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Unformatted text preview: Physics 509: Relativistic Quantum Field Theory Problem Set 8 Due Friday, 09 December 2011 28. Unitarity and scattering longitudinal vector states. Massive vectors have three polarization states: two transverse and one longitudinal. However, massless vectors have only transverse polarization states. Through introducing scalar degrees-of-freedom to a massless vector Lagrangian, we decouple the longitudinal modes from the transverse ones at energies large compared to the vector mass. However, when multiple self-interacting vectors are given the same mass, S-matrix elements for longitudinal vector scattering grow without bound for energies parametrically larger than the vector-mass. In this problem, we explore the tension between massive vectors and the unitarity of the S-matrix. a) Begin with a massive photon, L =- 1 4 g 2 F F + v 2 ( A A ) . (1) The mass term is not invariant under local gauge transformation A A + ( x )....
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