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Equation of state for pion-condensed neutron matter at finite temperature
Volume 78B, number 5
PHYSICS LETTERS
23 October 1978
EQUATION OF STATE FOR PION-CONDENSED NEUTRON MATTER AT FINITE TEMPERATURE * H. TOKI, Y. FUTAMI 1 and W. WEISE Institute of Theoretieal Physics, University of Regensburg, D-8400 Regensburg. W. Germany Received 26 July 1978
The equation of state for neutron matter in the presence of a pion condensate is investigated at finite, but small temperature within the a model. It is found that a transition of van der Waals type takes place at low temperature for sufficiently strong effective p-wave interaction, which disappears however beyond a critical temperature Tc. Within a wide variety of model assumptions, an upper limit of about 50 MeV is found for Tc.
The equation of state for dense matter at finite temperature enters in a variety of problems and speculations of current interest, e.g. in investigations of neutron star cooling [1], and in considerations of possible compression phenomena in high energy heavy-ion collisions [2,3]. Pion condensation [ 4 - 7 ] plays a particularly interesting role in such considerations, although no evidence for its existence has been established yet. So far, equations of state for pion condensed matter have been evaluated at zero temperature only (see refs. [8,9] for neutron matter and refs. [10,11] for symmetric nuclear matter). On the other hand, finite temperature plays an important role in high energy heaw-ion collisions, where temperatures + ' T > 50 MeV are reached if compression takes place f2. In the centre of a supernova, at the point where a neutron star is formed temperatures are supposed to be lower, of the order of T ~ 10 MeV. The study of the temperature dependent equation of state is motivated by the question up to which temperature a coherent effect like pion condensation can survive as a transition of the van der Waals-type such that the equation of state shows a region of
* Work supported in part by Deutsche Forschungsgemeinschaft. 1 Supported by DFG; on leave from Science University of Tokyo, Japan.
negative compressibility. Determinations of the phase boundary for pion condensation in the density/temperature plane [12, 13,19] show that an instability - if existent at zero temperature plane persists up to high temperatures, but the Green's function approach used in refs. [12, 13] does not allow to determine the nature of the phase transition and properties of the condensed phase. In this paper we would like to present a survey calculation of the equation of state for neutron matter at T=/= 0 using the o model of ref. [6]. A charged pion condensate appears in this model through formation of n e u t r o n - p r o t o n quasiparticles in the ground state together with a finite expectation value (Tr) for the pion field, which is accompanied by a a field of large mass such that (a) 2 + (Tr)2 = f 2 is a chiral invariant, with fTr = 94.5 MeV (pion decay constant). Concerning the finite temperature aspect, our investigation will be based on the following assumptions: (1) Thermal fluctuations of the o field are neglected; (2) Thermal fluctuations of the condensed pion field appear only through the coupling to the nucleon +l We use units such that the Boltzmann constant k B -= 1. +2 Assuming that the equilibrium concepts are meaningful, as claimed e.g. in hydrodynamic pictures. 547
Volume 78B, number 5
PHYSICS LETTERS
source function, i.e. through the thermal distribution of n e u t r o n - p r o t o n quasiparticles. The scale for fluctuations of the o field is set in the work of Baym and Grinstein [14], who find that the thermal expectation value (o) decreases with rising temperature and vanishes beyond a certain temperature T c. In their "modified Hartree" model in three dimensions, they find ire = x/'6f~r. Although the approximations used in ref. [14] tend to break down at temperatures larger than Tc, we take this value as an indication that, as long as we confine ourselves to T 2 ~ ~cc, such fluctuations will not matter very nmch. ha practice we shall work at temperatures T<½mrr, with mrr the pion mass. Assumption (1) is then saying that the radius fTr of the "chiral circle" remains unchanged at sufficiently low temperature. Assumption (2) is motivated by the fact that the driving force for the pion condensate is the p-wave coupling to a nucleon-hole pair, so that finite ten> perature will primarily enter in the nucleon Fermi distributions. It will then be shown that in the low temperature region, the leading terms in the thermal part of the pressure follow the standard Fermi gas result for the finite temperature extrapolation. With these assumptions, and according to the developments in ref. [6], the expectation values for the (negatively charged) pion and the o field are written in terms of the chiral angle 0: (7r) = 2
1/2frrsin0 e-ik'r,
(1)
(o) = f~ cos 0.
(2)
The hamiltonian density, reduced non-relativistically and supplemented by constraints for charge and baryon number conservation, is given by [6] c~+/.tpQ
_
2 up = l ( k 2 - / - t 2 ) f 2 sin20 - . f ~2 m~r COS0
+~(pll/.t--u+ P
+ kr2 sin 0 [p }apap,
(3)
where gA = 2fTrf/mTr (with f2/47r = 0.08). We have omitted here A-isobars and short-range correlations, which will be added later on. The p and pQ are baryon and total charge densities, and/~ plays the role of the pion chemical potential. Diagonalisation of the nucleon part of eq. (3) 548
gives the quasiparticle energies (relative to the ct~emical potential u):
E+(p) =p2 /2M- u + k2 cos20/8M + 1 (/.t + [(U -
p'k/M) 2 cos20 +g2k2 sin20l 1/2} (4)
Temperature is then introduced in terms of the grand partition function g = Tr exp [-(/} + P 0 -
u~)/T],
(5)
where/~ is the hamiltonian, 0 and N are the charge and baryon number operators. The thermodynamic potential, g2 = - ( T / V ) In g,
(6)
is obtained in the form a = l(k2
/12)j~ sin20
- f ~2m ~
2
cos 0
- 2 T ~ f d ~ P a ln[1 + exp(-E+(p)/T)l. - -
{z~7]')
(7)
~
Note that the explicit temperature dependence in this model, in the assumed absence of intrinsic thermal fluctuations of the meson fields, comes only in the distribution of quasiparticles. On the other hand, the strong coupling of the pion field to its nucleon sources introduces an inrplicit T dependence in (Tr). Equilibrium conditions require that g2 be at a minimum (i.e. the entropy is maximised) with respect to variation of the parameters k and 0. The chemical potentials/a and u are determined by the conditions of charge and baryon number conservation. In particular, ~gZ
-
d3p (_+)
.(& (p)/r) = -p,
(8)
w i t h the F e r m i d i s t r i b u t i o n f u n c t i o n
n(z) = (e z + 1) -1
1 m - l ( p _ lkT3 COS0)2
+ 1/jr 3 cos 0 1 ~gA ~"
23 October 1978
(9)
In order to proceed, we shall introduce the following approximations, valid for neutron matter, where the pion chemical potential H is of the order of the pion mass and larger. In eq. (4), we assume p • k/M'~ la and drop the k 2 cos20/8M term for similar reasons. (Note that such an approximation cannot be used for symmetric nuclear matter [11], where we have/~ = 0.) The conditions ~gZ/~k = ~g2/~/1 = ~2/~0 = 0 are then easily written down.
Volume 78B, number 5
PHYSICS LETTERS
Furthermore, at sufficiently low temperature T, only the lower quasiparticle states E _ ( p ) will be populated. In fact the gap between the two quasiparticle spectra (at T = 0) is given by
23 October 1978 p
i
i
i
i
[MeV fm 3] 20
6 = E'+(0) -- IJ- (PF) = M~2 COS20 +g2k2 sin20
- p 2 . / 2 M = (g2/2f2)p - (37rZp)2/3/ZM,
SO
(10) 15
where we have made use of the minimisation conditions at T = 0 arrive at the latter expression. For densities in our range of interest (p >~ 2P0), 6 is of the order of ma and larger. If we restrict ourselves to T < l m ~ , we can safely assume that only the E levels need to be taken into account. Then the free energy density, e=~+up,
¢0
10
(ll) 5
takes a simple form, namely:
g2 02
(g2 _ 1)sin2O
8f 2
1 + (g2 _ 1)sin20
e-
01
2 2 cos 0 + up f~rm~
4 ~.d3p p2 - 3 - ) ( 9 . ~ - 3 2M n((p2/2M \--..
X)/7),
(]2)
j
with
g2
x
20 0
= u +
(g2 _ 1)sin20
(13)
p
4f2
1+ (g2 _ 1)sin2O '
where u is the quasiparticle chemical potential, while X is determined by d3p n ( ( p 2 / 2 M - X)/T) = O.
(14)
(15)
Some more realistic features have to be added to eq. (12). First, short range correlations have to be introduced. This is done in terms o f a Fermi liquid parameter g', in the same way as in refs. [6,7]. Furthemmre, A-isobars have to be included. It is shown in ref. [8] that at T = 0 the combined effect o f g ' and/X-isobars can be absorbed to a very good degree of accuracy by an effective renormalisation o f g A : *2 = (1 - g')(1 + A)g 2, g 2 ___>gA
03
04
05 p[fm -3]
Fig. 1. Equation of state for pion condensed neutron matter using Pandharipande's equation of state [ 16] for the normal phase. The effective p-wave interaction strength is chosen to be g~ = 1.33. Numbers at the different isothermes denote temperature in MeV. The critical temperature for a first order transition of van der Waats form is Tc = 42 MeV in this example. T<~ l m a , g' and A are assumed to be independent of T. We shall simply use gA as our basic model parameter, which then replaces g A in eqs. (12) and (13). Tile energy density will be written in the form
e(p, T;O) = e(p, r = 0 ; 0 ) + Ae(7),
(17)
where e(p, T = 0; 0) = e0(p, 0 = 0)
Note that at zero temperature,
X ( T = O) - X 0 = (31r2p)2/3 /2M.
02
(16)
where realistic values for g' range between 0.5 and 0.6 [15], while A = 0 . 8 - 0 . 9 [8]. For temperatures
1 [g*A ] 2 --8~P]
(gA2 -- 1)sin20 1 + (gA2 -- l)sin20
2 2 sin20/2. + 2f~rnrr
(18)
In practice we shall use realistic equations of state for
Co(P, 0 = 0) to describe the normal phase at T = 0, alternatively those of Pandharipande (P) [16] and of Bethe and Johnson (B-J) [17]. Tire extrapolation to finite but small temperature in/xe is then done according to eqs. (12) and (14). After minimisation of e with respect to 0, the pressure P(p, 7) can be derived in the form
P(p, 7) = P(p, T = O) + PT(P) .
(19) 549
Volume 78B, number 5
PttYSICS LETTERS
k& [MeV}
B-J
20
0
13
135
14
g~
--
1/+5 '
I'5
Fig. 2. Critical temperature T c for first order transition into pion condensed neutron matter as a function of the effective p-wave interaction strength g ~ (see eq. (16)), for two different equations of state describing the normal phase: Pandharipande (P) [16]; Bethe and Johnson (B-J) It 7].
The standard low-temperature expansion yields [-Tr2 / T '~2
7r4
)4
While this is simply the finite temperature extrapolation for a free Fermi gas, corrections would arise from replacing M by the effective mass M*(p). At the moment, we shall omit these and present in fig. 1 an example for the isothermes in the pressure/ density plane. We observe that while a region of negative compression modulus exists at low temperature, it does not survive beyond a certain critical temperature Tc, which is somewhat larger than 40 MeV in the case shown here. We would now like to investigate the model dependence of this T c. For that purpose we compare values of T c obtained with a soft (P) and a stiff (B J) reference equation of state describing the normal system, as a function of the effective p-wave interaction strength g ; (see fig. 2). Clearly, the BJ equation of state, which has m&e repulsive correlations at high density, needs a larger g~, than the Pandharipande equation of state in order to drive the system into the pion condensate. It is interesting to note that irrespective of the particular model, we always obtain T c ~< 45 MeV. Next, we study the dependence of T c on the effective nucleon mass. We do this as follows: (a) The standard o model (at 0 = 0) [6], with hard core repulsion of radius r c added, is used to 550
23 October 1978
reproduce first the equation of state for normal neutron matter (either (P) or (B-J)) by appropriate adjustment of r c. This is possible for a wide range of densities. (b) The "chiral radius" f . (see eq. (2)) is treated as a variational parameter. Additional minimisation of f2 with respect to fTr determines the effective nucleon mass M* as a function of density. Unlike the procedure taken in ref. [18], we do not treat g A as an independent parameter, but use the identification gA = 2fcr(f/mTr) as in ref. [51. In fact, fTr (and therefore M*) turns out to be smoothly decreasing with increasing density. A decreasingfTr will effectively increase the repulsive s-wave 7r neutron interaction, while the p-wave attraction remains essentially the same. Hence incorporation of this degree of freedom shifts the curves in fig. 2 to larger values o f g ~ , without changing the upper limit for T c. The question about thernral fluctuations of the pion field (other than those taken into account through the coupling to the thermal distribution of nucleon sources) remains open. Such effects would probably lower the critical temperature T c even more. In summary, we find that the phase transition to pion condensed neutron matter resembles a van der Waals form of the equation of state at low temperature for sufficiently strong net attraction in the effective pion-nucleon interactions. This case would apply for neutron stars formed in a supernova at T ~ 10 MeV. However, we observe that a region of negative compression modulus exists only up to a critical temperature T c which turns out to be always smaller than 50 MeV for a wide variety of model assumptions. The low value of T c opens the question about the possible formation of density isomers in nuclear matter at temperatures as large as they are assumed to appear in high energy heavy ion collisions. An investigation of this problem for symmetric nuclear matter is in progress.
References [ 1 ] O. Maxwell et al., Astrophy s. J. 216 (1977) 77. [2] J. Hofmann et al., Phys. Rev. Lett. 36 (1976) 88. [3] V.M. Galitskii and I.N. Mishustin, Phys. Lett. 72 B (1978) 285. [4] A.B. Migdal, O. Markin and I. Mishustin, JETP 66 (1974) 443; A.B. Migdal, Rev. Mod. Phys. (1978), to be published.
Volume 78B, number 5
PHYSICS LETTERS
[5] C.K. Au and G. Baym, Nucl. Phys. A 236 (1974) 500. [6] D. Campbell, R. Dashen and J.R. Manassah, Phys. Rev. D 12 (1975) 919; G. Baym, D. Campbell, R. Dashen and J.R. Manassah, Phys. Lett. 58B (1975) 304. [7] G.E. Brown and W. Weise, Phys. Rept. 27 C (1976) 1. [8] W. Weise and G.E. Brown, Phys. Lett. 58 B (1975) 300. [9] C.K. Au, Phys. Lett. 61 B (1976) 300. [10] A.B. Migdal, G.A. Sorokin, O.A. Markin and I.N. Mishustin, Phys. Lett. 65 B (1976) 423. [ 11 ] tt.J. Pirner, M. Rho and K. Yazaki, to be published. [12] V. Ruck, M. Gyulassi and W. Greiner, Z. Phys. A 277 (1976) 391. [13] P. Hecking, W. Weise and R. Akhoury, to be published.
23 October 1978
[14] G. Baym and G. Grinstein, Phys. Rev. D 15 (1977) 2897. [15] S.O. Btickman and W. Weise, in: Mesons in nuclei, eds. M. Rho and D.H. Wilkinson (North-Holland, Amsterdam, to be published); G.E. Brown, S.O. B~ickman, E. Oset and W. Weise, Nucl. Phys. A 286 (1977) 191. [16] V. Pandharipande, Nucl. Phys. A 178 (1971) 123. [17] H.A. Bethe and M.B. Johnson, Nucl. Phys. A 230 (1974) 1. [18] M. Chanowitz and P.J. Siemens, Phys. Lett. 70 B (1977) 175. [19] M. Sano, Proc. Intern. Conf. on Nuclear Structure (Tokyo, 1977)p. 828.
551
PHYSICS LETTERS
23 October 1978
EQUATION OF STATE FOR PION-CONDENSED NEUTRON MATTER AT FINITE TEMPERATURE * H. TOKI, Y. FUTAMI 1 and W. WEISE Institute of Theoretieal Physics, University of Regensburg, D-8400 Regensburg. W. Germany Received 26 July 1978
The equation of state for neutron matter in the presence of a pion condensate is investigated at finite, but small temperature within the a model. It is found that a transition of van der Waals type takes place at low temperature for sufficiently strong effective p-wave interaction, which disappears however beyond a critical temperature Tc. Within a wide variety of model assumptions, an upper limit of about 50 MeV is found for Tc.
The equation of state for dense matter at finite temperature enters in a variety of problems and speculations of current interest, e.g. in investigations of neutron star cooling [1], and in considerations of possible compression phenomena in high energy heavy-ion collisions [2,3]. Pion condensation [ 4 - 7 ] plays a particularly interesting role in such considerations, although no evidence for its existence has been established yet. So far, equations of state for pion condensed matter have been evaluated at zero temperature only (see refs. [8,9] for neutron matter and refs. [10,11] for symmetric nuclear matter). On the other hand, finite temperature plays an important role in high energy heaw-ion collisions, where temperatures + ' T > 50 MeV are reached if compression takes place f2. In the centre of a supernova, at the point where a neutron star is formed temperatures are supposed to be lower, of the order of T ~ 10 MeV. The study of the temperature dependent equation of state is motivated by the question up to which temperature a coherent effect like pion condensation can survive as a transition of the van der Waals-type such that the equation of state shows a region of
* Work supported in part by Deutsche Forschungsgemeinschaft. 1 Supported by DFG; on leave from Science University of Tokyo, Japan.
negative compressibility. Determinations of the phase boundary for pion condensation in the density/temperature plane [12, 13,19] show that an instability - if existent at zero temperature plane persists up to high temperatures, but the Green's function approach used in refs. [12, 13] does not allow to determine the nature of the phase transition and properties of the condensed phase. In this paper we would like to present a survey calculation of the equation of state for neutron matter at T=/= 0 using the o model of ref. [6]. A charged pion condensate appears in this model through formation of n e u t r o n - p r o t o n quasiparticles in the ground state together with a finite expectation value (Tr) for the pion field, which is accompanied by a a field of large mass such that (a) 2 + (Tr)2 = f 2 is a chiral invariant, with fTr = 94.5 MeV (pion decay constant). Concerning the finite temperature aspect, our investigation will be based on the following assumptions: (1) Thermal fluctuations of the o field are neglected; (2) Thermal fluctuations of the condensed pion field appear only through the coupling to the nucleon +l We use units such that the Boltzmann constant k B -= 1. +2 Assuming that the equilibrium concepts are meaningful, as claimed e.g. in hydrodynamic pictures. 547
Volume 78B, number 5
PHYSICS LETTERS
source function, i.e. through the thermal distribution of n e u t r o n - p r o t o n quasiparticles. The scale for fluctuations of the o field is set in the work of Baym and Grinstein [14], who find that the thermal expectation value (o) decreases with rising temperature and vanishes beyond a certain temperature T c. In their "modified Hartree" model in three dimensions, they find ire = x/'6f~r. Although the approximations used in ref. [14] tend to break down at temperatures larger than Tc, we take this value as an indication that, as long as we confine ourselves to T 2 ~ ~cc, such fluctuations will not matter very nmch. ha practice we shall work at temperatures T<½mrr, with mrr the pion mass. Assumption (1) is then saying that the radius fTr of the "chiral circle" remains unchanged at sufficiently low temperature. Assumption (2) is motivated by the fact that the driving force for the pion condensate is the p-wave coupling to a nucleon-hole pair, so that finite ten> perature will primarily enter in the nucleon Fermi distributions. It will then be shown that in the low temperature region, the leading terms in the thermal part of the pressure follow the standard Fermi gas result for the finite temperature extrapolation. With these assumptions, and according to the developments in ref. [6], the expectation values for the (negatively charged) pion and the o field are written in terms of the chiral angle 0: (7r) = 2
1/2frrsin0 e-ik'r,
(1)
(o) = f~ cos 0.
(2)
The hamiltonian density, reduced non-relativistically and supplemented by constraints for charge and baryon number conservation, is given by [6] c~+/.tpQ
_
2 up = l ( k 2 - / - t 2 ) f 2 sin20 - . f ~2 m~r COS0
+~(pll/.t--u+ P
+ kr2 sin 0 [p }apap,
(3)
where gA = 2fTrf/mTr (with f2/47r = 0.08). We have omitted here A-isobars and short-range correlations, which will be added later on. The p and pQ are baryon and total charge densities, and/~ plays the role of the pion chemical potential. Diagonalisation of the nucleon part of eq. (3) 548
gives the quasiparticle energies (relative to the ct~emical potential u):
E+(p) =p2 /2M- u + k2 cos20/8M + 1 (/.t + [(U -
p'k/M) 2 cos20 +g2k2 sin20l 1/2} (4)
Temperature is then introduced in terms of the grand partition function g = Tr exp [-(/} + P 0 -
u~)/T],
(5)
where/~ is the hamiltonian, 0 and N are the charge and baryon number operators. The thermodynamic potential, g2 = - ( T / V ) In g,
(6)
is obtained in the form a = l(k2
/12)j~ sin20
- f ~2m ~
2
cos 0
- 2 T ~ f d ~ P a ln[1 + exp(-E+(p)/T)l. - -
{z~7]')
(7)
~
Note that the explicit temperature dependence in this model, in the assumed absence of intrinsic thermal fluctuations of the meson fields, comes only in the distribution of quasiparticles. On the other hand, the strong coupling of the pion field to its nucleon sources introduces an inrplicit T dependence in (Tr). Equilibrium conditions require that g2 be at a minimum (i.e. the entropy is maximised) with respect to variation of the parameters k and 0. The chemical potentials/a and u are determined by the conditions of charge and baryon number conservation. In particular, ~gZ
-
d3p (_+)
.(& (p)/r) = -p,
(8)
w i t h the F e r m i d i s t r i b u t i o n f u n c t i o n
n(z) = (e z + 1) -1
1 m - l ( p _ lkT3 COS0)2
+ 1/jr 3 cos 0 1 ~gA ~"
23 October 1978
(9)
In order to proceed, we shall introduce the following approximations, valid for neutron matter, where the pion chemical potential H is of the order of the pion mass and larger. In eq. (4), we assume p • k/M'~ la and drop the k 2 cos20/8M term for similar reasons. (Note that such an approximation cannot be used for symmetric nuclear matter [11], where we have/~ = 0.) The conditions ~gZ/~k = ~g2/~/1 = ~2/~0 = 0 are then easily written down.
Volume 78B, number 5
PHYSICS LETTERS
Furthermore, at sufficiently low temperature T, only the lower quasiparticle states E _ ( p ) will be populated. In fact the gap between the two quasiparticle spectra (at T = 0) is given by
23 October 1978 p
i
i
i
i
[MeV fm 3] 20
6 = E'+(0) -- IJ- (PF) = M~2 COS20 +g2k2 sin20
- p 2 . / 2 M = (g2/2f2)p - (37rZp)2/3/ZM,
SO
(10) 15
where we have made use of the minimisation conditions at T = 0 arrive at the latter expression. For densities in our range of interest (p >~ 2P0), 6 is of the order of ma and larger. If we restrict ourselves to T < l m ~ , we can safely assume that only the E levels need to be taken into account. Then the free energy density, e=~+up,
¢0
10
(ll) 5
takes a simple form, namely:
g2 02
(g2 _ 1)sin2O
8f 2
1 + (g2 _ 1)sin20
e-
01
2 2 cos 0 + up f~rm~
4 ~.d3p p2 - 3 - ) ( 9 . ~ - 3 2M n((p2/2M \--..
X)/7),
(]2)
j
with
g2
x
20 0
= u +
(g2 _ 1)sin20
(13)
p
4f2
1+ (g2 _ 1)sin2O '
where u is the quasiparticle chemical potential, while X is determined by d3p n ( ( p 2 / 2 M - X)/T) = O.
(14)
(15)
Some more realistic features have to be added to eq. (12). First, short range correlations have to be introduced. This is done in terms o f a Fermi liquid parameter g', in the same way as in refs. [6,7]. Furthemmre, A-isobars have to be included. It is shown in ref. [8] that at T = 0 the combined effect o f g ' and/X-isobars can be absorbed to a very good degree of accuracy by an effective renormalisation o f g A : *2 = (1 - g')(1 + A)g 2, g 2 ___>gA
03
04
05 p[fm -3]
Fig. 1. Equation of state for pion condensed neutron matter using Pandharipande's equation of state [ 16] for the normal phase. The effective p-wave interaction strength is chosen to be g~ = 1.33. Numbers at the different isothermes denote temperature in MeV. The critical temperature for a first order transition of van der Waats form is Tc = 42 MeV in this example. T<~ l m a , g' and A are assumed to be independent of T. We shall simply use gA as our basic model parameter, which then replaces g A in eqs. (12) and (13). Tile energy density will be written in the form
e(p, T;O) = e(p, r = 0 ; 0 ) + Ae(7),
(17)
where e(p, T = 0; 0) = e0(p, 0 = 0)
Note that at zero temperature,
X ( T = O) - X 0 = (31r2p)2/3 /2M.
02
(16)
where realistic values for g' range between 0.5 and 0.6 [15], while A = 0 . 8 - 0 . 9 [8]. For temperatures
1 [g*A ] 2 --8~P]
(gA2 -- 1)sin20 1 + (gA2 -- l)sin20
2 2 sin20/2. + 2f~rnrr
(18)
In practice we shall use realistic equations of state for
Co(P, 0 = 0) to describe the normal phase at T = 0, alternatively those of Pandharipande (P) [16] and of Bethe and Johnson (B-J) [17]. Tire extrapolation to finite but small temperature in/xe is then done according to eqs. (12) and (14). After minimisation of e with respect to 0, the pressure P(p, 7) can be derived in the form
P(p, 7) = P(p, T = O) + PT(P) .
(19) 549
Volume 78B, number 5
PttYSICS LETTERS
k& [MeV}
B-J
20
0
13
135
14
g~
--
1/+5 '
I'5
Fig. 2. Critical temperature T c for first order transition into pion condensed neutron matter as a function of the effective p-wave interaction strength g ~ (see eq. (16)), for two different equations of state describing the normal phase: Pandharipande (P) [16]; Bethe and Johnson (B-J) It 7].
The standard low-temperature expansion yields [-Tr2 / T '~2
7r4
)4
While this is simply the finite temperature extrapolation for a free Fermi gas, corrections would arise from replacing M by the effective mass M*(p). At the moment, we shall omit these and present in fig. 1 an example for the isothermes in the pressure/ density plane. We observe that while a region of negative compression modulus exists at low temperature, it does not survive beyond a certain critical temperature Tc, which is somewhat larger than 40 MeV in the case shown here. We would now like to investigate the model dependence of this T c. For that purpose we compare values of T c obtained with a soft (P) and a stiff (B J) reference equation of state describing the normal system, as a function of the effective p-wave interaction strength g ; (see fig. 2). Clearly, the BJ equation of state, which has m&e repulsive correlations at high density, needs a larger g~, than the Pandharipande equation of state in order to drive the system into the pion condensate. It is interesting to note that irrespective of the particular model, we always obtain T c ~< 45 MeV. Next, we study the dependence of T c on the effective nucleon mass. We do this as follows: (a) The standard o model (at 0 = 0) [6], with hard core repulsion of radius r c added, is used to 550
23 October 1978
reproduce first the equation of state for normal neutron matter (either (P) or (B-J)) by appropriate adjustment of r c. This is possible for a wide range of densities. (b) The "chiral radius" f . (see eq. (2)) is treated as a variational parameter. Additional minimisation of f2 with respect to fTr determines the effective nucleon mass M* as a function of density. Unlike the procedure taken in ref. [18], we do not treat g A as an independent parameter, but use the identification gA = 2fcr(f/mTr) as in ref. [51. In fact, fTr (and therefore M*) turns out to be smoothly decreasing with increasing density. A decreasingfTr will effectively increase the repulsive s-wave 7r neutron interaction, while the p-wave attraction remains essentially the same. Hence incorporation of this degree of freedom shifts the curves in fig. 2 to larger values o f g ~ , without changing the upper limit for T c. The question about thernral fluctuations of the pion field (other than those taken into account through the coupling to the thermal distribution of nucleon sources) remains open. Such effects would probably lower the critical temperature T c even more. In summary, we find that the phase transition to pion condensed neutron matter resembles a van der Waals form of the equation of state at low temperature for sufficiently strong net attraction in the effective pion-nucleon interactions. This case would apply for neutron stars formed in a supernova at T ~ 10 MeV. However, we observe that a region of negative compression modulus exists only up to a critical temperature T c which turns out to be always smaller than 50 MeV for a wide variety of model assumptions. The low value of T c opens the question about the possible formation of density isomers in nuclear matter at temperatures as large as they are assumed to appear in high energy heavy ion collisions. An investigation of this problem for symmetric nuclear matter is in progress.
References [ 1 ] O. Maxwell et al., Astrophy s. J. 216 (1977) 77. [2] J. Hofmann et al., Phys. Rev. Lett. 36 (1976) 88. [3] V.M. Galitskii and I.N. Mishustin, Phys. Lett. 72 B (1978) 285. [4] A.B. Migdal, O. Markin and I. Mishustin, JETP 66 (1974) 443; A.B. Migdal, Rev. Mod. Phys. (1978), to be published.
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PHYSICS LETTERS
[5] C.K. Au and G. Baym, Nucl. Phys. A 236 (1974) 500. [6] D. Campbell, R. Dashen and J.R. Manassah, Phys. Rev. D 12 (1975) 919; G. Baym, D. Campbell, R. Dashen and J.R. Manassah, Phys. Lett. 58B (1975) 304. [7] G.E. Brown and W. Weise, Phys. Rept. 27 C (1976) 1. [8] W. Weise and G.E. Brown, Phys. Lett. 58 B (1975) 300. [9] C.K. Au, Phys. Lett. 61 B (1976) 300. [10] A.B. Migdal, G.A. Sorokin, O.A. Markin and I.N. Mishustin, Phys. Lett. 65 B (1976) 423. [ 11 ] tt.J. Pirner, M. Rho and K. Yazaki, to be published. [12] V. Ruck, M. Gyulassi and W. Greiner, Z. Phys. A 277 (1976) 391. [13] P. Hecking, W. Weise and R. Akhoury, to be published.
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[14] G. Baym and G. Grinstein, Phys. Rev. D 15 (1977) 2897. [15] S.O. Btickman and W. Weise, in: Mesons in nuclei, eds. M. Rho and D.H. Wilkinson (North-Holland, Amsterdam, to be published); G.E. Brown, S.O. B~ickman, E. Oset and W. Weise, Nucl. Phys. A 286 (1977) 191. [16] V. Pandharipande, Nucl. Phys. A 178 (1971) 123. [17] H.A. Bethe and M.B. Johnson, Nucl. Phys. A 230 (1974) 1. [18] M. Chanowitz and P.J. Siemens, Phys. Lett. 70 B (1977) 175. [19] M. Sano, Proc. Intern. Conf. on Nuclear Structure (Tokyo, 1977)p. 828.
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