Subatomic Physics

372 The Weak Interaction
The total cross section is obtained by integration from 0 to −4p 2 ν ,
σ = G2 F V 2 ud Eq2 0
6π� 4 c 2 W
�
1 −
1
(1 + 4p 2 ν/q 2 0 )3
�
. (11.94)
This expression displays the essential features of the theoretical cross sections shown
in Fig. 11.16: At low energies, the term in the large parentheses can be expanded;
the result is identical to Eq. (11.90), and the cross section increases as p 2 ν .Athigher
energies, the term in the large parentheses becomes unity, and the cross section is
a constant.
The cross section, Eq. (11.94) has been derived for a superallowed 0 + → 0 +
transition, for which only a single vector form factor enters. Nucleons have spin
1/2, and at least three form factors are required to describe the cross section. Two
of these form factors are predicted from the CVC hypothesis to be identical to
those for the electromagnetic scattering of electrons, GE and GM introduced in
Eq. (6.38). The weak current, however, also contains an axial part, A, andasingle
form factor is sufficient to describe it. It is assumed that it has the same form as
GD, Eq. (6.42). Thus only one free parameter is left, q 2 0 ≡ M 2 A c2 . Figure 11.18
presents data for the elastic scattering νµn → µ − p and neutral current elastic
scatterings on protons. The theoretical curves are cross sections computed with
three form factors, GE,GM ,andG A F . GE and GM are given in Eq. (6.43) and
G A F by Eq. (6.42), with q2 0 ≡ M 2 A c2 and MA as indicated in Fig. 11.18. The data
show that the experimental results are compatible with these form factors and
with an axial mass MA =1.06 GeV/c 2 , somewhat larger than the vector mass
MV ≡ q0/c = √ 0.71 GeV/c 2 . This result is expected because axial vector mesons
have higher masses than their vector counterparts; the lowest axial vector meson is
the h1 with a mass of 1190 MeV/c 2 .
So far the discussion has been restricted to the elastic scattering due to charged
currents. The cross section for the true elastic scattering due to neutral currents
νµp −→ νµp
is more difficult to measure, but has been studied (44) to test the standard model
[Weinberg–Salam theory] (Chapter 13).
Both charged and neutral current weak interactions of neutrinos induce many
other reactions such as
νµp −→ µ − π + p.
Of particular interest are the inclusive reactions
νµp −→ µ − X, ¯νµp −→ µ + X,
νµp −→ νµX, ¯νµp −→ ¯νµX,
44 G. P. Zeller et al. Phys. Rev. Lett. 88, 091802 (2002).