#  $SU(2)_{L}$ triplet scalar as the origin of the $95\,$GeV excess?

Saiyad Ashanujjaman ID<EMAIL_ADDRESS>Department of Physics, SGTB Khalsa
College, Delhi 110007, India  Department of Physics and Astrophysics,
University of Delhi, Delhi 110007, India  Sumit Banik ID<EMAIL_ADDRESS>Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH–8057 Zürich,
Switzerland  Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland
Guglielmo Coloretti ID<EMAIL_ADDRESS>Physik-Institut,
Universität Zürich, Winterthurerstrasse 190, CH–8057 Zürich, Switzerland  Paul
Scherrer Institut, CH–5232 Villigen PSI, Switzerland  Andreas Crivellin ID
<EMAIL_ADDRESS>Physik-Institut, Universität Zürich,
Winterthurerstrasse 190, CH–8057 Zürich, Switzerland  Paul Scherrer Institut,
CH–5232 Villigen PSI, Switzerland  Bruce Mellado<EMAIL_ADDRESS>School
of Physics and Institute for Collider Particle Physics, University of the
Witwatersrand, Johannesburg, Wits 2050, South Africa  iThemba LABS, National
Research Foundation, PO Box 722, Somerset West 7129, South Africa  Anza-
Tshilidzi Mulaudzi<EMAIL_ADDRESS>School of Physics and
Institute for Collider Particle Physics, University of the Witwatersrand,
Johannesburg, Wits 2050, South Africa  iThemba LABS, National Research
Foundation, PO Box 722, Somerset West 7129, South Africa

###### Abstract

We explore the possibility that an $SU(2)_{L}$ triplet scalar with hypercharge
$Y=0$ is the origin of the $95\,$GeV di-photon excess. For a small mixing
angle with the Standard Model Higgs, its neutral component has naturally a
sizable branching ratio to $\gamma\gamma$ such that its Drell-Yan production
via $pp\to W^{*}\to HH^{\pm}$ is sufficient to obtain the desired signal
strength, where $H^{\pm}$ is the charged Higgs component of the triplet. The
predictions of this setup are: 1) The $\gamma\gamma$ signal has a $p_{T}$
spectrum different from gluon fusion but similar to associated production. 2)
Photons are produced in association with tau leptons and jets, but generally
do not fall into the vector-boson fusion category. 3) The existence of a
charged Higgs with $m_{H^{\pm}}\approx\\!(95\pm 5)\,$GeV leading to
$\sigma(pp\to\tau\tau\nu\nu)\approx 0.4\,$pb, which is of the same level as
the current limit and can be discovered with Run 3 data. 4) A positive
definite shift in the $W$ mass as suggested by the current global electroweak
fit.

††preprint: PSI-PR-23-20, ZU-TH 28/23, ICPP-71

##  I Introduction

The Standard Model (SM) is the currently accepted theoretical description of
the known constituents and interaction of matter. It has been successfully
tested in precision experiments [1, 2, 3] and the Brout-Englert-Higgs boson
[4, 5, 6, 7], the last missing piece, was finally discovered in 2012 at CERN
[8, 9, 10]. In fact, this $125\,$GeV particle has properties consistent with
the ones predicted by the SM [11, 12, 13, 14, 15]. However, this does not
exclude the existence of additional scalar bosons, as long as their role in
electroweak symmetry breaking is subleading and their production cross
sections are smaller than the ones of the SM-like Higgs [16, 17].

The minimality of the SM Higgs sector, i.e. the existence of a single
$SU(2)_{L}$ doublet scalar that simultaneously gives mass to the electroweak
(EW) gauge bosons and all fermions, is not guaranteed by any theoretical
principle or symmetry. A plethora of such extensions have been proposed in the
literature, including the addition of $SU(2)_{L}$ singlets [18, 19, 20],
doublets [21, 22, 23, 24, 25] and triplets [26, 27, 28, 29, 30, 31].

While Large Hadron Collider (LHC) searches for new particles did not lead to
any discovery (yet), there are interesting hints for new scalar bosons [32].
In particular, CMS [33, 34, 35] searches hint towards a neutral scalar $H$
decaying into two photons at 95 GeV. This is compatible with the latest ATLAS
result [36] and supported by $Z$-strahlung with $H\to b\bar{b}$ at LEP [37],
as well as by $\tau\tau$ [35] and $WW$ [38, 39, 40] searches.

So far, explanations of the $95\,$GeV excesses in terms of $SU(2)_{L}$
singlets and/or $SU(2)_{L}$ doublets were proposed in the literature [41, 42,
43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61,
62, 63, 64, 65], which all respect custodial symmetry at tree-level. However,
in case of a small vacuum expectation value (VEV), higher $SU(2)_{L}$
representations are compatible with the constraints from the $\rho$ parameter
[3]. Because the mass splitting among the $SU(2)_{L}$ components must be
small, i.e. of the order of the VEV of the new scalar, LHC searches for
multiply charged scalars [66, 67, 68, 69] would exclude a mass of $\approx
95\,$GeV for the neutral component. Therefore, we are left with the
$SU(2)_{L}$ triplet with hypercharge zero as the only alternative to singlets
and doublets. It is well known that this field provides a positive definite
shift in the $W$ mass (w.r.t. the SM prediction) [70, 71, 72, 73, 74, 75, 76,
77, 78, 79, 80, 81], as motivated by the current global electroweak fit [82,
83, 84] (driven by the CDF II result [85]). However, its collider
phenomenology has been barely studied. In this article, we study the viability
of $Y=0$ triplet as an alternative in addressing the hints for a
$\approx\\!95\,$GeV scalar.

##  II  The Model

The SM supplemented by a $SU(2)_{L}$ triplet scalar with hypercharge 0, is
commonly referred to as the $\Delta$SM [86, 87, 88, 89, 90, 91, 92, 93]. The
covariant derivative for $SU(2)_{L}$ is fixed, in the usual conventions, by
the generators, i.e. $T_{k}=\frac{\sigma_{k}}{2}$ for the doublet, with
$\sigma_{k}$ being the Pauli matrices. This then fixes the structure constants
$f_{ijk}=i\epsilon_{ijk}$ and the covariant derivative in the adjoint
representation for the triplet is

$D_{\mu}\Delta=\partial_{\mu}\Delta-
ig_{2}\left[\frac{\sigma_{k}}{2}\;W_{\mu}^{k},\;\Delta\right]\,,$ (1)

where the square bracket stands for the commutator. Note that therefore the
$SU(2)_{L}$ gauge boson interactions with the triplet are a factor of 2 higher
than for a doublet.

Since the triplet cannot have direct couplings to quarks or leptons, the
scalar potential

$\displaystyle V=$
$\displaystyle-\mu_{\Phi}^{2}\Phi^{\dagger}\Phi+\frac{\lambda_{\Phi}}{4}\left(\Phi^{\dagger}\Phi\right)^{2}-\mu_{\Delta}^{2}{\rm
Tr}\left(\Delta^{\dagger}\Delta\right)$ (2)
$\displaystyle+\frac{\lambda_{\Delta}}{4}\left[{\rm
Tr}\left(\Delta^{\dagger}\Delta\right)\right]^{2}+\mu\Phi^{\dagger}\Delta\Phi+\lambda_{\Phi\Delta}\Phi^{\dagger}\Phi{\rm
Tr}\left(\Delta^{\dagger}\Delta\right),$

describes its remaining interactions. Here $\Phi$ is the SM doublet, and we
define

$\Phi=\begin{pmatrix}\phi^{+}\\\
\phi^{0}\end{pmatrix},\quad\Delta=\frac{1}{2}\begin{pmatrix}\delta^{0}&\sqrt{2}\delta^{+}\\\
\sqrt{2}\delta^{-}&-\delta^{0}\end{pmatrix},$ (3)

with $\delta^{0}$ being real and $\delta^{-}=(\delta^{+})^{*}$. Note that Eq.
(2) has a global $O(4)_{H}\times O(3)_{\Delta}$ symmetry softly broken by the
$\mu$-term. After electroweak symmetry breaking, $\phi^{0}$ and $\delta^{0}$
acquire the VEVs $\langle\sqrt{2}\phi^{0}\rangle=v\approx 246\,$GeV and
$\langle\delta^{0}\rangle=v_{\Delta}$. The minimization conditions

$\displaystyle\begin{aligned}
\mu_{\Phi}^{2}&=-\mu\frac{v_{\Delta}}{2}+\frac{1}{4}v^{2}\lambda_{\Phi}+\frac{1}{2}\lambda_{\Phi\Delta}v_{\Delta}^{2},\\\
\mu_{\Delta}^{2}&=-\mu\frac{v^{2}}{4v_{\Delta}}+\frac{1}{2}v^{2}\lambda_{\Phi\Delta}+\frac{1}{4}\lambda_{\Delta}v_{\Delta}^{2},\end{aligned}$
(4)

can then be used to eliminate $\mu_{\Phi}^{2}$ and $\mu_{\Delta}^{2}$ in terms
of the other parameters of Eq. (2). The scalar mass matrices, in the bases
($\phi^{+},\delta^{+}$) and (Re$(\phi^{0})$, $\delta^{0}$), are

$\displaystyle\begin{aligned}
M^{2}_{\pm}&=\mu\begin{pmatrix}v_{\Delta}&\frac{v}{2}\\\\[2.84544pt]
\frac{v}{2}&\frac{v^{2}}{4v_{\Delta}}\end{pmatrix}\,,\\\
M^{2}_{0}&=\begin{pmatrix}\lambda_{\Phi}\frac{v^{2}}{2}&\lambda_{\Phi\Delta}vv_{\Delta}-\mu\frac{v}{2}\\\\[2.84544pt]
\lambda_{\Phi\Delta}vv_{\Delta}-\mu\frac{v}{2}&\lambda_{\Delta}\frac{v_{\Delta}^{2}}{2}+\mu\frac{v^{2}}{4v_{\Delta}}\end{pmatrix}\,.\end{aligned}$
(5)

The resulting mass eigenstates, in addition to Im$(\phi^{0})$, are

$\displaystyle\begin{aligned} G^{\pm}&=\cos\zeta\ \phi^{\pm}+\sin\zeta\
\delta^{\pm}\,,\\\ H^{\pm}&=-\sin\zeta\ \phi^{\pm}+\cos\zeta\
\delta^{\pm}\,,\\\ h&=\cos\alpha\ {\rm Re}(\phi^{0})+\sin\alpha\
\delta^{0}\,,\\\ H&=-\sin\alpha\ {\rm Re}(\phi^{0})+\cos\alpha\
\delta^{0}\,,\end{aligned}$ (6)

with the CP-even and charged Higgs mixing angles

$\displaystyle\tan 2\alpha$
$\displaystyle=\frac{4vv_{\Delta}\left(2\lambda_{\Phi\Delta}v_{\Delta}-\mu\right)}{2\lambda_{\Phi}v^{2}v_{\Delta}-2\lambda_{\Delta}v_{\Delta}^{3}-\mu
v^{2}}\,,\;\tan\zeta=-2\frac{v_{\Delta}}{v}\,.$ (7)

The massless states $G^{\pm}$ and Im$(\phi^{0})$ are the would-be Goldstone
bosons, eaten by the $W^{+}$ and $Z$. Among the massive states, $h$ is
identified as the 125 GeV (SM-like) Higgs, and $H$ and $H^{\pm}$ are the
triplet-like neutral and charged scalars with masses

$\displaystyle\begin{aligned}
m_{H}^{2}&=\lambda_{\Delta}\frac{v_{\Delta}^{2}}{2}+\mu\frac{v^{2}}{4v_{\Delta}}-\tan\alpha\left(\lambda_{\Phi\Delta}v_{\Delta}-\frac{\mu}{2}\right)v\,,\\\
m_{H^{\pm}}^{2}&=\mu\frac{v^{2}+4v_{\Delta}^{2}}{4v_{\Delta}}\,.\end{aligned}$
(8)

For vanishing $\alpha$, $m_{H^{\pm}}^{2}-m_{H}^{2}\simeq\mu
v_{\Delta}-\lambda_{\Delta}v_{\Delta}^{2}/2$, and thus the components are
nearly mass-degenerate for $v_{\Delta}\ll v$. However, for large $\alpha$ and
$v_{\Delta}$, vacuum stability and perturbative unitarity (see Sec. III.1)
allow a mass-splitting $\Delta_{m}=m_{H^{\pm}}-m_{H}$ of a few GeV.111In
addition, the EW radiative correction induces a mass-splitting of 160 MeV–170
MeV [94]. However, such a small splitting is of little consequence as far as
the LHC phenomenology and their contribution to the electroweak oblique
parameters are concerned [95, 72]. Note that in the end, all parameters of the
scalar potential can be expressed in terms of the physical masses and mixing
angles and the two VEVs $v$ and $v_{\Delta}$. In particular, the (dimension-
full) couplings of the neutral to the charged Higgses can be written as

$\displaystyle\begin{aligned}
A_{hH^{\pm}H^{\mp}}&\approx\frac{1}{2}\lambda_{\Delta}v_{\Delta}\sin\alpha+\lambda_{\Phi\Delta}v\cos\alpha\,,\\\
A_{HH^{\pm}H^{\mp}}&\approx\frac{1}{2}\lambda_{\Delta}v_{\Delta}\cos\alpha-\lambda_{\Phi\Delta}v\sin\alpha\,,\end{aligned}$
(9)

in the limit of small $v_{\Delta}$ and $m_{H}\approx m_{H^{\pm}}$.

Figure 1: Preferred regions ($1\sigma$ and $2\sigma$) by the
$h\to\gamma\gamma$ signal strength (blue) and the 95 GeV $H\to\gamma\gamma$
excess (green) in the $\alpha-\Delta_{m}$ plane for the two values of
$v_{\Delta}$ corresponding to the two $m_{W}$ benchmark points. The region
between the two red lines is allowed by vacuum stability and perturbative
unitarity. The dashed vertical line indicates the region preferred by the LEP
measurement of $Z+(H\to bb)$, and the region to the right of the solid
vertical line is preferred by the $h\to ZZ^{*}$ signal strength at $1\sigma$
level.

##  III Phenomenology

###  III.1 Vacuum stability and perturbative unitarity

The $\Delta$SM parameter space can be constrained by requiring the potential
in Eq. (2) to be bounded from below and that perturbative unitarity is
respected in all tree-level scattering processes. The former is a necessary
condition for vacuum stability while the latter limits the size of the quartic
interactions from $2\to 2$ scalar-scalar scattering.222Note that this is valid
as long as the $\mu$ parameter is not very large, which is satisfied for a
small VEV since $\mu\sim v_{\Delta}\ll v$ for $m_{H^{\pm}}\sim v/2$. The
vacuum stability conditions read [96, 70, 97]

$\displaystyle\lambda_{\Phi}>0,\quad\lambda_{\Delta}>0,\quad\sqrt{2}\lambda_{\Phi\Delta}+\sqrt{\lambda_{\Phi}\lambda_{\Delta}}>0,$
(10)

and from perturbative unitarity we obtain

$\displaystyle\begin{aligned} &|\lambda_{\Phi}|\leq
2\kappa\pi,\quad|\lambda_{\Delta}|\leq
2\kappa\pi,\quad|\lambda_{\Phi\Delta}|\leq\kappa\pi,\\\
&|6\lambda_{\Phi}+5\lambda_{\Delta}\pm\sqrt{(6\lambda_{\Phi}-5\lambda_{\Delta})^{2}+192\lambda_{\Phi\Delta}^{2}}|\leq
8\kappa\pi,\end{aligned}$ (11)

where $\kappa$ = 16 or 8 depending on whether one chooses $|(a_{0})|\leq 1$ or
$|{\rm Re}~{}(a_{0})|\leq\frac{1}{2}$ with $a_{0}$ denoting the leading
partial wave amplitude [98]. In order to be conservative, i.e. not to exclude
any potentially allowed parameter space, we opt for $\kappa=16$. Further, to
ensure perturbativity at all higher orders, we require all the trilinear and
quartic scalar couplings in (2) to be smaller than $4\pi$.333We checked
numerically using Vevacious [99, 100], SPheno [101, 102] and BSMArt [103] that
the inclusion of the one-loop effective potential and meta stability has only
a marginal effect on vacuum stability and perturbative unitarity. The region
between the red lines in Fig. 1 is allowed by both vacuum stability and
perturbative unitarity.

###  III.2 $W$ mass

The latest ATLAS update of $m_{W}=80.360(16)$ [104] (superseding the 2017
result [105]) as well as the LHCb result $m_{W}=80.354(32)$ [106] are
significantly smaller compared to $m_{W}=80.4335(94)\,$GeV obtained by CDF II.
The latter, when combined with the D0 result [85], leads to a new Tevatron
average of $m_{W}=80.427(89)\,$GeV. These, together with LEP [107], lead to a
naive global average of $m_{W}=80.406(7)\,$GeV. Because there is considerable
tension between these measurements ($\chi^{2}/{\rm dof}=4.3$), we inflate the
error on $m_{W}$ to 0.015 GeV to get a conservative average of [80]444This
naive average agrees well with the one obtained in a sophisticated fit
performed by HEPfit [82] prior to the ATLAS update.

$\displaystyle m_{W}^{\rm comb}=(80.406\pm 0.015){\rm\,GeV}.$ (12)

Comparing this with the SM prediction of $m_{W}^{\rm SM}=80.3499(56)\,$GeV
[82, 108, 109, 110, 111, 112, 113, 114, 3], with $m_{t}=172.5(0.7)\,$GeV [3],
the discrepancy of $56\,$MeV amounts to $3.7\sigma$. If we disregarded the CDF
II result, we find an average of

$\displaystyle m_{W}^{\rm comb~{}(w/o~{}CDF~{}II)}=(80.372\pm
0.010){\rm\,GeV},$ (13)

which corresponds to a discrepancy of 22 MeV ($2.2\sigma$).

In the $\Delta$SM, we have

$\displaystyle m_{W}^{2}$
$\displaystyle=\frac{g^{2}}{4}(v^{2}+4v_{\Delta}^{2}),\hskip
19.91692ptm_{Z}^{2}=\frac{g^{2}}{4\cos\theta_{W}^{2}}v^{2}.$ (14)

Therefore, $v_{\Delta}$ of a few GeV can easily alter the $m_{W}$ prediction
in the desired direction. As such, $m_{W}^{\rm comb}$ requires
$v_{\Delta}=4.60^{+0.58}_{-0.66}\,$GeV, while $m_{W}^{\rm
comb~{}(w/o~{}CDF~{}II)}$ requires $v_{\Delta}=2.89^{+0.59}_{-0.75}\,$GeV.

###  III.3 SM Higgs signal strength

Through the quartic interactions $\lambda_{\Delta}$ and $\lambda_{\Phi\Delta}$
in (2), $H^{\pm}$ contributes to the di-photon decay rate of the SM Higgs $h$
(see Fig. 2). The corresponding signal strength, w.r.t. the SM one, is given
by

$\displaystyle\mu_{h,\gamma\gamma}=\dfrac{\Gamma_{h\to\gamma\gamma}}{\Gamma_{h\to\gamma\gamma}^{\rm
SM}}=|\kappa_{\gamma}^{2}|\,,$ (15)

with

$\displaystyle\kappa_{\gamma}\approx\cos\alpha+\dfrac{A_{hH^{\pm}H^{\mp}}v}{2m_{H^{\pm}}^{2}}\dfrac{\beta^{0}_{H}\left(\frac{4m_{H^{\pm}}^{2}}{m_{h}^{2}}\right)}{\dfrac{4}{3}\beta^{1/2}_{H}\left(\frac{4m_{t}^{2}}{m_{h}^{2}}\right)+\beta^{1}_{H}\left(\frac{4m_{W}^{2}}{m_{h}^{2}}\right)},$
(16)

and the loop functions [115]

$\displaystyle\begin{aligned} \beta^{0}_{H}(x)&=-x\left[1-xf(x)\right],\\\
\beta^{1/2}_{H}(x)&=2x\left[1+(1-x)f(x)\right],\\\
\beta^{1}_{H}(x)&=-\left[2+3x+3x(2-x)f(x)\right].\end{aligned}$ (17)

Combining the most recent measurements of CMS [116] and ATLAS [117],
$\mu_{h,\gamma\gamma}^{\rm CMS}=1.12^{+0.09}_{-0.09}$ and
$\mu_{h,\gamma\gamma}^{\rm ATLAS}=1.04^{+0.10}_{-0.09}$, respectively, we get
the weighted average

$\displaystyle\mu_{h,\gamma\gamma}^{\rm exp}=1.08^{+0.07}_{-0.06}.$ (18)

The resulting preferred regions at the $1\sigma$ and $2\sigma$ level are shown
in blue in Fig. 1

$\gamma$$\gamma$$H^{\pm}$$h$ Figure 2: Feynman diagram showing the
modification of $h\to\gamma\gamma$ by the charged scalar $H^{\pm}$. Figure 3:
Feynman diagrams showing the DY processes $pp\to Z^{*},\gamma^{*}\to
H^{+}H^{-}\to\tau^{+}\tau^{-}\nu\bar{\nu}$ (left) and $pp\to W^{*}\to
H^{\pm}H^{0}$ (right).

While the $h\to\gamma\gamma$ signal strength is the most precise measured one,
it is affected by $h$-$H$ mixing and the $H^{\pm}$-loop contribution so that
cancellations occur. Therefore, the second-best measured SM Higgs signal $h\to
ZZ^{*}$ [118, 119] provides a complementary constraint of [3]

$\mu_{h,ZZ^{*}}^{\rm exp}=1.02\pm 0.08\,,$ (19)

which, to a very good approximation, is only sensitive to the mixing angle
$\alpha$. The region on the right of the solid vertical line in Fig. 1 is
compatible with $\mu_{h,ZZ^{*}}^{\rm exp}$ at the $1\sigma$ level.

###  III.4 Di-photon excess

While nearly all relevant decay modes of $H$ can be obtained from a rescaling
of the widths of a SM-like Higgs with a mass of 95 GeV by multiplying with
$\sin^{2}\alpha$, the decay $H\to WW^{*}$ is already generated at tree-level
via $v_{\Delta}$ and $H\to\gamma\gamma$ receives loop contributions from the
charged Higgs as well as from $W$ loops:555Only $Z\gamma$ also receives an
additional direct contribution from the $W$ loop, which is already present for
$\sin\alpha=0$, but the corresponding branching ratio is negligibly small.

$\displaystyle\Gamma(H\to\gamma\gamma)\approx\dfrac{\alpha_{\rm
em}^{2}g_{2}^{2}m_{H}^{3}}{1024\pi^{3}m_{W}^{2}}\left|-\dfrac{4}{3}\sin\alpha\,\beta_{H}^{1/2}\left(\frac{4m^{2}_{t}}{m^{2}_{H}}\right)+\left(-\sin\alpha+\dfrac{4v_{\Delta}}{v}\cos\alpha\right)\beta_{H}^{1}\left(\frac{4m^{2}_{W}}{m^{2}_{H}}\right)+\dfrac{A_{HH^{\pm}H^{\mp}}v}{2m_{H^{\pm}}^{2}}\beta_{H}^{0}\left(\frac{4m^{2}_{H^{\pm}}}{m^{2}_{H}}\right)\right|^{2}.$
(20)

Here, $\alpha_{\rm em}$ at $q^{2}=0$ numerically approximates well the NLO QED
corrections.

For a small mixing angle $\alpha$, $H$ is mainly produced via the Drell-Yan
(DY) process $pp\to W^{*}\to H^{\pm}H$ (see Fig. 3) with a leading order (LO)
cross-section of 1.77 pb for $m_{H^{\pm}}\approx m_{H}=95\,$GeV. While the QCD
corrections have not been estimated so far for the $\Delta$SM, it is obvious
that they pertain dominantly to the hadronic ends of the processes and are
thus expected to be the same as for sleptons or $SU(2)_{L}$ triplet leptons.
The latter has been calculated in Ref. [120], resulting in a correction factor
of 1.15, by which we naively rescale the LO cross-section (computed with
MadGraph5aMC@NLO [153]) to obtain $\approx 2\,$pb. In addition, $H$ is also
produced via gluon fusion (GF) and vector boson fusion (VBF) processes through
the mixing with $h$. The corresponding cross-section is calculated by
multiplying the production cross section of a SM-like 95 GeV Higgs by
$\alpha^{2}$. Neglecting the subdominant contribution from VBF, and using
$\sigma[pp\to h(95)]\approx 68\,$pb [121, 122, 123, 124, 125, 126, 127, 128,
129, 130, 131], we thus have

$\displaystyle\sigma[pp\to H\to\gamma\gamma]\approx{\rm
Br}[H\to\gamma\gamma]\times\left(2+68\alpha^{2}\right){\rm pb}.$ (21)

Normalizing the signal strength to the one of a hypothetical SM-like Higgs
with the same mass [121], we find numerically

$\displaystyle\mu_{H,\gamma\gamma}\approx(21.5+719\alpha^{2})\times{\rm
Br}[H\rightarrow\gamma\gamma].$ (22)

This has to be compared to the combination of the CMS and ATLAS analyses of a
low mass $\gamma\gamma$ searches of [63]

$\mu_{H,\gamma\gamma}^{\rm exp}=0.27^{+0.10}_{-0.09}\,.$ (23)

The resulting preferred regions are shown in green in Fig. 1.

###  III.5 $Zbb$, $WW$ and $\tau\tau$

While Br$[H\to WW]$ is large for a very small mixing angle $\alpha$, the
resulting effect in $\gamma\gamma$ would be too high if one aims at the
central value of the cross section of Ref. [40]. Therefore, $\alpha$ cannot be
too small, and it is possible to explain the $Zbb$ excess of LEP which
requires

$\mu_{bb}^{\exp}=\frac{\sigma^{\exp}\left(e^{+}e^{-}\rightarrow
ZH\right)}{\sigma^{\mathrm{SM}}\left(e^{+}e^{-}\rightarrow ZH\right)}{\rm
Br}\left(H\to b\bar{b}\right)=0.117\pm 0.057.$ (24)

For tau decays, the central values of the signal strength $\mu_{\tau\tau}^{\rm
exp}=1.2\pm 0.5$ cannot be fully explained, which is a general feature of most
SM extensions addressing the $95\,$GeV excess [57], the error is too large to
draw a conclusion here.

###  III.6 $pp\to H^{+}H^{-}\to\tau^{+}\tau^{-}\nu\bar{\nu}$

The charged Higgs in general dominantly decays to $\tau\nu$. Therefore, its
pair production and subsequent decays, i.e. $pp\to Z^{*},\gamma^{*}\to
H^{+}H^{-}\to\tau^{+}\tau^{-}\nu\bar{\nu}$, leads to a collider signature
searched for in the context of supersymmetric tau partners [132, 133, 134,
135]. While CMS [134] provides an upper bound on the cross section and
observes a weaker limit than expected, ATLAS [135] observes a stronger limit
than expected but does not provide a bound on the total cross section. Since
both bounds deviate from the expected limit by $\approx 1\,\sigma$ level, but
in opposite directions, we will thus use the expected limit on the cross
section provided by CMS [134] of $0.34^{+0.24}_{-0.12}\,$pb. Using once more
MadGraph5aMC@NLO at LO, we find a production cross-section of $0.86\,$pb which
we again multiply by a factor $1.15$ [136, 120] to include NLO QCD effects.
Taking into account that CMS and ATLAS assume a 100% branching ratio of the
stau to tau and neutralino, while we have
Br$[H^{\pm}\to\tau^{\pm}\nu_{\tau}]\approx 0.66\pm 0.03$ [121, 137, 138, 139,
140, 141, 142, 143, 144, 145, 146, 147, 148]666Since in our case the branching
ratio is dominated by $\tau\nu$ and $cs$, the error on Br$[H^{\pm}\to cs]$ is
dominating the error of Br$[H^{\pm}\to\tau^{\pm}\nu_{\tau}]$., a cross-section
of $\approx 0.44\pm 0.03\,$pb is predicted. This is in slight tension with the
95% exclusion limit.

Let us therefore consider the option to reduce Br$[H^{\pm}\to\tau\nu]$ by
increasing the mass splitting $\Delta m$ such that Br$[H^{\pm}\to HW^{*}]$
becomes sizable:777Note that Br$[H^{\pm}\to H^{*}W]$ is much smaller such that
it can be neglected.

$\displaystyle\Gamma(H^{\pm}\to
HW^{*})=\frac{9g^{4}m_{H^{\pm}}}{512\pi^{3}}\lambda_{HH^{\pm}W}^{2}G\left(\frac{m_{H}^{2}}{m_{H^{\pm}}^{2}},\frac{m_{W}^{2}}{m_{H^{\pm}}^{2}}\right),$
(25)

where $\lambda_{HH^{\pm}W}=2\cos\alpha\cos\beta-\sin\alpha\sin\beta$, and the
loop function $G(x,y)$ is given by

$\displaystyle
G(x,y)=\frac{1}{12y}\Bigg{[}2(-1+x)^{3}-9(-1+x^{2})y+6(-1+x)y^{2}$
$\displaystyle-6(1+x-y)y\sqrt{-\lambda(x,y)}\Bigg{\\{}\tan^{-1}\left(\frac{1-x+y}{\sqrt{-\lambda(x,y)}}\right)$
$\displaystyle+\tan^{-1}\left(\frac{1-x-y}{\sqrt{-\lambda(x,y)}}\right)\Bigg{\\}}-3\left(1+(x-y)^{2}-2y\right)y\log
x\Bigg{]},$

with $\lambda(x,y)=(1-x-y)^{2}-4xy$.

Here we choose $v_{\Delta}=0.86\,$GeV as a benchmark point, which allows for a
small region in parameter space with sizable mass splitting, as we see from
Fig. 4, that is allowed by the vacuum stability and perturbative
unitarity888Note that, for sizeable $\alpha$, the requirements of vacuum
stability and perturbative unitarity dictate that $\Delta_{m}\approx
22\alpha-3.75$ GeV. as well as compatible with $h\to\gamma\gamma,ZZ^{*}$,
$H\to\gamma\gamma$ and $Zbb$. Note that this scenario predicts a small
positive shift in the $W$ mass.

##  IV Conclusions and Outlook

Figure 4: Scenario with sizable $\Delta_{m}$ that is allowed by vacuum
stability due to $\Delta_{m}\approx 22\alpha-3.75$ [GeV] that suppresses
Br$[H^{\pm}\to\tau\nu]$ by enhancing Br$[H^{\pm}\to HW^{*}]$

In summary, the predictions of the neutral component of the $SU(2)_{L}$
triplet with hypercharge 0 being the origin of the 95 GeV excess are:

  * •

LHC Run 3 shows a stau-like excess.

  * •

Positive shift in the $W$ mass.

  * •

$H$ is produced in association with jets and $\tau$ leptons.

  * •

A charged Higgs with a mass below $\approx\\!100\,$GeV which could be very
well studied at future $e^{+}e^{-}$ colliders [149, 150, 151, 152].

  * •

A significantly broader $p_{T}$ spectrum of the di-photon system compared to
GF, as shown in Fig. 5.999While this information is currently not available,
it can be used in future analyses as a discriminator. To compare the $p_{T}$
of the di-photon system of the $\Delta$SM to the SM, we generated 100k events
at NLO using MadGraph5aMC@NLO with the parton shower performed by Pythia8
[154] and the detector simulation for the CMS detector [34], carried out with
Delphes3 [155]. The UFO model file at NLO of the $\Delta$SM was built using
FeynRules [156, 157, 158] and to increase the efficiency of the simulation,
the decay of $H$ to a photon pair was forced using MadSpin [159].

###### Acknowledgements.

S.B., G.C. and A.C. thank Michael Spira for useful discussions. S.A.
acknowledges the support via the Core Research Grant CRG/2018/004889 from
SERB, Government of India. The work of A.C. is supported by a professorship
grant from the Swiss National Science Foundation (No. PP00P21_76884). B.M.
gratefully acknowledges the South African Department of Science and Innovation
through the SA-CERN program, the National Research Foundation, and the
Research Office of the University of the Witwatersrand for various forms of
support. S.A. thanks Rameswar Sahu for useful discussions.

Figure 5: Transverse momentum normalized to the invariant mass of the photon
pair system for different production mechanisms of a $95\,$GeV scalar $H$.

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