Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Spin One Bottomonium Study in the Functional Formalism in the Feynman Gauge

Vladimir Sauli

TL;DR

This work computes spin-one bottomonium spectra via the covariant Bethe-Salpeter equation (BSE) using a ladder-rainbow kernel informed by lattice gluon dynamics, exploring infrared issues in nontrivial covariant gauges. An infrared regulator mass $m_L$ is introduced in the gauge term, and a two-pole gluon form factor is used to regulate the kernel; with $m_L \sim \Lambda_{QCD}$ the results below the $BB$ threshold reasonably match experimental data, while infrared divergences persist in the vector channel. The study highlights gauge dependence and the need for limited phenomenology to achieve stable solutions, suggesting nonperturbative infrared safety remains elusive in these setups. It also finds that bottom-quark dressing plays a modest role in mass spectra and that confinement is signaled by a pole-less heavy-quark propagator, though numerical limitations (e.g., absence of a running coupling) restrict the scope and reliability of above-threshold predictions.

Abstract

The Bethe-Salpeter equation for spin-one bottomonium coupled to quark and gluon propagators is solved in a class of non-trivial linear gauges. The interaction kernel is based on a known gluon propagator extrapolated from the lattice and contains only small amount of additional phenomenology. The first numerical results are obtained in the Feynman gauge where the associated problem with infrared divergences is circumvented by by introducing a symmetry breaking regulator mass $m_L$. The presence of this mass renders the bound state vertices of vector states finite but it is also necessary to prevent inconsistent solutions of the bound state equation. For $m_L\simeq Λ_{QCD}$ it gives us good agreement with the experimental data for vector and axial vector bottomonia below the $BB$ threshold. While the primary concern pertains to the physical ramifications of the proposed concept, a potential origin for the proposed interaction is also presented.

A Spin One Bottomonium Study in the Functional Formalism in the Feynman Gauge

TL;DR

This work computes spin-one bottomonium spectra via the covariant Bethe-Salpeter equation (BSE) using a ladder-rainbow kernel informed by lattice gluon dynamics, exploring infrared issues in nontrivial covariant gauges. An infrared regulator mass is introduced in the gauge term, and a two-pole gluon form factor is used to regulate the kernel; with the results below the threshold reasonably match experimental data, while infrared divergences persist in the vector channel. The study highlights gauge dependence and the need for limited phenomenology to achieve stable solutions, suggesting nonperturbative infrared safety remains elusive in these setups. It also finds that bottom-quark dressing plays a modest role in mass spectra and that confinement is signaled by a pole-less heavy-quark propagator, though numerical limitations (e.g., absence of a running coupling) restrict the scope and reliability of above-threshold predictions.

Abstract

The Bethe-Salpeter equation for spin-one bottomonium coupled to quark and gluon propagators is solved in a class of non-trivial linear gauges. The interaction kernel is based on a known gluon propagator extrapolated from the lattice and contains only small amount of additional phenomenology. The first numerical results are obtained in the Feynman gauge where the associated problem with infrared divergences is circumvented by by introducing a symmetry breaking regulator mass . The presence of this mass renders the bound state vertices of vector states finite but it is also necessary to prevent inconsistent solutions of the bound state equation. For it gives us good agreement with the experimental data for vector and axial vector bottomonia below the threshold. While the primary concern pertains to the physical ramifications of the proposed concept, a potential origin for the proposed interaction is also presented.

Paper Structure

This paper contains 12 sections, 49 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Quantum Mechanical $Q\bar{Q}$ models
  3. QCD functional formalism of gap equations
  4. DSE and BSE for spin one quarkonia
  5. Taming of Infrared Divergences in Quarkonium BSE Solutions
  6. Axial vector bottomonia
  7. Conclusion
  8. DSE Selfenergy evaluation
  9. The BSE
  10. Assorted Angular integrals
  11. Fit of DSE solution for the quark propagator
  12. Auxiliary normalization, the eigenvalue function

Figures (3)

  • Figure 1: Left panel: Bottom quark spectral function from DSE and its fit as described in the appendix. Right: The same as in the previous figure but in log scale for $p^2$.
  • Figure 2: $\Gamma_1(q_4,q_s=0.5GeV)$ for several vectors (left) and $\Gamma_5(q_4,q_s=0.5GeV)$ component for the axialvectors (right). Auxiliary normalization condition \ref{['normalization']} is used.
  • Figure 3: Sample of numerical search. The eigenvalue auxiliary function $N$ for CBSE2 and the integral error of iterations $\sigma$ for all approximations are shown against the total momentum of the quark-antiquark pair.