A numerical study of gluon scattering amplitudes in 𝒩 = 4 super Yang-Mills theory at strong coupling

Abstract

We study gluon scattering amplitudes in N = 4 super Yang-Mills theory at strong coupling via the AdS/CFT correspondence. We solve numerically the discretized Euler-Lagrange equations on the square worldsheet for the minimal surface with light-like boundaries in AdS spacetime. We evaluate the area of the surface for the 4, 6 and 8-point amplitudes using worldsheet and radial cut-off regularizations. Their infrared singularities in the cut-off regularization are found to agree with the analytical results near the cusp less than 5% at 520×520 lattice points. Recently Alday and Maldacena[1] proposed the prescription for the calculation of gluon scattering amplitudes in N = 4 super Yang-Mills (SYM) theory at strong coupling via the AdS/CFT correspondence. They showed that the gluon scattering amplitude is related to the area of a minimal surface in AdS spacetime surrounded by the Wilson loop with lightlike boundaries. They found the exact solution of the minimal surface corresponding to the 4-point amplitude and showed that it reproduces the perturbative formula conjectured by Bern, Dixon and Smirnov (BDS) [2]. The correspondence between gluon amplitudes and the Wilson loops has been examined also at weak coupling[3]. It is shown in [4] that the Wilson loops at weak coupling obey the anomalous conformal Ward identity, which determines the n = 4 and 5-point amplitudes completely. For n ≥ 6-point amplitudes, however, the conformal invariance of the amplitudes does not fix their dependence on the kinematical variables. Recently it is found that the 6-point 2-loop corrections to the Wilson loop agrees numerically with the gluon amplitudes but they differ from the BDS formula[5].In order to study gluon scattering amplitudes at strong coupling using the AdS/CFT correspondence, we need to find the solution of the minimal surface in AdS spacetime surrounded by the light-like Wilson loop. See [6,7,8,9,10,11,12,15] for references. The minimal surface is obtained by solving the Euler-Lagrange equations with the Dirichlet boundary conditions. These are non-linear partial differential equations and highly nontrivial to solve analytically for the polygon with n ≥ 5 boundaries. Any exact solution for n ≥ 5 is not yet known so far. In a previous paper [9], the authors including the two authors in the present paper, constructed solutions for the 6 and 8-point amplitudes by cutting and gluing the 4-point amplitude. The evaluation of the amplitudes shows that they do not agree with the BDS formula both for the infrared singularity and finite parts. In particular, the infrared singularities coincide with those of the 4-point amplitudes, which suggests that these simple solutions are not the minimal surfaces and might correspond to the other disconnected amplitudes.In this paper we will propose a practical approach to compute the minimal surface in AdS spacetime. We investigate numerically the solution of the Euler-Lagrange equations

Discretized minimal surface and the BDS conjecture in N=4 super-Yang–Mills theory at strong coupling

“…In the radial cut-off regularization scheme we introduce a cut-off r c in the radial direction [3,9]. For general (s, t) solution, the regularized area is surrounded by the cut-off curve C in the (y 1 , y 2 )-plane:…”

“…17) since π 2 3 = 3.28987.... Motivated from the analysis of the 4-point amplitude, the n-point amplitude is expected to have the structure [3,9]…”

“…Then we obtain 2 × (M − 1) 2 nonlinear simultaneous equations for y 0 [i, j] and r[i, j] and use Newton's method to find a numerical solution. In this paper we use the M = 520 lattice data [9], where the Newton method is repeatedly applied until the discrete equation The area S becomes large as M increases, which is due to the IR divergent behavior near cusps. In [9] we have defined the area of the surface in the radial cut-off regularization…”

“…In the case of the 4-point amplitude, Alday and Maldacena [1] showed that the dimensionally regularized area agrees with the formula conjectured by Bern, Dixon and Smirnov (BDS) [2] based on the perturbative analysis. See [3,4,5,6,7,8,9] for further developments.…”

“…In a previous paper [9], we constructed the minimal surfaces corresponding to the 4, 6 and 8 point amplitudes numerically and evaluate the area in the radial cut-off regularization. The light-like segments of the boundary is the same as the cut and glue type surface [8].…”