at $Q^2 = 50\,000\,$GeV$^2$. Since from Eqs. \[eqn:gen\_f2\] and \[eqn:gen\_xf3\] polarisation only enters the structure functions via terms proportional to $\chi_Z$ or $\chi_Z^2$, it is evident that beam polarisation predominantly affects the cross sections at high $Q^2$.

Although this analysis was performed at NLO in QCD, the dominant contributions of the data sets can be identified by considering the structure functions in the framework of the QPM. In this framework, the structure functions can be written in terms of sums and differences of the quark and anti-quark PDFs as

$$[F_2^{\gamma},F_2^{\gamma Z},F_2^{Z}] = 
\sum _q [e_{q}^{2}, 2e_{q}v_{q},v_{q}^{2}+a_{q}^{2}] 
 x (q + \bar{q}),
\label{eqn:struc1}$$

$$[xF_3^{\gamma Z},xF_3^{Z}] = 
\sum _q [e_{q}a_{q},v_{q}a_{q}] 
 2x (q - \bar{q}),
\label{eqn:struc2}$$

where $v_{q}$ and $a_{q}$ are the respective vector and axial-vector couplings of the quark $q$ to the $Z$ boson, and $e_{q}$ is the electric charge of the quark. The PDFs of the quarks and anti-quarks are denoted $q$ and $\bar{q}$, respectively. At any order in pQCD, all quarks kinematically accessible at HERA, i.e. all quarks except the top quark, have to be considered in Eqs. \[eqn:struc1\] and \[eqn:struc2\], but the sums are dominated by $u$- and $d$-quark contributions. It is assumed throughout the analysis that all $u$-type quarks 