## 2. Cooper-pair neutrino emissivityConsider 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 formalismFollowing Flowers et al. (1976) we will study the process (4) as annihilation of quasinucleons 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 () is given by (e.g., Friman & Maxwell 1979) where is the Fermi
weak-interaction constant, and the terms containing
and
describe contributions of the vector
and axial vector currents, respectively. The factors
and
are determined by quark composition
of nucleons (e.g., Okun' 1990), and they are different for is the neutrino 4-current ( is the nucleon 4-current and is the Pauli vector matrix. The nucleon current contains , the
secondary-quantized nonrelativistic spinor wave function of
quasi-nucleons in superfluid matter, and
, is its Hermitian conjugate.
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
where p is a quasiparticle momentum, is its energy with respect to the Fermi level (valid near the Fermi surface, at ), , and enumerate spin states, is a superfluid gap at the Fermi surface (), is the particle Fermi velocity; is a basic spinor (), and are, respectively, the annihilation and creation operators. and are matrix elements of the operators and which realize the Bogoliubov transformation from particle to quasiparticle states. In the cases of singlet and triplet pairing, one has For a singlet-state pairing, the gap is actually independent of p , so that and depend only on . For a triplet-state pairing, , and depend on orientation of p . Note general symmetry properties (e.g., Tamagaki 1970) Let and be 4-momenta of newly born neutrino and anti-neutrino, while and be 4-momenta of annihilating quasinucleons. Using the Fermi Golden rule one can easily obtain the neutrino emissivity in the form where =3 is the number of neutrino
flavors, an overall factor is
introduced to avoid double counting of the same
collisions, integration is meant to
be carried out over the domain in
which the process is kinematically allowed,
is the Fermi-Dirac distribution,
In this case denotes an initial state of the quasinucleon system (the individual states and are occupied) and is a final state of the system (the states and are empty). In Eq. (13) we neglected the interference terms proportional to since they vanish after subsequent integration over p and . Bilinear combinations of the neutrino current components (6) are calculated in the standard manner, and integration over and is taken with the aid of the Lenard integral. The result is: where is 4-momentum of a neutrino pair (, ) and is the metric tensor. Inserting (15) into (13) we obtain where . Now the problem reduces to 6-fold integration over quasinucleon momenta p and within the kinematically allowed domain . 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 and 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 . This allows us to set in all smooth functions in the integrand. For further integration in Eq. (16) we write
and
, where
d and
d are solid angle elements,
and
. Let us integrate over
first. For this purpose, we can fix
p and introduce a local reference frame where . Integration is performed
in the local coordinate frame Now we introduce dimensionless variables and the integration over and the most sophisticated term containing vanishes. ## 2.2. Practical formulaeInserting (21) into (20) and returning to the standard physical units we have where is an effective quasinucleon mass, is bare nucleon mass, , is the Boltzmann constant, and is a function to be determined. The integrand contains the functions and defined by Eqs. (14) with (see above). From Eqs. (10)-(12) for the singlet- and triplet-state pairings we obtain the same expression In the case of the singlet-state pairing the Bogoliubov operator possesses the properties and , and has the form (e.g, Tamagaki 1970) Then from Eqs. (14) we have ,
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 For the triplet-state pairing, according to Tamagaki (1970), , , and , where is a unitary (22) matrix. Using these relationships and Eq. (10), for the triplet case from (14) we obtain Contrary to the singlet-state paring, the axial-vector contribution does not vanish. The results (26) and (28) for where
Eqs. (22) and (26) for singlet-state pairing of neutrons with two
neutrino flavors () were obtained by
Flowers et al. (1976). Similar equations were derived by Voskresensky
& Senatorov (1986 , 1987). Note that the final expressions for
Now we obtain practical expressions for the function Let us introduce the notations where is the critical temperature of nucleon superfluidity. Using the standard equations of the BCS theory, Levenfish & Yakovlev (1994a, b) obtained analytic fits which relate to at any for all three superfluid types: One can easily see that the function Just after the superfluidity onset when the dimensionless gap parameter and , we obtain At temperatures 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
), and the neutrino emissivity
Finally, we have calculated
numerically in a wide range of The maximum fit error is about 1% at for ; about 3.4% at for ; and about 3% at for . © European Southern Observatory (ESO) 1999 Online publication: March 1, 1999 |