Abstract
We study finite N aspects of the O(m) × O(N − m) vector model with quartic interactions in general 2 ≤ d ≤ 6 spacetime dimensions. This model has recently been shown [1, 2] to display the phenomenon of persistent symmetry breaking at a perturbative Wilson-Fisher-like fixed point in d = 4 − ϵ dimensions. The large rank limit of the biconical model displays a conformal manifold and a moduli space of vacua. We find a set of three double trace scalar operators that are respectively irrelevant, relevant and marginal deformations of the conformal manifold in general d. We calculate the anomalous dimensions of the single and multi-trace scalar operators to the first sub-leading order in the large rank expansion. The anomalous dimension of the marginal operator does not vanish in general, indicating that the conformal manifold is lifted at finite N . In the case of equal ranks we are able to derive explicitly the scaling dimensions of various operators as functions of only d.
Highlights
JHEP05(2021)192 perturbatively in 1/N there are no signs of pathologies, e.g., the scaling dimensions of various operators satisfy unitarity bounds [15],2 the very presence of instability furnishes an obstacle to address the question of persistent symmetry breaking in the interacting φ6 model
In this paper we studied the critical O(m) × O(N − m) vector model (2.1) in general 2 ≤ d ≤ 6 dimensions
To reveal what happens with the conformal manifold in general dimension, we employed the large-N technique to identify an operator which happens to be exactly marginal in any d in the large-N limit
Summary
To identify the scaling dimensions of propagating fields at the fixed point, it is useful to diagonalize the matrix of couplings, h. It can be explicitly verified that σ+2 , σ−2 and σ+σ− correspond to the eigenvectors of ω This completes our construction of the primary operators which represent admissible nearly marginal deformations of the critical model in the vicinity of d = 4. We showed that the large-N scaling dimension of σ+σ− equals d in any number of dimensions, implying that to leading order in the 1/N expansion this operator is marginal. This provides an evidence for the existence of conformal manifold in general d. If the scaling dimension of the marginal operator σ+σ− acquires a non-trivial correction, the line of fixed points is lifted by the finite rank corrections in any number of dimensions
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