Cross sections and asymmetries

The measured raw asymmetries $\Delta$ are the physicl asymmetries $A_{phy}$ diluted by the beam polarization $P_b$, the target polarziation $P_t$, and the dilution factor $f$ due to the fact that we are using $^3$He instead of neutron target.:

$$A_{phy} = \frac{\Delta}{P_bP_tf}$$

Then the target asymmetry $A_1^{^3\rm {He}}$ can be extracted from the measured asymmetries $A_\perp$ and $A_\parallel$ as:

$$A_1 = \frac{1}{D(1+\eta\xi)}A_{\parallel}-\frac{\eta}{d(1+\eta\xi)}A_{\perp}$$ (1)

where $A_\parallel$ is the asymmetry when the beam polarization is parallel and antiparallel to the target polarization: $A_\parallel=\frac{\sigma^{\uparrow\downarrow} -\sigma^{\uparrow\uparrow}}{\sigma^{\uparrow\downarrow} +\sigma^{\uparrow\uparrow}}$. And $A_\perp$ is the asymmetry when the beam polarization is transverse to the target polarization: $A_\perp=\frac{\sigma^{\downarrow\leftarrow} -\sigma^{\uparrow\leftarrow}}{\sigma^{\downarrow\leftarrow} +\sigma^{\uparrow\leftarrow}}$.

Other variables are kinematics dependent, except $R={\sigma_L}/{\sigma_T}=\frac{1+\gamma^2}{2x}\frac{F_2}{F_1}-1$. $D$ is the depolarization factor $D=\frac{1-\epsilon E'/E}{1+\epsilon R}$.