Structural Origin of Square-Root Mass Scaling inQuantum Tunnelling

Quantum tunnelling phenomena across physics and chemistry are governed by an exponential sensitivity to the effective mass associated with the tunnelling coordinate. Within semiclassical WKB and instanton formalisms, this dependence takes a characteristic form, $\ln \mathcal{R} \propto -\sqrt{M_{\mathrm{eff}}}$, arising from the factorisation of the Euclidean action into mass-dependent and geometric contributions. Although this scaling is well established at the level of theory, its status as a unifying structural principle across distinct physical regimes has not been systematically clarified. In this work, we examine the extent to which the square-root effective-mass dependence functions as a universal semiclassical descriptor of tunnelling across radically different physical platforms. By reanalysing representative experimental datasets from molecular isotopologues, collective tunnelling in ultracold atomic systems, and macroscopic quantum escape in Josephson circuits, we demonstrate that the leading semiclassical trend persists over more than twenty orders of magnitude in tunnelling rate. When expressed in terms of $\ln \mathcal{R}$ versus $\sqrt{M_{\mathrm{eff}}}$, all platforms exhibit near-linear behaviour, indicating that the dominant contribution to the tunnelling exponent is controlled by a common Euclidean-action structure rather than system-specific microscopic details. We further identify the physical mechanisms—finite temperature, dissipation, many-body effects, and mass-dependent barrier reshaping—that delimit the regime of validity of this leading-order scaling. Taken together, the results support the view that the $\sqrt{M_{\mathrm{eff}}}$ dependence reflects a form of mechanical similarity in imaginary time, providing an ontologically neutral and cross-domain organising principle for quantum tunnelling phenomena from molecular systems to engineered macroscopic quantum devices.