Scattering amplitudes at strong coupling for 4K gluons

TL;DR

This paper addresses the problem of computing scattering amplitudes at strong coupling for $n=4K$ gluons in ${ m N}=4$ SYM, where a nontrivial monodromy around infinity arises. The authors develop a framework that combines the Hitchin/Y-system with a cutoff regularization and introduce ${T}$-functions, which are obtained as limits of ${Y}$-functions from higher-point systems, to fully determine the amplitudes. A key result is the explicit treatment of the cutoff part, including monodromy through $oldsymbol{\Delta_x}, oldsymbol{\Delta_y}$ and the necessary nonadjacent equations, along with a conjectured periods part on the relevant Riemann surface; these ingredients reproduce known ${ m AdS}_3$ results in the eight-point case and provide a consistent structure for ${ m AdS}_5$ as well. The work clarifies the amplitude/Wilson-loop duality at strong coupling for even $n$, highlights the nonuniqueness of the BDS-like piece in the $n=4K$ sector, and lays out a systematic approach for incorporating monodromy and nonadjacent data in the integrability framework.

Abstract

In this paper we study the scattering amplitudes at strong coupling for the case where the number of gluons is a multiple of four. This is an important missing piece in arXiv:1002.2459. The tricky point for n=4K is that there is some accidental degeneracy in such case. We explain this point in detail and show that a non-trivial monodromy around infinity was developed by the world-sheet coordinate transformation appearing in the computation. It turns out that besides solving the Y system, we also need to calculate T functions to compute the full amplitudes. We show that the T functions can be derived by taking a limit of Y functions of a higher-point case. As a check, we obtain the known result of eight-point in AdS_3 case.

Figures (3)

• Figure 1: The cutoff of the surface $\Sigma_0$. Fig (a) shows a portion of the surface in the $w$-plane. $L=-\log\epsilon_c$ is the cutoff. $\delta_i = -\log {\hat{X}}_i^+$. The origin should be chosen to be one of zeros of the polynomial $P(z)$. Fig (b) shows that for $n=4K$ cases the surface is not closed. There is a formal monodromy $\Delta = \Delta_x + i \Delta_y$, thus $\delta_{n+1} = \delta_1 + \Delta_x$, $\delta_{n+2} = \delta_2 + \Delta_y$. The total area is the sum of the area of various rectangles. Notice that we choose to treat the first cusp in a special way. Half of it from $\delta_1$ at the beginning, and half from the end of surface with $\delta_{n+1}$ which includes the effect of monodromy.
• Figure 2: The limit behavior of the WKB pattern. The crosses are zeros of the polynomial $p(z)$. The numbers indicate the various Stokes sectors. The dotted lines are WKB lines which connect different stokes sectors. The solid lines ending on the zeros separate different classes of WKB lines. We consider two different phases of $\zeta$, which show the contour formed by WKB lines for $Y_2$ and $Y_1$ respectively. By taking the rightmost zero to infinity, the structure of $n$=10 is reduced to that of $n$=8, and $Y_2$ and $Y_1$ of the higher-point case are reduced to $Y_1$ and $T_1$ of the lower-point case. Notice the change of labels of the Stokes sectors in the limit.
• Figure 3: The pattern of cycle structure for the Riemann surface. The crosses represent the zeros of polynomial $p(z)$. The wave lines indicate the branch cuts. There is a branch point at infinity for $n=4K+2$ case. Notice the the cycle $\gamma_m^\infty$ in $n=12$ may be taken as the the cycle $\gamma^1$ in $n=14$ by taking the rightmost zero to infinity.