Stars as Laboratories for Fundamental Physics - MPP Theory Group

320 Chapter 9
9.3 Neutral-Current Interactions
9.3.1 Hamiltonian
In order to make Eq. (9.13) explicit one must use a specific model for
the interactions between neutrinos and the medium. To this end I begin
with fermions which interact by virtue of an effective neutral-current
(NC) Hamiltonian,
H NC = G F ∑
∫
√ d 3 x B a µ (x) Ψ(x)G a γ µ (1 − γ 5 )Ψ(x) . (9.14)
2
a
Here, B a
µ typically is also a bilinear of the form ψ a γ µ ψ a or ψ a γ 5 γ µ ψ a
with a Dirac field ψ a which describes fermions of the medium. It is
assumed that all neutrino flavors scatter on a given species a in the
same way apart from overall factors which are given as a hermitian
n × n matrix G a of dimensionless coupling constants. In the absence of
flavor-changing neutral currents it is diagonal in the weak interaction
basis.
As a concrete example consider the ν e and ν µ flavors in a medium
of ultrarelativistic electrons which may be classified into a l.h. and a
r.h. “species,” a = L or R, so that B µ L,R = 1ψ 2 eγ µ (1 ∓ γ 5 )ψ e . With the
standard-model couplings given in Sect. 6.7.1 one finds in the weak
interaction basis G L = 2 sin 2 θ W + σ 3 and G R = 2 sin 2 θ W .
For the calculations it is convenient to write Eq. (9.14) in momentum
space,
H NC = G F ∑
∫
√ dp dp ′ B a µ (p − p ′ ) Ψ p G a γ µ (1 − γ 5 )Ψ p ′ , (9.15)
2
a
where B a µ (∆) = ∫ d 3 x B a µ (x)e −i∆·x is the Fourier transform of B a µ (x)
and Ψ p = a p u p + b † −pv −p in terms of the annihilation and creation
operators of Eq. (9.4).
A special case of NC interactions are those among the neutrinos
themselves with a Hamiltonian that is quartic in Ψ. In momentum
space these “self-interactions” are given by
H S = G ∫
F
√ dp dp ′ dq dq ′ (2π) 3 δ (3) (p + q − p ′ − q ′ )
2
× Ψ q G S γ µ (1 − γ 5 )Ψ q ′ Ψ p G S γ µ (1 − γ 5 )Ψ p ′ . (9.16)
In the standard model with three sequential neutrino families G S is the
3 × 3 unit matrix. For the evolution of a normal and a hypothetical
sterile flavor one would have G S = diag(1, 0).