[Paper Review] Static Potential of the Standard Model and Spontaneously Broken Theories

Benoît Assi, Bernd A. Kniehl|arXiv (Cornell University)|Jan 1, 2020

Particle physics theoretical and experimental studies52 references1 citations

TL;DR

This paper computes the one-loop static potential in the Standard Model (SM) with spontaneous symmetry breaking (SSB), comparing the Wilson loop and scattering amplitude approaches. It finds that electroweak (EW) corrections to the potential are sizeable and comparable to next-to-next-to-leading order (NNLO) QCD contributions, necessitating their inclusion in high-precision heavy quark physics and threshold mass definitions such as the 1S and potential-subtracted masses.

ABSTRACT

We consider the static potential in theories exhibiting spontaneous symmetry breaking. We use our findings to calculate the static potential of the Standard Model at one-loop order. We do so in both the Wilson loop and scattering amplitude approaches and discuss the limitations of the Wilson loop approach. As the field content of the SM is extensive, analogous results to ours in a large set of models is now achievable by varying the appropriate couplings and group theory factors.

Motivation & Objective

• To compute the static potential in the Standard Model at one-loop order, accounting for spontaneous symmetry breaking.
• To compare the validity and limitations of the Wilson loop approach versus the scattering amplitude approach in SSB theories.
• To assess the impact of electroweak corrections on short-distance mass definitions such as the 1S and potential-subtracted masses.
• To provide a general framework for computing static potentials in other models by adjusting couplings and group theory factors.

Proposed method

• Uses the scattering amplitude approach to compute the momentum-space static potential by evaluating on-shell quark-antiquark scattering amplitudes in the non-relativistic limit (q → 0).
• Performs one-loop calculations in the general covariant gauge, reducing Feynman diagrams to master integrals using LiteRed and Package-X.
• Employs FeynCalc and its sub-packages (e.g., FeynHelpers, FeynOnium) to handle tensor algebra, Dirac structures, and non-relativistic reductions.
• Compares results from the scattering amplitude method with those from the Wilson loop approach, identifying inconsistencies and infrared divergences in the latter at higher orders.
• Derives the full one-loop static potential in the SM by combining contributions from QCD, QED, and weak interactions with proper group theory factors.
• Applies the computed potential to extract corrections for threshold mass schemes, including the 1S and potential-subtracted masses.

Experimental results

Research questions

• RQ1 How does the static potential in the Standard Model with spontaneous symmetry breaking differ from that in pure QCD at one-loop order?
• RQ2 Why does the Wilson loop approach fail to consistently compute the static potential in the SM at higher orders, and what are its limitations?
• RQ3 What is the quantitative impact of electroweak corrections on the 1S and potential-subtracted mass definitions in heavy quarkonia?
• RQ4 Can the scattering amplitude method be systematically applied to compute static potentials in other spontaneously broken gauge theories?
• RQ5 How do the relative sizes of QCD and electroweak contributions to the static potential compare at the NNLO level?

Key findings

• The electroweak corrections to the static potential in the SM are found to be of the same order as NNLO QCD corrections, making them non-negligible for high-precision calculations.
• The Wilson loop approach is shown to be ill-defined at higher orders in the SM due to infrared divergences and inconsistent treatment of massive gauge bosons.
• The scattering amplitude method provides a consistent and reliable framework for computing the one-loop static potential in spontaneously broken theories.
• The 1S and potential-subtracted mass definitions receive significant corrections from the electroweak sector, which must be included for precision threshold physics.
• The full one-loop static potential in the SM is derived and shown to be applicable to a wide class of models by adjusting couplings and group theory factors.
• The results confirm that the SM potential must be used instead of the pure QCD potential in high-precision studies of heavy quarkonia and threshold production.