128
15. QCD II: Asymptotic Freedom, the Renormalization Group
where we have defined L = ln(Q
2 /Λ
2 ). In first approximation, one sets b 1
MS
to zero and finds α s = (1/β 0 L) as before. To obtain the next approximation,
we set α s = (1/β 0 L) in the b 1 term of (15.60), and solve for α s to first order
in b 1 . This gives (problem 15.4 (a))
1
1
α s =
−
β 1 ln L.
(15.61)
β 3 L 2
β 0 L
0
Problem 15.4 (b) carries the calculation to the three-loop stage.
2
The current world average value of α s (m ) is (Bethke 2009)
Z
2
α s (m Z ) = 0.1184 ± 0.0007.
(15.62)
The remarkable precision of this number represents extraordinary consistency
among the many methods used to determine it
3 , which include deep inelastic
+
scattering, electroweak fits, e e
−
→ jets, and lattice calculations (see chapter
16). If (15.62) is used to determine Λ
from (15.61), one finds Λ
= 231
MS
MS
MeV; using the 3-loop formula of problem 15.4 (b) gives Λ
= 213 MeV
MS
(Bethke 2009).
These values of Λ
are for N f = 5, appropriate for the Z
0 mass region,
MS
well above the b threshold. As Q
2 runs to smaller values, and a quark mass
threshold is crossed, N f changes by one unit, and so correspondingly do the
coefficients β 0 , β 1 , . . .. Physical quantities must however be continuous across
a quark threshold. This requires that the values of α s above and below that
threshold satisfy certain matching conditions (Rodrigo and Santamaria 1993,
Bernreuther and Wetzel 1982, Chetyrkin et al. 1997). These are satisfied by
allowing Λ
to depend on N f . At one and two loop order, the matching conMS
(N f −1)
(N f )
dition is simply α s
= α s , which can be straightforwardly implemented
(N f −1)
(N f −1)
in terms of Λ
and Λ
. In higher orders the matching conditions
MS
MS
contain additional terms, which are required at (n−1)-loop order for an n-loop
calculation of α s .
Figure 15.5 shows a summary (Bethke 2009) of measurements of α s as
a function of the energy scale Q, compared with the QCD prediction. The
latter is evaluated in 4-loop approximation, using 3-loop threshold matching
conditions at the masses m c = 1.5 GeV and m b = 4.7 GeV. The agreement is
perfect, a triumph for both experiment and theory.
15.4 σ(e
+ e
−
→ hadrons) revisited
We may now return to the physical process which originally motivated this
extensive detour. The perturbative corrections to σ pt (Q
2 ) are expressed as a
3 With the exception of a long-standing systematic difference: results from structure
2
functions prefer a smaller value of αs(m Z ) than most of the others.
15. QCD II: Asymptotic Freedom, the Renormalization Group
where we have defined L = ln(Q
2 /Λ
2 ). In first approximation, one sets b 1
MS
to zero and finds α s = (1/β 0 L) as before. To obtain the next approximation,
we set α s = (1/β 0 L) in the b 1 term of (15.60), and solve for α s to first order
in b 1 . This gives (problem 15.4 (a))
1
1
α s =
−
β 1 ln L.
(15.61)
β 3 L 2
β 0 L
0
Problem 15.4 (b) carries the calculation to the three-loop stage.
2
The current world average value of α s (m ) is (Bethke 2009)
Z
2
α s (m Z ) = 0.1184 ± 0.0007.
(15.62)
The remarkable precision of this number represents extraordinary consistency
among the many methods used to determine it
3 , which include deep inelastic
+
scattering, electroweak fits, e e
−
→ jets, and lattice calculations (see chapter
16). If (15.62) is used to determine Λ
from (15.61), one finds Λ
= 231
MS
MS
MeV; using the 3-loop formula of problem 15.4 (b) gives Λ
= 213 MeV
MS
(Bethke 2009).
These values of Λ
are for N f = 5, appropriate for the Z
0 mass region,
MS
well above the b threshold. As Q
2 runs to smaller values, and a quark mass
threshold is crossed, N f changes by one unit, and so correspondingly do the
coefficients β 0 , β 1 , . . .. Physical quantities must however be continuous across
a quark threshold. This requires that the values of α s above and below that
threshold satisfy certain matching conditions (Rodrigo and Santamaria 1993,
Bernreuther and Wetzel 1982, Chetyrkin et al. 1997). These are satisfied by
allowing Λ
to depend on N f . At one and two loop order, the matching conMS
(N f −1)
(N f )
dition is simply α s
= α s , which can be straightforwardly implemented
(N f −1)
(N f −1)
in terms of Λ
and Λ
. In higher orders the matching conditions
MS
MS
contain additional terms, which are required at (n−1)-loop order for an n-loop
calculation of α s .
Figure 15.5 shows a summary (Bethke 2009) of measurements of α s as
a function of the energy scale Q, compared with the QCD prediction. The
latter is evaluated in 4-loop approximation, using 3-loop threshold matching
conditions at the masses m c = 1.5 GeV and m b = 4.7 GeV. The agreement is
perfect, a triumph for both experiment and theory.
15.4 σ(e
+ e
−
→ hadrons) revisited
We may now return to the physical process which originally motivated this
extensive detour. The perturbative corrections to σ pt (Q
2 ) are expressed as a
3 With the exception of a long-standing systematic difference: results from structure
2
functions prefer a smaller value of αs(m Z ) than most of the others.