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Contrary to the hadron case, there is no necessity to introduce the parton-distribution function concept for leptons. A lepton can be considered as a point-like particle, with initial-state radiation handled by higher-order matrix elements. However, the parton distribution function approach offers a slightly simplified but very economical description of initial-state radiation effects for any hard process, also those for which higher-order corrections are not yet calculated.

Parton distributions for electrons have been introduced in PYTHIA, and are used also for muons and taus, with a trivial substitution of masses. Alternatively, one is free to use a simple `unresolved' $\mathrm{e}$, $f_{\mathrm{e}}^{\mathrm{e}}(x, Q^2) = \delta(x-1)$, where the $\mathrm{e}$ retains the full original momentum.

Electron parton distributions are calculable entirely from first principles, but different levels of approximation may be used. The parton-distribution formulae in PYTHIA are based on a next-to-leading-order exponentiated description, see ref. [Kle89], p. 34. The approximate behaviour is

    $\displaystyle f_{\mathrm{e}}^{\mathrm{e}}(x,Q^2) \approx \frac{\beta}{2}
(1-x)^{\beta/2-1} ~;$  
    $\displaystyle \beta = \frac{2 \alpha_{\mathrm{em}}}{\pi}
\left( \ln \frac{Q^2}{m_{\mathrm{e}}^2} -1 \right) ~.$ (48)

The form is divergent but integrable for $x \to 1$, i.e. the electron likes to keep most of the energy. To handle the numerical precision problems for $x$ very close to unity, the parton distribution is set, by hand, to zero for $x > 1-10^{-10}$, and is rescaled upwards in the range $1-10^{-7} < x < 1-10^{-10}$, in such a way that the total area under the parton distribution is preserved:
\left( f_{\mathrm{e}}^{\mathrm{e}}(x,Q^2) \right)_{\mathrm{m...
...10^{-10} \\ [4mm]
0 & x > 1-10^{-10} \, ~.
\end{array} \right.
\end{displaymath} (49)

A separate issue is that electron beams may not be monochromatic, more so than for other particles because of the small electron mass. In storage rings the main mechanism is synchrotron radiation. For future high-luminosity linear colliders, the beam-beam interactions at the collision vertex may induce a quite significant energy loss -- `beamstrahlung'. Note that neither of these are associated with any off-shellness of the electrons, i.e. the resulting spectrum only depends on $x$ and not $Q^2$. Examples of beamstrahlung spectra are provided by the CIRCE program [Ohl97], with a sample interface on the PYTHIA webpages.

The branchings $\mathrm{e}\to \mathrm{e}\gamma$, which are responsible for the softening of the $f_{\mathrm{e}}^{\mathrm{e}}$ parton distribution, also gives rise to a flow of photons. In photon-induced hard processes, the $f_{\gamma}^{\mathrm{e}}$ parton distribution can be used to describe the equivalent flow of photons. In the next section, a complete differential photon flux machinery is introduced. Here some simpler first-order expressions are introduced, for the flux integrated up to a hard interaction scale $Q^2$. There is some ambiguity in the choice of $Q^2$ range over which emissions should be included. The naïve (default) choice is

f_{\gamma}^{\mathrm{e}}(x,Q^2) = \frac{\alpha_{\mathrm{em}}}...
...x)^2}{x} \, \ln \left( \frac{Q^2}{m_{\mathrm{e}}^2} \right) ~.
\end{displaymath} (50)

Here it is assumed that only one scale enters the problem, namely that of the hard interaction, and that the scale of the branching $\mathrm{e}\to \mathrm{e}\gamma$ is bounded from above by the hard interaction scale. For a pure QCD or pure QED shower this is an appropriate procedure, cf. section [*], but in other cases it may not be optimal. In particular, for photoproduction the alternative that is probably most appropriate is [Ali88]:
f_{\gamma}^{\mathrm{e}}(x,Q^2) =
...{Q^2_{\mathrm{max}} (1-x)}{m_{\mathrm{e}}^2 \, x^2} \right) ~.
\end{displaymath} (51)

Here $Q^2_{\mathrm{max}}$ is a user-defined cut for the range of scattered electron kinematics that is counted as photoproduction. Note that we now deal with two different $Q^2$ scales, one related to the hard subprocess itself, which appears as the argument of the parton distribution, and the other related to the scattering of the electron, which is reflected in $Q^2_{\mathrm{max}}$.

Also other sources of photons should be mentioned. One is the beamstrahlung photons mentioned above, where again CIRCE provides sample parameterizations. Another, particularly interesting one, is laser backscattering, wherein an intense laser pulse is shot at an incoming high-energy electron bunch. By Compton backscattering this gives rise to a photon energy spectrum with a peak at a significant fraction of the original electron energy [Gin82]. Both of these sources produce real photons, which can be considered as photon beams of variable energy (see section [*]), decoupled from the production process proper.

In resolved photoproduction or resolved $\gamma\gamma$ interactions, one has to include the parton distributions for quarks and gluons inside the photon inside the electron. This is best done with the machinery of the next section. However, as an older and simpler alternative, $f_{\mathrm{q},\mathrm{g}}^{\mathrm{e}}$ can be obtained by a numerical convolution according to

f_{\mathrm{q},\mathrm{g}}^{\mathrm{e}}(x, Q^2) =
\int_x^1 \f...
...q},\mathrm{g}} \!
\left( \frac{x}{x_{\gamma}}, Q^2 \right) ~,
\end{displaymath} (52)

with $f^{\mathrm{e}}_{\gamma}$ as discussed above. The necessity for numerical convolution makes this parton distribution evaluation rather slow compared with the others; one should therefore only have it switched on for resolved photoproduction studies.

One can obtain the positron distribution inside an electron, which is also the electron sea parton distribution, by a convolution of the two branchings $\mathrm{e}\to \mathrm{e}\gamma$ and $\gamma \to \mathrm{e}^+ \mathrm{e}^-$; the result is [Che75]

f_{\mathrm{e}^+}^{\mathrm{e}^-}(x,Q^2) =
\frac{1}{2} \, \lef...
...\frac{4}{3} - x^2 - \frac{4}{3} x^3 +
2x(1+x) \ln x \right) ~.
\end{displaymath} (53)

Finally, the program also contains the distribution of a transverse $\mathrm{W}^-$ inside an electron

f_{\mathrm{W}}^{\mathrm{e}}(x,Q^2) = \frac{\alpha_{\mathrm{e...
...}{x} \,
\ln \left( 1 + \frac{Q^2}{m_{\mathrm{W}}^2} \right) ~.
\end{displaymath} (54)

next up previous contents
Next: Equivalent photon flux in Up: Parton Distributions Previous: Mesons and photons   Contents
Stephen Mrenna 2007-10-30