Mesons and photons

Data on meson parton distributions are scarce, so only very few sets have been constructed, and only for the $\pi^{\pm}$. PYTHIA contains the Owens set 1 and 2 parton distributions [Owe84], which for a long time were essentially the only sets on the market, and the more recent dynamically generated GRV LO (updated version) [Glü92a]. The latter one is the default in PYTHIA. Further sets are found in PDFLIB and can therefore be used by PYTHIA, just as described above for protons.

Like the proton was used as a template for simple hyperon sets, so also the pion is used to derive a crude ansatz for $\mathrm{K}^{\pm}/\mathrm{K}_{\mathrm{S}}^0/\mathrm{K}_{\mathrm{L}}^0$. The procedure is the same, except that now $f_{\mathrm{val}} = (f_{\u ,\mathrm{val}}^{\pi^+} + f_{\overline{\mathrm{d}},\mathrm{val}}^{\pi^+})/2$.

Sets of photon parton distributions have been obtained as for hadrons; an additional complication comes from the necessity to handle the matching of the vector meson dominance (VMD) and the perturbative pieces in a consistent manner. New sets have been produced where this division is explicit and therefore especially well suited for applications to event generation[Sch95]. The Schuler and Sjöstand set 1D is the default. Although the vector-meson philosophy is at the base, the details of the fits do not rely on pion data, but only on $F_2^{\gamma}$ data. Here follows a brief summary of relevant details.

Real photons obey a set of inhomogeneous evolution equations, where the inhomogeneous term is induced by $\gamma \to \mathrm{q}\overline{\mathrm{q}}$ branchings. The solution can be written as the sum of two terms,

$$f_a^{\gamma}(x,Q^2) = f_a^{\gamma,\mathrm{NP}}(x,Q^2;Q_0^2) + f_a^{\gamma,\mathrm{PT}}(x,Q^2;Q_0^2) ~,$$ (43)

where the former term is a solution to the homogeneous evolution with a (nonperturbative) input at $Q=Q_0$ and the latter is a solution to the full inhomogeneous equation with boundary condition $f_a^{\gamma,\mathrm{PT}}(x,Q_0^2;Q_0^2) \equiv 0$. One possible physics interpretation is to let $f_a^{\gamma,\mathrm{NP}}$ correspond to $\gamma \leftrightarrow V$ fluctuations, where $V = \rho^0, \omega, \phi, \ldots$ is a set of vector mesons, and let $f_a^{\gamma,\mathrm{PT}}$ correspond to perturbative (`anomalous') $\gamma \leftrightarrow \mathrm{q}\overline{\mathrm{q}}$ fluctuations. The discrete spectrum of vector mesons can be combined with the continuous (in virtuality $k^2$) spectrum of $\mathrm{q}\overline{\mathrm{q}}$ fluctuations, to give

$$f_a^{\gamma}(x,Q^2) = \sum_V \frac{4\pi\alpha_{\mathrm{em}}}... ...\, f_a^{\gamma,\mathrm{q}\overline{\mathrm{q}}}(x,Q^2;k^2) ~,$$ (44)

where each component $f^{\gamma,V}$ and $f^{\gamma,\mathrm{q}\overline{\mathrm{q}}}$ obeys a unit momentum sum rule.

In sets 1 the $Q_0$ scale is picked at a low value, 0.6 GeV, where an identification of the nonperturbative component with a set of low-lying mesons appear natural, while sets 2 use a higher value, 2 GeV, where the validity of perturbation theory is better established. The data are not good enough to allow a precise determination of $\Lambda_{\mathrm{QCD}}$. Therefore we use a fixed value $\Lambda^{(4)} = 200$ MeV, in agreement with conventional results for proton distributions. In the VMD component the $\rho^0$ and $\omega$ have been added coherently, so that $\u\overline{\mathrm{u}}: \d\overline{\mathrm{d}}= 4 : 1$ at $Q_0$.

Unlike the $\mathrm{p}$, the $\gamma$ has a direct component where the photon acts as an unresolved probe. In the definition of $F_2^{\gamma}$ this adds a component $C^{\gamma}$, symbolically

$$F_2^{\gamma}(x,Q^2) = \sum_{\mathrm{q}} e_{\mathrm{q}}^2 \le... ...f_{\mathrm{g}}^{\gamma} \otimes C_{\mathrm{g}} + C^{\gamma} ~.$$ (45)

Since $C^{\gamma} \equiv 0$ in leading order, and since we stay with leading-order fits, it is permissible to neglect this complication. Numerically, however, it makes a non-negligible difference. We therefore make two kinds of fits, one DIS type with $C^{\gamma} = 0$ and one $\overline{\mbox{\textsc{ms}}}$ type including the universal part of $C^{\gamma}$.

When jet production is studied for real incoming photons, the standard evolution approach is reasonable also for heavy flavours, i.e. predominantly the $\c$, but with a lower cut-off $Q_0 \approx m_{\c }$ for $\gamma \to \c\overline{\mathrm{c}}$. Moving to Deeply Inelastic Scattering, $\mathrm{e}\gamma \to \mathrm{e}X$, there is an extra kinematical constraint: $W^2 = Q^2 (1-x)/x > 4 m_{\c }^2$. It is here better to use the `Bethe-Heitler' cross section for $\gamma^* \gamma \to \c\overline{\mathrm{c}}$. Therefore each distribution appears in two variants. For applications to real $\gamma$'s the parton distributions are calculated as the sum of a vector-meson part and an anomalous part including all five flavours. For applications to DIS, the sum runs over the same vector-meson part, an anomalous part and possibly a $C^{\gamma}$ part for the three light flavours, and a Bethe-Heitler part for $\c$ and $\b$.

In version 2 of the SaS distributions, which are the ones found here, the extension from real to virtual photons was improved, and further options made available [Sch96]. The resolved components of the photon are dampened by phenomenologically motivated virtuality-dependent dipole factors, while the direct ones are explicitly calculable. Thus eq. () generalizes to

$$f_a^{\gamma^\star}(x,Q^2,P^2) = \sum_V \frac{4\pi\alpha_{\mathrm{em}}}{f_V^2} \left( \frac{m_V^2}{m_V^2 + P^2} \right)^2 \, f_a^{\gamma,V}(x,Q^2;\tilde{Q}_0^2)$$

$$+ \frac{\alpha_{\mathrm{em}}}{2\pi} \, \sum_{\mathrm{q}} 2 e_{\math... ... + P^2} \right)^2 \, f_a^{\gamma,\mathrm{q}\overline{\mathrm{q}}}(x,Q^2;k^2) ~.$$ (46)

In addition to the introduction of the dipole form factors, note that the lower input scale for the VMD states is here shifted from $Q_0^2$ to some $\tilde{Q}_0^2 \geq Q_0^2$. This is based on a study of the evolution equation [Bor93] that shows that the evolution effectively starts `later' in $Q^2$ for a virtual photon. Equation () is one possible answer. It depends on both $Q^2$ and $P^2$ in a non-trivial way, however, so that results are only obtained by a time-consuming numerical integration rather than as a simple parametrization. Therefore several other alternatives are offered, that are in some sense equivalent, but can be given in simpler form.

In addition to the SaS sets, PYTHIA also contains the Drees-Grassie set of parton distributions [Dre85] and, as for the proton, there is an interface to the PDFLIB library [Plo93]. These calls are made with photon virtuality $P^2$ below the hard process scale $Q^2$. Further author-recommended constrains are implemented in the interface to the GRS set [Glü99] which, along with SaS, is among the few also to define parton distributions of virtual photons. However, these sets do not allow a subdivision of the photon parton distributions into one VMD part and one anomalous part. This subdivision is necessary a sophisticated modelling of $\gamma\mathrm{p}$ and $\gamma\gamma$ events, see above and section . As an alternative, for the VMD part alone, the $\rho^0$ parton distribution can be found from the assumed equality

$$f^{\rho^0}_i = f^{\pi^0}_i = \frac{1}{2} \, (f^{\pi^+}_i + f^{\pi^-}_i) ~.$$ (47)

Thus any $\pi^+$ parton distribution set, from any library, can be turned into a VMD $\rho^0$ set. The $\omega$ parton distribution is assumed the same, while the $\phi$ and $\mathrm{J}/\psi$ ones are handled in the very crude approximation $f^{\phi}_{\mathrm{s},\mathrm{val}} = f^{\pi^+}_{\u ,\mathrm{val}}$ and $f^{\phi}_{\mathrm{sea}} = f^{\pi^+}_{\mathrm{sea}}$. The VMD part needs to be complemented by an anomalous part to make up a full photon distribution. The latter is fully perturbatively calculable, given the lower cut-off scale $Q_0$. The SaS parameterization of the anomalous part is therefore used throughout for this purpose. The $Q_0$ scale can be set freely in the PARP(15) parameter.

The $f_i^{\gamma,\mathrm{anom}}$ distribution can be further decomposed, by the flavour and the $p_{\perp}$ of the original branching $\gamma \to \mathrm{q}\overline{\mathrm{q}}$. The flavour is distributed according to squared charge (plus flavour thresholds for heavy flavours) and the $p_{\perp}$ according to $\d p_{\perp}^2 / p_{\perp}^2$ in the range $Q_0 < p_{\perp}< Q$. At the branching scale, the photon only consists of a $\mathrm{q}\overline{\mathrm{q}}$ pair, with $x$ distribution $\propto x^2 + (1-x)^2$. A component $f_a^{\gamma,\mathrm{q}\overline{\mathrm{q}}}(x,Q^2;k^2)$, characterized by its $k \approx p_{\perp}$ and flavour, then is evolved homogeneously from $p_{\perp}$ to $Q$. For theoretical studies it is convenient to be able to access a specific component of this kind. Therefore also leading-order parameterizations of these decomposed distributions are available [Sch95].