Trifundamental quartic model

We consider a multiscalar field theory either with short-range or long-range free action and with quartic interactions that are invariant under $O({N}_{1})\ifmmode\times\else\texttimes\fi{}O({N}_{2})\ifmmode\times\else\texttimes\fi{}O({N}_{3})$ transformations, of which the scalar fields form a trifundamental representation. We study the renormalization group fixed points at two loops at finite $N$ and in various large-$N$ scaling limits for small $\ensuremath{\epsilon}$, the latter being the deviation either from the critical dimension or from the critical scaling of the free propagator. In particular, for the homogeneous case ${N}_{i}=N$ for $i=1$, 2, 3, we study the subleading corrections to previously known fixed points. In the short-range model, for $\ensuremath{\epsilon}{N}^{2}\ensuremath{\gg}1$, we find complex fixed points with nonzero tetrahedral coupling that at leading order reproduce the results of Giombi et al. [Phys. Rev. D 96, 106014 (2017).]; the main novelty at next-to-leading order is that the critical exponents acquire a real part, thus allowing a correct identification of some fixed points as IR stable. In the long-range model, for $\ensuremath{\epsilon}N\ensuremath{\ll}1$, we find again complex fixed points with nonzero tetrahedral coupling that at leading order reproduce the line of stable fixed points of Benedetti et al. [J. High Energy Phys. 06 (2019) 053]; at next-to-leading order, this is reduced to a discrete set of stable fixed points. One difference between the short-range and the long-range cases is that in the former the critical exponents are purely imaginary at leading order and gain a real part at next-to-leading order, while for the latter the situation is reversed.

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