# Dynamical quantum phase transitions from quantum optics perspective

Jakub Zakrzewski  Instytut Fizyki Teoretycznej, Wydział Fizyki, Astronomii i
Informatyki Stosowanej, Uniwersytet Jagielloński, ulica profesora Stanisława
Łojasiewicza 11, PL-30-348 Kraków, Poland  Mark Kac Complex Systems Research
Center, Uniwersytet Jagielloński, Kraków, Poland

###### Abstract

In this work dedicated to Professor Iwo Białynicki-Birula on the occasion of
his 90th birthday, I attempt to show that dynamical quantum phase transitions
observed as singularities in the Loschmidt rate dynamics bear a close
resemblance to standard Rabi oscillations known from the dynamics of two-level
systems. For some many-body systems, this analogy may go even further, and the
behaviour observed for example for transverse Ising chain can be directly
mapped to such simple dynamics. A simple link between Loschmidt echo
singularities and quantum scars is also suggested.

##  I Introduction

The physics of complex systems may sometimes be understood (in particular
limiting cases) in a simple, enlightening form. This has been often
demonstrated in quantum optics, one of the many areas of Iwo Białynicki-Birula
outstanding contributions. As a scientific grandson of Iwo Białynicki-Birula,
I had relatively small overlap in scientific interests with him, our paths
crossed, for a moment, in the studies of nonspreading wave-packets [1, 2, 3,
4]. Still, however, I profited a lot from occasional conversations as well as
participation, from time to time, in unusually vivid seminars with his active
participation. Often his aim was to find a simple picture of the presented
effects. In this contribution I consider briefly two cases from studies of
nonequilibrium dynamics of many body systems which may be, in my opinion,
understood in simple terms: the dynamical quantum phase transitions (DQPT) [5,
6] and quantum many-body scars (QMBS) dynamics [7].

The simplest definition of DQPT consists of a sudden quench in which the
system is prepared in thr ground state $\Psi\rangle$ of a parameter dependent
Hamiltonian $H(\lambda=0)$, and $\lambda$ is suddenly changed to other value.
It has been observed that often, if the change of $\lambda$ moves the
Hamiltonian into a different phase, the time dynamics with $H(\lambda)$ of the
now nonstationary state after the quench reveals the so called Loschmidt echo
singularities. Their appearance is neither a necessary nor a sufficient
condition for the phase transition between $H(0)$ and $H(\lambda)$. Still
predominantly lack of singularities occurs if no phase transition is crossed
while changing $\lambda$ and vice versa.

Dynamical detection of quantum scars [7] is in some sense similar. One
prepares an initial nonstationary state for a many-body system described by
$H(\lambda)$ . When this initial state has a significant overlap with a few
almost equally spaced in energy eigenstates of $H(\lambda)$ \- the time
evolution of observables reveals oscillations even in a weakly ergodic regime,
i.e. when the dynamics of a typical generic state will lead to thermalization.

Both these phenomena, while of current interest, can be simply explained by
indentifying the “essential state model” i.e. a minimal approximate level
scheme allowing one to simulate the dynamics. Let us consider first consider
DQPT in the seminal example of a transverse Ising model.

##  II DQPT in transverse Ising model

The first work on DQPT [5] considers the transverse Ising model with
Hamiltonian of the form

$H=-\frac{1}{2}\sum_{i}\sigma^{z}_{i}\sigma^{z}_{i+1}-\frac{g}{2}\sum_{i}\sigma^{x}_{i}$
(1)

where $g$ is the strength of the magnetic field pointing in $Ox$ direction.
For small $g$ the interactions favor ferromagnetic (FM) (along $Oz$)
orientation, with two degenerate (in the thermodynamic limit) ground states
given for $g\rightarrow 0$ by
$\ket{\psi(\pm)^{z}}=\prod_{i}\ket{\pm}^{z}_{i}$, where $\ket{\pm}^{z}_{i}$
denotes eigenvectors of $\sigma^{z}_{i}$. The phase transition from FM to
paramagnetic order occurs for $g=1$, for large $g$, the unique ground state is
well approximated by $\ket{\varphi}=\prod_{i}\ket{+}^{x}_{i}$ with
$\ket{\pm}^{x}_{i}$ bering eigenvectors of $\sigma^{x}_{i}$.

Let $g$ serve as a parameter $\lambda$ and let us start with the ground state
of (1) for small $g$, say with $\ket{\Psi}=\ket{\psi(+)^{z}}$ and abruptly
change $g$ to large, positive value. In the new Hamiltonian the term
proportional to $g$ will dominate while the first interaction term will be a
small perturbation. The initial state can be then decomposed in the basis of
$\sigma^{x}_{i}$ eigenvectors as:

$\ket{\Psi(0)}=\prod_{i}\frac{1}{\sqrt{2}}(\ket{+}^{x}_{i}+\ket{-}^{x}_{i}).$
(2)

The initial state after a quench is, therefore, a product of two-state
combinations with coefficients of equal magnitude (the phase does not affect
the result). The subsequent time evolution, still neglecting interactions in
the final Hamiltonian yields

$\ket{\Psi(t)}=\prod_{i}\frac{1}{\sqrt{2}}\left(\ket{+}^{x}_{i}\exp(igt/2)+\ket{-}^{x}_{i}\exp(-igt/2)\right).$
(3)

By a survival probability (fidelity, return amplitude, or Loschmidt echo) one
calls (depending on the context) the squared overlap of initial and time
evolved state, ${\cal L}(t)\equiv|\langle\Psi(0)\ket{\Psi(t)}|^{2}$. Further
one may define [6] the rate function, $r(t)$ via ${\cal L}(t)=\exp{-Lr(t)}$
where $L$ is the system size (number of degrees of freedom). Such a measure
has a good thermodynamic limits. Singularities in $r(t)$ time dependence,
often referred to as Loschmidt echo singularities, are the defining features
of DQPT.

Let us immediately consider the simplified example above. The squared overlap
${\cal L}(t)$ becomes simply ${\cal L}(t)=\cos^{2L}(gt/2)$ and size-
independent rate $r(t)$ reveals singularities whenever the cosine function
vanishes i.e. for $t^{*}=(2k+1)\pi/g$ for integer $k$. This example clearly
shows that Rabi-type oscillations are the real origin of rate function
singularities in this case.

One can complain that the situation described above is too simplified,
singularities in form of finite cusps appear also for smaller changes of $g$
where the approximations made by us would not work fully. Then, however, one
can use Jordan-Wigner transformation into the noninteracting fermion system,
as in the original DQPT letter [5] and observe similar “two-level” dynamics
for a given $k$ as different $k$ decouple.

##  III Other examples

Our model, however, helps to explain also other situations. In fact, as
reviewed in [6] 2-band topological noninteracting models lead to exactly to
the same dynamics. Again here, due to lack of interactions, different $k$
values can be treated independently leading to similar estimate of critical
times at which singularities appear. Let us stress that while these
singularities are essential for phase-transition language application, they
seems just to be due to vanishing overlaps between the initial and time
evolved wavepacket.

Consider now the situation in which we make an abrupt quench within the same
phase then by definition the ground state changes slowly and continuously with
the change of the parameter for a finite system. So it is quite justified to
assume that the ground state at say $\lambda=0$ expands, in eigenstates
$\\{\ket{\psi_{k}}\\}$ of $H(\lambda)$ as

$\ket{\Psi}=\alpha_{0}\ket{\psi_{0}}+\sum_{k}\alpha_{k}\ket{\psi_{k}},$ (4)

with $|\alpha_{0}|\gg|\alpha_{k}$ for $k>0$. Then the survival probability
(Loschmidt echo) is dominated by a large $|\alpha_{0}|^{2}$ term. The
situation is more subtle in the thermodynamic limit due to Anderson
catastrophe. Still then one may expect that many eigenstates at final
parameter value contribute to the initial wavepacket, leading to many
superimposed oscillations at different frequencies. In such a situation rate
function should not reveal strong maxima (not speaking of singularities).

Note that situation is markedly different when the phase transition is crossed
in $\lambda$ as then, for an Ising system, via symmetry, as described above,
two eigenstates contribute significantly to the sum, leading to Rabi
oscillations at half of their energy difference (per site).

The discussion up till now was concentrated on spin-1/2 models leading to
simple Rabi oscillations. This might be a transverse Ising chain but also,
e.g. a quantum dot dynamics [8]. As known from quantum optics, Rabi
oscillations generalize to quantum revivals appearing when several equally
spaced levels are populated [9]. Here again one may expect that between
consecutive revivals minima of the survival probability lead to maxima (and
possibly cusps) of the Loschmidt rate functions. In a many body system, even a
more general sitation was experimentally realized many years ago for
interacting bosons in an optical lattice [10]. Initially bosons were kept in a
shallow lattice, then, abruptly, the lattice height was dramatically increased
separating different lattice sites. Within each site the initial, almost
coherenty state was a superposition of states with different site occupations
separated by the quadratic progression in the interaction strength $U$ (within
the tight binding Bose-Hubbard description) and the revivals were observed.
The corresponding Loschmidt rates reveal singularities (or maxima) as
discussed in detail recently [11] in the DQPT language at times when the
overlap between initial and time evolved state is minimal i.e. roughly in the
middle between two consecutive revivals.

##  IV Quantum scars

Recently an interesting manifestation of ergodicity breaking as persistent
oscillations for certain initial states was discovered experimentally with
ultracold Rydberg atoms [7]. This feature is due to the presence of few
atypical, almost equally spaced eigenstates – the so-called quantum many-body
scars (QMBS) [12, 13] that are embedded in the otherwise thermal spectrum of a
quantum many-body system. For initial states with high overlap with a few
QMBS, one observes long-lived oscillations of observables, whereas for generic
initial conditions the system quickly approaches the thermal equilibrium
state. The same oscillations should be present in the survival probability
leading, in turn, to maxima of Loschmidt echo rate function if the data are
interpretted in that way.

QMBS borrowed their name from single particle quantum chaos studies where
“quantum scar” described enhanced probability of eigenstates or wavepackets in
the regions of space occupied by unstable periodic orbits [14] \- in close
relation to semiclassical periodic orbits quantization [15, 16]. Then also the
concept of scarring by symetries were developed in the contex of hydrogen atom
in magnetic field studies [17]. Similar symmetry concepts were used for
nonergodic states construction in many-body case see, e.g. [18, 19].

Such QMBS may be easily imagined as having the origin in approximate
decoupling of a (not always apparent) single degree of freedom from the other
degrees of freedom. If this single degree is locally described by an harmonic
oscilator (or an angular momentum) then the corresponding eigenstates are
equidistant - their weak coupling to the remaining states preserves the energy
structure. Now, if by accident (or cleverness) the initial state is prepared
as a linear combination of those selected states (or if it has sufficiently
large overlap on at least a few of them) one may naturally expect a persistent
oscillation in the time dynamics. Let us mention also that quench dynamics and
Rabi oscillations resulting from excitations of two or more localized
integrals of motion in the context of many-body localization have been studied
recently [20, 21]. The localized, almost decoupled family of states may not be
easy to identify, one may try to identify it e.g., by adiabatic following from
some analytic limit [22] or via purely numerical approaches including
artificial intelligence [23].

##  V Conclusions

DQPT form a very intriguing interpretation of rapid quantum quenches. On the
other hand, the signatures of DQPT in the form of singularities of Loschmidt
echo rate functions, appear to a large extend due to the very definition of
this rate. The survival probability (Loschmidt echo) itself reveals no
singularities but rather smooth oscillations (or revivals in more complicated
cases).

Let us stress that the mechanism presented above considers rather simple
examples. For more complicated cases one may consider Loschmidt echo as coming
back not to a single ground state but to the degenerate manifold if it exists
[6]. After the first draft of this note was completed, a related work appeared
giving a more general picture of DQPT [24]. It has been brought also to our
attention that similar to DQPT cusp structures may appear in single particle
dynamics [25, 26].

##  VI Acknowledgments

This note was born as a reflection on a seminar given in Krakow by Tadeusz
Domański. His great presentation skills are acknowledged. I am grateful to
Dominique Delande, Mateusz Ła̧cki, Anatoly Polkovnikov, and Tommaso Roscilde
for discussions. The work of J.Z have been realized within the Opus grant
2019/35/B/ST2/00034, financed by National Science Centre (Poland). The
research has also been supported by a grant from the Priority Research Area
DigiWorld under the Strategic Programme Excellence Initiative at Jagiellonian
University.

## References

  * Bialynicki-Birula  _et al._ [1994] I. Bialynicki-Birula, M. Kaliński, and J. H. Eberly, Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven rydberg electrons, Phys. Rev. Lett. 73, 1777 (1994).
  * Delande  _et al._ [1995] D. Delande, J. Zakrzewski, and A. Buchleitner, A wave packet can be a stationary state, Europhysics Letters 32, 107 (1995).
  * Delande and Zakrzewski [1998] D. Delande and J. Zakrzewski, Spontaneous emission of nondispersive rydberg wave packets, Phys. Rev. A 58, 466 (1998).
  * Buchleitner  _et al._ [2002] A. Buchleitner, D. Delande, and J. Zakrzewski, Non-dispersive wave packets in periodically driven quantum systems, Physics Reports 368, 409 (2002).
  * Heyl  _et al._ [2013] M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical quantum phase transitions in the transverse-field ising model, Phys. Rev. Lett. 110, 135704 (2013).
  * Heyl [2018] M. Heyl, Dynamical quantum phase transitions: a review, Reports on Progress in Physics 81, 054001 (2018).
  * Bernien  _et al._ [2017] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature 551, 579 (2017).
  * Wrześniewski  _et al._ [2022] K. Wrześniewski, I. Weymann, N. Sedlmayr, and T. Domański, Dynamical quantum phase transitions in a mesoscopic superconducting system, Phys. Rev. B 105, 094514 (2022).
  * Eberly  _et al._ [1980] J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Periodic spontaneous collapse and revival in a simple quantum model, Phys. Rev. Lett. 44, 1323 (1980).
  * Greiner  _et al._ [2002] M. Greiner, O. Mandel, T. W. Hänsch, and I. Bloch, Collapse and revival of the matter wave field of a Bose-Einstein condensate, Nature 419, 51 (2002).
  * Lacki and Heyl [2019] M. Lacki and M. Heyl, Dynamical quantum phase transitions in collapse and revival oscillations of a quenched superfluid, Phys. Rev. B 99, 121107 (2019).
  * Turner  _et al._ [2018a] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Weak ergodicity breaking from quantum many-body scars, Nature Physics 14, 745 (2018a).
  * Turner  _et al._ [2018b] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Quantum scarred eigenstates in a rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations, Phys. Rev. B 98, 155134 (2018b).
  * Heller [1984] E. J. Heller, Bound-state eigenfunctions of classically chaotic hamiltonian systems: Scars of periodic orbits, Phys. Rev. Lett. 53, 1515 (1984).
  * Gutzwiller [1971] M. C. Gutzwiller, Periodic orbits and classical quantization conditions, Journal of Mathematical Physics 12, 343 (1971).
  * Bogomolny [1988] E. Bogomolny, Smoothed wavefunctions of chaotic quantum systems, Physica D 31, 169 (1988).
  * Delande and Gay [1987] D. Delande and J. C. Gay, Scars of symmetries in quantum chaos, Phys. Rev. Lett. 59, 1809 (1987).
  * Shiraishi and Mori [2017] N. Shiraishi and T. Mori, Systematic construction of counterexamples to the eigenstate thermalization hypothesis, Phys. Rev. Lett. 119, 030601 (2017).
  * Moudgalya  _et al._ [2018] S. Moudgalya, S. Rachel, B. A. Bernevig, and N. Regnault, Exact excited states of nonintegrable models, Phys. Rev. B 98, 235155 (2018).
  * Benini  _et al._ [2021a] L. Benini, P. Naldesi, R. A. Römer, and T. Roscilde, Loschmidt echo singularities as dynamical signatures of strongly localized phases, New Journal of Physics 23, 023030 (2021a).
  * Benini  _et al._ [2021b] L. Benini, P. Naldesi, R. A. Römer, and T. Roscilde, Quench dynamics of quasi-periodic systems exhibiting rabi oscillations of two-level integrals of motion, Annals of Physics 435, 168545 (2021b), special Issue on Localisation 2020\.
  * Aramthottil  _et al._ [2022] A. S. Aramthottil, U. Bhattacharya, D. González-Cuadra, M. Lewenstein, L. Barbiero, and J. Zakrzewski, Scar states in deconfined ${\mathbb{z}}_{2}$ lattice gauge theories, Phys. Rev. B 106, L041101 (2022).
  * Szołdra  _et al._ [2022] T. Szołdra, P. Sierant, M. Lewenstein, and J. Zakrzewski, Unsupervised detection of decoupled subspaces: Many-body scars and beyond, Phys. Rev. B 105, 224205 (2022).
  * Van Damme  _et al._ [2022] M. Van Damme, J.-Y. Desaules, Z. Papić, and J. C. Halimeh, The anatomy of dynamical quantum phase transitions (2022), arXiv:2210.02453 .
  * Zhang and Yang [2016] J. M. Zhang and H.-T. Yang, Cusps in the quench dynamics of a bloch state, Europhysics Letters 114, 60001 (2016).
  * Zhang and Liu [2016] J. M. Zhang and Y. Liu, Fermi’s golden rule: its derivation and breakdown by an ideal model, European Journal of Physics 37, 065406 (2016).


<EMAIL_ADDRESS>
# Search for the production of dark gauge bosons in the framework of Einstein-
Cartan portal in the simulation of proton-proton collisions at $\sqrt{s}=13.6$
TeV

S. Elgammal  [  Centre for theoretical physics, The British University in
Egypt.

(August 28, 2024)

###### Abstract

In the present work, we study the possible production of the heavy neutral
dark gauge boson (A′) candidates, which are originated from a simplified model
based on the Einstein-Cartan gravity, in association with dark matter. This
study has been performed by studying events with dimuon plus missing
transverse energy produced in the simulated proton-proton collisions at the
Large Hadron Collider, at 13.6 TeV center of mass energy and integrated
luminosity of 52 fb-1 corresponding to the LHC RUN III circumstances. These
provide the most stringent upper limits on the masses for various particles in
the model as, spin-1 (A′), as well as the heavy mediator (torsion field).

††preprint: APS/123-QED

##  I Introduction

The Standard Model of particle physics SM has been tested during more than 40
years [1], and its predictions agree very well with all experimental
observations. However, the SM is nowadays considered as a low energy
manifestation of other theories realized at high energy, generically known as
BSM (Beyond the Standard Model) theories [2]. One motivation for BSM physics
is to have a unified theory for the electromagnetic, weak and strong
interactions, in a unique Grand Unified Theory (GUT) [3]. The Super-Symmetry
(SUSY) attempts to also include gravitation lead to models with extra spatial
dimensions. These BSM models typically predict the existence of new dark
particles at the TeV scale and higher.

The existence of heavy neutral bosons (Z′) is a feature of many extensions of
the Standard Model. They arise in extended gauge theories, including grand
unified theories (GUT) [4], and other models like left-right symmetric models
(LRM) [5]. A specific case is the sequential standard model (SSM), in which
the Z′ boson has the same coupling as the SM Z′ [6]. Model of extra dimensions
like Randall and Sundrum model (RS) [7] predicts the existence of heavy
Kaluza-Klein gravitons. Searches for these heavy dark neutral gauge bosons
have been performed at the CMS and ATLAS experiments, at the Large Hadron
Collider (LHC), with no evidences of their existence using the full RUN II
period of the LHC data taking [8, 9].

Another alternative for Randall and Sundrum model could be achieved through
the Einstein-Cartan portal [10, 11, 12, 13, 14, 15, 16]. At which gravity
(represented by torsion field) can couple to the SM particles in addition to
dark sector fermions, it provides a mechanism of producing the dark sector
particles and allows a chance for probing dark gauge boson (A′), which
corresponds to a $U(1)_{D}$ symmetry, at LHC [17]. In this theory, the torsion
mass is in the TeV-scale regime, so that the A′ can be produced with the high
boost and missing transverse energy ($E^{miss}_{T}$) from dark-sector
fermions. The search for the A′ could be achieved at the LHC via its decay to
dilepton (i.e. A${}^{\prime}\rightarrow l^{+}l^{-}$) and large $E^{miss}_{T}$.

Many searches for DM have been performed via analysing the data collected by
the CMS experiment during RUN II. These searches rely on the production of a
visible object ”X”, which recoils against the large missing transverse energy
from the dark matter particles leaving a signature of
($\text{X}+E^{\text{miss}}_{T}$) in the detector [18]. The visible particle
could be a SM particle like W, Z bosons or jets [19], photon [20] or SM Higgs
boson [21].

In this analysis, we present a search for dark neutral gauge bosons (A′),
which are originated in a simplified model in Einstein-Cartan portal, at the
LHC simulated proton-proton collisions with 13.6 TeV center of mass energy
corresponding to the LHC RUN III circumstances [22]. The topology of the
studied simulated events is dimuon, from the decay of A′, plus large missing
transverse energy which attributes to dark matter. Similar search for dark
matter in this channel has been performed at the CMS experiment at the LHC
with the visible particle being a Z boson decaying to dimuon at $\sqrt{s}$ =
13 TeV [23].

In the following section II, the theoretical formalism of the $U(1)_{D}$
simplified model based on the Einstein-Cartan gravity and its free parameters
are presented. Then the simulation techniques used for events generation for
the signal and SM backrounds samples are displayed in section III. Afterwards,
the selection cuts and the strategy of the analysis are explained in section
IV. Finally, the results and the summary of this analysis are discussed in
sections V and VI respectively.

Figure 1: Feynman diagram for the simplified model based on Einstein-Cartan
gravity; for the production of dark gauge boson (A′) in association to dark
matter ($\chi$) pair [17].

##  II The simplified model in the framework of Einstein-Cartan gravity

The analyzed simplified model is based on the Einstein-Cartan gravity, which
has been discussed in [17], assumes the production of dark matters from
proton-proton collisions at the LHC in addition to a new heavy neutral dark
gauge boson A′.

The proposed dark gauge boson (A${}^{\prime}_{\mu}$) can be produced through
the process of pair annihilation of two quarks $q\bar{q}$ mediated by the
heavy torsion field ($S_{\rho}$, which is the axial–vector part of the torsion
tensor $T^{\lambda}_{\mu\nu}$ [17]), which then undergoes two dark matter
particles ($\chi$). Dark matter is heavy enough to decay to a
A${}^{\prime}_{\mu}$ and another dark matter ($\chi$) as shown in figure 1.

The interaction terms, in the effective Lagrangian, between the torsion field
and Dirac fermion $(\psi)$, is given by [17]

$\bar{\psi}i\gamma^{\mu}(\partial_{\mu}+i\texttt{g}_{\eta}\gamma^{5}S_{\mu}+...)\psi,$
(1)

where $\texttt{g}_{\eta}$ is the coupling of torsion field to Dirac fermions.

While the term in the effective Lagrangian at which the torsion field couples
to the dark matter, and between dark gauge boson (A${}^{\prime}_{\mu}$) and
dark matter are given by [17]

$\bar{\chi}(i\gamma^{\mu}D_{\mu}-M_{\chi})\chi,$ (2)

where
$D_{\mu}=\partial_{\mu}+i\texttt{g}_{\eta}\gamma^{5}S_{\mu}+i\texttt{g}_{D}A^{\prime}_{\mu}$,
$M_{\chi}$ is the dark matter mass and $\texttt{g}_{D}$ is the coupling of
dark gauge boson to dark matter.

The neutral dark gauge boson (A′) decays to the SM fermion pairs, in our case
we choose the muonic decay of A′. The highest significant branching ratio of
A${}^{\prime}\rightarrow\mu^{+}\mu^{-}$ could be reached if the following mass
assumption [17] is satisfied,

$M_{A^{\prime}}<2M_{\chi}.$ (3)

In this model, there are many free parameters including the torsion field mass
($M_{ST}$), the dark gauge boson mass ($M_{A^{\prime}}$), the mass of dark
matter ($M_{\chi}$) and the coupling constants ($\texttt{g}_{\eta}$ and
$\texttt{g}_{D}$). In this analysis, the values of these couplings are taken
to be $\texttt{g}_{\eta}=0.2$ and $\texttt{g}_{D}=1.2$, these values have been
chosen based on the results presented in [17].

Since we are interested by studying the possible production of heavy neutral
dark gauge boson at the LHC, with $M_{A^{\prime}}>100$ GeV, we have fixed the
mass of dark matter to be $M_{\chi}=500$ GeV in order to satisfy the mass
condition given in equation 3. In addition, a similar analysis [23] has shown
that, for axial-vector mediators, DM masses less than 300 GeV are excluded.

The typical signature of this process consists of a pair of opposite sign
muons from the decay of A′ plus a large missing transverse energy due to the
stable dark matter $\chi$. Since the CMS detector has been optimized to this
decay channel, which is a clean channel with respect to SM backgrounds. So
that our studied events are with the following topology
($\mu^{+}\mu^{-}+E^{miss}_{T}$).

$M_{A^{\prime}}$ $M_{ST}$ | 1250 | 1500 | 1750 | 1800 | 1970 | 2000 | 3000 | 4000 | 5000 | 6000 | 7000
---|---|---|---|---|---|---|---|---|---|---|---
200 | $0.00043$ | $0.00863$ | $0.0158$ | $0.017$ | $0.0180$ | $0.0181$ | $0.00895$ | $0.0026$ | $0.00061$ | $0.00012$ | $2.32\times 10^{-5}$
300 | $9.76\times 10^{-7}$ | $0.0040$ | $0.011$ | $0.012$ | $0.0142$ | $0.0145$ | $0.00874$ | $0.0027$ | $0.00066$ | $0.00014$ | $2.64\times 10^{-5}$
400 | $5.43\times 10^{-7}$ | $0.0010$ | $0.0065$ | $0.0075$ | $0.0101$ | $0.0105$ | $0.0079$ | $0.0026$ | $0.00067$ | $0.00014$ | $2.80\times 10^{-5}$
500 | $3.29\times 10^{-7}$ | $8.1\times 10^{-7}$ | $0.0032$ | $0.0041$ | $0.0066$ | $0.00695$ | $0.00688$ | $0.0019$ | $0.00066$ | $0.00014$ | $2.85\times 10^{-5}$
600 | $2.10\times 10^{-7}$ | $3.48\times 10^{-7}$ | $0.0011$ | $0.0017$ | $0.0039$ | $0.0042$ | $0.0058$ | $0.0023$ | $0.00063$ | $0.00014$ | $2.84\times 10^{-5}$
700 | $1.37\times 10^{-7}$ | $1.9\times 10^{-7}$ | $0.00012$ | $0.00041$ | $0.0019$ | $0.0022$ | $0.0048$ | $0.0020$ | $0.00059$ | $0.00014$ | $2.77\times 10^{-5}$
800 | $9.04\times 10^{-8}$ | $1.22\times 10^{-7}$ | $1.28\times 10^{-7}$ | $3.22\times 10^{-7}$ | $0.00071$ | $0.00091$ | $0.0039$ | $0.0018$ | $0.00054$ | $0.00013$ | $2.68\times 10^{-5}$
900 | $5.99\times 10^{-8}$ | $7.80\times 10^{-8}$ | $1.22\times 10^{-7}$ | $1.40\times 10^{-7}$ | $0.00011$ | $0.00021$ | $0.0030$ | $0.0016$ | $0.00050$ | $0.00012$ | $2.57\times 10^{-5}$
Table 1: The simplified model (based on Einstein-Cartan gravity) cross section
measurements times branching ratios (in pb) calculated for different sets of
the masses $M_{A^{\prime}}$ (in GeV), and $M_{ST}$ (in GeV), for the mass
assumption given in equation 3, with dark matter mass ($M_{\chi}=500$ GeV),
the following couplings constants $\texttt{g}_{\eta}=0.2,\leavevmode\nobreak\
\texttt{g}_{D}=1.2$ and at $\sqrt{s}=13.6$ TeV.

##  III Simulation of signal samples and SM backgrounds

The SM background processes yielding muon pairs in the signal region are
Drell-Yan (DY $\rightarrow\mu^{+}\mu^{-}$) production, the production of top
quark pairs ($\text{t}\bar{\text{t}}\rightarrow\mu^{+}\mu^{-}+2b+2\nu$) and
production of diboson ($W^{+}W^{-}\rightarrow\mu^{+}\mu^{-}+2\nu$,
$ZZ\rightarrow\mu^{+}\mu^{-}+2\nu$ and
$W^{\pm}Z\rightarrow\mu^{\pm}\mu^{+}\mu^{-}+\nu$). The second type of
background is the jets background, which comes from the misidentification of
jets as muons, where a jet or multijet pass the muons selection criteria. This
kind of background originates from two processes: W+jet and QCD multijet. The
contamination of single and multijet background in data is usually estimated
from data using a so called data driven method which is explained in [8],
nevertheless they are irrelevant for our study because our analysis is based
on MC simulations only.

The signal samples, for the simplified model (based on Einstein-Cartan
gravity), and the corresponding SM processes have been generated using
MadGraph5_aMC@NLO [24] interfaced to Pythia 8 for parton shower model and
hadronization [25], and DELPHES [26] for a fast detector simulation of CMS
experiment. They were generated from proton-proton collisions at the Large
Hadron Collider at 13.6 TeV center of mass energy, which corresponds to the
circumstances of RUN III, with muon $p_{T}>10$ GeV and $|\eta|<3$ rad.

For the simplified model, with the use of mass assumption given in equation 3,
table 1 indicates the cross section measurements times branching ratios
calculated for different sets of the dark gauge boson (A′) and torsion field
($ST$) masses. The simulated signals, used in this analysis, are private
production samples, at which we used the matrix element event generator
MadGraph5 aMC@NLO v2.6.7 [24]. We are grateful to Cao H. Nam, the authors of
[17], for sharing with us the Universal FeynRules Output (UFO) for the model.

All Monte Carlo samples used in this analysis and their corresponding cross
sections were calculated at next-to-leading order. Thus, the contributions of
the signal samples and the SM background processes have been estimated from
the Monte Carlo simulations, at which they are normalized to their
corresponding cross section and integrated luminosity of 52 fb-1 [22]. The
detector related systematic uncertainty is originated from the evaluation of
the integrated luminosity of the 2022-data, that are recorded by the CMS
during RUN III, was estimated to be 2.2% [27].

##  IV Event selection

The selection of event, for the analysis, has been designed to reconstruct a
final state with two high transverse momentum $(p_{T})$ muons in association
with missing transverse energy accounting for the DM candidate. The selection
is made in the form of cuts applied on different kinematic parameters. Each of
the two muons should pass the following preliminary selection:
$\bullet$ $p^{\mu}_{T}$ (GeV) $>30$,
$\bullet$ $\eta^{\mu}$ (rad) $<$ 3,
$\bullet$ $\text{IsolationVarRhoCorr}<0.1$,
”IsolationVarRhoCorr” represents the isolation cut in DELPHES software in
order to reject muons produced inside jets. In this cut, it is required that
the scalar $p_{T}$ sum of all muon tracks within a cone of $\Delta R=0.5$
around the muon candidate, excluding the muon candidate itself, should not
exceed 10% of the $p_{T}$ of the muon. This cut has been corrected for pileup
effect.

Thus, each event has been selected with two opposite charge muons, and the
invariant mass of the dimuon is bigger than 60 GeV, since we are looking for a
resonance in the high mass regime.

Figure 2: The measured dimuon invariant mass spectrum, after applying
preliminary cuts, for the estimated SM backgrounds and for different choices
of dark gauge boson (A′) masses generated based on the simplified model, with
mass of torsion field ($M_{ST}=2000$ GeV) and dark matter mass ($M_{\chi}=500$
GeV).

Figure 3: The distribution of the missing transverse energy, after the
preliminary selection; for the expected SM backgrounds, and different A′
masses produced by simplified model, with mass of torsion field ($M_{ST}=2000$
GeV) and dark matter mass ($M_{\chi}=500$ GeV).

Figure 2 shows the distribution of the dimuon invariant mass; the cyan
histogram represents the Drell-Yan background, the yellow histogram stands for
the vector boson pair backgrounds (WW, WZ and ZZ) and the $t\bar{t}$
background is represented by the red histogram. These histograms are stacked.
While the signals of the simplified model in the framework of Einstein-Cartan
gravity, which have been generated with different masses of the neutral dark
gauge boson A′ with fixed values of the torsion field mass ($M_{ST}=2000$ GeV)
and dark matter mass ($M_{\chi}=500$ GeV), are represented by different
colored lines, and are overlaid. The corresponding distribution of the missing
transverse energy is presented in figure 3. It is clearly shown from these
figures that, the signal samples are overwhelmed by the backgrounds. So that,
it is necessary to apply a more tighter set of cuts to discriminate signals
from SM backgrounds as will be explained in the next paragraph.

In addition to the preliminary selection, extra tighter cuts have been
applied. These tight cuts are based on three variables: the first variable is
related to the invariant mass of the dimuon, at which we restricted the
invariant mass of the dimuon to a small range around the mass of the dark
gauge boson A′, such that $(0.9\times
M_{A^{\prime}})<M_{\mu^{+}\mu^{-}}<(M_{A^{\prime}}+25)$. The second is the
relative difference between the transverse energy of dimuon
$(E_{T}^{\mu^{+}\mu^{-}})$ and the missing transverse energy
$(E^{\text{miss}}_{T})$, it has been selected to be less than 0.4. (i.e.
$|E_{T}^{\mu^{+}\mu^{-}}-E^{\text{miss}}_{T}|/E_{T}^{\mu^{+}\mu^{-}}<0.4$).
The third one is $\Delta\phi_{\mu^{+}\mu^{-},\vec{E}^{\text{miss}}_{T}}$,
which is defined as difference in the azimuth angle between the dimuon
direction and the missing transverse energy direction (i.e.
$\Delta\phi_{\mu^{+}\mu^{-},\vec{E}^{\text{miss}}_{T}}=|\phi^{\mu^{+}\mu^{-}}-\leavevmode\nobreak\
\phi^{miss}|$ ), it has been selected to be greater than 2.6 rad.

For dimuon events, with each muon passes the preliminary cuts, we present in
figure 4 the distributions of
$|E_{T}^{\mu^{+}\mu^{-}}-E_{T}^{\text{miss}}|/E_{T}^{\mu^{+}\mu^{-}}$ (a) and
$\Delta\phi_{\mu^{+}\mu^{-},\vec{E}_{T}^{\text{miss}}}$ (b) for the signal
presentation of the simplified model corresponding to Einstein-Cartan gravity,
which was generated with masses of dark gauge boson $M_{A^{\prime}}$ = 200
GeV, the torsion field mass $M_{ST}=2000$ GeV and dark matter mass
$M_{\chi}=500$ GeV and SM backgrounds. These distributions are scaled to one.
In these plots, the vertical dashed lines correspond to the chosen cut value
per each variable. These tight cuts have been applied in order to strongly
decrease the SM backgrounds. $ZZ$ background has been fully suppressed by the
mass window cut $(0.9\times
M_{A^{\prime}})<M_{\mu^{+}\mu^{-}}<(M_{A^{\prime}}+25)$.

(a)

(b)

Figure 4: Distributions of
$|E_{T}^{\mu^{+}\mu^{-}}-E_{T}^{miss}|/E_{T}^{\mu^{+}\mu^{-}}$ (a) and
$\Delta\phi_{\mu^{+}\mu^{-},\vec{E}_{T}^{\text{miss}}}$ (b) for the signal
presentation of the simplified model corresponding to the Einstein-Cartan
gravity with $M_{A^{\prime}}=200$ GeV and SM backgrounds, for dimuon events
with each muon is passing the preliminary cuts. The vertical dashed lines
correspond to the chosen cut value per each variable. All histograms are
normalized to unity to highlight qualitative features.

##  V Results

The shape-based analysis has been used based on the missing transverse energy
distributions ($E^{\text{miss}}_{T}$), which are good discriminate variable,
since the signals distributions are characterized by relatively large
$E^{\text{miss}}_{T}$ values compared to the SM backgrounds. The distribution
of the missing transverse energy, after the application of the final event
selection, is illustrated in figure 5. The event yields passing the analysis
final selection, for each of the SM backgrounds and the signal of simplified
model, which was generated with masses of dark gauge boson $M_{A^{\prime}}$ =
200 GeV, the torsion field mass $M_{ST}=2000$ GeV and dark matter mass
$M_{\chi}=500$ GeV; corresponding to an integrated luminosity of 52 fb-1 are
presented in table 2. Uncertainties include both statistical and systematic
components, summed in quadrature.

Figure 5: The distribution of the missing transverse energy, after final analysis selection cuts, for the expected SM background and one signal benchmark corresponding to the Einstein-Cartan gravity with $M_{A^{\prime}}=200$ GeV is superimposed. Process | No. of events
---|---
$\text{DY}\rightarrow\mu^{+}\mu^{-}$ | $86.8\pm 9.5$
$\text{t}\bar{\text{t}}$ | $1587.0\pm 53.0$
WW | $215.7\pm 15.4$
WZ | $22.1\pm 4.7$
Sum Bkgs | $1911.6\pm 60.7$
Simplified model signal | $575.8\pm 27.1$
(at $M_{A^{\prime}}$ = 200 GeV) |
Table 2: The number of events, satisfying the criteria of the final event
selection, are illustrated for each SM background and the the simplified model
in the Einstein-Cartan portal with coupling constants $g_{D}=1.2$,
$g_{\eta}=0.2$ and $M_{\chi}=500$ GeV, corresponding to a 52 fb-1 integrated
luminosity at 13.6 TeV centre of mass energy. The total uncertainty, including
the statistical and systematic components, is indicated.

In order to make a statistical interpretation for our results, we preformed a
statistical test based on the profile likelihood method, with the use of the
modified frequentist construction CLs [28, 29] used in the asymptotic
approximation [30] to derive exclusion limits on the product of signal cross
sections and branching fraction Br($A^{\prime}$ $\rightarrow\mu\mu$) at 95%
confidence level.

The 95% upper limit on the cross section times the branching ratio versus the
mass of torsion field $M_{ST}$, for the simplified model based on Einstein-
Cartan gravity, is presented in figure 6, with the muonic decay of the A′ and
coupling constant values of $\texttt{g}_{\eta}=0.2$ and $\texttt{g}_{D}=1.2$
and dark matter mass $M_{\chi}=500$ GeV. The black solid curve represents the
simplified model for $M_{A^{\prime}}=200$ GeV. Based on figure 6, we exclude
the torsion field $(ST)$ production in the mass range between 1375 - 6450 GeV
as shown from expected median.

For the simplified model (based on Einstein-Cartan gravity), the cross section
times the branching ratio limit is presented in figure 7 as a function of the
mediator’s masses $M_{ST}$ and the masses of the dark neutral gauge boson
$M_{A^{\prime}}$. The region between the respective pair of the expected 95%
dotted line is excluded. The results from the inclusive signal regions exclude
expected values of up to $1375<M_{ST}<6645$ GeV.

Figure 6: 95% CL upper limits on the cross section times the branching ratio
(expected), as a function of the mediator’s mass ($M_{ST}$) based on Einstein-
Cartan model, with the muonic decay of the A′. The black line represents the
Einstein-Cartan gravity with $M_{A^{\prime}}=200$ GeV.

Figure 7: The 95% CL upper limits on the product of the cross section and
branching fraction from the inclusive search, for variations of pairs of the
simplified model parameters ($M_{ST}$ and $M_{A^{\prime}}$). The filled region
indicates the upper limit. The dotted black curve indicates the expected
exclusions for the nominal A′ cross section.

##  VI Summary

A search for dark neutral gauge bosons (A′) produced in association with dark
matter $(\chi)$, in the framework of $U(1)_{D}$ simplified model based on the
Einstein-Cartan gravity, has been presented, using the simulated proton-proton
collisions corresponding to the LHC RUN III 13.6 TeV center of mass energy,
for an integrated luminosity of 52 fb-1. Results from muonic decay mode of A′
are discussed, with fixing the values of the coupling constants to be
$g_{D}=1.2$, $g_{\eta}=0.2$, and dark matter mass ($M_{\chi}=500$ GeV). We
have considered the variations of several parameters of the signal model: the
torsion field mass $(M_{ST})$ and the dark neutral gauge boson
mass$(M_{A^{\prime}})$. The general version of the search, which uses only
event-level kinematic variables, excludes models with $1375<M_{ST}<6645$ GeV
at 95% confidence level (CL).

###### Acknowledgements.

The author of this paper would like to thank Cao H. Nam, the author of [17],
for his useful discussions about the theoretical models, and sharing with us
the Universal FeynRules Output (UFO) for the model that were used for the
events generation.

## References

  * [1] F. Halzen and A. D. Martin, (1984), “Quarks And Leptons: An Introductory Course In Modern Particle Physics.” isbn: 0471887412, 9780471887416.
  * [2] P. Langacker, “The standard model and beyond”, (2010). isbn: 9781420079067.
  * [3] M. Cvetic and S. Godfrey, “Discovery and identification of extra gauge bosons”, arXiv:hep-ph/9504216.
  * [4] A. Leike, “The Phenomenology of extra neutral gauge bosons”, Phy. Rep. 317, 143 (1999) [arXiv:hepph/9805494].
  * [5] M. Cvetic, P. Langacker and B. Kayser, “Determination of g-R / g-L in left-right symmetric models at hadron colliders”, Phys. Rev. Lett. 68 (1992) 2871.
  * [6] S. Dimopoulos and H. Georgi, “Softly Broken Supersymmetry And SU(5)”, Nucl. Phys. B 193 (1981) 150.
  * [7] L. Randall and R. Sundrum, “A large mass hierarchy from a small extra dimension”, Phys. Rev. Lett. 83 (1999) 3370 [arXiv:hep-ph/9905221].
  * [8] CMS Collaboration, Search for resonant and nonresonant new phenomena in high-mass dilepton final state at $\sqrt{s}$ = 13 TeV, JHEP 07 (2021) 208 [arXiv:2103.02708v2] [hep-ex].
  * [9] ATLAS Collaboration, Search for new non-resonant phenomena in high-mass dilepton final states with the ATLAS detector, JHEP 11 (2020) 05.
  * [10] T.W. B. Kibble, Lorentz Invariance and the Gravitational Field. Journal of Mathematical Physics. 2 (2): 212–221 (1961).
  * [11] Sciama, D. W., The Physical Structure of General Relativity. Reviews of Modern Physics. 36 (1): 463–469 (1964-01-01).
  * [12] Hehl, Friedrich W.; von der Heyde, Paul; Kerlick, G. David; Nester, James M. General relativity with spin and torsion: Foundations and prospects. Reviews of Modern Physics. 48 (3): 393–416.
  * [13] D. Tsoubelis, Phys. Rev. Lett. 51, 2235 (1983).
  * [14] M. Shaposhnikov, A. Shkerin, I. Timiryasov, and S. Zell,
  * [15] J. High Energy Phys. 10 (2020) 177; 08 (2021) 162(E).
  * [16] M. Shaposhnikov, A. Shkerin, I. Timiryasov, and S. Zell, Phys. Rev. Lett. 126, 161301 (2021); 127, 169901(E) (2021).
  * [17] Cao H. Nam, Probing dark gauge boson via Einstein-Cartan portal. Phys. Rev. D 105, 075015 (2022) [arXiv:2112.10446] [hep-ph].
  * [18] Krovi AniruDF, Low Ian and Zhang Yue, Broadening dark matter searches at the LHC: mono-X versus darkonium channels. JHEP 10 (2018) 026 [arXiv:1807.07972] [hep-ph].
  * [19] CMS Collaboration, Search for new physics in final states with an energetic jet or a hadronically decaying W or Z boson and transverse momentum imbalance at $\sqrt{s}$ = 13 TeV, Phys. Rev. D 97 (2018) 092005. [arXiv:1712.02345] [hep-ex].
  * [20] CMS Collaboration, Search for new physics in the monophoton final state in proton-proton collisions at $\sqrt{s}$ = 13 TeV, JHEP. 10 (2017) 073, [arXiv:1706.03794v2] [hep-ex].
  * [21] CMS Collaboration, Search for dark matter particles produced in association with a Higgs boson in proton-proton collisions at $\sqrt{s}$ = 13 TeV, JHEP 03 (2020) 025, [arXiv:1908.01713v2] [hep-ex].
  * [22] https://twiki.cern.ch/twiki/bin/view/CMSPublic/
LumiPublicResults.

  * [23] CMS Collaboration, Search for dark matter produced in association with a leptonically decaying Z boson in proton-proton collisions at $\sqrt{s}$ = 13 TeV, Eur. Phys. J. C 81 (2021) 13 [arXiv:2008.04735] [hep-ex].
  * [24] Johan Alwall, Michel Herquet, Fabio Maltoni, Olivier Mattelaer, and Tim Stelzer. MadGraph 5 : Going Beyond. JHEP, 06:128, 2011.
  * [25] T. Sjöstrand, S. Ask, J.R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C.O. Rasmussen, P.Z. Skands. An Introduction to PYTHIA 8.2, Comput. Phys. Commun. 191 (2015) 159–177, arXiv:1410.3012 [hep-ph].
  * [26] J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, M. Selvaggi, DELPHES 3, A modular framework for fast simulation of a generic collider experiment, JHEP 1402 (2014).
  * [27] https://twiki.cern.ch/twiki/bin/viewauth/CMS/
LumiRecommendationsRun3.

  * [28] A. L. Read, Presentation of search results: the CLs technique, J. Phys. G: Nucl. Part.Phys. 28 (2002) 2693, doi:10.1088/0954-3899/28/10/313.
  * [29] T. Junk, Confidence level computation for combining searches with small statistics, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Volume 434, Issues 2–3, 1999, Pages 435-443, ISSN 0168-9002, https://doi.org/10.1016/S0168-9002(99)00498-2.
  * [30] G. Cowan et al., Asymptotic formulae for likelihood-based tests of new physics, Eur. Phys. J. C 71 (2011), p. 1554, doi: 10.1140/epjc/s10052-011-1554-0, arXiv: 1007.1727 [physics.data-an], Erratum: Eur. Phys. J. C 73 (2013) 2501.