Abstract
The standard nonperturbative approaches of renormalization group for tensor models are generally focused on a purely local potential approximation (i.e. involving only generalized traces and product of them) and are showed to strongly violate the modified Ward identities. This paper as a continuation of our recent contribution [Physical Review D 101, 106015 (2020)], intended to investigate the approximation schemes compatibles with Ward identities and constraints between $2n$-points observables in the large $N$-limit. We consider separately two different approximations: In the first one, we try to construct a local potential approximation from a slight modification of the Litim regulator, so that it remains optimal in the usual sense, and preserves the boundary conditions in deep UV and deep IR limits. In the second one, we introduce derivative couplings in the truncations and show that the compatibility with Ward identities implies strong relations between $\beta$-functions, allowing to close the infinite hierarchy of flow equations in the non-branching sector, up to a given order in the derivative expansion. Finally, using exact relation between correlations functions in large $N$-limit, we show that strictly local truncations are insufficient to reach the exact value for the critical exponent, highlighting the role played by these strong relations between observables taking into account the behavior of the flow; and the role played by the multi-trace operators, discussed in the two different approximation schemes. In both cases, we compare our conclusions to the results obtained in the literature and conclude that, at a given order, taking into account the exact functional relations between observables like Ward identities in a systematic way we can strongly improve the physical relevance of the approximation for exact RG equation.
Highlights
Random tensor models (RTMs) were initially introduced in the quantum gravity context at the beginning of the 1990s [1,2,3,4,5,6], as a natural extension of random matrix models (RMMs) used to quantize two-dimensional gravity [7,8,9,10,11,12]
Using an exact relation between correlations functions in large N limit, we show that strictly local truncations are insufficient to reach the exact value for the critical exponent, highlighting the role played by these strong relations between observables taking into account the behavior of the flow; and the role played by the multitrace operators, discussed in the two different approximation schemes
Dealing with this difficulty remains tractable for not so large truncations, and the first investigations, as for matrix models, provided encouraging results,discovering the critical fixed point corresponding to the double-scaling limit; having a single relevant direction with a critical exponent is in qualitative agreement with the exact analytic value θexact 1⁄4 d − 2
Summary
Random tensor models (RTMs) were initially introduced in the quantum gravity context at the beginning of the 1990s [1,2,3,4,5,6], as a natural extension of random matrix models (RMMs) used to quantize two-dimensional gravity [7,8,9,10,11,12]. The main difference between RMM and RTM in practice is the proliferation of the interactions, and of the beta functions with the rank of the truncation Dealing with this difficulty remains tractable for not so large truncations, and the first investigations, as for matrix models, provided encouraging results, (re)discovering the critical fixed point corresponding to the double-scaling limit; having a single relevant direction with a critical exponent is in qualitative agreement with the exact analytic value θexact 1⁄4 d − 2. The second strategy was to consider a modified regulator, including fine-tuned counterterms These counterterms do not change the UV and IR boundary conditions and are chosen to cancel the momentum-dependent terms in Ward identities using local potential approximation, such that the violation remains as small as possible in the considered range of couplings investigated by the RG flow (expecting that we remain not so far from the Gaussian fixed point, which is essentially the same assumption ensuring the validity of the truncation method). We introduce some useful definitions and properties that will be used to construct approximate solutions of the RG equation
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