A Generalized Nachtmann Theorem in CFT

Correlators of unitary quantum field theories in Lorentzian signature obey certain analyticity and positivity properties. For interacting unitary CFTs in more than two dimensions, we show that these properties impose general constraints on families of minimal twist operators that appear in the OPEs of primary operators. In particular, we rederive and extend the convexity theorem which states that for the family of minimal twist operators with even spins appearing in the reflection-symmetric OPE of any scalar primary, twist must be a monotonically increasing convex function of the spin. Our argument also applies to the OPE of nonidentical scalar primaries in unitary CFTs, constraining the twist of spinning operators appearing in the OPE. Finally, we argue that the same methods also impose constraints on the Regge behavior of certain CFT correlators.


Introduction
The operator product expansion (OPE), as introduced in [1][2][3], provides an algebraic structure in quantum field theory (QFT) which is of fundamental importance. The OPE is particularly useful in conformal field theory (CFT) where it enjoys several nice properties.
For example, two nearby scalar primary operators in any unitary CFT can be replaced by their OPE where the sum is over primary operators which transform homogeneously under the conformal group. The coefficient function D µ 1 µ 2 ··· p is completely fixed by conformal symmetry up to an overall real coefficient c O 1 O 2 Op which is commonly known as the OPE coefficient.
Clearly, the operators O p µ 1 µ 2 ··· that appear in the above OPE are also symmetric traceless representations of the Lorentz group and hence they can be labeled by scaling dimensions ∆ p and spins p . A key feature of the conformal OPE is that after insertion into any correlator, the OPE (1.1) converges absolutely even for finite separation x − y [4,5]. This fact plays a central role, both conceptually as well as technically, in the study of CFTs.
A CFT can be completely and uniquely defined by its OPEs. So, it is of interest to address the question of when a set of conformal OPEs defines a unitary and causal theory. In this paper, we answer a version of this question for interacting unitary CFTs in d > 2 dimensions by constraining the family of minimal twist operators that appears in the OPE (1.1), independent of the rest of the theory. The family of minimal twist operators is defined in the following way. First, consider all spin primary operators that appear in the OPE (1.1). Among these operators, we pick the operatorÕ which has the lowest twist. 1 We define the set of operators {Õ | ∈ Z ≥ } as the family of minimal twist operators.
Of course, any operator that appears in the OPE (1.1) must obey the unitarity bound which states that in d spacetime dimensions the twist τ p ≥ d − 2 for ≥ 1 and τ p ≥ d−2 2 for p = 0. Moreover, there is ample evidence suggestive of a stricter bound on the family of minimal twist operators appearing in the OPE (1.1). Most notably, a concrete example of such a bound was found by Komargodski and Zhiboedov in [6] by extending the Nachtmann theorem about QCD sum rules of [7] to CFT. In particular, it was argued in [6] that for a CFT to be unitary in d > 2 dimensions, twist must be a monotonically increasing convex function of the spin (above some lower cut-off ≥ c ) for the family of minimal twist operators with even spins appearing in any reflection-symmetric OPE OO † , where O is a scalar primary. The derivation of the convexity theorem in [6] depends on the assumptions that the CFT can flow to a gapped phase and the deep inelastic scattering amplitude in the gapped phase is polynomially bounded in the Regge limit.
However, these assumptions are not actually necessary since the convexity theorem, as shown in [8], can also be derived directly using the Lorentzian inversion formula of [9].
Moreover, the argument of [8] has the additional advantage that it implies the convexity property for the continuous even spin leading Regge trajectory above > 1.
In this paper, we provide an alternative derivation of the CFT Nachtmann theorem by using analyticity and positivity properties of CFT correlators in Lorentzian signature.
In fact, the theorem that we prove is slightly stronger than the convexity theorem of [6,8].
Furthermore, our derivation can be easily generalized to constrain the family of minimal twist operators that appears in the OPE of nonidentical scalar primaries as well. Our 1 The twist is defined in the usual way τ = ∆ − . main argument parallels the CFT-based derivation of the averaged null energy condition of [10]. It is well known that the OPE (1.1) is organized as an expansion in twist in the lightcone limit. Hence, the lightcone limit of a CFT four-point function is completely fixed by the spectrum of low-twist operators. Lorentzian correlators obey certain analyticity conditions which enable us to relate the lightcone limit of a CFT four-point function to a high energy Regge-like limit. Regge correlators are theory dependent, however, they are bounded because of Rindler positivity -which, in turn, constrains the family of minimal twist operators that contribute in the lightcone limit.
Let us now summarize our main results. Consider the family of minimal twist operators appearing in the OPE of both OO † and XX, where O is a scalar primary and X is a completely arbitrary operator with or without spin (not necessarily local or primary) in any interacting unitary CFT in d ≥ 3 dimensions. 2 The twistτ of these minimal twist operators as a function of the spin must obey the following conditions: • Monotonicity-For the family of minimal twist operators with even spins, twistτ is a monotonically increasing function of the spiñ for all even 1 and 2 .
• Global convexity-Twistτ for even spins is a non-concave function and hence for any even 1 , 2 and 3 • Local convexity-Operators in the family of minimal twist operators with odd spins obey a local convexity condition 3 for any odd o ≥ 3.
Furthermore, the same argument also imposes nontrivial constraints on OPE coefficients of minimal twist operators.  (12) which has twistτ (12) . The set of operators {Õ (12) | ∈ Z ≥ } is defined as the the family of minimal twist operators in the Similarly, we can define families of minimal twist operators for the OPEs Twists of these operators are denoted byτ (11) andτ (22) respectively. A Nachtmann-like theorem for nonidentical scalar primaries relates these three families of minimal twist operators. In particular, the twistτ (12) as a function of the spin must obey the following conditions in any interacting unitary CFT in d ≥ 3 dimensions.
• Lower bound for even spins-The family of minimal twist operators that appears in the OPE O 1 O 2 must obey a lower bound for even spins for even e ≥ 2 . (1.5) • Lower bound for odd spins-The family of minimal twist operators that appears in the OPE O 1 O 2 must obey a different lower bound for odd spins In recent years, the conformal bootstrap has made significant progress in constraining CFT data, both numerically and analytically, by using crossing symmetry, unitarity, and conformal symmetry. We closely examine some of the bootstrap results in d ≥ 3 dimensions to demonstrate that OPEs in unitary CFTs indeed obey the above conditions.
In particular, the numerical results of [12] enable us to study and compare families of minimal twist operators that appear in the OPEs σσ, , and σ of the 3d Ising CFT, where operators σ and are the lowest-dimension Z 2 -odd and Z 2 -even scalar primaries respectively (see figure 5). Furthermore, we also show that the lightcone bootstrap results of anomalous dimensions of various double-twist operators at large spins are consistent with the conditions (1.2)-(1.6).
Finally, we argue that the same methods can be deployed in the Regge limit to constrain the Regge behavior of certain CFT correlators. However, these constraints are expected to be theory dependent since the Regge limit is dominated by high spin exchanges which are in general non-universal. Nevertheless, this approach is still useful, particularly in the context of the AdS/CFT correspondence, where the constraints on the Regge correlator should be interpreted as bounds on various interactions of low energy effective field theories in AdS.
The outline of this paper is as follows. In section 2 we review analyticity and positivity properties of QFT correlators in Lorentzian signature. In section 3 we utilize these properties to derive a stronger version of the CFT Nachtmann theorem in unitary CFTs in d ≥ 3 dimensions. We then extend our analysis in section 4 to impose constraints on families of minimal twist operators that appear in the OPE of nonidentical scalar primaries. All these constraints are consistent with the conformal bootstrap results, as we discuss in section 5. Finally in section 6 we explain how the same methods could be useful in constraining the Regge behavior of certain CFT correlators.
as a function of complexified x i is analytic in the domain [10,17] Im where, Re x i ∈ R 1,d−1 and Im x i ∈ R 1,d−1 . The symbol x y represents that the point x is in the past lightcone of point y. This analyticity condition is simply a covariant version of the standard i prescription that computes Lorentzian correlators from Euclidean correlators. 5 Still, analyticity of Lorenzian correlators in the domain (2.1) is a nontrivial statement since it is intimately related to microcausality in Lorentzian signature [10].
Well behaved Euclidean correlators must also satisfy reflection positivity which is closely related to unitarity. In Lorentzian signature, there is a positivity property known as Rindler positivity which is physically very similar to reflection positivity [18,19] (see [10] for a review). It states that all Rindler reflection symmetric correlators must be positive in any unitary, Lorentz-invariant QFT. In order to be more specific, let us introduce the following notation for points x ∈ R 1,d−1 : where, x − = t − y and x + = t + y. The right and left Rindler wedges are defined as reflection takes a point on R to a point on L and vice versa. In general, the Rindler reflection of an operator is defined as where, P is the total number of t and y indices. Rindler positivity is the statement that Lorentzian correlators can be made precise in CFT [15,16].
the Lorentzian correlator where all x i ∈ L (or equivalently in R) and operators inside the correlator is ordered as written.
These analyticity and positivity conditions hold for any unitary Lorentzian QFT making them important tools that can be employed to derive very general results. In particular, these simple properties have far-reaching consequences for theories with conformal symmetry as we discuss next.

CFT in Lorentzian Signature
We now consider unitary CFTs in d > 2 spacetime dimensions. First we show that analyticity and positivity properties of Lorentzian correlators imply the CFT Nachtmann theorem above ≥ 2.
We start with Lorentzian correlators of two arbitrary CFT operators O 1 and O 2 (with or without spin), where operators inside correlators are ordered as written. All the points are restricted to be in a 2d subspace Analyticity and positivity properties of Lorentzian correlators, as discussed in [10], impose nontrivial constraints on G. These constraints can be conveniently described by with η > 0 which makes G ≡ G(η, σ) and G 0 ≡ G 0 (η, σ). In any unitary QFT, correlators G(η, σ) and G 0 (η, σ) in the regime 0 < η < 1 have the following properties: Null coordinates are defined as x ± = t ± y, where time is running upward.
Note that both these conditions are valid even when O 1 and O 2 are smeared.
Conditions (i) and (ii) together lead to nontrivial bounds on the spectrum of low-lying operators in interacting CFTs in d > 2 dimensions. The main argument is actually very intuitive. It is well known that CFT correlators in the lightcone limit η |σ| 1 have universal features because these correlators are completely fixed by the spectrum of low-twist operators. Now, the condition (i) which is basically causality in disguise relates the lightcone limit to a high energy Regge-like limit |σ| 1. In general, we do not have computational control on CFT correlators in the limit |σ| 1, however, these correlators are bounded because of the "unitarity" condition (ii) -which, in turn, imposes constraints on the spectrum of low-twist operators that contribute in the lightcone limit.

Lightcone Limit
We now restrict to a CFT correlator where O 1 = O is a scalar primary (real or complex) and O 2 = X is some arbitrary operator with or without spin (not necessarily local or primary). We are interested in the lightcone limit . This can be achieved by generalizing the lightcone OPE of [10] for complex scalar primaries. We start with the OPE where the sum is only over primary operators that have non-vanishing three-point functions with OO † . The contribution of the primary operator O p µ 1 µ 2 ··· with conformal dimension ∆ p and spin p and its descendants to the OPE OO † in the lightcone limit ρρ → 0 can be recast as where, twist τ p = ∆ p − p and λ p is a numerical coefficient exact value of which will not be important for us. 6 This lightcone OPE can be used inside arbitrary correlators.
In particular, we can use this OPE to derive the contribution of the OO † → O p → XX conformal block to the correlator X(1)O(ρ)O(ρ) X(1) in the Lorentzian lightcone limit (3.5). This Lorentzian correlator can be obtained from the Euclidean one by analytically continuing ρ along the path shown in figure 2. The corresponding contour for the ρintegral follows directly from the way operators are ordered inside the correlator. 7 Of course, we intend to utilize the analyticity and positivity conditions (i) and (ii). 6 The exact value of λ p can be computed from the three-point function where, c p is the coefficient of the two-point function of O p µ1µ2··· . Note that for real scalar primaries only operators with even spins can appear in the OPE. Hence, the factor of (−1) p is not there in [10] for real scalars.  Hence, it is more convenient to study the lightcone limit of normalized correlators G(η, σ) and G 0 (η, σ), as defined in (3.1), using variables (3.3). In these variables, the Lorentzian lightcone limit (3.5) can be equivalently defined as 0 < η |σ| 1. As an immediate consequence of the lightcone OPE (3.7), we can write the contribution of the operator O p to G(η, σ) − G 0 (η, σ) in the Lorentzian lightcone limit as an expansion where dots represent correction terms that are either suppressed by higher powers of η or decay for small σ. Notice that the sum in the above equation is only over even integers.
Moreover, coefficients C p, are real which can be computed from the lightcone OPE (3.7).
For example which is nonzero by definition and all other coefficients C p, ∝ C p, p . Since, however, these coefficients do not depend on η or σ, their actual values are not important to us.
By using the i prescription we can convince ourselves that that we do not need to cross any branch cuts to obtain a Rindler reflection symmetric correlator, even when some of the operators are time-like separated. Hence, the correlator G 0 (η, σ) can be computed by using the Euclidean OPE. This implies that G 0 (η, σ) is trivial in the lightcone limit (3.5) where it is dominated by the identity exchange where dots represent correction terms that are suppressed by positive powers of η and σ.
The OO † OPE converges only when all operators are spacelike separated in a correlator. On the other hand, the OO † OPE in a Lorentzian correlator such as G(η, σ) may diverge. Notice that contributions of individual operators in (3.8) become increasingly singular with increasing spin which is a manifestation of the fact that the lightcone OPE (3.7) does not converge on the second sheet. However, by considering scalar four-point functions it was argued in [20] that the second sheet conformal blocks still can be trusted in the lightcone limit. The same argument applies in general implying any finite number of terms in the lightcone OPE on the second sheet produce a reliable asymptotic expansion in the limit η → 0 [10]. Hence, G(η, σ) in the Lorentzian lightcone limit can be expressed as a divergent asymptotic series which is organized by twist 8 where, dots represent terms that are suppressed by positive powers of both η and σ.
The spectrum of operators that appear both in the OPE OO † and XX can have an accumulation point in the twist. In that case, it is generally expected that the above asymptotic expansion is not valid beyond the accumulation point in the twist spectrum.

Nachtmann Theorem in CFT
We now have all the ingredients to derive and generalize the CFT Nachtmann theorem.
Our starting point is the correlator and X is a completely arbitrary operator. The lightcone limit of this conformal four-point function, as we learned from equation (3.11), is an expansion in twist. So, in this limit only the family of minimal twist operators is important.
The family of minimal twist operators is defined in the usual way. In the present context, consider all spin operators that appear in the OPE of both OO † and XX.
Among these operators, we pick the operatorÕ which has the lowest twist. The family of minimal twist operators is then defined as {Õ | ∈ Z ≥ }. We adopt the notationτ to denote the twist ofÕ .
Clearly, the spectrum of operators that appear in the OPE of both OO † and XX can have both even and odd spins when O is a complex scalar. In that case, it is more convenient to discuss the even and odd (spin) family of minimal twist operators separately. First, we derive a general sum rule by integrating δG(η, σ), where As in [10], we choose an integration contour which is a half-disk near σ = 0, just below the real line in the complex-σ plane (see figure 3). The radius R of the semicircle is such that it satisfies η R 1. Strictly speaking, G 0 (η, σ) is only defined on the real line. However, we can still define a function G (−) 0 (η, σ) which is analytic on the lower half σ-plane and has the property Re G (−) 0 (η, σ) = G 0 (η, σ) (for a detailed discussion see appendix A). The analyticity property (i) then requires that for integer n ≥ 0. The above contour integral can be divided into two parts -one over the real line and one over the semicircle. The lightcone limit expression (3.8) is valid over the entire semicircle. Moreover, the semicircle integral can be further and crucially simplified using the identity where, S = {Re iθ , θ ∈ [0, −π]}. This enables us to write a sum rule The left hand side of the above relation is written as a formal sum which reflects the fact that any operatorÕ with = + 2N (for non-negative integer N ) can contribute to the correlator G(η, σ) a term that grows exactly as 1 σ −1 in the Lorentzian lightcone limit. However, it should be understood that for any only the term (or terms) with the lowestτ dominates.
The observant reader may have noticed that dots in equation (3.8) contain terms that decay with positive but non-integer powers of σ when operators with non-integer dimensions are exchanged. These terms appear to spoil the projection to a specific power of 1 σ by using the identity (3.14). However, these terms do not actually contribute to the sum rule (3.15), since these terms get exactly canceled by G (−) 0 (η, σ). This is explained in detail in appendix A.
Next we use the sum rule (3.15) to derive the monotonicity and convexity properties of the even spin family of minimal twist operators.

Monotonicity
The positivity condition (ii) immediately implies that the left hand side of (3.15) must be non-negative for any even . 9 Furthermore, the positivity condition (ii) also leads to the following inequality for any two even 2 > 1 ≥ 2 Hence, the sum rule (3.15) implies that Strictly speaking, the inequality (3.4) is valid only for real σ. There is a complexified version of Rindler positivity, as discussed in [10], which implies (3.4) on the straight line part of the contour 3 even when it lies just below the real axis in the complex-σ plane.
in the limit η → 0. Since, C , do not scale with η, the above inequality can be satisfied for all even 1 and 2 . The equality in the above relation holds only for free CFTs and 2d CFTs. In these CFTs, the relation (3.17) is trivially satisfied since they contain an infinite tower of higher spin conserved currents.
The monotonicity condition (3.17) greatly simplifies the sum rule (3.15) for interacting CFTs in d > 2 dimensions. For any even , clearly the first term C , ητ 2 dominates in the left hand side implying three-point functions which is consistent with the averaged null energy condition and its higher spin generalization as derived in [10].

Convexity
Similar to [6], we can actually derive a stronger constraint on the family of minimal twist operators with even spins. For any even ≥ 2, the Cauchy-Schwarz inequality of real integrable functions imposes In the regime 0 < η R 1, we can use the sum rule (3.15) and the monotonicity condition (3.17) to rewrite the above inequality as implying 2τ +2 ≥τ +τ +4 for any even ≥ 2. This local convexity condition can be applied iteratively to obtaiñ for all even 1 , 2 and 3 .
To summarize, twist is a monotonically increasing convex function of the spin for the even (spin) family of minimal twist operators appearing in the OPE of both OO † and XX, where O is a scalar primary and X is a completely arbitrary operator with or without spin (not necessarily local or primary) in any interacting unitary CFT in d ≥ 3 dimensions. This is, in general, stronger than the convexity theorem of [6] (with c = 2). In particular, On the other hand, twists of two odd spin minimal twist operators do not obey any such condition.
Odd spin minimal twist operators also satisfy a local convexity condition. For any odd 0 ≥ 3, the Cauchy-Schwarz inequality can be utilized to write

OPE of Nonidentical Operators
It is only natural to wonder if the preceding analysis can be extended to OPEs of nonidentical scalar primaries in unitary CFTs. As the discussion of section 2 leads us to expect, the family of minimal twist operators appearing in the OPE of nonidentical scalar primaries does obey some general constraints which we will discuss next. Before we proceed, we introduce the notation where operators are ordered as written. As remarked before, we do not need to cross any   First, we choose

Mixed Correlators and Rindler Positivity
where δ 1 and δ 2 are arbitrary real numbers that we will fix later. The Cauchy-Schwarz identity (4.2) now leads to an inequality which applies to any unitary QFT. This inequality can be further simplified for CFTs where some of the correlators are related because of conformal symmetry. Specifically for 0 <ρ < 1, ρ > 1 we obtain 10 for any real δ 1 and δ 2 , where operators inside correlators are ordered as written. In the above equation, we have defined ∆ 21 = ∆ 2 − ∆ 1 and δ 21 = δ 2 − δ 1 .

Inequality II
Likewise, we can derive a similar inequality for a different mixed correlator by choosing 10 Note that we are using the notation (4.1).
For 0 <ρ < 1, ρ > 1, the Cauchy-Schwarz identity (4.2) imposes for any real δ 1 and δ 2 . Of course, the above argument holds for any unitary QFT. However, conformal symmetry was used to simplify the Cauchy-Schwarz identity to obtain the final inequalities (4.4) and (4.6). So, for non-conformal QFTs, one must use the exact expression obtained from the Cauchy-Schwarz identity (4.2) for A and B given in (4.3) or (4.5).
The inequalities (4.4) and (4.6) together, as we explain below, impose constraints on the family of minimal twist operators appearing in the OPE O 1 O 2 . To derive any such constraint, first it is necessary to understand the Lorentzian lightcone limit of the mixed

Lightcone OPE
It is straightforward to generalize the lightcone OPE of [10] for nonidentical scalar primaries. The contribution of the primary operator O p µ 1 µ 2 ··· with conformal dimension ∆ p and spin p and its descendants to the OPE O 1 O 2 in the lightcone limit ρρ → 0 can be written as where, h p = ∆ p + p and twist τ p = ∆ p − p . Actual value of the coefficientλ p is not important for our purpose, however let us note it here anywaỹ where, c p is the coefficient of the two-point function of O p O p . 11 The lightcone OPE x we obtain 12 plus correction terms that are either suppressed by higher powers of η or decay for small σ. For our argument, as explained in appendix B, we can safely ignore these correction terms. In the above equation, actual values of coefficientsC p, ∝ |c O 1 O 2 Op | 2 are not important. These coefficients can be computed either from the OPE (4.7) or by using the lightcone conformal block derived by Dolan and Osborn in [23]. Finally, the mixed correlator in the Lorentzian lightcone limit can also be expressed as an asymptotic series which is organized by twist (4.10)

Constraints on the Family of Minimal Twist Operators
We are now in a position to derive constraints on the family of minimal twist operators that appears in the OPE O 1 O 2 . Consider the mixed Lorentzian correlator in a unitary CFT in d ≥ 3 spacetime dimensions (see figure 4). Moreover, in order to lighten the notation, we also define with I, J = 1, 2. Similarly, we can define δG mixed (η, σ) and δG IJ (η, σ) following (4.1).

Conditions (a) and (c) together suggest that in interacting CFTs in d ≥ 3 dimensions
there must be some relation between three families of minimal twist operators which we define next. First, consider all spin operators that appear in the OPE O 1 O 2 . Among these operators, we pick the lowest twist operatorÕ (12) which has twistτ (12) .  (11) ,τ (22) .
Lower bound onτ (12) for even ≥ 2 We can now use positivity properties (b) and (c) of CFT correlators to write the following relation for any two implying that the quantity on the left hand side must not grow in the limit η → 0.
As discussed in detail in appendices A and B, we can again define functions δG Re δG mixed (η, σ) and Re δG IJ (η, σ) to integrals over a semicircle by using the contour 3.
Hence, the semicircle integrals can be further simplified by utilizing the identity (3.14).

3d Ising Model and Other Examples
In this section we provide some simple examples to demonstrate that unitary CFTs in d ≥ 3 dimensions indeed obey the above constraints. 13 We are using the following notation Unitarity bound [σϵ] 0 even

3d Ising CFT
The first example that we consider is the 3d Ising CFT. This CFT contains operators σ and which are the lowest-dimension Z 2 -odd and Z 2 -even scalar primaries of the theory, respectively. In recent years, numerical bootstrap methods have led to significant progress in constraining the data of the 3d Ising CFT. For instance, the bootstrap has provided precise conformal dimensions for operators σ and just from crossing symmetry and unitarity [12,[24][25][26][27][28]. Furthermore, the same principles, as demonstrated in [12], are also sufficient to determine the spectrum of the 3d Ising CFT numerically in a systematic way.
In fact, the numerical data of [12] is so precise for several low-lying operators that we can actually study and compare families of minimal twist operators that appear in the OPE of σσ, , and σ .
Let us now examine the numerical 3d Ising data of [12]. The set of operators [σσ] 0 is of particular importance since these operators form the family of minimal twist operators for both σσ and OPEs. Clearly, the numerical data of [12] for the family [σσ] 0 is consistent with the Nachtmann theorem (see figure 5). Moreover, as we explained in the last section, the [σσ] 0 family also provides a lower bound for the twists of [σ ] 0 operators (both even and odd spins) which are minimal twist operators for the σ OPE. This lower bound is stronger than the unitarity bound, however, it is still relatively weak. Of course, the numerical data of [12] for the family [σ ] 0 , as we show in figure 5, is consistent with analytic bounds (4.17) and (4.19).
Interestingly, both even and odd spin operators in the family [σ ] 0 exhibit some nice features. For example, the 3d Ising data suggests that the twists of odd (or even) spin operators in the family [σ ] 0 obey some monotonicity and convexity conditions. However, we believe that these conditions are not true in generic CFTs.

Real Scalars
Constraints (4.17) and (4.19) are also visible from the large spin bootstrap of real scalar primaries [6,11]. Consider the CFT correlator φ 1 (x 1 )φ 1 (x 2 )φ 2 (x 3 )φ 2 (x 4 ) of two real scalar primaries φ 1 and φ 2 with dimensions ∆ 1 and ∆ 2 , respectively. The crossing relation in the traditional lightcone limit can be approximated as where each diagram represents a conformal block and O m is the lowest twist operator that appears both in φ 1 φ 1 and φ 2 φ 2 OPEs. Twists τ of the double twist operators [φ 1 φ 2 ] n, for large spin can be obtained by solving the above crossing equation. In particular, for the minimal twist tower (both even and odd spins) at large spin we get [21] τ [φ 1 φ 2 ] 0, where τ m is the twist and m is the spin of O m and c φ 1 φ 1 Om , c φ 2 φ 2 Om are OPE coefficients.
Similarly, by considering the u-channel, for large spin we obtain which agrees with (4.17) and (4.19) at large . In fact, if we consider exchange of multiple operators in the t-channel, contributions of each such , and τ satisfy the above inequality individually.

Complex Scalars
All the bounds discussed in this paper can be nicely demonstrated by studying the large spin behaviors of various double twist operators of a complex scalar primary O which is charged under a global U (1) symmetry. This scenario has been analyzed in detail in [6,21]. Consider the correlator O( . The crossing equation in the lightcone limit has the general form where T is the stress tensor, J is the U (1) symmetry current and S is a low dimensional scalar (if present). Similarly, we can write a slightly different crossing equation in the lightcone limit which already suggests that twists of [OO] n, and [OO † ] n, are not completely unrelated at large spin. Finally, let us also include the possibility of a low dimensional charged scalar C that can appear in the OO OPE. This leads to another crossing relation in the (5.6) One can solve these crossing equations simultaneously to obtain twists τ following [21] one can obtain for large even ( + ) and odd ( − ) spins. In the above equation, we have used the notation of [21] where C J and C T are central charges and S d = 2π d/2 Γ(d/2) . Note that ξ Om ∆ 1 ,∆ 2 is a positive quantity and hence τ for even spin is consistent with the Nachtmann theorem.
Whereas, for odd spin τ does not in general have to be a monotonically increasing convex function of . Moreover, the equation (5.7) also implies that for large spin τ Similarly, for the minimal twist tower [OO] 0, at large spin we obtain [21] τ does not obey a Nachtmann-like theorem in general. However, τ  [29][30][31][32] σ → 0 , with η = fixed (6.1) , where σ and η are defined in (3.3). Clearly, the Lorentzian lightcone limit is a special case of the Regge limit.
The main point we wish to emphasize in this section is that equations (3.16), (3.19), (3.24) as well as equations (4.14), (4.18) are valid even in the regime 0 < R η < 1.
Just like before, the analyticity condition of CFT correlators in Lorentzian signature now constraints the Regge behavior of certain CFT correlators by relating σ-integrals on the real line to an integral of the Regge limit of CFT correlators over the semicircle. However, these constraints are expected to be theory dependent because the Regge limit, for finite η, is dominated by high spin exchanges which are non-universal.
For the purpose of demonstration, we circumvent the intricacies of the Regge limit by assuming a specific Regge behavior. We consider CFTs in which the correlator

in the Regge limit admits an expansion
for some operators O 1 and O 2 with or without spin, where Λ is some cut-off scale and c L are σ independent real coefficients. This happens naturally in large-N CFTs. At first sight, the expansion (6.2) for L > 2 appears to be in contradiction with the chaos bound [19,33]. This suggests that coefficients c L are highly constrained. Alternatively, relations (3.16), (3.19) and (3.24) impose constraints on c L . These constraints ensure that the expansion (6.2) is consistent with the chaos bound.
Let us now be more precise. First, conditions (i) and (ii) lead to a positivity condition Moreover, the condition (ii) along with the relation (3.16) in the limit 1 Λ R η < 1 also imply the parametric bound for any even L ≥ 2. Similarly, equations (3.19) and (3.24) in the limit 1 Λ R η < 1 lead to the following quadratic relations Therefore, all coefficients with L > 2 must be parametrically suppressed in a systematic way implying that terms in (6.2) that grow faster than 1/σ can never dominate for 1 |σ| 1 Λ . This makes the expansion (6.2) consistent with the chaos bound. The above constraints are particularly useful in the context of the AdS/CFT correspondence where these constraints should be interpreted as bounds on various interactions of low energy effective field theories in AdS from UV consistency.
Of course, the discussion of this section can be extended to the mixed correlator simply by following the discussion of section 4. If the mixed correlator admits an expansion similar to (6.2) in the Regge limit, one can derive analogous bounds by exploiting equations (4.14) and (4.18) in the regime 1 Λ < R η < 1.
This concludes our discussion of various generalizations of the Nachtmann theorem in CFT.

Acknowledgments
in the lightcone limit. Clearly, the correlator G 0 (η, σ) is well defined only for real σ. In particular, for real positive σ 1, we can write the following expansion in the lightcone n are real coefficients and dots represent terms that are suppressed by higher powers of η. Obviously, the correction terms are suppressed by positive powers of σ as well. Moreover, notice that for negative σ Next, we consider the correlator in the lightcone limit. The goal is to figure out the behavior of G(η, σ) for complex σ with |σ| < 1. The contribution of the OO † → O p → XX conformal block to the above correlator in the Lorentzian lightcone limit can be computed using the OPE (3.7). For real positive σ 1, this contribution has the following structure where the real part of G(η, σ)| Op is exactly G 0 (η, σ)| Op which is given in (A.2). The rest of the terms in the above equation, for real positive σ 1, are completely imaginary. This follows from the fact that G(η, σ)| Op − G 0 (η, σ)| Op can be written as an integral of the discontinuity of some correlator across a branch cut in the ρ-plane. The leading imaginary contribution in the lightcone limit has two distinct parts G In fact, later we will argue that G (p) nint (η, σ) must be present in order to make G(η, σ) analytic on the lower half complex-σ plane. Finally, δG (p) int (η, σ) and δG (p) nint (η, σ) represent terms that are suppressed by higher powers of η. These correction terms are more difficult to compute since they depend on higher order terms of the lightcone OPE (3.7).
Nonetheless, it is easy to estimate the general behaviors of these correction terms. First of all, conformal invariance dictates that the correction terms with integer powers of σ cannot grow faster than 1/σ −1 . On the other hand, correction terms with non-integer powers of σ are fixed by the correction terms in (A.2) from analyticity and crossing. Thus, we conclude that where a is some positive number.
For real negative |σ| 1, we can write down a Lorentzian crossing equation. In particular, discussion of section 2 implies that Hence, if we rotate sigma σ → |σ|e −iπ in equation (A.5), that must be consistent with the above relation. The contribution G (p) int (η, σ) indeed satisfies this requirement. On the other hand, G 0 (η, σ)| Op in general does not obey the crossing relation. This implies that G(η, σ)| Op must contain an imaginary part with non-integer powers of σ, which we have denoted as G (p) nint (η, σ), such that the combination G 0 (η, σ)| Op + G (p) nint (η, σ) has the right behavior.
To be specific, let us consider a term C This relation is a manifestation of the fact that the lightcone limit conformal block (A.5) is valid even on the lower-half complex-σ plane. 16 So, G(η, σ) in the Lorentzian lightcone limit for Im σ < 0 can be expressed as an asymptotic series which is organized by twist where we have isolated the identity contribution. Note that this discussion applies to all correction terms in δG for any non-negative integer m. Hence, we can define the following generalized correlator on the lower-half σ plane where we have again isolated the identity contribution. Clearly, the generalized correlator 15 Note that n is an even integer. 16 Clearly, the relation (A.11) blows up when ∆ p + p is an odd integer. In that case, we should add terms like iσ ∆p log σ in G (p) nint (η, σ). Alternatively, we can treat ∆ p as a non-integer and take the integer limit at the end. where, S = {Re iθ , θ ∈ [0, −π]} with 0 < η R 1.
To be precise, we should also include the correction term δG (p) int (η, σ) in the right hand side. Since, however, terms in δG (p) int (η, σ) are always subleading compared to terms in G (p) int (η, σ), for our purpose we can safely ignore δG (p) int (η, σ). This immediately implies that we can use the identity (3.14) to project to different powers of 1/σ obtaining the sum rule

A.1 Scalar Example
For the purpose of demonstration, let us consider the special case where X = ψ is a scalar primary. We can use the explicit lightcone conformal block derived by Dolan  Similarly, we can analytic continue the Dolan-Osborn block along the path shown in figure   2 to obtain G(η, σ)| Op . In particular, using appendix B of [34], at the leading order in the Lorentzian lightcone limit we find for complex |σ| < 1 with Re σ > 0. This is completely consistent with the preceding discussion.

B Mixed Correlators in the Lightcone Limit
We can make a similar argument for the mixed correlator to show that terms that decay for small σ can be safely ignored even in section 4. However, since O 1 and O 2 are scalar primaries, we can provide a more direct argument. Again we only consider the non-trivial case where the dimensions of the exchanged operators are non-integers.