{"knowledge_schema": {"broad_category": "Physics \u2192 Quantum Field Theory \u2192 Renormalization", "refinement": "This problem involves analyzing the mass dimensions of parameters in a Lagrangian and determining the renormalizability of the theory.", "specific_scope": "The focus is on the Lagrangian involving fermionic fields and gauge fields, specifically examining the coupling constant \u03ba and its implications for the theory's renormalizability.", "goal": "Determine the mass dimension of the coupling constant \u03ba and assess whether the theory is renormalizable."}, "summary": "In this problem, we analyze the mass dimension of the coupling constant \u03ba in the given Lagrangian and evaluate the renormalizability of the theory. The mass dimension of \u03ba is found to be -1, indicating that it has negative mass dimension. This suggests that the theory is not renormalizable, as theories with negative mass dimensions for coupling constants typically lead to divergences that cannot be controlled through renormalization. Thus, the correct answer is that the mass dimension [\u03ba]_M=-1 and the theory is not renormalizable."}