High Precision Measurement of the Proton Elastic Form Factor Ratio at Low Q^{2}
Abstract

Experiment E08-007 measured the proton elastic form factor ratio in the range of by recoil polarimetry. Data were taken in 2008 at the Thomas Jefferson National Accelerator Facility in Virginia, USA. A 1.2 GeV polarized electron beam was scattered off a cryogenic hydrogen target. The recoil proton was detected in the left HRS in coincidence with the elasticly scattered electrons tagged by the BigBite spectrometer. The proton polarization was measured by the focal plane polarimeter (FPP).

In this low region, previous measurement from Jefferson Lab Hall A (LEDEX) along with various fits and calculations indicate substantial deviations of the ratio from unity. For this new measurement, the proposed statistical uncertainty () was achieved. These new results are a few percent lower than expected from previous world data and fits, which indicate a smaller at this region. Beyond the intrinsic interest in nucleon structure, the new results also have implications in determining the proton Zemach radius and the strangeness form factors from parity violation experiments.

High Precision Measurement of the Proton Elastic Form Factor Ratio at Low

by

Xiaohui Zhan

Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

January 2010

© Massachusetts Institute of Technology 2010. All rights reserved.


Author
Department of Physics
January 25, 2010

Certified by
William Bertozzi
Professor of Physics
Thesis Supervisor

Certified by
Shalev Gilad
Principle Research Scientist
Thesis Supervisor

Accepted by
Thomas J. Greytak
Associate Department Head for Education

High Precision Measurement of the Proton Elastic Form Factor Ratio at Low

by

Xiaohui Zhan

[]

\@normalsize

Submitted to the Department of Physics

on January 25, 2010, in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

\@normalsize

Thesis Supervisor: William Bertozzi
Title: Professor of Physics

\@normalsize

Thesis Supervisor: Shalev Gilad
Title: Principle Research Scientist

Acknowledgments

This work could not have be completed without the people who have supported and helped me along this long journey. I am extremely grateful for their care and considerations along these years for which I don’t have many opportunities to show my gratitude in front of them.

First, I would like to thank my advisor, Prof. William Bertozzi, for giving me the opportunity to start the graduate study at MIT, and for his continuing guidance, attention and support throughout these years. I won’t forget the “hard moments” he gave me during the preparation of the part III exam the same as the encouragement when I was frustrated. He helped me to understand how to become a physicist and at the mean time a happy person in life. I also would like to thank my another advisor Dr. Shalev Gilad for his valuable advices and suggestions during the whole analysis and the encouragement throughout my study and research at Jefferson Lab. Without their support, I would not complete the thesis experiment and finish the degree.

I would like to thank my academic advisor Prof. Bernd Surrow for his careful guidance in my graduate courses and the discussions for the future career. Many thanks to my thesis committee members: Prof. William Donnelly and Prof. Iain Stewart for their valuables comments and suggestions to this thesis.

Although it’s only been less than two years since I join the E08-007 collaboration, I had a wonderful experience and learned a lot in working with the spokespersons, post-docs and former graduate students. Individually, I sincerely appreciate Prof. Ronald Gilman for his guidance and support before and during the running of the experiment, and his helpful suggestions and discussions for the analysis afterwards. I would like to thank Dr. Douglas Higinbotham for being my mentor at Jefferson Lab and giving valuable advices in resolving different problems I encountered along the way. I would like to thank Dr. Guy Ron, for providing the first hand experimental running and analysis experience, the experiment would not run so smoothly without his effort. I would like to thank Dr. John Arrington for the valuable comments and discussions on the analysis and providing the form factors global fits. I also would like to thank Prof. Steffen Strauch, Prof. Eliazer Piasetzky, Prof. Adam Sarty, Dr. Jackie Glister and Dr. Mike Paolone for their guidance and inspiring discussions through the whole analysis process. In addition, I would like to thank the group former post-docs Dr. Nikos Sparveris and Dr. Bryan Moffit for their generous help on the experimental setup and assistance through the experiment. This work could not be done without the contribution from any one of them.

I would like to thank the Hall A staff members and the entire the Hall A collaboration for their commitment and shift efforts for this experiment. I would also like to thank the Jefferson Lab accelerator crew for delivering high quality beam for this experiment.

In the earlier days at Jefferson Lab, I worked with the saGDH/polarized He group. It was very special to me since that’s when I completed my first analysis assignment and learned quite some knowledge about the target system. I would like to thank Dr. Jian-Ping Chen for his supervision and guidance when I started the research in Jefferson Lab without any experience, his passion and rigorous attitude for physics have served as a model for me. I also would like to thank the former graduate students Vince Sulkosky, Jaideep Singh, Ameya Kolarkar, Patricia Solvignon and Aidan Kelleher for their patience and generous support on various things. I would like to thank the He lab/transversity fellow students: Chiranjib Dutta, Joe Katich, and Huan Yao for the great experience we had worked together.

Although I started my graduate life at MIT, I spent the last four years at Jefferson Lab. I am lucky to have friendships at both cities which made my graduate study and research an enjoyable experience. I would like to thank them for their support and encouragement: Bryan Moffit, Vince Sulkosky, Bo Zhao, Kalyan Allada, Lulin Yuan, Fatiha Benmokhtar, Ya Li, Jianxun Yan, Linyan Zhu, Ameya Kolarkar, Xin Qian, Zhihong Ye, Andrew Puckett, Peter Monaghan, Jin Huang, Navaphon Muangma, Kai Pan, Wen Feng, Wei Li, Feng Zhou,. Especially I would like to thank the group post-doc Vince Sulkosky for his effort in reading and correcting my thesis draft.

And Finally, I would like to show my deepest appreciation to my parents. I would not be anywhere without them and there is no words could ever match the love and support they gave since I was born. I also want to thank my fiance Yi Qiang, for the endless support over the past 8 years, and loving me for who I am.

Contents
List of Figures
List of Tables

Chapter 1 Introduction

When the proton and the neutron were discovered in 1919 and 1931 respectively, they were believed to be Dirac particles, just like the electron. They were expected to be point-like and to have a Dirac magnetic moment, expressed by:

(1.1)

where q, m, and s are the electric charge, mass and spin of the particle respectively. However, later measurements of these nucleons magnetic moments revealed the existence of the nucleon substructure. The first direct evidence that the proton has an internal structure came from the measurement of its anomalous magnetic moment 70 years ago by O. Stern [63],

(1.2)

where is the Bohr magneton. The first measurement of the charge radius of the proton by Hofstadter et al. [64, 65] yielded a value of 0.8fm, which is quite close to the modern value.

Starting from 1950s, electron scattering experiments were used to unravel the nucleon internal structure. Through the measurements of electromagnetic form factors and nucleon structure functions in elastic and deep inelastic lepton-nucleon scattering, it’s commonly accepted that in a simplistic picture, a nucleon is composed of three valence quarks interacting with each other through the strong force. The strong interaction theory, Quantum Chromodynamics (QCD) can make rigorous predictions when the four-momentum transfer squared, , is very large and the quarks become asymptotically free. However, predicting nucleon form factors in the non-perturbative regime is difficult due to the dominance of the soft scattering processes. As a consequence there are several phenomenological models which attempt to explain the data in this domain, and precise measurements of the nucleon form factors are desired to constrain and test these models.

In the one-photon-exchange (OPE) approximation, the elastic scattering cross section formalism is well known and can be parameterized by two form factors, and which are functions of . At low momentum transfer, the form factors can be interpreted as the fourier transform of the nucleon charge and magnetic densities. Earlier experiments measured the cross section of the elastic scattering which contains information about the internal structure responsible for the deviation from the scattering off point-like particles. However, after four decades of effort, there were still large kinematic regions where only very limited measurements of the form factors were possible, since the cross section of the unpolarized electron scattering is only sensitive to a specific combination of the form factors and the lack of a free neutron target.

In the last two decades, advances in the technology of intense polarized electron beams, polarized targets and polarimetry have ushered in a new generation of electron scattering experiments which rely on spin degrees of freedom. Compared to the cross section measurement, the polarization techniques have several distinct advantages. First, they have increased sensitivity to a small amplitude of interest by measuring an interference term. Second, spin-dependent experiments involve the measurement of polarizations or helicity asymmetries, and these quantities are independent of the cross section normalization, since most of the helicity independent systematic uncertainties can be canceled by measuring a ratio of polarization observable.

The first experiment to measure the recoil proton polarization observable in elastic scattering was done at SLAC by Alguard et al. [66], but the impact of the results was severely limited by the low statistics. Followed by that, the proton form factor measurements using recoil polarimetry were carried out at MIT-Bates [45, 67] and MAMI [68, 69]. Due to limited statistics and kinematics coverage, the ratio values were in agreement within uncertainties with the unpolarized measurements. More recent measurements of the proton form factor ratio using recoil polarimetry at Jefferson Lab [16, 17, 18], which have much better precision at high , deviated dramatically from the unpolarized data. This has prompted intense theoretical and experimental activities to resolve the discrepancy. The validity of analyzing data in the OPE approximation has been questioned, and two-photon-exchange (TWE) processes are now considered as an significant correction to the unpolarized data and mostly account for the discrepancy at high  [24].

While extending our knowledge at higher momentum transfer region is an ongoing endeavor, the proton form factor ratio behavior at low has also become the subject of considerable interest, especially, when potential discrepancy was observed from the most recent high precision measurements for . BLAST [19] did the first proton form factor ratio measurement via beam-target asymmetry at values of 0.15 to 0.65 , and the results are consistent with 1 in this region. LEDEX [20], which used the recoil polarimetry technique, observed a substantial deviation from unity at . However, the data quality of LEDEX was compromised due to the low beam polarization and background issues [70]. Hence, it was necessary to carry out a new high precision measurement to either confirm to refute the deviations at low momentum transfers. Beyond the intrinsic interest in the nucleon structure, an improved proton form factor ratio also impacts other high precision measurements such as parity violation experiments (HAPPEX) [71, 72], deeply virtual Compton scattering (DVCS) [73, 74], and also determination of other physical quantities such as the proton Zemach radius.

This thesis presents the analysis and results of experiment E08-007, which was conducted in 2008 at Jefferson Lab Hall A. In this experiment, the proton form factor ratio was measured at using recoil polarimetry.

1.1 Definitions and Formalism

1.1.1 Exclusive electron scattering

When scattered off a nuclear target, the electron exchanges virtual photons with the nucleus, which probes the electromagnetic structure of the nucleus. The electromagnetic coupling is small enough that it is valid to only consider the leading order. For the elastic scattering reaction off a proton, , the leading order diagram is shown in Fig. 1-1.

Figure 1-1: The leading order diagram of elastic scattering.

Initial and final electrons have four-momenta and respectively, and the initial and final protons and . The virtual photon has four-momentum , and the Lorentz-invariant four-momentum transfer squared is defined as:

(1.3)

where the last expression is valid in the Lab frame by neglecting the electron mass. The amplitude of is associated with the scale that the electromagnetic probe is sensitive to.

For exclusive elastic scattering, the recoil proton is also detected, so that can be defined from the proton momenta:

(1.4)

In the Lab frame,the initial proton is at rest, and Eq. 1.4 becomes:

(1.5)

where is the proton mass and is the kinetic energy of the final proton in the Lab frame.

1.1.2 Formalism

One of the advantages of the electromagnetic probe lies in the fact that the leptonic vertex is fully described by the theory of the electromagnetic interaction, Quantum ElectroDynamics (QED), and the information related to the unknown electromagnetic properties of the nucleon are contained by only the hadronic vertex . From the Feynman diagram in Fig. 1-1, the amplitude for elastic scattering can be written as:

(1.6)
(1.7)

where with the 0-th component as the time component, are the Dirac matrices in the chiral representation:

(1.8)

, and is the set of standard Pauli matrices:

(1.9)

and are the Dirac spinors for the initial and final electron, and , are the Dirac four-spinors for the initial and recoil proton respectively. In particular, the proton spinors enter in the plane-wave solution for a spin 1/2 particle which satisfies the Dirac equation:

(1.10)

and one can write:

(1.11)

with and is a normalized two-spinor, such that

(1.12)

While the leptonic current is fully described by QED, the hadronic current involves the factor , which contains the information about the internal electromagnetic structure of the proton. In general is some expression that involves and constants such as the proton mass , the electric charge . Since the hadronic current transforms as a vector, must be a linear combination of these vectors, where the coefficients can only be function of . It is convenient to write the current in the following way:

(1.13)

where is the proton anomalous magnetic moment and are the proton elastic form factors. They contain the information about the electromagnetic structure of the proton.

1.1.3 Nucleon Form Factors

and are distinguished according to their helicity characteristics, the projection of electron intrinsic spin along its direction of motion . is the Dirac form factor; it represents the helicity-preserving part of the scattering. On the other hand, the Pauli form factor represents the helicity-flipping part. and are defined in a similar way for the neutron. The form factors are normalized according to their static properties at . For the proton:

(1.14)

and for the neutron:

(1.15)

For reasons that will soon become obvious, it is more convenient to use the Sachs form factors [75]: and , which are defined as:

(1.16)

where is a kinematic factor. The Sachs form factors also have particular values at according to the static properties of the corresponding nucleon:

(1.17)
(1.18)

where and in units of nuclear magneton.

1.1.4 Hadronic Current in the Breit Frame

In the Breit frame, which is defined as the frame where the initial and final nucleon momenta are equal and opposite, the hadronic current has a simplified interpretation. A definition of variables in the Breit frame, which are noted with a subscript , is elaborated in Appendix A. Using the Gordon identity [76]

(1.19)

similarly, we can write:

(1.20)

In the Breit frame, the explicit expression of the hadronic current is simplified:

(1.21)
(1.22)

where is the Using , the time component can be expressed by:

(1.23)

By the definition of and in Eqs. 1.11 and 1.8, we now have:

(1.24)

Then, by the expressions:

(1.25)
(1.26)
(1.27)

we can finally get the simple relation:

(1.28)

The vector current can also be expressed in a similar way in the Breit frame:

(1.29)

Therefore, in the Breit frame, the electric form factor is directly related to the electric part of the interaction between the virtual photon and the nucleon, and the magnetic form factor is related to the magnetic part of this interaction. The electric and magnetic form factors can be associated with the Fourier transforms of the charge and magnetic current densities in this frame in the non-relativistic limit. The Fourier transforms can be expanded in powers of :

(1.30)
(1.31)

Notice that the first integral gives the total electric or magnetic charge, and the second integral defines the RMS electric or magnetic radii of the nucleon. However, the Breit frame is a mathematical abstraction, and for different value, the Breit frame experiences relativistic effect which is essentially a Lorentz contraction of the nucleon along the direction of motion. This results in a non-spherical distribution for the charge densities, and complicates the Fourier transform interpretation of the form factors in the rest frame.

1.2 Form Factor Measurements

1.2.1 Rosenbluth Cross Section

The differential cross section for scattering in the lab frame can be written as:

(1.32)

where we have neglected the electron mass, and is the amplitude defined in Eq. 1.7. Integrating over and gives:

(1.33)

where is the solid angle in which the electron is scattered, and has the form:

(1.34)

The hadronic and leptonic tensors are defined respectively as:

(1.35)
(1.36)

For unpolarized cross section, and are averaged over the incident particle spin states, and summed over the final particle spin states. Since the contraction of these two tensors is a Lorentz invariant, they can be calculated in any frame, as long as they are both calculated in the same frame.

In the single-photon exchange (Born) approximation, the formula for the differential cross section of electron scattering off nucleons is given by [77]:

(1.37)

where

(1.38)

is the Mott cross section for the scattering of a spin-1/2 electron from a spinless, point-like target, and is the recoil factor. This is the most general form for the electron elastic scattering cross section. Using Eq. 1.16, we can rewrite Eq. 1.37 without the interference term:

(1.39)

where is the electromagnetic fine structure constant, and this expression is known as the Rosenbluth formula.

Rosenbluth Separation

The Rosenbluth cross section has two contributions: the electric term and the magnetic term . As noted earlier, there is no interference term, so that the two contributions can be separated. We define the reduced cross section as:

(1.40)

where is the virtual photon polarization parameter. The quantity can be changed at a given , by changing the incident electron beam energy and the scattering angle. Therefore, at a fixed by varying , one can measure the elastic scattering cross section and separate the two form factors using a linear fit to the cross section. The slope is equal to and the intercept is equal to .

This method has been extensively used in the past 40 years to measure the elastic form factors and proved to be a very powerful method to measure the proton and the neutron magnetic form factors up to large . However, there are practical limitations. First, the neutron electric form factor is normalized to the static electric charge of the neutron, which is 0, and the cross section is completely dominated by the magnetic form factor. For the proton, the magnetic term will also dominate at large , since the factor increases quadratically as increases. As an example, at the magnetic term contributes about 95 of the total cross section. On the other hand, in the low region, the magnetic term is suppressed for the same reason and the electric term becomes dominant. Besides the difficulties from the physics side, the precision of Rosenbluth separation is also limited by the cross section measurements due to a widely different kinematic settings in order to cover a wide range of . Systematic errors are introduced by the inconsistent acceptance, luminosity, detector efficiency between different kinematics.

1.2.2 Polarization Transfer Measurements

In 1974, Akhiezer and Rekalo [78] discussed the interest of measuring an interference term of the form by measuring the transverse component of the recoiling proton polarization in the reaction . Thus, one could obtain in the presence of a dominating at large . Instead of directly measuring the separate form factors, the ratio can be accessed by measuring the polarization of the recoil nucleon. The virtue of the polarization transfer technique is that it is sensitive only to the ratio and does not suffer from the dramatically reduced sensitivity to a small component. Another way of measuring the interference term would be to measure the asymmetries in the scattering of a polarized beam off a polarized target.

By measuring the polarization of the recoil nucleon along a unit vector , we measure a preferential orientation of the spin along . In this case, when we average over initial proton spin states and sum over final proton spin states, the completeness relation

(1.41)

no longer holds. Instead we have to use:

(1.42)

so that the hadronic tensor becomes:

(1.43)

where is the unpolarized hadronic tensor and is the polarized one:

(1.44)

For recoil proton polarization measurements, a longitudinally polarized beam is required. The polarization of the beam is defined as:

(1.45)

where and are the number of electrons with their spin parallel and anti-parallel to their momentum respectively. Therefore, with a polarized electron beam, the leptonic tensor is modified and a new matrix is introduced:

(1.46)

The operator:

(1.47)

projects the spin along the momentum in a preferential direction. If the beam polarization is , and by further neglecting the electron mass, the leptonic tensor can be written as:

(1.48)

where is the Levi-Civita symbol. It is 0 if any two indices are identical, -1 under an even number of permutations and +1 under an odd number of permutations. Note that is anti-symmetrical.

In order to get the polarized amplitude, we contract the leptonic and the hadronic tensors:

(1.49)

where

  • is the amplitude squared of the unpolarized process.