Two-Loop integrals for CP-even heavy quarkonium production and decays: Elliptic Sectors

TL;DR

The paper addresses the analytic evaluation of the remaining elliptic sectors in two-loop master integrals for NNLO QCD corrections to CP-even heavy quarkonium production and decays. Using the differential equations method with carefully chosen canonical bases, it solves Sector I (sunrise-type) and Sector II (non-planar two-loop three-point) integrals, expressing results as Goncharov polylogarithms and iterative integrals over complete elliptic integrals, and establishing boundary conditions via regularity and analytic continuation. The master integrals, together with previously computed ones, complete the set needed for CP-even heavy quarkonium processes and are applicable to related exclusive and inclusive productions; the work also provides practical numerical representations (one- and two-fold integrals) for efficient evaluation. The findings enhance precision quarkonium phenomenology and offer a framework potentially extendable to other NNLO calculations involving elliptic sectors in heavy-quark production and decays.

Abstract

By employing the differential equations, we compute analytically the elliptic sectors of two-loop master integrals appearing in the NNLO QCD corrections to CP-even heavy quarkonium exclusive production and decays, which turns out to be the last and toughest part in the relevant calculation. The integrals are found can be expressed as Goncharov polylogarithms and iterative integrals over elliptic functions. The master integrals may be applied to some other NNLO QCD calculations about heavy quarkonium exclusive production, like $γ^*γ\rightarrow Q\bar{Q}$, $e^+e^-\rightarrow γ+ Q\bar{Q}$,~and~$H/Z^0\rightarrow γ+ Q\bar{Q}$, heavy quarkonium exclusive decays, and also the CP-even heavy quarkonium inclusive production and decays.

Figures (3)

• Figure 1: Typical two-loop Feynman diagrams for CP-even heavy quarkonium production.
• Figure 2: The set of 39 master integrals involve elliptic functions in sector I. The thin line denotes massless propagators and on-shell massless external particles; the thick line represents massive propagators and on-shell massive external particles; the dash line indicates off-shell external particles with momentum squared equal to $2 ss$. The internal lines with a dot mean the power of the propagators are raised to 2.
• Figure 3: The set of 8 master integrals that appear in sector II. Integrals $(C_1\ldots C_6)$ can be cast into canonical form, while integrals $(C_7,C_8)$ involve elliptic functions. The thin line denotes massless propagators and on-shell massless external particles; the thick line presents massive propagators and on-shell massive external particles; the dash line indicates off-shell external particles with momentum squared equal to $2 ss$. The internal lines with a dot mean the power of the propagators being raised to 2.