Self-consistent value of the electric radius of the proton from the Lamb shift in muonic hydrogen

Recently a high-precision measurement of the Lamb shift in muonic hydrogen has been performed. An accurate value of the proton charge radius can be extracted from this datum with a high accuracy. To do that a sufficient accuracy should be achieved also on the theoretical side, including an appropriate treatment of higher-order proton-structure effects. Here we consider a higher-order contribution of the finite size of the proton to the Lamb shift in muonic hydrogen. Only model-dependent results for this correction have been known up to date. Meanwhile, the involved models are not consistent either with the existing experimental data on the electron-proton scattering or with the value for the electric charge radius of the proton extracted from the Lamb shift in muonic hydrogen. We consider the higher-order contribution of the proton finite size in a model-independent way and eventually derive a self-consistent value of the electric radius of the proton. The reevaluated value of the proton charge radius is found to be $R_E=0.84022(56)\mathrm{fm}$.

Figure 1

Determination of the rms proton charge radius. For details see [6, 7].

Figure 2

Fractional contributions to the integrand in (3) as a function of $q/\mathrm{\Lambda }$ as follows from the dipole model. The red dot-dashed line is for the subtraction term with $G_E^{\prime }(0)$. The dashed line is for the subtraction term with 1 and the blue solid line is for the $G_E^2$ term, which should follow the experimental data.

Figure 3

The final relative uncertainty as the rms sum of contributions of (10) and (15) plotted as a function of $\nu =q_0/\mathrm{\Lambda }$. The partial uncertainties (10) and (15) for $I_{>}$ and $I_{<}$ are also presented. The red dashed line is for $\delta I_{<}$ and the blue solid line is for $\delta I_{>}$. The relative uncertainties are estimated assuming that the central value is determined by the standard dipole fit.

Figure 4

The electric-charge form factor of the proton $G_E(q^2)$. Top: the dipole parametrization and the fits from [9, 10, 11, 12, 13, 16] (see Appendix pp1 for details). Bottom: Relative deviation of the fits from the dipole form factor, $(G_E-G_{\text{dip}})/G_{\text{dip}}$. Horizontal axis: $q[\mathrm{Gev}/c]$. Blue dashed lines are for the chain fractions, green dot-dashed lines are for Padé approximations with $\tau =q^2/4m_p^2$, and the red solid one is for the Padé approximation with $q$. The standard dipole approximation is presented with a black bold solid line in the top graph.

Figure 5

Relative deviation of $(G_E^2-1)/q^4$, the integrand for $I_{3>}^{\mathrm{E}}$, for the dipole fit with $\mathrm{\Lambda }$, related to $R_E=0.8418\mathrm{fm}$ [1] (a bold black dotted line), from the standard dipole fit with ${\mathrm{\Lambda }}^2=0.71{\mathrm{GeV}}^2$. Effectively, the former fit was used in extraction in [1], while the latter dipole fit is better consistent with the data at high $q$. All the other lines are for the integrand from the fits used above and we use the same legend as in the previous plots.

Figure 6

Relative deviation of the electric form factor from the dipole form factor $(G_E-G_{\text{dip}})/G_{\text{dip}}$. The horizontal axis: $q[\mathrm{Gev}/c]$.