128
G. Altarelli and S. Forte
In fact there are no dim-2 Lorentz and gauge invariant operators. For example, g μ g μ
is not gauge invariant. In the massive theory, the ZERO is replaced by light quark
mass-squared m 2 . This is still negligible if m is taken as a lagrangian mass of a
few MeV. If on the other hand the mass were taken to be the constituent mass of
order QCD , this term would not be at all negligible and would substantially affect
the result (note that α s (m τ )/π ∼ 0.1 ∼ (0.6 GeV/m τ ) 2 and that QCD for three
flavours is large). For example, the PDG value and estimate of the error is [9]:
α s (m Z ) = 0.120 ± 0.003.
(4.111)
Most people believe the optimistic version. I am not convinced that the gap is
not filled up by ambiguities of 0(( 2
QCD /m 2
τ ) from δ pert [82]. In any case, one
can discuss the error, but it is true and remarkable, that the central value from
τ decay, obtained at very small Q 2 , is in good agreement with all other precise
determinations of α s at more typical LEP values of Q 2 .
Important determinations of α s (m Z ) are obtained from different infrared safe
observables related to event rates and jet shapes in e + e − annihilation. The main
problem of these measurements is the large impact of non perturbative hadronization
effects on the result and therefore on the theoretical error. The perturbative part is
known at NLO. One advantage is that the same measurements can be repeated at
different
√
s values (e.g. with the same detectors at LEP1 or LEP2) allowing for a
direct observation of the energy dependence. A typical result, from jets and event
shapes at LEP, quoted in Ref. [83], is given by:
α s (m Z ) = 0.121 ± 0.005.
(4.112)
Recently the rate of 4-jet events (proportional to α 2
s ) at LEP as function of y cut has
been used [84], for which a NLO theoretical calculation exists [85]. The quoted
result is α s (m Z ) = 0.1176 ± 0.0022 (the actual error could be somewhat larger
because the ambiguity from hadronisation modeling is always debatable).
4.6.2 α s from Deep Inelastic Scattering
QCD predicts the Q 2 dependence of F (x, Q 2 ) at each fixed x, not the x shape. But
the Q 2 dependence is related to the x shape by the QCD evolution equations. For
each x-bin the data allow to extract the slope of an approximately straight line in
dlogF (x, Q 2 )/dlogQ 2 : the log slope. The Q 2 span and the precision of the data
are not much sensitive to the curvature, for most x values. A single value of QCD
must be fitted to reproduce the collection of the log slopes. For the determination of
α s the scaling violations of non-singlet structure functions would be ideal, because
G. Altarelli and S. Forte
In fact there are no dim-2 Lorentz and gauge invariant operators. For example, g μ g μ
is not gauge invariant. In the massive theory, the ZERO is replaced by light quark
mass-squared m 2 . This is still negligible if m is taken as a lagrangian mass of a
few MeV. If on the other hand the mass were taken to be the constituent mass of
order QCD , this term would not be at all negligible and would substantially affect
the result (note that α s (m τ )/π ∼ 0.1 ∼ (0.6 GeV/m τ ) 2 and that QCD for three
flavours is large). For example, the PDG value and estimate of the error is [9]:
α s (m Z ) = 0.120 ± 0.003.
(4.111)
Most people believe the optimistic version. I am not convinced that the gap is
not filled up by ambiguities of 0(( 2
QCD /m 2
τ ) from δ pert [82]. In any case, one
can discuss the error, but it is true and remarkable, that the central value from
τ decay, obtained at very small Q 2 , is in good agreement with all other precise
determinations of α s at more typical LEP values of Q 2 .
Important determinations of α s (m Z ) are obtained from different infrared safe
observables related to event rates and jet shapes in e + e − annihilation. The main
problem of these measurements is the large impact of non perturbative hadronization
effects on the result and therefore on the theoretical error. The perturbative part is
known at NLO. One advantage is that the same measurements can be repeated at
different
√
s values (e.g. with the same detectors at LEP1 or LEP2) allowing for a
direct observation of the energy dependence. A typical result, from jets and event
shapes at LEP, quoted in Ref. [83], is given by:
α s (m Z ) = 0.121 ± 0.005.
(4.112)
Recently the rate of 4-jet events (proportional to α 2
s ) at LEP as function of y cut has
been used [84], for which a NLO theoretical calculation exists [85]. The quoted
result is α s (m Z ) = 0.1176 ± 0.0022 (the actual error could be somewhat larger
because the ambiguity from hadronisation modeling is always debatable).
4.6.2 α s from Deep Inelastic Scattering
QCD predicts the Q 2 dependence of F (x, Q 2 ) at each fixed x, not the x shape. But
the Q 2 dependence is related to the x shape by the QCD evolution equations. For
each x-bin the data allow to extract the slope of an approximately straight line in
dlogF (x, Q 2 )/dlogQ 2 : the log slope. The Q 2 span and the precision of the data
are not much sensitive to the curvature, for most x values. A single value of QCD
must be fitted to reproduce the collection of the log slopes. For the determination of
α s the scaling violations of non-singlet structure functions would be ideal, because