g-Functions and gluon scattering amplitudes at strong coupling

TL;DR

This work develops a CPT-based, integrable-model framework to compute the remainder function for gluon scattering amplitudes at strong coupling in N=4 SYM, by mapping AdS_3 minimal surfaces to the SU(n)_2/U(1)^{n-1} homogeneous sine-Gordon model. It leverages the deep connection between g-functions and T-functions to obtain analytic high-temperature expansions of the Y-/T-functions and the free energy, enabling explicit octagon and decagon remainder-function results and comparisons to two-loop predictions. The approach unifies boundary and bulk perturbations, clarifies how modular S-matrix data encode CFT limits, and provides predictive power for multi-point amplitudes, with near-coincidence to two-loop forms suggesting strong symmetry constraints from the Y-system and boundary CFT structure. The findings open avenues for extending to higher-point amplitudes and AdS_5 contexts, and for refining multi-parameter deformations of generalized parafermion CFTs in the strong-coupling regime.

Abstract

We study gluon scattering amplitudes/Wilson loops in N=4 super Yang-Mills theory at strong coupling by calculating the area of the minimal surfaces in AdS_3 based on the associated thermodynamic Bethe ansatz system. The remainder function of the amplitudes is computed by evaluating the free energy, the T- and Y-functions of the homogeneous sine-Gordon model. Using conformal field theory (CFT) perturbation, we examine the mass corrections to the free energy around the CFT point corresponding to the regular polygonal Wilson loop. Based on the equivalence between the T-functions and the g-functions, which measure the boundary entropy, we calculate corrections to the T- and Y-functions as well as express them at the CFT point by the modular S-matrix. We evaluate the remainder function around the CFT point for 8 and 10-point amplitudes explicitly and compare these analytic expressions with the 2-loop formulas. The two rescaled remainder functions show very similar power series structures.

Figures (4)

• Figure 1: (a) The scale parameter $l$-dependence of the free energy with fixed $\tilde{M}_1/\tilde{M}_2=1,1/2$ and $1/10$. Dashed lines represent the curve $\frac{\pi}{5}+\frac{1}{2}\tilde{M}_1\tilde{M}_2l^2+f^{(2)}_3l^{8/5}$. Deviation from the analytic formula at $l=0.5$ for $\tilde{M}_1/\tilde{M}_2=1/10$ comes from the next order correction $O(l^{12/5})$, which is estimated from the numerical fit as $0.08 l^{12/5}\approx 0.015$. (b) $\tilde{M}_1$-dependence of the coefficient of $l^{8/5}$ of the free energy for $\tilde{M}_2=1$. Dashed line corresponds to the curve $f^{(2)}_3$ given in \ref{['eq:f32-2']}.
• Figure 2: (a) The $l$-dependence of the Y-function $Y_1(0)$ for fixed $\tilde{M}_1/\tilde{M}_2=1,1/2$ and $1/10$ at $\varphi_1=\varphi_2=\pi/20$. Dashed lines represent the curve \ref{['eq:exp-Y']} at $\theta=0$ up to the order $l^{4/5}$. Deviation from the analytic formula comes from the next order correction $O(l^{6/5})$, which can be estimated from the numerical fit as $0.15 l^{6/5}\approx 0.004$ ($l=0.05$) for $\tilde{M}_1=\tilde{M}_2=1$. (b) Plots of the coefficient of $l^{4/5}$ in $Y_1(0)$ for $\varphi_1=\pi/20$ and various $\varphi_2$ at $2\tilde{M}_1=\tilde{M}_2=1$. Dashed line represents the curve $\frac{1}{2}\bigl(Y^{(2)}(\tilde{M}_1e^{-i\varphi_1},\tilde{M}_2e^{-i\varphi_2}) +Y^{(2)}(\tilde{M}_1e^{i\varphi_1},\tilde{M}_2e^{i\varphi_2})\bigr) =: y^{(2)}_{\rm (RSOS)_{3}} h(\tilde{M}_{j},\varphi_{j})$.
• Figure 3: The $l$-dependence of the remainder function with (a) equal phase $\varphi_1=\varphi_2={\pi/20}$ (b) different phase $\varphi_1=\pi/20$, $\varphi_2=\pi/5$. Dashed lines represent the curve $R_{10}^{(0)}+R_{10}^{(4)}l^{8/5}$.
• Figure 4: Plots of the $l$-dependence of the rescaled remainder functions at strong coupling (points) and at two loops (dashed lines). The functions are evaluated at $\tilde{M}_1=\tilde{M_2}=1$ and $\varphi_1=\varphi_2={\pi/20}$.