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

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We present the results of calculations analyzing nucleon Compton scattering to lowest order using perturbative QCD (pQCD) methods. Two scenarios are considered: (1) the incoming photon is real; and (2) the incoming photon is virtual. The case of a real photon has been previously analyzed at least 5 times using pQCD, but no two results are in agreement. Here it is shown that our result agrees with that of Brooks and Dixon published in 2000. The case of a virtual photon has been previously analyzed only once using pQCD. However, doubt has been cast on the validity of that result. The results presented here for virtual photon are believed to be more reliable. Some consideration is given of how to compare these results with experiment. Following the lead of Brooks and Dixon, for the proton, this involves normalizing the cross section using the Dirac proton form factor, which we also calculate. Finally, there is a comparison of our results with recent experiments.

Recent experiments by the HERMES group at DESY HERA are measuring semi-inclusive electroproduction of pions from deuterium. We point out that by comparing the production of ϩ and Ϫ from an isoscalar target, it is possible, in principle, to measure charge symmetry violation in the valence quark distributions of the nucleons. It is also possible in the same experiments to obtain an independent measurement of the quark fragmentation functions. We review the information which can be deduced from such experiments and show the ''signature'' for charge symmetry violation in such experiments. Finally, we predict the magnitude of the charge symmetry violation, from both the valence quark distributions and the pion fragmentation function, which might be expected in these experiments.

In the PQCD analyses of the exclusive production of higher generation hadrons, the quark distribution amplitude of the heavy quark system has often been approximated by a ␦ function from the nonrelativistic consideration. Going beyond the peaking approximation, the factorization of the covariant hard scattering amplitude from the nonperturbative quark distribution amplitude is no longer valid. We therefore use the light-cone time-ordered perturbation theory which is the step prior to the usual factorization formula and calculate the form factor of a pseudoscalar meson composed of a heavy quark and antiquark. However, we find that the numerical results for the cross section of exclusive heavy meson pair production in e ϩ e Ϫ annihilation are not much different from those of the peaking approximation. ͓S0556-2821͑97͒00103-3͔

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