Scale-relativistic corrections to the muon anomalous magnetic moment

The anomalous magnetic moment of the muon is one of the most precisely measured quantities in physics. Its experimental value exhibits a $4.2 , \sigma$discrepancy$\delta a_\mu=(251 \pm 59) \times 10^{-11}$ with its theoretical value calculated in the standard model framework, while they agree for the electron. The muon theoretical calculation involves a mass-dependent contribution which comes from two-loop vacuum polarization insertions due to electron-positron pairs and depends on the electron to muon mass ratio $x=m_e\mathrm{/}{m}_{\mu }$. In standard quantum mechanics, mass ratios and inverse Compton length ratios are identical. This is no longer the case in the special scale-relativity framework, in which the Planck length-scale is invariant under dilations. Using the renormalization group approach, we differentiate between the origin of $\ln x$ logarithmic contributions which depend on mass, and $x$ linear contributions which we assume to actually depend on inverse Compton lengths. By defining the muon constant ${\mathbb{C}}_{\mu }=\ln (m_{\mathbb{P}}\mathrm{/}{m}_{\mu })$ in terms of the Planck mass $m_{\mathbb{P}}$, the resulting scale-relativistic correction writes $\delta a_\mu= -\alpha^2 , (x :\ln^3 x)/(8 ; $\mathbb{C}$_\mu^2)$,where$\alpha$ is the fine structure constant. Its numerical value, $(230\pm 16)\times {10}^{-11}$, is in excellent agreement with the observed theory-experiment difference.