Stars as Laboratories for Fundamental Physics - MPP Theory Group

210 Chapter 6
Fig. 6.1. Polarization tensor as photon self-energy insertion.
formalism is not required because the only contribution is from lowestorder
forward scattering on charged particles. Moreover, because the
scattering amplitude involves nonrelativistically the inverse mass of the
targets one may limit one’s attention to the electrons.
Then one takes the standard (truncated) Compton-scattering matrix
element (e.g. Bjorken and Drell 1964; Itzykson and Zuber 1983)
and takes an average over the Fermi-Dirac distributions of the electrons.
To lowest order in α = e 2 /4π this yields (Altherr and Kraemmer
1992; Braaten and Segel 1993)
∫
Π µν d 3 (
)
p 1
(K) = 16πα
2E(2π) 3 e (E−µ)/T + 1 + 1
e (E+µ)/T + 1
× (P · K)2 g µν + K 2 P µ P ν − P · K (K µ P ν + K ν P µ )
(P · K) 2 − 1 4 (K2 ) 2 ,
(6.36)
where P = (E, p) and E = (p 2 + m 2 e) 1/2 , apart from refractive effects
for the electrons and positrons. The phase-space distributions represent
electrons and positrons at temperature T and chemical potential µ.
Over the years, the phase-space integration has been performed in
various limits (Silin 1960; Tsytovich 1961; Jancovici 1962; Klimov 1982;
Weldon 1982a; Altherr, Petitgirard, and del Río Gaztelurrutia 1993).
The most comprehensive analytic result is that of Braaten and Segel
(1993) which contains all previous cases in the appropriate limits.
The main simplification occurs from neglecting the (K 2 ) 2 term in
the denominator of Eq. (6.36). For light-like K’s this is exactly correct,
and in the nonrelativistic limit where m e is much larger than all
other energy scales the approximation is also trivially justified. In the
relativistic limit it is only justified if one is interested in Π(K) near the
light cone (ω = k) in Fourier space. In the relativistic limit both transverse
and longitudinal excitations have dispersion relations which are
approximately (ω 2 − k 2 ) 1/2 ≈ e T or eE F in the nondegenerate and degenerate
limits, respectively. As detailed by Braaten and Segel (1993),
this deviation from masslessness is small enough to justify the approximation
if one aims at the dispersion relations. Including 1 4 (K2 ) 2 yields
an O(α 2 ) correction—it can be ignored in an O(α) result.