QCD: The Theory of Strong Interactions
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Abstract
This chapter is devoted to a concise introduction to quantum chromodynamics (QCD), the theory of strong interactions (GellMann, Acta Phys Austriaca (suppl. IX):733, 1972; Gross and Wilczek, Phys Rev Lett 30:1343, 1973; Weinberg, Phys Rev Lett 31:494, 1973) [for a number of dedicated books on QCD, see Dokshitzer and Khoze (In: Frontieres (ed) Basics of perturbative QCD, 1991), and also Altarelli (QCD: the theory of strong interactions. In: LandoltBoernstein I 21A: Elementary particles 4. Springer, Berlin, 2008)]. The main emphasis will be on ideas without too many technicalities. As an introduction we present here a broad overview of the strong interactions [for reviews of the subject, see, for example, Altarelli (Phys Rep 81:1, 1982) and Altarelli (Ann Rev Nucl Part Sci 39:357, 1989)]. Then some methods of nonperturbative QCD will be briefly described, including both analytic approaches and simulations of the theory on a discrete spacetime lattice. Then we shall proceed to the main focus of the chapter, that is, the principles and applications of perturbative QCD, which will be discussed in detail.
Keywords
Deep Inelastic Scattering Renormalization Group Equation Splitting Function Parton Density Gluon Density2.1 Introduction
This chapter is devoted to a concise introduction to quantum chromodynamics (QCD), the theory of strong interactions [215, 234, 360] (for a number of dedicated books on QCD, see [173], and also [33]). The main emphasis will be on ideas without too many technicalities. As an introduction we present here a broad overview of the strong interactions (for reviews of the subject, see, for example, [29, 30]). Then some methods of nonperturbative QCD will be briefly described, including both analytic approaches and simulations of the theory on a discrete spacetime lattice. Then we shall proceed to the main focus of the chapter, that is, the principles and applications of perturbative QCD, which will be discussed in detail.

The group must admit complex representations because it must be able to distinguish a quark from an antiquark [214]. In fact, there are meson states made up of \(q\bar{q}\) but not analogous qq bound states. Among simple groups, this restricts the choice to SU(N) with N ≥ 3, SO(4N + 2) with \(N \geq 2\ \big[\) taking into account the fact that SO(6) has the same algebra as \(SU(4)\big]\), and E(6).

The group must admit a completely antisymmetric colour singlet baryon made up of three quarks, viz., qqq. In fact, from the study of hadron spectroscopy, we know that the lowlying baryons, completing an octet and a decuplet of (flavour) SU(3) (the approximate symmetry that rotates the three light quarks u, d, and s), are made up of three quarks and are colour singlets. The qqq wave function must be completely antisymmetric in colour in order to agree with Fermi statistics. Indeed, if we consider, for example, a N^{∗++} with spin zcomponent +3/2, this is made up of (u ↑ u ↑ u ↑) in an sstate. Thus its wave function is totally symmetric in space, spin, and flavour, so that complete antisymmetry in colour is required by Fermi statistics. In QCD this requirement is very simply satisfied by ε_{ abc }q^{ a }q^{ b }q^{ c }, where a, b, c are SU(3)_{colour} indices.

The choice of SU(N_{C} = 3)_{colour} is confirmed by many processes that directly measure N_{C}. Some examples are listed here.
There are many more experimental confirmations that N_{C} = 3. For example, the rate for Drell–Yan processes (see Sect. 2.9) is inversely proportional to N_{C}.
2.2 Nonperturbative QCD
The QCD Lagrangian in ( 1.28) has a simple structure, but a very rich dynamical content. It gives rise to a complex spectrum of hadrons, implies the striking properties of confinement and asymptotic freedom, is endowed with an approximate chiral symmetry which is spontaneously broken, has a highly nontrivial topological vacuum structure (instantons, U(1)_{A} symmetry breaking, strong CP violation which is a problematic item in QCD possibly connected with new physics, like axions, and so on), and an intriguing phase transition diagram (colour deconfinement, quark–gluon plasma, chiral symmetry restoration, colour superconductivity, and so on).
How do we get testable predictions from QCD? On the one hand there are nonperturbative methods. The most important at present is the technique of lattice simulations (for a recent review, see [272]): it is based on first principles, it has produced very valuable results on confinement, phase transitions, bound states, hadronic matrix elements, and so on, and it is by now an established basic tool. The main limitation is from computing power, so there is continuous progress and good prospects for the future.
Another class of approaches is based on effective Lagrangians, which provide simpler approximations than the full theory, valid in some definite domain of physical conditions. Typically at energies below a given scale L, particles with mass greater than L cannot be produced, and thus only contribute short distance effects as virtual states in loops. Under suitable conditions one can write down a simplified effective Lagrangian, where the heavy fields have been eliminated (one says “integrated out”). Virtual heavy particle short distance effects are absorbed into the coefficients of the various operators in the effective Lagrangian. These coefficients are determined in a matching procedure, by requiring that the effective theory reproduce the matrix elements of the full theory up to power corrections.
Chiral Lagrangians are based on soft pion theorems [362] and are valid for suitable processes at energies below 1 GeV (for a recent, concise review, see [212] and references therein). Heavy quark effective theories [178] are obtained by expanding in inverse powers of the heavy quark mass and are mainly important for the study of b and, to lesser accuracy, c decays (for reviews, see, for example, [301]).
Softcollinear effective theories (SCET) [84], are valid for processes where quarks have energies much greater than their mass. Light energetic quarks not only emit soft gluons, but also collinear gluons (a gluon in the same direction as the original quark), without changing their virtuality. In SCET, the logs associated with these soft and collinear gluons are resummed.
The approach using QCD sum rules [298, 325] has led to interesting results but now appears not to offer much potential for further development. On the other hand, the perturbative approach, based on asymptotic freedom, still remains the main quantitative connection to experiment, due to its wide range of applicability to all sorts of “hard” processes.
2.2.1 Progress in Lattice QCD
One of the main approaches to nonperturbative problems in QCD is by simulations of the theory on a lattice, a technique initiated by K. Wilson in 1974 [366] which has shown continuous progress over the last decades. In this approach the QCD theory is reformulated on a discrete space time, a hypercubic lattice of sites (in the simplest realizations) with spacing a and 4volume L^{4}. On each side, there are N sites with L = Na. Over the years we have learned how to efficiently describe a field theory on a discrete spacetime and how to implement gauge symmetry, chiral symmetry, and so on (for a recent review see, for example, [272]).
Gauge and matter fields are specified on the lattice sites and the path integral is computed numerically as a sum over the field configurations. Much more powerful computers than in the past now allow for a number of essential improvements. As one is eventually interested in the continuum limit a → 0, it is important to work with as fine a lattice spacing a as possible. Methods have been developed for “improving” the Lagrangian in such a way that the discretization errors vanish faster than linearly in a. A larger lattice volume (i.e., large L or N) is also useful since the dimensions of the lattice should be as large as possible in comparison with the dimensions of the hadrons to be studied. In many cases the volume corrections are exponentially damped, but this is not always the case. Lattice simulation is limited to large enough masses of light quarks: in fact, heavier quarks have shorter wavelengths and can be accommodated in a smaller volume. In general, computations are done for quark and pion masses heavier than in reality, and then extrapolated to the physical values, but at present one can work with smaller quark masses than in the past. One can also take advantage of the chiral effective theory in order to control the chiral logs log(m_{ q }∕4π f_{π}) and guide the extrapolation.
A big step that has been taken recently, made possible by the availability of more powerful dedicated computers, is the evolution from quenched (i.e., with no dynamical fermions) to unquenched calculations. In doing this, an evident improvement is obtained in the agreement between predictions and data. For example [272], modern unquenched simulations reproduce the hadron spectrum quite well. Calculations with dynamical fermions (which take into account the effects of virtual quark loops) involve evaluation of the quark determinant, which is a difficult task. Just how difficult depends on the particular calculation method. There are several approaches (Wilson, twisted mass, Kogut–Susskind staggered, Ginsparg–Wilson fermions), each with its own advantages and disadvantages (including the time it takes to run the simulation on a computer). A compromise between efficiency and theoretical purity is needed. The most reliable lattice calculations are today for 2 + 1 light quarks (degenerate up and down quarks and a heavier strange quark s). The first calculations for 2 + 1 + 1 including charm quarks are starting to appear.
Lattice QCD is becoming increasingly predictive and plays a crucial role in different domains. For example, in flavour physics it is essential for computing the relevant hadronic matrix elements. In high temperature QCD the most illuminating studies of the phase diagram, the critical temperature, and the nature of the phase transitions are obtained by lattice QCD: as we now discuss, the best arguments to prove that QCD implies confinement come from the lattice.
2.2.2 Confinement
The linearly increasing term in the potential makes it energetically impossible to separate a \(q\bar{q}\) pair. If the pair is created at one spacetime point, for example in e^{+}e^{−} annihilation, and then the quark and the antiquark start moving away from each other in the centerofmass frame, it soon becomes energetically favourable to create additional pairs, smoothly distributed in rapidity between the two leading charges, which neutralize colour and allow the final state to be reorganized into two jets of colourless hadrons that communicate in the central region by a number of “wee” hadrons with small energy. It is just like the familiar example of the broken magnet: if you try to isolate a magnetic pole by stretching a dipole, the magnet breaks down and two new poles appear at the breaking point.
Confinement is essential to explain why nuclear forces have very short range while massless gluon exchange would be long range. Nucleons are colour singlets and they cannot exchange colour octet gluons but only colourless states. The lightest colour singlet hadronic particles are pions. So the range of nuclear forces is fixed by the pion mass r ≃ m_{π}^{−1} ≈ 10^{−13} cm, since V ≈ exp(−m_{π}r)∕r.
A large investment is being made in heavy ion collision experiments with the aim of finding some evidence of the quark–gluon plasma phase. Many exciting results have been found at the CERN SPS in the past few years, more recently at RHIC and now at the LHC, in dedicated heavy ion runs [296] (the ALICE detector is especially designed for the study of heavy ion collisions).
2.2.3 Chiral Symmetry in QCD and the Strong CP Problem
In the QCD Lagrangian ( 1.28), the quark mass terms are of the general form [\(m\bar{\psi }_{\mathrm{L}}\psi _{\mathrm{R}} + \mathrm{h.c.}\)] (recall the definition of ψ_{L,R} in Sect. 1.5 and the related discussion). These terms are the only ones that show a chirality flip. In the absence of these terms, i.e., for m = 0, the QCD Lagrangian would be invariant under independent unitary transformations acting separately on ψ_{L} and ψ_{R}. Thus, if the masses of the N_{f} lightest quarks are neglected, the QCD Lagrangian is invariant under a global \(U(N_{\mathrm{f}})_{\mathrm{L}}\bigotimes U(N_{\mathrm{f}})_{\mathrm{R}}\) chiral group.
Consider N_{f} = 2. Then SU(2)_{V} corresponds to the observed approximate isospin symmetry and U(1)_{V} to the portion of baryon number associated with u and d quarks. Since no approximate parity doubling of light quark bound states is observed, the U(2)_{A} symmetry must be spontaneously broken (for example, no opposite parity analogues of protons and neutrons exist with a few tens of MeV separation in mass from the ordinary nucleons). The breaking of chiral symmetry is induced by the VEV of a quark condensate. For N_{f} = 2 this is [\(\bar{u}_{\mathrm{L}}u_{\mathrm{R}} +\bar{ d}_{\mathrm{L}}d_{\mathrm{R}} + \mathrm{h.c.}\)]. A recent lattice calculation [208] has given for this condensate the value [234 ± 18 MeV]^{3} (in \(\overline{\mathrm{MS}}\), N_{f} = 2 + 1, with the physical m_{ s } value, at the scale of 2 GeV). This scalar operator is an isospin singlet, so it preserves U(2)_{V}, but breaks U(2)_{A}. In fact, it transforms like (1/2,1/2) under \(U(2)_{\mathrm{L}}\bigotimes U(2)_{\mathrm{R}}\), but is a singlet under the diagonal group U(2)_{V}.
The pseudoscalar mesons are obvious candidates for the wouldbe Goldstone bosons associated with the breakdown of the axial group, in that they have the quantum number of the broken generators: the three pions are the approximately massless Goldstone bosons (exactly massless in the limit of vanishing u and d quark masses) associated with the breaking of three generators of \(U(2)_{\mathrm{L}}\bigotimes U(2)_{\mathrm{R}}\) down to \(SU(2)_{\mathrm{V}}\bigotimes U(1)_{\mathrm{V}}\bigotimes U(1)_{\mathrm{A}}\). The couplings of Goldstone bosons are very special: in particular only derivative couplings are allowed. The pions as pseudoGoldstone bosons have couplings that satisfy strong constraints. An effective chiral Lagrangian formalism [362] allows one to systematically reproduce the low energy theorems implied by the approximate status of Goldstone particles for the pion, and successfully describes QCD for energies at scales below ∼ 1 GeV.
Alternative solutions to the θproblem have also been suggested. Some of them can probably be discarded (for example, the idea that the up quark is exactly massless), while others are still possible: for example, in supersymmetric theories, if the smallness of θ could be guaranteed at the Planck scale by some feature of the more fundamental theory valid there, then the nonrenormalization theorems of supersymmetry would preserve its small value throughout the transition down to low energy.
2.3 Massless QCD and Scale Invariance
As discussed in Chap. 2, the QCD Lagrangian in ( 1.28) only specifies the theory at the classical level. The procedure for quantizing gauge theories involves a number of complications that arise from the fact that not all degrees of freedom of gauge fields are physical because of the constraints from gauge invariance which can be used to eliminate the dependent variables. This is already true for Abelian theories and one is familiar with the QED case. One introduces a gauge fixing term (an additional term in the Lagrangian density that acts as a Lagrange multiplier in the action extremization). One can choose to preserve manifest Lorentz invariance. In this case, one adopts a covariant gauge, like the Lorentz gauge, and in QED one proceeds according to the formalism of Gupta and Bleuler [102]. Or one can give up explicit formal covariance and work in a noncovariant gauge, like the Coulomb or the axial gauges, and only quantize the physical degrees of freedom (in QED the transverse components of the photon field).
While this is all for an Abelian gauge theory, in the nonAbelian case some additional complications arise, in particular the need to introduce ghosts for the formulation of Feynman rules. As we have seen, there are in general as many ghost fields as gauge bosons, and they appear in the form of a transformation Jacobian in the Feynman functional integral. Ghosts only propagate in closed loops and their vertices with gluons can be included as additional terms in the Lagrangian density, these being fixed once the gauge fixing terms and their infinitesimal gauge transformations are specified. Finally, the complete Feynman rules can be obtained in either the covariant or the axial gauges, and they appear in Fig. 2.1.
Once the Feynman rules are derived, we have a formal perturbative expansion, but loop diagrams generate infinities. First a regularization must be introduced, compatible with gauge symmetry and Lorentz invariance. This is possible in QCD. In principle, one can introduce a cutoff K (with dimensions of energy), for example, as done by Pauli and Villars [102]. But at present, the universally adopted regularization procedure is dimensional regularization, which we will describe briefly later on.
After regularization, the next step is renormalization. In a renormalizable theory (which is the case for all gauge theories in four spacetime dimensions and for QCD in particular), the dependence on the cutoff can be completely reabsorbed in a redefinition of particle masses, gauge coupling(s), and wave function normalizations. Once renormalization is achieved, the perturbative definition of the quantum theory that corresponds to a classical Lagrangian like ( 1.28) is completed.
In the QCD Lagrangian of ( 1.28), quark masses are the only parameters with physical dimensions (we work in the natural system of units ℏ = c = 1). Naively, we would expect massless QCD to be scale invariant. This is actually true at the classical level. Scale invariance implies that dimensionless observables should not depend on the absolute scale of energy, but only on ratios of energydimensional variables. The massless limit should be relevant for the large asymptotic energy limit of processes which are nonsingular for m → 0.
The naive expectation that massless QCD should be scale invariant is false in the quantum theory. The scale symmetry of the classical theory is unavoidably destroyed by the regularization and renormalization procedure, which introduce a dimensional parameter into the quantum version of the theory. When a symmetry of the classical theory is necessarily destroyed by quantization, regularization, and renormalization one talks of an “anomaly”. So in this sense, scale invariance in massless QCD is anomalous.
 All relevant energy variables must be large:$$\displaystyle{ E_{i} = z_{i}Q\;,\quad Q \gg m_{j}\;,\quad z_{i}\mbox{ scaling variables }O(1)\,. }$$(2.26)

There should be no infrared singularities (one talks of “infrared safe” processes).

The processes concerned must be finite for m → 0 (no mass singularities).
To have any chance of satisfying these criteria, processes must be as “inclusive” as possible: one should include all final states with massless gluon emission and add all mass degenerate final states (given that quarks are massless, \(q\bar{q}\) pairs can also be massless if “collinear”, that is moving together in the same direction at a common speed, the speed of light).
In perturbative QCD one computes inclusive rates for partons (the fields in the Lagrangian, that is, in QCD, quarks and gluons) and takes them as equal to rates for hadrons. Partons and hadrons are considered as two equivalent sets of complete states. This is called “global duality”, and it is rather safe in the rare instance of a totally inclusive final state. It is less so for distributions, like distributions in the invariant mass M (“local duality”), where it can be reliable only if smeared over a sufficiently wide bin in M.
The second one is the Kinoshita–Lee–Nauenberg theorem [265]: mass singularities connected with an external particle of mass m are canceled if all degenerate states (that is, with the same mass) are summed up. Hence, for a final state particle of mass m, we should add all final states that have the same mass in the limit m → 0, including also gluons and massless pairs. If a completely inclusive final state is taken, only the mass singularities from the initial state particles remain (we shall see that they will be absorbed inside the nonperturbative parton densities, which are probability densities for finding the given parton in the initial hadron).
At parton level, the final state is \(q\bar{q} + ng + n^{{\prime}}q^{{\prime}}\bar{q}^{{\prime}}\), and n and n^{ ′ } are limited at each order of perturbation theory. It is assumed that the conversion of partons into hadrons does not affect the rate (it happens with probability 1). We have already mentioned that, in order for this to be true within a given accuracy, averaging over a sufficiently large bin of Q must be understood. The binning width is larger in the vicinity of thresholds: for example, when one goes across the charm \(c\bar{c}\) threshold, the physical crosssection shows resonance bumps that are absent in the smooth partonic counterpart, which, however, gives an average of the crosssection.
2.4 The Renormalization Group and Asymptotic Freedom
In this section we aim to provide a reasonably detailed introduction to the renormalization group formalism and the concept of running coupling, which leads to the result that QCD has the property of asymptotic freedom. We start with a summary of how renormalization works.
We now become more specific, by concentrating on the case of massless QCD. If we start from a vanishing mass at the classical (or “bare”) level m_{0} = 0, the mass is not renormalized because it is protected by a symmetry, namely, chiral symmetry. The conserved currents of chiral symmetry are axial currents: \(\bar{q}\gamma _{\mu }\gamma _{5}q\). Using the Dirac equation, divergence of the axial current gives \(\partial ^{\mu }(\bar{q}\gamma _{\mu }\gamma _{5}q) = 2m\bar{q}\gamma _{5}q\). So the axial current and the corresponding axial charge are conserved in the massless limit. Actually, the singlet axial current is not conserved due to the anomaly, but since QCD is a vector theory, we do not have to worry about chiral anomalies in the present context. As there are no γ_{5} factors around, the chosen regularization preserves chiral symmetry as well as gauge and Lorentz symmetry, and the renormalized mass remains zero. The renormalized propagator has the form (2.30) with m = 0.
We could just as well use the quark–gluon vertex or any other vertex which coincides with e_{s0} in lowest order (even the ghost–gluon vertex, if we want). With a regularization and renormalization that preserves gauge invariance, we can be sure that all these different definitions are equivalent.
Here V_{bare} is what is obtained from computing the Feynman diagrams including, for example, the 1loop corrections at the lowest nontrivial order. V_{bare} is defined as the scalar function multiplying the 3gluon vertex tensor (given in Fig. 2.1), normalized in such a way that it coincides with e_{s0} in lowest order. V_{bare} contains the cutoff K, but does not know about μ. Z is a factor that depends both on the cutoff and on μ, but not on momenta. Because of infrared singularities, the defining scale μ cannot vanish. The negative value −μ^{2} < 0 is chosen to stay away from physical cuts (a gluon with negative virtual mass cannot decay). Similarly, in the massless theory, we can define Z_{ g }^{−1} as the inverse gluon propagator (the 1PI 2point function) at the same scale −μ^{2} (the vanishing mass of the gluon is guaranteed by gauge invariance).
If α(t) is small, we can compute β(α(t)) in perturbation theory. The sign in front of b then decides the slope of the coupling: α(t) increases with t (or Q^{2}) if β is positive at small α (QED), or α(t) decreases with t (or Q^{2}) if β is negative at small α (QCD). A theory like QCD in which the running coupling vanishes asymptotically at large Q^{2} is said to be (ultraviolet) “asymptotically free”. An important result that has been proven [145] is that, in four spacetime dimensions, all and only nonAbelian gauge theories are asymptotically free.
It is very important to note that QED and QCD are theories with “decoupling”, i.e., up to the scale Q, only quarks with masses m ≪ Q contribute to the running of α. This is clearly very important, given that all applications of perturbative QCD so far apply to energies below the top quark mass m_{ t }. For the validity of the decoupling theorem [60], the theory in which all the heavy particle internal lines are eliminated must still be renormalizable and the coupling constants must not vary with the mass. These requirements are satisfied for the masses of heavy quarks in QED and QCD, but they are not satisfied in the electroweak theory where the elimination of the top would violate SU(2) symmetry (because the t and b lefthanded quarks are in a doublet) and the quark couplings to the Higgs multiplet (hence to the longitudinal gauge bosons) are proportional to the mass.
In conclusion, in QED and QCD, quarks with m ≫ Q do not contribute to n_{f} in the coefficients of the relevant β function. The effects of heavy quarks are power suppressed and can be taken into account separately. For example, in e^{+}e^{−} annihilation for 2m_{ c } < Q < 2m_{ b }, the relevant asymptotics is for n_{f} = 4, while for 2m_{ b } < Q < 2m_{ t }, it is for n_{f} = 5. Going across the b threshold, the β function coefficients change, so the slope of α(t) changes. But α(t) is continuous, whence Λ changes so as to keep α(t) constant at the matching point at Q ∼ O(2m_{ b }). The effect on Λ is large: approximately Λ_{5} ∼ 0. 65Λ_{4}, where Λ_{4, 5} are for n_{f} = 4, 5.
Note the presence of a pole at ± bαt = 1 in (2.60) and (2.61). This is called the Landau pole, since Landau had already realised its existence in QED in the 1950s. For μ ∼ m_{ e } (in QED), the pole occurs beyond the Planck mass. In QCD, the Landau pole is located for negative t or at Q < μ in the region of light hadron masses. Clearly the issue of the definition and the behaviour of the physical coupling (which is always finite, when defined in terms of some physical process) in the region around the perturbative Landau pole is a problem that lies outside the scope of perturbative QCD.
Summarizing, we started from massless classical QCD which is scale invariant. But we have seen that the procedure of quantization, regularization, and renormalization necessarily breaks scale invariance. In the quantum QCD theory, there is a scale of energy Λ. From experiment, this is of the order of a few hundred MeV, its precise value depending on the definition, as we shall see in detail. Dimensionless quantities depend on the energy scale through the running coupling, which is a logarithmic function of Q^{2}∕Λ^{2}. In QCD the running coupling decreases logarithmically at large Q^{2} (asymptotic freedom), while in QED the coupling has the opposite behaviour.
2.5 More on the Running Coupling
In the last section we introduced the renormalized coupling α in terms of the 3gluon vertex at p^{2} = −μ^{2} (momentum subtraction). The Ward identities of QCD then ensure that the coupling defined from other vertices like the \(\bar{q}qg\) vertex are renormalized in the same way and the finite radiative corrections are related. But at present the universally adopted definition of α_{s} is in terms of dimensional regularization [333], because of computational simplicity, which is essential given the great complexity of present day calculations. So we now briefly review the principles of dimensional regularization and the definition of minimal subtraction (MS) [335] and modified minimal subtraction (\(\overline{\mathrm{MS}}\)) [82]. The \(\overline{\mathrm{MS}}\) definition of α_{s} is the one most commonly adopted in the literature, and values quoted for it normally refer to this definition.
Dimensional regularization (DR) is a gauge and Lorentz invariant regularization that consists in formulating the theory in D < 4 spacetime dimensions in order to make loop integrals ultraviolet finite. In DR one rewrites the theory in D dimensions (D is integer at the beginning, but then one realizes that the expression calculated from diagrams makes sense for all D, except for isolated singularities). The metric tensor is extended to a D × D matrix g_{ μ ν } = diag(1, −1, −1, …, −1) and 4vectors are given by k^{ μ } = (k^{0}, k^{1}, …, k^{ D−1}). The Dirac γ^{ μ } are f(D) × f(D) matrices and the precise form of the function f(D) is not important. It is sufficient to extend the usual algebra in a straightforward way like {γ_{ μ }, γ_{ ν }} = 2g_{ μ, ν }I, where I is the Ddimensional identity matrix, γ^{ μ }γ^{ ν }γ_{ μ } = −(D − 2)γ^{ ν }, or Tr(γ^{ μ }γ^{ ν }) = f(D)g_{ μ ν }.
2.6 On the Nonconvergence of Perturbative Expansions
It is important to keep in mind that, after renormalization, all the coefficients in the QED and QCD perturbative series are finite, but the expansion does not converge. Actually, the perturbative series is not even Borel summable (for reviews see, for example, [31]). After the Borel resummation, for a given process, one is left with a result that is ambiguous up to terms typically going as exp(−n∕bα), where n is an integer and b the absolute value of the first β function coefficient. In QED, these corrective terms are extremely small and not very important in practice. However, in QCD, α = α_{s}(Q^{2}) ∼ 1∕blog(Q^{2}∕Λ^{2}) and the ambiguous terms are of order (1∕Q^{2})^{ n }, that is, they are power suppressed. It is interesting that, through this mechanism, the perturbative version of the theory is somehow able to take into account the powersuppressed corrections. A sequence of diagrams with factorial growth at large order n is constructed by dressing gluon propagators by any number of quark bubbles together with their gauge completions (renormalons). The problem of the precise relation between the ambiguities of the perturbative expansion and the powersuppressed corrections has been discussed in recent years, also for processes without light cone operator expansion [31, 324].
2.7 e^{+}e^{−} Annihilation and Related Processes
2.7.1 \(R_{e^{+}e^{}}\)
2.7.2 The Final State in e^{+}e^{−} Annihilation
Experiments on e^{+}e^{−} annihilation at high energy provide a remarkable opportunity for systematically testing the distinct signatures predicted by QCD for the structure of the final state averaged over a large number of events. Typical of asymptotic freedom is the hierarchy of configurations emerging as a consequence of the smallness of α_{s}(Q^{2}). When all corrections of order α_{s}(Q^{2}) are neglected, one recovers the naive parton model prediction for the final state: almost collinear events with two backtoback jets with limited transverse momentum and an angular distribution 1 + cos^{2}θ with respect to the beam axis (typical of spin 1/2 parton quarks, while scalar quarks would lead to a sin^{2}θ distribution). To order α_{s}(Q^{2}), a tail of events is predicted to appear with large transverse momentum p_{T} ∼ Q∕2 with respect to a suitably defined jet axis (for example, the thrust axis, see below). This small fraction of events with large p_{T} consists mainly of threejet events with almost planar topology. The skeleton of a threejet event, to leading order in α_{s}(Q^{2}), is formed by three hard partons \(q\bar{q}g\), the third being a gluon emitted by a quark or antiquark line. To order α_{s}^{2}(Q^{2}), a hard perturbative nonplanar component starts to build up, and a small fraction of fourjet events \(q\bar{q}gg\) or \(q\bar{q}q\bar{q}\) appear, and so on.
Recently, motivated by the LHC experiments, there has been a flurry of improved jet algorithm studies: it is essential that correct jet finding should be implemented by LHC experiments for optimal matching of theory and experiment [185, 317]. In particular, existing sequential recombination algorithms like k_{T} [132, 172] and Cambridge/Aachen [174] have been generalized. In these recursive definitions, one introduces distances d_{ ij } between particles or clusters of particles i and j, and d_{ iB } between i and the beam (B). The inclusive clustering proceeds by identifying the smallest of the distances and, if it is a d_{ ij }, by recombining particles i and j, while if it is d_{ iB }, calling i a jet and removing it from the list. The distances are recalculated and the procedure repeated until no i and j are left.
For p = 1, one has the inclusive k_{ T } algorithm. It can be shown in general that for p ≥ 0 the behaviour of the jet algorithm with respect to soft radiation is rather similar to that observed for the k_{ T } algorithm. The case p = 0 is special, and it corresponds to the inclusive Cambridge/Aachen algorithm [174]. Surprisingly (at first sight), taking p to be negative also yields an algorithm that is infrared and collinear safe and has sensible phenomenological behaviour. For p = −1, one obtains the recently introduced “antik_{ T }” jetclustering algorithm [126], which has particularly stable jet boundaries with respect to soft radiation and is suitable for practical use in experiments.
2.8 Deep Inelastic Scattering
Deep inelastic scattering (DIS) processes have played, and still play, a very important role in our understanding of QCD and of nucleon structure. This set of processes actually provides us with a rich laboratory for theory and experiment. There are several structure functions F_{ i }(x, Q^{2}) that can be studied, each a function of two variables. This is true separately for different beams and targets and different polarizations. Depending on the charges of ℓ and ℓ^{ ′ } [see (2.28)], we can have neutral currents (γ, Z) or charged currents in the ℓ–ℓ^{ ′ } channel (Fig. 2.10). In the past, DIS processes were crucial for establishing QCD as the theory of strong interactions and quarks and gluons as the QCD partons.
At present DIS remains very important for quantitative studies and tests of QCD. The theory of scaling violations for totally inclusive DIS structure functions, based on operator expansion or diagrammatic techniques and renormalization group methods, is crystal clear and the predicted Q^{2} dependence can be tested at each value of x. The measurement of quark and gluon densities in the nucleon, as functions of x at some reference value of Q^{2}, which is an essential starting point for the calculation of all relevant hadronic hard processes, is performed in DIS processes. At the same time one measures α_{s}(Q^{2}), and the DIS values of the running coupling can be compared with those obtained from other processes. At all times new theoretical challenges arise from the study of DIS processes. Recent examples (see the following) are the socalled “spin crisis” in polarized DIS and the behaviour of singlet structure functions at small x, as revealed by HERA data. In the following we review the past successes and the present open problems in the physics of DIS.
In the 1960s the demise of hadrons from the status of fundamental particles to that of bound states of constituent quarks was the breakthrough that made possible the construction of a renormalizable field theory for strong interactions. The presence of an unlimited number of hadrons species, many of them with high spin values, presented an obvious deadend for a manageable field theory. The evidence for constituent quarks emerged clearly from the systematics of hadron spectroscopy. The complications of the hadron spectrum could be explained in terms of the quantum numbers of spin 1/2, fractionally charged u, d, and s quarks. The notion of colour was introduced to reconcile the observed spectrum with Fermi statistics.
However, confinement, which forbids the observation of free quarks, was a clear obstacle towards the acceptance of quarks as real constituents and not just as fictitious entities describing some mathematical pattern (a doubt expressed even by GellMann at the time). The early measurements of DIS at SLAC dissipated all doubts: the observation of Bjorken scaling and the success of Feynman’s “naive” (not so much after all) parton model imposed quarks as the basic fields for describing the nucleon structure (parton quarks).
In the language of Bjorken and Feynman, the virtual γ (or, in general, any gauge boson) sees the quark partons inside the nucleon target as quasifree, because their (Lorentz dilated) QCD interaction time is much longer than τ_{ γ } ∼ 1∕Q, the duration of the virtual photon interaction. Since the virtual photon 4momentum is spacelike, we can go to a Lorentz frame where E_{ γ } = 0 (Breit frame). In this frame q = (E_{ γ } = 0, 0, 0, Q) and the nucleon momentum, neglecting the mass m ≪ Q, is p = (Q∕2x, 0, 0, −Q∕2x). We note that this gives q^{2} = −Q^{2} and x = Q^{2}∕2( p ⋅ q), as it should.
Furthermore, recall that the helicity of a massless quark is conserved in a vector (or axial vector) interaction (see Sect. 1.5). So when the momentum is reversed, the spin must also flip. Since the process is collinear there is no orbital contribution, and only a photon with helicity ± 1 (transverse photon) can be absorbed. Alternatively, if partons were spin zero, only longitudinal photons would then contribute.
At higher orders, the evolution equations are easily generalized, but the calculation of the splitting functions rapidly becomes very complicated. For many years the splitting functions were only completely known at NLO accuracy [198], that is, α_{s}P ∼ α_{s}P_{1} +α_{s}^{2}P_{2} + ⋯ . But in recent years, the NNLO results P_{3} were first derived in analytic form for the first few moments, and then the full NNLO analytic calculation, a really monumental work, was completed in 2004 by Moch et al. [292].
Beyond leading order, a precise definition of parton densities should be specified. One can take a physical definition: for example, quark densities can be defined so as to keep the LO expression for the structure function F_{2} valid at all orders, the socalled DIS definition [42], and the gluon density can be defined starting from F_{L}, the longitudinal structure function. Alternatively, one can adopt a more abstract specification, for example, in terms of the \(\overline{\mathrm{MS}}\) prescription. Once the definition of parton densities is fixed, the coefficients that relate the different structure functions to the parton densities at each fixed order can be computed. Similarly, the higher order splitting functions also depend, to some extent, on the definition of parton densities, and a consistent set of coefficients and splitting functions must be used at each order.
2.8.1 The Longitudinal Structure Function
2.8.2 Large and Small x Resummations for Structure Functions
At values of x either near 0 or near 1 (with Q^{2} large), those terms of higher order in α_{s}, in both the coefficients or the splitting functions, which are multiplied by powers of log1∕x or log(1 − x) eventually become important and should be taken into account. Fortunately, the sequences of leading and subleading logs can be evaluated at all orders by special techniques, and resummed to all orders.
For x ∼ 1 resummation [329], I refer to the recent papers [202, 211] (the latter also involving higher twist corrections, which are important at large x), where a list of references to previous work can be found. More important is the small x resummation because, the singlet structure functions are large in this domain of x (while all structure functions vanish near x = 1). Here we briefly summarize the smallx case for the singlet structure function, which is the dominant channel at HERA, dominated by the sharp rise of the gluon and sea parton densities at small x.
The small x data collected by HERA can be fitted reasonably well, even at the smallest measured values of x, by the NLO QCD evolution equations, so that there is no dramatic evidence in the data for departures. This is surprising also in view of the fact that the NNLO effects in the evolution have recently become available and are quite large [292]. Resummation effects have been shown to resolve this apparent paradox. For the singlet splitting function, the coefficients of all LO and NLO corrections of order [α_{s}(Q^{2})log1∕x]^{ n } and α_{s}(Q^{2})[α_{s}(Q^{2})log1∕x]^{ n }, respectively, are explicitly known from the Balitski, Fadin, Kuraev, Lipatov (BFKL) analysis of virtual gluon–virtual gluon scattering [191, 284]. But the simple addition of these higher order terms to the perturbative result (with subtraction of all double counting) does not lead to a converging expansion (the NLO logs completely override the LO logs in the relevant domain of x and Q^{2}).
2.8.3 Polarized Deep Inelastic Scattering
The value of \(\Delta s \sim 0.11\) from totally inclusive data and SU(3) appears to be at variance with the value extracted from singleparticle inclusive DIS (SIDIS), where one obtains a nearly vanishing result for \(\Delta s\) in a fit to all data [159, 326] that leads to puzzling results. There is, in fact, an apparent tension between the first moments as determined by using the approximate SU(3) symmetry and from fitting the data on SIDIS (x ≥ 0. 001) (in particular for the strange density). But the adequacy of the SIDIS data is questionable (in particular the kaon data which fix \(\Delta s\)) and so is their theoretical treatment (for example, the application of parton results at too low an energy and the ambiguities in the kaon fragmentation function).
\(\Delta \varSigma\) is conserved in perturbation theory at LO (i.e., it does not evolve with Q^{2}). Regarding conserved quantities, we would expect them to be the same for constituent and for parton quarks. But actually, the conservation of \(\Delta \varSigma\) is broken by the axial anomaly and, in fact, in perturbation theory beyond LO, the conserved density is actually \(\Delta \varSigma ^{{\prime}} = \Delta \varSigma + \Delta g(n_{\mathrm{f}}/2\pi \alpha _{\mathrm{s}})\) [41]. Note also that \(\alpha _{\mathrm{s}}\Delta g\) is conserved in LO, that is \(\Delta g \sim \log Q^{2}\). This behaviour is not controversial, but it will be a long time before the log growth of \(\Delta g\) is confirmed by experiment! However, by establishing this behaviour, one would show that the extraction of \(\Delta g\) from the data is correct and that the QCD evolution works as expected.
If \(\Delta g\) were large enough, it could account for the difference between partons (\(\Delta \varSigma\)) and constituents (\(\Delta \varSigma ^{{\prime}}\)). From the spin sum rule it is clear that the log increase should cancel between \(\Delta g\) and L_{ z }. This cancelation is automatic, as a consequence of helicity conservation in the basic QCD vertices. \(\Delta g\) can be measured indirectly by scaling violations and directly from asymmetries, e.g., in SIDIS. Existing measurements by HERMES, COMPASS, and at RHIC are still crude, but show no hint of a large \(\Delta g\) at accessible values of x and Q^{2}. Present data, affected by large errors (see, in particular, [303] for a discussion of this point) are consistent [104, 159, 244, 277, 303, 326] with a sizable contribution of \(\Delta g\) to the spin sum rule in (2.112), but there is no indication that \(\alpha _{\mathrm{s}}\Delta g\) effects can explain the difference between constituents and parton quarks.
2.9 Hadron Collider Processes and Factorization
At least an NLO calculation of the reduced partonic crosssection σ_{ AB } is needed in order to correctly specify the scale, and in general also the definition of the parton densities and of the running coupling in the leading term. The residual scale and scheme dependence is often the most important source of theoretical error. It is important to ask to what extent the FT has been proven? In perturbation theory up to NNLO, it has been explicitly checked to hold for many processes: if corrections exist we already know that they must be small (we stress that we are only considering totally inclusive processes). At all orders, the most indepth discussions have been carried out in [146], in particular for Drell–Yan processes. The LHC experiments offer a wonderful opportunity for testing the FT by comparing precise theoretical predictions with accurate data on a wide variety of processes (for a recent review, see, for example, [119]).
A great effort has been and is being devoted to the theoretical preparation and interpretation of the LHC experiments. For this purpose very, difficult calculations are needed at NLO and beyond because the strong coupling, even at the large Q^{2} values involved, is not that small. Further powerful techniques for amplitude calculations have been devised.
An interesting development at the interface between string theory and QCD is twistor calculus. A precursor was the Parke–Taylor result in 1986 [305] on the amplitudes for n incoming gluons with given ± helicities [91]. Inspired by dual models, they derived a compact formula for the maximum nonvanishing helicityviolating amplitude (with n − 2 plus and 2 minus helicities) in terms of spinor products. In 2003, using the relation between strings and gauge theories in twistor space, Witten developed [368] a formalism in terms of effective vertices and propagators that allows one to compute all helicity amplitudes. The method, alternative to other modern techniques for the evaluation of Feynman diagrams [163], leads to very compact results.
Since then, there has been rapid progress (for reviews, see [128]). The method was extended to include massless external fermions [217] and also external EW vector bosons [96] and Higgs particles [167]. The level attained is already important for multijet events at the LHC. The study of loop diagrams came next. The basic idea is that loops can be fully reconstructed from their unitarity cuts. First proposed by Bern et al. [95], the technique was revived by Britto et al. [114] and then perfected by Ossola et al. [304] and further extended to massive particles in [186]. For a recent review of these new methods see [188].
In parallel with this, activity on event simulation has received a big boost from preparations at the LHC (see, for example, the review [130]). Powerful techniques have been developed to generate numerical results at NLO for processes with complicated final states: matrix element calculation has been matched with modeling of parton showers in packages like Black Hat [92] (onshell methods for loops), used in association with Sherpa [227] (for real emission), or POWHEG BOX [299] or aMC@NLO [203], the automated version of the general framework MC@NLO [206]. In a complete simulation, the matrix element calculation, improved by resummation of large logs, provides the hard skeleton (with large p_{T} branchings), while the parton shower is constructed by a sequence of factorized collinear emissions fixed by the QCD splitting functions. In addition, at low scales, a model of hadronization completes the simulation. The importance of all the components, matrix element, parton shower, and hadronization, can be appreciated in simulations of hard events compared with Tevatron and LHC data. One can say that the computation of NLO corrections in perturbative QCD has now been completely automated.
A partial list of examples of recent NLO calculations in pp collisions, obtained with these techniques is: W + 3 jets [187], Z, γ^{∗} + 3 jets [93], W, Z + 4 jets [94], W + 5 jets [97], \(t\bar{t}b\bar{b}\) [113], \(t\bar{t}\) + 2 jets [100], \(t\bar{t}\ W\) [129], WW + 2 jets [289], \(WWb\bar{b}\) [161], \(b\bar{b}b\bar{b}\) [232], etc. In the following we shall detail a number of the most important and simplest examples, without any pretension to completeness.
2.9.1 Vector Boson Production
Drell–Yan processes which include lepton pair production via virtual γ, W, or Z exchange, offer a particularly good opportunity to test QCD. This process, among those quadratic in parton densities with a totally inclusive final state, is perhaps the simplest one from a theoretical point of view. The large scale is specified and measured by the invariant mass squared Q^{2} of the lepton pair, which is not itself strongly interacting (so there are no dangerous hadronization effects). The improved QCD parton model leads directly to a prediction for the total rate as a function of s and τ = Q^{2}∕s. The value of the LO crosssection is inversely proportional to the number of colours N_{C}, because a quark of given colour can only annihilate with an antiquark of the same colour to produce a colourless lepton pair. The order α_{s}(Q^{2}) NLO corrections to the total rate were computed long ago [42, 273] and found to be particularly large, when the quark densities are defined from the structure function F_{2} measured in DIS at q^{2} = −Q^{2}. The ratio σ_{corr}∕σ_{LO} of the corrected and the Born crosssections was called the Kfactor [28], because it is almost a constant in rapidity. More recently, the NNLO full calculation of the Kfactor was completed in a truly remarkable calculation [240].
The rapidity distributions of the produced W and Z have also been measured with fair accuracy at the Tevatron and at the LHC, and predicted at NLO [55]. A representative example of great significance is provided by the combined LHC results for the W charge asymmetry, defined as A ∼ (W^{+} − W^{−})∕(W^{+} + W^{−}), as a function of the pseudorapidity η [340]. These data combine the ATLAS and CMS results at smaller values of η with those of the LHCb experiments at larger η (in the forward direction). This is very important input for the disentangling of the different quark parton densities.
2.9.2 Jets at Large Transverse Momentum
2.9.3 Heavy Quark Production
The top quark is really special: its mass is of the order of the Higgs VEV or its Yukawa coupling is of order 1, and in this sense, it is the only “normal” case among all quarks and charged leptons. Due to its heavy mass, it decays so fast that it has no time to be bound in a hadron: thus it can be studied as a quark. It is very important to determine its mass and couplings for different precision predictions of the SM. The top quark may be particularly sensitive to new heavy states or have a connection to the Higgs sector when we go beyond the SM theories.
The mass of the top (and the value of α_{s}) can be determined from the crosssection, assuming that QCD is correct, and compared with the more precise value from the final decay state. The value of the top pole mass derived in [27] from the crosssection data, using the best available parton densities with the correlated value of α_{s}, is m_{ t }^{pole} = 173. 3 ± 2. 8 GeV. This is to be compared with the value measured at the Tevatron by the CDF and D0 collaborations, viz., m_{ t }^{exp} = 173. 2 ± 0. 9 GeV. This quoted error is clearly too optimistic, especially if one identifies this value with the pole mass which it resembles. This error is only adequate within the specific procedure used by the experimental collaborations to define their mass (including Montecarlo, with assumptions about higher order terms, nonperturbative effects, etc.). The problem is how to export this value to other processes. Leaving aside the thorny issue of the precise relation between m_{ t }^{exp} with m_{ t }^{pole}, it is clear that there is good overall consistency.
The inclusive forward–backward asymmetry, A_{FB}, in the \(t\bar{t}\) rest frame has been measured by both the CDF [6] and D0 [9] collaborations, and found to be in excess of the SM prediction, by about 2σ [247]. For CDF the discrepancy increases at large \(t\bar{t}\) invariant mass, and reaches about 2. 5σ for \(M_{t\bar{t}} \geq 450\) GeV. Recently, CDF has obtained [7] the first measurement of the top quark pair production differential crosssection as a function of cosθ, with θ the production angle of the top quark. The coefficient of the cosθ term in the differential crosssection, viz., a_{1} = 0. 40 ± 0. 12, is found to be in excess of the NLO SM prediction, viz., 0. 15_{−0. 03}^{+0. 07}, while all other terms are in good agreement with the NLO SM prediction, and A_{FB} is dominated by this excess linear term. Is this a real discrepancy? The evidence is far from compelling, but this effect has received much attention from theorists [321]. A related observable at the LHC is the charge asymmetry A_{C} in \(t\bar{t}\) production. In contrast to A_{FB}, the combined value of A_{C} reported by ATLAS [1] and CMS [144] agrees with the SM, within the still limited accuracy of the data.
2.9.4 Higgs Boson Production
So far we have seen examples of resummation of large logs. This is a very important chapter of modern QCD. The resummation of soft gluon logs enter into different problems, and the related theory is subtle. The reader is referred here to some recent papers where additional references can be found [77]. A particularly interesting related development has to do with the socalled nonglobal logs (see, for example, [153]). If in the measurement of an observable some experimental cuts are introduced, which is very often the case, then a number of large logs can arise from the corresponding breaking of inclusiveness. It is also important to mention the development of software for the automated implementation of resummation (see, for example, [78]).
2.10 Measurements of α_{s}
2.10.1 α_{s} from e^{+}e^{−} Colliders
There is a vast and sophisticated literature on α_{s} from τ decay. Unbelievably small errors are obtained in one or the other of several different procedures and assumptions that have been adopted to end up with a specified result. With time there has been an increasing awareness of the problem of controlling higher orders and nonperturbative effects. In particular, fixed order perturbation theory (FOPT) has been compared with resummation of leading beta function effects in the socalled contourimproved perturbation theory (CIPT). The results are sizeably different in the two cases, and there have been many arguments in the literature about which method is best.
One important piece of progress comes from the experimental measurement of moments of the τ decay mass distributions, defined by modifying the weight function in the integral in (2.129). In principle, one can measure α_{s} from the sum rules obtained from different weight functions that emphasize different mass intervals and different operator dimensions in the light cone operator expansion. A thorough study of the dependence of the measured value of α_{s} on the choice of the weight function, and in general of higher order and nonperturbative corrections, has appeared in [89], and the interested reader is advised to look at that paper and the references therein.
2.10.2 α_{s} from Deep Inelastic Scattering
2.10.3 Recommended Value of α_{s}(m_{ Z })
 From Z decays and EW precision tests, i.e., (2.126):$$\displaystyle{ \alpha _{\mathrm{s}}(m_{Z}) = 0.1190 \pm 0.0026\;. }$$(2.143)
 From scaling violations in DIS, i.e., (2.142):$$\displaystyle{ \alpha _{\mathrm{s}}(m_{Z}) = 0.1165 \pm 0.0020\;. }$$(2.144)
 From R_{τ} (2.133):$$\displaystyle{ \alpha _{\mathrm{s}}(m_{Z}) = 0.1194 \pm 0.0021. }$$(2.145)
2.10.4 Other α_{s}(m_{ Z }) Measurements as QCD Tests
There are a number of other determinations of α_{s} that are important because they arise from qualitatively different observables and methods. Here I will give a few examples of the most interesting measurements.
A classic set of measurements comes from a number of infraredsafe observables related to event rates and jet shapes in e^{+}e^{−} annihilation. One important feature of these measurements is that they can be repeated at different energies in the same detector, like the JADE detector in the energy range of PETRA (most of the intermediate energy points in the righthand panel of Fig. 2.32 are from this class of measurements) or the LEP detectors from LEP1 to LEP2 energies. As a result, one obtains a striking direct confirmation of the running of the coupling according to the renormalization group prediction. The perturbative part is known at NNLO [213], and resummations of leading logs arising from the vicinity of cuts and/or boundaries have been performed in many cases using effective field theory methods. The main problem with these measurements is the possibly large impact of nonperturbative hadronization effects on the result, and therefore on the theoretical error.
2.11 Conclusion
We have seen that perturbative QCD based on asymptotic freedom offers a rich variety of tests, and we have described some examples in detail. QCD tests are not as precise as for the electroweak sector. But the number and diversity of such tests has established a very firm experimental foundation for QCD as a theory of strong interactions. The physics content of QCD is very large and our knowledge, especially in the nonperturbative domain, is still very limited, but progress both from experiment (Tevatron, RHIC, LHC, etc.) and from theory is continuing at a healthy rate. And all the QCD predictions that we have been able to formulate and to test appear to be in very good agreement with experiment.
The field of QCD appears to be one of great maturity, but also of robust vitality, with many rich branches and plenty of new blossoms. I may mention the very exciting explorations of supersymmetric extensions of QCD and the connections with string theory (for a recent review and a list of references, see [166]). In particular, N = 4 SUSY QCD (that is, with four spinor charge generators) has a vanishing beta function and is loopfinite. In the limit N_{C} → ∞ with λ = e_{s}^{2}N_{C} fixed, planar diagrams are dominant. There is progress towards a solution of planar N = 4 SUSY QCD. The large λ limit corresponds by the AdS/CFT duality (antide Sitter/conformal field theory), a string theory concept, to the weakly coupled string (gravity) theory on AdS_{5} × S_{5} (the 10 dimensions are compactified in a 5dimensional antide Sitter space times a 5dimensional sphere). By moving along this very tentative route, one can transfer some results (assumed to be of sufficiently universal nature) from the computable weak limit of the associated string theory to the nonperturbative ordinary QCD domain. Further along this line of investigation, there are studies of N = 8 supergravity, related to N = 4 SUSY Yang–Mills, which has been proven finite up to four loops. It could possibly lead to a finite field theory of gravity in four dimensions.
Footnotes
 1.
Actually, starting from order α_{s}^{2}, there are some “singlet” terms proportional to (∑_{ i }Q_{ i })^{2}. These small terms are included in F by dividing and multiplying by ∑_{ i }Q_{ i }^{2}.
 2.
The evolution equations are now often called the DGLAP equations (after Dokshitzer, Gribov, Lipatov, Altarelli, and Parisi). The first article by Gribov and Lipatov was published in 1972 [233] (even before the works by Gross and Wilczek and by Politzer!), and was followed in 1974 by a paper by Lipatov [283] (these dates correspond to the publication in Russian). All these articles refer to an Abelian vector theory (treated in parallel with a pseudoscalar theory). Seen from the point of view of the evolution equations, these papers, in the context of the Abelian theory, ask the right question and extract the relevant logarithmic terms from the dominant class of diagrams. But from their formal presentation, the relation to real physics is somewhat hidden (in this respect the 1974 paper by Lipatov makes some progress and explicitly refers to the parton model). The article by Dokshitser [171] was exactly contemporary to that by Altarelli and Parisi [40]. It now refers to the nonAbelian theory (with running coupling), and the discussion is more complete and explicit than in the Gribov–Lipatov articles. But, for example, the connection to the parton model, the notion of the evolution as a branching process, and the independence of the kernels from the process are not emphasized.
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