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Home , Kaon

JP0150781 JAERI-Conf 2001-012

4. Kaon condensation in hyperonic

Takumi Muto* 275-0023 =PHJIISSSP7ii£H 2-1-1

Department of Physics, Chiba Institute of Technology, 2-1-1 Shibazono, Narashino, Chiba 275-0023, Japan

In-medium kaon properties are studied in hyperonic matter, where are mixed in the ground state of -star matter. Kaon dis- persion relations and dependence of kaonic modes on the number density are obtained. P-wave kaon-baryon interactions as well as the s-wave interactions are taken into account within chiral effective Lagrangian, and the nonrelativistic effective baryon-baryon interactions are incorporated. It is shown that the system becomes unstable with respect to a spontaneous creation of a pair of the -hole modes with K+ and K~ quantum numbers, stemming from the p-wave kaon-baryon interaction. The onset density of this p-wave kaon condensation may be lower than that of the s-wave K~ condensation.

I. INTRODUCTION

Various hadronic phases in neutron stars have been considered both experimentally and theoretically. Possibility of kaon condensation is a long standing problem concerning a phase transition of high-density matter [1,2]. Motivated by kaon condensation, in-medium kaon properties have been also elaborated through kaon-baryon scattering, kaonic , and subthreshod K± production in heavy-ion collisions. On the other hand, it has been discussed based on the recent development of hypernulear experiments that hyperons (A, H~, E~ • • •) may be mixed as well as , , and in neutron-star matter. The kaon condensation and hyperonic matter have been investigated separately from theoretical points of view. But the possible existence region may overlap each other, so that the competition or coexistence problem of these phases have to be elaborated. Recently, we have pointed out a new mechanism of kaon condensation realized in hyperonic matter stemming from the p-wave kaon-baryon interaction [3]. In this talk, we have discussed in detail the interplay between kaon condensation and hyperonic matter by considering the kaon dispersion relations and dependence of kaonic modes on the baryon number density.

*Email address; [email protected]

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II. FORMULATION

A. Kaon-baryon interaction

We take into account the kaon-baryon interaction by the use of the effective chiral Lagrangian by Kaplan and Nelson [1], which includes both s-wave and ^-wave interac- tions. For simplicity, we take only the (p), neutron(n), A, £~, and E~ for the octet , and consider the chemically-equilibrated matter including the . The ef- fective Hamiltonian Hef! is obtained with the introduction of the charge chemical potential n(=/j,K = /ie) from the charge neutrality condition. After reducing to the nonrelativistic form for the baryonic part of Hes by the Foldy-Wouthuysen-Tani transformation, we add a potential term Vi (i=p, n, A, E~, E~) for each diagonal matrix element of the effective Hamiltonian for baryons. The effective ground state energy density of the system, £&, is obtained after diagonalization of the baryonic part o£Hes- We assume a plane-wave type for the classical kaon field as K~ = —^=0expi(k • r — fit), where / is the decay constant, 9 the chiral angle, and k is the kaon momentum. After expanding the effective energy density £,ff with respect to 9 around 9 = 0 as 2 f 1 2 A £eS = £eg(9 = 0) ——Dj{ (n,]ii\pB)9 + O(9 ), one obtains the kaon inverse propagator:

2 2 2 , k; pB) = u - k - m K + —

where S^t (i = p, n, A, S , H ) are the 'kaon-baryon sigma terms', which give the s- wave scalar attraction, 5Mjj the baryon mass differences, pi the number densities for the baryons i, gAp, g-£-n, and

B. Potential contribution

For the potential terms Vt in hyperonic matter, we refer to the nonrelativistic potential energy density £^ot by Balberg and Gal [4]. We define V as Vi = d£pot/dpi by neglecting the momentum dependence of baryon potentials. The coefficients are determined so as to reproduce the saturation properties of the symmetric nuclear matter: the symmetry energy ~ 30 MeV , the potential depths of the A(=-27 MeV) [4] and the E~ (=-16 MeV) [5] in nuclear matter. For V^-, we take the two cases: (I) Following recent analyses of £ hypernuclei [6] and S~ data [4], the potential depth of the E~ in nuclear matter is 3 taken to be repulsive, i.e., VE-{pB = p0, N = Z)=23.5 MeV (po=0.16 fm~ ). (II) As is conventionally used, it is taken to be attractive, i.e., V^-^PB = Po,N = Z)=—27 MeV.

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III. NUMERICAL RESULTS

Here we mainly discuss the results in Case I. The particle fractions as functions of the baryon number density ps are shown in Fig. 1 for the repulsive V%-.

3 Case I ( Repulsive Vi-) The A appears at pB ~ 0.37 fm , and its fraction rapidly increases with density, ex- 3 ceeding the proton fraction at pB ~ 0.4 fm~ . Soon after the appearance of A, the E~ are 3 mixed at pB ~ 0.4 fm~ , and its fraction in- creases with density. On the other hand, the electron fraction rapidly decreases after the appearance of the A and E~. Our results qualitatively reproduce the results in Ref. [4]. In Fig. 2, we show the dispersion relations of kaonic modes as functions of |k|. (a) is 3 for EKn=305 MeV , pB=0.50 fm~ (dashed line) and />B=0.57 fm~3 (solid line), (b) is 3 10"3 for E =403 MeV and p =0-48 fm- . 0.4 0.6 0.8 Kn B Fig. 1 PB (fm"

(a) Case I 3 (b) Case I PB = 0.50 fm" 3 SKn=305 MeV pB = 0.57 fm" 2Kn=403 MeV PB =0.48 fm-3

0 400 5 L- n-l 3 ?nn A E~ A"1 PTV 0 ?nn 190 180 -200 1170 • 3 160 ) 100200300400 -400 —C -400 k (MeV)

400 800 1200 400 800 1200 Fig. 2 k (MeV) g. 2 k(MeV)

The branches are characterized by a sign of the residue at their poles of DK, as is the case [7]: If dDj^/du; > 0 (dD^/du < 0) at the pole, the mode has the same quantum number as the K~ {K+). There are three collective particle-hole branches in addition to the K~ and K+ branches which reduce to the free kaons when the kaon-baryon interaction is turned off: the pA"1 which has the same as the K+, the H'A-1, and E~n~{ which have the same quantum number as the K~ (The superscript

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' — I1 denotes a hole state. ) The appearance of the pA l mode instead of the Ap 1 is due to the fact that the A is more abundant than the proton at this density (Fig. 1). We can see from Fig. 2 that the E'A"1 and pk~l branches merge at certain momentum kc(= 984 MeV) and density (PB—0.57 fm~3). The two excitation modes merge at the UJ axis, where the double-pole condition, £^1=0 and dDj^/do;=0, is satisfied. Physically, a pair of the two modes, [S-A"1] and [pA"1], are created spontaneously with no cost of energy because the energy of the [pA"1] mode with the quantum number K+ is to be reversed in sign [7]. Hence the system is unstable with respect to a pair creation of [S^A-1] and [pA-1] modes. This instability originates from the p-wave kaon-baryon interaction and is similar to that of pion condensation [7]. In Fig. 3, we show the dependence of the excitation energies of kaonic modes on the + baryon number density except for the K . (a) is for S^n=305 MeV and |k|=500 MeV, and (b) is for 5^=403 MeV and |k|=100 MeV.

(a) ZKn=305 MeV k = 500 MeV (b) ZKn=403 MeV k = 100 MeV 500 500

8.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 Fig. 3 3 Fig. 3 3 pB(fm- ) PB(fm- )

For (a), the K~ is repelled far from the remaining particle-hole modes over the relavant densities, and there is no level crossing. The E^A"1 and pA-1 modes merge at /9B — 0.60 fm~3, which corresponds to the instability with respect to p-wave condensation. The qualitative feature is also applied to the case at the critical momentum A;c which satisfies dDj}/d\k\ = 0, as well as the double-pole condition. The particle-hole branches such as the H^A"1 and pA"1 do not depend much on the value of |k|, nor does the critical point for the p-wave condensation. For E#n=403 MeV, on the other hand, the excitation energy of the K~ is small as compared with the E/cn=305 MeV at a given density due to the larger s-wave scalar attraction and the smaller momentum |k|, and one can see in Fig. 3 (b) that there are energy gaps between K~ and S~ra-1 branches and the Yrn~x and H-A"1 branches owing to the level crossings. As a result of the level crossings, the E~A-1 branch takes over the characteristics of the K~, the excitation energy of which changes appreciably depending on the magnitude of the s-wave scalar interaction simulated by £«•„. Thus, when the 1 level crossing occurs, the behavior of the E^A" branch is sensitive to the value of HKTI,

-21 - JAERI-Conf 2001-012 so that the critical point for the p-wave condensation, which is given by the merge point of the E~A~1 and the pA"1 branches, also depends on £#„. As seen from the K~ branch in Fig. 2, the K~ with k=0 has a minimum energy ^min(^'~)- Due to the substantial decrease of the charge chemical potential \x with density in the presence of hyperons as compared with the conventional neutron-star matter case, the condition giving a critical density of the s-wave K~ condensation, ujm{n(K~) = fj,, is not met at this density, and the onset of the p-wave condensation preceeds the s-wave 3 3 K~ condensation: p%{p)~ 0.57 far , p%{s)~ 0.67 far for S*rn=305 MeV, and p|(p)~ 3 3 0.48 far , pg(s)~ 0.50 fm- for EKn=403 MeV. 3 In Case II (the attractive Vs- ), the £~ appears first at pB ~ 0.3 far as a - mixing. The onset density of A becomes a little larger as compared with Case I, while the onset density of E~ is pushed up to a high density (pB ~ 0.68 far3). Due to the strong attraction of V^-, the E~n-1 branch is softer than the S~A-1 and K~ branches, so that the E^n"1 merges first with the pA"1 branch. In Case II, p%{v)~ 0.64 far3 and p^(s)^ 3 3 3 0.72 fm- for SKn=305 MeV, and p%(p)~ 0.53 far and pg(s)~ 0.54 far for EKn=403 MeV. It is to be noted that the pA"1, S~A~\ and T.'n'1 branches hardly depend on the magnitude of E#n, as far as T,Kn is not very large, so that the critical density of the p-wave pair condensation of these particle-hole modes changes little even for T,Kn=0.

TV. SUMMARY AND CONCLUDING REMARKS

We have shown that a collective pA"1 mode carrying the K+ quantum number appears over the densities where the A is more abundant than the proton. The system becomes unstable with respect to a creation of [E~A-1] and [pA-1] pair or [E~rrx] and [pA"1] pair (p-wave kaon condensation), which stems from the p-wave kaon-baryon interaction. The large mixing of A as compared with that of the proton is crucial for the appearance of the pA"1 mode. The onset density of this instability is lower than that of the s-wave K~ condensation for a standard value of the parameter HKn simulating the magnitude of the s-wave kaon-baryon scalar interaction, and it hardly depends on the value of E^n as long as Y,Kn is not too large. Our model used for the p-wave kaon-baryon interaction is based on the leading order expansion in the chiral perturbation theory. Higher order terms in chiral expansion are considered to be quantitatively important to kaon dynamics in highly dense matter. It has to be clarified whether the instability of the system with respect to the p-wave condensation leads to a fully condensed phase beyond the critical density. In this context, the EOS of the p-wave condensed phase and the characteristic features of the system have to be examined. Mixing of hyperons only already leads to appreciable softening of the EOS. Hence, further development of kaon condensates in hyperonic matter would make the EOS too soft to obtain the observed masses ~ 1.4M©. Relativistic effects may help weaken the softness of the EOS, since it has been shown that the energy gain of kaon condensation coming from the s-wave scalar attraction is suppressed by the relativistic effects [2]. In addition, in view of making the EOS consistent with observations, some realistic effects which reduce the p-wave kaon-baryon attraction should be taken into account: E.g., (1) vertex renormalization at the p-wave kaon-baryon vertices in terms of form factors, and (2) short-range correlations between baryons.

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We have not taken into account the effects of the other subthreshold resonances such as E(1385) (abbreviated to S*) on the dispersion relations of the kaonic modes. However, the excitation energies of these resonances are of the order ~450 MeV so that their branches lie far above the other particle-hole branches, pA"1, E^n"1, E~A-1. Furthermore, their coupling strengths to the K~N are not so large as compared with g^p, g%-n, and

[1] D. B. Kaplan and A. E. Nelson, Phys. Lett. B175 (1986), 57. [2] H. Fujii, T. Maruyama, T. Muto and T. Tatsumi, Nucl. Phys. A597 (1996), 645. [3] T. Muto, Proceedings of the VII International Conference on Hypernuclear and Strange , Torino, Italy, October 23-27,2000, to be published in Nucl. Phys. A. [4] S. Balberg and A. Gal, Nucl. Phys. A625 (1997), 435. [5] T. Fukuda et al, Phys. Rev. C58 (1998), 1306. [6] J. Dabrowski, Phys. Rev. C60 (1999), 025205. [7] A. B. Migdal et al., Phys. Rep. 192 (1990),179.

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