Greiner W., Schrammm S., Stein E. Quantum Chromodynamics
Подождите немного. Документ загружается.
278 5. Perturbative QCD I: Deep Inelastic Scattering
Example 5.7
The initial gluon wave function at Q
2
can be expanded into these quarks states:
|G
Q
2
(k) =
p, p
C
qq
( p, p
)|q
Q
2
( p)q
Q
2
( p
) , (2)
where the coefficients are given by
C
qq
( p, p
) =q
Q
2
( p)q
Q
2
( p
)|G
Q
2
(k) . (3)
Applying the kinetic part of the Hamiltonian (see (4) of Example 5.5) yields
ˆ
H
Q
2
|G
Q
2
(k) +∆
ˆ
H
Q
2
|G
Q
2
(k) =
p, p
C
qq
( p, p
)
ˆ
H
Q
2
|q
Q
2
( p)q
Q
2
( p
)
(4)
or
k|G
Q
2
(k) +∆
ˆ
H
Q
2
|G
Q
2
(k)
=
p, p
(|p|+|p
|) ×C
qq
( p, p
)|q
Q
2
( p)q
Q
2
( p
) . (5)
The coefficients can be obtained by projecting with q
Q
2
( p)q
Q
2
( p
)|:
C
qq
( p, p
) =
q
Q
2
( p)q
Q
2
( p
)|∆
ˆ
H
Q
2
|G
Q
2
(k)
|p|+|p
|−|k|
. (6)
To determine C
qq
( pp
) we need from the Hamiltonian ∆
ˆ
H
Q
2
(see (6) of
Example 5.5) the part that describes the quark–gluon interaction:
∆
ˆ
H
Q
2
=−g
2
(Q
2
)
a
d
3
x
f
q
f
(r) A
a
(r)γ
λ
a
2
q
f
(r). (7)
As in (15) in Example 5.5 it follows for the matrix element that
q
Q
2
( p)q
Q
2
( p
)|∆
ˆ
H
Q
2
|G
Q
2
(k)
=−
a
g(Q
2
)(2π)
3
δ
3
( p + p
−k)u( p, s)ε
a
·γ
λ
a
2
v( p
, s
). (8)
Again we can relate the squared coefficient C
qq
( p, p
) to the quark distri-
bution function by calculating the “decay probability” of a gluon if the re-
solving power Q
2
is increased. Following our previous calculation (see (14)
of Example 5.5) we start with a single gluon |G
Q
2
(k) and no quarks at the
resolution Q
2
. Increasing the resolution to Q
2
will generate quarks through
the process of pair production and we obtain the following quark distribution
function:
∆N
q
(c, Q
2
) =
1
8
a
1
2
ε
a
cs
c
s
1
2ω
d
3
p
(2π)
3
2E
d
3
p
(2π)
3
2E
×δ
x −
p ·k
|k|
2
|C
qq
( p, p
)|
2
. (9)
5.2 An Alternative Approach to the GLAP Equations 279
This is summed over spins (s, s
)andcolors(c, c
) (factor 1/8) of the outgoing
quarks and it is averaged over the spin ε
a
(factor 1/2) and color of the initial
gluon. Remenber that the gluon, being a vector particle, has spin 1. However,
being massless, the gluon has only two (transverse) spin directions, as is the case
for the photon. The δ function guarantees that the quark longitudinal momentum
is x times the gluon momentum:
x =
p
|k|
=
p ·k
|k|
2
. (10)
Inserting C
qq
( p, p
) from (6) into (9), we find that (9) becomes
∆N
q
(c, Q
2
) = g(Q
2
)
2
1
16
(2π)
3
d
3
p
(2π)
3
d
3
p
(2π)
3
×δ
x −
p ·k
|k|
2
δ
3
( p + p
−k)
1
8ωEE
1
|p|+|p
|−|k|
2
×
a
ε
a
cs
c
s
|u
c
( p, s)ε
a
·γ
λ
a
2
v
c
A( p
, s
)|
2
. (11)
We first perform the spin and color summations, again using ε/
a
=−γ ·ε for ε
a
=
(0, ε
a
):
a
ε
a
cs
c
s
|u
c
( p, s)ε
a
·γ
λ
a
2
v
c
( p
, s
)|
2
=
a,b
tr
λ
a
λ
b
4
ε
a
tr[p/(ε
a
·γ ) p/
(ε
b∗
·γ )]
=
a
1
2
ε
a
tr
p/ε/
a
p/
ε/
a∗
=
a
1
2
ε
a
4 ·[(p ·ε
a
)( p
·ε
a∗
) +( p
·ε
a
)( p ·ε
a∗
) −( p · p
)(ε
a
·ε
a∗
)]
=2
a
ε
a
[(p ·ε
a
)( p
·ε
a∗
)
+( p
·ε
a
)( p ·ε
a∗
) −( p
0
p
0
− p · p
)(ε
a
·ε
a∗
)]
=16
δ
ij
−
k
i
k
j
|k|
2
( p
i
p
j
+ p
i
p
j
+(|p||p
|−p · p
)δ
ij
)
=
32
|k|
2
|k|
2
|p||p
|−(k· p)(k · p
)
. (12)
Example 5.7
280 5. Perturbative QCD I: Deep Inelastic Scattering
Example 5.7
The summation over the colors a is performed exactly the same way as was done
in Example 5.5, (17). The quark distribution function can now be written as
∆N
q
(c, Q
2
) = g(Q
2
)
2
1
4
(2π)
3
d
3
pδ
x −
p ·k
|k|
2
1
8ωEE
(13)
×
1
|p|+|p
|−|k|
2
1
|k|
2
(|k|
2
|p||p
|−(k· p)(k · p
)) .
Here p
=k − p is due to the δ
3
( p + p
−k) function. This conservation of the
three-momentum will now be handled explicitly. We introduce explicit forms for
the gluon and quark momenta,
k =(0, 0, k),
p = (k
⊥
, xk),
p
=(−k
⊥
,(1 −x)k), (14)
so that the three-momentum is conserved: k = p + p
. As in the last sections we
take |k
⊥
| to lie between εQ and εQ
and assume that all momenta are much
greater than |k
⊥
|
2
:
|k||k
⊥
| , |p||k
⊥
| , |p
||k
⊥
| . (15)
The terms that enter (13) can now be approximated as in previous sections (see
(25) in Example 5.5):
1
|p|+|p
|−|k|
=
1
k
2
⊥
+x
2
k
2
+
k
2
⊥
+(1 −x)
2
k
2
−|k|
=
2k
|k
⊥
|
x(1 −x), (16)
1
ωEE
=
1
x(1 −x)k
3
,ω=k , (17)
1
|k|
2
(|k|
2
|p||p
|−(k· p)(k · p
)) =
1
2
|k
⊥
|
2
(1 −x)
2
+x
2
x(1 −x)
. (18)
Inserting these expressions into (13) and rewriting
d
3
p = d
2
k
⊥
dp
"
(19)
we can perform the integration over the longitudinal momentum, obtaining
∆N
q
(x, Q
2
) =
g(Q
2
)
2
8π
3
d
2
k
⊥
|k
⊥
|
2
1
2
(1 −x)
2
+x
2
=
g(Q
2
)
2
4π
2
dk
⊥
|k
⊥
|
1
2
(1 −x)
2
+x
2
. (20)
5.3 Common Parametrizations of the Distribution Functions and Anomalous Dimensions 281
Keeping in mind that |k
⊥
| is in the range εQ to εQ
, we can perform the last
integration. The result is
∆N
q
(x, Q
2
) =
g(Q
2
)
2
4π
2
1
2
(1 −x)
2
+x
2
ln |k
⊥
|
εQ
εQ
=
g(Q
2
)
2
8π
2
1
2
(1 −x)
2
+x
2
∆ ln Q
2
, (21)
since ε is an arbitrary constant ∆ ln " = 0. Comparing this result with the GLAP
equation (see (5.74)) yields, with the initial condition
N
G
(x, Q
2
) = δ(1 −x), N
q
(x, Q
2
) = 0 , (22)
the expression for the splitting function:
P
qG
(x) =
1
2
(1 −x)
2
+x
2
. (23)
5.3 Common Parametrizations of the Distribution Functions
and Anomalous Dimensions
Instead of dealing with the GLAP equations (5.76) we can transform the set of
integral equations into a set of linear equations by introducing the moments of
the structure functions. The nth moment is defined as
8
M
a
(n, Q
2
) =
1
0
dxx
n−1
N
a
(x, Q
2
), (5.84)
where N
a
is the structure function of the parton of kind a. Differentiating this
with respect to ln(Q
2
) and inserting the GLAP equations (5.76) results in
dM
a
(n, Q
2
)
dlnQ
2
=
1
0
dxx
n−1
dN
a
(x, Q
2
)
dlnQ
2
=
1
0
dxx
n−1
g
2
(Q
2
)
8π
2
b
1
x
dx
x
P
ab
x
x
N
b
(x
, Q
2
)
=
g
2
s
(Q
2
)
8π
2
b
1
0
dxx
n−1
1
x
dx
x
P
ab
x
x
N
b
(x
, Q
2
).
(5.85)
8
Usually one defines the nth moment as the expectation value of x
n
rather than x
n−1
,
which is commonly chosen in QCD.
Example 5.7
282 5. Perturbative QCD I: Deep Inelastic Scattering
The second integration over x
between x and 1 can be performed from x
=0
to x
= 1 if one introduces the Heavyside function in the integrand. Thus the
integrand vanishes for x
< x:
dM
a
(n, Q
2
)
dlnQ
2
=
g
2
(Q
2
)
8π
2
b
⎛
⎝
1
0
dxx
n−1
×
1
0
dx
x
P
ab
x
x
Θ(x
−x)N
b
(x
, Q
2
)
⎞
⎠
. (5.86)
Now the integrations can be exchanged. Substituting z = x/x
we get
dM
a
(n, Q
2
)
dlnQ
2
=
g
2
(Q
2
)
8π
2
b
⎛
⎜
⎝
1
0
dx
x
N
b
(x
, Q
2
)
×
1
0
dxx
n−1
P
ab
x
x
Θ(x
−x)
⎞
⎠
=
g
2
(Q
2
)
8π
2
b
⎛
⎜
⎝
1
0
dx
x
N
b
(x
, Q
2
)
×
1/x
0
dzx
(zx
)
n−1
P
ab
(z)Θ(x
−x
z)
⎞
⎟
⎠
=
g
2
(Q
2
)
8π
2
b
⎛
⎜
⎝
1
0
dx
x
n−1
N
b
(x
, Q
2
)
×
1/x
0
dzz
n−1
P
ab
(z)Θ(1 −z)
⎞
⎟
⎠
=
g
2
(Q
2
)
8π
2
b
1
0
dx
x
n−1
N
b
(x
, Q
2
)
1
0
dzz
n−1
P
ab
(z).
(5.87)
The last step can be performed since 1/x
≥1 (because x
≤ 1)andthe
Θ function contributes only for z = 1/x
≤1 . Therefore the upper bound of the
integration can be taken as 1.
5.3 Common Parametrizations of the Distribution Functions and Anomalous Dimensions 283
In the high-Q
2
regime α
s
(Q
2
) = g
s
(Q
2
)
2
/4π is given in leading-log order
(see (4.147) which has to be taken at µ
2
= Q
2
)by
α
s
(Q
2
) =
12
33 −2N
f
π
ln(Q
2
/Λ
2
)
, (5.88)
where N
f
denotes the number of quark flavors and Λ is the QCD cut-off
parameter. We obtain (see Example 4.4)
dM
a
(n, Q
2
)
dlnQ
2
=
6
33 −2N
f
1
ln(Q
2
/Λ
2
)
b
M
b
(n, Q
2
)
1
0
dzz
n−1
P
ab
(z)
=−
1
ln(Q
2
/Λ
2
)
b
M
b
(n, Q
2
)d
ab
(n), (5.89)
where we have abbreviated the moment of the splitting function P
ab
(z) by
d
ab
(n) =−
6
33 −2N
f
1
0
dzz
n−1
P
ab
(z). (5.90)
Thus we have derived a system of linear equations between the moments of the
structure functions and their derivatives. The moments d
ab
(n) of the splitting
functions are the anomalous dimensions. We shall come back to that in the dis-
cussion of the renormalization group (Sect. 5.4). Once the splitting functions are
given, the moments d
ab
(n) can easily be obtained (see Exercise 5.8):
d
Gq
(n) =−
8
33 −2N
f
n
2
+n +2
n(n
2
−1)
,
d
qq
(n) =
4
33 −2N
f
⎛
⎝
1 −
2
n(n +1)
+4
n
j=2
1
j
⎞
⎠
,
d
qG
(n) =−
3
33 −2N
f
n
2
+n +2
n(n +1)(n +2)
,
d
GG
(n) =
9
33 −2N
f
⎛
⎝
1
3
+
2N
f
9
−
4
n(n −1)
−
4
(n +1)(n +2)
+4
n
j=2
1
j
⎞
⎠
.
(5.91)
The moments M(n, Q
2
) of the parton distributions N(x, Q
2
) are given by experi-
ment. However, it is more difficult to calculate the parton distributions N(x) for
given moments M(n). Here we deleted the Q
2
dependence in the argument, be-
cause the relation between the distribution function N(x, Q
2
) and the momenta
M(n, Q
2
) is shown at the renormalization point Q
2
=µ
2
. At this special point
the Q
2
dependence drops out. After N(x, Q
2
=µ
2
) is known one may obtain
N(x, Q
2
) at any other Q
2
by applying the GLAP evolution; see, e.g., (5.86).
284 5. Perturbative QCD I: Deep Inelastic Scattering
Since the polynomials x
n
do not form an orthonormal system of functions in the
range 0 ≤ x ≤1, one has to transform the moments M(n) into coefficients
c
i
= f
M(n)
, (5.92)
so that
N(x) =
i
c
i
x
i
. (5.93)
The coefficients c
i
are functions of the moments N(n) and can be obtained from
the relation
M(n) =
i
1
i +n
c
i
, (5.94)
which is proven as follows.
Let f(x) be the function that is to be approximated by g(x) =
i
c
i
x
i
.Again
the moments of f(x) are denoted by
f
n
=
1
0
dxx
n−1
f(x). (5.95)
We require that
M =
1
0
dx
f(x) −g(x)
2
(5.96)
becomes minimal, in which case the variation of M with respect to the coeffi-
cients c
k
vanishes:
0 = δM =
∂M
∂c
k
δc
k
, (5.97)
from which it follows that
∂M
∂c
k
=0forallk . (5.98)
Inserting the series expansion for g(x) into (5.93), we can express M by the
coefficients c
i
and the moments f :
M =
1
0
dx
f(x)
2
+g(x)
2
−2 f(x)g(x)
=
1
0
dxf(x)
2
+
1
0
dxg(x)
2
−2
1
0
dxf(x)g(x)
5.3 Common Parametrizations of the Distribution Functions and Anomalous Dimensions 285
=C +
i, j
c
i
c
j
1
0
dxx
i+j
−2
i
c
i
1
0
dxx
i
f(x)
=C +
i, j
c
i
c
j
i + j +1
−2
i
c
i
f
i+1
. (5.99)
C =
+
1
0
dxf(x)
2
is a constant.
Performing the derivation with respect to c
k
we get the desired relation
between the coefficients c
i
and the moments f
i
of the function f :
0 =
∂M
∂c
k
=
∂
∂c
k
C +
i, j
c
i
c
j
i + j +1
−2
i
c
i
f
i+1
=
i, j
∂
∂c
k
c
i
c
j
i + j +1
−2
i
∂c
i
∂c
k
f
i+1
=
i, j
δ
ik
c
j
+c
i
δ
jk
i + j +1
−2
i
δ
ik
f
i+1
= 2
i
c
i
i +k +1
−2 f
k+1
, (5.100)
or
f
k
=
i
c
i
i +k
≡
i
A
ki
c
i
. (5.101)
In order to compute the coefficients c
i
from the known moments f
k
, one has to
determine the inverse of the matrix A
ki
:
c
i
=
A
−1
ik
f
k
. (5.102)
Obviously the coefficients c
i
change their values if one more moment f
k
is added
in the calculation.
It is possible to give an upper and lower bound to f(x) by employing the
Pad
´
e approximation method for the moments f
k
.
9
Recent parametrizations for
the parton distributions inside the nucleon are given by CTEQ. (The CTEQ group
provides continuously updated sets of distribution functions, obtainable via
WWW from http://www.phys.psu.edu/˜cteq/, which is regularly updated.) Here
we give another parametrization.
10
Writing for s in a leading-log approximation
s = ln
ln
Q
2
/(0.232 GeV)
2
ln
µ
2
/(0.232 GeV)
2
, (5.103)
9
W.W. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling: Numerical Recipes (Cam-
bridge University Press 1992)
10
M. Glück, E. Reya, A. Vogt: Z. Physik C 53, 127 (1992)
286 5. Perturbative QCD I: Deep Inelastic Scattering
where µ
2
= 0.25 GeV
2
, the parton distributions are parametrized as follows. The
distribution function for the u and d valence quarks in a proton are denoted by
v
p
(x, Q
2
). We use the parametrization
xv
p
(x, Q
2
) = Nx
a
1 + A
√
x +Bx
(1 −x)
D
, (5.104)
with
N =0.663 +0.191s −0.041s
2
+0.031s
3
,
a = 0.326 ,
A =−1.97 +6.74s −1.96s
2
,
B = 24.4 −20.7s +4.08s
2
,
D =2.86 +0.70s −0.02s
2
, (5.105)
for v = u
v
+d
v
and
N =0.579 +0.283s +0.047s
2
,
a = 0.523 −0.015s ,
A = 2.22 −0.59s −0.27s
2
,
B = 5.95 −6.19s +1.55s
2
,
D =3.57 +0.94s −0.16s
2
, (5.106)
for v = d
v
. The distributions for a neutron are obtained by interchanging u and
d quarks.
The gluon and light-sea-quark distributions are given by
xw
p
(x, Q
2
) =
'
x
α
A + Bx +Cx
2
ln
1
x
b
+s
α
exp
−E +
6
E
s
β
ln
1
x
(
(1 −x)
D
, (5.107)
with
α = 0.558 ,
β = 1.218 ,
a = 1.00−0.17s ,
b = 0 ,
A = 4.879s −1.383s
2
,
B = 25.92 −28.97s +5.596s
2
,
C =−25.69 +23.68s −1.975s
2
,
D =2.537 +1.718s +0.353s
2
,
E = 0.595 +2.138s ,
E
=4.066 (5.108)
5.3 Common Parametrizations of the Distribution Functions and Anomalous Dimensions 287
for w = G and
α = 1.396 ,
β = 1.331 ,
a = 0.412−0.171s ,
b = 0.566 −0.496s ,
A = 0.363 ,
B =−1.196 ,
C = 1.029 +1.785s −0.459s
2
,
D = 4.696 +2.109s ,
E = 3.838 +1.944s ,
E
= 2.845 (5.109)
for w =
¯
u and w =
¯
d, characterizing the light sea quarks. Finally for the s, c, b
quark distributions, we employ the relation
xw
p
(x, Q
2
) =
(s −s
w
)
α
[ln(1/x)]
a
1 + A
√
x +Bx
(1 −x)
D
×exp
−E +
6
E
s
β
ln
1
x
, (5.110)
where the s-quark distribution is given by
α = 0.803 ,
β = 0.563 ,
a = 2.082−0.577s ,
A =−3.055 +1.024s
0.67
,
B = 27.4 −20.0s
0.154
,
D = 6.22 ,
E = 4.33 +1.408s ,
E
= 8.27 −0.437s ,
s
w
= s
s
=0 . (5.111)
The c,
¯
c distribution is parametrized by
α = 1.01 ,
β = 0.37 ,
a = 0 ,
A = 0 ,
B = 4.24 −0.804s ,
D = 3.46 +1.076s ,