Optimal renormalization scale and scheme for exclusive processes

Abstract

We use the BLM method to fix the renormalization scale of the QCD coupling in exclusive hadronic amplitudes such as the pion form factor and the photon-to-pion transition form factor at large momentum transfer. Renormalization-scheme-independent commensurate scale relations are established which connect the hard scattering subprocess amplitudes that control exclusive processes to other QCD observables such as the heavy quark potential and the electron-positron annihilation cross section. The commensurate scale relation connecting the heavy quark potential, as determined from lattice gauge theory, to the photon-to-pion transition form factor is in excellent agreement with γe → π 0 e data assuming that the pion distribution amplitude is close to its asymptotic form √ 3f π x(1 − x). We also reproduce the scaling and normalization of the γγ → π + π − data at large momentum transfer. Because the renormalization scale is small, we argue that the effective coupling is nearly constant, thus accounting for the nominal scaling behavior of the data. However, the normalization of the space-like pion form factor F π (Q 2 ) obtained from electroproduction experiments is somewhat higher than that predicted by the corresponding commensurate scale relation. This discrepancy may be due to systematic errors introduced by the extrapolation of the γ

π−gluon exchange interaction between constituent quarks

“…The single gluon exchange interaction is weak because the effective quark-gluon coupling is weak in the infrared limit [6,7], and therefore the problem of a large gluonic spin-orbit interaction [3] is avoided. The pion exchange tensor interaction is in effect cancelled by the large tensor component of the π−gluon exchange interaction, and thus the incorrect (small) spin-orbit splitting of the low lying negative parity resonances is avoided.…”

“…Among these are the absence of gluonic degrees of freedom in cooled lattice calculations [4] and the very weak residual gluonic coupling in the valence-QCD approximation [5]. Phenomenological investigations of the behavior of the running coupling strength of quarks to gluons in the infrared limit also indicate that the coupling constant drops or freezes at a constant value [6,7]. The (possible) disconnection of gluons from the constituent quarks at long range may be described phenomenologically in several ways.…”

Publisher's Note: Pion form factor in QCD: From nonlocal condensates to next-to-leading-order analytic perturbation theory [Phys. Rev. D 70, 033014 (2004)]

“…While it may still be not possible to clarify all these theoretical issues conclusively, we believe that significant progress has quite recently been achieved in understanding the pion structure both from the theoretical side-perturbatively [26,27,28,29,30] and nonperturbatively [1, 31,32,33,34]-as well as from the experimental side [35,36,37] and associated data-processing techniques [38,39,40], a progress that could bring to a cleaner comparison between data and various theoretical QCD predictions [31,39,40,41,42,43,44,45,46,47,48,49]. 1 Moreover, a program to compute the electromagnetic and transition form factors of mesons on the lattice has been launched by two collaborations, [51,52] and [53], that may provide valuable insights when it is completed.…”

On confinement in a light-cone Hamiltonian for QCD

“…It has been used repeatedly in the past, see f.e. [42], as reviewed in [46]. Brodsky et al [46] have fixed the parameters by a fit to the non-relativistic heavy-quark lattice data [8] and get κ = 0.16 GeV and µ = 0.87 GeV.…”