We have discussed in the previous part several of the most popular QCD non-perturbative methods other than the QCD spectral sum rules (QSSR). Now, we shall dedicate this part of the book to the discussion of this non-perturbative approach, which has been used successfully for understanding the hadron properties and hadronic matrix elements, using those parameters (QCD coupling, quark masses and QCD condensates), derived from QCD first principles. This method was introduced by SVZ in 1979 [1] and reviewed in a book [3], numerous reviews and lecture notes [356–365]. Its basic concepts, based on the operator product expansion and dispersion relations, are well understood in quantum field theory, and is a fully relativistic approach in contrast to potential models, for example. Its applications are quite simple and transparent. However, on the one hand it has a limited accuracy (usually about 10–20% depending on the process), and some uses of the method in some QCD-like models show that its accuracy cannot be improved iteratively. On the other hand, confinement is not a result of the method but is put into it via the introduction of different QCD condensates. In practice, one has to introduce some assumptions, and the results are obtained from self-consistency.

However, in some cases, results obtained from the sum rules disagree with each other and have led to some polemics, which some people use to discredit the approach.

Introduction

Over a decade, a lot of experimental informations on heavy-quark decays and masses have been obtained from e+e– and hadron collider experiments. These have led to a detailed knowledge of the flavour sector of the standard model and to the discoveries of the B0 – 0 mixing, rare decays induced by penguin operators, … The experimental progress in the heavy flavour physics has been accompanied by some theoretical progress. Among other approaches, the discovery of the heavy-quark symmetry has led to the development of the heavy quark effective theory (HQET), which provides a systematic analysis of the properties of a hadron containing a heavy quark in terms of an expansion of the inverse of the heavy quark mass. Detailed discussions and references to the original works can be found in different reviews and lectures (see e.g. [545]).

Heavy-quark symmetry

When the mass of the heavy quark is much larger than the QCD scale ΛQCD, the QCD running coupling αs(mQ) is small, implying that at this scale of the order of the Compton wavelength λQ ∼ 1/mQ, one can safely use perturbative QCD for describing the hadrons. In this case the Q-bound states with the size λQ/αs(mQ) ≪ Rhad ∼ 1 fermi are like the hydrogen atom.

The anatomy of the SVZ expansion

For definiteness, let us illustrate our discussion from the generic two-point correlator:

where JH(x) is the hadronic current of quark and/or gluon fields. Here, the analysis is in principle much simpler than in the case of deep inelstic scatterings, because one has to sandwich the T-product of currents between the vacuum rather than between two proton states. Following SVZ [1], the breaking of ordinary perturbation theory at low q2 is due to the manifestation of non-perturbative terms appearing as power corrections in the operator product expansion (OPE) of the Green function à la Wilson [222]. In this way, one can write:

provided that m2 – q2 ≫ Λ2. For simplicity, m is the heaviest quark mass entering into the correlator; ν is an arbitrary scale that separates the long- and short-distance dynamics; C are theWilson coefficients calculable in perturbative QCD by means of Feynman diagrams techniques; 〈O〉 are the non-perturbative (non-calculable) condensates built from the quarks or/and gluon fields. Though, separately, C and 〈O〉 are (in principle) ν-dependent, this ν-dependence should (in principle) disappear in their product.

• The case D = 0 corresponds to the naïve perturbative contribution.

• […]

We review the present status for the determinations of the light and heavy quark masses, the light quark chiral condensate and the decay constants of light and heavy-light (pseudo)scalar mesons from QCD spectral sum rules (QSSR). Bounds on the light quark running masses at 2 GeV are found to be (see Tables 53.1 and 53.2): 6 MeV < (d + u)(2) < 11 MeV and 71 MeV < s(2) < 148 MeV. The agreement of the ratio ms/(mu + md) = 24.2 in Eq. (53.45) from pseudoscalar sum rules with the one (24.4 ± 1.5) from ChPT indicates the consistency of the pseudoscalar sum rule approach. QSSR predictions from different channels for the light quark running masses lead to (see Section 53.9.3): s(2) = (117.4 ± 23.4) MeV, (;d + u)(2) = (10.1 ± 1.8) MeV, (d – u)(2) = (2.8 ± 0.6) MeV with the corresponding values of the RG invariant masses. The different QSSR predictions for the heavy quark masses lead to the running masss values: c(c) = (1.23 ± 0.05) GeV and b(b) = (4.24 ± 0.06) GeV (see Tables 53.5 and 53.6), from which one can extract the scale independent ratio mb/ms = 48.8 ± 9.8.

Quantum Chromodynamics (QCD) continues to be an active field of research, which one can see from the number of publications in the field, as well as from the number of presentations at different QCD dedicated conferences, such as the regular QCD-Montpellier Conference Series. This continuous activity is due to the relative difficulty in tackling its non-perturbative aspects, although its asymptotic freedom property has facilated perturbative calculations of different hard and jet processes. Therefore, we think it is still useful to write a book on QCD in which, besides the usual pedagogical introduction to the field, some reviews of its modern developments, which have not yet been ‘compiled’ into a book, will be presented. Elementary introductions at the level of pre-Ph.D. in different specialized topics of QCD will be discussed, which may be useful for a future deeper research and for a guide in a given subject.

We start the book with a general elementary introduction to strong interactions, parton and quark models, …, and present the basic tools for understanding QCD as a gauge field theory (renormalization, operator product expansion, …). After, we present the usual hard processes (deep inelastic scattering, jets, …) calculable in perturbative QCD, and discuss the resummation (renormalons, …) of the perturbative series. Later, we discuss the different modern non-perturbative aspects of QCD (lattice, effective theories, …).

The QCD phases

We study here the uses of QCD spectral sum rules (QSSR) in a matter with non-zero temperature T and non-zero chemical potential ν (so-called Quark-Gluon Plasma (QGP)). Since the corresponding critical temperature for the colour deconfinement is expected to be rather small (Tc ≤ 1 GeV), these new states of matter can be investigated in high-energy hadron collisions. At high enough temperature T ≥ Tc ≈ 150 – 200 MeV corresponding to a vacuum pressure of about 500 MeV/fm3, QGP phase occurs and can be understood without confinement. In this phase, one also expects that chiral symmetry is restored (chiral symmetry restoration). However, it is a priori unclear, if the QGP phase and the chiral symmetry restoration occurs at the same temperature or not. Untuitively, one can expect that the deconfinement phase occurs before the chiral symmetry restoration. An attempt to show that the two phases are reached at the same temperature has been made in [839] using the FESR version of the Weinberg sum rules, which we shall discuss later on, where the constraint has been obtained by assuming that in the QGP phase, the continuum threshold starts from zero. In the QGP phase, the thermodynamics of the plasma is governed by the Stefan–Boltzmann law, as in an ordinary black body transition. This feature has been confirmed by a large number of lattice simulations.

In this part, we study different hard and jet processes in e+e–. These concern:

• one hadron inclusive production.

• γγ scatterings and the ‘spin’ of the photon.

• QCD jets.

• heavy quarkonia inclusive decays.

• e+e– → hadrons total cross-section.

• Z → hadrons inclusive decay

• τ → ν + hadrons semi-inclusive decays.

These processes are used as classical tests of perturbative QCD, where values of the running QCD coupling have been extracted. A pedagogical introduction to the physics of e+e– can be found in, for example, the book of [276]. More modern QCD phenomenology in e+e– can be found in different reviews and in the proceedings of the QCD-Montpellier series of conferences and many others.