112
G. Altarelli and S. Forte
Here the factor of t is a bit symbolic: it stands for log Q 2 /km 2 and what we exactly
put below Q 2 depends on the definition of the renormalised quark density, which
also fixes the exact form of the finite term f (z) in Eq. (4.75).
The effective parton density q(y, t) that we have defined is now scale dependent.
In terms of this scale dependent density we have the following relations, where we
have also replaced the fixed coupling with the running coupling according to the
prescription derived from the RGE:
F (x, t) =
1
x
dy
q(y, t)
y
e
2
[δ(
x
y
− 1) +
α s (t)
2π
f (
x
y
))] = e
2 q(x, t) + o(α s (t))
d
dt
q(x, t) =
α s (t)
2π
1
x
dy
q(y, t)
y
· P (
x
y
) + o(α s (t)
2 )
(4.79)
We see that in lowest order we reproduce the naive parton model formulae for the
structure functions in terms of effective parton densities that are scale dependent.
The evolution equations for the parton densities are written down in terms of kernels
(the “splitting functions”) that can be expanded in powers of the running coupling.
At leading order, we can interpret the evolution equation by saying that the variation
of the quark density at x is given by the convolution of the quark density at y times
the probability of emitting a gluon with fraction x/y of the quark momentum.
It is interesting that the integro-differential QCD evolution equation for densities
can be transformed into an infinite set of ordinary differential equations for Mellin
moments [2]. The moment f n of a density f (x) is defined as:
f n =
1
0
dxx
n−1 f (x)
(4.80)
By taking moments of both sides of the second of Eqs. (4.79) one finds, with
a simple interchange of the integration order, the simpler equation for the n-th
moment:
d
dt
q n (t) =
α s (t)
2π
· P n · q n (t)
(4.81)
To solve this equation we observe that:
log
q n (t)
q n (0)
=
P n
2π
t
0
α s (t)dt =
P n
2π
α s (t )
α s
dα
−bα
(4.82)
where we used Eq. (4.31) to change the integration variable from dt to dα(t)
(denoted as dα ) and β(α) −bα 2 + . . .. Finally the solution is:
q n (t) = [
α s
α s (t)
]
Pn
2πb · q n (0)
(4.83)
G. Altarelli and S. Forte
Here the factor of t is a bit symbolic: it stands for log Q 2 /km 2 and what we exactly
put below Q 2 depends on the definition of the renormalised quark density, which
also fixes the exact form of the finite term f (z) in Eq. (4.75).
The effective parton density q(y, t) that we have defined is now scale dependent.
In terms of this scale dependent density we have the following relations, where we
have also replaced the fixed coupling with the running coupling according to the
prescription derived from the RGE:
F (x, t) =
1
x
dy
q(y, t)
y
e
2
[δ(
x
y
− 1) +
α s (t)
2π
f (
x
y
))] = e
2 q(x, t) + o(α s (t))
d
dt
q(x, t) =
α s (t)
2π
1
x
dy
q(y, t)
y
· P (
x
y
) + o(α s (t)
2 )
(4.79)
We see that in lowest order we reproduce the naive parton model formulae for the
structure functions in terms of effective parton densities that are scale dependent.
The evolution equations for the parton densities are written down in terms of kernels
(the “splitting functions”) that can be expanded in powers of the running coupling.
At leading order, we can interpret the evolution equation by saying that the variation
of the quark density at x is given by the convolution of the quark density at y times
the probability of emitting a gluon with fraction x/y of the quark momentum.
It is interesting that the integro-differential QCD evolution equation for densities
can be transformed into an infinite set of ordinary differential equations for Mellin
moments [2]. The moment f n of a density f (x) is defined as:
f n =
1
0
dxx
n−1 f (x)
(4.80)
By taking moments of both sides of the second of Eqs. (4.79) one finds, with
a simple interchange of the integration order, the simpler equation for the n-th
moment:
d
dt
q n (t) =
α s (t)
2π
· P n · q n (t)
(4.81)
To solve this equation we observe that:
log
q n (t)
q n (0)
=
P n
2π
t
0
α s (t)dt =
P n
2π
α s (t )
α s
dα
−bα
(4.82)
where we used Eq. (4.31) to change the integration variable from dt to dα(t)
(denoted as dα ) and β(α) −bα 2 + . . .. Finally the solution is:
q n (t) = [
α s
α s (t)
]
Pn
2πb · q n (0)
(4.83)