Multi-Period Fermi-LAT and LHAASO Constraints on Axion-Like Particles from Mrk 421 via chi squared Profile Likelihood with Gaussian Copula Correlation

We present a joint multi-period analysis of axion-like particles (ALPs) using five sets of simultaneous Fermi-LAT and LHAASO observations of the TeV blazar Mrk 421. Photon-ALP oscillations are calculated self-consistently together with EBL absorption for two representative jet emission models: a two-zone hybrid model and a single-zone hadronic model. To account for weak correlations among different observational epochs, we adopt a χ squared-based profile analysis combined with a Gaussian copula (ρ=0.03). Nuisance parameters are optimized under the no-ALP hypothesis before profiling the ALP parameters m a and g aγ. In the low-mass regime (m a 1,neV) we obtain a 95% CL upper limit of g aγ=6.35 times10-13, GeV-1. Over the full mass range 0.1 -- 500,neV, the most conservative 95% CL upper limits are g aγ<6.69 times10-12, GeV-1 (two-zone) and g aγ<7.37 times10-12, GeV-1 (single-zone). Variations of the EBL template, the Galactic and intergalactic magnetic-field models, the spectral binning, and residual instrumental systematics change these limits by only a few percent. The jet magnetic field is the leading systematic and can shift the low-mass bound by a factor of a few; only a substantially weaker single-zone field degrades the constraint by about an order of magnitude. The analysis framework developed here provides a statistically consistent approach for future ALP searches with multi-messenger gamma-ray data.