  The proton charge radius inferred from muonic hydrogen spectroscopy is not
compatible with the previous value given by CODATA-2010, which, on its turn,
essentially relies on measurements of the electron-proton interaction. The
proton's new size was extracted from the 2S-2P Lamb shift in the muonic
hydrogen, which showed an energy excess of 0.3 meV in comparison to the
theoretical prediction, evaluated with the CODATA radius. Higher-dimensional
gravity is a candidate to explain this discrepancy, since the muon-proton
gravitational interaction is stronger than the electron-proton interaction and,
in the context of braneworld models, the gravitational potential can be hugely
amplified in short distances when compared to the Newtonian potential.
Motivated by these ideas, we study a muonic hydrogen confined in a thick brane.
We show that the muon-proton gravitational interaction modified by extra
dimensions can provide the additional separation of 0.3 meV between 2S and 2P
states. In this scenario, the gravitational energy depends on the
higher-dimensional Planck mass and indirectly on the brane thickness. Studying
the behavior of the gravitational energy with respect to the brane thickness in
a realistic range, we find constraints for the fundamental Planck mass that
solve the proton radius puzzle and are consistent with previous experimental
bounds.
