^{1}

^{1,2}

^{1}

^{3}

^{4}

^{3}.

We study the

In recent literature, there has been considerable interest in models where the degrees of freedom transform as tensors of rank 3 or higher. Such models with appropriately chosen interactions admit new kinds of large

It is also of interest to explore similar quantum theories of bosonic tensors

Such complex dimensions appear in various other large

In this paper, we continue the search for stable bosonic large

Diagrammatic representation of the eight possible

The theory

In Sec.

To study the large

If we added fermions to make the tensor model supersymmetric

Let us define the following propagators:

Diagrammatic representation of the Schwinger-Dyson equations. Solid lines denote

Multiplying the first equation by

For what range of

The gap equation is finally:

The condition that must be satisfied by

In position space, the IR two-point functions take the form

In terms of

It can be verified numerically that solutions to

Solving

For

Solving

There is an interesting transition in behavior which happens at

There are three types of scalar bilinears one can consider, which are of the schematic form:

Let us consider a bilinear of type

The integration kernel for type B bilinears.

When

We can solve equation

The spectrum of type B bilinears in

For spin

Processing the equation we have the following condition for the allowed twists of higher spin bilinears:

Solving equation

We find that the spectrum of type B bilinears appears to be real for all

Let us now study the spectrum of bilinear operators of type

The integration kernels

We now define the following kernels, depicted schematically in Fig.

The integration kernel gives rise to the following matrix:

The Schwinger-Dyson equations have a symmetry under

The spectrum of type A/C scalar bilinears in

The spectrum of type A/C scalar bilinears in

For

The spectrum of type A/C scalar bilinears in

Let us solve the Schwinger-Dyson equations in

Let us consider the

The next solution of the Schwinger-Dyson equation is

The subsequent eigenvalues may be separated into two sets. One of them has the form, for integer

We can also use

Let us also present the

The next two twists are

We can similarly derive explicit results for spinning operators in the type B sector using

In this section we use the renormalized perturbation theory to carry out the

To carry out the beta function calculation at finite

Our action is a special case of a general multifield

The two-loop contribution to the beta function.

In our case each index

The two-loop beta functions and anomalous dimensions for general

At finite

We have also calculated the

The scaling dimension of the marginal prism operator is

We have also performed two-loop calculations of the scaling dimensions of the tetrahedron and pillow operators; see the Appendix for the anomalous dimension matrix. In the large

We have also solved the equations for the fixed points of two-loop beta functions numerically for finite

This is similar, for example, to the situation in the

The numerical solutions for the coupling constants defined in

The action

A very similar

Solving for the scaling dimensions of type A/C bilinears in

The spectrum of scalar type A/C bilinears in 1d. Red vertical lines are asymptotes corresponding to

The smallest positive eigenvalue,

Of course, as observed in

Let us also list the type B scaling dimensions, i.e., the ones corresponding to operators

For large excitation numbers

In this paper we presented exact results for the

In this paper we analyzed the renormalization of the prismatic theory at the two-loop order, using the beta functions in

A

Another interesting extension of the

We are grateful to C.-M. Chang, M. Rangamani, D. Stanford, E. Witten and J. Yoon for useful discussions. I. R. K. thanks the Yukawa Institute for Theoretical Physics where some of his work on this paper was carried out during the workshop YITP-T-18-04 “New Frontiers in String Theory 2018”. S. P. thanks the Princeton Center for Theoretical Science for hospitality as well as the International Centre for Theoretical Sciences, Bengaluru where some of his work on this paper was carried out during the program “AdS/CFT at 20 and Beyond” (ICTS/adscft20/2018/05). The work of S. G. was supported in part by the US NSF under Grant No. PHY-1620542. The work of I. R. K. and F. P. was supported in part by the US NSF under Grant No. PHY-1620059. The work of S. P. was supported in part by a DST-SERB Early Career Research Award (ECR/2017/001023) and a DST INSPIRE Faculty Award. The work of G. T. was supported in part by the MURI Grant No. W911NF-14-1-0003 from ARO and by DOE Grant No. de-sc0007870.

In this Appendix we state our explicit two-loop results for the

We can study the anomalous dimensions for quartic operators

A consistent truncation of the system of eight coupling constants is to keep only