Sudakov Resummation and Finite Order Expansions of Heavy Quark Hadroproduction Cross Sections

TL;DR

This work develops Sudakov threshold resummation for heavy quark hadroproduction in both 1PI and PIM kinematics and translates the resummed expressions into analytic NNLO cross sections via NNLL expansions. It incorporates color-space matching, jet and soft function evolution, and exact scale-dependent terms from renormalization group methods, providing comprehensive results for gg and q qbar channels. The authors perform detailed numerical studies of partonic and hadronic cross sections, highlighting reductions in scale uncertainty and the effects of kinematics and factorization schemes on top and bottom production at Fermilab and HERA-B. The results offer improved predictions and practical NNLO estimates, with explicit formulas compiled for all relevant channels and kinematics, aiding precision QCD phenomenology and future LHC analyses.

Abstract

We resum Sudakov threshold enhancements in heavy quark hadroproduction for single-heavy quark inclusive and pair-inclusive kinematics. We expand these resummed results and derive analytical finite-order cross sections through next-to-next-to-leading order. This involves the construction of next-to-leading order matching conditions in color space. For the scale dependent terms we derive exact results using renormalization group methods. We study the effects of scale variations, scheme and kinematics choice on the partonic and hadronic cross sections, and provide estimates for top and bottom quark production cross sections.

Figures (24)

• Figure 1: Refactorized form of heavy quark partonic cross section near threshold.
• Figure 2: (a) The $\eta$-dependence of the scaling functions $f^{(k,0)}_{q{\overline{q}}}(\eta),\;k=0,1$ in the ${\overline{\rm{MS}}}$-scheme and 1PI kinematics. We show the exact results for $f^{(k,0)}_{q{\overline{q}}},\;k=0,1$ (solid lines), the LL approximation to $f^{(1,0)}_{q{\overline{q}}}$ (dotted line), the NLL approximation to $f^{(1,0)}_{q{\overline{q}}}$ (dashed line) and the NNLL approximation to $f^{(1,0)}_{q{\overline{q}}}$ (dashed-dotted line). (b) The same as (a) in PIM kinematics. The spaced-dotted curve corresponds to the approximation involving the leading two powers of $\ln(\beta)$.
• Figure 3: (a) The $\eta$-dependence of the scaling functions $f^{(k,0)}_{gg}(\eta),\;k=0,1$ in 1PI kinematics. We show the exact results for $f^{(k,0)}_{gg},\;k=0,1$ (solid lines), the LL approximation to $f^{(1,0)}_{gg}$ (dotted line), the NLL approximation to $f^{(1,0)}_{gg}$ (dashed line) and the the NNLL approximation to $f^{(1,0)}_{gg}$ (dashed-dotted line). (b) The same as (a) in PIM kinematics. The spaced-dotted curve corresponds to the approximation involving the leading two powers of $\ln(\beta)$.
• Figure 4: (a) The $\eta$-dependence of the scaling function $f^{(2,0)}_{q{\overline{q}}}(\eta)$ in the ${\overline{\rm{MS}}}$-scheme and 1PI kinematics. We show the LL approximation (dotted line), the NLL approximation (dashed line) and the NNLL approximation (dashed-dotted line). (b) The same as (a) in PIM kinematics. The spaced-dotted curve corresponds to the approximation involving the leading two powers of $\ln(\beta)$.
• Figure 5: (a) The $\eta$-dependence of the scaling function $f^{(2,0)}_{gg}(\eta)$ in 1PI kinematics. We show the LL approximation (dotted line), the NLL approximation (dashed line) and the NNLL approximation (dashed-dotted line). (b) The same as (a) in PIM kinematics. The spaced-dotted curve corresponds to the approximation involving the leading two powers of $\ln(\beta)$.