Photon Structure

In deep inelastic scattering the finite photon virtuality ensures that the timescale at which the process takes place is short and the photon acts like a point-like exchange boson. Towards smaller photon virtualities the photon becomes quasi-real and in photoproduction it can fluctuate into a hadronic final state. In resolved-photon processes (figs. 5b to d), the photon fluctuates into a hadronic state before the hard interaction and acts as a source of partons, one of which takes part in the hard interaction. Like for protons, the structure of resolved photons can be described by a photon structure function.

The size of these photon structure functions depends both on the approach in which the parton evolution is described (DGLAP, BFKL or CCFM equations) and on the order of the perturbative expansion. In the DGLAP approach, at next-to-leading order, most of the photon-structure is included in the hard matrix elements and the contributions from processes with resolved photons becomes small ( $\lesssim 5\%$). Calculations using $k_t$ factorization are able to give a reasonable description of the contributions from resolved photons already at leading order.

Parton density distributions for the photon have been extracted from measurements e.g.in $\gamma\gamma^*$ collisions at LEP [101]. In so-called Hadron-like processes a gluon from the photon interacts with a gluon from the proton (gluon-gluon-fusion, fig. 5b) to form a quark anti-quark final state. In contrast, in excitation processes (figs. 5c and d) the heavy quark is a constituent of the resolved photon. These contributions are relevant in the massless scheme where the heavy quarks are active partons in the photon. The two excitation diagrams differ mainly in the propagators of the hard matrix element. While the quark (i.e.fermion) propagator should cause the cross section to follow a $(1-\cos\theta^*)^{-1}$ behavior the gluon (i.e.boson) propagator defines a $(1-\vert\cos\theta^*\vert)^{-2}$ behavior, as in Rutherford scattering. Here, $\theta^*$ is the polar angle between the final state charm quark and the proton direction in the center-of-mass frame of the incoming hard partons.

Figure 5: Dominant diagrams in leading order pQCD for heavy quark photoproduction at HERA. The $c$ in the diagrams stands for both charm and beauty. Note that the distinction between resolved and direct processes becomes ambiguous at next-to-leading order as the next-to-leading order diagrams contain contributions in which a gluon is one of the two leading final state partons.

Experimentally, the signature for resolved photon processes is the presence of a photon remnant, i.e.a low momentum hadronic final state which carries away part of the initial photon energy which is not transfered to the parton participating in the hard process. In dijet events, the two leading jets provide a measure of the two hard final state partons and the fraction of the photon energy, in the proton rest frame, entering the hard interaction can be estimated using the observable

$$x_{\gamma}^{obs}= \frac{ \sum_{Jet_1}(E-p_z) + \sum_{Jet_2}(E-p_z)}{\sum_{h}(E-p_z)},$$ (3)

where the sums in the numerator run over the particles associated with the two jets and that in the denominator over all detected hadronic final state particles. The measured jet kinematics are used to approximate the kinematics of the partons before the hadronization.

For the direct process (fig.5a), $x_{\gamma}^{obs}$ approaches unity, as the hadronic final state consists of only the two hard jets and the proton remnant in the forward region which contributes little to $\sum_{h}(E-p_z)$.

Detailed studies of the heavy quark final states separately for resolved-type ( $x_{\gamma}^{obs}\lesssim 0.75$) and direct-type ( $x_{\gamma}^{obs}\gtrsim 0.75$) events allow to gain quantitative understanding of the size of the different contributions and allow to test the assumption of universality of the photon structure function. Data analyses in which jets from gluons and quarks can be distinguished and different final state topologies can be separated, may be able to provide further tests of the validity of these concepts¹.

Figure 6: Ratio of resolved-like to direct-like cross sections of dijet events containing a $D^*$ meson as measured by ZEUS (preliminary) as a function of photon virtuality $Q^2$ (taken from [102]). Also shown are the expectations from the Monte Carlo generator AROMA [103] and CASCADE [104].

The contribution from resolved-photon processes is expected to vanish towards larger photon virtualities. First measurements however, indicate that the suppression of resolved-type events towards larger photon virtualities occurs much more slowly for charm than for light quark events [102]. This is shown in fig. 6 where only a small or no suppression of the resolved-type contribution as a function of $Q^2$ is seen. The data tend to disprove the expectation from the AROMA Monte Carlo generator [103] which implements only the direct matrix element, in the massive scheme. In contrast, the CASCADE Monte Carlo program [104] describes the data rather well. In CASCADE, certain resolved-like contributions to the cross section are effectively implemented by use of the $k_t$ factorization ansatz, as mentioned above. AROMA and CASCADE, as well as other Monte Carlo generator programs commonly used at HERA, are described in detail in section 3. Measurements based on larger statistics are necessary to determine the contribution from resolved processes as a function of $Q^2$ more precisely, and to improve the understanding of the interplay between the various hard scales and the hadronic contributions from the photon.