The third Zemach moment and the size of the proton

Abstract

To resolve the puzzle of the proton size raised from the recent result of muonic hydrogen Lamb shift, De R\'{u}jula has proposed that a large value of the third Zemach moment $< r^3_{p}>{(2)}$ of the proton to be the solution. His suggestion has been criticized by many groups based on the $ep$ scattering data at low $Q^2$ regime. However, if there is a "thorn" or "lump" in the electric form factor of the proton $G_{E}(Q^2)$ at extremely low $Q^2$ regime, then the third Zemach moment $< r^3_{p}>{(2)}$ would be as large as De R\'{u}jula suggested. In this article, we show that the existence of such a "thorn" or "lump" has not been completely excluded, although tightly restricted, by the current data of $ep$ elastic scattering. We also suggest a more sophisticated global fitting procedure of $G_{E}(Q^2)$ for the future fitting.