Review
Precision physics with inclusive QCD processes

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Abstract

The inclusive production of hadrons through electroweak currents can be rigorously analysed with short-distance theoretical tools. The associated observables are insensitive to the involved infrared behaviour of the strong interaction, allowing for very precise tests of Quantum Chromodynamics. The theoretical predictions for σ(e+ehadrons) and the hadronic decay widths of the τ lepton and the Z, W and Higgs bosons have reached an impressive accuracy of O(αs4). Precise experimental measurements of the Z and τ hadronic widths have made possible the accurate determination of the strong coupling at two very different energy scales, providing a highly significant experimental verification of asymptotic freedom. A detailed discussion of the theoretical description of these processes and their current phenomenological status is presented. The most precise determinations of αs from other sources are also briefly reviewed and compared with the fully-inclusive results.

Introduction

Quantum Chromodynamics (QCD) [1], [2] provides a successful description of the strong interaction in terms of a single parameter: the strong coupling constant αs. This beautiful gauge theory has been precisely tested in may different processes and over a very broad range of mass scales. Although many aspects of the hadronic world need still to be better understood, the overwhelming consistency of all experimental results has established beyond any doubt that QCD is the right dynamical theory of the strong force.

At low energies, the growing of the effective running coupling generates a complicated non-perturbative regime, responsible for the hadronization of quarks and gluons into a rich variety of colour-singlet composite particles. A precise quantitative description of the hadron formation and dynamics remains unfortunately as an important open problem, which so far has been only partially approached through effective field theory descriptions and numerical tools. In fact, a complete analytical proof of confinement has not yet been accomplished, in spite of the many efforts performed along the years. Nevertheless, all theoretical studies and the large amount of data accumulated indicate that confinement is a truly fundamental property of QCD. Colourful objects have never been observed as asymptotic states.

Assuming that confinement is exact, one can perform very precise predictions for the inclusive production of hadrons in processes that do not contain strongly-interacting particles in the initial state, such as e+ehadrons, Zhadrons, W±hadrons, τντ+hadrons or Hhadrons. Since the separate identity of the produced hadrons is not specified, one just needs to compute the total production of quarks and gluons, summing over all possible configurations. Confinement guarantees that the computed QCD cross section or decay width will be identical to the corresponding inclusive hadronic production because the total probability that quarks and gluons hadronize is just one.

Pure perturbative calculations are usually enough to achieve accurate descriptions of high-energy inclusive processes. At low energies they need to be complemented with non-perturbative corrections that scale as powers of ΛQCD2nsn, starting with n=2. Using short-distance operator-product-expansion (OPE) [3] techniques, one can control rigorously these power corrections and determine above which scales their numerical impact becomes negligible.

The following sections present a detailed discussion of the theoretical tools involved in the analysis of inclusive processes and the current status of the resulting predictions. The running QCD coupling and quark masses are introduced in Section 2, which describes their associated β and γ functions that are currently known to O(αs5). Section 3 discusses the two-point correlation functions of the vector, axial-vector, scalar and pseudoscalar QCD currents, summarizing our present O(αs4) knowledge of these important dynamical objects. The inclusive high-energy observables are analysed in Section 4, which contains the QCD predictions for the e+e annihilation cross section into hadrons, and the hadronic widths of the electroweak Z, W and Higgs bosons. Section 5 reviews the theoretical analysis of the τ hadronic width, including perturbative and non-perturbative contributions, and updates its current phenomenological status. The highly-precise four-loop determinations of the strong coupling from the Z and τ hadronic widths are compared in Section 6 with the most accurate values of αs extracted from other sources, exhibiting the great success of QCD in correctly describing strong-interacting phenomena over a very broad range of energy scales. A few summarizing comments are finally given in Section 7. Some complementary technical details are compiled in appendices.

Section snippets

The QCD running coupling

The unique coupling constant of QCD, αsgs2(4π), is obviously the critical parameter governing all phenomena associated with the strong interaction. The renormalized coupling αs(μ2) depends on the chosen renormalization scheme and scale. The MS¯ scheme [4] is the conventionally adopted choice, while the dependence on the scale μ is determined by the renormalization-group equation μdαsdμ=αsβ(αs),β(αs)=n=1βnαsπn,which defines the so-called β function.1

Current correlators

Inclusive observables, such as σ(e+ehadrons), Γ(Zhadrons) or Γ(Whadrons) proceed through the colour-singlet vector Vijμ=q̄jγμqi  and axial-vector Aijμ=q̄jγμγ5qi  quark currents (i,j=u,d,s). The QCD dynamics is then encoded in the two-point correlation functions Πij,Jμν(q)id4xeiqx0|T(Jijμ(x)Jijν(0))|0=gμνq2+qμqνΠij,JT(q2)+qμqνΠij,JL(q2),where J=V,A and the superscript in the transverse and longitudinal components denotes the corresponding angular momentum J=1 (T) and J=0 (L) in the

Inclusive observables

At high energies, naive perturbation theory is usually adopted to predict quantities such as Re+e(s) and the hadronic decay widths of the Z, W and Higgs bosons. These observables correspond to hadronic spectral functions (correlator discontinuities) on the physical cut, where the OPE is not justified. However, non-perturbative effects are still assumed to be suppressed by the factor (ΛQCDs)4, provided the physical scale s is large enough and far away from thresholds and hadronic resonance

The hadronic width of the τ lepton

The τ lepton decays through the emission of a virtual W boson that generates four possible ντX final states, with X=eν̄e,μν̄μ,dū,sū (Fig. 7). If final fermion masses and QCD effects are ignored, the universality of the W couplings implies that the four decay modes have equal probabilities, except for an additional global factor NC|Vui|2 (i=d,s) in the two semileptonic channels. Since |Vud|2+|Vus|2=1|Vub|21, the total τ hadronic width is then predicted to be a factor of NC=3 larger

NNLO determinations of the strong coupling

The inclusive Z and τ hadronic widths provide a very important test of the Standard Model at an impressive N3LO precision, where LO refers to the first nontrivial QCD contribution. The strong coupling is determined in two completely different energy regimes and with very different experimental systematics, but the theoretical description of the two observables is based on similar current correlators, being the four-loop calculation of the Adler function the basic ingredient in both cases. The

Summary

A series of impressive four- and five-loop calculations has promoted the phenomenology of inclusive QCD processes into the realm of precision physics. The very accurate knowledge of the β and γ functions, which provides a powerful resummation of logarithmic corrections into the running coupling and quark masses, has been complemented with the O(αs4) computation of the two-point correlation functions of the vector, axial-vector, scalar and pseudoscalar currents, allowing us to obtain

Acknowledgments

This work has been supported in part by the Spanish Government and ERDF funds from the EU Commission [Grant FPA2017-84445-P], by the Generalitat Valenciana [Grant Prometeo/2017/053], by the EU H2020 research and innovation programme [Grant Agreement 824093] and by the EU COST Action CA16201 PARTICLEFACE .

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