###### Abstract

The sensitivity studies to observe long-lived particles predicted by a set of
beyond the Standard Model theoretical models are reported. The analysis is
based on a data sample of $e^{+}e^{-}$ collisions at $\sqrt{s}=3$ TeV,
simulated with the CLIC_ILD detector model and corresponding to an integrated
luminosity of 3 ab-1. Long-lived particle decay products are subsequently
combined to reconstruct the parent bosons employing secondary vertices
displaced from the beam axis. The upper limits on the production cross section
for the long-lived particle lifetimes from 1 to 300 ps, masses between 25 and
50 GeV/c2, and a parent Higgs mass of 126 GeV/c2 are determined.

##  1 Introduction

Among many theoretical descriptions of new phenomena beyond the Standard Model
(SM) there are a class of models predicting the existence of new massive Long-
Lived Particles (LLP) with a measurable flight distance, providing displaced
vertices (DV) which can be efficiently reconstructed by the tracking system of
the CLIC detector [1, 2]. One class of such models, the so-called Hidden
Valley, is a consequence of the string-theory, allowing for additional
$hidden$ gauge sector which couples to SM particles only at very high energies
[3, 4]. The Hidden Valley particles are predicted in some of these models to
have non-zero lifetime and to decay into $b\bar{b}$, having unobservable
partners that could serve as dark matter objects. In the present report the
analysis of the Standard Model Higgs boson decaying into two Hidden Valley
particles $h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ is described. The Higgs
boson is produced in WW-fusion, $e^{+}e^{-}\rightarrow
h^{0}\nu_{e}\bar{\nu}_{e}$, at $\sqrt{s}$ = 3 TeV, simulated with the CLIC_ILD
detector model [5], using an integrated luminosity of 3 ab-1. Similar searches
of the SM Higgs boson decaying into two LLP’s providing two $b\bar{b}$ di-jets
in the final state have already been reported by the D0 [6], CDF [7], ATLAS
[8], CMS [9] and LHCb [10] experiments.

##  2 Analysis strategy

As the Hidden Valley objects are predicted to be massive and have non-zero
lifetime, the possible presence of new long-lived particles is investigated
based on events with reconstructed vertices displaced from the beam axis. In
the process $h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$, in order to reconstruct
the parent Higgs boson, two $b$-tagged jets are assigned to each displaced
vertex [11]. A sensitivity of the CLIC_ILD detector to LLP production is
studied using Monte Carlo (MC) events generated with WHIZARD 1.95 [12] and
PYTHIA 6 [13], configured to produce Hidden Valley processes. The interaction
of the generated particles with the CLIC_ILD detector and its response are
implemented using the Geant4 [14] simulation package and the MOKKA [15]
detector description toolkit. Finally, the MARLIN software package [16] is
used for event reconstruction, with the track reconstruction as in Ref. [17].
The signal and background samples of $e^{+}e^{-}$ collisions at $\sqrt{s}$ = 3
TeV are generated, namely $h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ with
$\pi^{0}_{v}$ lifetimes from 1 to 300 ps, masses between 25 and 50 GeV/c2, and
parent Higgs mass of 126 GeV/c2, as well as $e^{+}e^{-}\rightarrow q\bar{q}$,
$e^{+}e^{-}\rightarrow q\bar{q}\nu\bar{\nu}$, $e^{+}e^{-}\rightarrow
q\bar{q}q\bar{q}$ and $e^{+}e^{-}\rightarrow q\bar{q}q\bar{q}\nu\bar{\nu}$.
For the signal samples a cross section of 0.42 pb is assumed, while for the
background samples it is 2.95 pb, 1.32 pb, 0.55 pb and 0.07 pb for
$e^{+}e^{-}\rightarrow q\bar{q}$, $e^{+}e^{-}\rightarrow
q\bar{q}\nu\bar{\nu}$, $e^{+}e^{-}\rightarrow q\bar{q}q\bar{q}$ and
$e^{+}e^{-}\rightarrow q\bar{q}q\bar{q}\nu\bar{\nu}$, respectively. The
distance of the generated $\pi^{0}_{v}$ to the primary vertex (PV) and the
radial distance to the beam axis are illustrated in Fig. 1, showing the
dependence on the $\pi^{0}_{v}$ lifetime.

Figure 1: Distance of the generated $\pi^{0}_{v}$ to (a) the PV and (b) its
radial distance to the beam axis for $\pi^{0}_{v}$’s generated with a mass of
50 GeV/c2 and with four different lifetimes: 1 ps (red line), 10 ps (blue
line), 100 ps (green line) and 300 ps (yellow line). Figures adopted from
[11].

The particles are reconstructed using default CLICdp packages [18], while the
jets are reconstructed using the longitudinally invariant $k_{t}$ algorithm
[19] as implemented in the FastJet package [20]. The jets are $b$\- and
$c$-tagged by passing a selection of parameters (including the impact
parameters that have been computed via the vertexing) through a Boosted
Decision Tree (BDT) [21]. The requirements on the particle impact parameter
components as well as the value of the $R$ parameter used for the jet
reconstruction are optimized according to the needs of the Hidden Valley
analysis. For the Hidden Valley analysis it was also required to find all
tracks from the $\pi^{0}_{v}$ decays, which are expected to be distant from
the beam axis. A dedicated procedure to reconstruct displaced vertices has
been developed and optimised for the Hidden Valley analysis (see [11] for
details), as the default CLICdp vertexing [22] could not efficiently find all
tracks from the $\pi^{0}_{v}$ decays. The comparison of the di-jet and four-
jet invariant mass distribution for the signal sample of $\pi^{0}_{v}$ with a
mass of 50 GeV/c2 and with a lifetime of 10 ps with the distributions for
$q\bar{q}$, $q\bar{q}\nu\bar{\nu}$, $q\bar{q}q\bar{q}$ and
$q\bar{q}q\bar{q}\nu\bar{\nu}$ backgrounds is shown in Fig. 2, where the jets
are reconstructed and $b$-tagged with $R=1.0$. Di-jets are constructed by
pairing two jets with an invariant mass closest to the $\pi^{0}_{v}$ mass.

Figure 2: Di-jet (a) and four-jet (b) invariant mass for the signal sample of
$\pi^{0}_{v}$ with a mass of 50 GeV/c2 and with a lifetime of 10 ps (black)
and the background of $q\bar{q}$ (red), $q\bar{q}\nu\bar{\nu}$ (blue),
$q\bar{q}q\bar{q}$ (green) and $q\bar{q}q\bar{q}\nu\bar{\nu}$ (yellow).
Figures adopted from [11].

Finally, two reconstructed jets which are positively tagged as $b$-jets are
assigned to each displaced vertex. The di-jets are assigned to a single
displaced vertex if the number of common charged tracks shared by this DV and
the two $b$-tagged jets is maximum. The efficiencies for the signal and
background samples with respect to the requirement to have at least two
reconstructed displaced vertices in the event are listed in Table 1.

Table 1: The efficiencies of the requirement to have at least two reconstructed displaced vertices in the event for signal and background samples. Process |  $\pi^{0}_{v}$ lifetime [ps] |  $\pi^{0}_{v}$ mass [GeV/c2] | fraction of events with $\geq$ 2 DVs [%]
---|---|---|---
$h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ | 1 | 25 | 68
$h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ | 10 | 25 | 86
$h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ | 100 | 25 | 93
$h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ | 300 | 25 | 80
$h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ | 1 | 35 | 70
$h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ | 10 | 35 | 86
$h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ | 100 | 35 | 94
$h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ | 300 | 35 | 82
$h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ | 1 | 50 | 72
$h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ | 10 | 50 | 89
$h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ | 100 | 50 | 90
$h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ | 300 | 50 | 86
$e^{+}e^{-}\rightarrow q\bar{q}$ | - | - | 6
$e^{+}e^{-}\rightarrow q\bar{q}\nu\bar{\nu}$ | - | - | 8
$e^{+}e^{-}\rightarrow q\bar{q}q\bar{q}$ | - | - | 9
$e^{+}e^{-}\rightarrow q\bar{q}q\bar{q}\nu\bar{\nu}$ | - | - | 11

##  3 Background reduction

In order to separate $\pi^{0}_{v}$ signal from the $q\bar{q}$,
$q\bar{q}\nu\bar{\nu}$, $q\bar{q}q\bar{q}$, $q\bar{q}q\bar{q}\nu\bar{\nu}$
background a multivariate analysis based on the Boosted Decision Tree Gradient
(BDTG) [23] is applied. It employs seven variables providing significant
signal to background separation, i.e. the number of tracks assigned to the
reconstructed DV, the number of reconstructed DVs in the event, the invariant
mass of the DV, the di-jet invariant mass of two jets assigned to the DV, the
four-jet invariant mass of two di-jets assigned to two DVs, the distance
$y_{n+1,n}$ at which the transition from a three jet event to a two jet event
takes place, the distance $y_{n-1,n}$ at which the transition from a four jet
event to a three jet event takes place. Fig. 3 shows as example the number of
tracks assigned to the reconstructed DV for the signal sample of $\pi^{0}_{v}$
with three different masses and the lifetime of 10 ps, and for the background
of $q\bar{q}$, $q\bar{q}\nu\bar{\nu}$, $q\bar{q}q\bar{q}$ and
$q\bar{q}q\bar{q}\nu\bar{\nu}$ events. The Boosted Decision Tree Gradient
method has been chosen as the most effective, where $\pi^{0}_{v}$ with a mass
of 50 GeV/c2 and lifetime of 10 ps has been used as a signal, and
$e^{+}e^{-}\rightarrow q\bar{q}\nu\bar{\nu}$ as a background. Distributions of
the expected number of events as a function of the cut on the BDTG response,
for the assumed integrated luminosity of 3 ab-1, with the cross sections
mentioned in Sec. 2 and selection efficiencies taken from Table 1, are shown
in Fig. 4 for $\pi^{0}_{v}$ mass hypotheses of 35 GeV/c2 and four different
lifetimes.

Figure 3: Number of tracks assigned to the reconstructed DV for signal samples
of $\pi^{0}_{v}$ with a mass of (a) 25 GeV/c2, (b) 35 GeV/c2 and (c) 50
GeV/c2, and with a lifetime of 10 ps (black) compared to $q\bar{q}$ (red),
$q\bar{q}\nu\bar{\nu}$ (blue), $q\bar{q}q\bar{q}$ (green) and
$q\bar{q}q\bar{q}\nu\bar{\nu}$ (yellow) background events. Figures adopted
from [11].

Figure 4: Distributions of the expected number of events as a function of the
cut on the BDTG response, for the assumed integrated luminosity of 3 ab-1, for
signal samples of $\pi^{0}_{v}$ with a mass of 35 GeV/c2 (black figures) and
four different lifetimes: (a) 1 ps, (b) 10 ps, (c) 100 ps, (d) 300 ps.
Distributions of four different types of background are also plotted:
$q\bar{q}$ (red), $q\bar{q}\nu\bar{\nu}$ (blue), $q\bar{q}q\bar{q}$ (green)
and $q\bar{q}q\bar{q}\nu\bar{\nu}$ (yellow). Figures adopted from [11].

##  4 Sensitivity and upper limits

For all the $\pi^{0}_{v}$ mass and lifetime hypotheses the sensitivity of the
CLIC_ILD detector to observe $\pi^{0}_{v}$ particles through the Higgs boson
decay $h^{0}\rightarrow\pi^{0}_{v}\pi^{0}_{v}$ has been estimated. The Higgs
boson production via WW-fusion process has been assumed,
$e^{+}e^{-}\rightarrow h^{0}\nu_{e}\bar{\nu}_{e}$, at $\sqrt{s}$ = 3 TeV and
integrated luminosity of 3 ab-1. Fig. 5 shows the sensitivity as a function of
the cut on the BDTG response for signal samples with different masses
andlifetimes, where four different sources of background are added:
$e^{+}e^{-}\rightarrow q\bar{q}$, $e^{+}e^{-}\rightarrow
q\bar{q}\nu\bar{\nu}$, $e^{+}e^{-}\rightarrow q\bar{q}q\bar{q}$,
$e^{+}e^{-}\rightarrow q\bar{q}q\bar{q}\nu\bar{\nu}$. The requirement on BDTG
response to be $>$ 0.95 for all the $\pi^{0}_{v}$ mass and lifetime
configurations is taken in order to almost completely remove the backgrounds,
keeping nearly the entire signal.

Figure 5: Sensitivity $S/\sqrt{S+B}$ for the expected number of events as a
function of the cut on the BDTG response, for signal samples of $\pi^{0}_{v}$
with a mass of (a) 25 GeV/c2, (b) 35 GeV/c2 and (c) 50 GeV/c2, and for four
different lifetimes: 1 ps (red line), 10 ps (blue line), 100 ps (green line)
and 300 ps (yellow line). Figures adopted from [11].

Finally, the upper limits on the product of the Higgs production cross-section
and the branching fraction of the Higgs boson decay into long-lived particles,
$\sigma(H)\times BR(H\rightarrow\pi^{0}_{v}\pi^{0}_{v})$ have been determined
employing the CL(s) method [24]. Under the assumption of a 100% branching
fraction for $\pi^{0}_{v}\rightarrow b\bar{b}$ and the cut on the BDTG
response $>0$, the 95% CL upper limits on $\sigma(H)\times
BR(H\rightarrow\pi^{0}_{v}\pi^{0}_{v})$ have been computed. Fig. 6 shows the
observed 95% CL cross-section upper limits on the $\sigma(H)\times
BR(H\rightarrow\pi^{0}_{v}\pi^{0}_{v})$, within the model [4], for three
different masses, as a function of $\pi^{0}_{v}$ lifetime.

Figure 6: Observed 95% CL cross-section upper limits on the $\sigma(H)\times
BR(H\rightarrow\pi^{0}_{v}\pi^{0}_{v})$, within the model [4], for three
different $\pi^{0}_{v}$ masses: 25 GeV/c2 (green), 35 GeV/c2 (yellow), 50
GeV/c2 (blue), as a function of $\pi^{0}_{v}$ lifetime. Figure adopted from
[11].

##  5 Conclusions

In the present report results of the search for the exotic LLPs within the
CLIC_ILD detector are described, for the SM Higgs boson decaying to a pair of
LLPs, using $e^{+}e^{-}$ event sample simulated at $\sqrt{s}$ = 3 TeV and
corresponding to an integrated luminosity of 3 ab-1. The analysis based on
reconstructed displaced vertices provides an immense reduction of the large
Standard Model background using a multivariate analysis approach. The upper
limits obtained are much more precise compared to those of the currently
operating detectors [8, 9, 10].

##  6 Acknowledgements

This work benefited from services provided by the ILC Virtual Organisation,
supported by the national resource providers of the EGI Federation. This
research was done using resources provided by the Open Science Grid, which is
supported by the National Science Foundation and the U.S. Department of
Energy’s Office of Science.

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