This book has covered many of the primary topics in perturbative QCD, with a focus on certain inclusive processes for which particularly systematic treatments are available. It should provide the reader with a sound conceptual framework for further study and research. However, hadronic interactions form a vast subject, and there is an enormous literature where perturbatively based methods have been applied.

This chapter gives a summary of a selection of important areas of further application of perturbative QCD.

One common theme, a prerequisite for actual perturbative calculations, is that the reactions have in some sense a controlling hard subprocess, occurring on a short distance scale, i.e., a distance scale significantly less than 1 fm, or, more-or-less equivalently, a momentum transfer significantly larger than the typical hadronic scale of a few hundred MeV.

Another recurring idea, perhaps the closest to a unifying motif, is the idea that one should try to separate (factor) phenomena on different scales of distance and momentum. This refers not just to scales of different virtuality, but also to a separation of phenomena at widely different rapidities. A characteristic here is that almost scattering processes examined in high-energy physics are ultra-relativistic. Thus time dilation and Lorentz contraction of fast-moving hadrons by themselves provide a wide range of distance scales. For example at the Tevatron collider we have proton and antiproton beams of energy almost 1 TeV. This allows the measurement of hard processes with momentum scales of several hundred GeV.

The theory of the strong interaction of hadrons – quantum chromodynamics, or QCD – is in many ways the most perfect and non-trivial of the established microscopic theories of physics. It is, as far as is known, a self-consistent relativistic quantum field theory. But, unlike the case of the electromagnetic and weak interactions, many primary phenomena governed by QCD are not amenable to direct calculation by weak-coupling perturbation theory. Moreover, QCD has few parameters.

To understand these assertions, first recall the classification of known microscopic interactions into strong, electromagnetic, weak, and gravitational. Precisely because the strong interaction is strong, it is useful to study QCD by itself, the other interactions being perturbations.

QCD is a quantum field theory of the kind called a non-abelian gauge theory (or a Yang-Mills theory). It has two types of field: quark fields and the gluon field. Particles corresponding to the quark fields form the basic constituents of hadrons, like the proton, with the gluon field providing the binding between quarks. There appear to be no states for isolated quarks and gluons; these particles are always confined into hadrons. This contrasts with quantum electrodynamics (QED), where instead of quarks and gluons, we have electrons and photons, which do exist in isolated single-particle states.

One key feature of QCD is “asymptotic freedom”: the effective coupling of QCD goes to zero at zero distance. Thus short-distance processes yield to the highly developed methods of Feynman perturbation theory.

A possible approach to a theory like QCD is just to state its definition, and then immediately proceed deductively. However, this begs the question of why we should use this theory and not some other. Moreover, the approach is quite abstract, and the initial connection to the real physical world is missing.

Instead, I will take a quasi-historical approach, after first stating the theory. Such an approach is suitable for newcomers since their background in QCD is like that of its inventors/discoverers, i.e., little or none. There were several lines of development, all of which powerfully converged on a unique theory from key aspects of experimental data. Of course, we see this much more readily in retrospect than was apparent at the time of the original work, and my account is selective in focusing on the issues now seen to be the most significant. A historical approach also enables the introduction of ideas and methods that do not specifically depend on QCD: e.g., deeply inelastic scattering and the parton model.

I have tried to make the presentation self-contained, in summarizing the relevant experimental phenomena and their consequences for theory. The reader is only assumed to have a working knowledge of relativistic quantum field theory. Inevitably there are issues, ideas, experiments, and historical developments which will be unfamiliar to many readers, and for which a complete treatment needs much more space. I give references for many of these.

An appealing interpretation of a parton density is that it is a number density of partons in a target hadron. As we saw in Sec. 6.7, a parton density in a simple theory is an expectation value of a light-front number operator, integrated over transverse momentum. A similar interpretation applies to fragmentation functions: Sec. 12.4.

As explained in Secs. 6.8 and 12.4, it is equally natural to define unintegrated, or transverse-momentum-dependent (TMD), parton densities and fragmentation functions, simply by omitting the integral over transverse momentum. In a sense, the TMD functions are more fundamental and present more information on non-perturbative phenomena than do the ordinary integrated functions. Therefore it is useful to find situations where TMD functions are needed.

In this chapter, I treat two characteristic cases. One is two-particle-inclusive e+e- annihilation when the detected hadrons are close to back-to-back. This process needs TMD fragmentation functions. Then I will extend this work to semi-inclusive DIS (SIDIS) with a detected hadron of low transverse momentum. In SIDIS, TMD parton densities are needed as well as fragmentation functions. A further extension to the Drell-Yan process at low transverse momentum will be covered in Sec. 14.5.

There are substantial complications in QCD. Although the discussion about light-front quantization and the associated definitions of number densities gives a general motivation, it does not work correctly in QCD (or any other gauge theory). The actual definitions are whatever is appropriate to consistently obtain a valid factorization theorem.

In this chapter, I treat inclusive hard processes in hadron-hadron collisions. These give some of the most important practical applications of factorization. But the actual derivation has substantial extra difficulties, compared with other processes we have examined.

Technically the extra difficulties concern the Glauber region. In e+e- annihilation or SIDIS, we deformed loop momenta out of the Glauber region in individual (cut) graphs. But this is no longer possible in hadron-hadron collisions. This situation results from interactions between the spectator parts of the beam hadrons, as I will illustrate by an example in Sec. 14.3. To get factorization, we will need a sum over cuts of the graphs, which in turn entails a sum over different unobserved final states in an inclusive cross section. The technical details will be explained in Sec. 14.4 for the case of the Drell-Yan process.

After that we will obtain factorization, including the version using TMD parton densities. I will summarize the situation for more general reactions with detected hadrons of high transverse momentum. There is a surprising lack of detailed published proofs. Although the statements of factorization are essentially trivial generalizations of those for Drell-Yan, there are underlying complications in the physics which makes the justification of the generalizations quite non-trivial.

This work also leads us to the frontiers of the factorization approach, beyond which more general methods are needed, e.g., in diffractive hadron-hadron scattering.

Basic ideas on the space-time structure of deeply inelastic scattering (DIS), symbolized in Figs. 2.2 and 2.5, led us to the parton model in Sec. 2.4. However, as we saw in Ch. 5, the leading regions can be more general than those that give the parton model. Indeed, the properties needed for the literal truth of the parton model are violated in any QFT that needs renormalization or that is a gauge theory, or both, like QCD.

Even so, the ideas that led to the parton model (the distance scales, time dilation and Lorentz contraction) are such basic properties that one should expect the parton model to be some kind of approximation to real QCD.

Because of the complications inherent to a sound treatment in QCD, it is useful to build up methodologies step by step. In this chapter, we treat situations where the parton model is correct, which happens in suitable model field theories. For these we will construct a strict field-theoretic implementation of the parton model.

One key result will be operator definitions of the parton distribution functions (parton densities or pdfs). Another result will be light-front quantization, whereby a probability interpretation of a pdf can be completely justified, in those model theories where the parton model is exact.

Field theory formulation of parton model

DIS concerns electron scattering off a hadronic target, e + P → e + X, to lowest order in electromagnetism, with kinematic variables and structure functions defined in Sec. 2.3.

In this chapter, I treat the complications caused by renormalizability of the underlying field theory when one analyzes the asymptotics of processes like DIS. There are four inter-related issues:

• The leading regions include hard-scattering subgraphs that can be of arbitrarily high order in the coupling.

• There are logarithmic unsuppressed contributions from momenta that interpolate between the different regions for a graph.

• The definitions of the parton densities are modified to remove their UV divergences. This we do by renormalization.

• The parton densities acquire a scale argument µ, the dependence on which is governed by renormalization-group (RG) equations, the famous Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations. In applications, we set µ of order Q, the large scale in the hard scattering.

I will give a derivation of factorization that in the absence of gauge fields is complete and satisfactory, and is also reasonably elementary. In QCD, the same factorization theorem is also valid for simple processes, like DIS, but its derivation needs enhancement, to be given in later chapters.

Factorization: overall view

To motivate the factorization idea, we still use the ideas about the space-time structure of DIS that motivated the parton model. As illustrated in the spatial diagram of Fig. 8.1, an electron undergoes a wide-angle hard scattering off a single parton in a high-energy target hadron. In the center-of-mass frame, the target is time-dilated and Lorentz contracted.

In this chapter we complete our treatment of inclusive structure functions for DIS in QCD. Our analysis so far started with the parton model, and we generalized it to a factorization property, for which we found a complete proof in a non-gauge theory in Sec. 8.9. We then formulated factorization in QCD (without a proof), using gauge-invariant definitions of parton densities from Sec. 7.6. This enabled us to make low-order calculations of the perturbative hard-scattering coefficients in Ch. 9

The methods of Ch. 10 allow us to complete the work for QCD. Compared with a non-gauge theory, there is no change in the form of factorization, i.e., (8.81) and (8.83). The DGLAP evolution equations, associated with the renormalization of parton densities, are also unchanged in structure.

One change in QCD is that the operators defining the parton densities acquire Wilson lines; we also need to justify the form of the gluon density. For the proof, the enhancements relative to Sec. 8.9 are caused by the extra gluons joining the hard and collinear subgraphs in leading regions. We need generalization beyond the related work in Ch. 10 because the gauge group of QCD is non-abelian. The subtractions in the hard scattering are more complicated than those with the ladder structures appropriate to a non-gauge theory. Finally, in generalizing DIS to an off-shell Green function instead of an on-shell matrix element, we need extra parton-density-like quantities involving gauge-variant operators.

In the previous chapter, we formalized the parton model in a simple quantum field theory. A number of further developments follow fairly simply, and this chapter's purpose is to give an account of them, before we go on to the full QCD treatment.

We first extend the parton model for DIS to the very important case of charged-current weak-interaction processes. Then we examine a particularly influential form of perturbation theory: light-front (or x+-ordered) perturbation theory. After that I present the light-front quantization of gauge theories, a natural extension of what we did earlier for non-gauge theories. We will thereby be able to introduce appropriate definitions for parton densities in a gauge theory, and to convert them to a gauge invariant form with the aid of what are known as Wilson-line operators.

DIS with weak interactions, neutrino scattering, etc.

We have extensively discussed DIS for the case of virtual photon exchange. The same principles apply equally to all lepto-production processes l + N → l′ + X, and thus they apply whether the exchanged electroweak boson is a W, Z or photon. There are a large number of different cases, and, as far as the theory by itself is concerned, all are a minor variation on the purely electromagnetic case, both at the parton-model level, and with all the QCD modifications.

Central assertions in setting up the parton model for DIS (Sec. 2.4) were that hard scattering occurs off a single parton constituent of the target, and that the hard scattering is just the Born approximation for electron-quark scattering. In fact, both assertions fail if taken literally. So in this chapter I show how to derive correct statements about the dominant configurations in DIS and the many other cases of interest. I will interleave a general treatment with a detailed discussion of specific examples.

Key insights were found by Sterman (1978) and Libby and Sterman (1978b), who systematized a correspondence between divergences in massless perturbative calculations and important configurations for high-energy processes. For any suitable process (like DIS) with an energy scale Q much larger than relevant particle masses, the main results are:

1. A one-to-one correspondence between mass divergences in massless perturbation theory and non-UV regions in loop-momentum space that give the large Q asymptote.

2. That mass divergences are at surfaces where the integral over loop momenta cannot be deformed away from singularities of propagators. These surfaces are called pinch singular surfaces (PSSs).

3. Simple and very general geometrical arguments in four-dimensional momentum space to locate the PSSs for a massless theory. The PSSs are in the typically higher-dimension space of all loop momenta.

4. Simple power-counting results for the strengths of the possible PSSs, and for the power dependence on Q of the contribution of the region associated with each PSS.

5. […]