130
15. QCD II: Asymptotic Freedom, the Renormalization Group
c n (Q
2 /μ
2 ) would cancel that of α s (μ
2 ). This requirement can be imposed
order by order in α s to fix the μ
2 -dependence of the coefficients, and is a
direct way of applying the RGE idea. Consider, for example, truncating the
series at the n = 2 stage:
(
)
α s (μ
2 )
σ(e
+ e
−
→ hadrons) ≈ σ pt (Q
2 ) 1 +
+ c 2 (Q
2 /μ
2 )(α s (μ
2 )/π)
2 .
π
(15.64)
Differentiating with respect to μ
2 and setting the result to zero we obtain
dc 2
πβ(α s (μ
2 ))
μ
2
= −
(15.65)
dμ 2
(α s (μ 2 )) 2
where an O(α
3 ) term has been dropped. Substituting the one-loop result
s
(15.48) – as is consistent to this order – we find
c 2 (Q
2 /μ
2 ) = c 2 (1) − πβ 0 ln(Q
2 /μ
2 ).
(15.66)
The second term on the right-hand side of (15.66) gives the contribution identified in (15.2).
In practice only a finite number of terms n = N will be available, and a μ
2
dependence will remain, which implies an uncertainty in the prediction of the
cross section (and similar physical observables), due to the arbitrariness of the
scale choice. This uncertainty will be of the same order as the neglected terms,
i.e. of order α
N +1 . Thus the scale dependence of a QCD prediction gives a
s
measure of the uncertainties due to neglected terms. For σ(e
+ e
−
→ hadrons)
the choice of scale μ
2 = Q
2 is usually made, so as to avoid large logarithms
in relations such as (15.66).
Before proceeding to our second main application of the RGE, scaling
violations in deep inelastic scattering, it is necessary to take another detour,
to enlarge our understanding of the scope of the RGE.
15.5 A more general form of the RGE: anomalous
dimensions and running masses
The reader may have wondered why, for QCD, all the graphs of figure 15.6
are needed, whereas for QED we got away with only figure 11.3. The reason
for the simplification in QED was the equality between the renormalization
constants Z 1 and Z 2 , which therefore cancelled out in the relation between
the renormalized and bare charges e and e 0 , as briefly stated before equation
(15.8) (this equality was discussed in section 11.6). We recall that Z 1 is the
field strength renormalization factor for the charged fermion in QED, and Z 1
is the vertex part renormalization constant; their relation to the counter terms
15. QCD II: Asymptotic Freedom, the Renormalization Group
c n (Q
2 /μ
2 ) would cancel that of α s (μ
2 ). This requirement can be imposed
order by order in α s to fix the μ
2 -dependence of the coefficients, and is a
direct way of applying the RGE idea. Consider, for example, truncating the
series at the n = 2 stage:
(
)
α s (μ
2 )
σ(e
+ e
−
→ hadrons) ≈ σ pt (Q
2 ) 1 +
+ c 2 (Q
2 /μ
2 )(α s (μ
2 )/π)
2 .
π
(15.64)
Differentiating with respect to μ
2 and setting the result to zero we obtain
dc 2
πβ(α s (μ
2 ))
μ
2
= −
(15.65)
dμ 2
(α s (μ 2 )) 2
where an O(α
3 ) term has been dropped. Substituting the one-loop result
s
(15.48) – as is consistent to this order – we find
c 2 (Q
2 /μ
2 ) = c 2 (1) − πβ 0 ln(Q
2 /μ
2 ).
(15.66)
The second term on the right-hand side of (15.66) gives the contribution identified in (15.2).
In practice only a finite number of terms n = N will be available, and a μ
2
dependence will remain, which implies an uncertainty in the prediction of the
cross section (and similar physical observables), due to the arbitrariness of the
scale choice. This uncertainty will be of the same order as the neglected terms,
i.e. of order α
N +1 . Thus the scale dependence of a QCD prediction gives a
s
measure of the uncertainties due to neglected terms. For σ(e
+ e
−
→ hadrons)
the choice of scale μ
2 = Q
2 is usually made, so as to avoid large logarithms
in relations such as (15.66).
Before proceeding to our second main application of the RGE, scaling
violations in deep inelastic scattering, it is necessary to take another detour,
to enlarge our understanding of the scope of the RGE.
15.5 A more general form of the RGE: anomalous
dimensions and running masses
The reader may have wondered why, for QCD, all the graphs of figure 15.6
are needed, whereas for QED we got away with only figure 11.3. The reason
for the simplification in QED was the equality between the renormalization
constants Z 1 and Z 2 , which therefore cancelled out in the relation between
the renormalized and bare charges e and e 0 , as briefly stated before equation
(15.8) (this equality was discussed in section 11.6). We recall that Z 1 is the
field strength renormalization factor for the charged fermion in QED, and Z 1
is the vertex part renormalization constant; their relation to the counter terms