Heavy Handed Quest for Fixed Points in Multiple Coupling Scalar Theories in the $\varepsilon$ Expansion

The tensorial equations for non trivial fully interacting fixed points at lowest order in the $\varepsilon$ expansion in $4-\varepsilon$ and $3-\varepsilon$ dimensions are analysed for $N$-component fields and corresponding multi-index couplings $\lambda$ which are symmetric tensors with four or six indices. Both analytic and numerical methods are used. For $N=5,6,7$ in the four-index case large numbers of irrational fixed points are found numerically where $||\lambda ||^2$ is close to the bound found by Rychkov and Stergiou in arXiv:1810.10541. No solutions, other than those already known, are found which saturate the bound. These examples in general do not have unique quadratic invariants in the fields. For $N \geqslant 6$ the stability matrix in the full space of couplings always has negative eigenvalues. In the six index case the numerical search generates a very large number of solutions for $N=5$.


Introduction
There is a huge literature devoted to analysing fixed points using the ε expansion for a very large number of physical systems and determining their critical exponents. A review covering many of the cases that have appeared in the literature is found in [2] and a wide range of known fixed points were also discussed by us with a different perspective in [3]. In many respects the ε expansion is a universal solvent for understanding critical phenomena and builds on and extends the historic analysis based on Landau mean field theory. In practice it reduces to determining β-functions and anomalous dimensions in a loop expansion based on Feynman graphs. Results have been recently extended to seven loops in [4], as applied in [5], and with O(N ) symmetry to six [6] which have been extended to when the symmetry is reduced to O(m) × O(n) [7] and also cubic symmetry [8]. Using sophisticated resummation techniques to extend the ε expansion to ε = 1 there is remarkable agreement with results obtained by the bootstrap in three dimensions [9], although some tension also exists [10,11].
The discovery of possible fixed points in any ε expansion reduces to finding the zeros of the one loop β-functions in 4 − ε dimensions. Higher loops provide perturbative corrections but do not generally eliminate the fixed point except at possible bifurcation points. Here the notionally small parameter ε scales out and it is necessary to solve a tensorial quadratic equation for the symmetric couplings λ ijkl where the indices range from 1 to N . The dimension of the space of symmetric 4-index couplings, 1 24 N (N +1)(N +2)(N +3), increases rapidly with N and the determination of possible fixed points consequently becomes non trivial for quite low N without additional assumptions such as imposing a symmetry to reduce the number of independent couplings. Of course for N = 1 the fixed point equations become totally trivial, giving rise to just the Ising fixed point. For N = 2, 3 historic discussions are contained in [12,13] and a detailed analysis for N = 2 is contained in [3]. More recently a careful analysis of the N = 3 case is contained in [14] and this has also been extended to N = 4 in [15].
There exist several examples of fixed points which appear for any N , most simply when there is O(N ) symmetry and just a single coupling, but there were until very recently no complete general results even when N = 3. Various theorems for fixed points and their stability properties have been obtained in [16,17,18,19]. More recently a fundamental bound was proved by Rychkov and Stergiou [1] which takes the form, after scaling out ε and the usual factors of 16π 2 which arise in a loop expansion, (1.1) For N = 2, 3 there are stronger bounds. When the bound is saturated there is a bifurcation point and the stability matrix develops a zero eigenvalue. Further bounds have been obtained by Hogervorst and Toldo [20,21] including tighter results for S N when N = 2, 3.
A related issue is the question of a lower bound for S N . Of course for the Gaussian theory with N free fields S N = 0. For two decoupled theories S N = S N 1 + S N 2 where N = N 1 + N 2 and S N 1 , S N 2 correspond to fixed points with N 1 , N 2 fields. For N decoupled Ising fixed points in our conventions S N = 1 9 N and assuming any perturbation of the N decoupled Ising fixed point theory so as to generate a fully interacting theory decreases A and hence increases S N , then at any resulting fixed point S N,fully interacting > 1 9 N . (1.3) However this is violated by S N for the fully interacting O(N ) symmetric fixed point if N 10 (as N → ∞ in this case S N → 3), which implies that for N 10 there is no RG flow from the decoupled Ising to the O(N ) theory. Without imposing any condition that the fixed point not contain decoupled free theories [21] obtained S N > 1 9 . In previous literature the starting point has usually been the determination of all quartic polynomials in the scalar fields φ i invariant under some subgroup H of O(N ) for particular N . The choice of H depends on the particular physical system for which critical exponents are to be found. For N = 4 [23,24] and N = 6 [25,26,27] detailed investigations for all possible subgroups of O(N ), the corresponding spaces of quartic polynomials and associated fixed points was undertaken. Further analysis for low N in 4 − ε and 3 − ε dimensions based on the symmetry groups for regular solids was described in [28]. In these discussions the condition that there is a unique quadratic polynomial, which may be taken as φ 2 = φ i φ i , is imposed. The fixed points found in this fashion are rational and generally have rational critical exponents. An emphasis in these papers is whether the fixed points in a particular symmetry class are stable or not. A fixed point is stable if there are no marginally relevant quartic operators, or equivalently the eigenvalues of the stability matrix, formed from the derivative of the β-function at the point where it vanishes, are all positive. Although not entirely evident in [23,24] and [25,26,27] different symmetry groups may lead to the same fixed point.
Fixed points may apparently have different couplings corresponding to restrictions to different bases of quartic polynomials but if they are related by an O(N ) rotation or reflection they are equivalent. The fixed points which are determined by the zeros of the multi coupling β-functions may also correspond to decoupled theories, reducible to fixed point theories, including free theories, with lower N . Although the fixed points may be the same, the number of couplings and hence the dimension of the stability matrix may differ. This ensures that the question as to whether a fixed point is stable depends on the symmetry group which is initially imposed. A particular fixed point has identical values of O(N ) invariants such as S N in (1.1) although equality of S N by itself does not suffice to guarantee the same fixed point. This will be shown by particular examples later.
In this paper we follow an orthogonal approach by looking for solutions of the basic equations numerically for low N where S N is close to the bound in (1.1). We focussed on such S N since we initially hoped to find fixed points where the bound was saturated. Although no such cases were found there are generically many fixed points wih increasing N where S N is very close to 1 8 N . Our numerical search used the optimization algorithm Ipopt [29] through pygmo, which provides Python bindings of the C++ library pagmo [30]. Ipopt can perform constrained non linear optimization. The quantity S N was given as an objective to Ipopt, while the β-function equations were given as constraints. Ipopt was then called many times on a cluster, with a random initial point provided by pygmo. In many runs the algorithm failed to find a solution, but feasible solutions were frequently found too. Since our problem is non convex, different solutions were generally obtained in different runs. In our runs Ipopt entered the so-called "restoration phase" [29], in which the objective function S N was ignored and only the β-function equation violations were minimized. Thus, for a feasible solution, S N was not attempted to be maximised by Ipopt. Only the β-function equations were numerically satisfied, with a tolerance we chose to be 10 −10 . We subsequently improved the solutions obtained using SymPy's nsolve or Mathematica's FindRoot, in some cases to a tolerance of 10 −200 . We note here that it is not guaranteed that our method will find all possible solutions.
This method reproduces the known fixed points where S N is rational but also produces many irrational fixed points. Of course it is necessary to isolate those which are decoupled theories. For decoupled interacting fixed points the stability matrix has the eigenvalue 1 with degeneracy greater than one and for a free theory there are eigenvalues −1. Although our search is restricted to fixed points with S N close to the bound, we believe that it is possible to find all non trivial fixed points for N = 3, 4 and perhaps N = 5, 6. In the space of all quartic couplings the stability matrix has negative eigenvalues always when N 6.
The irrational fixed points generally correspond to theories with two or more quadratic invariants. Such cases are not commonly considered but have recently been found to have relevance in discussions of conformal field theories (CFTs) at non zero temperature [31]. In many cases we are able to match the fixed points found numerically with results for fixed points in so called biconical theories, or various generalisations thereof. In the simplest case two theories which are separately O(n), O(m) invariant are linked by a product of two singlet quadratic operators so the symmetry O(n) × O(m) is preserved. The fixed points arising from RG flow starting from the decoupled theory perturbed by the product of quadratic operators include one with the maximal O(n + m) symmetry, but for n = m, and with suitable restrictions on n, m, they also lead to irrational fixed points with two quadratic invariants. The generalisations discussed here allow for several quadratic invariants. Such irrational fixed points were recognised long ago [32], for other literature see [33,34,35,36].
In this paper in the next section we discuss the general features of the lowest order equations for the couplings λ ijkl and their decomposition under O(N ) and introduce O(N ) invariants a 0 , a 2 , a 4 . Along with S N in (1.1) these serve to characterise different fixed points in an invariant fashion. Some bounds are obtained together with properties of stability matrix eigenvalues. In section 3 the fixed point equations are solved analytically for general N assuming various symmetries, in particular the case when the symmetry is S N , the permutation group of N objects. This includes previously known examples with cubic and tetrahedral symmetry. In section 4 fixed points which arise when two decoupled theories are perturbed by what can be regarded as double trace operators are described. These include so called biconical fixed points. They typically generate irrational fixed points although in some cases rational ones as well. Detailed results for all N up to N = 7 are presented in section 5. Where possible, numerical results are related to the analytic discussion earlier.
Irrational fixed points for N = 4 were found previously in [15]. This case is special in that fixed points which are degenerate at lowest order split at higher orders in ε. A similar discussion relating to the fixed point solutions for λ ijklmn is undertaken in section 6.

General N Fixed Points
The basic algebraic equation determining fixed points at lowest order in the ε expansion in 4 − ε dimensions is for λ ijkl the symmetric tensor determining the renormalisable couplings and we assume i, j, k, l = 1, . . . , N . For any tensor X ijkl , the action of the symmetriser S n is defined such S n,ijkl X ijkl denotes the sum over the n terms, with unit weight, which are the minimal necessary to ensure the sum is a fully symmetric four index tensor, if X ijkl is invariant under a subgroup G X ⊆ S 4 then n = 24/|G X |. With the potential (2.1) can be expressed more succinctly as The fixed point equation ( The symmetric coupling λ ijkl transforms as a four index tensor can be decomposed into the three possible O(N ) irreducible representations by writing where d 2,ij and d 4,ijkl are symmetric and traceless rank two and rank four tensors; d 2,ij is determined by and then d 4,ijkl is given in terms of λ ijkl by subtraction of the d 0 , d 2 pieces. Following Hogervorst and Toldo [20,21] If α r are the eigenvalues of λ ijkk then a 2 = r<s (α r − α s ) 2 /N and crucially it follows that For decoupled theories λ 1∪2,ijkl = λ 1,ijkl +λ 2,ijkl where λ 1 ·λ 2 = 0. In this case a 0 , a 1 , S N are additive though a 2 , a 4 are not, These determine the lowest order anomalous dimensions for the N -component fields φ i at the fixed point which are 1 12 γ ε. If (2.8) is satisfied then from (2.1) Conversely for a 2 non zero Γ ij should not be proportional to δ ij and there are necessarily different eigenvalues γ. For all eigenvalues γ equal then since [Γ ij ] is a positive matrix it is necessary that 0 γ 1 8 ; saturating the upper bound is equivalent to (1.1). Further invariants are also given by the 1 24 N (N + 1)(N + 2)(N + 3) eigenvalues {κ} of the stability matrix at the fixed point which are determined by (2.12) for v ijkl symmetric. In this case (2.13) Furthermore (κ + 1) 2 = 1 2 (N + 4)(N + 5)S N + 4(N + 4)a 1 + a 0 2 . (2.14) A solution is always obtained by taking v ijkl → λ ijkl giving, by virtue of (2.1), κ = 1. This is in general non degenerate except for decoupled theories. Directly from (2.12) and (2.1) gives solutions of (2.12) with κ = 0. In general ω ij corresponds to elements of the Lie algebra of O(N ). The number of zero modes (2.16) is given by 1 2 The eigenvalue equations may be extended to φ 2 and φ 3 operators where the anomalous dimensions are determined by where µ = a 0 , ν = (N + 2)a 0 . (2.18) A solution of the eigenvalue equation for ν is obtained with ν = 1 by taking v ijk → λ ijkl v l , for any vector v l , so this is N -fold degenerate.

Decomposition of Fixed Point Equation
Applying (2.5) in (2.1) gives rise to three separate equations Hence using (2.7), (2.22) we obtain the bounds This is just the Rychkov-Stergiou bound [1] as extended by Hogervorst-Toldo [20,21]; the Rychkov-Stergiou case arises for N 4 when a 2 = 0. This bound is saturated for a 0 = 1 2 N , which is necessary for the sum of two decoupled theories saturating the bound to also satisfy the bound since (2.9) requires N 2 a 0,1 − N 1 a 0,2 = 0. The bound on a 0 is saturated for the O(N ) symmetric theory. With this bound on a 0 then (2.13) requires κ < 0 when N 6. If (1.1) is saturated then necessarily a 2 = 0 and from (2.8) and (2.11) then Γ ij = 1 8 δ ij . The fixed point equations can be reduced for any basis of 4 and 2 index symmetric traceless tensors {d r,ijkl ,d a,ij }, r = 1, . . . , p, a = 1, . . . , q, satisfying where it is necessary that a rs , b rs t , c rs a are symmetric under r ↔ s and similarlyâ ab ,b ab r ,ĉ ab c for a ↔ b. Consistency between (2.24a), (2.24b) and (2.24c) requires The non associative algebra of symmetric matrices defined the symmetric product ∨ 2 is a Jordan algebra since, for U, V symmetric, the lowest order fixed point equations for the p + q + 1 couplings λ, g r , h a from (2.21a), (2.21b), (2.21c) are then λ = (N + 8)λ 2 + r,s a rs g r g s + (N + 16) a,bâab h a h b , where f sa r , e rb a can be eliminated using (2.25).

Further Bounds
A general analysis of (2.21a), (2.21b), (2.21c) is not straightforward. In general d 0 1 N +8 , which is equivalent to the a 0 bounds in (2.23). If (1.1) is to be saturated, then d 2,ij = 0 and the equations simplify to (2.34) The last two equations are equivalent to writing where w ijkl is a mixed symmetry tensor satisfying w ijkl = w (ij)kl = w klij , w i(jkl) = 0 and w ijkk = 0. With this definition for w ijkl Higher rank tensors can be formed from d 4,ijkl . There is a symmetric 6-index tensor and a corresponding mixed symmetry symmetry tensor Then and

Fixed Points Valid for Any N
For any N the maximal symmetry is of course O(N ) with a potential depending on a single coupling where (2.1) reduces to just λ = 1 N +8 . Invariably G contains a one dimensional submanifold with H sym = O(N ) but there may also be higher dimensional submanifolds with enhanced symmetry groups. For theories determined by a potential V (φ, g), g ∈ G, there is a group action on G given by The corresponding symmetry group is then the subgroup H sym ⊂ H defined by h ∈ H sym , g ·ĥ = g for arbitrary g ∈ G. H sym is a normal subgroup of H and the action of the quotient group Q G = H/H sym on G defines isomorphic theories and fixed points related by the action of Q G are equivalent. For an arbitrary potential depending on the complete set of 1 24 A standard procedure is to analyse the different subgroups H I ⊂ O(N ) and then determine all possible quartic polynomials invariant under each H I . However, differing H I may not have distinct quartic polynomials so that H sym may be the union of various H I . An alternative is to search for fixed points with a restricted symmetry where the number of couplings is such that analytic and irrational numerical solutions are possible for arbitrary N . Inevitably this approach will generate fixed points corresponding to decoupled theories but these can be identified by looking at the stability matrix eigenvalues or checking the additivity of S, a 0 , a 1 for decoupled theories.

Fixed Points with S N Symmetry
For an initial search for fixed points for general N we consider an ansatz with an overall S N × Z 2 symmetry which is obtained by taking no sums on i, j, k, l .
Hence if 3z−x = 3w−y = 2u there is a O(N −1) rotation symmetry which leaves φ i = φ j for all i, j invariant. For N = 3 it is only necessary that x − 3z = y − 3w for there to be a O (2) symmetry. When y, w, u are non zero there is just the overall Z 2 reflection symmetry. For y, w, u zero this extends to Z 2 N since the potential is invariant under φ i → −φ i for any i and this then corresponds to a theory with hypercubic symmetry Z 2 N S N . Otherwise theories in which the couplings are changed in sign for an odd number of reflections belonging to Z 2 N are equivalent. There are two S N singlet quadratic operators, i φ i 2 and i =j φ i φ j . For N = 3 the coupling u is irrelevant, for N = 2 both w, u can be dropped.
The associated lowest order fixed point equations for the five couplings from (2.1) become For N = 2 just the first three equations with w = u = 0 are relevant while for N 3 we need only keep the x, y, z, w equations and set u = 0. (3.9) and in the decomposition (2.5)

For this ansatz
For Γ ij defined in (2.10) then with this ansatz For v non zero there are two eigenvalues within this ansatz, namely γ = a 0 /N + (N − 1)v with degeneracy 1, and γ = a 0 /N − v with degeneracy N − 1.

O(N − 1) Basis
An alternative basis to that given in (3.4) is based on their transformation properties under the SO(N − 1) subgroup leaving the N -vector (1, 1, . . . , 1) invariant. A basis of generators is given by which satisfies the commutation relation 14) and the completeness relation We may define where ∆ is essentially the Casimir for SO(N − 1). Then for 19) The eigenvectors of M determine the linear combinations of x, y, z, w, u which correspond to different representations of O(N − 1). There are three eigenvalues 0, which is threefold degenerate, 3N and 4(N +1). Correspondingly we define the couplings σ, ρ, τ 0 , τ 1 , τ 2 where (3.20) In the decomposition (2.5) σ corresponds to d 0 , ρ to d 2,ij and τ 0 , τ 1 , τ 2 to d 4,ijkl .
In this basis instead of (3.8) When N = 3 the τ 2 equation is to be omitted and the remaining four equations are equivalent to those given in [14]. These equations correspond directly with the general form (2.33). Clearly a consistent truncation is to set τ 1 = 0 and for O(N − 1) symmetry τ 1 = τ 2 = 0. The equivalence relation (3.7) becomes just In terms of these couplings (3.23)

Fixed Points
Within this five coupling theory there are generically 18 real fixed points of (3.8) including the trivial Gaussian theory. Non trivial fixed points with rational S N at lowest order are given by Cases I,II and IV,V as well as XIII ± and XIV ± are equivalent in that the couplings are related by O(N ) rotations. Cases VIII, X and XII are decoupled theories.
At the fixed points given in (3.24) the results correspond essentially to theories with hypercubic, hypertetrahedral or O(N ) symmetry. For convenience we identify the hyperoctahedral group B n which is the symmetry group of a n-dimensional hypercube expressible as a wreath product where |B n | = 2 n n! and we have used the notation for a wreath product.
Results for the lowest order anomalous dimensions for φ 4 operators in the hypercubic and hypertetrahedral cases were given in [3] and extended in the hypercubic case in [37].
In the above (3.28) and for the cases with with hypertetrahedral symmetrŷ and As special cases Except for particular N the remaining fixed point is irrational. The couplings satisfy 3z − x = 3w − y = 2u and so there is a O (N − 1)

Fixed Points with Continuous Symmetry
When N factorises there are various fixed points which can be regarded as built from fixed points corresponding to the factors of N . For N = mn then for φ → ϕ ra , r = 1, . . . , n, a = 1, . . . , m, n > 1 there are non trivial fixed points obtained from the potential At the fixed point, after a suitable rescaling, the necessary equations corresponds to (2.1) become just .
The symmetry group is O(m) n S n = O(m) S n and this fixed point is denoted here by MN m,n . MN 1,n = C n , and MN 2,2 = O 4 . For these theories If m = 4 and the S-bound is saturated this is just n decoupled O 4 theories.

Tetragonal Fixed Points
In the condensed matter literature fixed points arising for systems with tetragonal symmetry are of interest [38,39,40,2]. These can be modelled by considering the potential 37) where N = 2n. This has the symmetry D 4 n S n . The potential is invariant under equivalence relation g ∼ (g + 3h)/2, h ∼ (g − h)/2. The fixed point equations for the three couplings are just λ = 2(n + 4)λ 2 + 2(g + h)λ , g = 12 λg + 3(g 2 + h 2 ) , h = 12 λh + 2(g + 2h)h , (3.38) which have 8 solutions but two pairs related the equivalence relation. The results all have a 2 = 0 since there is a unique quadratic invariant and the non trivial ones can be summarised by The tetragonal symmetry is enhanced at each fixed point and within this three coupling theory MN 2,n is stable for n > 2. At the MN 2,n fixed point g = 3h which is invariant under the equivalence on on the couplings. This agrees with Michel's theorem requiring that a stable fixed point is unique.

'Double Trace' Perturbations
A wide range of fixed points can be obtained by perturbations of decoupled theories. Assuming φ i = (ϕ a , ψ r ), a = 1, . . . , m, r = 1, . . . , n, N = m + n, the starting point is For the decoupled theory then We then consider a perturbed theory obtained by where for just single quadratic invariants ϕ 2 , ψ 2 , The additional term may be regarded as a double trace perturbation and the symmetry ensures the form (4.4) is preserved under any RG flow generated by the coupling h and we may require δµ 1 δ ab = δλ 1,abcc , δµ 2 δ rs = δλ 2,rstt . At any fixed point with m = n there are then two distinct invariant quadratic operators.
At lowest order the fixed point equation (2.3) requires Hence at this order we may take so that for a relevant perturbation h > 0. If the decoupled fixed points are such that is small we may set up a perturbation expansion in . At order h 2 a non trivial fixed point must satisfy To the extent that these equations can be inverted to define δV 1 , δV 2 it is possible to set up a series expansion in which should converge in the neighbourhood of = 0 so that there is a new non decoupled fixed point with symmetry H 1 × H 2 . In general the theory defined by (4.4) may have several fixed points some of which have an enhanced symmetry but for h non zero and small the fixed point is here denoted as B F P 1 * F P 2 .
As a special case we may impose m = n, V 1 = V 2 and then H 1 = H 2 = H. There is an additional Z 2 symmetry under ϕ ↔ ψ so that φ i = (ϕ a , ψ a ), a = 1, . . . , m and there is a single H 2 Z 2 invariant quadratic operator ϕ 2 + ψ 2 . Fixed points obtained with the additional Z 2 symmetry are generally rational.

Perturbed Cubic Theories
For the Z 2 symmetric case we assume V 1 = V 2 each correspond to a theory which generates the C m fixed point so that For the potential V 1 (ϕ) + V 1 (ψ) + 1 4 h ϕ 2 ψ 2 , after rescaling, the fixed point equations reduce to 2 , which is a decoupled theory as in this case h = g = 0.
For m = n there is no longer the Z 2 symmetry relating V 1 , V 2 and the couplings extend to λ 1 , λ 2 , g 1 , g 2 as well as h. The lowest order fixed point equations become λ 1 = (m + 8) λ 1 2 + 2 λ 1 g 1 + n h 2 , g 1 = 3 g 1 2 + 12 λ 1 g 1 , The symmetry group is B n ×B m . For g 1 = g 2 = 0, corresponding to a perturbed theory with O(m) × O(n) symmetry, the non trivial irrational fixed points are referred to as biconical.
For this case when m = n, λ 1 = λ 2 the symmetry group is O(m) 2 Z 2 and the fixed point is identical to MN m,2 .

Multi-conical Theories
To obtain fixed points with more than two quadratic invariants the biconical case is naturally extended to potentials where each ϕ r has m r components so that N = n r=1 m r . There are 1 2 n(n + 1) couplings and generically the symmetry is O(m 1 ) × · · · × O(m n ). If m r = m r , λ r = λ r , h rs = h r s for r, r ∈ S and all s there is an additional S dim S symmetry. Such fixed points in the triconical case, n = 3, were consider by Eichhorn et al [41]. Fixed points which are not reducible to decoupled or biconical theories with additional symmetry are here denoted by B Om 1 * Om 2 * ··· * Om n with O 1 = I. Extensions where the initial theories have cubic or tetrahedral symmetry are easily obtained.

Fixed Points for Theories containing S n Symmetries
A wider range of fixed points can be obtained using the results for S N symmetry obtained in 3.1 with additional fields. Many examples are encompassed by taking N = n + m and imposing S n × S m × Z 2 2 where theories with S n , S m symmetry, as discussed in subsection 3.1, are linked by products of quadratic operators which are singlets under S n and S m . The potential becomes For the potential (4.27) the fixed point equations reduce to together with those obtained by (x, y, z, w, u, t, n) ↔ (x , y , x , w , u , p, m) and where the β-functions are given in (3.8). For m = n we may reduce the couplings by imposing the symmetry condition (x , y , z , w , u , p) = (x, y, z, w, u, t). For small N results for fixed points which can not be reduced to decoupled products of fixed point theories (including free theories) with lower N are enumerated in various tables below. Analytic results have been added to complement the numerical ones. As already said our searches may not find all fixed points but we hope this to be the case for N = 3, 4, 5, 6.
For the T 4+ fixed point the O 4 eigenvalues for κ also split at O(ε 2 ) giving The degeneracies of course correspond to dimensions of representations of the respective symmetry groups B 4 and S 5 × Z 2 . In both cases there remains a representation with κ = 0 which should become non zero at higher order in the ε expansion.
The three irrational fixed points which were found in our numerical search were also obtained in [15]. One of these appears within the biconical framework described here, the other two appear to be special to N = 4. For the case with largest S 4 we follow [15] and consider the potential  For the other case 8 couplings are necessary

4) and
For h = 0 this corresponds to a triconical type theory with symmetry group of order 32, (Z 2 2 Z 2 ) × Z 2 × Z 2 where Z 2 2 Z 2 D 4 the two dimensional cubic symmetry group. For h non zero the symmetry is reduced to D 4 × Z 2 since a π/2 rotation of (φ 1 , φ 2 ) then requires also a reflection φ 3 → −φ 3 . There are clearly three quadratic invariants. Theories related by h → −h and also x 1 ↔ x 2 , h 1 ↔ h 2 are equivalent. The fixed point equations require Solutions with h non zero give the tetrahedral fixed point T 4− and also the case listed aŝ B O 2 * I * I above.
Rational fixed points for N = 4 were obtained in [24] by looking for fixed points arising from quartic potentials invariant under all possible subgroups of O(4) subject to there being a single invariant quadratic form. The subgroups of O(4) are non trivial [42,43,44]. Here we list their symmetry types discussed in [24] for which the group acting on the couplings defining equivalent theories is discrete, the number of couplings corresponding to the number of independent quartic potentials necessary to realise the required symmetry and the associated fixed points in our notation. These always involve the O(4) symmetric case but include those corresponding to decoupled theories which are omitted above. Each example is intended to correspond to a different chain of symmetry breaking of O(4) and therefore to be independent. The different symmetry groups in each case are described in Appendix B. However for a special restriction of the couplings the dipentagonal case reduces to the di-icosahedral which is identical with the hypertetrahedral theory restricted to N = 4. Each of the fixed points arising from solving the RG equations for the various potentials can be obtained starting from theories with just two couplings and also can in principle be extended to arbitrary N . The The lowest order invariants S N , a 0 , a 4 do not distinguish these new fixed points but they can be identified in terms of their stability matrix eigenvalues {κ} when calculated to O(ε 2 ) as was given in two cases with cubic and tetrahedral symmetry in (5.1) and (5.2) above. To identify the split fixed points we consider the β-functions for N = 4 to two loops in a canonical form

symmetry(dimension) number of couplings fixed points O(ε) fixed points O(ε
where g r are couplings for φ 4 operators which are symmetric traceless tensors and λ is, as previously, the coupling for the O(4) invariant (φ 2 ) 2 . Split solutions are obtained by requiring at lowest order in a ε expansion that λ = O(ε), g r = O(ε 2 ) and then These results are not affected by three loop contributions and it is straightforward to determine b st r in each case.

N = 6
There are more fixed points since 6 is non prime, rational fixed points are given by For N = 6 the total dimension of the space of slightly marginal deformations is 126. The dimension of so (6) is of course 15, for 14 zero modes there remains an O(2) continuous symmetry.

Mukamel and Krinsky Model
An example of an N = 6 theory intended to be related to phase transitions in antiferromagnets was considered by Mukamel and Krinsky long ago [38]. Taking φ i = (ϕ a , ψ a ), for a = 1, 2, 3, this is based on the potential, with slight changes of notation from [38], This has a Z 2 6 symmetry resulting from ϕ a → −ϕ a or ψ a → −ψ a for any individual a. There are also three Z 2 symmetries ϕ a ↔ ψ b , ϕ b ↔ ψ a , ϕ c ↔ ψ c for a = b = c = a and further the Z 3 cyclic symmetry generated by φ a → φ a+1 , ψ a → ψ a+1 mod 3. These form Z 2 3 Z 3 S 4 , the symmetry group of a tetrahedron. By considering just ψ a → ψ a+1 mod 3 and also ϕ a ↔ ψ a theories related by permutations of g 3 , g 4 , g 5 are equivalent.
For this theory Γ ij = g 1 2 + 6 g 2 2 + 3(g 3 2 + g 4 2 + g 5 2 ) δ ij . (5.13) There are 20 fixed points, the non trivial fixed points realised by this theory can be summarised, with previous notation and excluding equivalent fixed points related by per-mutations of g 3 , g 4 , g 5 , by Fixed Point g 1 g 2 g 3 g 4 g 5 {κ} These are ordered in terms of increasing S 6 . Even restricting to the five couplings in (5.12) all fixed points are unstable, as was realised by Mukamel and Krinsky. At non decoupled fixed points the symmetry is enhanced. For fixed points linked by a RG flow arising from a relevant perturbation the change in S 6 between the two fixed points must be positive. Hence there cannot be any RG flow from C 3 2 to C 6 or from O 3 2 to MN 2,3 although there may be a flow to MN 3,2 .
If we impose g 5 = g 4 = g 3 the potential reduces to that for perturbed cubic theories given by (4.4), (4.5) and V 1 = V 2 as in (4.16) where g 1 = g + 3λ, g 2 = λ, g 3  In this restricted case there is a stable fixed point MN 3,2 which provides a potential endpoint for the RG flow.

N = 7
The rational fixed points are more limited in this case  For this case the total dimension of the space of slightly marginal deformations is 210. We have not identified all fixed points as previously, in particular those with four and five quadratic invariants. There are no fixed points which have the maximal possible seven quadratic invariants, and also 21 zero κ, as would be expected if there were just Z 2 symmetry. Thus our numerical search fails to gain the prize promulgated in [1].

Six Index Case
In 3 − ε dimensions there may be fixed points starting from the renormalisable interaction V (φ) = 1 6! λ ijklmn φ i φ j φ k φ l φ m φ n . At lowest order possible fixed points in the ε expansion are determined, with a suitable rescaling, by finding solutions of λ ijklmn = S 10,ijklmn λ ijkpqr λ lmnpqr , (6.1) where S n,ijklmn here denotes the sum over the n permutations, with unit weight, necessary to ensure the sum is fully symmetric in ijklmn. This is equivalent to As before in (2.5) the coupling can be decomposed, with a similar notation for symmetrisation, as  The constraint (6.6) does not here lead to any modification for low N .
As was the case with the four index case the β-function equations can be simplified by imposing O(N − 1) symmetry. This requires leaving just four couplings. Defining

Fixed Points for Low N
In a similar fashion to previously we have looked numerically snd analytically for fixed points for low N . For the values of N considered here the fixed point with maximal ||λ|| 2 is always that with O(N ) symmetry as in (6.8). A pictorial representation of these fixed points (excluding the first one) is given in Figure  5. The distribution of these fixed points is similar to that of Figure 1  For N = 5 the number of solutions explode; those found by us are given in Appendix C.

Conclusion
The large number of potential fixed points which appear close together in the lowest order ε expansion equations for N = 6 and larger are presumably quite fragile. Reinstating ε, for two fixed points where ||λ 1 − λ 2 || = ξ ε at lowest order then what happens at higher orders in the ε expansion for ε ∼ ξ is far from clear. In Unless strong symmetry conditions are imposed then for any N 6 the stability matrix will have negative eigenvalues. This suggests according to standard lore that any phase transition is first order so that the apparent fixed point will not realise a CFT. Nevertheless the large numbers of solutions suggest that any classification of CFTs when d = 3 for instance is likely to be very non trivial.
The discussion here is also incomplete in that an implicit assumption made is that the expansion involves only integer powers of ε so that the lowest order contributions to the β-functions determine the leading contribution to the ε expansion. This assumption breaks down near a bifurcation point [3]. However, the analysis may be potentially more complicated away from bifurcation points in that in some theories the β-function corresponding to one, or more, particular coupling is zero to lowest order in the loop expansion but is non zero at the next order. This situation can arise in large N limits where couplings are rescaled by fractional powers of N . 4 The expansion of the equations for a fixed point then typically involves √ ε. As an illustration, for couplings h, g a , a = 1, . . . , n, then if to second order in a loop expansion the β-functions can be truncated to the form g) neglecting cubic terms requires h = 0 unless g a are constrained by k a g a = ε and then, subject to this constraint, the n equations ε g a = b a (g) + 1 2 k a h 2 potentially determine g a , h = O(ε), with the higher order terms generating the usual perturbative expansion in ε. However, if k a are zero, or can be scaled away in a large N limit, we may take h ∼ h 0 ε 1 2 , g a = O(ε). The higher order terms then involve an expansion in powers of √ ε. Similar scenarios arise in melonic theories [47,48,49]. The graphs relevant for the lowest order β-function in φ 4 and φ 6 theories may not be sufficient to generate the melonic interactions for fields with three or more indices.
The discussion of fixed points for the six index coupling undertaken in section 6 is less complete. All cases considered are such that ||λ|| 2 depends linearly on N for large N although there is no bound analogous to (1.1). Perhaps such a bound might follow from a more detailed analysis of the fixed point equations. The O(N ) symmetric fixed point is no longer the one with maximal ||λ|| 2 for N > 14. However, the relevance to physical theories is tenuous.
We are very grateful to Slava Rychkov who stimulated much of this work and who read this paper making many valuable suggestions.

A Alternative Formulation
An index free notation due to Michel [17] is convenient in many cases for analysing the fixed point equation (2.1). For symmetric four index tensors u ijkl , v ijkl , scalar products and symmetric tensor products defined by for P 4 the projector onto symmetric four index tensors. Clearly ||v|| 2 = v · v and the triple product The fixed point and eigenvalue equations are then The solutions {κ r , v r } can be chosen to form an orthonormal basis so that v r · v s = δ rs , In terms of this basis Directly from (A.2) Hence κ −1.
A proof that κ 1 in general is not immediately evident but should follow from bounds on the product. Using (A.2) twice then with where the symmetric tensor C ijk = r x i,r x j,r x k,r ||λ r || 2 . Conversely finding eigenvectors satisfying (A.9) implies the presence of two or more decoupled theories. For v 0 = λ/||λ||, and extending the index range for i to 0, 1, . . . , n − 1 with x 0,r = 1/||λ||, then we may define Crucially it is necessary to obtain (A.7) that C ijk = r x i,r x j,r x k,r ||λ r || 2 where C 0ij = δ ij /||λ||. Such a diagonalisation by essentially orthogonal matrices is not possible for arbitrary symmetric C ijk but depends on additional restrictions [50].

B Potentials and Symmetry Groups for N = 4
In [24] various potentials corresponding to subgroups of O(4) were identified. We revisit these from a different perspective. It is convenient here to adopt complex coordinates where The symmetries which are subgroups of O(4) are generated by a : ϕ,φ, ψ,ψ → e 1 5 π i ϕ, e − 1 5 π iφ , e − 3 5 π i ψ, e Clearly a ∈ Z 10 = Z 5 × Z 2 , b ∈ Z 4 and b generates the automorphisms of Z 5 . The symmetry group is then (Z 5 Z 4 ) × Z 2 with Z 2 = {e, a 5 }. For w = ±4v the symmetry is enhanced since which are solutions of (3.31) for N = 4. In terms of (B.2) but the full symmetry in (B.4) extends to S 5 × Z 2 , with a corresponding to a 5-cycle combined with a reflection and b a 4-cycle. Secondly The symmetries are generated by a : ϕ,φ, ψ,ψ → e where a ∈ Z 8 , b, c ∈ Z 2 . b, c generate Aut(Z 8 ) = Z 2 × Z 2 and the symmetry group is For a potential containing cubic symmetry The symmetry group is a subgroup of the permutations and reflections comprising the cubic symmetry group B 4 and is generated by x, y generate the quaternion group Q 8 and a, b, S 3 < Aut(Q 8 ) and the symmetry group is then Q 8 S 3 .
For the remaining case with diorthorhombic symmetry the symmetry group is obtained by combining two 2-cycles with reflections and can be generated by where a 4 = r 2 = s 2 = t 2 = e , rar = a 3 , st = ts , sas = a , srs = ra 2 , tat = a 3 , tst = s .
(B.14) The symmetry group is then D 4 (Z 2 × Z 2 ). A pictorial representation of these fixed points (excluding the first one) is given in Figure  6. The distribution of fixed points in d = 3 − ε is similar to that of Figure 1 of the d = 4 − ε case.

C Results for Six Indices and
1.12 1.14 1. 16