{"knowledge_schema": {"broad_category": "Quantum Field Theory \u2192 Scalar Field Theories \u2192 Renormalization", "refinement": "This problem involves analyzing the mass dimensions of terms in a Lagrangian density to determine the properties of the theory, specifically focusing on the coupling constant of a quartic interaction.", "specific_scope": "The task is to calculate the mass dimension of the coupling constant \u03bb in the given Lagrangian density and assess the renormalizability of the theory in four-dimensional spacetime.", "goal": "Determine the mass dimension of \u03bb and conclude whether the theory is renormalizable based on that dimension."}, "summary": "In this problem, we analyze a scalar field theory with a quartic interaction term to find the mass dimension of the coupling constant \u03bb and assess the renormalizability of the theory. By examining the mass dimensions of the terms in the Lagrangian density, we find that the mass dimension of \u03bb is 0, indicating that it is a dimensionless coupling constant. Consequently, the theory is renormalizable in four dimensions, as the interaction term does not introduce any additional divergences that cannot be handled by renormalization."}