Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 245
The absolute shift of the neutrino “masses” is rather negligible because
we are dealing with highly relativistic particles. Even in this limit,
however, the difference between the dispersion relation of different flavors
is important for oscillation effects. Hence the most noteworthy
medium effect is its flavor birefringence: ν e and ν µ,τ acquire different
effective masses because of the charged-current contribution from ν e -e
scattering. The difference of their potentials is V νe − V νµ,τ = √ 2G F n L
with n L = Y L n B the lepton-number density where the number fraction
of leptons is Y L = Y e + Y νe . Of course, neutrinos as a background
medium contribute only in a young supernova core.
6.7.2 Higher-Order Effects
In the early universe one has nearly equal densities of particles and antiparticles
with an asymmetry of about 10 −9 , leading to a near cancellation
of the refractive terms Eq. (6.106). One may think that the next
most important contribution is from ν-γ scattering, a process closely
related to the ν → ν ′ γγ decay briefly discussed in Sect. 7.2.2. If one
approximates the weak interactions by an effective four-fermion coupling
the relevant amplitude is given by the graph Fig. 6.16 which on
dimensional grounds should be of order αG F . However, electromagnetic
gauge invariance together with the left-handedness of the weak
interaction implies that it vanishes identically (Gell-Mann 1961). For
massive neutrinos the amplitude is proportional to αG F m ν , but even in
this case it vanishes in the forward direction (Langacker and Liu 1992).
Fig. 6.16. Neutrino-photon scattering with an effective four-fermion weak
interaction and a charged lepton l in the loop. This amplitude vanishes
entirely for massless neutrinos, and for massive ones it still vanishes in the
forward direction.
For the lowest-order ν-γ contribution to the refractive index one
must then use the full gauge-boson propagator and include all one-loop
amplitudes required by the standard model. Such calculations were
performed by Levine (1966) and by Cung and Yoshimura (1975) who
found that the scattering amplitude was proportional to αG F s/m 2 W
(center of mass energy √ s). Recently this problem was revisited by