CERN, EP Department, CH-1211 Geneva 23, Switzerland
The current world-average of the strong coupling at the Z pole mass, , is obtained from a comparison of perturbative QCD calculations computed, at least, at next-to-next-to-leading-order accuracy, to a set of 6 groups of experimental observables: (i) lattice QCD “data”, (ii) hadronic decays, (iii) proton structure functions, (iv) event shapes and jet rates in collisions, (v) Z boson hadronic decays, and (vi) top-quark cross sections in p-p collisions. In addition, at least 8 other extractions, usually with a lower level of theoretical and/or experimental accuracy today, have been proposed: pion, , W hadronic decays; soft and hard fragmentation functions; jets cross sections in pp, e-p and -p collisions; and photon F structure function in collisions. These 14 determinations are reviewed, and the perspectives of reduction of their present uncertainties are discussed.
The strong coupling , one of the fundamental parameters of the Standard Model, sets the
scale of the strength of the strong interaction between quarks and gluons, theoretically described by
Quantum Chromodynamics (QCD) .
Its current value at the reference Z pole mass amounts to = 0.1186 0.0013,
with a 1% uncertainty—orders of magnitude larger than that of the
gravitational (), Fermi (), and QED () couplings, making of the least
precisely known of all fundamental constants in nature. Improving our knowledge of is a
prerequisite to reduce the theoretical uncertainties in the calculations of all high-precision
perturbative QCD (pQCD) observables whose cross sections or decay rates depend on higher-order powers
of , as is the case for virtually all those accessible at the LHC. Chiefly, in the Higgs sector, the
uncertainty is currently the second major contributor (after the bottom mass) to the parametric
uncertainties of its dominant partial decay, and it’s the leading one for the
branching fractions. The running impacts also our understanding
of physics approaching the Planck scale, e.g. the stability of the electroweak vacuum
or the scale at which the interaction couplings unify.
The latest update of the Particle-Data-Group (PDG) world-average , obtained from a comparison of next-to-next-to-leading-order (NNLO) pQCD calculations to a set of 6 groups of experimental observables, has resulted in a factor of two increase in the uncertainty, compared to the previous (2014) PDG value . This fact calls for new independent approaches to determine from the data, with experimental and theoretical uncertainties different from those of the methods currently used, in order to reduce the overall uncertainty of the world-average. These proceedings provide a summary of all the determination methods described in detail in refs. where more complete lists of references can be found.
2 Current world average
The six methods used in the latest global extraction are shown in Fig. 2 (left, and top-right) roughly listed by increasing energy scale :
The comparison of NNLO pQCD predictions to computational lattice QCD “data” (Wilson loops, quark potentials, vacuum polarization,..) yields , and provides the most precise extraction today. Its 1% uncertainty (dominated by finite lattice spacing and statistics) has, however, doubled since the previous PDG pre-average due to a new calculation of the QCD static energy which is lower than the rest of lattice-QCD analyses. The expected improvements in computing power over the next 10 years would reduce the uncertainty down to 0.3%. Further reduction to the 0.1% level requires the computation of 4th-order pQCD corrections.
The ratio of hadronic to leptonic tau decays, known experimentally to within (), compared to pQCD at next-to-NNLO (NLO) accuracy, yields = 0.1192 0.0018, i.e. = 1.5%. This uncertainty has slightly increased (from ) compared to the previous PDG revision to cover the different results obtained by various pQCD approaches (FOPT vs. CIPT, with different treatments of non-pQCD corrections) . High-statistics spectral functions (e.g. from B-factories, or ILC/FCC-ee in the future) and solving CIPT–FOPT discrepancies (and/or NLO calculations, within a 10 years time scale) are needed to bring uncertainties below 1%.
The QCD coupling has been obtained from various analyses of proton structure functions (including NLO fits of , as well as global (approximately) NNLO fits of PDFs) yielding a central value lower than the rest of methods: = 0.1156 0.0023, with a moderate precision (slightly increased from the previous , driven by the spread of different theoretical extractions). Resolving the differences among fits, and/or full-NNLO global fits of DIS+hadronic data (including consistent treatment of heavy-quark masses) would yield an extraction with 1% uncertainty. Ultimate uncertainties in the 0.15% range require large-statistics studies at a future DIS machine (such as LHeC or FCC-eh) .
Combining the LEP data on event shapes and rates (thrust, C-parameter, N-jet cross sections) with NLO computations (matched, in some cases, with soft and collinear resummations at NLL accuracy), one obtains = 0.1169 0.0034. The = 2.9% uncertainty is mostly driven by the span of individual extractions which use different (Monte Carlo or more analytical) approaches to correct for hadronization effects. Reduction of the non-pQCD uncertainties, e.g. through new jet data at lower (higher) for the event shapes (jet rates), plus jet cross sections with improved resummation (beyond NLL), are needed to reach uncertainties below 1%.
Three closely-related Z hadronic decays observables measured at LEP (, , and ) compared to NLO calculations, yield with 2.5%. Uncertainties at the permil level will require high-precision and large-statistics measurements accessible e.g. with 10 Z bosons at the FCC-ee (and associated 5-loop calculations, with reduced parametric uncertainties).
Top-pair cross sections, theoretically known at NNLO+NNLL, are the first hadron collider measurements that constrain at NNLO accuracy. From the comparison of CMS data to pQCD, one obtains with a = 2.5% uncertainty (mostly dominated by the gluon PDF uncertainties) . Preliminary combination of all measurements at LHC and Tevatron increases its value to .
3 Other extractions
There exist at least 8 other classes of observables, often computed at a lower accuracy (NLO, or approximately-NNLO, aka. NNLO*), used to determine the QCD coupling (Fig. 2 right, bottom), but not yet included in the world-average. Ordered by their energy scale, those are:
The pion decay factor ( MeV) has been used to extract = 0.1174 0.0017. Although the calculation is (“optimized”) NNLO, the low scales involved challenge the validity of the pQCD approach.
The jet-energy dependence of the soft (low-) parton-to-hadron fragmentation functions (FF), provides = 0.1205 0.0022 at NNLO*+NNLL accuracy, with a 2% uncertainty , which could be halved including full-NNLO corrections.
- measurements of the photon structure function F have been used to obtain = 0.1198 0.0054 at NLO , with 4.5%. Extension to NNLO (and inclusion of new B-factories data) would reduce this uncertainty to 2%.
The decay ratio (with X = light hadrons) has been computed at NLO accuracy in the NRQCD framework. From the CLEO data one obtains = 0.119 0.007, with a 6%, uncertainty shared equally by experimental and theoretical systematics . NNLO corrections with improved long-distance matrix elements, and more precise measurements of the spectrum (and of the parton-to-photon FF) would allow for an extraction with 2% in a few years from now.
From the scaling violations of the hard (high-) parton-to-hadron FFs one extracts = 0.1176 0.0055 at NLO, with 5% uncertainties, mostly of experimental origin . Extension of the global FF fits at NNLO accuracy, and inclusion of new datasets (already available at B-factories) would allow reaching 2%.
The NNLO calculation of jet cross sections in DIS and photoproduction provides = 0.120 0.004 with 3% precision today . Upcoming full-NNLO analyses could reduce this uncertainty to the 1.5% level, whereas a future DIS machine (such as LHeC or FCC-eh) would further bring it below 1%.
Measurements of W hadronic decays, although computed at NLO, provide today a very imprecise = 0.117 0.030 with 25% uncertainty, due to the poor LEP data . A competitive extraction requires statistical samples of 10 W, available at FCC-ee, which (combined with NLO corrections) can ultimately yield 0.1%.
Various jet observables at hadron colliders (ratio of 3- to 2-jets, 3-jet mass, inclusive cross sections) have tested asymptotic freedom at TeV scales. Combining those, one obtains = 0.1179 0.0023 at NLO accuracy, with 2% dominated by theoretical uncertainties. The imminent incorporation of NNLO corrections and a consistent combination (including correlations) of the multiple datasets available at Tevatron and LHC, may reduce the uncertainties to the 1.5% level in the upcoming years.
Assuming all 14 extraction methods discussed here are computed at NNLO (or above) accuracy, and
provided that they yield consistent results, a simple weighted-average would have an uncertainty of
0.35%, 3 times better than the present value.
A permil-level uncertainty requires high-precision future colliders with very large Z and W samples, complemented with
4-order pQCD corrections, and improved parametric uncertainties.
Acknowledgments I am grateful to S. Bethke and G. Salam for useful discussions, and to R. Pérez-Ramos and M. Srebre for common work in two of the new extractions reported here.
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