An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 13.1

Critical Exponents and Scalar Field Theory — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · critical-exponents-scaling

Problem

Correction-to-scaling exponent. For critical phenomena in 4−ϵ4-\epsilon dimensions, the irrelevant contributions that disappear most slowly are those associated with the deviation of the coupling constant λ\lambda from its fixed-point value. This gives the most important nonuniversal correction to the scaling laws derived in Section 13.1. By studying the solution of the Callan-Symanzik equation, show that if the bare value of λ\lambda differs slightly from λ∗\lambda_*, the Gibbs free energy receives a correction G(M,t)→G(M,t)⋅(1+(λ−λ∗)tωνk^(tM−1/β)).G(M,t) \to G(M,t) \cdot \left(1 + (\lambda - \lambda_*) t^{\omega\nu} \hat{k}(tM^{-1/\beta})\right). This formula defines a new critical exponent ω\omega, called the *correction-to-scaling exponent*. Show that ω=ddλβ∣λ∗=ϵ+O(ϵ2).\omega = \frac{d}{d\lambda}\beta\bigg|_{\lambda_*} = \epsilon + \mathcal{O}(\epsilon^2).

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