Abstract
We study the properties of operators in a unitary conformal field theory whose scaling dimensions approach each other for some values of the parameters and satisfy von Neumann-Wigner non-crossing rule. We argue that the scaling dimensions of such operators and their OPE coefficients have a universal scaling behavior in the vicinity of the crossing point. We demonstrate that the obtained relations are in a good agreement with the known examples of the level-crossing phenomenon in maximally supersymmetric $\mathcal N=4$ Yang-Mills theory, three-dimensional conformal field theories and QCD.
Highlights
We demonstrate that the obtained relations are in a good agreement with the known examples of the level-crossing phenomenon in maximally supersymmetric N = 4 Yang-Mills theory, three-dimensional conformal field theories and QCD
JHEP03(2016)212 product given by two-point correlation function of Oi(x) on the cylinder and, its eigenvalues ∆i have to satisfy von Neumann-Wigner non-crossing rule [2] stating that the levels of the dilatation operator with the same symmetry cannot cross
We argued that the scaling dimensions of such operators and their OPE coefficients have a universal scaling behavior in the vicinity of the crossing point
Summary
Let us consider a pair of conformal operators O1 and O2 whose scaling dimensions depend on the coupling constant g and admit the 1/N expansion (1.1). Going to N → ∞ limit on the both sides of (2.4) and taking into account that ǫ = O(1/N ), we find c2 = −c1 + O(1/N ) together with φ x2 We can use this equation to express the coefficients c1 and c2 in terms of γ and obtain from (2.3) the conformal operators O± that take into account the leading nonplanar correction.. Matching this relation into (2.1) we obtain the leading O(1/N ) correction to the scaling dimension of the operator O−. Let us assume that the scaling dimensions ∆1 and ∆2 are continuous functions of the coupling constant g, intersecting in the planar limit at g = 0. Away from the crossing point, the resummed scaling dimensions approach their values in the planar limit, e.g. Away from the crossing point, the resummed scaling dimensions approach their values in the planar limit, e.g. ∆+ ≈ ∆1 at large negative g goes into ∆+ ≈ ∆2 at large positive g
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