312
20. Introduction to the Phenomenology of Weak Interactions
is then appropriate to average the ‘n’ and ‘p’ results to obtain an ‘isoscalar’
cross section σ
(νN) or σ
(¯ νN) :
d
2 σ
(νN)
= σ 0 x[q(x) + (1 y
dxdy
− )
2 q ¯(x)]
(20.134)
d
2 σ
(¯ νN)
= σ
2
0 x[(1 y
xd
− ) q(x) + q ¯(x)]
(20.135)
d y
¯
where q(x) = u(x) + d(x) and q ¯(x) = u ¯(x) + d(x).
Many simple and striking predictions now follow from these quark parton
results. For example, by integrating (20.134) and (20.135) over x we can write
dσ
(νN)
¯
= σ 0 [Q + (1 − y)
2 Q]
(20.136)
dy
dσ
(¯ νN)
¯
= σ 0 [(1 − y)
2 Q + Q]
(20.137)
dy
∫
where Q = xq(x)dx is the fraction of the nucleon’s momentum carried by
¯
quarks, and similarly for Q. These two distributions in y (‘inelasticity distributions’) therefore give a direct measure of the quark and antiquark composition of the nucleon. Figure 20.6 shows the inelasticity distributions as
reported by the CDHS collaboration (de Groot et al. 1979), from which the
authors extracted the ratio
¯
¯
Q/(Q + Q) = 0.15 ± 0.03
(20.138)
after applying radiative corrections. An even more precise value can be ob¯
tained by looking at the region near y = 1 for ν ¯N which is dominated by Q,
the small Q contribution (∝ (1 − y)
2 ) being subtracted out using νN data at
the same y. This method yields
¯
¯
Q/(Q + Q) = 0.15 ± 0.01.
(20.139)
Integrating (20.136) and (20.137) over y gives
σ
(νN)
1 ¯
= σ 0 (Q + Q)
(20.140)
3
σ
(¯ νN)
1
¯
= σ 0 ( Q + Q)
(20.141)
3
and hence
¯
Q + Q = 3(σ
(νN) + σ
(¯ νN) )/4σ 0
(20.142)
while
(
)
1 3r
¯
¯
Q/(Q + Q) =
− 1
(20.143)
2 1 + r