Conformal Invariance and Vector Operators in the O(N) Model

Abstract

It is widely expected that, for a large class of models, scale invariance implies conformal invariance. A sufficient condition for this to happen is that there exists no integrated vector operator, invariant under all internal symmetries of the model, with scaling dimension $-1$. In this article, we compute the scaling dimensions of vector operators with lowest dimensions in the O(N) model. We use three different approximation schemes: $\epsilon$ expansion, large N limit and third order of the derivative expansion of Non-Perturbative Renormalization Group equations. We find that the scaling dimensions of all considered integrated vector operators are always much larger than $-1$. This strongly supports the existence of conformal invariance in this model. For the Ising model, an argument based on correlation functions inequalities was derived, which yields a lower bound for the scaling dimension of the vector perturbations. We generalize this proof to the case of the O(N) model with $N\in \left\{ 2,3,4 \right\}$.

Appendices

Redundant Operators

This Appendix is devoted to a discussion of redundant vector operators, with an emphasis on conformal invariance. Redundant operators are of the form $(\delta S/\delta \phi (x)) {\mathcal {O}}(x)$. They are sometimes considered to be physically uninteresting. Indeed, they typically have short-range correlation functions.¹⁵

1.1 Redundant Operators and Breaking of Conformal Invariance

The aim of this section is to explain why redundant operators can be ignored as possible breakings of conformal invariance. On general grounds, a redundant operator, with short-range correlations can be responsible for the breaking of Ward identities of a typical symmetry. The mere existence of such an operator would have strong physical consequences because correlation functions for other fields would not display the corresponding invariance. We illustrate this in a very simple situation. Consider a model with two scalar fields $\phi _1$ and $\phi _2$ whose dynamics is given by a general action S which needs not be O(2)-symmetric. If we perform, in the path integral of the partition function, a change of variable $\phi _i\rightarrow \phi _i+\theta \epsilon _{ij}\phi _j$ (here $\epsilon _{ij}$ is the bidimensional Levi–Civita tensor and $\theta$ an infinitesimal angle) which corresponds to an infinitesimal rotation in internal space, we obtain:

$$\begin{aligned} \int d^dx\left( \epsilon _{ij}J_i\frac{\delta W}{\delta J_j}\right) =\int d^dx\left\langle \epsilon _{ij}\phi _i \frac{\delta S}{\delta \phi _j}\right\rangle . \end{aligned}$$

(55)

The brackets in the right-hand-side represent an average over the fields with the Boltzmann distribution in presence of sources $J_i$ for the fields $\phi _i$. Of course, if the action is O(2) symmetric, we recover the Ward identity for rotation in internal space. However, for a generic action S,  the right-hand-side does not vanish and the O(2) Ward identity is not satisfied. Now, what is of interest for us here is that the right-hand-side of the previous equation is the average of a redundant operator. The operator $\epsilon _{ij}\phi _i \frac{\delta S}{\delta \phi _j}$ appearing in the right-hand-side of Eq. (55), which has only contact terms in its correlation functions, is physically important because it induces a breaking of O(2) invariance, at the level of Ward identities.

To make an analogy with the strategy followed in this article to study conformal invariance, suppose we want to prove that a model is invariant under O(2) by searching for putative operators that could appear in the right-hand-side of Eq. (55). Suppose that we can discard the existence of such operators which are not redundant but that we have no control on redundant ones. Then, the previous example shows that we have no way to conclude on the O(2) invariance of the theory. If, instead, we can discard both non-redundant and redundant operators, then we conclude that the theory is indeed invariant.¹⁶

The situation is however different in the case of conformal invariance. Assume indeed that we find a redundant, integrated vector operator of dimension $-1$. Such an operator would be of the form¹⁷$\int d^dx (\delta {\varGamma }_k)/(\delta \phi (x)){\mathcal {O}}_\mu (x)$ where the operator ${\mathcal {O}}_\mu$ depends on x only through the field argument (as explained in Sect. 3, an explicit x-dependence would be inconsistent with the translational invariance of the operator ${\varSigma }_\mu$). This operator would yield a potential violation of the Ward identity:

$$\begin{aligned}&\int _{x} (K_{\mu }^{x}-2D_\star ^{\phi }x_{\mu }) \phi (x)\frac{\delta {\varGamma }_{k}}{\delta \phi (x)}- \frac{1}{2}\int _{x,y}\partial _{t}R_{k}(|x-y|)(x_{\mu }+y_{\mu })G_{k}(x;y) \nonumber \\&\quad =\int _x \frac{\delta {\varGamma }_k}{\delta \phi (x)}{\mathcal {O}}_\mu (x). \end{aligned}$$

(56)

The right-hand-side could actually be reabsorbed in a modification of the conformal transformation of the field $\phi$ [74]. At odds with the case of internal symmetries, the modified conformal transformation $\phi (x)\rightarrow \phi (x)+\epsilon _\mu [(K_{\mu }^{x}-2D_\star ^{\phi }x_{\mu })\phi (x)-{\mathcal {O}}_\mu (x)]$ is nontrivial because the x-independent term ${\mathcal {O}}_\mu$ cannot compensate the usual variations and the bracket is therefore non zero.

It can be shown that the modified conformal transformation, together with the usual translation, rotation, and scale transformation satisfy the conformal algebra.

To conclude, the existence of a putative redundant, integrated vector operator of dimension $-1$ would not lead to a breaking of conformal invariance but, instead, to a modification of the transformation of the field.

1.2 Exact Scaling Dimensions of Some Redundant Operators

In this section, we show that some redundant operators have simple scaling dimensions. We work in the framework of NPRG described in Sect. 2. We can choose the Hamiltonian (or action) to be of the Ginzburg–Landau type:

$$\begin{aligned} S[\phi ]=\int _x \frac{1}{2}(\nabla \phi )^2+\frac{1}{2}r_{\varLambda }\phi ^2+\frac{u_{\varLambda }}{4!}\phi ^4, \end{aligned}$$

(57)

where $\int _x=\int d^d x$. In order to determine the scaling dimension of an operator, we study the evolution of the corresponding coupling under the RG flow in the vicinity of the fixed point. To this end, we add to the action a part which couples to a vector operator:

$$\begin{aligned} S_{\text{ V }}[\phi ]=\int _x \frac{a^\mu _{\varLambda }}{3!}\phi ^3\partial _\mu {\varDelta }\phi . \end{aligned}$$

(58)

Up to integrations by parts, this operator is the same as the one considered in [38]. Moreover, it has been proved to be the most relevant integrated vector operator invariant under ${\mathbb {Z}}_2$ symmetry near $d=4$ [30].

We perform an infinitesimal transformation of the integration variable: $\phi \rightarrow \phi -a^\mu _{\varLambda }/u_{\varLambda }\partial _\mu {\varDelta }\phi$ in the path integral appearing in Eq. (3). It is readily found that the quadratic pieces in the action, including the regulating term ${\varDelta }S_k,$ are invariant under this transformation. The variation of the quartic part of the action is found to compensate exactly $S_{\text{ V }}$. We thus find that

$$\begin{aligned} W_{k}\left[ J,a^\mu _{\varLambda }\right] =W_{k}\left[ J+\frac{a^\mu _{\varLambda }}{u_{\varLambda }}\partial _\mu {\varDelta }J,0\right] +{\mathcal {O}}\left( a^\mu _{\varLambda }a^\nu _{\varLambda }\right) . \end{aligned}$$

(59)

At the level of the effective average action, this relation implies

$$\begin{aligned} {\varGamma }_{k}\left[ \phi ,a^\mu _{\varLambda }\right] ={\varGamma }_k\left[ \phi +a^\mu _{\varLambda }\partial _\mu {\varDelta }\phi ,0\right] +{\mathcal {O}}\left( a^\mu _{\varLambda }a^\nu _{\varLambda }\right) . \end{aligned}$$

(60)

This last equation states that the evolution of the effective action with an infinitesimal $a^\mu _{\varLambda }$ is related to the effective action at vanishing $a^\mu _{\varLambda },$ up to a modification of the field. This can be used in the following way. Defining the running coupling constants $u_k$ and $a^\mu _k$ as the prefactors of, respectively, $\int _x \frac{1}{4!}\phi ^4$ and $\int _x \frac{1}{3!}\phi ^3\partial _\mu {\varDelta }\phi$ in ${\varGamma }_k,$ we obtain that $a^\mu _k/u_k$ is constant along the flow. To obtain the scaling dimension of the vector operator, we introduce dimensionless, renormalized quantities (denoted with tilde) as

$$\begin{aligned} {\tilde{x}}&=k x, \end{aligned}$$

(61)

$$\begin{aligned} {\tilde{\phi }}({\tilde{x}})&=k^{-(d-2)/2}Z_k^{1/2}\phi (x), \end{aligned}$$

(62)

where $Z_k$ scales as $k^{-\eta }$ at the Wilson–Fisher fixed point with $\eta$ the anomalous dimension. The renormalized coupling constants are thus:

$$\begin{aligned} {\tilde{u}}_k&=k^{d-4} Z_k^{-2} u_k, \end{aligned}$$

(63)

$$\begin{aligned} {\tilde{a}}_k^\mu&=k^{d-1} Z_k^{-2} a_k^\mu . \end{aligned}$$

(64)

At the critical point, ${\tilde{u}}$ flows to a fixed point value $u_\star$. Consequently, when $k\rightarrow 0,$

$$\begin{aligned} {\tilde{a}}_k^\mu \sim a_{\varLambda }^\mu \frac{u_\star }{u_{\varLambda }}k^3 \end{aligned}$$

(65)

which shows that the scaling dimension of $a^\mu$ is exactly 3.

The proof given above relies strongly on the particular microscopic action given in Eq. (57). This gives interesting non-universal information on the flow of the coupling $a_k^\mu ,$ but confers a preeminent role to the peculiar form of the Hamiltonian. To overcome this issue, we now present an alternative proof of the same result, but which is based on the exact flow equation (6) for the effective average action, expressed in terms of dimensionless, renormalized, fields:

$$\begin{aligned} \begin{aligned} \partial _t {\varGamma }_k[{\tilde{\phi }}]=&\int _{{\tilde{x}}} \frac{\delta {\varGamma }_k}{\delta {\tilde{\phi }}({\tilde{x}})}\left( {\tilde{x}}^\rho \partial _{{\tilde{x}}^\rho }-\alpha \right) {\tilde{\phi }}({\tilde{x}})+\frac{1}{2}\int _{{\tilde{x}}{\tilde{y}}}\partial _t{\tilde{R}}({\tilde{x}}-{\tilde{y}}){\tilde{P}}_k({\tilde{x}},{\tilde{y}}), \end{aligned} \end{aligned}$$

(66)

where $R_k( x)=Z_k k^{d+2}{\tilde{R}}(k x),$$\alpha =-(d-2+\eta )/2,$$t=\log (k/{\varLambda })$ and ${\tilde{P}}_k$ is the dimensionless, renormalized, propagator:

$$\begin{aligned} \int _{{\tilde{y}}} {\tilde{P}}_k({\tilde{x}},{\tilde{y}})\left[ \frac{\delta ^2{\varGamma }_k}{\delta {\tilde{\phi }}({\tilde{y}})\delta {\tilde{\phi }}({\tilde{z}})}+{\tilde{R}}({\tilde{y}}-{\tilde{z}})\right] =\delta ({\tilde{x}}-{\tilde{z}}). \end{aligned}$$

(67)

We now identify an exact eigenvector of the linearized flow. To this end, we add to the Wilson–Fisher fixed-point effective action ${\varGamma }_\star$ a small perturbation

$$\begin{aligned} {\varGamma }_k={\varGamma }_\star +{\tilde{r}}_\mu (t)\int _{ {\tilde{x}}} \frac{\delta {\varGamma }_\star }{\delta {\tilde{\phi }}({\tilde{x}})} \tilde{\partial }_\mu {\tilde{{\varDelta }}} {\tilde{\phi }}({\tilde{x}}) \end{aligned}$$

(68)

and we compute the flow of this functional at linear order in ${\tilde{r}}_\mu$.

$$\begin{aligned} \begin{aligned}&\partial _t{{\tilde{r}}}_\mu (t) \int _{{\tilde{x}}} \frac{\delta {\varGamma }_\star }{\delta {\tilde{\phi }}({\tilde{x}})} \tilde{\partial }_\mu \tilde{{\varDelta }} {\tilde{\phi }}({\tilde{x}}) \\&\quad = {\tilde{r}}_\mu (t)\int _{{\tilde{x}}{\tilde{y}}} \frac{\delta \ }{\delta {\tilde{\phi }}({\tilde{y}})}\left[ \frac{\delta {\varGamma }_\star }{\delta {\tilde{\phi }}({\tilde{x}})} {\tilde{\partial }}_\mu {\tilde{{\varDelta }}} {\tilde{\phi }}({\tilde{x}})\right] \left( {\tilde{y}}^\rho \partial _{{\tilde{y}}^\rho }-\alpha \right) {\tilde{\phi }}({\tilde{y}})\\&\qquad -\frac{1}{2} {\tilde{r}}_\mu (t)\int _{{\tilde{x}}{\tilde{y}} {\tilde{z}}{\tilde{t}}{\tilde{w}}}\partial _t{\tilde{R}} ({\tilde{x}}-{\tilde{y}}){\tilde{P}}_\star ({\tilde{y}}, {\tilde{z}}){\varGamma }_\star ^{(3)}({\tilde{z}},{\tilde{t}}, {\tilde{w}}){\tilde{P}}_\star ({\tilde{w}},{\tilde{x}}) {\tilde{\partial }}_\mu {\tilde{{\varDelta }}}{\tilde{\phi }}({\tilde{t}}). \end{aligned} \end{aligned}$$

(69)

On the other hand, if we derive the fixed point equation with respect to ${\tilde{\phi }}({\tilde{x}}),$ multiply by ${\tilde{\partial }}_\mu {\tilde{{\varDelta }}} {\tilde{\phi }}({\tilde{x}})$ and integrate over ${\tilde{x}},$ we get

$$\begin{aligned} \begin{aligned} 0&=\int _{{\tilde{x}}{\tilde{y}}}{\tilde{\partial }}_\mu {\tilde{{\varDelta }}} {\tilde{\phi }}({\tilde{x}})\frac{\delta \ }{\delta {\tilde{\phi }}({\tilde{x}})}\left[ \frac{\delta {\varGamma }_\star }{\delta {\tilde{\phi }}({\tilde{y}})} \left( {\tilde{y}}^\rho \partial _{{\tilde{y}}^\rho }-\alpha \right) {\tilde{\phi }} ({\tilde{y}})\right] \\&\quad -\frac{1}{2} \int _{{\tilde{x}}{\tilde{y}}{\tilde{z}}{\tilde{t}} {\tilde{w}}}\partial _t{\tilde{R}}({\tilde{x}}-{\tilde{y}}) {\tilde{P}}_\star ({\tilde{y}},{\tilde{z}}){\varGamma }_\star ^{(3)} ({\tilde{z}},{\tilde{t}},{\tilde{w}}){\tilde{P}}_\star ({\tilde{w}}, {\tilde{x}}){\tilde{\partial }}_\mu {\tilde{{\varDelta }}}{\tilde{\phi }}({\tilde{t}}). \end{aligned} \end{aligned}$$

(70)

Combining the two equations, we obtain:

$$\begin{aligned} \begin{aligned} \partial _t {{\tilde{r}}}_\mu (t) \int _{{\tilde{x}}} \frac{\delta {\varGamma }_\star }{\delta {\tilde{\phi }}({\tilde{x}})} {\tilde{\partial }}_\mu {\tilde{{\varDelta }}}^n {\tilde{\phi }}({\tilde{x}})= {\tilde{r}}_\mu (t)\int _{{\tilde{x}}{\tilde{y}}} \frac{\delta {\varGamma }_\star }{\delta {\tilde{\phi }}({\tilde{x}})} \left[ {\tilde{\partial }}_\mu {\tilde{{\varDelta }}} ,{\tilde{x}}^\rho \partial _{{\tilde{x}}^\rho }\right] {\tilde{\phi }}({\tilde{x}}). \end{aligned} \end{aligned}$$

(71)

The commutator is easily evaluated to be equal to $3{\tilde{\partial }}_\mu {\tilde{{\varDelta }}}$. From this we deduce that the small perturbation introduced in Eq. (68) is an exact eigenoperator of the flow around the fixed point, with eigenvalue 3. This is consistent with the result found in the one-loop calculation of [30].

We can generalize the previous result in different ways. First, we can change the power of the Laplacian in Eq. (68) from unity to a positive integer n. The main change appears at the level of Eq. (71), where the commutator is now $[{\tilde{\partial }}_\mu {\tilde{{\varDelta }}}^n ,{\tilde{x}}^\rho \partial _{{\tilde{x}}^\rho }]=(2n+1){\tilde{\partial }}_\mu {\tilde{{\varDelta }}}^n$. This implies that the associated eigenvector has dimension $2n+1$. As a check of this result, we have considered the vector eigenoperators compatible with the ${\mathbb {Z}}_2$ symmetry whose scaling dimensions are 5 in $d=4$ and we have computed their first correction in $\epsilon =4-d$. There are 4 (independent) such operators: 1 ($O_6^3$) with 6 powers of the field and 3 derivatives and 3 ($O_{4,i}^5$ with $i\in \{1,2,3\}$) with 4 powers of the field and 5 derivatives. A one-loop calculation shows that $O_6^3$ has scaling dimension $5-5\epsilon /3+{\mathcal {O}}(\epsilon ^2)$. The eigenvectors $O_{4,i}^5$ have dimensions $5+{\mathcal {O}}(\epsilon ^2),$$5-4\epsilon /9+{\mathcal {O}}(\epsilon ^2)$ and $5-2\epsilon /3+{\mathcal {O}}(\epsilon ^2)$. The eigenoperator with scaling dimension $5+{\mathcal {O}}(\epsilon ^2)$ is found to be $\int _{{\tilde{x}}} {\tilde{\phi }}^3 {\tilde{\partial }}_\mu {\tilde{{\varDelta }}}^2{\tilde{\phi }},$ in agreement with the general result mentioned above. Other relations can be obtained if we consider in Eq. (68) an odd number of derivatives, with Lorentz indices not necessarily contracted together.

The present result also generalizes to the long-range Ising model, where the interaction between spins is not limited to nearest neighbors but decay as a power-law:

$$\begin{aligned} H=-\sum _{i,j}J(i-j)S_i S_j, \end{aligned}$$

(72)

where $J(i-j)\sim |i-j|^{-d-\sigma }$ and $\sigma$ is the exponent characterizing the decrease of the interactions. When $0<\sigma <2-\eta ,$ the model still has an extensive free-energy but belongs to a different universality class than the local Ising model. The Ginzburg–Landau Hamiltonian is identical to the one given in Eq. (57) except that the quadratic part is now, in Fourier space,

$$\begin{aligned} \int \frac{d^dq}{(2\pi )^d}\phi (-q) q^{\sigma }\phi (q). \end{aligned}$$

(73)

It is easy to verify that all the present analysis still applies to this case. We have checked that the one-loop calculation around the upper critical dimension $d_c=2\sigma$ gives that the most relevant integrated vector operator has scaling dimension $3+{\mathcal {O}}(\epsilon ^2)$. This result is important because it justifies the use of the conformal bootstrap program in this model [75], at least near the upper critical dimension.

We can also generalize the result to other internal groups. For O(N) theories, an exact eigenoperator can be found by adding a common O(N) index on both the functional derivative and the field appearing in Eq. (68) and summing over this index. The associated eigenvalue is again 3 (or $2n+1,$ if we change the power of the Laplacian). In [30] [see also Eq. (34)], we computed the scaling dimensions of the two vector operators of lowest dimension in an expansion in $\epsilon$ and found $3+{\mathcal {O}}(\epsilon ^2)$ and $3-6\epsilon /(N+8)+{\mathcal {O}}(\epsilon ^2)$. This result is consistent with the nonrenormalization theorem proven here. Let us stress, however, that in the O(N) model the non renormalization theorem does not constraint the leading vector operator but the next-to-leading, as can be seen already at one-loop level [30].

For completeness, we give other exact eigenvectors which can be obtained following the same idea. These are, however, not invariant under the internal symmetries of the theory, involve even derivatives and are not relevant to our discussion on conformal invariance. A well-known example is the eigenvector associated with the external magnetic field: $\int _{ {\tilde{x}}} \frac{\delta {\varGamma }_\star }{\delta {\tilde{\phi }}({\tilde{x}})}$ with eigenvalue $\alpha =-(d-2+\eta )/2$. Another exact eigenvector, valid for the O(N) model is $\int _{ {\tilde{x}}} \epsilon ^{ab}\frac{\delta {\varGamma }_\star }{\delta {\tilde{\phi }}^b({\tilde{x}})} {\tilde{{\varDelta }}}^n {\tilde{\phi }}^a({\tilde{x}})$ where $\epsilon ^{ab}$ is antisymmetric, which is associated with the eigenvalue 2n.

It was shown in [38] that there exists other redundant operators in $d=4-\epsilon$. In fact, all local operators of dimension up to 9 are redundant modulo total derivatives. Equivalently, all integrated vector operators of dimension up to 5 are redundant. For several of these, we were however not able to determine analytically their scaling dimensions.

$\epsilon$-Expansion

In this Appendix, we describe the 1-loop calculation of the two lowest scaling dimensions of vector operators. We present here the calculation in the framework of the NPRG, but of course, the calculation could be performed within more standard approaches, such as the Minimal Substraction scheme.

We start by observing that at tree-level (zero loop) the effective action takes its bare form:

$$\begin{aligned} {\varGamma }^{(tree)}_{k}=S=\int _x\Big \{r \rho +\frac{1}{3!}u\rho ^2+ \frac{1}{4}a_{\mu }\partial _{\mu }\rho \partial _{\nu }\phi _{i}\partial _{\nu }\phi _{i} +\frac{1}{2}b_{\mu }\partial _{\mu }\phi _{i}\partial _{\nu }\phi _{i}\partial _{\nu }\rho \Big \}. \end{aligned}$$

(74)

Differentiating successively with respect to $\phi _{n_{i}}\left( x_{i} \right)$ and Fourier transforming, we obtain the form of the non-zero vertices:

$$\begin{aligned} {{\varGamma }_{i_{1}i_{2}}^{(2,tree)}}(p_1)&= \delta _{i_1i_2}(r+p_1^2) \end{aligned}$$

(75)

$$\begin{aligned} {{\varGamma }_{i_{1}i_{2}i_{3}i_{4}}^{(4,tree)}}(p_1,p_2,p_3)&= \frac{u}{3}\Big [\delta _{i_{1}i_{2}}\delta _{i_{3}i_{4}} +\delta _{i_{1}i_{3}}\delta _{i_{2}i_{4}}+\delta _{i_{1}i_{4}} \delta _{i_{2}i_{3}}\Big ] +i \frac{a_{\mu }-b_\mu }{2} \nonumber \\&\quad \times \Big [\left( p_{1}+p_{2}\right) ^{\mu }\big (p_{1}\cdot p_{2}- p_{3}\cdot p_{4}\big )\delta _{i_{1}i_{2}}\delta _{i_{3}i_{4}} \nonumber \\&\quad + \left( p_{1}+p_{3}\right) ^{\mu }\big (p_{1}\cdot p_{3}- p_{2} \cdot p_{4}\big )\delta _{i_{1}i_{3}}\delta _{i_{2}i_{4}}\nonumber \\&\quad + \left( p_{1}+p_{4}\right) ^{\mu }\big (p_{1}\cdot p_{4}- p_{2}\cdot p_{3}\big )\delta _{i_{1}i_{4}}\delta _{i_{2}i_{3}}\Big ]\nonumber \\&\quad -i\frac{b_{\mu }}{2} \left( p_{1}^{\mu }p_{1}^2+p_{2}^{\mu }p_{2}^2 +p_{3}^{\mu }p_{3}^2+p_{4}^{\mu }p_{4}^2\right) \nonumber \\&\quad \times \left( \delta _{i_{1} i_{2}}\delta _{i_{3}i_{4}} +\delta _{i_{1}i_{3}} \delta _{i_{2}i_{4}}+ \delta _{i_{1}n_{i}}\delta _{i_{2}i_{3}}\right) . \end{aligned}$$

(76)

In the previous equations, we have omitted for notation simplicity the index k on the coupling constants. The momentum $p_4$ in ${\varGamma }^{(4)}$ is fixed by momentum conservation: $\sum _{i=1}^{n} p_i=0$. The flow of $a_{\mu }$ and $b_{\mu }$ are deduced from the flow equation of ${\varGamma }^{(4)}$ at zero external field, which is obtained by differentiating four times the RG equation (6) and evaluating it at $\phi =0$. We obtain:

$$\begin{aligned} \partial _{t}{\varGamma }^{(4)}_{i_{1}i_{2}i_{3}i_{4}}(p_1,p_2,p_3)&=\sum _{il}\int _{q}\partial _t R_k\left( q^2\right) G_k^2\left( q^2\right) \nonumber \\&\quad \bigg (-\frac{1}{2} {\varGamma }^{(6)}_{k,k,i_{1},i_{2},i_{3},i_{4}}(q,-q,p_1,p_2,p_3) \nonumber \\&\qquad + G_k\left( (q+p_{1}+p_{2})^2\right) {\varGamma }^{(4)}_{ki_{1}i_{2}l} (q,p_1,p_2) \nonumber \\&\quad {\varGamma }^{(4)}_{li_{3}i_{4}k}(q+p_1+p_2,p_3,p_4)+\text {2 perms.}\bigg ). \end{aligned}$$

(77)

At one loop, we can neglect the first term and replace the vertices in the right hand side of the flow equation by its tree-level form (76). Summing over i and l in the product of the ${\varGamma }_k^{(4,tree)},$ and keeping only terms which are at most linear in $a_\mu$ and $b_\mu ,$ we find that the result is independent of q:

$$\begin{aligned} \begin{aligned}&{{\varGamma }_{ki_{1}i_{2}l}^{(4,tree)}}(q,p_1,p_2) {{\varGamma }_{li_{3}i_{4}k}^{(4,tree)}}(-q,p_3,p_4)+2 \text {perms.} \\&\quad = \frac{u^{2}}{9}\left( N+8\right) \big (\delta _{i_{1}i_{2}} \delta _{i_{3}i_{4}}+\delta _{i_{1}i_{3}}\delta _{i_{2}i_{4}}+ \delta _{i_{1}i_{4}}\delta _{i_{2}i_{3}}\big )\\&\qquad +\frac{iua_{\mu }}{6}\Bigg \{\delta _{i_{1}i_{2}}\delta _{i_{3}i_{4}}\left[ (p_{1}+p_{2})^{\mu }(p_{1}\cdot p_{2}-p_{3}\cdot p_{4})\left( N+2\right) -2\sum _{k=1}^{4}(p_{k}^{\mu }p_{k}^{2})\right] \\&\qquad +2 \text { perms.}\Bigg \}\\&\qquad -\frac{iub_{\mu }}{6}\Bigg \{\delta _{i_{1}i_{2}}\delta _{i_{3}i_{4}} \left[ \left( p_{1}+p_{2}\right) ^{\mu }(p_{1}\cdot p_{2}-p_{3}\cdot p_{4})\left( N+2\right) +\sum _{k=1}^{4}(p_{k}^{\mu }p_{k}^{2})\left( N+6\right) \right] \\&\qquad +2 \text { perms.}\Bigg \}, \end{aligned} \end{aligned}$$

(78)

where cyclic permutations of the indices 2, 3 and 4 are not written explicitly.

The next step consists in identifying the prefactors of a given structure which involve both vector indices and momenta [see Eq. (76)] in the left hand side and right hand side of the flow equation (77). This implies that we must expand the r.h.s. in powers of the external momenta and extract terms of order zero and order three in momenta. However, Eq. (78) shows that the product of vertices already has a contribution with 0 and 3 powers of the external momenta. As a consequence, we can put, in the propagator, the external momenta to zero.

We can now extract the flows of $a_\mu$ and $b_\mu ,$ which read:

$$\begin{aligned} \partial _{t}u&=\frac{\left( N+8\right) }{3}u^{2}\int _{q}{{\dot{R}}\left( q\right) G^{3}\left( q\right) },\nonumber \\ \partial _{t}a_{\mu }&=\frac{u}{3}\left[ \left( N+4\right) a_{\mu }+4b_{\mu } \right] \int _{q}{{\dot{R}}\left( q\right) G^{3}\left( q\right) },\nonumber \\ \partial _{t}b_{\mu }&=\frac{u}{3}\left[ 2a_{\mu }+\left( N+6\right) b_{\mu } \right] \int _{q}{{\dot{R}}\left( q\right) G^{3}\left( q\right) }. \end{aligned}$$

(79)

The flow equations for the dimensionless variables are easily derived and correspond to those given in Eq. (29).

Large N Expansion

We discuss in this Appendix the large-N calculation of the two smallest scaling dimensions of the vector operators (which tend to 3 when $d\rightarrow 4$) as well as another one, (which tend to 5 when $d\rightarrow 4$). These can be deduced from the calculation of ${\varGamma }^{\left( 2\right) },$${\varGamma }^{\left( 4\right) }$ and ${\varGamma }^{\left( 6\right) }$ at vanishing external field in the large N limit. We recall that the large N limit is performed at fixed ${\hat{u}}=u N,$$\hat{a}_\mu =a_\mu N,$$\hat{b}_\mu =b_\mu N$ and $\hat{c}_\mu =c_\mu N^2$. Moreover, we only need the flow of $\hat{a}_\mu ,$$\hat{b}_\mu$ and $\hat{c}_\mu$ at linear order in $\hat{a}_\mu ,$$\hat{b}_\mu$ and $\hat{c}_\mu$.

$$\begin{aligned}&\int _x \left\{ \frac{{\hat{u}}^{\varLambda }}{4!N}\phi _{i}^{2}\phi _{j}^{2} +\frac{\hat{a}^{\varLambda }_{\mu }}{4N}\phi _{i}\partial _{\mu }\phi _{i} \partial _{\nu }\phi _{j}\partial _{\nu }\phi _{j} +\frac{\hat{b}^{\varLambda }_{\mu }}{2N}\phi _{i}\partial _{\nu }\phi _{i} \partial _{\nu }\phi _{j}\partial _{\mu }\phi _{j} \right. \nonumber \\&\quad \left. +\frac{\hat{c}^{\varLambda }_{\mu }}{4N^2}\phi _{i}\partial _{\nu } \phi _{i}\phi _{j}\partial _{\nu }\phi _{j}\phi _{l}\partial _{\mu } \phi _{l}\right\} . \end{aligned}$$

(80)

The bare propagators and 4-point vertex are easily deduced from those given in Eq. (76). The 6-point vertex is given by:

$$\begin{aligned}&{{\varGamma }_{i_{1}i_{2}i_{3}i_{4}i_{5}i_{6}}^{(6,tree)}} (p_1,p_2,p_3,p_4,p_5) \nonumber \\&\quad = -i\frac{c_{\mu }^{{\varLambda }}}{2}\bigg \lbrace \delta _{i_{1}i_{2}}\delta _{i_{3}i_{4}}\delta _{i_{5}i_{6}} \Big [\left( p_{1}+p_{2}\right) ^{\mu }(p_{1}+ p_{2})^2+\left( p_{3}+p_{4}\right) ^{\mu }(p_{3}+ p_{4})^2 \nonumber \\&\qquad +\left( p_{5}+p_{6}\right) ^{\mu }(p_{5}+ p_{6})^2\Big ]+perm\bigg \rbrace , \end{aligned}$$

(81)

where perm represents the 14 permutations which lead to different combinations of Kronecker delta.

1.1 A Source of Simplification

A major simplification occurs in the calculation, which is a consequence of the following property: the contribution linear in $a_\mu$ or $b_\mu$ in the 4-point vertex which is proportional to $\delta _{i_{1}i_{2}}\delta _{i_{3}i_{4}}$ vanishes in the exceptional configurations where the momenta are opposite by pairs in different delta’s (that is, if $p_1+p_3=p_2+p_4=0$ or if $p_1+p_4=p_2+p_3=0$). As a consequence, in a diagram made of a chain of bubbles (see Fig. 2), if a ${\varGamma }^{(4)}$ connecting two bubbles is replaced by a perturbation $a_\mu$ or $b_\mu ,$ the diagram vanishes. Otherwise stated, in a chain of bubbles diagram, the perturbations $a_\mu$ and $b_\mu$ only occur when attached to an external leg (see Fig. 6).

A somewhat similar situation occurs when a $c_\mu$ perturbation appears in a chain of bubble for the 4-point function. When this perturbation is inserted in a chain (whether in contact or not with external legs), given the structure of Eq. (81), the legs from the isolated loop cancels by itself and the remaining two pairs cancel with each other. This closely resembles the property mentioned previously that $a_\mu$ and $b_\mu$ don’t appear in an inner vertex of the chains. Moreover, the specific momentum structure appearing in the $c_\mu$ vertex, implies that it can’t appear neither attached to an external leg. We conclude that $c_\mu$ does not contribute to the flow of $a_\mu$ and $b_\mu$ at leading order in 1 / N.

The situation is even simpler in the calculation of ${\varGamma }^{(2)}$ because, by conservation of the momenta, the external legs have opposite momenta. In this situation, the perturbation 4-point vertex does not even contribute when attached to the external legs. As a consequence, the cactus diagrams for ${\varGamma }^{(2)}$ are independent of $a_\mu$ and $b_\mu$. This result is important because it implies that the inverse full propagator [which re-sums all cactus diagrams for ${\varGamma }^{(2)}$] is independent of $a_\mu ,$$b_\mu$ and $c_\mu$.

1.2 Computation of ${\varGamma }^{(2)}$

We first discuss the (standard) calculation of ${\varGamma }^{(2)}$ at leading order. As discussed above, we can remove $a_\mu$ and $b_\mu$ from this calculation. The sum of the cactus diagrams shown in Fig. 1 leads to

$$\begin{aligned} {\varGamma }_{ij}^{(2)}(p)=\delta _{ij}\Big \{p^2+r^{\varLambda }+\frac{\hat{u}^{\varLambda }}{6}\int _q\frac{1}{q^2+r^{\varLambda }+{\overline{{\varSigma }}}\left( r^{\varLambda }\right) }\Big \}=\delta _{ij}\Big \{p^2+r\Big \}, \end{aligned}$$

(82)

where $r=r^{{\varLambda }}+{\overline{{\varSigma }}}\left( r^{\varLambda }\right)$ and ${\overline{{\varSigma }}}\left( r^{\varLambda }\right)$ satisfies the gap equation:

$$\begin{aligned} {\overline{{\varSigma }}}\left( r^{\varLambda }\right) =\frac{\hat{u}^{\varLambda }}{6}\int _q\frac{1}{q^2+r^{\varLambda }+{\overline{{\varSigma }}}\left( r^{\varLambda }\right) }. \end{aligned}$$

(83)

As is well known, the only effect of the cactus diagrams is to modify the mass.

1.3 Computation of ${\varGamma }^{\left( 4\right) }$

In contrast to ${{\varGamma }}^{\left( 2\right) },$ the ${{\varGamma }}^{\left( 4\right) }$ vertex has corrections linear in $a^{\varLambda }_{\mu }$ and $b^{\varLambda }_{\mu }$ to leading order in the $\frac{1}{N}$ expansion (i.e. to order $\frac{1}{N}$). It is convenient to decompose the 4-point vertex function as ${\varGamma }^{\left( 4\right) }={\varGamma }^{\left( 4\right) }_u+{\varGamma }^{\left( 4\right) }_{a_{\mu }}+{\varGamma }^{\left( 4\right) }_{b_{\mu }},$ where the first term is independent of $a_\mu$ and $b_\mu ,$ the second term is linear in $a_\mu$ and the third is linear in $b_\mu$. We omit all other terms which do not enter into the calculation of the scaling dimensions we are interested in.

The term ${\varGamma }^{\left( 4\right) }_u$ is the simplest one since it corresponds to the usual theory with $a^{\varLambda }_{\mu }=b^{\varLambda }_{\mu }=0$. This gives the standard large-N result:

$$\begin{aligned}&{\varGamma }^{\left( 4\right) }_{u,i_1,i_2,i_3,i_4}(p_1,p_2,p_3) \nonumber \\&\quad =\frac{\hat{u}^{\varLambda }}{3N}\left[ \frac{\delta _{i_{1}i_{2}} \delta _{i_{3}i_{4}}}{1+\frac{\hat{u}^{\varLambda }}{6}{\varPi }\left( p_{1}+p_{2}\right) } \right. +\frac{\delta _{i_{1}i_{3}}\delta _{i_{2}i_{4}}}{1+\frac{\hat{u}^{\varLambda }}{6} {\varPi }\left( p_{1}+p_{3}\right) } +\left. \frac{\delta _{i_{1}i_{4}}\delta _{i_{2}i_{3}}}{1+\frac{\hat{u}^{\varLambda }}{6} {\varPi }\left( p_{1}+p_{4}\right) }\right] , \end{aligned}$$

(84)

where the function ${\varPi }\left( p\right)$ is defined as:

$$\begin{aligned} {\varPi }\left( p\right) =\int _q \frac{1}{q^2+r}\frac{1}{\left( q+p\right) ^2+r}. \end{aligned}$$

(85)

We now consider ${\varGamma }^{\left( 4\right) }_{a_{\mu }}$. For simplicity, we focus on the contribution proportional to $\delta _{i_1i_2}\delta _{i_3i_4}$. The other contribution are obtained by permutations of the external legs. The set of diagrams which contribute is easy to characterize because the perturbation ($a_\mu$ in this case) must be attached to the external legs (see Sect. C.1). The chain of bubbles diagrams for ${\varGamma }^{\left( 4\right) }_{a_{\mu }}$ are depicted in Fig. 6.

Fig. 6

Diagrams contributing to ${\varGamma }^{\left( 4\right) }_{a_\mu }$

The diagram with n couplings $\hat{u}^{\varLambda }$ and one $\hat{a}^{\varLambda }_{\mu }$ connected to $p_{1}$ and $p_{2}$ is equal to:

$$\begin{aligned}&\frac{i\hat{a}^{\varLambda }_{\mu }}{2N}\int _q\frac{\left( p_{1} +p_{2}\right) ^{\mu }\left[ p_{1}\cdot p_{2}+q\cdot \left( q+p_{1}+p_{2}\right) \right] }{\left[ q^{2}+r\right] \left[ \left( q+p_{1}+p_{2}\right) ^{2} +r\right] } \nonumber \\&\quad \times \left( \frac{-\hat{u}^{\varLambda }}{4!}\right) ^{n} \frac{4^{n}n!C^{n+1}_{n}}{\left( n+1\right) !} \left( {\varPi }\left( p_{1}+p_{2}\right) \right) ^{\left( n-1\right) }. \end{aligned}$$

(86)

If $\hat{a}^{\varLambda }_{\mu }$ is connected to $p_{3}$ and $p_{4},$ we get:

$$\begin{aligned}&\frac{i\hat{a}^{\varLambda }_{\mu }}{2N}\int _q\frac{\left( p_{1}+p_{2} \right) ^{\mu }\left[ -q\cdot \left( q+p_{1}+p_{2}\right) -p_{3}\cdot p_{4}\right] }{\left[ q^{2}+r\right] \left[ \left( q+p_{1}+p_{2}\right) ^{2}+ r\right] } \nonumber \\&\quad \times \left( \frac{-\hat{u}^{\varLambda }}{4!}\right) ^{n} \frac{4^{n}n!C^{n+1}_{n}}{\left( n+1\right) !} \left( {\varPi }\left( p_{1}+p_{2}\right) \right) ^{\left( n-1\right) }. \end{aligned}$$

(87)

When adding both diagrams we get the result for n couplings $\hat{u}^{\varLambda }$ and one $\hat{a}^{\varLambda }_{\mu }$:

$$\begin{aligned} \frac{i\hat{a}^{\varLambda }_{\mu }}{2N}\left( -\frac{\hat{u}^{\varLambda }}{6}{\varPi }\left( p_{1}+p_{2}\right) \right) ^{n}\left( p_{1}+p_{2}\right) ^{\mu }\left[ p_{1}\cdot p_{2}-p_{3}\cdot p_{4}\right] . \end{aligned}$$

(88)

Note that the previous construction does not make sense for $n=0$. However, it happens that Eq. (88) evaluated at $n=0$ indeed represents the contribution of the first diagram of Fig. 6 with one $a_\mu$ and no ${\hat{u}}$. It is straightforward to sum this general expression for all n to get:

$$\begin{aligned} {\varGamma }^{\left( 4\right) }_{a_{\mu }}= & {} \frac{i\hat{a}^{\varLambda }_{\mu }}{2N\left( 1+\frac{\hat{u}^{\varLambda }}{6}{\varPi }\left( p_{1}+p_{2}\right) \right) } \left( p_{1}+p_{2}\right) ^{\mu } \nonumber \\&\quad \times \left[ p_{1}\cdot p_{2}-p_{3}\cdot p_{4}\right] \delta _{i_1i_2}\delta _{i_3i_4}+\text {2 perms.} \end{aligned}$$

(89)

The calculation for ${\varGamma }_{b_\mu }^{(4)}$ proceeds in the same way. The contribution of diagrams with one $b_\mu$ and n couplings ${\hat{u}}$ (again focusing on the contribution proportional to $\delta _{i_1i_2}\delta _{i_3i_4}$) is:

$$\begin{aligned}&\frac{-i\hat{b}^{\varLambda }_{\mu }}{2N} \left( -\frac{\hat{u}^{\varLambda }}{6} {\varPi }\left( p_{1}+p_{2}\right) \right) ^{n} \nonumber \\&\quad \times \left[ p_{1}^{\mu }p_{1}^{2}+p_{2}^{\mu }p_{2}^{2} +p_{3}^{\mu }p_{3}^{2}+p_{4}^{\mu }p_{4}^{2}+\left( p_{1} +p_{2}\right) ^{\mu }(p_{1}\cdot p_{2}-p_{3}\cdot p_{4})\right] . \end{aligned}$$

(90)

To sum up, the four-point vertex with at most one $a_\mu$ or one $b_\mu$ is

$$\begin{aligned} \begin{aligned} {\varGamma }^{\left( 4\right) }_{i_1i_2i_3i_4}(p_1,p_2,p_3)&=\frac{\delta _{i_{1}i_{2}}\delta _{i_{3}i_{4}}}{N\big (1+ \frac{\hat{u}^{\varLambda }}{6}{\varPi }\left( p_1+p_2\right) \big )} \\&\quad \times \bigg \lbrace \frac{\hat{u}^{\varLambda }}{3}+i \frac{\hat{a}^{\varLambda }_{\mu }-\hat{b}^{\varLambda }_{\mu }}{2} \left( p_{1}+p_{2}\right) ^{\mu }\left[ p_{1} \cdot p_{2} - p_{3} \cdot p_{4}\right] \\&\qquad -i \frac{\hat{b}^{\varLambda }_{\mu }}{2} \left( p_{1}^{\mu }p_{1}^2+p_{2}^{\mu }p_{2}^2+p_{3}^{\mu } p_{3}^2+p_{4}^{\mu }p_{4}^2\right) \bigg \rbrace +\text {2 perms,} \end{aligned} \end{aligned}$$

(91)

where, again, the permutations are obtained by a cyclic permutation of the external indices 2, 3 and 4.

1.4 Computation of ${\varGamma }^{\left( 6\right) }$

The ${{\varGamma }}^{\left( 6\right) }$ vertex has corrections linear in $c^{\varLambda }_{\mu }$ to leading order in the $\frac{1}{N}$ expansion (i.e. to order $\frac{1}{N^2}$), but it may also have contributions coming from the types of diagrams shown in Fig. 7 where $a_\mu$ or $b_\mu$ is inserted at the core (i.e. the inner loop with three propagator) or, as before, attached to an external leg. However, these corrections do not contribute to the scaling dimensions of the operators under study because, as explained above, $c_\mu$ does not contribute to the flows of $a_\mu$ and $b\mu$ at leading order in 1 / N. We therefore do not compute these corrections.

Fig. 7

Left: a diagram contributing to ${\varGamma }^{\left( 6\right) },$ linear in $a_\mu$ or $b_\mu$. Right: a diagram contributing to ${\varGamma }^{\left( 6\right) }$ proportional to $c_\mu$

The diagrams to be computed are exceptionally simple since they have a $c_\mu$ at the core with no loop and then just chain of bubbles with u perturbations, these are schematically shown in Fig. 7.

The diagram (proportional to $\delta _{i_{1}i_{2}}\delta _{i_{3}i_{4}}\delta _{i_{5}i_{6}}$) with a chain with $n_1$ couplings $\hat{u}^{\varLambda }$ attached to the external momentums $p_1$ and $p_2,$ a chain with $n_2$ couplings $\hat{u}^{\varLambda }$ attached to the external momentums $p_3$ and $p_4,$ a chain with $n_3$ couplings $\hat{u}^{\varLambda }$ attached to the external momentums $p_5$ and $p_6$ and one $\hat{c}^{\varLambda }_{\mu }$ at the core is equal to:

$$\begin{aligned}&\frac{-i\hat{c}^{\varLambda }_{\mu }}{2N^2} \left( \frac{-\hat{u}^{\varLambda }}{4!}\right) ^{n_1}\left( \frac{-\hat{u}^{\varLambda }}{4!}\right) ^{n_2}\left( \frac{-\hat{u}^{\varLambda }}{4!}\right) ^{n_3}\frac{4^{n_1+n_2+n_3}(n_1+n_2+n_3)!C^{n_1+n_2+n_3+1}_{1}}{\left( n_1+n_2+n_3+1\right) !} \nonumber \\&\quad \times \left( {\varPi }\left( p_{1}+p_{2}\right) \right) ^{n_1}\left( {\varPi }\left( p_{3}+p_{4}\right) \right) ^{n_2}\left( {\varPi }\left( p_{5}+p_{6}\right) \right) ^{n_3} \nonumber \\&\quad \times \Big [\left( p_{1}+p_{2}\right) ^{\mu }(p_{1}+ p_{2})^2+\left( p_{3}+p_{3}\right) ^{\mu }(p_{3}+ p_{4})^2+\left( p_{5}+p_{6}\right) ^{\mu }(p_{5}+ p_{6})^2\Big ] \nonumber \\&\quad + \text {perms.} \end{aligned}$$

(92)

1.5 Running Couplings

A convenient way to deduce the scaling dimensions of the operators coupled to $\hat{a}^{\varLambda }_{\mu }$ and $\hat{b}^{\varLambda }_{\mu },$ is to introduce an infrared regulator in propagators:

$$\begin{aligned} \frac{1}{q^2+r^{\varLambda }}\rightarrow \frac{1}{q^2+r^{\varLambda }+R_k(q)} \end{aligned}$$

(93)

and study the running of the various couplings when varying the regulator. We thus define the renormalized couplings as:

$$\begin{aligned} {\varGamma }_{i_1i_2}^{(2)}(0)&=r_k \delta _{i_1i_2},\nonumber \\ {\varGamma }_{i_1i_2i_3i_4}^{(4)}(0,0,0)&=\frac{\hat{u}_k}{3N} \Big (\delta _{i_1i_2}\delta _{i_3i_4}+\delta _{i_1i_3} \delta _{i_2i_4}+\delta _{i_1i_3}\delta _{i_2i_4}\Big ),\nonumber \\ {\varGamma }^{\left( 4\right) , {\mathcal {O}}(p^3)}_{i_1i_2i_3i_4}(p_1,p_2,p_3)&=i\frac{\delta _{i_1i_2}\delta _{i_3i_4}}{2N} \bigg \lbrace (\hat{a}^k_{\mu }-\hat{b}^k_{\mu }) \left( p_{1}+p_{2}\right) ^{\mu }\left[ p_{1} \cdot p_{2} - p_{3} \cdot p_{4}\right] \nonumber \\&\quad -\hat{b}^k_{\mu }\left( p_{1}^{\mu }p_{1}^2+ p_{2}^{\mu }p_{2}^2+p_{3}^{\mu }p_{3}^2+p_{4}^{\mu }p_{4}^2\right) \bigg \rbrace \nonumber \\&\quad +\text {2 perms.}, \nonumber \\ {\varGamma }^{\left( 6\right) , \mathcal O(p^3)}_{i_1i_2i_3i_4i_5i_6}(p_1,p_2,p_3,p_4,p_5)&=-i\frac{\delta _{i_1i_2}\delta _{i_3i_4}\delta _{i_5i_6} \hat{c}^k_{\mu }}{2N^2} \nonumber \\&\quad \times \bigg \lbrace \left( p_{1}+p_{2}\right) ^{\mu }(p_{1}+ p_{2})^2 \nonumber \\&\quad +\left( p_{3}+p_{3}\right) ^{\mu }(p_{3}+ p_{4})^2+\left( p_{5}+p_{6}\right) ^{\mu }(p_{5}+ p_{6})^2 \bigg \rbrace \nonumber \\&\quad +\text {14 perms.} \end{aligned}$$

(94)

One can then conclude that the running couplings are:

$$\begin{aligned}&u_k =\frac{u}{1+\frac{u_{\varLambda }}{6}{\varPi }_k\left( 0\right) },\quad a^\mu _k=\frac{a^\mu _{\varLambda }}{1+\frac{u_{\varLambda }}{6}{\varPi }_k\left( 0\right) }, \quad b^\mu _k=\frac{b^\mu _{\varLambda }}{1+\frac{u_{\varLambda }}{6}{\varPi }_k\left( 0\right) } \nonumber \\&c^\mu _k =\frac{c^\mu _{\varLambda }}{\big (1+\frac{u_{\varLambda }}{6}{\varPi }_k\left( 0\right) \big )^3}+Y_a a^\mu _{\varLambda }+Y_b b^\mu _{\varLambda },\quad r_k=r^{{\varLambda }}+{\overline{{\varSigma }}}_{k}, \end{aligned}$$

(95)

where the functions ${\overline{{\varSigma }}}_{k}$ and ${\varPi }_{k}$ are calculated with the introduction of the infrared regulator:

$$\begin{aligned} {\overline{{\varSigma }}}_{k}\left( r_{\varLambda }\right)&=\frac{u_{\varLambda }}{6}\int _q\frac{1}{q^2+r_k+R_{k}\left( q^2\right) }, \end{aligned}$$

(96)

$$\begin{aligned} {\varPi }_{k}\left( 0\right)&=\int _q\frac{1}{\left( q^2+r_k+R_{k}\left( q^2\right) \right) ^{2}}. \end{aligned}$$

(97)

Taking this into account, we obtain the flow of the running couplings:

$$\begin{aligned} \partial _{t}r_k&=-\frac{\hat{u}_k}{6}\int _q \partial _{t}R_{k}(q) G_k^2(q),\nonumber \\ \partial _{t}\hat{u}_k&=-\frac{\hat{u}_k^3}{18}\int _q G_k^3(q)\int _{q'} \partial _{t}R_{k}(q') G_k^2(q') +\frac{\hat{u}_k^2}{3}\int _q \partial _{t}R_{k}(q) G_k^3(q),\nonumber \\ \partial _{t}\hat{a}_k^{\mu }&=-\frac{\hat{a}_k^\mu \hat{u}_k^2}{18} \int _q G_k^3(q)\int _{q'}\partial _{t}R_{k}(q') G_k^2(q') +\frac{\hat{a}_k^{\mu }\hat{u}_k}{3}\int _q \partial _{t} R_{k}(q) G_k^3(q),\nonumber \\ \partial _{t}\hat{b}_k^{\mu }&=-\frac{\hat{b}_k^\mu \hat{u}_k^2}{18} \int _q G_k^3(q)\int _{q'}\partial _{t}R_{k}(q') G_k^2(q') +\frac{\hat{b}_k^{\mu }\hat{u}_k}{3}\int _q \partial _{t}R_{k}(q) G_k^3(q), \nonumber \\ \partial _{t}\hat{c}_k^{\mu }&=-\frac{\hat{c}_k^\mu \hat{u}_k^2}{6}\int _q G_k^3(q)\int _{q'}\partial _{t}R_{k}(q') G_k^2(q') +\hat{c}_k^{\mu }\hat{u}_k\int _q \partial _{t}R_{k}(q) G_k^3(q) \nonumber \\&\quad +X_a \hat{a}_k^{\mu }+X_b \hat{b}_k^{\mu }, \end{aligned}$$

(98)

where in the previous equations,

$$\begin{aligned} G_k(q)= \frac{1}{q^2+r_k+R_{k}\left( q^2\right) }. \end{aligned}$$

Introducing dimensionless and renormalized variables as explained in the main text, we retrieve the flow equations (37).

Numerical Method

We describe in this section the details of the numerical method used to determine the scaling dimensions in the ${\mathcal {O}}(\partial ^3)$ approximation of NPRG.

We first determine the fixed point of the O(N) model at ${\mathcal {O}}(\partial ^2)$ in a ${\tilde{\rho }}$ grid with $N_\rho +1$ sites in a box ${\tilde{\rho }} \in [0,L_\rho ]$ (this corresponds to a step in the ${\tilde{\rho }}$-lattice ${\varDelta }\rho =L_\rho /N_\rho$). The derivatives are approximated by centered finite differences with five points (with the exception of the edges, i.e. the first two and last two sites, where we use lateral finite differences).

The internal momentum integrals that appear are one dimensional (the angular part is just a constant) and are calculated by a Legendre–Gauss quadrature with $N_q$ points in a box of size $|L_q|\equiv \frac{q_{max}}{k}$.

The normalization condition is fixed as $\tilde{Z}(\tilde{\rho }_i)|_{i=N_\rho /3}=1,$ where $\tilde{Z}(\tilde{\rho })$ is the dimensionless version of $Z_k(\rho )$ and $\tilde{\rho }_i$ is the value of $\tilde{\rho }$ at site i. On top of this, the size of the box $L_\rho$ is adjusted every time a new set of parameter is considered so that the minimum of the potential at the fixed point falls in the site $i=N_\rho /3$.

The parameters were chosen so as that the numerical error in the leading exponents ($\eta$ and $\nu$) is below one per mille in $d=3$. Then we varied the values of d for each N. In particular we used the exponential regulator $R_{k}(q^2)={\mathcal {Z}}_{k}q^2 r(q^2/k^2)$ with:

$$\begin{aligned} r(y)=\frac{\alpha }{e^{y}-1}, \end{aligned}$$

(99)

where ${\mathcal {Z}}_k$ is the field renormalization which is related to the running anomalous dimension by $\partial _t{\mathcal {Z}}_k=-\eta _k{\mathcal {Z}}_k$ (when approaching the fixed point $\eta _k$ approaches the field anomalous dimension). The parameter $\alpha$ was fixed in order to estimate the dependence of the results on the arbitrary regulating function $R_k(q)$. In order to do so, for each N we employ the criterium of minimal sensitivity (PMS) [76]. That is, we choose the value of $\alpha$ which minimizes the dependence on the regulator of some observable by varying the regulator in the family (99). This procedure has been tested before and shown to be very effective and predictive [58, 59]. In the present implementation we choose $\alpha$ to be an extremum of the critical exponent $\eta$ for $d=3$ (no significative difference was observed when doing PMS on another quantity such as critical exponent $\nu$).

After obtaining the fixed point, we computed the eigenvectors of the $3N_\rho$ linear system ($N_\rho$ variables for each function $\tilde{a}_{\mu },$$\tilde{b}_{\mu }$ and $\tilde{c}_{\mu }$) at the fixed points determined previously. To solve numerically the ${\mathcal {O}}(\partial ^3)$ of the DE for the O(N) model the following set of parameters was used:

$$\begin{aligned} N_\rho =60,\quad L_\rho =3\rho _0,\quad N_q=15,\quad L_q=4.2. \end{aligned}$$

(100)

Inequalities

In the present Appendix, we prove by induction the inequality (51) for $T\ge T_c$. We use generalizations of Griffiths and Lebowitz inequalities which were proven for $N=2,3,$ and 4 [41,42,43,44,45,46]. The inequalities obtained in those references are the following:

$$\begin{aligned}&\langle \phi _1(x^{(1)}_1)\ldots \phi _1(x^{(1)}_{n_1})\phi _2(x^{(2)}_1)\ldots \phi _2(x^{(2)}_{n_2})\ldots \phi _N(x^{(N)}_1)\ldots \phi _N(x^{(N)}_{n_N})\rangle \ge 0, \end{aligned}$$

(101)

$$\begin{aligned}&\langle \phi _\alpha (x_1)\ldots \phi _\alpha (x_n)\phi _\alpha (y_1)\ldots \phi _\alpha (y_m)\rangle \ge \langle \phi _\alpha (x_1)\ldots \phi _\alpha (x_n)\rangle \langle \phi _\alpha (y_1)\ldots \phi _\alpha (y_m)\rangle , \end{aligned}$$

(102)

$$\begin{aligned}&\langle \phi _\alpha (x_1)\ldots \phi _\alpha (x_n)\phi _\beta (y_1)\ldots \phi _\beta (y_m)\rangle \nonumber \\&\quad \le \langle \phi _\alpha (x_1)\ldots \phi _\alpha (x_n)\rangle \langle \phi _\beta (y_1)\ldots \phi _\beta (y_m)\rangle \,\mathrm {with}\,\alpha \ne \beta . \end{aligned}$$

(103)

Inequality (101) and (102) are very similar to the Griffiths inequalities I and II for a scalar field and (103) is very similar to the Lebowitz inequality.

First of all, it is clear that for $T\ge T_c$ we need only to consider correlations with n and m with the same parity. In order to begin the induction, we observe that the inequality is trivially true for $n+m=2$. We assume now the validity of (51) for all values of m and n such that $m+n<N$. Under this hypothesis, we then prove its validity for all $\{m,n\}$ such that $m+n=N$.

We first consider the case where n and m are odd. This case is simpler because the second term of the l.h.s. of (51) is zero because we are considering $T\ge T_{c}$ and (101) readily shows that $\nonumber G_{m,n}^{odd}\ge 0$. By symmetry, the structure of the two point function must take the form:

$$\begin{aligned} G_{m,n}^{odd}\left( x,y\right)&\equiv \left\langle \varphi _{i_1}\left( x\right) \cdots \varphi _{i_m} \left( x\right) \right. \left. \varphi _{j_1}\left( y\right) \cdots \varphi _{j_n}\left( y\right) \right\rangle = \sum _{l=0}^{\frac{n-1}{2}}f_{l}\left( x,y\right) \nonumber \\&\quad \times \Big [\delta _{i_{1}j_{1}}\cdots \delta _{i_{2l+1}j_{2l+1}} \delta _{i_{2l+2}i_{2l+3}}\delta _{j_{2l+2}j_{2l+3}}\cdots \delta _{i_{m-1}i_{m}}\delta _{j_{n-1}j_{n}} \nonumber \\&\quad +\left( \frac{m!n!}{\left( 2l+1\right) ! \left( m-2l-1\right) !!\left( n-2l-1\right) !!}-1\right) \text {perms.}\Big ], \end{aligned}$$

(104)

where without loss of generality we focused on the case $m\ge n$. In the previous sum, $2l+1$ corresponds to the number of Kronecker delta which connect i’s with j’s. In the first step of the proof, we consider the following configuration of indices:

$$\begin{aligned}&i_{k}= {\left\{ \begin{array}{ll} 1 &{} \text {for}\quad k=1, \\ 2 &{} \text {for}\quad k=2,\ldots ,m, \end{array}\right. } \\&j_{k}= {\left\{ \begin{array}{ll} 2 &{} \text {for}\quad k=1,\ldots ,2s, \\ 1 &{} \text {for}\quad k=2s+1,\ldots ,n \end{array}\right. } \end{aligned}$$

with s ranging from 0 to $\frac{n-1}{2}$. (For $s=0,$ all the $j_k$ are 1.) We can now apply inequality (103) for these index configurations to get:

$$\begin{aligned} {\tilde{G}}_{m,n}^{odd}\left( x,y\right)&\le \left\langle \varphi _{1}\left( x\right) \left( \varphi _{1}\left( y\right) \right) ^{n-2s}\right\rangle \left\langle \left( \varphi _{2}\left( x\right) \right) ^{m-1} \left( \varphi _{2}\left( y\right) \right) ^{2s}\right\rangle \nonumber \\&\le CG\left( x-y\right) , \end{aligned}$$

(105)

where the tilde on G indicates that it is taken in a particular configuration of indices and where we have used the recursion hypothesis (51) for $m'+n'<m+n$ and the fact that $\left\langle \left( \varphi _{2}\left( x\right) \right) ^{m-1}\left( \varphi _{2}\left( y\right) \right) ^{2s}\right\rangle$ is bounded by a constant to obtain the last line. Using, on the other hand the decomposition (104), we get:

$$\begin{aligned} \sum _{t=0}^{s} \frac{\left( 2s\right) !\left( n-2s\right) !!\left( m-1\right) !}{\left( 2t\right) !\left( 2s-2t\right) !!\left( m-1-2t\right) !!}f_{t}\left( x,y\right) \le CG\left( x-y\right) . \end{aligned}$$

(106)

By considering the different possible values of s,  we easily show that the $f_t$ are bounded by $G(x-y)$ (up to a multiplicative constant) which ensure that the inequality is valid for $G_{m,n}^{odd}$.

The even case is a bit different since the second term of the l.h.s. of (51) is nonzero. We again choose $m\ge n$ and look at the structure of the two point function:

$$\begin{aligned} G_{m,n}^{even}\left( x,y\right)&\equiv \left\langle \varphi _{i_1}\left( x\right) \cdots \varphi _{i_m} \left( x\right) \right. \left. \varphi _{j_1}\left( y\right) \cdots \varphi _{j_n}\left( y\right) \right\rangle = \sum _{l=0}^{\frac{n}{2}}g_{l}\left( x,y\right) \nonumber \\&\quad \times \left( \delta _{i_{1}j_{1}}\cdots \delta _{i_{2l}j_{2l}} \delta _{i_{2l+1}i_{2l+2}}\delta _{j_{2l+1}j_{2l+2}} \cdots \delta _{i_{m-1}i_{m}}\delta _{j_{n-1}j_{n}}\right. \nonumber \\&\quad +\left. \left( \frac{m!n!}{\left( 2l\right) !\left( m-2l\right) !! \left( n-2l\right) !!}-1\right) \text {perms.}\right) . \end{aligned}$$

(107)

Inequality (101) readily imposes that $G_{m,n}^{even}\left( x,y\right) \ge 0$. We now proceed in the same way as for the odd case. We take a configuration of indices:

$$\begin{aligned}&i_{k}= {\left\{ \begin{array}{ll} 1 &{} \text {for}\quad k=1, \\ 2 &{} \text {for}\quad k=2,\ldots ,m, \end{array}\right. } \\&j_{k}= {\left\{ \begin{array}{ll} 2 &{}\text {for}\quad k=1,\ldots ,2s-1, \\ 1 &{}\text {for}\quad k=2s,\ldots ,n \end{array}\right. } \end{aligned}$$

with s ranging from 1 to $\frac{n}{2}$. These configurations in combination with the inequality (103) impose an upper bound on a strict conical combinations of the $g_{i}$ functions not involving the $g_{0}$ function:

$$\begin{aligned} {\tilde{G}}_{m,n}^{even}\left( x,y\right)&=\sum _{t=1}^{t=s} \frac{\left( 2s-1\right) !\left( n-2s+1\right) !!\left( m-1\right) !}{\left( 2t-1\right) !\left( 2s-2t\right) !!\left( m-2t\right) !!}g_{t} \left( x,y\right) \nonumber \\&\le \left\langle \varphi _{1}\left( x\right) \left( \varphi _{1}\left( y\right) \right) ^{n-2s+1}\right\rangle \left\langle \left( \varphi _{2}\left( x\right) \right) ^{m-1} \left( \varphi _{2}\left( y\right) \right) ^{2s-1}\right\rangle \nonumber \\&\le CG^{2}\left( x-y\right) , \end{aligned}$$

(108)

where we have made use of the validity of recursion hypothesis (51) for all $m'+n'<m+n$ and the tilde is again used to recall that a particular configuration of indices was used. We, again, find a lower and an upper bound on a strict conical combination of the functions $g_{s},$ with $s=1,\ldots ,\frac{n}{2}$. This implies that the absolute value of each of these $g_{s}$ functions (with $s\ne 0$) is bounded by a constant times $G^{2}\left( x-y\right)$ (by arguments identical to the odd case). To complete the argument we need to study the function $g_{0}$. This one is clearly particular because it corresponds to the case where no i or j are connected through a Kronecker delta [see Eq. (104)]. To do this we consider the even simpler configuration where all indices $i=1$ and all $j=2,$ this yields for the two-point function:

$$\begin{aligned} 0 \le G_{m,n}^{even}\left( x,y\right) =g_{0}\left( x,y\right) \left( m-1\right) !!\left( n-1\right) !!\le \left\langle \varphi ^{m}_{1}\left( 0\right) \right\rangle \left\langle \varphi ^{n}_{1}\left( 0\right) \right\rangle \end{aligned}$$

(109)

from which immediately follows that:

$$\begin{aligned} g_{0}\left( x,y\right) -\frac{\left\langle \varphi ^{m}_{1}\left( 0\right) \right\rangle \left\langle \varphi ^{n}_{1}\left( 0\right) \right\rangle }{\left( m-1\right) !!\left( n-1\right) !!} \le 0. \end{aligned}$$

(110)

Let’s consider now all indices $i,j=1$ to obtain a lower bound on a strict conical combination of all the g functions by making use of the inequality (102):

$$\begin{aligned} \sum _{l=0}^{\frac{n}{2}}\frac{m!n!}{\left( 2l\right) !\left( m-2l \right) !!\left( n-2l\right) !!}g_{l}\left( x,y\right) \ge \left\langle \varphi ^{m}_{1}\left( 0\right) \right\rangle \left\langle \varphi ^{n}_{1}\left( 0\right) \right\rangle \end{aligned}$$

(111)

and combining this with (110) we obtain a lower bound:

$$\begin{aligned} \begin{aligned}&g_{0}\left( x,y\right) -\frac{\left\langle \varphi ^{m}_{1}\left( 0\right) \right\rangle \left\langle \varphi ^{n}_{1}\left( 0\right) \right\rangle }{\left( m-1\right) !!\left( n-1\right) !!} \\&\quad \ge -\frac{1}{\left( m-1\right) !!\left( n-1\right) !!}\sum _{l=1}^{\frac{n}{2}} \frac{m!n!}{\left( 2l\right) ! \left( m-2l\right) !!\left( n-2l\right) !!}g_{l}\left( x,y\right) \\&\quad \ge C G^{2}\left( x-y\right) \end{aligned} \end{aligned}$$

(112)

with a constant $C<0$.

So, we have bounded all the g functions with the exception of $g_{0}$ for which the bound involves $g_{0}$ minus a constant. It turns out that this constant has a simple interpretation: when inserted in Eq. (107), the multiplicative constants simplify and we are just left with $\left\langle \varphi ^{m}_{1}\left( 0\right) \right\rangle \left\langle \varphi ^{n}_{1}\left( 0\right) \right\rangle$, which exactly compensates the disconnected part appearing in (51). This concludes the proof of (51).