Accessibility of the second stable branch within perturbative RG

Determine whether the second real stable fixed-point branch near the physical forcing exponent \(\varepsilon=2\), if it exists, can be accessed within the perturbative renormalization-group framework or whether its emergence is intrinsically tied to the nonperturbative region of the RG flow.

Background

The proposed rational continuation of the five-loop large-d series produces a second real stable domain near $\varepsilon=2$, separated from the perturbative branch near $\varepsilon=0$ by an interval in which the reconstructed fixed point is complex. The paper explicitly cautions that this second branch has not been rigorously established and requires independent verification.

The unresolved issue is whether perturbative RG methods can reach or describe this putative physical branch, or whether it belongs only to a nonperturbative regime. Computing additional $\varepsilon$-expansion coefficients is identified as a direct way to test the conjectured continuation.