# Impact of non-zero strangeness on thermodynamics of finite volume quark
matter

Nisha Chahal<EMAIL_ADDRESS>Suneel Dutt<EMAIL_ADDRESS>Arvind
Kumar<EMAIL_ADDRESS>Department of Physics, Dr. B R Ambedkar National
Institute of Technology, Jalandhar - 144008, Punjab, India

###### Abstract

This paper investigates the impact of strangeness chemical potential and
finite volume on QCD critical end point by employing a (2+1) flavored Polyakov
quark meson model. Within the mean-field approximation, the model has been
extended to study the effect of vector interactions on the thermodynamics of
isospin-asymmetric quark matter. Susceptibilities of conserved charges like
the quark and strangeness number are analyzed using Taylor’s series expansion.
The chiral phase transition boundary in the QCD phase diagram is found to be
shifted towards higher values of the quark chemical potential ($\mu_{q}$) and
lower temperature (T) for decreasing system size. On the other hand, there is
an opposite change to lower quark chemical potential and higher temperature
for decreasing strangeness chemical potential.

##  I Introduction

Thermodynamics of strongly interacting matter, such as quark-gluon plasma
(QGP), is of tremendous interest to theoretical and experimental physicists
pandav2022search ; ohnishi2012phase . Heavy ion collision experiments like the
Large Hadron Collider (LHC) bruning2012large  at CERN in Switzerland and the
Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory
odyniec2010rhic ; kumar2013star  are crucial in recreating the conditions of
the early universe to understand the properties of QGP better. In addition to
the experimental facilities, lattice QCD simulations is also a powerful
technique for calculating the QCD partition function on a discrete space-time
lattice ratti2018lattice . It is a non-perturbative application of field
theory based on the Feynman path integral technique and can also be used to
study some important hadron properties gupta1999lattice ; detar2009qcd . The
pressure, energy density, and susceptibilities of conserved charges can be
obtained using lattice QCD simulations of thermodynamic phenomena. At high
temperatures and low baryonic densities, these computations have anticipated a
crossover transition from the confined to the deconfined state
guenther2021overview . On the other hand, due to the sign problem at higher
baryonic chemical potentials and low-temperature values, the phase
fluctuations derived from the complex fermion determinant are found to be very
large muroya2003lattice .

After formation, the QGP expands rapidly and cools down to form hadrons in a
small region of space. Due to the finite size of colliding nuclei and the
geometry of the collision, the fireball created in the initial stage is a
finite-sized system. In this case, the finite size refers to the influence of
the QGP’s finite volume on its properties and evolution. Finite-size effects
can affect particle flow, hadronization, and other variables sensitive to QGP
properties. As a result, the phase transition line and the QCD critical point
are modified due to finite volume considerations Braun:2011iz ;
MataCarrizal:2022gdr ; chahal2023effects . For low chemical potential values,
the impact of finite size effects on the curvature of the chiral transition
boundary has also been studied in lattice QCD de2007chiral ; karsch2004chiral
.

The effective models serve as an essential tool to study the properties of
matter formed in heavy-ion collisions at finite baryonic chemical potential
values tripolt2014effect ; palhares2010finite ; magdy2017influence ;
magdy2019influence . Using the quark-meson model, which serves as a low-energy
model for QCD, the effects of finite volume and long-range fluctuations have
been evaluated Braun:2010vd ; article . Based on the pion mass, it had been
observed that the curvature continually decreases for periodic and anti-
periodic boundary conditions as a function of system size, thus indicating
that the phase transition boundary may shift in a finite volume. The earlier
study of the Polyakov quark meson (PQM) model showed that the phase transition
boundary remains just a crossover for very small systems
bhattacharyya2017polyakov . The effect of finite volume on the QCD phase
diagram has also been analyzed considering different boundary conditions, such
as stationary, periodic, and antiperiodic for cubic and spherical regions
mata2022effects . Also, the concept of finite strangeness chemical potential
in quark matter is critical to our understanding of the fundamental building
blocks of the universe. Incorporating finite strangeness chemical potential
into the study of quark matter allows us to investigate the behavior of
strange quarks in this exotic condition. The large value of strangeness has
been shown to affect the generation of the neutron twin-stars, which can
predict the existence of first-order phase transition at lower temperature and
higher density values dexheimer2015role . In the framework of the Hadron
resonance gas (HRG) model, it has been highlighted that the critical
temperature decreases with the increase in the value of strangeness chemical
potential toublan2005qcd . Hence, the quark-hadron phase boundary is modified
as a consequence of finite strangeness chemical potential considerations and
is of great importance in studying thermodynamic properties of quark matter
mukherjee2019effects .

In the investigation of QCD matter, the susceptibilities of conserved charges
are particularly significant observables fan2019probing ; chahal2022quark ;
chatterjee2012fluctuations ; borsanyi2012fluctuations . These have been
recognized as observables that can be theoretically and experimentally
estimated to better understand the critical end point (CEP). Studying the
susceptibilities of conserved charges in finite volumes has gained popularity
recently since it can shed light on the nature of QCD phase transitions and
the equation of state of QCD matter in practical experimental settings
bhattacharyya2014fluctuations . A sizable shift in the chiral phase boundary
has been reported in finite volume studies of earlier PQM model
magdy2019influence . Including isospin chemical potential and vector
interactions in the PQM model have been found to affect the positioning of CEP
significantly chahal2022quark . In the present work, we use the three-flavored
Polyakov quark-meson model, extended by the inclusion of vector interactions
and isospin asymmetry, to study the thermodynamics of quark matter for finite
system size and non-zero value of strangeness chemical potential.

This paper is organized as follows: In Sect. II, we have described the PQM
model in detail and derived the grand canonical potential. In Sect. III, the
impact of the non-zero value of strangeness chemical potential and finite
volume on various thermodynamic variables and the QCD phase diagram has been
discussed. In Sect. IV, the critical findings of the current work have been
summarized.

##  II Polyakov quark meson model

The chiral quark meson model is an effective approach to study the strong
interactions between mesons and quarks. The spontaneous breaking of the chiral
symmetry in vacuum is described by including a scalar field that represents
the chiral condensate. It is a non-perturbative model based on the mean-field
approximation to study the thermodynamic properties of the system. The model
has been used to study various phenomena in QCD, such as the properties of
hadrons, the phase structure of QCD at finite temperature and density, and the
properties of quark-gluon plasma created in heavy-ion collisions
mintz2013phase ; skokov2010meson . It has also been used to study the behavior
of matter under extreme conditions, such as the properties of neutron stars
and the early universe zacchi2015compact ; otto2020hybrid . The total
Lagrangian of the model for $N_{f}$ flavors is given by PhysRevD.90.085001 ;
newvector

$\displaystyle\mathcal{L}=\bar{\Psi}i\gamma^{\mu}{\partial}_{\mu}\Psi+\text{Tr}\left(\partial_{\mu}\upvarphi^{\dagger}\partial^{\mu}\upvarphi\right)-m^{2}\text{Tr}\left(\upvarphi^{\dagger}\upvarphi\right)-\lambda_{1}\left[\text{Tr}\left(\upvarphi^{\dagger}\upvarphi\right)\right]^{2}-$
(1)
$\displaystyle\lambda_{2}\left[{Tr}\left(\upvarphi^{\dagger}\upvarphi\right)^{2}\right]+c\left(\text{det}(\upvarphi)+\text{det}\left(\upvarphi^{\dagger}\right)\right)+\text{Tr}\left[H\left(\upvarphi+\upvarphi^{\dagger}\right)\right]$
$\displaystyle+\mathcal{L}_{qm}-\frac{1}{4}\text{Tr}\left(V_{\mu\nu}V^{\mu\nu}\right)+\frac{m_{1}^{2}}{2}V_{a\mu}V_{\mu}^{a}.$

In the above equation, $\Psi$ = $(u,d,s)$ is the quark spinor for $N_{c}$ = 3,
color degrees of freedom and $\upvarphi$ =
$T_{a}(\sigma_{a}+i\gamma_{5}\pi_{a})$, where $T_{a}=\lambda_{a}$/2 are Gell-
Mann matrices. The first term in Eq. 1 accounts for the kinetic energy of
massless quarks. The next two terms describe scalar mesons kinetic energy and
mass term contributions. The terms involving $\lambda_{1}$ and $\lambda_{2}$
are quartic interaction terms, followed by the determinant term, which
corresponds to U(1)A anomaly in QCD vacuum t1976symmetry  and explicit
symmetry-breaking terms defined by $H=T_{a}h_{a}$. Through these symmetry-
breaking terms, the $\sigma$ meson has a finite vacuum expectation value (VEV)
and, consequently a finite quark mass beisitzer2014equation . The last two
terms in the Lagrangian are incorporated to define the vector meson
interaction with mesons. Due to $SU(3)_{L}\times SU(3)_{R}$ symmetry in the
effective Lagrangian, the term representing the quark-meson interaction can be
written as kovacs2016existence

$\mathcal{L}_{qm}=g_{s}\left(\bar{\Psi}_{L}\upvarphi\Psi_{R}+\bar{\Psi}_{R}\upvarphi^{\dagger}\Psi_{L}\right)-g_{v}\left(\bar{\Psi}_{L}\gamma^{\mu}L_{\mu}\Psi_{L}+\bar{\Psi}_{R}\gamma^{\mu}R_{\mu}\Psi_{R}\right).$
(2)

Here, $g_{v}$ and $g_{s}$ represent the coupling constants for the vector and
scalar mesons, respectively. In the above equation, $L_{\mu}$ and $R_{\mu}$
are defined in terms of pseudovector $(A_{\mu}^{a})$ and vector
$(V_{\mu}^{a})$ mesons as

$\displaystyle
L_{\mu}\left(R_{\mu}\right)=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{ccc}\frac{\omega+\rho^{0}}{\sqrt{2}}&\rho^{+}&K^{\star+}\\\
\rho^{-}&\frac{\omega-\rho^{0}}{\sqrt{2}}&K^{\star 0}\\\ K^{\star-}&K^{\star
0}&\phi\end{array}\right)^{\mu}\pm\frac{1}{2\sqrt{2}}\left(\begin{array}[]{ccc}\frac{f_{1}+a_{1}^{0}}{\sqrt{2}}&a_{1}^{+}&K_{1}^{+}\\\
{a_{1}^{-}}&\frac{f_{1}-a_{1}^{0}}{\sqrt{2}}&K_{1}^{0}\\\
K_{1}^{-}&\bar{K}_{1}^{0}&f_{1\phi}\end{array}\right)^{\mu}$ (9)

Using the total effective Lagrangian of the model, we obtain the thermodynamic
potential given as

$\begin{array}[]{cc}\Omega\left(\sigma_{u},\sigma_{d},\sigma_{s},\omega,\rho,\phi,\Phi,\bar{\Phi};T,\mu_{f}\right)=U\left(\sigma_{u},\sigma_{d},\sigma_{s}\right)+\Omega_{q\bar{q}}^{vac}\left(\sigma_{u},\sigma_{d},\sigma_{s}\right)+\\\
\mathcal{U}\left(\Phi,\bar{\Phi}:T,\mu_{f}\right)+\Omega_{q\bar{q}}^{th}\left(\sigma_{u},\sigma_{d},\sigma_{s},\Phi,\bar{\Phi}:T,\mu_{f}\right)+V(\omega,\rho,\phi)-U_{0}\left(\sigma_{u0},\sigma_{d0},\sigma_{s0}\right)\end{array}$
(10)

where the mesonic potential, including the chiral symmetry-breaking terms is
described as stiele2014thermodynamics

$\displaystyle
U\left(\sigma_{u},\sigma_{d},\sigma_{s}\right)=\frac{\lambda_{1}}{4}\left[\left(\frac{\sigma_{u}^{2}+\sigma_{d}^{2}}{2}\right)^{2}+\sigma_{s}^{4}+\left(\sigma_{u}^{2}+\sigma_{d}^{2}\right)\sigma_{s}^{2}\right]+\frac{\lambda_{2}}{4}\left(\frac{\sigma_{u}^{4}+\sigma_{d}^{4}}{4}+\sigma_{s}^{4}\right)-$
(11)
$\displaystyle\frac{c}{2\sqrt{2}}\sigma_{u}\sigma_{d}\sigma_{s}+\frac{m^{2}}{2}\left(\frac{\sigma_{u}^{2}+\sigma_{d}^{2}}{2}+\sigma_{s}^{2}\right)-\frac{h_{ud}}{2}\left(\sigma_{u}+\sigma_{d}\right)-h_{s}\sigma_{s}.$

Since vector-like gauge symmetries are not spontaneously broken in a vacuum,
$h_{ud}$ in the above equation represents the explicit symmetry breaking for
the $u$ and $d$ quarks VAFA1984173 . The six parameters $\lambda_{1}$,
$\lambda_{2}$, $m$, $c$, $h_{ud}$ and $h_{s}$ are determined by fitting the
known decay constants $f_{\pi}$ and $f_{K}$ along with the masses of mesons
such as $m_{\pi}$, $m_{K}$, $m_{\sigma}$ and squared masses of $\eta\prime$
and $\eta$ mesons lenaghan2000chiral . At vanishing temperature and baryonic
chemical potential values, the dynamical chiral symmetry breaking is taken
into account by the fermion vacuum term in the model represented by
$\Omega_{q\bar{q}}^{vac}\left(\sigma_{u},\sigma_{d},\sigma_{s}\right)$ in Eq.
11. Due to this term, the critical point is relocated to a lower temperature
and higher baryonic chemical potential values gupta2012revisiting ;
chatterjee2012including . Additionally, the vacuum potential energy,
$U_{0}(\sigma_{u0},\sigma_{d0},\sigma_{s0})$ is subtracted to obtain the
vanishing vacuum energy. The quark-antiquark interaction term is written as a
combination of the vacuum mesonic fluctuations and thermal terms,
$\Omega_{q\bar{q}}^{th}$, which is deduced from the fermionic determinant.
These terms can be described as PhysRevD.85.074018

$\Omega_{q\bar{q}}=\Omega_{q\bar{q}}^{vac}+\Omega_{q\bar{q}}^{th}$ (12)

where,

$\Omega_{q\bar{q}}^{vac}=-2N_{c}\sum_{f=u,d,s}\int\frac{d^{3}p}{(2\pi)^{3}}E_{f}^{*}=-\frac{N_{c}}{8\pi^{2}}\sum_{f=u,d,s}m_{f}^{*^{4}}\log\left[\frac{m_{f}^{*}}{\Lambda}\right],$
(13)

$\displaystyle\Omega_{q\bar{q}}^{th}=-2T\sum_{f=u,d,s}\int\frac{d^{3}p}{(2\pi)^{3}}\left[\ln{\bar{g}_{f}}+\ln
g_{f}\right].$ (14)

In above equation, $\bar{g}_{f}$ and $g_{f}$ are defined as

$\bar{g}_{f}=\left[1+3\Phi
e^{-(E_{f}^{*}-\mu_{f}^{*})/T}+3\bar{\Phi}e^{-2(E_{f}^{*}-\mu_{f}^{*})/T}+e^{-3(E_{f}^{*}-\mu_{f}^{*})/T}\right],\\\
$ (15)

and

$g_{f}=\left[1+3\bar{\Phi}e^{-(E_{f}^{*}+\mu_{f}^{*})/T}+3\Phi
e^{-2(E_{f}^{*}+\mu_{f}^{*})/T}+e^{-3(E_{f}^{*}+\mu_{f}^{*})/T}\right].$ (16)

In Eq. 13, $\Lambda$ is the regularisation scale parameter. The effective
single particle energy of the quarks is modified due to the interactions with
mesons and is defined as $E_{f}^{*}=\sqrt{p^{2}+m_{f}^{*2}}$. The $m_{f}^{*}$
represents the effective mass of constituent quarks given by

$m_{u}^{*}=\frac{g}{2}\sigma_{u},\quad
m_{d}^{*}=\frac{g}{2}\sigma_{d}\quad\text{and}\quad
m_{s}^{*}=\frac{g}{\sqrt{2}}\sigma_{s}.$ (17)

The Yukawa coupling constant $g$ value is determined by fixing the mass of
light quarks at $m_{l}$ = 300 MeV. In Eq. 10, the interaction term
incorporating the non-strange vector ($\omega$) and vector-isovector field
($\rho$), along with strange vector field ($\phi$) is written as
$V(\omega,\rho,\phi)=-\frac{1}{2}(m_{\omega}^{2}\omega^{2}+m_{\rho}^{2}\rho^{2}+m_{\phi}^{2}\phi^{2})$.
The effective chemical potential of the quarks is modified as a consequence of
vector-meson interactions KITAZAWA2003C289  and is defined in terms of quark
chemical potential, $\mu_{q}$, isospin chemical potential, $\mu_{I}$ and
strangeness chemical potential, $\mu_{S}$ as

$\begin{array}[]{l}\mu_{u}^{*}=\mu_{q}+\mu_{I}-g_{\omega u}\omega-g_{\rho
u}\rho\\\ \mu_{d}^{*}=\mu_{q}-\mu_{I}-g_{\omega d}\omega+g_{\rho d}\rho\\\
\mu_{s}^{*}=\mu_{q}-\mu_{S}-g_{\phi s}\phi.\end{array}$ (18)

To study the properties of chiral symmetry breaking and deconfinement within
the PQM framework, the model is extended by introducing gauge-invariant
Polyakov loop potential. It is defined as the trace of a Wilson loop in the
temporal direction, where the Wilson loop is a path-ordered exponential of the
gauge field. In the infinite mass limit of quarks, the order parameter that
contributes to confinement is determined by its expectation value. The
Polaykov loop is defined in a manner that obeys the center symmetry
polyakov1977thermal . In the current work, we use the polynomial form of the
Polyakov loop defined as grunfeld2018finite

$\frac{\mathcal{U}_{\text{poly
}}(\Phi,\bar{\Phi})}{T^{4}}=-\frac{b_{2}(T)}{2}\bar{\Phi}\Phi-\frac{b_{3}}{6}\left(\Phi^{3}+\bar{\Phi}^{3}\right)+\\\
\frac{b_{4}}{4}(\bar{\Phi}\Phi)^{2},$ (19)

and the temperature-dependent coefficient $b_{2}$ defined as

$b_{2}(T)=a_{0}+a_{1}\left(\frac{T_{0}}{T}\right)+a_{2}\left(\frac{T_{0}}{T}\right)^{2}+a_{3}\left(\frac{T_{0}}{T}\right)^{3}.$
(20)

The parameters are determined by fitting the data to lattice simulations,
which gives: $a_{0}=1.53$, $a_{1}=0.96$, $a_{2}=-2.3$, $a_{3}=-2.85$,
$b_{3}=13.34$ and $b_{4}=14.88$ ratti2006phases . To study the properties of
different scalar and vector fields, the total thermodynamic potential in Eq.
10 is minimized with respect to the fields as

$\frac{\partial\Omega}{\partial\sigma_{u}}=\frac{\partial\Omega}{\partial\sigma_{d}}=\frac{\partial\Omega}{\partial\sigma_{s}}=\frac{\partial\Omega}{\partial\omega}=\frac{\partial\Omega}{\partial\rho}=\frac{\partial\Omega}{\partial\phi}=\frac{\partial\Omega}{\partial\Phi}=\frac{\partial\Omega}{\partial\bar{\Phi}}.$
(21)

Using the total thermodynamic potential, pressure density is calculated by the
relation, $p$ = -$\Omega$. Further, the fluctuations of conserved charges are
calculated by Taylor’s series expansion method at zero value of the
corresponding chemical potential. The susceptibilities of $n_{th}$ order are
written as

$\chi^{qIS}_{ijk}=\frac{\partial^{i+j+k}[p/T^{4}]}{\partial\left(\mu_{q}/T\right)^{i}\partial(\mu_{I}/T)^{j}\partial(\mu_{S}/T)^{k}}$
(22)

Figure 1: The up quark condensate plotted as a function of baryon chemical
potential, $\mu_{B}$ for strangeness chemical potential, $\mu_{S}$ = 200, 0,
-100 and -160 MeV, value of system size, $R$ = $\infty$, 5 and 3 fm and
temperature, T = 30 and 60 MeV.

##  III Results and discussion

In this section, we have discussed the effect of the finite system size, $R$,
and varying values of the strangeness chemical potential, $\mu_{S}$, on the
asymmetric quark matter in the framework of the PQM model. The determination
of the parameters of the model is dependent on the inclusion of vacuum mesonic
fluctuations term. It has been highlighted that the expectation values of
curvature mass and sigma field are used as input to calculate the sigma-meson
mass, $m_{\sigma}$, and pion decay constant, $f_{\pi}$ 2016 . In the current
work, the value of the vector coupling constant has been fixed at $g_{v}$ =
6.5 and the isospin chemical potential, $\mu_{I}$ = 30 MeV for Figure 1 to
Figure 4. There is a finite value of isospin and strangeness chemical
potential in heavy-ion collisions; thus the field values and thermodynamic
quantities are calculated at finite values of these potentials
braun1999chemical .

In Figure 1, we have shown the variation of up quark condensate,
$\sigma_{u}/\sigma_{0}$, as a function of the baryonic chemical potential,
$\mu_{B}$ for $R$ = $\infty$, 5 and 3 fm, and the strangeness chemical
potential ranging from positive to negative for temperature values of T = 30
and 60 MeV. We have observed a sudden fall in the value of the quark
condensate with increasing $\mu_{B}$, which signifies the phase change at
higher chemical potential for all values of $\mu_{S}$ and $R$. For decreasing
volume and at a given value of $\mu_{S}$, this change in chiral condensate
values occurs at a higher value of $\mu_{B}$. On the contrary, for a given
system size and temperature, a drop in $\sigma_{u}/\sigma_{0}$ at lower
baryonic chemical potential values for decreasing $\mu_{S}$ is observed.
Hence, the critical chemical potential is found to be shifted towards higher
values for decreasing system volume and increasing strangeness chemical
potential. The nature of the phase transition from confined to deconfined
state with increasing $\mu_{B}$ is more accurately predicted through
derivatives of the strange and non-strange quark condensates.

Figure 2: The up quark density, $\rho_{u}$ and susceptibility, $\chi_{u}$
plotted as a function of baryon chemical potential, $\mu_{B}$ for temperature,
T = 30, 60, 90, 120 MeV, strangeness chemical potential, $\mu_{S}$ = 0, and
-160 MeV and value of system size, $R$ = 5 and 3 fm.  Figure 3: The strange
quark density, $\rho_{s}$ and susceptibility, $\chi_{s}$ plotted as a function
of baryon chemical potential, $\mu_{B}$ for temperature, T = 30, 60, 90, 120
MeV, strangeness chemical potential, $\mu_{S}$ = 0, and -160 MeV and value of
system size, $R$ = 5 and 3 fm.

The derivatives of the chiral condensates and the Polyakov loop parameters are
calculated to study the QCD phase diagram and, hence plotting the chiral phase
transition boundary and the deconfinement transition line, respectively. The
first-order phase transition line ends at a critical point and becomes a
crossover for the chiral limit stephanov2004qcd . The deconfinement transition
line remains a crossover for all the values of temperature. The first-order
phase transition line is deduced by plotting the susceptibility of the chiral
condensate as a function of baryonic chemical potential at a fixed temperature
value. In Figure 2, the variation in vector density, $\rho_{u}$ and
susceptibility, $\chi_{u}$ of up quark has been shown as a function of the
baryonic chemical potential for $R$ = 5 and 3 fm at $\mu_{S}$ = 200, 0, -100
and -160 MeV. The sharp rise in susceptibility values at low temperatures
suggests the occurrence of first-order phase transition, while the smooth
change signals the crossover. It is clear from Figure 2(a) and (c) that the
critical value of the baryonic chemical potential moves to a lower value at
finite $\mu_{S}$ and Figure 2(b) and (d) shows the shift to higher $\mu_{B}$
for decreasing volume. The sharp peak in the susceptibility of the up quark,
at T = 30 MeV, confirms the first-order phase transition. The susceptibility
peak smoothens with the increase in temperature values, indicating a crossover
transition. The change in the phase transition line from crossover to first-
order phase transition at varying values of $\mu_{S}$ and $R$ is highlighted
by the peak of $\chi_{u}$. The curve becomes smoother, and very little change
is observed at T = 120 MeV for all values of $R$ and $\mu_{S}$, showcasing the
crossover regime.

Figure 3 shows the dependence of the strange quark vector density, $\rho_{s}$,
and susceptibility, $\chi_{S}$, on the variation of strangeness chemical
potential and system size at different temperatures. For vanishing $\mu_{S}$
and given system size, $\rho_{S}$ is almost zero for $\mu_{B}$ $\approx$ 1050
MeV and T = 30 MeV. This shows that strange quarks are produced at the higher
baryonic chemical potential for the vanishing value of strangeness chemical
potential. With the increasing value of T and $\mu_{B}$, $\rho_{S}$ shows a
monotonically increasing trend. For the finite value of $\mu_{S}$, the density
of $s$ quarks in the system remains zero for finite $\mu_{B}$, and then a
sudden change is observed at fixed volume. This sudden rise may signify the
change in phase from a confined to a deconfined state. For high-temperature
values, $\rho_{S}$ increases uniformly with $\mu_{B}$. The increase in density
of $s$ quarks is observed to occur at a higher $\mu_{B}$ for $R$ = 3 fm at a
given temperature and strangeness chemical potential value. In comparison to
the up quark susceptibility in Figure 2, no peak is observed for the vanishing
value of $\mu_{S}$ at T = 30 MeV. Hence, the phase transition line of $s$
quark is a crossover at low-temperature values for given system sizes and zero
$\mu_{S}$, which indicates that the phase transition line of strange quarks
might not coincide with non-strange quarks at low-temperature values and
vanishing strangeness chemical potential. The appearance of the peak in the
$\chi_{S}$ at T = 30 and 60 MeV for both the given system volumes emphasizes
the shift of critical temperature to a higher point in the QCD phase diagram.

Figure 4: The derivatives of quark condensates and Polyakov loop variables
plotted as a function of temperature, T for quark chemical potential,
$\mu_{q}$ = 0 and 250 MeV, strangeness chemical potential, $\mu_{S}$ = 0, and
-160 MeV and value of system size, $R$ = 5 and 3 fm.

Figure 4 shows the derivatives of chiral strange and non-strange condensates
and Polyakov loop variables with varying temperature values for a given quark
chemical potential. The critical temperature is given by the peak of the
derivatives plotted. For the vanishing value of $\mu_{S}$, there are only
single peaks in the derivatives of chiral condensates and Polyakov loop
parameters. For the zero value of $\mu_{q}$, the curve for $\Phi$ coincides
with that of $\bar{\Phi}$ as these parameters have equal values in this
confined regime. The shift of peaks for different values of $\mu_{S}$ and
system size is the same as discussed earlier. For the finite value of
$\mu_{S}$ and $\mu_{q}$, two peaks are observed for $\Phi$. In this scenario,
the critical point is calculated by using the condition
$\Phi$(T)/$\Phi(T\to\infty)>$ 1/2 mao2010phase .

$R$ value |  $R$ = $\infty$ |  $R$ = 5 fm |  $R$ = 3 fm |  $R$ = $\infty$ |  $R$ = $\infty$
---|---|---|---|---|---
$\mu_{S}$ value |  $\mu_{S}$ = 0 |  $\mu_{S}$ = 0 |  $\mu_{S}$ = 0 |  $\mu_{S}$ = -100 MeV |  $\mu_{S}$ = -160 MeV
$T_{CP}$ (MeV) | 55 | 46 | 37 | 83 | 87
$\mu_{q(CP)}$ (MeV) | 306 | 321 | 352 | 268 | 238
Table 1: The value of critical temperature and quark chemical potential for
finite values of system size and strangeness chemical potential. Figure 5: The
QCD phase diagram for varying system sizes and strangeness chemical potential.

Using the critical temperature values derived from Figure 3 and Figure 4, the
QCD phase diagram for different system sizes and strangeness chemical
potential has been displayed in Figure 5. The chiral phase boundary and the
deconfinement transition line have been shown for $R$ = $\infty$, 5 and 3 fm
and $\mu_{S}$ = 0 in the right panel while for $\mu_{S}$ = 0, -100 and -160
MeV and $R$ = $\infty$ in the right panel. At the critical point, the first-
order phase transition changes to the crossover for the chiral phase
transition. The position of the critical point at the vanishing value of
$\mu_{S}$ and infinite size is $(\mu^{q}_{CP},T_{CP})$ = (306,55) MeV. For $R$
= 5 fm, the critical value of the temperature drops by 16.37 $\%$ and
increases by 4.91 $\%$ for quark chemical potential. A further drop of 19.57
$\%$ for the critical temperature and a 9.66 $\%$ increase in quark chemical
potential value is observed for $R$ = 3 fm. In the case of $\mu_{S}$ = -100
MeV, the critical temperature rises by 50.9 $\%$ whereas $\mu^{q}_{CP}$ falls
by 12.42 $\%$. For $\mu_{S}$ = -160 MeV, critical quark chemical potential
decreases by 11.2 $\%$, whereas $T_{CP}$ increases further by 4.81 $\%$.
Hence, from the above discussion, it is clear that the critical point shifts
to a lower temperature value and higher quark chemical potential with
decreasing system size. A similar change in the value of the critical point
has also been observed in the Polyakov loop modified Nambu-jona-lasinio (NJL)
model tripolt2014effect ; bhattacharyya2013thermodynamic , though opposite
change to a higher temperature for the critical point has also been reported
with a decrease in volume in the framework of Polyakov chiral quark mean field
model chahal2023effects . The inclusion of the fermion vacuum fluctuation term
in the current model significantly impacts the position of the QCD critical
point chatterjee2012including ; kovacs2023sensitivity . As the value of
$\mu_{S}$ approaches the positive value, the phase boundary becomes crossover
with decreasing $T_{CP}$ and increasing $\mu^{q}_{CP}$. The change in the
phase transition order at finite $\mu_{S}$ with increasing temperature has
also been highlighted in toublan2005qcd ; kumari2021quark . The values of the
critical points for varying volume and strangeness chemical potential has been
listed in Table 1. The deconfinement boundary remains crossover for all
temperatures and quark chemical potential values for all $R$ and $\mu_{S}$. A
very small shift to lower values is observed for the deconfinement transition
temperature for reduced system volume and negative values of strangeness
chemical potential.

Figure 6: The second order susceptibility, $\chi_{2}^{q}$ and kurtosis,
$\chi_{4}^{q}$/$\chi_{2}^{q}$ plotted as a function of temperature, T for
strangeness chemical potential, $\mu_{S}$ = 0, -100 and -160 MeV and value of
system size, $R$ = $\infty$, 5 and 3 fm. Figure 7: The second order
susceptibility, $\chi_{2}^{S}$, and kurtosis, $\chi_{4}^{S}$/$\chi_{2}^{S}$
plotted as a function of temperature, T for the value of system size, $R$ =
$\infty$, 5 and 3 fm.

Fluctuations and susceptibilities of conserved charges have been recognized as
observables, which helps to find the location and nature of the QCD critical
point. Figure 6 and Figure 7 show the second-order susceptibility and kurtosis
of quark number and strangeness number for varying values of system volume and
$\mu_{S}$. The susceptibilities have been calculated using Taylor’s series
expansion method for vanishing values of corresponding chemical potentials. In
Figure 6(a) and (c), for zero value of $\mu_{S}$ and changing system size,
$\chi_{2}^{q}$ and $\chi_{4}^{q}/\chi_{2}^{q}$ changes rapidly near the
transition regime. There is a slight shift in the critical temperature value
to a higher temperature with decreasing system size. But as discussed earlier,
for finite values of $\mu_{q}$, the critical temperature shifts towards lower
values for a reduced system volume. On the other hand, for finite values of
$\mu_{S}$ and infinite system size, the critical temperature is shifted to a
lower value for decreasing $\mu_{S}$, which contrasts with results discussed
at finite $\mu_{q}$. The kurtosis value drops nearly to zero for higher
temperatures, which signifies the change in degrees of freedom to quarks. The
value of second-order susceptibility for quarks increases monotonically with
the rise in temperature in both scenarios. The critical temperature for zero
value of $\mu_{q}$ appears to be around 155 MeV, which coincides with the
results of lattice QCD DING201452 . The value of susceptibilities depends on
the form of the Polyakov loop under consideration and vector interactions. The
comparison of these fluctuations of conserved charges with lattice data for
changing $g_{v}$ and the Polyakov loop has been discussed in earlier work for
the zero value of $\mu_{S}$ and infinite volume chahal2022quark .

The trend of second-order susceptibility of strangeness number is similar to
that of $\chi_{2}^{q}$. The kurtosis for strangeness number shows a peak
around the transition regime. The peak position is found to be shifted towards
a lower temperature value for increasing system volume. This is similar to the
change observed in susceptibilities of quark number. While investigating the
derivatives of the chiral condensates, the change observed in critical
temperature is opposite to that studied for finite chemical potential value.

##  IV Summary

To summarize, we have discussed the thermodynamic properties of asymmetric
quark matter using the Polyakov quark meson model extended by introducing
vector interactions. The effects of strangeness chemical potential, and finite
system size have been investigated by analyzing the variation of strange and
non-strange fields at varying temperatures and densities. The derivatives of
quark condensates and Polyakov loop variables have been studied to locate the
position of the QCD critical point. The impact of strangeness chemical
potential has been highlighted by studying the strangeness fraction for zero
and the negative value of $\mu_{S}$ at different temperatures. We have
observed that the production of s quarks gets saturated at high-temperature
values. The chiral phase boundary for $u$, $d$, and $s$ quarks coincides with
a crossover transition at lower $\mu_{q}$ before the critical point and a
first-order phase transition at higher values of quark chemical potential. The
deconfinement phase boundary remains a crossover for all temperature values.
With decreasing system volume, the critical point is found to shift to lower
values of temperature and higher values of quark chemical potential. The
positioning of the critical point is affected by the inclusion of the vacuum
term in the model. On the other hand, the critical point is repositioned to a
higher temperature and lower values of quark chemical potential for the
decreasing value of strangeness chemical potential. The susceptibilities of
conserved charges are enhanced in the transition region. The peak of the
kurtosis for the strangeness number gives the value of the critical point to
be $\approx$ 155 MeV, which is consistent with the lattice QCD studies at zero
chemical potential. In future work, the susceptibilities of conserved charges
can be studied at the finite value of chemical potential fan2019second . The
model can be further improvised using the functional renormalization approach
dupuis2021nonperturbative ; fu2022qcd .

## Acknowledgment

The authors sincerely acknowledge the support for this work from the Ministry
of Science and Human Resources (MHRD), Government of India, through an
Institute fellowship under the National Institute of Technology Jalandhar.
Arvind Kumar sincerely acknowledges the DST-SERB, Government of India, for
funding research project CRG/2019/000096.

## References

  * (1) A. Pandav, D. Mallick, B. Mohanty, Search for the qcd critical point in high energy nuclear collisions, Progress in Particle and Nuclear Physics (2022) 103960\.
  * (2) A. Ohnishi, Phase diagram and heavy-ion collisions: overview, Progress of Theoretical Physics Supplement 193 (2012) 1–10.
  * (3) O. Brüning, H. Burkhardt, S. Myers, The large hadron collider, Progress in Particle and Nuclear Physics 67 (3) (2012) 705–734.
  * (4) G. Odyniec, Rhic beam energy scan program—experimental approach to the qcd phase diagram, Journal of Physics G: Nuclear and Particle Physics 37 (9) (2010) 094028.
  * (5) L. Kumar and STAR Collaboration, Star results from the rhic beam energy scan-i, Nuclear Physics A 904 (2013) 256c–263c.
  * (6) C. Ratti, Lattice qcd and heavy ion collisions: a review of recent progress, Reports on Progress in Physics 81 (8) (2018) 084301.
  * (7) R. Gupta, Lattice qcd, in: AIP Conference Proceedings CONF-981188, Vol. 490, American Institute of Physics, 1999, pp. 3–9.
  * (8) C. DeTar, U. Heller, Qcd thermodynamics from the lattice, The European Physical Journal A 41 (3) (2009) 405–437.  doi:https://doi.org/10.1140/epja/i2009-10825-3.
  * (9) J. N. Guenther, Overview of the qcd phase diagram, The European Physical Journal A 57 (4) (2021) 1–23.
  * (10) S. Muroya, A. Nakamura, C. Nonaka, T. Takaishi, Lattice qcd at finite density: an introductory review, Progress of theoretical physics 110 (4) (2003) 615–668.
  * (11) J. Braun, B. Klein, B.-J. Schaefer, On the Phase Structure of QCD in a Finite Volume, Phys. Lett. B 713 (2012) 216–223.  arXiv:1110.0849, doi:10.1016/j.physletb.2012.05.053.
  * (12) N. B. Mata Carrizal, E. Valbuena Ordóñez, A. J. Garza Aguirre, F. J. Betancourt Sotomayor, J. R. Morones Ibarra, Effects of a Finite Volume in the Phase Structure of QCD, Universe 8 (5) (2022) 264.  doi:10.3390/universe8050264.
  * (13) N. Chahal, S. Dutt, A. Kumar, Effects of finite volume and magnetic fields on thermodynamic properties of quark matter and fluctuations of conserved charges, Physical Review C 107 (4) (2023) 045203.
  * (14) P. De Forcrand, O. Philipsen, The chiral critical line of nf= 2+ 1 qcd at zero and non-zero baryon density, Journal of High Energy Physics 2007 (01) (2007) 077\.
  * (15) F. Karsch, C. Allton, S. Ejiri, S. Hands, O. Kaczmarek, E. Laermann, C. Schmidt, Where is the chiral critical point in 3-flavor qcd?, Nuclear Physics B-Proceedings Supplements 129 (2004) 614–616.
  * (16) R.-A. Tripolt, J. Braun, B. Klein, B.-J. Schaefer, Effect of fluctuations on the qcd critical point in a finite volume, Physical Review D 90 (5) (2014) 054012\.
  * (17) L. Palhares, E. Fraga, T. Kodama, Finite-size effects and signatures of the qcd critical endpoint, Journal of Physics G: Nuclear and Particle Physics 37 (9) (2010) 094031.
  * (18) N. Magdy, M. Csanád, R. A. Lacey, Influence of finite volume and magnetic field effects on the qcd phase diagram, Journal of Physics G: Nuclear and Particle Physics 44 (2) (2017) 025101.
  * (19) N. Magdy, Influence of finite volume effect on the polyakov quark–meson model, Universe 5 (4) (2019) 94.
  * (20) J. Braun, B. Klein, P. Piasecki, On the scaling behavior of the chiral phase transition in QCD in finite and infinite volume, Eur. Phys. J. C 71 (2011) 1576\.  arXiv:1008.2155, doi:10.1140/epjc/s10052-011-1576-7.
  * (21) B. Klein, J. Braun, P. Piasecki, Scaling behavior in two-flavor qcd, finite quark masses and finite volume effects, PoS (11 2011).
  * (22) A. Bhattacharyya, S. K. Ghosh, R. Ray, K. Saha, S. Upadhaya, Polyakov–nambu–jona-lasinio model in finite volumes, EPL (Europhysics Letters) 116 (5) (2017) 52001.
  * (23) N. B. Mata Carrizal, E. Valbuena Ordóñez, A. J. Garza Aguirre, F. J. Betancourt Sotomayor, J. R. Morones Ibarra, Effects of a finite volume in the phase structure of qcd, Universe 8 (5) (2022) 264.
  * (24) V. Dexheimer, R. Negreiros, S. Schramm, Role of strangeness in hybrid stars and possible observables, Physical Review C 91 (5) (2015) 055808.
  * (25) D. Toublan, J. B. Kogut, The qcd phase diagram at nonzero baryon, isospin and strangeness chemical potentials: Results from a hadron resonance gas model, Physics Letters B 605 (1-2) (2005) 129–136.
  * (26) A. Mukherjee, A. Bhattacharyya, S. Schramm, Effects of a non-zero strangeness-chemical potential in strong interaction models, Physics Letters B 797 (2019) 134899.
  * (27) W. Fan, X. Luo, H. Zong, Probing the qcd phase structure with higher order baryon number susceptibilities within the njl model, Chinese Physics C 43 (3) (2019) 033103.
  * (28) N. Chahal, S. Dutt, A. Kumar, Quark matter properties and fluctuations of conserved charges in (2+ 1)-flavored quark model, Chinese Physics C (2022).
  * (29) S. Chatterjee, K. A. Mohan, Fluctuations and correlations of conserved charges in the (2+ 1) polyakov quark meson model, Physical Review D 86 (11) (2012) 114021\.
  * (30) S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, K. Szabo, Fluctuations of conserved charges at finite temperature from lattice qcd, Journal of High Energy Physics 2012 (1) (2012) 1–15.
  * (31) A. Bhattacharyya, R. Ray, S. Sur, Fluctuations at finite volume in strongley interacting matter, in: Proceedings of the DAE Symp. on Nucl. Phys, Vol. 59, 2014, p. 708.
  * (32) B. W. Mintz, R. Stiele, R. O. Ramos, J. Schaffner-Bielich, Phase diagram and surface tension in the three-flavor polyakov-quark-meson model, Physical Review D 87 (3) (2013) 036004.
  * (33) V. Skokov, B. Stokić, B. Friman, K. Redlich, Meson fluctuations and thermodynamics of the polyakov-loop-extended quark-meson model, Physical Review C 82 (1) (2010) 015206.
  * (34) A. Zacchi, R. Stiele, J. Schaffner-Bielich, Compact stars in a su (3) quark-meson model, Physical Review D 92 (4) (2015) 045022.
  * (35) K. Otto, M. Oertel, B.-J. Schaefer, Hybrid and quark star matter based on a nonperturbative equation of state, Physical Review D 101 (10) (2020) 103021.
  * (36) T. Beisitzer, R. Stiele, J. Schaffner-Bielich, Equation of state with an extended su(3) quark-meson model under conditions prevailing in core collapse supernovae, Phys. Rev. D 90 (2014) 085001.  doi:10.1103/PhysRevD.90.085001.
URL https://link.aps.org/doi/10.1103/PhysRevD.90.085001

  * (37) P. Kovács, Z. Szép, G. Wolf, Existence of the critical endpoint in the vector meson extended linear sigma model, Physical Review D 93 (11) (Jun 2016).  doi:10.1103/physrevd.93.114014.
URL http://dx.doi.org/10.1103/PhysRevD.93.114014

  * (38) G. t Hooft, Symmetry breaking through bell-jackiw anomalies, Physical Review Letters 37 (1) (1976) 8–11.
  * (39) T. Beisitzer, R. Stiele, J. Schaffner-Bielich, Equation of state with an extended su (3) quark-meson model under conditions prevailing in core collapse supernovae, Physical Review D 90 (8) (2014) 085001.
  * (40) P. Kovács, Z. Szép, G. Wolf, Existence of the critical endpoint in the vector meson extended linear sigma model, Physical Review D 93 (11) (2016) 114014\.
  * (41) R. Stiele, E. S. Fraga, J. Schaffner-Bielich, Thermodynamics of (2+ 1)-flavor strongly interacting matter at nonzero isospin, Physics Letters B 729 (2014) 72–78.
  * (42) C. Vafa, E. Witten, Restrictions on symmetry breaking in vector-like gauge theories, Nuclear Physics B 234 (1) (1984) 173–188.  doi:https://doi.org/10.1016/0550-3213(84)90230-X.
URL https://www.sciencedirect.com/science/article/pii/055032138490230X

  * (43) J. T. Lenaghan, D. H. Rischke, J. Schaffner-Bielich, Chiral symmetry restoration at nonzero temperature in the su (3) r$\times$ su (3) l linear sigma model, Physical Review D 62 (8) (2000) 085008.
  * (44) U. S. Gupta, V. K. Tiwari, Revisiting the phase structure of the polyakov-quark-meson model in the presence of vacuum fermion fluctuation, Physical Review D 85 (1) (2012) 014010.
  * (45) S. Chatterjee, K. A. Mohan, Including the fermion vacuum fluctuations in the (2+ 1) flavor polyakov quark-meson model, Physical Review D 85 (7) (2012) 074018\.
  * (46) S. Chatterjee, K. A. Mohan, Including the fermion vacuum fluctuations in the ($2+1$) flavor polyakov quark-meson model, Phys. Rev. D 85 (2012) 074018.  doi:10.1103/PhysRevD.85.074018.
URL https://link.aps.org/doi/10.1103/PhysRevD.85.074018

  * (47) M. Kitazawa, T. Koide, T. Kunihiro, Y. Nemoto, Effects of vector coupling on chiral and color-superconducting phase transitions - interplay among the scalar, pairing and vector interaction, Nuclear Physics A 721 (2003) C289–C292.  doi:https://doi.org/10.1016/S0375-9474(03)01052-2.
URL https://www.sciencedirect.com/science/article/pii/S0375947403010522

  * (48) A. M. Polyakov, Thermal properties of gauge fields and quark liberation, Tech. rep., International Centre for Theoretical Physics (1977).
  * (49) A. G. Grunfeld, G. Lugones, Finite size effects in strongly interacting matter at zero chemical potential from polyakov loop nambu-jona-lasinio model in the light of lattice data, The European Physical Journal C 78 (8) (2018) 1–13.
  * (50) C. Ratti, M. A. Thaler, W. Weise, Phases of qcd: Lattice thermodynamics and a field theoretical model, Physical Review D 73 (1) (2006) 014019.
  * (51) S. Carignano, M. Buballa, W. Elkamhawy, Consistent parameter fixing in the quark-meson model with vacuum fluctuations, Physical Review D 94 (3) (Aug 2016).  doi:10.1103/physrevd.94.034023.
URL http://dx.doi.org/10.1103/PhysRevD.94.034023

  * (52) P. Braun-Munzinger, I. Heppe, J. Stachel, Chemical equilibration in pb+ pb collisions at the sps, Physics Letters B 465 (1-4) (1999) 15–20.
  * (53) M. Stephanov, Qcd phase diagram and the critical point, Progress of Theoretical Physics Supplement 153 (2004) 139–156.
  * (54) H. Mao, J. Jin, M. Huang, Phase diagram and thermodynamics of the polyakov linear sigma model with three quark flavors, Journal of Physics G: Nuclear and Particle Physics 37 (3) (2010) 035001.
  * (55) A. Bhattacharyya, P. Deb, S. K. Ghosh, R. Ray, S. Sur, Thermodynamic properties of strongly interacting matter in a finite volume using the polyakov–nambu–jona-lasinio model, Physical Review D 87 (5) (2013) 054009.
  * (56) G. Kovács, P. Kovács, P. M. Lo, K. Redlich, G. Wolf, Sensitivity of finite size effects to the boundary conditions and the vacuum term, arXiv preprint arXiv:2307.10301 (2023).
  * (57) M. Kumari, A. Kumar, Quark matter within polyakov chiral su(3) quark mean field model at finite temperature, The European Physical Journal Plus 136 (1) (2021) 19.
  * (58) H.-T. Ding, Recent lattice qcd results and phase diagram of strongly interacting matter, Nuclear Physics A 931 (2014) 52–62, qUARK MATTER 2014.  doi:https://doi.org/10.1016/j.nuclphysa.2014.09.053.
URL https://www.sciencedirect.com/science/article/pii/S0375947414004357

  * (59) W. Fan, X. Luo, H. Zong, Second to tenth order susceptibilities of conserved charges within a modified nambu-jona-lasinio model, Chinese Physics C 43 (5) (2019) 054109.
  * (60) N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, N. Wschebor, The nonperturbative functional renormalization group and its applications, Physics Reports 910 (2021) 1–114.
  * (61) W.-j. Fu, Qcd at finite temperature and density within the frg approach: an overview, Communications in Theoretical Physics (2022).


# Traversable Wormhole Solutions in $f(R)$ Gravity Via Karmarkar Condition

M. Farasat Shamir<EMAIL_ADDRESS>National University of Computer
and Emerging Sciences,
Lahore Campus, Pakistan.  I. Fayyaz<EMAIL_ADDRESS>National University of
Computer and Emerging Sciences,
Lahore Campus, Pakistan.

###### Abstract

Motivated by recent proposals of possible wormhole shape functions, we
construct a wormhole shape function by employing the Karmarkar condition for
static traversable wormhole geometry. The proposed shape function generates
wormhole geometry that connects two asymptotically flat regions of spacetime
and satisfies the required conditions. Further, we discuss the embedding
diagram in three-dimensional Euclidean space to present the wormhole
configurations. The main feature of current study is to consider three well-
known $f(R)$ gravity models, namely exponential gravity model, Starobinsky
gravity Model and Tsujikawa $f(R)$ gravity model. Moreover, we investigate
that our proposed shape function provides the wormhole solutions with less (or
may be negligible) amount of exotic matter corresponding to the appropriate
choice of $f(R)$ gravity models and suitable values of free parameters.
Interestingly, the solutions obtained for this shape function generate stable
static spherically symmetric wormhole structure in the context of non-
existence theorem in $f(R)$ gravity. This may lead to a better analytical
representation of wormhole solutions in other modified gravities for the
suggested shape function.

Keywords: Modified Gravity, Wormhole Geometry, Karmarkar condition.
PACS: 04.50.Kd.

##  I Literature Survey

Wormholes are assumptive tube-like geometrical structures connecting two
widely separated asymptotically flat distant universes, or two different
asymptotically flat portions of the same universe and have no horizon. The
concept of a tube-like or bridge-like structure connecting two spacetimes was
firstly presented by Einstein and Rosen ein . Further, they explored the exact
solution that describe the geometrical structure of the bridge. The solution
presented by Einstein and Rosen is linked with the work of Flamm fla , who
first time developed the isometric embedding of Schwarzschild solution. Ellis
ell  introduced another term for wormholes which is known as $``Drainhole"$.
Wheeler whe  named them as $``Geons"$ and predicted the shape of a wormhole
which offers a twofold space. Static wormhole configurations have a constant
throat radius while non-static wormhole configurations have a variable throat
radius. Kar kar7  discussed the static wormhole and examined their properties
along with examples. Kar and Sahdev kar72  explored evolving Lorentzian
wormholes and discussed some results for exponential and Kaluza-Klein
inflation. The locally anisotropic wormhole and flux tube like solutions have
been studied using the technique of anholonomic frames vac . Dzhunushaliev et
al. Dzh  have investigated the linear stability analysis for wormhole
configurations with and without electric and/or magnetic fields. Visser et al.
Vis  have shown that the violation of null energy condition (NEC) can be
restricted to an arbitrarily small region by an appropriate choice of the
wormhole geometry. The static and spherically symmetric Lorentzian wormhole
solutions are discussed within the frame work of general theory of relativity
(GR) can7 .

Morris-Thorne mor1  were the first who gave the idea that human being can
travel through wormhole tunnels. They investigated the static spherically
symmetric wormholes by using the principles of GR and presented the
fundamental theory for traversable wormholes. According to Morris et al. mor2
investigation, for the formation of wormhole structure, the presence of exotic
matter plays an important role in the context of GR. Exotic matter is a form
of dark energy (having an EoS with $\omega<-1/3$), produces a repulsive force.
Recent observations have shown that the dark energy is the main reason for the
accelerated expansion of the universe. A few candidates have been proposed in
the literature to presenting dark energy, like, a positive cosmological
constant, Chaplygin gas, the quintessence fields etc. GR has been the most
successful theory of last century. One may simply needs to add a cosmological
constant or some other exotic source to explain the accelerated expansion.
However, there are models which may also explain dark energy by modifying
gravity. Recently, different researchers discussed wormholes geometry in
different modified theories of gravity nan ; lob ; ric ; meh1 ; dzh ; sha ;
meh2 . In $1970$, Buchdahl Buch  proposed one of the most well known and
simplest modified theory of gravity i.e., $f(R)$ gravity which can be obtained
by adopting a modification in the Einstein Hilbert action with an arbitrary
function of the Ricci scalar $R$. Following in this way, several modifications
of GR have been developed by considering different approaches. The geometry of
dark energy model can be depicted by $f(R)$ gravity cap1 ; cap2 . Harko et al.
har  discussed static spherically symmetric wormholes with non-exotic matter
(respected energy conditions) in the framework of $f(R)$ gravity. Rahaman and
collaborators rah ; jam  developed new solutions for static wormholes in
$f(R)$ gravity and acknowledged the existence of wormhole solutions that do
not require exotic matter. Bronnikov and Starobinsky bron7  discussed the very
well known non-existence theorem for wormhole geometry and observed that no
realistic wormhole can be formulated in scalar-tensor models for a positive
scalar function. Further, Bronnikov et al. bro  showed that for
$\frac{df}{dR}=F(R)<0$, the non-existence condition of wormhole could be
violated in $f(R)$ theory of gravity. Bahamonde et al. bah  discussed the
existence of wormholes in galactic halos. Moreover, they described a non-
static wormhole geometry that asymptotically impending towards the Friedmann
Lemaître Robertson Walker universe.

One interesting topic in traversable wormhole geometry is the study of
wormhole shape function (WSF) $\epsilon(r)$ for asymptotically flat wormhole
with their essential properties (see detail in Sec. $II$). Recently, many
authors have discussed various ansatz shape functions to describe the wormhole
structure. Godani and Samanta goda  explored asymptotically flat wormhole with
WSF $\epsilon(r)=\frac{r_{0}Log(r+1)}{Log(r_{0}+1)}$. Jahromi and Moradpour
jah  introduced a WSF $\epsilon(r)=a~{}tanhr$. The WSF
$\epsilon(r)=\alpha+\beta(r)$ is proposed by Cataldo and Liempi cata . Samanta
et al. sam  defined $\epsilon(r)=\frac{r}{e^{(r-r_{0})}}$, known exponential
WSF. Some authors jam2 ; shar2 ; sham3  used
$\epsilon(r)=(r_{0})^{n+1}r^{-n}$, for different values of $n$. Recently,
Golchin et al. gol  have considered a special class in $f(R)$ theory of
gravity ($F(R_{0})=0$ and $\frac{df}{dR}|_{(R_{0})}=0$, where $R_{0}$ is a
fixed quantity) and construct a WSF.

Motivating from this, in this article, we calculate a WSF by employing the
Karmarkar condition (KMc). Karmarkar karm  developed a mandatory condition for
a static and spherically symmetric line element to be of class one. In recent
years, different researchers have considered the KMc to discussed the
configurations of spherically symmetric compact objects 6abb ; 6ful ; 6kuh ;
6bah ; 6ged . Kuhfittig kuhf  developed wormhole geometry using KMc and shown
that the embedding theory may provide the basis for a complete wormhole
solution. Recently, Fayyaz and Shamir fay1  constructed a WSF by applying KMc
and analysed that it obeys all the required conditions and exhibit the
presence of exotic matter in the context of GR. To the best of our knowledge,
we first time apply this WSF in modified $f(R)$ theory of gravity. This paper
is composed as follows. In Sec. II, we construct a WSF by employing KMc and
discuss the embedding diagram for wormhole geometry into three-dimensional
Euclidean space. In Sec. III, we discuss the formulation of $f(R)$ theory of
gravity. Further, we consider three different well known viable $f(R)$ gravity
models and examined the NEC and weak energy condition (WEC) via graphical
representation. Conclusion is given in Sec. IV.

##  II The Geometry of Traversable Wormhole and Embedding Diagram

In the present work, our main focus is to develop a WSF using the KMc that
describes wormhole geometry. For this purpose, we assume static spherically
symmetric spacetime defined as:

$ds^{2}=-e^{\chi(r)}dt^{2}+e^{\eta(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}.$ (1)

The non-zero Riemann curvature coordinates corresponding to the above
spacetime (1), are given below:

$R_{1414}=\frac{e^{\chi}(2\chi^{\prime\prime}+\chi^{\prime
2}-\chi^{\prime}\eta^{\prime})}{4},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{1212}=\frac{r\eta^{\prime}}{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{1334}=R_{1224}Sin^{2}\theta,$
$R_{2323}=\frac{r^{2}Sin^{2}\theta(e^{\eta}-1)}{e^{\eta}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{3434}=\frac{rSin^{2}\theta\eta^{\prime}e^{\chi-\eta}}{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{1224}=0.$

These components satisfying the very well known Karmarkar relation

$R_{1414}=\frac{R_{1212}R_{3434}+R_{1224}R_{1334}}{R_{2323}},$

with $R_{2323}\neq 0$. The form of spacetime, satisfying the KMc is known as
embedding class one. By substituting the non-zero components of Riemann
curvature in Karmarkar relation, we get the following differential equation:

$\frac{\chi^{\prime}\eta^{\prime}}{1-e^{\eta}}=\chi^{\prime}\eta^{\prime}-2\chi^{\prime\prime}-\chi^{\prime
2},$

where $e^{\eta}\neq 1$. The solution of the above differential equation is
given as

$e^{\eta}=1+Ae^{\chi}\chi^{\prime 2}.$ (2)

Here, $A$ is an integrating constant. Now, to construct the WSF, we assume
Morris-Thorne metric defined as:

|  |
---|---|---
Figure 1: Evaluation of wormhole shape function $\epsilon(r)$ (Ist plot) and its required conditions $\epsilon(r)-r=0$ at $r=r_{0}$ (2nd plot), $\frac{\epsilon(r)}{r}\rightarrow 0$ for $r\rightarrow\infty$ (3rd plot) for $r_{0}=2$ and $C=1.9$ to present the realistic wormhole geometry. |
---|---
Figure 2: Condition
$\frac{\epsilon(r)-r\epsilon^{\prime}(r)}{\epsilon^{\prime}(r)}>0$ (Ist plot)
and $\epsilon^{\prime}(r)<1$ (2nd plot), for $r_{0}=2$ and $C=1.9$.

$ds^{2}=-e^{\chi(r)}dt^{2}+\frac{1}{1-\frac{\epsilon(r)}{r}}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}.$ (3)

The metric co-efficient $\chi(r)$ is known as redshift function such that
$\chi(r)\rightarrow 0$ as $r\rightarrow\infty$. Here, we assume the redshift
function $\chi(r)$, defined as anch ; sham3

$\chi(r)=\frac{-2\xi}{r},$ (4)

where $\xi$ is an arbitrary constant. By comparing the Eqns. (1) and (3), we
obtain

$\eta(r)=Log[\frac{r}{r-\epsilon(r)}].$ (5)

Here, $\epsilon(r)$ is a WSF. Now, by using Eqns. (2), (4) and (5), we
calculate the following WSF

$\epsilon(r)=r-\frac{r^{5}}{r^{4}+4\xi^{2}Ae^{\frac{-2\xi}{r}}}.$ (6)

According to Morris and Thorne mor1 , to get a traversable wormhole solution,
WSF should satisfy the following essential properties:

  1. 1.

$\epsilon(r)-r=0$ at $r=r_{0}$,

  2. 2.

The condition
$\frac{\epsilon(r)-r\epsilon^{\prime}(r)}{\epsilon^{\prime}(r)}>0$ must be
fulfilled at $r=r_{0}$,

  3. 3.

$\epsilon^{\prime}(r)<1$,

  4. 4.

$\frac{\epsilon(r)}{r}\rightarrow 0$ at $r\rightarrow\infty$,

where $r_{0}$ is known as a wormhole throat radius and $r$ is the radial
coordinate such that $r_{0}\leq r\leq\infty$. When we evaluate Eqn. (6) at the
throat i.e., $\epsilon(r_{0})-r_{0}=0$, we get a trivial solution $r_{0}=0$.
To handle this problem we add a free parameter $``C"$ in Eqn. (6). Now, Eqn
(6) takes the form
$\epsilon(r)=r-\frac{r^{5}}{r^{4}+4\xi^{2}Ae^{\frac{-2\xi}{r}}}+C$. Condition
($1$) provides
$A=\frac{r_{0}^{4}(r_{0}-C)}{4\xi^{2}e^{\frac{-2\xi^{2}}{r_{0}}}}$. After
substituting the value of $A$ in Eqn. (6), one can find the expression for WSF
$\epsilon(r)$ as

$\epsilon(r)=r-\frac{r^{5}}{r^{4}+r_{0}^{4}(r_{0}-C)}+C,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0<C<r_{0}.$
(7)

Conditions ($2$) and ($3$) also satisfy for the given range of $C$. Applying
condition ($4$) on Eqn. (7), we found that

$\lim_{r\rightarrow\infty}\frac{\epsilon(r)}{r}=0.$ (8)

|  |
---|---|---
Figure 3: Plot of embedding diagram for upper universe $h(r)>0$ with respect to radial coordinate with slice $t=const$ and $\theta=\pi/2$. For a full visualization of the wormhole surface, took a $2\pi$ rotation around the $h-$axis. |  |
---|---|---
Figure 4: Plot of embedding diagram for lower universe $h(r)<0$ with respect
to radial coordinate with slice $t=const$ and $\theta=\pi/2$. For a full
visualization of the wormhole surface, took a $2\pi$ rotation around the
$h-$axis.

Thus Eqn. (7) provides asymptotically flat traversable wormholes. We have
plotted all the mentioned conditions in Figs. 1 and 2. It can be clearly seen
from these figures that our proposed WSF fulfils all the required conditions.
To represent the wormhole geometry, we used embedding diagram and excerpt some
fruitful information. The spherical symmetry allows us to take an equatorial
slice $\theta=\frac{\pi}{2}$ with a rigorous part of time $t=const$. Using
these assumptions in Eqn. (3), we get

$ds^{2}=\frac{1}{1-\frac{\epsilon(r)}{r}}dr^{2}+r^{2}d\phi^{2}.$ (9)

To visualize this part, embedded Eqn. (9) into three-dimensional Euclidean
space with cylindrical coordinates ($r,~{}h,~{}\phi$), given by

$ds^{2}=dr^{2}+dh^{2}+r^{2}d\phi^{2}.$ (10)

The embedded surface $h\equiv h(r)$ in three-dimensional space with axially
symmetry can be written from last equation as

$ds^{2}=\bigg{[}1+\bigg{(}\frac{dh}{dr}\bigg{)}^{2}\bigg{]}dr^{2}+r^{2}d\phi^{2}.$
(11)

By comparing Eqns. (10) and (11), one can easily find that

$\frac{dh}{dr}=\pm\bigg{(}\frac{r}{\epsilon(r)}-1\bigg{)}^{\frac{-1}{2}}.$
(12)

From Eqn. (12), we investigate that at the throat the embedded surface is
vertical i.e. $\frac{dh}{dr}\rightarrow\infty$. We also examine that away from
the throat the space is asymptotically flat because $\frac{dh}{dr}$ tends to
$0$ as $r$ tends to infinity. We have shown the embedded diagram in Figs. 3
and 4. Moreover, one can visualize the upper universe for $h>0$ and the lower
universe $h<0$ in Figs. 3 and 4.

##  III $f(R)$ Gravity

Our starting point for the $f(R)$ theory of gravity is the Einstein-Hilbert
action for modified $f(R)$ gravity,

$\mathcal{S}=\int\sqrt{-g}(\frac{1}{16\pi G}[f(R)]+\mathcal{L}_{m})d^{4}x.$
(13)

Here, $\mathcal{L}_{m}$ stands for the matter Lagrangian field and $f(R)$ is a
function of the Ricci scalar $R$. One can derive the field equations by
varying the above action with respect to the metric $g_{\zeta\eta}$ as:

$F(R)R_{\zeta\eta}-\frac{1}{2}f(R)g_{\zeta\eta}-\nabla_{\zeta}\nabla_{\eta}F(R)+g_{\zeta\eta}\square
F(R)=8\pi T^{(m)}_{\zeta\eta},$ (14)

where $\square=\nabla_{\zeta}\nabla^{\zeta}$ and $\nabla_{\zeta}$ denotes the
covariant derivative. $T^{(m)}_{\zeta\eta}=(\rho,-p_{r},-p_{t},-p_{t})$
represents stress energy momentum tensor, where $\rho$, $p_{r}$ and $p_{t}$
are the radial and tangential pressures respectively, and
$F(R)=\frac{df}{dR}$. Now by using the trace of energy momentum tensor in
above equation and after some manipulations, the above equation can also be
written as

$G_{\zeta\eta}=R_{\zeta\eta}-\frac{1}{2}Rg_{\zeta\eta}=T^{(eff)}_{\zeta\eta}=\frac{8\pi}{F}T^{(m)}_{\zeta\eta}+\frac{1}{F}\bigg{[}\nabla_{\zeta}\nabla_{\eta}F(R)-\big{(}\square
F(R)+\frac{1}{2}RF(R)-\frac{1}{2}f(R)\big{)}g_{\zeta\eta}\bigg{]}.$ (15)

Now using Eqn. (15) along with eqn. (1), one can find the following field
equations as bah :

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{f}{2}-(1-\frac{\epsilon}{r})R^{\prime
2}f_{RRR}-\frac{f_{R}}{2r^{2}}\bigg{[}r\big{(}(\epsilon^{\prime}-4)+3\epsilon\big{)}\chi^{\prime}-2r(r-\epsilon)\chi^{\prime
2}+2r(\epsilon-r)\chi^{\prime\prime}\bigg{]}-\frac{f_{RR}}{2r^{2}}\bigg{[}2r(r-\epsilon)R^{\prime\prime}$
(16) $\displaystyle-(r(\epsilon^{\prime}-4)+3\epsilon)R^{\prime}\bigg{]}$
$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{-f}{2}+\frac{\epsilon
f_{R}}{2r^{3}}\bigg{[}2r^{2}\chi^{\prime
2}+2r^{2}\chi^{\prime\prime}-r\chi^{\prime}-2\bigg{]}+\frac{e^{-2\chi}f_{R}}{2r^{2}}\bigg{[}e^{2\chi}(\epsilon^{\prime}(r\chi^{\prime}+2)-2r^{2}(\chi^{\prime
2}+\chi^{\prime\prime}))\bigg{]}+f_{RR}R^{\prime}$ (17)
$\displaystyle(1-\frac{\epsilon}{r})(\chi^{\prime}+\frac{2}{r}),$
$\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{-f}{2}+\frac{f_{R}}{2r^{3}}\bigg{[}\epsilon(2r\chi^{\prime}+1)+(\epsilon^{\prime}-2r\chi^{\prime})r\bigg{]}+\frac{f_{RR}}{2r^{2}}\bigg{[}R^{\prime}(r(2r\chi^{\prime}-\epsilon^{\prime}+2)-\epsilon(2r\chi^{\prime}+1))+2r(r-\epsilon)R^{\prime\prime}\bigg{]}+$
(18) $\displaystyle f_{RRR}(R^{\prime 2}(1-\frac{\epsilon}{r})).$

Here prime denotes the derivative with respect to the radial coordinate $r$.
$f_{R},~{}f_{RR},~{}f_{RRR}$ are the first, second and third derivatives of
$f(R)$ with respect to the Ricci scalar $R$ respectively. Bronnikov and
Starobinsky bron7  discussed the stability condition for wormhole geometry
which is also free from ghosts. It is shown that no realistic wormhole can be
formulated in scalar-tensor models for a positive scalar function. In the
context of $f(R)$ theory of gravity, the non-existence of wormhole could be
violated if $\frac{df}{dR}=F(R)$ is negative bro . According to the classical
GR, presence of exotic matter in wormhole structure is the main cause of
violation of NEC denoted as $\rho+p_{r}\geq 0,~{}~{}\rho+p_{t}\geq 0$, and WEC
denoted as $\rho\geq 0,~{}~{}\rho+p_{r}\geq 0,~{}~{}\rho+p_{t}\geq 0$.
Hochberg and Visser hoc2 ; hoc4  extended the results for wormhole solutions,
which was earlier described by Morris-Thorne mor1 , with exotic matter and
provided the result that wormhole throat does not respect NEC. In recent work,
our main focus is to calculate the wormhole solutions that respects NEC and
also violates the non-existence theorem in the context of $f(R)$ theory of
gravity. In view of this approach, we further simplify our calculation for
$\rho+p_{r}$ and $\rho+p_{t}$. We found that to respect the NEC

$\displaystyle\rho+p_{r}$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{\prime}r-\epsilon}{2r^{3}}\bigg{(}f_{RR}R^{\prime}r+2fr\bigg{)},$
(19) $\displaystyle\rho+p_{t}$ $\displaystyle=$ $\displaystyle
f_{R}\frac{(\epsilon^{\prime}r+\epsilon)}{2r^{3}}-\frac{\chi^{\prime}(\epsilon^{\prime}r-\epsilon)}{2r^{2}},$
(20)

should be positive at the throat. One can notice that first part in Eqn. (19)
involves flaring out condition which gives $\epsilon^{\prime}r-\epsilon<0$.
Thus 2nd part must be negative for $\rho+p_{r}>0$, i.e,
$f_{RR}R^{\prime}r+2f_{R}<0$. Eqn. (20) involves term $f_{R}=df/dR$, as we
consider violation of non-existence theorem (i.e, $df/dR<0$) for spherically
symmetric wormhole structure, therefore, for $\rho+p_{t}>0$ the remaining term
should also be negative. This examination shows the importance of flaring out
condition and selection of $f(R)$ gravity model. In next Section, we discuss
wormhole solutions under the influence of three different viable $f(R)$
gravity models. It is important to mention that we use $\xi=-1$, $r_{0}=2$ and
$C=1.9$ throughout our further analysis.

###  III.1 The Exponential Gravity Model

|
---|---
|
Figure 5: The both terms of NEC $\rho+p_{r}$ (upper left) and $\rho+p_{t}$ (upper right) are respected at the throat for wormhole geometry for all the considering combinations of free parameters. The first term of WEC $\rho$ (lower left) also shows the validation at the throat. While, we get the violation of non-existence theorem when $\mu>0$, $R_{*}>0$ (lower right). |
---|---
|
Figure 6: $\rho+p_{r}>0$ (upper left), $\rho+p_{t}>0$ (upper left) and
$\rho>0$ (lower left) for all are considering combinations of $\lambda,R_{*}$
and $n$. Whereas, $F<0$ for $\lambda<0,R_{*}<0$ and $n>0$.

Cognola et al. cog7  introduced and investigated the exponential gravity
model. This model describes the inflation of early universe and accelerated
expansion of the current universe in a natural way. The exponential model is
defined as

$f(R)=R-\mu R_{*}[1-e^{\frac{-R}{R_{*}}}],$ (21)

where $R_{*}$ and $\mu$ are arbitrary constants. In particular, we use
$\mu=1.8$, $R_{*}=2$ and evaluate the graphical behaviour of
$F=\frac{df}{dR}$, $\rho$, $\rho+p_{r}$ and $\rho+p_{t}$. It can be clearly
seen from Fig. 5 that the existence of exotic matter can be avoided at the
throat for the traversable wormhole geometry. We itemize the analysis below,
in the light of exponential $f(R)$ gravity model as:

  * •

For $R_{*}<0$ and $\mu>0$, WEC is respected for throughout the wormhole
geometry, whereas, $F=df/dR>0$ i.e, there is no spherically symmetric wormhole
configurations. Such solutions may be interesting for non-static or thin-shell
wormholes due to its stable spherically symmetric perturbations bron7 .

  * •

For $R_{*}<0$ and $\mu<0$, we found that $\frac{df}{dR}<0$, whereas, violation
of NEC have shown that the wormhole space is filled with exotic matter.

  * •

For $R_{*}>0$ and $\mu<0$, we get the same result as for $R_{*}<0$ and
$\mu>0$.

  * •

For $R_{*}>0$ and $\mu>0$, we found that the existence of exotic matter can be
avoided at the throat for the traversable wormhole geometry. It can be clearly
seen from Fig. 5, the term $\rho+p_{r}$ is positive for $1\leq r/r_{0}<1.4$,
$\rho+p_{t}>0$ for $1\leq r/r_{0}<1.7$ and $\rho>0$ for $1\leq r/r_{0}<1.3$.
Therefore, NEC is respected for $r/r_{0}\in[1,1.4)$ and WEC is respected for
$r/r_{0}\in[1,1.3)$.

Thus, we conclude that $\mu=1.8$, $R_{*}=2$ are the appropriate values to get
wormhole solution which violates the non-existence theorem with the presence
of negligible amount of exotic matter by using exponential $f(R)$ gravity
model with our proposed shape function.

###  III.2 Starobinsky $f(R)$ gravity Model

One of the most well known $f(R)$ gravity model which is consistent with
cosmological conditions and satisfies solar system and laboratory tests,
proposed by Starobinsky star1 , defined as

$f(R)=R+\lambda R_{*}\big{[}(1+\frac{R^{2}}{R_{*}^{2}})^{-n}-1\big{]},$ (22)

where $\lambda,~{}R_{*}$ and $n$ are free parameters. Model $(\ref{719})$
contains properties of dark energy models and is consistent with cosmological
and local gravity constraints tsuj . In investigation, we get the following
analysis:

  * •

For $n<0$,

    * –

if $R_{*}<0$ and $\lambda<0$, we get the non-spherically symmetric wormhole
geometry with the presence of ordinary matter at the throat i.e., validation
of WEC,

    * –

we get the same result for $R_{*}<0$ and $\lambda>0$,

    * –

further, we analyze that for $R_{*}>0$ and $\lambda<0$, again WEC is respected
at the throat, whereas, $f>0$.

    * –

for $R_{*}>0$ and $\lambda>0$, we get some interesting results. For
$\lambda\in(0,~{}6)$ have shown an attractive geometry which is feasible for
traversable wormhole without exotic matter as $f<0$ with the validation of WEC
at the throat. Moreover, when $\lambda>6$, we found that $\rho+p_{r}>0$,
$\rho+p_{t}>0$ with $\rho<0$.

  * •

For $n>0$,

    * –

for the combinations, $R_{*}<0$, $\lambda<0$ and $R_{*}>0$, $\lambda<0$, we
get the non-existence condition i.e., $F>0$ with $\rho+p_{r}>0$,
$\rho+p_{t}>0$ and $\rho>0$,

    * –

for the combinations, $R_{*}<0$, $\lambda>0$ and $R_{*}>0$, $\lambda>0$, we
found the violation of non-existence theorem $F>0$. As, WEC is respected at
the throat for both the combinations, so, the traversable wormhole inner space
is filled with ordinary matter .

Thus, we can find the wormhole solutions that may have negligible amount of
exotic matter. In particular, we choose $R_{*}=-2,~{}~{}\lambda=-3.5$ and
$n=2$ to show the feasible traversable wormhole structure which have an
ordinary matter at the throat of the wormhole (see Fig. 6). Further, we have
also shown some other combinations which are respected the WEC at the throat
with non-asymptotically flat wormhole geometry by using our proposed WSF.

###  III.3 Tsujikawa $f(R)$ gravity Model

|
---|---
|
Figure 7: The both terms of NEC $\rho+p_{r}$ (upper left) and $\rho+p_{t}$
(upper right) are respected at the throat for wormhole geometry for all the
considering combinations of free parameters. The first term of WEC $\rho$
(lower left) also shows the validation at the throat. While, we get the
violation of non-existence theorem when $R_{*}<0$ and $\mu_{c}>0$ (lower
right).

Another viable $f(R)$ model that we consider is known as Tsujikawa model amen7
and it is defined as

$f(R)=R-\mu_{c}R_{*}Tanh(\frac{R}{R_{*}}).$ (23)

Here $R_{*}$ and $\mu_{c}$ are arbitrary constants. Tsujikawa amen7  described
that $\mu_{c}\in(0.905,1)$ to sustain the viability of the model i.e,
$F=\frac{df}{dR}>0$. Whereas for the violation of non-existence theorem of
static spherically symmetric wormhole $F=\frac{df}{dR}<0$. To get that
requirement we have chosen $\mu_{c}$ in the neighbourhood of defined range. We
assessed the geometric nature of wormhole structure through energy conditions
for $\mu_{c}=1.0135$ and $R_{*}=-2$. We analyzed the following points during
investigation:

  * •

For the combinations $R_{*}<0$, $\mu_{c}<0$ and $R_{*}>0$, $\mu_{c}<0$, WEC is
respected for $r/r_{0}\in(1,~{}\infty)$. This shows that the wormhole inner
space is filled with normal matter. Whereas $F>0$ shows the non-spherically
symmetric wormhole geometry

  * •

For the combinations $R_{*}<0$, $\mu_{c}>0$ and $R_{*}>0$, $\mu_{c}>0$, we
found that $F<0$ and WEC is at the throat. This shows the existence of an
attractive wormhole geometry whose vicinity of the throat is filled with
ordinary matter.

Therefor, it can be seen from the Fig. 7, for the chosen combination of
parameters, we get the violation for non-existence theorem i.e, $F<0$.
$\rho+p_{r}$ and $\rho+p_{t}$ both are positive at the wormhole throat
therefore NEC is respected which shows the absence of exotic matter at the
throat. Moreover, $\rho>0$ also reduces the presence of exotic matter and
provides stable and traversable wormhole structure.

##  IV Concluding Remarks

In recent years, different researchers proposed WSF that are either ansatz or
based on specific theory. Main purpose of this article is to construct a WSF
by employing the KMc. This condition is mandatory for the spacetime of class
one. Several authors extensively consider the KMc to discuss the
configurations of spherically symmetric compact objects. In Sec. II, we
discuss the construction of WSF in detail. Our proposed WSF fulfils all
necessary requirements and provide a viable relativistic configuration for
wormhole geometry. We have shown the graphical behaviour of shape funtion and
its properties in Figs. 1 and 2. Further, we discuss embedded diagram to
represent the wormhole structure. We consider equatorial slice
$\theta=\frac{\pi}{2}$, and a fixed moment of time i.e., $t=constant$ for
spherical symmetry, and for the visualization we embed it into three
dimensional Euclidean space. Moreover, one can visualize the upper universe
for $h>0$ and the lower universe $h<0$ in Figs. 4 and 3.

Further, we investigate exact solutions for static spherically symmetric
traversable wormhole geometry in the framework of $f(R)$ gravity. For this
purpose, we consider four viable $f(R)$ gravity models named as exponential
gravity model, Starobinsky gravity Model and Tsujikawa $f(R)$ gravity model to
discuss energy conditions and violation of non-existence theorem for wormhole
geometry.

  * •

Firstly, we consider exponential gravity model which shows the possibility of
static spherically symmetric traversable wormhole with violation of non-
existence theorem. Fig. 5 have shown that WEC is respected at the throat.
Further we investigate that with the variation of $\mu$ either we get the
violation of WEC and non-traversable wormhole structure i.e, $df/dr>0$. Thus
we conclude that $\mu>0$, $R_{*}>0$ is the appropriate combination to get
wormhole solution which violates the non-existence theorem with the presence
of negligible amount of exotic matter.

  * •

Secondly, we discuss wormhole structure using Starobinsky $f(R)$ gravity
model. We discuss different combinations of free parameters in detail and
investigate that for the maximum range of free parameters, non-spherically
symmetric wormhole geometry is respected the WEC at the vicinity of the
throat, such solutions may be interesting for non-static or thin-shell
wormholes due to its stable spherically symmetric perturbations bron7 .

  * •

Finally, we consider Tsujikawa gravity model. It can be seen from Fig. 7, for
the chosen combination of parameters, we get the violation for nonexistence
theorem i.e, $F<0$. $\rho+p_{r}$ and $\rho+p_{t}$ both are positive at the
wormhole throat therefore NEC is respected which shows the absence of exotic
matter at the throat. Moreover, $\rho>0$ for larger range of radial coordinate
$r$ which also reduce the presence of exotic matter and provides stable and
traversable wormhole structure.

Thus, in comparison with Fayyaz and Shamir fay1 , we have discussed the WSF
Eqn (7) in the context of GR and investigated the presence of exotic matter in
the formulation of wormhole. While in the framework of $f(R)$ theory of
gravity our results respects NEC and even WEC not at the throat but the larger
values of radial coordinate $r$. So, our proposed WSF shows the formation of
wormhole geometry with negligible amount of exotic matter. In future, it would
be interesting to check the stability of wormhole solution in other modified
gravities for this suggested shape function. As a first step, the study of
wormhole geometries using KMc in modified theories with matter coupling is
under process.

## Refrences

## References

  * (1) Einstein, A. and Rosen, N.: Phys. Rev. 48, (1935) 73.

  * (2) Flamm, L.: Comments on Einstein’s theory of gravity, Physikalische Zeitschrift, 17, (1916) 448.

  * (3) Ellis, H. G.: Journal of Mathematical Physics, 14(1), (1973) 104-118.

  * (4) Wheeler, J. A.: Phys. Rev. 97, (1955) 511.

  * (5) Kar, S.: Phys. Rev. D 49(2), (1994) 862.

  * (6) Kar, S., and Sahdev, D.: Phys. Rev. D 53(2), (1996) 722.

  * (7) Vacaru, S, I., Douglas, S., Vitalie, A. B., and Denis A. D.: Phys. Lett. B 519, (2001) 249.

  * (8) Dzhunushaliev, V., Vladimir, F., Douglas, S. and Ratbay, M.: Phys. Rev. D 82, (2010) 045032.

  * (9) Visser, M., Kar, S., and Naresh, D.: Phys. Rev. Lett. 90, (2003) 201102.

  * (10) Canfora, F., Dimakis, N. and Paliathanasis, A.: Phys. Rev. D 96, (2017) 025021.

  * (11) Morris, M. S. and Thorne, K. S.: Am. J. Phys. 56, (1988) 395.

  * (12) Morris, M. S. and Thorne, K. S. and Yurtsever, U.: Phys. Rev. Lett. 61, (1988) 1446.

  * (13) Nandi,K. K., Islam, A. and Evans, J.: Phys. Rev. D 55, (1997) 2497.

  * (14) Lobo, F. S. N. and Oliveira, M. A.: Phys. Rev. D 81, (2010) 067501.

  * (15) Richarte, M. and Simeone, C.: Phys. Rev. D 80, (2009) 104033.

  * (16) Mehdizadeh, M. R., Zangeneh, M. K. and Lobo, F. S. N.: Phys. Rev. D 91, (2015) 084004.

  * (17) Dzhunushaliev, V.D. and Singleton, D.: Phys. Rev. D 59, (1999) 064018.

  * (18) Shaikh, R., and Kar, S.: Phys. Rev. D 94, (2016) 024011.

  * (19) Mehdizadeh, M. R. and Ziaie, A.H.: Phys. Rev. D 95, (2017) 064049.

  * (20) Buchdahl, H. A.: Mon. Not. R. Astron. Soc. 150, (1970) 1.

  * (21) Capozziello, S. and Francaviglia, M.: Gen. Relt. Grav. 40, (2008) 357.

  * (22) Capozziello, S., Carloni, S. and Troisi, A.: Recent Res. Dev. Astron. Astrophys. 1, (2003) 625.

  * (23) Harko, T., Lobo, F. S. N., Mak, M. K. and Sushkov S. V.: Phys. Rev. D 87, (2013) 067504.

  * (24) Rahaman, F., Banerjee, A., Jamil, M., Yadav, A. K. and Idris, H., Int. J. Theor. Phys. 53, (2014) 1910.

  * (25) Jamil, M., Rahaman, F., Myrzakulov, R., Kuhfittig, P.K.F. and Ahmed, N., Mondal, U.F.: J. Korean Phys. Soc. 65, (2014) 917.

  * (26) Bronnikov, K. A. and Starobinsky, A. A.: JETP letters, 85(1), (2007) 1.

  * (27) Bronnikov, K. A., Skvortsova, M. V., and Starobinsky, A. A.: Gravitation and Cosmology, 16(3) (2010) 216-222.

  * (28) Bahamonde, S., Jamil, M., Pavlovic, P. and Sossich, M.: Phys. Rev. D 94, (2016) 044041.

  * (29) Godani, N. and Samanta, G. C.: Int. J. Mod. Phys. D 28, (2018) 1950039.

  * (30) Jahromi, A. S. and Moradpour, H.: Int. J. Mod. Phys. D 27, (2018) 1850024.

  * (31) Cataldo, M., Liempi, L. and Rodr´ıguez, P.: Phys. Lett. B 757, (2016) 130-135.

  * (32) Samanta, G. C., Godani, N. and Bamba, K.: Int. J. Mod. Phys. D 29(09), (2020) 2050068.

  * (33) Jamil, M., Momeni, D., Myrzakulov, R.: Eur. Phys. J. C 73, (2013) 2267.

  * (34) Sharif, M. and Zahra, Z.: Astrophysics and Space Science, 348(1), (2013) 275.

  * (35) Shamir, M. F. and Zia, S.: Astrophysics and Space Science 363, (2017) 247.

  * (36) Golchin, H. and Mehdizadeh, M. R.: Eur. Phys. J. C, 79(9), (2019) 777.

  * (37) Karmarkar, K. R.: Proc. Indian Acad. Sci. A 27 (1948) 56.

  * (38) Abbas, G., Qaisar, S., Javed, W., and Meraj, M. A.: IRAN J SCI TECHNOL A 42(3), (2018) 1659.

  * (39) Fuloria, P., and Pant, N.: Eur. Phys. J. A 53(11), (2017) 227.

  * (40) Kuhfittig, P. K.: Annals of Physics, 392, (2018) 63-70.

  * (41) Bhar, P., Singh, K. N., and Manna, T.: Int. J. Mod. Phys. D 26(09), (2017) 1750090.

  * (42) Gedela, S., Bisht, R. K., and Pant, N.: Eur. Phys. J. A 54(11), (2018) 207.

  * (43) Kuhfittig, P. K. F.: Pramana 92 (2019) 75.

  * (44) Fayyaz, I., Shamir, M. F.: Chin. J. Phys. 66, (2020) 553.

  * (45) Anchordoqui, L.A., Torres, D.F., Trobo, M.L., Bergliaffa, S.E.P.: Phys. Rev. D 57, (1998) 829.

  * (46) Hochberg, D. and Visser, M.: Phys. Rev. D58, (1998b) 044021 .

  * (47) Hochberg, D. and Visser, M: Phys. Rev. Lett. 81, (1998a) 746.

  * (48) Cognola, G., Elizalde, E., Nojiri, S., Odintsov, S. D., Sebastiani, L., and Zerbini, S.: Phys. Rev. D 77(4), (2008) 046009.

  * (49) Starobinsky, A. A., JETP letters, 86(3), (2007) 157.

  * (50) Tsujikawa, S., In Lectures on Cosmology, Springer, Berlin, Heidelberg, 2010, 99-145.

  * (51) S. Tsujikawa, Phys. Rev. D. 77, (2008) 023507.


# The Story of Bose, Photon Spin and Indistinguishability

Partha<EMAIL_ADDRESS>
Tagore Centre for Natural Sciences and Philosophy,
Rabindra Tirtha, New Town, Kolkata 700156, India

###### Abstract

As we approach the centenary of the discovery of quantum statistics in 1924,
it is important to revisit Bose’s original derivation of Planck’s law usually
ignored in most standard presentations of Bose-Einstein statistics. It
introduced not only the novel concept of the indistinguishability of photons
but also of their intrinsic spin, a fact unknown to most physicists.

By 1923 Satyendra Nath Bose (born 1st January 1894) had already made his mark
as a prodigious talent in the intellectual circles of Calcutta (now Kolkata),
the capital of British India, but was still a total stranger in the world of
physics at large. In the early 1920s he was teaching quantum theory to
students at the newly established Dacca University (now called Dhaka
University) in erstwhile East Bengal which eventually became Bangladesh in
1972. He was struggling with the derivation of Planck’s law – he found all
previous derivations, including those by Einstein, to be logically
unsatisfactory because of the simultaneous use of both classical and quantum
ideas. He was looking for a derivation that was entirely quantum theoretical.

On June 4, 1924 he sent a short paper ‘Planck’s Law and the Light-Quantum
Hypothesis’ to Albert Einstein with the humble request, ‘You will see that I
have tried to deduce the coefficient $8\pi\nu^{2}/c^{3}$ in Planck’s law
independent of the classical electrodynamics, only assuming that the ultimate
elementary regions in the Phase space have the content $h^{3}$. I do not know
sufficient German to translate the paper. If you think the paper worth
publication, I shall be grateful if you arrange its publication in Zeitschrift
für Physik’.

In a post card dated 2nd July, 1924 Einstein wrote to Bose, ‘Dear Colleaugue,
I have translated your work … It signifies an important step forward and I
liked it very much …… You are the first to derive the factor quantum
theoretically, even though because of the polarization factor 2 not wholly
rigorously. It is a beautiful step forward’. In a footnote to the translated
paper, Einstein wrote: ‘In my opinion Bose’s derivation signifies an important
advance. The method used here gives the quantum theory of an ideal gas as I
will work out elsewhere. -A Einstein’ [1]. Ten days later on 12th July he
wrote to Ehrenfest, ‘The Indian Bose has given a beautiful derivation of
Planck’s law including the constant $8\pi\nu^{2}/c^{3}$’.

With such a strong endorsement from Einstein, Bose instantly emerged as a
prominent figure in the world of quantum physics. His biography and the impact
of his paper on physics as a whole can be gleaned from [2, 3, 4]. It would be
instructive to revisit Bose’s original paper, particularly since standard
presentations of Bose-Einstein statistics gloss over two of its revolutionary
insights, both of which are contained in Bose’s ingenious derivation of the
phase space factor $8\pi\nu^{2}/c^{3}$, namely the idea of
indistinguishability of photons which is usually presented as an ad hoc
assumption and the idea of an additional internal degree of freedom of light-
quanta which is totally ignored.

Let’s begin with the Planck formula for the spectral density of
electromagnetic radiation emitted by a black body in thermal equilibrium at a
given temperature $T$,

$\rho(\nu,T)=\left(\frac{8\pi\nu^{2}d\nu}{c^{3}}\right)\left(\frac{h\nu}{e^{h\nu/kT}-1}\right).$
(1)

Planck tried very hard to find a theoretical basis for it, and finally ‘as an
act of desperation…. to obtain a positive result, under any circumstances and
at whatever cost’ introduced the idea of quanta of energy for the material in
the oven walls, not in the radiation itself which he took to be classical and
described by Maxwell’s theory. In mid-December 1900 he presented a statistical
derivation involving a distribution of the hypothetisized energy quanta among
the wall ‘resonators’ that had no more justification than that it gave the
right result.

With a hunch that the law implied a non-classical corpuscular nature of
radiation itself, Einstein used some results of Wien on the entropy of
radiation to calculate the volume dependence of the entropy of thermal
radiation, and from that he drew the following revolutionary conclusion:

> … We (further) conclude that monochromatic radiation of low density (within
> the range of validity of Wien’s radiation formula) behaves thermodynamically
> as if it consisted of mutually independent energy quanta of magnitude
> $R\beta\nu/N$. (italics added)

The factor $R\beta\nu/N=h\nu$. The idea of light-quanta was thus born, with
all its problems. First, there was a fundamental conflict between the
statistical independence of Einstein’s light-quanta and the Planck law which
implies correlated quanta [5, 6]. Second, there was no logically satisfactory
derivation of the Planck law as a whole despite several attempts by the
leading physicists of the time suh as Planck himself [7], Peter Debye [8],
Einstein [9], Pauli [10] and Einstein and Ehrenfest [11]. In all of them the
first factor $(8\pi\nu^{2}d\nu/c^{3})$ in the Planck law (1) was taken from
classical electrodynamics to be the number density of the modes of vibration
of radiation. The second factor was deduced by postulating various ad hoc
rules. Third, in all these attempts (except that of Debye) Planck’s
hypothesis, namely that quantization was restricted to the exchange of energy
between radiation and matter, was used. But by late 1923/early 24 the Compton
Effect had clearly shown that radiation itself consists of energy quanta.

Bose solved all these problems in one stroke by deriving the full Planck law,
including the first factor, from quantum theory. As we have seen, in his
letter to Einstein, he wrote, ‘You will see that I have tried to deduce the
coefficient $8\pi\nu^{2}/c^{3}$ independent of the classical electrodynamics.’
This is never mentioned in text books and other expositions, but is of crucial
importance. He did that by extending Planck’s method of quantization of
material resonators to radiation itself. Consider the phase space of a single
photon,

$\int dV.dp_{x}dp_{y}dp_{z}=V4\pi p^{2}dp=V\frac{4\pi
h^{3}\nu^{2}d\nu}{c^{3}}$ (2)

on using the Einstein relation $p=h\nu/c$. Since the irreducible elementary
regions of phase space are of size $h^{3}$, dividing by $Vh^{3}$ one obtains
the number density of phase space cells,

$\frac{4\pi\nu^{2}d\nu}{c^{3}}$ (3)

It is such a simple derivation, except that it is wrong by a factor of 2. The
paper is slightly hesitant at this point and says, ‘It seems, however,
appropriate to multiply this number once again by 2 in order to take into
account the fact of polarization.’ Therein hangs a tale. But before going into
that, suffice it to say here that the first factor now emerges with a totally
new meaning: it is the number of possible states of the photon, i.e., the
number of possible arrangements of a photon in the given volume. This number
is fixed, and hence permutations of the (identical) photons within a cell
cannot produce new cells, and this immediately implies that the photons are
indistinguishable particles. The cells are therefore labelled only by the
number of photons in them. Let $p^{s}_{i}$ be the number of cells with $i$
number of photons, each with frequency in the range $\nu^{s}$ and
$\nu^{s}+d\nu^{s}$. Then the probability of a state with type $s$ quanta is

$W^{s}=\frac{A^{s}!}{p^{s}_{0}!\,p^{s}_{1}!\,p^{s}_{2}!\,\cdots}$ (4)

with

$A^{s}=\frac{8\pi\nu^{s2}d\nu^{s}}{c^{3}}$ (5)

and

$N^{s}=0.p^{s}_{0}+1.p^{s}_{1}+2.p^{s}_{2}+\cdots$ (6)

The macroscopically defined probability of a state defined by all types of
quanta is then

$W=\Pi_{s}W^{s}=\Pi_{s}\frac{A^{s}!}{p^{s}_{0}!\,p^{s}_{1}!\,p^{s}_{2}!\,\cdots}.$
(7)

Maximizing the entropy $S=k{\rm log}W$ subject to the constraints that the
total energy is the sum of the energies of all the quanta,
$E=\Sigma_{s}N^{s}h\nu^{s}$, and the total number of quanta is the sum of the
numbers of quanta in all the states, $N^{s}=\Sigma_{r}rp^{s}_{r}$, leads then,
in the limit of large numbers $p^{s}_{r}$, to the statistical Planck factor
$(1-e^{-h\nu/kT})^{-1}$. QED. No hypothetical energy quanta, no wall
resonators, just physical indistinguishable photons! One of the foundations of
quantum mechanics was laid, and it followed rapidly.

Now to the story behind the factor of 2 which signals an additional degree of
freedom of the photon. I cannot recollect the exact date, but most probably
sometime in the late 70’s I had gone to meet Bose in his residence. We were
alone in his living room. He told me the following story in hush hush
confidence and made me promise never to divulge it publicly. Why then am I
breaking my promise? It’ll become clear shortly. ‘You know’, he said, ‘my
deduction of the Planck law had a factor of 2 missing. So I proposed that it
came from the fact that the photon had a spin, and that it can spin either
parallel or antiparallel to its direction of motion. That would give the
additional factor of 2. But the old man (meaning Einstein) crossed it out
(‘budho kété dilé’ in Bengali, his mother tongue) and said it was not
necessary to talk about spin, the factor of 2 comes from the two states of
polarization of light.’ Then he said rhetorically with a mischievous distant
look, ‘I can understand a spinning particle, but what can the polarization of
a particle mean?’ I was stunned!

Much later (again I cannot recollect the exact date, but certainly sometime
during 1993 when we were busy preparing for his birth centenary celebrations
in January 1994) I accidentally chanced upon a 1931 paper of Raman and
Bhagavantam with the title “Experimental proof of the spin of the photon”
[12]. I was intrigued and started reading it. I was amazed. Let me quote:

> In his well-known derivation of the Planck radiation formula from quantum
> statistics, Prof. S. N. Bose obtained an expression for the number of cells
> in phase space occupied by the radiation, and found himself obliged to
> multiply it by a numerical factor 2 in order to derive from it the correct
> number of possible arrangements of the quantum in unit volume. The paper as
> published did not contain a detailed discussion of the necessity for the
> introduction of this factor, but we understand from a personal communication
> by Prof. Bose that he envisaged the possibility of the quantum possessing
> besides energy $h\nu$ and momentum $h\nu/c$ also an intrinsic angular
> momentum $\pm h/2\pi$ around an axis parallel to he direction of its motion.
> The weight factor 2 thus arises from the possibility of the spin of the
> quantum being right-handed or left-handed, corresponding to the two
> alternative signs of the angular momentum. There is a fundamental difference
> between this idea, and the well-known result of classical electrodynamics to
> which attention was drawn by Poynting and more fully developed by Abraham
> that a beam of light may in certain circumstances possess angular momentum.
> Thus, according to the classical field theory, the angular momentum
> associated with a quantum of energy is not uniquely defined, while according
> to the view we are concerned with in the present paper, the photon has
> always an angular momentum having a definite numerical value of a Bohr unit
> with one or the other of the two possible alternative signs.

The Raman-Bhagavantam experiment conclusively showed that Bose’s view was the
correct one. It was the first ever experimental determination of photon spin,
but how many scientists and historians of science know this?

It should be clear now why I feel no compunction in breaking my promise–the
story has been in the public domain since 1931, though hardly noticed by
anyone! Did Bose forget about this paper? Unlikely. By the time I unearthed
it, he was no more, so I couldn’t check back with him. I guess it must have
been due to his deep respect for and gratitude to Einstein. After all he owed
his entire reputation to Einstein. There were strong initial objections to
Bose’s method. For example, Ehrenfest had objected to the new statistics on
the ground that the ‘mysterious influences’ among the molecules was
inconsistent with their statistical independence. Einstein’s response was
characteristic–Bose’s counting method was the only one consistent with
Nernst’s heat theorem for ideal gases. He wrote [13], ‘one recognizes at once
that the entropy must vanish at the absolute zero of temperature. The reason
is that then all molecules are in the first cell; and for this state there
exists only one distribution of molecules according to our counting method.
Hence our assertion is immediately proved to be correct.’ And again [14],
‘This result represents in itself a support of the view concerning the deep
natural relation between radiation and gas, since the same statistical
treatment which leads to Planck’s formula establishes – when applied to ideal
gases– the agreement with Nernst’s theorem.’ Thus was born the idea of Bose-
Einstein Condensates. Einstein therefore proposed to proceed with Bose’s
method, brushing aside all objections and leaving the problem of ‘mysterious
influences’ among the molecules to be sorted out in the future.

To conclude, the impact of Bose’s short paper turned out to be far-reaching.
In particular it gave birth to the field of quantum statistics– a seminal and
essential part of quantum theory. In his book ‘Subtle is the Lord’ Pais
states, ‘The paper by Bose is the fourth and last of the revolutionary papers
of the old quantum theory (the other three being by, respectively, Planck,
Einstein and Bohr). As noted above, Bose’s idea of “indistinguishability”,
followed by Einstein’s essential contributions, gave rise to the idea of Bose-
Einstein Condensates, which now has numerous applications. Last but not least,
as we now know, nature has only two types of particles - bosons and fermions -
depending upon whether they possess integer or half-integer spin respectively,
a fact that follows as a general consequence of relativistic quantum field
theory, and is of course amply verified empirically’.

A short paper by an unknown physicist from an unknown University in a far off
British Colony contained two seminal ideas – the idea of indistinguishability
of photons and the idea of an additional degree of freedom of photons which
later came to be called ‘spin’.

This is certainly a story worth telling.

I am grateful to Jogesh Pati for encouraging me to write this article and
suggesting several improvements of the original draft.

## References

  * [1] von Bose, ‘Planck’s Gesetz und Lichtquantenhypothese’, Zeit für Physik 26, 168-171, 1924; reproduced with English translation in S. N. Bose: The Man and His Work, Part 1: Collected Scientific Papers; see also Part II: Life, Lectures and Addresses, Miscellaneous Pieces, S. N. Bose archive, https://newweb.bose.res.in/Prof.S.N.Bose-Archive/BoseCentenaryPublications.
  * [2] Abraham Pais, Subtle is the Lord: The Science and the Life of Albert Einstein, Oxford University Press 2005.
  * [3] K. C. Wali, ‘Satyendra Nath Bose: His Life and Times’, World Scientific, 2009, and
  * [4] Partha Ghose, ‘Counting in the Dark’, Inference, vol. 7, No. 3, October 2022.
  * [5] L. Natanson, Phys. Zs. vol. 12, p. 659, 1911.
  * [6] P. Ehrenfest and H. Kamerlingh Onnes, 1914,Collected Papers, pp. 363-366.
  * [7] M. Planck, Verh. d. Deutsch. Phys. Ges. (2) vol. 2, p. 237, 1900.
Ann. d. Phys. (4) vol. 31, p. 758, 1910.

  * [8] P. Debye, Ann. d. Phys. vol. 88, p. 1427, 1910.
1923 Phys. Zs. vol. 24, p. 161, 1923.

  * [9] A. Einstein, Phys. ZS. vol. 18, p. 121, 1917.
  * [10] W. Pauli, Z. Phys. vol. 18, p. 272, 1923.
  * [11] A. Einstein and P. Ehrenfest, Z. Phys. vol. 19, p. 301, 1923.
  * [12] C. V. Raman and S. Bhagavantam, Ind. J. Phys. vol. 6, p. 353, 1931.
  * [13] A. Einstein, Sitz. ber. Preuss. Akad. Wiss. (Berlin), 261- 267(presented at the meeting of 10 July 1924).
  * [14] A. Einstein, Abhandlung, Sitz. ber. Preuss. Akad. Wiss. (Berlin), 3-14 (presented at the meeting of 8 January 1925).