have the same couplings, as do all $d$-type quarks. The SM predictions for the couplings are $v_{u} = 1/2 - 4/3 \sin^2\theta_W$, $a_{u} = 1/2$ and $v_{d} = -1/2 + 2/3 \sin^2\theta_W$, $a_{d} = -1/2$.

For most of the HERA phase space, $\chi^2_Z \ll \chi_Z$ and thus the influence of pure $Z$ exchange is small. In addition, $v_e \approx 0.04$ is small. Thus in Eqs. \[eqn:gen\_f2\] and \[eqn:gen\_xf3\] the axial-vector couplings are determined predominantly through the term $-a_e \chi_Z xF_3^{\gamma Z}$ and the vector couplings through the term $-P_e a_e  \chi_Z F_2^{\gamma Z}$. Thus the data obtained with polarised electron beams are crucial for a precise determination of the vector couplings. Nevertheless, it is the combination of all data that provides the final precision.

The charged current (CC) cross sections provide direct information on $M_W$. Taking polarisation into account, they can be written as

$${\frac{d^{2}\sigma_{\rm CC}(e^+p)}{dx_{\rm Bj} dQ^{2}}} = (1 + P_e)
  \frac{G_F^{2}M_W^4 }{2\pi x_{\rm Bj}(Q^2 + M_W^2)^{2}}
  \,x\, [ (\bar{u} + \bar{c}) + (1 -y)^2 (d + s + b)]\,,
\label{eqn:xsec_CC1}$$

$${\frac{d^{2}\sigma_{\rm CC}(e^-p)}{dx_{\rm Bj} dQ^{2}}} = (1 - P_e)
  \frac{G_F^{2}M_W^4 }{2\pi x_{\rm Bj}(Q^2 + M_W^2)^{2}}
  \,x\, [ (u + c) + (1 -y)^2 (\bar{d} + \bar{s} + \bar{b})]\,.
\label{eqn:xsec_CC2}$$

