ps8 - Physics 509 Relativistic Quantum Field Theory Problem...

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
μν
F
μν
+ v
2
( A
μ
A
μ
) . (1)
The mass term is not invariant under local gauge transformation A
μ
→ A
μ
+ ∂
μ
α ( x ).
Introduce a real scalar field θ ( x ), which transforms as θ ( x ) → θ ( x )+ α ( x ) under gauge-
transformations, into the action such that it is again gauge-invariant. This new scalar