Conformal symmetry and composite operators in the O(N )3 tensor field theory

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 φ 4 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 φ 4 and φ 2 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.

Instability of complex CFTs with operators in the principal series

“…Fishnet diagrams are very different from melonic diagrams in general. Nonetheless, as noticed in [73], the fishnet model has a number of similarities with the long-range O(N ) 3 model. For example, while melonic diagrams characteristically show up in the two-point function, there are no planar fishnet diagrams for the two-point function; however, for d < 4 the long-range nature of the kinetic term makes such difference irrelevant from the renormalization point of view, as there is no wave function renormalization in any case.…”

“…It is more convenient to illustrate the instability in long-range models. The long-range quartic O(N ) 3 model has been introduce and studied in [14,62,73,74] (see also [75] for a review).…”

“…The relation between λ and g, for real g is invertible up to the critical value 18), when evaluating the integral in the eigenvalue equation (2.40) the λ 2 term gives a vanishing contribution for Re(h) > d/2, where we expect to find the physical dimensions, and a divergent one for Re(h) < d/2; for h ∈ P, which is the important case corresponding to the basis of eigenfunctions, the result is indeterminate. This can be regulated by the same analytic regularization used for the perturbative computations in [73,74], that is, taking ζ = (d + )/4, or equivalently ∆ φ = (d − )/4; then, the λ 2 term does not contribute to the kernel eigenvalue, consistently with the Re(h) > d/2 half-plane where we wish to move the contour in (2.41). One finds two types of solutions of the equation k(h, J) = 1 at small renormalized tetrahedral coupling g. The first type,…”

Remarks on a melonic field theory with cubic interaction

“…In this respect, explicit calculations in the cubic O(N ) model of [44,45] have shown that indeed imaginary parts of scaling dimensions are (exponentially) suppressed at large N [58]. Moreover, along the lines of long-range O(N ) 3 tensor model [19][20][21][22], we will introduce and study also a long-range version of the AR model, for which the coupling is exactly marginal at large N , and we will find that in this case a real and unitary CFT can be identified at small coupling for either real or imaginary coupling, even at integer dimensions d < 6.…”

“…It is of course still possible to study SYK-like models in more than one dimension [8][9][10], but tensor models have some more appealing features, in particular in view of a potential AdS/CFT correspondence, such as the fact that they have a global symmetry from the start, which could then be gauged, whereas in SYK-like models the symmetry only emerges after quenching. As a consequence several tensor models have been studied, with a focus on the fact that their melonic large-N limit allows the identification of non-trivial fixed points of the renormalization group and the non-perturbative computation of the spectrum of bilinear operators [7,[11][12][13][14][15][16][17][18][19][20][21][22] (see also [23][24][25][26] for reviews and more references).…”