Abstract
In this note, we study a melonic tensor model in d dimensions based on three-index Dirac fermions with a four-fermion interaction. Summing the melonic diagrams at strong coupling allows one to define a formal large-N saddle point in arbitrary d and calculate the spectrum of scalar bilinear singlet operators. For d = 2 − ϵ the theory is an infrared fixed point, which we find has a purely real spectrum that we determine numerically for arbitrary d < 2, and analytically as a power series in ϵ. The theory appears to be weakly interacting when ϵ is small, suggesting that fermionic tensor models in 1-dimension can be studied in an ϵ expansion. For d > 2, the spectrum can still be calculated using the saddle point equations, which may define a formal large-N ultraviolet fixed point analogous to the Gross-Neveu model in d > 2. For 2 < d < 6, we find that the spectrum contains at least one complex scalar eigenvalue (similar to the complex eigenvalue present in the bosonic tensor model recently studied by Giombi, Klebanov and Tarnopolsky) which indicates that the theory is unstable. We also find that the fixed point is weakly-interacting when d = 6 (or more generally d = 4n + 2) and has a real spectrum for 6 < d < 6.14 which we present as a power series in ϵ in 6 + ϵ dimensions.
Highlights
One-dimensional theory, it appears natural to set one of the couplings, say g2, to zero, or, instead to set g2 = −g1
We find that the fixed point is weakly-interacting when d = 6 and has a real spectrum for 6 < d < 6.14 which we present as a power series in in 6 + dimensions
In the large N limit, the vector model can be more rigorously defined as a Legendre transform of the free fermionic theory, by introducing an Hubbard-Stratonovich auxiliary field σb, with the action
Summary
One would like to solve the exact Schwinger-Dyson equation carefully, at least numerically, to better understand if this strong-coupling limit is physical It would strengthen one’s confidence in the existence of the theories in d > 2 if there was an alternative description as an IR fixed points, similar to the Gross-Neveu-Yukawa model, even if both descriptions have a complex spectrum. Integrating out the gauge field in a tensorial theory would give rise to a “pillow” Such an interaction appears to be similar to a large flavor expansion, e.g., [69], and we expect that this would only affect 1/N corrections to the spectrum we have presented here. Perhaps the calculations here may serve as a useful warm-up for a study of these theories
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