One-loop amplitudes for four-point functions with two external massive quarks and two external massless partons up to O(epsilon^2)

TL;DR

This work delivers complete analytic O(ε^2) one-loop amplitudes for heavy-quark pair production via gluon fusion and quark-antiquark annihilation, within dimensional regularization. The authors develop a tensor decomposition framework with basis functions tied to scalar master integrals, and provide explicit renormalization prescriptions (on-shell heavy-quark scheme and fixed flavor) along with gauge-invariance checks. The results retain full spin and color information, enabling polarization-resolved and cross-section computations necessary for NNLO loop-by-loop contributions to hadroproduction and related photon-induced processes. The paper lays the groundwork for squaring these amplitudes to obtain NNLO cross sections and extends applicability to photoproduction and gamma-gamma reactions.

Abstract

We present complete analytical ${\mathcal O}(ε^2)$ results on the one-loop amplitudes relevant for the NNLO quark-parton model description of the hadroproduction of heavy quarks as given by the so-called loop-by-loop contributions. All results of the perturbative calculation are given in the dimensional regularization scheme. These one-loop amplitudes can also be used as input in the determination of the corresponding NNLO cross sections for heavy flavor photoproduction, and in photon-photon reactions.

Figures (6)

• Figure 1: Exemplary gluon fusion diagrams for the NNLO calculation of heavy hadron production.
• Figure 2: The $t$-, $u$- and $s$-shannel leading order (Born) graphs contributing to the gluon (curly lines) fusion amplitude. The thick solid lines correspond to the heavy quarks.
• Figure 3: The $t$-channel one-loop graphs contributing to the gluon fusion amplitude. Loops with dotted lines represent gluon, ghost and light and heavy quarks.
• Figure 4: The $s$-channel one-loop graphs contributing to the gluon fusion amplitude. Loops with dotted lines as in g1,h,j1 and j2 represent gluon, ghost and light and heavy quarks. The four-gluon coupling contribution appears in g2.
• Figure 5: The lowest order Feynman diagram contributing to the subprocess $q \bar{q} \rightarrow Q \overline Q$. The thick lines correspond to the heavy quarks.