# Revealing the supercritical dynamics of dusty plasmas and their liquid-like
to gas-like dynamical crossover

Dong Huang1 Matteo Baggioli2<EMAIL_ADDRESS>Shaoyu Lu1 Zhuang Ma1 Yan
Feng1<EMAIL_ADDRESS>1Institute of Plasma Physics and Technology, School
of Physical Science and Technology, Jiangsu Key Laboratory of Thin Films,
Soochow University, Suzhou 215006, China  2Wilczek Quantum Center, School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China &
Shanghai Research Center for Quantum Sciences, Shanghai 201315, China

###### Abstract

Dusty plasmas represent a powerful playground to study the collective dynamics
of strongly coupled systems with important interdisciplinary connections to
condensed matter physics. Due to the pure Yukawa repulsive interaction between
dust particles, dusty plasmas do not display a traditional liquid-vapor phase
transition, perfectly matching the definition of a supercritical fluid. Using
molecular dynamics simulations, we verify the supercritical nature of dusty
plasmas and reveal the existence of a dynamical liquid-like to gas-like
crossover which perfectly matches the salient features of the Frenkel line in
classical supercritical fluids. We present several diagnostics to locate this
dynamical crossover spanning from local atomic connectivity, shear relaxation
dynamics, velocity autocorrelation function, heat capacity, and various
transport properties. All these different criteria well agree with each other
and are able to successfully locate the Frenkel line in both 2D and 3D dusty
plasmas. In addition, we propose the unity ratio of the instantaneous
transverse sound speed $C_{T}$ to the average particle speed $\bar{v}_{p}$,
i.e., $C_{T}/\bar{v}_{p}=1$, as a new diagnostic to identify this dynamical
crossover. Finally, we observe an emergent degree of universality in the
collective dynamics and transport properties of dusty plasmas as a function of
the screening parameter and dimensionality of the system. Intriguingly, the
temperature of the dynamical transition is independent of the dimensionality
and it is found to be always $20$ times of the corresponding melting point.
Our results open a new path for the study of single particle and collective
dynamics in plasmas and their interrelation with supercritical fluids in
general.

##  I Introduction

A supercritical fluid Cowan:1988 ; Eckert:1996 ; Kessel:2005  typically refers
to a condensed state of matter in which the traditional liquid and gas phases
cannot be separated anymore by a sharp first-order phase transition. For
various substances, the supercritical fluid state can be achieved when the
temperature and the pressure are above the corresponding critical point
Cowan:1988 ; Eckert:1996 ; Kessel:2005 . Supercritical fluids have been
intensively investigated due to their widely applications in the nuclear
waste, petrochemical, food, and pharmaceutical industries Carleson:1996 ;
McHardy:1998 ; Kiran:2000 . Although there is no traditional liquid or gas
state within the supercritical regime, recently, several studies suggest that
a liquid-like to gas-like dynamical transition can be identified in
supercritical fluids using either the Frenkel line Brazhkin:2012 ;
Brazhkin:2013 ; Cockrell:2021 ; Yang:2015 ; Bryk:2017 ; Bolmatov:2013 ;
Lee:2021 , the Widom line Simeoni:2010 , or the Fisher-Widom line Fisher:1969
.

In the microscopic description of liquids proposed by Frenkel Frenkel:1946 ,
atomic particle motion is a combination of quasiharmonic vibrations around
potential minima and thermally induced jumps from an equilibrium position to a
new one. These hopping processes give the ability to flow in liquids and
happen at an average time $\tau$, also termed the liquid relaxation time
Brazhkin:2012 ; Frenkel:1946 ; Brazhkin:2013 . This view, deeply inspired by
the phenomenological ideas of Maxwell doi:10.1098/rstl.1867.0004 , implies
that liquids behave effectively as solids for time scales shorter than $\tau$,
or equivalently, for frequencies larger than the Frenkel frequency
$\omega_{F}=2\pi/\tau$. Following this picture inspired by solid state theory,
the minimal period of rigid-like vibrations is given by the Debye time
$\tau_{D}$, which is around $0.1-1$ ps in classical liquids. When
$\tau>\tau_{D}$, particles mainly vibrate at their equilibrium positions and
hop rarely, so that the typical liquid behavior is exhibited, often termed as
the “rigid liquid” state Brazhkin:2012 ; Brazhkin:2013 . In that regime,
liquids are expected to support propagating shear waves at frequencies
$\omega>\omega_{F}$. One can further derive a minimal cutoff wave-vector for
their propagation BAGGIOLI20201 , which implies a maximum propagation length
approximately equal to the sound speed times the relaxation time $\tau$.

Within this framework, the frequency of collective propagating shear waves in
liquids, responsible for the emergent ”rigidity”, has to fall in the range
$\omega_{F}<\omega<\omega_{D}$, with $\omega_{D}=2\pi/\tau_{D}$. The
relaxation time $\tau$ decreases with the temperature. Physically, this is
just reflected in a larger kinetic energy and therefore a stronger ability to
rearrange. When $\tau<\tau_{D}$, the particles’ hopping occurs more frequently
than solid-like vibrations, collective shear waves disappear, and the
“nonrigid” gas-like fluid state Brazhkin:2012 ; Brazhkin:2013  is approached.
Following this logic, in supercritical liquids, the concept of Frenkel line
Brazhkin:2012 ; Brazhkin:2013  has been introduced to discriminate the rigid
liquid from the nonrigid gas-like fluid state and it has been formally defined
by the condition $\tau\approx\tau_{D}$. In the past Brazhkin:2012 ;
Brazhkin:2013 ; Bolmatov:2013 ; Yang:2015 ; Cockrell:2021 , several
diagnostics, including the specific heat $c_{V}$, the velocity autocorrelation
function (VACF), and the mean squared displacement have been used to determine
the condition for the Frenkel line in various physical systems, as described
in detail later. Furthermore, when $\tau\rightarrow\tau_{D}$, i.e., at the
onset of the dynamical crossover, the propagation length of shear waves
approaches the minimum available value approximately given by the interatomic
distance. Therefore, the Frenkel line can be also defined as the disappearance
of collective shear waves in fluids BAGGIOLI20201  and it is related to
universal minimal values for different transport coefficients such as the
shear viscosity and the thermal diffusivity doi:10.1126/sciadv.aba3747 ;
PhysRevB.103.014311 .

Figure 1:  A putative phase diagram for dusty plasmas as a function of the
coupling parameter $\Gamma$ and the screening parameter $\kappa$. The solid
blue line indicates the first-order solid-liquid phase transition. The dashed
red line refers to a possible dynamical crossover in the supercritical regime
which is the subject of this work.

Dusty plasmas Thomas:1996 ; LI:1996 ; Melzer:1996 ; Merlino:2004 ; Kalman:2004
; Morfill:2009 ; Piel:2010 ; Bonitz:2010 , also termed complex plasmas, are
partially ionized gases containing micron-sized dust particles, and they
represent an excellent model system where the motion of individual dust
particles can be directly tracked. In the laboratory conditions, these dusts
have a typical charge of $-10^{5}e$ in the steady state Nosenko:2004 ;
Feng:2010 ; Feng:2012 , interacting with each other through the Yukawa
repulsion Konopka:2000 , and leading to a much higher potential energy between
neighboring dusts than their kinetic energy, i.e., these dusts are strongly
coupled Merlino:2004 ; Kalman:2004 ; Morfill:2009 ; Piel:2010 ; Bonitz:2010 ;
Ichimaru:1982  (cf. classical liquids). During experiments, these dusts can
form into either a single layer two-dimensional (2D) suspension or a three-
dimensional (3D) suspension, i.e., 2D Nunomura:2002 ; Nosenko:2004 ;
Nosenko:2008 ; Hartmann:2010 ; Couedel:2010 ; Kahlert:2012 ; Hartmann:2014 ;
Hartmann:2019 ; Feng:2010 ; Feng:2012 ; Du:2019 ; Gogia:2017  or 3D dusty
plasma Pramanik:2002 ; Tsai:2016 ; Hayashi:1999 ; Arp:2004 ; Bonitz:2006 ;
Zuzic:2000 ; Morfill:2006 ; Thomas:2005 , exhibiting collective solid and
liquid-like behaviors Thomas:1996 ; LI:1996 ; Melzer:1996 ; Merlino:2004 ;
Kalman:2004 ; Morfill:2009 ; Piel:2010 ; Bonitz:2010 ; Nunomura:2002 ;
Nosenko:2004 ; Nosenko:2008 ; Hartmann:2010 ; Couedel:2010 ; Kahlert:2012 ;
Hartmann:2014 ; Hartmann:2019 ; Feng:2010 ; Feng:2012 ; Hayashi:1999 ;
Arp:2004 ; Bonitz:2006 ; Zuzic:2000 ; Morfill:2006 ; Pramanik:2002 ; Tsai:2016
; Du:2019 ; Thomas:2005 ; Gogia:2017 , including the solid-liquid phase
transition or melting Melzer:1996 ; Nosenko:2004 ; Feng:2010 . Thus, dusty
plasmas provide an incredibly powerful platform to explore collective dynamics
of liquids and solids at the individual particle level Nunomura:2002 ;
Nosenko:2004 ; Nosenko:2008 ; Hartmann:2010 ; Couedel:2010 ; Kahlert:2012 ;
Hartmann:2019 ; Hartmann:2014 ; Feng:2010 ; Feng:2012 ; Du:2019 ; Gogia:2017 .

In the past thirty years, the solid-liquid phase transition of dusty plasmas
has been systematically investigated, in both experiments and simulations, as
well as theories Melzer:1996 ; Nosenko:2004 ; Feng:2010 . Commonalities
between fluid dusty plasmas and classical liquids, rooted in their shared
strongly coupled nature, have been also explored, creating a beneficial
exchange of ideas and results. As a concrete example, the dynamics of shear
waves and the existence of a critical wave-vector, typical of liquids
Trachenko_2016 ; BAGGIOLI20201 , have been the subject of several theoretical
and experimental studies in dusty plasmas PhysRevLett.85.2514 ; Goree:2012 ;
PhysRevLett.97.115001 . However, so far, one important aspect of the
collective dynamics of dusty plasmas remains elusive. In fact, due to the pure
repulsive interaction between dust particles, there is no liquid-vapor phase
transition Hansen:1986  in dusty plasmas. Thus, a melted or fluid dusty plasma
perfectly matches the definition of a supercritical fluid and should be
thought as such. To the best of our knowledge, the supercritical nature of
fluid dusty plasmas and the existence of a related liquid-like to gas-like
dynamical crossover (Frenkel line) have been never disclosed before. In this
work, we put forward this new interpretation and we further show that, as in
classical liquids, the supercritical regime of 2D and 3D dusty plasmas can
separated into a liquid-like and a gas-like phase by a dynamical crossover,
the Frenkel line (see cartoon in Fig. 1).

The rest of this paper is organized as follows. In Sec. II, we briefly
introduce our simulation method to mimic 2D and 3D dusty plasmas. In Sec. III,
we present the proposed supercritical nature of dusty plasmas and its
features. We also provide various diagnostics to discriminate the liquid-like
from the gas-like states, with the same resulting transition at $20$ times of
the melting point for both 2D and 3D dusty plasmas. In Sec. IV, we provide an
interpretation of the agreement between our newly introduced diagnostics and
the traditional Frenkel line criteria. In Sec. V, we give a summary of our
findings. In Appendix A, we provide the details of our MD and Langevin
simulations. In Appendix B, we present the calculated transport results from
our simulation data (self-diffusion constant, shear viscosity, and thermal
conductivity), as well as a new analysis of previous results from
Daligault:2012 ; Donko:2008 ; Daligault:2014 ; Donko:2004 ; Scheiner:2019 .

##  II Simulation method

We perform equilibrium molecular dynamics (MD) simulations of 2D and 3D Yukawa
liquids to mimic 2D and 3D fluid dusty plasmas as in Liu:2005 ; Goree:2012 .
The dynamics of each particle $i$ is governed by the following equation of
motion:

$m\ddot{\mathbf{r}}_{i}=-\nabla\Sigma_{j}\phi_{ij}$ (1)

where
$\phi_{ij}=Q^{2}\,\frac{\exp\left(-r_{ij}/\lambda_{D}\right)}{4\pi\epsilon_{0}r_{ij}}$
is the Yukawa repulsion between particles $i$ and $j$, separated by a distance
$r_{ij}$. Here, $\lambda_{D}$ is the Debye length, $Q$ the charge, and
$\epsilon_{0}$ the vacuum electric permittivity. To characterize dusty
plasmas, we use the screening parameter $\kappa=a/\lambda_{D}$ and the
coupling parameter $\Gamma=Q^{2}/\left(4\pi\epsilon_{0}ak_{B}T\right)$
Kalman:2004 ; Merlino:2004 ; Morfill:2009 ; Piel:2010 ; Bonitz:2010 , where
$a$ is the Wigner-Seitz radius of $(n\pi)^{-1/2}$ Kalman:2004  and
$(4n\pi/3)^{-1/3}$ Goree:2012 ; Silvestri:2019  as a function of the number
density $n$ in 2D and 3D systems, respectively. Clearly, by increasing the
screening parameter $\kappa$, the potential changes gradually from a long-
range Coulomb-like potential to a hard-sphere-like repulsion. Timescales are
normalized using the dusty plasma frequency
$\omega_{pd}=\sqrt{Q^{2}/\left(2\pi\epsilon_{0}ma^{3}\right)}$ Kalman:2004
and $\omega_{pd}=\sqrt{3Q^{2}/\left(4\pi\epsilon_{0}ma^{3}\right)}$ Goree:2012
for 2D and 3D dusty plasmas, respectively. For convenience, we will present
all our results as a function of the reduced coupling strength
$\Gamma/\Gamma_{m}$, where $\Gamma_{m}$ corresponds to the melting point
determined from the static structure measured from simulations Hartmann:2005 ;
Ohta:2000 . In analogy to thermal systems, we can think of the coupling
parameter $\Gamma$ as the effective inverse temperature for dusty plasmas, so
that a higher $\Gamma$ value corresponds to a lower temperature.

To simulate 2D and 3D dusty plasmas, we confine $N=4096$ and 8192 particles in
two simulation cells with the dimensions of $121.9a\times 105.6a$ and
$32.5a\times 32.5a\times 32.5a$, respectively, using periodic boundary
conditions. We always set the simulation conditions as melted dusty plasmas,
i.e., for each chosen $\kappa$ value between $0.5$ and $3$, we specify various
$\Gamma$ values lower than the corresponding melting point $\Gamma_{m}$ of 2D
Hartmann:2005  and 3D dusty plasmas Ohta:2000 . We also use the reduced
coupling strength $\Gamma/\Gamma_{m}$ Donko:2006 ; Khrapak:2020  to
characterize the relative temperature of the studied systems. The integration
time step is chosen to be $0.005\leavevmode\nobreak\ \omega_{pd}^{-1}$, small
enough as justified in Liu:2005 , while the interparticle Yukawa repulsion at
the radii beyond $22a$ Liu:2005  and $8a$ is truncated directly for 2D and 3D
systems, respectively. Other simulation details are the same as in Liu:2005 ;
Goree:2012 . As the output of our simulations, the obtained time series of
positions and velocities for all simulated particles are used to determine
various physical quantities reported in the main text. We also perform
Langevin dynamical simulations Feng:2008  of 2D and 3D dusty plasmas to
confirm that our reported results are valid. More details can be found in
Appendix A.

Figure 2:  Specific heat $c_{V}$ for 2D and 3D dusty plasmas as a function of
the reduced coupling strength $\Gamma/\Gamma_{m}$ for different values of the
screening parameter $\kappa$. The vertical dashed line indicates the critical
value $\Gamma/\Gamma_{m}=0.05$. The horizontal dotted line corresponds to the
value $0.5$.

##  III Results

###  III.1 Heat capacity

We start our discussion with the analysis of the heat capacity. In what
follows, we will refer to the specific heat per particle, so that the total
number of particles $N$ disappears from our expressions. Moreover, in dusty
plasmas, the temperature is expressed in units of energy, so that $k_{B}=1$ as
well Nosenko:2008 .

At lower temperatures, or equivalently larger coupling parameters, the Frenkel
frequency is much lower than the Debye one and collective shear waves display
an almost gapless dispersion as in solids. In that regime, for 3D systems, we
expect two transverse waves and one longitudinal wave there, each of which
contributes $k_{B}T/2$ potential energy from the equipartition theorem
Bolmatov:2013 ; Pathria:2011 , and $k_{B}T/2$ kinetic energy. For large values
of the coupling parameter, $\Gamma/\Gamma_{m}\gg 1$, the heat capacity
approaches therefore the solid-like value, $c_{V}=3$. When transferring from
the rigid liquid to the nonrigid gas-like state, the two transverse waves
cannot be sustained anymore, and the heat capacity reduces to $c_{V}=2$
Brazhkin:2013 ; Bolmatov:2013 . Qualitatively, this gradual decrease reflects
the disappearance of the solid like oscillations into the diffusive gas-like
motion. In other words, the number of transverse modes with frequency
$\omega>\omega_{F}$ decreases towards the gas-like regime. As a result, the
Frenkel line is determined by the simple condition $c_{V}^{\rm 3D}=2$ in 3D
systems Bolmatov:2013 . Following a similar argument, for 2D systems, there is
only one transverse wave, so that $c_{V}$ decreases from $2$ to $1.5$ when the
transverse wave cannot be sustained anymore. Thus, the Frenkel line for 2D
systems can be determined from the condition $c_{V}^{\rm 2D}=1.5$.

In the microcanonical ensemble, the specific heat $c_{V}$ can be derived from
the fluctuations of the kinetic energy (KE) using Lebowitz:1967 ;
Cancrini:2017

$\frac{2\,(\langle{\rm KE}^{2}\rangle-\langle{\rm
KE}\rangle^{2})}{Nd(k_{B}T)^{2}}=\frac{c_{V}-d/2}{c_{V}},$ (2)

where $d$ is the dimensionality of the simulated system and $k_{B}$ the
Boltzmann constant. Since the temperature is expressed in units of energy, the
obtained specific heat $c_{V}$ is dimensionless, as in Nosenko:2008 . The
numerical results from MD simulations are presented in Fig. 2 as a function of
the reduced coupling strength $\Gamma/\Gamma_{m}$, for different values of the
screening parameter $\kappa$ and for both 2D and 3D systems. Clearly, as the
reduced coupling strength $\Gamma/\Gamma_{m}$ increases, the specific heat
$c_{V}$ increases monotonically, for both 2D and 3D fluid dusty plasmas. Since
the coupling strength has to be interpreted as an inverse temperature, this
behavior is consistent with the expected heat capacity for a liquid, which
contrary to that of solids, decreases monotonically with temperature. At
$\Gamma\approx\Gamma_{m}$, the obtained values match the expectations from
solid state theory, $c_{V}^{\rm 3D}=3$ and $c_{V}^{\rm 2D}=2$. In the opposite
limit, $\Gamma/\Gamma_{m}\rightarrow 0$, the data approaches the ideal gas
result, $c_{V}^{\rm 3D}=3/2$ and $c_{V}^{\rm 3D}=1$.

From our simulation results in Fig. 2, $c_{V}^{\rm 2D}=1.5$ and $c_{V}^{\rm
3D}=2$ both occur at the same reduced coupling strength of
$\Gamma/\Gamma_{m}=\gamma_{c}=0.05$, with $\gamma_{c}$ the dimensionless
critical value. This clearly indicates that the Frenkel line for both 2D and
3D fluid dusty plasmas is located at the same $\gamma_{c}$, suggesting that
the liquid-like-gas-like transition in dusty plasmas or Yukawa systems is
independent of the dimensionality. Intuitively, this could be explained by
considering the dynamics of transverse shear waves. In absence of shear
excitations, the system is effectively isotropic and therefore insensitive to
the number of dimensions. This is further proved by the universal collapse of
the two curves in Fig. 2 for $\Gamma/\Gamma_{m}<\gamma_{c}$, where no
collective shear waves can be sustained any more. On the contrary, in the
rigid liquid phase, the heat capacity of the 3D system is consistently larger
than the 2D counterpart simply because of the larger number of emerging
propagating shear waves.

Intriguingly, all the 3D and 2D curves collapse into a universal one for
different values of the screening parameter $\kappa$. This hints towards a
possible universality of our results with respect to the particle interaction
potential chosen. As we will see, this universality pertains not only the
thermodynamic properties but also the transport ones, such as shear viscosity
and thermal conductivity.

Figure 3:  Calculated normalized velocity autocorrelation function (VACF)
$C_{v}(t)$ for 2D (a) and 3D (b) fluid dusty plasmas with $\kappa=1$ and
different values of the reduced coupling strength $\Gamma/\Gamma_{m}$. The
black lines correspond to the critical value $\gamma_{c}=0.05$.

###  III.2 Velocity autocorrelation function and transport

In order to verify further our results, we compute the normalized velocity
autocorrelation function (VACF):

$C_{v}(t)=\frac{\left<{\bf v}\left(t\right)\cdot{\bf
v}\left(0\right)\right>}{\left<{\bf v}\left(0\right)\cdot{\bf
v}\left(0\right)\right>}$ (3)

for 2D and 3D fluid dusty plasmas of $\kappa=1$. The numerical results are
shown in Fig. 3. Our obtained Frenkel line condition, $\gamma_{c}=0.05$ from
the heat capacity $c_{V}$ is further confirmed by our results of the VACF in
Fig. 3. In the rigid liquid state, the VACF contains significant oscillations
due to the rigidity of the underlying phase. However, in the nonrigid gas-like
state, the VACF does not contain oscillations any more, but rather a monotonic
decrease typical of gas-like systems. Thus, in Brazhkin:2013 , the Frenkel
line is proposed to be the critical temperature at which the oscillatory
behavior in the VACF just disappears. In Brazhkin:2013 , it is also mentioned
that, for some cases, such as systems with strong repulsive interactions, this
second criterion from the VACF may be not accurate.

In our systems, for 3D fluid dusty plasmas, as $\Gamma/\Gamma_{m}$ decreases,
the oscillations of VACF gradually decay until they completely disappear at
$\Gamma/\Gamma_{m}=0.05$, perfectly matching the Frenkel line extracted from
the heat capacity criterion in Fig. 2. Besides the results for $\kappa=1$
shown in Fig. 3, we have also confirmed that, for all other simulated $\kappa$
values, the critical point at which the oscillations just disappear is always
located at $\gamma_{c}=0.05$. Thus, the Frenkel line of 3D dusty plasmas
determined by the VACF is always at $\gamma_{c}=0.05$, in agreement with the
analysis of the heat capacity in Fig. 2. For 2D dusty plasmas, the dynamical
transition of the VACF oscillations at $\Gamma/\Gamma_{m}=0.05$ is not as
distinctive as for the 3D systems. This is consistent with the disclaimers
mentioned in Brazhkin:2013  and might be attributed to the anomalous diffusion
in 2D dusty plasmas Ott:2009 , which is absent in the 3D case.

Besides the VACF results presented here, in Appendix B, we show that the
various transport coefficients, such as the diffusion coefficient, the shear
viscosity, and the thermal conductivity, calculated from our current
simulation data and the previous results of Yukawa systems or one-component
plasmas Daligault:2012 ; Daligault:2014 ; Donko:2008 ; Scheiner:2019 ;
Donko:2004 , are also optimal diagnostics for the dynamical crossover.

###  III.3 Local atomic connectivity and shear relaxation

It is an important question to ask whether there exists any physical quantity
linked to the microscopic dynamics which can identify the Frenkel line
crossover. In the Frenkel/Maxwell theories, the re-arrangement time around
local minima $\tau$ plays a fundamental role. A more microscopic
characterization of such a process is given by the local atomic connectivity
$\tau_{LC}$, which is defined as the time for one particle to maintain its
surrounding neighbors, or equivalently the time for the atomic topological
structure change Egami:2013 ; Ashwin:2015 .

Within the Maxwell viscoelasticity theory, the shear stress relaxation time
plays an equivalently important role, specially in the collective dynamics of
gapped transverse waves. The shear stress relaxation time $\tau_{M}^{\rm ex}$,
first introduced in Ashwin:2015  to quantify the viscoelastic response of
dusty plasma liquids, controls the relaxation of the excess part (the particle
interaction portion) of the shear stress autocorrelation function. Following
Ashwin:2015 , we calculate the shear stress relaxation time $\tau_{M}^{\rm
ex}$ from the ratio of the excess parts of the shear viscosity $\eta^{\rm ex}$
to the infinite frequency shear modulus $G_{\infty}^{\rm ex}$ using

$\tau_{M}^{\rm ex}=\int_{0}^{\infty}G(t)^{\rm ex}dt/G_{\infty}^{\rm ex}\,,$
(4)

where $G(t)^{\rm ex}$ comes from the autocorrelation function of the particle
interaction portion of the shear stress time series Ashwin:2015 , while
$G_{\infty}^{\rm ex}$ is just the initial value of $G(t)^{\rm ex}$ Ashwin:2015
. The shear stress relaxation time $\tau_{M}^{\rm ex}$ is approximately equal
to the Maxwell relaxation time $\tau_{M}$ when $\Gamma$ is large, in the so-
called potential energy dominated regime Ashwin:2015 .

Figure 4:  Local atomic connectivity time $\tau_{LC}$ and shear stress
relaxation time $\tau_{M}^{\rm ex}$ multiplied with the Einstein frequency
$\omega_{E}$ for various conditions of 2D (a) and 3D dusty plasmas (b).

From Fig. 4, we discover that the Frenkel line condition can be expressed as
the product of the lifetime of the local atomic connectivity $\tau_{LC}$, or
the shear stress relaxation time $\tau_{M}^{\rm ex}$, and the Einstein
frequency $\omega_{E}$ to be unity, i.e., $\tau_{LC}\omega_{E}=1$ or
$\tau_{M}^{\rm ex}\omega_{E}=1$, for 2D or 3D fluid dusty plasmas,
respectively. Here, the Einstein frequency $\omega_{E}$ refers to the
oscillation frequency of one particle in the environment where all other
particles are assumed to be frozen stationary Kalman:2004 ; Feng:2008 .

As evident from the numerical data shown in Fig. 4, $\tau_{LC}\omega_{E}=1$ in
2D systems and $\tau_{M}^{\rm ex}\omega_{E}=1$ in 3D systems both occur at
$\Gamma/\Gamma_{m}=0.05$, corresponding to the location of Frenkel line
determined from $c_{V}$ in Fig. 2 and VACF in Fig. 3 above, as well as various
transport coefficients presented in Appendix B. Thus, both $\tau_{LC}$ and
$\tau_{M}^{\rm ex}$ can be used to quantify the relaxation time $\tau$ between
two consecutive hops of a single particle Frenkel:1946 . Furthermore, the
Einstein frequency $\omega_{E}$ is just proportional to the inverse of the
minimum particle vibration period. As a result, the Frenkel’s criterion
Frenkel:1946  of $\tau/\tau_{D}\approx 1$ is qualitatively equivalent to
$\tau_{LC}\omega_{E}\approx 1$ or $\tau_{M}^{\rm ex}\omega_{E}\approx 1$,
consistent with the findings in Fig. 4. Thus, we propose the conditions
$\tau_{LC}\omega_{E}=1$ and $\tau_{M}^{\rm ex}\omega_{E}=1$ as new diagnostics
to determine the Frenkel line for supercritical fluids. To the best of our
knowledge, these conditions have not been considered in classical liquids so
far.

In Fig. 4, it is clear that both $\tau_{LC}\omega_{E}$ in 2D and
$\tau_{M}^{\rm ex}\omega_{E}$ in 3D dusty plasmas exhibit a universal collapse
in the rigid liquid-like phase. On the contrary, this universal collapse seems
to be less precise in the gas-like regime, $\Gamma/\Gamma_{m}<0.05$ where the
trend of the data depends mildly on the screening parameter $\kappa$. We also
notice that the product $\tau_{M}^{\rm ex}\omega_{E}$ in 3D dusty plasmas
displays a discontinuous derivative at the Frenkel line which deserves further
investigations.

Finally, we also calculated $\tau_{LC}\omega_{E}$ in 3D and $\tau_{M}^{\rm
ex}\omega_{E}$ in 2D fluid dusty plasmas, but in that case the universal
collapse is less clear. Comparing with the other quantities presented in this
work, which all show a universal collapse as a function of $\kappa$, we might
speculate that this universality is an emergent property of the collective
dynamics which fails, or at least fades away, for microscopic quantities. As a
matter of fact, in the gas-like regime, the definition of $\tau_{LC}$ or
$\tau_{M}$ is not able to reflect the collective dynamics any more.

Figure 5:  The ratio of the instantaneous transverse sound speed $C_{T}$ to
the average particle speed $\bar{v}_{p}$ for various conditions in 2D and 3D
dusty plasmas. The vertical dashed line indicates the location of the Frenkel
line as derived from the specific heat and the other diagnostics above.

###  III.4 Instantaneous transverse sound speed

Besides the aforementioned conditions $\tau_{LC}\omega_{E}=1$ and
$\tau_{M}^{\rm ex}\omega_{E}=1$, we discover that the unity ratio of the
instantaneous transverse sound speed $C_{T}$ to the average particle speed
$\bar{v}_{p}$ can also discriminate the liquid-like and gas-like states and
therefore be an optimal diagnostic for the location of the Frenkel line. Here,
$C_{T}$ is the instantaneous speed of transverse sound which can be derived
from the infinite frequency shear modulus $G_{\infty}$ Egami:2013 ; Huang:2022
; Hansen:1986 . We observe that the unity ratio of $C_{T}/\bar{v}_{p}$ occurs
at the same value of $\Gamma/\Gamma_{m}=0.05$ corresponding to the proposed
Frenkel line. Some of these results for 2D dusty plasmas have already appeared
in Huang:2022 . Here, new results for 3D dusty plasmas, as well as more
results of 2D dusty plasmas under different conditions are presented.

When $C_{T}/\bar{v}_{p}>1$, the transverse sound speed is faster than the
motion of individual particles, corresponding to the “cooperative dynamics
regime” Huang:2022 , which is a defining property of rigid liquids. However,
when $C_{T}/\bar{v}_{p}<1$, in the “individual dynamics regime” Huang:2022
belonging to the nonrigid gas-like state, the average speed of individual
particles is larger than the transverse sound speed. Thus, the transition
between the cooperative and individual dynamics regimes at
$C_{T}/\bar{v}_{p}=1$ coincides exactly with the rigid liquid and nonrigid
gas-like states separated by the Frenkel line. Our newly proposed criteria for
the Frenkel line, $\tau_{LC}\omega_{E}=1$, $\tau_{M}^{\rm ex}\omega_{E}=1$ and
$C_{T}/\bar{v}_{p}=1$, may be verified in the future in other physical
systems, such as Lennard-Jones and soft-sphere systems Brazhkin:2012 ;
Brazhkin:2013 .

As compared with $\tau_{LC}\omega_{E}$, $\tau_{M}^{\rm ex}\omega_{E}$,
$c_{V}$, and others, the concept of the speed ratio of collective to
individual dynamics used in the diagnostic of $C_{T}/\bar{v}_{p}=1$ appears to
be superior due to at least two reasons. First, from Fig. 5, the results of
$C_{T}/\bar{v}_{p}$ for 2D and 3D fluid dusty plasmas collapse into one
universal “master curve”. This happens independently of the dimensionality of
the system and the value of the screening parameter $\kappa$. Second, the
concept of $C_{T}/\bar{v}_{p}$ can be generalized to different physical
processes. For example, during compressional shocks in 2D dusty plasmas, a
clear transition at the condition of $v_{left}/C_{l,preshock}=1$ has been
observed in Figs. 4 and 5 of Sun:2021 , where $v_{left}$ is the drift velocity
of particles after shocks and $C_{l,preshock}$ is the longitudinal wave speed.
Clearly, the main cooperative dynamics during compressional shocks Sun:2021
is represented by $C_{l,preshock}$, and not $C_{T}$ as above. At the same
time, for compressional related dynamics, the average particle speed should be
represented by the drift velocity $v_{left}$ in the postshock region along the
shock propagation direction, and not the thermal velocity $\bar{v}_{p}$. Thus,
in analogy to the $C_{T}/\bar{v}_{p}=1$ criterion, the predicted “phase”
transition should be located at $C_{l,preshock}/v_{left}=1$, which is exactly
what is observed in Sun:2021 . Therefore, when $C_{l,preshock}/v_{left}<1$,
the compressed system exhibits a gas-like behavior in which many particles can
penetrate the shock front to enter the preshock region, as confirmed in
Qiu:2022 . By extrapolation, we do expect that the speed ratio of the
cooperative and individual dynamics may be regarded as a universal criterion
valid for different physical processes.

##  IV Discussion

In this work, MD simulations are performed to investigate the collective and
microscopic dynamics in 2D and 3D fluid dusty plasmas. We propose that, below
the well-established solid-liquid phase transition (see Fig. 1), these systems
should be regarded as supercritical liquids. To support this hypothesis, we
reveal a dynamical transition between the rigid liquid and nonrigid gas-like
states in the supercritical regime of dusty plasma, using the concept of
Frenkel line. We probe this dynamical crossover with several microscopic and
macroscopic thermodynamic and transport quantities such as the heat capacity,
the VACF, the shear viscosity, and the thermal conductivity. Furthermore, we
propose several new criteria to identify the Frenkel line,

$\tau_{LC}\,\omega_{E}=\tau_{M}^{\rm ex}\,\omega_{E}=C_{T}/\bar{v}_{p}=1\,,$
(5)

providing a new perspective into this old debate. We expect these measures to
be generally valid beyond the dusty plasma system considered here.

Figure 6:  Obtained phase diagrams for 2D (a) and 3D (b) dusty plasmas. For
both 2D and 3D dusty plasmas, the melting point $\Gamma_{m}$ for the solid-
liquid phase transition (solid blue curves) is obtained from Hartmann:2005 ;
Ohta:2000 . The dashed line indicates the location of the dynamical Frenkel
crossover, which is universally given by $\Gamma/\Gamma_{m}=0.05$.

Starting from our conjecture, expressed as the cartoon in Fig. 1, we are now
in the position to fundamentally re-define the phase diagram for 2D and 3D
dusty plasmas, by adding a new dynamical crossover into it, as shown in Fig.
6. Importantly, from all our studied diagnostics, it is found that, for both
2D and 3D dusty plasmas, the transition point between the liquid-like and gas-
like states is always at $\Gamma/\Gamma_{m}=0.05$. This value corresponds to a
temperature of $20$ times of the melting point and coincides with the recently
proposed criterion to discriminate the strong and weak coupling regimes in
dusty plasmas in Huang:2022 , identified from the behavior of the shear
viscosity. Thus, the proposed weak coupling regime of dusty plasma in
Huang:2022  is just equivalent to the gas-like state, while the proposed
strong coupling regime in Huang:2022  corresponds to the liquid-like state.
From this point of view, the criterion of $\Gamma/\Gamma_{m}=0.05$ in
Huang:2022  does contain fundamental physical significance to discriminate the
strong and weak coupling regimes and it may supersede the traditional
criterion $\Gamma=1$ Ichimaru:1982 .

The supercritical dynamics for 2D and 3D dusty plasmas revealed in this work
presents an emergent degree of universality with respect to the dimensionality
of the system and the value of the screening parameter $\kappa$, which hints
towards the existence of a general fundamental origin. It would be fruitful to
extend this analysis to different systems, potentials and conditions to
ascertain, verify, and understand this conjectured universal character.

##  V Summary

In summary, we propose a fundamental re-definition of the phase diagram of 2D
and 3D dusty plasmas by introducing a new dynamical separation between liquid-
like and gas-like phases. Our results provide a strong evidence for the
supercritical collective dynamics in strongly coupled plasmas and open the
path towards a new interpretation of their fundamental nature which could be
fruitful for the modern understanding of plasmas, classical liquids and
supercritical phases of matter.

### Acknowledgments

This work was supported by the National Natural Science Foundation of China
under Grant Nos. 12175159 and 11875199, the 1000 Youth Talents Plan, startup
funds from Soochow University, and the Priority Academic Program Development
(PAPD) of Jiangsu Higher Education Institutions. We thank Xiaqing Shi for
helpful discussions. M.B. acknowledges the support of the Shanghai Municipal
Science and Technology Major Project (Grant No.2019SHZDZX01) and the
sponsorship from the Yangyang Development Fund. M.B. would like to thank
Chulalongkorn University for the warm hospitality during the completion of
this work.

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##  Appendix A Simulation details

###  A.1 MD simulations

All results presented in our paper are obtained using MD simulations similar
to those in Liu:2005 ; Goree:2012 ; Huang:2022 . For each simulation, first we
run $10^{6}$ steps with the Nosé-Hoover thermostat, so that the simulated
system reaches the equilibrium state under the specified conditions of
$\Gamma$ and $\kappa$. Then, the Nosé-Hoover thermostat is turned off, and we
integrate the next $10^{5}$ steps to confirm the equilibrium state is
unchanged. Finally, we integrate the last $10^{6}$ steps, which is the only
used for our data analysis reported in this paper. For each simulation run,
the output is just the obtained time series of positions and velocities for
all simulated particles, which we use to determine the various physical
quantities reported in the paper. Note, for our simulation data, when the
thermostat is turned off, we always make sure that the system temperature only
mildly fluctuates, without any overall drifts.

###  A.2 Langevin simulations

Besides the MD simulations presented in the main text, we also performed
Langevin dynamical simulations Feng:2008  of 2D and 3D dusty plasmas to
confirm that our reported results are accurate. In the Langevin dynamical
simulations, the equation of motion Feng:2008  for our simulated particles is
given by

$m\ddot{\mathbf{r}}_{i}=-\nabla\Sigma_{j}\phi_{ij}-\nu
m\dot{\mathbf{r}}_{i}+\xi_{i}(t),$ (6)

where the term of $-\nabla\Sigma_{j}\phi_{ij}$ is the Yukawa repulsion between
particles $i$ and $j$, the same as in the MD simulation. The second term $\nu
m\dot{\mathbf{r}}_{i}$ is the frictional gas drag, while the last term
$\xi_{i}(t)$ is the Langevin random noise obeying the fluctuation-dissipation
theorem Pathria:2011 ; Feng:2008 .

We simulate $N=4096$ and $8192$ dust particles for 2D and 3D systems, confined
in the simulation cells with same size and with periodic boundary conditions
as in our MD simulations. The specified values of the screening parameter
$\kappa$ and the coupling parameter $\Gamma$ in our Langevin simulations are
exactly the same as those used in the MD simulations. The time step and the
total temporal duration of our Langevin simulation are both chosen to be the
same as those for our MD simulation, which are $0.005\leavevmode\nobreak\
\omega_{pd}^{-1}$ and $10^{6}$ steps, respectively. The only new parameter in
the Langevin simulation is the gas damping rate $\nu$, which is chosen as
$\nu=0.03\leavevmode\nobreak\ \omega_{pd}$, a typical value in dusty plasma
experiments Feng:2008 .

The Langevin simulation data are also used to calculate various physical
quantities to confirm our findings in the main text. Different from the
microcanonical ensemble of MD simulations, Langevin simulations correspond to
the canonical ensemble, where the temperature, or the kinetic energy, of the
simulated system is unchanged. As a consequence, Eq. (2) in the main text
cannot be used to determine $c_{V}$ anymore. Here, we use the fluctuation of
the total energy ${\rm E}$ of the simulated system to derive the specific heat
$c_{V}$ as Pathria:2011

$c_{V}=\frac{\langle{\rm E}^{2}\rangle-\langle{\rm
E}\rangle^{2}}{N(k_{B}T)^{2}}.$ (7)

Except for the specific heat $c_{V}$, all other physical quantities are
calculated using the same methods as for our MD simulations. We find that the
physical quantities from our Langevin simulations are within numerical
uncertainties the same as those reported in this paper, further confirming the
reliability of all conclusions in this paper.

##  Appendix B Transport properties

Using our simulation data, and revisiting existing data in the literature, we
quantify transport properties such as the self-diffusion coefficient, the
shear viscosity using the Green-Kubo relation Pathria:2011 , and the thermal
conductivity. As we will show next, all these quantities are also very
efficient diagnostics to discriminate the rigid liquid phase from the gas-like
phase in the supercritical regime of 2D and 3D dusty plasmas.

###  B.1 Self-diffusion

For our 3D simulation data, we calculate the diffusion coefficients $D$ using
the corresponding Green-Kubo relation Pathria:2011

$D=\frac{1}{3}\int_{0}^{\infty}\left<{\bf v}\left(t\right)\cdot{\bf
v}\left(0\right)\right>dt.$ (8)

In fact, Eq. (8) is just the time integral of the velocity autocorrelation
function (VACF) $\left<{\bf v}\left(t\right)\cdot{\bf v}\left(0\right)\right>$
for the particle motion. We do not study the diffusion coefficient for our 2D
simulations. The reason is that as lower dimensional systems, 2D dusty plasmas
exhibit anomalous diffusion, as reported in Ott:2009 , which could strongly
affect our analysis and deserves a separate investigation.

Figure 7: Obtained normalized self-diffusion coefficient $D$ (a) and the
corresponding logarithmic derivative
$d[ln(D/(n^{-\frac{1}{3}}\bar{v}_{p}))]/d[ln(\Gamma/\Gamma_{m})]$ (b) for our
3D dusty plasma simulations, as well as those performed in Daligault:2012 .
The dashed line indicates the critical line, $\Gamma/\Gamma_{m}=0.05$, as
probed by other diagnostics.

Our calculated self-diffusion coefficients for 3D dusty plasma simulations
when $\kappa=1$, $2$, and $3$ are presented in Fig. 7(a) as a function of
$\Gamma/\Gamma_{m}$. Following the elementary kinetic theory for a dense
medium of particles Rosenfeld:2001 , we normalize the obtained self-diffusion
coefficient $D$ using $n^{-\frac{1}{3}}\bar{v}_{p}$. Besides our diffusion
results, in Fig. 7(a), we also plot the data of 3D Yukawa one-component
plasmas reported in Daligault:2012 , which well agree with our 3D dusty
results. Above the dynamical crossover line, in the rigid liquid phase,
$\Gamma/\Gamma_{m}>0.05$, all data for 3D Yukawa systems, and even for 3D
Coulomb system for $\kappa=0$, collapse into a single universal curve. This
universality appears less evident in the gas-like phase in which different
curves deviate from each other especially for very small values of
$\Gamma/\Gamma_{m}$.

The self-diffusion coefficient $D$ clearly exhibits a monotonic decreasing
trend as the coupling parameter $\Gamma$ increases. Given that $\Gamma$ has to
be interpreted as an inverse effective temperature, this trend is reasonable.
When the temperature is lowered, the diffusion is obviously suppressed. More
importantly, we observe that the Frenkel dynamical crossover corresponds to a
saddle point in the self-diffusion data as a function of $\Gamma$ which
locates the condition of lowest decreasing rate.

To better characterize this important feature, we use the logarithmic
derivative $d[ln(D/(n^{-\frac{1}{3}}\bar{v}_{p}))]/d[ln(\Gamma/\Gamma_{m})]$
obtained from the data points in panel (a) of Fig. 7. The resulting data are
presented in panel (b) of Fig. 7. We observe a clear maximum in the
logarithmic derivative which occurs exactly at $\Gamma/\Gamma_{m}=0.05$,
further confirming the lowest decreasing rate of the normalized diffusion
coefficient $D/(n^{-\frac{1}{3}}\bar{v}_{p})$ in Fig. 7(a). From both panels
in Fig. 7, we confirm that the self-diffusion coefficient represents as well
an efficient indicator for the liquid-like to gas-like dynamical crossover in
3D dusty plasmas, validating once more the universal value of
$\gamma_{c}=0.05$. To the best of our knowledge, this criterion has not been
discussed yet in the supercritical fluid community. Finally, notice that,
using the data of Daligault:2012 , also shown in Fig. 7 the supercritical
transition at $\Gamma/\Gamma_{m}=0.05$ is valid for 3D Coulomb systems as
well.

###  B.2 Shear viscosity

Using the Green-Kubo relation, the shear viscosity is obtained from the shear
stress fluctuations Pathria:2011 . First, we calculate the time series of the
shear stress from the velocities and locations of all simulated particles
using

$P_{xy}=\sum_{i=1}^{N}\left[mv_{ix}v_{iy}-\frac{1}{2}\sum_{j\neq
i}^{N}\frac{x_{ij}y_{ij}}{r_{ij}}\frac{\partial\phi\left(r_{ij}\right)}{\partial
r_{ij}}\right].$ (9)

Second, we obtain the autocorrelation function of the shear stress fluctuation
$C_{s}$ using

$C_{s}(t)=\left\langle P_{xy}(t)P_{xy}(0)\right\rangle.$ (10)

Finally, the shear viscosity is derived using the time integral below

$\eta=\frac{1}{Vk_{B}T}\int_{0}^{\infty}C_{s}(t)dt,$ (11)

where $V$ is the volume of the simulated 3D system, or the area of the
simulated 2D system.

Figure 8: The normalized shear viscosity $\eta$ for 2D and 3D dusty plasmas as
a function of the coupling parameter $\Gamma$ varies from our MD simulations.
The original viscosity results for 3D Yukawa fluids and one component plasmas
reported in Donko:2008 ; Daligault:2014  are also plotted here for comparison.
The minimum of the normalized shear viscosity always occurs at
$\Gamma/\Gamma_{m}=0.05$, as shown by the dashed line.

Our calculated results for the shear viscosity in 2D and 3D dusty plasmas are
presented in Fig. 8 as a function of $\Gamma/\Gamma_{m}$ and for different
values of $\kappa$. Following Rosenfeld:2001 , we normalize the obtained shear
viscosity $\eta$ using $m\bar{v}_{p}n^{\frac{1}{2}}$ and
$m\bar{v}_{p}n^{\frac{2}{3}}$ for 2D and 3D dusty plasmas, respectively.
Besides our viscosity results, in Fig. 8, we also plot the viscosity results
for 3D Yukawa fluids and one component plasmas reported in Donko:2008 ;
Daligault:2014 , all well agreeing with our results. From Fig. 8, all data for
both 2D and 3D dusty plasmas, as well as 3D Coulomb systems, collapse into one
universal curve in the liquid like phase, $\Gamma/\Gamma_{m}>0.05$. In the
gas-like phase, we observe mild deviations from this universal master curve
which deserve further investigations. More importantly, for all conditions in
the different systems in 2D and 3D, the minimum of the normalized shear
viscosity always occurs at $\Gamma/\Gamma_{m}=0.05$, the location of the
dynamical liquid-like to gas-like crossover.

In summary, we find that shear transport is a promising macroscopic quantity
to locate the Frenkel line in dusty plasmas as well. In order to rationalize
this finding, we resort to classical arguments already presented in
doi:10.1126/sciadv.aba3747 . From the standard kinetic theory, in the gas-like
regime, one expects the viscosity to be given by $\eta\sim\rho v_{p}L$ where
$\rho$ is the density, $v_{p}$ the average particle velocity, and $L$ the mean
free path. Since $v_{p}$ increases with temperature, $v_{p}\propto\sqrt{T}$,
the gas viscosity increases as well. On the contrary, in a liquid phase, the
viscosity arises from a quite different underlying dynamics and it decreases
with temperature as $\eta\sim\exp\left(U/T\right)$, where $U$ is the
activation energy. This immediately implies the existence of a minimum in the
viscosity which indicates the crossover between the liquid-like to gas-like
behavior. In addition, it turns out that the value of the viscosity at the
minimum is universal and given by simple fundamental constants
doi:10.1126/sciadv.aba3747 . It would be interesting to explore further the
value of the viscosity at the minimum in strongly coupled plasmas and compare
to the results of doi:10.1126/sciadv.aba3747 . A connection between the
minimum of the viscosity in classical liquids and in the strongly-coupled
quark-gluon plasma has indeed already been made in
10.21468/SciPostPhys.10.5.118 .

###  B.3 Thermal conductivity

A similar argument to the one just presented for the shear viscosity can be
made for energy transport and in particular thermal conductivity/diffusivity
PhysRevB.103.014311 . For this reason, we revisit the thermal conductivity
data for 3D Yukawa fluids and 3D one-component plasmas presented respectively
in Donko:2004  and Scheiner:2019 . In Scheiner:2019 , the thermal conductivity
is obtained from the fluctuation of the heat flux using the corresponding
Green-Kubo relation. In principle, we are also able to determine the thermal
conductivity with our simulation data. However, from the suggested time
duration needed for the thermal conductivity calculation Scheiner:2019 , our
simulation run is not long enough, which may lead to large uncertainties.
Thus, we only focus on the results reported in Donko:2004 ; Scheiner:2019 . As
we will see, those results reveal important physical lessons which were not
discussed in the original papers.

We conveniently reproduce the data in Fig. 9. Here, we normalize the thermal
conductivity $\lambda$ using $n^{\frac{2}{3}}k_{B}\bar{v}_{p}$ Rosenfeld:2001
.

Figure 9: The normalized thermal conductivity for 3D Yukawa fluids and 3D one-
component plasmas as the coupling parameter $\Gamma$ varies. The corresponding
original data are reported in Donko:2004 ; Scheiner:2019 . The minimum of the
normalized thermal conductivity always occurs at $\Gamma/\Gamma_{m}=0.05$, as
indicated by the dashed line.

From Fig. 9, the minimum of the normalized thermal conductivity also occurs at
$\Gamma/\Gamma_{m}=0.05$, which is the same value as the one extracted from
the shear viscosity in Fig. 8 and the diffusion coefficient in Fig. 7. This
value also corresponds to the proposed Frenkel line crossover probed with
several diagnostics in the main text.

In summary, we find that not only the self-diffusion coefficient and the shear
viscosity but also the thermal conductivity displays a distinct feature, i.e.,
a clear minimum, at the dynamical crossover between liquid-like and gas-like
states. Interestingly, the universal collapse observed for other quantities is
not so clear in the thermal conductivity data. A more detailed exploration in
this direction would be fruitful.


###### Abstract

The Compact Linear Collider (CLIC) is a proposed high-luminosity linear
electron-positron collider at the energy frontier. It is foreseen to be built
and operated in three stages, with a centre-of-mass energy ranging from a few
hundred GeV up to 3 TeV. The main beam-induced background impacting CLIC
physics analyses is produced by beamstrahlung radiation from the electron and
positron bunches traversing the high field of the opposite beam and converting
to hadrons, $\mathup{}\mathup{}\to\mathrm{hadron}$ events. The timing
selections are a powerful tool to discriminate signal from background events
at CLIC. For each CLIC stage, three sets of selections, Loose, Selected, and
Tight, are defined to allow the analysis of different signal topologies. The
selections are defined depending on the particle type and the reconstructed
polar angle of each particle flow object reconstructed using the particle flow
analysis. As a first step, the performance of the timing selections currently
defined for the CLIC $380\text{\,}\GeV$ energy stage are evaluated using a
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ sample decaying mainly
into light quarks as signal and $\mathup{}\mathup{}\to\mathrm{hadron}$ events
as background. As a result, after applying the selections the level of
background is significantly reduced, down to few GeV, while the shift of the
signal peak to lower energy is kept within a few percent. The cuts in the
Loose selection are then relaxed to search for possible improvements: in one
of the options considered, the signal component is partially recovered but
also the background energy mean remains unchanged with respect to not applying
any cuts. In conclusion, no change is foreseen on the timing selections.

##  1 Introduction

The Compact Linear Collider (CLIC) is a proposed high-luminosity linear
electron–positron collider at the energy frontier [1, 2]. It is foreseen to be
built and operated in three stages with increasing centre-of-mass energies. In
the first stage, CLIC will mainly operate at a centre-of-mass energy of
$380\text{\,}\mathrm{GeV}$ with the aim of measuring the properties of the top
quark and the Higgs boson with high precision. The accelerator parameters for
the first stage are reported in Table 1. The background at CLIC is produced by
beamstrahlung radiation from the electron and positron bunches traversing the
high field of the opposite beam. Photons then convert to two main types of
background, incoherent
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
pairs and $\mathup{}\mathup{}\to\mathrm{hadron}$ events. At the
$380\text{\,}\mathrm{GeV}$ energy stage, 0.18
$\mathup{}\mathup{}\to\mathrm{hadron}$ interactions occur on average per bunch
crossing with centre-of-mass energy more than $2\text{\,}\mathrm{GeV}$ [3].

To face the challenge of the beam-induced background and to fulfil the physics
requirements, a detector model with a highly segmented calorimetry system
optimized for particle flow techniques and timing capabilities has been
defined and optimized using a dedicated software suite [4]. The CLICdet layout
follows the typical collider detector scheme: the innermost part is composed
of a silicon pixel vertex detector and a silicon tracker; surrounding them, an
electromagnetic and hadronic calorimeter are placed, all embedded inside a
superconducting solenoid providing a $4\text{\,}\mathrm{T}$ field; in the
outermost part an iron yoke is interleaved with muon chambers. A longitudinal
cross section of CLICdet is shown in Figure 1.

CLICdet will operate in a triggerless readout mode, i.e. the entire bunch
train composed of 352 bunches at the $380\text{\,}\mathrm{GeV}$ energy stage
will be read out every $20\text{\,}\mathrm{ms}$. In the bunch train, the
bunches are separated by $0.5\text{\,}\mathrm{ns}$. At most, one hard
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
physics interaction in an entire bunch train will be produced, while most of
the bunch crossings will only produce background particles. To obtain
excellent detector performance in this triggerless readout mode, the
subdetectors must provide a precise hit timing information. This can be
achieved with a time-stamping capability of $10\text{\,}\mathrm{ns}$ for all
silicon tracking elements and a hit time resolution of $1\text{\,}\mathrm{ns}$
for all calorimeter hits. Studies have shown [1] that at the highest energy
stage of CLIC, the timing information coming from the subdetectors combined
with additional $p_{\text{T}}$ information can help to mitigate the impact of
the $\mathup{}\mathup{}\to\mathrm{hadrons}$ background. In the context of a
$380\text{\,}\mathrm{GeV}$ CLIC collider, a preliminary set of timing
selections are already defined but the performance is studied precisely for
the first time in this note. Therefore, as a first step the performance of the
timing selections currently used for the first CLIC stage are evaluated in
terms of absolute acceptance of the signal and rejection of the background;
while in a second step, the loosest selection available is relaxed to check if
further optimization is possible. Since a precise measurement of the top quark
is one of the most important parts in the CLIC physics programme at the first
stage, a pair of top quarks produced at the
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
interaction point is used as the signal.

The software framework and event simulation used in this study is briefly
described in Section 2. In Section 3 an introduction to the timing selections
is given with particular emphasis on those used for the first CLIC stage.
Further possible optimizations for the loose selection are described in
Section 4. In Section 5 this study is summarized and conclusions are given.

Table 1: Parameters for the first stage of CLIC. Parameter | Stage 1
---|---
Centre-of-mass energy [$\mathrm{GeV}$] | 380
Main tunnel length [$\mathrm{km}$] | 11.4
Repetition frequency [$\mathrm{Hz}$] | 50
Number of bunches | 352
Bunch separation [$\mathrm{ns}$] | 0.5
Number of particles per bunch [${10}^{9}$] | 5.2
IP beam size (horizontal) [$\mathrm{nm}$] |  $\simeq$149
IP beam size (vertical) [$\mathrm{nm}$] | 2.9
Bunch length [$\mathrm{\SIUnitSymbolMicro m}$] | 70
Total luminosity [${10}^{34}\text{\,}{\mathrm{cm}}^{-2}\text{\,}{\mathrm{s}}^{-1}$] | 1.5
Luminosity above 99% of $\sqrt{s}$ [${10}^{34}\text{\,}{\mathrm{cm}}^{-2}\text{\,}{\mathrm{s}}^{-1}$] | 0.9
Number of beamstrahlung photons per beam particle | 1.4
Number of $\mathup{}\mathup{}\to\mathrm{hadron}$ events per BX ($\sqrt{s}>$2\text{\,}\mathrm{GeV}$$) | 0.18
\begin{overpic}[width=346.89731pt]{figures/CLICdp_QuarterTop_view_HD_with_end_coils_Nov_2016.png}
\put(30.0,77.5){\huge{Yoke + muon chambers}} \put(20.0,54.0){\huge{Solenoid}}
\put(10.0,35.0){\huge{HCAL}} \put(12.0,22.5){\large{ECAL}}
\put(5.0,13.5){\color[rgb]{1,1,1}\large{Tracker}}
\put(0.0,0.0){\color[rgb]{1,1,1}\small{Vertex}} \end{overpic} Figure 1:
Longitudinal cross section of CLICdet [5].

##  2 Simulation and reconstruction

The CLICdet detector geometry is described with the DD4hep software framework
[6] and simulated in Geant4 [7] via the DDG4 [8] package of DD4hep.

The simulated events are generated mostly using the WHIZARD [9, 10] program
assuming zero polarization of the electron and proton beam. The initial states
are created in the GuineaPig [11] simulation of the CLIC collisions. The
performance of the timing selections for the $380\text{\,}\mathrm{GeV}$ CLIC
stage is assessed using a 6-fermion production, comprising predominantly
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events from hard
Standard Model
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
interactions. The six fermions in the final state correspond to
$\mathup{{{d}}}\mathup{{{d}}}\mathup{{{u}}}yy\mathup{{{u}}}$, where $y$ can be
b, d or s quarks. The simulation of the parton showering, hadronisation and
fragmentation is performed using PYTHIA [12]. This component of the event is
called signal. The $\mathup{}\mathup{}\to\mathrm{hadron}$ events are
hadronized in PYTHIA and the amount correspondent to 30 bunch crossings (BX)
is included. This component of the event is referred to as background.

The reconstruction software is implemented in the linear collider Marlin-
framework [13]. The first step of the reconstruction is to overlay the signal
event with the expected number of $\mathup{}\mathup{}\to\mathrm{hadron}$
background events for 30 bunch crossings. The signal event is placed in bunch
crossing 11 at $t=0\leavevmode\nobreak\ $\mathrm{ns}$$. At this stage, only
the energy deposits inside the timing window of $10\text{\,}\mathrm{ns}$ after
the signal event are selected. This time window matches the expected detector
timing resolutions and integration times. In the next step, the positions of
the energy deposits in the tracking detectors are smeared and the energy of
the calorimeter hits are scaled with the calibration constants. At this point
the tracking algorithms are run to obtain reconstructed tracks and the
particle flow approach by PandoraPFA [14, 15, 16] is used to build calorimeter
clusters and match them to tracks. All visible particles are reconstructed by
PandoraPFA using both reconstructed tracks and calorimeter hits. The particles
are called particle flow objects (PFOs). A PFO is marked as signal if any hit
in its cluster are coming from a MC Particle of the
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ event. If this
requirement is not satisfied, it is marked as background. A more detailed
description of the reconstruction of single particles and more complex events
is given in [5].

##  3 Timing selections

To reject the beam-induced background without impacting the physics
performance, the time-stamping capabilities of CLICdet described in Section 1
are used. Simply imposing tight timing selections at the hit level does not
provide a final solution due to the fact that it does not take into account
the hadronic shower development time and the time-of-flight corrections for
lower momentum particles. Only the combination of both timing capabilities of
the high-granularity calorimeters and particle flow reconstruction allows one
to obtain a precise time-stamp for the calorimeter clusters. Once the particle
flow clustering is finished, $p_{\text{T}}$ dependent timing selections are
applied. The timing selections are therefore also called $p_{\text{T}}$ vs.
time selections and their main aim is to reduce the background at the minimum
possible while keeping invariant the signal. They are imposed to each
reconstructed PFO and are defined depending on the particle type and its
reconstructed polar angle. The reconstructed polar angle is shown in Figure 2
for both the signal and the $\mathup{}\mathup{}\to\mathrm{hadrons}$.

Figure 2: Reconstructed polar angle of all PFO reconstructed in the event,
marked as signal (blue) and the $\mathup{}\mathup{}\to\mathrm{hadrons}$ (red).

PFOs are divided into three exclusive categories – photons, neutral hadrons
and charged particles – and accepted or rejected based on the time of the
calorimeter clusters and their reconstructed transverse momentum. The cluster
time is computed using the truncated energy weighted mean time of its
calorimeter hits. By default, the cluster time is computed using HCAL
clusters. ECAL clusters are used only in the following cases:

  * •

if the PFO is a photon;

  * •

if the number of ECAL hits reconstructed for the PFO is more than five;

  * •

if the number of ECAL hits reconstructed for the PFO is more or equal than
half of the number of HCAL hits.

The cluster time is corrected using the time-of-flight information coming from
the associated track in the case of a charged particle, and the time-of-flight
of a straight path in the case of neutrals.

In Figure 3 the cluster time as a function of the $p_{\text{T}}$ of the three
particle categories for both the signal and the background components of
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events is shown for the
$380\text{\,}\mathrm{GeV}$ CLIC collider. Given the large overlap of the
signal and background distributions, it is clear that the search for highly
performant selections is a challenging task. Three sets of $p_{\text{T}}$ vs.
time selections are therefore created to allow the analysis of different
signal topologies: Loose, Selected, and Tight. In all selections, no photon or
neutral hadron with $p_{\text{T}}$ more than $2\text{\,}\mathrm{GeV}$ is
rejected, and no timing cut is applied on charged particles if their
$p_{\text{T}}$ is more than $4\text{\,}\mathrm{GeV}$ in the case of Loose and
Selected selections, and more than $3\text{\,}\mathrm{GeV}$ in the case of
Tight one. The three selections for the $380\text{\,}\mathrm{GeV}$ CLIC stage
are listed in Tables 4, 4 and 4. The effect of the three timing selections on
the reconstructed energy for both the signal and the background components of
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events with overlay of
30 BX of $\mathup{}\mathup{}\to\mathrm{hadrons}$ background is shown in Figure
4.

Table 2: Loose selections used for the $380\text{\,}\mathrm{GeV}$ CLIC stage.

| $|\cos\theta|\leq 0.975$ |  | $|\cos\theta|>0.975$
---|---|---|---
|  $p_{\text{T}}$ range [$\mathrm{GeV}$] | time cut [$\mathrm{ns}$] |  |  $p_{\text{T}}$ range [$\mathrm{GeV}$] | time cut [$\mathrm{ns}$]
Photons | $0.75\leq p_{\text{T}}{}<2.00$ | $t<10.0$ |  | $0.75\leq p_{\text{T}}{}<2.00$ | $t<2.00$
$0.00\leq p_{\text{T}}{}<0.75$ | $t<2.50$ |  | $0.00\leq p_{\text{T}}{}<0.75$ | $t<1.00$
Neutral hadrons | $0.75\leq p_{\text{T}}{}<2.00$ | $t<10.0$ |  | $0.75\leq p_{\text{T}}{}<2.00$ | $t<10.0$
$0.00\leq p_{\text{T}}{}<0.75$ | $t<5.00$ |  | $0.00\leq p_{\text{T}}{}<0.75$ | $t<5.00$
| $0\leq|\cos\theta|\leq 1$
|  |  $p_{\text{T}}$ range [$\mathrm{GeV}$] |  | time cut [$\mathrm{ns}$] |
Charged particles |  | $0.75\leq p_{\text{T}}{}<4.00$ |  | $t<10.0$ |
| $0.00\leq p_{\text{T}}{}<0.75$ |  | $t<5.00$ |

Table 3: Selected selections used for the $380\text{\,}\mathrm{GeV}$ CLIC
stage.

| $|\cos\theta|\leq 0.975$ |  | $|\cos\theta|>0.975$
---|---|---|---
|  $p_{\text{T}}$ range [$\mathrm{GeV}$] | time cut [$\mathrm{ns}$] |  |  $p_{\text{T}}$ range [$\mathrm{GeV}$] | time cut [$\mathrm{ns}$]
Photons | $0.75\leq p_{\text{T}}{}<2.00$ | $t<5.0$ |  | $0.75\leq p_{\text{T}}{}<2.00$ | $t<2.0$
$0.00\leq p_{\text{T}}{}<0.75$ | $t<1.0$ |  | $0.00\leq p_{\text{T}}{}<0.75$ | $t<1.0$
Neutral hadrons | $0.75\leq p_{\text{T}}{}<2.00$ | $t<5.0$ |  | $0.75\leq p_{\text{T}}{}<2.00$ | $t<4.0$
$0.00\leq p_{\text{T}}{}<0.75$ | $t<2.5$ |  | $0.00\leq p_{\text{T}}{}<0.75$ | $t<2.0$
| $0\leq|\cos\theta|\leq 1$
|  |  $p_{\text{T}}$ range [$\mathrm{GeV}$] |  | time cut [$\mathrm{ns}$] |
Charged particles |  | $0.75\leq p_{\text{T}}{}<4.00$ |  | $t<10.0$ |
| $0.00\leq p_{\text{T}}{}<0.75$ |  | $t<3.00$ |

Table 4: Tight selections used for the $380\text{\,}\mathrm{GeV}$ CLIC stage.

| $|\cos\theta|\leq 0.975$ |  | $|\cos\theta|>0.975$
---|---|---|---
|  $p_{\text{T}}$ range [$\mathrm{GeV}$] | time cut [$\mathrm{ns}$] |  |  $p_{\text{T}}$ range [$\mathrm{GeV}$] | time cut [$\mathrm{ns}$]
Photons | $0.75\leq p_{\text{T}}{}<2.00$ | $t<1.0$ |  | $0.75\leq p_{\text{T}}{}<2.00$ | $t<2.0$
$0.00\leq p_{\text{T}}{}<0.75$ | $t<1.0$ |  | $0.00\leq p_{\text{T}}{}<0.75$ | $t<1.0$
Neutral hadrons | $0.75\leq p_{\text{T}}{}<2.00$ | $t<4.0$ |  | $0.75\leq p_{\text{T}}{}<2.00$ | $t<2.0$
$0.00\leq p_{\text{T}}{}<0.75$ | $t<2.0$ |  | $0.00\leq p_{\text{T}}{}<0.75$ | $t<2.0$
| $0\leq|\cos\theta|\leq 1$
|  |  $p_{\text{T}}$ range [$\mathrm{GeV}$] |  | time cut [$\mathrm{ns}$] |
Charged particles |  | $0.75\leq p_{\text{T}}{}<3.00$ |  | $t<4.0$ |
| $0.00\leq p_{\text{T}}{}<0.75$ |  | $t<2.00$ |

Figure 3: Cluster time as a function of the $p_{\text{T}}$ of the three
particle categories (photons at the top, neutral hadrons in the middle and
charged PFOs at the bottom) for both the signal (left) and the background
(right) components of $\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$
events with overlay of 30 BX of $\mathup{}\mathup{}\to\mathrm{hadrons}$
background for the $380\text{\,}\mathrm{GeV}$ CLIC stage. Figure 4:
Reconstructed energy distribution for all PFOs (solid line), and for PFOs
passing the different timing cuts for signal (dashed line) and background
(dotted line): without any selection (red), Loose (orange), Selected (green)
and Tight (blue). $\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$
events are used with overlay of 30 BX of
$\mathup{}\mathup{}\to\mathrm{hadrons}$ background for the
$380\text{\,}\mathrm{GeV}$ CLIC stage.

Figure 5: Signal and background reconstructed energy mean for different
applied timing selections: without any selection (red), Loose (orange),
Selected (green) and Tight (blue).
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events are used with
overlay of 30 BX of $\mathup{}\mathup{}\to\mathrm{hadrons}$ background for the
$380\text{\,}\mathrm{GeV}$ CLIC stage.

The mean of the energy distribution for the overall distribution of signal and
background is shown in Figure 5. The mean of the energy distribution after the
selections is shown in Figures 6 and 7 for each particle category and polar
angle group for the signal component and the overlay, respectively. It can be
noted that for PFOs with $\cos\theta>0.975$ the reduction of the background is
obtained with almost no loss in the signal, in particular in the case of
photons. The level of background is reduced more with the Tight selection than
with the other two selections, but more signal is lost. In the case of the
charged PFOs with $\cos\theta\leq 0.975$, the signal loss is almost
$10\text{\,}\mathrm{GeV}$ in the extreme case of the Tight selection. When
considering the Loose selection, the mean of the signal energy distribution is
shifted less than $1\text{\,}\mathrm{GeV}$ in the case of photons and neutral
hadrons, while in the case of charged PFOs it is almost
$2\text{\,}\mathrm{GeV}$. Therefore, in the next section, the $p_{\text{T}}$
vs. time selections on the charged PFOs are modified in the attempt of
recovering this loss.

Figure 6: Signal reconstructed energy mean for the three particle categories
(photons at the top, neutral hadrons in the middle and charged PFOs at the
bottom) for different applied timing selections: without any selection (red),
Loose (orange), Selected (green) and Tight (blue).
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events are used with
overlay of 30 BX of $\mathup{}\mathup{}\to\mathrm{hadrons}$ background for the
$380\text{\,}\mathrm{GeV}$ CLIC stage.

Figure 7: Background reconstructed energy mean for the three particle
categories (photons at the top, neutral hadrons in the middle and charged PFOs
at the bottom) for different applied timing selections: without any selection
(red), Loose (orange), Selected (green) and Tight (blue).
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events are used with
overlay of 30 BX of $\mathup{}\mathup{}\to\mathrm{hadrons}$ background for the
$380\text{\,}\mathrm{GeV}$ CLIC stage.

##  4 Optimization of timing selections for the $380\text{\,}\mathrm{GeV}$
CLIC stage

To check for possible optimizations, the following relaxed Loose selections
are applied:

  * •

increasing the time cut from $5\text{\,}\mathrm{ns}$ to
$6\text{\,}\mathrm{ns}$ in the charged PFO category, referred to as time$+$;

  * •

increasing the time cut from $5\text{\,}\mathrm{ns}$ to
$8\text{\,}\mathrm{ns}$ in the charged PFO category, referred to as time$++$;

  * •

decreasing the $p_{\text{T}}$ cut from $0.75\text{\,}\mathrm{GeV}$ to
$0.5\text{\,}\mathrm{GeV}$ for all categories, referred to as
$p_{\text{T}}$$-$;

  * •

decreasing the $p_{\text{T}}$ cut from $0.75\text{\,}\mathrm{GeV}$ to
$0\text{\,}\mathrm{GeV}$ for all categories, referred to as
$p_{\text{T}}$$--$.

They are compared to the case where no selection is applied and to the Loose
selections listed in Table 4, also referred to as vanilla selections.

The comparison of the energy mean of the relaxed cuts to the vanilla one can
be found in Figure 8 for all PFOs contained in each
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ event and in Figures 9
and 10 for the three particle categories divided into signal and background.
The effect of relaxing the cuts is an increase of the background component.
The $p_{\text{T}}$$--$ selection is the only one recovering the signal
component but also leaving the level of background essentially unchanged with
respect to not applying any cuts. This last observation is valid only for
photons and neutral hadrons, but not for charged PFOs. In fact, part of the
signal component of the charged PFOs suffers from a mis-correction of the PFO
time in the case where the track and the calorimeter clusters associated with
the charged PFO are not produced by the same simulated particle. Therefore,
this signal component cannot be recovered with a simple modification to the
$p_{\text{T}}$ vs. time selections. In conclusion, this study shows that no
further optimization is possible on top of the Loose selections currently
defined for the $380\text{\,}\mathrm{GeV}$ CLIC accelerator.

Figure 8: Signal and background reconstructed energy mean for different
relaxed cuts on the Loose selection: time$+$, time$++$, $p_{\text{T}}$$-$,
$p_{\text{T}}$$--$, from left to right respectively.
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events are used with
overlay of 30 BX of $\mathup{}\mathup{}\to\mathrm{hadrons}$ background for the
$380\text{\,}\mathrm{GeV}$ CLIC stage.

Figure 9: Signal reconstructed energy mean for the three particle categories
(photons at the top, neutral hadrons in the middle and charged PFOs at the
bottom) for different relaxed cuts on the Loose selection: time$+$, time$++$,
$p_{\text{T}}$$-$, $p_{\text{T}}$$--$, from left to right respectively.
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events are used with
overlay of 30 BX of $\mathup{}\mathup{}\to\mathrm{hadrons}$ background for the
$380\text{\,}\mathrm{GeV}$ CLIC stage.

Figure 10: Background reconstructed energy mean for the three particle
categories (photons at the top, neutral hadrons in the middle and charged PFOs
at the bottom) for different relaxed cuts on the Loose selection: time$+$,
time$++$, $p_{\text{T}}$$-$, $p_{\text{T}}$$--$, from left to right
respectively. $\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events
are used with overlay of 30 BX of $\mathup{}\mathup{}\to\mathrm{hadrons}$
background for the $380\text{\,}\mathrm{GeV}$ CLIC stage.

##  5 Conclusions

In this study the performance of the timing selections defined for the CLIC
$380\text{\,}\mathrm{GeV}$ energy stage is evaluated.
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events coming from hard
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
interactions and decaying mainly into light quarks are simulated and
reconstructed as signal. At the same time a realistic estimate of the
$\mathup{}\mathup{}\to\mathrm{hadrons}$ is included as background. The mean of
the total energy reconstructed in all events is chosen as a figure-of-merit
for this study. The results for the selection for the CLIC
$380\text{\,}\mathrm{GeV}$ collider are found to be satisfactory: after
applying the selections, the level of background is significantly reduced down
to a few GeV, while the energy mean of the signal distribution is only few
percent. In the extreme case of the Tight selection for example, the
background is reduced from about $45\text{\,}\mathrm{GeV}$ down to about
$8\text{\,}\mathrm{GeV}$ and less than 4% is lost in the signal energy mean.
In a second step of this study, the cuts in the Loose selection are relaxed to
search for possible improvements. Among all the options considered, only the
$p_{\text{T}}$$--$ relaxed selection partially recovers the signal component
but also leaves the background energy mean unchanged with respect to not
applying any cuts. Therefore, no change is foreseen on the timing selections
for the CLIC $380\text{\,}\mathrm{GeV}$ energy stage.

## Acknowledgements

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

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