What’s so bad about a negative coupling constant? A look at a QCD-like theory
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The scalar O(N) model in 3+1D can be solved non-perturbatively in the coupling, and has been found to be qualitatively similar to QCD. Namely, the theory has asymptotic freedom, bound states, and an equation of state similar to QCD. This makes it a nice testing ground for assumptions in heavy-ion physics. The asymptotic freedom of the model is a result of examining the beta function and determining that the coupling becomes negative in the UV above the Landau pole. To understand how a theory can still be defined for negative couplings, one can look for insight from PT-symmetric quantum theories, which have negative or even complex potentials, and yet can have physical partition functions. Specifically, we examine the massless ϕ4 scalar model with negative coupling in 0D and 0+1D, both for large N and for N=1. We test a recent conjecture by Ai, Bender, and Sarkar relating the partition function of the PT-symmetric field theory to that of the positive-coupling theory, and find that the conjecture fails for the massless case. However, we show that a related conjecture concerning the negative-coupling theory holds at large N. We claim that there is nothing wrong with negative couplings under certain conditions, and a Landau pole should not be seen as indicating that a theory has no interacting continuum limit. Lastly, we consider how this might apply to non-perturbative calculations in QCD, where the coupling may have a Landau pole in the IR.
