Simultaneous X-ray and optical polarization observations of the blazar Mrk 421

We present near-simultaneous X-ray and optical polarization measurements in the high synchrotron peaked (HSP) blazar Mrk 421. The X-ray polarimetric observations were carried out using {\it Imaging X-ray Polarimetry Explorer} ({\it IXPE}) on 06 December 2023. During {\it IXPE} observations, we also carried out optical polarimetric observations using 104cm Sampurnanand telescope at Nainital and multi-band optical imaging observations using 2m Himalayan Chandra Telescope at Hanle. From model-independent analysis of {\it IXPE} data, we detected X-ray polarization with degree of polarization ($\Pi_X$) of 8.5$\pm$0.5\% and an electric vector position angle ($\Psi_X$) of 10.6$\pm$1.7 degrees in the 2$-$8 keV band. From optical polarimetry on 06 December 2023, in B, V, and R bands, we found values of $\Pi_B$ = 4.27$\pm$0.32\%, $\Pi_V$= 3.57$\pm$0.31\%, and $\Pi_R$= 3.13$\pm$0.25\%. The value of $\Pi_B$ is greater than that observed at longer optical wavelengths, with the degree of polarization suggesting an energy-dependent trend, gradually decreasing from higher to lower energies. This is consistent with that seen in other HSP blazars and favour a stratified emission region encompassing a shock front. The emission happening in the vicinity of the shock front will be more polarized due to the ordered magnetic field resulting from shock compression. The X-ray emission, involving high-energy electrons, originates closer to the shock front than the optical emission. The difference in the spatial extension could plausibly account for the observed variation in polarization between X-ray and optical wavelengths. This hypothesis is further supported by the broadband spectral energy distribution modeling of the X-ray and optical data.