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Vector meson photoproduction and problem of gauge invariance

We discuss the problem of gauge invariance of the vector meson photoproduction at small x within the two-gluon exchange model. It is found that the gauge invariance is fulfilled if one includes the graphs with higher Fock states in the meson wave function. The obtained results are used to estimate the amplitudes with longitudinal and transverse photon and vector meson polarization. Investigation of vector meson photoproduction at small x is a problem of considerable interest. We are interested in the lowx region where the predominant contribution is determined by the two-gluon exchange and the vector meson is produced via the photon-two-gluon fusion. The factorization of diffractive vector meson production with longitudinally polarized photons into the hard part and parton distribution was shown in [1]. Thus, such processes, can be an excellent tool to study the generalized parton distribution [2]. Moreover, they should give important information on the vector meson wave function. The spin-density matrix elements which were studied at DESY (see [3] and references therein) should be sensitive to the vector meson wave function. To analyze spin effects in the γ → V transition, it is necessary to calculate the amplitude with transverse polarization of a vector meson. For the light meson production, this transition amplitude is not well defined because of the present end-point singularities [4]. One of the possible ways to regularize these end-point divergences is to include the transverse quark motion, as it was done, e.g., in [5, 6, 7, 8]. Unfortunately, such higher-twist effects can result in loss of the gauge invariance (GI) of the amplitude. In this report, we study the γ → V transition amplitude for different polarization of photon and vector meson at small x and check the GI of our results. The vector meson production can be described in terms of the kinematic variables which are follows: q = (L− L) = −Q, r P = (P − P ) = t, xP = q · (P − P ) q · P , s = (q + P ) , (1) where L,L and P, P ′ are the initial and final lepton and proton momenta, respectively, Q is the photon virtuality, rP is the momentum carried by the two-gluons, xP is part of proton momentum carried by the two-gluon system and s is the photon-proton energy squared. The vector meson is produced by the photon-two-gluon fusion and the momentum V = (q+ rP ) is on the mass shell. The xP variable which is equivalent to skewedness ζ is determined by xP ∼ ζ ∼ M V +Q 2 + |t| s . (2)