Elong-15-05-07

Going back through the literature, the NIKHEF measurements (for instance the Bouwhuis measurement) define the polarized D(e,e'd) cross section as

$$\sigma = \sigma_0\left[ 1 + \frac{A_d^T P_{zz}}{\sqrt{2}} \right]$$

where

$$ A^T_d = \sum_{i=0}^{2}d_{2i}T_{2i}$$

and

$$ d_{20} = \frac{3 \cos^2 \theta^* -1}{2},~~d_{21} = -\sqrt{\frac{3}{2}}\sin2\theta^*\cos\phi^*,~~d_{22}=\sqrt{\frac{3}{2}}\sin^2\theta^*\cos 2\phi^*$$

where $\theta^*$ and $\phi^*$ are in the frame where the $z$ axis is along the $\vec{q}$ (which for us is ~70deg from the beamline) and the $x$ axis is perpendicular to $z$ in the scattering plane. They take the same definitions given by Donnely and Raskin:

If I'm understanding that correctly, then for us $\theta^* = \theta_q = 70^{\circ}$ and $\phi^* = 0^{\circ}$ since our magnetic field will be oriented along the beamline.

If that's the case, the $d_{21}\approx -0.787,~~d_{22}\approx 1.08~~$ and $d_{20} \approx -0.3245$.

Rearranging their top equation to get to what we measure as $A_{zz} = \frac{2}{P_{zz}}\left( \frac{\sigma}{\sigma_0} - 1 \right)$,

$A_{zz} = \sqrt{2}\left[ \frac{\sqrt{2}}{P_{zz}}\left( \frac{\sigma}{\sigma_0}-1\right) \right] = \left[ (-0.3245) T_{20} -0.787 T_{21} + 1.08 T_{22}\right] \sqrt{2}$

$ A_{zz} = -0.458 T_{20} - 1.113T_{21} + 1.53T_{22}$.

The error from this would propagate to

$ \delta T_{20}^{stat} = 2.18\cdot \delta A_{zz}^{stat}$

$ \delta T_{20}^{sys} =\sqrt{ \left(2.18\cdot \delta A_{zz}^{sys}\right)^2 + \left(2.42\cdot\delta T_{21}\right)^2 + \left(3.33\cdot\delta T_{22}\right)^2 }$.

Looking at the world data for $T_{21}$ and $T_{22}$,

Garcon (1992)	Ferro-Luzzi (1996)	Abbott (2000)

Overlaying the plots on top of each other so they both read in GeV² looks like

In discussions with Doug, a reasonable uncertainty on T21 and T22 would be 10%. This was included in the systematics band, along with systematics from Azz, and is shown below.