One-loop fixed points of adjoint multi-scalar gauge theory in four dimensions

Abstract

We determine complete one-loop beta functions of the multi-scalar four-point couplings in four-dimensional $SU(N)$ gauge theories with $M$ adjoint scalar multiplets. For adjoints scalars, the sign of the one loop gauge coupling beta function depends solely on $M$, vanishing and changing sign precisely at $M=22$. For the multi-scalar potential at fixed gauge coupling we find several fixed points with different stability properties at large $N$. The analysis crucially involves the full set of four $SU(N)$ and $O(M)$ invariant single trace and double trace couplings. Taking the gauge coupling into account, there are asymptotically free RG flows for $M<22$ and non-trivial fixed points for $M=22$ at one loop, while $M>22$ appears to ruin the UV properties of the theory. Surprisingly, uniquely between $M=22$ and $M=21$ the number of fixed flows drops from eight to four in the large $N$ limit. There seems to be something very special about $M=22$. More speculatively, the $M=22$ one-loop conformal fixed point theory with $M$ adjoint scalars in $d=4$ suggests the possibility of an isolated non-supersymmetric, purely bosonic AdS${}_{4+1}\times$S${}^{22-1}$/CFT${}_4$ correspondence. Our example suggests that extending the potential to the complete set of terms allowed by symmetries may lead to real fixed points also in non-supersymmetric theories descending from $\mathcal{N}=4$ super-Yang-Mills theory.

Comments: The preprint is withrawn because of its erroneous conclusion on real fixed points. The determination of coefficients in equation 6.1 was mistaken, not to the algebraic approach, and correct results will be published in a different context