Model-independent determination of the magnetic radius of the proton from spectroscopy of ordinary and muonic hydrogen

To date the magnetic radius of the proton has been determined only by means of electron-proton scattering, which is not free of controversies. Any existing atomic determinations are irrelevant because they are strongly model dependent. We consider a so-called Zemach contribution to the hyperfine interval in ordinary and muonic hydrogen and derive a self-consistent model-independent value of the magnetic radius of the proton. More accurately, we constrain not a value of the magnetic radius by itself, but its certain combination with the electric-charge radius of the proton, namely, $R_E^2+R_M^2$. The result from the ordinary hydrogen is found to be $R_E^2+R_M^2=1.35(12){\mathrm{fm}}^2$, while the derived muonic value is $1.49(18){\mathrm{fm}}^2$. That allows us to constrain the value of the magnetic radius of proton $R_M=0.78(8)\mathrm{fm}$ at the 10% level.

Figure 1

Determination of the rms proton charge and magnetic radii. [Sick [2] has evaluated all the world data except the MAMI results [3]. Other evaluations of those data produced similar results (see, e.g., [4])] Ellipses for electron-proton scattering should be somewhat turned from a pure horizontal position because of a small correlation between $R_E$ and $R_M$. For details see [5, 6].

Figure 2

Fractional contributions to the integrand in (1) as a function of $q/\mathrm{\Lambda }$ as follows from the dipole model. The red dot-dashed line is the subtraction term with unity and the blue solid line is the $G_E{G}_M$ term, i.e., for the data (cf., [1, 15]).

Figure 3

The total fractional uncertainty ${\delta }_{2R}^{\prime }$ (a black solid line) as the rms sum of sources listed above as a function of $\nu =q_0/\mathrm{\Lambda }$. The partial uncertainties are also presented. The top plot is for ordinary hydrogen, while the bottom one is for $\mu \mathrm{H}$. The red dot-dashed lines are for the uncertainty of the low-momentum part of the $I_1^{\mathrm{EM}}$ integral and the blue solid ones are for high-momentum contribution to the uncertainty budget; the experimental uncertainties, which are different for muonic and ordinary hydrogen (see Table 1), are presented with green dashed lines.

Figure 4

Top: Magnetic form factor $G_M(q^2)/{\mu }_p$ of the proton: dipole parametrization and the fits. Bottom: Fractional deviation of fits from the dipole form factor $(G_M/{\mu }_p-G_{\mathrm{dip}})/G_{\mathrm{dip}}$. Horizontal axis: $q$ [$\mathrm{GeV}/c$]. The blue dashed lines are for the chain fractions, the 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 dipole fit in the top graph is presented with a black solid line.

Figure 5

Fractional deviation of the magnetic form factor from the electric one $(G_E-G_M/{\mu }_p)/G_E$ (from the same fitting procedures). Horizontal axis: $q$ [$\mathrm{GeV}/c$]. The blue dashed lines are for the chain fractions, the 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 $q$.

Figure 6

Determination of the rms proton charge and magnetic radii. Notation is the same as in Fig. 1. Two new striped belts are added from the HFS constraints in ordinary and muonic hydrogen above. The former is filled with vertical lines and the latter is with the horizontal ones. Their colors are the same as for the Lamb shift in the related atomic system. The squared area is the overlap of two striped areas.

Figure 7

Top: Fractional deviation of the magnetic form factor from the dipole form factor $(G_M/{\mu }_p–G_{\mathrm{dip}})/G_{\mathrm{dip}}$. Bottom: Fractional deviation of the magnetic form factor from the electric one $(G_E–G_M(q^2)/{\mu }_p)/G_E$. Horizontal axis: $q$ [$\mathrm{GeV}/c$] and the range $0.2–0.5\mathrm{GeV}/c$ is crucial for $q_0$ in all three cases considered.