Abstract
We continue the study of the bosonic O(N )3 model with quartic interactions and long-range propagator. The symmetry group allows for three distinct invariant \U0001d7194 composite operators, known as tetrahedron, pillow and double-trace. As shown in [1, 2], the tetrahedron operator is exactly marginal in the large-N limit and for a purely imaginary tetrahedron coupling a line of real infrared fixed points (parametrized by the absolute value of the tetrahedron coupling) is found for the other two couplings. These fixed points have real critical exponents and a real spectrum of bilinear operators, satisfying unitarity constraints. This raises the question whether at large-N the model is unitary, despite the tetrahedron coupling being imaginary.In this paper, we first rederive the above results by a different regularization and renormalization scheme. We then discuss the operator mixing for composite operators and we give a perturbative proof of conformal invariance of the model at the infrared fixed points by adapting a similar proof from the long-range Ising model. At last, we identify the scaling operators at the fixed point and compute the two- and three-point functions of \U0001d7194 and \U0001d7192 composite operators. The correlations have the expected conformal behavior and the OPE coefficients are all real, reinforcing the claim that the large-N CFT is unitary.
Highlights
In this paper, we first rederive the above results by a different regularization and renormalization scheme
The symmetry group allows for three distinct invariant φ4 composite operators, known as tetrahedron, pillow and double-trace
We discuss the operator mixing for composite operators and we give a perturbative proof of conformal invariance of the model at the infrared fixed points by adapting a similar proof from the long-range Ising model
Summary
We study the tensor model of [1, 2], that is, the O(N ) tensor model of Klebanov and Tarnopolsky [26] and Carrozza and Tanasa [44] (CTKT model) with a long-range covariance. Each interaction invariant is a 3-colored graph, and the Feynman propagators are represented as edges with a new color, which we call 0 (pictured in black), connecting the tensors. This leads to a representation of the perturbative expansion in terms of 4-colored graphs, as for example in figure 2. As a result of the combination of pillow, double-trace, and tetrahedron interactions, our model has a 1/N expansion dominated by melon-tadpole diagrams [1] with melons based on couples of tetrahedral vertices and tadpoles based on either pillow or double-trace vertices (see figure 4)
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