Recoil correction to the proton finite-size contribution to the Lamb shift in muonic hydrogen

The Lamb shift in muonic hydrogen was measured some time ago to a high accuracy. The theoretical prediction of this value is very sensitive to the proton finite-size effects. The proton radius extracted from muonic hydrogen is in contradiction with the results extracted from elastic electron-proton scattering. That creates a certain problem for the interpretation of the results from the muonic hydrogen Lamb shift. For the latter we need also to take into account the two-photon-exchange contribution with the proton finite size involved. The only way to describe it relies on the data from the scattering, which may produce an internal inconsistency of theory. Recently the leading proton-finite-size contribution to the two-photon exchange was found within the external field approximation. The recoil part of the two-photon-exchange has not been considered. Here we revisit calculation of the external-field part and take the recoil correction to the finite-size effects into account.

Figure 1

The diagram for the Friar correction to Lamb shift in muonic hydrogen. The appropriate subtractions are assumed.

Figure 2

Two-photon-exchange: elastic contribution. Necessary subtractions, which follow from an EDE consideration, are assumed. The circles are for the vertex with the appropriate form factors.

Figure 3

Two-photon-exchange: inelastic contribution (the proton polarizability).

Figure 4

Integrands for the individual contributions to $I_{\mathrm{eTPE}}$: for the Friar term (2) (the black line), the $\kappa$ term (the cyan dot-dot-dashed line), the $EF$ term (the red dashed line), the $M1$ term (the green dotted line), and the $M2$ term (blue dot-dashed line).

Figure 5

Top: Fractional contributions to the integrand of $I_{\mathrm{Fr}}$ in (2) as a function of $q/\mathrm{\Lambda }$ as follows from the dipole model. The red dot-dashed line is for the subtraction term with $G^{\prime }(0)$. The dashed line is for the subtraction term with 1 and the blue solid line is for the $G^2$ term, which represents the data. Bottom: the integrand (in arbitrary units) is presented with a dashed line and the integral ${\int }_0^{q}dq/q^4[(G_E(q^2){)}^2-1-2G_E^{\prime }(0)q^2]$ (in fractional units) in presented as a function of $q$ with a solid line. The shadow belt is for the area in the $q$ space from which 90% of the integral $I_{\mathrm{Fr}}$ come. See [9] for detail.

Figure 6

The value of the proton charge radius $R_E$ extracted [9] from measurement [1] of the muonic-hydrogen Lamb shift. The plot follows [9]. The horizontal belt is for the original value of the parameter. The other figure [of a more complicated shape] is for the radius determined in [9], presented as a function of the separation parameter $q_0$. Every area corresponds to one sigma. The internal and darker part of each figure is related to the partial uncertainty due to the calculation $I_{\mathrm{Fr}}$, while the whole area is for the total uncertainty of the radius. The final result of [9] $R_E=0.84022(56)\mathrm{fm}$ was obtained there at the optimal value $q_0\simeq 0.152\mathrm{GeV}$.

Figure 7

The continuity test for the proton electric form factor normalized against the standard dipole fit. The red area on the left with fast increasing (to the right) width is for the low momentum approximation, while the belt with approximately homogenous width on the right is for the empiric fit [14] we used, with the uncertainty of 1%.

Figure 8

The fractional deviation of the fits (see in App. pp1 for details) and the data for the electric form factor $G_E(q^2)$ from the standard dipole form factor. The data are taken from [12], where the proton form factors were extracted from experimental data on the elastic electron-proton scattering. The fits include the chain-fraction fits (a1) and (a3) (in $q^2$) [12] (the blue dashed line), the Padé-approximation (in $q^2$) fits (a5), (a7), (a9), and (18) [11, 12, 13] (the green dot-dashed lines) with the most recent one from [14] shown with a green bold solid line. An exceptional fit (a12) with an inverse polynomials in $q$ [39] is shown with a solid red line.

Figure 9

Individual contributions to the uncertainty $\delta I_{\mathrm{eTPE}}$ in relative units of the dipole value for various $q_0$. The total error $\delta I$ is presented by the thick solid line, other types of lines are described in the caption of Fig. 4. The recoil contributions are very small and to put all the contributions into the same picture we use different normalization. The values of $\delta I_{\mathrm{eTPE}}$ and $\delta I_{\mathrm{Fr}}$ are multiplied by 0.1, which allows us to use a linear scale (instead of the logarithmic one). The recoil corrections are [anti] correlated with the Friar term, which somewhat reduces the total uncertainty.