b_{1} technical note 2013-05 May 2013

Rates and Statistical Uncertainty Calculations for a Measurement of A_{zz}

E. Long (UNH), O. Rondon (INPP-UVA), P. Solvignon (Jefferson Lab)

Abstract

A proposal on measuring the deuteron structure function b_{1}^{d} was submitted to PAC 40. The asymmetry A_{zz} will be used to extract b_{1}^{d}. This tech note provides details on the calculations of the rates and statistical uncertainties completed for the proposal.

1 Method of Measuring A_{zz}

The deuteron cross section can be utilized to extract the tensor asymmetry, A_{zz}. The cross section takes the form

\frac{d^{2}σ_{D}}{dΩdE′} = σ_{D} = σ_{D}^{u} [1 − P_{z}P_{B}A_{∥} + \frac{1}{2}P_{zz}A_{zz}^{d}] ,  (1)

where σ_{D}^{u} is the unpolarized deuteron cross section, P_{B} is the polarization of the electron beam, P_{z} is the vector polarization of the target, and P_{zz} is the tensor polarization of the target. In the case of an unpolarized beam, the middle term vanishes leaving just

σ_{D} = σ_{D}^{u} [1 + P_{zz}A_{zz}] .  (2)

For the proposed measurement of A_{zz}, we are looking in the DIS region. The general form of DIS electron scattering off of a nucleus is described by

\frac{d^{2}σ_{X}^{u}}{dΩdE′} = σ_{X}^{u} = A_{X} (\frac{dσ}{dΩ})_{Mott_{p}} [\frac{2 · (\frac{F_{1}^{X}}{A_{X}} )}{m_{p}} tan^{2} (\frac{θ_{e′}}{2} ) + \frac{(\frac{F_{2}^{X}}{A_{X}} )}{ν}] ,  (3)

where A_{X} is the number of nucleons, F_{1}^{X} and F_{2}^{X} are the nuclear structure functions, θ_{e′} is the electron scattering angle, m_{p} is the mass of a proton, ν is the energy transfer, and (\frac{dσ}{dΩ})_{Mott_{p}} is the Mott cross section of a proton.

In order to reduce systematic effects, a ratio of cross section is used to determine A_{zz}. This ratio, in terms of the measured counts with the target tensor-polarized and unpolarized, is expressed as

\frac{N_{P ol}}{N_{u}} − 1 = f \frac{1}{2}A_{zz}P_{zz},  (4)

where N_{P ol} is the number of events recorded when the target is polarized, N_{u} is the number of events recorded when the target is unpolarized, and a dilution factor, f, is included since the target material will contain nitrogen and helium as well as deuterium. Thus, A_{zz} is measured through

A_{zz} = \frac{2}{f · P_{zz}} (\frac{N_{P ol}}{N_{u}} − 1) .  (5)

2 Rates

2.1 General Expressions

The total rates for ND3 are:

R_{Total} = A [L_{He}σ_{He} + L_{N}σ_{N} + L_{D}σ_{D}]  (6)
= A [L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u} (1 + \frac{1}{2}P_{zz}A_{zz}^{d})]  (7)

where A is the acceptance (ΔΩΔE′), σ_{A}^{u} is the unpolarized cross section of the given nucleus, and L_{A} is the luminosity. The general form of the luminosity is

L_{A} = N_{e} · N \frac{ρ_{A}}{M_{A}} z_{A}p_{f_{A}},  (8)

where N_{e} = I_{beam}/e is the rate of incident electrons, N is Avogadro’s number, ρ_{A} is the density, M_{A} is the atomic or molecular mass, z is the target thickness, and p_{f_{A}} is the packing fraction of the material. Since we are using solid deuterated ammonia beads surrounded by liquid helium, the luminosities come out to

L_{He} = [N \frac{ρ_{He}}{M_{He}} (1 − p_{f})] · (\frac{I_{beam}}{e} ) · z_{tgt},  (9)
L_{N} = [N \frac{ρ_{ND_{3}}}{M_{ND_{3}}} p_{f} ] · (\frac{I_{beam}}{e} ) · z_{tgt}, and  (10)
L_{D} = 3 [N \frac{ρ_{ND_{3}}}{M_{ND_{3}}} p_{f} ] · (\frac{I_{beam}}{e} ) · z_{tgt}.  (11)

The factor of 3 in the expression of L_{D} takes into account that there are three deuterium atoms in the ammonia molecule.

We will be detecting events from two different target states: polarized and unpolarized. The total rate when the target is tensor-polarized is written as

R_{Total}^{P ol} = A [L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u} (1 + \frac{1}{2}P_{zz}A_{zz}^{d})]  (12)

and when the target is unpolarized,

R_{Total}^{u} = A [L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u}] .  (13)

The total number of counts in each state would depend on the amount of time spent in each, such that

N_{P ol} = R_{Total}^{P ol}t_{P ol} and  (14)
N_{u} = R_{Total}^{u}t_{u}.  (15)

2.2 Expression of the Measured Asymmetry

From Refs. [1] and [2], the enhancement of the tensor polarization with solid polarized targets can be done via the ”hole burning“ method by pushing down either one of the |m_{z}| = 1 states moving its population to m_{z} = 0. But this necessarily enhances the absolute vector polarization, |m_{+} − m_{−}| because one of the m_{1}’s stays fixed. So, as it is said in Ref. [1], the improvement in P_{zz} comes only from better P_{z}.

In order to reduce systematic uncertainties, we find A_{zz} by measuring the ratio of events detected when the target is polarized with P_{zz} > 0 and when it is unpolarized. In terms of the rates for each state, this ratio is written

\frac{N_{P ol}}{N_{u}} = \frac{A [L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u} (1 + \frac{1}{2}P_{zz}A_{zz}^{d})] t_{P ol}}{A [L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u}] t_{u}} ,  (16)

\frac{N_{P ol}}{N_{u}} = (\frac{t_{P ol}}{t_{u}} ) [1 + (\frac{L_{D}σ_{D}^{u}}{L_{He}σ_{He}^{u} + L_{N}σ_{N}^{u} + L_{D}σ_{D}^{u}} ) \frac{1}{2}A_{zz}P_{zz}] .  (17)

The luminosity terms make up the dilution factor, such that f = \frac{L_{D}σ_{D}^{u}}{L_{He}σ_{He}^{u}+L_{N}σ_{N}^{u}+L_{D}σ_{D}^{u}} and

\frac{N_{P ol}}{N_{u}} = (\frac{t_{P ol}}{t_{u}} ) [1 + f \frac{1}{2}A_{zz}P_{zz}] .  (18)

If equal time is spent on each polarization state such that t_{P ol} ≈ t_{u}, then

\frac{N_{P ol}}{N_{u}} = 1 + f \frac{1}{2}A_{zz}P_{zz}.  (19)

This can be split up into a measured ratio, A_{meas}, defined as

A_{meas} = \frac{N_{P ol}}{N_{u}} − 1.  (20)

From this, A_{zz} can be extracted by

A_{zz} = \frac{2}{f · P_{zz}} A_{meas}.  (21)

Table 1: Values used in the rate estimates.

ρ_{ND_{3}} 1.007 g.cm^{−3}
M_{ND_{3}}  20 g.mol^{−1}
p_{f}(ND_{3})  0.65
z_{tgt}  3 cm
P_{zz}  0.25

3 Statistical Uncertainty

Given the form of A_{zz} in terms of A_{meas},

A_{zz} = \frac{2}{f · P_{zz}} A_{meas},  (22)

we can find the uncertainty from

δA_{zz} = \sqrt{(\frac{∂A_{zz}}{∂A_{meas}} δA_{meas})^{2} + (\frac{∂A_{zz}}{∂f} δf)^{2} + (\frac{∂A_{zz}}{∂P_{zz}} δP_{zz})^{2}}.  (23)

In this case, we can break it down into statistical and systematic uncertainties,

δA_{zz} = \sqrt{(δA_{zz}^{Stat})^{2} + (δA_{zz}^{Sys})^{2}}.  (24)

The statistical component relates as

δA_{zz}^{Stat} = \frac{∂A_{zz}}{∂A_{meas}} δA_{meas} = \frac{2}{f · P_{zz}} δA_{meas}.  (25)

Since A_{meas} = \frac{N_{P ol}}{N_{u}} − 1, the uncertainty in A_{meas} depends on the number of events counted in each state as described by

δA_{meas} = \sqrt{(\frac{∂A_{meas}}{∂N_{P ol}} δN_{P ol})^{2} + (\frac{∂A_{meas}}{∂N_{u}} δN_{u})^{2}},  (26)
δA_{meas} = \sqrt{(\frac{1}{N_{u}} \sqrt{N_{P ol}})^{2} + (−\frac{N_{P ol}}{N_{u}^{2}} \sqrt{N_{u}})^{2}},  (27)
δA_{meas} = \sqrt{\frac{N_{P ol}}{N_{u}^{2}} + \frac{N_{P ol}^{2}}{N_{u}^{3}}} .  (28)

Because A_{zz} is very small, we can assume that N_{P ol} ≈ N_{u} ≈ \frac{N}{2} , then

δA_{meas} = \sqrt{\frac{N/2}{N^{2}/4} + \frac{N^{2}/4}{N^{3}/8}},  (29)

δA_{meas} = \frac{2}{\sqrt{N}} .  (30)

Equation (30) can be put back into equation (24) to get

δA_{zz}^{Stat} = \frac{2}{f · P_{zz}} (\frac{2}{\sqrt{N}} ) ,  (31)

δA_{zz}^{Stat} = \frac{4}{f · P_{zz}\sqrt{t · R_{Total}}} .  (32)

4 Kinematics

In order to determine the kinematics to be used, a balance was struck between the physical constraints of the detector, increasing the rate by moving to a lower angle, and maximizing the number of events occurring from DIS. The kinematics selected for this technote are described in Table 2.

Table 2: Central values of the kinematics chosen for the b_{1} proposal.

Detector x Q^{2} W E_{e′} θ_{e′}  θ_{q}
SHMS 0.15 1.21 2.78 6.70 7.35 11.13
SHMS 0.30 2.00 2.36 7.45 8.96 17.66
SHMS 0.45 2.58 2.00 7.96 9.85 23.31
HMS 0.55 3.81 2.00 7.31 12.50 22.26

Estimates for the total kinematic range were calculated in a FORTRAN code, discussed in Section 5, which looked at the entire acceptance range of the HMS and SHMS. These ranges are displayed in Figure 1.

Figure 1: Kinematics chosen for the HMS and SHMS. The darker colored points indicate the central HMS or SHMS value, the red (blue) dotted lines represent the limits of the central value of the HMS (SHMS), and the light grey represent events that will be cut to ensure that DIS is held by maintaining W > 1.85 GeV, shown as a solid black line.

Figure 2: The calculated rates are presented alongside the amount of time, assuming 100% efficiency, that was used to estimate the uncertainty in A_{zz} and b_{1}^{d}.

5 Calculated Uncertainties

Code was written by P. Solvignon and E. Long to automate the method described in Sections 2 and 3 using the kinematics discussed in Section 4. The FORTRAN code used is available at

http://nuclear.unh.edu/~elong/analysis_files/2013-05-09/b1_rates-2013-05-09.tar.

gz. It takes the weighted cross-sections for each kinematics setting and uses it to estimate both the rates and the statistical uncertainty in A_{zz}. The rates and the amount of time spent at each setting, assuming 100% efficiency, are presented in Figure 2.

Table 3: Contributions to systematic uncertainty in A_{zz}.

Source  Relative Uncertainty
Polarimetry  8.0%
Dilution/Packing Fraction  4.0%
Radiative Corrections  1.5%
Charge Determination  1.0%
Detector Resolution and Efficiency  1.0%
Total  9.2%

Since many of the spectrometer settings overlap in x, as shown in Figure 1, the number of events in each x bin were integrated so that a distinct measurement of A_{zz} against x could be made. The calculated uncertainty in A_{zz} is shown in Figure 3 along with a band representing the systematic uncertainties. Contributions to the systematic uncertainty are given in Table 3. An additional absolute uncertainty due to charge and efficiency drift,

δA_{zz}^{drift} = ±\frac{2}{fP_{zz}} δξ,  (33)

is included in the systematic estimate, where δξ = δQ + δϵ. Table 4 lists δA_{zz}^{drift} for each spectrometer setting. It was included in the systematics as an average weighted by the number of events each setting contributes to a particular x bin. Using the Kumano fit, the total uncertainties are calculated as shown in Table 5.

Table 4: Estimated in A_{zz} caused by charge and efficiency drift. Details on these estimates can be found in [3].

\begin{matrix}⟨x⟩ & δA_{zz}^{drift}(×10^{−3}) \\ 0.15 & 4.6 \\ 0.30 & 3.7 \\ 0.45 & 2.8 \\ 0.55 & 2.1 \end{matrix}

Table 5: Calculated uncertainties in A_{zz}. The systematic uncertainties are based on A_{zz} from the Kumano fit at each ⟨x⟩.

\begin{matrix}⟨x⟩ & A_{zz}^{Kumano} ± δA_{zz}^{stat} ± δA_{zz}^{sys}(×10^{−3}) \\ 0.16 & −4.39 ± 1.98 ± 4.53 \\ 0.28 & 3.40 ± 2.90 ± 3.88 \\ 0.36 & 7.97 ± 3.19 ± 3.28 \\ 0.49 & 13.6 ± 3.00 ± 2.96 \end{matrix}

Additionally, the code estimates the uncertainty on b_{1}^{d}, which is extracted from A_{zz} by

b_{1}^{d} = −\frac{3}{2}A_{zz} (\frac{F_{1}^{d}}{A_{D}} ) = −\frac{3}{2}A_{zz} (\frac{F_{1}^{d}}{2} ) .  (34)

The uncertainty on b_{1}^{d} is dependent on the uncertainty in A_{zz} such that

δb_{1}^{d} = \sqrt{[−\frac{3}{2} (\frac{F_{1}^{d}}{2} ) δA_{zz}]^{2}}.  (35)

This uncertainty was also calculated, based on the uncertainty on A_{zz}, and is show in Table 6 and Figure 4.

Figure 3: The estimated uncertainty in A_{zz} is plotted against previous data by HERMES and a number of theoretical calculations.

Table 6: Calculated uncertainties in b_{1}^{d}. The systematic uncertainties are based on b_{1}^{d} from the Kumano fit at each ⟨x⟩.

\begin{matrix}⟨x⟩ & b_{1}^{Kumano} ± δb_{1}^{stat} ± δb_{1}^{sys}(×10^{−3}) \\ 0.16 & 5.81 ± 2.46 ± 5.61 \\ 0.28 & −2.54 ± 2.03 ± 2.72 \\ 0.36 & −3.79 ± 1.47 ± 1.51 \\ 0.49 & −2.73 ± 0.68 ± 0.67 \end{matrix}

Figure 4: The estimated uncertainty in b_{1}^{d} is plotted against previous data by HERMES and a number of theoretical calculations.

References

[1] T.W. Meyer and E.P. Schilling, Tensor polarized deuteron targets for intermediate energy physics experiments, BONN-HE-85-06 (1985)

[2] S. Bueltmann, D. Crabb, Y. Prok. UVa Target Studies, UVa Polarized Target Lab technical note, 1999.

[3] D. Keller, et al. The b_{1} Collaboration Response to TAC and iTAC Comments, b1 technical note, 2013.