Negative kaons in dense baryonic matter - Niels Bohr Institutet

PHYSICAL REVIEW C 68, 015803 共2003兲

Negative kaons in dense baryonic matter Evgeni E. Kolomeitsev* The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark and ECT*, Villa Tambosi, I-38050 Villazzano (TN), INFN G.C. Trento, Italy

Dmitri N. Voskresensky Moscow Engineering Physical Institute, Kashirskoe shosse 31, RU-115409 Moscow, Russia and GSI, Planck Str. 1, D-64291 Darmstadt, Germany 共Received 26 November 2002; published 29 July 2003兲 The kaon polarization operator in dense baryonic matter of arbitrary isotopic composition is calculated including s- and p-wave kaon-baryon interactions. The regular part of the polarization operator is extracted from the realistic kaon-nucleon interaction based on the chiral and 1/N c expansion. Contributions of ⌳(1116), ⌺(1195), ⌺ * (1385) resonances are taken explicitly into account in the pole and regular terms with the inclusion of mean-field potentials. The baryon-baryon correlations are incorporated and fluctuation contributions are estimated. Results are applied for K ⫺ in neutron star matter. Within our model a second-order phase transition to the s-wave K ⫺ condensate state occurs at ␳ c ⲏ4 ␳ 0 with baryon-baryon correlations included. We show that a second-order phase transition to the p-wave K ⫺ condensate state may occur at densities ␳ c ⬃(3 –5) ␳ 0 dependent on the parameter choice. We demonstrate that a first-order phase transition to a protonenriched 共approximately isospin-symmetric兲 nucleon matter with a p-wave K ⫺ condensate can occur at smaller densities, ␳ ⱗ2 ␳ 0 . The transition is accompanied by the suppression of hyperon concentrations. DOI: 10.1103/PhysRevC.68.015803

PACS number共s兲: 26.60.⫹c, 13.75.Jz, 21.65.⫹f, 97.60.Jd

I. INTRODUCTION

Strangeness modes in compressed hadronic matter have been the focus of interests during the last decade. Strangeness is considered to be a good probe of the dynamics of heavy-ion collisions. A vast amount of data is already accumulated for different collision energy regimes at GANIL, GSI, CERN, and BNL facilities 关1兴, and new data are in advent 关2兴. Understanding strangeness production requires a systematic study of the evolution of virtual strangeness modes in the quark-gluon plasma and in the soup of virtual hadrons. At the breakup stage, these virtual modes are redistributed between real strange particles which can be observed in experiment. Another interesting topic is the strangeness content of neutron stars. With an increasing density, strangeness shows up in the filling of the hyperon Fermi seas and/or in the creation of a kaon condensate. A better understanding of the topics mentioned above needs more in-depth knowledge of the kaon-baryon interaction in dense baryonic matter. The possibility of kaon condensation in dense nuclear matter was risen in Refs. 关3,4兴. Kaon condensation in neutron star interiors may have interesting observational consequences: 共i兲 The softening of the equation of state 共EoS兲 due to the appearance of a kaon condensate phase lowers the maximum neutron star mass and can induce the transformation of neutron stars into low-mass black holes 关5兴; 共ii兲 Kaon condensation is predicted to be accompanied with the change of the nucleon isospin composition from a neutron-enriched star (NⰇZ) to a ‘‘nuclear’’ star (N⬃Z) or even to a ‘‘proton’’

*Email address: [email protected] 0556-2813/2003/68共1兲/015803共31兲/$20.00

star (NⱗZ), where the electric charge of protons is compensated by the charge of the condensed kaons 关6兴; 共iii兲 The enhanced neutrino-emission processes occurring on protons in proton-enriched matter and on the kaon condensate field lead to substantially faster cooling of the star 关7兴. The K ⫺ condensate is created in neutron stars through weak multiparticle processes e ⫺ ⫹X→K ⫺ ⫹X ⬘ ,

n⫹X→p⫹K ⫺ ⫹X ⬘ ,

共1兲

in which electrons are replaced by K ⫺ mesons and neutrons are converted into protons and K ⫺ 关8兴. The symbolic writing 共1兲 assumes that surrounding baryons (X,X ⬘ ) assure the momentum conservation, and thus, the critical point is determined by the energy balance only. These processes become possible if the electron chemical potential ␮ e exceeds the minimal K ⫺ energy,

␻ min共 kជ m 兲 ⫽min兵 ␻ min共 kជ 兲其 , kជ

where ␻ min(兩kជ兩) is the K ⫺ energy at the lowest quasiparticle branch of the spectrum of K ⫺ excitations in neutron star matter. In Refs. 关4,8,9兴 it has been postulated that there is only one kaon branch, for which ␻ (kជ ⫽0)→m K (m K is the kaon mass兲 as the baryon density ␳ →0, and the minimum is achieved at kជ m ⫽0. The critical point of the s-wave K ⫺ condensation in a second-order phase transition is determined by the condition ␻ (kជ ⫽0)⫽ ␮ e . A first-order phase transition to the kaon condensate state was investigated in Ref. 关10兴 applying the Maxwell construction principle and in Ref. 关11兴

68 015803-1

©2003 The American Physical Society


EVGENI E. KOLOMEITSEV AND DMITRI N. VOSKRESENSKY

according to the Gibbs criteria. The p-wave kaon-nucleon interaction, which changes the kaon spectrum at finite momenta was disregarded in those works. The p-wave ⌳(1116) –nucleon-hole and ⌺(1195)-nucleon hole contributions to the kaon polarization operator were introduced in Ref. 关12兴 in the framework of the chiral SU共3兲 symmetry. However, the authors of Ref. 关12兴 focused on the discussion of the s-wave kaon condensation only, and considered the polarization operator at zero momentum. In Ref. 关6兴, we worked out a possibility for the p-wave kaon condensation. The kaon polarization operator was constructed with the inclusion of the ⌳(1116) –nucleon-hole and ⌺(1195) –nucleon-hole contributions in the p-wave part of the kaon polarization operator and the kaon-pion and kaonkaon interactions. The multibranch spectrum of K ⫺ mesons was found and the possibility of the p-wave kaon condensation related to the population of hyperon–nucleon-hole modes was demonstrated. Possibilities of the first-order phase transitions in neutron star interiors to a protonenriched matter with a p-wave K ⫺ condensate and to a ¯ 0 condensate were neutron-enriched matter with a p-wave K investigated. The case of a large hyperon admixture in the neutron star core was considered in Ref. 关13兴. In such a medium both the K ⫺ and the K ⫹ spectra possess extra branches associated with the particle-hole excitations ⌶⫺⌳ ⫺1 , ⌶⫺⌺ ⫺1 for K ⫺ and N⫺⌳ ⫺1 , N⫺⌳ ⫺1 for K ⫹ 关hole states are labeled here by (⫺1)]. At large kaon momenta, the branches of K ⫹ and K ⫺ spectra merge, signaling an instability with respect to K ⫹ K ⫺ pair creation. The analysis in Refs. 关6,13兴 relied heavily on the pole approximation for the particle-hole diagrams. Final width effects were thereby disregarded too. The presence or absence of quasiparticle branches in the K ⫺ spectrum depends on the kaon energy and on the strength of the s- and p-wave attraction 关14兴. The short-range baryon-baryon correlations, which, as a rule, suppress the attraction, were not included in Refs. 关6,13兴. The role of the correlations for the p wave has been investigated in Ref. 关15兴. However, the relative strength of the s- and p-wave attraction remained model dependent because no systematic investigation of the kaon-nucleon interaction including s and p waves was available at that time. Recently, the kaon-nucleon scattering has been studied in the framework of a relativistic chiral SU共3兲 Lagrangian imposing constraints from the K ⫹ -nucleon and pion-nucleon sectors 关16兴. The covariant coupled-channel Bethe-Salpeter equation was solved with the interaction kernel truncated to the third chiral order including the terms which are leading in the large N c limit of QCD. All SU共3兲 symmetry-breaking effects are well under control by the combined chiral and large N c expansions. This analysis gives an opportunity to extend the results of Refs. 关6,15兴 taking into account off-pole 共regular background兲 contributions to the kaon self-energy. The accurate fit to experimental data achieved in Ref. 关16兴 fixes the values of the kaon-nucleon-hyperon coupling constants. Particularly, the ⌺ * (1385)-pole contribution to the kaon-nucleon scattering was proved to be sizable, and was not included in Refs. 关6,13兴.

The discussion of the s-wave and, especially, of the p-wave kaon-baryon interactions in nuclear matter is important for the kaon production in heavy-ion collisions 关17兴. The momentum dependence of kaon yields is experimentally measured 关18兴. Also, the multibranch K ⫺ spectrum can be tested via ¯␯ scattering on atomic nuclei 关15兴. Peculiarities of the K ⫺ -nucleon interaction near the mass shell are of great importance for the physics of K ⫺ atoms 关19,20兴. This paper is structured as follows. In Sec. II, we describe baryon matter within a relativistic mean-field model. In Sec. III, we introduce the kaon-nucleon interaction in vacuum following the partial-wave analysis of Ref. 关16兴. Then, we separate the pole contributions of ⌳(1116), ⌺(1195), and ⌺ * (1385) hyperons in p waves. Sections IV through VII are devoted to the construction of the kaon polarization operator. In Sec. IV, we build the polarization operator in the gas approximation, but including the mean-field potentials that act on baryons. Besides the ⌳(1116), ⌺(1195), and ⌺ * (1385)-nucleon-hole contributions the polarization operator contains a regular attractive part that is weakly dependent on the kaon energy. In Sec. V, we separate the s- and p-wave parts of the kaon polarization operator. The occupation of hyperon Fermi seas is incorporated in Sec. VI. Repulsive baryon-baryon correlations are evaluated and included in the hyperon-nucleon particle-hole channels and in the regular part of the polarization operator in Sec. VII. In each of these sections, we illustrate the strength of new terms included into the polarization operator and suggest effective parametrizations. We relegate the discussion of contributions from kaon fluctuations 共baryon self-energies, multiloop corrections兲 to Appendix C. We argue that these effects do not modify substantially the kaon polarization operator at zero temperature in the region of small kaon energies and momenta, which is of our interest here. In Sec. VIII, we analyze different possibilities for the second- and first-order phase transitions to the s- and p-wave K ⫺ condensates. Particularly, we argue for the p-wave K ⫺ condensation at ␳ ⱗ2 ␳ 0 ( ␳ 0 ⯝0.17 fm⫺3 is the density of nuclear saturation兲 arising via a first-order phase transition. In this phase transition, all the hyperon Fermi seas are melted and neutron star matter becomes proton enriched with an approximately symmetric-isospin composition, N ⯝Z. In Appendix A, we discuss how the results depend on the specifics of the EoS. Some technical information on the Green’s functions is provided in Appendix B. Throughout this paper, we use units of ប⫽c⫽1. Although a number of new effects are incorporated in our scheme some other effects not included here might be also important. The present calculations suffer from many uncertainties, most of which are due to the lack of experimental information on the coupling constants, the absence of unambiguous way for going off-mass shell and the lack of study of more complicated in-medium fluctuation effects, which we just roughly estimated in the present work. Among them, there are the pion softening effects 关21兴, which can significantly affect the results at finite temperatures, and the contribution of nonlinear meson-meson interactions. The latter may partially suppress the condensate contribution to the energy at densities above the critical one.

015803-2


NEGATIVE KAONS IN DENSE BARYONIC MATTER II. BARYON INTERACTION IN RELATIVISTIC MEAN-FIELD MODEL

E l共 ␮ l 兲 ⫽

A. Lagrangian of the model

We consider a dense system consisting of baryons and leptons, which we describe by a Lagrangian density containing a baryon and a lepton contribution L⫽LB ⫹Ll . It is convenient to describe the baryon matter at densities relevant for neutron star interiors by the mean-field solution of the Lagrangian 关22兴, LB ⫽

兺B ¯B 共 i ⳵” ⫺g ␻ B ␻” ⫺g ␳ B ␳ជ” • ជt B ⫺m B ⫹g ␴ B ␴ 兲 B ⫹

⳵ ␮ ␴⳵ ␮ ␴ m ␴2 ␴ 2 m N b 共 g ␴ N ␴ 兲 3 c 共 g ␴ N ␴ 兲 4 ⫺ ⫺ ⫺ 2 2 3 4

⫺

␻ ␮ ␯ ␻ ␮ ␯ m ␻2 ␻ ␮ ␻ ␮ ␳ជ ␮ ␯ ␳ជ ␮ ␯ m 2␳ ␳ជ ␮ ␳ជ ␮ ⫹ ⫺ ⫹ , 4 2 4 2

共2兲

where all states of the baryon (J P ⫽ 21 ⫹ ) octet B ⫽(n, p,⌳,⌺ ⫾,0,⌶ ⫺0 ) interact via exchanges of scalar, vector, and isovector mesons ␴ , ␻ ␮ , ␳ជ ␮ . Heavier baryons do not appear at baryon densities under consideration ( ␳ ⭐6 ␳ 0 ) and, therefore, are not included. In Eq. 共2兲, ជt B denotes the isospin operator acting on the baryon B. The fieldstrength tensors for the vector mesons are given by ␻ ␮ ␯ ⫽ ⳵ ␮ ␻ ␯ ⫺ ⳵ ␯ ␻ ␮ for the ␻ mesons and ␳ជ ␮ ␯ ⫽ ⳵ ␮ ␳ជ ␯ ⫺ ⳵ ␯ ␳ជ ␮ for the ␳ mesons. Equations of motion for the baryons follow from Eq. 共2兲, and give

⑀ B 共 pជ 兲 ⫽ 冑m B* 2 ⫹pជ 2 ,

E B 共 pជ 兲 ⫽ ⑀ B 共 pជ 兲 ⫹V B ,

共3兲

where m B* ⫽(m B ⫺g ␴ B ␴ ) is the baryon effective mass and V B ⫽g ␻ B ␻ 0 ⫹g ␳ B ␳ 03t 3B , with ␴ , ␻ 0 , ␳ 03 being the meanfield solutions of the equations of motion for the meson fields. The composition of the cold neutron star matter at baryon density ␳ is determined by the ␤ -equilibrium conditions E B (p FB )⫽ ␮ n ⫺q B ␮ e . Here, p FB is the Fermi momentum, q B is the electric charge of a given baryon species B, and ␮ n , ␮ e are the chemical potentials of neutrons and electrons determined by the total baryon density ␳ and the electroneutrality condition. The lepton Lagrangian density is the sum of the electron and the ␮ ⫺ meson contributions Ll ⫽Le ⫹L␮ . In ␤ equilibrium ␮ ␮ ⫽ ␮ e , and muons appear in the system only when ␮ e exceeds the muon mass m ␮ . The energy density of the system is given by E tot⫽E mes⫹

兺B

E Bkin共 p FB 兲 ⫹

兺

l⫽e ⫺ , ␮ ⫺

E l共 ␮ e 兲 ,

2 ⫹ 12 m ␳2 ␳ 03 ,

冕 ␲ 1

2

p FB

0

␲

2

␪ 共 ␮ l ⫺m l 兲

冕

␮l

ml

d ⑀⑀ 2 冑⑀ 2 ⫺m 2l ,

共4兲

where ␪ (x) is the step function. B. Coupling constants

The coupling constants in Eq. 共2兲 are adjusted such as to reproduce properties of the equilibrium nuclear matter: saturation density ␳ 0 , binding energy E bind , compressibility modulus K, and effective nucleon mass m N* ( ␳ 0 ). In the following, we use the values ␳ 0 ⫽0.17 fm⫺3 and E bind ⫽⫺16 MeV. For the nuclear compressibility modulus, we take the value K⫽210 MeV, which follows from the variational calculation 关23兴. Following Ref. 关22兴, we adopt the symmetry energy a sym⫽36.8 MeV which lies within the interval allowed by microscopic calculations 关24兴. We take the effective nucleon mass m N* ( ␳ 0 )⫽0.85m N , cf. argumentation in Ref. 关21兴. For discussion of uncertainties in the choices of parameters, we refer the interested reader to Ref. 关25兴. The corresponding coupling constants of Lagrangian 共2兲 are g ␻2 N m N2 m ␻2

⫽54.60,

g ␴2 N m N2 m ␴2

b⫽0.02028,

⫽164.5,

g ␳2 N m N2 m 2␳

⫽121.7,

c⫽0.04716.

共5兲

In order to verify the sensitivity of the results to details of the EoS, we explore another set of the parameters in Appendix A, which is fitted to reproduce the microscopic calculations of the Urbana-Argonne group 关26兴. When including hyperons, one also has to specify the hyperon couplings to the meson fields, x M H ⫽g M H /g M N with M ⫽( ␴ , ␻ , ␳ ) and H⫽(⌳,⌺,⌶). Couplings to the vector mesons are estimated from the quark counting as x ␻ ⌳(⌺) ⫽x ␳ ⌺ ⫽ 32 and x ␻ ⌶ ⫽x ␳ ⌶ ⫽ 31 . Alternatively, relying on the SU共3兲 symmetry 关27兴, one would find x ␳ ⌺ ⫽x ␳ ⌶ ⫽1 for the ␳ meson couplings. The scalar meson couplings can be constrained by making use of hyperon binding energies in infinite nuclear matter at saturation 关28兴, extrapolated from hyH , pernucleus data. For the given hyperon binding energy E bind we have the following relation between the scalar and vector couplings: H ⫽ 共 g ␻2 N ␳ 0 /m ␻2 兲 x ␻ H ⫺ 共 m N ⫺m N* 兲 x ␴ H E bind

⫽ 共 80.73x ␻ H ⫺140.70x ␴ H 兲 MeV.

E mes⫽ 31 bm N 共 g ␴ N ␴ 兲 3 ⫹ 41 c 共 g ␴ N ␴ 兲 4 ⫹ 21 m ␴2 ␴ 2 ⫹ 21 m ␻2 ␻ 20

E Bkin共 p FB 兲 ⫽

1

dpp 2 ⑀ B 共 p 兲 ,

There is convincing evidence from the systematic study of ⌳ ⯝⫺30 MeV 关29兴. For hypernuclei that for ⌳ particles E bind ⌺ hyperons in nuclei, on the other hand, the data are still ⌺ controversial, giving the broadband, ⫺10 MeV⬍E bind ⬍30 MeV , from a slight attraction to a strong repulsion ⌶ 关30兴. Following Ref. 关31兴, we adopt for ⌶ the value E bind ⯝⫺18 MeV advocated also in Ref. 关27兴. To cover different possibilities we consider four cases. 015803-3

PHYSICAL REVIEW C 68, 015803 共2003兲 6

1.0 0.8

n

(a)

n

(b)

n

(c)

(d)

n

60

0.6

0.2 0.0

I=0

I=1

40 Σ

p

Ξ

--

Λ

--

Ξ

--

Λ

Ξ

p

p

Ξ

--

Λ

p

4

π

0.4

FS(s) [1/m ]

concentration


--

Λ

0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 6

ρ /ρ 0

20 0 2 -20 -40

FIG. 1. Concentration of baryon species in neutron star matter. Four panels correspond to the four choices of the hyperon-nucleon interaction constants 共cases I–IV兲 specified in text.

Case I: vector meson couplings are taken according to the ⌳ quark counting, x ␻ ⌳(⌺) ⫽x ␳ ⌺ ⫽2x ␻ ⌶ ⫽2x ␳ ⌶ ⫽ 32 and E bind ⌶ ⌺ ⫽⫺30 MeV, E bind⫽⫺18 MeV, E bind⫽30 MeV. ⌺ Case II: the same as in case I but E bind ⫽⫺10 MeV. Case III: the same as in case I but the ␳ meson couplings are taken according to the SU共3兲 symmetry, i.e., x ␳ ⌺,⌶ ⫽1. ⌺ Case IV: the same as in case III, but for E bind ⫽⫺10 MeV. C. Particle concentrations

In Fig. 1, we show the resulting concentrations of different baryon species as function of the baryon density. Panels 共a兲–共d兲 correspond to the four choices of the hyperon-meson coupling constants specified above. We see that hyperons appear in neutron star matter in all cases at density ␳ ⬎ ␳ c,H ⯝(2.5–3) ␳ 0 . The latter value is rather insensitive to various choices of hyperon-nucleon interactions. However, the order in which hyperons populate the Fermi seas depends crucially on the details of hyperon-nucleon interactions. The ⌺ ⫺ hyperons do not appear at least up to 6 ␳ 0 , except for case II. However, even in case II their concentration is very small. The place of ⌺ ⫺ is readily taken by ⌶ ⫺ and ⌳ hyperons. This observation is in line with the results of Ref. 关32兴. In ⌺ ⬍0, but ⌺ hyperons do not appear case IV, we choose E bind due to the increasing repulsion mediated by ␳ mesons with larger coupling constants than in case II. We see that in cases I–IV, the proton concentration saturates when hyperons appear in the system. We also see that none of the choices I–IV support effects observed in Ref. 关13兴, where ⌳ hyperons become more abundant than protons already at 3 ␳ 0 . Therefore, the p-wave K ⫹ K ⫺ condensation discussed in Ref. 关13兴 does not show up in the framework of our model for all four parameter choices.

1000

1100

1200

s

1/2

1300

1400

0 1500 1000

[MeV]

1100

1200

s

1/2

1300

1400

1500

[MeV]

FIG. 2. Real parts of the s-wave KN scattering amplitudes in isospin-zero and isospin-one channels 关16兴.

Here, the triangle is the full meson-baryon scattering amplitude with strangeness ⫺1 and the circle stands for the interaction kernel. This equation involves rescatterings through all possible meson-baryon intermediate states allowed by the strangeness conservation. The interaction kernel can be derived from the SU共3兲 chiral Lagrangians 关33–36兴 or can be phenomenologically adjusted to fit the data within the K-matrix formalism 关37,38兴. In view of the general interest to the s-wave kaon condensation prevailing in the literature so far, most of the attention has been paid to the kaon-nucleon interaction in the s wave. Although far below the threshold the s-wave kaon-nucleon scattering amplitude is a rather smooth function of the kaon energy, close to the threshold the amplitudes vary strongly due to the ⌳(1405) resonance 共see Fig. 2兲. Therefore, the extrapolation into the subthreshold region of the scattering amplitude, adjusted to fit the data above the K ⫺ N threshold, crucially depends on the microscopic model applied. Its inmedium modification is a matter of debate too 关39– 43兴. Up to recently, only scarce information on the p-wave kaon-nucleon interaction was available. In the isospin-zero channel, the small p-wave amplitudes were not separated from the large contribution of the ⌳(1405) resonance in the s wave, which dominates near threshold energies. In the isospin-one channel, determination of the p-wave amplitudes remained also uncertain due to a lack of direct experimental information on the K ⫺ n scattering at low energies. This gap ¯ N interaction was obwas filled in Ref. 关16兴, where the K tained as a solution of the covariant coupled-channel BetheSalpeter equation with a kernel derived from a relativistic chiral SU共3兲 Lagrangian with extra constraints from the K ⫹ nucleon and pion-nucleon sector. This analysis provides reliable estimates for both the s- and p-wave K ⫺ N scattering amplitudes, which we will use in the following.

III. K À -NUCLEON INTERACTION IN VACUUM A. Forward scattering amplitudes

The kaon-nucleon interaction in vacuum results as the solution of the coupled-channel Bethe-Salpeter equation

¯ N forward scattering amplitudes in a given The vacuum K isospin channel I have the following contributions from sand p-partial waves: T (I) 共 s 兲 ⫽T S(I) 共 s 兲 ⫹T P(I) 共 s 兲 ⫽

共6兲 015803-4

¯E 共 s,m N2 ,m K2 兲 ⫹m N 2m N

⫻ 关 F S(I) 共 s 兲 ⫹Q 2 共 s,m N2 ,m K2 兲 F P(I) 共 s 兲兴 ,


NEGATIVE KAONS IN DENSE BARYONIC MATTER

I=0

4

2

2

0

0

-2

-2

-4

-4

π

FP(s) [1/m 3]

4

1000

1100

1200

s

1/2

1300

1400

1500 1000

the pole contributions from the p-wave amplitudes. We define the pole and pole-subtracted amplitudes

I=1

(I) ␦ F P(I) 共 s 兲 ⫽F P(I) ⫺F pole 共 s 兲,

(0) F pole 共 s 兲 ⫽⫺2

1100

[MeV]

1200

s

1/2

1300

1400

共 s⫺p 2 ⫺k 2 兲 2 ⫺4k 2 p 2 , 4s

F P(I) 共 s 兲 ⫽

8 ␲ 冑s ¯E ⫹ 共 s,m N2 ,m K2 兲

8 ␲ 冑s/Q 2 共 s,m N2 ,m K2 兲 ¯E ⫹ 共 s,m N2 ,m K2 兲

2 2 ¯E ⫹ ⫹i0 共 s,m N2 ,k 2 兲 s⫺m ⌳

,

共10兲

1500

(1) F pole 共 s 兲 ⫽⫺4

(I) f 0⫹ 共 s 兲,

(I) (I) 关 f 1⫺ 共 s 兲 ⫹2 f 1⫹ 共 s 兲兴 . 共8兲

The partial amplitudes f l⫾ ⫽ f l,J⫽l⫾ 21 in Eq. 共8兲 are the scattering amplitudes for given angular momentum l and total ¯ (s, p 2 ,k 2 )⫾ 冑p 2 . The real momentum J, and ¯E ⫾ (s, p 2 ,k 2 )⫽E parts of the partial amplitudes T S(I) and T P(I) are shown in Figs. 2 and 3 by solid lines. The pronounced peak structures in the p-wave amplitude are due to the ⌳(1116) pole in the isospin-zero channel, and the ⌺(1195) and ⌺ * (1385) poles in the isospin-one channel. B. Separation of the pole terms

The hyperon s-channel exchanges are responsible for the strongest variation of the polarization operator at low frequencies and momenta. Therefore, they deserve a special consideration in view of the baryon modifications by the mean-field potentials. To treat them explicitly, we separate

2 m⌺ C KN⌺

共 m ⌺ ⫹m N 兲 2

2 ¯E ⫹ 共 s,m N2 ,m K2 兲 s⫺m ⌺2 ⫹i0

8 ⫺ 3

共7兲

where s⫽(p⫹k) 2 , and p⫽„⑀ N (pជ ), pជ …, k⫽( ␻ ,kជ ) are the four-momenta of the incoming nucleon and kaon, respectively. For nucleons and kaons on mass shell, the quantity Q 2 (s,m N2 ,m K2 ) is the square of the center-of-mass momentum in the kaon-nucleon channel, ¯E (s,m N2 ,m K2 ) ⫽ 冑m N2 ⫹Q 2 (s,m N2 ,m K2 ) is the nucleon energy in the centerof-mass frame, ¯E (s, p 2 ,k 2 )⫽(s⫹p 2 ⫺k 2 )/2冑s, m N and m K are the free nucleon and kaon masses. We shall neglect the isospin-symmetry breaking within kaon and nucleon isospin multiplets, which is irrelevant in dense nuclear matter. The invariant partial-wave amplitudes F S(I) and F P(I) are related to the standard partial-wave amplitudes 共see Ref. 关44兴 Sec. 3.1 for definitions兲 as F S(I) 共 s 兲 ⫽

共 m ⌳ ⫹m N 兲 2

[MeV]

FIG. 3. Real parts of the p-wave KN scattering amplitudes F P in isospin-zero and isospin-one channels 共solid lines兲关16兴. The dashed lines represent the corresponding pole-subtracted scattering amplitudes 共9兲.

Q 2 共 s,p 2 ,k 2 兲 ⫽

2 2m ⌳ C KN⌳

共9兲

C KN⌺ * 共 s/m ⌺ * 兲

2

¯E ⫹ 共 m 2 ,m N2 ,m K2 兲 ⌺*

i 2 s⫺m ⌺ * ⫹ ␥ ⌺ * 共 s 兲 2

¯E ⫹ 共 s,m N2 ,m K2 兲

.

共11兲 The ⌺ * width, ␥ ⌺ * (s)⫽( 冑s⫹m ⌺ * ) 关 ␥ ␲ ⌳ (s)⫹ ␥ ␲ ⌺ (s) ⫹ ␥ KN (s) 兴 , includes contributions from ␲ ⌳, ␲ ⌺, and KN 2 2 3 )兩 / channels, ␥ ␾ B (s)⫽C ␾ B⌺ * ¯E ⫹ (s,m B2 ,m ␾2 ) 兩 Q(s,m B2 ,m G (12␲ 冑s), where m B is the mass of the corresponding baryon B⫽(n,p,⌳,⌺ ⫾,0,⌶ ⫺0 ) and m ␾ is the mass of the corresponding light meson ␾ ⫽( ␲ ,K). The coupling constants C ␲ ⌳(⌺)⌺ * can be extracted from the partial width of the ⌺ * hyperon. The values of the KN⌳(⌺) coupling constants C KN⌳ ⯝⫺0.68/m ␲ , C KN⌺ ⯝0.34/m ␲ entering Eqs. 共10兲 and 共11兲 follow directly from the amplitudes shown in Fig. 3. The values of C KN⌳ and C KN⌺ are defined as couplings of ⌳ to (K † N) and ⌺ ␣ to (K † ␶ ␣ N) isospin states, where K, N stand for the isospin doublets, ⌺ ␣ is the isospin triplet, and ␶ ␣ are the isospin Pauli matrices. Note that the value of C KN⌳ , obtained on the basis of Ref. 关16兴, is smaller than that fitted by the Ju¨lich group 关45兴 and used in Refs. 关12,6兴. The scattering amplitude near the ⌺ * resonance can be only approximately described by the last term in Eq. 共11兲, since the corresponding amplitude results from a multichannel dynamics, which generates the energy-dependent selfenergy of the ⌺ * resonance. This self-energy, however, can be neglected at 冑s relevant for the calculations below. Reference 关16兴 gives for the ⌺ * resonance coupling C KN⌺ * ⯝0.84/m ␲ defined by the Lagrangian term ¯ * ␣ ⳵ ␮ K † ␶ ␣ N. The pole subtracted amplitudes ⌺ C KN⌺ *

␮

␦ F (I) (s) are shown in Fig. 3 by dashed lines. We see that the

amplitudes in both isospin channels are smooth functions of 冑s in the subthreshold region 冑s⬍1300 MeV of our interest. Note that the procedure of pole subtraction is not unambiguous. Forms 共10兲 and 共11兲 are chosen here for a later convenience, since they allow direct comparison with the analysis of Refs. 关6,14,13兴. To consider the kaon-nucleon interaction in dense matter, we also need to take into account the mean-field potentials acting on baryons according to Lagrangian 共2兲.

015803-5


EVGENI E. KOLOMEITSEV AND DMITRI N. VOSKRESENSKY IV. K À POLARIZATION OPERATOR FROM SCATTERING AMPLITUDES

as the sum of the pole and the regular parts. The pole part is generated by hyperon exchanges

A. Gas approximation and baryonic mean fields

pole pole pole I p-pole 共 ␻ ,kជ 兲 ⫽I ⌳ 共 ␻ ,kជ 兲 ⫹I ⌺pole共 ␻ ,kជ 兲 ⫹I ⌺ * 共 ␻ ,kជ 兲 ,

Our next aim is to construct the retarded K ⫺ polarization operator in baryonic matter, ⌸ Rtot( ␻ ,kជ ), related to the kaon propagator as D K⫺1 ( ␻ ,kជ )⫽ ␻ 2 ⫺kជ 2 ⫺m K2 ⫺⌸ Rtot( ␻ ,kជ ). The spectral function is determined as A K ( ␻ ,kជ ) ⫽⫺2ImD K ( ␻ ,kជ ). Quasiparticle branches of the spectrum appear in the energy-momentum region where the kaon width ⌫ K ⫽⫺2Im⌸ Rtot( ␻ ,kជ ) is much smaller than any other typical energy scale. Then, one can put ⌫ K →0 in the kaon Green’s function and the quasiparticle branches are given by the dispersion equation ReD K⫺1 ( ␻ ,kជ )⫽0. From the vacuum scattering amplitudes, we can construct the causal polarization operator ⌸ C(0) ( ␻ ,kជ ) related to the partial-wave amplitudes, F S(I) and F P(I) in the gas approximation ⌸ C(0) 共 ␻ ,kជ 兲 ⫽I s wave共 ␻ ,kជ 兲 ⫹I p wave共 ␻ ,kជ 兲 ,

pole 2 I⌳ 共 ␻ ,kជ 兲 ⫽⫺C KN⌳

⫻

⫽⫺

冕

冑

共 2 ␲ 兲 3 2 m N2 ⫹pជ 2

再

2

共12兲

⫻

冎

⫽⫺

⫻

冕

再

1 2

共2␲兲

3

2

冑

m N2 ⫹pជ 2

冑m N2 ⫹ pជ 2

关 n p 共 pជ 兲 ⫹2n n 共 pជ 兲兴 , 共17兲

冕

冑

2 m N2 ⫹ pជ 2

共 2␲ 兲3

Q 2 共 s,m N2 ,k 2 兲 i

s m ⌺*

关 n p 共 pជ 兲 ⫹2n n 共 pជ 兲兴 .

共18兲 The regular part I reg p wave can be expressed as

ជ I pregwave共 ␻ ,kជ 兲 ⫽¯I pregwave共 ␻ ,kជ 兲 ⫹ ␦ I reg p wave共 ␻ ,k 兲 ,

共19兲

including the part of integral 共14兲 evaluated with ␦ F P(I) from Eq. 共9兲, ¯I reg ជ p wave共 ␻ ,k 兲 ⫽⫺

共14兲

where n i (pជ ) are the nucleon Fermi occupations, i⫽(n,p). At zero temperature n p (pជ )⫽ ␪ (p F,p ⫺ 兩 pជ 兩 ) and n n (pជ ) ⫽ ␪ (p F,n ⫺ 兩 pជ 兩 ), s⫽( ␻ ⫹ 冑m N2 ⫹pជ 2 ) 2 ⫺(kជ ⫹pជ ) 2 . There are simple relations between the causal 共‘‘⫺,⫺’’兲 and the retarded 共‘‘R’’兲 Green’s functions and polarization operators. For zero temperature and positive frequencies, they coincide. For T⫽0 their real parts are still the same, whereas the imaginary parts are different, but can be interrelated. Bearing this in mind, we will further suppress the subscripts R and C for brevity. Exploiting decomposition 共9兲 of the p-wave scattering amplitude, we write reg ជ ជ I p wave共 ␻ ,kជ 兲 ⫽I pole p wave共 ␻ ,k 兲 ⫹I p wave共 ␻ ,k 兲 ,

s 2 ⫺m ⌺2 ⫹i0

2

关 F P(0) 共 s 兲 ⫹F P(1) 共 s 兲兴 n p 共 pជ 兲

冎

共 2␲ 兲3

s⫺m ⌺ * ⫹ ␥ ⌺ * 共 s 兲 2

共13兲

Q 2 共 s,m N2 ,k 2 兲

⫹F P(1) 共 s 兲 n n 共 pជ 兲 ,

共16兲

2 2 2 2d 3 pជ ¯E ⫹ 共 m ⌺ * ,m N ,k 兲

⫻

I p wave共 ␻ ,kជ 兲 ⬅I p wave,p 共 ␻ ,kជ 兲 ⫹I p wave,n 共 ␻ ,kជ 兲 2d 3 pជ ¯E ⫹ 共 s,m N2 ,k 2 兲

冕

n p 共 pជ 兲 ,

2d 3 pជ ¯E ⫺ 共 m ⌺2 ,m N2 ,k 2 兲

共 m ⌺ ⫹m N 兲 2 m ⌺

关 F S(0) 共 s 兲

⫹F S(1) 共 s 兲兴 n p 共 pជ 兲 ⫹F S(1) 共 s 兲 n n 共 pជ 兲 ,

2 s 2 ⫺m ⌳ ⫹i0

4 2 pole I ⌺ * 共 ␻ ,kជ 兲 ⫽⫺ C KN⌺ * 3

I s wave共 ␻ ,kជ 兲 ⬅I s wave,p 共 ␻ ,kជ 兲 ⫹I s wave,n 共 ␻ ,kជ 兲 2d 3 pជ ¯E ⫹ 共 s,m N2 ,k 2 兲 1

冑m N2 ⫹ pជ 2

共 2␲ 兲3

共 m ⌳ ⫹m N 兲 2 m ⌳

2 I ⌺pole共 ␻ ,kជ 兲 ⫽⫺C KN⌺

⫻

冕

2 ,m N2 ,k 2 兲 2d 3 pជ ¯E ⫺ 共 m ⌳

⫻

冕

再

1 2

2d 3 pជ ¯E ⫹ 共 s,m N2 ,k 2 兲共2␲兲

3

2

冑

m N2 ⫹ pជ 2

Q 2 共 s,m N2 ,k 2 兲

关 ␦ F P(0) 共 s 兲 ⫹ ␦ F P(1) 共 s 兲兴 n p 共 pជ 兲

冎

⫹ ␦ F P(1) 共 s 兲 n n 共 pជ 兲 ,

共20兲

and the nonpole contributions from the hyperon exchanges

共15兲 015803-6

ជ ជ ជ ␦ I reg p wave共 ␻ ,k 兲 ⫽ ␦ I ⌳ 共 ␻ ,k 兲 ⫹ ␦ I ⌺ 共 ␻ ,k 兲 , 2 ␦ I ⌳ 共 ␻ ,kជ 兲 ⫽⫺C KN⌳

⫻

冕

2d 3 pជ 共 2␲ 兲3

m ⌳ 冑s⫺m N2 ⫹k 2 共 m ⌳ ⫹m N 兲 2

冑

共21兲

2 冑s m N2 ⫹ pជ 2

冑s⫹m ⌳

n p 共 pជ 兲 , 共22兲


NEGATIVE KAONS IN DENSE BARYONIC MATTER 2 ␦ I ⌺ 共 ␻ ,kជ 兲 ⫽⫺C KN⌳

⫻

⫻

冕

larization operator generated by the hyperon poles I Ppole depends strongly on baryonic mean fields, which change the pole positions in the amplitude. This part will be treated explicitly in the course of our consideration.

2d 3 pជ m ⌺ 冑s⫺m N2 ⫹k 2

冑

共 2 ␲ 兲 3 2 冑s m N2 ⫹pជ 2

共 m ⌺ ⫹m N 兲 2

冑s⫹m ⌺

B. Pole part of the polarization operator

关 n p 共 pជ 兲 ⫹2n n 共 pជ 兲兴 .

共23兲

To obtain the last relation, we used 2 ¯E 共 s,m N2 ,k 2 兲 ⫺E ¯ 共mH ,m N2 ,k 2 兲

⫽

m H 冑s⫺m N2 ⫹k 2 2m H 冑s

共冑s⫺m H 兲 ,

H⫽ 兵 ⌳,⌺ 其 .

共24兲

The above construction of the polarization operator corresponds to a gas approximation, and does not take into account either the mean-field potentials acting on baryons or the vertex corrections due to baryon-baryon correlations, or possible modifications of the scattering amplitudes in medium. The modification of the baryon propagator on the mean-field level can be easily incorporated in integrals 共13兲 and 共14兲 by the replacement m N →m N* . Effects induced by this modification in the kinematic prefactors in Eq. 共13兲 and 共14兲 can be easily traced back. The scaling of the nucleon mass in s is a more subtle issue. Solving the coupled-channel Bethe-Salpeter equation, one sums all two-particle reducible diagrams for the part of the s plane corresponding to a K ⫺ N scattering. This approach is explicitly crossing noninvariant and a continuation of amplitudes far below the K ⫺ N threshold can generate artificial singularities in the scattering amplitude. In Ref. 关16兴, from where we borrow the amplitudes, the approximation scheme for solution of the Bethe-Salpter equation was furnished in such a way that the K ⫺ N and K ⫹ N scattering amplitudes exhibit the approximate crossing symmetry, smoothly matching each other for 冑s⬃m N . Therefore, amplitudes depicted in Figs. 2 and 3 are still physically well constrained in the corresponding intervals of 冑s shown there. However, for somewhat smaller 冑s, the K ⫺ N s-wave scattering amplitude gets unphysical poles. To cure this problem, the complete solution of the BetheSalpeter equation for K ⫺ N scattering has to be redone with medium modified baryon masses. Fortunately, there are some indications that this would not drastically change the results. We demonstrate here that the final results for integrals with the s-wave scattering amplitudes can be nicely modeled with the polarization operator following from a leading-order chiral Lagrangian, which has no explicit dependence on the baryon masses. Loop corrections due to the iteration of the interaction kernel should be suppressed for small kaon frequencies to keep the approximate crossing invariance of the amplitude. The pole subtracted p-wave amplitude is a rather smooth function of 冑s, as it is shown in Fig. 3, being mainly determined by the contact terms of the chiral Lagrangian and thus, has a weak baryon mass dependence. Hence, extrapolating the amplitude to somewhat smaller 冑s, we do not expect its strong variation. On contrary, the part of the po-

Here, we find the contributions to the K ⫺ polarization operator from hyperon poles in the K ⫺ N scattering amplitude determined by Eqs. 共16兲–共18兲. Relying on the explicit calculations of Refs. 关6,15兴 we can easily incorporate the scalar and vector mean fields acting on baryons. The scalar field is taken into account with the help of the replacement m B →m B* . The baryon vector potentials are included in the pole terms by the shift of the kaon frequency ␻ → ␻ ⫹ ␦ v iH , with ␦ v iH ⫽V i ⫺V H , i⫽ 兵 n,p 其 , H⫽ 兵 ⌳,⌺ 其 . This follows from the difference of the baryon energies, see Eq. 共3兲. Please notice that this energy shift is obvious only for the pole contribution to the polarization operator. Generally, due to the absence of the gauge invariance for massive vector fields, such a shift is not motivated for more complicated diagrams. Writing explicitly all contributions, we cast

ជ ⌸ (pole,0) 共 ␻ ,kជ 兲 ⬅I pole p wave共 ␻ ,k 兲 (pole,0) ⫽⌸ (pole,0) 共 ␻ ,kជ 兲 ⫹⌸ p⌺ 0 共 ␻ ,kជ 兲 p⌳ (pole,0) (pole,0) ⫹2⌸ n⌺ ⫺ 共 ␻ ,kជ 兲 ⫹⌸ p⌺ * 0 共 ␻ ,kជ 兲

⫹2⌸ n⌺ * ⫺ 共 ␻ ,kជ 兲 , (pole,0)

共25兲

where each term is equivalent to the pole contribution of the hyperon particle-nucleon-hole loop diagram 共Schro¨dinger picture兲

共26兲 1

written in terms of the Lindhard function as (pole,0) 2 * ⫺m N* 兲 2 ⫺ 共 ␻ ⫹ ␦ v iH 兲 2 ⌸ iH 共 ␻ ,kជ 兲 ⫽C KNH 关共 m H 2 ⫹kជ 2 兴 ␩ NH ⌽ iH 共 ␻ ,kជ 兲 , (pole,0)

⌸ i⌺ *

共 ␻ ,kជ 兲 ⫽C KN⌺ * ␩ i⌺ * 共 ␻ ,kជ 兲 kជ 2 ⌽ i⌺ * 共 ␻ ,kជ 兲 , 2

␩ NH ⫽

1

2

* m N* ⫹m H 2m N*

,

For further convenience, we introduce notation i⫽(n,p) and continue to use N when the quantity does not depend on the nucleon isospin.

015803-7



2 ជ ␩ i⌺ * 共 ␻ ,k 兲 ⫽

共 m ⌺** ⫹m N* 兲 2 ⫺ 共 ␻ ⫹ ␦ v iH 兲 2 ⫹kជ 2 2

6m ⌺**

.

We reserved the superscript ‘‘0’’ for each term in Eq. 共25兲 to indicate that neither the baryon self-energy corrections beyond a mean-field approximation nor the vertex corrections due to baryon-baryon correlations are included. The 共retarded兲 Lindhard function ⌽ accounting for the relativistic kinematics is defined as ⌽ iH 共 ␻ ,kជ 兲 ⫽

冕

4m N* 2

2d 3 p

2 ⫹i0 共 2 ␲ 兲 3 2 ⑀ i 共 p 兲 s⫺m H

m N* 2 2 ␲ 2 兩 kជ 兩

冕

p Fi

0

In our discussion, we would like to put particular emphasis on in-medium effects, which modify the K ⫺ spectrum at finite momenta. For this purpose, we define the momentum independent part, called the s-wave part of the polarization operator, and the momentum dependent part, called the p-wave part of the polarization operator: ⌸ C(0) 共 ␻ ,kជ 兲 ⫽⌸ S(0) 共 ␻ 兲 ⫹⌸ P(0) 共 ␻ ,kជ 兲 ⬅⌸ C(0) 共 ␻ ,0兲 ⫹ 关 ⌸ C(0) 共 ␻ ,kជ 兲 ⫺⌸ C(0) 共 ␻ ,0兲兴 .

n i 共 pជ 兲 . 共28兲

The term ⌸ C(0) ( ␻ ,0) does not depend on kជ , whereas the term 关 ⌸ C(0) ( ␻ ,kជ )⫺⌸ C(0) ( ␻ ,0) 兴 depends on kជ , and vanishes at 兩 kជ 兩 ⫽0. In order to avoid misunderstanding, we point out that the s- and p-wave scattering amplitudes contribute to both parts of the polarization operator

For zero temperature, we have ⌽ iH 共 ␻ ,kជ 兲 ⫽

V. s- AND p-WAVE PARTS OF THE POLARIZATION OPERATOR

共27兲

冋

册

⫹ ⌬ iH 共 ␻ ,kជ , pជ 兲 dpp ln ⫺ , ⑀ i 共 p 兲 ⌬ iH 共 ␻ ,kជ , pជ 兲

⌸ S(0) 共 ␻ 兲 ⫽I s wave共 ␻ ,0兲 ⫹I p wave共 ␻ ,0兲 ,

⫾ 2 ជ ⌬ iH 共 ␻ ,kជ ,pជ 兲 ⫽ 关 ␻ ⫹ ␦ v iH ⫹ ⑀ i 共 pជ 兲兴 2 ⫺ ⑀ H 共兩 p 兩 ⫿ 兩 kជ 兩兲 , 共29兲

where p Fi is the Fermi momentum of the nucleon species i . The nonrelativistic form of the Lindhard function used, e.g., in Ref. 关21兴共with a different normalization兲 is recovered after expanding ⑀ B (p)⬇m B* ⫹p 2 /(2m B* ) in Eq. 共29兲. The imaginary part of the 共retarded兲 Lindhard function is obtained as an analytical continuation ln(x)⫽ln兩x兩⫹i␲␪(⫺x) leading to nonzero contribution for ⫺ ជ ⫹ ជ ␻ iH 共 k 兲 ⬍ ␻ ⬍ ␻ iH 共 k 兲.

共30兲

⌸ P(0) 共 ␻ ,kជ 兲 ⫽ 关 I s wave共 ␻ ,kជ 兲 ⫺I s wave共 ␻ ,0兲兴 ⫹ 关 I p wave共 ␻ ,kជ 兲 ⫺I p wave共 ␻ ,0兲兴 .

In the following, we discuss the s- and p-wave parts of the polarization operator. A. p-wave part

Following decomposition 共15兲, we split the p-wave part of the polarization operator into the pole and the regular contributions

⫾ are the upper and the lower borders of the correHere, ␻ iH sponding particle-hole continuum

⫹ ជ ␻ iH 共 k 兲⫽

冦

冑共 m H* ⫺m N* 兲 2 ⫹kជ 2 ⫹ ␦ v iH , E H 共兩 kជ 兩 ⫹p Fi 兲 ⫺E i 共 p Fi 兲 ,

冉冊冉冊

k⬍p Fi

k⬎p Fi

* mH

m N*

* mH

m N*

⫺ ជ ␻ iH 共 k 兲 ⫽E H 共兩 kជ 兩 ⫺p Fi 兲 ⫺E i 共 p Fi 兲 .

⫺1

⬇⫺

m N*

2

8 ␲ 2 兩 kជ 兩 3 ⑀ Fi

冋

⫹˜⫺ ˜ iH ⌬ ⌬ iH

2

ln

冉冊 ⫹ ˜ iH ⌬

⫺ ˜ iH ⌬

⫺

⫹ 2 ⫺ 2 ˜ iH ˜ iH 兲 ⫺共 ⌬ 兲共⌬

4

册

共34兲

For the pole p-wave part, we have ⌸ P(pole,0) 共 ␻ ,kជ 兲 ⫽I P(pole) 共 ␻ ,kជ 兲 ⫺I P(pole) 共 ␻ ,0 兲

ជ ជ ជ ⫽⌸ (P,0) p⌳ 共 ␻ ,k 兲 ⫹⌸ p⌺ 0 共 ␻ ,k 兲 ⫹2⌸ n⌺ ⫺ 共 ␻ ,k 兲 (P,0)

(P,0)

(P,0) (P,0) ⫹⌸ p⌺ * 0 共 ␻ ,kជ 兲 ⫹2⌸ n⌺ * ⫺ 共 ␻ ,kជ 兲 ,

共31兲

Baryon energies include vector potentials according to Eq. 共3兲. An approximate expression for ⌽ iH renders ⌽ iH 共 ␻ ,kជ 兲

⌸ P(0) 共 ␻ ,kជ 兲 ⫽⌸ P(pole,0) 共 ␻ ,kជ 兲 ⫹⌸ P(reg,0) 共 ␻ ,kជ 兲 .

⫺1 ,

共32兲

共33兲

共35兲

(P,0) (pole,0) (pole,0) ⌸ iH 共 ␻ ,kជ 兲 ⫽⌸ iH 共 ␻ ,kជ 兲 ⫺⌸ iH 共 ␻ ,0兲 ,

共36兲

where we used Eqs. 共16兲–共18兲 and 共25兲. Expanding the p-wave pole part of the polarization operator for small kaon momenta, we have (P,0) 2 P kជ 2 ␾ iH ⌸ iH 共 ␻ ,kជ 兲 ⫽C KNH 共 ␻ 兲 ⫹O 共 kជ 4 兲 , P 2 2 * ⫺m N* 兲 2 ␾ iH ⌽ iH 共 ␻ ,0兲 ⫹ ␩ NH 共 ␻ 兲 ⫽ ␩ NH 关共 m H

,

⫾ ⫾ ˜ iH with ⌬ ⫽⌬ iH ( ␻ ,kជ , p Fi ) and ⑀ Fi ⫽ ⑀ i (p Fi ) , being valid for ⫹ ⫺ ␻ ⬍( ␻ iH ⫹ ␻ iH )/2.

015803-8

⫺ 共 ␻ ⫹ ␦ v iH 兲 2 兴

⳵ ⌽ iH 共 ␻ ,kជ 兲 ⳵ kជ 2

冏

,

⌽ i⌺ * 共 ␻ ,kជ 兲 ⫽C KN⌺ * kជ 2 ␩ i⌺ * 共 ␻ ,0兲 ⌽ i⌺ * 共 ␻ ,0兲 . 2

2

共37兲

兩 kជ 兩 ⫽0

共38兲



b(S) p

0.1

b

b(S) m* /mN p

0.1

(S)

FIG. 4. Left panel: Coefficients 共40兲 of expansion 共39兲 of the polarization operator. Solid, dashed, and dotted lines correspond to nucleon densities ␳ n, p ⫽1 ␳ 0 , 3 ␳ 0 , and 5 ␳ 0 , respectively. Right panel: the same as on the left plane but integrals evaluated with effective nucleon mass in kinematical prefactors and P) coefficients b (S, scaled by p m N* /m N .

N

n

b(S) (P)

0.0

(S)

b ,b

(S)

b ,b

(P)

n

0.0

b(P) n

-0.1

b(P) n

-0.1

b(P) m* /mN p

b(P)

N

p

-0.2

-0.2 0

50

100

150

200

0

250

50

100

The regular part of the p-wave polarization operator is defined by

ជ ⌸ P(reg,0) 共 ␻ ,kជ 兲 ⫽ 关 I s wave共 ␻ ,kជ 兲 ⫺I s wave共 ␻ ,0兲兴 ⫹ 关 I reg p wave共 ␻ ,k 兲 ⫺I reg p wave共 ␻ ,0 兲兴 . At small kaon momenta, the real part of ⌸ P(reg ,0) can be written as

冉

Re⌸ P(reg,0) 共 ␻ ,kជ 兲 ⫽kជ 2 b p 共 ␻ 兲

150

200

250

ω [MeV]

ω [MeV]

冊

␳p ␳n ⫹b n 共 ␻ 兲 ⫹O 共 kជ 4 兲 , ␳0 ␳0 共39兲

p-wave K ⫺ condensate amplitude in the most general case, one needs to deal with momenta up to 兩 kជ 兩 ⬃ p F,n ⬃m K . In this case, we have to extrapolate our result for the regular part of the polarization operator to such momenta. Fortunately, within our approach the critical points of the s- and p-wave condensations do not deviate much from each other and the kaon condensate momentum in the vicinity of the critical density remains small. Also, the main contribution to the kaon polarization operator comes from the pole terms, which are written explicitly for arbitrary momenta in Eq. 共25兲. Thereby, the ambiguity of the mentioned interpolation should not significantly affect our conclusions.

(P) b i 共 ␻ 兲 ⫽b (S) i 共 ␻ 兲 ⫹b i 共 ␻ 兲 ,

冏冏

␳0 ⳵ I b (S) 共 ␻ ,kជ 兲 i 共 ␻ 兲⫽ ␳ i ⳵ kជ 2 s wave,i

␳ 0 ⳵ reg I b (P) 共 ␻ ,kជ 兲 i 共 ␻ 兲⫽ ␳ i ⳵ kជ 2 p wave,i

B. s-wave part

The kaon-nucleon interaction results in the following contributions to the s-wave part of the K ⫺ meson polarization operator

, kជ ⫽0

(reg,0) ⌸ S(0) 共 ␻ 兲 ⫽I s wave共 ␻ ,0兲 ⫹¯I reg 共␻兲 p wave共 ␻ ,0 兲 ⫹ ␦ ⌸

,

共40兲

⫹⌸ (pole,0) 共 ␻ ,0兲 ,

kជ ⫽0

where we used the fact that the real part of the kaon polarization operator is an even function of the kaon momentum. and b (S,P) are shown in Fig. 4 for T The quantities b (S,P) n p ⫽0 and for several values of densities ␳ i as functions of the kaon energy. In these calculations, the integrals I S and I P have been evaluated with the free nucleon masses. We see that these coefficients are almost density independent and only weakly dependent on the kaon energy in the interval 100 MeV⬍ ␻ ⬍250 MeV . As we have discussed in the beginning of this section, we replace the baryon masses by the effective masses, as they follow from the mean-field solutions, only in the kinematical prefactors in Eqs. 共13兲 and 共14兲. The results are shown in the right plane of Fig. 4. The coefficients b (S,P) n depend moderately on the density whereas the coefficients exhibit a stronger density dependence, which can be b (S,P) p parametrized by the factor m N /m N* as demonstrated in Fig. 4. The energy dependence is still weak in the 100 MeV⬍ ␻ ⬍250 MeV interval. Our result 共39兲 is derived for rather small values of a kaon momentum, 兩 kជ 兩 Ⰶm K . In order to find the actual value of the

共41兲

where the last two terms correspond to nonpole and pole parts of the hyperon exchange terms in the amplitude, respectively. The ⌸ (pole,0) term is given by Eq. 共25兲. Using Eq. 共21兲, we present ␦ ⌸ (reg,0) as follows:

␦ ⌸ (reg,0) 共 ␻ 兲 ⬅ ␦ I pregwave共 ␻ ,0兲 ⫽ ␦ ⌸ (reg,0) 共 ␻ ,0兲 p⌳ ⫹ ␦ ⌸ p⌺ 0 共 ␻ ,0兲 ⫹2 ␦ ⌸ n⌺ ⫺ 共 ␻ ,0兲 , (reg,0)

(reg,0)

with (reg,0) 2 ␦ ⌸ iH 共 ␻ ,0兲 ⫽⫺C KNH

⫻

冕

2d 3 pជ 共 2␲ 兲3

* 冑s 0 ⫺m N* 2 ⫹ ␻ 2 兲共 m H* ⫹m N* 兲 2 共mH *兲 2 ⑀ i 共 p 兲冑s 0 共冑s 0 ⫹m H

n i 共 pជ 兲 , 共42兲

where s 0 ⫽ 关 ␻ ⫹ ⑀ i (pជ ) 兴 2 ⫺ pជ 2 . We also included the dependence of the effective masses on the mean field. For T⫽0

015803-9



according to case I. In order to illustrate the strength of different contributions to the s-wave part of the polarization operator 共41兲, we consider several test polarization operators

500

neutron matter

case I

ωS [MeV]

400

-- -1

Σ n

Λp

300

Π (1,0)

-1

S

Π (2,0)

Im Π =0

200 proton matter

Π (0)

µe

100

(a)

S

(b)

0 0

1

2

3

4

ρ/ρ 0

0

⌸ S(1,0) 共 ␻ 兲 ⫽I s wave共 ␻ ,0兲 ⫹¯I reg p wave共 ␻ ,0 兲 ,

共46兲

⌸ S(2,0) 共 ␻ 兲 ⫽⌸ S(1,0) 共 ␻ 兲 ⫹ ␦ ⌸ (reg,0) 共 ␻ 兲 ,

共47兲

⌸ S(0) 共 ␻ 兲 ⫽⌸ S(2,0) 共 ␻ 兲 ⫹⌸ (pole,0) 共 ␻ ,0兲 .

共48兲

S

1

2

3

ρ/ρ 0

4

5

FIG. 5. Panel 共a兲: The energy of the lowest K ⫺ branch of the dispersion equation 共45兲 at kជ ⫽0 calculated with ⌸ S(1,0) 共solid lines兲 for proton and neutron matter. Dashed lines present approximate spectra given by Eq. 共49兲. Panel 共b兲: The energy of the lowest K ⫺ branch of Eq. 共45兲 at kជ ⫽0 for neutron star matter 共case I of the hyperon-nucleon interaction兲 corresponding to the s-wave polarization operators ⌸ S(1,0) ,⌸ S(2,0) ,⌸ S(0) given by Eqs. 共46兲–共48兲共solid lines兲. Dashed lines are solutions for approximate relations 共43兲 and 共50兲. Dotted line shows the electron chemical potential. Dash-dotted line depicts the border of the imaginary part of the K ⫺ polarization operator 共border of hyperonization兲.

In Fig. 5共a兲 solid lines show the energy of the lowest branch of the solution of Eq. 共45兲 with ⌸ S(1,0) for the cases of pure proton and neutron matter. The contribution of ¯I reg p wave( ␻ ,0) is very small, at the level of few percent. It is instructive to compare this result with the one given by the frequently used parametrization of the K ⫺ spectrum motivated by the leading-order chiral perturbation theory ( ␹ PT) 关8兴,

␻ S␹ PT共 ␳ n , ␳ p 兲 ⫽ 冑m K2 ⫺S K ⫹V K2 ⫺V K , S K⫽

1 f2

scal scal scal 关 ⌺ KN 共 ␳ scal p ⫹ ␳ n 兲 ⫹C 共 ␳ p ⫺ ␳ n 兲兴 ,

and for small kaon energies the integral in Eq. 共42兲 can be well approximated by the following expression: (reg,0) 2 ␦ ⌸ iH 共 ␻ ,0兲 ⬇⫺C KNH

⫺

冉

* 2 ⫺m N* 2 mH

␻2

* ⫹m N* mH

2m N*

冊

␳i ,

␳ scal i ⫹ ␻␳ i 共43兲

where ␳ scal stands for the scalar density of nucleons defined i by

␳ scal i ⫽

冕

p F,i

0

2d 3 pជ

m N*

共 2 ␲ 兲 3 ⑀ i 共 pជ 兲

.

In this section, we illustrate the strength of the different terms in Eq. 共41兲, applying the polarization operator to the problem of the s-wave kaon condensation. Neutron star matter becomes unstable with respect to reactions 共1兲 producing K ⫺ mesons at the zero momentum when the solution 关 ␻ S⫽ ␻ min(kជ) at kជ ⫽0] of the dispersion equation 共45兲

related to the lowest branch of the spectrum, meets the electron chemical potential. Then, the s-wave K ⫺ condensation may occur via a second-order phase transition. In Fig. 5, we present the energy of the lowest K ⫺ branch of the dispersion equation 共45兲 with momentum kជ ⫽0 as a function of the density. The hyperon interactions are taken

共 2 ␳ p⫹ ␳ n 兲

4f2

共49兲

,

where f ⯝90 MeV is the pion decay constant in the chiral limit 关16兴, and ⌺ KN and C stand for the isoscalar and the isovector kaon-nucleon ⌺ terms, and are related to explicit chiral symmetry breaking. The SU共3兲 symmetry predicts C ⫽m K2 (2m ⌶ ⫺3m ⌺ ⫹m ⌳ )/ 关 16(m K2 ⫺m ␲2 ) 兴 ⬇66 MeV 关4兴. A model polarization operator leading to the dispersion relation 共49兲 can be written as 关8兴 ⌸ S(␹ PT,0) 共 ␻ 兲 ⫽⫺

共44兲

C. Energy of the lowest branch of the dispersion equation at kᠬ Ä0

␻ S2⫺m K2 ⫺Re⌸ S共 ␻ S兲 ⫽0,

V K⫽

⫺

⌺ KN f

2

scal 共 ␳ scal p ⫹␳n 兲⫺

2 ␳ p⫹ ␳ n 2f2

␻.

C f2

scal 共 ␳ scal p ⫺␳n 兲

共50兲

Spectrum 共49兲, calculated using ⌺ KN ⫽150 MeV, is shown in Fig. 5 关panel 共a兲兴 by dashed lines. We observe a good agreement of the model spectrum with the one following from the numerical evaluation of the integrals 关 I s wave( ␻ ,0) ⫹¯I pregwave( ␻ ,0) 兴 . The obtained value of the effective kaonnucleon ⌺ term is two to three times smaller than what was used as an ad hoc parameter in Ref. 关8兴 with the same parametrization 共50兲. The results for the realistic composition of neutron star matter 共Fig. 1, case I兲 are presented in Fig. 5共b兲 as a representative example. Solid lines depict the energy of the lowest branch of the dispersion equation calculated for kជ ⫽0 with ⌸ S(1,0) , ⌸ S(2,0) , and ⌸ S(0) . Dashed lines show solutions obtained with the approximate expressions 共50兲 in ⌸ S(1,0) and 共43兲 for ␦ ⌸ (reg,0) in ⌸ S(2,0) . The excellent coincidence of the curves justifies the accuracy of Eqs. 共50兲 and 共43兲.

015803-10



The crossing point of the solid and dotted lines corresponds to the critical density of the s-wave condensation. We observe that the lines corresponding to ⌸ S(1,0) do not cross the chemical potential 共dotted line兲. Therefore, the s-wave kaonnucleon interaction, following from Ref. 关16兴, would not support a second-order phase transition into the s-wave K ⫺ condensate state due to the small value of the kaon-nucleon 兺 term. However, an additional attraction comes from the (2,0) . It makes the condensation term ␦ I p(reg) wave included in ⌸ S possible at densities ⭓4.5␳ 0 . Another attractive piece is the (0) pole term I ppole wave( ␻ ,0) taken into account in ⌸ S . The significance of these terms, originating both from the hyperon ¯ N interaction, was pointed first in exchange diagram in K Ref. 关6兴. Both of these contributions were disregarded in works 关8,9,5,7,10–12兴 discussing the s-wave K ⫺ condensation. The curves ␻ S calculated with the full s-wave polarization operator ⌸ S(0) have cuts. In the region between the cuts, Eq. 共45兲 has no solutions with positive residues 关6兴. The dasheddotted line depicts the border of the imaginary part of the K ⫺ polarization operator. We see that within our approach, the curve calculated with ⌸ S(0) and ␮ e meet at an energy below the region of the imaginary part. Thus, with the full polarization operator 共41兲 and 共48兲, we recovered the results of previous works 关8,9,5,7,10–12兴共where, however, the two to three times larger ⌺ term was used兲 about the possibility of the K ⫺ condensate production in reaction 共1兲 at rather moderate densities. In our case, the critical density is ␳ c,S ⯝2.7␳ 0 . The reader should bear in mind that the baryonbaryon correlations are still not included in the above analysis. They will increase ␳ c,S . This issue is addressed in Sec. VII.

VI. CONTRIBUTIONS OF THE HYPERON FERMI SEAS TO THE POLARIZATION OPERATOR

When the nucleon density exceeds the critical density of hyperonization ␳ c,H , the Fermi sea of hyperon H begins to grow, and the K ⫺ polarization operator receives new contributions.

2 2 2 ␩ iH ⌽ iH 共 ␻ ,kជ 兲 → ␩ NH ⌽ iH 共 ␻ ,kជ 兲 ⫹ ␩ HN ⌽ Hi 共 ⫺ ␻ ,⫺kជ 兲 ,

共52兲

where the last term implies interchange of all indices i↔H in Eqs. 共28兲, 共29兲, and 共34兲. The result of such a replacement can be cast as (pole,0) (pole,0) (pole,0) ␦ ⌸ hyp 共 ␻ ,kជ 兲 ⫽⌸ ⌳p 共 ⫺ ␻ ,⫺kជ 兲 ⫹⌸ ⌺ 0 p 共 ⫺ ␻ ,⫺kជ 兲 (pole,0) ⫹2⌸ ⌺ ⫺ n 共 ⫺ ␻ ,⫺kជ 兲 ,

and is to be added to the total polarization operator. B. Regular terms

There are no experimental constraints on the hyperon contribution to the regular part of the polarization operator. As a rough estimation, we suggest to extend the model polarization operator 共50兲 to the hyperon sector according to the leading-order terms of a chiral Lagrangian ( ␹ PT,0) ␦ ⌸ S,hyp 共 ␻ 兲 ⫽⫺

f2

scal

scal

scal ⫹ ␳ ⌺⫺⫹ ␳ ⌶⫺兲⫺ 共␳⌳

scal

scal

冊

C⌳ f

2

␳ ⌳scal⫹

冉

C 1 scal ␳⌳ f2 3

␳ ⌺ ⫺ ⫹2 ␳ ⌶ ⫺ 2f2

␻. 共54兲

In this expression, we utilize the value of ⌺ KN from the fit with formula 共49兲 to the numerical results in Fig. 5 (⌺ KN ⯝150 MeV), whereas the values of coefficients C ⬇66 MeV and C ⌳ ⫽m K2 (m ⌶ ⫺m ⌳ )/ 关 12(m K2 ⫺m ␲2 ) 兴 ⬇34 MeV and f ⯝90 MeV are predicted by the chiral SU共3兲 symmetry. To estimate the nonpole contribution from the nucleon u-channel exchange 共analogous to ␦ ⌸ (reg, 0) ), we use the approximate relation 共43兲 (reg,0) 2 ␦ ⌸ S,hyp 共 ␻ 兲 ⫽⫺C KN⌳

⫺

The hyperon contribution to the K ⫺ polarization operator is related to the diagrams 共in Schro¨dinger picture兲

where hyperons and nucleons interchange their roles compared to diagram 共26兲. The hyperon contributions to the pole part of the polarization operator ⌸ (pole,0) can be simply included in Eq. 共27兲 with the help of the replacement

⌺ KN

⫺ ␳ ⌺⫺⫹ ␳ ⌶⫺ ⫹

A. Pole terms

共51兲

共53兲

冋

*2 m N* 2 ⫺m ⌳ * 2m ⌳

␻2

* ⫹m N* m⌳

⫺ ␻␳ ⌺ ⫺ ⫺

册

␳ ⌳scal⫺ ␻␳ ⌳

2 ␳ ⌳ ⫺2C KN⌺

␻2 m ⌺* ⫹m N*

册

冋

m N* 2 ⫺m ⌺* 2 2m ⌺*

␳ ⌺⫺ .

scal

␳ ⌺⫺

共55兲

In our estimation, we do not take into account contributions to the regular p-wave part of the polarization operator ⬀kជ 2 ␳ H . C. The energy of the lowest branch of the dispersion equation at kᠬ Ä0

Solutions related to the lowest branch of the dispersion equation 共45兲 for kជ ⫽0 calculated with the polarization operator

015803-11



with the correlation function C ii ⬘ (r)⬍0 and it is reduced in comparison to the product of two single-particle densities. The correlation function can be approximately written as

500

case I

ωS [MeV]

400

C ii ⬘ 共 r 兲 ⬇C core共 r 兲 ⫹ ␦ ii ⬘ C Pauli 共 r 兲关 1⫹C core共 r 兲兴 i

300 200

Π(0)

S hyp

µe

100 0 0

1

2

3

ρ/ρ0

4

5

FIG. 6. The energy of the lowest branch of the dispersion equa(0) tion at kជ ⫽0 calculated with the polarization operator ⌸ S,hyp given by Eq. 共56兲 is depicted by solid lines. The dashed line shows the corresponding solution with the polarization operator 共48兲. The dotted line depicts the electron chemical potential. (0) (0) ⌸ S,hyp 共 ␻ 兲 ⫽⌸ S(0) 共 ␻ 兲 ⫹ ␦ ⌸ S,hyp 共 ␻ 兲, (0) ( ␹ PT,0) (reg,0) (pole,0) ␦ ⌸ S,hyp 共 ␻ 兲 ⫽ ␦ ⌸ S,hyp 共 ␻ 兲 ⫹ ␦ ⌸ hyp 共 ␻ 兲 ⫹ ␦ ⌸ hyp 共 ␻ ,0 兲

共57兲

with contributions from the hard core, C core, and the Pauli exclusion principle, C Pauli, assuming that both correlations contribute multiplicatively. The former can be taken from the description of the nuclear matter with the realistic nucleonnucleon interaction. A convenient parametrization C core(r) ⬇⫺ j 0 (m 0 r) with m 0 ⬇5.6m ␲ was suggested in Ref. 关49兴. Here, j l (x) is the spherical Bessel function. For the Pauli correlation, we use the expression for the ideal fermion gas 2 2 (r)⫽⫺9 j 21 (p Fi r)/(2p Fi r ) . 关50兴, C Pauli i A. Correction of s-wave and regular p-wave terms

A general derivation of corrections to the meson propagation in dense nuclear matter due to nucleon-nucleon correlations 共so-called Ericson-Ericson-Lorentz-Lorenz corrections兲 can be found in Refs. 关51,52兴 for pions and in Refs. 关53,54兴 for kaons. Correlation processes can be presented by symbolic diagrammatic equation

共56兲

are shown in Fig. 6 by solid lines in comparison with the corresponding solutions obtained without inclusion of hyperons 共dashed lines兲. Calculations are done for the hyperon coupling constants corresponding to case I. We see that the presence of hyperons produces an additional small attraction only at rather high densities (⬎4 ␳ 0 ). The net effect is small ( ␹ PT,0) term due to the partial cancellation of the attractive ⌸ S,hyp (reg,0) and the repulsive ␦ ⌸ hyp term and because in our model the hyperon concentrations are much smaller than the neutron concentration and even smaller than the proton one. This allows us not to care much about the hyperon Fermi-sea occupations when considering the kជ ⫽0 case. VII. BARYON-BARYON CORRELATIONS

Working with the polarization operator constructed by integrating the meson-nucleon scattering amplitude over the nucleon Fermi sea, e.g., as in Eqs. 共13兲 and 共14兲, one assumes that all multiple meson-nucleon interactions are independent from each other and have the same probability proportional to the local nucleon density ␳ (rជ ). However, successive meson-nucleon scatterings in dense nuclear matter are not independent due to the presence of a repulsive core in nucleon-nucleon interactions, and due to the Pauli exclusion principle 关46 – 48兴. The probability to find two nucleons i and i ⬘ at the positions rជ 1 and rជ 2 , respectively, is proportional to the two-particle density

共58兲 The wavy line represents the kaon, and the sum goes over baryon species. The absence of arrows on the solid fermion lines means that both particles and holes are treated on equal footing 共the conservation of charges, e.g., strangeness, baryonic number, etc., in each vertex is implied兲. The hatched triangle is the bare scattering amplitude 共scattering on a particle or on a hole兲 and the full triangle stands for the amplitude including baryon-baryon correlations. The dotted line symbolically depicts the two-baryon correlation function C BB ⬘ due to the BB ⬘ correlations through the core and the Pauli principle. There are neither experimental informations nor theoretical estimations for the hyperon-nucleon and hyperon-hyperon correlations. Since the latter ones are less relevant for our discussion below we will neglect them. Thus, we include only minimal correlations given by the C ii ⬘ nucleon-nucleon correlation functions. First, we consider correlation corrections to the s-wave part of the kaon polarization operator ⌸ S(0) given by Eq. 共41兲 and the regular p-wave part ⌸ P(reg,0) from Eq. 共39兲. We separate the contributions induced by the scattering on a nucleon species i⫽ 兵 n,p 其 (0) (0) ⌸ S(0) 共 ␻ 兲 ⫽⌸ S,n 共 ␻ 兲 ⫹⌸ S,p 共 ␻ 兲, (0) ⌸ S,i 共 ␻ 兲 ⫽I s wave,i 共 ␻ ,0兲 ⫹I p wave,i 共 ␻ ,0兲 ,

␳ ii ⬘ 共 rជ 1 ,rជ 2 兲 ⫽ 关 1⫹C ii ⬘ 共兩 rជ 1 ⫺rជ 2 兩兲兴 ␳ i 共 rជ 1 兲 ␳ i ⬘ 共 rជ 2 兲 , 015803-12



FIG. 7. Left panel— contribution for the Pauli spin correlations to the p-wave correlation function 共62兲 vs the Fermi momentum for different kaon energies (p F(0) is the Fermi momentum for ␳ ⫽ ␳ 0 ); Right panel—the s-wave 共dashed lines兲 and p-wave 共solid lines兲 correlation functions 共61兲 and 共62兲 evaluated for ␻ ⫽ ␮ e in neutron star matter 共case I兲 as function of the total baryon density.

cf. Eqs. 共13兲 and 共14兲. Then, adopting results from Refs. 关55,54兴, we may present the polarization operator terms corrected by baryon-baryon correlations as

⌸ S共 ␻ 兲 ⫽

˜ 共 ␻ 兲 ⫹⌸ ˜ 共 ␻ 兲 ⫹2⌸ ˜ 共 ␻ 兲⌸ ˜ 共 ␻ 兲 ␰ (S) 共 ␻ 兲 ⌸ S,n S,p S,n S,p pn ˜ 共 ␻ 兲⌸ ˜ 共 ␻ 兲共 ␰ (S) 共 ␻ 兲兲 2 1⫺⌸ S,n S,p pn

˜ 共 ␻ 兲⫽ ⌸ S,i

⌸ Preg共 ␻ ,kជ 兲 ⫽kជ 2

(0) ⌸ S,i 共␻兲 (0) 1⫺ ␰ (S) ii 共 ␻ 兲 ⌸ S,i 共 ␻ 兲

,

共59兲

(P) ˜b p 共 ␻ 兲 ⫹b ˜ n 共 ␻ 兲 ⫹2b ˜ p 共 ␻ 兲˜b n 共 ␻ 兲 ␰ np 共␻兲 (P) ˜ p 共 ␻ 兲˜b n 共 ␻ 兲共 ␰ np 1⫺b 共 ␻ 兲兲 2

˜b i 共 ␻ 兲 ⫽

b i 共 ␻ 兲共 ␳ i / ␳ 0 兲 1⫺b i 共 ␻ 兲 ␳ i ␰ (P) ii 共 ␻ 兲 / ␳ 0

(S)

.

,

,

共60兲

(S)

␰ ii ⬘ 共 ␻ 兲 ⫽ (P)

1 3

冕

冕

d 3 rC ii ⬘ 共 r 兲 D K0 共 ␻ ,r 兲 ,

共61兲

to the p wave ␰ ’s and with

m 20 1 3 共 m 20 ⫹m K2 ⫺ ␻ 2 兲

共64兲

We turn now to the consideration of correlation effects in the particle-hole channel. ¯ N scattering amplitude If we approximate the free K 共hatched triangle兲 in Eq. 共58兲 by the hyperon-exchange diagram—the same that produces the particle-hole diagrams 共26兲—we can see that the account of correlations via Eq. 共58兲 is equivalent to the replacement2

共62兲

with the Pauli and hard-core contributions from Eq. 共57兲, i,i ⬘ ⫽ 兵 n,p 其 , and D K0 ( ␻ ,r)⫽⫺exp(⫺冑m K2 ⫺ ␻ 2 r)/(4 ␲ r) as the free kaon propagator. Using Eq. 共58兲, one finds that the repulsive core contributes with (P,core) ␰ ii ⬘ ⫽ ␰ K(P,core) 共 ␻ 兲 ⫽

1 共 m 20 ⫹m K2 ⫺ ␻ 2 兲

B. Correction of p-wave pole terms

d 3 rC ii ⬘ 共 r 兲 ⵜ 2 D K0 共 ␻ ,r 兲

1 (S) ⫽⫺ 关 C ii ⬘ 共 0 兲 ⫺ 共 m K2 ⫺ ␻ 2 兲 ␰ ii ⬘ 共 ␻ 兲兴 , 3

⫽ ␰ K(S,core) 共 ␻ 兲 ⫽

to the s-wave ones. The contribution from the Pauli spincorrelation 共second term in C ii ⬘ ) to the p-wave correlation function ␰ (P) ii is shown in Fig. 7 共left panel兲 as a function of the Fermi momentum for different kaon energies. We see that the correlation parameter decreases with the density since the core correlations hold baryons apart from each other and suppresses, thereby, the effect of the Pauli exclusion principle. The right panel of Fig. 7 presents the values of the correlation function 共61兲 and 共62兲 calculated for ␻ ⫽ ␮ e in the neutron star matter with hyperon coupling constants according to case I. In view of their smallness, we leave the contributions from the hyperon Fermi seas to the regular part of the polar(0) without corrections for baryonization operator ␦ ⌸ S,hyp baryon correlations.

(P)

The functions ␰ ii ⬘ and ␰ ii ⬘ are defined as

␰ ii ⬘ 共 ␻ 兲 ⫽

(S,core)

␰ ii ⬘

共65兲 with a modified vertex 共fat point兲 obeying equation The replacement 共65兲 and 共66兲 can be explicitly proven in the nonrelativistic limit. Working with relativistic kinematics, we apply it only to the pole part of diagram 共26兲 written in terms of the Lindhard function 共28兲. Thereby, we preserve the correct transition to the nonrelativistic limit. 2

共63兲

015803-13



共66兲 The particle-hole irreducible box

loc T HN ⬘

共the square兲 can be

expressed in terms of ␰ P and kaon-nucleon-hyperon coupling constants. Below we would like to put the discussion on a more phenomenological level. According to the arguments of the Fermi-liquid theory 关56兴, the particle-hole irreducible box loc T HN ⬘ has a weak dependence on incoming energies and momenta and, therefore, can be parametrized in terms of phenomenological Landau-Migdal parameters:

共67兲 with ␣ ⫽1,2,3. The amplitudes are normalized with C 0 ⫽300 MeV fm3 allowing for a comparison of the values for the hyperon-nucleon correlation parameters with those for the nucleon-nucleon correlations introduced in Ref. 关21兴. In Eq. 共67兲, (1/2) ⫽(1⫺ ជt ⌺ • ␶ជ )/3 is the projection operator onto the ⌺N state with isospin 1/2, ជt ⌺ are the isospin-one matrices and ␶ជ are the P ⌺N (1/2) Pauli matrices of the nucleon isospin. The projector P ⌺ * N is defined analogously. The spin-spin operators in the particle-hole ជ 1 ␴ជ 2 ) and S 12⫽(Sជ 1 Sជ 2 † ), with Sជ standing for the spin operator, which couples spin- 21 and spin- 23 channel are given by s 12⫽( ␴ states. The inclusion of correlations according to Eq. 共66兲 brings the pole polarization operator 共25兲 into the form ⌸ pole共 ␻ ,kជ 兲 ⫽

˜ 共 ␻ ,kជ 兲 ⫹⌸ ˜ 共 ␻ ,kជ 兲 ⫹2c f ⬘ ⌸ ˜ ជ ˜ ជ ⌸ p⌳ ⌺ ⌳⌺ p⌳ 共 ␻ ,k 兲 ⌸ ⌺ 共 ␻ ,k 兲 /3 ˜ 共 ␻ ,kជ 兲 ⌸ ˜ 共 ␻ ,kជ 兲 /3 ⬘2 ⌸ 1⫺c 2 f ⌳⌺ p⌳ ⌺

˜ 共 ␻ ,kជ 兲 ⫽ ⌸ p⌳

ជ ⌸ (0) p⌳ 共 ␻ ,k 兲

⬘ C 0 ⌽ p⌳ 共 ␻ ,kជ 兲 1⫺ f ⌳

,

⌸ ⌺ * 共 ␻ ,kជ 兲 ⫽

c⫽C 0 / 共 C KN⌳ C KN⌺ 兲 ,

⌸ p⌺ 0 共 ␻ ,kជ 兲 ⫹2⌸ n⌺ ⫺ 共 ␻ ,kជ 兲 (0)

˜ 共 ␻ ,kជ 兲 ⫽ ⌸ ⌺

⫹⌸ ⌺ * 共 ␻ ,kជ 兲 ,

(0)

1⫺g ⌺⬘ C 0 关 ⌽ p⌺ 0 共 ␻ ,kជ 兲 ⫹2⌽ n⌺ ⫺ 共 ␻ ,kជ 兲兴 /3 (0) (0) ⌸ p⌺ * 0 共 ␻ ,kជ 兲 ⫹2⌸ n⌺ * ⫺ 共 ␻ ,kជ 兲

,

1⫺g ⌺⬘ * C 0 关 ⌽ p⌺ * 0 共 ␻ ,kជ 兲 ⫹2⌽ n⌺ * ⫺ 共 ␻ ,kជ 兲兴 /3

C. Correlation parameters

To the best of our knowledge, there is no direct experimental information about the values of the Landau-Migdal ⬘ *) , ⬘ , g ⌺(⌺ parameters for the hyperon-nucleon interactions f ⌳ ⬘ . In principle, this information could be extracted and f ⌳⌺ from multistrange hypernucleus data, which, however, are ⬘ rather poor. In Ref. 关15兴, the Landau-Migdal parameter f ⌳

.

共68兲

was estimated in line with Ref. 关57兴, where these parameters for the nucleon-nucleon interaction were calculated within the Ericson-Ericson-Lorentz-Lorenz approach. We follow this approach below. Further corrections can be computed as in Ref. 关49兴. Following 关57兴, we assume that the squared block in Eq. 共67兲 is determined by exchanges of kaon and heavy strange vector meson K * with mass m K * ⯝892 MeV. This can be

015803-14



depicted in diagrams as

500 Π(2)

ωS [MeV] 共69兲 In this approximation, correlation parameters are equal for the HN ⫺1 and the NH ⫺1 interactions. Including hyperon Fermi seas means that the pole term of the polarization operator is corrected through replacement 共52兲 in Eq. 共68兲. Block 共69兲, being evaluated at the zero momentum and energy transfer, contributes to the local interaction in Eq. 共67兲 as (P,core)

loc ⬇C KNH C KNH ␰ K(P,core) 共 0 兲 ⫹C K * NH C K * NH ␰ K * T HN

共 0 兲,

共70兲

H⫽⌳,⌺,⌺ * , loc T ⌳⌺ 共 ␻ 兲 ⬇C KN⌳ C KN ⬘ ⌺ ␰ K(P,core) 共 0 兲 (P,core)

⫹C K * N⌳ C K * N ⬘ ⌺ ␰ K *

共71兲

共 0 兲.

For shortness, we do not write here explicitly the spin and isospin operators and note that those are exactly the same as in Eq. 共67兲. The vector-meson coupling constants C K * NH in ជ ⫻kជ 兴 . Eq. 共70兲 correspond to the nonrelativistic vertex ⬀ 关 ␴ ¨ The coupling constants can be taken from the Julich model of the hyperon-nucleon interaction via meson exchanges C K * N⌺ ⫽(0.07/m ␲ ), and 关45兴: C K * N⌳ ⫽⫺(1.3/m ␲ ), C K * N⌺ * ⫽(0.7/m ␲ ). These values account for the form factors used in Ref. 关45兴. Particularly, the very soft form factor is responsible for a strong suppression of the C K * N⌺ vertex. Thus, the Landau-Migdal correlation parameters 共67兲 can be cast as 2 ⬘ ⫽C KN⌳ C0 f ⌳ 关 ␰ K(P,core) 共 0 兲 ⫹R ⌳⌳ ␰ K *

(P,core)

2 C 0 g ⌺⬘ ⫽3C KN⌺ 关 ␰ K(P,core) 共 0 兲 ⫹R ⌺⌺ ␰ K *

共 0 兲兴 ,

(P,core)

共 0 兲兴 ,

⬘ ⫽C KN⌳ C KN⌺ 关 ␰ K(P,core) 共 0 兲 ⫹R ⌳⌺ ␰ K * C 0 f ⌳⌺

共 0 兲兴 ,

C 0 g ⌺⬘ * ⫽3C KN⌺ * 关 ␰ K(P兲,core共 0 兲 ⫹R ⌺ * ⌺ * ␰ K *

共 0 兲兴 ,

(P,core)

2

(P),core

where the first term was introduced in Eq. 共63兲, ␰ K(P, core) (0) (P,core) ⯝0.24, the second one is equal to ␰ K * (0)⫽ 32 m 20 /(m 20 2 ⫹m K * )⯝0.28, cf. Ref. 关57兴, and R HH ⬘ ⫽C K * NH C K * NH ⬘ /(C KNH C KNH ⬘ ) with R ⌳⌳ ⯝3.7, R ⌺⌺ ⯝0.04, R ⌳⌺ ⯝0.39, and R ⌺ * ⌺ * ⯝0.69. The additional factor (P,core) 2 in ␰ K * compared to ␰ K(P,core) originates from the reducជ 2 ⫻kជ 兴关 ␴ជ 1 ⫻kជ 兴 →s 12 . Finally, we estimate the followtion 关 ␴ ing values for the correlation parameters of Eq. 共67兲 as

⬘ ⯝0.9, g ⌺⬘ ⯝0.1, f⌳

⬘ ⯝⫺0.1, f ⌳⌺

g ⌺⬘ * ⯝1.2.

共72兲

case I

S

400

Π(2,0) S

300 Π

200

Π(0)

µe

100 0

S

0

S

1

2

3

ρ/ρ0

4

5

FIG. 8. Solid lines present the energy of the lowest branch of the dispersion equation at kជ ⫽0 calculated with the polarization operators ⌸ S(2) and ⌸ S including effects of baryon-baryon correlations. Dashed-dotted continuations of solid curves demonstrate effect of the filling of the hyperon Fermi seas on the s-wave polarization operator. Correlation parameters are taken according to Fig. 7 共right panel兲, 共61兲 and 共72兲. Dashed lines show solutions ⌸ S(2,0) and ⌸ S(0) without correlation effects, as in Fig. 5, panel 共b兲. The dotted line depicts the electron chemical potential.

Compared to Ref. 关15兴, we obtained a smaller value of the ⬘ ⯝0.6/m ␲2 , since we included here the form parameter C 0 f ⌳ factors mentioned above.3

D. Energy of the lowest branch of the dispersion relation at kᠬ Ä0

Figure 8 illustrates how strongly the kaon spectrum changes after including the short-range correlations. We show the lowest branch of the kaon spectrum at kជ ⫽0. This is calculated for the polarization operator ⌸ S(2) constructed from ⌸ S(2,0) according to Eq. 共59兲 and for the polarization operator ⌸ S( ␻ )⫽⌸ S(2) ( ␻ )⫹⌸ pole( ␻ ,0), where ⌸ pole( ␻ ,0) follows from Eq. 共68兲 with parameters 共72兲. The hyperon couplings are chosen according to case I. At ␳ ⬎ ␳ c,H , we have to include correlations in the term (0) (pole,0) ␦ ⌸ S,hyp in Eq. 共56兲. The pole term ␦ ⌸ hyp ( ␻ ,0) is included in Eq. 共68兲 with the help of replacement 共52兲. The ( ␹ PT,0) (0) ( ␻ ) and ␦ ⌸ S,h can be corrected in the other terms ␦ ⌸ S,hyp same manner as the ⌸ S(0) term. However, these terms are rather small as it is demonstrated by Fig. 6. Therefore, we omit correlations in them. From Fig. 8, we see that baryon-baryon correlations suppress the s-wave part of the polarization operator, in agreement with the statements 关53,54兴. It results in an increase of the critical density of the s-wave condensation from 2.7␳ 0 to 4.3␳ 0 for case I, chosen as an illustration. 3 Note that different normalizations of Landau-Migdal parameters are used here and in Ref. 关15兴.

015803-15


EVGENI E. KOLOMEITSEV AND DMITRI N. VOSKRESENSKY VIII. K À CONDENSATION IN NEUTRON STARS

300

␻ 2 ⫺kជ 2 ⫺m K2 ⫺Re⌸ tot共 ␻ ,kជ 兲 ⫽0,

共73兲

where the complete polarization operator is given by ( ␹ PT,0) ⌸ tot共 ␻ ,kជ 兲 ⫽⌸ S共 ␻ 兲 ⫹⌸ Preg共 ␻ ,kជ 兲 ⫹⌸ Ppole共 ␻ ,kជ 兲 ⫹ ␦ ⌸ hyp 共␻兲 reg,0 ⫹ ␦ ⌸ hyp 共 ␻ 兲.

共74兲

It contains the s-wave part, the regular p-wave, and the pole parts of the polarization operator given by Eqs. 共59兲, 共60兲, and 共68兲, respectively, and the terms determined by the hyperon populations 共54兲 and 共55兲. The correlation parameters are taken according to Eqs. 共61兲, 共62兲, and 共72兲. There are two different possibilities: K ⫺ condensation may occur in neutron star matter via a second-order phase transition or a first-order phase transition. The dynamics of these phase transitions is quite different. Both possibilities might be realized at different physical conditions related to different stages of a neutron star evolution. In case of a second-order phase transition, at the moment, when the density in the neutron star center achieves the critical density ␳ IIc , reactions 共1兲 become operative. At this transition, the isospin composition and the density of the system change smoothly. During the time ␶ ⬀ ␶ react冑␳ IIc / 冑␳ ⫺ ␳ IIc , where ␶ react is the typical time of the weak processes 共1兲, the system creates an energetically favorable condensate state. The condensate appears within the region where ␳ ⬎ ␳ IIc . If this happens during a supernova explosion, the typical size of the condensate region might become of the order of the neutron star radius. Due to energy conservation, the gained energy must be released in such a transition. When the condensate region is heated up to the temperatures T⭓T opac ⬃(1 –2)-MeV neutrinos are trapped. At this stage, the cooling time is determined by the neutrino heat transport from the condensate interior to the star surface 关21兴. When the star cools down to temperatures T⬍T opac , the neutron star becomes transparent for neutrinos, and these can be directly radiated away. A part of the energy is also radiated by photons from the star surface. In binary long-living systems, the critical density in the neutron star center can be achieved by accretion. Then, the transition is characterized by the typical accretion time. In case of a first-order phase transition, the final state might significantly differ from the initial one in its isospin composition and its density. Thus, this new state cannot be prepared in microscopic processes. Small-size droplets of the new phase are not energetically favorable due to a positive surface energy. When the density in the star center exceeds the value ␳ Ic ⬍ ␳ IIc , the system arrives at a metastable state. In I this state, a droplet with the density ␳ fin c ⬎ ␳ c and a suffi-

(I)

(II)

(III)

(IV)

ωs [MeV]

In Secs. IV–VII, we have constructed the K ⫺ polarization operator. Now we use it to study a possible instability of the system with respect to a phase transition into a state with K ⫺ condensate. First, we investigate the solutions of the K ⫺ dispersion relation

200 100 0 2

3

4

5

2

3

4

5

2

3

4

5

2

3

4

5

6

ρ/ρ 0

FIG. 9. Solid lines are the energy of the lowest K ⫺ branch of the dispersion relation 共73兲 for kជ ⫽0. Dotted lines show electron chemical potentials. Dashed lines are calculated without baryon-baryon correlations. Different panels correspond to the interactions given by cases I–IV.

ciently large 共overcritical兲 size, created by fluctuations, will continue growing. At zero temperature, the probability for the creation of such a droplet via quantum fluctuations is very small, but it increases greatly with the temperature 关21兴. Thus, a first-order phase transition is most likely at an initial stage of a neutron star formation or cooling when the temperature is sufficiently large. If the density in the center of a star exceeds the value ␳ IIc , a second-order phase transition occurs. In long-living binary stellar systems, where the neutron star slowly accretes the mass from a star companion and the temperature is already very small, a second-order phase transition is a more probable one 共depending on the accretion rate兲. A. Second-order phase transition to the s-wave condensate state

Let us first analyze the possibility of the s-wave K ⫺ condensation. In Fig. 9, we summarize the results of Secs. V C, VI C, and VII D, showing the energy of the lowest K ⫺ branch of the kaon dispersion relation 共73兲 for kជ ⫽0 together with the electron chemical potential. For the given parameter choice, the crossing points of the lines indicate the critical density of the s-wave condensation via a second-order phase transition. We see that for all models the neutron star matter is unstable with respect to the reactions in Eq. 共1兲 at densities (3 –5.2) ␳ 0 . The critical density depends on the choice of the correlation parameters and the parameters of the hyperonnucleon interactions. As long as correlations are not included, we support the conjecture of previous works on the (II) ⬃3 ␳ 0 . Howpossibility of the s-wave condensation at ␳ c,S ever, the baryon-baryon correlations shift the condensation critical density to values larger than those ones discussed in Refs. 关5,7–12兴. B. Second-order phase transition to the p-wave condensate state

In this section, we study a principal possibility for the p-wave K ⫺ condensation in neutron star matter via a secondorder phase transition. We investigate whether the p-wave condensation can occur before the s-wave condensation and to what extent this depends on the parameters of hyperonnucleon interactions and correlations. Let us first assume that the s-wave K ⫺ condensation is indeed possible at some critical density ␳ c,S . The lowest en-

015803-16


␻ ⬇ ␻ S⫹ ␣ 共 ␻ S兲 Z S共 ␻ S兲 kជ 2 ,

共75兲

where ␻ S is, as before, ␻ (kជ ⫽0) for the lowest energy branch of the dispersion law given by the solution of the equation ␻ S2⫽m K2 ⫹Re⌸ S( ␻ S ,kជ ⫽0), and

冉

⳵ Re⌸ S共 ␻ ,kជ 兲 Z S⫺1 共 ␻ 兲 ⫽ 2 ␻ S⫺ ⳵␻

冏

⳵ Re⌸ P共 ␻ ,kជ 兲 ␣ 共 ␻ 兲 ⫽1⫹ ⳵ kជ 2

冏冊 kជ ⫽0

⬎0,

共76兲

⳵ (0) (0) Re⌸ (pole,0) 共 ␻ ,kជ 兲兩 kជ ⫽0 ⫽ ␣ ⌳p ⫹ ␣ ⌺(0) ⫹ ␣ ⌺ * , ⳵ kជ 2 (0)

(0)

1 0.5

0

3

4

5

2

3

4

-0.2

1

-0.6 -0.8

0

3

0 0.5

5

2

0.5

5

2

3

2

3

4

1

5

6

(IV)

0 0.5

0

5

4

0.5

(III)

1

4

3

ρc,S/ρ0

(II)

(I)

-0.4

0

4

1

0.5

5

2

3

ρc,S/ρ0

4

0

5

2

3

0.5

4

1

5

6

˜b p 共 ␻ 兲 ⫽

˜b n 共 ␻ 兲 ⫽

b p共 ␻ 兲 ␳ p/ ␳ 0 1⫺b p 共 ␻ 兲 ␳ p ␰ (P) pp 共 ␻ 兲 / ␳ 0 b n共 ␻ 兲 ␳ n/ ␳ 0 (P) 1⫺b n 共 ␻ 兲 ␳ n ␰ nn 共 ␻ 兲/␳0

,

.

共78兲

(0)

(0) 2 2 P 2 P ␣ iH ␾ iH ␾ Hi 共 ␻ 兲 ⫽C KNH 关 ␩ NH 共 ␻ 兲 ⫹ ␩ HN 共 ⫺ ␻ 兲兴 , (0) 2 2 ␣ i⌺ * 共 ␻ 兲 ⫽C KN⌺ * ␩ N⌺ * ⌽ i⌺ * 共 ␻ ,0兲 ,

H⫽ 兵 ⌳,⌺ 其 .

2 P The term ␩ HN ␾ iN (⫺ ␻ ) accounts for the contribution of the hyperon Fermi sea. Once the baryon-baryon correlations are included in the pole part of the polarization operator accord(0) is to be replaced by ing to Eq. 共68兲, ␣ pole

冏

⳵ Re⌸ pole共 ␻ ,kជ 兲 ⳵ kជ 2

. kជ ⫽0

(0) (␻) The regular part follows from Eq. 共39兲, ␣ reg ⫽b p ( ␻ ) ␳ p / ␳ 0 ⫹b n ( ␻ ) ␳ n / ␳ 0 , with the coefficients defined in Eq. 共40兲. The suppression of the regular part ␣ reg due to the baryon-baryon correlations can be taken into account, according to Eq. 共60兲

␣ reg共 ␻ 兲 ⫽

1

-0.8 2

(IV)

(III)

FIG. 10. ␣ pole( ␮ e ), ␻ S⫽ ␮ e , 共solid lines兲, and ⫺1⫺ ␣ reg 共dashed lines兲 vs critical density of the s-wave K ⫺ condensation for ⬘ various correlation parameter choices. Labels show values of f ⌳ ⬘ ⫽0, and f ⌳⬘ ⫽g ⌺⬘ ⫽0,0.5,1. ⫽g ⌺⬘ . Solid lines are drown for f ⌳⌺ Dashed lines are for three choices of the correlation parameters used in calculation of ␣ reg , ␰ nn ⫽ ␰ p p ⫽ ␰ n p ⫽0,0.5,1 共from the upper line to the lower one兲. Cases I–IV correspond to those in Fig. 1. In the upper plot, ⌺ * is taken with g ⌺⬘ * ⫽g ⌺ , V ⌺ * ⫽V ⌺ , and g ␴ ⌺ * ⫽g ␴ ⌺ . In the lower plot, V ⌺ * ⫽0 and g ␴ ⌺ * ⫽0. Hyperon populations are included in the mean fields but not in the polarization operator.

共77兲

with ␣ ⌺(0) ⫽ ␣ ⌺ 0 p ⫹2 ␣ ⌺ ⫺ n and ␣ ⌺ * ⫽ ␣ p⌺ * 0 ⫹2 ␣ n⌺ * ⫺ . From Eqs. 共37兲 and 共38兲, we have

␣ pole共 ␻ 兲 ⫽

0.5

0

1

. kជ ⫽0

Without baryon-baryon correlations, the contribution of the pole part is

(0)

-0.6

2

␣ 共 ␻ 兲 ⫽1⫹ ␣ pole共 ␻ 兲 ⫹ ␣ reg共 ␻ 兲 .

(0)

(II)

(I)

-0.4

-1.0

If at ␳ c,S , we have ␣ ( ␻ S)⬍0, then instead of the s-wave condensation we have the p-wave condensation at a somewhat smaller density. The aim of this section is to find the value ␣ ( ␻ S) in Eq. 共75兲 at the critical point of the s-wave condensation, i.e., when ␻ S⫽ ␮ e . Using the p-wave kaon polarization operator determined in Sec. V, we write

(0) ␣ pole ⫽

-0.2

-1.0

αpole, -1-αreg

ergy branch of the K ⫺ spectrum at small momenta is given by

αpole, -1-αreg


(P) ˜b p 共 ␻ 兲 ⫹b ˜ n 共 ␻ 兲 ⫹2b ˜ p 共 ␻ 兲˜b n 共 ␻ 兲 ␰ np 共␻兲 (P) 2 ˜ p 共 ␻ 兲˜b n 共 ␻ 兲共 ␰ np 1⫺b 兲共␻兲

,

(P)

Although the ␰ ii ⬘ ( ␻ ) functions can be evaluated as in Eq. 共62兲, we will treat them here as free energy-independent parameters ␰ ii ⬘ in order to investigate the sensitivity of the results to their variation. In Figs. 10 and 11 we show the values of ␣ pole( ␮ e ) 共solid lines兲 and ⫺1⫺ ␣ reg 共dashed lines兲, calculated as a function of ␳ c,S for various baryon-baryon correlation parameters and hyperon couplings. The ⌺ hyperon contribution is proved to be very small. The major contribution to the strength is due to ⌳ p ⫺1 and ⌺ * n ⫺1 loops. The hyperon Fermi seas are not incorporated for the moment 共we drop terms ⬀⌽ Hi in ⌸ pole). Inclusion of ⌺ * into the mean-field model 共2兲 is quite uncertain due to the absence of any empirical constraint on the coupling constants. We consider two different cases: In the upper plot in Fig. 10, we assume that ⌺ * couples to the mean field with the same strength as the ⌺ hyperon (V ⌺ * ⫽V ⌺ , g ␴ ⌺ * ⫽g ␴ ⌺ ); in the lower plot in Fig. 10, we detach ⌺ * from the mean-field potentials (V ⌺ * ⫽0, g ␴ ⌺ * ⫽0). At the crossing point of the solid line with the corresponding dashed line, we have ␣ pole⫽⫺1⫺ ␣ reg and therefore, ␣ ⫽0. This means that the given density is the critical density for both the s- and the p-wave condensations. For values of ␳ c,S , for which the solid line is below the corresponding dashed line ␣ pole⬍⫺1⫺ ␣ reg . This means that ␣ ⬍0 and for

015803-17


-0.2

(I)

-0.4

(II)

1

-0.6

0

1 0.5

0.5

0.5

3

4

5

2

3

0.5

0

0

2

(IV)

1

-0.8 -1.0

αpole, -1-αreg

(III)

4

5

2

3

ρc,S/ρ0

4

αpole, -1-αreg

αpole, -1-αreg


1

0

5

2

3

4

5

-0.2

(I)

-0.4

(II)

(III)

(IV)

-0.6 -0.8 -1.0

6

2

3

4

5

2

3

4

5

2

3

4

5

2

3

4

5

6

ρc,S/ρ0

-0.2

(I)

-0.4

(II)

1

-0.6

0.5 0

2

3

(IV)

1

-0.8 -1.0

(III)

0.5

0

4

5

2

3

4

5

2

0 0.5

1

3

5

ρc,S/ρ0

4

0 0.5

2

3

4

1

5

6

FIG. 11. Same as in Fig. 10, but with account for the hyperon population in the polarization operator.

such a parameter set K ⫺ condensation occurs in the p-wave state at a density somewhat smaller than that assumed for the s-wave condensation. Contributions from the hyperon Fermi seas are included in Fig. 11. Comparing Figs. 10 and 11, we see that the hyperon population affects noticeably the p-wave attraction. Figures 10 and 11 also show that the ⌺ * hyperons contribute significantly to the K ⫺ polarization operator, increasing the attraction in the p wave. The most favorable case for the p-wave condensation is realized when ⌺ * is detached from the mean-field potentials. In Fig. 12, we show the results for the values of the correlation parameters evaluated in Sec. VII. For ⌺ * hyperons detached from the mean-field potentials our model predicts a preference for the p-wave condensation. The p-wave condensation is preferred at ␳ c,S⭐3.5␳ 0 for the cases II and IV, and at ␳ c,S⯝4 ␳ 0 and ␳ c,S⯝4.5␳ 0 for cases III and I, provided the s-wave softening of the spectrum is also rather high. For ⌺ * coupled with the same strength as ⌺, s-wave condensation might be preferable to the p-wave condensation. C. First-Order phase transition to the K À condensate state

In the previous two sections, we have studied the possibility of K ⫺ condensation assuming that this occurs via a second-order phase transition. Here, we investigate the properties of K ⫺ excitations in baryonic matter of different particle compositions, in order to understand whether an abrupt 共 of a first order兲 phase transition into a state with new particle composition and new baryon density can be energetically favorable. We consider 共i兲 nucleon-hyperon matter 共NHM兲 with a composition which we have discussed above 共see Fig. 1兲; 共ii兲 neutron-proton matter 共NPM兲 consisting only of protons and neutrons in ␤ equilibrium; 共iii兲 isospin-symmetrical nuclear matter 共ISM兲 consisting of protons and neutrons with ␳ p ⫽ ␳ n and leptons that compensate for the electric charge of the protons; and 共iv兲 proton-enriched matter 共PEM兲 consisting of protons and neutrons with concentration Y p ⫽ ␳ p / ␳ ⫽0.7 and charge compensated by the leptons. Thus, in cases 共ii兲–共iv兲, we switch off the hyperons in our mean-field model

FIG. 12. Values ␣ pole 共solid lines兲 and ⫺1⫺ ␣ reg 共dashed lines兲, ␻ S⫽ ␮ e , calculated for the correlation parameters 共72兲. Solid lines are calculated with ⌺ * hyperons coupled to the mean field as strong as ⌺ hyperons. Dotted lines are with ⌺ * detached from the meanfield potentials. Cases I–IV correspond to various sets of hyperon coupling constants. Contributions from the hyperon populations are included.

共2兲 and in cases 共iii兲 and 共iv兲 we also freeze the value of the proton concentration. For each case, we calculate the total energy of the system with and without K ⫺ condensation. In Fig. 13, we show the energy of the lowest K ⫺ branch for the dispersion equation 共73兲, ␻ min(kជm ,Y p ,␳) minimized with respect to the momentum as a function of baryon density for different proton concentrations. We see that the more protons exist in the matter, the smaller is the value of the density at which the kaon energy ␻ min(kជm ,Y p ,␳) meets the electron chemical potential. Therefore, if the energy gain due to the condensation is large enough to compensate an energy loss due to fermion kinetic energies, the system undergoes a first-order phase transition from the NHM state to the state with a proton-enriched composition and a different density. Let us first compare the energies of NHM, NPM, ISM, and PEM taking into account the possibility for the K ⫺ condensation in each case. For densities ␻ min(kជm ,Y p ,␳)⬎␮e , there is no condensation and the energy density of the system is given by Eq. 共4兲. When ␻ min(kជm ,Y p ,␳)⬍␮e , the K ⫺ excitations appear, replacing partially the leptons. These excitations occupy a single state with the lowest energy and form a K ⫺ condensate. The electron chemical potential is fixed now as ␮ e ⫽ ␻ min(kជm ,Y p ,␳). The energy density of the system with the K ⫺ condensate reads (K) ⫽E mes⫹ E tot

(K) , 兺B E Bkin⫹ l⫽兺␮ e E l共 ␻ min共 kជ m ,Y p , ␳ 兲兲 ⫹E cond

共79兲

(K) where E cond is the energy density of the condensate field related to the mean-field Lagrangian

LK ⫽ 关 ␻ 2 ⫺m K2 ⫺kជ 2 ⫺⌸ 共 ␻ ,kជ 兲兴兩 ␾ kជ 兩 2 .

共80兲

The ␾ kជ is the K ⫺ condensate mean field with the wave vector kជ . This field component should be found from the minimization of an appropriate thermodynamical potential. For simplicity, we neglect higher-order terms, such as ⬀ 兩 ␾ kជ 兩 4 , which represent an effective kaon-kaon interaction in dense baryonic matter. The effective kaon-kaon interaction depends on the structure of the mean field. In the absence of a nonlinear effective kaon-kaon interaction, the

015803-18



FIG. 13. The energy of the lowest K ⫺ branch ␻ min(kជm ,Y p ,␳) minimized in 兩 kជ 兩 , and the electron chemical potential, calculated for the NHM, NPM, ISM, and PEM. In the right and left panels results with and without baryon-baryon correlation effects are shown. In the NHM case, the hyperon couplings of case I are used and the ⌺ * contributions are taken with g ⌺⬘ * ⫽g ⌺ , V ⌺ * ⫽V ⌺ , and g ␴ ⌺ * ⫽g ␴ ⌺ .

condensate field is of the running plane-wave type, ␾ kជ ⫽exp关⫺i␻min(kជ m ,Y p , ␳ )t⫹ikជ m rជ 兴 . The dispersion relation 共73兲 is fulfilled for kជ ⫽kជ m and ␻ ⫽ ␻ min(kជm ,Y p,␳), and the density of the charged kaon condensate ␳ K is fixed by the electroneutrality condition

␳ K⫽

兺B q B ␳ B ⫺ ␳ e ⫺ ␳ ␮

⫺

preference of ISM. From Fig. 14, we see that PEM (Y p ⯝0.7) has a larger energy than ISM. As it is also seen from Fig. 14, the resulting isospin composition can be only slightly above Y p ⫽1/2. In the following, we neglect this small difference and consider ISM as the final configuration. In Fig. 15, we plot the lowest branch of the K ⫺ excitation spectrum ␻ min(兩kជ兩) in the ISM at various densities. On the left panel, baryon-baryon correlations are switched off and on right panel, switched on. We see that for densities ⭓(2.5–3.5) ␳ 0 , the spectra have minima 共dots in Fig. 15兲 at finite values of the kaon momentum kជ m ⫽0. This signals that the transition from NHM to a dense ISM would occur as a first-order phase transition into the state with the p-wave kaon condensate. The calculations shown in Figs. 13–15 are done for g ⌺⬘ * ⫽g ⌺ , V ⌺ * ⫽V ⌺ , and g ␴ ⌺ * ⫽g ␴ ⌺ . The results obtained with the ⌺ * hyperons detached from the mean-field potentials are checked to be of minimal difference. Assuming that the surface tension is large enough, the initial and final state densities can be determined by the double-tangent Maxwell construction 关58,59兴. In Fig. 16, we show such a construction between NHM and ISM states. We see that the critical density for the beginning of a first-order phase transition is equal to ␳ Ic⯝1.4␳ 0 without correlations and ⯝2.1␳ 0 with correlations. The critical densities of the I ⯝5 ␳ 0 and ⯝6 ␳ 0 , respectively. final state are ␳ fin If the surface tension is smaller than some critical value then the phase transition results in the mixed phase 关60,58,61,59兴. In such a case, the local charge-neutrality condition is to be replaced by the global charge-neutrality con-

,

and thus, the energy density of the kaon condensate equals (K) ⫽ ␻ min共 kជ m ,Y p , ␳ 兲 E cond

冉兺 B

冊

q B␳ B⫺ ␳ e⫺ ␳ ␮⫺ .

In Fig. 14, we show the energy per baryon in various baryonic systems, NHM, NPM, ISM, and PEM (Y p ⯝0.7), with and without the kaon condensate 共dashed and solid lines, respectively兲. We see that for a density ␳ c⯝3 ␳ 0 without baryon-baryon correlations and ⯝4 ␳ 0 with correlations, the condensate state in ISM becomes energetically favorable compared to NHM. The transition to the new, more symmetrical isospin configuration increases the Fermi energies of the leptons 共the latter ones are needed to compensate a larger charge of protons兲, but reduces the symmetry energy of nuclear matter and also the total Fermi energy of nucleons. Without a K ⫺ condensate, the energy loss is larger than the gain and the system chooses NHM with the composition shown in Fig. 1 共in Fig. 13, the ISM lines lie above the NPM lines兲. With the K ⫺ condensation, the energy of the system decreases significantly since leptons are replaced by kaons. The latter energy gain is large enough to support the 250

250

PEM

PEM

NPM 200

ISM

Etot/ρ-mN [MeV]

Etot/ρ-mN [MeV]

200

150

100

50

150

100

50

NHM

NHM

NPM

ISM

0

0 1

2

3

ρ /ρ 0

4

5

6

1

2

3

ρ /ρ 0

4

015803-19

5

6

FIG. 14. Total energy per baryon of matter with different particle compositions: without 共solid lines兲 and with 共dashed lines兲 the K ⫺ condensate. Abbreviations are the same as in Fig. 13. The curves in the right and the left panels are drawn with and without inclusion of correlations, respectively. In the NHM case, the parameters are the same as in Fig. 13.


EVGENI E. KOLOMEITSEV AND DMITRI N. VOSKRESENSKY 300

350

2.0 200 150

3.0 100

3.5 50

4.0

0 0.0

0.5

2.0

300

2.5

ωmin(k) [MeV]

ωmin(k) [MeV]

250

3.0

250

3.5

200 150

4.0

100

4.5

50

5.0 5.5

1.0

1.5

0 0.0

2.0

0.5

1.0

1.5

2.0

k [mπ]

k [mπ]

dition. The critical density for the appearance of kaon condensate droplets within a mixed phase is still smaller than the value given by Maxwell construction. The presence of the mixed phase may have interesting observable consequences 共see Ref. 关62兴, and references therein兲. Thus, relying on the analysis above, we argue that the critical density of a first-order phase transition can be even smaller than 2 ␳ 0 , and that such transition occurs into the p-wave condensate state.

In this work, we constructed the K ⫺ polarization operator in dense baryonic matter of arbitrary isotopic composition, including both the s- and p-wave K ⫺ -baryon interactions. We used a relativistic mean-field model to describe the baryon properties. The polarization operator was applied then to study the s- and p-wave K ⫺ condensations in neutron star interiors. The results are presented for two different models of the equation of state, cf. Sec. II and Appendix A. Finite temperature effects can easily be incorporated in our general scheme. To describe the kaon-nucleon interaction, we used the kaon-nucleon scattering amplitude obtained as a solution of the coupled-channel Bethe-Salpeter equation with an interaction kernel derived from a relativistic chiral SU共3兲 Lagrangian with the large N c constraints of QCD 关16兴. The ⌳p ⫺1 , ⌺N ⫺1 , ⌺ * N ⫺1 particle-hole contributions were taken into account in the polarization operator. Effects of the filling of 150

125

125

3

3

Etot-mNρ [MeV fm ]

150

100

NHM ISM

75 50

•

25

•

0 0

1

NHM

100

•

75

ISM 50 25

•

0

2

3

ρ /ρ 0

4

5

6

0

1

2

FIG. 15. The energy of the lowest branch of the dispersion equation 共73兲 as function of the momentum for the ISM at various densities 共labels are densities in ␳ 0 ). The polarization operator is taken according to Eq. 共74兲. Calculations without and with baryon-baryon correlations are shown on left and right panels, respectively. Full circles mark the position of the minimum.

the hyperon H⫽(⌳,⌺,⌶) Fermi seas at densities above the hyperonization point ␳ ⬎ ␳ c,H ⯝(2.5–3) ␳ 0 are analyzed. In Fig. 5, we compared the regular s-wave part of the polarization operator with the simplified form widely used in the literature. The kaon-nucleon ⌺ term extracted from this comparison (⌺⯝150 MeV) was found to be two to three times smaller than what allows for the s-wave K ⫺ condensation in ordinary neutron star matter composed mostly of neutrons. However, we found a sizable attractive support from the hyperon exchange terms of the p-wave scattering amplitude contributing to the s-wave part of the polarization operator. Inclusion of these terms , which were omitted in previous works, makes a second-order phase transition to the s-wave K ⫺ condensate state possible at densities ⯝3 ␳ 0 when the correlation effects are not included. We evaluated baryon-baryon short-range correlation parameters and corrected all the s- and p-wave terms of the polarization operator, accordingly. The correlations increased the critical point of a second-order phase transition to the s-wave K ⫺ condensate state to densities ⲏ(4 –5) ␳ 0 , 共see Fig. 9兲. We estimated 共see Appendix C兲 the effects of the kaon fluctuations. Their contributions are small at low kaon energies, and as first approximation at zero temperature, they can be neglected. Our next observation 共see Appendix C兲 was that at kជ ⫽0, the imaginary part of the pole term of the polarization operator is finite only in a rather narrow interval of kaon energies. If the electron chemical potential crosses the K ⫺ branch within this energy interval, the s-wave condensation

IX. CONCLUSION

Etot-mNρ [MeV fm ]

2.5

3

ρ /ρ 0

015803-20

4

5

6

FIG. 16. Energy densities of NHM and ISM with and without the K ⫺ condensate shown by dashed and solid lines, respectively. For NHM, the forces of case I are used. Calculations with and without baryon-baryon correlations are shown in left and right panel, respectively. The dashdotted lines represent doubletangent Maxwell constructions between two phases.


NEGATIVE KAONS IN DENSE BARYONIC MATTER 100

200 175 3

75

Etot-mNρ [MeV fm ]

3

Etot-mNρ [MeV fm ]

NHM ISM

50

25

150

NHM ISM

125

•

100

FIG. 19. Same as in Fig. 16 but for parameters 共A1兲.

75 50

•

25 0

0 0

1

2

3

ρ /ρ 0

4

5

6

0

1

2

will not occur. However, this possibility is not realized for the parameter choice used in our model. Further, we have investigated the possibility of a secondorder phase transition to the p-wave K ⫺ condensate state. We showed that in the vicinity of the critical point of the s-wave K ⫺ condensation the p-wave part of the polarization operator, induced mainly by ⌳ –proton holes and ⌺ * –nucleon holes, and some regular terms, is large and attractive. This may change the sign of the momentum derivative of the energy at the lowest K ⫺ spectrum branch at the origin. If this occurred, it would mean that there is a p-wave condensate instead of an s-wave one appearing at somewhat smaller density. This statement, although rather model dependent, holds for a wide range of varying parameters. The results depend essentially on the ⌺ * hyperon coupling to the mean-field potentials. In the most favorable case, when ⌺ * is detached from the mean-field potentials, the second-order phase transition to the p-wave condensate state may occur already at ␳ ⬃3 ␳ 0 共with correlations included兲, cf. Fig. 12. This result is also sensitive to the details of the equation of state and to the parametrization of the hyperon-nucleon interaction. For an equation of state with the parameters from Eq. 共A1兲, the critical density is increased with respect to the one calculated with the parameters defined in Eq. 共5兲. We have also discussed the possibility of a first-order phase transition to a K ⫺ condensate state. We have found that in the presence of a K ⫺ condensation, the isospinsymmetrical neutron-proton matter is energetically favorable than the standard nucleon-hyperon-lepton matter for densities ⲏ(3 –4) ␳ 0 共depending on the values of parameters of baryon-baryon correlations兲. This yields a possibility for a first-order phase transition. At such a transition, hyperons are replaced by nucleons and electrons are replaced by the condensate K ⫺ mesons. In dense, isospin-symmetrical nuclear matter K ⫺ excitations are condensed in the p-wave state. With the help of Maxwell construction, we found that the critical density at the beginning of the phase transition is about 2 ␳ 0 with the baryon-baryon correlations included, cf. Figs. 16 and 19. The final state density is about (5 –6) ␳ 0 . Occurrence of such a strong first-order phase transition may have interesting observable consequences: blowing-off a part of the exterior of the neutron star, strong neutrino pulls, gravitational waves, strong pulsar glitches, etc. These effects have been previously discussed in relation to a first-order phase transition to the pion condensate state 关21兴. Here, we

3

ρ /ρ 0

4

5

6

may expect a stronger energy release compared to the pion condensate phase transition since the typical energy scale is larger, m K Ⰷm ␲ . Our derivations can be helpful not only for the description of neutron star interiors, but also for discussion of kaonic effects in other nuclear systems, such as atomic nuclei and in the systems formed in heavy-ion collisions. Therefore, of particular interest is the further more detailed analysis of the p-wave effects on K ⫺ spectra in nucleus-nucleus collisions, also motivated by present SIS and SPS experiments and the future SIS200 program at GSI. ACKNOWLEDGMENTS

The authors acknowledge J. Knoll, T. Kunihiro, M. F. M. Lutz, A. Mocsy, T. Muto, G. Ripka, T. Tatsumi, and W. Weise for stimulating discussions. D.N.V. highly appreciates hospitality and support of GSI Darmstadt. This work was supported in part by DFG 共Project Nos. 436 Rus 113/558/0 and 436 Rus 113/558/0-2兲 and by RFBR Grant No. NNIO-00-0204012. APPENDIX A: VARIATION OF THE PARAMETERS OF THE BARYON INTERACTION

In this section, we investigate the sensitivity of the results of Sec. VIII to the particular choice of the EoS. For comparison, we adjust the parameters of the mean-field model to reproduce the EoS from Ref. 关63兴, which is a good fit to the optimal EoS of the Urbana-Argonne group 关26兴 up to four times the nuclear saturation density and smoothly incorporates the causality limit at higher densities. The corresponding coupling constants of Lagrangian 共2兲 are g ␻2 N m N2 m ␻2

⫽91.25,

g ␴2 N m N2 m ␴2

b⫽0.08675,

⫽195.6,

g ␳2 N m N2 m ␳2

c⫽0.08060.

⫽77.50, 共A1兲

These correspond to the following bulk parameters of nuclear matter at saturation: ␳ 0 ⫽0.16 fm⫺3 , binding energy E bind⫽⫺15.8 MeV, compression modulus K⫽250 MeV, symmetry energy a sym⫽28 MeV, and the effective nucleon mass m N* ( ␳ 0 )⫽0.8m N .

015803-21



The calculations show that a second-order phase transition to the s-wave K ⫺ condensate state occurs at densities (3.5–6) ␳ 0 , depending on the choice of the parameters. A second-order phase transition to the p-wave state may occur in model 共A1兲 for ␳ ⭓(4.5–5.5) ␳ 0 in cases II–IV, and it does not occur for case I up to 6 ␳ 0 . In Fig. 19, we show the double-tangent construction for the EoS with the parameters of Eq. 共A1兲. The first-order phase transition starts at the density ␳ Ic ⯝2.5␳ 0 共with correlations included兲. This value is only slightly larger than 2.1␳ 0 given by the EoS with parameters 共5兲. The final state density I ⯝5.5␳ 0 , i.e., slightly less than the value 5.9␳ 0 given by is ␳ fin model 共5兲. Thus, the main conclusion is that all the general trends in the behavior of the kaon condensation are the same for both models of the EOS. The critical densities of the sand p-wave condensations are only slightly higher for the parameters of Eq. 共A1兲. FIG. 17. The EoS for ISM (N⫽Z) and neutron matter (Z⫽0) calculated with the parameter set 共5兲共dotted lines兲 and 共A1兲共dashed lines兲. Solid lines show the EoS from Ref. 关63兴.

concentration

In Fig. 17, we show energies for the nucleon isospinsymmetrical matter and the pure neutron matter for two choices of the mean-field EoS, model 共5兲, simulating a softer EoS and 共A1兲 simulating a stiffer Urbana-Argonne EoS. Even though parameters 共5兲 and 共A1兲 are rather different, energies and other thermodynamic characteristics of the neutron star matter are rather close to each other for both parameter choices in the absence of a K ⫺ condensate. For the ISM case, the EoS with parameters 共A1兲 is significantly stiffer than the one calculated with parameters 共5兲 at ␳ ⲏ3 ␳ 0 . In Fig. 18, we show concentrations of the baryon species in neutron star matter corresponding to the EoS given by choice 共A1兲 for the four choices of the hyperon-nucleon interaction 共cases I–IV兲 which we have used throughout this paper. We see that the critical density of the hyperonization is ⬇3 ␳ 0 for all choices. The general trends are the same as the ones in Fig. 1. The most essential difference is that the proton concentrations in Fig. 18 are smaller than those in Fig. 1. This should have consequences for the neutrino cooling of neutron stars. Indeed, when the proton concentration exceeds 11% –14% an efficient cooling mechanism becomes operative via the direct Urca processes p→n⫹e⫹¯␯ . This difference might be used, in principle, to select the more appropriate EoS. 1.0 0.8

n

(a)

n

(b)

n

(c)

(d)

n

0.6 0.4 0.2

Ξ

p

--

Λ

Σ

--

Ξ

--

Λ

Ξ

p

p

0.0

Λ

Ξ

--

p

APPENDIX B: NONEQUILIBRIUM GREEN’S FUNCTIONS

Two-point 共2P兲 functions 共a Green’s function or a selfenergy兲 are introduced within the Schwinger-BaymKadanoff-Keldysh 共SBKK兲 approach as, see Ref. 关64兴, iF 共 x,y 兲 ⫽ ⫽

冉

冉

iF ⫺⫺ 共 x,y 兲

iF ⫺⫹ 共 x,y 兲

iF ⫹⫺ 共 x,y 兲

iF ⫹⫹ 共 x,y 兲

冊

具 TAˆ 共 x 兲 Bˆ 共 y 兲典

⫿ 具 Bˆ 共 y 兲 Aˆ 共 x 兲典

具 Aˆ 共 x 兲 Bˆ 共 y 兲典

具 T ⫺1 Aˆ 共 x 兲 Bˆ 共 y 兲典

冊

,

共B1兲

where T and T ⫺1 are the usual time and antitime ordering operators. Note that in notations of Ref. 关65兴, in contrast to the Green’s functions, the ‘‘⫾,⫿’’ self-energies would have an extra sign ‘‘⫺,’’ since they contain the vertices V ⫺ ⫽ ⫺iV 0 and V ⫹ ⫽⫹iV 0 . Not all four components of F are independent. The useful Kubo-Schwinger-Martin relations among them can be found in Ref. 关64兴. We denote the fermionic Green’s functions and selfˆ i, j , respectively, and the bosonic ones ˆ i, j and ⌺ energies by G i, j i, j as D and ⌸ with i, j⫽⫾ . The hats on the fermionic 2P ˆ R,A ⫽ 关 p” ⫺m f functions point on their spin structure, e.g., G R,A ⫺1 ⫺⌺ˆ (p) 兴 . For systems in equilibrium, the ‘‘⫺⫹’’ Green’s functions are connected to the spectral functions and occupation numbers by the Kubo-Schwinger-Martin relations ˆ ⫺⫹ 共 p 兲 ⫽⫺Aˆ f共 p 兲 n f共 ⑀ 兲 , iG

ˆ ⫹⫺ 共 p 兲 ⫽Aˆ f共 p 兲关 1⫺n f共 ⑀ 兲兴 , iG

iD ⫺⫹ 共 p 兲 ⫽A b共 p 兲 n b共 ⑀ 兲 , iD ⫹⫺ 共 p 兲 ⫽A b共 p 兲关 1⫹n b共 ⑀ 兲兴 , 共B2兲

--

Λ

0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 6

ρ /ρ 0

i⌺ˆ ⫺⫹ 共 p 兲 ⫽ ␥ˆ f共 p 兲 n f共 ⑀ 兲 ,

FIG. 18. Concentration of baryon species in neutron star matter for EoS 共A1兲. Four panels correspond to four choices of the hyperon-nucleon interaction parameters specified in text. 015803-22

i⌺ˆ ⫹⫺ 共 p 兲 ⫽⫺ ␥ˆ f共 p 兲关 1⫺n f共 ⑀ 兲兴 ,

i⌸ ⫺⫹ 共 p 兲 ⫽⫺ ␥ b共 p 兲 n b共 ⑀ 兲 , i⌸ ⫹⫺ 共 p 兲 ⫽⫺ ␥ b共 p 兲关 1⫹n b共 ⑀ 兲兴 ,

共B3兲



ˆ fR (p), A b(p)⫽⫺2ImD Rb (p) are the where Aˆ f(p)⫽⫺2ImG fermion and the boson spectral functions, ␥ˆ f(p) ˆ R (p), ␥ ⫽⫺2Im⌸ R are the corresponding ⫽⫺2Im⌺ b f b widths, and n f,b共 ⑀ 兲 ⫽ 兵 exp关共 ⑀ ⫺ ␮ f,b兲 /T 兴 ⫾1 其 ⫺1

共B4兲

are the fermion/boson occupation numbers, and ␮ f,b stands for the fermionic and bosonic chemical potentials. In the quasiparticle approximation ( ␥ b→0 in the bosonic Green’s functions兲, we have A b共 q 兲 ⬇2 ␲ ␦ 关 q 20 ⫺qជ 2 ⫺m 2b ⫺Re⌸ R 共 q 0 ,qជ 兲兴 ⫽

兺i 2 ␲ Z qb,iជ ␦ „q 0 ⫺ ␻ ib共 qជ 兲 …,

共B5兲

1. Correction of the baryon Green’s functions due to kaon fluctuations

In Refs. 关19,20,39– 41,43兴, it has been argued that fluctuation contributions to the hyperon self-energy are essential at low densities and at kaon energies not far away from the mass shell. This is, mainly, due to the presence of dynamically generated ⌳(1405) resonance close to the kaonnucleon threshold. This resonance dominates the s-wave kaon polarization operator at ␻ ⬃m K and it is very sensitive to the Pauli-blocking effect and to the modification of the kaon spectral density. At lower kaon frequencies, corresponding to our main interest, the influence of the ⌳(1405) resonance is small. Thus, what remains to be analyzed are the self-energy contributions of ⌳(1116), ⌺(1195), and ⌺ * (1385) resonances. Here, we demonstrate that in the low energy region and at baryon densities of our interest these contributions are rather suppressed. Let us first show this on the example of the diagram

b,i

ReD Rb 共 q 兲 ⬇

Z qជ

兺i

q 0 ⫺ ␻ ib共 qជ 兲

,

b,i

where Z qជ ⫽1/关 2q 0 ⫺ ⳵ Re⌸ R / ⳵ q 0 兴 q 0 ⫽ ␻ i (qជ ) are the quasiparb ticle normalization factors corresponding to a given spectrum branch ␻ ib(qជ ). As a step towards the nonrelativistic limit, it is convenient to approximate the spin structure of the fermionic Green’s ˆ fR (p)⫽( p” ⫹m f)G fR (p) and Aˆ f(p)⫽(p” functions as G ⫹m f)A f(p). In the quasiparticle approximation ( ␥ f→0), we have A f共 p 兲 ⬇2 ␲ ␦ 关 ⑀ 2 ⫺ ⑀ 20 共 pជ 兲 ⫺Re⌺ R 共 ⑀ , pជ 兲兴 ⫽Z pជ f

ReG fR 共 ⑀ ,pជ 兲 ⬇

1

␲

⑀ pជ

␦ 共 ⑀ ⫺ ⑀ pជ 兲 , 共B6兲

This is the self-energy insertion to the full hyperon Green’s function due to the KN intermediate states. In Eq. 共C1兲, we draw the full vertices. The nucleon line represents the full Green’s function. To be specific, let us concentrate on the kaon-proton self-energy insertion for the ⌳ hyperon. This corresponds to H⫽⌳, K⫽K ⫹ 共according to selected arrow direction兲, N⫽ p in diagram 共C1兲. Within the SBKK diagram technique, using the relations between the retarded and ‘‘⫺,⫹’’ Green’s functions and the self-energies, we obtain ⫺i⌺ˆ R 共 p 兲 ⫽

f

Z pជ

2 ⑀ pជ ⑀ ⫺ ⑀ pជ

,

冕

ˆ ⫺⫹ 共 p⫹q 兲 D R 共 q 兲 ⫹G ˆ R 共 p⫹q 兲 D ⫺⫹ 共 q 兲 Vˆ 关 G

ˆ ⫺⫹ 共 p⫹q 兲 2ImD R 共 q 兲 ⫺iG

Z pជ ⫽ 关 1⫺ ⳵ Re⌺ R / ⳵ ⑀ 2 兩 ⑀ ⫽ ⑀ pជ 兴 ⫺1 , f

obeys the dispersion equation ⑀ pជ ⫽ ⑀ 20 (pជ ) ⫹Re⌺ R ( ⑀ pជ ,pជ ) with ⑀ 0 (pជ )⫽ 冑m f2 ⫹pជ 2 . The self-energy ⌺ R includes averaging over the spin structure ⌺ R ⫽ 21 Tr兵 ⌺ˆ R •(p” ⫹m f) 其 . The mean-field solutions 共3兲 of the equation of motion for a baryon B, following from Eq. 共2兲, can be parametrized by the self-energy ⌺ R ( ⑀ , pជ )⫽2V B ⑀ ⫹V B2 ⫺2m N g ␴ ␴ ⫹g ␴2 B ␴ 2 . Then, we have ⑀ pជ ⫽E B (pជ ), with E B (pជ ) defined in Eq. 共3兲 B and Z pជ ⫽ 关 1⫺V B /E B (pជ ) 兴 ⫺1 ⫽1⫹V B / ⑀ B (pជ )⬇1. where ⑀ pជ

共C1兲

ˆ R 共 p⫹q 兲 D R 共 q 兲兴 Vˆ ⫹G

2

d 4q 共 2␲ 兲4

共C2兲

.

Here, ⫺iVˆ ⫽⫺iVˆ 0 ⌫ and ⫺iVˆ 0 are the full and bare ‘‘⫺’’ vertices, and iVˆ ⫽iVˆ 0 ⌫, iVˆ 0 are the corresponding ‘‘⫹’’ vertices, ⌫ is a scalar form factor, which includes short-range nucleon-baryon correlations and simulates, thereby, the difference between the bare and the full vertices. The last term in Eq. 共C2兲 vanishes, since both retarded Green’s functions have their poles in the same complex q 0 semiplane. To separate the contributions from particles K ⫹ and antiparticles K ⫺ , we may use the following decompositions for the retarded Green’s functions and the Wigner’s densities

APPENDIX C: EVALUATION OF FLUCTUATION TERMS R

We will estimate fluctuation effects to the baryon selfenergies and the feedback from the in-medium kaon modification to the polarization operator. 015803-23

A

D R 共 q 兲 ⫽ ␪ 共 q 0 兲 D K ⫹ 共 q 兲 ⫹ ␪ 共 ⫺q 0 兲 D K ⫺ 共 ⫺q 兲 , ⫺⫹

⫹⫺

共C3兲

D ⫺⫹ 共 q 兲 ⫽ ␪ 共 q 0 兲 D K ⫹ 共 q 兲 ⫹ ␪ 共 ⫺q 0 兲 D K ⫺ 共 ⫺q 兲 . 共C4兲



This allows us to reduce the integration over q 0 in Eq. 共C2兲 to positive values only. For the ⌳ self-energy, we have 1 ˆ R 共 p” ⫹m * 兲其 ⫽ ⌺ R 共 p 兲 ⫽ Tr兵 ⌺ ⌳ 2

冕

R

d4 q ␪ 共 q 0 兲共 2␲ 兲4

* V 2 兵 iG ⫺⫹ 共 p 2m ⌳

In the region interesting for us, ␻ K ⫺ ⱗE ⌳ (0)⫺E p (0), the ␦ functions in Eq. 共C7兲 allow only for the contribution of the

lowest branch of the K ⫺ spectrum. Further evaluation can be easily done for pជ ⫽0. For positive ⑀ , the imaginary part differs from zero only for ⑀ ⬎ ␻ K ⫺ (0)⫹E p (0), and it is equal to

R

⫹q 兲关 D K ⫹ 共 q 兲 ⫺2iImD K ⫹ 共 q 兲兴 ⫹G R 共 p

Im⌺ R 共 ⑀ ,0 兲

⫺⫹ R ⫹q 兲 iD K ⫹ 共 q 兲 ⫹iG ⫺⫹ 共 p⫺q 兲关 D K ⫺ 共 q 兲

* 2m ⌳

⫺⫹

R

⫺2iImD K ⫺ 共 q 兲兴 ⫹G R 共 p⫺q 兲 iD K ⫺ 共 q 兲其 ,

共C5兲

where we used the notations V 2 ⫽V 20 ⌫ 2 共 q 兲 ,

再

V 20

冎

* 2m ⌳

⫽⫺

兺i 2 冕 1

d q 共2␲兲

K ⫺ ,i

2

v 20 qជ 2 ⌫ 2 兵 Z qជ

4␲␣

共C8兲

⳵ ⌬ ⫺ 共 ⑀ ,0,q 兲 ⳵ qជ 2

冏

Ⰶ1, qជ 2 ⫽0

¯ 0 ⫽ 冑⌬ ⫺ ( ⑀ ,0,0)/ ␣ 0 , where we may approximate ¯q ⬇q

Having in mind that we are interested in the self-energy insertion to the ⌳⫺p ⫺1 loop, after expanding the denominators of the fermionic Green’s functions near the poles, and assuming that fermions are nonrelativistic, we may insert * ⫺m N* )Ⰶm ⌳* ,m N* . ( pq)⯝m N* ␻ , p 2 ⯝m N* 2 , and ␻ , 兩 qជ 兩 ,(m ⌳ As a result, we obtain at the nonrelativistic p-wave vertex 2 * ⯝(0.3–0.5)/m ␲2 . For V 20 ⯝ v 20 qជ 2 , with v 20 ⫽C KN⌳ m N* /m ⌳ 2 ⌺ hyperon, we would have v 20 ⫽C KN⌺ m N* /m ⌺* 2 ⯝(0.07–0.09)/m ␲ , and for ⌺ * the corresponding coupling 2 is v 20 ⫽ 32 C KN⌺ * m N* /m ⌺** ⯝(0.3–0.5)/m ␲2 . To evaluate Im⌺ R and Re⌺ R , we use the quasiparticle approximation for the spectral functions 共B6兲. As we shall see below, nonquasiparticle corrections to the intermediate fermion Green’s functions, although exist, produce small contributions to the self-energies at the kaon energies of interest. We first evaluate Im⌺ R . Using Eqs. 共B2兲, 共B5兲, and 共B6兲, we get from Eq. 共C2兲 3

K⫺

Z ¯q ⌫ 2 共 ¯q 兲¯q 3 ␪ 共 ¯q ⫺ p F,p 兲 ,

¯ )⫽0 and ␣ where ¯q is a solution of the equation ⌬ ⫺ ( ⑀ ,0,q 2 ⫽⫺ ⳵ ⌬ ⫺ ( ⑀ ,0,qជ )/ ⳵ qជ 兩 qជ 2 ⫽q¯ 2 ⬎0. If ¯q is small, namely,

共C6兲

Im⌺ 共 ⑀ ,pជ 兲

v 20

⫺⌬ ⫺ 共 ⑀ ,0,0 兲

*兲共 p” ⫹m ⌳ 共 p” ⫺q” ⫹m N* 兲 1 2 ⫽C KN⌳ q” ␥ 5 q” ␥ 5 . Tr 2 * 2m ⌳ 2m N*

R

⫽⫺

关1

p i ⫺n pជ ⫺qជ 兴 ␦ „⌬ ⫺ 共 ⑀ , pជ ,qជ 兲 … K ⫹ ,i p i n pជ ⫹qជ ␦ „⌬ ⫹ 共 ⑀ , pជ ,qជ 兲 …其 ,

⫹Z qជ

i ⫻⌬ ⫾ 共 ⑀ , pជ ,qជ 兲 ⫽ ⑀ ⫾ ␻ K ⫾ 共 qជ 兲 ⫺E p 共 pជ ⫾qជ 兲 , i

共C7兲

␣ 0 ⫽⫺

⳵ ⌬ ⫺ 共 ⑀ ,0,q 兲 ⳵ qជ 2

冏

. qជ 2 ⫽0

From the ␪ function in Eq. 共C8兲, it follows that the fluctuations contribute to the imaginary part of the given diagram 2 ␣ 0. only, if ⑀ ⬎ ␻ K ⫺ (0)⫹E p (0)⫹ p F,p From Fig. 15, we see that ␻ K ⫺ (qជ ) is a very flat function of qជ 2 , and we can, therefore, neglect the momentum dependence of the K ⫺ spectrum for 0⬍ 兩 qជ 兩 ⱗ2 m ␲ . The imaginary part of the diagram at finite pជ is Im⌺ R 共 ⑀ ,pជ 兲

* 2m ⌳

⫽⫺ v 20

冕

¯ 兩 pជ 兩 ⫹q 0

¯ 兩兩兩 pជ 兩 ⫺q 0

d兩 qជ 兩兩 qជ 兩 3 K ⫺ 2 Z ជ ⌫ 关 m N* 4 ␲ 兩 pជ 兩 q

⫹⌬ ⫺ 共 ⑀ ,0,0 兲兴 ␪ 关 ⌬ ⫺ 共 ⑀ ,p F,p ,0兲兴 ⬇⫺

v 20 m N*

2␲

K⫺

Z ¯q ⌫ 2 共 ¯q 0 兲¯q 0 共 ¯q 20 ⫹pជ 2 兲 0

⫻ ␪ 关 ⌬ ⫺ 共 ⑀ ,p F,p ,0兲兴 . The energy and momentum of the ⌳ within the ⌳p ⫺1 loop, contributing to the K ⫺ polarization operator at 兩 qជ 兩 ⫽0 are ⑀ ⫽E p (pជ )⫹ ␻ K ⫺ (0) and 兩 pជ 兩 ⭐ p F,p . Therefore, the conditions for nonzero ␪ functions in Eqs. 共C8 and C9兲 are not fulfilled within the momentum integration interval of the ⌳ p ⫺1 loop. For Re⌺ R , using Eqs. 共B2兲, 共B6兲, 共C3兲, and 共C6兲, we get from Eq. 共C2兲

where i runs over all the K ⫺ and K ⫹ branches. Here, ␻ Ki (qជ ) p are functions of qជ 2 and n pជ ⫽ ␪ (p F,p ⫺ 兩 pជ 兩 ) is the proton occupation function. We used also that occupations of real K ⫹ and K ⫺ mesons are absent at T⫽0 共we do not consider here the processes on the kaon condensate field兲. 015803-24

Re⌺ R 共 ⑀ ,pជ 兲

* 2m ⌳

⫽⫺

兺i p

⫹

冕

d 3 qជ 共2␲兲 K ⫹ ,i

n pជ ⫹qជ Z qជ

i ⌬⫹ 共 ⑀ ,pជ ,qជ 兲

冎

v 2 qជ 2 ⌫ 2 3 0

.

再

p

K ⫺ ,i

n pជ ⫺qជ Z qជ

i ⌬⫺ 共 ⑀ ,pជ ,qជ 兲

共C9兲



For relevant ⑀ , the main contribution is given by the first term with the lowest K ⫺ branch. For 兩 pជ 兩 ⫽0 the integral in ¯ , and we can expand Eq. 共C9兲 is determined by 兩 qជ 兩 ⬃q K,i 2 2 ¯ ⫺qជ ). If Z ជ and ⌫ depend weakly on the ⌬ ⫺ ( ⑀ ,0,qជ )⬇ ␣ (q q momentum, the remaining integration is straightforward and gives Re⌺ R 共 ⑀ ,0 兲

* 2m ⌳

K⫺

⬇

v 20 ⌫ 2 共 ¯q 兲 Z ¯q

2 ␲ 2␣

冏

冋

3 p F,p

3

¯ 2 p F,p ⫹q

冏册

* 2m ⌳

⬇

␲

2

K⫺

⌫ 2 共 ¯q 0 兲 Z ¯q

0

冋

3 p F,p

3

⫹ 共 ¯q 20 ⫹pជ 2 兲 p F,p

冏册

冏

1 兩¯q 0 兩 ⫹p F,p ⫺ 共 ¯q 20 ⫹pជ 2 兲兩¯q 0 兩 ln . 2 兩¯q 0 兩 ⫺p F,p

* 2m ⌳

⫺

⬇c⬅ v 20 m N* ⌫ 2 共 0 兲 Z K0 ␳ p ⬃0.3m ␲ ⌫ 2 共 0 兲 ⫺

⬘ C 0 ⌽ p⌳ „␻ K ⫺ 共 0 兲 ,0… 1⫺ f ⌳

div

3 p F,p

␲

2

ln

p F,p ⫺ 兩 pជ 兩 p F,p

p F,p ⫺ 兩 pជ 兩 , p F,p

共C14兲

and estimate the variation of the Lindhard function 共29兲:

␦ ⌽ p⌳ ⯝

冕

ជ 兩兩 pជ 兩 2 p F,p d 兩 p ␲

0

2

冋

册

1 1 ⫺ . ⌬⫺3c ln共 1⫺ p/p F,p 兲 ⌬

␦ ⌽ p⌳ /⌽ p⌳ ⯝F 共 3c/⌬ 兲 , F 共 a 兲 ⫽a

冕

1

0

dx 共 1⫺x 兲 2 / 共 1⫺a ln x 兲 ,

共C15兲

and find 兩 ␦ ⌽ p⌳ /⌽ p⌳ 兩 ⱗ0.2 for 兩 3c/⌬ 兩 ⱗ0.6.

␳p , ␳0

K⫺

1

⫺

F,p

⯝3c ln

共C11兲

with m N* ⯝0.6m N and Z K0 ⫽Z ¯q ⫽0 ⬃1/关 2 ␻ K ⫺ (0) 兴 ⬃1/(2 ␮ e ) . Assuming that the modification of the p-wave KN⌳ vertex is determined by the graphical equation 共66兲 with ⌳p ⫺1 intermediate states, we can use the corresponding part of Eq. 共68兲 and write ⌫共 0 兲⯝

⯝ v 20 m N* ⌫ 2 共 p F,p 兲 Z Kp

Using Eq. 共C12兲, we write

Now, we estimate the real part of the hyperon self-energy at pជ ⫽0 and ⑀ ⫽E p (0)⫹ ␻ K ⫺ (0). We have ¯q 0 →0 and Re⌺ R

* 2m ⌳

共C10兲

The extension for finite pជ can be easily done if we neglect the qជ dependence of the kaon spectrum. This is justified by our numerical analysis for 兩 qជ 兩 ⱗ2m ␲ . Then, ⌬ ⫺ ( ⑀ ,pជ ⫺qជ ,0) ⬇ 关 ¯q 20 ⫺(pជ ⫺qជ ) 2 兴 /2m N* , and the integration gives v 20 m N*

冉冊 Re⌺ R

1 兩¯q 兩 ⫹p F,p ¯ 2 兩¯q 兩 ln ⫺q . 2 兩¯q 兩 ⫺p F,p

Re⌺ R 共 ⑀ ,pជ 兲

for ␳ p ⱗ3 ␳ 0 . We conclude that the absolute value of the fluctuation contribution is small for pជ →0 and this can be mimicked by a variation of the weakly constrained parameters of Lagrangian 共2兲. From Eqs. 共C10兲 and 共C11兲, one can see that Re⌺ R is logarithmically divergent at 兩 pជ 兩 →p F,p when ¯q →p F,p . In order to analyze which effects this can produce for the ⌳p ⫺1 contribution to the polarization operator, we separate the leading divergent term

The above estimations prove that the self-energy corrections of the ⌳ propagator induced by the kaon fluctuations 关diagram 共C1兲兴 do not modify the properties of K ⫺ excitations with small energies and momenta, which we consider in the main part of this paper. The same estimation can be ¯ 0 n contribution to the ⌳ self-energy and done also for the K for the ⌺ and ⌺ * self-energies. 2. Correction of the baryon Green’s functions due to pion fluctuations

.

There is another type of diagrams

The Lindhard function can be estimated as ⌽ p⌳ „␻ K ⫺ 共 0 兲 ,0…⬇

2m N* ␳ p ⯝ ⫺ ⌬ p⌳ „␻ K ⫺ 共 0 兲 ,0,p F,p …

␳p , 共C12兲 ⌬ 共C16兲

where ⌬⫽ ␻ K ⫺ (0)⫺E ⌳ (0)⫹E p (p F,p ). The value of E ⌳ (0) ⫺E p (p F,p ) can be estimated from Fig. 5 where this is shown by the dashed-dotted line. We estimate ⌬⬃⫺m ␲ and therefore, ⌽ p⌳ ⬃⫺0.5m ␲2 ␳ p / ␳ 0 . Thus, we obtain ⌫(0)⯝1/(1 ⫹0.3␳ p / ␳ 0 ) and

* 兲 ⱗ0.2m ␲ , Re⌺ R / 共 2m ⌳

共C13兲

relating to pion fluctuations. Pions are soften in the nucleon matter already at densities ⬃ ␳ 0 . The softening effect may result in pion condensation at densities ␳ ⬎ ␳ c ␲ , where the critical density for pion condensation ␳ c ␲ might be smaller than that for the kaon condensation 关66,21兴. However, in this work, we concentrate only on kaon polarization effects. Thus, we disregard such a possibility assuming ␳ ⬍ ␳ c ␲ .

015803-25



The main contribution to Eq. 共C16兲 comes from the width of soft pions due to the Landau damping. The maximum of the pion spectral function is achieved at a small pion energy q 0 ⬍m ␲ and a finite momentum 兩 qជ 兩 ⫽ 兩 qជ m兩 . In the diagram with the ␲ 0 intermediate states the typical momenta of hyperons are ⬃p Fn for Hn ⫺1 loops in the the K ⫺ polarization operator or ⬃p Fp for the H p ⫺1 loops. With a simplifying assumption 兩 qជ m兩 Ⰶ2 p Fn ,2p Fp , the nucleon Green’s function can be factorized out from the integral. The result is then reduced to the calculation of a pionic tadpole 关67兴

From Eq. 共C18兲, we obtain the advanced polarization operator

␦ ⌸ A共 k 兲 ⫽

冕

关 ⫺iP ⫹⫺ 共 k⫹q 兲 D A 共 q 兲 ⫹P A 共 k⫹q 兲 iD ⫹⫺ 共 q 兲

⫺P ⫹⫺ 共 k⫹q 兲 2ImD A 共 q 兲 ⫹iP A 共 k⫹q 兲 iD A 共 q 兲兴

d 4q 共 2␲ 兲4

,

共C19兲

where P corresponds to the NN ⫺1 loop 共including vertices兲. The last term in Eq. 共C19兲 is zero. Let us consider first the real part of the diagram. Using Eqs. 共B2兲, 共B3兲, we obtain 共C17兲 At zero temperature, the in-medium contributions from pion fluctuations given by Eq. 共C17兲 are numerically small for all densities except in the closest vicinity of the pion condensation critical point 关67,21兴. For finite temperatures such contributions are substantially increased 关68兴. The fermion self-energy insertions discussed here enter the loop diagrams of the kaon polarization operator. Since we found them to be rather small, we may still work with the quasiparticle fermion Green’s functions in the hyperon– nucleon-hole loop diagrams treating fermions on the meanfield level, as we did it in main part of the text. 3. Fluctuation contributions to the K

À

␦ Re⌸ A 共 k 兲 ⫽

冕兵

关 1⫹n K 共 k 0 ⫺q 0 兲兴 ␥ P共 k⫺q 兲 ReD A 共 ⫺q 兲

⫹ 关 1⫹n K 共 q 0 兲兴 A K 共 q 兲 ReP A 共 k⫹q 兲其

d 4q 共 2␲ 兲4

,

共C20兲 where ␥ P⫽2ImP A and in the first term we have replaced b ⫽⫺n qb and 共C3兲, we req→⫺q. Using relations 1⫹n ⫺q 0 0 duce the integration in Eq. 共C20兲 to the positive energies:

␦ Re⌸ A 共 k 兲 ⫽

冕

d4 q 共2␲兲

4

␪ 共 q 0 兲兵关 1⫹n K 共 k 0 ⫺q 0 兲兴 A

⫻ ␥ P共 k⫺q 兲 ReD K ⫺ 共 q 兲 ⫹ 关 1⫹n K 共 k 0 ⫹q 0 兲兴

polarization operator

One of the important fluctuation processes in the kaon polarization operator is given by the diagram

A

⫻ ␥ P共 k⫹q 兲 ReD K ⫹ 共 q 兲 ⫹ 关 1⫹n K 共 q 0 兲兴 A K ⫹ 共 q 兲 ⫻ReP A 共 k⫹q 兲 ⫹n K 共 q 0 兲 A K ⫺ 共 q 兲 ReP A 共 k⫺q 兲其 . 共C21兲

共C18兲 The diagram with free vertices was extensively discussed in the context of kaonic atoms 关19,20兴. In Ref. 关19兴 both vertices were taken constants and equal to V 0 ⯝4 ␲ (1 I⫽0 I⫽1 ⫹m K /m N ) 冑(a K¯ N ) 2 ⫹3(a K¯ N ) 2 for isospin-symmetrical I ¯ N scattering length for isospin I. In case, where a ¯ is the K

The NN ⫺1 loop is suppressed at large frequencies ␻ ⬎ 兩 kជ 兩 v F,N ⫹kជ 2 /2m N* . Therefore, the second and the third terms give a small contribution in the energy region of our interest. The fourth term is identically zero 关 n K (q 0 )⫽0 for q 0 ⬎0 at T⫽0]. Thus, we may keep only the first term in Eq. 共C21兲

␦ Re⌸ A 共 k 兲 ⬇

d 4q 共 2␲ 兲4

␪ 共 q 0 兲 V 20 ⌫ 2 关 1⫹n K 共 k 0 ⫺q 0 兲兴 A

⫻ ␥ 0 共 k⫺q 兲 ReD K ⫺ 共 q 兲 .

KN

the threshold region, a large value of the vertex V 0 ⬃4 ␲ /m ␲ was obtained. We will also use the energyindependent bare vertex 共hatched box兲 but estimate V 0 using our amplitudes of Figs. 2 and 3 in the far off-mass-shell region. In this energy region, V 0 ⬃1/m ␲ , i.e., a much smaller value than that in the threshold region. Accordingly, we expect a significant suppression of the contribution from this diagram to the kaon polarization operator. Another suppression comes from short-range correlations taken into account in the full box 共right vertex兲.

冕

共C22兲

Here, we introduced ␥ P⫽V 20 ⌫ 2 ␥ 0 , using the fact that the correlation factor ⌫ is expressed through the Lindhard function ⌽ A , which leads to a ⌫ 2 factor. In the spacelike region, ␥ 0 (k)⫽ ␥ 0 (k 0 ,kជ ) has a compact analytic presentation 关69兴, which at T⫽0 gives

015803-26

␥ 0 共 k 0 ,kជ 兲 ⫽

m N* 2 k 0 2 ␲ 兩 kជ 兩

,

0⬍k 0 ⬍⍀ ⫺ 共 kជ 兲 ,

共C23兲



␥ 0 共 k 0 ,kជ 兲 ⫽

m N* 3 4 ␲ 兩 kជ 兩 3

Re␦ ⌸ A1 共 k 0 ,0兲

关 ⍀ ⫺ 共 kជ 兲 ⫹k 0 兴关 ⍀ ⫹ 共 kជ 兲 ⫺k 0 兴 ,

0⬍⍀ ⫺ 共 kជ 兲 ⬍k 0 ⬍⍀ ⫹ 共 kជ 兲 , ⍀ ⫾ 共 kជ 兲 ⫽

2 兩 kជ 兩 p F,N ⫾kជ 2 2m N*

共C24兲

⫹ .

Let us discuss the low energy contribution 共C23兲 to the polarization operator which we label below by subscript ‘‘1.’’ The q 0 integration is easily done using max共 ␯ ,0兲 ⬍q 0 ⬍k 0 ,

V 20 m N* 2 共2␲兲

2

⌫ 2 共 0 兲 Z K0

冕冊 2p F,N

d 兩 qជ 兩兩 qជ 兩

q ⫹ (k 0 )

2␲2

⫺

冋冉冕

q ⫺ (k 0 )

0

⍀ ⫺ 共 qជ 兲 ⫹

冕


q ⫹ (k 0 )

2␲2

q ⫺ (k 0 )

册

k0 ,

2 bordered by q ⫾ (k 0 )⫽p F,N ⫾ 冑p F,N ⫺2m N* k 0 which follows from the equation k 0 ⫽⍀ ⫺ (q ⫾ ) . The integration results in

Re␦ ⌸ A1 共 k 0 ,0兲 ⯝⫺

Cp F,N 4␲2

␯ ⫽k 0 ⫺⍀ ⫺ 共 kជ ⫺qជ 兲 .

冋冉

1⫺ 1⫺

2k 0 m N* 2 p F,N

冊册 3/2

. 共C28兲

Now, we discuss the contribution from the energy region 共C24兲 which we indicate by subscript ‘‘2.’’

We obtain

Re␦ ⌸ A1 共 k 兲 ⬇

⯝⫺

V 20 m N* 2 共 2␲ 兲2

冋

K⫺

冕

兩 kជ ⫺qជ 兩 ⬍2p F,N

2 d 3 qជ ⌫ Z qជ 共 2 ␲ 兲 3 兩 kជ ⫺qជ 兩

Re␦ ⌸ A2 共 k 0 ,0兲 ⯝

关 k 0 ⫺ ␻ K ⫺ 共 qជ 兲兴 ⫻ 关 k 0 ⫺ ␻ K ⫺ 共 qជ 兲兴 ln max共 ␯ ,0兲 ⫺ ␻ K ⫺ 共 qជ 兲

册

⫺k 0 ⫹max共 ␯ ,0兲 .

⫻

⫻

共C25兲

Re␦ ⌸ A2 共 k 0 ,0兲 ⯝

0

2␲

2

4␲2

k 0 ⫺⍀ ⫺ (qជ )

k 0 ⫺⍀ ⫹ (qជ )

dq 0

关 ⍀ ⫺ 共 qជ 兲 ⫹k 0 ⫺q 0 兴关 ⍀ ⫹ 共 qជ 兲 ⫺k 0 ⫹q 0 兴

q 0⫺ ␻ K⫺

V 20 m N* 3 共2␲兲

4

冕

ជ兩 2p F,N d 兩 q

.

兩 qជ 兩

0

K⫺

⌫ 2 Z qជ

冋

册

⫺

K ⌫ 2 Z qជ min„k 0 ,⍀ ⫺ 共 qជ 兲 ….

We have neglected terms ⬀ 关 k 0 ⫺ ␻ K ⫺ (qជ ) 兴 in the integrand, since these are small due to the flatness of the K ⫺ spectrum. 2 /(2m N* ), we find For k 0 ⬃ ␻ K ⫺ (0)⬎ p F,N

2 /(2m N* ) there is always k 0 ⬎⍀ ⫺ (qជ ), For k 0 ⬃ ␻ K ⫺ (0)⬎p F,N and we find

V 20 m N*

兩 qជ 兩

0

K⫺

⌫ 2 Z qជ ␪ „k 0 ⫺⍀ ⫺ 共 qជ 兲 …

3 2 1 2 ⫺ ⍀⫺ 共 qជ 兲 ⫹2⍀ ⫺ 共 qជ 兲 ⍀ ⫹ 共 qជ 兲 ⫺ ⍀ ⫹ 共 qជ 兲 . 2 2

共C26兲

Re␦ ⌸ A1 共 k 0 ,0兲 ⯝⫺

ជ兩 2p F,N d 兩 q

⍀ ⫺ 共 qជ 兲 ⫻ ␪ „k 0 ⫺⍀ ⫺ 共 qជ 兲 … ⍀ ⫺ 共 qជ 兲 ⍀ ⫹ 共 qជ 兲 ln ⍀ ⫹ 共 qជ 兲

共 2␲ 兲2

冕

冕

冕

For k 0 ⬃ ␻ K ⫺ (0) and kជ ⫽0 this contributes to

V 20 m N* 2

ជ 兩兩 qជ 兩 2p F,N d 兩 q

共 2␲ 兲4

⫻

This contribution can be easily estimated for 兩 k 0 ⫺ ␻ K ⫺ (0) 兩 /k 0 Ⰶ ␻ K ⫺ (0) and kជ ⫽0. We note that due to the flatness of the kaon spectrum ␻ K ⫺ (qជ )⯝ ␻ K ⫺ (0). Hence, for k 0 ⬃ ␻ K ⫺ (0), we can neglect the first term in Eq. 共C25兲. Then,

Re␦ ⌸ A1 共 k 0 ,0兲 ⯝⫺

V 20 m N* 3

⫺

⌫ 2 共 0 兲 Z K0 ␳ N p F,N ⬅⫺

C p F,N

.

4␲2 共C27兲

2 /(2m N* ), the inequalities k 0 ⬎⍀ ⫺ (qជ ) For k 0 ⬃ ␻ K ⫺ (0)⬍p F,N or k 0 ⬍⍀ ⫺ (qជ ) together with 0⬍ 兩 qជ 兩 ⬍2 p F,n determine three regions for momentum integration,

Re␦ ⌸ A2 共 k 0 ,0兲 ⯝3

Cp F,N 4␲2

G共 0 兲,

共C29兲

2 /(2m N* ), we have and for k 0 ⬃ ␻ K ⫺ (0)⬍ p FN

Re␦ ⌸ A2 共 k 0 ,0 兲 ⯝3

Cp F,N 4␲2

2 G 共 x 兲 , x⫽ 冑1⫺2k 0 m N* / p F,N .

Here, we introduced functions

015803-27

共C30兲



G共 x 兲⫽

1 16

冉冕冕冊冋 1⫺x

0

⫹

册

␥ 1 Im␦ ⌸ A ⫽ ⫽ 2 2

2⫺t ⫺4t 2 ⫹4t , dtt 共 4⫺t 2 兲 ln 2⫹t

2

1⫹x

2 5 G 共 0 兲 ⫽⫺ , G 共 x→1 兲 ⬇⫺ 共 1⫺x 兲 . 3 4

共C31兲

冕

k 0 dq

d 3q

0

2␲ 共 2␲ 兲3

0

V 20 ⌫ 2 ␥ 0 共 k⫺q 兲 A K ⫺ 共 q 兲 . 共C35兲

The K ⫺ spectral function can be presented in the form ⫺

The suppression of the nucleon–nucleon-hole loop in the scalar-isoscalar channel can be taken into account as in Ref. 关21兴关for recent review, see Ref. 关70兴, Eqs. 共4兲–共8兲兴 ⌫ ⫽1/关 1⫺2( f ⫹ f ⬘ )C 0 A NN ( ␻ ⫽0,q⫽p F,N ) 兴 , where A NN is the nucleon–nucleon-hole loop 共without the spin degeneracy factor 2兲, and A NN ( ␻ ⫽0,q⫽p F,N )⯝⫺m N p F,N /(2 ␲ 2 ). The Landau-Migdal parameters of the NN interaction are f ⯝0 and f ⬘ ⯝0.5–0.6 关71兴. Thus, we have ⌫⬃1/关 1 ⫹0.3( ␳ N / ␳ 0 ) 1/3兴 . For ␳ N ⱗ(3 –5) ␳ 0 we have C⬃m ␲ . 2 /(2m N* ), the attractive contribution of When ␻ K ⫺ (0)⬎p F,N diagram 共C18兲 is estimated to be ⫺Re␦ ⌸ A ⱗ0.3m ␲2 . The Re␦ ⌸ A remains attractive also for larger densities, when 2 ␻ K ⫺ (0)⬍p F,N /(2m N* ). However, its absolute value is addi2 tionally suppressed by the ratio ␻ K ⫺ (0)m N* /p F,N . The imaginary part of the diagram under consideration describes the processes in which a kaon excitation dissolves into multiparticle nucleon–nucleon-hole modes. If the K ⫺ energy meets the electron chemical potential in the region where the imaginary part of this diagram is, the K ⫺ condensation does not occur via a second-order phase transition. Using Eq. 共C19兲, we obtain Im␦ ⌸ A ⫽

冕

d4 q 共2␲兲

4

共C32兲

Im␦ ⌸ A ⫽

冕␪

␥ 共 k 0 ,kជ 兲 ⫽

兩 kជ ⫺qជ 兩 ⬍2p F,N

⫹

共 2 ␲ 兲 3 2 ␲ 兩 kជ ⫺qជ 兩

再

关 k 0 ⫺ ␻ K ⫺ 共 qជ 兲兴

m N* 2 兩 kជ ⫺qជ 兩 2

关 ⍀ ⫺ 共 kជ ⫺qជ 兲 ⫹k 0 ⫺ ␻ K ⫺ 共 qជ 兲兴

⫻ 关 ⍀ ⫹ 共 kជ ⫺qជ 兲 ⫺k 0 ⫹ ␻ K ⫺ 共 qជ 兲兴 ␪ „k 0 ⫺⍀ ⫺ 共 kជ ⫺qជ 兲

冎

As before, we assume that ␻ K ⫺ (qជ ) is a very flat function in qជ 2 . Then, we obtain

⫹

2m N*

8␲3

⯝␪共 z 兲␪

d q 1 兵 ␥ P共 k⫹q 兲 A K ⫹ 共 q 兲关 n b共 q 0 兲共 2␲ 兲4 2

冉

2 p F,N

m N* 3 V 20

b

冉

冕

共C34兲

The second term gives the main contribution. Taking into account that at T⫽0 there is (1⫹n qK ⫹n kK ⫺q )⫽ ␪ (k 0 0 0 0 ⫺q 0 ), we obtain 015803-28

⫺z

冊

m N* 2 V 20 4␲3

2p F,N

⫺q ⫺ (⫺z)

2 p F,N

2m N*

冉

⫹␪共 z 兲␪ 4

⫺n b共 k 0 ⫹q 0 兲兴 ⫹ ␥ P共 k⫺q 兲 A K ⫺ 共 q 兲关 1⫹n 共 q 0 兲 ⫹n 共 k 0 ⫺q 0 兲兴其 .

冕

m N* 2 V 20

d 3q

⫻ ␪ „k 0 ⫺ ␻ K ⫺ 共 qជ 兲 …␪ „␻ K ⫺ 共 qជ 兲 ⫹⍀ ⫺ 共 kជ ⫺qជ 兲 ⫺k 0 …

4

共q0兲

,

where we separated two different contributions: The first term is the ordinary ␦ function from the spectral branch; the second term is a possible nonquasiparticle contribution, which can be obtained only by the self-consistent solution of Eq. 共C35兲. Let us now consider the quasiparticle contribution to the spectral function given by the first term in Eq. 共C36兲.

共C33兲

Using relations 1⫹n (⫺q 0 )⫽⫺n (q 0 ) and 共C3兲, we reduce the integral in Eq. 共C33兲 to the positive energies: b

␥ 2共 q 兲

共C36兲

␥⫽␪共 z 兲␪

d 4q 1 ␥ P共 k⫹q 兲 A K 共 q 兲关 n b共 q 0 兲共 2␲ 兲4 2

⫺n b共 k 0 ⫹q 0 兲兴 .

␥共 q 兲关 q 20 ⫺m K2 ⫺Re⌸ A 共 q 兲兴 2 ⫹ 41

⫺ ␻ K ⫺ 共 qជ 兲 …␪ „⍀ ⫹ 共 kជ ⫺qជ 兲 ⫹ ␻ K ⫺ 共 qជ 兲 ⫺k 0 … .

With the help of relations 共B2兲 and 共B3兲, we find

冕

⫹

关 iP ⫹⫺ 共 k⫹q 兲 ImD A 共 q 兲

⫹ImP A 共 k⫹q 兲 iD ⫹⫺ 共 q 兲兴 .

Im␦ ⌸ A ⫽

A K ⫺ 共 q 兲 ⫽2 ␲ Z Kq ␦ „0 ⫺ ␻ K ⫺ 共 qជ 兲 …

⫺z

2 p F,N

m N*

H共 x 兲⫽

冊

冕

冕

q ⫹ (z)

q ⫺ (z)

K Z qជ ⌫ 2 兩 qជ 兩 d 兩 qជ 兩

d 兩 qជ 兩 K 2 Z ជ ⌫ 关 ⍀ ⫺ 共 qជ 兲 ⫹z 兴关 ⍀ ⫹ 共 qជ 兲 ⫺z 兴兩 qជ 兩 q

m N* 2 V 20 2␲

⫺z

z

冊

3

2 zZ K0 ⌫ 2 p F,N 冑p F,N ⫺2m N* z ⫺

m N* V 20 32␲

2

冑1⫹x⫺1

3

⫺

4 Z K0 ⌫ 2 p F,N H 共 2m N* z/ p 2F,N 兲 ,

dt 关 4t 2 ⫺ 共 t 2 ⫺x 兲 2 兴 , t

1 H 共 x→0 兲 ⬇4 共 1⫹x 兲 ,H 共 x→8 兲 ⬇ 共 x⫺8 兲 2 , 3

共C37兲



where z⫽k 0 ⫺ ␻ K ⫺ (0) . As follows from the ␪ functions there is no width for energies k 0 ⬍minq兵␻K⫺(qជ )其⫽␻K⫺(qជ m) and for k 0 ⬍ ␻ K ⫺ (0). The width exists only above the branch. Exactly at the branch, i.e., for k 0 ⫽ ␻ K ⫺ (kជ ), the ␦ -function contributes only for kជ ⫽0. Thus, the quasiparticle part of the K ⫺ spectral function generates no width at kជ ⫽0 at the critical point of the second-order phase transition to the s-wave K ⫺ condensation, and k 0 ⫽ ␮ e ⫽ ␻ K ⫺ (0). At low energies, k 0 ⬍ ␻ K ⫺ (kជ ), where the ␦ -function term does not contribute, there might appear another contribution from the self-consistent solution given by the second term of 共C36兲. In the following, we find this self-consistent solution and demonstrate that the width exists even for kជ ⫽0 affecting the critical condition for the s-wave condensation. In order to avoid rather cumbersome expressions we make several simplifying assumptions. We assume that k 0 ⬃ ␻ K ⫺ (kជ )Ⰶ ⑀ FN , and ( ⳵␻ K ⫺ / ⳵ qជ 2 ) 0 is very small in the interval 0⬍ 兩 qជ 兩 ⬍k 0 / v FN . Then, the spectral function simplifies as ⫺

¯ 共 q0兲⫽ A K ⫺ 共 q 0 兲 ⯝A

共 Z K0 兲 2 ␥ 共 q 0 兲

, ⫺ 关 q 0 ⫺ ␻ K ⫺ 共 0 兲兴 2 ⫹ 41 关 Z K0 ␥ 共 q 0 兲兴 2 共C38兲

and the self-consistent solution of Eq. 共C35兲 for the width ␥ is determined by

␥共 k0兲⬇

m N* 2 V 20 ⌫ 2 8␲4 ⫹

冕

2p F,N

0

m N* 3 V 20 ⌫ 2 16␲ 4

冕


冕

0


2p F,N

0

min兵 k 0 ,⍀ ⫺ (qជ ) 其

冕

¯ 共 k 0 ⫺z 兲 dzzA

min兵 k 0 ,⍀ ⫹ (qជ ) 其

⍀ ⫺ (qជ )

␥共 k0兲⫽

m N* 2 V 20 ⌫ 2 8␲4 ⫹ ⫹

⫹

冕

q ⫹ (k 0 )

q ⫺ (k 0 )

再冋冕

0


m N* 3 V 20 ⌫ 2 16␲ 4

q ⫺ (k 0 )

冋冕

冕

⫹

k0

0

冕册 2p F,N

q ⫹ (k 0 )

冎


⫹ ⫻ ⫻

⍀ ⫺ (qជ )

d兩q兩

q ⫹ (k 0 )

兩q兩

⍀ ⫺ (qជ )

⫻关 ⍀ ⫹ 共 qជ 兲 ⫺z 兴 ¯A 共 k 0 ⫺z 兲 ,

0

2p F,N

* /p F,N ] 2p F,N ⫺[k 0 m N

冕冋冕

2p F,N

0


* ]/p F,N [k 0 m N

0


冕

冕冕

⫹

兩 qជ 兩 d 兩 qជ 兩

k0

0

册

冊冕

* ⫺qជ 2 /2m N* [ 兩 qជ 兩 p F,N ] /m N

dz

0

¯ 共 k 0 ⫺z 兲 ⫹ dz zA

2p F,N

* /p F,N ] 2p F,N ⫺[k 0 m N

min兵 k 0 ,⍀ ⫹ (qជ ) 其

⍀ ⫺ (qជ )

兩 qជ 兩 d 兩 qជ 兩

册

m N* 3 V 20 ⌫ 2 16␲ 4

dz 关 ⍀ ⫺ 共 qជ 兲 ⫹z 兴

To solve this equation, we assume ␥ (k 0 )⯝ ␣ k 0 for small values of k 0 which we consider here. The main contribution comes from the third integral of the first line ⬀k 0 . In the first integral 兩 qជ 兩 ⬃k 0 and z⬃k 0 . In the second integral 2p F,N ⫺ 兩 qជ 兩 ⬃k 0 and again z⬀k 0 . Thus, the first two integrals ⬀k 20 . In the fourth integral after the replacement z⫽ 兩 qជ 兩 p F,N /m N* ⫺qជ 2 /(2m N* )⫹ ␰ , we see that ␰ ⬃qជ 2 ⬃k 20 and the integral is ⬀k 40 . In the fifth integral, besides this replacement, we introduce 兩 qជ 兩 ⫽2 p F,N ⫺k 0 m N* /p F,N ⫹y and observe that ␰ ⬀y ⬀k 0 . This integral is ⬀k 20 . Thus, keeping only the third integral, we obtain

␣ k 0⯝

m N* 2 V 20 4␲ ⫻

冕

4

k0

0

0

⫺

2 共 Z K0 兲 2 ⌫ 2 p F,N

z ␣ 共 k 0 ⫺z 兲 dz ⫺

关 k 0 ⫺z⫺ ␻ K ⫺ 共 0 兲兴 2 ⫹ 41 共 Z K0 兲 2 ␣ 2 共 k 0 ⫺z 兲 2

␥ ⯝ ␣ k 0 ⯝m N* V 0 ⌫p F,N k 0 / ␲ 2 ,

¯ 共 k 0 ⫺z 兲 dz zA

min兵 k 0 ,⍀ ⫹ (qជ ) 其

* ]/p F,N [k 0 m N

,

which has the nontrivial solution

⫺

0

2p F,N

冋冉冕

dz

q ⫺ (k 0 )

冕册 ជជ 冕

冕

4

⫻ 关 ⍀ ⫹ 共 qជ 兲 ⫺z 兴 ¯A 共 k 0 ⫺z 兲 .

dz 关 ⍀ ⫺ 共 qជ 兲

冕

8␲ ⫹

¯ 共 k 0 ⫺z 兲 ⫹z 兴关 ⍀ ⫹ 共 qជ 兲 ⫺z 兴 ␪ „k 0 ⫺⍀ ⫺ 共 qជ 兲 …A ⫽

m N* 2 V 20 ⌫ 2

dz 关 ⍀ ⫺ 共 qជ 兲 ⫹z 兴共C39兲

2 where we replaced q 0 ⫽k 0 ⫺z . For k 0 ⬃ ␻ K ⫺ Ⰶp F,N /2m N* , we have q ⫺ ⬇k 0 m N* /p F,N and q ⫹ ⬇2 p F,N ⫺k 0 m N* /p F,N . Then, Eq. 共C39兲 reduces to

共C40兲

for ␣ ⬎2/Z K0 . As one may expect, this inequality indeed ⫺ holds for rather large densities since Z K0 ⬃1/关 2 ␻ K ⫺ (0) 兴 Ⰷ1/m ␲ for small values k 0 ⬃ ␻ K ⫺ (0). A principal question is whether there is a width for K ⫺ at low energies. A second-order phase transition with a K ⫺ condensation cannot occur if ␻ K ⫺ (0) crosses ␮ e in the energy region, where imaginary part of the polarization operator exists. The condensation is possible when the electron chemical potential exceeds ␻ K ⫺ (0) reaching the upper border of the 2 /(2m N* ), we region with the width. At least for ␻ K ⫺ (0)⬎ p FN cannot find self-consistent solution for ␥ . We would like to stress that a second-order phase transition to the s-wave K ⫺ condensation may not occur only if ␻ K (0)⫽ ␮ e 2 ⬍ p Fn /(2m N* ). For realistic values of parameters, we have

015803-29


EVGENI E. KOLOMEITSEV AND DMITRI N. VOSKRESENSKY 2 ␻ K (0)⫽ ␮ e ⬎p Fn /(2m N* ) and there is no problem for the ⫺ s-wave K condensation. Thus, the above examples demonstrate that in spite of many new peculiarities associated with the fluctuation processes we may drop the fluctuation contributions for a rough analysis and treat baryons on the mean-field level. A diagram similar to Eq. 共C17兲共with kaon lines instead of hyperon lines and a different vertex兲 describes the K ⫺ ␲ -␲ K ⫺ interaction. Within the simplifying assumption for the pion momentum 兩 qជ m 兩 Ⰶ2 p F,n the diagram is reduced to

共C41兲 At zero temperature, the in-medium contribution of pion fluctuations given by Eq. 共C41兲 is numerically small except for a narrow region near the pion condensation critical point 关67,21兴. For finite temperature such contributions are substantially increased in the vicinity of the pion condensation critical point 关68兴.

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