Prospect for Measuring ${G_E^n}$ at High Momentum Transfers

TL;DR

This paper discusses the design and potential of an experiment to measure the neutron electric form factor at high momentum transfers, utilizing advanced detection equipment and polarized helium-3 targets to improve precision.

Contribution

It introduces a new experimental setup with optimized parameters and discusses innovative methods for systematic calibration using existing proton form factor data.

Findings

Projected measurement up to \\qsq~= 3.4 \\gvsq, doubling previous data range.

Design of the experimental apparatus including BigBite spectrometer and neutron detectors.

Conceptual framework for calibration using proton form factor data.

Abstract

Experiment E02-013, approved by PAC21, will measure the neutron electric form factor at \qsq~up to 3.4 \gvsq, which is twice that achieved to date. The main features of the new experiment will be the use of the electron spectrometer BigBite, a large array of neutron detectors, and a polarized \he3~target. We present the parameters and optimization of the experimental setup. A concept of an experiment for \GEN~where precision \GEP~data is used for calibration of the systematics of a Rosenbluth type measurement is also discussed. A concept of the circulating gas flow in the polarized He$^3$ target is presented.

Equations

$$g^2=\tau \cdot \frac{F_{{ϵ}_1}^2{ϵ}_2^{-1}-F_{{ϵ}_2}^2{ϵ}_1^{-1}}{F_{{ϵ}_2}^2-F_{{ϵ}_1}^2}$$

$$\sigma (g^2)\approx \frac{\sigma (F_{ϵ}^2)}{F_{ϵ}^2}\frac{\sqrt{2}\cdot \tau }{{ϵ}_1-{ϵ}_2}$$

$$F^2=N_{events}/[I_{beam}\cdot d_{target}\cdot t_{DAQ}\cdot {\sigma }_{Mott}\cdot {\Omega }_e\cdot {\eta }_e]$$

$${\left(\frac{F_{{ϵ}_2}^n}{F_{{ϵ}_1}^n}\right)}^2={\left(\frac{F_{{ϵ}_2}^p}{F_{{ϵ}_1}^p}\right)}^2\cdot \frac{N_2^{e,e^{\prime }n}}{N_1^{e,e^{\prime }n}}\cdot \frac{N_1^{e,e^{\prime }p}}{N_2^{e,e^{\prime }p}}\cdot \frac{{\Omega }_{{ϵ}_2}^n}{{\Omega }_{{ϵ}_1}^n}\cdot \frac{{\Omega }_{{ϵ}_1}^p}{{\Omega }_{{ϵ}_2}^p}\cdot \frac{{\eta }_{{ϵ}_2}^n}{{\eta }_{{ϵ}_1}^n}\cdot \frac{{\eta }_{{ϵ}_1}^p}{{\eta }_{{ϵ}_2}^p}$$

Full text

1 Introduction

Elastic electron scattering, which in the one-photon approximation is characterized by two form factors, is the simplest exclusive reaction on the nucleon. It provides important ingredients to our knowledge of nucleon structure. There are well-founded predictions of pQCD for the Q2 dependence of the form factors and their ratio in the limit of large momentum transfer [1]. Predictions of a fundamental theory always attract substantial attention from experimentalists. Recent surprising results on ${G_{E}^{p}}$ show that the ratio ${G_{E}^{p}/G_{M}^{p}}$ declines sharply as Q2 increases, and therefore pQCD is not applicable up to 10 (GeV/c)2. According to [2, 3] the electric and magnetic form factors behave differently, starting at Q2 $\approx 1$ (GeV/c)2. The same mechanisms causing this deviation should also be present in the neutron. It is an intriguing question, how the ratio ${G_{E}^{n}/G_{M}^{n}}$  develops in this Q2  regime, where confinement plays an important role.

2 World data on ${G_{E}^{n}}$

The study of ${G_{E}^{n}}$  has been a priority in electromagnetic labs for the last 15 years. Figure 1 presents recent data [4, 5, 6, 7, 8, 9] along with points representing the accuracy of JLab experiments [10, 11] which have already collected data, and the expected statistical accuracy of experiment E02-013. Presently published results can be fitted by the Galster approximation [12]. The double polarization technique used in these experiments was introduced more than 20 years ago [14, 15, 16]. The experiments used a polarized electron beam and three different targets: unpolarized deuterium (together with a neutron polarimeter), polarized ND3, and polarized 3He.

3 Experiment E02-013

The steady progress of the E93-028 [10] and the E93-026 [11] experiments has made possible the accurate determination of ${G_{E}^{n}}$ up to 1.47 (GeV/c)2. The next step in Q2 requires an experimental approach with much higher Figure-of-Merit (FOM).

In E02-013 [13] we optimized the setup in several respects:

• •

the solid angle of the electron spectrometer,

• •

the neutron detector efficiency and the trigger logic,

• •

the type of polarized target.

A recent addition in Hall A at JLab, the BigBite spectrometer developed by NIKHEF [17], has a 76 msr solid angle for a 40 cm long target. We found that for the identification of quasi-elastic scattering, the momentum resolution of BigBite ($\approx$ 1%) is sufficient for electron momenta up to 1.5 GeV/c. The luminosity available with the ${}^{3}\vec{He}$ target is about $10^{36}$ Hz/cm2. According to our calculations it can be used with BigBite in spite of the direct view of the target by the detectors. Neutrons with kinetic energy above 1 GeV with which we have to deal at the proposed momentum transfers, can be efficiently detected with a relatively high detector threshold, which allows to suppress background and is crucial for the operation at the expected luminosity, which is about a factor of 10 higher than used in a recent JLab experiment [10] with a polarized ND3 target.

In the last several years the theoretical development of the Generalized Eikonal Approximation (GEA) [18] has provided a framework for taking into account nuclear effects in the extraction of ${G_{E}^{n}}$ from the experimental asymmetry. The GEA prediction for the asymmetry as a function of the missing transverse momenta $p_{miss,perp}$ is shown in Fig. 2. The GEA calculations and experimental data from JLab Hall B for the unpolarized reaction 3He(e,e’p) have demonstrated the dominance of quasi-elastic scattering at $p_{miss,perp}$ below 0.15 GeV/c, when a modest cut of 0.5 GeV/c is applied on $p_{miss,parallel}$.

Table 3 summarizes the contributions to the error budget for the highest Q2 point. For each Q2 the measurement will be done with $\sim$14% statistical accuracy for three intervals of $p_{miss,perp}$. As a result the systematics will be evaluated by comparison of an experimental asymmetry and the GEA prediction vs $p_{miss,perp}$.

4 Future considerations

Experiment E02-013 is based on presently achieved parameters of the ${}^{3}\vec{He}$ target and the existing electron spectrometer. With additional developments the FOM of the experiment can be increased by a factor of 5 and a measurement of ${G_{E}^{n}}$ will be feasible at Q2 up to 5 (GeV/c)2.

4.1 Luminosity with the ${}^{3}\vec{He}$ target

The present configuration of the ${}^{3}\vec{He}$ target has the highest FOM at a beam current of 12-15 $\mu$A, when the beam-induced depolarization time is on the order of 30 hours. The use of the higher beam current requires a higher rate of polarizing and faster delivery of the polarized gas to the target cell. Advances in solid-state laser technology have made available 100 and even 200 W light power suitable for polarizing Rb atoms. Fig. 3 shows the target cell where the polarized gas flows through two tubes connecting the pumping and target cells. The flow will dramatically reduce the time for exchange of the polarized atoms between the pumping cell and the target cell.

4.2 High momentum spectrometer

The FOM of the experiment is approximately proportional to $E_{f}^{2}/E_{i}^{2}\,=\,(E_{i}-Q^{2}/2M)^{2}/E_{i}^{2}$, where the $E_{i(f)}$ is the initial(final) electron energy. By using a beam energy of 7.8 GeV it is possible to increase the FOM by a factor of 2.7 in comparison to the plan in E02-013 [13] for Q2=3.4 (GeV/c)2. It requires a new spectrometer for scattered electrons with a momentum 6 GeV/c and a solid angle of 75 msr. For Q2= 5 (GeV/c)2 the gain of FOM is 3.4. The relative momentum resolution should be of 0.5% to keep a W resolution sufficient for identification of the quasi-elastic events. The base component of the spectrometer is a dipole magnet with a 4.5 T$\cdot$m field integral and a 35 cm open gap. The scheme of a spectrometer based on such a dipole magnet is shown in Fig. 4 [19]. We call it Super BigBite. Its characteristics are similar to BigBite, but with the momentum range extended by a factor of 5-8. As in the case of BigBite, the detector will be open to the target, so it can be used mainly with a polarized target luminosity.

5 Rosenbluth approach

In the Rosenbluth method the form factors ratio g\,=\,$${G_{E}/G_{M}}  is obtained from two (or more) measurements at different beam energies at a fixed value of Q2. The following equation is used to find $g$:

[TABLE]

where $F$ is the total form factor measured experimentally, $F^{2}=\left(G_{E}^{2}+\frac{\tau}{\epsilon}G_{M}^{2}\right)/(1+\tau)$, $\tau=\frac{Q^{2}}{4M^{2}_{N}}$ and $\epsilon$ is virtual photon polarization. The uncertainty of $g$, which is growing with Q2, can be estimated from the equation

[TABLE]

where we neglect uncertainties in $\epsilon$ and $\tau$. The total form factor is calculated from the event rates and other parameters of the experiment as

[TABLE]

Each of these experimental parameters - the beam current $I_{beam}$, the target density $d_{target}$, the data taking time $t_{DAQ}$, the Mott cross section $\sigma_{Mott}$, the detector solid angle $\Omega_{e}$, and the detection efficiency $\eta_{e}$ - is known with limited accuracy, which contributes to the systematics of the measurement. Some of them cancel in the calculation of the ratio $g$, because of the good stability of the target and detectors. A sufficiently accurate determination of the beam energy, the detector solid angle, and the scattering angle present a big challenge for the experiment. In the best case the overall systematic error is on the level of a few percent. By detecting the recoiling proton, as was suggested in LOI99-103 [20], the acceptance of the detector can be excluded from the list of problems, because at a given value of the proton momentum the solid angle of the detector is fixed. Experiment E01-001[21], which used such an approach, recently took data in JLab Hall A.

Quasi-elastic electron scattering from the deuteron $D(e,e^{\prime}n)p$, with the ratio method suggested by Durand [22], has been used for determination of the neutron magnetic form factor in recent experiments at Bonn [23], Mainz [24], and JLab [25]. The same reaction can be used for measurement of the ratio ${G_{E}^{n}/G_{M}^{n}}$ even with less stringent requirements on the knowledge of the absolute neutron detection efficiency. The small value of ${G_{E}^{n}}$ made such measurement quite difficult; however, as we are proposing here, the problem can be solved by using the complementary $D(e,e^{\prime}p)n$ reaction for calibration of the experiment. We will use the fact that in the Q2 region of 5 (GeV/c)2 the ratio of the proton form factors ${G_{E}^{p}/G_{M}^{p}}$ is already well known from JLab experiments [2, 3]. In a dedicated experiment the accuracy of $g_{p}$ can be improved to the level of 2-3%.

The proposed scheme will use the magnetic spectrometer as an electron arm and a non-magnetic detector as a hadron arm. The last one will consist of a large array of plastic scintillators and veto detectors. At a few (GeV/c)2 momentum transfer the kinetic energy of the recoiling nucleon is above 1 GeV and proton and neutron interactions with the detector are similar (nuclear interaction dominates). The neutron detection efficiency of different measurements will be similar to each other because of equal kinetic energy of the neutron in both measurements. Most of the remaining variations of the detector efficiency and solid angle will affect in the same way the complementary reaction $D(e,e^{\prime}p)n$.

The ratio $F^{2}_{\epsilon_{2}}/F^{2}_{\epsilon_{1}}$, which defines as the value of $g_{n}$, can be expressed in the proposed experiment as

[TABLE]

Several parameters such as the beam current, the electron-arm solid angle and efficiency, the Mott cross section, the data taking time, and the target parameters all cancel out from the final ratio of the form factors at two different values of $\epsilon$. The remaining parameters are the neutron/proton detector solid angle $\Omega$ and efficiency $\eta$, whose variations for different $\epsilon$ need to be controlled.

For the proposed non-magnetic detector at large nucleon energy the neutron and proton detector efficiency will be almost equally affected by any change of rates and drifts of detector parameters, so it will be compensated. The detector solid angle is defined by the detector size. It can be well controlled and small changes will be the same for both proton and neutron channels.

The prospect of the Rosenbluth approach for a measurement of ${G_{E}^{n}}$ depends on the high rate capability of the neutron detector. The potential FOM is higher than that possible in the double polarization approach by a factor 10-20, when it operates at a luminosity of $10^{38}$ Hz/cm2. Experiment E93-038[11], which was done at a similar luminosity, developed the appropriate techniques for background reduction.

Conclusion

The experimental field of neutron electromagnetic form factors made very good progress in recent years. The present frontier for ${G_{E}^{n}}$ is $Q^{2}$ above 2 (GeV/c)2. JLab experiment E02-013 will do the measurement of ${G_{E}^{n}}$ up to $Q^{2}=3.4$ (GeV/c)2. There are possibilities of the further enhancements of the luminosity and polarization of the polarized ${}^{3}\vec{He}$ target. The Rosenbluth approach may also be revived by using calibration on the proton ${G_{E}^{p}/G_{M}^{p}}$ ratio.

Acknowledgments

It is pleasure to thank G. Cates, B. Reitz, and K. McCormick for collaboration in developing the E02-013 experiment. I would like also to acknowledge the crucial contributions made by P. Degterenko, V. Nelyubin, M. Sargsyan, and by members of the Hall A collaboration. Discussions with W. Brooks, K. de Jager, and B. Mecking on the possibility of the Rosenbluth approach for the ${G_{E}^{n}}$ measurement are greatly appreciated. The Southeastern Universities Research Association operates the Thomas Jefferson National Accelerator Facility under Department of Energy contract DE-AC05-84ER40150.