15.6. QCD corrections to the parton model predictions: scaling violations 143
partially with respect to μ
2
F , and setting the result to zero, we obtain (to order
α s on the right-hand side)
2 ∫
2 ∂q(x, μ
2
F )
α
1
s (μ )
dy
μ F
=
P
(y, μ
2
qq (x/y)q
F ).
(15.108)
∂μ 2
F
2π
x y
This equation is the analogue of equation (15.35) describing the running of
the coupling α s with μ
2 , and is a fundamental equation in the theory of
perturbative applications of QCD. It is called the DGLAP equation, after
Dokshitzer (1977), Gribov and Lipatov (1972), and Altarelli and Parisi (1977).
The above derivation is not rigorous: a more sophisticated treatment (Georgi
and Politzer 1974, Gross and Wilczek 1974) confirms the result and extends
it to higher orders.
Equation (15.108) shows that, although perturbation theory cannot be
used to calculate the distribution function q(x, μ
2
F ) at any particular value
μ
2
F = μ
2
0 , it can be used to predict how the distribution changes (or ‘evolves’) as
μ
2
F varies. (We recall from (15.105) that q(x, μ
2
0 ) can be found experimentally
via xq(x, μ
2 ) = 2F (x, Q
2
2
0
2
= μ 0 )/e
2
i .) As in the case of σ(e
+ e
−
2
→ hadrons)
and the scale μ , the choice of factorization scale is arbitrary, and would
cancel from physical quantities if all powers in the perturbation series were
(N +1)
included. Truncating at N terms results in an ambiguity of order α s
. In
deep inelastic predictions, the standard choice for scales is μ
2 = μ
2
F = Q
2 .
The way the non-singlet distribution changes can be understood qualitatively as follows. The change in the distribution for a quark with momentum
fraction x, which absorbs the virtual photon, is given by the integral over y of
the corresponding distribution for a quark with momentum fraction y, which
radiated away (via a gluon) a fraction x/y of its momentum with probability (α s /2π)P qq (x/y). This probability is high for large momentum fractions:
high-momentum quarks lose momentum by radiating gluons. Thus there is
a predicted tendency for the distribution function q(x, μ
2 ) to get smaller at
large x as μ
2 increases, and larger at small x (due to the build-up of slower
partons), while maintaining the integral of the distribution over x as a constant. The effect is illustrated qualitatively in figure 15.14. In addition, the
radiated gluons produce more qq ¯ pairs at small x. Thus the nucleon may be
pictured as having more and more constituents, all contributing to its total
momentum, as its structure is probed on ever smaller distance (larger μ
2 )
scales.
In general, the right-hand side of (15.108) will have to be supplemented
by terms (calculable from figure 15.10) in which quarks are generated from
the gluon distribution; the equations must then be closed by a corresponding
one describing the evolution of the gluon distributions (Altarelli 1982). In the
now commonly used notation, this generalization of (15.108) reads
∫
∂f i/p (x, μ
2
∑
F )
2
α s (μ
2
F )
1 dy (1)
μ F
=
P
2
i
j/
∂μ 2
F
2π
x
←j (x/y)f p (y, μ ),
(15.109)
y
F
j=q,g
partially with respect to μ
2
F , and setting the result to zero, we obtain (to order
α s on the right-hand side)
2 ∫
2 ∂q(x, μ
2
F )
α
1
s (μ )
dy
μ F
=
P
(y, μ
2
qq (x/y)q
F ).
(15.108)
∂μ 2
F
2π
x y
This equation is the analogue of equation (15.35) describing the running of
the coupling α s with μ
2 , and is a fundamental equation in the theory of
perturbative applications of QCD. It is called the DGLAP equation, after
Dokshitzer (1977), Gribov and Lipatov (1972), and Altarelli and Parisi (1977).
The above derivation is not rigorous: a more sophisticated treatment (Georgi
and Politzer 1974, Gross and Wilczek 1974) confirms the result and extends
it to higher orders.
Equation (15.108) shows that, although perturbation theory cannot be
used to calculate the distribution function q(x, μ
2
F ) at any particular value
μ
2
F = μ
2
0 , it can be used to predict how the distribution changes (or ‘evolves’) as
μ
2
F varies. (We recall from (15.105) that q(x, μ
2
0 ) can be found experimentally
via xq(x, μ
2 ) = 2F (x, Q
2
2
0
2
= μ 0 )/e
2
i .) As in the case of σ(e
+ e
−
2
→ hadrons)
and the scale μ , the choice of factorization scale is arbitrary, and would
cancel from physical quantities if all powers in the perturbation series were
(N +1)
included. Truncating at N terms results in an ambiguity of order α s
. In
deep inelastic predictions, the standard choice for scales is μ
2 = μ
2
F = Q
2 .
The way the non-singlet distribution changes can be understood qualitatively as follows. The change in the distribution for a quark with momentum
fraction x, which absorbs the virtual photon, is given by the integral over y of
the corresponding distribution for a quark with momentum fraction y, which
radiated away (via a gluon) a fraction x/y of its momentum with probability (α s /2π)P qq (x/y). This probability is high for large momentum fractions:
high-momentum quarks lose momentum by radiating gluons. Thus there is
a predicted tendency for the distribution function q(x, μ
2 ) to get smaller at
large x as μ
2 increases, and larger at small x (due to the build-up of slower
partons), while maintaining the integral of the distribution over x as a constant. The effect is illustrated qualitatively in figure 15.14. In addition, the
radiated gluons produce more qq ¯ pairs at small x. Thus the nucleon may be
pictured as having more and more constituents, all contributing to its total
momentum, as its structure is probed on ever smaller distance (larger μ
2 )
scales.
In general, the right-hand side of (15.108) will have to be supplemented
by terms (calculable from figure 15.10) in which quarks are generated from
the gluon distribution; the equations must then be closed by a corresponding
one describing the evolution of the gluon distributions (Altarelli 1982). In the
now commonly used notation, this generalization of (15.108) reads
∫
∂f i/p (x, μ
2
∑
F )
2
α s (μ
2
F )
1 dy (1)
μ F
=
P
2
i
j/
∂μ 2
F
2π
x
←j (x/y)f p (y, μ ),
(15.109)
y
F
j=q,g