A note on the glueball mass spectrum

TL;DR

This work translates holographic predictions for glueball spectra in $QCD_3$ and $QCD_4$ into tractable eigenvalue problems for a dilaton in black-hole geometries. By constructing two local solution bases around singular points and evaluating the $p$-dependent Wronskian, the authors locate eigenvalues as zeros of the function $r(p)$, achieving arbitrary precision with convergent series. The calculated spectra reveal negative eigenvalues that refine the naïve WKB expectation, providing quantitative tests of the Maldacena–Witten duality and illustrating both the method's power and its numerical limits for high-lying states. The results include explicit first states for both $QCD_3$ and $QCD_4$ along with orthogonality relations for the corresponding eigenfunctions, offering a robust framework for spectral tests of holographic gauge/gravity duality.

Abstract

A conjectured duality between supergravity and $N=\infty$ gauge theories gives predictions for the glueball masses as eigenvalues for a supergravity wave equations in a black hole geometry, and describes a physics, most relevant to a high-temeperature expansion of a lattice QCD. We present an analytical solution for eigenvalues and eigenfunctions, with eigenvalues given by zeroes of a certain well-computable function $r(p)$, which signify that the two solutions with desired behaviour at two singular points become linearly dependent. Our computation shows corrections to the WKB formula $m^2= 6n(n+1)$ for eigenvalues corresponding to glueball masses QCD-3, and gives the first states with masses $m^2=$ 11.58766; 34.52698; 68.974962; 114.91044; 172.33171; 241.236607; 321.626549, ... . In $QCD_4$, our computation gives squares of masses 37.169908; 81.354363; 138.473573; 208.859215; 292.583628; 389.671368; 500.132850; 623.97315 ... for $O++$. In both cases, we have a powerful method which allows to compute eigenvalues with an arbitrary precision, if needed so, which may provide quantative tests for the duality conjecture. Our results matches with the numerical computation of [5] well withing precision reported there in both $QCD_3$ and $QCD_4$ cases. As an additional curiosity, we report that for eigenvalues of about 7000, the power series, although convergent, has coefficients of orders ${10}^{34}$; tricks we used to get reliably the function $r(p)$, as also the final answer gets small, of order ${10}^{-6}$ in $QCD_4$. In principle we can go to infinitely high eigenavalues, but such computations maybe impractical due to corrections.