Conformal constraints for anomalous dimensions of leading twist operators

TL;DR

This work demonstrates the feasibility of extracting two-loop anomalous dimensions of leading-twist operators from one-loop conformal data using a light-ray operator framework in the $\varphi^3$ model at its conformal fixed point. By combining a carefully defined scalar product on conformal eigenfunctions, a controlled divergence analysis, and $sl(2)$-based diagrammatic techniques, the authors compute the two-loop anomalous dimensions across isotopic sectors and verify consistency with known two-loop evolution kernels. The approach yields a nontrivial cross-check of standard perturbative results, while clarifying that the apparent simplifications are offset by the need to handle finite parts in the MS scheme. Overall, the method provides a robust alternative route to anomalous dimensions that can serve as a stringent consistency check for higher-order calculations, with the caveat that practical gains depend on enabling higher-order automation via computer algebra.

Abstract

Leading-twist operators have a remarkable property that their divergence vanishes in a free theory. Recently it was suggested that this property can be used for an alternative technique to calculate anomalous dimensions of leading-twist operators and allows one to gain one order in perturbation theory so that, i.e., two-loop anomalous dimensions can be calculated from one-loop Feynman diagrams, etc. In this work we study feasibility of this program on a toy-model example of the $\varphi^3$ theory in six dimensions. Our conclusion is that this approach is valid, although it does not seem to present considerable technical simplifications as compared to the standard technique. It does provide one, however, with a very nontrivial check of the calculation as the structure of the contributions is very different.

Figures (6)

• Figure 1: The leading order diagrams for the correlator of two conformal operators, $\langle{\mathcal{O}^{(n)}_j(x)\mathcal{O}_j^{(\bar{n})}(0)}\rangle$.
• Figure 2: The LO diagrams for the correlator of divergence of conformal operators, $\langle{\partial\mathcal{O}^{(n)}_j(x)\partial\mathcal{O}_j^{(\bar{n})}(0)}\rangle$.
• Figure 3: NLO correction to the correlator of conformal operators $\langle{\mathcal{O}^{(n)}_j(x)\mathcal{O}_j^{(\bar{n})}(0)}\rangle$. The parameter $\delta=\epsilon/2$.
• Figure 4: The "sl(2)" diagram: an arrow line from $w$ to $z$ with index $\alpha$ stands for the propagator $(1-z\bar{w})^{-\alpha}$. The indices have the following values: $\alpha=2-3\epsilon/2$, $\beta=1-\epsilon/2$ and $\gamma=1$. The black circle denote an integration vertex with the $sl(2)$ invariant measure $\mu_{s+\delta}$, $s+\delta=3/2-\epsilon/4$.
• Figure 5: One loop correction to the correlator of the divergence of conformal operators $\langle{\partial\mathcal{O}^{(n)}_j(x)\partial\mathcal{O}_j^{(\bar{n})}(0)}\rangle$.