Stars as Laboratories for Fundamental Physics - MPP Theory Group

330 Chapter 9
In the second-order term H int appears quadratic so that one obtains
expressions like ⟨Υ l Υ k ⟩. However, because the medium is assumed to
be in an eigenstate of L l they do not contribute for l ≠ k. Thus,
in the final result the contributions of different flavors can be added
incoherently.
The rates of production P∆ l and absorption A l ∆ of a ν l are functions
of the energy-momentum transfer ∆ to the medium,
P l ∆ = 1 2 G2 F
A l ∆ = 1 2 G2 F
∫ +∞
−∞
∫ +∞
−∞
dt e −i∆ 0t ⟨ Υ l (∆, t)γ µ ∆ µ Υ l (∆, 0) ⟩ ,
dt e −i∆ 0t ⟨ Tr [ γ µ ∆ µ Υ l (∆, 0)Υ l (∆, t) ]⟩ . (9.45)
These expressions are defined for both positive and negative energy
transfer ∆ 0 because P−P
l plays the role of an absorption rate for antineutrinos
with physical (P 0 > 0) four momentum while A l −P plays
that of a production rate. Put another way, A l ∆ and P∆ l represent the
rate of absorption or production of lepton number of type l, independently
of the sign of ∆ 0 .
It is useful to define a flavor matrix of production rates which in
the weak basis has the form
⎛
P ∆ ≡ 1 P e ⎞
∆ 0 0
⎜
⎝ 0 P µ ⎟
∆ 0 ⎠ . (9.46)
2
0 0 P∆
τ
An analogous definition pertains to A ∆ . Then one finds for the CC
collision integrals (Sigl and Raffelt 1993)
˙ρ p,CC = {P P , (1 − ρ p )} − {A P , ρ p } ,
˙ρ p,CC = {A −P , (1 − ρ p )} − {P −P , ρ p } , (9.47)
where {·, ·} is an anticommutator. The kinetic term for ρ p is related to
that for ρ p by the crossing relation Eq. (9.26).
The r.h.s. of Eq. (9.47) for ρ p is the difference between a gain and a
loss term corresponding to the production or absorption of a ν p . For a
single flavor they take on the familiar form P P (1−f p ) and A P f p where
(1 − f p ) is the usual Pauli blocking factor.
9.4.3 Weak-Damping Limit
The meaning of Eq. (9.47) becomes more transparent if one makes various
approximations which are justified for the conditions of a SN core.
As discussed in Sect. 9.3.5 one may ignore the antineutrino degrees