Stars as Laboratories for Fundamental Physics - MPP Theory Group

248 Chapter 6
Moreover, if s is measured from the point of closest approach one has
s = (r 2 − b 2 ) 1/2 and so ds = (r 2 − b 2 ) −1/2 r dr. Altogether one finds
∫ R
∆α = −2 dr √ b ∂ rn refr
b r2 − b . (6.121)
2
The refraction by the surface of the body was ignored: it is assumed
that it is “soft” with ∂ r n refr = 0 at r = R. The absolute sign was chosen
such that ∆α is positive if the spherical body acts as a “focussing lens.”
This is the case, for example, for light passing through the atmosphere
of the Earth where ∂ r n refr < 0 and so the Sun near the horizon appears
“lifted” (Sommerfeld 1958).
With the results for the neutrino refractive index of Sect. 6.7.1 one
has for a beam of energy ω
√
2
∫ GF n R B,c
∆α = 2
dr b ∂ r[n B (− 1Y 2 n + Y e )]
√ , (6.122)
ω b n B,c r2 − b 2
where n B,c is the baryon density at the center of the lens. This expression
applies to ν e while for ν µ or ν τ the term Y e is absent, and for
antineutrinos the overall sign changes.
For the Sun with a central density of about 150 g cm −3 the overall
coefficient is 2.3×10 −18 (10 MeV/ω). The integral expression is dimensionless
and thus of order unity. Therefore, the focal length of the
Sun as a neutrino lens is of order 10 18 R ⊙ (solar radius) for 10 MeV
neutrinos, or about the radius of the visible universe!
6.8 Majoron Decay
As a first application for the neutrino dispersion relation I consider the
interaction of neutrinos with the hypothetical majorons (Sect. 15.7).
These particles are Nambu-Goldstone bosons of a symmetry which is
spontaneously broken by a Higgs fields which gives the neutrinos Majorana
masses. For the present purposes it is enough to specify a pseudoscalar
interaction
L int = ih ψ ν γ 5 ψ ν χ (6.123)
between the neutrinos of a given family and the massless majorons.
Here, ψ ν is a Majorana neutrino field while h is a dimensionless Yukawa
coupling constant.
Majorana neutrinos are fermions with only two degrees of freedom.
They correspond to a helicity-minus ν and a helicity-plus ν. In fact, if