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Astron. Astrophys. 343, 650-660 (1999)

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2. Cooper-pair neutrino emissivity

Consider neutrino emission due to Cooper pair formation (4) of nucleons. In the absence of superfluidity the process is strictly forbidden by energy-momentum conservation: a neutrino pair cannot be emitted by a free nucleon. Superfluidity introduces the energy gap into the nucleon dispersion relation near the Fermi surface which opens the reaction.

2.1. General formalism

Following Flowers et al. (1976) we will study the process (4) as annihilation [FORMULA] of quasinucleons [FORMULA] in a Fermi liquid with creation of a neutrino pair. The process goes via electroweak neutral currents; neutrinos of any flavors can be emitted. We will assume that nucleons are nonrelativistic and degenerate, and we will use the approximation of massless neutrinos. Let us derive the neutrino energy generation rate (emissivity) due to Cooper pairing of protons or neutrons in a triplet or singlet state. In this way we will generalize the results by Flowers et al. (1976) to the case of Cooper pairing of protons, and to the most important case of triplet pairing. The interaction Hamiltonian ([FORMULA]) is given by (e.g., Friman & Maxwell 1979)

[EQUATION]

where [FORMULA] is the Fermi weak-interaction constant, and the terms containing [FORMULA] and [FORMULA] describe contributions of the vector and axial vector currents, respectively. The factors [FORMULA] and [FORMULA] are determined by quark composition of nucleons (e.g., Okun' 1990), and they are different for n and p. For the reactions with neutrons, one has [FORMULA] and [FORMULA], while for those with protons, [FORMULA] and [FORMULA], where [FORMULA] is the axial-vector constant and [FORMULA] is the Weinberg angle ([FORMULA]). Notice that similar interaction Hamiltonian describes the neutrino emission due to pairing of hyperons in neutron star matter. The latter process has been discussed in the literature (Balberg & Barnea 1998, Schaab et al. 1998) and will be considered briefly in Sect. 2.2.

Furthermore, in Eq. (5)

[EQUATION]

is the neutrino 4-current (µ=0,1,2,3), [FORMULA] is a Dirac matrix, [FORMULA] is a neutrino bispinor amplitude,

[EQUATION]

is the nucleon 4-current and [FORMULA] is the Pauli vector matrix.

The nucleon current contains [FORMULA], the secondary-quantized nonrelativistic spinor wave function of quasi-nucleons in superfluid matter, and [FORMULA], is its Hermitian conjugate. [FORMULA] is determined by the Bogoliubov transformation. The transformation for singlet-state pairing is well known (e.g., Lifshitz & Pitaevskii 1980). The generalized Bogoliubov transformation for a triplet-state 3P2 pairing was studied in detail, for instance, by Tamagaki (1970). In the both cases [FORMULA] can be written as

[EQUATION]

where p is a quasiparticle momentum,

[EQUATION]

is its energy with respect to the Fermi level (valid near the Fermi surface, at [FORMULA]), [FORMULA], [FORMULA] and [FORMULA] enumerate spin states, [FORMULA] is a superfluid gap at the Fermi surface ([FORMULA]), [FORMULA] is the particle Fermi velocity; [FORMULA] is a basic spinor ([FORMULA]), [FORMULA] and [FORMULA] are, respectively, the annihilation and creation operators. [FORMULA] and [FORMULA] are matrix elements of the operators [FORMULA] and [FORMULA] which realize the Bogoliubov transformation from particle to quasiparticle states. In the cases of singlet and triplet pairing, one has

[EQUATION]

where

[EQUATION]

For a singlet-state pairing, the gap [FORMULA] is actually independent of p , so that [FORMULA] and [FORMULA] depend only on [FORMULA]. For a triplet-state pairing, [FORMULA], [FORMULA] and [FORMULA] depend on orientation of p . Note general symmetry properties (e.g., Tamagaki 1970)

[EQUATION]

Let [FORMULA] and [FORMULA] be 4-momenta of newly born neutrino and anti-neutrino, while [FORMULA] and [FORMULA] be 4-momenta of annihilating quasinucleons. Using the Fermi Golden rule one can easily obtain the neutrino emissivity in the form

[EQUATION]

where [FORMULA]=3 is the number of neutrino flavors, an overall factor [FORMULA] is introduced to avoid double counting of the same [FORMULA] collisions, integration is meant to be carried out over the domain [FORMULA] in which the process is kinematically allowed, [FORMULA] is the Fermi-Dirac distribution, T is the temperature, [FORMULA]=1,2,3, and

[EQUATION]

In this case [FORMULA] denotes an initial state of the quasinucleon system (the individual states [FORMULA] and [FORMULA] are occupied) and [FORMULA] is a final state of the system (the states [FORMULA] and [FORMULA] are empty). In Eq. (13) we neglected the interference terms proportional to [FORMULA] since they vanish after subsequent integration over p and [FORMULA].

Bilinear combinations of the neutrino current components (6) are calculated in the standard manner, and integration over [FORMULA] and [FORMULA] is taken with the aid of the Lenard integral. The result is:

[EQUATION]

where [FORMULA] is 4-momentum of a neutrino pair ([FORMULA], [FORMULA]) and [FORMULA] is the metric tensor. Inserting (15) into (13) we obtain

[EQUATION]

where [FORMULA].

Now the problem reduces to 6-fold integration over quasinucleon momenta p and [FORMULA] within the kinematically allowed domain [FORMULA]. Since the nucleon Fermi liquid is assumed to be strongly degenerate, only narrow regions of momentum space near the nucleon Fermi-surface contribute into the reaction. Thus we can set [FORMULA] and [FORMULA] in all smooth functions under the integral. One can prove that the presence of superfluidity (of energy gaps) makes the process kinematically allowed in a small region of momentum space where p is almost antiparallel to [FORMULA]. This allows us to set [FORMULA] in all smooth functions in the integrand.

For further integration in Eq. (16) we write [FORMULA] and [FORMULA], where d[FORMULA] and d[FORMULA] are solid angle elements, [FORMULA] and [FORMULA]. Let us integrate over [FORMULA] first. For this purpose, we can fix p and introduce a local reference frame XYZ with the Z-axis antiparallel to p . Let [FORMULA] and [FORMULA] be, respectively, azimuthal and polar angles of [FORMULA] with respect to XYZ. Since the space allowed for [FORMULA] is small, we have [FORMULA], [FORMULA], [FORMULA], [FORMULA], where [FORMULA]. In this case [FORMULA], [FORMULA], and [FORMULA], where [FORMULA]. The quantities [FORMULA] and [FORMULA] are smooth functions of p and [FORMULA]. In these functions, we set [FORMULA] which makes them independent of q . Therefore, [FORMULA] and [FORMULA] are the only variables which depend on [FORMULA]. The integration over [FORMULA] in Eq. (16) contains the term,

[EQUATION]

where [FORMULA]. Integration is performed in the local coordinate frame XYZ, but the result is transformed to the basic coordinate frame using tensor character of [FORMULA]. Subsequent integration over [FORMULA] from 0 to [FORMULA] is easy and yields

[EQUATION]

Now we introduce dimensionless variables

[EQUATION]

which give

[EQUATION]

and the integration over x and [FORMULA] is restricted by the domain where [FORMULA]. The outer integration is over orientations of nucleon momentum p . Performing the inner integration over x and [FORMULA] we can assume that this orientation is fixed (and the vector n is constant). Then the superfluid gap [FORMULA] is fixed as well. Introducing [FORMULA] we have [FORMULA] and [FORMULA]. The integration domain can be rewritten as [FORMULA]. In the nonrelativistic limit, we are interested in, [FORMULA], and the domain transforms to the narrow strip in the [FORMULA] plane near the [FORMULA] line. It is sufficient to set [FORMULA] and [FORMULA] in smooth functions and integrate over [FORMULA] in the narrow range [FORMULA]. In this way we come to a simple equation

[EQUATION]

and the most sophisticated term containing [FORMULA] vanishes.

2.2. Practical formulae

Inserting (21) into (20) and returning to the standard physical units we have

[EQUATION]

where [FORMULA] is an effective quasinucleon mass, [FORMULA] is bare nucleon mass, [FORMULA], [FORMULA] is the Boltzmann constant, and

[EQUATION]

is a function to be determined. The integrand contains the functions [FORMULA] and [FORMULA] defined by Eqs. (14) with [FORMULA] (see above). From Eqs. (10)-(12) for the singlet- and triplet-state pairings we obtain the same expression

[EQUATION]

In the case of the singlet-state pairing the Bogoliubov operator [FORMULA] possesses the properties [FORMULA] and [FORMULA], and has the form (e.g, Tamagaki 1970)

[EQUATION]

Then from Eqs. (14) we have [FORMULA], i.e., the axial-vector contribution vanishes for the singlet-state pairing in accordance with the result by Flowers et al. (1976) (to be exact, the main term in the nonrelativistic expansion of Q over [FORMULA] vanishes). In this case the gap is isotropic and integration over d[FORMULA] is trivial:

[EQUATION]

For the triplet-state pairing, according to Tamagaki (1970), [FORMULA], [FORMULA], and [FORMULA], where [FORMULA] is a unitary (2[FORMULA]2) matrix. Using these relationships and Eq. (10), for the triplet case from (14) we obtain

[EQUATION]

which yields

[EQUATION]

Contrary to the singlet-state paring, the axial-vector contribution does not vanish.

The results (26) and (28) for 1S0 and 3P2 superfluids can be written in a unified manner:

[EQUATION]

where F stands for [FORMULA] or [FORMULA], while [FORMULA] or [FORMULA] is a dimensionless reaction constant that depends on the particle species and superfluid type. In Table 1 we list the values of a for singlet-state and triplet-state superfluids of neutrons and protons (calculated using the values of [FORMULA] and [FORMULA] cited in Sect. 2.1). (A) denotes 1S0 pairing, while (B) and (C) are two types of 3P2 pairing with total projection of the Cooper-pair momentum onto the z-axis equal to [FORMULA] and 2, respectively. One can hardly expect triplet-state pairing of protons in a neutron star core but we present the corresponding value for completeness of discussion. We give also the values of a for singlet-state pairing of hyperons. Hyperon superfluidity has been discussed recently by Balberg & Barnea (1998) and incorporated into calculations of the neutron star cooling by Schaab et al. (1998). The value of a for hyperons is determined by the vector constant [FORMULA] of weak neutral currents in Eq. (5) as a sum of contributions of corresponding quarks (e.g., Okun' 1990). One can see that the efficiency of the neutrino emission due to singlet-state pairing of various particles is drastically different. The emission is quite open for n, [FORMULA], [FORMULA] but strongly (by two orders of magnitude) reduced for p and [FORMULA], and vanishes for [FORMULA] and [FORMULA] hyperons. Notice that the values of [FORMULA], [FORMULA] and a can be renormalized by manybody effects in dense matter which we ignore, for simplicity. Notice also that in the cases of singlet-state pairing of protons and [FORMULA] the first non-vanishing relativistic corrections to the emissivity Q due to axial-vector neutral currents ([FORMULA]) could be larger than the small ([FORMULA]) zero-order contribution of the vector currents. Since we neglect relativistic corrections, our expressions for Q in these cases may be somewhat inaccurate (give reliable lower limits of Q).


[TABLE]

Table 1. Reaction constant a in Eq. (29)


Eqs. (22) and (26) for singlet-state pairing of neutrons with two neutrino flavors ([FORMULA]) were obtained by Flowers et al. (1976). Similar equations were derived by Voskresensky & Senatorov (1986 , 1987). Note that the final expressions for Q obtained by the latter authors contain a misprint: there is [FORMULA] instead of [FORMULA] in the denominator although the numerical formula includes the correct factor [FORMULA]. In addition, the expressions by Voskresensky & Senatorov (1986 , 1987) are written for one neutrino flavor and erroneously contain the axial-vector contribution which is actually negligible.

Now we obtain practical expressions for the function F in Eq. (29). Following Levenfish & Yakovlev (1994a, b) we consider three types of BCS superfluid: (A), (B) and (C) as described above. In case (A) the superfluid gap is isotropic, [FORMULA]. In cases (B) and (C) the gap is anisotropic and depends on angle [FORMULA] between quasinucleon momentum p and the z-axis: [FORMULA], [FORMULA], respectively, where [FORMULA] is a temperature-dependent amplitude. Therefore, at given T the gap [FORMULA] has minimum equal to [FORMULA] for quasinucleons at the equator of the Fermi-sphere, whereas the gap [FORMULA] has maximum [FORMULA] at the equator and nodes at the poles of the Fermi-sphere. In the cooling theories of neutron stars one commonly considers the nodeless pairing (B) of neutrons. However, thermodynamics of nucleon superfluid is very model-dependent and one cannot exclude that the C-type superfluid appears in the neutron star cores instead of the B-type, at least at some temperatures and densities.

Let us introduce the notations

[EQUATION]

where [FORMULA] is the critical temperature of nucleon superfluidity. Using the standard equations of the BCS theory, Levenfish & Yakovlev (1994a, b) obtained analytic fits which relate [FORMULA] to [FORMULA] at any [FORMULA] for all three superfluid types:

[EQUATION]

One can easily see that the function F in Eq. (29) depends actually on the only parameter v and on the superfluid type. An analysis of this function from Eqs. (26) and (28) is quite similar to that carried out by Levenfish & Yakovlev (1994b) in their study of the effect of superfluidity on the heat capacity of nucleons. Therefore we will omit technical details and present the final results.

Just after the superfluidity onset when the dimensionless gap parameter [FORMULA] and [FORMULA], we obtain

[EQUATION]

At temperatures T much below [FORMULA] one has [FORMULA] and

[EQUATION]

Therefore the neutrino emission due to the nucleon pairing differs significantly from the majority of other neutrino reactions. The process has a threshold (becomes allowed at [FORMULA]), and the neutrino emissivity Q is a nonmonotonic function of temperature. It grows rapidly with decreasing T just below [FORMULA] which does not happen in other reactions. With further decrease of T the emissivity Q reaches maximum and then decreases. According to Eqs. (33) the decrease of Q is exponential for the nodeless superfluids of (A) or (B), and it is power-law ([FORMULA]) for the superfluid (C). The power-law behaviour of Q in case (C) occurs due to the presence of nodes in the superfluid gap. At [FORMULA] superfluids (A), (B), and (C) suppress the Cooper-pair neutrino emission in the same manner in which they suppress the heat capacity and the direct Urca process (Levenfish & Yakovlev 1994a, b).

Finally, we have calculated [FORMULA] numerically in a wide range of v and fitted the results by simple expressions which reproduce also the asymptotes (32) and (33):

[EQUATION]

The maximum fit error is about 1% at [FORMULA] for [FORMULA]; about 3.4% at [FORMULA] for [FORMULA]; and about 3% at [FORMULA] for [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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