15.4. σ(e
+ e
−
hadrons) revisited
129
→
FIGURE 15.5
Comparison between measurements of α s and the theoretical prediction, as a
function of the energy scale Q (Bethke 2009). (See color plate I.)
power series in α s ,
[
(
) ]
∑ ∞
n
+
2
α
2
s (μ )
σ(e e
−
→ hadrons) = σ pt (Q ) 1 +
c n (Q
2 /μ
2 )
, (15.63)
π
n=1
where μ is the renormalization scale. (A similar expansion can be written for
many other physical quantities too.) The coefficients from c 2 onwards depend
on the renormalization scheme (see appendix O), and are usually quoted in
the MS scheme. c 1 is the leading order (LO) coefficient, and we already know
that c 1 = 1 from (15.1). c 2 is the next-to-leading (NLO) coefficient; c 2 (1)
was calculated by Dine and Sapirstein (1979), Chetyrkin et al. (1979) and by
Celmaster and Gonsalves (1980), and has the value 1.9857 − 0.1152N f . The
next-to-next-to-leading (NNLO) coefficient c 3 (1) was calculated by Samuel
and Surguladze (1991) and by Gorishnii et al. (1991), and is equal to -12.8
for five flavours. The N3LO coefficient c 4 (1) (which requires the evaluation of
some twenty thousand diagrams) may be found in Baikov et al. (2008) and
Baikov et al. (2009).
The physical cross section σ(e
+ e
−
→ hadrons) must be independent of the
renormalization scale μ
2 , and this would also be true of the series in (15.63) if
an infinite number of terms were kept: the μ
2 -dependence of the coefficients
+ e
−
hadrons) revisited
129
→
FIGURE 15.5
Comparison between measurements of α s and the theoretical prediction, as a
function of the energy scale Q (Bethke 2009). (See color plate I.)
power series in α s ,
[
(
) ]
∑ ∞
n
+
2
α
2
s (μ )
σ(e e
−
→ hadrons) = σ pt (Q ) 1 +
c n (Q
2 /μ
2 )
, (15.63)
π
n=1
where μ is the renormalization scale. (A similar expansion can be written for
many other physical quantities too.) The coefficients from c 2 onwards depend
on the renormalization scheme (see appendix O), and are usually quoted in
the MS scheme. c 1 is the leading order (LO) coefficient, and we already know
that c 1 = 1 from (15.1). c 2 is the next-to-leading (NLO) coefficient; c 2 (1)
was calculated by Dine and Sapirstein (1979), Chetyrkin et al. (1979) and by
Celmaster and Gonsalves (1980), and has the value 1.9857 − 0.1152N f . The
next-to-next-to-leading (NNLO) coefficient c 3 (1) was calculated by Samuel
and Surguladze (1991) and by Gorishnii et al. (1991), and is equal to -12.8
for five flavours. The N3LO coefficient c 4 (1) (which requires the evaluation of
some twenty thousand diagrams) may be found in Baikov et al. (2008) and
Baikov et al. (2009).
The physical cross section σ(e
+ e
−
→ hadrons) must be independent of the
renormalization scale μ
2 , and this would also be true of the series in (15.63) if
an infinite number of terms were kept: the μ
2 -dependence of the coefficients