Unpolarized and polarized structure functions

Cross section:

$$\frac{\mathrm{d}^2\sigma}{\mathrm{d}\Omega\mathrm{d}E^\prime} = \Big(\frac{\mathrm{d}^2\sigma}{\mathrm{d}\Omega}\Big)_{Mott} \Big[W_2(Q^2,\nu)+2W_1(Q^2,\nu)\tan^2\frac{\theta}{2}\Big]$$ (1)

Scaling at $Q^2\rightarrow\infinity$, with fixed $x$: $M W_1(\nu,Q^2)\rightarrow F_1(x)$, $\nu W_2(\nu,Q^2)\rightarrow F_2(x)$, then

$$\frac{\mathrm{d}^2\sigma}{\mathrm{d}x\mathrm{d}y} = \frac{2\pi x^2}{Q^4}s\big[1+(1-y)^2\big]F_2(x)$$ (2)

$$F_1(x,Q^2) = \frac{F_2(x,Q^2)(1+\gamma^2)}{(2x(1+R(x,Q^2))}$$ (3)

The cross section difference between a target polarized along $\alpha$ and $\alpha +\pi$ is

$$\frac{\mathrm{d}^3\Big({\sigma(\alpha)-\sigma(\alpha +\pi)}\Big)} {\mathrm{d}x\mathrm{d}y\mathrm{d}z} = \frac{e^4}{4\pi^2 Q^2} \Big\{\cos{\alpha}\big\{{(1-\frac{y}{2} -\frac{y^2}{4}\gamma^2}-\frac{y}{2}\gamma^2 g_2(x,Q^2)\big\}$$

$$~ -\sin\alpha \cos\phi \sqrt{\gamma^2(1-y-\frac{y^2}{4}\gamma^2} \big\{{\frac{y}{2}g_1(x,Q^2)+g_2(x,Q^2)}\big\}\Big\}$$ (4)

Within the quark-parton model:

$$F_1(x) = \frac{1}{2}\sum_i e_i^2\big[q_i^\uparrow(x)+q_i^\downarrow(x)\big]$$ (5)

$$g_1(x) = \frac{1}{2}\sum_i e_i^2\big[q_i^\uparrow(x)-q_i^\downarrow(x)\big]$$ (6)

Ignoring quark mass effects, $g_2^n(x,Q^2)$ can be decomposed into the sum of two terms:

$$g_2(x,Q^2) = g_2^{WW}(x,Q^2) +\bar g_2(x,Q^2)$$ (7)

where $g_2^{WW}(x,Q^2)$ is purely twist-2 and can be expressed in terms of $g_1(x,Q^2)$ as follows:

$$g_2^{WW}(x,Q^2) = -g_1(x,Q^2) + \int_x^1 \frac{g_1(y,Q^2)}{y}\mathrm{d}y$$ (8)