277
9.2. Bjorken scaling and the parton model
weighted by the probability f i (x) for the parton of type i to have a fraction x of
momentum. These probability distributions – or parton distribution functions
(PDFs) – are not predicted by the model and are, in this parton picture,
fundamental parameters of the proton. The structure function W 2 becomes
∫
∑ 1
2
W 2 (ν, Q
2 ) =
dx f i (x)e i δ(ν − Q
2 /2M x).
(9.31)
0
i
Using the result for the Dirac δ-function (see appendix E, equation (E.34))
δ(x − x 0 )
δ(g(x)) =
(9.32)
|dg/dx| x=x0
where x 0 is defined by g(x 0 ) = 0, we can rewrite
δ(ν − Q
2 /2M x) = (x/ν)δ(x − Q
2 /2M ν)
(9.33)
under the x integral. Hence we obtain
∑
2
νW 2 (ν, Q
2 ) =
e i xf i (x) ≡ F 2 (x)
(9.34)
i
which is the desired scaling behaviour. Similar manipulations lead to
M W 1 (ν, Q
2 ) = F 1 (x)
(9.35)
where
2xF 1 (x) = F 2 (x).
(9.36)
This relation between F 1 and F 2 is called the Callan–Gross relation (see
Callan and Gross 1969): it is a direct consequence of our assumption of spin1 partons. The physical origin of this relation is best discussed in terms of
2
virtual photon total cross sections for transverse (λ = ±1) virtual photons
and for a longitudinal/scalar (λ = 0) virtual photon contribution. The lon2
gitudinal/scalar photon is present because q / 0 for a virtual photon (see
=
comment (4) in section 8.3.1). However, in the discussion of polarization
vectors a slight difference occurs for space-like q
2 . In a frame in which
0
q
μ = (q , 0, 0, q
3 )
(9.37)
the transverse polarization vectors are as before
∈
μ (λ = ±1) = ∓2
−1/2 (0, 1, ±i, 0)
(9.38)
with normalization (see equation (7.87))
∈
∗
· ∈ = −1.
(9.39)
To construct the longitudinal/scalar polarization vector, we must satisfy
q · ∈ = 0
(9.40)
9.2. Bjorken scaling and the parton model
weighted by the probability f i (x) for the parton of type i to have a fraction x of
momentum. These probability distributions – or parton distribution functions
(PDFs) – are not predicted by the model and are, in this parton picture,
fundamental parameters of the proton. The structure function W 2 becomes
∫
∑ 1
2
W 2 (ν, Q
2 ) =
dx f i (x)e i δ(ν − Q
2 /2M x).
(9.31)
0
i
Using the result for the Dirac δ-function (see appendix E, equation (E.34))
δ(x − x 0 )
δ(g(x)) =
(9.32)
|dg/dx| x=x0
where x 0 is defined by g(x 0 ) = 0, we can rewrite
δ(ν − Q
2 /2M x) = (x/ν)δ(x − Q
2 /2M ν)
(9.33)
under the x integral. Hence we obtain
∑
2
νW 2 (ν, Q
2 ) =
e i xf i (x) ≡ F 2 (x)
(9.34)
i
which is the desired scaling behaviour. Similar manipulations lead to
M W 1 (ν, Q
2 ) = F 1 (x)
(9.35)
where
2xF 1 (x) = F 2 (x).
(9.36)
This relation between F 1 and F 2 is called the Callan–Gross relation (see
Callan and Gross 1969): it is a direct consequence of our assumption of spin1 partons. The physical origin of this relation is best discussed in terms of
2
virtual photon total cross sections for transverse (λ = ±1) virtual photons
and for a longitudinal/scalar (λ = 0) virtual photon contribution. The lon2
gitudinal/scalar photon is present because q / 0 for a virtual photon (see
=
comment (4) in section 8.3.1). However, in the discussion of polarization
vectors a slight difference occurs for space-like q
2 . In a frame in which
0
q
μ = (q , 0, 0, q
3 )
(9.37)
the transverse polarization vectors are as before
∈
μ (λ = ±1) = ∓2
−1/2 (0, 1, ±i, 0)
(9.38)
with normalization (see equation (7.87))
∈
∗
· ∈ = −1.
(9.39)
To construct the longitudinal/scalar polarization vector, we must satisfy
q · ∈ = 0
(9.40)