Indirect new physics effects on $\sigma_{\rm had}$ confront the $(g-2)_\mu$ window discrepancies and the CMD-3 result

Recent lattice determinations of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment $a_{\mu}^{\rm HVP}$ have confirmed the discrepancy with the data-driven dispersive method. In the meanwhile the CMD-3 collaboration has reported a result for the $e^+e^-\to \pi^+\pi^-$ cross section considerably larger than previous experimental results (and close to the lattice determinations) exacerbating the discordance between different $e^+e^-$ datasets. We explore to what extent these disagreements can be accounted for by some new physics effect altering selectively the individual experimental determinations of $\sigma(e^+e^- \to\;$hadrons). We find that specific effects of GeV-scale new particles are able to shift upwards the KLOE and BaBar results in the low and intermediate energy windows, while leaving unaffected the CMD-3 energy scan. Although these new physics effects cannot fully explain all the discrepancies among the different $\sigma(e^+e^- \to\;$hadrons) datasets, they succeed in mitigating the overall tension between data-driven and lattice estimates of $a_{\mu}^{\rm HVP}$. Remarkably, the additional loop corrections involving the new particles concur to solve the residual discrepancy with the experimental value of $(g-2)_\mu$.