# Loop-induced Higgs decays in two Higgs doublet scenarios: some observations

Snehit Panghal<EMAIL_ADDRESS>Biswarup Mukhopadhyaya
<EMAIL_ADDRESS>
###### Abstract

We study the decay of three neutral Higgs bosons into $\gamma\gamma$ and
$Z\gamma$ in the four major types of two Higgs doublet models with CP
conserved. Particular focus is made on the possibility of the width of the
latter decay dominating over the former. For the 125-GeV scalar, such a
possibility is mostly excluded when one imposes all theoretical and
experimental constraints, other than the Higgs data at the Large Hadron
Collider. For the other CP-even neutral scalar, there remain regions in each
kind of model where the $Z\gamma$ mode dominates. For the CP-odd state, this
possibility is restricted to only the models of Types II and Y, in certain
regions of the parameter space.

##  1 Introduction

Ever since the discovery of a 125-GeV scalar at the Large Hadron Collider
(LHC)[1, 2], we have accumulated enough data to be convinced that this scalar
plays a significant role in executing spontaneous breakdown of electroweak
symmetry, as laid down in the standard model (SM) of particle physics[3, 4,
5]. Whether the role is exclusive is, however, not completely clear yet. In
principle, there may exist additional scalars belonging to doublets (or even
higher representations) of SU(2), which participate in electroweak symmetry
breaking (EWSB) following the same paradigm. While the issue could be largely
settled by measuring precisely the quartic self-interaction strength of the
125-GeV scalar, such measurement is not an immediate possibility yet[6]. It
is, therefore, hardly surprising that additional scalar multiplets should be
postulated and examined in the light of the available data. The minimal
templates of such scalar sectors beyond the standard model (BSM) are two Higgs
doublet models (2HDM)[7].

A crucial test for any 2HDM scenario lies in the prediction of loop-induced
decays of scalars, especially those of the 125-GeV particle, which we denote
by $h$. Such decays which are not allowed at the tree-level are
$h\rightarrow\gamma\gamma$ and $h\rightarrow Z\gamma$. While the former has
already been measured to have a branching ratio (BR) of $(2.27\pm 2.1\%)\times
10^{-3}$, the latter is decisively found to have smaller rates[8, 9], thus
confirming that the diphoton channel has larger decay width. In this paper, we
shall scan the parameter space of each of four major types of 2HDM, studying
the BR for the diphoton mode vis-à-vis the $Z\gamma$ mode for not only $h$ but
also the other neutral spin-0 field in each scenario.

The four types of 2HDM considered here are those of Type-I, Type-II, Type-X
and Type-Y (the flipped) kind111There are some other scenarios, in whose
context the relative strength of $Z\gamma$ and $\gamma\gamma$ decay channels
have been explored. See, for example, [10].. There are various motivations for
such extensions, ranging from supersymmetry[11] to the anomalous magnetic
moment of the muon[12, 13].The main difference among them lies in the nature
of their Yukawa couplings to quarks and leptons. In each of them, there exist
in addition to $h$, an additional neutral scalar $H$, a neutral pseudoscalar
$A$, and a pair of mutually conjugate charged scalars $H^{\pm}$. While the
additional scalars may be beyond current experimental reach, there are
contributions of the charged scalars even in
$h\rightarrow\gamma\gamma,Z\gamma$. Thus, the 2HDM parameter space in each
case plays a role in deciding the BR of each of these decays, as well as their
relative strength. Moreover, the 2HDM parameter space is additionally limited
by theoretical and experimental constraints. While the former category
comprises vacuum stability as well as perturbative unitarity, the latter
includes direct search limits on the additional scalars, electroweak
constraints (EWPC), and also those from rare decays in flavor physics, such as
$b\rightarrow s\gamma$ and $B_{s}\rightarrow\mu^{+}\mu^{-}$. And, last but not
the least, the existing data on the particle $h$ imposes crucial on any BSM
scenario including 2HDM.

The detailed study reported here takes all the above constraints into account.
There emerges, in particular, a rather striking observation: the diphoton
channel for $h$ is automatically predicted over most of the parameter space to
have larger BR than $Z\gamma$ in all four 2HDM types, once all constraints
other than those from $h$-data are taken into account. One thus concludes that
the thus allowed regions for the four types of 2HDM are consistent with the
observations on loop induced decays of the 125-GeV scalar. In addition, we
also identify regions where $Z\gamma$ can dominate over diphotons in the
decays of $H$ and $A$. This can help in identifying to which category of 2HDM
such scalars belong, if they are revealed in the upcoming data.

##  2 The major types of 2HDM

The 2HDM is the simplest extension of the SM Higgs, wherein there is an
additional scalar $SU(2)_{L}$ doublet. The doublets can be expanded about
their vacuum expectation values(vev) as[7, 14, 15, 16]

$\Phi_{1}=\left(\begin{array}[]{c}\Phi_{1}^{+}\\\
\frac{v_{1}+\rho_{1}+i\eta_{1}}{\sqrt{2}}\end{array}\right)\quad,\quad\Phi_{2}=\left(\begin{array}[]{c}\Phi_{2}^{+}\\\
\frac{v_{2}+\rho_{2}+i\eta_{2}}{\sqrt{2}}\end{array}\right),$ (1)

where SM vacuum expectation value, $v=246$ GeV is expressed as
$v=\sqrt{v_{1}^{2}+v_{2}^{2}}$. Thus, unlike SM, there are eight real scalar
fields in the model, rather than four. From Goldstone’s theorem, three of the
fields gives mass to the W and Z bosons because of the spontaneous symmetry
breaking[17], and the rest of the five correspond to different Higgs boson
particles. Three of them are neutral bosons, two scalars($h$ and $H$) and one
pseudoscalar($A$), and the other two fields correspond to positively and
negatively charged Higgs($H^{\pm}$).

Experimentally, Flavor-Changing Neutral currents(FCNC) are highly suppressed
[18, 19]. However, these are allowed at tree-level in 2HDM. The model is thus
constrained using the Paschos-Glashow–Weinberg condition, which states that
the requirement of naturally flavor conserving currents implies that fermions
of a given charge couple with a single scalar doublet only[20, 21]. A simple
way to ensure such an arrangement is to impose discrete symmetries on the
Lagrangian. One such Group is the $Z_{2}$ group. Under the imposition of this
discrete symmetry on the Lagrangian, the scalar-field doublets transform as
$\Phi_{1}\rightarrow\Phi_{1}$, $\Phi_{2}\rightarrow-\Phi_{2}$ or vice versa,
and the quarks and leptons can have odd or even transformations over the same
groups. Combination of different sets of these transformations lead to
different variations in coupling of fermions with scalars. In general, these
combinations can be classified into four major types of 2HDM, depending on how
the fermions couple to the two scalar doublets. The Yukawa Lagrangian for
these four major types of 2HDM can be expressed as

$\displaystyle\mathcal{L}_{\text{Yukawa }}^{2\mathrm{HDM}}=$
$\displaystyle-\sum_{f=u,d,\ell}\frac{M_{f}}{v}\left(\kappa_{h}^{f}\bar{f}fh+\kappa_{H}^{f}\bar{f}fH-i\kappa_{A}^{f}\bar{f}\gamma_{5}fA\right)$
(2)
$\displaystyle-\left\\{\frac{\sqrt{2}V_{ud}}{v}\bar{u}\left(m_{u}\kappa_{A}^{u}\mathrm{P}_{L}+m_{d}\kappa_{A}^{d}\mathrm{P}_{R}\right)dH^{+}+\frac{\sqrt{2}m_{\ell}\kappa_{A}^{\ell}}{v}\overline{\nu_{L}}\ell_{R}H^{+}+\text{H.c.
}\right\\},$

where the coupling constant values are mentioned in Table 1. These values can
be understood in the context of the requirement to constrain the scalar
potential

In the strictest versions of these theories, the $Z_{2}$ symmetry applies to
all terms in the potential. However, one can get away without any tree-level
FCNC even if $Z_{2}$ is softly broken in the pure scalar sector[22, 23], which
just yield finite corrections to masses. On the other hand, there can be
quartic terms that break the $Z_{2}$ symmetry in the ’hard’ way. These induce
logarithmically divergent loop contributions (see Figure 1) to, say,
$\bar{d}d\Phi_{2}$ coupling in a scenario where $\bar{d}d\Phi_{1}$ terms are
set down at the tree level. This results in the appearance of a divergent part
in the $\bar{d}d\Phi_{1}$ coupling, which effectively violates natural flavor
conservation at higher scales, via renormalization group(RG) evolution. The
FCNC processes thus induced at different scales can take a toll on the 2HDM
phenomenology[24, 25]. Additionally, the RG analysis of 2HDM also shows that
soft $Z_{2}$ symmetry breaking extends the valid parameter space in the model,
whereas an exact symmetry makes the parameter space of the model too
constrained[26, 27]. The calculations and analyses here will thus be on the
CP-conserving 2HDM with softly-breaking $Z_{2}$ symmetry.

(a) Soft $Z_{2}$ breaking term

(b) Hard $Z_{2}$ breaking term

Figure 1: Example of an effective Lagrangian terms which generate FCNC. These
terms from soft $Z_{2}$ symmetry breaking 1(a) have finite radiative
corrections, whereas the ones from hard $Z_{2}$ symmetry breaking 1(b) have
infinite radiative corrections.[24]

Thus, for a CP-conserving 2HDM with softly breaking $Z_{2}$ symmetry, the
general potential is

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

When expressed in the terms of mass eigenstates of the charged and neutral
Higgs bosons, the parameters $m_{11}^{2},m_{22}^{2}$, and
$\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}$ in the quartic potential can
be written in terms of $\alpha$: the neutral CP-even Higgs diagonalizing
mixing angle, tan$\beta=\frac{v_{2}}{v_{1}}$: the ratio of the two vev, and
the masses $m_{h}$, $m_{H}$, $m_{A}$, $m_{H\pm}$ of all the Higgs Bosons. The
mass $m_{h}$ is taken to be the 125 GeV Higgs boson. The Higgs decay width
values will thus depend on these parameters, and have important
phenomenological implications. The coupling constants of the three neutral
Higgs bosons with the different fermions and bosons for the four types of 2HDM
in the mass basis, as given in Table 1 below.

| Type I | Type II | Type Lepton-specific(X) | Type Flipped(Y)
---|---|---|---|---
$\kappa^{u}_{h}$ | $\frac{\cos{\alpha}}{\sin{\beta}}$ | $\frac{\cos{\alpha}}{\sin{\beta}}$ | $\frac{\cos{\alpha}}{\sin{\beta}}$ | $\frac{\cos{\alpha}}{\sin{\beta}}$
$\kappa^{d}_{h}$ | $\frac{\cos{\alpha}}{\sin{\beta}}$ | $-\frac{\sin{\alpha}}{\cos{\beta}}$ | $\frac{\cos{\alpha}}{\sin{\beta}}$ | $-\frac{\sin{\alpha}}{\cos{\beta}}$
$\kappa^{l}_{h}$ | $\frac{\cos{\alpha}}{\sin{\beta}}$ | $-\frac{\sin{\alpha}}{\cos{\beta}}$ | $-\frac{\sin{\alpha}}{\cos{\beta}}$ | $\frac{\cos{\alpha}}{\sin{\beta}}$
$\kappa^{u}_{H}$ | $\frac{\sin{\alpha}}{\sin{\beta}}$ | $\frac{\sin{\alpha}}{\sin{\beta}}$ | $\frac{\sin{\alpha}}{\sin{\beta}}$ | $\frac{\sin{\alpha}}{\sin{\beta}}$
$\kappa^{d}_{H}$ | $\frac{\sin{\alpha}}{\sin{\beta}}$ | $\frac{\cos{\alpha}}{\cos{\beta}}$ | $\frac{\sin{\alpha}}{\sin{\beta}}$ | $\frac{\cos{\alpha}}{\cos{\beta}}$
$\kappa^{l}_{H}$ | $\frac{\sin{\alpha}}{\sin{\beta}}$ | $\frac{\cos{\alpha}}{\cos{\beta}}$ | $\frac{\cos{\alpha}}{\cos{\beta}}$ | $\frac{\sin{\alpha}}{\sin{\beta}}$
$\kappa^{u}_{A}$ | $\cot{\beta}$ | $\cot{\beta}$ | $\cot{\beta}$ | $\cot{\beta}$
$\kappa^{d}_{A}$ | $-\cot{\beta}$ | $\tan{\beta}$ | $-\cot{\beta}$ | $\tan{\beta}$
$\kappa^{l}_{A}$ | $-\cot{\beta}$ | $\tan{\beta}$ | $\tan{\beta}$ | $-\cot{\beta}$
$\kappa^{W/Z}_{h}$ | $\sin{(\beta-\alpha)}$
$\kappa^{W/Z}_{H}$ | $\cos{(\beta-\alpha)}$
$\kappa^{H\pm}_{h}$ | $\sin(\beta-\alpha)(m_{h}^{2}+2m_{H^{\pm}}^{2}-2\frac{m_{12}^{2}}{\sin\beta\cos\beta})+2\cos(\beta-\alpha)(m_{h}^{2}-\frac{m_{12}^{2}}{\sin\beta\cos\beta})\cot(2\beta)$
$\kappa^{H\pm}_{H}$ | $\cos(\beta-\alpha)(m_{h}^{2}+2m_{H^{\pm}}^{2}-2\frac{m_{12}^{2}}{\sin\beta\cos\beta})+2\sin(\beta-\alpha)(m_{h}^{2}-\frac{m_{12}^{2}}{\sin\beta\cos\beta})\cot(2\beta)$
Table 1: Quark, lepton, vector boson and self Higgs couplings to neutral
Higgs- h, H and A in the four types of 2HDM[7]. $\alpha$ is the mass matrix
diagonalising mixing angle, and $\tan\beta$ is the ratio of the two vev of the
scalar potential of 2HDM.

##  3 The $\gamma\gamma$ and $Z\gamma$ decay modes in SM and 2HDM

###  3.1 Standard Model

In the SM, the Higgs field doesn’t interact with photons, which makes it
massless. Thus, a Higgs boson can decay to photons only via loop induced
diagrams where fermions as well as gauge bosons participate.

The expression for Higgs to the $\gamma\gamma$ partial decay width at leading
order is given by

$\Gamma(h\rightarrow\gamma\gamma)=\frac{G_{F}\alpha_{em}^{2}M_{H}^{3}}{128\sqrt{2}\pi^{3}}\left|\sum_{f}N_{c}Q_{f}^{2}T_{1/2}^{H}\left(\tau_{f}\right)+T_{1}^{H}\left(\tau_{W}\right)\right|^{2}.$
(4)

where $G_{F}$ is the Fermi constant, $\alpha_{em}$ is the fine structure
constant, $N_{c}$ is the number of colored charges, $Q_{f}$ is the charge of a
particular fermion, $M_{H},M_{f},M_{W}$ are the Higgs mass, fermion masses and
the W-boson mass, respectively. The summation over $f$ is for all fermion
loops, and $\tau_{f}=\frac{M^{2}_{H}}{4M^{2}_{f}}$. The functions used are

$\displaystyle T_{1/2}^{H}(x)$
$\displaystyle=2\left[\frac{1}{x}+\left(\frac{1}{x}-\frac{1}{x^{2}}\right)f(x)\right]$
(5) $\displaystyle T_{1}^{H}(x)$
$\displaystyle=-\left[2+\frac{3}{x}+3(2x-1)\frac{f(x)}{x^{2}}\right]$

where $f(x)$ is defined as

$f(x)=\begin{cases}\arctan^{2}\sqrt{\frac{x}{1-x}}&x\leq 1\\\
-0.25\left[\log\frac{1+\sqrt{1-x^{-1}}}{1-\sqrt{1-x^{-1}}}-i\pi\right]^{2}&x>1\end{cases},\text{
and}$ (6)

The expression for Higgs to $Z\gamma$ partial decay width at leading order is
given by

$\displaystyle\Gamma(h\rightarrow Z\gamma)=$
$\displaystyle\frac{G_{F}^{2}M_{W}^{2}\alpha_{em}M_{H}^{3}}{64\pi^{4}}\left(1-\frac{M_{Z}^{2}}{M_{H}^{2}}\right)^{3}\left|\sum_{f}N_{f}\frac{Q_{f}\hat{v}_{f}}{c_{W}}T_{1/2}^{H}\left(\tau_{f},\lambda_{f}\right)+T_{1}^{H}\left(\tau_{W},\lambda_{W}\right)\right|^{2}$
(7)

where $\tau_{i}=4M_{i}^{2}/M_{H}^{2}\text{ and
}\lambda_{i}=4M_{i}^{2}/M_{Z}^{2}$, $M_{Z}$ being the mass of Z-boson, and the
functions used are

$\displaystyle T_{1/2}^{H}(x,y)$
$\displaystyle=\left[I_{1}(x,y)-I_{2}(x,y)\right]$ (8) $\displaystyle
T_{1}^{H}(x,y)$
$\displaystyle=\cos{\theta_{W}}\left\\{4\left(3-\tan^{2}{\theta_{W}}\right)I_{2}(x,y)+\left[\left(1+\frac{2}{x}\right)\tan^{2}{\theta_{W}}-\left(5+\frac{2}{x}\right)\right]I_{1}(x,y)\right\\}$

with $\hat{v}_{f}=2I_{f}^{3}-4Q_{f}\sin^{2}\theta_{W}$. The functions $I_{1}$
and $I_{2}$ are,

$\displaystyle
I_{1}(x,y)=\frac{xy}{2(x-y)}+\frac{x^{2}y^{2}}{2(x-y)^{2}}\left[f\left(x^{-1}\right)-f\left(y^{-1}\right)\right]+\frac{x^{2}y}{(x-y)^{2}}\left[g\left(x^{-1}\right)-g\left(y^{-1}\right)\right]$
(9) $\displaystyle
I_{2}(x,y)=-\frac{xy}{2(x-y)}\left[f\left(x^{-1}\right)-f\left(y^{-1}\right)\right]$

where the function $g(x)$ is

$g(x)=\begin{cases}\sqrt{x^{-1}-1}\sin^{-1}\sqrt{x}&x<1\\\
\frac{\sqrt{1-x^{-1}}}{2}\left[\log\frac{1+\sqrt{1-x^{-1}}}{1-\sqrt{1-x^{-1}}}-i\pi\right]&x\geq
1\end{cases}$ (10)

Note that the fermion loops interfere destructively with the W-boson loops
because of their antisymmetric properties. The dominant contribution comes
from the W-boson loops followed by the top-quark fermion loop, because of its
heavy mass. All constants and mass values needed to calculate these partial
decay widths are experimentally known. The SM thus gives a unique value for
these partial decay widths.

###  3.2 Two Higgs doublet models

In 2HDM, what may yield insights into details of the model parameters include
the $\gamma\gamma$, $Z\gamma$ decay rates of not only $h$, but also of $H$ and
$A$. While the fermion loops in general contribute to all the decays, the
$W$-loops are present for $h$\- and $H$-decays only. In addition,
$H^{\pm}$-driven loops contribute to the loop-induced decay widths of $h$ and
$H$, but not $A$. These features lend richness to the study of the
$\gamma\gamma$ and $Z\gamma$ channels, as we shall see below.
The partial decay width of $h$ and $H$ to $\gamma\gamma$ in the leading order
are given by

$\Gamma(h/H\rightarrow\gamma\gamma)=\frac{G_{F}\alpha_{em}^{2}M_{H}^{3}}{128\sqrt{2}\pi^{3}}\left|\sum_{f}N_{c}\kappa_{h/H}^{f}Q_{f}^{2}T_{1/2}^{H}\left(\tau_{f}\right)+\kappa^{W}_{h/H}T_{1}^{H}\left(\tau_{W}\right)+\kappa_{h/H}^{H\pm}T_{0}^{H}(\tau_{H\pm})\right|^{2}$
(11)

where

$\displaystyle T_{0}^{H}(x)$
$\displaystyle=-\left[\frac{1}{x}-\frac{f(x)}{x^{2}}\right]$ (12)

Also, $\tau_{i}=\frac{M^{2}_{H}}{4M^{2}_{i}}$.
The partial decay width of CP-even scalar to Z$\gamma$ in the leading order is
given by

$\displaystyle\Gamma(h/H\rightarrow Z\gamma)=$
$\displaystyle\frac{G_{F}^{2}M_{W}^{2}\alpha_{em}M_{H}^{3}}{64\pi^{4}}\left(1-\frac{M_{Z}^{2}}{M_{H}^{2}}\right)^{3}$
(13)
$\displaystyle\times\left|\sum_{f}N_{c}\kappa_{h/H}^{f}\frac{Q_{f}\hat{v}_{f}}{c_{W}}T_{1/2}^{H}\left(\tau_{f},\lambda_{f}\right)+\kappa_{h/H}^{W}T_{1}^{H}\left(\tau_{W},\lambda_{W}\right)+\kappa_{h/H}^{H^{\pm}}T_{0}^{H}(\tau_{H\pm},\lambda_{H\pm})\right|^{2}$

where

$T_{0}^{H}(x,y)=\frac{\cos{2\theta_{W}}}{\sin{\theta_{W}}}I_{1}(x,y)\frac{M_{W}^{2}}{M_{H^{\pm}}^{2}}$
(14)

$\theta_{W}$ being the electroweak mixing angle. The expressions are very
similar to the SM result, with an important difference in the coupling
constants. From Table 1, it can be seen that the magnitude of the contribution
of fermions and W-boson loops depend on the parameters $\alpha$ and $\beta$.
The contribution from charged Higgs loop depend additionally on the parameter
$m_{12}^{2}$ and the unknown mass of charged Higgs boson, $m_{H^{\pm}}$.

For the CP-odd scalar $A$, only the fermion loops contribute to the in the
leading order of the decay width, yielding

$\Gamma(A\rightarrow\gamma\gamma)=\frac{G_{F}\alpha_{em}^{2}M_{A}^{3}}{128\sqrt{2}\pi^{3}}\left|\sum_{f}N_{c}\kappa_{A}^{f}Q_{f}^{2}T_{1/2}^{A}\left(\tau_{f}\right)\right|^{2},\text{
and}$ (15) $\Gamma(A\rightarrow
Z\gamma)=\frac{G_{F}^{2}M_{W}^{2}\alpha_{em}M_{H}^{3}}{64\pi^{4}}\left(1-\frac{M_{Z}^{2}}{M_{H}^{2}}\right)^{3}\left|\sum_{f}N_{f}\kappa_{A}^{f}\frac{Q_{f}\hat{v}_{f}}{c_{W}}I_{2}\left(\tau_{f},\lambda_{f}\right)\right|^{2}$
(16) $\text{where }T_{1/2}^{A}(x)=\frac{2f(x)}{x}$ (17)

The only unknown parameters controlling the decay width of $A$ are thus
$m_{A}$ and $\tan\beta$.

##  4 Results for the SM-like $h$

In the case of 125 GeV Higgs, the SM gives a unique prediction for the loop
induced decays, namely, $B(h\rightarrow\gamma\gamma)>B(h\rightarrow Z\gamma)$.
In 2HDM, however, with four unknown parameters-
$\alpha,\beta,m_{12}^{2},m_{H^{\pm}}$ for the light Higgs, this inequality
reverses in some parameter space regions. This happens for parameters where
the $W$-loop and $H^{\pm}$-loop contributors begin to get suppressed, but at a
particular point $\Gamma(h\rightarrow\gamma\gamma)$ is subject to a stronger
suppression than $\Gamma(h\rightarrow Z\gamma)$ and the latter thus becomes
larger in value in comparison.
The above statement can be understood by considering figure 2, where the
branching ratios of $\gamma\gamma$ and $Z\gamma$ decays are plotted against
$\alpha$ for other fixed parameter values, $\tan\beta$=2, $m_{12}^{2}$=800,
$m_{H\pm}$=1500. Both have a profile similar to the decay of $h\rightarrow WW$
as the dominant loop contributions are from W-bosons. $h$ couples to $W$ boson
with a coupling strength proportional to $\sin(\beta-\alpha)$(Table 1), and
$\Gamma(h\rightarrow WW)$ thus vanishes for $\beta=\alpha$. The $\gamma\gamma$
and $Z\gamma$ decays, however, have small yet significant contributions of
fermion loops with different amplitudes as shown in Equation 11 and 13. The
two decay widths thus dip at different values of $\alpha$. This gives us a
point in the parameter space where the dip for $\gamma\gamma$ precedes or
succeeds the point of dip for $Z\gamma$, leading to the possibility of
$B(h\rightarrow\gamma\gamma)<B(h\rightarrow Z\gamma)$ in the nearby region of
values of $\alpha$ corresponding to the dip in $h\rightarrow\gamma\gamma$
rates (see Figure 2). Thus, there are regions in the parameter space where
$B(h\rightarrow Z\gamma)>B(h\rightarrow\gamma\gamma)$ for all four types of
2HDM. (Figure 3)

Figure 2:  $B(h\rightarrow Z\gamma)$ and $B(h\rightarrow\gamma\gamma)$ plotted
against the mixing angle $\alpha$. Other parameter values are fixed with the
following values: $\tan\beta$=2, $m_{12}^{2}$=800, $m_{H\pm}$=1500.

Figure 3: The ratio of the two branching ratios $B(h\rightarrow Z\gamma)$ and
$B(h\rightarrow\gamma\gamma)$ plotted against the mixing angle $\alpha$, for
different values of $\tan\beta,m_{12}^{2},m_{H^{\pm}}$.

However, such a prospect is disallowed over most of the parameter space by
constraints on the scalar sector itself. These constraints are applied using
ScannerS[28]. The results are confirmed independently using 2HDMC[29], Super-
iso[30], HiggsBounds[31] and HiggsSignals[32]. The sample regions of parameter
space considered are the following:
$\alpha=[-\pi/2,\pi/2],\tan\beta=[0.1,70],m_{12}^{2}=[0.001,700000]GeV^{2},m_{H^{\pm}}=[20GeV,2000GeV]$.

When subjected to constraints of, Vacuum stability[33], Perturbative
Unitarity[34], Potential positivity, Electroweak precision
constraints(EWPC)[35], flavor constraints[36], HiggsBounds and HiggsSignal
measurement constraints, all regions of the parameter space where
$B(h\rightarrow\gamma\gamma)<B(h\rightarrow Z\gamma)$ are excluded for all
types of 2HDM (Figure 4). This is mainly because the majority of the parameter
space regions where $Z\gamma$ decay exceeds $\gamma\gamma$ are nearby points
where the major decay channels of the 125 GeV scalar, $h\rightarrow WW$ and
$h\rightarrow ZZ$, vanish or are highly suppressed. Experimental constraints
thus exclude the parameter values.

Figure 4: Sample regions of parameter space of the four different 2HDM types
allowed by all theoretical and experimental constraints of 2HDM.
$B(h\rightarrow\gamma\gamma)>B(h\rightarrow Z\gamma)$ everywhere.

However, even without considering the available data on the 125 GeV scalar,
the theoretical and EWPC results, together with flavor constraints as well as
analysis based on HiggsBounds ensure that the overwhelming majority of regions
where $B(h\rightarrow\gamma\gamma)<B(h\rightarrow Z\gamma)$ are ruled out,
with exception in a restricted parameter space regions for Type I, II and Type
X 2HDM (Figure 5).

(a)

(b)

Figure 5: (a)Sample regions of parameter space allowed by theoretical and
experimental constraints of 2HDM, excluding the HiggsSignals data for 125 GeV
scalar. (b)The subset of points in “(a)” where
$B(h1\rightarrow\gamma\gamma)<B(h\rightarrow Z\gamma)$, when data on $h$ are
not used.

The theoretical and HiggsBounds constraints restrict $m_{H}$ and $m_{12}^{2}$
to an upper limit of $\sim$ 310 GeV and, $\sim$ 15000 $GeV^{2}$, respectively.
The EWPC thus also restricts $m_{A}$ and $m_{H^{\pm}}$ to an upper limit
(Figure 6). This is the reason exceptionally few points are allowed in Type II
and none for Type-Y as flavor constraints restrict $m_{H^{\pm}}\geq 600$ GeV
for these two model types.

Figure 6: (Left) Other regions of the parameter space:
$m_{H},m_{H}\pm,m_{A},m_{12}^{2}$, allowed by theoretical and experimental
constraints of 2HDM, excluding the HiggsSignals data for 125 GeV scalar.
(Right) The corresponding plots are subset points where $B(h\rightarrow
Z\gamma)>B(h\rightarrow\gamma\gamma)$.

The above feature, namely most of the parameter spaces of 2HDM being
disallowed by constraints excluding the 125-GeV scalar data, has been
illustrated above for the state of brevity. However, it has been checked that
a more exhaustive scan brings out the same feature.

##  5 Results for $H$ and $A$

###  5.1 Decays of the CP-even heavy higgs $H$

The branching ratios of the heavy scalar Higgs depend on five parameters of
the model- $\alpha$, $\beta$, $m_{12}^{2}$, $m_{H^{\pm}}$, and $m_{H}$. The
decay channels $H\rightarrow\gamma\gamma$ and $H\rightarrow Z\gamma$ become
less significant with higher Higgs mass, as there are many other tree-level
Higgs decays like $H\rightarrow tt$, $H\rightarrow hh$ with much higher
partial decay widths that become significant. For $M_{H}\leq 250GeV$, the loop
decays are relatively more significant in the regions of the parameter space,
where the couplings to the gauge boson or heavy fermion pairs come with
suppression factors arising out of parameters in the scalar potential.

Figure 7: The ratio of the two branching ratios $B(H\rightarrow Z\gamma)$ and
$B(H\rightarrow\gamma\gamma)$ plotted against $m_{H}$ for different values of
$\tan\beta,m_{12}^{2},m_{H^{\pm}},\alpha$.

For heavy scalar Higgs, there are possible regions in the parameter space
where $B(H\rightarrow Z\gamma)>B(H\rightarrow\gamma\gamma)$ for all four types
of 2HDM, as can be seen in Figure 7 and 8. The five unknown parameters allow
for varied possibilities in the values of decay amplitudes in equation 11 and
13.

Figure 8: The ratio of the two branching ratios $B(H\rightarrow Z\gamma)$ and
$B(H\rightarrow\gamma\gamma)$ plotted against $m_{H}$ for different values of
$\tan\beta,m_{12}^{2},m_{H^{\pm}},\alpha$.

Note that in Figure 7 and 8, the profile of $\frac{B(H\rightarrow
Z\gamma)}{B(H\rightarrow\gamma\gamma)}$ are similar for all four types of
2HDM, particularly for large values of $m_{H}$. This is because the dominant
contribution of the decay width comes from W-bosons loops, where the coupling
strength is proportional to $\cos(\beta-\alpha)$(Table 1) in all model types.
The contribution further increases when $m_{H}$ increases, thereby making the
profile practically independent of the model type. The trilinear couplings
with $H$ driving the charged Higgs loops, too, have the same parameter
dependence in all four models. The slight differences for low values of
$m_{H}$ is due to fermion loops having significant contribution to the decay
amplitude, which vary for different model type.
Applying the theoretical and experimental constraints on the model, including
HiggsSignals, allows $\Gamma(H\rightarrow
Z\gamma)>\Gamma(H\rightarrow\gamma\gamma)$ for Higgs masses
$m_{H}=[126GeV,2000GeV]$ in all four types of 2HDM. This is showcased by
plotting the five parameters against one another in Figure 9.

Figure 9: (Left) The regions of parameter space of
$m_{H},\tan\beta,m_{12}^{2},m_{H^{\pm}},\alpha$ allowed by theoretical and
experimental constraints for all 4 types of 2HDM plotted against one another.
The corresponding image on right are scatter points for which
$\Gamma(H\rightarrow Z\gamma)>\Gamma(H\rightarrow\gamma\gamma)$.

###  5.2 CP-odd Higgs

The CP-odd Higgs doesn’t couple to the pair of vectors or charged scalar
bosons. This makes its decay width such that major tree level and one-loop
level decays depend only on two parameters of the 2HDM, namely, $\tan\beta$
and $m_{A}$. $A\rightarrow Zh$ and $A\rightarrow ZH$, if kinematically
allowed, are the only major decay channels that depend on the parameter
$\alpha$, in addition to $\beta$.

Figure 10: The ratio of the two branching ratios $B(A\rightarrow Z\gamma)$ and
$B(A\rightarrow\gamma\gamma)$ plotted vs $m_{A}$ for different values of
$\tan\beta$

The loop decays being mediated only by the fermions explains why at low Higgs
mass these decays don’t have high branching ratios at fermio-phobic regions.

For the pseudoscalar Higgs, there are no regions in the parameter space where
$B(A\rightarrow Z\gamma)>B(A\rightarrow\gamma\gamma)$ for Type I and Type-X
2HDM. This happens because the major fermionic contributions to the loop-
induced decays are the heavier quarks. For all such quarks, the couplings to
$A$ are proportional to $\cot\beta$ in Type I and X (see Table 1). Thus, there
is no scope for some particular values of $\beta$ enhancing the contribution
to $\Gamma(A\rightarrow Z\gamma)$ to values greater than those of
$\Gamma(A\rightarrow\gamma\gamma)$. In Types II and Y, on the other hand, the
coupling to the top quark goes as $\cot\beta$ whereas that to the bottom quark
is proportional to $\tan\beta$. This makes the relative strengths of the
contributions to both the decay widths variable, depending on $\beta$. It is
found that some regions thus allow bigger contributions to
$\Gamma(A\rightarrow Z\gamma)$ as compared to
$\Gamma(A\rightarrow\gamma\gamma)$, for $\tan\beta>20$, once all constraints
are imposed (Figure 11).

Figure 11: (Left) The regions of parameter space spanned by $\tan\beta$ and
$m_{A}$ allowed by theoretical and experimental constraints for all four types
of 2HDM. The remaining parameters in the scalar sector have been marginalized
over. (Right) Points in the parameter space, in Types 1 and Y, where
$\Gamma(A\rightarrow Z\gamma)>\Gamma(A\rightarrow\gamma\gamma)$.

This leads us to an interesting conclusion: if a spin-0, CP-odd particle is
identified in future experiments, which is found to have larger branching
ratio to $Z\gamma$ than to $\gamma\gamma$, then one may identify the decaying
particle as one belonging to Types II or Y, as opposed to Types I or X.

##  6 Summary and Conclusions

We have considered the four major kinds of 2HDM, namely, those of Type-I,
Type-II, Type-X and Type-Y, and have studied the relative values of the
$\gamma\gamma$ and $Z\gamma$ branching ratios of all the neutral scalar
particles in the electroweak symmetry breaking sector of each one. We have
scanned the parameter space of each scenario over rather extensive ranges of
values of the scalar potential in each case. The scan has duly taken care of
theoretical constraints such as the vacuum stability requirement as well as
perturbative unitarity, electroweak precision constraints, and also
experimental constraints from the Higgs data, searches for heavy scalars, and
flavor constraints.

As far as the already discovered 125-GeV scalar is concerned, the diphoton
channel events are already under scrutiny and are clearly more abundant than
events in the $Z\gamma$ channel. Thus, any region of the parameter space in
either model, which corresponds to, $\Gamma(h\rightarrow
Z\gamma)\geq\Gamma(h\rightarrow\gamma\gamma)$ is automatically ruled out.
Remarkably, the overwhelming majority of allowed regions automatically ensure
that the $\gamma\gamma$ branching ratio dominates over that of $Z\gamma$. This
happens even without taking recourse to the analysis of data on the 125-GeV
scalar; the HiggsBounds and flavor constraints do it together with Electroweak
precision constraints. The limited regions of exceptions in Types I, II and X
are eliminated by the 125-GeV data, using the code HiggsSignals.

As for the heavier scalar H, there are regions in all four types of models
where the branching ratio in the $Z\gamma$ channel can exceed that into
$\gamma\gamma$. This statement is true even after all constraints are taken
into account.

The pseudoscalar A, however, always dominantly decays into $\gamma\gamma$ in
models of Type I and Type X. In Type II or Type Y, however, $Z\gamma$ rates
may exceed those for diphotons after all constraints are applied, for
$\tan\beta\geq 20$. Thus, for any discovered pseudoscalar which decays more
frequently into $Z\gamma$ than to $\gamma\gamma$, the candidature of Types I
or Type X will have to be cancelled.

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