No-go result for supersymmetric gauge theories in the causal approach

Abstract

We consider the massless supersymmetric vector multiplet in a purely quantum framework and propose a power counting formula. Then we prove that the interaction Lagrangian for a massless supersymmetric non-Abelian gauge theory (SUSY-QCD) is uniquely determined by some natural assumptions, as in the case of Yang-Mills models. The result can be easily generalized to the case when massive multiplets are present, but one finds out that the massive and the massless Bosons must be decoupled, in contradiction with the standard model. Going to the second order of perturbation theory produces an anomaly which cannot be eliminated. We make a thorough analysis of the model working only with the component fields.

Anti-BRST in the Causal Approach

“…We remark that our consideration is not based on supersymmetry. To have a supersymmetric model one would have to include the supersymmetric partners of the ghost fields and this could lead to no-go results as in [8].…”

“…Proof: The generic expression for T depending non-trivially on the variables (Ψ µ , χ, χ) is a summ of the following contributions: (7) aǫ ⊗ γ µ γ ǫ Ψ µ , T 8 = u a Ψµ t (8) aǫ ⊗ γ µ γ ǫ χ T 9 = ũa Ψµ t (9) aǫ ⊗ γ µ γ ǫ χ, T 10 = ũa χ t (10) aǫ ⊗ γ µ γ ǫ Ψ µ .…”

“…Ψν s (8) aǫ ⊗ γ µ γ ρ γ ν γ ǫ χ T µ 9 = u a Ψρ s (9) aǫ ⊗ γ µ γ ǫ Ψ ρ , T µ 10 = u a Ψµ s (10) aǫ ⊗ γ ρ γ ǫ Ψ ρ T µ 11 = u a Ψρ s (11) aǫ ⊗ γ ρ γ ǫ Ψ µ , T µ 12 = u a χ s (12) aǫ ⊗ γ µ γ ǫ χ T µ 13 = u a χ s (13) aǫ ⊗ γ µ γ ǫ χ, T µ 14 = u a Ψν s (14) aǫ ⊗ γ ν γ µ γ ρ γ ǫ Ψ ρ T 15 = ũa χ s (15) aǫ ⊗ γ µ γ ǫ χ.…”

Wick Theorem and Hopf Algebra Structure in Causal Perturbative Quantum Field Theory

“…There are, at least to our knowledge, three rigorous ways to do that. For completeness, we reiterate them as follows [13]:…”