T-functions and multi-gluon scattering amplitudes

TL;DR

This work develops a unified framework connecting gluon scattering amplitudes in ${\cal N}=4$ SYM at strong coupling to integrable systems through the Y-/T-system, enabling a concise expression of the remainder function for $2n$-point amplitudes in terms of T-functions. The authors derive a leading-order high-temperature expansion around the CFT/regular-polygon limit, with the momentum dependence captured by a single mass-coupling function $t_s^{(2,0)}$ (or $\kappa_n G$) and explicit forms in single-mass cases. They express all cross-ratios and the remainder via Y-/T-functions, provide detailed odd/even $n$ formulas, and show cancellation mechanisms that simplify the expansion. A thorough comparison with two-loop results reveals a striking closeness between strong-coupling and two-loop remainders across many $n$, and the large-$n$ analysis indicates a converging behavior of rescaled remainders, suggesting robust structural constraints on amplitudes across coupling regimes.

Abstract

We study gluon scattering amplitudes/Wilson loops in N=4 super Yang-Mills theory at strong coupling which correspond to minimal surfaces with a light-like polygonal boundary in AdS_3. We find a concise expression of the remainder function in terms of the T-function of the associated thermodynamic Bethe ansatz (TBA) system. Continuing our previous work on the analytic expansion around the CFT/regular-polygonal limit, we derive a formula of the leading-order expansion for the general 2n-point remainder function. The T-system allows us to encode its momentum dependence in only one function of the TBA mass parameters, which is obtained by conformal perturbation theory. We compute its explicit form in the single mass cases. We also find that the rescaled remainder functions at strong coupling and at two loops are close to each other, and their ratio at the leading order approaches a constant near 0.9 for large n.

Figures (5)

• Figure 1: Examples of sequential cross-ratios. $c_{1,6}$ for $n=7$ is shown in (a). $c_{2,6}^{aux}$ for $n=8$ is shown in (b). In (c), the dotted and dashed line stand for $c_{1,6}^{left}$ and $c_{1,6}^{right}$ for $n=8$, respectively. In (d), the dotted and dashed line stand for $d_{1,5}^{left}$ and $d_{1,5}^{right}$ for $n=8$, respectively. Superscripts $\pm$ are suppressed here for simplicity.
• Figure 2: Graphical representation of Y-functions for $n=7$. $Y_s^{[k]}$ are represented by tetragons in the heptagon formed by the cusp coordinates $x^\pm_{i}$$(i=1, ..., 7)$. Here, the $i$-th vertex stands for $x_{i}^{+}$. The $+$ sign indicates factors appearing in the numerator of the cross-ratios, whereas the $-$ sign indicates those in the denominator.
• Figure 3: Graphical representation of a recursion relation for $c_{i,j}$. The dashed line represents $c_{1,-2}$. The dotted line stands for $c_{2,-3}$. The bold line represents $Y_3$.
• Figure 4: Plots of the rescaled remainder functions at strong coupling $(\times)$ and at two loops $(+)$ for 12-point amplitudes (left) and for 14-point amplitudes (right). The functions are evaluated for $m_{s} = l e^{\frac{\pi i }{20}s}$. At $l=2$, $\bar{R}_{12}^{\rm strong} =-0.538$, $\bar{R}_{12}^{\rm2\hbox{-}loop} =-0.542$, whereas $\bar{R}_{14}^{\rm strong} =-0.559$, $\bar{R}_{14}^{\rm2\hbox{-}loop} =-0.565$.
• Figure 5: Decomposition of $c^{aux \, +}_{7,9}$ for $n=10$. The $i$-th vertex stands for $x_{i}^{+}$. The dotted line represents $c^{aux \, +}_{7,9}$ and the bold line represents the tetragon corresponding to $c_{3,9}^{aux \, +}$. The dashed line represents the fan-shaped tetragons corresponding to $T_{2(2+p)-1}^{[2+2p]}/T_{2(2+p)-3}^{[2p]}$ for $p=1,2$.