272
9. Deep Inelastic Electron–Nucleon Scattering and the Parton Model
Yet another choice of variables is sometimes used instead of these, namely the
dimensionless variables
x = Q
2 /2M ν
(9.17)
whose significance we shall see in the next section, and
y = ν/k
(9.18)
which is the fractional energy transfer in the ‘laboratory’ frame. Note that
relation (8.224) shows that x = 1 for elastic scattering. The Jacobian for the
transformation from Q
2 and ν to x and y is (see problem 9.2(b))
dQ
2 dν = 2M k
2 y dx dy.
(9.19)
We emphasize that the foregoing – in particular (9.3), (9.12) and (9.16) – is all
completely general, given the initial one-photon approximation. The physics
is all contained in the ν and Q
2 dependence of the two structure functions W 1
and W 2 .
A priori, one might expect W 1 and W 2 to be complicated functions of ν
and Q
2 , reflecting the complexity of the inelastic scattering process. However, in 1969 Bjorken predicted that in the ‘deep inelastic region’ – large ν
and Q
2 , but Q
2 /ν finite – there should be a very simple behaviour. He predicted that the structure functions should scale, i.e. become functions not of
Q
2 and ν independently but only of their ratio Q
2 /ν. It was the verification
of approximate ‘Bjorken scaling’ that led to the development of the modern
parton model. We therefore specialize our discussion of inelastic scattering to
the deep inelastic region.
9.2 Bjorken scaling and the parton model
From considerations based on the quark model current algebra of Gell-Mann
(1962), Bjorken (1969) was led to propose the following ‘scaling hypothesis’:
in the limit
)
Q
2
→ ∞
with x = Q
2 /2M ν fixed
(9.20)
ν → ∞
the structure functions scale as
M W 1 (Q
2 , ν) → F 1 (x)
(9.21)
νW 2 (Q
2 , ν) → F 2 (x).
(9.22)
9. Deep Inelastic Electron–Nucleon Scattering and the Parton Model
Yet another choice of variables is sometimes used instead of these, namely the
dimensionless variables
x = Q
2 /2M ν
(9.17)
whose significance we shall see in the next section, and
y = ν/k
(9.18)
which is the fractional energy transfer in the ‘laboratory’ frame. Note that
relation (8.224) shows that x = 1 for elastic scattering. The Jacobian for the
transformation from Q
2 and ν to x and y is (see problem 9.2(b))
dQ
2 dν = 2M k
2 y dx dy.
(9.19)
We emphasize that the foregoing – in particular (9.3), (9.12) and (9.16) – is all
completely general, given the initial one-photon approximation. The physics
is all contained in the ν and Q
2 dependence of the two structure functions W 1
and W 2 .
A priori, one might expect W 1 and W 2 to be complicated functions of ν
and Q
2 , reflecting the complexity of the inelastic scattering process. However, in 1969 Bjorken predicted that in the ‘deep inelastic region’ – large ν
and Q
2 , but Q
2 /ν finite – there should be a very simple behaviour. He predicted that the structure functions should scale, i.e. become functions not of
Q
2 and ν independently but only of their ratio Q
2 /ν. It was the verification
of approximate ‘Bjorken scaling’ that led to the development of the modern
parton model. We therefore specialize our discussion of inelastic scattering to
the deep inelastic region.
9.2 Bjorken scaling and the parton model
From considerations based on the quark model current algebra of Gell-Mann
(1962), Bjorken (1969) was led to propose the following ‘scaling hypothesis’:
in the limit
)
Q
2
→ ∞
with x = Q
2 /2M ν fixed
(9.20)
ν → ∞
the structure functions scale as
M W 1 (Q
2 , ν) → F 1 (x)
(9.21)
νW 2 (Q
2 , ν) → F 2 (x).
(9.22)