Seeking fixed points in multiple coupling scalar theories in the ε expansion

Fixed points for scalar theories in 4 − ε, 6 − ε and 3 − ε dimensions are discussed. It is shown how a large range of known fixed points for the four dimensional case can be obtained by using a general framework with two couplings. The original maximal symmetry, O(N), is broken to various subgroups, both discrete and continuous. A similar discussion is applied to the six dimensional case. Perturbative applications of the a-theorem are used to help classify potential fixed points. At lowest order in the ε-expansion it is shown that at fixed points there is a lower bound for a which is saturated at bifurcation points.


Introduction
In seeking conformal field theories, whether through finding zeros of β-functions or applying bootstrap consistency conditions, it is natural to impose as much symmetry as possible. This then restricts the number of couplings and correspondingly non zero operator product coefficients so that the number of channels to be considered in a bootstrap analysis is reduced. In the cardinal exemplar of the success of the numerical bootstrap, the three dimensional Ising model with two relevant fields σ, ǫ, there is a Z 2 symmetry with σ Z 2 odd.

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The imposition of symmetries might introduce a selection bias in finding non trivial CFTs. On the other hand the fixed points of non symmetric theories might have an emergent symmetry. According to the notion of universality fixed points are determined by the associated symmetry group. To an extent it is often tacitly assumed that fixed points without a high degree of symmetry may be neglected. Nevertheless for any potential classification of CFTs in three or more dimensions it is necessary to determine all possible fixed points which may have smaller symmetry groups than have been hitherto considered. Even with knowledge of the fixed points in quantum field theories in particular dimensions, an important question is how they are linked under possible RG flows induced by perturbations by adding relevant operators. Determining the spectrum of relevant operators is part of the data defining a CFT or can be found by solving linear eigenvalue equations. What happens under RG flow, and whether new fixed points are attained, is a non linear problem. For slightly relevant perturbations it is possible to use conformal perturbation theory but this becomes difficult beyond lowest order. In a perturbative context the ε expansion can be taken to higher order by determining β-functions and anomalous dimensions in a time honoured fashion. This may be applied, as originally suggested by Wilson and Fisher [1], in 4 − ε dimensions but can also be used in 6 − ε and 3 − ε dimensions.
In two dimensions, at least for minimal models, there is a well understood framework of CFTs which are linked by RG flows. In two dimensions conformal perturbation theory can be used systematically, as originally proposed by Zamolodchikov [2] and extended subsequently [3][4][5][6][7][8]. A crucial organising principle is the c-theorem which requires that the Virasoro central charge c for any unitary CFT at fixed points reached by RG flow arising from a relevant perturbation has a lower value of c than the original unperturbed CFT. Minimal models, with c < 1, then flow ultimately to the Ising model with c = 1 2 . There is no such simple picture in three or other dimensions of current interest. Here we attempt to discuss potential fixed points and RG flows using the ε-expansion with various additional assumptions for simplicity. For determining critical exponents in many different condensed matter systems with a wide range of symmetries there is a large literature from the 1970's and later. A review discussing large number such models which are applicable to critical phenomena for condensed matter systems is [9]. The ε-expansion is a version of conformal perturbation theory starting from free field theories and which allows an extension beyond lowest order. An assumption made here is that it is qualitatively correct in determining possible CFTs in three dimensions starting from 4 − ε dimensions through essentially one loop calculations. Of course in particular applications loop calculations to three or more loops and sophisticated resummation methods may be used to obtain more accurate results for critical exponents.
Results in the ε-expansion depend critically on the number N of scalar fields. The symmetry of the kinetic term is O(N ). In 4−ε dimensions there is a fixed point with O(N ), N = N symmetry. This is stable for N 4. For N > 4 there are relevant perturbations which lead to a variety of fixed points. Whether there are fixed points which cannot be reached by simple RG flows from free fields is not yet clear. A related question is how many couplings are necessary in order to find particular fixed points. For the Ising model it is sufficient to consider just N = 1 and the φ 4 perturbation with a single coupling. In 6 − ε JHEP05(2018)051 dimensions if N = N + 1 an O(N ) renormalisable theory is formed by fields forming an Nvector and a singlet. There are then fixed points with O(N ) symmetry if N is large enough.
In ε-expansions the starting point is a potential of the form where {P I (φ)} are polynomials of degree 4, or 3 or 6, depending on the dimension. The symmetry group H g for V is determined by For each polynomial P I (φ) there is a corresponding symmetry group H I defined as in (1.2). For generic couplings H g ⊇ H ≃ ∩ H I , but H g may be larger than H. Under RG flow the symmetry is in general ∩ H g for all g, but the symmetry may be enhanced at particular points in the space of couplings. It is also important to note that the larger symmetry defined by V (gφ; g) = V (φ; g g) , (g g) I = g J g J I , for all g ∈ G g ⊂ O(N ) , (1.3)

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is positive definite. In four dimensions an a-function satisfying the requirements for a strong a-theorem can be constructed directly from the one and two loop β-functions [17], the same is true in six dimensions [18] and, as will be shown later, in three dimensions. In four [19] and six [20] there are arguments that these results, expressing the β-functions as a gradient flow, can be extended to arbitrary perturbative order. This imposes non trivial constraints on the higher loop β-functions, even for purely scalar theories. Using minimal subtraction the expressions for a(g) can be easily extended to 4 − ε and 6 − ε dimensions and so the existence of an a-function is relevant for fixed points accessible through an εexpansion. There are however no comparable arguments for theories starting from three dimensions. In four dimensions at a fixed point a, determined by the trace anomaly for a curved background or the three point function for the energy momentum tensor on flat space, is necessarily positive. However in a perturbative discussion in 4 − ε dimensions the results for a(g) away from fixed points are not necessarily bounded below. Under RG flow a(g) decreases but it may be that no fixed point is attained and the assumption that the couplings are O(ε) breaks down. In particular a(g) could become negative in situations where the potential V (φ; g) is also unbounded below and there is no stable ground state. These conclusions hold even more strongly in perturbative discussions in 6 − ε dimensions, since cubic potentials of course are never bounded below. In our discussions in 6−ε and 4−ε dimensions we consider perturbative expressions for a(g), neglecting the free field contributions which are irrelevant here and also allowing for convenience an arbitrary overall normalisation. The most stable fixed point g * is then the one at which a(g * ) is the least local minimum. In the framework of the ε-expansion we are able to show that a(g * ) has a lower bound, of order ε 2 , ε 3 in 6 − ε, 4 − ε dimensions. This bound is attained when two fixed points collide and then disappear from the set of fixed points for real couplings, so that there is a saddle-node bifurcation. If the initial couplings are small but are such that a(g) is less than the bound then the RG flow cannot reach perturbative fixed points calculable in the ε-expansion. The existence of a(g) nevertheless avoids the possibility of more complicated bifurcations [21].
In this paper we discuss a range of fixed points present in scalar field theories when the number of fields N > 1 and there are two or more couplings. For simplicity we first consider in sections 2 and 3 the cases when N = 2 in 6 − ε and 4 − ε dimensions. For an arbitrary renormalisable theory a complete analysis of potential fixed points is possible. In the four dimensional case there is just the O(2) symmetric point and two decoupled Ising fixed points. In both cases the couplings transform under O(2) and potential fixed points are related to the different irreducible components.
For general N there are many more possibilities of finding fixed points where the initial symmetry is broken by slightly relevant perturbations. In section 4 we consider perturbations of the O(N ) symmetric theory in six dimensions. The unperturbed theory has a critical point in 6−ε dimensions so long as N > N crit [22,23]

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In section 5 we analyse breakings of the O(N ) symmetric theory in 4 − ε dimensions due to perturbations formed by a symmetric traceless 4-tensor. At lowest order in the ε-expansion this becomes relevant for N > 4. We consider a general framework in which the RG flow may be restricted to a two dimensional space of couplings. In the simplest case to three loops there are three coefficients a, b, c, but overall normalisation is arbitrary and there are bounds which constrain the allowed values. This formalism covers a range of theories considered in the literature, including ones with hypercubic and hypertetrahedral symmetry, and also cases where there is an unbroken O(m) symmetry, N = n m. It is shown how the a-function has a lower bound at fixed points which is attained at bifurcation points where new fixed points emerge. The general formalism is shown to give identical results to three loops with previous calculations for particular models.
We also discuss in section 6 scalar theories in 3 − ε dimensions with a φ 6 interaction using results for β-functions for a general renormalisable potential at two and four loops. Such theories have an O(N ) invariant fixed point but, by determining the anomalous dimensions of scalar operators with dimensions less than three, there is no apparent large N theory valid for dimensions away from three (the O(N ) theories in four and six dimensions at large N can be related to an O(N ) model for any d with an interaction σ φ 2 ). The O(N ) fixed point is unstable against perturbations for any N and we consider the simple case leading to a fixed point with hypercubic symmetry.
In section 7 we extend the framework of section 5 to multiple couplings corresponding to several slightly relevant operators formed by symmetric traceless 4-tensors. The algebraic conditions necessary for a closed RG flow are generalised to this case and it is shown how the bound on the a-function remains valid to lowest order in ε and that this bound is realised at the bifurcation point where there is an exactly marginal operator. An example with three couplings, in which there are five inequivalent fixed points, is considered. A theory with arbitrarily many couplings with a closed RG flow is also considered.
Various details are contained in five appendices. In appendix A we analyse the bifurcation point for the O(N ) symmetric theory in 6 − ε dimensions. Appendix B describes a different 4 − ε fixed point than the ones considered in section 5. In appendix C we obtain bounds on the coefficients a, b which appear in the general discussion in 4 − ε dimensions. The bounds are saturated in the case of hypercubic symmetry. Appendix D considers perturbations of the theory with hypertetrahedral symmetry. The analysis shows in general the presence of relevant operators. Finally in appendix E we show how the perturbative expression for C T , the coefficient of the energy momentum tensor two point function, at the fixed point can be constrained by large N results. This allows C T to be found for arbitrary theories in the ε expansion in 4 − ε dimensions to order ε 3 .
In general the different sections are more or less independent and may be read, if desired, separately. Detailed results for many cases have already appeared in the literature but are here presented in a hopefully unified framework.

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2 Two flavour scalar theory in 6 − ε dimensions A general two component φ 3 theory is described by a potential depending on couplings G I = (λ 1 , λ 2 , g 1 , g 2 ). Results for β-functions for such renormalisable φ 3 theories, with fields φ i , in 6 − ε dimensions are succinctly given in terms of for γ ij the symmetric anomalous dimension matrix andβ V corresponding to the contributions of one particle irreducible graphs. Taking and at two loops 1 and also at three loops

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Including O(φ 2 ) contributions in V (φ) then results for (2.2) immediately determine the associated anomalous dimensions for such operators. For the two component theory determined by (2.1) the maximal symmetry is determined by the transformations with V θ expressed in terms of transformed couplings defines an equivalent theory, related by a field redefinition. The SO(2) transformations of the couplings given by (2.8) and (2.9) may be extended to O(2) by including reflections generated by φ 1 ↔ φ 2 which give an equivalence under λ 1 ↔ λ 2 , g 1 ↔ g 2 . The couplings may be decomposed in terms of real two-dimensional SO(2) irreducible representations where under G I → G θ,I v n → R n (θ)v n , R n (θ) = cos nθ − sin nθ sin nθ cos nθ . (2.11) The four couplings can be reduced to three invariants where I 3 2 +I 4 2 = I 1 I 2 3 . Clearly I 4 is odd under λ 1 ↔ λ 2 , g 1 ↔ g 2 . The invariants determine the couplings up to an SO(2) equivalence. From (3.9) invariant subspaces under SO(2) are obtained by imposing either v 1 or v −3 to be zero giving the conditions a) λ 1 = −g 1 , λ 2 = −g 2 and b) λ 1 = 3g 1 , λ 2 = 3g 2 . (2.13) Case a) corresponds to λ ijk in (2.1) being symmetric traceless and case b) to taking λ ijk = 3 δ (ij v k) with v i = (g 1 , g 2 ). With the restrictions in (2.13) the only invariants are I 1 or I 2

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respectively. If λ 2 = g 2 = 0 and g 1 = −λ 1 then the O(2) symmetry is reduced to Z 3 ; in this case V (φ) = 1 12 λ 1 (φ 1 + iφ 2 ) 3 + (φ 1 − iφ 2 ) 3 . If g 1,θ = g 2,θ = 0 for some θ, so that the theory is equivalent to two decoupled theories, then it is necessary that I 1 = I 2 , or (2.14) The symmetric 2×2 anomalous dimension matrix for this theory is γ(G) and transforms as Hence tr γ, det γ, and the eigenvalues of γ, are scalars under the equivalence transformations of the couplings and are functions of the invariants I 1,2,3 . For the β-function This ensures that for a fixed point when β I (G * ) = 0 then also β I (G * θ ) = 0. For the theory defined by (2.1) the β-functions to one loop are given from (2.3): The positive contributions in the second line of each expression arise from the 2 × 2 anomalous dimension matrix which at lowest order is, from (2.3), (2.18) Finding zeros of the β-functions giving fixed points requires relations between the couplings. Requiring that the couplings satisfy a) in (2.13) the perturbative expansion of the β-function to all orders takes the form 2 The perturbative results to three loops give The fixed point in this case then arises from the zeros of f (x) Other fixed points require complex couplings and so are not considered here. Following from (2.15) the perturbative expansion the anomalous dimension matrix then takes the form To three loops As a consequence of (2.16) at a fixed point satisfying a) in (2.13) by considering the derivative of (2.16) at θ = 0 which gives in this case As a consequence det[∂ I β J (G)] G=G * = 0 so that the matrix [∂ I β J (G)]| G=G * has a zero eigenvalue. At lowest order

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An alternative reduction to that in (2.13) arises if we impose with C, C ′ real invariants such that I 2 3 = C 2 I 1 , I 3 = C I 1 = C ′ I 2 , I 4 = 0. Subject to (2.29) with F, F ′ functions of invariants. To find fixed points it is then sufficient to require just F = F ′ = 0 where F, F ′ depend on two independent variables. For a perturbative expansion valid away from a fixed point it is necessary that C, C ′ are expressible as a series expansion in the couplings. This can be achieved in (2.29) by requiring In this case F = F ′ and As (2.31) implies (2.14) this corresponds to a decoupled free field theory combined with a single field φ 3 theory. Note that since det γ = 0 in this case there is always a zero anomalous dimension. To three loops, Of course F (x) has no perturbative zeros for real couplings for ε > 0.
3 Two flavour scalar theory in 4 − ε dimensions The structure of fixed points can be fully analysed for just two scalars [26]. A general two component φ 4 theory is determined by the potential (3.1) with couplings G I = (λ 1 , λ 2 , λ 0 , g 1 , g 2 ). The transformations of the couplings G I → G θ,I , generating equivalent theories as in (2.8), induced by (2.6), can be decomposed into a singlet and two real two dimensional vectors. The transformations of the couplings may be reduced to irreducible components as

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Finding fixed points with multiple couplings is a non trivial exercise. Analysing first the zeros for real couplings determined by the one loop β-functions imposes various constraints. These constraints reduce the representation content of the couplings to a single invariant so that higher order contributions to the β-functions are determined by a function of a single variable. There are three cases The zeros determine fixed points. Case a) is equivalent to two decoupled Ising models, within the ε-expansion, corresponding to taking λ 0 * = g 1 * = g 2 * = 0, λ 1 * = 1 3 ε + O(ε 2 ) and also to the theory obtained by taking λ 0 * = λ 1 * , g 1 * = g 2 * = 0, λ 1 * = 1 6 ε + O(ε 2 ). Case b) is described by an Ising model and a free field theory, it is equivalent to taking λ 2 * = λ 0 * = g 1 * = g 2 * = 0 when φ 2 has no interaction. It is easy to check that these cases satisfy (3.5). Case c) is the O(2) symmetric Heisenberg fixed point. Although the original theory defined by (3.1) has in general just a reflection Z 2 symmetry, the fixed points all have at least a Z 2 × Z 2 reflection symmetry.
For case a) the left eigenvectors and eigenvalues of [∂ I β J (G)]| G=G * always include a zero mode given by and to O(ε 3 ) the results for the others are given by

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with γ * the lowest order contribution to the anomalous dimension at the critical point. For the three coupling theory in (4.1) the general form (2.3) determines and also the anomalous dimensions for σ, ϕ become Of course for h = 0 these reduce to standard results for the O(N ) symmetric theory. For h = 0 the fixed points for were analysed by Fei et al. [23]. In the ε expansion possible fixed points are in general determined by the lowest order one loop contributions to the β-functions, higher orders give an expansion in powers of ε for the positions of the fixed points. At large N there are three inequivalent O(N ) fixed points, obtained by the vanishing of β λ , β h in (4.5), with non zero real (λ * , g * ) ≃ (−λ * , −g * ), given by, to first order in ε, Defining x = λ/g and then writing the one loop β-functions as the fixed point equations require For N > 0 this cubic equation has coincident roots, so that For N ′ crit < N < N crit there is only one real solution of f (x) = 0 with x < 0 for any N > 0. For N > N crit there are two roots for x > 0, which merge at x = x crit when (λ 2 , N g 2 ) = (0.4581, 6.218)ε, and one root with x < 0. For real couplings it is necessary that at the fixed point f (x * ) = 0 then f g (x * ) > 0 or (x * − 6) 2 > 44 − N . For large N the roots are 6, ± N 5 which correspond to the three inequivalent fixed points given by (4.7). For N = 1 f (x) has roots at x = ±1, 6 5 but there is just one inequivalent fixed point with

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real couplings when λ * = −g * and λ * 2 = ε. The roots at x = 1, 6 5 correspond to the fixed points which merge when N → N ′ crit and give g 2 < 0, and so both λ, g are imaginary. In contrast to four dimensions, as a consequence of the reflection symmetry (λ, g) ∼ −(λ, g), the stable fixed point is not unique.
A discussion of the ε, ε 2 corrections to the bifurcation point in (4.10) is given in appendix A.
The stability matrix in this case is and, with the β-functions given by (4.5) for h = 0.
This determines the eigenvalues of M to lowest order in ε For (λ * , g * ) to be a stable fixed point the eigenvalues must be positive which requires For any x * the corresponding g * is determined to this order by and at a fixed point in accord with (4.4). This gives and we may then obtain The minimum is attained when the fixed points merge. At the large N fixed points in (4.7) with A * lower at the stable fixed point.

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The h = 0 and N = 1 results here are related to the discussion in section 4 by taking λ 1 = λ, g 1 = g and λ 2 = g 2 = 0. The fixed point obtained with λ * = −g * has a Z 3 symmetry as described there. For λ = g the condition (2.14) is satisfied and the theory becomes the sum of two decoupled N = 0 theories, with λ 1 = λ 2 = λ/ √ 2, g 1 = g 2 = 0. However from (4.5) the large N fixed point is unstable for non zero h since so that the coupling h corresponds to a relevant operator. To find fixed points which are reached for non zero h we consider the linear transformation and the β-functions become (4.23) For h ′ = 0 the β-functions are just those of the unperturbed O(m) theory, for m an integer, and so have the same fixed points, while n fields become free. So long as m g ′ 2 + λ 2 > ε the h ′ coupling is irrelevant so under RG flow the couplings are attracted to the sub manifold corresponding to h ′ = 0. A similar reduction arises for g ′ = 0 taking m ↔ n. There are also fixed points for g ′ , h ′ non zero. For x = g ′ /λ, y = h ′ /λ the vanishing of the lowest order β-functions in (4.22) requires with then, for any solution of (4.24), λ at the fixed point is given by Consistency of (4.24) requires x = y or x + y = − 3 2 . In the first case then from (4.21) h = 0 and g ′ = h ′ = g and the theory reduces to the unbroken O(N ) theory and has the same fixed points as in (4.7) for large N . For the second case if m ≫ n then we may take y ≈ − 3 2 and m x 2 ≈ 5(1 − 9 2 n). With n taking integer values this does not lead to real solutions. The two cases coincide for x = y = − 3 4 which requires N = 2 9 . To determine the possible values of α which are given in terms of b in (4.21) it is necessary to find the various solutions of (4.2). To this end we consider eigenvectors JHEP05(2018)051 Other solutions are obtained for r → N − r. The solutions for α are then for δ ± projection operators given by (4.30) The eigenvalues of γ ϕ ij are 1 6 g ′ 2 , 1 6 h ′ 2 according to whether the eigenvalue of d ij is α, −1/α. The results for the O(m) × O(n) theory can be extended to higher loops. Thus the two loop β-functions are given by applying (2.4) and for the two loop anomalous dimensions By letting φ i = (σ, Φ ar , ρ ab ), a, b = 1, . . . m, r = 1, . . . n and where ρ ab = ρ ba , ρ aa = 0 are 1 2 (m − 1)(m + 2) additional scalars, the O(N ) symmetric theory can be broken to O(m) × O(n)/Z 2 , N = m n, by extending the potential to In general there are then five couplings. For m = 2 tr(ρ 3 ) = 0 so that the λ ρ coupling is irrelevant. By considering reflections of σ and ρ it is clear that there are equivalence relations The restricted theory with g σ , λ σ andλ set to zero has been discussed in [28] and the general theory in [29]. For this case there is a tractable large n limit valid for arbitrary d which links perturbative results in six and four dimensions. The potential (4.33) defines a renormalisable theory in d = 6 − ε dimensions and the β-functions have the form As a consequence of (4.34) the β-functions satisfy the identities

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with anomalous dimensions At two loops the results for the β-functions are rather non trivial but can be obtained from (2.2) For the anomalous dimensions The corresponding three-loop results, obtained using (2.5), for the anomalous dimensions and β-functions of the theory defined by (4.33) are included in an ancillary file. When m < 4 there are fixed points with just λ ρ non zero Fixed points determined by vanishing of the β-functions for the couplings g σ , g ρ , λ σ , λ ρ ,λ belong to equivalence classes as a consequence of (4.36). For large n, to leading order in 1/n, there are three inequivalent fixed points similar to those in the O(N ) model where the non zero couplings are O(n − 1 2 ). To O(ε 2 ) these are given by The eigenvalues of the stability matrix to first order in ε at large n are then Clearly the last fixed point is stable and there are in general three real roots, for large m, The corresponding leading non zero contributions to JHEP05(2018)051 the anomalous dimensions for each case are given by, to order 1/n and O(ε 3 ) from three loops, using (2.5), The anomalous dimensions for σ, ρ in the same limit are These results are in accord with large n expansions [30][31][32]. The operators σ, ρ have canonical dimension 1 2 (d − 2) and the 1 2 ε terms in (4.46) ensure that they have leading dimension 2 at large n.

Scalar theories with reduced symmetry in 4 − ε dimensions
For N -flavour scalar theories in 4−ε dimensions the fixed point structure even with just two essential couplings can be rather non trivial. Many such theories with different symmetry groups have been discussed in the literature [33]. A large class can be encompassed by considering a quartic potential with two couplings of the form where d ijkl is symmetric and traceless on any pair of indices and is assumed to be the unique tensor invariant under some subgroup H ⊂ O(N ) (in general there are, for arbitrary N ,  [34]. In general there are bounds such that , which lead to constraints on λ, g to ensure the potential is bounded below. For λ positive then for g > 0 stability requires 3λ/g > δ − while for g < 0, −3λ/g > δ + . A general quartic potential would contain a term 1 4 φ 2 d ij φ i φ j , with d ij symmetric and traceless, in addition to (5.1). For cases considered here the only quadratic in φ invariant under H is φ 2 and so such terms can be dropped. The restriction to (5.1) does not exclude any fixed points relevant for condensed matter systems considered in the literature but an JHEP05(2018)051 example in which the perturbation is extended by a further coupling and where the RG flow realises new IR fixed points is described in appendix B.
To two loop order it is sufficient to ensure the RG flow is restricted to the two dimensional space parameterised by λ, g to impose Tensors d ijkl related by O(N ) transformations define equivalent theories, manifestly H always contains Z 2 induced by φ i → −φ i for all i. For any non zero d ijkl , a > 0 and we may choose b > 0 by changing the sign of d ijkl if necessary, the value of b 2 /a is independent of the choice of normalisation. As a consequence of (5 There is then only one strongly relevant Z 2 invariant operator which needs to be tuned to zero to attain an IR fixed point under RG flow. For (5.2) the subgroup H is discrete. More general possibilities than (5.2) allowing continuous H and also more than one such tensor are considered later.
For N = 2 (5.2) requires b = 0, which can be seen by taking For N = 3 the inequality is saturated so that a = 4 b 2 . Possible solutions for a, b are further constrained by considering the eigenvalue problem For solutions µ ± then for d + eigenvectors with eigenvalue µ + there must be The traceless condition then leads, as in the discussion in section 4, to using the inequality (5.3). This requires d ± are restricted to the finite range The spaces of dimension d ± must form representations of H.

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For the general scalar theory the lowest order β-function is determined, after rescaling λ ijkl → (4π) 2 λ ijkl , by the gradient flow equation [17] and at any fixed point at this order [16] A with γ * the lowest order φ anomalous dimension matrix at the fixed point as given in (3.6).
For positive λ ijkl A is bounded below so that A has at least one non zero perturbative minimum in the ε expansion so long as the RG flow does not destabilise the vacuum. With the constraint (5.2) the lowest order β-functions become The associated a-function is given by specialising (5.7) so that, with the β-functions in (5.9), To lowest order in the O(N ) invariant theory, obtained when g = 0, the Heisenberg fixed point is given by For theories described by the lowest order β-functions in (5.9) there are potentially, besides the trivial Gaussian fixed point and the O(N ) symmetric fixed point with g = 0, two additional fixed points. To analyse these fixed points it is convenient to write in (5.9) Then the fixed point condition requires The cubic polynomial f (x) has simultaneous roots for N = 4 or b 2 /a = To leading order in ε the additional fixed points are given by For the stability matrix At a fixed point λ * = ε/f λ (x * ). From (5.18) the eigenvalues of M to first order in ε are just just as in (5.19). In consequence the fixed point is The bound arises when The bound is attained at the Heisenberg fixed point (5.12) when N = 4. In general F (N ) vanishes when the fixed points merge, for small F so that for N > 4 the least A * , and a stable fixed point, is expected to be achieved for the smallest F (N ) and for the fixed point correspond to X − , as observed above. In general for N > 4, A * ,H > A * ± . At two loops and from (5.10) to lowest order in ε At the Heisenberg fixed point given by (5.12) γ H = N + 2 4(N + 8) 2 ε 2 . (5.26) At three loops results further conditions are necessary to restrict the RG flow to two couplings. In addition to (5.2) we require A δ pq δ rs + δ pr δ qs + δ ps δ qr + c d pqrs , and The two and three loop results for β-functions determine corrections as an expansion in ε to the structure of fixed points found at one loop. The perturbation given by the coupling g in (5.1) becomes relevant in the O(N ) symmetric theory, and so induces an RG flow in which the O(N ) symmetry is broken, when ∂ g β g λ=λ * ,g=0 < 0. With results to three loops this requires

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The results in (5.12) may be straightforwardly extended to higher order, for large N they simplify to In the two coupling theory considered here the broken symmetry fixed points merge at critical point N = N crit . To lowest order in an ε-expansion N crit = N 0 with N 0 determined by F (N 0 ) = 0, with F is defined in (5.14). In the neighbourhood of the critical point, N ≈ N crit , the straightforward ε-expansion breaks down but if F (N ) has a simple zero at N = N 0 we can write for N > 4, N ≈ N 0 , where F 1 , F 2 are determined by ensuring that any singular terms in the ε-expansion as F → 0 are cancelled. Including higher loop corrections the critical N is then determined by iteratively solvingF as an expansion in ε. For fixed points to be presentF (N, ε) > 0, for N > N crit,H and F (N, ε) < 0 the RG flow does attain fixed points accessible in the ε-expansion. Using the two and three loop results for the β functions 4 The general theory described by (5.1) may be extended to include relevant operators quadratic in φ by letting
For application here it is necessary to consider possible tensors d ijkl satisfying (5.2) as well as (5.27). If b = 0 solving the fixed point equations at lowest order using (5.9) requires N < 4 or g = 0.

Hypercubic fixed poins
An extension of the simple N = 2 case is obtained by taking, as was considered long ago [35][36][37] and more recently in [38], N , which ensures no other couplings than λ, g are necessary for renormalisability. If λ = 1 N +2 g in (5.1) this reduces to N decoupled φ 4 theories. In (5.2) and (5.27) 39) and the solutions of (5.4) give Clearly (5.39) saturates the inequality (5.3). In (5.14) which is positive for any N > 1.

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For this case the roots of f (x) = 0 are x − = 1 2 (N − 4)(N + 2)/(N − 1), x + = N + 2 as well as x = 0. At these roots f ′ (x ± ) = ±x ± and f ′ (0) = N − 4. Thus for N > 4 x * = x − is the stable fixed point, for N < 4 x * = 0. At lowest order there are then fixed points for with the anomalous dimension at the fixed point obtained from (5.24). From (5.19) the eigenvalues of the stabilty matrix to lowest order are given by For the second case the (φ 2 ) 2 interaction is cancelled so this describes N decoupled φ 4 theories. For N = 2 the two fixed points are equivalent and correspond to case a) in (3.9) with λ 0 = λ 1 or λ 0 = 0. For N = 4 the results in (5.42) corresponding to x * = x − are identical with (5.12). The higher order results for β-functions allow an extension of (5.43) as an expansion in ε. The results for κ 0,+ , κ 1,+ match those for the Ising model as expected and as given in (3.12). To the next order The critical point is stable for κ 1,− > 0. Including three loop contributions this arises for exactly the same N as in (5.30) when the O(N ) fixed point is unstable. In consequence the O(N ) theory with a hypercubic perturbation has a unique stable fixed point for all N .
Although the fixed point at x * = x − is stable for N 4 there may be slightly relevant φ 4 perturbations. At this fixed point the anomalous dimensions to lowest order are determined by the eigenvalues of the operator The 1 24 N (N + 1)(N + 2)(N + 3) φ 4 operators may be decomposed into irreducible repre-

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e α or the vertices of the hypertetrahedron and Z 2 to reflections φ i → −φ i . This expression for d ijkl satisfies (5.2) with , (5.49) and in (5.27) The solutions of (5.4) give and for In this case from (5.19) For the a-function

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so that for N ≥ 5 the tetrahedral fixed point has a lower A and also a lower symmetry than the cubic one. The difference vanishes when N = 3 since in this case the cubic and tetrahedral theories coincide. The marginal perturbations at this fixed point are analysed in appendix D.
The result (5.52) has a double zero at N = 5 when (5.53) and (5.54) coincide. Taking into account higher order contributions for N ≈ 5 the two fixed points have an expansion where

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Such tensors may be present for N ≥ 4. For calculations beyond one loop we require There are various consistency conditions such as The eigenvalue equation (5.4) is extended to A particular example arises when d ijkl is the symmetric traceless rank 4 tensor for a Lie group when there is an independent quartic Casimir. For a simple Lie algebra with a basis {t i } where we assume, with i, j, · · · = 1, . . . , N ,

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then we may take Similarly we may define, if tr(t i {t j , t k }) = 0, For the case of O(n) then t i are a basis of n × n antisymmetric matrices so that the theory reduces to antisymmetric tensor fields ϕ ab = (t φ ) ab , a, b = 1, . . . , n, as discussed in [40,41]. In this case in (5.68) so that y = −2(2n − 1)/(N + 2), z = −8/(n + 1). In this case for  [30,42]. This case can be discussed in the present context by taking φ i = Φ ar with a = 1, . . . , m, r = 1, . . . , n and, with the usual summation convention for repeated paired indices, This vanishes for m = 1 or n = 1 and for m ≤ n, satisfies the bounds In this case there are two tensors w u,ijkl , u = 1, 2, which are determined by and in (5.60)

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From (5.25) For a fixed point R mn > 0 and so this respects the bound (5.20). In application to condensed matter systems the two fixed points, denoted by +, −, are referred to as chiral and anti-chiral [42]. Since A * + < A * − the chiral fixed point is therefore the stable one. At higher order the results are complicated but for n ≫ m and to order ε 3 , taking into account two and three loop contributions, Correspondingly in this limit, with notation from (4.45), There are also similar results for γ ρ which decomposes into four cases depending on the eigenvalues in (5.81) At the fixed point ∆ σ,ρ = d − 2 + γ σ,ρ * . For case (ii) ∆ ρ * ± , ∆ σ * + = 2 + O(1/n), the large n analysis is based on couplings ρ ab Φ ar Φ br , with ρ ab symmetric and traceless, and σ Φ 2 which have dimension d, for the anti-chiral fixed point only the ρ ab coupling is relevant. The results in (5.89) show the universal behaviour at large n expected by large n results [30][31][32].

M N fixed points
An alternative breaking of O(N ) symmetry (referred to in some literature as the M N model [9]) is realised when O(N ) → S n ⋉ O(m) n for N = m n, φ i is decomposed as n m-vectors ϕ r and This expression reduces to the hypercubic model in (5.38) for m = 1 and if in (5.1) λ = m+2 3(N +2) g to n decoupled O(m) theories. Past discussions include [44][45][46]. The β-functions for such theories have been determined to four loops [47] for m = 2 and are physically relevant for m = 2, n = 2, 3. For m = n = 2 this case with O(2) 2 symmetry is identical with the O(2) × O(2) symmetric theory, taking The β-functions in the two cases are related by In the framework described here there is a single mixed symmetry tensor For eigenvectors of the d, w tensors with eigenvalues (µ, µ ′ ) there are now three cases The results in (5.93) are sufficient to determine the β-functions in this case to three loops for general m, n. 10

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The polynomial f (x), as defined in (5.13), now has roots, apart from x = 0, given by x − = (N − 4)(N + 2)/2m(n − 1), x + = 3(N + 2)/(m + 2), x + > x − for m < 4. In this case f ′ (x ± ) = ± 1 3 (4 − m)x ± . According to (5.19) a stable fixed point requires f ′ (x * ) < 0. For N > 4, when the O(N ) symmetric fixed point has relevant perturbations, the stable fixed point is given by x * = x − or x * = x + depending on whether m < 4 or m > 4. In (5.14) so that for this breaking there are always non trivial additional fixed points which coincide when m = 4. The two fixed points extending (5.42) are then given by The stability matrix eigenvalues at lowest order are then The fixed point realised for x * = x + corresponds to decoupled O(m) theories, this is the stable fixed point for m > 4. Using (5.8) For a positive potential at the x * = x − fixed point we should require (m + 8)N > 16(m − 1) and then stability requires m < 4 or m = 1, 2, 3.

Scalar theories in 3 − ε dimensions
An ε expansion starting from three dimensional renormalisable theories determines the properties of tricritical (since there are at least two relevant operators if Z 2 symmetry is imposed for a single component field) fixed points [48,49]. Such φ 6 theories have been considered from a CFT perspective in [50,51].
For our discussion we assume N -component scalar fields φ i and initially a general sextic potential V (φ) = 1 6! λ iklmnp φ i φ j φ k φ l φ m φ n + . . . . After rescaling V → (8π) 2 V the lowest order, two loop, result for the β-functions for the couplings contained in V is determined by . For the next order, at four loops, 11 with the four loop anomalous dimension matrix For the O(N ) symmetric case the potential is restricted to the form and from (6.3) The four loop result gives Equivalent results were obtained in [53] and extended to six loops in [54,55]. At the fixed point to lowest order , γ φ * = (N + 2)(N + 4) 24(3N + 22) 2 ε 2 . (6.8) For large N the perturbative contributions to the β-function and the anomalous dimension γ φ at higher orders have a leading N dependence of the form (N λ) p N ⌊ 1 2 (p−3)⌋ and (N λ) p N ⌊ 1 2 (p−2)⌋ , p = 2, 3, . . . , respectively. 12 Hence with a rescaled coupling A large N IR fixed point is given, at leading order, by (6.10) The bound [53] determines the radius of convergence of the ε-expansion. At the fixed point
Extending the discussion to lower dimension operators we may consider with ρ ij , τ ijk , µ ij , ν ijkl symmetric and traceless. The results in (6.1) and (6.2) determine the anomalous dimensions. For the O(N ) symmetric theory determined by (6.4) For large N the leading terms in γ σ , γ ξ are expected to be of the form (N λ) p N ⌊ 1 2 (p−1)⌋ , (N λ) p N ⌊ 1 2 p⌋ . In the large N limit for fixedλ these results, together with [55], imply 13 A UV fixed point for large N was also proposed in [56,57]. However [58,59] showed for N → ∞ and d = 3, when the β-function vanishes, that a one loop effective potential was bounded below only when η = 8π 2λ ≤ 16π 2 orλ ≤ 2. The UV fixed point in (6.12) for ε = 0 is then in an unstable regime and is no longer relevant in this limit and there is a new scale invariant fixed point atλ = 2. The situation for large but finite N is less clear [60].

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At the fixed point the leading results for these anomalous dimensions are given by taking λ →λ * as in (6.10). The large N limit here is very different from that which interpolates between four and six dimensions. 14 Only a finite number of terms contribute to any given order in 1/N for fixed N ε. As is clear from (6.10) the O(N ) tricritical IR fixed point is not present for N ε > 36 π 2 . As argued in [63] there is a critical line N = N c (d) which involves another fixed point and [63] showed N c (d) ∝ 1/ε as ε → 0. With tensorial scalar fields transforming under O(N ) 5 there may be large N limiting theories which may be analysed directly [64].
For two flavours the discussion in sections 2, 3 can be extended by taking a general potential of the form with seven couplings G I = (λ 1 , λ 2 , g 1 , g 2 , h 1 , h 2 , h). Decomposing into representations of O (2) give where I 1 is an invariant and v 2 , v 4 , v 6 are two-vectors transforming as in (3.9). The O(2) invariant theory as in (6.4) corresponds to taking v 2 = v 4 = v 6 = 0 implying h 1 = h 2 = h = 0, λ 1 = λ 2 = 5g 1 = 5g 2 = 15λ. In a similar fashion to (3.3) there are five O(2) invariants, in addition to I 1 , which can be constructed from v 2 , v 4 , v 6 using the transformation (2.11). Imposing restrictions on the O(2) representation space formed by the couplings may reduce the RG flow to a sub manifold. Thus setting v 2 = v 4 = 0 gives, since tensor products of v 6 cannot generate vectors transforming as v 2 , v 4 , β I (G) = G I a(g 1 + g 2 , h 2 − g 1 g 2 ) + G 0 I b(g 1 + g 2 , h 2 − g 1 g 2 ) , G I = (2g 1 + 3g 2 , 3g 1 + 2g 2 , g 1 , g 2 , −h , −h, h) , G 0 I = (5, 5, 1, 1, 0, 0, 0) , (6.18) where g 1 + g 2 , h 2 − g 1 g 2 are two O(2) invariants formed from I 1 , v 6 2 . Due to the restriction on the representations of the couplings the anomalous dimension matrix is diagonal and we must have 14 However with the large N dependence assumed above there might be an alternative large N limit where The positive square root is chosen to ensure a stable potential and this fixed point is identical with (6.12) continued to ε < 0 and taking −N ε ≫ 1. In [62] it was suggested that f (λ ′2 ) should have a zero leading to an IR fixed point in the vicinity of three dimensions. Such a fixed point is not accessible with perturbative arguments but is motivated by the requirement that the UV fixed point should be annihilated by merging with an IR point for some 0 < −ε < 1 so as to ensure a trivial theory in four dimensions.

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In the large N limit corresponding to (6.33) where [a uv ] is positive definite and It is convenient to use a uv , and its matrix inverse, to lower and raise indices for the couplings {g u }. Of course (7.1) may be extended as in (5.59), but this is not pursued here. The lowest order β-functions are just a simple modification of (5.9) β λ = −ε λ + (N + 8)λ 2 + g 2 , g 2 = a uv g u g v , and associated a-function from (5.10) becomes At two loops we can easily extend (5.23) to the multi-coupling case and also at three loops (5.28) with an appropriate generalisation of (5.27). The couplings (g 1 , . . . , g p ) ∈ R p define a vector space spanned by the polynomials d u,ijkl φ i φ j φ k φ l . If there is a subspace V ⊂ R p such that for any g u , g ′v ∈ V then g u g ′v b uv w ∈ V the RG flow may be restricted to couplings belonging to V . It is easy to see that there are no fixed points with λ * = 0 other than the trivial Gaussian one. Hence it is sufficient to consider x u = g u /λ. Under RG flow from (7.3) to lowest order d dt Higher order contributions in (7.5) involve contributions at ℓ loops of the form λ ℓ+1 f ℓ,λ (x) and λ ℓ f ℓ u (x). Perturbative fixed points are then obtained iteratively by first solving At any fixed point, where the lowest order β-functions in (7.3) vanish, there is a similar bound to (5.20) Any {λ, g u } such that A < − 1 48 N ε 3 cannot then flow to a fixed point accessible in the ε-expansion. The bound in (7.7) arises when and so requires N > 4. To analyse this further we consider the stability matrix at the fixed point The eigenvalue problem may be simplified by noting that so that ε is always an eigenvalue for this class of theories, and also (7.11) where κ and v r , u r are solutions of the reduced eigenvalue problems for f u (x) given in (7.6) and where x * is determined by f u (x * ) = 0, corresponding to a fixed point where (7.3) vanish. The results (7.10) and (7.11) with (7.12) generalise (5.19) in the two coupling case. In general there are p eigenvalues κ and associated eigenvectors.
To verify (7.11) it is necessary to show that (7.12) implies In consequence when (7.8) is satisfied there is a zero eigenvalue of the stability matrix as expected when two fixed points annihilate. Assuming λ * > 0 it is easy to see that stable fixed points correspond to finding local maxima of (7.14)

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For N < 4 manifestly there is a maximum at x = 0. In general when N > 4 and the origin is a minimum and F (x * ) = 1 The function F (x) on R p can reduced to one on S p ≃ R p ∪ {∞} by modifying F (x) for very large x 2 so that as x → ∞ F (x) → C, a finite constant, and then F (∞) becomes the maximum of F . On S p F (x) is a smooth bounded function with at least one maximum and one minimum. We may use baby Morse theory to constrain the stationary points of F (in embryonic form analogous results were found in [71,72]). Assuming F has C t stationary points at finite x with non degenerate Hessians and t = 0, 1, . . . , p negative eigenvalues (C 0 , C p are the number of minima, maxima) then where we add in the second identity the contribution of the maxima at infinity separately. A perfect Morse function on S p , saturating the inequalities in (7.15), has just one minimum and one maximum and no other stationary points. This is realised if there is just a single minimum for finite x in (7.14) giving C 0 = 1, C p = 0. If b max = max(b rst e r e s e t )| e 2 =1 this holds if N > 4 and b max 2 < 4 9 (N − 4). For a stable fixed point it is necessary that C p ≥ 1. If a new fixed point is generated by varying the parameters this must correspond to ∆C q = ∆C q+1 = 1 for some q.

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under (7.17) and correspond to N decoupled Ising models, the next two are also equivalent and represent the hypercubic fixed point discussed in the two coupling case. The last two fixed points are not equivalent since each is invariant under (7.17). In this case the d-terms combine to r (ϕ r 2 + ψ r 2 ) 2 − 2 n+1 (ϕ 2 + ψ 2 ) 2 and the symmetry is enhanced to S n ⋉ O(2) n . These fixed points are identical to the M N theory starting from (5.90) when m = 2, the results for A * are identical to (5.98) with m = 2. The final fixed point is stable and has the least value of A * . Results for the stability matrix eigenvalues for each fixed point above are given to three loop order in [69], of course for equivalent theories the expressions are the same. All the fixed points in this three coupling theory correspond to ones obtained by RG flow starting from the trivial Gaussian fixed point in just single or double coupling theories.
An extension to models with arbitrarily many couplings which have stable fixed points can be achieved by adapting results due to Michel [74] to the formalism described here (a related discussion is contained in [75]). We take φ i = ϕ a 1 ,...,ap, r , a u = 1, 2, . . . , m u , where r = 1, . . . , n and u = 1, . . . , p and define d u,ijkl φ i φ j φ k φ l = a u+1 ,...,ap,r a 1 ,...,au ϕ a 1 ,...,ap, r There are then p couplings g u which, along with λ, define a closed manifold under RG flow. Each (m 1 , . . . , m p ) represents a different theory with different symmetries although as will be shown they may flow to common fixed points. For m 1 = 1, m 2 = 2 d 1 , d 2 are equivalent to (7.16), the couplings are related to the previous case by g 1 → g 1 − g 2 , g 2 → g 2 . Scalar potentials formed by linear combinations of terms of the form (7.21) include various theories which have been considered in the literature. In (7.1) the non zero contributions are given by It is easy to verify (7.2). From (7.6)

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Hence the fixed points at lowest order in ε are given by Setting all x u to zero gives the O(N ) symmetric Heisenberg fixed point. If x u 1 , x u 2 , . . . , x uq , u 1 < u 2 < · · · < u q , are non zero, solutions of the fixed point equations can be found by taking where B satisfies In consequence there are two solutions for B B = S q , S q − 1 6 (N + 2) , (7.28) giving at the fixed point For stability it is necessary that λ * 1 or λ * 2 are positive, thus for the first fixed point, assuming N > 4, S q < 0 or 12 S q > N − 4 . For the two cases in (7.28) In general M u 1 < M u 2 < · · · < M uq < N so that if M u 1 > 4 then S q < 0. For S q > 0 it is necessary that M u 1 < 4 requiring m 1 < 4. For q even Besides the Heisenberg fixed point there are for each q = 1, . . . , p p q choices and hence 2(2 p − 1) fixed points with some non zero x * i . These results for p = 1 are identical to (5.96) and (5.98) for m 1 → m.
For each fixed point given by (7.26) and (7.28), q = 1, . . . , p we may determine the p eigenvalues of the stability matrix by solving (7.12) with (7.24). The results take the simple form for the two cases in (7.28)

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The number of negative modes is given by t = p − u 1 + u 2 − · · · − u q , q odd , Thus for q = 1, t = p−u 1 , for q = 2, t = u 2 −u 1 −1 and for q = p, t = ⌊ 1 2 (p−1)⌋. Assuming 12S q > N − 4 then A * 1 < A * 2 and λ * 1 corresponds to a stable fixed point in this multi coupling theory when t = 0. This is possible only for q = 1 or 2 when u 1 = p or u 2 = u 1 + 1 respectively. For S q < 0 a stable fixed point arises for t + 1 = p and A * 2 < A * 1 . It is then necessary to take q = 1, u 1 = 1 and m 1 > 4. For q = 1, u 1 = p, so that only x * p is non zero, so that this positive for M p = 1, 2, 3 which may be realised when p = 2 for m 1 = 1, m 2 = 2, 3. If q = 2, u 2 = u 1 + 1 then and for positivity we may take M 1 < 4, M 2 > 4. For p = 2 and m 1 = 1, m 2 = 2, 3 this is negative in accord with the result that there should be just one stable fixed point. It is easy to verify that these results are in accord with (7.20) when m 1 = 1, m 2 = 2.

Conclusion
Finding fixed points in the ε-expansion is hardly terra incognita, nevertheless there is as yet no mappa mundi. At one level this amounts to determining quartic, or cubic or sextic, polynomials in N variables invariant under all possible subgroups of O(N ). Nevertheless understanding RG flows between different fixed points depends to a large extent on constructing an a-function which decreases monotonically under RG flow. Here we considered just the lowest order expression although this can be extended, at least perturbatively, to higher orders. The discussions here suggest that fixed points are stable against RG flow to othe fixed points near a bifurcation point. However the RG flow may also lead to a decreasing indefinitely, leading in unitary theories to a trivial IR limit such as when all fields are massive.

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is the discriminant for the cubic f (x), for d(N ) > 0 there are three real roots. When Crucially d 1 (N ), d 2 (N ) are finite as N → N 1 , although the two terms in d 2 (N ) separately have poles. Hence where For the three roots of d(N ) = 0 the above formalism gives The results agree precisely with [23]. Correspondingly, extending (4.10), We here discuss perturbations of the O(N ) theory by operators 1 4 φ 2 d ij φ i φ j with d ij symmetric and traceless. At the O(N ) fixed point this is not relevant but we find new fixed points by considering a potential where d ij satisfies (4.2) to ensure a closed RG flow. The one loop β-functions are then A simpler form is obtained by taking b = α − 1/α with α = n m , N = m + n and defining
For g 2 = 0 there is the O(N ) symmetric fixed point with g 1 * = g 2 * = g 3 * = 1 N +8 ε for which the stability matrix eigenvalues are (1,8 N +8 , 4−N N +8 )ε so that this is unstable for N > 4. Finding other fixed points with g 2 non zero, realising a coupled theory with O(m) × O(n) symmetry, is more involved. For n = m there is a symmetry ϕ ↔ ψ and it is easy to see by requiring β g 1 − β g 3 = 0 that we must take g 1 = g 3 or g 1 + g 3 = 1 m+8 ε. For the latter there are no real fixed points for m > 1. For the former we may recover the O(N ), N = 2m, fixed point or g 1 * = g 3 * = For g 2 = 0 and m = n finding fixed points is simplified by letting x = g 1 /g 2 , y = g 3 /g 2 and then it is sufficient to solve (m + 2)x − 6 y + 4 y = m , (n + 2)y − 6 x + 4 x = n . (B.6) The couplings at a g 2 non zero fixed point to O(ε) are given by g 2 * = ε 4 + (m + 2)x + (n + 2)y , g 1 * = x g 2 * , g 3 * = y g 2 * . (B.7)

C Bounds for tensor products
The tensorial relation (5.2) necessary at lowest order for a simple two coupling renormalisable theory in four dimensions introduces two parameters a, b. Here we demonstrate the inequality (5.3) valid so long as d ijkl are real. To this end we consider the decomposition of the rank six tensor d ijkp d lmnp into irreducible components. A symmetric traceless tensor may be defined by

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A mixed symmetry tensor corresponding to a [4,2] Young tableaux is given by From (5.62) a e u h u = e u e v a ′ uv ≥ 0. Setting the w tensor contributions to zero positivity for N > 2 is equivalent to (5.3). For N = 3 M ijklmn = 0 and we must take e u h u = 0 so that a = 4b 2 then. For N < 16 the w-tensor contributions ensure a more stringent bound on b 2 /a. The 8 index tensor formed from two d ijkl may also be decomposed into representations of O(N ). The mixed symmetry [4,4] representation gives the bound a ≤ 15b 2 in the absence of w-tensor contributions when N = 4.
As an illustration of similar constraints we consider a similar discussion for a three index symmetric traceless tensor d ijk which may define a single coupling renormalisable theory in six dimensions if at one loop order it satisfies [24,77] d ikl d jkl = α δ ij , α > 0 , d ilm d jmn d knl = β d ijk . (C.5) In this case analogous symmetric traceless and mixed symmetry tensors are formed by where the lower and upper bounds require S ijkl = 0 and M ijkl = 0 respectively. At two loop order it is necessary that Identities similar to (C.8) are required at higher order for graphs of new topology, the coefficients may be calculated for the two limiting cases in a similar fashion [78]. M ijkl = 0 is realised by taking d ijk = α e i α e j α e k α when α = N −1 N +1 , β = N −2 N +1 . The case S ijkl = 0, with additional assumptions, reduces to the F 4 family considered in [79] where N = 5, 8, 14, 26. Other solutions with a single tensor d ijk , satisfying (C.5), (C.8) and the various higher order extensions, are provided by the invariant symmetric tensors for SU(n), N = n 2 − 1.

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E C T for scalar theories For a CFT obtained at an RG fixed point a crucial observable is C T , the coefficient of the energy momentum tensor two point function. In even dimensions this is related to particular contributions to the anomaly under Weyl recalling of the metric for a curved space background. For interacting theories the leading contributions to the anomaly have been calculated in both four and six dimensions as well as C T directly in both cases. Higher order terms in the perturbation expansion have not so far been determined. Here we consider some constraints obtained by matching the perturbation expansion with O(N ) results for large N in any dimension. A similar argument is described in [80].
In four dimensions for a potential as in (3.1) then the general form for C T to four loops is given by, after rescaling the coupling λ ijkl → 16π 2 λ ijkl , has the form C T C T,scalar = N − 5 36 λ ijkl λ ijkl + α λ ijkl λ klmn λ mnij , (E.1) where C T,scalar is the result for a single free scalar and the four loop coefficient α has not been obtained directly. For the O(N ) symmetric theory given by (5.1) with g = 0, so that λ ijkl = λ(δ ij δ kl + δ ik δ jl + δ il δ jk ), then (E.1) becomes C T C T,scalar = N 1 − in accord with [84,85]. In an ancillary file we include results for the central charge for the theories of section 5.
A similar approach may be adopted in six dimensions. In this case a renormalisable scalar theory, for N component φ i , with interaction 1 6 λ ijk φ i φ j φ k the general form for C T to three loops in perturbation theory is given by, after rescaling λ ijk → (4π) λ ijk λ ijk + α λ ijk λ ilm λ jln λ kmn + β λ ijk λ ijl λ mnk λ mnl , (E.7)
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