# Single-qubit-gate error below 10-4 in a trapped ion

K. R. Brown<EMAIL_ADDRESS>present address: Georgia Tech Research
Institute, 400 10th Street Northwest, Atlanta, Georgia 30318, USA.  A. C.
Wilson  Y. Colombe  C. Ospelkaus  Present address: QUEST, Leibniz Universität
Hannover, Im Welfengarten 1, D-30167 Hannover and PTB, Bundesallee 100,
D-38116 Braunschweig, Germany.  A. M. Meier  E. Knill  D. Leibfried  D. J.
Wineland  National Institute of Standards and Technology, 325 Broadway,
Boulder, CO 80305, USA

###### Abstract

With a 9Be+ trapped-ion hyperfine-states qubit, we demonstrate an error
probability per randomized single-qubit gate of 2.0(2) $\times$ 10-5, below
the threshold estimate of 10-4 commonly considered sufficient for fault-
tolerant quantum computing. The 9Be+ ion is trapped above a microfabricated
surface-electrode ion trap and is manipulated with microwaves applied to a
trap electrode. The achievement of low single-qubit-gate errors is an
essential step toward the construction of a scalable quantum computer.

In theory, quantum computers can solve certain problems much more efficiently
than classical computers Nielsen and Chuang (2000). This has motivated
experimental efforts to construct and to verify devices that manipulate
quantum bits (qubits) in a variety of physical systems Ladd et al. (2010). The
power of quantum computers depends on the ability to accurately control
sensitive superposition amplitudes by means of quantum gates, and errors in
these gates are a chief obstacle to building quantum computers DiVincenzo
(2000). Small gate errors would enable fault-tolerant operation through the
use of quantum error correction protocols Preskill (1998). While the maximum
tolerable error varies between correction strategies, there is a consensus
that $10^{-4}$ is an important threshold to breach Preskill (1998); Knill
(2010). Single-qubit gates with errors slightly above this level have been
achieved with nuclear spins in liquid-state nuclear-magnetic resonance
experiments Ryan et al. (2009) and with neutral atoms confined in optical
lattices Olmschenk et al. (2010); here we demonstrate single-qubit error
probabilities of $2.0(2)\times 10^{-5}$, substantially below the threshold.
Reaching fault-tolerance still requires reducing two-qubit-gate errors from
the current state of the art (7 $\times$ 10-3 for laser-based Benhelm et al.
(2008) and 0.24 for microwave-based gates Ospelkaus et al. (2011)) to similar
levels.

To determine the average error per gate (EPG), we use the method of randomized
benchmarking Knill et al. (2008). Compared to other methods for evaluating
gate performance, such as quantum process tomography Poyatos et al. (1997),
randomized benchmarking offers the advantage that it efficiently and
separately can determine the EPG and the combined state-preparation and
measurement errors. Because it involves long sequences of random gates, it is
sensitive to errors occurring when gates are used in arbitrary computations.
In randomized benchmarking, the qubit, initialized close to a pure quantum
state, is subjected to predetermined sequences of randomly selected Clifford
gates Bravyi and Kitaev (2005) for which, in the absence of errors, the
measurement outcome is deterministic and efficiently predictable. Clifford
gates include the basic unitary gates of most proposed fault-tolerant quantum
computing architectures. Together with certain single-qubit states and
measurements, they suffice for universal quantum computing Bravyi and Kitaev
(2005); Knill (2005). To establish the EPG, the actual measurement and
predicted outcome are compared for many random sequences of different lengths.
Under assumptions presented in Ref. Knill et al. (2008), this yields an
average fidelity as a function of the number of gates that decreases
exponentially to $1/2$ and determines the EPG. Randomized benchmarking has
been used to quantify single-qubit EPGs in a variety of systems as summarized
in Table 1.

Reference | System | Gate error
---|---|---
This Rapid Communication (2011) | Single trapped ion | $2.0(2)\times 10^{-5}$
Reference Ryan et al. (2009) (2009) | Nuclear magnetic resonance | $1.3(1)\times 10^{-4}$
Reference Olmschenk et al. (2010) (2010) | Atoms in an optical lattice | $1.4(1)\times 10^{-4}$
Reference Biercuk et al. (2009) (2009) | Trapped-ion crystal | $8(1)\times 10^{-4}$
Reference Knill et al. (2008) (2008) | Single trapped ion | $4.8(2)\times 10^{-3}$
Reference Chow et al. (2010) (2010) | Superconducting transmon | $7(5)\times 10^{-3}$
Table 1: Reported average EPG for Pauli-randomized $\pi/2$ gates in different
systems as determined by randomized benchmarking.

To improve on the results of Ref. Knill et al. (2008), we integrated a
microwave antenna into a surface-electrode trap structure Ospelkaus et al.
(2008). The use of microwave radiation instead of optical stimulated-Raman
transitions to drive qubit rotations suppresses decoherence from laser beam
pointing instability and power fluctuations and eliminates decoherence from
spontaneous emission. The microwave amplitude can be stabilized more easily
than laser power, and because the antenna is integrated into the trap
electrodes, unwanted motion of the trap does not affect the microwave-ion-
coupling strength. The small distance (40 $\mu$m) between the trap surface and
the ion permits transition rates comparable to those based on lasers. Improved
shielding from ambient magnetic-field fluctuations was achieved by locating
the trap inside a copper vacuum enclosure held at 4.2 K by a helium-bath
cryostat. The thickness of the walls, combined with the increase in electrical
conductivity of copper at 4.2 K, effectively shields against the ambient
magnetic field fluctuations that typically limit coherence in room-temperature
ion-trap experiments Knill et al. (2008). This shielding is evident when we
change the magnetic field external to the cryostat; the accompanying response
in ion fluorescence lags the change with an exponential time constant of
3.8(2) s. In addition, cryogenic operation decreases the background gas
pressure to negligible levels, thereby enabling long experimental runs with
the same ion, and it suppresses ion heating Deslauriers et al. (2006);
Labaziewicz et al. (2008); Brown et al. (2011).

The 9Be+ ion is trapped $40$ $\mu$m above a surface-electrode trap Seidelin et
al. (2006) constructed of 8-$\mu$m-thick gold electrodes electroplated onto a
crystalline quartz substrate and separated by 5-$\mu$m gaps (Fig. 1).

Figure 1: (Color online) Micrograph of the ion trap, showing radio-frequency
(rf) electrodes and control electrodes. The red sphere indicates the
approximate ion position in the $x$-$y$ plane. Also shown are the directions
of the static magnetic field $B_{0}$ and of the microwave current used to
drive hyperfine transitions. (Inset) Energy level diagram (not to scale) of
the 2s 2S1/2 hyperfine states in 9Be+. Blue dashed lines indicate the
transitions used to prepare and measure $|\downarrow\rangle$. The solid black
line indicates the qubit transition, and the red dashed line indicates one of
the transitions used to shelve $|\uparrow\rangle$ into a dark state.

A static magnetic field $B_{0}$, parallel to the trap surface and collinear
with a Doppler cooling laser beam, is applied to break the degeneracy of the
ground-state Zeeman sublevels (Fig. 1 inset). We drive 2s 2S1/2 hyperfine
transitions with microwave pulses near 1.25 GHz, coupled with a 4-nF capacitor
to one end of a trap control electrode. The microwave current is shunted to
ground at the other end of the electrode by the 4-nF capacitor of an RC
filter. Microwave pulses are created by frequency quadrupling the output of a
direct-digital synthesizer whose frequency and phase can be updated in less
than 1 $\mu$s by a field-programmable gate array [(FPGA), 16-ns timing
resolution]. An rf switch creates approximately rectangular-shaped pulses.
This signal is amplified and is delivered via a coaxial cable within the
cryostat and a feedthrough in the copper vacuum enclosure. In this Rapid
Communication, we use the clock transition
($|F=2,m_{F}=0\rangle\equiv|\downarrow\rangle\leftrightarrow|1,0\rangle\equiv|\uparrow\rangle$)
in 9Be+ for the qubit instead of the previously used
$|2,-2\rangle\leftrightarrow|1,-1\rangle$ transition Knill et al. (2008) (Fig.
1 inset). The clock transition is a factor of 20 less sensitive to magnetic-
field fluctuations (950 MHz/T at field $B_{0}=1.51\times 10^{-3}$ T, compared
to 21 GHz/T).

A benchmarking experiment proceeds as follows. The ion is Doppler cooled and
optically pumped to the $|2,-2\rangle$ state with $\sigma^{-}$-polarized laser
radiation near the
$|^{2}$S${}_{1/2},2,-2\rangle\leftrightarrow|^{2}$P${}_{3/2},3,-3\rangle$
cycling transition at 313 nm. Then, the qubit is initialized in
$|\downarrow\rangle$ with two microwave $\pi$ pulses, resonant with the
$|2,-2\rangle\leftrightarrow|1,-1\rangle$ and
$|1,-1\rangle\leftrightarrow|\downarrow\rangle$ transitions (blue lines in
Fig. 1 inset). Pulse duration is then controlled by a digital delay generator,
which has a 5-ps timing resolution. The frequency, phase, and triggering of
each pulse remain under control of the FPGA.

A predetermined sequence of randomized computational gates is then applied.
Each computational gate consists of a Pauli gate ($\pi$ pulse) followed by a
(non-Pauli) Clifford gate ($\pi/2$ pulse). The gate sequence is followed by
measurement randomization consisting of a random Pauli gate and a Clifford
gate chosen deterministically to yield an expected measurement outcome of
either $|\uparrow\rangle$ or $|\downarrow\rangle$. The Pauli gates are chosen
with equal probability from the set $e^{-i\pi\sigma_{\mathrm{p}}/2}$, where
$\sigma_{\mathrm{p}}\in$ {$\pm\sigma_{x}$, $\pm\sigma_{y}$, $\pm\sigma_{z}$,
$\pm I$}. The Clifford gates are chosen with equal probability from the set
$e^{-i\pi\sigma_{\mathrm{c}}/4}$, where $\sigma_{\mathrm{c}}\in$
{$\pm\sigma_{x}$, $\pm\sigma_{y}$}. In practice, a Clifford gate is
implemented as a single (rectangular-shaped) $\pi/2$ pulse of duration
$\tau_{\pi/2}\approx 21$ $\mu$s with appropriate phase. For calibration
simplicity, a Pauli gate is implemented for $\sigma_{\mathrm{p}}\in$
{$\pm\sigma_{x}$, $\pm\sigma_{y}$} as two successive $\pi/2$ pulses, and an
identity gate $\pm I$ is implemented as an interval of the same duration
without the application of microwaves. A gate $e^{-i\pi\sigma_{z}/2}$ is
implemented as an identity gate, but the logical frame of the qubit and
subsequent pulses are adjusted to realize the relevant change in phase. All
pulses are separated by a delay of 0.72 $\mu$s.

To detect the final qubit state, $\pi$ pulses implement the transfer
$|\downarrow\rangle\rightarrow|1,-1\rangle\rightarrow|2,-2\rangle$ (blue lines
in Fig. 1 inset). Two additional pulses implement the transfer
$|\uparrow\rangle\rightarrow|\downarrow\rangle\rightarrow|1,1\rangle$ (black
and red lines in Fig. 1 inset). The ion is then illuminated for 400 $\mu$s by
313-nm light resonant with the cycling transition, and the resulting
fluorescence is detected with a photomultiplier. The entire sequence
experiment (from initialization through detection) is repeated 100 times (for
each sequence) to reduce statistical uncertainty. On average, approximately 13
photons are collected from an ion in the bright $|2,-2\rangle$ state, but only
$0.14$ are collected from an ion in the dark $|1,1\rangle$ state (due largely
to laser light scattered from the trap surface). To normalize the detection
and to eliminate errors due to slow fluctuations in laser power, each sequence
experiment is immediately followed by two reference experiments, where the ion
is prepared in the $|\downarrow\rangle$ and $|\uparrow\rangle$ states,
respectively, and the above detection protocol is implemented. From the
resulting bright and dark histograms [inset to Fig. 2(b)], we take the median
to establish a threshold for $|\downarrow\rangle$ and $|\uparrow\rangle$
detection.

Results are shown in Fig. 2.

Figure 2: (Color online) Results of the single-qubit benchmarking experiments.
(a) Histogram of sequences of a given length with a given fidelity. Fidelity
is discretized to 0.01 precision because 100 experiments were performed for
each sequence. (b) Mean fidelity for each sequence length with error bars. The
black trace is a least-squares fit to Eq. (1) yielding an EPG of 2.0(2)
$\times$ 10-5. (Inset) Summed histogram of bright and dark calibration
experiments with a red line indicating the detection threshold.

Sequence length refers to the number of computational gates in a sequence. We
implement sequences of lengths 1, 3, 8, 21, 55, 144, 233, 377, 610, and 987,
with 100 different sequences at each length, for a total of 1000 unique
sequences. With the 21-$\mu$s $\pi/2$ duration used here, a sequence of 987
computational gates requires approximately 64 ms to complete. Our current
software limits the experiment to sequences of length $\lesssim 1,300$ gates.

Theoretically, the average probability for obtaining a correct measurement
result (the fidelity) after a sequence of length $l$ is Knill et al. (2008)

$\bar{\mathcal{F}}=\frac{1}{2}+\frac{1}{2}(1-d_{\mathrm{if}})(1-2\mathcal{E}_{\mathrm{g}})^{l},$
(1)

where $d_{\mathrm{if}}$ describes errors in initialization and measurement and
$\mathcal{E}_{\mathrm{g}}$ is the EPG. A least-squares fit of the observed
decay in fidelity to Eq. (1) yields $\mathcal{E}_{\mathrm{g}}=2.0(2)\times
10^{-5}$ and $d_{\mathrm{if}}=2.7(1)\times 10^{-2}$. Here, $d_{\mathrm{if}}$
is limited by imperfect laser polarization caused by inhomogeneities in the
birefringence of the cryogenic windows of the vacuum enclosure.

The following systematic effects may contribute to the EPG: magnetic-field
fluctuations, microwave phase and frequency instability and resolution limits,
ac Zeeman shifts, pulse amplitude and duration fluctuations, microwave-ion-
coupling strength fluctuations, decoherence caused by unintended laser
illumination of the ion, and off-resonant excitation to other levels in the
ground-state hyperfine manifold.

During the benchmarking, we calibrate the qubit transition frequency
approximately every 60 s. The difference between each frequency recalibration
and the first calibration is plotted in Fig. 3(a) for the time period
corresponding to the data in Fig. 2.

Figure 3: Changes in (a) qubit transition frequency and (b) $\pi/2$ duration
during the benchmarking experiments. Change is defined as the difference
between the recalibrated value and the first calibration. Typical transition
frequencies and $\pi/2$ durations are approximately 1.250 7385 GHz and 20.50
$\mu$s, respectively.

Monte Carlo simulations of the sequences indicate an EPG contribution of
$\mathcal{E}_{\mathrm{g}}=\beta\Delta^{2}$, where $\beta=1.91\times
10^{-8}/$Hz2 and $\Delta$ is the detuning of the microwave frequency from the
qubit frequency (assumed constant for all of the sequences). In the absence of
recalibrations, the root-mean-square (rms) difference of 25 Hz would give a
predicted EPG of $1.2\times 10^{-5}$. However, with regular recalibration, the
rms difference in frequency between adjacent calibration points (15 Hz) gives
a predicted contribution to the EPG of $0.4\times 10^{-5}$. The microwave
frequency and phase resolution are 0.37 Hz and $1.5$ mrad, respectively,
leading to a predicted EPG contribution of less than $10^{-7}$.

A theoretical estimate for the expected ac Zeeman shift of the clock (qubit)
transition yields a value of less than $1$ Hz. In principle, this shift can be
determined by comparing the qubit frequency measured in a Ramsey experiment
with that of a Rabi experiment. Such back-to-back comparisons yielded values
ranging from +14 Hz to -10 Hz, each with errors of approximately 2 Hz. The
source of this variation and of the discrepancy with theory is not known, but
if we assume, as a worst-case scenario, a miscalibration of 15 Hz for the
frequency, we estimate an EPG contribution of $0.4\times 10^{-5}$.

One measure of errors caused by qubit-frequency fluctuations (e.g., from
fluctuating magnetic fields) is to characterize decoherence as an exponential
decay through a T2 process Olmschenk et al. (2010); Knill et al. (2008). To
check this, we implement a Ramsey experiment. The ion is prepared in
$|\downarrow\rangle$, and we apply a single $\pi/2$ pulse. After waiting for
an interval $\tau/2$, we apply a $\pi$ pulse to refocus the qubit, and
following another interval $\tau/2$, we apply an additional $\pi/2$ pulse,
ideally restoring the qubit back to $|\downarrow\rangle$. An exponential fit
of the resulting decay in the $|\downarrow\rangle$ state probability over
periods $\tau\lesssim 100$ ms gives T${}_{2}=0.38(4)$ s. Assuming this value
of T2 also describes frequency fluctuations at times on the order of the gate
pulses, we predict an EPG contribution of $9\times 10^{-5}$. Because this
exceeds the benchmark value, we believe that the noise at shorter periods, in
this experiment, is smaller than that predicted by a simple exponential fitted
at longer durations.

We recalibrate the $\pi/2$ duration approximately every 120 s with a sequence
of 256 in-phase $\pi/2$ pulses [Fig. 3(b)]. Monte Carlo simulations indicate
an EPG contribution of $\mathcal{E}_{\mathrm{g}}=\gamma(\Delta\tau)^{2}$,
where $\gamma=2.7\times 10^{-3}/\mu$s2 and $\Delta\tau$ represents a
miscalibration in the $\pi/2$ time (assumed constant for all sequences). In
the absence of recalibration, the 23-ns rms drift would correspond to an EPG
of $0.1\times 10^{-5}$; from the estimated residual miscalibration between
points of 5 ns, we predict an EPG contribution of less than $10^{-7}$.

We characterize pulse-to-pulse microwave power fluctuations by turning on the
microwaves continuously and sampling the power every 10 ns. The integral of
the sampled power over a 25-$\mu$s interval is proportional to the total
rotation angle during a pulse of the same duration. We perform this integral
12 times, with each 25-$\mu$s interval following the previous one by 10 s.
Within 120 s after turning on the microwaves, we observe a 1% drift in the
power. If the pulse-to-pulse variation in microwave power is, in fact, this
large, it corresponds to an EPG contribution of $3\times 10^{-5}$. However,
after a 20-min warm-up interval, we measure a pulse-to-pulse power variation
of only 0.1%, corresponding to an EPG contribution of $0.03\times 10^{-5}$.
Because the duty cycle of the benchmarking experiment is not constant, with
sequences of different lengths and calibration experiments interspersed
throughout, it is difficult to assign a specific EPG contribution to this
effect. However, we do observe larger EPG at higher microwave powers,
consistent with temperature effects playing a role at these higher powers.

To investigate unintended laser light as a source of decoherence, (e.g., from
optical pumping), the ion is prepared in $|\downarrow\rangle$ and is allowed
to remain in the dark for varying durations. We observe no decay in the
$|\downarrow\rangle$ state probability with an uncertainty of $2\times
10^{-7}$/$\mu$s, corresponding to the absence of gate errors during the
65-$\mu$s randomized gate interval with an uncertainty of $1\times 10^{-5}$.
Similar results are obtained for an ion prepared in $|\uparrow\rangle$.

Microwave-induced transitions from the qubit levels into other Zeeman levels
within the ground-state hyperfine manifold can be inferred by observing an
asymmetry between sequences ending in $|\downarrow\rangle$ and those ending in
$|\uparrow\rangle$. While the $|2,-2\rangle$ state fluoresces with 13 photons
detected on average, other hyperfine states yield, at most, 1.3 photons during
the 400-$\mu$s detection period. Therefore, transitions from the qubit
manifold to other levels would show up as a loss of fidelity for sequences
ending in $|\downarrow\rangle$, while they would not affect the apparent
fidelity of sequences ending in $|\uparrow\rangle$. For the bright sequences
in Fig. 2, the EPG is $2.2(5)\times 10^{-5}$, while for the dark sequences it
is $2.0(5)\times 10^{-5}$. We conclude that qubit leakage contributes an EPG
of $<0.2(7)\times 10^{-5}$. Similarly, if ion heating contributes to the EPG,
it should appear as a deviation from exponential decay in the benchmarking
data, which we do not observe.

For future work, it seems likely that microwave power fluctuations could be
controlled passively through a suitable choice of amplifiers and switching
circuitry or actively via feedback. Shorter pulses at higher microwave powers
would diminish errors associated with fluctuating qubit frequency, but errors
due to off-resonant transitions become more of a concern in this regime. Off-
resonant transitions could be suppressed with the use of appropriately shaped
pulses, which concentrate the microwave spectrum near the qubit transition
frequency. Self-correcting pulse sequences Levitt (1986) could be used to
reduce the effects of errors in $\pi/2$ duration and transition frequency. In
a multizone trap array, single-qubit gates implemented with microwaves will be
susceptible to cross talk between zones; however, this effect can be mitigated
with careful microwave design, the use of nulling currents in spectator zones
Ospelkaus et al. (2008), and the use of composite pulses Levitt (1986). A
demonstration of two-qubit gates with errors small enough to enable scalable
quantum computing remains challenging, but high-fidelity single-qubit gates
should make this task easier. For example, many refocusing and decoupling
techniques are based on single-qubit gates and can reduce errors during two-
qubit gate operations Viola et al. (1999).

###### Acknowledgements.

This work was supported by IARPA, NSA, DARPA, ONR, and the NIST Quantum
Information Program. We thank U. Warring, M. Biercuk, A. VanDevender, J.
Amini, and R. B. Blakestad for their help in assembling parts of the
experiment, and we thank J. Britton, S. Glancy, A. Steane, and C. Bennett for
comments. This article is a contribution of the U.S. Government, not subject
to U.S. copyright.

## References

  * Nielsen and Chuang (2000) M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information_ (Cambridge University Press, Cambridge, UK, 2000).
  * Ladd et al. (2010) T. D. Ladd et al., Nature (London) 464, 45 (2010).
  * DiVincenzo (2000) D. P. DiVincenzo, Fortschr. Phys. 48, 771 (2000).
  * Preskill (1998) J. Preskill, Proc. R. Soc. Lond., Ser. A 454, 385 (1998).
  * Knill (2010) E. Knill, Nature (London) 463, 441 (2010).
  * Ryan et al. (2009) C. A. Ryan, M. Laforest, and R. Laflamme, New J. Phys. 11, 013034 (2009).
  * Olmschenk et al. (2010) S. Olmschenk et al., New J. Phys. 12, 113007 (2010).
  * Benhelm et al. (2008) J. Benhelm et al., Nat. Phys. 4, 463 (2008).
  * Ospelkaus et al. (2011) C. Ospelkaus et al., Nature (London) 476, 181 (2011).
  * Knill et al. (2008) E. Knill et al., Phys. Rev. A 77, 012307 (2008).
  * Poyatos et al. (1997) J. F. Poyatos, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 78, 390 (1997).
  * Bravyi and Kitaev (2005) S. Bravyi and A. Kitaev, Phys. Rev. A 71, 022316 (2005).
  * Knill (2005) E. Knill, Nature (London) 434, 39 (2005).
  * Biercuk et al. (2009) M. J. Biercuk et al., Quantum Inf. Comput. 9, 920 (2009).
  * Chow et al. (2010) J. M. Chow et al., Phys. Rev. A 82, 040305 (2010).
  * Ospelkaus et al. (2008) C. Ospelkaus et al., Phys. Rev. Lett. 101, 090502 (2008).
  * Deslauriers et al. (2006) L. Deslauriers et al., Phys. Rev. Lett. 97, 103007 (2006).
  * Labaziewicz et al. (2008) J. Labaziewicz et al., Phys. Rev. Lett. 100, 013001 (2008).
  * Brown et al. (2011) K. R. Brown et al., Nature (London) 471, 196 (2011).
  * Seidelin et al. (2006) S. Seidelin et al., Phys. Rev. Lett. 96, 253003 (2006).
  * Levitt (1986) M. H. Levitt, Prog. Nucl. Magn. Reson. Spectrosc. 18, 61 (1986).
  * Viola et al. (1999) L. Viola, S. Lloyd, and E. Knill, Phys. Rev. Lett. 83, 4888 (1999).


aainstitutetext: Department of Physics, Birla Institute of Technology and
Science-Pilani, K. K. Birla Goa campus, NH-17B, Zuarinagar, Goa 403726,
Indiabbinstitutetext: Theoretical Physics Division, Physical Research
Laboratory, Shree Pannalal Patel Marg, Ahmedabad 380009, Gujarat, India

# Jet substructure probe to unfold singlet-doublet dark matter in the presence
of non-standard cosmology

Prasanta Kumar Das b Partha Konar a Saumyen Kundu  b and Sudipta Show
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We examine the singlet-doublet fermionic dark matter model, where the non-
thermal production of the dark matter in light of a non-standard cosmology
demands a significantly large interaction rate than the typical radiation-
dominated Universe. Despite being a model of freeze-in light dark matter and
heavy mediator, the characteristic long-lived particle searches at the
collider experiment and the displaced vertex signature do not help in probing
such a dark sector since this non-standard interaction mandates nearly prompt
decay. We make a counterproposal to probe such signal with di-fat-jets
generated from the boosted decays of massive vector bosons and SM Higgs, along
with the substantial missing transverse momentum to probe the dark matter at
LHC. Interestingly, substructure variables associated with these fat jets have
an additional handle to tackle the extensive QCD background as it encodes
implicit footmarks of their origin. We adopt the multivariate analysis with
the booted decision tree to constrain the measured relic density allowed
parameter space of dark matter in the presence of the modified cosmological
scenario. Our study shows how the non-trivial expansion affects dark matter
production in the early Universe and alters the required search strategies at
colliders. This probe provides the best discovery prospect at the HL-LHC for
extended parameter space now opened up in the dark sector.

###### Keywords:

Non-standard Cosmology, Freeze-in Dark Matter, Large Hadron Collider, Boosted
Jet, Jet-substructure

##  1 Introduction

Numerous indirect pieces of evidence at diverse length scales of our Universe
have conclusively established the omnipresence of some non-luminous, non-
baryonic mysterious matter commonly known as dark matter (DM). It is
interesting that so far, all such clues that are based on different celestial
observations, such as the flatness of the galaxy rotation curve Sofue:2000jx ,
the studies of bullet cluster collision Clowe:2006eq , gravitational lensing
of distant galaxies where only the gravitational interaction of DM is
involved. Another indirect hint comes from experiments like WMAP Hinshaw_2013
and PLANCK Planck:2018vyg , probing the anisotropy of the cosmic microwave
background, which quantifies the present amount of dark matter of the Universe
with great precision. Despite all these, the primary quest on the particle
nature of dark matter, its interactions with different SM or other dark-sector
particles and the production mechanism at an early stage of the Universe still
remain elusive. Based on these observed properties, there exists a plethora of
models for DM. Among all such variants, the weakly interacting massive
particle (WIMP) paradigm Green:2002ht ; Chang:2017gla ; Chang:2017dvm ;
Visinelli:2017qga ; Arcadi:2017wqi ; Choubey:2017yyn ; Reinert:2017aga ;
Evans:2017kti ; Garny:2018icg ; Blanco:2019hah ; Bhardwaj:2018lma ;
Bhardwaj:2019mts ; Konar:2020wvl ; Konar:2020vuu ; Heurtier:2019beu ;
Habermehl:2020njb ; Xing:2021pkb ; Borah:2022byb ; Belanger:2022qxt ;
Gines:2022qzy ; Bernal:2022wck ; Kundu:2021cmo ; Medina:2021ram ;
Tallman:2022nts ; Kang:2022zqv ; Dutta:2022wdi  is the most explored one till
date. Non-observation of any DM signature so far in direct detection
Akerib:2016vxi ; Zhang:2018xdp ; Aprile:2018dbl  and indirect detection
MAGIC:2016xys  experiments as well as in the collider frontiers
Chatrchyan:2012xdj ; Aad:2012tfa  severely constrain this scenario. Null
results, so far, from all these experiments, especially the stringent
constraints from direct detection, put a question mark on the scale of the DM
interaction and encourage us to examine alternative paradigms of DM production
mechanism. Non-thermal origin of DM is one such exciting scenario that
naturally explains the null results thanks to its feeble interaction with the
visible sector. Due to this same rationale, the DM never gets thermalized with
the primordial bath in the early Universe; it is instead produced gradually
from the scattering and/or the decay of visible particles in the early
Universe until the production mechanism freezes in. Recently the feebly
interacting massive particle (FIMP) Hall:2009bx ; Konar:2021oye ;
Ghosh:2021wrk ; Chakrabarty:2022bcn  paradigm of DM has also got significant
attention in the literature.

Since dark matter production is the phenomenon of the early Universe, the
history of the early Universe is crucial. Typically, most DM studies consider
the radiation-dominated early Universe, although there is no direct evidence
to date about the energy content of the Universe at higher temperatures. So,
it is prudent to take a holistic view considering the possibility of
domination in species other than radiation in the energy budget DEramo:2017gpl
; DEramo:2017ecx  during the pre-BBN era. In this work, we adopt the scenario
of fast expansion where the spices ($\phi$), red-shifting faster than
radiation, dominates the energy content of the early Universe.

In this work, we focus on the freeze-in production of dark matter where direct
and indirect searches can hardly provide any constraint due to the feeble
interaction of DM with the particles of the visible sector. One cannot
directly produce such DM at the collider too. However, exploring the
production of other dark sector particles 111Freeze-in production requires
additional dark sector particle(s) with enough interaction with the SM to stay
at thermal equilibrium and produce DM from decay. in high-energy collision
events can provide important inputs on this sector and DM. Due to this weaker
interaction strength, interesting constraints can emerge, especially in the
parameter space where the decay of the long-lived particle to DM with an SM
partner. Thus, a viable freeze-in dark matter scenario can be characterized by
signatures like disappearing tracks and displaced vertex at LHC if the decay
process is localized within a specific range of the detector. This requires DM
mass of the order of keV Hall:2009bx ; Calibbi:2018fqf ; No:2019gvl .
Interestingly, this is also a typical mass scale that astrophysical
experiments can probe in the context of warm dark matter Viel:2013fqw ;
Yeche:2017upn ; Irsic:2017ixq . Such warmness suppresses the structure
formation at a short distance, which solves the small-scale problem in the
$\Lambda$CDM (the standard picture of cosmology) Bode:2000gq ; Lovell_2012 .
Authors in the refs Calibbi:2018fqf ; No:2019gvl  have studied the frozen-in
warm dark matter and demonstrated the connection between the long-lived
particles along with the displaced vertex signature as a powerful tool in the
context of singlet doublet dark matter scenarios.

In this work, we consider the singlet-doublet model Yaguna:2015mva ;
Fiaschi:2018rky ; Restrepo:2019soi ; Arcadi:2018pfo ; Esch:2018ccs ;
Calibbi:2018fqf ; Maru:2017pwl ; Maru:2017otg ; Xiang:2017yfs ; Abe:2017glm ;
Banerjee:2016hsk ; Horiuchi:2016tqw ; Calibbi:2015nha ; Cheung:2013dua ;
Cohen:2011ec ; Enberg:2007rp ; DEramo:2007anh ; No:2019gvl ; Barman:2019aku ;
DuttaBanik:2018emv ; Barman:2019tuo ; Bhattacharya:2018fus ;
Bhattacharya:2015qpa ; Bhattacharya:2017sml ; Restrepo:2015ura ;
Freitas:2015hsa ; Cynolter:2015sua ; Bhattacharya:2016lts ;
Bhattacharya:2016rqj ; Wang:2018lhk ; Abe:2019wku ; Barman:2019oda ;
Konar:2020wvl ; Konar:2020vuu ; Konar:2021oye  to demonstrate the impact of
non-standard cosmology both in the dark matter phenomenology as well as in the
prospect of collider probes. We exhibit that the introduction of modified
cosmology demands a larger interaction strength than the usual (standard
cosmological) scenario, and naturally, it also alters the probing methodology
at the collider. A larger interaction rate demands prompt decays rather than
giving a displaced vertex signature. We suggest probing this final state with
the production of $Z/W/h$-bosons, along with a substantial missing transverse
momentum from DM particles at the LHC. The bosons are also expected to be
significantly _boosted_ due to their production from heavy mediators.
Identities of their individual decay products are mostly undetectable, being
submerged in localized pixels at the detector along the boosted direction, and
such detections are expected to have poor efficiency. On the other hand,
directly looking at total hadronic decay products in a large radius jet can be
highly advantageous, especially if one looks at the formation of substructures
inside such jets to assess their origin. Substructure variables can
significantly improve the detection of such events compared to QCD jet
background events. We look at the jet substructure of the boosted $Z/W/h$-jets
coupled with sophisticated multivariate analysis (MVA) to probe the
significant part of the parameter space, inaccessible with the displaced
vertex searches of this model.

This paper is organized as follows. In section 2, we sketch the singlet-
doublet dark matter model discussing different decay modes of the heavier
fermions, which have an essential role both in frozen-in DM production and at
the LHC search. In section 3, we briefly review the non-standard cosmology
that arises due to the existence of another species in the early Universe. In
the next section (section 4), we discuss the phenomenology of the frozen-in DM
in the context of non-standard cosmology. In section 5, we addressed the
collider setup in detail and the search strategy using the substructure
variables. We also include the multivariate analysis in this section before
presenting our main results. Finally, we summarise and conclude in section 6.

##  2 Structure of the model

We execute our analysis in the singlet-doublet model No:2019gvl ;
Konar:2020wvl ; Konar:2020vuu ; Konar:2021oye , which consists of a
hyperchargeless singlet Dirac fermion ($\chi$) and an $SU(2)_{L}$ Dirac
fermion doublet ($\Psi$)

$\Psi=\begin{pmatrix}\psi^{+}\\\ \psi^{0}\end{pmatrix}$

with hypercharge $1/2$. Here, we employ the discrete $\mathcal{Z}_{2}$
symmetry under which the BSM fields change sign while all the SM fields are
assigned with even $\mathcal{Z}_{2}$ charges. The Lagrangian for the BSM
fermions can be written as

$\mathcal{L}_{f}=i\overline{\chi}\gamma^{\mu}\partial_{\mu}\chi+i\overline{\Psi}\gamma^{\mu}D_{\mu}\Psi+m_{\chi}\overline{\chi}\chi+m_{\Psi}\overline{\Psi}\Psi+y\overline{\Psi}H\chi+h.c.$
(1)

Both the neutral particles $\psi^{0}$ and $\chi$ get mixed after the EW
symmetry breaking because of the Yukawa type interaction term in Eq. (1).
After mixing, the new mass eigenstates are $\chi_{1}$ and $\chi_{2}$ having
masses $m_{1}$ and $m_{2}$, respectively. We make simple assumptions $m_{2}\gg
m_{1}$ and $y\ll 1$, which are needed for the freeze in production to yield a
correct amount of dark matter relic density. In this limit, one gets
$m_{1}\approx m_{\chi}$, $m_{2}\approx m_{\Psi}$. Hence, the simplified form
of the singlet-doublet mixing angle becomes

$\sin\theta\approx\frac{yv}{\sqrt{2}(m_{2}-m_{1})},$ (2)

where $v$ is the vacuum expectation value of the SM Higgs. A radiative mass
splitting between the charge states ($\psi^{\pm}$) and neutral state
($\psi^{0}$) is generated due to the loops of the EW gauge boson Thomas:1998wy
; Cirelli:2009uv , so that mass splitting $\mathcal{O}[260-340]$ MeV can be
produced for the considered range of $m_{\Psi}\in$ $[100-2000]$ GeV. For
simplicity, we would neglect the mass splitting for the dark matter
phenomenology and assume that the masses of all the heavy states are equal
such that $m_{\psi^{\pm}}=m_{2}=m_{\Psi}$. The interactions of the dark matter
after the EW symmetry breakings are $h-\chi_{2}-\chi_{1}$,
$Z-\chi_{2}-\chi_{1}$, $W^{\pm}-\psi^{\mp}-\chi_{1}$ where the first one
responsible for the Yukawa interaction expressed in Eq. (1) and rest of the
interactions come from the singlet doublet mixing originated by the Yukawa
interaction after the symmetry breaking. Corresponding decays widths for these
heavy BSM states can be given by,

$\displaystyle\Gamma(\chi_{2}\to h\chi_{1})=\frac{y^{2}}{32\pi
m_{\Psi}^{2}}\left[(m_{\Psi}+m_{\chi})^{2}-m_{h}^{2}\right]\lambda(m_{\Psi},m_{\chi},m_{h}),$
(3) $\displaystyle\Gamma(\chi_{2}\to Z\chi_{1})=\frac{y^{2}}{32\pi
m_{\Psi}^{2}}\frac{\left[(m_{\Psi}-m_{\chi})^{2}-m_{Z}^{2}\right]\left[(m_{\Psi}+m_{\chi})^{2}+2m_{Z}^{2}\right]}{m_{\Psi}^{3}(m_{\Psi}-m_{\chi})^{2}}\lambda(m_{\Psi},m_{\chi},m_{Z}),$
(4) $\displaystyle\Gamma(\psi^{\pm}\to W^{\mp}\chi_{1})=\frac{y^{2}}{32\pi
m_{\Psi}^{2}}\frac{\left[(m_{\Psi}-m_{\chi})^{2}-m_{W}^{2}\right]\left[(m_{\Psi}+m_{\chi})^{2}+2m_{W}^{2}\right]}{m_{\Psi}^{3}(m_{\Psi}-m_{\chi})^{2}}\lambda(m_{\Psi},m_{\chi},m_{W})$
(5)

where,
$\lambda(a,b,c)=\sqrt{a^{4}+b^{4}+c^{4}-2a^{2}b^{2}-2b^{2}c^{2}-2c^{2}a^{2}}$.

##  3 A framework with fast expanding Universe

The standard model of cosmology assumes that the energy budget of the early
Universe was radiation-dominated prior to the Big Bang Nucleosynthesis (BBN)
era. Although, the lack of any observational evidence can not rule out the
possibility that some other field could have dictated the evolution of the
Universe rather than radiation, where the corresponding field redshifts
smaller or larger than the radiation. Many previous studies DEramo:2017gpl ;
DEramo:2017ecx ; Konar:2020vuu ; Konar:2021oye  have demonstrated the non-
trivial impact of such modified cosmology on DM phenomenology.

The non-standard cosmology can be simply realized by considering the
contribution of the extra new species ($\phi$) along with the radiation
component in the energy content of the Universe. Here, we consider the non-
standard scenario where the parameter of the equation of state ($\omega$) is
larger than that of the radiation ($\omega_{rad}$=1/3). One can parameterize
the same by writing the corresponding energy density as $\rho_{\phi}\propto
a^{-3(1+\omega)}$, which can be re-expressed as $\rho_{\phi}\propto
a^{-(4+n)}$ with $\omega=\frac{1}{3}(n+1)$ provided $n>0$. Hence, the modified
version of the Hubble rate DEramo:2017gpl ; DEramo:2017ecx ; Konar:2020vuu ;
Konar:2021oye  of the Universe can be delineated as

$\displaystyle H^{2}=\frac{\rho_{rad}+\rho_{\phi}}{3M_{P}^{2}},$ (6)

where $M_{P}$ represents the reduced Planck mass and the $\rho_{rad}$ and
$\rho_{\phi}$ stand for the radiation energy density and the energy density of
the species ($\phi$), respectively. The total energy density as a function of
temperature can be expressed as

$\displaystyle\rho(T)$ $\displaystyle=\rho_{\rm rad}(T)+\rho_{\phi}(T)$ (7)
$\displaystyle=\rho_{\rm
rad}(T)\left[1+\frac{g_{*}(T_{r})}{g_{*}(T)}\left(\frac{g_{*s}(T)}{g_{*s}(T_{r})}\right)^{(4+n)/3}\left(\frac{T}{T_{r}}\right)^{n}\right],$
(8)

with $\rho_{rad}=\frac{\pi^{2}}{30}g_{*}(T)T^{4}$, $g_{*}$ and $g_{*s}$ denote
the relativistic degrees of freedom for energy and entropy, respectively.
Here, $T_{r}$ represents the temperature at the end of the fast-expanding
Universe era, after which the Universe enters into a radiation-dominated
phase. Thus the non-standard behaviour of the early Universe can be described
simply by the set of two parameters $(n,T_{r})$. A larger value of $n$ or a
smaller value of $T_{r}$ entails that the Universe is red shifting faster. It
is important to note that one can get back the standard picture in the absence
of the extra species ($\phi$) with energy density, $\rho=\rho_{\text{rad}}$.
The BBN constraints on the number of relativistic degrees of freedom put a
lower bound on $T_{r}(\geq(15.4)^{1/n}$ MeV$)$. An interesting scenario, $n=2$
($\omega=1$), is very well accepted as the kination domination DEramo:2017gpl
; DEramo:2017ecx ; Konar:2020vuu ; Konar:2021oye , which we consider our case
study throughout our whole analysis. In the next section, we discuss the
effects of the modified cosmology on the DM production mechanism.

##  4 Dark Matter Phenomenology

Figure 1:  Contours of the measured dark matter relic density $\Omega
h^{2}=0.12$ are shown for different non-standard cosmology parameters
($T_{r}$) with a fixed value of $n=2$ on the Yukawa coupling (y) and doublet
mass ($m_{\psi}$) plane. Non-standard cases are plotted with dashed lines for
corresponding $T_{r}$ values, which opened up a vast region of parameter space
over and above the standard cosmology ($n=0~{}or~{}T_{r}\to\infty$) confined
on a solid line in this plot. Dark matter mass is also fixed at $m_{\chi}=12$
keV.

In our setup, the doublet ($\Psi$) becomes part of the thermal bath because of
its gauge interaction, while the dark matter ($\chi$) never gets thermalized
with the bath due to the feeble Yukawa coupling ($y\ll 1$). In the early
Universe, the dark matter is produced gradually via freeze-in from the
scattering and the decay of the bath particles starting from zero (or
negligibly small) number density. Since the DM production crucially depends on
the early history of the Universe, the freeze-in number density of DM is built
upon the Yukawa coupling ($y$) along with the non-standard parameters $n$ and
$T_{r}$.

Benchmark | $y$ | $m_{\psi}$ | $m_{\chi}$ | $n$ | $T_{r}$ | $\Omega_{\chi}h^{2}$ | $\Gamma_{\Psi}$ | $c\tau_{\Psi}$
---|---|---|---|---|---|---|---|---
Points |  | [GeV] | [KeV] |  | [GeV] |  | [GeV] | [cm]
BP-1 | $1\times 10^{-6}$ | 860 | 12 | 2 | 0.031 | 0.12 | $2.5\times 10^{-11}$ | 0.0008
BP-2 | $5\times 10^{-8}$ | 810 | 12 | 2 | 11.24 | 0.12 | $5.9\times 10^{-14}$ | 0.3
BP-3 | $1.3\times 10^{-8}$ | 865 | 12 | 0 | - | 0.12 | $4.3\times 10^{-15}$ | 4.6
Table 1: Benchmark Points considered in the analysis.

Here, the dark matter production takes place in two steps: before and after
the _Electroweak Symmetry breaking_ (EWSB). _Before_ EWSB, the dark matter
production is carried out by different scatterings ($HH\to\chi\chi$,
$\Psi\Psi\to\chi\chi$) and the decay ($\Psi\to H\chi$) processes. Since the
scattering is proportional to $y^{4}$ while the decay varies as $y^{2}$, one
can safely neglect the scattering contributions in the limit $y\ll 1$. _After_
EWSB, some additional channels open up due to the singlet doublet mixing.
Here, the processes contributing to the DM productions are the scatterings
($SMSM\to\chi_{1}\chi_{1}$, $SMSM\to\chi_{2}\chi_{1}$) with both the processes
mediated by the SM Higgs and $SU(2)_{L}$ gauge bosons and decays ($\chi_{2}\to
h(Z)\chi_{1}$, $\psi^{\pm}\to W^{\pm}\chi_{1}$). If $y\ll 1$, once again, all
the scattering processes remain subdominant and would have a negligible impact
compared to decays in producing DM. Since decays significantly dominate over
the scattering in both regimes, it is very important to define the thermally
average decay width,

$\langle\Gamma_{\Psi}\rangle=\langle\Gamma_{\Psi\to
H\chi}\rangle~{}\Theta(T-T_{\rm EW})+\langle\Gamma_{\chi_{2}\to
h\chi}+\Gamma_{\chi_{2}\to h\chi}+\Gamma_{\psi^{\pm}\to
h\chi}\rangle~{}\Theta(T_{\rm EW}-T)$ (9)

with $T_{EW}(=$160 GeV$)$ is the temperature of the EWSB. Now, one can track
the dark matter abundance by solving the following Boltzmann equation,

$\displaystyle\frac{dY_{\chi}}{dz}=\frac{{\langle\Gamma_{\Psi}\rangle}}{Hz}Y_{\Psi}^{\rm
eq}(T),$ (10)

where $Y_{i}$ represents the abundance of the $i^{\text{th}}$ species,
$z(=m_{\chi}/T)$ is a dimensionless variable and
${\langle\Gamma_{\Psi}\rangle}(=\Gamma_{\Psi}\frac{K_{1}(x)}{K_{2}(x)})$
refers to the thermally averaged decay rate. One can obtain the relic density
of the dark matter by using the following relation

$\displaystyle\Omega_{\chi}h^{2}=2.755\times 10^{8}\times m_{\chi}\times
Y_{\chi}({x\to\infty})$ (11)

In Figure 1, we have shown the contour of the correct relic density of the
dark matter in the plane of the Yukawa coupling and doublet mass. When the
value of the parameter ($T_{r}$) reduces, then one needs larger Yukawa
coupling ($y$) for the fixed values of ($n$), and dark matter mass to satisfy
the relic density constraints as the effects of the fast expansion lasts for a
longer time. Hence, in this modified scenario, the entire parameter space
above the standard case becomes allowed for different values of $T_{r}$ (and
$n$). It is important to mention that any large Yukawa coupling is not allowed
for our considered mass range of the heavy fermion since there is an upper
limit on $T_{r}$ from BBN. We have depicted the BBN excluded region by the
shaded red area in the upper portion of the plot.

Before moving further to describe the possible probe at the LHC, let us
construct relevant benchmarks for our current study. We demonstrate our
results considering the benchmark points in Table 1, which satisfy all the
dark matter constraints. The first two benchmark points are relevant for non-
standard cosmology, while the last one (BP-3) is placed overlapping the
standard case for presentation purposes. All these points satisfy the relic
density constraint for suitable values in $T_{r}$ (and $n$). It is to be noted
that the decay length for BP-3 is around 5 cm, so this can be probed better
through the DV signature as the length of the tracker is around 1cm. Although,
for BP-1 and BP-2, the decay length is sub-centimetre since the presence of
alternative cosmology dictates very rapid decay. As a result, the DV signature
can not be a better tool to probe our interesting region of parameter space;
rather, the study of jet substructure plays a pivotal role in collider
searches.

##  5 Probing dark sector at the LHC

Figure 2: Left plot shows different modes of heavy BSM fermions pair
production cross-section at the LHC as a function of doublet mass $m_{\psi}$.
In the right plot branching ratio of different decay modes of heavy neutral
fermion $\chi_{2}$ are shown as a function of fermion doublet mass $m_{\psi}$.

This section discusses the possible collider signature to probe the model. As
mentioned already, tiny coupling makes direct dark matter production at the
LHC difficult. Still, the presence of other dark sector particles with
relatively significant coupling makes a case for their pair production which
eventually decays to produce the DM candidate at the collider. The
conventional and popular search strategy of FIMPs at colliders has been to
look for the displaced signatures of the mediators owing to their large decay
length. Other search channels, such as disappearing tracks or mono-jet search,
are less sensitive at LHC in the standard cosmological scenarios. Now, with a
larger interaction rate as necessitated by the alternative cosmological
scenario discussed in section 3, the displaced vertex search loses its
significance, and prompt searches take over. In such a scenario, decay of the
heavy mediators $\Psi^{\pm}$ and $\chi_{2}$ exert a substantial boost on the
decay products leading to the emission of large radius hadronic clusters. Here
we intend to examine the possible substructure of these clusters in the
reconstructed final states of the events.

Figure 3: Representative diagram for the production of heavy BSM fermions pair
at the LHC, finally generating a pair of DM candidates along with massive
bosons. Hadronic decay of these bosons produces boosted jets $J_{i=0,1}$
observed at a hadron calorimeter.

The production channels are s-channel pair-production of charged and neutral
fermions through Z-boson or photon exchange or associated production of
charged and neutral fermions through s-channel W-bosons.

$pp\to\chi_{2}\overline{\chi}_{2},\;\;\;\;\;\;pp\to\psi^{+}\psi^{-},\;\;\;\;\;\;pp\to\psi^{\pm}\overline{\chi}_{2}$
(12)

The production cross-section plotted in Figure 2 as a function of the doublet
mass $m_{\Psi}$ at 14 TeV LHC. These production channels result in final state
of the form

$\displaystyle\chi_{2}\overline{\chi}_{2}$ $\displaystyle\to
Z(h)Z(h)\chi_{1}\overline{\chi}_{1},$ (13) $\displaystyle\psi^{+}\psi^{-}$
$\displaystyle\to W^{+}W^{-}\chi_{1}\overline{\chi}_{1},$ (14)
$\displaystyle\psi^{\pm}\chi_{2}$ $\displaystyle\to
W^{\pm}Z(h)\chi_{1}\overline{\chi}_{1}$ (15)

These heavy fermions promptly decay into a DM candidate with a boson following
Eqs. (3), (4) and (5). Depending upon the mass of this heavy fermion, the
boson’s outcomes can be observed in the large radius hadronic clusters in the
hardon calorimeter, termed fatjets. Notice the additional advantage of looking
into the dominant hadronic channel if the QCD background is controlled
effectively. So, essentially our signal topology will assume the form

$pp\to 2J_{\rm{CA8}}+\not{E}_{T}$ (16)

where, $J_{\rm{CA8}}$ are jets clustered with Cambridge-Aachen algorithm with
jet radius $R=0.8$, whose 2-prong substructure we intend to probe. The Feynman
diagram for the signal processes is shown in Figure 3 where $J_{0}$ and
$J_{1}$ are the two fatjets originating from the hadronic decay of the gauge
bosons $Z/h/W$.

###  5.1 Analysis setup and event simulation

We implement the model in FeynRules Alloul:2013bka  to obtain the UFO
Degrande:2011ua  library, which we feed to MadGraph5v2.9.9 Alwall:2014hca  to
generate the events at leading order at $14$ TeV CM energy. Pythia8 is used
for showering and hadronization with matching up to two to four extra jets
(depending upon the processes) with MLM matching scheme Mangano:2006rw  with
virtuality-ordered parton showers. The detector effects are simulated using
Delphes3 deFavereau:2013fsa  with the default CMS configuration card. The
particle-flow objects are then clustered using FastJet Cacciari:2011ma  with
the Cambridge-Aachen algorithm Dokshitzer:1997in  with the radius-parameter
$R=0.8$ and minimum transverse momentum of $180$ GeV to form the fatjets.

### Backgrounds

Different SM processes can imitate the signature of di-fat-jets with missing
transverse momentum. We extensively generated and studied these backgrounds
from the following processes.

  * •

V+jets: QCD jets mimic the boosted jet with the invisible decay of vector
boson. Contribution from this background can be extremely large.

    1. (i)

Z+jets – This process gives the most dominant background where the Z-boson
decays invisibly, resulting in large missing energy.

    2. (ii)

W+jets – This process also contributes considerably when the W-boson decays
leptonically, and the leptons are missed in the detector. These missed
leptons, along with the neutrinos, contribute to missing energy.

  * •

VV+jets: This background can be divided into three types, namely, $WZ,WW$ and
$ZZ$ matched up to two extra jets.

    1. (i)

WZ+jets – This is the most dominant of the three where $W$-boson decays
hadronically and $Z$-boson decays invisibly.

    2. (ii)

WW+jets – This background contributes when the leptons from the semileptonic
decay of two W-bosons are missed in the central tracker.

    3. (iii)

ZZ+jets – In this process, one of the Z-bosons decays hadronically while the
other invisibly.

  * •

Single top: Three processes, namely $tW,\,tb,\,tj$, contribute to this
background. Of the three single top quark production associated with a W-boson
contributes most significantly.

  * •

$t\bar{t}$+jets: Semileptonic decay of top-pair contributes as a significant
background to our signal process. This background process can be controlled
substantially by applying a $b$-tag veto.

###  5.2 Event Selection

The detector simulated datasets are analyzed in the ROOT BRUN199781
framework, where we use the following baseline-selection criteria.

  * •

Events are selected with at least two fatjets with cone-radius $R=0.8$ and
transverse momentum $p_{T}(J_{i})>180$ GeV.

  * •

The missing transverse momentum is required to be larger than $100$ GeV.

  * •

Events that have lepton with transverse momentum $p_{T}(\ell)>10$ GeV within
pseudorapidity range $|\eta(\ell)|<2.4$ are vetoed. This veto controls SM
events with gauge boson decaying leptonically.

  * •

We also apply a condition that the azimuthal separation between any of the
fatjets and missing momentum to be less than 0.2, i.e.,
$|\Delta\phi(\not{p}_{T},J_{0,1})|>0.2$. This eliminates possible contribution
to missing transverse momenta coming from jet mismeasurement.

On top of the baseline-selection criteria, we applied some extra selection
cuts to reduce the overwhelming backgrounds before going for the multivariate
analysis (MVA) using the TMVA package Hocker:2007ht  integrated into the ROOT
framework. The additional cuts are in the following:

  * •

Events containing b-tagged small radius jets are rejected. These slim jets are
clustered with an anti-$k_{T}$ algorithm with radius parameter $R=0.4$. This
cut particularly helps in reducing the top-pair background.

  * •

Events with the mass of both the pruned prune  fatjets higher than $40$ GeV
are accepted. Although these fatjets are expected to be even heavier,
acceptance cuts are kept low, assuming further improvements in machine
learning models.

###  5.3 Substructure Variables

Jet substructure observables have proved to be very efficient tools for
analyzing datasets in the boosted topology. The heavier particles in our
model, $\psi^{\pm},\chi_{2}$, while decaying, impart a significant boost to
the daughter gauge bosons ($h,Z,W^{\pm}$), which can be reconstructed by
clustering its hadronic decay signature with larger cone radius with a
sequential algorithm. These clusters, when analyzed with suitable observables,
reveal their two-prong substructure. In this analysis, we have used the
following substructure variables,

Figure 4: Normalized distributions of the important input high-level variables
characterizing fat-jets used in the multivariate analysis for the
discrimination of signal (blue-shaded lines) and the backgrounds (red-hatched
lines) considering the first signal benchmark point BP-1.

  * •

Pruned jet mass — Systematic effects in the substructure sourced from the
recombination algorithm can often be significant. Application of pruning prune
improves identification of heavy particle decays by removing soft and wide
angle proto-jets. Pruning betters the mass resolution of the heavy particles.
We performed the pruning with the standard method and as implemented in the
Delphes modules with $z_{\rm{cut}}=0.1$222RcutPrun in Delphes3 and
$D_{\rm{cut}}=0.5$ and checked for the condition $z={\rm
min}(p_{T_{i}},p_{T_{j}})/p_{T_{i+j}}<z_{\rm cut}$ and $\Delta
R_{ij}>D_{\rm{cut}}$. If the condition is satisfied, the merging is vetoed.
Otherwise, a pruned jet is formed. In this analysis, we have used the pruned
jet mass of the two fatjets, i.e. $M(J_{0})$ and $M(J_{1})$, whose
distribution is shown in Figure 4.

  * •

Energy correlator — The generalized energy correlation functions have proved
to be very efficient in probing the N-prong substructure of a boosted jet just
using the energy and pairwise angles of the particles without requiring
explicit identification of sub-jet regions. We use here the _energy
correlation double ratio_ defined as Larkoski:2013eya ,

$C_{N}^{(\beta)}=\frac{{\rm ECF}(N+1,\beta){\rm ECF}(N-1,\beta)}{{\rm
ECF}(N,\beta)^{2}}$ (17)

where, ${\rm ECF}(N,\beta)$ is defined as

${\rm ECF}(N,\beta)=\sum_{i_{1}<i_{2}<...<i_{N}\in
J}\left(\prod_{a=1}^{N}p_{T_{i_{a}}}\right)\left(\prod_{b=1}^{N-1}\prod_{c=b+1}^{N}R_{i_{b}i_{c}}\right)$

$R_{ij}$ being the distance between $i$ and $j$ in the rapidity-azimuthal
angle plane, $R_{ij}=\sqrt{(y_{i}-y_{j})^{2}+(\phi_{i}-\phi_{j})^{2}}$ and
$\beta$, the angular exponent that helps in optimization of the discrimination
power. As discussed in Larkoski:2013eya , for 2-prong substructures like the
$W/Z/h$, as in our case, one needs a 3-point energy correlator, i.e.,
$C_{2}^{\beta}$. So we used the variables $C_{2}(J_{0})$ and $C_{2}(J_{1})$
respectively for the leading and sub-leading fatjets.

  * •

N-subjettiness ratio — N-subjettiness is an excellent jet-shape variable for
boosted object identification. This variable is essentially a measure of how
well the jet energies are aligned into different constituent subjets in an
N-prong fatjet. It is defined as Thaler:2010tr ,

$\tau_{N}^{(\beta)}=\frac{1}{N_{0}}\sum_{k}p_{T_{k}}\,\rm{min(\Delta
R_{1,k},\Delta R_{2,k},...,\Delta R_{N,k})}.$ (18)

Here, $k$ runs over constituent particles in a given jet and $\Delta
R_{J,k}=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$ is the distance in the
$\eta-\phi$ plane between a candidate subjet J and a constituent particle k.
The normalization factor is defined as $N_{0}=\sum_{k}p_{T_{k}}R_{0}$, where
$p_{T_{k}}$ is the transverse momentum of the jet of radius $R_{0}$. As shown
in Thaler:2010tr , rather than $\tau_{N}$, a better variable can be the ratio
$\tau_{N}/\tau_{N-1}$ for identification of an N-prong fatjet, so we chose the
subjettiness ratio $\tau_{21}=\tau_{2}/\tau_{1}$ of the two fatjets as part of
our substructure variables in the analysis.

NTrees | 390
---|---
MaxDepth | 4
MinNodeSize | 5%
BoostType | AdaBoost
AdaBoosBeta | 0.4
BaggedSampleFraction | 0.6
nCuts | 10
SeparationType | GiniIndex
PruneMehod | CostComplexity
PruneStrenth | 0.5
Table 2: List of the parameters used in the adaptive BDT analysis.

Other than these substructure variables, we use other kinematic variables that
appear sensitive in isolating the signal from the vast background.

  * •

The missing transverse momentum, $\not{p}_{T}$.

  * •

The effective mass of the process $M_{eff}=\sum_{vis}|p_{T}|+|\not{p}_{T}|$.

  * •

Transverse momentum of the leading and sub-leading fatjets, $p_{T}(J_{0})$,
$p_{T}(J_{1})$

  * •

The distance between the two fatjets in the $\eta-\phi$ plane, $\Delta
R(J_{0},J_{1})$.

  * •

The difference in the azimuthal angle between the missing transverse momentum
and the leading fatjets, $\Delta\phi(\not{p}_{T},J_{0})$.

In Figure 4, the normalized distributions of all the variables are shown where
the first two rows contain the substructure variables, and the other two rows
present the rest of the variables. The distributions for the background (red
hatched) consists of weighted contributions from all the variables discussed
in subsection 5.2.

###  5.4 Multivariate Analysis

We analyze the datasets using the multivariate analysis (MVA) method, which
produces non-linear decision boundaries more efficiently than the univariate
cut-based analysis. We employ the adaptive BDT method among the other
different MVA methods since it results in better performance in terms of the
Receiver Operating Characteristic (ROC). The configuration we used is shown in
Table 2.

Figure 5: Plots showing the Linear Correlation between the input variables
used in the MVA analysis for signal (left panel) and background (right panel)
for benchmark point BP-1.

$\not{p}_{T}$ | $M_{eff}$ | $\Delta R(J_{0},J_{1})$ | $C_{2}(J_{0})$ | $M(J_{0})$ | $M(J_{1})$ | $C_{2}(J_{1})$ | $p_{T}(J_{0})$ | $\tau_{21}(J_{0})$ | $\tau_{21}(J_{1})$ | $p_{T}(J_{1})$ | $\Delta\phi(J_{0},\not{p}_{T})$
---|---|---|---|---|---|---|---|---|---|---|---
45.38 | 25.30 | 25.30 | 23.61 | 20.29 | 16.66 | 15.78 | 13.19 | 12.66 | 12.63 | 10.60 | 4.91

Table 3: Method unspecific relative separation power of the different input
variables according to their rank before using at MVA.

The preprocessing of the datasets is done by selecting a subset of variables
from a larger collection of variables. Such choice is based on the linear
correlation among them and their relative importance as a discriminator
between the signal and background. For evaluating the linear correlation
coefficient, we used the formula

$\rho(x,y)=\frac{\rm{cov}(x,y)}{\sigma_{x}\sigma_{y}}$ (19)

where, $\rm{cov}(x,y)=\langle xy\rangle-\langle x\rangle\langle y\rangle$ is
the covariance between $x$ and $y$ and $\sigma_{x}$ ($\sigma_{y}$) is the
standard deviation of the the variable $x$ ($y$). We illustrated the
correlation between different variables in Figure 5 for signal (left panel)
and background (right panel) where we see that most of the variables used are
uncorrelated. Some variables like $p_{T}(J_{0})$, $M_{eff}$ show a higher
correlation between them. But considering their high separation power, we have
still used them. In Table 3, the relative importance of the variables for
discriminating the signal from the background in the classification process.
The parameter setup we used for BDT training is given in Table 2. Despite
being a robust classifier, BDT can suffer from overtraining. We have checked
this by looking at the Kolmogorov-Smirnov probability as shown in the left
panel of Figure 6 along with BDT output for the benchmark point. In the right
panel of Figure 6, we have shown the signal efficiency (blue line), background
efficiency (red line) and the signal significance (green line) as functions of
BDT cut values. We have evaluated the signal significance using the formula,
$sig=S/\sqrt{S+B}$, where $S$ and $B$ are the signal and background events
after the application of the BDT cut. We see that a significance of 2.0 is
achieved by applying a cut at 0.43 for the benchmark points considered. Note
that the value of the optimal cut may differ for a different point in the
parameter space we are working.

###  5.5 Results

In this section, we have presented the results of our MVA analysis. As
discussed in section 2, the main independent parameters of the model are the
doublet mass ($m_{\psi}$), coupling ($y$) and the dark matter mass
($m_{\chi}$). So for a fixed dark matter mass, the parameter space of the
model can be defined with $y$ and $m_{\Psi}$. Based on these parameters, we
considered two non-standard benchmark points as shown in Table 1 to
demonstrate the performance of our BDT analysis in two different regions based
on accessibility by previous searches. One of those points lies in a region
probed by the displaced vertex search performed in ref. Calibbi:2018fqf ,
whereas the other point lies outside the region excluded by the displaced
vertex search.

Figure 6: Normalized distributions of the BDT output (left panel) for the
signal in the background with the result of the overtraining test (KS value)
and efficiency curve (right panel) for signal and background as a function of
the BDT cut value. We also show the variations of the significance obtained in
the analysis with the cut value. Both plots are generated considering the
first benchmark point BP-1.

Type | Z+jets | W+jets | ZZ+jets | WZ+jets | WW+jets | single-top | $t\overline{t}+jets$ | Total
---|---|---|---|---|---|---|---|---
Events | 97415 | 88496 | 5500 | 3220 | 774 | 8115 | 29711 | 233231
before BDT
Benchmark | Events | Optimal | Background | Background | Signal | Signal | Significance
before BDT | BDT cut | efficiency | events | efficiency | events
BP-1 | 45 | 0.41 | 3.98$\times 10^{-6}$ | 1 | 0.1027 | 5 | 2.0
BP-2 | 49 | 0.40 | 7.99$\times 10^{-6}$ | 2 | 0.1098 | 5 | 2.0

Table 4: Cut-flow table for the signal and backgrounds. The first two rows
give the event numbers before applying the BDT cut, and the rest of the rows
contain information on signal events before and after the BDT cut (along with
the cut efficiencies), the optimal BDT cut values, the background events after
BDT cut along with the cut efficiency and the obtained significance for the
two benchmark points at $\mathcal{L}_{int}=300$fb-1.

In the first two rows of Table 4, we gave the event yields for each background
before BDT analysis and then we presented the event yields for signal (both
benchmark points) along with the numbers after BDT analysis, the optimal BDT
cut value and the obtained significance.

Based on this optimized setup we scanned the model parameter space to obtain a
$2\sigma$ exclusion limit. We note that the production of the heavy mediators
($\chi_{2},\;\psi^{\pm}$) occurs through electroweak processes, but the
dependence of the coupling $y$ in the production cross-section of the dark
matter candidate ($\chi_{1}$) is induced via the decay widths (or, branching
fractions) of the mediators as shown in Eqs. (3),(4) and (5). In section 4, we
discussed that the dark matter phenomenology can be presented in the plane of
the coupling ($y$) and the doublet mass ($m_{\psi}$). In Figure 7, we compiled
our results from the collider analysis along with the results from DM
phenomenology in the same plane to estimate the current limits on our model
parameters in the prompt search context. In Figure 7, the region on the left
of the brown dashed contour represents the region excluded by our analysis at
$300$ fb-1 of luminosity. We also presented the results from the prompt search
as performed in CMS:2018szt  in the context of supersymmetric charginos and
neutralinos in the purple dashed contour. For comparison, we also presented
the analysis results from the displaced vertex + missing momentum search from
Calibbi:2018fqf . We see that our analysis can probe a larger area in the
parameter space when the coupling is substantially large to trigger a prompt
decay of the mediators. We also presented the result for a luminosity of
$3000$ fb-1 to project the searching prospect of HL-LHC. We found that while
at $300$ fb-1 luminosity the bound on the doublet mass can reach up to $880$
GeV, with the increased luminosity of 3000 fb-1 it goes beyond $1235$ GeV.

Figure 7: Exclusion limits at 14 TeV c.m. energy at LHC with
$\mathcal{L}_{int}=300$ fb-1 (orange-shaded region) and $3000$ fb-1 (blue-
shaded region) for a DM mass of 12 keV. The shaded region in cyan and purple
are excluded by the previous displaced vertex search (with $300$ fb-1) and
prompt search at LHC (with 35.9 fb-1). Also, contour lines from Figure 1 are
also shown to indicate the permissible region from the relic density
constraint in the non-standard cosmological scenario in the SDDM model
parameter space. The blue-colored stars represents the three benchmark points
discussed in section 4.

##  6 Summary and conclusion

In this work, we examine a simple extension of the SM with a singlet and a
doublet fermion, where the lightest neutral particle serves as a dark matter
candidate. Due to its gauge interaction, the doublet part thermalizes with the
SM particles in the early Universe. On the contrary, the dark matter candidate
cannot do the same and remains in the primordial soup with negligible or no
presence. In this scenario, gradual dark matter production occurs via the
decay of heavy doublet fermion through a tiny Yukawa coupling or the singlet
doublet mixing angle such that the DM never reaches thermal equilibrium. Non-
thermal freeze-in dark matter is one of the exciting possibilities explaining
the non-observation of any signature yet at the different ground or space-
based DM detection experiments, owing to this feeble interaction.

Interestingly, the long-lived particle with displaced vertex signature
provides some handle for the BSM models with the heavy mediator and light dark
matter. In addition, the lightweight dark matter with typical mass scale
$\mathcal{O}(10)$ keV can alleviate the small scale issue of $\Lambda$CDM by
suppressing the structure formation at that scale as a result of the warmness
of dark matter. Moreover, such dark matter offers the platform to be tested in
various cosmological and astrophysical probes.

Lacking evidence of the energy content of the Universe in the pre-BBN era, we
consider the possibility of non-standard cosmology and discuss its effect on
the evolution of dark matter. We have shown that the fast-expanding Universe
demands a larger interaction rate than the standard scenario (radiation-
dominated Universe) to satisfy the relic density constraint.

As a fallout of this requirement, not only the dark matter phenomenology but
one needs to relook at the corresponding collider search strategy.
Characteristic long-lived particle searches at the collider experiment and the
displaced vertex signature can no more constrain this scenario effectively. In
this work, we construct the extended parameter space that can satisfy the dark
matter relic density and build a suitable search at the LHC that can
effectively explore a large part of these parameters.

After copious production of dark sector heavy fermions at the LHC, they
promptly decay into the dark matter candidate associated with a heavy gauge
boson or a Higgs boson. The heaviness of these mother particles with
lightweight dark matter ensures the production of boosted hadronic jets from
the final decay of such bosons. We capitalize on such inherent features to
build our search strategy comprised of two boosted jets and the substantial
MET. These boosted fat jets bears characteristic substructure pattern
different from single prong large radius QCD jets, which can help us probe
through this channel, prevailing over the extensive QCD background. We have
also adopted multivariate analysis based on the boosted decision tree with the
help of twelve kinematic and boosted jet variables to provide the optimum
separation between signal and background. Our analysis puts severe constraints
on the vast plane of Yukawa coupling and heavy particle mass at 14 TeV LHC,
for $\mathcal{L}_{\text{int}}$ = 300 fb-1 and 3000 fb-1. Most of the region of
our parameter space remains inaccessible for the displaced vertex constraint.

###### Acknowledgements.

This work is supported by the Physical Research Laboratory (PRL), Department
of Space, Government of India. Part of the computational work is performed
using the HPC resources (Vikram-100 HPC) and TDP project at PRL. Part of the
work of SK is performed using the HPC resources (Koshambi) at the BITS Pilani,
K.K. Birla Goa Campus, mentained by the CSIS department. SK would like to
acknowledge the fellowship support of the BITS Pilani.

## References

  * (1) Y. Sofue and V. Rubin, _Rotation curves of spiral galaxies_ , _Ann. Rev. Astron. Astrophys._ 39 (2001) 137–174, [astro-ph/0010594].
  * (2) D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones et al., _A direct empirical proof of the existence of dark matter_ , _Astrophys. J. Lett._ 648 (2006) L109–L113, [astro-ph/0608407].
  * (3) G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley et al., _Nine-year wilkinson microwave anisotropy probe ( wmap ) observations: Cosmological parameter results_ , _The Astrophysical Journal Supplement Series_ 208 (Sep, 2013) 19.
  * (4) Planck collaboration, N. Aghanim et al., _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6, [1807.06209].
  * (5) A. M. Green, _Effect of halo modeling on WIMP exclusion limits_ , _Phys. Rev. D_ 66 (2002) 083003, [astro-ph/0207366].
  * (6) C.-F. Chang, X.-G. He and J. Tandean, _Two-Higgs-Doublet-Portal Dark-Matter Models in Light of Direct Search and LHC Data_ , _JHEP_ 04 (2017) 107, [1702.02924].
  * (7) C.-F. Chang, X.-G. He and J. Tandean, _Exploring Spin-3/2 Dark Matter with Effective Higgs Couplings_ , _Phys. Rev. D_ 96 (2017) 075026, [1704.01904].
  * (8) L. Visinelli, _(Non-)thermal production of WIMPs during kination_ , _Symmetry_ 10 (2018) 546, [1710.11006].
  * (9) G. Arcadi, M. Lindner, F. S. Queiroz, W. Rodejohann and S. Vogl, _Pseudoscalar Mediators: A WIMP model at the Neutrino Floor_ , _JCAP_ 03 (2018) 042, [1711.02110].
  * (10) S. Choubey, S. Khan, M. Mitra and S. Mondal, _Singlet-Triplet Fermionic Dark Matter and LHC Phenomenology_ , _Eur. Phys. J. C_ 78 (2018) 302, [1711.08888].
  * (11) A. Reinert and M. W. Winkler, _A Precision Search for WIMPs with Charged Cosmic Rays_ , _JCAP_ 01 (2018) 055, [1712.00002].
  * (12) J. A. Evans, S. Gori and J. Shelton, _Looking for the WIMP Next Door_ , _JHEP_ 02 (2018) 100, [1712.03974].
  * (13) M. Garny, J. Heisig, M. Hufnagel and B. Lülf, _Top-philic dark matter within and beyond the WIMP paradigm_ , _Phys. Rev. D_ 97 (2018) 075002, [1802.00814].
  * (14) C. Blanco, M. Escudero, D. Hooper and S. J. Witte, _Z’ mediated WIMPs: dead, dying, or soon to be detected?_ , _JCAP_ 11 (2019) 024, [1907.05893].
  * (15) A. Bhardwaj, A. Das, P. Konar and A. Thalapillil, _Looking for Minimal Inverse Seesaw scenarios at the LHC with Jet Substructure Techniques_ , _J. Phys. G_ 47 (2020) 075002, [1801.00797].
  * (16) A. Bhardwaj, P. Konar, T. Mandal and S. Sadhukhan, _Probing the inert doublet model using jet substructure with a multivariate analysis_ , _Phys. Rev. D_ 100 (2019) 055040, [1905.04195].
  * (17) P. Konar, A. Mukherjee, A. K. Saha and S. Show, _Linking pseudo-Dirac dark matter to radiative neutrino masses in a singlet-doublet scenario_ , _Phys. Rev. D_ 102 (2020) 015024, [2001.11325].
  * (18) P. Konar, A. Mukherjee, A. K. Saha and S. Show, _A dark clue to seesaw and leptogenesis in a pseudo-Dirac singlet doublet scenario with (non)standard cosmology_ , _JHEP_ 03 (2021) 044, [2007.15608].
  * (19) L. Heurtier and H. Partouche, _Spontaneous Freeze Out of Dark Matter From an Early Thermal Phase Transition_ , _Phys. Rev. D_ 101 (2020) 043527, [1912.02828].
  * (20) M. Habermehl, M. Berggren and J. List, _WIMP Dark Matter at the International Linear Collider_ , _Phys. Rev. D_ 101 (2020) 075053, [2001.03011].
  * (21) C.-Y. Xing and S.-H. Zhu, _Dark Matter Freeze-Out via Catalyzed Annihilation_ , _Phys. Rev. Lett._ 127 (2021) 061101, [2102.02447].
  * (22) D. Borah, S. Jyoti Das, A. K. Saha and R. Samanta, _Probing WIMP dark matter via gravitational waves’ spectral shapes_ , _Phys. Rev. D_ 106 (2022) L011701, [2202.10474].
  * (23) G. Belanger, A. Mjallal and A. Pukhov, _WIMP and FIMP dark matter in the inert doublet plus singlet model_ , _Phys. Rev. D_ 106 (2022) 095019, [2205.04101].
  * (24) E. U. Ginés, O. Mena and S. J. Witte, _Revisiting constraints on WIMPs around primordial black holes_ , _Phys. Rev. D_ 106 (2022) 063538, [2207.09481].
  * (25) N. Bernal and Y. Xu, _WIMPs during reheating_ , _JCAP_ 12 (2022) 017, [2209.07546].
  * (26) S. Kundu, A. Guha, P. K. Das and P. S. B. Dev, _EFT analysis of leptophilic dark matter at future electron-positron colliders in the mono-photon and mono-Z channels_ , _Phys. Rev. D_ 107 (2023) 015003, [2110.06903].
  * (27) A. D. Medina, N. I. Mileo, A. Szynkman and S. A. Tanco, _Elusive muonic WIMP_ , _Phys. Rev. D_ 106 (2022) 075018, [2112.09103].
  * (28) B. Tallman, A. Boone, A. Vijayakumar, F. Lopez, S. Apata, J. Martinez et al., _Potential for definitive discovery of a 70 GeV dark matter WIMP with only second-order gauge couplings_ , 2210.15019.
  * (29) S. Kang, A. Kar and S. Scopel, _Halo-independent bounds on the non-relativistic effective theory of WIMP-nucleon scattering from direct detection and neutrino observations_ , 2212.05774.
  * (30) K. Dutta, A. Ghosh, A. Kar and B. Mukhopadhyaya, _MeV to multi-TeV thermal WIMPs are all observationally allowed_ , 2212.09795.
  * (31) LUX collaboration, D. S. Akerib et al., _Results from a search for dark matter in the complete LUX exposure_ , _Phys. Rev. Lett._ 118 (2017) 021303, [1608.07648].
  * (32) PandaX collaboration, H. Zhang et al., _Dark matter direct search sensitivity of the PandaX-4T experiment_ , _Sci. China Phys. Mech. Astron._ 62 (2019) 31011, [1806.02229].
  * (33) XENON collaboration, E. Aprile et al., _Dark Matter Search Results from a One Ton-Year Exposure of XENON1T_ , _Phys. Rev. Lett._ 121 (2018) 111302, [1805.12562].
  * (34) MAGIC, Fermi-LAT collaboration, M. L. Ahnen et al., _Limits to Dark Matter Annihilation Cross-Section from a Combined Analysis of MAGIC and Fermi-LAT Observations of Dwarf Satellite Galaxies_ , _JCAP_ 02 (2016) 039, [1601.06590].
  * (35) CMS collaboration, S. Chatrchyan et al., _Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC_ , _Phys. Lett._ B716 (2012) 30–61, [1207.7235].
  * (36) ATLAS collaboration, G. Aad et al., _Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC_ , _Phys. Lett._ B716 (2012) 1–29, [1207.7214].
  * (37) L. J. Hall, K. Jedamzik, J. March-Russell and S. M. West, _Freeze-In Production of FIMP Dark Matter_ , _JHEP_ 03 (2010) 080, [0911.1120].
  * (38) P. Konar, R. Roshan and S. Show, _Freeze-in dark matter through forbidden channel in U(1)B-L_ , _JCAP_ 03 (2022) 021, [2110.14411].
  * (39) P. Ghosh, P. Konar, A. K. Saha and S. Show, _Self-interacting freeze-in dark matter in a singlet doublet scenario_ , _JCAP_ 10 (2022) 017, [2112.09057].
  * (40) N. Chakrabarty, P. Konar, R. Roshan and and S. Show, _Thermally corrected masses and freeze-in dark matter: a case study_ , 2206.02233.
  * (41) F. D’Eramo, N. Fernandez and S. Profumo, _When the Universe Expands Too Fast: Relentless Dark Matter_ , _JCAP_ 05 (2017) 012, [1703.04793].
  * (42) F. D’Eramo, N. Fernandez and S. Profumo, _Dark Matter Freeze-in Production in Fast-Expanding Universes_ , _JCAP_ 02 (2018) 046, [1712.07453].
  * (43) L. Calibbi, L. Lopez-Honorez, S. Lowette and A. Mariotti, _Singlet-Doublet Dark Matter Freeze-in: LHC displaced signatures versus cosmology_ , _JHEP_ 09 (2018) 037, [1805.04423].
  * (44) J. M. No, P. Tunney and B. Zaldivar, _Probing Dark Matter freeze-in with long-lived particle signatures: MATHUSLA, HL-LHC and FCC-hh_ , _JHEP_ 03 (2020) 022, [1908.11387].
  * (45) M. Viel, G. D. Becker, J. S. Bolton and M. G. Haehnelt, _Warm dark matter as a solution to the small scale crisis: New constraints from high redshift Lyman- $\alpha$ forest data_, _Phys. Rev. D_ 88 (2013) 043502, [1306.2314].
  * (46) C. Yèche, N. Palanque-Delabrouille, J. Baur and H. du Mas des Bourboux, _Constraints on neutrino masses from Lyman-alpha forest power spectrum with BOSS and XQ-100_ , _JCAP_ 06 (2017) 047, [1702.03314].
  * (47) V. Iršič et al., _New Constraints on the free-streaming of warm dark matter from intermediate and small scale Lyman- $\alpha$ forest data_, _Phys. Rev. D_ 96 (2017) 023522, [1702.01764].
  * (48) P. Bode, J. P. Ostriker and N. Turok, _Halo formation in warm dark matter models_ , _Astrophys. J._ 556 (2001) 93–107, [astro-ph/0010389].
  * (49) M. R. Lovell, V. Eke, C. S. Frenk, L. Gao, A. Jenkins, T. Theuns et al., _The haloes of bright satellite galaxies in a warm dark matter universe_ , _Monthly Notices of the Royal Astronomical Society_ 420 (jan, 2012) 2318–2324.
  * (50) C. E. Yaguna, _Singlet-Doublet Dirac Dark Matter_ , _Phys. Rev._ D92 (2015) 115002, [1510.06151].
  * (51) J. Fiaschi, M. Klasen and S. May, _Singlet-doublet fermion and triplet scalar dark matter with radiative neutrino masses_ , _JHEP_ 05 (2019) 015, [1812.11133].
  * (52) D. Restrepo, A. Rivera and W. Tangarife, _Singlet-Doublet Dirac Dark Matter and Neutrino Masses_ , _Phys. Rev._ D100 (2019) 035029, [1906.09685].
  * (53) G. Arcadi, _2HDM portal for Singlet-Doublet Dark Matter_ , _Eur. Phys. J._ C78 (2018) 864, [1804.04930].
  * (54) S. Esch, M. Klasen and C. E. Yaguna, _A singlet doublet dark matter model with radiative neutrino masses_ , _JHEP_ 10 (2018) 055, [1804.03384].
  * (55) N. Maru, N. Okada and S. Okada, _Fermionic Minimal Dark Matter in 5D Gauge-Higgs Unification_ , _Phys. Rev._ D96 (2017) 115023, [1801.00686].
  * (56) N. Maru, T. Miyaji, N. Okada and S. Okada, _Fermion Dark Matter in Gauge-Higgs Unification_ , _JHEP_ 07 (2017) 048, [1704.04621].
  * (57) Q.-F. Xiang, X.-J. Bi, P.-F. Yin and Z.-H. Yu, _Exploring Fermionic Dark Matter via Higgs Boson Precision Measurements at the Circular Electron Positron Collider_ , _Phys. Rev._ D97 (2018) 055004, [1707.03094].
  * (58) T. Abe, _Effect of CP violation in the singlet-doublet dark matter model_ , _Phys. Lett._ B771 (2017) 125–130, [1702.07236].
  * (59) S. Banerjee, S. Matsumoto, K. Mukaida and Y.-L. S. Tsai, _WIMP Dark Matter in a Well-Tempered Regime: A case study on Singlet-Doublets Fermionic WIMP_ , _JHEP_ 11 (2016) 070, [1603.07387].
  * (60) S. Horiuchi, O. Macias, D. Restrepo, A. Rivera, O. Zapata and H. Silverwood, _The Fermi-LAT gamma-ray excess at the Galactic Center in the singlet-doublet fermion dark matter model_ , _JCAP_ 1603 (2016) 048, [1602.04788].
  * (61) L. Calibbi, A. Mariotti and P. Tziveloglou, _Singlet-Doublet Model: Dark matter searches and LHC constraints_ , _JHEP_ 10 (2015) 116, [1505.03867].
  * (62) C. Cheung and D. Sanford, _Simplified Models of Mixed Dark Matter_ , _JCAP_ 1402 (2014) 011, [1311.5896].
  * (63) T. Cohen, J. Kearney, A. Pierce and D. Tucker-Smith, _Singlet-Doublet Dark Matter_ , _Phys. Rev._ D85 (2012) 075003, [1109.2604].
  * (64) R. Enberg, P. J. Fox, L. J. Hall, A. Y. Papaioannou and M. Papucci, _LHC and dark matter signals of improved naturalness_ , _JHEP_ 11 (2007) 014, [0706.0918].
  * (65) F. D’Eramo, _Dark matter and Higgs boson physics_ , _Phys. Rev._ D76 (2007) 083522, [0705.4493].
  * (66) B. Barman, D. Borah, P. Ghosh and A. K. Saha, _Flavoured gauge extension of singlet-doublet fermionic dark matter: neutrino mass, high scale validity and collider signatures_ , _JHEP_ 10 (2019) 275, [1907.10071].
  * (67) A. Dutta Banik, A. K. Saha and A. Sil, _Scalar assisted singlet doublet fermion dark matter model and electroweak vacuum stability_ , _Phys. Rev._ D98 (2018) 075013, [1806.08080].
  * (68) B. Barman, S. Bhattacharya, P. Ghosh, S. Kadam and N. Sahu, _Fermion Dark Matter with Scalar Triplet at Direct and Collider Searches_ , _Phys. Rev._ D100 (2019) 015027, [1902.01217].
  * (69) S. Bhattacharya, P. Ghosh, N. Sahoo and N. Sahu, _Mini Review on Vector-Like Leptonic Dark Matter, Neutrino Mass, and Collider Signatures_ , _Front.in Phys._ 7 (2019) 80, [1812.06505].
  * (70) S. Bhattacharya, N. Sahoo and N. Sahu, _Minimal vectorlike leptonic dark matter and signatures at the LHC_ , _Phys. Rev._ D93 (2016) 115040, [1510.02760].
  * (71) S. Bhattacharya, N. Sahoo and N. Sahu, _Singlet-Doublet Fermionic Dark Matter, Neutrino Mass and Collider Signatures_ , _Phys. Rev._ D96 (2017) 035010, [1704.03417].
  * (72) D. Restrepo, A. Rivera, M. Sánchez-Peláez, O. Zapata and W. Tangarife, _Radiative Neutrino Masses in the Singlet-Doublet Fermion Dark Matter Model with Scalar Singlets_ , _Phys. Rev._ D92 (2015) 013005, [1504.07892].
  * (73) A. Freitas, S. Westhoff and J. Zupan, _Integrating in the Higgs Portal to Fermion Dark Matter_ , _JHEP_ 09 (2015) 015, [1506.04149].
  * (74) G. Cynolter, J. Kovács and E. Lendvai, _Doublet–singlet model and unitarity_ , _Mod. Phys. Lett._ A31 (2016) 1650013, [1509.05323].
  * (75) S. Bhattacharya, B. Karmakar, N. Sahu and A. Sil, _Unifying the flavor origin of dark matter with leptonic nonzero $\theta_{13}$_, _Phys. Rev._ D93 (2016) 115041, [1603.04776].
  * (76) S. Bhattacharya, B. Karmakar, N. Sahu and A. Sil, _Flavor origin of dark matter and its relation with leptonic nonzero $\theta_{13}$ and Dirac CP phase $\delta$_, _JHEP_ 05 (2017) 068, [1611.07419].
  * (77) J.-W. Wang, X.-J. Bi, P.-F. Yin and Z.-H. Yu, _Impact of Fermionic Electroweak Multiplet Dark Matter on Vacuum Stability with One-loop Matching_ , _Phys. Rev._ D99 (2019) 055009, [1811.08743].
  * (78) T. Abe and R. Sato, _Current status and future prospects of the singlet-doublet dark matter model with CP-violation_ , _Phys. Rev._ D99 (2019) 035012, [1901.02278].
  * (79) B. Barman, A. Dutta Banik and A. Paul, _Singlet-doublet fermionic dark matter and gravitational waves in a two-Higgs-doublet extension of the Standard Model_ , _Phys. Rev. D_ 101 (2020) 055028, [1912.12899].
  * (80) S. D. Thomas and J. D. Wells, _Phenomenology of Massive Vectorlike Doublet Leptons_ , _Phys. Rev. Lett._ 81 (1998) 34–37, [hep-ph/9804359].
  * (81) M. Cirelli and A. Strumia, _Minimal Dark Matter: Model and results_ , _New J. Phys._ 11 (2009) 105005, [0903.3381].
  * (82) A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, _FeynRules 2.0 - A complete toolbox for tree-level phenomenology_ , _Comput. Phys. Commun._ 185 (2014) 2250–2300, [1310.1921].
  * (83) C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter, _UFO - The Universal FeynRules Output_ , _Comput. Phys. Commun._ 183 (2012) 1201–1214, [1108.2040].
  * (84) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., _The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations_ , _JHEP_ 07 (2014) 079, [1405.0301].
  * (85) M. L. Mangano, M. Moretti, F. Piccinini and M. Treccani, _Matching matrix elements and shower evolution for top-quark production in hadronic collisions_ , _JHEP_ 01 (2007) 013, [hep-ph/0611129].
  * (86) DELPHES 3 collaboration, J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens et al., _DELPHES 3, A modular framework for fast simulation of a generic collider experiment_ , _JHEP_ 02 (2014) 057, [1307.6346].
  * (87) M. Cacciari, G. P. Salam and G. Soyez, _FastJet User Manual_ , _Eur. Phys. J. C_ 72 (2012) 1896, [1111.6097].
  * (88) Y. L. Dokshitzer, G. D. Leder, S. Moretti and B. R. Webber, _Better jet clustering algorithms_ , _JHEP_ 08 (1997) 001, [hep-ph/9707323].
  * (89) R. Brun and F. Rademakers, _Root — an object oriented data analysis framework_ , _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 389 (1997) 81–86.
  * (90) A. Hocker et al., _TMVA - Toolkit for Multivariate Data Analysis_ , physics/0703039.
  * (91) S. D. Ellis, C. K. Vermilion and J. R. Walsh, _Recombination algorithms and jet substructure: Pruning as a tool for heavy particle searches_ , _Phys. Rev. D_ 81 (May, 2010) 094023.
  * (92) A. J. Larkoski, G. P. Salam and J. Thaler, _Energy Correlation Functions for Jet Substructure_ , _JHEP_ 06 (2013) 108, [1305.0007].
  * (93) J. Thaler and K. Van Tilburg, _Identifying Boosted Objects with N-subjettiness_ , _JHEP_ 03 (2011) 015, [1011.2268].
  * (94) CMS collaboration, A. M. Sirunyan et al., _Combined search for electroweak production of charginos and neutralinos in proton-proton collisions at $\sqrt{s}=$ 13 TeV_, _JHEP_ 03 (2018) 160, [1801.03957].