in a decrease (increase) of the $e^{+} p$ ($e^{-} p$) cross sections, respectively.

The data were published as reduced cross sections which were defined for $e^-p$ and $e^+p$ NC scattering as

$$\sigma^{e^{\pm} p}_{r,{\rm NC}} 
  =
  \frac {x_{\rm Bj}Q^{4}} {2 \pi \alpha^{2}_0}
  \frac {1} {Y_{+}}
  {\frac{d^{2}\sigma(e^{\pm}p)}{dx_{\rm Bj} dQ^{2}}}
  =
  \tilde{F_{2}}(x_{\rm Bj},Q^{2}) \mp \frac {Y_{-}} {Y_{+}} x \tilde{F_{3}}
  (x_{\rm Bj},Q^{2})- \frac {y^2} {Y_{+}} F_{L}(x_{\rm Bj},Q^{2}).
\label{eqn:red}$$

In this definition, the fine-structure constant is fixed to $\alpha_0$, i.e.at scale zero. The QED corrections applied to the data use a running $\alpha$ to correct the data accordingly.

The generalised structure functions depend on the longitudinal polarisation of the electron beam, which is defined as

$$P_{e}=\frac{N_{R}-N_{L}}{N_{R}+N_{L}}, 
\label{eqn:pol}$$

where $N_{R}$ and $N_{L}$ are the numbers of right- and left-handed electrons in the beam.

In all orders, the functions $\tilde{F_2}^\pm$ and $x\tilde{F_3}^\pm$ can be split into structure-function terms depending on $\gamma$ exchange ($F_2^{\gamma}$), $Z$ exchange ($F_2^Z$, $xF_3^Z$) and $\gamma/Z$ interference ($F_2^{\gamma Z}$, $xF_3^{\gamma Z}$) as

$$\tilde{F_2}^\pm = F_2^{\gamma} - (v_e \pm P_e a_e) \chi_{Z} F_2^{\gamma Z} +
  (v_e^2 + a_e^2 \pm 2 P_e v_e a_e) {\chi_{Z}^{2}} F_2^{Z}  ,
\label{eqn:gen_f2}$$

$$x\tilde{F_3}^\pm =  - (a_e \pm P_e v_e) \chi_{Z} xF_3^{\gamma Z} + (2 v_e a_e 
\pm P_e(v_e^2 + a_e^2)) 