QFT_Solution_II-3.pdf - Dylan J Temples Solution Set Three Quantum Field Theory II QFT and the Standard Model M Schwartz March 6 2017 Contents 1

QFT_Solution_II-3.pdf - Dylan J Temples Solution Set Three...

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Dylan J. Temples: Solution Set Three Quantum Field Theory II QFT and the Standard Model - M. Schwartz March 6, 2017 Contents 1 Renormalization of Scalar QED. 2 2 Photon-Photon Scattering. 12 2.1 Quantum Electrodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Effective Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Terms from ( F μν F μν ) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 Terms from ( F μν ˜ F μν ) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Euler-Heisenberg Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1
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Dylan J. Temples Quantum Field Theory II : Solution Set Three 1 Renormalization of Scalar QED. Consider QED coupled to a scalar field. Calculate the running of the electric charge in this theory. Show that the optical theorem is satisfied. Determining the Counter-term Lagrangian. In terms of the bare fields, denoted by tildes, the Lagrangian for scalar QED is L 0 = - μ ˜ φ * μ ˜ φ - ˜ m 2 ˜ φ * ˜ φ - 1 4 ˜ F μν ˜ F μν + i ˜ e n ˜ φ ˜ A μ μ ˜ φ * - ˜ φ * ˜ A μ μ ˜ φ o + ˜ e 2 ˜ φ * ˜ φ ˜ A μ ˜ A μ , (1) where ˜ F μ ν = μ ˜ A ν - ν ˜ A μ , ˜ m is the renormalized scalar mass, and ˜ e is the renormalized coupling. We then define the renormalized fields from the bare fields, starting with the photon: ˜ A μ = Z 3 A μ , so ˜ F μν = p Z 3 h μ ˜ A ν - ν ˜ A μ i , (2) thus - 1 4 ˜ F μν ˜ F μν = - 1 4 Z 3 F μν F μν = - 1 4 F μν F μν - ( Z 3 - 1) 1 4 F μν F μν . (3) We continue with the definition of the renormalized scalars fields ˜ φ ( * ) = p Z 2 φ ( * ) , (4) so - μ ˜ φ * μ ˜ φ = - Z 2 μ φ * μ φ = - μ φ * μ φ - ( Z 2 - 1) μ φ * μ φ . (5) For now, we ignore the bare constants ˜ e and ˜ m , so that: ˜ φ ˜ A μ μ ˜ φ * = Z 2 p Z 3 φA μ μ φ * (6) ˜ φ * ˜ φ ˜ A μ ˜ A μ = Z 2 Z 3 φ * φA μ A μ . (7) Now we define the bare constants as ˜ m = r 1 + δ Z 2 m (8) ˜ e = Z 1 Z 2 Z 3 e . (9) Then the second term in the bare Lagrangian is - ˜ m 2 ˜ φ * ˜ φ = - 1 + δ Z 2 m 2 ( Z 2 φ * φ ) = - m 2 φ * φ - δm 2 φ * φ , (10) and the fourth is i ˜ e n ˜ φ ˜ A μ μ ˜ φ * - ˜ φ * ˜ A μ μ ˜ φ o = i Z 1 Z 2 Z 3 e ( Z 2 p Z 3 ) { φA μ μ φ * - φ * A μ μ φ } = ie { φA μ μ φ * - φ * A μ μ φ } + ie ( Z 1 - 1) { φA μ μ φ * - φ * A μ μ φ } (11) Page 2 of 16
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Dylan J. Temples Quantum Field Theory II : Solution Set Three and the final term is ˜ e 2 ˜ φ * ˜ φ ˜ A μ ˜ A μ = Z 2 1 Z 2 2 Z 3 e 2 ( Z 2 Z 3 ) φ * φA μ A μ (12) = Z 2 1 Z 2 e 2 φ * φA μ A μ = e 2 φ * φA μ A μ + Z 2 1 Z 2 - 1 e 2 φ * φA μ A μ . (13) Collecting these results, we have the QED Lagrangian: L QED = - μ φ * μ φ - m 2 φ * φ - 1 4 F μν F μν + ie { φA μ μ φ * - φ * A μ μ φ } + e 2 φ * φA μ A μ . (14) and the counter-term Lagrangian: L ct = - ( Z 2 - 1) μ φ * μ φ - δm 2 φ * φ - ( Z 3 - 1) 1 4 F μν F μν + ie ( Z 1 - 1) { φA μ μ φ * - φ * A μ μ φ } + Z 2 1 Z 2 - 1 e 2 φ * φA μ A μ , (15) and so L 0 = L QED + L ct . (16) Correction to the photon propagator. We are interested in the running of the coupling constant in the scalar QED theory, so as seen for the spinor QED theory, we only must concern ourselves with the correction to the self-energy of the photon, i.e. , the counter-term containing Z 3 . Quickly, we will review the Feynman rules for scalar QED. Let us work in the Feynman gauge, so the propagators are
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