Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 249
they are massless there is no operational distinction between a Majorana
neutrino and the two active degrees of freedom of a Dirac neutrino.
Therefore, according to Eq. (6.107) the dispersion relation for the helicity
± states of a Majorana neutrino is
E ± = (m 2 + p 2 ) 1/2 ∓ V, (6.124)
where the medium-induced potential V was given in Eq. (6.108). For
ν e it is V = √ 2G F (n e − 1n 2 n) with the electron and neutron densities
n e and n n , respectively.
This dispersion relation implies that the medium is “optically active”
with regard to the neutrino helicities, just as some media are
birefringent with regard to the photon circular polarization. In the
optical case the left-right symmetry (parity) is broken by the medium
constituents which must have a definite handedness; sugar molecules
are a well-known example. In the neutrino case parity is broken by the
structure of the interaction; the medium itself is unpolarized.
Because there is an energy difference between the Majorana helicity
states ν ± for a given momentum, decays ν − → ν + χ are kinematically
allowed. For relativistic neutrinos the squared matrix element is found
to be |M| 2 = 4h 2 P 1 · P 2 with the four-momenta P 1,2 of the initial and
final neutrino state. The differential decay rate is then
dΓ = 4h2
2E 1
d 3 p 2
2E 2 (2π) 3
d 3 k
2ω(2π) 3 P 1 · P 2 (2π) 4 δ 4 (P 1 − P 2 − K)
(6.125)
with the majoron four-momentum K. Integrating out the d 3 k variable
removes the momentum δ function. The remaining differential decay is
∫
dΓ
+1
= α χ dx P 1 · P 2 p 2 2
dE 2 −1 E 1 E 2 ω δ(E 1 − E 2 − ω), (6.126)
where x = cos θ for the angle between p 1 and p 2 and α χ ≡ h 2 /4π
is the majoron “fine-structure constant.” In the δ function one must
use ω = k = |p 1 − p 2 | = (p 2 1 + p 2 2 − 2p 1 p 2 x) 1/2 . With ∫ dx δ[f(x)] =
|df/dx| −1 = ω/p 1 p 2 and with energy-momentum conservation which
yields P 1 − P 2 = K and thus P 1 · P 2 = 1(P 2 2 1 + P2 2 ) one finds
dΓ
dE 2
= α χ
(E 2 1 + E 2 2 − p 2 1 − p 2 2) p 2
2E 1 E 2 p 1
. (6.127)