{"knowledge_schema": {"broad_category": "Physics \u2192 Quantum Field Theory \u2192 Scalar Field Theory", "refinement": "This problem involves analyzing the mass dimensions of parameters in a Lagrangian density and determining the renormalizability of the theory based on these dimensions.", "specific_scope": "The focus is on the quartic interaction term in the Lagrangian density of a scalar field theory and its implications for the coupling constant \u03bb.", "goal": "Determine the mass dimension of the coupling constant \u03bb and assess whether the theory is renormalizable in four dimensions."}, "summary": "In this problem, we analyze the mass dimension of the coupling constant \u03bb in the given Lagrangian density for a scalar field theory. By examining the dimensions of the terms in the Lagrangian, we find that the mass dimension of \u03bb is 0, indicating that it is a dimensionless coupling constant. Furthermore, since the theory is based on a scalar field with a quartic interaction in four dimensions, it is classified as renormalizable. Therefore, the correct answer is that [\u03bb]_M = 0, and the theory is renormalizable."}