Nonidentical protons

We have calculated the proton charge radius by assuming that the real proton radius is not unique and the radii are randomly distributed in a certain range. This is performed by averaging the elastic electron-proton differential cross section over the form factor cutoff. By using a dipole form factor and fitting the middle value of the cutoff to the low-$Q^2$ Mainz data, we found the lowest ${\chi }^2/N$ for a cutoff $\Lambda =0.8203\pm 0.0003$ GeV, which corresponds to a proton charge radius $r_E=0.8333\pm 0.0004$ fm. The result is compatible with the recent precision measurement of the Lamb shift in muonic hydrogen as well as recent calculations using more sophisticated techniques. Our result indicates that the relative variation of the form factor cutoff should be around 21.5$\%$. Based on this result we have investigated effects of the nucleon radius variation on the symmetric nuclear matter (SNM) and the neutron star matter (NSM) by considering the excluded volume effect in our calculation. The mass-radius relation of a neutron star is found to be sensitive to this variation. The nucleon effective mass in the SNM and the equation of state of both the SNM and the NSM exhibit a similar sensitivity.

Figure 1

Square root of the averaged $G_{E,p}^2$ calculated from Eq. (5) for different values of the relative variation $\Delta \Lambda /{\Lambda }_1$ (shown here) compared with experimental data. Experimental data are taken from Refs. [3] (open circles) and [42] (filled squares).

Figure 2

As Fig. 1, but limited to $Q^2\le 0.25$ GeV${}^2$. The results obtained for two values of $\Delta \Lambda /{\Lambda }_1$ are compared with the standard dipole form factor and experimental data. Note the linear scale for the $Q^2$ axis.

Figure 3

(a) The obtained ${\chi }^2/N$ values as functions of the proton charge radius $r$ and $\Delta \equiv \Delta \Lambda /{\Lambda }_1$. (b) The projection of ${\chi }^2/N$ on the $r-\Delta$ plane for ${\chi }^2/N\le 16$. The intersection of the vertical and horizontal dashed lines locates the minimum position of ${\chi }^2/N$, which is accurately shown in Fig. 4.

Figure 4

${\chi }^2/N$ as a function of the obtained proton (a) charge and (b) magnetic radii. The lowest and highest values of relative variation in the form factor cutoff, $\Delta \equiv \Delta \Lambda /{\Lambda }_1$, are indicated. Vertical lines indicate the minimum positions of ${\chi }^2/N$. The corresponding $\Delta$ values are also shown.

Figure 5

Square root of the averaged $G_{E,p}^2$ obtained from different calculations compared with experimental data for very low $Q^2$. Notation for experimental data is as in Fig. 1. The result obtained in the present work is represented by the solid curve. Parameters for the double-dipole and Friedrich-Walcher form factors are taken from Ref. [31].

Figure 6

Comparison between proton bag radii as a function of the ratio between nucleon and nuclear saturation densities obtained from the QMC models [34] and Eq. (18) with two $\beta$ values.

Figure 7

The neutron star mass as a function of its radius for four nucleon radius assumptions. The calculation with (a) $\beta =0.0005$ and (b) $\beta =0.01$ in Eq. (18). Gray horizontal bars show the mass of the SRJ164-2230 pulsar, which is believed to be the heaviest observed neutron star [46]. Notation of the curves in both (a) and (b) is given in Fig. 8.

Figure 8

Effective nucleon mass as a function of the ratio between nucleon and nuclear saturation densities. Results were obtained with different values of the nucleon radius $r_N\left(0\right)$ as indicated (in fm).

Figure 9

Equation of states of the symmetric nuclear matter obtained from calculations with different values of the nucleon radius as a function of (a) energy and (b) density. Notation of the curves is as in Fig. 8.

Figure 10

As Fig. 9a, but for neutron star matter at (a) lower and (b) higher energies. (a) The dash-dot-dotted line in shows the causality constraint, and the filled circles indicate the center pressures and energy densities in the neutron star with maximum mass.

Figure 11

Neutron star mass as a function of its center pressure for different nucleon radii obtained with $\beta =0.01$. Filled circles indicate the maximum masses with the corresponding center pressures.

Figure 12

Proton, neutron, electron, and muon fractions in the neutron star matter as a function of the ratio between nucleon and nuclear saturation densities. Notation of the curves is as in Fig. 8.