# Classical integrability for three-point functions: cognate structure at weak and strong couplings

Kazama, Yoichi (Research Center for Mathematical Physics, Rikkyo University, Toshima-ku, Tokyo, 171-8501, Japan) (Quantum Hadron Physics Laboratory, RIKEN Nishina Center, Wako, 351-0198, Japan) (Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo, 153-8902, Japan) ; Komatsu, Shota (Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada) ; Nishimura, Takuya (Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo, 153-8902, Japan)

13 October 2016

Abstract: In this paper, we develop a new method of computing three-point functions in the SU(2) sector of the N = 4 $\mathcal{N}=4$ super Yang-Mills theory in the semi-classical regime at weak coupling, which closely parallels the strong coupling analysis. The structure threading two disparate regimes is the so-called monodromy relation, an identity connecting the three-point functions with and without the insertion of the monodromy matrix. We shall show that this relation can be put to use directly for the semi-classical regime, where the dynamics is governed by the classical Landau-Lifshitz sigma model. Specifically, it reduces the problem to a set of functional equations, which can be solved once the analyticity in the spectral parameter space is specified. To determine the analyticity, we develop a new universal logic applicable at both weak and strong couplings. As a result, compact semi-classical formulas are obtained for a general class of three-point functions at weak coupling including the ones whose semi-classical behaviors were not known before. In addition, the new analyticity argument applied to the strong coupling analysis leads to a modification of the integration contour, producing the results consistent with the recent hexagon bootstrap approach. This modification also makes the Frolov-Tseytlin limit perfectly agree with the weak coupling form.

Published in: JHEP 1610 (2016) 042