Bounds on multiscalar CFTs in the epsilon expansion

We study fixed points with N scalar fields in $4 - \varepsilon$ dimensions to leading order in $\varepsilon$ using a bottom-up approach. We do so by analyzing O(N) invariants of the quartic coupling $\lambda_{ijkl}$ that describes such CFTs. In particular, we show that $\lambda_{iijj}$ and $\lambda_{ijkl}^2$ are restricted to a specific domain, refining a result by Rychkov and Stergiou. We also study averages of one-loop anomalous dimensions of composite operators without gradients. In many cases, we are able to show that the O(N) fixed point maximizes such averages. In the final part of this work, we generalize our results to theories with N complex scalars and to bosonic QED. In particular we show that to leading order in $\varepsilon$, there are no bosonic QED fixed points with N<183 flavors.


Introduction
Second-order phase transitions are described by scale-invariant (or rather conformally invariant) quantum field theories. There are two fundamentally different ways to approach the study of phase transitions. On the one hand, one can try to construct specific fixed points, using the logic of effective field theory. That is to say that one writes down a Hamiltonian for the order parameters of the system, including all possible relevant operators that are consistent with its global symmetry group. Given an effective Hamiltonian, one has to analyze whether it admits one or several fixed points that can describe the phase transition of interest.
An orthogonal way of thinking of this problem is bottom-up: one could attempt to make a list of all conformal field theories together with their spectrum (i.e. their critical exponents). Given such a list, it would be possible to look for theories with a specific global symmetry group in two or three spacetime dimensions. The problem of classifying CFTs is known as the conformal bootstrap [1][2][3] -see Ref. [4] for a comprehensive review.
Although both approaches appear to be radically different, they lead to the same experimental predictions. For instance, the Ising CFT in three dimensions has been studied for decades via renormalization group methods, yielding predictions for its critical exponents up to four significant digits [5]. In recent years, these predictions were recovered and even significantly improved by establishing rigorous bounds on the space of unitary, Z 2 -symmetric CFTs with two relevant perturbations [6,7].
Unfortunately, there is at present no simple method to classify all CFTs, even after fixing a spacetime dimension d and a symmetry group G. To make progress, we analyze in this paper a simplified classification problem that has been studied since the 1970s [8][9][10][11][12]. To be precise, we consider N scalars φ i that interact via a general quartic interaction, (1.1) without imposing any constraints on the couplings λ ijkl , except for the fact that they must be real-valued (otherwise, the interaction (1.1) can flow to a non-unitary CFT). In two or three dimensions, any fixed point obtained by turning on the interaction (1.1) is strongly coupled. However, in d = 4 − ε dimensions with ε 1, the RG flow of the couplings {λ ijkl } can be studied perturbatively. The beta functions of the λ ijkl are of the form β(λ ijkl ) = −ελ ijkl + P ijkl (λ) (1.2) where P ijkl (λ) is an analytic function of the couplings that starts at order λ 2 . Consequently, Eq. (1.2) has roots that can be expanded order by order in the small parameter ε. To study CFTs of this type, one can initially discard two-and higher-loop contributions to the beta function (1.2).
At this stage, it is worth mentioning that the study of CFTs in the epsilon expansion is an active field of research. Let us briefly sketch some different approaches, without attempting to be exhaustive. Conformal symmetry strongly restricts perturbative fixed points, to the extent that certain anomalous dimensions can be determined using representation theory ideas [13][14][15]. Moreover, the ε expansion lends itself well to analytic bootstrap methods [16][17][18][19]. For the Wilson-Fisher CFT, modern tools have been developed to compute anomalous dimensions systematically without Feynman diagrams [20,21]. In other recent work [22], equations of motion were combined with conformal symmetry to provide many quantitative results to leading order in ε. We also point to Ref. [23], which recently discussed thermal phase transitions for a large class of multiscalar theories of the form (1.1).
The problem of classifying one-loop fixed points in the ε expansion is mathematically welldefined, but it is highly non-trivial: it consists of finding zeroes of d N quadratic polynomials in d N variables, where d N is the number of couplings λ ijkl (for instance, d 2 = 5 and d 3 = 15). Even for the case of N = 2 fields, the classification of fixed points was completed only recently, by Osborn and Stergiou [24]. For N ≥ 3 however, no such classification is known. Progress has been made in a different direction: in Ref. [25], Rychkov and Stergiou were able to show that the norm |λ| 2 = λ 2 ijkl of fixed point couplings is bounded from above. This development is interesting because it rules out large swaths of theory space T , the d N -dimensional space of couplings. Since d N ≈ 1 24 N 4 , already for moderate values of N it's extremely complicated to find fixed points inside the high-dimensional space T . By showing that fixed points are instead restricted to a compact domain D ⊂ T , the task of hunting for interacting CFTs becomes much easier. 1 The main goal of this work is to further shrink the allowed region D where CFTs can live. This paper is organized as follows. In Sec. 2, we review the problem of finding one-loop fixed points with N scalars in the epsilon expansion, and we discuss some families of known fixed points and their symmetry groups. For N = 3, 4 we provide a list of known CFTs along with quantitative data about their couplings λ ijkl . In Sec. 3, we define O(N ) invariants of λ ijkl , such as λ iijj , and in turn we prove upper and lower bounds on these invariants. After that, in Sec. 4 we consider the spectrum of order-ε anomalous dimensions γ (1) of operators of the type O r ∼ φ i 1 · · · φ ir . A key result is that their averages γ r and γ 2 r are strongly constrained -in particular, we prove that the O(N ) fixed point maximizes these averages for many values of N and r. In the final part of this manuscript, the above strategy is applied to theories with N complex scalars as well as to bosonic QED. In particular, we show that there are no QED fixed points with fewer than 183 flavors. Several appendices provide proofs and explicit computations.
Note: During the completion of this paper we became aware of [27] which obtains a number of new fixed points for N = 5, 6, 7 via numerical and analytical methods, and which derives further bounds on O(N ) invariants that are consistent with ours.

Review of the epsilon expansion
In this section, we will briefly review the classification problem at hand. As announced in the introduction, we consider a theory of N real scalars φ i with a Z 2 symmetry φ i → −φ i described by the Lagrangian we have introduced an arbitrary factor B > 0 to normalize the couplings λ ijkl . Without imposing any further restrictions, the coupling λ ijkl can be any symmetric tensor. In d = 4−ε dimensions with ε small, the quartic couplings have mass dimension ε 1 so they are weakly relevant. Working in minimal subtraction, the beta function for the couplings reads (2.2) Cubic and higher-order terms in λ can be computed in perturbation theory, by taking into account two-and higher-loop Feynman diagrams. In this scheme, the small parameter ε only appears in the leading term −ελ ijkl : all higher-order terms are independent of ε. From now on, we will set B = 16π 2 to simplify further formulas.
Consequently, we can look for fixed points β(λ ) = 0 of the form 3) The leading-order contribution λ ijkl is necessarily a root of the one-loop beta function ijkl that are subleading in ε can be determined by taking into account n-loop contributions to the beta function. At least when N = 1, these subleading contributions are completely fixed 2 by the leading piece λ ijkl .
It is therefore an interesting problem to classify solutions of the one-loop beta function (2.4). Once such a one-loop CFT λ ijkl is found, it's possible to compute its spectrum of anomalous dimensions (again, to leading order in ε), which gives a rough estimate for its critical exponents in d = 3, and its global symmetry group G can be determined as well. If more precise quantitative predictions for 3d physics are needed, it becomes necessary to perform a higher-loop computation and to resum its results.

Global symmetries
By construction, any Lagrangian (2.1) has a Z 2 global symmetry φ i → −φ i . A coupling λ ijkl can however have a larger global symmetry G ⊂ O(N ). In particular, there exists a maximally symmetric fixed point with an O(N ) global symmetry: (2.5) 2 A sufficient condition for this to happen for N ≥ 2 is found in [12].
At the same time, there exist CFTs with discrete symmetry groups, like the hypercubic fixed point: 3 The above potential is invariant under G = Z N 2 S N , where S N permutes the φ i and the k-th copy of Z 2 flips the sign of φ k . Finally, there are fixed points with symmetry groups that are products of discrete and continuous groups. Consider for instance where u i is an arbitrary unit vector. This is the well-known Wilson-Fisher CFT, which in two and three dimensions describes the Ising model. Its symmetry group is acts on fields orthogonal to u i . Concretely, we can e.g. set u i = (1, 0, . . . , 0) such that the potential only depends on a single field φ 1 . It is easy to generalize to k ≤ N copies of the Wilson-Fisher fixed point: Mathematically, the symmetry group of a coupling λ ijkl is its stabilizer under the action of O(N ). Concretely, an element R of the orthogonal group acts as so its symmetry group is the set of matrices that leave λ ijkl invariant, that is to say

Orbits
Two Lagrangians related by a field redefinition φ i → R i j φ j are physically equivalent. However, they have different-looking couplings. For concreteness, consider two copies of the Wilson-Fisher fixed point: Now rotate the two fields by an angle θ. This gives rise to a one-dimensional family of potentials all of which describe the same physics. For a general coupling λ ijkl , the set of equivalent couplings is known as its orbit: The "size" of the orbit of a tensor λ ijkl is closely related to its symmetry group: a version of the orbit-stabilizer theorem shows that (2.14) For instance, the theory (2.11) has a discrete symmetry group, so the previous formula correctly predicts that its orbit is one-dimensional.
As an important consequence, individual components of the coupling (λ 1111 , λ 1112 , . . .) don't have a physical meaning by themselves: observables like critical exponents should not depend on a choice of frame, but instead they should be constant on orbits. It is therefore convenient to discuss invariants of a given coupling, like |λ| 2 = λ 2 ijkl . This idea will be explored in more detail in Section 3.

Role of isotropy
In much of the RG literature, fixed points are studied based on their putative symmetry group. That is to say that a global symmetry group G ⊂ O(N ) is chosen, and fixed points are then sought inside the subset of couplings which are invariant under G. If G is sufficiently "large", the number of invariant couplings I 4 will be of order unity, so it is relatively straightforward to construct fixed points. For instance, the hypercubic symmetry group Z N 2 S N has I 4 = 2, hence the fixed point (2.6) could be found by solving for two coefficients.
Even though (2.1) contained only a kinetic term and a quartic coupling, the full φ 4 Lagrangian also contains mass terms, i.e. couplings of the form m ij φ i φ j . Such terms are not generated in minimal subtraction, but they appear in more general renormalization schemes, and they definitely must be considered when doing RG computations directly in three dimensions. Enforcing invariance under some symmetry group G also limits the number I 2 of such mass terms.
Much of the literature has focused on QFTs with I 2 = 1, said to be isotropic. From a group theory point of view, this means investigating subgroups of O(N ) that have a unique quadratic invariant, δ ij . When N is small, it is possible to classify such groups and to find all possible CFTs in the corresponding universality classes. This work has been done for N = 2, 3 in [28] and for N = 4 in [29]. For N = 2, 3, the only isotropic fixed points turn out to be the O(N ) CFT and N copies of the Ising model; for N = 4 the solution is much richer.
In the rest of this work, we will not impose isotropy, and instead look for completely general fixed points. Throughout the next section, we will however comment on the role of isotropy.
3 Bounds on λ ijkl at fixed points The number of couplings λ ijkl in a theory with N fields is Instead, we can use a bottom-up method to prove theorems about the space of solutions. Assuming that λ ijkl is a real zero of the one-loop beta function (2.4) for a given number of fields N , what can we say about λ ijkl ? Both in the older literature [9] and in more recent papers [24,25] this question was explored, and in this section we will pursue this strategy further.

Bounds on |λ|
Let us start by reviewing one of the main results of Ref. [25] by Rychkov and Stergiou. They found that if λ ijkl is a one-loop fixed point, then its norm |λ| 2 = λ 2 ijkl obeys the following inequality: The proof of Eq. (3.2) relies on a clever application of the Cauchy-Schwarz inequality, and we refer to [25] for details.
We will now present a new result, namely that the norm of any interacting fixed point λ ijkl is bounded from below as well: Of course, λ ijkl = 0 is also a fixed point -it describes N decoupled free scalars. To show (3.3), it will be useful to define an operator ∨ on the space of tensors. If A ijkl and B m 1 ···mr are two symmetric tensors of rank 4 resp. r ≥ 2, then let 4 where X (i 1 ···ir) denotes symmetrization over the indices {i k }. For instance, for r = 4 this ∨ product is given by In particular, the one-loop β function (2.4) can be written as omitting the superscript on β (1) from now on. Two key properties of the ∨ product are stated below.
Fact. The ∨ product (3.4) satisfies where |B| 2 = B 2 i 1 ···ir is the usual norm on tensors. If (3.7) is saturated, then there exist constants c, c and a unit vector u i such that The proof of (3.7) and (3.8) is given in Appendix A.
Coming back to the β function, we can use the above inequality to get as promised. When the inequality (3.9) is saturated, Eq. (3.8) implies that λ factorizes as follows: The overall normalization c is fixed to be 1/3 by requiring that λ ijkl is a fixed point. In conclusion, any fixed point obeying |λ| = 1/3 is equal the Wilson-Fisher CFT (2.7).

Bounds on O(N ) invariants of λ
We can move ahead and further restrict the space of fixed points, by carefully analyzing the solutions of β(λ) = 0. As announced before, it will be useful to consider O(N ) invariants of the coupling λ ijkl , since such invariants don't depend on a choice of frame.
To define such invariants, we recall that symmetric tensors are reducible under O(N ). If V r denotes the vector space of rank-r symmetric tensors, then we have the following decomposition in terms of irreducible representations: writing [ ] for the traceless symmetric rank-irrep of O(N ). Let us make the decomposition (3.11) explicit for the r = 4 case. For an arbitrary tensor A ijkl the decomposition in invariant components is determined by projectors P : V 4 → [ ] ⊂ V 4 as follows: We note that the projector P 2 has a simple physical interpretation. In an isotropic quantum field theory, the coupling λ ijkl must obey for some z ∈ R -after all, δ ij is the only allowed invariant. A tensor satisfying (3.13) then necessarily has P 2 λ = 0.
To proceed, we will can define O(N ) invariants a (λ) by means of the projectors (3.12). There will be one linear invariant a 0 (λ) and two quadratic invariants a 2,4 (λ) which by construction obey a (P λ) = a (λ) and a (P λ) = 0 if = . (3.14) Explicitly, they are given by Notice that the quantity a 0 (λ) can have either sign (since a 0 (λ) is odd under λ → −λ) while a 2 (λ) and a 4 (λ) are positive, These invariants are related to |λ| 2 via the equation which can for example be used to eliminate a 4 (λ). Following the previous discussion, a 2 (λ) always vanishes in isotropic theories.
For concreteness, let us compute the above invariants for the Wilson-Fisher CFT tensored with N − 1 free fields (2.7). We get which obeys In the tables below, we have collected a list of known fixed points of N = 3 and N = 4 fixed points from the literature (see for instance Ref. [29], and in particular Ref. [26] for three new fixed points) and computed their invariants.
In what follows, we will derive bounds on the invariants a (λ) for that are valid at any fixed point. This idea is not new: essentially, we are generalizing results due to Brézin et al. from Ref. [9]. The new ingredient in our work is that we allow for anisotropic theories; in their work, they only considered couplings of the form where W ijkl is a traceless tensor. Couplings of the form (3.19) automatically have a 2 (λ) = 0.
Going back to our computation, we start by applying a 0 to the fixed point condition β(λ) = 0, making use of the identity 5 This is an important relation, showing that the invariant a 2 (λ) is redundant. Now, notice that the RHS of (3.21) is bounded from below by 1/9, assuming that λ = 0. This restricts the range of a 0 : This bound is not yet optimal. To refine it, we can simply replace |λ| 2 in (3.21) by the RHS of (3.16), which leads to Since the RHS of (3.23) is positive, this immediately implies that The upper bound is optimal, since + N is precisely the value of a 0 of the O(N ) fixed point from (3.18b).
We can also obtain an upper bound on |λ| 2 , starting from the relation (3.21). Since a 2 (λ) ≥ 0, it follows that As a function of a 0 (λ) the RHS attains a maximum at a 0 (λ) = 1 2 N , so in particular |λ| 2 ≤ 1 8 N , which is the previously announced result (3.2). However, for low values of N , the maximum 1 2 N is larger than + N . This is the case for N = 2, 3, and consequently for these values of N , |λ| 2 is maximal at the O(N ) fixed point. In summary, we find that For N = 2 and N = 3 this is a slight improvement compared to Eq. (3.2): for instance 45/121 ≈ 0.3719 is slightly smaller than 0.3720.
We already proved that |λ| ≥ 1 3 for any interacting fixed point. This lower bound can be refined using the condition a 4 (λ) ≥ 0. Using (3.16) and (3.21), we find and therefore The two functions appearing on the RHS are equal if For − N < a 0 (λ) ≤ N , the norm |λ| 2 is bounded from below by 1 9 , whereas for N < a 0 (λ) ≤ + N the relevant bound is given by the non-trivial function of a 0 (λ) appearing in (3.28). For N = 3 and N = 4, the allowed regions in the plane spanned by a 0 (λ) and |λ| 2 are shown in Figures 1 and 2, along with the location of all known fixed points. Finally, we can turn to the anisotropy a 2 (λ), which by construction is non-negative. To obtain an upper bound, we start by rewriting Eq. (3.21): Depending on whether a 0 (λ) is smaller or larger than ρ # N , the RHS can then be bounded by (3.28). For small a 0 (λ), we get the inequality  Meanwhile, for larger a 0 (λ) we find This allowed region in the plane parametrized by a 2 (λ) and |λ| 2 is plotted for N = 3, 4 in Scanning over all allowed values of a 0 (λ), we conclude in particular that In physics terms, this means that the anisotropy of a coupling λ ijkl is at most of order unity, although naively one could expect that there are fixed points with a 2 (λ) of order |λ| 2 ∼ 1 8 N . Finally, let us compare our results to those of Ref. [9] by Brézin et al. In that seminal work, a version of (3.21) was found with a 2 (λ) = 0, along with a counterpart of (3.23). The authors of [9] correctly found that for isotropic theories a 0 (λ) ≤ + N , and later Michel [12] pointed out that for N < 4, the O(N ) model maximizes |λ| 2 , again assuming isotropy. Interestingly, both of these results are ultimately independent of the isotropy assumption.

Anomalous dimensions
So far, we derived bounds on the couplings λ ijkl describing a general CFT at order ε in the epsilon expansion. In what follows, we will instead focus on the the spectrum of such a CFT, that is to say its one-loop anomalous dimensions. We will start by briefly reviewing the problem of computing one-loop anomalous dimensions and proceed to derive explicit bounds for operators without derivatives.

One-loop anomalous dimensions
To start, we briefly review the computation of anomalous dimensions in perturbative quantum field theory. Local operators in the free scalar CFT are of the form free theory : where ∂ is shorthand for a product of derivatives, say ∂ = ∂ µ 1 · · · ∂ µ . Such operators can be organized into irreducible representations of the rotation group O(d) and of the global symmetry group, O(N ). In d = 4 − ε dimensions, the operator (4.1) has scaling dimension ε is the dimension of a free scalar field. If the theory is instead deformed by a coupling λ ijkl such that an IR fixed point is reached, the scaling dimensions change. There will be a new set of scaling operators [O α ] with non-zero anomalous dimensions, so where the γ The scaling operators [O α ] can be expressed as linear combinations of free theory operators of the form (4.1). The computation of the relevant change-of-basis matrix at general orders in perturbation theory is complicated, and we will not discuss in this work. However, it is known that expressions for scaling operators at one loop can be computed by solving a diagonalization problem that only involves free-theory CFT data [30]. To be precise, let be the operator that perturbs the Gaussian theory, and let O ρ be a generic operator in the free theory, say a Lorentz scalar, of 4d scaling dimension ∆ free = r, which is necessarily an integer, r ∈ {1, 2, 3, . . .}. In the 4d Gaussian CFT, we have the following OPE: where C σ ρ is a matrix of OPE coefficients that can e.g. be determined using Wick contractions. It can be shown that scaling operators of the IR CFT are precisely of the form where v ρ is an eigenvector of the matrix C: Moreover, the one-loop anomalous dimension γ Notice that the eigenvalue problem (4.6) only involves operators of 4d dimension r, even though the OPE V × O ρ also contains operators with dimensions other than r.
The formulas (4.5), (4.6) and (4.7) provide an explicit recipe to compute one-loop anomalous dimensions of scalar operators of dimension r. It is perhaps illustrative to spell this out for the case r = 2. All operators with 4d dimension ∆ free = 2 are of the form O ij = : φ i φ j : or ∂ µ φ i , but the latter have have non-trivial Lorentz spin so they don't mix with the O ij . Using Wick's theorem, we have the following OPE: This result is well-known and can be obtained using other means, e.g. [31]. It is straightforward to generalize to N ≥ 2 fields.
For operators of classical dimension r ≥ 4, one might expect mixing of operators of the form : φ r : (without derivatives) with composite operators built out of m derivatives and r − m fields, say : φ r−4 ∂ µ φ∂ µ φ : having m = 2. At one-loop order, it can be shown [32] that such mixing does not arise, based on a careful analysis of the matrix C appearing in Eq. (4.5). It is therefore possible to study operators built out of r fields φ i without gradients by a very modest generalization of the r = 2 diagonalization problem treated above. If we let As a consequence, if v = v i 1 ···ir solves the eigenvalue problem v . In (4.12) we have allowed for a general normalization of the couplings, to facilitate comparison to the literature. 7 Notice that for fixed r, the number of such operators is equal to the number of symmetric tensors of rank r, that is to say which behaves as N r /r! at large N . For a given coupling λ ijkl , finding anomalous dimensions γ (1) v for rank-r operators is therefore equal to diagonalizing a matrix of size d N,r × d N,r .
The cases r = 0, 1 and r = 3, 4 deserve special treatment. The unique operator with r = 0 is the identity operator, which does not get renormalized. The fundamental fields φ i only get renormalized at order ε 2 , by a two-loop diagram. This explains the prefactor r(r − 1) in the eigenvalue equation (4.12). For r = 3, there are N operators of the form which have anomalous dimension γ O i = 1. These operators therefore satisfy as required because of multiplet recombination: in the IR theory, the field φ i combines with O i to form a long multiplet [13]. The perturbing operator V with r = 4 has γ (1) V = 2, so it has scaling dimension ∆ V = 4 + O(ε 2 ). 8

Bounds and sum rules
We will now use Eq. (4.12) to prove properties of one-loop anomalous dimensions γ (1) v in general multiscalar CFTs. First, we will argue that the individual γ It turns out that sums of the form (4.16) can be expressed in terms of n-th order invariants of the coupling λ ijkl and as such they can be related to the bounds derived in Sec. 3.
First, let us consider the individual anomalous dimensions γ (1) v . Setting B = 16π 2 , we can simply appeal to (3.7) to find that Moreover, equality in (4.17) only holds for the Wilson-Fisher CFT. To be precise, the bound is saturated for the following potential and operators: where u i can be any unit vector. In other words, the well-known one-loop anomalous dimensions 1 6 r(r − 1) of the Wilson-Fisher fixed point play a special role.
If N is large, this result is significantly stronger than (4.17), but formula (4.19) does reflect the ∼ r 2 scaling at large r of (4.17).
Next, we turn to the averages γ r and γ 2 r from (4.16). For specific theories, such sums can be computed explicitly. In appendix B, we solve the eigenvalue problem Moreover, the spectrum of the N -field hypercubic fixed point was recently analyzed by Antipin and Bersini in Ref. [33], where explicit results were given for scalar operators with r ≤ 5. They find for instance that γ hypercubic r = 2r(r − 1)(N − 1) 3N (N + 1) for r = 2, 3, 4, 5. (4.22) The quantity γ 2 hypercubic r can be computed as well, and the result is printed in Eq. (B.12).
The above formulas look simple, even though they arise from the complicated-looking eigenvalue problem (4.12). Indeed, we claim that both of these quantities can be expressed in terms of the invariants a k (λ) that we have encountered previously: where the function S N,r is given by S N,r (x, y) = 4(r − 2)(N + r)x + (r − 2)(r − 3)x 2 − (N + r)(7r − 17 − N )y (4.24) using the shorthand notation x = a 0 (λ), y = |λ| 2 . The derivation of these identities is straightforward but slightly cumbersome, and we defer the proof of Eq. (4.23) to Appendix C.
In the derivation of the second sum rule the relation (3.21) was used, so it only holds at fixed points. It's easy to check that (4.23) is valid for the Wilson-Fisher, the O(N ) and the hypercubic CFTs (for r ≤ 5). In fact, the first sum rule shows that the formula (4.22) holds for any r ≥ 2, as was already conjectured in [33].
Above, we showed that individual anomalous dimensions γ (1) were bounded in terms of |λ|. In particular, it follows from (4.17) and the Rychkov-Stergiou bound (3.26) that This is a statement about the spectrum of any one-loop CFT in the epsilon expansion, regardless of any information about its coupling λ ijkl . However, the bound (4.25) is far from optimal, at least in the examples considered above. With some additional work, a sharper statement can be derived.
Theorem. Consider any one-loop CFT with N ≥ 2 scalar fields, labeled by a coupling λ ijkl . For any r ∈ {2, 3, . . .}, the quantity γ λ r is bounded from above by its value in the O(N ) theory:

(4.26a)
For r = 2, we have in addition If N is sufficiently small, the following analog of (4.26a) holds for r ≥ 3: The domain of r and N for which γ 2 r is maximal at the O(N ) fixed point is shown in Fig. 5.
The above result is a clear improvement compared to the naive bound (4.25): in particular, the average of the anomalous dimensions γ r scales at most as 1/N at large N -as can be seen from eq. (4.21a) -compared to √ N . We also remark that the restriction on N from (4.26c) is not very severe; even for r = 3, it only requires N ≤ 14 (and in fact the result holds for N ≤ 18 in that case).
The proof of the bounds (4.26a) is straightforward but long. It relies on the fact that the quantities γ 1,2 λ r can be expressed in terms of the invariants a 0 (λ) and |λ| 2 , which we know are restricted to a compact domain D N ⊂ R 2 . Therefore and determining the supremum of S N,r is an exercise in calculus. Some details are provided in Appendix D. for these values of N . The bound (4.28) can only be saturated by an isotropic CFT with a 0 = 1 2 N (N + 4)/(N − 4); it is an interesting question whether such a one-loop fixed point exists. Similar bounds can be obtained for r ≥ 4, but we will not discuss this further.
Using the same ideas, it is possible to derive uniform lower bounds on γ r and γ 2 r . Such lower bounds don't appear to involve any know fixed points, and we will not discuss them in the present work either.
Let us finally comment on the case where λ ijkl is isotropic. In that case, the order-ε contribution to the critical exponent ν is given by the anomalous dimension of the operator φ · φ, which can be computed as a special case of (4.12). Using this information, the authors of [9] derived a bound on ν valid for general isotropic theories, comparing it to the value of ν of the O(N ) fixed point.

Bounds for complex scalars and bosonic QED
So far, we studied in detail the theory of N real scalars φ i interacting by means of a quartic interaction. The landscape of theories with N complex scalars is not as well-explored. In this section, we will adapt the methods from Sec. 3 to analyze two types of Lagrangians with complex degrees of freedom: first, the theory of N complex scalars with aφφφφ interaction, and second the U (1) gauge theory of N charged scalars, also known as bosonic QED.

Interactions with N complex scalars
To proceed, let us consider the theory of N complex scalars φ i and their conjugatesφ i in 4 − ε dimensions, interacting by means of the following quartic interaction: Here B is an arbitrary constant that will be fixed later. It is possible to allow for a more general quartic interaction, including terms of the form h ijkl φ i φ j φ k φ l , h ijklφ i φ j φ k φ l and their conjugates. However, the action (5.1) is invariant under the U (1) symmetry Q that assigns a charge +1 to φ i and −1 toφ i , and we will later gauge precisely this U (1).
By construction, the couplings g ijkl are invariant under the Z 2 × Z 2 symmetry g ijkl = g jikl = g ijlk (5.2) contrary to the full S 4 permutation invariance of the real coupling λ ijkl . In addition, the reality of L enforces that g * ijkl = g klij . Setting B = 24π 2 , the one-loop beta function of g ijkl is given by Even though V is not manifestly positive for an arbitrary coupling g ijkl , it is indeed positive at fixed points. To check this, we simply rewrite the above equation as As an example, take the most symmetric interaction of the form (5.1): This potential is invariant under a U (N ) symmetry that acts on the fields as follows: The prefactor 1/(N +4) in (5.1) is fixed by imposing that (5.7) is a fixed point. For N = 1, this is the theory of a single doublet with a (φφ) 2 interaction (sometimes known as the XY model).
As a side note, the U (N ) fixed point (5.7) is equal to the O(2N ) fixed point from (2.5), as can be seen by decomposing φ i andφ i in terms of 2N real degrees of freedom.

Invariants and bounds
Just as before, we can try to restrict the space of zeroes of the one-loop beta function (5.4). By analogy with the result (3.3), we can manipulate Eq. (5.4) to show that at a fixed point The proof of this inequality is given in Appendix E. This bound is optimal, since the U (1) fixed point from Eq. (5.7) has |g| = 1 5 . To obtain more refined bounds, we can introduce three projection operators Q i as follows: which indeed obey Q i Q j = δ ij Q i . Associated to these projectors, we can define invariants b 1 (g) = g ijij , b (g) = g klij Q (g) ijkl = |Q i (g)| 2 for = 1, 2. (5.11) Owing to the reality condition (5.3) b 1 (g) ∈ R, and by construction b 2 (g), b 3 (g) ≥ 0. These invariants are related to the norm |g| 2 = g ijkl g klij as follows: For example, the U (N ) fixed point (5.7) has invariants In passing, remark the complex analog of the isotropy requirement (3.13) is (5.14) If Eq. (5.14) is satisfied, b 2 (g) = 0. In particular, the U (N ) fixed point has this isotropy property. Just like in the real case, we will not impose (5.14).
Let us make a brief comment on the group-theoretical meaning of the projectors Q . The group U (N ) acts on the tensor g ijkl through cf. the action (5.8) on the fields φ i ,φ i . Under this action the tensor g ijkl is reducible: its irreducible components can be identified with a trivial representation, the adjoint rep of dimension N 2 − 1 and a third representation of dimension 1 4 N 2 (N − 1)(N + 3), all of which are self-conjugate. These three irreps are precisely the image of the projectors Q 1 , Q 2 and Q 3 .
In order to find an additional relation between the invariants that holds at fixed points, let us introduce the matrix γ ij = g ikjk , which is Hermitian (γ † = γ). From (5.4), it follows that b 1 (β(g)) = −b 1 (g) + 3|g| 2 + 2tr(γ 2 ). (5.16) At the same time, Consequently, at a fixed point we can write We are now in a very similar situation as before: the (in)equalities (5.9), (5.12) and (5.18) are nearly identical to their real counterparts (3.3), (3.16) and (3.21), the only difference being that some coefficients differ. It is therefore straightforward to show that e.g. b 1 (g) and |g| 2 can be restricted to a compact domain. We will not provide a complete derivation, since it would be almost identical to Sec. 3.2. Instead, we will provide a summary of the results.
Theorem. Given the action (5.1) with N ≥ 1 fields, if g ijkl describes a non-trivial CFT at one loop, then The upper bound in (5.19a) is optimal, because σ + N coincides with b 1 of the U (N ) CFT. Inside this interval, In particular The inequality (5.21) is the counterpart of the Rychkov-Stergiou bound (3.2) and its refinement (3.26) in the case of N real scalars.
Let us make a final comment on the role of charge conjugation C, the Z 2 symmetry that exchanges fields and their conjugates, C : φ i ↔φ i . From the point of view of (5.1), C is a discrete symmetry that may be imposed, although we do not do so in this work. Concretely, charge conjugation implies that g ijkl = g klij or equivalently g ijkl ∈ R, recalling the reality condition (5.3). From a practical point of view, imposing C reduces the number of real couplings to 1 8 It is an open question whether C puts interesting constraints on the space of theories -we only remark that "simple" CFTs like the U (N ) fixed point (5.7) are manifestly invariant under C.

Bosonic QED
In this subsection we discuss what happens to the Lagrangian (5.1) if we gauge the U (1) symmetry under which φ i resp.φ i have charges ±1. The resulting action describes bosonic versions of QED in 4 − ε dimensions, allowing for the most general coupling g ijkl that is consistent with gauge invariance and unitarity. Concretely, we minimally couple the matter fields (φ i ,φ i ) from (5.1) to a photon A µ : where D µ = ∂ µ + ieA µ is the gauge covariant derivative and F µν = ∂ µ A ν − ∂ ν A µ . For later convenience, we can redefine the gauge coupling as follows: α := e 2 24π 2 . (5.23) The one-loop beta functions of α and g ijkl are given by 9 where and are two tensor contractions defined in Eq. (E.2). The gauge coupling α has two possible values at a fixed point: α = 0 or α = α := ε/N . If α = 0, the photon decouples and the scalars interact according to the ungauged interaction (5.1). If α = α however, the beta function of the scalar couplings is modified. Rescaling g ijkl by a factor 1/ε, these modified beta functions are given by where β 0 (g) denotes the ungauged one-loop beta function (5.4). Remark that in the limit N → ∞, the two terms of order 1/N and 1/N 2 vanish, and the ungauged beta function β 0 (g) is recovered.
As an example, we can look for maximally symmetric fixed points, having P SU (N ) 10 as a global symmetry group: If the number of flavors N is larger than 183, there are two unitary fixed points with different values of c, namely 11 The two solutions are referred to as bQED + and bQED in [35]: bQED + is known as the abelian Higgs or the non-compact CP N −1 model, whereas bQED is unstable. The fact that the solutions (5.26) only exist for N ≥ 183 (at leading order in ε!) has been known for a long time [36]. For even values of N , Refs. [37,35] also describe two families of one-loop CFTs with symmetry group SU ( 1 2 N ) × SU ( 1 2 N ), which exist for N ≥ 197. These results suggest that it is difficult to construct one-loop fixed points of the Langrangian (5.22) with a large symmetry group but a finite number of flavors, say N 100. However, it could be possible that there exist CFTs at small N that have a small or even trivial symmetry group. In what follows we will show that this is not the case, by proving that there are no one-loop CFTs at all with N < 183 flavors. To do so, we employ the by now familiar strategy of carefully analyzing relations between the invariants b (g) and |g| 2 . For example, we can apply the invariant b 1 to (5.25), using the fact that b 1 (T ) = 1 2 N (N + 1) as well as (5.16). This leads to the identity Using (5.12), |g| 2 can be eliminated in favor of b 3 (g): By construction, b 2 (g) and b 3 (g) are positive. Therefore we must have  10 The global symmetry group is not equal to U (N ), since the U (1) factor is gauged and must be modded out. In addition, Eq. (5.26) has a Z2 charge conjugation symmetry C. 11 These solutions have the following invariants: b1(g ± ) = c± 2 N (N + 1), b2(g ± ) = b3(g ± ) = 0 and |g ± | 2 = c 2 ± 2 N (N + 1).
as we claimed above.
More quantitatively, we can obtain upper and lower bounds on |g|. To wit, if we use that b 2 ≥ 0 then it follows from (5.27) that Notice that in the limit N → ∞, the inequality (5.31) is asymptotically equal to (5.21) from the ungauged case. Finally, we can analyze the beta function (5.25) directly, rewriting it as Using the triangle inequality, the inequalities from Sec. E and the fact that |T | 2 = 1 2 N (N + 1), it follows that For N ≥ 728 this leads to a new constraint on |g|, namely that either |g| ≤ ξ − (N ) or |g| ≥ ξ + (N ) for two algebraic functions ξ ± (N ) that can be determined from (5.33). Approximating ξ ± (N ), it can be shown that

Discussion
In this work we showed that one-loop fixed points with N real or complex scalars are strongly constrained by unitarity alone. Hopefully, these results provide a stepping stone towards further numerical work along the lines of [26]. In our work we used linear and quartic invariants of the coupling λ ijkl , but it is also possible to construct higher-order O(N ) invariants. If it's possible to improve our bounds using e.g. cubic or quartic invariants, it would certainly be worthwhile to do so. In addition, it would be interesting to find a deeper explanation why there seem to be many fixed points close to the O(4) theory, as we mentioned in Sec. 3.2.
We expect that the methods used in this work generalize to many other situations. For instance, it is possible to study φ n -type interactions with n = 4, which are marginal in d = 2n/(n − 2) -see e.g. [24] for a discussion of fixed points with n = 3, 6 or [22] for generic even n and n = 3. Likewise, one can add N fermions and attempt to constrain fixed points with quartic and Yukawa couplings. In the latter case, it could be interesting to hunt for supersymmetric fixed points.
In the second part of our work, we studied averages of anomalous dimensions of operators built out of r fields without gradients. For one, it is an obvious question to extend these results to spinning composite operators. Moreover, the quantities γ 1,2 are similar to averages of anomalous dimensions and OPE coefficients that are encountered in analytic bootstrap computations (see e.g. [19]), and it would be interesting to make contact with these results, either in general or for specific theories.
Stergiou for discussions and correspondence. In particular we thank A. Stergiou for sharing a draft of [27] before publication.
A Proof of equations (3.7) and (3.8) In this appendix, we prove the two equations Eqs. (3.7), (3.8) from the main text. To prove the first inequality, we remark that The tensor C is simply a contraction of A and B which equals A ∨ B after symmetrizing its indices. Therefore We will now prove two facts: (i) any symmetric rank-4 tensor A of the above form factorizes as A = c u ⊗4 for some constant c, and (ii) if a symmetric rank-k tensor T can be written as T = T ⊗ u for some vector u i = 0, then T = c u ⊗k for some constant c ∈ R.
Setting B = T and B = T ⊗ u, these two facts yield the desired result.
In order to establish (i), notice that the symmetry of A in its indices requires that A is proportional to M . Hence A ijkl = c M ij M kl for some constant c. But the fact that A is symmetric under all permutations also implies that A ijkl = A lijk i.e. M ij M kl = M il M kl . By taking traces, this implies that M 2 − tr(M )M = 0. Yet M is diagonalizable (since it's symmetric), and the last equation shows that any non-zero eigenvalue ν must satisfy ν = tr(M ). This can only be satisfied if M has at most one non-zero eigenvalue, or in other words the image of M is at most one-dimensional. This establishes that there exists a vector u i such that M ij = ±u i u j .
The proof of (ii) proceeds by induction. For k = 1 it's a tautology. Suppose that the claim holds for all k ≤ n. For k = n + 1, the symmetry of T implies that for all indices i 1 , . . . , i n+1 . By contracting with u i n+1 , it follows that But T has rank k = n, so it T factorizes completely. This completes the proof.

B Sum rules for the Wilson-Fisher, O(N ) and hypercubic CFTs
In what follows we will compute the quantities γ r and γ 2 r for some infinite families of known fixed points. To this end, it will be useful to write an index-free version of (4.12). To wit, if we let O = v i 1 ···ir φ i 1 · · · φ ir and V (φ) = λ ijkl φ i φ j φ k φ l , then (4.12) becomes To be clear, here the φ i are treated simply as auxiliary vectors, and not as operators, so there is no need to normal order.
For example, we can consider the Wilson-Fisher theory tensored with N − 1 free fields Any polynomial of degree r can be written as Such a polynomial obeys Consequently, the sums over anomalous dimensions are given by and as shown in Eq. (4.20).
For the O(N ) model, any rank-r polynomial can be written in the form where P (φ) is a harmonic polynomial of degree = r − 2k. The number of such polynomials is This decomposition reflects the decomposition of symmetric tensors in terms of traceless symmetric tensors. Now from which it follows that Let us first turn to the proof of (4.23). We will first choose a basis {e α } of symmetric tensors of rank r, normalized such that Working in this basis, the anomalous dimensions {γ α } are eigenvalues of the d N,r ×d N,r matrix In particular, we can express sums over eigenvalues as Such traces are easiest to evaluate in an index-free formalism. For the k = 1 case, let us introduce auxiliary vectors u i resp. v j , such that This is nothing but a rewriting of the second equation of (C.1). Applying the operator D u,v r − 2 times, we find (D u,v ) r−2 (u · v) r = (r!) 2 d N,r 2N (N + 1) (u · v) 2 (C. 5) and performing the final four derivatives, we obtain tr M (λ) = d N,r 2N (N + 1) λ ijkl ∂ 4 ∂u i ∂u j ∂v k ∂v l (u · v) 2 = 2d N,r N (N + 1) λ iijj . (C.6) Inserting this into (C.3), the first equation of (4.23) is recovered. For the second sum rule, we need to evaluate a slightly more complicated expression: Working out the second line, we get tr M 2 (λ) = d N,r 4r(r − 1)(N ) 4 λ ijkl λ abcd ∂ 4 ∂u i ∂u j ∂w k ∂w l ∂ 4 ∂v a ∂v b ∂t c ∂t d P(u, v, w, t) (C.8a) where P(u, v, w, t) = (N + r)(N + r − 1)(u · v) 2 (w · t) 2 + 4(r − 2)(N + r)(u · w)(u · v)(w · t)(v · t) Computing the remaining derivatives, we obtain 1 d N,r tr M 2 (λ) = 4 r(r − 1)(N ) 4 (N + r)(N + r + 1)λ 2 ijkl + 4(r − 2)(N + r)λ 2 ijkk + (r − 2)(r − 3)λ 2 iijj . (C.9) Plugging this into (C.3) with k = 2, and using Eq. (3.21), we finally obtain the second equation of (4.23).
D Bound on γ r and γ 2 r for general theories In this section, we derive the bounds appearing in Eq. (4.26). Let's first obtain (4.26a) involving γ λ r . According to (4.23), this quantity is proportional (by a positive constant) to a 0 (λ). But the latter quantity is bounded from above by its value at the O(N ) fixed point, as derived in Eq. (3.24). Therefore (4.26a) follows immediately.
For r ≥ 3, the function S N,r (x, y) depends both on x = a 0 (λ) and y = |λ| 2 . Schematically, for fixed N , the domain D N is of the form where f ± (x) and ± N depend only on N . The functions f ± (x) are equal at x = ± N , and we also know that + For N = N (r), the function S N,r does not depend on y; for N > N (r) the sup of S N,r will be on the upper boundary curve f + , whereas for N < N (r) it will be on the lower curve f − .
(D. 5) If α N,r > 0 the maximum will be attained either at x = − N , + N or at x = β N,r /2. The maximum will be attained at + N iff β N,r /2 ≥ + N . For r = 3, this requires that where ζ(r), ζ (r) are two algebraic functions of r. The function ζ(r) will play a role later, and an explicit formula is included in the source code of this T E X file.
If α N,r < 0 then this function is a negative parabola, and it can only have a local minimum. Therefore the sup of S N,r will be attained at either x = − N or x = + N . It turns out that the sup is always attained at x = + N . Hence S N,r is also saturated by the O(N ) fixed point for r ≥ 4, N (r) < N < ζ (r). (D.8) Next we consider the case N = N (r), where S N (r),r (x, y) =: S r (x) does not depend on y, and in fact d dx S r (x) = 4(r − 2) N (r) + r + 1 2 (r − 3)x > 0 (D.9) for all r ≥ 3 and all allowed values of x. Therefore the maximum will be attained at x = + N . Finally we can consider the case N < N (r). In this case, we have to look for a supremum of the function S N,r (x, f − (x)). The computation goes along the same lines as before. We find that for all N < N (r), the maximum is reached at x = + N . Summarizing, sup D N S N,r is attained at + N , that is to say at the O(N ) fixed point, when This leads to the restriction on N that appears in (4.26c).