Multi Messenger Study of GRB 221009A with VHE Gamma-ray and Neutrino Afterglow from a Gaussian Structured Jet

Recent detections of very-high-energy (VHE; $\gtrsim 100~{\rm GeV}$) emission from GRB afterglows, most notably the unprecedented brightness of GRB~221009A observed by LHAASO, reveal components beyond the standard electron synchrotron model. The multi-TeV photons motivate synchrotron self-Compton and possible hadronic contributions, while the non-detection of coincident neutrinos by IceCube/KM3NeT/GRAND200k constrains the microphysical parameters, jet kinetic energy, and ambient-medium density. We model the VHE afterglow of GRB~221009A with an external forward shock from a Gaussian structured jet in a uniform-density medium. This angular structure reproduces the extreme TeV output at an off-axis angle but without demanding large energies as in a top-hat jet. We compute the corresponding $p\gamma$ neutrino flux in the PeV-EeV range and derive a time-integrated upper limit using the effective areas of IceCube-Gen2 and GRAND200k. This provides insight into the contribution of individual GRBs to neutrino events. The predicted neutrino flux for GRB~221009A, using parameters inferred from the multi-wavelength spectral energy distribution, lies below the sensitivities of these detectors. Even our correlation analysis, optimized for neutrino searches with the upcoming GRAND200k, indicates that the expected number of events from this GRB is of order $\sim 0.1$ under a highly optimistic microphysical parameter regime. We also compare neutrino-flux variations from on-axis and off-axis viewing geometries and find an approximately order of magnitude difference in the signal. Thus, our study concludes that a brighter burst closer than GRB~221009A would be crucial for neutrino detection by upcoming telescopes. Future GRB detections by the Cherenkov Telescope Array will provide important constraints on their geometry, radiation mechanisms, and possible associated neutrino signals.