Abstract
We study the $O(N)^3$ symmetric quantum field theory of a bosonic tensor $\phi^{abc}$ with sextic interactions. Its large $N$ limit is dominated by a positive-definite operator, whose index structure has the topology of a prism. We present a large $N$ solution of the model using Schwinger-Dyson equations to sum the leading diagrams, finding that for $2.81 < d < 3$ and for $d<1.68$ the spectrum of bilinear operators has no complex scaling dimensions. We also develop perturbation theory in $3-\epsilon$ dimensions including eight $O(N)^3$ invariant operators necessary for the renormalizability. For sufficiently large $N$, we find a "prismatic" fixed point of the renormalization group, where all eight coupling constants are real. The large $N$ limit of the resulting $\epsilon$ expansions of various operator dimensions agrees with the Schwinger-Dyson equations. Furthermore, the $\epsilon$ expansion allows us to calculate the $1/N$ corrections to operator dimensions. The prismatic fixed point in $3-\epsilon$ dimensions survives down to $N\approx 53.65$, where it merges with another fixed point and becomes complex. We also discuss the $d=1$ model where our approach gives a slightly negative scaling dimension for $\phi$, while the spectrum of bilinear operators is free of complex dimensions.
Highlights
In recent literature, there has been considerable interest in models where the degrees of freedom transform as tensors of rank 3 or higher
Even if we ignore this and consider the large N limit formally, we find that in d < 4 the OðNÞ3 invariant operator φabcφabc has a complex dimension of the form d 2 þ iαðdÞ
Þ g1 φ φ a1b1c1 a1b2c2 φ φ a2b1c2 a3b3c1 φa3b2c3 φa2b3c3 6!. This quantum field theory (QFT) is super-renormalizable in d < 3
Summary
There has been considerable interest in models where the degrees of freedom transform as tensors of rank 3 or higher. We study the OðNÞ3 symmetric theory of scalar fields φabc with a sixth-order interaction, whose This QFT is super-renormalizable in d < 3. II and III, we will show that there is a smooth large N limit where g1N3 is held fixed and derive formulas for various operator dimensions in continuous d We will call this large N limit the “prismatic” limit: the leading Feynman diagrams are not the same as the melonic diagrams, which appear in the OðNÞ5 symmetric φ6 QFT for a tensor φabcde [15]. For N > Ncrit, where Ncrit ≈ 53.65, we find a prismatic RG fixed point where all eight coupling constants are real At this fixed point, ε expansions of various operator dimensions agree in the large N limit with those obtained using the Schwinger-Dyson equations. Our large N solution gives a slightly negative scaling dimension, Δφ ≈ −0.09, while the spectrum of bilinear operators is free of complex scaling dimensions
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