At this point we recall lemma 4.2 from section 4.1. That lemma implies that Schwartz functions of the form $g(H)$ are dense in the Schwartz space. In particular, for any Schwartz $F$, we can find a sequence of functions $\{H_r\}_{r=1}^{\infty}$ such that $g(H_r)(p_2, p_3) \to \hat{F}(p_2 - p_3, p_3)$ in the Schwartz space. Then it follows from (6.92), (6.95), (6.97) that

$$ \begin{aligned} \langle \Psi(H_r) | \Psi(H_r) \rangle &\to \langle \Psi_M(F) | \Psi_M(F) \rangle, \tag{6.98} \\ \langle \Psi(H_r) | \Psi_M(F) \rangle &\to \langle \Psi_M(F) | \Psi_M(F) \rangle. \end{aligned} $$

From the first equation we conclude that $\langle \Psi_M(F) | \Psi_M(F) \rangle \ge 0$, proving Wightman positivity. The two equations taken together imply that

$$ \langle \Psi(H_r) - \Psi_M(F) | \Psi(H_r) - \Psi_M(F) \rangle \to 0, \tag{6.99} $$

i.e. OS states can approximate Wightman states in norm.

## 6.7.2 OS states for complexified times

Let us discuss next the states (6.88) obtained by putting operators at complexified time positions. In these states we don't take the limit to Minkowski space, so they are defined without smearing. Using the Fourier-Laplace representation, their inner product (6.90) is expressed as

$$ \begin{aligned} \langle \mathcal{O}(x_1) \mathcal{O}(x_2) | \mathcal{O}(x_3) \mathcal{O}(x_4) \rangle &= G_4(x_2^\theta, x_1^\theta, x_3, x_4) \\ &= \int dp \, W(p_1, p_2, p_3) \overline{f_{x_1, x_2}(p_2, p_1)} f_{x_3, x_4}(p_2, p_3), \end{aligned} \tag{6.100} $$

where $f_{x,y}(p, q)$, where $0 > \mathrm{Re}(x^0) > \mathrm{Re}(y^0)$, is any Schwartz function which agrees with

$$ e^{p^0 x^0 - i \mathbf{p} \cdot \mathbf{x} - q^0 (x^0 - y^0) + i \mathbf{q} \cdot (\mathbf{x} - \mathbf{y})}. \tag{6.101} $$

for $p, q$ in the forward light cone (where this function is exponentially decreasing) and extends it somehow outside the light cones (it does not matter how because $W$ has support in the forward light cones).

Since $f_{x,y}$ is a Schwartz function, it can be approximated by Schwartz functions of the form $g(H)$ as in the preceding subsection. This implies that non-smeared complexified OS states can be approximated in norm by Euclidean OS states smeared with compactly supported test functions. In particular, the inner product (6.90) is positive definite, providing an extension of pointwise OS positivity to complexified times:

$$ G_4(y^\theta, x^\theta, x, y) \ge 0, \quad (0 > \mathrm{Re} x^0 > \mathrm{Re} y^0). \tag{6.102} $$

As usual, positive-definite inner product implies a Cauchy-Schwarz inequality for the complexified times:

$$ |G_4(x_1, x_2, x_3, x_4)|^2 \le G_4(x_1, x_2, x_2^\theta, x_1^\theta) G_4(x_4^\theta, x_3^\theta, x_3, x_4), \tag{6.103} $$

valid for $\mathrm{Re} x_1^0 > \mathrm{Re} x_2^0 > 0 > \mathrm{Re} x_3^0 > \mathrm{Re} x_4^0$. The analogues of these properties for conformal blocks will be useful in section 7.
**Remark 6.3.** We can extend the reflection operation further for points with complexified both time and space coordinates, as

$$x = (\varepsilon + it, \mathbf{x} + i\mathbf{y}) \mapsto x^{\theta} = (-\varepsilon + it, \mathbf{x} - i\mathbf{y}). \qquad (6.104)$$

With this definition, we can show by the same arguments as above that $G_4(y^\theta, x^\theta, x, y) \ge 0$ (pointwise OS positivity) remains true for $0 > (\mathrm{Re} x^0, \mathrm{Im} \mathbf{x}) > (\mathrm{Re} y^0, \mathrm{Im} \mathbf{y})$ where $\eta_1 > \eta_2$ means $\eta_1 - \eta_2 > 0$ (i.e. in the forward light cone). We can then show that the states $|O(x)O(y)\rangle$ make sense for such $x, y$ and can be approximated in norm by integrated Euclidean OS states.

## 6.8 Wightman clustering

### 6.8.1 2 + 2 split

In this section we will derive clustering (2.4) for Wightman 4-point functions (see [2], section 4.4). As in section 3.3 for the OS case, we will consider 2+2 and 3+1 splits separately. The property we need to prove in the 2+2 case can be written conveniently in the language of Wightman states $|\Psi_M(F)\rangle$, at our disposal by the discussion in section 6.7:

$$\langle \Psi_M(F_1) | \Psi_M(U_{\lambda a} F_2) \rangle \rightarrow \langle \Psi_M(F_1) | \Omega \rangle \langle \Omega | \Psi_M(F_2) \rangle \qquad (6.105)$$

as $\lambda \to \infty$ for any spacelike vector $a$ and any Schwartz test functions $F_1, F_2$, where $U_{\lambda a}$ is translation: $(U_{\lambda a} F_2)(x, y) = F_2(x - \lambda a, y - \lambda a)$, and $\Omega$ is the vacuum state corresponding to inserting the unit operator. By Lorentz invariance it's enough to prove this for $a = (0, \mathbf{a})$, purely spatial vector. In section 3.3 we showed the OS clustering, which we can also write using the integrated OS states (6.93), as

$$\langle \Psi(H_1) | \Psi(U_{\lambda a} H_2) \rangle \rightarrow \langle \Psi(H_1) | \Omega \rangle \langle \Omega | \Psi(H_2) \rangle. \qquad (6.106)$$

As explained in section 6.7, we can find states $|\Psi(H_1)\rangle$ and $|\Psi(H_2)\rangle$ which approximate $|\Psi_M(F_1)\rangle$ and $|\Psi_M(F_2)\rangle$ in norm within any $\varepsilon > 0$. Moreover it's obvious from that construction that the norm is invariant under shifts in purely spatial direction (i.e. the operator $U_{\lambda a}$ is unitary). Hence we have $\|\Psi(U_{\lambda a} H_2) - \Psi_M(U_{\lambda a} F_2)\| = \|\Psi(H_2) - \Psi_M(F_2)\| \le \varepsilon$ for any $\lambda$. By these properties, (6.106) implies (6.105).$^{58}$

### 6.8.2 3 + 1 split

Let us first restate the Euclidean 3+1 clustering argument from section 3.3 in a somewhat more explicit form, and specializing to scalars. So let $\varphi(x_1), \chi(x_2, x_3, x_4)$ be two smooth functions with compact support$^{59}$

$$\mathrm{supp}(\varphi) \subset \{x_1^0 > 0\}, \quad \mathrm{supp}(\chi) \subset \{0 > x_2^0 > x_3^0 > x_4^0\}. \qquad (6.107)$$

<footnote>$^{58}$Indeed we have $|\langle\Psi_M(F_1)|\Psi_M(U_{\lambda a} F_2)\rangle - \langle\Psi(H_1)|\Psi(U_{\lambda a} H_2)\rangle| \le C\varepsilon$ with some $C$ independent of $\lambda$. Now passing to the limit $\lambda \to \infty$ and using (6.106) we obtain $\limsup_{\lambda \to \infty} \langle\Psi_M(F_1)|\Psi_M(U_{\lambda a} F_2)\rangle \le \langle\Psi_M(F_1)|\Omega\rangle\langle\Omega|\Psi_M(F_2)\rangle + C'\varepsilon$, and an analogous lower bound on $\liminf_{\lambda \to \infty}$. Since $\varepsilon > 0$ is arbitrary we obtain (6.105).</footnote>

<footnote>$^{59}$For simplicity, in this section we prove clustering for compactly-supported, as opposed to Schwartz, test functions. We expect that it should be possible to find a proof for Schwartz test functions as well. In any case, the most natural proof would use positivity and the OPE similarly to 2+2 split, provided positivity for higher-point functions is proven (which we don't do in this paper).</footnote>
We would like to show

$$ \lim_{\lambda \to \infty} (G, \varphi_\lambda \otimes \chi) = 0, \quad \varphi_\lambda := \varphi(\cdot - \lambda \hat{e}_1) \qquad (6.108) $$

where $G = G_4^E$ is the Euclidean 4-point function of four identical scalars, and $\hat{e}_1$ is the $x^1$ unit vector. The main idea is that we can find a conformal transformation which moves the point at infinity as well as all the other points to some finite positions. The suppression of the integral then comes from the Jacobian of this transformation. Consider a special conformal transformation $f(x) = \frac{x^\mu+x^2b^\mu}{1+2x\cdot b+x^2b^2} = \mathcal{J} \circ T_b \circ \mathcal{J}$, where $\mathcal{J}$ is inversion and $T_b$ is a translation by $b = \hat{e}_1$. We have $f(-\hat{e}_1) = \infty$, while $f$ is non-singular on supp($\varphi_\lambda$) and supp($\chi$). We also have $f(\infty) = \hat{e}_1$. By conformal invariance we have (compare (6.78)) $(G, \Phi) = (G, \Phi^f)$ where $\Phi^f(x_1, \dots, x_4) = \Phi(f^{-1}(x_1), \dots, f^{-1}(x_4)) \prod_{i=1}^4 J(f^{-1}(x_i))^{\Delta\circ -d}$, where $J(x) = \frac{1}{1+2x\cdot b+x^2b^2}$. We apply this equation with $\Phi = \varphi_\lambda \otimes \chi$. The function $\chi$ is mapped by this transformation to some smooth function. Suppression of the integral in the limit $\lambda \to \infty$ will come from the transformation of $\varphi_\lambda$, which is mapped to

$$ \varphi_{\lambda}^{f}(x_1) := \varphi(f^{-1}(x_1) - \lambda \hat{e}_1) J(f^{-1}(x_1))^{\Delta\circ -d}. \qquad (6.109) $$

Namely we have

$$ |(G, \varphi_{\lambda}^{f} \otimes \chi^{f})| \le C(\lambda)I, \quad I = \int dx_1 |\varphi_{\lambda}^{f}(x_1)|, \quad C(\lambda) = \sup_{x_1 \in \text{supp} \varphi_{\lambda}^{f}} |(G(x_1, \cdot), \chi^f)|. \qquad (6.110) $$

The function $\varphi_\lambda^f$ is nonzero for $f^{-1}(x_1) \in \text{supp}(\varphi) + \lambda \hat{e}_1$, which is a point near infinity for $\lambda$ large. We conclude that $\varphi_\lambda^f$ is supported in a small neighborhood, order $1/\lambda$, of $f(\infty) = \hat{e}_1$. Since $G$ is real-analytic at nonzero point separation, this implies that $C(\lambda)$ is bounded by some constant for $\lambda \ge \lambda_0$. To compute $I$, we do the change of variables $x_1 = f(y)$:

$$ I = \int dy |\varphi(y - \lambda \hat{e}_1)| J(y)^{\Delta\circ} \sim \frac{\text{const}}{\lambda^{2\Delta\circ}}. \qquad (6.111) $$

This finishes the proof of Euclidean 3+1 clustering, eq. (6.108).

Let us proceed next to show Wightman 3+1 clustering. We will show the same equation as (6.108), namely

$$ \lim_{\lambda \to +\infty} (G, \varphi_\lambda \otimes \chi) = 0, \qquad (6.112) $$

where now $G = G_4^M$ is the Minkowski 4-point function, which is a tempered distribution, and $\varphi(x_1)$ and $\chi(x_2, x_3, x_4)$ are arbitrary compactly supported test functions (i.e. no support requirements analogous to (6.107)).$^{60}$ The proof will be based on the same idea of moving the point at infinity to a finite position, paying attention to $G$ now being distribution, and to the requirement (6.79) on invariance under finite Minkowski conformal transformations.

<footnote>$^{60}$The method described below cannot be straightforwardly generalized to the case of Schwartz test functions.</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 5. Location of supports of $\varphi_\lambda$ and $\chi$ with respect to the singularity light cone of $f$.</figcaption></figure>

We will use the same transformation $f(x) = \frac{x^\mu+x^2b^\mu}{1+2x\cdot b+x^2b^2}$, $b = \hat{e}_1$. By translation invariance, we may assume that $\text{supp}(\chi)$ lies at larger $x^1$ values than of the singularity light cone $x^0 = \pm|x + \hat{e}_1|$ of this transformation (see section 6.6). For sufficiently large $\lambda$, $\text{supp}(\varphi_\lambda)$ will also satisfy this condition. As we scale $b$ to zero to connect $f$ to the identity, the singularity light cone moves away to infinity along the negative $x^1$ direction, without touching $\text{supp}(\varphi_\lambda)$ nor $\text{supp}(\chi)$, see figure 5. Hence requirement (6.79) is satisfied and we may apply invariance (6.78), which says $(G, \varphi_\lambda \otimes \chi) = (G, \varphi_\lambda^f \otimes \chi^f)$.

Now, using translation invariance of the 4-point function, $G(x_1, x_2, x_3, x_4) = \tilde{G}(x_2 - x_1, x_3 - x_1, x_4 - x_1)$ we may write

$$ (G, \varphi_{\lambda}^{f} \otimes \chi^{f}) = \int dx_{1} \varphi_{\lambda}^{f}(x_{1}) F(x_{1}), \qquad (6.113) $$

where $F(x_1) = (\tilde{G}, T_{x_1} \cdot \chi^f)$ and $T_a$ is a translation. $\tilde{G}$ is a distribution, but since translation is a continuous operation in the space of test functions, we know that $F(x_1)$ is a continuous function of $x_1$. When $\lambda$ goes to $+\infty$, the support of $\varphi_\lambda^f$ shrinks to the point $\hat{e}_1$.61 Hence for $\lambda \ge \lambda_0$ we can bound $|F|$ on $\text{supp}(\varphi_\lambda^f)$ by a constant, and estimate (6.113) in absolute value by $\text{const} \times \int dx_1 |\varphi_\lambda^f(x_1)|$. This remaining integral is computed via the change of variables as the Euclidean one, and goes to zero as $\lambda^{-2\Delta\phi}$, completing the proof.

## 6.9 Local commutativity

Let us show that the constructed Minkowski correlators satisfy local commutativity. This follows by a robust argument which uses only Lorentz invariance, analyticity in the forward tube, existence of the boundary distribution, together with real analyticity and permutation symmetry of the Euclidean correlators away from coincident points (OS [2], section 4.5). Here for completeness we will provide this argument for $n$-point functions which is its natural setting. In section 6.9.1 below we will make some remarks specific to CFT 4-point functions.

So, we start from the Euclidean correlator $G^E(x_1, \dots, x_n)$ at $x_1^0 > x_2^0 > \dots > x_n^0$ and its analytic continuation $G(x_1, \dots, x_n)$ to the forward tube $\mathcal{T}_n$ which is the set of points

<footnote>61It is important for the argument that, as one can easily check, $\text{supp}(\varphi_\lambda^f)$ shrinks to a compact set (in fact, a point) and not, say, spreads out along some light cone.</footnote>
$x_k \in \mathbb{C}^d$ such that their differences $y_k = x_k - x_{k+1}$ satisfy $\text{Re} y_k^0 > |\text{Im} y_k|$ or equivalently $\eta_k > 0$ in terms of $\zeta_k = (iy_k^0, y_k) = \xi_k + i\eta_k$, $\xi_k, \eta_k \in \mathbb{R}^{1,d-1}$. We will write $G$ instead of $G_n$. We know by theorem 4.1 that this analytic continuation is invariant under Lorentz transformations $\zeta_k \to \Lambda\zeta_k$ where $\Lambda \in L_+^\uparrow$, the identity component of the real Lorentz group. Since $G$ is translationally invariant, it depends only on $\zeta_k$, and we will abuse of notation by sometimes writing $G(\zeta_1, \dots, \zeta_{n-1})$ and $(\zeta_1, \dots, \zeta_{n-1}) \in \mathcal{T}_n$ instead of $G(x_1, \dots, x_n)$ and $(x_1, \dots, x_n) \in \mathcal{T}_n$.

**Step 1.** We will extend domain of analyticity of $G$ using the complex Lorentz group $L(\mathbb{C})$, defined as the set of complex matrices $A$ preserving the Minkowski metric, i.e. $A^T g A = g$ where $g = \text{diag}(-1, 1, \dots, 1)$. We will only need the component of $L(\mathbb{C})$ connected to the identity, denoted $L_+(\mathbb{C})$. For any $\Lambda \in L_+(\mathbb{C})$ consider the equation

$$G(\zeta_1, \dots, \zeta_{n-1}) = G(\Lambda^{-1}\zeta_1, \dots, \Lambda^{-1}\zeta_{n-1}). \qquad (6.114)$$

The two sides of this equation coincide for real $\Lambda \in L_+^\uparrow$ (by Lorentz invariance of $G_n$), and hence by analyticity in the components of $\Lambda$ also for complex $\Lambda \in L_+(\mathbb{C})$, at least for $\Lambda$ close to 1. In other words, eq. (6.114) is just an identity if $\Lambda \approx 1$ and the arguments of $G_n$ on both sides are in the forward tube. But a general $\Lambda \in L_+(\mathbb{C})$ does not preserve the forward tube. For such $\Lambda$, eq. (6.114) extends analytically $G$ from the forward tube to the set

$$\mathcal{T}'_n = \bigcup_{\Lambda \in L_+(\mathbb{C})} \Lambda \cdot \mathcal{T}_n, \qquad (6.115)$$

called the extended tube. The Bargmann-Hall-Wightman theorem shows that no further topological obstructions arise in this analytic continuation; see [75], p. 78 for details. Call this extension $\tilde{G}$.

**Step 2.** Let us consider $\tilde{G}(x_1, \dots, x_n)$ for

$$\epsilon_1 > \dots > \epsilon_{k-1} > 0 > \epsilon_{k+2} > \dots > \epsilon_n, \qquad (6.116)$$

while assuming that $\epsilon_k, \epsilon_{k+1}$ are near zero and much smaller than other $\epsilon_i$'s, and $|x_k - x_{k+1}| < |x_k - x_{k+1}|$, $x_k, x_{k+1}$ real, so that $x_k - x_{k+1}$ approaches a spacelike separation. For $\epsilon_k > \epsilon_{k+1}$ this configuration is in the forward tube, so we know $\tilde{G}$ is analytic there and agrees with $G(x_1, \dots, x_n)$. Let us show that the configurations with $\epsilon_k < \epsilon_{k+1}$ are in the extended tube. We may set $x_{k+1} = 0$ for this argument, so that

$$\zeta_k = (t_k + i\epsilon_k, \mathbf{x}_k). \qquad (6.117)$$

We may assume without loss of generality that $\mathbf{x}_k = (x_k^1, 0, \dots, 0)$, $x_k^1 > |t_k|$. Then acting on $\zeta_k$ with the complexified Lorentz transformation

$$\Lambda_\theta = \begin{pmatrix} \cosh(i\theta) & \sinh(i\theta) \\ \sinh(i\theta) & \cosh(i\theta) \end{pmatrix} \in L_+(\mathbb{C}), \qquad (6.118)$$

with small $\theta$ we get, using $\Lambda_\theta \approx \begin{pmatrix} 1 & i\theta \\ i\theta & 1 \end{pmatrix}$, $\zeta'_k = \Lambda_\theta \zeta_k \approx (t_k, x_k^1) + i(\theta x_k^1 + \epsilon_k, \theta t_k)$, and thus $\eta'_k \approx (\theta x_k^1 + \epsilon_k, \theta t_k)$. If $\epsilon_k$ is negative but very small, we can can achieve $\eta'_k > 0$ by choosing
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Projection of the set $Q_{n,k}$, where the function $\tilde{G}$ is holomorphic, to the plane $(\epsilon_k - \epsilon_{k+1}, t_k - t_{k+1})$. The vertical extent of this region is determined by the condition $|t_k - t_{k+1}| < |\mathbf{x}_k - \mathbf{x}_{k-1}|$. The horizontal extent is determined, among other things, by the condition that $\epsilon_k, \epsilon_{k+1}$ have to be much smaller than all the other $\epsilon_i$'s.</figcaption></figure>

an appropriate small $\theta$. We need $\theta$ small so that all the other $\zeta'_i$ remain in the forward light cone, and this will work because we are assuming that $\epsilon_k$ is very much smaller than all the other $\epsilon_i$'s.

The bottom line is that the extended tube contains an open set of configurations as above, with $|t_k - t_{k+1}| < |\mathbf{x}_k - \mathbf{x}_{k+1}|$ and $\epsilon_k, \epsilon_{k+1}$ small, with $\epsilon_k - \epsilon_{k+1}$ of any sign. Let us call this set $Q_{n,k}$. By restricting this set a bit, we may assume that $Q_{n,k}$ is invariant under permutations of $x_k$ and $x_{k+1}$. By Step 1 we know that function $\tilde{G}$ is holomorphic in the extended tube and hence also in $Q_{n,k}$. In particular, it is analytic if we set $\epsilon_k = \epsilon_{k+1} = 0$. This already has an interesting consequence: the Minkowski correlator is analytic with respect to a pair of spacelike-separated points (while it remains a distribution with respect to all the other points). The projection of $Q_{n,k}$ to the plane $(\epsilon_k - \epsilon_{k+1}, t_k - t_{k+1})$ is shown schematically in figure 6.

The set $Q_{n,k}$ contains real configurations (horizontal axis in figure 6, setting other $t_i \to 0$ as well). Restriction of $\tilde{G}$ to the real part of $Q_{n,k}$ agrees with the Euclidean correlator $G^E$. (They agree for $\epsilon_k > \epsilon_{k+1}$ by construction and for $\epsilon_k < \epsilon_{k+1}$ by the uniqueness of analytic continuation. Recall that the Euclidean correlator $G^E$ is real analytic everywhere away from coincident points, i.e. for $\epsilon_k - \epsilon_{k+1}$ of any sign as long as $\mathbf{x}_k \neq \mathbf{x}_{k-1}$.) One consequence of this fact is that $\tilde{G}$ restricted to the real part $Q_{n,k}$ is permutation invariant w.r.t. $x_k \leftrightarrow x_{k+1}$:

$$ \tilde{G}(\dots x_k, x_{k+1} \dots) = \tilde{G}(\dots x_{k+1}, x_k \dots), \qquad (6.119) $$

because the Euclidean correlator has this property. Finally, since $Q_{n,k}$ is connected to the real configurations (see figure 6), we conclude that permutation invariance (6.119) holds everywhere in $Q_{n,k}$.<fnref n="62" />

We now see the meaning of $\tilde{G}$ for configurations with $\epsilon_k < \epsilon_{k+1}$. Via permutation invariance (6.119), such configurations are mapped to the forward tube and hence can be evaluated as $G$ for the permuted configurations.

<footnote n="62">⁶²In fact, $\tilde{G}$ can be extended to a single-valued holomorphic function on the “permuted extended tube” $\bigcup_{\pi \in S^n} \pi T'_n$, and satisfied permutation invariance (6.119) on this large set. See [75], appendix II, [76] and [77], section 9.D. However for our purposes analyticity and permutation invariance on $Q_{n,k}$ will suffice.</footnote>
**Step 3.** We are now ready to show local commutativity. We have to prove that boundary value limits of two holomorphic functions agree:

$$ \lim_{\epsilon_i \to 0} G(\dots x_k, x_{k+1} \dots) = \lim_{\epsilon_i \to 0} G(\dots x_{k+1}, x_k \dots), \qquad (6.120) $$

when approaching a Minkowski configuration in which $x_k - x_{k+1}$ is spacelike. Note that, by the original definition, the two limits are from different forward tubes: the first one must respect the condition $\epsilon_k > \epsilon_{k+1}$, while the second $\epsilon_{k+1} > \epsilon_k$. By theorem 4.1, Part 3, we can take the limits $\epsilon_i \to 0$ in any order, so let us send $\epsilon_k, \epsilon_{k+1} \to 0$ first, while keeping other $\epsilon_i$ fixed for the moment. For very small $\epsilon_k, \epsilon_{k+1}$, the configurations on both sides will be in $Q_{n,k}$ where both sides are restrictions of the function $\tilde{G}$ analytic around $\epsilon_k, \epsilon_{k+1} = 0$ and satisfying permutation invariance (6.119). It follows that the two sides of (6.120) agree in the limit $\epsilon_k, \epsilon_{k+1} \to 0$. Sending the remaining $\epsilon_i \to 0$ we recover the local commutativity.

### 6.9.1 Local commutativity for CFT 4-point functions

In this paper we analytically continued the CFT 4-point function $\langle O(x_1)O(x_2)O(x_3)O(x_4) \rangle$ to the forward tube using $\rho, \bar{\rho}$ coordinates. We would like to indicate here that this provides an alternative path to understanding local commutativity. We have shown previously that $0 < |\rho|, |\bar{\rho}| < 1$ in the forward tube. Since the extended tube is obtained from the forward tube by complexified Lorentz transformations and $\rho, \bar{\rho}$ are invariant under such transformations, it follows that $0 < |\rho|, |\bar{\rho}| < 1$ also in the extended tube. Below we will show this explicitly for the configurations used in the proof of local commutativity. We consider separately $k = 1$ and $k = 2$ ($k = 3$ being analogous to $k = 1$).

**k = 1:** here $x_1, x_2$ approach spacelike-separated Minkowski points. We know that $|\rho|, |\bar{\rho}| < 1$ in $D_4$, $\epsilon_1 > \epsilon_2 > \epsilon_3 > \epsilon_4$. Extended tube analyticity suggests that this must remain true also for $\epsilon_1 = \epsilon_2 > \epsilon_3 > \epsilon_4$. Indeed, this follows from critical rereading of the proof of lemma 6.1 (section 6.4, eq. (6.38) and below). (That proof does not use the condition $\eta_1 > \eta_2$ but only $\eta_1, \eta_2 > 0$. $^{63}$) It is also important for analyticity that $\rho, \bar{\rho}$ not vanish. In the forward tube $\rho, \bar{\rho}$ do not vanish because $x_{ij}^2 \neq 0$, $i < j$ (lemma 5.1). When $\epsilon_1 = \epsilon_2$ we have $x_{12}^2 > 0$ (spacelike separation), hence also nonzero. These observations show that the CFT 4-point function can be analytically extended, using the $\rho, \bar{\rho}$ expansion, to a neighborhood of points with $\epsilon_1 = \epsilon_2 > 0 > \epsilon_3 > \epsilon_4$, $x_{12}^2 > 0$, in agreement with the general QFT arguments given above.

Let us now permute the first two points: $(\epsilon_1 + it_1, \mathbf{x}_1) \leftrightarrow (\epsilon_2 + it_2, \mathbf{x}_2)$. In the Euclidean region, this transformation maps $\rho \to -\rho, \bar{\rho} \to -\bar{\rho}$ and leaves the 4-point function of identical scalars invariant because the expansion (6.11) contains only even $m$. The same transformation remains true for complexified times for spacelike separation. Taking the limit $\epsilon_1, \epsilon_2 \to 0$, we recover local commutativity very explicitly.

<footnote>$^{63}$An alternative argument is as follows. In section 7 we will show the Cauchy-Schwarz inequality for $\rho, \bar{\rho}$, theorem 7.2, which bounds $\rho, \bar{\rho}$ for any configuration in the forward tube with $\epsilon_1 > \epsilon_2 > 0 > \epsilon_3 > \epsilon_4$ in terms of $\rho, \bar{\rho}$ of “reflection-symmetric” configurations having $\epsilon_3 = -\epsilon_2, \epsilon_4 = -\epsilon_1$. The proof of lemma 7.1, eq. (7.7) shows that $\rho, \bar{\rho}$ remain less than 1 for the latter configurations in the limit $\epsilon_1 \to \epsilon_2$.</footnote>
k = 2: now we are interested in the limit $\epsilon_2 \to \epsilon_3$ from inside $\epsilon_1 > \epsilon_2 > \epsilon_3 > \epsilon_4$.<fnref n="64" /> As for k = 1, critical rereading of the proof of lemma 6.1 shows that $|\rho|, |\bar{\rho}|$ remain less than 1. (We put in that proof $\zeta_2 = \xi_2 + i\eta_2$, $\xi_2 = (t_2, x_2)$ spacelike, and $\eta_2 = (\epsilon_2, 0)$, $\epsilon_2 > 0$. The proof does not use the condition $\eta'_2 > 0$ but only $\eta'_{24} > 0$. The latter condition remains true for $\epsilon_2 \to \epsilon_3 = 0$, as $\zeta'_2$ goes to a finite real vector.) Hence, the CFT 4-point function can be analytically extended, using the $\rho, \bar{\rho}$ expansion, to a neighborhood of points with $\epsilon_1 > \epsilon_2 = \epsilon_3 > \epsilon_4$, $x_{23}^2 < 0$.

To finish the proof of local commutativity, we fall back on the general argument, appealing to the permutation invariance of the (real-analytic) CFT 4-point function under $x_2 \leftrightarrow x_3$. (Unlike for $k=1$, the s-channel OPE expansion (6.11) cannot be used to make this step more explicit, as it does not manifestly have this invariance.)

## 6.10 Generalization to non-identical scalars

In the previous subsections we proved that the 4-point function of identical scalars has analytic continuation to the forward tube $\mathcal{T}_4$, and its boundary value in the Minkowski region is a tempered distribution. Then Minkowski conformal invariance, Wightman positivity, Wightman clustering and local commutativity follow from their Euclidean analogues.

In this section we will indicate how to generalize analytic continuation and temperedness to 4-point functions of non-identical scalars. The proof of the other properties is the same as in the case of identical scalars.

We consider the 4-point function of scalar primary operators $O_i$ with scaling dimensions $\Delta_i$,

$$ G_{1234}^{E}(c_E) := \langle O_1(x_1)O_2(x_2)O_3(x_3)O_4(x_4) \rangle \\ = \frac{1}{(x_{12}^2)^{\frac{\Delta_1+\Delta_2}{2}} (x_{34}^2)^{\frac{\Delta_3+\Delta_4}{2}}} \left(\frac{x_{24}^2}{x_{14}^2}\right)^{\frac{\Delta_1-\Delta_2}{2}} \left(\frac{x_{14}^2}{x_{13}^2}\right)^{\frac{\Delta_3-\Delta_4}{2}} g_{1234}(c_E), \quad (6.121) $$

which reduces to (6.1) when $\Delta_i$'s are identical. The analytic continuation of the prefactor to the forward tube $\mathcal{T}_4$ is straightforward. The function $g_{1234}(c_E)$ only depends on the conformal equivalence class of $c_E$, i.e. $g_{1234}(c_E) = g_{1234}(\rho(c_E), \bar{\rho}(c_E))$. By the similar argument to that in section 6.2, the function $g_{1234}(c_E)$ has the following series expansion

$$ g_{1234}(c_E) = \left[ \frac{(1-\rho)(1-\bar{\rho})}{(1+\rho)(1+\bar{\rho})} \right]^{\frac{\Delta_1-\Delta_2-\Delta_3+\Delta_4}{2}} \sum_{\delta,m} a_{12}(\delta,m) a_{\overline{43}}(\delta,m)^* r^\delta e^{im\theta}, \quad \rho(c_E) = r e^{im\theta}, \quad (6.122) $$

where the sum runs over a discrete set of pairs $(\delta, m)$ with $\delta \ge 0$, $m \in \mathbb{Z}$ (not necessarily even for non-identical scalars), $|m| \le \delta$. Analogously to the case of identical scalars, the

<footnote n="64">⁶⁴The discussion on the local commutativity of this type can also be found in the study of causality in a shockwave background (see section 5 of [4]). In [4], the 2-point function in a shockwave background is defined by $\langle O(x)O(y) \rangle_{\Psi} := \frac{\langle \Psi(i\delta)O(x)O(y)\Psi(-i\delta) \rangle}{\langle \Psi(i\delta)\Psi(-i\delta) \rangle}$, where $x, y$ are Minkowski points and “$i\delta$” means the Euclidean point $(\delta, 0, \dots, 0)$. In our language it corresponds to the 4-point function $\langle \Psi(x_1)O(x_2)O(x_3)\Psi(x_4) \rangle$ with $\epsilon_1 = -\epsilon_4 = \delta > 0$ and $\epsilon_2 = \epsilon_3 = 0$. We know that the 4-point function is regular analytic at such configurations. So the commutator $[O(x), O(y)]$ vanishes in the shockwave background when $x$ and $y$ are spacelike separated.</footnote>
sum is absolutely convergent when $|\rho(c_E)| < 1$ (see below). Also, when $d \ge 3$ we have $p_{\delta,-m} = p_{\delta,m}$, where $p_{\delta,m} = a_{12}(\delta, m)a_{43}(\delta, m)^*$. Analogously to section 6.3, the analytic continuation of $g_{1234}(c)$ in $d \ge 3$ will be given by the formula (compare (6.27))

$$g_{1234}(c) = \left( \frac{x_{14}^2 x_{23}^2}{x_{13}^2 x_{24}^2} \right)^{\frac{\Delta_1 - \Delta_2 - \Delta_3 + \Delta_4}{4}} \sum_{m, \delta, 0 \le m \le \delta} p_{\delta,m} R_{\delta/2-m/2}(c) \Phi_m(c). \quad (6.123)$$

In $d=2$, $p_{\delta,m} \ne p_{\delta,-m}$ but the functions $\rho(c)^m$ and $\bar{\rho}(c)^m$ are individually holomorphic. In this case the analytic continuation of $g_{1234}(c)$ is given by the formula (compare (6.31)):

$$g_{1234}(c) = \left( \frac{x_{14}^2 x_{23}^2}{x_{13}^2 x_{24}^2} \right)^{\frac{\Delta_1 - \Delta_2 - \Delta_3 + \Delta_4}{2}} \sum_{m, \delta, 0 \le m \le \delta} R_{\delta/2-m/2}(c) [p_{\delta,m}\rho(c)^m + p_{\delta,-m}\bar{\rho}(c)^m]. \quad (6.124)$$

We would like to show that

(a) when $r = \max\{|\rho|, |\bar{\rho}|\} < 1$, the series

$$\tilde{g}_{1234}(\rho, \bar{\rho}) = \sum_{\delta, m} a_{12}(\delta, m) a_{43}(\delta, m)^* \rho^{(\delta+m)/2} \bar{\rho}^{(\delta-m)/2} \quad (6.125)$$

is absolutely convergent;

(b) the remainder $\tilde{g}_{1234}(\rho, \bar{\rho}; \delta_*) := \sum_{\delta \ge \delta_*, m} a_{12}(\delta, m) a_{43}(\delta, m)^* \rho^{(\delta+m)/2} \bar{\rho}^{(\delta-m)/2}$ has a power law bound, uniform in $\delta_*$:

$$|\tilde{g}_{1234}(\rho, \bar{\rho}; \delta_*)| \le C(1-r)^{-\Delta_1 - \Delta_2 - \Delta_3 - \Delta_4}. \quad (6.126)$$

This is done as follows (compare [36], section 4.2). Consider the 4-point functions $\langle O_1 O_2 O_2^\dagger O_1^\dagger \rangle$, $\langle O_4^\dagger O_3^\dagger O_3 O_4 \rangle$, and let $\tilde{g}_{12\overline{21}}$, $\tilde{g}_{4\overline{3}34}$ be the analogues of (6.125):

$$\tilde{g}_{12\overline{21}}(\rho, \bar{\rho}) = \sum_{\delta, m} |a_{12}(\delta, m)|^2 \rho^{(\delta+m)/2} \bar{\rho}^{(\delta-m)/2}, \quad (6.127)$$

$$\tilde{g}_{4\overline{3}34}(\rho, \bar{\rho}) = \sum_{\delta, m} |a_{43}(\delta, m)|^2 \rho^{(\delta+m)/2} \bar{\rho}^{(\delta-m)/2}. \quad (6.128)$$

Noticing that $|m| \le \delta$, we estimate (6.125) by absolute value and apply Cauchy-Schwarz inequality:

$$|\tilde{g}_{1234}(\rho, \bar{\rho}; \delta_*)| \le \sum_{\delta, m} |a_{12}(\delta, m)| |a_{43}(\delta, m)| r^\delta \le [\tilde{g}_{12\overline{21}}(r, r) \tilde{g}_{4\overline{3}34}(r, r)]^{1/2}. \quad (6.129)$$

The functions $\tilde{g}_{12\overline{21}}(r, r)$ and $\tilde{g}_{4\overline{3}34}(r, r)$ correspond to the 4-point functions at the Euclidean configurations with $\rho = \bar{\rho} = r < 1$, hence their series expansions (6.127), (6.128) are convergent by the Euclidean OPE axiom. Therefore, (6.125) is absolutely convergent when $|\rho|, |\bar{\rho}| < 1$. This finishes the proof of part (a).

Using the t-channel OPE, we can show that for $0 \le r < 1$,

$$\tilde{g}_{12\overline{21}}(r, r) \le C(1-r)^{-2\Delta_1 - 2\Delta_2}, \quad (6.130)$$

$$\tilde{g}_{4\overline{3}34}(r, r) \le C(1-r)^{-2\Delta_3 - 2\Delta_4}, \quad (6.131)$$

with some $C > 0$. Combining (6.130), (6.131) with (6.129) we get (6.126). This finishes the proof of part (b).
## 7 Optimal powerlaw bound from Cauchy-Schwarz $\rho, \bar{\rho}$ inequality

In section 6.5 we provided a powerlaw bound for the 4-point function, based on the inequality (6.53) for $\max(|\rho(c)|, |\bar{\rho}(c)|)$. That did the job of allowing us to apply theorem 4.1 and prove that the Minkowski 4-point function is a distribution, but the actual bound (6.53) is not optimal. It is interesting to get a better bound on $|\rho(c)|, |\bar{\rho}(c)|$, because this will translate into a better powerlaw bound for the 4-point function, allowing us to get a better idea about the regularity of the Minkowski 4-point function as a distribution, i.e. how many derivatives test functions must have. In the proof of theorem 4.1, parameters $A_n$ and $B_n$ of the powerlaw bound enter into eq. (C.6) which provides an upper bound on the regularity.

In this section we will provide such an optimal bound on $|\rho(c)|, |\bar{\rho}(c)|$. The main idea of the bound and of its proof is inspired by section 6.7.2. Let us denote by $D_4^{(0)}$ the subset of configurations $c \in D_4$ satisfying the condition $\mathrm{Re}\,x_1^0 > \mathrm{Re}\,x_2^0 > 0 > \mathrm{Re}\,x_3^0 > \mathrm{Re}\,x_4^0$. We showed that the 4-point functions for complexified times satisfy the Cauchy-Schwarz inequality (6.103) for $c = (x_1, x_2, x_3, x_4) \in D_4^{(0)}$. For a general configuration $c \in D_4^{(0)}$ we define two configurations

$$c_{12} = (x_1, x_2, x_2^\theta, x_1^\theta), \quad c_{34} = (x_4^\theta, x_3^\theta, x_3, x_4), \qquad (7.1)$$

where $\theta$ is the operation in (6.91) which generalizes the OS reflection to complexified times. We will call such configurations, for obvious reasons, reflection-symmetric. It is clear that both $c_{12}, c_{34} \in D_4^{(0)}$. Eq. (6.103) can now be written as

$$|G_4(c)|^2 \le G_4(c_{12})G_4(c_{34}) \quad (c \in D_4^{(0)}). \qquad (7.2)$$

Since we know that $G_4$ can be written as a convergent power series in $\rho, \bar{\rho}$, eq. (7.2) suggests that there should be a corresponding bound for the $\rho, \bar{\rho}$ coordinates. This is indeed the case, as we have the following couple of results:

**Lemma 7.1.** Any reflection-symmetric configuration $c \in D_4^{(0)}$ has $\rho(c), \bar{\rho}(c) \in (0, 1)$.

**Theorem 7.2** (Cauchy-Schwarz inequality for $\rho, \bar{\rho}$). For any configuration $c \in D_4^{(0)}$ we have the inequality:

$$\max\{|\rho(c)|, |\bar{\rho}(c)|\}^2 \le \max\{\rho(c_{12}), \bar{\rho}(c_{12})\} \times \max\{\rho(c_{34}), \bar{\rho}(c_{34})\}. \qquad (7.3)$$

We will next prove lemma 7.1. We will then show how, combined with theorem 7.2, this implies an optimal bound on $\rho, \bar{\rho}$. Finally we will present a proof of theorem 7.2, which is surprisingly subtle.

### 7.1 Proof of lemma 7.1

To prove the lemma, consider a reflection-symmetric configuration $c$ as in (7.1) with:

$$x_1 = (\epsilon_1 + it_1, \mathbf{x}_1), \quad x_2 = (\epsilon_2 + it_2, \mathbf{x}_2), \quad \epsilon_1 > \epsilon_2 > 0, \qquad (7.4)$$

$$x_3 = x_2^\theta = (-\epsilon_2 + it_2, \mathbf{x}_2), \quad x_4 = x_1^\theta = (-\epsilon_1 + it_1, \mathbf{x}_1).$$
We will compute $z(c)$, $\bar{z}(c)$ explicitly. We can use translations in the $x$ direction, as well as spatial rotations to simplify these computations. All these transformations do not change the conformal class of configuration, hence preserve $u, v$ and $z, \bar{z}$. They also commute with time reflection, and so map reflection-symmetric configurations to reflection-symmetric ones. By using this freedom, we get an equivalent configuration $c'$ with the same $z, \bar{z}$:

$$x_1' = (\epsilon_1 + it_1, 0), \quad x_2' = (\epsilon_2 + it_2, |\mathbf{x}_2 - \mathbf{x}_1|, 0, \dots, 0), \quad x_3' = (x_2')^\theta, \quad x_4' = (x_1')^\theta. \quad (7.5)$$

This is an effectively two-dimensional configuration. The $z, \bar{z}$ variables of a two-dimensional 4-point configuration $x_k = (x_k^0, x_k^1)$ are given by eq. (6.30), which we copy here

$$z = \frac{(z_1 - z_2)(z_3 - z_4)}{(z_1 - z_3)(z_2 - z_4)}, \quad \bar{z} = \frac{(\bar{z}_1 - \bar{z}_1)(\bar{z}_3 - \bar{z}_4)}{(\bar{z}_1 - \bar{z}_3)(\bar{z}_2 - \bar{z}_4)}, \quad z_k = x_k^0 + ix_k^1, \quad \bar{z}_k = x_k^0 - ix_k^1. \quad (7.6)$$

Applying this to the configuration $c'$, we get $z, \bar{z}$ for $c'$ (which are the same as for $c$). It's easy to see that $z_3 - z_4 = (z_1 - z_2)^*$, $z_1 - z_3 = (z_2 - z_4)^*$ as a consequence of reflection symmetry, and similarly for $\bar{z}$'s. So we get $z(c), \bar{z}(c)$ both real and positive. Explicit expressions come out to be

$$z(c) = \frac{(\epsilon_1 - \epsilon_2)^2 + (t_1 - t_2 - |\mathbf{x}_1 - \mathbf{x}_2|)^2}{(\epsilon_1 + \epsilon_2)^2 + (t_1 - t_2 - |\mathbf{x}_1 - \mathbf{x}_2|)^2}, \quad (7.7)$$

$$\bar{z}(c) = \frac{(\epsilon_1 - \epsilon_2)^2 + (t_1 - t_2 + |\mathbf{x}_1 - \mathbf{x}_2|)^2}{(\epsilon_1 + \epsilon_2)^2 + (t_1 - t_2 + |\mathbf{x}_1 - \mathbf{x}_2|)^2}.$$

In particular we see that $0 < z(c), \bar{z}(c) < 1$. The function $f(\zeta)$ in the definition of $\rho$ variables maps the interval $(0, 1)$ to itself. Hence also $0 < \rho(c), \bar{\rho}(c) < 1$, and the lemma and proved.

## 7.2 Optimal bound for $\rho, \bar{\rho}$

We wish to derive a powerlaw bound on $\frac{1}{1-r}$, $r = \max(|\rho|, |\bar{\rho}|)$, since by the arguments in section 6.5 this implies a powerlaw bound for the 4-point function. Our aim here is to improve on (6.53), (6.54).

Consider first a configuration $c \in D_4^{(0)}$. For such a configuration, by theorem 7.2, we have

$$r(c) \le \sqrt{r(c_{12})r(c_{34})} \le \max(r(c_{12}), r(c_{34})), \quad (7.8)$$

and hence

$$\frac{1}{1 - r(c)} \le \max \left( \frac{1}{1 - r(c_{12})}, \frac{1}{1 - r(c_{34})} \right). \quad (7.9)$$

We are thus reduced to study $r(c)$ for reflection-symmetric configurations, like in (7.4). By definition (6.9) of $\rho$ variables, we have

$$\frac{1}{1 - \rho} = \frac{1 + \sqrt{1 - z}}{2\sqrt{1 - z}} \le \frac{1}{\sqrt{1 - z}}, \quad z \in [0, 1), \quad (7.10)$$
so it suffices to study $1/(1-z)$ and $1/(1-\bar{z})$. Using $z, \bar{z}$ for reflection-symmetric configurations computed in eqs. (7.7) we have

$$ \frac{1}{1-z(c_{12})} = \frac{(\epsilon_1 + \epsilon_2)^2 + (t_1 - t_2 - |\mathbf{x}_1 - \mathbf{x}_2|)^2}{4\epsilon_1 \epsilon_2}, \qquad (7.11) $$

and an analogous relation for $\frac{1}{1-\bar{z}(c_{12})}$. From these equations, using $\epsilon_2 < \epsilon_1$, and estimating $\epsilon_1 - \epsilon_2, |t_1 - t_2|, |\mathbf{x}_1 - \mathbf{x}_2|$ from above by $|x_1 - x_2|$ (see (4.10)), we easily get

$$ \frac{1}{1-z(c_{12})}, \frac{1}{1-\bar{z}(c_{12})} \le \left(1 + \frac{1}{\epsilon_2}\right)^2 (1 + |x_1 - x_2|)^2, \qquad (7.12) $$

Putting together this relation, an analogous relation for $z(c_{34}), \bar{z}(c_{34})$, eqs. (7.9) and (7.10), we get

$$ \frac{1}{1-r(c)} \le \max \left\{ \left(1 + \frac{1}{\epsilon_2}\right) (1 + |x_1 - x_2|), \left(1 + \frac{1}{|\epsilon_3|}\right) (1 + |x_3 - x_4|) \right\} \quad (c \in \mathcal{D}_4^{(0)}). \qquad (7.13) $$

This was for $c \in \mathcal{D}_4^{(0)}$. For a general configuration $c \in \mathcal{D}_4$, we will shift the coordinates by a translation in time direction (which of course does not change $\rho, \bar{\rho}$), arranging so that the shifted configurations $c'$ has $\epsilon_2 > 0 > \epsilon_3$, i.e. $c' \in \mathcal{D}_4^{(0)}$. Specifically we will choose

$$ \epsilon_2(c') = \frac{1}{2}(\epsilon_2(c) - \epsilon_3(c)), \quad \epsilon_3(c') = -\frac{1}{2}(\epsilon_2(c) - \epsilon_3(c)). \qquad (7.14) $$

Then, using (7.13) for $c'$, we obtain a bound on $\frac{1}{1-r(c)}$ which for example can be expressed as

$$ \frac{1}{1-r(c)} \le 2 \left(1 + \frac{1}{\epsilon_2 - \epsilon_3}\right) \left(1 + \max\{|x_1 - x_2|, |x_3 - x_4|\}\right) \quad (c \in \mathcal{D}_4). \qquad (7.15) $$

This is a powerlaw bound of the type we were looking for. By considering reflection-symmetric configurations, it's easy to see that the exponents in this bound cannot be improved. Eq. (7.15) is much stronger than our previous suboptimal bound (6.53); in fact it implies a bound of the same form as (6.53) with the power exponent 12 replaced by 2.

## 7.3 Proof of theorem 7.2

Although (7.3) looks like a simple-enough geometric inequality, we do not know an elementary proof of this fact. We essentially guessed this inequality, checked it numerically, and then looked for a proof. Our guess started in the Euclidean region, where $\bar{\rho} = \rho^*$, and (7.3) takes the form

$$ |\rho(c)|^2 \le \rho(c_{12})\rho(c_{34}) \quad (c \in \mathcal{D}_4^{(0)} \text{ Euclidean}). \qquad (7.16) $$

Even in this case we don't know an elementary proof. We guessed that this must hold, because otherwise it was hard to imagine that the 4-point function itself would satisfy a Cauchy-Schwarz inequality. Indeed (7.16) implies the Euclidean version of (6.103). We then guessed (7.3) as a generalization of (7.16) for complexified times.
Our proof of (7.3) reverses this logic, by deriving it from (6.103). There exist many explicit CFT 4-point functions, e.g. mean field theories (MFT). One could imagine that by considering (6.103) for a family of such 4-point functions, and passing to some limit (e.g. of scaling dimension of the mean field going to infinity), one could recover (7.3). We haven't managed to make this work using MFTs, but a closely related strategy does work. Namely we will apply this sort of argument not to the full 4-point function, but to a single conformal block, since the latter also satisfy (6.103) (as we will explain).

Now that we explained the main idea, let us supply the details. By applying a translation, we may set $\mathbf{x}_1 = 0$. The remaining spacial component vectors $\mathbf{x}_2, \mathbf{x}_3, \mathbf{x}_4$ span at most three-dimensional subspace of $\mathbb{R}^{d-1}$. This shows that it is enough to prove the inequality (7.3) in the case $d = 4$. We assume that the readers are familiar with the conformal blocks, which encode contributions of a primary into a 4-point function. In the considered case of 4 identical Hermitean scalar, the relevant OPE has the form (simplifying the general case considered in section 3.1)

$$ \varphi(x_1)\varphi(x_2) = f_{\varphi\varphi}C_{(\lambda)}(x_1, x_2, x_0, \mathcal{D}_0)\mathcal{O}^{(\lambda)}(x_0) \quad (7.17) $$

where $\mathcal{O}^{(\lambda)}$ is a dimension $\Delta$, spin $\ell$ symmetric traceless primary. The conformal block then can be computed by

$$ g_{\Delta,\ell}(c) = C_{(\lambda)}(x_1, x_2, x_0, \mathcal{D}_0)C_{(\mu)}(x_3^{\theta}, x_4^{\theta}, x_0^{\theta}, \mathcal{D}_0^{\theta})\langle \mathcal{O}^{(\lambda)}(x_0)\mathcal{O}^{\dagger(\mu)}(x_0^{\theta}) \rangle. \quad (7.18) $$

The 4d Euclidean conformal blocks are known explicitly [63, 64]:

$$ g_{\Delta,\ell}(c) = \frac{z\bar{z}}{z-\bar{z}}[k_h(z)k_{\bar{h}-1}(\bar{z}) - k_h(\bar{z})k_{\bar{h}-1}(z)], \quad (7.19) $$

where $h, \bar{h} = (\Delta \pm \ell)/2$, and $k_\beta(z) = z^{\beta_2}F_1(\beta, \beta, 2\beta; z)$. (We only cite the result for external operators with equal dimensions.) We will assume that the exchanged operator $\mathcal{O}$ satisfies the 4d unitarity bound $\Delta \ge \ell+2$. As eq. (7.19) shows, Euclidean conformal blocks are real-analytic functions whenever $|z| < 1$. We can also use this formula to analytically continue them to the forward tube. We wish to show that this analytic continuation satisfies some properties. This is best shown not from the explicit formula, but by adapting the robust 4-point function arguments from section 6. Indeed, conformal blocks allow an expansion of the same form as (6.11), with non-negative coefficients which are fixed by conformal invariance. This can be shown by arguments similar to those in section 6.2. The existence of the representation (7.18) guarantees Hilbert space unitarity. Then, by the arguments of section 6.3, conformal blocks admit an analytic extension to the forward tube (which is of course the same as the one following from the explicit formula (7.19)). The point of the current construction is that it shows that the analytic extension satisfies an inequality analogous to (6.46):

$$ |g_{\Delta,\ell}(c)| \le g_{\Delta,\ell}(c_*) \quad (7.20) $$

Then, by the arguments in section 6.5, conformal blocks satisfy the powerlaw bound in the forward tube. (As is easy to see from (7.19), 4d Euclidean conformal blocks grow
as $1/(1-z)$ as $z \to 1^-$ along the real axis, which replaces eq. (6.48).) Finally, by the arguments analogous to section 6.7.2 we conclude that the analytically continued conformal blocks satisfy Cauchy-Schwarz inequality:

$$|g_{\Delta,l}(c)|^2 \le g_{\Delta,l}(c_{12})g_{\Delta,l}(c_{34}) \quad \text{for any } c \in \mathcal{D}_4^{(0)}. \qquad (7.21)$$

(Euclidean reflection positivity of conformal blocks follows from the representation (7.18), which we assume to be valid in the Euclidean region.)

In the rest of the argument we will only need two facts, the Cauchy-Schwarz inequality (7.21) and the explicit Dolan-Osborn formula (7.19). We will apply (7.21) to the blocks of spin $\ell \ge 1$ at the unitarity bound, i.e. with $\bar{h} = 1, h = \ell + 1$. The Cauchy-Schwarz inequality for $\rho, \bar{\rho}$ will follow by extracting the asymptotics in the limit $h \to +\infty$. The asymptotic behavior of $k_h$ is given by the following lemma:

**Lemma 7.3.** For any fixed $z \in \mathbb{C} \setminus [1, +\infty)$, the function $k_h(z)$ has the following asymptotic behavior in terms of the $\rho$ variable defined in (6.9):

$$k_h(z) = (4\rho)^h \left[ \frac{1}{\sqrt{1-\rho^2}} + o(1) \right], \quad h \to +\infty. \qquad (7.22)$$

*Proof.* We have the following identity for $k_h(z)$ [66]:

$$k_h(z) = (4\rho)_2^h F_1(1/2, h; h + 1/2; \rho^2). \qquad (7.23)$$

The region $z \notin [1, +\infty)$ corresponds to $|\rho| < 1$, where the hypergeometric function $_2F_1$ has the power series representation

$$_2F_1(1/2, h; h + 1/2; \rho^2) = \sum_{n=0}^{\infty} \frac{(1/2)_n (h)_n}{n!(h + 1/2)_n} \rho^{2n}. \qquad (7.24)$$

When $h \to +\infty$, each coefficient of the series increases monotonically, and tends to the coefficients of the convergent in the disk $|\rho| < 1$ series

$$\sum_{n=0}^{\infty} \frac{(1/2)_n}{n!} \rho^{2n} = \frac{1}{\sqrt{1-\rho^2}}.$$

This implies the statement of the lemma. $\square$

Consider now inequality (7.21) for the blocks with $\bar{h} = 1, h = \ell + 1$. Since $k_0 \equiv 1$, it reads:

$$|w \cdot [k_h(z) - k_h(\bar{z})]|^2 \le w_{12}w_{34} \cdot [k_h(z_{12}) - k_h(\bar{z}_{12})][k_h(z_{34}) - k_h(\bar{z}_{34})], \qquad (7.25)$$

where we denoted $w = \frac{z\bar{z}}{z-\bar{z}}$, and similarly $w_{12}, w_{34}$. Let us assume that the configuration $c \in \mathcal{D}_4^{(0)}$ is such that

$$|\rho| \neq |\bar{\rho}|, \quad \rho_{12} \neq \bar{\rho}_{12}, \quad \rho_{34} \neq \bar{\rho}_{34}. \qquad (7.26)$$
Then, using lemma 7.3, for large h inequality (7.25) becomes:

$$ (A + o(1)) \max\{|\rho|, |\bar{\rho}|\}^{2h} \le (B + o(1)) \max\{\rho_{12}, \bar{\rho}_{12}\}^h \max\{\rho_{34}, \bar{\rho}_{34}\}^h, \quad (7.27) $$

where A, B are some positive h-independent quantities. Now, taking the limit $h \to +\infty$, we obtain inequality (7.3).

It's easy to see that configurations which violate the condition (7.26) are non-generic. They can be approached by configurations which do satisfy (7.26). Therefore, by continuity inequality (7.3) is valid also for such exceptional configurations.

# 8 OPE convergence in the forward tube and in Minkowski space

We have several OPE convergence statements scattered throughout the paper. The Euclidean CFT axioms assume convergence of the OPE series for $O_1(x_1)O_2(x_2)$ whenever the two points $x_1, x_2$ are closer to the OPE center than any other point. Then we established OPE convergence in the Hilbert space sense (section 3.2) in the Euclidean region for states generated by two operators in the half-space. Then in section 6.2 we used Hilbert space language to derive the power series representation (6.11) for the 4-point function, whose convergence is thus morally equivalent to OPE convergence (for the 4-point functions). We then used this power series representation to analytically continue the 4-point function to the forward tube, and then define the Minkowski 4-point function as a boundary value in the sense of distributions. Finally, in section 6.7 we showed, by arguments not using conformal invariance, that the OS states $|O_1(x_1)O_2(x_2)\rangle$ can be extended to the forward tube and (when integrated against test functions) to the Minkowski region, and that the so obtained states can be arbitrarily well approximated by (integrated) OS states. Therefore, OPE convergence holds for these states, as for the OS states.

In this section we will give a more explicit discussion of the OPE convergence for the Minkowski 4-point function and for the 2-operator states in the forward tube and Minkowski space. We will also explain how our approach and results compare to the classic paper by Mack [41].

## 8.1 Convergence of conformal block decomposition for 4-point functions

Let us consider the 4-point function of identical scalars (6.1):

$$ G(x_1, x_2, x_3, x_4) \equiv G(c) = \langle O(x_1)O(x_2)O(x_3)O(x_4) \rangle = \frac{g(\rho, \bar{\rho})}{(x_{12}^2 x_{34}^2)^{\Delta_O}}. \quad (8.1) $$

The discussion below can be easily extended to non-identical scalars using the same ideas as in section 6.10.

We know that in the Euclidean region the function $g(\rho, \bar{\rho})$ has a convergent conformal block decomposition

$$ g(\rho, \bar{\rho}) = \sum_{\Delta, l} C_{\Delta, l}^2 g_{\Delta, l}(\rho, \bar{\rho}). \quad (8.2) $$
As in section 7.3, we will assume that the reader is familiar with conformal blocks. The main point is that the conformal block decomposition is obtained by separating the series (6.11) into parts corresponding to the conformal multiplets of primary operators $O_{\Delta,l}$ occurring in the $O \times O$ OPE with coefficients $C_{\Delta,l}$. Conformal blocks in the Euclidean region by themselves have power series expansions like (6.11) with positive coefficients (fixed by conformal symmetry). As in section 6.3, we can use this expansion to analytically continue conformal blocks to the forward tube. By an analogue of the bound (6.28) we know that conformal block expansion remains convergent everywhere in the forward tube, since $|\rho|, |\bar{\rho}| < 1$ there. Since individual conformal blocks are smaller than the 4-point function in the Euclidean region, by the arguments in section 6.5 we know that they satisfy a powerlaw bound, and hence they become tempered distributions when going to the Minkowski region.<fnref n="65" />,<fnref n="66" />,<fnref n="67" />

By the arguments like in section 6.5, $g(\rho, \bar{\rho})$, the partial sums of the conformal block decomposition $g(\rho, \bar{\rho}; \Delta_*) = \sum_{\Delta \le \Delta_*, l} C_{\Delta,l}^2 g_{\Delta,l}(\rho, \bar{\rho})$, and the corresponding remainders satisfy in the forward tube a uniform bound:

$$|g(\rho, \bar{\rho}; \Delta_*)|, |g(\rho, \bar{\rho}) - g(\rho, \bar{\rho}; \Delta_*)| \le C(1-r)^{-4\Delta}, \quad r = \max\{|\rho|, |\bar{\rho}|\}. \quad (8.3)$$

Consider the 4-point partial sums including the prefactor $G(c; \Delta_*) = \frac{1}{(x_{12}^2 x_{34}^2)^{\Delta_O}} g(\rho, \bar{\rho}; \Delta_*)$. By the powerlaw bound of $(1-r(c))^{-1}$, we have the powerlaw bounds

$$|G(c; \Delta_*)|, |G(c) - G(c; \Delta_*)| \le C \left(1 + \max_k \frac{1}{\epsilon_k - \epsilon_{k+1}}\right)^A \left(1 + \max_i |x_i - x_{i+1}|\right)^B \quad (8.4)$$

for all $c \in \mathcal{D}_4$ and $\Delta_* \ge 0$. Consider the boundary value of $G(c; \Delta_*)$, call it $G^M(x_1, x_2, x_3, x_4; \Delta_*)$, where $x_i \in \mathbb{R}^{1,d-1}$; it is a tempered distribution by theorem 4.1. The following theorem establishes distributional convergence of conformal block decomposition.

**Theorem 8.1.** We have $G^M(x; \Delta_*) \to G^M(x)$ in the sense of tempered distributions.

*Proof.* Denote $H(c; \Delta_*) = G(c) - G(c; \Delta_*)$. We have to show that, as $\Delta_*$ goes to infinity, the boundary value of $H(c; \Delta_*)$ converges to 0 in the sense of tempered distributions, i.e,

<footnote n="65">⁶⁵This argument shows that any conformal block which occurs in a reflection-positive CFT 4-point function satisfies a powerlaw bound. E.g. conformal blocks for $l \ge 0$, $\Delta \ge l + d - 2$ occur in a 4-point function $\langle \varphi_1 \varphi_2 \varphi_1 \varphi_2 \rangle$ where $\varphi_1, \varphi_2$ are two GFFs of appropriately chosen equal dimension. It should be also possible to show that conformal blocks satisfy a powerlaw bound without relying on a fiducial 4-point function. E.g. for $d = 4$ conformal blocks this follows from their explicit Dolan-Osborn expressions. For general $d$, powerlaw bound on the diagonal $z = \bar{z}$ can be shown using the differential equation found in [78] and extended to $z \ne \bar{z}$ by the usual arguments.</footnote>

<footnote n="66">⁶⁶It should be noted that away from light cones conformal blocks are better than distributions: they are real-analytic there (although this fact won't play a role for us). In even $d$ this is obvious from their explicit expressions in terms of hypergeometric functions. For general $d$ this follows from a first-order matrix ODE satisfied by a finite-length vector including the conformal block and its low-order derivatives. Such an ODE exists for a length-8 vector and can be built using the quadratic and quartic Casimir equations [79].</footnote>

<footnote n="67">⁶⁷Also “conformal partial waves” $\frac{g_{\Delta,l}(\rho, \bar{\rho})}{(x_{12}^2 x_{34}^2)^{\Delta_O}}$ are tempered distributions in the Minkowski space. Therefore their Fourier transforms are well defined. Explicit expressions for these Fourier transforms were found recently in [80].</footnote>
for any Schwartz test function $f \in S(\mathbb{R}^{4d})$

$$ \lim_{\Delta_* \to \infty} \lim_{\lambda \to 0^+} \int H(\lambda \epsilon + it, \mathbf{x}; \Delta_*) f(t, \mathbf{x}) dt d\mathbf{x} = 0, \quad (\epsilon_1 > \epsilon_2 > \epsilon_3 > \epsilon_4), \quad (8.5) $$

where we write for brevity $t$ instead of $t_1, t_2, t_3, t_4$ etc. The proof is the same as in our paper [36], theorem 3.1. We will retrace here the main steps for completeness and because we will need it to establish a stronger result below. Define $L_f(\lambda; \Delta_*) := \int H(\lambda \epsilon + it, \mathbf{x}; \Delta_*) f(x) dx$ with $x = (t, \mathbf{x})$. Since $H$ is holomorphic in $\tau = \lambda \epsilon + it$ we have

$$ \begin{aligned} L_f^{(n)}(\lambda; \Delta_*) &= \int \left( \left( \epsilon \cdot \frac{\partial}{i\partial t} \right)^n H(\lambda \epsilon + it, \mathbf{x}; \Delta_*) \right) f(x) dx \quad (8.6) \\ &= \int H(\lambda \epsilon + it, \mathbf{x}; \Delta_*) \left( \left( i\epsilon \cdot \frac{\partial}{\partial t} \right)^n f(x) \right) dx, \end{aligned} $$

which by the powerlaw bound (8.4) implies

$$ L_f^{(n)}(\lambda; \Delta_*) \le \frac{C_n}{\lambda^A} |f|_{p_n}, \quad \lambda \in (0, 1], \quad p_n = \max\{n, \lceil B \rceil + 4d + 1\}. \quad (8.7) $$

These bounds blow up in the $\lambda \to 0$ limit, but by using the Newton-Leibniz repeatedly one can get bounds which do not blow up:

$$ L_f^{(n)}(\lambda; \Delta_*) \le D_n |f|_{p_{n+\lceil B \rceil+1}}, \quad \lambda \in (0, 1]. \quad (8.8) $$

Using this for $n=1$ one proves that the limit $L_f(0; \Delta_*) = \lim_{\lambda \to 0^+} L_f(\lambda; \Delta_*)$ exists and

$$ |L_f(0; \Delta_*) - L_f(\lambda; \Delta_*)| \le D_1 \lambda |f|_{\max\{[A]+2, \lceil B \rceil+4d+1\}}. \quad (8.9) $$

By Lebesgue's dominated convergence theorem, for any fixed $\lambda$ in $(0, 1]$, $L_f(\lambda; \Delta_*)$ tends to zero as $\Delta_* \to +\infty$. Thus the previous bound implies

$$ \begin{aligned} \varlimsup_{\Delta_* \to \infty} |L_f(0, \Delta_*)| &\le \varlimsup_{\Delta_* \to \infty} |L_f(0, \Delta_*) - L_f(\lambda; \Delta_*)| + \varlimsup_{\Delta_* \to \infty} |L_f(\lambda; \Delta_*)| \\ &\le D_1 \lambda |f|_{\max\{[A]+2, \lceil B \rceil+4d+1\}}. \end{aligned} \quad (8.10) $$

Since $\lambda$ can be arbitrarily small, we get $\varlimsup_{\Delta_* \to \infty} |L_f(0, \Delta_*)| = 0$. This finishes the proof. $\square$

### 8.1.1 Convergence rate for compactly supported test functions

Because of the use of Lebesgue's theorem on dominated convergence, theorem 8.1 does not give the rate of convergence. We will now give the rate in an important special case of compactly supported test functions. This provides an explicit example for the remark in [36], the last paragraph of section 3.3.

The idea is that not only $H(c, \Delta_*) \to 0$ pointwise but it does so exponentially fast. We will first derive the exponential convergence bound, upgrading the Euclidean argument from [65], to the forward tube, and then use it. Let $F(t)$ be the Laplace transform of a positive measure $\mu(E)$ on $E \ge 0$:

$$ F(t) = \int_{0}^{\infty} \mu(E) e^{-Et} dE, \quad \mu(E) \ge 0. \quad (8.11) $$
We assume that the integral is convergent for $t > 0$ and $F(t) \sim t^{-\alpha}$ as $t \to 0^+$, and $\alpha > 0$. Then by the Hardy-Littlewood tauberian theorem we know that

$$M(E) = \int_{0}^{E} \mu(E') dE' \sim \frac{E^{\alpha}}{\Gamma(\alpha + 1)} \quad \text{as } E \to +\infty. \qquad (8.12)$$

We can now estimate the remainder $F_{E_*}(t) := \int_{E_*}^{\infty} \mu(E)e^{-Et} dE$, via

$$F_{E_*}(t) = \int_{E_*}^{\infty} e^{-Et} dM(E) = -M(E_*)e^{-E_*t} + t \int_{E_*}^{\infty} M(E)e^{-Et} dE, \qquad (8.13)$$

which gives $|F_{E_*}(t)| \le C_1 e^{-E_*t} E_*^\alpha + C_2 e^{-E_*t} \left(\frac{1+E_*t}{t}\right)^\alpha$, and finally

$$|F_{E_*}(t)| \le \text{const} \times e^{-E_*t} (t^{-1} + E_*)^\alpha. \qquad (8.14)$$

for any $E_* \ge 1$ (say).

Now let's go back to the 4-point function of four identical scalars $G(c) = \langle \mathcal{O}(x_1)\mathcal{O}(x_2)\mathcal{O}(x_3)\mathcal{O}(x_4) \rangle$, $c \in \mathcal{D}_4$. The remainder $H(c; \Delta_*)$ can clearly be bounded by replacing $\rho, \bar{\rho}$ with $r = \max\{|\rho|, |\bar{\rho}|\}$:

$$|H(c; \Delta_*)| \le \frac{1}{|x_{12}^2|^{|\Delta_O|} |x_{34}^2|^{|\Delta_O|}} [g(r, r) - g(r, r; \Delta_*)], \qquad (8.15)$$

By setting $t = \log(1/r)$, $g(r, r)$ with its representation (6.11) is in the same form as (8.11) and by eq. (6.48) we have that the corresponding $F(t) \sim t^{-\alpha}$ with $\alpha = 4\Delta_O$. Bounding the remainder $g(r, r) - g(r, r; \Delta_*)$ by (8.14), and using (8.15), we get a bound on the remainder $|H(c; \Delta_*)|$ for any $\Delta_* \ge 1$ (say):

$$|H(c; \Delta_*)| \le \frac{\text{const}}{|x_{12}^2|^{|\Delta_O|} |x_{34}^2|^{|\Delta_O|}} \times r(c)^{\Delta_*} \left( \frac{1}{\log(r(c)^{-1})} + \Delta_* \right)^{4\Delta_O}. \qquad (8.16)$$

Let us now convert this into an explicit estimate for the distributional convergence rate, improving on theorem 8.1 for compactly supported test functions. Recall that we have an upper bound on $r(c) = \max\{|\rho(c)|, |\bar{\rho}(c)|\}$ ($c = (x_1, x_2, x_3, x_4)$), eq. (7.15), which we copy here:

$$\frac{1}{1 - r(c)} \le 2 \left( 1 + \frac{1}{\epsilon_2 - \epsilon_3} \right) \left( 1 + \max\{|x_1 - x_2|, |x_3 - x_4|\} \right). \qquad (8.17)$$

This bound tells us how much $r(c)$ is separated from 1. In turn, by (8.16) this translates into an explicit bound on $|H(c; \Delta_*)|$. Let us retrace the proof of theorem 8.1, replacing eq. (8.10) by

$$\begin{aligned} |L_f(0, \Delta_*)| & \le |L_f(0, \Delta_*) - L_f(\lambda; \Delta_*)| + |L_f(\lambda; \Delta_*)| \\ & \le D_1 \lambda |f|_{\max\{|A|+2, \lfloor B \rfloor+4d+1\}} + |L_f(\lambda; \Delta_*)|. \end{aligned} \qquad (8.18)$$

We will now choose $\lambda$ small, as a function of $\Delta_*$, so that the second term in the r.h.s. is smaller than the first one. Let us choose and fix $\epsilon_1 > \epsilon_2 > \epsilon_3 > \epsilon_4$. For $x_k^\lambda = (\lambda\epsilon_k + it_k, x_k)$, $x_k^M = (it_k, x_k)$, bound (8.17) gives

$$\frac{1}{1 - r(c^\lambda)} \le \frac{C_\epsilon}{\lambda} (1 + \max\{|x_1^M - x_2^M|, |x_3^M - x_4^M|\}) \quad (0 < \lambda \le 1), \qquad (8.19)$$
where $C_\epsilon$ is a constant which depends only on $\epsilon_i$ but not on $\lambda$ or $x_k^M$. If $f \in C_0^\infty(\mathbb{R}^{4d})$, a compactly supported test function, then (8.19) implies

$$ \frac{1}{1 - r(c^\lambda)} \le \frac{A_f}{\lambda} \quad (0 < \lambda \le 1, (x_k^M) \in \text{supp}(f)), \qquad (8.20) $$

where $A_f = 2C_\epsilon \sup_{(x_k^M) \in \text{supp}(f)} (1 + \max\{|x_1^M - x_2^M|, |x_3^M - x_4^M|\})$.

Now by (8.16), (8.20), and lemma 5.2(b) we have a bound for a compactly supported test function $f$:

$$ |L_f(\lambda; \Delta_*)| \le \frac{\text{const}}{\lambda^{4\Delta_o}} e^{-\frac{\lambda}{A_f} \Delta_*} \left(\frac{A_f}{\lambda} + \Delta_*\right)^{4\Delta_o} \int |f| dx. \qquad (8.21) $$

By this and (8.18) we have

$$ |L_f(0; \Delta_*)| \le A_1 \lambda + \frac{A_2}{\lambda^{4\Delta_o}} e^{-\frac{\lambda}{A_f} \Delta_*} \left( \frac{A_f}{\lambda} + \Delta_* \right)^{4\Delta_o}, \qquad (8.22) $$

where all constants $A_1, A_2, A_f$ are $f$-dependent. If we choose $\lambda = 1/\Delta_*^\gamma$ then for any $\gamma \in (0,1)$ the first term dominates (the second term decays exponentially fast for large $\Delta_*$). It is easy to see that the first term still dominates for $\lambda = \kappa \frac{\log \Delta_*}{\Delta_*}$ with sufficiently large $\kappa$. We therefore obtain the following promised strengthening of theorem 8.1.

**Theorem 8.2.** For any compactly supported $C^\infty$ test function $f$, we have $(G^M(\Delta_*), f) - (G^M, f) \to 0$ as $O(\frac{\log \Delta_*}{\Delta_*})$.

It is somewhat surprising that conformal block decomposition converges so slowly in the Minkowski region, while it converges exponentially fast in the Euclidean region.

## 8.2 OPE convergence in the sense of $\mathcal{H}^{\text{CFT}}$

We will now rephrase the question of OPE convergence from the point of view of states generated by two operators (rather than 4-point functions which represent inner products of such states). We already discussed these questions to some extent in section 3.2 in the Euclidean region, and section 6.7 in the forward tube and in the Minkowski region. We will now update that discussion.

Recall that in section 6.7 we defined states (see eq. (6.88) and remark 6.3)

$$ \psi(x_1, x_2) = |\mathcal{O}(x_1)\mathcal{O}(x_2)\rangle, \quad x_k = (\epsilon_k + it_k, x_k + i\mathbf{y}_k), \quad 0 \succ (\epsilon_1, \mathbf{y}_2) \succ (\epsilon_1, \mathbf{y}_2), \qquad (8.23) $$

as elements of a vector space with inner products (6.90):

$$ \langle \mathcal{O}(x_1)\mathcal{O}(x_2)|\mathcal{O}(x_3)\mathcal{O}(x_4)\rangle = G_4(x_2^\theta, x_1^\theta, x_3, x_4), \qquad (8.24) $$

where $x^\theta = (-\epsilon + it, \mathbf{x} - i\mathbf{y})$ for $x = (\epsilon + it, \mathbf{x} + i\mathbf{y})$. We then proved that this inner product was positive definite, and that these states could be approximated in norm by Euclidean states, and so belong to the same Hilbert space. The arguments of section 6.7 did not use conformal symmetry.
So, by arguments of section 6.7 we have a map $\psi(x_1, x_2)$ from $x_1, x_2$ as in (8.23) into $\mathcal{H}^{\text{CFT}}$. We claim that this map is holomorphic. To begin with, this map is continuous with respect to the $\mathcal{H}^{\text{CFT}}$ norm, and in particular bounded on compact subsets. This follows easily from the continuity of $G_4$. To show holomorphicity, we will use Morera's theorem and Osgood's lemma (which remain valid for Hilbert-space-valued functions of complex variables). Morera's theorem says that a locally continuous function of one complex variable is holomorphic if its integral over any small contour is zero. Let $C$ be a small 1d contour in the region of $\xi = (x_1, x_2)$ where $\psi$ is defined (we assume that one of the components of $\xi$ goes around the contour while the others stay fix). We have

$$ \left\| \int_C d\xi \, \psi(\xi) \right\|^2 = \int_{\xi' \in C} d\bar{\xi}' \int_{\xi \in C} d\xi \, G_4(\xi'^{\theta}, \xi) = 0, \quad (8.25) $$

the last integral being zero because $G_4$ is holomorphic in $\xi$. Hence $\int_C d\xi \, \psi(\xi) = 0$ and by Morera's theorem $\psi(x_1, x_2)$ is holomorphic in each component separately. Finally, by Osgood's lemma [68] $\psi(x_1, x_2)$ is holomorphic in all variables jointly.

Let us connect this discussion with the OPE. In the Euclidean region, OPE says

$$ \psi(x_1, x_2) = |\mathcal{O}(x_1)\mathcal{O}(x_2)\rangle = \sum_{k, \lambda} f_{\mathcal{O}\mathcal{O}k} C_{k, \lambda}(x_1, x_2, x_S, \mathcal{D}) |(\mathcal{O}_k^\dagger)^{(\lambda)}(x_S)\rangle, \quad (8.26) $$

$$ C_{k,\lambda}(x_1, x_2, x_S, \mathcal{D}) = \sum_{\alpha} f_{\mathcal{O}\mathcal{O}k} C_{k,\lambda,\alpha}(x_1, x_2, x_S) \mathcal{D}^{\alpha}, $$

where $x_1, x_2$ are two Euclidean points in the lower halfspace, $0 > x_1^0, x_2^0, x_S = (-1, 0, \dots)$ is the south pole, and $\mathcal{D} = \mathcal{D}_{x_S}$ is the image of $\partial/\partial x|_{x=0}$ under a conformal transformation which maps $0, \infty$ to $x_S, x_N$. We proved in section 3.2, from the Euclidean CFT axioms, that the series in the r.h.s. converges in $\mathcal{H}^{\text{CFT}}$. As discussed in section 3.2, convergence holds provided that the series is summed in a certain manner: for each $\Lambda$ we define a partial sum $\psi_{\Lambda}(x_1, x_2)$ over all terms with $\Delta_k + |\alpha| < \Lambda$, and then tend $\Lambda \to \infty$. This procedure is needed because, although the states $\mathcal{D}^{\alpha}|(\mathcal{O}_k^\dagger)^{(\lambda)}(x_S)\rangle$ are orthogonal for different $|\alpha|$ (because they are eigenvectors of $\frac{K^0-P^0}{2}$ with different eigenvalues), they are not orthonormal. This can be corrected as follows. For each $k$, let us orthonormalize the infinite multiplet of states $\mathcal{D}^{\alpha}|(\mathcal{O}_k^\dagger)^{(\lambda)}(x_S)\rangle$. Let $e_{k,n}, n \in \mathbb{Z}_{\ge 0}$, be the corresponding orthonormal basis of states (there is obviously a lot of arbitrariness in this basis). We then can write

$$ \psi(x_1, x_2) = |\mathcal{O}(x_1)\mathcal{O}(x_2)\rangle = \sum_k f_{\mathcal{O}\mathcal{O}k} \sum_n \tilde{C}_{k,n}(x_1, x_2) e_{k,n}, \quad (8.27) $$

where $\tilde{C}_{k,n}$'s are some finite linear combinations of $C_{k,\lambda,\alpha}(x_1, x_2, x_S)$. Since $e_{k,n}$'s with different $k$ are also orthogonal, this equation is an expansion of the state $\psi(x_1, x_2)$ in an orthonormal basis. Convergence of the series is now equivalent to the finiteness of the norm of $\psi(x_1, x_2)$:

$$ \|\psi(x_1, x_2)\|_{\mathcal{H}^{\text{CFT}}} = \sum_{k,n} |f_{\mathcal{O}\mathcal{O}k} \tilde{C}_{k,n}(x_1, x_2)|^2 < \infty. \quad (8.28) $$

Moreover, by definition, this norm is nothing but the 4-point function $\langle \mathcal{O}(x_2^\theta)\mathcal{O}(x_1^\theta)\mathcal{O}(x_1)\mathcal{O}(x_2)\rangle$.
Eqs. (8.26)-(8.28) were all in the Euclidean region, but we claim that they continue to make sense in the forward tube. The argument is as follows. We know by the arguments around (8.25) that $\psi(x_1, x_2)$ have analytic continuation to the region (8.23). The inner product $\langle e_{k,n} | \psi(x_1, x_2) \rangle$ is then the analytic continuation of $f_{\mathcal{O}\mathcal{O}k} \tilde{C}_{k,n}(x_1, x_2)$ from the Euclidean to the same region. (This inner product is a finite linear combination of $x_N$-derivatives of the 3-point function $\langle (\mathcal{O}_k)^{(\lambda)}(x_N) \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle$, hence holomorphic in the forward tube.) We thus obtain the following fact:

**Theorem 8.3.** Expansion (8.27), analytically continued from the Euclidean region to the forward tube term by term, converges in the sense of $\mathcal{H}^{\text{CFT}}$ to the same states $\psi(x_1, x_2)$ in the region (8.23) that we defined in section 6.7.

For the subsequent discussion, we also define the states

$$ \psi_k(x_1, x_2) = f_{\mathcal{O}\mathcal{O}k} \sum_n \tilde{C}_{k,n}(x_1, x_2) e_{k,n}. \qquad (8.29) $$

The norms of these states is given by conformal blocks (up to prefactor $f_{\mathcal{O}\mathcal{O}k}^2/(x_{12}^2)^{4\Delta\phi}$). Just as the state $\psi(x_1, x_2)$, each state $\psi_k(x_1, x_2)$ is an $\mathcal{H}^{\text{CFT}}$-valued holomorphic function in the region (8.23), moreover in this region we have

$$ \psi(x_1, x_2) = \sum_k \psi_k(x_1, x_2), \qquad (8.30) $$

the series convergent in the sense of $\mathcal{H}^{\text{CFT}}$. The norm of the tail of this series is given by the function $H(c, \Delta_*)$ from the proof of theorem 8.1:

$$ \left\| \sum_{\Delta_k > \Delta_*} \psi_k(x_1, x_2) \right\| = H(c, \Delta_*), \quad c = (x_2^\theta, x_1^\theta, x_1, x_2). \qquad (8.31) $$

Vladimirov's theorem 4.1 remains true for Hilbert-space-valued holomorphic functions, whose norm satisfies a powerlaw bound in the forward tube. Applying such a version of theorem 4.1, as well as arguments from the proof of theorem 8.1 and from section 8.1.1 (in particular using the bound (8.16)), it's easy to obtain the following result (we omit the proof).

**Theorem 8.4.** (a) The boundary value $\text{bv}(\psi) = \lim_{\epsilon \to 0} \psi(x_1, x_2)$ exists as $\mathcal{H}^{\text{CFT}}$-valued tempered distributions, and similarly for each $\text{bv}(\psi_k)$: $\text{bv}(\psi), \text{bv}(\psi_k) \in S'(\mathbb{R}^{2d}) \otimes \mathcal{H}^{\text{CFT}}$. Explicitly, the limit

$$ (\text{bv}(\psi), f) = \lim_{\epsilon \to 0} \int \psi(\epsilon + it, \mathbf{x}) f(t, \mathbf{x}) \, d^2t \, d^{2(d-1)}\mathbf{x} \qquad (8.32) $$

exists as a vector in $\mathcal{H}^{\text{CFT}}$ for any Schwartz function $f \in S(\mathbb{R}^{2d})$, and is a continuous linear operator from $S(\mathbb{R}^{2d})$ to $\mathcal{H}^{\text{CFT}}$, and analogously for $\text{bv}(\psi_k)$;

(b) (Distributional OPE convergence in $\mathcal{H}^{\text{CFT}}$) For each Schwartz test function $f \in S(\mathbb{R}^{2d})$, the series $\sum_k (\text{bv}(\psi_k), f)$ converges in $\mathcal{H}^{\text{CFT}}$ norm to $(\text{bv}(\psi), f)$;

(c) (Convergence rate) For compactly supported test functions, the series in (b) summed over $\Delta_k \le \Delta_*$ converges with rate $O(\sqrt{\log \frac{\Delta_*}{\Delta_*}})$.
**Remark 8.1.** The following more finegrained version of theorem 8.4(b) also holds. Denote

$$E_{k,n}(x_1,x_2) = f_{\mathcal{O}\mathcal{O}_k} \tilde{C}_{k,n}(x_1,x_2)e_{k,n}. \qquad (8.33)$$

As explained above, $\tilde{C}_{k,n}(x_1,x_2)$ is a finite sum of terms like $(\mathcal{D}^\theta)^\alpha \langle (\mathcal{O}_k)^{(\lambda)}(x_N)\mathcal{O}(x_1)\mathcal{O}(x_2) \rangle$ (all having the same $|\alpha|$), so $\text{bv}(E_{k,n})$ exists by Vladimirov's theorem. Then for each Schwartz test function $f \in S(\mathbb{R}^{2d})$, the series $\sum_{k,n} (\text{bv}(E_{k,n}), f)$ converges in $\mathcal{H}^{\text{CFT}}$ norm to $(\text{bv}(\psi), f)$.

It would be interesting to prove a version theorem 8.4(c), truncating the series $\sum_{k,n} (\text{bv}(E_{k,n}), f)$ to $k, n$ such as the corresponding $\Delta_k + |\alpha| \le \Delta_*$. To do so one would have to find an analogue of the bounds (8.31), (8.16) valid for such a truncation. This is not straightforward because the partial sums of the series (8.28) truncated to $\Delta_k + |\alpha| \le \Delta_*$ do not correspond to a simple truncation of the $\rho, \bar{\rho}$ series of the full 4-point function (basically because the transformation which maps $x_1, x_2, x_1^\theta, x_2^\theta$ to their $\rho, \bar{\rho}$ conformal frame does not necessarily map $x_S, x_N$ to $0, \infty)$.

## 8.3 Comparison to Mack's work on OPE convergence

To assume OPE convergence as an axiom in Euclidean CFT, and to derive Minkowski physics from it, as we did in this paper, seems natural from the modern perspective. On the contrary, in the early days of CFTs it was considered natural to assume standard Minkowski physics (such as Wightman axioms). The OPE convergence was not assumed at the time, but was something to be derived.

This was the underlying philosophy of the works by Lüscher and Mack [71] and of Mack [41, 81]. Written 45 years ago, these papers remain widely cited, but not everyone is familiar with what precisely has been derived there. Here we will present a short review for the benefit of the modern audience.

These works make two main assumptions. First, that we have a unitary quantum field theory in the Minkowski signature which satisfies Wightman axioms (in particular has a Hilbert space $\mathcal{H}$ on which the Poincaré group acts unitarily and quantum fields are operator-valued distributions). Correlators then have the usual analyticity properties of Wightman functions, in particular they are real-analytic in the Euclidean. The second main assumption is that these Euclidean correlators are invariant under the action of the Euclidean conformal group.

Using these two assumptions, Lüscher and Mack [71] proved that the Hilbert space $\mathcal{H}$ carries a unitary representation not just of the Poincaré but of the group $G^* = \text{universal cover of the Lorentzian conformal group SO}(d, 2)^{68}$. Mack [81] then classified all unitary positive energy representations of $G^*$. It should be mentioned that refs. [41, 71, 81]

<footnote n="68">⁶⁸One also often quotes Lüscher and Mack [71] for proving that CFT correlation functions may be continued to the Minkowski cylinder $S^{d-1} \times \mathbb{R}$. This is a misquotation as they did not prove this, but posed it as a conjecture. What they did prove was that CFT correlation functions can be analytically continued to a domain of which $S^{d-1} \times \mathbb{R}$ is a real boundary. One still needs to establish a powerlaw bound to take the boundary limit and obtain a distribution. We plan to derive this fact in our future work [72], for 4-point functions, from the Euclidean CFT axioms, using the $\rho, \bar{\rho}$ expansion.</footnote>
only consider $d = 4$ spacetime dimensions. Many explicit group theoretic calculations are done only for this value of $d$. The results should however generalize to arbitrary $d$ with appropriate modifications.

Continuing this program, Mack [41] studied distributional OPE convergence in Minkowski CFT. Since we also have results of this kind (section 8.2), it will be particularly interesting to compare with Mack’s discussion. So let us review his argument. Compared to [71, 81], ref. [41] includes one extra assumption: that the OPE $\varphi_i(x_1)\varphi_j(x_2)$ of two fields acting on the Minkowski vacuum is valid in an asymptotic sense. Namely that for some dense set of states $\psi$ we have$^{69}$

$$ \langle \psi | \varphi_i(x_1) \varphi_j(x_2) \rangle \sim \sum_k C_{ijk}(x_1, x_2) \langle \psi | \varphi_k(0) \rangle, \qquad (8.34) $$

as an asymptotic expansion for rescaling $x_1, x_2 \to \lambda x_1, \lambda x_2$, where $C_{ijk}(x_1, x_2)$ are some $\psi$-independent homogeneous functions: $C_{ijk}(\lambda x_1, \lambda x_2) = \lambda^{\Delta_k - \Delta_i - \Delta_j} C_{ijk}(x_1, x_2)$. $^{70}$ Asymptotic means that if we truncate the expansion at $\Delta_k = \Delta_*$ and take $\lambda \to 0$ limit for any fixed $x_1, x_2$ then the error is $o(\lambda^{\Delta_* - \Delta_i - \Delta_j})$. Note that there are both primaries and descendants among $\varphi_k$’s. The main result of [41] is to convert this asymptotic expansion to an expansion convergent in the Hilbert space sense.

The first step is to use a general result that any Hilbert space carrying a unitary representation of a semisimple Lie group can be decomposed as a direct integral of unitary irreducible representations [82]. Since, by the above-mentioned result of [71], $\mathcal{H}$ carries a unitary representation of $G^*$, we thus have

$$ \mathcal{H} = \int^{\oplus} d\mu_{\chi} d\tilde{\mu}_{\nu} \mathcal{H}^{\chi\nu}, \qquad (8.35) $$

where $\chi$ labels different unitary irreps of $G^*$, $\chi = (\Delta_{\chi}, \rho_{\chi})$ with $\Delta_{\chi}$ the scaling dimension and $\rho_{\chi}$ a Lorentz group irrep, and $\nu$ labels different copies of the same irrep. Only positive energy irreps may occur, since all states of Wightman theory have positive energy. By definition, eq. (8.35) identifies every vector $\psi \in \mathcal{H}$ with a Hilbert-space-valued function $(\chi, \nu) \mapsto \psi_{\chi\nu} \in \mathcal{H}^{\chi}$, some standard realization of the irrep $\chi$. It is assumed that $\langle \psi | \psi \rangle < \infty$, inner products being given by the following integral:

$$ \langle \psi | \psi' \rangle = \int d\mu_{\chi} d\tilde{\mu}_{\nu} \langle \psi_{\chi\nu} | \psi'_{\chi\nu} \rangle_{\mathcal{H}^{\chi}}, \qquad (8.36) $$

Also $G^*$ acts on $\psi$ by acting on each $\psi_{\chi\nu}$. Furthermore, ref. [81] realized $\mathcal{H}^{\chi}$ as a space of distributions $\psi(x)$ on Minkowski space$^{71}$ taking values in the representation space of $\rho_{\chi}$, with Fourier transform supported in the forward light cone, and for which the following inner product (defined initially on a dense subset of smooth $\psi, \psi'$) is finite:

$$ \langle \psi | \psi' \rangle_{\mathcal{H}^{\chi}} = \int dx dy \overline{\psi(x)} I^{\chi}(x - y) \psi'(y), \qquad (8.37) $$

<footnote>$^{69}$ Mack assumes $x_2 = -x_1$ but here for simplicity we will assume that this is valid for any $x_1, x_2$.</footnote>

<footnote>$^{70}$ In fact $C_{ijk}$ is a distribution so homogeneity should be understood in the sense of pairing with a rescaled test function.</footnote>

<footnote>$^{71}$ More properly $\psi(x)$ is a distribution on the Lorentzian cylinder on which the group $G^*$ acts naturally, but due to a periodicity condition it may be reconstructed from its values on the Poincaré patch.</footnote>
where $I^\chi$ is an “intertwining kernel”. Physically, $I^\chi$ is the Minkowski CFT 2-point function of the primary in the “shadow irrep” of $\chi$.

The above integration measure $d\mu_\chi d\tilde{\mu}_\nu$ depends on the theory; from the abstract arguments alone it may be continuous or discrete. Ref. [41] then proceeds to show that (a) this measure is actually discrete (a sum of delta-functions), so that the direct integral is a direct sum; (b) that the state $|O_i(x_1)O_j(x_2)\rangle$ produced by two Minkowski primary operators acting on the Minkowski vacuum can be written as

$$|O_i(x_1)O_j(x_2)\rangle = \sum_{k,a} f_{ijk}^a \int dx B_{k,ij}^a(x, x_1, x_2)|O_k(x)\rangle, \quad (8.38)$$

where $B_{ijk}^a(x, x_1, x_2)$, $a = 1...N_{ijk}$, are some kinematically determined distributions, the convergence is in the Hilbert space sense after integrating with an arbitrary test functions $f(x_1, x_2)$, and the local primary operators $O_k$ occurring in this sum have quantum numbers in the discrete set where the integration measure $d\mu_\chi d\tilde{\mu}_\nu$ is supported.

To show how this comes about, let us focus on the case of scalar identical $O_i = O_j = O$ for simplicity. In this case expansion (8.38) will end up being precisely our expansion (8.30) (although derived under very different assumptions), with $B(x, x_1, x_2)$ related to the OPE kernel in (8.26).

Applying (8.36) with $|\psi'\rangle = |O(x_1)O(x_2)\rangle$ gives (eq. (2.6) in [41]):

$$\langle\psi|O(x_1)O(x_2)\rangle = \int d\mu_\chi d\tilde{\mu}_\nu c_{\chi\nu} \int dx \overline{\psi_{\chi\nu}(x)} B_\chi(x, x_1, x_2), \quad (8.39)$$

where we denoted $\int dy I^\chi(x-y) \psi'_{\chi\nu}(y) = c_{\chi\nu} B_\chi(x, x_1, x_2)$ where $c_{\chi\nu}$ is a proportionality factor, and $B_\chi(x, x_1, x_2)$ is a kinematically determined distribution (it is basically the amputated Minkowski 3-point function $\langle O_\chi(x)O(x_1)O(x_2)\rangle$). Mack then undertakes a meticulous study of $B_\chi(x, x_1, x_2)$ and of its Fourier transform with respect to the first argument $\hat{B}_\chi(p, x_1, x_2)$. This actually takes most of his paper, and involves many explicit nontrivial calculations (e.g. it involves the first ever explicit characterization of the most general 3-point function of CFT primaries in arbitrary irreps). One of the main results is that $\hat{B}_\chi(p, x_1, x_2)$ are entire functions of $p$:

$$\hat{B}_{\chi}(p, x_1, x_2) = \sum_{|\alpha| \ge 0} b_{\chi}^{\alpha}(x_1, x_2) (-ip)_{\alpha}, \quad (8.40)$$

where $b_\chi^\alpha(x_1, x_2) = (x_{12}^2)^{-\Delta_\phi+\Delta_\chi/2}$ times a polynomial in $x_1, x_2$ of degree $|\alpha|$, in particular $b_\chi^\alpha(\lambda x_1, \lambda x_2) = \lambda^{\Delta_\chi+|\alpha|-2\Delta_\phi} b_\chi^\alpha(x_1, x_2)$.

Let us now specialize to states $\psi$ for which the function $\psi_{\chi\nu}(x)$ has Fourier transform of compact support (one can show that such states are dense in $H$). Then the previous equations imply the following convergent expansion for the integrand in (8.39):

$$\int dx \overline{\psi_{\chi\nu}(x)} B_\chi(x, x_1, x_2) = \sum_{\alpha} b_\chi^\alpha(x_1, x_2) \overline{\partial^\alpha \psi_\chi(0)}, \quad (8.41)$$

Mack then claims (before eq. (2.11')) that if, for each $\chi$, this convergent expansion is truncated at $\Delta_\chi + |\alpha| = \Delta_*$, and inserted back into (8.39), this results in an asymptotic
expansion for the l.h.s. of (8.39). I.e. for any $\Delta_*$ (Mack does not write this equation explicitly):

$$ \langle \psi | \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle = \left\{ \int d\mu_{\chi} d\tilde{\mu}_{\nu} c_{\chi\nu} \sum_{\Delta_{\chi} + |\alpha| \le \Delta_*} b_{\chi}^{\alpha}(x_1, x_2) \overline{\partial^{\alpha} \psi_{\chi}(0)} \right\} + E(x_1, x_2; \Delta_*), \quad (8.42) $$

where the error term $E(\lambda x_1, \lambda x_2; \Delta_*) = \mathcal{O}(\lambda^{\Delta_* - 2\Delta_\alpha})$ as $\lambda \to 0$ for any fixed $x_1$. Unfortunately, Mack does not give any justification of this claim, which to us does not appear self-evident. The difficulty is that although for every $\chi, \nu$ the truncated series has error $\mathcal{O}(\lambda^{\Delta_* - 2\Delta_\alpha})$, the constant will certainly depend on $\chi, \nu$. How do we know that the error estimate survives after the integration in $\chi, \nu$? It might be possible to close this omission in Mack's reasoning using normalizability of $|\psi\rangle$, but this needs extra arguments compared to what is given in his paper, and we have not investigated this in detail.<fnref n="72" /> Researchers relying on Mack's result should keep this caveat in mind.

Assuming that (8.42) is true, the argument is completed as follows. We now have two asymptotic expansions for the l.h.s. of $\langle \psi | \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle$, one coming from (8.42), and another from (8.34). The second one is discrete (by assumption), so the first one also must be discrete. This establishes that the measure $d\mu_{\chi} d\tilde{\mu}_{\nu}$ is discrete, a sum of delta functions, hence we can write (8.39) with the r.h.s. as a sum, not an integral:

$$ \langle \psi | \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle = \sum_n c_{\chi_n} \int dx \overline{\psi_{\chi_n}(x)} B_{\chi_n}(x, x_1, x_2). \quad (8.43) $$

A more detailed comparison of this equation with (8.34) leads us to conclude that (a)

$$ c_{\chi_n} \overline{\psi_{\chi_n}(x)} = f_n \langle \psi | \mathcal{O}_{\chi_n}(x) \rangle, \quad (8.44) $$

where $\mathcal{O}_{\chi_n}$ are primary operators related by rescaling to a subset of the local operators $\varphi_k$, we choose them unit-normalized (hence a coefficient $f_n$); (b) that all the other operators $\varphi_k$ are the descendants $\mathcal{O}_{\chi_n}$'s; and (c) that all coefficients $C_k(x_1, x_2)$ are basically the expansion coefficients $b_{\chi_n}^{\alpha}(x_1, x_2)$ in (8.40). From (8.43) and (8.44), we have

$$ \langle \psi | \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle = \sum_n f_n \int dx \langle \psi | \mathcal{O}_{\chi_n}(x) \rangle B_{\chi_n}(x, x_1, x_2), \quad (8.45) $$

for a dense set of states $\psi$. Because of the orthogonality of different $|\mathcal{O}_{\chi_n}(x)\rangle$'s, this implies that

$$ |\mathcal{O}(x_1) \mathcal{O}(x_2)\rangle = \sum_n f_n \int dx B_{\chi_n}(x, x_1, x_2) |\mathcal{O}_{\chi_n}(x)\rangle, \quad (8.46) $$

the sum convergent in the Hilbert space sense after integrating out with any test function $f(x_1, x_2)$. This is eq. (8.38) in the considered case $\mathcal{O}_i = \mathcal{O}_j = \mathcal{O}$.

<footnote n="72">⁷²We also tried, but unfortunately we did not manage, to get feedback from Prof. Gerhard Mack concerning this matter.</footnote>
### 8.3.1 Relating Mack's kernel B to the Euclidean OPE kernel C

Now we would like to relate Mack's OPE kernel B to our OPE kernel $C_{a,(\lambda)}^{(\mu)(\nu)}(x_1, x_2, x_0, \partial_0)$ defined by eqs. (2.35) and (2.36). We only consider the OPE kernel for the scalar external operators for simplicity, i.e. $C_{\chi}(x_1, x_2, x_0, \partial_0)$, $\chi = (\Delta, \ell = 0)$; similar remarks apply in the general case. We first give the conclusion:

$$C_{\chi}(x_1, x_2, x_0, \partial_0) = \sum_{\mu} b_{\chi}^{\mu}(x_1 - x_0, x_2 - x_0) \partial_{0}^{\mu}, \qquad (8.47)$$

where the coefficient functions $b_{\chi}^{\mu}$ are the same as in (8.40). One could “derive” this by using (8.38) and formally manipulating the integral in the momentum space:<fnref n="73" />

$$\begin{aligned} |\mathcal{O}(x_1)\mathcal{O}(x_2)\rangle_{\chi} &= \int dx B_{\chi}(x, x_1, x_2)|\mathcal{O}_{\chi}(x)\rangle \\ &= \int dp \hat{B}_{\chi}(p, x_1, x_2)|\hat{\mathcal{O}}_{\chi}(p)\rangle = \sum_{\mu} b_{\chi}^{\mu}(x_1, x_2)|\partial^{\mu} \mathcal{O}_{\chi}(0)\rangle, \end{aligned} \qquad (8.48)$$

which shows (8.47) in the case when $x_0 = 0$. The general $x_0$ case follows by translation invariance. This derivation is not rigorous for various reasons: (a) we did not clarify the meaning of $\hat{\mathcal{O}}_{\chi}(p)$; (b) why can we swap the order of summation and integration? (c) the above derivation is done in Minkowski region, how do we match the coefficients $b_{\chi}^{\mu}(x_1, x_2)$ with the Euclidean coefficients in (2.36)?

Below we will give a rigorous justification of (8.47), using only the two- and 3-point functions which are kinematically determined by conformal invariance. Recall that on the Euclidean side, the formal power series of $C_{\chi}$ (the scalar version of (2.36))

$$C_{\chi}(x_1, x_2, x_0, \partial_0) = \frac{1}{(x_{12}^2)^{\Delta_0 - \Delta_{\chi}/2}} \sum_{\mu} c_{\chi}^{\mu}(x_{10}, x_{20}) \partial_{0}^{\mu} \qquad (8.49)$$

is determined by the Euclidean two- and 3-point functions:

$$\langle \mathcal{O}_{\chi}^{\dagger}(y) \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle_E = \frac{1}{(x_{12}^2)^{\Delta_0 - \Delta_{\chi}/2}} \sum_{\alpha} c_{\chi, \alpha}(x_{10}, x_{20}) \langle \mathcal{O}_{\chi}^{\dagger}(y) \partial^{\alpha} \mathcal{O}_{\chi}(x_0) \rangle_E.$$

Here we already used translation invariance, which implies $C_{a,(\lambda),\alpha}^{(\mu)(\nu)}(x_1, x_2, x_0) = c_{a,(\lambda),\alpha}^{(\mu)(\nu)}(x_{10}, x_{20})$ on the r.h.s. of (2.36). One can match the coefficients $c_{\chi}^{\mu}(x_{10}, x_{20})$ with the Taylor expansion of $\langle \mathcal{O}_{\chi}^{\dagger}(y) \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle_E$ around $x_1 = x_2 = x_0$. In Euclidean one can always find a proper region for the matching: let $y$ be sufficiently far from the $(x_0, x_1, x_2)$ cluster, so that the Taylor expansions of $[(y - x_1)^2]^{\#}$ around $x_1 = x_0$ and $[(y - x_2)^2]^{\#}$ around $x_2 = x_0$ are convergent.

On the Minkowski side, the OPE kernel $B_{\chi}$ is kinematically determined by the equality

$$\hat{G}_{\mathcal{O}_{\chi}\mathcal{O}}(p, x_1, x_2) = \hat{B}_{\chi}(p, x_1, x_2) \hat{G}_{\chi}(p), \qquad (8.50)$$

<footnote n="73">⁷³Since here we are only interested in the OPE kernels $B_{\chi}$ and $C_{\chi}$, we set $f_{\chi} = 1$ (the overall coefficient) for convenience.</footnote>
where $\hat{G}_{\mathcal{O}_{\chi}\mathcal{O}}(p, x_1, x_2) = \int dy \langle \mathcal{O}_{\chi}^{\dagger}(y) \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle_M e^{-ip \cdot y}$ and $\hat{G}_{\chi}(p) = \int dy \langle \mathcal{O}_{\chi}^{\dagger}(y) \mathcal{O}_{\chi}(0) \rangle_M e^{-ip \cdot y}$ (see [41], eq. (8.2)).$^{74}$ All Fourier transforms here are in the sense of distributions. To get an equation valid in the sense of functions we pick a test function $\varphi$ with compactly supported Fourier transform, and integrate (8.50) against $\hat{\varphi}$, which gives:

$$ \int \langle \mathcal{O}_{\chi}^{\dagger}(x) \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle_M \varphi(x) dx = \int dp \hat{\varphi}(p) \hat{B}_{\chi}(p, x_1, x_2) \hat{G}_{\chi}(p). \quad (8.51) $$

The variable $x$ ranges over the Minkowski space, while we will pick $x_1, x_2$ complex, in the forward tube region

$$ \mathrm{Im}(x_1), \mathrm{Im}(x_2) < 0. \quad (8.52) $$

Then the 3-point function $\langle \mathcal{O}_{\chi}^{\dagger}(x) \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle_M$ is nonsingular as a function of $x$ and the l.h.s. of (8.51) is a finite number. To transform the r.h.s. of (8.51) we will use the fact that $\hat{B}_{\chi}$ has the following form (a more detailed version than (8.40)):

$$ \hat{B}_{\chi}(p, x_1, x_2) = \frac{e^{-ip \cdot x_1}}{(x_{12}^2)^{\Delta_{\mathcal{O}} - \Delta_{\chi}/2}} E_{\chi}(x_{12} \cdot p, x_{12}^2 p^2), \quad (8.53) $$

where $E_{\chi}(z_1, z_2)$ is some entire function on $\mathbb{C}^2$. Hence as long as $x_{12}^2 \neq 0$ (not necessarily real), $\hat{B}_{\chi}(p, x_1, x_2)$ has the following convergent expansion:

$$ \hat{B}_{\chi}(p, x_1, x_2) = \frac{e^{-ip \cdot x_1}}{(x_{12}^2)^{\Delta_{\mathcal{O}} - \Delta_{\chi}/2}} \sum_{\alpha} a_{\chi}^{\alpha}(x_{12}) (-ip)_{\alpha}, \quad (8.54) $$

where $a_{\chi}^{\alpha}(x)$ is some $\mathrm{SO}(1, d-1)$-covariant, homogeneous, symmetric polynomial of degree $|\alpha|$. Plugging this into (8.51), using that the expansion (8.54) converges uniformly on the support of $\hat{\varphi}$ (assumed compact), and the fact that $\hat{G}_{\chi}$ is a tempered measure,$^{75}$ we obtain:

$$ \int \langle \mathcal{O}_{\chi}^{\dagger}(x) \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle_M \varphi(x) dx \\ = \frac{1}{(x_{12}^2)^{\Delta_{\mathcal{O}} - \Delta_{\chi}/2}} \sum_{\alpha} a_{\chi}^{\alpha}(x_{12}) \int \langle \mathcal{O}_{\chi}^{\dagger}(x) \partial_{\alpha} \mathcal{O}_{\chi}(x_1) \rangle_M \varphi(x) dx. \quad (8.55) $$

At this stage we have established that for any $x_1, x_2$ as in (8.52), and for any test $\varphi$ with compact $\hat{\varphi}$, the series in the r.h.s converges to the l.h.s.

Now the key point is that the r.h.s. of (8.55) is a convergent power series in $x_{21}$, while the l.h.s. can be expanded in such a convergent power series. Indeed we know the explicit form of $\langle \mathcal{O}_{\chi}^{\dagger}(x) \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle_M$:

$$ \langle \mathcal{O}_{\chi}^{\dagger}(x) \mathcal{O}(x_1) \mathcal{O}(x_2) \rangle_M = \frac{1}{(x_{12}^2)^{\Delta_{\mathcal{O}} - \Delta_{\chi}/2}} \frac{1}{[(x - x_1)^2 (x - x_2)^2]^{\Delta_{\chi}/2}}. \quad (8.56) $$

<footnote>$^{74}$In the unitary CFTs, $\hat{G}_{\overline{n}\mathcal{O}}(p, x_1, x_2)$ and $\hat{G}_{\chi_n}(p)$ vanish unless $p \in \overline{V}_+$, so the behavior of $\hat{B}_{\chi_n}(p, x_1, x_2)$ outside the forward light cone is not important.</footnote>

<footnote>$^{75}$This is a consequence of the Bochner-Schwartz theorem: any positive tempered distribution is the Fourier transform of some positive tempered measure (see [83], section 8.2).</footnote>
For all $x$ in the Minkowski space, the function $[(x-x_1)^2(x-x_2)^2]^{-\Delta_\chi/2}$ is holomorphic in $x_1, x_2$ as long as $\mathrm{Im}(x_1), \mathrm{Im}(x_2) < 0$. It's easy to show that this remains true after integration in $\varphi$. At this point we can match the expansions for the two sides of (8.55), and get

$$ \sum_{|\alpha|=n} \int dx \varphi(x) \left\{ \frac{(x_{21})_\alpha}{\alpha!} \partial_{x_2}^\alpha [(x-x_1)^2(x-x_2)^2]^{-\Delta_\chi/2} \Big|_{x_2=x_1} - a_\chi^\alpha(x_{21}) \langle \mathcal{O}_\chi^\dagger(x) \partial_\alpha \mathcal{O}_\chi(x_1) \rangle_M \right\} = 0. \quad (8.57) $$

Up to this point it was crucial to keep the function $\varphi$ in the game to keep convergence issues under control, but now we can get rid of it. Indeed $\sum_{|\alpha|=n}$ is a finite sum, also $\langle \mathcal{O}_\chi^\dagger(x) \partial_\alpha \mathcal{O}_\chi(x_1) \rangle_M$ is a holomorphic function in the forward tube $\mathcal{T}_2$. For any Minkowski point $x_0$, we choose a sequence of test functions $\varphi_k$ of compact support $\hat{\varphi}_k$ such that $\varphi_k$ tends to $\delta(x-x_0)$. Passing to the limit, (8.57) implies the same equality for the integrand. I.e. for any fixed $n \in \mathbb{N}$, and any Minkowski $x$,

$$ \frac{(x_{21})_\alpha}{\alpha!} \partial_{x_2}^\alpha [(x-x_1)^2(x-x_2)^2]^{-\Delta_\chi/2} \Big|_{x_2=x_1} - a_\chi^\alpha(x_{21}) \langle \mathcal{O}_\chi^\dagger(x) \partial_\alpha \mathcal{O}_\chi(x_1) \rangle_M = 0. \quad (8.58) $$

Now as promised we are reduced to an equation which only involves 2-point and 3-point functions which are holomorphic. E.g. we can take $x = 0$ and $x_1, x_2$ in Euclidean. Then this equation is the same one as the equation which determines the Euclidean OPE kernel for $x_0 = x_1$, i.e. $C_\chi(x_1, x_2, x_1, \partial)$. For convenience in this discussion we use Minkowski coordinates for Euclidean correlators (i.e. we write the Euclidean correlators as $\langle\langle\mathcal{O}(-i\tau, \mathbf{x})\dots\rangle\rangle_M$). Under this convention we have

$$ C_{\chi}(x_1, x_2, x_1, \partial) = \frac{1}{(x_{12}^2)^{\Delta_{\mathcal{O}} - \Delta_{\chi}/2}} \sum_{(\mu)} a_{\chi}^{\mu}(x_{21}) \partial_{\mu} = \sum_{(\mu)} b_{\chi}^{\alpha}(0, x_{21}) \partial_{\alpha}. \quad (8.59) $$

This establishes (8.47) for $x_0 = x_1$. The general case reduces to this one by noticing that $c_\chi^\alpha$ satisfies the relation:

$$ c_{\chi}^{\alpha}(x_{10}, x_{20}) = \sum_{\beta \le \alpha} \frac{1}{\beta!} c_{\chi}^{\alpha-\beta}(0, x_{21}) x_{10}^{\beta}, \quad (8.60) $$

where $\beta \le \alpha$ means $\beta_i \le \alpha_i$ for all $i$; and $b_\chi$ satisfies the identical relation with $c_\chi \to b_\chi$. For $c_\chi^\alpha$ this follows by translation invariance and analyticity of CFT two- and 3-point functions, and for $b_\chi^\alpha$ from $\hat{B}_\chi(p, x_{10}, x_{20}) = e^{-ip \cdot x_{10}} \hat{B}_\chi(p, 0, x_{21})$.

# 9 Review of Osterwalder-Schrader theorem

In this section we review the results of [2, 3] and, in particular, discuss the linear growth condition and why it was necessary for establishing Wightman axioms in [3].

In [2] Osterwalder and Schrader formulated an equivalence theorem which stated that a set of axioms for Euclidean correlation functions (a version of the Osterwalder-Schrader axioms described in section 2.2) is equivalent to Wightman axioms for Euclidean correlation
functions. Unfortunately, later a technical error was discovered in their proof, and in [3] Osterwalder and Schrader gave two new results.

The first result of [3] is a revised equivalence theorem, which shows that a stronger version of Euclidean axioms is in fact equivalent to Wightman axioms. The proof of this theorem is rather simple. However, as we will review, this is at the expense of the new version of Euclidean axioms being rather hard to verify.

The second result of [3] shows that the original OS axioms, plus a “linear growth condition,” imply Wightman axioms and a growth condition on Wightman distributions. A partial result in the reverse direction is also valid. It assumes a stronger growth condition on the Wightman distributions than follows from the direct result, and it yields a growth condition on Euclidean correlators which is weaker than the linear growth condition. Therefore, these latter results do not establish an equivalence of any two systems of axioms, but they do allow to establish Wightman axioms from OS axioms in some situations.

In what follows we will review the general structure of the arguments of [2, 3]. For our purposes it will suffice to ignore the space coordinates and focus only on the time arguments of the fields. We will not completely reproduce all arguments of [2, 3], and in some of the omitted steps the space arguments and Lorentz symmetry are important. We will also work with correlation functions involving a single hermitian scalar field $\phi$, similarly to [2, 3]. In CFT applications we are interested in correlation functions of all local operators. It should be relatively straightforward to adapt the discussion of [2, 3] to this more general setup.

Our main goal is to construct an analytic continuation of the Euclidean correlation functions

$$G_n^E(t_1, \dots, t_n) = \langle \phi(t_1) \dots \phi(t_n) \rangle \quad (9.1)$$

from real to complex $t_k$, and to establish that the Wightman functions recovered in the limit of pure imaginary $t_k$ (real Lorentzian times) are tempered distributions. This is the most non-trivial part of the argument. Other Wightman axioms such as positivity, spectrum condition, etc., follow relatively easily and have been reviewed in section 6.

## 9.1 The argument of [2]

Physically, the analyticity of position-space correlation functions is due to positivity of energy. More concretely, the Euclidean evolution operator $e^{-Ht}$ is well-defined and holomorphic in $t$ for $Re\,t > 0$ due to the spectrum of $H$ being non-negative. The first step to establishing analyticity is thus to construct the operator $H$, and for this we first need to construct a Hilbert space on which it acts.

The Hilbert space $\mathcal{H}^{\text{OS}}$ is constructed, as we discussed in section 2.2, by considering the vector space $\mathcal{H}_{0}^{\text{OS}}$ of formal linear combinations of states$^{76}$

$$|\phi(t_1)\phi(t_2)\dots\phi(t_n)\rangle \quad (9.2)$$

<footnote>$^{76}$In section 2.2 the states are introduced as integrals of these quantities. This is also what is done in [2, 3], since they assume only that the Euclidean correlators are distributions. Here, for simplicity of discussion, we use the knowledge that correlators are functions and use states evaluated at points. The arguments easily generalize to distributions and smeared states, but become more technical.</footnote>
with $0 > t_1 > t_2 > \dots > t_n$. A Hermitian inner product is introduced on $\mathcal{H}_0^{\text{OS}}$ by

$$ \langle \phi(s_1)\phi(s_2)\dots\phi(s_m)|\phi(t_1)\phi(t_2)\dots\phi(t_n)\rangle = G_n^E(-s_m, \dots, -s_1, t_1, \dots, t_n). \quad (9.3) $$

By OS reflection positivity-axiom this inner product is positive-semidefinite. The Hilbert space $\mathcal{H}_0^{\text{OS}}$ is obtained from $\mathcal{H}_0^{\text{OS}}$ by modding out null states and completing the resulting quotient space with respect to the above inner product. We can naturally think of $|\phi(t_1)\phi(t_2)\dots\phi(t_n)\rangle$ as states in $\mathcal{H}^{\text{OS}}$.

Physically, to construct the Hamiltonian $H$, we first define it by its action on (9.2). Then we note that $H$ has to be positive, otherwise the correlation functions would grow exponentially at large distances. Formally, one first defines for $t > 0$ an operator $U_t$ on $\mathcal{H}_0^{\text{OS}}$ by

$$ U_t|\phi(t_1)\phi(t_2)\dots\phi(t_n)\rangle = |\phi(t_1-t)\phi(t_2-t)\dots\phi(t_n-t)\rangle. \quad (9.4) $$

The usual care must be taken to ensure that this defines an operator on $\mathcal{H}^{\text{OS}}$. For this one notes that for any $\Psi \in \mathcal{H}_0^{\text{OS}}$ we have $|\langle\Psi|U_t|\Psi\rangle| \le P(t)$ for some polynomial $P(t)$ since the Euclidean correlation functions are assumed to be powerlaw-bounded when groups of points are separated to infinity. Then a simple estimate gives

$$ |\langle\Psi|U_t|\Psi\rangle| \le \|\Psi\|\|U_t\Psi\| = \|\Psi\|\|\langle\Psi|U_{2t}|\Psi\rangle|^{1/2} \le \dots \le \|\Psi\|^{1+1/2+\dots+1/2^{n-1}}|\langle\Psi|U_{2nt}|\Psi\rangle|^{1/2^n}. \quad (9.5) $$

Using $|\langle\Psi|U_t|\Psi\rangle| \le P(t)$ we get in the limit $n \to \infty$

$$ |\langle\Psi|U_t|\Psi\rangle| \le \|\Psi\|^{1+1/2+\dots+1/2^{n-1}} (P(2^nt))^{1/2^n} \to \|\Psi\|^2. \quad (9.6) $$

This shows that $U_t$ maps null states to null states and thus is defined on (a dense subset of) $\mathcal{H}^{\text{OS}}$. By the above, it is also a bounded operator, so it extends in a unique way to all of $\mathcal{H}^{\text{OS}}$. Furthermore, noting that it is symmetric, of norm at most 1, and we have the semigroup law $U_tU_s = U_{t+s}$, we find that $U_t = e^{-Ht}$ for a non-negative self-adjoint Hamiltonian $H$ (see, e.g., [84] section 141).

Since the domain in which we need to construct the analytic continuation of $G_n^E$ is awkward to define in $t_k$ variables, we introduce the difference variables $y_k = t_k - t_{k+1}$. Due to translation invariance, $G_n^E(t_1 \dots t_n)$ can be rewritten as

$$ G_n^E(t_1 \dots t_n) = S_{n-1}(y_1 \dots y_{n-1}) \quad (9.7) $$

for some functions $S_n$. Similarly, we will use the following notation for states in terms of $y_k$ variables,

$$ |\Psi_n(-t_1; y_1 \dots y_{n-1})\rangle \equiv |\phi(t_1)\phi(t_2)\dots\phi(t_n)\rangle. \quad (9.8) $$

Note that

$$ \langle\Psi_m(t'; y'_1 \dots y'_{m-1})|\Psi_n(t; y_1 \dots y_{n-1})\rangle = S_{m+n-1}(y'_{m-1}, \dots, y'_1, t' + t, y_1, \dots, y_{n-1}). \quad (9.9) $$
In terms of $S_{n-1}(y_1 \dots y_{n-1})$, our goal is to construct an analytic continuation to $\text{Re } y_k > 0$ and show that the limit of all $\text{Re } y_k \to 0^+$ exists in the sense of tempered distributions.

With a positive $H$ now constructed, we can define a holomorphic family of bounded operators $U_\tau = e^{-H\tau}$ for $\text{Re } \tau > 0$, which will be our main tool for analytically continuing the correlation functions $S_n$. In particular, we can now consider the matrix elements

$$ \langle \Psi_m(t'; y'_1 \dots y'_{m-1}) | U_\tau | \Psi_n(t; y_1 \dots y_{n-1}) \rangle = S_{m+n-1}(y'_{m-1}, \dots, y'_1, t' + t + \tau, y_1, \dots, y_{n-1}), \quad (9.10) $$

which are analytic for $\text{Re } \tau > 0$. This establishes the desired analyticity of $S_{n-1}(y_1 \dots y_{n-1})$ in each variable $y_k$ separately. In [2] they additionally establish some growth conditions on these individual holomorphic functions which then imply that for fixed $y_k, y'_k$ and $\text{Re } \tau > 0$ the above function can be represented as the Fourier-Laplace transform

$$ S_{m+n-1}(y'_{m-1}, \dots, y'_1, \tau, y_1, \dots, y_{n-1}) = \int d\alpha e^{-\alpha\tau} \check{S}(\alpha) \quad (9.11) $$

for some tempered distribution $\check{S}(\alpha)$. In other words, $S_{m+n-1}$ can be extended to a holomorphic function in the right-half plane in each variable separately, and each such holomorphic function can be represented as a Fourier-Laplace transform of a tempered distribution. The erroneous lemma 8.8 of [2] states that under these conditions, the full function $S_{m+n-1}$ is a simultaneous Fourier-Laplace transform in all its variables of a tempered distribution,

$$ S_{m+n-1}(\tau_1 \dots \tau_{m+n-1}) = \int d\alpha e^{-\alpha_1 \tau_1 - \dots - \alpha_{m+n-1} \tau_{m+n-1}} \check{S}_{m+n-1}(\alpha_1 \dots \alpha_{m+n-1}). \quad (9.12) $$

From this, the tempered Wightman distributions are obtained immediately by setting $\text{Re } \tau_k \to 0^+$ in which case the Fourier-Laplace transform above becomes a Fourier transform of a tempered distribution. Fourier transform of a tempered distribution is, of course, itself tempered.

## 9.2 The argument of [3]

### 9.2.1 Fixing the equivalence theorem

Unfortunately, lemma 8.8 of [2] is wrong. As explained in [3], the function $S_2(y_1, y_2) = e^{-y_1y_2}$ gives a simple counter-example. For fixed $y_2 > 0$, we find that $S_2(y_1, y_2)$ is holomorphic for $\text{Re } y_1 > 0$ and is there the Fourier-Laplace transform of the tempered distribution $\delta(\alpha - y_2)$. The same statements hold with $y_1$ and $y_2$ exchanged. However, $S_2(y_1, y_2)$ is not a Fourier-Laplace transform of a tempered distribution in both variables simultaneously. For if this were the case, the corresponding Wightman function $S_2(ix_1, ix_2) = e^{ix_1x_2}$ would be a tempered distribution, which it is not since it grows faster than any power in some directions.

The first result of [3] (see also the review in [43]) rescues lemma 8.8 by making a stronger assumption about $S_n(y_1 \dots y_n)$ which they denote by $\check{\mathcal{E}}_0$. Concretely, let $\mathbb{R}_+^n$ be the set of points $(y_1, \dots, y_n)$ with $y_k > 0$. Let $\mathcal{S}(\mathbb{R}_+^n)$ be the subspace of the space of Schwartz functions, consisting of functions supported on $\mathbb{R}_+^n$ with the induced topology.
The functions $S_n(y_1 \dots y_n)$ can be viewed as distributions in the continuous dual space $S'(\mathbb{R}_+^n)$ defined by, for $f \in S(\mathbb{R}_+^n)$

$$S_n(f) = \int dy_1 \dots dy_n S_n(y_1 \dots y_n) f(y_1 \dots y_n). \qquad (9.13)$$

Note that smoothness of $f$ together with its support properties ensures that $f(y_1 \dots y_n)$ vanishes with all derivatives whenever $y_k = y_j$ for $k \neq j$. The assumption that $S_n$ has at most powerlaw singularities at coincident points and at infinity means that $S_n(f)$ is continuous in $f$ in the topology of $S(\mathbb{R}_+^n)$. The additional assumption $\tilde{E}0$ is that it is also continuous in $f$ in a weaker topology. This weaker topology is defined by the usual Schwartz norms on $\mathbb{R}_+^n$

$$|g|_{p,+} = \sup_{x \in \mathbb{R}_+^n, |\alpha| \le p} |(1+x^2)^{p/2} \partial^{(\alpha)} g(x)| \qquad (9.14)$$

but applied not to $f$ and instead to its Fourier-Laplace$^{77}$ transform $\tilde{f}$

$$\tilde{f}(q_1 \dots q_n) = \int dy_1 \dots dy_n e^{-q_1 y_1 - \dots - q_n y_n} f(y_1 \dots y_n). \qquad (9.15)$$

One establishes that $\tilde{f} = 0$ iff $f = 0$ (injectivity) and that the set of all images $\tilde{f}$ is dense in an appropriate space of Schwartz functions (denseness). The proof of (9.12) then becomes straightforward: one first defines $\tilde{S}_n$ by $\tilde{S}_n(\tilde{f}) = S_n(f)$. This definition makes sense due to the injectivity property. The assumption $\tilde{E}0$ ensures that $\tilde{S}_n$ is continuous. The denseness property just mentioned then allows to extend $\tilde{S}_n$ to an appropriate space of Schwartz functions by continuity, establishing temperedness of $\tilde{S}_n$ and allowing one to define tempered Wightman distributions as Fourier transforms of $\tilde{S}_n$. It is similarly not difficult to show that Wightman axioms imply $\tilde{E}0$.

### 9.2.2 Wightman axioms from linear growth condition

As we can see, the axiom $\tilde{E}0$ is not very different from directly assuming temperedness of Wightman distributions, even though it is formulated for Euclidean correlators. It is also unclear how to verify this axiom in practice.$^{78}$ For this reason, [3] introduced an alternative “linear growth condition” on the correlation functions $S_n$ which is easier to verify and has been established in some models (see below), yet is also sufficient to establish temperedness of Wightman functions (though this condition is not known to follow from Wightman axioms). The construction of the analytic continuation of the functions $S_n$ as

<footnote>$^{77}$As written, this is a Laplace transform. It is a Fourier transform in the spatial variables which we are omitting.</footnote>

<footnote>$^{78}$We would also like to mention related work by Zinoviev [85]. Zinoviev replaces axiom $\tilde{E}0$ by an axiom E5 which imposes that certain limits exist which allow to compute the inverse Laplace transform of $S_n$. While E5 may look like a more constructive version of $\tilde{E}0$, in practice its verification appears just as hard as assuming outright that $S_n$ is a Laplace transform (which is what $\tilde{E}0$ essentially does). We are grateful to David Brydges for an enlightening explanation of Zinoviev’s construction, and in particular for pointing out that it represents a generalization of Post’s Laplace transform inversion formula [86] to the case of distributions.</footnote>
well the proof of the temperedness of the resulting Wightman distributions is much more complicated than using $\tilde{E}0$. Therefore, our review of these arguments will be even more schematic than the above, and we will only try to illustrate the key ideas and explain why and how the linear growth condition is used. We are not aware of any previous attempt to review this part of [3].

First of all, let us state the linear growth condition of [3]. Note that the correlation functions $G_n^E$ can be viewed as distributions in $(^0S)'(\mathbb{R}^{d \cdot n})$, where $^0S(\mathbb{R}^{d \cdot n})$ is the space of Schwartz functions of $n$ arguments in $\mathbb{R}^d$ which vanish with all derivatives at coincident points, by

$$G_n^E(f) = \int d^d x_1 \dots d^d x_n f(x_1, \dots, x_n) G_n^E(x_1, \dots, x_n). \qquad (9.16)$$

Here we have temporarily restored the spatial coordinates. In fact, [2, 3] do not assume that $G_n^E$ are functions, and only that they are distributions in $(^0S)'(\mathbb{R}^{d \cdot n})$. It follows, however, from the OS axioms (without the linear growth condition) that $G_n^E$ are real-analytic functions, as shown in [3, 46].

Note that the assumption that $G_n^E \in (^0S)'(\mathbb{R}^{d \cdot n})$ means $G_n^E$ is sufficiently continuous as a linear functional or, equivalently, is sufficiently bounded. That is,

$$|G_n^E(f)| \le \sigma_n |f|_{q_n} \qquad (9.17)$$

for all $f \in {}^0S(\mathbb{R}^{d \cdot n})$ and some $\sigma_n > 0$ and $q_n \in \mathbb{Z}_{\ge 0}$, where $|f|_p$ denotes the Schwartz norms on $^0S(\mathbb{R}^{d \cdot n})$. The linear growth condition requires $q_n$ to grow at most linearly, and $\sigma_n$ at most as a power of a factorial. In other words, the linear growth condition is the statement that there exists a positive integer $s$ and a sequence $\sigma_n$ such that

$$|G_n^E(f)| \le \sigma_n |f|_{n \cdot s} \qquad (9.18)$$

for any $n$ and $f \in {}^0S(\mathbb{R}^{d \cdot n})$, and $\sigma_n \le \alpha(n!)^\beta$ for some constants $\alpha, \beta$.

The unusual feature of the linear growth condition is that this is a condition on all $n$-point correlation functions $G_n^E$. It has to hold for all $n$ in order for the result of [3] to imply, say, even just the temperedness of 3-point Wightman distribution. In order to understand why this is required, below we will review the basic strategy behind the proof of [3]. There are two steps in the argument. In the first step, one establishes analyticity of $S_n(y_1, \dots, y_n)$ in the region $\text{Re } y_k > 0$. This does not require the linear growth condition [3, 46]. In the second step, which does use the linear growth condition, one proves a bound on $S_n$ in this region, which allows the application of Vladimirov's theorem and thus the construction of tempered Wightman distributions.

We conclude this section with additional comments about the linear growth condition. First of all, appendix of [3] shows that the linear growth condition follows from requiring that $G_n^E \in S'(\mathbb{R}^{d \cdot n})$ and imposing

$$|G_n^E(f_1 \otimes \dots \otimes f_n)| \le \sigma_n \prod_{i=1}^{n} |f_i|_r, \qquad (9.19)$$
for any $n$, where $f_i \in S(\mathbb{R}^d)$, $|\cdot|_r$ is some fixed Schwartz space norm, and $\sigma_n \le \alpha(n!)^\beta$. In other words, while in (9.18) the $n$-point function variables are smeared jointly, here each variable is smeared separately. Note that the total smearing $f_1 \otimes \dots \otimes f_n$ does not necessarily exclude coincident points, that's why we need to assume $G_n^E \in S'(\mathbb{R}^{d\cdot n})$ and not $G_n^E \in ({}^0S)'(\mathbb{R}^{d\cdot n})$ as above.

Although (9.19) is stronger than (9.18), it is easier to verify in particular models. E.g. it holds for any gaussian scalar field $O$ with a two point function $G_2$ having a powerlaw asymptotics in the UV.<fnref n="79" /> It has been also established in some non-gaussian models.<fnref n="80" />

More generally, bound (9.19) is natural for field theories realizable as random distributions.<fnref n="81" /> Imagine that there is a measure $d\mu$ in the space of distributions $\phi \in S'(\mathbb{R}^d)$ such that for every test function $f \in S(\mathbb{R}^d)$ the following expectation value is finite:

$$S(f) = \int d\mu e^{\phi(f)}. \qquad (9.20)$$

Such measures make rigorous sense of the Feynman path integral. Eq. (9.20) is a rigorous version of generating functional, and differentiating with respect to $f$ one defines correlation functions $\langle\phi(x_1) \dots \phi(x_n)\rangle$ which are in this framework automatically distributions in $S'(\mathbb{R}^{d\cdot n})$. Bound (9.19) in this case can be reduced to an estimate on the growth of $S(f)$. The Osterwalder-Schrader and Wightman axioms then follow.

The field $\phi$ in (9.20) is naturally a “fundamental field” of some model, such as $P(\phi)_2$ [88] or $(\phi^4)_3$ (see [88], section 23.1 for references). Sometimes this framework can be extended to generating functionals $\int d\mu e^{\phi'(f)}$ where $\phi'$ is a composite operator. E.g. $\phi' = :\phi^n:$, $n < \deg P$, in $P(\phi)_2$ is treated in [88]. See also [89] for the general problem to construct $:\phi^2:$ as a random distribution given $\phi$.

### 9.2.3 Analytic continuation

There are three tricks used together to construct the analytic continuation of $S_n$. The first trick was already used above: it is the observation that if the states $|\Psi_n(t; y_1, \dots, y_n)\rangle$ and $|\Psi_m(t'; y'_1, \dots, y'_m)\rangle$ are defined for some values of $t, y_k$ and $t', y'_k$, then we can compute the matrix elements

$$\langle \Psi_m(t'; y'_1 \dots y'_{m-1}) | U_\tau | \Psi_n(t; y_1 \dots y_{n-1}) \rangle \\ = S_{m+n-1}(\overline{y'_{m-1}}, \dots, \overline{y'_1}, t' + t + \tau, y_1, \dots, y_{n-1}) \qquad (9.21)$$

with $\text{Re } \tau > 0$, thus potentially extending the domain of analyticity of $S_{m+n-1}$.

<footnote n="79">⁷⁹Then $G_n^E(f_1 \otimes \dots \otimes f_n)$ is a sum of $(n-1)!!$ terms, products of Wick contractions $G_2(f_i \otimes f_j)$, which can be bounded by $A|f_i|_r|f_j|_r$ where $r$ depends on the UV dimension of $O$. We thus get (9.19) with $\sigma_n = (n-1)!!A^{n/2}$.</footnote>

<footnote n="80">⁸⁰See e.g. [87], theorem 1.1.8, which establishes eq. (9.19) for Schwinger functions of arbitrarily high normal-ordered powers $:\phi^n:$ of the fundamental field $\phi$ in weakly coupled $P(\phi)_2$ theories.</footnote>

<footnote n="81">⁸¹See [88], section 6. This book introduced axioms for random distributions, numbered OS0-OS5. This chosen name is a bit unfortunate because these axioms are quite different in spirit from the original Osterwalder-Schrader axioms described in section 2.2, and appear much stronger. E.g. they make the recovery of Wightman axioms a relatively trivial task. We don't know how to derive the axioms of [88] from the Euclidean CFT axioms.</footnote>
The second trick, intuitively, says that we can write

$$ \langle \Psi_n(t; y_1 \dots y_{n-1}) | \Psi_n(t; y_1 \dots y_{n-1}) \rangle = S_{2n-1}(\overline{y_{n-1}}, \dots, \overline{y_1}, 2t, y_1, \dots, y_{n-1}), \quad (9.22) $$

and so the state $|\Psi_n(t; y_1 \dots y_{n-1})\rangle$, whose norm appears in the left-hand side, should be well-defined as long as the correlation function in the right-hand side is well-defined. That is, while we start with the states $|\Psi_n(t; y_1 \dots y_{n-1})\rangle$ defined for positive real $y_k$, we should be able to analytically continue them in $y_k$ if we manage to analytically continue the correlators $S_{2n-1}$. Of course, this is not a proof that $|\Psi_n(t; y_1 \dots y_{n-1})\rangle$ is well-defined. We will give the proof below, after we get more information about the domain in which we wish to construct it.

The final trick is the idea of analytic completion for functions of several complex variables. Recall that for $n > 1$ not every domain in $\mathbb{C}^n$ is the domain of holomorphy of some holomorphic function: there exist domains $\mathcal{D} \subset \mathbb{C}^n$ such that any $f$ holomorphic in $\mathcal{D}$ can be extended to a function holomorphic in a strictly larger domain $\mathcal{D}' \supset \mathcal{D}$. For our applications the relevant theorem is Bochner's tube theorem, which states that any holomorphic function in a tube domain of the form $\mathcal{D} = \mathbb{R}^n + iX$, where $X$ is a connected open subset of $\mathbb{R}^n$, can be extended to a holomorphic function on $\mathcal{D}' = \text{ch}(\mathcal{D}) = \mathbb{R}^n + i\text{ch}(X)$, where $\text{ch}$ denotes the convex hull. Note that since $\mathcal{D}'$ is a convex set, it is a domain of holomorphy$^{82}$ and so $f$ cannot be extended any further by analytic completion alone. The requirement that $X$ is open is a bit too restrictive and we'll need also a degenerate case of this theorem, as described below.

These three tricks are applied one by one infinitely many times in order to construct the full analytic continuation of $S_n$. Instead of setting up the procedure in its full glory, we will only follow the first steps to see how it works in principle. The full analysis is performed in [3].

First, it helps to introduce new variables $w_i$ by

$$ e^{w_i} = y_i. \quad (9.23) $$

Our domains of analyticity in terms of $w_i$ will always be tubes of the form $(w_1 \dots w_n) \in \mathcal{D}(X) = \mathbb{R}^n + iX$ for various $X \subset \mathbb{R}^n$, and so we'll often just describe $X$. For example, we start with $S_n$ and $\Psi_n$ defined for real positive $y_i$, which corresponds to the domain $\mathcal{D}(\{0\})$ in $w_i$.

Consider the 2-point function $S_1(y_1)$. We start with the domain $\mathcal{D}(\{0\}) = \mathbb{R}$ in $w_1$, corresponding to real positive $y_1$. Next, we apply the first trick. Specifically, we write

$$ \langle \Psi_1(t') | U_{\tau} | \Psi_1(t) \rangle = S_1(t' + t + \tau), \quad (9.24) $$

<footnote>$^{82}$To see this, it suffices to show that for any point $z$ on the boundary of $\mathcal{D}'$ there exists a function holomorphic in $\mathcal{D}'$ but singular at $z$. In general, such functions might not exist since the set of singularities of a holomorphic function cannot be arbitrary. However, it is easy to construct a function singular on any complex codimension-1 hyperplane in $\mathbb{C}^n$ (take the reciprocal of an affine-linear function). For a convex $\mathcal{D}'$ one can always find such a hyperplane passing through a given boundary point but staying away from the interior of $\mathcal{D}'$.</footnote>
<figure><img src="image_8.png" /><figcaption>**Figure 7.** Left: set $X_2$. Right: domain $X'_2$ which defines the envelope of holomorphy $\mathcal{D}(X'_2)$ of $\mathcal{D}(X_2)$.</figcaption></figure>

and since we are free to choose $t > 0$ and $t' > 0$ arbitrarily small, while $U_\tau$, as discussed above, is a well-defined bounded operator for $\mathrm{Re} \tau > 0$, we obtain an analytic continuation of $S_1(y_1)$ to the right half-plane.

We are now done with the analytic continuation of $S_1$, since our goal was to continue all $y_k$ to the right-half plane. In terms of $w_i$, this corresponds to the domain $w_1 \in \mathcal{D}((-\frac{\pi}{2}, +\frac{\pi}{2}))$, i.e. a strip. For higher-point functions, in terms of $w_i$, we should stop when our domain of analyticity is $\mathcal{D}((-\frac{\pi}{2}, +\frac{\pi}{2}) \times \cdots \times (-\frac{\pi}{2}, +\frac{\pi}{2}))$.

Consider now the 3-point function $S_2(y_1, y_2)$. We can again use the first trick and define it on $(w_1, w_2) \in \mathcal{D}(X_2)$, where $X_2 = \{0\} \times (-\frac{\pi}{2}, +\frac{\pi}{2}) \cup (-\frac{\pi}{2}, +\frac{\pi}{2}) \times \{0\}$ (see figure 7, left). In more detail, we write the following two equations for $S_2(y_1, y_2)$, representing it as an inner product in two ways, and inserting a $U_\tau$,

$$ \langle \Psi_2(t'; y'_1) | U_\tau | \Psi_1(t) \rangle = S_2(y'_1, t' + t + \tau), \quad (9.25) $$

$$ \langle \Psi_1(t') | U_\tau | \Psi_2(t, y_1) \rangle = S_2(t' + t + \tau, y_1), \quad (9.26) $$

where the left-hand sides are well-defined (at this point) for $t, t', y_1, y'_1 > 0$ and $\mathrm{Re} \tau > 0$. We see that the first equation defines $S_2(y_1, y_2)$ for real $y_1 > 0$ and $\mathrm{Re} y_2 > 0$ as a holomorphic function of $y_2$. The second equation does the same, but with $y_1$ and $y_2$ exchanged. In terms of $(w_1, w_2)$ this corresponds to the “analyticity domain” $\mathcal{D}(X_2)$ described above. We write “analyticity domain” in quotes because $\mathcal{D}(X_2)$ is not open (and has empty interior), and thus is not a domain. Correspondingly, we cannot say that $S_2$ is an holomorphic function of two variables on $\mathcal{D}(X_2)$. We will deal with this problem momentarily.

To proceed with the analytic continuation of $S_2(y_1, y_2)$, we want to use the third trick, the tube theorem, to extend the analyticity domain from $\mathcal{D}(X_2)$ to $\mathcal{D}(X'_2)$, with $X'_2 = \mathrm{ch}(X_2)$ (figure 7, right).

The problem with this is that $X_2$ is not open, as mentioned above, so the tube theorem does not apply. Instead, for this step one has to use Malgrange-Zerner theorem [90], which allows $X_2$ to be a union of intervals, with $S_2(y_1, y_2)$ separately holomorphic in one variable on each of these intervals, as is the case in our setup. The conclusion is still that $S_2(y_1, y_2)$ can be analytically continued to $\mathcal{D}(X'_2)$.

Note that the domain $\mathcal{D}(X'_2)$ is not yet the full analyticity domain $\mathcal{D}((-\frac{\pi}{2}, +\frac{\pi}{2}) \times (-\frac{\pi}{2}, +\frac{\pi}{2}))$ that we are aiming for. In particular, $X'_2$ is a proper
subset of the square $(-\frac{\pi}{2}, +\frac{\pi}{2}) \times (-\frac{\pi}{2}, +\frac{\pi}{2})$, see the right panel of figure 7. Importantly, it doesn't approach the corners $(\pm\frac{\pi}{2}, \pm\frac{\pi}{2})$, which correspond to pure imaginary $y_1, y_2$. Pure imaginary $y_1, y_2$ is, in turn, where we want to recover the Wightman distributions.

To extend the domain of analyticity of $S_2(y_1, y_2)$ even further, we need to first extend the domain of $\Psi_2(t, y_1)$, which can be done by the second trick above — via the equality

$$ \langle \Psi_2(t, y_1) | \Psi_2(t, y_1) \rangle = S_3(\overline{w_1}, 2t, y_1). \qquad (9.27) $$

Note that we are not interested in the analytic continuation in $t$ here — it is automatic when we act on $\Psi_2$ with $e^{-Ht}$ — so we can assume $t$ is real. For $S_3$ we can run the same argument as we just did for $S_2$ and conclude that it is holomorphic in $\mathcal{D}(X'_3)$, where $X'_3$ is the convex hull of three intersecting intervals on coordinate axes (an octahedron). As discussed above, we expect that $\Psi_2(t, y_1)$ is defined whenever $t$ and $y_1$ are such that the arguments of $S_3$ above are in its analyticity domain. This happens whenever

$$ (\overline{w_1}, \log 2t, w_1) \in \mathcal{D}(X'_3), $$

which is equivalent to

$$ (\operatorname{Im} \overline{w_1}, \operatorname{Im} \log 2t, \operatorname{Im} w_1) \in X'_3. \qquad (9.28) $$

Since we take $t$ to be real and positive, we have $\operatorname{Im} \log 2t = 0$ and so $t$ is otherwise unconstrained. By construction of $X'_3$ and $X'_2$, $(\operatorname{Im} \overline{w_1}, 0, \operatorname{Im} w_1) \in X'_3$ is equivalent to $(\operatorname{Im} \overline{w_1}, \operatorname{Im} w_1) \in X'_2$. Using $\operatorname{Im} \overline{w_1} = -\operatorname{Im} w_1$, we conclude that $w_1$ is constrained by

$$ (-\operatorname{Im} w_1, \operatorname{Im} w_1) \in X'_2. \qquad (9.29) $$

This is equivalent to $|\operatorname{Im} w_1| < \frac{\pi}{4}$, which is the same as $w_1 \in \mathcal{D}((-\frac{\pi}{4}, +\frac{\pi}{4}))$. To conclude, we expect $\Psi_2(t, y_1)$ to be defined and holomorphic in $y_1$ for $t > 0$ and $w_1 \in \mathcal{D}((-\frac{\pi}{4}, +\frac{\pi}{4}))$.

We can now apply the first trick to $S_2(y_1, y_2)$ again, writing it as inner product of $\Psi_1$ and $\Psi_2$ in the two ways (9.25) and (9.26). However, this time we can use $\Psi_2(t, y_1)$ in a wider domain of $y_1$, as computed above, equivalent to $w_1 \in \mathcal{D}((-\frac{\pi}{4}, +\frac{\pi}{4}))$. From (9.25) we conclude that $S_2(y_1, y_2)$ is analytic for

$$ (w_1, w_2) \in \mathcal{D} \left( \left( -\frac{\pi}{4}, +\frac{\pi}{4} \right) \times \left( -\frac{\pi}{2}, +\frac{\pi}{2} \right) \right), \qquad (9.30) $$

where the domain of analyticity in $w_1$ comes from that of $\Psi_2(t, y_1)$, and in $w_2$ from $e^{-H\tau}$. Similarly, (9.26) now implies analyticity in the domain

$$ (w_1, w_2) \in \mathcal{D} \left( \left( -\frac{\pi}{2}, +\frac{\pi}{2} \right) \times \left( -\frac{\pi}{4}, +\frac{\pi}{4} \right) \right). \qquad (9.31) $$

Combining the two together, we find that $S_2(y_1, y_2)$ is analytic for $(w_1, w_2) \in \mathcal{D}(X''_2)$, where

$$ X''_2 \equiv \left( -\frac{\pi}{4}, +\frac{\pi}{4} \right) \times \left( -\frac{\pi}{2}, +\frac{\pi}{2} \right) \cup \left( -\frac{\pi}{2}, +\frac{\pi}{2} \right) \times \left( -\frac{\pi}{4}, +\frac{\pi}{4} \right), \qquad (9.32) $$

see the left panel of figure 8.
<figure><img src="image_9.png" /><figcaption>**Figure 8.** Left: set $X_2''$. Right: domain $X_2'''$ which defines the envelope of holomorphy $\mathcal{D}(X_2''')$ of $\mathcal{D}(X_2'')$.</figcaption></figure>

Using the tube theorem, we can now extend the analyticity domain from $\mathcal{D}(X_2'')$ further to $\mathcal{D}(X_2''')$, where $X_2''' = \text{ch}(X_2'')$ is the convex hull of $X_2''$ shown in the right panel of figure 8.

We see that in order to analytically continue the 3-point function $S_2$, it was useful to split it into an inner product of one-operator and two-operator states $\Psi_1$ and $\Psi_2$, and use the information about the latter that is provided by its norm, the 4-point function $S_3$. Still, we have not yet managed to analytically continue $S_2$ to the entire region of interest (we still have the corners missing in the right panel of figure 8). The only way to fix this is to extend the region of analyticity of $S_3$. For that, we have to split it into a product of two states, and extend the region of analyticity of these states. It is useless to split it as a product of two $\Psi_2$ states, since their norm is computed by $S_3$ itself and we won't learn anything new in this way. Instead, we have to split it as a product of $\Psi_1$ and $\Psi_3$. This will lead us to consider the norm of $\Psi_3$, which is computed by the six-point function $S_5$. Following this logic, eventually, we will be forced to consider $S_n$ with arbitrarily high $n$ just in order to construct the analytic continuation of $S_2$. Fortunately, it can be shown that this procedure converges to the desired domain $\mathcal{D}((-\frac{\pi}{2}, +\frac{\pi}{2}) \times \cdots \times (-\frac{\pi}{2}, +\frac{\pi}{2}))$ for all $S_n$, see [3] for details.

To finish the discussion of the analytic continuation of $S_n$, let us justify the second trick, which constructs the states $\Psi_n$ based on analyticity of their norm $S_{2n-1}$. Let $C$ be the domain of analyticity of $S_{2n-1}(y_1 \dots y_{2n-1})$ known to us, expressed in terms of $w_i$, and let $D$ be the domain of the arguments $t, w_1 \dots w_{n-1}$ of $\Psi_n(t; y_1 \dots y_{n-1})$ for which the arguments of $S_{2n-1}$ in the right-hand side of

$$ \langle \Psi_n(t; y_1 \dots y_{n-1}) | \Psi_n(t; y_1 \dots y_{n-1}) \rangle = S_{2n-1}(\overline{y_{n-1}}, \dots, \overline{y_1}, 2t, y_1, \dots, y_{n-1}), \quad (9.33) $$

belong to $C$. As is clear from the above discussion, $C$ (expressed in terms of $w_i$) is always of the form $C = \mathcal{D}(X)$ for some $X$. We similarly have $D = \mathcal{D}(Y)$ for some $Y$. By the tube theorem (or Malgrange-Zerner theorem), we can assume that $X$ (and thus also $Y$) is open, non-empty, and convex. Furthermore, it is easy to convince oneself that $X$, and thus $Y$, is invariant under reflections along any of the coordinate real axes (i.e. sending $w_i$ to $\overline{w_i}$ for some $i$).

Suppose now that we have a point $(t; w_1^0 \dots w_{n-1}^0) \in D$. Then by definition of $D$ we have

$$ p = (\overline{w_{n-1}^0} \dots \overline{w_1^0}, \log 2t, w_1^0 \dots w_{n-1}^0) \in C. \quad (9.34) $$
<figure><img src="image_10.png" /><figcaption>Figure 9. Schematic picture of the tube C and polydisk P.</figcaption></figure>

The above properties imply that there are $r_i > 0$ such that the polydisk

$$P = \{(w_1 \dots w_{2n-1}) \mid |w_i| < r_i\} + \text{Re } p \quad (9.35)$$

is contained in C, $P \subset C$, and moreover $p \in P$. Indeed, since $C = \mathcal{D}(X)$, this will be true if $\text{Im } P \subset X$ and $\text{Im } p \in \text{Im } P$.<fnref n="83" /> By construction, $\text{Im } P$ is a box with sides $2r_i$ centered at 0. On the other hand, the properties of $X$ imply that together with any point $x$, $X$ contains such a box with $x$ being one of its vertices. We can then find an $\varepsilon > 0$ such that $(1 + \varepsilon) \text{Im } p \in X$, and take $\text{Im } P$ to be the box defined by the vertex $x = (1 + \varepsilon) \text{Im } p$. See figure 9 for an intuitive picture.

Writing temporarily the state $\Psi_n$ as a function of $w_k$ instead of $y_k$, we define it at $w_k$ by the Taylor series

$$|\Psi_n(t; w_1 \dots w_{n-1})\rangle \equiv \sum_{\alpha} \frac{(w - \text{Re } w^0)^{\alpha}}{\alpha!} \partial^{\alpha} |\Psi_n(t; \text{Re } w_1^0 \dots \text{Re } w_{n-1}^0)\rangle \quad (9.36)$$

($\alpha$ is a multiindex so $w^\alpha = w_1^{\alpha_1} w_2^{\alpha_2} \dots$ etc.). Note that the state in the right-hand side is well defined since the corresponding $y_k = e^{\text{Re } w_k^0} > 0$. To check whether this Taylor series converges, we look at its remainder

$$\sum_{|\alpha|>N} \frac{(w - \text{Re } w^0)^{\alpha}}{\alpha!} \partial^{\alpha} |\Psi_n(t; \text{Re } w_1^0 \dots \text{Re } w_{n-1}^0)\rangle, \quad (9.37)$$

whose norm squared is

$$\sum_{|\alpha|>N} \sum_{|\beta|>N} \frac{(w - \text{Re } w^0)^{\alpha}}{\alpha!} \frac{(\bar{w} - \text{Re } w^0)^{\beta}}{\beta!} \times \partial^{\alpha} \partial^{\beta} S_{2n-1}(\text{Re } w_{n-1}^0 \dots \text{Re } w_1^0, \log 2t, \text{Re } w_1^0 \dots \text{Re } w_{n-1}^0), \quad (9.38)$$

where $\beta$-derivatives act on the first $n-1$ arguments of $S_{2n-1}$, while $\alpha$-derivatives act on the last $n-1$ arguments. Here we also temporarily write $S_{2n-1}$ as function of $w_k$. This

<footnote n="83">⁸³The latter is because $\text{Im } \text{ of the section of } P$ by $\text{Re } x = \text{Re } p$ is $\text{Im } P$.</footnote>
norm is clearly just the tail of the Taylor series of $S_{2n-1}$ expanded around the point Re $p$, and evaluated at $(\overline{w_{n-1}}, \dots, \overline{w_1}, \log 2t, w_1, \dots, w_{n-1})$. (We are not expanding in $t$.) Since $S_{2n-1}$ is holomorphic in the polydisk $P$ centered at Re $p$, this Taylor series converges in $P$ and thus this remainder tends to 0 there.

Since $p = (\overline{w_{n-1}^0} \dots \overline{w_1^0}, \log 2t, w_1^0 \dots w_{n-1}^0) \in P$, the remainder tends to 0 at $p$, and thus (9.36) converges at $(t; w_1^0 \dots w_{n-1}^0)$. Furthermore, since $P$ is open, it follows that (9.36) converges in some neighborhood of $(t; w_1^0 \dots w_{n-1}^0)$, defining $|\Psi_n(t; w_1 \dots w_{n-1})\rangle$ as a holomorphic $\mathcal{H}^{\text{OS}}$-valued function in that neighborhood. Since the choice of $(t; w_1^0 \dots w_{n-1}^0) \in D$ was arbitrary, we have defined $|\Psi_n(t; w_1 \dots w_{n-1})\rangle$ as a holomorphic function of $w_i$ for all points in $D$.

### 9.2.4 Temperedness bound

Now that the correlation functions $S_n(y_1 \dots y_n)$ have been analytically continued from $y_k > 0$ to Re $y_k > 0$, we only need to establish a bound on their growth as Re $y_i \to 0$ in order to construct tempered Wightman distributions by an application of Vladimirov's theorem. The logic proceeds by establishing a bound on $S_n(y_1 \dots y_n)$ for real $y_k$, and then repeating the analytic continuation described above, while keeping track of this bound. We will only sketch this rather technical argument in very general terms.

The final temperedness bound that we want to establish is

$$|S_n(y_1 \dots y_n)| \le c_n \left( \left( 1 + \sum_k |y_k| \right) \left( 1 + \sum_k (\text{Re } y_k)^{-1} \right) \right)^{p_n}, \quad (9.39)$$

for some sequences $c_n$ and $p_n$.<fnref n="84" /> We would like (9.39) to hold for all $y_k$, Re $y_k > 0$. For real positive $y_k$ (i.e. in the Euclidean) this holds as a consequence of (2.15). As discussed in remark 2.4, the original OS axioms did not include (2.15), so their first step was to derive (9.39) for $y_k > 0$ using (9.17).

In principle at fixed $n$, (9.39) looks reasonable given (9.17): both say, intuitively, that the correlation functions cannot be too singular at coincident points or grow too fast at infinity. However, (9.39) imposes this in a much more direct way. It turns out that in general one cannot derive direct bounds such as (9.39) from averaged statements such as (9.17), even if we know that $S_n$ is real analytic.

Consider the real-analytic function $\sin(e^x)$, $x \in \mathbb{R}$. It is a bounded function, hence a tempered distribution. Thus its first derivative $e^x \cos(e^x)$ is also a tempered distribution. This is an example of a real-analytic tempered distribution which is not polynomially bounded. So some further assumptions are needed beyond real analyticity.<fnref n="85" />

In our case, the functions $S_n(y_1 \dots y_n)$ are real-analytic and satisfy (9.17). In addition, they satisfy OS positivity. We already used OS positivity to show real-analyticity, and we will now have to invoke it again to prove (9.39) for $y_k > 0$. The full argument is rather

<footnote n="84">⁸⁴Here, for simplicity, we again ignore spatial arguments of the correlation functions, although they need to be taken care of at this step in order to establish “temperedness in spatial directions.” Furthermore, note that Osterwalder and Schrader establish additional bounds on $c_k$, etc., which are not important for the application of Vladimirov’s theorem.</footnote>

<footnote n="85">⁸⁵Incidentally, our example shows that the Corollary of lemma 1 in [46] is wrong.</footnote>
technical; we will explain the main idea on the example of $S_1(y)$. Since we know that $S_1$ is holomorphic, in particular harmonic, by the mean value theorem for harmonic functions we can write it as a radially symmetric average

$$ S_1(y) = \int dx dt S_1(y + x + it)k_\rho(x, t) = \int_{|t|,|t'|<\rho} dt dt' T(t|g_\rho(\cdot, t+t'), g_\rho(\cdot, t')) \quad (9.40) $$

$$ T(t|\varphi_1, \varphi_2) := \int dx dx' S_1(y + x + it)\varphi_1(x + x')\varphi_2(x'), $$

where $k_\rho$ is a $C_0^\infty$ radial function supported in a ball of radius $\rho$ and of integral 1, and we choose $\rho$ sufficiently small so that all points under the integral sign are where $S_2$ is analytic. We also chose

$$ k_{\rho}(x, t) = \int dx' dt' g_{\rho}(x + x', t + t') g_{\rho}(x', t'), \quad (9.41) $$

a convolution of another radial $C_0^\infty$ function with itself (and hence a radial function). The point of this construction is that, for generic $\varphi_1, \varphi_2$, $T(0|\varphi_1, \varphi_2)$ is an inner product $\langle \Psi_1 | \Psi_2 \rangle$ of two OS states:

$$ \langle \Psi_1 | = \int dx \mathcal{O}(y/2 + x)\varphi_1(x), \quad |\Psi_2\rangle = \int dx \mathcal{O}(-y/2 + x)\varphi_2(x). \quad (9.42) $$

The norm of these states, and hence their inner product, can be bounded using (9.17). Furthermore $T(t|\varphi_1, \varphi_2) = \langle \Psi_1 | e^{-iHt} | \Psi_2 \rangle$ satisfies the same bound. Using this bound for $\varphi_1 = g_\rho(\cdot, t+t')$, $\varphi_2 = g_\rho(\cdot, t')$, eq. (9.40) gives a bound on $S_1(y)$. The same idea works for higher point functions. We first have to estimate the norm of some states using (9.17).$^{86}$ We then analytically continue separately in each time, and then use Malgrange-Zerner theorem to extend the bound on $T$ to an open set. A single use of Malgrange-Zerner theorem suffices here, like in figure 7. We refer the reader to section VI.1 of [3] for full details.

Once (9.39) is established for $y_k > 0$, one repeats the analytic continuation procedure that we described above, keeping track of the implications of (9.39). The analytic continuation used three tricks: (1) analytically continuing $S_n$ by representing it in the form (9.21) (as $e^{-H\tau}$ inserted between two states), (2) expressing the norms of these states in terms of higher-point $S_n$ as in (9.22), and (3) analytic completion.

The bound (9.39) propagates through the tricks (1) and (2) by the use of Cauchy-Schwarz inequality, as well as by using the fact that the norm of $e^{-H\tau}$ is bounded from above by 1 (i.e. eq. (9.6)).

To propagate the bound through trick (3), the following simple idea is used. Suppose we have domains $\mathcal{D}' \supset \mathcal{D}$ such that any holomorphic function $f$ on $\mathcal{D}$ can be extended to a holomorphic function on $\mathcal{D}'$. Then we have the equality of images

$$ f(\mathcal{D}') = f(\mathcal{D}), \quad (9.43) $$

<footnote>$^{86}$Note that the linear growth condition is not needed at this point: eq. (9.17) with some $\sigma_n$ and $q_n$ suffices to establish (9.39) with some $c_n$ and $p_n$. The linear growth condition gives in addition $c_n$ of factorial growth and $p_n$ growing at most linearly. This turns out important later in the proof, see below.</footnote>
and in particular

$$ \sup_{z \in \mathcal{D}'} |f(z)| = \sup_{z \in \mathcal{D}} |f(z)|. \qquad (9.44) $$

To see this, suppose $a \in \mathbb{C}$ is a value which $f$ assumes in $\mathcal{D}'$ but not in $\mathcal{D}$. Then the function $(f(z) - a)^{-1}$ is holomorphic in $\mathcal{D}$ but has a singularity in $\mathcal{D}'$, which is a contradiction. This shows that if we have a bound on $f$ in $\mathcal{D}$, it is also valid in $\mathcal{D}'$.

Finally, recall that in order to construct the analytic continuation of $S_{n_0}$ for some fixed $n_0$, we had to use $S_n$ with arbitrarily high $n$ in the process. This means that in order to establish the bound (9.39) on $S_{n_0}$ for all $Re y_k$, we have to use (9.39) for $y_k > 0$ for $S_n$ with arbitrarily high $n$. These bounds need to combine in a way that is strong enough to establish (9.39) for $S_{n_0}$. For this, it is important that $c_n$ is of factorial growth and $p_n$ grows at most linearly. This requires the same of the sequence $\sigma_n$ and the index of the seminorm in (9.17), explaining the need for the linear growth condition.

# 10 Conclusions

In this paper we studied the relationship between the modern Euclidean CFT axioms (which we formulated in section 2.3) and the more traditional Osterwalder-Schrader and Wightman axioms. We showed that at least for ($n \le 4$)-point functions, both OS and Wightman axioms follow from the Euclidean CFT axioms. Our Euclidean CFT axioms are quite modest. In particular, beyond the minimal assumptions of regularity of correlators and the standard constraints of unitarity, we assumed only a very weak form of the convergent OPE.

Our derivation of Wightman axioms is of particular importance: it shows that the conformal Wightman 4-point functions are well-defined tempered distributions for arbitrary configurations of the 4 points, even when no OPE channel is convergent in the sense of functions. We have furthermore shown that these tempered distributions can always be computed by a conformal block expansion which is convergent in the sense of distributions, generalizing our previous results in cross-ratio space [36], and giving a derivation of Mack's results [41] from Euclidean CFT axioms.

For example, consider the configuration in figure 10, where the operators in a 4-point function are inserted at $x_1 = 0, x_3 = \hat{e}_1, x_4 = \infty \hat{e}_1$, while $x_2 = t \hat{e}_0 + x \hat{e}_1$ is allowed to move in a plane parametrized by $(t, x)$. The cross-ratios for this configuration are $z, \bar{z} = x \pm t$. It is then easy to see that for $x_2$ in the blue region of figure 10 $|\rho|, |\bar{\rho}| < 1$ and the s-channel OPE converges in the sense of functions. Our results imply that the s-channel OPE also converges in the red region where $|\rho|$ and/or $|\bar{\rho}| = 1$, but now the convergence is in the sense of distributions. In particular, the 4-point function is at least a distribution for all values of $x_2$. Of course, in some regions of figure 10 this was obviously true — for example, in the red part of the regions 9, 12 (labeling according to the classification in [37]), one can show that the 4-point function is real-analytic using the convergent t-channel OPE. One may hope to establish real-analyticity also in the region $\mathcal{J}$ using u-channel OPE. This would indeed be the case for the ordering $\langle \mathcal{O}(x_2)\mathcal{O}(x_1)\mathcal{O}(x_3)\mathcal{O}(x_4) \rangle$. However, for the ordering $\langle \mathcal{O}(x_1)\mathcal{O}(x_2)\mathcal{O}(x_3)\mathcal{O}(x_4) \rangle$ that we are discussing here, it turns out that no OPE
<figure><img src="image_11.png" /><figcaption>**Figure 10.** Minkowski configurations with $x_1 = 0, x_3 = \hat{e}_1, x_4 = \infty \hat{e}_1$ and $x_2 = t \hat{e}_0 + x \hat{e}_1$. Blue: configurations where $|\rho|, |\bar{\rho}| < 1$ and the 4-point function is analytic. Red: configurations where $|\rho|$ and/or $|\bar{\rho}| = 1$. Boldface numbers $X = 7, 9, 12$ denote the causal type of the configuration according to [37] (excluding lightlike separations). $S_{1,2}$ are double light-cone singularities.</figcaption></figure>

channel converges in region 7 in the sense of functions.<fnref n="87" /> Therefore, before our work it was not at all clear whether this correlator makes any sense in region 7 if we assume only the Euclidean CFT axioms.

While we have shown that the correlator is *at least* distributional in region 7, we have not excluded the possibility of it being real-analytic there. For example, in 2 dimensions Virasoro symmetry implies that the 4-point function is analytic everywhere away from light-cone singularities [57]. This is perhaps too much to expect in higher dimensions, but one can still ask whether analyticity can be established in a larger domain. One approach is to ask for the envelope of holomorphy of the known domain of analyticity. Since the 4-point function is essentially only a function of two cross-ratios, this might be a tractable question [91]. We leave working out the full consequences of this idea for future work.<fnref n="88" />

In an upcoming paper [42], we will generalize our results to external operators with spins. In addition, there are many other fundamental open questions which we believe are important to understand. First of all, this paper is concerned with properties of CFT Wightman functions in Minkowski space. However, it is expected that Lorentzian CFTs should be naturally defined on Minkowski cylinder [71], which is the smallest physically-sensible space on which finite conformal transformations can act. Yet, it is not known whether CFT Wightman functions can be defined as tempered distributions on Minkowski

<footnote n="87">⁸⁷For a reader comfortable with cuts in $z, \bar{z}$ plane the intuitive argument is simple: we have $z < 0$ (on what we'll call s-channel cut), $\bar{z} > 1$ (on t-channel cut). Furthermore according to the operator ordering, when $z$ crosses 0 we need to make $Im\,t$ slightly negative (and thus $Im\,z$ slightly negative), because $x_2$ at this point crosses the null cone of $x_1$, and when $\bar{z}$ crosses 1 we need to make $Im\,t$ slightly positive (and thus $Im\,\bar{z}$ slightly negative) because it corresponds to $x_2$ crossing the null cone of $x_3$. Thus both $z$ and $\bar{z}$ end on lower sides of their respective cuts, and so one of them must have crossed the $u$-channel cut at (0, 1) when analytically continuing from a Euclidean configuration. We conclude that s- and t- channel OPEs are only distributionally convergent, while $u$-channel is badly divergent.</footnote>

<footnote n="88">⁸⁸Another approach could be via alternative representations of the 4-point function having an extended region of analyticity, e.g. [92].</footnote>
cylinder (see note 68). Answering this question in the positive for CFT ($n \le 4$)-point functions is the main goal of our forthcoming paper [72].

An important problem is to extend our results to ($n > 4$)-point functions. As we discuss in appendix B, even deriving the OS axioms might require some strengthening of the OPE axiom. Another interesting possibility is to formulate Euclidean CFT axioms as OS axioms supplemented with a very weak form of the OPE (for example, asymptotic OPE in Euclidean space). This is perhaps less attractive, since it is desirable to formulate CFT axioms directly in terms of the CFT data (scaling dimensions and OPE coefficients). However, it will still be interesting to establish an equivalence between OS+(weak OPE axiom) and (possibly a stronger version of) our Euclidean CFT axioms, perhaps using arguments similar to those of [41]. Once OS axioms are established, it is likely that a strategy similar to that of the present paper can be pursued to establish Wightman axioms, using a comb-like OPE channel.

In this paper we only considered Wightman functions, but in practice one often needs time-ordered Minkowski correlators. Textbook definition of time-ordered correlators involves multiplying Wightman functions by $\theta$-functions implementing time ordering. Since Wightman functions are in general distributions, this definition does not make rigorous sense at coincident points. As a matter of fact, time-ordered correlators have not been rigorously defined just from Wightman axioms alone (see e.g. [77, p.505]) in a general QFT. In a general QFT setting, it is known that defining time-ordered Minkowski correlators is closely related to defining Euclidean correlators at coincident points [93]. In the future, it would be interesting to define time-ordered CFT Minkowski correlators as distributions just from Euclidean CFT axioms.<fnref n="89" />

A more ambitious goal is to understand the relationship of CFT axioms to Haag-Kastler axioms. This appears to be considerably harder since these axioms deal with operator algebras rather than local correlation functions, and some qualitatively new ideas seem to be required.

## Acknowledgments

Some of our results were first presented in a talk at the Simons Foundation [94], in lectures at the IPHT Saclay [69], and in an online seminar [95]. SR thanks Riccardo Guida and Antti Kupiainen for organizing the last two events.

SR thanks Gian Michele Graf for mentioning Zinoviev's work [85] and to David Brydges for the explanation of this work. We thank Marc Gillioz and Tom Hartman for communications concerning their work. We are grateful to Malek Abdesselam, Simon Caron-Huot, Tom Hartman and David Simmons-Duffin for comments on the draft.

PK is supported by DOE grant DE-SC0009988 and by the Adler Family Fund at the Institute for Advanced Study. The work of SR and JQ is supported by the Simons Foundation grant 488655 and 733758 (Simons Collaboration on the Nonperturbative Bootstrap). SR is supported by Mitsubishi Heavy Industries as an ENS-MHI Chair holder.

<footnote n="89">⁸⁹Time-ordered Minkowski correlators are expected to be to some extent ambiguous at coincident points, just as the Euclidean correlators. For external CFT operators of non-integer external dimension one may choose to fix this ambiguity imposing scale invariance. Ambiguity will remain for external fields of integer dimensions, like the stress tensor.</footnote>
## A Lorentzian CFT literature

Recent years have seen an explosion of the uses of Lorentzian CFT, motivated in particular by the conformal bootstrap applications. In this appendix we will mention some of these works, and comment on their underlying assumptions. See also [96] for a modern pedagogical introduction to Lorentzian CFT.

**Conformal collider bounds.** One of the first “modern” Lorentzian CFT results was obtained in [97]. This work considered a thought experiment, creating a CFT state via a (smeared) local operator and measuring energy coming out at null infinity in a particular direction, integrated over time. On physical grounds, one expects $\langle \Psi | \int dx^- T_{--} | \Psi \rangle \ge 0$ for any state (“averaged null energy condition” - ANEC). One interesting case is of 3-point functions $\langle O^\dagger T_{\mu\nu} O \rangle$ where $O$ has nontrivial spin, when there are several independent OPE coefficients multiplying different tensor structures allowed by conformal symmetry. In this case ANEC implies that certain linear combinations of these OPE coefficients must be non-negative (“conformal collider bounds”). Interference effects can be used to strengthen conformal collider bounds to provide explicit lower bounds [98], while combining conformal collider bounds with stress-tensor Ward identities leads to constraints on operator dimensions which are sometimes stronger than standard unitarity bounds [99]. See below for work aiming to justify ANEC, or to derive conformal bounds directly without using ANEC.

**Light-cone bootstrap.** Refs. [67, 100] were the first to notice that some bootstrap constraints become more visible in the Lorentzian signature. These references pioneered the “analytic light-cone bootstrap” which studies conformal four point functions in the regime of $0 < z, \bar{z} < 1$ real, i.e. in the kinematics of figure 10 when the point $x_2$ is spacelike with respect to $x_1, x_3$. By studying the light cone limit $z \to 0$ at fixed $\bar{z}$ of one OPE channel and requiring that it should be reproduced by the crossed channel, they argued that, in any CFT for $d > 2$, the OPE should contain a series of operators of arbitrarily large spin and twist asymptoting to a particular value. The original argument has some caveats (see the discussion in [101], appendix F) and a mathematically rigorous proof is lacking. It would be nice to provide such a proof, given the extreme importance of the light-cone bootstrap in the modern bootstrap literature. There is little doubt that the light-cone bootstrap results are correct. Numerical bootstrap studies of the critical 3d Ising and the $O(2)$ models [102, 103] have found the series of operators predicted by the light-cone bootstrap see also [104]). Ref. [105] used the light-cone bootstrap to derive the conformal collider bounds of [97] without using ANEC.

**Causality constraints.** Refs. [4, 45, 106] pioneered the study of causality constraints for CFT 4-point functions. In particular ref. [4] pointed out that the $z, \bar{z}$ and $\rho, \bar{\rho}$ expansions are sufficient to construct Lorentzian 4-point functions for many kinematic configurations and show local commutativity (i.e. that spacelike-separated operators commute). See also note 64. These techniques led to a proof of ANEC [45]. As mentioned in footnote 29, some steps in these papers are not completely rigorous. See appendix A.1 below for a more detailed review of [4].
**Bulk point singularity.** Ref. [57] studied the CFT 4-point function on the Lorentzian cylinder focusing on “bulk-point” configurations which correspond to scattering events in AdS/CFT [107–111]. Using a local AdS dual description, one may suspect that the 4-point function should be singular at such configurations. However, on the boundary CFT side, one does not see this singularity in perturbation theory in $d=2$ and $d=3$ dimensions [57]. In $d=2$, ref. [57] showed non-perturbatively that the CFT 4-point function is analytic everywhere away from light cones (in particular regular at bulk-point configurations). This assumes Virasoro symmetry and unitarity and uses Zamolodchikov’s $q$-variables [112]. What happens non-perturbatively in $d \ge 3$ (or in $d=2$ in the absence of the local stress tensor) is still an open problem. Note that at bulk-point configurations, the $\rho$-expansion of the CFT 4-point function does not absolutely converge in $s$-channel (as $|\rho| = |\bar{\rho}| = 1$ there) and diverges in $t$-, $u$-channels [37]. In this paper we only considered the CFT 4-point functions in flat space, but by the same strategy we will show in [72] that the Wightman axioms also hold for CFT 4-point functions on Lorentzian cylinder. In particular, this will show that the CFT 4-point functions are well defined at bulk-point configurations in the sense of tempered distributions (but it will not settle the question of their analyticity there).

**Lorentzian inversion formula.** Ref. [56] introduced an analogue of Froissart-Gribov formula in the context of conformal field theory, which is now known as the Lorentzian inversion formula (LIF). This formula computes the OPE data of a scalar 4-point function in terms of a Lorentzian integral of this 4-point function. The OPE data is extracted in the form of a function $C(\Delta, \ell)$. For integer $\ell$, the function $C(\Delta, \ell)$ encodes the scaling dimensions of exchanged primary operators of spin $\ell$ in the positions of poles in $\Delta$, and the corresponding OPE coefficients are encoded in residues. Lif has many interesting properties, such as analyticity in $\ell$, and suppression of double-twist operators when a cross-channel conformal block expansion is used under the integral. The original derivation of [56] was done in cross-ratio space. The formula was re-derived in position space in [113]. The derivation was further simplified and generalized in [55].

Among other applications, LIF has been used to systematize and extend many of the results of light-cone bootstrap (see, e.g., [103, 114–120]). Similarly to light-cone bootstrap, this application is not completely rigorous simply due to the fact that LIF expresses $C(\Delta, \ell)$ in terms of an integral, and the local operators correspond to singularities of $C(\Delta, \ell)$. In other words, the integral has no chance of converging near the values of $\Delta, \ell$ relevant to local operators, except perhaps for leading-twist operators (see [104] for steps in this direction). This necessarily makes any conclusions about anomalous dimensions of local operators reliant on additional assumptions. These are easy to justify in some perturbative expansions, but in non-perturbative setting do not appear to have been solidly understood.

**Light-ray operators.** Ref. [55] generalized LIF to external operators with spin and uncovered an interesting relation to Knapp-Stein intertwining operators, especially to what they called the light transform. They interpreted the analyticity of LIF in $\ell$ in terms of families of non-local non-integer-spin operators, the light-ray operators. These operators are defined for generic complex $\ell$ and reduce to light-transforms (null integrals) of local
operators for integer spins. More recently, light-ray operators have been used to understand an OPE for event-shape observables such as energy-energy correlators in CFT [121–123] (see also [124, 125]). The light-ray operators correspond to poles in $\Delta$ of $C(\Delta, \ell)$, and the issues with convergence of LIF described above prevent a simple rigorous proof of their non-perturbative existence. (E.g., for generic $\ell$, $C(\Delta, \ell)$ could have cuts or a natural boundary of analyticity in $\Delta$.) It would be interesting to find such a proof. In addition to clarifying the nature of light-ray operators, it would probably also have a bearing on the light-cone bootstrap results discussed above.

**Conformal Regge theory** provides a way to understand Minkowski correlators in Regge limit, and was developed in refs. [126–129]. Regge limit in CFT is a limit of a 4-point function in Lorentzian signature in which $O_2$ approaches the “image of $O_3$ in the next Poincaré patch,” in 4-point function with the ordering

$$ \langle O_4 O_3 O_2 O_1 \rangle. \qquad (A.1) $$

The operators $O_1$ and $O_4$ are kept spacelike separated, with $O_1$ in past of $O_2$ and $O_4$ in the future of $O_3$.<fnref n="90" /> The image of $O_3$ in the next Poincaré patch is the first point on Minkowski cylinder where all future-directed null geodesics from $O_3$ meet. A lot of interest in Regge limit comes from its interpretation as bulk high-energy scattering through AdS/CFT. Kinematically, this limit is somewhat similar to the $O_2 \to O_3$ limit because $O_3$ and its image in the next Poincaré patch transform in the same way under conformal group. For example, the cross-ratios $z_t, \bar{z}_t \to 0$ in Regge limit. (Here by $z_t, \bar{z}_t$ we mean the cross-ratios for t-channel $O_2 \times O_3$.) However, they do so after $\bar{z}_t$ crosses the cut $[1, \infty)$, and so in terms of $\rho_t, \bar{\rho}_t$ we have $\rho_t \to 0$ and $\bar{\rho}_t \to \infty$. Therefore, the $O_2 \times O_3$ OPE is divergent. Conformal Regge theory gives a way to resum the $O_2 \times O_3$ OPE in a way that exhibits a dominant contribution from a “Reggeon” exchange, which is an example of a light-ray operator. Justification for this resummation, which involves analytic continuation of OPE data in spin, comes from LIF (which historically was understood after Conformal Regge theory was established). In the context of our paper, it would be interesting to understand whether such resummations can be made rigorous enough (in axiomatic sense) and used to prove that Minkowski correlators are functions in regions where so far only temperedness has been proven.<fnref n="91" /> For this it might not be necessary to understand the Reggeon or more general light-ray operators, since the resummation procedure can be stopped at a point where the correlator is expressed as an integral of $C(\Delta, \ell)$ over a region where LIF converges. See [92] for progress on these questions.

**Works of Gillioz, Luty et al.** Papers by this group of authors are characterized by the systematic use of momentum space in Lorentzian CFT. So, refs. [60, 80] computed Lorentzian momentum space 3-point functions (3 scalars and scalar-scalar-spin $\ell$) by solving

<footnote n="90">⁹⁰In a symmetric version of the limit, which is related to the one described here by a conformal transformation, the operators $O_1$ and $O_4$ approach each other’s images in the same way as $O_2$ and $O_3$ do.</footnote>

<footnote n="91">⁹¹In the classic Regge limit there is a channel in which the OPE converges regularly, but it is possible that some causal orderings can be relaxed while keeping the resummation procedure valid.</footnote>
the conformal Ward identities. In momentum space, it's also possible to form conformal blocks by gluing 3-point functions [80]. See also notes 40, 67.

Ref. [130] carried out this program quite explicitly in 2d CFT, with an eye towards eventual conformal bootstrap applications. They stressed that the momentum conformal block expansion generally converges only in the sense of distributions — one of the first mentions of distributional convergence in the modern CFT literature. For some momenta configurations, they argued that the momentum conformal blocks can be pointwise bounded by the position conformal blocks with an appropriately chosen real $z \in (0, 1)$. For such configurations the momentum expansion converges in the ordinary sense of functions. The same work also proposes a bootstrap equation in the momentum space, obtained by transforming the local commutativity constraint multiplied by a test function selecting configurations with a spacelike pair of points (however, examples of test functions chosen in [130] may be too singular).

Refs. [131–133] studied the Fourier transform of the time-ordered Minkowski 4-point function in relation to various interesting physics questions. Note that, as mentioned in the conclusions, time-ordered Minkowski CFT 4-point functions have not yet been rigorously defined as distributions. The Fourier transform depends on 4 momenta $p_i$, and to reduce functional complexity it is interesting to take some or all of these momenta lightlike, $p_i^2 \to 0$.

So, ref. [131] considered the Fourier transform of the connected time-ordered 4-point function $\langle T\{O_1O_2O_3O_4\}\rangle_c$. Here they worked with operators of scaling dimension $\Delta_i > d/2$ for which the Fourier transform is expected to have a finite limit as $p_i^2 \to 0$.<fnref n="92" /> Ref. [131] proposed a Lorentzian CFT analogue of the optical theorem:

$$ \mathrm{Im} \mathcal{M}_{1234}(s, t) = \sum_{O \neq 1} f_{12O} f_{43O}^* \mathcal{N}_O(q) \langle \bar{T} \{ \hat{O}_1(p_1) \hat{O}_2(p_2) \} O^\dagger(0) \rangle \langle O(0) T \{ \hat{O}_3(p_3) \hat{O}_4(p_4) \} \rangle, \quad (A.2) $$

where $\mathcal{M}_{1234}$ is proportional to the Fourier transform of $\langle T\{O_1O_2O_3O_4\}\rangle_c$, $f_{ijk}$ is the same as in (2.33), $\mathcal{N}_O(q)$ is some normalization factor at $q = p_1 + p_2 (= -p_3 - p_4)$, and $\hat{O}$ denotes the Fourier transform of $O$. Eq. (A.2) is supposed to hold in the following kinematic region in the momentum space:

$$ p_i^2 = 0, \quad s = (p_1 + p_2)^2 > 0, \quad t = (p_1 + p_3)^2 \le 0, $$

and was derived from a combinatorial operator identity

$$ \sum_{k=0}^{n} (-1)^k \sum_{\sigma \in S_n} \frac{1}{k!(n-k)!} \bar{T}\{\mathcal{O}_{\sigma_1}(x_{\sigma_1}) \cdots \mathcal{O}_{\sigma_k}(x_{\sigma_k})\} T\{\mathcal{O}_{\sigma_{k+1}}(x_{\sigma_{k+1}}) \cdots \mathcal{O}_{\sigma_n}(x_{\sigma_n})\} = 0, \quad (A.3) $$

summing over all permutations, with $T(\bar{T})$ time ordering (anti-time ordering). Note that the use of this identity may be not fully safe in the distributional context, as it arises from a non-smooth partition of unity.

<footnote n="92">⁹²We thank Marc Gillioz for explanations of his work and in particular of the distinction between the high dimension case discussed here and the low dimension case below.</footnote>
The CFT optical theorem (A.2) was used in ref. [131] to study the scale anomalies that appear in a specific class of CFT correlation functions. In fact, unlike Wightman functions which are conformally invariant distributions, time-ordered correlator distributions may, for certain scaling dimensions, contain pieces which violate scale invariance. Thus the scale anomaly describes the violation of dilatation Ward identities, and in position space it is an ultralocal term, located at coincident points. In Fourier space, scale anomaly translates into a nonzero imaginary part of $M_{1234}(s,t)$ at $t=0$. Eq. (A.2) then computes the scale anomaly coefficient through a positive definite sum rule (in particular predicts that it is positive). These scale anomalies also appear in the Euclidean signature, and a similar sum rule for anomaly coefficients can also be found in ref. [131]. However, the Euclidean sum rule is not positive definite unlike the Lorentzian case.

Ref. [131] tested the above ideas for the scalar 4-point function of external dimensions $\Delta = 3d/4$. Ref. [132] then studied the more interesting case of the stress tensor 4-point function $\langle \mathcal{T}\{T_{\mu_1\nu_1}T_{\mu_2\nu_2}T_{\mu_3\nu_3}T_{\mu_4\nu_4}\} \rangle$ whose scale anomaly is proportional to the stress-tensor 2-point function coefficient $c_T$. Assuming that the $t \to 0$ limit is finite, the CFT optical theorem expresses $c_T$ as a sum of positive contributions of all operators in the $T \times T$ OPE apart from the identity (the stress tensor contribution is known, proportional to $c_T$, and can be moved to the l.h.s.). The contributions from the scalars and the spin-2 operators are computed explicitly in [132].

The more recent ref. [133] studied instead the Fourier transform of the time-ordered 4-point function (or Euclidean 4-point function) in the opposite case of the low external dimensions $\Delta_{\phi} < d/2$. Unlike in [131, 132], in this low dimension case the Fourier transform is singular as $p_i^2 \to 0$, and one obtains a finite quantity multiplying it by $(p_i^2)^{d/2-\Delta_{\phi}}$ before taking the limit, a CFT analogue of LSZ reduction. Doing so, they defined a “CFT scattering amplitude” $A(s, t, u)$ ($p_i^2 \to 0$ for all $i$) and a closely related “form factor” $F(s, t, u)$ where $p_i^2 \to 0$ only for $i = 1, 2, 3$. Because the limit $p_i^2 \to 0$ has to be taken one momentum at a time, crossing symmetry is not obvious. Ref. [133] also gave an alternative derivation, starting from the Mellin representation of the CFT 4-point function, where crossing symmetry of $F(s, t, u)$ and $A(s, t, u)$ follows from the crossing symmetry of the Mellin amplitude. In the future, crossing symmetric quantities $A(s, t, u)$ and $F(s, t, u)$ may turn out useful in a bootstrap analysis. It should be stressed that the results of [133] in no way contradict the usual lore that there are no S-matrices in interacting CFTs. In spite of the name adopted in [133], the existence of the quantity $A(s, t, u)$ does not imply that we can set up a wave-packet scattering experiment in a CFT. Wave-packets would quickly diffuse before reaching the interaction region, the singularity of $(p_i^2)^{\Delta_{\phi}-d/2}$ being a cut rather than a pole.

## A.1 Review of Hartman et al. [4]

**Relating different orderings via analytic continuations.** Here we will comment on some of the results of [4] in more details. The first part of this paper considers the Lorentzian CFT 4-point functions with operators $O_1, O_3, O_4$ fixed at zero time and the spatial positions 0, $\hat{e}_1$ and $\infty$, while the operator $O_2$ is inserted at Minkowski position
$t_2\hat{e}_0 + y_2\hat{e}_1$. They consider four different operator orderings

$$ \langle O_2 O_1 O_3 O_4 \rangle, \langle O_1 O_2 O_3 O_4 \rangle, \langle O_3 O_2 O_1 O_4 \rangle, \langle O_1 O_3 O_2 O_4 \rangle \quad (A.4) $$

in the region of $0 < y_2 < 1/2$ and $t_2$ positive. As $t_2$ is increased from zero, the operator $O_2$, initially spacelike with respect to all other insertions, crosses the light cone first of $O_1$ and then of $O_3$. With $z, \bar{z} = y_2 \pm t_2$, denoting $G(z, \bar{z}) = \langle O_2 O_1 O_3 O_4 \rangle$, they give the following prescription to compute the correlators for the other orderings (ref. [4], eq. (3.22)):

$$ \langle O_1 O_2 O_3 O_4 \rangle = G(z, \bar{z})_{z \to e^{-2\pi i z}}, \quad (A.5) $$

$$ \langle O_3 O_2 O_1 O_4 \rangle = G(z, \bar{z})_{(\bar{z}-1) \to e^{-2\pi i (\bar{z}-1)}}, $$

$$ \langle O_1 O_3 O_2 O_4 \rangle = G(z, \bar{z})_{z \to e^{-2\pi i z}, (\bar{z}-\bar{z}_0) \to e^{-2\pi i (\bar{z}-\bar{z}_0)}}. $$

Their justification of this prescription relied on some presumed analyticity properties of $G(z, \bar{z})$ which, to our knowledge, have never been shown in a general QFT context. Nevertheless we will see below that for CFTs eq. (A.5) turns out to be true (with $\bar{z}_0 = 1$).

The real parameter $\bar{z}_0$ was introduced in [4] as the position of the first $\bar{z}$ singularity of $G(z, \bar{z})_{z \to e^{-2\pi i z}}$. Their goal was to show that $\bar{z}_0 \ge 1$, which using (A.5) then implies local commutativity $\langle O_1[O_2, O_3]O_4 \rangle = 0$ for $\bar{z} < 1$ i.e. when $O_2$ is spacelike to $O_3$. In our paper (section 6.9) we presented a different way to understand and derive local commutativity which is closer to the classic literature: as we have reviewed there, it is a robust consequence of the existence of the analytic continuation to the forward tube, which we constructed.

Let's see what it would take to justify (A.5). We define the function $G(z, \bar{z})$ by the $O_2 \times O_1$ OPE expansions, which converges for $|\rho|, |\bar{\rho}| < 1$, i.e. as long as $z, \bar{z}$ stay away from $z, \bar{z} \in (+1, \infty)$. The points $z = 0$ and $\bar{z} = 0$ are branch points singularities with cuts which we put along the negative real axis. Note that the contours in (A.5) are all within the analyticity domain of $G(z, \bar{z})$. So the prescription (A.5) is meaningful. We still have to see if it agrees with the rigorous definition which computes the Minkowski 4-point function by analytically continuing from the Euclidean region staying in the forward tube corresponding to the chosen operator ordering. We will see that it will indeed agree, but showing it for the last ordering will be subtle.

For definiteness we will focus on the region $t_2 > 1 - y_2$ i.e. $\bar{z} > 1$, where $O_2$ crossed both light cones. The end point of the analytic continuation contour is always the same while the initial point depends on the operator ordering. E.g. for the ordering $\langle O_2 O_1 O_3 O_4 \rangle$ we have to pick initial Euclidean times $\varepsilon_2 > \varepsilon_1 > \varepsilon_3$, while for $\langle O_1 O_2 O_3 O_4 \rangle$ we have $\varepsilon_1 > \varepsilon_2 > \varepsilon_3$ etc. For any of these orderings, we denote

$$ z_1, \bar{z}_1 = \pm i\varepsilon_1, \quad z_2, \bar{z}_2 = y_2 \pm i(\varepsilon_2 + it_2), \quad z_3, \bar{z}_3 = 1 \pm i\varepsilon_3 \quad (A.6) $$

and compute (in the limit $z_4, \bar{z}_4 = \infty$)

$$ z = \frac{z_1 - z_2}{z_1 - z_3}, \quad \bar{z} = \frac{\bar{z}_1 - \bar{z}_2}{\bar{z}_1 - \bar{z}_3}. \quad (A.7) $$
<figure><img src="image_12.png" /><figcaption>**Figure 11.** $z, \bar{z}$ curves for the analytic continuation from the Euclidean; the first 3 orderings.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 12.** $z, \bar{z}$ curves for the analytic continuation from the Euclidean for the $\langle O_1 O_3 O_2 O_4 \rangle$ ordering.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 13.** $z, \bar{z}$ curves for computing the $\langle O_1 O_3 O_2 O_4 \rangle$ ordering via (A.5).</figcaption></figure>

We are interested in the curves which $z, \bar{z}$ trace as the Euclidean times are scaled to zero and the Minkowski time $t_2$ from 0 to its final value. For the first three orderings the resulting curves are shown in figure 11.

We see that in all these cases, the curves lie in the analyticity domain of $G(z, \bar{z})$ i.e. they don't cross $(1, +\infty)$. For the first two cases this was guaranteed by our results that $|\rho|, |\bar{\rho}| < 1$ for the s-channel OPE expansion. For the third case it was not guaranteed but it also turns out to be true, by inspection. We also see that in all these 3 cases, the curves go around $z = 0$ and $\bar{z} = 1$ in agreement with (A.5).<fnref n="93" />

For the fourth ordering $\langle O_1 O_3 O_2 O_4 \rangle$ when we have to assign $\varepsilon_1 > \varepsilon_3 > \varepsilon_2$, the analytic continuation inside the forward tube gives the $z, \bar{z}$ curves shown in figure 12, while prescription (A.5) would correspond to figure 13.

<footnote n="93">⁹³To see this more clearly in the third case, deform the curves continuously moving the initial $z$ into the upper half plane and the initial $\bar{z}$ into the lower half plane.</footnote>
<figure><img src="image_15.png" /><figcaption>Figure 14. Deforming the $z, \bar{z}$ curves for the $\langle O_1 O_3 O_2 O_4 \rangle$ ordering.</figcaption></figure>

The two figures are clearly not the same. Moreover the curves in the first figure cross $(+1, \infty)$ where the definition of $G(z, \bar{z})$ via the $O_1 \times O_2$ channel OPE expansion stops converging. Can we show that the analytic continuation in figure 12 exists and that it agrees with the one in figure 13?

For this, let us bring in the $O_2 \times O_3$ OPE expansion, which correspond to expanding in $z_t = 1-z$, $\bar{z}_t = 1-\bar{z}$ or in the corresponding $\rho_t, \bar{\rho}_t$. In the Euclidean region the two expansions agree. The $O_2 \times O_3$ expansion converges away from $z_t, \bar{z}_t \in (+1, \infty)$ i.e. $z, \bar{z} \in (-\infty, 0)$. Thus the curves in both figures 12, 13 lie within the range of analyticity of the $O_2 \times O_3$ expansion, so we can compare the analytic continuations. Since only integer spins occur in the expansion, the analytic continuation does not change under $\rho_t \to e^{2\pi i} \rho_t, \bar{\rho}_t \to e^{-2\pi i} \bar{\rho}_t$ (such arguments were systematically exploited in [37]).$^{94}$ So let us add extra loops to the blue and the red curves in the opposite directions around 1, see figure 14. Adding the loops and deforming the curves continuously (the first step is shown in figure 14) we can bring them to those in figure 13. This finishes the proof that the prescription (A.5) is correct also for the fourth ordering.

**Positivity constraints.** We wish to comment on another result of [4]: an argument for positivity of certain conformal block expansion coefficients. We present the argument exchanging the role of s and t channels w.r.t. [4]. Let $G(z, \bar{z}) = 1 + \dots$ be the holomorphic function defined by the s-channel OPE expansion (i.e. $(z\bar{z})^{\Delta_1+\Delta_2}$ times the $G(z, \bar{z})$ discussed above). We will define a certain analytic continuation of the function $G(z, \bar{z})$. Let us start with $\bar{z}$ and $z$ close to zero, $\bar{z}$ in the upper half plane and $z$ in the lower half plane. In this range all three channels s,t,u converge. We wish to analytically continue $G(z, \bar{z})$ by taking $z$ through $(1, +\infty)$ and bring it back close to zero, in the upper half plane (see figure 15), while we don't touch $\bar{z}$. This analytic continuation can be performed using the t-channel or u-channel expansions, with the same result (but not the s-channel since it stops converging on $(1, +\infty)$). We denote the result of this analytic continuation by $\hat{G}(z, \bar{z})$, with $\mathrm{Im}\,z, \mathrm{Im}\,\bar{z} > 0$.

Although $\hat{G}(z, \bar{z})$ is so defined with both $z, \bar{z}$ in the upper half plane, it has continuous limits when they both approach positive real axis, or both approach negative real axis, since in the first case the t-channel and in the second case the u-channel remains convergent. We will be interested in the situation when $O_1 = O_2, O_3 = O_4$. In this case all expansion

<footnote>$^{94}$Sometimes this property is called “Euclidean single-valuedness”.</footnote>
<figure><img src="image_16.png" /><figcaption>Figure 15. Definition of $\hat{G}(z, \bar{z})$. Red: integration contour in eq. (A.11).</figcaption></figure>

coefficients in the t and u channels are positive. This implies that the Euclidean correlator $G_E(z, \bar{z})$ will be positive for real $z, \bar{z} > 0$ (using t-channel) and for real $z, \bar{z} < 0$ (using u-channel). The difference between $G_E(z, \bar{z})$ and $\hat{G}(z, \bar{z})$, for real $z, \bar{z} > 0$ or $z, \bar{z} < 0$ is that in the first case $z, \bar{z}$ approach the real axis from the opposite sides while in the second case from the same one. When we take $z$ through $(1, +\infty)$ cut, this only brings in some phases in the t and u-channel expansion of $\hat{G}(z, \bar{z})$ with respect to $G_E(z, \bar{z})$. This implies that we have a bound for real $z, \bar{z} > 0$ or $z, \bar{z} < 0$:

$$|\hat{G}(z, \bar{z})| \le G_E(z, \bar{z}) \qquad (A.8)$$

In what follows $\hat{G}(z, \bar{z})$ will be used as a holomorphic function with $z, \bar{z}$ in the upper half plane satisfying the bound (A.8) on its boundary, while $G_E(z, \bar{z})$ will be used only with real $z, \bar{z}$.

In particular, since $G_E \approx 1$ for $z, \bar{z}$ near zero up to small corrections, eq. (A.8) says that $\hat{G}$ is bounded, for small real $z, \bar{z} > 0$ or $z, \bar{z} < 0$, by 1 up to small corrections. This argument can be generalized to show that $\hat{G}(z, \eta z)$ for $\eta > 0$ real and $z$ near zero in the upper half plane is bounded by a constant.<fnref n="95" />

We now pass to the non-rigorous part of the argument. Although the s-channel stops converging when crossing $(1, +\infty)$, ref. [4] proposed that, in the regime $|\bar{z}| \ll |z| \ll 1$, the behavior $\hat{G}(z, \bar{z})$ can nevertheless be predicted from the s-channel expansion, by organizing it in $\bar{z}^{\tau/2}$ where $\tau = \frac{1}{2}(\Delta - \ell)$ is the twist. The typical term is

$$\bar{z}^{\tau/2} k_{\frac{1}{2}(\Delta+\ell)}(z), \qquad (A.9)$$

where $k_h(x) = {}_2F_1(h, h, 2h, x)$ is the collinear conformal block. This is the same expansion as used in the light-cone bootstrap, which has its own problems of rigor, but here it is proposed to use it after $z-1 \to e^{2\pi i}(z-1)$. Under this continuation the collinear conformal block, which has a $\log(1-z)$ behavior near $z=1$, picks up an imaginary piece which, for $z$ small, behaves as $\sim iz^{1-\frac{1}{2}(\Delta+\ell)}$ (see [4], eq. (4.28)). Considering $\bar{z} = \eta z$, $\eta \ll 1$, $|z| \ll 1$, we then have, according to the proposal of ref. [4],

$$\hat{G}(z, \eta z) \approx 1 - B(\Delta, \ell) p_{\Delta, \ell} \times i \frac{\eta^{\tau/2}}{z^{\ell-1}}, \qquad (A.10)$$

<footnote n="95">⁹⁵Let $z = re^{i\varphi}$, $r \ll 1$. We consider $0 \le \varphi \le \pi/2$, when the argument uses the t-channel, the case $\pi/2 \le \varphi \le \pi$ is analogous using the u-channel. The key point is that the $\rho$ variable in the t-channel $\rho_t \approx 1 - \sqrt{r}e^{i\varphi/2}$, $|\rho_t| \approx 1 - \sqrt{r}\cos(\varphi/2)$. This allows to compare the function $\hat{G}(z, \eta z)$ to $G_E(z', \eta z')$ with real $z' = r\cos^2(\varphi/2)$, times a factor $\sim (z\bar{z})^{\Delta_1}/(z'\bar{z}')^{\Delta_1}$ from the crossing kernel, which is bounded by a constant.</footnote>
where $\Delta, \ell$ are the dimension and spin of the leading twist operator (which may e.g. be the stress tensor), $p_{\Delta,\ell}$ its conformal block coefficient, and $B(\Delta, \ell) \ge 0$ some explicitly known constant. The spin $\ell$ is even since we are assuming $O_1 = O_2$. Eq. (A.10) assumes that the limit $\eta \to 0$ is taken before $z \to 0$.<fnref n="96" />

Now, let us consider the holomorphic function $f(z) = 1 - \hat{G}(z, \eta z)$, and integrate $z^{\ell-2}f(z)$ along the contour shown in figure 15.<fnref n="97" /> We have

$$ \int_{\text{arc}} z^{\ell-2} f(z) dz + \int_{-\varepsilon}^{\varepsilon} x^{\ell-2} f(x) dx = 0. \qquad (A.11) $$

Using (A.10) and that the integral over the arc of $1/z$ is $\pi i$, we get in particular:<fnref n="98" />

$$ \begin{aligned} \pi B(\Delta, \ell) p_{\Delta, \ell} &\approx \eta^{-\tau/2} \int_{-\varepsilon}^{\varepsilon} x^{\ell-2} \operatorname{Re}[1 - \hat{G}(z, \eta z)] dx \\ &\approx \eta^{-\tau/2} \int_{-\varepsilon}^{\varepsilon} x^{\ell-2} \operatorname{Re}[G_E(z, \eta z) - \hat{G}(z, \eta z)] dx, \end{aligned} \qquad (A.12) $$

where in the final step we replaced 1 by $G_E(z, \eta z)$. Since $G_E(z, \eta z)$ has a rigorously convergent expansion for small $z$, it satisfies the bound:

$$ G_E(z, \bar{z}) = 1 + O(\eta^{\tau/2}). \qquad (A.13) $$

So the last replacement was legitimate if e.g. $\ell \ge 2$. By (A.8), the r.h.s. of (A.12) is a positive quantity. This equation then implies that $p_{\Delta,\ell}$ must be positive as well.

As already mentioned, the weak point of this argument is that the s-channel expansion stops converging when we cross $(1, +\infty)$. It is therefore not at all obvious that analytic continuations of the individual conformal block expansion terms have anything to do with the asymptotics of $\hat{G}(z, \bar{z})$. Ref. [4] was of course aware of this, and provided some arguments, inspired by the light-cone bootstrap, why nevertheless the asymptotics from the leading twist terms can be trusted. We don't know how to make those arguments rigorous. It would be interesting to understand if asymptotics (A.10) can be justified using just Euclidean CFT axioms of section 2.3 or requires additional assumptions. The same question also looms over the proofs of ANEC [45] and ANEC commutativity [122] which involved similar “light-cone limit on the second sheet” considerations.

## B OS axioms for higher-point functions

In this appendix we discuss the modifications necessary to derive from the Euclidean CFT axioms the OS axioms (positivity and cluster property) for $n$-point functions with $n > 4$,

<footnote n="96">⁹⁶In fact, in the opposite limit $z \to 0$ for fixed $\eta$, eq. (A.10) would violate the discussed above rigorous bound that $\hat{G}(z, \eta z)$ is bounded by a constant. There is no paradox because that's not the limit we are supposed to be taking.</footnote>

<footnote n="97">⁹⁷We can also take an intermediate step of adding a small semicircle of radius $\varepsilon'$ around zero, but since $\hat{G}$ is bounded for small $z$, the limit $\varepsilon' \to 0$ is not problematic.</footnote>

<footnote n="98">⁹⁸Note that the quantity $\operatorname{Re}[G_E(z, \eta z) - \hat{G}(z, \eta z)]$ appearing in (A.12) is essentially the double discontinuity considered in [56]. Similarly, eq. (A.12) can be formally obtained from the Lorentzian inversion formula of [56] by expanding the integrand in a light-cone limit. We thank Tom Hartman for pointing this out.</footnote>
compared to the $n \le 4$ case considered in section 3.1. As we explain below, it appears that there is no simple proof of OS positivity for $n > 4$ from the Euclidean CFT axioms of section 2.3. Since the reason for this is rather technical, let us first discuss the conceptual implications of this.

Ideally, one would like to have a set of Euclidean CFT axioms that would imply Wightman axioms (and therefore also OS axioms) and also be powerful enough to derive all the usual CFT lore such as OPEs, radial quantization, operator-state correspondence, crossing symmetry, etc. These statements, as we saw in the main text, make sense and can be non-trivial even when we restrict our attention to $n$-point functions with bounded $n$.

In particular, we have found that the axioms we formulated in section 2.3 achieve the above goal for $n \le 4$. Extending our results to $n > 4$ using the same strategy would require a solution to two problems: first, we need to derive OS axioms (specifically, positivity and cluster property) for $n > 4$, and, second, we need to prove that OS axioms together with the OPE imply Wightman axioms.

Conceptually, it seems plausible that OS axioms + OPE imply Wightman axioms for $n > 4$ because we expect that for $n > 4$ there is again an OPE channel which is convergent in the entire forward tube (i.e. the one given by taking the OPE in the same order as the operators appear in the Wightman ordering). This question clearly merits further study but is beyond the scope of this paper.

However, it is less clear to us how to even attempt a derivation of OS positivity for $n > 4$ from Euclidean CFT axioms of section 2.3. Let us first explain why this is the case, and then we will discuss the possible modifications to these CFT axioms.

Suppose we want to prove the positivity

$$ \langle \Psi | \Psi \rangle \ge 0, \qquad (B.1) $$

where $\Psi$ is a state created by a product of three local operators, $|\Psi\rangle = |\varphi_1(x_1)\varphi_2(x_2)\varphi_3(x_3)\rangle$.

To prove this positivity the natural idea would be to use the OPE expansion repeatedly for the two copies of $\Psi$ and then use the positivity of the 2-point function. However, for this we need our OPE approximation for $\langle \Psi |$ to be conjugate to our approximation for $|\Psi\rangle$. This is non-trivial to achieve because we have to perform the OPEs one at a time. For example, we can first construct an approximation of $|\Psi\rangle$ in terms of a state $|\Psi'\rangle$, created by single operator insertions, such that

$$ |\langle \Psi | \Psi \rangle - \langle \Psi | \Psi' \rangle| < \varepsilon. \qquad (B.2) $$

Similarly, we can construct a state $\langle \Psi''|$ such that

$$ |\langle \Psi'' | \Psi' \rangle - \langle \Psi | \Psi' \rangle| < \varepsilon \qquad (B.3) $$

and thus

$$ |\langle \Psi'' | \Psi' \rangle - \langle \Psi | \Psi \rangle| < 2\varepsilon. \qquad (B.4) $$
These approximations are possible by the repeated use of the OPE (3.2). Note, however, that since the OPE axiom is formulated for correlation functions, the number of terms we have to include in the OPE for a given $\epsilon$ depends on the correlation function in which the OPE is performed. It then follows that the state $\langle\Psi''|$ depends on $|\Psi'\rangle$ (because in order to construct it we use the OPE in the correlation function $\langle\Psi|\Psi'\rangle$) and is in general different from it. It is therefore not obvious that $\langle\Psi''|\Psi'\rangle \ge 0$, which is what we would like to use in order to prove $\langle\Psi|\Psi\rangle \ge 0$ with the help of the above inequalities.

In the case when $n = 4$ and $|\Psi\rangle$ is created by 2 operators we were able to solve this difficulty. This was because in this case the only difference between $|\Psi'\rangle$ and $\langle\Psi''|$ can be in the number of OPE terms included in the approximation, and we were able to use an orthogonality property of the 2-point function to show $\langle\Psi''|\Psi'\rangle = \langle\Psi'|\Psi'\rangle$ by arguing that we can assume that $\langle\Psi''|$ contains more terms than $|\Psi'\rangle$ and that those terms which are in $\langle\Psi''|$ but not in $|\Psi'\rangle$ do not contribute to the product $\langle\Psi''|\Psi'\rangle$.

This argument does not work in the case at hand, $|\Psi\rangle = |\varphi_1(x_1)\varphi_2(x_2)\varphi_3(x_3)\rangle$. The reason for this is that in order to construct $|\Psi'\rangle$ or $\langle\Psi''|$ we need to perform two OPE's in each case. For example, the first one can be $\varphi_1 \times \varphi_2 = \sum_k O_k$ and the second one can be $\varphi_3 \times O_k$. Both OPE's have to be truncated at some point, and while the truncation of the second OPE affects only the set of terms that are present in $|\Psi'\rangle$ or $\langle\Psi''|$, where we truncate the first $\varphi_1 \times \varphi_2$ OPE affects the coefficients of these terms. Since now $|\Psi'\rangle$ and $\langle\Psi''|$ contain terms with differing coefficients, we cannot use orthogonality to argue $\langle\Psi''|\Psi'\rangle = \langle\Psi'|\Psi'\rangle$ anymore. There is no way to ensure that $\varphi_1 \times \varphi_2$ OPEs are truncated in the same way in the construction of both states because the truncation in $\langle\Psi''|$ depends, through our OPE axiom, on $|\Psi'\rangle$, and thus might happen to be always at a higher order than the truncation used for $|\Psi'\rangle$.

This all is to say that due to a rather technical reason it appears that there is no simple proof of OS positivity of higher-point functions from the Euclidean CFT axioms as stated in section 2.3. Importantly, this doesn't mean that there is no proof at all. Indeed, the Euclidean CFT axioms are sufficient to derive the standard crossing-symmetry equations for 4-point functions. It could happen that in all solutions to these crossing-symmetry equations the OPE coefficients have such asymptotics that a stronger form of the OPE axiom holds and allows us to prove the OS positivity for $n > 4$. However, it is not clear how to implement this line of reasoning in practice.

It is therefore interesting to look for a stronger version of Euclidean CFT axioms. We discuss below some simple modifications of the OPE axiom which avoid the above problem and allow to prove OS positivity for higher-point functions.

Morally, we want some kind of statement of uniformity for the convergence rate of the OPE: it should make $|\Psi''\rangle$ above independent of the truncation made in $|\Psi'\rangle$, as long as this truncation is done at a sufficiently high order. This would allow us to make both truncations at a high order and ensure $\langle\Psi''|\Psi'\rangle = \langle\Psi'|\Psi'\rangle \ge 0$.

One option is to assume a stronger form of the OPE, which allows us to perform two OPE's simultaneously. Specifically, we can assume that the double sum

$$ \langle O_1 O_2 O_3 O_4 \dots \rangle = \sum_{k,l} \langle O_k O_l \dots \rangle, \qquad (B.5) $$
is convergent, where we wrote the two OPEs schematically as $O_1O_2 = \sum_k O_k$ and $O_3O_4 = \sum_l O_l$. Convergence of the double sum means that

$$ \left| \langle O_1 O_2 O_3 O_4 \dots \rangle - \sum_{k,l} \langle O_k O_l \dots \rangle \right| < \varepsilon, \qquad (B.6) $$

when the sums are truncated in a way that includes some $\varepsilon$-dependent finite set of terms, but is otherwise arbitrary. In particular, both sums can be truncated in the same way, and this solves the problem that we encountered above. A disadvantage of this approach is that it is unclear how to derive this axiom from OS axioms and the usual single OPE axiom (however, a heuristic argument based on cutting the Euclidean path integral can be made). This is somewhat subtle and is related to the question of whether the path integral over a spherical layer ($r_1 < r < r_2$) with operator insertions in the interior represents a bounded operator. We can't say with confidence whether or not this is the case.

Another option is to assume resumed repeated OPE, i.e. that the following sum converges, schematically,

$$ \langle O_1 \dots O_m O_{m+1} \dots O_n \rangle = \sum_k c_k \langle O_k O_{m+1} \dots O_n \rangle, \qquad (B.7) $$

where the coefficients $c_k$ are chosen so that

$$ \langle O_1 \dots O_m O_k^\theta \rangle = c_k \langle O_k O_k^\theta \rangle, \qquad (B.8) $$

assuming $\langle O_k O_l^\theta \rangle \propto \delta_{kl}$. This version of the axiom is essentially the statement that one-operator states form a basis of the CFT Hilbert space, formulated without explicitly introducing the Hilbert space. In other words, above we are approximating the state $\langle O_1 \dots O_m |$ in terms of an orthonormal basis of states $\langle O_k |$, and the coefficients are computed by inner products. This form of the axiom is easy to derive from OS + convergent OPE, and also easily allows us to solve our problem by using the same strategy as in the case $n = 4$. However, it does appear to be an overly strong assumption, making our axioms not very different from assuming OS + convergent OPE outright.

Finally, an interesting prospect might be, instead of formulating an entirely new set of axioms, to add an *asymptotic* OPE axiom (and conformal invariance) to OS axioms. It is likely that using logic very similar to that of Mack [41], which we reviewed in section 8.3, one can prove that (OS axioms)+(asymptotic OPE)+(conformal invariance) imply convergent OPE.

# C Details on Vladimirov's theorem

## C.1 Limit in the sense of distributions

Let us start with a reminder of what the limit in the sense of tempered distributions means. Let $f(u)$, $u = (t_k, \mathbf{x}_k) = (t_1, \mathbf{x}_1, \dots, t_n, \mathbf{x}_n) \in \mathbb{R}^{nd}$, be a Schwartz test function, i.e. an infinitely differentiable function decreasing at infinity faster than any power together with all its derivatives. This can be also stated as finiteness of all Schwartz norms:

$$ |f|_N = \sup_{u \in \mathbb{R}^{nd}, |\alpha| \le N} (1 + u^2)^{N/2} |\partial_u^\alpha f| < \infty \quad \forall N \ge 0. \qquad (C.1) $$
That the limit (4.8) exists in the sense of distributions means two requirements. First, that the r.h.s. of (4.8) has a finite limit integrated against any $f$ as above:

$$ (G_n^M, f) := \lim_{\epsilon_k \to 0} \int dt \, d\mathbf{x} \, G_n(\epsilon_k + it_k, \mathbf{x}_k) f(t_k, \mathbf{x}_k) \quad \text{exists for any Schwartz } f. \quad (C.2) $$

The $G_n^M$ defined by this equation is a linear functional on the Schwartz space. The second requirement is that this functional should be continuous (and thus is itself a tempered distribution). Continuity means that it should be bounded by one of the norms (C.1) with a sufficiently large $N$, i.e.:

$$ |(G_n^M, f)| \le C|f|_{N_*}, \quad (C.3) $$

with $f$-independent $C$ and $N_*$.

Note that by eq. (C.3), $G_n^M$ can be extended from the Schwartz space to a larger space of test functions, which are required to be differentiable only $N_*$ times and have a finite $|f|_{N_*}$. Parameter $N_*$ thus characterizes regularity of the distribution $G_n^M$. The proof of theorem 4.1 will determine $N_*$ in terms of $A_n$ and $B_n$, see eq. (C.6).

## C.2 Proof of theorem 4.1

Unfortunately, we do not know a reference where theorem 4.1 is stated and proved succinctly in the form we need. Such results are considered standard in the theory of several complex variables. For similar statements see [134], Chapter 5, and [5], theorem 2-10. For the convenience of the reader, we present here a proof based on these sources.

The usefulness of Vladimirov's theorems for establishing distributional properties of CFT correlators was recognized in our recent work [36]. There, we considered expansions of the CFT 4-point function $g(\rho, \bar{\rho})$ in terms of conformally invariant cross-ratios $\rho, \bar{\rho}$. It is well known that such expansions converge in the interior of the unit disk $|\rho|, |\bar{\rho}| < 1$. Using Vladimirov's theorems, we showed in [36] that they also converge on the boundary of this disk, in the sense of distributions. In this paper we are interested in CFT correlators as functions of positions $x_k$, not of cross-ratios, but the basic principle is the same as in [36]: a powerlaw bound on an holomorphic function near a boundary implies temperedness of the limiting distribution.

By translation invariance it's enough to study the function $G_n$ expressed in terms of the differences $y_k = x_k - x_{k+1}$ which we denote by $\mathcal{G}(y)$, $y = (y_1, \dots, y_{n-1})$. We also denote $y_k = (y_k^0, \mathbf{y}_k)$, $y_k^0 = \varepsilon_k + is_k$, $\varepsilon_k > 0$, $\mathbf{y}_k \in \mathbb{R}^{d-1}$.

Consider first the case when all $\varepsilon_k$ go to zero together along a fixed direction: $\varepsilon_k = rv_k$ where $r \to 0$ and $v = (v_k)$ is a vector with positive components. Later on we will show that the limit continues to exist if $\varepsilon_k \to 0$ independently (as well as the more general statement about the limit from inside the forward tube).

So, let us prove that $\mathcal{G}(y)$ has a limit as $r \to 0$ which is a tempered distribution in variables $s_k, \mathbf{y}_k$. As in (C.2), we fix a Schwartz test function $f$ and consider the integral (we will omit index $k$ on $\varepsilon, s, v, \mathbf{y}$ if no confusion may arise)

$$ h(r) = \int ds \, d\mathbf{y} \, \mathcal{G}(rv + is, \mathbf{y}) f(s, \mathbf{y}). \quad (C.4) $$
The problem is analogous to theorems used in [36], so we will be brief. As in [36], section 3.3 and appendix C, using analyticity in $y^0$, integration by parts, and the powerlaw bound one can show that derivatives of $h$ in $r$ satisfy the bound:

$$|\partial_r^j h(r)| \le \frac{C}{r A_n} |f|_N, \qquad (C.5)$$

where $|f|_N$ is a Schwartz norm (C.1) of a sufficiently large order $N$ depending on $j$ and $B_n$. The constant $A_n$ is the same as in (4.9), in particular the same $A_n$ works for all $j$. In what follows we only need this equation for finitely many $j$ (up to $[A_n] + 1$). Using the Newton-Leibniz formula in the $r$ direction several times, one then proves that the same bound as (C.5) holds in fact without $1/r^{A_n}$ singularity in the r.h.s. It then follows that, first of all, $\lim_{r \to 0} h(r)$ exists, and second, it is a continuous linear functional of $f$, that is, a distribution. The limit holds uniformly when the components $v_k$ vary on any fixed compact interval contained in $(0, +\infty)$. Its $v$-independence is shown exactly as in [36], eq. (C.7). Let us denote the limiting distribution $\mathcal{G}(is, y) = \mathcal{G}^M(s, y)$.

It is of some interest to know the precise regularity of the distribution $\mathcal{G}^M$ (i.e. how many derivatives the test function must have to be pairable with $\mathcal{G}^M$) and the rate of its growth at infinity. Following the above argument in detail, one can show the following bound which contains this information:

$$|(\mathcal{G}^M, f)| \le \text{Const.} \int ds \, dy (1 + |s| + |y|)^{B_n} \max_{|\alpha| \le [A_n] + 1} |\partial_s^\alpha f(s, y)|. \qquad (C.6)$$

This in particular implies (C.3) with $N_* = \max([A_n] + 1, B_n + nd + 1)$.

Parts 2,3 of theorem 4.1 are new compared to [36], since such questions do not arise in the cross-ratio space.

Lorentz invariance is easy to show, as follows. Rotation invariance of $G_n^E$ implies that $\mathcal{G}(y)$ satisfies for real $y$ the differential equations

$$\{y^a \partial_{y^b} - y^b \partial_{y^a}\} \mathcal{G}(y) = 0, \quad a, b \in \{0, 1, \dots, d-1\} \qquad (C.7)$$

(as usual $y = (y_k)$, summation in $k$ understood). By the uniqueness of analytic continuation, these equations continue to hold for complex $y_0$. That's the only place where we use real-analyticity in the spatial direction.<fnref n="99" /> By taking the limit $\epsilon \to 0$ in (C.7), we recover precisely the differential equations expressing the Lorentz invariance of $\mathcal{G}^M$. Let us explain in more detail how the limit is taken and why it exists. Consider for definiteness $a = 0$, $b = 1$, other cases being similar. Eq. (C.7) then says $\{(\epsilon + it)\partial_{y^1} + iy^1\partial_t\}\mathcal{G}(\epsilon + it, y) = 0$, in the sense of functions, and hence integrating by parts in the sense of distributions acting on test functions $\varphi(t, y)$:

$$(\mathcal{G}_\epsilon, \{(\epsilon + it)\partial_{y^1} + iy^1\partial_t\}\varphi) = 0, \qquad (C.8)$$

where we denoted $\mathcal{G}_\epsilon(t, y) = \mathcal{G}(\epsilon + it, y)$. Now we take the limit $\epsilon \to 0$. We know that (a) $\mathcal{G}_\epsilon \to \mathcal{G}^M$ in the sense of distributions, and also that (b) $|\mathcal{G}_\epsilon, \varphi|$ is uniformly

<footnote n="99">⁹⁹With some extra tricks, it's possible to replace it by the assumption of mere continuity in $y$, as in [3], theorem 4.3. In the CFT applications we have in mind, real analyticity appears a more natural assumption.</footnote>