15.6. QCD corrections to the parton model predictions: scaling violations 135
ables. Here the γ i are the anomalous dimensions relevant to the quantity R,
and γ m is an analogous ‘anomalous mass dimension’, arising from finite shifts
in the mass parameter when the scale μ
2 is changed. Just as with the solution
(15.76) of (15.73), the solution of (15.79) is given in terms of a ‘running mass’
m(|q
2
|). Formally, we can think of γ m in (15.79) as analogous to β(α s ) and
ln m as analogous to α s . Then equation (15.41) for the running α s ,
∂α s (|q
2
|) = β(α s (
∂t
|q
2
|))
(15.80)
where t = ln(|q
2
|/μ
2 ), becomes
∂(ln m(|q
2
|)) = γ m (α s (
∂t
|q|
2 )).
(15.81)
Equation (15.81) has the solution
∫ ln
m(|q
2
|q
2
|) = m(μ
2
|
) exp
d ln |q
′2 γ m (α s ( q
′2 ).
(15.82)
ln μ 2
|
| |
To one-loop order in QCD, γ m (α s ) turns out to be −
1 α s (Peskin and
π
Schroeder 1995, section 18.1). Inserting the one-loop solution for α s in the
form (15.53), we find
[
ln(μ
2 /Λ
2
] 1
)
πβ 0
m(|q
2
|) = m(μ
2 )
,
(15.83)
ln(|q 2 |/Λ 2 )
where (πβ )
−1
0
= 12/(33 − 2N f ). Thus the quark masses decrease logarithmically as |q
2
| increases, rather like α s (|q
2
|). It follows that, in general, quark
mass effects are suppressed both by explicit m
2 /|q
2
| factors, and by the logarithmic decrease given by (15.83). Further discussion of the treatment of
quark masses is contained in Ellis, Stirling and Webber (1996), section 2.4;
see also the review by Manohar and Sachrajda in Nakamura et al. 2010.
15.6 QCD corrections to the parton model predictions
for deep inelastic scattering: scaling violations
As we saw in section 9.2, the parton model provides a simple intuitive explanation for the experimental observation that the nucleon structure functions
in deep inelastic scattering depend, to a good first approximation, only on
the dimensionless ratio x = Q
2 /2M ν, rather than on Q
2 and ν separately;
this behaviour is referred to as ‘scaling’. Here M is the nucleon mass, and Q
2
and ν are defined in (9.7) and (9.8). In this section we shall show how QCD
ables. Here the γ i are the anomalous dimensions relevant to the quantity R,
and γ m is an analogous ‘anomalous mass dimension’, arising from finite shifts
in the mass parameter when the scale μ
2 is changed. Just as with the solution
(15.76) of (15.73), the solution of (15.79) is given in terms of a ‘running mass’
m(|q
2
|). Formally, we can think of γ m in (15.79) as analogous to β(α s ) and
ln m as analogous to α s . Then equation (15.41) for the running α s ,
∂α s (|q
2
|) = β(α s (
∂t
|q
2
|))
(15.80)
where t = ln(|q
2
|/μ
2 ), becomes
∂(ln m(|q
2
|)) = γ m (α s (
∂t
|q|
2 )).
(15.81)
Equation (15.81) has the solution
∫ ln
m(|q
2
|q
2
|) = m(μ
2
|
) exp
d ln |q
′2 γ m (α s ( q
′2 ).
(15.82)
ln μ 2
|
| |
To one-loop order in QCD, γ m (α s ) turns out to be −
1 α s (Peskin and
π
Schroeder 1995, section 18.1). Inserting the one-loop solution for α s in the
form (15.53), we find
[
ln(μ
2 /Λ
2
] 1
)
πβ 0
m(|q
2
|) = m(μ
2 )
,
(15.83)
ln(|q 2 |/Λ 2 )
where (πβ )
−1
0
= 12/(33 − 2N f ). Thus the quark masses decrease logarithmically as |q
2
| increases, rather like α s (|q
2
|). It follows that, in general, quark
mass effects are suppressed both by explicit m
2 /|q
2
| factors, and by the logarithmic decrease given by (15.83). Further discussion of the treatment of
quark masses is contained in Ellis, Stirling and Webber (1996), section 2.4;
see also the review by Manohar and Sachrajda in Nakamura et al. 2010.
15.6 QCD corrections to the parton model predictions
for deep inelastic scattering: scaling violations
As we saw in section 9.2, the parton model provides a simple intuitive explanation for the experimental observation that the nucleon structure functions
in deep inelastic scattering depend, to a good first approximation, only on
the dimensionless ratio x = Q
2 /2M ν, rather than on Q
2 and ν separately;
this behaviour is referred to as ‘scaling’. Here M is the nucleon mass, and Q
2
and ν are defined in (9.7) and (9.8). In this section we shall show how QCD