Physical Foundations of Cosmology

3.5 Nucleosynthesis 99

p

e−

n

ν

Fig. 3.5.

where

GF=

παw

√

2 MW^2

 1. 17 × 10 −^5 GeV−^2

is the Fermi coupling constant and (pi·pj) are the scalar products of the 4-momenta
entering the vertex. The factorgA 1 .26 corrects the axial vector “weak charge”
of the nucleon by accounting for the possibility that gluons inside the nucleon split
into quark–antiquark pairs, thus contributing to the weak coupling. Note that the
Fermi constant can be determined to very high accuracy by measuring the lifetime
of the muon, whilegAcan be measured only in interactions involving nucleons.
For the processa+b→c+d, the differential cross-section is

dσab

d

=

1

( 8 π)^2

|M|^2

(pa+pb)^2

(

(pc·pd)^2 −m^2 cm^2 d

(pa·pb)^2 −m^2 am^2 b

) 1 / 2

. (3.98)

This expression is manifestly Lorentz invariant and can be used in any coordi-
nate system. The 4-momenta of the outgoing particlescanddare related to the
4-momenta of the colliding particlesaandbby the conservation law:pc+pd=
pa+pb.
Let us now consider the particular reaction

n+ν→p+e−.

At temperatures of order a few MeV the nucleons are nonrelativistic and we have

(pn+pν)^2 m^2 n, (pn·pν)=mnν,

√

(pp·pe)^2 −m^2 pm^2 empe

√

1 −(me/e)^2 =mpeve,

(3.99)

whereνis the energy of the incoming neutrino andeν+Qis the energy of the
outgoing electron. The energyQ 1 .293 MeV,introduced in (3.84), is released
when the neutron is converted into the proton. Expression (3.98) is valid only in
empty space. At temperatures above 0.5 MeV there are many electron–positron
pairs and the allowed final states for the electron are partially occupied. As a result,