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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.

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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...

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  • $\begingroup$ Can you point to some specific equations / page numbers perhaps? $\endgroup$ – Thomas Bakx Mar 25 at 18:08

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