## 2. Quantum formalismIn 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 In this case, ,
erg cm 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 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,
, The next step consists in integrating over the 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
Keeping in the where the functions 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 |