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Does kaon condensation occur in neutron stars in the relativistic mean-field theory?
29 May 1997
PHYSICS LETTERS 6 Physics Letters B 401 (1997) 219-223
Does kaon condensation occur in neutron stars in the relativistic mean-field theory? Z.G. Daia*b, K.S. Chengb a Department of Astronomy, Nanjing University, Nanjing 210093, China b Department of Physics, University of Hong Kong, Hong Kong
Received 13 January 1997; revised manuscript received 18 March 1997 Editor: W. Haxton
Abstract We discuss whether kaon condensation occurs in neutron star matter by using the Zimanyi-Moszkowski model in the relativistic mean-field theory. As a comparison, we also discuss the results from the Boguta-Bodmer model. We show that even though hyperons which may increase the critical condensation density are not included, kaon condensation might not occur in stable neutron stars for the Zimanyi-Moszkowski
1. Introduction Kaon condensation in neutron stars has been of growing interest since its possibility was suggested by Kaplan and Nelson [ 11. The basic idea is that the energy of an antikaon (K-) is lowered by interaction with nucleons. In neutron star matter in beta equilibrium, therefore, one expects that antikaons are present if the energy to create one is less than the electron chemical potential. In addition to softening of the equation of state, kaon condensation plays important roles in the evolution of neutron stars. First, if the condensation occurs at lower densities (e.g., IV 3~0, where po is the nuclear matter density), the consequence of hypercritical accretion of neutron stars left by supernovae [ 21 may form black holes with mass of N 1.5Ma [ 31. Second, the threshold density for kaon condensation is moderately high, this exotic state may accelerate cooling of neutron stars, reducing the stellar surface temperature [4]. This effect can be compared with X-ray observations. Third, if the condensa0370-2693/97/$17.00
model. @ 1997 Elsevier Science B.V.
tion takes place at higher densities (e.g., 2 5~0)) then accretion of neutron stars in low-mass X-ray binaries may result in phase transitions from normal nuclear matter to kaon-condensation matter in the stellar interiors, which may produce gravitational wave bursts and/or cosmological gamma-ray bursts [ 51. Now one may ask a question: At which density do kaons start to condensate? It is still a controversial issue. The calculations based on chiral perturbation theory showed that kaon condensation may set in at densities 3-4~0 [ 61. Pandharipande, Pethick and Thorsson [ 71 discussed the role of kaon-nucleon and nucleonnucleon correlations and found that these correlations raise the threshold desity for kaon condensation to much higher densities. The relativistic mean-field (RMF) theory has also been applied to the studies of the above issue. It has been proposed [ 8,91 to use the mean-field approach for the baryon interactions and show that the critical density for kaon condensation may be lifted to higher densities (> 4~0) when hyperons are included. These
8 1997 Elsevier Science B.V. All rights reserved.
PII SO370-2693(97)00416-4
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Cheng/
Physics Letters B 401 (1997) 219-223
studies adopted the Boguta-Bodmer (BB) model in the RMF theory [ IO]. It is known that in the standard model of Walecka [ 1 I] the incompressibility of nuclear matter is overestimated. There are two ways to solve this question. First, the cubic and quartic terms for the scalar field were introduced into the Lagrangian [ 10 1. These nonlinear terms shift the incompressibility to reasonable values in comparison with empirical data. Zimanyi and Moszkowski (ZM) proposed an alternative nonlinear model [ 121, in which the non-linearity is contained in the connection between the effective nucleon mass and the scalar field. Thus the Lagrangian of this model has no extra terms, and consequently deals with fewer parameters as compared with the BB model. The ZM model also yields reasonable values of incompressibility and a satisfactory effective nucleon mass for nuclear matter [ 131. This model was recently used to study the structure of neutron stars and properties of supernova matter in [ 141, where it was assumed that the kaon meson is neglected in studying the properties of neutron star matter. In this letter we discuss whether kaon condensation occurs in neutron star matter by using the ZM model. As a comparison, we also discuss the results from the BB model.
r
I
Fig. I. Kaon energy (dashed lines) and electron chemical potential (solid lines) in neutron star matter based on the Boguta-Bodmer model [ IO] and the Zimanyi-Moszkowski model ( 12 ] as functions of density for Case A.
Here $, 4, VP”, Rp and D denote the fields of the baryon, isoscalar scalar meson ((T), isoscalar vector meson (u), isovector vector meson ( p) and isovector scalar meson (6) with masses of mq, m,, mP and ma (980 MeV), respectively. In this letter we consider the contribution of the S meson, which has also been studied in [9]. m* is the effective nucleon mass. For the ZM model, m mr =
1 +
&N(b/m
and
U(4)
=O;
’
for the BB model, 2. Models and results
m* = m - ga,&,
In the RMF theory, the strong interaction is described by the exchange of mesons between nucleons through the Yukawa couplings. We follow the notation of [ 131. The Lagrangian density of a nuclear system is given by
where m is the free nucleon mass. The parameters &N, &N, gpN and g6N are the coupling constants for interactions between mesons and nucleons. b and c are also constants. The above parameters except for mg and gsN have been listed in Table 1 in [ 141. In our calculations, gsN is set to be 5.95, which is taken from the Bonn model [ 151. In addition, we consider the contribution of the muon to the composition of neutron star matter. The parameter sets used here can yield reasonable nuclear structures of 40Ca and 208Pb [ 131 and satisfactory properties of nuclear matter at the saturation density [ 141. According to the standard procedure [ 11,141, we calculate the electron chemical potentials (CL,) of neutron star matter based on the two models. The results are shown by the solid lines in Figs. 1 and 2. Only in Fig. 2 is the contribution of the delta meson included. Comparing these two figures, we find that the
L = c
cC;b[Yp(ia~-gwhrYpVCL-gpNYCLT3bRCL)
--m*
b=y,n
Fclv = $Vv - a, VP , G,,
= a, R, - a, R, .
(2) (3)
U(4)
= ibqS3 +
fcc$”,
(5)
Z.G. Doi, KS
Cheng/Physics
221
Letters B 401 (1997) 219-223
Table 1 The coupling constants for interactions between the kaon meson and the scalar mesons Parameter
Case A
RUK
&SK B
‘SUK f%K
baryon
number
density
(fm-‘j
Fig. 2. The same as in Fig. 1 but for Case B.
effect of the delta meson on the electron chemical potential is insignificant. It is also seen that the electron chemical potential for the ZM model is much smaller than that for the BB model. It should be pointed out that these two models give different properties of neutron stars, as shown by Dai and Cheng [ 141. First, because the proton fraction at a high density for the BB model is larger than the value at which the direct Urea process starts to work, and because the proton fraction in the interior of a stable neutron star for the ZM model is less than this value, the result of neutron star cooling due to neutrino emission is that the stellar surface temperature for the BB model should be much lower than that for the ZM model at t < lo6 years. This may provide an observational signature for the ZM model. Second, these two models result in different structures of neutron stars: for the ZM model the maximum mass of neutron stars is 1.9Mo at a central density of N 6.4~0 with a radius of - 11.Okm; for the BB model the maximum mass is 2.2Mo at a central density of N 5.1~0 with a radius of N 11.9 km. We now calculate effective kaon energies in neutron star matter by using the RMF theory. In the following we adopt the meson-exchange picture for the kaonnucleon interaction simply because we use it also for parametrizing nucleon interactions. After considering the contribution of the delta meson, we adopt the simple kaon-meson Lagrangian density
& = (8~ +
i&K&&
+ &KnR/d
+ i&xVRp)*R(afi + ig,Kv,
K - rn;EK + &,KmKitKCfl
- &wmdK73D,
(6)
ZM
BB 1.931 0.0
1.902 0.0
1.366 10.85
0.397 2.854
where g&, g&X, gpK and gaK are the coupling constants for interactions between kaon mesons and other mesons. It should be noted that this Lagrangian density fulfills the Ward identity in the medium, requiring that the vector field should be coupled to a conserved current [9]. Eq. (6) is similar to that given in [ 91 except for the term of the 6 meson. From the equation of motion for the vector field, one further finds m$‘pVp = 0. Thus, the equation of motion for kaons in neutron star matter is then given by [&a’ + mi -
&K4OmK
+ &XDO~3~K
+2(g,KVo+g,K73Ro)iaC-(goKVo+g,K73Ro)21K =o.
(7)
Decomposing the kaon field into plane waves, we obtain the following dispersion relation for antikaons in neutron star matter UK = -
m2Kc&K&
&KmK+O + &KQRo)
+ gSKmK@o
.
+
k2
(8)
The mean fields, (PO,DO, 6 and Ro, are calculated from the standard procedure based on Eq. ( 1) . The coupling constants of the vector mesons, goK and &K, are chosen from the SU(3) relations: g,K = gpK = 3.02. As for choices of the coupling constants gqK and gsK, we would adopt two possible scenarios: (i> Case A, in which the-contribution of the S meson is neglected (viz., gSK = g&v = 0), and &K is taken from the Bonn model (viz., g&g&/m: = 62.8 GeV2) [ 161. Thus, we can obtain the values of g@Kfor the ZM and BB models, which are listed in Table 1. (ii) Case B, in which g& and g&Kare fixed to the KN-scattering lengths. For a given isospin I, the KN-scattering lengths in the tree approximation are [ 171
Z.G. Dai, KS. Cheng/Physics
222 /=I a0
mK
= h(
1 f
gaKguN -
mK/mN)
_2gwKgwN --
2 -gp
m20 =
-0.31
m$
+
gsKgSN 4
KgpN
Letters B 401 (1997) 219-223
glected. According to Eq. (8), thus, the kaon energy can be approximated by wk(k=O)
“mk-qp,
(11)
mi
fm,
(9)
where --
and
+ ----+~(I gwKgoN
-2Y,).
mZ, mK
I=0 a0 =
1+
h( _2goKgwN mZ, =
-O.O9fm,
g,K gaN ~-
mK/mN)
m:
3 gsKgsN 4
+ 6g~Kg~N m:
>
( 10)
where the experimental values have been taken from [ 18 1. From the above two equations, we can &O obtain the values of&K and &?SK for the ZM and BB models, which are shown in Table 1. The onset of s-wave kaon condensation is determined by the condition, pe = tiIy( k = 0). Once this condition is satisfied, a subsequent reaction will occur: e- + K- + v,. The kaon energies and the electron chemical potentials in neutron star matter based on the ZM and BB models are shown in Figs. 1 and 2 for Cases A and B respectively. It can be seen from Fig. 2 that for the ZM model the kaon energy is higher than the electron chemical potential in the density range p < 6.3~0. Because the central density of the neutron star with the maximum mass based on the ZM model is about 6.4~0 [ 141, kaon condensation might not occur in stable neutron stars. For the BB model, however, we can see from Fig. 2 that the electron chemical potential exceeds the kaon energy at densities higher than 3.5~0, and thus kaon condensation might take place at these densities. Because the central density of the neutron star with the maximum mass based on the BB model is about 5.1~0, kaon condensation might exist in massive neutron stars. This conclusion is similar to that from chiral perturbation theory in [ 61. We now discuss the origin of the difference between the kaon energy for the ZM model and that for the BB model. In the RMF theory, the w- and pmeson fields are written as VO= g,Npfmt and Ro = &N( 2q, - 1) p/2mi where YP is the proton fraction. At low densities, the c-meson field is approximated by 40 M g&p/m: and the &meson field can be ne-
(12)
P
Clearly, the kaon energy is not only determined by the proton fraction, but also, what is more important, by the coupling constants. From the values of the coupling constants given in [ 141 and in Table 1, we obtain 7 as a function of proton fraction: in Case A, 77= 511.5 - 54.7$
(MeVfm’)
for
ZM,
(13)
7 = 679.5-149.9$
(MeVfm’)
for
BB;
(14)
(MeVfm”)
for
ZM,
(15)
(MeVfm’)
for
BB.
(16)
in Case B,
77= 320.6-
54.7Y,
7 = 608.9-149.9$
Since Y, is less than 0.1 at low densities for the two models, the influence of the proton fraction on the kaon energy is insignificant. Comparing Eq. ( 13) with (14) or Eq. (15) with (16), we can see that v for the ZM model is smaller than that for the BB model. This leads to a difference of the kaon energies for these two models. Furthermore, because the constant for coupling between the a-meson field and the kaonmeson field (&K) significantly decreases from Case A to Case B for the ZM model, the value of 77decreases and thus the kaon energy increases. Therefore, the slope of the kaon energy in Fig. 2 is significantly larger than that in Fig. 1 for this model. Another important reason why kaon condensation might not occur in neutron stars for the ZM model rather than for the BB model is that the electron fraction in neutron star matter based on the ZM model is always smaller than that based on the BB model at the same density.
3. Brief discussion In this letter we have applied the ZM model in the RMF theory to discuss whether kaon condensation oc-
Z.G. Dai, KS. Cheng/Physics
curs in neutron stars. For this purpose, we have compared kaon energies with the electron chemical potentials at different densities. In calculating kaon energies, we have used two different cases to fix the coupling constants for interactions between the kaon meson and the scalar mesons. In the first case these constants are taken from the Bonn model and the contribution of the isovector scalar meson is neglected, while in the second case the constants are fixed to the experimantal K/V-scattering lengths in the two isospin channels. We do need not to consider the effect of the offshell term on kaon energies in neutron star matter. This is because the off-shell term of the kaon-nucleon scattering amplitude is ultimately not relevant to the determination of in-medium kaon energies, as shown by Lee [ 61 and Thorsson and Wirzba [ 191. In our study we have neglected the contribution of hyperons, which has been studied by other authors [ 8,9,20]. Here we would refer to the results in [ 91. In [9] the Boguta-Bodmer model is used to study the properties of neutron star matter and the hyperonhyperon interaction is implemented by introducing additional meson exchanges. It is demonstrated that within the RMF approach the presence of hyperons inside neutron stars may make the onset of kaon condensation occur at densities higher than that without including hyperon-hyperon interaction. In conclusion, even though hyperons are not included, kaon condensation might not occur in stable neutron stars based on the ZM model in the relativistic mean-field theory. Acknowledgments
We would like to thank the referee for helpful comments and suggestions. This work was supported in part by the National Natural Science Foundation of China and in part by a RGC grant of Hong Kong.
Letters B 401 (1997) 219-223
223
References [ 1 ] D.B. Kaplan and A.E. Nelson, Phys. Lett. B 175 ( 1986) 57; A.E. Nelson and D.B. Kaplan, Phys. Lea. B 192 (1987) 193. [2] R.A. Chevalier, Astrophys. .I. 346 (1989) 847; GE. Brown, Astrophys. J. 440 (1995) 270. ]3] GE. Brown and H.A. Bethe, Astrophys. J. 423 ( 1994) 659. [4] D. Page and E. Baron, Astrophys. J. 354 ( 1990) L17; Astrophys. J. 382 (1991) Llll (E). [S] KS. Cheng and Z.G. Dai, Phys. Rev. Lett. 77 ( 1996) 1210; Astrophys. J., submitted ( 1996). [6] GE. Brown, K. Kubodera, M. Rho and V. Thomson, Phys. Lett. B 291 (1992) 355; G.E. Brown, C.-H. Lee, M. Rho and V. Thorsson, Nucl. Phys. A 567 (1994) 937; V. Thomson, M. Prakash and J.M. Lattimer, Nucl. Phys. A 572 (1994) 693; for a detailed review see, C.-H. Lee, Phys. Rep. 275 ( 1996) 255. [7] V.R. Pandharipande, C.J. Pethick and V. Thorsson, Phys. Rev. Len. 75 ( 1995) 4567. 18 I PJ. Ellis, R. Knormn and M. Prakash, Phys. Lett. B 349 (1995) 11. ]9] J. Schaffner and I.N. Mishustin, Phys. Rev. C 53 (1996) 1416. [lOI J. Boguta and A.R. Bodmer, Nucl. Phys. A 292 ( 1977) 413. 1111 J.D. Walecka, Ann. Phys. 83 (1974) 491. 1121J. Zimanyi and S.A. Moszkowski, Phys. Rev. C 42 ( 1990) 1416. [I31 M. Centelles, X. Vifias, M. Barranco S. Marcos and R.I. Lombard, Nucl. Phys. A 537 (1992) 486. [I41 K.S. Cheng, Z.G. Dai and C.C. Yao, Astrophys. J. 464 ( 1996) 348; Z.G. Dai and K.S. Cheng, Astron. Astrophys., in press (1996). 1151 R. Machleidt, K. Holinde and C. Elster, Phys. Rep. 149 (1987) 1. iI61 R. Bthtgen, K. Holinde, A. Mtiller-Groeling, J. Speth and P Wybomy, Nucl. Phys. A 506 (1990) 586; A. Miiller-Groeling, K. Holinde and J. Speth, Nucl. Phys. A 513 (1990) 557. [ 171 J. Cohen, Phys. Rev. C 39 (1989) 2285. [ 181 T. Barnes and ES. Swanson, Phys. Rev. C 39 (1994) 1166. [ 191 V. Thorsson and A. Wirzba, Nucl. Phys. A 589 (1995) 633. [20] T. Muto, Prog. Theor. Phys. 55 (1993) 415.
PHYSICS LETTERS 6 Physics Letters B 401 (1997) 219-223
Does kaon condensation occur in neutron stars in the relativistic mean-field theory? Z.G. Daia*b, K.S. Chengb a Department of Astronomy, Nanjing University, Nanjing 210093, China b Department of Physics, University of Hong Kong, Hong Kong
Received 13 January 1997; revised manuscript received 18 March 1997 Editor: W. Haxton
Abstract We discuss whether kaon condensation occurs in neutron star matter by using the Zimanyi-Moszkowski model in the relativistic mean-field theory. As a comparison, we also discuss the results from the Boguta-Bodmer model. We show that even though hyperons which may increase the critical condensation density are not included, kaon condensation might not occur in stable neutron stars for the Zimanyi-Moszkowski
1. Introduction Kaon condensation in neutron stars has been of growing interest since its possibility was suggested by Kaplan and Nelson [ 11. The basic idea is that the energy of an antikaon (K-) is lowered by interaction with nucleons. In neutron star matter in beta equilibrium, therefore, one expects that antikaons are present if the energy to create one is less than the electron chemical potential. In addition to softening of the equation of state, kaon condensation plays important roles in the evolution of neutron stars. First, if the condensation occurs at lower densities (e.g., IV 3~0, where po is the nuclear matter density), the consequence of hypercritical accretion of neutron stars left by supernovae [ 21 may form black holes with mass of N 1.5Ma [ 31. Second, the threshold density for kaon condensation is moderately high, this exotic state may accelerate cooling of neutron stars, reducing the stellar surface temperature [4]. This effect can be compared with X-ray observations. Third, if the condensa0370-2693/97/$17.00
model. @ 1997 Elsevier Science B.V.
tion takes place at higher densities (e.g., 2 5~0)) then accretion of neutron stars in low-mass X-ray binaries may result in phase transitions from normal nuclear matter to kaon-condensation matter in the stellar interiors, which may produce gravitational wave bursts and/or cosmological gamma-ray bursts [ 51. Now one may ask a question: At which density do kaons start to condensate? It is still a controversial issue. The calculations based on chiral perturbation theory showed that kaon condensation may set in at densities 3-4~0 [ 61. Pandharipande, Pethick and Thorsson [ 71 discussed the role of kaon-nucleon and nucleonnucleon correlations and found that these correlations raise the threshold desity for kaon condensation to much higher densities. The relativistic mean-field (RMF) theory has also been applied to the studies of the above issue. It has been proposed [ 8,91 to use the mean-field approach for the baryon interactions and show that the critical density for kaon condensation may be lifted to higher densities (> 4~0) when hyperons are included. These
8 1997 Elsevier Science B.V. All rights reserved.
PII SO370-2693(97)00416-4
220
Z.G. Dai, KS.
Cheng/
Physics Letters B 401 (1997) 219-223
studies adopted the Boguta-Bodmer (BB) model in the RMF theory [ IO]. It is known that in the standard model of Walecka [ 1 I] the incompressibility of nuclear matter is overestimated. There are two ways to solve this question. First, the cubic and quartic terms for the scalar field were introduced into the Lagrangian [ 10 1. These nonlinear terms shift the incompressibility to reasonable values in comparison with empirical data. Zimanyi and Moszkowski (ZM) proposed an alternative nonlinear model [ 121, in which the non-linearity is contained in the connection between the effective nucleon mass and the scalar field. Thus the Lagrangian of this model has no extra terms, and consequently deals with fewer parameters as compared with the BB model. The ZM model also yields reasonable values of incompressibility and a satisfactory effective nucleon mass for nuclear matter [ 131. This model was recently used to study the structure of neutron stars and properties of supernova matter in [ 141, where it was assumed that the kaon meson is neglected in studying the properties of neutron star matter. In this letter we discuss whether kaon condensation occurs in neutron star matter by using the ZM model. As a comparison, we also discuss the results from the BB model.
r
I
Fig. I. Kaon energy (dashed lines) and electron chemical potential (solid lines) in neutron star matter based on the Boguta-Bodmer model [ IO] and the Zimanyi-Moszkowski model ( 12 ] as functions of density for Case A.
Here $, 4, VP”, Rp and D denote the fields of the baryon, isoscalar scalar meson ((T), isoscalar vector meson (u), isovector vector meson ( p) and isovector scalar meson (6) with masses of mq, m,, mP and ma (980 MeV), respectively. In this letter we consider the contribution of the S meson, which has also been studied in [9]. m* is the effective nucleon mass. For the ZM model, m mr =
1 +
&N(b/m
and
U(4)
=O;
’
for the BB model, 2. Models and results
m* = m - ga,&,
In the RMF theory, the strong interaction is described by the exchange of mesons between nucleons through the Yukawa couplings. We follow the notation of [ 131. The Lagrangian density of a nuclear system is given by
where m is the free nucleon mass. The parameters &N, &N, gpN and g6N are the coupling constants for interactions between mesons and nucleons. b and c are also constants. The above parameters except for mg and gsN have been listed in Table 1 in [ 141. In our calculations, gsN is set to be 5.95, which is taken from the Bonn model [ 151. In addition, we consider the contribution of the muon to the composition of neutron star matter. The parameter sets used here can yield reasonable nuclear structures of 40Ca and 208Pb [ 131 and satisfactory properties of nuclear matter at the saturation density [ 141. According to the standard procedure [ 11,141, we calculate the electron chemical potentials (CL,) of neutron star matter based on the two models. The results are shown by the solid lines in Figs. 1 and 2. Only in Fig. 2 is the contribution of the delta meson included. Comparing these two figures, we find that the
L = c
cC;b[Yp(ia~-gwhrYpVCL-gpNYCLT3bRCL)
--m*
b=y,n
Fclv = $Vv - a, VP , G,,
= a, R, - a, R, .
(2) (3)
U(4)
= ibqS3 +
fcc$”,
(5)
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Cheng/Physics
221
Letters B 401 (1997) 219-223
Table 1 The coupling constants for interactions between the kaon meson and the scalar mesons Parameter
Case A
RUK
&SK B
‘SUK f%K
baryon
number
density
(fm-‘j
Fig. 2. The same as in Fig. 1 but for Case B.
effect of the delta meson on the electron chemical potential is insignificant. It is also seen that the electron chemical potential for the ZM model is much smaller than that for the BB model. It should be pointed out that these two models give different properties of neutron stars, as shown by Dai and Cheng [ 141. First, because the proton fraction at a high density for the BB model is larger than the value at which the direct Urea process starts to work, and because the proton fraction in the interior of a stable neutron star for the ZM model is less than this value, the result of neutron star cooling due to neutrino emission is that the stellar surface temperature for the BB model should be much lower than that for the ZM model at t < lo6 years. This may provide an observational signature for the ZM model. Second, these two models result in different structures of neutron stars: for the ZM model the maximum mass of neutron stars is 1.9Mo at a central density of N 6.4~0 with a radius of - 11.Okm; for the BB model the maximum mass is 2.2Mo at a central density of N 5.1~0 with a radius of N 11.9 km. We now calculate effective kaon energies in neutron star matter by using the RMF theory. In the following we adopt the meson-exchange picture for the kaonnucleon interaction simply because we use it also for parametrizing nucleon interactions. After considering the contribution of the delta meson, we adopt the simple kaon-meson Lagrangian density
& = (8~ +
i&K&&
+ &KnR/d
+ i&xVRp)*R(afi + ig,Kv,
K - rn;EK + &,KmKitKCfl
- &wmdK73D,
(6)
ZM
BB 1.931 0.0
1.902 0.0
1.366 10.85
0.397 2.854
where g&, g&X, gpK and gaK are the coupling constants for interactions between kaon mesons and other mesons. It should be noted that this Lagrangian density fulfills the Ward identity in the medium, requiring that the vector field should be coupled to a conserved current [9]. Eq. (6) is similar to that given in [ 91 except for the term of the 6 meson. From the equation of motion for the vector field, one further finds m$‘pVp = 0. Thus, the equation of motion for kaons in neutron star matter is then given by [&a’ + mi -
&K4OmK
+ &XDO~3~K
+2(g,KVo+g,K73Ro)iaC-(goKVo+g,K73Ro)21K =o.
(7)
Decomposing the kaon field into plane waves, we obtain the following dispersion relation for antikaons in neutron star matter UK = -
m2Kc&K&
&KmK+O + &KQRo)
+ gSKmK@o
.
+
k2
(8)
The mean fields, (PO,DO, 6 and Ro, are calculated from the standard procedure based on Eq. ( 1) . The coupling constants of the vector mesons, goK and &K, are chosen from the SU(3) relations: g,K = gpK = 3.02. As for choices of the coupling constants gqK and gsK, we would adopt two possible scenarios: (i> Case A, in which the-contribution of the S meson is neglected (viz., gSK = g&v = 0), and &K is taken from the Bonn model (viz., g&g&/m: = 62.8 GeV2) [ 161. Thus, we can obtain the values of g@Kfor the ZM and BB models, which are listed in Table 1. (ii) Case B, in which g& and g&Kare fixed to the KN-scattering lengths. For a given isospin I, the KN-scattering lengths in the tree approximation are [ 171
Z.G. Dai, KS. Cheng/Physics
222 /=I a0
mK
= h(
1 f
gaKguN -
mK/mN)
_2gwKgwN --
2 -gp
m20 =
-0.31
m$
+
gsKgSN 4
KgpN
Letters B 401 (1997) 219-223
glected. According to Eq. (8), thus, the kaon energy can be approximated by wk(k=O)
“mk-qp,
(11)
mi
fm,
(9)
where --
and
+ ----+~(I gwKgoN
-2Y,).
mZ, mK
I=0 a0 =
1+
h( _2goKgwN mZ, =
-O.O9fm,
g,K gaN ~-
mK/mN)
m:
3 gsKgsN 4
+ 6g~Kg~N m:
>
( 10)
where the experimental values have been taken from [ 18 1. From the above two equations, we can &O obtain the values of&K and &?SK for the ZM and BB models, which are shown in Table 1. The onset of s-wave kaon condensation is determined by the condition, pe = tiIy( k = 0). Once this condition is satisfied, a subsequent reaction will occur: e- + K- + v,. The kaon energies and the electron chemical potentials in neutron star matter based on the ZM and BB models are shown in Figs. 1 and 2 for Cases A and B respectively. It can be seen from Fig. 2 that for the ZM model the kaon energy is higher than the electron chemical potential in the density range p < 6.3~0. Because the central density of the neutron star with the maximum mass based on the ZM model is about 6.4~0 [ 141, kaon condensation might not occur in stable neutron stars. For the BB model, however, we can see from Fig. 2 that the electron chemical potential exceeds the kaon energy at densities higher than 3.5~0, and thus kaon condensation might take place at these densities. Because the central density of the neutron star with the maximum mass based on the BB model is about 5.1~0, kaon condensation might exist in massive neutron stars. This conclusion is similar to that from chiral perturbation theory in [ 61. We now discuss the origin of the difference between the kaon energy for the ZM model and that for the BB model. In the RMF theory, the w- and pmeson fields are written as VO= g,Npfmt and Ro = &N( 2q, - 1) p/2mi where YP is the proton fraction. At low densities, the c-meson field is approximated by 40 M g&p/m: and the &meson field can be ne-
(12)
P
Clearly, the kaon energy is not only determined by the proton fraction, but also, what is more important, by the coupling constants. From the values of the coupling constants given in [ 141 and in Table 1, we obtain 7 as a function of proton fraction: in Case A, 77= 511.5 - 54.7$
(MeVfm’)
for
ZM,
(13)
7 = 679.5-149.9$
(MeVfm’)
for
BB;
(14)
(MeVfm”)
for
ZM,
(15)
(MeVfm’)
for
BB.
(16)
in Case B,
77= 320.6-
54.7Y,
7 = 608.9-149.9$
Since Y, is less than 0.1 at low densities for the two models, the influence of the proton fraction on the kaon energy is insignificant. Comparing Eq. ( 13) with (14) or Eq. (15) with (16), we can see that v for the ZM model is smaller than that for the BB model. This leads to a difference of the kaon energies for these two models. Furthermore, because the constant for coupling between the a-meson field and the kaonmeson field (&K) significantly decreases from Case A to Case B for the ZM model, the value of 77decreases and thus the kaon energy increases. Therefore, the slope of the kaon energy in Fig. 2 is significantly larger than that in Fig. 1 for this model. Another important reason why kaon condensation might not occur in neutron stars for the ZM model rather than for the BB model is that the electron fraction in neutron star matter based on the ZM model is always smaller than that based on the BB model at the same density.
3. Brief discussion In this letter we have applied the ZM model in the RMF theory to discuss whether kaon condensation oc-
Z.G. Dai, KS. Cheng/Physics
curs in neutron stars. For this purpose, we have compared kaon energies with the electron chemical potentials at different densities. In calculating kaon energies, we have used two different cases to fix the coupling constants for interactions between the kaon meson and the scalar mesons. In the first case these constants are taken from the Bonn model and the contribution of the isovector scalar meson is neglected, while in the second case the constants are fixed to the experimantal K/V-scattering lengths in the two isospin channels. We do need not to consider the effect of the offshell term on kaon energies in neutron star matter. This is because the off-shell term of the kaon-nucleon scattering amplitude is ultimately not relevant to the determination of in-medium kaon energies, as shown by Lee [ 61 and Thorsson and Wirzba [ 191. In our study we have neglected the contribution of hyperons, which has been studied by other authors [ 8,9,20]. Here we would refer to the results in [ 91. In [9] the Boguta-Bodmer model is used to study the properties of neutron star matter and the hyperonhyperon interaction is implemented by introducing additional meson exchanges. It is demonstrated that within the RMF approach the presence of hyperons inside neutron stars may make the onset of kaon condensation occur at densities higher than that without including hyperon-hyperon interaction. In conclusion, even though hyperons are not included, kaon condensation might not occur in stable neutron stars based on the ZM model in the relativistic mean-field theory. Acknowledgments
We would like to thank the referee for helpful comments and suggestions. This work was supported in part by the National Natural Science Foundation of China and in part by a RGC grant of Hong Kong.
Letters B 401 (1997) 219-223
223
References [ 1 ] D.B. Kaplan and A.E. Nelson, Phys. Lett. B 175 ( 1986) 57; A.E. Nelson and D.B. Kaplan, Phys. Lea. B 192 (1987) 193. [2] R.A. Chevalier, Astrophys. .I. 346 (1989) 847; GE. Brown, Astrophys. J. 440 (1995) 270. ]3] GE. Brown and H.A. Bethe, Astrophys. J. 423 ( 1994) 659. [4] D. Page and E. Baron, Astrophys. J. 354 ( 1990) L17; Astrophys. J. 382 (1991) Llll (E). [S] KS. Cheng and Z.G. Dai, Phys. Rev. Lett. 77 ( 1996) 1210; Astrophys. J., submitted ( 1996). [6] GE. Brown, K. Kubodera, M. Rho and V. Thomson, Phys. Lett. B 291 (1992) 355; G.E. Brown, C.-H. Lee, M. Rho and V. Thorsson, Nucl. Phys. A 567 (1994) 937; V. Thomson, M. Prakash and J.M. Lattimer, Nucl. Phys. A 572 (1994) 693; for a detailed review see, C.-H. Lee, Phys. Rep. 275 ( 1996) 255. [7] V.R. Pandharipande, C.J. Pethick and V. Thorsson, Phys. Rev. Len. 75 ( 1995) 4567. 18 I PJ. Ellis, R. Knormn and M. Prakash, Phys. Lett. B 349 (1995) 11. ]9] J. Schaffner and I.N. Mishustin, Phys. Rev. C 53 (1996) 1416. [lOI J. Boguta and A.R. Bodmer, Nucl. Phys. A 292 ( 1977) 413. 1111 J.D. Walecka, Ann. Phys. 83 (1974) 491. 1121J. Zimanyi and S.A. Moszkowski, Phys. Rev. C 42 ( 1990) 1416. [I31 M. Centelles, X. Vifias, M. Barranco S. Marcos and R.I. Lombard, Nucl. Phys. A 537 (1992) 486. [I41 K.S. Cheng, Z.G. Dai and C.C. Yao, Astrophys. J. 464 ( 1996) 348; Z.G. Dai and K.S. Cheng, Astron. Astrophys., in press (1996). 1151 R. Machleidt, K. Holinde and C. Elster, Phys. Rep. 149 (1987) 1. iI61 R. Bthtgen, K. Holinde, A. Mtiller-Groeling, J. Speth and P Wybomy, Nucl. Phys. A 506 (1990) 586; A. Miiller-Groeling, K. Holinde and J. Speth, Nucl. Phys. A 513 (1990) 557. [ 171 J. Cohen, Phys. Rev. C 39 (1989) 2285. [ 181 T. Barnes and ES. Swanson, Phys. Rev. C 39 (1994) 1166. [ 191 V. Thorsson and A. Wirzba, Nucl. Phys. A 589 (1995) 633. [20] T. Muto, Prog. Theor. Phys. 55 (1993) 415.