ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 8, с. 1113-1118
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
SUPERFLUID STATES IN ^-STABLE NUCLEAR MATTER
© 2014 J. M. Dong1), U. Lombardo2)*, W. Zuo1)
Received July 26, 2013
Two superfluid states of nuclear matter, which are supposed to play an important role in neutron stars, are discussed: the first one due to the proton—proton 150 pairing in ^-equilibrium nuclear matter; the second one due to the anisotropic neutron—neutron 3PF2 pairing in neutron matter. Since the two phases appear at high density of nuclear matter, the three-body forces were added to the pairing interaction and the strong correlation effects in the single-paricle spectrum. The energy gaps, obtained solving the extended BCS equations, significantly deviate from the values without medium effects so as to limit the role of these two superfluid states in the interpretation of phenomena occurring in the neutron-star core.
DOI: 10.7868/S0044002714080054
1. INTRODUCTION
The extension of the pairing theory [1] to nuclear matter [2] was stimulated by the belief that pairing is a universal phenomenon of Fermi systems. Nuclear matter was only a theoretical benchmark of the pairing with nuclear interaction, until Migdal addressed the idea that "superfluidity of nuclear matter may lead to some interesting cosmic phenomena if stars exist which have neutron cores. A star of this type would be in superfluid state with a transition temperature corresponding to 1 MeV" [3]. Ginzburgand Kirzhnits later estimated a gap exp(-1/N(0)V) w 1 MeV [4]. Over the last three decades, this idea has been consolidating during the investigation on the dynamical and thermodinamical properties of pulsars. From the very beginning three types of superfluidity have been supposed to set on neutron stars (NS) [5]. In the the inner crust, made of a nuclear lattice in equilibrium with a gas of neutrons dripped out from the neutron-rich nuclei, neutrons are superfluid paired in a 1S0 state with energy gap of the order of 1.5 MeV. The core is made of asymmetric nuclear matter in f3 equilibrium with electrons and muons. There the proton gas is so much diluted that superconducting 1S0 proton pairs, embedded in a dense neutron fluid, can survive until deep inside the star. The neutron density in the core is too high for 1S0 neutron pairs, because the nuclear interaction is repulsive at short distances. However, a strong attractive component of the interaction appears at high density, that favors
^Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, China.
2)Universita di Catania and Laboratori Nazionali del Sud (INFN), Catania, Italy. E-mail: lombardo@lns.infn.it
neutron—neutron coupling in the 3PF2 anisotropic pairing state. Recently the interest has been focused on the NS interior, where both the vortex pinning responsible for the observed period glitches [6] and the nucleon superfluidity responsible for the main cooling mechanisms [7, 8] are supposed to be located. In particular, the recent observations of cooling in the NS of Cassiopeia A [9] have been considered to be a direct evidence of the anisotropic 3PF2 neutron— neutron pairing in NS core, and the energy gap needed to explain the data has been estimated to be about 0.1 MeV [7].
The theoretical justification for the three superfluid states was suggested by estimates based on the weak coupling approximation (f w eF and A < f) in the pure BCS approach without any medium effects. At variance with electron pairing in metals, the nucleons, which experience pairing coupling, also generate pairing screening. Therefore, it appears necessary to go beyond the pure BCS approximation properly adding medium polarization effects [10]. In addition, since the two superfluid phases occur in the NS core, one can easily predict a strong influence by the nuclear three-body force which is increasingly stronger in the higher-density domain [11 — 13].
2. FORMALISM
The single-particle (s.p.) spectrum of the nuclear matter is derived in the framework of the Brueckner— Bethe—Goldstone theory [14], where the nucleon interaction is described by realistic two-body forces (2BF) and phenomenological or microscopic three-body force (3BF). The 2BF adopted in the present calculations is the Bonn B meson-exchange interaction, whose meson parameters (masses, coupling
1 ►
N* = A, R ц = | n, p, a, œ
2 ►
Fig. 1. Diagrams contributing to nuclear three-body forces.
constants, and momentum cut-offs) are determined by the fit with the experimental phase shifts of nucleon—nucleon scattering [15]. The 3BF, originally introduced to reproduce the saturation properties, is expected to dominate the high-density behavior of nuclear matter. The 3BF microscopic model [16] embodies physical processes that are not contained in the polarization corrections, like excitation of nucleon resonances or nucleon—antinucleon pairs, as shown in Fig. 1. The 3BF here adopted is consistent with the 2BF in that it is based on the exchange of mesons, whose parameters are the same as in the Bonn B 2BF [17]. Therefore, there are no free parameters in the present model of nucleon—nucleon interaction.
2.1. Normal Phase in the BHF Approximation
The propagator in the normal phase is given by
G 1 (p,u ) = и
p^ 2m
E(p,w ),
(1
where E(p,u) is the self-energy. The self-energy and the related quantities are derived from the Brueckner—Hartree—Fock (BHF) approximation of the Brueckner— Bethe—Goldstone (BBG) theory [14]. Grace to the inclusion of the 3BF in the Brueckner theory, the hole-line expansion for the energy and the self-energy can be extended up to high densities. The self-energy, truncated to the second order (see [18] for details), provides us with a good reproduction of the empirical nuclear mean field [19] and the optical-model potential [20], so that we are quite confident that the next orders are irrelevant for the present calculation.
Expanding the latter in a series of powers of the quasi-particle energy around the Fermi surface, we obtain
G-1 (p,u) « Z(p)-1 (u - ep), (2)
where the quasi-particle energy and the quasi-particle strength are
2 P2
eP = — + S(p,eP) - eF,
Z (p) =
1 -
дЩи)
ди
1
respectively.
The Z(p) factor is related to the depletion of the occupation number n(p) around the Fermi surface. According to the Migdal—Luttinger theorem [21] its value ZF = Z(pF) (Z factor) equals the discontinuity of the occupation number at the Fermi surface, i.e.,
lim [n(p — s) — n(p + s)] £—► 0
Z(pF), (3)
where pF is the Fermi momentum. In our approximation E(p,u) = E1 (p,u) + E2(p,u), where E1 (p,u) determines the left limit and S2 (p,u) the right limit of the preceding equation for e 0.
In Fig. 2a, we display the calculated occupation probability in pure neutron matter at density p = = 0.3 fm-3. One easily observes the remarkable deviation from the ideal Fermi gas (solid line) due to the strong short-range correlations. As expected, the deviation is slightly enhanced by 3BF.
In Fig. 2b the calculated Z factor is displayed vs. density in the two different approximations for the self-energy, i.e., E « E1 and E « E1 + E2, respectively. The calculation of ZF from Eq. (3) requires a high numerical accuracy: increasing the accuracy the calculated ZF gets lower and lower until the converging value is reached. It is noticed that without 3BF, the Z factors decrease slowly as a function of density. Adding the contribution of E2 leads to an overall reduction of the Z factor. The 3BF reduces further the Z factor and its effect increases rapidly with density, much more than that obtained by adopting only 2BF. Therefore 3BF induces a strong extra deviation from the ideal Fermi-gas model.
2.2. Generalized BCS Theory
The spectrum of a superfluid homogeneous Fermi system is derived from the generalized gap equation [22]:
A(p) = (4)
p'
where Gs (p,u) is the propagator in the superfluid phase. V is the sum of all irreducible nucleon— nucleon interaction diagrams in the static limit. In principle, it includes the bare interaction (2BF and 3BF) and the medium polarization terms. The effect of the latter on the pairing gap will not be discussed, but its introduction does not affect the main conclusions of the present contribution (see [23] for details).
The propagator in the presence of pairing is given
by
G-1 (p, и) = G-1 (p, и) + A2(p)G-1 (—p, —и). (5)
3
Ш=6
Р
k, fm 1 p, fm 3
Fig. 2. Occupation numbers vs. momentum at a given density (a) and ZF factors vs. density (b) in pure neutron matter. The effect of 3BF is shown in both panels. In panel b the calculations are reported for two approximations of the self-energy.
All energies are measured from the chemical potential f. The latter is determined from the constraint on the number of particle conservation
N = gJ2ns(p)= (6)
p
p
where ns(p) denotes the occupation number of quasi-particles in the state p.
In the one-pole approximation [22] the energy integration in Eq. (4) can be performed analytically. Since the analytical structure of the abnormal propagator is not modified by the presence of the Z factor, the gap equation takes the same form as in the pure BCS case, in which Z = 1. One easily obtains:
A(p) = -iM/V„ /Z(")Z('/)A"' ■ (7, 2 J ^(v - ,.)2 + AJ
where ep is the on-shell self-energy. The preceding gap equation is equivalent to the BCS version except for the Z2 factor that is modeling the effect of the interaction around the Fermi surface. Since the value of the Z factor, though depending on the ground-state correlations, is always less than unity in the vicinity of the Fermi surface, inevitably the energy gap will turn out quenched in this respect.
The two coupled equations, Eq. (6) and Eq. (7), determine the properties of the superfluid states.
3. SUPERFLUID STATE IN DENSE NUCLEAR MATTER
3.1.1S0 Proton—Proton Pairing in ¡3-Stable Nuclear Matter
The proton superfluidity in the 1S0 two-body channel ranges in the low-density nuclear matter
around p w 0.02 fm
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