Constraining Glueball Couplings

TL;DR

This work extends the dual S-matrix bootstrap to constrain three-point glueball couplings in SU(3) Yang-Mills by treating bound-state poles and their residues within a rigorous, dispersion-based framework. It formulates and solves a semidefinite program that yields rigorous upper bounds from unitarity, crossing, and Roy equations, and constructs the glue-hedron—the allowed region for the couplings of light glueballs. The analysis covers single-scalar and single-spin-2 bound states and then the full glueball spectrum, revealing structure such as kink and cusp features and Regge-like trajectories in extremal amplitudes. The results provide first-principles bounds that can inform lattice estimates and motivate further extensions to more complex spectra and additional lattice inputs.

Abstract

We set up a numerical S-matrix bootstrap problem to rigorously constrain bound state couplings given by the residues of poles in elastic amplitudes. We extract upper bounds on these couplings that follow purely from unitarity, crossing symmetry, and the Roy equations within their proven domain of validity. First we consider amplitudes with a single spin 0 or spin 2 bound state, both with or without a self-coupling. Subsequently we investigate amplitudes with the spectrum of bound states corresponding to the estimated glueball masses of pure SU(3) Yang-Mills. In the latter case the 'glue-hedron', the space of allowed couplings, provides a first-principles constraint for future lattice estimates.

Figures (14)

• Figure 1: Singularity structure of the $GG\to GG$ amplitude in the $s$-plane at fixed $t=0$. In red, we denote the $s$-channel poles and cut, in blue the corresponding crossed $u$-channel singularities.
• Figure 2: Bounds on the maximum residue at a scalar bound state pole of mass $m_b^2$. In green, the primal bound obtained by constructing maximal analytic, crossing, and unitary amplitudes. In red, the rigorous dual excluded region.
• Figure 3: Bound on the maximum residue $|g_\text{max}|$ at the spin-two bound state of mass $m_b^2$. The red region is rigorously excluded. Different colours correspond to a different number of constraints.
• Figure 4: Mass of the resonance $m^2$ as a function of the spin $J$ extracted from the extremal amplitudes in figure \ref{['fig:spin2_pole']} for $m_b^2>2$. Resonances extracted from the same amplitude are denoted with the same colour. Dashed lines delimitate the window where we impose Roy equations and extract the resonances.
• Figure 5: In purple, bound on the residue at the scalar bound state, panel $a)$, and at the spin-two bound state, panel $b)$, in presence of the self-coupling pole with mass $m=1$. In red, the same bound in absence of the self-coupling pole given respectively in fig. \ref{['fig:scalar_pole']}, and in fig. \ref{['fig:spin2_pole']}.