Electromagnetic Form Factors at Large Momentum Transfer

Recent improvements of the hard scattering picture for the large p⊥ behaviour of electromagnetic form factors, namely the inclusion of both Sudakov corrections and intrinsic transverse momentum dependence of the hadronic wave function, are reviewed. On account of these improvements the perturbative contributions to the pion’s and the nucleon’s form factor can be calculated in a theoretically self-consistent way for momentum transfers as low as about 2 and 3GeV, respectively. This is achieved at the expense of a substantial suppression of the perturbative contribution in the few GeV region. Eventual higher twist contributions are discussed in some detail. Supported in part by BMFT, FRG under contract 06 WU 737 E-mail: [email protected] Invited talk presented at the Intern. Conf. on Physics with GeV-Particle Beams, Jülich (1994) 1. The hard scattering picture There is general agreement that perturbative QCD in the framework of the hard-scattering picture (HSP) is the correct description of form factors at asymptotically large momentum transfer (see [1] and references therein). In the HSP a form factor is expressed by a convolution of distribution amplitudes (DA) with hard scattering amplitudes calculated in collinear approximation within perturbative QCD. The universal, process independent DAs, which represent hadronic wave functions integrated over transverse momenta, are controlled by long distance physics in contrast to the hard scattering amplitudes which are governed by short distance physics. The DAs cannot be calculated by perturbative means, we have to rely on models. In principle lattice gauge theory offers a possibility to calculate the DAs but with the present-day computers a sufficient accuracy can not be achieved, only a few moments of the pion and the proton DA have been obtained [2]. It is of utmost phenomenological interest whether or not the asymptotic perturbative result can already be applied at experimentally accessible momentum transfers. The major topic of this talk is to answer that question. In order to keep the technical effort simple I am going to discuss the electromagnetic form factor of the pion mainly. The generalization to the phenomenological more important case of the nucleon form factor is straightforward. Now let us consider the electromagnetic form factor of the pion. To lowest order pertubative QCD the hard scattering amplitude TH is to be calculated from the two one-gluon exchange diagrams. Working out the diagrams one finds TH(x1, y1, Q,~k⊥,~l⊥) = 16π αs(μ)CF x1y1Q + (~k⊥ +~l⊥)2 , (1.1) where Q(≥ 0) is the momentum transfer from the initial to the final state pion. x1 (y1) is the longitudinal momentum fraction carried by the quark and ~k⊥ (~l⊥) its transverse momentum with respect to the initial (final) state pion. The momentum of the antiquark is characterized by x2 = 1 − x1 (y2 = 1 − y1) and −~k⊥ (−~l⊥). CF (= 4/3) is the colour factor and αs is the usual strong coupling constant to be evaluated at a renormalization scale μ. The expression (1.1) is an approximation in so far as only the most important ~k⊥and ~l⊥-dependences have been kept. Denoting the wave function of the pion’s valence Fock state by Ψ0, the form factor is given by Fπ(Q ) = ∫ dx1 d 2k⊥ 16π ∫ dy1 d 2l⊥ 16π Ψ 0 (y1,~l⊥)TH(x1, y1, Q,~k⊥,~l⊥)Ψ0(x1, ~k⊥). (1.2) Strictly speaking Ψ0 represents only the soft part of the pion wave function, i.e. the full wave function with the perturbative tail removed from it [1]. Contributions from higher Fock states are neglected in (1.2) since, at large momentum transfer, they are suppressed by powers of αs/Q . At large Q one may neglect the k⊥and l⊥-dependence in the gluon propagator as well; TH can then be pulled out of the transverse momentum integrals, and these integrations apply only to the wave functions. Defining the DA by fπ 2 √ 6 φ(x1, μF ) = ∫ d2k⊥ 16π Ψ0(x1, ~k⊥), ∫ 1 0 dx1 φ(x1, μF ) = 1, (1.3)