{\chi_{Z}^{2}} xF_3^{Z}~,
\label{eqn:gen_xf3}$$

where $\chi_{Z}$ is the relative strength of $Z$ exchange with respect to photon exchange. These structure functions depend on the vector and axial-vector couplings of the $Z$ boson to the electron. The SM predictions for these couplings are $v_{e} = -1/2 + 2\sin^2\theta_W$ and $a_{e} = -1/2$. The on-shell definition of $\sin^2\theta_W = 1 - M_W^2/M_Z^2$ was chosen for the analysis. In the on-shell scheme, this definition is valid to all orders and $M_W$ becomes

$$M_W = \frac{A_0}{\sin^2\theta_W \sqrt{(1 - \Delta R)}} ~,
\label{eq:rewrite1}$$

where $A_0=\sqrt{\pi \alpha_0 / \sqrt{2} {G_{F}}} = 37.28039\,$GeV is a constant [@PDG14], $G_{F}$ is the Fermi coupling constant and $\Delta R$ accounts for radiative corrections, the running of $\alpha$ and bosonic loop corrections dominated by the influence of the mass of the top quark [@PDG14].

The relative strength of $Z$ exchange with respect to $\gamma$ exchange depends on the on-shell $\sin^2\theta_W$ and $\Delta R$ as

$$\chi_{Z}=\frac{1}{\sin^2{2\theta_W}} \frac{Q^{2}}{M_{Z}^{2}+Q^{2}} \frac{1}{1 - \Delta R} ~,
  \label{eqn:chiz}$$

where $M_Z$ is the pole mass of the $Z$ boson. The value of $\chi_Z$ is 0.03 at $Q^2 = 185\,$GeV$^2$, the lowest value of $Q^2$ for which ZEUS published inclusive NC cross sections with polarised beams and increases to 1.1 