Kaon effective mass and energy in dense nuclear matter

Kaon effective mass and energy in dense nuclear matter

18 August 1994 PHYSICS LETTERS B ELSEVIER Physics Letters B 334 (1994) 268-274 Kaon effective mass and energy in dense nuclear matter Jtirgen Schaf...

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18 August 1994 PHYSICS LETTERS B

ELSEVIER

Physics Letters B 334 (1994) 268-274

Kaon effective mass and energy in dense nuclear matter Jtirgen Schaffner, Avraham Gal 1, Igor N. Mishustin 2,3, Horst St6cker, Walter Greiner Institutfiir TheorettschePhys~k,J W. Goethe Umversitiit,D-60054 Frankfurtam Main, Germany Received 11 March 1994, revised manuscriptreceived 7 June 1994 Editor' C Mahaux

Abstract Within the framework of the relativistic mean field model we study the properties of kaons and antikaons in dense nuclear matter. The kaon effective mass tends to saturate at high densities. Kaons do not undergo condensation. Antikaon condensation may occur m chiral models, only for relatively high values of the nucleon effective mass, m*/mN > 0.75, and does not occur in models based on one boson exchange underlying KN interactions.

1. Introduction

Nuclear matter can become highly compressed in relativistic heavy ion collisions. This rames the possibility of studying hadron properties under extreme conditions. Medium modifications of kaons have attracted much attention recently since Kaplan and Nelson suggested the likelihood o f kaon condensation in dense matter [ 1 ]. The onset o f kaon condensation will have serious implications for the composition of interior cores o f neutron stars [ 2 - 4 ] , for the equation of state of nuclear matter [ 5 - 7 ] , and for central collisions of relativistic heavy ions, as e.g. for the transverse spectra of kaons and antikaons [ 8,9]. Here we wish to examine the properties of kaons and antikaons in dense matter using the relativistic mean field ( R M F ) model [ 10]. The nucleon coupling con1 Permanent address Racah Institute of Physics, The Hebrew Umverslty, Jerusalem 91904, Israel 2 Current address. The Niels Bohr Institute, Blegdamsvej 17, DK2100 Copenhagen, Denmark 3 Permanent address: I.V Kurchatov Institute of Atomic Energy, Moscow 123182, Russia 0370-2693/94/$07 00 © 1994 Elsevier Science B.V All rights reserved SSDIO370-2693(94)OO749-W

stants are fitted to the ground state properties o f nuclear matter, here taken as E / A = - 16 MeV, Po = 0.15 f m - 3 and a compressibility K = 300 MeV. One parameter, the effective mass m * of the nucleon at normal nuclear density Po, is left free. Fits to the properties o f finite nuclei suggest m * / m N = 0.55 in order to get a correct spin-orbit splitting [ 11 ], while an optical potential analysis indicates m * / m N ~ 0.7 _ 0.05 [ 12]. To cover the whole range we use in the following the parameter sets given in [ 13] for m~,/mN=0.55, 0.65, 0.75, 0.85. W e consider two different coupling schemes for the kaon-nucleus interaction as follows: (i) couplings given by chiral models; and (ii) couplings motivated by underlying one-boson exchanges ( O B E ) in the K N system.

2. K N interaction from chiral models First we add to the free Lagrangian kaon-nucleon interaction terms motivated by chiral models. Kaplan and Nelson [ 1 ] considered

- ~ ~rN = ~

NNKK,

( 1)

J. Schaffner et al / Physics Letters B 334 (1994) 268-274

269

wherefK = 93 MeV is the kaon decay constant and ~rN is the K N sigma term. This term gives a scalar attraction in s-waves and shifts the kaon mass to an effective mass given by [ 1 ]

Assuming Ps---PB, m* vanishes for the critical value

Ps m . 2 = rnZr - 2~KNf---~r.

For P>~Pc, kaons (as well as antikaons) form a condensate. Derivative corrections are uncertain in magnitude, but will not change this value of Pc [ 3]. In Fig. la we plot the kaon effective mass, Eq. (2), as a function of the baryon density Ps, where the scalar nucleon density is taken from the RMF model [ 10].

[9] 2

2

~ 1.0 fm-a--~6po .

mrfr

Pc= ~KN

(2)

The value ,YKN= 2m~, consistent with KN scattering data, is often used in calculations [ 3 ]. The quantity Ps denotes the scalar density of nucleons which depends implicitly on the baryon density Pa of the medium.

Chlral interaction 500 450 400

~ " 300

~

""

--.

250

" - " ""

200

X

\

150 ,

\

100 .... 50 __

o

m*/m=0 55 * m,/m=0.65 m./m---0.75 m /m=0.85 ps~pB 1

\\ \ \ I

2

3

4

5

2

3

4

5

6

7

8

9

7

8

9

b)

~ 6oo

20O

0

'~"

0

1

6

"

10

P/P0 Fig 1 (a) The effecUve mass of the kaon (and antikaon) tn nuclear matter versus the baryon denmty, using a scalar interaction from chn'al models m * / m den•tes the effe•twe mass •f the nuc•e•n at n•rma• nuc•ear denslty. The ••wer dash-d•tted hne ls calcu•ated f•r the appr•ximati•n Ps = Pa (b) The energy of the kaon (upper group of curves) and antikaon (lower group of curves) m nuclear matter versus the baryon density, using interaction terms from chiral models Same labelhng as in (a)

J. Schaffner et aL / Physics Letters B 334 (1994) 268-274

270

The approximation ps =PB (dash-dotted line) results in kaon condensation at a density of about 6/90as shown above. However, the situation changes significantly when Eq. (2) with correct Ps is used instead. Here kaon condensation occurs only for a nucleon effective mass (at Po) of m*/mN > 0.75. For lower effective masses kaon condensation does not occur as the corresponding curves saturate at higher densities. This is a relativistic effect related to the reduction of the nucleon effective mass at high densities. The effect is more pronounced for stronger fields (i.e. for lower nucleon effective mass at Po) and for higher densities. Additional terms have also been considered in chiral models [ 1,3]

"$a~iv=

3i

8f 2 1QT~,N(RO~'K- KO~'K') ,

(3)

which gives a vector interaction for kaons. This term gives repulsion for kaons in nuclear matter while it is attractive for antikaons [3] due to G parity. The wave equation for kaons with scalar and vector interactions now reads 3i

-

(O~,O~'+mZx- ~ K N Ps + 4--~KNT~NO~')K=O, -f"ffK

(4)

after accounting for the conservation of the baryon current. Fourier transformation yields for zero momentum _ w2+rn 2

~KN

f~

3 p~+ 4f-----~KpB oJ = 0 .

(5)

The energy oJ(K) of kaons in the nuclear medium is given by =

m

+ ~

PB,

(6)

where m* is defined by Eq. (2). We note that o~(K) > 0 always, even when the effective mass crosses zero, so that kaon condensation does not occur in nuclear matter. It is important to recognize that the vector term appears twice [ 14]. It shifts both the energy and the effective mass term under the square root. Some authors (e.g. [ 15] ) have discarded the p~ term above on grounds that the Lagrangian (3) amounts to a linear density approximation. A definitive resolution of this question must await a systematic treatment of higher

density terms. For antikaons, reversing the sign of the last term in Eq. (6) yields 3 o) (/~) =

m~2 "[-k~K PB ]

2_

3 -~K PB .

(7)

Note that the energy of antikaons will always be positive unless the effective mass of kaons crosses zero. Fig. lb shows the energy of kaons and antikaons in nuclear matter according to Eqs. (6), (7). The nucleon scalar density is taken from the RMF model. The dashdotted lines stand for the approximation Ps = PB- Antikaon condensation sets in only for nucleon effective masses (at Po) m*/mN > 0.75, in accordance with Fig. la. For nucleon effective masses m*/mN <0.75 the kaon effective mass saturates at a finite positive value as PB ~ oo (see Fig. 1a) and antikaon condensation does not occur for any density. This is so because ~o(k) ~ 0 only for PB ~ oo. We emphasize again that the energy of antikaons does not depend linearly on the density in the dispersion relation (7). Finally, the energy of a/~K pair is given by twice the square root m Eqs. (6), (7). Fig. lb makes it clear that this is always (i.e. for all densities) larger than the free kaon mass, and therefore above the energy of the two pion system, in contrast to the expectation raised m

[151. 3. K N interaction from one-boson exchange models

Up to now we considered a hybrid model, calculating the nuclear environment by an effective meson field theory while using dispersion relations from chiral models. A consistent treatment couples kaons directly to meson fields generated by the nucleons. The scalar interaction term in the Lagrangian proposed in [ 16] is

2 , m = - go~cmKo'K,K .

(8)

It shifts the kaon effective mass according to m . 2 =m2K+go,KrnKo r ,

(9)

where the attractive scalar field o-is negative. Note that this definition is different from Eq. (2) due to the selfinteraction terms of the scalar field. From [ 16] one finds the value

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J. Schaffner et al. / Physlcs Letters B 334 (1994) 268-274

= 62.8 G e V - z . GKN ~ go~cgo~v 2 m~

attraction is m a x i m a l , resulting in a m i n i m a l kaon e f f e c t i v e mass

(10)

m,Z(min) =m 2_

The use of GrN instead of g,~Kensures that the potential felt by kaons is the same for any parametrization of the N N interaction. The nucleon effective mass in the RMF model is given by m * = mN + goTvCr•

go¢: m x m u .

(12)

gou This yields asymptotic values of m * (rain) -- 420, 400, 366, 243 MeV for the parameter sets with m * / m u = 0.55, 0.65, 0.75, 0.85, respectively [13]. Fig. 2a shows the density dependence of the kaon effective mass Eq. (9) for the various parameter sets. There is no zero crossing at all; in fact, the kaon effective mass is always higher than 400 MeV, up to densities of 10

(11)

It has been shown [ 10] that m * approaches zero at high densities. In this limit, the corresponding scalar

OBE mteractaon 500 450 400 350

:~ 25o .K-

N 2oo 150 100 - - m*/m--0.55 - - m*/m--0.65 50 . . . . m*/m--0.75 m*/m=0.85 0 1

2

3

4

5

6

7

8

9

:

9

10

1600 1400 1200 ~0 1000

K

2-2

" ......

"

600 400 200 0

0

1

2

3

4

5

6

7

8

plpo

Fig. 2 (a) The effective mass of the kaon (and antikaon) in nuclearmatter versus the baryondensity,using a scalar interactionmotwated by one-bosonexchange models. (b) The energy of the kaon (upper group of curves) and antikaon (lower group of curves) in nuclear matter versus the baryon density,using interactionsmotwated by one-bosonexchangemodels.

J. Schaffner et al ~Physics Letters B 334 (1994) 268-274

272

Po. This behaviour is independent of the parametrization of the N N force. Note that this result is consistent with K N scattering [ 16]. Furthermore, there are also interaction terms generated by vector meson exchanges. For a charge symmetric nuclear matter the additional term in the Lagrangian reads Sa,oK

=

- ig~,K( k,O~,KW'-- KOt, K W ' ) ,

(13)

where V~ denotes the o~-meson vector field. It gives rise to repulsion for kaons and attraction for antikaons. Note that this term can be easily transformed into that of the chiral model Eq. (3) [ 9 ]. The dispersion relation for an antikaon with zero momentum is now modified to ~o( R ) = ~/m2 + g~,Ko'rnK + (g,oKVo) z - g,olcVo .

(14)

For kaons, reverse the sign of the last term. In the RMF model the time component of the vector field is given by 2

m ~oVo -- g~oNPB ,

(15)

while the vector components V vanish in the medium rest frame. Note that the vector potential appears twice in Eq. (14). Therefore, in this model too KK condensation does not occur, since the sum energies for a kaon and an antikaon will always be positive. The coupling of a kaon to the ~omeson can be related to that of a pion to the p meson via SU (3). Ideal mixing gives gKmo = g ~ o / 2 = 3.02 where g~,w is known from the partial decay width of the p meson. The energy of an antlkaon is plotted in Fig. 2b versus the baryon density. The/~ energy drops slowly to 10(O200 MeV at 10po, but never crosses zero energy. This saturation is quite robust against the choice of parametrization. Hence antikaon condensation does not occur at any density. This behaviour can be easily understood on general grounds as discussed above (see discussion of Eqs. (6), (7)): the vector field grows linearly with the baryon density in nuclear matter. We also plot in Fig. 2b the energy of a kaon in the medium. As expected, it increases monotonically with PB. Only for m * / m l v > 0.75 does the kaon energy show a small dip at low density. 4. Discussion We examined the in-medium properties of kaons and antikaons in the RMF models for two classes of models

for interaction terms: chiral and OBE models. Positive strangeness kaon condensation does not occur, even in a precursor form. Antikaon condensation is possible only for a limited range of parameters in chiral models when the effective mass goes down to zero at sufficiently high density. In OBE models the kaon effective mass is higher than 400 MeV up to 10190and antikaon condensation does not occur at all. This relativistic effect results from the difference between the nucleon scalar and vector densities in dense matter. There is also a saturation effect for the energy of antikaons due to a cancellation at high densities between the two terms through which the attractive vector interaction contributes to the/~ dispersion relation. Therefore, antikaon condensation at zero chemical potential is only possible for a zero kaon effective pass in both approaches. KK condensation is not possible at all. Calculations using the Nambu-Jona-Lasinio model [ 14] also find a moderate medium dependence of the effective kaon mass up to 2t9o in line with our result. However, their extrapolation to high density depends crucially on the density behaviour of the quark condensate. These authors assume a linear dependence which might be appropriate only for the low density regime considered by them. Most recently it was pointed out in [ 17] that the offshell behaviour of low energy scattering strongly influences the in-medium effective masses of the mesons whereas the basic assumption of the original work [ 1 ] was an on-shell approximation. This makes the issue of kaon condensation more model dependent than hitherto recognized, in line with our discussion. Among the various off-shell correction terms we mention the range correction [ 3 ] to the zero range Lagrangian ( 1 ) which may be incorporated into our calculations by multiplying 2frN by ( 1 - D w 2 ( K ) / m ~ ) as done in [14] with D = 0.5. This leads to an increase of the kaon effective mass and a similar increase for the energy, rendering antikaon condensation even less favorable than discussed above. The kaon dispersion relation in dense matter is of vital importance both for relativistic heavy ion collisions (where high baryon densities are achieved for short times) and in astrophysics. For the first topic, however, both the real and the imaginary part (scattering cross section which was not discussed here) of the

J Schaffneret al. /Phystcs Letters B 334 (1994) 268-274

optical potential will influence the spectra and yields o f K + and K - mesons [8,18,19]. The k dispersion law in the nuclear medium will be affected by coupling to nucleon hole (N - l)-hyperon (Y) excitations lying below mK(p = 0). This has been recently discussed in [ 20 ]. First, the/~ nuclear potential Ve will not vary linearly with p near any of the energies of these coupled channels. Indeed, near threshold (to = mr, only 27 MeV above N - IA(1405) ), and for nuclear densities (P<~Po), the wealth of K - atomic data is well fitted by a potential of a more involved density dependence [21], with a depth Re V £ ( p = p o ) = - (180-200) MeV.

(16)

This strong attraction suggests that each of its scalar and vector components is individually large. The scalar component in both models discussed above is of a depth of about - 4 0 MeV, so that the vector part must be particularly strong. Assuming that the to meson couples universally to nonstrange quarks, one gets the following estimate [ 15] for the vector component: VxCV)(po)

=

_

~ u ( v ) ~ . ~'oJ~ = - 110 MeV ~-N

(17)

where V~v) is taken from our RMF calculation for nucleons. On the other hand, the chiral-model vector term of Eq. (3), with fK = 93 MeV, gives a depth of only half of that on the r.h.s, of Eq. (17). It would be premature, however, to conclude in favor of (17), and subsequently extrapolate it to higher densities, in view of the role which N - 1 A ( 1 4 0 5 ) excitations play in producing the non linear density dependence of Ve for p<~Po [21]. The second effect of low lying coupled channels is to introduce a rather strong (absorptive) imaginary part into Ve. From K - atoms one finds [21] Im V e ( p = p o ) = - (80-100) MeV,

(18)

reflecting primarily the dominance of the available /~V~ 7r~, ~'A free-space decay modes. It is clear that a test R meson implanted into a nucleus will get strongly absorbed as long as its effective energy does not fall below about 310 MeV, placing the/~N channel below the TrYchannels (assuming that the pion effective energy hardly changes with the density [ 17,22], and neglecting baryonic binding energy differences for final states consisting of a bound hyperon). Suppose now that the nuclear density is increased to several times Po, as expected in the interior of neutron stars.

273

The energy of the/~ may, as in some of the curves of Figs. lb, 2b above, have decreased sufficiently so that the free decay channels become closed and densely packed electrons, estimated to have a chemical potential /ze- =20(0300 MeV, may undergo [3] a weak interaction e - ---,K - re. A buildup of such K - mesons is by no means ensured since the nuclear medium offers more involved decay modes that require at least two nucleons, viz. k N N ~ AN, ~ N .

(19)

This nonmesonic decay branch has been estimated for K - nuclear interactions at rest [ 23 ] to constitute about 20% of the K - absorption rate. The rate for (19) will not decrease substantially as the phase space decreases, since it is expected to behave roughly as p2 upon increasing the density. To eliminate the decay modes (19), one would have to lower the energy of the K below to* < to3 - to~*

(20)

which is m ( A ) - m ( N ) = 176 MeV in free space. This requires an unrealistically large density for the curves of Fig. 2b, for example. It therefore appears plausible that kaons will decay into hyperons via the reactions (19). As shown in [24], other hyperons such as E are also expected to be formed. A somewhat analogous situation, again within the RMF model, was critically discussed in [ 25 ]. More detailed calculations are obviously needed to pin down the composition of strange matter at high densities, including both kaon and hyperon degrees of freedom.

Acknowledgement We thank Maria Berenguer for fruitful and stimulating discussions. This work was supported by the Graduiertenkolleg "Theoretische und Experimentelle Schwerionenphysik" of the Deutsche Forschungsgemeinschaft (DFG), by the Gesellschaft ftir Schwerionenforschung Darmstadt (GSI), and by the Bundesminesterium f/Jr Forschung und Technologie (BMFT). A.G. Acknowledges an award by the Alexander von Humboldt Foundation and I.N.M. acknowledges support by the DFG for a guest professorship.

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J. Schaffner et al /Physics Letters B 334 (1994) 268-274

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[ 15] G.E. Brown, C M. Ko and K. Kubodera, Z Phys. A 341 (1992) 301. [ 16] A. MiJller-Groeling, K. Holinde and J. Speth, Nucl. Phys A 513 (1990) 557 [17] H. Yabu, S Nakamura and K Kubodera, Phys. Lett B 317 (1993) 269; J Delorme, M Encson and TE.O. Ericson; Phys Lett B 291 (1992) 379 [ 18] C Hartnack, J J~inickeand J. Aichelln, Umversit6 de Nantes, prepnnt LPN 93-10; C Hartnack, L Sehn, J Janicke, H. Stocker and J Aichehn, submitted to Nucl Phys A. [ 19] C Spieles, A Jahns, H Sorge, H. St6cker and W. Greiner' Proc XXXI Intern. Winter Meeting on Nuclear Physics (Bormio, Italy, 1994) ed I. Iori, m press [20] H. Yabu, S Nakamura, F. Myhrer and K. Kubodera, Phys Lett B 315 (1993) 17 [ 21 ] E. Friedman, A Gal and C J Batty, Phys Lett. B 308 (1993) 6; Nucl. Phys A, in press. [22] M. Lutz, S. Khmt and W. Welse, Nucl. Phys A 542 (1992) 521 [23] See for example Table 2 in: C Vander Velde-Wilquet, J Sacton, D N. Tovee and D.H. Davis, Nuovo Cimento 39 A (1977) 538 [24] J. Schaffner, C.B. Dover, A Gal, H. St~cker and C. Gremer, Phys. Rev Lett 71 (1993) 1328, Ann. Phys (NY), in press [25] B W Lynn, A E Nelson and N. Tetradls, Nucl Phys. B 345 (1990) 186