Searching for hadronic scale baryonic and dark forces at $(g-2{)}_{\mu }$'s lattice-vs-dispersion front

The anomalous magnetic moment of the muon ($a_{\mu }$) provides a stringent test of the quantum nature of the Standard Model (SM) and its extensions. To probe beyond the SM physics, one needs to be able to subtract the SM contributions, which consists of a non-perturbative part, namely, the hadronic vacuum polarization (HVP) of the photon. The state of the art is to predominantly use two different methods to extract this HVP: lattice computation, and dispersion relation-based, data-driven method. Thus one can construct different forms of the $a_μ$ test" which compares the precise measurement of $a_μ$ to its theory prediction. Additionally, this opens the possibility for another subtle test, where these two theory" predictions themselves are compared against each other, which is denoted as the ``HVP-test". This test is particularly sensitive to hadronic scale new physics. Therefore, in this work, we consider a SM extension consisting of a generic, light $\sim (100\mathrm{M}\mathrm{e}\mathrm{V}-1\mathrm{G}\mathrm{e}\mathrm{V})$ vector boson and study its impact on both tests. We develop a comprehensive formalism for this purpose. We find that in the case of data-driven HVP being used in the $a_{\mu }$ test, the new physics contributions effectively cancels for a flavor-universal vector boson. As an illustration of these general results, we consider two benchmark models: i)~the dark photon ($A^{\mathrm{\prime }}$) and ii)~a gauge boson coupled to baryon-number ($B$). Using a combination of these tests, we are able to constrain the parameter space of $B$ and $A^{\mathrm{\prime }}$, complementarily to the existing limits. As a spin-off, our preliminary analysis of the spectrum of invariant mass of $3\pi$ in events with ISR at the $B-$ factories (BaBar, Belle) manifests the value of such a study in searching for $B\to 3\pi$ decay, thus motivating a dedicated search by experimental collaborations.