Bifundamental Multiscalar Fixed Points in $d=3-\epsilon$

We study fixed-points of scalar fields that transform in the bifundamental representation of $O(N)\times O(M)$ in $3-\epsilon$ dimensions, generalizing the classic tricritical sextic vector model. In the limit where $N$ is large but $M$ is finite, we determine the complete beta function to order $1/N$ for arbitrary $M$. We find a rich collection of large-$N$ fixed-points in $d=3$, as well as fixed-points in $d=3-\epsilon$, that can be studied to all orders in the parameter $\hat{\epsilon}=N\epsilon$. With the goal of defining a large-$N$ nonsupersymmetric conformal field theory dominated by a web of planar diagrams, we also study fixed-points in the ``bifundamental'' large-$N$ limit, in which $M$ and $N$ are both large, but the ratio $M/N$ is held fixed. We find a unique infrared fixed-point in $d=3-\epsilon$, which we determine to order $\epsilon^2$. When $M/N \ll 1$, we also find an ultraviolet fixed-point in $d=3$ and $d=3-\epsilon$ that merges with the infrared fixed-point at $\epsilon \sim O(M/N)$. We expect at least one of two candidate fixed-points in integer dimensions -- the infrared fixed-point in $d=2$ and the ultraviolet fixed-point in $d=3$ -- to survive for finite values of $M/N$.


Introduction
In this paper, we study multiscalar fixed-points in 3 − dimensions with O(N ) × O(M )/Z 2 symmetry. Our primary motivation for studying large-N fixed-points with bifundamental matter originates from string theory and the AdS/CFT correspondence, which we describe below, although we believe such fixed-points could be of more general interest in other contexts as well.
The N = 6 supersymmetric U (N ) k × U (M ) −k Chern-Simons theory with matter in the bifundamental representation known as ABJ theory [1], plays an important and unique role in our understanding of the AdS/CFT correspondence, as first observed in [2]. When N is large, but M N , the theory is a large-N vector model [3], and as per general expectations in [4], is a dual to a higher-spin gauge theory [5][6][7][8][9][10], where the gauge fields are dressed with Chan-Paton factors transforming under U (M ). When N = M , the theory becomes the ABJM theory [11], whose large-N limit consists of a web of planar diagrams dual to type IIA string theory on AdS 4 × CP 3 when λ = N k is held fixed. The dual reduces to type IIA supergravity when λ is taken to strong coupling. In the language of the higher-gauge theory, the parameter M/N plays the role of a gravitational 't Hooft coupling. When M/N is small, the theory contains an infinite tower of higher-spin gauge fields, and as M/N increases, the tower of higher-spin gauge fields somehow coalesce to form strings. (See [12,13] for more discussion.) Does a similar phenomenon occur for other examples of conformal field theories with higher-spin duals, with less (see [14]) or no supersymmetry? For example U (N ) k Chern-Simons theory coupled to fundamental fermions (or bosons) is believed to be dual to a oneparameter family of party-violating higher-spin gauge theories [4,15]. One can generalize these theories to U (N ) k ×U (M ) −k or O(N ) k ×O(M ) −k Chern-Simons theories with bifundamental matter [16,17] -when M/N is small these theories would be virtually identical to their vector model counterparts, and therefore dual to a higher-spin gauge theory -but when M/N is order 1, the large-N limit of such theories, like the ABJM theory, becomes a web of planar diagrams suggesting that they would be dual to some sort of string theory [18]. If the behaviour of these theories when α = 1 and the 't Hooft coupling λ = N k is large is similar to that of ABJM, it is natural to conjecture that such theories could conceivably be dual to simple non-supersymmetric theories of Einstein gravity in AdS 4 .
In an extension of the weak gravity conjecture [19] as part of the swampland program, the authors of [20] (see also [21]) conjecture that non-supersymmetric AdS vacuua do not exist in a consistent quantum theory of gravity. This translates into the statement that a strongly interacting 1 non-supersymmetric conformal field theory whose large-N limit consists of a web of planar diagrams should not exist, or at least there should be some reason for it to not have a simple holographic dual. Therefore a simple way to test of the conjecture of [20] is to construct examples of such large-N CFT's and attempt to demonstrate or rule out their existence at strong coupling.
If the conjecture of [20] is correct, then it should be impossible for the bifundamental Chern-Simons theories in [16,17] to attain strong coupling. Unfortunately, it is hard to say anything about non-supersymmetric bifundamental Chern-Simons theories in the limit where M/N = 1 and λ is large. But there are other simpler examples of conformal field theories with higher-spin duals. Specifically, one can ask, is it possible to generalize the critical O(N ) vector model in 3 dimensions [22] to a critical O(N ) × O(M )/Z 2 model, with M and N both large, and α ≡ M N held fixed? As α → 0, this theory would reduce to the critical O(N ) model, whose dual is conjectured to be a higher-spin gauge theory [23][24][25][26][27], modified slightly, as per the discussion in [2], to include Chan-Paton factors transforming in O(M ). In the dual description, α = M/N is again the 't Hooft coupling for the gravitational coupling 1/N , (which evidently possesses a sort of S-duality as our field theory is dual under interchange of M and N ) and one can ask whether the higher-spin gauge fields coalesce into string like objects as α increased to 1.
γ φ attains its maximum at α = 1, confirming the expectation that α = 1 represents strong coupling for the theory. We therefore conclude that, for this theory, the fixed-point becomes complex before strong coupling is attained, in accordance with the conjecture of [20]. A physical interpretation for complex fixed-points was put forth in [39,40], so formally, one could attempt to define a holographic dual for the complex α = 1 fixed-point. This dual might contain a very small number of massless fields (possibly only the graviton) if a gap develops when d = 3, but also a few fields with masses below the Breitenlohner-Freedman bound, corresponding to primary operators of complex dimension [41].
Similar questions can also be asked about the existence of bifundamental multiscalar critical fixed-points starting from sextic interactions in d = 3 − and cubic interactions in d = 6 − dimensions. In this paper, we focus our attention on bifundamental φ 6 fixed-points in 3 − dimensions.
Is it possible to construct a strongly interacting conformal field theory whose large-N limit is dominated by a web of planar diagrams, via a bifundamental multiscalar theory in d = 3 − ? We have discovered two candidate fixed-points in integer dimensions -the infrared bifundamental fixed-point in d = 3− for = 1, and the ultraviolet bifundamental fixed-point in d = 3. Our calculations were perturbative -we require 1 for the IR fixed-point, and in α 1 for the UV fixed-point -so we are unable to conclusively demonstrate the existence of a strong-coupling limit. Nevertheless, we expect that, for any value of α, at least one of these two candidate fixed-points exist.
The IR fixed-point always exists for d sufficiently close to 3. For α 1, the UV fixedpoint also exists in and near d = 3, and merges with the IR fixed-point at a critical value of d * = 3 − 9 2π 2 α + O(α 2 ) from Eq. (3.24). Therefore, at small α, the IR fixed-point does not extend to d = 2, but the UV bifundamental fixed-point exists in d = 3. Our arguments for the existence of the UV fixed-point require α 1 (or α 1) -therefore the UV bifundamental fixed-point may or may not exist d = 3 at any finite value of α. However, if the UV fixed-point does not exist for some finite value of α, then the IR fixed-point almost certainly survives until d = 2, because there is no other candidate fixed-point for it to merge with at finite .

A brief review of the sextic O(N ) vector model
The large-N limit of the sextic O(N ) vector model in 3 − dimensions is quite different from the the large-N limit of the quartic O(N ) vector model in 4 − dimensions. Classic references include, e.g., [42][43][44][45][46][47][48][49][50][51]. In particular, Pisarski [46] showed that it is possible to demonstrate the existence of a large-N interacting ultraviolet fixed-point for the sextic O(N ) vector model in d = 3. The argument of [46] leading to a fixed-point in d = 3 is essentially as follows. In the large-N limit, the beta function for the sextic triple-trace interaction vanishes to all orders in the 't Hooft coupling. Therefore, in order to determine large-N fixed-points, one must consider the first 1/N correction to the beta function. A simple graph theoretic argument shows that the first 1/N correction to the beta function also vanishes beyond fourloops [46]. Let us briefly review this argument. Consider a O(N ) symmetric scalar theory with sextic interaction vertex ∼ g(φ i φ i ) 3 . Note that, unlike for quartic scalar theories, an L-loop correction to the interaction vertex is proportional to g n where n = L/2 + 1. Any such diagram contains exactly e int = 3(n − 1) internal edges (i.e., propagators), as seen in Fig.  1. The largest possible power of N for such a diagram is the total number of disconnected cycles formed by the internal edges, following O(N ) index-contractions. (For example, in the four-loop diagram in Fig. 1, there are three disconnected cycles.) Since our graph can contain no tadpoles (or it would vanish), each such disconnected cycle must contain at least 2 internal edges, so the diagram is proportional to, at most, N e int /2 = N 3 2 (n−1) . Using the 't Hooft coupling λ = gN 2 , we find each the L-loop correction to the vertex is proportional to gλ L/2 N −L/4 . Therefore the only contributions to the beta function at order 1/N come The interaction vertices can be chosen so that the two-loop and four-loop diagrams in the top row are proportional to α in the bifundamental large-N limit, or 1/N in the vector model large-N limit. However, any diagram with six or more loops, such as the diagram in the second row, is suppressed by at least α 2 in the bifundamental large-N limit, or 1/N 2 in the vector model large-N limit. from two-loop and four-loop diagrams. An analogous result applies to the bifundamental O(N ) × O(M ) theories we will study in this paper, in the limit M N .
For sextic theories at finite N , the four-loop beta function is cubic in the coupling constants, and therefore only allows one to determine next-to-leading order corrections, (i.e., O( 2 )), in the epsilon expansion. However, in the vector model large-N limit, the four-loop beta function is the complete beta function, to first nontrivial order in 1/N . Any higher-order corrections to the beta function are proportional to 1/N 2 and can be made arbitrarily small by making N sufficiently large -so any interacting fixed-point (with no marginal directions) of the O(1/N ) four-loop beta function will be reliable in the large-N limit, even those that do not become free when d = 3. Extending this analysis to d = 3 − , it is easy to see that the four-loop beta function allows one to determine fixed-points valid to all orders in the "rescaled" epsilon parameterˆ = N .
[46] finds two fixed-points in d = 3− -an IR fixed-point which becomes free in d = 3 and a UV fixed-point, which is not free when d = 3. These two merge into a complex fixed-point at a critical value ofˆ ,ˆ C = 36π 2 + O(1/N ), and there is no real fixed-point for <ˆ C /N . Therefore, at least when N is large, the sextic O(N ) fixed-point does not extend to a CFT in 2 dimensions. The UV fixed-point, however, appears to be an example of an ultraviolet-stable large-N fixed-point in d = 3, that may survive for N sufficiently large but finite.
An important aspect of sextic O(N ) model is the phenomena of spontaneous breaking of conformal invariance, [48,52,53,51] and this continues to receive attention in the literature: see, e.g., [54][55][56]. We do not provide a review of this here. Let us also remark that more recent discussion about the asymptotic safety and flows between this UV fixed-point and the (quartic) Wilson Fisher fixed-point in d = 3 appears in [57]. Some operator-productexpansion coefficients and anomalous dimensions in the UV fixed-point were computed more recently, using the background field method, in [58]. A similar fixed-point arising from a triple-trace deformation to ABJM was studied in [59].
When we study the bifundamental O(N ) × O(M )/Z 2 model below, we will use similar arguments to [46] to determine interacting fixed-points in d = 3 when N is large, and M is finite. These arguments also apply when M and N are both large, but the ratio M/N 1. However, if M/N ∼ O(1), higher loop contributions to the beta function are no longer suppressed by 1/N (or M/N ), and it is not possible to determine interacting fixed-points in d = 3 from the four-loop beta function.

Summary of results
In this paper, we generalize the sextic O(N ) vector model in 3 − dimensions, to a bifundamental O(N ) × O(M )/Z 2 model. We study this theory in two natural large-N limits. The first is a bifundamental large-N limit motivated by ABJ theory in which both M and N are large, and the ratio α = M N is held fixed. The second is the conventional "vector model" large-N limit, where N is large and M is held finite. We also briefly discuss fixed-points when M = 2 and N is finite.
In the bifundamental limit, we find a unique, nontrivial infrared fixed-point in d = 3− for the appropriate 't Hooft couplings, in the expansion. We compute anomalous dimensions of all the spin-zero sextic, quartic and quadratic operators that are invariant under O(N )×O(M ) symmetry group at this fixed-point. Unlike the quartic theory in 4 − dimensions, the fixedpoint we study here exists as a real fixed-point for all finite values of α, including α = 1. However, as α → 0, the large-N beta function vanishes in d = 3 (as expected from [46]).
To better understand the α → 0 limit, we study the bifundamental theory in the limit where α 1, to first order in (M/N ). Diagrammatically, this limit is very similar to the vector model large-N limit. All our sextic couplings are exactly marginal when α = 0, and the four-loop beta function is the complete beta function for the theory to first order in α but all orders in the sextic 't Hooft couplings. In this limit, we are therefore able to determine an interacting ultraviolet fixed-point in d = 3 − , which remains non-interacting in d = 3, in addition to the infrared fixed-point which becomes free when d = 3. Both fixed-points can be determined to all orders in the parameter˜ = /α, For d < 3 the ultraviolet fixedpoint is ultraviolet-stable, and may be used to provide an asymptotically safe definition of the theory. For d = 3, however, one of the couplings is marginal, so higher-order corrections are needed to determine its existence. Finite M calculations indicate that the ultraviolet fixed-point does exist when d = 3. It would be interesting to know whether or not the d = 3 ultraviolet fixed-point survives at small but finite α, or, when α = 1. If so, it might constitute an interacting on-supersymmetric large-N CFT in 3 dimensions, with a potentially simple gravitational dual.
Does the infrared bifundamental fixed-point in d = 3 − extend to d = 2? Using the results from studying the theory at small α, we show that the infrared fixed-point merges with the ultraviolet fixed-point and becomes complex at C = 9 2π 2 α + O(α 2 ). For small α, we can therefore conclude that the fixed-point does not extend to d = 2. But, unlike the φ 4 theory, the range of for which the fixed-point is real increases with α. It is therefore conceivable, but not-at-all certain, that the fixed-point could extend to d = 2 for α = 1. It may be possible to calculate the order α 2 term in C with higher loop calculations, which could perhaps shed more light on this question. Of course, a merger is only possible if the ultraviolet fixed-point exists in d = 3. Therefore we expect that, at any value of α, either the UV fixed-point exists in d = 3, or the IR fixed-point survives until d = 2. Of course, our computations are perturbative in nature, and it remains to be seen whether either of these fixed-points are strongly interacting at α = 1.
We then study the theory in the more familiar vector model large-N limit, in which N is large while M is held finite. Again, as in [46], the large-N beta function is exactly computable to first nontrivial order in 1/N , and we are able to use it to determine large-N interacting fixed-points in d = 3 for any finite value of M . For sufficiently large values of M , we find a total of 6 nontrivial interacting fixed-points in d = 3, including two ultraviolet-stable fixedpoints that may provide asymptotically safe definitions of the theory when N is large. We can extend these fixed-points to d = 3 − to all orders in the parameterˆ = N , and thereby determine the range of for which the fixed-points exist in the large-N limit. We find a rich collection of infrared and ultraviolet fixed-points, generalizing the fixed-points of [46], that generically merge into complex fixed-points at various finite values ofˆ . However, unlike the case of M = 1 in [46], we also find that there also exist two (four, in the case of M = 3) real fixed-points that persist for arbitrarily large values ofˆ , and therefore could conceivably extend to d = 2.
While the primary focus of this paper is on large-N fixed-points, we also study perturbative fixed-points for M = 2 and N finite in section 5.
Let us mention some closely-related work. For the special case of M = N , the theory we study was recently also studied perturbatively in [60], where the focus was on generating non-conventional fixed-points by studying the theory at non-integer values of N , and a closely related supersymmetric model was recently studied in [61]. O(N ) × O(M ) fixed-points can also be considered in the larger context of tensor models, e.g., [41,[62][63][64][65][66], and in particular, fixed-points of O(N ) 3 tensor models with sextic interactions were discussed in [67][68][69][70][71]. The theory we consider can also be coupled to a Chern-Simons gauge field in d = 3 -some two-loop computations in the resulting theory appear in [72].
Let us also remark that the analysis of [73][74][75], who study the sextic O(N ) theory in 3 − using functional renormalization group techniques, suggests certain subtleties in the N → ∞ limit of the sextic O(N ) model. Such subtleties could also be present in the N → ∞ limit of the O(N ) × O(M ) models when M is finite, although the analysis of [73][74][75] would need to be redone in this case.
We study a theory that contains M N real scalar fields, denoted as φ ab where a = 1, 2, . . . , N and b = 1, 2, . . . , M , that is invariant under the symmetry group O(N ) × O(M )/Z 2 . We can write the action for a scalar theory in 3 − dimensions with this symmetry group as: Near d = 3, the most general renormalizable potential can be written as a sum of sextic, quadratic and quartic terms: and a triple-trace operator, O The most general quartic interaction invariant under O(N ) × O(M )/Z 2 , is the sum of two quartic operators, and a double-trace quartic quartic operator, O 3 In Eq. (2.2), each coupling constant is divided by 6!, which is a convenient choice for our purposes.
The only quadratic operator invariant under our symmetry group is the usual, single-trace, mass term: At any fixed-point, the two relevant quartic couplings and the mass term must be tuned to zero, in addition to any sextic couplings which turn out to be relevant when quantum corrections are taken into account.
Using the results for a general sextic multiscalar model in [35], we can obtain the four-loop beta function for this theory for finite N and M . These expressions, which are rather long, are presented in Appendix A. The results of [35] also allow us to obtain expressions for the anomalous dimensions for the field φ ab , φ 2 , O In this paper, we are primarily interested in determining fixed-point solutions to these beta functions in the limit when N is large. We consider two natural large-N limits: a bifundamental large-N limit, in which both M and N are large, and a vector-model large-N limit, in which N is large but M is finite. Results for the bifundamental large-N limit are presented in section 3. Results for the vector-model large-N limit are presented in section 4. For completeness, we also study the perturbative fixed-points of our model for M = 2 and N finite in section 5.
Note that, for a sextic theory, a two-loop beta function allows one to determine fixedpoints up to order , and a four-loop beta function allows one to determine fixed-points up to order 2 . However, as we will see below, in the limit where N is large and M is finite, or when M/N 1, the four loop beta function is the complete beta function to first nontrivial order in 1/N and M/N . In analogy with [2], we define the bifundamental large-N limit of the theory as follows: We take the limit N → ∞, keeping the ratio α = M/N fixed. We will always assume that N > M .
In this large-N limit, the following 't Hooft couplings for the single-trace, double-trace and triple-trace interactions are held fixed: (3.1) These definitions follow from a standard graph-theoretical analysis of the leading contribution to free energy graphs, as in, e.g., [84,63]. Alternatively, the definitions in (3.1) can be obtained by imposing the requirement that two-loop corrections to the scalar propagator remain finite in our large-N limit. The factors of (8π) 2 are chosen for convenience.
This choice of 't Hooft couplings does not preserve the symmetry between N and M . An alternative choice of couplings which preserves the symmetry between N and M is given bȳ λ 1 = αλ 1 ,λ 2 = √ αλ 2 , andλ 3 = λ 3 . In the regime where α ∼ O(1), the difference between λ i and λ i is simply a matter of notation. However, in the regime where α 1, these two choices of 't Hooft couplings may define different large-N limits. The large-N limit defined by holding λ i fixed is clearly non-singular for all α < 1 and reduces to the familiar vector model large-N limit when α is very small, so it appears most suitable for our present discussion.
To two loops, the beta functions in this large-N limit are: The system of equations β λ i = 0 possess a unique nonzero solution, which is: We have also computed four-loop and 1/N corrections to this fixed-point in Appendix C. In particular, we will make use of the four-loop corrections presented in Appendix C.1 in the next subsection, when presenting anomalous dimensions at this fixed-point.

Stability and anomalous dimensions
In the large-N limit, at the four-loop level, the anomalous dimension for the field and the mass operator at the fixed-point are independent of α: The quartic single-trace and double-trace operators given in (2.6) do not mix at this order, and have the anomalous dimensions The anomalous dimension of the double-trace quartic operator, is just twice that of (φ ab ) 2 , as expected in the large-N limit, while the single-trace quartic operator acquires a nontrivial anomalous dimension.
The stability matrix for the sextic couplings at the fixed-point is given by To four-loops, the eigenvectors of the matrix in (3.8) are: respectively. So, the fixed-point (3.5) is stable in two directions, and unstable in one direction. (3.12) The scaling dimensions of quartic and quadratic operators φ n i are related to their anomalous dimensions via We thus see that the scaling dimensions of the double-trace and triple-trace operators satisfy ∆ (λ 2 ) = ∆ ν 1 + ∆ φ 2 , and ∆ (λ 3 ) = 3∆ φ 2 , in the large-N limit, as expected. We have calculated O(1/N ) corrections to the stability matrix and anomalous dimensions in Appendix C.
Let us briefly discuss the anomalous dimensions of the unprotected single-trace operators. Although γ φ and γ φ 2 are independent of α to the order we were able to compute, we expect that subsequent higher-order corrections would depend on α in a nontrivial way. Anomalous dimensions of the quartic and sextic single-trace operators do show explicit dependence on α, which are plotted in Fig. 2. As expected, both are symmetric under interchange of α and α −1 , and the anomalous dimensions attain their maximum value at α = 1. This result is consistent with the expectation that α acts as a measure of the strength of interactions, and that anomalous dimensions of generic unprotected operators could diverge when α → 1 and → 1, as required for a simple gravitational dual description.
We also observe that the 2 terms in the anomalous dimensions of O   anomalous dimensions can become negative, and arbitrarily large in magnitude, leading to a violation of unitarity. This suggests that, at least in the limit α 1 (or α 1), our fixed-point only exists up to some critical value of , which we denote as C (α), that vanishes as α → 0. This is consistent with our expectations -when α → 0 our model reduces to the classic tricritical vector model studied in [46]. In the large-N limit, [46] shows that the tricritical vector model fixed-point only exists for < 36π 2 N , which tends to 0 as N → ∞.

The M N → 0 limit
In this subsection, we study our theory in the limit of N large and α 1 in more detail, and compute the critical value of C (α) to first order in α. Of course, the results of this section also determine C to leading order in α −1 , since our theory has a symmetry α ↔ 1 α . Observe that the large-N beta-functions in (3.2)-(3.4) vanish when α → 0, as expected from [46,35]. Let us now understand this limit better by keeping only those terms linear in α in the four-loop large-N beta function: The complete beta function of our theory can be expanded as a power series in α and λ i . We now argue that the 4-loop beta function given in equations (3.14)-(3.16) is actually the complete beta function, valid to all orders in λ i , but only to first order in α.
Recall that an expansion in powers of α = M/N , near α = 0 for a bifundamental theory is closely related to an expansion in powers of 1/N for the vector model obtained by setting M = 1. In [46], it was argued that the only contributions to the 1/N terms in the beta function for the vector model come from diagrams with 4 loops or fewer. Applying the discussion in [46] regarding the 1/N terms in the sextic vector model to our bifundamental model, we observe that the diagrams which contribute to the beta function at order α p in the bifundamental model are a proper subset of the diagrams that contribute to a vector model at order 1/N p , i.e., those diagrams which are also proportional to M p and therefore planar in the usual graph-theoretic sense. It therefore follows that the only terms in the beta function proportional to α originate from two-loop and four-loop diagrams. So, the 4-loop beta function given in equations (3.14)-(3.16) is actually the complete beta function, valid to all orders in λ i but only to first order in α.
Because this is beta function contains all terms linear in α, we can use it to compute fixed-points that are valid to all orders in the parameter˜ ≡ /α. Moreover, we can also use the beta function to determine interacting fixed-points in d = 3, since any higher-loop corrections to the beta function can be made arbitrarily small by reducing α. Higher-loop contributions to the beta function will therefore only give rise to corrections to these fixedpoints that are suppressed by powers of α; unless a fixed-point has a marginal direction, in which case it may or may not survive higher-order corrections. Higher-loop corrections could also lead to new fixed-points, at sufficiently strong coupling.
Let us now look for zeros of the beta function. For any fixed-point, λ 2 and λ 3 are uniquely determined from the value of λ 1 via equations (3.15) and (3.16), which depend linearly on λ 2 and λ 3 respectively: Any fixed-point lies on the one dimensional "critical curve", determined by the solutions to these two equations. For˜ < 4/π 2 , there are two poles at , so this critical curve consists of three disconnected components, while for˜ > 4/π 2 it consists of a single connected component, as can be seen from the plot of λ * 2 in Fig. 3. As in [46], we find two solutions. The first is a generalization of the conventional IR fixed-point of [46], given by using which we can determine λ 2 and λ 3 from Eq. (3.17) to be:  Figure 3: A plot of λ * 2 (λ 1 ) determined using (3.17). For < 4 π 2 α, there are two poles, and the critical curve determined by (3.17) consists of three disconnected components. For > 4 π 2 α, there are no poles, and the critical curve consists of a single component.
If we truncate this solution to order 2 , we recover the small α limit of the fixed-point in the previous subsection; but, because we know the complete beta function in this limit, the solutions are valid to all orders in˜ .
The second solution is a generalization of the UV fixed-point of [46], and is given by using which we can determine λ 2 and λ 3 from Eq. (3.17) to be: Ordinarily, one would regard the "UV" fixed-point as an artifact of perturbation theory that would not survive higher-loop corrections -but, because we know the complete beta function to first order in α, this solution is physically meaningful. Notice that this solution is only valid when˜ = 0, because we had to divide by˜ in Eq. (3.17). We discuss the case of d = 3 in subsection 3.2.1, in which case λ 3 is undetermined and marginal.
The beta function for λ 1 is shown in Fig. 4 for various values of /α. Fig. 4 illustrates that, as in [46], we find both the IR and UV fixed-points merge and become complex (i.e., cease to exist) for > C , where This implies that, for small α, the bifundamental fixed-point does not extend to d = 2, much like the vector model.  .14), is plotted for various values of /α. When 0 < < 9 2π 2 α, the UV fixed-point flows to the IR fixed-point. When > 9 2π 2 the fixed-points disappear.
If we possessed higher-order corrections in α to the beta functions, we would be able to determine higher-order corrections to the UV and IR fixed-points above, and the corresponding corrections to Eq. (3.24). For α ∼ O(1), we therefore expect the bifundamental fixed-point to remain real up to some unknown function of α, C (α) ∼ O(α).
If we use Eq. (3.24) (which is only valid to first order in α) to obtain a crude estimate for the range of for which the fixed-point is real, we find that C ≈ 0.45, when α = 1. However, because C (α) must be symmetric in α ↔ α −1 , we can attempt to improve this estimate using a two-sided Padé approximation of the form C = A 0 +A 2 α+A 0 α 2 B 0 +B 1 α+B 0 α 2 . Substituting the order α and order α 2 results, we find: .
To determine B 1 we would need to know C (α) to order α 2 . This requires, in principle, an eight-loop calculation. If we simply set B 1 = 0, we find C ≈ .2, as shown in Fig. 5. This seems to suggest that the fixed-point may not extend to d = 2, even for α = 1.
Let us now discuss anomalous dimensions at this fixed-point. At the leading order in α and four-loop level, the anomalous dimensions are: , and γ ν 2 = 4α 2 λ 2 1 675 . (3.26) Diagrammatic arguments, similar to those given earlier for the corrections to the vertex, imply that higher-loop corrections to these anomalous dimensions are suppressed by additional powers of α [46]. Hence these expressions are valid to all orders in the 't Hooft couplings, but to leading nontrivial order in α.
Our results determine C (α) to first order in α and 1/α, shown as dashed black lines. With an α 2 calculation, one could determine a Padé approximation that interpolates between both of these curves, schematically shown in red.
The anomalous dimensions for φ and φ 2 at the IR fixed-point are given by The quartic single-trace and double-trace operators decouple, with the anomalous dimensions (3.29) and For the UV fixed-point, the anomalous dimensions are: and γ ν 2 = 2γ φ 2 .
Let us now discuss stability of the fixed-points. The eigenvalues of the stability matrix (3.8), which is lower triangular, at the IR fixed-point are: corresponding to λ 1 , λ 2 and λ 3 respectively. For α < 4 π 2 , the first two of these are positive. Hence, the IR fixed-point has two stable and one unstable directions. When α > 4 π 2 , the second eigenvalue becomes negative, and there are two unstable directions. Some of the matrix elements become singular at α = 4 π 2 because λ 2 becomes singular, but eigenvalues of the stability matrix remain finite.
The eigenvalues of the stability matrix (3.8), at the UV fixed-point are: All eigenvalues are negative when > 0, which seems to suggest that the theory is asymptotically safe in this limit. However, in exactly three dimensions, the triple-trace coupling is marginal -higher-order corrections would be needed to determine whether the operator is stable, unstable, or if the fixed-point ceases to exist, as discussed in subsection 3.2.1.
Let us now discuss flows between these fixed-points. In Fig. 4, there is an apparent flow from the "UV" fixed-point to the IR fixed-point along the critical curve defined by (3.17), which is stable with respect to deformations in λ 1 if is nonzero. For 9 2π 2 >˜ > 4 π 2 the critical curve consists of a single connected component, and the flow from the UV fixed-point to the IR fixed-point implied by Fig. 4 does exist. However, for˜ < 4 π 2 the critical curve actually consists of three disconnected components, with the two fixed-points on different components, and the "apparent flow" along the critical curve implied by Fig. 4, does not actually exist. In this range of , which includes d = 3, there do, however, exist more complicated flows between the UV and IR fixed-point, which exit the critical curve of Eq. (3.17).
It is actually possible to solve for flows in the λ 1 -λ 2 plane exactly, even when˜ = 0. We have: (3.37) When = 0, this simplifies to The free fixed-point (black) also flows to the IR fixed-point. λ 3 is tuned so that β λ 3 vanishes along these flows. appropriately. We assume λ 3 is tuned to solve β λ 3 = 0. Some flows between the UV and IR fixed-points for small˜ and for d = 3 are presented in Fig. 6 and 7.
In the small α limit, flows from the UV fixed-point to the IR fixed-point may be affected by the spontaneous breaking of conformal invariance, as discussed for the sextic vector model in [48,52]. Investigation of this effect is left for future work.

Ultraviolet fixed-point in d = 3
Here we discuss the fixed-points to the beta functions (3.18)-(3.20) when d = 3 exactly.
This then implies that β λ 3 vanishes identically for all λ 3 . Because λ 3 is marginal, we cannot be certain of the existence of such a fixed-point until higher-order corrections in α or 1/N are included in the beta function. Following the discussion in [46], an order α 2 calculation would require an 8-loop calculation. Notice that, when > 0 it is possible to determine λ U V 3 and, using this result, we obtain a finite value for λ U V 3 = − 1080 π 2 in the limit → 0; however, as far as we can tell, this result is not reliable in d = 3 and would be modified when α 2 or 1/N 2 corrections to the beta function are included.
To leading order in the 1/N expansion, anomalous dimensions do not depend on λ 3 , so they can be determined for this putative fixed-point. Their values when d = 3 can be read off from (3.31), (3.32), (3.34) and (3.35). In the large-N limit, we know that ∆ λ 3 = 3∆ φ 2 . Because ∆ φ 2 > 0 we expect that in d = 3, deformations to the UV fixed-point corresponding to λ 3 become irrelevant when higher-order corrections in α are taken into account.
Another nontrivial solution in exactly d = 3 can be obtained by with λ 3 again marginal and undetermined.
Note also that, we can also obtain a third class of potentially nontrivial solutions by setting which leaves λ 3 marginal and undetermined to this order in α.
In the next section we study fixed-points of the theory when N is large and M is finite. We are able to determine several fixed-points without marginal directions that approach one of the possible forms above when M → ∞. In particular, the d = 3 fixed-point [A + ] defined in the next subsection approaches a fixed-point of the form given in Eq. The finite M results indicate that an ultraviolet bifundamental fixed-point in d = 3 exists, but it would be nice to confirm this by a higher-order calculation in α. Assuming the fixedpoint does exist, we can then speculate about the possibility of it extending to finite α. There would be no in-principle obstacle to computing arbitrarily high order corrections in α, to the fixed-point. The power series in α obtained in this way approximates a function that interpolates between two large-N saddle points -hence it is rather different from the 1/N expansion, and one might hope that it has a finite radius of convergence. Because the theory has a symmetry in α ↔ α −1 , we could use a few terms of a power series in α to construct a two-sided Padé approximation as in Fig. 5, which could be used to estimate the radius of convergence, or to estimate various observables in the theory near α = 1.

Fixed points for N large and M finite
In this section, we describe fixed-points of the O(N ) × O(M )/Z 2 theory in the limit where N is large and M is finite. This large-N limit is a conventional vector model large-N limit, as the three interactions O 1 , O 2 and O 3 are effectively triple-trace interactions with respect to O(N ) index contractions. Our results and analysis are therefore an extension classic results for the tricritical vector model presented in [46], which is a special case of our model corresponding to M = 1.
The natural 't Hooft couplings in this limit remain those given in the previous section. To zeroth order in the 1/N expansion the beta function vanishes, as in [46], essentially due to the fact that our interactions are all effectively triple-trace in this limit. Keeping terms up to first order 1/N , the beta functions in this limit are: Again, as argued in [46] and reviewed in the introduction, this four-loop beta function is actually the full beta function to order 1/N . All higher loop corrections to this beta function are suppressed by 1/N 2 . Therefore, we can use this beta function to determine interacting fixed-points in d = 3 as well as fixed-points that are valid to all orders in the parameter = N .
These beta-functions can be written as a gradient of a potential β a = T ab ∂U ∂λ b , given in Eq. (D.3) in Appendix D, with the inverse metric: While we are mainly interested in integer values of M , theories with O(M ) symmetry for non-integer M may be of interest in some contexts, as described in [85]. For M ≤ 2, this metric is not positive definite. In particular, theories with O(M ) symmetry group for non-integer M may not be unitary, and could display unusual RG behaviour, such as limit cycles [60,61]. Because the beta functions are the gradient of a potential, it appears that limit cycles should be impossible even for non-integer values of M . However, the metric in Eq. (4.2) is only positive definite for M > 2. when M = 2 the metric possesses a zero eigenvalue, due to the fact that the three interaction terms O 1 , O 2 and O 3 are not all independent. When −3 < M < 2, the metric contains positive and negative eigenvalues, and therefore gives rise to non-unitary flows, as in [60]. When M < −3 the metric is again negative definite, so no unusual RG behaviour is possible. We briefly study fixed-points in our model for non-integer −3 < M < 2, in Appendix E, with a view to find unconventional fixed-points similar to those in [60,61] -however, we find that our model does not give rise to limit cycles.
When solving for the zeros of the beta function, we find fixed-points which become free in the limit → 0. Following [46] we refer to these as IR fixed-points. There are also several fixed-point solutions of the beta function that do not vanish asˆ → 0. Following [46] we refer to these as "UV" fixed-points. These correspond to interacting fixed-points in d = 3.
We are able to study the existence of fixed-points for all values of M ∼ O(1) and˜ ∼ O(1), i.e., study fixed-points in the M -˜ plane. This analysis is reminiscent of the analysis of [73][74][75] which studies fixed-points of the sextic O(N ) model in the N − plane, via functional renormalization. However, the O(N ) model is not solvable when N and are both finite, any conclusions about the dynamics of the theory in that regime necessarily involve uncontrolled approximations (which may still be self-consistent and physically justified). In contrast, our analysis of fixed-points in the M -˜ plane is controlled by the small parameter 1/N , which can be made arbitrarily small. Interestingly, in our results for the theory when N is large and M is finite, we also find that the˜ → 0 limit and the M → ∞ limit do not commute.
At finiteˆ , some of these fixed-points merge -for example, when M = 1, there are exactly two nontrivial fixed-points which merge atˆ * = 36 π 2 + O(1/N ). This implies that, the merger occurs at d * = 3 − 36 π 2 N + O(1/N 2 ), which is extremely close to d = 3 for N large. To determine the order 1/N 2 correction to d * would require an 8-loop calculation. 4 Of course, the location of the merger at finite N remains unknown. An interesting feature of the new fixed-points we determine below for M ≥ 2 not present in the M = 1 theory, is that some fixed-points persist for all values ofˆ -these fixed-points could conceivably survive to d = 2 although more work would be needed to demonstrate this conclusively.
In subsection 4.1.1 we present two simple fixed-points that are valid for all M . These fixed-points are a straightforward generalization of the fixed-points of [46]. In subsection 4.1.2 we present two additional solutions which are valid for all M > 3, but we are only able to determine their analytic form near d = 3.
The remaining zeros of the beta functions in Eq. (4.1) are rather complicated, and depend on M in a nontrivial way. In addition, so this case must be treated separately. In subsection 4.2, we discuss the fixed-points of the theory for M = 2, which, as mentioned previously, must be treated separately as the three operators O 1 , O 2 and O 3 are not independent. In subsection 4.3 we discuss the fixed-points of the theory for M = 3, and in subsection 4.4 we discuss the fixed-points for M > 3.

Some fixed-points for arbitrary M
While we are forced to resort to numerics to determine the fixed-point solutions of the beta functions (4.1), we are able to determine some solutions analytically for arbitrary M . These are presented in this section.

[A + ] and [A − ]
There are two fixed-points -a "UV" fixed-point and an IR fixed-point -that are valid for arbitrary M and arbitraryˆ . These are fixed-points for which λ 1 = λ 2 = 0, and λ 3 is given by one of, For M = 1, these solutions are those given in [46,35]. When M = 1, there is only one independent coupling, so g 1 and g 2 should be set to zero, and these are the only fixed-points, as discussed in [46]. Note that, when M is large the IR solution does not approach the The beta function for λ 3 in (4.1) is plotted in Fig. 8, for various values of M , when λ 1 = λ 2 = 0. From Fig. 8, it is easy to see that there is a flow from the UV fixed-point to the IR fixed-point for which λ 1 = λ 2 = 0. (This flow, however, could be affected by spontaneous breaking of conformal invariance [48].) The UV and IR fixed-points cease to exist when > 36 π 2 M N . For the IR fixed-point, the anomalous dimension of φ is: For the UV fixed-point, The quartic anomalous dimension matrix in this limit is: Thus we obtain the following anomalous dimensions: For smallˆ the stability matrix ∂ λ b β a at the IR fixed-point is: This is stable in one direction and unstable in two directions for smallˆ . For arbitraryˆ , the eigenvalues of the stability matrix are: (4.11) We observe that atˆ = 32 π 2 M , the second eigenvalue changes sign, so the IR fixed-point becomes stable in two directions and unstable in one direction.
For smallˆ , the stability matrix at the UV fixed-point is: Thus, the UV fixed-point is unstable in two directions and stable in one direction. (When = 0 one of the unstable directions becomes marginal direction, and higher-order corrections in 1/N will be required to determine stability.) For arbitraryˆ , the eigenvalues of the stability matrix are: (4.13) These do not change sign asˆ is varied from 0 to 36 π 2 M .

[A + 2 ] and [B + ]
Here we present two UV fixed-points, denoted [A + 2 ] and [B + ], which exist for all M ≥ 3. We present these solutions as power series inˆ .
The fixed-point [A + 2 ] is given by The field anomalous dimension for [A + 2 ] is: The quartic anomalous dimension matrix is: with the eigenvalues: The stability matrix is: with the eigenvalues: For smallˆ , we see [A + 2 ] is unstable in one direction and stable in the other two directions. The fixed-point [B + ] is given by The field anomalous dimension is: The stability matrix is: For smallˆ , one can show that [B + ] is unstable in two directions and stable in the other direction.
We had to resort to numerics to determine the behaviour of these solutions whenˆ is finite, as discussed below for various values of M .

M = 2
In this section, we present all the fixed-points of the theory when M = 2. We must study this case separately because, when M = 2, the single-trace, double-trace and triple trace operators are not independent, and obey the relation We choose the following two independent 't Hooft couplings: The beta-functions for these redefined couplings reduce to These beta functions can be written as the gradient of the following potential,  . (4.33) In terms of the redefined 't Hooft couplings, the anomalous dimension of φ is given by In the leading order, γ φ 2 = 32γ φ for all the fixed-points. The quartic anomalous dimension matrix is Numerically, we find that the total number of distinct fixed-points n fixed of the beta function varies withˆ as follows:  A schematic plot illustrating how the fixed-points merge and vary as a function ofˆ is presented in Fig. 9.
In the next subsection we discuss the behaviour of these fixed-points near d = 3. In subsection 4.2.2, we discuss the fixed-points which persist forˆ arbitrarily large, as these could conceivably correspond to fixed-points near d = 2 or 4. (However, our large-N limit is defined such thatˆ = N ∼ O(1), so our analysis does not apply to ∼ O(1).)

Fixed points near d = 3
There are a total of 5 fixed-pointsˆ very small but nonzero. Apart from the free fixed-point, two of these are the IR and UV solutions [A − ], and [A + ] given in (4.3) and (4.4).
The remaining two solutions are UV fixed-points, given by the following expressions near = 0:  Table 1. We observe that the fixed-point [B + (2) ] is unstable in all directions, and acts as a stable UV fixed-point that may provide an asymptotically safe definition of the theory in this large-N limit. Numerically determined flows in d = 3 between these fixed-points in theλ 1 -λ 3 plane are shown in Fig. 10. Figure 9: A schematic plot of the number of fixed-points as a function ofˆ , for M = 2. The horizontal axis is schematic -for any given value ofˆ fixed-points are arranged in order of increasing U , so RG flow is only possible from right to left. The colour of each line specifies the stability of each fixed-point: red, orange, and green respectively denote fixed-points with zero, one, and two unstable directions. Black dots indicate mergers of fixed-points.
≈ 808.83 Table 1: We were able to determine a formal analytic expression for the flows in d = 3, given by where the sum is over the three roots of the polynomial x 3 − 6x 2 − x + 1 = 0, and C is an integration constant.

Fixed points at largeˆ
The fixed-point solutions [B + (2) ] and [C + (2) ] are presented as power series inˆ , but we also studied their behaviour for arbitraryˆ numerically, which is depicted schematically in Fig. 9. The fixed-point, [B (2) ] appears to only exist up toˆ = 3.98, while [C + (2) ] appears to survive for allˆ . There is another solution [C − (2) ] that emerges at finiteˆ , and exists for arbitrarily largeˆ . Here we present the asymptotic form of the fixed-points that survive whenˆ large. (But, recall that we still requireˆ N for our beta functions to be valid). We find two nontrivial fixed-points with the following asymptotic expressions whenˆ 1: In the limit of largeˆ both fixed-points have the same anomalous dimensions, which are: The quartic anomalous dimension matrix is The eigenvalues of the quartic anomalous dimension matrix are − 25.87ˆ N and 2ˆ N .
The eigenvalues of the stability matrix are − 56.15ˆ N and 4ˆ N , so in this limit, the fixed-points are stable in one direction and unstable in the other. The value of the potential at either fixed-points approaches: whenˆ → ∞. Flows for largeˆ are shown in Fig. 11.

M = 3
We next study the theory when M = 3. Although the same beta functions apply for M = 3 and M > 3, the structure of the solutions is slightly different for M = 3, so we present this case separately.
Numerically, for M = 3, we find that the total number of distinct solutions n fixed to the beta function varies withˆ as follows:  A schematic plot of these fixed-points, according by potential, as a function of epsilon is given in Fig. 12 When d = 3, we find the anomalous dimensions

Fixed points at largeˆ
Four fixed-points with M = 3 survive whenˆ large, as shown in Fig. 12 The anomalous dimensions and the value of the potential U for these solutions are: The

M > 3
We now discuss solutions for M > 3.

Fixed points in d = 3
In d = 3 the number of UV fixed-points as a function of M , which we treat as a continuous parameter, is depicted schematically in Fig. 13. Fig. 13 also depicts the stability of each fixed-point, and arranges the fixed-points in order of increasing value of the potential, which indicates which flows are possible.  Table 2. A similar table for M = 14 is given in Table 3. We also determined asymptotic expressions for these fixed-points in the limit of M 1, which are given in Table 4.  Fixed Point λ 1  Table 2: Fixed points, stability and anomalous dimensions when M = 4 and d = 3. Eigenvalues of sextic and quartic anomalous dimension matrices presented in an arbitrary order.
Here we find a UV-stable fixed-point.  Table 3: Fixed points, stability and anomalous dimensions when M = 14 and d = 3. Eigenvalues of sextic and quartic anomalous dimension matrices presented in an arbitrary order.      Tables 5 and 6.
The fixed-point in Table 5, which we denote as [A − 2 ], is stable in all directions. For large M , the coupling constants obey the following asymptotic form: with higher-order corrections inˆ and 1/M . The field anomalous dimension approaches as M grows large. The quartic anomalous dimensions are: and all three eigenvalues of the stability matrix approach 2ˆ N + O(1/M ). The fixed-point in Table 6, which we denote as [A − 3 ] approaches the bifundamental fixedpoint described in Eq. For M ≥ 11, this fixed-point is stable in two directions and unstable in one direction. The anomalous dimensions approach those of (3.5) as M grows large, as expected.      For any M > 3 there are two nontrivial fixed-points that survive for largeˆ which we denote by [F ± ]. Both are unstable in two directions and stable in one direction. For M = 14, whenˆ is large, these two fixed-points have the following asymptotic behaviour:

Fixed points at largeˆ
It is possible to generalize this largeˆ asymptotic solution to an analytic expression valid for arbitrary M ≥ 3, but the expression is too unwieldy to reproduce here. For large M , the fixed-point solutions [F ± ] have the following asymptotic form: The anomalous dimensions for [F ± ] when M is large are both given by the following expressions. The anomalous dimension of φ is (4.60) The anomalous dimensions of quartic operators are: 61) and the eigenvalues of the stability matrix are:  [60]. We restrict attention to M = 2, since this is presumably the most physical example.
When M = 2, the three couplings are not independent, as discussed in section 4.2. We use the two independent couplingsg 1 andg 2 given by The two-loop beta functions for these couplings when M = 2, for finite N are: It is possible to solve the zeros of this beta function analytically, for arbitrary N . There are in general four fixed-points -the free fixed-point and three interacting fixed-points that we denote by [

Discussion
It would be interesting to explore whether bootstrap techniques, as described in [38] and references therein, could shed more light on the existence or non-existence of these CFT's for α = 1. It may also be interesting to study the small α limit, or the finite M , large M fixed-points in greater detail using analytic bootstrap [87][88][89][90][91]38]. Fixed  Table 7: Fixed points, stability and anomalous dimensions when M = 2 and N = 3 in d = 3 − . Eigenvalues of sextic and quartic anomalous dimension matrices presented in an arbitrary order.
Fixed   Fixed  Table 9: Fixed points, stability and anomalous dimensions when M = 2 and N = 12 in d = 3 − . Eigenvalues of sextic and quartic anomalous dimension matrices presented in an arbitrary order.
For the infrared fixed-point, we found that the anomalous dimensions of the unprotected single-trace quartic and sextic operators attained their maximum at α = 1. This is consistent with the general expectation that scaling dimensions of unprotected operators become large when one approaches the matrix-like large-N limit from the vector model. The scaling dimensions of φ and φ 2 were independent of α at this order, but we suspect that higherorder corrections would display similar α dependence as the quartic and sextic single-trace operators. It should be possible to confirm this by computing 6-loop corrections to the large-N anomalous dimensions following [92]. It would also be of interest to compute the spectrum of higher-spin operators, to four or six loops, of the theory. To determine the order α 2 correction to Eq. (3.24), which would allow a Padè approximation of the critical dimension, an 8-loop calculation is required.
It is worth mentioning the non-supersymmetric sextic tensor model of [67] in d = 3 − also ceased to exist at a finite value of . There, a real melonic fixed-point with was found to exist perturbatively in d = 3 − , but large-N techniques showed that it ceased to be real for a finite, but small critical value of . However, for the supersymmetric tensor model of [68], smooth interpolation from d = 3 − to d = 1 + was possible, though as noted there, the existence of the theory in d = 2 exactly is still uncertain. 5 Perhaps it would be of interest to consider fixed-points N = 1 supersymmetric version of the O(N )×O(M ) theory, particularly in the limit M/N 1 and compare results to our non-supersymmetric model.
We also studied the theory in the limit where N is large and M is finite. In this limit, it is possible to compute the complete beta function of the theory to first order in 1/N . Studying the zeros of this beta function, we found a rich collection of large-N fixed-points in d = 3 and in d = 3 − , which are determined to all orders in the parameterˆ = N . For most values of M , the fixed-points include at least one ultraviolet stable fixed-point, that could potentially serve as an asymptotically safe definition of the theory in the large-N limit. Unlike the case of M = 1, [47], we also found fixed-points which survive to arbitrarily large values ofˆ . fixed-points in d = 3, these fixed-points are ultraviolet fixed-points, and require a reasonably high degree of fine-tuning, as all quartic interactions must be tuned to zero, in addition to any relevant sextic interaction terms. Therefore, it seems unlikely that these fixed-points would be relevant to frustrated anti-ferromagnetic systems, or other real world condensed matter/statistical systems, unless those systems possess a high degree of fine tuning. Nevertheless, it would be interesting to construct such statistical models which would presumably be generalizations of the tricritical Ising model that possess several chemical potentials that need to be tuned to critical values. Some of our fixed-points in sections 4 and 5 do appear to extend to d = 2 without any obstruction, and it would also be interesting identify if they correspond to any known d = 2 conformal field theories.
There are several additional computations one could carry out related to the theory we study in this paper. We mention a few possibilities for future work below.
It is possible to study flows between the large-N fixed-points determined for finite M , or at small α. We determined some flows analytically, in the limit of small α and for M = 2, as shown in Fig.s 3.38 and 10. It would be interesting to better study the flows between the large-N fixed-points for various values of M . We have not discussed the phenomena of spontaneous breaking of conformal invariance, [48,52,53,51], which could occur along these flows. We hope to investigate the Bardeen-Moshe-Bender phenomenon in the theories considered here in future work, both in the limit where M and N are both large, but α = M/N is small; or for the theories in which N is large and M is finite. Such an analysis is important for confirming the existence of the UV fixed-points identified in this paper.
One could determine the higher-spin spectrum of this and related multiscalar CFT's perturbatively, or using the technique of [93]. In the absence of a gauge field, the scaling dimensions of spin-s operators, (represented schematically, as φ∂ s φ), are not expected to grow as log s when s → ∞, and instead approach a finite value as s → ∞, suggesting that the dual is not string-like. (This objection does not apply for the non-supersymmetric bi-fundamental Chern-Simons theories, where the Chern-Simons gauge field, although nondynamical, still gives rise to a log s dependence of anomalous dimensions of spin s operators for large s [94,17,95].) When M is finite and N is large, the theory is a large-N vector model, so there are various computations possible in principle that may be worth exploring. One may also be able to study these fixed-points in the presence of a Chern-Simons gauge field as in [4,15,[96][97][98]. For M = 1 the fixed-points of the sextic interaction were studied in [99] -however, many of the terms in the beta function at present remain uncomputed, due to the difficulty of 1/N calculations in Chern-Simons vector models. Generalizing these discussions to higher values of M is possible in principle. 6 Let us also point out that the sextic O(N ) theory was studied in the presence of a boundary in [101], and similar computations could be attempted for arbitrary M .
[73] studies the existence and merger of the UV fixed-point with IR fixed-point in the sextic O(N ) theory as a function of N and using the non-perturbative renormalization group technique -they seem to find that a nonperturbtive UV fixed-point and IR fixed-point merge at N (d) ≈ 3.6/(3−d) in apparent agreement with [46]. The authors of [73] also discuss quartic but not sextic O(N ) × O(2)/Z 2 fixed-points. The same group use similar techniques to also point out potential subtleties associated with the N → ∞ limit [74,75]. It may be interesting and worthwhile to repeat the non=perturbative renormalization group analysis of [73][74][75] for the theories considered in this paper -both for the large-N bifundamental fixed-points of section 3.2 in the α − d plane, as well as the finite M fixed-points of section 4 in the N − d plane.
In addition, drawing inspiration from the discussion in [73,102], one could also study bifundamental fixed-points with φ 2m interactions for m > 3. Such interactions become relevant for d < 2 1− 1 m , and could define multi-critical interacting conformal field theories in d = 2.
Another avenue for future research arises from the observation that, for non-integer M in the −3 < M < 2, our theory can be used to define large-N non-unitary fixed-points in d = 3, that could exhibit unconventional behaviour, as in [60,61]. Unfortunately, as described in Appendix E, we did not find a fixed-point whose stability matrix contains purely imaginary eigenvalues, so our particular model does not contain (perturbative) limit cycles. However, it may be possible to construct slightly more complicated theories, with sextic interactions that are effectively triple-trace, (e.g. O(M 1 ) × O(M 2 ) × O(N ) tensor models with N large and M 1 and M 2 finite) to generate non-unitary large-N fixed-points with unconventional behaviour in d = 3.
We briefly discussed the perturbative fixed-points of the O(N ) × O(2) theory when N is finite. As the focus of the paper was mainly on large-N limits, we have not studied fixedpoints with O(N ) × O(M )/Z 2 symmetry when both M and N are finite and greater than 2. Such fixed-points were studied when M = N in [60], but the case of M = N appears not to have yet been considered in the literature, and could be investigated in order to better map out the space of multiscalar fixed-points.
The anomalous dimensions of sextic operators are determined from the eigenvalues of the stability matrix ∂ λ b β a .

C Four-loop and 1/N corrections to the bifundamental fixedpoint
In this appendix, we discuss four-loop and 1/N corrections in the bifundamental large-N limit, described in section 3.1.

C.2 1/N corrections
The complete beta function allows us to compute 1/N and 1/N 2 corrections to the fixed-point at two-loops and four-loops, without much difficulty. Here, however, we content ourselves with 1/N and 1/N 2 corrections to the two-loop fixed-point.
We find that the 1/N 2 corrections to the fixed-point are: The anomalous dimension of the scalar field in this limit is (C.10) The anomalous dimensions of quartic operators mix, when 1/N corrections are taken into account. The corrections to the anomalous dimension matrix are: 11) The eigenvalues of the above matrix are: The corrections to the stability matrix are: When terms of order 1/N 2 are included in the two-loop beta function, we find the beta function admits a new fixed-point, not present for the leading-order beta function, which is: When terms of order 1/N 4 are included in the two-loop beta function, we find that the beta function admits another new fixed-points, which is: We mention these fixed-points only because, when α is small, these correspond to large M limits of the two infrared fixed-points [A − ] and [A − 2 ] determined in section 4.1.1. Note, however, that, at finite α, the fixed-points determined in this appendix are only perturbative fixed-points. Unlike the fixed-points of section 4.1.1, we expect these fixed-points to receive corrections from all higher loop contributions to the beta function , which, in the bifundamental large-N limit, are not suppressed by powers of N .

D Beta-functions as the gradient of a potential
In a general scalar theory in 3 − dimensions, the beta-function up to four loops may be written as a gradient, of the following potential, where g ijklmn is assumed to be a symmetric tensor, U = − g ijklmn g ijklmn + 1 3 g ijklmn β This expression is similar to the ones used in [66] and [110]. The inverse metric is trivial at this order: (T −1 ) ab = ∂g ijklmn ∂g a ∂g ijklmn ∂g b . (D. 2) The expression for the potential when M and N are both finite is too long to present here. We will mainly make use of this potential in the limit where N is large and M is finite. eigenvalues, or a pair of purely imaginary eigenvalues. Such fixed-points were dubbed as unconventional fixed-points in [60]. If a fixed-point were found in which the eigenvalues were purely imaginary, then we would expect small deformations around the fixed-point to result in limit cycles -i.e., closed RG flows which start and end at the same point in coupling constant space. Such a fixed-point is known as a Hopf point. For fixed-points with complex eigenvalues, RG trajectories form spirals in coupling constant space, which are unusual, but, admittedly much less interesting than limit cycles.
Our fixed-points differ from those in [60] in that our fixed-points are large-N saddle points in integer dimension d = 3, while the fixed-points there were perturbative finite N fixed-points studied in d = 3 − . So if we were to find limit cycles in our model, it would be quite interesting, and there could be a formal holographic dual to such RG flows.
The fixed-point solutions are depicted schematically in Fig. 18. We find non-unitary fixed-points with whose stability matrix ∂ λ i β j has a pair of complex eigenvalues, similar to some of the fixed-points in [60]. Unfortunately, however, we do not find any fixed-points with purely imaginary eigenvalues, hence we do not expect limit cycles in the theory. 7 We also studied the fixed-points in the range −3 < M < 0, and did not find any fixed-point with purely imaginary eigenvalues. For M < −3, the inverse metric T in Eq. (4.2) is negative definite, so no unconventional fixed-points are possible.
the fixed-points in the main text correspond to symmetry groups with integer values of M and N , and therefore may describe phase transitions of some as-yet unknown, but not unconventional, statistical models that generalize the classic models such as the tricritical Ising model in a straightforward way -they may be considered exotic only to the extent that presumably require the tuning of several chemical potentials, rather than just one. Figure 18: A plot of the fixed-points that exist in d = 3, for non-integer M between 0 and 2. We find that, for some ranges of M fixed-points become "spooky" or unconventional [60] -i.e., the stability matrix ∂ λ i β j contains complex eigenvalues. These ranges are highlighted in black. The colour of each line denotes the number of unstable directions as in Fig. 13, where directions corresponding to complex eigenvalues of the stability matrix are considered unstable if their real part is negative. Unlike Fig. 13, fixed-points are not arranged according to the value of the potential.