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Physics 509: Relativistic Quantum Field TheoryProblem Set 8Due Friday, 09 December 201128. 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 introducingscalar degrees-of-freedom to a massless vector Lagrangian, we decouple the longitudinalmodes from the transverse ones at energies large compared to the vector mass. However,when multipleself-interactingvectors are given thesamemass, S-matrix elements forlongitudinal vector scattering grow without bound for energies parametrically larger thanthe vector-mass. In this problem, we explore the tension between massive vectors and theunitarity of the S-matrix.a) Begin with a massive photon,L=-14g2FμνFμν+v2(AμAμ).(1)The mass term is not invariant under local gauge transformationAμ→Aμ+∂μα(x).Introduce a real scalar fieldθ(x), which transforms asθ(x)→θ(x)+α(x) under gauge-transformations, into the action such that it is again gauge-invariant. This new scalar