Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 327
physically most transparent to work always in the weak interaction
basis.
In order to derive an equation of motion for ˜ρ p one evaluates the
collision term in Eq. (9.32) by inserting ˜ρ p ’s under the integral. Expanding
the result in Pauli matrices leads to an expression of the form
1
(a 2 p + A p · σ). In general, the polarization vector A p produced by the
collision term is not parallel to v p = (s p , 0, c p ) because the collision
term couples modes with different mixing angles. However, the assumed
fast oscillations average to zero the A p component perpendicular to v p .
Therefore, the r.h.s. of Eq. (9.32) is of the form 1[a 2 p+(v p·A p ) (v p·σ)].
With a matrix of coupling constants in the weak basis of
( ) ge 0
G =
(9.36)
0 g x
the collision integral Eq. (9.32) becomes explicitly
∫
f˙
p x = 1 dp { ]
′ w(ν x 4
p → νp x ′)[ (4 − s 2 p)gx 2 + (2 − s 2 p)t p t p ′g e g x
]
+ w(νp e → νp x ′)[ −s 2 pgx 2 − s 2 pt p t p ′g e g x
where
+ w(ν x p → ν e p ′)[ s 2 pg 2 e − (2 − s 2 p)t p t p ′g e g x
]
+ w(ν e p → ν e p ′)[ s 2 pg 2 e + s 2 pt p t p ′g e g x
]}
, (9.37)
w(νp a → νp b ′) ≡ W P ′ P fp b ′(1 − f p) a − W P P ′fp(1 a − fp b ′) (9.38)
(Raffelt and Sigl 1993). The equation for fp e is the same if one exchanges
e ↔ x everywhere. In the absence of mixing s p = t p = 0, leading to
the usual collision integral for each species separately.
If ν x is neither ν µ nor ν τ but rather some hypothetical sterile species
its coupling constant is g x = 0 by definition. In this case the collision
integral simplifies to
˙ f x p = 1 4 s2 pg 2 e
∫
dp ′[ W P ′ P fp e ′(2 − f p e − fp)
x
− W P P ′(f e p + f x p)(1 − f e p ′)] . (9.39)
If the ν e stay approximately in thermal equilibrium, detailed balance
yields
∫
f˙
p x = 1 4 s2 pge
2 dp ′[ W P ′ P fp e ′(1 − f p) x − W P P ′ fp(1 x − fp e ′)] . (9.40)
If in addition the mixing angle is so small that the ν x freely escape one
may set f x p = 0 on the r.h.s. so that the integral expression becomes
∫ dp ′ W P ′ P f e p ′.