It follows from Eq. \[eq:rewrite1\] that the coupling $G_F$ can be rewritten in terms of $\sin^2\theta_W$ and $M_W$ as

$$G_F = \frac{\pi \alpha_0}{\sqrt{2}\, \sin^2\theta_W \,M_W^2}\, 
        \frac{1}{1 - \Delta R}~.
\label{eqn:GF}$$

Substituting $G_F$ into Eqs. \[eqn:xsec\_CC1\] and \[eqn:xsec\_CC2\] parameterises the dependence of the CC cross sections on $\sin^2\theta_W$.

Experimental setup {#sec-setup}
==================

The analysis is based on inclusive cross sections for $ep$ scattering published by the H1 [@H1allhQ2; @H1FL1; @H1FL2; @Adloff:1999ah; @Adloff:2000qj; @Adloff:2003uh; @Collaboration:2009kv; @Collaboration:2009bp] and ZEUS [@ZEUS2NCe; @ZEUS2NCp; @ZEUS2CCe; @ZEUS2CCp; @ZEUSFL; @Breitweg:1998dz; @zeuscc97; @Chekanov:2001qu; @Chekanov:2002zs; @Chekanov:2002ej; @Chekanov:2003vw; @Chekanov:2003yv] collaborations for both the HERAII and HERAI periods. A description of all data sets, including their respective integrated luminosities was published previously [@HERAPDF20]. All data sets were taken as input individually; data sets were not combined, in contrast to the HERAPDF2.0 analysis.

Polarised beams were available for the HERAII period from 2003 to 2007 when the electron beam energy was $E_e = 27.5\,{{\,\text{Ge}{\kern-0.06667em}\text{V\/}}}$ and the proton beam was $E_p = 920\,{{\,\text{Ge}{\kern-0.06667em}\text{V\/}}}$, corresponding to a centre-of-mass energy of 318GeV. The information on beam polarisation was used in this analysis for the corresponding ZEUS HERAII data sets; the H1 HERAII data sets were used as published for zero polarisation. The kinematic range of these 