# Are there QFTs that are UV free but do not show confinement?

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Hi!

My question is pretty neat: are there examples of theories that are UV free but are not confined (in the sense of QCD)?  I am asking this since I would like to better understand the relationship between asymptotic freedom and confinement.

It would be great if you could provide explicit references and articles!

Thank you!

There are seemingly no renormalizable, Lagrangian asymptotically free relativstic QFTs in 4 space-time dimensions except nonabelian gauge theories, which have confinement. But there is no obvious relation between confinement and asymptotic freedom.

What?? Is QCD non renormalizable?

Of course it is. I edited my previous comment to say what I had intended to say.

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$\mathcal{N}=2$ supersymmetric pure Yang-Mills theory with gauge group $SU(2)$ (in spacetime dimension 4) is a non-abelian gauge theory, asymptotically free, without confinement. Remark that to give a precise definition of "confinement" can be tricky, but this example is probably without confinement whatever your precise definition is: $SU(2)$ is Higgsed to $U(1)$ by the scalar field, the low energy theory is some $U(1)$ gauge theory, including some stable massive $U(1)$-charged particles.

answered Mar 26, 2019 by (5,140 points)

Just for clarity, why is there a scalar field in a pure Yang-Mills theory?

It is a "pure Yang-Mills theory" only as a N=2 theory, i.e. it contains only one N=2 vector multiplet, i.e. the smallest set of fields containing a gauge field and invariant under N=2 supersymmetry (and without gravity). In plain words, we have in the Lagrangian: a gauge field, a massless Dirac fermion and a massless complex scalar field, all in the adjoint representation of the gauge group.

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