Amplitude for neutron-proton conversion from first principles

Question

I'm reading Dodelson's Modern Cosmology, and in one of the exercises on Big Bang nucleosynthesis it is quoted without reference that the amplitude for neutron-to-proton conversion is given by

$$|mathcal{M}|^2 _{n+nu_e rightarrow  p^ + e^-} = 32G_F ^2(1+3g_A ^2)m_p^2 p_nu p_e $$
with $g_A$ the axial-vector coupling of the nucleon.

Does anyone know a reference for a derivation of this amplitude from first principles?

Many thanks in advance.

Cosmas Zachos · Answer

You just go to a reputable particle physics text, like Chapter 5.1-5.6 pp 30,31,35,36 of L B Okun's legendary workhorse Leptons & Quarks, North Holland, 1982, ISBN-13: 978-0444869241 , and slug through it. Might also do Prob 10.4 for eqn (10.43) and finally eqn (10.44), p 310 of D Griffiths' Introduction to Elementary Particles, Wiley 2008, ISBN-13: 978-3527406012.

Further, there are dozens of write-ups like this on the internet, or else.

I have not line-reversed the antineutrino of neutron decay to your quasielastic neutrino-neutron scattering, but Okun deals with it on pp143-144 Ch 17.  An early classic by C.H.Llewellyn Smith, Physics Reports 3 (5), June 1972, pp 261-379, has all the stuff on pp 301-302, but you'd cloy yourself in a surfeit of form-factors...

Amplitude for neutron-proton conversion from first principles

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