}\Phi_{1})-m_{2}^{2}(\Phi^{\dagger}_{2}\Phi_{2})-\left[m_{12}^{2}(\Phi^{\dagger}_{1}\Phi_{2})+\text{h.c.}\right]$
$\displaystyle+\lambda_{1}(\Phi^{\dagger}_{1}\Phi_{1})^{2}+\lambda_{2}(\Phi^{\dagger}_{2}\Phi_{2})^{2}+\lambda_{3}(\Phi^{\dagger}_{1}\Phi_{1})(\Phi^{\dagger}_{2}\Phi_{2})+\lambda_{4}(\Phi^{\dagger}_{1}\Phi_{2})(\Phi^{\dagger}_{2}\Phi_{1})$
$\displaystyle+\left[\lambda_{5}(\Phi^{\dagger}_{1}\Phi_{2})^{2}+\lambda_{6}(\Phi^{\dagger}_{1}\Phi_{1})(\Phi^{\dagger}_{1}\Phi_{2})+\lambda_{7}(\Phi^{\dagger}_{2}\Phi_{2})(\Phi^{\dagger}_{1}\Phi_{2})+\text{h.c.}\right]\,.$

The parameters $m_{1}^{2},m_{2}^{2},\lambda_{1},\lambda_{2},\lambda_{3}$ and
$\lambda_{4}$ are real while $m_{12}^{2},\lambda_{5},\lambda_{6}$ and
$\lambda_{7}$ are in general complex.

In the $\Phi$-basis, both Higgs doublets get a vev. Without loss of
generality, one can write

$\left<\Phi_{1}^{\dagger}\Phi_{1}\right>=\dfrac{v_{1}^{2}}{2}\,,\qquad\left<\Phi_{2}^{\dagger}\Phi_{2}\right>=\dfrac{v_{2}^{2}}{2}\,,\qquad\xi\equiv\arg\left<\Phi_{1}^{\dagger}\Phi_{2}\right>$
(A.2)

where, $v_{1},v_{2},\xi$ are real,
$v^{2}=v_{1}^{2}+v_{2}^{2}=(246\,\text{GeV})^{2}$ and

$\tan\beta\equiv\frac{v_{2}}{v_{1}}\,.$ (A.3)

Under the $\mathbb{Z}_{2}$ symmetry the doublets transform differently as

$\Phi_{1}\to+\Phi_{1}\qquad\text{and}\qquad\Phi_{2}\to-\Phi_{2}$ (A.4)

which in turn enforces the condition

$\lambda_{6}=\lambda_{7}=0$ (A.5)

We allow the discrete symmetry to be softly broken in the potential by
$m^{2}_{12}$.

The Higgs basis is defined by the condition that only one doublet admits a
vev, i.e.

$\left<H_{1}^{\dagger}H_{1}\right>=\dfrac{v^{2}}{2}\,,\qquad\left<H_{2}^{\dagger}H_{2}\right>=0\,.$
(A.6)

It can be reached by a rotation of the doublets

$\begin{pmatrix}e^{-i\xi/2}H_{1}\\\ H_{2}\\\
\end{pmatrix}=\begin{pmatrix}c_{\beta}&s_{\beta}e^{-i\xi}\\\
-s_{\beta}e^{i\xi}&c_{\beta}\\\ \end{pmatrix}\begin{pmatrix}\Phi_{1}\\\
\Phi_{2}\\\ \end{pmatrix}$ (A.7)

As the $\mathbb{Z}_{2}$-symmetry is not defined in this basis, $\lambda_{6}$\-
and $\lambda_{7}$-like terms are again developed. The potential reads

$\displaystyle V(H_{1},H_{2})=\,$
$\displaystyle\widetilde{m}_{1}^{2}H_{1}^{\dagger}H_{1}+\widetilde{m}_{2}^{2}H_{2}^{\dagger}H_{2}-\left(\widetilde{m}_{12}^{2}H_{1}^{\dagger}H_{2}+\text{h.c.}\right)+\dfrac{1}{2}\widetilde{\lambda}_{1}\left(H_{1}^{\dagger}H_{1}\right)^{2}+$
$\displaystyle+\dfrac{1}{2}\widetilde{\lambda}_{2}\left(H_{2}^{\dagger}H_{2}\right)^{2}+\widetilde{\lambda}_{3}\left(H_{1}^{\dagger}H_{1}\right)\left(H_{2}^{\dagger}H_{2}\right)+\widetilde{\lambda}_{4}\left(H_{1}^{\dagger}H_{2}\right)\left(H_{2}^{\dagger}H_{1}\right)+$
(A.8)
$\displaystyle+\left[\dfrac{1}{2}\widetilde{\lambda}_{5}\left(H_{1}^{\dagger}H_{2}\right)^{2}+\widetilde{\lambda}_{6}\left(H_{1}^{\dagger}H_{1}\right)\left(H_{1}^{\dagger}H_{2}\right)+\widetilde{\lambda}_{7}\left(H_{2}^{\dagger}H_{2}\right)\left(H_{1}^{\dagger}H_{2}\right)+\text{h.c.}\right]\,.$
(A.9)

The tilded parameters are defined in terms of the original ones via

$\displaystyle\widetilde{m}_{1}^{2}$
$\displaystyle=m_{1}^{2}c_{\beta}^{2}+m_{2}^{2}s_{\beta}^{2}+s_{2\beta}{\texttt{Re}}(m_{12}^{2}e^{i\xi})\,,$
(A.10) $\displaystyle\widetilde{m}_{2}^{2}$
$\displaystyle=m_{1}^{2}s_{\beta}^{2}+m_{2}^{2}c_{\beta}^{2}-s_{2\beta}{\texttt{Re}}(m_{12}^{2}e^{i\xi})\,,$
$\displaystyle\widetilde{m}_{12}^{2}e^{i\xi/2}$
$\displaystyle=(m_{2}^{2}-m_{1}^{2})c_{\beta}s_{\beta}+{\texttt{Re}}(m_{12}^{2}e^{i\xi})c_{2\beta}+i{\texttt{Im}}(m_{12}^{2}e^{i\xi})\,,$

and

$\displaystyle\widetilde{\lambda}_{1}$
$\displaystyle=\lambda_{1}c_{\beta}^{4}+\lambda_{2}s_{\beta}^{4}+\lambda_{345}c^{2}_{\beta}s^{2}_{\beta}\,,$
(A.11) $\displaystyle\widetilde{\lambda}_{2}$
$\displaystyle=\lambda_{1}s_{\beta}^{4}+\lambda_{2}c_{\beta}^{4}+\lambda_{345}c^{2}_{\beta}s^{2}_{\beta}\,,$
$\displaystyle\widetilde{\lambda}_{3}$
$\displaystyle=\lambda_{3}+(\lambda_{1}+\lambda_{2}-\lambda_{345})\frac{s^{2}_{2\beta}}{4}\,,$
$\displaystyle\widetilde{\lambda}_{4}$
$\displaystyle=\lambda_{4}+(\lambda_{1}+\lambda_{2}-\lambda_{345})\frac{s^{2}_{2\beta}}{4}\,,$
$\displaystyle\widetilde{\lambda}_{5}e^{i\xi/2}$
$\displaystyle=\dfrac{s^{2}_{2\beta}}{4}(\lambda_{1}+\lambda_{2}-\lambda_{345})+{\texttt{Re}}(\lambda_{5}e^{2i\xi})+ic_{2\beta}{\texttt{Im}}(\lambda_{5}e^{2i\xi})\,,$
$\displaystyle\widetilde{\lambda}_{6}e^{i\xi/2}$
$\displaystyle=-\frac{s_{2\beta}}{2}(\lambda_{1}c_{\beta}^{2}-\lambda_{2}s_{\beta}^{2}-\lambda_{345}c_{2\beta}-i{\texttt{Im}}(\lambda_{5}e^{2i\xi}))\,,$
$\displaystyle\widetilde{\lambda}_{7}e^{i\xi/2}$
$\displaystyle=-\frac{s_{2\beta}}{2}(\lambda_{1}s_{\beta}^{2}-\lambda_{2}c_{\beta}^{2}+\lambda_{345}c_{2\beta}+i{\texttt{Im}}(\lambda_{5}e^{2i\xi}))$

where
$\lambda_{345}=\lambda_{3}+\lambda_{4}+{\texttt{Re}}(\lambda_{5}e^{2i\xi})$.

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