Engineering Confining Dilatons: A WKB Inverse Problem in Holographic QCD

Abstract

This work presents a WKB-based inverse problem approach within the framework of holographic bottom-up QCD to engineer confining dilatons from hadronic mass spectra. Starting from a general parameterization of nonlinear radial Regge trajectories, $M_n^2=a(n+b{)}^{\hat{I}}½$, we apply the Rydberg-Klein-Rees (RKR) formula to derive the large-z behavior of the corresponding holographic confining potential. This potential is inversely related to the dilaton field profile, leading naturally to a non-quadratic dilaton $\hat{I}¦(z)=(\hat{I}ºz{)}^{2-\hat{I}\pm }$, where the parameters ($\hat{I}º$, $\hat{I}\pm$) are uniquely determined by the spectral parameters ($a$,$\hat{I}½$). We successfully test this method by fitting the spectra of heavy quarkonia ($c\bar{c}$ and $b\bar{b}$), achieving good agreement with experimental data. Furthermore, we extend this formalism to describe the spectroscopy of tetraquark states by superimposing an additional potential term, derived from the Bethe-Salpeter equation for diquarks, onto the standard mesonic confining potential. This work establishes a powerful and flexible bottom-up framework for deriving confinement directly from spectral data, applicable to both conventional and exotic hadrons.

13 pages, 5 tables