Prismatic Large Models for Bosonic Tensors
We study the symmetric quantum field theory of a bosonic tensor with sextic interactions. Its large limit is dominated by a positive-definite operator, whose index structure has the topology of a prism. We present a large solution of the model using Schwinger-Dyson equations to sum the leading diagrams, finding that for and for the spectrum of bilinear operators has no complex scaling dimensions. We also develop perturbation theory in dimensions including eight invariant operators necessary for the renormalizability. For sufficiently large , we find a “prismatic” fixed point of the renormalization group, where all eight coupling constants are real. The large limit of the resulting expansions of various operator dimensions agrees with the Schwinger-Dyson equations. Furthermore, the expansion allows us to calculate the corrections to operator dimensions. The prismatic fixed point in dimensions survives down to , where it merges with another fixed point and becomes complex. We also discuss the model where our approach gives a slightly negative scaling dimension for , while the spectrum of bilinear operators is free of complex dimensions.
Prismatic Large Models for Bosonic Tensors
Simone Giombi, Igor R. Klebanov, Fedor Popov,
[10pt] Shiroman Prakash, Grigory Tarnopolsky
- 1 Introduction
- 2 Large Limit
- 3 Bilinear Operators
- 4 Large results in dimensions
- 5 Renormalized perturbation theory
- 6 Bosonic Quantum Mechanics
- 7 Discussion
- A The two-loop beta functions and anomalous dimensions
In recent literature, there has been considerable interest in models where the degrees of freedom transform as tensors of rank or higher. Such models with appropriately chosen interactions admit new kinds of large limits, which are not of ’t Hooft type and are dominated by the so-called melonic Feynman diagrams [1, 2, 3, 4, 5]. Much of the recent activity has been on the quantum mechanical models of fermionic tensors [4, 5], which have large limits similar to that in the Sachdev-Ye-Kitaev (SYK) model [6, 7, 8, 9, 10, 11, 12, 13].
This QFT is super-renormalizable in and is formally solvable using the Schwinger-Dyson equations in the large limit where is held fixed. However, this model has some instabilities. One problem is that the “tetrahedral” operator is not positive definite. Even if we ignore this and consider the large limit formally, we find that in the invariant operator has a complex dimension of the form .111Such complex dimensions appear in various other large theories; see, for example, [16, 17, 18, 19]. From the dual AdS point of view, such a complex dimension corresponds to a scalar field whose is below the Breitenlohner-Freedman stability bound [20, 21]. The origin of the complex dimensions was elucidated using perturbation theory in dimensions: the fixed point was found to be at complex values of the couplings for the additional invariant operators required by the renormalizability . In  a symmetric theory for tensor and sextic interactions was also considered. It was found that the dimension of operator is real in the narrow range , where . However, the scalar potential of this theory is again unstable, so the theory may be defined only formally. In spite of these problems, some interesting formal results on melonic scalar theories of this type were found recently .
In this paper, we continue the search for stable bosonic large tensor models with multiple symmetry groups. Specifically, we study the symmetric theory of scalar fields with a sixth-order interaction, whose Euclidean action is
This QFT is super-renormalizable in . When the fields are represented by vertices and index contractions by edges, this interaction term looks like a prism (see figure 11 in ); it is the leftmost diagram in figure 1. Unlike with the tetrahedral quartic interaction (1.1), the action (1.2) is positive for . In sections 2 and 3, we will show that there is a smooth large limit where is held fixed and derive formulae for various operator dimensions in continuous . We will call this large limit the “prismatic” limit: the leading Feynman diagrams are not the same as the melonic diagrams, which appear in the symmetric QFT for a tensor .
The theory (1.2) may be viewed as a tensor counterpart of the bosonic theory with random couplings, which was introduced in section 6.2 of . Since both theories are dominated by the same class of diagrams in the large limit, they have the same Schwinger-Dyson equations for the 2-point and 4-point functions. We will confirm the conclusion of  that the theory does not have a stable IR limit; this is due to the appearance of a complex scaling dimension. However, we find that in the ranges and , the large prismatic theory does not have any complex dimensions for the bilinear operators. In section 5 we use renormalized perturbation theory to develop the expansion of the prismatic QFT. We have to include all eight operators invariant under the symmetry and the symmetry permuting the groups; they are shown in figure 1 and written down in (A.1). For , where , we find a prismatic RG fixed point where all eight coupling constants are real. At this fixed point, expansions of various operator dimensions agree in the large limit with those obtained using the Schwinger-Dyson equations. Futhermore, the expansion provides us with a method to calculate the corrections to operator dimensions, as shown in (5.8), (5.9). At the prismatic fixed point merges with another fixed point, and for both become complex.
2 Large Limit
To study the large limit of this theory, we will find it helpful to introduce an auxiliary field so that222If we added fermions to make the tensor model supersymmetric [5, 15, 25, 26] then would be interpreted as the highest component of the superfield .
where . Integrating out gives rise to the action (1.2). The advantage of keeping explicitly is that the theory is then a theory with symmetry dominated by the tetrahedral interactions, except it now involves two rank-3 fields; this shows that it has a smooth large limit. Thus, a prismatic large limit for the theory with one 3-tensor may be viewed as a tetrahedral limit for two 3-tensors.
Let us define the following propagators:
In the free theory , and . In the strong coupling limit the self-energies of the fields are given by the inverse propagators: and . The Schwinger-Dyson equations for the exact two-point functions can be written as:
and represented in figure 2.
Multiplying the first equation by on the left and on the right, and likewise for the second equation we obtain:
where . We have to take the large limit keeping fixed. In the IR limit, let us assume
is related to the scaling dimension of , via
For what range of and can we drop the free terms in the gap equations above? In the strong coupling limit we require and . Since , we have . In terms of , we then find
Notice that, if we had the usual kinetic term for the field, the allowed range for would be larger. Therefore, our solution may also apply to a model with two dynamical scalar fields interacting via the particular interaction given above.
The gap equation is finally:
Dimensional analysis of the strong coupling fixed point actually does not fix : we get from the first equation and from the second equation. Let us try to solve the above equations, in the hope that numerical factors arising from the Feynman integrals may determine . The overall constant is not determined from this procedure, but note that , and therefore . This procedure is analogous to solving an eigenvalue equation, and perhaps it is not surprising that we have to do this, since a solution for also determines the anomalous dimension of a composite operator . We then find
The condition that must be satisfied by is then:
In position space, the IR two-point functions take the form
In terms of , (2.9) may be written as
2.1 The scaling dimension of
The solution lies within the allowed range (2.5), while the one with the other branch of the square root is outside it.
For we find multiple solutions within the allowed range (2.5), as shown for . One of the solutions gives ; this produces singularities in the large dimensions of scalar bilinears, and we will not use it. The other solution,
appears to be acceptable. Although is negative, it lies above the unitarity bound. We note that there is also a positive solution , which lies outside of the allowed range (although it would be allowed if the field was dynamical).
There is an interesting transition in behavior which happens at where there is a double root at . The critical dimension is the solution of
Its numerical value is . For slightly above one of the solutions for is zero, while the other is positive; we have to pick the positive one. However, for slightly below one of the solutions for is zero, while the other is negative. Special care may be needed for continuation to ; in particular, for studying the case.
3 Bilinear Operators
There are three types of scalar bilinears one can consider, which are of the schematic form: , and , where is an auxiliary null vector introduced to encode the spin of the operators, , and . We note that there is mixing of operators of type and . It is easy to convince oneself that there is no mixing with the operators by drawing a few diagrams.
3.1 Bilinears of type B
where is the twist of the bilinear, is the null polarization vector, is the conformally invariant tensor structure defined in [27, 28] and we took the limit in the second line. The eigenvalue equation, obtained using the integration kernel depicted schematically in figure 5, is
When , we have:
which translates into
We can solve equation 3.5 numerically to find the allowed scaling dimensions for type B operators in various dimensions. In the type B scaling dimensions are
as shown in figure 6. In the pure language, the first one can be identified with the tetrahedral operator. The type B scaling approach the asymptotic formula
For example, for we numerically find , which is very close to (3.7).
For spin the eigenvalue equation is:
Note that the spectrum of type bilinears does not contain the stress tensor, which is of type .
Processing the equation we have the following condition for the allowed twists of higher spin bilinears:
Using this equation one can find the allowed twists of spin- type B bilinears. For example, the spectrum when and is found from figure 7 to be , which approach from above.
We find that the spectrum of type B bilinear appears to be real for all . However, there are ranges of where the spectrum of type A/C operators do contain complex eigenvalues, as we discuss in the next section.
3.2 Mixing of bilinears of type A and C
Let us now study the spectrum of bilinear operators of type and . As mentioned earlier, by drawing a a few diagrams (see figure 8) one can see that these operators mix, in the sense that the two-point function . Let be the twist of mixture of and operators, which we denote as . As in the previous subsection, from the three-point functions and , we define
We now define the following kernels, depicted schematically in figure 8:
Note the factor of , which appears from a careful counting of the Wick contractions.
The integration kernel gives rise to the following matrix
The condition for it to have eigenvalue 1, which determines the allowed values of , is
Luckily, this condition is independent of the constant , as one can see from the following expressions,
Thus, the equation we need to solve is:
One can check that the stress-tensor, which has and , appears in this spectrum for any .
The spectrum contains complex modes for . In the graphical solution for the scaling dimensions in the type A/C sector is shown in figure 9. The lowest few are
The eigenvalue at is exact, and in general is an eigenvalue for any . There is also a solution , which corresponds to the shadow dimension of . As is further lowered, the part of the graph between and moves up so that the two solutions become closer. In , where , the two solutions merge into a single one at (for discussions of mergers of fixed points, see [29, 30, 31]). For , the solutions become complex and the prismatic model becomes unstable. The plot for is shown in figure 10.
For , the spectrum of bilinears is again real. The plot for , where , is shown in figure 11. At this critical value of there are two solutions at ; one is the shadow of the other.
4 Large results in dimensions
Let us solve the Schwinger-Dyson equations in . The results will be compared with renormalized perturbation theory in the following section. The scaling dimension of is found to be
This is within the allowed range (2.5) and is close to its upper boundary. The scaling dimension of is
Let us consider type A/C bilinears. For the first scalar eigenvalue we have,
The next eigenvalue is . This is due to the fact that the Schwinger-Dyson equations have a symmetry under . In a given CFT, only one of this pair of solutions corresponds to a primary operator dimension, while the other one is its “shadow.” In the present case, the lowest dimension corresponds to operator , as we will show in the next section.
The next solution of the S-D equation is for all . While this seems to correspond to an exactly marginal operator, we believe that the corresponding operator is redundant: it is a linear combination of and . Similar redundant operators with showed up in the Schwinger-Dyson analysis of multi-flavor models [11, 32]. They decouple in correlation functions  and were shown to vanish by the equations of motion . The next eigenvalue is
It should correspond to the sextic prism operator (1.2), which is related by the equations of motion to a linear combination of and .
The subsequent eigenvalues may be separated into two sets. One of them has the form, for integer ,
For large this approaches , as expected for an operator of the form . The other set of eigenvalues has the form, for integer ,
For large this approaches , as expected for an operator of the form . These simple asymptotic forms suggest that for large the mixing between operators and approaches zero.
We can also use (3.5) to derive the expansions of the dimensions of type B operators,
where the additional terms are there to make them conformal primaries. For we find
This scaling dimension corresponds to the operator , which in the original language is the tetrahedron operator . For the higher operators we get
Using the equations of motion, we can write , up to a total derivative, as a sum of the three -particle operators shown in the leftmost column of figure 9 in . In general, for ,
which agrees for large with the expected asymptotic behavior
4.1 Higher Spin Spectrum
Let us also present the expansions for the higher spin bilinear operators which are mixtures of type and . The lowest eigenvalue of twist for spin is
where is the harmonic number and the last two terms (as well as all higher-order terms) vanish when as expected. In the large limit, this becomes:
Comparing with (5.8), we see that
The next two twists are