Subatomic Physics

11.14. The Weak Current of Hadrons at High Energies 371
In the c.m. all momenta have the same magnitude so that the square of the
momentum transfer becomes
−q 2 =(p ν − p µ) 2 =2p 2 ν(1 − cos ϑ), (11.87)
where ϑ is the c.m. scattering angle. With Eq. (11.87), the solid-angle element
dΩ =2π sin ϑdϑ can be written as
so that
dΩ =− π
p2 dq
ν
2 ,
dσ = −1
4π�4c2 E
W |〈µ−N ′ |Hw|νN〉| 2 dq 2 . (11.88)
The central problem is now the matrix element. At low energies, where the structure
of the particles can be neglected, we have already considered weak 0 + → 0 +
transitions caused by charged weak currents.
The matrix element is given by Eq. (11.68), and the differential cross section in
this case is
dσ = − G2F V 2 ud
2π�4c2 E
W dq2 . (11.89)
The total cross section is obtained by integrating over dq 2 . The minimum
squared momentum transfer is −4p 2 ν, the maximum as given by Eq. (11.87) is 0,
and the integration from 0 to −4p 2 ν yields
σtot = 2G2 F V 2 ud
π� 4 c 2
E
W p2 ν
(11.90)
For the case of spin zero considered here, the cross section is modified by a weak
form factor, Fw, and Eq. (11.89) becomes
dσ = − G2F V 2 ud
2π�4c2 E
W |Fw(q 2 )| 2 dq 2 . (11.91)
The weak form factor Fw is predicted by the CVC hypothesis. Feynman and Gell-
Mann postulated that the vector form factors appearing in the electromagnetic and
in the weak currents must have the same form. For our simplified example CVC
states that for the vector interaction
Fw(q 2 )=Fem(q 2 ). (11.92)
No spinless nucleons exist, and the form factor Fem for our specific example cannot
be determined. However, we can assume that Fem has the same form as the form
factors that appear in the nucleon structure. In particular we can identify Fem with
GD as given in Eq. (6.42): The weak cross section then becomes, with Eq. (11.92),
dσ = − G2F V 2 ud
2π�4c2 E
W
dq 2
(1 + |q2 |/q2 . (11.93)
0 )4