Inverting the Quark Self-Energy from Chiral Low-Energy Constants: A Possible L8 Tension in the Leading-Nc Rainbow Framework

We pose and study the inverse problem of the chiral low-energy constants (LECs): with our re-implemented and checked Yang large-Nc forward operator (reproducing all eight LECs of Yang’s Table I within 30%, four within 15%), we use the experimental LECs of Pan et al. and the lattice quark mass function of Oliveira et al. as data to invert for the quark self-energy ∑(p2). (1) Treating ∑ as a free function results in a multimodal posterior; two independent analyses locate the LEC information window at ∼0.5–1.75GeV and distinguish the profile-sensitive L1–L3, L5 from the comparatively profile-rigid L7, L8 under the tested changes in ∑. (2) Parametrizing ∑ through the kernel of the gap equation (Maris–Tandy/Qin–Chang) collapses the inversion to three kernel parameters (D,ω,m): the resulting posterior is unimodal, the lattice data and LECs are fitted simultaneously, and the uncertainties in ∑ at representative momenta are reduced by factors of 2–4. (3) The fitted result contains four apparent deviations from Pan/FLAG. A proper-time regulator scan examines their response to the forward prescription. The apparent tensions in L5, L7 and the condensate are not particularly stable under the checks performed, whereas L8 remains above the Pan central value throughout the scan. It also varies little across the two kernel realizations and the exploratory M=B/A profile proxy. At the representative Λreg=2GeV point, 103L8=0.99 differs by 2.6σ from the quoted Pan value 103L8Pan=0.60±0.15; the condensate-based B0 comparison and the scalar resonance-saturation estimate show the same tendency. These results support a possible tension associated with the scalar-channel LEC L8 within the tested leading-Nc rainbow construction.