Photon-neutrino interactions Page: 4 of 7
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It is also of interest to examine the annihilation channel -y-y -- yvi, which provides an
energy loss mechanism for stellar processes. In this case, the double differential cross section
d20/ sin Bddw' is
d2cr 1 Gjca3
nd 3, 11 0 Ww' 3(w - w')[2224w2 - w(592w + 520w') sin2o
sin Bdodw' 3,110, 400 gr4me
+139w'2 sin4o], (7)
where 0 is the scattering angle of the final photon in the center of mass, w' is its energy and
w is the initial photon energy. When integrated, Eq. (7) gives the total cross section
67 G23 w 8
-) 67- GSoa)W=w2 . (8)
1, 275, 750 r4 \me (
Unlike yv -> y-yv, the final photon in the annihilation channel does not acquire any circular
3. Discussion and conclusions
A direct comparison of the elastic and inelastic cross sections, Eqs. (3), (6) and (8), is
given in Table 1. The w10 behavior of the inelastic cross sections versus the w6 behavior of
the elastic cross section is evident as is the 12 orders of magnitude difference at w = 2me.
Certainly, the effective Lagrangian Eq. (4) provides an adequate description of the inelastic
processes for w < me. The application of Eq. (7) to stellar energy loss is therefore completely
justified. At some point beyond w = me, the cross section ceases to grow as the tenth
power, begins a transition to a 'soft' behavior and eventually decreases. The precise range
of applicability of the power law is somewhat subjective in the sense that numerical factors
resulting from the loop integrals which define Leff are often included in the definition of the
scale factor. Including the denominator factors of Eq. (6), the effective scale is ~ 47rme.
Table 1: v(w) in cm2 for w in keV
As a rough indication of the importance of the -yv -- yyv process in cosmology, consider
the mean number of collisions in an expansion time t. Assuming that there is an effective
scale greater than me, Eq. (6) can be written
o(yv - yyv) = 3.5 x 10-55T1 , cm2 , (9)
L 1.46 x _-__ 2.39 x___. _ -_
1 1.46 x 10-" 2.39 x 10-84 2.05 x 10-82
100 1.46 x 10-64 2.39 x 10-64 2.05 x 10-72
511 1.77 x 10-57 3.30 x 10-57 3.65 x 10-68
1022 1.81 x 10-5" 2.96 x 10-4 2.34 x 10-66
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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/m1/4/: accessed April 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.