Beyond Borel Windows: A Systematic Optimization Framework for QCD Sum Rules

We propose a systematic optimization framework for determining the sum rule window in QCD Laplace sum rules. Instead of relying primarily on fixed convergence percentages and visual plateau selection, the procedure combines an OPE entropy criterion, local mass stationarity, and a correlated analysis of the Laplace parameter $\tau$, the continuum threshold $t_c$, and the interpolating-current mixing angle $\theta$. The normalized OPE entropy is introduced to quantify the redistribution of the QCD contributions between the perturbative and nonperturbative sectors and to delimit an initial Working Region. This domain is subsequently refined by minimizing the residual variation of the mass estimator and requiring weak sensitivity to the continuum threshold and to the current composition. As an application, we study the lowest fully charmed tetraquark state with quantum numbers $J^{PC}=2^{++}$ using a mixed diquark--antidiquark and meson--meson interpolating current. The optimization selects $\theta=17.2^\circ\pm0.4^\circ$, $\sqrt{t_c}=7.08\pm0.08~\mathrm{GeV}$, and $\tau=0.410\pm0.010~\mathrm{GeV}^{-2}$, leading to the mass prediction $M_{c\bar c c\bar c}^{2^{++}}=(6.32\pm0.11)~\mathrm{GeV}$. The predicted state lies in the near-threshold region of the di-$J/\psi$ spectrum and is compatible, within uncertainties, with the lowest resonant component reported by ATLAS and with the enhancement parametrized as $BW_0$ in the recent CMS publication. The proposed framework provides a reproducible way of identifying finite domains of reduced auxiliary-parameter sensitivity and of incorporating the remaining dependence into the final uncertainty.