Static potential inN=4supersymmetric Yang-Mills theory at weak coupling

Abstract

We compute the static potential associated to the locally 1/2 BPS Wilson loop in N =4 supersymmetric Yang-Mills theory with O(λ 2 /r) accuracy. We also resum the leading logarithms, of O(λ n+1 ln n λ/r), and show the structure of the renormalization group equation at next-to-leading order in the multipole expansion. In order to obtain these results it is crucial the use of an effective theory for the ultrasoft degrees of freedom. We develop this theory up to next-to-leading order in the multipole expansion. Using the same formalism we also compute the leading logarithms, of O(λ n+3 ln n λ/r), of the static potential associated to an ordinary Wilson loop in the same theory.PACS numbers: 11.30. Pb, 11.10.Hi, 12.39.Hg There is a huge interest in the study of the YangMills theory with N = 4 supersymmetry in four dimensions. One of the reasons is the conjectured existence of a correspondence between the N =4 supersymmetric Yang-Mills and the type IIA superstring theory on an AdS 5 × S 5 background [1]. This duality is known as the AdS/CFT correspondence and has importance consequences, since it allows to compute the strong 't Hooft coupling limit (λ ≡ Ncg 2 4π ) of certain correlators in the N =4 supersymmetric Yang-Mills theory for large N c . This is so because this limit becomes equivalent to the classical limit on the string theory side for which computational techniques exist.Checks of this conjecture are difficult to obtain, since the only quantitative approach to gauge theories is based on computations at weak coupling. In some cases it is possible to check the conjecture with the result at weak coupling. This usually happens when non-renormalization theorems exists that permit to perform the computation on the perturbative side exactly.In other cases no such checks exist and, usually, one can only study both the weak and strong coupling limit with increasing degree of accuracy hoping to gain further input on how the extrapolation from weak to strong coupling takes place. In this paper we concentrate in one of such correlators, the locally 1/2 BPS static Wilson loop, or, more specifically, the large time limit of its logarithm. Its strong 't Hooft coupling limit for large N c has been computed using the AdS/CFT correspondence in [2,3]. On the other hand the question whether the understanding of the infrared structure of the static potential in the weak coupling regime may shed some light on the AdS/CFT correspondence has been addressed in Refs. [4,5]. In these references some infrared divergences were found, which allowed the authors to obtain the leading logarithmic correction to the tree level result. These infrared divergences have a similar origin to those found in the QCD static potential at weak coupling [6], which, however, in this case first appear at three loops. They are due to the existence of degrees of freedom with energy of O(λ/r), which we will call ultrasoft in what follows. This scale is much smaller than the soft scale ∼ 1/r at weak coupling. In any case, it is somewhat surprising that, af...

Quark-anti-quark potential in N $$ \mathcal{N} $$ = 4 SYM

“…Currently the function Ω(λ) is known at 3 loops at weak coupling [1,[3][4][5][6][7][8] and at one loop at strong coupling [9][10][11][12]. In fact even at low orders the weak coupling expansion is rather involved and requires using a nontrivial low-energy effective theory.…”

“…We will first demonstrate this procedure at the leading order in the coupling and then present our result to a high order in perturbation theory. 3 To study the 2nd scale we start from the 4th order Baxter equation (2.9) and expand it at large u (notice that in the 2nd scale u is large as it is ∼ 1/g 2 ). By doing this we obtain a finite difference equation of the form…”

Generalized quark‐antiquark potential in AdS/CFT

“…where V (2) is written as a sum of the contribution of ladder 4 and interacting graphs. The analytic expressions (5) undergo various checks.…”

“…Here we have extended the expressions to θ = 0 and computed the integrals in closed form. 4 After subtracting the exponentiation of the O(λ) term. 5 Note the uniform transcendentality three (when e 2iφ is considered rational) of both interacting and ladder graphs at this order.…”

“…Above, L is the distance between the lines, K is the complete elliptic integral of the first kind and the weakcoupling expansion is the field-theoretical calculation of [2,3,4]. On the string theory (strong coupling) side, the question of evaluating the first quantum string correction a 1 to the classical result of [1] 1 is a hard mathematical problem.…”