Distribution Functions of the Nucleon and Pion in the Valence Region

Distribution Functions of the Nucleon and Pion in the Valence Region

Roy J. Holt holt@anl.gov Physics Division, Argonne National Laboratory,
Argonne, Illinois 60439, USA
   Craig D. Roberts cdroberts@anl.gov Physics Division, Argonne National Laboratory,
Argonne, Illinois 60439, USA
Department of Physics, Peking University, Beijing 100871, China
Abstract

We provide an experimental and theoretical perspective on the behavior of unpolarized distribution functions for the nucleon and pion on the valence-quark domain; namely, Bjorken-. This domain is key to much of hadron physics; e.g., a hadron is defined by its flavor content and that is a valence-quark property. Furthermore, its accurate parametrization is crucial to the provision of reliable input for large collider experiments. We focus on experimental extractions of distribution functions via electron and muon inelastic scattering, and from Drell-Yan interactions; and on theoretical treatments that emphasize an explanation of the distribution functions, providing an overview of major contemporary approaches and issues. Valence-quark physics is a compelling subject, which probes at the heart of our understanding of the Standard Model. There are numerous outstanding and unresolved challenges, which experiment and theory must confront. In connection with experiment, we explain that an upgraded Jefferson Lab facility is well-suited to provide new data on the nucleon, while a future electron ion collider could provide essential new data for the mesons. There is also great potential in using Drell-Yan interactions, at FNAL, J-PARC and GSI, to push into the large- domain for both mesons and nucleons. We argue furthermore that explanation, in contrast to modeling and parametrization, requires a widespread acceptance of the need to adapt theory: to the lessons learnt already from the methods of nonperturbative quantum field theory; and a fuller exploitation of those methods.

pacs:
12.38.Qk, 12.38.Bx, 12.38.Lg, 13.60.Hb
Contents

I Introduction

From the first deep inelastic scattering experiments Bloom et al. (1969); Breidenbach et al. (1969) and the advent of the parton model Bjorken (1967); Bjorken and E.A.Paschos (1969); Feynman (1969) there has been a tremendous effort to deduce the parton distribution functions of the most stable hadrons – the proton, neutron and pion. The long sustained and thriving interest in these structure functions is motivated by the necessity to understand hadron structure at a truly fundamental level. While it is anticipated that quantum chromodynamics (QCD) will provide the explanation of hadron structure, a quantitative description of hadrons within QCD is not yet at hand. An explanation and prediction of the behavior of parton distribution functions in the valence-quark region;111As will become clear, for a given hadron in its infinite momentum frame, a parton’s Bjorken- value specifies the fraction of the hadron’s momentum carried by this parton. There is no unambiguous beginning to the valence-quark domain. We choose because thereafter the gluon distribution can be said to be much smaller than the valence -quark distribution in the proton. viz., Bjorken-, poses an important challenge for QCD and related models of hadron structure. This is the domain on which the transition takes place from deep inelastic scattering, with incoherent elastic scattering from numerous loosely-correlated small- partons, to scattering from dressed-quarks that become increasingly well correlated as . We will focus our attention herein on the behavior of unpolarized structure functions on the valence-quark domain.

The parton distributions are also essential to an understanding of QCD’s role in nuclear structure. It has been known Aubert et al. (1983) for more than two decades that the parton distribution functions in a nucleus cannot simply be obtained by adding together the distributions within the constituent nucleons. This mismatch is the so-called EMC effect and, although concerted efforts have led to the identification of some of the ingredients necessary to an explanation, we still lack a completely satisfactory understanding of the nuclear dependence of parton distribution functions. This hampers us enormously. For example, an accounting for the EMC effect in light nuclei is key to extracting the neutron structure function. That is a necessary precursor to a veracious determination of the differences between the parton distribution functions of the light-quarks.

An accurate determination of the pointwise behavior of distribution functions in the valence region is also important to very high-energy physics. Particle discovery experiments and Standard Model tests with colliders are only possible if the QCD background is completely understood. QCD evolution, apparent in the so-called scaling violations by parton distribution functions, entails that with increasing center-of-mass energy, , the support at large- in the distributions evolves to small- and thereby contributes materially to the collider background.

Deep inelastic scattering (DIS) of electrons from protons and bound neutrons at the Stanford Linear Accelerator Center (SLAC) led to the discovery of quarks. These experiments observed more electrons scattering with high energy at large angles than could be explained if protons and neutrons were uniform spheres of matter.222In both method and results this series of experiments, conducted from 1966-1978 and for which Taylor, Kendall and Friedman were awarded the 1990 Nobel Prize in Physics, was kindred to that which led Rutherford to discovery of the nucleus in 1911. In the approximately forty intervening years, electron DIS has played a central role in measuring structure functions, and SLAC has been especially effective in mapping the proton’s structure functions in the valence-quark region.

More recently, muon scattering experiments – performed by the European Muon Collaboration (EMC), the New Muon Collaboration (NMC) and the Bologna-CERN-Dubna-Munich-Saclay Collaboration (BCDMS) – have contributed to our store of information. There have also been a substantial number of neutrino scattering experiments. However, they have generally used nuclear targets rather than pure hydrogen targets. Drell-Yan experiments have been effective at measuring the anti-quark distributions in the proton and nuclei. Putting all this together, it can be said that as a consequence the proton structure function is extremely well known, at least for .

Herein we will discuss the status of charged-lepton and Drell-Yan experiments, as well as prospects for new experiments at, e.g., the Thomas Jefferson National Accelerator Faclity (JLab) and FermiLab (FNAL). On the other hand, although a number of experiments at the Conseil Européen pour la Recherche Nucléaire (CERN) and Deutsches Elektronen-Synchrotron (DESY) have also provided measurements of the proton structure function, many of these efforts focused on the low- behavior and hence they will not be discussed.

An experimental determination of the neutron structure function at high Bjorken- has proved especially troublesome, the main reason being that most of our information about the neutron structure function is obtained from DIS experiments on a deuteron target, for which the nature of the EMC effect is simply unknown. Consequently, inference of the neutron structure function from proton and deuteron measurements is model-dependent. We will treat this topic in some depth and, in addition, canvass prospects for the future.

On the theoretical side the challenge is first to parametrize and ultimately to calculate the parton distribution functions. We write ultimately because the distribution functions are essentially nonperturbative and therefore cannot be calculated in perturbation theory. Thus, absent a truly-accurate, quantitative and predictive nonperturbative tool, the QCD calculation of parton distribution functions remains an alluring but distant prospect. Herein we will provide both an historical perspective on past attempts at a theoretical interpretation and an overview of recent progress towards this goal. Notably, today, there are no computations of the pointwise behavior of the nucleon’s valence-quark distribution function that agree with the predictions of the QCD parton model.

There has, on the other hand, been much success to date with parametrization. A number of independent groups have analyzed and performed a global fit to the vast body of extant DIS and Drell-Yan nucleon structure function data. In this approach the physical processes are factorized into the product of short- and long-distance contributions. The so-called short-distance parts are calculable in perturbation theory, whereas the long-distance parts are determined by the parton distribution functions. In this way, via the operator product expansion in QCD, scattering processes involving hadrons are connected to subprocesses involving partons. The utility of this approach is grounded on the fact that so long as factorization is valid, the parton distribution functions are universal; namely, all hadron-level interactions for which factorization applies are described by the same small body of parton distribution functions. This being true, the nonperturbative problem is reduced to that described above: instead of calculating all cross-sections from scratch, one has merely to calculate the distribution functions. Herein we will introduce the modern parametrizations.

At its simplest, the nucleon is a three-body problem. As ostensibly a two-body problem, it would appear theoretically simpler to calculate properties of the pion and hence the parton distributions therein. However, the pion is in fact both a bound state and the Goldstone mode associated with dynamical chiral symmetry breaking (DCSB) in QCD. This amplifies the importance of understanding its properties. However, it also significantly complicates the calculation of the pion’s distribution functions and places additional constraints on any framework applied to the task: the pion simply cannot veraciously be described as a constituent quark-antiquark pair. Instead, key features of nonperturbative quantum field theory must be brought to bear.

Although it is not presently possible to perform deep inelastic scattering on a free pion, the pion structure function has been measured in pionic Drell-Yan experiments at CERN and FNAL. Owing to predictions from the QCD parton model, the form of the pion’s valence-quark distribution function provides a stringent test of our understanding of both QCD and QCD-based approaches to hadron structure. The FNAL Drell-Yan experiment Conway et al. (1989) measured the pion structure function up to high Bjorken-, where the QCD predictions were expected to be realized. However, they were not manifest, so that a very puzzling discrepancy remains at this time. The status of structure function measurements of the pion as well as prospects for future experiments at JLab and a possible future electron ion collider (EIC) will be presented, as will a full description of the theoretical perspective.

The purpose of this article is not to present a comprehensive review of all the issues involved in extracting structure functions from high energy data, as in Sterman (1995), nor an exhaustive digest of theoretical developments in the field. Rather the focus will be on structure functions in the valence region, which have chiefly been extracted from electron and muon inelastic scattering, as well as from Drell-Yan interactions. Moreover, we will focus on the theoretical treatments that emphasize an explanation of these structure functions and summarize the major contemporary approaches and issues. To end this Introduction, a brief remark on notation: unless denoted otherwise, when addressing a distribution function, it will be that associated with the proton; e.g., will mean the proton’s valence -quark distribution.

Ii Nucleon structure functions from electron and muon scattering

ii.1 Kinematics

Figure II.1: Feynman diagram representing deep inelastic lepton scattering from a hadron.

The Feynman diagram for lepton inelastic scattering from a hadron is given in Fig. II.1. Practically, the leptons in a laboratory scattering process can be electrons, muons or neutrinos, as illustrated in the figure, and the current can be a photon, a charged weak current , or a neutral weak current of four momentum transfer, . The incident lepton has incoming and outgoing four momenta of and , respectively, so that the momentum- and energy-transfers are given by

(II.1)
(II.2)

where the incident momentum and mass of the hadron are given by and , while the outgoing debris after the inelastic event has a momentum .

The invariants in the scattering process are: the square of the four momentum transfer; the square of the invariant mass; , which is related to the energy transfer in the target rest-frame; and the square of the total energy in the center-of-mass frame. The square of the four-momentum transfer depends on the energy of the beam and scattered lepton as well as the lepton scattering angle: neglecting the lepton’s mass,

(II.3)

The square of the invariant mass, W, is given by

(II.4)

where is the mass of the hadron and

(II.5)

is the Bjorken variable. The square of the total energy in the center of mass is given by the Mandelstam variable, :

(II.6)

A final, useful invariant is

(II.7)

which, in the target rest-frame, is a measure of the fractional energy loss by the incident lepton.

The Bjorken limit is theoretically defined as

(II.8)

In this limit, can be shown to equal the fraction of the hadron’s momentum carried by the struck quark. Empirically, in order to avoid complications associated with the production of hadron resonances, the Bjorken scaling regime is explored via

(II.9)

In principle, the deep inelastic approximation is only good when

(II.10)

However, it is empirically well established that scaling in deep inelastic scattering (DIS) sets in at tractable values of and . As a practical matter, evaluations of data typically impose the requirement that and . As discussed and explored, e.g., in Malace et al. (2009); Accardi et al. (2009), data at high Bjorken but not meeting these kinematic requirements would be subject to large target-mass corrections, which are kinematic and owe to binding of partons in the hadron. Before discussing these topics in more detail, the cross section for inelastic electron scattering will be presented.

ii.2 The cross section for charged lepton scattering

The cross section for inelastic charged-lepton scattering from a hadron via a one-photon exchange process, as indicated in Fig. II.1, is well known and given by

(II.11)

where is the leptonic tensor and is the hadronic tensor. The leptonic tensor has the following straightforward dependence on the kinematic variables:

(II.12)

where is the lepton’s mass. Conserving current and parity, the hadronic tensor for a spin- charged particle depends on two structure functions, and ; viz.,

(II.13)

Contracting the leptonic tensor with the hadronic tensor yields the well-known cross section of the form:

(II.14)

The quantities and in Eq. (II.14) are often written in terms of dimensionless structure functions, and , for the nucleon:

(II.15)
(II.16)

In turn, these structure functions are frequently written in terms of the transverse () and longitudinal () structure functions, which correspond to absorption of transverse and longitudinal virtual photons, respectively:

(II.17)
(II.18)

They can, in principle, be measured separately by performing a Longitudinal/Transverse (L/T) separation, as will be discussed in Sec. II.8. The most intuitive physical picture of the interaction in the deep inelastic scattering regime is provided by the parton model.

ii.3 The parton model

It was discovered at SLAC in the late 1960s that the structure functions appeared to scale; i.e., their evolution with is nearly independent of over a very large kinematic range. These important points are well documented, not only in the original papers Bloom et al. (1969); Breidenbach et al. (1969); Bjorken (1967); Bjorken and E.A.Paschos (1969); Feynman (1969) but also in the Nobel lectures summarized in Friedman (1991); Kendall (1991); Taylor (1991), and here will only be reviewed briefly.

The observed scaling feature means that the cross-section is not separately a function of the two kinematic variables, energy transfer, , and . Instead, the cross-section’s behavior can be expressed in a dependence on only one scaling variable; namely, Bjorken . Indeed, in the Bjorken-limit, Eq. (II.9), the proton structure functions and assume a form consistent with elastic scattering from a point fermion; viz.,

(II.19)
(II.20)

where is given by Eq. (II.5). This near-dependence on only a single dimensionless variable is referred to as Bjoken scaling. Assuming this to be the case, then the expression for the cross section, Eq. (II.11), becomes

(II.21)

Capitalizing on these results and assumptions, further analysis leads to the relation Callan and Gross (1969):

(II.22)

This “Callan-Gross relation” is observed to be approximately valid, a feature that is often cited as evidence that the pointlike constituents of the proton; i.e., the quarks, are spin- degrees-of-freedom.

An intuitive understanding of the observed scaling behavior is provided by the quark-parton-model (QPM) or QCD parton model. In this model the electron collides with a collection of partons within the hadron. Owing to Lorentz contraction of the hadron at extremely high momentum, these partons are practically frozen in time during the extremely brief collision process. Moreover, the probability of finding two partons near enough to interact with each other drops as . Thus, initial and final state interactions are suppressed in the DIS regime. Within this model, the photon interacts incoherently with individual partons and the cross section depends in a simple way on the probability of finding a quark of flavor “” with fraction of the proton’s momentum. These properties are realized in the following simple expression for the electromagnetic structure function

(II.23)

where the are the charges of the individual quarks and are the probability densities for finding quark- with fraction- of the hadron’s momentum. Assuming that only the relatively light quarks (, , , ) and their respective antiquarks contribute to the proton structure function, then the proton structure function can be written as

(II.24)
Figure II.2: (Color online) Quark flavor dependence of the proton structure function according to the CTEQ6L (leading order) evaluation at a scale GeV Stump et al. (2003); Martin et al. (2004a). We plot the quantities in order to emphasize the contribution to the structure function from from the various quark flavors.

In Fig. II.2 we depict the contributions of these quark flavors to the structure function in the valence region. The CTEQ6L leading order (LO) evaluation Stump et al. (2003); Martin et al. (2004a), a modern global evaluation of existing data, was used for illustration. The distribution of up and down quark flavors is fixed primarily by electron and muon scattering from the proton and neutron, while the distribution of the and quark sea is largely determined by Drell-Yan processes. The strange and charm quark sea is determined by neutrino interactions. The -quark distribution is the dominant contribution to the proton structure function, while the - and anti--quarks are significant. The charm and strange quark distributions are small and are not easily distinguishable in the figure. We note that semi-inclusive electron scattering from the proton and deuteron is emerging as an important tool for determining the strange quark distribution, thereby providing an important complement to the method of neutrino scattering from complex nuclei as a probe of non-valence proton structure functions. Of course, the and quark distributions in the valence region are particularly important for testing descriptions of nucleon structure, and it is therefore vital to obtain accurate and precise data, and a reliable flavor separation in this region.

A primary goal of modern theory is the computation of valence-quark distribution functions, using QCD-motivated or QCD-based models and nonperturbative methods in QCD. Naturally, valence quarks have no strict empirical meaning because experiment cannot readily distinguish a valence- from a sea-quark. However, valence-quark distributions for the - and -quarks in the nucleon can be defined as the following flavor-nonsinglet combinations:

(II.25)
(II.26)

Conservation of charge leads to important normalization conditions:

(II.27)

Naturally,

(II.28)

with the same result for all heavier quarks. However, it does not follow that must be identically zero.

Consistent with nomenclature, the combinations in Eqs. (II.25) and (II.26) have nonzero values of various flavor quantum numbers, such as isospin and baryon number. Singlet distributions also play a role and are constituted from a sum of quark distribution functions: . Under QCD evolution, discussed below, singlet distributions mix amongst themselves and with the gluon distribution, but the nonsinglet distributions do not.

ii.4 QCD and scaling violations

The proton structure function in the valence region is shown in Fig. II.3 as a function of Bjorken and for three values of for the CTEQ6L LO evaluation. Although the range illustrated is relatively large, from 2 GeV to 100 GeV, the structure function does not exhibit a strong dependence on . This result validates the parton model as an approximation to a QCD description of the structure function. Of course, the structure functions have a dependence on , indicating the necessity of QCD rather than a parton model to describe the data. This relatively small dependence is known as a scaling violation.

The validity of the QPM led to the idea of factorization in deep inelastic scattering. In factorization, the cross section is separated into two distinct contributions: a short-distance part, described by perturbative QCD (pQCD); and a long distance part. The -dependence driving scaling violation originates in the pQCD part and owes to the same phenomena that generate running of the strong coupling constant, , and hence asymptotic freedom. This factorization for the moments of the structure functions can be shown rigorously using the operator product expansion and the renormalization group equations. Alternatively, a distribution function where the moments can be written as products can be expressed in terms of a convolution of two distributions. Naturally, the long-distance part is only calculable using nonperturbative methods and presents a challenge for modern hadron theory.

Figure II.3: (Color online) Structure function of the proton at three values of , computed from the CTEQ6L leading-order evaluation.

At small distances or, equivalently, large momentum transfers in QCD, the effective coupling constant becomes small. This feature is known as asymptotic freedom Gross and Wilczek (1974); Politzer (1974). In terms of the renormalization scale, , the strong coupling constant has the perturbative form:

(II.29)

where: ; and is the number of quark flavors involved in the process. From Eq. (II.29) it is apparent that as , so that a parton-model limit is contained within QCD; e.g., as will become clear below, Eq. (II.23) is obtained from Eq. (II.37) in this limit.

Plainly, the definition of short- and long-distance phenomena is ambiguous, and associated with a separation of the cross-section into these two contributions is a factorization scale; namely, a mass-scale chosen by the practitioner as the boundary between hard and soft. If that scale is large enough, then the hard part of the scattering cross-section can be calculated in pQCD. This, however, introduces another scale, associated with renormalization. For convenience and economy, the factorization and renormalization scales are usually chosen to be the same. It follows, in addition, that factorization is scheme-dependent because the choice of renormalization scheme implicitly specifies a division of the finite pieces of the cross-section into those that are retained in the hard contribution and those understood to be contained in the soft piece. The part identified as owing to long-distance effects is basically the parton density distribution function.

In fitting data to construct PDFs, it is usual to use the (modified minimal subtraction) renormalization scheme whereas, in regard of factorization, there are two commonly used schemes. The DIS scheme Altarelli et al. (1978) was designed to ensure there are no higher-order corrections to the expression for the structure function in terms of the quark PDFs; i.e., all finite contributions are absorbed into the PDF. On the other hand, in the more widely used scheme Bardeen et al. (1978), in addition to the divergent piece, only the usual combination is absorbed into and hence the expression for exhibits explicit O corrections. [See Brock et al. (1995) for more on these points.]

The CTEQ6L (leading-order) structure functions shown in Fig. II.3 were evaluated using the -scheme (modified minimal subtraction). At leading-order there is no difference between factorization schemes. As we will subsequently see (Fig. II.4), radiated gluons give rise to the scaling violation or -dependence of the structure functions. They, and the quark distribution functions defined therefrom, then depend on both and . The dependence is now routinely described within the framework of next-to-leading-order (NLO) QCD evolution.

An important and extremely useful feature of factorization is that a measurement of a structure function at relatively-low permits the calculation, through the use of pQCD, of the structure function at high . In leading order, a set of integro-differential equations, now known as the DGLAP evolution equations Dokshitzer (1977); Gribov and Lipatov (1972); Lipatov (1975); Altarelli and Parisi (1977), are used for this purpose. Intuitively, one may think that as increases, a parton can sometimes be resolved into two partons; e.g., a quark can split into a quark and a gluon, or a gluon into two gluons – see Fig. II.4 for a graphical representation of the leading order DGLAP equations. The two resolved partons then share in the fraction of the nucleon momentum carried by the initial unresolved parton at lower . The parton distribution thus becomes a function of . Such evolution of the parton distribution function (PDF) can be seen in Fig. II.3. At the highest values of the structure function is shifted toward lower values of . This work has been generalized to other reaction processes Furmanski and Petronzio (1982).

Figure II.4: Graphical representation of the leading order DGLAP equations.

We note that in principle the domain of “relatively-low ” means, nevertheless, . Notwithstanding this, in some empirical determinations of PDFs through data fitting [see Sec. V.3] and in the application of models to their calculation [see Sec. VI], evolution is sometimes applied from low . In such cases an interpretation of the low- distribution as a parton model density is questionable owing to the probable importance of nonperturbative corrections. However, contributions from such corrections are suppressed by DGLAP evolution to larger- and hence one anticipates that in these applications there is always a whereafter the desired interpretation becomes valid.

As stated above, the evolution equations enable the accurate calculation of the PDFs at a general , provided they are known at another scale, , so long as pQCD is a valid tool at both scales. At next-to-leading order the QCD evolution equations have the form Herrod and Wada (1980)

(II.30)

where the convolution is defined as

(II.31)

and

(II.32)

In these formulae, and or are the non-singlet PDFs, while is the singlet combination. The splitting functions are given by

(II.33)
(II.34)

where

(II.35)

The leading order DGLAP equations are obtained simply by eliminating the terms in the equations above.

It will be apparent from Eqs. (II.30) and (II.31) that is a fixed point; namely, that the value of each distribution function is invariant under evolution – it is the same at every value of the resolving scale . This is because the right-hand-side of each evolution equation vanishes at . Naturally, when Bjorken- is unity, then and hence one is strictly dealing with the situation in which the invariant mass of the hadronic final state is precisely that of the target: in Fig. II.1; viz., elastic scattering. The structure functions inferred experimentally in the neighborhood of are therefore determined by the target’s elastic form factors, which all vanish in the limit . This fact per se is not interesting. However, the rate at which distribution functions vanish with is, because it can lead to nonzero renormalization-scale-invariant distribution-function-ratios at . For example, the value of in the nucleon is an unambiguous, scale invariant feature of QCD and hence a discriminator between models. [For a concrete illustration of this point, see also Fig. VI.42, which depicts .] This is an important consequence, and one much discussed and explored in experiment and theory. (See Secs. II.7 and VI.)

We reiterate, however, that the neighborhood of poses difficulties for experiment, and for theory, too, owing, e.g., to the importance on this domain of target-mass corrections and higher-twist contributions (explained below). Whilst the equivalence between and elastic scattering cannot be avoided, one might choose to approach the limit obliquely. For example, noting from Eq. (II.9) that

(II.36)

one would ideally choose a path such that both and are simultaneously kept as large as kinematically possible. This is experimentally very difficult, as discussed in connection with Fig. II.7. Future experimental possibilities are outlined in Sec. IV.

A great triumph of QCD is the good agreement between the evolved structure function for the proton and experiment over many orders of magnitude in . Of course, the structure function should be calculated at the same order as the PDFs. The equation for the structure function at NLO in the DIS scheme is given by

(II.37)

where is the gluon distribution and the Wilson coefficients, , are given in Furmanski and Petronzio (1982); Glück et al. (1995a).

Figure II.5: (Color online) Comparison of the proton’s distributions in LO, NLO and NNLO at

During the past several years rather complete NNLO fits Martin et al. (2007) in the scheme have been performed for the data. These fits have made use of NNLO splitting functions Moch et al. (2004); Vogt et al. (2004). The LO, NLO and NNLO -quark distribution functions are plotted at a scale of in Fig. II.5 for comparison. Naturally, the structure functions themselves are measured quantities and do not depend on the order of the fit. In the valence region, the largest difference occurs between LO and NLO as expected. There is some evidence Yang and Bodek (2000) from early NNLO analyses that evolution at NNLO largely offsets the effect of high twist in an NLO analysis on the large- domain. However, a more recent analysis Blumlein et al. (2007) performed up to N3LO contradicts this finding.

ii.5 High twist effects and target mass corrections

As we have seen, the interactions between partons at short distances that give rise to scaling violations are reasonably well described in a pQCD approach through the -evolution equations. Probes of a hadron at intermediate values of might expose correlations among the partons. Thus far, we have only considered the case in DIS where structure functions are governed by incoherent scattering from the partons. As we move to lower where nucleon resonances could dominate or to very high where the elastic scattering limit and complete coherence dominates, then correlations between the partons must be taken into account. Ultimately a theoretical treatment of these correlations demands an understanding of the way that quarks and gluons bind to form hadrons. Although perturbative QCD cannot explain binding effects, it does permit one to construct a model of power corrections to the perturbative result. It was found Ellis et al. (1982, 1983) that the first power correction, , to the structure function is governed by the intrinsic transverse components of the parton momentum.

The operator product expansion is normally used to order contributions to a DIS structure function according to their twist, , which is defined as the difference between the naive mass-dimension of an operator and its “spin”. Here “spin” means the number of vector indices contracted with the configuration-space four-vector, . For example Sterman (1995), consider the expansion of a dimension-six current-current correlator, typical of DIS,

(II.38)

where is the factorization scale. One is interested in the behavior of the coefficient function because the more highly singular is this function on the light-cone, the more important is the associated operator at large . Now, suppose the operator has dimension , then, in order to balance dimensions on both sides of Eq. (II.38), one must have , since each factor of on the right-hand-side contributes mass dimension negative-one. Plainly, the largest value of is obtained with the smallest value of twist , and the associated operators are those which dominate the large- behavior of the correlator. As Eq. (II.38) indicates, numerous operators with different dimensions have the same twist and all such operators are associated with the same degree of light-cone singularity.

As evident in the example above, the leading-twist contributions in DIS are twist-2. In unpolarized DIS, the higher twist components; i.e., twist …, are suppressed by , , …, respectively. Thus, in general a structure function should be expressed in the form Alekhin et al. (2004)

(II.39)

where refers to the leading-twist part. As a practical matter, the magnitudes of higher-twist terms are generally unknown and the estimates are therefore somewhat controversial. Before one can claim discovery of higher twist components, it is essential that these be distinguished from target mass corrections and -evolution effects.

Figure II.6: (Color online) Empirical nonsinglet higher-twist coefficient, denoted , as a function of Bjorken-, for the proton (left panel) and deuteron (right panel), as obtained when the twist-2 contributions are treated at NLO, NNLO and N3LO. [Figure adapted from Blumlein et al. (2007).]

Several analyses have shown that higher-twist coefficients can become quite large at high- and relatively low . An analysis Virchaux and Milsztajn (1992) of BCDMS and SLAC data indicated that the twist-4 coefficient is sizable for the proton and deuteron. Here, however, the inclusion of target mass corrections reduced the magnitude of the twist-4 coefficient. This finding was supported in an analysis that included NNLO Yang and Bodek (2000). It was found that the NNLO contribution can partially compensate for a large higher-twist effect. More recently, an analysis Blumlein et al. (2007) of world data indicated that the twist-4 coefficient is sizable for NLO, NNLO and N3LO for the proton and deuteron. While increasing the order of the QCD analysis of the data reduces the twist-4 coefficient, the coefficient is significant at high . This is displayed in Fig. II.6, which plots an empirically determined correction defined via

(II.40)

where describes inclusion of target mass corrections of the twist-2 contributions to the structure function. At present, there exists no statistically significant determination of the magnitude of the twist-6 coefficient.

In connection with the size of higher-twist contributions, it is relevant to consider their impact on Bloom-Gilman duality Bloom and Gilman (1971). This is an expectation that the nucleon structure function, when measured in the region dominated by low-lying nucleon resonances, should follow a global scaling curve that is determined by high energy data. NB. Whilst it has been observed that the standard application of DGLAP evolution to deep inelastic structure functions can appear to be inconsistent with Bloom-Gilman duality at fixed , this apparent conflict is resolved if one takes into account the fact that the struck quark is far-off shell in the domain Lepage and Brodsky (1980); Brodsky (2005).

Underlying Bloom-Gilman duality is the assumption of quark-hadron duality; namely, a belief that one obtains equivalent descriptions of physical phenomena irrespective of whether one uses partonic or hadronic degrees-of-freedom Shifman (2000); Melnitchouk et al. (2005). While this assumption is probably correct in principle, in comparisons between computations that use finite-order truncations, it can and is violated in practice. The presence of large higher-twist contributions can therefore entail an apparently strong violation of Bloom-Gilman duality.

Experiments, particularly at the highest values of , might not be performed at sufficiently large momentum transfer to avoid high-twist effects and target-mass corrections. A pQCD treatment is truly valid only in the limit that the mass of the hadron is negligible in comparison with the dominant scale, . The target-mass corrections are kinematic and owe to binding of partons in the hadron. Nachtmann first pointed out Nachtmann (1973) that at finite and non-vanishing target mass, the scaling variable should be the fraction, , of the nucleon’s light front momentum carried by the parton; viz.,

(II.41)

Naturally, this Nachtmann variable reduces to Bjorken- for .

It was subsequently shown Georgi and Politzer (1976) that at leading-order, if the quantity is reasonably small, then it is straightforward to apply corrections for finite mass to the structure function. If the leading-twist structure function is denoted by and the target-mass corrected structure function by , then one has the expressions:

(II.42)
(II.43)

A concern with the Nachtmann or the Georgi-Politzer approach is the so-called “threshold problem” It results from the fact that the maximum kinematic value of is less than unity, which means that the corrected leading twist structure function does not vanish at .

Target-mass corrections have been extended to NLO QCD DeRújula et al. (1977), and generalized to include charge and neutral weak current deep inelastic scattering Kretzer and Reno (2004). De Rújula et al argued that the threshold problem could be solved by considering higher-twist effects.

A recent interesting approach Steffens and Melnitchouk (2006) to the “threshold problem” followed the original Georgi-Politzer approach but with one critical difference. Namely, the upper limit of the integrals in Eq. (II.42) becomes the upper limits of and that are permitted by kinematics at rather than merely unity. In terms of existing data, this solution does not result in a large effect for , but can have a significant effect below where the underlying assumption of factorization becomes questionable anyway.

In addition to target-mass corrections, there are also effects from nonzero current-quark mass Barbieri et al. (1976). Such corrections will not be discussed herein because of our focus on high-, whereat heavy quarks have little effect in nucleons and Goldstone bosons. These effects, in addition to those arising from target-mass corrections, are discussed in a recent review Schienbein et al. (2008). Furthermore, fresh work Accardi and Qiu (2008) has pointed out that changing the upper limit of the integrals in the Georgi-Politzer equations leads to an abrupt cutoff at , and proposed a “quark jet” mechanism to give a more reasonable approach to the very large behavior.

ii.6 The proton structure function

Measurements of structure functions at very high values of are extremely challenging. The main problems can be seen at a glance from Fig. II.7. Here the is plotted from Eq. (II.4) as a function of . Typical evaluations of the parton distribution functions require that . Clearly, the necessary to meet this demand is extremely high and impractical, at present, for .

Figure II.7: vs.  for three values of . Measurements at very high demand very high to remain in the DIS region.

The first DIS measurements, performed at SLAC, were limited to a beam energy of 24.5 GeV. Although this beam energy limit is a serious problem for very high measurements, the SLAC experiments had good control of systematic errors. Relatively high luminosities were possible, the incident electron energy was very well known and the spectrometer properties were relatively well understood. A consistent re-analysis of all the early SLAC experiments was performed in Whitlow et al. (1992), and the result is displayed in Fig. II.8.

Figure II.8: Proton structure function vs. Bjorken- with . For ease of representation: the upper band of data were multiplied by a factor of 10 and correspond to the domain ; the next band are scaled by a factor of 3 and cover the domain: ; the third-lowest band are unscaled and cover: ; and the lowest band are scaled by a factor of 1/5 and cover: .

After the importance of the early SLAC experiments was realized, the quest for additional data, particularly at high , led to muon scattering experiments at CERN and FNAL. At CERN a series of experiments was performed at muon energies between 90 and 280 GeV by the NMC collaboration Amaudruz et al. (1992), between 120 and 280 GeV by the BCDMS collaboration Benvenuti et al. (1990b) and at 280 GeV by the EMC collaboration Aubert et al. (1987). In this case, the combination of very high beam energies and relatively low luminosities makes it difficult to obtain structure function data at very high . The practical limit is about as shown in Fig. II.8. The FNAL E665 experiment provides data at even lower than the CERN experiments since the beam energy was 470 GeV.

Data for various ranges from the BCDMS Benvenuti et al. (1990b), EMC Aubert et al. (1983, 1987), NMC Amaudruz et al. (1992); Arneodo et al. (1997a) and SLAC Whitlow et al. (1990) experiments are plotted in Fig. II.8. The effect of the kinematic constraints on the reach in is evident throughout the relatively wide range of values for . The SLAC data persist out to the highest values of , but only for relatively low values of .

If the condition on is relaxed to , then data from SLAC exist up to as indicated in Fig. II.9. Clearly, the CTEQ6L evaluation at the very highest values of deviate from the data. However, if target mass corrections are taken into account, as shown by the dashed curve in the figure, then good agreement is restored. This illustrates that target mass corrections are indispensable in order to fully understand and extract parton distributions at very high .

Figure II.9: (Color online) The proton structure functions vs. Bjorken- with GeV. The dashed black curve represents the CTEQ6L evaluation with , while the solid red curve indicates the result obtained when target mass corrections are incorporated Malace et al. (2009).

ii.7 The neutron structure function

Interest in the neutron structure function at very high has burgeoned during the past decade. The neutron structure function, or at least the ratio of the neutron structure function to that of the proton, is believed to be one of the best methods to determine the ratio. A knowledge of the this ratio in the valence region would provide an important constraint on models for the nucleon Isgur (1999); Brodsky et al. (1995). For example, if one were to assume that a simple flavor symmetric model for the nucleon were valid, then the ratio would just be 1/2 for the valence quark distributions. In this model, the nucleon and would be degenerate, so this is clearly not a correct description of the nucleon. Other interesting limits occur, for example, where the -quark in the proton is “sequestered” in a pointlike scalar diquark. In this model, and . Furthermore, in a world where pQCD could be applied naively and scattering at large- involves only quarks with the same helicity as the target hadron, then and . [An extensive discussion of these issues is provided in Sec. VI, where it is argued, e.g., that and only with reliable data at will one be empirically able to determine the behavior.]

The parton model is a good starting point for introducing the neutron structure function. Under the assumption of isospin symmetry and recalling Eq. (II.27), the neutron structure function is simply obtained from Eq. (II.24) by the operation ; viz.,

(II.44)

Neglecting the -quarks, the expression for the neutron-to-proton structure function ratio is

(II.45)

From this equation it is readily apparent that the ratio is: when u quarks dominate; 4 when d quarks dominate; and, if the sea dominates, the ratio approaches unity. Within the parton model, Eq. (II.45) leads to the Nachtmann limit Nachtmann (1973),

(II.46)

As interest has grown, a number of new methods for measuring the ratio have been proposed. One approach Fenker et al. (2003) is to tag very low momentum protons emerging from the deuteron when deep inelastic scattering from the neutron is performed. This technique minimizes off-shell effects since at very low momentum the neutron is practically a spectator in the deuteron. Another method Afnan et al. (2000, 2003); Bissey et al. (2001); Petratos et al. (2006) is to perform deep inelastic scattering from He and H at very high . In forming the ratio of the scattering rates from these two nuclei, the nuclear effects cancel to a high degree in extracting the ratio.

Two proposed methods avoid nuclear effects altogether. Both a ratio of charge-current neutrino to antineutrino scattering from the proton and parity-violating deep inelastic electron scattering at high from the proton are sensitive to the ratio. Nevertheless, both of these methods are fraught with technical difficulties, particularly at high .

The primary problem with measuring the neutron structure function is that there exists no practical free neutron target. Typically, a deuteron target is employed in experiments. The neutron structure function is then extracted from a measurement of the proton and deuteron structure functions by employing a model for the deuteron wave function. As an example, we consider the covariant approach of Thomas and Melnitchouk (1998). In this case, the proton and neutron structure functions are convoluted with a nucleon density function in the deuteron. The neutron is also off shell, so that a model for the off-shell behavior is necessary. Then the expression for the deuteron structure function is given by

(II.47)

where is the probability of finding a nucleon of momentum in the deuteron and is the off shell correction. The extraction process is iterative. First, the off-shell effect is subtracted from the measured deuteron structure function. Then the proton structure function, convoluted as indicated by the first term in Eq. (II.47), is subtracted from the remainder, leaving the convoluted neutron structure function. The neutron structure function is then deconvoluted to give the neutron structure function. The last three steps are then repeated until convergence is achieved.

The results Thomas and Melnitchouk (1998) of this procedure for the ratio of the neutron to proton structure functions are shown in Fig. II.10. In this work a covariant deuteron wave function and a consistent off-shell correction were used Melnitchouk et al. (1994). Another application Burov et al. (2004) of the covariant approach indicates that data for the deuteron structure function at very high are essential for constraining the high neutron structure function. Although the convolution approach has been used by a number of authors as a step toward explaining the EMC effect, the convolution method has no firm theoretical basis.

Figure II.10: The ratio of the neutron to proton structure functions as a function of . The results of three extraction methods are shown.

Two other extractions, also shown in the figure, yield very different results at high . The extraction, labeled “Whitlow et al (Paris)” Whitlow et al. (1992) made use of light-front kinematics with the null vector along the incident beam direction and an early deuteron wave function. The extraction labeled “Whitlow et al (EMC)” Whitlow et al. (1992) made a density-dependent extrapolation of the EMC effect in heavy nuclei to that in the deuteron. There is significant controversy Yang and Bodek (1999); Melnitchouk et al. (2000); Yang and Bodek (2000) surrounding the density-dependent extrapolation method.

A recent approach Arrington et al. (2009) made use of the light-front, with null-plane dynamics and modern deuteron wavefunctions. This framework, which can be applied even when the the constraints of DIS kinematics are not met, greatly simplifies the convolution equation in terms of the nucleon structure functions. The analysis established that it is important to account properly for the -dependence of the data and incorrect to use a simple convolution formula to analyze data, if that data does not satisfy the DIS kinematic constraints. When the CD-Bonn potential is used to determine the deuteron wave function, the method of Arrington et al. (2009) produces a ratio similar to that labeled “Whitlow et al (Paris)” in Fig. II.10, which was obtained using a light-front approach with the null vector aligned along the direction of the incident electron beam Frankfurt and Strikman (1981). Whilst this particular choice of null-vector orientation is consistent with DIS kinematics, the very high SLAC data, included in the analysis, are not. They are usually excluded when fitting PDFs.

ii.8 The longitudinal structure function

Before the structure function can be extracted from data, one must either know or measure the structure function. By combining Eqs. (II.14), (II.15) and (II.16), the expression for the cross section in terms of and is given by

(II.48)

This cross section is often expressed in terms of the and structure functions in the following way:

(II.49)

where

(II.50)

and

(II.51)

Here, and refer to the longitudinal and transverse cross sections, respectively. Since the quarks are pointlike spin- objects, the cross-section for absorption of a longitudinal photon is small in comparison with that for a transverse photon. Furthermore, a vector-vector interaction preserves chirality, so the longitudinal cross-section will depend on violations of this chirality. Violations of order would be expected, where is the struck quark’s current-mass. Intrinsic transverse quark momentum, , as well as higher orders in from the gluon distribution also can give rise to an increase in the longitudinal cross section. For example, the Bjorken-Feynman model gives the result Feynman (1972) that

(II.52)

The QCD contribution to , and consequently to , of order is given by Reya (1981); Altarelli (1982)

(II.53)

where is the gluon distribution function, and only four flavors of quarks are considered in the second term. An excellent extraction Whitlow et al. (1990) of and was performed from a global analysis of the SLAC data.

It has long been believed that a tight constraint on the gluon structure function can be built from good measurements of . The best extractions for in the valence region are from the lower energy SLAC and JLab data Whitlow et al. (1990); Yang and Bodek (2000); Tvaskis et al. (2007). The best extractions of are from HERA data, but at lower than the subject of this review. Nevertheless, the deduced gluon distribution is very sensitive to the order of used in the analysis Martin et al. (2006).

Figure II.11: (Color online) Dressed-quark mass function, : solid curves – Dyson-Schwinger equation (DSE) results, obtained as explained in Bhagwat et al. (2003); Bhagwat and Tandy (2006), “data” – numerical simulations of unquenched lattice-QCD Bowman et al. (2005). (NB. MeV is the uppermost curve and current-quark mass decreases from top to bottom.) One observes the current-quark of perturbative QCD evolving into a constituent-quark as its momentum becomes smaller. The constituent-quark mass arises from a cloud of low-momentum gluons attaching themselves to the current-quark. This is dynamical chiral symmetry breaking (DCSB): an essentially nonperturbative effect that generates a quark mass from nothing; namely, it occurs even in the chiral limit. [Adapted from Bhagwat et al. (2007).]

ii.9 The gluon structure function

Interest in the gluon structure function has grown markedly during the past decade. Part of the interest resides in the fact that gluons comprise more than 98% of the rest-mass of the nucleon as well as more than half of it’s light-front momentum. This interesting aspect of the role of glue in the nucleon can be seen from Fig. II.11 . Here the mass of a light-quark is plotted as a function of the quark’s momentum. The “data” in the plot are results from lattice simulations, while the curves are from Dyson-Schwinger equation (DSE) calculations. The curve labeled “chiral limit” is the DSE result obtained with the current-quark mass set to zero. Clearly, even if the current-quark mass vanishes, the quark mass rises rapidly to near 0.3 GeV at low quark momentum. This is dynamical chiral symmetry breaking (DCSB) and a clear demonstration in QCD of the effect on the quark mass produced by the presence of gluons with strong self-interactions.

Sensitivity to the gluon structure function via lepton beams can be produced by a careful measurement of scaling violations; i.e., measurements of the longitudinal structure function or the partial derivative of with respect to . This is particularly effective at low values of where this derivative is directly proportional to the gluon distribution function at first order.

Iii Distribution functions from Drell-Yan interactions

Informative reviews on the Drell-Yan process exist Kenyon (1982); Reimer (2007). In this section we emphasize that the Drell-Yan interaction provides: (i) the cleanest access to the antiquark distributions in the valence region in hadrons; (ii) a means to determine the quark distributions in the proton at very high ; and (iii) the most information on the parton distributions in mesons. Naturally, in order to isolate the valence quark distributions in the proton, it is imperative to measure the antiquark distributions.

iii.1 The Drell-Yan interaction

The Drell-Yan process was devised Drell and Yan (1970a, 1971) to explain hadron-hadron collisions where an anti-lepton lepton pair is produced. For example, two hadrons, and , collide and produce the lepton pair :

(III.1)

At relatively large values of momentum transfer, say greater than a few GeV, the underlying process is believed to be dominated by antiquark-quark annihilation:

(III.2)
Figure III.12: Feynman diagram representing the Drell-Yan interaction

The Feynman diagram for this underlying Drell-Yan interaction is given in Fig. III.12. Here, two hadrons collide and a time-like photon of mass is formed from the annihilation of a quark in one of the hadrons with an antiquark in the other. The photon then decays by emitting an antilepton-lepton pair. In a typical experiment, the lepton pair is often an antimuon-muon pair, as a matter of technical convenience.

In this process the square of the four momentum transfer is just given by the square of the dilepton mass, . At leading-order, the Drell-Yan cross-section can be determined in a straightforward manner to yield:

(III.3)

where refers to the structure function of the quark of flavor in the beam (b) or target (t) hadron, respectively. It has been shown Altarelli et al. (1979) that if the structure functions have the same definition as those in deep inelastic scattering; namely, that if one employs the DIS factorization scheme, then the NLO part is calculable and becomes a multiplicative factor to the expression above, as discussed in the next subsection.

Figure III.13: Feynman graphs of NLO corrections to the Drell-Yan process. Top row – Compton-like scattering in QCD; viz., (and also , not shown explicitly). Middle row – QCD radiative correction; viz., . Bottom diagram – QCD vertex correction. [Figure adapted from Reimer (2007).]

iii.2 QCD and higher order corrections

The NLO QCD processes that contribute to Drell-Yan scattering are depicted in Fig. III.13. These processes lead to a modification of the Drell-Yan cross section by introducing the so-called factor:

(III.4)

With PDFs defined in the DIS factorization scheme, the -factor is given approximately by Altarelli et al. (1979)

(III.5)

and assumes a value between 1.5 and 2. The consideration of NNLO, as well as NLO diagrams, also leads to a simple factorization of the cross-section and an approximate factor of two for . The factorization scheme dependence of the K-factor is described at length in van Neerven and Zijlstra (1992). We note in addition that the factor depends on kinematics, a fact shown in Wijesooriya et al. (2005) to be important at very high for pionic Drell-Yan studies.

iii.3 High- quark distribution functions

The Drell-Yan process presents a valuable method for measuring parton distribution functions in hadrons at very high . For example, it can be used to probe the quark distribution in the beam proton. To see how, consider that if - and -quarks are neglected and the beam-target kinematics are chosen such that is large, then for protonproton collisions, Eq. (III.4) can be rewritten Webb (2003)

(III.6)

because and for large-. Now suppose that the target is a deutron with a similar kinematic setup, then Eq. (III.4) can be written Webb (2003)

(III.7)

where, as usual herein, means the distribution of flavor--quarks in the proton; and one has assumed isospin symmetry and neglected nuclear binding effects in the weakly-bound deuteron. It is thus apparent that this kinematic setup produces Drell-Yan cross sections that are primarily sensitive to the valence distributions in the proton beam, and the antiquark distributions at small in the proton and deuteron targets.

Figure III.14: (Color online) Ratios of the measured Drell-Yan (squares) and (triangles) cross-sections to NLO calculations based on the MRST 2001 PDF fit Martin et al. (2002a), plotted as a function of beam momentum fraction, here labeled , with averaging over the target momentum fraction. The shaded area between the solid lines represents the experimental uncertainty ranges Martin et al. (2003) on in the MRST 2001 PDF fit. The data are from the FNAL E866 experiment. [Figure adapted from Webb (2003); Webb et al. (2003).]

Some results from analysis of the FNAL E866 Drell-Yan data are reproduced in Fig. III.14. In this figure the E866 results for and collisions are divided by the appropriate differential cross-section computed using the MRST 2001 PDFs Martin et al. (2002a). Analogous ratios plotted as a function of Webb et al. (2003) indicate that the MRST 2001 PDFs provide a very good description of the cross-section’s -dependence on the complete -domain, and a good description of the cross-section for . In this light, consider first the data, which hints that the plotted ratio is smaller than one at . Given that the -quark is responsible for roughly 80% of the cross-section, this observation can be interpreted as an indication that the MRST 2001 PDFs overestimate the proton’s valence -quark distribution. There is a strong signal from the cross-section that the plotted ratio is less than unity for . Given that this cross-section is proportional to and the greater suppression, one can argue that the PDFs overestimate the proton’s valence -quark distribution. A consideration of the impact of this and other recent data on PDF fits is presented in Owens et al. (2007).

iii.4 ratio and the Gottfried sum rule

One of the most celebrated applications of the Drell-Yan process is the measurement of the flavor dependence of the antiquark sea in the proton. This was first suggested more than twenty years ago Bickerstaff et al. (1984, 1986); Garvey (1987); Ellis and Stirling (1991), at which time it was usually assumed that the light-quark sea was flavor symmetric. Such experiments led subsequently to a demonstration that this is not true; namely, the light-quark sea is flavor asymmetric.

One of the first indications that the light-quark sea might be flavor asymmetric was observation of the violation of the Gottfried sum rule. The Gottfried sum rule is defined by Gottfried (1967)

(III.8)

Using Eqs. (II.25) – (II.27) and assuming isospin invariance, becomes:

(III.9)

If the sea is flavor symmetric, then the Gottfried sum just evaluates to . Any other value is termed a “violation” of the sum rule.

The earliest hints of a possible violation of the sum rule can be found in data from SLAC Bodek et al. (1973) and Fermilab Ito et al. (1981). The NMC experiment at CERN gave the result Amaudruz et al. (1991); Arneodo et al. (1994, 1997a)

(III.10)

significantly different from . A more recent re-evaluation of the Gottfried sum, using a neural network parametrization of all then available data on the nonsinglet structure function Benvenuti et al. (1990a, 1989); Arneodo et al. (1997b), yielded Abbate and Forte (2005), in agreement with the earlier analyses.

Of course, it was pointed out long ago that Pauli blocking would give some enhancement of the ratio Field and Feynman (1977). This mechanism might have been sufficient to explain the early SLAC data but Drell-Yan experiments have since given far more information about the magnitude of the effect. Pion cloud models also generate such an effect Thomas (1983).

 Experiment  
 NMC 0.147 0.039
 HERMES 0.16  0.03
 E866 0.118 0.011
Table III.1: Values of , which appears in the Gottfried sum rule, Eq. (III.9), compiled in Peng (2003). The original sources are: NMC (DIS) Amaudruz et al. (1991); Arneodo et al. (1994); HERMES (semi-inclusive DIS) Ackerstaff et al. (1998); and E866 (Drell-Yan) Hawker et al. (1998); Peng et al. (1998); Towell et al. (2001). NB. At NNLO, is almost scale-independent for GeV, as illustrated in Fig. 3 of Abbate and Forte (2005).

A compilation Peng (2003) of values for , which characterizes the second term in Eq. (III.9), are given in Table III.1. The relative agreement is reasonable among these determinations. The most accurate result quoted is from the Drell-Yan experiment, FNAL E866, and the comparison is meaningful because is almost scale-independent for