Two Dimensional Renormalization Group Flows in Next to Leading Order
Rubik Poghossian
TL;DR
The paper advances Zamolodchikov’s analysis of the RG flow from the 2D minimal model $M_p$ to $M_{p-1}$ by incorporating second-order perturbative corrections in the coupling to the nearly marginal field $φ_{1,3}$. Using a careful second-order perturbation framework, conformal symmetry to fix insertion points, and the large-$p$ (small $ε$) expansion, it computes the beta function, the $c$-function, and the anomalous dimensions for several operator families, establishing a precise UV→IR map. The IR fixed point is shown to coincide with $M_{p-1}$, with anomalous-dimension spectra matching the IR dimensions and explicit, scheme-consistent maps of UV fields to linear combinations of IR fields; strikingly, the mixing coefficients remain ε- and ε^2-free in the adopted Zamolodchikov scheme. The results also corroborate Gaiotto’s RG domain wall concept within this higher-order context, clarifying the operator-mixing structure along the flow and reinforcing the robustness of Zamolodchikov’s leading-order picture at next-to-leading order in $1/p$. Overall, the work provides concrete, high-precision UV–IR identifications for multiple operator families and strengthens the understanding of 2D RG flows in the near-marginal regime.
Abstract
Zamolodchikov's famous analysis of the RG trajectory connecting successive minimal CFT models $M_p$ and $M_{p-1}$ for $p\gg 1$, is improved by including second order in coupling constant corrections. This allows to compute IR quantities with next to leading order accuracy of the $1/p$ expansion. We compute in particular, the beta function and the anomalous dimensions for certain classes of fields. As a result we are able to identify with a greater accuracy the IR limit of these fields with certain linear combination of the IR theory $M_{p-1}$. We discuss the relation of these results with Gaotto's recent RG domain wall proposal.