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=Data Analysis=
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This chapter describes the techniques used to analyze the data collected during the EG1b experiment and calculate semi-inclusive cross sections for the following reactions: <math>\overrightarrow{e}^{-} \overrightarrow{N} \rightarrow e^{-} \pi^{+} X</math> and <math>\overrightarrow{e}^{-} \overrightarrow{N} \rightarrow e^{-} \pi^{-} X</math> using <math>\mbox{NH}_{3}$ and $\mbox{ND}_{3}</math> polarized targets respectively. The goal of this work is to measure charged pion asymmetries defined according to the incident electron helicity and the target polarization. The measured asymmetries may be used to determine if the fragmentation function is independent of the observed final state hadron fractional energy (<math>z</math>) \cite{Christova}. The quantity (<math>\Delta R_{np}^{\pi^{+} + \pi^{-}}</math>) is defined in terms of the ratio of the difference of polarized semi-inclusive deep inelastic scattering cross sections for proton and neutron targets to unpolarized cross sections:  <br>
 
 
In this chapter, we discuss the techniques used to analyze the data collected during the EG1b experiment and calculate semi-inclusive cross sections for the following reactions: <math>e^-p \rightarrow e^- \pi^+ n</math> and <math>e^-n \rightarrow e^- \pi^- p</math> for NH3 and ND3 polarized targets respectively. The goal of this work is measure the fragmentation function depends on the Bjorken scaling variable (<math>x_b</math>) and the four momentum transfer squared (<math>Q^2</math>) as well as evaluate the independence of the fractional energy of the observed final state hadron (<math>z</math>). The fragmentation function (<math>\Delta R_{np}^{\pi^+ + \pi^-}</math> ) can be expressed in terms of the ratio of the difference of polarized to unpolarized cross sections for the semi inclusive deep inelastic scattering for proton and neutron targets. <br>
 
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
 
|<math>\Delta R_{np}^{\pi^+ + \pi^-} = \frac{\Delta \sigma_p^{\pi^+ + \pi^-} - \Delta \sigma_{n}^{\pi^+ + \pi^-}}{\sigma_p^{\pi^+ + \pi^-} - \sigma_{n}^{\pi^+ + \pi^-}}</math>
 
|<math>\Delta R_{np}^{\pi^+ + \pi^-} = \frac{\Delta \sigma_p^{\pi^+ + \pi^-} - \Delta \sigma_{n}^{\pi^+ + \pi^-}}{\sigma_p^{\pi^+ + \pi^-} - \sigma_{n}^{\pi^+ + \pi^-}}</math>
|}
+
|}<br>
 
+
A measure of <math>\Delta R</math> can be used to test for independent fragmentation when it is compared to inclusive structure functions as shown in Eq. (1.54).
There are couple of main steps in the data analysis, which will be discussed bellow in this chapter: <br>
 
*Data Selection
 
*Radiative Corrections
 
*Electron Identification
 
** EC cuts
 
** Cerenkov cuts
 
** Fidcucial Cuts
 
* Pion Identification
 
*Specific Event Reconstruction Efficiency(Inclusive, semi-inclusive and exclusive)
 
*Asymmetries
 
**Beam Charge Asymmetry
 
**Semi-inclusive Deep Inelastic Scattering Asymmetry(SIDIS)
 
*<math>x_b</math> and <math>z</math> binning
 
*Statistical and Systematic Errors
 
*The Fragmentation Test
 
*Models
 
  
 
==The CLAS Data Selection==
 
==The CLAS Data Selection==
  
The data files from the EG1b run were chosen for this analysis are listed in Table 1.1. During the experiment, 2.2 GeV, 4.2 GeV and 5.7 GeV longitudinally polarized electron beams were used on the polarized frozen ammonia NH3 and ND3 targets. This work will discuss the analysis of 4.2 GeV electron beam on hydrogen and deuteron targets. The collected data has been filtered by applying restrictions, which will be discussed in tin this chapter below. For the final results, the run sets were combined for each target type.
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The data files from the EG1b experiment chosen for this analysis are listed in Table~\ref{tab:EG1bRuns}. During the experiment, 2.2 GeV, 4.2 GeV and 5.7 GeV longitudinally polarized electron beams were used to probe the polarized frozen ammonia NH<math>_{3}</math> and ND<math>_{3}</math> targets. This work will discuss the analysis of the 4.2 GeV energy electron beam data set as this energy provided the most statistics. The collected data have been tested by applying restrictions discussed later in this chapter. <br>
 
 
 
{| border="1" style="text-align: center;" align="center"
 
{| border="1" style="text-align: center;" align="center"
 
|-
 
|-
| Run Set || Target Type || Torus Current(A) || Target Polarization || Half Wave Plane(HWP)
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| Run Set || Target Type || Torus Current(A) || Target Polarization || HWP
 
|-
 
|-
 
| 28100 - 28102 || ND3 || +2250 || -0.18 || +1
 
| 28100 - 28102 || ND3 || +2250 || -0.18 || +1
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{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|'''Table 1.1.''' EG1b Runs used for Analysis (Run Sets, Target Type, Torus Current, Target Polarization, HWP).
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|'''Table 1.1.''' EG1b Runs analyzed for this work.
 
|}<br>
 
|}<br>
  
==Radiative Corrections==
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=Particle Identification=
In double spin asymmetry analysis the electron nucleon scattering process is given as an one photon exchange event, so called the Born approximation([https://wiki.iac.isu.edu/images/f/f2/TheBornApproximation.png Fig 1.1.1]). In reality, there are multiple photon effects during the experiments. These high order processes, also called radiative effects, can be calculated and used to correct the cross section data.<br>
+
Additional tests were performed on the electron and a pion candidates reconstructed using the standard CLAS software package on the raw data collected during the EG1b experiment. Electrons are identified by matching the charged particle hits in the Cherenkov counter, electromagnetic calorimeter, and Time of Flight system. Geometrical and timing cuts are applied to improve electron identification. In addition, cuts are applied on the energy deposited by the particle into the calorimeter and the number of photoelectrons produced in the Cherenkov counter. Charged pions are identified by matching the hits in the drift chamber and ToF counter, along with a Cherenkov cut requiring that the number of photons for pions be less than two.  
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|
 
[[File:TheBornApproximation.png|350px|thumb|'''Figure 1.1.1.''' The Born Approximation]]
 
|}<br>
 
There are two types of radiative corrections, internal and external. Internal radiative corrections describe the contributions, which took place during the lepton-hadron interaction.  In first order approximation they include vertex photon exchange, self energy and vacuum polarization Fig. 1.1.2 .<ref name="RadiativeCorrections"> Nucleon Form Factors. In Scholarpedia, from http://www.scholarpedia.org/article/Nucleon_Form_factors#History </ref><br>
 
{| border="2"  align="center"
 
|-
 
|[[File:InetranRadiativeCor1.png|350px|thumb|Vertex Photon Exchange]]|| [[File:InetranRadiativeCor2.png|350px|thumb|Vacuum Polarization]]
 
|-
 
|[[File:InetranRadiativeCor3.png|350px|thumb|Self Energy]]|| [[File:InetranRadiativeCor4.png|350px|thumb|Self Energy]]
 
|}<br>
 
{| border="2"  align="center"
 
|-
 
|'''Figure 1.1.2''' Internal Radiation
 
|}<br>
 
On the other hand, the external radiative corrections account for the Bremsstrahlung by the incoming and scattered electron and by the recoiling target nucleon [https://wiki.iac.isu.edu/images/0/04/ExternalRadiativeCor.png Fig. 1.1.3].<br>
 
  
{| border="0" style="background:transparent;"  align="center"
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==Electron Identification==
|-
 
|
 
[[File:ExternalRadiativeCor.png|350px|thumb|'''Figure 1.1.3''' Externall Radiation]]
 
|}<br>
 
One of the major advantages of the double polarization experiments is the minimum contribution from the radiative corrections. For the <math>~4.2 GeV</math> incident electron beam data the radiative corrections are less than 5%<ref>http://www.jlab.org/Hall-B/secure/eg1/EG2000/fersch/QUALITY_CHECKS/file_quality/runinfo.txt</ref>. Due to negligible contributions from the radiative corrections, they are not included in the double spin asymmetry analysis.<br>
 
  
=Particle Identification=
+
The CLAS trigger system required the particle to deposit energy in the electromagnetic calorimeter and illuminate the Cherenkov counter within a <math>150</math> ns time window (Figure~\ref{fig:ElectronTrack}). Unfortunately, this trigger suffers from a background of high energy negative pions that may be misidentified as electrons. The pion contamination of the electron sample is reduced using cuts on the energy deposited in the electromagnetic calorimeter and the momentum measured in the track reconstruction for the known magnetic field. The energy deposition mechanism for the pions and electrons in the electromagnetic calorimeter is different. The total energy deposited by the electrons in the EC is proportional to their kinetic energy, whereas pions are minimum ionizing particles and the energy deposition is independent of their momentum (Figure~\ref{fig:ECTOTvsP}). The pion background is further suppressed using geometrical and time matching between the Cherenkov counter hit and the measured track in the drift chamber.<br>
The correct identification of an electron and a pion is the main requirement for the semi-inclusive analysis. Even though, the CLAS trigger system is is a coincidence of the negatively charged particle track detected in the drift chamber with hits in the electromagnetic calorimeter and cherenkov counter in <math>150 ns</math> time window [https://wiki.iac.isu.edu/index.php/File:Electron_track.png Fig. 2.1]. Unfortunately, the background negatively charged pions passing through the drift chamber that coincide with the noise signal in the cherenkov counter not related to the path in the wire chamber, but observed in the same CLAS sector can be misidentified as electrons. These background pions are the result of the quasi-real photoproduction, when the polar angle of the scattered lepton is approximately zero and is not accounted by the CLAS detector. In order to remove contamination due to those pions, geometrical and time matching between the cherenkov counter hit and the measured track in the drift chamber has been implemented in the data analysis. In addition, the pion contamination of the electron sample is reduced using the cuts on the energy deposited in the electromagnetic calorimeter and the momentum measured in the track reconstruction for the known magnetic field. The energy deposition mechanism for the pions and electrons in the electromagnetic calorimeter is different. The total energy deposited by the electrons in the EC is proportional to their kinetic energy, whereas pions are minimum ionizing particles and the energy deposition is independent of their momentum.<br>
 
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
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[[File:electron_track.png|400px|thumb|'''Figure 2.1''' Example of electron passing through the drift chambers and creating the signal in the cherenkov counter and electromagnetic calorimeter. Electron track is highlighted by the blue line (Run number 27095, Torus Current +2250 (inbending)).]]
 
[[File:electron_track.png|400px|thumb|'''Figure 2.1''' Example of electron passing through the drift chambers and creating the signal in the cherenkov counter and electromagnetic calorimeter. Electron track is highlighted by the blue line (Run number 27095, Torus Current +2250 (inbending)).]]
 
|}<br>
 
|}<br>
 
==Electron Identification==
 
  
 
===EC CUTS===
 
===EC CUTS===
  
The CLAS electromagnetic calorimeter was used to separate electrons from the pions. Electromagnetic calorimeter contains <math>13</math> layers of lead-scintillator sandwichs composed of <math>~2mm</math> thick lead and <math>10 mm</math> thick scintilltaor. Each set of <math>13</math> layers are subdivided into <math>5</math> inner and <math>8</math> outer layers that are named the inner and outer calorimeter respectively.<br>
+
The CLAS electromagnetic calorimeter was used to reduce the misidentification of electron and negative pion candidates. The electromagnetic calorimeter contains thirteen layers of lead-scintillator sandwiches composed of <math>\sim 2</math> mm thick lead and <math>10</math> mm thick scintillator. Each set of thirteen layers are subdivided into five inner and eight outer layers that are named the inner and outer calorimeter respectively.<br>
Electrons interacting with the calorimeter produce electromagnetic showers and release the energy into the calorimeter. The deposited energy is proportional to the momentum of the electrons. [https://wiki.iac.isu.edu/index.php/File:E_total_vs_e_inner1_before_cuts_file_dst27070.gif Fig. 2.2.1] shows the correlation of the inner and outer calorimeter electron candidate's energy measured by the calorimeter and divided by the electron's momentum reconstructed by the drift chamber. As shown in the figure, there is an island near <math>E/p = 0.2</math>, which contains most of the electron candidates as well as some regions below <math>0.2</math> which will be argued are negative pions misidentified as electrons.<br>
+
Electrons interact with the calorimeter producing electromagnetic showers that release energy into the calorimeter. The deposited energy is proportional to the momentum of the electrons. Figure~\ref{fig:ECCuts} shows the correlation of the inner and outer calorimeter electron candidate's energy measured by the calorimeter and divided by the particles momentum reconstructed by the drift chamber. As shown in the Figure~\ref{fig:ECCuts}, there is an island near <math>E/p = 0.2</math>, which contains most of the electron candidates as well as some regions below <math>0.2</math> which will be argued to be negative pions misidentified as electrons.<br>
Pions entering the calorimeter are typically minimum ionizing particles, loosing little of their incident energy in the calorimeter at a rate of <math>2 MeV g^{-1} cm^2</math>. Electrons, on the other hand, deposit a larger fraction of their momentum into the calorimeter. Energy deposition into the electromagnetic calorimeter is different for electrons and pions. Passing through the calorimeter pions loose about <math>0.08 GeV</math> energy independent their momentum, producing the constant signal in the calorimeter around <math>0.08 GeV</math>. In order to eliminate misidentified pions from the electron sample, following cut has been applied:<br>
+
Pions entering the calorimeter are typically minimum ionizing particles, loosing little of their incident energy in the calorimeter at a rate of 2 <math>\mbox{MeV g}^{-1} \mbox{cm}^{2}</math>. Electrons, on the other hand, deposit a larger fraction of their momentum into the calorimeter. As a result, the energy deposited into the electromagnetic calorimeter is different for electrons and pions. Pions loose about <math>0.08</math> GeV of energy traversing the calorimeter independent their momentum thereby producing the constant signal in the calorimeter around <math>0.08</math> GeV. In order to reduce misidentified pions from the electron sample, the following cut has been applied:<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
 
|
 
|
<math>EC_{inner}> 0.08 \times p</math>
+
<math>EC_{inner}> 0.08 \times p,</math>
 
|}<br>
 
|}<br>
where <math>p</math> represents particle momentum and <math>EC_{innner}</math>- inner part of the calorimeter. <br>
+
where <math>p</math> represents particle momentum and <math>EC_{innner}</math> the inner part of the calorimeter. <br>
 
Since the energy loss of pions is related to the detector thickness the correlation can be established between the energy deposition into the inner and outer layers of the detector:  
 
Since the energy loss of pions is related to the detector thickness the correlation can be established between the energy deposition into the inner and outer layers of the detector:  
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
 
|
 
|
<math>\frac{EC_{tot}}{EC_{inner}} = \frac{13}{5}</math>
+
<math>\frac{EC_{tot}}{EC_{inner}} = \frac{13}{5},</math>
 
|}<br>
 
|}<br>
 
which gives the following cut for the energy deposition into the outer layer of the calorimeter:<br>
 
which gives the following cut for the energy deposition into the outer layer of the calorimeter:<br>
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|-
 
|-
 
|
 
|
<math>EC_{tot} > 0.2 \times p</math>
+
<math>EC_{tot} > 0.2 \times p.</math>
 
|}<br>
 
|}<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
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|-
 
|-
 
|
 
|
'''Figure 1.2.2''' <math>EC_{inner}/p</math> vs <math>EC_{tot}/p</math> before and after EC cuts (<math>EC_{tot}>0.2p</math>, for EC inner - <math>EC_{inner}>0.08p</math>).
+
'''Figure 2.2''' <math>EC_{inner}/p</math> versus <math>EC_{tot}/p</math> before and after EC cuts (<math>EC_{tot}>0.2p</math> and  <math>EC_{inner}>0.08p</math>). After applying EC cuts about <math>46 \%</math> of the events have been removed from the electron sample.
 
|}
 
|}
  
 
===Cherenkov Counter Cut===
 
===Cherenkov Counter Cut===
  
The Cherenkov counter has been used to separate electrons from the background negatively charged pions. When the velocity of a charged particle is greater than the local phase velocity of light or when it enters a medium  with different optical properties the charged particle will emit photons. The Cherenkov light is emitted under a constant angle <math>\theta_c</math> - the angle of Cherenkov radiation relative to the particle's direction. It can be shown geometrically that the cosine of the Cherenkov radiation angle is anti-proportional to the velocity of the  charged particle<br>
+
The Cherenkov counter has been used to further reduce the negatively charged pion background in the reconstructed electron sample. When the velocity of a charged particle is greater than the local phase velocity of light or when it enters a medium  with different optical properties, the charged particle will emit photons. Cherenkov light is emitted at the critical angle <math>\theta_{c}</math> representing the angle of Cherenkov radiation relative to the particle's direction. It can be shown that the cosine of the Cherenkov radiation angle is inversely proportional to the velocity of the  charged particle <br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|<math>cos \theta_c=\frac{1}{n \beta}</math>
+
|<math>cos \theta_c=\frac{1}{n \beta},</math>
 
|}<br>
 
|}<br>
where <math>\beta c</math> is the particle's velocity and <math>n</math> - index of refraction of the medium. The charged particle in time <math>t</math> travels <math>\beta c t</math> distance, while the electromagnetic waves - <math>\frac{c}{n}t</math>. For a medium with given index of refraction <math>n</math>, there is a threshold velocity <math>\beta_{thr}=\frac{1}{n}</math>, below which no radiation can take place. This process may be used to observe the passage of charged particles in a detector, which can measure the produced photons.<br>
+
where <math>\beta c</math> is the particle's velocity and <math>n</math> the index of refraction of the medium. The charged particle in time <math>t</math> travels a distance <math>\beta c t</math>, while the electromagnetic waves travel <math>\frac{c}{n}t</math>. For a medium with given index of refraction <math>n</math>, there is a threshold velocity <math>\beta_{thr}=\frac{1}{n}</math>, below which no radiation is emitted. This process may be used to distinguish between the highly relativistic electrons and the less relativistic pions based on the number of photons produced. The number of photons produced per unit path length of a particle with charge <math>Ze</math> and per unit energy interval of the photons is proportional to the sine of the Cherenkov angle \cite{Nakamure} <br>
The number of photons produced per unit path length of a particle with charge <math>Ze</math> and per unit energy interval of the photons is proportional to the sine of the Cherenkov angle<ref name="Nakamure"> Nakamure, K., ''et al.''. (2010). The Review of Particle Physics. Particle Data Group. J. Phys. G '''37''', 075021.</ref> <br>
 
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
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{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|<math>\beta=\frac{v}{c}=\frac{pc}{\sqrt{(pc)^2 + (mc^2)^2}}</math>
+
|<math>\beta=\frac{v}{c}=\frac{pc}{\sqrt{(pc)^2 + (mc^2)^2}}.</math>
 
|}<br>
 
|}<br>
After deriving the Taylor expansion of our function and considering only the first two terms, we get following<br>
+
Taylor expanding Eq. 3.6 and keeping only the first two terms we get following<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
 
|<math>\frac{d^2 N}{dEdx}=\frac{\alpha z^2}{\hbar c}sin ^2 \theta_c=\frac{\alpha z^2}{\hbar c}[\beta^2 n^2 (E) - 1]</math>
 
|<math>\frac{d^2 N}{dEdx}=\frac{\alpha z^2}{\hbar c}sin ^2 \theta_c=\frac{\alpha z^2}{\hbar c}[\beta^2 n^2 (E) - 1]</math>
 
|}<br>
 
|}<br>
The gas used in the CLAS Cerenkov counter is perfluorobutane <math>C_4 F_{10}</math> with index of refraction equal to <math>1.00153</math>. The number of photoelectrons emitted by electrons is about <math>13</math>. On the other hand, calculations show that the number of photons produced by the negatively charged pions in the Cherenkov detector is approximately <math>2</math>. The theoretical results of the number of photons produced by the electrons and pions when passing through the Cherenkov counter are shown on Fig. 1.2.1.  
+
The gas used in the CLAS Cerenkov counter is perfluorobutane <math>C_{4}F_{10}</math> with index of refraction equal to <math>1.00153</math>. The number of photoelectrons emitted by electrons is about thirteen. On the other hand, calculations show that the number of photons produced by the negatively charged pions in the Cherenkov detector is approximately <math>2</math>. The theoretical results of the number of photons produced by the electrons and pions when passing through the Cherenkov counter are shown on Figure~\ref{fig:NPHETheoretical}. <br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|[[Image:e_momentum_vs_numb_of_photoelectrons_27095_theory.gif|300px|thumb|Electrons.]]
+
|[[Image:e_momentum_vs_numb_of_photoelectrons_27095_theory.gif|300px|thumb|For electrons.]]
||[[Image:pi_momentum_vs_numb_of_photons_27095_theory.gif|300px|thumb|Pions.]]
+
||[[Image:pi_momentum_vs_numb_of_photons_27095_theory.gif|300px|thumb|For pions.]]
 
|}<br>
 
|}<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|'''Figure 2.2.1''' Theoretical Calculation of the Number of Photoelectrons for Electrons and Pions.
+
|'''Figure 2.3''' Theoretical Calculation of the Number of Photoelectrons for electrons and pions.
|}
+
|}<br>
The number of photoelectron distribution measured in the cherenkov detector and the energy deposition dependence on number of photoelectrons are shown on [https://wiki.iac.isu.edu/images/4/48/Electrons_nphe_without_cuts_all_data_with_fits.gif Fig. 2.2.2] and [https://wiki.iac.isu.edu/images/2/20/E_total_vs_nphe_momentum_cut_file_dst27070.gif Fig. 2.2.3]. One can see, that a single photoelectron peak is caused by the misidentified pions as electrons. <br>
+
The distribution of the number of photoelectrons measured in the Cherenkov detector and the energy deposition dependence on number of photoelectrons are shown on Figure~\ref{fig:nphewithoutcuts} and Figure~\ref{fig:nphewithcuts}. One can see that a single photoelectron peak is caused by misidentifying pions as electrons. <br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
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{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|'''Figure 2.2.2''' The number of photoelectrons without cuts.
+
|'''Figure 2.4''' The number of photoelectrons without cuts.
 
|}<br>
 
|}<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
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{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|'''Figure 2.2.3''' The total energy deposited into the Calorimeter vs The Number of Photoelectrons.
+
|'''Figure 2.5''' The total energy deposited into the Calorimeter vs The Number of Photoelectrons.
 
|}<br>
 
|}<br>
These negative pions are produced when lepton goes at polar angle close to zero and is not measured by the detector. In order to reduce the electron sample contamination due to those pions, geometrical cuts on the location of the particle at the entrance to the cerenkov detector and time matching cuts has been developed by Osipenko. <ref name="Osipenko"> Osipenko, M., Vlassov, A.,& Taiuti, M. (2004). Matching between the electron candidate track and the Cherenkov counter hit. '''CLAS-NOTE 2004-020'''. </ref> For each CLAS Cherenkov detector segment the following cut has been applied<br>
+
 
 +
===Geometric and Timing cuts===
 +
Negative pions may be produced when the lepton scatters at a polar angle close to zero and is not observed by the detector. In order to reduce the electron sample contamination due to those pions, geometrical cuts on the location of the particle at the entrance to the Cherenkov detector and time matching cuts have been developed by Osipenko, so called OSI cuts \cite{Osipenko}. For each CLAS Cherenkov detector segment the following cut has been applied<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|<math>|\theta_p - {\theta_p}^{center} - {\theta_p}^{offset}| < 3 \sigma_p</math>  
+
|<math>|\theta_p - {\theta_p}^{center} - {\theta_p}^{offset}| < 3 \sigma_{p},</math>  
 
|}<br>
 
|}<br>
where <math>\theta_p</math> represents the measured polar angle in projective plane for each electron event. Cherenkov counter projective plane is an imaginary plane behind the Cherenkov detector where cherenkov radiation would arrive in the case if it moved the same distance from emission point to PMT, without reflections in the mirror system, <math>{\theta_p}^{center}</math> - the polar angle from the CLAS center to the image of Cherenkov counter segment center  and <math>{\theta_p}^{offset}</math> - the shift in the segment center position. In addition to geometrical cuts, timing cuts has been applied in order to perform time matching between Cherenkov counter and time of flight system. <br>
+
where <math>\theta_{p}</math> represents the measured polar angle with respect to a projectile plane for each electron event. The Cherenkov counter's projective plane is an imaginary plane behind the Cherenkov detector where Cherenkov radiation would have arrived if it had moved the same distance from emission point to the PMT, without reflections in the mirror system. ${\theta_{p}}^{center}$ is the polar angle from the CLAS detector center to the image of Cherenkov counter segment center  and <math>{\theta_{p}}^{offset}</math> is the shift in the segment center position. In addition to geometrical cuts, timing cuts have been applied to match the time between a Cherenkov counter hit and time of flight system. <br>
The pion contamination in electron sample was estimated by fitting the number of photoelectron distribution with two Gaussian distributions convoluted with a Landau distribution, which is presented below<ref name="Lanczos"> Lanczos, C. (1964). A precision approximation of the Gamma function. SIAM Journal of Numerical Analysis, B1 '''86'''. </ref>:
+
The pion contamination in an electron sample was estimated by fitting the number of photoelectron distribution using two Gaussian distributions convoluted with a Landau distribution \cite{Lanczos}: <br>
 +
 
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|<math>N_{pe}= p_0 e^{-0.5 \left (\frac{x-p_1}{p_2} \right )^2} + p4\frac{1}{1-\left(\frac{x-p5}{p6}\right )} + p_6 e^{-0.5 \left (\frac{x-p_7}{p_8} \right )^2}</math>  
+
|<math>N_{pe}= p_0 e^{-0.5 \left (\frac{x-p_1}{p_2} \right )^2} + p4\frac{1}{1-\left(\frac{x-p5}{p6}\right )} + p_6 e^{-0.5 \left (\frac{x-p_7}{p_8} \right )^2}.</math>  
 
|}<br>
 
|}<br>
It appears, that pion contamination in electron sample is <math>9.63 % \pm 0.01 %</math> before applying the hard cut on the number of photoelectrons produced in the cherenkov counter and after <math>nphe>2.5</math> cut contamination is about <math>4.029% \pm 0.003</math>(Fig. 2.2.4).<br>
+
 
 +
The fits in Figure~\ref{fig:OSICuts}.(a) suggest that the pion contamination in the electron sample is <math>9.63 % \pm 0.01 %</math> before applying the OSI cuts and after the OSI cuts the contamination is about <math>4.029% \pm 0.003 %</math> (Figure~\ref{fig:OSICuts}.(b)).<br>
 +
 
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
Line 207: Line 167:
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|'''Figure 2.2.4''' The number of photoelectrons before and after OSI Cuts.
+
|'''Figure 2.6''' The number of photoelectrons before and after OSI Cuts.
 
|}<br>
 
|}<br>
  
 +
==Pion Identification==
  
==Pion Identification==
+
Charged pions are identified using a coincidence hit in the drift chamber and Time-of-Flight (ToF) counter. Pions are separated from the other charged particles by looking at the particle momentum versus the <math>\beta</math> distribution. The particle velocity, <math>\beta=\frac{v}{c}</math>, is calculated from the difference of the RF time and the time-of-flight measurement in the ToF system with the path length from the vertex to the ToF counters. The mass of the charged particle can be identified by combining the particle's <math>\beta</math> with the particle momentum obtained from the track measured by the drift chamber in the known magnetic field. The particle mass in a magnetic field is given as<br>
  
Charged pions are identified by a coincidence hit in the drift chamber and TOF counter. Pions are separated from the other charged particles by looking at the particle momentum versus the <math>\beta</math> distribution. The particle velocity, <math>\beta=\frac{v}{c}</math>, is calculated from the difference of the vertex start time and the time-of-flight measurement in the TOF system with the path length from the vertex to the Time-of-Flight counters. The mass of the charged particle can be identified by combining the particle <math>\beta</math> with the particle momentum obtained from the tracking passing through the drift chamber in the known magnetic field. The particle mass in a magnetic field is given as<br>
+
{| border="0" style="background:transparent;"  align="center"
{| border="0" style="background:transparent;"  align="center"
 
 
|-
 
|-
 
|<math>p=\frac{m\beta}{\sqrt{1-\beta^{2}}}</math>
 
|<math>p=\frac{m\beta}{\sqrt{1-\beta^{2}}}</math>
Line 220: Line 180:
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|<math>m=p\sqrt(\beta^{2} - 1)</math>
+
|<math>m=p\sqrt{(\beta^{2} - 1)}</math>
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|<math>\beta=\frac{L_{path}}{t_{flight}},</math>
 
|}<br>
 
|}<br>
where<br>
 
<math>Insert formula here</math>
 
where m is the mass of the charged particle, <math>\beta</math> - its velocity
 
  
=Specific Event Reconstruction Efficiency=
+
where <math>m</math> is the mass of the charged particle, <math>\beta</math> its velocity, <math>p</math> particle momentum, <math>L_{path}</math> the path length from the vertex to scintillators and <math>t_{flight}</math> the time of flight from the interaction vertex to the ToF system.<br>
==Inclusive Event Reconstruction Efficiency==
+
Using the above information (particle momentum from the drift chambers and the timing information from the ToF system), the mass squared of the charged particle was calculated and is shown on Figure~\ref{fig:PvsM2}. The pion mass band is around <math>\sim 0</math>. To isolate charged pions from the rest of the particles, a <math>3\sigma</math> cut on the momentum versus <math>\beta</math> distribution has been applied \cite{Park2008}.<br>
The goal of this thesis is to measure the semi-inclusive asymmetry when an electron and a pion are detected in the final state.  Pions of opposite charge will be observed using the same scintillator by flipping the CLAS Torus magnetic field direction. Although the pions will be detected by the same detector elements, the electrons will intersect different detector elements.  As a result, the electron efficiency will need to be evaluated in terms of the electron rate observed in two different scintillator paddles detecting the same electron kinematics.  The pairs of scintillator paddles that had been chosen have the highest semi-inclusive rates.<br>
+
{| border="0" style="background:transparent;" align="center"
The electron efficiency of individual scintillator detectors using the 4.2 GeV data for ND3 and NH3 targets is investigated below. Only the electron is detection in the final state (inclusive case). The pion contamination in the electron sample was removed by the applying cuts described above. The electron paddle number 10 (B<0) and 5 (B>0) were chosen respectively because they contained the most electron events in a first pass semi-inclusive pion analysis of the data set. The electron kinematics(Momentum, scattering angle and invariant mass) for these scintillators is shown on Fig. 3.1. <br>
+
|-
{| border="1" |cellpadding="20" cellspacing="0
+
|[[File:PionMomentumvsBeta.png|300px]]
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|'''Figure 2.7''' The charged particle momentum versus <math>\beta</math> distribution. The pion and proton bands are clearly separated.  
 +
|}<br>
 +
{| border="0" style="background:transparent;" align="center"
 
|-
 
|-
|[[File:EmomInclusiveoverlay4-2GeVfcup.gif|300px]] ||[[File:EthetaInclusiveoverlay4-2GeVfcup.gif|300px]] || [[File:WInclusiveoverlay4-2GeVfcup.gif|300px]]
+
|[[File:PionMomentumvsMass.png|300px]]
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
| Electron Momentum((NH3,B>0), (NH3,B<0), (ND3,B>0) && (ND3,B<0))|| Electron Scattering Angle <math>\theta</math>((NH3,B>0), (NH3,B<0), (ND3,B>0) && (ND3,B<0))|| W Invariant mass((NH3,B>0), (NH3,B<0), (ND3,B>0) && (ND3,B<0))
+
|'''Figure 2.8''' The charged particle momentum versus mass squared distribution for the <math>\overrightarrow{e}p\rightarrow\overrightarrow{e}^{\prime}  \pi^{+} n</math> electroproduction process. The bands around 0 and 1 represent pions and protons respectively <ref name="Park2008" >Park, K., Burkert, V. D., & Kim, W. (The CLAS Collaboration). (2008). Cross sections and beam asymmetries for <math>\vec{e} p</math>-><math>en\pi^+</math> in the nucleon resonance region for <math>1.7 < Q^2 < 4.5</math> <math>(GeV)^2</math>. Phys. Rev., C77, 015208.</ref>.
|}
+
|}<br>
 +
In addition to the charged particle velocity (<math>\beta</math>), the fiducial volume cuts have been applied for the charged pion identification. Since the drift chambers and scintillators are used for pion detection, the polar angle range where pions are detected is much larger than for electrons. For the EG1b experiment, pions were detected from <math>8^{\circ}</math> to <math>180^{\circ}</math> \cite{Park2008}. The pion identification code has been developed by Joshua Pierce \cite{JoshuaPierce}.
  
'''Figure 3.1. Electron Kinematics.'''
+
=Event Reconstruction Efficiency=
  
 +
The goal of this work is to measure the semi-inclusive asymmetry when an electron and a pion are detected in the final state. For this analysis, pions of opposite charge will be observed using the same scintillator paddles by flipping the CLAS torus magnetic field direction. Although the pions will be detected by the same detector elements, the electrons will intersect different detector elements. As a result, the electron reconstruction efficiency was evaluated in terms of the electron rate observed in two different scintillator paddles detecting the same electron kinematics.
  
Ratios of the inclusive electron rate, normalized using the FC, of scintillator paddles 5 and 10 were measured.  The two ratios are designed to measure the CLAS detectors ability to observed the same electron kinematics using different detector elements positioned for opposite Torus polarities. 
+
==Inclusive Electron Event Reconstruction Efficiency==
  
<math>\frac{ND3,B>0, PaddleNumber^{e^-}=5}{ND3,B<0, PaddleNumber^{e^-}=10}=1.57 \pm 0.16</math>
+
The electron reconstruction efficiency for individual scintillator detectors using the 4.2 GeV EG1b data is investigated below. Only the electron is detected in the final state (inclusive case). The pion contamination in the electron sample was removed by applying the cuts described above. The electron paddle numbers 10 (B<0)  and 5 (B>0) were chosen respectively, because they contained the most electron events in a first pass semi-inclusive pion analysis of the data set. The electron kinematics (momentum, scattering angle and invariant mass) for these scintillators is shown on Figure~\ref{fig:ElectronKinematics}.<br>
 +
Ratios of the inclusive electron rate, normalized using the gated Faraday cup, detected in scintillator paddles # 5 and # 10 were measured. The two ratios are constructed to quantify the CLAS detector's ability to reconstruct electrons in scintillator paddle #5 using a positive Torus polarity and scintillator #10 using the negative Torus polarity.<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|<math>\frac{ND3,B>0, PaddleNumber^{e^-}=5}{ND3,B<0, PaddleNumber^{e^-}=10}=1.57 \pm 0.16</math>
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|<math>\frac{NH3,B>0, PaddleNumber^{e^-}=5}{NH3,B<0, PaddleNumber^{e^-}=10}=1.76 \pm 0.17</math>
 +
|}<br>
 +
Notice the above ratios are statistically the same. The semi-inclusive analysis to be performed in this thesis will be taking ratios using an <math>ND_{3}</math> and <math>NH_{3}</math> target.  Below is the observed ratio comparing the inclusive electrons observed in scintillator #5 for a positive torus polarity and an ND<math>_{3}</math> target to the electrons observed in scintillator #10 when the torus polarity is negative and the target is <math>NH_{3}</math>. <br>
 +
{| border="1"  |cellpadding="20" cellspacing="0
 +
|-
 +
|[[File:EmomInclusiveoverlay4-2GeVfcup.gif|300px]] ||[[File:EthetaInclusiveoverlay4-2GeVfcup.gif|300px]] || [[File:WInclusiveoverlay4-2GeVfcup.gif|300px]]
 +
|-
 +
| Electron Momentum((NH3,B>0),  (NH3,B<0), (ND3,B>0) and (ND3,B<0))|| Electron Scattering Angle <math>\theta</math>((NH3,B>0),  (NH3,B<0), (ND3,B>0) and (ND3,B<0))|| W Invariant mass((NH3,B>0),  (NH3,B<0), (ND3,B>0) and (ND3,B<0))
 +
|}
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|'''Figure 2.9.''' Electron Kinematics.
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|<math>\frac{ND3,B>0, PaddleNumber^{e^-}=5}{NH3,B<0, PaddleNumber^{e^-}=10}=1.55 \pm 0.15</math>
 +
|}<br>
 +
The above ratios, which have been observed to be ammonia target independent, indicate a difference in an electron detector efficiency when the torus polarity is flipped.  An electron detection efficiency  "correction coefficient" is defined in terms of the above ratio and measured to be <math>\frac{ND_{3},B>0,E_{PaddleNumber}=5}{NH_{3},B<0,E_{PaddleNumber}=10} = 0.645</math> and <math>\frac{ND_{3},B<0,E_{PaddleNumber}=10}{NH_{3},B>0,E_{PaddleNumber}=5} = 1.82</math>. The impact of these corrections on the data is illustrated in the next section.<br>
  
<math>\frac{NH3,B>0, PaddleNumber^{e^-}=5}{NH3,B<0, PaddleNumber^{e^-}=10}=1.76 \pm 0.17</math>
+
==Exclusive and Semi-Inclusive Event Reconstruction Efficiencies==
  
 
+
After determining the electron reconstruction efficiency for the selected paddle numbers, the measured single pion electroproduction rate was compared to the MAID 2007 unitary model that has been developed using the world data of pion photo and electro-production to determine the impact of using the above "correction coefficient". The model is well adopted for predictions of the observables for pion production, like five fold cross section, total cross section,  etc.<br>
Notice the ratio is statistically the same if the NH3 and ND3 targets are used for this ratio in a manner which will be similar to the semi-inclusive analysis.
+
The MAID 2007 model has predictions of the total cross section for the following two cases that are related to our work:<br>
 
+
{| border="0" style="background:transparent;"  align="center"
<math>\frac{ND3,B>0, PaddleNumber^{e^-}=5}{NH3,B<0, PaddleNumber^{e^-}=10}=1.55 \pm 0.15</math>
+
|-
 
+
|<math>\gamma^*</math> + proton<math>(NH_3) \rightarrow</math> <math>\pi^+</math> + neutron  
 
+
|}<br>
 
+
{| border="0" style="background:transparent;"  align="center"
The above ratios, which have been observed to be ammonia target independent, indicate a difference in an electron detector efficiency when the Torus polarity is flipped.  In order to make detector efficiency the same for electrons, the ratios were used as a "correction coefficient".  The "correction coefficient" for the case <math>\frac{ND3,B>0,E_{PaddleNumber}=5}{NH3,B<0,E_{PaddleNumber}=10}</math> is <math>0.645</math> and for the <math>\frac{ND3,B<0,E_{PaddleNumber}=10}{NH3,B>0,E_{PaddleNumber}=5}</math> it is <math>1.82</math>.
+
|-
 
+
|<math>\gamma^*</math> + neutron<math>(ND_3) \rightarrow</math> <math>\pi^-</math> + proton.
 
+
|}<br>
== Exclusive and Semi-Inclusive Event reconstruction efficiencies==
+
The ratio of the pions detected in the scintillator paddles, located between the Cherenkov counter and electromagnetic calorimeter, is shown in Figure~\ref{fig:PionPaddleNumbervsRatio}. The ratios were taken for four different cases. The intrinsic assumption is that, for the inbending case, positive pions and for the outbending case negative pions have the same trajectories with the same kinematics. In addition, negatively charged pions in the inbending field and positively charged pions in the outbending fields are detected by the same detector elements.<br>
After establishing the electron efficiency for the selected paddle numbers, the measured single pion electroproduction rate was compared to the MAID 2007 unitary model that has been developed using the world data of pion photo and electro-production to determine the impact of using the above "correction coefficient". The model is well adopted for predictions of the observables for pion production, like five fold cross section, total cross secton and etc.
+
{| border="0" style="background:transparent;"  align="center"
 
+
|-
The MAID 2007 model has predictions of the total cross section for the following two cases, that are related to our work:<br>
+
|[[File:paddenumbvsratio.jpg|350px]]
 
+
|}
:::::::::::<math>\gamma^*</math> + proton<math>(NH_3) \rightarrow</math> <math>\pi^+</math> + neutron <br>
+
{| border="0" style="background:transparent;"  align="center"
 
+
|-
:::::::::::<math>\gamma^*</math> + neutron<math>(ND_3) \rightarrow</math> <math>\pi^-</math> + proton <br>
+
|'''Figure 2.10.''' Pion paddle number versus Ratio for Semi-Inclusive case.
 
+
|}<br>
 
+
Using MAID 2007, the total cross section was calculated for the following invariant mass and four momentum transferred square values: 1.7 GeV < W<1.8 GeV and <math>Q^{2}=1.1 GeV^{2}</math> \cite{MAID2007}.<br>
 
+
{| border="0" style="background:transparent;"  align="center"
The ratio of the pions detected in the scintillator paddles, located between the Cherenkov counter and electromagnetic calorimeter, is shown on Fig. 3.2. The ratios were taken for four different cases. Assuming that, for the inbending case positive pions and for the outbending case negative pions have the same trajectories(the same kinematics) and vice versa((the inbending,negative pion) and (the outbending, positive pions)).<br>
+
|-
 
+
|<math>\sigma = \sigma_{T} + \epsilon \sigma_{L} + \sqrt{2\epsilon(1 + \epsilon)}\sigma_{LT} cos{\phi_{\pi}}^{CM} + \epsilon \sigma{TT} cos2{\phi_{\pi}}^{CM} + h \sqrt{2\epsilon (1-\epsilon)}\sigma_{LT^{\prime}}sin{\phi_{\pi}}^{CM}, </math>
[[File:paddenumbvsratio.jpg|500px]]<br>
+
|}<br>
'''Figure 3.2. Pion paddle number vs Ratio.'''
+
where <math>{\phi_{\pi}}^{CM}</math> is the pion azimuthal angle in the CM frame, <math>\epsilon = (1 + 2(1 + \frac{\nu^{2}}{Q^{2}})\tan^{2}\frac{\theta_{e}}{2})^{-1}</math> is the virtual photon polarization, <math>\nu=E_{i} - E_{f}</math> the energy difference of the initial and final state electron, <math>Q^{2} = 4 E_{i} E_{f} \sin^{2}\frac{\theta_{e}}{2}</math> the four momentum transferred squared, <math>\theta_{e}</math> the electron scattering angle and $h$ the electron helicity. After applying corrections from the inclusive cases, the ratios have been compared to the results from MAID2007. The difference of the measured and MAID2007 model ratios for each pion paddle number is shown in Figure~\ref{fig:MAID2007Experiment}. One can conclude from Figure~\ref{fig:MAID2007Experiment} that the "inclusive corrections" do not impact single pion production rates for the exclusive cases.<br>
 
+
{| border="0" style="background:transparent;"  align="center"
Using MAID 2007 total cross section was calculated for the following invariant mass and four momentum transferred square: <math>1.7< W<1.8</math> and <math>Q^2=1.1 GeV^2</math>. <ref name="MAID2007" > http://wwwkph.kph.uni-mainz.de/MAID//maid2007/maid2007.html</ref>. After applying correction coefficients from inclusive cases, the ratios have been compared to the results from MAID2007.
+
|-
 
+
|[[File:maid2007minusexperiment.png|400px]]
::<math>\sigma = \sigma_{T} + \epsilon \sigma_{L} + \sqrt{2\epsilon(1 + \epsilon)}\sigma_{LT} cos{\phi_{\pi}}^{CM} + \epsilon \sigma{TT} cos2{\phi_{\pi}}^{CM} + h \sqrt{2\epsilon (1-\epsilon)}\sigma_{LT^{\prime}}sin{\phi_{\pi}}^{CM} </math>
+
|}<br>
 
+
{| border="0" style="background:transparent;"  align="center"
After applying correction coefficients from inclusive cases, the ratios have been compared to the results from MAID2007.
+
|-
 
+
| Pion Paddle Number versus MAID2007 - Experiment<math>(N(\pi^{-},ND_{3})/N(\pi^{+},NH_{3}))</math>. The Black and red data represent B>0/B<0 and B<0/B>0 cases respectively before corrections. The green and blue points represent the ratios for B>0/B<0 and B<0/B>0 after inclusive corrections.
[[File:black_red_maid_inclusiveerror.png|350px]] [[File:green_blue_inclusiveerror.png|350px]]
+
|}<br>
 
 
'''Figure 3.3.  Pion paddle number vs Ratio after correction.'''
 
 
 
Applied corrections are following:
 
 
 
<math>\frac{N(\pi^+,ND_3,B<0)}{N(\pi^-,NH_3,B>0)} \times 1.82</math>
 
 
 
<math>\frac{N(\pi^+,ND_3,B>0)}{N(\pi^-,NH_3,B<0)} \times 0.645</math>
 
 
 
<math>\frac{N(\pi^-,ND_3,B>0)}{N(\pi^+,NH_3,B<0)} \times 0.645</math>
 
 
 
<math>\frac{N(\pi^-,ND_3,B<0)}{N(\pi^+,NH_3,B>0)} \times 1.82</math>
 
 
 
 
 
 
 
 
 
Exclusive cases  
 
 
 
[[File:negativepionND3beforecorrection.png|350px]] [[File:negativepionND3aftercorrection.png|350px]]
 
 
 
[[File:positivepionND3beforecorrection.png|350px]] [[File:positivepionND3aftercorrection.png|350px]]
 
  
 
=Asymmetries=
 
=Asymmetries=
==NES Asymmetry==
 
  
 +
The double spin asymmetry measurements in this thesis are performed by comparing scattering events that occur when the incident probe spin and nuclear target spin are parallel to the scattering events that occur when the spins are anti-parallel. <br>
  
 +
==Beam Charge Asymmetry==
  
The double spin asymmetry measurements performed in this thesis are performed by comparing scattering events that occur when the incident probe spin and nuclear target spin are parallel to scattering events that occur when the spins are anti-parallel. The helicity of the electron beam was flipped at a rate of 1 HZ. The helicity is prepared at the source such that helicity pairs are produced pseudo randomly. If the first electron bunch is pseudo randomly chosen to be positive (negative) then it is labeled as the original helicity state and denoted in software by a 2 (1).  The next helicity state is prepared to be a complement to the first state and labeled in the software as either a 4, if the original helicity state was a 1 (negative), or 3 if the original helicity state was a 2 (positive).  The helicity process is then repeated.  Figure NES.1 illustrates the signals used to label helicity state.  The clock pulse (SYNC) is used to indicate that a change in the pockel cell used to define the helicity state may have occurred.  The helicity bit indicates the helicity state that was set.  The original/complement pulse identifies if the state is an original or complement helicity state.  All three bits are recorded in the raw data file for each event and then converted to the labels 1,2,3,4 during DST file production once the particles have been reconstructed.
+
The helicity of the electron beam was flipped at a rate of 1 Hz. The helicity is prepared at the source such that helicity pairs are produced pseudo randomly. <br>
 
 
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
 
|
 
|
[[File:Helicity_state.png|400px|thumb|'''Figure NES.1. The Helicity State: A one bit signal from the beam injector gives the helicity information, whereas a sync bit with a 2 HZ frequency is generated at the same time and is equal to the helicity flip time.''']]
+
[[File:Helicity_state.png|400px]]
 
|}<br>
 
|}<br>
 
 
Two scalers were used to record several ancillary detectors, such as a Faraday cup and several PMTs mounted on the beam line,  according to their helicity label.  One of the scalers was gated by the DAQ live time in order to record beam conditions when the DAQ was able to take data and not busy recording data.  The second scaler remained ungated.  Both scalers recorded the SYNC and Helicity signals from the injector along with the counts observed from ancillary detectors during the SYNC interval.  The Faraday cup signal recorded by the gated  helicity scaler is used to normalize the events reconstructed during the same helicity interval.  The beam charge asymmetry measured by the gated helicity scaler is shown in Figure NES. 2. as a function of run number.  For each run number the gaussian fit was used to extract the mean values of the asymmetry and corresponding error(Figure NES. 3).
 
 
:::::::::<math>A_{BeamCharge} = \frac{ \Sigma FC^{hel1,hel2}-\Sigma FC^{hel4,hel3}}{\Sigma FC^{hel1,hel2}+\Sigma FC^{hel4,hel3}}</math>
 
 
 
 
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
|'''Figure 2.12.''' The Helicity State: A one bit signal from the beam injector gives the helicity information, whereas a sync bit with a 2 Hz frequency is generated at the same time and is equal to the helicity flip time.
[[File:RunNumber_vs_BeamChargeAsymmetryPulsePair07_03_2012.png|400px]]
+
|}<br>
|}
+
 
 +
If the first electron bunch is pseudo randomly chosen to be positive (negative) then it is labeled as the original helicity state and denoted in software by a 2 (1). The next helicity state is prepared to be a complement to the first state and labeled in the software as either a 4, if the original helicity state was a 1 (negative), or 3 if the original helicity state was a 2 (positive). The helicity selection process is then repeated.<br>
 +
Figure~\ref{fig:HelicityState} illustrates the signals used to label the helicity states. The clock pulse (SYNC) is used to indicate that a change in the pockel cell used to define the helicity state may have occurred. The helicity bit identifies the helicity state that was set. The original/complement pulse identifies if the state is an original or complement helicity state. All three bits are recorded in the raw data file for each event and then converted to the labels 1, 2, 3, 4 during DST file production once the particles have been reconstructed.<br>
 +
Two scalers were used to record several ancillary detectors, such as a Faraday cup and several PMTs mounted on the beam line,  according to their helicity label. One of the scalers was gated by the DAQ live time in order to record beam conditions when the DAQ was able to take data and not busy recording data. The second scaler remained ungated. Both scalers recorded the SYNC and Helicity signals from the injector along with the counts observed from ancillary detectors during the SYNC interval. The Faraday cup signal recorded by the gated  helicity scaler is used to normalize the events reconstructed during the same helicity interval. The beam charge asymmetry below is measured by the gated helicity scaler. For each run number, a gaussian fit was used to fit the beam charge asymmetry distributions as in Figure~\ref{fig:BeamAsym28101}. The beam charge asymmetry is defined as<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
|<math>A_{BeamCharge} = \frac{ \Sigma FC^{hel1,hel2}-\Sigma FC^{hel4,hel3}}{\Sigma FC^{hel1,hel2}+\Sigma FC^{hel4,hel3}},</math>
'''Figure NES. 2. Run Number vs the beam charge Asymmetry'''
+
|}<br>
|}
+
where <math>FC^{hel1,hel2}</math> (<math>FC^{hel4,hel3}</math>) represents the Faraday cup counts observed for the original helicity 1 and 2 states (the complement helicity 3 and 4 states).<br>
 
 
 
 
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
 
|
 
|
 
[[File:FC_ChargeAsymmetry_282101RunNumber_14_23HelPairs_03_12_12.png|400px]]
 
[[File:FC_ChargeAsymmetry_282101RunNumber_14_23HelPairs_03_12_12.png|400px]]
|}
+
|}<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
 
|
 
|
'''Figure NES. 3. Beam charge asymmetry for run #28101 using the gated faraday cup counts for two helicity pairs(1-4 and 2-3 helicity pairs).<math>A_{1-4}=(11.5 \pm 4.4) \times 10^{-5}</math> and <math>A_{2-3}=(-2.3 \pm  4.4) \times 10^{-5}</math>'''
+
'''Figure 2.14.''' Beam charge asymmetry for run #28101 using the gated Faraday cup counts for two helicity pairs(1-4 and 2-3 helicity pairs).<math>A_{1-4}=(11.5 \pm 4.4) \times 10^{-5}</math> and <math>A_{2-3}=(-2.3 \pm  4.4) \times 10^{-5}</math>.
|}
+
|}<br>
 
+
EG1b data sets, with the same half wave plane, target type, target polarization and beam torus, have been combined. The beam charge asymmetries have been calculated for the each run group and are listed in Table~\ref{tab:RunvsBeamAsym}.<br>
{| border="1" |cellpadding="20" cellspacing="0
+
{| border="1" style="text-align: center;" align="center"
 
|-
 
|-
 
|Run Group || Half wave plane(HWP) || <math>A_{1-4}</math> || <math>A_{2-3}</math>
 
|Run Group || Half wave plane(HWP) || <math>A_{1-4}</math> || <math>A_{2-3}</math>
Line 361: Line 329:
 
|-
 
|-
 
| 28449 - 28479 || +1 ||<math> (-11.6 \pm 16.5) \times 10^{-4}</math> || <math>(-21.6 \pm 16.5) \times 10^{-4}</math>
 
| 28449 - 28479 || +1 ||<math> (-11.6 \pm 16.5) \times 10^{-4}</math> || <math>(-21.6 \pm 16.5) \times 10^{-4}</math>
|}
+
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|'''Table 4.1.''' Run Group versus Beam Charge Asymmetry.
 +
|}<br>
  
 +
==Electron Asymmetry==
  
A measurement of the electron cross section helicity difference needs to account for the possible helicity dependence of the incident electron flux ( Charge Asymmetry). Fig. NES. 4 shows the reconstructed electron asymmetry before it is normalized by the gated Faraday Cup as a function of the run number for the 4.2 GeV data set. The reconstructed electron asymmetry can be defined following way:
+
A measurement of the electron cross section helicity difference needs to account for a possible helicity dependence of the incident electron flux (charge Asymmetry). Figure~\ref{fig:RunvsEAsym} shows the reconstructed electron asymmetry before it is normalized by the gated Faraday Cup as a function of the run number for the 4.2 GeV data set. The reconstructed electron asymmetry can be defined following way:<br>
 
+
{| border="0" style="background:transparent;"  align="center"
:::::<math>A_{NES}^{+-} = \frac{NES^{+} - NES^{-}}{NES^{+} + NES^{-}}  \equiv (2-3)</math> or <math>A_{NES}^{-+} = \frac{NES^{-} - NES^{+}}{NES^{-} + NES^{+}} \equiv (1-4)</math>
+
|-
 +
|<math>A_{NES}^{+-} = \frac{NES^{+} - NES^{-}}{NES^{+} + NES^{-}}  \equiv (2-3)</math> or <math>A_{NES}^{-+} = \frac{NES^{-} - NES^{+}}{NES^{-} + NES^{+}} \equiv (1-4)</math>
 +
|}<br>
  
where <math>NES^+</math>(<math>NES^-</math>) represents number of reconstructed electrons in the final state for the positive(negative) beam helicity.  
+
where <math>NES^{+}</math> (<math>NES^{-}</math>) represents number of electron scattered for the positive (negative) beam helicity. <br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
 
|
 
|
[[File:NES_Asymmetry_Before_FCNormalization.png|400px|thumb|'''Run Number vs NES Asymmetry before FC normalization. The red points represent NES asymmetry for the helicity 2-3 pair and the black points - the helicity pair 1-4. The green line shows the sing of the half wave plane and purple line - the sign of the target polarization. ''']]
+
[[File:NES_Asymmetry_Before_FCNormalization.png|400px]]
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|'''Figure''' Run Number versus Electron Asymmetry before FC normalization. The black and red points represent reconstructed electron asymmetry for the helicity 1-4 pair for ND<math>_{3}</math> and NH<math>_{3}</math> target respectively. The blue and green points represent the helicity pair 2-3 for ND<math>_{3}</math> and NH<math>_{3}</math> respectively. The green line shows the sign of the half wave plane (HWP) and the purple line is the sign of the target polarization (TPol).
 
|}<br>
 
|}<br>
 +
Systematic effects on the asymmetry measurement may be investigated  by separating the data into two groups based on which helicity state is set first. The first group (black data points) represents the electron asymmetry observed when the first (original) helicity state is negative and its complement state is positive (helicity state #1 - state #4). The second group (red data points) represents the asymmetry observed when the first state is positive and the complement state is negative (helicity state #2 - #3). Both groups were divided into two subgroups based the target type used. The diamond points on the histogram represent the data for the NH<math>_{3}</math> target and the squares for the ND<math>_{3}</math> target. Two lines on the histogram are used to identify the sign of the half wave plate (HWP) and the target polarization (TPol). The relative spin orientation can be changed by either inserting a half wave plane (HWP) or by populating a different target polarization state with a different RF frequency. One would expect the asymmetry to change sign if either the HWP is inserted or the target polarization is rotated 180 degrees. As one can see for Figure~\ref{fig:WvsNESAsym} and Figure~\ref{fig:RunvsEAsym}, the electron asymmetry (sign(hel1-hel4), sign(hel3-hel2) and sign(hel42-hel13)) changes sign if the HWP or Target Polarization sign is changed. <br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
| [[File:28205.png|300px]] || [[File:28222.png|300px]]
'''Figure NES.4. Run Number vs NES Asymmetry for All Runs before FC normalization'''
+
|-
|}
+
| NH3, Tpol>0 and HWP>0. || NH3, Tpol<0 and HWP>0
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
| W versus <math>(NES^{hel42} - NES^{hel13})</math>. The electron asymmetry (sign(hel42-hel13)) changes sign when the HWP or Target Polarization sign is changed.
 +
|}<br>
 +
The un-normalized reconstructed electron asymmetry has been calculated as:
 +
<math>A_{NES} = \frac{NES^{hel1,hel2} - NES^{hel4,hel3}}{NES^{hel1,hel2} + NES^{hel4,hel3}},</math><br>
 +
and normalized by the Faraday cup
 +
<math>A_{NES}^{FCnormalized} = \frac{\frac{NES^{hel1,hel2}}{FC^{hel1,hel2}} - \frac{NES^{hel4,hel3}}{FC^{hel4,hel3}}}{\frac{NES^{hel1,hel2}}{FC^{hel1,hel2}} + \frac{NES^{hel4,hel3}}{FC^{hel4,hel3}}}.</math><br>
  
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
| [[File:NES_Asymmetry_After_FCNormalization.png|400px]]
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
| '''Figure''' Run Number versus Electron Asymmetry after applying FC normalization. The black and red points represent the reconstructed electron asymmetry for the helicity 1-4 pair for ND<math>_{3}</math> and NH<math>_{3}</math> target respectively. The blue and green points are the helicity pair 2-3 for ND<math>_{3}</math> and NH<math>_{3}</math> respectively. The green line shows the sign of the half wave plane (HWP) and the purple line is the sign of the target polarization (TPol).
 +
|}<br>
  
Systematic effects on the asymmetry measurement may be investigated  by separating the data into two groups based on which helicity state is set first.  The first group(black data points) represents the electron asymmetry observed when the first (original) helicity state is negative and its complement state is positive(helicity state #1 - state #4). The second group(red data points) represents the asymmetry observed, when the first state is positive and the complement state is negative(helicity state #2 - #3). Both groups were divided into two subgroups based the target type used. The diamond points on the histogram represent the data for the NH3 target and the squares for the ND3 target.  On the same histograms are presented the signs of the half wave plane(HWP) and the target polarization(TPol).  The relative spin orientation can be changed by either inserting a half wave (HWP) or by populating a different target polarization state with a different RF frequency.  One would expect the asymmetry to change sign if either the HWP is inserted or the target polarization is rotated 180 degrees.  As one can see, the electron asymmetry sign ( sign(hel1-hel4) && sign(hel3-hel2) ) is opposite of the sign of (HWP<math>\times</math>Target_Polarization). The NES asymmetry has been calculated the following way before accounting for the Faraday cup:
+
==Semi-Inclusive Asymmetries==
  
:::::::::<math>A_{NES} = \frac{NES^{hel1,hel2} - NES^{hel4,hel3}}{NES^{hel1,hel2} + NES^{hel4,hel3}}</math>
+
The asymmetries from semi-inclusive pion electroproduction using proton or deuteron targets can be written in terms of the difference of the yield when the electron spin is parallel and antiparallel to the spin of the nucleon. There are four combinations of semi-inclusive asymmetries: <math>A_{NH_{3}}^{\pi^{+}}</math>, <math>A_{NH_{3}}^{\pi^{-}}</math>, <math>A_{ND_{3}}^{\pi^{+}}</math> and <math>A_{ND_{3}}^{\pi^{-}}</math><br>
  
The NES asymmetry after the gated faraday cup normalization is defined as:
+
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
| <math>A_{NH_{3}}^{\pi^{+}}=\frac{N_{NH_{3},\pi^{+}}^{\uparrow \uparrow} - N_{NH_{3}, \pi^{+}}^{\uparrow \downarrow}}{N_{NH_{3},\pi^{+}}^{\uparrow \uparrow} + N_{NH_{3}, \pi^{+}}^{\uparrow \downarrow}}</math>
 +
|}<br>
  
:::::::::<math>A_{NES}^{FCnormalized} = \frac{\frac{NES^{hel1,hel2}}{FC^{hel1,hel2}} - \frac{NES^{hel4,hel3}}{FC^{hel4,hel3}}}{\frac{NES^{hel1,hel2}}{FC^{hel1,hel2}} + \frac{NES^{hel4,hel3}}{FC^{hel4,hel3}}}</math>
+
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
| <math>A_{NH_{3}}^{\pi^{-}}=\frac{N_{NH_{3},\pi^{-}}^{\uparrow \uparrow} - N_{NH_{3}, \pi^{-}}^{\uparrow \downarrow}}{N_{NH_{3},\pi^{-}}^{\uparrow \uparrow} + N_{NH_{3}, \pi^{-}}^{\uparrow \downarrow}}</math>
 +
|}<br>
  
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
| <math>A_{ND_{3}}^{\pi^{+}}=\frac{N_{ND_{3},\pi^{+}}^{\uparrow \uparrow} - N_{ND_{3}, \pi^{+}}^{\uparrow \downarrow}}{N_{ND_{3},\pi^{+}}^{\uparrow \uparrow} + N_{ND_{3}, \pi^{+}}^{\uparrow \downarrow}}</math>
[[File:NES_Asymmetry_After_FCNormalization.png|400px|thumb|''' Run Number vs NES Asymmetry after applying FC normalization. The red points represent NES asymmetry for the helicity 2-3 pair and the black points - the helicity pair 1-4. The green line shows the sing of the half wave plane and purple line - the sign of the target polarization.  ''']]
 
 
|}<br>
 
|}<br>
 +
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
| <math>A_{ND_{3}}^{\pi^{-}}=\frac{N_{ND_{3},\pi^{-}}^{\uparrow \uparrow} - N_{ND_{3}, \pi^{-}}^{\uparrow \downarrow}}{N_{ND_{3},\pi^{-}}^{\uparrow \uparrow} + N_{ND_{3}, \pi^{-}}^{\uparrow \downarrow}},</math>
'''Figure NES.5. Run Number vs NES Asymmetry for All Runs after FC normalization'''
+
|}<br>
|}
 
  
On the Fig. NES. 6 data runs are combined for the same target type, target polarization, beam torus and half wave plane.  
+
where <math>N_{NH_{3},\pi^{+},\pi^{-}}^{\uparrow \downarrow}</math>(<math>N_{NH_{3},\pi^{+},\pi^{-}}^{\uparrow \uparrow}</math>) and <math>N_{ND_{3},\pi^{+},\pi^{-}}^{\uparrow \downarrow}</math>(<math>N_{ND_{3},\pi^{+},\pi^{-}}^{\uparrow \uparrow}</math>) represent the number of <math>\pi^{+}</math> and <math>\pi^{-}</math> hadrons detected in the final state with the scattered electron, when the spin of the initial electron beam was antiparallel (parallel) to the spin of the proton and neutron respectively. <br>
 +
The kinematic coverage for the events used in the measured asymmetries are shown on Figure~\ref{fig:InvariantMassvsQ2} and Figure~\ref{fig:MissingMassM_X}. The semi-inclusive asymmetries are listed in Table~\ref{tab:RunvsSIDISAsym}. The first group <math>A_{hel1-hel4}</math> represents the asymmetry measured when the first original electron spin (hel1) is antiparallel to the target nucleon spin and its complement state (hel4) is parallel, whereas the second group <math>A_{hel2-hel3}</math> represents measuring the SIDIS asymmetry when the first helicity state is parallel (hel2) and its complement state is antiparallel to the spin of the nucleon. For the final measurement, the two groups are combined into one <math>A_{hel42-hel13}</math> set. Instead of looking at original and complement electron helicity states, they have been combined into the positive (hel42) and negative (he13) helicity states, which are parallel and antiparallel to the spin of the target nucleon respectively. <br>
  
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
| [[File:InvariantMass_vsQ2.png|300px]] || [[File:M_X.png|300px]]
[[File:NES_Asymmetry_BeforeandAfter_FCNormalizationgrouped.png|400px]]
+
|-
|}
+
| Invariant Mass versus <math>Q^{2}</math>. || Missing Mass.
 +
|}<br>
 +
 
 +
{| border="1" style="text-align: center;" align="center"
 +
|-
 +
| Target type, Beam Torus || <math>A_{hel1-hel4} \times 10^{-4}</math> || <math>A_{hel2-hel3} \times 10^{-4}</math> || <math>A^{raw} \times 10^{-4}</math>
 +
|-
 +
| NH<math>_{3}</math>, B>0, <math>\pi^{+}</math> || <math>-139.84 \pm 81.52</math> || <math>143.15 \pm 81.78</math> || <math>136.2 \pm 57.74</math>
 +
|-
 +
| NH<math>_{3}</math>, B<0, <math>\pi^{+}</math> || <math>-223.76 \pm 117.10</math> || <math>247.65 \pm 116.59</math> || <math>237.69 \pm 82.65</math>
 +
|-
 +
| ND<math>_{3}</math>, B>0, <math>\pi^{-}</math> || <math>-6.37 \pm 188.73</math> || <math>-98.11 \pm 188.03</math> || <math> 9.21 \pm 127.22</math>
 +
|-
 +
| ND<math>_{3}</math>, B<0, <math>\pi^{-}</math> || <math>-63.73 \pm 105.14</math> || <math>-30.34 \pm 6085.54</math> || <math>12.37 \pm 71.10</math>
 +
|-
 +
| NH<math>_{3}</math>, B>0, <math>\pi^{-}</math> || <math>-155.45 \pm 128.21</math> || <math>-72.55 \pm 128.92</math> || <math> 35.11 \pm 90.91</math>
 +
|-
 +
| NH<math>_{3}</math>, B<0, <math>\pi^{-}</math> || <math>9.60 \pm 119.31</math> || <math>72.94 \pm 119.36</math> || <math>32.39 \pm 84.38</math>
 +
|-
 +
| ND<math>_{3}</math>, B>0, <math>\pi^{+}</math> || <math>-76.59 \pm 126.60</math> || <math>110.28 \pm 126.13</math> || <math>92.25 \pm 85.38</math>
 +
|-
 +
| ND<math>_{3}</math>, B<0, <math>\pi^{+}</math> || <math>-29.22 \pm 107.53</math> || <math>123.98 \pm 106.86</math> || <math>92.25 \pm 85.38</math>
 +
|}<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
|'''Figure''' Run Number versus SIDIS Asymmetry for Each Type Target material and Beam Torus.
'''Figure NES. 6. Run groups vs NES Asymmetry before and after FC Normalization'''
+
|}<br>
|}
 
  
 +
The ratio of the combined semi-inclusive deep inelastic asymmetries (<math>A^{raw}</math>) from Table~\ref{tab:RunvsSIDISAsym}, for two different torus settings have been corrected for the electron reconstruction efficiency. The Figure~\ref{fig:TheRatioOfSIDIS} represents the asymmetry ratios before and after electron reconstruction efficiency corrections. The ratios have been calculated for each target and charged pion type. The SIDIS asymmetries before and after the correction are statistically the same. The result indicates that the electron reconstruction efficiency does not change the asymmetries.
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
| [[File:SIDIS_Asymmetry_Before_After_InclusiveCorrections.png|400px]]
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
| '''Figure''' The ratio of the SIDIS asymmetries for two torus field settings <math>\frac{A^{raw}(B>0)}{A^{raw}(B<0)}</math> versus target and the charged pion type. The black squares represent the data before electron reconstruction efficiency and the red data - after electron reconstruction efficiency has been applied.
 +
|}<br>
 +
The data have been combined for each target type and asymmetries measured for two <math>x_{B}</math> values (Table~\ref{tab:SIDISAsymx03x04}). The SIDIS asymmetries were calculated for the following kinematic range: electron scattering angle <math>17.5<\theta_{e}<18.5</math>, electron momentum 2.55 GeV<<math>P_{e}</math><2.75 GeV, invariant mass 1.6 GeV<W<1.8 GeV and momentum transferred squared 0.9 GeV<math>^{2}</math><<math>Q^{2}</math><1.3 GeV<math>^{2}</math>.<br>
  
==SIDIS Asymmetries==
+
{| border="1" style="text-align: center;" align="center"
 +
|-
 +
|  SIDIS Asymmetry  ||  <math>x_{B}=0.3</math>  ||  <math>x_{B}=0.4</math>
 +
|-
 +
| <math>A_{NH_{3}}^{\pi^+,raw}</math> || <math>(150.08 \pm 65.96) \times 10^{-4}</math>  ||  <math>(217.20 \pm 69.06) \times 10^{-4}</math>
 +
|-
 +
| <math>A_{ND_{3}}^{\pi^-,raw}</math> || <math>(39.04 \pm 81.44) \times 10^{-4}</math> ||  <math>(91.90 \pm 96.14) \times 10^{-4}</math>
 +
|-
 +
| <math>A_{NH_{3}}^{\pi^-,raw}</math> || <math>(100.81 \pm 83.49) \times 10^{-4}</math>  ||  <math>(-23.98 \pm 94.92) \times 10^{-4}</math>
 +
|-
 +
| <math>A_{ND_{3}}^{\pi^+,raw}</math> || <math>(53.17 \pm 74.89) \times 10^{-4}</math>  ||  <math>(85.17 \pm 82.97) \times 10^{-4}</math>
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
| '''Figure''' SIDIS Asymmetries for <math>x_{B}=0.3</math> and <math>x_{B}=0.4</math>.
 +
|}<br>
  
SIDIS asymmetries:
+
In addition to <math>x_{B}</math>, the data have been subdivided according to the fraction energy of the observed final state pion(<math>z</math>). <math>A_{NH_{3}}^{\pi^+,raw}</math> asymmetries for two <math>z</math> are shown in Table~\ref{tab:SIDISz}.
  
 +
{| border="1" style="text-align: center;" align="center"
 +
|-
 +
| <math>z</math>  ||  <math>x_{B}=0.3</math>  ||  <math>x_{B}=0.4</math>
 +
|-
 +
| 0.4  || <math>(125.62 \pm 81.92) \times 10^{-4}</math>  || <math>( 165.81 \pm 86.76) \times 10^{-4}</math>
 +
|-
 +
| 0.7  || <math>(140.32 \pm 158.98) \times 10^{-4}</math> || <math>(238.34 \pm 156.79) \times 10^{-4}</math>
 +
|}<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
| <math>A_{NH_{3}}^{\pi^+,raw}</math> SIDIS Asymmetry.
<math>A_{NH3}^{\pi^+}=\frac{N_{NH3,\pi^+}^{hel1,hel2} - N_{NH3, \pi^+}^{hel4,hel3}}{N_{NH3,\pi^+}^{hel1,hel2} + N_{NH3, \pi^+}^{hel4,hel3}}</math>
+
|}<br>
|}
 
 
 
  
 +
==Dilution Factor==
 +
To exclude the contributions of polarized nucleons from the non-hydrogen nuclei in the ammonia target (Nitrogen) and the cooling material of the target (Helium) cell to the semi-inclusive rates, the measured raw double spin asymmetries have to be divided by a dilution factor. The dilution factor accounts for the fraction of events coming from the desired polarized target nucleon. The dilution factors are calculated by combining the data from runs using different target types. During the EG1b experiment, several runs were taken with Carbon C<math>^{12}</math> and an empty target cell (He). All the runs in this thesis used liquid Helium as the coolant during the experiment. The Carbon and empty target runs were used to estimate the dilution of the data by the noise attributed to the interaction of the incident electron beam with the Nitrogen or Helium nucleons present in the target cell \cite{Fersch2008}.<br>
 +
In order to calculate the dilution factor, we need to define the number of counts <math>N_{A,C,MT}^{hel13,hel24}</math> with beam helicity negative and positive for ammonia, Carbon and empty target runs. All are weighted by the corresponding gated Faraday cup counts \cite{Khun2010}. Normalized rates are defined as <br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|<math>n_{C,MT} = \frac{N^{+}_{C,MT} + N^{-}_{C,MT}}{FC^{+} + FC^{-}}</math>
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|<math>n_{A} = \frac{1}{2}(\frac{N^{+}_{A}}{FC^{+}} + \frac{N^{-}_{A}}{FC^{-}})</math>,
 +
|}<br>
 +
where <math>n_{C}</math> is the rate from a Carbon target, <math>n_{MT}</math> is from an empty target and <math>n_{A}</math> is from one of the NH<math>_{3}</math> or ND<math>_{3}</math> ammonia targets.<br>
 +
The counts for all four targets (empty, Carbon and ammonia) can be expressed as the sum of counts from the entrance and exit window foils (<math>\rho_{F}</math>, <math>l_{F}</math>, <math>\sigma_{F}</math>), liquid Helium coolant (<math>\rho_{He}</math>, <math>l_{He}</math>, <math>\sigma_{He}</math>), Carbon(<math>\rho_{C}</math>, <math>l_{C}</math>, <math>\sigma_{C}</math>), Nitrogen (<math>\rho_{N}</math>, <math>l_{N}</math>, <math>\sigma_{N}</math>), Hydrogen (Deuterium) (<math>\rho_{H(D)}</math>, <math>l_{H(D)}</math>, <math>\sigma_{H(D)}</math>).<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|<math>n_{MT}=\rho_{F}l_{F}\sigma_{F} + \rho_{He}L\sigma_{He}=f\rho_{C}l_{C}\sigma_{C} + \rho_{He}L\sigma_{He}</math>,
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|<math>n_{C}=\rho_{F}l_{F}\sigma_{F} + \rho_{C}l_{C}\sigma_{C} + \rho_{He}(L - l_{C})\sigma_{He}</math>,
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|<math>n_{A}=\rho_{F}l_{F}\sigma_{F} + \rho{He}(L-l_{A})\sigma_{He} + \rho_{A}l_{A}(\sigma_{N} + 3\sigma_{H(D)})</math>,
 +
|}<br>
 +
where <math>f = \frac{\rho_{F}l_{F}\sigma_{F}}{\rho_{C}l_{C}\sigma_{C}}</math>, <math>\sigma_{H(D)}</math> represents the Hydrogen (Deuteron) cross section.<br>
 +
Using above system of equations, we define two new spectra to account for the Carbon target and the difference in the amount of Helium in ammonia targets vice versa the Carbon target. For a target cell with length L and the Carbon target with length <math>l_{C}</math>, the Carbon and LHe contributions can be written following way:<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|<math>n_{12C}^{\prime} = \frac{L}{L + fl_{C}}n_{C} - \frac{L-l_{C}}{L + fl_{C}}n_{MT} = \rho_{C}l_{C}\sigma_{C}</math>
 +
|}<br>
 +
and<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
|<math>n_{4He}^{\prime} = \frac{(1 + f)l_{He}}{L + fl_{C}}n_{MT} - \frac{fl_{He}}{L + fl_{C}}n_{C} = \rho_{He}l_{He}\sigma_{He}</math>,
<math>A_{ND3}^{\pi^-}=\frac{N_{ND3,\pi^-}^{hel1,hel2} - N_{ND3, \pi^-}^{hel4,hel3}}{N_{ND3,\pi^-}^{hel1,hel2} + N_{ND3, \pi^-}^{hel4,hel3}}</math>
+
|}<br>
|}
+
where <math>n_{12C}^{\prime}</math> is the rate from the Carbon nucleus only and <math>n_{He}^{\prime}</math> is the rate from liquid Helium only. The length and densities are listed in Table~\ref{tab:LengthandDensityTarget}.<br>
 
+
We need to establish how the rate from the Carbon target is related to the rate from the N<math>^{15}</math> in NH<math>_{3}</math> and ND<math>_{3}</math>. The cross section for N<math>^{15}</math> can be written in terms of the cross sections on C<math>^{12}</math> target material (<math>\sigma_{12C}</math>) and on a bound neutron in N<math>^{15}</math> (<math>\sigma_{n}^{\prime}</math>).<br>
 
 
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
|<math>\sigma_{15N} \approx \frac{7}{6}\sigma_{12C} + \sigma_{n}^{\prime} = \left(\frac{7}{6} + \frac{\sigma_{n}^{\prime}}{\sigma_{12C}}\right)\sigma_{12C}</math>.
[[File:SIDIS_Asymmetry_Before_FCNormalization03_12_12.png|400px]]
+
|}<br>
|}
+
It is assumed that when the scattering occurs on protons inside the target material, the ratio of <math>\frac{\sigma_{n}^{\prime}}{\sigma_{12C}} = 0</math> and when the reaction happens on neutrons inside the target, the value of the ratio is <math>\frac{\sigma_{n}^{\prime}}{\sigma_{12C}} = \frac{1}{6}</math>, because there are six bound neutrons in C<math>^{12}</math> \cite{Khun2010}.<br>
 +
Using above quantities the background represented by the number of counts due to the non-hydrogen and non-deuterium parts of the ammonia target can be expressed as<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
|<math>n_{B}  =  \left[ \frac{\rho_{A}l_{A}}{\rho_{C}l_{C}}\left(\frac{7}{6} + \frac{\sigma_{n}^{\prime}}{\sigma_{12C}}\right) + f \right] n_{12C}^{\prime} + (L - l_{A})n_{4He}^{\prime}</math>
'''Figure NES. 6. Run Number vs Semi inclusive asymmetry before FC Normalization.'''
+
|}<br>
|}
 
 
 
 
 
 
 
 
 
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
| <math>=  n_{MT} + l_{A} \left[ \frac{\rho_{A}}{\rho_{C}l_{C}}\left(\frac{7}{6} + \frac{\sigma_{n}^{\prime}}{\sigma_{12C}}\right)n_{12C}^{\prime} - n_{4He}^{\prime}\right]</math>.
[[File:SIDIS_Asymmetry_After_FCNormalization03_12_12.png|400px]]
+
|}<br>
|}
+
The dilution factor <math>d_{f}</math> is<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
|<math>d_{f} = \frac{n_{A} - n_{B}}{n_{A}}</math>.
'''Figure NES. 6. Run Number vs Semi inclusive asymmetry after FC Normalization.'''
+
|}<br>
|}
+
{| border="1" style="text-align: center;" align="center"
 
+
|-
 
+
|  Item  ||  Description  ||  Value
{| border="1" |cellpadding="20" cellspacing="0  
+
|-
 +
| <math>\rho_{F} l_{F}</math>  ||  Density times target length for empty target.  || Al: 167 <math>\mu</math>m; 0.045 g/cm<math>^{2}</math>. Kapton: 384 <math>\mu</math>m; 0.055 g/cm<math>^{2}</math>. Total=Al + Kapton=0.0996 g/cm<math>^{2}</math>
 +
|-
 +
| <math>\rho_{C} l_{C}</math> ||  Density times target length for Carbon target.  ||  0.498 g/cm<math>^{2}</math>
 +
|-
 +
| f  ||  The ratio of counts from foils to the C<math>^{12}</math> slab in the Carbon target.  ||  <math>0.200</math>
 +
|-
 +
| <math>\rho_{He}</math> ||  He density.  ||  0.145 g/cm<math>^{3}</math>
 +
|-
 +
| <math>L</math>  ||  The length of the target cell from the entrance to exit foil.  ||  1.90 cm
 
|-
 
|-
| Target type, Beam Torus sign (B) || <math>A_{hel1-hel4}</math> || <math>A_{hel2-hel3}</math>  || <math>A_{hel13-hel42}</math>
+
| <math>\rho_{C}</math> || C<math>^{12}</math> density. || 2.17 g/cm<math>^{3}</math>
 
|-
 
|-
| NH3, B>0, <math>\pi^+</math> || <math>(-139.84 \pm 81.52) \times 10^{-4}</math>
+
| <math>l_{C}</math> || Carbon target length. || 0.23 cm
|| <math>(143.15 \pm 81.78) \times 10^{-4}</math> ||<math> (-136.2 \pm 57.74) \times 10^{-4}</math>
 
 
|-
 
|-
| NH3, B<0, <math>\pi^+</math> ||<math> (-223.76 \pm 117.10) \times 10^{-4}</math>
+
| <math>\rho_{NH_{3}}</math> || NH<math>_{3}</math> density || 0.917 g/cm<math>^{3}</math>
  || <math>(247.65 \pm 116.59) \times 10^{-4}</math> || <math>(-237.69 \pm 82.65) \times 10^{-4}</math>
 
 
|-
 
|-
| ND3, B>0, <math>\pi^-</math> || <math>(-6.37 \pm 188.73) \times 10^{-4}</math>
+
| <math>\rho_{ND_{3}}</math> || ND<math>_{3}</math> density || 1.056 g/cm<math>^{3}</math>
  || <math>(-98.11 \pm 188.03) \times 10^{-4}</math> ||<math> (-9.21 \pm 127.22) \times 10^{-4}</math>
 
 
|-
 
|-
| ND3, B<0, <math>\pi^-</math> || <math>(-63.73 \pm 105.14) \times 10^{-4}</math>
+
| <math>l_{A}</math>  || Ammonia target length. || 0.6 cm
  || <math>(-30.34 \pm 6085.54) \times 10^{-4}</math> ||<math> (-12.37 \pm 71.10) \times 10^{-4}</math>
+
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
| NH3, B>0, <math>\pi^-</math> || <math>(-155.45 \pm 128.21) \times 10^{-4}</math>
+
|Length and density values for different types of target material reproduced from the EG1b experiment \cite{Khun2010}.}
|| <math>(-72.55 \pm 128.92) \times 10^{-4}</math> ||<math>(-35.11 \pm 90.91) \times 10^{-4}</math>
+
|}<br>
 +
{| border="1" style="text-align: center;" align="center"
 
|-
 
|-
| NH3, B<0, <math>\pi^-</math> ||<math>(9.60 \pm 119.31) \times 10^{-4}</math>|| <math>(72.94 \pm 119.36) \times 10^{-4}
+
| The fractional energy of the observed final state hadron(z) || Dilution Factor(<math>d_{f}</math>)
</math> || <math>(-32.39 \pm 84.38) \times 10^{-4}</math>
 
 
|-
 
|-
| ND3, B>0, <math>\pi^+</math> || <math>(-76.59 \pm 126.60) \times 10^{-4}</math>
+
| NH<math>_{3}</math>, <math>\pi^{+}</math> && <math>z=0.4</math> || <math>0.160 \pm 0.02</math>
|| <math>(110.28 \pm 126.13) \times 10^{-4}</math> ||<math> (-92.25 \pm 85.38) \times 10^{-4} </math>
 
 
|-
 
|-
| ND3, B<0, <math>\pi^+</math> || <math>(-29.22 \pm 107.53) \times 10^{-4}</math>
+
| NH<math>_{3}</math>, <math>\pi^{+}</math> && <math>z=0.7</math> || <math>0.152 \pm 0.03</math>
|| <math>(123.98 \pm 106.86) \times 10^{-4}</math> ||<math> (-92.25 \pm 85.38) \times 10^{-4}</math>
+
|}<br>
|}
 
 
 
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
|Calculated dilution Factor for NH<math>_{3}</math> target type.
[[File:SIDIS_Asymmetry_After_FCNormalizationRunsGrouped03_13_12.png|400px]]
+
|}<br>
|}
+
The dilution factor is compared below in Table~\ref{tab:CompareDilution}:<br>
 +
{| border="1" style="text-align: center;" align="center"
 +
|-
 +
| Reaction || Dilution Factor(<math>d_{f}</math>)
 +
|-
 +
|    Resonance region || 0.11 - 0.13
 +
|-
 +
|   Inclusive || 0.14 - 0.17
 +
|-
 +
|    Semi-Inclusive || 0.122 - 0.182
 +
|}<br>
 
{| border="0" style="background:transparent;"  align="center"
 
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|
+
|{Dilution Factor compared with other results  \cite{Prok2004} \cite{Fersch2008}.
'''Figure NES. 6. Run groups vs Semi inclusive asymmetry after FC Normalization.'''
+
|}<br>
|}
 
  
 +
==Fragmentation <math>\Delta R_{np}^{\pi^{+} + \pi^{-}}</math>==
  
{| border="1"  |cellpadding="20" cellspacing="0  
+
A test of fragmentation can be performed by calculating the ratio of the difference of polarized to unpolarized cross sections for proton and neutron targets <math>\Delta R_{np}^{\pi^{+} + \pi^{-}}</math> and showing that it is independent of <math>z</math>. The fragmentation function can be written following way<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
|<math>\Delta R_{np}^{\pi^{+} + \pi^{-}}(x, z, Q^{2}) = \frac{\Delta \sigma_{p}^{\pi^{+} + \pi^{-}} - \Delta \sigma_{n}^{\pi^{+} + \pi^{-}}}{\sigma_{p}^{\pi^{+} + \pi^{-}} - \sigma_{n}^{\pi^{+} + \pi^{-}}}</math>
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
| = \frac{\Delta \sigma_{p}^{\pi^{+} + \pi^{-}}}{\sigma_{p}^{\pi^{+} + \pi^{-}} - \sigma_{n}^{\pi^{+} + \pi^{-}}} - \frac{\Delta \sigma_{n}^{\pi^{+} + \pi^{-}}}{\sigma_{p}^{\pi^{+} + \pi^{-}} - \sigma_{n}^{\pi^{+} + \pi^{-}}}
 +
|}<br>
 +
{| border="0" style="background:transparent;align="center"
 +
|-
 +
| <math>= \frac{(\Delta u +\Delta \bar{u}) - (\Delta d + \Delta \bar{d})}{(u+\bar{u}) - (d+\bar{d})}(x,Q^{2})</math>
 +
|}<br>
 +
{| border="0" style="background:transparent;"  align="center"
 +
|-
 +
| <math>=  \frac{g_{1}^{p} - g_{1}^{n}}{F_{1}^{p} - F_{1}^{n}}(x,Q^{2})</math>.
 +
|}<br>
 +
The left side of the fragmentation function is calculated using the measured double spin asymmetries for the ammonia targets. However, the right side can be extracted using the model. <br>
 +
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
| SIDI Asymmetry  || <math>X_B=0.3</math> || <math>X_B=0.4</math>  
+
|[[File:Fragfunc.png|400px|thumb|<math>x_{B}</math> versus <math>\Delta R_{np}^{\pi^{+} + \pi^{-}}</math>. Black data points represent the Model, red and green data points represent fragmentation function for <math>z=0.4</math> and <math>z=0.7</math> respectively.]]
 +
|}<br>
 +
{| border="1" style="text-align: center;" align="center"
 
|-
 
|-
|<math> A_{NH3}^{\pi^+,hel13-hel24}</math>||<math>(-150.08 \pm 65.96) \times 10^{-4}</math> || <math>(-217.20 \pm 69.06) \times 10^{-4}</math>  
+
| <math>z</math> || <math>x_{B}=0.3</math> || <math>x_{B}=0.4</math>
 
|-
 
|-
|<math> A_{ND3}^{\pi^-,hel13-hel24}</math>||<math>(-39.04 \pm 81.44) \times 10^{-4}</math> || <math>(-91.90 \pm 96.14) \times 10^{-4}</math>
+
|   0.4 || 14 % || 12 %
 
|-
 
|-
|<math> A_{NH3}^{\pi^-,hel13-hel24}</math>||<math>(-100.81 \pm 83.49) \times 10^{-4}</math> || <math>(23.98 \pm 94.92) \times 10^{-4}</math>
+
|   0.7 || 21 % || 32 %
 +
|}<br>  
 +
{| border="0" style="background:transparent;"  align="center"
 
|-
 
|-
|<math> A_{ND3}^{\pi^+,hel13-hel24}</math>||<math>(-53.17 \pm 74.89) \times 10^{-4}</math> || <math>(-85.17 \pm 82.97) \times 10^{-4}</math>  
+
| Statistical Z - test for the data comparison with the model. The probability of not observing the data point for each <math>z</math> and <math>x_{B}</math> values within the Inclusive Model.
|}
+
|}<br>
 +
The pion asymmetries, <math>A_{NH3}^{\pi^-}</math>, <math>A_{{ND3}^{\pi^+}</math>, <math>A_{ND3}^{\pi^-}</math> are statistically consistent with zero as shown in Table~\ref{tab:SIDISAsymx03x04}. As a result, the contributions from <math>\Delta \sigma_{p}^{\pi^{-}}</math>, <math>\Delta \sigma_{n}^{\pi^{+}}</math> and <math>\Delta \sigma_{n}^{\pi^{-}}</math> to the fragmentation function test (Eq. 1.54) are negligible. Only the first term is non-zero in <math>\Delta R_{np}^{\pi^{+} + \pi^{-}}</math>. After correcting the cross section difference for the target polarization, beam polarization, and dilution factor, the <math>\Delta R_{np}^{\pi^{+} + \pi^{-}}</math> function for two values of <math>z</math> and <math>x_{B}</math> is shown on Figure~\ref{fig:xBvsDelta} and is compared to the Model.<br>
  
<math> A_{NH3}^{\pi^+,hel13-hel24}</math>
 
  
{| border="1" |cellpadding="20" cellspacing="0
+
==Systematic errors==
 +
Systematic errors from the experimental setup and the cuts used for particle identification and background elimination were estimated. The systematic errors associated with the electromagnetic cuts and fiducial cuts for electron identification were calculated by comparing the SIDIS asymmetries before and after cuts were applied. Systematic effects related to the dilution factor and polarization were estimated following way: First, the SIDIS asymmetries were calculated for the standard values of the dilution factor and polarization. Then asymmetries were recalculated by changing the value of the each parameter by the amount of its uncertainty. The difference between these two values is the systematic effect. The systematic errors for the SIDIS asymmetries are shown in Table~\ref{tab:SystematicErrors}.<br>
 +
{| border="1" style="text-align: center;" align="center"
 +
|-
 +
| Source || Systematic Error
 +
|-
 +
| Geometrical and timing cuts || <math>5.80 \times 10^{-2}</math>
 +
|-
 +
| Electromagnetic calorimeter cut || <math>4.07 \times 10^{-2}</math>
 
|-
 
|-
| <math>z</math> || <math>X_B=0.3</math> || <math>X_B=0.4</math>  
+
| Dilution Factor || <math>5.03 \times 10^{-2}</math>
 
|-
 
|-
| 0.45 ||<math>(-125.62 \pm 81.92) \times 10^{-4}</math> ||<math>( -165.81 \pm 86.76) \times 10^{-4}</math>
+
| Target and beam polarization || <math>1.10 \times 10^{-2}</math>
 
|-
 
|-
| 0.7 ||<math>(-140.32 \pm 158.98) \times 10^{-4}</math>||<math>(-238.34 \pm 156.79) \times 10^{-4}</math>
+
| Total || <math> 8.76  \times 10^{-2}</math>
 +
|}<br>
 +
{| border="0" style="background:transparent;" align="center"
 
|-
 
|-
| 0.9 ||<math>(-355.44 \pm 200.23) \times 10^{-4}</math> ||<math>(-438.76 \pm 189.03) \times 10^{-4}</math>
+
|The systematic errors for the <math>A_{NH_{3}}^{\pi^+}</math> asymmetry.
|}
+
|}<br>
 
 
 
 
 
 
<math>\Delta R_{np}^{\pi^+ + \pi^-} = \frac{\Delta \sigma_p^{\pi^+ + \pi^-} - \Delta \sigma_{n}^{\pi^+ + \pi^-}}{\sigma_p^{\pi^+ + \pi^-} - \sigma_{n}^{\pi^+ + \pi^-}}=</math>
 
 
 
<math>=\frac{\Delta \sigma_p^{\pi^+ + \pi^-}}{\sigma_p^{\pi^+ + \pi^-} - \sigma_{n}^{\pi^+ + \pi^-}} - \frac{\Delta \sigma_{n}^{\pi^+ + \pi^-}}{\sigma_p^{\pi^+ + \pi^-} - \sigma_{n}^{\pi^+ + \pi^-}}=</math>
 
 
 
<math> = \frac{(\sigma_p^{\pi^+,hel13} - \sigma_p^{\pi^+,hel24}) + (\sigma_p^{\pi^-,hel13} - \sigma_p^{\pi^-,hel24})}{\sigma_p^{\pi^+ + \pi^-} - \sigma_{n}^{\pi^+ + \pi^-}}  - \frac{(\sigma_n^{\pi^+,hel13} - \sigma_n^{\pi^+,hel24}) + (\sigma_n^{\pi^-,hel13} - \sigma_n^{\pi^-,hel24})}{\sigma_p^{\pi^+ + \pi^-} - \sigma_{n}^{\pi^+ + \pi^-}} = </math>
 
 
 
 
 
 
 
<math>=\frac{(\Delta u +\Delta \bar{u}) - (\Delta d + \Delta \bar{d})}{(u+\bar{u}) - (d+\bar{d})}(x,Q^2)=</math>
 
 
 
 
 
 
 
<math>= \frac{g_1^p - g_1^n}{F_1^p - F_1^n}(x,Q^2)</math>
 
 
 
=Results=
 
 
 
 
 
<math>F(d) = \frac{\Delta d_v}{d_v} = \frac{Y}{Z}= \frac{\Delta \sigma_p^{\pi^+ \pm \pi^-} - 4\Delta \sigma_{2H}^{\pi^+ \pm \pi^-}}{\sigma_p^{\pi^+ \pm \pi^-} - 4\sigma_{2H}^{\pi^+ \pm \pi^-}} (x,Q^2) = </math> <br>
 
 
 
:<math> = \frac{([({\sigma_p}^{\pi^+})_{1/2}-({\sigma_p}^{\pi^+})_{3/2}] - [({\sigma_p}^{\pi^-})_{1/2}-({\sigma_p}^{\pi^-})_{3/2}]) - 4 \times ([({\sigma_{2H}}^{\pi^+})_{1/2}-({\sigma_{2H}}^{\pi^+})_{3/2}] - [({\sigma_{2H}}^{\pi^-})_{1/2}-({\sigma_{2H}}^{\pi^-})_{3/2}])}{([({\sigma_p}^{\pi^+})_{1/2}+({\sigma_p}^{\pi^+})_{3/2}] - [({\sigma_p}^{\pi^-})_{1/2}+({\sigma_p}^{\pi^-})_{3/2}]) - 4 \times ([({\sigma_{2H}}^{\pi^+})_{1/2} + ({\sigma_{2H}}^{\pi^+})_{3/2}] - [({\sigma_{2H}}^{\pi^-})_{1/2} + ({\sigma_{2H}}^{\pi^-})_{3/2}])} = </math> <br>
 
 
 
:: <math>= \frac{[(N_1 - N_2) - (N_3 - N_4)] - 4 \times [(N_5 - N_6) - (N_7 - N_8)]}{[(N_1 + N_2) - (N_3 + N_4)] - 4 \times [(N_5 + N_6) - (N_7 + N_8)]}</math> <br>
 
 
 
 
 
<math>\partial F(d)= \{ \frac{\partial}{\partial N_1} \times \partial N_1 + \frac{\partial}{\partial N_2} \times \partial N_2 + \frac{\partial}{\partial N_3} \times \partial N_3 + \frac{\partial}{\partial N_4} \times \partial N_4 + \frac{\partial}{\partial N_5} \times \partial N_5 + \frac{\partial}{\partial N_6} \times \partial N_6 + \frac{\partial}{\partial N_7} \times \partial N_7 + \frac{\partial}{\partial N_8} \times \partial N_8 \}^2 F </math> <br>
 
 
 
:: <math>= \frac{1}{Z^4} \{ Y ( - \partial N_1 - \partial N_2 + \partial N_3 + \partial N_4 + 4\partial N_5 + 4\partial N_6 - 4\partial N_7 - 4\partial N_8) + </math><br>
 
 
 
:: <math>+ Z (\partial N_1 - \partial N_2 - \partial N_3 + \partial N_4 - 4\partial N_5 + 4\partial N_6 + 4\partial N_7 - 4\partial N_8) \}^2</math>
 
 
 
==Notes==
 
  
<references/>
+
[[Delta_D_over_D]]
  
[[http://wiki.iac.isu.edu/index.php/Delta_D_over_D Go Back]]
+
[https://wiki.iac.isu.edu/index.php/SIDIS_PionAsym_EG2000 Go Back]

Latest revision as of 06:48, 10 December 2012

This chapter describes the techniques used to analyze the data collected during the EG1b experiment and calculate semi-inclusive cross sections for the following reactions: [math]\overrightarrow{e}^{-} \overrightarrow{N} \rightarrow e^{-} \pi^{+} X[/math] and [math]\overrightarrow{e}^{-} \overrightarrow{N} \rightarrow e^{-} \pi^{-} X[/math] using [math]\mbox{NH}_{3}$ and $\mbox{ND}_{3}[/math] polarized targets respectively. The goal of this work is to measure charged pion asymmetries defined according to the incident electron helicity and the target polarization. The measured asymmetries may be used to determine if the fragmentation function is independent of the observed final state hadron fractional energy ([math]z[/math]) \cite{Christova}. The quantity ([math]\Delta R_{np}^{\pi^{+} + \pi^{-}}[/math]) is defined in terms of the ratio of the difference of polarized semi-inclusive deep inelastic scattering cross sections for proton and neutron targets to unpolarized cross sections:

[math]\Delta R_{np}^{\pi^+ + \pi^-} = \frac{\Delta \sigma_p^{\pi^+ + \pi^-} - \Delta \sigma_{n}^{\pi^+ + \pi^-}}{\sigma_p^{\pi^+ + \pi^-} - \sigma_{n}^{\pi^+ + \pi^-}}[/math]


A measure of [math]\Delta R[/math] can be used to test for independent fragmentation when it is compared to inclusive structure functions as shown in Eq. (1.54).

The CLAS Data Selection

The data files from the EG1b experiment chosen for this analysis are listed in Table~\ref{tab:EG1bRuns}. During the experiment, 2.2 GeV, 4.2 GeV and 5.7 GeV longitudinally polarized electron beams were used to probe the polarized frozen ammonia NH[math]_{3}[/math] and ND[math]_{3}[/math] targets. This work will discuss the analysis of the 4.2 GeV energy electron beam data set as this energy provided the most statistics. The collected data have been tested by applying restrictions discussed later in this chapter.

Run Set Target Type Torus Current(A) Target Polarization HWP
28100 - 28102 ND3 +2250 -0.18 +1
28106 - 28115 ND3 +2250 -0.18 -1
28145 - 28158 ND3 +2250 -0.20 +1
28166 - 28190 ND3 +2250 +0.30 +1
28205 - 28217 NH3 +2250 +0.75 +1
28222 - 28236 NH3 +2250 -0.68 +1
28242 - 28256 NH3 +2250 -0.70 -1
28260 - 28275 NH3 +2250 +0.69 -1
28287 - 28302 ND3 -2250 +0.28 +1
28306 - 28322 ND3 -2250 -0.12 +1
28375 - 28399 ND3 -2250 +0.25 -1
28407 - 28417 NH3 -2250 +0.73 -1
28456 - 28479 NH3 -2250 -0.69 +1
Table 1.1. EG1b Runs analyzed for this work.


Particle Identification

Additional tests were performed on the electron and a pion candidates reconstructed using the standard CLAS software package on the raw data collected during the EG1b experiment. Electrons are identified by matching the charged particle hits in the Cherenkov counter, electromagnetic calorimeter, and Time of Flight system. Geometrical and timing cuts are applied to improve electron identification. In addition, cuts are applied on the energy deposited by the particle into the calorimeter and the number of photoelectrons produced in the Cherenkov counter. Charged pions are identified by matching the hits in the drift chamber and ToF counter, along with a Cherenkov cut requiring that the number of photons for pions be less than two.

Electron Identification

The CLAS trigger system required the particle to deposit energy in the electromagnetic calorimeter and illuminate the Cherenkov counter within a [math]150[/math] ns time window (Figure~\ref{fig:ElectronTrack}). Unfortunately, this trigger suffers from a background of high energy negative pions that may be misidentified as electrons. The pion contamination of the electron sample is reduced using cuts on the energy deposited in the electromagnetic calorimeter and the momentum measured in the track reconstruction for the known magnetic field. The energy deposition mechanism for the pions and electrons in the electromagnetic calorimeter is different. The total energy deposited by the electrons in the EC is proportional to their kinetic energy, whereas pions are minimum ionizing particles and the energy deposition is independent of their momentum (Figure~\ref{fig:ECTOTvsP}). The pion background is further suppressed using geometrical and time matching between the Cherenkov counter hit and the measured track in the drift chamber.

Figure 2.1 Example of electron passing through the drift chambers and creating the signal in the cherenkov counter and electromagnetic calorimeter. Electron track is highlighted by the blue line (Run number 27095, Torus Current +2250 (inbending)).


EC CUTS

The CLAS electromagnetic calorimeter was used to reduce the misidentification of electron and negative pion candidates. The electromagnetic calorimeter contains thirteen layers of lead-scintillator sandwiches composed of [math]\sim 2[/math] mm thick lead and [math]10[/math] mm thick scintillator. Each set of thirteen layers are subdivided into five inner and eight outer layers that are named the inner and outer calorimeter respectively.
Electrons interact with the calorimeter producing electromagnetic showers that release energy into the calorimeter. The deposited energy is proportional to the momentum of the electrons. Figure~\ref{fig:ECCuts} shows the correlation of the inner and outer calorimeter electron candidate's energy measured by the calorimeter and divided by the particles momentum reconstructed by the drift chamber. As shown in the Figure~\ref{fig:ECCuts}, there is an island near [math]E/p = 0.2[/math], which contains most of the electron candidates as well as some regions below [math]0.2[/math] which will be argued to be negative pions misidentified as electrons.
Pions entering the calorimeter are typically minimum ionizing particles, loosing little of their incident energy in the calorimeter at a rate of 2 [math]\mbox{MeV g}^{-1} \mbox{cm}^{2}[/math]. Electrons, on the other hand, deposit a larger fraction of their momentum into the calorimeter. As a result, the energy deposited into the electromagnetic calorimeter is different for electrons and pions. Pions loose about [math]0.08[/math] GeV of energy traversing the calorimeter independent their momentum thereby producing the constant signal in the calorimeter around [math]0.08[/math] GeV. In order to reduce misidentified pions from the electron sample, the following cut has been applied:

[math]EC_{inner}\gt 0.08 \times p,[/math]


where [math]p[/math] represents particle momentum and [math]EC_{innner}[/math] the inner part of the calorimeter.
Since the energy loss of pions is related to the detector thickness the correlation can be established between the energy deposition into the inner and outer layers of the detector:

[math]\frac{EC_{tot}}{EC_{inner}} = \frac{13}{5},[/math]


which gives the following cut for the energy deposition into the outer layer of the calorimeter:

[math]EC_{tot} \gt 0.2 \times p.[/math]


Before cuts.
After EC cuts.


Figure 2.2 [math]EC_{inner}/p[/math] versus [math]EC_{tot}/p[/math] before and after EC cuts ([math]EC_{tot}\gt 0.2p[/math] and [math]EC_{inner}\gt 0.08p[/math]). After applying EC cuts about [math]46 \%[/math] of the events have been removed from the electron sample.

Cherenkov Counter Cut

The Cherenkov counter has been used to further reduce the negatively charged pion background in the reconstructed electron sample. When the velocity of a charged particle is greater than the local phase velocity of light or when it enters a medium with different optical properties, the charged particle will emit photons. Cherenkov light is emitted at the critical angle [math]\theta_{c}[/math] representing the angle of Cherenkov radiation relative to the particle's direction. It can be shown that the cosine of the Cherenkov radiation angle is inversely proportional to the velocity of the charged particle

[math]cos \theta_c=\frac{1}{n \beta},[/math]


where [math]\beta c[/math] is the particle's velocity and [math]n[/math] the index of refraction of the medium. The charged particle in time [math]t[/math] travels a distance [math]\beta c t[/math], while the electromagnetic waves travel [math]\frac{c}{n}t[/math]. For a medium with given index of refraction [math]n[/math], there is a threshold velocity [math]\beta_{thr}=\frac{1}{n}[/math], below which no radiation is emitted. This process may be used to distinguish between the highly relativistic electrons and the less relativistic pions based on the number of photons produced. The number of photons produced per unit path length of a particle with charge [math]Ze[/math] and per unit energy interval of the photons is proportional to the sine of the Cherenkov angle \cite{Nakamure}

[math]\frac{d^2 N}{dEdx}=\frac{\alpha z^2}{\hbar c}sin ^2 \theta_c=\frac{\alpha z^2}{\hbar c}[1-\frac{1}{\beta^2 n^2 (E)}][/math]


[math]\frac{d^2 N}{d\lambda dx} = \frac{2 \pi \alpha z^2}{\lambda^2}[1-\frac{1}{\beta^2 n^2 (\lambda)}][/math]


[math]\beta=\frac{v}{c}=\frac{pc}{\sqrt{(pc)^2 + (mc^2)^2}}.[/math]


Taylor expanding Eq. 3.6 and keeping only the first two terms we get following

[math]\frac{d^2 N}{dEdx}=\frac{\alpha z^2}{\hbar c}sin ^2 \theta_c=\frac{\alpha z^2}{\hbar c}[\beta^2 n^2 (E) - 1][/math]


The gas used in the CLAS Cerenkov counter is perfluorobutane [math]C_{4}F_{10}[/math] with index of refraction equal to [math]1.00153[/math]. The number of photoelectrons emitted by electrons is about thirteen. On the other hand, calculations show that the number of photons produced by the negatively charged pions in the Cherenkov detector is approximately [math]2[/math]. The theoretical results of the number of photons produced by the electrons and pions when passing through the Cherenkov counter are shown on Figure~\ref{fig:NPHETheoretical}.

For electrons.
For pions.


Figure 2.3 Theoretical Calculation of the Number of Photoelectrons for electrons and pions.


The distribution of the number of photoelectrons measured in the Cherenkov detector and the energy deposition dependence on number of photoelectrons are shown on Figure~\ref{fig:nphewithoutcuts} and Figure~\ref{fig:nphewithcuts}. One can see that a single photoelectron peak is caused by misidentifying pions as electrons.

Electrons nphe without cuts all data with fits.gif
Figure 2.4 The number of photoelectrons without cuts.


E total vs nphe momentum cut file dst27070.gif


Figure 2.5 The total energy deposited into the Calorimeter vs The Number of Photoelectrons.


Geometric and Timing cuts

Negative pions may be produced when the lepton scatters at a polar angle close to zero and is not observed by the detector. In order to reduce the electron sample contamination due to those pions, geometrical cuts on the location of the particle at the entrance to the Cherenkov detector and time matching cuts have been developed by Osipenko, so called OSI cuts \cite{Osipenko}. For each CLAS Cherenkov detector segment the following cut has been applied

[math]|\theta_p - {\theta_p}^{center} - {\theta_p}^{offset}| \lt 3 \sigma_{p},[/math]


where [math]\theta_{p}[/math] represents the measured polar angle with respect to a projectile plane for each electron event. The Cherenkov counter's projective plane is an imaginary plane behind the Cherenkov detector where Cherenkov radiation would have arrived if it had moved the same distance from emission point to the PMT, without reflections in the mirror system. ${\theta_{p}}^{center}$ is the polar angle from the CLAS detector center to the image of Cherenkov counter segment center and [math]{\theta_{p}}^{offset}[/math] is the shift in the segment center position. In addition to geometrical cuts, timing cuts have been applied to match the time between a Cherenkov counter hit and time of flight system.
The pion contamination in an electron sample was estimated by fitting the number of photoelectron distribution using two Gaussian distributions convoluted with a Landau distribution \cite{Lanczos}:

[math]N_{pe}= p_0 e^{-0.5 \left (\frac{x-p_1}{p_2} \right )^2} + p4\frac{1}{1-\left(\frac{x-p5}{p6}\right )} + p_6 e^{-0.5 \left (\frac{x-p_7}{p_8} \right )^2}.[/math]


The fits in Figure~\ref{fig:OSICuts}.(a) suggest that the pion contamination in the electron sample is [math]9.63 % \pm 0.01 %[/math] before applying the OSI cuts and after the OSI cuts the contamination is about [math]4.029% \pm 0.003 %[/math] (Figure~\ref{fig:OSICuts}.(b)).

Before Cuts.
After OSI Cuts.


Figure 2.6 The number of photoelectrons before and after OSI Cuts.


Pion Identification

Charged pions are identified using a coincidence hit in the drift chamber and Time-of-Flight (ToF) counter. Pions are separated from the other charged particles by looking at the particle momentum versus the [math]\beta[/math] distribution. The particle velocity, [math]\beta=\frac{v}{c}[/math], is calculated from the difference of the RF time and the time-of-flight measurement in the ToF system with the path length from the vertex to the ToF counters. The mass of the charged particle can be identified by combining the particle's [math]\beta[/math] with the particle momentum obtained from the track measured by the drift chamber in the known magnetic field. The particle mass in a magnetic field is given as

[math]p=\frac{m\beta}{\sqrt{1-\beta^{2}}}[/math]


[math]m=p\sqrt{(\beta^{2} - 1)}[/math]


[math]\beta=\frac{L_{path}}{t_{flight}},[/math]


where [math]m[/math] is the mass of the charged particle, [math]\beta[/math] its velocity, [math]p[/math] particle momentum, [math]L_{path}[/math] the path length from the vertex to scintillators and [math]t_{flight}[/math] the time of flight from the interaction vertex to the ToF system.
Using the above information (particle momentum from the drift chambers and the timing information from the ToF system), the mass squared of the charged particle was calculated and is shown on Figure~\ref{fig:PvsM2}. The pion mass band is around [math]\sim 0[/math]. To isolate charged pions from the rest of the particles, a [math]3\sigma[/math] cut on the momentum versus [math]\beta[/math] distribution has been applied \cite{Park2008}.

PionMomentumvsBeta.png


Figure 2.7 The charged particle momentum versus [math]\beta[/math] distribution. The pion and proton bands are clearly separated.


PionMomentumvsMass.png


Figure 2.8 The charged particle momentum versus mass squared distribution for the [math]\overrightarrow{e}p\rightarrow\overrightarrow{e}^{\prime} \pi^{+} n[/math] electroproduction process. The bands around 0 and 1 represent pions and protons respectively <ref name="Park2008" >Park, K., Burkert, V. D., & Kim, W. (The CLAS Collaboration). (2008). Cross sections and beam asymmetries for [math]\vec{e} p[/math]->[math]en\pi^+[/math] in the nucleon resonance region for [math]1.7 \lt Q^2 \lt 4.5[/math] [math](GeV)^2[/math]. Phys. Rev., C77, 015208.</ref>.


In addition to the charged particle velocity ([math]\beta[/math]), the fiducial volume cuts have been applied for the charged pion identification. Since the drift chambers and scintillators are used for pion detection, the polar angle range where pions are detected is much larger than for electrons. For the EG1b experiment, pions were detected from [math]8^{\circ}[/math] to [math]180^{\circ}[/math] \cite{Park2008}. The pion identification code has been developed by Joshua Pierce \cite{JoshuaPierce}.

Event Reconstruction Efficiency

The goal of this work is to measure the semi-inclusive asymmetry when an electron and a pion are detected in the final state. For this analysis, pions of opposite charge will be observed using the same scintillator paddles by flipping the CLAS torus magnetic field direction. Although the pions will be detected by the same detector elements, the electrons will intersect different detector elements. As a result, the electron reconstruction efficiency was evaluated in terms of the electron rate observed in two different scintillator paddles detecting the same electron kinematics.

Inclusive Electron Event Reconstruction Efficiency

The electron reconstruction efficiency for individual scintillator detectors using the 4.2 GeV EG1b data is investigated below. Only the electron is detected in the final state (inclusive case). The pion contamination in the electron sample was removed by applying the cuts described above. The electron paddle numbers 10 (B<0) and 5 (B>0) were chosen respectively, because they contained the most electron events in a first pass semi-inclusive pion analysis of the data set. The electron kinematics (momentum, scattering angle and invariant mass) for these scintillators is shown on Figure~\ref{fig:ElectronKinematics}.
Ratios of the inclusive electron rate, normalized using the gated Faraday cup, detected in scintillator paddles # 5 and # 10 were measured. The two ratios are constructed to quantify the CLAS detector's ability to reconstruct electrons in scintillator paddle #5 using a positive Torus polarity and scintillator #10 using the negative Torus polarity.

[math]\frac{ND3,B\gt 0, PaddleNumber^{e^-}=5}{ND3,B\lt 0, PaddleNumber^{e^-}=10}=1.57 \pm 0.16[/math]


[math]\frac{NH3,B\gt 0, PaddleNumber^{e^-}=5}{NH3,B\lt 0, PaddleNumber^{e^-}=10}=1.76 \pm 0.17[/math]


Notice the above ratios are statistically the same. The semi-inclusive analysis to be performed in this thesis will be taking ratios using an [math]ND_{3}[/math] and [math]NH_{3}[/math] target. Below is the observed ratio comparing the inclusive electrons observed in scintillator #5 for a positive torus polarity and an ND[math]_{3}[/math] target to the electrons observed in scintillator #10 when the torus polarity is negative and the target is [math]NH_{3}[/math].

EmomInclusiveoverlay4-2GeVfcup.gif EthetaInclusiveoverlay4-2GeVfcup.gif WInclusiveoverlay4-2GeVfcup.gif
Electron Momentum((NH3,B>0), (NH3,B<0), (ND3,B>0) and (ND3,B<0)) Electron Scattering Angle [math]\theta[/math]((NH3,B>0), (NH3,B<0), (ND3,B>0) and (ND3,B<0)) W Invariant mass((NH3,B>0), (NH3,B<0), (ND3,B>0) and (ND3,B<0))
Figure 2.9. Electron Kinematics.


[math]\frac{ND3,B\gt 0, PaddleNumber^{e^-}=5}{NH3,B\lt 0, PaddleNumber^{e^-}=10}=1.55 \pm 0.15[/math]


The above ratios, which have been observed to be ammonia target independent, indicate a difference in an electron detector efficiency when the torus polarity is flipped. An electron detection efficiency "correction coefficient" is defined in terms of the above ratio and measured to be [math]\frac{ND_{3},B\gt 0,E_{PaddleNumber}=5}{NH_{3},B\lt 0,E_{PaddleNumber}=10} = 0.645[/math] and [math]\frac{ND_{3},B\lt 0,E_{PaddleNumber}=10}{NH_{3},B\gt 0,E_{PaddleNumber}=5} = 1.82[/math]. The impact of these corrections on the data is illustrated in the next section.

Exclusive and Semi-Inclusive Event Reconstruction Efficiencies

After determining the electron reconstruction efficiency for the selected paddle numbers, the measured single pion electroproduction rate was compared to the MAID 2007 unitary model that has been developed using the world data of pion photo and electro-production to determine the impact of using the above "correction coefficient". The model is well adopted for predictions of the observables for pion production, like five fold cross section, total cross section, etc.
The MAID 2007 model has predictions of the total cross section for the following two cases that are related to our work:

[math]\gamma^*[/math] + proton[math](NH_3) \rightarrow[/math] [math]\pi^+[/math] + neutron


[math]\gamma^*[/math] + neutron[math](ND_3) \rightarrow[/math] [math]\pi^-[/math] + proton.


The ratio of the pions detected in the scintillator paddles, located between the Cherenkov counter and electromagnetic calorimeter, is shown in Figure~\ref{fig:PionPaddleNumbervsRatio}. The ratios were taken for four different cases. The intrinsic assumption is that, for the inbending case, positive pions and for the outbending case negative pions have the same trajectories with the same kinematics. In addition, negatively charged pions in the inbending field and positively charged pions in the outbending fields are detected by the same detector elements.

Paddenumbvsratio.jpg
Figure 2.10. Pion paddle number versus Ratio for Semi-Inclusive case.


Using MAID 2007, the total cross section was calculated for the following invariant mass and four momentum transferred square values: 1.7 GeV < W<1.8 GeV and [math]Q^{2}=1.1 GeV^{2}[/math] \cite{MAID2007}.

[math]\sigma = \sigma_{T} + \epsilon \sigma_{L} + \sqrt{2\epsilon(1 + \epsilon)}\sigma_{LT} cos{\phi_{\pi}}^{CM} + \epsilon \sigma{TT} cos2{\phi_{\pi}}^{CM} + h \sqrt{2\epsilon (1-\epsilon)}\sigma_{LT^{\prime}}sin{\phi_{\pi}}^{CM}, [/math]


where [math]{\phi_{\pi}}^{CM}[/math] is the pion azimuthal angle in the CM frame, [math]\epsilon = (1 + 2(1 + \frac{\nu^{2}}{Q^{2}})\tan^{2}\frac{\theta_{e}}{2})^{-1}[/math] is the virtual photon polarization, [math]\nu=E_{i} - E_{f}[/math] the energy difference of the initial and final state electron, [math]Q^{2} = 4 E_{i} E_{f} \sin^{2}\frac{\theta_{e}}{2}[/math] the four momentum transferred squared, [math]\theta_{e}[/math] the electron scattering angle and $h$ the electron helicity. After applying corrections from the inclusive cases, the ratios have been compared to the results from MAID2007. The difference of the measured and MAID2007 model ratios for each pion paddle number is shown in Figure~\ref{fig:MAID2007Experiment}. One can conclude from Figure~\ref{fig:MAID2007Experiment} that the "inclusive corrections" do not impact single pion production rates for the exclusive cases.

Maid2007minusexperiment.png


Pion Paddle Number versus MAID2007 - Experiment[math](N(\pi^{-},ND_{3})/N(\pi^{+},NH_{3}))[/math]. The Black and red data represent B>0/B<0 and B<0/B>0 cases respectively before corrections. The green and blue points represent the ratios for B>0/B<0 and B<0/B>0 after inclusive corrections.


Asymmetries

The double spin asymmetry measurements in this thesis are performed by comparing scattering events that occur when the incident probe spin and nuclear target spin are parallel to the scattering events that occur when the spins are anti-parallel.

Beam Charge Asymmetry

The helicity of the electron beam was flipped at a rate of 1 Hz. The helicity is prepared at the source such that helicity pairs are produced pseudo randomly.

Helicity state.png


Figure 2.12. The Helicity State: A one bit signal from the beam injector gives the helicity information, whereas a sync bit with a 2 Hz frequency is generated at the same time and is equal to the helicity flip time.


If the first electron bunch is pseudo randomly chosen to be positive (negative) then it is labeled as the original helicity state and denoted in software by a 2 (1). The next helicity state is prepared to be a complement to the first state and labeled in the software as either a 4, if the original helicity state was a 1 (negative), or 3 if the original helicity state was a 2 (positive). The helicity selection process is then repeated.
Figure~\ref{fig:HelicityState} illustrates the signals used to label the helicity states. The clock pulse (SYNC) is used to indicate that a change in the pockel cell used to define the helicity state may have occurred. The helicity bit identifies the helicity state that was set. The original/complement pulse identifies if the state is an original or complement helicity state. All three bits are recorded in the raw data file for each event and then converted to the labels 1, 2, 3, 4 during DST file production once the particles have been reconstructed.
Two scalers were used to record several ancillary detectors, such as a Faraday cup and several PMTs mounted on the beam line, according to their helicity label. One of the scalers was gated by the DAQ live time in order to record beam conditions when the DAQ was able to take data and not busy recording data. The second scaler remained ungated. Both scalers recorded the SYNC and Helicity signals from the injector along with the counts observed from ancillary detectors during the SYNC interval. The Faraday cup signal recorded by the gated helicity scaler is used to normalize the events reconstructed during the same helicity interval. The beam charge asymmetry below is measured by the gated helicity scaler. For each run number, a gaussian fit was used to fit the beam charge asymmetry distributions as in Figure~\ref{fig:BeamAsym28101}. The beam charge asymmetry is defined as

[math]A_{BeamCharge} = \frac{ \Sigma FC^{hel1,hel2}-\Sigma FC^{hel4,hel3}}{\Sigma FC^{hel1,hel2}+\Sigma FC^{hel4,hel3}},[/math]


where [math]FC^{hel1,hel2}[/math] ([math]FC^{hel4,hel3}[/math]) represents the Faraday cup counts observed for the original helicity 1 and 2 states (the complement helicity 3 and 4 states).

FC ChargeAsymmetry 282101RunNumber 14 23HelPairs 03 12 12.png


Figure 2.14. Beam charge asymmetry for run #28101 using the gated Faraday cup counts for two helicity pairs(1-4 and 2-3 helicity pairs).[math]A_{1-4}=(11.5 \pm 4.4) \times 10^{-5}[/math] and [math]A_{2-3}=(-2.3 \pm 4.4) \times 10^{-5}[/math].


EG1b data sets, with the same half wave plane, target type, target polarization and beam torus, have been combined. The beam charge asymmetries have been calculated for the each run group and are listed in Table~\ref{tab:RunvsBeamAsym}.

Run Group Half wave plane(HWP) [math]A_{1-4}[/math] [math]A_{2-3}[/math]
28100 - 28105 +1 [math](5.88 \pm 34.40) \times 10^{-4}[/math] [math](4.03 \pm 34.36) \times 10^{-4}[/math]
28106 - 28115 -1 [math] (7.53 \pm 22.30) \times 10^{-4}[/math] [math] (8.28 \pm 22.30) \times 10^{-4}[/math]
28145 - 28240 +1 [math](31.70 \pm 7.99) \times 10^{-4}[/math] [math](30.40 \pm 7.99) \times 10^{-4}[/math]
28242 - 28284 -1 [math] (49.6 \pm 10.8) \times 10^{-4}[/math] [math] (47.9 \pm 10.8) \times 10^{-4}[/math]
28286 - 28324 +1 [math] (36.3 \pm 11.6) \times 10^{-4}[/math] [math] (37.0 \pm 11.5) \times 10^{-4}[/math]
28325 - 28447 -1 [math] (21.1 \pm 13.4) \times 10^{-4}[/math] [math](22.2 \pm 13.4) \times 10^{-4}[/math]
28449 - 28479 +1 [math] (-11.6 \pm 16.5) \times 10^{-4}[/math] [math](-21.6 \pm 16.5) \times 10^{-4}[/math]


Table 4.1. Run Group versus Beam Charge Asymmetry.


Electron Asymmetry

A measurement of the electron cross section helicity difference needs to account for a possible helicity dependence of the incident electron flux (charge Asymmetry). Figure~\ref{fig:RunvsEAsym} shows the reconstructed electron asymmetry before it is normalized by the gated Faraday Cup as a function of the run number for the 4.2 GeV data set. The reconstructed electron asymmetry can be defined following way:

[math]A_{NES}^{+-} = \frac{NES^{+} - NES^{-}}{NES^{+} + NES^{-}} \equiv (2-3)[/math] or [math]A_{NES}^{-+} = \frac{NES^{-} - NES^{+}}{NES^{-} + NES^{+}} \equiv (1-4)[/math]


where [math]NES^{+}[/math] ([math]NES^{-}[/math]) represents number of electron scattered for the positive (negative) beam helicity.

NES Asymmetry Before FCNormalization.png


Figure Run Number versus Electron Asymmetry before FC normalization. The black and red points represent reconstructed electron asymmetry for the helicity 1-4 pair for ND[math]_{3}[/math] and NH[math]_{3}[/math] target respectively. The blue and green points represent the helicity pair 2-3 for ND[math]_{3}[/math] and NH[math]_{3}[/math] respectively. The green line shows the sign of the half wave plane (HWP) and the purple line is the sign of the target polarization (TPol).


Systematic effects on the asymmetry measurement may be investigated by separating the data into two groups based on which helicity state is set first. The first group (black data points) represents the electron asymmetry observed when the first (original) helicity state is negative and its complement state is positive (helicity state #1 - state #4). The second group (red data points) represents the asymmetry observed when the first state is positive and the complement state is negative (helicity state #2 - #3). Both groups were divided into two subgroups based the target type used. The diamond points on the histogram represent the data for the NH[math]_{3}[/math] target and the squares for the ND[math]_{3}[/math] target. Two lines on the histogram are used to identify the sign of the half wave plate (HWP) and the target polarization (TPol). The relative spin orientation can be changed by either inserting a half wave plane (HWP) or by populating a different target polarization state with a different RF frequency. One would expect the asymmetry to change sign if either the HWP is inserted or the target polarization is rotated 180 degrees. As one can see for Figure~\ref{fig:WvsNESAsym} and Figure~\ref{fig:RunvsEAsym}, the electron asymmetry (sign(hel1-hel4), sign(hel3-hel2) and sign(hel42-hel13)) changes sign if the HWP or Target Polarization sign is changed.

28205.png 28222.png
NH3, Tpol>0 and HWP>0. NH3, Tpol<0 and HWP>0


W versus [math](NES^{hel42} - NES^{hel13})[/math]. The electron asymmetry (sign(hel42-hel13)) changes sign when the HWP or Target Polarization sign is changed.


The un-normalized reconstructed electron asymmetry has been calculated as: [math]A_{NES} = \frac{NES^{hel1,hel2} - NES^{hel4,hel3}}{NES^{hel1,hel2} + NES^{hel4,hel3}},[/math]
and normalized by the Faraday cup [math]A_{NES}^{FCnormalized} = \frac{\frac{NES^{hel1,hel2}}{FC^{hel1,hel2}} - \frac{NES^{hel4,hel3}}{FC^{hel4,hel3}}}{\frac{NES^{hel1,hel2}}{FC^{hel1,hel2}} + \frac{NES^{hel4,hel3}}{FC^{hel4,hel3}}}.[/math]

NES Asymmetry After FCNormalization.png


Figure Run Number versus Electron Asymmetry after applying FC normalization. The black and red points represent the reconstructed electron asymmetry for the helicity 1-4 pair for ND[math]_{3}[/math] and NH[math]_{3}[/math] target respectively. The blue and green points are the helicity pair 2-3 for ND[math]_{3}[/math] and NH[math]_{3}[/math] respectively. The green line shows the sign of the half wave plane (HWP) and the purple line is the sign of the target polarization (TPol).


Semi-Inclusive Asymmetries

The asymmetries from semi-inclusive pion electroproduction using proton or deuteron targets can be written in terms of the difference of the yield when the electron spin is parallel and antiparallel to the spin of the nucleon. There are four combinations of semi-inclusive asymmetries: [math]A_{NH_{3}}^{\pi^{+}}[/math], [math]A_{NH_{3}}^{\pi^{-}}[/math], [math]A_{ND_{3}}^{\pi^{+}}[/math] and [math]A_{ND_{3}}^{\pi^{-}}[/math]

[math]A_{NH_{3}}^{\pi^{+}}=\frac{N_{NH_{3},\pi^{+}}^{\uparrow \uparrow} - N_{NH_{3}, \pi^{+}}^{\uparrow \downarrow}}{N_{NH_{3},\pi^{+}}^{\uparrow \uparrow} + N_{NH_{3}, \pi^{+}}^{\uparrow \downarrow}}[/math]


[math]A_{NH_{3}}^{\pi^{-}}=\frac{N_{NH_{3},\pi^{-}}^{\uparrow \uparrow} - N_{NH_{3}, \pi^{-}}^{\uparrow \downarrow}}{N_{NH_{3},\pi^{-}}^{\uparrow \uparrow} + N_{NH_{3}, \pi^{-}}^{\uparrow \downarrow}}[/math]


[math]A_{ND_{3}}^{\pi^{+}}=\frac{N_{ND_{3},\pi^{+}}^{\uparrow \uparrow} - N_{ND_{3}, \pi^{+}}^{\uparrow \downarrow}}{N_{ND_{3},\pi^{+}}^{\uparrow \uparrow} + N_{ND_{3}, \pi^{+}}^{\uparrow \downarrow}}[/math]


[math]A_{ND_{3}}^{\pi^{-}}=\frac{N_{ND_{3},\pi^{-}}^{\uparrow \uparrow} - N_{ND_{3}, \pi^{-}}^{\uparrow \downarrow}}{N_{ND_{3},\pi^{-}}^{\uparrow \uparrow} + N_{ND_{3}, \pi^{-}}^{\uparrow \downarrow}},[/math]


where [math]N_{NH_{3},\pi^{+},\pi^{-}}^{\uparrow \downarrow}[/math]([math]N_{NH_{3},\pi^{+},\pi^{-}}^{\uparrow \uparrow}[/math]) and [math]N_{ND_{3},\pi^{+},\pi^{-}}^{\uparrow \downarrow}[/math]([math]N_{ND_{3},\pi^{+},\pi^{-}}^{\uparrow \uparrow}[/math]) represent the number of [math]\pi^{+}[/math] and [math]\pi^{-}[/math] hadrons detected in the final state with the scattered electron, when the spin of the initial electron beam was antiparallel (parallel) to the spin of the proton and neutron respectively.
The kinematic coverage for the events used in the measured asymmetries are shown on Figure~\ref{fig:InvariantMassvsQ2} and Figure~\ref{fig:MissingMassM_X}. The semi-inclusive asymmetries are listed in Table~\ref{tab:RunvsSIDISAsym}. The first group [math]A_{hel1-hel4}[/math] represents the asymmetry measured when the first original electron spin (hel1) is antiparallel to the target nucleon spin and its complement state (hel4) is parallel, whereas the second group [math]A_{hel2-hel3}[/math] represents measuring the SIDIS asymmetry when the first helicity state is parallel (hel2) and its complement state is antiparallel to the spin of the nucleon. For the final measurement, the two groups are combined into one [math]A_{hel42-hel13}[/math] set. Instead of looking at original and complement electron helicity states, they have been combined into the positive (hel42) and negative (he13) helicity states, which are parallel and antiparallel to the spin of the target nucleon respectively.

InvariantMass vsQ2.png M X.png
Invariant Mass versus [math]Q^{2}[/math]. Missing Mass.


Target type, Beam Torus [math]A_{hel1-hel4} \times 10^{-4}[/math] [math]A_{hel2-hel3} \times 10^{-4}[/math] [math]A^{raw} \times 10^{-4}[/math]
NH[math]_{3}[/math], B>0, [math]\pi^{+}[/math] [math]-139.84 \pm 81.52[/math] [math]143.15 \pm 81.78[/math] [math]136.2 \pm 57.74[/math]
NH[math]_{3}[/math], B<0, [math]\pi^{+}[/math] [math]-223.76 \pm 117.10[/math] [math]247.65 \pm 116.59[/math] [math]237.69 \pm 82.65[/math]
ND[math]_{3}[/math], B>0, [math]\pi^{-}[/math] [math]-6.37 \pm 188.73[/math] [math]-98.11 \pm 188.03[/math] [math] 9.21 \pm 127.22[/math]
ND[math]_{3}[/math], B<0, [math]\pi^{-}[/math] [math]-63.73 \pm 105.14[/math] [math]-30.34 \pm 6085.54[/math] [math]12.37 \pm 71.10[/math]
NH[math]_{3}[/math], B>0, [math]\pi^{-}[/math] [math]-155.45 \pm 128.21[/math] [math]-72.55 \pm 128.92[/math] [math] 35.11 \pm 90.91[/math]
NH[math]_{3}[/math], B<0, [math]\pi^{-}[/math] [math]9.60 \pm 119.31[/math] [math]72.94 \pm 119.36[/math] [math]32.39 \pm 84.38[/math]
ND[math]_{3}[/math], B>0, [math]\pi^{+}[/math] [math]-76.59 \pm 126.60[/math] [math]110.28 \pm 126.13[/math] [math]92.25 \pm 85.38[/math]
ND[math]_{3}[/math], B<0, [math]\pi^{+}[/math] [math]-29.22 \pm 107.53[/math] [math]123.98 \pm 106.86[/math] [math]92.25 \pm 85.38[/math]


Figure Run Number versus SIDIS Asymmetry for Each Type Target material and Beam Torus.


The ratio of the combined semi-inclusive deep inelastic asymmetries ([math]A^{raw}[/math]) from Table~\ref{tab:RunvsSIDISAsym}, for two different torus settings have been corrected for the electron reconstruction efficiency. The Figure~\ref{fig:TheRatioOfSIDIS} represents the asymmetry ratios before and after electron reconstruction efficiency corrections. The ratios have been calculated for each target and charged pion type. The SIDIS asymmetries before and after the correction are statistically the same. The result indicates that the electron reconstruction efficiency does not change the asymmetries.

SIDIS Asymmetry Before After InclusiveCorrections.png


Figure The ratio of the SIDIS asymmetries for two torus field settings [math]\frac{A^{raw}(B\gt 0)}{A^{raw}(B\lt 0)}[/math] versus target and the charged pion type. The black squares represent the data before electron reconstruction efficiency and the red data - after electron reconstruction efficiency has been applied.


The data have been combined for each target type and asymmetries measured for two [math]x_{B}[/math] values (Table~\ref{tab:SIDISAsymx03x04}). The SIDIS asymmetries were calculated for the following kinematic range: electron scattering angle [math]17.5\lt \theta_{e}\lt 18.5[/math], electron momentum 2.55 GeV<[math]P_{e}[/math]<2.75 GeV, invariant mass 1.6 GeV<W<1.8 GeV and momentum transferred squared 0.9 GeV[math]^{2}[/math]<[math]Q^{2}[/math]<1.3 GeV[math]^{2}[/math].

SIDIS Asymmetry [math]x_{B}=0.3[/math] [math]x_{B}=0.4[/math]
[math]A_{NH_{3}}^{\pi^+,raw}[/math] [math](150.08 \pm 65.96) \times 10^{-4}[/math] [math](217.20 \pm 69.06) \times 10^{-4}[/math]
[math]A_{ND_{3}}^{\pi^-,raw}[/math] [math](39.04 \pm 81.44) \times 10^{-4}[/math] [math](91.90 \pm 96.14) \times 10^{-4}[/math]
[math]A_{NH_{3}}^{\pi^-,raw}[/math] [math](100.81 \pm 83.49) \times 10^{-4}[/math] [math](-23.98 \pm 94.92) \times 10^{-4}[/math]
[math]A_{ND_{3}}^{\pi^+,raw}[/math] [math](53.17 \pm 74.89) \times 10^{-4}[/math] [math](85.17 \pm 82.97) \times 10^{-4}[/math]


Figure SIDIS Asymmetries for [math]x_{B}=0.3[/math] and [math]x_{B}=0.4[/math].


In addition to [math]x_{B}[/math], the data have been subdivided according to the fraction energy of the observed final state pion([math]z[/math]). [math]A_{NH_{3}}^{\pi^+,raw}[/math] asymmetries for two [math]z[/math] are shown in Table~\ref{tab:SIDISz}.

[math]z[/math] [math]x_{B}=0.3[/math] [math]x_{B}=0.4[/math]
0.4 [math](125.62 \pm 81.92) \times 10^{-4}[/math] [math]( 165.81 \pm 86.76) \times 10^{-4}[/math]
0.7 [math](140.32 \pm 158.98) \times 10^{-4}[/math] [math](238.34 \pm 156.79) \times 10^{-4}[/math]


[math]A_{NH_{3}}^{\pi^+,raw}[/math] SIDIS Asymmetry.


Dilution Factor

To exclude the contributions of polarized nucleons from the non-hydrogen nuclei in the ammonia target (Nitrogen) and the cooling material of the target (Helium) cell to the semi-inclusive rates, the measured raw double spin asymmetries have to be divided by a dilution factor. The dilution factor accounts for the fraction of events coming from the desired polarized target nucleon. The dilution factors are calculated by combining the data from runs using different target types. During the EG1b experiment, several runs were taken with Carbon C[math]^{12}[/math] and an empty target cell (He). All the runs in this thesis used liquid Helium as the coolant during the experiment. The Carbon and empty target runs were used to estimate the dilution of the data by the noise attributed to the interaction of the incident electron beam with the Nitrogen or Helium nucleons present in the target cell \cite{Fersch2008}.
In order to calculate the dilution factor, we need to define the number of counts [math]N_{A,C,MT}^{hel13,hel24}[/math] with beam helicity negative and positive for ammonia, Carbon and empty target runs. All are weighted by the corresponding gated Faraday cup counts \cite{Khun2010}. Normalized rates are defined as

[math]n_{C,MT} = \frac{N^{+}_{C,MT} + N^{-}_{C,MT}}{FC^{+} + FC^{-}}[/math]


[math]n_{A} = \frac{1}{2}(\frac{N^{+}_{A}}{FC^{+}} + \frac{N^{-}_{A}}{FC^{-}})[/math],


where [math]n_{C}[/math] is the rate from a Carbon target, [math]n_{MT}[/math] is from an empty target and [math]n_{A}[/math] is from one of the NH[math]_{3}[/math] or ND[math]_{3}[/math] ammonia targets.
The counts for all four targets (empty, Carbon and ammonia) can be expressed as the sum of counts from the entrance and exit window foils ([math]\rho_{F}[/math], [math]l_{F}[/math], [math]\sigma_{F}[/math]), liquid Helium coolant ([math]\rho_{He}[/math], [math]l_{He}[/math], [math]\sigma_{He}[/math]), Carbon([math]\rho_{C}[/math], [math]l_{C}[/math], [math]\sigma_{C}[/math]), Nitrogen ([math]\rho_{N}[/math], [math]l_{N}[/math], [math]\sigma_{N}[/math]), Hydrogen (Deuterium) ([math]\rho_{H(D)}[/math], [math]l_{H(D)}[/math], [math]\sigma_{H(D)}[/math]).

[math]n_{MT}=\rho_{F}l_{F}\sigma_{F} + \rho_{He}L\sigma_{He}=f\rho_{C}l_{C}\sigma_{C} + \rho_{He}L\sigma_{He}[/math],


[math]n_{C}=\rho_{F}l_{F}\sigma_{F} + \rho_{C}l_{C}\sigma_{C} + \rho_{He}(L - l_{C})\sigma_{He}[/math],


[math]n_{A}=\rho_{F}l_{F}\sigma_{F} + \rho{He}(L-l_{A})\sigma_{He} + \rho_{A}l_{A}(\sigma_{N} + 3\sigma_{H(D)})[/math],


where [math]f = \frac{\rho_{F}l_{F}\sigma_{F}}{\rho_{C}l_{C}\sigma_{C}}[/math], [math]\sigma_{H(D)}[/math] represents the Hydrogen (Deuteron) cross section.
Using above system of equations, we define two new spectra to account for the Carbon target and the difference in the amount of Helium in ammonia targets vice versa the Carbon target. For a target cell with length L and the Carbon target with length [math]l_{C}[/math], the Carbon and LHe contributions can be written following way:

[math]n_{12C}^{\prime} = \frac{L}{L + fl_{C}}n_{C} - \frac{L-l_{C}}{L + fl_{C}}n_{MT} = \rho_{C}l_{C}\sigma_{C}[/math]


and

[math]n_{4He}^{\prime} = \frac{(1 + f)l_{He}}{L + fl_{C}}n_{MT} - \frac{fl_{He}}{L + fl_{C}}n_{C} = \rho_{He}l_{He}\sigma_{He}[/math],


where [math]n_{12C}^{\prime}[/math] is the rate from the Carbon nucleus only and [math]n_{He}^{\prime}[/math] is the rate from liquid Helium only. The length and densities are listed in Table~\ref{tab:LengthandDensityTarget}.
We need to establish how the rate from the Carbon target is related to the rate from the N[math]^{15}[/math] in NH[math]_{3}[/math] and ND[math]_{3}[/math]. The cross section for N[math]^{15}[/math] can be written in terms of the cross sections on C[math]^{12}[/math] target material ([math]\sigma_{12C}[/math]) and on a bound neutron in N[math]^{15}[/math] ([math]\sigma_{n}^{\prime}[/math]).

[math]\sigma_{15N} \approx \frac{7}{6}\sigma_{12C} + \sigma_{n}^{\prime} = \left(\frac{7}{6} + \frac{\sigma_{n}^{\prime}}{\sigma_{12C}}\right)\sigma_{12C}[/math].


It is assumed that when the scattering occurs on protons inside the target material, the ratio of [math]\frac{\sigma_{n}^{\prime}}{\sigma_{12C}} = 0[/math] and when the reaction happens on neutrons inside the target, the value of the ratio is [math]\frac{\sigma_{n}^{\prime}}{\sigma_{12C}} = \frac{1}{6}[/math], because there are six bound neutrons in C[math]^{12}[/math] \cite{Khun2010}.
Using above quantities the background represented by the number of counts due to the non-hydrogen and non-deuterium parts of the ammonia target can be expressed as

[math]n_{B} = \left[ \frac{\rho_{A}l_{A}}{\rho_{C}l_{C}}\left(\frac{7}{6} + \frac{\sigma_{n}^{\prime}}{\sigma_{12C}}\right) + f \right] n_{12C}^{\prime} + (L - l_{A})n_{4He}^{\prime}[/math]


[math]= n_{MT} + l_{A} \left[ \frac{\rho_{A}}{\rho_{C}l_{C}}\left(\frac{7}{6} + \frac{\sigma_{n}^{\prime}}{\sigma_{12C}}\right)n_{12C}^{\prime} - n_{4He}^{\prime}\right][/math].


The dilution factor [math]d_{f}[/math] is

[math]d_{f} = \frac{n_{A} - n_{B}}{n_{A}}[/math].


Item Description Value
[math]\rho_{F} l_{F}[/math] Density times target length for empty target. Al: 167 [math]\mu[/math]m; 0.045 g/cm[math]^{2}[/math]. Kapton: 384 [math]\mu[/math]m; 0.055 g/cm[math]^{2}[/math]. Total=Al + Kapton=0.0996 g/cm[math]^{2}[/math]
[math]\rho_{C} l_{C}[/math] Density times target length for Carbon target. 0.498 g/cm[math]^{2}[/math]
f The ratio of counts from foils to the C[math]^{12}[/math] slab in the Carbon target. [math]0.200[/math]
[math]\rho_{He}[/math] He density. 0.145 g/cm[math]^{3}[/math]
[math]L[/math] The length of the target cell from the entrance to exit foil. 1.90 cm
[math]\rho_{C}[/math] C[math]^{12}[/math] density. 2.17 g/cm[math]^{3}[/math]
[math]l_{C}[/math] Carbon target length. 0.23 cm
[math]\rho_{NH_{3}}[/math] NH[math]_{3}[/math] density 0.917 g/cm[math]^{3}[/math]
[math]\rho_{ND_{3}}[/math] ND[math]_{3}[/math] density 1.056 g/cm[math]^{3}[/math]
[math]l_{A}[/math] Ammonia target length. 0.6 cm


Length and density values for different types of target material reproduced from the EG1b experiment \cite{Khun2010}.}


The fractional energy of the observed final state hadron(z) Dilution Factor([math]d_{f}[/math])
NH[math]_{3}[/math], [math]\pi^{+}[/math] && [math]z=0.4[/math] [math]0.160 \pm 0.02[/math]
NH[math]_{3}[/math], [math]\pi^{+}[/math] && [math]z=0.7[/math] [math]0.152 \pm 0.03[/math]


Calculated dilution Factor for NH[math]_{3}[/math] target type.


The dilution factor is compared below in Table~\ref{tab:CompareDilution}:

Reaction Dilution Factor([math]d_{f}[/math])
Resonance region 0.11 - 0.13
Inclusive 0.14 - 0.17
Semi-Inclusive 0.122 - 0.182


{Dilution Factor compared with other results \cite{Prok2004} \cite{Fersch2008}.


Fragmentation [math]\Delta R_{np}^{\pi^{+} + \pi^{-}}[/math]

A test of fragmentation can be performed by calculating the ratio of the difference of polarized to unpolarized cross sections for proton and neutron targets [math]\Delta R_{np}^{\pi^{+} + \pi^{-}}[/math] and showing that it is independent of [math]z[/math]. The fragmentation function can be written following way

[math]\Delta R_{np}^{\pi^{+} + \pi^{-}}(x, z, Q^{2}) = \frac{\Delta \sigma_{p}^{\pi^{+} + \pi^{-}} - \Delta \sigma_{n}^{\pi^{+} + \pi^{-}}}{\sigma_{p}^{\pi^{+} + \pi^{-}} - \sigma_{n}^{\pi^{+} + \pi^{-}}}[/math]


= \frac{\Delta \sigma_{p}^{\pi^{+} + \pi^{-}}}{\sigma_{p}^{\pi^{+} + \pi^{-}} - \sigma_{n}^{\pi^{+} + \pi^{-}}} - \frac{\Delta \sigma_{n}^{\pi^{+} + \pi^{-}}}{\sigma_{p}^{\pi^{+} + \pi^{-}} - \sigma_{n}^{\pi^{+} + \pi^{-}}}


[math]= \frac{(\Delta u +\Delta \bar{u}) - (\Delta d + \Delta \bar{d})}{(u+\bar{u}) - (d+\bar{d})}(x,Q^{2})[/math]


[math]= \frac{g_{1}^{p} - g_{1}^{n}}{F_{1}^{p} - F_{1}^{n}}(x,Q^{2})[/math].


The left side of the fragmentation function is calculated using the measured double spin asymmetries for the ammonia targets. However, the right side can be extracted using the model.

[math]x_{B}[/math] versus [math]\Delta R_{np}^{\pi^{+} + \pi^{-}}[/math]. Black data points represent the Model, red and green data points represent fragmentation function for [math]z=0.4[/math] and [math]z=0.7[/math] respectively.


[math]z[/math] [math]x_{B}=0.3[/math] [math]x_{B}=0.4[/math]
0.4 14 % 12 %
0.7 21 % 32 %


Statistical Z - test for the data comparison with the model. The probability of not observing the data point for each [math]z[/math] and [math]x_{B}[/math] values within the Inclusive Model.


The pion asymmetries, [math]A_{NH3}^{\pi^-}[/math], [math]A_{{ND3}^{\pi^+}[/math], [math]A_{ND3}^{\pi^-}[/math] are statistically consistent with zero as shown in Table~\ref{tab:SIDISAsymx03x04}. As a result, the contributions from [math]\Delta \sigma_{p}^{\pi^{-}}[/math], [math]\Delta \sigma_{n}^{\pi^{+}}[/math] and [math]\Delta \sigma_{n}^{\pi^{-}}[/math] to the fragmentation function test (Eq. 1.54) are negligible. Only the first term is non-zero in [math]\Delta R_{np}^{\pi^{+} + \pi^{-}}[/math]. After correcting the cross section difference for the target polarization, beam polarization, and dilution factor, the [math]\Delta R_{np}^{\pi^{+} + \pi^{-}}[/math] function for two values of [math]z[/math] and [math]x_{B}[/math] is shown on Figure~\ref{fig:xBvsDelta} and is compared to the Model.


Systematic errors

Systematic errors from the experimental setup and the cuts used for particle identification and background elimination were estimated. The systematic errors associated with the electromagnetic cuts and fiducial cuts for electron identification were calculated by comparing the SIDIS asymmetries before and after cuts were applied. Systematic effects related to the dilution factor and polarization were estimated following way: First, the SIDIS asymmetries were calculated for the standard values of the dilution factor and polarization. Then asymmetries were recalculated by changing the value of the each parameter by the amount of its uncertainty. The difference between these two values is the systematic effect. The systematic errors for the SIDIS asymmetries are shown in Table~\ref{tab:SystematicErrors}.

Source Systematic Error
Geometrical and timing cuts [math]5.80 \times 10^{-2}[/math]
Electromagnetic calorimeter cut [math]4.07 \times 10^{-2}[/math]
Dilution Factor [math]5.03 \times 10^{-2}[/math]
Target and beam polarization [math]1.10 \times 10^{-2}[/math]
Total [math] 8.76 \times 10^{-2}[/math]


The systematic errors for the [math]A_{NH_{3}}^{\pi^+}[/math] asymmetry.


Delta_D_over_D

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