Greiner W., Schrammm S., Stein E. Quantum Chromodynamics
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288 5. Perturbative QCD I: Deep Inelastic Scattering
E = 4.61 +1.49s ,
E
=2.555 +1.961s ,
s
w
=s
c
=0.888 , (5.112)
and the b,
¯
b distribution by
α =1.00 ,
β = 0.51 ,
a = 0 ,
A = 0 ,
B = 1.848 ,
D =2.929 +1.396s ,
E = 4.71 +1.514s ,
E
=4.02 +1.239s ,
s
w
=s
b
= 1.351 . (5.113)
The parametrizations are valid in the x range
10
−5
≤ x < 1 (5.114)
and the scale
µ
2
≤ Q
2
≤ 10
8
GeV
2
(5.115)
with an averaged inaccuracy of 2% −3%.
EXERCISE
5.8 Calculation of Moments of Splitting Functions
(Anomalous Dimensions)
Problem. Calculate the moments of the splitting functions P
Gq
, P
qq
, P
qG
,and
P
GG
. The splitting functions are explicitly (see (5.70)–(5.72))
(a) P
Gq
=
4
3
1 +(1 −x)
2
x
,
(b) P
qq
=−
4
3
(1 +x) +2 δ(1−x)
+ lim
ε→0
8
3
1
1 −x +ε
−δ(1 −x)
1
0
dy
1
1 − y +ε
,
5.3 Common Parametrizations of the Distribution Functions and Anomalous Dimensions 289
(c) P
qG
=
1
2
x
2
+(1 −x)
2
,
(d) P
GG
=6
1
x
−2 +x(1 −x)
+
11
2
−
N
f
3
δ(1 −x)
+ lim
ε→0
6
1
1 −x +ε
−δ(1 −x)
1
0
dy
1
1 − y +ε
.
Solution. (a) Inserting the explicit expression for the splitting function P
Gq
we
get for the nth moment
d
Gq
(n) =−
6
33 −2N
f
1
0
dxx
n−1
P
Gq
(x)
=−
6
33 −2N
f
1
0
dxx
n−1
4
3
1 +(1 −x)
2
x
=−
8
33 −2N
f
1
0
dx
2x
n−2
−2x
n−1
+x
n
=−
8
33 −2N
f
2
n −1
x
n−1
−
2
n
x
n
+
1
n +1
x
n+1
1
0
. (1)
Since n ≥ 1weget
d
Gq
(n) =−
8
33 −2N
f
2
n −1
−
2
n
+
1
n +1
=−
8
33 −2N
f
n
2
+n +2
n(n
2
−1)
. (2)
(b) The nth moment of the splitting function P
qq
is given by
d
qq
(n) =−
6
33 −2N
f
1
0
dxx
n−1
P
qq
(x)
=−
6
33 −2N
f
1
0
dxx
n−1
−
4
3
(1 +x) +2δ(1 −x)
+ lim
ε→0
8
3
1
1 −x +ε
−δ(1 −x)
1
0
dy
1
1 − y +ε
Exercise 5.8
290 5. Perturbative QCD I: Deep Inelastic Scattering
Exercise 5.8
=−
6
33 −2N
f
−
4
3
1
0
dx
x
n−1
+x
n
+2
+ lim
ε→0
8
3
1
0
dx
x
n−1
1 −x +ε
−
1
0
dy
1
1 − y +ε
. (3)
With the expansion
x
n−1
1 −x +ε
=−
n−2
i=0
x
i
+
1
1 −x +ε
(4)
this equation can be put into a simple form:
d
qq
(n) =−
6
33 −2N
f
−
8
3
n
i=0
1
0
dxx
i
+
4
3
1
0
dx
x
n−1
+x
n
+2
=−
6
33 −2N
f
−
8
3
n
i=0
1
i +1
+
4
3
1
n
+
1
n +1
+2
=−
6
33 −2N
f
−
8
3
n
i=2
1
i
−
8
3
1
n +1
−
8
3
+
4
3
1
n
+
4
3
1
n +1
+2
=
4
33 −2N
f
4
n
i=2
1
i
+1 −
2
(n +1)n
. (5)
(c) For the moment d
qG
(n) we get
d
qG
(n) =−
6
33 −2N
f
1
0
dxx
n−1
P
qG
(x)
=−
6
33 −2N
f
1
0
dxx
n−1
1
2
x
2
+(1 −x)
2
=−
3
33 −2N
f
1
0
dx
2x
n+1
−2x
n
+x
n−1
=−
3
33 −2N
f
2n +2x
n+2
−
2
n +1
x
n+1
+
1
n
x
n
1
0
=−
3
33 −2N
f
n
2
+n +2
(n +2)(n +1)n
. (6)
5.3 Common Parametrizations of the Distribution Functions and Anomalous Dimensions 291
(d) The remaining moments d
GG
(n) of the splitting function P
GG
are obtained
in the same way as in (b):
d
GG
(n) =−
6
33 −2N
f
1
0
dxx
n−1
P
GG
(x)
=−
6
33 −2N
f
6
1
0
dxx
n−1
1
x
−2 +x(1 −x)
+
11
2
−
f
3
+6 lim
ε→0
1
0
dx
x
n−1
1 −x +ε
−
1
0
dy
1
1 − y +ε
. (7)
Again we employ the relation
x
n−1
1 −x +ε
=−
n−2
i=0
x
i
+
1
1 −x +ε
(8)
and get, after integrating and rearranging,
d
GG
(n) =−
6
33 −2N
f
6
1
n −1
−
2
n
+
1
n +1
−
1
n +2
+
11
2
−
f
3
−6
n−2
i=0
1
0
dxx
i
=−
6
33 −2N
f
6
1
n −1
−
2
n
+
1
n +1
−
1
n +2
−
n−2
i=0
1
i +1
+
11
2
−
N
f
3
=−
6
33 −2N
f
−6
n
i=2
1
i
−6
+6
1
n −1
−
1
n
+
1
n +1
−
1
n +2
+
11
2
−
N
f
3
=
9
33 −2N
f
4
n
i=2
1
i
+
1
3
+
2N
f
9
−4
1
n −1
−
1
n
+
1
n +1
−
1
n +2
=
9
33 −2N
f
4
n
i=2
1
i
+
1
3
+
2N
f
9
−
4
n(n −1)
−
4
(n +2)(n +1)
. (9)
Exercise 5.8
292 5. Perturbative QCD I: Deep Inelastic Scattering
5.4 Renormalization and the Expansion Into Local Operators
The Q
2
dependence of the distribution functions in the previous sections was de-
scribed with the help of the GLAP equations. Starting with an – experimentally
measured – input distribution function at low Q
2
the evolution to a higher scale
Q
2
was determined by perturbative QCD which entered into the calculation of
the four splitting functions. However, there exists a more general approach of
wider applicability which allows us to rederive the GLAP equations in a more
formal manner. This formalism will be discussed in this section. This particular
approach, the so-called operator product expansion (OPE), is of general valid-
ity and not restricted to the context of deep inelastic lepton nucleon scattering.
In fact, we shall encounter it again during the discussion of QCD sum rules in
Chap. 7.
The following analysis again starts with (5.1):
W
µµ
=
1
2π
d
4
x exp
(
iq ·x
)
1
2
pol.
N|
ˆ
J
µ
(x)
ˆ
J
µ
(0)|N . (5.116)
In the rest system of the nucleon (see, e.g., Exercise 3.6, (2) and (8))
P
µ
=(M, 0, 0, 0), q
µ
=
(
q
0
, 0, 0, q
3
)
=
⎛
⎝
ν
M
, 0, 0,
6
ν
2
M
2
+Q
2
⎞
⎠
,
(5.117)
where ν = q · p = q
0
M and Q
2
=−q
2
=q
2
3
−q
2
0
=q
2
3
−ν
2
/M
2
.Herewe
are interested in the asymptotic region (the Bjorken limit), ν, Q
2
M
2
,
Q
2
/(2νM ) = x, and the exponential factor in (5.116) is
exp
iq
0
x
0
−q
3
x
3
=exp
⎡
⎣
i
ν
M
x
0
−
6
ν
2
M
2
+Q
2
x
3
⎤
⎦
=exp
⎡
⎣
i
ν
M
⎛
⎝
x
0
−
6
1 +
Q
2
M
2
ν
2
x
3
⎞
⎠
⎤
⎦
≈exp
i
ν
M
x
0
−x
3
−
1
2
Q
2
ν
2
M
2
x
3
=exp
i
ν
M
x
0
−x
3
−x
M
2
ν
x
3
≈exp
i
ν
M
x
0
−x
3
. (5.118)
Provided that
x
0
−x
3
is much larger than M/ν, its contribution to (5.116)
averages to zero:
x
0
−x
3
∼ O
M
ν
. (5.119)
5.4 Renormalization and the Expansion Into Local Operators 293
Therefore only x
µ
values with
x
2
= x
2
0
−x
2
1
−x
2
2
−x
2
3
≤ x
2
0
−x
2
3
∼ O
M
ν
(5.120)
contribute to the integral (5.116). On the other hand, one can prove that the ma-
trix element N|J
µ
(x)J
µ
(0)|N only contributes for x
2
≥ 0. To this end recall
that this matrix element can be written as
X
d
4
xN|
ˆ
J
µ
(x)|X X|
ˆ
J
µ
(0)|N exp
(
iq ·x
)
(5.121)
with physical final states X. The crucial point is that the expression with
exchanged coordinates
X
d
4
xN|
ˆ
J
µ
(0)|X X|
ˆ
J
µ
(x)|N exp
(
iq ·x
)
∼
X
...δ
4
(
P
X
− P
N
+q
)
(5.122)
is equal to zero, since there is no physical state |X with baryon number
one and energy E
X
= P
0
N
−q
0
= M
N
−ν/M
N
< M
N
. Hence the matrix elem-
ent in (5.116) can be replaced by the matrix element of the commutator
ˆ
J
µ
(x)
ˆ
J
µ
(0)
−
:
N|
ˆ
J
µ
(x)
ˆ
J
µ
(0)|N −N|
ˆ
J
µ
(0)
ˆ
J
µ
(x)|N =N|
ˆ
J
µ
(x)
ˆ
J
µ
(0)
−
|N .
(5.123)
The commutator vanishes for spacelike distances x
2
< 0 due to the requirements
of causality: the commutator is proportional to the Green function, which has to
vanish for space like distances. Therefore only time and lightlike x
µ
contribute
to (5.116):
W
µµ
=
1
2π
x
2
≥0
d
4
x exp
(
iqx
)
1
2
pol.
N|[
ˆ
J
µ
(x)
ˆ
J
µ
(0)]
−
|N . (5.124)
It follows with the help of (5.119) that W
µµ
is dominated by the matrix elem-
ent of the current commutator for lightlike x values. Therefore the following
expansion is also known as the light-cone expansion.
It is well known that a commutator of currents with a lightlike distance
diverges. This divergence also occurs in the free case
ˆ
J
µ
(x) =
ˆ
¯
Ψ(x)γ
µ
ˆ
Ψ(x), (5.125)
294 5. Perturbative QCD I: Deep Inelastic Scattering
where it can be reduced to the divergence of the free propagator S(x) for x
2
→0:
[
ˆ
J
µ
(x)
ˆ
J
ν
(0)]
−
=[
ˆ
¯
Ψ(x)γ
µ
ˆ
Ψ(x),
ˆ
¯
Ψ(0)γ
ν
ˆ
Ψ(0)]
−
=
ˆ
¯
Ψ(x)γ
µ
ˆ
Ψ(x)
ˆ
¯
Ψ(0)γ
ν
ˆ
Ψ(0)
+[
ˆ
¯
Ψ(x)γ
µ
]
α
ˆ
¯
Ψ
β
(0)
ˆ
Ψ
α
(x)[γ
ν
ˆ
Ψ(0)]
β
−
ˆ
¯
Ψ(0)γ
ν
ˆ
Ψ(0)
ˆ
¯
Ψ(x)γ
µ
ˆ
Ψ(x)
−
ˆ
¯
Ψ
β
(0)[
ˆ
¯
Ψ(x)γ
µ
]
α
[γ
ν
ˆ
Ψ(0)]
β
ˆ
Ψ
α
(x). (5.126)
Here we have added the second term and substracted it again (fourth term). In-
deed, owing to the anticommutation properties of
ˆ
Ψ and
ˆ
¯
Ψ , the second and the
fourth term cancel. The first two and the last two terms are combined and the
anticommutators that occur are replaced by the propagator:
[
ˆ
¯
Ψ(x)γ
µ
]
α
S
αβ
(x)[γ
ν
ˆ
Ψ(0)]
β
−[
ˆ
¯
Ψ(0)γ
ν
]
β
S
βα
(−x)[γ
µ
ˆ
Ψ(x)]
α
,
{
ˆ
Ψ
α
(x),
ˆ
¯
Ψ
β
(0)}=S
αβ
(x) =
iγ
µ
∂
µ
+m
αβ
·i∆(x). (5.127)
In the case of massless fields and x
2
→0, S(x) is given by
S(x) →γ
µ
∂
x
µ
1
2π
ε(x
0
)δ
x
2
(5.128)
with
11
ε(x
0
) = sgn(x
0
), (5.129)
[
ˆ
J
µ
(x),
ˆ
J
ν
(0)]
−
→
Ψ(x)γ
µ
γ
λ
γ
ν
Ψ(0) −Ψ(0)γ
ν
γ
λ
γ
µ
Ψ(x)
×
1
2π
∂
λ
ε(x
0
)δ(x
2
). (5.130)
In the following, only the leading divergence of S(x), i.e., the free propagator,
is considered. The remaining terms are discussed later. For the electromagnetic
interaction treated here W
µν
is symmetric in µ and ν (see (3.18)). Therefore we
can write the current commutator as
1
2
ˆ
J
µ
x
2
,
ˆ
J
ν
−
x
2
−
+
ˆ
J
ν
x
2
,
ˆ
J
µ
−
x
2
−
=
ˆ
Ψ
x
2
1
2
(γ
µ
γ
λ
γ
ν
+γ
ν
γ
λ
γ
µ
)
ˆ
Ψ
−
x
2
−
ˆ
Ψ
−
x
2
1
2
(γ
ν
γ
λ
γ
µ
+γ
µ
γ
λ
γ
ν
)
ˆ
Ψ
x
2
1
2π
∂
λ
ε(x
0
)δ(x
2
)
= s
µλνβ
ˆ
Ψ
x
2
γ
β
ˆ
Ψ
−
x
2
−
ˆ
Ψ
−
x
2
γ
β
ˆ
Ψ
x
2
×
1
2π
∂
λ
ε(x
0
)δ(x
2
), (5.131)
11
W. Greiner and J. Reinhardt: Field Quantization (Springer, Berlin, Heidelberg 1996).
5.4 Renormalization and the Expansion Into Local Operators 295
with
s
µλνβ
= g
µλ
g
νβ
+g
µβ
g
νλ
−g
µν
g
λβ
. (5.132)
Here we have shifted the position arguments of the current operators from “x”
and “0” to “±x/2”. This operation does not change anything, because (5.124) is
invariant under such translations, but it emphasizes the symmetry properties of
the current operator. In the case of neutrino–nucleon scattering, i.e., for a parity-
violating process, W
µµ
is no longer symmetric in µ and µ
(see Exercise 3.3).
Let us see how the commutator
ˆ
J
µ
(x),
ˆ
J
ν
(0)
−
can be calculated in this case.
Because
γ
µ
γ
λ
γ
ν
=(s
µλνβ
+iε
µλνβ
γ
5
)γ
β
(5.133)
(as we shall show in Exercise 5.9), we obtain the following additional terms:
ˆ
J
µ
x
2
,
ˆ
J
ν
−
x
2
−
=
ˆ
Ψ
x
2
(s
µλνβ
+iε
µλνβ
γ
5
)γ
β
ˆ
Ψ
−
x
2
−
ˆ
Ψ
−
x
2
(s
νλµβ
+iε
νλµβ
γ
5
)γ
β
ˆ
Ψ
x
2
1
2π
∂
λ
ε(x
0
)δ(x
2
)
=
s
µλνβ
ˆ
Ψ
x
2
γ
β
ˆ
Ψ
−
x
2
−
ˆ
Ψ
−
x
2
γ
β
ˆ
Ψ
x
2
+iε
µλνβ
ˆ
Ψ
x
2
γ
5
γ
β
ˆ
Ψ
−
x
2
+
ˆ
Ψ
−
x
2
γ
5
γ
β
ˆ
Ψ
x
2
×
1
2π
∂
λ
ε(x
0
)δ(x
2
). (5.134)
Here the full antisymmetry of the ε tensor, i.e. ε
µλνβ
=−ε
νλµβ
, has been
utilized. Inserting (5.134) into the matrix element (5.124) yields
N|
ˆ
J
µ
x
2
,
ˆ
J
ν
−
x
2
−
|N (5.135)
=
1
2π
∂
λ
ε(x
0
)δ(x
2
)
×
s
µλνβ
N|
ˆ
Ψ
x
2
γ
β
ˆ
Ψ
−
x
2
−
ˆ
Ψ
−
x
2
γ
β
ˆ
Ψ
x
2
|N
+ iε
µλνβ
N|
ˆ
Ψ
x
2
γ
5
γ
β
ˆ
Ψ
−
x
2
+
ˆ
Ψ
−
x
2
γ
5
γ
β
ˆ
Ψ
x
2
|N
,
which is a typical form for the operator-product expansion (OPE). The name
OPE will become obvious in the following.
The first factor on the right-hand side of (5.132) is a divergent expression,
which can be derived from perturbative QCD and which does not depend on
the hadron considered. The terms in the curly brackets are finite matrix elem-
ents, which contain all the nonperturbative information. The divergent factors are
called Wilson coefficients.
The Wilson coefficients can be calculated using perturbation theory and writ-
ten as a systematic expansion in the coupling constant α
s
. The determination of
296 5. Perturbative QCD I: Deep Inelastic Scattering
the matrix elements requires nonperturbative methods such as lattice calcula-
tions or QCD sum rules. This can be compared to the approach of the GLAP
equations where the splitting functions are calculated within perturbative QCD
but the input distribution functions – as pure nonperturbative quantities – have
to be taken from models or from experiment.
EXERCISE
5.9 Decomposition Into Vector and Axial Vector Couplings
Problem. Prove relation (5.133),
γ
µ
γ
α
γ
ν
= (s
µανβ
+iε
µανβ
γ
5
)γ
β
. (1)
Solution. γ
µ
γ
α
γ
ν
anticommutes with γ
5
. On the other hand, it must be a linear
combination of the 16 matrices which span the Clifford space, namely 1, γ
5
, γ
µ
,
γ
µ
γ
5
,andσ
µν
. Obviously this implies
γ
µ
γ
α
γ
ν
=a
µανβ
γ
β
+b
µανβ
γ
5
γ
β
. (2)
Multiplying by γ
δ
and taking the trace we find with
tr{γ
µ
γ
α
γ
ν
γ
δ
}=4(g
µα
g
νδ
+g
µδ
g
να
−g
µν
g
αδ
)
= 4a
µανδ
(3)
and
tr{γ
β
γ
δ
}=g
β
δ
, tr{γ
5
γ
β
γ
δ
}=0
that
g
µα
g
νδ
+g
µδ
g
να
−g
µν
g
αδ
=a
µανδ
. (4)
The second coefficient is determined by multiplying (2) by γ
δ
γ
5
and taking the
trace
4b
µανδ
= 4iε
µανδ
. (5)
Thus we have proven (1):
γ
µ
γ
α
γ
ν
= g
µα
γ
ν
+g
να
γ
µ
−g
µν
γ
α
+iε
µανδ
γ
5
γ
δ
. (6)
5.4 Renormalization and the Expansion Into Local Operators 297
We have shown above that in the Bjorken limit short (or more precisely lightlike)
distances give the dominant contribution to the scattering tensor. For later pur-
poses it is convenient to expand the bilocal operator
Ψ(x/2)Ψ(−x/2) into a sum
of local operators. We easily find
ˆ
Ψ
x
2
ˆ
Ψ
−
x
2
=
ˆ
Ψ(0)
1 +
←−
∂
µ
1
x
µ
1
2
+
1
2!
←−
∂
µ
1
←−
∂
µ
2
x
µ
1
2
x
µ
2
2
+...
×
1 −
x
ν
1
2
∂
ν
1
+
x
ν
1
2
x
ν
2
2
1
2!
∂
ν
1
∂
ν
2
−...
ˆ
Ψ(0)
=
n
(−1)
n
n!
x
2
µ
1
x
2
µ
2
...
x
2
µ
n
×
ˆ
Ψ(0)
←→
∂
µ
1
←→
∂
µ
2
...
←→
∂
µ
n
ˆ
Ψ(0), (5.136)
←→
∂
µ
= ∂
µ
−
←−
∂
µ
. (5.137)
With the free current operator, (5.121) therefore becomes
W
µµ
=−
1
2π
x
2
≥0
dx
4
exp
(
iq ·x
)
n
(−1)
n
n!
x
2
µ
1
x
2
µ
2
...
x
2
µ
n
×[1 −(−1)
n
]
1
2
pol.
N|
ˆ
O
n
µ
1
... µ
n
β
|N s
µαµ
β
1
2π
∂
α
ε(x
0
)δ(x
2
)
=
1
2π
x
2
≥0
dx
4
exp
(
iq ·x
)
n odd
1
n!
x
2
µ
1
x
2
µ
2
...
x
2
µ
n
s
µαµ
β
×
1
2
pol.
N|
ˆ
O
n
µ
1
... µ
n
β
|N
1
π
∂
α
ε(x
0
)δ(x
2
)
, (5.138)
where
ˆ
O
n
µ
1
... µ
n
β
(0) =
ˆ
Ψ(0)
←→
∂
µ
1
←→
∂
µ
2
...
←→
∂
µ
n
γ
β
ˆ
Ψ(0). (5.139)
Although the product of operator
ˆ
J
µ
x
2
,
ˆ
J
ν
−x
2
is singular for x
2
→0, this
is not the case for the operators
ˆ
O
n
µ
1
µ
2
...µ
n
(0), which are now well defined. All
singularities of the current commutator are contained in the coefficient function
∂
α
ε(x
0
)δ(x
2
)
(see (5.138)). To calculate the structure function we write
1
2
pol.
N(P)|
ˆ
O
n
µ
1
... µ
n
β
(0)|N(P) =A
(n)
P
µ
1
P
µ
2
...P
µ
n
P
β
+ trace terms , (5.140)
where the A
(n)
are some constants. This equation follows simply from Lorentz
covariance. Because the operator is a Lorentz tensor with n +1 indices we have
to parameterize its matrix element accordingly. Furthermore, since we sum over