Generalized BChS Model with Group Interactions: Shift in the Critical Point and Mean-Field Ising Universality

We introduce a generalized version of the Biswas-Chatterjee-Sen (BChS) model \cite{Biswas} with group interactions of size $q$, extending the original pairwise interaction dynamics. Within a mean-field framework, we derive an exact expression for the critical noise $p_c(q)$, showing that it increases monotonically with $q$ and approaches $1/2$ in the large-$q$ limit, consistent with a Gaussian approximation. Despite this shift in the phase boundary, the critical behavior remains unchanged across all $q$: the order parameter scales as $(p_c(q)-p)^{1/2}$, and the relaxation timescale diverges as $|p-p_c(q)|^{-1}$, identical to the original BChS model \cite{Biswas}. Finite-size scaling of the Binder cumulant, order parameter, and its fluctuations confirm that the system belongs to the mean-field Ising universality class for all $q$. Our results demonstrate that higher-order interactions modify the location of the transition without altering its universality class.