An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 12.3

The Renormalization Group — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · renormalization-group-fixed-points

Problem

Asymptotic symmetry. Consider the following Lagrangian, with two scalar fields ϕ1\phi_1 and ϕ2\phi_2: L=12((∂μϕ1)2+(∂μϕ2)2)−λ4!(ϕ14+ϕ24)−2ρ4!(ϕ12ϕ22).\mathcal{L} = \frac{1}{2}((\partial_\mu \phi_1)^2 + (\partial_\mu \phi_2)^2) - \frac{\lambda}{4!}(\phi_1^4 + \phi_2^4) - \frac{2\rho}{4!}(\phi_1^2\phi_2^2). Notice that, for the special value ρ=λ\rho = \lambda, this Lagrangian has an O(2)O(2) invariance rotating the two fields into one another. (a) Working in four dimensions, find the β\beta functions for the two coupling constants λ\lambda and ρ\rho, to leading order in the coupling constants. (b) Write the renormalization group equation for the ratio of couplings ρ/λ\rho/\lambda. Show that, if ρ/λ<3\rho/\lambda < 3 at a renormalization point MM, this ratio flows toward the condition ρ=λ\rho = \lambda at large distances. Thus the O(2)O(2) internal symmetry appears asymptotically. (c) Write the β\beta functions for λ\lambda and ρ\rho in 4−ϵ4 - \epsilon dimensions. Show that there are nontrivial fixed points of the renormalization group flow at ρ/λ=0,1,3\rho/\lambda = 0, 1, 3. Which is the most stable? Sketch the pattern of coupling constant flows. This flow implies that the critical exponents are those of a symmetric two-component magnet.

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