[Paper Review] Threshold corrections to the gamma t anti-t vertex at O(alpha alpha(s))

Dirk Seidel|ArXiv.org|Feb 18, 2009

Medical Imaging Techniques and Applications37 references3 citations

TL;DR

This paper presents the complete calculation of O(ααs) threshold corrections to the γt̄t vertex from W-boson and gluon exchange at the t̄t threshold, a critical missing piece for precision top quark pair production cross-section predictions at e⁺e⁻ colliders. Using non-relativistic QCD and Mellin-Barnes integration techniques, the authors achieve high-precision numerical evaluation of two-loop vertex diagrams, yielding corrections of order 0.1% to the cross section, with strong cancellations between one- and two-loop electroweak contributions indicating good perturbative convergence.

ABSTRACT

In these proceedings a recent calculation of the last missing piece of the two-loop O(alpha alpha(s)) corrections to gamma t anti-t vertex at the t anti-t threshold due to the exchange of a W boson and a gluon is summarised. The calculation constitutes a building block of the top quark threshold production cross section at electron positron colliders.

Motivation & Objective

• To compute the final missing O(ααs) two-loop corrections to the γt̄t vertex at the t̄t threshold, specifically from W-boson and gluon exchange, to complete the theoretical framework for top quark pair production at e⁺e⁻ colliders.
• To enable high-precision determination of top quark mass, width, and strong coupling αs via threshold scans at the International Linear Collider (ILC).
• To ensure gauge invariance by completing the set of vertex, self-energy, and box diagrams at O(ααs), as required for a consistent prediction of the total cross section.
• To develop and apply advanced numerical techniques—specifically Mellin-Barnes integration with contour deformation—to handle the highly oscillatory and challenging two-loop integrals arising in the calculation.

Proposed method

• The calculation is performed within the framework of non-relativistic QCD (NRQCD), which separates hard, potential, soft, and ultrasoft scales in t̄t production.
• Two-loop vertex diagrams involving W-boson and gluon exchange are evaluated using dimensional regularization and the method of regions to handle the threshold kinematics.
• Mellin-Barnes integrals are employed to represent the loop integrals, with contour deformation applied to improve numerical stability and convergence.
• The integration contours are optimized to avoid singularities and ensure well-behaved integrands, enabling high-precision numerical evaluation with the Divonne algorithm from the Cuba library.
• The results are cross-checked using sector decomposition via the FIESTA program to validate initial conditions and ensure numerical reliability.
• The final corrections are combined with known one-loop electroweak and QCD contributions to compute the full helicity amplitudes hI,V and hI,A at O(ααs).

Experimental results

Research questions

• RQ1 What is the size and structure of the O(ααs) two-loop corrections to the γt̄t vertex from W-boson and gluon exchange at the t̄t threshold?
• RQ2 How do these corrections affect the total cross section for top quark pair production at e⁺e⁻ colliders?
• RQ3 To what extent do cancellations between one-loop and two-loop electroweak contributions stabilize the perturbative series?
• RQ4 Can Mellin-Barnes integration with contour deformation be effectively applied to two-loop threshold integrals with oscillatory behavior?
• RQ5 What is the impact of these corrections on the precision of top quark mass and αs measurements at future e⁺e⁻ colliders?

Key findings

• The O(ααs) correction from W-boson and gluon exchange to the γt̄t vertex is found to be 0.2×10⁻³, contributing approximately 0.1% to the total cross section.
• For a Higgs boson mass of 120 GeV, the O(ααs) correction from Higgs exchange is -17.6×10⁻³, showing strong cancellation with the one-loop contribution of 21.1×10⁻³.
• The corrections from W and Z boson exchanges at O(ααs) are small—0.2×10⁻³ and -1.0×10⁻³, respectively—indicating good perturbative behavior in the electroweak sector.
• The numerical evaluation of the two-loop integrals is achieved with high precision using Mellin-Barnes techniques and the Divonne integration algorithm, with results stable to four significant digits.
• The results complete the O(ααs) vertex corrections, forming a necessary component for achieving a theory uncertainty below 3% in the total t̄t cross section at the ILC.
• The strong cancellation between one-loop and two-loop Higgs contributions suggests that the electroweak sector is well-behaved perturbatively, in contrast to the more divergent behavior seen in pure QCD corrections.