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Possibility of ρ Meson Condensation in Neutron Star: Unified

Approach of Chiral SU(3) Model and QCD Sum Rules

Shivam∗ and Arvind Kumar†

Department of Physics, Dr. B R Ambedkar National Institute

of Technology Jalandhar, Jalandhar – 144011,Punjab, India

Abstract

In the present work the conjunction of chiral SU(3) model with QCD sum rules is employed to

explore the possibility of ρ meson condensation in neutron stars. The quark and gluon condensates

in terms of which the in-medium masses of ρ 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 ρ meson condensation is satisfied for the density of about 7ρ0, where ρ0 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 ρ meson condensation.

∗Electronic address: shivamchoudhary413@gmail.com†Electronic address: iitd.arvind@gmail.com, kumara@nitj.ac.in

1

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, Rus-

sia 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

[1] and this has triggered much more theoretical calculations in the field [2–6]. The data

obtained on the masses and radii of the neutron star from the Neutron Star Interior Com-

position Explorer (NICER) experiment at international space station impose the significant

constraint on the EoS [7–12]. In order to explain the observed data from various other lab-

oratories like LIGO-Virgo [13–15], LIGO-India [16], GEO [17] we need to mould our models

and the existence of these high mass neutron stars will also hold out restriction on theoreti-

cal calculations [18]. 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 [19].

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 [20–25], Polyakov Quark-Meson Coupling model [26, 27], chiral effective mod-

els [28–30], QCD sum rules, coupled channel approach [31–34], NJL and PNJL models

[35–38] etc., are employed to investigate the properties of hadrons in dense interacting en-

vironment. The idea of condensate formation in dense matter was first proposed a long ago

by Kaplan and Nelson [39]. The inspiration for the meson condensation comes from strong

attractive interaction of mesons with nucleons. Due to this attractive interaction, the effec-

tive energy of the mesons becomes low in the matter as compared to that in the vacuum.

2

This notion provides a base for the kaon-antikaon condensation in dense matter [40]. 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 [41]. 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

[40, 42].

The in-medium mass reduction of light vector ρ mesons predicted by various studies[43–

45] motivate to investigate the possibility of ρ meson condensation in neutron star [46, 47].

The [48] results on ρ0 → π+π− production in the peripheral Au-Au collisions at√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 [49] provides the possibility in the mass reduction of ω

mesons but width broadening in the ρ mesons [50]. Future heavy-ion collision experiments

may help to clarify the situation on the mass shift of ρ mesons further. In order to look for

the possibility of ρ 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 ρ mesons. The presence of these antikaon or ρ meson condensates not

only soften the EoS which will impose the constraint on the mass of star [51, 52] but will also

affect the restoration of chiral symmetry [29]. Thus, the possibility of ρ 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 ρ meson by

assuming a linear density approximation and a considerable amount of decrease is observed

in the in-medium masses of ρ mesons [43]. Consequently, different studies on using QCD

sum rules for investigating the properties of vector mesons evolved[30, 53, 54]. In QCD sum

rule analysis, in-medium properties of different mesons are evaluated in terms of medium

modification of quark and gluon condensates[55]. 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) model[56, 57] to calculate

the density dependence of quark and gluon condensates. Within the model, we have non-

3

linear interaction terms of the scalar fields σ, ζ , δ, χ and also for vector fields ω and ρ.

The scalar iso-vector field δ and vector iso-vector field ρ 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 χ 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 ρ mesons. The in-medium mass of ρ meson and electron chemical potential are then

compared to check out for any possibility of ρ 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

[28, 56–59].

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 ρ mesons and its

condensation. The chiral SU(3) model in neutron star matter including the necessary condi-

tions of charge neutrality and beta equilibrium is discussed in subsection IIA. In subsection

IIB, we will discuss the QCD sum rules used to calculate the in-medium masses of ρ mesons

in the present work. The result obtained from the present investigation are discussed in

section III.

II. FORMALISM

A. 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 [58, 60]. Therefore, effective

models constrained by various properties of QCD are formulated to probe the physics of non-

perturbative regime [28, 56–59, 61]. Chiral SU(3) model is one such effective field theoretical

model based on broken scale invariance and spontaneous chiral symmetry breaking [62]. The

property of broken scale invariance of QCD is incorporated by adding a glueball field χ at

tree level through logarithmic potential [58, 63]. The Lagrangian density in chiral SU(3)

4

model consists of various interaction terms among different field particles and is given by

L = LKin + Li,Scal + Li,V ec + LScal + LV ec + LSB (1)

where the first term LKin in the above Lagrangian represents the kinetic energy of nucleons

and scalar (σ, ζ , δ and χ) and vector (ω and ρ) mesons, Li,Scal+Li,V ec represents the interac-

tion of nucleons with scalar and vector mesons, Lscal represents the meson-meson interaction

which is responsible for the spontaneous breaking of chiral symmetry, Lvec 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, LSB simply explains the ex-

plicit 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 [28, 56]

Li,V ec + Li,Scal = −ψi[m∗i + gωiγ0ω + gρiγ0τ3ρ]ψi, (2)

LScal =− 1

2k0χ

2(

σ2 + ζ2 + δ2)

+ k1(

σ2 + ζ2 + δ2)2

+k2

(

σ4

2+δ4

2+ 3σ2δ2 + ζ4

)

+ k3χ(

σ2 − δ2)

ζ

−k4χ4 − 1

4χ4ln

χ4

χ40

+d

3χ4ln

(

(

(σ2 − δ2) ζ

σ20ζ0

)(

χ

χ0

)3)

, (3)

LV ec =1

2

(

m2

ωω2 +m2

ρρ2

)

(

χ

χ0

)2

+ g4(ω4 + 6ω2ρ2 + ρ4), (4)

LSB = −(

χ

χ0

)2(

1

2m2

πfπ (σ + δ) +1

2m2

πfπ (σ − δ) +(√2m2

KfK − 1√2m2

πfπ)

ζ

)

, (5)

where m∗i is the effective mass of ith nucleon. The values of various coupling constants,

i.e., gfi (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 [28, 58, 60].

Also, mπ, fπ and mK , fK are masses and decay constants of π and K meson, respectively.

The values of these constants are listed in table I. The thermodynamical potential per unit

volume in grand canonical ensemble for the medium of neutron stars is given by

Ω

V= −Lvec −L0 −LSB − Vvac − βi − αe. (6)

5

In above, Vvac is the vacuum energy which is subtracted to obtain the zero vacuum energy,

αe and βi are the energy terms arising from the electrons and nucleons. The corresponding

expressions are given as

βi =∑

i

γi48π2

[(

2k3fi − 3m∗2i kfi

)

µ∗i + 3m∗4

i lnkfi + µ∗

i

m∗i

]

, (7)

and

αe =µ4e

12π2, (8)

where kfi represents the fermi momentum of the nucleon species and γi is the fermionic spin-

isospin degeneracy factor which is taken to be equal to 2. Furthermore, E∗fi=√

k2fi +m∗2i

and µ∗i = µi − gωiω − gρiτ3ρ, are the single particle energy and effective chemical potential,

respectively [56, 57]. 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 [64]. We have following equations for σ, ζ, δ, χ, ω and ρ:

k0χ2σ − 4k1

(

σ2 + ζ2 + δ2)

σ − 2k2(

σ3 + 3σδ2)

− 2k3χσζ

−d3χ4

(

2σ

σ2 − δ2

)

+

(

χ

χ0

)2

m2

πfπ −∑

i

gσiρsi = 0, (9)

k0χ2ζ − 4k1

(

σ2 + ζ2 + δ2)

ζ − 4k2ζ3 − k3χ

(

σ2 − δ2)

−d3

χ4

ζ+

(

χ

χ0

)2 [√2m2

KfK − 1√2m2

πfπ

]

−∑

i

gζiρsi = 0, (10)

k0χ2δ − 4k1

(

σ2 + ζ2 + δ2)

δ − 2k2(

δ3 + 3σ2δ)

+ 2k3χδζ

+2

3dχ4

(

δ

σ2 − δ2

)

−∑

i

gδiτ3ρsi = 0, (11)

k0χ(

σ2 + ζ2 + δ2)

− k3(

σ2 − δ2)

ζ + χ3

[

1 + ln

(

χ4

χ40

)]

+ (4k4 − d)χ3

−4

3dχ3ln

(

(

(σ2 − δ2) ζ

σ20ζ0

)(

χ

χ0

)3)

+2χ

χ20

[

m2

πfπσ +

(√2m2

KfK − 1√2m2

πfπ

)

ζ

]

− χ

χ20

(m2

ωω2 +m2

ρρ2) = 0, (12)

(

χ

χ0

)2

m2

ωω + 4g4(ω3 + 3ωρ2)−

∑

i

gωiρi = 0, (13)

6

(

χ

χ0

)2

m2

ρρ+ 4g4(ρ3 + 3ω2ρ)−

∑

i

gρiτ3ρi = 0. (14)

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 [56] and is written as

m∗i = gσiσ + gζiζ + gδiτ3δ + δm, (15)

where δm is a constant term added in order to get the vacuum mass of the nucleon. Moreover,

ρi and ρsi are vector and scalar densities of ith nucleon [60]. 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 ρ

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 [62]. The scalar quark condensates are related to

explicit symmetry breaking through relation [65]

∑

i

mi〈qiqi〉 = −LSB, (16)

where LSB is given by eq. (5). From this one can identify following expressions for light

quark condensates [62]:

mu〈uu〉 =1

2m2

πfπ(σ + δ)

(

χ

χ0

)2

, (17)

md〈dd〉 =1

2m2

πfπ(σ − δ)

(

χ

χ0

)2

, (18)

ms〈ss〉 =(√

2m2

KfK − 1√2m2

πfπ

)

(

χ

χ0

)2

ζ. (19)

The energy-momentum tensor θµν in terms of dilaton field in effective chiral model is given

as

θµν = (∂µχ)

(

∂Lχ

∂(∂νχ)

)

− gµνLχ, (20)

7

where Lχ is the Lagrangian due to the dilaton field and is given by [66]

Lχ =1

2(∂µχ)(∂

µχ)− k4χ4 − 1

4χ4ln

(

χ4

χ40

)

+d

3χ4ln

(

(

(σ2 − δ2) ζ

σ20ζ0

)(

χ

χ0

)3)

. (21)

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 θµν is known as the

trace anomaly which arises due to the broken scale invariance property of QCD [62]. Using

the broken scale invariance Lagrangian, the trace of energy-momentum tensor in the limit

of vanishing quark masses is given by

θµµ = (∂µχ)

(

∂Lχ

∂(∂µχ)

)

− 4Lχ = −(1 − d)χ4. (22)

Comparing the trace of of energy-momentum tensor in QCD with that calculated in effective

chiral model, the gluon condensate found out to be [67, 68]⟨

αs

πGa

µνGaµν

⟩

=8

9(1− d)χ4. (23)

The scalar gluon condensate can be modified by incorporating the finite mass of quarks and

its expression becomes [67]⟨

αs

πGa

µνGaµν

⟩

=8

9

[

(1− d)χ4 +(

m2

πfπσ +(√

2m2

KfK − 1√2m2

πfπ

)

ζ)

(

χ

χ0

)2]

. (24)

B. 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 ρ mesons in charge neutral matter [43, 68–70].

To obtain the QCD sum rules for ρ mesons, we start with the retarded two-point correlation

function defined in terms of vector current jρµ corresponding to ρ mesons at finite baryonic

density as [70]:

ΠρRµν (q; ρB) = i

∫

d4xeiqx〈Rjρµ(x)jρν (0)〉ρB , (25)

where q = (q0,q), 〈...〉ρB is the grand canonical ensemble average, ρB is the baryonic density,

Rjρµ(x)jρν (0) ≡ Θ(x0)[jρµ(x), jρν(0)] with j

ρµ=

1

2(uγµu - dγµd) and Θ(x0) is the step function.

In the limit q → 0 the longitudinal and transverse invariant relevant for the medium will

merge into single one given by [43]

ΠρR = ΠρRµµ/(−3q2)|q→0. (26)

8

Moreover, since the complex q0(= ω) plane is analytic in the upper half plane, following

dispersion relation can be written for the correlation function in the medium [70, 71]

ReΠρR(ω2) = P

∫ ∞

0

du2Rρ

h

u2 − ω2+ subtractions. (27)

In above, on right side (known as phenomenological side) Rρh = 1

πImΠρR(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

[70]. Considering these three contributions, we can write

12πImΠρR(u) = δ(u2)Rsc + F ∗ρ δ(u

2 −m∗2ρ ) + c0θ(u

2 − s∗ρ0 ), (28)

where Rsc = 3π2 ρBmN

is Landau damping contribution [71]. Also, m∗ρ is the in-medium mass

of ρ meson, F ∗ρ is the pole residue and s∗ρ0 is the continuum threshold which separates the

resonance part and the perturbative part of spectral density. The parameters m∗ρ, F

∗ρ and

s∗ρ0 are known as phenomenological parameters. The subtraction term in eq. (27) accounts

for the convergence of the integral. If ImΠρR(u) in the dispersion integral diverges then

subtraction term exists otherwise it will be zero [72]. Assuming the integral to be converging

in the present investigation, we used subtraction term to be zero [70].

For the large values of Q2 (i.e. Q2 ≡ −q20 > 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

ReΠρR(Q2) = −c0lnQ2 +

∞∑

n=1

cnQ2n

, (29)

where cn includes the Wilson coefficients [73] 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 [68, 70, 74]:

F ∗ρ = (c0s

∗ρ0 + c1)− Rsc, (30)

F ∗ρm

∗2ρ =

(s∗ρ0 )2c02

− c∗2, (31)

9

gσN gζN gδN gωN gρN

-9.83 1.22 -2.34 12.13 4.03

k0 k1 k2 k3 k4

2.37 1.40 -5.55 -2.65 -0.23

mπ(MeV) mK (MeV) fπ (MeV) fK(MeV) g4

139 498 93.3 122.143 38.9

σ0 (MeV) ζ0 (MeV) χ0 (MeV) d ρ0 (fm−3)

-93.3 -106.8 401.91 0.06 0.15

mω(MeV) mρ(MeV) mN (MeV) δm(MeV) kq

783 770 939 150 2.0850

Table I: Values of various parameters.

and

F ∗ρm

∗4ρ =

(s∗ρ0 )3c03

+ c∗3. (32)

The Wilson coefficients cn appearing in the above given sum rule equations upto n = 3 are

given by

c0 =3

2

[

1 +αs(Q

2)

π

]

, c1 =−9

2(m2

u +m2

d), (33)

c∗2 =π2

2

⟨αs

πGa

µνGaµν⟩

+ 6π2〈muuu+mddd〉, (34)

c∗3 = −αsπ3224

27kq(〈uu〉2 + 〈dd〉2), (35)

where αs is the coupling constant, mu and md are the masses of up and down quarks with

respective values of 4 and 7 MeV and kq 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 kq which is calculated first by assuming the vacuum case only

[68]. In our calculations, we have used a value of 2.0850 for kq.

III. RESULTS AND DISCUSSION

Here, we will present the results of our present investigation of ρ mesons and possibility

of its condensation in neutron star medium. As discussed earlier, the quark and gluon

10

140

160

180

200

220

240

260

280

(-<q

q>M

eV)3

1 2 3 4 5 6 7 8

B/ 0

<u u>, mu=4 MeV<d d>, md=7 MeV<u u>, mu=5.5 MeV<d d>, md=5.5 MeV

Figure 1: The scalar quark condensates are plotted against the increasing baryonic density in

neutron star matter.

c∗2↓, m∗

ρ → ρB = ρ0 ρB = 2ρ0 ρB = 4ρ0

c∗2=0 738.7 (MeV), (11.5%) 679.3 (MeV), (18.7%) 638.9 (MeV), (23.4%)

c∗2 6= 0 653.9 (MeV),(15.8%) 554.8 (MeV), (28.5%) 433.1 (MeV), (44.2%)

Table II: Variation of mass with c∗2

condensates to be used in QCD sum rules to obtain the in-medium masses of ρ 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 ρB

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 ρB = 0, ρ0 and 4ρ0 the values of 〈uu〉 are observed to be (-279.6

MeV)3, (-243.6 MeV)3 and (-192.9 MeV)3, respectively. For 〈dd〉 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), 〈uu〉

11

365

370

375

380

385

390

395

<s/

Ga

Ga

>1/

4(M

eV)

0 1 2 3 4 5 6 7 8

B/ 0

mq= 0mq= 0

Figure 2: The gluon condensates are plotted against the increasing baryonic density in neutron

star matter.

(or 〈dd〉) condensate is directly proportional to the scalar field σ (note that the magnitude of

δ field is very small as compared to σ field and also, the scalar dilaton field χ 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 σ 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 (δ field will be zero at ρB =0) of 〈uu〉 and 〈dd〉 quark condensates quoted above is

because we considered different current quark masses of u and d quarks. Having considered

average value of two masses, mq = mu = md = 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 δ appearing with opposite sign in eqs. (17) and (18). In [75–79], the density

dependence of quark condensates is evaluated for nuclear medium relevant for heavy-ion

collisions. Calculations of [80, 81] within linear Walecka model and of [82, 83] using Born

potential in Dirac-Brueckner approach show that quark condensates first decreases with

density and then start increasing at higher baryonic density.

12

0.0

0.5

1.0

1.5

2.0

2.5

3.0

F*

(GeV

)2

1 2 3 4 5 6 7 8

B/ 0

Figure 3: Variation of pole residue F ∗ρ with the increasing baryonic density in neutron star matter

is shown in above figure.

As can be seen from fig. 2, the value of scalar gluon condensate 〈αs

πGa

µνGaµν〉 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 ρB = 0, ρ0, 4ρ0 and 6ρ0 is (391.3 MeV)4,

(389.6 MeV)4, (386.7 MeV)4 and (387.1 MeV)4, respectively. The corresponding values of

〈αs

πGa

µνGaµν〉, 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 χ 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. [42, 84].

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

ρB=ρ0, quark condensate drops to ∼13% while gluon condensate to only ∼0.4% of respective

vacuum values.

In the following now, we will discuss the modification of the masses of ρ mesons in

13

0.30.40.50.60.70.80.91.01.11.21.3

s*

(GeV

)2

1 2 3 4 5 6 7 8

B/ 0

Figure 4: Variation of continuum threshold parameter s∗ρ with the increasing baryonic density in

neutron star matter.

charge neutral matter using the above discussed medium modified scalar quark and gluon

condensates. Solving coupled finite energy sum rules given by eqs. (30) to (32), we obtain the

density dependence of pole residue F ∗ρ , continuum threshold parameter s∗ρ and in-medium

vector ρ meson mass m∗ρ plotted in figs. 3 to 5. As can be seen from figs. 3 and 4, there is a

continuous decrease in the value of F ∗ρ and s∗ρ as a function of baryonic density. At density,

ρB= 0, ρ0 and 4ρ0 the values of F ∗ρ (s∗ρ) 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 ∗ρ will decrease

more at higher density [71].

The dependence of the in-medium masses of ρ 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. [68, 71], 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 table II

and fig. 5 that the mass of ρ meson depends largely on c∗2 at higher density. The drop in the

14

0

100

200

300

400

500

600

700

800

900

m*

,e

(MeV

)

0 1 2 3 4 5 6 7 8

B/ 0

m*

e

m*, c2= 0

Figure 5: The in-medium mass of ρ meson, m∗ρ and the electron chemical potential, µe are plotted

against the increasing baryonic density in neutron star matter.

mass of ρ meson in the absence of c∗2 at ρB = ρ0, 2ρ0 and 4ρ0 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 4ρ0. 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 ρ meson decrease with increase in the density of

the medium. At ρB = 0, ρ0 and 4ρ0, it is found to have a value of 776.8, 653.9 and 433.1

MeV, respectively. The drop in the mass of ρ mesons with increasing density is consistent

with earlier QCD sum rule calculations [70, 71, 85]. 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 ρ 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

15

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 ρ meson is appreciable which could make them a better contestant to exist

in place of electrons. Thus, when the in-medium mass of ρ mesons becomes comparable to

the effective electron chemical potential these electrons may get replaced by the negatively

charged ρ mesons. As can be seen from fig. 5, at baryonic density ρB = 7.3ρ0, the condition

ωρ = µe− is satisfied and hence possibility of ρ meson condensation arise at this density.

Furthermore, the possibility of ρ meson to be found at the core of a neutron star is also

supported by few other works which includes [46, 47]. In ref. [47], a more simpler approach

is used to calculate the in-medium masses of ρ mesons by considering the linear dependence

on σ field.

The presence of strong magnetic field at the interior core of the neutron star may also

effect the threshold density at which the ρ meson condensation takes place [20, 87, 89–98].

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 1014-1015G at the surface [86]

which becomes even more intense at the interior core. According to the virial theorem, it

is approximated to be about 1018-1019G in the interior core of the star [87, 88]. 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 [20]. 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 [96–98]. In case of magnetized nuclear

matter it was observed that the drop in the medium modified mass of ρ meson is a bit

low as compared to the non-magnetized neutron matter [42] which may slightly diminish

the possibility of ρ meson condensation in neutron star also. Furthermore, calculations of

refs. [20, 96, 99] 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 [100–102] 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

16

EoS and increase in the mass of neutron star. Similar kind of consequences may arise due to

ρ 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. Calcu-

lations of refs. [56, 103] shows that as a consequence of increased number of neutrinos the

electron density should also increase and therefore demanding the charge neutrality condi-

tion 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 ρ mesons.

In brief, we have investigated the possibility of ρ 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 ρ mesons. The effect of

scattering between the nucleons is incorporated through the Landau damping term, Rsc.

An appreciable decrease in the mass of ρ 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 ρ meson condensation is satisfied is found to be 7.3ρ0.

17

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