Testing the Amp\`ere-Maxwell law on the photon mass and Lorentz-Poincar\'e symmetry violation with MMS multi-spacecraft data

We investigate possible evidence from Extended Theories of Electro-Magnetism by looking for deviations from the Amp\`ere-Maxwell law. The photon, main messenger for interpreting the universe, is the only free massless particle in the Standard-Model (SM). Indeed, the deviations may be due to a photon mass for the de Broglie-Proca (dBP) theory or the Lorentz Symmetry Violation (LSV) in the SM Extension (SME), but also to non-linearities from theories as of Born-Infeld, Heisenberg-Euler. With this aim, we have analysed six years of data of the Magnetospheric Multi-Scale mission, which is a four-satellite constellation, crossing mostly turbulent regions of magnetic reconnection and collecting about $95\%$ of the downloaded data, outside the solar wind. We examined 3.8 million data points from the solar wind, magnetosheath, and magnetosphere regions. In a minority of cases, for the highest time resolution burst data and optimal tetrahedron configurations drawn by the four spacecraft, deviations have been found ($2.2\%$ in modulus and $4.8\%$ in Cartesian components for all regions, but raising up in the solar wind alone to $20.8\%$ in modulus and $29.7\%$ in Cartesian components and up to 45.2\% in the extreme low-mass range). The deviations might be due to unaccounted experimental errors or, less likely, to non-Maxwellian contributions, for which we have inferred the related parameters for the dBP and SME cases. Possibly, we are at the boundaries of measurability for non-dedicated missions. We discuss our experimental results (upper limit of photon mass of $2.1 \times 10^{-51}$ kg, and of the LSV parameter $|\vec{k}^{\rm AF}|$ of $6 \times 10^{-9}$ m$^{-1}$), as the deviations in the solar wind, versus more stringent but model-dependent limits.