Photon-neutrino interactions Page: 3 of 7
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where F, is the photon field tensor and 0 is the neutrino field. The resulting cross section
for elastic scattering is
3G a2 4 m w 12 4 2
-(yv - yv) = 4 G 1 + - In m2 (mWIw . (3)
4 7r3 m 1Imnr~]rw~
Despite the enhancement of the In2 (m~w/me) factor, this cross section is exceedingly small
and likely to be of little practical importance in astrophysics.
2. Inelastic processes
The source of the large suppression in the yv elastic amplitude is the Yang theorem
prohibition of a two photon coupling to a J = 1 state. There is no similar restriction on the
coupling of three photons. This suggests an examination of the inelastic process -yv -+ yyv
to determine if the scale of the loop integrals resulting from the set of diagrams represented
by Fig. 1 is set by the electron mass rather than mw. This turns out to be the case, and one
can obtain an effective Lagrangian of the form
=1 GF a3/2 1 [5 (N ,)-(FAFFAF14 (4)
2 47/ m4 e 180 F 180 F
where N ~ is
Nun = 8 ( 7~(1 + -Y5)0) - av ( _Yp(1 + -Y5)0) . (5)
The numerical factors 5/180 and -14/180 are familiar from the Euler-Heisenberg  expan-
sion of the photon-photon scattering amplitude. This occurs because after replacing the W
propagator by mr and performing a Fierz transformation, the vp -+ 'y-y amplitude is the
product of a neutrino current and the photon to three photon amplitude. Using Eq. (4), the
cross section for yv -> yyv is
131 G 2 a3 w 8
a-(yv - yv) 0 w2. (6)
This cross section is illustrated in Fig. 2 for 1 keV < w < me.
Further details of the scattering process are presented in Fig. 3. By retaining the polar-
ization vector of one of the final state photons, it is possible to obtain the cross section for
this photon to be produced with either positive or negative helicity. This is illustrated in
the left panel of Fig.3 by the dashed and dot-dashed lines. The solid line in this panel is
the angular distribution for unpolarized scattering and 9 is the angle between the outgoing
photon and the incident photon. The difference between the positive and negative helicity
cross sections, which are fifth order polynomials in cos 0, results in the polarization P(9)
illustrated in the right panel of Fig. 3. The existence of a net circular polarization is possible
because the weak interaction violates parity.
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Dicus, D.A. & Repko, W.W. Photon-neutrino interactions, report, December 1, 1997; United States. (digital.library.unt.edu/ark:/67531/metadc694914/m1/3/: accessed April 25, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.