Abstract
In this paper, we develop a new method of computing three-point functions in the SU(2) sector of the \( \mathcal{N}=4 \) 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.
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ArXiv ePrint: 1603.03164
An erratum to this article can be found online at http://dx.doi.org/10.1007/JHEP02(2018)047.
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Kazama, Y., Komatsu, S. & Nishimura, T. Classical integrability for three-point functions: cognate structure at weak and strong couplings. J. High Energ. Phys. 2016, 42 (2016). https://doi.org/10.1007/JHEP10(2016)042
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DOI: https://doi.org/10.1007/JHEP10(2016)042