# Analysis of the $gg\to H\to hh\to 4\tau$ process
in the 2HDM lepton specific model at the LHC

Yan Ma<EMAIL_ADDRESS>College of Physics and Electronic
Information, Inner Mongolia Normal University, Hohhot 010022, PR China A.
Arhrib<EMAIL_ADDRESS>Abdelmalek Essaadi University, Faculty of Sciences
and Techniques, B.P. 2117 Tétouan, Tanger, Morocco &
Department of Physics and Center for Theory and Computation, National Tsing
Hua University, Hsinchu, Taiwan 300 S. Moretti<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Physics & Astronomy, University of
Southampton, Southampton SO17 1BJ, UK &
Department of Physics & Astronomy, Uppsala University, Box 516, 75120 Uppsala,
Sweden S. Semlali<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Southampton, Southampton SO17 1BJ, UK &
Particle Physics Department, Rutherford Appleton Laboratory, Chilton, Didcot,
Oxon OX11 0QX, United Kingdom Y. Wang<EMAIL_ADDRESS>College of Physics
and Electronic Information, Inner Mongolia Normal University, Hohhot, 010022,
China &
Key Laboratory for Physics and Chemistry of Functional Materials, Inner
Mongolia Normal University, Hohhot, 010022, China Q.S. Yan
<EMAIL_ADDRESS>Center for Future High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, PR China &
School of Physics Sciences, University of Chinese Academy of Sciences, Beijing
100039, PR China

###### Abstract

We analyse the signature of a light Higgs boson pair in the 2-Higgs Doublet
Model(2HDM) Type-X (or lepton specific) over the parameter spaces allowed by
theoretical self-consistency requirements as well as the latest experimental
constraints from the Large Hadron Collider (LHC), precision test data and $B$
physics. Over the viable regions of the latter, wherein the Standard Model
(SM)-like Higgs boson discovered at the LHC in 2012 is the heavier CP-even
state of the 2HDM, $H$, it is found that the SM-like Higgs boson can decay
into a pair of the lighter CP-even Higgs boson, $h$, via the process $H\to hh$
with a Branching Ratio (BR) of $5\%-10\%$ or so, (with $2m_{h}<m_{H}=125$
GeV). Furthermore, in the Type-X scenario, the lighter Higgs bosons $h$ can
dominantly decay into two $\tau$’s due to a large $\tan\beta$. Therefore, the
pair of lighter Higgs bosons can altogether decay into a 4 $\tau$ final state.
In order to suppress the huge SM background events, we confine ourself to
consider the fraction of signal events with two Same-Sign (SS) $\tau$’s
further decaying into same sign leptons while the other two $\tau$’s decay
hadronically. By using Monte Carlo (MC) and Machine Learning (ML) tools, we
thus focus on the analysis of the signal process $pp\to H\to
hh\to\tau^{+}\tau^{-}\tau^{+}\tau^{-}\to\ell v_{\ell}\ell
v_{\ell}\tau_{h}\tau_{h}$ (where $\ell=e,\mu$ and $\tau_{h}$ means a hadronic
decay of the $\tau$) and explore the feasibility of such a search at the LHC
for a collision energy $\sqrt{s}=~{}\text{14 TeV}$ and a luminosity
$\text{300}~{}\text{fb}^{-1}$.

KEYWORDS: 2-Higgs Doublet Model, Type-X Scenario, Higgs Physics

###### Contents

  1. I Introduction
  2. II The 2HDM
  3. III Bounds and constraints in the parameter space scans
  4. IV Collider phenomenology
    1. IV.1 Event generation and selection
    2. IV.2 Event reconstruction
    3. IV.3 Multi-variate analysis
    4. IV.4 Significances
  5. V Conclusions

##  I Introduction

The LHC has played an irreplaceable role in our understanding of the
microcosmic subatomic structure, particularly with the discovery of a 125 GeV
scalar particle by the ATLAS and CMS experimental teams in 2012 Aad:2012tfa ;
Chatrchyan:2012ufa . This was the last particle predicted by the Standard
Model (SM) and its properties measured in the above experiments are currently
consistent with the particle predicted by the SM at the $5\sigma$ level
ATLAS:2022vkf ; CMS:2022dwd . To investigate the Higgs sector at the LHC,
wherein the presence of additional (pseudo)scalar particles might exist,
remains an intriguing task, particularly in new physics models with additional
doublet fields. Therefore, to discover extra Higgs bosons at the LHC is one of
the prime targets of Beyond the SM (BSM) signal searches.

One of the simplest extensions of the SM is the 2HDM, which contains two
complex Higgs doublets Lee:1973iz ; Deshpande:1977rw ; Branco:2011iw . After
Electro-Weak Symmetry Breaking (EWSB), there are three Goldstone bosons, which
are ‘eaten’ by the $W^{\pm}$ and Z bosons while the remaining five degrees of
freedom incarnate into physical Higgs bosons: two neutral CP-even scalars
(${h}$ and ${H}$, with $M_{{h}}\textless M_{{H}}$), one CP-odd pseudoscalar
(${A}$) and two charged Higgs states ${H}^{\pm}$ (with mixed CP status).

In order to be consistent with the stringent experimental constraints from
Flavour Changing Neutral Currents (FCNCs) at the tree level, typically, a
$Z_{2}$ symmetry is introduced into the Yukawa sector such that each type of
fermion only couples to one of the doublets of the 2HDM Glashow:1976nt .
Depending on the $Z_{2}$ charge assignments of the Higgs doublets, we can
define four basic scenarios, known as (Yukawa) Types. In the lepton specific
scenario (or Type-X), the extra Higgs doublet $\phi_{2}$ only couples to the
lepton sector and not to the quark sector, where the light Higgs boson mainly
decays to $\tau\tau$ for the wide range of parameter spaces Aoki:2009ha .

There have been several experiments for searching the SM Higgs exotic decays
through four fermion final states $pp\to H\to XX\to f_{1}f_{2}f_{3}f_{4}$.
Such process benefits from the large cross-section of $\sigma(pp\to H)$, and
the branching ratio of the SM-like Higgs exotic decay $Br(H\to hh)$ could
reach up to 10% in some BSM models. Motivated by the recent searches for
$\tau\tau\tau\tau$ by CMS experiments CMS:2015twz ; CMS:2017dmg ; CMS:2019spf
, we will study the process $pp\to H\to hh\tau\tau\tau\tau$ in the 2HDM type-X
model. In such a scenario, $h\to\tau^{+}\tau^{-}$ decay could be enhanced,
which can lead to a sizable $4\tau$ final state. Therefore, it is worth to
examine whether this statement is robust enough after taking into account the
effects of parton shower, hadronization, heavy flavor decays and detector
resolution.

In this paper, we assume that the heavier CP-even Higgs boson $H$ is the
observed SM-like Higgs boson whose properties agree with the LHC measurements.
In such a parameter space, a heavier Higgs boson can decay into two light
Higgs bosons $h$ when kinematically allowed, i.e.,
$2M_{{h}}<125~{}\text{GeV}$. We perform a detailed MC study on the signal
process $pp\to H\to hh\to 4\tau\to lv_{l}lv_{l}\tau_{h}\tau_{h}$ (with
$l=e,\mu$ and $\tau_{h}$ ) and relevant background processes, in order to
examine its feasibility at the LHC. We do so after taking into account
relevant theoretical and experimental conditions. In order to improve the
experimental significance, we will require two leptons to be the same sign
(SS) leptons. Our analysis demonstrates that, by using this signature, the
relevant parameter space can be either discovered or ruled out by using the
current integrated luminosity at the LHC.

The paper is organised as follows. In the next section, we briefly describe
the 2HDM and its Yukawa scenarios, then introduce a few Benchmark Points (BPs)
for our MC analysis which pass all present theoretical and experimental
constraints. In the following section, we perform a detailed collider analysis
of these BPs and examine the potential to discover the aforementioned
signature within the 2HDM Type-X scenario. Finally, we present some
conclusions.

##  II The 2HDM

The scalar sector of the 2HDM consists of two weak isospin doublets with
hyper-charge $Y=1$. The most general Higgs potential for the 2HDM that
complies with the $\rm{SU(2)_{L}\times U(1)_{Y}}$ gauge structure of the EW
sector of the SM has the following form Branco:2011iw :

$\displaystyle V(\phi_{1},\phi_{2})$ $\displaystyle=$ $\displaystyle
m_{11}^{2}(\phi_{1}^{\dagger}\phi_{1})+m_{22}^{2}(\phi_{2}^{\dagger}\phi_{2})-[m_{12}^{2}(\phi_{1}^{\dagger}\phi_{2})+\text{h.c.}]~{}$
(1)
$\displaystyle+\frac{1}{2}\lambda_{1}(\phi_{1}^{\dagger}\phi_{1})^{2}+\frac{1}{2}\lambda_{2}(\phi_{2}^{\dagger}\phi_{2})^{2}+\lambda_{3}(\phi_{1}^{\dagger}\phi_{1})(\phi_{2}^{\dagger}\phi_{2})+\lambda_{4}(\phi_{1}^{\dagger}\phi_{2})(\phi_{2}^{\dagger}\phi_{1})~{}$
$\displaystyle+\frac{1}{2}\left[\lambda_{5}(\phi_{1}^{\dagger}\phi_{2})^{2}+\rm{h.c.}\right]+~{}\left\\{\left[\lambda_{6}(\phi_{1}^{\dagger}\phi_{1})+\lambda_{7}(\phi_{2}^{\dagger}\phi_{2})\right](\phi_{1}^{\dagger}\phi_{2})+\rm{h.c.}\right\\},$

where $\phi_{1}$ and $\phi_{2}$ are the two Higgs doublet fields. By
hermiticity of such a potential, $\lambda_{1,2,3,4}$ as well as
$m_{11,22}^{2}$ are real parameters while $\lambda_{5,6,7}$ and $m_{12}^{2}$
can be complex, in turn enabling possible Charge and Parity (CP) violation
effects in the Higgs sector. Upon enforcing two minimization conditions of the
potential, $m^{2}_{11}$ and $m^{2}_{22}$ can be replaced by $v_{1,2}$, which
are the Vacuum Expectation Values (VEVs) of the Higgs doublets $\phi_{1,2}$,
respectively. Moreover, the coupling $\lambda_{1,2,3,4,5}$ can be substituted
by the four physical Higgs masses (i.e., $M_{{h}},M_{{H}},M_{{A}}$ and
$M_{{H}^{\pm}}$) and the parameter $\text{sin}(\beta-\alpha)$, where $\alpha$
and $\beta$ are, respectively, the mixing angles between CP-even and the angle
related to the VEV values, i.e., $\tan\beta=v_{1}/v_{2}$. Thus, the
independent input parameters can be taken as $M_{{h}},M_{{H}},M_{{A}}$ and
$M_{{H}^{\pm}}$, $\lambda_{6}$, $\lambda_{7}$, $\text{sin}(\beta-\alpha)$,
$\text{tan}(\beta)$ and $m^{2}_{12}$.

If both Higgs doublet fields of the general 2HDM couple to all fermions, the
ensuing scenario can induce FCNCs in the Yukawa sector at tree level. As
intimated, to remedy this, a $Z_{2}$ symmetry is imposed on the Lagrangian
such that each fermion type interacts with only one of the Higgs doublets
Glashow:1976nt . As a consequence, there are four possible types of 2HDM,
namely Type-I, Type-II, Type-X (or lepton specific) and Type-Y (or flipped).
However, such a symmetry is explicitly broken by the quartic couplings
$\lambda_{6,7}$ and softly broken by the (squared) mass term $m^{2}_{12}$. In
what follows, we shall consider a CP-conserving (i.e., $m^{2}_{12}$ and
$\lambda_{5}$ are real) 2HDM Type-X and assume that
$\lambda_{6}=\lambda_{7}=0$ to forbid the explicit breaking of $Z_{2}$, while
also taking $m^{2}_{12}$ to be generally small, thereby preventing large FCNCs
at tree level, which are incompatible with experiment.

In general, the couplings of the neutral and charged Higgs bosons to fermions
can be described by the Yukawa Lagrangian given by Branco:2011iw

$\displaystyle-{\mathcal{L}}_{\rm
Yukawa}=\sum_{f=u,d,l}\left(\frac{m_{f}}{v}\kappa_{f}^{h}\bar{f}fh+\frac{m_{f}}{v}\kappa_{f}^{H}\bar{f}fH-i\frac{m_{f}}{v}\kappa_{f}^{A}\bar{f}\gamma_{5}fA\right)+$
$\displaystyle\left(\frac{V_{ud}}{\sqrt{2}v}\bar{u}(m_{u}\kappa_{u}^{A}P_{L}+m_{d}\kappa_{d}^{A}P_{R})dH^{+}+\frac{m_{l}\kappa_{l}^{A}}{\sqrt{2}v}\bar{\nu}_{L}l_{R}H^{+}+\text{h.c.}\right),$
(2)

where $\kappa_{f}^{S}$ ($S=h,H$ and $A$) are the Yukawa couplings in the 2HDM,
which are illustrated in Tab. 1 for the Type-X under consideration. Here,
$V_{ud}$ refers to a Cabibbo-Kobayashi-Maskawa (CKM) matrix element and
$P_{L,R}$ denote the left- and right-handed projection operators. The coupling
of the two CP-even states $h$ and $H$ to gauge bosons $VV$ ($V=W^{\pm},Z$) are
proportional to $\sin(\beta-\alpha)$ and $\cos(\beta-\alpha)$, respectively.
If we assume that either $h$ or $H$ can be the observed SM-like Higgs boson,
the coupling to gauge bosons is obtained for $h$ when
$\cos(\beta-\alpha)\rightarrow 0$ and for $H$ when
$\sin(\beta-\alpha)\rightarrow 0$. Therefore, each scenario can explain the
125 GeV Higgs signal at the LHC. However, following our works Arhrib:2021xmc ;
Wang:2021pxc ; Arhrib:2021yqf ; Li:2023btx , though, we shall focus in the
present paper on the scenario where $H$ mimics the observed signal with mass
$\sim\,125$ GeV (as previously intimated).

| $\kappa_{u}^{S}$ | $\kappa_{d}^{S}$ | $\kappa_{\ell}^{S}$
---|---|---|---
$h$ |  $\cos\alpha/\sin\beta$ |  $\cos\alpha/\sin\beta$ |  $-\sin\alpha/\cos\beta$
$H$ |  $\sin\alpha/\sin\beta$ |  $\sin\alpha/\sin\beta$ |  $\cos\alpha/\cos\beta$
$A$ |  $\cot\beta$ |  $-\cot\beta$ |  $\tan\beta$
Table 1: Yukawa couplings of the fermions $f=u,d$ and $\ell$ to the neutral
Higgs bosons $S=h,H$ and $A$ in the 2HDM Type-X.

##  III Bounds and constraints in the parameter space scans

In order to identify regions that satisfy both theoretical requirements and
experimental observations, the following numerical explorations of the
possible parameter spaces have been studied:

$\displaystyle M_{{h}}\in[15,60]~{}(\text{GeV}),\quad M_{{H}}=125,\quad
M_{{A}}=~{}245(\text{GeV}),\quad M_{{H}^{\pm}}=258~{}(\text{GeV})$ (3)
$\displaystyle\text{sin}(\beta-\alpha)\in[-0.25,-0.05],\quad\text{tan}(\beta)\in[2,13],\quad
m_{12}^{2}\in[0,500](\text{GeV})^{2}$

The program 2HDMC-1.8.0 Eriksson:2009ws  was used for calculating the cross
sections ($\sigma$’s) and BRs as well as testing the following theoretical and
experimental constraints.

  * •

Vacuum stability Deshpande:1977rw : the scalar potential should be bounded by
$\lambda_{1}>0$, $\lambda_{2}>0$,
$\lambda_{3}>-\sqrt{\lambda_{1}\lambda_{2}}$,
$\lambda_{3}+\lambda_{4}-\lambda_{5}>-\sqrt{\lambda_{1}\lambda_{2}}$.

  * •

Perturbativity constraints Kanemura:1993hm ; Akeroyd:2000wc , which imply the
condition $|\lambda_{i}|\textless 8\pi(i=,...5)$.

  * •

Tree-level perturbative unitarity Kanemura:1993hm ; Arhrib:2000is ;
Akeroyd:2000wc : requiring unitary when $2\to 2$ scattering processes
involving Higgs and gauge bosons at high energy.

  * •

Oblique EW parameters ($S$, $T$ and $U$) Peskin:1990zt ; Peskin:1991sw , which
control the mass splitting between the Higgs states, with the following
measured values ParticleDataGroup:2022pth :

$\displaystyle S=-0.02\pm 0.10,\quad T=0.03\pm 0.12.$ (4)

The correlation factor between $S$ and $T$ is set as $0.92$ for consistency at
$95\%$ Confidence Level (CL).

To account for potential additional Higgs bosons, exclusion bounds at $95\%$
CL are enforced using the HiggsBounds-5.9.0 program Bechtle:2020pkv , which
systematically checks each parameter point against the $95\%$ CL exclusion
limits derived from Higgs boson searches conducted by LEP, Tevatron and LHC
experiments.

To ensure agreement with the measurements of the SM-like Higgs state,
constraints are enforced using the HiggsSignals-2.6.0 program Bechtle:2020uwn
, which incorporates the combined measurements of the SM-like Higgs boson from
LHC Run-1 and Run-2 data.

Constraints from flavor physics are incorporated using SuperIso v4.1
Mahmoudi:2008tp  by utilizing the following observables:

  * •

BR$(B\to X_{s}\gamma)=(3.32\pm 0.15)\times 10^{-4}$ HFLAV:2022esi ,

  * •

${\rm
BR}(B^{0}\to\mu^{+}\mu^{-})_{\text{(LHCb)}}$=$\left(1.2^{+0.8}_{-0.7}\right)\times
10^{-10}$ LHCb:2021awg ; LHCb:2021vsc ,

  * •

${\rm
BR}(B^{0}\to\mu^{+}\mu^{-})_{\text{(CMS)}}$=$\left(0.37^{+0.75}_{-0.67}\right)\times
10^{-10}$ CMS:2022mgd ,

  * •

${\rm BR}(B_{s}\to\mu^{+}\mu^{-})_{\text{(LHCb)}}$ =
$\left(3.09^{+0.46}_{-0.43}\right)\times 10^{-9}$ LHCb:2021awg ; LHCb:2021vsc
,

  * •

${\rm BR}(B_{s}\to\mu^{+}\mu^{-})_{\text{
(CMS)}}$=$\left(3.83^{+0.38}_{-0.36}\right)\times 10^{-9}$ CMS:2022mgd ,

  * •

BR$(B\to\tau\nu)=(1.09\pm 0.20)\times 10^{-4}$ HFLAV:2022esi .

In the left panel of Fig. 1, we show the branching ratio of the light Higgs
decaying into pairs of $\tau$ leptons within the allowed parameter space of
the 2HDM Type-X. The total width of the $h$ state ($\Gamma_{h}$) is obviously
dominated by $h\to\tau\tau$. Such a significant branching ratio is attributed
to the enhancement of the light Higgs couplings to leptons at large
$\tan\beta$. The branching ratio of the SM-like Higgs boson decaying into a
pair of light Higgs bosons is displayed on the right panel. The precise
measurements of the Higgs boson couplings ATLAS:2022vkf ; CMS:2022dwd  tighten
the constraints on Higgs decay to non-SM particles and force the $\rm{BR}(H\to
hh)$ to be below 12%. Both ATLAS and CMS collaborations have reported an upper
limit of 12% ATLAS:2022vkf  and 16 %CMS:2022dwd , respectively, on
$\rm{BR}(H\to BSM)$ at 95%CL.

Figure 1: $\rm{BR}(h\to\tau\tau)$ as a function of $m_{h}$ vs. $\tan\beta$
(left panel) and $\rm{BR}(H\to hh)$ (right panel) in the 2HDM type-X.

To test the allowed parts of the parameter space, twelve BPs are given in Tab.
2, in terms of the 2HDM Type-X parameters entering our production and decay
process, including the corresponding $pp\to H\to 2h\to 4\tau$ cross sections
at the LHC with $\sqrt{s}=14$ TeV. As one can see from such a table, the light
Higgs boson mass varies from about $15~{}\text{GeV}$ to $60~{}\text{GeV}$,
which can be paired produced from a $125~{}\text{GeV}$ Higgs boson. The cross
sections of the BPs are around $2-5$ pb, which are large enough to be detected
at the collider.

| $m_{h}$ | $\sin(\beta-\alpha)$ | $\tan\beta$ |  $\sigma_{4\tau,~{}14~{}{\rm TeV}}$ (pb)
---|---|---|---|---
BP1 | 19.04 | –0.12 | 8.06 | 2.53
BP2 | 21.8 | –0.162 | 6.35 | 3.29
BP3 | 25.11 | –0.16 | 5.82 | 2.18
BP4 | 28.00 | –0.17 | 5.61 | 2.72
BP5 | 32.14 | –0.15 | 6.36 | 4.01
BP6 | 36.02 | –0.107 | 9.72 | 3.31
BP7 | 44.00 | –0.188 | 5.10 | 2.43
BP8 | 46.88 | –0.094 | 10.29 | 4.74
BP9 | 47.30 | –0.10 | 9.92 | 3.88
BP10 | 50.98 | –0.13 | 7.57 | 2.81
BP11 | 55.10 | –0.149 | 6.45 | 2.65
BP12 | 58.16 | –0.11 | 8.91 | 1.80
Table 2: 2HDM Type-X input parameters and Leading Order (NLO) cross sections
(at parton level with $\sqrt{s}=14$ TeV) for each BP are presented. The unit
of all masses is GeV and we fix the SM-like Higgs boson mass as $M_{H}$ = 125
GeV.

In this paper, we mainly focus on two SS taus further leptonicly decay, while
the other two tau hadronicly decay (tau-jets). Thus, the signal will be SS
leptons and two ($\tau$-)jets. The leptons are used for triggering purposes,
so its kinematics is bound to comply with the trigger requirements.

##  IV Collider phenomenology

In this section, we present a detailed MC analysis for both signal and
background events at the detector level. In the $pp\to H\to
hh\to\tau^{+}\tau^{-}\tau^{+}\tau^{-}\to lv_{l}lv_{l}\tau_{h}\tau_{h}$
process, a light Higgs boson decays into two $\tau$ leptons and one of the
$\tau$ decays into a charged lepton and neutrino, while the other $\tau$
decays hadronically (thus labelled as $\tau_{h}$). As a result, the final
states consist of two SS leptons and two jets. The SM backgrounds are:
$t\overline{t}\to lv_{l}lv_{l}b\overline{b}$, $W^{\pm}tb\to
lv_{l}lv_{l}b\overline{b}$ (where the $W^{\pm}$ boson and $b$-jet are not from
a top quark resonance), $WWjj\to lv_{l}lv_{l}jj)$, $Zjj\to lv_{l}lv_{l}jj$,
$ZZ\to lv_{l}lv_{l}\tau_{h}\tau_{h}$, $t\overline{t}Z\to
lv_{l}lv_{l}b\overline{b}\tau\tau$, $t\overline{t}ZZ\to
lv_{l}lv_{l}lv_{l}b\overline{b}\tau\tau$ and $t\overline{t}WW\to
lv_{l}lv_{l}lv_{l}jj$. In order to suppress the huge SM background process
$Zjj$, we deliberately choose two SS leptons plus two hadronic $\tau$’s in the
signal events, which is about $10\%$ of the total number of the $4\tau$ ones.

###  IV.1 Event generation and selection

We use MadGraph5-aMC@NLO v3.4.0 Alwall:2014hca  to compute the cross sections
and generate both signal and background events at parton level. The following
kinematic cuts are adopted for the signal and background generations:

$\displaystyle|\eta(l,j)|<2.5,~{}p_{\text{T}}(l,j)>10~{}\text{GeV},~{}E_{T}^{\text{miss}}>5~{}\text{GeV},~{}\Delta
R(ll,lj,jj)>0.4.$ (5)

Then parton level events are then passed to Pythia Sjostrand:2006za ;
Sjostrand:2014zea  to simulate initial and final state radiation (i.e., the
QED and QCD emissions), parton shower, hadronization and heavy flavor decays.
We further use Delphes v3.4.2 deFavereau:2013fsa  to simulate the detector
effects. For each event, the anti-$k_{t}$ jet algorithm Cacciari:2008gp  is
used to cluster jets with jet parameter $\Delta R=0.4$ in the FastJet package
Cacciari:2011ma . The following kinematic cuts are adopted in order to emulate
the detector acceptance:

$\displaystyle|\eta(l,j)|<2.5,~{}p_{T}(l,j)>10~{}\text{GeV}.$ (6)

The leptons are used for triggering purposes, so its kinematics is bound to
comply with the trigger requirements. 111The CMS developed a double muon
trigger to search for low pt muons from B meson decays that are close
together. The same approach has been adopted recently to search for a pairs of
electron from B meson decays, by tightening the topological selection on the
two electrons L1 objects. In our recent paper Arhrib:2023apw , we discussed
the possibility of developing an electron-muon trigger to search for low pt
leptons (pt 10 GeV)

Then, we select two leptons and two jets final states from detector simulated
events. In order to improve the signal event rate, the tau tagging is not
applied. In order to further suppress the huge SM backgrounds, the SS leptons
are selected. The cross sections for the signal and background processes with
detector acceptance cuts, with $2l2j$ selection and with two SS lepton
selection are shown in the Tab. 3 and in Tab. 4, respectively.

$\sigma$ (fb) | BP1 | BP2 | BP3 | BP4 | BP5 | BP6 | BP7 | BP8 | BP9 | BP10 | BP11 | BP12
---|---|---|---|---|---|---|---|---|---|---|---|---
parton level generation | 8.95 | 11.37 | 7.38 | 9.05 | 13.32 | 11.33 | 10.43 | 23.16 | 19.26 | 15.81 | 17.00 | 12.75
selection of 2l2j | 0.67 | 0.88 | 0.59 | 0.76 | 1.10 | 0.96 | 0.88 | 1.97 | 1.62 | 1.46 | 1.70 | 1.38
selection of SS 2l | 0.337 | 0.438 | 0.31 | 0.41 | 0.59 | 0.52 | 0.50 | 1.14 | 0.92 | 0.83 | 0.93 | 0.75
Table 3: Cross sections for signals in the parton level, detector level, and after selection of SS leptons at $\sqrt{s}=~{}\text{14 TeV}$ for $\text{300}~{}\text{fb}^{-1}$.  $\sigma$ (fb) | $t\bar{t}$ | $W^{\pm}tb$ | $W^{+}W^{-}jj$ | $Zjj$ | $ZZ$ | $t\bar{t}Z$ | $t\bar{t}ZZ$ | $t\bar{t}W^{+}W^{-}$
---|---|---|---|---|---|---|---|---
parton level generation | 16060 | 518.3 | 1053 | 317600 | 18.89 | 0.49 | $1.14\times 10^{-4}$ | 0.02
selection of 2l2j | 8787.7 | 289.9 | 530.1 | 151086 | 10.0 | 0.33 | $1.1\times 10^{-4}$ | 0.018
selection of SS 2l | 19.43 | 0.62 | 1.99 | 0 | 2.51 | 0.079 | $3.3\times 10^{-5}$ | $7.6\times 10^{-3}$
Table 4: Background rates after the acceptance cuts in Eq. (6) at
$\sqrt{s}=~{}\text{14 TeV}$ for $\text{300}~{}\text{fb}^{-1}$.

In Fig. 2, the transverse momentum distributions of the leading and the
second-leading jets and leptons as well as the transverse missing energy (MET)
are presented for, e.g., BP2 and background events. It is clear that the
transverse momenta of the signal final state are always much softer than for
background events, because they emerge from a rather light Higgs bosons, which
energies are capped at a mass of 125 GeV.

(a)

(b)

(c)

(d)

(e)

Figure 2: Distributions in (a)-(d) transverse momenta of the leading/second-
leading jets/leptons and (e) MET for BP2 and backgrounds when
$\sqrt{s}=~{}\text{14 TeV}$ and L = $\text{300}~{}\text{fb}^{-1}$.

###  IV.2 Event reconstruction

To further improve the signal-to-background discrimination, we reconstruct
some kinematic features of signal events. Specifically, because the light
Higgs boson decays into two $\tau$ and one of the $\tau$’s decays to a lepton
and two neutrinos plus two $\tau$ neutrinos emerge from hadronic $\tau$
decays, there will be six neutrinos in a signal event. It is thus extremely
hard to completely reconstruct the momentum of each light Higgs.

However, since the lepton momenta and MET are all very small, we can
approximately reconstruct the light Higgs boson with only one lepton and one
jet. Thus, there are two possible combinations with two leptons and two
$\tau$-jets in each event. By defining a suitable $\chi^{2}$,

$\chi^{2}={(M_{{lj}}^{1}-M_{{h}})}^{2}+{(M_{lj}^{2}-M_{h})}^{2},$ (7)

where $M_{h}$ is the light Higgs mass, we pair the two leptons and two jets to
find a combination which minimizes it and then assign these to $M_{lj}^{1}$
and $M_{lj}^{2}$, respectively. Taking again BP2 as an example, the
reconstructed light Higgs mass distributions are shown in Fig. 3.

In order to further separate signal from background events, we can also
combine the two leptons and two jets together, denoting their invariant masses
as $M_{ll}$ and $M_{jj}$, respectively, as shown in Fig. 4. Since, for signal
events, all particles come from the SM-like Higgs boson decaying, any such
combination only yield a smaller invariant mass, typically, about half of the
SM-like Higgs boson mass. In contrast, for background events, these lepton and
jet pairs always come from a heavy resonance, a top (anti)quark and a
$W^{\pm}$ or $Z$ boson, so the background spectra are somewhat harder in
general.

Figure 3: Distributions in invariant mass of a lepton and a jet for BP2 and
backgrounds. The reconstructed $M_{h}$ values, corresponding to the best
$\chi^{2}$ fit, is labelled as $M_{lj}^{1}$ (left) and $M_{lj}^{2}$ (right),
respectively, when $\sqrt{s}=~{}\text{14 TeV}$ and L =
$\text{300}~{}\text{fb}^{-1}$.

Figure 4: Distributions in invariant mass of lepton (left) and jet (right)
pairs for BP2 and backgrounds when $\sqrt{s}=~{}\text{14 TeV}$ and L =
$\text{300}~{}\text{fb}^{-1}$.

We can further reconstruct the invariant mass of the SM-like Higgs boson
$M_{H}$ with two leptons and two jets but without any MET. It is expected that
the sampled values of it will be lower than the real SM-like Higgs boson mass.
Nevertheless, it is helpful enough to distinguish the signal and background
events further, see Fig. 5 (left).

Another useful kinematic variable is $H_{T}$, which is the scalar sum of all
visible final states transverse momenta, which are shown in Fig. 5 (right).

Figure 5: Distributions in invariant mass of two leptons and two jets (left)
and in scalar sum of visible momenta (right) for BP2 and backgrounds when
$\sqrt{s}=~{}\text{14 TeV}$ and L = $\text{300}~{}\text{fb}^{-1}$.

###  IV.3 Multi-variate analysis

To optimize the signal and background distinction, a Gradient-Boosted Decision
Tree (GBDT) approach is further applied, which is implemented in the Toolkit
for Multi-Variate Analysis (TMVA) within ROOT Therhaag:2010zz .

Ten input variables in total are used for the GBDT/TMVA analysis, which are
shown in Tab. 5. In addition to the above invariant masses and $H_{T}$, we
also calculate five angles between pairs of objects in the final state,
$\cos(\theta_{l_{1}j_{1}})$, $\cos(\theta_{l_{1}j_{2}})$,
$\cos(\theta_{l_{2}j_{1}})$, $\cos(\theta_{l_{2}j_{2}})$ and
$\cos(\theta_{h_{1}h_{2}})$, which is defined as the cosine of the angle
between the two reconstructed final states states. The distributions are shown
in Fig. 6. As we expect these to be useful, since in the signal they all tend
to be collinear to each other (because of the boost structure herein) unlike
the background case.

Energy variables | $M_{l\tau_{j}}^{1}$ | $M_{l\tau_{j}}^{2}$ | $M_{ll}$ | $M_{jj}$ | $H_{T}$
---|---|---|---|---|---
Angular variables | $\cos(\theta_{h_{1}h_{2}})$ | $\cos(\theta_{l_{1}j_{1}})$ | $\cos(\theta_{l_{1}j_{2}})$ | $\cos(\theta_{l_{2}j_{1}})$ | $\cos(\theta_{l_{2}j_{2}})$

Table 5: The input observables used in the GBDT analysis.

(a)

(b)

(c)

(d)

(e)

Figure 6: Distributions in the (cosine of the) angle between (a)-(d) pairs of
jets/leptons and (e) the two reconstructed light Higgs states for BP2 and
backgrounds when $\sqrt{s}=~{}\text{14 TeV}$ and L =
$\text{300}~{}\text{fb}^{-1}$.

In the training stage, it is found that the invariant mass variables are
powerful, but angular variables are still useful. Ultimately, for the GBDT
output, a very good separation between the signal and backgrounds can be
achieved, which is shown in Fig. 7.

Figure 7: GBDT output for the BP2 signal and backgrounds.

###  IV.4 Significances

The process of searching for signals at high energy colliders using ML
techniques is ultimately a statistical process. Therefore, to look for
signals, a large number of events need to be collected and the kinematic
variables of the final state particles need to be analyzed to find differences
from the noise, but then a sizeable statistical sample needs to be collected,
so has to have enough significance.

The latter is defined as $\Sigma=\frac{N_{S}}{\sqrt{(N_{S}+N_{B})}}$, where
$N_{S}$ and $N_{B}$ represent the number of events for the signal ($B$) and
(total) background ($B$), respectively, at a given luminosity $L$.

To distinguish the signal from the background and to compute the ensuing
significance, we apply six kinematic cuts on reconstructed observables and one
MVA cut on GBDT outputs, the details of which are listed in Tab. 6, for BP2
and where $L=\text{300}~{}\text{fb}^{-1}$ (with a CM energy
$\sqrt{s}=~{}\text{14 TeV}$, as previously mentioned). Following the
acceptance cuts and upon applying the SS lepton criterion (the combination of
which we refer to as pre-selection), we see a strong decrease of the initially
huge SM backgrounds to the same order as the signal. For example, the
initially troublesome $Zjj$ is almost eliminated after this cut (thus, it is
no longer included in this table). As Figs. 3–5 show, the kinematic selection
is already quite efficient is separating signal from background, as –
depending on the BP adopted – $\Sigma$ values can reach values around 4, even
with rather loose cuts. However, exploitation of the GBDT output more than
double the final significance.

$L=\text{300}~{}\text{fb}^{-1}$ | $S$ | $t\bar{t}$ | $t\bar{t}W^{+}W^{-}$ | $t\bar{t}Z$ | $t\bar{t}ZZ$ | $Wtb$ | $W^{+}W^{-}jj$ | $ZZ$ | $B$ | $\Sigma$
---|---|---|---|---|---|---|---|---|---|---
Acceptance | 262.6 | 2.17$\times 10^{6}$ | 2.3 | 63 | 0.01 | 53751.6 | 119699 | 2790.8 | 2.35$\times 10^{6}$ | 0.17
SS Leptons | 131.5 | 5829.8 | 1.5 | 23.6 | 0.01 | 186.6 | 595.5 | 754.1 | 7391.1 | 1.52
$H_{T}\in[40,200]$ | 129.9 | 1040.7 | 0.002 | 0.5 | 1e-05 | 40.4 | 127.9 | 379.1 | 1588.7 | 3.13
$M_{ll}\in[10,80]$ | 129.8 | 905.8 | 0.001 | 0.5 | 1e-05 | 32.7 | 58.4 | 283.8 | 1281.1 | 3.45
$M_{h_{1}}\in[0,150]$ | 128.7 | 809.4 | 0.001 | 0.4 | 1e-05 | 31.1 | 56.9 | 271.4 | 1169.2 | 3.57
$M_{h_{2}}\in[0,150]$ | 126.9 | 732.3 | 0.001 | 0.4 | 8e-06 | 28 | 55.3 | 260.5 | 1076.5 | 3.66
$M_{H}\in[20,230]$ | 96.9 | 269.8 | 0 | 0.2 | 5e-06 | 10.9 | 26.9 | 124.3 | 432 | 4.21
${\rm GBDT}\in[0.5,1]$ | 92.6 | 0 | 0 | 0.02 | 0 | 0 | 3.2 | 7.1 | 10.2 | 9.13
Table 6: Response to our selection cuts for the signal (e.g., BP2) and
background (separately and total) rates computed at $\sqrt{s}=~{}\text{14
TeV}$ for $\text{300}~{}\text{fb}^{-1}$.

The results for all twelve BPs are shown in Tab. 7. With only the acceptance
and SS lepton criteria, the significances can reach 3 for BP8, BP9 and BP11.
After applying the kinematic cuts, the significances can be all larger than 3,
except for BP3. Upon applying the MVA selection, significances can reach
values as large as 14 for BP8. In short, it is clear that a judicious
combination of pre-selection and kinematic cuts combinaed with ML analysis can
result in large sensitivity to a significant expanse of 2HDM Type-X parameter
space through our chosen signal.

$\Sigma$ | BP1 | BP2 | BP3 | BP4 | BP5 | BP6 | BP7 | BP8 | BP9 | BP10 | BP11 | BP12
---|---|---|---|---|---|---|---|---|---|---|---|---
after selecting SS leptons | 1.17 | 1.52 | 1.06 | 1.40 | 2.04 | 1.79 | 1.71 | 3.89 | 3.15 | 2.86 | 3.18 | 2.58
after kinematic cuts, w/o GBDT | 3.09 | 4.21 | 2.66 | 3.79 | 5.20 | 4.50 | 4.50 | 8.84 | 7.58 | 6.65 | 7.41 | 6.24
after kinematic cuts, w/ GBDT | 7.65 | 9.13 | 6.27 | 8.61 | 10.41 | 9.45 | 9.34 | 14.32 | 12.91 | 11.25 | 12.14 | 10.74
Table 7: Significances following our different selections for all signals
(BP1–BP12) at $\sqrt{s}=~{}\text{14 TeV}$ for $\text{300}~{}\text{fb}^{-1}$.

##  V Conclusions

In this paper, we have investigated the feasibility of producing the SM-like
Higgs boson ($H$) in single mode at the LHC decaying via a resonance into two
lighter neutral CP-even Higgs bosons ($hh$) in the 2HDM Type-X. We have
focused on $4\tau$ final states, where two $\tau$’s decay leptonically to form
a pair of SS leptons (electrons and/or muons) and two $\tau$’s decay
hadronically, thus producing two jets. In order to motivate future studies of
this signature, we have also proposed 12 BPs and processed these through a
realistic MC analysis for a CM (proton-proton) energy $\sqrt{s}=~{}\text{14
TeV}$ and (integrated) luminosity $L=\text{300}~{}\text{fb}^{-1}$. Eventually,
we have proven that this kind of signal could be found at the end of Run 3 of
the LHC, following a dedicated selection based on kinematic and TMVA analysis.

## Acknowledgments

SM is supported in part through the NExT Institute and the STFC Consolidated
Grant ST/L000296/1. SS is supported in full through the NExT Institute and
acknowledges the use of the IRIDIS High Performance Computing Facility, and
associated support services, at the University of Southampton, in the
completion of this work. YW’s work is supported by the Natural Science
Foundation of China Grant No. 12275143, the Inner Mongolia Science Foundation
Grant No. 2020BS01013 and the Fundamental Research Funds for the Inner
Mongolia Normal University Grant No. 2022JBQN080. QSY is supported by the
Natural Science Foundation of China under the Grants No. 11875260 and No.
12275143.

## References

  * (1) G. Aad et al. [ATLAS], Phys. Lett. B 716 (2012) 1, [arXiv:1207.7214 [hep-ex]].
  * (2) S. Chatrchyan et al. [CMS], Phys. Lett. B 716 (2012) 30, [arXiv:1207.7235 [hep-ex]].
  * (3) G. Aad et al. [ATLAS], Nature 607, no.7917, 52-59 (2022) [erratum: Nature 612, no.7941, E24 (2022)] doi:10.1038/s41586-022-04893-w [arXiv:2207.00092 [hep-ex]].
  * (4) A. Tumasyan et al. [CMS], Nature 607, no.7917, 60-68 (2022) doi:10.1038/s41586-022-04892-x [arXiv:2207.00043 [hep-ex]].
  * (5) T. D. Lee, Phys. Rev. D 8 (1973), 1226-1239.
  * (6) N. G. Deshpande and E. Ma, Phys. Rev. D 18 (1978), 2574.
  * (7) G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Phys. Rept. 516 (2012), 1-102 [arXiv:1106.0034 [hep-ph]].
  * (8) S. L. Glashow and S. Weinberg, Phys. Rev. D 15 (1977), 1958
  * (9) M. Aoki, S. Kanemura, K. Tsumura and K. Yagyu, Phys. Rev. D 80, 015017 (2009) doi:10.1103/PhysRevD.80.015017 [arXiv:0902.4665 [hep-ph]].
  * (10) V. Khachatryan et al. [CMS], JHEP 01, 079 (2016) doi:10.1007/JHEP01(2016)079 [arXiv:1510.06534 [hep-ex]].
  * (11) V. Khachatryan et al. [CMS], JHEP 10, 076 (2017) doi:10.1007/JHEP10(2017)076 [arXiv:1701.02032 [hep-ex]].
  * (12) A. M. Sirunyan et al. [CMS], Phys. Lett. B 800, 135087 (2020) doi:10.1016/j.physletb.2019.135087 [arXiv:1907.07235 [hep-ex]].
  * (13) A. Arhrib, R. Benbrik, M. Krab, B. Manaut, S. Moretti, Y. Wang and Q. S. Yan, JHEP 10, 073 (2021) doi:10.1007/JHEP10(2021)073 [arXiv:2106.13656 [hep-ph]].
  * (14) Y. Wang, A. Arhrib, R. Benbrik, M. Krab, B. Manaut, S. Moretti and Q. S. Yan, JHEP 12, 021 (2021) doi:10.1007/JHEP12(2021)021 [arXiv:2107.01451 [hep-ph]].
  * (15) A. Arhrib, R. Benbrik, M. Krab, B. Manaut, S. Moretti, Y. Wang and Q. S. Yan, Symmetry 13, no.12, 2319 (2021) doi:10.3390/sym13122319 [arXiv:2110.04823 [hep-ph]].
  * (16) Z. Li, A. Arhrib, R. Benbrik, M. Krab, B. Manaut, S. Moretti, Y. Wang and Q. S. Yan, [arXiv:2305.05788 [hep-ph]].
  * (17) D. Eriksson, J. Rathsman and O. Stål, Comput. Phys. Commun. 181 (2010), 189-205 [arXiv:0902.0851 [hep-ph]].
  * (18) S. Kanemura, T. Kubota and E. Takasugi, Phys. Lett. B 313, 155-160 (1993) doi:10.1016/0370-2693(93)91205-2 [arXiv:hep-ph/9303263 [hep-ph]].
  * (19) A. G. Akeroyd, A. Arhrib and E. M. Naimi, Phys. Lett. B 490, 119-124 (2000) doi:10.1016/S0370-2693(00)00962-X [arXiv:hep-ph/0006035 [hep-ph]].
  * (20) A. Arhrib, [arXiv:hep-ph/0012353 [hep-ph]].
  * (21) S. Kanemura, T. Kubota and E. Takasugi, Phys. Lett. B 313 (1993), 155-160 [arXiv:hep-ph/9303263 [hep-ph]].
  * (22) A. G. Akeroyd, A. Arhrib and E. M. Naimi, Phys. Lett. B 490 (2000), 119-124 [arXiv:hep-ph/0006035 [hep-ph]]. A. Arhrib, [arXiv:hep-ph/0012353 [hep-ph]].
  * (23) M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65 (1990), 964-967.
  * (24) M. E. Peskin and T. Takeuchi, Phys. Rev. D 46 (1992), 381-409.
  * (25) R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022) doi:10.1093/ptep/ptac097
  * (26) P. Bechtle, D. Dercks, S. Heinemeyer, T. Klingl, T. Stefaniak, G. Weiglein and J. Wittbrodt, Eur. Phys. J. C 80 (2020) no.12, 1211 [arXiv:2006.06007 [hep-ph]].
  * (27) P. Bechtle, S. Heinemeyer, T. Klingl, T. Stefaniak, G. Weiglein and J. Wittbrodt, Eur. Phys. J. C 81 (2021) no.2, 145 [arXiv:2012.09197 [hep-ph]].
  * (28) F. Mahmoudi, Comput. Phys. Commun. 180 (2009), 1579-1613 [arXiv:0808.3144 [hep-ph]].
  * (29) Y. S. Amhis et al. [HFLAV], Phys. Rev. D 107, no.5, 052008 (2023) doi:10.1103/PhysRevD.107.052008 [arXiv:2206.07501 [hep-ex]].
  * (30) R. Aaij et al. [LHCb], Phys. Rev. D 105, no.1, 012010 (2022) doi:10.1103/PhysRevD.105.012010 [arXiv:2108.09283 [hep-ex]].
  * (31) R. Aaij et al. [LHCb], Phys. Rev. Lett. 128, no.4, 041801 (2022) doi:10.1103/PhysRevLett.128.041801 [arXiv:2108.09284 [hep-ex]].
  * (32) A. Tumasyan et al. [CMS], Phys. Lett. B 842, 137955 (2023) doi:10.1016/j.physletb.2023.137955 [arXiv:2212.10311 [hep-ex]].
  * (33) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli and M. Zaro, JHEP 07 (2014), 079 [arXiv:1405.0301 [hep-ph]].
  * (34) T. Sjostrand, S. Mrenna and P. Z. Skands, JHEP 05 (2006), 026 [arXiv:hep-ph/0603175 [hep-ph]].
  * (35) T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen and P. Z. Skands, Comput. Phys. Commun. 191 (2015), 159-177 [arXiv:1410.3012 [hep-ph]].
  * (36) J. de Favereau et al. [DELPHES 3], JHEP 02 (2014), 057 [arXiv:1307.6346 [hep-ex]].
  * (37) M. Cacciari, G. P. Salam and G. Soyez, JHEP 04, 063 (2008) doi:10.1088/1126-6708/2008/04/063 [arXiv:0802.1189 [hep-ph]].
  * (38) M. Cacciari, G. P. Salam and G. Soyez, Eur. Phys. J. C 72, 1896 (2012) doi:10.1140/epjc/s10052-012-1896-2 [arXiv:1111.6097 [hep-ph]].
  * (39) A. Arhrib, S. Moretti, S. Semlali, C. H. Shepherd-Themistocleous, Y. Wang and Q. S. Yan, [arXiv:2310.02736 [hep-ph]].
  * (40) J. Therhaag, PoS ICHEP2010 (2010), 510