Possibility of $\rho$ Meson Condensation in Neutron Star: Unified Approach of Chiral SU(3) Model and QCD Sum Rules

TL;DR

This paper investigates the potential for $ ho$ meson condensation in neutron stars by combining chiral SU(3) models with QCD sum rules, finding it possible at high densities around 7 times nuclear saturation density.

Contribution

It introduces a unified approach using chiral SU(3) models and QCD sum rules to assess $ ho$ meson condensation in neutron star matter, a novel combination in this context.

Findings

$ ho$ meson condensation occurs at about 7$ ho_0$ density.

Quark and gluon condensates are calculated within the chiral SU(3) model.

Magnetic field effects are qualitatively discussed.

Abstract

In the present work the conjunction of chiral SU(3) model with QCD sum rules is employed to explore the possibility of $\rho$ meson condensation in neutron stars. The quark and gluon condensates in terms of which the in-medium masses of $\rho$ mesons can be expressed are calculated using the chiral SU(3) model in the charge neutral matter which is relevant for neutron stars. It is observed that condition of $\rho$ meson condensation is satisfied for the density of about 7$\rho_{0}$, where $\rho_{0}$ is the nuclear saturation density. In the end, a brief qualitative discussion of the magnetic field is also involved to check out for the further possibility of $\rho$ meson condensation.

Tables

Table 1. Table 1: Values of various parameters.

$g_{\sigma N}$	$g_{\zeta N}$	$g_{\delta N}$	$g_{\omega N}$	$g_{\rho N}$
-9.83	1.22	-2.34	12.13	4.03
$k_0$	$k_1$	$k_2$	$k_3$	$k_4$
2.37	1.40	-5.55	-2.65	-0.23
$m_{\pi }$(MeV)	$m_K$ (MeV)	$f_{\pi }$ (MeV)	$f_K$(MeV)	$g_4$
139	498	93.3	122.143	38.9
${\sigma }_0$ (MeV)	${\zeta }_0$ (MeV)	${\chi }_0$ (MeV)	$d$	${\rho }_0$ (${\text{fm}}^{-3}$)
-93.3	-106.8	401.91	0.06	0.15
$m_{\omega }$(MeV)	$m_{\rho }$(MeV)	$m_N$(MeV)	$\delta m$(MeV)	$k_q$
783	770	939	150	2.0850

Table 2. Table 2: Variation of mass with c 2 ∗ superscript subscript 𝑐 2 c_{2}^{*}

$c_{2\downarrow }^{\ast },m_{\rho }^{\ast }\to$	${\rho }_B={\rho }_0$	${\rho }_B=2{\rho }_0$	${\rho }_B=4{\rho }_0$
$c_2^{\ast }$=0	738.7 (MeV), (11.5$\%$)	679.3 (MeV), (18.7$\%$)	638.9 (MeV), (23.4$\%$)
$c_2^{\ast }\ne 0$	653.9 (MeV),(15.8$\%$)	554.8 (MeV), (28.5$\%$)	433.1 (MeV), (44.2$\%$)

Equations

$$\mathcal{L}={\mathcal{L}}_{Kin}+{\mathcal{L}}_{i,Scal}+{\mathcal{L}}_{i,Vec}+{\mathcal{L}}_{Scal}+{\mathcal{L}}_{Vec}+{\mathcal{L}}_{SB}$$

$${\mathcal{L}}_{i,Vec}+{\mathcal{L}}_{i,Scal}=-{\bar{\psi }}_i[m_i^{\ast }+g_{\omega i}{\gamma }_0\omega +g_{\rho i}{\gamma }_0{\tau }_3\rho ]{\psi }_i,$$

$${\mathcal{L}}_{Scal}=$$

$$+$$

$$-$$

$$\displaystyle\mathcal{L}_{Vec}=\frac{1}{2}\Big{(}m_{\omega}^{2}\omega^{2}+m_{\rho}^{2}\rho^{2}\Big{)}\Bigg{(}\frac{\chi}{\chi_{0}}\Bigg{)}^{2}+g_{4}(\omega^{4}+6\omega^{2}\rho^{2}+\rho^{4}),$$

$$\displaystyle\mathcal{L}_{SB}=-\left(\frac{\chi}{\chi_{0}}\right)^{2}\left(\frac{1}{2}m_{\pi}^{2}f_{\pi}\left(\sigma+\delta\right)+\frac{1}{2}m_{\pi}^{2}f_{\pi}\left(\sigma-\delta\right)+\big{(}\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\big{)}\zeta\right),$$

$$\frac{\Omega }{V}=-{\mathcal{L}}_{vec}-{\mathcal{L}}_0-{\mathcal{L}}_{SB}-{\mathcal{V}}_{vac}-{\beta }_i-{\alpha }_e.$$

$$\displaystyle\beta_{i}=\sum_{i}\frac{\gamma_{i}}{48\pi^{2}}\Big{[}\Big{(}2k_{fi}^{3}-3m_{i}^{*2}k_{fi}\Big{)}\mu_{i}^{*}+3m_{i}^{*4}\ln\Big{\{}\frac{k_{fi}+\mu_{i}^{*}}{m_{i}^{*}}\Big{\}}\Big{]},$$

$${\alpha }_e=\frac{{\mu }_e^4}{12{\pi }^2},$$

$$k_0{\chi }^2\sigma -4k_1\left({\sigma }^2+{\zeta }^2+{\delta }^2\right)\sigma -2k_2\left({\sigma }^3+3\sigma {\delta }^2\right)-2k_3\chi \sigma \zeta$$

$$-$$

$$k_0{\chi }^2\zeta -4k_1\left({\sigma }^2+{\zeta }^2+{\delta }^2\right)\zeta -4k_2{\zeta }^3-k_3\chi \left({\sigma }^2-{\delta }^2\right)$$

$$-$$

$$k_0{\chi }^2\delta -4k_1\left({\sigma }^2+{\zeta }^2+{\delta }^2\right)\delta -2k_2\left({\delta }^3+3{\sigma }^2\delta \right)+2k_3\chi \delta \zeta$$

$$+$$

$$k_0\chi \left({\sigma }^2+{\zeta }^2+{\delta }^2\right)-k_3\left({\sigma }^2-{\delta }^2\right)\zeta +{\chi }^3\left[1+\ln \left(\frac{{\chi }^4}{{\chi }_0^4}\right)\right]+(4k_4-d){\chi }^3$$

$$-$$

$$\displaystyle\Bigg{(}\frac{\chi}{\chi_{0}}\Bigg{)}^{2}m_{\omega}^{2}\omega+4g_{4}(\omega^{3}+3\omega\rho^{2})-\sum_{i}{g_{\omega i}\rho_{i}}=0,$$

$$\displaystyle\Bigg{(}\frac{\chi}{\chi_{0}}\Bigg{)}^{2}m_{\rho}^{2}\rho+4g_{4}(\rho^{3}+3\omega^{2}\rho)-\sum_{i}{g_{\rho i}\tau_{3}\rho_{i}}=0.$$

$$m_i^{\ast }=g_{\sigma i}\sigma +g_{\zeta i}\zeta +g_{\delta i}{\tau }_3\delta +\delta m,$$

$$\sum _i{m}_i⟨{\bar{q}}_i{q}_i⟩=-{\mathcal{L}}_{SB},$$

$$m_{u}\langle\bar{u}u\rangle=\frac{1}{2}m_{\pi}^{2}f_{\pi}(\sigma+\delta)\Bigg{(}\frac{\chi}{\chi_{0}}\Bigg{)}^{2},$$

$$m_{d}\langle\bar{d}d\rangle=\frac{1}{2}m_{\pi}^{2}f_{\pi}(\sigma-\delta)\Bigg{(}\frac{\chi}{\chi_{0}}\Bigg{)}^{2},$$

$$m_{s}\langle\bar{s}s\rangle=\Big{(}\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\Big{)}\Bigg{(}\frac{\chi}{\chi_{0}}\Bigg{)}^{2}\zeta.$$

$$\theta_{\mu\nu}=(\partial_{\mu}\chi)\Bigg{(}\frac{\partial{\mathcal{L_{\chi}}}}{\partial(\partial^{\nu}\chi)}\Bigg{)}-g_{\mu\nu}\mathcal{L_{\chi}},$$

$$\mathcal{L_{\chi}}=\frac{1}{2}(\partial_{\mu}\chi)(\partial^{\mu}\chi)-k_{4}\chi^{4}-\frac{1}{4}\chi^{4}{\rm{ln}}\Bigg{(}\frac{\chi^{4}}{\chi_{0}^{4}}\Bigg{)}+\frac{d}{3}\chi^{4}{\rm{ln}}\Bigg{(}\bigg{(}\frac{\left(\sigma^{2}-\delta^{2}\right)\zeta}{\sigma_{0}^{2}\zeta_{0}}\bigg{)}\bigg{(}\frac{\chi}{\chi_{0}}\bigg{)}^{3}\Bigg{)}.$$

$$\theta_{\mu}^{\mu}=(\partial_{\mu}\chi)\Bigg{(}\frac{\partial{\mathcal{L_{\chi}}}}{\partial(\partial_{\mu}\chi)}\Bigg{)}-4{\mathcal{L_{\chi}}}=-(1-d)\chi^{4}.$$

$$\Bigg{\langle}\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G^{a\mu\nu}\Bigg{\rangle}=\frac{8}{9}(1-d)\chi^{4}.$$

$$\Bigg{\langle}\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G^{a\mu\nu}\Bigg{\rangle}=\frac{8}{9}\Bigg{[}(1-d)\chi^{4}+\Big{(}m_{\pi}^{2}f_{\pi}\sigma+\Big{(}\sqrt{2}m_{K}^{2}f_{K}-\frac{1}{\sqrt{2}}m_{\pi}^{2}f_{\pi}\Big{)}\zeta\Big{)}\Bigg{(}\frac{\chi}{\chi_{0}}\Bigg{)}^{2}\Bigg{]}.$$

$${\Pi }_{\mu \nu }^{\rho R}(q;{\rho }_B)=i\int d^{4}x{e}^{iqx}⟨\mathcal{R}{j}_{\mu }^{\rho }(x)j_{\nu }^{\rho }(0){⟩}_{{\rho }_B},$$

$${\Pi }^{\rho R}={\Pi }_{\mu \mu }^{\rho R}/(-3q^2){|}_{\text{q}\to 0}.$$

$$\text{Re}{\Pi }^{\rho R}({\omega }^2)=P{\int }_0^{\infty }d{u}^2\frac{R_h^{\rho }}{u^2-{\omega }^2}+\text{subtractions}.$$

Full text

I Introduction

The study of astrophysical compact objects like neutron stars plays a very important role in understanding the strong interaction physics at high baryonic densities. The heavy ion collision experiments, for example, Large Hadron Collider (LHC) and Relativistic Heavy Ion Collider (RHIC) aims to explore the Equation of State (EoS) of dense matter at high temperature and very low baryonic density whereas Compressed Baryonic Matter (CBM) experiment of FAIR project, Nuclotron-based Ion Collider Facility (NICA) at Dubna, Russia and Japan Proton Accelerator Research Complex (J-PARC) will focus on high baryonic density and relatively moderate temperature. The recent detection of gravitational waves by LIGO from neutron star merger puts more strong constraint on the EoS of dense medium Annala2018 and this has triggered much more theoretical calculations in the field Abbot2017 ; Pal ; Benjamin ; Kyu ; Ho2018 . The data obtained on the masses and radii of the neutron star from the Neutron Star Interior Composition Explorer (NICER) experiment at international space station impose the significant constraint on the EoS Gendreau ; Baubock ; Miller ; Ozel ; Grindlay ; Watts . In order to explain the observed data from various other laboratories like LIGO-Virgo BP ; Abb ; BPA , LIGO-India Iyer , GEO Hough we need to mould our models and the existence of these high mass neutron stars will also hold out restriction on theoretical calculations Baym . Furthermore, the importance of studying the neutron star like objects lies in the fact that at the core of these objects the density can be up to many times of the nuclear saturation density which is not achieved in human-made laboratories and ultimately makes these objects a natural laboratory to understand the strong interaction physics Lattimer . Whereas, the strongly interacting matter in heavy ion collisions exists at finite temperature and baryonic density, the matter in the neutron star is at high baryonic density and zero temperature in the ground state.

Numerous theoretical non-perturbative approaches, for example, Quark Meson Coupling (QMC) model Yue ; Guichon ; HW ; Panda ; Tsushima ; Sibirtsev , Polyakov Quark-Meson Coupling model Schaefer ; Mohan , chiral effective models Dexheimer ; Mallik ; Thorsson , QCD sum rules, coupled channel approach Tol763 ; Tol635 ; Tol77 ; Hofmann , NJL and PNJL models Nambu ; Kukushima ; Kashiwa ; Ghosh etc., are employed to investigate the properties of hadrons in dense interacting environment. The idea of condensate formation in dense matter was first proposed a long ago by Kaplan and Nelson Kaplan . The inspiration for the meson condensation comes from strong attractive interaction of mesons with nucleons. Due to this attractive interaction, the effective energy of the mesons becomes low in the matter as compared to that in the vacuum. This notion provides a base for the kaon-antikaon condensation in dense matter Pal20 . The reason for the possibility of kaon condensation in a neutron star is its attractive interaction with nucleons both in vector and axial vector channels Kaplan2 . With the increasing baryonic density of the medium, the in-medium energy of the antikaon decrease whereas the chemical potential of electron increases. Beyond a particular density at which both become equal, antikaon can play the role of charge neutrality and energetically antiakons will be favored over electrons. This will lead to the phenomenon of antikaon condensation in neutron stars and will have the consequences on EoS and hence, the mass and radius of these objects Pal20 ; AK .

The in-medium mass reduction of light vector $\rho$ mesons predicted by various studiesLee ; Hayacizaki ; Asakawa motivate to investigate the possibility of $\rho$ meson condensation in neutron star Mallick2015 ; EE . The starc1 results on $\rho^{0}\rightarrow\pi^{+}\pi^{-}$ production in the peripheral Au-Au collisions at $\sqrt{S}=200$ GeV were reported and attractive mass-shift of about $70$ MeV was observed. The enhancement in the production of low mass invariant dileptons in the heavy ion collision experiment like CERES at CERN Agakichiev provides the possibility in the mass reduction of $\omega$ mesons but width broadening in the $\rho$ mesons Ko . Future heavy-ion collision experiments may help to clarify the situation on the mass shift of $\rho$ mesons further. In order to look for the possibility of $\rho$ meson condensation, an approach similar to the antikaons can be used where we try to anticipate a comparison between effective electron chemical potential and in-medium masses of $\rho$ mesons. The presence of these antikaon or $\rho$ meson condensates not only soften the EoS which will impose the constraint on the mass of star Schaffner ; Glendenning but will also affect the restoration of chiral symmetry Thorsson . Thus, the possibility of $\rho$ meson condensation in the highly dense matter cannot be simply ignored.

Hatsuda and Lee used QCD sum rules to calculate the in-medium masses of $\rho$ meson by assuming a linear density approximation and a considerable amount of decrease is observed in the in-medium masses of $\rho$ mesons Lee . Consequently, different studies on using QCD sum rules for investigating the properties of vector mesons evolvedMallik ; Stefan ; KK . In QCD sum rule analysis, in-medium properties of different mesons are evaluated in terms of medium modification of quark and gluon condensatesWeise . Generally, these calculations employ linear density approximation and results are normally valid upto nuclear saturation density. In our present paper, we will use non-linear effective chiral SU(3) modelSchramm2008 ; HANAUSKE2000 to calculate the density dependence of quark and gluon condensates. Within the model, we have non-linear interaction terms of the scalar fields $\sigma$, $\zeta$, $\delta$, $\chi$ and also for vector fields $\omega$ and $\rho$. The scalar iso-vector field $\delta$ and vector iso-vector field $\rho$ contribute in isospin asymmetric matter and since, in the charge neutral matter of neutron stars the density of proton and neutrons are different, these fields are important to consider in the model calculations. Also, as we will see in next section the dilaton field $\chi$ will be used to write the expressions for gluon condensates through the trace anomaly property of QCD. The scalar quark and gluon condensates calculated in charge neutral matter relevant for neutron stars using chiral model will be further used in the QCD sum rules to find the density dependence of the masses of vector $\rho$ mesons. The in-medium mass of $\rho$ meson and electron chemical potential are then compared to check out for any possibility of $\rho$ meson condensation in a neutron star. The chiral SU(3) model has already been widely used by many authors to study the hadronic properties in nuclear and strange hadronic matter and also the properties of neutron stars Dexheimer ; Sanyal ; Schramm2008 ; HANAUSKE2000 ; Greiner .

This paper is organised in the following way: In section II, we will discuss the formalism opted for the present investigation of calculating the in-medium masses of $\rho$ mesons and its condensation. The chiral SU(3) model in neutron star matter including the necessary conditions of charge neutrality and beta equilibrium is discussed in subsection II.1. In subsection II.2, we will discuss the QCD sum rules used to calculate the in-medium masses of $\rho$ mesons in the present work. The result obtained from the present investigation are discussed in section III.

II Formalism

II.1 Chiral SU(3) Model in Neutron Star

Quantum Chromodynamics, the theory of strong interaction due to large coupling constant is not directly applicable in non-perturbative regime Greiner ; Papazoglou . Therefore, effective models constrained by various properties of QCD are formulated to probe the physics of non-perturbative regime Balazs ; Greiner ; Dexheimer ; Sanyal ; Schramm2008 ; HANAUSKE2000 . Chiral SU(3) model is one such effective field theoretical model based on broken scale invariance and spontaneous chiral symmetry breaking kumar2015 . The property of broken scale invariance of QCD is incorporated by adding a glueball field $\chi$ at tree level through logarithmic potential Schechter1980 ; Greiner . The Lagrangian density in chiral SU(3) model consists of various interaction terms among different field particles and is given by

[TABLE]

where the first term $\mathcal{L}_{Kin}$ in the above Lagrangian represents the kinetic energy of nucleons and scalar ($\sigma$, $\zeta$, $\delta$ and $\chi$) and vector ($\omega$ and $\rho$) mesons, $\mathcal{L}_{i,Scal}+\mathcal{L}_{i,Vec}$ represents the interaction of nucleons with scalar and vector mesons, $\mathcal{L}_{scal}$ represents the meson-meson interaction which is responsible for the spontaneous breaking of chiral symmetry, $\mathcal{L}_{vec}$ represents the self interaction of vector mesons and is responsible for generating the mass of spin-1 mesons through the interaction with spin-0 mesons and the last term, $\mathcal{L}_{SB}$ simply explains the explicit symmetry breaking of U(1), SU(2) and chiral symmetry, and is responsible for the masses of pseudoscalar mesons. Explicitly, various terms are given by Dexheimer ; Schramm2008

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $m_{i}^{*}$ is the effective mass of $i^{th}$ nucleon. The values of various coupling constants, $i.e.$, $g_{fi}$ ($f$ represents the field and $i$ represents the nucleon) and other constants are chosen in such a way so as to reproduce the vacuum masses of hadrons, hyperon potentials, nuclear saturation properties like binding energy per nucleon ($B/A$), asymmetry energy Greiner ; Papazoglou ; Dexheimer . Also, $m_{\pi}$, $f_{\pi}$ and $m_{K}$, $f_{K}$ are masses and decay constants of $\pi$ and $K$ meson, respectively. The values of these constants are listed in table 1. The thermodynamical potential per unit volume in grand canonical ensemble for the medium of neutron stars is given by

[TABLE]

In above, $\mathcal{V}_{vac}$ is the vacuum energy which is subtracted to obtain the zero vacuum energy, $\alpha_{e}$ and $\beta_{i}$ are the energy terms arising from the electrons and nucleons. The corresponding expressions are given as

[TABLE]

and

[TABLE]

where $k_{fi}$ represents the fermi momentum of the nucleon species and $\gamma_{i}$ is the fermionic spin-isospin degeneracy factor which is taken to be equal to 2. Furthermore, $E_{fi}^{*}$=$\sqrt{k_{fi}^{2}+m_{i}^{*2}}$ and $\mu_{i}^{*}=\mu_{i}-g_{\omega i}\omega-g_{\rho i}\tau_{3}\rho$, are the single particle energy and effective chemical potential, respectively Schramm2008 ; HANAUSKE2000 . The equations of motion corresponding to scalar and vector fields can be obtained by minimizing the grand potential given by eq. 6 because this will correspond to an equilibrium condition Wang2003 . We have following equations for $\sigma,\zeta,\delta,\chi,\omega$ and $\rho$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The effective nucleon masses $m_{i}^{*}$ can be generated with the help of scalar fields except for a small explicit mass term and this effective mass will decrease with the increasing baryonic density due to the partial restoration of chiral symmetry Schramm2008 and is written as

[TABLE]

where $\delta m$ is a constant term added in order to get the vacuum mass of the nucleon. Moreover, $\rho_{i}$ and $\rho_{i}^{s}$ are vector and scalar densities of $i^{th}$ nucleon Papazoglou . Equation 9 to eq. 14 are solved simultaneously along with the conditions of charge neutrality and beta-equilibrium to obtain density dependent values of various scalar and vector fields relevant for medium of neutron stars.

Now, we discuss the strategy to evaluate the scalar quark and gluon condensates, in terms of above obtained scalar fields, to be used later in QCD sum rule calculations for $\rho$ mesons. In chiral effective model, the explicit symmetry breaking is used to calculate the scalar quark condensates while the scalar gluon condensates can be obtained through the broken scale invariance property of QCD kumar2015 . The scalar quark condensates are related to explicit symmetry breaking through relation Kumar2010

[TABLE]

where $\mathcal{L}_{SB}$ is given by eq. 5. From this one can identify following expressions for light quark condensates kumar2015 :

[TABLE]

[TABLE]

[TABLE]

The energy-momentum tensor $\theta_{\mu\nu}$ in terms of dilaton field in effective chiral model is given as

[TABLE]

where $\mathcal{L_{\chi}}$ is the Lagrangian due to the dilaton field and is given by Rajesh

[TABLE]

The scalar gluon condensate is equal to the trace of energy-momentum tensor which is non-vanishing in the present case. This property of non-vanishing trace of $\theta_{\mu\nu}$ is known as the trace anomaly which arises due to the broken scale invariance property of QCD kumar2015 . Using the broken scale invariance Lagrangian, the trace of energy-momentum tensor in the limit of vanishing quark masses is given by

[TABLE]

Comparing the trace of of energy-momentum tensor in QCD with that calculated in effective chiral model, the gluon condensate found out to be Mishra ; Kumar

[TABLE]

The scalar gluon condensate can be modified by incorporating the finite mass of quarks and its expression becomes Kumar

[TABLE]

II.2 QCD sum rules

In this subsection, we will discuss briefly the QCD sum rules used in the present work to calculate the in-medium masses of vector $\rho$ mesons in charge neutral matter Zschocke2002 ; Lee ; Mishra ; Kwon . To obtain the QCD sum rules for $\rho$ mesons, we start with the retarded two-point correlation function defined in terms of vector current $j^{\rho}_{\mu}$ corresponding to $\rho$ mesons at finite baryonic density as Zschocke2002 :

[TABLE]

where $q=(q^{0},\textbf{q})$, $\langle...\rangle_{\rho_{B}}$ is the grand canonical ensemble average, $\rho_{B}$ is the baryonic density, $\mathcal{R}j^{\rho}_{\mu}(x)j^{\rho}_{\nu}(0)\equiv\Theta(x^{0})[j^{\rho}_{\mu}(x),j^{\rho}_{\nu}(0)]$ with $j^{\rho}_{\mu}$=$\frac{1}{2}$($\bar{u}\gamma_{\mu}u$ - $\bar{d}\gamma_{\mu}d$) and $\Theta(x^{0})$ is the step function.

In the limit $\textbf{q}\rightarrow 0$ the longitudinal and transverse invariant relevant for the medium will merge into single one given by Lee

[TABLE]

Moreover, since the complex $q^{0}(=\omega)$ plane is analytic in the upper half plane, following dispersion relation can be written for the correlation function in the medium Hatsuda ; Zschocke2002

[TABLE]

In above, on right side (known as phenomenological side) $R_{h}^{\rho}=\frac{1}{\pi}$Im$\Pi^{\rho R}(u)$ (where $h$ stands for hadrons) is the spectral density. In the free space, the spectral density is generally written as sum of resonance pole term and continuum term, whereas in the charge neutral matter in which we are interested in the present work, additional contribution, known as Landau damping contribution, from the scattering of mesons with nucleons will need to consider Zschocke2002 . Considering these three contributions, we can write

[TABLE]

where $R_{\text{sc}}$ = $3\pi^{2}\frac{\rho_{B}}{m_{N}}$ is Landau damping contribution Hatsuda . Also, $m_{\rho}^{*}$ is the in-medium mass of $\rho$ meson, $F_{\rho}^{*}$ is the pole residue and $s_{0}^{*\rho}$ is the continuum threshold which separates the resonance part and the perturbative part of spectral density. The parameters $m_{\rho}^{*}$, $F_{\rho}^{*}$ and $s_{0}^{*\rho}$ are known as phenomenological parameters. The subtraction term in eq. 27 accounts for the convergence of the integral. If $\text{Im}\Pi^{\rho R}(u)$ in the dispersion integral diverges then subtraction term exists otherwise it will be zero AV . Assuming the integral to be converging in the present investigation, we used subtraction term to be zero Zschocke2002 .

For the large values of $Q^{2}$ (i.e. $Q^{2}\equiv-q_{0}^{2}>0$), we perform the operator product expansion (OPE) of the product of vector current appearing in eq. 25. Thus, for the left hand side of eq. 27, we have

[TABLE]

where $c_{n}$ includes the Wilson coefficients HL and include the contribution in terms of non-zero quark and gluon condensates. On comparing the phenomenological (right side) and OPE side (left side) of eq. 27 in asymptotic limit, we obtain following set of finite energy sum rule equations Zschocke2002 ; Mishra ; Kraniskov :

[TABLE]

[TABLE]

and

[TABLE]

The Wilson coefficients $c_{n}$ appearing in the above given sum rule equations upto $n=3$ are given by

[TABLE]

[TABLE]

[TABLE]

where $\alpha_{s}$ is the coupling constant, $m_{u}$ and $m_{d}$ are the masses of up and down quarks with respective values of 4 and 7 MeV and $k_{q}$ is the factorization parameter. In the above discussion, the value of $c_{3}^{*}$ represent the 4-quark condensates and in calculating this we need the factorization parameter $k_{q}$ which is calculated first by assuming the vacuum case only Mishra . In our calculations, we have used a value of 2.0850 for $k_{q}$.

III Results and Discussion

Here, we will present the results of our present investigation of $\rho$ mesons and possibility of its condensation in neutron star medium. As discussed earlier, the quark and gluon condensates to be used in QCD sum rules to obtain the in-medium masses of $\rho$ mesons are calculated using chiral SU(3) model. In figs. 1 and 2 we have shown the variation of scalar quark condensates and gluon condensates, respectively as a function of baryonic density $\rho_{B}$ of charge neutral matter.

As can be seen from fig. 1, as the density of matter is increased the magnitude of quark condensate decrease. At $\rho_{B}=0,\rho_{0}$ and $4\rho_{0}$ the values of $\langle\bar{u}u\rangle$ are observed to be (-279.6 MeV)3, (-243.6 MeV)3 and (-192.9 MeV)3, respectively. For $\langle\bar{d}d\rangle$ above values changes to (-232.1 MeV)3, (-202.3 MeV)3 and (-160.2 MeV)3, respectively. The drop in the magnitude of quark condensates as a function of density is consistent with the expectation of chiral symmetry restoration at high baryonic density. As can be seen from eqs. 17 and 18, $\langle\bar{u}u\rangle$ (or $\langle\bar{d}d\rangle$) condensate is directly proportional to the scalar field $\sigma$ (note that the magnitude of $\delta$ field is very small as compared to $\sigma$ field and also, the scalar dilaton field $\chi$ vary very little as function of density) whose in-medium value is calculated by solving coupled equations of scalar and vector fields in charge neutral matter, as was discussed earlier also. The magnitude of scalar field $\sigma$ is observed to decrease with increase in the density of medium and hence causes decrease in the magnitude of scalar quark condensates. The difference in the vacuum values ($\delta$ field will be zero at $\rho_{B}$ =0) of $\langle\bar{u}u\rangle$ and $\langle\bar{d}d\rangle$ quark condensates quoted above is because we considered different current quark masses of $u$ and $d$ quarks. Having considered average value of two masses, $m_{q}=m_{u}=m_{d}=5.5$ MeV, vacuum value of these quark condensates will coincide whereas in the medium they still have different values because of iso-vector field $\delta$ appearing with opposite sign in eqs. 17 and 18. In Bali ; SS ; Hong ; DH ; Sho , the density dependence of quark condensates is evaluated for nuclear medium relevant for heavy-ion collisions. Calculations of Walecka ; Serot within linear Walecka model and of NK ; DELFINO using Born potential in Dirac-Brueckner approach show that quark condensates first decreases with density and then start increasing at higher baryonic density.

As can be seen from fig. 2, the value of scalar gluon condensate $\langle\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G^{a\mu\nu}\rangle$ also shows a decrease with respect to vacuum value. We calculated the gluon condensates in charge neutral medium, with and without quark mass term. We found that the value of gluon condensate, without quark mass term, at density $\rho_{B}=0,\rho_{0}$, $4\rho_{0}$ and $6\rho_{0}$ is (391.3 MeV)4, (389.6 MeV)4, (386.7 MeV)4 and (387.1 MeV)4, respectively. The corresponding values of $\langle\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G^{a\mu\nu}\rangle$, considering finite quark mass term, are (372.9 MeV)4, (372.5 MeV)4, (369.9 MeV)4 and (370.2 MeV)4, respectively. The dilaton field $\chi$ varies very little as a function of density of medium and this behaviour is further reflected in the density dependence of scalar gluon condensates which is also in agreement with the calculations in refs. Thomas ; AK . We should notice here that the variation of quark condensate is more in comparison to the gluon condensate. This can also be concluded from the drop in the two condensates at $\rho_{B}$=$\rho_{0}$, quark condensate drops to $\sim$13% while gluon condensate to only $\sim$0.4% of respective vacuum values.

In the following now, we will discuss the modification of the masses of $\rho$ mesons in charge neutral matter using the above discussed medium modified scalar quark and gluon condensates. Solving coupled finite energy sum rules given by eqs. 30, 31 and 32, we obtain the density dependence of pole residue $F_{\rho}^{*}$, continuum threshold parameter $s_{\rho}^{*}$ and in-medium vector $\rho$ meson mass $m_{\rho}^{*}$ plotted in figs. 5, 3 and 4. As can be seen from figs. 3 and 4, there is a continuous decrease in the value of $F_{\rho}^{*}$ and $s_{\rho}^{*}$ as a function of baryonic density. At density, $\rho_{B}$= 0, $\rho_{0}$ and 4$\rho_{0}$ the values of $F_{\rho}^{*}$ ($s_{\rho}^{*}$) are observed to be 2.817 (1.285), 1.994 (0.926) and 0.929 (0.489) GeV2, respectively. Due to more scattering of nucleons at higher density the effect of Landau damping term becomes significant and thus the value of $F_{\rho}^{*}$ will decrease more at higher density Hatsuda .

The dependence of the in-medium masses of $\rho$ mesons on the condensates is introduced through the Wilson coefficients, $c_{2}^{*}$ and $c_{3}^{*}$ as given by eqs. 34 and 35. As also discussed in earlier calculations of ref. Hatsuda ; Mishra , the coefficient $c_{2}^{*}$ in eq. 34 can be written in terms of gluon condensates and the twist-two quark condensates, while the coefficient $c_{3}^{*}$ of eq. 35 can be written in terms of twist-four quark condensates. It can be inferred from the tables 2 and 5 that the mass of $\rho$ meson depends largely on $c_{2}^{*}$ at higher density. The drop in the mass of $\rho$ meson in the absence of $c_{2}^{*}$ at $\rho_{B}=\rho_{0}$, $2\rho_{0}$ and $4\rho_{0}$ is found to be 11.5$\%$, 18.7$\%$ and 23.4$\%$ of the vacuum value. On the other hand, in presence of $c_{2}^{*}$ the drop is more at higher density which is about 45$\%$ of the vacuum value, near the density of $4\rho_{0}$. In absence of $c_{2}^{*}$, the dependence is solely on the twist-four quark condensates which contributes positively to the mass, while in contrary to this, the presence of the $c_{2}^{*}$ at higher density will contribute more negatively through the twist-two quark condensates.

From fig. 5, we observe that the mass of $\rho$ meson decrease with increase in the density of the medium. At $\rho_{B}=0$, $\rho_{0}$ and 4$\rho_{0}$, it is found to have a value of 776.8, 653.9 and 433.1 MeV, respectively. The drop in the mass of $\rho$ mesons with increasing density is consistent with earlier QCD sum rule calculations Hatsuda ; Koike ; Zschocke2002 . In fig. 5, we also plotted the chemical potential of electron as a function of density. On one hand, there is a drop in the in-medium mass of $\rho$ meson with the increasing baryonic density while on the other hand, the chemical potential of the electron increases with the increased density. The baryonic decay will give rise to the presence of electrons and these released electrons will require a larger amount of energy to exist at higher densities. Thus, at such high baryonic densities, it is not energetically favorable for electrons to be present and also in parallel to this effect, the drop in the mass of $\rho$ meson is appreciable which could make them a better contestant to exist in place of electrons. Thus, when the in-medium mass of $\rho$ mesons becomes comparable to the effective electron chemical potential these electrons may get replaced by the negatively charged $\rho$ mesons. As can be seen from fig. 5, at baryonic density $\rho_{B}=7.3\rho_{0}$, the condition $\omega_{\rho}=\mu_{e^{-}}$ is satisfied and hence possibility of $\rho$ meson condensation arise at this density. Furthermore, the possibility of $\rho$ meson to be found at the core of a neutron star is also supported by few other works which includes EE ; Mallick2015 . In ref. Mallick2015 , a more simpler approach is used to calculate the in-medium masses of $\rho$ mesons by considering the linear dependence on $\sigma$ field.

The presence of strong magnetic field at the interior core of the neutron star may also effect the threshold density at which the $\rho$ meson condensation takes place Prakash ; Cardall ; Chakrabarty ; Zhang ; Mao ; Wei ; Iwamoto ; Khalilov ; Yue ; Rabhi ; S ; Lattimer2 . The observations of pulsars and gamma repeaters clearly indicate the presence of a strong magnetic field in neutron stars. It is estimated to be around $10^{14}$-$10^{15}$G at the surface Thompson which becomes even more intense at the interior core. According to the virial theorem, it is approximated to be about $10^{18}$-$10^{19}$G in the interior core of the star Lai ; Zhang . The dense quark matter at the center is assumed to be the reason for such a high magnetic field. It was observed that the composition of neutron star could change efficaciously in the presence of such a high magnetic field Yue . As discussed earlier, in various papers the effect of the Landau quantization due to such a high magnetic field on anomalous magnetic moments of nucleons could affect the EoS of the neutron star Prakash ; S ; Lattimer2 . In case of magnetized nuclear matter it was observed that the drop in the medium modified mass of $\rho$ meson is a bit low as compared to the non-magnetized neutron matter AK which may slightly diminish the possibility of $\rho$ meson condensation in neutron star also. Furthermore, calculations of refs. Yuan ; Prakash ; Yue have also shown the sharp decrease in the masses of nucleons in presence of magnetic field. As a result, the Fermi momenta of protons will fall and proton fraction will increase and to maintain the beta equilibrium condition electron chemical potential also which results in increase of electron density. Moreover, calculations in QMC model, Modified Quark-Meson Coupling (MQMC) and RMF model Shen ; Ryu ; Dey show that the threshold density for kaon and antikaon condensation shift to higher values in presence of magnetic field which results in the large fraction of electrons at higher densities. This causes more stiffness in the EoS and increase in the mass of neutron star. Similar kind of consequences may arise due to $\rho$ meson condensation in neutron stars when finite magnetic field effects will be considered in future detailed calculations.

Contrary to the neutron stars, proto-neutron star which is the initial stage of the neutron star just after the supernova explosion include neutrinos as an integral part of it. Calculations of refs. Pons ; Schramm2008 shows that as a consequence of increased number of neutrinos the electron density should also increase and therefore demanding the charge neutrality condition proton density should also increase. The increasing proton density will decrease the isospin asymmetry in the star which will soften the EoS. Due to the large population of electron at lower density, it may also diminish the possibility of presence of $\rho$ mesons.

In brief, we have investigated the possibility of $\rho$ meson condensation in neutron star using a combined approach of chiral SU(3) model and QCD sum rules. In this, chiral SU(3) model is employed to calculate the scalar quark and gluon condensates through scalar and vector fields in charge neutral matter. The condensates are used in QCD sum rules through the Wilson coefficients, to determine the in-medium masses of $\rho$ mesons. The effect of scattering between the nucleons is incorporated through the Landau damping term, $R_{sc}$. An appreciable decrease in the mass of $\rho$ meson is observed and alongside to this, the electron chemical potential as calculated by solving the chiral model in neutron star shows an increasing behaviour with increasing baryonic density. The threshold density at which condition of $\rho$ meson condensation is satisfied is found to be 7.3$\rho_{0}$.