Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particles Interacting with Electrons and Baryons 95
3.2.4 Neutrino Pairs
The photoneutrino process was first studied by Ritus (1961) and by
Chiu and Stabler (1961). The effective neutral-current Hamiltonian is
H int = G F
√ ψ e γ µ (C V − C A γ 5 )ψ e ψ ν γ µ (1 − γ 5 )ψ ν , (3.11)
2
where G F is the Fermi constant, and the dimensionless couplings C V
and C A are given in Appendix B. Of course, in the early sixties neutral
currents were not known—one used Fierz-transformed charged currents
which gave C V = C A = 1. For general C V ’s and C A ’s the cross section
was first calculated by Dicus (1972) who found 15 (Fig. 3.3)
σ = σ 0
[
(C
2
V + C 2 A) ˆσ + − (C 2 V − C 2 A) ˆσ −
]
,
ˆσ + = 49 5 (13 ŝ − 7) 15 − 117 ŝ − 55 ŝ3
+ +
12 (ŝ − 1) 2 12 ŝ 2
ˆσ − = −39 +
+ 25 − 28 ŝ − 27 ŝ2 − 2 ŝ 3 + 2 ŝ 4
(ŝ − 1) 3 ln(ŝ),
120 ŝ
(ŝ − 1) − 8 ŝ − 1 + 12 (2 + 2 ŝ + 5 ŝ2 + ŝ 3 )
ln(ŝ),
2 ŝ 2 (ŝ − 1) 3
σ 0
= α G2 F m 2 e
9 (4π) 2 . (3.12)
In the nonrelativistic (NR) limit this is (CM photon energy ω)
σ NR = σ 0
6
35 (C2 V + 5C 2 A) (ŝ − 1) 4
= σ 0
96
35 (C2 V + 5C 2 A) (ω/m e ) 4 . (3.13)
In the extreme relativistic (ER) limit it is
σ ER = σ 0 (C 2 V + C 2 A) 2ŝ [ ln(ŝ) − 55
24 ]
= σ 0 (CV 2 + CA) 2 16 (ω/m e ) 2 [ ln(2ω/m e ) − 55 ], (3.14)
48
where σ − does not contribute.
15 With CV 2 = C2 A = 1 this result agrees with that of Ritus (1961) while Chiu and
Stabler (1961) appear to have an extra factor 2ŝ/(ŝ+1). In the nonrelativistic limit
with ŝ → 1 this deviation makes no difference while in the extreme relativistic limit
their result is a factor of 2 larger than that of Ritus (1961) and Dicus (1972).