282
9. Deep Inelastic Electron–Nucleon Scattering and the Parton Model
These are equivalent to the sum rules
∫ 1
2 =
dx [u(x) − u ¯(x)]
(9.66)
0
∫ 1
1 =
dx [d(x) − d ¯ (x)]
(9.67)
0
which are, of course, just the excess of u and d quarks over antiquarks inside
the proton. Testing these sum rules requires neutrino data to separate the
various structure functions, as we shall explain in volume 2, chapter 20.
One can gain some further insight if one is prepared to make a model. For
example, one can introduce the idea of ‘valence’ quarks (those of the elementary constituent quark model) and ‘sea’ quarks (q¯ q pairs created virtually).
Then, in a proton, the u and d quark distributions would be parametrized by
the sum of valence and sea contributions
u = u V + q S
(9.68)
d = d V + q S
(9.69)
while the antiquark and strange quark distributions are taken to be pure sea
u ¯ = d ¯ = s = ¯
s = q S
(9.70)
where we have assumed that the ‘sea’ is flavour-independent. Such a model
replaces the six unknown functions now in play by three, and is consequently
more predictive. The strangeness sum rule (9.63) is now satisfied automatically, while (9.66) and (9.67) are satisfied by the valence distributions alone:
∫ 1
dx u V (x) = 2
(9.71)
0
∫ 1
dx d V (x) = 1.
(9.72)
0
One more important sum rule emerges from the picture of xf i (x) as the
fractional momentum carried by quark i. This is the momentum sum rule
∫ 1
dx x[u(x) + u ¯(x) + d(x) + d ¯ (x) + s(x) + s ¯(x)] = 1 − ∈
(9.73)
where ∈ is interpreted as the fraction of the proton momentum that is not
carried by quarks and antiquarks. The integral in (9.73) is directly related
to ν and ¯
ν cross sections, and its evaluation implies ∈ ≃
1 (the CHARM
2
(1981) result was 1 − ∈ = 0.44 ± 0.02). This suggests that about half the
total momentum is carried by uncharged objects. These remaining partons
are identified with the gluons of QCD. They have their own PDF, g(x).
An enormous effort, both experimental and theoretical, has gone into determining the parton distribution functions. The subject is regularly reviewed
0
9. Deep Inelastic Electron–Nucleon Scattering and the Parton Model
These are equivalent to the sum rules
∫ 1
2 =
dx [u(x) − u ¯(x)]
(9.66)
0
∫ 1
1 =
dx [d(x) − d ¯ (x)]
(9.67)
0
which are, of course, just the excess of u and d quarks over antiquarks inside
the proton. Testing these sum rules requires neutrino data to separate the
various structure functions, as we shall explain in volume 2, chapter 20.
One can gain some further insight if one is prepared to make a model. For
example, one can introduce the idea of ‘valence’ quarks (those of the elementary constituent quark model) and ‘sea’ quarks (q¯ q pairs created virtually).
Then, in a proton, the u and d quark distributions would be parametrized by
the sum of valence and sea contributions
u = u V + q S
(9.68)
d = d V + q S
(9.69)
while the antiquark and strange quark distributions are taken to be pure sea
u ¯ = d ¯ = s = ¯
s = q S
(9.70)
where we have assumed that the ‘sea’ is flavour-independent. Such a model
replaces the six unknown functions now in play by three, and is consequently
more predictive. The strangeness sum rule (9.63) is now satisfied automatically, while (9.66) and (9.67) are satisfied by the valence distributions alone:
∫ 1
dx u V (x) = 2
(9.71)
0
∫ 1
dx d V (x) = 1.
(9.72)
0
One more important sum rule emerges from the picture of xf i (x) as the
fractional momentum carried by quark i. This is the momentum sum rule
∫ 1
dx x[u(x) + u ¯(x) + d(x) + d ¯ (x) + s(x) + s ¯(x)] = 1 − ∈
(9.73)
where ∈ is interpreted as the fraction of the proton momentum that is not
carried by quarks and antiquarks. The integral in (9.73) is directly related
to ν and ¯
ν cross sections, and its evaluation implies ∈ ≃
1 (the CHARM
2
(1981) result was 1 − ∈ = 0.44 ± 0.02). This suggests that about half the
total momentum is carried by uncharged objects. These remaining partons
are identified with the gluons of QCD. They have their own PDF, g(x).
An enormous effort, both experimental and theoretical, has gone into determining the parton distribution functions. The subject is regularly reviewed
0