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Gauge independent form factors \rho^(e; e) and \hat{\kappa}^(e; e)(q^2) forMoller scattering at s << m_W^2 are derived. It is pointed out that\hat{\kappa}^(e; e) is very different from its counterparts in other processes.The relation between the effective parameter \hat{\kappa}^(e; e)(q^2,\mu)\sin^2 \hat{\theta}_W(\mu) and \sin^2 \theta_eff is derived in ascale-independent manner. A gauge and process-independent running parameter\sin^2 \hat{\theta}_W (q^2), based on the pinch-technique self-energy a_{\gammaZ} (q^2), is discussed for all q^2 values. At q^2=0 it absorbs very accuratelythe Czarnecki-Marciano calculation of the Moller scattering asymmetry at low svalues, and at q^2 = m^2_Z it is rather close to \sin^2 \theta_eff. The q^2dependence of \sin^2 \hat{\theta}_W (q^2) is displayed in the space andtime-like domains.Comment: A new paragraph has been inserted at the beginning of the discussion  in Section

The electroweak form factor $\hat{\kappa}(q^2)$ and the running of $\sin^2\hat{\theta}_{\mathrm {W}}$