Dispersive determination of electroweak-scale masses

Abstract: We demonstrate that the Higgs boson mass can be extracted from the dispersion relation obeyed by the correlation function of two $b$-quark scalar currents. The solution to the dispersion relation with the input from the perturbative evaluation of the correlation function up to next-to-leading order in QCD and with the $b$ quark mass $m_b=4.43$ GeV demands a specific Higgs mass 114 GeV. Our observation offers an alternative resolution to the long-standing fine-tuning problem of the Standard Model (SM): the Higgs mass is determined dynamically for the internal consistency of the SM. The similar formalism, as applied to the correlation function of two $b$-quark vector currents with the same $m_b$, leads to the $Z$ boson mass 90.8 GeV. This solution exists only when the $Z$ and $W$ boson masses are proportionate, conforming to the Higgs mechanism of the electroweak symmetry breaking. We then consider the mixing between the $Q\bar u$ and $\bar Qu$ states for a fictitious heavy quark $Q$ and a $u$ quark through the $b\bar b$ channel, inspired by our earlier analysis of neutral meson mixing. Its dispersion relation, given the perturbative input from the responsible box diagrams and the same $m_b$, fixes the top quark mass 176 GeV. It is highly nontrivial to predict the above electroweak-scale masses with at most 9\% deviation from their measured values using the single parameter $m_b$. More accurate results are expected, as more precise perturbative inputs are adopted.