Coupled Scalar–Tensor Flow in History-Dependent Gravity: Resolving Residual Lattice Deviations in the Glueball Spectrum

We address the long-standing discrepancy between continuum QCD approaches and lattice QCD in the glueball spectrum, in particular the mass splitting between the scalar (0++) and tensor (2++) states. In previous works, this discrepancy was parametrized through an effective mixing coefficient c_mix, which either vanished in strict truncations or required phenomenological tuning. In this work, we demonstrate that such a parameter is not fundamental, but rather an emergent consequence of an incomplete truncation. By extending the functional renormalization group (FRG) framework to a coupled scalar–tensor truncation, we promote the corresponding operators O_S ~ F² and O_T ~ FF - trace to independent running couplings c_S(k) and c_T(k). Projecting the Wetterich equation onto this basis, we derive a closed system of coupled beta functions. Crucially, the History-Dependent Gravity (HDG) framework provides a fundamental non-local extension of the effective action, whose infrared manifestation in the Yang–Mills sector is realized through these coupled FRG flows. The HDG-induced memory kernel K_k(0) ~ k^(-α) (α>0) generates a non-trivial infrared source term that dynamically couples the scalar and tensor flows. As a result, the effective mixing arises naturally as c_mix^eff ~ b_S c_T + b_T c_S, eliminating the need for any ad hoc input. We identify a novel mixed infrared fixed point characterized by c_S* ≠ 0 and c_T* ≠ 0, which corresponds to dynamical tensor enhancement. Using these IR values as input to the Bethe–Salpeter equation (BSE), we show that the enhanced tensor coupling directly increases the spin-2 eigenvalue, yielding the correct mass splitting ΔM = M(2++) - M(0++) in agreement with lattice QCD. The resulting spectrum shows quantitative agreement with lattice benchmarks: individual masses agree within a few percent (relative deviations below 2%), and the mass splitting is reproduced within 50 MeV (ΔM = 0.73 GeV vs. lattice 0.68 ± 0.09 GeV). This establishes a fully self-consistent and predictive framework for non-perturbative glueball spectroscopy, directly linking the macroscopic glueball spectrum to the microscopic temporal memory postulates of the HDG framework. To our knowledge, this is the first derivation in which the scalar–tensor glueball splitting is obtained as a consequence of a fully dynamical RG flow without introducing phenomenological mixing parameters. The mechanism identified here suggests that apparent deficiencies of minimal truncations may originate from missing operator mixing rather than missing degrees of freedom, and that channel mixing in non-perturbative QCD is an unavoidable consequence of the renormalization group structure when the theory is extended by fundamental temporal non-locality.