Inverse Problem Approach for Non-Perturbative QCD: Applications

In arXiv:2211.13753 we proposed a novel theoretical approach for non-perturbative QCD, named the inverse problem approach. It extracts non-perturbative quantities by solving a dispersion integral equation as an inverse problem, using perturbative QCD and operator product expansion as inputs. In this work, we apply this approach to the pion and $a_1(1260)$ decay constants, as well as the continuum spectral function. The extracted $f_π= 130.8 \pm 8.1$ MeV agrees with the experimental value, and the reconstructed spectral function reproduces the ALEPH $τ$-decay data from threshold up to $s \simeq 3$ GeV$^2$. The advantages of this approach are as follows: (i) the whole non-perturbative range is solved simultaneously, yielding the ground state, excited states, and the continuum spectrum within a single framework; (ii) it is based on rigorous mathematics, with not only statistical but also systematic uncertainties well controlled and quantifiable; (iii) the regularized solutions exhibit convergence, allowing the precision to be systematically improved with better inputs; (iv) quark-hadron duality emerges naturally at high energy scales, avoiding the low-energy duality assumption and the Borel transformation, and eliminating their associated uncertainties. Finally, the successful physical application of the inverse problem approach demonstrates its broad applicability to non-perturbative quantities.