Stars as Laboratories for Fundamental Physics - MPP Theory Group

246 Chapter 6
Dicus and Repko (1993) who worked out explicitly the matrix elements
and cross sections. From their results one can extract the forward
scattering amplitude which leads to a refractive index for ν l (l = e, µ, τ)
in a photon bath,
[
)]
n refr − 1 = α 4π
G F
m 2 W
1 + 4 3 ln ( m
2
W
m 2 l
⟨E γ ⟩n γ , (6.114)
where ⟨. . .⟩ means an average. Numerically, the term in square brackets
is 32.9 for l = e and thus not small, but with 4π in the denominator the
whole expression is still of order αG F /m 2 W , i.e. of order 41 G 2 F. (See also
Nieves, Pal, and Unger 1983; Nieves 1987; Langacker and Liu 1992.)
Because the ν-γ term is so small a larger refractive index arises
if one includes the second term in the expansion Eq. (6.104) of the
gauge-boson propagators. Of course, in graphs (a) and (c) of Fig. 6.15
forward scattering implies Q = 0 so that only the first term contributes,
except when f = ν where the exchange graph has Q ≠ 0. This case
and graphs (b) and (d) yield a second-order contribution (Nötzold and
Raffelt 1988). For a neutrino of flavor l which is either e, µ, or τ it is
n refr − 1 = 8√ 2 G F
3m 2 Z
+ 8√ 2 G F
3m 2 W
(
⟨Eνl ⟩n νl + ⟨E νl ⟩n νl
)
(
⟨El −⟩n l − + ⟨E l +⟩n l +
)
. (6.115)
In this case the contributions from background fermions and antifermions
add with the same sign, and the global sign remains the same for
ν and ν as test particles. In practice, only an electron-positron background
is of relevance in the early universe so that ν τ , for example,
only feels a second-order contribution from other ν τ ’s and ν τ ’s. These
results are of order G 2 F/α and thus they are the dominant contribution
in a CP-symmetric plasma.
One-loop corrections to the amplitudes of Fig. 6.15 yield other
higher-order terms which are of order G 2 F like Eq. (6.114). They are
still interesting because the loops involve charged leptons with a mass
depending on their flavor. Therefore, the universality of the effective
neutral-current interaction is broken on this level, leading to different
refractive indices for different ν l . Between ν e and ν µ or ν τ the medium
is already birefringent to lowest order from ν e -e charged-current interactions.
Between ν µ and ν τ the one-loop correction dominates. Assum-
41 Note that m −2
Z
= cos2 Θ W sin 2 Θ W
√
2 GF /πα and m −2
W
= sin2 Θ W
√
2 GF /πα.