Determine whether a third relevant direction exists in three dimensions

Determine whether the candidate three-dimensional \(\mathcal N=1\) Gross–Neveu–Yukawa fixed point with two superfields possesses a third relevant direction in the full six-coupling space by computing higher-order corrections to the unresolved stability eigenvalue.

Background

The six-coupling stability analysis of the coupled fixed point yields two positive resummed eigenvalues at $d=3$: one supersymmetric relevant direction and one nonsupersymmetric direction. A third eigenvalue has a poorly controlled two-loop continuation because its contributions are comparable and have opposite signs.

Depending on whether one uses a raw truncation or a Padé approximant, this eigenvalue appears either positive or negative near $d=3$. Thus the number of relevant deformations, and consequently the infrared accessibility and universality properties of the proposed three-dimensional theory, cannot be settled at the order studied.