[Paper Review] First radial excitations of heavy quarkonium in a contact interaction

Marco A. Bedolla, E. Santopinto|arXiv (Cornell University)|Jul 5, 2018

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper computes the masses and weak decay constants of the first radial excitations in heavy quarkonium states (ηc,b(2S), ψ(2S), Υ(2S), χc0,b0(2P), χc1,b1(2P)) using a symmetry-preserving vector×vector contact interaction within the Schwinger-Dyson and Bethe-Salpeter equation framework. The model, based on a simplified QCD-inspired interaction with proper-time regularization, reproduces experimental masses with good accuracy, validating its use for studying heavy quarkonium spectroscopy without full QCD complexity.

ABSTRACT

For the flavor-singlet heavy quark systems of quarkonia, we compute the masses of the first radial excitation of mesons in four different channels: pseudo-scalar ($η_{c,b}(2S)$), vector ($ψ(2S),Υ(2S)$), scalar ($χ_{c_0,b_0}(2P)$) and axial vector ($χ_{c_1,b_{1}}(2P)$), as well as the weak decay constants of the $η_{c,b}(2S)$ and $ψ(2S),Υ(2S)$. The framework for this analysis is provided by a symmetry-preserving Schwinger-Dyson equations treatment of a vector$ imes$vector contact interaction. The results found for the meson masses are in good agreement experimental data and earlier model calculations based upon Schwinger-Dyson and Bethe-Salpeter equations (BSEs) involving sophisticated interaction kernels.

Motivation & Objective

• To compute the masses and weak decay constants of the first radial excitations in heavy quarkonium systems (ηc,b(2S), ψ(2S), Υ(2S), χc0,b0(2P), χc1,b1(2P)).
• To test the predictive power of a symmetry-preserving vector×vector contact interaction in modeling heavy quarkonium spectroscopy.
• To assess whether a simplified contact interaction model can reproduce experimental results comparable to full QCD-based calculations and sophisticated kernel models.
• To extend previous studies on ground-state heavy quarkonia to include radial excitations using the same contact interaction framework.

Proposed method

• Uses a vector×vector contact interaction as a simplified QCD kernel, preserving chiral symmetry and confinement via proper-time regularization.
• Applies the rainbow-ladder truncation to the Schwinger-Dyson equations (SDEs), ensuring consistency with the axial-vector Ward-Takahashi identity.
• Solves the Bethe-Salpeter equation (BSE) for meson bound states in four different quantum number channels: pseudoscalar, vector, scalar, and axial vector.
• Employs Nakanishi-like perturbation theory integral representations for Bethe-Salpeter amplitudes to access large momentum transfer regions.
• Computes masses via the eigenvalue equation of the BSE and decay constants from matrix elements of the current operator.
• Uses regularization schemes to remove quadratic and logarithmic divergences, ensuring renormalizability and symmetry preservation.

Experimental results

Research questions

• RQ1 Can a symmetry-preserving contact interaction model accurately predict the masses of the first radial excitations in heavy quarkonium?
• RQ2 How do the computed weak decay constants of ηc,b(2S) and ψ(2S), Υ(2S) compare with experimental data?
• RQ3 Does the contact interaction model reproduce the observed mass splitting between ground states and their radial excitations?
• RQ4 To what extent does this simplified model capture the dynamics of heavy quarkonia compared to full QCD or complex kernel models?

Key findings

• The computed masses of the first radial excitations (ηc,b(2S), ψ(2S), Υ(2S), χc0,b0(2P), χc1,b1(2P)) are in good agreement with experimental data.
• The weak decay constants of ηc,b(2S) and ψ(2S), Υ(2S) are computed and found to be consistent with expectations from other models and data.
• The model successfully describes the mass spectrum of heavy quarkonia, including radial excitations, with a minimal set of parameters.
• The contact interaction model, despite its simplicity, yields results quantitatively comparable to those from sophisticated QCD-based models and full SDE/BSE approaches.
• The framework preserves key symmetries (e.g., axial-vector Ward-Takahashi identity) and correctly implements confinement through proper-time regularization.