118
G. Altarelli and S. Forte
Fig. 4.14 The dominant
singlet splitting function
xP (x, α s (Q 2 )) for
α s (Q 2 ) ∼ 0.2. The
resummed result from
Ref. [29] is compared with
the LO, NLO and NNLO
perturbative results
0.5
0.4
0.3
0.2
0.1
0
10
0
10
1
10
2
10
3
10
4
10
5
10
6
x
-1
LO
NLO GLAP
NLO rc res
NNLO Glap
P
x
4.5.3.2 Polarized Deep Inelastic Scattering
In polarized DIS one main question is how the proton helicity is distributed among
quarks, gluons and orbital angular momentum: 1/2 + g + L z = 1/2 (for a
recent review, see, for example, [31]). For a parton density p (either a quark or a
gluon) p indicates the first moment of the net helicity difference p + − p − in a
polarized proton with helicity +1/2 or:
2 ) =
1
0
dx[p + (x, Q
2 ) − p − (x, Q
2 )]
(4.96)
Experiments have shown that the quark moment is small (the “spin crisis”):
values from a recent fit [32] are exp ∼ 0.21 ± 0.14 and g exp ∼ 0.50 ± 1.27
at Q 2 = 1 GeV 2 (see also [33]). This is surprising because is conserved in
perturbation theory at LO (i.e. it does not evolve in Q 2 ). For conserved quantities
we would expect that they are the same for constituent and for parton quarks. But
actually the conservation of is broken by the axial anomaly. In perturbation
theory the conserved density is actually = + n f /2πα s g [34]. Note
that also α s g is conserved in LO, that is g ∼ logQ 2 . This behaviour is not
controversial but it will take long before the log growth of g will be confirmed by
experiment! But to establish this behaviour would show that the extraction of g
from the data is correct and that the QCD evolution works as expected. If g is large
enough it could account for the difference between partons () and constituents (
). From the spin sum rule it is clear that the log increase should cancel between
g and L z . This cancelation is automatic as a consequence of helicity conservation
in the basic QCD vertices. From the spin sum rule one obtains that either g + L z is
large or there are contributions to at very small x outside of the measured region.
G. Altarelli and S. Forte
Fig. 4.14 The dominant
singlet splitting function
xP (x, α s (Q 2 )) for
α s (Q 2 ) ∼ 0.2. The
resummed result from
Ref. [29] is compared with
the LO, NLO and NNLO
perturbative results
0.5
0.4
0.3
0.2
0.1
0
10
0
10
1
10
2
10
3
10
4
10
5
10
6
x
-1
LO
NLO GLAP
NLO rc res
NNLO Glap
P
x
4.5.3.2 Polarized Deep Inelastic Scattering
In polarized DIS one main question is how the proton helicity is distributed among
quarks, gluons and orbital angular momentum: 1/2 + g + L z = 1/2 (for a
recent review, see, for example, [31]). For a parton density p (either a quark or a
gluon) p indicates the first moment of the net helicity difference p + − p − in a
polarized proton with helicity +1/2 or:
2 ) =
1
0
dx[p + (x, Q
2 ) − p − (x, Q
2 )]
(4.96)
Experiments have shown that the quark moment is small (the “spin crisis”):
values from a recent fit [32] are exp ∼ 0.21 ± 0.14 and g exp ∼ 0.50 ± 1.27
at Q 2 = 1 GeV 2 (see also [33]). This is surprising because is conserved in
perturbation theory at LO (i.e. it does not evolve in Q 2 ). For conserved quantities
we would expect that they are the same for constituent and for parton quarks. But
actually the conservation of is broken by the axial anomaly. In perturbation
theory the conserved density is actually = + n f /2πα s g [34]. Note
that also α s g is conserved in LO, that is g ∼ logQ 2 . This behaviour is not
controversial but it will take long before the log growth of g will be confirmed by
experiment! But to establish this behaviour would show that the extraction of g
from the data is correct and that the QCD evolution works as expected. If g is large
enough it could account for the difference between partons () and constituents (
). From the spin sum rule it is clear that the log increase should cancel between
g and L z . This cancelation is automatic as a consequence of helicity conservation
in the basic QCD vertices. From the spin sum rule one obtains that either g + L z is
large or there are contributions to at very small x outside of the measured region.