Factorization theorems in perturbative QCD give a justification for and improvement of parton-model predictions. In the "QCD improved" parton model, physically observed cross sections involving hadrons can be written as convolutions of perturbatively calculable partonic hard parts with parton distributions, which summarize uncalculable non-perturbative effects (Owens and Tung, 1992; see Sec. IV above).
The parton distributions are often presented as functions of x and
f;
and are customarily interpreted as the probability densities for
finding a parton within a hadron, with its momentum fraction between
x and x+dx. Below we denote the factorization scale by
f.
Although perturbative QCD cannot predict the absolute normalization of
these parton distributions, their evolution with the factorization
scale can be calculated
(Sec. IV.B.3).
More precisely, the scale dependence is governed by a set of coupled
integro-differential evolution equations, valid to all orders in
s
(Gribov and Lipatov, 1972a;
Altarelli and Parisi, 1977),
where t=ln(f2 /
2 ),
and the subscript q denotes quark flavors. The kernels,
Pij(z), have the physical interpretation of probability
densities for obtaining a parton of type i from one of type
j with a fraction z of the parent parton's momentum. At
the leading order, the Pij(z) are given in
Eq. (6.7)
above. The next-to-leading-order (or two-loop) expressions for
Pij(z) were calculated by several groups
[30]. Up until recently, there had
been an unresolved minor discrepancy for Pgg(z)
between results obtained in different gauges. This has now been clarified
(Hamberg and van
Neerven, 1992).
This set of equations can be solved exactly in moment space
(Reya, 1981;
Altarelli, 1982),
once a set of input distributions is specified at an initial value
0.
One can then invert the moments to get the x and
f-dependent parton distributions.
However, this method requires the knowledge of initial parton
distributions at all values of x from 0 to 1, and no
experimental measurements at fixed
f
can reach all the way to x=0. In current global analysis
of parton distributions, one solves this set of equations
numerically. Note that one needs input distributions only for x
greater than or equal to the smallest momentum fraction at which
parton distributions are desired.
Global analysis of parton distributions involves making use of experimental data from many physical processes and using the parton evolution equations to extract a set of universal parton distributions which best fit the existing data. These distributions can then be used in predicting all other physical observables at energy scales far beyond those presently achievable. Herein lies the wide-ranging usefulness of the QCD improved parton model. Beyond this, however, the very possibility of a global fit tests the internal consistency of our fundamental theoretical picture of hard scattering, based on factorization and the universality of parton distributions.
A typical procedure for global analysis involves following necessary steps:
In all high-energy data, deeply inelastic scattering of leptons on
nucleon and nuclear targets remains the primary source of information
on parton distributions, because of its high-statistics. Such data
are known to be mostly sensitive to certain combinations of quark
distributions. Drell-Yan lepton pair production, and direct photons
at large transverse momenta provide important complementary
information on antiquark and gluon distributions. Most data used in
obtaining recent parton distributions are at fixed target energies.
Collider results have not reached the accuracy necessary to be
included in global fits, but they will eventually offer a significant
opportunity to probe the small-x region (say x
10-3).
Parton distributions defined in different factorization schemes are
different. The commonly used factorization schemes in the literature are
the DIS and
schemes. In principle, parton distributions obtained in one scheme
can be directly transformed into the other scheme. However, the
transformation is not reliable in certain kinematic regions where the
perturbation series expansion has abnormal behavior
(Owens and Tung, 1992).
It is preferable to perform independent analyses in these schemes.
The truncation of the perturbation series invariably leads to renormalization and factorization scale dependence for QCD predictions. Consequently, parton distributions obtained from global analysis will depend on the choice of scales. If significant scale dependence is found to exist in a particular kinematic region for some processes, then the usefulness of such data is limited, until new theoretical techniques are developed to reduce that dependence.
There is considerable freedom in choosing the parametric form of the
input parton distributions at scale
0. The parameterization must be general
enough to accommodate all the possible x and quark-flavor
dependence, but it should not contain so many parameters that the
fitting procedure becomes very much underdetermined. In practice, for
each flavor it is common to use a functional form
where P(x) is a smooth function. In above expression, xA1 dominates the small x feature and (1-x)A2 determines the large x behavior.
When calculating the
2
for a given fit, both statistical and systematic errors should be
taken into account. The most expedient, and hence the most often
used, method is to combine these errors in quadrature
(Morfín and Tung, 1991).
However, real systematic errors are correlated; they must eventually
be incorporated in that way when the analysis reaches a truly
quantitative stage.
After minimizing the
2 (e.g., using the MINUIT package of CERN
library), the resultant parton distributions can be presented in two
ways. One way is just to give the relevant QCD parameters and the
parameterization of input parton distributions at scale
0
The user can then produce the parton distributions at another value of
f
by using this information as input to a reliable QCD evolution
program. The other, more commonly used, is to approximate the
outcome of a global fit over
(x,
f)
by a set of parameterized functions. Such parameterization varies
widely between the available distributions sets, ranging from a simple
interpolation formula over a large three-dimensional array
(x,
f, and flavor),
to Chebeschev polynomial expansions, to simple
f-dependent parameterizations of the form of
the above equation with an appropriately chosen smooth function
P(x). It has been found that a logarithmic factor of the form
logA3(1/x)
is particularly effective in rendering the
f-dependence of the coefficient functions
Ai smooth for the QCD evolved distributions.
Although, in principle, the form of the parameterization is arbitrary
so long as the approximated distributions still fit the data,
extrapolation of the distributions out of the fitting region (e.g.,
into the small-x region) will give very different predictions.
It has been demonstrated that good fits to data can be obtained with
the coefficient A1 (which controls the small x
behavior) varying, say, from -0.5 to 0.2. Such uncertainty should be
regarded as evidence of our lack of knowledge of the uncharted
region. It is not meaningful to take the extrapolation of any
particular set of parton distributions as "predictions". This
uncertainty can be reduced either by new experimental measurements or
by theoretical advances which allow true predictions extending to
small x along the same way the usual evolution equation does
for the f variable.
The first generation parton distribution sets, based on leading-order evolution and data of the early 1980s, were widely used in calculations of high-energy processes (Glück et al., 1982; Duke and Owens, 1984; Eichten, Quigg, Hinchliffe, and Lane, 1984). However, since then experimental data have been dramatically improved (and substantially changed, in some cases), and these distributions are no longer able to fit the new data.
Second-generation global analyses, based on next-to-leading order evolution and more recent data, have been carried out by several groups in recent years. Some of the groups perform specialized analyses focusing on some specific issue or process, such as the gluon distributions and direct photon production (Aurenche, Baier, Fontannaz, Owens, and Werlen, 1989), neutrino scattering (Diemoz et al., 1988), etc.; and others study a wide range of processes (Martin, Roberts, and Stirling, 1988; Martin et al., 1989; Harriman, Martin, Roberts, and Stirling, 1990; Kwiecinski et al., 1990; Morfín and Tung, 1991). These analyses differ considerably on various issues, such as the range of data used, the way experimental errors are treated, the choice of schemes, assumptions on the input distributions, and so on.
A compilation of currently available parton distribution sets, both old and new, have been made at CERN and it has been distributed as a program package PDFLIB (Plothow-Besch, 1991). Because most of the older distributions are seriously inconsistent with current data, and because of the differences mentioned above, indiscriminate use of all the distributions in this collection can lead to meaningless results.
For example, it is important to compare only correctly corresponding
objects. Thus, the leading-order, next-to-leading-order DIS, and
next-to-leading-order
distributions are different objects, and should not be compared or mixed.
When calculating physical quantities (such as cross sections or
structure functions), one must convolve leading-order,
next-to-leading-order DIS, and next-to-leading-order
distributions with the corresponding hard-scattering parts in order to
yield meaningful predictions.
We are about to enter yet another era of precision in QCD global analysis. Recently released NMC data (Amaudruz et al., 1992) on F2n/F2p, F2p - F2n, and F2p,d using a muon beam and CCFR data (Mishra et al., 1992; Leung et al., 1993; Quintas et al., 1993) on F2,3Fe using (anti-) neutrinos should have a significant impact on QCD global analyses because of their extended kinematic coverage (particularly at small x), their high statistics and minimal systematic errors. The precision of the current generation of DIS experiments (including the previously published SLAC, BCDMS, and CDHSW data) now far exceeds the size of next-to-leading order QCD contributions to these processes; thus they probe the full complexity of QCD mixing effects between quarks and gluons in a properly conducted QCD analysis. At the same time, data being accumulated at the Fermilab Tevatron on many hadron collider processes (such as W and Zproduction, lepton pair production, direct-photon production, jet production, and heavy flavor production) are beginning to be quantitative enough to provide complementary information and constraints on parton distributions. Finally, the HERA electron-proton collider (H1 Collaboration, Abt et al., 1993, 1994; ZEUS Collaboration, Derrick et al., 1993b) is now providing direct measurements of structure functions at very small x.
The new DIS data have been incorporated in two recent global analysis efforts (Botts et al., 1993; Martin et al., 1993). The most notable result from each of these new global analyses is the apparent extraordinary quantitative agreement of the next-to-leading-order QCD parton framework with the very-high-statistics DIS experiments over the entire kinematic range covered, and the consistency of this framework with all available experiments on lepton pair and direct-photon production as well. The parton distributions are determined with much more precision than before.
On the other hand, these analyses also are calling into question, for the first time, the ultimate consistency of the existing theoretical framework with all existing experimental measurements! (This can be regarded as testimony to the progress made in both theory and experiment - considering the fact that contradictions come with precision, and they are a necessary condition for discovering overlooked shortcomings and/or harbingers of new physics.) When all available total inclusive DIS data and their associated errors are taken seriously in the latest analysis, the CTEQ Collaboration (Botts et al., 1993) found a good global fit only if the strange quark has a much softer distribution than the non-strange ones and rises above the latter in the small x region below x = 0.1. This result is unexpected, and it also appears to be in conflict with the dedicated measurement of s(x) done with dimuon final states in neutrino scattering (Rabinowitz et al., 1993). (The latter is not available in a form that can be included in any of the existing global analyses.) Thus, either there are unknown theoretical flaws in the next-to-leading order QCD analysis or some of the experimental data sets need to be reexamined both in their measured values and in the assessed systematic errors. In the analysis of Martin, Roberts, and Stirling (1993), the strange-quark content of the nucleon is assumed to be consistent with the dimuon result; reasonable fits are obtained only by letting the normalization of the data sets vary freely, unconstrained by the stated experimental errors, and by increasing some experimental errors attributed to other sources.
The emergence of the apparent contradictions has already spurred efforts by both theorists and experimentalists to examine the existing assumptions rigorously and to institute new improvements in their respective analyses. These efforts, aided by data from the hadron collider experiments and from HERA, herald an exciting new era in global QCD analysis. We expect, on the one hand, vigorous study of small-x behavior, and on the other hand, much more stringent tests of the pQCD framework from the many overlapping lepton-hadron and hadron-hadron processes which can now by studied quantitatively.