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Unformatted text preview: UNIVERSITY OF CALIFORNIA, SANTA BARBARA Department of Physics Physics 221A Quantum Field Theory Fall 2007 http://www.kitp.ucsb.edu/~joep/Web221A/221A.html FINAL EXAM SOLUTIONS 1. a) The propagator for field i = A,B,C is i/ ( k 2 + m 2 i i ). There is an ABB vertex ig and a CBB vertex ih . b) This is identical to Srednicki Ex. 11.1, from HW#5, except that g is replaced by g/ 2. Hence (see HW solutions) Γ = g 2 q m 2 A 4 m 2 B 32 πm 2 A . c) The graph is a loop of B fields with one A and two C propagators attached. An example of this type from the Standard Model is the decay of the Higgs to two gluons through a top quark loop. d) This is identical to the calculation in Chapter 16, except that (1) we are in d = 4, and (2) there are two kinds of vertex, couplings g and h . Therefore, start with Eq. 16.1, with d = 4 and with g 3 replaced by gh 2 . Carrying through to 16.8 at ε = 2 we have T = gh 2 32 π 2 Z dF 3 D 1 . Note that there are no factors of ˜ μ because the couplings are defined in d = 4 from the start. Also, from 16.5 we have D = x 1 x 3 k 2 1 + x 2 x 3 k 2 2 + x 1 x 2 k 2 3 + m 2 B = m 2 B x 1 x 3 m 2 A x 2 ( x 1 + x 3 ) m 2 C . e) In this limit D → m 2 B and we can use R dF 3 = 1 to find T = gh 2 32 π 2 m 2 B . A coupling 1 2 ˜ gAC 2 would give T = ˜ g , so ˜ g = gh 2 32 π 2 m 2 B . The rate, from part (b), is then Γ = ˜ g 2 q m 2 A 4 m 2 C 32 πm 2 A = g 2 h 4 q m 2 A 4 m 2 C 2 15 π 5 m 2 A m 4 B . 1 It is good to check units. The threescalar couplings g and h have units of m 1 in d = 4, so Γ has units of m 1 as it should....
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This note was uploaded on 02/04/2012 for the course PHYS 253A at Harvard.
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 Quantum Field Theory

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