Tamar AnalysisChapt

Data Analysis

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] using NH3 and ND3 polarized targets respectively. The goal of this work is to measure the charged pion asymmetries defined according to the incident electron helicity and the target polarization. The measured asymmetries are used to evaluate the fragmentation function dependence 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.

[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]

The CLAS Data Selection

The data files from the EG1b experiment 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 to probe the polarized frozen ammonia NH3 and ND3 targets. This work will discuss the analysis of the 4.2 GeV energy electron beam data set on hydrogen and deuteron targets as this energy provided the most statistics. The collected data have been tested by applying restrictions discussed in section XXX below.

Run Set	Target Type	Torus Current(A)	Target Polarization	Half Wave Plane(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 thesis (Run Sets, Target Type, Torus Current, Target Polarization, HWP).

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, which can be achieved with geometrical and timing cuts on the electron sample. In addition, the cuts has been applied onto the energy deposited into the calorimeter and number of photoelectrons produced in the Cherenkov counter. Whereas, charged pions are identified by matching the hits in the drift chamber and TOF counter. a coincidence hit in the drift chamber and TOF counter

Electron Identification

The CLAS trigger system is a coincidence of the negatively charged particle track detected in the electromagnetic calorimeter and cherenkov counter in [math]150 ns[/math] time window Fig. 2.1. Unfortunately, a background of high energy negative pions may be misidentified as electrons. These pions are mis-identified as electrons due to quasi-real photoproduction, when the polar angle of the scattered lepton is approximately zero and is not accounted by the CLAS detector. 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. The pion background is further supressed 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 [math]13[/math] layers of lead-scintillator sandwichs composed of [math]~2mm[/math] thick lead and [math]10 mm[/math] thick scintillator. 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.
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. Fig. 2.2 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.
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. As a result, the energy deposited into the electromagnetic calorimeter is different for electrons and pions. Pions loose about [math]0.08 GeV[/math] energy traversing the calorimeter independent their momentum thereby producing the constant signal in the calorimeter around [math]0.08 GeV[/math]. 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] vs [math]EC_{tot}/p[/math] before and after EC cuts ([math]EC_{tot}\gt 0.2p[/math], for EC inner - [math]EC_{inner}\gt 0.08p[/math]). After applying EC cuts about 46 % 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 anti-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] - 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.
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>

[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]

After deriving the Taylor expansion of our function and considering 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 [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. 2.3.

Electrons.

Pions.

Figure 2.3 Theoretical Calculation of the Number of Photoelectrons for Electrons and Pions.

The number of photoelectron distribution measured in the cherenkov detector and the energy deposition dependence on number of photoelectrons are shown on Fig. 2.4 and Fig. 2.5. One can see, that a single photoelectron peak is caused by the misidentified pions as electrons.

Figure 2.4 The number of photoelectrons without cuts.

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

Geometric and TIming cuts

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

[math]|\theta_p - {\theta_p}^{center} - {\theta_p}^{offset}| \lt 3 \sigma_p[/math]

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.
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>:

[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]

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\gt 2.5[/math] cut contamination is about [math]4.029% \pm 0.003[/math](Fig. 2.6).

Before Cuts.

After OSI Cuts.

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

Pion Identification

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

[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 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 Fig. 2.8. The pion mass band is around [math]~0[/math]. To isolate charged pions from the rest of the particles a [math]3\sigma[/math] on [math]\beta[/math] vs momentum has been applied<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>.

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

Figure 2.8 The charged particle momentum vs mass squared distribution for the [math]\overrightarrow{e}p\rightarrow \overrightarrow{e}^{\prime} \pi^{+} n [/math] electroproduction process. The bands around [math]0[/math] and [math]1[/math] 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 of the charged particle velocity([math]\beta[/math]), the fiducial volume cuts has 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 EG1b experiment, pions were detected from [math]8^{\circ}[/math] to [math]180^{\circ}[/math]<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>. The pion identification code has been developed by Josh Pierce<ref name="PionCodeJosh"> Joshua Pierce. Pion Identification. From EG1 Hall-B http://www.jlab.org/Hall-B/secure/eg1/EG2000/josh/pion.cc </ref>

Specific 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. 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 efficiency will need to be evaluated in terms of the electron rate observed in two different scintillator paddles detecting the same electron kinematics.

Inclusive Event Reconstruction Efficiency

The electron efficiency of individual scintillator detectors using the 4.2 GeV data for ND3 and NH3 targets is investigated below. Only the electron detection is required in the final state (inclusive case). The pion contamination in the electron sample was removed by applying 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 Fig. 2.9.

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 2.9. Electron Kinematics.

Ratios of the inclusive electron rate, normalized using the gated faraday cup, of scintillator paddles 5 and 10 were measured. The two ratios are constructed to quantify the CLAS detectors ability to observe the same electron kinematics in scintillator paddle #5 for a positive Torus polarity and scintillator #10 using the negatie 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 ND3 and and NH3 target. Below is the observed ratio comparing the inclusive electrons observed in scintillator #5 for a positive torus polarity and ND3 target to the electrons observed in scintillator #10 when the torus polarity is negative and the target is NH3.

[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{ND3,B\gt 0,E_{PaddleNumber}=5}{NH3,B\lt 0,E_{PaddleNumber}=10} = 0.645[/math] and [math]\frac{ND3,B\lt 0,E_{PaddleNumber}=10}{NH3,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 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.
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 on Fig. 2.10. 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 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.

Figure 2.10. Pion paddle number vs 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: [math]1.7\lt W\lt 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>.

[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 pion azimuthal angle in CM frame, [math]\epsilon = (1 + 2(1 + \frac{\nu^2}{Q^2})tan^2\frac{\theta_e}{2})^{-1}[/math] - 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] - Four momentum transferred square, [math]\theta_e[/math] - electron scattering angle and [math]h[/math] - 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 on Fig. 2.11. One can conclude from the Fig. 3.3 that the inclusive corrections do not impact a single pion production for the exclusive cases.

Pion Paddle Number vs MAID2007 - Experiment([math]N(\pi^{-},ND3)/N(\pi^{+},NH3)[/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. 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 Fig. 2.12 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.

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.

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 Fig. 2.13 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(Fig. 2.14).

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

EG1b runs, 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 4.1.

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 vs Beam Charge Asymmetry.

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. 2.15 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 reconstructed electrons in the final state for the positive(negative) beam helicity.

Figure 2.15. Run Number vs Electron Asymmetry before FC normalization. The black and red points represent reconstructed electron asymmetry for the helicity 1-4 pair for ND3 and NH3 target respectively. The blue and green points - the helicity pair 2-3 for ND3 and NH3 respectively. The green line shows the sign of the half wave plane(HWP) and the purple line - 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 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 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, the electron asymmetry sign ( sign(hel1-hel4) && sign(hel3-hel2) ) is opposite of the sign of (HWP[math]\times[/math]Target_Polarization). The reconstructed electron asymmetry has been calculated the following way before and after accounting for the Faraday cup:

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

and

[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]

Figure 2.16. Run Number vs Electron Asymmetry after applying FC normalization. The black and red points represent reconstructed electron asymmetry for the helicity 1-4 pair for ND3 and NH3 target respectively. The blue and green points - the helicity pair 2-3 for ND3 and NH3 respectively. The green line shows the sign of the half wave plane(HWP) and the purple line - the sign of the target polarization(TPol).

Semi-Inclusive Asymmetries

The ratio of the combined semi-inclusive deep inelastic asymmetries ([math]A_{hel13-hel42}[/math]) from the Table. 4.3 , for two different torus settings have been corrected for the inclusive rate corrections. The Fig. 4.7 represents asymmetry ratios before and after inclusive rate corrections. The ratios have been calculated for each target and charged pion type. SIDIS asymmetries are statistically the same. The results suggest inclusive rate corrections do not make significant change on the 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 electron spin is parallel and antiparallel to the spin of the nucleon. There are totally four combinations of semi inclusive asymmetries: [math]A_{NH3}^{\pi^+}[/math], [math]A_{NH3}^{\pi^-}[/math], [math]A_{ND3}^{\pi^+}[/math] and [math]A_{ND3}^{\pi^-}[/math]

[math]A_{NH3}^{\pi^+}=\frac{N_{NH3,\pi^+}^{hel13} - N_{NH3, \pi^+}^{hel24}}{N_{NH3,\pi^+}^{hel13} + N_{NH3, \pi^+}^{hel24}}[/math]

[math]A_{NH3}^{\pi^-}=\frac{N_{NH3,\pi^-}^{hel13} - N_{NH3, \pi^-}^{hel24}}{N_{NH3,\pi^-}^{hel13} + N_{NH3, \pi^-}^{hel24}}[/math]

[math]A_{ND3}^{\pi^+}=\frac{N_{ND3,\pi^+}^{hel13} - N_{ND3, \pi^+}^{hel24}}{N_{ND3,\pi^+}^{hel13} + N_{ND3, \pi^+}^{hel24}}[/math]

[math]A_{ND3}^{\pi^-}=\frac{N_{ND3,\pi^-}^{hel13} - N_{ND3, \pi^-}^{hel24}}{N_{ND3,\pi^-}^{hel13} + N_{ND3, \pi^-}^{hel24}}[/math]

where [math]N_{NH3,\pi^+,\pi^-}^{hel13}[/math]([math]N_{NH3,\pi^+,\pi^-}^{hel24}[/math]) and [math]N_{ND3,\pi^+,\pi^-}^{hel13}[/math]([math]N_{ND3,\pi^+,\pi^-}^{hel24}[/math]) represent the number of [math]\pi^{+}[/math] and [math]\pi^{-}[/math] hadron type 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 kinematics for the events used in the measured asymmetries are

Figure . Kinematic plots(Invariant Mass(W), [math]Q^{2}[/math] and Missing Mass([math]M_{X}[/math]))

The semi inclusive asymmetries are listed in Table 4.3 (Fig. 2.18). The first group [math]A_{hel1-hel4}[/math] represents asymmetry when the first original electron spin(1) is antiparallel to the target nucleon spin and its complement state(hel4) is parallel, whereas the second group [math]A_{hel2-hel3}[/math] represents measured SIDIS asymmetry when the first helicity state is parallel(hel2) and its complement state is antiparallel to the spin of the nucleon. In order to decrease statistical errors due to small number of events, the two groups have been combined into one [math]A_{hel13-hel42}[/math] set. Instead of looking at original and compliment electron helicity states, they have been combined into the positive(hel24) and negative(he13) helicity states, which are parallel and antiparallel to the spin of the target nucleon respectively.

Target type, Beam Torus sign (B)	[math]A_{hel1-hel4}[/math]	[math]A_{hel2-hel3}[/math]	[math]A_{hel13-hel42}[/math]
NH3, B>0, [math]\pi^+[/math]	[math](-139.84 \pm 81.52) \times 10^{-4}[/math]	[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](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](-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](-30.34 \pm 6085.54) \times 10^{-4}[/math]	[math] (-12.37 \pm 71.10) \times 10^{-4}[/math]
NH3, B>0, [math]\pi^-[/math]	[math](-155.45 \pm 128.21) \times 10^{-4}[/math]	[math](-72.55 \pm 128.92) \times 10^{-4}[/math]	[math](-35.11 \pm 90.91) \times 10^{-4}[/math]
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} [/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]	[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]	[math](123.98 \pm 106.86) \times 10^{-4}[/math]	[math] (-92.25 \pm 85.38) \times 10^{-4}[/math]

Table 4.3. Run Number vs SIDIS Asymmetry for Each Type Target material and Beam Torus.

The ratio of the combined semi-inclusive deep inelastic asymmetries ([math]A_{hel13-hel42}[/math]) from the Table. 4.3 , for two different torus settings have been corrected for the inclusive rate corrections. The Fig. 4.7 represents asymmetry ratios before and after inclusive rate corrections. The ratios have been calculated for each target and charged pion type. SIDIS asymmetries are statistically the same. The results suggest inclusive rate corrections do not make significant change on the asymmetries.

Figure 2.18. The ratio of the SIDIS asymmetries for two torus field settings [math]\frac{A_{hel13-hel42}(B\gt 0)}{A_{hel13-hel42}(B\lt 0)}[/math] vs target and the charged pion type. The black squares represent the data before inclusive rate corrections and the red data - after inclusive rate corrections has been applied.

The data has been combined for each target type and asymmetries were measured for two [math]x_{B}[/math] values(Table 4.4). 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 [math]2.55\lt P_{e}\lt 2.75[/math], invariant mass[math]1.6\lt W\lt 1.8[/math] and four momentum transferred square [math]0.9\lt Q^{2}\lt 1.3[/math].

SIDI Asymmetry	[math]x_{B}=0.3[/math]	[math]x_{B}=0.4[/math]
[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] 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]
[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]
[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]

Table 4.4. 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 has been subdivided for the fraction energy of the observed final state pion([math]z[/math]). [math] A_{NH3}^{\pi^+,hel13-hel24}[/math] asymmetries for three [math]z[/math] are shown in Table 4.5.

[math]z[/math]	[math]X_B=0.3[/math]	[math]X_B=0.4[/math]
0.45	[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]
0.9	[math](-355.44 \pm 200.23) \times 10^{-4}[/math]	[math](-438.76 \pm 189.03) \times 10^{-4}[/math]

Table 4.5. SIDIS Asymmetries vs [math]z[/math].

Dilution Factor

To exclude the contributions from the polarized nucleons of the non-hydrogen nuclei of ammonia target(nitrogen) and the cooling material of the target(helium) cell in semi inclusive channels, the measured raw double spin asymmetries have to be divided by the so called dilution factor. The dilution factor accounts for the fraction of all events coming from the desired polarized target nucleon. The dilution factors are calculated by combining the data from the runs for different target type. During EG1b experiment, several runs were taken with carbon [math]C^{12}[/math] target material and empty target cell. All the runs in this thesis include liquid helium, that has been used 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. <ref name="Robfersch2004"> Fersch, R. G. (2008). Measurement of Inclusive Proton Double Spin Asymmetries and polarized Structure Functions. Doctoral dissertation. The College of William and Mary, Williamsburg, VA.</ref>
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 of them are weighted by the corresponding gated faraday cup counts.<ref name="Kuhn2010"> Khun, S. E. (2010). Dilution Factor for Exclusive Channels. Old Dominion University, Norfolk, VA.</ref>

[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]

Using above quantities 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.

[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}F[/math]

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

where

[math]f = \frac{\rho_{F}l_{F}\sigma_{F}}{\rho_{C}l_{C}\sigma_{C}}[/math]

We need to establish how the cross section from the carbon target is related to the cross from the [math]N^{15}[/math] in ammonia. The cross section for [math]N^{15}[/math] can be written in terms of the cross sections on [math]C^{12}[/math] target material ([math]\sigma_{12C}[/math]) and on a bound neutron in [math]N^{15}[/math] ([math]\sigma_{n}^{\prime}[/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 neutron 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 [math]C^{12}[/math].<ref name="Kuhn2010"> Khun, S. E. (2010). Dilution Factor for Exclusive Channels. Old Dominion University, Norfolk, VA.</ref>

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

Using above quantities, the number of counts due to the non-hydrogen and non-deuterium parts of the ammonia target can be expressed as

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

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

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

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

Item	Description	Value
[math]\rho_{F} l_{F}[/math]	Density times target length for empty target.	Al: [math]167 \mu m[/math]; [math]0.045 g/cm^{2}[/math]. Kapton: [math]384 \mu m[/math]; [math]0.055 g/cm^{2}[/math]. [math]Total=Al + Kapton=0.0996 g/cm^{2}[/math]
[math]\rho_{C} l_{C}[/math]	Density times target length for Carbon target.	[math]0.498 g/cm^{2}[/math]
f	The ratio of counts from foils to the [math]C^{12}[/math] slab in the Carbon target.	[math]0.200[/math]
[math]\rho_{He}[/math]	[math]He[/math] density.	[math]0.145 g/cm^{3}[/math]
[math]L[/math]	The length of the target cell from the entrance to exit foil.	[math]1.90 cm[/math]
[math]\rho_{C}[/math]	[math]C^{12}[/math] density.	[math]2.17 g/cm^{3}[/math]
[math]l_{C}[/math]	Carbon target length.	[math]0.23 cm[/math]
[math]\rho_{NH3}[/math]	[math]NH3[/math] density	[math]0.917 g/cm^{3}[/math]
[math]\rho_{ND3}[/math]	[math]ND3[/math] density	[math]1.056 g/cm^{3}[/math]
[math]l_{A}[/math]	Ammonia target length.	[math]0.6 cm[/math]

Table 4.6. Length and density values for different types of target material reproduced from EG1b experiment.<ref name="Kuhn2010"> Khun, S. E. (2010). Dilution Factor for Exclusive Channels. Old Dominion University, Norfolk, VA.</ref>

The fractional energy of the observed final state hadron(z)	Dilution Factor([math]d_{f}[/math])
[math]NH3, \pi^{+}[/math] && [math]z=0.45[/math]	[math]0.160 \pm 0.07[/math]
[math]NH3, \pi^{+}[/math] && [math]z=0.85[/math]	[math]0.152 \pm 0.08[/math]

Table 4.7. Calculated dilution Factor for NH3 target type.

Radiative Corrections

In double spin asymmetry analysis the electron nucleon scattering process is given as an one photon exchange event, so called the Born approximation(Fig 2.19). 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.

Figure 2.19. The Born Approximation

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. 2.20.<ref name="RadiativeCorrections"> Nucleon Form Factors. In Scholarpedia, from http://www.scholarpedia.org/article/Nucleon_Form_factors#History </ref>

Vertex Photon Exchange	Vacuum Polarization
Self Energy	Self Energy

Figure 2.20 Internal Radiation

On the other hand, the external radiative corrections account for the Bremsstrahlung by the incoming and scattered electron and by the recoiling target nucleon Fig. 2.21.

Figure 2.21 Externall Radiation

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.

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 depends on [math]x_{B}[/math], whereas is independent of [math]z[/math]. The fragmentation function can be written following way

[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{\Delta \sigma_p^{\pi^+}}{\sigma_{p}^{\pi^+ + \pi^-} - \sigma_{n}^{\pi^+ + \pi^-}} + \frac{\Delta \sigma_p^{\pi^-}}{\sigma_p^{\pi^+ + \pi^-} - \sigma_{n}^{\pi^+ + \pi^-}} - \frac{\Delta \sigma_n^{\pi^+}}{\sigma_p^{\pi^+ + \pi^-} - \sigma_{n}^{\pi^+ + \pi^-}} - \frac{\Delta \sigma_n^{\pi^-}}{\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]

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. At the end the last two are compared.

Systematic errors

You need to show systematic errors for the asymmetry measurements

sources of such error are polarization, dilution factor, cuts,...

Results

insert a paragraph comparing the dilution factor(don't forget polarization dilution factor) corrected positive pion asymmetry with another measurement.

I think you max Z bin should be 0.8

In the fragmentation function contributions from the cross sections of [math]\Delta \sigma_p^{\pi^-}[/math], [math] \Delta \sigma_n^{\pi^+}[/math] and [math]\Delta \sigma_n^{\pi^-}[/math] are negligible. Only non zero in [math]\Delta R_{np}^{\pi^+ + \pi^-} [/math] is the first term. After correcting the cross section difference for the target polarization, beam polarization and dilution factor, the fragmentation function for two values of [math]z[/math] and [math]x_{B}[/math] is shown on Fig. 2.22 and is compared to the Model.

Figure 2.22 [math]x_{B}[/math] vs [math]\Delta R_{np}^{\pi^+ + \pi^-} [/math] Fragmentation function. Black data points represent the Model, red and green data points represent fragmentation function for [math]z=0.45[/math] and [math]z=0.85[/math] respectively.

Variables	Inclusive	z=0.45	z=0.85
Mean of [math]x_{B}[/math]	0.35	0.35	0.35
Mean of [math]\Delta R_{np}^{\pi^+ + \pi^-} [/math]	0.958	0.765	1.215
Standard Deviation	0.016	0.029	0.177
Slope	-0.22	0.41	-2.5
Intercept	1.04	0.62	2.09

Table 4.8. Statistical test for the data comparison with the model.

Notes

<references/>

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