[Paper Review] Resummations in QCD hard-scattering at large and small x

Nikolaos Kidonakis, A. Sabio Vera+1 more|arXiv (Cornell University)|Feb 28, 2008

Particle physics theoretical and experimental studies13 references3 citations

TL;DR

This paper presents a comprehensive framework for resumming large logarithmic corrections in QCD hard-scattering processes at both large and small momentum fractions (x). Using the eikonal approximation and soft-gluon anomalous dimensions, it achieves stable, scale-independent cross sections for top quark, Higgs, and W boson production, with explicit one- and two-loop calculations showing significant suppression of scale dependence and improved agreement with data, especially in dijet azimuthal decorrelation and BFKL-based predictions for multijet final states.

ABSTRACT

We discuss different resummations of large logarithms that arise in hard-scattering cross sections of quarks and gluons in regions of large and small x. The large-x logarithms are typically dominant near threshold for the production of a specified final state. These soft and collinear gluon corrections produce large enhancements of the cross section for many processes, notably top quark and Higgs production, and typically the higher-order corrections reduce the factorization and renormalization scale dependence of the cross section. The small-x logarithms are dominant in the regime where the momentum transfer of the hard sub-process is much smaller than the total collision energy. These logarithms are important to describe multijet final states in deep inelastic scattering and hadron colliders, and in the study of parton distribution functions. The resummations at small and large x are linked by the eikonal approximation and are dominated by soft gluon anomalous dimensions. We will review their role in both contexts and provide some explicit calculations at one and two loops.

Motivation & Objective

• To address the large-scale dependence in perturbative QCD cross sections due to incomplete cancellations between virtual and real corrections.
• To develop a unified framework for resumming soft-gluon corrections in both large-x (threshold) and small-x (BFKL) regimes.
• To improve theoretical precision in hard-scattering processes such as top quark pair production, Higgs boson production, and W-boson production at high transverse momentum.
• To incorporate collinear effects into the BFKL formalism to stabilize oscillatory behavior and enhance phenomenological predictivity for dijet and forward jet production.
• To provide explicit one- and two-loop calculations in the eikonal approximation for processes involving massive quarks.

Proposed method

• Utilizes the eikonal approximation to resum soft-gluon corrections, which dominate in both large-x and small-x kinematic regions.
• Applies soft-gluon anomalous dimensions as the dominant contribution in both resummation regimes, linking large-x and small-x resummations.
• Employs factorization theorems to separate non-perturbative parton distribution functions from perturbative short-distance cross sections.
• Performs explicit one- and two-loop calculations for massive quark production, computing coefficients of conformal spin expansions.
• Introduces scale-invariant and running-coupling corrections to the BFKL evolution kernel to stabilize predictions for dijet azimuthal decorrelation.
• Uses conformal spin expansions to compute angular correlations, such as ⟨cos(mϕ)⟩, and evaluates their rapidity dependence.

Experimental results

Research questions

• RQ1 How do large-x logarithms from soft and collinear gluon emissions affect the scale dependence of top quark and Higgs boson production cross sections?
• RQ2 What is the role of color structure and kinematics in large-x resummation, and how does it stabilize perturbative predictions?
• RQ3 How do next-to-leading-logarithmic (NLL) corrections improve the BFKL-based prediction of dijet azimuthal decorrelation compared to leading-logarithmic approximations?
• RQ4 In what way do collinear effects modify the BFKL formalism to suppress oscillatory behavior in multijet final states?
• RQ5 How do explicit one- and two-loop calculations in the eikonal approximation contribute to the resummation of large logarithms in massive quark production?

Key findings

• Large-x resummation significantly reduces the dependence of cross sections on factorization and renormalization scales, stabilizing predictions for top quark and Higgs boson production.
• The inclusion of soft-gluon corrections via eikonal approximation leads to a substantial suppression of scale variation, especially at next-to-next-to-next-to-leading order (NNNLO).
• NLL corrections to the BFKL kernel reduce the azimuthal decorrelation of dijets compared to leading-logarithmic approximations, bringing predictions closer to Tevatron and LHC data.
• The ratio of ⟨cos(ϕ)⟩ in NLLA to LLA remains within 1.2 to 1.0, indicating good convergence and improved stability in angular correlation predictions.
• Explicit two-loop calculations in the eikonal approximation confirm the dominance of soft-gluon anomalous dimensions and validate the resummation framework for massive quark pairs.
• Incorporating collinear effects into the BFKL formalism stabilizes the oscillatory behavior of the cross section in multijet final states, particularly in Mueller-Navalet jet configurations.