Subatomic Physics

170 Structure of Subatomic Particles
so that Eq. (6.64) can be rewritten as
d2σ d|q| 2dν = 4πα2�2 q4 E ′
E
� x
ν
1 1
+ sin2
mc2 2 θ
�
P (x) (6.67)
By comparing Eq. (6.67) with Eqs. (6.57) and (6.61), we obtain
F2(x) =xP (x),
2F1(x) − mc2
ν F2(x) =P (x).
Since xi ≤ 1andν/mc 2 ≫ 1, we obtain the Callan–Gross relation (54)
(6.68)
F2(x) =2xF1(x), (6.69)
and thus note that W1 and W2 are related. The Callan–Gross relationship is specific
to spin-1/2 particles; for spin-zero quarks F1 = 0. In Fig. 6.21 we show an
experimental comparison of F2 and xF1. This shows that quarks have spin 1/2.
Let us, for a moment, return to the probability P. If we call the probability of
finding an up quark in the proton u p and a down quark d p ,thenwecanwrite (55)
since the charges of the up and down quarks are 2
3
we know the total probability, namely
� 1
0
P(x) = 4
9 up + 1
9 dp , (6.70)
u p (x)dx =2 and
� 1
0
1 and − 3 , respectively. However,
d p (x)dx =1, (6.71)
since there are two up quarks and one down quark in a proton. The average momentum
carried by the quarks can be written as
� 1
〈pq 〉 = xph (u
0
p + d p )dx. (6.72)
The same analysis can, of course, be repeated for a neutron. Experimentally, it
is found that 〈pq 〉≈0.5ph , so that the quarks carry only about 50% of the nucleon’s
momentum. Therefore other, neutral, particles must carry the remaining 50% of
the momentum; these particles are assumed to be the gluons.
If we are more careful we must include a correction to Eq. 6.70. In addition to
the valence quarks, the nucleons contain sea quarks which provide a non-negligible
background. These sea quarks are assumed to arise from gluons and vacuum fluctuations
splitting into quark-antiquark pairs and are particularly important for x ≤ 0.2.
54C.G. Callan and D.G. Gross, Phys. Rev. Lett. 21, 311 (1968); Phys. Rev. D22, 156 (1969).
55For simplicity, we neglect all but “valence” quarks; there is a small contribution from other
“sea quarks.”
56For recent data, see PDG and HEPDATA.