On BLM scale fixing in exclusive processes

The investigation of the quark-gluon dynamics through perturbative calculations is most useful to extract from experimentally measurable observables quantities such as parton distributions, generalized parton distributions (GPDs) and (generalized) distribution amplitudes (GDAs, DAs). Factorization theorems allow to calculate scattering amplitudes in a perturbative way, provided a renormalization scale and a factorization scale are choosen. Whereas the observables in the extensively studied inclusive reactions are in general related to an amplitude (as the case of inclusive DIS expressed as the imaginary part of the forward virtual Compton scattering reaction), exclusive cross sections which have been much studied recently are based on a factorization theorem at the amplitude level, and thus require to square the amplitude given by its perturbative expansion. Renormalization scale fixing has been the subject of intense studies and different strategies [1, 2] have been put forward to maximize the predictivity of theoretical studies through ensuring the smallness of corrections related to higher orders terms in the perturbative series. The phenomenological success of these proposals is quite impressive in a number of cases, mostly related to inclusive cross sections or jet physics. With the advent of next to leading order results, it has been advocated to use these procedures also in hard exclusive processes [3, 4]. Even at the Born level, the hard meson electroproduction amplitude contains αS . Therefore the choice of the renormalization scale μR is very crucial for the practical estimations of the observables related to meson electroproduction. The first study of a hard electroproduction amplitude including the analysis of the next-to-leading orders (NLO) has been implemented by Belitsky and Muller [3] for π production, i.e. for γp → πn process. This study used an appropriate continuation of the NLO calculations known for the electromagnetic pion form factor [5] onto the case of the meson electroproduction process. In this work, focusing on the Brodsky-Lepage-Mackenzie (BLM) procedure [2] (but we expect our remarks to be quite general, so that they should apply also to other optimization procedures), we examine in detail