Antikaon condensation in strongly magnetized dense matter Debraj Kundu1Vivek Baruah Thapa1 2yand Monika Sinha1z 1Indian Institute of Technology Jodhpur Jodhpur 342037 India

2025-04-30 0 0 861.38KB 16 页 10玖币

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(Anti)kaon condensation in strongly magnetized dense matter

Debraj Kundu,1, ∗Vivek Baruah Thapa,1, 2, †and Monika Sinha1, ‡

1Indian Institute of Technology Jodhpur, Jodhpur 342037, India

2National Institute of Physics and Nuclear Engineering (IFIN-HH), RO-077125, Bucharest, Romania

(Dated: March 23, 2023)

Recent observations of several massive pulsars, with masses near and above 2M, point towards

the existence of matter at very high densities, compared to normal matter that we are familiar with

in our terrestrial world. This leads to the possibility of appearance of exotic degrees of freedom

other than nucleons inside the core of the neutrons stars (NS). Another signiﬁcant property of NSs

is the presence of high surface magnetic ﬁeld, with highest range of the order of ∼1016 G. We

study the properties of highly dense matter with the possibility of appearance of heavier strange

and non-strange baryons, and kaons in presence of strong magnetic ﬁeld. We ﬁnd that the presence

of a strong magnetic ﬁeld stiﬀens the matter at high density, delaying the kaon appearance and,

hence, increasing the maximum attainable mass of NS family.

I. INTRODUCTION

The state of matter inside neutron stars (NSs) is an

unsolved mystery of modern science. Born from the

remnants of a supernova explosion, a neutron star ex-

hibits a range of densities inside its structure, the density

at the core possibly being several times that of nuclear

saturation density [1–6]. Many recent astrophysical ob-

servations indicate that the possible lower limit of NS

maximum mass is above 2M, viz. PSR J1614-2230

(M= 1.97 ±0.04M) [7,8], MSP J0740+6620 (M=

2.14+0.20

−0.18Mwith 95% credibility) [9], PSR J0348+0432

(M= 2.01 ±0.04M) [10] and PSR J0952-0607 (M=

2.35 ±0.17M) [11]. These ﬁndings strengthen the idea

of the existence of highly dense matter in the core of

NSs. Thus, investigation of the matter inside NSs pro-

vides us with a unique opportunity to study matter under

extreme conditions that cannot be attained in any of the

terrestrial laboratories.

The gravitational pull inside the NS is balanced mostly

by the Fermi degeneracy pressure of neutrons, along

with some amounts of protons and leptons (electrons and

muons). In addition, the extreme matter density inside

NSs can lead to energetically favorable conditions for ex-

otic particles to appear. Hyperons are one such species

of particles that might appear inside the NS if the baryon

chemical potential becomes high enough. The possibility

of their occurrence was ﬁrst suggested in [12]. Another

class of particle species that might make its appearance

is ∆-resonances. Its appearance pushes the threshold for

the onset of hyperons to higher densities [13–15].

Similarly, another possible addition to the degrees of

freedom can come from the appearance of meson con-

densates [16] if the lepton chemical potential becomes

high enough. However, for the lowest massive meson π

(pion), the repulsive s-wave pion-nucleon scattering po-

∗kundu.1@iitj.ac.in

†thapa.1@iitj.ac.in

‡ms@iitj.ac.in

tential increases the eﬀective ground state mass of π-

meson [17,18]. However, a few works [19,20] have ar-

gued the possibility of pion condensation due to the fact

that p−wave scattering potential is attractive in nature.

On the other hand, (anti)kaon ( ¯

K≡K−,¯

K0) mesons

may appear in the form of s-wave Bose condensates due

to the attractive nature of (anti)kaon optical potential.

K+and K0kaons have repulsive optical potentials and

their presence in nuclear matter increases their eﬀective

masses. Thus, the occurrence of K+and K0in NS mat-

ter is discouraged. The threshold density for the onset of

Kis highly sensitive to its optical potential and whether

or not hyperons are present [21]. The presence of ¯

Kin

NS matter has been extensively studied in past literature

[22–28].

As already mentioned, the veriﬁcation of the theoreti-

cal models of highly dense matter can only be done with

the observations from NSs. The astrophysical observ-

able properties of NSs should be studied to constrain the

dense matter models. For example, one should note that

the appearance of hyperons tends to soften the equation

of state (EoS) and, consequently, results in lowering of

the maximum mass of NSs. Studies [13,14] have indi-

cated that the inclusion of ∆-resonances does not aﬀect

the implied maximum mass signiﬁcantly, but it reduces

the radius and thereby increases the compactness of the

stars. The appearance of ¯

K, similar to hyperons, softens

the EoS and, thus, lowers the maximum mass of NSs.

The theoretical model of dense matter can be ob-

tained from terrestrial laboratory data by extrapolating

the nuclear matter properties at nuclear saturation den-

sity and it can be further constrained from the recent

mass-radius measurements of NSs, viz. the NICER mis-

sion observations give the mass-radius measurements of

PSR J0030+0451 as 1.44+0.15

−0.14 M,13.02+1.24

1.06 km [29]

and 1.34+15

−16 M,12.71+1.14

−1.19 km [30], respectively. An-

other important constraint on highly dense matter inside

NSs comes from the gravitational wave detection obser-

vations which provide us with the estimate of maximum

limit of tidal deformability of the star made of highly

dense matter.

arXiv:2210.14565v2 [astro-ph.HE] 22 Mar 2023

Another salient feature of NSs is their strong surface

magnetic ﬁeld in the range 108−1016 G. A particular

class of NSs which have ultra strong surface magnetic

ﬁeld of 1014 −1016 G [31,32] are called magnetars. The

matter inside NSs also experiences Pauli paramagnetism

and Landau diamagnetism. Pauli paramagnetism is ap-

plicable for both charged and uncharged particles while

the Landau diamagnetism aﬀects only charged particles,

being particularly strong for light particles like leptons.

In our present work, we ﬁrst note down the constraint

on model parametrizations from the astrophysical obser-

vations of mass-radius measurements of many pulsars as

well as tidal deformability from GW observations. Then,

with the constrained model, we study the properties of

dense matter and NSs with strong magnetic ﬁeld.

Previous studies have been conducted on NS mat-

ter containing hyperons and ∆-resonances without

(anti)kaon condensates [13,14] and with (anti)kaon

condensates [33]. The study without (anti)kaons was

also extended to accommodate strong magnetic ﬁelds

in [34]. Work has also been done on NS matter con-

taining (anti)kaon condensates, but no hyperons or ∆-

resonances, under the eﬀect of strong magnetic ﬁelds

[35,36]. In this paper, we present the novel study

of matter inside NS having a strong magnetic ﬁeld

(magnetar) containing hyperons, (anti)kaon condensates

and ∆-resonances (as exotic degrees of freedom) in β-

equilibrium. We have used the relativistic mean ﬁeld

(RMF) model to describe the interactions between the

particles. As the soft matter with hyperons attains

the lower limit of maximum mass with density depen-

dent baryon-meson interactions, we use density depen-

dent RMF (DD-RMF) model to study the eﬀect of strong

magnetic ﬁeld on NS composed of matter with (anti)kaon

condensates along with ∆-resonances and hyperons.

In the next section (sec. II) we discuss the matter

model under the eﬀect of magnetic ﬁeld. Then, in sec.

III, we discuss the results with model parameters com-

patible with the the astrophysical observations. Section

IV presents a brief summary of our work.

II. FORMALISM

A. DD-RMF Model

Here, we lay down the formulation for the DD-

RMF model. We consider the NS matter to be com-

posed of nucleons (n, p), leptons (e−, µ−), hyperons

(Λ,Ξ,Σ), (anti)kaons ( ¯

K≡K−,¯

K0) and ∆-resonances

(∆−,∆0,∆+,∆++). In this model, the strong interac-

tions between the nucleons, hyperons, (anti)kaons and

∆-resonances are mediated by the following meson ﬁelds:

isoscalar-scalar σ, isoscalar-vector ωµand isovector-

vector ρµ. We have also considered the strange isoscalar-

vector meson ﬁeld φµas a mediator of hyperon-hyperon

and (anti)kaon-hyperon interactions. Throughout our

work, we have used the natural units, ~=c=G= 1.

The total Lagrangian density is given by: [13,22,23,

35,37–40]

L=Lm+Lem (1)

where Lmand Lem are the matter and the electro-

magnetic ﬁeld contributions, respectively.

For the matter part of the Lagrangian density, We have

Lm=X

ψb(iγµDµ

(b)−m∗

b)ψb+X

ψdν (iγµDµ

(d)−m∗

d)ψν

ψl(iγµDµ

(l)−ml)ψl+D(¯

K)∗

µ¯

KDµ

(¯

K)K−m∗2

K¯

2(∂µσ∂µσ−m2

σσ2)−1

4ωµν ωµν +1

2m2

ωωµωµ

−1

4ρµν ·ρµν +1

2m2

ρρµ·ρµ−1

4φµν φµν +1

2m2

φφµφµ

(2)

where ψb,ψν

dand ψlrepresent the ﬁelds of octet baryons,

∆-resonances and leptons, respectively. And ¯

Krepre-

sents the (anti)kaon condensate ﬁelds. ∆−resonances,

being spin-3/2 particles, are governed by the Schwinger-

Rarita ﬁeld equations [41]. mb,md,mland mKstand for

the masses of octet baryons, ∆-resonances, leptons and

(anti)kaons, respectively. σ,ωµ,ρµand φµare the me-

son ﬁelds with masses mσ,mω,mρand mφ, respectively.

The covariant derivatives in Eqn.(2) are given by:

Dµ(j)=∂µ+igωj ωµ+igρj τj.ρµ+igφj φµ+ieQAµ

Dµ(l)=∂µ+ieQAµ

(3)

with jrepresenting the octet baryons (b),∆-resonances

(d)and (anti)kaons ( ¯

K), and lrepresenting leptons. τj

is the isospin operator for the ρµmeson ﬁelds. eQ is

the charge of the particle with ebeing unit positive

charge. We choose the direction of magnetic ﬁeld as

the z-axis with the ﬁeld four-vector potential as Aµ≡

(0,−yB, 0,0), with Bbeing the magnetic ﬁeld magni-

tude. Under the eﬀect of this magnetic ﬁeld, the motion

of the charged particles is Landau quantized in the plane

perpendicular to the direction of ﬁeld, the momentum in

the perpendicular direction being p⊥= 2νe|Q|B, where

νis the Landau level. Here, the baryon-meson coupling

parameters are considered density dependent.

The gauge mesonic contributions in Eqn.(2) contain

the ﬁeld strength tensors:

ωµν =∂νωµ−∂µων

ρµν =∂νρµ−∂µρν

φµν =∂νφµ−∂µφν

(4)

The eﬀective masses of the baryons and (anti)kaons

used in Eqn.(2) are given by:

m∗

b=mb−gσbσ

m∗

d=md−gσdσ

m∗

K=mK−gσK σ

(5)

gσj in Eqn.(5) and gωj ,gρj ,gφj in Eqn.(3) are density

dependent coupling parameters.

The electro-magnetic ﬁeld part of the Lagrangian den-

sity in Eqn.(1) is given by:

Lem =−1

16πFµν Fµν (6)

where Fµν is the electro-magnetic ﬁeld tensor.

In the relativistic mean ﬁeld approximation, the me-

son ﬁelds acquire the following ground state expectation

values:

σ=X

gσbns

b+X

gσdns

d+X

gσK ns

ω0=X

gωbnb+X

gωdnd−X

gωK n¯

φ0=X

gφbnb−X

gφK n¯

ρ03 =X

gρbτb3nb+X

gρdτd3nd

gρK τ¯

K3n¯

(7)

where the scalar density ns

j=h¯

ψψiand the vector

(baryon) number density nj=h¯

ψγ0ψi.

The scalar density, baryon number density and the ki-

netic energy density of the uncharged baryons at the tem-

perature T= 0 limit are given by:

u=2Ju+ 1

2π2m∗

upFuEFu−m∗2

uln pFu+EFu

m∗

u

nu=(2Ju+ 1) p3

6π2

εu=2Ju+ 1

2π2pFuE3

Fu−m∗2

8pFuEFu

+m∗2

uln pFu+EFu

m∗

u

(8)

where J,pFand EFrepresent the spin, Fermi momen-

tum and Fermi energy, respectively. Here the uncharged

baryons are denoted by subscript u.

The scalar density, baryon number density and the ki-

netic energy density of the charged baryons at the tem-

perature T= 0 limit are given by:

•For spin- 1/2 baryons:

c=e|Q|B

2π2m∗

νmax

ν=0

(2 −δν,0) ln pc(ν) + EFc

pm∗2

c+ 2νe|Q|B!

(9)

nc=e|Q|B

2π2

νmax

ν=0

(2 −δν,0)pc(ν)(10)

εc=e|Q|B

4π2

νmax

ν=0

(2 −δν,0)"pc(ν)EFc+m∗2

c+ 2νe|Q|B

ln pc(ν) + EFc

pm∗2

c+ 2νe|Q|B!# (11)

•For spin- 3/2 baryons:

c=e|Q|B

2π2m∗

νmax

ν=0

(4 −δν,1−2δν,0)

ln pc(ν) + EFc

pm∗2

c+ 2νe|Q|B!(12)

nc=e|Q|B

2π2

νmax

ν=0

(4 −δν,1−2δν,0)pc(ν)(13)

εc=e|Q|B

4π2

νmax

ν=0

(4 −δν,1−2δν,0)pc(ν)EFc+

m∗2

c+ 2νe|Q|Bln pc(ν) + EFc

pm∗2

c+ 2νe|Q|B!(14)

where p(ν) = √pF−2νeB. The charged baryons are

denoted by subscript c. The maximum value of νis

given by:

νmax =Int p2

2e|Q|B.(15)

In case of Dirac particles, the degeneracy of the lowest

Landau level is unity and 2 for all other levels [42]. While

for the Schwinger-Rarita particles, the same is 2 for the

lowest, 3 in the second and 4 for the other remaining

Landau levels [43].

The number density of (anti)kaon ( ¯

K) condensates is

given by [35]:

nK−= 2qm∗2

K+|qK−|B¯

KK (16)

n¯

K0= 2m∗

K¯

KK (17)

where |qK−|is the charge of K−.

In the case of leptons, the number density and kinetic

energy density are given by:

nl=e|Q|B

2π2

νmax

ν=0

(2 −δν,0)pl(ν)(18)

εl=e|Q|B

4π2

νmax

ν=0

(2 −δν,0)pl(ν)EFl+m2

l+ 2νe|Q|B

ln pl(ν) + EFc

pm2

l+ 2νe|Q|B!(19)

The leptons are denoted by subscript l. Throughout

Eqns.(8)-(19), we have

pF=qE2

F−m∗2(20)

The chemical potentials of octet baryons (b)with spin-

1/2 and ∆-resonances (d)with spin- 3/2 are given by:

µb=qp2

Fb+m∗2

b+gωbω0+gρbτb3ρ03+

gφbφ0+ Σr(21)

µd=qp2

Fd+m∗2

d+gωdω0+gρdτd3ρ03 + Σr(22)

Σrrepresents the self-energy re-arrangement term and is

given by :

Σr=X

b∂gωb

∂n ω0nb−∂gσb

∂n σns

b+∂gρb

∂n ρ03τb3nb

+∂gφb

∂n φ0nb+X

d∂gωd

∂n ω0nd−∂gσd

∂n σns

+∂gρd

∂n ρ03τd3nd(23)

where n=Pbnb+Pdndis the total vector (baryon)

number density. Σris required in case of density de-

pendent coupling models in order to maintain thermody-

namic consistency [26]. The chemical potential of s-wave

condensates of (anti)kaons is given by:

µK−=qm∗2

L+|qK−|B−gωK ω0−1

2gρK ρ03 +gφK φ0

(24)

µ¯

K0=m∗

K−gωK ω0+1

2gρK ρ03 +gφK φ0(25)

Threshold condition for the onset of the ith baryon is

given by:

µi=µn−qiµe(26)

with µe=µn−µpbeing the electron chemical potential.

qirefers to the charge of the ith baryon.

The threshold condition for the appearance of

(anti)kaons is given by:

µK−=µe=µn−µp(27)

µ¯

K0= 0 (28)

where µK−and µ¯

K0are the chemical potentials of K−

and ¯

K0, respectively. Muons (µ−)appear when the

chemical potential of electrons reaches the rest mass of

muons [µe=mµ].

The matter inside NS is electrically neutral, with the

charge neutrality condition gven by:

qbnb+X

qdnd−ne−nµ−nK−= 0 (29)

The total energy density of the nuclear matter is given

by:

ε=X

εb+X

εd+X

εl+1

2m2

σσ2+1

2m2

ωω2

2m2

ρρ2

03 +1

2mφφ2

0+ε¯

K(30)

where ε¯

Kis the kaonic contribution to the total energy

density and is given by:

ε¯

K=m∗

K(nK−+n¯

K0)(31)

From the Gibbs-Duhem relation, we get the matter pres-

sure as:

P=X

µbnb+X

µdnd+X

µlnl−ε(32)

(Anti)kaons, being s-wave condensates, do not contribute

explicitly to the matter pressure. Σrcontributes explic-

itly only to the matter pressure.

B. Star structure

The solution for the Einstein’s equations for general

relativity for a static and spherically symmetric star gives

us the Tolman-Oppenheimer-Volkoﬀ (TOV) equations.

These equations are then numerically solved for a par-

ticular EoS to obtain the mass-radius relationship of the

NS. The TOV equations are as follows [1]:

dP (r)

dr =−[P(r) + ε(r)][M(r)+4πr3P(r)]

r[r−2M(r)]

dM(r)

dr = 4πr2ε(r)

(33)

where M(r)is the gravitational mass included within

radius r. The TOV equations are solved with the bound-

ary conditions M(0) = 0 and P(R)=0, where Ris

the radius of the NS. The presence of strong magnetic

ﬁeld, however, distorts the spherical symmetry of the star

structure. The most general coupled set of equations de-

termining the spherically symmetric star structure, as

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(Anti)kaoncondensationinstronglymagnetizeddensematterDebrajKundu,1,VivekBaruahThapa,1,2,yandMonikaSinha1,z1IndianInstituteofTechnologyJodhpur,Jodhpur342037,India2NationalInstituteofPhysicsandNuclearEngineering(IFIN-HH),RO-077125,Bucharest,Romania(Dated:March23,2023)Recentobservationsofseveralmassiv...

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Antikaon condensation in strongly magnetized dense matter Debraj Kundu1Vivek Baruah Thapa1 2yand Monika Sinha1z 1Indian Institute of Technology Jodhpur Jodhpur 342037 India

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