ps5 - Physics 509 Relativistic Quantum Field Theory Problem...

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Physics 509: Relativistic Quantum Field Theory Problem Set 5 Due Friday, 11 November 2011 20. Coulomb Scattering Peskin, Problem 5.1. Further, under what limit does one recover the result obtained for non-relativistic Coulomb scattering? Show that in this limit, one correctly reproduces the Rutherford formula. 21. Gauge Invariance for Compton Scattering a) Consider QED of a complex scalar φ of mass m and charge e : L = - 1 4 F μν F μν + ( D μ φ * )( D μ φ ) - m 2 φ * φ , where D μ = μ + ieA μ is the covariant derivative. Consider Compton scattering φ ( p 1 ) γ ( k 1 ) φ ( p 2 ) γ ( k 2 ), where the polarization vectors of the photons γ ( k 1 ), γ ( k 2 ) are given by 1 , 2 . Write down the tree level amplitude for this process; there are three diagrams. Gauge invariance in the form of the Ward identity dictates that this amplitude should be invari- ant under the change 1 1 + λk 1 , for any constant λ , and likewise for 2 2 + λk 2 . Check that this is indeed the case; you need only consider the case 1 1 + λk 1 .
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