Axial-vector contributions in two-photon reactions: Pion transition form factor and deeply-virtual Compton scattering at NNLO in QCD

Using the approach based on conformal symmetry we calculate the two-loop coefficient function for the axial-vector contributions to two-photon processes in the $\overline{\mathrm{MS}}$ scheme. This is the last missing element for the complete next-to-next-to-leading order (NNLO) calculation of the pion transition form factor ${\gamma }^{*}\gamma \to \pi$ in perturbative QCD. The corresponding high-statistics measurement is planned by the Belle II collaboration and will allow one to put strong constraints on the pion light-cone distribution amplitude. The calculated NNLO corrections prove to be rather large and have to be taken into account. The same coefficient function determines the contribution of the axial-vector generalized parton distributions to deeply virtual Compton scattering (DVCS) which is investigated at the JLAB 12 GeV accelerator, by COMPASS at CERN, and in the future will be studied at the Electron Ion Collider EIC.

Figure 1

The ratios $T_{\mathrm{NLO}}/T_{\mathrm{LO}}$, $T_{\mathrm{NNLO}}/T_{\mathrm{LO}}$, and $T_{\mathrm{NNLO}}/T_{\mathrm{NLO}}$ at the scale $Q^2={\mu }^2=4{\mathrm{GeV}}^2$. Here $T_{\mathrm{LO}}=T^{(0)}$, $T_{\mathrm{NLO}}=T^{(0)}+a_s{T}^{(1)}$ and $T_{\mathrm{NNLO}}=T^{(0)}+a_s{T}^{(1)}+a_s^2{T}^{(2)}$ are the CFs defined in (4) truncated at the first (tree level), second (one-loop) and third (two-loop) terms, respectively. For this calculation we used ${\alpha }_s(\mu =2\mathrm{GeV})=0.3009$ and $n_f=4$.

Figure 2

The NNLO/LO and NLO/LO ratios for the integral $I(\alpha )$ in Eq. (13) as a function of $\alpha$.

Figure 3

The $\pi {\gamma }^{*}\gamma$ form factor at the NNLO (solid curves) and NLO (dashed curves) in QCD perturbation theory for the two models of the pion LCDA in Eq. (16). The experimental data are from CLEO [34] (green, open triangles), BABAR [1] (light blue, circles) and Belle [2] (dark blue, squares). In addition, the expectation for the error bars achievable at Belle II [3] is shown in red. The central value for the red boxes is arbitrary.

Figure 4

The axial-vector Compton form factor $\mathcal{H}(\xi )$ in Eq. (21), calculated to NNLO and NLO accuracy. Shown are the ratios for the absolute values (top panel) and the phase differences (bottom panel), both with respect to the tree-level.