Origin of dressing phase in N = 4 Super Yang – Mills

We derive the phase factor proposed by Beisert, Eden and Staudacher for the Smatrix of planar N = 4 Super Yang–Mills, from the all-loop Bethe ansatz equations without the dressing factor. We identify a configuration of the Bethe roots, from which the closed integral formula of the phase factor is reproduced in the thermodynamic limit. This suggests that our configuration describes the “physical vacuum” in the sense that the dressing phase is nothing but the effective phase for the scattering of fundamental excitations above this vacuum, clarifying the physical origin of the dressing phase. March 2007 [email protected] [email protected] Integrability has become of increasing importance in the study of N = 4 Super Yang–Mills (SYM) and of the dual superstrings in AdS5 ×S5. The spectral problem of the dilatation operator at one loop was identified with that of a conventional integrable spin-chain [1,2], which can be systematically solved by using Bethe ansatz. Integrability beyond one loop has also been extensively studied and, in particular, the all-loop Bethe ansatz equations were postulated [3]. Note, however, that the spin-chain picture does not fully apply at higher loops due to several new features yet unknown in the field of integrable models, such as length fluctuation. Nevertheless, conventional integrability revives by converting the picture into a particle model, at least in the limit of infinite length of operators, or the large-spin limit [4–6]. Asymptotic particle states were realized in terms of SYM operators [7]. They exhibit the factorized scattering property and thus all the multi-particle scattering processes are governed by the elementary two-particle S-matrix. This S-matrix was determined up to an overall scalar factor by purely algebraic consideration of the centrally extended su(2|2) symmetry [8, 9]. As is expected from the AdS/CFT correspondence, this S-matrix with a pair of the su(2|2) symmetry also emerges on the string theory side [10–13]. The choice of the gauge breaks the conformal invariance in two dimensions and one obtains a massive worldsheet theory, where S-matrix is naturally defined as the scattering of elementary excitations. As the symmetry completely constrains the form of the matrix, what is left to be determined is again the overall scalar factor. The determination of the scalar factor, as a function of two momenta and the coupling, is important in two aspects: Firstly, it is the last missing element for the systematic construction of the spectrum of the scaling dimension/energy on the Yang– Mills/string side. Secondly, identification of the scalar factors on both sides serves as a strong quantitative check of the AdS/CFT correspondence. The form of the scalar factor was first studied on the string side, based on the data of classical string spectrum [14]. Succeedingly 1/ √ λ corrections were analyzed and an all-order form was postulated [15]. This form was shown [16] to be consistent with the constraint from the crossing symmetry [17]. On the other hand, the form of scalar factor was rather obscure on the Yang–Mills side, since it stays trivial up to three loops. However, it turned out to deviate from the unity at four loops [18]. Meanwhile, Beisert, Eden and Staudacher managed to construct a closed integral formula [19] consistent with the above four-loop result as well as a sort of analytic continuation of the proposal on the string side. The integral formula is highly intricate, but does not seem totally hopeless to handle analytically. We refer to a reference [20] for recent investigations.