108
G. Altarelli and S. Forte
Structure functions are defined starting from the general form of W μν given
Lorentz invariance and current conservation. For example, for EW currents between
unpolarized nucleons we have:
W μν = (−g μν +
q μ q ν
q 2 ) W 1 (ν, Q
2 ) + (p μ −
mν
q 2 q μ )(p ν −
mν
q 2 q ν )
W 2 (ν, Q 2 )
m 2
−
−
i
2m 2 μνλρ p
λ q
ρ W 3 (ν, Q
2 )
W 3 arises from VA interference and is absent for pure vector currents. In the limit
Q 2 >> m 2 , x fixed, the structure functions obey approximate Bjorken scaling
which in reality is broken by logarithmic corrections that can be computed in QCD:
mW 1 (ν, Q
2 ) → F 1 (x)
νW 2,3 (ν, Q
2 ) → F 2,3 (x)
(4.67)
The γ − N cross-section is given by (W i = W i (Q 2 , ν)):
dσ γ
dQ 2 dν
=
4πα 2 E
Q 4 E
· [2 sin
2 θ
2
W 1 + cos
2 θ
2
W 2 ]
(4.68)
while for the ν − N or ¯
ν − N cross-section one has:
dσ ν,¯ ν
dQ 2 dν
=
G 2
F E
2πE
(
m 2
W
Q 2 + m 2
W
)
2
· [2 sin
2 θ
2
W 1 + cos
2 θ
2
W 2 ±
E + E
m
sin
2 θ
2
W 3 ]
(4.69)
(W i for photons, ν and ¯
ν are all different, as we shall see in a moment).
In the scaling limit the longitudinal and transverse cross sections are given by:
σ L ∼
1
s
[
F 2 (x)
2x
− F 1 (x)]
σ RH,LH ∼
1
s
[F 1 (x) ± F 3 (x)]
σ T = σ RH + σ LH
(4.70)
where L, RH, LH refer to the helicity 0, 1, −1, respectively, of the exchanged gauge
vector boson.
In the ‘60’s the demise of hadrons from the status of fundamental particles to that
of bound states of constituent quarks was the breakthrough that made possible the
construction of a renormalisable field theory for strong interactions. The presence
of an unlimited number of hadrons species, many of them with large spin values,
presented an obvious dead-end for a manageable field theory. The evidence for
constituent quarks emerged clearly from the systematics of hadron spectroscopy.
G. Altarelli and S. Forte
Structure functions are defined starting from the general form of W μν given
Lorentz invariance and current conservation. For example, for EW currents between
unpolarized nucleons we have:
W μν = (−g μν +
q μ q ν
q 2 ) W 1 (ν, Q
2 ) + (p μ −
mν
q 2 q μ )(p ν −
mν
q 2 q ν )
W 2 (ν, Q 2 )
m 2
−
−
i
2m 2 μνλρ p
λ q
ρ W 3 (ν, Q
2 )
W 3 arises from VA interference and is absent for pure vector currents. In the limit
Q 2 >> m 2 , x fixed, the structure functions obey approximate Bjorken scaling
which in reality is broken by logarithmic corrections that can be computed in QCD:
mW 1 (ν, Q
2 ) → F 1 (x)
νW 2,3 (ν, Q
2 ) → F 2,3 (x)
(4.67)
The γ − N cross-section is given by (W i = W i (Q 2 , ν)):
dσ γ
dQ 2 dν
=
4πα 2 E
Q 4 E
· [2 sin
2 θ
2
W 1 + cos
2 θ
2
W 2 ]
(4.68)
while for the ν − N or ¯
ν − N cross-section one has:
dσ ν,¯ ν
dQ 2 dν
=
G 2
F E
2πE
(
m 2
W
Q 2 + m 2
W
)
2
· [2 sin
2 θ
2
W 1 + cos
2 θ
2
W 2 ±
E + E
m
sin
2 θ
2
W 3 ]
(4.69)
(W i for photons, ν and ¯
ν are all different, as we shall see in a moment).
In the scaling limit the longitudinal and transverse cross sections are given by:
σ L ∼
1
s
[
F 2 (x)
2x
− F 1 (x)]
σ RH,LH ∼
1
s
[F 1 (x) ± F 3 (x)]
σ T = σ RH + σ LH
(4.70)
where L, RH, LH refer to the helicity 0, 1, −1, respectively, of the exchanged gauge
vector boson.
In the ‘60’s the demise of hadrons from the status of fundamental particles to that
of bound states of constituent quarks was the breakthrough that made possible the
construction of a renormalisable field theory for strong interactions. The presence
of an unlimited number of hadrons species, many of them with large spin values,
presented an obvious dead-end for a manageable field theory. The evidence for
constituent quarks emerged clearly from the systematics of hadron spectroscopy.