The Πγ Transition Form Factor and the Pion Wave Function

The pion wave function is discussed in the light of the recent CLEO data on the πγ transition form factor. It turns out that the wave function is close to the asymptotic form whereas wave functions strongly concentrated in the end-point regions are disfavoured. Consequences for other exclusive quantities, as for instance the pion’s electromagnetic form factor, are also discussed. [email protected] Supported by the Deutsche Forschungsgemeinschaft. The theoretical description of large momentum transfer exclusive reactions is based on a factorization of longand short-distance physics. The latter physics is contained in the so-called hard scattering amplitude to be calculated within perturbation theory. Universal, process-independent (light cone) wave functions interpolating between hadronic and partonic degrees of freedom, comprise the long-distance physics. The wave functions are not calculable with sufficient degree of accuracy at present. However, for the pion which is the hadron of interest in this letter, the valence Fock state wave function is theoretically rather well constrained. The main uncertainty of the wave function lies in the x-dependent part of it, the so-called distribution amplitude φ(x). x denotes the usual momentum fraction the valence quark carries. Previous studies of the pion’s electromagnetic form factor as well as other large momentum transfer exclusive reactions, as for instance γγ → ππ, seemed to indicate that the pion distribution amplitude is much broader than the so-called asymptotic one, φAS(x) = 6x(1 − x). Chernyak and Zhitnitsky [1] proposed such a broad distribution amplitude which is strongly end-point concentrated and leads to a leading twist contribution to the pion’s electromagnetic form factor in apparently fair agreement with the admittedly poor data [2]. This result is, however, obtained at the expense of the dominance of contributions from the endpoint regions, x → 0 or 1, where the use of perturbative QCD is unjustified as has been pointed out by several authors [3, 4]. Now the prevailing opinion is that the pion’s electromagnetic form factor is controlled by soft physics (e.g. the overlap of the initial and final state wave functions, occasionally termed the Feynman contribution) for momentum transfer less than about 10 GeV [3, 5, 6, 7]. There is another exclusive quantity namely the πγ transition form factor which, for experimental and theoretical reasons, allows a more severe test of our knowledge of the pion wave function than the pion’s electromagnetic form factor. Recently the πγ form factor has been measured in the momentum transfer region from 2 to 8 GeV [8] with rather high precision. Together with previous CELLO measurement [9] we now have at our disposal much better data above 1 GeV for the πγ transition form factor than for the pion form factor. From the theoretical point of view the analysis of the πγ transition form factor is much simpler than that of the pion form factor: It is, to lowest order, a QED process, QCD only provides corrections of the order of 10% in the momentum transfer region of interest [10]. The difficulties with the end-point regions where the gluon virtuality becomes small, do not occur. Higher Fock state contributions, suppressed by powers of αs/Q , are expected to be small (see the discussion in [11]). Moreover, and in contrast to the pion form factor, the Feynman contribution, which may arise through vector meson dominance, is presumably very small due to a helicity mismatch. The purpose of this letter is to extract information on the pion wave function from a perturbative analysis of the πγ form factor. It will turn out that the new CLEO data [8] allow a fairly precise determination of that wave function. It will