What are the anomalies that arise with quarks and leptons being extended to N=2 multiplets?

Question

In his blog post, Could Nature, LHC, prefer \(\mathcal{N}=2\) supersymmetry?, @LubošMotl remarks that:

Before the experts leave this article with the word "obvious bullshit", let me say that they only talk about the extended supersymmetry of the gauge sector. It would really lead to contradictions if you tried to extend the quarks and leptons to N=2 multiplets.

What exactly are the anomalies that arise when we extend gravitons and fermionic particles to \(\mathcal{N}=2\) (32 supercharges)?

Also, how is it possible to have a different number of supercharges for different sectors?

Answer 1

The main problem with $\mathcal{N}=2$ supersymmetric quantum field theories in four dimensions with gauge group $G$ is that they are not chiral. As you most probably know, the SM is chiral as they are its various extensions that admit supersymmetry such as the MSSM  (which is already almost excluded but nevertheless). Any theory describing the real world must be chiral. Another thing to notice is that the $\mathcal{N}=2$ theory has no superpotential something that would give rise to interactions between fermions but not for the gauginos, something forbidden by the R-symmetry. Another inconsistency is that all fields transform in the adjoint representation of the gauge group $G$. Another thing to notice is that this theory is one-loop exact unlike theories with less supersymmetry. In specific $\mathcal{N}=1$ theories receive corrections to the gauge coupling at all orders of perturbation theory.

As for your second question note that while the fermions $\psi, \lambda$ transform as a doublet under the R-symmetry group $SU(2)_R$ the bosonic fields $A_{\mu}, D,F$ are singlets. This is due to the $SU(2)_R$ rotation of the supercharges $Q_{\alpha}^i$, $i=1,2$ that transform in the fundamental of the $SU(2)_R$ (as they do the fermions).

Comments

+1 Thanks for the answer, this clarifies my questions well!

I'm slightly confused by your remark that the MSSM is excluded - surely, the MSSM is  found in the field-theory limit of the heterotic string theories and M-theory, for example? I was under the impression that the MSSM is supported by a number of experimental observations (e.g. the prediction of the Higgs mass at 125 GeV/c²).

Or did you mean that the MSSM is only useful as a field theory limit of superstring theory, and not as a complete ToE model on it's own (which is somewhat obvious, as it does not describe gravity)?

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No, I mean that the MSSM's (minimal supersymmetric model) parameter space is almost excluded by LHC. At least this is what my pheno friends support. You can try to look up for the data (there are some nice graphs) that actually show it. I don't think that heterotic strings reproduce the MSSM exactly. But, as I said, I am not sure.

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@conformal_gk Ok, thanks for the clarification...