Next-to-next-to-leading order QCD corrections to the photon's parton structure

TL;DR

This work delivers the first NNLO QCD corrections to the photon's parton structure, deriving NNLO photonic coefficient functions and six even Mellin moments for photon–quark and photon–gluon splitting functions. By applying the operator product expansion to forward γγ scattering amplitudes and using both MS-bar and DIS_gamma factorization schemes, the authors obtain NNLO photon anomalous dimensions and coefficient functions, providing x-space approximations with explicit transformations between schemes. The study then propagates these NNLO inputs through the photon evolution equations, yielding NNLO-consistent predictions for F2^gamma and the inhomogeneous evolution kernels, and quantifying the numerical impact across x. Although constrained to six Mellin moments (limiting precise x-space results at very small x), the parametrizations and numerical analyses show perturbative stability and furnish practical inputs for NNLO analyses of photon structure in eγ processes. The work also delivers Fortran routines for the NNLO coefficients and split-function approximations to facilitate phenomenological applications.

Abstract

The next-to-next-to-leading order (NNLO) corrections in massless perturbative QCD are derived for the parton distributions of the photon and the deep inelastic structure functions F_1^gamma and F_2^gamma. We present the full photonic coefficient functions at order alpha alpha_s and calculate the first six even-integer moments of the corresponding O(alpha alpha_s^2) photon-quark and photon-gluon splitting functions together with the moments of the alpha alpha_s^2 coefficient functions which enter only beyond NNLO. These results are employed to construct parametrizations of the splitting functions which prove to be sufficiently accurate at least for momentum fractions x >= 0.05. We also present explicit expressions for the transformation from the MS_bar to the DIS_gamma factorization scheme and write down the solution of the evolution equations. The numerical impact of the NNLO corrections is discussed in both schemes.

Figures (9)

• Figure 1: Approximations of the $N_{\rm f}^0$ parts $P_{\rm p\gamma,1}^{(2)}(x)$ of the NNLO photon-quark (p = q) and photon-gluon (p = g) splitting functions, as obtained from the six moments calculated in Sect. 3. The selected representatives (\ref{['qp1']}) and (\ref{['gp1']}) are shown by the full curves.
• Figure 2: As Fig. 1, but for the $N_{\rm f}^1$ contributions $P_{\rm p\gamma,2}^{(2)}(x)$, with p = ns, ps (left) and p = g (right). The full curves represent the parametrizations in Eqs. (\ref{['qp2']}), (\ref{['ps2']}) and (\ref{['gp2']}).
• Figure 3: The perturbative expansion (\ref{['Pexp']}) of the photon-quark splitting function $P_{\rm q\gamma}(x, a_{\rm s})$ in the $\overline{\hbox{MS}}$ scheme for $\alpha_{\rm s} = 0.2$ and $N_{\rm f} = 4$. The relative NNLO corrections in the right part are also shown for the non-singlet splitting function $P_{\rm ns\gamma}$. Here and in the following three figures the subscripts 'A' and 'B' indicate the respective approximations in Eqs. (\ref{['qp1']}), (\ref{['ps2']}) and (\ref{['gp2']}).
• Figure 4: As Fig. 3, but for the photon-quark splitting functions (\ref{['Pdsg']}) in the DIS$_\gamma$ scheme.
• Figure 5: The NLO and NNLO approximations (\ref{['Pexp']}) for the photon-gluon splitting function $P_{\rm g\gamma}(x, a_{\rm s})$ in the $\overline{\hbox{MS}}$ scheme (left) and the DIS$_\gamma$ scheme (right) for $\alpha_{\rm s} = 0.2$ and $N_{\rm f} = 4$. Note that $x\!\cdot\! P$ is displayed here, unlike in Fig. 3 and Fig. 4.