Title: Photometric Re-calibration of VPHAS+ $u$-band Photometry with the Stellar Colour Regression Method and Gaia DR3

Abstract: The u band magnitude is vital for determining stellar parameters and investigating specific astronomical objects. However, flux calibration in the u band for stars in the Galactic disk presents significant challenges. In this study, we introduce a comprehensive re-calibration of $u$-band photometric magnitudes of the VPHAS+ Data Release 4 (DR4), employing the Stellar Colour Regression (SCR) technique. By leveraging the expansive set of XP spectra and $G_{\rm BP}$ photometry from Gaia Data Release 3 (DR3), as well as the individual stellar extinction values provided by the literature, we have obtained precise model magnitudes of nearly 3 million stars. Our analysis identifies systematic magnitude offsets that exhibit a standard deviation of 0.063 mag across different observational visits, 0.022 mag between various CCDs, and 0.009 mag within pixel bins. We have implemented precise corrections for these observational visits, CCD chips, and pixel bins-dependent magnitude offsets. These corrections have led to a reduction in the standard deviation between the observed magnitudes and the model magnitudes from 0.088 mag to 0.065 mag, ensuring that the calibrated magnitudes are independent of stellar magnitude, colour, and extinction. The enhanced precision of these magnitudes substantially improves the quality of astrophysical research and offers substantial potential for furthering our understanding of stellar astrophysics.