Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 161
According to the discussion of Sect. 4.6 the transition probability is
given as
W (K 1 , K 2 ) = G2 F
8
n B
2ω 1 2ω 2
S µν N µν , (4.88)
where S µν is an effective structure function given in terms of the vector,
axial-vector and mixed structure function of the medium defined in
Eq. (4.38), and n B is the baryon density. The tensor N µν is the squared
matrix element of the neutrino current; it was given in Eq. (4.17) in
terms of K 1 and K 2 . In an isotropic medium the tensorial composition
of the dynamical structure function can be expressed in terms of the
energy-momentum transfer K and the four-velocity U of the medium;
U = (1, 0, 0, 0) in its rest frame. Thus, in analogy to Eqs. (4.40) and
(4.41) one may write
S µν = S 1 U µ U ν + S 2 (U µ U ν − g µν ) + S 3 K µ K ν
+ S 4 (K µ U ν + U µ K ν ) + i S 5 ϵ µναβ U α K β , (4.89)
where the functions S l (l = 1, . . . , 5) depend on medium properties
and on the energy-momentum transfer K = (ω, k). The transition
probability is then found to be (Raffelt and Seckel 1995)
W (K 1 , K 2 ) = G2 Fn B
4
[
(1 + cos θ)S1 + (3 − cos θ)S 2
− 2(1 − cos θ)(ω 1 + ω 2 )S 5
]
, (4.90)
where θ is the neutrino scattering angle. The terms proportional to S 3
and S 4 vanish identically.
Next, one may consider processes involving massive Dirac neutrinos
with specified helicities. Gaemers, Gandhi, and Lattimer (1989)
showed that in this case the same expressions apply with N µν as constructed
in Eq. (4.17) if one substitutes K i → 1 2 (K i ± m ν S i ), i = 1
or 2, where the plus sign refers to ν, the minus sign to ν, and S i is
the covariant spin vector. For relativistic neutrinos one may consider
a noncovariant lowest-order expansion in terms of m ν . In this limit K i
remains unchanged for left-handed states while
K i = (ω i , k i ) → ˜K i = (m ν /2ω i ) 2 (ω i , −k i ) (4.91)
for right-handed ones. After this substitution has been performed all
further effects of m ν are of higher order so that one may neglect m ν
everywhere except in the global “spin-flip factor.”