Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 323
After a lengthy but straightforward calculation one arrives at the
NC collision term (Sigl and Raffelt 1993)
∫
˙ρ p,coll = 1 dp ′[ W
2
P ′ ,P Gρ p ′G(1 − ρ p ) − W P,P ′ρ p G(1 − ρ p ′)G
+ W −P ′ ,P (1 − ρ p )G(1 − ρ p ′)G − W P,−P ′ρ p Gρ p ′G + h.c. ] ,
(9.22)
where P and P ′ are neutrino four momenta with physical (positive) energies
P 0 = |p| and P ′ 0 = |p ′ |. The nonnegative transition probabilities
W K ′ ,K = W (K ′ , K) are Wick contractions of medium operators of the
form
W (K ′ , K) = 1 8 G2 F S µν (K ′ − K) N µν (K ′ , K) , (9.23)
where K and K ′ correspond to neutrino four-momenta with K 0 and K 0
′
positive or negative. The “medium structure function” is
S µν (∆) ≡
∫ +∞
−∞
dt e i∆ 0t ⟨B µ (t, ∆)B ν (0, −∆)⟩ , (9.24)
where the energy transfer ∆ 0 can be both positive and negative. In the
ultrarelativistic limit the neutrino tensor can be written as
N µν = 1 2 (U µ U ′ν + U ′µ U ν − U · U ′ g µν − iϵ µναβ U α U ′ β), (9.25)
where U ≡ K/K 0 and U ′ ≡ K ′ /K 0 are the neutrino four velocities.
Therefore, N µν is an even function of K and K ′ . Note that the definition
Eq. (9.25) differs slightly from the corresponding Eq. (4.17).
The first two terms of the collision integral Eq. (9.22) are due to
neutrino scattering off the medium. The positive term represents gains
from scatterings ν p ′ → ν p while the negative one is from losses by the
inverse reaction. The third and fourth expressions account for pair
processes, i.e. the creation or absorption of ν p ν p ′ by the medium. The
pair terms are found by direct calculation or from the scattering ones
by “crossing,”
P → −P and ρ p → (1 − ρ p ) . (9.26)
For example, the reaction ν p X → X ′ ν p ′ transforms to X → X ′ ν p ν p ′
under this operation where X and X ′ represent medium configurations.
The collision integral for ρ p is found by direct calculation or by
applying the crossing operation Eq. (9.26) to all neutrinos and antineutrinos
appearing in Eq. (9.22). The neutrino gain terms then transform
to the antineutrino loss terms and vice versa.