# Distributions in CFT. Part II. Minkowski space

Petr Kravchuk, Jiaxin Qiao, Slava Rychkov

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Petr Kravchuk, Jiaxin Qiao, Slava Rychkov. Distributions in CFT. Part II. Minkowski space. Journal of High Energy Physics, 2021, 2021 (08), pp.094. 10.1007/JHEP08(2021)094. hal-03203628

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# Distributions in CFT. Part II. Minkowski space

Petr Kravchuk,ª Jiaxin Qiaoᵇ,ᶜ and Slava Rychkovᶜ,ᵇ

E-mail: pkravchuk@ias.edu, jiaxin.qiao@phys.ens.fr

**ABSTRACT:** CFTs in Euclidean signature satisfy well-accepted rules, such as the convergent Euclidean OPE. It is nowadays common to assume that CFT correlators exist and have various properties also in Lorentzian signature. Some of these properties may represent extra assumptions, and it is an open question if they hold for familiar statistical-physics CFTs such as the critical 3d Ising model. Here we consider Wightman 4-point functions of scalar primaries in Lorentzian signature. We derive a minimal set of their properties solely from the Euclidean unitary CFT axioms, without using extra assumptions. We establish all Wightman axioms (temperedness, spectral property, local commutativity, clustering), Lorentzian conformal invariance, and distributional convergence of the s-channel Lorentzian OPE. This is done constructively, by analytically continuing the 4-point functions using the s-channel OPE expansion in the radial cross-ratios $ρ, \bar{ρ}$. We prove a key fact that $|ρ|, |\bar{ρ}| < 1$ inside the forward tube, and set bounds on how fast $|ρ|, |\bar{ρ}|$ may tend to 1 when approaching the Minkowski space.

We also provide a guide to the axiomatic QFT literature for the modern CFT audience. We review the Wightman and Osterwalder-Schrader (OS) axioms for Lorentzian and Euclidean QFTs, and the celebrated OS theorem connecting them. We also review a classic result of Mack about the distributional OPE convergence. Some of the classic arguments turn out useful in our setup. Others fall short of our needs due to Lorentzian assumptions (Mack) or unverifiable Euclidean assumptions (OS theorem).

**KEYWORDS:** Conformal and W Symmetry, Conformal Field Theory, Field Theories in Higher Dimensions

ARXIV ePRINT: 2104.02090

I wanted to learn about elementary particles
by studying boiling water.
Alexander Polyakov [1]

<footnote>ªInstitute for Advanced Study,
Princeton, New Jersey 08540, U.S.A.</footnote>

<footnote>ᵇLaboratoire de Physique de l'Ecole normale supérieure, ENS,
Université PSL, CNRS, Sorbonne Université, Université de Paris,
F-75005 Paris, France</footnote>

<footnote>ᶜInstitut des Hautes Études Scientifiques,
91440 Bures-sur-Yvette, France</footnote>
# Contents

1 Introduction 1

1.1 Executive summary of results for CFT experts 4

2 Axioms 7

2.1 Wightman axioms 7

2.2 Osterwalder-Schrader axioms 10

2.3 Euclidean CFT axioms 14

3 Euclidean CFT ⇒ Osterwalder-Schrader 18

3.1 OS reflection positivity 19

3.2 Denseness and Hilbert space implications 22

3.3 OS clustering 23

4 Euclidean CFT ⇒ Wightman: basic strategy 24

4.1 Recovering Minkowski averages from Euclidean averages 28

5 Two- and 3-point functions 31

5.1 Comparison with the $i\epsilon$-prescription 33

6 Scalar 4-point function 35

6.1 Informal introduction to basic issues 35

6.2 Euclidean 4-point function in radial coordinates 37

6.3 Analytic continuation 39

6.4 Proof of $z, \bar{z} \notin [1, +\infty)$ 43

6.5 4-point function powerlaw bound 45

6.5.1 A powerlaw bound on $\frac{1}{1-r(c)}$ 46

6.6 Conformal invariance 50

6.6.1 Conformal invariance in terms of cross ratios 52

6.6.2 Fixing points 56

6.7 Wightman positivity 57

6.7.1 Wightman states 58

6.7.2 OS states for complexified times 59

6.8 Wightman clustering 60

6.8.1 2 + 2 split 60

6.8.2 3 + 1 split 60

6.9 Local commutativity 62

6.9.1 Local commutativity for CFT 4-point functions 65

6.10 Generalization to non-identical scalars 66
<table>
  <tbody>
    <tr>
      <td>7</td>
      <td>Optimal powerlaw bound from Cauchy-Schwarz ρ, ρ̄ inequality</td>
      <td>68</td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>Proof of lemma 7.1</td>
      <td>68</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>Optimal bound for ρ, ρ̄</td>
      <td>69</td>
    </tr>
    <tr>
      <td>7.3</td>
      <td>Proof of theorem 7.2</td>
      <td>70</td>
    </tr>
    <tr>
      <td>8</td>
      <td>OPE convergence in the forward tube and in Minkowski space</td>
      <td>73</td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>Convergence of conformal block decomposition for 4-point functions</td>
      <td>73</td>
    </tr>
    <tr>
      <td>8.1.1</td>
      <td>Convergence rate for compactly supported test functions</td>
      <td>75</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>OPE convergence in the sense of H<sup>CFT</sup></td>
      <td>77</td>
    </tr>
    <tr>
      <td>8.3</td>
      <td>Comparison to Mack's work on OPE convergence</td>
      <td>80</td>
    </tr>
    <tr>
      <td>8.3.1</td>
      <td>Relating Mack's kernel B to the Euclidean OPE kernel C</td>
      <td>84</td>
    </tr>
    <tr>
      <td>9</td>
      <td>Review of Osterwalder-Schrader theorem</td>
      <td>86</td>
    </tr>
    <tr>
      <td>9.1</td>
      <td>The argument of [2]</td>
      <td>87</td>
    </tr>
    <tr>
      <td>9.2</td>
      <td>The argument of [3]</td>
      <td>89</td>
    </tr>
    <tr>
      <td>9.2.1</td>
      <td>Fixing the equivalence theorem</td>
      <td>89</td>
    </tr>
    <tr>
      <td>9.2.2</td>
      <td>Wightman axioms from linear growth condition</td>
      <td>90</td>
    </tr>
    <tr>
      <td>9.2.3</td>
      <td>Analytic continuation</td>
      <td>92</td>
    </tr>
    <tr>
      <td>9.2.4</td>
      <td>Temperedness bound</td>
      <td>98</td>
    </tr>
    <tr>
      <td>10</td>
      <td>Conclusions</td>
      <td>100</td>
    </tr>
    <tr>
      <td>A</td>
      <td>Lorentzian CFT literature</td>
      <td>103</td>
    </tr>
    <tr>
      <td>A.1</td>
      <td>Review of Hartman et al. [4]</td>
      <td>107</td>
    </tr>
    <tr>
      <td>B</td>
      <td>OS axioms for higher-point functions</td>
      <td>112</td>
    </tr>
    <tr>
      <td>C</td>
      <td>Details on Vladimirov's theorem</td>
      <td>115</td>
    </tr>
    <tr>
      <td>C.1</td>
      <td>Limit in the sense of distributions</td>
      <td>115</td>
    </tr>
    <tr>
      <td>C.2</td>
      <td>Proof of theorem 4.1</td>
      <td>116</td>
    </tr>
    <tr>
      <td>D</td>
      <td>Intuition about lemma 4.2</td>
      <td>120</td>
    </tr>
  </tbody>
</table>

# 1 Introduction

Quantum Field Theory (QFT) can be studied via constructive or axiomatic approaches. The former use microscopic formulations, while the latter rely on axioms. There are many constructive approaches, e.g. using Hamiltonian, path integral, lattice, etc. There are also many axiomatic approaches, corresponding to various sets of axioms (Wightman [5], Osterwalder-Schrader [2, 3], Haag-Kastler [6, 7], etc.). Historically, axiomatic approaches played an important role in clarifying general QFT properties, but they did not have a
tremendous success in making predictions about concrete theories in $d > 2$ dimensions.<fnref n="1" />
This started to change recently, with the revival of the bootstrap philosophy [10]. Our
focus here will be on conformal field theories (CFTs) in dimension $d \ge 2$, i.e. QFTs invari-
ant under the action of conformal group, which are nowadays studied via the conformal
bootstrap. This axiomatic approach led to precise determinations of many experimentally
measurable quantities, such as the critical exponents of the 3d Ising [11–15], $O(N)$ [15–19]
and other critical points (see review [20]).<fnref n="2" />

The numerical conformal bootstrap relies on the “Euclidean CFT axioms”,<fnref n="3" /> which
specify properties of correlation functions in any unitary CFT in $\mathbb{R}^d$ via a set of simple and
commonly accepted rules, such as the unitarity bounds on primary operator dimensions,
conformally invariant and convergent Operator Product Expansion (OPE), and reality
constraints on OPE coefficients.

On the other hand, correlation functions in a general unitary QFT (and in particular
in a CFT) should satisfy Osterwalder-Schrader (OS) and Wightman axioms. It is then
interesting and important to inquire what is the relation of Euclidean CFT axioms to these
other sets of axioms.<fnref n="4" /> To carry out this analysis will be the main goal of our paper. Our
main conclusion will be that the Euclidean CFT axioms imply OS axioms and Wightman
axioms for 2, 3 and 4-point functions. In this paper we only consider bosonic operators.

The relation of Euclidean CFT and OS axioms is perhaps not so surprising since they
both deal with the Euclidean correlation functions. It is more interesting that we are
able to construct Minkowski $n$-point functions (for $n = 2, 3, 4$), and show that they satisfy
Wightman axioms, such as temperedness, spectral condition, and unitarity. Temperedness
(being a tempered distribution) is a crucial property of Minkowski correlation functions:
it shows that in a certain averaged sense they are meaningful everywhere including the
light-cone and double light-cone singularities. One might be tempted to think that in CFT
this question is relatively trivial due to OPE. However, as discussed in [36, 37], already
for 4-point functions there exist causal configurations of points in Minkowski space, away
from the null cones, for which no OPE channel is convergent in the conventional sense. We
briefly discuss one such example in the conclusions (section 10).

A theorem of Osterwalder and Schrader [2, 3] says that, under some extra assumption,
OS axioms imply Wightman axioms. Unfortunately this extra assumption, the so called
“linear growth condition”, which involves the Euclidean $n$-point functions with arbitrarily
high $n$, appears impossible to verify from the Euclidean CFT axioms (see section 9). For
this reason we cannot appeal to the OS theorem, and we will give an independent derivation
of Wightman axioms for CFT correlators.

The study of distributional properties of CFT correlators started in our recent
work [36]. There, we considered expansions of the CFT 4-point function $g(\rho, \bar{\rho})$ in terms of

<footnote n="1">¹In *d* = 2 significant progress has been achieved axiomatically for massive integrable models using the S-matrix bootstrap [8] as well as for rational CFTs [9].</footnote>

<footnote n="2">²There is also an ongoing revival of the S-matrix bootstrap applicable to nonintegrable massive QFTs in $d \ge 2$ [21–35].</footnote>

<footnote n="3">³The term “Euclidean bootstrap axioms” is also sometimes used.</footnote>

<footnote n="4">⁴Clarifying the relation to the Haag-Kastler axioms appears more challenging as those axioms do not deal with correlation functions but with operator algebras.</footnote>
conformally invariant cross-ratios $\rho, \bar{\rho}$. While such expansions converge in the usual sense for $|\rho|, |\bar{\rho}| < 1$, in [36] we showed that they also converge for $|\rho|, |\bar{\rho}| = 1$ in the sense of distributions. As explained in [36], results of this sort follow naturally from the theory of functions of several complex variables (namely Vladimirov's theorem), given some apriori information about the growth of the analytically continued correlator. That key insight of [36], "Look for a powerlaw bound!", will be transported here from the cross-ratio to the position space.

The readers interested in our main technical result — analytic continuation of a scalar Euclidean CFT 4-point function to the forward tube and showing that the Minkowski 4-point function is a tempered distribution — may follow the **fast track**: start with the executive summary in section 1.1, proceed to sections 4 and 5 (skipping 4.1 and 5.1), then continue with sections 6.1–6.5 (optionally including section 7) and finish with the discussion in section 10. This is only about 20-25 pages.

On the other hand, we made an effort to make the exposition self-contained and to review main ideas and results of the axiomatic quantum field literature, directly or tangentially related to our discussion. This explains the great total length of our work. The reader will find here:

* A review of classic QFT axioms: Wightman (section 2.1), OS (section 2.2). A review of main implications among these axioms: how OS reflection positivity robustly implies Wightman positivity (section 6.7). A review of the Osterwalder-Schrader theorem about how OS axioms imply Wightman axioms under the additional assumptions of the linear growth condition (section 9).

* A formulation of ‘Euclidean CFT axioms’ for unitary CFT in Euclidean space $\mathbb{R}^d$ (section 2.3). We consider bosonic fields in arbitrary tensor representations. Our axioms encode in a consistent and non-redundant manner the main properties used in the numerical conformal bootstrap calculations.<fnref n="5" /> They are applicable to any globally conformally invariant theory in $d \ge 2$. We do not include the axioms involving the local stress tensor and the conserved currents. In particular our axioms would be too weak (but valid) when applied to local 2d CFTs, as they know nothing about the Virasoro algebra.<fnref n="6" /> See remark 2.6 for a comparison between our axioms and the CFT rules gathered in the conformal bootstrap review [20].

* A derivation of OS axioms from Euclidean CFT axioms for 4-point function (section 3). A notable result is a rigorous proof that the state produced by two operators in lower Euclidean half-space belongs to the CFT Hilbert space generated by single-point operator insertions. The higher-point case is discussed in appendix B, where we need a somewhat stronger form of the OPE axiom than in section 2.3.

<footnote n="5">⁵See also [38] for a recent informal exposition of Euclidean CFT axioms (incomplete as it omits tensor fields) for mathematical physics audience. Ref. [39] attempted the axiomatization of Euclidean CFT in $d > 2$ dimensions similar to Segal’s axioms in $d = 2$ [40]. It is not immediately obvious if the axioms of ref. [39] are equivalent to ours, or how to connect them to practical CFT calculations.</footnote>

<footnote n="6">⁶Recall that while in $d = 2$ assuming the existence of a local stress tensor immediately implies Virasoro symmetry, no such dramatic statements are currently known in $d > 2$.</footnote>
* A derivation of Wightman axioms from Euclidean CFT axioms for scalar 4-point functions (section 6). As mentioned above, this is the main technical result of the paper. The key observation is that the analytic continuation from Euclidean to Minkowski can be done in a way which keeps the s-channel $ρ, \bar{ρ}$ less than 1 in absolute value along the continuation contour. When we take the Minkowski limit $|ρ|, |\bar{ρ}|$ stay less than 1 for some causal orderings and approaches 1 for others (see [37] for a classification) but even if $|ρ|, |\bar{ρ}| \to 1$ they approach this limit sufficiently slowly (“powerlaw bound”) which guarantees that the Minkowski 4-point function is a tempered distribution everywhere. E.g. using this argument we can show for the first time that the CFT 4-point function is a tempered distribution on the double light-cone singularity.

* We include also a derivation of other expected properties of Minkowski 4-point functions, such as conformal invariance, unitarity, clustering, and local commutativity (sections 6.6–6.9). The reader may find it curious how some of the steps do not use conformal invariance at all but follow simply from analyticity and/or OS positivity.

* Section 7 proves a curious geometric “Cauchy-Schwarz” inequality for $ρ, \bar{ρ}$ variables which provides an alternative way of understanding why $|ρ|, |\bar{ρ}| < 1$ in the forward tube. It bounds $ρ, \bar{ρ}$ for a generic configuration by $ρ, \bar{ρ}$ for reflection-symmetric configurations. It would be interesting to find an elementary proof of this inequality (our proof uses some facts about conformal blocks).

* Section 8 shows that the (s-channel) conformal block expansion of 4-point Wightman functions converges in the sense of distributions for all configurations of points in Minkowski space. It is also shown that the OPE for the state-valued distributions $|O(x_1)O(x_2)\rangle$ with $x_1, x_2 \in \mathbb{R}^{d-1,1}$ converges in the sense of distributions. We discuss the relationship of these results to the classic work of Mack [41] and prove estimates for the convergence rates of these expansions.

* Section 9 contains a review of the papers [2, 3] by Osterwalder and Schrader. In particular, we discuss the gap in the arguments of [2] which precludes the derivation of Wightman axioms from the OS axioms of [2], and explain in detail how this gap is filled in [3] with the addition of new axioms.

* Appendix A is a guide to the modern Lorentzian CFT literature: conformal collider bounds, light-cone bootstrap, causality constraints, the Lorentzian OPE inversion formula, light-ray operators, etc. Our results help put some of these considerations on a firmer footing. We indicate the most critical remaining steps, which still wait to be rigorously derived from the Euclidean CFT axioms.

We conclude in section 10. Some additional technical details are given in appendices B)–(D).

## 1.1 Executive summary of results for CFT experts

This paper is rather lengthy as a result of our attempt to make it self-contained. In this section we give a brief summary of the main technical results, aimed at the more expert
readers who may not wish to read the expository parts of this paper. Note, however, that here we omit many secondary results, some of which are mentioned above.

The basic question we address in this paper is the question of the distributional properties of Wightman 4-point functions in CFTs. As is well-known, Wightman n-point distributions are recovered from the boundary values of functions holomorphic in the forward tube $\mathcal{T}_n$. For an n-point function

$$ \langle 0 | \mathcal{O}_1(x_1) \cdots \mathcal{O}_n(x_n) | 0 \rangle \quad (1.1) $$

the forward tube is defined as the set of $x_i \in \mathbb{C}^{1,d-1}$ subject to

$$ \operatorname{Im} x_1 < \operatorname{Im} x_2 < \cdots < \operatorname{Im} x_n, \quad (1.2) $$

where $<$ denotes the causal ordering in $\mathbb{R}^{1,d-1}$. Analyticity in $\mathcal{T}_n$ and existence of the boundary value as $\operatorname{Im} x_i \to 0$ is usually derived from Wightman or OS axioms (with extra assumptions in the latter case). In this paper we want to understand this question from the point of view of CFT axioms.

With the cases $n = 2, 3$ being relatively trivial in a CFT, our main observation is that a particular OPE channel for 4-point functions converges everywhere in the interior of $\mathcal{T}_4$. Specifically, we take the OPE $\mathcal{O}(x_1) \times \mathcal{O}(x_2)$ in the Wightman function of identical scalar operators

$$ \langle 0 | \mathcal{O}(x_1) \mathcal{O}(x_2) \mathcal{O}(x_3) \mathcal{O}(x_4) | 0 \rangle. \quad (1.3) $$

This OPE is expected to converge, at least distributionally for real $x_i$, from the results of Mack [41]. However, his work assumes Wightman axioms from the beginning, and our goal here is to clarify the implications of Euclidean CFT axioms, which only assume convergence of the Euclidean OPE.

To see that this OPE channel converges, we show in lemma 6.1 that for any configuration of $x_i$ in $\mathcal{T}_4$ the radial cross-ratios $\rho$ and $\bar{\rho}$ for this OPE belong to the open unit disk (for the definition of radial cross-ratios, see section 6.2):

$$ |\rho|, |\bar{\rho}| < 1. \quad (1.4) $$

This implies convergence of the conformal block expansion in $\mathcal{O}(x_1) \times \mathcal{O}(x_2)$ channel in the interior of $\mathcal{T}_4$. A technical way to see this is to note that the expansion in descendants

$$ g(\rho, \bar{\rho}) = \sum_{h, \bar{h} > 0} p_{h, \bar{h}} \rho^h \bar{\rho}^{\bar{h}}, \quad (1.5) $$

where $g$ is the conformally-invariant part of the 4-point function and $p_{h,\bar{h}} > 0$, can be bounded term-by-term by

$$ |p_{h,\bar{h}} \rho^h \bar{\rho}^{\bar{h}}| \le p_{h,\bar{h}} r^h r^{\bar{h}}, \quad (1.6) $$

where $r = \max(|\rho|, |\bar{\rho}|) < 1$. The right-hand side of this inequality is a term in the expansion of $g(r, r)$, a Euclidean configuration in which the OPE is known to converge, so (1.5) is dominated by a convergent series. Therefore, (1.5) is convergent for $r < 1$, and moreover uniformly so on compact subsets, since each term $p_{h,\bar{h}} r^h r^{\bar{h}}$ is monotonic. We can then conclude that the sum $g(\rho, \bar{\rho})$ is a holomorphic function.
This reasoning also gives us the inequality

$$|g(\rho, \bar{\rho})| \le g(r, r). \qquad (1.7)$$

So, we find that the correlator can be recovered inside of $\mathcal{T}_4$ from the $\mathcal{O}(x_1) \times \mathcal{O}(x_2)$ OPE, is analytic there, and is bounded by a Euclidean configuration.

In section 6.5.1 we establish a stronger form of lemma 6.1, schematically,

$$1 - r(c) \ge \mathrm{dist}(c, \partial \mathcal{T}_4)^a \qquad (1.8)$$

for some $a > 0$, where $c \in \mathcal{T}_4$ is a configuration of 4 points in $\mathcal{T}_4$. (The more precise form also bounds $1 - r(c)$ as $c$ goes to infinity.) This immediately implies a powerlaw bound on $g(r, r)$ near the boundary of $\mathcal{T}_4$. Indeed, near $r \to 1$ the correlator is dominated by the identity operator in the crossed channel, and so

$$g(r(c), r(c)) \le C(1 - r(c))^{-4\Delta_{\varphi}}, \qquad (1.9)$$

and thus

$$|g(c)| \le C \mathrm{dist}(c, \partial \mathcal{T}_4)^b \qquad (1.10)$$

for some real $b$. This allows us to use Vladimirov's theorem 4.1, which implies that the boundary limit (as $x_i$ approach real Minkowski values) of (1.3) exists in the space of tempered distribution. (We establish a more refined bound for $x_i \to \infty$ to claim temperedness.)

The above bounds hold just as well for the truncated conformal block expansion as for the full correlator. A variant of theorem 4.1 then allows us to conclude that the conformal block expansion, while converging in the sense of functions in the interior of $\mathcal{T}_4$, converges in the space of tempered distributions on the Minkowski boundary.

We extend the above results to correlators of non-identical scalars by replacing the term-by-term bound (1.6) with a standard Cauchy-Schwarz argument, bounding the correlator in terms of a product of two reflection-symmetric Euclidean correlators. While it is intuitively obvious that similar arguments should also work for operators with spin, we found that the extension to spinning operators, due to the complexity of tensor structures, requires enough additional work to warrant a separate paper [42].

Finally, in section 7 we prove theorem 7.2, which gives an optimal bound of the form (1.8). Specifically, it is

$$r(c)^2 \le r(c_{12})r(c_{34}), \qquad (1.11)$$

together with a bound for the right-hand side. Here, if $c = (x_1, x_2, x_3, x_4)$ (where $x_i$ are real in Minkowski space), then $c_{12} = (x_1, x_2, x_2^*, x_1^*)$ and $c_{34} = (x_4^*, x_3^*, x_3, x_4)$. The bound for the right-hand side is easier to obtain because the configurations $c_{12}$ and $c_{34}$ are reflection symmetric. This is done in section 7.2. The bound (1.11) looks like a Cauchy-Schwarz-type inequality, and is indeed derived from the Cauchy-Schwarz inequality for unitary conformal blocks (7.21). The latter is true because of the unitarity of conformal representations corresponding to these blocks. In the limit $\Delta + \ell \to \infty$, $\Delta - \ell$ fixed, conformal blocks are dominated by $r^{(\Delta+\ell)/2}$, which reduces the conformal block Cauchy-Schwarz inequality to (1.11).
# 2 Axioms

## 2.1 Wightman axioms

In this section we will state the properties of Wightman correlation functions in a unitary QFT, to which we will refer here as “Wightman axioms.” These axioms appear as “properties of the vacuum expectation values” in [5], section 3-3, and as (W1)–(W5) in [43], theorem II.1. Refs. [5, 43] give in addition another set of axioms (called Gårding-Wightman axioms in [43]) saying that fields are operator-valued distributions in the Hilbert space on which the Lorentz group acts, etc. This other set of axioms will not be used in this work. In any case, the Wightman reconstruction theorem [5] says that the two sets of axioms are equivalent.

A unitary QFT in Minkowski space studies n-point correlators

$$ \langle \varphi_1(x_1) \dots \varphi_n(x_n) \rangle, \qquad (2.1) $$

(Wightman functions) of local operators $\varphi_i(x)$, $x \in \mathbb{R}^{1,d-1}$.$^7$ For simplicity in this paper we will only consider bosonic operators, although more generally one should allow fermionic operators and spinor representations. Wightman functions are translation and $SO(1, d-1)$ invariant.$^8$ We will choose a basis of local operators $O_i$ transforming in irreducible $SO(1, d-1)$ representations $\rho_i$. Then, Wightman functions remain invariant when

$$ O_i^\alpha(x) \rightarrow \rho_i(g)^\alpha \partial_\beta O_i^\beta(g^{-1}x), \qquad (2.2) $$

where $g \in SO(1, d-1)$, and $\rho_i(g)$ are finite-dimensional matrices of the representation $\rho_i(\alpha, \beta = 1 \dots \dim \rho_i)$. Let $C$ be the complex vector space whose elements are arbitrary components of $O_i$'s, and their finite linear combinations with constant complex coefficients. Operators $\varphi_i$ in (2.1) can be arbitrary elements of $C$, and Wightman function (2.1) is multi-linear in $\varphi_i$. Note that in this and the next section derivatives of local operators (of any order) are counted as independent operators, while in the CFT section 2.3 we will start making distinction between primaries and their derivatives.

Wightman functions (2.1) are required to be tempered distributions, i.e. can be paired with Schwartz class test functions $f(x_1, \dots, x_n)$. For this reason they are sometimes referred to as “Wightman distributions”. Note that the test functions $f(x_1, \dots, x_n)$ with which Wightman functions are paired do not have to vanish at coincident points (unlike for the Schwinger functions discussion in the next section). This means that, in a distributional sense, Wightman functions have meaning for all configurations, including coincident points and light-cone singularities. Translation and Lorentz invariance of Wightman functions are

<footnote>$^7$In this paper, we use the term “operator” according to CFT terminology, where the term “field” is used in Wightman and Osterwalder-Schrader terminology. For example, in refs. [5, 43], the term “operator” is reserved for the actual unbounded Hilbert space operator $\phi(f)$ defined on a dense domain of the Hilbert space (which justifies the terminology “operator-valued distribution” $\phi$ in the Gårding-Wightman formulation).</footnote>

<footnote>$^8$Here and everywhere in this paper, $SO(1, d-1)$ means the connected component of the group. E.g. $\mathrm{diag}(-1, -1, 1, \dots, 1)$ is not included. Of course, it is possible for the theory to also possess discrete space-time symmetries such as time inversion or space reflection. The consequences of such symmetries are straightforward to figure out and are not discussed in this paper.</footnote>
also understood not pointwise but in the sense of distributions (i.e. that the pairing should remain invariant if the test function is transformed in the dual way).<fnref n="9" />

We will not consider here other Minkowski correlators, such as retarded, advanced, or time ordered, which are obtained from Wightman functions multiplying by theta-functions of time coordinate differences, and whose distributional properties require a separate discussion.

Limiting to the bosonic case as we are, local commutativity (also called microcausality) morally says that operators commute at spacelike separation. Wightman axioms impose this as a constraint on Wightman functions:

$$ \langle \varphi_1(x_1) \dots \varphi_p(x_p) \varphi_{p+1}(x_{p+1}) \dots \varphi_n(x_n) \rangle = \langle \varphi_1(x_1) \dots \varphi_{p+1}(x_{p+1}) \varphi_p(x_p) \dots \varphi_n(x_n) \rangle \quad (2.3) $$

whenever $x_p - x_{p+1}$ is spacelike: $(x_p - x_{p+1})^2 > 0$ (in this paper we use the $- + \dots +$ convention). Since we are talking about distributions, this constraint means that (2.3) holds when paired with any test function whose support is contained in $(x_p - x_{p+1})^2 > 0$.

Clustering says that correlators should factorize if two groups of points are far separated in a spacelike direction:

$$ \langle \varphi_1(x_1) \dots \varphi_p(x_p) \varphi_{p+1}(x_{p+1} + \lambda a) \dots \varphi_n(x_n + \lambda a) \rangle \\ \rightarrow \langle \varphi_1(x_1) \dots \varphi_p(x_p) \rangle \langle \varphi_{p+1}(x_{p+1}) \dots \varphi_n(x_n) \rangle \quad (2.4) $$

as $\lambda \to \infty$ for any spacelike vector $a$, limit understood in the sense of distributions.

We next discuss the spectral condition. By translation invariance we can write

$$ \langle \varphi_1(x_1) \dots \varphi_n(x_n) \rangle = W(\xi_1, \dots, \xi_{n-1}), \quad \xi_k = x_k - x_{k+1}, \quad (2.5) $$

where $W$ is a tempered distribution in one less variable. Consider its Fourier transform:

$$ \widehat{W}(q_1, \dots, q_{n-1}) = \int W(\xi_1, \dots, \xi_{n-1}) e^{i \sum_{k=1}^{n-1} q_k \cdot \xi_k} d\xi_1 \dots d\xi_{n-1}, \quad (2.6) $$

where $q_k = (E_k, q_k)$, $\xi_k = (t_k, \xi_k)$, $q_k \cdot \xi_k = -E_k t_k + q_k \cdot \xi_k$. Since $W$ is a tempered distribution, the Fourier transform $\widehat{W}$ is well defined and is also a tempered distribution. The spectral condition then says that $\widehat{W}$ must be supported in the product of closed forward light cones, i.e. in the region

$$ E_k \ge \mathbf{q}_k, \quad k = 1, 2, \dots, n-1. \quad (2.7) $$

For the two remaining conditions we need to discuss conjugation. Physically, each operator $\varphi$ should have a conjugate $\varphi^\dagger$. In the discussed framework we cannot define $\varphi^\dagger$ as an adjoint of an operator acting on a Hilbert space, since we do not have a Hilbert space. Instead, we will simply assume that there is a rule which associates $\varphi^\dagger$ to $\varphi$, and impose the expected relations at the level of correlation functions (eq. (2.10) below). This rule, conjugation

<footnote n="9">⁹Although Wightman functions can be shown to be real-analytic at some totally spacelike-separated configurations (Jost points), in general they may be singular even away from light cones (in particular when there are timelike separations).</footnote>
map $\dagger : \mathcal{C} \to \mathcal{C}$, associates to each independent component $\mathcal{O}_i^\alpha(\alpha = 1...\dim \rho_i)$ of the above-mentioned basis of $\mathcal{C}$ a conjugate operator $(\mathcal{O}_i^\alpha)^\dagger$. This map is required to be an involution, i.e. $\dagger\dagger = 1$. Furthermore, it is extended to the whole of $\mathcal{C}$ by anti-linearity, i.e. $(c_1\varphi_1 + c_2\varphi_2)^\dagger = c_1^*\varphi_1^\dagger + c_2^*\varphi_2^\dagger$.<fnref n="10" />

Let us group operators $(\mathcal{O}_i^\alpha)^\dagger$ in a multiplet which we denote by $\mathcal{O}_i^\dagger$, i.e. $(\mathcal{O}_i^\dagger)^\alpha = (\mathcal{O}_i^\alpha)^\dagger$. We will see below that $(\mathcal{O}_i^\alpha)^\alpha$ transform under $g \in \text{SO}(1, d-1)$ with matrices complex-conjugate to those of $\mathcal{O}_i^\alpha$:

$$ \mathcal{O}_i^\alpha \to \rho_i(g)^{\alpha}_{\beta} \mathcal{O}_i^\beta \quad \Rightarrow \quad (\mathcal{O}_i^\dagger)^\alpha \to \overline{\rho_i(g)^{\alpha}_{\beta}} (\mathcal{O}_i^\dagger)^\beta. \qquad (2.8) $$

In other words, $\mathcal{O}_i^\dagger$ transforms in the conjugate representation $\overline{\rho_i}$.

Since we are considering only bosonic operators, the relevant representations $\rho_i$ are tensors $T^{\mu_1...\mu_l}$, on which $g \in \text{SO}(1, d-1)$ act as:

$$ T^{\mu_1...\mu_l} \to (\rho_i(g)T)^{\mu_1...\mu_l} = g^{\mu_1}_{\nu_1} \dots g^{\mu_l}_{\nu_l} T^{\nu_1...\nu_l}. \qquad (2.9) $$

Depending on $\rho_i$, these tensors have some fixed rank and mixed symmetry properties. In addition, in even $d$, for tensors with $d/2$ antisymmetric indices, (anti-)chirality<fnref n="11" /> constraints must be imposed. All tensor representations of $\text{SO}(1, d-1)$ are real (i.e. matrices $\rho_i(g)^{\alpha}_{\beta}$ in (2.8) can be chosen real), except for (anti-)chiral representations in $d=0 \mod 4$ which are complex-conjugate to each other. For operators in real representations we can choose a basis such that $\mathcal{O}_i = \mathcal{O}_i^\dagger$.

After this intermezzo we are ready to formulate hermiticity and positivity conditions. Hermiticity says that complex conjugate correlators equal correlators of conjugated operators in inverted order:

$$ \overline{\langle\varphi_1(x_1) \dots \varphi_n(x_n)\rangle} = \langle\varphi_n^\dagger(x_n) \dots \varphi_1^\dagger(x_1)\rangle. \qquad (2.10) $$

This would be true of course if $\varphi$'s were operators acting on a Hilbert space, with $\varphi^\dagger$'s their adjoints. In the present framework without Hilbert space it is imposed as an axiom. This axiom implies in particular (2.8), i.e. that $\mathcal{O}_i^\dagger$ transforms in the conjugate irrep $\overline{\rho_i}$.

The last Wightman axiom, positivity, is most conveniently written down using the language of states. One considers basic ket states $|\psi(f, \varphi_1, \dots, \varphi_n)\rangle$, associated with $n$ local operators $\varphi_1, \dots, \varphi_n \in V$ and a complex Schwartz test function of $n$ variables $f$. One defines the inner product on basic ket states by

$$ \langle\psi(g, \chi_1, \dots, \chi_m)|\psi(f, \varphi_1, \dots, \varphi_n)\rangle := \int dx dy \overline{g(x_1, \dots, x_m)} f(y_1, \dots, y_n) \qquad (2.11) \\ \times \langle\chi_m^\dagger(x_m) \dots \chi_1^\dagger(x_1) \varphi_1(y_1) \dots \varphi_n(y_n)\rangle. $$

The vector space of ket states $\mathcal{H}_0$ consists of finite linear combinations $|\Psi\rangle$ of basic ket states, with the inner product extended to it by (anti)linearity. Positivity then says that the so defined inner product is positive semidefinite:

$$ \langle\Psi|\Psi\rangle \ge 0 \quad \forall |\Psi\rangle \in \mathcal{H}_0. \qquad (2.12) $$

<footnote n="10">¹⁰The $\dagger$ operation is denoted by $*$ in [5].</footnote>

<footnote n="11">¹¹Chiral and anti-chiral representations are sometimes also called “self-dual” and “anti-self-dual”. We use “chiral” and “anti-chiral” to avoid the clash with “dual representation” in mathematician’s sense.</footnote>
**Remark 2.1.** A comment is in order concerning the meaning of these states. They may be seen as just a convenient notation, since eq. (2.12) can be rewritten without ever using the word “state” (see [5], eq. (3-35)). But they are more than that: the vector space of states $\mathcal{H}_0$ is “almost” the Hilbert space $\mathcal{H}$ of our QFT. The only difference between $\mathcal{H}_0$ and $\mathcal{H}$ is that $\mathcal{H}_0$ is not complete and may contain some states of zero norm. However, since $\mathcal{H}_0$ has a positive semidefinite inner product, as expressed by eq. (2.12), we can obtain from it a Hilbert space $\mathcal{H}$ via a standard procedure of completion and modding out by states of zero norm. This is the first step of the Wightman reconstruction theorem [5], and the resulting Hilbert space $\mathcal{H}$ turns out to be (possibly a superselection sector of) the Hilbert space of the QFT, on which fields can then be realized as operator valued distributions.

**Remark 2.2.** Although we included hermiticity as a separate axiom because of its suggestive form, it can be derived from positivity, considering the states of the form $|\Psi\rangle = |\psi(f_0, 1)\rangle + |\psi(f, \varphi_1, \dots, \varphi_n)\rangle$ where $f_0 \in \mathbb{C}$ and 1 is the unit operator.

**Remark 2.3.** Another interesting positivity property of Wightman functions is called Rindler Reflection positivity, or Wedge Reflection positivity [44]. A restricted version of this property (with wedge-ordered points) can be derived from Wightman axioms, while a stronger version (no wedge-ordering) follows from Tomita-Takesaki theory which relies on Haag-Kastler axioms [44]. In CFT context this property has been discussed, e.g., in [45]. We will not discuss these properties in this paper. However, it would be interesting to see whether the stronger form of Rindler positivity (including distributional information) can be derived from CFT axioms without the appeal to Tomita-Takesaki theory (the weaker version following from our results on Wightman axioms and [44]). We believe this can be done, and it could be a nice exercise for someone wishing to master our techniques.

## 2.2 Osterwalder-Schrader axioms

We next describe a version of the Osterwalder and Schrader axioms [2, 3] of Euclidean unitary QFT (see the end of the section about the relation to the original OS axioms). The setup is similar to Wightman axioms with $SO(d)$ replacing $SO(1, d-1)$. We consider a basis of local bosonic operators $O_i$ transforming<fnref n="12" /> in $SO(d)$ irreps $\rho_i$, counting derivatives as independent operators. Finite linear combinations of their components span a complex vector space $\mathcal{C}$ of local operators. The axioms specify properties of translation and $SO(d)$ invariant $n$-point correlators (often called Schwinger functions)

$$ \langle \varphi_1(x_1) \dots \varphi_n(x_n) \rangle, \quad \varphi_i \in \mathcal{C}, \quad x_i \in \mathbb{R}^d. \qquad (2.13) $$

These correlators are defined away from coincident points (i.e. whenever $x_i \neq x_j$ for each $i, j$). We will assume that<fnref n="13" />

$$ \text{correlators are real-analytic,} \qquad (2.14) $$

<footnote n="12">¹²In the sense of eq. (2.2) where now $g \in SO(d)$.</footnote>

<footnote n="13">¹³Recall that a $C^\infty$ function of $m$ real variables is called real-analytic in a domain $D \subset \mathbb{R}^m$ if it has a convergent Taylor series expansion in a small ball around every point of this domain. Equivalently, such a function has an analytic extension to a small open neighborhood of this domain inside $\mathbb{C}^m$.</footnote>
and grow not faster than some power when some points approach each other or go to infinity, i.e.

$$|\langle\varphi_1(x_1) \dots \varphi_n(x_n)\rangle| \le C \left(1 + \max_{i \ne j} \left(\frac{1}{|x_i - x_j|}, |x_i - x_j|\right)\right)^p \quad (2.15)$$

with some correlator-dependent positive constants $C, p$. Unlike Wightman axioms, OS axioms do not bother what happens *precisely* at coincident points (not even in the sense of distributions).

As we are limiting to the bosonic case, correlators remain invariant when operators are permuted:<fnref n="14" />

$$\langle\varphi_1(x_1) \dots \varphi_n(x_n)\rangle = \langle\varphi_{\pi(1)}(x_{\pi(1)}) \dots \varphi_{\pi(n)}(x_{\pi(n)})\rangle. \quad (2.16)$$

To formulate the Euclidean version of hermiticity and positivity, we will need some simple facts about SO(d) representations. Abstractly, for any irrep $\rho$ acting $T^\alpha \to \rho(g)^{\alpha_\beta} T^\beta$, the conjugate representation $\bar{\rho}$ acts with complex conjugate matrices $\overline{\rho(g)^{\alpha_\beta}}$. Since SO(d) is compact, we have $\bar{\rho} \simeq \rho^*$, the dual representation. The SO(d) irreps $\rho$ are again tensors $T^{\mu_1 \dots \mu_l}$ like in (2.9), of in general mixed symmetry, and with (anti)-chirality constraints if having $d/2$ antisymmetric indices in even $d$. All of them are real, except for chiral representations in $d=2 \mod 4$ which are complex-conjugate to the anti-chiral ones.<fnref n="15" />

We will also need the reflected representation $\rho^R$ with matrices $\rho^R(g) = \rho(g^R)$, where $g \to g^R = \Theta g \Theta$, $\Theta = \text{diag}(-1, 1, \dots, 1)$, is an automorphism of SO(d). For tensor representations, we can consider the map

$$T^{\mu_1 \dots \mu_l} \to \Theta^{\mu_1}_{\nu_1} \dots \Theta^{\mu_l}_{\nu_l} T^{\nu_1 \dots \nu_l}, \quad (2.17)$$

which preserves rank and mixed symmetry properties. It also maps chiral to anti-chiral tensors in any even $d$. Whenever the representation space is preserved, this map serves as an intertwiner between $\rho^R$ and $\rho$. This means that $\rho^R \simeq \rho$ for all tensor representations without chirality constraints, while this operation interchanges chiral and antichiral irreps in any even $d$.<fnref n="16" />

Applying both conjugation and reflection we get the conjugate reflected representation $\bar{\rho}^R$ (isomorphic to dual reflected). From the above it follows that $\bar{\rho}^R \simeq \rho$ for all SO(d) irreps, except for (anti-)chiral tensors in $d=0 \mod 4$ which are interchanged.

Just as for Wightman axioms, we will need a conjugation operation $\dagger : C \to C$ on the vector space of local operators, which is involutive, anti-linear, and associates to each independent component $O_i^\alpha(\alpha = 1 \dots \dim \rho_i)$ a conjugate operator $(O_i^\dagger)^\alpha := (O_i^\alpha)^\dagger$. Then the hermiticity axiom takes the form<fnref n="17" />

$$\overline{\langle\varphi_1(x_1) \dots \varphi_n(x_n)\rangle} = \langle\varphi_n^\dagger(x_1^\theta) \dots \varphi_1^\dagger(x_n^\theta)\rangle, \quad (2.18)$$

<footnote n="14">¹⁴In particular one can sort all operators so that the Euclidean time coordinates are ordered $x_1^0 \ge x_2^0 \ge \dots \ge x_n^0$, and Euclidean correlator for any other ordering can be obtained by trivially permuting field labels.</footnote>

<footnote n="15">¹⁵This well-known shift from $d=0 \mod 4$ for SO(1, $d-1$) is induced by raising the indices of the $\varepsilon$-tensor. E.g. $\varepsilon^{01}\varepsilon_{10} = -1$ for SO(2), while it is 1 for SO(1, 1).</footnote>

<footnote n="16">¹⁶In odd $d$, $\Theta$ is a product of $-1$ and an SO(d) matrix, so that $g \to g^R$ is an inner automorphism. This provides another argument why $\rho^R \simeq \rho$ for all irreducible SO(d) representations in odd $d$.</footnote>

<footnote n="17">¹⁷Although we write the operators in the r.h.s. in the inverted order like in (2.10), permutation invariance renders this detail unimportant for the OS axioms.</footnote>
similar to the Minkowski counterpart (2.10) but with an important difference that the operators in the r.h.s. are put at reflected positions

$$x^{\theta} := \Theta x. \qquad (2.19)$$

This change has a consequence that $O_i^\dagger$ transforms in the conjugate reflected representation $\bar{\rho}_i^R$, explaining why we introduced this concept in the first place.<fnref n="18" /> For self-conjugate-reflected representations we may choose a basis such that

$$(O_i^\dagger)^{(\mu)} = \Theta_{(\nu)}^{(\mu)} O_i^{(\nu)}, \qquad (2.20)$$

where $\Theta_{(\nu)}^{(\mu)} := \Theta_{\nu_1}^{\mu_1} \dots \Theta_{\nu_l}^{\mu_l}$ is the intertwiner (2.17).

To write positivity, basic ket states $|\psi(f, \varphi_1, \dots, \varphi_n)\rangle$ are associated with n local operators $\varphi_1, \dots, \varphi_n \in C$ and a complex compactly supported Schwartz test function of n variables $f(x_1, \dots, x_n)$ which vanishes unless all points are in the lower half space and have time variables ordered: $0 > x_1^0 > x_2^0 > \dots > x_n^0$. These support requirements were absent in the Wightman case. The inner product on the basic ket states is defined by

$$\langle\psi(g, \chi_1, \dots, \chi_m)|\psi(f, \varphi_1, \dots, \varphi_n)\rangle := \int dx dy \overline{g(y_1^\theta, \dots, y_m^\theta)} f(x_1, \dots, x_n) \times \langle\chi_m^\dagger(y_m) \dots \chi_1^\dagger(y_1) \varphi_1(x_1) \dots \varphi_n(x_n)\rangle, \qquad (2.21)$$

and is extended by (anti)linearity to the vector space $\mathcal{H}_0^{\text{OS}}$ of finite linear combinations $|\Psi\rangle$ of basic ket states. In this notation, positivity takes the same form as (2.12), i.e. that the so defined inner product must be positive semidefinite:

$$\langle\Psi|\Psi\rangle \ge 0 \quad \forall |\Psi\rangle \in \mathcal{H}_0^{\text{OS}}. \qquad (2.22)$$

This is referred to as “OS reflection positivity” because of the reflected g arguments in (2.21), differently from the Wightman case. Because of this reflection and the above test function support requirements, all operators in (2.21) sit at separated positions. This is one reason why the OS axioms involve ordinary functions, without worrying about coincident points. In contrast, Wightman positivity integrates operator insertions over coincident points and makes sense only for distributions.

Just as in the Wightman case (remark 2.1), we can complete the vector space $\mathcal{H}_0^{\text{OS}}$, mod out by states of zero norm, and obtain a Hilbert space $\mathcal{H}^{\text{OS}}$ of the Euclidean theory.

Although we included hermiticity as an independent axiom, it can be derived from positivity, just as in remark 2.2 in the Wightman case.

One simple consequence of OS reflection positivity is pointwise positivity of 2n-point functions at reflection invariant configurations of points:

$$\langle\varphi_n^\dagger(x_n^\theta) \dots \varphi_1^\dagger(x_1^\theta) \varphi_1(x_1) \dots \varphi_n(x_n)\rangle \ge 0 \qquad (2.23)$$

for any $x_1, \dots, x_n$ in the lower half space.<fnref n="19" /> This follows from (2.22) by taking $|\Psi\rangle = |\psi(f, \varphi_1, \dots, \varphi_n)\rangle$ and localizing f near one configuration of points. In general, imposing (2.23) for all $\varphi$'s and $x$'s would be clearly weaker than full OS reflection positivity.

<footnote n="18">¹⁸Indeed we have $\langle(\mathcal{O}_i^\dagger)^\alpha(x)\dots\rangle = \overline{\langle\mathcal{O}_i^\alpha(x^\theta)\dots\rangle} = \overline{\rho(g)^\alpha}_\beta \langle\mathcal{O}_i^\beta(g^{-1}x^\theta)\dots\rangle = \overline{\rho(g)^\alpha}_\beta \langle(\mathcal{O}_i^\dagger)^\beta((g^R)^{-1}x)\dots\rangle$.</footnote>

<footnote n="19">¹⁹For tensor operator in self-conjugate-reflected representations, choosing the real basis (2.20), this becomes $\langle\dots\Theta_{(\nu)}^{(\mu)}\mathcal{O}^{(\nu)}(x^\theta)\mathcal{O}^{(\mu)}(x)\dots\rangle \ge 0$ (no sum on $\mu$), i.e. tensor indices are also reflected.</footnote>
E.g. (2.22), but not (2.23), can be used to bound 3-point functions in terms of 2- and 4-point functions, or non-reflection-invariant 4-point functions by reflection-invariant ones. However for CFTs we will see below that OS reflection positivity can be reduced to a form of (2.23) for 2-point functions plus a form of (2.18) for 3-point functions.

Finally, the OS clustering asserts that

$$ \lim_{\lambda \to \infty} \int dx dy \overline{g(y_1^\theta, \dots, y_m^\theta)} f(x_1, \dots, x_n) \langle \chi_m^\dagger(y_m) \dots \chi_1^\dagger(y_1) \varphi_1(x_1 + \lambda a) \dots \varphi_n(x_n + \lambda a) \rangle \\ = \int dx dy \overline{g(y_1^\theta, \dots, y_m^\theta)} f(x_1, \dots, x_n) \langle \chi_m^\dagger(y_m) \dots \chi_1^\dagger(y_1) \rangle \langle \varphi_1(x_1) \dots \varphi_n(x_n) \rangle \tag{2.24} $$

for any Schwartz test functions $f(x_1, \dots, x_n)$ and $g(y_1, \dots, y_m)$ supported for $0 > x_1^0 > x_2^0 > \dots > x_n^0$ and $0 > y_1^0 > y_2^0 > \dots > y_n^0$, for any local fields $\varphi_1, \dots, \varphi_n$ and $\chi_1, \dots, \chi_m$, and for any $a \in \mathbb{R}^d$ which is parallel to the $x^0$ plane ($a^0 = 0$, called “purely spatial” elsewhere). The latter requirement is somewhat analogous to having the Wightman cluster property (2.4) to be satisfied only for spacelike $a$.<fnref n="20" />

Note that the Minkowski operators can be mapped to Euclidean operators. In particular any $SO(1, d-1)$ irrep can be mapped to an $SO(d)$ irrep. This map of irreps originates from the map between the two Lie algebras which have the same complexification. It can then be shown that a pair of conjugate $SO(1, d-1)$ irreps is mapped to a pair of $SO(d)$ irreps which are conjugate-reflected to each other. This gives another rationale for the appearance of reflected irreps in the OS axioms.

**Remark 2.4.** The stated version of OS axioms includes the assumption of real analyticity (2.14) and the bound (2.15). These assumptions are natural from physics perspective; they also easily follow from Wightman axioms. The original OS axioms did not include (2.14) nor (2.15), but included instead a differently stated assumption:

$$ \text{correlators are distributions on } {}^0S, \tag{2.25} $$

where ${}^0S$ is the space of Schwartz test function vanishing at coincident points with all their derivatives.

We would like to discuss here the relation between (2.14)+(2.15) and (2.25). In one direction this is easy: clearly (2.15) implies (2.25). In the other direction it can be shown that (2.25) and other OS axioms (in particular OS positivity and rotation invariance) imply real analyticity (2.14). This is a result of [2, 3] and [46]. It is also possible to derive (2.15) from (2.25) and other OS axioms [3]. These issues will be reviewed further in section 9.

**Remark 2.5.** In ref. [3], Osterwalder and Schrader introduced an extra assumption on Euclidean correlators, the linear growth condition. We will come back to this in section 9. Here we would like to stress that this condition is not needed for the implications discussed in the previous paragraph. On the other hand, the linear growth condition was used in ref. [3] when showing that the Euclidean correlators can be Wick-rotated to Lorentzian signature, resulting in Wightman distributions.

<footnote n="20">²⁰This is axiom E4 in [2]. Ref. [2] also mentions a stronger axiom E4', but we will be content here with checking the easier axiom E4.</footnote>
## 2.3 Euclidean CFT axioms

Wightman and OS axioms stated in the previous two sections are standard. We took care to present them for general operator representations and in general $d$. We will now present axioms for Euclidean unitary CFT. Just as OS axioms, these concern correlators in Euclidean signature, but there is an extra assumption of conformal invariance. Another feature of the CFT axioms is that assumptions are imposed on simple building blocks (2- and 3-point functions) from which more complicated correlators can be constructed. Properties of these complicated correlators then follow. The point of our paper is how one can recover OS axioms and (after Wick rotation) Wightman axioms in this setup.

A Euclidean unitary CFT in $\mathbb{R}^d$ ($d \ge 2$) deals with local primary operators $O_i(x)$ and with their $n$-point correlation functions $\langle O_{i_1}(x_1) \dots O_{i_n}(x_n) \rangle$. Correlators are real-analytic functions defined away from coincident points, which are permutation-invariant as in (2.16). Each primary is characterized by its scaling dimension $\Delta_i$ and is an $SO(d)$ tensor transforming in an irreducible representation $\rho_i$.<fnref n="21" /> The scaling dimensions are real and nonnegative, with the unit operator having dimension zero. The set of scaling dimensions (“spectrum”) is assumed to be discrete, by which we mean that there are finitely many $\Delta_i$'s in any finite interval $[a, b] \subset \mathbb{R}$.<fnref n="22" />

The set of all local operators of a CFT consists of primaries $O_i(x)$ and their space-time derivatives $\partial_{\mu_1} \dots \partial_{\mu_n} O_i(x)$, often referred to as descendants. The correlation functions of the descendant operators are simply the derivatives of the correlation functions of primary operators. They are well-defined since the correlators of primaries are assumed to be real-analytic.

Parameters $\Delta_i$ and $\rho_i$ determine transformation properties of $O_i(x)$ under the conformal group $SO(d+1, 1)$, and correlators remain invariant under these standard transformations which we will not write down. These constraints determine the functional form of 1,2,3-point functions. In particular, the unit operator is the only one with a nonzero 1-point function. See, e.g., [20] for a review of these facts.

An important fact that follows from the conformal invariance of correlation functions is that one is allowed to insert an operator at spatial infinity. The primary operator at spatial infinity is defined as

$$ \langle O_i(\infty) \cdots \rangle \equiv \lim_{L \to +\infty} L^{2\Delta_i} \langle O_i(L\hat{e}_0) \cdots \rangle. \qquad (2.26) $$

To see that this limit exists one can use a conformal map that takes $\infty$ to a finite point and moves no other operators to infinity. After applying this map the limit (2.26) turns into a limit in which all points approach finite values. We conclude that (2.26) exists, and is then of course independent of the concrete conformal map that we chose. In the definition (2.26) we have chosen a particular direction $(\hat{e}_0)$ for the limit. Using conformal symmetry it is

<footnote n="21">²¹Operators can also be grouped into multiplets of the global symmetry group $G$ which a CFT might have, but we will not discuss global symmetry here. For simplicity we will only consider bosonic operators. More generally one should allow fermionic operators and spinor representations.</footnote>

<footnote n="22">²²There exist 2d unitary CFTs, such as the Liouville theory, with a continuous spectrum of scaling dimensions. In this case axioms need to be modified. All known unitary CFTs in $d \ge 3$ have a discrete spectrum.</footnote>
easy to show that (2.26) is independent of this direction, up to a rotation on the indices of $O_i$. In what follows we will always allow for Euclidean CFT correlators to have one of the operators to be at $\infty$.

A Euclidean unitary CFT comes equipped with a conjugation operation $\dagger$, an involutive antilinear operator on the vector space of local operators (including both primaries and descendants), similarly to the OS axioms. Every primary $O_i$ is mapped by $\dagger$ to a conjugate primary $O_i^\dagger$ such that the 2-point function $\langle O_i^\dagger O_i \rangle$ does not vanish. The $O_i$ and $O_i^\dagger$ have equal scaling dimensions, and transform in the conjugate-reflected irreps. Recall that in section 2.2 we saw that most SO(d) irreps are self-conjugate-reflected, $\rho_i \simeq \bar{\rho}_i^R$, the only exception being (anti-)chiral tensors in $d = 0 \mod 4$ which are exchanged by this operation. For operators in self-conjugate-reflected irreps we may choose operator basis such that eq. (2.20) holds, which we copy here:

$$ (\mathcal{O}_i^\dagger)^{(\mu)} = \Theta_{(\nu)}^{(\mu)} \mathcal{O}_i^{(\nu)}. \qquad (2.27) $$

The functional form of the $\langle \mathcal{O}_i^\dagger(x) \mathcal{O}_i(y) \rangle$ 2-point function is fixed by conformal symmetry:

$$ \langle (\mathcal{O}_i^\dagger)^{(\mu)}(x) \mathcal{O}_i^{(\nu)}(y) \rangle = \mathcal{N}_i I^{(\mu),(\nu)}(x-y), \qquad (2.28) $$

where $(\mu)$, $(\nu)$ are collections of tensor indices (of equal length), $I^{(\mu),(\nu)}(x-y)$ is a tensor function depending only on $\Delta_i$, $\rho_i$, and $\mathcal{N}_i$ is a constant.

We are free to rescale the basis of primaries, multiplying them by some positive constants which should be the same for both $O_i$ and $O_i^\dagger$. This transformation clearly preserves the above conditions on $\dagger$, as well as further conditions which will be discussed below, notably (2.32). Using this freedom we can rescale $\mathcal{N}_i$ by a positive real number and fix $|\mathcal{N}_i|$ in some arbitrary unimportant way, e.g. so that some component of the 2-point function is of absolute value one at unit separation. On the other hand the phase of $\mathcal{N}_i$ cannot be changed in this way. Instead, it is uniquely determined by the positivity condition discussed below.

Positivity is imposed in Euclidean CFT axioms only on 2-point functions. We write it again using the language of states. Basic ket states are $|\partial^{(\beta)} \mathcal{O}_i^{(\nu)}\rangle$ where $\mathcal{O}_i^{(\nu)}$ is a primary component and $\partial^{(\beta)}$ an arbitrary derivative. The inner product is defined as

$$ \langle \partial^{(\alpha)} \mathcal{O}_i^{(\mu)} | \partial^{(\beta)} \mathcal{O}_i^{(\nu)} \rangle = \Theta_{(\alpha')}^{(\alpha)} \langle \partial^{(\alpha')} (\mathcal{O}_i^\dagger)^{(\mu)}(x_N) \partial^{(\beta)} \mathcal{O}_i^{(\nu)}(x_S) \rangle, \qquad (2.29) $$

i.e. as the value of the shown 2-point function inserting the operators at $x_S = (-1, 0, \dots, 0)$ and $x_N = (1, 0, \dots, 0) = (x_S)^\theta$ (where N,S stands for north, south). For ket states with $i \neq j$ the inner product vanishes since the 2-point function is zero. This inner product is extended by (anti)linearity to the vector space $\mathcal{H}_0^{\text{CFT}}$ of finite complex linear combinations of basic ket states. In this language, Euclidean CFT *positivity* reads exactly as the Wightman and OS positivity: $\langle \Psi | \Psi \rangle \ge 0$ for all states of this restricted form. More prosaically, this can also be stated that the infinite matrices $M_i^{(\alpha)(\mu),(\beta)(\nu)}$ built out of 2-point functions in the r.h.s. of (2.29) are all positive semidefinite when restricted to finite subspaces.
CFT positivity can be analyzed primary by primary, and it depends only on the primary 2-point function, eq. (2.28) which determines the full matrix $M_i^{(\alpha)(\mu),(\beta)(\nu)}$. Clearly, only one phase of the normalization constant $N_i$ in eq. (2.28) can give rise to a positive definite matrix, so that phase is uniquely fixed. Once the phase of $N_i$ is fixed, positivity for a given primary depends only on its $\Delta$, $\rho$. It then can be shown that CFT positivity holds if and only if every $\Delta$, in addition to being real and non-negative, lies above a certain minimal $\rho$-dependent value ("unitarity bound"):

$$ \Delta \ge \Delta_{\min}(\rho). \qquad (2.30) $$

These unitarity bounds are documented in the literature, e.g. we have $\Delta_{\min} = d/2 - 1$ for scalars, and $d + \ell - 2$ for spin-$\ell$, $\ell \ge 1$. For arbitrary SO(d) representations see [47].$^{23}$

For future uses, we wish to define the CFT Hilbert space $\mathcal{H}^{\text{CFT}}$ via completion of $\mathcal{H}_0^{\text{CFT}}$, after modding out by zero norm states (for operators saturating the unitarity bounds, some descendants have zero norm). This can be done abstractly, or explicitly using a basis as we now describe. Throwing out zero-norm descendants, the remaining states can be organized choosing an orthonormal basis. We may choose such a basis independently among descendants of each primary, and then combine all these bases, e.g. in the order of non-decreasing scaling dimensions. The elements of $\mathcal{H}^{\text{CFT}}$ are then formal linear combinations $\sum_n c_n|n\rangle$, where $|n\rangle$ are orthonormal basis elements, and $c_n$ is an arbitrary complex $\ell_2$ sequence. The norm on $\mathcal{H}^{\text{CFT}}$ is the $\ell_2$ norm of the sequence $c_n$. Restricting to sequences $c_n$ which have only a finite number of nonzero elements, we get elements of $\mathcal{H}_0^{\text{CFT}}$ (modulo the zero-norm states).

Let us continue with the axioms. CFT hermiticity condition is imposed only on the 2-point and 3-point functions, namely:

$$ \langle (\mathcal{O}_i^\dagger)^{(\mu)}(x_1) \mathcal{O}_i^{(\nu)}(x_2) \rangle = \overline{\langle \mathcal{O}_i^{(\mu)}(x_1^\theta) (\mathcal{O}_i^\dagger)^{(\nu)}(x_2^\theta) \rangle}, \qquad (2.31) $$

which is also a consequence CFT positivity and in particular fixed the phase of $N_i$ up to a sign,$^{24}$ and

$$ \langle \mathcal{O}_i^{(\mu)}(x_1) \mathcal{O}_j^{(\nu)}(x_2) \mathcal{O}_k^{(\lambda)}(x_3) \rangle = \overline{\langle (\mathcal{O}_i^\dagger)^{(\mu)}(x_1^\theta) (\mathcal{O}_j^\dagger)^{(\nu)}(x_2^\theta) (\mathcal{O}_k^\dagger)^{(\lambda)}(x_3^\theta) \rangle}. \qquad (2.32) $$

<footnote>$^{23}$We chose to express CFT positivity inserting operators at the points $(\pm 1, 0, \dots, 0)$ which corresponds to the N-S quantization (see [48]) and will facilitate the comparison with the Osterwalder-Schrader reflection positivity. Equivalently, one could go via a conformal transformation to the more familiar radial quantization corresponding to inserting the operators at 0 and $\infty$. CFT positivity is then equivalent to radial quantization states having positive norm on every level, which is how the unitarity bounds are usually worked out in Euclidean CFTs [47]. In mathematical language, this latter condition corresponds to having a positive-definite Shapovalov form on the parabolic Verma module. Recent work [49, 50] explained how the determinant formulas by Jantzen [51] provide a rigorous justification of the Euclidean unitarity bounds (both in the necessary and sufficient directions).</footnote>

<footnote>$^{24}$For the special case $x_1 = x_N$, $x_2 = x_S$, eq. (2.31) is nothing but hermiticity of the matrix $M_i^{(\mu),(\nu)}$, a consequence of positive-semidefiniteness. The general case reduces to the special one mapping $x_1, x_2$, to $x_N$, $x_S$ by a conformal transformation (both sides of (2.31) have the same conformal transformation properties).</footnote>
for any 3 primaries $O_i, O_j, O_k$.<fnref n="25" /> Similarly to the 2-point function case, this condition can be simplified using the conformally invariant tensor structures, with an important difference that the normalization of operators has already been fixed. Conformal invariance constrains the 3-point functions to take the form

$$ \langle O_i^{(\mu)}(x_1) O_j^{(\nu)}(x_2) O_k^{(\lambda)}(x_3) \rangle = \sum_{a=1}^{N_{ijk}} f_{ijk}^a \langle O_i^{(\mu)}(x_1) O_j^{(\nu)}(x_2) O_k^{(\lambda)}(x_3) \rangle_a, \quad (2.33) $$

where $\langle O_i^{(\mu)}(x_1) O_j^{(\nu)}(x_2) O_k^{(\lambda)}(x_3) \rangle_a$ span the finite-dimensional space (of dimension $N_{ijk}$) of solutions of conformal invariance constraints on the 3-point functions of the operators with given $\Delta_s, \rho_s$ ($s = i, j, k$). On the other hand the coefficients $f_{ijk}^a \in \mathbb{C}$ are not fixed by conformal symmetry (no sum on $i, j, k$ in the r.h.s. of (2.33)). We often refer to $f_{ijk}^a$ as the “OPE coefficients.” It is always possible to choose the basis structures $\langle O_i^{(\mu)}(x_1) O_j^{(\nu)}(x_2) O_k^{(\lambda)}(x_3) \rangle_a$ to satisfy the hermiticity constraint (2.32) individually, in which case the OPE coefficients must satisfy

$$ (f_{ijk}^a)^* = f_{ijk}^a, \quad (2.34) $$

where the barred indices refer to the conjugate operators $O_i^\dagger, O_j^\dagger, O_k^\dagger$. In particular, when all three operators are self-conjugate-reflected i.e. satisfy (2.27), the OPE coefficients $f_{ijk}^a$ must be real.

Finally, unitary CFTs enjoy a convergent operator product expansion (OPE). This means that any correlation function<fnref n="26" /> satisfies

$$ \langle O_i^{(\mu)}(x_1) O_j^{(\nu)}(x_2) O_m^{(\rho)}(x_3) \cdots \rangle = \sum_k \sum_{a=1}^{N_{ijk}} f_{ijk}^a C_{a,(\lambda)}^{(\mu)(\nu)}(x_1, x_2, x_0, \partial_0) \langle (O_k^\dagger)^{(\lambda)}(x_0) O_m^{(\rho)}(x_3) \cdots \rangle, \quad (2.35) $$

where the first sum runs over all primary operators $O_k$ in the theory, and $C_{a,(\lambda)}^{(\mu)(\nu)}(x_1, x_2, x_0, \partial_0)$ is a formal sum of the form

$$ C_{a,(\lambda)}^{(\mu)(\nu)}(x_1, x_2, x_0, \partial_0) = \sum_{\alpha} C_{a,(\lambda),\alpha}^{(\mu)(\nu)}(x_1, x_2, x_0) (\partial/\partial x_0)^{\alpha}. \quad (2.36) $$

This differential operator is determined by conformal symmetry<fnref n="27" /> and depends only on $\Delta_s, \rho_s$ ($s = i, j, k$). Here $(\partial/\partial x)^\alpha = (\partial/\partial x^0)^{\alpha_0} \cdots (\partial/\partial x^{d-1})^{\alpha_{d-1}}$ with $\alpha = (\alpha_0, \dots, \alpha_{d-1}) \in (\mathbb{Z}_{\ge 0})^d$ a multiindex. Convergence of OPE means that the sum (2.35), with $C$ expanded as in (2.36), converges whenever

$$ x_1, x_2 \in B(x_0, R), \quad R = \min(|x_3 - x_0|, |x_4 - x_0|, \dots) \quad (2.37) $$

<footnote n="25">²⁵Note that since $\dagger$ is involutive, this covers the case when an operator and its conjugate are interchanged between the two sides of this equation.</footnote>

<footnote n="26">²⁶Importantly, we allow here for one of the operators to be inserted at spatial infinity.</footnote>

<footnote n="27">²⁷There is an ambiguity when $O_k$ is in a short conformal representation (in unitary theories this happens only if $O_k$ is a conserved current or a free field). This subtlety will not play any role in this paper.</footnote>
where $B(x_0, R)$ is an open ball centered at $x_0$ and of radius $R$. In other words, OPE converges whenever $x_1$ and $x_2$ are the two closest operator insertions to $x_0$ (in Euclidean distance). Convergence should be understood carefully as follows. For each $O_k^\dagger$ in the r.h.s. of (2.35), and for each $n \in \mathbb{Z}_+$, we perform finite summation over $a, \lambda$, and all multiindices $\alpha$ with $|\alpha| = n$. We are left then with the doubly infinite sum

$$ \sum_k \sum_{n=0}^{\infty} g_{k,n}(\{x_i\}). \qquad (2.38) $$

This doubly infinite sum has to converge absolutely for every $x_1, x_2$ as in (2.37).$^{28}$

That the same coefficients $f_{ijk}^a$ appear in the OPE (2.35) and the 3-point function (2.33) follows immediately by using the OPE inside the latter 3-point function.

Local Euclidean CFTs contain the conserved stress tensor operator $T_{\mu\nu}$ of dimension $d$, and in case of continuous global symmetry, conserved global symmetry currents $J_\mu$ of dimensions $d-1$. We will not discuss here additional axioms involving 3-point functions and OPE coefficients of these operators, related to their conservation and Ward identities, see e.g. [52, 53].

**Remark 2.6.** The just given Euclidean CFT axioms are more careful in what concerns reality constraints than the set of CFT rules gathered in the conformal bootstrap review [20]. They are also more economical: e.g. Ref. [20] assumed OS reflection positivity and clustering for $n$-point functions, which for us will be theorems to prove, not assumptions. Ref. [20] also included some constraints on the CFT data which emerge when considering CFT in Minkowski signature, most notably the Averaged Null Energy Condition (ANEC). In this paper we will establish all Wightman axioms for scalar Minkowski CFT 4-point functions from the Euclidean CFT axioms, but we will not discuss ANEC. A proof of ANEC [54] has been given using the Haag-Kastler axioms for general QFT. CFT arguments have also been given in [45, 55], but they rely on some assumptions which have not been rigorously proven from axioms. It would be interesting to fill these gaps and establish ANEC as a theorem from Euclidean CFT axioms.$^{29}$

## 3 Euclidean CFT ⇒ Osterwalder-Schrader

In this section we will discuss some simple consequences of CFT axioms and in particular will show that they imply OS axioms for 4-point functions (the case of higher-point functions is more subtle and is discussed in appendix B). Here we will prove only the OS reflection-positivity and the cluster property. The “Euclidean temperedness” bound (2.15)

<footnote>$^{28}$The requirement of absolute convergence can be somewhat relaxed, see section 3.1.</footnote>

<footnote>$^{29}$ The argument in [45] uses an OPE asymptotic expansion on the second sheet, outside of the range of convergence of the OPE rigorously implied by the Euclidean CFT axioms (see appendix A.1 where we review this method going back to [4]). In [55] ANEC is derived using manipulations with a generalization of the Lorentzian inversion formula of [56], of which some have not been rigorously justified. For example, the derivation starts with the Euclidean inversion formula, which is readily justified from Harmonic analysis only for external scaling dimensions on principal series and square-integrable correlators, none of which is generically the case in the required setup.</footnote>
will follow from our arguments in the following sections, where we establish power-law bounds on CFT correlation functions. The remaining OS axioms are a subset of the CFT axioms.

## 3.1 OS reflection positivity

In this section we prove OS positivity for compactly-supported test functions. The extension to Schwartz functions is easy once we establish (2.15) for CFT correlators, see remark 4.2.

To someone familiar with OPE in CFTs, the goal of this section may seem straightforward to achieve: in usual CFT treatments, one can express any state as a sum of “single operator states”, created by action of a single insertion of primaries and their descendants on the vacuum, and our two-point positivity axiom makes sure that the inner products of such states are positive-semidefinite. The part of this wisdom that doesn’t work immediately in our setup is being able to express states in the OS Hilbert space as infinite OPE sums. Our OPE axiom is much weaker than this statement, in particular the truncation order for the OPE needed to achieve a given precision $\epsilon$ in (2.35) may a priori depend on everything in the left-hand side, including coordinates $x_i$ and the “spectator” operators $O_m, \dots$, in an arbitrary fashion. (On the contrary, for a Hilbert space OPE statement, the truncation order is good for all choices of spectator operators as long as an appropriate norm remains bounded by a fixed constant.) In spite of this difficulty, in this and the next section we will be able to recover reflection positivity for $n$-point functions up to $n \le 4$ and Hilbert space OPE convergence for states created by up to 2 operator insertions, but doing so requires some care.

First let us slightly reformulate the OPE convergence property. Consider an $n$-point correlation function with operators inserted at $x_1 \dots x_n$. Let $S$ be a hyperplane and $x_0$ be a point such that $x_1, x_2, x_0$ are on one side of $S$ while all the other points $x_i$, $i > 2$, are on the other side. Using a conformal transformation we can map $S$ to a sphere $S'$ so that $x_0$ is mapped to the center of $S'$ which we denote by $x'_0$. Let $x'_i$ denote the positions of all the other points $x_i$ after this map. We can then use the OPE (2.35) for the correlation function evaluated at $x'_i$,

$$ \langle O_i^{(\mu)}(x'_1) O_j^{(\nu)}(x'_2) O_m^{(\rho)}(x'_3) \dots \rangle = \sum_k \sum_{a=1}^{N_{ijk}} f_{ijk}^a C_{a,(\lambda)}^{(\mu)(\nu)}(x'_1, x'_2, x'_0, \partial_{0'}) \langle (O_k^\dagger)^{(\lambda)}(x'_0) O_m^{(\rho)}(x'_3) \dots \rangle. \quad (3.1) $$

Transforming this expansion term-by term to the original coordinates $x_i$ we find the convergent expansion (with convergence understood in the same sense as in the previous section$^{30}$)

$$ \langle O_i^{(\mu)}(x_1) O_j^{(\nu)}(x_2) O_m^{(\rho)}(x_3) \dots \rangle = \sum_k \sum_{a=1}^{N_{ijk}} f_{ijk}^a \mathfrak{e}^{(\mu)(\nu)}(x_1, x_2, x_0, \mathcal{D}_0) \langle (\mathcal{O}_k^\dagger)^{(\lambda)}(x_0) \mathcal{O}_m^{(\rho)}(x_3) \dots \rangle. \quad (3.2) $$

<footnote>$^{30}$Careful reading of the argument below shows that, in the 4-point case, the requirement of absolute convergence of (2.38) could be replaced by a weaker requirement that we can find any subsequence of partial sums of (2.38) which approximates the correlator pointwise.</footnote>
Here the differential operators $\mathcal{D}^{(\alpha)}$ are simply the derivatives $(\frac{\partial}{\partial x'})^{(\alpha)}$ expressed in the original coordinates $x$, and conjugated by the conformal transformation factor of $\mathcal{O}_k^\dagger$. The functions $\mathcal{E}_{a,(\lambda)}^{(\mu)(\nu)}$ are obtained from $C_{a,(\lambda)}^{(\mu)(\nu)}$ by our conformal transformation. The important point is that truncation of $C_{a,(\lambda)}^{(\mu)(\nu)}$ in order of derivatives $\partial^{(\alpha)}$ corresponds to truncation of $\mathcal{E}_{a,(\lambda)}^{(\mu)(\nu)}$ in order of operators $\mathcal{D}^{(\alpha)}$.

We now specialize to $S$ being the $x^0 = 0$ plane, $x_0 = x_S = (-1, 0, \dots)$, and take $S'$ to be the unit sphere with the center $x'_0 = 0$. Then the derivatives $(\frac{\partial}{\partial x'})^{(\alpha)} \mathcal{O}_k^{(\mu)}(x'_0)$ are eigenstates of the standard dilatation generator $D$ with eigenvalues $\Delta_k + |\alpha|$. Note that $D$ has two fixed points: $x'_0 = 0$ and infinity. Applying our conformal map, we find that the derivatives $\mathcal{D}^{(\alpha)}\mathcal{O}_k^{(\mu)}(x_S)$ are in turn eigenstates, with the same eigenvalues, of the conformal generator $D' = (K^0 - P^0)/2$ that preserves $x_S$ and $x_N = x_S^\theta$ (which is the image of infinity under our conformal map) and acts by dilatations near these two points. This, together with the conformal invariance and diagonality of 2-point functions, implies

$$ \langle (\mathcal{D}^{(\alpha)})^{\theta} \mathcal{O}_{j}^{\dagger(\mu)}(x_{N}) \mathcal{D}^{(\beta)} \mathcal{O}_{k}^{(\nu)}(x_{S}) \rangle \propto \delta_{|\alpha|,|\beta|} \delta_{j,k}, \qquad (3.3) $$

where $(\mathcal{D}^{(\alpha)})^{\theta}$ is obtained from $\mathcal{D}^{(\alpha)}$ by replacing $x \to x^\theta$.

The OPE (3.2) gives an expansion for ket states $|\Psi\rangle \in \mathcal{H}_0^{\text{OS}}$ created by two local operators in terms of ket states created by a single operator, which are elements of $\mathcal{H}_0^{\text{CFT}}$. We would like to have a dual version of this expansion for bra states $\langle\Psi|$ . For this we need to understand how the OPE transforms under the conjugation. Note that the formal differential operators $\mathcal{E}_{a,(\lambda)}^{(\mu)(\nu)}(x_1, x_2, x_0, \mathcal{D}_0)$ can be uniquely determined by the equation

$$ \langle \mathcal{O}_1^{(\mu)}(x_1) \mathcal{O}_2^{(\nu)}(x_2) \mathcal{O}_3^{(\rho)}(x_3) \rangle_a = \mathcal{E}_{a,(\lambda)}^{(\mu)(\nu)}(x_1, x_2, x_0, \mathcal{D}_0) \langle (\mathcal{O}_3^{\dagger})^{(\lambda)}(x_0) \mathcal{O}_3^{(\rho)}(x_3) \rangle, \qquad (3.4) $$

where it is understood that the points are arranged as above, so that the formal sum defined by $\mathcal{E}_{a,(\lambda)}^{(\mu)(\nu)}(x_1, x_2, x_0, \mathcal{D}_0)$ actually converges. By applying complex conjugation on both sides and using the 2-point and 3-point hermiticity constraints (2.31) and (2.32) we find

$$ \langle (\mathcal{O}_1^{\dagger})^{(\mu)}(x_1^{\theta}) (\mathcal{O}_2^{\dagger})^{(\nu)}(x_2^{\theta}) (\mathcal{O}_3^{\dagger})^{(\rho)}(x_3^{\theta}) \rangle_a = [\mathcal{E}_{a,(\lambda)}^{(\mu)(\nu)}(x_1, x_2, x_0, \mathcal{D}_0)]^* \langle \mathcal{O}_3^{(\lambda)}(x_0^{\theta}) (\mathcal{O}_3^{\dagger})^{(\rho)}(x_3^{\theta}) \rangle, \qquad (3.5) $$

which implies that

$$ [\mathcal{E}_{a,(\lambda)}^{(\mu)(\nu)}(x_1^{\theta}, x_2^{\theta}, x_0^{\theta}, \mathcal{D}_0)]^* = \tilde{\mathcal{E}}_{a,(\lambda)}^{(\mu)(\nu)}(x_1, x_2, x_0, \mathcal{D}_0), \qquad (3.6) $$

where $\tilde{\mathcal{E}}_{a,(\lambda)}^{(\mu)(\nu)}(x_1, x_2, x_0, \mathcal{D}_0)$ is the formal sum that appears in the OPE for operators with conjugate-reflected quantum numbers.

We are now ready to prove OS positivity for 4-point functions. Let $|\Psi_0\rangle$ be an OS ket state involving at most two local operators, i.e.<fnref n="31" />

$$ |\Psi_0\rangle = \sum_{i,j,\alpha,\beta} \int dx_1 dx_2 f_{i,j,(\alpha)(\beta)}(x_1, x_2) |\mathcal{O}_i^{(\alpha)}(x_1) \mathcal{O}_j^{(\beta)}(x_2)\rangle, \qquad (3.7) $$

<footnote n="31">³¹In the notation of section 2.2 this could be written as $\sum_{i,k,\alpha,\beta} |\psi(f_{i,j,(\alpha),(\beta)}, \mathcal{O}_i^{(\alpha)}, \mathcal{O}_j^{(\beta)})\rangle$.</footnote>
where $f(x_1,x_2)$ is a compactly supported test function vanishing unless $0 > x_1^0 > x_2^0$. (Terms with one or no operators are realized by setting one or both operators to the identity.) Since by Euclidean CFT axioms the correlation functions are real-analytic, the integrals that appear in the expression for $\langle\Psi_0|\Psi_0\rangle$ can be approximated by finite Riemann sums, reflection-symmetric if necessary. This implies that, for any $\varepsilon > 0$, we can pass from $|\Psi_0\rangle$ to a ket state $|\Psi\rangle$ which is created by a finite linear combination of insertions of up to two local operators with $x_1^0, x_2^0 < 0$:

$$|\Psi\rangle = \sum_{i,j,\alpha,\beta,x_1,x_2} c_{i,j,(\alpha)(\beta),x_1,x_2} |\mathcal{O}_i^{(\alpha)}(x_1)\mathcal{O}_j^{(\beta)}(x_2)\rangle, \quad (3.8)$$

and has the property that

$$|\langle\Psi_0|\Psi_0\rangle - \langle\Psi_0|\Psi\rangle| < \varepsilon, \quad |\langle\Psi_0|\Psi\rangle - \langle\Psi|\Psi\rangle| < \varepsilon, \quad (3.9)$$

so that as a result

$$|\langle\Psi_0|\Psi_0\rangle - \langle\Psi|\Psi\rangle| < 2\varepsilon \quad (3.10)$$

We are therefore reduced to proving the nonnegativity of $\langle\Psi|\Psi\rangle$.

Now, the OPE convergence axiom implies that, for any $\varepsilon > 0$, starting from $|\Psi\rangle$ and using the OPE (3.2) in the half-space $x^0 < 0$, we can construct a state $|\psi\rangle = |\psi_\Lambda\rangle \in \mathcal{H}_0^{\text{CFT}}$ created by a finite linear combination of local operators at $x_S$ such that

$$|\langle\Psi|\psi\rangle - \langle\Psi|\Psi\rangle| < \varepsilon. \quad (3.11)$$

Here $\Lambda$ is an OPE truncation cutoff which we need to increase appropriately as $\varepsilon$ gets smaller. Namely, we will obtain $|\psi\rangle$ by keeping in the OPE all terms with $\Delta_k + |\alpha| < \Lambda$, where $\Delta_k$ is the dimension of a primary $\mathcal{O}_k$ appearing in the OPE, and $\alpha$ is the order of the descendant $\mathcal{D}^{(\alpha)}\mathcal{O}_k$.

We can then repeat this procedure in the upper half-plane $x^0 > 0$ and construct a state $\langle\psi'| = \langle\psi_{\Lambda'}|$ of local operators inserted at $x_N$ such that

$$|\langle\Psi|\psi\rangle - \langle\psi'|\psi\rangle| < \varepsilon. \quad (3.12)$$

Eq. (3.6) and the reality constraint (2.34) for the OPE coefficients imply that the state $\langle\psi'|$ differs from $\langle\psi|$ at most by where the OPE expansion was truncated. Furthermore, we can always assume that $\langle\psi'|$ contains at least all the terms that $\langle\psi|$ does (i.e. $\Lambda' \ge \Lambda$), since adding more OPE terms to $\langle\psi'|$ can only improve (3.12).

Eq. (3.3) then implies that $\langle\psi'|\psi\rangle = \langle\psi|\psi\rangle$ and therefore

$$|\langle\Psi|\psi\rangle - \langle\psi|\psi\rangle| < \varepsilon. \quad (3.13)$$

Combining this with (3.11) we conclude:

$$|\langle\Psi|\Psi\rangle - \langle\psi|\psi\rangle| < 2\varepsilon, \quad (3.14)$$

Since by the CFT positivity axiom $\langle\psi|\psi\rangle$ is non-negative, we conclude that $\langle\Psi|\Psi\rangle$ is also non-negative. This completes the proof of OS positivity for states created by up to two operator insertions.
## 3.2 Denseness and Hilbert space implications

Here we will describe some useful byproducts of the just given argument. Note that (3.11) and (3.13), in addition to (3.14), also implies

$$ \|\Psi - \psi_{\Lambda}\| \equiv \langle \Psi - \psi_{\Lambda} | \Psi - \psi_{\Lambda} \rangle = \langle \Psi | \Psi \rangle - \langle \Psi | \psi_{\Lambda} \rangle + \langle \psi_{\Lambda} | \psi_{\Lambda} \rangle - \langle \psi_{\Lambda} | \Psi \rangle < 2\varepsilon. \quad (3.15) $$

This means that any $|\Psi\rangle$ can be approximated arbitrarily well by a $|\psi_{\Lambda}\rangle \in \mathcal{H}_0^{\text{CFT}}$. In other words, $\mathcal{H}_0^{\text{CFT}}$ is a dense subspace of the vector space of $\Psi$'s.

This fact has a simple but quite powerful consequence involving the CFT Hilbert space $\mathcal{H}^{\text{CFT}}$, defined in section 2.3 as the completion of $\mathcal{H}_0^{\text{CFT}}$. Eq. (3.15) implies, using the triangle inequality $\|\psi_{\Lambda_1} - \psi_{\Lambda_2}\| \le \|\Psi - \psi_{\Lambda_1}\| + \|\Psi - \psi_{\Lambda_2}\|$, that the states $|\psi_{\Lambda}\rangle$ corresponding to smaller and smaller $\varepsilon$ form a Cauchy sequence. Therefore, these states have a limit in $\mathcal{H}^{\text{CFT}}$ as $\Lambda \to \infty$, which we call $|\psi_{\infty}\rangle$. This $\psi_{\infty}$ is nothing but the full, untruncated, OPE expansion of the state $\Psi$. We claim that the map mapping $\Psi$'s to the corresponding $\psi_{\infty}$'s is isometric, i.e. it preserves the inner products:

$$ \langle \Phi | \Psi \rangle = \langle \varphi_{\infty} | \psi_{\infty} \rangle. \quad (3.16) $$

Here the inner product on the l.h.s. is the OS inner product, computed using CFT 4-point functions with operators inserted in the lower and upper half-spaces, while the inner product in the r.h.s. is the $\mathcal{H}^{\text{CFT}}$ inner product, defined as the limit of $\mathcal{H}_0^{\text{CFT}}$ inner product removing the cutoff:

$$ \langle \varphi_{\infty} | \psi_{\infty} \rangle := \lim_{\Lambda \to \infty} \langle \varphi_{\Lambda} | \psi_{\Lambda} \rangle. \quad (3.17) $$

The proof of (3.16) is straightforward. We write:

$$ \langle \varphi_{\Lambda} | \psi_{\Lambda} \rangle = \langle \Phi + (\varphi_{\Lambda} - \Phi) | \Psi + (\psi_{\Lambda} - \Psi) \rangle = \langle \Phi | \Psi \rangle + \text{err}(\Lambda), \quad (3.18) $$

$$ \text{err}(\Lambda) = \langle \varphi_{\Lambda} - \Phi | \Psi \rangle + \langle \Phi | \psi_{\Lambda} - \Psi \rangle + \langle \varphi_{\Lambda} - \Phi | \psi_{\Lambda} - \Psi \rangle. \quad (3.19) $$

By eq. (3.15) we know that $\|\Psi - \psi_{\Lambda}\|, \|\Phi - \varphi_{\Lambda}\|$ go to zero as $\Lambda \to \infty$. Hence, $\text{err}(\Lambda) \to 0$ and (3.16) is proved.

Eqs. (3.16) and (3.17) mean that OPE converges in the sense of the CFT Hilbert space. This property is often used in the CFT literature (see section 6.2). Note that CFT axioms in section 2.3 only assume pointwise OPE convergence, which is a weaker statement. Curiously, by the given arguments, pointwise OPE convergence plus CFT positivity imply Hilbert space convergence, at least for the 4-point functions.

In the above argument we used the Hilbert space $\mathcal{H}^{\text{CFT}}$, the completion of $\mathcal{H}_0^{\text{CFT}}$. We may introduce a second Hilbert space as the completion of the space of $\Psi$'s, call it $\mathcal{H}^{(2)}$. This Hilbert space contains e.g. all $\Psi_0$ states (3.7). (Similarly to (3.15), eqs. (3.9) and (3.10) imply that $\Psi$ states are dense in the $\Psi_0$ states.) Although $\mathcal{H}^{(2)}$ may look like a “bigger” space than $\mathcal{H}^{\text{CFT}}$, actually it’s not. Indeed, the map from $\Psi$ to $\psi_{\infty}$ extends to an isometric map from $\mathcal{H}^{(2)}$ to $\mathcal{H}^{\text{CFT}}$. In other words, eq. (3.16) remains true for any $\Phi, \Psi \in \mathcal{H}^{(2)}$. Since we can view $\mathcal{H}_0^{\text{CFT}}$ as a subspace of the space of $\Psi$'s, we also have that this map is surjective. The Hilbert spaces $\mathcal{H}^{(2)}$ and $\mathcal{H}^{\text{CFT}}$ are thus unitarily equivalent and may be identified.
## 3.3 OS clustering

Here we will derive the OS clustering (2.24) from CFT axioms. We will consider only $m+n=4$, i.e. when the left-hand side of (2.24) can be written in terms of a 4-point function (this also covers $m+n<4$ since we can choose some of $\varphi$'s or $\chi$'s to be the trivial identity field). We can assume that all $\chi$'s and $\varphi$'s in (2.24) are primary fields, since any derivatives can be integrated by parts.

First as a general remark, assuming OS positivity, clustering (for any $m, n$) only needs to be established point-wise, i.e.

$$ \lim_{\lambda \to \infty} \langle \chi_m^\dagger(y_m) \dots \chi_1^\dagger(y_1) \varphi_1(x_1 + \lambda a) \dots \varphi_n(x_n + \lambda a) \rangle = \langle \chi_m^\dagger(y_m) \dots \chi_1^\dagger(y_1) \rangle \langle \varphi_1(x_1) \dots \varphi_n(x_n) \rangle. \quad (3.20) $$

This follows from the dominated convergence theorem. Indeed, OS positivity and translation invariance (recall that we only consider $a^0 = 0$!) implies a uniform in $\lambda$ bound<fnref n="32" />

$$ \begin{gathered} \left| \langle \chi_m^\dagger(y_m) \dots \chi_1^\dagger(y_1) \varphi_1(x_1 + \lambda a) \dots \varphi_n(x_n + \lambda a) \rangle \right|^2 \quad (3.21) \\ \le \langle \chi_m^\dagger(y_m) \dots \chi_1^\dagger(y_1) \chi_1(y_1^\theta) \dots \chi_n(y_n^\theta) \rangle \times \langle \varphi_n^\dagger(x_n^\theta) \dots \varphi_1^\dagger(x_1^\theta) \varphi_1(x_1) \dots \varphi_n(x_n) \rangle. \end{gathered} $$

It then follows that the integrand in (2.24) is bounded by a $\lambda$-independent integrable function, and the dominated convergence theorem is applicable.

Going back to the 4-point function case which is our focus in this section, let us start with $m=n=2$. Since we already proved OS positivity for states created by at most two operators (section 3.1), in this case we can rely on the above observation and we only need to check the point-wise limit:

$$ \lim_{\lambda \to \infty} \langle \chi_2^\dagger(y_2) \chi_1^\dagger(y_1) \varphi_1(x_1 + \lambda a) \varphi_2(x_2 + \lambda a) \rangle = \langle \chi_2^\dagger(y_2) \chi_1^\dagger(y_1) \rangle \langle \varphi_1(x_1) \varphi_2(x_2) \rangle. \quad (3.22) $$

To see this, we apply the OPE (3.2) to $\chi_2^\dagger(y_2)\chi_1^\dagger(y_1)$ in the left-hand side. The results of the previous section imply that this OPE can be interpreted as expanding the state in the Hilbert space $\mathcal{H}$ created by $\chi_2^\dagger(y_2)\chi_1^\dagger(y_1)$ in terms of eigenstates of $(K^0 - P^0)/2$. This implies that the OPE converges uniformly in $\lambda$ since the norm of $|\varphi_1(x_1 + \lambda a)\varphi_2(x_2 + \lambda a)\rangle$ is independent of $\lambda$ due to translation invariance ($a^0 = 0$!). We can thus use the OPE to approximate $\langle \chi_2^\dagger(y_2)\chi_1^\dagger(y_1)\varphi_1(x_1 + \lambda a)\varphi_2(x_2 + \lambda a)\rangle$ for any $\lambda$ to within any $\varepsilon > 0$ by a finite sum of 3-point functions of the form $\langle (\mathcal{D}^{(\alpha)})^\theta \mathcal{O}_i^{(\nu)}(x_N)\varphi_1(x_1 + \lambda a)\varphi_2(x_2 + \lambda a)\rangle$ times some $\lambda$-independent coefficients. Of these, only the term corresponding to the identity operator, i.e. the one with $(\mathcal{D}^{(\alpha)})^\theta \mathcal{O}_i^{(\nu)} = 1$, does not decay with $\lambda$. It is easily verified that the contribution of this term is precisely equal to $\langle \chi_2^\dagger(y_2)\chi_1^\dagger(y_1)\rangle \langle \varphi_1(x_1)\varphi_2(x_2)\rangle$. This finishes the proof of clustering for $m=n=2$.

In the remaining case $m=3, n=1$, we will consider the limit for the integral (since we have not yet proved OS positivity for states involving 3 operators). Note that $\langle \varphi_1(x) \rangle = 0$

<footnote n="32">³²This follows, similarly to (2.23), by applying OS positivity to the state $|\Psi\rangle = |\psi(F, \varphi_1 \dots \varphi_n)\rangle + e^{i\alpha}|\psi(G, \chi_1 \dots \chi_n)\rangle$ where $F, G$ tend to delta functions localizing the operators at points $x_1 + \lambda a, \dots, x_n + \lambda a$ and $y_1^\theta, \dots, y_n^\theta$ respectively, and choosing the phase $\alpha$ appropriately.</footnote>
unless $\varphi_1 \propto 1$, in which case the cluster property becomes trivial. This means that we only need to prove

$$ \lim_{\lambda \to \infty} \int dx dy \overline{g(y_1^\theta, y_2^\theta, y_3^\theta)} f(x_1) \langle \chi_3^\dagger(y_3) \chi_2^\dagger(y_2) \chi_1^\dagger(y_1) \varphi_1(x_1 + \lambda a) \rangle = 0 \quad (3.23) $$

for $\varphi_1 \neq 1$. This in turn is a very simple consequence of conformal invariance, and of the fact that $\Delta_\varphi > 0$ for all operators but the identity. We will be somewhat schematic. The main point is that the configuration $(y_3, y_2, y_1, \infty)$ is nonsingular from the conformal kinematics point of view (for which the conformal compactification $S_d$ of the Euclidean space $\mathbb{R}^d$ is the appropriate arena). One way to see it is that the cross ratios are finite in this limit. Thinking in a pedestrian way, we can find a conformal transformation $g_\lambda$ which will move points $(y_3, y_2, y_1, x_1 + \lambda a)$ to some points which have finite limits as $\lambda \to \infty$. Transforming the integral (3.23) to this coordinate system, the only singular behavior at large $\lambda$ comes from the Weyl transformation factor as the operator $\varphi_1$ is moved from near infinity to a finite position. This factor implies that the integral (3.23) will go to zero as $\lambda^{-2\Delta_\varphi}$, proving clustering in this particular case. See section 6.8.2 for additional details. More generally, the same argument will also work for arbitrary $m$ as long as $n = 1$.

# 4 Euclidean CFT $\Rightarrow$ Wightman: basic strategy

We will now pass to the main task of our paper: given a Euclidean unitary CFT, recover Minkowski correlators and show that they satisfy Wightman axioms.

Let us first discuss this problem without assuming conformal invariance. Suppose we know correlators $G_n^E(x_1, \dots, x_n)$ of a scalar field in a Euclidean QFT which is translationally and rotationally invariant, but not necessarily conformally invariant. We are assuming, as discussed above, that the correlators $G_n^E$ are defined and real-analytic (see footnote 13) for non-coincident Euclidean points $(x_k \in \mathbb{R}^d, x_i \neq x_j)$.

We would like to recover correlators in Minkowski signature. We are only interested here in Wightman correlation functions, where the operator ordering is fixed while the Minkowski time coordinates vary independently. We will call them simply “Minkowski correlators”. Starting from this section we will focus on correlators of scalar primaries; correlators of fields in general $SO(d)$ representations will be considered in our future publication [42].

To understand the equations below, it helps to keep in mind the basic heuristic. If we had a Hilbert space, field operators $\phi$, and a Hamiltonian $H$, then the Minkowski correlators would be given by

$$ G_n^M(x_1^M, \dots, x_n^M) = \langle 0 | \phi(0, \mathbf{x}_1) e^{-iH(t_1-t_2)} \phi(0, \mathbf{x}_2) e^{-iH(t_2-t_3)} \dots | 0 \rangle, \\ x_k^M = (t_k, \mathbf{x}_k). \quad (4.1) $$

while the Euclidean correlators by

$$ G_n^E(x_1, \dots, x_n) = \langle 0 | \phi(0, \mathbf{x}_1) e^{-H(\epsilon_1 - \epsilon_2)} \phi(0, \mathbf{x}_2) e^{-H(\epsilon_2 - \epsilon_3)} \dots | 0 \rangle, \\ x_k = (\epsilon_k, \mathbf{x}_k), \quad \epsilon_k > \epsilon_{k+1}, \quad (4.2) $$
We stress that the r.h.s. of these two equations will never be used in this paper. We just use them to illustrate the intuitive property that $G_n^M$ can be recovered from $G_n^E$ by analytic continuation $\epsilon_k \to \epsilon_k + it_k$ and sending $\epsilon_k \to 0$ while respecting $\epsilon_k > \epsilon_{k+1}$. In other words, there is a holomorphic function $G_n$ which reduces to $G_n^M$ in one limit and to $G_n^E$ in another. The precise domain of analyticity of $G_n$ can be clarified from the Wightman axioms [5]. Their basic consequence is that Minkowski correlators can be analytically continued to the “forward tube” (see below), which contains the configuration space of Euclidean time-ordered points as a section. In this paper we will derive Wightman axioms, rather than assume them. In particular, we will carry out analytic continuation to the forward tube just from the properties of the Euclidean correlators.

Let us put these observations into a definition of what it means to recover $G_n^M$ from $G_n^E$. We consider $n$-point configurations with complexified coordinates:

$$c = (x_1, \dots, x_n), \quad x_k = (x_k^0, \mathbf{x}_k) \in \mathbb{C}^d. \qquad (4.3)$$

The “forward tube” $\mathcal{T}_n$ is defined as the set of all such configurations for which the differences $y_k = x_k - x_{k+1} = (y_k^0, \mathbf{y}_k) \in \mathbb{C}^d$ satisfy the constraint:

$$\mathrm{Re} y_k^0 > |\mathrm{Im} \mathbf{y}_k|, \quad k = 1, \dots, n-1. \qquad (4.4)$$

Equivalently, this means that vectors $\mathrm{Im}(iy_k^0, \mathbf{y}_k)$ belong to the open forward light cone of $\mathbb{R}^{1,d-1}$, explaining the name “forward tube”.<fnref n="33" />

Let $\mathcal{D}_n$ be the subset of the forward tube consisting of the configurations with real spatial parts $\mathbf{x}_k$. Equivalently, we have:

$$\mathcal{D}_n = \{ c \mid x_k^0 = \epsilon_k + it_k, \mathbf{x}_k \in \mathbb{R}^{d-1}, \epsilon_1 > \epsilon_2 > \dots > \epsilon_n \}. \qquad (4.5)$$

Finally, we denote by $\mathcal{D}_n^E$ the Euclidean part of $\mathcal{D}_n$ obtained by setting all $t_k = 0$.

Minkowski correlators are then defined by the following two-step procedure:

**Step 1.** One finds an extension $G_n^E$ from $\mathcal{D}_n^E$ to a function $G_n(x_1, \dots, x_n)$ such that one of the two conditions is satisfied:

$$G_n \text{ is defined on } \mathcal{T}_n, \text{ and holomorphic in all variables } x_k^0, \mathbf{x}_k, \qquad (4.6)$$

or

$$G_n \text{ is defined on } \mathcal{D}_n, \text{ is holomorphic in variables } x_k^0 \text{ and is real-analytic in } \mathbf{x}_k. \qquad (4.7)$$

Real analyticity in $\mathbf{x}_k$ means that $G_n$ can be extended from $\mathcal{D}_n$ to a holomorphic function defined on a neighborhood of $\mathcal{D}_n$ which allows small imaginary parts for $\mathbf{x}_k$. This neighborhood can be arbitrarily small. Condition (4.7) is thus weaker than (4.6) and may be easier to check, although theorem 4.1 below shows that the two conditions are equivalent under the “powerlaw bound” assumption.

<footnote n="33">³³The just given definition of the forward tube is adapted to the Euclidean coordinates. In section 1.1, eq. (1.2), we wrote the same definition in terms of Minkowski coordinates $x_k^M = (-ix_k^0, \mathbf{x}_k)$.</footnote>
**Step 2.** Minkowski correlators are defined as the limits of $G_n$ from inside $D_n$ by sending $\epsilon_i \to 0$:<fnref n="34" />

$$G_n^M(x_1^M, \dots, x_n^M) = \lim_{\epsilon_i \to 0} G_n(x_1, \dots, x_n), \quad x_k^M = (t_k, \mathbf{x}_k), \quad k = 1 \dots n. \quad (4.8)$$

As mentioned several times, Minkowski correlators are expected to be tempered distributions, and therefore this limit has to be understood in the distributional sense. To show that the limit exists and has properties required by Wightman axioms, one relies on the following powerful theorem of several complex variables:

**Theorem 4.1** (Vladimirov's theorem). Suppose that the function $G_n$ is translation- and rotation-invariant, satisfies (4.7) and in addition satisfies everywhere on $D_n$ the following 'powerlaw bound' with some positive constants $C_n, A_n, B_n$:

$$|G_n(x_1, \dots, x_n)| \le C_n \left(1 + \max_k \frac{1}{\epsilon_k - \epsilon_{k+1}}\right)^{A_n} (1 + \max_i |x_i - x_{i+1}|)^{B_n}, \quad (4.9)$$

$$|x_i - x_j|^2 = |\epsilon_i + it_i - \epsilon_j - it_j|^2 + |\mathbf{x}_i - \mathbf{x}_j|^2. \quad (4.10)$$

Then:

1. Limit (4.8) exists in the sense of tempered distributions. The limiting value $G_n^M$ is a tempered distribution.<fnref n="35" />

2. The distribution $G_n^M$ is Poincaré-invariant and satisfies the Wightman spectral condition. I.e. its Fourier transform $W(p_1, \dots, p_{n-1})$ with respect to the differences $x_k^M - x_{k+1}^M$ has support in the product of closed forward light cones, which is the region $E_k \ge |\mathbf{p}_k|$, $p_k = (E_k, \mathbf{p}_k)$.

3. The function $G_n$ can be extended from a holomorphic function on the whole forward tube $\mathcal{T}_n$. The limit (4.8) exists also from the forward tube, i.e. when $\Re y_k^0 \to 0$, $|\mathrm{Im} \mathbf{y}_k| \to 0$, satisfying (4.4).

See appendix C for the proof of Vladimirov's theorem and a reminder of what the limit in the sense of distributions means. In the process of the proof, it will be established that the holomorphic function $G_n$ on $\mathcal{T}_n$ can be written as a "Fourier-Laplace" transform

$$G_n(x_1, \dots, x_n) = \int dp_1 \dots dp_{n-1} W(p_1, \dots, p_{n-1}) e^{\sum_{k=1}^{n-1} (-E_k(x_k^0 - x_{k+1}^0) + i \mathbf{p}_k \cdot (\mathbf{x}_k - \mathbf{x}_{k+1}))}, \quad (4.11)$$

where $W$ is a tempered distribution which is the Fourier transform of the tempered distribution $G_n^M$, mentioned in Part 2 of the theorem.

<footnote n="34">³⁴We will see in theorem 4.1 that this limit has to be taken along a fixed direction and is independent of direction. If the stronger condition (4.6) holds, the limit can in fact be taken along any direction in the forward null cone.</footnote>

<footnote n="35">³⁵This part of the theorem does not need translation- and rotation-invariance of $G_n$.</footnote>
To use theorem 4.1, one needs to verify the powerlaw bound (4.9). This strategy was first developed by Osterwalder and Schrader (OS) [2, 3].<fnref n="36" /> Their full list of assumptions included, in addition to reflection positivity and other OS axioms listed in section 2.2, a less widely known linear growth condition, which roughly says that $G_n^E$ (appropriately integrated) grows with $n$ not faster than a power of $n!$ and the degree of its singularities grows not faster than linearly in $n$. The proof of the powerlaw bound was the most technical part of the OS construction, and it crucially relied on the linear growth condition. See appendix 9 for the review.

In this paper we aim to define Minkowski correlators of a conformal field theory, given Euclidean correlators satisfying the CFT axioms of section 2.3. As seen in section 3.1, reflection positivity is robustly encoded in CFT via the positivity requirements for 2-point functions and reality constraints on the OPE coefficients. On the other hand, not much is known about how CFT $n$-point functions grow with $n$. In particular, we are unable to justify the OS growth condition in our setup, hence we cannot appeal to the OS theorem.

In this paper we will be able to circumvent this difficulty, by giving an alternative proof of the powerlaw bound for the most important in applications cases of 2, 3 and 4-point functions. Then, by theorem 4.1, these correlators exist in Minkowski space and are tempered distributions. Our proof of the powerlaw bound uses only the Euclidean CFT axioms. In fact, the two- and 3-point function case is almost trivial, these correlators being fixed by conformal invariance. The 4-point function case is much deeper and is one of our main results. Remaining Wightman axioms not mentioned in theorem 4.1 (positivity and clustering) will also be shown to hold.

**Remark 4.1.** In practice, to compute the Minkowski correlator function one may connect a Minkowski configuration to a Euclidean configuration by a curve $c(s)$, $0 \le s \le 1$, where $c(0)$ is Euclidean, $c(1)$ Minkowskian, and $c(s)$, $0 < s < 1$, belong to the forward tube. In general, the curve should remain in the forward tube except for the endpoint $c(1)$. This means that we must have strict inequalities:

$$ \mathrm{Re} x_1^0(s) > \mathrm{Re} x_2^0(s) > \dots > \mathrm{Re} x_n^0(s) \qquad (4.12) $$

everywhere along the analytic continuation contour, except for $s = 1$. See figure 1.

In the literature, one sometimes encounters a different prescription for computing the Minkowski correlators (see e.g. [4], section 3.1), where one puts all points but one at Minkowski positions, and considers correlators as a holomorphic function of the complexified coordinate of the remaining point. One then imagines that Wightman functions are holomorphic functions branching at light-cone separation, and that one can access different operator orderings by going around branch points. We would like to warn the reader that this prescription has to be taken with a grain of salt. To our knowledge there is no general result that the only Wightman functions singularities are branch cuts on the light cones. This is known to be true only in some special cases, e.g. for CFT 2-point and 3-point functions, as well as for CFT 4-point functions in $d = 2$ [57]. While some analytic continuation

<footnote n="36">³⁶OS used a slightly stronger version of theorem 4.1 with real analyticity in $x_k$ replaced by the weaker assumption of continuity in these variables, but this difference is not essential.</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1. Inequalities (4.12) should be satisfied along the analytic continuation contour.</figcaption></figure>

beyond the forward tube can be done in a general QFT (to the so called permuted extended tube), it does not suffice to justify the analytic continuation prescription of [4] in a general QFT. In CFTs in higher dimensions, the prescription of ref. [4] has some applicability, with the understanding that the correlator is analytic along the continuation contour but may stop being analytic at the endpoint (see appendix A.1).

**Remark 4.2.** A powerlaw bound in the forward tube (4.9) of course implies a powerlaw bound for the Euclidean 4pt function itself. Together with rotation invariance, this will imply the remaining OS axiom, the “Euclidean temperedness bound” (2.15). Indeed, by rotation invariance, we can choose the direction of the $x^0$ axis before applying the Euclidean powerlaw bound. Let us choose the $x^0$ direction so that, after ordering the operators according to $\epsilon_1 > \epsilon_2 > \epsilon_3 > \epsilon_4$, we have $\epsilon_k - \epsilon_{k+1} \ge \alpha|x_k - x_{k+1}|$ for each $k$. Such a direction exists for a sufficiently small positive $\alpha$, depending on $d$ and the number of points but not on $x_i$.<fnref n="37" /> Applying the Euclidean case of the powerlaw bound (4.9) in this frame we obtain (2.15).

## 4.1 Recovering Minkowski averages from Euclidean averages

Minkowski correlators provided by theorem 4.1, being tempered distributions, can be paired with a Schwartz test function $F$:

$$ (G_n^M, F) = \int dx G_n^M(x_1 \dots x_n) F(x_1 \dots x_n). \qquad (4.13) $$

Here we will discuss how these pairings can be computed given the Euclidean correlators (compare [2], section 4.3). This discussion will be needed in section 6.7 below and may be skipped on the first reading.

Eq. (4.13) can equivalently be expressed via the Fourier transform $W$ of $G_n^M$ with respect to $x_k - x_{k+1}$:

$$ (G_n^M, F) = \int dp W(p_1, \dots, p_{n-1}) f(p_1 \dots p_{n-1}), \qquad (4.14) $$

<footnote n="37">³⁷For each pair of points $(x_i, x_j)$ we consider the set of direction $\hat{e}_0$ such that $|(x_i - x_j) \cdot \hat{e}_0| \le |x_i - x_j| \sin \delta$. This gives a subset $U_\delta$ of the sphere $S^{d-1}$ with $\text{Vol}(U_\delta) \le 2\delta \text{Vol}(S^{d-2})$. If we choose $\delta_* = \frac{\text{Vol}(S^{d-1})}{2n(n-1)\text{Vol}(S^{d-2})}$, then the total volume excluded by considering all possible $(x_i, x_j)$ pairs is less than $\text{Vol}(S^{d-1})/2$. Therefore, we can find a direction $\hat{e}_0$ such that the opposite inequality $|(x_i - x_j) \cdot \hat{e}_0| \ge |x_i - x_j| \sin \delta_*$ holds for all pairs. Then, renumbering the points in the order of decreasing $x_i^0$, we obtain $x_k^0 - x_{k+1}^0 \ge |x_k - x_{k+1}| \sin \delta_*$.</footnote>
where $f(p_1 \dots p_{n-1}) = \hat{F}(-p_1, p_1 - p_2, \dots, p_{n-2} - p_{n-1}, p_{n-1})$. Natural pairings for Euclidean correlators are

$$ (G_n^E, \varphi) = \int dx G_n^E(x_1 \dots x_n) \Phi(x_1 \dots x_n). \qquad (4.15) $$

where $\Phi$ is a $C^\infty$ test function compactly supported in $x_1^0 > \dots > x_n^0$. We wish to discuss how pairings (4.13) or (4.14) can be found given (4.15).

By the Fourier-Laplace representation (4.11), we can write (4.15) as

$$ (G_n^E, \varphi) = \int dp \, W(p_1, \dots, p_{n-1}) g(p_1, \dots, p_{n-1}), \qquad (4.16) $$

where $g$ is any Schwartz class function which agrees inside the forward light cones with $\tilde{\varphi}$, the Fourier-Laplace transform of $\varphi(y_1, \dots, y_{n-1}) = \int dx_n \Phi(x_n + \sum_{i=1}^{n-1} y_i, x_n + \sum_{i=2}^{n-1} y_i, \dots, x_n)$:

$$ g \in S, \quad g(p) = \tilde{\varphi}(p) \quad (p_k \in \overline{V_+}). \qquad (4.17) $$

$$ \tilde{\varphi}(p_1, \dots, p_{n-1}) = \int dy \, \varphi(y_1 \dots y_{n-1}) e^{\sum_{k=1}^{n-1} (-p_k^0 y^0 + i \mathbf{p}_k \cdot \mathbf{y}_k)}, $$

Note that we cannot just put $g = \tilde{\varphi}$ because $\tilde{\varphi}$ is by itself not a Schwartz function (it may grow exponentially in the negative $p_k^0$ directions, although it will decrease exponentially in the positive one, since $\varphi$ is supported at $y_k^0 > 0$). On the other hand the values of $g$ outside the light cones, where $W$ is supported, are unimportant. We can for example take

$$ g(p_1, \dots, p_{n-1}) = \chi(p_1^0) \dots \chi(p_{n-1}^0) \tilde{\varphi}(p_1, \dots, p_{n-1}), \qquad (4.18) $$

where $\chi(s)$ is a $C^\infty$ function which equals identically 1 for $s \ge 0$ and 0 for $s < -1$.

Suppose then that we find a sequence of $C^\infty$ functions $\{\varphi_r\}_{r=1}^\infty$ compactly supported at $y_k^0 > 0$, the corresponding functions $g_r \in S$ such that $g_r = \tilde{\varphi}_r$ inside the light cones, and in addition that $g_r \to f$ in the sense of the Schwartz space (i.e. that all Schwartz space seminorms of the difference go to zero), where $f$ is the function in (4.14). Let us put $\Phi_r(x_1, \dots, x_n) = \varphi(x_1 - x_2, x_2 - x_3, \dots, x_{n-1} - x_n)\omega(x_n)$ where $\omega$ is any $C_0^\infty$ function of integral one. Then we will have

$$ (G_n^E, \Phi_r) = (W, g_r) \longrightarrow (W, f) = (G_n^M, F) \quad (r \to \infty), \qquad (4.19) $$

and so we will solve the problem of computing Minkowski averages given Euclidean averages. The following lemma, loosely related to lemma 8.2 in [2], shows that it is indeed possible to find such sequences $\varphi_r$ and $g_r$ for any Schwartz class $f$.

**Lemma 4.2.** The set of functions $g \in S(\mathbb{R}^{d(n-1)})$ which satisfy (4.17) for some $\varphi$ a $C^\infty$ test function compactly supported in $y_k^0 > 0$ is dense in the Schwartz space.

We will give a formal proof; see appendix D for some intuition. We will consider the case $n=2$ as $n>2$ is no more complicated. For each $\varphi$ we consider the set $A_\varphi$ of Schwartz functions $g$ which satisfy (4.17):

$$ A_\varphi := \{g \in S(\mathbb{R}^d) \mid g|_{\overline{V_+}} = \tilde{\varphi}\}. \qquad (4.20) $$
We know that $A_\varphi$ is non-empty, e.g. we can take $g$ from (4.18) (it is not hard to show that this is a Schwartz function). Our lemma says that $A \equiv \text{“the union of } A_\varphi \text{ over all } \varphi\text{”}$ is dense in $S(\mathbb{R}^d)$. The proof will be by contradiction. Note that $A$ is a linear subspace of $S(\mathbb{R}^d)$. If $\bar{A} \neq S(\mathbb{R}^d)$, then there exists a tempered distribution $T \in S'(\mathbb{R}^d)$ such that $T$ vanishes on all test functions from $A$ but does not vanish identically.<fnref n="38" />

So $T$ in particular vanishes on $A_0 = \{g \in S(\mathbb{R}^d) | g|_{\bar{V}_+} = 0\}$ (take $\varphi = 0$). This means that the support of the distribution $T$ is contained inside $\bar{V}_+$. Consider the Fourier transform of $T$,

$$ \hat{T}(x) := \int \frac{d^d p}{(2\pi)^d} T(p) e^{ip^0 x^0 - i \mathbf{p} \cdot \mathbf{x}}. \qquad (4.21) $$

We can consider $\hat{T}(x)$ for real $x$ where it is a distribution. Since $\text{supp}(T) \subseteq \bar{V}_+$ it is also natural to consider $\hat{T}(\xi + i\eta)$ where $\xi, \eta$ are real and $\eta$ is in the forward cone. We know that $\hat{T}(x)$ is a holomorphic function for such $x = \xi + i\eta$. We also know that the distribution $\hat{T}(x)$ for real $x$ can be obtained as a limit of the holomorphic function $\hat{T}(\xi + i\eta)$ as $\eta \to 0$.

Let us now come back to the assumption that $(T, g) = 0$ for any $g \in A_\varphi$. We will apply this to a function $g$ of the form $g = X(p)\tilde{\varphi}$ where $X(p)$ is a $C^\infty$ function identically 1 on the forward light cone and such that $X(p)e^{-p^0 x^0 + i \mathbf{p} \cdot \mathbf{x}}$ is in Schwartz class for any $x^0 > 0$. It is easy to see that such functions $X(p)$ exist. Writing $(T, g)$ in full we get:

$$ \begin{aligned} 0 = (T, g) &= \int dp T(p) X(p) \int dx e^{-p^0 x^0 + i \mathbf{p} \cdot \mathbf{x}} \varphi(x) \\ &= \int dx \varphi(x) \int dp T(p) X(p) e^{-p^0 x^0 + i \mathbf{p} \cdot \mathbf{x}} \\ &= \int dx \varphi(x) \hat{T}(ix^0, \mathbf{x}). \end{aligned} \qquad (4.22) $$

The swap of the order of integration between the first and the second line can be justified as follows. Since $T(p)$ is a tempered distribution, we can write it as a finite sum of derivatives of continuous functions of power growth: $T(p) = \sum_\alpha \partial_\alpha^\alpha F_\alpha(p)$. Using distributional integration by parts, we can rewrite the first line of (4.22) as a sum of ordinary integrals, apply Fubini's theorem to swap the integration order, and integrate by parts back to express the answer in terms of $T(p)$.

Because $\varphi(x)$ has compact support in the region $x^0 > 0$, the argument of $\hat{T}$ in the last line of (4.22) is of the form $\xi + i\eta$ with $\eta = (x^0, 0)$ in the forward light cone, where we know $\hat{T}$ is analytic. So, from the fact that the last line of (4.22) vanishes for any $\varphi$ we conclude that

$$ \hat{T}(\xi + i\eta) = 0, \quad \xi = (0, \mathbf{x}), \quad \eta = (ix^0, 0), \quad x^0 > 0. \qquad (4.23) $$

The set of these points is a totally real submanifold, and so by analyticity we conclude that $\hat{T}(\xi + i\eta)$ is identically zero for any $\xi \in \mathbb{R}^d$ and any $\eta$ in the forward cone. Furthermore,

<footnote n="38">³⁸The corresponding statement for normed spaces is standard, being a well-known consequence of the Hahn-Banach theorem (see e.g. [58], Corollary 1.8). For the Schwartz space, we can first find a Schwartz norm $| \cdot |_n$, such that $\bar{A}$ is not everywhere dense with respect to this norm, and then apply the standard statement with the norm $| \cdot |_n$. This gives a linear functional $T$ on $S(\mathbb{R}^d)$ continuous with respect to $| \cdot |_n$, hence $T \in S'(\mathbb{R}^d)$.</footnote>
as mentioned above, $\hat{T}(x)$ for real $x$ is a boundary value of $\hat{T}(\xi + i\eta)$. Therefore, $\hat{T} = 0$ in the sense of distributions. However we assumed above that $T$ was not identically zero. The reached contradiction shows that $A$ is dense in $S$.

## 5 Two- and 3-point functions

Let us see how the strategy from section 4 works for the CFT 2-point and 3-point functions. The Euclidean 2-point and 3-point correlators of scalar primaries are given by $[x_{ij}^2 = (x_i - x_j)^2]$

$$G_2^E(x_1, x_2) = \frac{1}{(x_{12}^2)^\Delta}, \qquad (5.1)$$

$$G_3^E(x_1, x_2, x_3) = \frac{c_{123}}{(x_{12}^2)^{h_{123}}(x_{13}^2)^{h_{132}}(x_{23}^2)^{h_{23}}}, \quad h_{ijk} = (\Delta_i + \Delta_j - \Delta_k)/2. \qquad (5.2)$$

In this case, the standard way to obtain the Wightman functions is to write these Euclidean correlators in terms of $x_{ij}^2$ with $i < j$ (as we did). Substituting the analytic continuation of $x_{ij}^2$,

$$x_{ij}^2 = (x_i - x_j)^\mu (x_i - x_j)_\mu = (x_i^0 - x_j^0)^2 + (\mathbf{x}_i - \mathbf{x}_j)^2, \quad x_i = (x_i^0, \mathbf{x}_i) \in \mathbb{C}^d. \qquad (5.3)$$

into the Euclidean 2-point and 3-point functions expressions, we obtain their analytic continuations. Suppose further that $x_{ij}^2 \neq 0$, $i < j$, in the forward tube (this will be shown below). Then the functions

$$c \mapsto x_{ij}^2 \quad (i < j) \qquad (5.4)$$

are holomorphic functions from the forward tube to $\mathbb{C}\setminus\{0\}$, the universal covering of the complex plane minus the origin. On the other hand $z \mapsto z^h$ is holomorphic from this universal covering to $\mathbb{C}$. Composing these two holomorphic functions, we conclude that $(x_{ij}^2)^h$, $i < j$, are holomorphic on the forward tube. Hence this procedure analytically extends the Euclidean 2-point and 3-point functions to the whole forward tube $\mathcal{T}_n$ ($n = 2, 3$).

We will now give a simple lemma which proves that indeed $x_{ij}^2 \neq 0$, $i < j$, in the forward tube. Actually the lemma says that a bit more is true, namely $x_{ij}^2 \in \mathbb{C}\setminus(-\infty, 0]$. This has the following practical consequence. In general, to compute the analytic continuation of $(F(c))^h$, where $F(c)$ is a nonzero holomorphic function, we need to know the phase of $F(c)$, i.e. to which sheet of the Riemann surface $\mathbb{C}\setminus\{0\}$ it belongs. To compute the phase we need to connect $c$ to some $c_E$ by a curve and analytically continue along this curve, following the phase. However, this is unnecessary for $(x_{ij}^2)^h$. Indeed, by the lemma below $x_{ij}^2$ always belongs to the principal sheet. So there is no need to use a curve to compute the phase: it can be computed unambiguously just by plugging the coordinates into (5.3).

**Lemma 5.1.** Let $y = (y^0, \mathbf{y}) \in \mathbb{C}^d$ satisfy $\mathrm{Re} y^0 > |\mathrm{Im} \mathbf{y}|$. Then

$$y^2 = (y^0)^2 + \mathbf{y}^2 \in \mathbb{C}\setminus(-\infty, 0]. \qquad (5.5)$$
Proof. We will denote by Greek letters $\xi, \eta$, etc., vectors of Minkowski space $\mathbb{R}^{1,d-1}$ with the Minkowski inner product $\xi^2 = -(\xi^0)^2 + \xi^2$.39 Decomposing the vector $(iy^0, \mathbf{y})$ into its real and imaginary parts:

$$(iy^0, \mathbf{y}) = \xi + i\eta, \qquad (5.6)$$

condition Re $y^0 > |\mathrm{Im}\,\mathbf{y}|$ means that $\eta^0 > 0$ and $-\eta^2 > 0$, i.e. $\eta$ is in the open forward light cone, which we will denote by $\eta > 0$. In this notation, we have to prove that

$$(\xi + i\eta)^2 = \xi^2 - \eta^2 + 2i(\xi\eta) \notin \mathbb{C}\setminus(-\infty, 0]. \qquad (5.7)$$

where by our conventions all inner products involving $\xi, \eta$ are Minkowski. Suppose this is violated, i.e.

$$(\xi\eta) = 0, \quad \xi^2 - \eta^2 < 0, \qquad (5.8)$$

for some $\xi, \eta$. Since $\eta$ is timelike, $(\xi\eta) = 0$ implies that $\xi$ is spacelike. But then $\xi^2 - \eta^2 = \xi^2 + (-\eta^2) > 0$. Thus the two conditions in (5.8) cannot both be true, and (5.7) is proved. $\square$

As the next step of implementing the strategy from section 4, we need to check that the constructed analytic continuations satisfy a powerlaw bound so that we can apply theorem 4.1. Although we already constructed analytic continuation to the whole $\mathcal{T}_n$, we only need to check the bound on $\mathcal{D}_n$ which is somewhat easier. The powerlaw bound follows from the following lemma.

**Lemma 5.2.** (a) Let $y = (\varepsilon + is, \mathbf{y})$, $\varepsilon, s \in \mathbb{R}$, $\mathbf{y} \in \mathbb{R}^{d-1}$. Then $y^2$ is bounded above and below in the absolute value, as follows:

$$\varepsilon^2 \le |y^2| \le |y|^2 = |\varepsilon + is|^2 + \mathbf{y}^2; \qquad (5.9)$$

(b) On $\mathcal{D}_n$ ($n = 2,3$), each $1/(x_{ij}^2)^h$ factor in (5.1), (5.2) ($h \in \mathbb{R}$) satisfies a powerlaw bound:

$$\left| \frac{1}{(x_{ij}^2)^h} \right| \le \frac{|x_i - x_j|^B}{(\epsilon_i - \epsilon_j)^A} \quad (i > j), \qquad (5.10)$$

where $A = 2h$, $B = 0$ for $h$ positive and $A = 0$, $B = -2h$ for $h$ negative.

Proof. (a) The upper bound is obvious. Let us show the lower bound by an explicit computation (see lemma 6.3 below for an alternative proof). We have:

$$|y^2|^2 = |(\varepsilon + is)^2 + \mathbf{y}^2|^2 = (\varepsilon^2 - s^2 + \mathbf{y}^2)^2 + 4\varepsilon^2 s^2, \qquad (5.11)$$

Minimizing this in $\mathbf{y}$, we get

$$\min_{\mathbf{y}} |y^2|^2 = \begin{cases} (\varepsilon^2 - s^2)^2 + 4\varepsilon^2 s^2 = (\varepsilon^2 + s^2)^2, & |s| \le \varepsilon, \\ 4\varepsilon^2 s^2, & |s| \ge \varepsilon. \end{cases}$$

Minimizing this next in $s$, we find that the absolute minimum is located at $\mathbf{y} = 0$, $s = 0$, and is equal to $\varepsilon^4$. Part (b) follows from (a). $\square$

<footnote>39We remind the readers that everywhere in this paper we are using $-, +, \dots, +$ Minkowski signature.</footnote>
Now that we have the powerlaw bound, we can apply theorem 4.1. We conclude that the Minkowski 2-point and 3-point functions, defined as $\epsilon_i \to 0$ limits of the analytically continued Euclidean correlators, exist, are Lorentz-invariant tempered distributions, and satisfy the spectral condition.<fnref n="40" />

## 5.1 Comparison with the $i\epsilon$-prescription

Here we will comment on the “$i\epsilon$-prescription” often used in the literature to define Minkowski 2-point and 3-point correlators, and how it compares with our definition. We will focus on the 2-point case for definiteness (same remarks hold for the 3-point case).

The $i\epsilon$-prescription defines the Minkowski 2-point correlator $G_2^M(x_1^M, x_2^M)$ as

$$ \frac{1}{(-s - i\epsilon)^2 + \mathbf{y}^2)^\Delta} \qquad (5.12) $$

with $s = t_1 - t_2$, $\mathbf{y} = \mathbf{x}_1 - \mathbf{x}_2$ and taking the $\epsilon \to 0^+$ limit. The precise meaning of the limit is often left implicit in the physics literature. Away from the light cone the 2-point correlator is an ordinary function, the limit can be understood pointwise and it agrees with our definition. Clearly, on the light cone the limit must be understood in distributional sense, integrating against a test function $f(s, \mathbf{y})$. That is what we showed above: Vladimirov's theorem guarantees that the limit $\epsilon \to 0$ exists as a tempered distribution and can be therefore integrated against any Schwartz test function. In physics literature, one instead often hears that such integrals should be defined by “shifting the integration contour”. Note however that this alternative way of understanding the $\epsilon \to 0$ limit would only work for analytic test functions. Let us discuss the consequences of this limitation.

It is helpful to recall that the theory of distributions commonly uses three classes of test functions, denoted $S, K, Z$ [61]. Here $S$ is the space of Schwartz functions, $K$ (denoted sometimes by $\mathcal{D}$) is the space of compactly supported $C^\infty$ functions, and $Z$ consists of entire holomorphic functions decreasing faster than any power in the real directions and bounded by some fixed exponential in the imaginary directions. Note that $K, Z \subset S$. The corresponding distribution spaces thus satisfy the opposite inclusion: $S' \subset K', Z'$. The elements of $S'$ are precisely the tempered distributions discussed above, $K'$ are distributions on the compactly supported test functions$^{41}$, while $Z'$ is yet another distributional class.

Importantly, the Fourier transform $\mathcal{F}$ leaves $S$ invariant. Since the Fourier transform is defined in the theory of distributions by duality, we also have $\mathcal{F}(S') = S'$: the Fourier transform of a tempered distribution is also a tempered distribution. On the other hand, one can show (see [61]) that $\mathcal{F}(K) = Z$. This is the rationale behind introducing the space $Z$, and this also implies that $\mathcal{F}$ maps $K'$ to $Z'$ and vice versa.

Coming back to (5.12), shifting the integration contour defines this distribution as an element of $Z'$. The pairing with a test function $f \in Z$ is thus defined by

$$ \int_C dz \int d\mathbf{y} \frac{1}{(-z^2 + \mathbf{y}^2)^\Delta} f(z, \mathbf{y}) \qquad (5.13) $$

<footnote n="40">⁴⁰Since these are tempered distributions, their Fourier transforms are well defined. Explicit expressions for these Fourier transforms are known in many cases. See [59] for $\langle O_{\Delta,l}(p)O_{\Delta,l}(-p) \rangle$ and [60] for $\langle O_{\Delta_1}(p_1)O_{\Delta_2}(p_2)O_{\Delta,l}(p_3) \rangle$.</footnote>

<footnote n="41">⁴¹They are briefly mentioned in the proof of theorem 4.1, appendix C.2, after eq. (C.13).</footnote>
with the contour C running parallel to the real axis in the lower half plane.<fnref n="42" /> By the previous paragraph, this is then sufficient to define the Fourier transform of the 2-point function as an element of $K'$. By moving the contour far away from the real axis, one shows that the Fourier transform vanishes for negative energies, and by Lorentz invariance one concludes that the support must belong to the forward light cone. These arguments have parallels in the proof of Part 2 of theorem 4.1 (see appendix C.2).

Compared to this simple and almost elementary discussion, theorem 4.1 proves a stronger statement that the 2-point distribution (5.12) can be extended to test functions of Schwartz class and, furthermore, to functions which have only a finite number of derivatives as expressed by eq. (C.3). This can be seen as a finer characterization of the singularity structure at short distances. The Fourier transform is then also a tempered distribution, thus bounded by some power, which is a stronger statement than it being an element of $K'$ since those can grow arbitrarily fast at infinity.

Since the 2-point and 3-point correlators are known in closed form, one can in principle verify that their Fourier transform does not grow too fast at infinity by an explicit computation. This would provide an alternative proof of temperedness. Our point here is that theorem 4.1 reaches this conclusion without any computations. For the 4-point correlators considered below, the Fourier transform cannot be evaluated easily, and theorem 4.1 appears to be the only realistic way to show temperedness.

It is instructive to discuss why we insist so much on temperedness. In other words, why Wightman axioms require that the Minkowski $n$-point correlators must be tempered distributions, and not of some other class? There is a simple reason why temperedness is a natural requirement, while $K'$ or $Z'$ would not suffice. The point is that Wightman axioms include both commutativity at spacelike separation and the spectral condition (the Fourier transform supported in the forward tube). Both these conditions need compactly supported test functions: the former in position space, the latter in momentum space. The space $S$ is large enough to write both these conditions, while $K$ or $Z$ are inadequate as we would lose one of them.<fnref n="43" />

Finally, sometimes by the $i\epsilon$-prescription one means the following simplified form of (5.12):

$$ \frac{1}{(-s^2 + i0^+ \operatorname{sign}(s) + \mathbf{y}^2)^\Delta}, \qquad (5.14) $$

which agrees with (5.12) away from the light cone. By Vladimirov's theorem, this defines a distribution for $s > 0$ (including the light cone) and another distribution for $s < 0$, but it is not an adequate starting point for defining the distribution around $(s, \mathbf{y}) = (0, 0)$.

<footnote n="42">⁴²We can somewhat relax the condition $f \in Z$. At the very least, $f$ must be holomorphic in the lower half-plane close to the real axis and decrease sufficiently fast at infinity for the integral to be convergent.</footnote>

<footnote n="43">⁴³For completeness it should be noted that one can reduce $S$ a bit and still be able to formulate both these axioms, as for Jaffe fields [62], which may have stronger-than-powerlaw singularities at short distances. For CFTs and for any theory which asymptotes to a CFT at short distances, there is no reasons to consider such modifications, and $S$ remains the natural choice.</footnote>
# 6 Scalar 4-point function

This section is the heart of our paper. In it we will show how to define Minkowski 4-point functions starting from Euclidean 4-point functions of a unitary CFT. We will follow the strategy of section 4 and in particular will rely on theorem 4.1. To avoid inessential details, we will focus on the case of four identical scalars. Non-identical scalars can be treated by the same argument (see section 6.10). Additional complications arise for spinning operators; this case is postponed to a future publication [42].

So, we consider the Euclidean CFT 4-point function of four identical scalar Hermitian primaries, which by conformal invariance can be written as:

$$G_4^E(c_E) = \langle \mathcal{O}(x_1)\mathcal{O}(x_2)\mathcal{O}(x_3)\mathcal{O}(x_4) \rangle = \frac{1}{(x_{12}^2 x_{34}^2)^{\Delta_O}} g(c_E). \quad (6.1)$$

Here $c_E = (x_1, x_2, x_3, x_4)$ denotes an ordered configuration of four Euclidean non-coincident points ($x_k \in \mathbb{R}^d$, $x_i \neq x_j$), $\Delta_O$ is the scaling dimension of $\mathcal{O}$, and $g(c_E)$ is a real function which depends only on the conformal class of $c_E$. It can be written as a function of two conformally invariant cross-ratios $u, v$:

$$g(c_E) = g(u, v), \quad u = \frac{x_{12}^2 x_{34}^2}{x_{13}^2 x_{24}^2}, \quad v = \frac{x_{14}^2 x_{23}^2}{x_{13}^2 x_{24}^2}. \qquad (6.2)$$

Our plan is as follows. After a discussion of the basic issues involved in the analytic continuation of the 4-point function (section 6.1), we will introduce a representation in terms of the radial coordinates $\rho, \bar{\rho}$ (section 6.2), and use it to construct the analytic continuation to the whole forward tube $\mathcal{T}_4$ (section 6.3). This construction works because $\rho, \bar{\rho}$ remain strictly inside the unit disc everywhere in the forward tube (lemma 6.1 and eq. (6.21)), a fundamental fact proved in section 6.4. We then briefly review the well-established powerlaw bound on $g(\rho, \bar{\rho})$ with respect to $\rho, \bar{\rho}$, and prove a powerlaw bound on $|\rho(c)|, |\bar{\rho}(c)|$ with respect to $c \in \mathcal{T}_4$. Combining these powerlaw bounds together, we will get a powerlaw bound on the analytically continued 4-point function $G_4(c)$, which by theorem 4.1 implies (as $c$ approaches the Minkowski region) the existence of the boundary limit of $G_4(c)$ as a tempered distribution (section 6.5). After establishing temperedness, we will derive the Minkowski conformal invariance (section 6.6), Wightman positivity (section 6.7), Wightman clustering (section 6.8) and local commutativity (section 6.9). Some of them do not rely on conformal properties: for these we will use the standard arguments given by Osterwalder and Schrader [2]. In section 6.10, we will generalize the above results to non-identical scalars by using Cauchy-Schwarz arguments.

## 6.1 Informal introduction to basic issues

Here we wish to outline a few basic difficulties which must be dealt with when analytically continuing the 4-point function. We will be using $u, v$ coordinates as an example, although we will see below that other coordinates will be more suitable for our task. Readers uninterested in philosophical discussions may skip directly to section 6.2.

Given any point $c$ of the forward tube, we can connect it to a Euclidean point $c_E$ by a curve, and analytically continue the 4-point function along the curve (see figure 2, left).
<figure><img src="image_3.png" /><figcaption>Figure 2. Illustration of the discussion in section 6.1.</figcaption></figure>

The forward tube being simply connected, the analytic continuation (if it exists) does not depend on the curve. Furthermore, let us take into account that our conformal 4-point function effectively only depends on two variables $u, v$. Applying lemma 5.1, we see that $u, v$ are both nonzero holomorphic functions on the forward tube. Consider the map:

$$ \omega : c \to (u, v). \qquad (6.3) $$

Since the forward tube is simply connected, we can consider this map as acting from $\mathcal{T}_4$ to $(\mathbb{C}\setminus\{0\})^2$, where tilde denotes the universal cover. Denote by $\omega(\mathcal{T}_4)$ and $\omega(\mathcal{D}_4^E)$ the images of the forward tube and of its Euclidean part under this map (see figure 2, right).

Now suppose that we found an analytic continuation of $g(u, v)$ from $\omega(\mathcal{D}_4^E)$, where it is initially defined, to the whole of $\omega(\mathcal{T}_4)$. Then we could immediately write down the analytic continuation of the 4-point function to the forward tube as follows:⁴⁴

$$ G_4(c) = \frac{1}{(x_{12}^2 x_{34}^2)^{\Delta_O}} g(u(c), v(c)). \qquad (6.4) $$

This formula defines the analytic continuation to the forward tube as a composition of two holomorphic functions:

$$ \mathcal{T}_4 \xrightarrow{\omega} \omega(\mathcal{T}_4) \xrightarrow{g} \mathbb{C}. \qquad (6.5) $$

We would like to use this strategy, but its direct implementation is hindered by a couple of difficulties:

* We don't know much about the shape or even topology of $\omega(\mathcal{T}_4)$. E.g. we don't know if this set is simply connected. The continuous image of a simply connected set, such as the forward tube, does not have to be simply connected (figure 3). If $\omega(\mathcal{T}_4)$ is not simply connected, there is no guarantee that $g(u, v)$ will be single-valued on it. And if $g(u, v)$ has branch cuts, then a simple formula like (6.4) using only the endpoint values $(u(c), v(c))$ will not work; we will need to know in addition “from which side of the cut” we got to this point along the analytic continuation contour (see figure 3).

* To be sure, we don't know if the above difficulty is actually realized. Perhaps the set $\omega(\mathcal{T}_4)$ is, after all, simply connected, and $g(u, v)$ has a single-valued analytic continuation to it. Even if this is the case, how can we construct this extension starting from $g(u, v)$ in the Euclidean region?
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Illustration of a potential difficulty if the set $\omega(\mathcal{T}_4)$ were not simply connected (see section 6.1).</figcaption></figure>

In this paper we will circumvent these difficulties rather than attacking them head-on. In the Euclidean region, one often uses different variables to parametrize the cross-ratios $u, v$, such as the Dolan-Osborn variables $z, \bar{z}$, or the radial variables $\rho, \bar{\rho}$. As one can imagine, a smart choice of Euclidean variables can greatly simplify the analytic continuation. We will see that the radial variables are ideally suited for this task, allowing a natural resolution of the above-mentioned difficulties.

## 6.2 Euclidean 4-point function in radial coordinates

We first recall the well-known Dolan-Osborn variables $z, \bar{z}$ [63, 64], which are two complex variables related to $u, v$ by

$$u = z\bar{z}, \quad v = (1-z)(1-\bar{z}), \quad \text{or} \tag{6.6}$$

$$z, \bar{z} = \frac{1}{2} \left( 1 + u - v \pm \sqrt{(1+u-v)^2 - 4u} \right). \tag{6.7}$$

Since in the Euclidean case we only consider non-coincident points, we have $u, v \neq 0$, and hence $z, \bar{z} \neq 0, 1$. It is possible to fix a Euclidean conformal frame by setting the four points to positions

$$x_1 = 0, \quad x_2 = a\hat{e}_0 + b\hat{e}_1, \quad x_3 = \hat{e}_0, \quad x_4 = \infty\hat{e}_0, \tag{6.8}$$

where $\hat{e}_\mu$ is the standard orthonormal basis of $\mathbb{R}^d$. Using this frame, we obtain $z, \bar{z} = a \pm ib$. This shows that in the Euclidean, the variables $z, \bar{z}$ are complex-conjugate ($\bar{z} = z^*$).

Euclidean configurations with real $z = \bar{z}$ correspond to four points lying on a circle, which maps in the frame (6.8) to four points on a line. The three possibilities $z < 0$, $z \in (0, 1)$, $z \in (1, +\infty)$ are then realized for different cycling orderings.

The radial variables $\rho, \bar{\rho} \in \mathbb{C}$ [65, 66] are defined in terms of the Dolan-Osborn variables by the formula:

$$\rho = f(z), \quad \bar{\rho} = f(\bar{z}), \quad f(w) := \frac{w}{(1 + \sqrt{1 - w})^2}. \tag{6.9}$$

The function $f(w)$ in this definition<fnref n="45" /> is the uniformization map for the complex plane minus the cut $(1, +\infty)$, i.e. it is a one-to-one map of $\mathbb{C}\setminus[1, +\infty)$ onto the unit disk. Eq. (6.9)

<footnote n="44">⁴⁴The prefactor analytically continues just as the 2-point and 3-point functions in section 5.</footnote>

<footnote n="45">⁴⁵The definition assumes the standard branch of the square root function.</footnote>
thus associates with any Euclidean configuration a pair of complex conjugate $\rho, \bar{\rho}$ ($\bar{\rho} = \rho^*$) belonging to the unit disk: $|\rho| \le 1$. Moreover we have $|\rho| < 1$ except for the Euclidean configurations with $z = \bar{z} \in (1, +\infty)$. As explained above, this happens when four points lie on a circle in the cyclic order 1324. For such exceptional configurations one may define $\rho, \bar{\rho}$ by continuity so that $|\rho| = 1, \bar{\rho} = \rho^*$.

The meaning of the coordinate $\rho$ is clarified by mapping the 4-point configuration to a conformal frame (compare (6.8))

$$x_1 = -\alpha \hat{e}_0 - \beta \hat{e}_1, \quad x_2 = \alpha \hat{e}_0 + \beta \hat{e}_1, \quad x_3 = \hat{e}_0, \quad x_4 = -\hat{e}_0, \qquad (6.10)$$

Using this frame, we obtain $\rho, \bar{\rho} = \alpha \pm i\beta$.

There is a small difference between $d=2$ and $d \ge 3$ dimensions. In $d \ge 3$, conformal frames (6.8) and (6.10) are unique only up to a sign of $b$ and $\beta$ (flipped rotating by $\pi$ in the 12 plane), which implies that pairs $(z, \bar{z})$ and $(\rho, \bar{\rho})$ are defined only up to permutation. On the other hand in $d=2$ flipping the sign of $b$ or $\beta$ is a parity transformation, which is not in the identity component of the conformal group. Hence the conformal frames are unique and $z, \bar{z}$ as well as $\rho, \bar{\rho}$ are individually meaningful.

In a unitary Euclidean CFT, the 4-point function admits a power-series expansion in the $\rho$ coordinate, absolutely convergent whenever $|\rho| < 1$ [65]. Specifically, the function $g(c_E)$ appearing in the 4-point function (6.1) of four identical scalar Hermitean primaries has a series expansion of the form

$$g(c_E) = \sum_{\delta, m} p_{\delta, m} r^{\delta} e^{im\theta}, \qquad (6.11)$$

where the sum runs over a discrete set of pairs $(\delta, m)$ with $\delta \ge 0$, $m \in 2\mathbb{Z}$, and the variables $r, \theta \in \mathbb{R}$ are the modulus and the phase of $\rho(c_E) = re^{i\theta}$. The sum is absolutely convergent when $r = |\rho(c_E)| < 1$. In addition, we know that $|m| \le \delta$ and $p_{\delta, m} \ge 0$ for all terms in (6.11). Finally, when $d \ge 3$ we have $p_{\delta, -m} = p_{\delta, m}$, so that the r.h.s. of (6.11) is uniquely defined in spite of $\rho(c_E)$ being defined only up to complex conjugation.

The readers familiar with this fact may skip to section 6.3 where we will use it to perform analytic continuation. In the rest of this section we recall how it follows from the CFT axioms [65, 67].

We consider the 4-point function in the conformal frame configuration (6.10) and write it as the inner product of two states created by the operators outside and inside a unit sphere $S$ centered at the origin:

$$\langle \mathcal{O}(1, 0, \mathbf{0}) \mathcal{O}(-1, 0, \mathbf{0}) | \mathcal{O}(\alpha, \beta, \mathbf{0}) \mathcal{O}(-\alpha, -\beta, \mathbf{0}) \rangle \qquad (6.12)$$

We can find a conformal transformation which maps the sphere $S$ to $x^0 = 0$ plane, its center 0 to $x_S$ and the infinity to $x_N$. This is the setup in which we developed the CFT Hilbert space picture in section 3.2. Applying the inverse transformation, we are allowed to use the Hilbert space language in the frame (6.12), which is the familiar setting of radial quantization. We decompose the radial quantization Hilbert space, produced by local
operators inserted at the origin, in orthonormalized eigenstates $|\delta, m\rangle$ of the dilatation D and the planar rotation $M_{01}$. The ket state is expanded in this basis as

$$|\mathcal{O}(\alpha, \beta, \mathbf{0})\mathcal{O}(-\alpha, -\beta, \mathbf{0})\rangle = \sum_{\delta, m} c_{\delta, m} r^{\delta - 2\Delta_{\phi}} e^{im\theta} |\delta, m\rangle. \qquad (6.13)$$

The dependence of the expansion coefficients in this formula on $r$ and $\theta$ is fixed by knowing how the state in the l.h.s. transforms under rotations and dilatations. The transformation $\theta \to \theta + \pi$ swaps the two operators leaving the state invariant for the considered case of identical operators. Hence the state in the r.h.s. also must remain invariant, proving that $m$ must be even.

Setting $r = 1, \theta = 0$ in (6.13), we get

$$|\mathcal{O}(1, 0, \mathbf{0})\mathcal{O}(-1, 0, \mathbf{0})\rangle = \sum_{\delta, m} c_{\delta, m} |\delta, m\rangle. \qquad (6.14)$$

In the considered frame the OS reflection is the inversion with respect to the sphere S: $x^\mu \to x^\mu/x^2$. In particular, this leaves $x_3$ and $x_4$ invariant. Applying this transformation to (6.14), we get

$$\langle \mathcal{O}(1, 0, \mathbf{0}) \mathcal{O}(-1, 0, \mathbf{0}) | = \sum_{\delta, m} c_{\delta, m}^* \langle \delta, m |. \qquad (6.15)$$

Taking the inner product of (6.13) and (6.15), we get

$$\langle \mathcal{O}(1, 0, \mathbf{0}) \mathcal{O}(-1, 0, \mathbf{0}) | \mathcal{O}(\alpha, \beta, \mathbf{0}) \mathcal{O}(-\alpha, -\beta, \mathbf{0}) \rangle = \sum_{\delta, m} |c_{\delta, m}|^2 r^{\delta - 2\Delta_{\phi}} e^{im\theta}. \qquad (6.16)$$

Comparing this with eq. (6.1), and using that $x_{12}^2 = 4r^2$, $x_{34}^2 = 4$ in the considered conformal frame, we obtain (6.11) with $p_{\delta,m} = 16^{\Delta_{\phi}}|c_{\delta,m}|^2 \ge 0$.

In the above argument we chose for simplicity the sphere of radius 1, but any sphere of radius $r < r_0 < 1$ would work equally well and give rise to the same expression. Absolute convergence for $r < 1$ follows, because both the bra and the ket states are normalizable for such $r_0$ (while for $r_0 = 1$ as above the bra state $\langle\varphi(x_3)\varphi(x_4)|$ is not normalizable).

The restriction $|m| \le \delta$ follows from the 2d unitarity bounds. The 2d unitarity bound applies, as any $d$-dimensional CFT restricted to a plane can be seen as a unitary 2d CFT. For 2d primaries of spin $J$ and dimension $\Delta$, the 2d unitarity bound says $|J| \le \Delta$. The descendants at level $n \in \mathbb{Z}_{\ge 0}$ have $\delta = \Delta + n$, $|m - J| \le n$, hence $|m| \le \delta$ follows.

Finally, let us prove that $p_{\delta,m} = p_{\delta,-m}$ in $d \ge 3$. We consider eq. (6.14) and perform a $\pi$ rotation in the 12 plane. In the r.h.s. $|\delta, m\rangle \to |\delta, -m\rangle$ because $M_{01} \to -M_{01}$ under such a rotation. On the other hand the l.h.s. does not change. This implies that we must have $c_{\delta,m} = c_{\delta,-m}$, and hence $p_{\delta,m} = p_{\delta,-m}$. (In $d = 2$, these properties also hold under the additional assumption of parity invariance.)

## 6.3 Analytic continuation

In this section we will construct the analytic continuation of the Euclidean 4-point function (6.1) to the forward tube $\mathcal{T}_4$ (recall the forward tube definition (4.4)). Analytic continuation to $\mathcal{D}_4 \subset \mathcal{T}_4$ has already been given in [37], section 3.4, and we will use a somewhat
streamlined version of that construction. We will analytically continue to the full forward tube $\mathcal{T}_4$, since this does not lead to additional complications.

The analytic continuation will be given by the formula

$$G_4(c) = \frac{1}{(x_{12}^2 x_{34}^2)^{\Delta_c}} g(c), \quad c \in \mathcal{T}_4. \qquad (6.17)$$

Here the prefactor trivially analytically continues to $\mathcal{T}_4$ similarly to the 2-point and 3-point functions discussed in section 5. We will construct $g(c)$, analytic continuation of $g(c_E)$, starting from eq. (6.11).

First we have to define the variables $z(c)$, $\bar{z}(c)$ on the forward tube, which is naturally done as follows. Given a configuration $c \in \mathcal{T}_4$, we evaluate $u = u(c)$, $v = v(c)$ via (6.2). By lemma 5.1, $u(c)$ and $v(c)$ are nonzero holomorphic functions on the forward tube. We then define $z(c)$, $\bar{z}(c)$ via (6.7):

$$z(c), \bar{z}(c) = \frac{1}{2} \left( 1 + u(c) - v(c) \pm \sqrt{[1 + u(c) - v(c)]^2 - 4u(c)} \right). \qquad (6.18)$$

Unlike for Euclidean configurations, for a general configuration $c \in \mathcal{T}_4$ these are two complex numbers unrelated by conjugation. Since $u(c)$ and $v(c)$ are nonzero, eq. (6.6) implies $z(c), \bar{z}(c) \in \mathbb{C}\setminus\{0, 1\}$.

Since eq. (6.18) only defines $z(c)$, $\bar{z}(c)$ up to permutation, we view it as a map from the forward tube to $\mathbb{C}^2/\mathbb{Z}_2$, the set of unordered pairs of complex numbers. This map is continuous, and is analytic everywhere except on $\Gamma \subset \mathcal{T}_4$ where the expression under the square root vanishes:

$$\Gamma = \{c \in \mathcal{T}_4 : [1 + u(c) - v(c)]^2 - 4u(c) = 0\}. \qquad (6.19)$$

Actually, it turns out that in $d=2$ one can resolve the ambiguity inherent in eq. (6.18) and define $z(c)$, $\bar{z}(c)$ as individually globally holomorphic functions on $\mathcal{T}_4$. We will bring up this fact below when we need it. Ref. [37], appendix A, showed that such an improvement is impossible in $d \ge 3$.

The following result is fundamental for our construction. The proof is elementary but a bit tricky and is postponed to section 6.4.

**Lemma 6.1.** For any $c \in \mathcal{T}_4$ we have $z(c), \bar{z}(c) \notin [1, +\infty)$.

We next define $\rho(c), \bar{\rho}(c)$ on $\mathcal{T}_4$, via

$$\rho(c) = f(z(c)), \quad \bar{\rho}(c) = f(\bar{z}(c)), \qquad (6.20)$$

where $f$ is the same function as in (6.9), mapping $\mathbb{C}\setminus[1, +\infty)$ onto the unit disk. By lemma 6.1, we then have$^{46}$

$$0 < |\rho(c)|, |\bar{\rho}(c)| < 1 \quad \text{for any} \quad c \in \mathcal{T}_4. \qquad (6.21)$$

Moreover, $\rho(c)$ and $\bar{\rho}(c)$ are locally holomorphic away from $\Gamma$. Because of this, and since eq. (6.11) for the 4-point function converges in the Euclidean for any $|\rho| < 1$, we may hope

<footnote>$^{46}$Note that the converse is not true: the region in which $0 < |\rho|, |\bar{\rho}| < 1$ is larger than the forward tube. For example, it includes the extended forward tube (see section 6.9.1).</footnote>
to use eq. (6.11) to analytically continue $g(c)$ to the whole forward tube. We will now carry out this strategy. Note that some extra care is needed, because $\rho(c)$ and $\bar{\rho}(c)$ are, just as $z(c)$ and $\bar{z}(c)$, not globally holomorphic and are defined only up to permutation (except in $d=2$, see below), and because (6.11) contains in general non-integer powers.

To begin with, we rewrite eq. (6.11) equivalently as

$$g(c_E) = \sum_{\delta,0 \le m \le \delta} (\rho\bar{\rho})^{\delta/2-m/2} (p_{\delta,m}\rho^m + p_{\delta,-m}\bar{\rho}^m), \qquad (6.22)$$

Various pieces of this formula need to be analytically continued to the forward tube. Consider first

$$R(c) = \rho(c)\bar{\rho}(c), \qquad (6.23)$$

which is a candidate for the analytic continuation of $\rho\bar{\rho}$ from the Euclidean region. We can view it as a composition of two functions: $c \mapsto (\rho(c), \bar{\rho}(c))$ which is a continuous function from $\mathcal{T}_4$ to $\mathbb{C}^2/\mathbb{Z}_2$ analytic away from $\Gamma$, followed by $(\rho, \bar{\rho}) \mapsto \rho\bar{\rho}$ which is a continuous holomorphic function from $\mathbb{C}^2/\mathbb{Z}_2$ to $\mathbb{C}$. Hence $R(c)$ is a continuous function on the forward tube, analytic everywhere except perhaps on $\Gamma$. However, manifold $\Gamma$ has complex codimension one, and by an analogue of Riemann's theorem about removable singularities we conclude that $R(c)$ is in fact analytic also on $\Gamma$, and thus on the whole $\mathcal{T}_4$.<fnref n="47" />

In addition, $R(c)$ is nonzero in the forward tube. Thus we can lift $R(c)$ to a holomorphic function $\tilde{R}(c)$ from the forward tube to the universal cover $\tilde{\mathbb{C}}\setminus\{0\}$. Composing this function with $z^h : \tilde{\mathbb{C}}\setminus\{0\} \to \mathbb{C}$, we obtain an analytic continuation of $(\rho\bar{\rho})^h$ for any $h \in \mathbb{R}$, which we denote by $R_h(c)$. This discussion mirrors the one around eq. (5.4) in section 5. However, unlike $x_{ij}^2$ in that discussion, it is not true that $\tilde{R}(c)$ always belongs to the principal sheet of $\tilde{\mathbb{C}}\setminus\{0\}$. So, in general, to compute the phase of the analytically continued function, one should follow the phase of $\rho\bar{\rho}$ along a curve joining $c_E$ to $c$.

Following a curve is perfectly fine as a theoretical device. For practical computations of the phase, one may wish to use instead the following trick which avoids having to look at the curve. (The reader happy to follow the curve may skip the trick and go directly to eq. (6.26).) Consider the identity:

$$\rho\bar{\rho} = \frac{1}{16}u(1+\rho)^2(1+\bar{\rho})^2 = \frac{1}{16}\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}Y^2, \quad Y = (1+\rho)(1+\bar{\rho}), \qquad (6.24)$$

which follows by using $z = \frac{4\rho}{(1+\rho)^2}$, the inverse of the relation (6.9) between $\rho$ and $z$, as well as $u = z\bar{z}$ and the expression for $u$. The function $Y(c) = (1+\rho(c))(1+\bar{\rho}(c))$ is holomorphic on $\mathcal{T}_4$ by the same “analyticity on $\mathcal{T}_4\setminus\Gamma$ plus Riemann’s theorem” argument as used above

<footnote n="47">⁴⁷The precise argument is as follows. Let us keep all complex coordinates fixed and vary just one, say $x_1^0$. There are two cases: either (6.19) is identically zero as a function of $x_1^0$, or it is a nonzero polynomial of $x_1^0$. In the first case $R(c)$ is trivially holomorphic in $x_1^0$. In the second case (6.19) vanishes at most for a few isolated values of $x_1^0$. We can then apply 1d Riemann's theorem to say that $R(c)$ is also analytic at those isolated points. By these arguments, we conclude that $R(c)$ is holomorphic in each variable separately. Finally, a continuous function of several complex variables holomorphic in each variable separately is jointly holomorphic [68].</footnote>
for $\rho\bar{\rho}$. In addition, and this is the key point, because $|\rho(c)|, |\bar{\rho}(c)| < 1$, we know that $Y(c) \in \mathbb{C}\setminus(-\infty, 0]$. The upshot of the trick is that eq. (6.24) expresses $\rho\bar{\rho}$ as a product of factors which all remain on the principal sheet of $z^h$ upon the analytic continuation. Hence we can compute the analytic continuation of $(\rho\bar{\rho})^h$ by

$$R_h(c) = \frac{1}{16^h} \frac{(x_{12}^2)^h (x_{34}^2)^h}{(x_{13}^2)^h (x_{24}^2)^h} Y(c)^{2h} \quad (h \in \mathbb{R}), \qquad (6.25)$$

This determines the phase of $R_h(c)$ unambiguously without having to look at the curve joining $c_E$ to $c$.

Next, we consider for an integer $m$ a function

$$\Phi_m(c) = \rho(c)^m + \bar{\rho}(c)^m. \qquad (6.26)$$

Just as $\rho\bar{\rho}$ and $Y$, it is continuous on $\mathcal{T}_4$ and holomorphic on $\mathcal{T}_4\setminus\Gamma$, and thus holomorphic on the whole $\mathcal{T}_4$.

We can now define the analytic continuation of (6.22). Consider first $d \ge 3$, when $p_{\delta,-m} = p_{\delta,m}$. In this case the analytic continuation is given by the formula

$$g(c) = \sum_{m, \delta, 0 \le m \le \delta} p_{\delta,m} R_{\delta/2-m/2}(c) \Phi_m(c). \qquad (6.27)$$

This series consists of holomorphic functions, and it reduces to (6.22) in the Euclidean region. Furthermore, every term in the series can be bounded in absolute value by:

$$\begin{aligned} |p_{\delta,m} R^{\delta/2-m/2}(c) \Phi_m(c)| &\le p_{\delta,m} |\rho(c) \bar{\rho}(c)|^{\delta/2-m/2} (|\rho(c)|^m + |\bar{\rho}(c)|^m) \\ &\le p_{\delta,m} r^{\delta-m} (r^m + r^m), \end{aligned} \qquad (6.28)$$

where $r = r(c) = \max(|\rho(c)|, |\bar{\rho}(c)|)$, which is $< 1$ by eq. (6.21). Here we used $p_{\delta,m} \ge 0$ in the first line, and $\delta - m \ge 0$ in the second line. The terms in the r.h.s. of (6.28) comprise a positive convergent series whose sum is the Euclidean 4-point function (6.22) evaluated at $\rho = \bar{\rho} = r(c)$. This proves that (6.27) converges uniformly on compact subsets of $\mathcal{T}_4$, and hence defines a holomorphic function in $\mathcal{T}_4$.

It remains to consider $d = 2$. As anticipated above, in this case the functions $z(c), \bar{z}(c)$ are individually globally holomorphic on $\mathcal{T}_4$. This can be seen introducing coordinates (see [37], section 3.5)

$$z_k = x_k^0 + i x_k^1, \quad \bar{z}_k = x_k^0 - i x_k^1, \quad k = 1, 2, 3, 4. \qquad (6.29)$$

Then the explicit formulas for $z(c), \bar{z}(c)$ are given by:

$$z(c) = \frac{(z_1 - z_2)(z_3 - z_4)}{(z_1 - z_3)(z_2 - z_4)}, \quad \bar{z}(c) = \frac{(\bar{z}_1 - \bar{z}_2)(\bar{z}_3 - \bar{z}_4)}{(\bar{z}_1 - \bar{z}_3)(\bar{z}_2 - \bar{z}_4)}. \qquad (6.30)$$

The functions $\rho(c), \bar{\rho}(c)$ defined by (6.20) are also individually globally holomorphic on $\mathcal{T}_4$. As a consequence, the functions $\rho(c)^m$ and $\bar{\rho}(c)^m$ are individually holomorphic in $d = 2$,
and not just their sum (6.26). We can therefore define the analytic continuation of $g(c)$ by the formula (compare (6.27)):

$$g(c) = \sum_{m, \delta, 0 \le m \le \delta} R_{\delta/2-m/2}(c) [p_{\delta,m} \rho(c)^m + p_{\delta,-m} \bar{\rho}(c)^m]. \quad (6.31)$$

This formula would be appropriate for non-parity invariant 2d CFTs which may have $p_{\delta,m} \ne p_{\delta,-m}$. Analyticity follows from the uniform convergence on compact subsets, by the same argument as for $d \ge 3$.

Finally, we wish to explain how the above construction may be translated into the language of section 6.1, to see how the issues raised there are resolved. This is instructive but not strictly speaking necessary, so we will be schematic. In the 2d case, when $\rho(c)$, $\bar{\rho}(c)$ are individually defined, the translation is in terms of the map

$$\Omega : c \mapsto (\rho(c), \bar{\rho}(c)) \in (\mathbb{D} \setminus \{0\})^2, \quad (6.32)$$

where $\mathbb{D}$ is the open unit disk, and we lifted each of the maps $\rho(c)$, $\bar{\rho}(c)$ to the universal cover of $\mathbb{D} \setminus \{0\}$. This map is the present analogue of $\omega$ in (6.3). The function $g(\rho, \bar{\rho})$ extends analytically to the whole $(\mathbb{D} \setminus \{0\})^2$, which makes it unnecessary to understand the precise shape of $\Omega(\mathcal{T}_4)$.

For $d \ge 3$, $\rho(c)$, $\bar{\rho}(c)$ are defined only up to permutation. Translation can then be done in terms of their symmetric combinations $\rho\bar{\rho}$, $\rho + \bar{\rho}$. Any symmetric polynomial in $\rho$, $\bar{\rho}$, such as the r.h.s. of (6.26), can be expressed as a polynomial in these coordinates. Let then $X$ be the image of $(\mathbb{D} \setminus \{0\})^2$ under the map $(\rho, \bar{\rho}) \mapsto (\rho\bar{\rho}, \rho + \bar{\rho})$. The following map is holomorphic on $\mathcal{T}_4$:

$$\Omega : c \mapsto (\rho(c)\bar{\rho}(c), \rho(c) + \bar{\rho}(c)) \in \tilde{X}, \quad (6.33)$$

where we lifted to the universal cover. The above argument can be interpreted as showing that the function $g(\rho, \bar{\rho})$ extends analytically to the whole $\tilde{X}$. Understanding the precise shape of $\Omega(\mathcal{T}_4)$ is once again unnecessary.

## 6.4 Proof of $z, \bar{z} \notin [1, +\infty)$

Here we will prove lemma 6.1 which played such a fundamental role in the previous section. Just as for lemma 5.1, it will be helpful to use the Minkowski metric. Thus we pass from Euclidean complex coordinates $x_k \in \mathbb{C}^d$ to Minkowski complex coordinates $\zeta_k = (ix_k^0, x_k) \in \mathbb{C}^{1,d-1}$. Definitions of $u, v$ are then rewritten equivalently as

$$u = \frac{\zeta_{12}^2 \zeta_{34}^2}{\zeta_{13}^2 \zeta_{24}^2}, \quad v = \frac{\zeta_{23}^2 \zeta_{14}^2}{\zeta_{13}^2 \zeta_{24}^2}, \quad (6.34)$$

where $\zeta_{ij} = \zeta_i - \zeta_j$ and $\zeta^2 = -(\zeta^0)^2 + \zeta^2$. We denote

$$\zeta_k = \xi_k + i\eta_k, \quad \xi_k, \eta_k \in \mathbb{R}^{1,d-1}. \quad (6.35)$$

We will thus use Minkowski norm for $\xi$'s, $\eta$'s and their differences. The forward tube condition on $x_k$ is rewritten as $\eta_k - \eta_{k+1} > 0$ which is the notation for

$$\eta_k^0 - \eta_{k+1}^0 > 0 \quad \text{and} \quad -(\eta_k - \eta_{k+1})^2 > 0. \quad (6.36)$$
We will need the following lemma which is related to lemma 5.1 (see the proof at the end of the section).

**Lemma 6.2.** Let $\zeta = \xi + i\eta$ and $\eta^2 < 0$. Then

(a) $\zeta^2 \neq 0$;

(b) Define $\zeta' = \xi' + i\eta'$ by

$$ \zeta' = \zeta/\zeta^2, \qquad (6.37) $$

which is finite by Part (a). Then $\eta'$ belongs to the same causal part of the light cone (future or past) as $\eta$. I.e. $\eta \succ 0 \Rightarrow \eta' \succ 0$. Analogously, $\eta \prec 0 \Rightarrow \eta' \prec 0$.

Let us start the proof of lemma 6.1. The $z, \bar{z}$ are defined from $u, v$ via (6.6). It is not hard to see from the first line of (6.6) that $z, \bar{z}$ are precisely the two solutions of the quadratic equation

$$ z^2 - (1 + u - v)z + u = 0. \qquad (6.38) $$

We thus have to show that, assuming (6.36), this equation has no solutions which are real and belong to the interval $[1, +\infty)$.

Without loss of generality, we can assume that $\zeta_3 = 0$.<fnref n="48" /> Then we have $\eta_1, \eta_2 \succ 0$ while $\eta_4 \prec 0$. Then we apply lemma 6.2 and map the configuration $(\zeta_1, \zeta_2, 0, \zeta_4)$ to the configuration $(\zeta'_1, \zeta'_2, \infty, \zeta'_4)$ with $\eta'_1, \eta'_2 \succ 0$ while $\eta'_4 \prec 0$. These relations imply $\eta'_{14} \succ 0, \eta'_{24} \succ 0$ which will be used below.

Since $u, v$ are invariant under the inversion, we have (this can be checked by a direct computation)

$$ u = \frac{(\zeta'_{12})^2}{(\zeta'_{24})^2}, \quad v = \frac{(\zeta'_{14})^2}{(\zeta'_{24})^2}, \qquad (6.39) $$

and eq. (6.38) reduces to

$$ (\zeta'_{24})^2 z^2 - [(\zeta'_{24})^2 + (\zeta'_{12})^2 - (\zeta'_{14})^2]z + (\zeta'_{12})^2 = 0. \qquad (6.40) $$

Using that $\zeta'_{12} = \zeta'_{14} - \zeta'_{24}$, this equation can be written equivalently as

$$ (\zeta'_{14} + (z - 1)\zeta'_{24})^2 = 0. \qquad (6.41) $$

Now let us suppose that $z \in [1, +\infty)$. Then

$$ \mathrm{Im}[\zeta'_{14} + (z - 1)\zeta'_{24}] = \eta'_{14} + (z - 1)\eta'_{24} \succ 0. \qquad (6.42) $$

Then eq. (6.41) is in contradiction with lemma 5.1. lemma 6.1 is demonstrated.

<footnote n="48">⁴⁸It is important to move $\zeta_3$ (or $\zeta_2$) to zero rather than $\zeta_1$ or $\zeta_4$, because only then, after applying the inversion, one gets causal information not only on $\eta'_k$'s but also on some of their differences.</footnote>
**Proof of lemma 6.2.** This was shown in [55], footnote 74, and we reproduce the argument here for completeness. Part (a) is a partial case of lemma 5.1 (for $\eta \prec 0$ we should apply it to the complex conjugate vector $\zeta^* = \xi - i\eta$). Let us show Part (b). To show that $\eta \succ 0 \Rightarrow \eta' \succ 0$, we write

$$ \zeta' = \frac{\xi + i\eta}{\xi^2 - \eta^2 + 2i(\xi, \eta)} = \frac{(\xi + i\eta)(\xi^2 - \eta^2 - 2i(\xi, \eta))}{(\xi^2 - \eta^2)^2 + 4(\xi, \eta)^2}. \quad (6.43) $$

So, up to a positive factor, $\eta'$ is given by

$$ (\xi^2 - \eta^2)\eta - 2(\xi, \eta)\xi. \quad (6.44) $$

For $\xi = 0$ this is given by $(-\eta^2)\eta \succ 0$. More generally, this squares to

$$ (\xi^2 - \eta^2)^2 \eta^2 + 4(\xi, \eta)^2 \xi^2 - 4(\xi, \eta)^2 (\xi^2 - \eta^2) = \eta^2 ((\xi^2 - \eta^2)^2 + 4(\xi, \eta)^2) < 0. \quad (6.45) $$

Therefore, for all $\xi$, we have that $\eta'$ is timelike. Since we have shown that $\eta' \succ 0$ for $\xi = 0$, by continuity it follows that $\eta' \succ 0$ for all $\xi$.

Finally, the implication $\eta \prec 0 \Rightarrow \eta' \prec 0$ follows by complex conjugation.

## 6.5 4-point function powerlaw bound

We wish to show next that the analytically continued 4-point function satisfies a powerlaw bound, so that we can apply theorem 4.1. The prefactor in eq. (6.17) satisfies a powerlaw bound by lemma 5.2. Furthermore, eq. (6.28) implies that the analytic continuation $g(c)$ constructed in section 6.3 is bounded by a Euclidean 4-point function, namely:

$$ |g(c)| \le g_E(c_*), \quad (6.46) $$

where $c_*$ is any Euclidean 4-point function configuration having $\rho(c_*) = \bar{\rho}(c_*) = r = r(c) = \max(|\rho(c)|, |\bar{\rho}(c)|)$. We choose the conformal frame (6.10):

$$ c_* : \quad x_1 = -r\hat{e}_0, \quad x_2 = r\hat{e}_0, \quad x_3 = \hat{e}_0, \quad x_4 = -\hat{e}_0. \quad (6.47) $$

Using the convergent OPE in the $x_2 \to x_3$, $x_1 \to x_4$ channel, we have the asymptotics

$$ G_4^E(c_*) \sim \frac{1}{(1-r)^{4\Delta_\varphi}} \quad (r \to 1). \quad (6.48) $$

The function $g_E(c_*)$ satisfies the same asymptotics up to a constant, being related to $G_4^E(c_*)$ via eq. (6.1) by a factor which is non-singular in the $r \to 1$ limit. Since $g_E(c_*)$ is a positive monotonically increasing function for $0 \le r < 1$ (see eq. (6.11)), we conclude that it has a bound

$$ g_E(c_*) \le \frac{\text{const.}}{(1 - r(c))^{4\Delta_\varphi}}, \quad (6.49) $$

and $|g(c)|$ by (6.46) satisfies the same bound.

The upshot of this discussion is that we will have a powerlaw bound on $G_4(c)$ if we manage to get a powerlaw bound on $\frac{1}{1-r(c)}$. We will next state and prove such a bound.
Before launching into the technical discussion, let us discuss intuitively why a result like this is expected to be true. We know (lemma 6.1) that $|\rho(c)|, |\bar{\rho}(c)| < 1$ and now we wish to prove that $|\rho(c)|, |\bar{\rho}(c)|$ do not approach 1 too quickly as $c$ goes to the Minkowski boundary of the forward tube. This may remind the reader of the Schwarz-Pick lemma, which says that if $f(w)$ is a function from a unit disk to itself and $f(0) = 0$, then $|f(w)| \le |w|$, hence providing a bound on how fast $|f(w)|$ can approach 1 as $|w| \to 1$. In the 2d case, when $\rho(c)$ and $\bar{\rho}(c)$ are individually defined holomorphic functions in the forward tube, it is indeed possible to use the Schwarz-Pick lemma to prove a powerlaw bound on $\max(|\rho(c)|, |\bar{\rho}(c)|)$ [69]. It should be possible to generalize the Schwarz-Pick argument to any $d$, although we have not worked it out in full details.<fnref n="49" /> The proof below will be different and more direct: it will simply mimic the proof of lemma 6.1, replacing all “> 0” inequalities by “$\ge \varepsilon$” with an explicit positive $\varepsilon$.

### 6.5.1 A powerlaw bound on $\frac{1}{1-r(c)}$

Let us introduce some notation. We will measure the size of a complex vector $\zeta \in \mathbb{C}^{1,d-1}$ by $|\zeta|,$

$$|\zeta|^2 = |\zeta^0|^2 + |\zeta^1|^2 + \dots + |\zeta^{d-1}|^2. \qquad (6.50)$$

Clearly $|\zeta_1, \zeta_2| \le |\zeta_1||\zeta_2|$. We also define for $\zeta = \xi + i\eta$, $\xi, \eta \in \mathbb{R}^{1,d-1}$, and $\eta^2 < 0$ (i.e. timelike)

$$S(\zeta) = \max\left(\frac{1}{\sqrt{-\eta^2}}, |\zeta|\right). \qquad (6.51)$$

Thus $S(\zeta)$ is large either if some component of $\zeta$ (real or imaginary) is large or if $\eta$ approaches the light cone. Note that $S(\zeta) \ge 1$ for any $\zeta$. We will never need $S(\xi + i\eta)$ for spacelike $\eta$.

Finally we consider an analogous function on $\mathcal{T}_4$:

$$S(c) = \max_{i<j} S(\zeta_{ij}), \qquad (6.52)$$

which becomes large if any of $S(\zeta_{ij})$ become large. We claim that there is the following bound (recall $r(c) = \max(|\rho(c)|, |\bar{\rho}(c)|)$)

$$\frac{1}{1-r(c)} \le 720S(c)^{12} \quad (c \in \mathcal{T}_4). \qquad (6.53)$$

This bound will be shown for any $c$ in the forward tube, which is the natural setting. When we specify to $c \in \mathcal{D}_4 \subset \mathcal{T}_4$ [see eq. (4.5)], we have

$$S(c) = \max_{i<j} \max \left\{ \frac{1}{|\epsilon_i - \epsilon_j|}, |x_i - x_j| \right\}. \qquad (6.54)$$

Eq. (6.53) then becomes a powerlaw bound for $\frac{1}{1-r(c)}$ on $\mathcal{D}_4$ of the form (4.9), precisely as needed for applying theorem 4.1.

<footnote n="49">⁴⁹For any $d$, the Schwarz-Pick lemma allows a natural generalization to holomorphic functions in the forward tube [70].</footnote>
The proof of the bound (6.53) will build upon the proof of $z, \bar{z} \notin [1, +\infty)$ given in section 6.4. There we showed that $z$ solves eq. (6.41), which however is inconsistent for $z \in [1, +\infty)$ and $c$ in the forward tube. Here we will use the same eq. (6.41), but make the rest of the argument quantitative, by showing that if $c$ stays away from the boundary or infinity of the forward tube, so that $S(c)$ is bounded, then both $z(c)$ and $\bar{z}(c)$ must stay a finite distance away from $[1, +\infty)$, as measured by an upper bound on $\frac{1}{1-r(c)}$ expressed by eq. (6.53). The proof is straightforward but somewhat technical and we split it into a series of lemmas.

**Lemma 6.3.** Let $\zeta = \xi + i\eta$, $\eta^2 < 0$. Then for any $\xi$

$$|\zeta|^2 \geq (-\eta^2). \qquad (6.55)$$

*Proof.* This is a generalization of lemma 5.2(a) and could be proven analogously. We give a slightly different proof for a change. We have

$$|\zeta|^2 = (\xi^2 - \eta^2)^2 + 4(\xi, \eta)^2 = (\xi^2)^2 + (\eta^2)^2 + 2[2(\xi, \eta)^2 - \xi^2\eta^2]. \qquad (6.56)$$

The lemma now follows from the inequality:

$$2(\xi, \eta)^2 - \xi^2\eta^2 \geq 0. \qquad (6.57)$$

Eq. (6.57) is obvious for $\xi^2 \geq 0$, so let us consider $\xi^2 < 0$. By Lorentz invariance and homogeneity it's enough to consider $\xi = (\pm 1, 0, \dots, 0)$ in which case the l.h.s. of (6.57) becomes $(\eta^0)^2 + \eta^2 \geq 0$. $\square$

Then we have the following strengthening of lemma 6.2(b):

**Lemma 6.4.** Let $\zeta = \xi + i\eta$, $\eta^2 < 0$, and $\zeta' = \zeta/\zeta^2$. Then

$$S(\zeta') \leq [S(\zeta)]^3. \qquad (6.58)$$

*Proof.* We have

$$|\zeta'| = \frac{|\zeta|}{|\zeta|^2} \leq \text{[by lemma 6.3]} \quad \frac{|\zeta|}{-\eta^2} \leq S(\zeta)^3. \qquad (6.59)$$

We also have (see the proof of lemma 6.2, in particular eq. (6.45)) that $\eta'^2 < 0$ and

$$\frac{1}{-\eta'^2} = \frac{|\zeta^2|^2}{-\eta^2} \leq \text{[by lemma 6.3]} \quad S(\zeta)^6. \qquad \square$$

**Lemma 6.5.** Let $\zeta_i \in \mathbb{C}^{1,d-1}$, $\eta_i > 0$ ($i = 1, 2$). Then

$$S(\zeta_1 + \zeta_2) \leq S(\zeta_1) + S(\zeta_2). \qquad (6.60)$$

*Proof.* We have $|\zeta_1 + \zeta_2| \leq |\zeta_1| + |\zeta_2|$ and $-(\eta_1 + \eta_2)^2 \geq -\eta_1^2 - \eta_2^2$ (since $\eta_1 \cdot \eta_2 < 0$). $\square$
**Lemma 6.6.** Let $\Upsilon_i = \Phi_i + i\Psi_i \in \mathbb{C}^{1,d-1}$, $\Phi_i, \Psi_i \in \mathbb{R}^{1,d-1}$, $\Psi_i > 0$ ($i=1,2$), and $z$ solves the equation

$$ (\Upsilon_1 + (z-1)\Upsilon_2)^2 = 0. \qquad (6.61) $$

Then

$$ 1 - |\rho(z)| \ge \delta_0 := \frac{1}{45S^4}, \qquad S = \max(S(\Upsilon_1), S(\Upsilon_2)). \qquad (6.62) $$

*Proof.* Note that $z = 4\rho/(1+\rho)^2$, and so eq. (6.61) can be rewritten as

$$ ((\rho+1)^2\Upsilon_1 - (\rho-1)^2\Upsilon_2)^2 = 0. \qquad (6.63) $$

For $\rho = e^{i\alpha}$, multiplying this equation by $e^{-2i\alpha}$, it becomes

$$ (\Upsilon)^2 = 0, \qquad \Upsilon = \left(2 \cos \frac{\alpha}{2}\right)^2 \Upsilon_1 + \left(2 \sin \frac{\alpha}{2}\right)^2 \Upsilon_2, \qquad (6.64) $$

which contradicts lemma 6.2(a), since $\mathrm{Im} \, \Upsilon > 0$. So $\rho$ cannot lie precisely on the unit circle (as we already knew). It should then not be surprising that it also cannot get too close to the unit circle, which is what (6.62) says. This can be shown by a straightforward although somewhat technical generalization of the above argument.

Denoting $\rho = re^{i\alpha} = e^{i\alpha} - \delta e^{i\alpha}$, $\delta = 1-r > 0$, and multiplying (6.63) by $e^{-2i\alpha}$, it becomes

$$ \left( \left( 2 \cos \frac{\alpha}{2} - \delta e^{i\alpha/2} \right)^2 \Upsilon_1 + \left( 2 \sin \frac{\alpha}{2} + i\delta e^{i\alpha/2} \right)^2 \Upsilon_2 \right)^2 = 0, \qquad (6.65) $$

or

$$ (\Upsilon + \Upsilon')^2 = 0 \qquad (6.66) $$

with

$$ \Upsilon = 4 \cos^2 \frac{\alpha}{2} \Upsilon_1 + 4 \sin^2 \frac{\alpha}{2} \Upsilon_2, \qquad (6.67) $$

$$ \Upsilon' = \kappa_1 \Upsilon_1 + \kappa_2 \Upsilon_2, \qquad (6.68) $$

$$ \kappa_1 = -4 \cos \frac{\alpha}{2} \delta e^{i\alpha/2} + \delta^2 e^{i\alpha}, \quad \kappa_2 = 4i \sin \frac{\alpha}{2} \delta e^{i\alpha/2} - \delta^2 e^{i\alpha}. \qquad (6.69) $$

So for $\delta$ small, $\Upsilon'$ is a small correction to $\Upsilon$. We write $\mathrm{Im}(\Upsilon + \Upsilon') = \Psi + \Psi'$, where

$$ \Psi = \mathrm{Im} \, \Upsilon = 4 \cos^2 \frac{\alpha}{2} \Psi_1 + 4 \sin^2 \frac{\alpha}{2} \Psi_2, \quad \Psi' = \mathrm{Im} \, \Upsilon'. $$

We know that $\Psi > 0$. In addition we also have a lower bound on $-\Psi^2$:

$$ -\Psi^2 \ge 16 \cos^4 \frac{\alpha}{2} (-\Psi_1^2) + 16 \sin^4 \frac{\alpha}{2} (-\Psi_2^2) \ge \frac{1}{S^2} \times 16 \min \left\{ \cos^4 \frac{\alpha}{2}, \sin^4 \frac{\alpha}{2} \right\} = \frac{4}{S^2}. \qquad (6.70) $$
We will now show that $-(\Psi + \Psi')^2$ remains strictly positive if $\delta < \delta_0$. This will imply, by lemma 6.2(a), that eq. (6.66) cannot hold, and hence we must have $\delta \ge \delta_0$, i.e. eq. (6.62), proving the lemma.

To implement this natural strategy, we will need only crude estimates of the size of various terms. Note that $\delta_0 < 1$ since $S \ge 1$, so in particular we have $\delta^2 \le \delta$. Using this we have the bounds $|\kappa_i| \le 5\delta$, and hence an upper bound

$$|\Psi'| \le |\Upsilon'| \le 10\delta S. \qquad (6.71)$$

We also have an upper bound $|\Psi| \le 4S$. Using these, (6.70), and $\delta^2 \le \delta$, we have:

$$\begin{aligned} -(\Psi + \Psi')^2 &= -\Psi^2 - 2(\Psi, \Psi') - (\Psi')^2 \\ &\ge \frac{4}{S^2} - 2|\Psi||\Psi'| - |\Psi'|^2 \\ &\ge \frac{4}{S^2} - 80\delta S^2 - 100\delta^2 S^2 \\ &\ge \frac{4}{S^2} - 180\delta S^2 = \frac{4(1 - \delta/\delta_0)}{S^2}, \end{aligned} \qquad (6.72)$$

which is strictly positive for $\delta < \delta_0$. As explained above this proves the lemma. $\square$

Finally we can prove (6.53). We repeat the proof of lemma 6.1 given in section 6.4. As there, we reduce to configuration having $\zeta_3 = 0$ and obtain that $z$ (as well as $\bar{z}$) is a solution of eq. (6.41), which has the form (6.61) with

$$\Upsilon_1 = \zeta'_{14} = \zeta'_1 - \zeta'_4, \quad \Upsilon_2 = \zeta'_{24} = \zeta'_2 - \zeta'_4, \quad \zeta'_i = \zeta_i/\zeta^2 \quad (i = 1, 2, 4). \qquad (6.73)$$

Let us write $\Upsilon_i = \Phi_i + i\Psi_i \in \mathbb{C}^{1,d-1}$, $\Phi_i, \Psi_i \in \mathbb{R}^{1,d-1}$. As was already pointed out in section 6.4, we have $\Psi_i > 0$ ($i = 1, 2$). Furthermore, by lemma 6.4 we know that $S(\zeta'_i) \le S(c)^3$, and then applying lemma 6.5 that $S(\Upsilon_i) \le 2S(c)^3$. Thus lemma 6.6 implies (6.53) (note that $720 = 45 \times 16$).

**Remark 6.1.** The bound (6.53) is not optimal. We will prove a better bound in section 7, by a different argument.

Let us recap. In section 6.3 we have analytically continued the Euclidean 4-point function to the forward tube, and here we showed that this analytic continuation satisfies a powerlaw bound. Then by theorem 4.1, the Minkowski 4-point function defined as the limit (4.8) exists, is a Lorentz-invariant tempered distribution, and satisfies Wightman spectral condition. In the remainder of this section we will show that this distribution is also conformally invariant (section 6.6), that it satisfies the remaining Wightman axioms (positivity in section 6.7, clustering in section 6.8, and local commutativity in section 6.9). Later in section 8 we will also show that it can be computed by a convergent (in the sense of distributions) OPE.

Now that we know that the Minkowski 4-point function is a distribution everywhere, one may inquire about the regularity of this distribution. E.g. for some configurations the 4-point function is actually real-analytic [37]. We will come back to this question in the conclusion section.
## 6.6 Conformal invariance

Conformal invariance of Euclidean 4-point function (6.1) can be described as invariance under finite conformal transformations $x \to x' = f(x)$,

$$ \Omega_1\Omega_2\Omega_3\Omega_4 G_4^E(x'_1, x'_2, x'_3, x'_4) = G_4^E(x_1, x_2, x_3, x_4), \quad (6.74) $$

where $\Omega_i = J(x_i)^{\Delta_O}$ and $J(x) = \det(\partial f^\mu / \partial x^\nu)^{1/d}$ is the local scale factor. Alternatively, and equivalently, this can be expressed as invariance under infinitesimal conformal transformations, a conformal Ward identity, which says that the Euclidean correlator is annihilated by a sum of differential operators, one per point:

$$ \sum_{i=1}^{4} \mathcal{D}(x_i, \partial_{x_i}) G_4^E(x_1, x_2, x_3, x_4) = 0. \quad (6.75) $$

There is a differential operator per conformal group generator $(\partial^\mu$ for $P_\mu$, $x^\mu \partial^\nu - x^\nu \partial^\mu$ for $M_{\mu\nu}$, $x \cdot \partial + \Delta_O$ for $D$, $x^2 \partial^\mu - 2x^\mu (x \cdot \partial) - 2x^\mu \Delta_O$ for $K_\mu$).

Since all these differential operators have polynomial coefficients, Ward identities (6.75) continue to hold in the forward tube for the function $G(x_1, x_2, x_3, x_4)$. Taking the limit to the Minkowski boundary, we obtain that the Minkowski 4-point function satisfies infinitesimal Minkowski conformal invariance expressed by the Ward identities.

The possibility to take the limit is guaranteed by the standard result that distributional limits commute with derivatives. Indeed, suppose that we have, in the sense of distributions, $\lim_{\epsilon \to 0} f_\epsilon = g$. This means that for any test function $\varphi$, we have $\lim_{\epsilon \to 0} (f_\epsilon, \varphi) = (g, \varphi)$. But then for any derivative $\partial$,

$$ (\partial g, \varphi) = -(g, \partial \varphi) = -\lim_{\epsilon \to 0} (f_\epsilon, \partial \varphi) = \lim_{\epsilon \to 0} (\partial f_\epsilon, \varphi), \quad (6.76) $$

which implies that $\lim_{\epsilon \to 0} \partial f_\epsilon = \partial g$. A similar argument shows that the limit commutes with multiplication of distributions by polynomials. All this is analogous to how we prove Lorentz invariance of the Minkowski correlator in appendix C.

So we have shown that the Minkowski 4-point function satisfies Lorentzian conformal Ward identities. This means that

$$ \sum_{i=1}^{4} (\mathcal{D}_i G_4^M, \varphi) = 0, \quad (6.77) $$

where $\mathcal{D}_i$ are the analytic continuations of the Euclidean differential operators to Minkowski space, and the pairing with the Schwartz test functions is defined by integration by parts. Note that the conformal Ward identities in Minkowski space hold also at coincident points (i.e. the test function $\varphi$ does not have to be zero at coincident points).

Now let us discuss invariance of Minkowski 4-point function under finite Lorentzian conformal transformations. Since $G_4^M$ is a distribution, the appropriate form of writing is to transform the test function:

$$ (G_4^M, \varphi) = (G_4^M, \varphi^f), \quad (6.78) $$
<figure><img src="image_5.png" /><figcaption>**Figure 4.** In the 2d case, the special conformal transformation (6.80) is singular on the blue light cone $x^0 = \pm|x^1 + \beta^{-1}|$. Suppose $\varphi$ is supported as shown on the right of the light cone. As $\beta \to 0$, the light cones moves towards the left infinity and does not touch supp($\varphi$). Therefore, such a $\varphi$ satisfies the condition for the invariance under a finite special conformal transformation (6.80).</figcaption></figure>

where $\varphi^f(x_1, \dots, x_4) = \varphi(f^{-1}(x_1), \dots, f^{-1}(x_4)) \prod_{i=1}^4 J(f^{-1}(x_i))^{\Delta\varphi^{-d}}$. However we have to be careful. This invariance is true not for every test function $\varphi$ but only for an $f$-dependent subset of test functions.

Let $f_t$ be a smooth family of Lorentzian conformal transformations connecting $f$ to the identity: $f_0 = \text{id}$, $f_1 = f$. Suppose that

$$ \varphi^{f_t} \text{ is a Schwartz function for any } f_t \text{ in the family.} \qquad (6.79) $$

Then we can integrate infinitesimal conformal invariance and prove that (6.78) is true. For translations, Lorentz transformations and dilatations, eq. (6.79) is clearly satisfied and eq. (6.78) holds for any $\varphi$. However, for general conformal transformations, (6.79) may not necessarily be true. The problems will appear if $f$ is singular on the support of $\varphi$, as $\varphi^f$ may then not be a Schwartz function. As a concrete example, consider the Lorentzian special conformal transformation:

$$ f(x) = \frac{x^{\mu} + x^{2}b^{\mu}}{1 + 2x \cdot b + x^{2}b^{2}}. \qquad (6.80) $$

The corresponding scale factor is $J(x) = \frac{1}{1+2x \cdot b+x^2b^2}$. Take for definiteness spacelike $b = \beta \hat{e}_1$, where $\beta > 0$ and $\hat{e}_1$ is the unit vector in the $x^1$ direction. The transformation (6.80) is then singular for $x^0 = \pm|x + \beta^{-1}\hat{e}_1|$, where the scale factor blows up, i.e. on the light cone whose vertex is at $x = -\beta^{-1}\hat{e}_1$.<fnref n="50" /> Scaling $\beta$ to zero we can connect the transformation (6.80) to the identity. Under this scaling the light cone of singularities moves away to infinity along the negative $x^1$ direction. Requirement (6.79), and hence finite invariance (6.78), will hold if the light cone of singularities, while moving away, does not touch the support of $\varphi$ (see figure 4 for the 2d case).

Note that such a support requirement still leaves the possibility for both spacelike and timelike separations among the points $x_i$ in the support of $\varphi$. For $x_i \in \text{supp} \varphi$, the points $f(x_i)$ will have the same causal structure as the points $x_i$, i.e. $(f(x_i) - f(x_j))^2$ will have the

<footnote n="50">⁵⁰Recall that we are using $-$, $+$, $\cdots$ + Minkowski signature.</footnote>
same sign as $(x_i - x_j)^2$. This follows from the fact that $J(x) > 0$ on supp($\varphi$), as guaranteed by being able to continuously connect to the identity without singularities.

In the early CFT days, it was considered puzzling that Lorentzian special conformal transformations may change the causal structure of a point configuration. As we see here, the puzzle can be avoided by either limiting to infinitesimal conformal invariance, or by restricting the class of test functions so that the causal structure is preserved. A third way to deal with the puzzle is to consider the Lorentzian conformal transformations acting on the Lorentzian cylinder as opposed to the Minkowski space [71]. We will revisit the Lorentzian cylinder in our future publication [72].

**Remark 6.2.** We would like to contrast the Minkowski conformal Ward identities (6.77) with conformal Ward identities valid for Euclidean correlators. Euclidean correlators are real-analytic away from coincident points and naturally satisfy conformal Ward identities for such configurations. Although in this paper we don't need it, in some questions it might be useful to extend Euclidean correlators, in the sense of distributions, also to coincident points. One may ask if such an extension can be done in a way so that the resulting distributional correlators satisfy conformal Ward identities analogous to (6.77). In general the answer is no, already for 2-point functions. Namely 2-point functions of primaries of dimension $\Delta$ such that $2\Delta - d \in \mathbb{Z}$ will in general not allow even a scale invariant extension at coincident points, let alone conformally invariant one. E.g. this feature will always be present for the stress tensor 2-point function.

### 6.6.1 Conformal invariance in terms of cross ratios

So as we have just seen, Minkowski correlator $G_4^M$ is conformally invariant. If it were a function, conformal invariance would imply that we could write it as the usual prefactor times a function of the cross-ratios. Since it is a distribution, one might hope that it can be written as the prefactor times a distribution of the cross-ratios. We will now develop this point of view and show that it indeed works, at least locally.<fnref n="51" />

Our goal in this section will be to make sense of the formula:

$$G_4(c) = \frac{g(z(c), \bar{z}(c))}{(x_{12}^2 x_{34}^2)^{\Delta\varphi}}, \qquad (6.81)$$

where $g(z, \bar{z})$ will be in general a distribution in two variables, and $g(z(c), \bar{z}(c))$ its pullback to the space $\mathbb{R}^{4d}$ of Minkowski 4-point configurations $c$. This equation will be understood in the sense of integrating both parts with a test function. Because of the difficulty described in footnote 51, we will only consider compactly supported $C^\infty$ test functions $\varphi(c)$, with the

<footnote n="51">⁵¹We note right away that one does not expect a very nice global statement. Indeed, the cross-ratio space is morally the moduli space of four points on Minkowski cylinder $M$. (Note that we have not yet constructed the Wightman functions as distributions on Minkowski cylinder. However, it is not important for the point that we are trying to convey.) This is a quotient space ($M^4)/G$ where $G$ is the universal cover of Lorentzian conformal group. This quotient space is rather singular, which has to do with different configurations in $M^4$ having different stability subgroups (light-cones, $z = \bar{z}$). The quotient space ($M^4)/G$ is not only not smooth, it is not even Hausdorff. So away from some regular regions of ($M^4)/G$ one shouldn't expect a simple statement of the form (6.81), unless one finds a smoother model of this moduli space.</footnote>
additional requirement that all $c \in \text{supp} \varphi$ have the same causal ordering. In particular, this implies that $\text{supp} \varphi$ contains no $c$'s with lightlike separated pairs. The causal ordering of a configuration $c = (x_1, x_2, x_3, x_4)$ is encoded by the directed graph with vertices 1, 2, 3, 4 and edges $i \to j$ if $x_j$ belongs to the open future light cone of $x_i$ (no edge if two points are spacelike).

Since $u, v$ are real in Minkowski space, $z, \bar{z}$ are either both real (excluding 0, 1) or complex conjugate. Ref. [37] divided all causal orderings into 4 classes according to possible values of $(z, \bar{z})$:

* Class S: $(z, \bar{z}) \in (0, 1) \times (-\infty, 0)$ or the other way around

* Class T: $(z, \bar{z}) \in (0, 1) \times (1, +\infty)$ or the other way around

* Class U: $(z, \bar{z}) \in (-\infty, 0) \times (1, +\infty)$ or the other way around

* Class E causal orderings which contain configurations realizing the remaining possibilities:

- $E_{\text{su}}: (z, \bar{z}) \in (-\infty, 0) \times (-\infty, 0)$

- $E_{\text{st}}: (z, \bar{z}) \in (0, 1) \times (0, 1)$

- $E_{\text{tu}}: (z, \bar{z}) \in (1, +\infty) \times (1, +\infty)$

- $E_{\text{stu}}: z, \bar{z}$ are complex-conjugate and not real

Some class E causal orderings realize only one of the four subclasses, while others contain configurations in each subclass. In the latter case different subclasses are connected along configurations with $z = \bar{z}$ (see figure 2 in [37]).

To simplify the discussion, we will assume that $\text{supp} \varphi$ does not include any configurations with $z = \bar{z}$. In particular, this implies that all configurations from $\text{supp} \varphi$ fall into a single class S, T, U or a single subclass $E_{\text{su}}, E_{\text{st}}, E_{\text{tu}}, E_{\text{stu}}$. Below we will comment how one can add the $z = \bar{z}$ configurations.

If $\text{supp} \varphi$ falls into class S, $E_{\text{su}}, E_{\text{st}}, E_{\text{stu}}$, we will have $|\rho|, |\bar{\rho}| < 1$. These cases do not require special treatment, since the correlator is a function, and eq. (6.81) is true in the ordinary sense of functions.

If $\text{supp} \varphi$ falls into class T or U, we will have $|\rho| < 1, |\bar{\rho}| = 1$ or the other way around. Then $g(z, \bar{z})$ will be a function in $z$ and a distribution in $\bar{z}$.<fnref n="52" /> This case can be treated analogously, and simpler, than the $|\rho|, |\bar{\rho}| = 1$ case discussed below.

Finally, if $\text{supp} \varphi$ falls into class $E_{\text{tu}}$, we will have $|\rho|, |\bar{\rho}| = 1$. Then $g(z, \bar{z})$ will generally be a distribution in two variables. This is the case we will focus on. E.g. it is realized for the causal ordering $1 \to 3 \to 2 \to 4$.

Let us define the distribution $g(z, \bar{z})$ for $z, \bar{z} \in (1, +\infty)$. We first define the distribution $g(\rho, \bar{\rho})$ with $|\rho|, |\bar{\rho}| = 1$. This is done using the series in the r.h.s. of eq. (6.22), which we now consider as a function of two independent variables $\rho, \bar{\rho}$. To be precise we consider the series:

$$g(\rho, \bar{\rho}) = \sum_{\delta, 0 \le m \le \delta} e^{i\Phi(\delta/2-m/2)} (\rho\bar{\rho})^{\delta/2-m/2} (p_{\delta,m}\rho^m + p_{\delta,-m}\bar{\rho}^m), \quad (6.82)$$

<footnote n="52">⁵²For some (but not all) of these causal orderings, it can be shown using another OPE channel that $g(z, \bar{z})$ is actually a function of both variables. See ref. [37].</footnote>
which we view as a holomorphic function on $(\mathbb{D}\setminus(-1, 0))^2$. Here $e^{i\Phi}$, $\Phi \in \{0, \pm 2\pi, \pm 4\pi\}$, is the phase acquired by $\rho(c)\bar{\rho}(c)$ upon analytic continuation from Euclidean space (as discussed in section 6.3 this phase is the same as for $u(c)$). This phase is constant for each causal ordering and it may be determined by following a path from $c_E$ to $c$ for any particular $c$. Alternatively, the phase can also be determined from (6.24). E.g. the causal ordering $1 \to 3 \to 2 \to 4$ has $\Phi = 0$.<fnref n="53" />

It's easy to see that function (6.82) satisfies a powerlaw bound as $|\rho|, |\bar{\rho}| \to 1$. This is a baby version of the problems studied in this paper, which was considered in [36]. The limit of $g(re^{i\theta_1}, re^{i\theta_2})$ as $r \to 1$ defines a tempered distribution on the boundary of the domain of analyticity, parametrized by the two angles $\theta_1, \theta_2$. We can write it as $g(\rho, \bar{\rho})$, with $\rho, \bar{\rho} \in S^1$.<fnref n="54" />

In fact we are interested only in a part of this distribution, because $\rho, \bar{\rho} \neq \pm 1$ for each fixed causal ordering. The points $-1, 1$ divide the circle into two open arcs, and within supp $\varphi$, $\rho$ and $\bar{\rho}$ will each live in one or the other arc. Each arc is mapped smoothly and one-to-one to $(1, +\infty)$ by the $\rho \mapsto z$ map. Thus we obtain the distribution $g(z, \bar{z})$ defined for $z, \bar{z} > 1$. Although in general $z(c), \bar{z}(c)$ are defined only up to permutation, let us define them in the case at hand, with real $z \neq \bar{z}$, so that $\bar{z}(c) > z(c)$.

Now let us go back to making sense of (6.81). Suppose first $g(z(c), \bar{z}(c))$ were a function. Integrating (6.81) against a test function we have:

$$ \int d^{4d}c G_4(c)\varphi(c) = \int d^{4d}c g(z(c), \bar{z}(c))\tilde{\varphi}(c), \quad \tilde{\varphi}(c) = \frac{\varphi(c)}{(x_{12}^2 x_{34}^2)^{\Delta\varphi}}. \quad (6.83) $$

Note that $\tilde{\varphi}(c)$ is still $C^\infty$ since we are away from light cones. We would like to continue by expressing the r.h.s. of the previous equation as an integral of $g(z, \bar{z})$ against a two-dimensional test function:

$$ \begin{aligned} \int d^{4d}c g(z(c), \bar{z}(c))\tilde{\varphi}(c) &= \int dz d\bar{z} g(z, \bar{z})\psi(z, \bar{z}), \\ \psi(x_1, x_2) &= \int d^{4d}c \delta(x_1 - z(c))\delta(x_2 - \bar{z}(c))\tilde{\varphi}(c). \end{aligned} \quad (6.84) $$

We would like to know if $\psi(x_1, x_2)$ is a smooth function. By our assumptions, $\tilde{\varphi}(c)$ is supported away from $\bar{z}(c) = z(c)$. In this region the map $c \to (z(c), \bar{z}(c))$ is a submersion, which means that the Jacobian has maximal rank (i.e. 2). Alternatively, this means that the form $dz \wedge d\bar{z}$ does not vanish anywhere away from $z = \bar{z}$. Showing this is a matter of an easy computation.<fnref n="55" />

<footnote n="53">⁵³We have that $x_{12}^2, x_{34}^2, x_{13}^2, x_{24}^2$ all acquire phase $-\pi$, hence $u = \frac{x_{12}^2 x_{34}^2}{x_{13}^2 x_{24}^2}$ acquires phase 0.</footnote>

<footnote n="54">⁵⁴In [36] we also discussed a more general distribution defined on the product of universal covers of two circles. Here eq. (6.82) with fixed $\alpha$ will be sufficient for our purposes.</footnote>

<footnote n="55">⁵⁵Start by noting that, away from $z = \bar{z}$, we have $dz \wedge d\bar{z} \propto du \wedge dv$ with a nonvanishing prefactor. We need to understand where $\nabla u$ can become proportional to $\nabla v$. Using the embedding space formalism [73] we write $u = \frac{(X_1 X_2)(X_3 X_4)}{(X_1 X_3)(X_2 X_4)}$, $v = \frac{(X_1 X_2)(X_3 X_4)}{(X_1 X_3)(X_2 X_4)}$ where $X_i$ are null cone $d+2$ dimensional vectors. For any $X_i, X_j, X_k$ the direction $R_{i,jk} = X_j(X_i X_k) - X_k(X_i X_j)$ is tangent to the null cone at $X_i$. Imposing $R_{i,jk} \cdot \nabla X_i (u - \alpha v) = 0$ for all unequal $i, j, k$ where $\alpha$ is a constant, one gets a set of simple algebraic constraints on $u, v$. These constraints can be easily solved to show that $\alpha = \frac{2u}{-1+u+v}$ while $(1+u-v)^2 - 4u = 0$. The latter is precisely the constraint characterizing $z = \bar{z}$.</footnote>
Using the fact that $c \to (z(c), \bar{z}(c))$ is a submersion, it's easy to show that $\psi(x_1, x_2)$ is smooth for $\tilde{\varphi}(c)$ supported away from $\bar{z}(c) = z(c)$ (see Chapter III.1 of [61] for such arguments). To summarize, for every smooth function $\varphi(c)$ compactly supported away from $\bar{z}(c) = z(c)$ and from the light cones, we constructed a smooth function $\psi(z, \bar{z})$ compactly supported in $1 < z < \bar{z}$ such that

$$ \int d^{4d}c G_4(c)\varphi(c) = \int dz d\bar{z} g(z, \bar{z})\psi(z, \bar{z}) \quad (6.85) $$

holds in case $g(z, \bar{z})$ is a function. We now claim that this equation continues to hold, with the same $\psi$, in case $g(z, \bar{z})$ is a distribution. The point is that we can find a sequence of functions $g_n(z, \bar{z})$ which tend to $g(z, \bar{z})$ in the sense of distributions, so that the corresponding $\frac{g_n(z(c), \bar{z}(c))}{(x_{12}^2 x_{34}^2)^{\Delta_O}}$ tend to $G_4(c)$ in the sense of distributions on $\mathbb{R}^{4d}$. Since both $\varphi$ and $\psi$ are smooth, we are allowed to pass to the limit on both sides of the equation, proving the claim. The functions $g_n(z, \bar{z})$ are given e.g. by the partial sums of the series (6.82), transformed from the $\rho$ to the $z$ coordinates.

Let us now discuss how configurations where $z = \bar{z}$ can be included into this discussion. The basic difficulty is that the map $c \mapsto (z, \bar{z})$ fails to be a submersion near such configurations. So in general the function $\psi(z, \bar{z})$ will not be smooth. Consider e.g. the causal ordering $1 \to 3 \to 2 \to 4$. In this case it's possible to show (we omit the proof) that the function $\psi(z, \bar{z})$ behaves like

$$ |z - \bar{z}|^{d-2} \text{ times a smooth function near } z = \bar{z}, \quad (6.86) $$

which in general is not smooth unless $d$ is even.

We need to be able to make sense of the r.h.s. of (6.85) for such non-fully-smooth test functions. This is possible due to the following observation. Above we explained, following the arguments first presented in [36], that $g(\rho, \bar{\rho})$ is a distribution for $|\rho|, |\bar{\rho}| = 1$. But in fact it's a bit better than that (the fact not mentioned in [36]): it is a distribution in $\rho$ for each fixed value of $\bar{\rho}/\rho = e^{i\alpha}$! Indeed if we substitute $\bar{\rho} = e^{i\alpha}\rho$ with a fixed $\alpha$ into (6.82), we get a holomorphic function in the unit disk of $\rho$, which satisfies a powerlaw bound, hence its boundary value is a distribution. This can be generalized to holomorphic maps $\bar{\rho} = f(\rho)$ which maps the unit disk into itself (or at least a portion of the unit disk near $\rho = \rho_0$ into the unit disk). Translating to $z, \bar{z}$, this implies in particular that $g(z, z+t)$ is a distribution for any fixed $t$. In fact, the map $\bar{z} = z+t$ corresponds to a map $\bar{\rho} = f_t(\rho)$ to which the previous argument is applicable. So $g(z, \bar{z})$ is by no means the most general distribution in two variables, as it allows the restriction to the submanifold $\bar{z} = z+t$ for any $t$. E.g. $\delta(z - \bar{z})$ is not allowed by this property, while $\delta(z + \bar{z})$ is allowed. Following this logic a bit more carefully, it can be shown (we omit the proof) that $g(z, \bar{z})$ can be paired with test functions $\psi(z, \bar{z})$ which, when expressed in terms of $s = z + \bar{z}, t = \bar{z} - z$, have the following property: $\psi(s, t)$ is $C^\infty$ with respect to $s$ for any fixed $t$, with bounds on derivatives in the $s$ direction which are integrable in the $t$ direction. Eq. (6.86) is compatible with this requirement.

A further complications arises near the $z = \bar{z} > 1$ locus for the causal orderings which include configurations in both $E_{\text{tu}}$ and $E_{\text{stu}}$ subclasses. In this case the function $\psi(z, \bar{z})$
defined in (6.84) will consist of two functions $\psi_1(z, \bar{z})$ and $\psi_2(z, \bar{z})$: one defined for real $z, \bar{z}$, another for complex-conjugate $z, \bar{z}$. The two functions $\psi_i$ will be glued along the $z = \bar{z} > 1$ line. The resulting glued function will not in generally be smooth on the $z = \bar{z} > 1$ line (while it will be smooth away from it). However the directions orthogonal to the line turn out analogous to the $t$ direction in the previous paragraph, i.e. the test function is actually not required to be smooth in these directions for the pairing to be defined. This allows to make sense of the formula (6.85) also in this case. We omit the details.

## 6.6.2 Fixing points

We would like to put the results of the previous section in the context of a general question of “fixing points” in a distribution. E.g. we know that the Minkowski 4-point function is a translationally invariant distribution. Using translation invariance we can always fix one of the 4 points to a given position, e.g. zero, and consider it as a distribution with respect to the remaining 3 positions. One could ask if one can do better than that, i.e. to fix $n$ points to given positions and consider the 4-point function as a distribution with respect to the remaining $4-n$ positions. Where the 4-point function is real-analytic we can of course consider all four points as fixed.

Now, results of section 6.6.1 show that, if one excludes lightlike separations limiting to configurations having some fixed causal ordering, one can fix a conformal frame, i.e. fix three points to some fixed positions, and the fourth point to a position characterized by two conformal cross ratios, and consider the distribution as a distribution in only two variables (cross ratios). It is not clear if results of section 6.6.1 can be generalized to cover lightlike separations.

In some cases it is possible to argue that one can fix more than one point without using conformal invariance. E.g. we may always fix a consecutive pair of points, i.e. $(x_k, x_{k+1})$, where $k = 1, 2$ or $3$, to spacelike-separated positions in Minkowski space, while allowing the remaining two points to approach Minkowski limit from the forward tube. The proof of lemma 6.1 can be slightly modified to show that $|\rho|, |\bar{\rho}| < 1$ for such configurations (see section 6.9.1 below). Moreover, a powerlaw bound also holds, by a slight modification of the argument after eq. (6.73).$^{56}$ Then our arguments show that the Minkowski 4-point function is a distribution with respect to the two unfixed coordinates, which depends analytically on the fixed coordinates. In this case the unfixed coordinates may have any causal orderings and also lightlike separation.

One interesting case is that of the double light cone (DLC) singularity, i.e. the region close to $x_1 = 0, x_3 = \hat{e}_1, x_4 = \infty$, while $x_2$ on the light cones of $x_1, x_3$. Our results are the

<footnote>$^{56}$Since $S(c) = \infty$ in these cases, we cannot rely on (6.53). Instead we directly show powerlaw bounds on $S(\Upsilon_1), S(\Upsilon_2)$ defined in eq. (6.73). Then the powerlaw bound on $|\rho|, |\bar{\rho}|$ holds by lemma 6.6. For $k=1$, by fixing $\zeta_3 = 0$, and using lemmas 6.4 and 6.5, we have $S(\Upsilon_i) \le S(\zeta_i') + S(-\zeta_i') \le S(x_{i3})^3 + S(x_{i4})^3$ ($i=1,2$). This is the desired powerlaw bound with respect to $x_3$ and $x_4$. For $k=2$, $S(\Upsilon_1)$ is bounded as for $k=1$, while for $S(\Upsilon_2)$ we argue as follows. Since $x_2$ and $x_3$ are spacelike separated, after fixing $\zeta_3 = 0$, $\zeta_2$ is a spacelike Minkowski point, hence so is $\zeta'_2$, i.e. $\text{Im}(\zeta'_4 + \zeta'_2) = \text{Im}(\zeta'_4)$. Then by lemma 6.4, $S(\Upsilon_2) \le S(\zeta'_4) + |\zeta'_2| \le S(\zeta_4)^3 + |\zeta'_2|$, which is the needed bound. Case $k=3$ follows by similar arguments or by mapping it to $k=1$ via $(x_1, x_2, x_3, x_4) \rightarrow (x'_1 = x'_4, x'_2 = x'_3, x'_3 = x'_2, x'_4 = x'_1)$ which maps $\rho$ and $\bar{\rho}$ are to their complex conjugates.</footnote>
first ones which establish the existence of the Wightman 4-point function in a neighborhood of DLC. However, there is a difference between restricting to one causal ordering near DLC or studying an open neighborhood of DLC which includes several causal orderings (see figure 10 in Conclusions). In the former case we can use directly the results of section 6.6.1 and represents the 4-point function as a distribution in two variables $z, \bar{z}$. In the latter case we can fix, by the above argument, two successive spacelike points $x_3$ and $x_4$. We are left with a distribution depending on $x_1, x_2$, i.e. $2 \times d$ coordinates. This distribution still satisfies conformal invariance Ward identities w.r.t. infinitesimal conformal transformations preserving $x_2$. It would be interesting to understand how this constrains the distribution at the DLC.

Although it is not directly related, we would also like to mention here the classic result of Borchers [74] which says that it is enough to smear Wightman functions $G_M(x_1, \dots, x_n = 0)$ with respect to the time variables only, i.e. integrating with respect to $h_1(x_1^0) \dots h_{n-1}(x_{n-1}^0)$ where $h_i \in S(\mathbb{R})$, after which they become $C^\infty$ functions in the remaining spatial variables $\mathbf{x}_i$. This result is valid in any QFT satisfying Wightman axioms. It holds because smearing in time, which acts as an energy cutoff, is effectively also a momentum cutoff because $|\mathbf{p}| \le E$.

## 6.7 Wightman positivity

Recall that in section 3.1 we showed that CFT axioms imply OS reflection positivity for 4-point functions. That discussion gives us access to OS states $|O(x)O(y)\rangle$ with $0 > x^0 > y^0$, with finite norm, and inner products measured by the Euclidean 4-point function. We know that these states belong to the CFT Hilbert space, i.e. can be arbitrarily well approximated in norm by states produced by inserting finite linear combinations of CFT local operators at one point in the half-space $x^0 < 0$, e.g. the south pole $x_S = (-1, 0)$.

Now that we analytically continued the 4-point function, we can consider other states involving operators at complexified coordinates. We wish to prove that those states belong to the CFT Hilbert space and have a positive definite inner product. This can be shown by a robust argument, going back to Osterwalder and Schrader [2], section 4.3. The argument uses only OS positivity and the Fourier-Laplace representation, but not directly the CFT axioms.

We will consider two new kinds of states. First, states generated by a pair of Minkowski operators smeared with respect to an arbitrary Schwartz test function:

$$|\Psi_M(F)\rangle = \int dx dy F(x, y)|O(ix^0, \mathbf{x})O(iy^0, \mathbf{y})\rangle, \qquad (6.87)$$

and second, states generated by a pair of Euclidean operators at complexified time positions:

$$|O(x_3)O(x_4)\rangle, \quad x_i = (\epsilon_i + it_i, \mathbf{x}_i), \quad 0 > \epsilon_3 > \epsilon_4. \qquad (6.88)$$

The inner products of states (6.87) are given by integrals of the Minkowski 4-point function

$$\langle\Psi_M(F_1)|\Psi_M(F_2)\rangle = \int dx G_4^M(x_1, x_2, x_3, x_4) \overline{F_1(x_2, x_1)} F_2(x_3, x_4), \qquad (6.89)$$
while the natural inner product on the states (6.88) is:

$$ \langle \mathcal{O}(x_1)\mathcal{O}(x_2)|\mathcal{O}(x_3)\mathcal{O}(x_4)\rangle = G_4(x_2^\theta, x_1^\theta, x_3, x_4), \qquad (6.90) $$

where the OS reflection operation extends to points with complex time coordinates by:

$$ x = (\varepsilon + it, \mathbf{x}) \mapsto x^\theta = (-\varepsilon + it, \mathbf{x}). \qquad (6.91) $$

The states (6.87) also have a natural inner product $\langle \mathcal{O}(x_1)\mathcal{O}(x_2)|\Psi(F)\rangle$ with the OS states.<fnref n="57" />

We wish to show that all these new inner products are positive definite and, moreover, that the new states can be approximated in norm by the smeared OS 2-operator states at Euclidean positions. Note that the positive definiteness of (6.89) is precisely Wightman positivity for the 4-point case.

### 6.7.1 Wightman states

Let us start with (6.89). Rewriting the inner product in terms of $W(p_1, p_2, p_3)$, the (distributional) Fourier transform of $G_4^M$ with respect to $y_k = x_k - x_{k+1}$, we obtain

$$ \langle \Psi_M(F_1) | \Psi_M(F_2) \rangle = \int dp W(p_1, p_2, p_3) [\hat{F}_1(p_2 - p_1, p_1)]^* \hat{F}_2(p_2 - p_3, p_3). \qquad (6.92) $$

We will also need the inner products of the (smeared) OS states

$$ |\Psi(H)\rangle = \int dx dy H(x, y) |\mathcal{O}(x)\mathcal{O}(y)\rangle \qquad (6.93) $$

where $H$ is any $C^\infty$ function compactly supported at $0 > x^0 > y^0$. Their inner products are given by

$$ \langle \Psi(H_1) | \Psi(H_2) \rangle = \int dx G_4^E(x_1, x_2, x_3, x_4) \overline{H_1(x_2^\theta, x_1^\theta)} H_2(x_3, x_4). \qquad (6.94) $$

This can be expressed using the Fourier-Laplace representation (4.11). We obtain

$$ \langle \Psi(H_1) | \Psi(H_2) \rangle = \int dp W(p_1, p_2, p_3) \overline{g(H_1)(p_2, p_1)} g(H_2)(p_2, p_3), \qquad (6.95) $$

where $g(H)(p, q)$ is a Schwartz class function related to $H(x, y)$ as follows. First we form the function $h(y_1, y_2) = H(-y_1, -y_1 - y_2)$ which has support at $y_1^0, y_2^0 > 0$. Next we consider $\tilde{h}$, the Fourier-Laplace transform of $h(y_1, y_2)$:

$$ \tilde{h}(p_1, p_2) = \int dy_1 dy_2 e^{-p_1^0 y_1^0 + i \mathbf{p}_1 \cdot \mathbf{y}_1 - p_2^0 y_2^0 + i \mathbf{p}_2 \cdot \mathbf{y}_2} h(y_1, y_2). \qquad (6.96) $$

Finally, $g(H)$ is an arbitrary Schwartz class function which coincides with $\tilde{h}$ inside the forward light cones. We also have an analogous formula for the inner product between states of two types:

$$ \langle \Psi(H) | \Psi_M(F) \rangle = \int dp W(p_1, p_2, p_3) \overline{g(H)(p_2, p_1)} \hat{F}_2(p_2 - p_3, p_3). \qquad (6.97) $$

<footnote n="57">⁵⁷We start from the analytically continued Euclidean 4-point function $G_4(x_1, x_2, x_3, x_4)$ and take the limit where $x_1, x_2$ are kept at fixed Euclidean positions, while $x_3, x_4$ approach the Minkowski space. By theorem 4.1, the limit is a distribution in $x_3, x_4$, and the inner product is its pairing with the test function $F$.</footnote>