Bootstrapping the long-range Ising model in three dimensions

An essential property of the model is the observation that the propagator does not receive loop corrections. The intuitive justification for this is that a nonlocal kinetic term cannot be modified by local divergences. A useful test is evaluating the first two-loop diagram that could potentially influence $\gamma_\phi$. Choosing an analytic regularization scheme, the diagram in figure 1 yields

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle G(k) &= \frac{\lambda^2}{6(4\pi)^d} \frac{\Gamma \left ( \frac{3s}{2} - d \right) \Gamma \left ( \frac{d - s}{2} \right)^3}{\Gamma \left ( \frac{3d - 3s}{2} \right) \Gamma \left ( \frac{s}{2} \right)^3} \left | \frac{k}{\mu} \right |^{2d - 3s} \nonumber \\ &= \frac{\lambda^2}{6(4\pi)^d} \frac{\Gamma \left ( \frac{3\varepsilon - d}{4} \right) \Gamma \left ( \frac{d - \varepsilon}{4} \right)^3}{\Gamma \left ( \frac{3d - 3\varepsilon}{4} \right) \Gamma \left ( \frac{d + \varepsilon}{4} \right)^3} \left | \frac{k}{\mu} \right |^{\frac{d - 3\varepsilon}{2}} \; . \label{two-loop-phi} \nonumber \end{align} \tag{ 2.3 }$$

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This requires no minimal subtraction as the $\varepsilon \rightarrow 0$ limit is finite for all $1 \leqslant d < 4$. Indeed, $\Gamma \left ( -\frac{d}{4} \right)$ is the only gamma function with a non-positive argument. It is reassuring that a pole appears for d  =  4, which is precisely the value that turns (2.1) into a local theory⁴. The upshot is that the dimension of $\phi$ has the exact expression

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta_\phi = \frac{d - s}{2} \; , \label{non-renorm1} \nonumber \end{align} \tag{ 2.4 }$$

a relation which has been proven rigorously in [44]. By demanding that the short-range scaling dimensions are approached continuously, we can take $\Delta_\sigma^{\mathrm{SRI}} = \frac{d - s_*}{2}$ as the definition of s_*. This is a correction to [21] which initially predicted s_*  =  2. The correct behaviour at the crossover was explicitly demonstrated in [22] by taking $d \rightarrow 4$ to make the entire flow perturbative. When this is done, one sees that the weakly irrelevant operator $\phi \partial^2 \phi$ is responsible for the breakdown of (2.4).

The second operator with an exactly known scaling dimension is $\phi^3$. This is obvious in the Wilson–Fisher fixed point as the equation of motion $\partial^2 \phi = \frac{\lambda}{3!} \phi^3$ places $\phi^3$ squarely in the $\phi$ multiplet⁵. In the LRI, this protected dimension is instead a consequence of the nonlocal equation of motion

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \partial^s \phi = \frac{\lambda}{3!} \phi^3 \; , \label{nonlocal-eom} \nonumber \end{align} \tag{ 2.5 }$$

with $\phi^3$ remaining an independent primary. From (2.5), we read off

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta_{\phi^3} = \frac{d + s}{2} \; . \label{non-renorm2} \nonumber \end{align} \tag{ 2.6 }$$

These operators, satisfying $\Delta_\phi + \Delta_{\phi^3} = d$, are often said to form a shadow pair [25].

Bootstrap methods, which take (2.4) and (2.6) as input, are useful for improving the estimates of unprotected scaling dimensions. The study of these was initiated in [21] as well, which found the exponent

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta_{\phi^2} = \frac{d - \varepsilon}{2} + \frac{\varepsilon}{3} + \left [ \psi(1) - 2 \psi \left ( \frac{d}{4} \right) + \psi \left ( \frac{d}{2} \right) \right ] \left ( \frac{\varepsilon}{3} \right)^2 + O(\varepsilon^3) \; . \label{dim-eps} \nonumber \end{align} \tag{ 2.7 }$$

One quantity, which to our knowledge has not been computed yet, is the dimension of the leading spin-2 operator $T_{\mu\nu}$. We can use $\Delta_T$ to distinguish between different long-range Ising models since it gives a rough measure of how nonlocal a theory is. This makes it important for the bootstrap which is agnostic to microscopic parameters like s. Constructing the operator

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{\mu\nu} = \left ( \phi \partial_\mu \partial_\nu \phi - \frac{1}{d} \delta_{\mu\nu} \phi \partial^2 \phi \right) + y \left ( \partial_\mu \phi \partial_\nu \phi - \frac{1}{d} \delta_{\mu\nu} (\partial \phi)^2 \right) \; , \label{t-def} \nonumber \end{align} \tag{ 2.8 }$$

we choose $y = -\frac{\Delta_\phi + 1}{\Delta_\phi}$ in order to make it a conformal primary⁶. The one-loop diagram that one might expect to give anomalous scaling to $\left < \phi(-k_1 - k_2) \phi(k_1) T_{\mu\nu}(k_2) \right >$ is shown in figure 2.

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After Feynman parameters are first invoked, the integral takes the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle G(k_1, k_2) &= \lambda \mu^\varepsilon \frac{\Gamma(s)}{\Gamma \left ( \frac{s}{2} \right)^2} \int_0^1 \int_{\mathbb{R}^d} \frac{x^{\frac{s}{2} - 1} (1 - x)^{\frac{s}{2} - 1}}{[(k_2 + p)^2 x + p^2 (1 - x)]^s} \label{1loop-t1} \nonumber \\ & \left [ (k_2 + p)_\nu ((k_2 + p)_\mu - yp_\mu) + p_\nu(\,p_\mu - y(k_2 + p)_\mu) - \mathrm{trace} \right ] \frac{{\rm d}p}{(2\pi)^d} {\rm d}x \; . \nonumber \end{align} \tag{ 2.9 }$$

After making the denominator spherically symmetric in p , the piece in brackets becomes $(\,p + (1 - x) k_2)_\nu [(\,p + (1 - x) k_2) - y (\,p - x k_2)_\mu] + (\,p - x k_2)_\nu [(\,p - x k_2)_\mu - y (\,p + (1 \!-\! x) k_2)_\mu]$ with the trace subtracted. Expanding this, all parts that are odd in a given component of $p_\mu$ must vanish. This reveals an overall factor of $k_{2\mu} k_{2\nu} - \frac{1}{d} \delta_{\mu\nu} k_2^2$ which we omit in what follows. The other parts of (2.9) become

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle G(k_1, k_2) &\propto \frac{\lambda}{(4\pi)^{\frac{d}{2}}} \frac{\Gamma \left ( s - \frac{d}{2} \right)}{\Gamma \left ( \frac{s}{2} \right)^2} \left [ \frac{\Gamma \left ( \frac{d - s}{2} \right)^2}{\Gamma(d - s)} - 2(1 - y)\frac{\Gamma \left ( \frac{d - s}{2} + 1 \right)^2}{\Gamma(d - s + 2)} \right ] \left | \frac{k_2}{\mu} \right |^{-\varepsilon} \nonumber \\ &= \frac{1}{\varepsilon} \frac{\lambda}{(4\pi)^{\frac{d}{2}} \Gamma \left ( \frac{d}{2} \right)} \left [ 2 - (1 - y) \frac{d}{d + 2} \right ] \left | \frac{k_2}{\mu} \right |^{-\varepsilon} + O(1) \; . \label{1loop-t2} \nonumber \end{align} \tag{ 2.10 }$$

After inserting the leading order value of y , it becomes clear that (2.10) is completely regular. This vanishing one-loop anomalous dimension, which had to be the case in d  =  4, is actually true for all other values of d as well⁷. A two-loop diagram is therefore necessary to see perturbative corrections in $\Delta_T$.

The two-loop $\gamma_T$ comes from the diagram in figure 3. All other candidates either involve a scaleless loop or have figure 2 as a subdiagram. While standard methods are not enough to evaluate this integral in general, it is sufficient to set k₁  =  0 if we are only interested in the pole term. Starting with the q integral already evaluated,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle G(0, k_2) =&~ \frac{\lambda^2 \mu^{2\varepsilon}}{(4\pi)^{\frac{d}{2}}} \frac{\Gamma \left ( s - \frac{d}{2} \right) \Gamma \left ( \frac{s - \varepsilon}{2} \right)^2}{\Gamma(s - \varepsilon) \Gamma \left ( \frac{s}{2} \right)^2} \int_{\mathbb{R}^d} \frac{1}{|p|^{s + \varepsilon} |k_2 + p|^s} \nonumber \\ & \left [ (k_2 + p)_\nu ((k_2 + p)_\mu - yp_\mu) + p_\nu(\,p_\mu - y(k_2 + p)_\mu) - \mathrm{trace} \right ] \frac{{\rm d}p}{(2\pi)^d} \nonumber \\ &\propto \frac{\lambda^2 \mu^{2\varepsilon}}{(4\pi)^{\frac{d}{2}}} \frac{\Gamma \left ( s - \frac{d}{2} \right) \Gamma \left ( \frac{s - \varepsilon}{2} \right)^2 \Gamma \left ( s + \frac{\varepsilon}{2} \right)}{\Gamma(s - \varepsilon) \Gamma \left ( \frac{s}{2} \right)^3 \Gamma \left ( \frac{s + \varepsilon}{2} \right)} \nonumber \\ & \int_0^1 \int_{\mathbb{R}^d} \frac{x^{\frac{s}{2} - 1} (1 - x)^{\frac{s + \varepsilon}{2} - 1}}{[\,p^2 + x (1 - x) k_2^2]^{s + \frac{\varepsilon}{2}}} [1 - 2x(1 - x)(1 - y)] \frac{{\rm d}p}{(2\pi)^d} {\rm d}x \; . \label{2loop-t1} \nonumber \end{align} \tag{ 2.11 }$$

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Once there is only a Feynman paremeter left to deal with, its integral can be expanded as a series in $\varepsilon$. The first term vanishes, as it must by consistency, but the second term does not. This gives the correlator an overall $\frac{1}{\varepsilon}$ dependence⁸.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle G(0, k_2) &\propto \frac{\lambda^2}{(4\pi)^d} \frac{\Gamma \left ( s - \frac{d}{2} \right) \Gamma \left ( \frac{s - \varepsilon}{2} \right)^2 \Gamma \left ( s + \frac{\varepsilon - d}{2} \right)}{\Gamma(s - \varepsilon) \Gamma \left ( \frac{s}{2} \right)^3 \Gamma \left ( \frac{s + \varepsilon}{2} \right)} \left | \frac{k_2}{\mu} \right |^{-2\varepsilon} \nonumber \\ & \int_0^1 x^{\frac{s}{2} - \varepsilon - 1} (1 - x)^{\frac{s - \varepsilon}{2} - 1} [1 - 2x(1 - x)(1 - y)] {\rm d}x \nonumber \\ &= \frac{1}{\varepsilon} \frac{\lambda^2}{(4\pi)^d \Gamma \left ( \frac{d}{2} \right)^2} \frac{4}{d(d + 2)} \left | \frac{k_2}{\mu} \right |^{-2\varepsilon} + O(1) \; . \label{2loop-t2} \nonumber \end{align} \tag{ 2.12 }$$

From this expression, $\gamma_T$ can easily be read off. Substituting the fixed point, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta_T = \frac{d + 4 - \varepsilon}{2} - \frac{8}{d(d + 2)} \left ( \frac{\varepsilon}{3} \right)^2 + O(\varepsilon^3) \; . \label{dim-t} \nonumber \end{align} \tag{ 2.13 }$$

If one wishes to approximate critical exponents near values of s where they are known exactly, the flow emanating from (2.1) only solves half of the problem. An ideal scenario would also explain how $\Delta_T$ and $\Delta_{\phi^2}$ approach d and $\newcommand{\e}{{\rm e}} \Delta_\epsilon^{\mathrm{SRI}}$ respectively. In [23], it was realized that the LRI at s  =  s_* must contain more than just the SRI. Exact results in 2D and numerical results in 3D have established that the Ising model does not contain scalar primaries that form a shadow pair, nor does it have a vector that is able to recombine with $T_{\mu\nu}$. These problems were solved by introducing a generalized free scalar $\chi$ which plays the role of $\phi^3$. This also makes it possible to write down the (not unit-normalized) conformal primary

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle V_\mu = \Delta_\sigma \sigma \partial_\mu \chi - \Delta_\chi \chi \partial_\mu \sigma \; , \label{recombining-vector} \nonumber \end{align} \tag{ 2.14 }$$

which has the same dimension as $\partial^\nu T_{\mu\nu}$.

The newly introduced kinetic term can be tuned such that $\sigma \chi$ has dimension $d - \delta$ where $\newcommand{\e}{{\rm e}} \delta \equiv \frac{s_* - s}{2}$. Viewing this operator as a deformation analogous to $\phi^4$, [23] conjectured that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S = S_{\mathrm{SRI}} + \int -\frac{1}{2} \chi \partial^{-s} \chi + g\sigma\chi {\rm d}x \label{flow2-action} \nonumber \end{align} \tag{ 2.15 }$$

flows to the same LRI fixed point as (2.1). The statement of (2.15) is that one can deform an exactly known correlation function by inserting $\newcommand{\e}{{\rm e}} \exp \left ( g \int \sigma\chi {\rm d}x \right)$ and applying the rules of conformal perturbation theory [46]. Encouraging results also follow from deforming correlators that are known with high precision from the numerical bootstrap [47]. Reviewing the results of [23, 24], the beta function is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \beta(g) = \left\{\begin{array}{@{}ll@{}} -\delta g + 1.268\,404 g^3 + O(g^5) & d = 2 \nonumber \\ -\delta g + 12.26(3) g^3 + O(g^5) & d = 3 \end{array}\right. \label{flow2-beta} \nonumber \end{align} \tag{ 2.16 }$$

and the system in the IR preserves the relations (2.4) and (2.6). The expansion dual to (2.7) is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta_\epsilon = \left\{\begin{array}{@{}ll@{}} 1 + O(g^4) & d = 2 \nonumber \\ 1.412\,625(10) + 3.3(5) g^2 + O(g^4) & d = 3 \end{array}\right. \label{dim-eps2} \nonumber \end{align} \tag{ 2.17 }$$

and the one dual to (2.13) is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta_T = \left\{\begin{array}{@{}ll@{}} 2 + \frac{15}{32} \pi^2 g^2 + O(g^4) & d = 2 \nonumber \\ 3 + 28.605\,55(6) g^2 + O(g^4) & d = 3 \end{array}\right. \; . \label{dim-t2} \nonumber \end{align} \tag{ 2.18 }$$

Some expressions in terms of g are exact, which is why we have refrained from plugging in the fixed point. The reason why $\Delta_T$ is known much more precisely than $\newcommand{\e}{{\rm e}} \Delta_\epsilon$ in three dimensions is that anomalous dimensions of broken currents must be compatible with multiplet recombination. Specifically, the formula in [48] allows (2.18) to be calculated purely from the contact term in the three-point function $\left < T_{\mu\nu} V_\rho \sigma \chi \right >$. From now on, we will use the $(\phi, \phi^2, \phi^3)$ and $\newcommand{\e}{{\rm e}} (\sigma, \epsilon, \chi)$ notation interchangeably.

In addition to the non-renormalization theorems we have discussed, the nonlocal equation of motion can be used to derive an infinite family of ratios between OPE coefficients. Equations like (2.5) are statements about fields in a Lagrangian rather than unit-normalized operators in a CFT. We will therefore equip them with s-dependent prefactors which we call $n_\sigma$ and $n_\chi$. Writing the integral expression for (2.5) yields

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle n_\chi(s) \chi(x) = \int \frac{n_\sigma(s) \sigma(y)}{|x - y|^{d + s}} {\rm d}y \; , \label{better-eom} \nonumber \end{align} \tag{ 2.19 }$$

which is recognizable as the shadow transform. The idea is to use (2.19) in three-point functions containing $\sigma$ and $\chi$ in order to find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\lambda_{12\chi}}{\lambda_{12\sigma}} = \frac{n_\sigma(s)}{n_\chi(s)} R_{12} \; , \label{ratio-schematic} \nonumber \end{align} \tag{ 2.20 }$$

where R₁₂ is a known function of the quantum numbers. If we repeat this for a second pair of operators, we can arrange to have the normalizations cancel out, leaving us with

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\lambda_{12\chi} \lambda_{34\sigma}}{\lambda_{12\sigma} \lambda_{34\chi}} = \frac{R_{12}}{R_{34}} \; . \label{ratio-cancelled} \nonumber \end{align} \tag{ 2.21 }$$

The explicit R_ij was computed for scalars in [25]. Subsequently, [23] used it to test the IR duality between the flows that generate the $\varepsilon$-expansion and the $\delta$-expansion. In what follows, we aim to generalize this result to the case of spinning operators.

A natural language for this is the embedding formalism [49] which associates to each $x^\mu \in \mathbb{R}^d$ a null ray $X^M \sim \lambda X^M \in \mathbb{R}^{d + 1, 1}$. If one chooses representatives such that X⁺   =  1 (the Poincaré section of the null cone), a Lorentz transformation on $X^M = (1, x^2, x^\mu)$ precisely implements a conformal transformation on $x^\mu$. Conformally invariant quantities can therefore be built out of the Lorentz scalars

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle X_{ij} \equiv -2 X_i \cdot X_j = x_{ij}^2 \; . \label{dot-product} \nonumber \end{align} \tag{ 2.22 }$$

An especially useful incarnation of the embedding space was developed in [50, 51] which used polarization vectors to make the formalism index-free. One such vector Z, in addition to being null, must satisfy a transversaity condition with X:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle X_{ii} = 0 \; , \; X_i \cdot Z_i = 0 \; , \; Z_{ii} = 0 \; . \label{drop-these} \nonumber \end{align} \tag{ 2.23 }$$

This is because tracelessness in $\mathbb{R}^{d + 1, 1}$ is stronger than tracelessness in $\mathbb{R}^d$. An important result is that correlation functions may only depend on polarization vectors through the combinations

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H_{ij} &\equiv -2 [(Z_i \cdot Z_j)(X_i \cdot X_j) - (X_1 \cdot Z_2)(X_2 \cdot Z_1)] \nonumber \\ V_{i, jk} &\equiv \frac{(Z_i \cdot X_j)X_{ik} - (Z_i \cdot X_k)X_{ij}}{X_{jk}} \; . \label{hv-structures} \nonumber \end{align} \tag{ 2.24 }$$

We will begin with the three-point function $\newcommand{\e}{{\rm e}} \left < \phi_1(x_1) \mathcal{O}_2^{\mu_1, \dots, \mu_\ell}(x_2) \sigma(x_3) \right >$, which has a single tensor structure. The most straightforward extension of it to the projective null cone is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left < \Phi_1(X_1) \mathcal{O}_2(X_2, Z_2) \sigma(X_3) \right > = \frac{\lambda_{12\sigma} V_{2,13}^\ell}{X_{13}^{\frac{\Delta_\sigma + \Delta_{12} - \ell}{2}} X_{23}^{\frac{\Delta_\sigma - \Delta_{12} + \ell}{2}} X_{12}^{\frac{\Delta_1 + \Delta_2 - \Delta_\sigma + \ell}{2}}} \; . \label{lifted-3pt} \nonumber \end{align} \tag{ 2.25 }$$

As a check, this is a degree-$\newcommand{\e}{{\rm e}} \ell$ polynomial in Z₂ which is invariant under $Z_2 \mapsto Z_2 + X_2$. It also transforms with the correct weights when X₁, X₂ and X₃ are scaled individually. Lifting the equation of motion to the embedding space as well,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left < \Phi_1(X_1) \mathcal{O}_2(X_2, Z_2) \chi(X_3) \right > = \frac{n_\sigma(s)}{n_\chi(s)} \frac{\lambda_{12\sigma}}{X_{12}^{\frac{\Delta_1 + \Delta_2 - \Delta_\sigma + \ell}{2}}} \int \frac{V_{2,10}^\ell DX_0}{X_{01}^{\frac{\Delta_\sigma + \Delta_{12} - \ell}{2}} X_{02}^{\frac{\Delta_\sigma - \Delta_{12} + \ell}{2}} X_{03}^{d - \Delta_\sigma}} \; . \label{cintegral1} \nonumber \end{align} \tag{ 2.26 }$$

This type of object, which has exponents adding up to d, is called a conformal integral. Suitable technology for treating conformal integrals in the embedding space, including the formula

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \int \frac{X^{A_1} \dots X^{A_n}}{(-2 X \cdot Y)^{d + n}} DX = \frac{\pi^{\frac{d}{2}} \Gamma \left ( \frac{d}{2} + n \right)}{\Gamma(d + n)} \frac{Y^{A_1} \dots Y^{A_n}}{(-Y^2)^{\frac{d}{2} + n}} - \mathrm{traces} \; , \label{cintegral-trick} \nonumber \end{align} \tag{ 2.27 }$$

was developed in [52]. For similar integrals with a slight excess in the exponents, see [53]. Before returning to (2.26), it is worth expanding the tensor structure as

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle V_{2,10}^\ell = X_{01}^{-\ell} \sum_{n = 0}^{\infty} \binom{\ell}{n} (Z_2 \cdot X_1)^n (Z_2 \cdot X_0)^{\ell - n} X_{02}^n X_{12}^{\ell - n} \; . \label{v-expansion} \nonumber \end{align} \tag{ 2.28 }$$

The result of (2.26) will contain $\newcommand{\e}{{\rm e}} V_{2,13}^\ell$ and in particular $\newcommand{\e}{{\rm e}} (Z_2 \cdot X_3){}^\ell$, which can only come from the first term of (2.28). It is therefore enough to focus on the n  =  0 term and infer the others from conformal invariance. Introducing Schwinger parameters and using (2.27), we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left < \Phi_1(X_1) \mathcal{O}_2(X_2, Z_2) \chi(X_3) \right > &= \frac{n_\sigma(s)}{n_\chi(s)} \frac{\lambda_{12\sigma} Z_2^{A_1} \dots Z_2^{A_\ell}}{X_{12}^{\frac{\Delta_1 + \Delta_2 - \Delta_\sigma - \ell}{2}}} \frac{\pi^{\frac{d}{2}} \Gamma \left ( \frac{d}{2} + \ell \right)}{\Gamma(d - \Delta_\sigma) \Gamma \left ( \frac{\Delta_\sigma + \Delta_{12} + \ell}{2} \right) \Gamma \left ( \frac{\Delta_\sigma - \Delta_{12} + \ell}{2} \right)} \nonumber \\ & \int_0^\infty \int_0^\infty \frac{(X_3 + \alpha X_1 + \beta X_2)^{A_1} \dots (X_3 + \alpha X_1 + \beta X_2)^{A_\ell}}{\alpha^{-\frac{\Delta_\sigma + \Delta_{12} + \ell}{2}} \beta^{-\frac{\Delta_\sigma - \Delta_{12} + \ell}{2}} [\alpha X_{13} + \beta X_{23} + \alpha \beta X_{12}]^{\frac{d}{2} + \ell}} \frac{{\rm d}\alpha}{\alpha} \frac{{\rm d}\beta}{\beta} \nonumber \\ & + O(Z_2 \cdot X_1) \; . \label{cintegral2} \nonumber \end{align} \tag{ 2.29 }$$

If we again discard $Z_2 \cdot X_1$ terms, we can evaluate the integral to arrive at

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle R_{12} = \pi^{\frac{d}{2}} \frac{\Gamma \left ( \Delta_\sigma - \frac{d}{2} \right) \Gamma \left ( \frac{d - \Delta_\sigma + \Delta_{12} + \ell}{2} \right) \Gamma \left ( \frac{d - \Delta_\sigma - \Delta_{12} + \ell}{2} \right)}{\Gamma(d - \Delta_\sigma) \Gamma \left ( \frac{\Delta_\sigma + \Delta_{12} + \ell}{2} \right) \Gamma \left ( \frac{\Delta_\sigma - \Delta_{12} + \ell}{2} \right)} \; . \label{ratio} \nonumber \end{align} \tag{ 2.30 }$$

This logic can be repeated for correlators that have arbitrary spin in both positions. The difference here is that there is no longer a unique tensor structure:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left < \mathcal{O}_1(X_1, Z_1) \mathcal{O}_2(X_2, Z_2) \sigma(X_3) \right > = \sum_{m = 0}^{\mathrm{min}(\ell_1, \ell_2)} \lambda_{12\sigma}^{(m)} \frac{V_{1,23}^{\ell_1 - m} V_{2,13}^{\ell_2 - m} H_{12}^m}{X_{13}^{\frac{\Delta_\sigma + \tau_{12}}{2}} X_{23}^{\frac{\Delta_\sigma - \tau_{12}}{2}} X_{12}^{\frac{\tau_1 + \tau_2 - \Delta_\sigma}{2}}} \; . \label{lifted-general} \nonumber \end{align} \tag{ 2.31 }$$

However, the factor of $H_{12}^m$ is untouched by the integration. This means that $\lambda_{12\chi}^{(m)}$ is proportional to $\lambda_{12\sigma}^{(m)}$ and R_ij acquires one extra index instead of two. Carrying out the computation, we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle R_{12}^{(m)} = \pi^{\frac{d}{2}} \frac{\Gamma \left ( \Delta_\sigma - \frac{d}{2} \right) \Gamma \left ( \frac{d - \Delta_\sigma + \Delta_{12} + \ell_1 + \ell_2 - 2m}{2} \right) \Gamma \left ( \frac{d - \Delta_\sigma - \Delta_{12} + \ell_1 + \ell_2 - 2m}{2} \right)}{\Gamma(d - \Delta_\sigma) \Gamma \left ( \frac{\Delta_\sigma + \Delta_{12} + \ell_1 + \ell_2 - 2m}{2} \right) \Gamma \left ( \frac{\Delta_\sigma - \Delta_{12} + \ell_1 + \ell_2 - 2m}{2} \right)} \; . \label{ratio-general} \nonumber \end{align} \tag{ 2.32 }$$

As a further generalization, one could consider mixed-symmetry tensors using the formalism of [54].

We will first consider a primary $\mathcal{O}$ whose spin is odd. In this case, the OPE coefficients on the right hand side of (3.1) vanish by Bose symmetry. For the left hand side to be nonzero, the dimension of $\mathcal{O}$ must be a pole of the following expression.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{R_{\sigma\mathcal{O}}}{R_{\chi\mathcal{O}}} = \frac{\Gamma \left ( \frac{d - \Delta + \ell}{2} \right)^2 \Gamma \left ( \frac{d - 2\Delta_\sigma + \Delta + \ell}{2} \right) \Gamma \left ( \frac{2\Delta_\sigma - d + \Delta + \ell}{2} \right)}{\Gamma \left ( \frac{\Delta + \ell}{2} \right)^2 \Gamma \left ( \frac{2\Delta_\sigma - \Delta + \ell}{2} \right) \Gamma \left ( \frac{2d - 2\Delta_\sigma - \Delta + \ell}{2} \right)} .\label{super-ratio-expression} \nonumber \end{align} \tag{ 3.2 }$$

The poles above the unitarity bound, which come entirely from the squared gamma function in the numerator, are

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta = d + \ell + 2n = \Delta_\sigma + \Delta_\chi + \ell + 2n \; . \label{super-ratio-poles} \nonumber \end{align} \tag{ 3.3 }$$

These are nothing but the dimensions of the double-twist operators $\newcommand{\e}{{\rm e}} [\sigma\chi]_{n, \ell} \sim \chi \partial_{\mu_1} \dots \partial_{\mu_\ell} \partial^{2n} \sigma$ at s  =  s_*. The first element of this list, $[\sigma\chi]_{0, 1}$, is known to renormalize upon lowering the value of s. As its dimension leaves the pole (3.3), the left hand side of (3.1) is able to remain nonzero. This is because $[\sigma\chi]_{0,1}$ recombines with the stress–energy tensor and Bose symmetric OPEs are allowed to contain odd-spin descendants. All of the other double-twist operators stay primary (at least in three dimensions) and thus face a radically different situation. Bose symmetry continues to enforce $\lambda_{\sigma\sigma\mathcal{O}} = \lambda_{\chi\chi\mathcal{O}} = 0$ which means that $\Delta$ can only change continuously if $\lambda^2_{\sigma\chi\mathcal{O}}$ jumps to zero discontinuously. We therefore arrive at the following proposal.

In the long-range Ising model given by a generic $\frac{d}{2} < s < s_*$, all odd-spin primaries in $\sigma \times \chi$ (the double-twist operators other than the first one) have a scaling dimension that is independent of s.

This result, which should strictly be called a conjecture, would represent the most natural scenario even if we did not have continuity in s. It also follows from the earlier form of the OPE ratio (2.20) after a simple check that the normalizations $n_\sigma(s)$ and $n_\chi(s)$ are nonzero. One possibility that we cannot rule out is that these odd-spin primaries are only protected within a finite interval starting at s  =  s_*. This is because we used the fact that $\lambda^2_{\sigma\chi\mathcal{O}}$ was strictly positive. If this coefficient were to smoothly approach zero at some value of s, we would have to worry about the behaviour in figure 4.

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Running this argument around the other side of the duality is more subtle since $\phi$ and $\phi^3$ cannot be treated as independent fields. At small values of the spin, however, it is clear how the protected operators in $\phi \times \phi^3$ match up with those in $\sigma \times \chi$. Taking n  =  0 for example,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{\mu\nu} \ni \chi \partial_\mu \sigma & \phi^3 \partial_\mu \phi \in \phi^4 \nonumber \\ \chi \partial_\mu \partial_\nu \partial_\rho \sigma & \leftrightarrow \phi^3 \partial_\mu \partial_\nu \partial_\rho \phi \nonumber \\ \chi \partial_\mu \partial_\nu \partial_\rho \partial_\sigma \partial_\tau \sigma & \leftrightarrow \phi^3 \partial_\mu \partial_\nu \partial_\rho \partial_\sigma \partial_\tau \phi \nonumber \\ &\dots& \label{initial-matching} \nonumber \end{align} \tag{ 3.4 }$$

Starting at $\newcommand{\e}{{\rm e}} \ell = 7$, the conformal primary where four copies of $\phi$ are saturated by derivatives is not unique. A generating function counting the number of such primaries for all $\newcommand{\e}{{\rm e}} \ell$ can be found in [55]. Looking at one example, the primary subspace for operators of the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{O}_7 =&~ c_0 \phi^3 \partial^7 \phi + c_1 \phi^2 \partial \phi \partial^6 \phi + c_2 \phi^2 \partial^2 \phi \partial^5 \phi + c_3 \phi \partial \phi \partial \phi \partial^5 \phi \nonumber \\ & + c_4 \phi^2 \partial^3 \phi \partial^4 \phi + c_5 \phi \partial \phi \partial^2 \phi \partial^4 \phi + c_6 \partial \phi \partial \phi \partial \phi \partial^4 \phi + c_7 \phi \partial \phi \partial^3 \phi \partial^3 \phi \nonumber \\ & + c_8 \phi \partial^2 \phi \partial^2 \phi \partial^3 \phi + c_9 \partial \phi \partial \phi \partial^2 \phi \partial^3 \phi + c_{10} \partial \phi \partial^2 \phi \partial^2 \phi \partial^2 \phi \label{ansatz-7} \nonumber \end{align} \tag{ 3.5 }$$

is two-dimensional. We should therefore only expect one linear combination to be protected. Given a basis consisting of $\left \{\mathcal{O}_7^{(1)}, \mathcal{O}_7^{(2)} \right \}$, we may choose t such that $t \mathcal{O}_7^{(1)} + \mathcal{O}_7^{(2)}$ decouples from $\phi \times \phi^3$. This gives an operator that is free to renormalize since it has vanishing OPE coefficients on either side of (3.1). It is only the orthogonal operator that maintains its exact double-twist dimension. More generally, when the subspace is N-dimensional, the solutions to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle t_1 \left < \phi(x_1) \phi^3(x_2) \mathcal{O}_\ell^{(1)} \right > + \dots + t_{N - 1} \left < \phi(x_1) \phi^3(x_2) \mathcal{O}_\ell^{(N - 1)} \right > + \left < \phi(x_1) \phi^3(x_2) \mathcal{O}_\ell^{(N)} \right > = 0 \label{decoupling-subspace} \nonumber \end{align} \tag{ 3.6 }$$

are $(N - 1)$-dimensional, pointing us to a unique protected operator once again. It is not surprising that $\phi \times \phi^3$ contains only one leading-twist operator of each spin in a suitable basis. Indeed if the degeneracy could not be removed, one would be able to repeat our non-renormalization argument based on the nonlocal equation of motion $\partial^s \phi \propto \phi^3$ in the local fixed point governed by $\partial^2 \phi \propto \phi^3$. It is clear that the Wilson–Fisher fixed point does not have an odd-spin protected tower. Instead, the odd-spin operators which have the dimensions (3.3) in d  =  4 are able to avoid the Bose symmetry constraints for primaries by recombining with higher spin currents. It would be interesting to see how this structure is reproduced in perturbation theory.

The 2D case would also be very interesting to study further. This describes a fixed line obtained by deforming a Virasoro symmetric theory. As a result, the definition of $\newcommand{\e}{{\rm e}} [\sigma\chi]_{n, \ell}$ is ambiguous just like the double-twist operator $\newcommand{\e}{{\rm e}} [\phi\phi^3]_{n, \ell}$. This time, rotating the bases to remove the maximal number of operators from $\sigma \times \chi$ is probably not the right solution to the mixing problem. Consider the operators with $(h, \bar{h}) = (4, 1)$ where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{O}_3^{(1)} &= \sigma L_{-1}^3 \chi - 93 L_{-1} \sigma L_{-1}^2 \chi + \frac{713}{3} L_{-1}^2 \sigma L_{-1} \chi - \frac{3565}{51} L_{-1}^3 \sigma \chi \nonumber \\ \mathcal{O}_3^{(2)} &= L_{-3} \sigma \chi - \frac{128}{153} L_{-1}^3 \sigma \chi \label{ansatz-3} \nonumber \end{align} \tag{ 3.7 }$$

is a valid basis⁹. The previous logic would suggest splitting these into one combination with $\lambda_{\sigma\chi\mathcal{O}} = 0$ and another combination with $\lambda_{\sigma\chi\mathcal{O}} \neq 0$. This presents a problem as both operators in such a splitting would have nonzero overlap with

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Lambda = \left ( L_{-4} - \frac{5}{3} L_{-2}^2 \right) I \; . \label{current-4} \nonumber \end{align} \tag{ 3.8 }$$

There is only room for a higher spin current to recombine with one multiplet so we must demand that the protected spin-3 operator is the one that fails to give an anomalous dimension to (3.8). To solve for this operator, we have evaluated the three-point functions

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left < \Lambda(z_1, \bar{z}_1) \mathcal{O}_3^{(1)}(z_2, \bar{z}_2) \sigma\chi(z_3, \bar{z}_3) \right > &= \frac{124\,775 / 544}{z_{12}^7 z_{13} z_{23} \bar{z}_{23}^2} \nonumber \\ \left < \Lambda(z_1, \bar{z}_1) \mathcal{O}_3^{(2)}(z_2, \bar{z}_2) \sigma\chi(z_3, \bar{z}_3) \right > &= \frac{1225 / 3264}{z_{12}^7 z_{13} z_{23} \bar{z}_{23}^2} \label{ward-id-correlators} \nonumber \end{align} \tag{ 3.9 }$$

using the Virasoro Ward identity¹⁰. This allows us to repeat the calculation that [23] did for the stress–energy tensor and say that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle & \int \left < \bar{\partial}_1 \Lambda(z_1, \bar{z}_1) \left [ \frac{544}{124\,775} \mathcal{O}_3^{(1)} - \frac{3264}{1225} \mathcal{O}_3^{(2)} \right ](z_2, \bar{z}_2) \sigma\chi(z_3, \bar{z}_3) \right > {\rm d}z_3 {\rm d}\bar{z}_3 = 0 \nonumber \\ & \int \left < \bar{\partial}_1 \Lambda(z_1, \bar{z}_1) \left [ \frac{713}{1513} \mathcal{O}_3^{(1)} + \frac{3422\,400}{10\,591} \mathcal{O}_3^{(2)} \right ](z_2, \bar{z}_2) \sigma\chi(z_3, \bar{z}_3) \right > {\rm d}z_3 {\rm d}\bar{z}_3 \propto \gamma_\Lambda \; . \label{hs-recombination} \nonumber \end{align} \tag{ 3.10 }$$

Based on this, one might hope that all $\sigma \times \chi$ operators in the 2D theory are either protected or eaten. The first counter-example to this appears at the next level which has three $\newcommand{\e}{{\rm e}} \ell = 5$ operators built from $\sigma$ and $\chi$ but only one $\newcommand{\e}{{\rm e}} \ell = 6$ current that needs to be broken.

We will now discuss the treatment of $\newcommand{\e}{{\rm e}} [\sigma\chi]_{n,\ell}$ operators with even spin. Looking at (3.1) for s  =  s_*, we again have a removable singularity since $\mathcal{O}$ has a dimension given by (3.3) while decoupling from the $\sigma \times \sigma$ and $\chi \times \chi$ OPEs. The difference is that the coefficients $\lambda_{\sigma\sigma\mathcal{O}}$ and $\lambda_{\chi\chi\mathcal{O}}$ turn on for s  <  s_* as they are not constrained by any kinematic principle. This allows the right hand side of (3.1) to become a ratio of finite numbers. The shadow relation then becomes a statement about operators of an unknown dimension still having constrained OPE coefficients.

This situation is ubiquitous in the superconformal bootstrap. It allows four-point functions to be decomposed into blocks that include the contributions of many conformal primaries. These superconformal blocks have been computed in [56] and many subsequent works. In the long-range Ising model which is non-supersymmetric, it is the nonlocal operator in (2.5) rather than a supercharge, which allows certain conformal blocks to be combined.

To make this precise, consider the general form of the four-point function for scalar primaries

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left < \phi_i(x_1) \phi_j(x_2) \phi_k(x_3) \phi_l(x_4) \right > = \left ( \frac{|x_{24}|}{|x_{14}|} \right)^{\Delta_{ij}} \left ( \frac{|x_{14}|}{|x_{13}|} \right)^{\Delta_{kl}} \frac{G^{ijkl}(u, v)}{|x_{12}|^{\Delta_i + \Delta_j}|x_{34}|^{\Delta_k + \Delta_l}} \; . \label{4pt-form} \nonumber \end{align} \tag{ 3.11 }$$

The unknown function, depending on the cross-ratios $u = \frac{x_{12}^2 x_{34}^2}{x_{13}^2 x_{24}^2}$ and $v = \frac{x_{14}^2 x_{23}^2}{x_{13}^2 x_{24}^2}$, has the conformal block expansion

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle G^{ijkl}(u, v) = \sum_{\mathcal{O}} \lambda_{ij\mathcal{O}} \lambda_{kl\mathcal{O}} G^{\Delta_{ij} , \Delta_{kl}}_{\mathcal{O}}(u, v) \; . \label{block-sum-form} \nonumber \end{align} \tag{ 3.12 }$$

By demanding crossing symmetry for the $\left < \sigma\sigma\chi\chi \right >$ correlator, we derive the crossing equations

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle & \sum_{\mathcal{O}^+_{2 | \ell}} \lambda_{\sigma\sigma\mathcal{O}} \lambda_{\chi\chi\mathcal{O}} F^{\sigma\sigma ; \chi\chi}_{-,\Delta,\ell}(u, v) + \sum_{\mathcal{O}^+_{2 | \ell}} \lambda_{\sigma\chi\mathcal{O}}^2 F^{\chi\sigma ; \sigma\chi}_{-,\Delta,\ell}(u, v) - \sum_{\mathcal{O}^+_{2 \nmid \ell}} \lambda_{\sigma\chi\mathcal{O}}^2 F^{\chi\sigma ; \sigma\chi}_{-,\Delta,\ell}(u, v) = 0 \label{change0} \nonumber \\ & \sum_{\mathcal{O}^+_{2 | \ell}} \lambda_{\sigma\sigma\mathcal{O}} \lambda_{\chi\chi\mathcal{O}} F^{\sigma\sigma ; \chi\chi}_{+,\Delta,\ell}(u, v) - \sum_{\mathcal{O}^+_{2 | \ell}} \lambda_{\sigma\chi\mathcal{O}}^2 F^{\chi\sigma ; \sigma\chi}_{+,\Delta,\ell}(u, v) + \sum_{\mathcal{O}^+_{2 \nmid \ell}} \lambda_{\sigma\chi\mathcal{O}}^2 F^{\chi\sigma ; \sigma\chi}_{+,\Delta,\ell}(u, v) = 0 \; , \nonumber \end{align} \tag{ 3.13 }$$

where we have used a shorthand for the convolved conformal block

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F_{\pm, \Delta, \ell}^{ij ; kl}(u, v) = v^{\frac{\Delta_j + \Delta_k}{2}} G_{\Delta, \ell}^{\Delta_{ij}, \Delta_{kl}}(u, v) \pm u^{\frac{\Delta_j + \Delta_k}{2}} G_{\Delta, \ell}^{\Delta_{ij}, \Delta_{kl}}(v, u) \; . \label{convolved-block} \nonumber \end{align} \tag{ 3.14 }$$

We can modify (3.13) to account for the protected operators

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle & \sum_{\mathcal{O}^+_{2 | \ell}} \lambda_{\sigma\sigma\mathcal{O}} \lambda_{\chi\chi\mathcal{O}} F^{\sigma\sigma ; \chi\chi}_{-,\Delta,\ell}(u, v) + \sum_{\mathcal{O}^+_{2 | \ell}} \lambda_{\sigma\chi\mathcal{O}}^2 F^{\chi\sigma ; \sigma\chi}_{-,\Delta,\ell}(u, v) - \sum_{\ell = 1, 3, \dots} \sum_{n = 0}^\infty \lambda_{\sigma\chi\mathcal{O}}^2 F^{\chi\sigma ; \sigma\chi}_{-,d + \ell + 2n,\ell}(u, v) = 0 \nonumber \\ & \sum_{\mathcal{O}^+_{2 | \ell}} \lambda_{\sigma\sigma\mathcal{O}} \lambda_{\chi\chi\mathcal{O}} F^{\sigma\sigma ; \chi\chi}_{+,\Delta,\ell}(u, v) - \sum_{\mathcal{O}^+_{2 | \ell}} \lambda_{\sigma\chi\mathcal{O}}^2 F^{\chi\sigma ; \sigma\chi}_{+,\Delta,\ell}(u, v) + \sum_{\ell = 1, 3, \dots} \sum_{n = 0}^\infty \lambda_{\sigma\chi\mathcal{O}}^2 F^{\chi\sigma ; \sigma\chi}_{+,d + \ell + 2n,\ell}(u, v) = 0 \; , \nonumber \\ & \label{change1} \nonumber \end{align} \tag{ 3.15 }$$

but this only imposes the odd-spin case of the shadow relation. Imposing the even-spin case as well leads to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle & \sum_{\mathcal{O}^+_{2 | \ell}} \lambda_{\sigma\sigma\mathcal{O}} \lambda_{\chi\chi\mathcal{O}} \mathcal{F}_{-,\Delta,\ell}(u, v) - \sum_{\ell = 1, 3, \dots} \sum_{n = 0}^\infty \lambda_{\sigma\chi\mathcal{O}}^2 F^{\chi\sigma ; \sigma\chi}_{-,d + \ell + 2n,\ell}(u, v) = 0 \nonumber \\ & \sum_{\mathcal{O}^+_{2 | \ell}} \lambda_{\sigma\sigma\mathcal{O}} \lambda_{\chi\chi\mathcal{O}} \mathcal{F}_{+,\Delta,\ell}(u, v) + \sum_{\ell = 1, 3, \dots} \sum_{n = 0}^\infty \lambda_{\sigma\chi\mathcal{O}}^2 F^{\chi\sigma ; \sigma\chi}_{+,d + \ell + 2n,\ell}(u, v) = 0 \; , \label{change2} \nonumber \end{align} \tag{ 3.16 }$$

where we have defined the convolved superblocks

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{F}_{\pm,\mathcal{O}}(u, v) = F^{\sigma\sigma ; \chi\chi}_{\pm,\Delta,\ell}(u, v) \mp \frac{R_{\sigma\mathcal{O}}}{R_{\chi\mathcal{O}}} F^{\chi\sigma ; \sigma\chi}_{\pm,\Delta,\ell}(u, v) \; . \label{superblocks} \nonumber \end{align} \tag{ 3.17 }$$

It is easy to read off what the non-convolved superblocks are.

In contrast to other known superblocks, e.g. the 4D $\mathcal{N} = 1$ classification in [57], the relative coefficient in (3.17) is not a rational function of $\Delta$. Indeed (3.2) has infinitely many poles. It may therefore be of interest to compute rational approximations for the coefficient, similar to what is already standard practice for the conformal blocks themselves. We discuss both problems together in appendix.

It is easiest to start with the results that can be obtained from the $\left < \sigma\sigma\sigma\sigma \right >$ correlator alone. In this case, there is no compelling reason to restrict our analysis to three dimensions. Our bound on the spin-2 gap $\Delta_T$, which we plot for 2D and 3D, has been known since the early work in [11].

From figure 5, it appears that the $\Delta_T$ bound is saturated by generalized free field theory. This gives us an idea of how the allowed region in $\newcommand{\e}{{\rm e}} (\Delta_\sigma, \Delta_\epsilon)$ space must behave. Not only must it become more restrictive as $\Delta_T$ is increased—its boundary must move from left to right at a known rate.

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It is straightforward to derive an upper bound of this type. In figure 6, we have done this for six different values of the spin-2 gap. In 2D, the minimum $\Delta_T$ values we sample are $\{2, 2.2, 2.4, 2.6, 2.8, 3\}$, while in 3D they are $\{3, 3.1, 3.2, 3.3, 3.4, 3.5\}$. If the previously observed saturation is correct, the edges of these plots must continue moving left as our computational power is increased. For instance, we expect a $5\%$ change for the 2D red region and a $3\%$ change for the 3D red region.

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Once we disallow $\Delta_T$ at the unitarity boumd, the kink corresponding to the SRI quickly disappears. The point $\newcommand{\e}{{\rm e}} (\Delta_\sigma, \Delta_\epsilon) = \left ( \frac{d}{4}, \frac{d}{2} \right)$, marking the first LRI to be described by mean-field theory, does not display any feature. We may therefore conclude that a single correlator gives very little information about the spectrum of non-trivial long-range Ising models.

Before proceeding to our multi-correlator results in three dimensions, we should comment on the fact that some theories can saturate numerical bootstrap bounds even when there is no kink. For a judiciously chosen quantity, there is some evidence that this is the case for the LRI in two dimensions¹². Instead of bounding a gap, one can maximize the OPE coefficient of an operator in the spectrum. From Monte Carlo data [19], it is clear that in every 2D long-range Ising model, one such operator has a dimension close to 1. In some sense, this is explained by the perturbative calculation (2.17) around s  =  s_*, since the leading order anomalous dimension of $\newcommand{\e}{{\rm e}} \epsilon$ happens to vanish. By setting $\newcommand{\e}{{\rm e}} \Delta_\epsilon = 1$ and maximizing $\newcommand{\e}{{\rm e}} \lambda_{\sigma\sigma\epsilon}^2$, we have extracted the low-lying spectrum using the extremal functional method of [73–75]. At $\Delta_\sigma = \frac{1}{8}$, this is guaranteed to agree with the spectrum of the SRI. However, figure 7 also shows very interesting behaviour at $\Delta_\sigma = \frac{1}{2}$ as we now explain¹³.

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Here, the leading spin-0 and spin-2 dimensions are both close to 3. We can see that this is exactly what happens in a generalized free field theory if we write the operators as $\sigma \partial^2 \sigma$ and $\sigma \partial_\mu \partial_\nu \sigma$ respectively. The dimensions of the next double-twist operators, found by inserting extra powers of $\partial^2$, appear somewhat too high but this could easily be an effect of the numerics. This makes it tempting to conjecture that given a long-range Ising model with dimensions $\newcommand{\e}{{\rm e}} (\Delta_\sigma, \Delta_\epsilon)$, all other crossing symmetric four-point functions $\left < \sigma\sigma\sigma\sigma \right >$ have a smaller value of $\newcommand{\e}{{\rm e}} \lambda_{\sigma\sigma\epsilon}^2$. This approach to studying the LRI is ultimately perturbative since it requires the dimension of $\newcommand{\e}{{\rm e}} \epsilon$ as input. Nevertheless, it could be useful for reducing the number of Feynman diagrams one encounters. Instead of computing separate diagrams for each anomalous dimension, the conjecture would enable us to compute only diagrams for $\newcommand{\e}{{\rm e}} \gamma_\epsilon$ and then feed these into the bootstrap machinery to learn about other observables.

The sparseness of the spectrum in figure 7 hints at another significant limitation. By perturbing around s  =  s_* or $s = \frac{d}{2}$, it becomes clear that several additional families of operators enter the $\sigma \times \sigma$ OPE in a generic LRI. In particular, the number of scalars having $\Delta < 9$ should be much more than 4. Table 1 shows 16 such operators that can be constructed with the deformation of [23, 24].

Table 1. Some operators having $\Delta < 9$ in the CFT obtained by coupling the 2D SRI to a generalized free field of dimension $\frac{15}{8}$. We only show the ones where both parts are scalars. Anything involving $\chi$ decouples from $\sigma \times \sigma$ at s  =  s_* where the quoted dimensions hold exactly. At slightly smaller values of s however, these operators acquire a nonzero OPE coefficient as long as they are even with respect to the diagonal $\mathbb{Z}_2$ from the two theories.

Operator	Dimension
$\newcommand{\e}{{\rm e}} \epsilon$	1
$\sigma\chi$	2
$\chi^2$	$\frac{15}{4}$
$\newcommand{\e}{{\rm e}} \epsilon^\prime$	4
$\newcommand{\e}{{\rm e}} \epsilon \chi^2$	$\frac{19}{4}$
$\sigma \chi^3$	$\frac{23}{4}$
$\chi \partial^2 \chi$	$\frac{23}{4}$
$\sigma^\prime \chi$	6
$\newcommand{\e}{{\rm e}} \epsilon \chi \partial^2 \chi$	$\frac{27}{4}$
$\chi^4$	$\frac{15}{2}$
$\newcommand{\e}{{\rm e}} \epsilon^\prime \chi^2$	$\frac{31}{4}$
$\sigma \chi^2 \partial^2 \chi$	$\frac{31}{4}$
$\chi \partial^4 \chi$	$\frac{31}{4}$
$\newcommand{\e}{{\rm e}} \epsilon^{\prime\prime}$	8
$\newcommand{\e}{{\rm e}} \epsilon \chi^4$	$\frac{17}{2}$
$\newcommand{\e}{{\rm e}} \epsilon \chi \partial^4 \chi$	$\frac{35}{4}$

The tendency for the extremal functional method to miss several operators was discussed in [76], which noticed that the numerical spectrum is dominated by double-twist families. These happen to be the families required to match the crossed-channel singularity produced by a unique minimal-twist operator in the analytic bootstrap of [77, 78]. A loose conjecture arising from this is that in any crossing equation with a twist gap, several multi-twist operators with significant OPE coefficients will nevertheless provide a negligible contribution in the numerical boostrap. This is supported, for instance, by the test of the extremal functional method in [79], showing essential differences between the 2D and 3D Ising models. So far, the most reliable numerical bootstrap spectra all come from special cases involving a higher spin symmetry. It is worth mentioning that the analytic bootstrap has recently been extended to handle these cases as well in [80, 81]. What this means for the present case is that we only get a clear picture of the low-lying operators at $\Delta_\sigma = \frac{1}{8}$ i.e. the 2D Ising spectrum. As soon as we raise $\Delta_\sigma$, the theory maximizing $\newcommand{\e}{{\rm e}} \lambda_{\sigma\sigma\epsilon}^2$ becomes nonlocal. Even though the nonlocality is small, we never see operators involving $\chi$ because their contributions in the $\sigma \times \sigma$ OPE are small as well.

In order to improve upon our single correlator results, the next logical step is to bootstrap the correlators $\left <\sigma\sigma\sigma\sigma \right >$, $\newcommand{\e}{{\rm e}} \left < \sigma\sigma\epsilon\epsilon \right >$ and $\newcommand{\e}{{\rm e}} \left < \epsilon\epsilon\epsilon\epsilon \right >$. For this system, it makes a difference whether there are two relevant primary operators or three. Even if we did not know about the shadow relation (2.6), we would be able to infer the existence of a third relevant primary from the well known island of [13]. The assumptions that lead to an island are incompatible with the LRI because there must be a continuous line of fixed points that lead away from the SRI.

Imposing the existence of three relevant primaries, as we should, we find a reassuring exclusion plot in which all regions are connected. Figure 9 shows how they change as a function of $\Delta_T \in \{3, 3.1, 3.2, 3.3, 3.4, 3.5 \}$. Because $3 - \Delta_\sigma$ is different from $3\Delta_\sigma$, many of the generalized free theory solutions from figure 6 are excluded this time.

The boundary of each region has an upper branch and a lower branch. For the intermediate values $\Delta_T \in \{3.1, 3.2, 3.3, 3.4 \}$, we do not observe any evidence that either branch contains a point corresponding to an LRI. If intermediate LRI models do saturate one of the branches, existing estimates for the critical exponents suggest that this should be the upper one. Lower branches for these values of $\Delta_T$ all have $\newcommand{\e}{{\rm e}} \Delta_\epsilon < 1.4$. While there is no candidate LRI kink in figure 9, there is a 'concave kink' for some of the lower branches at $\Delta_\sigma \approx 0.58$. It appears to be a coincidence that the leftmost edge of the $\Delta_T \geqslant 3.1$ region of figure 8 is also near this value of $\Delta_\sigma$. If any bound in figure 9 were to intersect the region where CFTs can exist without $\chi$, a vanishing $\newcommand{\e}{{\rm e}} \lambda_{\sigma\epsilon\chi}^2$ would signal the presence of a kink. Instead, we have found that this OPE coefficient decreases slightly at the special point without going to zero. In figure 10, we maximize $\newcommand{\e}{{\rm e}} \lambda_{\sigma\epsilon\chi}^2$ for $\Delta_T \in \{3, 3.05, 3.1, 3.15 \}$¹⁴. The fact that $\chi$ decouples at a single point in the local case supports the proposal in [23]. It also agrees with the expectation that there is only one irreducible CFT with the same critical exponents as the Ising model.

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In order to gain non-perturbative information about the LRI critical exponents, we will need to examine the minimal system of four-point functions that allows access to (4.2). This consists of $\newcommand{\e}{{\rm e}} \left < \sigma\sigma\epsilon\epsilon \right >$, $\left < \sigma\sigma\chi\chi \right >$, $\newcommand{\e}{{\rm e}} \left < \epsilon\epsilon\chi\chi \right >$ and the three identical correlators. This system yields a much more restrictive region than figure 9 and it will turn out to have interesting features. In order to check that they are in the right place, we have summarized our perturbative data about the LRI in table 2. The last row resums the expansions around $s = \frac{d}{2}$ and s  =  s_* using the $[3, 3]$ Padé approximant. What this means is that we start with the ansatz

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta_{\mathcal{O}}(s) = \frac{a_0 + a_1 s + a_2 s^2}{1 + b_1 s + b_2 s^2} \; , \label{pade-ansatz} \nonumber \end{align} \tag{ 4.4 }$$

and fix the coefficients by demanding that (4.4) have the correct Taylor expansion around the two solvable points.

Table 2. Restating our perturbative results for unprotected operators in the LRI. These expressions are specializations of (2.7), (2.13), (2.17) and (2.18) to d  =  3. To interpolate between the two expansions, we have calculated the symmetric Padé approximant.

	$\newcommand{\e}{{\rm e}} \Delta_\epsilon$	$\Delta_T$
$\varepsilon$-expansion	$\frac{3}{2} - \frac{1}{6} \varepsilon + 0.181\,22 \varepsilon^2$	$\frac{7}{2} - \frac{1}{2} \varepsilon - 0.059\,26 \varepsilon^2$
$\delta$-expansion	$1.412\,63 + 0.269 \delta$	$3 + 2.333 \delta$
$[3, 3]$-Padé	$\frac{1.3759 - 1.7116s + 0.4013s^2}{1 - 1.2086s + 0.2758s^2}$	$\frac{4.7026 - 4.3183s + 0.8191s^2}{1 - 0.7812s + 0.0850s^2}$

Excluding points for $\Delta_T \in \{3, 3.1, 3.2, 3.3, 3.4, 3.5 \}$ again reveals the boundaries in figure 11. The lower branches are much more restrictive than those in figure 9 even though we have not made use of the superblocks yet. As an example, the plot for $\Delta_T = 3$ already appears to single out the onset of mean-field theory—the blue region reaches a very narrow throat at $\newcommand{\e}{{\rm e}} \left ( \Delta_\sigma, \Delta_\epsilon \right) = \left ( \frac{3}{4}, \frac{3}{2} \right)$. The lower branch for this plot also experiences a jump at $\Delta_\sigma \approx 0.65$¹⁵. These features persist for higher values of $\Delta_T$ as well until the allowed region splits into two lobes. The bottom lobe of the $\Delta_T = 3.5$ plot stays very narrow to the left of the throat and ends at $\Delta_\sigma \approx 0.73$. This leftmost edge continues to recede as the number of derivatives is increased.

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The regions shown here start to look more promising after we increase the number of derivatives. The $\Delta_T = 3.3$ region, for instance, moves to the right of the jump and develops two lobes that are connected by a narrow bridge. This makes it possible to plot a comparison between the bottom lobes and the results of table 2 for $\Delta_T > 3.25$. Instead of performing this check ceteris paribus, we have removed the assumption that the $\newcommand{\e}{{\rm e}} [\sigma\chi]_{n, \ell}$ operators are protected. The main conjecture of this work can be tested after the fact by computing an extremal spectrum at several points using the script in [76].

Allowing a continuum of odd-spin operators in the $\sigma \times \chi$ OPE we have plotted the allowed regions for $\Delta_T \in \{3.25, 3.3, 3.35, 3.4, 3.45 \}$ in figure 12. Points on the edge, where we have extracted the spectrum, have been highlighted if they exhibit one of the following interesting properties.

• 1.  
  A point is yellow if it contains a vector suitably close to $[\sigma\chi]_{1,1}$. Our threshold is that its dimension must be within $5\%$ of 6.
• 2.  
  A point is green if it additionally contains a symmetric tensor in $\sigma \times \chi$ whose dimension is within $5\%$ of $\Delta_T$. Note that this is always true for the OPEs with Bose symmetry.

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Points in the first set are likely to survive when we impose the non-renormalization of the double-twist tower. Points in the second set are likely to survive when we use superblocks or the nine correlator system described in appendix¹⁶. Further significance to these points can be seen by plotting the dimension of the first irrelevant scalar. In all long-range Ising models, we expect a value reasonably close to 3 since $\sigma\chi$ is marginally irrelevant at s  =  s_* and $\phi^4$ is marginally irrelevant at $s = \frac{d}{2}$. Figure 13 shows that this is predominantly achieved at the green and yellow points which cluster around the local minimum. Several other exercises along these lines are possible, e.g. checking that the $\mathbb{Z}_2$-odd OPEs $\newcommand{\e}{{\rm e}} \sigma \times \epsilon$ and $\newcommand{\e}{{\rm e}} \epsilon \times \chi$ have low-lying operators in common as well¹⁷. One could also imagine a comparison involving OPE coefficients, in order to see that versions of (2.21) hold with multiple spinning operators. In practice, we have found this difficult as some of the gamma functions are highly sensitive to small errors in the exchanged dimension.

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Looking at the $[3, 3]$ Padé approximant line in figure 12, we see that it comes remarkably close to the green and yellow points. Even after a lobe becomes blunt enough that it can no longer be considered a kink, it is apparently worthwhile to locate desirable features in the extremal spectrum. This approach, advocated in [75], could perhaps reveal useful information about the smooth $\Delta_T < 3.25$ boundaries in figure 11. The crosses in figure 12, being offset from the green and yellow points, simply reflect the fact that convergence is noticeably slower near the $\newcommand{\e}{{\rm e}} \left ( \Delta_\sigma, \Delta_\epsilon \right) = \left ( \frac{3}{4}, \frac{3}{2} \right)$ mean-field theory. They would appear very close to the boundaries if we continued to plot them down to $\Delta_T = 3$.

It should come as no surprise that we have not seen any islands yet. If a point satisfies crossing symmetry, unitarity and the shadow relation for some spin-2 gap $\Delta_T$, it clearly continues to satisfy these criteria when $\Delta_T$ is made less restrictive. To produce an island in this situation, one must resort to imposing whichever additional gaps appear to be most plausible [82]. It is natural to ask if an LRI island can be produced by applying this logic in the spin-2 sector—i.e.by setting the continuum to begin at some $\Delta_{T^\prime} > \Delta_T$ so that the leading spin-2 operator is isolated.

It turns out that this problem demonstrates the power of the superblocks (3.17). Spectral plots analogous to figure 13 tell us that the region carved out by ordinary conformal blocks is perfectly compatible with a large spin-2 gap. It is only the extra OPE relations encoded by the superblocks that ensure a more restrictive region as $\Delta_{T^\prime}$ is increased. Figure 14 shows our attempt to isolate the $\Delta_T = 3.1$ LRI by imposing $\Delta_{T^\prime} \geqslant 4.5$. The result is a fairly large island in which the perturbative prediction from [23] can be found near the bottom¹⁸. There are most likely other allowed points outside this island that we have not attempted to find. It would be interesting to check how small the spin-2 gap can be made before the two regions reunite.

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To approximate conformal blocks for the numerical bootstrap, it is most efficient to use the radial co-ordinate $\rho$ [58]. This is defined by using a conformal transformation to map our four points to the Euclidean configuration $\left < \phi_i(-1) \phi_j(-\rho, -\bar{\rho}) \phi_k(\rho, \bar{\rho}) \phi_l(1) \right >$. Writing $r = |\rho|$ and $\newcommand{\e}{{\rm e}} \eta = \cos \arg \rho$, we convert the block to a meromorphic function of $\Delta$ and apply the recursion relation found in [60]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H^{\Delta_{ij}, \Delta_{kl}}_{\Delta, \ell}(r, \eta) &= r^{-\Delta} G^{\Delta_{ij}, \Delta_{kl}}_{\Delta, \ell}(r, \eta) \nonumber \\ H^{\Delta_{ij}, \Delta_{kl}}_{\Delta, \ell}(r, \eta) &= H^{\Delta_{ij}, \Delta_{kl}}_{\infty, \ell}(r, \eta) + \sum_i \frac{c_i^{\Delta_{ij}, \Delta_{kl}} r^{n_i}}{\Delta - \Delta_i(\ell)} H^{\Delta_{ij}, \Delta_{kl}}_{\Delta_i(\ell) + n_i, \ell_i}(r, \eta) \; . \label{meromorphic-recursion} \nonumber \end{align} \tag{ A.1 }$$

The entire piece and the residue expression, both found in [13], are:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle h_{\infty, \ell}^{\Delta_{12}, \Delta_{34}}(r, \eta) &= \frac{\ell!}{(2\nu)_{\ell}} \frac{(-1)^{\ell} C_{\ell}^{\nu}(\eta) (1 - r^2)^{-\nu}}{(1 + r^2 + 2r\eta)^{\frac{1}{2}(1 + \Delta_{12} - \Delta_{34})}(1 + r^2 - 2r\eta)^{\frac{1}{2}(1 - \Delta_{12} + \Delta_{34})}} \label{longer-data} \nonumber \\ c_1^{\Delta_{12}, \Delta_{34}}(\ell, k) &= -\frac{k(-4)^k}{(k!)^2} \frac{(\ell + 2\nu)_k}{(\ell + \nu)_k} \left ( \frac{1}{2} (1 - k + \Delta_{12}) \right)_k \left ( \frac{1}{2} (1 - k + \Delta_{34}) \right)_k \nonumber \\ c_2^{\Delta_{12}, \Delta_{34}}(\ell, k) &= \frac{k (\nu + 1)_{k - 1} (-\nu)_{k + 1}}{(k!)^2} \frac{\ell + \nu - k}{\ell + \nu + k} \left ( \frac{\ell + \nu - k + 1}{2} \right)_k^{-2} \left ( \frac{\ell + \nu - k}{2} \right)_k^{-2} \nonumber \\ & \quad\left ( \frac{1}{2} (1 - k + \ell - \Delta_{12} + \nu) \right)_k \left ( \frac{1}{2} (1 - k + \ell + \Delta_{12} + \nu) \right)_k \nonumber \\ & \quad \left ( \frac{1}{2} (1 - k + \ell - \Delta_{34} + \nu) \right)_k \left ( \frac{1}{2} (1 - k + \ell + \Delta_{34} + \nu) \right)_k \nonumber \\ c_3^{\Delta_{12}, \Delta_{34}}(\ell, k) &= -\frac{k(-4)^k}{(k!)^2} \frac{(\ell + 1 - k)_k}{(\ell + \nu + 1 - k)_k} \left ( \frac{1}{2} (1 - k + \Delta_{12}) \right)_k \left ( \frac{1}{2} (1 - k + \Delta_{34}) \right)_k \; . \nonumber \end{align} \tag{ A.2 }$$

With this data, (A.1) can be used to generate derivatives with respect to $(z, \bar{z})$ defined such that u  =  |z|² and $v = |1 - z|^2$. However, a more efficient approach is to switch to the variables $a = z + \bar{z}$ and $b = (z - \bar{z}){}^2$ and compute derivatives around a  =  1. Derivatives around b  =  0 can then be found from the second-order Casimir differential equation satisfied by the blocks as explained in [59]. Due to the structure of this equation, the pattern exhibited by $\newcommand{\e}{{\rm e}} \partial_a^m \partial_b^n G^{\Delta_{ij}, \Delta_{kl}}_{\Delta, \ell}(a, b)$ is:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle n &\in \{0, \dots, n_{\mathrm{max}} \} \nonumber \\ m &\in \{0, \dots, 2(n_{\mathrm{max}} - n) + m_{\mathrm{max}} \} \; . \label{ab-derivs} \nonumber \end{align} \tag{ A.3 }$$

Our choices for $(m_{\mathrm{max}}, n_{\mathrm{max}})$ are stated alongside the results in section 4. The other cutoff parameters used in this work are $k_{\mathrm{max}} = 40$ and $\newcommand{\e}{{\rm e}} \ell_{\mathrm{max}} = 20$.

If we wish to combine conformal blocks according to (3.17), the correct rational approximations will include extra poles apart from those captured in table A1. The coefficient $\newcommand{\e}{{\rm e}} R(\Delta, \ell)$ in the superblock

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathcal{G}_{\Delta, \ell}(u, v) = G_{\Delta, \ell}^{0, 0}(u, v) + R(\Delta, \ell) v^{\Delta_\sigma - \frac{d}{2}} G_{\Delta, \ell}^{\Delta_{\chi\sigma}, \Delta_{\sigma\chi}}(u, v) \label{repeated-block} \nonumber \end{align} \tag{ A.4 }$$

can be written as an infinite product since the numerator and denominator of (3.2) have the same sum of gamma function arguments.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle R(\Delta, \ell) = \prod_{k = 0}^{\infty} \frac{(\Delta + \ell + 2k)^2 (\Delta - 2\Delta_\sigma - \ell - 2k) (\Delta + 2\Delta_\sigma - 2d - \ell - 2k)}{(\Delta - d - \ell - 2k)^2 (\Delta - 2\Delta_\sigma + d + \ell + 2k) (\Delta + 2\Delta_\sigma - d + \ell + 2k)} . \label{fortunate-product} \nonumber \end{align} \tag{ A.5 }$$

Table A1. The data required to apply (A.1) where $\nu = \frac{d - 2}{2}$. Each series of poles is labelled by k which in practice needs to be truncated at some $k_{\mathrm{max}}$. There is also a maximum spin in our problem which we label $\newcommand{\e}{{\rm e}} \ell_{\mathrm{max}}$.

ni	$\newcommand{\e}{{\rm e}} \Delta_i(\ell)$	$\newcommand{\e}{{\rm e}} \ell_i$	$\newcommand{\e}{{\rm e}} c_i^{\Delta_{12}, \Delta_{34}}(\ell)$
k	$\newcommand{\e}{{\rm e}} 1 - \ell - k$	$\newcommand{\e}{{\rm e}} \ell + k$	$\newcommand{\e}{{\rm e}} c_1^{\Delta_{12}, \Delta_{34}}(\ell, k)$
2k	$1 + \nu - k$	$\newcommand{\e}{{\rm e}} \ell$	$\newcommand{\e}{{\rm e}} c_2^{\Delta_{12}, \Delta_{34}}(\ell, k)$
k	$\newcommand{\e}{{\rm e}} 1 + \ell + 2\nu - k$	$\newcommand{\e}{{\rm e}} \ell - k$	$\newcommand{\e}{{\rm e}} c_3^{\Delta_{12}, \Delta_{34}}(\ell, k)$

To rule out solutions to crossing, we use the semidefinite program solver SDPB [61]. This requires a 'positive-times-polynomial' expression for the derivative of each conformal block, or more precisely, convolved conformal block (3.14) as these are what enter in crossing equations. To find these expressions with the above algorithm, we use the helper program PyCFTBoot [64]. Note that the existence of a positive prefactor for each polynomial is usually attributed to the fact that poles in table A1 are below the unitarity bound. For the superblocks in this work, we have poles on the whole real line coming from (A.5). This time, positivity is a consequence of the fact that every pole above the unitarity bound is a double pole. Although this seems forunate, numerical stability is an additional obstacle to using the rational approximation (A.5). We will return to this issue at the end of this appendix.

Here, we write the statement of crossing symmetry used to find kinks in this work. Clearly, $\left < \sigma\sigma\sigma\sigma \right >$, $\newcommand{\e}{{\rm e}} \left < \epsilon\epsilon\epsilon\epsilon \right >$ and $\left < \chi\chi\chi\chi \right >$ each have one crossing equation. By repeating the analysis of [13], one sees that the mixed correlators $\newcommand{\e}{{\rm e}} \left < \sigma\sigma\epsilon\epsilon \right >$, $\left < \sigma\sigma\chi\chi \right >$ and $\newcommand{\e}{{\rm e}} \left < \epsilon\epsilon\chi\chi \right >$ each have three. The full system is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sum_{\mathcal{O}^+_{2 | \ell}} \left ( \lambda_{\sigma\sigma\mathcal{O}} \; \lambda_{\epsilon\epsilon\mathcal{O}} \; \lambda_{\chi\chi\mathcal{O}} \right) V^{(0)}_{\Delta, \ell} \left ( \begin{array}{@{}c@{}} \lambda_{\sigma\sigma\mathcal{O}} \nonumber \\ \lambda_{\epsilon\epsilon\mathcal{O}} \nonumber \\ \lambda_{\chi\chi\mathcal{O}} \end{array} \right) + \sum_{\mathcal{O}^-} \lambda^2_{\sigma\epsilon\mathcal{O}} V^{(1)}_{\Delta, \ell} + \lambda^2_{\epsilon\chi\mathcal{O}} V^{(3)}_{\Delta, \ell} + \sum_{\mathcal{O}^+} \lambda^2_{\sigma\chi\mathcal{O}} V^{(2)}_{\Delta, \ell} = 0 \label{6corr-crossing1} \nonumber \end{align} \tag{ A.6 }$$

where each vector has twelve components. We first write these components for the sums with no restriction on spin.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle V^{(1)}_{\Delta, \ell} = \left [ \begin{array}{@{}c@{}} 0 \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ F_{-,\Delta,\ell}^{\sigma\epsilon ; \sigma\epsilon} \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ (-1)^\ell F_{-,\Delta,\ell}^{\epsilon\sigma ; \sigma\epsilon} \nonumber \\-(-1)^\ell F_{+,\Delta,\ell}^{\epsilon\sigma ; \sigma\epsilon} \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ 0 \end{array} \right ], V^{(2)}_{\Delta, \ell} = \left [ \begin{array}{@{}c@{}} 0 \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ F_{-,\Delta,\ell}^{\sigma\chi ; \sigma\chi} \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ (-1)^\ell F_{-,\Delta,\ell}^{\chi\sigma ; \sigma\chi} \nonumber \\ -(-1)^\ell F_{+,\Delta,\ell}^{\chi\sigma ; \sigma\chi} \nonumber \\ 0 \nonumber \\ 0 \end{array} \right ], V^{(3)}_{\Delta, \ell} = \left [ \begin{array}{@{}c@{}} 0 \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ F_{-,\Delta,\ell}^{\epsilon\chi ; \epsilon\chi} \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ 0 \nonumber \\ (-1)^\ell F_{-,\Delta,\ell}^{\chi\epsilon ; \epsilon\chi} \nonumber \\ -(-1)^\ell F_{+,\Delta,\ell}^{\chi\epsilon ; \epsilon\chi} \end{array} \right ]. \label{6corr-crossing2} \nonumber \end{align} \tag{ A.7 }$$

The first sum, whose operators must have even spin, involves components that are $3 \times 3$ matrices. We write them individually as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle && V^{(0)}_1 = \left ( \begin{array}{@{}ccc@{}} F_{-,\Delta,\ell}^{\sigma\sigma ; \sigma\sigma} & 0 & 0 \nonumber \\ 0 & 0 & 0 \nonumber \\ 0 & 0 & 0 \end{array} \right) \; , \; V^{(0)}_{7, 8} = \left ( \begin{array}{@{}ccc@{}} 0 & \frac{1}{2} F_{\mp,\Delta,\ell}^{\sigma\sigma ; \epsilon\epsilon} & 0 \nonumber \\ \frac{1}{2} F_{\mp,\Delta,\ell}^{\sigma\sigma ; \epsilon\epsilon} & 0 & 0 \nonumber \\ 0 & 0 & 0 \end{array} \right) \nonumber \\ && V^{(0)}_2 = \left ( \begin{array}{@{}ccc@{}} 0 & 0 & 0 \nonumber \\ 0 & F_{-,\Delta,\ell}^{\epsilon\epsilon ; \epsilon\epsilon} & 0 \nonumber \\ 0 & 0 & 0 \end{array} \right) \; , \; V^{(0)}_{9, 10} = \left ( \begin{array}{@{}ccc@{}} 0 & 0 & \frac{1}{2} F_{\mp,\Delta,\ell}^{\sigma\sigma ; \chi\chi} \nonumber \\ 0 & 0 & 0 \nonumber \\ \frac{1}{2} F_{\mp,\Delta,\ell}^{\sigma\sigma ; \chi\chi} & 0 & 0 \end{array} \right) \nonumber \\ && V^{(0)}_3 = \left ( \begin{array}{@{}ccc@{}} 0 & 0 & 0 \nonumber \\ 0 & 0 & 0 \nonumber \\ 0 & 0 & F_{-,\Delta,\ell}^{\chi\chi ; \chi\chi} \end{array} \right) \; , \; V^{(0)}_{11, 12} = \left ( \begin{array}{@{}ccc@{}} 0 & 0 & 0 \nonumber \\ 0 & 0 & \frac{1}{2} F_{\mp,\Delta,\ell}^{\epsilon\epsilon ; \chi\chi} \nonumber \\ 0 & \frac{1}{2} F_{\mp,\Delta,\ell}^{\epsilon\epsilon ; \chi\chi} & 0 \end{array} \right) \label{6corr-crossing3} \nonumber \end{align} \tag{ A.8 }$$

with the rest being zero.

As we have already mentioned, these crossing equations do not account for the correlators $\left < \sigma\sigma\sigma\chi \right >$, $\left < \chi\chi\chi\sigma \right >$ and $\newcommand{\e}{{\rm e}} \left < \epsilon\epsilon\sigma\chi \right >$. Even though (A.6) is enough to impose both constraints from the nonlocal equation of motion, the more general system makes it clear that we do not need multiple sums for operators in the same representation. The sum rule

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle && \sum_{\mathcal{O}^+_{2 | \ell}} \left ( \lambda_{\sigma\sigma\mathcal{O}} \; \lambda_{\epsilon\epsilon\mathcal{O}} \; \lambda_{\chi\chi\mathcal{O}} \; \lambda_{\sigma\chi\mathcal{O}} \right) \tilde{V}^{(0)}_{\Delta, \ell} \left ( \begin{array}{@{}c@{}} \lambda_{\sigma\sigma\mathcal{O}} \nonumber \\ \lambda_{\epsilon\epsilon\mathcal{O}} \nonumber \\ \lambda_{\chi\chi\mathcal{O}} \nonumber \\ \lambda_{\sigma\chi\mathcal{O}} \end{array} \right) \nonumber \\ && + \sum_{\mathcal{O}^-} \left ( \lambda_{\sigma\epsilon\mathcal{O}} \; \lambda_{\epsilon\chi\mathcal{O}} \right) \tilde{V}^{(1)}_{\Delta, \ell} \left ( \begin{array}{@{}c@{}} \lambda_{\sigma\epsilon\mathcal{O}} \nonumber \\ \lambda_{\epsilon\chi\mathcal{O}} \end{array} \right) + \sum_{\mathcal{O}^+_{2 \nmid \ell}} \lambda^2_{\sigma\chi\mathcal{O}} \tilde{V}^{(2)}_{\Delta, \ell} = 0 \label{9corr-crossing1} \nonumber \end{align} \tag{ A.9 }$$

now has sixteen rows. It is trivial to determine the first twelve by demanding that the equations of (A.6) are captured in (A.9). Therefore, we will simply write

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle & \tilde{V}^{(0)}_{13} = \left ( \begin{array}{@{}cccc@{}} 0 & 0 & 0 & \frac{1}{2} F_{-,\Delta,\ell}^{\sigma\sigma ; \sigma\chi} \nonumber \\ 0 & 0 & 0 & 0 \nonumber \\ 0 & 0 & 0 & 0 \nonumber \\ \frac{1}{2} F_{-,\Delta,\ell}^{\sigma\sigma ; \sigma\chi} & 0 & 0 & 0 \end{array} \right) , \tilde{V}^{(0)}_{14} = \left ( \begin{array}{@{}cccc@{}} 0 & 0 & 0 & 0 \nonumber \\ 0 & 0 & 0 & 0 \nonumber \\ 0 & 0 & 0 & \frac{1}{2} F_{-,\Delta,\ell}^{\chi\chi ; \sigma\chi} \nonumber \\ 0 & 0 & \frac{1}{2} F_{-,\Delta,\ell}^{\chi\chi ; \sigma\chi} & 0 \end{array} \right) \nonumber \\ & \tilde{V}^{(0)}_{15} = \left ( \begin{array}{@{}cccc@{}} 0 & 0 & 0 & 0 \nonumber \\ 0 & 0 & 0 & \frac{1}{2} F_{-,\Delta,\ell}^{\epsilon\epsilon ; \sigma\chi} \nonumber \\ 0 & 0 & 0 & 0 \nonumber \\ 0 & \frac{1}{2} F_{-,\Delta,\ell}^{\epsilon\epsilon ; \sigma\chi} & 0 & 0 \end{array} \right) , \tilde{V}^{(0)}_{16} = \left ( \begin{array}{@{}cccc@{}} 0 & 0 & 0 & 0 \nonumber \\ 0 & 0 & 0 & \frac{1}{2} F_{+,\Delta,\ell}^{\epsilon\epsilon ; \sigma\chi} \nonumber \\ 0 & 0 & 0 & 0 \nonumber \\ 0 & \frac{1}{2} F_{+,\Delta,\ell}^{\epsilon\epsilon ; \sigma\chi} & 0 & 0 \end{array} \right) \label{9corr-crossing2} \nonumber \end{align} \tag{ A.10 }$$

for $\newcommand{\e}{{\rm e}} \mathcal{O}^+_{2 | \ell}$,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tilde{V}^{(1)}_{13, 14} = \left ( \begin{array}{@{}cc@{}} 0 & 0 \nonumber \\ 0 & 0 \end{array} \right) \; , \; \tilde{V}^{(1)}_{15, 16} = \left ( \begin{array}{@{}cc@{}} 0 & \frac{1}{2} F_{\mp,\Delta,\ell}^{\sigma\epsilon ; \epsilon\chi} \nonumber \\ \frac{1}{2} F_{\mp,\Delta,\ell}^{\sigma\epsilon ; \epsilon\chi} & 0 \end{array} \right) \label{9corr-crossing3} \nonumber \end{align} \tag{ A.11 }$$

for $\mathcal{O}^-$ and $\tilde{V}^{(2)}_{13, 14, 15, 16} = 0$ for $\newcommand{\e}{{\rm e}} \mathcal{O}^+_{2 \nmid \ell}$. The presence of extra equations and larger matrices certainly increases the computation time when using the full system (A.9). However, the main impact on performance comes from the fact that conformal blocks with vanishing and non-vanishing dimension differences can now be found in the same matrix. In simpler problems, the former blocks can be approximated with rational functions of a much lower degree, owing to the fact that their residues vanish for half of the poles in the table A1. In the present case, this advantage is lost as all entries of the $4 \times 4$ matrix need to be accompanied by a common denominator.

After trying a few examples, the bounds from (A.9) appear indistinguishable from those obtained with (A.6). This is consistent with a piece of lore stating that additional four-point functions only help if they increase the number of gaps / OPE constraints that can be imposed. An earlier example of this was noticed in [13] which first studied the 3D Ising model using three correlators. Without a constraint on the number of relevant $\mathbb{Z}_2$-odd operators, the allowed region turned out to be the same as what was already known from a single correlator.

This common denominator mentioned in the ten correlator bootstrap appears in the six correlator case as well if the equations are formulated with superblocks. It also needs to be extended to include the poles of (A.5), leading to a four-fold increase in the degree of each rational approximation. An even more severe problem concerns the stability of SDPB. For normal operation, the user constructs a measure out of a given block's poles and then uses it to compute a basis of orthogonal polynomials [61]. These polynomials only exist for the ordinary blocks because our superblocks are singular above the unitarity bound. In other words, the double poles do not interfere with positivity but they lead to a measure that is not normalizable.

Because of this, we have been unable to complete a full run of SDPB by applying $k_{\mathrm{max}} = 40$ to the rational approximation (A.5). Instead, figure 14 was produced by evaluating superblocks on a discrete $\Delta$ grid which allows us to use the exact gamma functions. It would be interesting to see if alternative solvers are able to remove this complication. Table A2 shows our discretization choices for odd-spin operators (which are protected) and even-spin operators (which should approximate a continuum) when using the information in (3.16).

Table A2. The grid spacing $\Delta_{N + 1} - \Delta_N$ and number of points used for operators in $\sigma \times \chi$ once all consequences of the shadow relation are imposed. After the last point, we use rational approximation to demand positivity on individual sum rule vectors in a continuum. In other words, we no longer impose the shadow relation after a high enough cutoff in $\Delta$.

	Step size	Number of points
$\newcommand{\e}{{\rm e}} \ell = 0$	0.01	1700
$\newcommand{\e}{{\rm e}} \ell = 2$	0.01	1550
$\newcommand{\e}{{\rm e}} 2 \nmid \ell$	2	10

This results in file sizes of about 1GB. We have checked that shrinking the step size and including superblocks for $\newcommand{\e}{{\rm e}} \ell = 4$ does not lead to a significant change in the allowed region of figure 14.