Stars as Laboratories for Fundamental Physics - MPP Theory Group

250 Chapter 6
If one ignores the vacuum mass relative to V one has p 1,2 = E 1,2 ∓ V
so that to lowest order in V
dΓ V (E 1 − E 2 )
= α χ . (6.128)
dE 2
E 2 1
Therefore, the final-state neutrino spectrum has a triangular shape
where E 2 varies between 0 and E 1 . The integrated decay rate is
Γ = 1 2 α χV (6.129)
as first shown by Berezhiani and Vysotsky (1987). Various subleties
were covered in the detailed discussion by Giunti et al. (1992).
Interestingly, under the same circumstances the decay χ → ν + ν − is
equally possible (Rothstein, Babu, and Seckel 1993) and proceeds with
the same rate Eq. (6.129)—see Berezhiani and Rossi (1994).
In these calculations it was assumed that the majoron is massless
as it behooves a Nambu-Goldstone boson. Does this remain true in a
medium The majorons could have a Yukawa coupling g to electrons
in which case one would expect on dimensional grounds that they develop
a medium-induced “mass” of order g(n e /m e ) 1/2 in analogy to the
photon plasma mass. Indeed, if one works with a pseudoscalar coupling
analogous to Eq. (6.123) one finds such a result. Even if they did
not couple to electrons, in a supernova core there is a background of
neutrinos to which majorons couple by assumption.
However, a pseudoscalar coupling is not appropriate for a Nambu-
Goldstone boson as it is not invariant under a shift χ → χ + χ 0 . The
pseudoscalar expression is only the lowest-order expansion of an exponential
coupling which respects the symmetry. Equivalently, a derivative
coupling of the sort (1/2f) ψγ µ γ 5 ψ ∂ µ χ can be used which satisfies
the symmetry explicitly (Sect. 14.2.3). Either way one finds that the
forward scattering amplitude between Nambu-Goldstone bosons and
fermions vanishes—there is no refractive index. The same conclusion
was reached by Flynn and Randall (1988) on more general grounds.
For a suitable choice of parameters the medium-induced decay of
electron neutrinos can deplete the solar neutrino flux before it leaves
the Sun. However, it is doubtful if this effect could explain all current
solar neutrino measurements (Sect. 10.8). For h ∼ > 10 −6 a radical modification
of the neutrino signal from a supernova collapse is expected
(Sect. 15.7.2).