6533b824fe1ef96bd12816d3

RESEARCH PRODUCT

Brodsky-Lepage-Mackenzie scale for the pion transition form factor

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description

The next-to-leading order (NLO) Brodsky-Lepage-Mackenzie (BLM) scale for the pion transition form factor is determined. To achieve that, a consistent calculation up to ${n}_{f}$-proportional next-to-next-to-leading order contributions to both the hard-scattering amplitude and the perturbatively calculable part of the pion distribution amplitude is performed. By combining and matching the results obtained for these two amplitudes, a proper cancellation of collinear singularities is established and the ${\ensuremath{\gamma}}_{5}$ ambiguity problem (related to the use of the dimensional regularization method) is resolved by using the naive-${\ensuremath{\gamma}}_{5}$ as well as the 't Hooft--Veltman (HV) schemes. It is demonstrated that the prediction for the pion transition form factor is independent of the factorization scale ${\ensuremath{\mu}}_{F}^{2}$ at every order in the strong coupling constant, making it possible to use the simplest choice ${\ensuremath{\mu}}_{F}^{2}{=Q}^{2}$ at the intermediate steps of the calculation. Assuming the pion asymptotic distribution amplitude and working in the $\overline{\mathrm{MS}}$ scheme, we find the BLM scale to be ${\ensuremath{\mu}}_{R}^{2}={\ensuremath{\mu}}_{\mathrm{BLM}}^{2}\ensuremath{\approx}{Q}^{2}/9.$ Based on the same distribution, the complete NLO prediction for the pion transition form factor is calculated in the ${\ensuremath{\alpha}}_{V}$ definition of the QCD coupling renormalized at ${\ensuremath{\mu}}_{R}^{2}={\ensuremath{\mu}}_{V}^{2}{=e}^{5/3}{\ensuremath{\mu}}_{\mathrm{BLM}}^{2}\ensuremath{\approx}{Q}^{2}/2.$ It is in good agreement with the presently available experimental data.