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Parton Distributions

The cross section for a process $ij \to k$ is given by

\sigma_{ij \to k} = \int \d x_1 \int \d x_2 \, f^1_i(x_1) \,
f^2_j(x_2) \, \hat{\sigma}_{ij \to k} ~.
\end{displaymath} (1)

Here $\hat{\sigma}$ is the cross section for the hard partonic process, as codified in the matrix elements for each specific process. For processes with many particles in the final state it would be replaced by an integral over the allowed final-state phase space. The $f^a_i(x)$ are the parton-distribution functions, which describe the probability to find a parton $i$ inside beam particle $a$, with parton $i$ carrying a fraction $x$ of the total $a$ momentum. Actually, parton distributions also depend on some momentum scale $Q^2$ that characterizes the hard process.

Parton distributions are most familiar for hadrons, such as the proton, which are inherently composite objects, made up of quarks and gluons. Since we do not understand QCD, a derivation from first principles of hadron parton distributions does not yet exist, although some progress is being made in lattice QCD studies. It is therefore necessary to rely on parameterizations, where experimental data are used in conjunction with the evolution equations for the $Q^2$ dependence, to pin down the parton distributions. Several different groups have therefore produced their own fits, based on slightly different sets of data, and with some variation in the theoretical assumptions.

Also for fundamental particles, such as the electron, is it convenient to introduce parton distributions. The function $f^{\mathrm{e}}_{\mathrm{e}}(x)$ thus parameterizes the probability that the electron that takes part in the hard process retains a fraction $x$ of the original energy, the rest being radiated (into photons) in the initial state. Of course, such radiation could equally well be made part of the hard interaction, but the parton-distribution approach usually is much more convenient. If need be, a description with fundamental electrons is recovered for the choice $f_{\mathrm{e}}^{\mathrm{e}}(x, Q^2) = \delta(x-1)$. Note that, contrary to the proton case, electron parton distributions are calculable from first principles, and reduce to the $\delta$ function above for $Q^2 \to 0$.

The electron may also contain photons, and the photon may in its turn contain quarks and gluons. The internal structure of the photon is a bit of a problem, since the photon contains a point-like part, which is perturbatively calculable, and a resolved part (with further subdivisions), which is not. Normally, the photon parton distributions are therefore parameterized, just as the hadron ones. Since the electron ultimately contains quarks and gluons, hard QCD processes like $\mathrm{q}\mathrm{g}\to \mathrm{q}\mathrm{g}$ therefore not only appear in $\mathrm{p}\mathrm{p}$ collisions, but also in $\mathrm{e}\mathrm{p}$ ones (`resolved photoproduction') and in $\mathrm{e}^+\mathrm{e}^-$ ones (`doubly resolved 2$\gamma$ events'). The parton distribution function approach here makes it much easier to reuse one and the same hard process in different contexts.

There is also another kind of possible generalization. The two processes $\mathrm{q}\overline{\mathrm{q}}\to \gamma^* / \mathrm{Z}^0$, studied in hadron colliders, and $\mathrm{e}^+\mathrm{e}^-\to \gamma^* / \mathrm{Z}^0$, studied in $\mathrm{e}^+\mathrm{e}^-$ colliders, are really special cases of a common process, $\mathrm{f}\overline{\mathrm{f}}\to \gamma^* / \mathrm{Z}^0$, where $\mathrm{f}$ denotes a fundamental fermion, i.e. a quark, lepton or neutrino. The whole structure is therefore only coded once, and then slightly different couplings and colour prefactors are used, depending on the initial state considered. Usually the interesting cross section is a sum over several different initial states, e.g. $\u\overline{\mathrm{u}}\to \gamma^* / \mathrm{Z}^0$ and $\d\overline{\mathrm{d}}\to \gamma^* / \mathrm{Z}^0$ in a hadron collider. This kind of summation is always implicitly done, even when not explicitly mentioned in the text.

A final comment on parton distributions is that, in general, the composite structure of hadrons allow for multiple parton-parton scatterings to occur, in which case correllated parton distributions should be used to describe the multi-parton structure of the incoming beams. This will be discussed in section [*].

next up previous contents
Next: Initial- and Final-State Radiation Up: Hard Processes and Parton Previous: Resonance Decays   Contents
Stephen Mrenna 2007-10-30