Self-consistent modeling of energy-dependent synchrotron polarization in blazars with application to Mrk 421

Abstract: Observations of black hole jets closely aligned with our line of sight, known as blazars, have revealed that their X-ray emission can be significantly more polarized than in the optical bands. This chromatic polarization is often attributed to an energy-stratified jet. In blazars with X-ray emission dominated by electron synchrotron, the pitch angle between the magnetic field lines and electron velocity might also affect the frequency dependence of the polarization degree, but this effect is generally neglected in state-of-the-art models. In this work, we model the polarization and spectral energy distribution of blazars, accounting for the system's temporal evolution and pitch-angle-dependent electron synchrotron cooling. Our analysis investigates the impact of the magnetic field structure and electron distribution on polarization and tests whether this can help explain its observed frequency dependence. By extending the numerical simulation code AM3, we incorporate a self-consistent calculation of electron synchrotron emission and simulate various magnetic-field structures. We apply these models to Mrk 421 and compare them with observations. Finally, we present a time-dependent shock-in-jet model to investigate frequency-dependent polarization angle swings. Our results show how different magnetic-field structures affect energy-dependent polarization signatures. In the case of Mrk 421, the data can be explained by a helical field in the jet. In the shock-in-jet scenario, the X-ray polarization angle changes by 70 degrees, while remaining almost constant in the optical/infrared. Our results demonstrate that a self-consistent treatment of the temporal evolution of electrons is essential to model the polarized synchrotron emission from blazars and can naturally explain chromatic polarization signatures without requiring an energy-stratified jet.