Following Bruenn (1985) the contribution of thermal pair production
and absorption of neutrino-antineutrino pairs to the right
hand side of the Boltzmann equation (TP collision term) is

where
()
is the neutrino (antineutrino) invariant distribution
function, () is the neutrino (antineutrino)
energy in the frame comoving with the matter, ()
the cosine of the angle between the neutrino
(antineutrino) momentum and the polar axis, is the azimutal angle,
and is the
angle between neutrino and antineutrino directions.
In what follows, the explicit dependence on *t* and *r* of
the distribution functions will be omitted
and we will assume axial symmetry with respect to the polar axis.
The superscripts *a* and *p* refer to absorption and production,
respectively.

Assuming electrons and positrons in equilibrium
at a temperature *T*, the following
relation between the absorption and production kernels is satisfied

where the temperature is measured in units of energy.
Expanding as follows

where are the Legendre polynomials and
are the Legendre moments of the production kernel,
the TP collision term can be written as

In order to obtain the expressions for ,
we need to evaluate the following integral over electron energy *E*

where and are summarized in Table 1 (click here) for the
different neutrino types and is the Fermi-Dirac
distribution function

being the electron degeneracy parameter,
and

In the previous expression stands for the step function.

Although , and
are complicated
functions of , and , the integrals *J*_{l}
can be done analytically, and the results for *l*=0,1 were first
obtained by Bruenn (1985). We have calculated the
integrals for *l*=2,3 and
their dependence on *E* is simply a polynomial law of degree 2*l*+5
with coefficients being functions of and .

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Then, taking into account the following relation

the functions can be expressed in a simple way
in terms of the dimensionless variables and
.

where

The explicit expressions for *a*_{ln}, *c*_{ln} and *d*_{ln} coefficients are in
Appendix A and the method to evaluate the *G*_{n}(*a*,*b*) integrals
is detailed in Appendix B.