Astron. Astrophys. 342, 192-200 (1999)

2. Quantum formalism

In this section we obtain a general formula for the neutrino energy emission rate in the direct Urca reactions valid at any magnetic field G (at higher fields protons become relativistic). The calculation is done in the Born approximation within the standard quantum mechanical framework with weak interactions described by the Weinberg-Salam-Glashow theory. We adopt the conventional assumption that reacting electrons are relativistic, while protons and neutrons are nonrelativistic. All these particles are strongly degenerate. The rate of transition from an initial state to a final state is , where V is the normalization volume and describes weak interaction in the second-quantization formalism. For the neutron decay, we have ()

In this case, , erg cm³ is the Fermi weak coupling constant, is the Cabibbo angle, is the axial-vector coupling constant, is a Pauli matrix, and is a Dirac matrix. Finally, is a field operator in the coordinate representation, i.e., the expression of the form , where q denotes full set of quantum numbers, is an annihilation operator, and is an eigenstate.

To evaluate the total neutrino energy loss rate (emissivity) we need to sum times the energy of the newly born antineutrino over all initial and final states. First of all, we can sum the matrix element over the electron spin states. The energy-conserving delta-function is not affected by this summation, since all, except the lowest, electron states are spin-degenerate. Choosing the Landau gauge of the vector potential and performing a tedious calculation, we get

where

In these equations, is a cartesian component of a particle momentum, is a particle energy, and are, respectively, the doubled proton and neutron spin projections onto the magnetic field direction, , n is the electron Landau level number, , is the antineutrino wave vector, and . and are the normalization lengths. Finally, and are the Laguerre functions (e.g., Kaminker & Yakovlev, 1981), is the proton Landau level number, and . If any index (n or ) is negative, .

The next step consists in integrating over the x components of proton and electron momenta, which specify the y coordinates of the Larmor guiding centers of these particles. This operation gives the factor and removes the second delta-function in Eq. (3). Thus, we may write a general formula for the neutrino emissivity (including the inverse reaction which doubles the emission rate) as

In this case, is a Fermi-Dirac distribution, and the particle energies are given by the familiar expressions:

with the proton and neutron gyromagnetic factors and . In principle, the factors , , can be renormalized in dense matter which we ignore, for simplicity.

Since the electron and proton distributions are independent of signs of and we can simplify the expression for M by omitting the terms which would anyway yield zero after the integration:

Keeping in the z component of the momentum conserving delta-function in Eq. (5) would lead to a subtle thermal effect: it would mollify the resulting functions on a temperature scale. We will not pursue the accurate description of the effect here, both because the calculation would be quite complex, and because the temperature scale is assumed to be small. Therefore, we will neglect the neutrino momentum in the delta-function and, for the same reason, omit it from the definition of the vector q . Then, using the isotropy of the neutron distribution, we can further simplify the expression for M:

where the functions F and depend now on
.

Notice that the results of this and subsequent sections are equally valid for direct Urca processes involving hyperons. The results for hyperons are easily obtained by changing the values of reaction constants (, etc.) as described, for instance, by Prakash et al. (1992).

© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
helpdesk.link@springer.de