In view of its physical importance in predicting the order of chiral phase transitions in QCD and frustrated spin systems, we perform the conformal bootstrap program of [$O(n)\times O(2)$]-symmetric conformal field theories in $d=3$ dimensions with a special focus on $n=3$ and 4. The existence of renormalization group fixed points with these symmetries has been controversial over the years, but our conformal bootstrap program provides nonperturbative evidence. In both the $n=3$ and 4 cases, we find singular behaviors in the bounds of scaling dimensions of operators in two different sectors, which we claim correspond to chiral and collinear fixed points, respectively. In contrast to the cases with larger values of $n$, we find no evidence for the antichiral fixed point. Our results indicate the possibility that the chiral phase transitions in QCD and frustrated spin systems are continuous with the critical exponents that we predict from the conformal bootstrap program.

• Received 13 August 2014

DOI:https://doi.org/10.1103/PhysRevD.91.021901