Note on the electromagnetic radius of proton

Conventionally, the elastic electron-proton scattering process is depicted in Fig. 1, where the one-photon exchange process is assumed. The kinematical variable required in the following discussion is only the opposite of the virtual photon momentum squared $Q^{2}=-q^{2}$.

The cross section is given in most standard textbooks. It is usually expressed in terms of the electric and magnetization (Sachs) form factors $G_{E}$ and $G_{M}$, which depend solely on $Q^{2}$, i.e.,

where $\tau={Q^{2}}/{4M^{2}}$, $M$ is the proton mass, and the Mott cross section is given by

with $E$ and $E^{\prime}$ the initial and final electron energies, respectively, defined in Fig. 1, whereas the virtual photon polarization reads

For completeness, we have also to mention that $\theta$ is the electron scattering angle and in the limit of $Q^{2}\to 0$ the form factors are normalized to

where $\mu$ is the proton magnetic moment in the unit of nuclear magneton.

In the non-relativistic view it is common to interpret the form factors $G_{E}$ and $G_{M}$ as the Fourier transforms of charge and magnetization distributions inside the proton

However, it is well known that this is not true in the relativistic case, where the relativistic wave functions are frame dependent and, therefore, the interpretation is only valid in the case of Breit frame. There have been solutions proposed to solve this problem. For instance, see Refs. [28, 13]. Nevertheless, for the operational definition of the form factor slope at photon point, we can Taylor expand the form factors as [13]

from which we can calculate the corresponding root mean square (rms) radius as

It is important to emphasize here that the squared radius defined by Eq. (7) is, up to the factor of $-6$, nothing but the slope of the form factor at $Q^{2}=0$. Thus, although the recent and modern data bring us much closer to this photon point, the determination of radius by using this method is merely an estimate, which is strongly influenced by the data at both small and large $Q^{2}$. The latter is the main interest in this note and will be shown in the next section, when we discuss the result of our analysis.

For decades the standard dipole form factor

has been used to fit the available experimental data, from very low to very high $Q^{2}$. The impressive agreement between Eq. (8) and experimental data has led to the question whether the special parameter $0.71$ GeV² has a physical meaning related to the structure of the proton. Unfortunately, there is none [14]. Moreover, it was soon realized that the $Q^{2}$ distribution of experimental data are not smooth; they form a structure at $Q^{2}\approx 0.2$ GeV², which is clearly not reproducible by using Eq. (8). Nevertheless, it is always interesting to use the dipole form factor

in the study of general baryon form factors and to fit the parameter $\Lambda$, since it is the simplest possible form factor to this end. Furthermore, we are also curios to investigate the limit of dipole form factor in view of the recent available experimental data in the low $Q^{2}$ region. Note that by using Eq. (7) to the dipole parameterization given by Eq. (9) we obtain the rms radius of

with error given by

Note that using the form factor parameter given by the standard dipole of Eq. (8), $\Lambda=0.84~{}{\rm GeV}=4.26~{}{\rm fm}^{-1}$, the corresponding radius according to Eq. (10) is 0.81 fm, much smaller than that obtained from present experiments using elastic electron-proton scattering.

A modification to the dipole form factor, Eq. (9), is known as the double dipole (DD) one. The DD form factor is certainly more flexible than the dipole one due to the addition of two new parameters $a_{0}$ and $a_{2}$, whereas $a_{1}\equiv\Lambda$, i.e.,

Note that $G_{\rm DD}$ is properly normalized at photon point, i.e., $G_{\rm DD}(0,a_{0},a_{1},a_{2})=1$. It is straightforward to calculate the rms radius given by Eq. (12). The obtained radius reads

with the corresponding error

To account for the small structure at $Q^{2}\approx 0.2$ GeV² Friedrich and Walcher (FW) added a non-smooth term $G_{\rm ns}$ to the DD form factor given by Eq. (12), with

Thus, the complete expression for the FW form factor is [14]

It is obvious that the corresponding expressions of radius and its error can be obtained from Eqs. (13) and (14) with the addition of the non-smooth contribution. Explicitly, they read

and

The last model investigated in this work is obtained from the Arrington et al. ansatz [5]. In this ansatz the form factor is expanded in a polynomial form, i.e.,

with $\tau={Q^{2}}/{4M^{2}}$ and $M$ is the proton mass. Although the summation in Eq. (19) could be performed up to infinity, Arrington et al. limited it to $n=3$. Thus, Eq. (19) reduces to

Since only parameters $a_{1}$ and $b_{1}$ are proportional to the $Q^{2}$, only these parameters survive in the limit of photon point. As a consequence, the corresponding radius and error are given only in terms of $a_{1}$ and $b_{1}$. Explicitly, they are given by

with the corresponding error

As stated in the Introduction, the latest measurement by the PRad collaboration [6] is very interesting to analyze, since the corresponding data yield a smaller proton radius, compared to those obtained by different collaborations. Furthermore, the extracted radius is consistent with the muonic-hydrogen atom experiment [8]. It is notoriously known that the data points with very small $Q^{2}$ have the form factors very close to one and, therefore, difficult to distinguish. To overcome this problem, in Fig. 2 we plot the relative electric form factor of proton to the standard dipole one, $G_{E,p}-G_{\rm SD}$. As will be explained in the next figures, for this purpose the use of $G_{E,p}-G_{\rm SD}$ is superior than that of $G_{E,p}/G_{\rm SD}$ or $G_{E,p}$ itself. Furthermore, for the sake of brevity, let us define $\Delta\equiv G_{E,p}-G_{\rm SD}$.

At first glance we may say that both the 1.1 and 2.2 GeV PRad data are randomly scattered around the standard dipole $G_{\rm SD}$ curve, i. e., around $\Delta=0$. Only for $Q^{2}\geq 0.005$ GeV² more data start to deviate from the standard dipole parameterization with negative $\Delta$, albeit with relatively large error bars.

In their report, PRad collaboration used the parameterization similar to the Arrington model, i.e., Eq. (19) with $n=1$ both in the numerator and denominator. Furthermore, a more complicated treatment was also used in order to achieve an optimal $\chi^{2}$. In this analysis, we will only use the dipole form factor, since it is simpler and more under-control. The result of fitting this dipole form factor to different data sets is shown in Fig. 2, where for comparison we also show the result of fitting to all available low $Q^{2}$ data that include older experiments.

It is apparent from Fig. 2 that the 1.1 GeV data set yields slightly larger slope, i.e., larger radius and smaller $\Lambda$. The more scattered data points of the 2.2 GeV data set, especially for $Q^{2}\geq 0.005$ GeV², tend to decrease the extracted radius. Thus, different data sets from the same experiment can produce different proton radii. Furthermore, since fitting both 1.1 and 2.2 GeV data sets simultaneously yields almost similar result to fitting to 2.2 GeV data set (compare the dotted and dashed curves in Fig. 2), this indicates that the data with higher $Q^{2}$ values are more decisive for determination of the proton radius. This is obvious, since the slope is strongly dependent on the higher $Q^{2}$ behavior of form factor.

The extracted radii obtained from different data sets given in Fig. 2 are shown in Fig. 3, where we also show the original value extracted by the PRad collaboration and the recommended values from CODATA 2014 [30] and CODATA 2018 [31]. As originally reported by the PRad collaboration [6], they obtained the radius consistent with the later CODATA value. As shown in Fig. 3, in spite of the use of different parameterization, within the obtained error bars all values extracted in the present work by using different PRad data sets are consistent with the original one. The most consistent radius is obtained from fitting the 1.1 GeV data. The use of 2.2 GeV data (individually and simultaneously) yields smaller values and error bars. This is in line with the previous finding in the discussion of Fig. 2. Therefore, we confirm the finding of the PRad collaboration that their data lead to smaller radius, closer to the value obtained from the muonic hydrogen atom spectroscopy.

We also note that fitting to all other data with small $Q^{2}$, i.e., up to 0.06 GeV², results in a larger radius. Nevertheless, it is still closer to the value from muonic hydrogen spectroscopic measurement. This is caused by the fact that there are data smaller than the PRad data in the $0.01\lesssim Q^{2}\lesssim 0.05$ GeV² region, which tend to shift the $\Delta$ curve down, increasing the slope and, therefore, the extracted radius. We have also investigated the effect of the PRad data on this fitting. We found that the effect is small, the form factor cut-off slightly increases from $0.8012\pm 0.0010$ to $0.8024\pm 0.0010$ GeV after the inclusion of the PRad data. The small error indicates that the dipole form factor has very small degree of freedom if we include all available data. This is understood since the number of free parameter is only one. As we will see later, this is not the case, when we use different form factors with more free parameters.

Having shown that the higher $Q^{2}$ data may significantly influence the result of extracting the proton charge radius at the $Q^{2}\approx 0$ region, we are in the position to investigate the effect of higher $Q^{2}$ data to this end. As stated in Sec. 2, there is a visible structure at $Q^{2}\approx 0.2$ GeV² which forces us to introduce some physics to account this phenomenon. Obviously, the use of dipole form factor is inadequate to fit this structure. An alternative solution has been proposed by Friedrich and Walcher [14].

A list of references to the available experimental data is given in Table 3.2. From this Table it is apparent that the high $Q^{2}$ data originate from relatively older experiments. The newest experiments are racing to reach the lowest possible $Q^{2}$ region.

The results of fitting free parameters of the dipole, double-dipole, Friedrich-Walcher, and Arrington form factors are given in Tables 3.2, 3.2, 3.2, and 3.2, respectively. From these Tables it is obvious that the new data deviate significantly from the standard dipole parameterization. The dipole parameterization can give a better agreement with data, but it is also clear that the double-dipole form factor yields a much better agreement due to the larger number of free parameters. This is also true for the Friedrich-Walcher and Arrington form factors, except in these form factors more physics is involved. Nevertheless, due to the large data errors in the large $Q^{2}$ region, the smaller $\chi^{2}$ does not directly show the best parameterization, especially when we discuss the structure at $Q^{2}\approx 0.2$ GeV².

A quick glance at Tables 3.2-3.2 reveals that the extracted radii from all refitted models are smaller than the values obtained from the conventional elastic electron-proton scattering. This result is corroborated by our result in the previous discussion on the new PRad data and shown in Fig. 3. However, we will discuss this later, when we compare our present result with those given by previous works in Fig. 6, because it is easier to analyze them by using this figure.

For the electric form factor $G_{E,p}$, comparisons between the fitted models and experimental data are shown in Fig. 4, where we display the electric form factor $G_{E,p}$, the electric form factor in the unit of the standard dipole one $G_{E,p}/G_{\rm SD}$, and the difference between the electric and the standard dipole form factors $G_{E,p}-G_{\rm SD}$, as functions of $Q^{2}$. These three methods of presenting the proton electric form factor are intended to look for the best and sensitive way to compare the performance of different parameterizations.

From panel (a) of Fig. 4 we can see that the plot of $G_{E,p}$ cannot clearly distinguish the difference between the 7 parameterizations discussed above. Only at very high $Q^{2}$ the difference among them shows up. Unfortunately, the presently available data cannot single out the best parameterization from our calculation. Moreover, the structure at $Q^{2}\approx 0.2$ GeV² cannot be seen here.

The electric form factor in the unit of the standard dipole one $G_{E,p}/G_{\rm SD}$ shown in panel (b) of Fig. 4 starts to display the difference between the parameterizations at $Q^{2}\gtrsim 0.2$ GeV². However, since the interesting structure is shown by the data at $Q^{2}\approx 0.2$ GeV², the plot of $G_{E,p}/G_{\rm SD}$ does not help too much in this case. Furthermore, as shown in this panel, relative to the standard dipole one, most of the parameterizations quickly decrease as the $Q^{2}$ increases from 1 GeV². Therefore, comparison between data and models is less useful for the present purpose.

Only the third method, which uses $\Delta\equiv G_{E,p}-G_{\rm SD}$ shown in panel (c), gives the clearest information for our present purpose, since the structure around $0.2$ GeV² is clearly revealed by experimental data and model calculations. Interestingly, panel (c) of Fig. 4 also indicates that the proton electric form factor $G_{E,p}$ is approaching the standard dipole one $G_{\rm SD}$ not only at low $Q^{2}$, but also at high $Q^{2}$ region, although in the latter experimental data are wildly scattered with large error bars. The fact that all form factors approaches the standard dipole one does not appear in panel (b) since all form factors decrease faster than the standard dipole one. However, since at high $Q^{2}$ all form factors are very tiny and inaccessible through experiment, we may conclude that the third method is the best one to compare all form factor models. From panel (c) of Fig. 4 it is clear that the structures near $0.2$ GeV² and the differences between the presented models could be investigated if we had experimental data with comparable accuracies to those at low $Q^{2}$.

To investigate the structure near $Q^{2}=0.2$ GeV² let us focus our attention on panel (c) of Fig. 4, especially in the range of $0.001\lesssim Q^{2}\lesssim 10$ GeV². This is shown in Fig. 5, where we compare the analyzed models with experimental data. By looking at the data in the range of $0.05\lesssim Q^{2}\lesssim 5$ GeV² we can infer that there are at least two minima located at $Q^{2}\approx 0.1$ and 0.2 GeV². Another possible minimum is seen at $Q^{2}\approx 0.4$ GeV², which is less visible than the previous two. Clearly, these minima cannot be produced by the dipole form factor, since as shown in Fig. 5, this form factor can only create one minimum, due to its smooth behavior. The same phenomenon is also displayed by the double dipole one.

As stated by Friedrich and Walcher [14], the structure can be best described by using the non-smooth term $G_{\rm ns}$ given by Eq. (15). As shown in Fig. 5, by using this term, the original F-W model is able to produce two minima at $Q^{2}\approx 0.15$ and 2 GeV². The refitted F-W model shifts these two minima closer to experimental data, i.e., at $Q^{2}\approx 0.1$ and 0.3 GeV².

The original Arrington model also produces two minima with their positions very close to those of the original F-W one. The two models differ only in their form factor magnitudes. As in the case of refitted F-W model, the refitted Arrington model is much closer to experimental data. The remarkable difference here is that the refitted Arrington model produces three minima at $Q^{2}\approx 0.1,0.3$ and 2 GeV². It is of course somewhat strange that both original F-W and Arrington models have a minimum at 2 GeV², whereas almost no experimental data support this phenomenon. Even the refitted Arrington model has also a minimum at this position. Presumably, it is the way of the models to produce a peak at 0.8 GeV², where a number of experimental data points are located.

From Fig. 5 we observe that for $Q^{2}\to 0$ the softest slope (relative to the horizontal $\Delta=0$ line) is obtained for the Dipole and refitted F-W models, whereas the steepest one is given by the original Arrington model. From this observation, we can estimate that the smallest radius is obtained by the Dipole and refitted F-W models, while the largest radius is obtained by the original Arrington model. This is proven in Fig. 6 and will be discussed later. At this stage it is important to emphasize that the different slopes obtained from different models shown in Fig. 5 are strongly related to different descriptions of the structure near $Q^{2}\approx 0.2$ GeV². In other words, the experimental data at this kinematics significantly influence the calculated proton radius. Therefore, our present work recommends that future experiments should focus on this kinematics instead of going to lower $Q^{2}$. We note that similar conclusion can be also found in previous works [58, 59, 60]. By comparing the models shown in Fig. 5, we believe that experimental data with the same accuracy as the latest data from JLab, for instance, would significantly help to clarify whether the three minima really exist and, therefore, to single out the best parameterization that globally fits the form factor data from 0.0001 to 10 GeV².

Now, let us consider the extracted proton radius from the different models that we have discussed above. The result is shown in Fig. 6 by the open circles. There are many analyses of this radius in the literature and, as a consequence, only recent results are displayed in Fig. 6.

A newer radius obtained by using initial state radiation (ISR) in MAMI Mainz and formally published in 2021 is smaller than the Mainz 2010 result. However, the quoted error is significantly large, making it consistent with both values obtained from the conventional elastic electron-proton scattering experiments and muonic hydrogen spectroscopic measurement. On the other hand, instead of measuring the cross section, the JLab Hall A experiment [62] utilized the polarization transfer of electron-proton elastic scattering in the region of $Q^{2}=0.3-0.7$ GeV² and obtained the radius which is consistent with the value obtained from the Mainz-2010 electron-proton scattering experiment [2].

More recently, the PRad collaboration performed an experiment at JLab and succeeded to reach much lower $Q^{2}$ by using a new electromagnetic calorimeter that could cover a larger area, with a hole in the center to allow the non-scattered electrons to pass through, and therefore could reach much smaller scattering angles [6]. It is widely known that the problem to reach smaller $Q^{2}$ is related to the problem of dealing with very forward scattering angles. The result of this experiment is more precise than that of the polarization transfer experiment at Hall A. As shown in Fig. 6, a smaller radius consistent with the value from muonic hydrogen spectroscopic measurement [8] was obtained. As discussed in the previous Section we have refitted the PRad data by means of a dipole form factor and obtained a consistent result as shown in the same figure. The smaller error obtained in this case originates from the nature of dipole form factor, which has only one parameter and, as a consequence, a very simple error formula given by Eq. (11), where $\Delta\Lambda$ is obtained from MINUIT. This phenomenon was also found in our previous study [69, 70, 71, 72].

The next interesting result is shown by four theoretical calculations, with two of them consistent with the muonic hydrogen value, whereas the other two consistent with the value from electron-proton scattering experiment. By using an improved two-pion continuum, Lin et al. found smaller radius [63] consistent with the muonic hydrogen value. A similar result was obtained by Cui et al., who analyzed the PRad data by means of a statistical sampling approach. Different result was obtained by Arrington and Sick [64]. By using a global fit to elastic electron-proton data they obtained a radius of $0.879\pm 0.011$ fm, which is identical to the CODATA-2014 value. Sick [66] continued the investigation by constraining the extrapolation to very low $Q^{2}$ region, where no experimental data are available, by using experimental data at finite $Q^{2}$ in order to reduce model dependence. He obtained a radius value of $0.887\pm 0.012$ fm, which is consistent with the value from electron-proton scattering experiment too, as shown in Fig. 6.

It is interesting to see that our fits (indicated by open circles in Fig. 6) using dipole form factor to both low $Q^{2}$ and all $Q^{2}$ data are mostly closer to the value obtained from the muonic hydrogen atom spectroscopy. The relatively large difference obtained from fit to PRad and fit to all low $Q^{2}$ data is a clear indication of the difference between the two data sets, as has been stated in the previous section. It is also interesting that, by using all data, including the highest $Q^{2}$ ones, the extracted radius decreases and approaches the muonic hydrogen atom value. This is the clear evidence that the high $Q^{2}$ data have significant effect on the extracted radius. As a consequence, high $Q^{2}$ data could become a stringent constraint to the existing models that try to extract the proton charge radius.

Surprisingly, fitting to all data but with a double dipole form factor results in a larger radius, consistent with the value obtained from the conventional electron-proton scattering experiment. Once again, this result indicates that the extraction of radius also depends strongly on the form factor model. Compared to the dipole form factor, the larger error in this case originates from the larger number of degrees of freedom (free parameters). However, if we look back to Fig. 5, we observe that the double dipole form factor yields only one minimum at $Q^{2}\approx 0.2$ GeV² and, therefore, does not precisely reproduce the structure shown by experimental data at this kinematics.

As previously discussed, the more precise models that reproduce the structure around $Q^{2}\approx 0.2$ GeV² are the F-W and Arrington models. Original calculations of these models yield also different radii. Whereas the original F-W model predicted a smaller radius than the muonic hydrogen atom one, the Arrington model corroborates the conventional electron-proton scattering radius.

Interestingly, refitting the two models to new data shifts their radii closer to each other, i.e., increases the F-W radius and decreases the Arrington one. The origin of this phenomenon can be traced back to Fig. 5. Refitting the F-W model shifts the second minimum to a lower $Q^{2}$, where more experimental data show a minimum. This process makes the corresponding slope to the $Q^{2}=0$ point steeper and, as a consequence, makes the extracted radius larger. In the case of Arrington model, the refitting process increases the number of minima from two to three. This is possible since, as shown in Eq. (20), the Arrington model has more complex functions of $Q^{2}$. In this case, the new minimum is located at $Q^{2}\approx 0.4$ GeV². This new minimum makes the slope of form factor, relative to the standard dipole one, softer and, therefore, makes the extracted radius smaller.

Note that as shown in Tables 3.2 and 3.2, the difference between the $\chi^{2}$’s obtained from F-W and Arrington models is almost negligible (608 and 607). However, by carefully looking at Fig. 5, we believe that the F-W model is closer to reproduce experimental data at two minima near 0.1 and 0.4 GeV², whereas the third minimum of Arrington model near 2 GeV² seems to be unsupported by experimental data. Therefore, our present study prefers the refitted Friedrich-Walcher model that yields a radius of $0.836\pm 0.011$ fm, consistent with the muonic hydrogen atom value. Nevertheless, a more quantitative conclusion should wait for more precise experimental data in this kinematical region. As stated before, experimental data with accuracy comparable to the latest data in the low $Q^{2}$ region would be very decisive to this end.

To emphasize the importance of including higher $Q^{2}$ data in the extraction of proton charge radius, in Fig. 7 we show the effect of the data range used in the fit on the extracted form factor derivatives at photon point along with the corresponding proton charge radii. For this purpose we divide the available data into five data ranges written in detail in the caption of Fig. 7.

From the data shown in Fig. 4(c) we would expect that the largest effect could be obtained if we included the data near the structure, i.e., with $Q^{2}$ between 0.01 and 1 GeV², which corresponds to the use of the data ranges 3 and 4. This is proven by both the form factor derivatives at photon point and the corresponding proton charge radii shown in Fig. 7. All form factors show significant derivative or radius changes once we switch from the data range 2 to 3, as well as from the data range 3 to 4. Furthermore, as expected from the data shown in Fig. 4(c), the use of data range 1 or 2 does not produce different derivatives or radii. The same phenomenon is also observed in the case of data range 4 or 5. This result emphasizes our previous finding that the inclusion of data with $Q^{2}$ between 0.01 and 1 GeV² significantly changes the form factor slope to the $Q^{2}=0$ point, as discussed above.

The large error bars obtained from fitting the data range 1 or 2, especially in the case of Arrington and Friedrich-Walcher form factors, originate from the number of used free parameters, which is unnecessarily too large to fit the relatively few data points. This is not the case for the dipole form factor, since it has only one free parameter. To conclude this section, we have shown that the use of higher $Q^{2}$ data has a significant effect on the extracted proton charge radius.

For completeness, in Fig. 8 we also present the proton magnetic form factor $G_{M,p}/\mu$, where we display the behavior of the form factor in the form of $G_{M,p}/\mu G_{\rm SD}$ and $G_{M,p}/\mu-G_{\rm SD}$, similar to the case of electric form factor discussed above. It is important to note here that the ratio $G_{M,p}/\mu G_{\rm SD}$ reveals the second structure (peak or maximum) clearly at $Q^{2}\approx 2$ GeV² while the difference $G_{M,p}/\mu-G_{\rm SD}$ display both structures at $Q^{2}\approx 0.2$ and 1-2 GeV² moderately.

Different from the electric form factor, here we see that the refitted F-W and Arrington models agree with each other, while the difference between the original models is more obvious in panel (c) of Fig. 8, since the difference only exists in the first structure (minimum) at 0.2 GeV².

The extracted proton magnetic radii $r_{M}$ obtained in the present work are shown in Fig. 9 by open circles, along with those previously calculated and available in literature. In this figure we compare the radii with that of the Particle Data Group [68] (PDG), i.e., $r_{M}=0.851\pm 0.026$ fm. Here we can see that most of the calculations agree with the PDG estimate. This is very different from the case of charge radius. The proton magnetic radii extracted from refitting the F-W and Arrington models are also moving closer to each other, as expected from the behavior of their form factors for small $Q^{2}$ shown in Fig. 8. While the refitted F-W model yields $r_{M}=0.821\pm 0.004$ fm, the Arrington refitted one yields $r_{M}=0.826\pm 0.002$ fm. Thus, the extracted radii from the two models are consistent to each other as well as to the PDG estimate. The present calculation indicates that refitting models with the new data from very low to very high $Q^{2}$ can improve the agreement of the extracted proton magnetic radii.