# Thermodynamics of a rotating hadron resonance gas with van der Waals
interaction

Kshitish Kumar Pradhan  Bhagyarathi Sahoo  Dushmanta Sahu  Raghunath
Sahoo111Corresponding author<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Indore, Simrol, Indore 453552, India

###### Abstract

Studying the thermodynamics of the systems produced in ultra-relativistic
heavy-ion collisions is crucial in understanding the QCD phase diagram.
Recently, a new avenue has opened regarding the implications of large initial
angular momentum and subsequent vorticity in the medium evolution in high-
energy collisions. This adds a new type of chemical potential into the
partonic and hadronic systems, called the rotational chemical potential. We
study the thermodynamics of an interacting hadronic matter under rotation,
formed in an ultra-relativistic collision. We introduce attractive and
repulsive interactions through the van der Waals equation of state.
Thermodynamic properties like the pressure ($P$), energy density
($\varepsilon$), entropy density ($s$), trace anomaly
($(\varepsilon-3P)/T^{4}$), specific heat ($c_{\rm v}$) and squared speed of
sound ($c_{\rm s}^{2}$) are studied as functions of temperature ($T$) for zero
and finite rotation chemical potential. The charge fluctuations, which can be
quantified by their respective susceptibilities, are also studied. The
rotational (spin) density corresponding to the rotational chemical potential
is explored. In addition, we explore the possible liquid-gas phase transition
in the hadron gas with van der Waals interaction in the $T$ – $\omega$ phase
space.

##  I Introduction

There have been intense investigations has been going on to understand the
behavior of the strongly interacting matter produced in the ultra-relativistic
heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) at BNL and
the Large Hadron Collider (LHC) at CERN. Such matter can be described by
Quantum Chromodynamics (QCD). According to QCD, a smooth crossover phase
transition is expected at high temperatures ($T$) and vanishing baryochemical
potentials ($\mu_{\rm B}$) region in the QCD phase diagram, which is described
by the RHIC and LHC experiments. As one goes towards low temperatures, high
baryochemical potential region, the phase transition becomes a first-order
one. These phase transition lines meet at a hypothesized critical endpoint
(CEP), which has been one of the most exciting topics of discussion in the
high-energy physics community. To gather valuable and reliable information
about the QCD matter, lattice QCD (lQCD) has been the most successful theory
based on first principles. However, at non-vanishing chemical potential, the
lQCD breaks down because of the fermion sign problem Borsanyi:2013bia ;
HotQCD:2014kol . Nevertheless, there have been significant attempts to bypass
this problem indirectly Allton:2005gk ; deForcrand:2002hgr ; Fodor:2001pe ,
but the issue still largely persists. The more simplistic Hadron Resonance Gas
(HRG) model effectively explains the QCD matter behavior and matches the lQCD
data up to temperature ($T\simeq 150$ MeV) Bellwied:2013cta ; HotQCD:2012fhj ;
Bellwied:2017ttj . The HRG model works at both zero baryochemical potential
and baryon-rich environments and successfully explains the hadron yields from
heavy-ion collisions using only two parameters, the temperature, and the
baryochemical potential. However, after $T\simeq 150$ MeV, the hadrons start
to melt, and the results from the HRG model start deviating from the lQCD
estimations. The HRG model also breaks down while estimating the higher-order
fluctuations and correlations of conserved charges Bazavov:2013dta . It has
been observed that repulsive interactions between the hadrons can
substantially affect the behavior of thermodynamic and transport properties,
particularly the higher order fluctuations Pal:2020ucy ; Bhattacharyya:2013oya
. The repulsive interactions can be incorporated into the hadron gas by a van
der Waals type repulsion where the hardcore radius of the hadrons serves as
the repulsion in the Excluded Volume Hadron Resonance Gas (EVHRG) model or
through a repulsive mean field potential which is introduced in the hadron gas
in the Relativistic Mean-Field Hadron Resonance Gas (RMFHRG) model. Recently,
an interacting hadron resonance gas model was introduced where both long-
distance attraction and short-distance repulsion between the hadrons was taken
into account through the van der Waals equation of state Vovchenko:2016rkn .
The van der Waals-type interaction in the hadronic medium delays the melting
of hadrons, and thus, the VDWHRG model can explain the lQCD data even up to
$T\sim$ 180 MeV. This model, which has a liquid-gas phase transition, is very
effective in estimating a variety of thermodynamic and transport properties
Pradhan:2022gbm ; Sarkar:2018mbk ; Samanta:2017yhh .

In order to understand the medium formed in ultra-relativistic collisions, it
is essential to study its thermodynamic properties. The fundamental
thermodynamic quantities, such as pressure ($P$), energy density
($\varepsilon$), and entropy density ($s$), can give us necessary information
about the system. The scaled pressure, energy density, and entropy density
provide information about the degrees of freedom in the medium. Similarly, the
speed of sound tells about the interaction in the medium. A massless ideal gas
gives $c_{\rm s}^{2}$ to be 1/3, whereas the value for a hadron gas is 1/5.
Studying $c_{\rm s}^{2}$ can help us understand whether the medium is partonic
or hadronic or approaches a massless ideal gas limit. On the other hand, the
specific heat of a system is estimated via the temperature fluctuations in the
system. It gives the measure of the amount of heat energy needed to raise the
system’s temperature by one unit. It is also expected to diverge near the
critical point and thus is an excellent observable to study the phase
transition. Similarly, trace anomaly also plays a vital role in the QCD
dynamics and phase transition. It measures the deviation from the masslessness
of the constituents in the medium. In recent studies Sahu:2020nbu ;
Sahu:2020swd ; Sahu:2020mzo ; Sahu:2021cdl , various thermodynamic observables
were studied as functions of final state charged particle density in
pseudorapidity ($\langle dN_{\rm ch}/d\eta\rangle$), which shed light on a
possible change in dynamics of the system after a threshold in $\langle
dN_{\rm ch}/d\eta\rangle$. This suggests that after $\langle dN_{\rm
ch}/d\eta\rangle\simeq$ 10 – 20, small systems like pp and p-Pb mimic the
behavior of heavy-ion collisions.

In addition, studying the correlations and fluctuations of conserved charges
is a reliable method for comprehending the physics of the phase transition of
strongly interacting matter. As the fluctuation-dissipation theorem relates
susceptibilities to fluctuations in a system near thermal equilibrium, the
related susceptibilities indicate inherent statistical fluctuations. Changes
in conserved charges at finite temperatures and chemical potential are
sensitive signs that hadronic matter is changing into quark-gluon plasma
(QGP). Moreover, divergent fluctuations can also indicate the presence of the
CEP; however, the shift from the hadronic to QGP phase is continuous for the
vanishing net baryon chemical potential.

Recently, it was found that there is a finite hyperon polarization in
relativistic heavy-ion collisions at the STAR experiment, which led to the
conclusion that finite vorticity is present in the medium STAR:2017ckg . This
opens up a whole new window of exciting consequences. The vorticity or
rotation gets coupled with the temperature in the medium, thus changing the
entire dynamics of the system evolution Singh:2018bih ; Sahoo:2023xnu . The
fundamental Euler’s thermodynamic equation gets modified in the presence of a
finite rotation, adding a new rotation chemical potential into the system
Becattini:2010 ; Florkowski:2017ruc . Apart from the vorticity coming from the
initial global orbital angular momentum of the colliding heavy ions, there are
several other sources from which vorticity can be generated in a system. The
smoke-loop type vortex created in the vicinity of fast-moving jets in an
expanding fireball also contributes to the vorticity of the system, although
they are not responsible for hyperon polarization Betz:2007kg . Inhomogeneous
transverse expansion of fireball may produce transverse vorticity circling the
longitudinal axis in the system Xia:2018tes ; Jiang:2016woz ; Wei:2018zfb ;
Becattini:2017gcx ; Pang:2016igs ; Voloshin:2017kqp . Additionally, vorticity
can be generated from the Einstein-de Haas effect Einstein:1915 , where a
magnetized medium creates a finite rotation. This indicates that the huge
magnetic field produced in heavy ion collisions due to fast-moving spectators
may magnetize the medium and it may generate a large amount of vorticity in
the system, whereas the reverse effect is the famous Barnett effect Barnett .
Thus, the analogy between rotation and the magnetic field is a well-known
phenomenon studied in many physical systems McInnes:2016dwk ; Yang:2023zqe .
Similarly, viscosity in the medium is also responsible for generating finite
vorticity and vice versa Fu:2021pok ; Sahoo:2023xnu . Thus, it is necessary to
include the effect of rotation while studying the medium formed in an ultra-
relativistic collision. Recently, many studies have been conducted with the
introduction of rotation to understand the QCD phase structure. Vorticity
formation in the ultra-relativistic heavy-ion collision has been studied from
hydrodynamic models such as ECHO-QGP, PICR, vHLLE, MUSIC, 3-FD, CLVisc in
(3+1) dimensional model Becattini:2015ska ; Csernai:2013bqa ; Csernai:2014ywa
; Ivanov:2019ern ; Karpenko:2016jyx . Event generators, such as AMPT, UrQMD,
and HIJING, have also been used to estimate kinematic and thermal vorticity
Jiang:2016woz ; Deng:2016gyh ; Li:2017slc ; Wei:2018zfb ; Deng:2020ygd ;
Vitiuk:2019rfv . Moreover, the non-zero local vorticity can help us probe the
chiral vortical effect (CVE), which is a non-trivial consequence of
topological quantum chromodynamics Rogachevsky:2010ys ; Kharzeev:2007tn . This
effect is the vortical analog of the chiral magnetic effect (CME)
Kharzeev:2007jp ; Fukushima:2008xe  and chiral separation effect (CSE)
Son:2004tq ; Metlitski:2005pr . It represents the vector, and axial currents
generation along the vorticity Banerjee:2008th ; Erdmenger:2008rm ; Son:2009tf
; Jiang:2015cva . CVE is extremely important because it induces baryon charge
separation along the vorticity direction, which can be experimentally probed
by two-particle correlations Csernai:2013vda .

There are several studies on the effect of magnetic fields on the QCD phase
diagram. In Mizher:2010zb , the authors have coupled the linear sigma model to
quarks to study the chiral transition as well as to Polyakov loops to consider
the confinement. Taking a constant uniform magnetic field, they investigated
how the magnetic field affects the chiral and deconfinement transitions. It
was shown that the chiral condensate is enhanced by the magnetic field, and
the transition temperature rises as a result. This phenomenon is commonly
known as magnetic catalysis. Numerous studies that involve the Nambu-Jona-
Lasinio model and its extended versions, such as the PNJL Kashiwa:2011js , and
EPNJL Gatto:2010pt  models came to similar conclusions regarding the rise in
$T_{c}$ and the strength of the transitions. However, in contrast to this, the
lattice QCD results Bali:2011qj ; Bali:2012zg , showed that the magnetic field
actually suppresses rather than increases the critical temperature for the
chiral phase transition of QCD. Since rotation in the medium adds another kind
of chemical potential, it can affect the phase transition, and hence, it will
be intriguing to observe how rotation affects the QCD phase diagram. In ref.
Fujimoto:2021xix , the authors explore the deconfinement from a rapidly
rotating hot and dense hadronic matter, similarly, in ref. Jiang:2016wvv ;
Wang:2018sur , the authors investigate the chiral phase transition in a system
of fermions under rotation using the Nambu-Jona Laisino (NJL) model. Here the
authors have shown the importance of angular velocity in determining the phase
transition from hadronic to quark degrees of freedom. Their results are
presented by a phase diagram in the temperature-angular momentum $T-\omega$
plane along with the phase diagram in the temperature-chemical potential
$T-\mu_{B}$ plane. Similarly, the quark deconfinement has been studied in
rotating neutron stars in Mellinger:2017dve . The authors in ref. Sun:2021hxo
have studied the chiral phase transition along with the spin polarization in a
three-flavor NJL model. Apart from this, the rotation effect has been explored
in the mesonic condensation of the isospin matter Zhang:2018ome . The authors
in Liu:2017spl  has also studied a combined effect of rotation and magnetic
field on pion condensation. They have demonstrated increased condensation upon
increasing the rotation (angular velocity). These kinds of studies evident
that introducing angular velocity in the medium adds another kind of chemical
potential known as rotational chemical potential. Similar to baryon chemical
potential, this rotational chemical potential can also lead to a phase
transition.

Given the above information, it would be interesting to understand what effect
rotation plays in quantifying a hadronic system’s thermodynamic properties. In
this work, we take rotation into account and estimate various thermodynamic
properties and charge fluctuations in an interacting hadron gas for the first
time. We also look for a possible criticality in the rotating medium for a
liquid-gas phase transition. The structure of this work is as follows. The
section II gives a detailed calculation of the thermodynamic observables and
the susceptibilities within the scope of a VDWHRG model that includes
rotation. We briefly examine the results in section III and provide a summary
in section IV.

##  II Formulation

In this work, we have assumed a system of relativistic gas of massive fermions
and bosons having half-integral and integral spin ($S$), rotating with a
constant angular velocity vector $\omega$. The density operator for rotational
grand canonical ensemble having large volume $V$ and angular momentum ${\bf
J}$ is given as Becattini:2007nd ; Becattini:2009wh ; landau ; beca1 ; beca2 ;
Becattini:2013fla

$\widehat{\rho}_{\omega}=\frac{1}{Z_{\omega}}\exp[(-\widehat{H}+\mu\widehat{Q}+{\bf\omega}\cdot\widehat{\bf
J})/T]{\sf P}_{V},$ (1)

where $\widehat{H}$ is the hamiltonian operator, $\widehat{Q}$ is a generic
conserved charges, $\mu$ is the relevant chemical potential, and $Z_{\omega}$
is the partition function of the rotating system given as

$Z_{\omega}={\rm tr}\exp[(-\widehat{H}+\mu\widehat{Q}+\omega\cdot\widehat{\bf
J})/T]{\sf P}_{V}.$ (2)

The ${\sf P}_{V}$ is the projector onto localized states $|h_{V}\rangle$,
${\sf P}_{V}=\sum_{h_{V}}|h_{V}\rangle\langle h_{V}|$. The partition function
in Eq. (2) can be written as a product of single-particle partition functions.
The calculation of Eq. (2) then reduces to compute matrix elements of single-
particle compatible operators $\widehat{h},\widehat{q},\widehat{\bf j}$, like
this Becattini:2007nd ; Becattini:2009wh :

$\begin{split}&\langle
p_{\pm},\tau_{\pm}|\exp[-(\widehat{h}+\mu\widehat{q}+\omega\cdot\widehat{\bf
j})/T]{\sf P}_{V}|p_{\pm},\sigma_{\pm}\rangle\\\ &=({\rm e}^{(\epsilon-\mu
q)/T}\pm 1)^{-1}\langle p_{\pm},\tau_{\pm}|\exp[\omega\cdot\widehat{\bf
j}/T]{\sf P}_{V}|p_{\pm},\sigma_{\pm}\rangle,\end{split}$ (3)

where $p_{\pm}$ is the single particle four-momentum. The $\sigma_{\pm}$ and
$\tau_{\pm}$ are level of polarization states. The $\pm$ corresponds to
fermions and bosons, respectively. An analytical extension from imaginary
values of $\omega$ can be used to derive the matrix elements on the right-hand
side of Eq. (3). After replacing $\omega/T$ with $-i\phi$, we have a rotation
${\sf R}_{\hat{\omega}}(\phi)$ around the axis $\omega$ of an angle $\phi$. A
more detailed explanation can be found in the Ref. Becattini:2007nd ;
Becattini:2009wh . The matrix element can be rewritten in terms of the
rotational matrix ${\sf R}_{\hat{\omega}}(\phi)$ as

$\langle p_{\pm},\tau_{\pm}|\exp[\omega\cdot\widehat{\bf j}/T]{\sf
P}_{V}|p_{\pm},\sigma_{\pm}\rangle=\langle p_{\pm},\tau_{\pm}|{\sf
R}_{\hat{\omega}}(\phi){\sf P}_{V}|p_{\pm},\sigma_{\pm}\rangle.$ (4)

Expanding the matrix element of the right-hand side,

$\langle p_{\pm},\tau_{\pm}|{\sf R}_{\hat{\omega}}(\phi){\sf
P}_{V}|p_{\pm},\sigma_{\pm}\rangle=\sum_{\sigma^{\prime}_{\pm}}\int d^{3}{\rm
p}^{\prime}\;\langle p_{\pm},\tau_{\pm}|\widehat{\sf
R}_{\hat{\omega}}(\phi)|p^{\prime}_{\pm},\sigma^{\prime}_{\pm}\rangle\langle
p^{\prime}_{\pm},\sigma^{\prime}_{\pm}|{\sf
P}_{V}|p_{\pm},\sigma_{\pm}\rangle.$ (5)

The matrix element representation of the rotation involves a Dirac delta and a
Wigner matrix. Thus,

$\langle p_{\pm},\tau_{\pm}|\widehat{\sf
R}_{\hat{\omega}}(\phi)|p^{\prime}_{\pm},\sigma^{\prime}_{\pm}\rangle=\delta^{3}\left({\bf
p_{\pm}}-{\sf R}_{\hat{\omega}}(\phi)({\bf
p_{\pm}}^{\prime})\right)D^{S}([{\sf
R}_{\hat{\omega}}(\phi)(p^{\prime}_{\pm})]^{-1}{\sf
R}_{\hat{\omega}}(\phi)[p^{\prime}_{\pm}])_{\tau_{\pm}\sigma^{\prime}_{\pm}}.$
(6)

It’s difficult to determine the ${\sf P}_{V}$ matrix element over momentum
eigenstates. A theoretical quantum field framework can be used to perform the
calculation beca2

$\langle p^{\prime}_{\pm},\sigma^{\prime}_{\pm}|{\sf
P}_{V}|p_{\pm},\sigma_{\pm}\rangle=\frac{1}{2}\sqrt{\varepsilon\over\varepsilon^{\prime}}\,\int_{V}d^{3}{\rm
x}\;{\rm e}^{i{\bf x}\cdot({\bf p_{\pm}}-{\bf
p_{\pm}}^{\prime})}\left(D^{S}([p^{\prime}_{\pm}]^{-1}[p_{\pm}])+D^{S}([p^{\prime}_{\pm}]^{\dagger}[p_{\pm}]^{\dagger-1})\right)_{\sigma^{\prime}_{\pm}\sigma_{\pm}}\langle
0|{\sf P}_{V}|0\rangle,$ (7)

where $\langle 0|{\sf P}_{V}|0\rangle$ is the vaccum expectation of the
projector ${\sf P}_{V}$ and tends to 1 for large volume. Substituting the Eqs.
(6) and (7) in Eq. (5), we have

$\langle p_{\pm},\tau_{\pm}|\widehat{\sf R}_{\hat{\omega}}(\phi){\sf
P}_{V}|p_{\pm},\sigma_{\pm}\rangle=\int_{V}d^{3}{\rm x}\;{\rm e}^{i{\bf
x}\cdot({\bf p_{\pm}}-{\sf R}_{\hat{\omega}}(\phi)^{-1}({\bf
p_{\pm}}))}\;\;\frac{1}{2}\left(D^{S}([p_{\pm}]^{-1}{\sf
R}_{\hat{\omega}}(\phi)[p_{\pm}])+D^{S}([p_{\pm}]^{\dagger}{\sf
R}_{\hat{\omega}}(\phi)[p_{\pm}]^{\dagger-1})\right)_{\tau_{\pm}\sigma_{\pm}}$
(8)

Taking advantage of the unitarity of the Wigner rotation, i.e.,

$D^{S}([{\sf R}_{\hat{\omega}}(\phi)(p^{\prime}_{\pm})]^{-1}{\sf
R}_{\hat{\omega}}(\phi)[p^{\prime}_{\pm}])=D^{S}([{\sf
R}_{\hat{\omega}}(\phi)(p^{\prime}_{\pm})]^{\dagger}{\sf
R}_{\hat{\omega}}(\phi)[p^{\prime}_{\pm}]^{\dagger-1})$ (9)

and the unitarity of ${\sf R}$ itself as an SL(2,C) matrix. The analytical
prolongation of equation (8) to imaginary angles yields the final expression
for the matrix element in Eq. (3) as:

$\begin{split}\langle p_{\pm},\tau_{\pm}|\exp[\omega\cdot\widehat{\bf
j}/T]{\sf P}_{V}|p_{\pm},\sigma_{\pm}\rangle=&\int_{V}d^{3}{\rm x}\;{\rm
e}^{i{\bf x}\cdot({\bf p_{\pm}}-{\sf R}_{\hat{\omega}}(i\omega/T)^{-1}({\bf
p_{\pm}}))}\\\ &\times\frac{1}{2}\left(D^{S}([p_{\pm}]^{-1}{\sf
R}_{\hat{\omega}}(i\omega/T)[p_{\pm}])+D^{S}([p_{\pm}]^{\dagger}{\sf
R}_{\hat{\omega}}(i\omega/T)[p_{\pm}]^{\dagger-1})\right)_{\tau_{\pm}\sigma_{\pm}}\end{split}$
(10)

The equilibrium single-particle phase space distribution can be calculated
with the help of the matrix element in Eq. (10). The spacial integral form in
Eq. (3) allows us to write the phase-space distribution as:

$\begin{split}f({\bf x},{\bf
p})_{\tau_{\pm}\sigma_{\pm}}=&(e^{(\varepsilon-\mu q)/T}\pm 1)^{-1}{\rm
e}^{i{\bf x}\cdot({\bf p_{\pm}}-{\sf R}_{\hat{\omega}}(i\omega/T)^{-1}({\bf
p_{\pm}}))}\\\ &\times\frac{1}{2}\left(D^{S}([p_{\pm}]^{-1}{\sf
R}_{\hat{\omega}}(i\omega/T)[p_{\pm}])+D^{S}([p_{\pm}]^{\dagger}{\sf
R}_{\hat{\omega}}(i\omega/T)[p_{\pm}]^{\dagger-1})\right)_{\tau_{\pm}\sigma_{\pm}}\end{split}$
(11)

A non-relativistic thermodynamic equilibrium system with given angular
momentum $\omega$ can have a rigid velocity according to
$\mathbb{v}=\boldsymbol{\omega}\times\mathbb{R}$, which in the relativistic
system adds another constraint of $|\boldsymbol{\omega}\times\mathbb{R}|\ll
1$. Therefore the ratio between $\omega$ and $T$ is very small for a proper
macroscopic system (and in fact, for the majority of practical reasons), i.e.:
$\frac{\hbar\omega}{KT}\ll 1$. As a result, the lowest order term in
$\omega/T$ is a good approximation for the difference between the momenta in
the exponent of Eq. (11).

${\bf p_{\pm}}-{\sf R}_{\hat{\omega}}(i\omega/T)^{-1}({\bf p_{\pm}})={\bf
p_{\pm}}-\left[\cosh\frac{\omega}{T}\,{\bf
p_{\pm}}-i\sinh\frac{\omega}{T}\,\hat{\omega}\times{\bf
p_{\pm}}+(1-\cosh\frac{\omega}{T})\,{\bf
p_{\pm}}\cdot\hat{\omega}\hat{\omega}\right]\simeq
i\frac{\omega}{T}\hat{\omega}\times{\bf p_{\pm}}$ (12)

This results in the phase-space distribution function in Eq. (11) becoming;

$\displaystyle f({\bf x},{\bf p_{\pm}})_{\tau_{\pm}\sigma_{\pm}}$
$\displaystyle=$ $\displaystyle(e^{(\varepsilon-\mu q)/T}\pm 1)^{-1}e^{-{\bf
x}\cdot(\omega\times{\bf p_{\pm}})/T}\frac{1}{2}\left(D^{S}([p_{\pm}]^{-1}{\sf
R}_{\hat{\omega}}(i\omega/T)[p_{\pm}])+D^{S}([p_{\pm}]^{\dagger}{\sf
R}_{\hat{\omega}}(i\omega/T)[p_{\pm}]^{\dagger-1})\right)_{\tau_{\pm}\sigma_{\pm}}$
(13) $\displaystyle=$ $\displaystyle(e^{(\varepsilon-\mu q)/T}\pm
1)^{-1}e^{-{\bf p_{\pm}}\cdot(\omega\times{\bf
x})/T}\frac{1}{2}\left(D^{S}([p_{\pm}]^{-1}{\sf
R}_{\hat{\omega}}(i\omega/T)[p_{\pm}])+D^{S}([p_{\pm}]^{\dagger}{\sf
R}_{\hat{\omega}}(i\omega/T)[p_{\pm}]^{\dagger-1})\right)_{\tau_{\pm}\sigma_{\pm}}$
$\displaystyle=$ $\displaystyle(e^{(\varepsilon-\mu q)/T}\pm 1)^{-1}e^{-({\bf
p_{\pm}}\cdot\bf v)/T}\frac{1}{2}\left(D^{S}([p_{\pm}]^{-1}{\sf
R}_{\hat{\omega}}(i\omega/T)[p_{\pm}])+D^{S}([p_{\pm}]^{\dagger}{\sf
R}_{\hat{\omega}}(i\omega/T)[p_{\pm}]^{\dagger-1})\right)_{\tau_{\pm}\sigma_{\pm}}$

where we have used the definition of ${\bf v}=\omega\times\bf{x}$

The single-particle phase-distribution in Eq. (13) for the ideal rotating
relativistic fermions and bosons particles is the unnormalized one, and we
need to take the trace of the matrix in Eq. (13) to obtain the so-called
phase-space density in $({\bf x},{\bf p})$:

$\begin{split}f({\bf x},{\bf p})&=\sum_{\sigma_{\pm}}f({\bf x},{\bf
p})_{\sigma_{\pm}\sigma_{\pm}}\\\ &=(e^{(\varepsilon-\mu q)/T}\pm
1)^{-1}e^{-({\bf p}\cdot\bf
v)/T}\chi\\!\left(\frac{\omega}{T}\right)\end{split}$ (14)

being:

$\chi\\!\left(\frac{\omega}{T}\right)\equiv{\rm tr}D^{S}({\sf
R}_{\hat{\omega}}(i\omega/T))=\frac{\sinh(S+\frac{1}{2})\frac{\omega}{T}}{\sinh(\frac{\omega}{2T})}$
(15)

For fluid rest frame ${\bf v}$ = 0, hence the single-particle distribution
function is given as

$f({\bf x},{\bf p})=\frac{1}{e^{(\varepsilon-\mu q)/T}\pm
1}\bigg{(}\frac{\sinh(S+\frac{1}{2})\frac{\omega}{T}}{\sinh(\frac{\omega}{2T})}\bigg{)},$
(16)

where the q is the conserved charge.

###  II.1 van der Waals HRG model

In contrast to the QGP phase, where the degrees of freedom are basically
quarks and gluons, the hadronic phase is described by the confined state of
the quarks and gluons. The ideal HRG model deals with a system of non-
interacting point particles with hadronic degrees of freedom. The basic
quantity required to calculate the hadron yields and thermodynamic properties
is the partition function of the ensemble. The partition function for ith
particle species in a Grand Canonical Ensemble (GCE) of ideal HRG can be
written as Andronic ,

$lnZ^{id}_{i}=\pm\frac{Vg_{i}}{2\pi^{2}}\int_{0}^{\infty}p^{2}dp\
ln\\{1\pm\exp[-(E_{i}-\mu_{i})/T]\\}.$ (17)

Here, the degeneracy of ith hadronic species is given by $g_{i}$, whereas
$E_{i}=\sqrt{p^{2}+m_{i}^{2}}$ gives the free particle energy of the $i^{th}$
hadron, $m_{i}$ being the mass of the ith hadron. The $\pm$ sign corresponds
to baryons and mesons, respectively. The chemical potential is denoted by
$\mu_{i}$, and is given by,

$\mu_{i}=B_{i}\mu_{B}+S_{i}\mu_{S}+Q_{i}\mu_{Q},$ (18)

where $\mu_{B}$, $\mu_{S}$, and $\mu_{Q}$, respectively, represent the baryon
chemical potential, strangeness chemical potential, and charge chemical
potential. The baryon number, strangeness, and electric charge of the ith
hadron are denoted by $B_{i}$, $S_{i}$, and $Q_{i}$, respectively.

In a rotating medium of hadron gas, the partition function in Eq. (2) for a
single hadronic species is equivalent to the one defined in Eq. (17)
multiplied by a factor $\chi(\frac{\omega}{T})$, given as

$lnZ^{id}_{i}=\pm\frac{Vg_{i}}{2\pi^{2}}\int_{0}^{\infty}p^{2}dp\
ln\\{1\pm\exp[-(E_{i}-\mu_{i})/T]\\}\chi(\frac{\omega}{T}),$ (19)

where $\chi(\frac{\omega}{T})$ is given by the Eq. (15). The partition
functions for all hadrons and resonances, $\ln Z_{i}^{id}$, can then be added
to form the overall partition function of the hadron gas;

$\ln Z=\sum_{i}\ln Z_{i}^{id}.$ (20)

This partition function may now be used to derive the different thermodynamic
quantities for a single hadronic species, such as pressure $P_{i}$, energy
density $\varepsilon_{i}$, number density $n_{i}$, and entropy density
$s_{i}$, as

$\begin{split}P^{id}_{i}(T,\mu_{i})=&\pm\frac{Tg_{i}}{2\pi^{2}}\int_{0}^{\infty}p^{2}dp\
ln\\{1\pm\exp[-(E_{i}-\mu_{i})/T]\\}\\\
&\times\chi(\frac{\omega}{T})\end{split}$ (21)
$\varepsilon^{id}_{i}(T,\mu_{i})=\frac{g_{i}}{2\pi^{2}}\int_{0}^{\infty}\frac{E_{i}\
p^{2}dp}{\exp[(E_{i}-\mu_{i})/T]\pm 1}\chi(\frac{\omega}{T})$ (22)
$n^{id}_{i}(T,\mu_{i})=\frac{g_{i}}{2\pi^{2}}\int_{0}^{\infty}\frac{p^{2}dp}{\exp[(E_{i}-\mu_{i})/T]\pm
1}\chi(\frac{\omega}{T})$ (23) $\displaystyle s^{id}_{i}(T,\mu_{i})=$
$\displaystyle\pm\frac{g_{i}}{2\pi^{2}}\int_{0}^{\infty}p^{2}dp\Big{[}\ln\\{1\pm\exp[-(E_{i}-\mu_{i})/T]\\}$
$\displaystyle\pm\frac{(E_{i}-\mu_{i})/T}{\exp[(E_{i}-\mu_{i})/T]\pm
1}\Big{]}\chi(\frac{\omega}{T})$ (24)

Along with this, we also compute another density related to angular velocity
(the so-called rotational chemical potential $\omega$) known as spin density.
In the presence of rotation in the medium, the Euler equation of thermodynamic
variables becomes Florkowski:2017ruc ,

$\varepsilon+P=sT+n\mu+\rm{w}\omega.$ (25)

Therefore the new spin density $\rm{w}$ can be calculated as,

$\begin{split}\rm{w}&=\frac{\partial P}{\partial\omega}\bigg{|}_{T,\mu}\\\
=&\pm\frac{Tg_{i}}{2\pi^{2}}\int_{0}^{\infty}p^{2}dp\
ln\\{1\pm\exp[-(E_{i}-\mu_{i})/T]\\}\frac{\partial}{\partial\omega}\chi\\!\left(\frac{\omega}{T}\right),\end{split}$
(26)

Moreover, The nth-order susceptibilities of conserved charges can be
calculated from the relation,

$\chi_{x}=\frac{\partial^{n}(P/T^{4})}{\partial(\frac{\mu_{x}}{T})^{n}},$ (27)

where the corresponding conserved charges, such as the baryon number, electric
charge, and strangeness number, are represented by the letter x.

The Ideal HRG model does not include interactions among the hadrons and,
therefore, is unable to explain different thermodynamic quantities estimated
from lQCD at high temperatures and baryon densities. Now to introduce the
interaction in the medium, we start with the van der Waals equation, which, in
canonical ensemble representation, can be written as Samanta:2017yhh ;
W.Greiner123Stocker

$\Bigg{(}P+\bigg{(}\frac{N}{V}\bigg{)}^{2}a\Bigg{)}\big{(}V-Nb\big{)}=NT$ (28)

where the VDW parameters $a$ and $b$, both positive, describe the attractive
and repulsive interactions, respectively, among the hadrons. P, V, T, and N,
respectively, stand for pressure, volume, temperature, and the number of
particles in the system.

Writing number density, $n\equiv N/V,$ the above equation can be simplified as

$P(T,n)=\frac{nT}{1-bn}-an^{2},$ (29)

The two terms in the above equation represent the correction factor to the
ideal case due to repulsion and attraction separately. The excluded volume
correction, or the correction for repulsive interactions, is incorporated in
the first term by changing the total volume V to an effective volume that is
accessible to particles using the appropriate volume parameter $b=16\pi
r^{3}/3$, where $r$ is the particle’s hardcore radius. In contrast, the second
term accounts for the attractive interactions between particles. For $a=0$,
Eq. (29) reduces to the EVHRG equation of state, where only repulsive
interactions are included. And for both $a=0$, $\&$ $b=0$, it reduces to the
ideal HRG.

This method is then applied to the GCE, where the VDW equation of state takes
the form Samanta:2017yhh ; Vovchenko:2015vxa ; Vovchenko:2015pya

$P(T,\mu)=P^{id}(T,\mu^{*})-an^{2}(T,\mu),$ (30)

where, $P(T,\mu)$ is the VDW pressure and reduces to ideal one,
$P^{id}(T,\mu)$ when there is no interaction. The particle number density of
the VDW hadron gas, n(T,$\mu$) is given by

$n(T,\mu)=\frac{\sum_{i}n_{i}^{id}(T,\mu^{*})}{1+b\sum_{i}n_{i}^{id}(T,\mu^{*})}.$
(31)

Here, $i$ runs over all hadrons and resonances in the interacting medium, and
$\mu^{*}$ is the modified chemical potential given by

$\mu^{*}=\mu-bP(T,\mu)-abn^{2}(T,\mu)+2an(T,\mu).$ (32)

Using Eq. (30), the $\mu^{*}$ can also be written as

$\mu^{*}=\mu-\frac{bn(T,\mu)T}{1-bn(T,\mu)}+2an(T,\mu).$ (33)

Additional thermodynamical variables like energy density $\varepsilon(T,\mu)$
and entropy density $s(T,\mu)$ can now be calculated as,

$\varepsilon(T,\mu)=\frac{\sum_{i}\epsilon_{i}^{id}(T,\mu^{*})}{1+b\sum_{i}n_{i}^{id}(T,\mu^{*})}-an^{2}(T,\mu).$
(34) $s(T,\mu)=\frac{s^{id}(T,\mu^{*})}{1+bn^{id}(T,\mu^{*})},$ (35)

As formulated initially, the VDWHRG model includes interactions confined to
all pairings of baryons or anti-baryons Samanta:2017yhh ; Vovchenko:2015vxa ;
Vovchenko:2015pya ; Vovchenko:2016rkn . Considering the fact that annihilation
processes dominate short-range interactions between baryon-antibaryon pairs,
the interaction between them was neglected Andronic2012 ; Vovchenko:2016rkn .
Previously, all meson-related interactions, such as meson-meson or
meson-(anti)baryon interactions, are neglected as the inclusion of these
interactions suppresses the thermodynamic quantities in the crossover region
at vanishing baryochemical potential in comparison with LQCD data
Vovchenko:2016rkn . However, by assuming a hard-core radius $r_{M}$ for
mesons, a more realistic formalism that takes into account meson-meson
interactions was developed by selecting the VDW parameters that best fit the
LQCD data Sarkar:2018mbk . As a result, the VDWHRG model’s total pressure is
expressed as Samanta:2017yhh ; Pradhan:2022gbm ; Vovchenko:2015vxa ;
Vovchenko:2015pya ; Vovchenko:2016rkn ; Sarkar:2018mbk

$P(T,\mu)=P_{M}(T,\mu)+P_{B}(T,\mu)+P_{\bar{B}}(T,\mu),$ (36)

where the pressure contributions made by mesons and (anti)baryons,
respectively, are denoted by $P_{M}(T,\mu),P_{B(\bar{B})}(T,\mu)$, and are
given by,

$P_{M}(T,\mu)=\sum_{i\in M}P_{i}^{id}(T,\mu^{*M}),$ (37)
$P_{B}(T,\mu)=\sum_{i\in B}P_{i}^{id}(T,\mu^{*B})-an^{2}_{B}(T,\mu),$ (38)
$P_{\bar{B}}(T,\mu)=\sum_{i\in\bar{B}}P_{i}^{id}(T,\mu^{*\bar{B}})-an^{2}_{\bar{B}}(T,\mu).$
(39)

Here, mesons, baryons, and anti-baryons are each represented by $M$, $B$, and
$\bar{B}$. Due to the excluded volume correction, mesons have a modified
chemical potential of $\mu^{*M}$, whereas baryons and anti-baryons have
modified chemical potentials of $\mu^{*B}$ and $\mu^{*\bar{B}}$, respectively,
as a result of VDW interactions Sarkar:2018mbk . Taking a simple case of
vanishing electric charge and strangeness chemical potentials, where
$\mu_{Q}=\mu_{S}=0$, the modified chemical potential for mesons and
(anti)baryons can be determined from Eq. 18 and Eq. 32 as;

$\mu^{*M}=-bP_{M}(T,\mu),$ (40)
$\mu^{*B(\bar{B})}=\mu_{B(\bar{B})}-bP_{B(\bar{B})}(T,\mu)-abn^{2}_{B(\bar{B})}+2an_{B(\bar{B})},$
(41)

where $n_{M}$, $n_{B}$ and $n_{\bar{B}}$ are the modified number densities of
mesons, baryons, and anti-baryons, respectively, which are given by

$n_{M}(T,\mu)=\frac{\sum_{i\in M}n_{i}^{id}(T,\mu^{*M})}{1+b\sum_{i\in
M}n_{i}^{id}(T,\mu^{*M})},$ (42) $n_{B(\bar{B})}(T,\mu)=\frac{\sum_{i\in
B(\bar{B})}n_{i}^{id}(T,\mu^{*B(\bar{B})})}{1+b\sum_{i\in
B(\bar{B})}n_{i}^{id}(T,\mu^{*B(\bar{B})})}.$ (43)

There are different approaches to estimate the VDW parameters. They can be
obtained by reproducing the ground state of the nuclear matter
Vovchenko:2015vxa . Alternatively, one can obtain the parameters by fitting
lattice QCD results for different thermodynamic quantities Samanta:2017yhh ;
Sarkar:2018mbk . The parameters in the model are now set to $a=0.926$ GeV fm3
and $b=(16/3)\pi r^{3}$, where the $r$ being the hardcore radius of each
hadron, given as $r_{M}=0.2$ fm for mesons, and $r_{B,(\bar{B})}=0.62$ fm, for
(anti)baryons Sarkar:2018mbk . Using this information, we now proceed to
estimate various thermodynamic quantities in a rotating hadron resonance gas
with VDW interactions.

Figure 1: (Color Online) Variation of different thermodynamical quantities as
functions of temperature for $\mu_{\rm B}$ = 0 GeV, and for certain values of
rotation chemical potentials. The red triangles are the Wuppertal-Budapest
lattice QCD data WBcoll , and the shaded region are the HotQCD lattice data
HotQCD .

##  III Results and Discussion

We explore the effect of rotation in the hadron gas by considering an
interacting hadron gas model, namely, the VDWHRG model, with attractive and
repulsive interactions between the hadrons. The model takes into the
contributions of all hadrons and resonances up to a mass cut-off of 2.25 GeV
available in the particle data group PDG2016 . The van der Waals parameters
are obtained by fitting thermodynamic quantities like energy density and
pressure in the VDWHRG model to the available lattice QCD data Sarkar:2018mbk
. It is noteworthy to mention that the van der Waals parameters should, in
principle, change with respect to a change in rotation. However, as it is non-
trivial to have $a$ and $b$ as functions of $\omega$, we neglect the
dependency in the current study. We then estimate various thermodynamic
quantities at finite rotation by taking the obtained $a$ and $b$ values from
the fitting.

Figure 2: (Color Online) Pressure for different spin particles as a function
of $\omega$ at a constant temperature, $T$ = 155 MeV.

Fig. 1 shows the variation of $P/T^{4}$, $\varepsilon/T^{4}$, $s/T^{3}$,
$(\varepsilon-3P)/T^{4}$, $c_{v}/T^{3}$, and $c_{\rm s}^{2}$ with temperature
at zero baryochemical potential for certain values of $\omega$. The red
triangles are the lQCD results from the Wuppertal-Budapest collaboration
WBcoll , and the shaded region shows the lattice results from Hot QCD
collaboration HotQCD , at $\mu_{B}=0.0$ GeV. All the calculations are done for
$\mu_{B}$ = 0.0 GeV. Our results at $\omega=0~{}\rm fm^{-1}$, represented by a
solid black line, are in good agreement with the lQCD estimations. From the
upper-left panel of fig.1, we observe that $P/T^{4}$ increases with
temperature for all $\omega$ values. At a given temperature, $P/T^{4}$ is
higher for a higher value of $\omega$. Similar trends can be observed in
$\varepsilon/T^{4}$, $s/T^{3}$, $(\varepsilon-3P)/T^{4}$ as well as in
$c_{v}/T^{3}$ plots. However, the slopes of the spectra differ for each
observable. In the trace anomaly plot, we observe a peak that shifts towards
low temperatures with an increase in $\omega$. This peak signifies the
conformal symmetry breaking at which the constituent particles become
massless. The behavior of $c_{s}^{2}$ is also crucial to understand the phase
transition region. In VDWHRG, there appears a minimum in $c_{s}^{2}$, which is
in agreement with lQCD, and this minimum can be interpreted as a signature of
the transition from hadrons to quark degrees of freedom. By increasing the
value of rotational chemical potential, the minima shift towards the lower
temperature regime, suggesting that the phase transition temperature decreases
in the presence of rotation.

In order to understand the effect of spin and rotation on the basic
thermodynamic quantities, we plot the scaled pressure as a function of
$\omega$ for various spin particles in fig. 2. The temperature is taken to be
constant at $T$ = 155 MeV. We observe that at $\omega=0$ fm-1, the
contribution to pressure is dominated by the spin-0 particles, followed by the
spin-1, spin-1/2, and spin-3/2 particles. This is due to the fact that at
$\omega=0$ fm-1, the contribution to pressure comes only from the Boltzmann
factor in the distribution function. Thus, the lesser massive particles will
dominantly contribute. The spin-0 particles, which consist of pions, kaons,
etc., contribute the most, followed by the vector mesons such as $\rho$ and
$\eta$. Finally, the spin-1/2 baryons, such as protons and neutrons, and
spin-3/2 baryons, such as $\Lambda$, $\Xi$, will contribute to the pressure,
respectively. As the rotation increases, the contribution of spin-1 particles
dominates due to the $\omega$ term in the distribution function, and the
contribution from the spin-0 particles remains almost the same. The slight
decrease in the spin-0 particles trend at high $\omega$ is due to the van der
Waals effect. In addition, the spin-3/2 particles contribute more than
spin-1/2 due to the effect of rotation.

Figure 3: (Color Online) Variation of scaled entropy density at low
temperature and higher angular velocity values is shown.  Figure 4: (Color
Online) Net spin density = $\frac{\partial P}{\partial\omega}$ as a function
of temperatures for different values of $\omega$.

Figure 5: (Color Online) The baryon number susceptibility (left panel), charge
susceptibility (middle panel), and strangeness susceptibility (right panel) as
functions of temperature for different values of $\omega$.

In addition to the speed of sound, the entropy density, number density, etc.,
are observables that show discontinuities at the first-order phase transition.
Our study deals with the van der Waals interaction in the hadronic phase.
Therefore a liquid-gas phase transition is expected at $T-\mu_{B}$ plane,
which is estimated in various works Vovchenko:2015pya ; Samanta:2017yhh ;
Sarkar:2018mbk  with different van der Waals parameters. Since rotation adds
another chemical potential to the system, it is useful to see if the angular
velocity alone can lead to a phase transition. Fig. 3 shows the behavior of
entropy density in the $T-\omega$ phase space. Here, the angular velocity
($\omega$) is taken to take into account small iterations. The temperature is
taken with an interval of 1 MeV for the calculation. One can observe a smooth
trend of the scaled entropy density at high temperatures and low rotational
chemical potential. However, the smooth curve at comparatively low $\omega$
starts changing its shape as one approaches high $\omega$ values. At around
$T\simeq 69$ MeV, a discontinuity appears for $\omega\simeq 0.65$ GeV. A clear
first-order phase transition is observed as one approaches higher $\omega$
values. This suggests that the rotation has the same effect in achieving a
liquid-gas phase transition as the baryon chemical potential Samanta:2017yhh .
Therefore a hadron gas can be liquified either by increasing baryon density
and lowering the temperature or by increasing the angular velocity while
decreasing the temperature of the gas. Compared to Ref. Sarkar:2018mbk , where
the critical point is around $T=65$ MeV for $T-\mu_{B}$ plane, here the
temperature for the critical point is slightly higher, though the VDW
parameters are the same. This shows that the phase transition in the presence
of a rotational chemical potential appears more quickly than that for the
baryochemical potential case. Analogous to how the magnetic field affects the
chiral phase transition in raising the critical temperature Mizher:2010zb ,
here, the angular velocity raises the critical temperature for the liquid-gas
phase transition as compared to that of baryochemical potential.

Fig. 4 shows the temperature dependence of normalized dimensionless spin
density estimated in the VDWHRG model using Eq. 26. Similar to number density,
which can be defined as the change in pressure (or free energy) with respect
to chemical potential, spin density can also be defined as the change in the
pressure as a function of rotational chemical potential. The net spin density
in a system is defined as the density of hadrons of positive spin minus the
density of hadrons of negative spin. It is observed that much like other
thermodynamic densities, such as number density and entropy density, spin
density also increases with an increase in temperature. Moreover, at a
particular temperature, the value of spin density increases with increased
rotational chemical potential.

Figure 6: (Color Online) Various susceptibilities as functions of $\omega$ at
a constant temperature, $T$ = 155 MeV.

We also estimate the susceptibilities of various conserved quantities to show
their dependence on rotational chemical potential. Fluctuations of conserved
charges like net baryon density, electric charge, and strangeness are
essential probes for hadronization and can help us locate the phase boundary.
Large fluctuations in these quantities are one of the essential signatures of
the critical endpoint. Since the rotation can affect the phase transition and
hence the critical point, it is essential to see its effect on the
fluctuations of different conserved charges. We have used Eq. 27 to calculate
the second-order susceptibilities of different conserved quantities. Fig. 5
shows the temperature dependence of second-order fluctuations of conserved
quantities, namely, baryon density, electric charge, and strangeness,
respectively, from left to right. The red triangles are the results of lQCD
calculations from the Wuppertal-Budapest collaboration WBcoll  whereas the
shaded region represents the lQCD results of HotQCD collaboration HotQCD . The
solid black line, calculated in the VDWHRG model at zero baryochemical
potential and zero rotation, agrees with lattice results for baryon and charge
susceptibility. However, the results for strangeness susceptibility are
slightly suppressed than those of lattice results. The fluctuations in every
conserved quantity increase with an increase in rotational chemical potential.
It is observed that for higher $\omega$, a more prominent peak appears in the
case of baryon density fluctuations. However, the trends seem to be saturated
in the case of charge susceptibilities, and a monotonic increasing behavior is
observed for strangeness fluctuations within the range of studied $\omega$.
Fig. 6 shows the variation of all three susceptibilities as a function of
rotational chemical potential at a fixed temperature $T=155$ MeV. It is
observed that the baryon number and charge susceptibilities increase sharply
with $\omega$ after a particular value. However, the increase in strangeness
susceptibility is a bit slower. This is because the first two susceptibilities
are more sensitive to the microscopic structure of the matter Asakawa:2000wh
and can provide helpful information about the structural changing as the
quarks get deconfined at adequately high temperatures.

##  IV Summary

In this work, we estimate the effect of rotation on the thermodynamic
properties of an interacting hadron resonance gas. We observe that rotation
has a similar effect on the thermodynamic properties as the baryon chemical
potential. The rotational chemical potential enhances all observables like
pressure, energy density, entropy density, etc. We also observe that the
rotation in a system could lead to a first-order liquid-gas phase transition,
although the initial angular momentum required for it would be so high that
within LHC energy, it may not be possible. In addition, we estimate the spin
density associated with the rotational chemical potential and its behavior as
a function of temperature. The effect of rotation on fluctuations in conserved
quantities is also explored, and one can find that it enhances the second-
order fluctuations in all conserved quantities. In view of our study, we must
pay attention to the effect of rotation produced in a non-central heavy-ion
collision while studying the particle dynamics and the thermodynamics of the
system.

Recent studies focusing on vorticity and polarization in the medium formed in
ultra-relativistic collisions lead us to an exciting pathway. As the
scientific community shifts its attention to rotational dynamics in the
evolving QCD medium, myriad unique consequences can be unraveled. Moreover, it
will be interesting to see the results of the lattice calculation by taking
care of the rotation into the system.

## Acknowledgement

K.K.P. and B.S. acknowledge the financial aid from UGC and CSIR, Government of
India, respectively. The authors gratefully acknowledge the DAE-DST,
Government of India funding under the mega-science project “Indian
participation in the ALICE experiment at CERN” bearing Project No.
SR/MF/PS-02/2021-IITI (E-37123).

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