Abstract
Assuming locality, Lorentz invariance and parity conservation we obtain a set of differential equations governing the 3-point interactions of massless bosons, which in turn determines the polynomial ring of these amplitudes. We derive all possible 3-point interactions for tensor fields with polarisations that have total symmetry and mixed symmetry under permutations of Lorentz indices. Constraints on the existence of gauge-invariant cubic vertices for totally symmetric fields are obtained in general spacetime dimensions and are compared with existing results obtained in the covariant and light-cone approaches.Expressing our results in spinor helicity formalism we reproduce the perhaps mysterious mismatch between the covariant approach and the light cone approach in 4 dimensions. Our analysis also shows that there exists a mismatch, in the 3-point gauge invariant amplitudes corresponding to cubic self-interactions, between a scalar field ϕ and an antisymmetric rank-2 tensor field Aμν . Despite the well-known fact that in 4 dimensions rank-2 anti-symmetric fields are dual to scalar fields in free theories, such duality does not extend to interacting theories.
Highlights
For massless spin-1, 2, 3 fields and interactions of two scalars with a spin-s boson using the Fronsdal fields
Boels and Medina [7] have obtained the threepoint amplitudes using the constraints of on-shell gauge invariance. Their results are expressed in terms of polarization tensors with the gauge transformations manifest; these have been done for polarization vectors and rank-2 polarization tensors, but not for general polarizations
There emerges a series of works combining these two approaches to study the spinor helicity amplitudes [19, 22, 23], the most notable discovery being a mismatch between the light-cone and covariant approaches: there are cubic vertices existing in light-cone approach but are absent in the covariant approach [19, 21,22,23]
Summary
In the case of a rank-r tensor field φμ1···μr describing a massless spin s particle, the condition (2.1) is fulfilled if the field operator is of the form φμ1···μr =. The above equations cannot be simultaneously satisfied In such cases, we require only Equation (2.3) to hold and that any physical quantities (for example, the scattering amplitudes) must be invariant under translations of the ISO(2) (which we loosely call the “gauge transformation”): μ1···μr (k, σ) → D [S(α, β)]μ1···μr μ1···μr μ1···μr (k, σ) which, ensures Lorentz invariance of the amplitudes. The tensor condition (2.3) and the gauge transformation (2.7) become (Jz ⊗I ⊗· · ·⊗I +I ⊗Jz ⊗· · ·⊗I +· · ·+I ⊗· · ·⊗I ⊗Jz) (k, σ) = σ (k, σ),.
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