Scale relativity of the proton radius: solving the puzzle

Abstract: The proton size has been found, with a $6\,\sigma$ statistical significance, to be larger by 4% when it is measured relatively to the electron than to the muon [Pohl2010,Antognini2013]. We solve this proton radius puzzle by accounting for the relativity of the proton scale. The proton to electron and proton to muon scale ratios are obtained by direct measurement, but their comparison requires a conversion to electron reference which is currently made by assuming the usual law of scale ratio composition, $\rho_ {pe}=\rho_ {p\mu} \times \rho_ {\mu e}$. Using instead the special scale relativistic law $$\ln \rho_ {pe}=(\ln\rho_ {p\mu}+\ln\rho_ {\mu e})/(\ln\rho_ {p\mu} \ln\rho_ {\mu e}/(\ln\rho_ {\mathbb{P} e})<sup>2$$, where $\mathbb{P}$ denotes the Planck length-scale, the two determinations of the proton radius, showing now a ratio $1.009 \pm 0.008$, recover their agreement within about $1\sigma$. The proton radius puzzle therefore provides one with a highly significant test of the special scale relativity theory.