Tailoring Three-Point Functions and Integrability III. Classical Tunneling

TL;DR

This work derives the classical-limit form of three-point functions in $\mathcal{N}=4$ SYM with one non-BPS operator and two BPS operators by mapping to spin-chain overlaps and condensing Bethe roots into cuts. The authors obtain continuum expressions for the Gaudin norm $\mathcal{B}$ and the vacuum-descendent overlap $\mathcal{A}$, including a solvable long-range Ising model as a by-product, and express the structure constant ratio $r = \frac{C_{123}^{\circ\bullet\circ}(\{u\})}{C_{123}^{\circ\circ\circ}}$ as a compact functional of the cut density $\rho$ and quasi-momenta $q(u)$ via $r = \exp\left[ \int_0^1 dt \oint_{\cup\mathcal{C}_k} \frac{du}{2\pi i} q(u) \log(1-e^{iq(u)t}) - \int_{\cup\mathcal{C}_k} du \; \rho(u) \log(2\sinh(\pi t \rho(u))) \right]$. The approach combines determinant methods, path-integral saddle points, and a careful treatment of UV/IR anomalies, yielding a result with potential connections to strong-coupling analyses and fermionic-like phase-space interpretations. A notable by-product is the exact solution of a long-range Ising model in the thermodynamic limit, with a closed-form function $F(\rho)$ entering the anomaly. These findings illuminate a classical tunneling picture for three-point functions and motivate further generalizations in both gauge theory and integrable many-body contexts.

Abstract

We compute three-point functions between one large classical operator and two large BPS operators at weak coupling. We consider operators made out of the scalars of N=4 SYM, dual to strings moving in the sphere. The three-point function exponentiates and can be thought of as a classical tunneling process in which the classical string-like operator decays into two classical BPS states. From an Integrability/Condensed Matter point of view, we simplified inner products of spin chain Bethe states in a classical limit corresponding to long wavelength excitations above the ferromagnetic vacuum. As a by-product we solved a new long-range Ising model in the thermodynamic limit.

Figures (10)

• Figure 1: A classical extended string decays into two protected classical strings through what we call a classical tunneling process.
• Figure 2: Three-point function of $SU(2)$ operators at tree level considered in paper1. This is the simplest non-extremal non-trivial configuration. We consider two protected operators ${\mathcal{O}}_1$ and ${\mathcal{O}}_3$ and one non-protected operator ${\mathcal{O}}_2$. All operators are classical, i.e. made of a large number of fields. The nontrivial contraction (see arrows in the figure) involves $L'$ fields of ${\mathcal{O}}_2$. The non-trivial part of the structure constants is given by the universal ratio $\mathcal{A}(\bf u)/\mathcal{B}({\bf u})$, see (\ref{['c123disc']}). This ratio is in a sense an intrinsic property of the non-protected operator ${\mathcal{O}}_2$. It only knows about the (length of the) protected operators through $L'$ which only appears in $\mathcal{A}({\bf u})$, see (\ref{['AdiscInt']}).
• Figure 3: Two examples of real configurations of Bethe roots (pink dots in the complex $u$ plane) and corresponding densities $\rho(u)$ (blue dots above the complex $u$ plane). The depicted configurations are the so called two cut solutions. A general classical solution can have an arbitrary large number of cuts. The results presented in the main text are valid for all such configurations.
• Figure 4: In an interval $$u-du/2,u+du/2$$ one has $\rho(u) du$ roots described by an approximately constant density $\rho(u)$. As $du \to 0$ the approximation becomes exact. We can use this trick to study exactly quantities which are roughly diagonal as described in the main text. A very illustrative example is the toy model (\ref{['toyB']}).
• Figure 5: The second main ingredient in the construction is the function $\mathcal{A}({\bf u})$ in (\ref{['Adisc']}) graphically depicted in this figure. It is given by a sum over all possible ways of partitioning the Bethe rapidities $\{u_j\}$ into the partitions ${\color{blue}\alpha\color{black}}$ (blue empty circles) and ${\color{red}\bar{\alpha}\color{black}}$ (filled red circles). For example, for the configuration of seven roots displayed in the figure one sums over the $2^7=128$ way of splitting the Bethe roots into two groups. The purpose of the current section is to address the computation of $\mathcal{A}({\bf u})$ in the classical limit, i.e. when the number of roots becomes very large.