Nesting and dressing

Abstract

We compute the anomalous dimensions of field strength operators Tr F L in N = 4 SYM from an asymptotic nested Bethe ansatz to all-loop order. Starting from the exact solution of the one-loop problem at arbitrary L, we derive a single effective integral equation for the thermodynamic L → ∞ limit of these dimensions. We also include the recently proposed phase factor for the S-matrix of the planar AdS/CFT system. The terms in the effective equation corresponding to, respectively, the nesting and the dressing are structurally very similar. This hints at the physical origin of the dressing phase, which we conjecture to arise from the hidden presence of infinitely many auxiliary Bethe roots describing a nontrivial "filled" structure of the theory's BPS vacuum. We finally show that the mechanism for creating effective nesting/dressing kernels is quite generic by also deriving the integral equation for the all-loop dimension of a certain one-loop so(6) singlet state.There is much evidence that planar N = 4 SYM theory is integrable and that its spectral problem is therefore exactly solvable. It was shown by Minahan and Zarembo that the dilatation operator in the scalar matter sector at one loop can be mapped to the Hamiltonian of an integrable so(6) spin chain and hence its eigenvalues can be found with a Bethe ansatz [1]. This extends to the full set of operators, leading to an integrable non-compact nearest-neighbor super magnet [2]. The special "solvable" properties of the N = 4 model under dilatation were already hinted at by Lipatov in [3], and extend a rather generic if incomplete phenomenon in more general gauge theories such as QCD, as first shown by Belitsky, Braun, Derkachov, Korchemsky and Manashov [4].The concept of factorized scattering, one of the hallmarks of integrability, can be extended to higher loop orders [5], and to strong coupling [6], where the gauge theory is expected to be more suitably described by a superstring theory in a curved AdS 5 × S 5 background. Various analyses of this topic [7] led to a set of asymptotic all-loop Bethe equations for the full theory [8]. It should be stressed that quantum integrability remains to be proven in both gauge and string theory. In particular, on the gauge side one phenomenologically finds an integrable long-range spin chain, and the all-loop factorization of the multi-body magnon S-matrix into two-body processes currently has to be assumed. Correspondingly, symmetry fixes the magnon S-matrix only up to an overall phase factor [5,8,9]. The latter encodes our lack of understanding of the underlying microscopic integrable structure 1 . However, as was argued by Janik, the dressing phase may be constrained by invoking crossing-invariance 2 [11]. And indeed the string S-matrix satisfies crossing to the known [6, 12] orders [13]. A proposal for the complete structure of the dressing phase has recently been made in [14] by combining Bethe ansatz techniques for the all-order perturbative large spin limit of Wilson twist operators [15] with conjectures by Beisert, H...

A generalized scaling function for AdS/CFT

“…For a possible mechanism generating this type of convolution structure see [22]. The novel contributions generated by a non-vanishing j are encoded in the kernel 15) as well as the explicit, rightmost term of the first line of (1.10), and the further convolution in the second line of that equation.…”

“…Similarly, the one-loop anomalous dimension γ 1 , see (2.2), may be rewritten as 22) where γ E is Euler's constant. Note that the NLIE (2.18) in conjunction with the Bethe equations for the hole roots (2.14) is fully equivalent, for the ground-state, to the algebraic Bethe equations (2.1) for arbitrary finite values of M and L. (The generalization to the case of excited states is fairly straightforward but will not be discussed in this paper.)…”

Constructing the AdS/CFT dressing factor

“…Fortunately it is possible to choose m i in such a way that it is so. For example m1 = m , m4 = −m (35) and all the others m i are zero. This amounts to a different prescription for the mode numbers comparatively to [21].…”

Wrapping interactions at strong coupling: The giant magnon

“…Finally let us note that the complexity of the dressing phase has prompted several groups to suggest that it can be obtained in some simpler setting by eliminating other degrees of freedom/higher levels [30,31,32,33,34]. We hope that the formalism of finite size corrections may be a strong test on the proposed constructions as it is sensitive to all virtual particles of the theory.…”