16. Structure functions
1
16. STRUCTURE FUNCTIONS
Updated July 2009 by B. Foster (University of Oxford), A.D. Martin (University of
Durham), and M.G. Vincter (Carleton University).
16.1.
Deep inelastic scattering
Highenergy leptonnucleon scattering (deep inelastic scattering) plays a key role in
determining the partonic structure of the proton. The process
`N
→
`
0
X
is illustrated in
Fig. 16.1. The ±lled circle in this ±gure represents the internal structure of the proton
which can be expressed in terms of structure functions.
k
k
q
P, M
W
Figure 16.1:
Kinematic quantities for the description of deep inelastic scattering.
The quantities
k
and
k
0
are the fourmomenta of the incoming and outgoing
leptons,
P
is the fourmomentum of a nucleon with mass
M
,and
W
is the mass
of the recoiling system
X
. The exchanged particle is a
γ
,
W
±
,o
r
Z
; it transfers
fourmomentum
q
=
k
−
k
0
to the nucleon.
Invariant quantities:
ν
=
q
·
P
M
=
E
−
E
0
is the lepton’s energy loss in the nucleon rest frame (in earlier
literature sometimes
ν
=
q
·
P
). Here,
E
and
E
0
are the initial and ±nal
lepton energies in the nucleon rest frame.
Q
2
=
−
q
2
=2(
EE
0
−
−→
k
·
k
0
)
−
m
2
`
−
m
2
`
0
where
m
`
(
m
`
0
) is the initial (±nal) lepton mass.
If
0
sin
2
(
θ/
2)
À
m
2
`
,
m
2
`
0
,then
≈
4
0
sin
2
(
2), where
θ
is the lepton’s scattering angle with respect to the lepton
beam direction.
x
=
Q
2
2
Mν
where, in the parton model,
x
is the fraction of the nucleon’s momentum
carried by the struck quark.
y
=
q
·
P
k
·
P
=
ν
E
is the fraction of the lepton’s energy lost in the nucleon rest frame.
W
2
=(
P
+
q
)
2
=
M
2
+2
−
Q
2
is the mass squared of the system
X
recoiling against
the scattered lepton.
s
k
+
P
)
2
=
Q
2
xy
+
M
2
+
m
2
`
is the centerofmass energy squared of the leptonnucleon
system.
K. Nakamura
et al.
(Particle Data Group), JP G
37
, 075021 (2010) (http://pdg.lbl.gov)
December 17, 2010
12:57
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16. Structure functions
The process in Fig. 16.1 is called deep (
Q
2
À
M
2
) inelastic (
W
2
À
M
2
) scattering
(DIS). In what follows, the masses of the initial and scattered leptons,
m
`
and
m
`
0
,a
re
neglected.
16.1.1.
DIS cross sections
:
d
2
σ
dx dy
=
x
(
s
−
M
2
)
d
2
σ
dx dQ
2
=
2
πMν
E
0
d
2
σ
d
Ω
Nrest
dE
0
.
(16
.
1)
In lowestorder perturbation theory, the cross section for the scattering of polarized
leptons on polarized nucleons can be expressed in terms of the products of leptonic and
hadronic tensors associated with the coupling of the exchanged bosons at the upper and
lower vertices in Fig. 16.1 (see Refs. 1–4)
d
2
σ
dxdy
=
2
πyα
2
Q
4
X
j
η
j
L
μν
j
W
j
μν
.
(16
.
2)
For neutralcurrent processes, the summation is over
j
=
γ,Z
and
γZ
representing
photon and
Z
exchange and the interference between them, whereas for chargedcurrent
interactions there is only
W
exchange,
j
=
W
. (For transverse nucleon polarization, there
is a dependence on the azimuthal angle of the scattered lepton.)
L
μν
is the lepton tensor
associated with the coupling of the exchange boson to the leptons. For incoming leptons
of charge
e
=
±
1 and helicity
λ
=
±
1,
L
γ
μν
=2
³
k
μ
k
0
ν
+
k
0
μ
k
ν
−
k
·
k
0
g
μν
−
iλε
μναβ
k
α
k
0
β
´
,
L
μν
=(
g
e
V
+
eλg
e
A
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 Spring '11
 Kutter
 Energy, Quark, Quantum chromodynamics, Nucl

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