Proton radius reconstruction from simulated electron-proton elastic scattering cross sections at low transfer momenta

TL;DR

This paper discusses a planned low-momentum transfer electron-proton scattering experiment at Mainz MAMI aiming to precisely measure the proton charge radius using an innovative detection method in a specific low Q^2 range.

Contribution

It introduces a new experimental approach for measuring ep scattering cross sections at low Q^2 to improve proton radius determination.

Findings

Simulation results support the feasibility of the experiment.

Expected sub-percent precision in proton radius measurement.

Potential to resolve existing proton radius discrepancies.

Abstract

This note is motivated by preparations of a new $ep$ elastic scattering experiment in the low transfer momentum region to be carried out in the 720 MeV electron beam of the Mainz Microtron MAMI. This experiment will use an innovative method allowing for detection of recoil protons in coincidence with the scattered electrons. The goal is to measure the $ep$ differential cross sections in the $Q^2$ range from 0.001 GeV$^2$ to 0.04 GeV$^2$ and to determine the proton charge radius with sub-percent precision.

Tables

Table 1. Table 2: Mean values of the R ​ 2 𝑅 2 R2 and R ​ 4 𝑅 4 R4 parameters determined from the fits of 1000 independently generated G E 2 superscript subscript 𝐺 𝐸 2 G_{E}^{2} ( Q 2 superscript 𝑄 2 Q^{2} ) distributions (as shown in Fig.  2 ) for various options of the fixed parameters R ​ 6 𝑅 6 R6 and R ​ 8 𝑅 8 R8 . FF* denotes the form factor parameters used to generate the G E 2 superscript subscript 𝐺 𝐸 2 G_{E}^{2} ( Q 2 superscript 𝑄 2 Q^{2} ) distribution. In all shown fits, the mean values of parameters A 𝐴 A and χ 2 superscript 𝜒 2 \chi^{2} /ndf are equal to 1.0 with 10 − 4 superscript 10 4 10^{-4} and 10 − 3 superscript 10 3 10^{-3} accuracy, respectively.

Fit	* fm2	* fm4	* fm6	$R8$ fm8
FF*	0.7700*	2.63*	25.98*	373.5*
Fit1	0.7703(5) +0.0003(5)	2.61$\pm$0.03 -0.03(3)	26$\pm$1 0$\pm$1	374 fixed
Fit2	0.7693(5) -0.0007(5)	2.49$\pm$0.03 -0.14(3)	23.4$\pm$1.0 -1.6$\pm$1.0	0 fixed
Fit3	0.7727(2) +0.0027(2)	2.743(7) +0.113(7)	26 fixed	0 fixed
Fit4	0.7665(2) -0.0035(2)	2.284(7) -0.346(7)	10 fixed	0 fixed
Fit5	0.7761(2) +0.0061(2)	3.00(7) +0.37	35 fixed	0 fixed

Table 2. Table 6: The values of R4 and R6 in different presentations of the proton Form Factor, corresponding to R2 = 0.7700 fm 2

Form Factor	$R4$, fm4	$R6$, fm6	$\eta =R6/R4$, fm2
Dipole FF	1.49	5.3	3.6
Di$\chi$EFT [7,8]	1.6	9.0	5.6
Bernauer [5]	2.63	26	9.9

Equations

$$\frac{d\sigma }{dt}=\frac{\pi {\alpha }^2}{t^2}\left\{G_E^2\left[\frac{{\left(4M+t/{\epsilon }_e\right)}^2}{4M^2-t}+\frac{t}{{\epsilon }_e^2}\right]-\frac{t}{4M^2}{G}_M^2\left[\frac{{\left(4M+t/{\epsilon }_e\right)}^2}{4M^2-t}-\frac{t}{{\epsilon }_e^2}\right]\right\}{\text{GeV}}^{-4},\text{vspace}.4cm$$

$$G_M(Q^2)={\mu }_p\cdot G_E(Q^2)=2.793G_E(Q^2).$$

$$G_E(Q^2)=1-R2\cdot B_2\cdot Q^2/C_2+R4\cdot B_4\cdot Q^4/C_4-R6\cdot B_6\cdot Q^6/C_6+R8\cdot B_8\cdot Q^8/C_8,$$

$$d\sigma /dt(R_p=0.8775\text{ fm})/d\sigma /dt(R_p=0)=G_E^2(Q^2).$$

$$(G_E{)}_i^2=N_i(R_p=0.8775\text{ fm})/N_i(R_p=0).$$

$$(G_E{)}_i^2=N_i(R_p=0.8775\text{ fm})/0.01N_i(R_p=0).$$

$$G_E(Q^2{)}_{fit}=A\cdot (1-R2\cdot B_2\cdot Q^2/C_2+R4\cdot B_4\cdot Q^4/C_4-R6\cdot B_6\cdot Q^6/C_6+R8\cdot B_8\cdot Q^8/C_8)$$

$$G_E(Q^2{)}_{\text{fit}}=A\cdot (1-R2\cdot B_2\cdot Q^2/C_2+R4\cdot B_4\cdot Q^4/C_4-\eta \cdot R_4\cdot B_6\cdot Q^6/C_6+R8\cdot B_8\cdot Q^8/C_8),$$

Full text

In the $ep$ elastic scattering experiments, the proton charge radius is extracted from the slope of the electric form factor at the momentum transfer squared $Q^{2}\rightarrow$0. In order to estimate the level of statistical and systematic errors in the extracted proton radius, we simulated the $ep$ elastic scattering differential cross section using the proton form factor available from analysis of the experimental data from the A1 experiment at Mainz. Then the proton radius was extracted from fitting the simulated pseudo-data with the cross section calculated using a $Q^{2}$ power series expansion of the proton electric form factor up to the $Q^{8}$ term. About 70 million of the $ep$ elastic scattering events were generated in the $Q^{2}$ range from 0.001 GeV2 to 0.04 GeV2, that corresponds to the statistics to be collected in our experiment in 45 days. For the considered $Q^{2}$ range and statistics, the main conclusions of these studies are as follows:

• •

The extracted value of the proton charge radius is not sensitive to the $Q^{8}$ term, so this term can be neglected in the fits.

• •

The fits with four free parameters ($A,<r_{p}^{2}>,<r_{p}^{4}>,<r_{p}^{6}>$) determine the proton charge $rms$-radius $R_{p}$ = $<r_{p}^{2}>^{1/2}$ with the errors $\Delta R_{p}$ (stat)= 0.0085 fm (sigma) and $\Delta R_{p}(syst)\leq$ 0.001 fm.

• •

The statistical error can be reduced by a factor of two down to $\Delta R_{p}(stat)$ = 0.0042 fm by fixing parameter $<r_{p}^{6}>$ to some value determined in the experiments performed at larger transfer momenta. As an example, we have used the published value of $<r_{p}^{6}>$ = 29.8 (7.6)(12.6) fm6 determined in such experiments. Unfortunately, this value suffers from rather large systematic uncertainty that resulted in a systematic error in the extracted proton radius : $\Delta R_{p}$ (syst) = 0.0025 fm. Another promising approach is to use a theoretical value for $<r_{p}^{6}>$ in the fits .

1 Introduction

The striking difference in the proton charge $rms$-radius extracted from the two types of experiments, the elastic $ep$ scattering experiments ( $R_{p}$ = 0.879 (5)(6) fm [1], Rp = 0.875(10) fm [2]) ) and the muonic Lamb shift experiments( $R_{p}$ = 0.8409 (4) fm [3] ), so called “proton radius puzzle”, is widely discussed. As it is generally agreed, new high precision measurements of the $ep$ scattering differential cross sections in the low momentum transfer region are needed to resolve this puzzle. Recently, a new experiment was proposed by our collaboration [4] to be carried out in the 720 MeV electron beam of the Mainz Microtron MAMI. An innovative method will be used allowing for detection of recoil protons in coincidence with the scattered electrons. The goal of this experiment is to measure the $ep$ differential cross sections in the $Q^{2}$ range from 0.001 GeV2 to 0.04 GeV2 with 0.1 % relative and 0.2% absolute precision and to determine the proton charge radius with sub-percent precision. In this $Q^{2}$ range, about 70 million $ep$ elastic scattering events should be collected in 45 days of the beam time.

This note considers possible algorithms of analysis of the experimental data from this experiment. In order to estimate the level of statistical and systematic errors in the extracted proton radius, we simulated the $ep$ elastic scattering differential cross section using the proton form factor available from analysis of the experimental data from the A1 experiment at Mainz. Then the proton radius was extracted from fitting the simulated pseudo-data with the cross section calculated using various approximations for the $Q^{2}$ dependence of the proton form factor.

2 Generation of $ep$ scattering events

For this analysis, the $ep$ scattering events were generated according to the following function describing the $ep$ elastic scattering differential cross section:

[TABLE]

where -t = $Q^{2}$; $\alpha$ = 1/137.036; $M$ is the proton mass ($M$ = 938.272 MeV); $\varepsilon_{e}$ is the total electron energy ($\varepsilon_{e}$ =720.5 MeV); $G_{E}(Q^{2}$) and $G_{M}(Q^{2}$) are the electric and magnetic form factors, respectively. We have accepted the following approximation valid for the small $Q^{2}$ region:

[TABLE]

$G_{E}(Q^{2}$) is taken as a power series expansion:

[TABLE]

where $B_{n}$ = $(5.06773)^{n}$, $C_{n}$ = (n+1)!, n=2,4,6,8; $R2$ = $<r_{p}^{2}>$, $R4$ = $<r_{p}^{4}>$ , $R6$ = $<r_{p}^{6}>$, and $R8$ = $<r_{p}^{8}>$. The rms-radius $R_{p}$ = $(R2)^{1/2}$. In such presentation, $<r_{p}^{n}>$ and $Q^{n}$ are expressed in fmn and in GeVn, respectively. 1 fm = 5.06773 GeV*-1*; 1 GeV*-2* = 0.389379 mb.

The $ep$ scattering events were generated in the $Q^{2}$ range from 0.001 GeV2 to 0.04 GeV2 using the values of $R2$, $R4$, $R6$, and $R8$ obtained by J.C.Bernauer [5,6] from analysis of the cross sections measured in the A1 experiment:

$R2$ = 0.7700 fm2, $R4$ = 2.63 fm4, $R6$ = 26 fm6, $R8$ = 374 fm8 .

The corresponding proton rms-radius is $R_{p}$ =$(R2)^{1/2}$ = (0.7700 fm${}^{2})^{1/2}$ = 0.8775 fm.

The $ep$ scattering cross sections integrated over the $Q^{2}$ range   0.001 GeV${}^{2}\leq Q^{2}\leq$ 0.04   GeV 2 are:

$\sigma$($R_{p}$ = 0.8775 fm) = 0.248703 mb and $\sigma$($R_{p}$ = 0) = 0.254724 mb.

The ratio of these cross sections is $K$ = 0.976363.

As it follows from eqs.(1) and (2), the ratio of the differential cross sections gives the form factor squared in function of $Q^{2}$:

[TABLE]

We find this ratio by generating two similar samples of the $ep$ scattering events: one for $R_{p}$=0.8775 fm and another one for $R_{p}$ = 0. These samples should correspond to the same luminosity. That means that the number of generated events for $R_{p}$ = 0.8775 fm should be by a factor of $K$ = 0.976363 less than that for $R_{p}$ = 0. Then the value of $(G_{E})_{i}^{2}$ in each bin can be obtained by the ratio of the numbers of generated events in that bin:

[TABLE]

In order to reduce contribution of fluctuations in $N_{i}$($R_{p}$=0) to the statistical error in $(G_{E})_{i}^{2}$, the ($R_{p}$=0) sample is generated with 100 times larger statistics, therefore eq. ( 5) is transformed to:

[TABLE]

The $ep$ scattering events were generated using the ROOT framework. Besides the analytical function of $d\sigma/dt$, we use as the input parameters: the $Q^{2}$ range, the binning within this range, and the total number of generated events. At the level of the events generation, we use 1000 bins of equal width in the $Q^{2}$ range 0.001 GeV${}^{2}\leq Q^{2}\leq 0.04$ GeV2 with a possibility of further re-binning of the generated $G_{E}^{2}$ ($Q^{2}$) distribution. For each bin, the program gives the numbers of events integrated over the bin width,

$N_{i}$ ($R_{p}$ = 0.8775 fm) and $N_{i}$ ($R_{p}$ = 0), and determines $(G_{E})_{i}^{2}$ according to eq.( 6). About 70 million events generated in the $Q^{2}$ range from 0.001 GeV2 to $Q^{2}$ = 0.04 GeV2 correspond to the expected number of events to be collected in our experiment in 45 days of continuous running with integrated luminosity $L_{int}$ = $2.8\cdot 10^{8}$ mb*-1*. As an example, Figure 1 presents the simulated differential cross sections. The total number of generated events was $N_{ev}$($R_{p}$ = 0.8775 fm) = $6.96369\cdot 10^{7}$ events and $N_{ev}$($R_{p}$ = 0) = 7.13227$\cdot 10^{9}$ events. Figure 2 (left panel) shows the $G_{E}^{2}$ ($Q^{2}$) distribution determined according to eq.( 6). The right panel shows the same spectrum after re-binning the generated spectrum to 100 bins in the same $Q^{2}$ range.

3 Fitting of the $G_{E}^{2}$ ($Q^{2}$) distributions

To fit the generated $G_{E}^{2}$ ($Q^{2}$) distributions, we use the power series expansion of the form factor:

[TABLE]

with the constants $B_{n}$ and $C_{n}$ as in eq.( 3). The goal was to see how many $Q^{2}$ terms should be retained in this expression to provide minimal combined statistical plus systematic error in determination of the proton radius. The following options have been tested:

Option 1: $A$, $R2$, $R4$, $R6$ are free parameters, $R8$ is a fixed variable.

Option 2: $A$, $R2$, $R4$ are free parameters, $R6$ and $R8$ are fixed variables.

Statistical errors in measurements of the proton radius

Table 1 compares the statistical errors in $R2$ and $R4$ obtained by fitting the generated $G_{E}^{2}$ ($Q^{2}$) with $G_{E}(Q^{2})_{fit}$ represented by eq.( 7) with four or three free parameters for statistics planned to collect in 45 days of continuous running of the experiment.

From comparison of Fit 1 and Fit 2 in Table 1, one can see that reduction of the number of free parameters by fixing R6 to some fixed value reduces the statistical error in determination of the proton radius by a factor of two ( from $\pm$ 0.0085 fm to $\pm$ 0.0041 fm). Also, the $R4$ parameter is determined with 8% precision in this fit.

**Systematic biases in measurement of the proton radius

**

We have performed a number of fitting sets with various fixed values of $R6$ and $R8$ to study possible systematic biases related to this procedure. In each fitting set the fit was repeated 1000 times with independently generated $G_{E}^{2}$ ($Q^{2}$) distributions. Figures 3 and 4 show the examples of such fits with four free parameters and with three free parameters, respectively.

The distributions shown in Figs. 3, 4 were obtained with 1000 bins in the $G_{E}^{2}$ ($Q^{2}$) distributions.

The re-binning of these distributions to 100 bins gives identical fitting results, except the $\chi^{2}$ distribution becomes wider by a factor of three (Fig. 5).

As it follows from Fig. 4, the fits with three free parameters can provide 0.0072/0.770 = 0.94% statistical precision in determination of $R2$ (0.47% precision in $R_{p}$). In addition, $R4$ is measured with 8% statistical precision. In these fits, $R6$ and $R8$ were fixed to 26 fm6 and to zero, respectively. To see the sensitivity of obtained values of $R2$ and $R4$ to the chosen value of $R6$, the fits were repeated with $R6$ = 10 fm6 and 35 fm6. The results are presented in Figs. 6, 7 and in Table 2.

As concerns the influence of parameter $R8$ on measurement of $R2$, it is proved to be practically negligible, as it follows from comparison of Fit1 with Fit2 in Table 2. The variation of $R8$ from 374 fm8 to zero shifts the value of $R2$ by 0.13% (0.065% shift in $R_{p}$). On the other hand, the sensitivity of the extracted value of $R2$ to the fixed values of $R6$ is more essential (Fits 3,4,5). The variation of $R6$ from 10 fm6 to 35 fm6 resulted in a systematic shift of $R2$ by 1.2 % (0.6% in $R_{p}$).

The systematic biases were studied also by another method when the simulated cross sections were generated with 1000 times higher statistics: ($N_{ev}$($R_{p}$ = 0.8775 fm) = 6.96369$\cdot 10^{10}$ events and

$N_{ev}$ ($R_{p}$ = 0) = 7.13227$\cdot 10^{10}$ events). The results are presented in Table 3. As it follows from Fits 1,2,3 in Table 3, variation of $R8$ from $R8$ = 0 to $R8$ =700 fm8 resulted in a 0.2% shift in the extracted $R2$ value. Therefore, it is safe to fix $R8$ at $R8$ = 374 fm8 and consider the systematic error in $R2$ due to uncertainties in $R8$ to be on a level of $\pm$ 0.1 % ( 0.05% in $R_{p}$).

While fixing the $R6$ parameter, it is natural to take into account the results of previous analyses of the $ep$ scattering data. According to [6], $R6$ = 29.8 (7.6)(12.6) fm6 and $R4$ = 2.59 (19)(04) fm4 . Therefore, we can fix $R6$ at $\simeq$ 26 fm6 with uncertainty of $\pm$ 15 fm6. As one can see from Fits 4, 5, 6 in Table 3, such uncertainty in $R6$ leads to $\pm$ 0.8% systematic errors in $R2$ ($\pm$ 0.4% in $R_{p}$). As to the $R4$ parameter, it can be determined directly from our experimental data, and comparison with the A1 data could be used as a cross check.

Additional study of the systematic shifts in the $R2$ values was done by fitting the ratio of the differential cross sections $d\sigma/dt$($R_{p}$) / $d\sigma/dt$($R_{p}$ =0) generated with high statistics for three options of the polynomial Form Factor, FF1, FF2, and FF3, with variations of the $R4$, $R6$, and $R8$ values consistent with the uncertainties of the A1 data. The fitting function contained three free parameters ($A$, $R2$, $R4$), while the $R6$ and $R8$ parameters were fixed to 26 fm6 and to 374 fm8, respectively. The results are presented in Table 4.

Table 4 shows that the fits with a fixed $R6$ parameter ($R6$ =26 fm6) reproduce $R2$ with $\pm$ 0.56 % systematic error ($\pm$ 0.28% error in the proton radius), assuming that the $R6$ value in the real experimental data will be in the limits 11 fm6 $<R6<$ 41 fm6.

4 Summary

We have analyzed the simulated $ep$ scattering differential cross section expected from an experiment aimed at high precision measurement of the proton charge rms-radius $R_{p}$ = $<r_{p}^{2}>^{1/2}$ . Following the Proposal of our experiment, it was accepted that 70 million of the $ep$ elastic scattering events will be collected in the $Q^{2}$ range 0.001 GeV${}^{2}\leq Q^{2}\leq$ 0.04 GeV2. The $ep$ elastic scattering events were generated with the polynomial proton charge form factor determined by J.C.Bernauer et al. in the data analysis of the A1 experiment [5,6], with an additional assumption that $G_{M}$($Q^{2}$) = $\mu_{p}\cdot G_{E}(Q^{2}$) in the considered $Q^{2}$ range. The generated pseudo-data were fitted with a polynomial function:

$G_{E}(Q^{2})=A\cdot(1-<r_{p}^{2}>\cdot B_{2}\cdot Q^{2}/C_{2}+<r_{p}^{4}>\cdot B_{4}\cdot Q^{4}/C_{4}-<r_{p}^{6}>\cdot B_{6}\cdot Q^{6}/C_{6}+<r_{p}^{8}>\cdot B_{8}\cdot Q^{8}/C_{8})$ , where $B_{n}$ = (5.06773)n, $C_{n}$ = (n+1)!, n = 2,4,6,8; $<r_{p}^{n}>$ and $Q^{n}$ are expressed in fmn and in GeVn, respectively. Two options have been tested:

Option 1: $A$, $<r_{p}^{2}>$, $<r_{p}^{4}>$, $<r_{p}^{6}>$ are free parameters, $<r_{p}^{8}>$ is a fixed variable;

Option 2: $A$, $<r_{p}^{2}>$, $<r_{p}^{4}>$ are free parameters, $<r_{p}^{6}>$ and $<r_{p}^{8}>$ are fixed variables.

The results of the analysis can be summarized as follows:

• •

The $Q^{8}$ term plays very little role in determination of $R_{p}$. The variation of $<r_{p}^{8}>$ from zero to 700 fm8 leads to increasing the $R_{p}$ value by 0.001 fm. Therefore, one can fix $<r_{p}^{8}>$, for example, at the value from the A1 analysis ( $<r_{p}^{8}>$ = 374 fm8 [5]). This may introduce a systematic error in $R_{p}$ due to uncertainties in $<r_{p}^{8}>$ on a negligible level of $\pm$ 0.0005 fm.

• •

The statistical error in $R_{p}$ in the fits with four free parameters ($A$, $<r_{p}^{2}>$, $<r_{p}^{4}>$, $<r_{p}^{6}>$)

is $\pm$ 0.0085 fm. The advantage of such fit is a negligibly small systematic bias.

• •

The statistical error in $R_{p}$ can be reduced by a factor of two (down to $\pm$ 0.0042 fm) in the fit with three free parameters ($A$, $<r_{p}^{2}>$, $<r_{p}^{4}>$) by fixing $<r_{p}^{6}>$ to some value followed from the analysis of the $ep$ scattering data in the higher $Q^{2}$ region. However, in this case some systematic bias may be introduced because of uncertainties in the $<r_{p}^{6}>$ value. The sensitivity of $R_{p}$ to variations in $<r_{p}^{6}>$, as determined in our analysis, is as follows: a shift in $<r_{p}^{6}>$ by 6 fm6 produces a shift in $R_{p}$ by 0.001 fm.

• •

The existing polynomial fits to the available $ep$ scattering data determined various moments of the proton form factor $<r_{p}^{n}>$ [5,6]. In particular, it was found that $<r_{p}^{6}>$ = 29.8 (7.6)(12.6) fm6. Unfortunately, this result suffers from a large systematic error, which corresponds to

a $\pm$ 0.0025 fm systematic bias in the extracted value of the proton radius $R_{p}$.

• •

Another approach to the proton form factor was demonstrated recently by J.$M$. Alarcon et al. [7,8]. On the basis of the Dispersive Improved Chiral Effective Field Theory, they calculated various FF moments from $<r_{p}^{2}>$ to $<r_{p}^{20}>$ with remarkably small error bars. Their predictions for the lowest moments of the charge FF are: $<r_{p}^{2}>$ = (0.701, 0.768) fm2, $<r_{p}^{4}>$ = (1.47, 1.6) fm4, $<r_{p}^{6}>$ = (8.5, 9.0) fm6, $<r_{p}^{8}>$ = (127, 130) fm8. Note that precision of the calculations is higher for higher FF moments in this approach, so it looks safe to take the predicted values of $<r_{p}^{6}>$ = 9.0 fm6 and $<r_{p}^{8}>$ = 130 fm8 for our fits. The systematic bias will be negligible in this case, even assuming the real error in $<r_{p}^{6}>$ will be an order of magnitude larger than that quoted above.

• •

Besides the proton radius $R_{p}$, the $<r_{p}^{4}>$ parameter will be also determined with 8% statistical errors in the fits with fixed $<r_{p}^{6}>$ and $<r_{p}^{8}>$ .

In conclusion, Table 5 presents the statistical and systematic errors related to the procedure of extraction of the proton charge radius from the experimental data expected in our experiment.

Some other options of the analysis are presented in the ANNEXes to this note.