100
G. Altarelli and S. Forte
Note the presence of a pole in Eqs. (4.46, 4.47) at ±bαt = 1, called the Landau
pole, who realised its existence in QED already in the ‘50’s. For μ ∼ m e (in QED)
the pole occurs beyond the Planck mass. In QCD the Landau pole is located for
negative t or at Q < μ in the region of light hadron masses. Clearly the issue of the
definition and the behaviour of the physical coupling (which is always finite, when
defined in terms of some physical process) in the region around the perturbative
Landau pole is a problem that lies outside the domain of perturbative QCD.
The non leading terms in the asymptotic behaviour of the running coupling can in
principle be evaluated going back to Eq. (4.36) and computing b at 2-loops and so
on. But in general the perturbative coefficients of β(α) depend on the definition of
the renormalised coupling α (the renormalisation scheme), so one wonders whether
it is worthwhile to do a complicated calculation to get b if then it must be repeated
for a different definition or scheme. In this respect it is interesting to remark that
actually both b and b are independent of the definition of α, while higher order
coefficients do depend on that. Here is the simple proof. Two different perturbative
definitions of α are related by α ∼ α(1 + c 1 α + . . .). Then we have:
β(α
) =
dα
d log μ 2 =
dα
d log μ 2 (1 + 2c 1 α + . . .)
= ±bα
2 (1 + b
α + . . .)(1 + 2c 1 α + . . .)
= ±bα
(1 + b
α
+ . . .)
(4.49)
which shows that, up to the first subleading order, β(α ) has the same form as β(α).
In QCD (N C = 3) one has calculated:
b
=
153 − 19n f
2π(33 − 2n f )
(4.50)
By taking b into account one can write the expression of the running coupling at
next to the leading order (NLO):
α(Q
2 ) = α LO (Q
2 )[1 − b
α LO (Q
2 ) log log
Q 2
2 + . . .]
(4.51)
where α
−1
LO = b log Q 2 // 2 is the LO result (actually at NLO the definition of is
modified according to b log μ 2 // 2 = 1/α + b log bα).
Summarizing, we started from massless classical QCD which is scale invariant.
But we have seen that the procedure of quantisation, regularisation and renormalisation necessarily breaks scale invariance. In the quantum QCD theory there is
a scale of energy, , which from experiment is of the order of a few hundred
MeV, its precise value depending on the definition, as we shall see in detail.
Dimensionless quantities depend on the energy scale through the running coupling
which is a logarithmic function of Q 2 // 2 . In QCD the running coupling decreases
G. Altarelli and S. Forte
Note the presence of a pole in Eqs. (4.46, 4.47) at ±bαt = 1, called the Landau
pole, who realised its existence in QED already in the ‘50’s. For μ ∼ m e (in QED)
the pole occurs beyond the Planck mass. In QCD the Landau pole is located for
negative t or at Q < μ in the region of light hadron masses. Clearly the issue of the
definition and the behaviour of the physical coupling (which is always finite, when
defined in terms of some physical process) in the region around the perturbative
Landau pole is a problem that lies outside the domain of perturbative QCD.
The non leading terms in the asymptotic behaviour of the running coupling can in
principle be evaluated going back to Eq. (4.36) and computing b at 2-loops and so
on. But in general the perturbative coefficients of β(α) depend on the definition of
the renormalised coupling α (the renormalisation scheme), so one wonders whether
it is worthwhile to do a complicated calculation to get b if then it must be repeated
for a different definition or scheme. In this respect it is interesting to remark that
actually both b and b are independent of the definition of α, while higher order
coefficients do depend on that. Here is the simple proof. Two different perturbative
definitions of α are related by α ∼ α(1 + c 1 α + . . .). Then we have:
β(α
) =
dα
d log μ 2 =
dα
d log μ 2 (1 + 2c 1 α + . . .)
= ±bα
2 (1 + b
α + . . .)(1 + 2c 1 α + . . .)
= ±bα
(1 + b
α
+ . . .)
(4.49)
which shows that, up to the first subleading order, β(α ) has the same form as β(α).
In QCD (N C = 3) one has calculated:
b
=
153 − 19n f
2π(33 − 2n f )
(4.50)
By taking b into account one can write the expression of the running coupling at
next to the leading order (NLO):
α(Q
2 ) = α LO (Q
2 )[1 − b
α LO (Q
2 ) log log
Q 2
2 + . . .]
(4.51)
where α
−1
LO = b log Q 2 // 2 is the LO result (actually at NLO the definition of is
modified according to b log μ 2 // 2 = 1/α + b log bα).
Summarizing, we started from massless classical QCD which is scale invariant.
But we have seen that the procedure of quantisation, regularisation and renormalisation necessarily breaks scale invariance. In the quantum QCD theory there is
a scale of energy, , which from experiment is of the order of a few hundred
MeV, its precise value depending on the definition, as we shall see in detail.
Dimensionless quantities depend on the energy scale through the running coupling
which is a logarithmic function of Q 2 // 2 . In QCD the running coupling decreases