Revisiting Very High Energy Gamma-Ray Absorption in Cosmic Propagation under the Combined Effects of Axion-Like Particles and Lorentz Invariance Violation

Very-high-energy (VHE; $E\gtrsim 100$ GeV) gamma rays are expected to experience strong attenuation during cosmological propagation due to electron-positron pair production on the extragalactic background light (EBL). Recent observations of GRB 221009A (z = 0.151), including photons up to $\sim 18$ detected by LHAASO and a $\sim 300\mathrm{T}\mathrm{e}\mathrm{V}$ event reported by Carpet-3, suggest a higher-than-expected transparency of the Universe at extreme energies. These observations cannot be explained by standard EBL absorption alone; moreover, neither Lorentz invariance violation (LIV) nor photon-axion-like particle (ALP) oscillations, when considered in isolation, appear sufficient to account for the survival of such photons over cosmological distances. In this work, we propose a joint propagation scenario that incorporates photon-ALP mixing in astrophysical magnetic fields together with subluminal quadratic LIV corrections to the $\gamma \gamma$ pair-production threshold. Applying this framework to the broadband gamma-ray spectrum of GRB 221009A, we show that ALPs with coupling ($g_{a\gamma }=1.685\times {10}^{-10}{\mathrm{G}\mathrm{e}\mathrm{V}}^{-1}$ ) and mass ($m_a=9.545\times {10}^{-8}\mathrm{e}\mathrm{V}$), combined with a quadratic LIV energy scale ($E_{\mathrm{L}\mathrm{I}\mathrm{V},2}=1.30\times {10}^{-7}{E}_{\mathrm{P}\mathrm{l}}$) adopted from the literature, can significantly enhance the photon survival probability in the energy range (10$\text{-}$300) TeV. The resulting enhancement exceeds that obtained from either ALP mixing or LIV effects alone. These results indicate that a combined ALP-LIV scenario may provide a viable interpretation of the extreme-energy gamma-ray observations of GRB 221009A and highlight the potential of VHE gamma-ray measurements as probes of physics beyond the Standard Model.