A scaling theory for the long-range to short-range crossover and an infrared duality^*

The need to resolve the above-mentioned difficulties leads us to the following modified picture of the LRFP to SRFP crossover (referred to as 'our picture' below). Like in the standard picture, the crossover in our picture does happen continuously and at $s=s_*$. However, and this is where we differ, we posit that LRFP crosses over not to SRFP, but to a larger theory, which consists from SRFP and a decoupled sector: the mean-field theory of a gaussian field χ. This larger theory will be referred to as 'SRFP+χ'. The two point (2pt) function of χ will be taken unit-normalized: $\newcommand{\D}{\Delta} 1/\vert x\vert ^{2\Delta_\chi}$.

Assuming our picture, we can construct the flow from the 'SRFP+χ' theory to the LRFP by turning on the perturbation

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\calO}{{{\mathcal O}}} \displaystyle \label{eq:ourpert} g_0 \int {\rm d}^dx\, \calO(x),\quad {{\mathcal O}}= \sigma\cdot \chi. \nonumber \end{align} \tag{ 1.8 }$$

The sign of g₀ is arbitrary since it can be flipped by the $\mathbb{Z}_2^\chi$ symmetry $\chi\to-\chi$. In fact the decoupled SRFP+χ theory has an enlarged $\mathbb{Z}_2^\sigma\times \mathbb{Z}_2^\chi$ symmetry which is broken to the diagonal when the perturbation $\newcommand{\calO}{{{\mathcal O}}} \calO$ is turned on. This is as it should be, since the LRFP has only a single $\mathbb{Z}_2$ symmetry $\phi\to-\phi$. The enlarged symmetry of SRFP+χ leads to selection rules, which will appear many times in the RG calculations below.

Connection to the standard picture is established by integrating out χ, which should generate precisely Sak's non-local perturbation (1.6)¹³. This fixes the dimension $[\chi]=(d+s)/2$, so that

$$\begin{align} \renewcommand{\d}{{\rm d}} \newcommand{\calO}{{{\mathcal O}}} \displaystyle [\calO]=[\chi]+[\sigma]=d-\delta,\quad \delta = (s_*-s)/2. \nonumber \end{align} \tag{ 1.9 }$$

This crosses from relevant to irrelevant at the same location as before. We emphasize however that χ is not simply a theoretical construct introduced to represent $\newcommand{\calO}{{{\mathcal O}}} \calO_{\rm Sak}$, but is a physical field.

One way to think about χ is that it's a remnant of the long-range interactions present in the original model (1.2). Notice also that $[\chi]$ coincides with $[\phi]_{\rm dis}$, the dimension of ϕ in the disordered phase of the long-range model; see appendix A. This suggests a similarity between the disordering effects of temperature and of the short-range critical fluctuations on the long-range correlations.

It should be possible to verify the existence of χ via lattice measurements. This is true even in the short-range regime $s>s_*$, where it is decoupled. The point is that it is decoupled from the SRFP scaling fields, but not from the lattice operators. It should thus be possible to detect χ by measuring the spin-spin correlation function $\renewcommand{\l}{\ell} \langle S_i S_j\rangle$ on the lattice. At the critical point and at large distances, this function has a power law expansion of the form

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \langle S_i S_j\rangle \sim \sum c_k/r^{2\Delta_k},\quad r=\vert i-j\vert, \nonumber \end{align} \tag{ 1.10 }$$

where $\newcommand{\D}{\Delta} \Delta_k$ are the dimensions of $\mathbb{Z}_2$-odd scalar operators. We predict that at the LRFP the dimension of χ should appear among $\newcommand{\D}{\Delta} \Delta_k$.

The existence of χ allows to resolve the difficulties concerning the crossover description. First of all, since χ and ϕ satisfy the shadow relation $[\chi]+[\phi]=d$, χ can be identified with $\phi^3$ at the crossover point. This identification and its consequences will be discussed in detail below. We will also see that using χ one can construct a vector operator playing the role of $V_\mu$. Finally, since $\newcommand{\calO}{{{\mathcal O}}} \calO$ is a local operator, we will be able to use the well-developed framework of conformal perturbation theory to compute the long-range critical exponents near the crossover point.

The RG flow diagram of our picture is shown in figure 2. We predict that in the intermediate regime $d/2 < s < s_*$ the LRFP is the common IR endpoint of two distinct RG flows:

• 1.  
  Flow (1.2) from the mean field theory, which is weakly coupled near the lower end of the intermediate regime $\newcommand{\eps}{\epsilon} (\epsilon \to 0$). We will call this '$\phi^4$-flow'.
• 2.  
  Our newly proposed flow emanating from the SRFP+χ theory, which is weakly coupled near the crossover $\renewcommand{\d}{{\rm d}} (\delta \to 0$). We will call this '$\sigma\chi$-flow'.

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In quantum-field theoretic parlance, this situation—when the same IR theory can be reached from two different UV descriptions—is referred to as 'infrared duality'. A famous example is the Seiberg duality which establishes the IR equivalence of UV-distinct $\newcommand{\calN}{{{\mathcal N}}} \calN=1$ supersymmetric gauge theories [7]. Another example is the particle/vortex duality between the XY model and the $U(1)$ Abelian Higgs model in 3d, both flowing to the same $O(2)$ Wilson-Fisher critical point [8, 9]. The novelty of our example is that the IR fixed point does not have a local stress tensor.

We will start by investigating the structure of the LRFP close to the crossover ($\renewcommand{\l}{\ell} \renewcommand{\d}{{\rm d}} \delta\ll 1$). In this region our $\sigma\chi$-flow provides a weakly coupled description of the LRFP, allowing a number of explicit computations. In section 2 we compute the beta-function, at the leading nontrivial order, and establish the fixed point existence. In section 3 we compute the leading anomalous dimensions for the most interesting operators. In particular, we demonstrate that the stress tensor acquires anomalous dimension, and exhibit the eaten operator $V_\mu$, thus resolving the paradox of missing states. These two sections include also a self-consistent presentation of necessary conformal perturbation theory techniques.

In section 4 we review what is known about the $\phi^4$-flow, mostly following [6]. We discuss a number of results valid to all orders in perturbation theory (the absence of the anomalous dimension of ϕ, the shadow relation between ϕ and $\phi^3$). We also give a streamlined exposition of the proof of the conformal invariance of the LRFP. Finally, we discuss the relation between the OPE coefficients of ϕ and $\phi^3$ which can be proved using the non-local equation of motion (EOM).

We start section 5 with some all-order results for the $\sigma\chi$-flow, established by analogy with the $\phi^4$-flow. We then show that the results about the LRFP obtained from the $\phi^4$-flow and the $\sigma\chi$-flow match together beautifully, providing compelling evidence for the infrared duality. At this high point, we conclude.

We also have a number of appendices. Appendix A is devoted to the discussion of the long-range Ising model off criticality. Appendix C is an (admittedly incomplete) review of our favorite works on the long-range Ising model, both in the physics and the mathematics literature. Appendix D evaluates some integrals arising in the conformal perturbation theory calculations. Appendix E checks explicitly a curious conclusion, reached in [6] and reviewed in section 4.3, that scale invariance of the IR fixed point of the $\phi^4$ flow implies that the kernel of the $\phi\times\phi^4$ OPE should integrate to zero. Finally, appendix F provides some sanity checks of the relations between the OPE coefficients which follow from the non-local EOM, both for the $\phi^4$ and for the $\sigma\chi$-flow.

The 2d SRFP is the minimal model CFT $\newcommand{\calM}{{{\mathcal M}}} \calM(3, 4)$ [10] and everything about it is known exactly. In particular, we have $\newcommand{\D}{\Delta} \Delta_\sigma=1/8$, $\newcommand{\vareps}{\varepsilon} \newcommand{\D}{\Delta} \Delta_ \varepsilon=1$, $\newcommand{\vareps}{\varepsilon} C_{\sigma\sigma \varepsilon}=1/2$. The 4pt function of σ is given by

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{eq:4sigma2d} \langle \sigma(0)\sigma(1)\sigma(z)\sigma(\infty)\rangle = \frac{\vert 1+\sqrt{1-z}\vert +\vert 1-\sqrt{1-z}\vert }{2 \vert z\vert ^{1/4}\vert 1-z\vert ^{1/4}}. \nonumber \end{align} \tag{ 2.15 }$$

Comparing the notation to (2.11), here we have fixed $\hat e$ at 1 on the real axis, while $y=z$ runs over the full complex plane. In spite of the appearance the 4pt function is smooth across $z\in(1, +\infty)$.

The 4pt function of χ is gaussian, given by the sum of three Wick contractions. In the same kinematics,

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \label{eq:4chi2d} \langle \chi(0)\chi(1)\chi(z)\chi(\infty)\rangle = 1+\frac 1{\vert z\vert ^{2\Delta_\chi}} +\frac 1{\vert 1-z\vert ^{2\Delta_\chi}},\quad \Delta_\chi=2-\Delta_\sigma=15/8. \nonumber \end{align} \tag{ 2.16 }$$

The 4pt function of $\newcommand{\calO}{{{\mathcal O}}} \calO$ is given by the product of (2.15) and (2.16):

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{eq:F2d} F(z,\bar z) = \left [1 + \frac{1}{\vert z\vert ^{\frac{15}{4}}} + \frac{1}{\vert z - 1\vert ^{\frac{15}{4}}} \right ]\frac{\vert 1 + \sqrt{1 - z}\vert + \vert 1 - \sqrt{1 - z}\vert }{2\vert z\vert ^{\frac{1}{4}}\vert z - 1\vert ^{\frac{1}{4}}}. \nonumber \end{align} \tag{ 2.17 }$$

We were not able to evaluate the integral of $F(z)$ analytically, so we will report the results of the numerical evaluation. We employ method 2, so that we have to integrate over the region $\mathcal{R}$.

As discussed after equation (2.9), the integral is not convergent around 0. If we expand $F(z, \bar z)$ around $z=0$, we encounter several terms responsible for the non-convergence. The terms $\vert z\vert ^{-4}$ and $\vert z\vert ^{-2}$ correspond to contributions of the identity operator and energy density ε respectively. Other terms, such as $z/\vert z\vert ^3$ and $z^2/\vert z\vert ^4$ ($+$h.c.), are the contributions of $\newcommand{\vareps}{\varepsilon} \partial_\mu \varepsilon$ and $T_{\mu\nu}$ in the $\newcommand{\calO}{{{\mathcal O}}} \calO\times\calO$ OPE; however, they will vanish upon angular integration.

To deal with the divergences, we remove from the region $\mathcal{R}$ a small disk $\vert z\vert <a$ around the origin, and divide the rest into two regions: the annulus $\mathcal{A}(a<\vert z\vert <r_0$) centered around zero and its complement $\bar{\mathcal{A}}$, see figure 3. Here r₀ is arbitrary subject to $a<r_0<1/2$. In $\mathcal{A}$ we expand $F(z)$ functions as a series in z and $\newcommand{\zb}{\bar{z}} \zb$ up to some high order. We can then drop the terms that vanish upon angular integration, and we integrate exactly the remaining terms. The power-divergent, as $a\to0$, part of the answer is dropped. In the complement of the annulus we integrate $F(z)$ numerically. The so regulated integral over $\newcommand{\calR}{{{\mathcal R}}} \calR$ is then:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\calR}{{{\mathcal R}}} \displaystyle I_{d=2}=\int_{\calR} {\rm d}^2z\, F(z,\bar z) = -0.403\,746\ldots \nonumber \end{align} \tag{ 2.18 }$$

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All the shown digits are exact, and we checked that the result is stable against changes of r₀. This implies

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{eq:beta3d=2} \beta_3= -3(2\pi) I_{d=2}/3!= 1.268\,404\quad (d=2). \nonumber \end{align} \tag{ 2.19 }$$

The $d=3$ SRFP is not yet exactly solved; however, high precision results are available thanks to the progress of the numerical conformal bootstrap [17–21]. Recently, the approximate critical 3d Ising 4pt function extracted from the bootstrap data was used in [16] to study the random bond Ising critical point. It was also used in [22] to qualify the non-gaussianity of the 3d Ising model.

Here we proceed analogously and will use the OPE coefficients $\newcommand{\calO}{{{\mathcal O}}} C_{\sigma \sigma \calO}$ and dimensions $\newcommand{\calO}{{{\mathcal O}}} \newcommand{\D}{\Delta} \Delta_\calO$ of the lowest lying operators (such that $\newcommand{\calO}{{{\mathcal O}}} \newcommand{\D}{\Delta} \Delta_\calO$ is smaller than some cutoff in the spectrum $\newcommand{\D}{\Delta} \Delta_*$) to construct an approximate 4pt function for the σ field:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \newcommand{\calO}{{{\mathcal O}}} \displaystyle \langle \sigma(0) \sigma(\hat e)\sigma(y)\sigma(\infty) \rangle\simeq \frac{1}{\vert x\vert ^{2 \Delta_\sigma}} \sum_{\calO:\Delta_\calO<\Delta_*}C_{\sigma \sigma \calO}^2\, g_{\Delta_\calO, \ell_\calO}(z,\bar z), \nonumber \end{align} \tag{ 2.20 }$$

where $\newcommand{\D}{\Delta} g_{\Delta, \ell}$ are the conformal blocks. Let us fix $\renewcommand{\l}{\ell} \hat e=(1, 0, \ldots, 0)$. Then z is the complex coordinate related to y by

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle z= y_1 + {\rm i} \vert y_\perp\vert,\quad y_\perp = (y_2,\ldots,y_d). \nonumber \end{align} \tag{ 2.21 }$$

The 4pt function only depends on $\vert y_\perp\vert$ because of rotation invariance around the x₁ axis. The usual conformal cross ratios u, v are $u=\vert z\vert ^2$, $v=\vert 1-z\vert ^2$. Instead of z, it will be convenient to work with the radial coordinate ρ [23]

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \rho(z)=\frac{z}{\left(1+\sqrt{1-z} \right)^2}. \nonumber \end{align} \tag{ 2.22 }$$

In three dimensions, the conformal blocks are not known exactly; however, they can be computed efficiently as a series in r and $\eta=\cos \theta$, where $\rho=r {\rm e}^{{\rm i}\theta}$, using a recursion relation [24, 25]. The conformal block expansion converges for $r<1$ [26], while in the integration region $\mathcal{R}$ the maximum value of r is $2-\sqrt{3}\simeq 0.27<1$, so our series expansion will converge exponentially fast.

When approximating the 4pt function, we have to take into account three different sources of error:

• 1.  
  We do not know the OPE coefficients and the operator dimensions exactly, as they are obtained through the numerical conformal bootstrap. The uncertainty due to this will turn out to be subleading;
• 2.  
  We compute the conformal blocks as a series expansion in r. Here we did it up to order $O(r^{12})$, which provided sufficient accuracy, but it would be straightforward to compute them to a higher order;
• 3.  
  We know the dimensions and the OPE coefficients of primary operators only up to a dimension $\newcommand{\D}{\Delta} \Delta_*$. The error introduced is of order $\newcommand{\D}{\Delta} r^{\Delta_*}$ [26]. We use data from the numerical conformal bootstrap on operators up to dimension $\newcommand{\D}{\Delta} \Delta_*=8$.

We will focus on the last source of error, since it will be the dominant one. The error one introduces when truncating the conformal block expansions of a 4pt function of identical scalars σ to some dimension $\newcommand{\D}{\Delta} \Delta_*$ was estimated in [26] (see also [27]) to be

$$\begin{align} \newcommand{\zb}{\bar{z}} \newcommand{\G}{\Gamma} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \newcommand{\calO}{{{\mathcal O}}} \displaystyle \left \vert \sum_{\calO: \Delta_ {{\mathcal O}} \geqslant \Delta_*} C_{\sigma \sigma \calO}^2\, g_{\Delta_\calO, \ell_\calO}(z,\zb) \right \vert \lesssim \frac{2^{4 \Delta_\sigma}}{\Gamma(4 \Delta_\sigma + 1)} \Delta_*^{4 \Delta_\sigma} \vert \rho(z)\vert ^{\Delta_*} \label{eq:trunc}\,. \nonumber \end{align} \tag{ 2.23 }$$

This error estimate is essentially optimal for real $0<z<1$, when the 4pt function is in a reflection positive configuration, and all conformal blocks are positive. This corresponds to the configuration with $\eta=1$ in the ρ plane. For configurations with $\eta<1$, conformal blocks decrease in absolute value by unitarity, and hence the same estimate (2.23) applies, although it's no longer optimal. When we integrate the 4pt function over the η coordinate, we will not be in a reflection positive configuration, but we will nonetheless bound the truncated operators contribution by its largest possible value, obtained for $\eta=1$. Clearly, the obtained error estimate will be overly conservative, since it does not take into account cancelations due to the varying sign of contributions of operators with spin.

Once we have constructed the approximated 4pt function, we integrate it in the region $\mathcal{R}$¹⁷. We follow the procedure outlined in appendix C of [16]: this consists in expanding the 4pt function as a power series in r and η, then integrating over r and dropping the diverging contributions of the identity and the energy operator. Finally we series-expand again with respect to η and we integrate the result exactly.

The data concerning the operator dimensions up to $\newcommand{\D}{\Delta} \Delta_*=8$ and their OPE coefficients can be found in table 2 of [21] (our $\newcommand{\calO}{{{\mathcal O}}} C_{\sigma\sigma\calO}=f_{\sigma\sigma\calO}$ given in that table). The OPE coefficients given there are in the normalization for which the small r limit of the conformal block is $\newcommand{\D}{\Delta} \renewcommand{\l}{\ell} g_{\Delta, \ell}\simeq\frac{\ell!}{(\nu)_ \ell} (-1){\hspace{0pt}}^ \ell C_ \ell^{\nu}(\eta) (4 r){\hspace{0pt}}^\Delta+\ldots$, where $C_ \ell^{\nu}$ is a Gegenbauer polynomial, $\nu=\frac{d}{2}-1=1/2$ and $(\nu)_\ell$ is the Pochhammer symbol. Using these values, we obtain $I_{d=3}=-1.950 \pm 0.005$. The error is dominated by the truncation error, which we estimate by integrating (2.23)¹⁸. The g³ term of the beta-function is then

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{eq:beta3d=3} \beta_{3}=12.26\pm 0.03\quad (d=3). \nonumber \end{align} \tag{ 2.24 }$$

If $\renewcommand{\l}{\ell} \renewcommand{\d}{{\rm d}} 0<\delta\ll 1$, the flow that we are studying will reach a fixed point at

$$\begin{align} \renewcommand{\d}{{\rm d}} \renewcommand{\l}{\ell} \displaystyle \label{eq:FP} g^2 = g_*^2= \delta/\beta_3. \nonumber \end{align} \tag{ 2.25 }$$

This fixed point is naturally identified with LRFP. Notice that for our picture to be correct, we must have $\beta_3>0$ (otherwise the fixed point at real g does not exist). The sign of $\beta_3$ was not manifest in the above calculations, since the regulated integrals are not sign-definite¹⁹. Still, we have seen that $\beta_3$ is positive in both $d=2$ and $d=3$. This provides a nontrivial check on our picture.

That $\beta_3>0$ means that the operator $\sigma\chi$ is marginally irrelevant at the crossover. The flow at the crossover will be affected by logarithmic corrections to scaling due to $\sigma\chi$. One must be aware of this fact when interpreting Monte Carlo simulation data in the crossover region. See the discussion in appendix C.1.

We will now apply the developed formalism to determine the anomalous dimension of a few selected operators, first in $d=2$ and then in $d=3$.

3.1.1. χ, σ and ${ \newcommand{\calO}{{{\mathcal O}}} \calO}$.

The arguments and the results of this section work for any dimension, so we keep d general. We will see in section 5.1 that the anomalous dimensions of the three operators we consider here can be discussed to all orders. As a check, we will reproduce here the lowest-order versions of those results using the general formalism.

We consider first the field χ. It clearly plays a very special role the $\sigma\chi$-flow, being described by a non-local action in the UV. As a consequence, we expect that χ does not get anomalous dimension to all orders in δ. This is similar to what happens for the ϕ field in the $\phi^4$-flow. Here we will check, by an explicit computation, that the anomalous dimension of χ vanishes at order g².

To see this, observe that the integral (3.6), with $\Phi=\chi$ and $\newcommand{\calO}{{{\mathcal O}}} \calO=\sigma\chi$, only has power divergences, and no logarithmic divergences. Indeed its integrand is

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \frac{1}{\vert x_1-x_2\vert ^{2\Delta_\sigma}}\left(\frac{1}{\vert x_1\vert ^{2\Delta_\chi}}+ \frac{1}{\vert x_1\vert ^{2\Delta_\chi}}+ \frac{1}{\vert x_1-x_2\vert ^{2\Delta_\chi}}\right). \nonumber \end{align} \tag{ 3.13 }$$

The integral only has power divergences by the same argument as that given for the beta-function subtraction terms (2.10). If we were to apply method 1 to this integral, we would end up with an identically vanishing integrand. Notice that in this case the OPE $\newcommand{\calO}{{{\mathcal O}}} \Phi\times \calO$ contains an operator σ with dimension $\newcommand{\D}{\Delta} \Delta_\sigma<\Delta_\Phi$, So more subtraction terms are needed than the ones given in equation (3.11). After these subtractions, the integrand is identically zero.

Next we consider the field σ. It is also special, because it acts as a source for χ, and so the classical equation of motion (EOM) of χ sets a linear non-local relation between the two. In quantum theory, this non-local EOM implies that the IR dimensions of σ and χ satisfy the shadow relation:

$$\begin{align} \newcommand{\D}{\Delta} \displaystyle \Delta_\chi+\Delta_\sigma=d. \nonumber \end{align} \tag{ 3.14 }$$

This should be compared with the shadow relation (1.7) for the $\phi^4$-flow. The two relations suggest that in the IR limit of the two flows, ϕ has to be identified with σ, and $\phi^3$ with χ. This fits nicely our proposed IR duality and will be discussed further in section 5.2.

Here we will check the shadow relation at the leading order in g. The anomalous dimension of the spin field σ can be reduced by a trick to the g³ term of the beta-function, which we already computed. Let us consider the original integral (3.6) for $\Phi=\sigma$. It's easy to see that this integral (multiplied by the overall volume) is exactly one third of the integral (2.6) in the beta-function calculation. Indeed, the integrand in both cases involves the 4pt function of σ multiplied by a correlation function of χ, which has one term in the first case and three terms in the second one. These three terms all contribute equally, and so we obtain $B=A/3$. This fact is not manifest the expressions provided by methods 1 and 2, but we checked it numerically. Given $B=A/3$, the anomalous dimension of σ is found to be:

$$\begin{align} \renewcommand{\d}{{\rm d}} \displaystyle \gamma_\sigma= \delta+ O(\delta^2). \nonumber \end{align} \tag{ 3.15 }$$

This checks the shadow relation at the lowest order.

We pause to notice that, as a consequence of the discussed RG equations, the 2pt function $\renewcommand{\l}{\ell} \langle\sigma\sigma\rangle$ at the crossover ($\renewcommand{\d}{{\rm d}} \delta=0$) exhibits a $\renewcommand{\l}{\ell} 1/\log r$ suppression in presence of the marginally irrelevant $\sigma\chi$ perturbation. See appendix B.

Finally, we discuss the operator $\newcommand{\calO}{{{\mathcal O}}} \calO=\sigma\chi$ which drives the flow. By the general RG arguments, the anomalous dimension of this operator should be given by:

$$\begin{align} \newcommand{\calO}{{{\mathcal O}}} \displaystyle \gamma_\calO(g)=\beta'(g). \nonumber \end{align} \tag{ 3.16 }$$

Using the leading beta-function expression, at the fixed point this becomes

$$\begin{align} \renewcommand{\d}{{\rm d}} \renewcommand{\l}{\ell} \newcommand{\calO}{{{\mathcal O}}} \displaystyle \label{eq:gammaO} \gamma_\calO(g_*)=3\delta, \nonumber \end{align} \tag{ 3.17 }$$

up to order g² corrections. As expected, $\newcommand{\calO}{{{\mathcal O}}} \calO$ becomes irrelevant at the IR fixed point:

$$\begin{align} \renewcommand{\d}{{\rm d}} \newcommand{\calO}{{{\mathcal O}}} \displaystyle [\calO]_{\rm LRFP}=d+2\delta+O(\delta^2), \nonumber \end{align} \tag{ 3.18 }$$

Equation (3.17) can also be obtained in the formalism of the previous section. It follows by noticing that $B=A$ for the renormalization of $\newcommand{\calO}{{{\mathcal O}}} \calO$.

3.1.2. ε.

To compute the anomalous dimension of ε, we need to use the result from the Ising minimal model (see e.g. [28])

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\vareps}{\varepsilon} \displaystyle \label{eq:Mattis} \langle \varepsilon(0) \sigma(z) \sigma(1) \varepsilon(\infty) \rangle =\frac{\vert 1+z\vert ^2}{4 \vert z\vert \vert 1-z\vert ^{1/4}}. \nonumber \end{align} \tag{ 3.19 }$$

To obtain the correlation function $\newcommand{\vareps}{\varepsilon} \newcommand{\calO}{{{\mathcal O}}} \renewcommand{\l}{\ell} \langle \varepsilon {{\mathcal O}}\calO \varepsilon\rangle$ we multiply (3.19) by $\renewcommand{\l}{\ell} \langle \chi(z)\chi(1)\rangle$. This time we will use method 1, and we will be able to carry out the integration analytically. We need to subtract the divergent terms due to the relevant operators, as shown in (3.11). The ε subtraction term is absent since, thanks to the Kramers–Wannier duality, $\newcommand{\vareps}{\varepsilon} C_{\varepsilon \varepsilon \varepsilon}=0$ in two dimensions. We obtain the integral

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \int {\rm d}^2 z \frac{1}{\vert 1-z\vert ^{4}}\left(\frac{\vert 1+z\vert ^2}{4 \vert z\vert }-1\right), \label{eq:epsInt} \nonumber \end{align} \tag{ 3.20 }$$

to be computed with circular cutoffs around 0, 1 and infinity. Careful evaluation shows that this integral is zero, see appendix D for the proof. Unfortunately, the only proof we found was by brute force, and it would be nice to find an underlying reason. We also checked this result by numerical evaluation. Numerically, this was also previously observed in [16], equations (6.14) and (5.28), in an unrelated computation which led to the same integral.

Therefore the anomalous dimension of ε vanishes at order $g_*^2$, while order $g_*^3$ will be zero by the $\mathbb{Z}_2$ selection rules. Therefore

$$\begin{align} \renewcommand{\d}{{\rm d}} \renewcommand{\l}{\ell} \newcommand{\vareps}{\varepsilon} \displaystyle \label{eq:gammaepsd=2} \gamma_\varepsilon = O(g_*^4)=O(\delta^2)\qquad (d=2). \nonumber \end{align} \tag{ 3.21 }$$

3.1.3. ${T_{\mu\nu}}$.

We now come to the discussion of the stress tensor operator, the source of some paradoxes discussed in the introduction. The LRFP is a non-local theory, and we do not expect it to contain a conserved local stress tensor operator. Let us examine this issue from the RG point of view. The UV theory SRFP+χ consists of two decoupled sectors. The SRFP is a local theory, with a conserved local stress tensor which we call $T_{\mu\nu}$. The χ sector is non-local, without a local stress tensor²⁰. When we perturb the UV theory with the operator $\sigma \chi$, the two sectors are no longer decoupled and locality of the SRFP is lost. This implies that, at the IR fixed point, the operator $T_{\mu\nu}$ will acquire an anomalous dimension. We will still call it $T_{\mu\nu}$ and will sometimes refer to it as the 'stress tensor', but it has to be kept in mind that this operator will not be conserved at the IR fixed point.

We will compute the $T_{\mu\nu}$ anomalous dimension in two ways, first directly and then using the multiplet recombination which will clarify the puzzle of missing states.

For the direct computation, it is sufficient to consider only one tensor component, say $T \equiv T_{zz}$, as all the components will acquire the same anomalous dimension. The stress tensor in the UV is conventionally normalized as

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle T(z,\bar{z}) T(0)\rangle = \frac{c}{2} \frac{1}{z^4}. \nonumber \end{align} \tag{ 3.22 }$$

In the case of the two dimensional Ising model, the central charge is $c=\frac{1}{2}$. The 4pt function $\renewcommand{\l}{\ell} \langle T(0) \sigma(z) \sigma(1) T(\infty) \rangle$ is then recovered in the standard way using the Ward identity twice on the 2pt function of σ. For the 4pt function involving two $\newcommand{\calO}{{{\mathcal O}}} \calO$ insertions we obtain:

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle T(0)\sigma\chi(z)\sigma\chi(1)T(\infty) \rangle=\frac{1}{\vert 1-z\vert ^{4}}\left(\frac{1}{4}+\frac{(1-z)^{2}(z^{2}+30z+1)}{256z^{2}}\right). \nonumber \end{align} \tag{ 3.23 }$$

Although the stress tensor is not a scalar operator, the discussion of section 3 on how to compute the anomalous dimensions still applies. We aim for an analytic result and use method 1. Since $\newcommand{\vareps}{\varepsilon} C_{T T \varepsilon}=0$ in $d=2$, we only need to subtract the contribution of the identity in (3.11). Note that since the stress tensor is not unit-normalized, subtracting the contribution of the identity means subtracting $(4\vert z-1\vert ^4){\hspace{0pt}}^{-1}$. The resulting integral can be evaluated exactly:

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \int {\rm d}^2 z \frac{1}{(1-\bar{z})^{2}}\frac{(z^{2}+30z+1)}{256z^{2}}=-\frac{15}{128}\pi. \label{eq:TInt} \nonumber \end{align} \tag{ 3.24 }$$

There is a subtlety in this computation related to the contribution of the region near $z=1$. This is explained in appendix D.

Equation (3.8) as written is valid for the unit-normalized operators. To make up for the fact that T is not, we need to multiply its rhs by an extra factor $2/c$. Finally, we obtain:

$$\begin{align} \renewcommand{\d}{{\rm d}} \renewcommand{\l}{\ell} \displaystyle \gamma_T=\frac{15}{32}\pi^2 g_*^2+O(g_*^4)\approx 3.65\, \delta+O(\delta^2)\quad (d=2), \label{eq:gammaT1} \nonumber \end{align} \tag{ 3.25 }$$

where we used (2.25) and (2.19).

We will now recompute the same anomalous dimension using the recombination of multiplets²¹. As we will see, this method requires only the integration of a 3pt function fixed by a Ward identity, and gives $\gamma_T$ as a function of $\newcommand{\D}{\Delta} \Delta_\sigma$ and of the central charge c for arbitrary d. So we switch to general d until the end of this section.

The stress tensor at the SRFP satisfies conservation equation $\partial^\mu T_{\mu \nu}=0$, meaning that some of his descendants are zero. As we say, it belongs to a short multiplet. The same operator taken to the IR, to the LRFP, which is a non-local theory, is not expected to be conserved: $\partial^\mu T_{\mu \nu} \propto V_\nu\ne 0$. In other words, the stress tensor multiplet becomes long by eating the $V_\nu$ multiplet. The vector $V_\nu$ must exist in the UV theory as well; this was puzzling in the standard picture. The puzzle is neatly resolved in our picture, since this multiplet can be easily constructed with the help of the χ field. Namely, we have:

$$\begin{align} \newcommand{\D}{\Delta} \displaystyle V_{\nu} = \sigma (\partial_\nu \chi) -\frac{\Delta_\chi}{\Delta_\sigma}(\partial_\nu \sigma)\chi. \nonumber \end{align} \tag{ 3.26 }$$

This is clearly a vector field and of dimension $d+1$ at the crossover point. The relative coefficient between the two terms is fixed by requiring that $V_\nu$ be a (non-unit normalized) vector primary at the crossover. For this it is sufficient to check that the 2pt function of $V_\nu$ and of the descendant $\partial_\nu(\sigma\chi)$ vanishes.

Since $V_\mu$ given above is the only candidate to be eaten, at the IR fixed point we expect

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{eq:recomb} \partial^\mu T_{\mu \nu}=b(g) V_\nu, \nonumber \end{align} \tag{ 3.27 }$$

where $b(g) \to 0$ as $g \to 0$.

We will be interested in the first nontrivial order: $b(g) = b_1 g+O(g^2)$. The value of b₁ can be determined by studying the 2pt function of $V_{\mu}$ with $T_{\mu \nu}$, computed at first order in perturbation theory. It will be more convenient to utilise the descendant $\partial^\mu T_{\mu \nu}$, as this will allow us to use the Ward identity. On the one hand from multiplet recombination (3.27) we expect at the lowest order in $g_*$:

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle \partial^\mu T_{\mu \nu}(x) V_{\rho}(y) \rangle_g \approx b_1 g_* \langle V_{\nu}(x) V_{\rho}(y)\rangle_0. \label{eq:geta} \nonumber \end{align} \tag{ 3.28 }$$

Here and below we mark with subscript g the IR fixed point correlators, while with subscript 0 the correlators in the UV theory SRFP+χ. The 2pt function of $V_\mu$ entering this equation can be computed explicitly given its definition:

$$\begin{align} \renewcommand{\d}{{\rm d}} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \langle V_{\mu}(x) V_{\nu}(0) \rangle_0=2d \frac{\Delta_\chi}{\Delta_\sigma}\frac{I_{\mu\nu}(x)}{\vert x\vert ^{2d+2}},\quad I_{\mu \nu}(x)=\delta_{\mu \nu} -2\frac{x_\mu x_\nu}{x^2}. \label{eq:normV} \nonumber \end{align} \tag{ 3.29 }$$

Notice that this functional form is consistent with the conformal primary nature of $V_\mu$.

On the other hand perturbation theory predicts for the correlator in the lhs of (3.28)

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\calO}{{{\mathcal O}}} \displaystyle \langle \partial^\mu T_{\mu \nu}(x) V_{\rho}(y) \rangle_g = g_* \int {\rm d}^d z \langle \partial^\mu T_{\mu \nu}(x) V_{\rho}(y) \calO(z)\rangle_0. \label{eq:geta1} \nonumber \end{align} \tag{ 3.30 }$$

The 3pt function that we need to integrate is the sum of two factorized terms:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \newcommand{\calO}{{{\mathcal O}}} \displaystyle \nonumber \langle \partial^\mu T_{\mu \nu}(x) V_{\rho}(y) \calO(z) \rangle_0=&~\langle \partial^\mu T_{\mu \nu}(x) \sigma (y) \sigma(z)\rangle \langle \partial_\rho \chi(y) \chi(z) \rangle_0 \nonumber \\ &- \frac{\Delta_\chi}{\Delta_\sigma}\langle \partial^\mu T_{\mu \nu}(x) \partial_\rho \sigma (y) \sigma(z)\rangle \langle \chi(y) \chi(z) \rangle_0. \nonumber \end{align} \tag{ 3.31 }$$

When the 3pt functions in the rhs are expressed using the Ward identity²² of the unperturbed theory, we get terms proportional to $\renewcommand{\d}{{\rm d}} \delta(x-y)$ and to $\renewcommand{\d}{{\rm d}} \delta(x-z)$. We assume that $x \ne y$, so only $\renewcommand{\d}{{\rm d}} \propto\delta(x-z)$ terms are important. They yield a non-zero contribution when we integrate over z. We obtain

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \newcommand{\calO}{{{\mathcal O}}} \displaystyle \nonumber &\int {\rm d}^d z \langle \partial^\mu T_{\mu \nu}(x) V_{\rho}(0) \calO(z)\rangle_0\nonumber \\ &=-\langle \sigma(x) \partial_\nu \sigma(0) \rangle \langle \partial_\rho \chi(x)\chi(0) \rangle+\frac{\Delta_\chi}{\Delta_\sigma} \langle \partial_\nu \sigma(x) \partial_\rho \sigma(0) \rangle \langle \chi(x) \chi(0)\rangle =2\Delta_\chi \frac{I_{\nu \rho}(x)}{\vert x\vert ^{2d+2}}. \nonumber \end{align} \tag{ 3.32 }$$

Using (3.28)–(3.30), the value of b₁ is fixed:

$$\begin{align} \newcommand{\D}{\Delta} \displaystyle b_1=\Delta_\sigma/d. \nonumber \end{align} \tag{ 3.33 }$$

Now let us compute the anomalous dimension of $T_{\mu\nu}$. The 2pt function normalization customary for d dimensional CFT is [33] (see also [34])

$$\begin{align} \renewcommand{\d}{{\rm d}} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \label{eq:TTSRFP} \langle T_{\mu \nu}(x) T_{\rho \sigma}(0) \rangle = \frac{c_T}{2{\rm S}_{d}^2} \frac{1}{\vert x\vert ^{2\Delta_T}} \left[I_{\mu \rho}(x) I_{\nu \sigma}(x)+\left(\mu \leftrightarrow \nu \right) -\frac{2}{d}\delta_{\mu \nu} \delta_{\lambda \sigma}\right]. \nonumber \end{align} \tag{ 3.34 }$$

In this normalization, and assuming the Ward identities are normalized as in footnote ²², the free massless scalar has $c_T=d/(d-1)$.

Equation (3.34) follows just from conformal invariance and the fact that $T_{\mu\nu}$ transforms as a rank 2 symmetric traceless primary. So it's valid both at the SRFP in the UV, and at the LRFP in the IR. In this argument we assumed conformal invariance of the LRFP, which will be discussed below in section 4²³. In the UV we have $c_T=c_T^{\rm SRFP}$ and $\newcommand{\D}{\Delta} \Delta_T=d$, corresponding to the conserved local stress tensor. In the IR both c_T and $\newcommand{\D}{\Delta} \Delta_T$ receive $O(g_*^2)$ corrections. At the intermediate distances there is some interpolating behavior which will not be important.

Let $\newcommand{\D}{\Delta} \Delta_T=d+\gamma_T$ in the IR, where $\gamma_T$ is the anomalous dimension. The quantity of interest is the 2pt function of the divergence of $T_{\mu\nu}$ at the LRFP which can be found by an explicit differentiation of (3.34). This vanishes for $\gamma_T=0$, consistent with the fact that $T_{\mu\nu}$ is conserved in the UV, and for nonzero $\gamma_T$ is given by:

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle \partial^\mu T_{\mu \nu}(x) \partial^\rho T_{\rho \sigma}(0) \rangle_{g} \approx \frac{c_T}{{\rm S}_{d}^2} \gamma_T \left(d+1-\frac{2}{d}\right) \frac{I_{\nu \sigma }}{\vert x\vert ^{2d+2}}, \label{eq:rec1} \nonumber \end{align} \tag{ 3.35 }$$

In (3.35) we dropped terms higher order in $g_*^2$. One such higher order term is the correction to c_T which will not play any role, so in all subsequent equations $c_T=c_T^{\rm SRFP}$.

At the same order, using the recombination of multiplets equation (3.27), we expect:

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle \partial^\mu T_{\mu \nu}(x) \partial^\rho T_{\rho \sigma}(0) \rangle_{g}\approx b_1^2 g_*^2\langle V_{\nu}(x) V_\sigma(0)\rangle_{0}.\label{eq:rec2} \nonumber \end{align} \tag{ 3.36 }$$

From the last two equations, the 2pt function of $V_\mu$, and the value of b₁ we find the lowest-order anomalous dimension of the stress tensor:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \boxed{\gamma_T=\frac{2{\rm S}_{d}^2}{c_T}\frac{\Delta_\sigma (d-\Delta_\sigma)}{{\rm d}^2+d-2} g_*^2+O(g_*^4)}. \label{eq:gammaT2} \nonumber \end{align} \tag{ 3.37 }$$

Let us now specialize to $d=2$. In the usual 2d normalization, the 2d critical Ising has central charge $1/2$, half that of the free massless scalar. As mentioned, $c_T=\frac{d}{d-1}=2$ for the free massless scalar in the normalization of (3.34) and of footnote ²², and so $c_T=1$ for the 2d Ising in the same normalization. It is then easy to see that (3.37) agrees with the result (3.25) obtained via the integration of the 4pt function.

The anomalous dimensions of χ, σ and $\newcommand{\calO}{{{\mathcal O}}} \calO$ were already discussed in section 3.1.1 for any d.

3.2.1. $\boldsymbol{ \newcommand{\vareps}{\varepsilon} \varepsilon}$.

Recall that order $g_*^2$ anomalous dimension of the energy operator was zero in $d=2$, for mysterious reasons unexplained by any obvious symmetry. As we will see this does not happen in three dimensions. We set up a numerical computation for this anomalous dimension using the CFT data from the numerical conformal bootstrap. In order to compute the 4pt function $\newcommand{\vareps}{\varepsilon} \renewcommand{\l}{\ell} \langle \varepsilon \sigma \sigma \varepsilon \rangle$, we will need the operator dimensions and the OPE coefficients of the operators appearing in the $\sigma \times \sigma$, $\newcommand{\vareps}{\varepsilon} \sigma \times \varepsilon$ and $\newcommand{\vareps}{\varepsilon} \varepsilon \times \varepsilon$ OPEs. For operators up to $\newcommand{\D}{\Delta} \Delta_*=8$, these can be found in table 2 of [21]. We will use method 2. We construct the 4pt function in the region where one ε is close to one $\newcommand{\calO}{{{\mathcal O}}} \calO$ and the region where the $\newcommand{\calO}{{{\mathcal O}}} \calO$'s are close together:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \newcommand{\calO}{{{\mathcal O}}} \newcommand{\vareps}{\varepsilon} \displaystyle \langle\varepsilon(0)\sigma(z)\sigma(1)\varepsilon(\infty)\rangle=\frac{1}{\vert z\vert ^{\Delta_\sigma+\Delta_\varepsilon}}\sum_{\calO:\Delta_\calO<\Delta_*}C_{\sigma \varepsilon \calO}^2 g^{\Delta_{\varepsilon \sigma},\Delta_{\sigma \varepsilon}}_{\Delta_\calO, \ell_\calO}(z,\bar{z}), \label{eq:eps3d1} \nonumber \end{align} \tag{ 3.38 }$$

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \newcommand{\calO}{{{\mathcal O}}} \newcommand{\vareps}{\varepsilon} \displaystyle \langle\sigma(0)\sigma(z)\varepsilon(1)\varepsilon(\infty)\rangle=\frac{1}{\vert z\vert ^{2\Delta_\sigma}}\sum_{\calO:\Delta_\calO<\Delta_*}C_{\sigma \sigma \calO} C_{\varepsilon \varepsilon \calO} g^{0,0}_{\Delta_\calO, \ell_\calO}(z,\bar{z}). \label{eq:eps3d2} \nonumber \end{align} \tag{ 3.39 }$$

Here $\newcommand{\calO}{{{\mathcal O}}} \newcommand{\D}{\Delta} g_{\Delta_\calO, \ell_\calO}$ with upper indices are the conformal blocks for the external scalars with unequal dimensions, which we compute via recursion relations from [35]. The operators entering the sum in the first (resp. second) equation are $\mathbb{Z}_2$ odd (resp. even). In both cases, we will be integrating over the region $\mathcal{R}$ defined in section 2.

Once again, the largest error contribution when approximating the 4pt function will come from the truncation of the spectrum at dimension $\newcommand{\D}{\Delta} \Delta_*$. The same line of reasoning used to obtain the truncation error for four identical scalar in [26] will go through in the case of equation (3.38). Analyzing the proof in [26], it's possible to see that the truncation error will be given by (2.23) with the change $\newcommand{\vareps}{\varepsilon} \newcommand{\D}{\Delta} \Delta_\sigma \to (\Delta_\sigma+\Delta_ \varepsilon)/2$ in all occurrences in the rhs

For equation (3.39), however, we cannot map the 4pt function onto a reflection positive configuration, and therefore we cannot find a bound on the contribution of the truncated operators in the same way. We need to first use Cauchy's inequality so that the tail of $\newcommand{\vareps}{\varepsilon} \renewcommand{\l}{\ell} \langle \sigma \sigma \varepsilon \varepsilon \rangle$ can be bounded by the tails of $\renewcommand{\l}{\ell} \langle \sigma \sigma \sigma \sigma \rangle$ and $\newcommand{\vareps}{\varepsilon} \renewcommand{\l}{\ell} \langle \varepsilon \varepsilon \varepsilon \varepsilon \rangle$. At this point we can use again the result of [26], and we obtain

$$\begin{align} \newcommand{\G}{\Gamma} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \newcommand{\calO}{{{\mathcal O}}} \newcommand{\vareps}{\varepsilon} \displaystyle \left\vert \sum_{\calO:\Delta_\calO>\Delta_*}C_{\sigma \sigma \calO} C_{\varepsilon \varepsilon \calO} g^{0,0}_{\Delta_\calO, \ell_\calO}(z,\bar{z})\right\vert \lesssim \frac{2^{2\Delta_\sigma +2 \Delta_\varepsilon}}{\sqrt{\Gamma(4\Delta_\sigma+1) \Gamma(4\Delta_\varepsilon+1)}} \Delta_*^{2\Delta_\sigma +2 \Delta_\varepsilon} \vert \rho(z)\vert ^{\Delta_*}. \nonumber \end{align} \tag{ 3.40 }$$

Truncating the CFT data up to $\newcommand{\D}{\Delta} \Delta_*=8$²⁴, and carrying out the integration in the region $\mathcal{R}$, we obtain a nonzero value, unlike in $d=2$. The order $g_*^2$ anomalous dimension is

$$\begin{align} \renewcommand{\d}{{\rm d}} \renewcommand{\l}{\ell} \newcommand{\vareps}{\varepsilon} \displaystyle \label{eq:gammaeps} \gamma_\varepsilon\approx 3.3 g_*^2+O(g_*^4)\approx 0.27\,\delta+O(\delta^2)\quad (d=3), \nonumber \end{align} \tag{ 3.41 }$$

where in the second equality we used (2.25) and (2.24). The total truncation error on the coefficient 3.3, estimated as above, is $\pm 0.5$. So we are confident that ε gets a nonzero anomalous dimension in $d=3$ already at the lowest order allowed by the $\mathbb{Z}_2$ selection rules.

3.2.2. $\boldsymbol{T_{\mu\nu}}$.

In $d=3$ the data needed to compute the 4pt function $\renewcommand{\l}{\ell} \langle T \sigma\sigma T \rangle$ are not yet available. So we cannot compute the anomalous dimension of $T_{\mu\nu}$ using the formalism of section 3. However, we can still use the general d expression (3.37) obtained by using the recombination method.

Using the spin field dimension $\newcommand{\D}{\Delta} \Delta_\sigma = 0.518\, 1489(10)$ [20] and the central charge $c_T/c_T^{\rm free}=0.946\, 539(1)$ [16, 18] with the free scalar central charge $c_T^{\rm free}=d/(d-1)=3/2$, we get

$$\begin{align} \renewcommand{\d}{{\rm d}} \displaystyle \gamma_T=28.605\,55(6)g_*^2+O(g_*^4)\approx 2.33\, \delta+O(\delta^2)\,\quad (d=3). \nonumber \end{align} \tag{ 3.42 }$$

We will be following mostly recent work [6] by some of us, where further references and details can be found. For this discussion we normalize the action and the coupling as

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\calN}{{{\mathcal N}}} \displaystyle S=\frac{\calN_s}2 \int {\rm d}^dx\, \phi\ (- \partial^2)^{s/2} \phi+\frac{g_0}{4!} \int {\rm d}^d x\, \phi^4, \label{eq:S} \nonumber \end{align} \tag{ 4.2 }$$

choosing $\newcommand{\calN}{{{\mathcal N}}} \calN_s$ so that the free 2pt function of ϕ is unit-normalized.

All-order statements are proven by setting up perturbation theory within the analytic regularization scheme, where one defines the correlation functions via analytic continuation of position-space Feynman integrals in $\newcommand{\eps}{\epsilon} \epsilon$. This automatically subtracts UV divergences. Also this scheme is an example of a mass independent scheme, implying that the mass term does not renormalize and the IR fixed point is reached for zero UV mass. For this reason we dropped the $\phi^2$ term from the original action (1.2).

Although there are no UV divergences, some diagrams contain poles in $\newcommand{\eps}{\epsilon} \epsilon$. It helps to reorganize perturbation theory by absorbing these poles into a coupling redefinition. Namely, one defines a dimensionless 'renormalized coupling' g related to the original coupling g₀ by

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\eps}{\epsilon} \displaystyle \label{eq:g0} g_0= Z_g(g, \epsilon)g\mu^ \epsilon \nonumber \end{align} \tag{ 4.3 }$$

where μ is an energy scale, and Z_g contains a series of ascending poles in $\newcommand{\eps}{\epsilon} \epsilon$:

$$\begin{align} \newcommand{\eps}{\epsilon} \displaystyle Z_g(g, \epsilon)=1+\sum _{k=1}^\infty \frac{f_k(g)}{\epsilon^k}. \nonumber \end{align} \tag{ 4.4 }$$

The factor Z_g is chosen in such a way that perturbative expansion of correlation functions at fixed distances and for a fixed g and μ no longer contains any poles in $\newcommand{\eps}{\epsilon} \epsilon$. That this can be done means that the theory at $\newcommand{\eps}{\epsilon} \epsilon=0$ is renormalizable. Renormalizability can be proven similarly to how it's done for the local theories with marginal couplings (see note 6 in [6] for references). Our case is even simpler than the local case, in that there is no wavefunction renormalization, due to the fact that the kinetic term for ϕ is non-local. This fact implies that the gamma-function $\gamma_\phi(g)$ vanishes. We now review the standard argument that this also implies vanishing of the anomalous dimension at the fixed point²⁵.

Since $\gamma_\phi(g)=0$, the Callan–Symanzik (CS) equation for the 2pt function of ϕ takes a simplified form:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \label{CS} [r\, \partial_r+\beta(g) \partial_g ]F(r,g)=0,\qquad \langle\phi(x)\phi(0)\rangle=F(r,g)/\vert x\vert ^{2\Delta_\phi},\qquad r=\mu\vert x\vert. \nonumber \end{align} \tag{ 4.5 }$$

The beta-function is

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\eps}{\epsilon} \displaystyle \beta(g)=\mu\, \partial_\mu g\vert _{g_0}=-\frac{\epsilon g}{1+g(\log Z_g)^{\prime}}=- \epsilon g+g^2 f_1^{\prime}(g) \nonumber \end{align} \tag{ 4.6 }$$

(for deriving the last equation the absence of poles in $\newcommand{\eps}{\epsilon} \epsilon$ is crucial). Integrating the CS equation we find

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{Fsq} F(r,g)=\hat F (\bar g(r,g)) \nonumber \end{align} \tag{ 4.7 }$$

where $\bar g(r, g)$ is the 'running coupling' obtained by solving the beta-function equation with a boundary condition at $r=1$:

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{eq:running} r\, \partial_r g=-\beta(\bar g),\quad \bar g(1,g)=g. \nonumber \end{align} \tag{ 4.8 }$$

Note that we are not interested in the limit $\newcommand{\eps}{\epsilon} \epsilon\to0$, but in the case when $\newcommand{\eps}{\epsilon} \epsilon$ is finite and small. So the reorganization of perturbation theory in terms of the coupling g is not a physical necessity, but a technical tool which disentangles the structure of the perturbative expansion. In principle, the correlation functions obtained by solving the CS equations could be obtain by taking the original perturbation theory in g₀ and carefully resumming terms enhanced by poles in $\newcommand{\eps}{\epsilon} \epsilon$. The RG technology is nothing but a systematic way to perform such resummations. See [37], p 90, for a nice discussion of this point.

The beta-function at the lowest non-trivial order takes the form

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\eps}{\epsilon} \displaystyle \label{eq:beta-stan} \beta(g)=- \epsilon g+K g^2+\ldots,\quad K=(3/2) {S}_d, \nonumber \end{align} \tag{ 4.9 }$$

so that the IR fixed point is at $\newcommand{\eps}{\epsilon} g_*= \epsilon/K$. From (4.7) we conclude that in the IR, where $\bar g\to g_*$, the 2pt function of ϕ behaves as

$$\begin{align} \newcommand{\D}{\Delta} \displaystyle \hat F(g_*)/\vert x\vert ^{2\Delta_\phi}. \nonumber \end{align} \tag{ 4.10 }$$

with the same power law as in the UV (although a different constant prefactor).

The function $\hat F$ can be found by matching with fixed order perturbation theory. At the lowest nontrivial order we have (see [6], equation (3.26))

$$\begin{align} \newcommand{\G}{\Gamma} \renewcommand{\l}{\ell} \displaystyle \hat F(g)=1+ Q g^2+\ldots,\qquad Q=(\pi^d/6) {\Gamma\left(-\frac d4\right)}/{\Gamma\left(\frac{3d}{4}\right)}<0. \label{eq:Q} \nonumber \end{align} \tag{ 4.11 }$$

This will be useful in section 4.4.

We have a non-local equation of motion (EOM) which relates the ϕ and $\phi^3$ operators:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\calN}{{{\mathcal N}}} \displaystyle \label{eq:NLEOM} \calN_s (- \partial^2)^{s/2}\phi+\frac{g_0^2}{3!} \phi^3 = 0. \nonumber \end{align} \tag{ 4.12 }$$

This equation implies that in the IR the dimensions of ϕ and $\phi^3$ should be related by

$$\begin{align} \displaystyle [\phi^3]=[\phi]+s \nonumber \end{align} \tag{ 4.13 }$$

which, given that the IR dimension of $[\phi]$ is known, can be rewritten as the 'shadow relation'

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{eq:sh} [\phi]+[\phi^3]=d. \nonumber \end{align} \tag{ 4.14 }$$

A formal proof of the shadow relation can be given as follows (see [6] for another proof). In perturbation theory, the non-local EOM (4.12) simply means that the diagrams contributing to correlation functions of $\phi^3$ are the diagrams of ϕ amputated by one propagator²⁶. In other words correlators of $\phi^3$ are related to those of ϕ by (in momentum space)

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \frac{g_0}{3!} \langle\phi^3(\,p)\ldots\rangle = \vert\,p\vert ^s \langle\phi(\,p)\ldots\rangle. \nonumber \end{align} \tag{ 4.15 }$$

This can be rewritten as

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\eps}{\epsilon} \displaystyle \label{eq:phi3ren} \frac{g \mu^ \epsilon}{3!} \langle (\phi^3)_{\rm R}(\,p)\ldots\rangle = \vert\,p\vert ^s \langle\phi(\,p)\ldots\rangle. \nonumber \end{align} \tag{ 4.16 }$$

where we expressed g₀ via g via (4.3) and defined the renormalized $\phi^3$ operator via

$$\begin{align} \displaystyle \phi^3 = Z_{\phi^3}\cdot (\phi^3)_{\rm R},\qquad Z_{\phi^3} = Z_g^{-1}. \nonumber \end{align} \tag{ 4.17 }$$

Equation (4.16) implies that the correlation functions of the so defined $(\phi^3)_{\rm R}$ are free of poles in $\newcommand{\eps}{\epsilon} \epsilon$. Since $Z_{\phi^3}$ is related to $Z_g^{-1}$, a short computation allows to express $\gamma_{\phi^3}$ via the beta-function all along the flow:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\eps}{\epsilon} \displaystyle \gamma_{\phi^3}(g)=\mu\, \partial_\mu \log Z_{\phi^3}\vert _{g_0} = \epsilon+\beta(g)/g. \nonumber \end{align} \tag{ 4.18 }$$

In particular in the IR we have $\newcommand{\eps}{\epsilon} \gamma_{\phi^3}\to \epsilon$, which given the dimension of ϕ is equivalent to the shadow relation (4.14). This argument proves the shadow relation to all orders in perturbation theory.

It has been shown in [6] that the IR fixed point of the $\phi^4$-flow has $SO(d+1, 1)$ conformal invariance to all orders in perturbation theory. Here we will present a somewhat streamlined version of the same argument.

Let $\renewcommand{\l}{\ell} G(x_1\ldots x_N)$ be an N-point correlation function of ϕ. We would like to show that at large distances it is conformally invariant. Large here means compared to the distance scale $\mu_c^{-1}$ where, roughly, the RG flow transitions from the UV to IR behavior. For example, we can define the energy scale $\mu_c$ as where the running coupling is half way between the UV and IR fixed points.

The first step is to show that correlation functions of the $\phi^4$-flow satisfy broken conformal Ward identities (for dilatations and special conformal transformations). These identities are valid at all distances and take the form:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \sum _{i=1}^N [x_i. \partial_{x_i} +\Delta_\phi] G(x_1\ldots x_N) = B_D=b_D \beta(g)\, \nonumber \end{align} \tag{ 4.19 }$$

$$\begin{align} \renewcommand{\d}{{\rm d}} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \sum _{i=1}^N \left[(2x_i^\mu x_i^\lambda-\delta^{\mu\lambda}x_i^2)\frac{\partial}{\partial{x^\mu_i}} +2\Delta_\phi x_i^\lambda\right]G(x_1\ldots x_N) = B_k = b_K \beta(g), \nonumber \end{align} \tag{ 4.20 }$$

where the breaking terms $B_D, B_K$ are, as shown, proportional to $\beta(g)$, with the following proportionality coefficients:

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle b_D = - \partial_g G(x_1\ldots x_N) \nonumber \end{align} \tag{ 4.21 }$$

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\eps}{\epsilon} \displaystyle =\frac{\mu^{\epsilon}}{4!} \int {\rm d}^d x \,G(x_1\ldots x_N;(\phi^4)_{\rm R}(x)), \label{eq:scaleLRI1} \nonumber \end{align} \tag{ 4.22 }$$

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\eps}{\epsilon} \displaystyle b_K =2 \frac{\mu^{\epsilon}}{4!} \int {\rm d}^d x\, x^\lambda \,G(x_1\ldots x_N;(\phi^4)_{\rm R}(x)). \label{eq:scLRI1} \nonumber \end{align} \tag{ 4.23 }$$

As shown, b_D can be represented in two equivalent ways. The first way is via the derivative of the correlator with respect to g. This is the way leading to the CS equation. The second way is via an integral of $\renewcommand{\l}{\ell} G(x_1\ldots x_N;(\phi^4)_{\rm R}(x))$, which is the correlator of N ϕ's with one insertion of the renormalized $\phi^4$ field. On the other hand, b_K has only one representation, of the second type.

It may be surprising that the considered theory satisfies the above Ward identities, given that it does not have a local stress tensor. These identities have been derived in [6] by using the 'Caffarelli–Silvestre trick'. This trick consists in formally representing the non-local part of the Lagrangian as a free massless scalar theory living in (in general non-integer) $D=d+2-s$ dimensions. The original d-dimensional space where the interaction $\phi^4$ is concentrated is a plane in this D-dimensional ambient space. The D-dimensional theory being local, it has a stress tensor and enjoys D-dimensional scale and conformal invariance. The interaction plane breaks some of the ambient symmetries, but one can still derive Ward identities for the transformations 'tangential to the plane', and restricting those to the plane, one gets the above Ward identities. This is done in detail in [6] and we will not repeat the derivation here.

The second step is to show that the above Ward identities imply conformal invariance. For this we need to show that the breaking terms B_D, B_K are negligible at large distances. Instead of keeping $\mu_c$ fixed and tending $x_i\to\infty$ we will, equivalently, do the opposite. Namely, let us send $g\to g_*$ (for fixed μ). In this limit $\mu_c\to \infty$: the RG flow transitions from UV to IR at very short distances. So at all finite distances x_i the correlators tend to those of LRFP. In this situation, we want to show that the breaking terms approach zero. This will establish that the LRFP correlators satisfy conformal Ward identities.

The breaking terms are proportional to $\beta(g)$. Since $\beta(g_*) = 0$, it's enough to show that the coefficients b_D, b_K remain finite in the considered limit. All known to us prior literature on the related matters tacitly assumed this fact without any justification²⁷.

However, as we explained in [6], this issue actually deserves discussion. The end result is that b_D and b_K do remain finite, and so we do have conformal invariance. The proof is a bit technical and we relegate it to a separate section below. The key idea is to use the fact that b_D has two representations.

4.3.1. Proof that b_D and b_K remain finite.

That b_D remains finite is actually obvious using the first representation: since correlation functions have a good power series expansion in g, the derivative with respect to g at $g=g_*$ is finite, order by order in perturbation theory. This is clearly true when all distances $x_i=O(\mu^{-1})$, so that perturbation theory does not contain large logarithms. For smaller or larger distances we can use CS equation which expresses the correlator in terms of the running coupling. The end result is that $\partial_g G$ for fixed x_i, has a finite limit as $g\to g_*$. So b_D is finite.

However, let us examine the finiteness of b_D and b_K for $g\to g_*$ using the second representation (which is the only one available for b_K). One realizes that the finiteness is not so obvious from this point of view. Since the field $\phi^4$ is irrelevant at the LRFP, the correlators in the rhs will have non-integrable powerlike singularities for x near x_i. So for $g=g_*$ the integrals in the rhs are formally powerlike divergent. On the other hand for g somewhat below $g_*$ the integrals are convergent because at $\renewcommand{\l}{\ell} \vert x-x_i\vert \lesssim \mu_c^{-1}$ we go to the UV regime where $\phi^4$ is relevant.

A moment's thought shows that the only potentially divergent term in the integrals for b_D and b_K can be due to the $\phi\times \phi^4 \supset \phi$ OPE (other terms in the OPE will give rise to convergent integrals):

$$\begin{align} \displaystyle \phi(0) (\phi^4)_{\rm R}(x) \sim C(\mu_c, x) \phi(0).\nonumber \end{align} \tag{ 4.24 }$$

Namely, b_D and b_K will remain finite if and only if

$$\begin{align} \displaystyle I=\int {\rm d}^d x\, C(\mu_c, x) \nonumber \end{align} \tag{ 4.25 }$$

remains finite in the considered limit²⁸. We are using here the fact that the OPE kernel C is rotationally invariant, being the OPE of two scalars, and so when we integrate in the neighborhood of x_i in b_K, we will get a contribution proportional to $\renewcommand{\l}{\ell} x_i^\lambda b_K$, while the term proportional to

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \int {\rm d}^d x\, (x-x_i)^\lambda C(\mu_c, x-x_i) \nonumber \end{align} \tag{ 4.26 }$$

vanishes by rotational invariance.

Now we can complete the argument. We have seen that b_D can only remain finite if I is finite. But we know that b_D remains finite, from the first representation. So I remains finite. But then b_K also remains finite. So we are done.

Let us make a couple more remarks. Notice that we can write, by dimensional analysis

$$\begin{align} \displaystyle C(\mu_c, x)= \frac{f(x \mu_c)}{\vert x\vert ^{[\phi^4]_{\rm IR}}}. \nonumber \end{align} \tag{ 4.27 }$$

Then

$$\begin{align} \displaystyle I=\int {\rm d}^d x \,C(\mu_c, x)= \mu_c^{[\phi^4]_{\rm IR}-d} \int {\rm d}^d x\, C(1, x). \nonumber \end{align} \tag{ 4.28 }$$

So the only way $I$ can remain finite is if the integral is actually zero! This is a nontrivial constraint on the OPE kernel, and a curious example of UV/IR conspiracy in quantum field theory. In appendix E we present an explicit check of this fact at the lowest nontrivial order in g. As we find there, the OPE kernel crosses from negative in the UV to positive in the IR, so that the total integral is zero.

If we assume that $I=0$, then we can justify the following prescription for evaluating the integrals in b_D and b_K, directly at $g=g_*$. As mentioned, those integrals at $g=g_*$ are formally UV divergent, as a power of UV cutoff. The prescription consists in just dropping this divergence. Our argument that $I=0$ can be seen as a formal justification that this naive prescription is correct.

As explained in appendix C of [6], the non-local EOM (4.12), together with the conformal symmetry, lead to a relation between the OPE coefficients of the operators ϕ and $\phi^3$. Let us review this result. The non-local EOM can be written in position space as

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \label{eq:nleomapp} \int {\rm d}^d y \frac{1}{\vert x-y\vert ^{2(d - \Delta_\phi)}} \phi(y) = C\phi^3(x). \nonumber \end{align} \tag{ 4.29 }$$

The precise value of C will be unimportant.

We are interested in the large distance behavior of correlators of ϕ and $\phi^3$, which is described by the LRFP. As discussed in the previous section, LRFP is a CFT. The operators ϕ and $\phi^3$ are primary operators in this CFT (this is obvious for ϕ and also true for $\phi^3$ since there is no operator of which it could be a descendant).

From the CFT point of view, the overall normalization of operators is not so important. So we introduce operators $\tilde\phi$, $\tilde{\phi^3}$, unit-normalized in the IR. How is their normalization related to that of ϕ, $\phi^3$? It's convenient to write the 2pt function of ϕ in the form:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \newcommand{\eps}{\epsilon} \displaystyle \label{eq:rho-def} {\langle \phi(x) \phi(0) \rangle} = \frac{1 + \rho(\epsilon)}{\vert x\vert ^{2 \Delta_\phi}}. \nonumber \end{align} \tag{ 4.30 }$$

This and all subsequent equations in this section are at large distances so that we are at the LRFP. The coefficient $\newcommand{\eps}{\epsilon} \rho(\epsilon)$ is a nontrivial function of which is $\newcommand{\eps}{\epsilon} O(\epsilon^2)$ as $\newcommand{\eps}{\epsilon} \epsilon \to 0$. From (4.11) we have

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\eps}{\epsilon} \displaystyle \label{eq:rho0} \rho(\epsilon)= \epsilon^2 Q/K^2+\ldots. \nonumber \end{align} \tag{ 4.31 }$$

We can now act with the non-local EOM twice on (4.30) and derive the normalization of $\phi^3$ in the IR. It's important to realize though that the 1 in the numerator does not contribute to the 2pt function of $\phi^3$. This is obvious by looking at which diagrams contribute (see [6], (3.37) and footnote ²⁶). So we obtain:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \newcommand{\eps}{\epsilon} \displaystyle \label{eq:phi3phi3app} {\langle \phi^3(x) \phi^3(0) \rangle} = \frac{\rho(\epsilon) M_2 / C^2}{\vert x\vert ^{2 \Delta_{\phi^3}}}, \nonumber \end{align} \tag{ 4.32 }$$

where $\newcommand{\D}{\Delta} \Delta_{\phi^3} = d - \Delta_\phi$ consistently with the shadow relation, and

$$\begin{align} \newcommand{\G}{\Gamma} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \label{eq:M2} M_2 = \frac{\pi^d \Gamma(d/2 - \Delta_\phi) \Gamma(\Delta_\phi - d/2)}{\Gamma(d- \Delta_\phi) \Gamma(\Delta_\phi)}. \nonumber \end{align} \tag{ 4.33 }$$

Notice that the integral in the non-local EOM is over all distances, while the 2pt functions of ϕ used in the calculation holds only at large distances. We are taking a reasonable assumption that the large distance limit commutes with the non-local EOM. This has been shown in [6] for this 2pt function calculation, but not for the 3pt function calculation performed below. It would be interesting to prove this assumption in full generality.

As discussed in [6], we expect LRFP to be described by a unitary CFT. Indeed, in the considered range of s the field ϕ satisfies the unitarity bound in the UV, so we have an RG flow starting at a unitary UV theory perturbed by a hermitian operator²⁹. Unitarity of the LRFP implies that we should have nonnegative 2pt functions. Notice also that $M_2<0$ for $0<s<2$. This means that we expect:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\eps}{\epsilon} \displaystyle \label{eq:expect} -1< \rho(\epsilon) < 0. \nonumber \end{align} \tag{ 4.34 }$$

For small $\newcommand{\eps}{\epsilon} \epsilon$, this is in agreement with the explicit result (4.31), since $Q<0$.

Provided (4.34) holds, the relation with the unit-normalized fields is as follows:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\eps}{\epsilon} \displaystyle \label{eq:normal} \tilde \phi(x) \equiv \frac{1}{\sqrt{1 + \rho(\epsilon)}} \phi(x), \qquad \qquad \tilde \phi^3(x) \equiv \frac{1}{\sqrt{\rho(\epsilon) M_2 / C^2}} \phi^3(x). \nonumber \end{align} \tag{ 4.35 }$$

The non-local EOM relation between the unit-normalized fields is the same as (4.29) but with C replaced by

$$\begin{align} \newcommand{\eps}{\epsilon} \displaystyle \tilde C= \sqrt{\frac{\rho(\epsilon) M_2}{1 + \rho(\epsilon)}}. \nonumber \end{align} \tag{ 4.36 }$$

Not surprisingly, C has canceled out from this relation.

Now let $\newcommand{\calO}{{{\mathcal O}}} \calO_1$ and $\newcommand{\calO}{{{\mathcal O}}} \calO_2$ be two scalar primary operators of the LRFP CFT (which may or may not coincide with $\tilde\phi$ or $\tilde\phi^3$). By conformal invariance, 3pt functions of $\tilde\phi$ and $\tilde\phi^3$ with these operators take the form:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle {\langle \tilde\phi(x) {\mathcal{O}}_1(y) {\mathcal{O}}_2(z) \rangle} = \frac{\lambda_{12\tilde\phi}}{\vert x-y\vert ^{\Delta_\phi + \Delta_1 - \Delta_2} \vert x-z\vert ^{\Delta_\phi - \Delta_1 + \Delta_2} \vert y-z\vert ^{- \Delta_\phi + \Delta_1 + \Delta_2}}, \label{eq:phi12} \nonumber \end{align} \tag{ 4.37 }$$

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle {\langle \tilde\phi^3(x) {\mathcal{O}}_1(y) {\mathcal{O}}_2(z) \rangle} = \frac{\lambda_{12 \tilde\phi^3}}{\vert x-y\vert ^{\Delta_{\phi^3} + \Delta_1 - \Delta_2} \vert x-z\vert ^{\Delta_{\phi^3} - \Delta_1 + \Delta_2} \vert y-z\vert ^{- \Delta_{\phi^3} + \Delta_1 + \Delta_2}}. \nonumber \end{align} \tag{ 4.38 }$$

We would like to use the non-local EOM to relate these two 3pt functions. The diagrams which contribute to the first 3pt function are of two types. Either ϕ is connected to an interaction vertex, or it's connected directly to one of the fields inside $\newcommand{\calO}{{{\mathcal O}}} \calO_1$ and $\newcommand{\calO}{{{\mathcal O}}} \calO_2$. The non-local EOM maps the diagrams of the first type into those of $\phi^3$. Diagrams of the second type depend only on one of the two distances $x-y$, $x-z$. The non-local EOM maps them into local terms proportional to (derivatives of) $\renewcommand{\d}{{\rm d}} \delta(x-y), \delta(x-z)$, irrelevant for the purposes of comparison with the IR behavior.

So, we act with the non-local EOM on the first 3pt function and use Symanzik's star integral formula [41]. We thus obtain an expression of the same functional form as the second 3pt function. This provides a further consistency check of the shadow relation and allows to deduce relative normalization of the 3pt function coefficients [6]:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\eps}{\epsilon} \displaystyle \label{eq:ratio} \frac{\lambda_{12\tilde \phi^3}}{\lambda_{12 \tilde \phi}}= \frac{M_3}{\tilde C}=M_3 \sqrt{\frac{1 + \rho(\epsilon)} {\rho(\epsilon) M_2}}, \nonumber \end{align} \tag{ 4.39 }$$

where

$$\begin{align} \newcommand{\G}{\Gamma} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \label{eq:M3} M_3 = \pi^{d/2}\frac{\Gamma(\frac{d - \Delta_\phi + \Delta_{12}}{2}) \Gamma(\frac{d - \Delta_\phi - \Delta_{12}}2) \Gamma(\Delta_\phi - d/2)} {\Gamma(d-\Delta_\phi) \Gamma(\frac{\Delta_\phi + \Delta_{12}}2) \Gamma(\frac{\Delta_\phi - \Delta_{12}}2)},\quad \Delta_{12} =\Delta_1 - \Delta_2. \nonumber \end{align} \tag{ 4.40 }$$

Even though this equation depends on $\newcommand{\eps}{\epsilon} \rho(\epsilon)$ which is an unknown function of $\newcommand{\eps}{\epsilon} \epsilon$ (although it can be computed perturbatively for small $\newcommand{\eps}{\epsilon} \epsilon$), notice that it predicts a nontrivial dependence of the ratio on $\newcommand{\D}{\Delta} \Delta_{12}$. In appendix F.1 we perform some checks of this predicted dependence. Taking further ratios we can get relations independent of $\newcommand{\eps}{\epsilon} \rho(\epsilon)$. Namely, if we consider two OPEs $\newcommand{\calO}{{{\mathcal O}}} \calO_1\times\calO_2$ and $\newcommand{\calO}{{{\mathcal O}}} \calO_1^{\prime}\times\calO_2^{\prime}$, then

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{eq:relative3pt1} \frac{\lambda_{12 \tilde \phi^3}/\lambda_{12 \tilde \phi}} {\lambda_{1^{\prime}2^{\prime} \tilde \phi^3}/\lambda_{1^{\prime}2^{\prime} \tilde \phi}} = \frac{M_3}{M_3^{\prime}}. \nonumber \end{align} \tag{ 4.41 }$$

We have seen in the previous section how many nontrivial facts about the $\phi^4$-flow can be proved to all orders in the $\newcommand{\eps}{\epsilon} \epsilon$-expansion. Here we will give a parallel discussion for the $\sigma\chi$-flow. Arguing by analogy, we will motivate a number of all order results in the δ-expansion. In the next section we will see how it all fits together with the infrared duality.

Compared to the standard perturbation theory of a Lagrangian field theory, whose structural properties are well-understood to all orders, conformal perturbation theory is an underdeveloped subject. The usual discussion starts from perturbing a CFT by a weakly relevant operator $\newcommand{\calO}{{{\mathcal O}}} \calO$ of scaling dimension $\renewcommand{\d}{{\rm d}} d-\delta$, where $\renewcommand{\l}{\ell} \renewcommand{\d}{{\rm d}} \delta\ll 1$. To define perturbation theory, one needs a regulator, and point splitting is a natural choice. However point splitting is awkward to implement at higher orders. Dimensional regularization or analytic regularization in δ are not viable in general, because the CFT may exist or be tractable only for a fixed spacetime dimension, and because the relevant perturbation usually exists only for a fixed, physical value of δ. This is unlike Lagrangian perturbation theory, where correlation functions can be analytically continued to arbitrary d.

Fortunately, the case of the $\sigma\chi$-flow is better than this generic situation, since the dimension of χ is a continuously varying parameter—the gaussian action which governs the dynamics of χ is defined for any $\newcommand{\D}{\Delta} \Delta_\chi$. So in our case we can consider analytic continuation in δ as a way to regulate integrals. This provids a potential pathway to an all-order discussion.

We will now discuss how things might plausibly work out in this all-order perturbation theory. The results will be in agreement with the finite order computations that we performed in sections 2, 3 using the point-splitting regulator, and with further subsequent checks. Still, our all-order discussion of the $\sigma\chi$-flow will not reach the level of rigor which was possible for the $\phi^4$-flow.

The basic object to study are the correlators of χ, defined in perturbation theory by series-expanding the interaction, evaluating the correlation functions in the factorized theory, and integrating using the above-mentioned analytic regulator. Some integrals will produce poles in δ. We conjecture that, to all orders, such poles can be removed by defining the renormalized coupling g related to the bare coupling by the usual relation:

$$\begin{align} \renewcommand{\d}{{\rm d}} \displaystyle g_0 =Z_g(g,\delta)\mu^\delta g\, \nonumber \end{align} \tag{ 5.1 }$$

(the function Z_g is of course different from that of the $\phi^4$-flow). This conjecture seems reasonable because SRFP does not contain any marginal operator. Our computations in sections 2, 3 can be seen as a low-order test. It would be nice to find a full proof³⁰. Assuming the conjecture, we can derive analogues of the all-order $\phi^4$-flow statements by an almost verbatim repetition of the arguments.

A part of the conjecture is that the gamma-function of χ is zero. This is motivated in the same way as for ϕ in the $\phi^4$ flow. Namely, that poles in δ correspond to short-distances divergences of the integral for $\renewcommand{\d}{{\rm d}} \delta=0$, the divergences are local, and the action of χ is non-local, so it can't be renormalized. We then obtain that the anomalous dimension of χ at the fixed point is identically zero. This is an all-order generalization of the lowest-order result in section 3.1.1.

For future use, notice that if χ is unit-normalized in the UV, then in the IR we will have

$$\begin{align} \renewcommand{\d}{{\rm d}} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \label{eq:chinormIR} \langle \chi(x) \chi(0) \rangle = \frac{1 + \kappa (\delta) }{\vert x\vert ^{2\Delta_\chi}}\,, \nonumber \end{align} \tag{ 5.2 }$$

It's clear that κ has an expansion in even powers of g so $\renewcommand{\d}{{\rm d}} \kappa(\delta)=O(\delta)$ for small δ. In fact from the lowest order diagram we can easily obtain (using equation (3.14) in [6]):

$$\begin{align} \newcommand{\G}{\Gamma} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \kappa = g_*^2 \pi^d \frac{\Gamma \left(\frac d2- \Delta _{\sigma }\right) \Gamma \left(\Delta _{\sigma }-\frac{d}{2}\right)}{\Gamma \left(\Delta _{\sigma }\right) \Gamma \left(d-\Delta _{\sigma }\right)}+O(g_*^4). \nonumber \end{align} \tag{ 5.3 }$$

This is negative, similarly to how $\newcommand{\eps}{\epsilon} \rho(\epsilon)$ starts out negative for small $\newcommand{\eps}{\epsilon} \epsilon$.

We can argue that the IR fixed point of the $\sigma\chi$-flow should be conformally invariant. Indeed, we can derive the broken conformal Ward identities for the $\sigma\chi$-flow by the same Caffarelli–Silvestre trick. We can then show that these Ward identities imply the conformal invariance in the IR.

We also have a non-local EOM:

$$\begin{align} \newcommand{\D}{\Delta} \displaystyle \int {\rm d}^d y \frac{1}{\vert x-y\vert ^{2(d - \Delta_\chi)}} \chi(y) = \ C^{\prime} \sigma (x)\,. \nonumber \end{align} \tag{ 5.4 }$$

From this we can see that the shadow relation $\newcommand{\D}{\Delta} \Delta_\sigma+\Delta_\chi = d$ holds at the IR fixed point, generalizing the lowest-order result in section 3.1.1.

Finally, we can repeat verbatim the calculation of section 4.4. Given (5.2), we obtain

$$\begin{align} \renewcommand{\d}{{\rm d}} \renewcommand{\l}{\ell} \displaystyle \label{dualratio} \frac{\lambda_{12 \tilde \sigma }} {\lambda_{12 \tilde \chi } } = \hat M_3 \sqrt{\frac{{1 + \kappa(\delta)}} {{\kappa(\delta) \hat M_2}} }\,, \nonumber \end{align} \tag{ 5.5 }$$

where $\hat M_2$ and $\hat M_3$ are the 'shadow' quantities obtained from (4.33) and (4.40) replacing $\newcommand{\D}{\Delta} \Delta_\phi$ with $\newcommand{\D}{\Delta} \Delta_\chi=d-\Delta_\phi$. See appendix F.2 for checks of this relation in conformal perturbation theory.

We have seen that the both $\phi^4$-flow and $\sigma\chi$-flow have a conformally invariant IR fixed point. The dimensions of two operators at the fixed point are exactly known (one by non-renormalization, another by the shadow relation):

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{eq:phi4res} [\phi]=(d-s)/2,\quad [\phi^3]=(d+s)/2, \nonumber \end{align} \tag{ 5.6 }$$

and

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{eq:sigmachi4res} [\chi]=(d+s)/2,\quad [\sigma]=(d-s)/2. \nonumber \end{align} \tag{ 5.7 }$$

The $\phi^4$-flow relations have been proved to all orders in $\newcommand{\eps}{\epsilon} \renewcommand{\l}{\ell} \epsilon=2s-d\ll 1$, near the crossover to mean field³¹.

Under the reasonable assumption of renormalizability, the $\sigma\chi$-flow relations hold to all orders in $\renewcommand{\l}{\ell} \renewcommand{\d}{{\rm d}} \delta=2(s_*-s)\ll 1$, near the crossover to short range.

The most natural interpretation of these results is that there is only one CFTs for each s, which describes the fixed points of both flows (infrared duality). The fields $\phi, \phi^3$ for the first flow have to be identified in the IR with $\sigma, \chi$ for the second flow (up to proportionality coefficients). Finally, the above equations for the IR field dimensions are valid exactly and not just in perturbation theory. Indeed, if there were nonperturbative corrections, say, to the first set of equations, they would presumably become largest near the short-range crossover, but this is where the second set of equations becomes accurate and shows that there are no corrections.

See figure 4 for the predicted dependence of the most important LRFP operator dimensions on s between the mean-field and the short-range crossovers. The solid lines joining ϕ to σ and χ to $\phi^3$ are straight lines. The other lines are known only approximately in the $\newcommand{\eps}{\epsilon} \epsilon$ and δ expansion around the crossovers. The shown shape of the lines is the simplest consistent with these asymptotics. The line joining $\phi^2$ to ε deserves a comment. We have $\newcommand{\eps}{\epsilon} [\phi^2]=(d- \epsilon)/2+\gamma=d- \epsilon/6+O(\epsilon^2)$ near the mean-field crossover, so that the line starts going linearly down. In $d=3$ it joins to $\newcommand{\vareps}{\varepsilon} \newcommand{\D}{\Delta} \Delta_ \varepsilon\approx 1.41<d/2$ with the negative first derivative, see (3.41). However, in $d=2$ it rises back up to $\newcommand{\vareps}{\varepsilon} \newcommand{\D}{\Delta} \Delta_ \varepsilon=1=d/2$ and has a zero first derivative at the crossover, see (3.21).

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As a further check of the duality, we will show that the relations (4.39) and (5.5) for the OPE coefficients can be made compatible with each other. We must have

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{eq:musthave} \frac{\lambda_{12 \tilde \phi^3}}{\lambda_{12 \tilde \phi}} \equiv \frac{\lambda_{12 \tilde \chi }} {\lambda_{12 \tilde \sigma } } \,. \nonumber \end{align} \tag{ 5.8 }$$

It's not trivial that the two sides can agree, because the dependence on $\newcommand{\D}{\Delta} \Delta_{1, 2}$ must match. Fortunately it does, thanks to the following identity,

$$\begin{align} \displaystyle M_3 \hat M_3 = M_2 = \hat M_2. \nonumber \end{align} \tag{ 5.9 }$$

Equation (5.8) holds provided that ρ and κ obey, at the same value of s, the relation:

$$\begin{align} \renewcommand{\d}{{\rm d}} \newcommand{\eps}{\epsilon} \displaystyle \frac{\kappa(\delta) }{1 + \kappa(\delta)} = \frac{1 + \rho(\epsilon)}{\rho(\epsilon)}. \nonumber \end{align} \tag{ 5.10 }$$

This leads to nontrivial predictions for the behavior of the two functions near the mean-field and short-range crossovers. Since as we have seen $\renewcommand{\d}{{\rm d}} \kappa(\delta)=O(\delta)$ for small δ, we conclude that the normalization of the 2pt function of ϕ must vanish linearly in s close to the short-range crossover:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\eps}{\epsilon} \displaystyle 1+ \rho(\epsilon)=O(s_*-s). \label{eq:normvan} \nonumber \end{align} \tag{ 5.11 }$$

This was previously discussed as a necessary condition for a smooth crossover in [6], Appendix C³². The same vanishing of the 2pt function normalization has been previously argued in [43], via a completely different argument. See also appendix B for a discussion of the $\renewcommand{\l}{\ell} \langle\phi\phi\rangle$ correlator precisely at the crossover, where it exhibits a logarithmic suppression with respect to the naive scaling. We derive the suppression via RG, but it can also be thought of as a remnant of the vanishing of the normalization (5.11) [43].

Analogously, we must have

$$\begin{align} \renewcommand{\d}{{\rm d}} \displaystyle 1 + \kappa(\delta) = O((2s-d)^2) \nonumber \end{align} \tag{ 5.12 }$$

when approaching the crossover to the mean-field regime. In this case the vanishing is expected to be quadratic since $\newcommand{\eps}{\epsilon} \rho(\epsilon)=O(\epsilon^2)$.

In summary, we have accumulated strong evidence for a novel IR duality: the $\phi^4$-flow and the $\sigma \chi$-flow end in the same IR fixed point. Unlike all previously studied examples of IR dualities, our theories lack a local stress tensor. Non-locality, which is an essential feature of our construction, comes with a surprising bonus: remarkable computation power! One of its immediate consequences is the non-renormalization theorem for $\newcommand{\D}{\Delta} \Delta_\phi$ and $\newcommand{\D}{\Delta} \Delta_\chi$. The non-local equations of motions were then used to show that $\newcommand{\D}{\Delta} \Delta_{\phi^3}$ and of $\newcommand{\D}{\Delta} \Delta_\sigma$ obey shadow relations at the IR fixed point, and that OPE coefficients involving the shadow pairs must come in precise ratios. In the paradigmatic examples of IR dualities (such as 4d Seiberg duality and 3d mirror symmetry), it is supersymmetry that gives analytic control. Curiously, we were able to achieve significant analytic control in our non-supersymmetric setting, thanks precisely to the non-local nature of the problem.

The study of the long-range Ising model has started in earnest in [3], where also the effective description based on the $\phi^4$-flow has been proposed. That reference has erroneously put the crossover to short range at $s=2$. This was corrected by [2, 4], leading to what we called the 'standard picture', which has since been supported by theoretical studies [56, 57] and by lattice Monte Carlo simulations [58].

More recently some debate restarted about the nature of the long-range to short-range crossover. Lattice Monte Carlo simulations in [59] observed deviations from the standard picture near the crossover (see also [60]) However, [59] may have underestimated systematic errors due to possible logarithmic corrections to scaling near the crossover [61]. From our perspective these logarithmic corrections are associated with the operator $\sigma\chi$, marginally irrelevant at the crossover (see section 2.3).

Defenu et al [62] analyzed the problem using the functional renormalization group and also found support for the standard picture.

As a side remark, some of this recent literature likes to phrase the conclusions in terms of the so called 'effective dimension' $D_{\rm eff}(s)$ such that the LRFP in d dimensions is supposed to have the same critical exponents as the SRFP in $D_{\rm eff}$ dimensions. We would like to use this occasion to stress that this 'effective dimension' is clearly not a fundamental notion, bound to work only for a few exponents and only in low orders of perturbation theory.

Many of the above-mentioned papers also considered the $O(N)$ generalization of the long-range Ising model. Recently, [43] analyzed the crossover in the large N approximation. They argued that the IR normalization of the 2pt function of ϕ vanishes as $s\to s_*$. While we do not fully understand the details of their argument, the conclusion agrees with our picture, as discussed in section 5.2. See also [63] for a recent discussion of the large N limit in this model.

The $\phi^4$-flow has been studied via rigorous renormalization group analysis in $d=1, 2, 3$ in [64–67]. These works show that an infrared fixed point exists nonperturbatively at least for sufficiently small $\newcommand{\eps}{\epsilon} \epsilon>0$. For some critical exponents, dependence on $\newcommand{\eps}{\epsilon} \epsilon$ in this region of small $\newcommand{\eps}{\epsilon} \epsilon$ has also been rigorously investigated. Mitter [66] announced a proof that ϕ does not acquire anomalous dimension. Slade [67] showed that the susceptibility and the specific heat critical exponents take, at leading order in $\newcommand{\eps}{\epsilon} \epsilon$, values expected from the $\newcommand{\eps}{\epsilon} \epsilon$-expansion predictions for $\gamma_\phi$ and $\gamma_{\phi^2}$ (this reference considers the general $O(n)$ case, including $n=0$ corresponding to self-avoiding walks).

Long-range Ising model can also be studied directly from the lattice Hamiltonian, without relying on the renormalization group. It is known that the model has a phase transition separating a low-β phase with vanishing magnetization from a high-β phase where the magnetization is nonzero. Moreover the transition is continuous, in the sense that magnetization vanishes as $\beta_c$ is approached from above. The above statements have been proved rigorously for $d=1$, $0<s<1$ and for $d=2, 3$, $0<s<2$, which includes the range $0<s<2-\eta_{\rm SR}$ we are interested in ([51], section 1.4). Incidentally, the same paper also proved for the first time continuity of the phase transition in the short-range Ising model for $d=3$.

Another rigorous result worth mentioning is that the spin-spin correlator of the long-range Ising model decays at $\beta<\beta_c$ with the exponent $d+s$, as in our equation (A.6). See [36], equation (2.8). Many other rigorous results about the long-range Ising model on the lattice are reviewed in that paper.

Equation (4.39) predicts the following dependence of the OPE coefficient ratio on $\newcommand{\D}{\Delta} \Delta_{1}$ and $\newcommand{\D}{\Delta} \Delta_2$:

$$\begin{align} \newcommand{\G}{\Gamma} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \label{ratiogphi4} R_{12} \equiv \frac{\lambda_{12 \tilde \phi}} {\lambda_{12 \tilde \phi^3 } } = g \cdot \frac{\Gamma(\frac{\Delta_\phi + \Delta_{12}}{2}) \Gamma(\frac{\Delta_\phi - \Delta_{12}}{2}) }{\Gamma(\frac{\Delta_{\phi^3} + \Delta_{12}}{2}) \Gamma(\frac{\Delta_{\phi^3} - \Delta_{12}}{2})}\cdot A(g). \nonumber \end{align} \tag{ F.1 }$$

The function $A(g)$ has a finite limit for $g \to 0$. It is independent of $\newcommand{\D}{\Delta} \Delta_1$ and $\newcommand{\D}{\Delta} \Delta_2$ (but it does depend on $\newcommand{\D}{\Delta} \Delta_\phi$ and d). Here we would like to perform some very simple checks of this relation.

Scalar conformal primaries of the mean field theory schematically have the form:

$$\begin{align} \newcommand{\calO}{{{\mathcal O}}} \displaystyle \calO\sim \partial^{2k}\phi^{n}, \nonumber \end{align} \tag{ F.2 }$$

where derivatives have to be distributed and contracted to get a primary. We take two primaries of this form, assuming $n_1 \geqslant n_2$ without loss of generality. To get a possibility for nonzero 3pt functions we assume that $n_1-n_2$ is odd.

Equation (F.1) generically predicts $R_{12}=O(g)$. However, there are exceptions if

• 1.  
  $n_1=n_2+1$, $k_1 \geqslant k_2$,
• 2.  
  $n_1=n_2+3$, $k_1 \geqslant k_2$.

In case 1 (resp. 2) the second Gamma functions in the numerator, resp. denominator, is near a pole. Assuming that $\newcommand{\calO}{{{\mathcal O}}} \calO_1$ and $\newcommand{\calO}{{{\mathcal O}}} \calO_2$ get unequal anomalous dimensions $O(g)$ we expect $\newcommand{\G}{\Gamma} \Gamma\sim 1/g$. This predicts

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{eq:predict} {\rm{case~1:~}} R_{12}=O(1), \qquad {\rm{case~2:~}} R_{12}=O(g^2). \nonumber \end{align} \tag{ F.3 }$$

We would like to check, in several simple examples, how this agrees with the perturbation theory.

First consider $k_1=k_2=0$. Let's start with $\newcommand{\calO}{{{\mathcal O}}} \calO_1=\phi^{n+1}$, $\newcommand{\calO}{{{\mathcal O}}} \calO_2=\phi^n$ (case 1). In perturbation theory, 3pt functions $\renewcommand{\l}{\ell} {\langle \phi \phi^{n+1}\phi^n \rangle}$ and $\renewcommand{\l}{\ell} {\langle \phi^3 \phi^{n+1}\phi^n \rangle}$ exist already in the mean field theory. Here are the corresponding diagrams for $n=4$:

Thus $R_{12}=O(1)$, in agreement with the above prediction.

Next let us examine $\newcommand{\calO}{{{\mathcal O}}} \calO_1=\phi^{n+3}$, $\newcommand{\calO}{{{\mathcal O}}} \calO_2=\phi^n$ (case 2). In perturbation theory, 3pt functions $\renewcommand{\l}{\ell} {\langle \phi^3 \phi^{n+3}\phi^n \rangle}$ appears at $O(1)$, while $\renewcommand{\l}{\ell} {\langle \phi \phi^{n+3}\phi^n \rangle}$ needs one coupling insertion. E.g. for $n=2$:

Moreover, the integral over the position of this insertion (the red subdiagram) gives an extra $\newcommand{\eps}{\epsilon} O(\epsilon)$ suppression, thanks to the formula:

$$\begin{align} \newcommand{\G}{\Gamma} \renewcommand{\l}{\ell} \displaystyle \label{eq:key} \int {\rm d}^d x \, \frac{1}{\vert x + y\vert ^{2a} \vert x\vert ^{2b} } \sim \frac{1}{\Gamma (d - a - b)} \,, \nonumber \end{align} \tag{ F.6 }$$

where in the considered case $\newcommand{\eps}{\epsilon} a + b= d+O(\epsilon)$³⁷. So all in all we have $\newcommand{\eps}{\epsilon} R_{12}=O(\epsilon g)=O(g^2)$, in agreement with (F.3).

Now let's consider $\newcommand{\calO}{{{\mathcal O}}} \calO_2=\phi^{n}$, $\newcommand{\calO}{{{\mathcal O}}} \calO_1=\phi^{n+5+2r}$. In perturbation theory, both 3pt functions require coupling insertions:

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\calO}{{{\mathcal O}}} \displaystyle {\langle \phi \calO_1\calO_2 \rangle}_g=O(g^{r+2}),\qquad {\langle \phi^3 \calO_1\calO_2 \rangle}_g=O(g^{r+1}). \nonumber \end{align} \tag{ F.7 }$$

Both have just one power of $\newcommand{\eps}{\epsilon} \epsilon$ suppression due to 'red subdiagrams'. So $R_{12}=O(g)$ as expected for the generic case. Here's an example for $r=0$:

Let us now examine a more complicated example. We would like to understand the origin of the restriction $k_1 \geqslant k_2$. We will only study $n_1=n_2+1$ but $k_1, k_2$ general. The 3pt function $\newcommand{\calO}{{{\mathcal O}}} \renewcommand{\l}{\ell} {\langle \phi \calO_1\calO_2 \rangle}$ is nonzero in the mean field theory if and only if $\newcommand{\calO}{{{\mathcal O}}} \calO_1$ occurs in the OPE $\newcommand{\calO}{{{\mathcal O}}} \phi(x)\times \calO_2(0)$. To pick up terms with $n_1=n_2+1$ ϕ's in the OPE, we are not allowed to take any Wick contractions, only to expand $\phi(x)$ around $x=0$. This will produce terms with a non-negative number of derivatives acting on ϕ, i.e. all $\newcommand{\calO}{{{\mathcal O}}} \calO_1$'s of this type have $k_1 \geqslant k_2$. Analogously it's easy to see that $\newcommand{\calO}{{{\mathcal O}}} \calO_1\in \phi^3\times \calO_2$ is only possible if $k_1 \geqslant k_2$.

So we get the following picture. If $k_1 \geqslant k_2$, then both 3pt functions $\newcommand{\calO}{{{\mathcal O}}} \renewcommand{\l}{\ell} {\langle \phi \calO_1\calO_2 \rangle}$, $\newcommand{\calO}{{{\mathcal O}}} \renewcommand{\l}{\ell} {\langle \phi^3 \calO_1\calO_2 \rangle}$ exist in MFT, we expect $R_{12}=O(1)$, and this is what (F.3) predicts in this case. On the other hand if $k_1 < k_2$, then we need coupling insertions to generate the 3pt functions. Generically we expect $\newcommand{\calO}{{{\mathcal O}}} \renewcommand{\l}{\ell} {\langle \phi^3 \calO_1\calO_2 \rangle}_g=O(g)$. On the other hand, $\newcommand{\calO}{{{\mathcal O}}} \renewcommand{\l}{\ell} {\langle \phi \calO_1\calO_2 \rangle}_g=O(g^2)$, since the diagrams with just one coupling insertions will vanish, as they involve $\newcommand{\calO}{{{\mathcal O}}} \renewcommand{\l}{\ell} {\langle \phi^3 \calO_1\calO_2 \rangle}_{\rm MFT}=0$. So we indeed expect $R_{12}=O(g)$, consistently with (F.3).

This section is the analogue of the previous one for the $\sigma\chi$-flow at $s\to s_*$. The OPE coefficient ratio is predicted by (5.5) to have dependence of the ratio on $\newcommand{\D}{\Delta} \Delta_{1}$ and $\newcommand{\D}{\Delta} \Delta_2$ of the form:

$$\begin{align} \newcommand{\G}{\Gamma} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \label{ratiog} R_{12} \equiv \frac{\lambda_{12 \tilde \chi }} {\lambda_{12 \tilde \sigma } } = g \cdot \frac{\Gamma(\frac{\Delta_\chi + \Delta_{12}}{2}) \Gamma(\frac{\Delta_\chi - \Delta_{12}}{2}) }{\Gamma(\frac{\Delta_\sigma + \Delta_{12}}{2}) \Gamma(\frac{\Delta_\sigma - \Delta_{12}}{2})}\cdot \tilde A(g). \nonumber \end{align} \tag{ F.9 }$$

To avoid awkward square roots we have expressed the answer as a function of the coupling g, to be evaluated at the fixed point $\renewcommand{\d}{{\rm d}} g_* \sim \sqrt{\delta}$. The function $\tilde A(g)$ has a finite limit for $g \to 0$. It is independent of $\newcommand{\D}{\Delta} \Delta_1$ and $\newcommand{\D}{\Delta} \Delta_2$ (but it does depend on $\newcommand{\D}{\Delta} \Delta_\sigma$, $\newcommand{\D}{\Delta} \Delta_\chi=d-\Delta_\sigma$ and d). In particular, since we will be focussing on the dependence on $\newcommand{\D}{\Delta} \Delta_{1}$ and $\newcommand{\D}{\Delta} \Delta_2$, we will not have to keep track of the IR normalization of σ and χ (as long as it's the same for all considered correlators).

For generic ${{\mathcal O}}_1$ and ${{\mathcal O}}_2$, the χ OPE coefficient is suppressed by a power of g, reflecting the factorization of the theory into the product of the SRFP and of the Gaussian χ model as $s \to s_*$. As we will see, there are however interesting cases where one of the Gamma functions develops a pole, corresponding to a non-vanishing χ correlator in the factorized theory (if the pole is in one of the numerator Gamma functions) or to a further suppression of the χ correlator in conformal perturbation theory (if the pole is in one of the denominator Gamma functions).

Let $\newcommand{\D}{\Delta} \{{{\mathcal O}}^{\rm SR}_{\Delta_i, \ell_i, \alpha_i} \}$ be the conformal primary operators of the SRFP, where $\newcommand{\D}{\Delta} \Delta_i$ and $\ell_i$ are their conformal dimensions and spins, and $\alpha_i = \pm 1$ their $\mathbb{Z}_2$ quantum numbers. The general scalar primary of the factorized theory at $s = s_*$ takes the schematic form

$$\begin{align} \renewcommand{\l}{\ell} \newcommand{\D}{\Delta} \displaystyle \label{generic} \partial^{\ell + 2k}[ {{\mathcal O}}^{\rm SR}_{\Delta_, \ell, \alpha} \chi^n] \,, \nonumber \end{align} \tag{ F.10 }$$

where the derivatives are contracted to give a scalar and distributed in such a way to give a primary. The χ parity $\beta = (-1){\hspace{0pt}}^n$ is an exact $\mathbb{Z}_2$ symmetry in the factorized theory. The interaction $g \int \chi \sigma$ preserves the diagonal $\mathbb{Z}_2$ symmetry, whose quantum number we denote by $\nu = \alpha \cdot \beta$. As we turn on g, two operators with different ${{\mathcal O}}^{\rm SR}$ and different numbers of χ's can a priori mix, provided that they have the same conformal dimension and that $\alpha_1 \beta_1 = \alpha_2 \beta_2$. In such a case, the correct dilation eigenstates will be linear combinations of states of the form (F.10). This is however a very rare and perhaps impossible phenomenon, which will discuss at the end. In the bulk of the analysis, we will assume that both ${{\mathcal O}}_1$ and ${{\mathcal O}}_2$ take the form (F.10)³⁸,

$$\begin{align} \newcommand{\D}{\Delta} \displaystyle {{\mathcal O}}_1 = \partial^{\ell_1 + 2k_1} [{{\mathcal O}}^{\rm SR}_{\Delta_1, \ell_1, \alpha_1} \chi^{n_1}] \,, \quad {{\mathcal O}}_2 = \partial^{\ell_2 + 2k_2} [{{\mathcal O}}^{\rm SR}_{\Delta_2, \ell_2, \alpha_2} \chi^{n_2}] \,. \nonumber \end{align} \tag{ F.11 }$$

In order for the 3pt functions $\renewcommand{\l}{\ell} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle$ and $\renewcommand{\l}{\ell} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle$ to have a chance of being non-zero, the diagonal $\mathbb{Z}_2$'s of ${{\mathcal O}}_1$ and ${{\mathcal O}}_2$ must be opposite, $\nu_1 = - \nu_2$, and we will assume that this is always the case.

In the factorized theory at $s = s_*$, σ and χ have different $\mathbb{Z}_2 \times \mathbb{Z}_2$ quantum numbers, respectively $(-, +)$ and $(+, -)$, so if $\renewcommand{\l}{\ell} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle$ is non-zero, $\renewcommand{\l}{\ell} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle$ is zero, and viceversa. We then expect the ratio (F.9) to be either zero or infinite as $g\to0$.

With no loss of generality, we will assume that $n_1 \geqslant n_2$. Let's enumerate the various cases:

• 1.  
  $n_1 = n_2$.In this case, the scaling in g as $g \to 0$ expected from $\mathbb{Z}_2 \times \mathbb{Z}_2$ selection rules is given, generically³⁹, by

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle_g = O(1) \,, \quad \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle_g = O(g) \,. \nonumber \end{align} \tag{ F.12 }$$

  Indeed $\alpha_1 = - \alpha_2$, making $\renewcommand{\l}{\ell} \langle {{\mathcal O}}^{\rm SR}_1 {{\mathcal O}}^{\rm SR}_2 \sigma \rangle \neq 0$ in the SRFP. While $\renewcommand{\l}{\ell} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle$ is already non-zero at $g=0$, a non-zero $\renewcommand{\l}{\ell} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle$ requires one insertion of the interaction $g \int \chi \sigma$. We have $\newcommand{\D}{\Delta} \Delta_{12} = \Delta^{\rm SR}_1 - \Delta^{\rm SR}_2 + 2k_1 - 2k_2$ at $g=0$. In the generic case, there are no poles coming from the Gamma functions of (F.9), and the $O(g)$ behavior of the ratio expected from conformal perturbation theory is correctly reproduced. We have checked in simple examples that the numerical coefficients are also correctly reproduced.No two operators in the SRFP have dimensions differing by $\newcommand{\D}{\Delta} \Delta_\chi$ plus an integer, so for $n_1 = n_2$ there are never poles from the numerator Gamma functions. It is possible however to have a pole from the denominator Gamma functions, resulting in a zero in $R_{12}$, if $\newcommand{\D}{\Delta} \Delta_{12} = \Delta_\sigma + 2k_1 - 2 k_2$ with $k_1 \geqslant k_2$, or $\newcommand{\D}{\Delta} \Delta_{12} = -\Delta_\sigma + 2k_1 - 2 k_2$ with $\renewcommand{\l}{\ell} k_1 \leqslant k_2$. Given that operators with given $\mathbb{Z}_2 \times \mathbb{Z}_2$ quantum numbers acquire an $O(g^2)$ anomalous dimension, this zero should be interpreted as an additional factor of g², so all in all $R_{12} = O(g^3)$ in these cases. Let's see in an example how this can be compatible with conformal perturbation theory. Take

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{firstdimreg} {{\mathcal O}}_1 = \sigma \chi^n \,,\quad {{\mathcal O}}_2 = \chi^n \,. \nonumber \end{align} \tag{ F.13 }$$

  We still have $\renewcommand{\l}{\ell} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle_g = O(1)$, but we are going to argue that $\renewcommand{\l}{\ell} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle_g = O(g^3)$. Indeed, the $O(g)$ contribution to $\renewcommand{\l}{\ell} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle_g$ arises from the integral

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{renormalizediszero} g \int {\rm d}^d y \; \langle \sigma \chi^n (x_1) \; \chi^n(x_2) \; \chi(x_3) \; \sigma \chi (y) \rangle_0 \,, \nonumber \end{align} \tag{ F.14 }$$

  which needs to be regulated and renormalized. We claim that the renormalized integral is actually $\renewcommand{\d}{{\rm d}} O(\delta)$, so that the net contribution is $\renewcommand{\d}{{\rm d}} O(g \delta) = O(g^3)$. A quick and dirty way to see this is to evaluate the integral in dimensional regularization, using equation (F.6). Proving this for examples more complicated than (F.13) requires a version of (F.6) with an extra $(x+z){\hspace{0pt}}^{2c}$ in the numerator.
• 2.  
  $n_1 = n_2+1$.Since we require $\nu_1 = - \nu_2$, in this case we have $\alpha_1 = \alpha_2$. In the generic case, ${{\mathcal O}}^{\rm SR}_1$ and ${{\mathcal O}}^{\rm SR}_2$ are two different operator with the same SR $\mathbb{Z}_2$ quantum number, and then

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle_g = O(g) \,, \quad \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle_g = O(g^2) \,. \nonumber \end{align} \tag{ F.15 }$$

  Indeed, for the first correlator one needs a single insertion of the interaction, while the second correlator vanishes at $g=0$ (because ${{\mathcal O}}^{\rm SR}_1 \neq {{\mathcal O}}^{\rm SR}_2$) and then selection rules force the insertion of two interactions. We observe again the generic behavior $O(g)$ for the ratio of OPE coefficients, in agreement with (F.9).The more interesting case is when ${{\mathcal O}}^{\rm SR}_1 = {{\mathcal O}}^{\rm SR}_2$. Then selection rules would predict

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{funnycasegeneric} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle_g = O(g) \,, \quad \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle_g = O(1) \,. \nonumber \end{align} \tag{ F.16 }$$

  In this case, $\newcommand{\D}{\Delta} \Delta_{12} = \Delta_\chi + 2k_1 - 2k_2$ for $g=0$. If $k_1 \geqslant k_2$, we encounter in a pole from the second Gamma function in the numerator of (F.9). Operators with definite $\mathbb{Z}_2 \times \mathbb{Z}_2$ quantum numbers acquire anomalous dimension at order $O(g^2)$, so for small g the pole is regulated to $\sim 1/g^2$, and $R_{12} \sim 1/g$, in agreement with (F.16). On the other hand, if $k_1 < k_2$, we seem to have a problem. There is no pole in the Gamma function, so (F.9) predicts $R_{12} \sim O(g)$, in contradiction with (F.16).To understand the resolution of this puzzle, let's consider the following example. Let

  $$\begin{align} \displaystyle {{\mathcal O}}_1 = \sigma \chi^3 \,, \quad {{\mathcal O}}^A_2 = \sigma (\partial_\mu \chi \partial^\mu \chi + a \chi \Box \chi) \,, \quad {{\mathcal O}}^B_2 = \partial_\mu \sigma \partial_\mu \chi \chi + b \Box \sigma \chi^2 \,, \nonumber \end{align} \tag{ F.17 }$$

  where the coefficients a and b are fixed such that ${{\mathcal O}}^A_2$ and ${{\mathcal O}}^B_2$ are conformal primaries. It is important to realize that ${{\mathcal O}}^A_2$ and ${{\mathcal O}}^B_2$ mix in conformal perturbation theory, indeed the $O(g^2)$ correction to the dilation operator contains an off-diagonal term arising from the non-vanishing correlator $\renewcommand{\l}{\ell} \langle {{\mathcal O}}^A_2 \, {{\mathcal O}}^B_2 \, \sigma \chi \sigma \chi \rangle_0$. The eigenstates of the dilation operator take the form

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{dilationeigenstates} {{\mathcal O}}^I_2 = {{\mathcal O}}^A_2 + c {{\mathcal O}}^B_2 \, \quad {{\mathcal O}}^{II}_2 = {{\mathcal O}}^A_2 + d {{\mathcal O}}^B_2, \nonumber \end{align} \tag{ F.18 }$$
• where the coefficients c and d have no g dependence. We claim that

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle = O(g) \,, \quad \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle = O(g^2) \,, \nonumber \end{align} \tag{ F.19 }$$

  where ${{\mathcal O}}_2$ is either one of the dilation eigenstates in (F.18), so that $R_{12} = O(g)$ in agreement with (F.9). Indeed

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{samereason} \langle {{\mathcal O}}_1 (x_1) {{\mathcal O}}_2 (x_2) \chi (x_3) \rangle_0 \sim \langle \chi(x_1) \chi(x_3) \rangle_0 \, \langle \sigma \chi^2(x_1) {{\mathcal O}}_2 (x_2) \rangle_0= 0 \,, \nonumber \end{align} \tag{ F.20 }$$

  where in the last step we have used orthogonality of conformal primaries of different dimension. We then need to go to $O(g^2)$ to find a non-zero contribution to $\renewcommand{\l}{\ell} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle_g$. On the other hand,

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{itsnotzero} \langle {{\mathcal O}}_1 (x_1) {{\mathcal O}}_2 (x_2) \sigma (x_3) \sigma \chi (x_4) \rangle_0 \sim \langle \chi(x_1) \chi(x_4) \rangle_0 \, \langle \sigma \chi^2(x_1) {{\mathcal O}}_2 (x_2) \sigma (x_3) \sigma (x_4) \rangle_0 \neq 0 \,, \nonumber \end{align} \tag{ F.21 }$$

  so $\renewcommand{\l}{\ell} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle_g = O(g)$. It is essential that the dilation eigenstate ${{\mathcal O}}_2 \neq {{\mathcal O}}^A_2$, i.e. that it does not factorize into σ times a primary of the χ theory, otherwise the correlator (F.21) would vanish.
• 3.  
  $n_1 = n_2+ 2 m$, $m \geqslant 1$. We must have $\alpha_1 = -\alpha_2$. Selection rules and the requirement that we have enough χ insertions would naively give

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle_g = O(g^{2m}) \,, \quad \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle_g = O(g^{2m-1}) \,, \nonumber \end{align} \tag{ F.22 }$$

  but the actual behavior is

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle_g = O(g^{2m}) \,, \quad \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle_g = O(g^{2m+1}) \,. \nonumber \end{align} \tag{ F.23 }$$

  One can convince oneself in examples that the $O(g^{2m-1})$ contribution to $\renewcommand{\l}{\ell} \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle_g$ vanishes—the corresponding renormalized correlator turns out to be $\renewcommand{\d}{{\rm d}} O(\delta)$, a phenomenon we already encountered in (F.14). It would be nice to find a general argument. Conformal perturbation theory is then consistent with the generic behavior $R_{12} = O(g)$ predicted by (F.9).
• 4.  
  $n_1 = n_2+ 2 m +1$, $m \geqslant 1$. Now $\alpha_1 = \alpha_2$. The naive scaling is

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle_g = O(g^{2m+1}) \,, \quad \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle_g = O(g^{2m}) \,, \nonumber \end{align} \tag{ F.24 }$$

  and the correct one

  $$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle_g = O(g^{2m+1}) \,, \quad \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle_g = O(g^{2m+2}) \,, \nonumber \end{align} \tag{ F.25 }$$

  for the same reason as the previous case. Again, we find agreement with (F.9).

Finally, let's consider the possibility of mixing of states of the form (F.10) with different $\mathbb{Z}_2 \times \mathbb{Z}_2$ quantum numbers. This requires a conspiracy of quantum numbers that is hard to arrange. A naive candidate is the following. Take $d=2$ and consider

$$\begin{align} \displaystyle {{\mathcal O}}_A = (\partial_\mu \chi \partial_\mu \chi + a\, \chi \Box \chi) \chi^{n-2} \,,\quad {{\mathcal O}}_B = \sigma \chi^{n+1}\,, \nonumber \end{align} \tag{ F.26 }$$

where the coefficient a is fixed to make ${{\mathcal O}}_A$ a conformal primary. Since $\newcommand{\D}{\Delta} \Delta_\chi + \Delta_\sigma = 2$, both states have dimension $\newcommand{\D}{\Delta} 2 + n \Delta_\chi$. At first sight, it might seem that dilation operator has an off-diagonal component at order $O(g)$ arising from

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \label{AB} g \int {\rm d}^2 y \, \langle {{\mathcal O}}_A (x_1) {{\mathcal O}}_B (x_2)\, \chi \sigma (y) \rangle\,. \nonumber \end{align} \tag{ F.27 }$$

However, this correlator is of order $\renewcommand{\d}{{\rm d}} O(\delta)$ for the same reason as the correlator in (F.20). An off-diagonal component would arise at $O(g^3)$, but there are diagonal anomalous dimensions already at order $O(g^2)$, which presumably lift the degeneracy between ${{\mathcal O}}_A$ and ${{\mathcal O}}_B$. So it appears that there is no mixing after all, barring some coincidence. It would be nice to decide this one way or another by a detailed computation.

Let's consider the power counting in the scenario that the two states (F.27) do mix to leading order. (Maybe a more intricate example would actually work along these lines). Let ${{\mathcal O}}_1$ be one of the two dilation eigenstates, of the form

$$\begin{align} \displaystyle {{\mathcal O}}_1 = {{\mathcal O}}_A + b\, {{\mathcal O}}_B \nonumber \end{align} \tag{ F.28 }$$

for some coefficient $b = O(1)$, and choose

$$\begin{align} \displaystyle {{\mathcal O}}_2 = \chi^n \sigma. \nonumber \end{align} \tag{ F.29 }$$

Then we have

$$\begin{align} \renewcommand{\l}{\ell} \displaystyle \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \sigma \rangle_g = O(1) \,, \quad \langle {{\mathcal O}}_1 {{\mathcal O}}_2 \chi \rangle_g = O(1) \,, \nonumber \end{align} \tag{ F.30 }$$

so $R_{12}$ would be finite as $g \to 0$ limit in this scenario. Note that $\newcommand{\D}{\Delta} \Delta_{12} = \Delta_\chi$ for $g=0$, which gives rise to a pole in (F.9). Now however the anomalous dimension of ${{\mathcal O}}_1$ is of order $O(g)$. The pole is regulated to $\sim 1/g$ and (F.9) predicts $R_{12} = O(1)$. So we win again.