Mass estimates of the SU(2) $0^{++}$ glueball from spectral methods

TL;DR

The paper proposes a spectral-density approach to extract hadron masses from lattice QCD correlators through an ill-posed Laplace inversion regularized by Tikhonov with a positivity constraint. It validates the method on a meson toy-model and then applies it to SU(2) glueballs with $0^{++}$ quantum numbers, obtaining ground-state masses in good agreement with traditional large-time fits and revealing hints of excited states. The method uses the Schwinger function and a non-negative NNLS solution with the Morozov discrepancy principle to control regularization, providing a smear-independent cross-check of spectral information. This approach offers a practical, first-principles pathway to spectroscopy from lattice data with potential broader applications to hadron structure.

Abstract

The estimation of the Källén-Lehmann spectral density from gauge invariant lattice QCD two point correlation functions is proposed, and explored via an inversion strategy based on Tikhonov regularisation. We test the method on a mesonic toy model, showing that our methodology is competitive with the traditional Maximum Entropy Method. As proof of concept the SU(2) glueball spectrum for the quantum numbers $J^{PC}=0^{++}$ is investigated, for various values of the lattice spacing, using the published data of arXiv:1910.07756. Our estimates for the ground state mass are in good agreement with the traditional approach, which is based on the large time exponential behaviour of the correlation functions. Furthermore, the spectral density also contains hints of excites states in the spectrum. Spectroscopic analysis of glueball two-point functions therefore provides a straightforward and insightful alternative to the traditional method based on the large time exponential behaviour of the correlation functions.

Figures (5)

• Figure 1: Reconstruction the toy-model spectral density function for various $d$ and $N$. The dashed black line is the original spectral function, while the blue full curve is the reconstructed spectral function as given by the Tikhonov regularized NNLS method.
• Figure 2: $C(\tau)/C(\tau_1)$ for all data sets.
• Figure 3: $\rho(\omega)$ for all data sets, using the $ip$-method. The $y$-axis has been normalized to give the last peak an intensity of 1. Clearly positivity constraints should be imposed.
• Figure 4: $\rho(\omega)$ for all datasets, subject to $\rho(\omega)~\geq~0$. The $y$-axis has been normalized to give the maximum an intensity of 1.
• Figure 5: Maxima of \ref{['fig:rho_all']} in order of increasing $\omega$/, with corresponding left and right Half Width at Half Maximum (HWHM).