# Higgs production at next generation $e^{+}e^{-}$ colliders

Deniz YILMAZ1 , Mehmet SAHIN2, Dogukan Hazar YAVUZ1
1Physics Engineering Department, Ankara University, Ankara,Turkey
2Department of Computer Engineering, Usak University, Usak,Turkey
<EMAIL_ADDRESS>
###### Abstract

In this study, Higgs production processes, Higgsstrahlung and vector boson (W
and Z) fusion processes, were investigated for four different future lepton
colliders (CEPC, ILC, CLIC, and FCC-ee). The cross sections for each
production process and corresponding backgrounds were calculated considering
the ISR and beamstrahlung effects. Various cuts and the b-tagging method were
used to reduce the background. Finally, the number of events for each collider
was determined, and significance calculations were performed. In our
calculations, high event numbers were obtained for all four colliders for the
Higgsstrahlung, W, and Z fusion process. This shows that electron-positron
colliders will play an important role in future Higgs physics research.

##  1 Introduction

The discovery of the Higgs boson at the Large Hadron Collider (LHC) [1, 2]
confirmed the electroweak symmetry breaking mechanism of the Standard Model
(SM) [3, 4, 5, 6]. However, there is still some unknown about the observed
Higgs boson: is it the fundamental scalar of the SM, or a more complex object,
or part of an extended Higgs sector? Studying the properties of the Higgs
boson at the LHC and in future colliders is crucial to understanding its true
nature. Up to now, some properties of the Higgs boson have been measured at
the LHC with an accuracy of about 10% [7, 8, 9, 10]. Although the LHC Run 2 to
be developed will examine it with higher data, because of the complexity of
the internal structure of the proton, the LHC will not be sensitive enough to
examine the properties of Higgs.

Electron-positron colliders, which will be installed to precisely measure the
properties of the Higgs particle, have unique capabilities for the measurement
of Higgs boson parameters, including the Higgs total cross section, decay
width, branching ratios, Higgs width, and determination of Higgs couplings.

Therefore, today, four $e^{+}e^{-}$ colliders are being designed to study the
properties of the Higgs boson and other standard model (SM) particles with
high precision: the International Linear Collider (ILC) [11], with a center of
mass energy of 250 – 500 GeV, Compact Linear Collider (CLIC) [12] with center
of mass energies of 380 – 1500 – 3000 GeV, Circular Electron Positron Collider
(CEPC) with center of mass energies between 90 and 250 GeV [13] and the Future
$e^{+}e^{-}$ Circular Collider (FCC-ee) [14], which will be located in a new
tunnel at CERN at 240 GeV center of mass energy. The main beam parameters of
these colliders [11, 12, 13, 14] are given in Table 1. The integrated
luminosities given here are annual values.

Table 1: The main collider parameters Parameters | CEPC | FCC-ee | ILC | CLIC
---|---|---|---|---
Center of mass energy (GeV) | 240 | 240 | 250 | 500 | 380 | 1500 | 3000
Number of particles per bunch ($10^{10}$) | 15 | 18 | 2 | 2 | 0.52 | 0.37 | 0.37
Horizontal beam size at IP ($\sigma_{x}$) ($\mu$m) | 20.9 | 13.7 | 0.516 | 0.474 | 0.149 | 0.06 | 0.04
Vertical beam size at IP ($\sigma_{y}$) (nm) | 60 | 36 | 7.66 | 5.86 | 2.9 | 1.5 | 1
Bunch length (mm) | 4.4 | 5.3 | 0.3 | 0.3 | 0.07 | 0.044 | 0.044
Luminosity ($10^{5}pb^{-1}$ ) | 6 | 17 | 1.35 | 1.8 | 1.5 | 3.7 | 5.9

In the electron-positron collider, Higgs bosons are produced by the
Higgsstrahlung and vector boson (W and Z) fusion processes [15, 16, 17, 18,
19, 20, 21]. In this study, these three processes were examined and
calculations were performed using CalcHEP [22, 23]. In the electron-positron
collider, it is important to consider the effects of ISR and Beamstrahlung
[24, 25]. The parameters listed in Table 1 were used to calculate the ISR and
the beamstraghlung effects. In section 2 cross sections are given for these
three processes. Section 3 provides signal and background analyses, the number
of events for each collider, and the significance calculations. Finally,
conclusion is provided in the section 4.

##  2 Higgs Production at the electron - positron colliders

The main production processes of Higgs at the $e^{+}e^{-}$ colliders are the
Higgsstrahlung and W/Z fusion mechanisms given below, as shown in Figure 1.

$\displaystyle Higgs-strahlung$ $\displaystyle e^{+}e^{-}\rightarrow ZH$
$\displaystyle Wfusion$ $\displaystyle
e^{+}e^{-}\rightarrow\overline{\nu_{e}}\nu_{e}H$ $\displaystyle Zfusion$
$\displaystyle e^{+}e^{-}\rightarrow e^{+}e^{-}H$

Figure 1: The Feynmann diagrams of the Higgs production processes Figure 2:
The cross sections of the Higgs production mechanisms as a function of center-
of-mass energy.

The cross section for the Higgsstrahlung process can be written as

$\sigma(e^{+}e^{-}\rightarrow ZH)=\dfrac{G^{2}_{F}M^{4}_{Z}}{96\pi
s}(\eta^{2}_{e}+a^{2}_{e})\kappa^{1/2}\dfrac{\kappa+12M^{2}_{Z}/s}{(1-M^{2}_{Z}/s)^{2}}$
(1)

where $a_{e}=-1$, $\eta_{e}=-1+4sin^{2}\theta_{W}$ are the Z charges of the
electron and $\kappa=(1-(M_{H}+M_{Z})^{2}/s)(1-(M_{H}-M_{Z})^{2}/s)$ is the
usual two particle phase space function. The total cross section for the
vector boson fusion mechanism is

$\sigma(e^{+}e^{-}\rightarrow VV\rightarrow
l\overline{l}H)=\dfrac{G^{2}_{F}m^{4}_{V}}{64\sqrt{2}\pi^{3}}\int_{x_{H}}^{1}dx\int_{x}^{1}\dfrac{dyT(x,y)}{[1+(y-x)/x_{V}]^{2}}.$
(2)
$T(x,y)=(\dfrac{2x}{y^{3}}-\dfrac{3x+1}{y^{2}}+\dfrac{x+2}{y}-1)[\dfrac{z}{z+1}-log(z+1)]+\dfrac{xz^{2}(1-y)}{y^{3}(z+1)},$

where $V$ denotes the vector bosons W or Z and $x_{H}=m^{2}_{H}/s$,
$x_{V}=m^{2}_{V}/s$ and $z=y(x-x_{H})/xx_{V}$ ($\sqrt{s}$ is the center-of-
mass energy) [26].

The behavior of the production cross-sections of the Higgs boson calculated by
the Higgsstrahlung and the W/Z fusion mechanisms using the CalcHEP simulation
program, depending on the center of mass energy, are shown in Figure 2 and
Figure 3. The relevant production cross sections as a function of the center
of mass energy are shown in Figure 2. As shown in Figure 2, the Higgsstrahlung
suppresses the vector boson production processes for moderate values of the
energy due to the additional electroweak coupling. With the increase in
energy, the cross sections of the vector boson procecesses increase
logarithmically and become dominant. At a center of mass energy of about 250
GeV, Higgs bosons are predominantly produced from the ZH process as seen in
the same figure. In the Figure 3, the cross sections are shown as a function
of the center of mass energy for each production mechanisms for four electron-
positron colliders with the ISR and the beamstrahlung effects of each
colliders.

Figure 3: The cross section comparison for Higgsstrahlung, W Fusion and Z
Fusion processes for four $e^{+}e^{-}$colliders.

##  3 Signal and Background Analyses

Figure 4: Transverse momentum distribution plots for the $e^{-}/e^{+}$ (upper
left ), $\mu^{-}/\mu^{+}$ (upper right) and $J/J$ (bottom) final state
particles of signal 1 and the corresponding bacground processes in FCC-ee
collider with 240 GeV center of mass energy.

Because the Higgs boson’s decay rate to $b\overline{b}$ is greater than the
decay rate to other quarks and leptons [27, 28, 29, 30, 31], the
$b\overline{b}$ decay mode of Higgs ($H\longrightarrow b\overline{b}$) is
considered in all production processes in this study. Since the cross sections
of the background processes corresponding to the leptonic decays of the Z
boson are less than the background cross sections corresponding to the other
decays, the leptonic decays of the Z boson in the Higgsstrahlung process are
taken into account. The signal processes are given below.

Signal 1: $\displaystyle\quad\quad Higgsstrahlung$ $\displaystyle
e^{+}e^{-}\rightarrow ZH\rightarrow l\overline{l}b\overline{b}$
$\displaystyle\quad\quad Zfusion$ $\displaystyle e^{+}e^{-}\rightarrow
e^{+}e^{-}b\overline{b}$ Signal 2: $\displaystyle\quad\quad Wfusion$
$\displaystyle e^{+}e^{-}\rightarrow\overline{\nu_{e}}\nu_{e}b\overline{b}$

Here, $l$ and $\overline{l}$ are $e^{-},\mu^{-}$ and $e^{+},\mu^{+}$,
respectively. The corresponding background processes analysed here are as
follows:

For signal 1: $\displaystyle i)$ $\displaystyle\quad\quad
e^{+}e^{-}\rightarrow ZZ\rightarrow l\overline{l}JJ,$ $\displaystyle ii)$
$\displaystyle\quad\quad e^{+}e^{-}\rightarrow e^{+}e^{-}Z\rightarrow
e^{+}e^{-}JJ,$ $\displaystyle iii)$ $\displaystyle\quad\quad
e^{+}e^{-}\rightarrow t\overline{t}\rightarrow W^{+}JW^{-}J\rightarrow
l\overline{l}JJ\nu_{l}\overline{\nu_{l}},$ For signal 2:
$\displaystyle\quad\quad e^{+}e^{-}\rightarrow JJ,$

here, $J$ represents the quark and antiquark:
$J=d,\overline{d},u,\overline{u},s,\overline{s},c,\overline{c},b,\overline{b}$.
The transverse momentum ($P_{T}$), pseudo rapidity ($\eta$) and invariant mass
($M_{inv}$) distributions of the final state particles were investigated by
using CalcHEP program in order to find the cut values to distinguish the
signal from the background in the FCC-ee collider with a center of mass energy
of 240 GeV. The background $iii$ process corresponding to Signal 1 is not
included in the calculations for 240 GeV, as it starts to contribute at 350
GeV and greater center of mass energies. Because the transverse momentum,
pseudo rapidity, and invariant mass distributions of the final state particles
in the signal and background processes will exhibit similar behavior for other
colliders, the cut values obtained can be used for CEPC, ILC, and CLIC.

Figure 5: Transverse momentum distribution plots for the $b/\overline{b}$ and
$J/J$ final state particles of signal 2 and the corresponding bacground
processes in FCC-ee collider with 240 GeV center of mass energy.

Transverse momentum distribution plots for the final state particles of signal
1 and the corresponding background processes i and ii are shown in Figure 4,
while the graphs of signal 2 are shown in Figure 5. As can be seen from Figure
4 and 5, when a transverse momentum cut of 35 GeV is applied to the $e^{-}$,
$e^{+}$, $\mu^{-}$, $\mu^{+}$, and two jets ($J$) in the final state particles
of signal 1 and signal 2 and the corresponding background processes, the
signal will almost not change, but the background will be significantly
reduced.

Pseudorapidity plots for signal 1, signal 2, and the corresponding backgrounds
are shown in Figure 6 and 7. As can be seen from the figures, cut regions of
$-2.5<\eta_{J,J}<2.5$, $-2.5<\eta_{e^{-},e^{+}}<2.5$,
$-2.5<\eta_{\mu^{-},\mu^{+}}<2.5$ will be appropriate for $e^{-}$, $e^{+}$,
$\mu^{-}$, $\mu^{+}$ and two jets (J) in the final state particles of signal 1
and signal 2 and the corresponding background processes.

Figure 6: Pseudorapidity distribution plots for the $e^{-}/e^{+}$ (upper left
), $\mu^{-}/\mu^{+}$ (upper right) and $J/J$ (bottom) final state particles of
signal 1 and the corresponding bacground processes in FCC-ee collider with 240
GeV center of mass energy.  Figure 7: Pseudorapidity distribution plots for
the $b/\overline{b}$ and $J/J$ final state particles of signal 2 and the
corresponding bacground processes in FCC-ee collider with 240 GeV center of
mass energy.

An $E_{T}^{miss}$ cut value of $>$15 GeV was also used for neutrinos in our
calculations.

Invariant mass distribution plots for signal 1, signal 2 and their
corresponding background processes are shown in Figure 8. As can be seen from
the figures, in the calculations, it would be appropriate to exclude the 80
GeV $<M_{inv}(e^{-},e^{+})<$100 GeV and 80 GeV $<M_{inv}(\mu^{-},\mu^{+})<$
100 GeV regions for the $l\overline{l}$ final states in signal and background
processes. In addition, only the 115 GeV $<M_{inv}(J,J)<$135 GeV region was
included in the calculations for two final jet states in the signal and
background processes. These included and excluded invariant mass regions allow
the signal to be distinguished from the background.

In addition to these cut values, the separation cuts of $\Delta R(l,J)>$0.5
and $\Delta R(\overline{l},J)>$0.5 distinguish the final state leptons and
antileptons from the jets, while the $\Delta R(J,J)>$0.5 separation cut was
used to distinguish the final state jets from each other.

All the cut values obtained are listed in Table 2, and these cut values were
used in the calculations for the four colliders. In addition to the cut values
in Table 2, because the Higgs boson decays to $b\overline{b}$ in our signal
processes, it is possible to further reduce the background cross section value
using the b-tagging method [27]: 68% is used for the b-tagging identification
rate, and a 1% ratio is used for misidentification rate with light quarks as b
quarks.

Figure 8: Invariant mass plots for the signal 1 (left) and signal 2 (right)
and the corresponding bacground processes in FCC-ee collider with 240 GeV
center of mass energy. Table 2: Cut values
$E^{miss}_{T}(\nu_{l},\overline{\nu_{l}})>$ 15 GeV
---
$P_{T}(l,\overline{l})>$ 35 GeV
$P_{T}(J)>$ 35 GeV
-2.5 $<\eta(l,\overline{l})<$ 2.5
-2.5 $<\eta(J)<$ 2.5
80 GeV $<M_{inv}(l,\overline{l})<$ 100 GeV region is excluded
115 GeV$<M_{inv}(J,J)<$ 135 GeV region is included
$\Delta R(l,J)>$ 0.5
$\Delta R(\overline{l},J)>$ 0.5
$\Delta R(J,J)>$ 0.5

The following equation is used to calculate the significance of the obtained
data:

$\mathcal{S}=\sqrt{2((s+b)\ln(1+s/b)-s)}$ (3)

where $s$ and $b$ represent signal and background events, respectively [32].
Cross-sections, event rates, and significance values were calculated for the
signal and background processes using the cut values in Table 2, b-tagging
method, and nominal integrated luminosity given in Table 1. The event rates
and significance values of the signals and corresponding backgrounds are
obtained for four future lepton colliders. The numerical results are given in
Table 3-6. The abbreviations used in the tables are: Sg CS (signal cross-
section), Bg CS ( background cross-section),$\mathcal{L}$ (integrated
luminosity), No. SgE (number of signal events) and No. BgE (number of
background events).

Table 3: Cross sections, number of events and the significance values for CEPC. Colliders | Processes |  | Sg CS
---
($pb$)
| Bg CS
---
($pb$)
| $\mathcal{L}$
---
($pb^{-1}$)
No. SgE | No.BgE | $\mathcal{S}$
CEPC (240 GeV) | Signal 1 | 9.91$\times 10^{-5}$ | 2.38$\times 10^{-4}$ | 6$\times 10^{5}$ | 59.5 | 142.8 | 4.7
| Signal 1
---
(with b-tagging)
6.72$\times 10^{-5}$ | 3.68$\times 10^{-5}$ | 40.32 | 22.08 | 7
Signal 2 | 1.24$\times 10^{-2}$ | 5.22$\times 10^{-1}$ | 7440 | 313200 | 13.24
| Signal 2
---
(with b-tagging)
8.45$\times 10^{-3}$ | 7.17$\times 10^{-2}$ | 5070 | 43020 | 23.98
Table 4: Cross sections, number of events and the significance values for FCC-ee. Colliders | Processes |  | Sg CS
---
($pb$)
| Bg CS
---
($pb$)
| $\mathcal{L}$
---
($pb^{-1}$)
No. SgE | No.BgE | $\mathcal{S}$
FCC-ee (240 GeV) | Signal 1 | 9.98$\times 10^{-5}$ | 2.36$\times 10^{-4}$ | 1.7$\times 10^{6}$ | 169.7 | 401.2 | 7.95
| Signal 1
---
(with b-tagging)
6.79$\times 10^{-5}$ | 3.67$\times 10^{-5}$ | 115.4 | 62.4 | 11.9
Signal 2 | 1.25$\times 10^{-2}$ | 5.37$\times 10^{-1}$ | 21250 | 912900 | 22.15
| Signal 2
---
(with b-tagging)
8.53$\times 10^{-3}$ | 7.38$\times 10^{-2}$ | 14501 | 125460 | 40.18
Table 5: Cross sections, number of events and the significance values for ILC. Colliders | Processes |  | Sg CS
---
($pb$)
| Bg CS
---
($pb$)
| $\mathcal{L}$
---
($pb^{-1}$)
No. SgE | No.BgE | $\mathcal{S}$
ILC (250 GeV) | Signal 1 | 1.33$\times 10^{-4}$ | 2.9$\times 10^{-4}$ | 1.35$\times 10^{5}$ | 17.95 | 39.15 | 2.68
| Signal 1
---
(with b-tagging)
9.04$\times 10^{-5}$ | 4.34$\times 10^{-5}$ | 12.2 | 5.86 | 4.03
Signal 2 | 1.3$\times 10^{-2}$ | 5.33$\times 10^{-1}$ | 1755 | 71955 | 6.51
| Signal 2
---
(with b-tagging)
8.82$\times 10^{-3}$ | 7.32$\times 10^{-2}$ | 1190 | 9882 | 11.74
ILC (500 GeV) | Signal 1 | 1.41$\times 10^{-3}$ | 1.7$\times 10^{-3}$ | 1.8$\times 10^{5}$ | 253.8 | 306 | 12.98
| Signal 1
---
(with b-tagging)
9.57$\times 10^{-4}$ | 3.32$\times 10^{-4}$ | 172.3 | 59.8 | 16.88
Signal 2 | 2.86$\times 10^{-2}$ | 1.13$\times 10^{-1}$ | 5148 | 20340 | 34.71
| Signal 2
---
(with b-tagging)
1.94$\times 10^{-2}$ | 1.56$\times 10^{-2}$ | 3492 | 2808 | 56.54
Table 6: Cross sections, number of events and the significance values for CLIC. Colliders | Processes |  | Sg CS
---
($pb$)
| Bg CS
---
($pb$)
| $\mathcal{L}$
---
($pb^{-1}$)
No. SgE | No.BgE | $\mathcal{S}$
CLIC (380 GeV) | Signal 1 | 6.44$\times 10^{-4}$ | 1.08$\times 10^{-3}$ | 1.5$\times 10^{5}$ | 96.6 | 162 | 6.98
| Signal 1
---
(with b-tagging)
4.38$\times 10^{-4}$ | 2.53$\times 10^{-4}$ | 65.7 | 37.95 | 8.76
Signal 2 | 1.7$\times 10^{-2}$ | 2.11$\times 10^{-1}$ | 2550 | 31650 | 14.14
| Signal 2
---
(with b-tagging)
1.16$\times 10^{-2}$ | 2.9$\times 10^{-2}$ | 1740 | 4350 | 24.86
CLIC (1500 GeV) | Signal 1 | 1.95$\times 10^{-3}$ | 1.47$\times 10^{-3}$ | 3.7$\times 10^{5}$ | 721.5 | 543.9 | 26.34
| Signal 1
---
(with b-tagging)
1.32$\times 10^{-3}$ | 2.34$\times 10^{-4}$ | 488.4 | 86.58 | 36.64
Signal 2 | 1.03$\times 10^{-1}$ | 9.22$\times 10^{-3}$ | 38110 | 3411 | 362
| Signal 2
---
(with b-tagging)
6.99$\times 10^{-2}$ | 1.27$\times 10^{-3}$ | 25863 | 470 | 400
CLIC (3000 GeV) | Signal 1 | 6.01$\times 10^{-4}$ | 8.47$\times 10^{-4}$ | 5.9$\times 10^{5}$ | 355 | 499.7 | 14.38
| Signal 1
---
(with b-tagging)
4.09$\times 10^{-4}$ | 1.32$\times 10^{-4}$ | 241 | 77.8 | 20.44
Signal 2 | 1.61$\times 10^{-1}$ | 2.72$\times 10^{-3}$ | 94990 | 1605 | 776
| Signal 2
---
(with b-tagging)
1.09$\times 10^{-1}$ | 3.74$\times 10^{-4}$ | 64310 | 221 | 777

##  4 Conclusion

After the discovery of the Higgs particle, precise measurements of the Higgs
properties became an important step forward for future research in particle
physics. Electron positron colliders to be installed for this purpose have
unique capabilities for the measurement of Higgs boson parameters, including
the Higgs total cross section of production processes, decay width, branching
rates and determination of Higgs couplings. In this study, the Higgsstrahlung
and W and Z fusion processes were examined, and the data obtained are
presented in graphs and tables for four different electron-positron colliders.
The production cross-sections for each process and additionally cross-sections
for various final state backgrounds were calculated. In the calculations, we
attempted to reduce the background by transverse momentum, pseudo rapidity,
invariant mass, cone-angle constraints, and the b-tagging method. Significance
calculations were performed by determining the number of events related to the
production processes and the background for each collider. The values are
listed in Table 3-6.

When the results are examined in Table 5 , it is seen that the desired
significance value for Signal 1 cannot be reached at the luminosity value
given for ILC – 250 GeV. For Signal 1 processes to be observed in the ILC-250
GeV, the collider needs to accumulate data for a longer period of time. Again,
at the end of one year, it was seen that the statistical significance value of
$5\sigma$ would be reached after the b-tagging method for the Signal 1
processes in the CEPC collider. Therefore, the CEPC collider will enable the
properties of the Higgs boson to be investigated precisely through Signal 1
processes. It is seen that at the end of 1 year in the FCC-ee collider, a
significance value of 7.95 will be reached without b-tagging and a high
significance value of 11.9 can be reached by using b-tagging. This shows that
FCC-ee will be more advantageous than ILC-250 GeV and CEPC 250 GeV colliders
for investigating Higgs boson properties through the Signal 1 group around
these center of mass energies (240-250 GeV). In the ILC-500 GeV and
CLIC-380-1500-3000 GeV colliders, results well above the desired significance
value can be obtained for signal 1 processes, even without the b-tagging.
Therefore, the properties of the Higgs boson through Signal 1 processes can be
studied with precision in colliders other than the ILC-250 GeV collider. Since
the results obtained for the Signal 2 process are greater than 5 significance
values, the properties of the Higgs boson can be studied precisely for all
colliders through this channel.

As a result, in future lepton colliders, the Higgs boson can be observed with
high event rates via Higgsstrahlung, W and Z fusion. Thus, electron–positron
colliders can precisely measure the properties of the Higgs boson.

## Acknowledgment

We would like to thank Professor Dr Inanc Sahin for his suggestions.

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# The Vector Dark Matter, LHC Constraints Including a 95 GeV Light Higgs Boson

Seyed Yaser Ayazi<EMAIL_ADDRESS>Physics Department, Semnan
University, P.O. Box. 35131-19111, Semnan, Iran  Mojtaba Hosseini
<EMAIL_ADDRESS>Physics Department, Semnan University, P.O. Box.
35131-19111, Semnan, Iran  Saeid Paktinat Mehdiabadi<EMAIL_ADDRESS>Department of Physics, Yazd University, P.O. Box 89195-741, Yazd, Iran
Rouzbeh Rouzbehi<EMAIL_ADDRESS>Physics Department, Semnan
University, P.O. Box. 35131-19111, Semnan, Iran

###### Abstract

We study LHC searches for an extension of the Standard Model (SM) by
exploiting an additional Abelian $U_{D}(1)$ gauge symmetry and a complex
scalar Higgs portal. As the scalar is charged under this gauge symmetry, a
vector dark matter (VDM) candidate can satisfy the observed relic abundance
and limits from direct dark matter (DM) searches. The ATLAS and CMS
experiments have developed a broad search program for the DM candidates,
including associate production of Higgs boson, Z boson and top quark which
couple to DM. In this paper, we perform an extensive analysis to constrain the
model by using these experiments at LHC. It can be seen that the LHC results
can exclude some parts of the parameter space which are still allowed by relic
density and the direct detection searches. Furthermore, exclusion limits on
the parameter space of the model by using the new results of CMS and ATLAS
collaborations for new light Higgs boson with mass $\sim 95~{}\rm GeV$ are
provided.

##  1 Introduction

The existence of DM is largely known through a variety of astrophysical and
cosmological experiments[1]. Although it is measured to be about $26.4\%$ of
the total energy density of Universe via indirect observations, the true
nature of DM particles is still unknown. However, we know that DM must be
electrically neutral and does not interact with ordinary baryonic matter.

The main effects caused by DM are gravitational, but the existence of a Weakly
Interacting Massive Particle (WIMP) is often hypothesised, since it leads to
the correct relic density for non-relativistic matter in the early universe
(WIMP miracle)[2]. In general, the dark sector can interact with the SM
particles through the Higgs portal [3]. We consider a $U(1)$ extension of SM
in which a scalar mediated between SM and the dark sector and a vector can
play the role of DM particle. Experiments around the world seek to unravel the
nature of DM with different strategies. One of these strategies is to search
at particle colliders. Such dark particles could be produced in particle
colliders and detected as the large missing transverse momentum produced in
association with the SM particles. To have a realistic model, it must satisfy
different available constraints from the DM abundance via standard thermal
freeze-out processes, direct detection experiments and the LHC searches for
the DM particles. In this regard, we perform an analysis on the model,
focusing on relic density, direct detection and several searches on associate
production of Higgs boson, Z gauge boson and top quark with the DM particle.
In particular, we determine the region of parameter space that can be excluded
by the searches at LHC. Some of the LHC analyses have reported the data and
background event yields in different signal regions. It enables us to find the
limits on the DM signal yields by some statistical methods. By implementing
the signal selections in an event generator, one can find the DM signal yields
in different points of the parameter space. This yield is compared to the
limits extracted from the LHC analyses to decide whether a point is allowed or
excluded.

The recent results, reported by the CMS and ATLAS collaborations, indicate an
excess in the $\gamma$$\gamma$ and $\tau$$\tau$ final states consistent with a
light Higgs boson ($\sim 95~{}\rm GeV$) [4]-[5]. They provide interesting
hints for new physics. Motivated by these results, we perform a study to
constrain the model by considering a $95~{}\rm GeV$ light Higgs boson.

To explain the model and analysis, the paper is organized as follows. In
Section 2, we introduce the model under consideration. In Sections 3 and 4,
the allowed parameter space of the model is probed by considering the relic
density of DM and direct detection experiments. We present the method and the
main results for the LHC constraints on the model by a list of processes in
Section 5. In Section 6, the excluded region in the light of the newly
observed $95~{}\rm GeV$ light Higgs boson is reported. We also study indirect
constraint from H.E.S.S experiment on the DM candidate in Section 7. Finally,
Section 8 concludes the paper.

##  2 The Model

For the reader’s convenience, we briefly introduce the main characteristics of
the model which is fully developed in Ref.[6]. We extend SM by a dark sector
where the $U_{D}(1)$ Abelian gauge symmetry is spontaneously broken by the
dark Higgs mechanism. We introduce a complex scalar $S$ which has a unit
charge under $U_{D}(1)$ and the $U_{D}(1)$ gauge field $V_{\mu}$. We also
consider an additional $Z_{2}$ symmetry, under which the vector field
$V_{\mu}$ and the scalar field $S$ transform as follows:

$V_{\mu}\rightarrow-V_{\mu},~{}~{}~{}S\rightarrow S^{*}$ (2.1)

In order for DM to be stable, we suppose the kinetic mixing between the vector
field $V_{\mu}$ and the SM $U(1)$ gauge boson $B_{\mu}$, i.e.,
$V_{\mu\nu}B^{\mu\nu}$, is forbbiden. Therefore, the vector field $V_{\mu}$
can be considered as a DM candidate. The total Lagrangian of the model is
given by:

${\cal L}={\cal
L}_{SM}-\frac{1}{4}V_{\mu\nu}V^{\mu\nu}+(D^{\prime}_{\mu}S)^{*}(D^{\prime\mu}S)-V(H,S),$
(2.2)

where ${\cal L}_{SM}$ is the SM Lagrangian without the Higgs potential term
and the covariant derivative and field strength of $V_{\mu}$ are given as:

$\displaystyle D^{\prime}_{\mu}S=(\partial_{\mu}+ig_{v}V_{\mu})S,$
$\displaystyle V_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu},$

The potential which is renormalizable and invariant under gauge and $Z_{2}$
symmetry is:

$V(H,S)=-\mu_{H}^{2}H^{\dagger}H-\mu_{S}^{2}S^{*}S+\lambda_{H}(H^{\dagger}H)^{2}+\lambda_{S}(S^{*}S)^{2}+\lambda_{SH}(S^{*}S)(H^{\dagger}H).$
(2.3)

The quartic interaction, $\lambda_{SH}(S^{*}S)(H^{\dagger}H)$, is the only
connection between the dark sector and the SM. After the electroweak symmetry
breaking [6], the parameters of the model are given by:

$\displaystyle\nu_{2}=\frac{M_{V}}{g_{v}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sin\alpha=\frac{\nu_{1}}{\sqrt{\nu_{1}^{2}+\nu_{2}^{2}}}$
$\displaystyle\lambda_{H}=\frac{\cos^{2}\alpha M_{H_{1}}^{2}+\sin^{2}\alpha
M_{H_{2}}^{2}}{2\nu_{1}^{2}}$ $\displaystyle\lambda_{S}=\frac{\sin^{2}\alpha
M_{H_{1}}^{2}+\cos^{2}\alpha M_{H_{2}}^{2}}{2\nu_{2}^{2}}$
$\displaystyle\lambda_{SH}=\frac{\left(M_{H_{2}}^{2}-M_{H_{1}}^{2}\right)\sin\alpha\cos\alpha}{\nu_{1}\nu_{2}}$
(2.4)

where $\alpha$ is the mixing angle between weak eigenstates and mass
eigenstates. $\nu_{1}$ and $\nu_{2}$ are vacuum expectation values of Higgs
fields. The mass eigenstates of Higgs fields can be written as following:

$\displaystyle
M^{2}_{H_{2},H_{1}}=\lambda_{H}\nu_{1}^{2}+\lambda_{S}\nu_{2}^{2}\pm\sqrt{(\lambda_{H}\nu_{1}^{2}-\lambda_{S}\nu_{2}^{2})^{2}+\lambda_{SH}^{2}\nu_{1}^{2}\nu_{2}^{2}},$
(2.5)

where, we suppose $M_{H_{1}}=125~{}\rm GeV$ and $\nu_{1}=246~{}\rm GeV$. It is
remarkable that this construction is rather minimal, in the sense that only 3
new free parameters are added: coupling $g_{v}$ and new mass parameters
$M_{H_{2}}$ and $M_{V}$.

We also have studied various theoretical and experimental constraints on the
parameter space of the model. For theoretical constraints, we must consider
constraints such as perturbativity, unitarity and positivity of the potential.
In this regard, we suppose all theoretical conditions which have been
discussed in [6]. For experimental constraints, we consider invisible Higgs
decay, relic density and direct detection bounds. The constraints for
invisible Higgs decay is completely similar to [6] but for two other
constraints, we extend the studied parameter space.

##  3 Relic density

The evolution of the number density of DM particles $(n_{X})$ with time is
governed by the Boltzmann equation:

$\dot{n}_{X}+3Hn_{X}=-\langle\sigma_{ann}\nu_{rel}\rangle[n_{X}^{2}-(n_{X}^{eq})^{2}],$
(3.1)

where $H$ is the Hubble parameter and
$n_{X}^{eq}\sim(m_{X}T)^{3/2}e^{-m_{X}/T}$ is the particle density before
particles get out of equilibrium. The dominant Feynman diagrams for DM
production processes are shown in the Fig. 1. In this regard, we calculate the
relic density numerically for the VDM particle by implementing the model into
micrOMEGAs [7]. The allowed range of parameter space corresponding to observed
DM relic density( according to the data of Planck collaboration [8]) is
depicted in the Fig. 2. As can be seen, the most allowed points of the model
are in the region of $60\rm~{}GeV<M_{V}<2000\rm~{}GeV$ and $0.1<g_{v}<1$. For
$M_{V}<60\rm~{}GeV$, because of $sin\alpha<0.44$ [9, 10] the parameter space
is limited.

Figure 1: The dominant Feynman diagrams for DM relic density production cross
section.

Figure 2: The allowed range of parameter space consistent with the DM relic
density.

##  4 Direct Detection

Before examining the constraints of LHC on the various channels, let us turn
our attention to the direct detection(DD) of VDM in the model. At the tree
level, a VDM particle can interact elastically with a nucleon either through
$H_{1}$ or via $H_{2}$ exchange[11]. We use the XENONnT experiment results to
constrain the parameter space[12]. Also for $M_{V}<30~{}\rm GeV$, the results
of the XENON1T experiment are used[13]. Figure 3 shows the parameter space of
the model in agreement with the DD and relic density constraints. As it can be
seen, for different values of the DM mass, there are points that are protected
from all the presented phenomenological constraints.

Figure 3: The allowed range of parameter space consistent with the DM relic
density and DD.

##  5 The LHC searches and the DM constraints

The LHC experiments at CERN have done a long list of searches for the new
physics beyond SM (BSM) and also measurements of SM cross sections and
variables. Up to this date, no sign of new physics or significant deviation
from the SM predictions is reported. This lack of experimental evidence for
new physics can be used to constrain any newly introduced extension of SM,
e.g., the proposed DM model. To produce the results of this section, the event
yields and uncertainties reported by the LHC experiments are used to find the
95% and 68% Confidence Level upper limit on the event yields of BSM. The
procedure is done using the tools provided in ROOT [14] analysis framework. It
is based on a Likelihood ratio semi-bayesian method [15]-[16]. The idea is to
find the maximum room for events from BSM, by comparing the expected number of
events from SM and the observed event yields in data. The authors have used a
similar method to constrain other BSM’s in Ref. [17].

To find the upper limits, 20% total uncertainty is assumed on DM signals. The
found maximum number of BSM events is divided by the corresponding integrated
luminosity to find the upper limit on the cross section times the acceptance
times the experimental efficiency of the BSM in question. In continue, this
quantity is called the upper limit of the visible cross section
($\sigma^{UL}_{vis}$). For various channels $\sigma^{UL}_{vis}$ values are
shown in Table 1.

Table 1: The upper limit of the visible pp cross section ($\sigma^{UL}_{vis}$) for different channels based on the LHC searches at $\sqrt{s}=13$ TeV Channel | $\sigma^{UL}_{vis}$ (fb)
---|---
CMS | ATLAS
95%CL | 68%CL | 95%CL | 68%CL
$VVZ,Z\to\ell^{+}\ell^{-}$ | 1.22 | 0.66 | 4.43 | 3.41
$H_{1}VV,H_{1}\to\tau^{+}\tau^{-}$ | 0.75 | 0.39 | 0.06 | 0.03
$H_{1}VV,H_{1}\to b\bar{b}$ | 1.75 | 0.92 | 2.6 | 1.36
$t\bar{t}$VV | 85.91 | 44.1 | 0.66 | 0.39

To generate the events<EMAIL_ADDRESS>[18] (MG) is used. But, to make
the model understandable by MG, the introduced model is implemented in
LanHEP[19]. The output of this package can be used in MG, easily. The analysis
selections are implemented in MG to simulate the experimental acceptance as
accurate as possible. The generated events are passed to PYTHIA8[20] for
patron showering and hadronization. The experimental efficiency within the
acceptance is assumed to be 100%, which is an optimistic assumption and ends
up to conservative results in the shown plots. To have a more realistic
estimation of the efficiencies, one needs to pass the generated events through
the full detector simulation and reconstruct the events in detail, which is
beyond the scope of this analysis. In most of the LHC analyses, there are
several signal regions (bins). The event yields from SM prediction in
different bins are summed up and treated as a single bin analysis. The same is
done for the collision data, also. A special attention is paid to have
completely separated bins and one event has no chance to appear in two
different bins and be taken into account for two times. For each test point,
the input parameters are set in MG and 10000 events are generated to measure
the production cross section of that test point. The measured cross section by
MG is compared with $\sigma^{UL}_{vis}$. Only a test point with a measured
production cross section less than $\sigma^{UL}_{vis}$ is considered as an
allowed point, otherwise, it is marked as excluded.

###  5.1 LHC Constraints on H+DM

Both multipurpose LHC experiments have reported results on search for
associate production of Higgs boson and DM. Some of the leading order Feynman
diagrams, contributing in this process, are shown in Fig. 4.

$q$$\bar{q}$$V$$\bar{V}$$X$$X$$h_{1}/h_{2}$$q$$\bar{q}$$V$$\bar{V}$$q$$X$$h_{1}/h_{2}$
Figure 4: Feynman diagrams relevant for the production of X=Z/$H_{1}$ in
association with DM at the LHC in the process pp $\to$XVV.

One of the most interesting decay modes of Higgs boson is its decay to two
photons via a top quark mediated loop. Since the proposed model is implemented
in leading order, it can not handle such decays and this final state is
ignored in this analysis. In continue, the decays of Higgs boson to two $\tau$
leptons or two b-quarks are considered. The CMS experiment has reported the
events yields of the SM prediction and collision data after applying the cuts
in events with a SM Higgs boson and the missing transverse momentum in
Ref.[21]. The analysis uses $35.9~{}\rm fb^{-1}$ of data in $\sqrt{s}=13~{}\rm
TeV$. The paper contains the results on Higgs boson decay to two b-quarks. A
special algorithm is used to identify the Higgs bosons with medium Lorentz-
boost, where b-jets produced in the decay of the Higgs boson are merged and
appear as a large-radius jet. The yields are reported in four non-overlapping
bins of the missing transverse momentum. The ATLAS experiment has done a
similar search using $139~{}\rm fb^{-1}$ of data in $\sqrt{s}=13~{}\rm TeV$
[22]. They also use a variable-radius track-jet algorithm to treat the boosted
Higgs bosons properly. For events with exactly two b-jets, five exclusive bins
of the missing transverse momentum are defined to report the yields.

Figure 5(left) shows the excluded regions in 95% CL in ($M_{V}$, $M_{H_{2}}$)
plane for different $g_{v}$ values using the CMS results. Similar result by
using the ATLAS measurements is shown in Fig. 5(right).

(a) CMS (b) ATLAS

Figure 5: The colored regions are excluded at 95% CL by CMS (left) and ATLAS
(right) measurements in $pp\to H_{1}VV,(H_{1}\to b\bar{b})$ channel for
different $g_{v}$ values.

In Ref. [23], the CMS experiment considers the events consistent with the
decay of Higgs boson to two $\tau$ leptons to search for DM produced next to
Higgs boson. The analysis uses $35.9~{}\rm fb^{-1}$ of data in
$\sqrt{s}=13~{}\rm TeV$. Due to different decay modes of the $\tau$ lepton,
three final states are defined. Events with both $\tau$ leptons decay
hadronically or one of them decays hadronically and the other one decays to an
electron or a muon characterise these final states. No further binning is
applied. The ATLAS experiment has reported the results of a similar search
based on $139~{}\rm fb^{-1}$ of data in the same center-of-mass energy in Ref.
[24]. Only the hadronically decaying $\tau$ leptons are considered in this
paper. The analysis defines two mass ranges. The high mass range is optimised
for the booted Higgs bosons, but it is not completely separated from the low
mass range. To avoid the complexities due to overlapping signal regions, only
the low mass range is used in our analysis. Figure 6(left)

(a) CMS (b) ATLAS

Figure 6: The colored regions are excluded at 95% CL by CMS (left) and ATLAS
(right) measurements in $pp\to H_{1}VV,(H_{1}\to\tau^{+}\tau^{-})$ channel for
different $g_{v}$ values.

shows the excluded regions in 95% CL in ($M_{V}$, $M_{H_{2}}$) plane for
different $g_{v}$ values using the CMS results. Similar result by using the
ATLAS measurements is shown in Fig. 6(right). The reaches in $M_{H_{2}}$
direction are comparable but, ATLAS is more powerful in $M_{V}$ direction. The
ATLAS experiment uses more data, and this can be the source of this
superiority.

###  5.2 The LHC upper bounds for $pp\rightarrow t\bar{t}$+DM

Associate production of DM and top quark pair ($t\bar{t}$) can happen in
Feynamn diagrams like Fig. 7.

$g$$\bar{t}$$g$$t$$\bar{V}$$V$$h_{1}/h_{2}$ Figure 7: Feynman diagrams
relevant for the production of $t\bar{t}$ in association with DM at the LHC in
the process pp $\to t\bar{t}$VV.

The LHC experiments have reported several analysis on search for DM production
in association with $t\bar{t}$. In this paper, only the latest analyses are
considered. The CMS experiment has used 2.2$fb^{-1}$ of data in
$\sqrt{s}=13~{}\rm TeV$ for this search [25]. The analysis combines the
results from different $t\bar{t}$ final states, including zero, one or both
top quarks decay hadronically. In the all hadronic $t\bar{t}$ events, a
special discriminator is used to categorize the events based on the number of
the resolved top tagged. It helps to include in the analysis, the boosted top
quarks which may have some overlaps between the jets. On the other hand, the
ATLAS experiment has used 36.1$fb^{-1}$ of data in the same center-of-mass
energy [26]. They have presented the results for three sets of selection
criteria, when only fully hadronic and fully leptonic decays of $t\bar{t}$ are
considered. They also have special treatment for highly boosted top quarks, by
using the large-radius jets. Figure 8(left) shows the parts of ($M_{V}$,
$M_{H_{2}}$) parameter space, excluded at 95% CL using CMS results when
different $g_{v}$ values are assumed. Same result by using the ATLAS
measurements is shown in Fig. 8(right). Similar to previous section, ATLAS
uses more data and is more powerful in $M_{V}$ direction.

(a) CMS (b) ATLAS

Figure 8: The colored regions are excluded at 95% CL by CMS (left) and ATLAS
(right) measurements in $pp\to t\bar{t}VV$ channel for different $g_{v}$
values.

###  5.3 LHC Constraints on Z+DM

The Feynman diagrams like Fig. 4 can contribute in associate production of the
Z boson and DM. The CMS experiment has reported the events yields of the
backgrounds and data after applying the cuts in events with a Z boson and the
missing transverse momentum [27]. The analysis uses 137$fb^{-1}$ of data in
$\sqrt{s}$=13 TeV for search for DM produced in association with a
leptonically decaying Z boson. The results for dielectron and dimuon final
states are summed up and reported in two jet multiplicity, zero and one jet,
categories. Moreover, the ATLAS experiment has reported results on same
channel with applying different selection cuts [28]. The analysis uses
36.1$fb^{-1}$ of data in the center-of-mass energy of 13 TeV. The results are
reported for dimuon and dielectron final states separately, and no further
binning is done. Figure 9(left)

(a) CMS (b) ATLAS

Figure 9: The colored regions are excluded at 95% CL by CMS (left) and ATLAS
(right) measurements in $pp\to ZVV,(Z\to\ell^{+}\ell^{-})$ channel for
different $g_{v}$ values.

shows the parts of ($M_{V}$, $M_{H_{2}}$) parameter space, excluded at 95% CL
using CMS results when different $g_{v}$ values are assumed. Same result by
using the ATLAS measurements is shown in Fig. 9(right). Two experiments have
similar reaches in low $M_{H_{2}}$ but, the CMS experiment, that uses more
data in this channel, has a better exclusion for $M_{V}$ in high $M_{H_{2}}$.
Comparing with other LHC constraints, this channel gives the best exclusion.
It can even increase the exclusion in $M_{V}$ direction comparing to DD
results shown in Fig. 3.

##  6 LHC constraints on Vector Dark Matter Model with a $95~{}\rm GeV$ Light
Higgs

Recently, CMS collaboration has reported new results, indicating a signal
excess with a local (global) significance of $2.9\sigma$ ($1.3\sigma$) in the
di-photon final state around $95~{}\rm GeV$, using all LHC data collected at
$\sqrt{s}=13~{}\rm TeV$ [4]. The ATLAS group also has reported a similar
excess with a local significance of $1.7~{}\sigma$ [5]. Further anomalies in
the $\tau\tau$ [29] and $b\bar{b}$ [30] final states have been reported by the
CMS and LEP collaborations, respectively. The impacts of these reports have
been studied in a variety of different BSM models[31, 32, 33, 34, 35, 36]. In
the proposed model, new Higgs emerges as a promising candidate. In this
regard, $H_{2}$ is assumed to be the newly observed particle and
$m_{H_{2}}=95~{}\rm GeV$. Looking at Figs. 5-9, shows only the processes
$pp\to t\bar{t}VV$ and $pp\to ZVV,(Z\to\ell^{+}\ell^{-})$ have sensitivity in
$m_{H_{2}}$ close to $95~{}\rm GeV$. Therefor the results of these channels
are used to constrain the proposed model in the light of the newly observed 95
GeV excess. Figures 10 and 11 show the excluded regions in $M_{V}$-$g_{v}$
plane, for $pp\to ZVV,(Z\to\ell^{+}\ell^{-})$ and $pp\to t\bar{t}VV$,
respectively. The results based on the analysis of both CMS and ATLAS
collaborations are shown. Apart from very low values of $g_{v}$ both channel
and both collaborations, exclude $M_{V}$ below 60 GeV. Dependence on $M_{V}$
is small but, in low values of $g_{v}$, dependence on this parameter is large
and small changes in coupling can switch between allowed and excluded region.
Similar behaviour was seen in the previous shapes (Figs. 5-9) also, when the
sensitive area is for $g_{v}$ between 0.01 and 0.1 and beyond this range, no
sensitivity is seen.

(a) CMS (b) ATLAS

Figure 10: The colored regions are excluded at 95% CL by CMS (left) and ATLAS
(right) measurements in $pp\to ZVV,(Z\to\ell^{+}\ell^{-})$ channel for
different $g_{v}$ values. Mass of $H_{2}$ is fixed to 95 GeV.

(a) CMS (b) ATLAS

Figure 11: The colored regions are excluded at 95% CL by CMS (left) and ATLAS
(right) measurements in $pp\to t\bar{t}VV$ channel for different $g_{v}$
values. Mass of $H_{2}$ is fixed to 95 GeV.

##  7 The H.E.S.S. upper bound

Indirect detection experiments of DM can be an attractive avenue for signs of
DM. Photons, neutrinos and positrons are the most important products from the
annihilation of DM, among which photons are the best option due to their
larger cross section and clearer signals. The H.E.S.S. experiment checks the
cosmic gamma-rays in the photon energy range of 0.03 to 100 TeV so it can
observe high energy processes in the universe[37]. The monochromatic lines of
photon are one of the promising messengers in the indirect search for DM. In
our model, DM can annihilate to two photons through quantum loop corrections.
The dominant Feynman diagrams contributing to this process are shown in Fig.
12. The cross section is as follows:

Figure 12: The dominant Feynman diagrams for the annihilation of a DM pair
into monochromatic gamma-ray lines.

$\sigma(s)\nu=\frac{1}{4\pi s}|M_{XX\longrightarrow\gamma\gamma}|^{2}$ (7.1)

where $\sqrt{s}$ is the center of mass energy. The amplitude
$M_{XX\longrightarrow\gamma\gamma}$ is given by

$M_{XX\longrightarrow\gamma\gamma}(s)=ig_{v}^{2}\nu_{2}[\frac{i\sin\theta}{s-m_{H_{1}}^{2}-im_{H_{1}}\Gamma_{H_{1}}}M_{{H_{1}}\longrightarrow\gamma\gamma}(s)+\frac{i\cos\theta}{s-m_{H_{2}}^{2}-im_{H_{2}}\Gamma_{H_{2}}}M_{{H_{2}}\longrightarrow\gamma\gamma}(s)]$
(7.2)

where

$M_{{H_{1}}\longrightarrow\gamma\gamma}=\frac{\alpha g_{e}s\cos\theta}{8\pi
M_{W}}[3(\frac{2}{3})^{2}F_{t}+F_{W}]$ (7.3)
$M_{{H_{2}}\longrightarrow\gamma\gamma}=\frac{\alpha g_{e}s\sin\theta}{8\pi
M_{W}}[3(\frac{2}{3})^{2}F_{t}+F_{W}].$ (7.4)

Here, $g_{e}=2\sqrt{\pi\alpha}/sin\theta_{w}$ and

$F_{t}=-2\tau[1+(1-\tau)f(\tau)]$ (7.5) $F_{W}=2+3\tau+3\tau(2-\tau)f(\tau)$
(7.6)

where $\tau=\frac{4m_{i}^{2}}{s}$ with $i=t,W$ and

$\displaystyle
f(\tau)=\bigg{\\{}\begin{array}[]{cc}{(\sin^{-1}\sqrt{\frac{1}{\tau}})^{2}}&{if~{}~{}~{}\tau\geq
1}\\\
{\frac{-1}{4}(ln\frac{1+\sqrt{1-\tau}}{1-\sqrt{1+\tau}}-i\pi)^{2}}&{if~{}~{}~{}\tau<1}\end{array},$
(7.9)

Therefore, the annihilation cross section of a pair of DM into two photons is
as follows:

$\sigma(s)\nu=\frac{g_{v}^{4}\nu_{2}^{2}}{4\pi s}|\frac{\sin\theta
M_{{H_{1}}\longrightarrow\gamma\gamma}}{s-m_{H_{1}}^{2}-im_{H_{1}}\Gamma_{H_{1}}}+\frac{\cos\theta
M_{{H_{2}}\longrightarrow\gamma\gamma}}{s-m_{H_{2}}^{2}-im_{H_{2}}\Gamma_{H_{2}}}|^{2}$
(7.10)

Finally, the average thermal cross section is equal to:

$\langle\sigma(s)\nu\rangle=\frac{1}{8M_{V}^{4}TK_{2}^{2}(\frac{M_{X}}{T})}\int_{4M_{V}^{2}}^{\infty}ds(s-4M_{V}^{2})\sqrt{s}K_{1}(\frac{\sqrt{s}}{T})\sigma(s)$
(7.11)

where $K_{1}$ and $K_{2}$ are modified Bessel functions and $T$ is the freeze-
out temperature. Figure 13 shows the allowed parameter space in agreement with
the H.E.S.S. experiment. From the comparison of figures 13 and 2, it is clear
that only for $1GeV\leq M_{V}\leq 10GeV$ and $2GeV\leq M_{H_{2}}\leq 40GeV$,
the parameter space of our model is compatible with H.E.S.S. results.
Comparing this figure with relic density and direct detection results
indicates that allowed range of parameters space, which can simultaneously
explain relic density, direct detection and H.E.S.S experiment is fairly small
for $g_{v}$ coupling constant.

Figure 13: The allowed range of parameter space consistent with H.E.S.S data.

##  8 Conclusions

In this paper, we revisit an extension of the SM by an extra $U_{D}(1)$
symmetry. The model consists of an extra scalar as the Higgs portal and a
vector gauge field as a DM candidate. It is shown that the model can satisfy
the observed cosmological DM abundance via the standard freeze-out mechanism
and, at the same time, fulfill the theoretical constraints such as
perturbativity, unitarity and positivity of the potential and also
experimental constraint such as invisible Higgs decay and DD bounds. In the
following, we focus on associate production of Higgs boson, Z gauge boson and
t quark with the VDM. The results of the LHC experiments on similar final
states are used to constrain the parameter space of the model. A statistical
method is used to find the maximum room for the events from this DM model in
LHC data. It is shown that the results of the associate production of Z gauge
boson and DM can exclude some parts of parameters space which are still
allowed by relic density and the DD searches. Inspired by the newly observed
light Higgs boson close to 95 GeV, the model is tested if its new scalar is
the light Higgs boson. The excluded parameter space in the light of the LHC
constraints and this assumption are reported also. At the end, the constraints
from the indirect searches are also examined for the completeness.

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