# Uncovering novel phase structures in \(\Box ^k\) scalar theories with the renormalization group

## Abstract

We present a detailed version of our recent work on the RG approach to multicritical scalar theories with higher derivative kinetic term \(\phi (-\Box )^k\phi \) and upper critical dimension \(d_c = 2nk/(n-1)\). Depending on whether the numbers *k* and *n* have a common divisor two classes of theories have been distinguished. For coprime *k* and \(n-1\) the theory admits a Wilson-Fisher type fixed point. We derive in this case the RG equations of the potential and compute the scaling dimensions and some OPE coefficients, mostly at leading order in \(\epsilon \). While giving new results, the critical data we provide are compared, when possible, and accord with a recent alternative approach using the analytic structure of conformal blocks. Instead when *k* and \(n-1\) have a common divisor we unveil a novel interacting structure at criticality. \(\Box ^2\) theories with odd *n*, which fall in this class, are analyzed in detail. Using the RG flows it is shown that a derivative interaction is unavoidable at the critical point. In particular there is an infrared fixed point with a pure derivative interaction at which we compute the scaling dimensions and, for the particular example of \(\Box ^2\) theory in \(d_c=6\), also some OPE coefficients.

## 1 Introduction

Quantum field theory (QFT) has proven remarkably effective at describing physical systems close to their critical point where the correlation length tends to infinity. One of the essential tools in this regard is the renormalization group (RG) which governs the effective description of physical systems in terms of the length scales at which they are probed. In particular universality of physical properties in the vicinity of a scale invariant point which had long remained a conundrum owes its resolution to this notion [1]. RG has not only brought conceptual insight but through epsilon expansion has provided a perturbative framework to compute critical properties [2].

Applicability of RG methods relies solely on scale symmetry which is an inherent feature of physical systems at criticality. However, scale invariance often comes along with conformal invariance. In particular unitary scale invariant theories have been shown to be conformal in two dimensions and perturbatively also in four dimensions. The presence of conformal symmetry allows to take advantage of the whole apparatus of conformal field theory (CFT). Indeed developing CFT methods and applying them in the study of critical systems has been an active line of research in the recent years [3, 4, 5, 6, 7, 8, 9, 10, 11].

Well-known examples that have been studied extensively with quantum field theory methods include the Ising model as a unitary theory described by a \(\phi ^4\) potential and the Lee-Yang edge singularity which is nonunitary and is characterized by a cubic interaction \(\phi ^3\) [12]. Built upon these results and their extensions [13] such models have been generalized to all multicritical theories with both RG [14, 15, 16] and CFT techniques [7, 8, 10]. This includes theories with even [7, 8, 14, 15] and odd potentials [10, 16].

Another class of scalar theories that further generalizes the above models and has recently received more attention in the literature includes those with a higher derivative kinetic term of the form \(\phi (-\Box )^k\phi \) [17, 18, 19, 20, 21, 22] or \(\Box ^k\) theories for short, where *k* is a positive integer. Despite their nonunitary nature, theories with higher derivatives have found interesting physical applications. The theory of elasticity [23] provides such an example. Furthermore particular quartic derivative models have been shown to describe the isotropic phase of Lifshitz critical theories, and may be relevant for the physics of certain polymers [24]. The latter has also been studied with \(\epsilon \)-expansion techniques [25, 26] and more recently with non perturbative functional RG methods [27, 28] as well.

Apart from possible physical applications, \(\Box ^k\) theories are interesting also at the theoretical level as they provide a framework to build new universality classes and serve as a testing ground for methods and ideas in higher dimensional CFTs. Moreover, they demonstrate new features not present in standard QFTs and in this sense they are instructive from the point of view of perturbation theory.

At the free theory level \(\Box ^k\) theories have been previously studied in [17] where the coefficient of the energy-momentum tensor two-point function for \(k=2,3\) is extracted, and in [18, 19, 20] where general \(\Box ^k\) theories have been investigated with particular global symmetries, mostly motivated by possible links to quantum gravity in de Sitter space. At the interacting level multicritical \(\Box ^k\) theories, with and without global symmetries, that demonstrate a generalized Wilson-Fisher fixed point have been analyzed in [7, 8] using the analytic structure of conformal blocks. In [21] *O*(*N*) symmetric \(\Box ^k\) theories with quartic interaction have been explored in the large *N* limit.

In a recent letter [22] we have reported the results of our study of general multicritical \(\Box ^k\) single-scalar theories with \(\mathbb {Z}_2\) symmetric critical interactions using standard perturbative RG methods and extracted critical information by performing \(\epsilon \) expansion below an upper critical dimension \(d_c = 2nk/(n-1)\). This upper critical dimension is fixed by the requirement that \(\phi ^{2n}\) be a marginal operator. We have divided such theories into two classes which we have referred to as *first* and *second* type theories. The first class consists of theories in which the numbers *k* and \(n-1\) are relatively prime. These theories admit a generalized Wilson-Fisher fixed point with a single marginal interaction \(\phi ^{2n}\) as in [7, 8]. In the second class *k* and \(n-1\) have a common divisor. The phase structure of these theories is more involved compared to that of the first type theories and is characterized by the presence of fixed points with derivative interactions. Contrary to theories of the first type there are no fixed points with pure potential interactions.

In this work we present the details of our approach which relies on functional perturbative RG [14, 15, 16, 29] but give some new results as well. In particular we extend the beta function of the potential for \(\Box ^2\) theories of the second type, that was reported in [22] at quadratic level, to cubic order in the couplings. This is necessary for the calculation of OPE coefficients beyond free theory for non derivative operators. For this purpose we need not only the counter-terms cubic in the potential but also those with one power of *Z* couplings that parameterize two-derivative operators. The Euclidean higher-derivative scalar theories that we consider here are free of an explicit scale and therefore do not suffer from inconsistencies discussed in [30].

The paper is organized as follows. In the first part of Sect. 2 we present the general setup and basic definitions. The ordering of couplings/operators according to their canonical dimension in \(\Box ^k\) theories is different from that of standard theories. We devote a separate subsection to clarifying this issue and determining couplings of the same dimension that can possibly mix together. In the last part of this section we resort to dimensional analysis to constrain as much as possible the structure of beta functionals before getting into perturbative loop calculations. In Sect. 3 we analyze theories of the first type and report the cubic beta functional of the potential as well as the beta functional of *Z* at quadratic order. We obtain critical data such as the field and coupling anomalous dimensions and OPE coefficients in terms of \(\epsilon \) in the neighbourhood of the generalized Wilson-Fisher fixed point and compare with the literature when possible. Section 4 discusses \(\Box ^2\) theories of the second type which correspond to odd values of *n*. The beta functional of the potential as well as the two and four-derivative couplings are computed at quadratic level in terms of *V* and *Z*. We describe the phase diagram of these theories and present the field anomalous dimension and some critical exponents in the vicinity of the infrared fixed point. Finally, for the particular case of \(n=3\) we extend the beta function of the potential in Sect. 5 by including cubic corrections that are required for the computation of OPE coefficients at order \(\epsilon \). We then conclude in Sect. 6 and devote several appendices to details of the computations.

## 2 General \(\Box ^k\) scalar theories

### 2.1 Setup and definitions

*k*is a positive integer. We will calculate critical properties of such deformations using the renormalization group in dimensional regularization and \(\overline{\mathrm {MS}}\) scheme and perform \(\epsilon \)-expansion in \(d=d_c-\epsilon \), that is below a critical dimension \(d_c\) which will be determined shortly. The canonical dimension of the field is

*d*-dimensional Dirac delta function at the spacetime point

*x*

*x*dependence of the propagator is fixed by translation and scale invariance. The solution to this equation in coordinate space is

*r*propagators

*n*the value of

*r*at which the first pole occurs. This satisfies \((n-1)\delta _c = k\). Then \(\delta _c\) and the space dimension at criticality \(d_c\) will depend on

*k*,

*n*in the following way

*n*exists is equivalent to the fact that leading quantum corrections induced by non-derivative couplings are quadratic. This follows from the fact that, apart from numerical factors and couplings, (2.6) shows the expression for a diagram with two non-derivative vertices. The upper critical dimension \(d_c\) can also be fixed by the requirement that \(\phi ^{2n}\) be a marginal operator. However, as we will see, this is not necessarily the only marginal interaction.

### 2.2 General pattern of coupling mixing

The pattern of coupling mixing in \(\Box ^k\) theories is more involved than in the standard case with a 2-derivative kinetic term. Before getting into the perturbative calculations it is therefore very useful to sort the operators according to their canonical dimension and determine those of the same dimension which can potentially mix together.

*k*must be a multiple of \(\ell _*\), because \(\Delta \ell = k\) admits a solution, and that is \(\sigma _*=2(n-1)\). So the set of operators is partitioned into equivalence classes, where two operators belong to the same equivalence class iff their number of derivatives differ by multiples of \(\ell _*\). Couplings of operators belonging to different equivalence classes do not mix together. Therefore, schematically, and without distinguishing different operators of the same dimension and the same number of derivatives, the mixing of operator couplings is described by the following table where each row is understood to be multiplied by the operator on the right and collects operators with the same number of derivatives. Operators belonging to the same column have the same dimension and their couplings can therefore potentially mix together. There are \(\ell _*\) of such tables corresponding to \(\ell =0,1,\dots , \ell _*-1\). For different \(\ell \) belonging to this set the couplings of operators in the tables do not mix together, and altogether the \(\ell _*\) tables describe the mixing of all possible couplings. For each \(\ell _*\), Table (2.11) sorts operators according to their canonical dimensions. The table for \(\ell _* =0\), which includes non-derivative operators in its first row, is shown as

Unlike the standard case of \(k=1\) where couplings with positive dimension (relevant) correspond to operators with no derivatives and do not mix with other couplings, for \(k>1\) this is not always the case.

*k*and \(n-1\) have a non-trivial common divisor.

*k*and \(n-1\) are relatively prime numbers. In this case if

*k*is also an odd number it is clear that no intermediate values of \(\sigma \) solve (2.13). The \(\ell _*=0\) pattern of mixing, which includes non-derivative couplings as well, in this case is given by the following table If instead

*k*is even then \(\sigma =n-1\) will also solve the equation and this corresponds to operators of the form \(\partial ^k\phi ^{n+1}\). In principle, such operators can therefore be present in a scale invariant deformation of (2.1). However if

*k*is even and

*k*and \(n-1\) are relatively prime then \(n+1\) is an odd number and the operators \(\partial ^k\phi ^{n+1}\) are \(\mathbb {Z}_2\) odd. Therefore it is consistent to set their couplings to zero and consider only deformations of the form \(\phi ^{2n}\). The \(\ell _*=0\) table describing the mixing pattern in this case is

*k*and \(n-1\) have a common divisor. In this case it is easily seen that intermediate values of

*m*that solve (2.13) always exist, and among them \(\mathbb {Z}_2\) even operators are always present. Consequently one cannot a priori omit such operators. In this case a truncation of the action to non-derivative interactions is inconsistent. For the special case of \(k=2\),

*n*must be an odd number \(n=2m+1\). The \(\ell _*=0\) mixing pattern in this case is given as Based on this reasoning we distinguish theories of the

*first type*where

*k*and \(n-1\) are relatively prime from those of the

*second type*where

*k*and \(n-1\) have a common divisor. The two type of theories have qualitatively different features. In the next section we will study each type in turn, but before getting into perturbative RG calculations let us see what we can infer just from dimensional analysis about possible terms that can appear in the beta functions.

### 2.3 Structure of beta functions

In order to perform a RG analysis of scalar \(\Box ^k\) theories, instead of dealing with a finite truncation of the renormalized action, following [14, 15, 16, 29] we find it more convenient to adopt a functional framework in which at a given order in a derivative expansion infinitely many couplings are collected into a finite number of functions that parameterize operators of the given derivative order (see e.g [15] for further details).

One of the advantages of using the functional framework is that it allows us to extract many critical quantities with a single computation. Apart from that the functional approach gives more insight into the structure of the flow equations as it shows how the beta functions of many couplings are related to each other. In fact dimensional analysis alone fixes the structure of the beta functions to a high extent. To see this explicitly suppose that we are interested in the beta functional of couplings corresponding to a 2*a*-derivative operator, which we call \(\beta _a\). In the functional approach such couplings are collected into a function that we refer to as \(W_a(\phi )\). For \(a\ge 2\) there is more than one function of this sort as the basis of 2*a*-derivative operators has more than one element . However for the purpose of the argument of this section we do not need to distinguish such functions.

*N*vertices of which the

*i*th vertex represents interactions parameterized by \(W_{a_i}(\phi )\) with \(i=1,2,\dots , N\). In the most general case any two vertices are connected by a bunch of propagators which we refer to as an edge. Figures 2, 3, 6 and 7 provide examples of such diagrams with two and three vertices. The number of edges in the diagram is \(I\equiv N(N-1)/2\). If we regard each edge as a single internal line the resulting graph will have \(L\equiv I -N+1\) independent loops. Let us also denote the total number of propagators by

*R*. Now, a diagram gives rise to an \(\epsilon \) pole only if (omitting vertex couplings, external lines and their derivatives) it is dimensionless. Imposing this condition gives the following constraint

*a*vertex. For instance one (but not the only) possibility is that out of all the derivatives in the vertices a number \(2(\sum _ia_i-a)\) act on internal lines and 2

*a*remain external. The above equation can be put in a more useful form

*k*,

*n*and the total number of propagators in a diagram, this equation constrains the sum of derivatives of vertices that can contribute to the 2

*a*-derivative vertex. This will prove very useful in determining possible terms in a beta functional and what diagrams to compute. In this work we are interested in quadratic and cubic contributions in the couplings, that is \(N=2,3\).

## 3 Theories of the first type

*V*. To compute the functional betas it is convenient to adopt the background-field method where the the field \(\phi \) is shifted as \(\phi \rightarrow \phi +f\) and

*f*is considered as the quantum field. We will therefore only deal with vacuum diagrams with vertices that are functions of \(\phi \). As the first step we need to evaluate the quadratic and cubic counter-terms relevant for our purpose.

### 3.1 Counter-term diagrams

The counter-term diagrams for \(V(\phi )\) at quadratic and cubic order in the *V*-couplings are discussed in Appendix. A for theories with general *k*, *n*. Here we pick those for theories of the first type and extract their divergences. At quadratic level the only diagram that has a pole contributing to the potential is an \((n-1)\)-loop diagram given by

*n*in the superscript of \(V^{(n)}\) refers to the number of field derivatives of the potential. The divergent part of this diagram is already given in the appendix and is nothing but Eq. (A.2) for \(r=n\), which corresponds to \(l=0\). We call this \(V_{c.t.2}\)

*r*,

*s*,

*t*is positive and for compactness of the expression we have defined the following quantity

*n*propagators. We take \(r=n\) so the diagram will be

*n*propagators each

*n*propagators. In this case one of the vertices is the counter-term vertex \(U_{0,c.t.}\) discussed in (A.1) which is nothing but the quadratic counter-term for the potential \(V_{c.t.2}\)

### 3.2 Beta function of the potential

*r*,

*s*,

*t*subject to the conditions \(r+s+t=2n\) and \(r,s,t \ne n\), and the second sum runs over positive integers with a fixed sum \(s+t=n\). The anomalous dimension \(\eta \) is still unspecified at this point. This is fixed by our assumption that the field \(\phi \) is always in the canonically normalized form. Let us now calculate this quantity.

### 3.3 Fixed point and field anomalous dimension

The function \(V(\phi )\) induces a flow for the coefficient of the kinetic term as well. The counter-term appears at \(2(n-1)\) loops and comes from the diagram

### 3.4 Coupling anomalous dimensions

*g*. At this order the anomalous dimensions read

*k*and in agreement with [7, 8]. These anomalous dimensions vanish at order \(\epsilon \) unless \(i\ge n\). For instance the anomalous dimension \(\tilde{\gamma }_2\) is of order \(\epsilon ^2\) for \(n>2\). This can be obtained from the general formula (3.23). Notice that, independent of the value of

*k*, \(\phi ^2\) is always a relevant operator, and hence (3.23) will not be affected my mixing at any order in \(\epsilon \). For theories with \(n>2\) the second and third terms in the first line and the first terms in the brackets in the second and third lines vanish. Therefore \(\tilde{\gamma }_2\) takes the following form

### 3.5 OPE coefficients

In this section we compute what we call the \(\overline{\mathrm {MS}}\) OPE coefficients, which can be calculated by expanding the coupling beta functions around the fixed point and reading off the coefficient of the quadratic terms in the coupling deformations. We distinguish them from the standard OPE coefficients by a tilde \(\tilde{C}^l{}_{ij}\). These quantities which from now on we refer to as OPE coefficients for short depend on the renormalization scheme. However, the non zero OPE coefficients that we find with this method and using dimensional regularization are those that are dimensionless at \(\epsilon =0\) [15]. These OPE coefficients can be shown to be less sensitive to changes of scheme and in this sense universal (we refer the reader to section 2 of [15] for further details).

*l*is fixed by the universality condition for \(\tilde{C}^l_{\;\,ij}\)

*k*. These OPE coefficients have been extracted from the cubic beta functional after the rescaling (3.15). As pointed out earlier, this rescaling affects the OPE coefficients by a global overall factor and has been fixed such that the OPE coefficients are consistent with the CFT normalization. This is done by requiring any of the leading terms that is independent of \(\epsilon \) in (3.27) to match the value found from simple Wick counting.

### 3.6 Beta function of two-derivative couplings

*Z*affects the flow of the potential starting from cubic terms in the flow and leaves the quadratic flow unaltered. Let us calculate the contribution of the potential and the wavefunction

*Z*to the flow of

*Z*at quadratic level. The contribution that is quadratic in the potential is present only for the case \(k=1\). This is seen from the argument of Sect. A.1 and the expression for the counter-term can be found from Eq. (A.2) for \(l=1\) and \(r=2n-1\). This comes from the diagram in Fig. 2 but applied to the standard case with \(k=1\). The contribution to the beta function is found by multiplying the coefficient of \((\partial \phi )^2\) by \(2(n-1)\epsilon \)

*V*and

*Z*couplings. The relevant diagram is

*Z*-vertex there are three possibilities depending on whether the propagators are connected to one or both fields in \((\partial \phi )^2\) or only to fields in \(Z(\phi )\). The counter-term corresponding to this diagram is given as

*i*because the stability matrix is lower triangular. A simple calculation gives

## 4 Theories of the second type: the \(k=2\), \(n=2m+1\) example

*k*and \(n-1\) are relatively prime. We will now turn our attention to theories of the second type in which

*k*and \(n-1\) have a common divisor. As argued in Sect. 2.2 these theories are considerably more involved. Instead of complicating the calculations, as a first encounter with such theories we find it more instructive to concentrate on the special case of \(k=2\) which already shows the novel features of second-type theories. For the \(k=2\) case, theories with even values of

*n*fall in the first class already discussed. We therefore consider here \(\Box ^2\) theories with \(n=2m+1\), where

*m*is a natural number. For such theories, apart from the kinetic operator there are two marginal operators \(\phi ^{2(2m+1)}\) and \(\phi ^{2m}(\partial \phi )^2\) at the critical dimension. According to arguments of Sect. A potential approximation is therefore inconsistent as a lowest-order derivative expansion and one has to take into account 2-derivative interactions as well. Therefore, we consider the Lagrangian

*m*in terms of which the field dimension at criticality and the upper critical dimension are

*V*and

*Z*for general

*m*. For this purpose we need the quadratic counter-terms.

### 4.1 Counter-terms for *V* of the form \(V^{(a)}\,^2\)

Let us first consider counter-terms that are quadratic in the *V* couplings. According to the arguments of Sect. A.1 these can contribute at any order in the number of derivatives. In fact Eq. (A.3) shows that the diagram of Fig. 2 with \(r = (2+l)m+1\) contribute to operators with 2*l* derivatives. Here we are interested in the counter-term \(U_{0,c.t.}\) which contribute to the potential *V*. We thus need to choose \(l=0\). This gives a melon diagram with \(r=2m+1\) propagators

### 4.2 Counter-terms for *Z* of the form \(V^{(a)}\,^2\)

The contribution to the function *Z* comes from (A.2) with \(l=1\). This corresponds to \(r=3m+1\) giving rise to the diagram

### 4.3 Counter-terms for *V* of the form \(V^{(a)}Z^{(b)}\)

Let us now consider quadratic contributions to the potential counter-term with a *V*-coupling and a *Z*-coupling. The relevant diagram is a melon diagram with \(m+1\) propagators.

*r*number of propagators is found in Sect. B. The diagram of relevance here is evaluated by setting \(r=m+1\) in Eq. (B.4). The first two terms in this equation involve \(G^{m+1}_{xy}\) which is finite for \(k=2\) and \(n=2m+1\). We are therefore left with the last term, the pole of which can be easily calculated. This gives

### 4.4 Counter-terms for *Z* of the form \(V^{(a)}Z^{(b)}\)

A similar diagram but with \(2m+1\) propagators contribute to the wavefunction *Z*

### 4.5 Counter-terms for *Z* of the form \(Z^{(a)}Z^{(b)}\)

There is also a counter-term that is quadratic in *Z*. This contributes only to *Z*. The diagram is an *m*-loop melon diagram

*r*in (C.1). To complete the calculation we need to know the divergences of the combinations of propagators and their derivatives in (C.1) for \(r=m+1\). The quantity \(G^{m+1}\) is finite so the first three terms do not contribute. For the last three terms we need to extract the divergences of (C.5), (C.6), (C.7) and (B.4). Setting \(r=m+1\) these are found to be

*Z*

### 4.6 Beta functions at quadratic order

*V*and

*Z*at quadratic order

*Z*. Let us move to dimensionless variables

*v*is compatible with the

*v*-rescaling introduced earlier in (3.15) for the first-type theories, which is made to match the CFT normalization. For the dimensionless variables the beta functions reduce to

### 4.7 Fixed points

*v*,

*z*functions as

*m*.

### 4.8 Spectrum of couplings

*i*th element is given, for \(i=1,\dots , 2m+1\), as

*g*is present only in the last component \(i=2m+1\). The anomalous dimension for the relevant coupling \(g_i\) is therefore

*i*th block of the matrix \({\mathcal {M}}_0^{(2)}\) governs the \(\phi ^i - \phi ^{i-2m-2}(\partial \phi )^2\) mixing, i.e. the mixing between the couplings \(g_i\) and \(h_{i-2m-2}\), for \(i=2m+2,\dots , 4m+1\). These two by two matrices are given by

*h*coupling, which corresponds to a two-derivative operator. Then, from (2.18) it is seen that in a quadratic term in the beta function of \(g_i\), the coupling

*h*can be coupled only to non-derivative couplings \(g_j\), and in a quadratic term in the beta function of \(h_i\), the coupling

*h*can be coupled either to non-derivative couplings \(g_j\) or to two-derivative couplings \(h_j\). Therefore the stability matrix governing the mixing in the \(g_i\)-\(h_j\) sector is lower triangular. The above argument is in fact general: If we collectively denote the coupling of 2

*l*-derivative operators by \(c_l\), then the beta function \(\beta _{c_i}\) can include a quadratic term of the form \(h c_j\) only for \(j\le i\). This constraint on

*i*,

*j*indeed follows from (2.18) which in this case translates to

#### 4.8.1 Some scaling relations

*S*as \(\beta _S\)

*x*, it follows from the definition (4.31) that the beta function of

*s*is related to \(\beta _S\) through

*s*and its field derivatives, but there is also an explicit \(\varphi \) dependence in (4.33). Let us denote the scaling solution by \(s_*(\varphi )\) so that \(\beta _{s_*}=0\). Taking the field derivative of the fixed point equation \(\beta _{s_*}=0\) we find

*g*,

*h*are understood to take their fixed point values. This is consistent with the eigenvalue \(\theta _r\) in (4.35). For the particular fixed point (4.21) with \(g=0\), this eigenvalue equation reduces to \(\tilde{\omega }_{2m-1} = 2m\epsilon \) which can be verified independently using the general formula (4.29).

### 4.9 Counter-terms of four-derivative operators

*r*(to be determined shortly) give the complete counter-terms, quadratic in the potential and the wavefunction, for the four-derivative operators. As a byproduct we can find apart from the field anomalous dimension the quadratic flows of all four derivative couplings. These are introduced into the Lagrangian as

*V*,

*Z*.

#### 4.9.1 Counter-terms of the form \(V^{(a)}\,^2\)

Let us start with the quadratic contributions in the potential. This is found immediately from (A.2) by setting \(l=2\) and hence \(r=4m+1\). The relevant diagram is

*m*-loops. After an integration by parts in \(U_{2,c.t.}\) this evaluates to

#### 4.9.2 Counter-terms of the form \(V^{(a)}Z^{(b)}\)

The contributions of the form \(V^{(a)}Z^{(b)}\) come from the diagram of Fig. 6 with \(r=3m+1\)

*r*can easily be found from the constraint relation (2.18) for \(N=2\), \(a_1+a_2=1\), and \(a=2\). This 3

*m*-loop diagram is directly calculated from (B.2) by setting \(r=3m+1\)

#### 4.9.3 Counter-terms of the form \(Z^{(a)}Z^{(b)}\)

Counter-terms quadratic in *Z* are also present for \(W_i\). These come from the 2*m*-loop diagram

### 4.10 Beta functions of four-derivative couplings and \(\gamma _\phi \)

## 5 The \(k=2\), \(n=3\) example: including higher order corrections

The quadratic beta functions calculated in the previous section have been used to obtain the field and coupling anomalous dimensions at leading order in \(\epsilon \). For the OPE coefficients instead they are capable of giving only the free theory value which can be obtained by Wick counting. If we wish to extend this and find some OPE coefficients beyond free theory we need some cubic corrections to the beta functions as well. For instance if we are interested in computing the OPE coefficients of non-derivative operators we must find cubic corrections to \(\beta _V\) that are at least quadratic in the potential. These include corrections of the form \(V^{(a)}V^{(b)}V^{(c)}\) which are cubic in the potential. However as we saw earlier the function *Z* is non zero at the scale invariant point so corrections of the form \(V^{(a)}V^{(b)}Z^{(c)}\) will also contribute to such OPE coefficients. In this section we extend the results of the previous section by calculating cubic corrections to the beta function of the potential that are of the two forms mentioned above. Although straightforward to extend to general values of *m*, for this calculation we find it more instructive to restrict ourselves to the special case of \(m=1\).

### 5.1 Counter-terms of the form \(V^{(a)}V^{(b)}V^{(c)}\)

Here we present the diagrams that contribute to cubic terms of the form \(V^{(a)}V^{(b)}V^{(c)}\) in \(\beta _V\). Each diagram is followed by the expression for its UV divergence. We do not enter into the details of the calculations here and instead as an example we provide in the appendix the computational steps for one of the diagrams. For cubic contributions to the counter-term of the potential the number of propagators in the relevant diagram is \(2n=6\). There are three triangle diagrams with six propagators which we will now consider. The first diagram has two propagators in each edge. This falls in the first class described in App. A.2 for which non of the edges is divergent and therefore it has a simple pole in \(\epsilon \). It can be evaluated using equations (A.4) and (A.5)

*y*,

*z*in the Mellin–Barnes representation (A.4) that give rise to divergences in this case are \((y,z)=(0,0)\). The next triangle diagram has an edge with three propagators. This leads to a subdivergence and a double \(\epsilon \)-pole. Again the relevant poles in the Mellin–Barnes integral are \((y,z)=(0,0)\) which give

### 5.2 Counter-terms of the form \(V^{(a)}V^{(b)}Z^{(c)}\)

Apart from cubic corrections in the potential it is necessary for the computation of OPE coefficients to obtain also cubic corrections of the form \(V^{(a)}V^{(b)}Z^{(c)}\) in \(\beta _V\). This is because for \(\Box ^2\) theories of the second type the function *Z* does not vanish at the fixed point and therefore these terms can also contribute to the OPE coefficients. Here we present the diagrams leading to such terms along with the expression for their UV divergence. Computational details for a sample diagram are presented in the Appendix D. To extract the divergences we continue using the Mellin–Barnes representation. Compared to the cubic case in the potential the computation is more involved as the function *Z* can appear in three different ways in each vertex. In order not to increase the number of diagrams these are not distinguished here. Furthermore for a diagram with given topology the function *Z* can appear on different vertices which leads to different diagrams. By dimensional analysis a diagram with two *V*-vertices and a *Z*-vertex contributes to the flow of the potential if its number of propagators is \(2n-1=5\). There are four triangle diagrams of this sort, which come in two different topologies. In the first diagram the *Z* function appears at a four-vertex. In this case it turns out that only the cases in which both derivatives at the *Z*-vertex act on the propagator contribute to the divergence. The results is

*Z*function appears at a three-vertex. Similar to the previous case, the divergence receives contributions only when derivatives at the

*Z*-vertex act on the propagators. We get

*Z*appears on the two-vertex while in the other it appears on the four-vertex. In both cases the diagram has a non-local divergence. In the first case the pole term is

*Z*can appear in either of the three vertices, hence there are three diagrams of this type altogether. When the

*Z*function appears on the two-vertex the pole terms is given as

*Z*can also appear on the five-vertex which leads to the \(\epsilon \) pole

*Z*function then the pole term will be

*Z*vertex. The counter-term vertex will then have to be quadratic in the potential. Apart from the case with a single propagator just considered, To provide a counter-term for the potential the only possibility is that the diagram have two propagators and there be no derivatives on the counter-term vertex. The counter-term quadratic in

*V*and with no derivatives is given in (5.6). The pole term for such a diagram is then given as

*V*vertex. Its quadratic counter-term vertex will then be of the form \(V^{(a)}Z^{(b)}\). Apart from the trivial single propagator case considered above which can be omitted there are two other cases. The first has two propagators and a counter-term vertex with two derivatives. The counter-term vertex can be obtained from Eq. (4.6) setting \(m=1\)

*V*, we expect here that no non-local term be present in the final result. Although this is indeed the case, manipulating these terms a bit shows that the situation is slightly more involved in this case. In fact the non-local terms of the \(V^2Z\) diagrams do not cancel out among the diagrams but sum up to local terms instead. The cancellation of terms proportional to \(\gamma \) can be verified here as well, similar to the \(V^3\) case. Summing up different contributions and simplifying, the total counter-term of the form \(V^2Z\) reads

### 5.3 Beta functions

*V*and

*Z*at quadratic level for general values of

*m*. Here we only need the \(m=1\) results. For \(m=1\) the quadratic counter-terms are

*V*. This is related to the quadratic and cubic counter-terms \(V_{c.t.2}\) and \(V_{c.t.2}\) as

*Z*. After the rescalings (4.16) the beta function reads

*V*we need the quadratic flow of

*Z*, given in the last term of (5.29), and this is crucial for the cancellation of \(\epsilon \) poles in the beta function (5.28). This is a general feature of second type theories, that the potential contributions to the flow of the potential is affected by the flow of the couplings of higher derivative operators. For completeness we report here also the \(m=1\) version of the quadratic flows of

*Z*and \(W_i\) functions after the rescaling (4.16). For the

*Z*function this is

### 5.4 Critical data

*m*the beta functions of

*V*and

*Z*at quadratic level and obtained the field and coupling anomalous dimensions in terms of the couplings and at the non-trivial IR fixed point also in terms of \(\epsilon \). Here for the particular case of \(m=1\) we first present the anomalous dimensions for all three non-trivial fixed points in terms of \(\epsilon \), which requires (5.32) and (5.33) and only the quadratic terms in the beta function (5.31). Next we take advantage of the cubic terms in (5.31) to extract some OPE coefficients as well. For \(m=1\) the general leading order anomalous dimensions given in Sects. 4.8 and 4.9 reduce to

*g*and

*h*. Instead of giving the complete formula which is slightly involved, for simplicity we report here the result only for the infrared fixed point with \(g=0\)

*i*,

*j*and

*h*is understood to take its fixed point value \(3\epsilon /8\). Also the dimensionless condition fixes \(l=i+j-6\). It may still be useful to give a few explicit examples for the OPE coefficients in terms of \(\epsilon \) at the infrared fixed point

## 6 Conclusions

In this work we have shown from one side that renormalization group techniques are still fundamental tools to investigate novel critical properties of QFTs, which in recent years has been addressed with alternative powerful CFT techniques, either analytical or numerical. On the other hand, agreement of CFT results (for instance [7, 8, 22]) with our RG analysis, which relies neither on unitarity nor conformal symmetry, can provide further evidence for the enhancement of scale invariance to conformal invariance for nonunitary single-scalar higher dimensional theories with higher derivative kinetic terms investigated here. Indeed we show how some non trivial results obtained both with RG and CFT techniques, such as scaling dimensions and a class of OPE coefficients, match in the first non trivial order in \(\epsilon \) expansion.

We have taken advantage of the functional form of the flow equations to extract critical information and give compact formulas encompassing (sometimes infinitely) many quantities. The general pattern of coupling mixing is presented which is different from the standard case and depends on the number of derivatives in the kinetic term and the space dimension, or equivalently *k*, *n*. We have given a constraint formula that is a result of dimensional analysis and determines to a high extent the structure of terms, i.e combination of functions and their field derivatives, that appear in the beta functionals. Based on the possible marginal operators we have distinguished two types of theories and argued that they have qualitatively different features.

Theories of the first type have a generalized Wilson-Fisher fixed point. We have extracted critical information around this fixed point and shown that when available from both approaches our results for these theories agree with those of [7, 8], which is based on the structure of conformal blocks in a CFT framework. Beyond known results we give the next to leading order values for the coupling anomalous dimensions corresponding to the relevant and marginal non-derivative operators and the leading order anomalous dimensions for the couplings of all two-derivative operators. We also give an infinite set of OPE coefficients at order \(\epsilon \).

The phase diagram for second-type theories has a richer structure. We have analyzed those with a \(\Box ^2\) kinetic term which correspond to odd values of \(n=2m+1\) and discovered that apart from the Gaussian fixed point there are three nontrivial fixed points all of which include derivative interactions. In particular there is a fixed point with a pure derivative interaction that is infrared attractive. We have computed the quadratic flow, in terms of *V* and *Z*, of couplings corresponding to operators up to four derivatives for general *m*. The field anomalous dimension and some critical exponents for the potential and *Z* couplings are given at leading order in terms of the fixed point couplings, and for the infrared fixed point in terms of \(\epsilon \) in which case the range of validity extends to all *V*, *Z* couplings.

In a previous letter [22], where some of the results of this paper were anticipated, we have shown how constraints from conformal symmetry together with a knowledge of the Schwinger-Dyson equations, can be used to obtain some of these results, providing in this way indications, at least at the leading order \(\epsilon \)-expansion, that scale invariance could be enhanced to conformal invariance, also for second-type theories. A more comprehensive investigation along these lines is in progress [31].

For the special case of \(\Box ^2\) theories in \(6-\epsilon \) dimensions the beta functional of the potential is extended to cubic order. This enables us to compute some OPE coefficients of non-derivative operators at order \(\epsilon \). For this purpose we need not only cubic contributions in the potential of the form \(V^{(a)}V^{(b)}V^{(c)}\) but also cubic terms linear in the *Z* coupling, i.e. contributions of the form \(V^{(a)}V^{(b)}Z^{(c)}\).

Several novel features appear in this computation. For instance the computation of the pure potential contribution to the flow of *V* at cubic level requires the quadratic flow of *Z*. This is another indication of the fact that derivative couplings cannot be ignored in theories of the second type even at leading order in derivative expansion. Another particular feature is that nonlocal ultraviolet divergences show up in separate diagrams that finally combine to a local term in physical amplitudes. This signals the fact that it would be too naive to simply drop the nonlocal divergences in the diagrams.

A higher derivative theory of the first type may be considered as a possible construction to describe the isotropic phase of Lifshitz critical theories which is relevant for physics of certain polymers. It is not yet clear if multicritical theories of the novel second type can play a role in understanding new phases. We also note that these theories and in particular the second type ones have critical points, characterized by a finite dimensional UV critical surface, spanned by the UV attractive directions, and may be seen as a non unitary realization of asymptotically safe scalar theories [1, 32, 33, 34]. \(\Box ^2\) theories are non trivial in dimensions less than 6 but it is likely that also critical theories with higher number of derivatives (\(k > 2\)) demonstrate at least in some cases a similar behavior.

The results of this work can be extended in several directions. We have concentrated here on \(\Box ^k\) theories with \(\mathbb {Z}_2\)-even critical interactions. It would be interesting to extend these results to the case where the scale invariant theory includes \(\mathbb {Z}_2\)-odd marginal operators. Another line of investigation would be including multiple scalar fields and studying various global symmetries [29], or possibly considering fields of higher spin such as \(\frac{1}{2}\) and 1. Theories of the second type in particular have not been studied with other methods. They therefore provide a setting to apply and test alternative approaches that do not rely on RG, especially those that are solely based on conformal symmetry, which may possibly also have the potential to improve these findings.

## Notes

### Acknowledgements

We would like to thank A. Petkou for discussions.

## Supplementary material

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