Glueballs and k-strings in SU(N) gauge theories : calculations with improved operators

TL;DR

The study tackles precise spectroscopy of glueballs and k-strings in SU($N$) gauge theories up to $N=8$, introducing improved blocking and smearing operators and employing anisotropic lattices to control systematic errors. It demonstrates that these enhanced operators yield significantly better overlaps onto the lightest states and enables reliable continuum and large-$N$ extrapolations for both the glueball spectrum and k-string tensions. The key finding is that k-string tensions lie between the Casimir Scaling and MQCD predictions, with the leading finite-$N$ correction characterized by an exponent in the range $1 ightarrow 2$, while glueball masses extrapolate consistently to $N oinity$ consistent with prior work and factorization at large $N$. The results also reveal quasi-stable/unstable strings and near-degeneracies among excited string states, and show that correlators used for mass extraction remain tractable despite large-$N$ factorization, supporting the viability of these methods for high-precision nonperturbative QCD-like studies.

Abstract

We test a variety of blocking and smearing algorithms for constructing glueball and string wave-functionals, and find some with much improved overlaps onto the lightest states. We use these algorithms to obtain improved results on the tensions of k-strings in SU(4), SU(6), and SU(8) gauge theories. We emphasise the major systematic errors that still need to be controlled in calculations of heavier k-strings, and perform calculations in SU(4) on an anisotropic lattice in a bid to minimise one of these. All these results point to the k-string tensions lying part-way between the `MQCD' and `Casimir Scaling' conjectures, with the power in 1/N of the leading correction lying in [1,2]. We also obtain some evidence for the presence of quasi-stable strings in calculations that do not use sources, and observe some near-degeneracies between (excited) strings in different representations. We also calculate the lightest glueball masses for N=2, ...,8, and extrapolate to N=infinity, obtaining results compatible with earlier work. We show that the N=infinity factorisation of the Euclidean correlators that are used in such mass calculations does not make the masses any less calculable at large N.

Figures (15)

• Figure 1: Effective mass of the SU(4) $k=2$ string as a function of $n_t$ for a $16^4$ lattice with old blocking, $\bullet$, and improved blocking, $\circ$. (The value of $a$ differs by about 3% in the two calculations.)
• Figure 2: Effective mass of the SU(4) $k=2$ string as a function of $n_t$ for an anisotropic lattice, $\circ$, and an isotropic lattice, $\bullet$, both with the same value of $a_s$, $a_s\surd\sigma \simeq 0.198$, and $L_s=16$. The value of $n_t$ is for the anisotropic lattice, and the isotropic values of $n_t$ have been scaled up by the calculated value of $a_s/a_t$.
• Figure 3: The lightest glueball masses expressed in units of the fundamental string tension, in the continuum limit, plotted against $1/N^2$. The $0^{++}$, $\bullet$, the $2^{++}$, $\star$, and the first excited $0^{++\star}$, $\circ$. Dotted lines are extrapolations to $N=\infty$.
• Figure 4: The lightest glueball masses expressed in units of the fundamental string tension plotted against $1/N^2$. The $0^{++}$, $\bullet$, the $2^{++}$, $\star$, and the first excited $0^{++}$, $\circ$. Dotted lines are extrapolations to $N=\infty$. On $16^4$ lattices with a nearly common value of the lattice spacing, $a\sqrt{\sigma}\in[0.195,0.210]$, except for SU(2) where $a\sqrt{\sigma}\simeq 0.24$.
• Figure 5: Ratio of the tension of the $k=2$ string to that of the fundamental ($k=1$) string in SU(4), for the anisotropic ($\circ$) and isotropic ($\bullet$) calculations. Plotted against the square of the (spatial) lattice spacing. Continuum extrapolations are shown.