[Paper Review] Next-to-leading order in the large $N$ expansion of the multi-orientable random tensor model

Matti Raasakka, Adrian Tanasă|arXiv (Cornell University)|Oct 11, 2013

Black Holes and Theoretical Physics17 references4 citations

TL;DR

This paper analyzes the next-to-leading order (NLO) in the large $N$ expansion of the multi-orientable (MO) random tensor model, identifying NLO graphs as those containing a non-orientable jacket. It derives the radius of convergence $\lambda_c$ and susceptibility exponent $\gamma_{\text{NLO}} = \frac{3}{2}$, establishing a key step toward defining a double-scaling limit for the MO model, distinct from the colored tensor model due to its broader class of allowed topologies.

ABSTRACT

In this paper we analyze in detail the next-to-leading order (NLO) of the recently obtained large $N$ expansion for the multi-orientable (MO) tensor model. From a combinatorial point of view, we find the class of Feynman tensor graphs contributing to this order in the expansion. Each such NLO graph is characterized by the property that it contains a certain non-orientable ribbon subgraph (a non-orientable jacket). Furthermore, we find the radius of convergence and the susceptibility exponent of the NLO series for this model. These results represent a first step towards the larger goal of defining an appropriate double-scaling limit for the MO tensor model.

Motivation & Objective

• To characterize the structure of next-to-leading order (NLO) graphs in the large $N$ expansion of the multi-orientable (MO) random tensor model.
• To determine the critical behavior of the NLO series, including the radius of convergence and susceptibility exponent.
• To establish whether the MO model admits a double-scaling limit, analogous to that in colored tensor models, by analyzing the NLO sector.
• To contrast the NLO structure of the MO model with that of the colored tensor model, particularly regarding orientability of jackets.
• To lay the foundational groundwork for future non-perturbative studies of the MO model via double-scaling limits.

Proposed method

• Combinatorial classification of Feynman tensor graphs contributing to the NLO in the large $N$ expansion of the MO tensor model.
• Identification of NLO graphs as those containing a non-orientable ribbon subgraph, specifically a non-orientable jacket.
• Use of Schwinger-Dyson equations to relate the connected two-point function $G_{\text{NLO}}$ to the free energy $E_{\text{NLO}}$.
• Derivation of the critical behavior via the relation $G_{\text{NLO}} \sim (1 - \lambda^2/\lambda_c^2)^{-1/2}$, leading to the susceptibility exponent $\gamma_{\text{NLO}} = \frac{3}{2}$.
• Application of the large $N$ expansion formalism to the MO tensor model, where the partition function is defined with a Gaussian kinetic term and a quartic interaction term $S_p$.
• Use of the loop-vertex expansion and resummation techniques as a reference for non-perturbative behavior, comparing with perturbative results.

Experimental results

Research questions

• RQ1 What is the combinatorial structure of next-to-leading order graphs in the large $N$ expansion of the multi-orientable tensor model?
• RQ2 How does the presence of non-orientable jackets in NLO graphs affect the critical behavior of the model?
• RQ3 What is the radius of convergence and susceptibility exponent of the NLO series in the MO tensor model?
• RQ4 How does the NLO critical behavior of the MO model compare to that of the colored tensor model?
• RQ5 Can a double-scaling limit be consistently defined for the MO tensor model based on its NLO sector?

Key findings

• The NLO graphs in the MO tensor model are characterized by the presence of a non-orientable jacket, distinguishing them from the NLO graphs of the colored tensor model, which only contain orientable jackets.
• The radius of convergence of the NLO series is found to be $\lambda_c$, the same critical coupling constant as in the leading order (LO) series.
• The susceptibility exponent for the NLO series is $\gamma_{\text{NLO}} = \frac{3}{2}$, indicating a distinct critical behavior from the LO order.
• The critical behavior of the NLO two-point function is $G_{\text{NLO}} \sim (1 - \lambda^2/\lambda_c^2)^{-1/2}$, consistent with the derived exponent.
• The free energy at NLO scales as $E_{\text{NLO}} \sim (1 - \lambda^2/\lambda_c^2)^{1/2}$, confirming the critical exponent $\gamma_{\text{NLO}} = \frac{3}{2}$.
• These results suggest the possibility of defining a double-scaling limit for the MO tensor model, analogous to that in colored tensor models, though the role of half-integer degree sectors remains an open question.