8.324 Relativististic Field Theory II, Assignment 7 - MIT ...

Quantum Field Theory II (8.324) Fall 2010Assignment 7Readings• Peskin & Schroeder chapters 10, 12, 13.• Weinberg vol 1 chapter 12 and Vol 2 chapter 18.Problem Set 71. Renormalization group properties (30 points)(a) Consider a coupling constant λ and a redefined coupling constant λ¯(λ).Find the general transformation law for the beta function, namely therelation between β(λ) and β(λ). If we think of λ as a coordinate we seethat β transforms as a tensor. What kind of tensor ?(b) Assume thatβ(λ) = b 2 λ 2 + b 3 λ 3 + b 4 λ 4 + ···and consider the perturbatively defined and invertible coupling constantredefinition:λ¯(λ) = λ+ a 2 λ 2 + a 3 λ 3 + ··· .Calculate β¯(λ¯) writing it in the formVerify that:β¯(λ¯) =¯b 2 λ¯2+¯b 3 λ¯3+¯b 4 λ¯4+ ···i. b 2 = b 2 and b 3 = b 3 .ii. It is possible to make ¯ b 4 anything you want by such a coupling redefinition.iii. Let λ = λ F denote a fixed point. Show that λ¯ = λ¯F is also a fixed′ ′point. How are the derivatives β and β¯ related at the fixed point?1

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