Heavy quark pair production in gluon fusion at next-to-next-to-leading $\mathcal{O}({\alpha }_s^4)$ order: One-loop squared contributions

We calculate the next-to-next-to-leading-order $\mathcal{O}({\alpha }_s^4)$ one-loop squared corrections to the production of heavy-quark pairs in the gluon-gluon fusion process. Together with the previously derived results on the $q\overline{q}$ production channel, the results of this paper complete the calculation of the one-loop squared contributions of the next-to-next-to-leading-order $\mathcal{O}({\alpha }_s^4)$ radiative QCD corrections to the hadroproduction of heavy flavors. Our results, with the full mass dependence retained, are presented in a closed and very compact form, in dimensional regularization.

Figure 1

Exemplary gluon fusion diagrams for the NNLO calculation of heavy-hadron production.

Figure 2

The $t$-, $u$-, and $s$-channel LO 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 the gluon, ghost, and light and heavy quarks.

Figure 4

The $s$-channel one-loop graphs contributing to the gluon fusion amplitude. Loops with the dotted lines as in g1, h, j1, and j2 represent the gluon, ghost, and light and heavy quarks. The four-gluon coupling contribution appears in g2.