Stars as Laboratories for Fundamental Physics - MPP Theory Group

Oscillations of Trapped Neutrinos 329
conversion could occur. The deviation of θ from its vacuum value in
a medium is entirely from charged-current interactions with electrons,
even though they may be written in an effective NC form. The coherent
neutrino energy shifts by an electron background are enough to allow
true NC collisions with, say, neutrons to achieve flavor equilibrium!
9.4 Charged-Current Interactions
9.4.1 Hamiltonian
Besides neutrino scattering or pair processes one must also include
charged-current (CC) reactions where neutrinos are absorbed or produced
by the medium (converted into or from charged leptons) such
that the total lepton number of the neutrino ensemble changes by one
unit. The corresponding interaction Hamiltonian can be written in the
form
H CC = G ∫
F
√
2
d 3 x Υ(x)Ψ(x) + h.c. , (9.43)
where the neutrino field Ψ is, again, a column vector in flavor space with
the entries Ψ l , l = e, µ, τ in the standard model. Further, Υ is a row of
Dirac operators representing the medium. In the interaction basis Υ l
carries the lepton number corresponding to the flavor l. For example,
in a medium of nucleons and electrons the field Υ e corresponding to the
electron lepton number can be written for standard-model couplings as
Υ e = γ µ (1 − γ 5 )ψ e ψ n γ µ (C V − C A γ 5 )ψ p , (9.44)
where ψ p , ψ n , and ψ e are the proton, neutron, and electron Dirac fields,
respectively, while C V = 1 and C A = 1.26 are the dimensionless CC
vector and axial-vector nucleon coupling constants.
9.4.2 Kinetic Terms
One may now insert H CC into Eq. (9.13) in order to derive the explicit
CC collision integral for the evolution of ρ p and ρ p . The operators
Υ l violate the lepton number L l corresponding to flavor l. Therefore,
⟨Υ l ⟩ = 0 at all times if the medium is in an eigenstate of L l (l = e, µ,
τ, or additional exotic flavors). This assumption implies that the CC
interaction Eq. (9.43) does not contribute to refractive effects given by
the first-order term in Eq. (9.13).