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On the possibility of the condensation of the charged rho-meson field in dense isospin asymmetric baryon matter
. li!B
30 January 1997
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PHYSICS
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LETTERS B
Physics Letters B 392 (1997) 262-266
On the possibility of the condensation of the charged rho-meson field in dense isospin asymmetric baryon matter D.N. Voskresensky GSI, Darmstadt,
Planckstr.
’
1, D-64291 Darnwtadt,
Germany
Received 2 August 1996; revised manuscript received 21 November 1996 Editor: C. Mahaux
Abstract It is shown that at sufficiently large density the isospin-asymmetric baryon matter may undergo the phase transition to the p-condensate state being characterized by the finite values of the charged p-meson mean fields. The appearance of such a condensate diminishes the asymmetry energy of the baryon matter. It is shown that in the neutron stars the corresponding phase transition is of the first order. Possible consequences are enumerated. PACS: 21.65.+f; 21.90.Sf Keywords: Rho meson condensation; Asymmetrical nuclear matter; Neutron stars
According to Refs. [l] the p-meson can be treated as a dynamical gauge boson of a hidden local symmetry in the nonlinear chit-al Lagrangian. The resulting p-meson-baryon Lagrangian density renders
-
+& + $m@,plL,
(1)
where pyp = d,p,--&p,+g[ p,, p,l, the summation is over the baryon species, g,,B and KpB are the pbaryon coupling constants and g is the p-p coupling constant. The problem of determination of the p-meson Green’s function and the polarization operator in 1Permanent address: Moscow Institute for Physics and Engineering, Kashirskoe shosse 31, 115409 Moscow, Russia. 0370-2693/97/$17.00
nuclear matter has been widely discussed in the literature, see [2-lo] and refs. therein. Refs. [25,7] analyze p-meson interaction with A-isobarnucleon hole, nucleon-nucleon hole, pion, sigma and nucleon-antinucleon intermediate states. Tensor part of p-baryon interaction (being proportional to the KP~ coupling) has been studied in Refs. [ 6,9, lo]. All above mentioned contributions to the p-meson polarization operator are q-dependent and arise beyond the mean field level. The neutral p-meson field appears already on the mean field level as the response on the presence of the source of the baryon density for asymmetrical nuclear matter, see [ 1 I]. Ref. [ 121 discussed an influence of the neutral p-meson field on the pion condensate state of the neutron matter. According to our knowledge a part of the q-independent non-abelian interaction between various p-meson species was not so far considered. In this letter we will study a part of the non-abelian
Copyright 0 1997 Published by Elsevier Science B.V. All rights reserved.
PII SO370-2693(96>01561-4
D.N. Voskresensky/Physics
p-p-interaction in a dense isospin asymmetrical nuclear matter. Although this interaction manifests itself both on the mean field level and beyond the mean field, we will restrict ourselves by consideration of the mean fields. Therefore all mentioned above effects related to the loop corrections, being very important for determination of the q-dependent part of the polarization operator, do not alter the mean field solutions found below. Thus, we drop the tensor part of p-Binteraction and also neglect the other effects arising beyond the mean field. We will find mean field static solutions for charged p-meson condensate fields in rather dense isospin asymmetrical nuclear matter and discuss physical consequences of the possible charged p-meson condensation phenomenon. For the sake of simplicity we will not consider other baryons besides the nucleons neglecting of the possible filling of the A-isobar and hyperon states in sufficiently dense nucleon matter, see [ 131. Then, we deal with the SU( 2) -massive Yang-Mills field of p-mesons interacting with the two flavored massive fermion (neutron-proton) field. The analogy to the SU(2) QCD is straightforward. The p-mesons take a part of the (massive) gluons and the nucleons, of the two flavored quarks. In the following we use the parameter choice (cf. [1,13]): mP = 769 MeV, &a = gpN = g = m,/F,, F, = 132 MeV. According to the idea of Brown and Rho [ 141 we correct the pion decay coupling constant and accordingly the p-meson mass in the baryon matter replacing them by the corresponding effective values FG and rn;N m,FGIF,, whereas the value g&,N= g remains unchanged. The linear interpolation formula used in Refs. [ 141 yields mz/mp N (1 - ~/3ea) 1/3, where eo is the nuclear saturation density. Using QCD sum rules Ref. [ 151 predicts even stronger decrease of the p-meson mass with increasing the density. However, there is considerable uncertainty in this result implied by our poor knowledge of the density dependence of the 4-quark condensates, see Ref. [ 161. E.g., if one suggests that 4-quark condensates do not depend on the density the p-meson mass is found almost independent of the density 2 . 2Please notice that the effects of the p-meson polarization, see Refs. [ 2,3,5-7,9,10], lead to a renormalization of the q-dependent terms in the polarization operators of p-mesons and to the shifts of the frequencies in the corresponding dispersion laws different
263
Letters B 392 (1997) 262-266
We will use the mean field approximation and in accordance with the energy minimization consider the static fields. Since the isospin enters the nonlinear terms of pPy in the form of vector products of isospin vectors, the field p: does not enter the equations for pz, n = 1,2,3, and to explain the instability it is natural to assume pz = 0. Analysis of these non-linear terms also shows that there are no terms in the energy relating to the non-linear coupling of the sourceless field p; with the field p: and the coupling of the fields p: and p;, m = 1,2,3, leads to the repulsion increasing the system energy. Thus we may use the field ansatz Pp = {p;,pfi,P:}T Introducing
P;=Pi=o* the charged fields
(3) and using the gauge condition a,& = 0, after the averaging of Eq. ( 1) over the nucleon states the energy density for the remaining static mean p-meson fields renders
where es = i&N( 4 - en) is the source particle density, pP is the proton density, en is the neutron density, and the summation over the repeated indices implies. Variation of this functional over the fields gives the corresponding motion equations (cf. [ 171 for the corresponding quark-gluon case) :
= 0, APi
- “z;*p:
= -es
+ 2g2P$P,‘12.
(5)
The problem is reduced to the description of a nonlinear charged boson field ( p,‘) in the massive electriclike field (gp:) of the charged source particles (of density es). Analogous problem of the behavior of the for various p-meson species [ 101. However these .effects do not alter p-meson masses in the sense of Refs. [ 14-161 related to a density dependence of the quark condensates.
D.N. Voskresensky/Physics
264
charged pion field in an electric field of a supercharged nucleus was considered in [ 181. We may choose p$p; = p,;pk, that corresponds to the minimum of the energy density (4). Considering the homogeneous nucleon matter we find two constant solutions of the motion equations 3 . The solution Pi = eslm;2,
Pn‘=O
(6)
corresponds to the presence in the asymmetric nucleon matter of the mean neutral p-meson field (~2) and to the absence of the charged p-meson fields. With such a solution one gets an appropriate asymmetry energy of nucleon matter at the saturation nuclear density e = ~0, cf. [ 131. However at sufficiently large densities of the isospin asymmetric nucleon matter, e > @c(V) = 2m;3/(gg,Nlvl), v=(N-Z)/(N+Z),
(7)
there also exists other nontrivial equations g”(pz)”
= mz2,
solution of the motion
2g2p,+p, = -m;2 + es/p&
(8)
that corresponds to the saturation of the neutral pmeson field in the presence of the non-zero charged p-meson fields. For the solution (6) the energy density (4) renders E = &(ep
- @A2 8mz2 ’
whereas the solution
(9) (8) corresponds
to
Comparing the latter two equations we see that for e > pc ( Y) the solution (8) is energetically more favorable compared to the solution (6). For e = am the energy densities (9) and ( 10) coincide together with their first density derivatives, whereas the corresponding second derivatives differ. Thus (ignoring the Coulomb forces) we may conclude that for the fixed nucleon isospin configuration (V ) the charged 3 the problem of the matching of these solutions with the corresponding solutions in vacuum can be solved quite analogously to that in [ 181. We consider here a big system and the surface effects are not of our interest.
Letters B 392 (1997) 262-266
p-meson field would arise at p = &(v) by the second order phase transition. The value of the critical density (7) decreases with increase of the coefficient Iv] of the isospin asymmetry of the nucleon matter and appreciably decreases (N (mf/mp)3) with the decrease of the effective pmeson mass. For the neutron (proton) matter and for rn;= m,,one would have ec as large as 20~0. However with the help of the estimate of the effective pmeson mass based on the QCD sum-rules [ 151, rn;N 0.45m, for p N 2~0, we instead find that ec N 2~0 for the neutron (proton) matter. Using mentioned above estimate of Ref. [ 141 one obtains ec Y 2.6~0. We should also mention that the p-meson mass may decrease with increase of the temperature leading to an extra diminishing of the critical density. Thus, discussed above phase transition to the condensate of the charged p-meson fields may, indeed, occur at appropriate densities in the neutron star matter, whether the effective p-meson mass being substantially decreased. We see from (9) and ( 10) that both the energy densities behave quite differently with the change of the p-meson mass. The asymmetry energy (9) increases with the decrease of the p-meson mass for e < ec, whereas the energy density ( 10) decreases for e > &. In the limit case rnz--f0 the asymmetry energy ( 10) tends to zero. In the neutron stars the discussed phase transition is of the first order since the minimum of the total energy, E,,(e) , (including electron contribution) corresponds to the presence of somewhat larger protonelectron fraction in the old phase (for e < ec), than in the new one (for Q > ec). The critical density, ec, is now determined by the condition E,,( v:t), ec) = &,(v,$~), ec), w h ere superscripts 1 and 2 mean the old and the new phases, and subscript m means that the isotopic composition corresponds to the energy minimum. Due to a jump of v in the critical point, the pressure also undergoes some jump acquiring a typical van der Waals form. When the density in the central region of the star enlarges the value ~~1 < ec ( vnl), where ~~1 is given by the equal Maxwell constructions in the dependence of the pressure on the density at V~ (e) , the star may jump after some time passage to a more favorable isospin configuration with e > ec and v = v,$F) in the central region, cf. discussion [ 191 of an analogous pion condensate phase transition. In order to demonstrate more closely the gain in
D.N. Voskresensky/Physics
the total energy due to the phase transition let us for simplicity consider the limit, when rn;being sufficiently small. Then according to (9) the energetically favorable in the old phase is the isospin symmetrical matter. It has the energy density E,,( N = Z) = 27’,‘,(N = Z) +T,(N = Z), where 7”(N = Z) = T,(N = Z) is the energy density of the protons (neutrons) and Te is the energy density of ultrarelativistic electrons. In the new phase the more favorable is the neutron matter since the asymmetry energy (10) is equal to zero. Then one has E,,( Z = 0) = T,(Z = 0).Using that neutron, proton and electron Fermi momenta are connected by the relations pan (N = Z) = PQ,(N = Z) = PF~(N = Z) = PF~(Z = 0)/2’j3, in the nonrelativistic limit rn$ > J?FNone obtains
T,(Z = 0) 21 (3r2Q)5/3/(10r2m$),T,(N = Z) = (31r*~)~/~/(2~/~4~-~)2),T~(N= Z) =T,,(Z=0)/25/3 and for appropriate densities one has E,,(Z = 0) < ~~~~ (N = Z) . In utrarelativistic limit, rn&< PFN one obtains E,,(Z = 0) = 24/3T,(N= Z) < E,,,(N= Z) = 3T,(N = Z) and one again sees that the neutron matter is more energetically favorable than the isospin symmetrical one. Thus, for Q > eC we argue for a more neutron enriched neutron star matter than it occurs in traditional calculations4, that might be important for the problems related to the equation of state of the neutron stars. The net proton charge is now compensated not only by the corresponding electron charge but also by the electric charge of the charged p-meson field that also diminishes the energy. The appearance of the first order phase transition may have some consequences for the neutron star dynamics as a star quake, reheating, a jump in neutrino emission, etc., quite analogously to those discussed previously for the pion and kaon condensate phase transitions [ 19,211. The preference in a less isospin asymmetric composition in dense matter due to the decrease of the p-meson mass for .Q < eC and in a more neutron enriched composition for Q > eC due to the charged p meson condensation change conditions for the neu-
4This conclusion may be changed if the density is such that there are the pion or kaon condensates, cf. [ 20,19,21]. Therefore, we for simplicity suggest that the nucleon density that we deal with is still smaller than those values for the corresponding phase transitions.
Letters B 392 (1997) 262-266
265
trino radiation reactions in the neutron star interiors. The reactions e +p +-+ n + v from the dense region in supernova explosions are now in favor of a decrease of the neutronization rate for e < eC, whereas they are in favor of an increase of this rate for Q > eC. A more proton enriched matter of a neutron star that we argue for Q < eC allows now for the direct Urea processes n --f p + e + i; starting from smaller densities resulting in a substantial increase of the neutrino radiation from the corresponding star regions, whereas the same processes being suppressed for e > PC. In conclusion we have demonstrated that the isospin-asymmetric nucleon matter undergoes at e = eC the phase transition and becomes a condensate of the charged p-meson field, being coupled with the mean neutral p-meson field. An optimistic estimate of the critical value of the density for such a transition obtained with taking into account of the decrease of the p-meson mass in accordance with the results of Refs. [ 14,15,2] is (2-3) ea for the neutron star matter. Nevertheless, due to the mentioned above uncertainties in estimates of the p-meson mass we can’t conclude that the value of the critical density is very robust. More accurate estimates of the effective pmeson mass are still necessary in order to give a more definite conclusion on the possibility of the charged p-meson condensation at appropriate densities and temperatures. The asymmetry energy of the nucleon matter diminishes in the presence of the charged p-meson field condensate in favor of a more neutron enriched neutron star matter that also corresponds to a more soft equation of state at high densities. In neutron stars the condensate arises by the first order phase transition that may have some dynamical consequences for the neutron star evolution analogous to those for the pion and kaon condensates. It may affect the neutrino radiation rate from supernova explosions and subsequent neutron star cooling. We should also mention that the effects of nonabelian interaction lead to the corresponding contributions to the p-meson polarization operators. E.g., for the charged mesons according to Eq. (5) one has 6II* = -g”(p:)’ - 2((pg)2). The first term is via the mean field, whereas the second one is given by the temperature and density dependent part of the tadpole graph. One should add these terms to the corresponding contributions calculated in Refs. [ 2,3,5-7,9,10].
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Thus one can hope that effects of the non-abelian pp-interaction may manifest itself also in the heavy ion collisions. I acknowledge the hospitality and support of GSI, which made this work possible. The discussions of the results with G.E. Brown, B. Friman, J. Knoll and S.H. Lee are gratefully acknowledged.
References [II M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. 54 (1985) 1215; M. Harada, F. Sannino and J. Schechter, Preprint SU-4240624 (1995). PI M. Asakawa and C.M. Ko, Phys. Rev. C 48 (1993) R526. [31 M. Herrmann, B. Friman and W. Nbrenbetg, Nucl. Phys. A 560 (1993) 411. [41 G. Chanfray and F! Schtick, Nucl. Phys. A 545 (1992) 271~. r51 H.C. Jean, J. Piekarewicz and A.G. Williams, Phys. Rev. C 49 (1994) 1981. [61 H. Shiomi and T. Hatsuda, Phys. Lett. B 334 ( 1994) 281. [71 B. Friman, Preprint GSI-95-24; B. Friman and M. Sayeur, Preprint GSI-95-68.
Letters B 392 (1997) 262-266
[8] R. Machleidt, Adv. in Nucl. Phys. 19 (1989) 189. [9] A.K. Dutt-Mazumber, B. Dutta-Roy, A. Kundu and T. De, Phys. Rev. C 53 (1996) 790. [lo] A.K. Dutt-Mazumber, A. Kundu, T. De and B. Dun&Roy, Phys. Lett. B 378 (1996) 35. [ 111 B.D. Serot and J.D. Walecka, Ad. Nucl. Phys. 16 (1986) 1. [ 121 I. Kunihiro, Prog. Theor. Phys. 60 (1978) 1229. [ 131 N.K. Glendenning, Nucl. Phys. A 493 (1989) 521. [ 141 G.E. Brown and M. Rho, Phys. Rev. Lett. 66 (1991) 2720; Phys. Rep. 269 (1996) 334. [ 151 T. Hatsuda and S.H. Lee, Phys. Rev. C 46 (1992) R34. [ 161 R. Brockmann and W. Weise, Phys. Lett. B 367 (1996) 40. [ 171 A.B. Migdal, Pisma v ZhBTF 28 (1978) 35; in Engl. [JETP Len 27 (1978) 351. [18] D.N. Voskresensky and A.I. Chernoutsan, Yad. Fiz. 27 (1978) 1411; in Engl. [Sov. J. Nucl. Phys. 27 (1978) 7421. [ 191 A.B. Migdal, E.E. Saperstein, M.A. Troitsky and D.N. Voskresensky, Phys. Rep. 192 (1990) 179. [20] A.B. Migdal, Rev. Mod. Phys. 50 (1978) 107; G.E. Brown and W. Weise, Phys. Rep. 27 (1976) 1. [21] G.E. Brown, C.H. Lee, M. Rho and V. Thorsson, Nucl. Phys. A 567 (1994) 937; C.H. Lee, G.E. Brown, D.-P. Min and M. Rho, Nucl. Phys. A 585 (1995) 401; E.E. Kolomeitsev, D.N. Voskresensky and B. Kiimpfer, Nucl. Phys. A 588 (1995) 889.
30 January 1997
cm3 _a
e
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PHYSICS
EJ_SEXVIER
LETTERS B
Physics Letters B 392 (1997) 262-266
On the possibility of the condensation of the charged rho-meson field in dense isospin asymmetric baryon matter D.N. Voskresensky GSI, Darmstadt,
Planckstr.
’
1, D-64291 Darnwtadt,
Germany
Received 2 August 1996; revised manuscript received 21 November 1996 Editor: C. Mahaux
Abstract It is shown that at sufficiently large density the isospin-asymmetric baryon matter may undergo the phase transition to the p-condensate state being characterized by the finite values of the charged p-meson mean fields. The appearance of such a condensate diminishes the asymmetry energy of the baryon matter. It is shown that in the neutron stars the corresponding phase transition is of the first order. Possible consequences are enumerated. PACS: 21.65.+f; 21.90.Sf Keywords: Rho meson condensation; Asymmetrical nuclear matter; Neutron stars
According to Refs. [l] the p-meson can be treated as a dynamical gauge boson of a hidden local symmetry in the nonlinear chit-al Lagrangian. The resulting p-meson-baryon Lagrangian density renders
-
+& + $m@,plL,
(1)
where pyp = d,p,--&p,+g[ p,, p,l, the summation is over the baryon species, g,,B and KpB are the pbaryon coupling constants and g is the p-p coupling constant. The problem of determination of the p-meson Green’s function and the polarization operator in 1Permanent address: Moscow Institute for Physics and Engineering, Kashirskoe shosse 31, 115409 Moscow, Russia. 0370-2693/97/$17.00
nuclear matter has been widely discussed in the literature, see [2-lo] and refs. therein. Refs. [25,7] analyze p-meson interaction with A-isobarnucleon hole, nucleon-nucleon hole, pion, sigma and nucleon-antinucleon intermediate states. Tensor part of p-baryon interaction (being proportional to the KP~ coupling) has been studied in Refs. [ 6,9, lo]. All above mentioned contributions to the p-meson polarization operator are q-dependent and arise beyond the mean field level. The neutral p-meson field appears already on the mean field level as the response on the presence of the source of the baryon density for asymmetrical nuclear matter, see [ 1 I]. Ref. [ 121 discussed an influence of the neutral p-meson field on the pion condensate state of the neutron matter. According to our knowledge a part of the q-independent non-abelian interaction between various p-meson species was not so far considered. In this letter we will study a part of the non-abelian
Copyright 0 1997 Published by Elsevier Science B.V. All rights reserved.
PII SO370-2693(96>01561-4
D.N. Voskresensky/Physics
p-p-interaction in a dense isospin asymmetrical nuclear matter. Although this interaction manifests itself both on the mean field level and beyond the mean field, we will restrict ourselves by consideration of the mean fields. Therefore all mentioned above effects related to the loop corrections, being very important for determination of the q-dependent part of the polarization operator, do not alter the mean field solutions found below. Thus, we drop the tensor part of p-Binteraction and also neglect the other effects arising beyond the mean field. We will find mean field static solutions for charged p-meson condensate fields in rather dense isospin asymmetrical nuclear matter and discuss physical consequences of the possible charged p-meson condensation phenomenon. For the sake of simplicity we will not consider other baryons besides the nucleons neglecting of the possible filling of the A-isobar and hyperon states in sufficiently dense nucleon matter, see [ 131. Then, we deal with the SU( 2) -massive Yang-Mills field of p-mesons interacting with the two flavored massive fermion (neutron-proton) field. The analogy to the SU(2) QCD is straightforward. The p-mesons take a part of the (massive) gluons and the nucleons, of the two flavored quarks. In the following we use the parameter choice (cf. [1,13]): mP = 769 MeV, &a = gpN = g = m,/F,, F, = 132 MeV. According to the idea of Brown and Rho [ 141 we correct the pion decay coupling constant and accordingly the p-meson mass in the baryon matter replacing them by the corresponding effective values FG and rn;N m,FGIF,, whereas the value g&,N= g remains unchanged. The linear interpolation formula used in Refs. [ 141 yields mz/mp N (1 - ~/3ea) 1/3, where eo is the nuclear saturation density. Using QCD sum rules Ref. [ 151 predicts even stronger decrease of the p-meson mass with increasing the density. However, there is considerable uncertainty in this result implied by our poor knowledge of the density dependence of the 4-quark condensates, see Ref. [ 161. E.g., if one suggests that 4-quark condensates do not depend on the density the p-meson mass is found almost independent of the density 2 . 2Please notice that the effects of the p-meson polarization, see Refs. [ 2,3,5-7,9,10], lead to a renormalization of the q-dependent terms in the polarization operators of p-mesons and to the shifts of the frequencies in the corresponding dispersion laws different
263
Letters B 392 (1997) 262-266
We will use the mean field approximation and in accordance with the energy minimization consider the static fields. Since the isospin enters the nonlinear terms of pPy in the form of vector products of isospin vectors, the field p: does not enter the equations for pz, n = 1,2,3, and to explain the instability it is natural to assume pz = 0. Analysis of these non-linear terms also shows that there are no terms in the energy relating to the non-linear coupling of the sourceless field p; with the field p: and the coupling of the fields p: and p;, m = 1,2,3, leads to the repulsion increasing the system energy. Thus we may use the field ansatz Pp = {p;,pfi,P:}T Introducing
P;=Pi=o* the charged fields
(3) and using the gauge condition a,& = 0, after the averaging of Eq. ( 1) over the nucleon states the energy density for the remaining static mean p-meson fields renders
where es = i&N( 4 - en) is the source particle density, pP is the proton density, en is the neutron density, and the summation over the repeated indices implies. Variation of this functional over the fields gives the corresponding motion equations (cf. [ 171 for the corresponding quark-gluon case) :
= 0, APi
- “z;*p:
= -es
+ 2g2P$P,‘12.
(5)
The problem is reduced to the description of a nonlinear charged boson field ( p,‘) in the massive electriclike field (gp:) of the charged source particles (of density es). Analogous problem of the behavior of the for various p-meson species [ 101. However these .effects do not alter p-meson masses in the sense of Refs. [ 14-161 related to a density dependence of the quark condensates.
D.N. Voskresensky/Physics
264
charged pion field in an electric field of a supercharged nucleus was considered in [ 181. We may choose p$p; = p,;pk, that corresponds to the minimum of the energy density (4). Considering the homogeneous nucleon matter we find two constant solutions of the motion equations 3 . The solution Pi = eslm;2,
Pn‘=O
(6)
corresponds to the presence in the asymmetric nucleon matter of the mean neutral p-meson field (~2) and to the absence of the charged p-meson fields. With such a solution one gets an appropriate asymmetry energy of nucleon matter at the saturation nuclear density e = ~0, cf. [ 131. However at sufficiently large densities of the isospin asymmetric nucleon matter, e > @c(V) = 2m;3/(gg,Nlvl), v=(N-Z)/(N+Z),
(7)
there also exists other nontrivial equations g”(pz)”
= mz2,
solution of the motion
2g2p,+p, = -m;2 + es/p&
(8)
that corresponds to the saturation of the neutral pmeson field in the presence of the non-zero charged p-meson fields. For the solution (6) the energy density (4) renders E = &(ep
- @A2 8mz2 ’
whereas the solution
(9) (8) corresponds
to
Comparing the latter two equations we see that for e > pc ( Y) the solution (8) is energetically more favorable compared to the solution (6). For e = am the energy densities (9) and ( 10) coincide together with their first density derivatives, whereas the corresponding second derivatives differ. Thus (ignoring the Coulomb forces) we may conclude that for the fixed nucleon isospin configuration (V ) the charged 3 the problem of the matching of these solutions with the corresponding solutions in vacuum can be solved quite analogously to that in [ 181. We consider here a big system and the surface effects are not of our interest.
Letters B 392 (1997) 262-266
p-meson field would arise at p = &(v) by the second order phase transition. The value of the critical density (7) decreases with increase of the coefficient Iv] of the isospin asymmetry of the nucleon matter and appreciably decreases (N (mf/mp)3) with the decrease of the effective pmeson mass. For the neutron (proton) matter and for rn;= m,,one would have ec as large as 20~0. However with the help of the estimate of the effective pmeson mass based on the QCD sum-rules [ 151, rn;N 0.45m, for p N 2~0, we instead find that ec N 2~0 for the neutron (proton) matter. Using mentioned above estimate of Ref. [ 141 one obtains ec Y 2.6~0. We should also mention that the p-meson mass may decrease with increase of the temperature leading to an extra diminishing of the critical density. Thus, discussed above phase transition to the condensate of the charged p-meson fields may, indeed, occur at appropriate densities in the neutron star matter, whether the effective p-meson mass being substantially decreased. We see from (9) and ( 10) that both the energy densities behave quite differently with the change of the p-meson mass. The asymmetry energy (9) increases with the decrease of the p-meson mass for e < ec, whereas the energy density ( 10) decreases for e > &. In the limit case rnz--f0 the asymmetry energy ( 10) tends to zero. In the neutron stars the discussed phase transition is of the first order since the minimum of the total energy, E,,(e) , (including electron contribution) corresponds to the presence of somewhat larger protonelectron fraction in the old phase (for e < ec), than in the new one (for Q > ec). The critical density, ec, is now determined by the condition E,,( v:t), ec) = &,(v,$~), ec), w h ere superscripts 1 and 2 mean the old and the new phases, and subscript m means that the isotopic composition corresponds to the energy minimum. Due to a jump of v in the critical point, the pressure also undergoes some jump acquiring a typical van der Waals form. When the density in the central region of the star enlarges the value ~~1 < ec ( vnl), where ~~1 is given by the equal Maxwell constructions in the dependence of the pressure on the density at V~ (e) , the star may jump after some time passage to a more favorable isospin configuration with e > ec and v = v,$F) in the central region, cf. discussion [ 191 of an analogous pion condensate phase transition. In order to demonstrate more closely the gain in
D.N. Voskresensky/Physics
the total energy due to the phase transition let us for simplicity consider the limit, when rn;being sufficiently small. Then according to (9) the energetically favorable in the old phase is the isospin symmetrical matter. It has the energy density E,,( N = Z) = 27’,‘,(N = Z) +T,(N = Z), where 7”(N = Z) = T,(N = Z) is the energy density of the protons (neutrons) and Te is the energy density of ultrarelativistic electrons. In the new phase the more favorable is the neutron matter since the asymmetry energy (10) is equal to zero. Then one has E,,( Z = 0) = T,(Z = 0).Using that neutron, proton and electron Fermi momenta are connected by the relations pan (N = Z) = PQ,(N = Z) = PF~(N = Z) = PF~(Z = 0)/2’j3, in the nonrelativistic limit rn$ > J?FNone obtains
T,(Z = 0) 21 (3r2Q)5/3/(10r2m$),T,(N = Z) = (31r*~)~/~/(2~/~4~-~)2),T~(N= Z) =T,,(Z=0)/25/3 and for appropriate densities one has E,,(Z = 0) < ~~~~ (N = Z) . In utrarelativistic limit, rn&< PFN one obtains E,,(Z = 0) = 24/3T,(N= Z) < E,,,(N= Z) = 3T,(N = Z) and one again sees that the neutron matter is more energetically favorable than the isospin symmetrical one. Thus, for Q > eC we argue for a more neutron enriched neutron star matter than it occurs in traditional calculations4, that might be important for the problems related to the equation of state of the neutron stars. The net proton charge is now compensated not only by the corresponding electron charge but also by the electric charge of the charged p-meson field that also diminishes the energy. The appearance of the first order phase transition may have some consequences for the neutron star dynamics as a star quake, reheating, a jump in neutrino emission, etc., quite analogously to those discussed previously for the pion and kaon condensate phase transitions [ 19,211. The preference in a less isospin asymmetric composition in dense matter due to the decrease of the p-meson mass for .Q < eC and in a more neutron enriched composition for Q > eC due to the charged p meson condensation change conditions for the neu-
4This conclusion may be changed if the density is such that there are the pion or kaon condensates, cf. [ 20,19,21]. Therefore, we for simplicity suggest that the nucleon density that we deal with is still smaller than those values for the corresponding phase transitions.
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trino radiation reactions in the neutron star interiors. The reactions e +p +-+ n + v from the dense region in supernova explosions are now in favor of a decrease of the neutronization rate for e < eC, whereas they are in favor of an increase of this rate for Q > eC. A more proton enriched matter of a neutron star that we argue for Q < eC allows now for the direct Urea processes n --f p + e + i; starting from smaller densities resulting in a substantial increase of the neutrino radiation from the corresponding star regions, whereas the same processes being suppressed for e > PC. In conclusion we have demonstrated that the isospin-asymmetric nucleon matter undergoes at e = eC the phase transition and becomes a condensate of the charged p-meson field, being coupled with the mean neutral p-meson field. An optimistic estimate of the critical value of the density for such a transition obtained with taking into account of the decrease of the p-meson mass in accordance with the results of Refs. [ 14,15,2] is (2-3) ea for the neutron star matter. Nevertheless, due to the mentioned above uncertainties in estimates of the p-meson mass we can’t conclude that the value of the critical density is very robust. More accurate estimates of the effective pmeson mass are still necessary in order to give a more definite conclusion on the possibility of the charged p-meson condensation at appropriate densities and temperatures. The asymmetry energy of the nucleon matter diminishes in the presence of the charged p-meson field condensate in favor of a more neutron enriched neutron star matter that also corresponds to a more soft equation of state at high densities. In neutron stars the condensate arises by the first order phase transition that may have some dynamical consequences for the neutron star evolution analogous to those for the pion and kaon condensates. It may affect the neutrino radiation rate from supernova explosions and subsequent neutron star cooling. We should also mention that the effects of nonabelian interaction lead to the corresponding contributions to the p-meson polarization operators. E.g., for the charged mesons according to Eq. (5) one has 6II* = -g”(p:)’ - 2((pg)2). The first term is via the mean field, whereas the second one is given by the temperature and density dependent part of the tadpole graph. One should add these terms to the corresponding contributions calculated in Refs. [ 2,3,5-7,9,10].
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Thus one can hope that effects of the non-abelian pp-interaction may manifest itself also in the heavy ion collisions. I acknowledge the hospitality and support of GSI, which made this work possible. The discussions of the results with G.E. Brown, B. Friman, J. Knoll and S.H. Lee are gratefully acknowledged.
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