# Leading order CFT analysis of multi-scalar theories in \(d>2\)

## Abstract

We investigate multi-field multicritical scalar theories using CFT constraints on two- and three-point functions combined with the Schwinger–Dyson equation. This is done in general and without assuming any symmetry for the models, which we just define to admit a Landau–Ginzburg description that includes the most general critical interactions built from monomials of the form \(\phi _{i_1} \dots \phi _{i_m}\). For all such models we analyze to the leading order of the \(\epsilon \)-expansion the anomalous dimensions of the fields and those of the composite quadratic operators. For models with even *m* we extend the analysis to an infinite tower of composite operators of arbitrary order. The results are supplemented by the computation of some families of structure constants. We also find the equations which constrain the nontrivial critical theories at leading order and show that they coincide with the ones obtained with functional perturbative RG methods. This is done for the case \(m=3\) as well as for all the even models. We ultimately specialize to \(S_q\) symmetric models, which are related to the *q*-state Potts universality class, and focus on three realizations appearing below the upper critical dimensions 6, 4 and \(\frac{10}{3}\), which can thus be nontrivial CFTs in three dimensions.

## 1 Introduction

The concept of universal behavior in physical systems is very fruitful and has been successfully spread to other quantitative sciences. At the theoretical level quantum and statistical field theories are important tools to study the approach to criticality of physical systems, where most of the details of the microscopic interactions are washed off by the presence of a second order phase transition and universal features emerge. In fact, only the degrees of freedom involved, the symmetries and the dimensionality of the system play a crucial role. In the so-called theory space of all quantum field theories (QFTs) these universal features appear as special points corresponding to critical theories in which the correlation length diverges and physics is nontrivial at all scales. These special QFTs are often lifted to become conformal field theories (CFTs) i.e. scale invariance together with Poincare symmetry are promoted to full conformal invariance [1].

Below their upper critical dimension, which is defined as the dimensionality above which a QFT exhibits Gaussian exponents, these QFTs are generally interacting. In order to investigate these systems nonperturbatively one can resort to numerical techniques, when such an approach is feasible. These investigations might, for example, take the form of Monte-Carlo simulations which are based on the application of the Metropolis algorithm on a lattice (see for example [2] and references therein). From our – admittedly theoreticians’ – point of view simulations are useful to benchmark the results obtained with alternative analytical, sometimes approximate, methods.

Historically the most recognized analytical method to investigate theories which exhibit a second order phase transition is the renormalization group (RG) approach [3], especially after the impetus given by the pioneering work of Wilson [4, 5]. Along this line both perturbative and nonperturbative RG approaches can be employed. The latter has acquired the name of exact RG, which is typically formulated at functional level in terms of flow equations for the Wilsonian action [6, 7] or for the 1PI effective average action [8, 9]. Perturbative investigations have been around since the early days [10, 11] and have mainly lead to results expressed in the \(\epsilon \)-expansion and its resummation below the upper critical dimension of a QFT [12, 13].

In general RG methods, especially if nonperturbative, can give access to global flow, i.e. flows which cover the full theory space. Consequently such flows have an enormous amount of information but require an equal amount of difficult computations. One usually needs to resort to some approximations in order to obtain tractable RG equations in the full theory space, but reasonably precise results can be nevertheless obtained, as shown for example in the computation of the critical exponents of the Ising universality class and its *O*(*N*)-symmetric extensions [14]. The investigations of several different universality classes – new and old – continues to this day and certainly will not lose momentum any time soon [15, 16, 17, 18, 19].

Soon after the early developments of the Wilsonian action, it has been observed that the perturbative RG too can be conveniently formulated at functional level [20, 21]. In this approach, later referred to as the functional perturbative RG, one constructs beta functions which encode the scale dependence of several couplings at the same time and obtains results for several quantities in a more efficient way. This method has also another important advantage which is relevant for this paper: it can be used with a lot of generality in that very little of the system under investigation must be specified a priori. For example, it can be used for theories with rather arbitrary interacting potential (as we will do in this paper), as well as for families of scalar theories with both unitary and nonunitary interactions [22, 23], and even for higher derivative theories in which the derivative interactions are present at criticality [24, 25]. Since it may also happen that internal symmetries emerge at critical points [26, 27, 28, 29, 30], this method can be taken as a starting point to investigate theories not constrained a priori by any symmetry, even including supersymmetry [31, 32].

In the past few years alternative methods based on conformal invariance have gained considerable popularity and shown increasing success. The general strategy of these methods is to focus on the critical points in the space of all theories, assume that scale invariance is promoted to be local, and consequently exploit the enhanced conformal symmetry of the system. It is not an understatement to say that if one assumes that critical points are CFTs, even in \(d>2\), there is a significant advantage when computing close-to-criticality quantities because of the constraints on correlators imposed by conformal symmetry. These ideas are at the base of the conformal bootstrap approach [33, 34, 35], which follows an early suggestion by A. Polyakov and is based on the consistency conditions that are obtained by rewriting the conformal partial wave expansion in two of the *s*, *t* and *u* channels thanks to operator product expansion (OPE) associativity. This method was employed in the analysis of some critical theories and is currently giving the most precise evaluation of the critical exponents of the universality class of the Ising model [36], and is able to deal with various symmetry groups (see for example [37]). Nevertheless, in order to push the analysis to the best accuracy, a good amount of computing power is required even for the conformal bootstrap. In this light, CFT methods have taken the stage as consistent and numerically effective substitutes of both lattice and RG methods at criticality.

Besides the numerical achievements of the conformal bootstrap, several analytic realizations of the underlying idea have been developed, including some which involve perturbative expansions in small parameters such as \(\epsilon \). Among these methods we mention those based on the singularity structure of conformal blocks [38, 39] and their Mellin representation [40, 41], and on the large spin expansions [42, 43, 44, 45].

In this work we concentrate on a CFT-based method which determines the conformal data of a theory in the \(\epsilon \)-expansion by requiring consistency between the Schwinger–Dyson equation (SDE), related to a general action at criticality, and conformal symmetry in the Gaussian limit \(\epsilon \rightarrow 0\) [46]. We refer to this method as SDE+CFT for brevity and very briefly discuss the some properties of both SDE and CFT in the following paragraphs. The interplay of these properties is the essential bulding block of this paper’s analysis.

*S*is the conformally invariant action. Furthermore, conformal symmetry greatly constrains the correlators appearing in the above equation, even in \(d>2\). Adopting a basis \(O_a\) of normalized scalar primary operators with scaling dimensions \(\Delta _a\), the two point correlators have the following form:

Thanks to the power of conformal symmetry, a CFT is completely and uniquely determined by providing a basis of primary operators \(O_a(x)\), the scaling dimensions \(\Delta _a\) and the structure constants \(C_{abc}\), which together are known as *CFT data*. The idea of the SDE+CFT approach is to move below the upper critical dimension \(d_c\), above which the theory is Gaussian, and interpolate the nontrivial correlators shown above with those of the trivial Gaussian theory as a function of the critical coupling. The consistency of this interpolation determines the leading order corrections in \(\epsilon \) of some conformal data when one exploits further relations between operators that are primary only in the Gaussian limit. In this work we restrict ourselves to the information we can extract from the analysis of two- and three-point functions, which is the current state-of-the-art of the approach. Investigations based on this approach have been applied up to now to scalar theories with and without *O*(*N*) symmerty [46, 47, 48, 49, 50], and extended to unitary and nonunitary families of multicritical single-scalar theories [23, 24, 51].

*N*different) fields \(\phi _i\), and with a generic interaction encoded in the potential

*SO*(

*N*) field transformation \(\phi \rightarrow R\phi \), which lead to equivalent theories describing the same physics, imposes further constraints on the couplings associated to inequivalent theories. This can be analyzed with group theoretical methods [27, 28, 29], also introducing invariants on the space of couplings under such field redefinitions [30] in terms of which any universal quantity is expected to be expressed. Discrete symmetries such as permutations can also be taken into account.

In the main text we are able to write explicit eigenvalue equations that depend functionally on the potential and from which many universal conformal data at the leading nontrivial perturbative order can be extracted. In particular we derive expressions for the anomalous dimensions of the fields, the anomalous dimensions of the quadratic composite operators, and for several classes of structure constants. When treating even unitary models with \(m=2n\) we are able to extend the procedure to an infinite tower of higher order composite operators besides the quadratic ones. For all even models we write the equations which fix, at the leading order, possible critical potentials in terms of the parameter \(\epsilon \). It turns out that these equations coincide with the fixed point equations obtained for a generic potential using the functional perturbative RG approach. This provides further insight on how some information of one approach (RG) is encoded in the other (CFT) and viceversa.

Likewise, for the cubic nonunitary model with \(m=3\) we obtain, in complete analogy to the single field case, results for anomalous dimensions of the fields, for the quadratic composite operators, and for some structure constants. We are also able to fix as a function of \(\epsilon \) the critical potentials. Similar to the even case, the critical conditions coincide wih those of the functional perturbative RG approach. For higher order nonunitary models with \(m=2n-1\) and \(n>2\) we are not able to find enough constraints on the critical potential to set it in terms of \(\epsilon \), which is again a situation in complete analogy to the single-field case.

We then specialize our very general results by giving a more explicit form to the potential that is constrained by symmetry. As an interesting example, we choose the symmetry to be the permutation group \(S_q\) acting on the fields with \(q=N+1\) and we study it in detail. This symmetry group corresponds to Potts-like field theories, which include as special cases the standard field-theoretical cubic realization of the Potts universality class, the reduced Potts model, and – in principle – infinitely many generalizations. Despite being much less constraining than *O*(*N*) symmetry (which nevertheless emerges as an effective symmetry for some fixed points), the group tensor structures appearing in the potential can be naturally factorized thus reducing strongly the number of independent parameters. The Potts models [52, 53, 54, 55, 56, 57, 58, 59] are quite ubiquitous in statistical mechanics: Several interesting models can be obtained if one takes analytic continuations of *q*. The most relevant continuations for this paper are to the value \(q=1\), which is related to models of percolation, and to \(q=0\), which is related to the random cluster model known as spanning forest [60].^{1} The easiest way to construct an \(S_q\)-invariant potential interaction is to follow a standard vector representation of the \(S_q\) group. We concentrate on the Landau–Ginzburg description of Potts models which have upper critical dimensions \(d_c>3\), and therefore can be nontrivial in \(d=3\). This restricts our specific investigation to the cubic [61], quartic [63], and quintic potentials for which we obtain some universal conformal data, recovering as usual several RG results.

“Multi-field generality” and “functional description” are ingredients that bring this work close, in spirit, to that of Osborn and Stergiou [30] in which several similar questions are addressed using multiloop perturbative RG methods instead of the CFT+SDE technique. Our work should also be regarded as a companion to a forthcoming paper devoted to the study of multi-field multicritical Potts models with functional perturbatve RG techniques [64]. While the CFT+SDE methods used in this work are still limited to the leading order of the \(\epsilon \)-expansion, their value lies in the fact that they outline the importance of conformal invariance at criticality and they facilitate the computation of conformal data and, more generally, of the OPE.

In the next section, which is the most important of the paper, we apply the CFT+SDE technique to a multi-field multicritical model with a general potential, providing in several subsections general expressions for the conformal data in terms of the potential. We shall then introduce in Sect. 3 the Potts model and discuss its field representations and group invariants, along with some useful relations, and introduce the relevant Landau–Ginzburg representation we shall later use. Having imposed the \(S_q\) symmetry we also introduce operators on the space of quadratic fields that project them into irreducible representations with definite anomalous dimensions, which we also give. In Sect. 4 we present the analysis and the results for the specialized cubic, (restricted) quartic, and quintic Potts universality classes. We then present our conclusions. The paper ends with two appendices. The first contains some useful relations for free theory correlators which are used extensively in the text. The second includes three parts reporting in order: the reduction relations for \(S_q\) symmetric tensors, some computational details for the quintic model, and few useful RG results [64] needed for the quintic model.

## 2 CFT data from classical equations of motion: general results

*N*fields, where no symmetry is imposed on the model. In the Landau–Ginzburg description, these models are expressed by the following action

*m*fields, one has the relation \(d_c=2m/(m-2)\). We shall in general adopt the perturbative \(\epsilon \)-expansion technique below the critical dimension \(d=d_c-\epsilon \). All the fields have the same canonical dimension \(\delta =d/2-1=2/(m-2)-\epsilon /2\). The method we employ is based on the use of the Schwinger–Dyson equations (SDE) combined with the assumption of conformal symmetry of the critical model. Critical information is extracted from the study of two and three point correlators, whose functional form is completely fixed in terms the conformal data parameters. Our analysis gives access to some of these conformal data at leading order in the \(\epsilon \)-expansion (different quantities can have a different power in \(\epsilon \) at leading order).

We devote separate subsections to the computation of the field anomalous dimension, the critical exponents of the mass operators, critical exponents of all higher order operators for even models, and some structure constants (or OPE coefficients). Finally we shall show for the case of \(m=3\) corresponding to \(d_c=6\), the case \(m=4\) corresponding to \(d_c=4\), and then for general models with even \(m>4\), how the CFT constraints together with the Schwinger–Dyson equations can be used to fix the critical theory, i.e. the \(\epsilon \) dependence of the couplings present in the potential. In particular we show that these constraints are exactly the same as the fixed point conditions of beta functions which appear in the functional perturbative RG approach [21, 22, 23, 30].

We stress that the results given in this section are general and, as such, depend on the generic potential *V* which defines a multicritical model. We shall then restrict ourselves in the next sections to specific models having in particular the \(S_q\) symmetry.

### 2.1 Field anomalous dimension

*n*is the multicriticality label corresponding to the power of the classically marginal potential \(\phi ^{2n}\) in the theory [21, 22, 23]. Notice that for the cubic and quintic models which we shall study in detail later in Sect. 4

*n*is a half odd number.

*N*different anomalous dimensions associated to the true scaling fields, which correspond to the defining primaries of the CFT. These are related to the scaling fields \( \tilde{\phi }_i \) through a linear transformation which leaves the kinetic term invariant

*R*can always be chosen such as to diagonalize also the space of fields with the same dimension. In terms of the scaling fields one may also define \(V(\phi _i)=\tilde{V}(\tilde{\phi }_i)\).

*c*is the free theory value of \(\tilde{c}\) given by Eq. (A.2). For the calculation of the l.h.s one can Taylor expand the potential and use (A.3) to get at leading order

*n*integer or half integer) this expression picks only the \(\ell =2n-1\) term in the sum. In this case, noticing that \(2\delta _c+4=2(2n-1)\delta _c\), Eq. (2.6) at leading order gives the anomalous dimension of the field \(\tilde{\phi }_a\)

*R*is the matrix that diagonalizes it. In the rest of the paper we drop the tilde on the fields and the potential and always assume, unless otherwise stated, to work in the diagonal basis.

Let us make an aside comment here. In the RG analysis of physical systems close to criticality the approach to criticality is controlled by parameters such as the temperature, the simplest example being the Ising model with quartic interaction. In this model using for example dimensional regularization one has to tune to zero the mass operator, which is relevant at criticality. In the multi-field case in order to reach such a condition for all fields, insisting to tune only one parameter, one is forced to require that all the “bare masses” coincide. This requirement is equivalent to the so called zero trace property on generic quartic interactions \(v_{ijkk}=v \delta _{ij}\) [26], which implies that at the fixed point all the anomalous dimensions are equal. In single-field multicritical models where there as more than one relevant operator, the approach to criticality can be controlled by introducing other tuning parameters. Requiring the same number of parameters, in the multi-field case, to control the approach to criticality, one is forced to introduce other conditions on the critical potential to have the same bare mass and bare couplings of all the relevant operators which we would like to tune to zero. In our general CFT approach of this paper we are not concerned with these extra requirements and keep the arguments as general as possible, imposing no symmetry on the model.

### 2.2 Quadratic operators

*n*is either an integer or a half odd number. For \(n\ne 2\) this is done by applying the operator \(\Box _x\Box _y\) to the three-point function \(\langle \phi (x) \,\phi (y)\, \phi ^{2}(z) \rangle \) and calculating it at leading order in two ways: first by using the SDE

*S*refers to the particular choice of \(S_{pq}\). Then the two equations for the single-field case (2.12) and (2.13) are generalized respectively to

*g*has been replaced by the 2

*n*th derivative of the potential, which is evaluated at \(\phi =0\), as will be understood in the rest of the paper. The two free structure constants in (2.15) and (2.16) are defined respectively as the coefficients in the correlation functions

*p*,

*q*indices are separately symmetrized. These notations are introduced in Appendix A in the more general sense. Equating the two Eqs. (2.15) and (2.16) and simplifying a bit leads to

### 2.3 Higher order composite operators: recurrence relation and its solution

*n*. For the unitary “even” critical theories with integer

*n*we move on and seek scaling operators of arbitrary order

*k*and their anomalous dimensions. For simplicity of notation we sometimes use the following abbreviation

*k*for which the r.h.s of (2.34) does not vanish is \(k=n-1\), which means that the anomalous dimensions are linear in the couplings starting from \(\gamma ^S_n\), while all the lower ones, \(\gamma ^S_k\) with \(k<n\), are at least quadratic. So we start from Eq. (2.34) with \(k=n-1\). The anomalous dimensions \(\gamma ^S_{n-1}\) and \(\gamma ^i\) are of higher order and can be omitted, giving

*n*fields. Let us now move to the next case \(k=n\). This time Eq. (2.34) gives

The anomalous dimensions \(\gamma ^S_{2n-1}\) corresponding to descendant operators are obtained from an identity relating the scaling dimensions of the descendant operators to those of the fields \(\phi ^i\), and can be shown to satisfy (2.48) for \(k=2n-2\). Finally, the second Eq. (2.51) gives the missing eigenvalue equation for \(\gamma ^S_{2n}\). In other words we will obtain a relation for \(\gamma ^S_{2n}\) where instead of the tensor \(S_{i_1\cdots i_{2n-1}}\) the descendent structure \(V_{i i_1\cdots i_{2n-1}}\) is present with the corresponding anomalous dimension. This is exactly what is needed to complete the space of composite operators of order \(2n-1\) in the recurrence relation (2.34). These will be discussed in detail in Sect. 2.6.

### 2.4 Structure constants

Apart from the anomalous dimensions of the quadratic and higher order operators that we have discussed so far, conformal symmetry along with the equations of motion provide information on several classes of leading order structure constants. This has been shown in the single-field case in [51]. It is straightforward to extend the computation of structure constants of single-field theories to the multi-field case. In this section we make such a generalization and provide compact formulas for some structure constants in multicritical and multi-field even or odd models.

#### 2.4.1 Generalization of \(C_{1,2p,2q-1}\)

*n*is an integer. Several set of structure constants had been computed for the single-field case in [51], for example

*q*,

*p*are constrained as in the single-field case, and the integers

*r*,

*s*,

*t*satisfy the relation

*c*factors in the free structure constants are removed and, defining \(\hat{V}(\hat{\phi })=V(\phi )\), in the new equation of motion a factor of \(c^{-1}\) will appear on the r.h.s as if \(V\rightarrow V/c\) in (2.2). Also, in terms of the rescaled field the 2

*n*th field derivative of the potentials will be \(c^n\) times the original one. More explicitly

*c*factors in the free structure constants and make the replacement \(V_{i_1\cdots i_{2n}}\rightarrow c^{n-1} V_{i_1\cdots i_{2n}}\). We shall make such a choice of normalization in Sect. 4, when studying some Potts models, but only when we give explicit \(\epsilon \) dependent expressions for the structure constants.

#### 2.4.2 Generalization of \(C_{1,2p,2q}\) and \(C_{1,2p-1,2q-1}\)

*q*,

*p*constrained as in the single-field case and where the integers

*r*,

*s*,

*t*satisfy the relation

*q*,

*p*fall in the range \(q+p\ge \ell +1\) and \(|q-p|\le \ell \), and the integers

*r*,

*s*,

*t*satisfy the relation

#### 2.4.3 Generalization of \(C_{1,1,1}\)

#### 2.4.4 Generalization of \(C_{1,1,2k}\)

*k*fields is a scaling operator satisfying (2.48). Using the result of [51] and following the arguments of the previous sections this is straightforwardly calculated As in the single-field case, in this equation

*n*is either an integer or a half-odd number and

*k*is constrained to the range \(2\le k \le 2n-1\) and \(k\ne n,n-1\).

### 2.5 “Fixed point” equation from CFT

We conclude this section showing in general how the constraints imposed by conformal symmetry on two and three point functions together with the use of the Schwinger–Dyson equations can fix the possible critical theories at leading order in \(\epsilon \). We shall follow a path which is slightly different from the one employed in [49, 51], and do not directly rely on the conditions on the scaling dimensions of descendant operators from the equation of motion (when the interactions are turned on below the critical dimension). Interestingly enough we find conditions which can be simplified to match exactly the fixed point condition of the RG approach in its functional form [21, 22, 23, 30, 64] which we dubbed functional perturbative RG approach. It is well known that in general, fixed point equations admit solutions which are characterized by some internal symmetries not necessarily realized away from criticality, giving a scenario where critical theories can have a higher level of symmetry, or an emergent symmetry. Therefore all the discussions in the literature with RG techniques regarding possible symmetry enhancements at criticality [27, 28, 29, 30] are directly applicable also in this CFT perturbative framework, at least in the cases shown below, i.e. all unitary multicritical models and the one with a cubic potential.

#### 2.5.1 The \(d_c=6\) case

#### 2.5.2 The \(d_c=4\) case

*kl*. Simplifying the result one obtains Making the rescaling \(V\rightarrow 4V/c\) to match the RG normalization removes the

*c*/ 4 factors and we finally get

#### 2.5.3 General even models

*n*derived from RG. It is obtained by taking the 2

*n*th field derivative of the leading order beta functional

*n*th derivative of the potential persists. The tensor coefficient of \(\delta v_{i_1\cdots i_{n+l}}\) then gives the stability matrix that coincide with (2.48), of course after suitable rescaling of the potential. It might be worth mentioning that since \(V_{i_1 \dots i_m}\) constitute the most general set of couplings at criticality the Eqs. (2.73), (2.82) and (2.98) are completely general at leading order and admit all possible fixed points, while at higher orders in perturbation theory these equations are corrected by higher powers of the interactions \(V_{i_1 \dots i_m}\).

### 2.6 The missing pieces in the recurrence relation

#### 2.6.1 The case \(k=2n-2\)

*i*

*i*, and denote them from now on as \(\gamma ^i_{2n-1}\). Let us now insert this into Eq. (2.48) (setting also \(l=n-1\)) to see what we get

*i*index is taken into the symmetrizing parenthesis. This can be done because

#### 2.6.2 The case \(k=2n-1\)

## 3 Potts models

*q*different values \(\sigma _l=1,\dots ,q\). The model can be characterized by the microscopic Hamiltonian

*q*objects which acts globally on the set of

*q*spin states. The model is a fundamental actor in the theory of phase transitions because for \(J>0\) it can exhibit either first or second order phase transition according to both the value of

*q*and the dimensionality

*d*of the lattice.

There is an alternative formulation on the lattice based on random clusters [53], equivalent for \(q\ge 2\), which has the advantage of allowing for an analytic continuation in *q*. A straightforward expectation is that the critical physics of the *q*-states Potts model can be captured by an opportune field theoretic realization of an \(S_q\)-invariant model, and that the renormalization group flow of such model admits either a Gaussian fixed point if the phase transition is first order (for values of *q* above a certain dimensional dependent threshold \(q_c(d)\)), or a non-Gaussian fixed point if the phase transition is second order. For this latter case one expects the universal features of the model also for \(d>2\) to be described by a CFT, if scale invariance is lifted to conformal invariance.^{2}

Several RG analysis of the Potts model, also for the specific analytic continuations to \(q=1\) (percolation) or \(q=0\) (spanning forests), are available in the literature. The analytic continuation can be performed within a chosen representation of the \(S_q\) discrete symmetry group. Perturbatively the standard approach is based on the \(\epsilon \)-expansion below the upper critical dimension[54]. A first attempt to study within wilsonian exact RG was made in [55]. In \(d=2\) several exact results are available [56, 57, 58]. See also [59] for a review of the Potts models.

### 3.1 Zoology of \(S_q\)-invariant interactions

*q*vectors \(e^\sigma \) which point in the directions of the vertices of a

*N*-simplex, i.e. a simplex in \({\mathbb R}^N\), for \(N=q-1\). The set of vectors satisfies the following properties

*N*-dimensional space \({\mathbb R}^N\).

*a*. Expressing these invariants in terms of the basic fields \(\phi _i\) allows one to write down the most general \(S_q\)-invariant actions.

*q*(and possibly analytically continue it) we shall not require any further property, although it is possible to treat the cases \(q=1\), 2 and 3, for which some simplification occurs, separately.

*N*fields \(\phi _i\) is

*V*can be written as

*i*refers to the fact that each element \(\mathcal{I}_{i,j}\) is a fully \(S_q\)-invariant products of

*i*copies of the field components \(\phi \), while

*j*parametrizes the increasing size of the tensors \(q^{(i)}\) in its construction (the presence of the tensors \(q^{(i)}\) instead of the Kronecker delta represents, to some extent, the departure from an

*O*(

*N*) invariant theory). The first few invariants are

*O*(

*N*) symmetry (in which the only allowed invariants are powers of \(\phi _i\phi _i\)): notice that while the basis operators chosen in this paper are the same as the one of [55], the two bases differ in the way the label

*j*is assigned. By construction some invariants are algebraically related, for example

*q*it is possible to find even more relations among the invariants. In particular, given a natural value of

*q*there is a finite number of independent invariants that we can build out of the field multiplet. We come back to this point later when specializing some results to the first few low values of

*q*.

One can write useful relations to simplify contractions of such \(q^{(i)}\) tensor. We present some of them in the Appendix B.1.

### 3.2 Quadratic operators: imposing \(S_{N+1}\) invariance

*N*-dimensional space of fields \(\phi _i\) or equivalently \(\psi ^\alpha \) carry the standard representation of \(S_{N+1}\) which in the Young-Tableaux notation is nothing but the following diagram with \(N+1\) boxes From this, one can determine the decomposition of the symmetric product \(\phi _{i}\phi _{j}\), or equivalently \(\psi ^{\alpha }\psi ^{\beta }\), of two fields into irreducible representations. These irreducible representations are the representations carried by the quadratic scaling operators. Indeed the symmetric product of two standard representations is decomposed as

^{3}

^{4}

*N*-dimensional subspace characterized by \(\sum _\alpha a^\alpha =0\). A convenient basis that spans this subspace is \(e^\alpha _i\), \(i=1,\ldots ,N\), the elements of which are ensured by (3.10) to lie on the

*N*-dimensional subspace. This choice makes the eigenvectors transform covariantly under rotations. Therefore there are

*N*eigenvectors \(u_p\) with eigenvalue \(\gamma ^2_2\), which can most conveniently be written as

*N*-dimensional representation labelled by a single index is equivalent to the double-index redundant description (3.25). One can also verify that \(q^{(3)}_{ijk}(P_2)_{jk,lm}=q^{(3)}_{ilm}\) and \(\delta _{jk}(P_2)_{jk,lm}=0\).

*p*,

*q*) back into (3.37) leads to the set of \((N+1)(N-2)/2\) independent eigenvectors \(u_{pq,ij}\) labeled by

*p*,

*q*and given explicitly by

## 4 Potts models with \(d_c=6,4,\frac{10}{3}\)

### 4.1 Cubic Potts model

#### 4.1.1 Anomalous dimension

#### 4.1.2 Quadratic operators

#### 4.1.3 Structure constants

*c*is simply given as \(\zeta _3 \rightarrow \zeta _3 \sqrt{c}/8\). Agreement between CFT and RG results is then verified immediately at this level. In order to obtain the explicit form of these OPE coefficients we choose the scaling operators \(\mathcal{S}_2\) and \(\tilde{\mathcal{S}}_2\) among (3.50), but excluding the descendant operator \(\mathcal {O}^{(2)}_{2,ij}\). This gives

#### 4.1.4 Critical coupling \(\zeta _3(\epsilon )\)

*N*limit these critical data tend to those of the Lee-Yang model [63]

### 4.2 Quartic (restricted) Potts model

#### 4.2.1 Anomalous dimension

#### 4.2.2 Quadratic operators

#### 4.2.3 Critical couplings \(\zeta _{4,1}(\epsilon )\) and \(\zeta _{4,2}(\epsilon )\)

*g*is the coefficient of \(\frac{1}{4!}\phi ^4\) in the Lagrangian. Equating the two and using the known result \(\gamma _2 = cg/4\), which may be found by applying only one \(\Box _x\) to the same correlator as the above, and the fact that \(\eta =\mathcal {O}(\epsilon ^2)\), one finds \(g = 4c\epsilon /3\).

#### 4.2.4 Quartic scaling operators

*N*vanishes and also the last expression blows up in the limit \(N\rightarrow -1\) which is a sign that the norm of the operators inside parenthesis vanishes in these limits. At the second nontrivial fixed point the scaling operators and their anomalous dimensions are

#### 4.2.5 Structure constants: some examples

*V*and \(\mathcal{S}_4\) which vanishes if they have different eigenvalues. This can also be checked explicitly for the three \(\mathcal{O}^{(4)}_2\) operators given in the previous section.

### 4.3 Quintic Potts model

#### 4.3.1 Anomalous dimension

#### 4.3.2 Quadratic operators

#### 4.3.3 Structure constants

*ij*provide a redundant description of the set of operators. Using the nonredundant descriptions given by (3.51) and (3.52) one may re-express the above structure constants apart from the last one. These are

#### 4.3.4 Some universal results

## 5 Conclusions

We have employed a general method based on the use of conformal symmetry and Schwinger–Dyson equations to investigate multicritical multi-field scalar QFTs, characterized by a critical potential of order *m*. At the leading order in the perturbative \(\epsilon \)-expansion, this method gives access, in a simple way, to some universal data which includes both nontrivial anomalous dimensions and structure constants. These results generalize the method applied to generic multicritical models of a single scalar field presented in [51]. Even without assuming any symmetry and considering only two- and three-point functions, one can already find a considerable amount of information which includes the anomalous dimensions of the fields, the scaling dimension of the quadratic composite operators, a tower of all-order composite operators for the “even” (*m*=2*n*) unitary multicritical theories, and the explicit form of several structure constants. For \(m=2n\) and \(m=3\) one can also find the equations that constrain the interaction potential at criticality. In particular we show how these constraints can be cast in exactly the same form as fixed-point conditions which could be obtained from a functional perturbative RG analysis. Most of the general results and computational strategies presented in this work are new: while part of the results could in principle be obtained from more standard perturbative RG methods, one of the objectives of this paper has been to show how conformal symmetry alone can give access to such critical information.

We remark here some interesting general features of our investigation. The results obtained in the first part of the paper for a general class of multi-field scalar QFTs are derived without the use of any internal symmetry, but of course can be specialized to cases characterized by any (continuous or discrete) symmetry. We have focused on the derivation of many universal quantities, but we stress that also the criticality conditions that we have obtained on the set of possible couplings of the potential in the Landau–Ginzburg description are important by themselves. In fact these conditions, which coincide with fixed point of functional beta functions determined with standard perturbative RG, are in many cases expected to lead to the emergence of some symmetries at criticality, which is a fact that has already been observed in the literature (see [30] and references therein). Clearly, with a growing number of fields the pattern of solutions is increasingly complex, but in principle any multicritical multi-field scalar theory can be analyzed specializing our general framework. Pursuing such an approach one can access – at the perturbative level – all the possible internal continuous or discrete symmetries of a theory with given critical dimension \(d_c\) and number of fields *N* that is a CFT at criticality. In other words one can expect constrained emergent symmetries at a critical point.

We have then specialized the analysis to potentials characterized by \(S_q\) invariance, which encompass the Potts model and some of its multicritical generalizations. We have used the standard representation theory of \(S_q\) to construct all the symmetric interaction terms in the multicritical potential. Even before embarking on explicit calculations for particular models, we have explored how far we could get from a knowledge of the symmetry group alone. We have given explicit expressions for the decomposition of the quadratic operators into scaling operators, which carry irreducible representations of \(S_q\), and we have presented model independent formulas for anomalous dimensions of such operators.

The formalism gives the possibility to perform an analytic continuations of *q* to some specific values, e.g. the ones of special interest in statistical mechanics: \(q=1\) (percolation) and \(q=0\) (spanning forest). Therefore, as an application we have analyzed in detail all the theories which have an upper critical dimension \(d_c>3\), which are nontrivial critical models in any integer dimensions *d* in the range \(3 \le d<d_c\), and specialized *q* to the values of interest. The results we have found match with the ones that will be presented in a companion paper [64] which is devoted to the study of Potts-like field theories with functional perturbative renormalization group methods and which puts an emphasis on those with quintic interactions (\(d_c=10/3\)). The results found there are confirmed by the present investigation for all critical and multicritical Potts-like models. In the cubic (standard Potts) and quartic (restricted Potts) critical models one can fix, with the aid of relations based on conformal invariance, the critical values of the couplings as a function of \(\epsilon \). Unfortunately this is not possible in the quintic case, therefore we completed the analysis of the quintic model using the RG results of the companion paper which for convenience we have included and briefly discussed in Appendix B.3. Likewise in the main text, we have also focused on the generalization of percolation and spanning forests universality classes, for which we show that critical solutions associated to second order phase transition do exist, depending on the kind of multicritical theory considered.

There are several directions and extensions of this work that one can take for future investigations. One such direction would be the inclusion of large-*N* types of analyses. It is not immediately clear if in this very general framework the large-*N* expansion can be of help because we lack the constraints offered by a symmetry such as *O*(*N*), but we expect that there can be intermediate semi-general situations in which it could be useful. We also expect that extending the analysis to higher order correlation functions one can have access to further relations and informations on the conformal data. In particular one could wish to extend the results to the next-to-leading nontrivial order in the \(\epsilon \) expansion. Furthermore, one could change the structure and the degrees of freedom of the theories: e.g. considering general tensor models or theories with fermion [67] or vector fields. Again interesting constraints on the symmetries should be investigated both in this framework and in the functional perturbative RG to get access to their universal data. Among these possible extensions we would like to include the study of nonunitary higher-derivative theories, which have recently been investigated in the single-field case in [24, 25] and which are important to extend critical theories to higher dimensions.

A final future direction that we would like to point out is inspired by the works [65, 66] and involves the study of “walking transitions”. In some systems scale invariance can be approximately realized because the renormalization group runs “close” to complex fixed points. These complex CFTs are nonunitary, but otherwise fully consistent conformal theories with complex conformal data. For these models too we expect that the general conditions of criticality derived in Sect. 2.5 select the allowed internal symmetries compatible with the number of fields and the upper critical dimensions. Furthermore, the *q*-states Potts model considered in this paper is a prototypical example for the investigation of complex CFTs because, as a function of *q*, pairs of fixed points that are related by the reflection \(\phi ^i \rightarrow -\phi ^i\) annihilate and morph into pairs of purely imaginary complex CFTs (structurally similar to the PT-invariance of the Lee-Yang model’s potential [68]). Therefore, by opportunely tuning *q* and *d* it is realistically possible to encounter scenarios in which walking transitions are realized.

## Footnotes

- 1.
In absence of better nomenclature, we address all continuum field-theoretical models with \(S_q\) symmetry as “Potts model”. As it will be shown in the paper, these include the standard universality class that describes the standard lattice Potts model at criticality, as well as some of its multicritical generalizations which we classify by the order of the critical interaction in the corresponding potentials.

- 2.
- 3.
We keep the summation on

*a*,*b*implicit, while summations on \(\alpha ,\beta \) are made explicit. - 4.
Note that summing on \(\alpha \) in (3.31) gives \(S_{aa}=0\).

- 5.
The analytic continuation in

*N*of the results will also play a pivotal role in the discussion of [64].

## Notes

### Acknowledgements

OZ acknowledges support from the DFG under Grant No. Za 958/2-1.

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