{"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 based on those dimensions.", "specific_scope": "The focus is on the Lagrangian involving fermionic fields and gauge fields, specifically examining the mass dimension of the coupling constant \u03ba and the implications for renormalizability.", "goal": "Determine the mass dimension of \u03ba and assess whether the theory is renormalizable based on that dimension."}, "summary": "In this problem, we analyze the mass dimension of the coupling constant \u03ba in the given Lagrangian, which involves fermionic fields and gauge fields. The mass dimension of \u03ba is crucial for understanding the behavior of the theory under renormalization. By calculating the dimensions of the other components in the Lagrangian, we find that [\u03ba]_M = 1, indicating that the theory is renormalizable. Therefore, the correct answer is that the mass dimension of \u03ba is 1 and the theory is renormalizable."}