Remarks on a melonic field theory with cubic interaction

Abstract

We revisit the Amit-Roginsky (AR) model in the light of recent studies on Sachdev-Ye-Kitaev (SYK) and tensor models, with which it shares some important features. It is a model of N scalar fields transforming in an N-dimensional irreducible representation of SO(3). The most relevant (in renormalization group sense) invariant interaction is cubic in the fields and mediated by a Wigner 3jm symbol. The latter can be viewed as a particular rank-3 tensor coupling, thus highlighting the similarity to the SYK model, in which the tensor coupling is however random and of even rank. As in the SYK and tensor models, in the large-N limit the perturbative expansion is dominated by melonic diagrams. The lack of randomness, and the rapidly growing number of invariants that can be built with n fields, makes the AR model somewhat closer to tensor models. We review the results from the old work of Amit and Roginsky with the hindsight of recent developments, correcting and completing some of their statements, in particular concerning the spectrum of the operator product expansion of two fundamental fields. For 5.74 < d < 6 the fixed-point theory defines a real CFT, while for smaller d complex dimensions appear, after a merging of the lowest dimension with its shadow. We also introduce and study a long-range version of the model, for which the cubic interaction is exactly marginal at large N , and we find a real and unitary CFT for any d < 6, both for real and imaginary coupling constant, up to some critical coupling.

Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: phi3 QFT in 6 Dimensions

“…This resonant property has profound implications for the structure of the trans-series beyond the leading exponential order, leading to an even richer structure. 11 In a resonant case, the exponent coefficient ( ⃗ k • ⃗ λ) in (5.5) may take the same value for different integer-valued vectors ⃗ k. For example, e −2/x appears through one power of the seed term with λ 2 = 2, but it also appears via the square of the seed term with λ 1 = 1. Therefore, when we grade the solution by its exponential order, e −(integer)/x , a given order can have contributions from different ⃗ k vectors, and when they mix there can also appear logarithmic terms in the solution to the Dyson-Schwinger equation.…”

“…On the non-perturbative side, this QFT has a real Lipatov instanton when g is real, for which the conventional one-instanton semi-classical analysis [21,66,78,93] of the fluctuation determinant has been studied [76,77]. Further extensions to multi-dimensional cubic interactions have many interesting applications and implications for conformal quantum field theories in general [11,19,38,39,49,52,53,54,55,56,57]. For other analyses of resurgence properties of renormalization group and Dyson-Schwinger equations see [5,6,7,8,9,13].…”

The F-theorem in the melonic limit

“…In this paper, we test the F-theorem in the long-range O(N ) 3 bosonic tensor model introduced in [22]. This model is one of many examples of field theories with a melonic large-N limit [23][24][25][26][27][28][29][30][31][32][33] (see also [34,35] for reviews). The model of [22], which we consider here, is one of the most extensively studied, and it has numerous interesting features:…”

Instability of complex CFTs with operators in the principal series

“…where we introduced µ∆(h, J) from equation ( 92) of [59] (or A.5 of [17]), which was derived from the formulas in [38,39]. The squared prefactor originates from the different normalization we use here for G (X1, X2) (see (2.7), and remember that for a free theory we have K = 0 and Fs(X1, X2, X3, X4) = G (X1, X3)G (X2, X4) + G (X1, X4)G (X2, X3)), while the 1/2 factor cancels with the factor 2 in (2.41) (originating from the 1/2 in (2.33)).…”

“…Many examples of such merging of fixed points are known, see for example the list of references in [3]. In some instances, a special case of complex scaling dimension is found, one whose real part is equal to d/2, d being the space(-time) dimension; examples include non-supersymmetric orbifolds of N = 4 super Yang-Mills [4][5][6], gauge theories with matter in the Veneziano limit [7,8], and large-N theories dominated by melonic diagrams [9][10][11][12][13][14][15][16][17], or by fishnet diagrams [18][19][20]. The typical large-N mechanism leading to complex dimensions of such type is the following [6]: due to the large-N simplifications, the beta function for the coupling of a double-trace operator O 2 turns out to be governed by a quadratic beta function, and hence the reality of the fixed points depends on its discriminant D; in the case D < 0, the fixed points are complex, and the scaling dimension of O 2 is ∆ O 2 = d + 2 i |D|; lastly, the large-N limit implies that the scaling dimension of O is half that of O 2 , and thus it is of the claimed form.…”