The momentum distribution functions of the partons within the proton are called Parton Distribution Functions.

Contents [hide] 1 DefinitionofPDFs 2 Methodofdeterminations 3 ExamplesofPDFs 4 Datasetsinfit 5 Treatment of experimental errors and estimation of uncertainties 6 Modeluncertainties 7 Theoryuncertainties 8 Results 9 Applicationsandprospects 10 References 11 Externallinks 12 Seealso

Definition of PDFs

The Parton name was proposed by Richard Feynman in 1969 as a generic description for any particle constituent within the proton, neutron and other hadrons. These particles are referred today as quarks and gluons.

There are six types of quarks, known as flavours: up (u), down (d), charm (c), strange (s), top (t) and bottom (b). The antiparticles of quarks are antiquarks. Quarks have various intrinsic properties, including electric charge, spin, mass and colour charge.

Quarks carry a fractional electric charge of value, either -1/3 or +2/3 times the elementary charge (where the electron has -1 unit), depending on flavour. Up, charm and top quarks have a charge +2/3, while down, strange and bottom quarks have -1/3. The quarks which determine the quantum numbers of hadrons are called constituent or valence quarks. For example the proton is composed of two up quarks (referred below as u_v quark) and one down quark (referred below as d_v quark), and the neutron of two down quarks and one up quark.

Quarks are spin 1/2 particles. The spin direction is called polarisation.

Quarks possess a property called colour charge. There are three types of colour charge. Each quark carries a colour. The system of attraction and repulsion between coloured quarks is called strong interaction, which is mediated by force carrying particles known as gluons. Gluons, like the photons are massless, have a spin of 1 and no electric charge but carry colour charge. The theory that describes strong interactions is called Quantum Chromodynamics (QCD).

The three-quark model assuming that a proton or a neutron is made of three free non-interacting quarks in a bag is too simple. It cannot match a scattering process like the inelastic scattering of electrons off protons. Those valence quarks are imbedded in a sea of virtual quark-antiquark pairs generated by the gluons which hold the quarks together in the proton. All of these particles - valence quarks, sea quarks and gluons- are partons.

The partonic structure of a nucleon is best probed in scattering processes like Deep Inelastic Scattering (DIS) of leptons (electrons, muons or neutrinos) off nucleons, where the lepton acts as a probe which transfers a four momentum of modulus q to the nucleon in the collision. The Nobel Prize was awarded to Jerome Friedman, Henry Kendall and Richard Taylor in 1990 for their pioneering electron-proton DIS experiment at SLAC in 1966 which first provided evidence for a partonic structure of the nucleon.

In DIS the resolving power of the probe is approximately ℏ /q and so the level of structure revealed increases with q. For q=100GeV, the resolution is roughly 0.002fm, sufficient to probe the internal structure of the nucleon. It is convenient to consider a frame in which the target nucleon has a very large momentum. In such a frame the momentum of the parton is almost collinear with the nucleon momentum, so that the target can be seen as a stream of partons, each carrying a fraction x of the longitudinal momentum. The momentum distribution functions of the partons within the proton are simply called Parton Distribution Functions (PDFs) when the spin direction of the partons is not considered. They represent the probability densities (strictly speaking they rather represent number densities as they are normalised to the number of partons) to find a parton carrying a momentum fraction x at a squared energy scale Q2 (= − q 2). DIS experiments have shown that the number of partons goes up at low x with Q2, and falls at high x. At low Q2 the three valence quarks become more and more dominant in the nucleon. At high Q2 there are more and more quark-antiquark pairs which carry a low momentum fraction x. They constitute the sea quarks. A salient finding of the DIS experiments is that the quarks and antiquarks only carry about half of the nucleon momentum, the remainder being carried by the gluons. The fraction carried by gluons increases with increasing Q2.

The central feature of QCD is the asymptotic freedom discovered in 1973 by David Gross, David Politzer and Frank Wilczek (Nobel Prize in 2004). It implies that interactions between partons within a nucleon becomes arbitrarily weak at shorter distances. QCD gives quantitative predictions about the rate of change of parton distributions when the Q2 energy scale varies. It is governed by the QCD evolution equations for parton densities from (Gribov and Lipatov 1972), (Altarelli and Parisi 1977) and (Dokshitzer 1977) (DGLAP) in the domain where perturbative calculations can be applied, that is in the limit where the running coupling constant of α s (Q2) of QCD is much smaller than one (α s (Q2) ≪ 1). The equations have been formulated at different level of approximations, relative to different power of α s (Q2) in the perturbative development, usually named as Leading-Order (LO), i.e. first order in α s (Q2), Next-to-Leading-Order (NLO) and Next-to-Next–Leading-Order (NNLO). In the following we will consider Parton Distributions Functions obtained with evolution equations at the most widely used order.

The DGLAP differential equations give the Q2 dependence but cannot make a definitive prediction of the x dependence of the parton distributions at a given Q2. It has to be extracted from the data. The parton distributions are related to the observable cross sections by the QCD factorisation theorems (see for example (Collins, Soper and Sterman 2009)). The cross section of a hard process can be written as a calculable parton interaction convoluted with the parton densities. The factorisation theorems are the whole basis to extract the PDFs from some processes and to apply perturbative calculations to many important processes involving hadrons. In DIS it reads

σ (x,Q2) ≈ Σ aCa ⊗ fa /A(x,Q2)+remainder

Here Ca is the calculable part and fa/A is the parton distribution of parton a in a hadron of type A. The sum is over all type of partons, a. It is conventional to call the first term on the right of the above equation the leading twist contribution. The remainder is called the higher twist correction. It is formally of order 1/Q2 but not precisely known. The correction is often neglected in extracting PDFs from the cross sections. The convolution of the Ca coefficient functions with the parton distributions is not uniquely defined at NLO. Usually the Ca coefficient functions and the parton distributions are written in the Modified Minimal Subtraction Scheme of factorisation called MS-bar (see MS-bar definition of parton distribution functions).

Method of determinations

PDFs sets are obtained by a fit on a large number of cross section data points in a large grid of Q2 and x values from many experiments. The most commonly used procedure consists of parameterising the dependence of the parton distributions (quarks, antiquarks, gluon) on the variable x at some low value of Q2=Q20, which is large enough that the unknown terms of the perturbative equations are assumed to be negligible, and evolving these input distributions up in Q2 through the DGLAP equations. The number of unknown parameters is typically between 10 and 30. The factorisation theorems allow to derive predictions for the cross sections.. These predictions are then fitted to as much of the experimental data together as possible, to determine the parameters and to provide parton distributions.

Examples of PDFs

An overview of parton distributions in the proton is shown in the figures below at two scales Q=2GeV (Figure 1) and Q=100GeV (Figure 2).

Figure 1: Overview of the CTEQ6M parton distribution at Q = 2 GeV (Pumplin et al. 2002).

Figure 2: Overview of the CTEQ6M parton distribution at Q = 100 GeV (Pumplin et al. 2002).

As naively expected, at small Q2 and large x values above 0.1, the u quarks are the dominant distributions, more than twice as large as the d quarks at high x and much larger than the heavy quarks. At low x value the sea is not flavour symmetric. There are significantly less strange quarks than up and down quarks. The charm density is null below the charm threshold (mc=Q ≈ 1.5GeV) and increases slowly as energy increases. At higher Q2 (Figure 2) the shape of the quark and gluon distributions changes quickly at very low x. The sea becomes more flavour symmetric, since at low x the evolution is flavour-independent, and there are more and more sea quarks and gluons. The rise of the parton densities at low x and high Q2 values is a foundational prediction of QCD (DeRujula et al., 1974) which was clearly verified at the HERA electron-proton collider at DESY in 1993.

There is however not a unique set of Parton Distribution Functions commonly accepted. There are several groups in competition to provide the best parametrisation of parton distributions. The groups do not use the same input data. They differ mainly in the way the PDFs are parameterised, in the treatment of heavy quarks and in the value of the coupling constant α s as well in the way the experimental errors are treated and the theoretical errors are estimated.

Data sets in fit

The very extensive and precise DIS data from fixed-target lepton-nucleon scattering experiments at SLAC, FNAL, CERN and from the electron-proton HERA collider at DESY provide the backbone of parton distribution analysis. The lepton-nucleon data include electron, muon and neutrino DIS measurements on hydrogen, deuterium and nuclear targets. DIS data however are insufficient to determine accurately flavour decomposition of the quark and antiquark sea or the gluon distribution at large x. In inclusive DIS, the gluon is only probed via the rate of evolution. The additional physical processes which are used in the fits are:

• The single jet inclusive production in nucleon-nucleon interactions, selecting jets with large transverse energy; this quantity is dependent on the gluon distribution.
• Dilepton production in the virtual photon Drell-Yan process pN → μ + μ − +X, which is a probe of the sea quark distribution.
• Electroweak Z and W boson production pp¯ → W +(W −)+X at the Tevatron collider which is sensitive to the up and down quark and antiquark distributions.

The determination of the most global fits like CTEQ6.6 (Nadolsky et al. 2008), MSTW08 (Martin et al. 2008), GJR08 (Glück et al. 2008) and more recently NNPDF2.0 (Ball et al. 2010) are based on data from DIS and proton-nucleon fixed target experiments as well as results from the HERA and Tevatron colliders. The four groups do not fit exactly the same data sets, e.g. GJR has no W and Z production data. ABKM09 (Alekhin et al. 2009) fit combines DIS and fixed target Drell-Yan data. HERAPDF1.0 (Aaron et al. 2010) fit uses only HERA data.