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hw31 - and m are the bare coupling constant and the bare...

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Homework Set No. 1, Physics 880.08 Deadline – Wednesday, April 13, 2011 1. (15 pts) Show that the light-by-light scattering QED diagram pictured below does not contain a UV divergence (even thought naively counting powers of momenta it seems like it is logarithmically divergent). The ellipsis in the figure represent all possible permutations of the vertices along the loop. (Hint: you may find formulas (A.41) and (A.42) in Appendix A.4 of Peskin and Schroeder useful.) 2. Using dimensional regularization find the one-loop beta-function β ( λ ) = μ 2 d λ d μ 2 of the real scalar ϕ 3 theory with the Lagrangian density L = 1 2 μ ϕ 0 μ ϕ 0 - m 2 0 2 ϕ 2 0 - λ 0 3! ϕ 3 0 (1) in six (!) space-time dimensions. Here ϕ 0 is the bare field, while
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Unformatted text preview: λ and m are the bare coupling constant and the bare mass correspondingly. a. (20 pts) Similar to how we did it in class for φ 4 theory (see also HW 4 from last quarter), rewrite the Lagrangian (1) in terms of renormalized physical Felds ϕ , coupling λ , mass m and the counterterms. By calculating the one-loop propagator and vertex corrections Fnd the divergent ( ∼ 1 /ǫ with ǫ = 6-d ) parts of the coe±cients of the counterterms δ λ and δ Z . (Ignore the tadpole graph.) Note that we do not need the Fnite parts of the counterterms to Fnd β ( λ )! b. (10 pts) Using the results of part a Fnd β ( λ ). 1...
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