Reconcilement of VHE $gamma$-ray/X-ray correlation studies in Mrk 421 and break-down at high fluxes

Multi-wavelength campaigns have been carried out to study the correlation between the very high energy (VHE) $gamma$-ray and the X-ray emissions in blazars but, no conclusive results have been achieved yet. In this paper, we add Milagro data to the existing VHE $gamma$-ray data from HEGRA-CT1 and Whipple and test the consistency and robustness of the reported correlation between VHE $gamma$-ray and X-ray fluxes in Mrk 421. We found that at monthly time scale the correlation is robust, consistent between instruments and described as a linear function. Furthermore, most of the fluxes on shorter time scales are consistent with the correlation within 3 $sigma_A$ even, where $sigma_A$ is an estimated intrinsic scatter. However, a break-down of the correlation becomes clearly evident at high states of activity with fluxes $rm gtrsim 2.5times 10^{-10}, cm^{-2}s^{-1}$ at energies above 400 GeV independently of the time scale, observational period or instrument, even for single flares, the X-ray and VHE $gamma$-ray emissions lie on the correlation until the VHE $gamma$-ray flux reaches values higher than the one mentioned above. We have interpreted our results within the one-zone synchrotron self-Compton model. We found that describing a single and unique $gamma$-ray/X-ray correlation strongly narrows the range of possible values of the magnetic field $B$ when a constant value of the spectral index along the correlation is assumed.