QCD sum rules with spectral densities solved in inverse problems

We construct QCD sum rules for nonperturbative studies without assuming the quark-hadron duality for the spectral density at low energy on the hadron side. Instead, both resonance and continuum contributions to the spectral density are solved with the operator-product-expansion input on the quark side by treating sum rules as an inverse problem. This new formalism does not involve the continuum threshold, does not require the Borel transformation and stability analysis, and can be extended to extract properties of excited states. Taking the two-current correlator as an example, we demonstrate that the series of $\rho$ resonances can emerge in our formalism, and the decay constants $f_{\rho(770)} (f_{\rho(1450)},f_{\rho(1700)},f_{\rho(1900)})\approx$ 0.22 (0.19, 0.14, 0.14) GeV for the masses $m_{\rho(770)} (m_{\rho(1450)},m_{\rho(1700)},m_{\rho(1900)})\approx$ 0.78 (1.46, 1.70, 1.90) GeV are determined. We also show that the decay width $\Gamma_{\rho(770)}\approx 0.17$ GeV can be obtained by substituting a Breit-Wigner parametrization for the $\rho(770)$ pole on the hadron side. It is observed that quark condensates of dimension-six on the quark side are crucial for establishing those $\rho$ resonances. Handling the conventional sum rules with the duality assumption as an inverse problem, we find that the multiple pole sum rules widely adopted in the literature do not describe the $\rho$ excitations reasonably. The precision of our theoretical outcomes can be improved systematically by including higher-order and higher-power corrections on the quark side. Broad applications of this formalism to abundant low energy QCD observables are expected.