A Colour-Casimir Adjacency Matrix Approach to Fully-Heavy Tetraquarks

Abstract

We present a phenomenological framework for fully-heavy tetraquark spectroscopy based on spectral graph theory. The four valence partons are the vertices of \(K_4\), with edge weights fixed by the colour-Casimir factors of the two colour-singlet diquark--antidiquark channels, \(\bar3\otimes3\) and \(6\otimes\bar6\); a physical state is a coherent mixture of the two, controlled by one mixing angle. Unlike an earlier version, which passed the colour-weighted adjacency matrix through an absolute-degree graph Laplacian, here the spectrum is obtained by \emph{directly} diagonalising the adjacency matrix; we show analytically that the Laplacian step inverts the physical correspondence between colour attraction and mass ordering. Fixing the model's three parameters (mixing angle, energy scale, effective charm mass) against \(X(6900)\), \(X(7100)\), and the more tentative \(X(7200)\) gives \(α=0.740\), \(γ=23.2\,\mathrm{MeV}\), \(m_c=1.77\,\mathrm{GeV}\); with three parameters fixed by three inputs this has zero residual degrees of freedom and is not itself a statistical test. The framework's value rests instead on two genuinely predictive, zero-additional-parameter applications: the all-bottom ground state falls at \(18.7\)--\(18.9\,\mathrm{GeV}\) for standard \(m_b\), with no retuning; and for \(T_{cc}^+\) the same mechanism under-binds by \(\approx93\,\mathrm{MeV}\), a quantitative diagnostic of missing long-range (molecular) dynamics. We discuss the sensitivity to the tentative status of \(X(7200)\), the tension between this static colour-configuration picture and the diquark radial-excitation interpretation favoured by recent CMS data, and what further data would make the model falsifiable in a stronger sense.