Bounds on hadronic contributions to the anomalous magnetic moment of the $\tau$ lepton from QCD Sum Rules

A QCD sum-rule study is performed to place bounds on the leading-order (LO) hadronic vacuum polarization (HVP) contribution to the anomalous magnetic moment of the $\tau$ lepton ($a_{\tau }^{\mathrm{H}\mathrm{V}\mathrm{P},\mathrm{L}\mathrm{O}},a_{\tau }=(g-2{)}_{\tau }\mathrm{/}2$). The QCD contributions to $a_{\tau }^{\mathrm{H}\mathrm{V}\mathrm{P},\mathrm{L}\mathrm{O}}$ are modeled as a combination of Laplace QCD sum rules (LSRs) and finite-energy QCD sum rules (FESRs) of various weights, which are bounded by Hölder and related fundamental inequalities. The light-quark ($u,d,s$) contributions are evaluated up to five-loop order in perturbative contributions in the chiral limit, LO in light quark mass corrections, next-to-leading order (NLO) in dimension-four QCD condensates and to LO in dimension-six QCD condensates. Extending previous work on $a_{\mu }^{\mathrm{H}\mathrm{V}\mathrm{P},\mathrm{L}\mathrm{O}}$, we present a robust methodology for using QCD sum rules to constrain $a_{\tau }^{\mathrm{H}\mathrm{V}\mathrm{P},\mathrm{L}\mathrm{O}}$, and more generally any lepton $g-2$. A criterion is developed to extract from the bounds analysis a final result of $a_{\tau }^{\mathrm{Q}\mathrm{C}\mathrm{D}}=(3.385\pm 0.048)\times {10}^{-6}$, which after inclusion of charmonium and bottomonium contributions gives a final result $a_{\tau }^{\mathrm{H}\mathrm{V}\mathrm{P},\mathrm{L}\mathrm{O}}=(3.501\pm 0.048)\times {10}^{-6}$, comparable to data-driven and lattice predictions. Lattice QCD results along with our value for $a_{\tau }^{\mathrm{H}\mathrm{V}\mathrm{P},\mathrm{L}\mathrm{O}}$ are used to obtain a combined QCD determination from lattice and sum-rule methods of $a_{\mu }^{\mathrm{H}\mathrm{V}\mathrm{P},\mathrm{L}\mathrm{O}}=(7.217\pm 0.233)\times {10}^{-8}$. Our analysis provides fundamental QCD sum-rule inputs into the theoretical determinations of $a_{\tau }$, and establishes new methodologies for QCD sum-rule determinations of low-energy observables.