130
G. Altarelli and S. Forte
In both analyses the dominant error is theoretical and, in my opinion, should be
somewhat larger than quoted. An interesting perspective on theoretical errors can
be obtained by comparing analyses with different methods. We add the following
examples. From truncated moments (but with a limited set of proton data and NLO
kernels) [91]: α s (m Z ) = 0.122 ±0.006, from Nachtmann moments (which take into
account some higher twist terms) and proton data [92]: α s (m Z ) = 0.1188 ± 0.0017.
A combination of measurements at HERA by H1 and Zeus, also including final state
jet observables, leads to α s (m Z ) = 0.1186 ± 0.0051 [93], most of the error being
theoretical. Finally, to quote a number that appears to me as a good summary of
the situation of α s (m Z ) from DIS one can take the result from a NNLO analysis of
available data by the MRST group [94] as quoted by Particle Data Group, W.-M.
Yao et al. [9]:
α s (m Z ) = 0.1167 ± 0.0036
(4.118)
If we compare these results on α s from DIS with the findings at the Z, given by
Eq. (4.105), we see that the agreement is good, with the value of α s from the most
precise DIS measurements a bit on the low side with respect to e + e − .
4.6.3 Summary on α s
There are a number of other determinations of α s which are important because they
arise from qualitatively different observables and methods. For example [9, 83],
some are obtained from the Bjorken sum rule and the scaling violations in polarized
DIS, from ϒ decays, from the 4-jet rate in e + e − . A special mention deserves the
“measurement” of α s from lattice QCD [95]. A number of hadronic observables,
in particular ϒ − ϒ splitting, pion and kaon decay constants, the B s mass and the
baryon mass are used to fix the lattice spacing and to accurately tune the QCD
simulation. The value of α s is then obtained by computing non perturbatively a
number of quantities related to Wilson loops that can also be given in perturbation
theory. The result is then evolved with state of the art beta functions to obtain
α s (m Z ) = 0.1170 ± 0.0012. This result is interesting for its really special nature
but it is not clear that the systematics due to the lattice technology is as small as
claimed.
Summarising: there is very good agreement among many different measurements
of α s . In Fig. 4.22 [83], a compilation of the data is reported with each measurement
plotted at the scale of the experiment, which shows the consistency of the measurements and the running of α s . This is a very convincing, quantitative test of QCD.
If I take the values of α s (m Z ) from precision electroweak data, Eq. (4.105), from τ
decay with the central value as in Eq. (4.109) but the larger error as in Eq. (4.111),
from jets in e + e − , Eq. (4.112), and from DIS, Eq. (4.118), the average is :
α s (m Z ) = 0.119 ± 0.002
(4.119)
G. Altarelli and S. Forte
In both analyses the dominant error is theoretical and, in my opinion, should be
somewhat larger than quoted. An interesting perspective on theoretical errors can
be obtained by comparing analyses with different methods. We add the following
examples. From truncated moments (but with a limited set of proton data and NLO
kernels) [91]: α s (m Z ) = 0.122 ±0.006, from Nachtmann moments (which take into
account some higher twist terms) and proton data [92]: α s (m Z ) = 0.1188 ± 0.0017.
A combination of measurements at HERA by H1 and Zeus, also including final state
jet observables, leads to α s (m Z ) = 0.1186 ± 0.0051 [93], most of the error being
theoretical. Finally, to quote a number that appears to me as a good summary of
the situation of α s (m Z ) from DIS one can take the result from a NNLO analysis of
available data by the MRST group [94] as quoted by Particle Data Group, W.-M.
Yao et al. [9]:
α s (m Z ) = 0.1167 ± 0.0036
(4.118)
If we compare these results on α s from DIS with the findings at the Z, given by
Eq. (4.105), we see that the agreement is good, with the value of α s from the most
precise DIS measurements a bit on the low side with respect to e + e − .
4.6.3 Summary on α s
There are a number of other determinations of α s which are important because they
arise from qualitatively different observables and methods. For example [9, 83],
some are obtained from the Bjorken sum rule and the scaling violations in polarized
DIS, from ϒ decays, from the 4-jet rate in e + e − . A special mention deserves the
“measurement” of α s from lattice QCD [95]. A number of hadronic observables,
in particular ϒ − ϒ splitting, pion and kaon decay constants, the B s mass and the
baryon mass are used to fix the lattice spacing and to accurately tune the QCD
simulation. The value of α s is then obtained by computing non perturbatively a
number of quantities related to Wilson loops that can also be given in perturbation
theory. The result is then evolved with state of the art beta functions to obtain
α s (m Z ) = 0.1170 ± 0.0012. This result is interesting for its really special nature
but it is not clear that the systematics due to the lattice technology is as small as
claimed.
Summarising: there is very good agreement among many different measurements
of α s . In Fig. 4.22 [83], a compilation of the data is reported with each measurement
plotted at the scale of the experiment, which shows the consistency of the measurements and the running of α s . This is a very convincing, quantitative test of QCD.
If I take the values of α s (m Z ) from precision electroweak data, Eq. (4.105), from τ
decay with the central value as in Eq. (4.109) but the larger error as in Eq. (4.111),
from jets in e + e − , Eq. (4.112), and from DIS, Eq. (4.118), the average is :
α s (m Z ) = 0.119 ± 0.002
(4.119)