The leading trajectory in the 2+1D Ising CFT
PPrepared for submission to JHEP
The leading trajectory in the 2+1D Ising CFT
Simon Caron-Huot, a Yan Gobeil, a Zahra Zahraee, a a Department of Physics, McGill University, Montr´eal, QC, Canada
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We study the scattering of lumps in the 2+1-dimensional Ising CFT, indirectly,by analytically continuing its spectrum using the Lorentzian inversion formula. We findevidence that the intercept of the model is below unity: j ∗ ≈ .
8, indicating that scattering isasymptotically transparent corresponding to a negative Lyapunov exponent. We use as inputthe precise spectrum obtained from the numerical conformal bootstrap. We show that thetruncated spectrum allows the inversion formula to reproduce the properties of the spin-twostress tensor to 10 − accuracy and we address the question of whether the spin-0 operators ofthe model lie on Regge trajectories. This hypothesis is further supported by analytics in thelarge-N O(N) model. Finally, we show that anomalous dimensions of heavy operators decreasewith energy at a rate controlled by ( j ∗ − a r X i v : . [ h e p - t h ] J u l ontents Z -even family 12 (cid:15) Z -odd twist family 24 σ(cid:15) ] family 274.3 Continuing to spin 0 29 J = 0 in the large- N O ( N ) model 335.3 Implications for anomalous dimensions heavy operators? 395.3.1 Comments on conformal Regge theory 44 A.1 Dimensional reduction for cross-channel blocks 46A.2 New closed-form expression for radial expansion coefficients 48A.3 Lorentzian inversion in 2d 48A.4 Collinear expansion ¯ z → B Compact approximations from large-spin perturbation theory 50C Convexity of the leading trajectory 53 – 1 –
Introduction
Recent results in conformal field theories open the possibility to answer questions about theirreal time dynamics [1]. Thanks to the venerable Wick rotation, a d -dimensional EuclideanCFT is equivalent to a ( d −
1) + 1 dimensional one with a time direction, a map which canbe used in either direction. Intuition about real-time processes, in particular lightcone andhigh-energy limits, underlies many recent analytic results about CFTs [2–6]. On the otherhand, the currently most precise numerical results rely on Euclidean methods. In this paperwe attempt to use these numerical results to learn about the real-time dynamics of the 2+1-dimensional Ising CFT.The most basic question we would like to answer is whether high-energy scattering in thistheory is transparent or opaque. A typical physical experiment we have in mind consists ofpreparing a pair of lumps, regions of positive spins and some given transverse size, to whichwe apply a large boost, see fig. 1. Do the lumps pass through each other, or disperse intooblivion?This information is contained in the Regge limit of the four-point correlator of the spinfield σ : (cid:104) σ σ σ σ (cid:105) − (cid:104) σ σ (cid:105)(cid:104) σ σ (cid:105) ∝ G − . (1.1)CFT four-point functions depend on two real variables. The Regge limit is attained byapplying a large relative boost between (1 ,
2) and (3 , boost η →∞ ( G − ∝ e ( j ∗ − η (1.2)The Regge intercept j ∗ is interpreted as the spin of an effective Reggeized particle exchangedbetween the lumps. It is known that j ∗ ≤ j ∗ < j ∗ = 1: scatter-ing can be asymptotically transparent or opaque, respectively. Transparency, for the 2+1-dimensional Ising CFT, would mean that highly boosted lumps pass through each otherwithout interacting. This is to be contrasted with the strong interactions (which is not aCFT, but high-energy forward scattering can be discussed very generally) where protonsappear increasingly opaque at high energies, as witnessed experimentally by the increasingelastic and inelastic cross sections. For CFTs, since high-energy scattering can be viewed aslate-time evolution in Rindler space, the question of transparency versus opacity is equivalentto the question of whether the theory thermalizes on Rindler space, transparency meaninglack of thermalization, see section 9 of [9]. – 2 – x
132 4
Figure 1 : Scattering of lumps. We probe this process by correlating four local measurements.In the opaque case, one may expect to see transient exponential growth: j transient ∗ >
1. The bound of chaos states that, in any unitary theory, j transient ∗ ≤ j transient ∗ −
1) as a Lyapunov exponent is discussed in Appendix A there).A plausible, standard scenario is that in theories with j transient ∗ >
1, opacity is first reachedat small impact parameters, leading to a black disc whose radius grows with energy.Examples of theories with either type of behavior exist. To give a few examples, two-dimensional minimal models have j ∗ < j ∗ ≈
2, thus nearlysaturating the chaos bound. This reflects graviton exchange in the dual gravitational picture[12, 13]. This feature is also observed in the Sachdev-Ye-Kitaev (SYK) model [14]. Thetwo-dimensional version of SYK studied in [9] also exhibits transient growth, albeit with anon-maximal exponent: 1 < j transient ∗ <
2. For QCD, fits of hadron scattering data suggesta Pomeron intercept j transient ∗ ≈ .
09 [15]. In the perturbative regime of a four-dimensionalperturbative gauge theory, the famous BFKL analysis shows, very generally, that j transient ∗ ≈ O ( α s ) >
1. This conclusion is very much tied to the gluon having spin 1, and in weaklycoupled quantum field theories without vector bosons, we thus generally expect j ∗ < ε -expansionaround d = 4. This would suggest transparency. On the other hand, the theory lacks atuneable coupling constant, and it is unclear whether d = 3 is “close enough” to d = 4 forthis argument to be convincing. The main goal of this paper is to study this question usingnumerical data on the excited states of the 3D Ising CFT. We will find numerical evidencethat the model is indeed in the category of vector-free perturbative theories: j ∗ ≈ . < j ∗ (∆), which reduces tothe intercept at a special point.Let us briefly review the 3D Ising CFT. It is characterized by having Z symmetry and– 3 –nly two relevant operators, called σ and (cid:15) , which are respectively odd and even under Z .(From the bootstrap perspective, this defines the theory.) They are scalars and their scalingdimensions and OPE coefficients have been determined using Monte Carlo simulations andthe numerical bootstrap. The best numbers available, including the errors, are∆ σ = 0 . , ∆ (cid:15) = 1 . ,f σσ(cid:15) = 1 . , f (cid:15)(cid:15)(cid:15) = 1 . . (1.3)The spectrum also contains multi-twist families made out of these operators. The leadingtrajectory can be viewed as composites [ σσ ] ,J (defined below), which can be identified un-ambigously for J ≥
2. We will also study the leading odd trajectory [ σ(cid:15) ] ,J . We will benefitfrom the high-accuracy data and analysis for these families and other operators reported in[5]. Note that the stress tensor is a member of the leading trajectory: T = [ σσ ] , .The intercept is but one point on a continuous curve, j ∗ ≡ j ∗ (∆ = ). Our maintool to study the full curve will be the Lorentzian inversion formula, which reconstructs thedimensions and OPE coefficients in one channel, as a continuous function of spin, in termsof operators exchanged in cross-channels. A well-understood large-spin expansion has beenknown to work well even down to J = 2 [2–5]. We will approach the intercept in two steps:First we will establish numerical convergence of the operator sum by reproducing the knownstress tensor dimension, ∆ ∗ (2) = 3, to high accuracy. From there we will gradually reduce ∆.In addition to the leading trajectory and intercept, we will discuss the following simplequestion: do the spin-0 operators σ and (cid:15) lie on Regge trajectories? We will find numericalevidence that σ lies on the shadow of the leading odd trajectory. Within the ε -expansion, itis know that (cid:15) resides on an analytically continued branch of the leading trajectory [16]; wewill find that the 3D numerical data, while compatible with this hypothesis, does not add tothe evidence.This paper is organized as follows. In section 2 we first review the Lorentzian inversionformula and how it can be used to extract low twist CFT data in a general theory. Then wefocus on three dimensions and discuss the accurate numerical evaluation of the formula usingthe method of dimensional reduction [17, 18], comparing the result with large-spin approxi-mations. In section 3 and 4 we apply this method to the 3d Ising model. We specifically workon the (cid:104) σσσσ (cid:105) and the (cid:104) σ(cid:15)(cid:15)σ (cid:105) correlators to extract data for spin-two operators in the [ σσ ] and [ σ(cid:15) ] families, the intercept, and we describe our attempts to reach spin 0. In section5 we discuss various aspects relevant to the interpretation of the results. In subsection 5.1we comment on general distinctions between theories with intercept above and below 1. Insubsection 5.2 we analyze the leading trajectories of the critical O ( N ) model at large N inboth bilinears [ φ i φ j ] and [ φ i S ] , which we will find to be analogous to the [ σσ ] and [ σ(cid:15) ] trajectories in 3D Ising. Finally, in subsection 5.3 we propose a formula which relates theintercept being less than unity to regularity of the heavy spectrum. Section 6 contains ourconcluding remarks. Appendix A contains explicit inversion integrals utilized in the paper,appendix B records compact approximations to large-spin operators, and appendix C pro-– 4 –ides a short proof that the leading trajectory is convex. Note:
While this work was beingcompleted, closed related methods have been applied to the critical O(2) model [19].
We consider a correlation function of 4 scalar primary operators (cid:104) φ ( x ) φ ( x ) φ ( x ) φ ( x ) (cid:105) = 1( x ) ∆1+∆22 ( x ) ∆3+∆42 (cid:18) x x (cid:19) a (cid:18) x x (cid:19) b G ( z, ¯ z ) , (2.1)where x ij = x i − x j , a = ∆ − ∆ , b = ∆ − ∆ and the conformal cross-ratios z , ¯ z are defined as z ¯ z = x x x x , (1 − z )(1 − ¯ z ) = x x x x . (2.2)We can use the OPE for operators 1 and 2 together and for operators 3 and 4 to decomposethe correlator in s-channel conformal blocks as follows G ( z, ¯ z ) = (cid:88) ∆ ,J f O f O G ( a,b )∆ ,J ( z, ¯ z ) , (2.3)where f ij O is the OPE coefficient and G ( a,b )∆ ,J ( z, ¯ z ) is the s-channel conformal block, whichresums the contribution of the primary with dimension ∆ and spin J and all of its descendants.More explicitly, conformal blocks are special functions that are the eigenfunctions of thequadratic and quartic Casimir equation. They admit closed form in even spacetime dimension,for instance the conformal blocks in d = 2 dimension can be written as follows: G ( a,b )∆ ,J ( z, ¯ z ) = k ( a,b )∆ − J ( z ) k ( a,b )∆+ J (¯ z ) + k ( a,b )∆+ J ( z ) k ( a,b )∆ − J (¯ z )1 + δ J, , ( d = 2) , (2.4)where k is the hypergeometric function k ( a,b ) β ( z ) = z β/ F ( β/ a, β/ b, β, z ) . (2.5)Here β = ∆ + J is the conformal spin. We also introduce τ = ∆ − J which we refer to astwist. We will use ∆ , J, β, τ in different contexts to specify the operators in the spectrum.Conformal blocks do not accept a simple closed-form expression in odd spacetime dimen-sions and one must resort to various approximations. The main approximation we will use isto write 3d blocks as sums over 2d blocks [18], as reviewed in appendices A.1.In general, we normalize the blocks so that: lim z (cid:28) ¯ z (cid:28) G ∆ ,J ( z, ¯ z ) = z τ ¯ z β . The leadingterm as z, ¯ z → z, ¯ z → G ( a,b )∆ ,J ( z, ¯ z ) = ( z ¯ z ) ∆2 C J (cid:18) z + ¯ z √ z ¯ z (cid:19) , where C J ( η ) ≡ Γ (cid:0) d − (cid:1) Γ( J + d − d − (cid:0) J + d − (cid:1) F (cid:0) − J, j + d − , d − , − η (cid:1) . (2.6)The function C J ( η ) is a multiple of a Gegenbauer function, C J ( η ) ∝ C ( d/ − J ( η ), satisfyinglim η →∞ C J ( η ) = (2 η ) J . – 5 – .1 Lorentzian inversion formula A good starting point for analytics is an alternate form of the OPE in which one integratesover operators dimensions along the principal series, but where spin is still discrete and needsto be summed over: G ( z, ¯ z ) = ∞ (cid:88) J =0 (cid:90) d/ i ∞ d/ − i ∞ d ∆2 πi c (∆ , J ) F ( a,b )∆ ,J ( z, ¯ z ) + (non-normalizable) . (2.7)where non-normalizable modes describe operators with ∆ < d (which includes, notably, theidentity). The CFT data is then encoded in the poles of the analytic function c (∆ , J ). Theseare located at the position of the physical operators in the conformal block expansion, andthe residues give the OPE coefficients in the following way f O f O = − Res ∆ (cid:48) =∆ c (∆ (cid:48) , J ) . (2.8)The harmonic function F ∆ ,J is a single-valued, shadow-symmetric combination of the blockand its shadow [20]: F ( a,b )∆ ,J ( z, ¯ z ) = 12 G ( a,b )∆ ,J ( z, ¯ z ) + K ( a,b ) d − ∆ ,J K ( a,b )∆ ,J G ( a,b ) d − ∆ ,J ( z, ¯ z ) , (2.9)where K ( a,b )∆ ,J = Γ(∆ − (cid:0) ∆ − d (cid:1) κ ( a,b )∆+ J , κ ( a,b ) β = Γ (cid:16) β − a (cid:17) Γ (cid:16) β + a (cid:17) Γ (cid:16) β − b (cid:17) Γ (cid:16) β + b (cid:17) π Γ( β − β ) . (2.10)The functions F ∆ ,J satisfy an orthogonality relation which allows to read off the OPEdata from the correlator (Euclidean inversion formula). The Lorentzian inversion formulareconstructs the same data using less information, the double discontinuity [6, 21, 22]: c t (∆ , J ) = κ ( a,b )∆+ J (cid:90) (cid:90) dzd ¯ z µ ( z, ¯ z ) G ( − a, − b ) J + d − , ∆+1 − d ( z, ¯ z ) dDisc [ G ( z, ¯ z )] , (2.11)which needs to be summed with the contribution of the u-channel to give the full coefficients: c (∆ , J ) = c t (∆ , J ) + ( − J c u (∆ , J ) . (2.12) c u (∆ , J ) is obtained from c t (∆ , J ) by exchanging the operators 1 and 2. The measure is µ ( z, ¯ z ) = 1( z ¯ z ) (cid:12)(cid:12)(cid:12)(cid:12) z − ¯ zz ¯ z (cid:12)(cid:12)(cid:12)(cid:12) d − . (2.13) This form agrees with ref. [6] using the identity: G ( − a, − b ) ( z, ¯ z ) J, ∆ = ((1 − z )(1 − ¯ z )) a + b G ( a,b ) ( z, ¯ z ) J, ∆ . – 6 –he double discontinuity of the correlator is a certain linear combination of analytic contin-uation around ¯ z = 1 which computes the expectation value of a double commutator [6]:dDisc [ G ( z, ¯ z )] = cos[ π ( a + b )] G ( z, ¯ z ) − e iπ ( a + b ) G (cid:8) ( z ¯ z ) − e − iπ ( a + b ) G (cid:9) ( z ¯ z ) . (2.14)This combination is positive definite and is analogous to the absorptive (imaginary) part ofa scattering amplitude. The coefficient function which comes out of Lorentzian inversion isautomatically shadow-symmetric: c (∆ , J ) K ( a,b )∆ ,J = c ( d − ∆ , J ) K ( a,b ) d − ∆ ,J . (2.15)Along the principal series, Re(∆) = d , convergence of the Lorentzian inversion formulais controlled by the Regge limit and requires J > j ∗ , where j ∗ is the intercept defined ineq. (1.2). The Lorentzian inversion formula then manifests the analyticity of the spectrumin spin, giving an organizing principle for operators of spin J > j ∗ . In a unitary CFT thisalways include all operators with J ≥ ∗ ( J ). For integer J ≥
2, eq. (2.8)shows that this is the pole nearest to the principal series, and positivity of the integrand (forreal ∆) implies convergence in a strip: d − ∆ ∗ ( J ) < Re(∆) < ∆ ∗ ( J ). For non-integer spin,the leading trajectory answers a simple question: when does the integral (2.11) converge?The resulting smooth curve can also be parametrized as j ∗ (∆), where convergence issatisfied for J > j ∗ (∆). With this definition, it is easy to show using positivity of the dDiscthat j ∗ (∆) is a real and convex function, see [23], extending the integer-spin convexity provedin ref. [2, 24] using Nachtmann’s theorem (we give an alternative proof in appendix C). Sincethe leading trajectory is also manifestly shadow-symmetrical, j ∗ (∆) = j ∗ ( d − ∆), it followsthat its minimum, the intercept must be at the symmetrical point: j ∗ ≡ j ∗ ( d ). This agreeswith the physical definition of the intercept given earlier in eq. (1.2) since convergence of theLorentzian inversion formula at that point is controlled by the Regge limit of correlator.Two practical points worth mentioning are as follows: first, at the cost of a factor of two,we can restrict the integration range in the Lorentzian inversion formula to z < ¯ z . Second,when we are interested in extracting s-channel data from poles at ∆ > d/
2, we can decomposethe s-channel blocks follows and restrict to the first term, g pure (defined to have a single towerof terms in the limit 0 (cid:28) z (cid:28) ¯ z (cid:28) G ( a,b ) J + d − , ∆+1 − d ( z, ¯ z ) = g ( a,b )pure J + d − , ∆+1 − d ( z, ¯ z ) + Γ(∆ − − ∆ + d )Γ(∆ − d )Γ( − (∆ + 1 − d )) g ( a,b )pure J + d − , − ∆+1 ( z, ¯ z ) . (2.16)This is because the second term does not contribute to the poles ∆ > d/ ∼ d/ .2 Extracting low-twist OPE data For generic β , we will only be interested in the poles and residues of c ( z, β ), which will comefrom the small- z limit of the integrand. In particular, for the pole corresponding to theoperator of smallest twist it suffices to take z → c t (∆ , J ) = (cid:90) dz z z − τ C t ( z, β ) + (collinear descendents) , (2.17)where we have defined a generating function C t ( z, β ): C t ( z, β ) = (cid:90) z d ¯ z ¯ z κ β k ( − a, − b ) β (¯ z ) dDisc [ G ( z, ¯ z )] . (2.18)The generating function encodes the spectrum through power laws. More precisely, if weexpand it as C ( z, β ) = (cid:88) m C m ( β ) z τm , (2.19)it is easy to see that each power will produce in eq. (2.17) a pole C m τ m − τ , interpreted as anoperator of twist τ m following eq. (2.8). Note that eq. (2.17) does not subtract collineardescendants, since neglected corrections by integer powers of z affect the residues at shiftedvalued τ m + 2 , τ m + 4 , . . . . In this paper we will restrict ourselves to the lowest twist familyfor which m = 0 and collinear descendants play no role (for more information on how to treathigher twist families see [25]).The exponents τ m give the twist of operators in the spectrum. The coefficient C m ( β )are related to OPE coefficients but the relation is slightly subtle because eq. (2.8) requiresresidues computed at constant spin J , whereas C m ( β ) gives residues at constant β . The exactrelation includes a Jacobian [5, 6, 26]: f O f O = (cid:18) − d τ m ( β )d β (cid:19) − C m ( β ) (cid:12)(cid:12)(cid:12)(cid:12) β − τ =2 J . (2.20)Our strategy to gain knowledge from the inversion formula is to insert the t-channeldecomposition of the correlator (obtained from the s-channel by swapping operators 1 and 3,equivalent to fusing 1 with 4 and 2 with 3) into the generating function in eq. (2.18): C t ( z, β ) = (cid:88) ∆ (cid:48) ,J (cid:48) f O (cid:48) f O (cid:48) c ∆ ··· ∆ ∆ (cid:48) ,J (cid:48) ( z, β ) (2.21)where c ∆ ··· ∆ ∆ (cid:48) ,J (cid:48) ( z, β ) ≡ (cid:90) z d ¯ z ¯ z κ ( a,b ) β k ( − a, − b ) β (¯ z )dDisc (cid:34) ( z ¯ z ) ∆1+∆22 [(1 − z )(1 − ¯ z )] ∆2+∆32 G ( a (cid:48) ,b (cid:48) )∆ (cid:48) ,J (cid:48) (1 − z, − ¯ z ) (cid:35) . (2.22)where a (cid:48) = ∆ − ∆ , b (cid:48) = ∆ − ∆ . To obtain the u -channel generating function C u we inter-change ∆ with ∆ wherever they appear in eq. (2.21).– 8 –e only know closed-form expressions for the integral (2.21) in special cases. An impor-tant one is the t -channel identity (which can only be physically realized when ∆ = ∆ and∆ = ∆ ): C ( z, β ) (cid:12)(cid:12)(cid:12) t − channel identity = z ∆1+∆22 (1 − z ) ∆2+∆32 I (∆ , ∆ ) ( β ) , (2.23)where I (∆ , ∆ ) ( β ) = (cid:90) d ¯ z ¯ z κ β k ( − a, − a ) β (¯ z )dDisc (cid:34) ¯ z ∆1+∆22 (1 − ¯ z ) ∆ (cid:35) = Γ (cid:16) β +∆ − ∆ (cid:17) Γ (cid:16) β +∆ − ∆ (cid:17) Γ (∆ ) Γ (∆ ) Γ( β −
1) Γ (cid:16) β +∆ +∆ − (cid:17) Γ (cid:16) β − ∆ − ∆ + 1 (cid:17) . (2.24)Another important analytic result pertains to the case where we insert a two-dimensionalblock in the t -channel, where the integral reduces to a 1d 6j symbol [27–29]. Although we areinterested in d = 3, we will use this result by writing the 3d blocks as sums over 2d blocks.A brief review on this method of dimensional reduction is given in appendix A.1, here wequote the final result. A conformal block in d dimension can be expanded as a sum over( d − G ( a,b )∆ ,J ( z, ¯ z ; d ) = (cid:88) A ( a,b ) m,n (∆ , J ) G ( a,b )∆+ m,J − n ( z, ¯ z ; d −
1) 0 ≤ n ≤ J, m = 0 , , ... (2.25)where the coefficients A are determined recursively using the Casimir differential equation.Inserting this expansion into the inversion integral (2.21) for each individual block, we obtainthree-dimensional inversion integrals as a sum over two-dimensional inversion integrals givenanalytically in eq. (A.8) c ∆ ··· ∆ ∆ (cid:48) ,J (cid:48) ( z, β ; d =3) = (cid:88) m =0 J (cid:48) (cid:88) n =0 A ( a (cid:48) ,b (cid:48) ) m,n (∆ (cid:48) , J (cid:48) ) c ∆ ··· ∆ ∆ (cid:48) + m,J (cid:48) − n ( z, β ; d =2) . (2.26)In practice, the error in this method can be reduced to zero by including as many terms asneeded, since exponential convergence rapidly sets in. In our analysis, for most values of J (cid:48) and ∆ (cid:48) , going to m = 15 is more than enough. The analytic formulas for 2d integrals requiresthe ¯ z integral in eq. (2.21) to have lower bound 0 instead of z ; the difference is negligiblecompared to other sources of error as long as we are away from the intercept (however forcompleteness in fig. 3.2 the twist of stress-tensor resulting from the inversion formula with¯ z > z is given as well). In section 3.2 we will use a different approximation when we approachthe intercept. In theory, the exponents in eq. (2.19) are obtained by analyzing the z → t -channel sum (2.21). In particular, the leading twist and OPE coefficient is equal to the– 9 –ollowing limit: τ ( β ) = lim z → z∂ z C ( z, β ) C ( z, β ) , C ( β ) = lim z → C ( z, β ) z z∂zC ( z,β ) C ( z,β ) . (2.27)In practice, however, we only have access to a finite number of terms in the t -channel sum,which prevents us from taking z arbitrarily small: the limit lies at the boundary of convergenceof the t -channel OPE. In some previous analyses, a convenient value of z was simply fixed[5, 17]; one could also consider fitting the z -dependence to a power law.Our approach in this paper will be to plot the quantity τ = z∂ z C ( z,β ) C ( z,β ) as a function of z and look for a plateau. If z is chosen too large, we expect errors due to neglected higher-twist s -channel operators, while if z is too small, we expect truncation errors from the t -channelsum. By restricting our attention to a plateau region we hope to simultaneously minimizeboth sources of error (in addition to getting rough error estimates). The effectiveness of the analytical bootstrap in extracting large spin data is well established[2–4]. These results are typically obtained by considering the double lightcone limit ( z, ¯ z ) → (0 , t -channel must be reproduced by large-spin-tails in the s -channel. Let us briefly review how these results relate to the formulas justreviewed, highlighting the ways in which our analysis will differ.The basic physical picture is that large spin (or large β ) pushes the integral (2.21) to the¯ z → k -function (defined in eq. (2.5)). At sufficiently largespin, the t -channel identity given in eq. (2.23) thus dominates. Its particular z dependencethen implies the existence of so-called double-twist families [ φ φ ] n,J , where J denotes thespin of operator. Their twist approximates the naive dimensional analysis [2, 3, 30]: τ [ φ φ ] n,J = ∆ φ + ∆ φ + 2 n + γ, (2.28)where γ , the anomalous dimension of the operator, vanishes in the large-spin limit. Fromthe scaling relation √ − ¯ z ∝ β , one can easily see that the correction, due to exchange of t -channel operator of lowest nontrivial twist τ min , decays like γ ( n, (cid:96) ) (cid:39) γ n J τ min . (2.29)These corrections are found by analyzing the collinear ¯ z → t -channel blocks.lim ¯ z → G ( a (cid:48) ,b (cid:48) )∆ (cid:48) ,J (cid:48) (1 − ¯ z, − z ) → (1 − ¯ z ) ∆ (cid:48)− J (cid:48) k ( a (cid:48) ,b (cid:48) )∆ (cid:48) + J (cid:48) (1 − z ) + O (1 − ¯ z ) . (2.30)One can readily see that using this approximation the inversion integral over ¯ z is greatlysimplified and can be performed analytically; the result is particularly simple if one expandsinstead in powers of − ¯ z ¯ z , see eq. (2.24). For the leading trajectory in the Ising CFT, taking– 10 –he coefficient of log z in eq. (2.30) then gives a simple pocket-book formula for large-spincorrections: τ [ σσ ] ,J ≈ σ − (cid:88) O = (cid:15),T λ σσ O Γ(∆ σ ) Γ (cid:0) ∆ σ − τ O (cid:1) Γ(∆ O + J O )Γ (cid:0) ∆ O + J O (cid:1) (cid:18) β − (cid:19) τ O . (2.31)We have chosen β (cid:55)→ β − β − J = 2, although theerrors are somewhat difficult to estimate a priori. Analogous formulas for OPE coefficientsand for [ σσ ] and [ (cid:15)(cid:15) ] trajectories are recorded in appendix B.One could try to estimate errors by studying further 1 /J corrections, but let us reporthere on a more straightforward exercise which is to simply compare the approximation ineq. (2.30) with the actual integrand entering the Lorentzian inversion formula. We do thishere for a single t -channel block ( (cid:15) ), reserving discussion of the sum over blocks to the nextsection. The ¯ z -dependence of the integrand of eq. (2.21) comes from two factors: the s -channel block k β (¯ z ) and the t -channel block. Their product is shown for (cid:15) -exchange in fig. 2for β = 5 and β = 10. We show three approximations for the t -channel blocks: the 3d to 2dexpansion (called “exact” since terms beyond the third one are invisible on the plot), and thecollinear series in powers of (1 − ¯ z ) whose first two terms are given for reference in eq. (A.16).One can see that at the larger value β = 10 (corresponding roughly to J = 4) even theleading collinear term matches the integrand very well. At the integrated level, it underes-timates the (cid:15) contribution by only 3%. For β = 5 (corresponding to the stress tensor) theerror is up to 10%, coming mostly from the region of ¯ z not close to 1. Because this multipliesa small coefficient, this corresponds to a 4 × − error on the twist of the stress tensor.Replacing the power of (1 − ¯ z ) by − ¯ z ¯ z produces similar numbers. Including up to the thirdterm in the (1 − ¯ z ) series reduces the errors to 0 .
3% and 2%, respectively. Since our goal willbe to do much better than this, we need to employ formulas which are valid at all ¯ z . Weachieve this in the next section by using the 3d to 2d expansion of blocks, which convergesmuch faster. – 11 – (cid:2)(cid:3)(cid:4) (cid:1) (cid:5)(cid:6) Figure 2 : Integrand of the inversion formula (2.21) with z = 10 − , comparing the exactcross-channel block for (cid:15) -exchange (using the 3d to 2d series) with its collinear series in(1 − ¯ z ). For β = 5, the collinear limit in eq. (2.30) approximates the dominant region well butunderestimates the integrand at small ¯ z . At larger values of β this region becomes negligible.Note that we rescaled the integrand by 2 β β √ − ¯ z to make features more visible. Z -even family In this section we study the leading Regge trajectory in the 3D Ising model (i.e., [ σσ ] family)by applying the formalism developed in section 2, focusing on low spins.We begin with the stress tensor, which is the spin 2 operator of [ σσ ] family. This willserve as a benchmark case: while its twist is known from conservation laws, reproducing itas an infinite sum over cross-channel operators is nontrivial. We show that in order to getbest control over systematic errors, we need to work at significantly lower values of z thanpreviously considered, which is feasible by including subleading families ([ σσ ] and [ (cid:15)(cid:15) ] ) inthe cross-channel and resumming their large-spin tails. We obtain both the twist and OPEcoefficient of the stress-tensor with error at the level 10 − which is compatible with the errorin the numerical data used in inversion formula (see the numerical error for spin 6 operatorof [ (cid:15)(cid:15) ] family in table. 4 of [5]).In subsection 3.2 we apply a similar analysis to the intercept, where convergence in thes-channel twist will be found to be slower. In subsection 3.3 we briefly discuss attempts toreach the operator (cid:15) itself through an analytic continuation of the trajectory close to theintercept. Here we calculate the twist and the OPE coefficient of the stress-energy tensor in the [ σσ ] family using the inversion formula. Since it is a conserved operator it saturates the genericspin unitarity bound: ∆ = J + d − . (3.1)– 12 –o when we are in 3 dimension stress tensor has scaling dimension 3, twist τ = 1 and conformalspin β = 5. This operator belong to [ σσ ] family with asymptotic twist at large spin equalto 2∆ σ ≈ . γ T ≈ − . z . This is important and allows us to sum overinfinite families. This is because the z → z dependence is necessary to accuratelycut off the sums at J (cid:48) ∼ / √ z (see [5, 6]). The 3d to 2d series converges rapidly and wealways include sufficiently many terms that we can neglect this source of error, effectivelytreating the blocks as “exact”. We will comment on the small- z expansion for the exchangeof a single operator and its region of validity later in this section.First, let us show the effect of various t -channel truncations to twist τ = 2 z∂ z C ( z,β ) C ( z,β ) ,evaluated at β = 5 and for various values of z . This will illustrate the relative importance ofsubleading families depending on the value of z . The data used for exchanged operators isfrom tables in the appendix of[5] We see in fig. 3 that including the subleading twist families ���� �� - - - - - - - - - - - Figure 3 : The effect of different t -channel truncations in eq. (2.21) on the extracted stress-tensor anomalous dimension γ T = τ T − φ . Including more operators enables us to reachlower values of z where we find a stable z -independent plateau.[ σσ ] and [ (cid:15)(cid:15) ] significantly improves the result. In addition, we also observe once we multiply C ( z, β ) with (1 − z ) ∆ σ , the plateau extends to larger value of z for each of truncation in thecross channel. This is illustrated for (cid:15) exchange in fig. 3. The reason for this is that theregion of large z is contaminated both by collinear descendants of the leading trajectory,and by higher-twist trajectories. Since ∆ σ is close to the unitarity bound, the latter aremuch smaller, and the former are largely canceled by multiplying by the mean-field factor– 13 – ��� ��
10 20 30 400.0010.0020.0030.0040.0050.0060.007 (a) [ σσ ] family ���� ��
10 20 30 400.0010.0020.0030.0040.0050.0060.007 (b) [ σσ ] family ���� ��
10 20 30 400.0010.0020.0030.004 (c) [ (cid:15)(cid:15) ] family (d) large-spin diagram for [ (cid:15)(cid:15) ] exchange Figure 4 : Partial sum contributions to C ( z, β = 5) for the first three leading families asa function of the maximal spin. The [ σσ ] family converges for any z , but other families,especially [ (cid:15)(cid:15) ] , are very sensitive to large spins. Figure (d) shows how the exchange of [ (cid:15)(cid:15) ] can contribute as log z (1 − z ) ∆ σ . One salient point is that for sufficiently small z all the curves eventually departfrom the correct stress-tensor twist. This is because at smaller z the OPE converges moreslowly and operators with both higher spin and twist need to be included. It is also apparentthat summing up to a finite spin cutoff is not sufficient to create a plateau, since operatorswith quite large spin are also important (see fig. 4). By resumming the higher spin tails in allthe families in two different independent way, we bypass this problem and produce two curveswhich as can be seen are the most successful curves in reproducing the anomalous dimensionof stress-tensor, both in terms of accuracy and stability, given the publicly available numericaldata. We will discuss how we performed these resummations and obtained the stable curvesin the paragraphs below.Fig. 4 shows that high spin tails are strongly needed for [ (cid:15)(cid:15) ] family, which has not yetconverged at spin 40 at the shown values of z . In addition, it is also required for [ σσ ] family– 14 –ven though this family is converging more quickly. This is in contrast with [ σσ ] family forwhich the sum is fully convergent for the whole region we are considering, so resumming itshigh spin operator has a negligible effect as can be seen in fig. 4.A strong tail for the [ (cid:15)(cid:15) ] family was to be expected physically since, in large-spin pertur-bation theory, single- (cid:15) exchange produces a log( z ) term which accounts for a large fractionof the stress tensor anomalous dimension (see eq. (2.31)). One may thus expect the box-likediagram in fig. 4(d) to contain a log z term which exponentiates single- (cid:15) exchange. Sincesuch a log z cannot be generated by individual t -channel operators and must necessarilycome from a large-spin tail [5, 32].Accurate numerical data for large spin tails at spins J >
40 is unavailable. In principleone could obtain good analytic approximations for this region using large spin perturbationtheory, where the couplings between σσ and [ (cid:15)(cid:15) ] follow from mixed correlators, as was alsostudied in [5]. We derive this analytic approximation with the inversion integrals having lowerbound z (this is done by subtracting 0 to z integral from the 2 d integrals in eq. A.8. Seeappendix. A.3 for more details). However, in the spirit of the data-driven approach followedin this paper, we adopt a simple modelling and fitting strategy as well. We will compare thetwo methods for estimation of the error in the tail. We do not directly fit the OPE data oflarge-spin operators (twist and OPE coefficients) since all we will need is their contributionto the ¯ z -integrated double-discontinuity. The important advantage of this method is thatthe difficulties related to performing the inversion formula for large spin blocks, such as theexpensive 3 d to 2 d expansion is avoided. The cross channel block has a simple z -dependence,as can be seen from the large spin and small z expansion (see appendix A in [3]): G τ (cid:48) ,(cid:96) (cid:48) (1 − ¯ z, − z ) → k β (1 − z ) v τ (cid:48) / F ( τ (cid:48) , ¯ z ) + O (1 / √ (cid:96), / √ z ) , (3.2)where k β (1 − z ) = (1 − z ) β/ F ( β/ a, β/ / + b, β, − z ) was defined in eq. (2.5) andthe function F won’t be important to us. The prime notation is associated with the cross-channel operators. The next thing we want to estimate is the large spin expansion of theOPE coefficients. At large spin the OPE coefficient of operators converges to their values inmean field theory [3], f σσ [ σσ ] n,(cid:96) ∼ [1 + ( − (cid:96) ] P ∆ σ σ +2 n,(cid:96) where P ∆ σ σ +2 n,(cid:96) ≡ (∆ σ − d/ n (∆ σ ) n + (cid:96) n ! (cid:96) !( (cid:96) + d/ n (2∆ σ + n − d +1) n (2∆ σ +2 n + (cid:96) − (cid:96) (2∆ σ + n + (cid:96) − d/ n (3.3)and where ( a ) b denotes the Pochhammer symbol which is defined as ( a ) b ≡ Γ( a + b )Γ( a ) . Thisallows us to estimate the inversion formula in (2.21) at large spin as follows: C β ( z, β (cid:48) ) ∼ C (cid:48) P ∆ σ σ +2 n,(cid:96) β (cid:48)− τ (cid:48) k β (cid:48) (1 − z ) z ∆1+∆22 (1 − z ) ∆2+∆32 (3.4)This fitting can be done for all of the three families included in the cross-channel. Howeveras just explained it only has an impact for subleading families. To account for the mixing of– 15 – ��� ��
50 100 150 200 250 3000.0000.0020.0040.0060.0080.0100.012
Figure 5 : Partial sums contributing to C ( z =10 − . , β =5), extrapolated to very large spins.the [ σσ ] and [ (cid:15)(cid:15) ] family, we will fit their sum for each spin to the function given in the RHSof eq. 3.4, for which we will have n = 1 and β = 2 J + τ [ (cid:15)(cid:15) ] . The fit is done with data havingspin 18 and higher for both families . The parameter of the fit and their covariance matrixfor each family is given as follows: C (cid:48) [ σσ ] +[ (cid:15)(cid:15) ] = 0 . , τ (cid:48) [ σσ ] +[ (cid:15)(cid:15) ] = 0 . , COV = (cid:32) . . . . (cid:33) . (3.5)Note that τ (cid:48) ∼ σ which is the expected value. Using this fit we can estimate the contri-bution of the tail of the aforementioned families to C ( z, β ) and make the sum over families aconvergent sum as can be seen in the fig. 5. By adding Gaussian noise to the fitted values C (cid:48) and τ (cid:48) with the quoted covariance, we find branching curves results in an error of order 10 − in the final answer for the stress-tensor twist (the size of the branching is compatible withthe difference between the analytic tail and the fit manifested in a magnified version fig. 5 ).We note that the fit uncertainties are highly correlated, and varying C (cid:48) and τ (cid:48) independentlywould generate very different curves! We see that once the contribution of higher spin istaken into account in the sum over families, the flatness of the curve and thus in dependencyof γ T from z is restored (see fig. 6 and fig. 3).The order ∼ − error in the tail at small z in fig. 6 is comparable with the error onthe numerical data of the [ (cid:15)(cid:15) ] family (see for instance the data for spin 6 operator table. 4 in[5]). So one cannot hope to reduce the error just by improving the tail. In addition, the erroron the numerical data as opposed to the error caused by truncation do not have a definitesign, this makes it impossible to give an upper bound on the result.The final issue to be addressed is the range of z accessible to us. As predicted we cannotget arbitrarily close to zero as the error of the tails would eventually become significant (seefig. 6). However, we are allowed to take any z such that in the range depicted in fig. 6. Wechoose the decade in which the curve has the smallest error (in terms of the standard deviation– 16 – ��� �� - - - - - - - - - Figure 6 : Magnified version of fig. 3, showing the importance of resumming large-spin tailsto extract a z -independent stress-tensor twist. The difference between the two tails which isof order 10 − and is a result of the difference between blue and orange curves in fig. 5, givesan estimate of the error for the tail contribution) ���� ��
200 400 600 800 1000 - - - - - (a) Evaluated at z = 10 − , . ���� ��
50 100 150 200 250 300 - - - - - - (b) Evaluated at z = 10 − . Figure 7 : Individual high twist operators 2 z∂ z C τ (cid:48) ,j (cid:48) ( z,β ) C τ (cid:48) ,j (cid:48) ( z,β ) − σ push the curve of γ β =5 down.with respect to the average of the function in the decade) which is log z ∈ ( − . , − . τ = 1 . ± × − (cid:16) error from deviation fromflatness and differences between two tails (cid:17) (3.6)It is crucial to address the question of removing the residual gap between this result andthe actual twist of the stress-tensor. We argue that this can be done by including highertwist family in the OPE. To understand whether this is the right resolution, the first thingto check would be whether the high twist operators push the curve down or up. We do thatby looking at 2lim z → z∂ z C τ (cid:48) ,J (cid:48) ( z,β ) C τ (cid:48) ,J (cid:48) ( z,β ) − σ for individual operators in fig. 7 and checking thatthe contribution of each is well below the average derived in eq. 3.6.Now that we have a reliable way of computing the twist for the β as small as β of the– 17 –tress-tensor, we can calculate it for other points in the vicinity of stress-tensor by repeatingthis procedure. We can then use these point and get the function τ ( β ) by interpolation. Thefunction we get is demonstrated in fig. 8. ���� �� Figure 8 : τ ( β ) in the vicinity of β = 5. The point that the line J = 2 crosses c ( β ) gives thelocation of the stress-tensor.Now we know by eq. (2.20) that the squared of the OPE coefficient is related to C ( β = 5)with a Jacobian factor which we can then calculate with the function τ ( β ) derived above.Thus we arrive at the value of the OPE coefficient.Again confining ourselves to the decade log z ∈ [ − . , − . f σσT = 0 . ± × − (3.7)We summarize the result for the stress-tensor in the table. 1. Remarkably we are able toobtain the twist and OPE coefficient of the stress-tensor with accuracy 10 − !∆ T f σσT Inversion Formula (separate fit) 1.00013(5) 0.326077(12)Numerical Result 1 0.32613776(45)
Table 1 : Twist and OPE of the stress-tensor, the spin 2 operator in [ σσ ] , derived from theinversion integral (all the 3 families and the high spin tail from the fit is included) comparedwith the value derived from numerical bootstrap. Comparison between z → and finite z As mentioned in previous sections, when we are interested in the exchange of the first fewleading twist blocks in the cross-channel, we have the luxury of taking z → F function. However, one need to take into account the errors introduced both by– 18 –runcation in the cross-channel OPE expansion of the correlator in dDisc as well as errorintroduced by higher order terms in the z expansion. As an illustration, in fig. 9, we com-pare the anomalous dimension of the stress-tensor derived by using the z → (cid:15) and T are exchanged in the cross-channel with the one derived withby keeping the full z dependence (plotted in fig. 3 in cyan color) as well as the subtracted full z dependant one (plotted in fig. 3 in gray color) . We emphasize again that one is allowed todo that because there is no infinite sum involved. ���� �� - - - - - - - - - - Figure 9 : Comparison between the twist derived by evaluating exactly individual blocks(using 2d expansion) and their small- z limit. The agreement extends to larger values of z when the former is multiplied by (1 − z ) ∆ σ .From fig. 9 one can observe that the z → z dependant one up to z ∼ − . However the twist obtained by the unsubtracted integralstarts to differ at z = 10 − . . We can then conclude that z → z < − for the exchange of (cid:15) and T. Comparison between ¯ z → and finite ¯ z When β is large enough, most of the contribution to the integral in C ( z, β ) comes from ¯ z → − ¯ z expansion with the result non-perturbative in 1 − ¯ z . In section 2.4 we have already seen the error of the collinear expansionfor the stress-tensor is not negligible. However, it is worthwhile to compare the final resultsderived with this expansion with the non-perturbative one. In figs. 10 we compare the finalanswer for the twist.One can see from this plot that the relative error on the anomalous dimension is approx-imately 6% which is compatible with the analysis in subsection 2.4.– 19 – ��� �� - - - - - - - - - - - - - Figure 10 : Comparison between the twist derived from collinear expansion versus the exactevaluation of individual blocks using 2d expansion.
To understand the extent of validity of inversion formula for low spin it is crucial to study[ σσ ] regge trajectory at spin below two. Similar to stress-tensor, we extract the informationfrom (cid:104) σσσσ (cid:105) correlator.In free field theory, or the UV fixed point, we know that operators in [ φφ ] family lie on astraight line in J − ∆ plane with ∆ − J = 2∆ φ . In addition due to the shadow symmetry(∆ ↔ d − ∆), we have the straight line trajectory for the shadow family as well. The trajectoryand its shadow are plotted in fig. 11.Note that in fig. 11 the two curves must intersect each other at the shadow symmetricpoint with ∆ = d/
2, the spin at this point is d/ − φ .Moving on the RG flow from this Gaussian fixed point to Wilson Fischer fixed point,operators acquire anomalous dimension and they move away from the straight line trajec-tories and lie on a smoother curve. Analyticity of the mentioned curve for J ≥ ε expansion) operators with spin smaller than two, has also been shown to be analytic inspin and lie on the Regge trajectory (see [16]).What we are interested in this section is to extend our methods to spin smaller than 2and capture the non-perturbative characteristic of the leading Regge trajectory, [ σσ ] and itsintercept, j ∗ .There are subtleties associated with going to such low β . One for instance is that asdiscussed in section. 2, close to the shadow-symmetric point, ∆ = d/ J Figure 11 : The Regge trajectory of [ φφ ] and its shadow in free field theory. The twotrajectories intersect at the shadow symmetric point.of complex variables ρ and ¯ ρ which is related to z and ¯ z as follows (see [33]) ρ = z (1 + √ − z ) ↔ z = 4 ρ (1 + ρ ) ¯ ρ = ¯ z (1 + √ − ¯ z ) ↔ ¯ z = 4 ¯ ρ (1 + ¯ ρ ) (3.8)We expand the s-channel block in eq. 2.11 in ρ, ¯ ρ → z ∼ ¯ z ∼ ∼ d/ z and ¯ z . This is theappropriate range for the vicinity of the intercept. The generating function replacing eq. 2.21is then as follows: C ( z, β ) = κ ( β ) (cid:90) z zz ) ( z − ¯ z )( z ¯ z ) z ( τ/ G β, − τρ ( z, ¯ z )dDisc[ G ( z, ¯ z )] (3.9)For calculating eq. 3.9, the s-channel block is expanded in ρ and ¯ ρ to 6th order. Inaddition, in order to perform the integral, we expand the correlator in cross-channel expansion.Since we are interested in very small β , it would be beneficial to use the full cross-channelblocks. By numerically integrating the integral form of the conformal block for spin 0 which– 21 –s introduced in [34–36], we obtain the (cid:15) exchange. G ∆ , ( z, ¯ z ) = Γ(∆)Γ( ∆+∆ )Γ( ∆ − ∆ ) u ∆2 (cid:90) dσ (1 − (1 − (1 − z )(1 − ¯ z )) σ ) − ∆+∆122 × σ ∆+∆34 − (1 − σ ) ∆ − ∆34 − F (cid:18) ∆ + ∆ , ∆ − ∆ , ∆ − d − , z ¯ zσ (1 − σ )1 − (1 − (1 − z )(1 − ¯ z )) σ (cid:19) (3.10)This integral representation is exploited in different contexts in the literature, see for instance[37]. In addition, one can also derive similar integral representation for the exchange ofconserved current: G j + d − ,j ( z, ¯ z ) = (cid:90) dt (2 j Γ(1 + j ))( √ π Γ(1 / j )) √ z ¯ z ( √ − t √ t √ − tz − ¯ z + t ¯ z ) (cid:32) − √ − tz − ¯ z + t ¯ z (cid:112) − tz − ¯ z + t ¯ z ] (cid:33) j (3.11)specifying to spin 2 gives us the exchange of stress tensor. Now we have all the ingredientto perform the inversion formula for the exchange of (cid:15) and T which are the leading twistoperators. Once the integral of the blocks are done, one can perform the inversion integral ineq. 3.9 numerically as well to obtain the generating function. We can then use this generatingfunction to obtain the twist at different values of conformal spin using eq. 2.27. Now triviallythe function τ ( β ), gives us the function J (∆), from which we can read of the intercept as canbe seen in fig. 12.In our analysis, we find that the smallest value of β for which eq. (2.21) agrees witheq. (3.9) is β min ∼
3. For β s smaller than this value we must use the latter.However, note that since we are not exchanging the twist families, precision of our resultwill be moderate. To quantify our error, we compare the twist derived at different values of z with exchange of only (cid:15) as well as exchange of both (cid:15) and T. We see that our results doesnot change drastically in any of the mentioned cases. This comparison is plotted in fig. 12Remarkably one can see in fig. 12, that the intercept of the leading Regge trajectory, [ σσ ] is below one, j ∗ ≈ .
8, which conclusively shows that 3D Ising theory is transparent at highenergies. We also estimate the (shadow symmetrical) residue at the intercept to be:Res J = j ∗ c t (∆ , J ) K (∆ , J ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆= ≈ . . (3.12)Recently a similar estimate j O (2) ∗ ≈ .
82 was obtained in [19] for the O (2) model using arelated method. It would be interesting to compare the details. See section 5.1 for a detaileddiscussion on the implication of this result. (cid:15) It was shown in perturbation theory (see [38] ), i.e., ε -expansion, one can obtain an analyticcurve for the leading Regge trajectory and analytically continue it to recover (cid:15) and its shadowon the continued curve. – 22 – ��� �� - - - Figure 12 : The function J (∆) is plotted at low spin, for the exchange of (cid:15) and for theexchange of (cid:15) and stress-tensor at two different values of z . The important takeaway is thespin of intercept is below 1. The width created by these curves estimates the error of theanalysis.In this subsection we study the possibility of finding (cid:15) operator on the curve obtainedby the analytic continuation of the Regge trajectory to spin below intercept in the full non-perturbative 3D Ising CFT. In order to perform the analytic continuation, we need an ansatzfor the leading trajectory near the intercept which reproduces the data obtained from inversionformula properly. The ansatz we use for the function J (∆) needs to have two importantcharacteristic. First, it should be symmetric under the shadow transform (∆ ↔ d − ∆).Second, asymptotically it must approach the lines J = ∆ − σ and J = − ∆ + d − σ . Onesimple ansatz that satisfies both of these condition is as follows.( J − d/ σ ) = (∆ − d/ + A (3.13)This corresponds to the following τ ( β ) function: τ ( β ) = Aβ − β + B − β (3.14)It follows that we must have β = d − σ and B = d − β , however, we keep their valuesunidentified. The data points are fitted with this ansatz for each of the different cases con-sidered above to obtain the value for A , β and B in each case. As an example the values forthe fit of data obtained from (cid:15) + T at z = 10 − is given (blue lines in fig. 13): A = − . β = 1 . B = 2 . j < j ∗ isstraightforward (see fig. 13 for the trajectory and its analytic continuation). Evaluating thefunction at J = 0 gives us an estimation of the conformal dimension of (cid:15) operator and its– 23 – ��� �� - - - Figure 13 : The interpolation of data points and their fit for two sets of points consideredin fig. 12 is given. The analytic continuation is obtained once the fit is found. The (cid:15) and itsshadow are represented by the two red dots. We see the prediction for ∆ (cid:15) by looking at theintersection of the curves and the ∆ axis.shadow. As an example the fit of data obtained from (cid:15) + T at z = 10 − gives the followingestimates : ∆ (cid:15) = 1 . ˜ (cid:15) = 1 .
868 (3.16)These values are not close to (cid:15) operator. However the analysis predict their existence as wasseen before in perturbation theory. Note that a slight shift in the vertical axis of this curvecan land us on a curve which includes (cid:15) . However it should be obvious that using this methodto determine quantitative properties of (cid:15) would be numerically unstable. Z -odd twist family In this section we study the low spin operators in [ σ(cid:15) ] family by analysing the correlator (cid:104) σ(cid:15)(cid:15)σ (cid:105) , which in our notation corresponds to ∆ = ∆ σ and ∆ = ∆ (cid:15) . According to our previ-ous discussion this correlator leads to data about the [ σ(cid:15) ] family. The operators exchangedin the t-channel, where we fuse σ with σ and (cid:15) with (cid:15) are (cid:15) and the families [ σσ ] , [ σσ ] , [ (cid:15)(cid:15) ] .The u-channel involves the same fusion as the s-channel, σ with (cid:15) so σ and the [ σ(cid:15) ] familyare exchanged. We will use data provided in [5] for the dimensions and OPE coefficients.There are many interesting physical facts about 3D Ising model that one can understandby studying this family of operators. For instance, the absence of a global conserved currentwith spin 1. According to the unitarity bound in eq. 3.1, this spin one operator, if existed,must have scaling dimension 2 and thus, must belong to [ σ(cid:15) ] family. Our goal is to verifythis fact analytically by extending the inversion formula to conformal dimension of order ∼ C even ( z, β )– 24 – ��� ��
10 20 30 400.050.100.150.200.250.30 (a) For the value of z = 10 − . ���� ��
10 20 30 400.20.40.60.81.0 (b) For the value of z = 10 − . Figure 14 : Partial sum over the contributions of different families for the spin 2 operatorwith β = 6 .
18 .to low spin. We show that indeed this family contains a spin 0 operator and the operator iscompatible with being the shadow of σ operator. This is showed by comparing the scalingdimension and the OPE coefficient and showing that they are in the right neighbourhood.As a warm up example, we start by studying the spin 2 operator, for which we havecontrolled error and accurate data. Again, we apply the procedure explained in section 2.3to derive the twist and the OPE coefficient. To control the error of the result same as section3, we illustrate the importance of including the subleading twist families to obtain a stableanswer with controlled error. In addition, we see the range of z in which the twist expansionis consistent, the sum over families are convergent and no resummation is required. However,the overall accuracy of the result for the spin 2 operator in [ σ(cid:15) ] is less compared to the spin2 operators in [ σσ ] and this is due to the fact that there are families of higher twist thatneed to be exchanged in the cross-channel (for instance [ σ(cid:15) ] , [ (cid:15)(cid:15) ] and [ σσ ] ) to get less errorand a more stable result.Once we have familiarized ourselves with the procedure, we move on to the spin 1 andspin 0 operators by following the same steps. As expected, the result will be less stable andwe have less control over the errors. In this section we verify with what accuracy the anomalous dimension and OPE coefficient, f σ(cid:15)O , of spin 2 operator in [ σ(cid:15) ] family, O , can be derived from inversion formula withvarious truncation in the t-channel expansion. The procedure and the steps taken here arethe same as the calculation for stress-tensor. However, the details of the calculation are ofcourse different. Again since we are dealing with low spin, d-1 expansion of the block is thecorrect tool to use. But before we proceed to that, we need to find the relevant range of z inwhich eq. 2.27 can be applied. This would be the range in which the twist expansion of theargument of the dDisc breaks down. One can indeed confirm that indeed convergence of thetwist expansion breaks in relatively large z , which is shown in fig. 14.– 25 –ith the same analysis performed in fig. 14, we can conclude that the range of z in whichwe have a valid twist expansion of the dDisc argument begins at 10 − . Thus this will be thelowest z for evaluation of the twist and the OPE coefficient.In fig. 15 we show the result for the anomalous dimension of the spin 2 operator. Thecorrelator in the argument of the dDisc in C t is built by exchanging (cid:15) , [ σσ ] , [ σσ ] and [ (cid:15)(cid:15) ] inthe t-channel and the argument for C u is built by exchanging σ and [ σ(cid:15) ] for the u-channel.We sum the contribution of the t-channel and u-channel since we are interested in the spin 2operator in the s-channel (see eq. 2.12). We will call this function C even ( z, β ) (in section 4.2,where we are interested in odd spin operators of [ σ(cid:15) ] , we subtract the u-channel contributionfrom the t-channel and subsequently the function derived this way will be called C odd ( z, β ).In order to understand the importance of the subleading families we compare the resultwith when only (cid:15) , [ σσ ] and σ are exchanged in fig. 15. ���� �� - - - - Figure 15 : γ ( β = 6 .
18) is given for comparison between different truncation in the cross-channel expansion of the dDisc. We see the importance of including the subleading familiesto get a plateau. In addition, multiplying C even ( z, β ) with (1 − z ) ∆1+∆22 has been shownto extend the range of z in which we have a flat curve (see the paragraph below fig. 3 forexplanation).Following the same procedure as the one for stress-tensor, we choose the decade inwhich we have the most stable result (the smallest standard deviation) which is log z ∈ [ − . , − . z in this region, we get the following result for the twistof the spin 2 operator of [ σ(cid:15) ] : τ = 2 . ± . . (4.1)We can calculate τ ( β ) for a number of operators in the vicinity of the spin 2 operator, i.e., β = 6 .
18 for a fixed z . This will help us derive the function τ ( β ) by interpolation. As anothermethod of getting the anomalous dimension of the spin 2 operator, we again intersect the spin2 line with τ ( β ) as done in figure. 16. The point of intersection is where the spin 2 operatoris located. – 26 – ��� �� Figure 16 : The intersection point of the line J = 2 and the curve τ ( β ) is where the spin 2operator is located.Same as [ σσ ] , once we have the function τ ( β ) , we can compute the OPE coefficient bycalculating C ( β ) according to eq. 2.27 and multiplying it by the relevant Jacobian factor ineq. 2.20 to get: f σ(cid:15)O = 0 . ± . . (4.2)In the table. 2, we summarised our analytical result for the scaling dimension and OPEcoefficient of spin 2 operator along with values predicted by numerical bootstrap (see [5]).∆ O f σ(cid:15)O Inversion Formula 2.1845(35) 0.3907(14)Numerical Result 2.180305(18) 0.38915941(81)
Table 2 : Twist and OPE of spin 2 operator in [ σ(cid:15) ] derived analytically compared with thevalue derived from numerical bootstrap. [ σ(cid:15) ] family Now that we have familiarized ourselves with the basic process for extracting the low spin in[ σ(cid:15) ] family, we try to extend the analysis to study the absence of spin 1 conserved current inthe odd spin sector of this family. Note that throughout this subsection all of the operatorsdiscussed in section 4.1 is exchanged in the cross-channel expansion of the double discontinuity.From the unitarity bound in eq. 3.1 we know that the conserved spin 1 operator, ifexisted, has dimension 2. Thus we start the analysis by examining the function C odd ( z, β ) inthe vicinity of β = 3.The first step is to realize the range of z in which we can trust the twist expansionfollowing the procedure illustrated for O operator in fig. 14. We find that the inversionformula is safe to use for z > − . . Thus all of the analysis in this section is done with3 different values of z in this range, z = 10 − . , z = 10 − and z = 10 − . , to evaluate thestability. – 27 –n fig. 17 C odd ( z, β ) multiplied by the factor (1 − z ) ∆2+∆32 /z ∆ σ +∆ (cid:15) is depicted. The(1 − z ) ∆2+∆32 factor subtract the collinear descendants and higher twist contamination (seethe paragraph below fig. 3 for explanation) and division by z ∆ σ +∆ (cid:15) reduces the z -dependenceof C odd ( z, β ) to z γ β / . ���� �� - - - - - Figure 17 : (1 − z ) ∆2+∆32 C odd ( z, β ) /z ∆ σ +∆ (cid:15) for β in the vicinity of spin 1 operator for threedifferent values of z . One can observe the vanishing of the OPE coefficient for β ∼ .
3A few notes must follow: we emphasize much alike section 3.2 the stability in z is mod-erate. The difference in the curve is suggestive of what the error should be.We see that all three curves cross zero at β ∼ .
3. This implies the vanishing of the OPEcoefficient for that conformal spin.Theoretically, the J = 1 operator is absent from the spectrum if and only if the vanishingof C occurs precisely when the trajectory crosses J = 1. Because of C vanishing, the numericalevaluation of the twist using z∂ z log C is however unstable. To assess whether the vanishingof C and J = 1 occur at the same point, we consider the following combination: f ( z, β ) = (cid:0) β/ − − z∂ z (cid:1) C odd ( z, β ) . (4.3)Using eq. (2.27) it can also be written as: f ( z, β ) = ( J − C odd ( z, β ) . (4.4)Note that this function vanishes when the trajectory contains an operator with spin 1 orwhen C odd ( z, β ) is zero. If the residue at the spin 1 point vanishes (which in turn implies theabsence of spin 1 operator), then the two zeros in f ( z, β ) must be at the same place and weexpect to find a curve tangent to the x -axis. By examining the function f ( z, β ) in fig. 18, we– 28 – ��� �� Figure 18 : f ( z, β ) /z ∆ σ +∆ (cid:15) = ( J − C odd ( z, β ) /z ∆ σ +∆ (cid:15) for β close to ∆ σ + ∆ (cid:15) + 2 for threedifferent values of z . Within errors, this function seems compatible with having a double zerotouching the real axis.see that the curves are almost tangential but not completely! We speculate this to be causedby the truncation of the t-channel OPE.Lastly, to recognize whether vanishing of C odd ( z, β ) is due to a subtle cancellation betweenthe t-channel and u-channel, we study (1 − z ) ∆2+∆32 C even ( z, β ) /z ∆ σ +∆ (cid:15) for the same values of β in fig. 19.In fig. 19, we can explicitly observe that indeed when the t-channel and the u-channelare added instead of subtracted as it is done in C even ( z, β ), there is no vanishing of the OPEcoefficient.To sum up, we studied the odd-sector of [ σ(cid:15) ] family through constructing C odd ( z, β )and its derivative. We showed that the analytic calculation is consistent with the absence ofglobal conserved current with spin 1 operator and this absence results from a very interestingconspiracy between the u-channel and t-channel terms. In this section, we try to push the analysis to see whether [ σ(cid:15) ] trajectory can contain a spin 0operator. Again the accuracy of our analysis is moderate since the range of z accessible to usis small (by an analysis similar to what has been done in section 4.1 we realize that z < − cannot be used) and even in this region the result varies quite a bit since the expansion of thedDisc converges more slowly for such a small value of β and more subleading twist operatorsneed to be exchanged as the contribution of each operator falls with its twist as 1 /β τ (cid:48) .– 29 – ��� �� Figure 19 : (1 − z ) ∆2+∆32 C even ( z, β ) /z ∆ σ +∆ (cid:15) for β close to ∆ σ + ∆ (cid:15) + 2 for three differentvalues of z . One can observe that the function does not vanish for any β . This is in contrastwith (1 − z ) ∆2+∆32 C odd ( z, β ) /z ∆ σ +∆ (cid:15) in fig. 17 for the same range of β .That being said, we can still proceed with extracting the twist of such low β and intersectit with J = 0 line to find out if the trajectory admits a spin 0 operator and if it does, whatthe twist of such spin 0 operator is. Once again, the difference of the result for different valueof z , gives us an estimation of the error. This analysis is done in fig. 20We see in fig. 20 that the error of the analysis is indeed not negligible meaning that wecannot pin down the operator with great accuracy. However what is completely manifestfrom this analysis is that [ σ(cid:15) ] trajectory does include a spin 0 operator with dimensionin the neighbourhood of ∆ ∼ .
5. If one calculates the squared of OPE coefficient of thisoperator, one gets f σ(cid:15) ˜ σ ∼ .
15 We make the conjecture that this operator is indeed shadowof σ operator. As a support for this conjecture we remind the reader of the scaling dimensionof shadow of sigma, ˜ σ which is ∆ ˜ σ = 3 − ∆ σ (cid:39) .
48 and its OPE coefficient, f σ(cid:15) ˜ σ (cid:39) .
44 (thisis calculated using eqref. 2.15 which relates c (∆ , J ) and c ( d − ∆ , J )). For convenience, ourresults along with what is expected from numerical bootstrap is summarised in the table. 3.Our analysis predicts that the spin 0 operator of [ σ(cid:15) ] is in the vicinity of shadow of σ operator. However, the gap between the OPE coefficient of shadow of σ and the resultobtained from the inversion formula indicates that even though our analysis is compatiblewith shadow of sigma belonging to [ σ(cid:15) ] trajectory, using this method to predict quantitativelyproperties of the operator would not be numerically very effective. This is similar to what weobserved for (cid:15) operator in section. 3.3.We also get the Chew-Frautschi plot for this analytic trajectory [ σ(cid:15) ] , even (as we did for[ σσ ] family in fig. 12) in fig. 19 which has been placed in section 5.2 for compariosn with– 30 – ��� �� Figure 20 : The intersection between the three curves τ ( β ) and the line J = 0 is the predictedlocation of the spin 0 operator. The black dot on the J = 0 line is the actual location ofshadow of σ operator. ∆ ˜ σ f σ(cid:15) ˜ σ Inversion Formula at z = 10 − z = 10 − . z = 10 − Table 3 : Twist and OPE of spin 2 operator in [ σ(cid:15) ] derived analytically compared with thevalue derived from numerical bootstrapsimilar trajectories in O ( N ) model. – 31 – Extended discussion
To shed light on the results presented in this paper, we give an extended discussion on thefollowing aspects. First, we discuss the qualitative distinctions between the Regge trajectoriesof transparent and opaque theories, we compare 3D Ising with the critical O ( N ) model atlarge N (which is in the transparent class), and we work out a novel formula showing thattransparency implies regularity of the heavy spectrum. When do we expect the spectrum to be analytic down to J = 0? Here we argue that, in manysituations, this is closely related to asymptotic transparency. Let us try to sketch, more generally, what singularities we expect in the complex (∆ , J )-plane. We begin with the region of large spin and dimension. There we certainly find double-twist trajectories, which have approximately constant twist τ ≈ ∆ i + ∆ j + 2 n and lie near to45 ◦ in the figure. More generally we also expect multi-twist operators, built of products ofmany primaries and derivatives, and it is interesting to try and track their trajectories. Sincethe number of local operators grows with spin, we expect the number of trajectories to beinfinite, likely accumulating at discrete twist values (with, presumably, only a finite numberof them having a nonzero OPE coefficient at a given integer spin). These are the solid linesshown in fig. 21. As explained in ref. [22], Regge trajectories represent non-local operators,which reduce to line integrals of local operators at the position of the crosses.Although general classification of nonlocal operators is still lacking, we also expect near-horizontal trajectories. In a weakly coupled gauge theory these are well-known to arise ascolor-singlet combinations of null-infinite Wilson lines U ( x ⊥ ) ∝ P e i (cid:82) ∞−∞ dx + A + ( x + , − ,x ⊥ ) where x ± = t ± x are lightcone coordinates. The simplest such trajectory, the BFKL Pomeron, islabelled by the positions of two Wilson lines, where the quantum number ∆ is conjugate totheir transverse separation (see [12, 39–41] for various distinct perspectives). Notice that sincethe gauge fields have spin 1 the integral (cid:82) dx + A + is formally boost-invariant (momentarilyneglecting the need to introduce rapidity cutoff); products of multiple Wilson lines thushave the same spin (boost) quantum number. More generally, non-local operators satisfy thestandard addition law from Regge theory: J [ O O ] ≈ J O + J O − , (5.1)where the offset is due to the mismatching number of dx + on both sides.The sharp difference between Lagrangian theories which contain vector bosons, and thosewhich do not (“matter-like” theories) is where these near-horizontal trajectories lie. A non-local composite of two scalars would give a single trajectory near J ≈ −
1, and the firstaccumulation point of trajectories is delayed to J ≈ −
2; a composite of two fermions may A discussion along these lines was first presented by one of the authors at the 2018 Azores workshop onthe analytic bootstrap. – 32 –roduce a single trajectory near J ≈
0, but it is still effectively isolated from more compli-cated composites. In contrast, in gauge theories one immediately runs into infinitely manytrajectories that mix with each other. One reason BFKL were able to make progress is thatmixing between states of different number of elementary Reggeized gluons is suppressed bytwo effects: by weak coupling and/or the planar limit, see [41]. Quantum corrections move thetwo-Reggeon intercept above 1 in both limits: the intercept is j transient ∗ = 1 + O ( α s ) at weakcoupling, and at strong coupling j transient ∗ ≈ e η ( j transient ∗ − at large boost, onenaturally expects double exchange to grow twice as fast, giving an effective excitation of spin2 j transient ∗ − > j transient ∗ . This argument seems rather unavoidable due to cluster decomposi-tion in spacetime dimensions d >
2, since excitations can be widely separated in the transverseplane. It is not possible to have just a single trajectory with j ∗ >
1, there must be an infinitetower! The growth of course can only be transient because the correlator is bounded; it isgenerally expected that the higher trajectories stop the growth rather than speed it up, in thesame way that the higher-order Taylor coefficients of the function (1 − e − x ) limit its initiallinear growth. See [12] for further discussions; we do not have anything to add here about howsaturation works, if only to note that convexity requires that all singularities cancel belowthe red line in fig. 21.An important lesson from this discussion is that while in asymtptotically transparenttheories it seems perfectly reasonable, if numerically challenging, to analytically continuetrajectories to J = 0, in opaque theories there may be much more serious obstructions tocrossing J = 1. J = 0 in the large- N O ( N ) model O ( N ) models at large N is a theory for which we have analytical control by using 1 /N expansion. This theory is a specifically suitable theory for testing the ideas put forward inthis paper as its operator’s contents resembles the one in 3d Ising. This is because 3d Isingis given by O ( N ) model at N = 1. The leading O ( N )-Bilinear twist families of the O ( N )model has been studied at large N to order 1 /N in [38]. We will reproduce their result of O ( N )-Bilinear twist family with slightly different approach up to order 1 /N and comparewith our results in section 3. In addition, we study the O ( N )-Fundamental twist family upto order 1 /N to study our conjecture in this model and compare with the results obtained in3d Ising model in section 4. O ( N ) model is a theory of N scalar fields φ i that transform in the fundamental rep-resentation of O ( N ). The OPE of these fields can be separated into three different tensorstructures: φ i × φ j = (cid:88) S δ ij O + (cid:88) T O ( ij ) + (cid:88) A O [ ij ] , (5.2)where S stands for singlet of even spin, T stands for symmetric traceless of even spin and A stands for anti-symmetric tensors of odd spin. Similar to previous sections, we want to– 33 – - ⨯ ⨯⨯ ⨯ ...? (a) Asymptotically transparent ⨯ ⨯ ? ? ... (b) Asymptotically opaque Figure 21 : Chew-Frautschi sketches for transparent and opaque theories. Solid lines indicatemulti-twist trajectories, and dashed lines show possible BFKL-like horizontal trajectories.Complicated behavior could occur where accumulation points of trajectories intersect. (a)In scalar-like theories, most serious complications would seem restricted to
J < J = 0 plausible. (b) In nonabelian gauge theories, here at weakcoupling, the perturbative leading trajectory has intercept j transient ∗ > Leading O ( N ) -Bilinear Twist Family First we review how this works for the low spin operators in [ φ i φ j ] double-twist families,which are the leading twist families of the O ( N ) theory. This discussion will follow closely[38]. The data for the spectrum will be derived in the limit that N is large and is thus givenas an analytic expansion in 1 /N . We look at a 4-point function of these scalar fields, whichcan be again separated in three independent tensor structures: x φ x φ (cid:104) φ i ( x ) φ j ( x ) φ k ( x ) φ l ( x ) (cid:105) = δ ij δ kl G S ( u, v ) + (cid:18) δ il δ jk + δ ik δ jl − N δ ij δ kl (cid:19) G T ( u, v ) + (cid:16) δ il δ jk − δ ik δ jl (cid:17) G A ( u, v ) . (5.3)The functions that appear are just the usual conformal block expansion but with the sumsonly over the operator in the given sector. We can also expand this correlator in the t and u– 34 –hannels to obtain the following crossing symmetry equation: f S ( u, v ) = 1 N f S ( v, u ) + N + N − N f T ( v, u ) + 1 − N N f A ( v, u ) → N f S ( v, u ) + 12 f T ( v, u ) + 12 f A ( v, u ) ,f T ( u, v ) = f S ( v, u ) + N − N f T ( v, u ) + 12 f A ( v, u ) ,f A ( u, v ) = − f S ( v, u ) + 2 + N N f T ( v, u ) + 12 f A ( v, u ) . (5.4)The leading equations at large N are the main tool of this section. The crossing equationsinto the u-channel are essentially the same but with minus signs everywhere in the A sector.The functions that appear are defined by f ( u, v ) = u − ∆ φ G ( u, v ) to incorporate the factorscoming from crossing. The full correlation function can be expanded in 1 /N and the explicitexpression at N = ∞ follows from Wick contraction and is as follows : G ijkl ( u, v ) = δ ij δ kl + u d − δ ik δ jl + (cid:16) uv δ il (cid:17) d − δ jk . (5.5)This equations means that we have the following expansion for each decomposition: G S ( z, ¯ z ) = 1 + 1 N G (1) S ( u, v ) + . . . G T ( z, ¯ z ) = u d − (cid:18) v d − (cid:19) + 1 N G (1) T ( u, v ) + . . . G A ( z, ¯ z ) = u d − (cid:18) − v d − (cid:19) + 1 N G (1) A ( u, v ) + . . . (5.6)Now we are equipped to look at the N scaling of different terms in the crossing. First wesee that for T and A operators, there is a whole tower of double twist operators exchangedat N → ∞ limit, so their OPE coefficients are of order 1 and are the ones from generalizedfree fields. The double twist operators in the S sector do not appear at this order so theirOPE coefficients must have term that scale as a negative power of N ( 1 /N / ). Fromdimensional analysis we can find that the leading scaling dimension of φ is 1 /
2. This meansthat double twist operators of spin J have a leading dimension of 1 + J . There is howeverstill the possibility that the scalars that appear in the OPE are shadows of the double twists,similar for the σ operator in the [ σ(cid:15) ] family of the 3d Ising model. This possibility will beincompatible with the leading N behaviour of the correlator for the T sector. However, forthe S sector, if the shadow does not appear we indeed run into trouble, as the crossing wouldimply that T and A operator do not have anomalous dimension of order 1 /N . We will thencall the operator appearing in the OPE S and it has a leading dimension of d − ∆ [ φφ ] S, = 2.The idea for studying this theory using the inversion formula is to use the crossingsymmetry equations (5.4) to understand how to combine the different elements that appearin the generating function C ( z, β ). Crossing then dictates what combination of f R ( v, u )– 35 –ppears in t-channel correlator of the inversion formula for [ φφ ] R double twist operators ineach sector.One can see from from crossing eqs. (5.4), the leading behaviour of the OPE coefficients isthen given by the identity contribution. This can be found by evaluating (2.24) at β = 1 + 2 J and multiplying by 2 because the identity appears both in the t and u channels. The resultfor S can be deduced from evaluating the answer at spin 0 and transforming to the shadowwith (2.15). We find: f φφS = 4 π N + O (cid:18) N (cid:19) ,f φφ [ φφ ] T/A ( J ) = N f φφ [ φφ ] S ( J ) = 2Γ( J )Γ (cid:0) J + (cid:1) π Γ(2 J )Γ( J + 1) + O (cid:18) N (cid:19) . (5.7)The next step is to calculate anomalous dimensions. One important point to emphasizewith this setup is that whether we are summing over the whole twist family or consideringonly the exchange of a single operator, taking z → /N prevents problems caused byloss of the log terms when the wrong order of limits are taken and this problem is simplyresolved because log terms appear only in next order in 1 /N .The simplest anomalous dimensions to first calculate are those in the T and A familiesbecause they receive contribution only from S at order 1 /N . This is done using the inversionformula at z → f φφφ S calculated above and it leads to γ [ φφ ] T/A ( β ) = − π N (2 J + 1)(2 J − . (5.8)One nice use for this result comes from the fact that we should recover that the spin 1 operatorin the A sector is a conserved current with dimension 2. This can be used to fix the correctionto the dimension of φ since the dimension of the spin 1 operator is given by 2∆ φ +1+ γ [ φφ ] A (1).The result is ∆ φ = 12 + 43 π N , (5.9)which is consistent with the previous calculations summarized in [38, 42, 43].We can continue by calculating the anomalous dimensions of the singlet double twists.Here the double twist operators in the other sectors contribute. We could again use eq. A.9but it gets complicated for general spin. We instead lean on the fact that conformal blocksfor conserved currents, which the double twists are at leading order, are very simple. Usingthe method described in Appendix A.4 we find that for conserved currents the coefficient ofthe log is (cid:114) ¯ z − ¯ z G ∆ ,J (1 − z, − ¯ z ) (cid:12)(cid:12)(cid:12)(cid:12) log = − J + 1)Γ (cid:0) J + (cid:1) . (5.10)This can be used directly in (2.21) along with the data already found for the T and A doubletwist operators. The contributions are exactly the same for both sectors, except that they– 36 –ontribute with even and odd spins. The result for the sum of the contributions from φ S (same as for T and A sectors) and the double twists is γ [ φφ ] S ( J ) = − π N (2 J − . (5.11)We have now come to a point where an consistency check is possible. Indeed the spin 2operator in this family should be the stress tensor and it should have a dimension of 3, whichwe find to be the case.As a concluding remark we plot the Chew-Frautschi plot of [ φφ ] ,S at N = 1000 infig. 22. The analogous plot for [ σσ ] is fig. 12. At N → ∞ the intercept approaches 1 /
2. AsN decreases we see that the value of the intercept increases: j ∗ = 12 − π N + (cid:114) π N (5.12)This result can be compared with our result for the intercept in section 3.2. We emphasizethat for small N we do not expect this formula to capture the true physics. As mentionedabove the intercept for O (2) model is recently calculated to be ∼ .
82 in [19], where eq. 5.12would predict 1.00152. The difference for 3D Ising is of course is more drastic; eq. 5.12 for N = 1 predicts the intercept to be 1.13013, we see indeed in section 3.2 that it is ∼ . N .Another remarkable fact is that if we analytically continue the Chew-Frautschi plot to J smaller than the intercept, we can recover the spin 0 operator of [ φφ ] and its shadow (S) onthe continued curve (dashed line in fig. 22). Leading O ( N ) -fundamental twist family Now we turn to the leading O ( N )-Fundamental twist family, [ φ i S ] . We study this familyby considering the 4-point function (cid:104) φ i SSφ j (cid:105) . The index structure of crossing equations istrivial for this correlator as there is one possible option, δ ij . This means that we can discardthe indices from crossing and consequently from inversion formula and use its stripped versionas: G ( u, v ) = u ∆1+∆22 v ∆2+∆32 G ( v, u ) (5.13)To obtain information about [ φ i S ] by using the inversion formula, we first need to calculatethe t-channel and u-channel dDisc of the correlator. For the 1 /N expansion dDisc t we havethe identity operator which is the only operator exchanged at O ( N ), then at the next orderwe have the exchange of operator S which is a single twist operator and its 1 /N suppression Analytic continuation of the leading Z even trajectory to J < J has also been studied in (cid:15) -expansion in[16]. There it was shown that one can discover (cid:15) operator on the analytically curve as the shadow of spin 0operator in [ σσ ] family. One might hope to apply the same procedure to 3d Ising and analytically continuethe trajectory in fig. 12 to find (cid:15) operator, however since we do not have an analytical expression for thetrajectory, this continuation cannot be done in a numerically controlled convincing way. – 37 – ��� �� - - - Figure 22 : The Chew-Frautschi plot for [ φφ ] S and its analytic continuation (dashed lines)of the [ φφ ] ,S is plotted for N = 1000 and N = 10. We see that as N increase the interceptapproaches 1 / /N .Dots indicate the spin-0 S operator.comes from its OPE coefficient, f φφS f SSS (In fact there is additional 1 /N suppression becauseof sin(∆ S − S ) /
2, however this is cancelled with the factor of Γ(1 − ∆ S /
2) from the inver-sion formula as in eq. 5.15). Contribution of all the double twist operators are additionallysuppressed with a factor of 1 /N due to the sine factors of dDisc (see eq. 2.14). To calculatethe u-channel discontinuity we exchange operators 1 and 2. Then the leading contributionto this expansion comes from the exchange of φ . The large N scaling of this contributioncan be assumed to firstly comes from the OPE coefficient f φφS . In addition, there is anadditional 1 /N suppression because S has integer scaling dimension: the sine factors whichcan be calculated to be sin[ π ∆ φ − ∆ S − ∆ φ , (5.14)scales as 1 /N . However, only one of these will contribute as the other one gets cancelledwith a factor of Γ(1 − ∆ S /
2) in the inversion integral using the following identity:Γ( z ) γ (1 − z ) = π sin( πz ) (5.15)At the end we observe that the exchange of φ is not suppressed by 1 /N as might have firstguessed but it is suppressed with a factor of 1 /N . Now, we have enough information toobtain the leading order OPE coefficient and anomalous dimension of [ φ i S ] operator. Forour purpose which is to compare with section 4, we would like obtain the data for spin0 operator of this family and verify whether this operator can be the shadow of φ . This– 38 –ould be analogues to our conjecture that shadow of σ belongs to [ σ(cid:15) ] family. The leadingcontribution to the OPE coefficient of [ φS ] family comes from the identity using eq.(2.24): f φS [ φS ] ,j = I (∆ φ , ∆ S ) (2 j + ∆ φ + ∆ S ) (5.16)If one uses this formula for the spin 0 operator, i.e., β = 5 / O ( N ) to be 1. Now we can use the shadow transform given in eq. (2.15) to obtainthe f φS ˜[ φS ] , to be : f φS ˜[ φS ] , = 0 + 4 π N (5.17)Notice that f φS ˜[ φS ] , vanishes at order N , this is because the shadow transform gives zeroat leading order for operator with dimension d − as can be easily seen from eq. (2.15). Thefact that f φS ˜[ φS ] , matches with f φφS up to order 1 /N is an evidence for our conjecture that φ i operator is the shadow of the spin 0 operator in the [ φ i S ] family.Interestingly, once we assume analyticity at spin 0, we can have a prediction for the OPEcoefficient f SSS at leading order which is otherwise not trivial to obtain. This can be doneby imposing that the leading order anomalous dimension of spin 0 operator in [ φ i S ] , is suchthat d − ∆ (0)[ φS ] , − γ (1)[ φS ] , = ∆ (0) φ + γ (1) φ + ∆ (0) S + γ (1) S (5.18) γ (1)[ φS ] , comes from the exchange of S in the t-channel and is proportional f φφS f SSS . Thisallows us to predict the value of f SSS in terms of the other know variables to be: f SSS = 0 + 2 π √ N (5.19)The Chew-Frautschi plots of [ φ i S ] as well as [ σ(cid:15) ] are plotted in fig. 23a and 23b forcomparison of their similarity. What does transparency imply for the spectrum of heavy operators? Naively one may expectthat it implies a certain regularity in the spectrum, to prevent different components of thewavefunctions from arriving with random phases. This subsection summarizes our attempt toderive a quantitative version of this statement. The argument is similar to that used alreadyin [44] to show that, in holographic CFTs, anomalous dimensions of large-(∆ (cid:48) , J (cid:48) ) OPE data inthe t -channel grows with energy ∆ (cid:48) in a way controlled by exchange of a s -channel Reggeizedgraviton. Our proposed formula, eq. (5.33), conversely shows that an intercept j ∗ < regular spectrum .The results in ref. [44] were obtained pre-Conformal Regge theory using the so-calledimpact parameter representation, and were derived in the context the language of holography.We try to give them a fresh look in light of ref. [7], which allows us to strip the formula fromits holographic context. We proceed in two steps: first we work out implications for the Regge– 39 – ��� �� (a) [ σ(cid:15) ] , even family obtained at z = 10 − . . ���� �� (b) [ φ i S ] family for N = 6 including correction oforder 1 /N . Figure 23 : Chew-Frautschi plot for the tow families mentioned above. See fig. 12 and fig. 22for similar plots of the leading trajectory in 3D Ising and O ( N ) model respectively.limit of the correlator in ( z, ¯ z )-space, then we convert those to heavy cross-channel operators.The second step will rely on an unproven identity about blocks in eq. (5.32), but otherwisewe believe that all steps are rigorous. The first step is achieved by the Watson-Sommerfeldresummation of our eq. (2.7), as given in eqs. (5.21) and (5.22) of ref. [22] (restricting theblocks to the leading power given in eq. (2.6); see also [23]): e iπ ( a + b ) G ( z, ¯ z ) (cid:9) −G ( z, ¯ z ) → d + i ∞ (cid:90) d − i ∞ d ∆2 πi C − ∆ (cid:18) z + ¯ z √ z ¯ z (cid:19) Res J = j ∗ (∆) (cid:20) c t (∆ , J ) + e − iπJ c u (∆ , J ) κ (∆ + J )( e − iπJ −
1) ( z ¯ z ) − J (cid:21) (5.21)The formula simplifies significantly when considering the dDisc in eq. (2.14). We need toadd the complex conjugate conjugation path, which gives the same thing with just iπ (cid:55)→ − iπ inside the square bracket, and eq. (5.21) reduces to:lim z, ¯ z → dDisc G ( z, ¯ z ) = d + i ∞ (cid:90) d − i ∞ d ∆2 πi C − ∆ (cid:18) z + ¯ z √ z ¯ z (cid:19) Res J = j ∗ (∆) (cid:20) c t (∆ , J )2 κ (∆ + J ) ( z ¯ z ) − J (cid:21) . (5.22)To our knowledge, this formula has not appeared in print before. Notice that the u -channelcoefficient has canceled, as well as the integer-spin poles: the t -channel dDisc is directlyrelated to the t -channel contribution to the Lorentzian inversion formula. This is perhaps In our conventions, eq. (5.22) of ref. [22] reads, to leading power: e iπ ( a + b ) F ∆ ,J ( z, ¯ z ) (cid:9) − F ∆ ,J ( z, ¯ z ) → C − ∆ (cid:18) z + ¯ z √ z ¯ z (cid:19) πiκ (∆ + J ) ( z ¯ z ) − J . (5.20) – 40 –ot too surprising given the form of the Lorentzian inversion formula in eq. (2.11): as aconsistency check, we tried inserting eq. (5.22) back into the latter, and we indeed recover c t (∆ , J ) using the orthogonality relation between C − ∆ along the principal series. In otherwords, to leading power, eq. (5.22) is just the inverse of the inversion formula.Because of this interpretation, we can assume that eq. (5.22) is valid even for correlatorswhich do not grow in the Regge limit, even though the validity of eq. (5.21) in this case doesnot strictly follow from the works [7, 22, 23] and may require further discussion [45].What does eq. (5.22) imply for heavy t -channel operators? We follow the logic of refs. [44,46], where the z ∼ ¯ z → t -channel operators with largedimension and spin ∆ (cid:48) ∼ J (cid:48) ∼ / √ z . We review the Euclidean case [46]. The starting pointis the fact that the unit operator in the s -channel is reproduced by an infinite sum over t -channel operators:1 = ∞ (cid:88) n,J (cid:48) =0 (1 + ( − J ) P (0) (∆ (cid:48) , J (cid:48) ) (cid:18) z ¯ z (1 − z )(1 − ¯ z ) (cid:19) ∆ σ G ∆ (cid:48) ,J (cid:48) (1 − z, − ¯ z ) (cid:124) (cid:123)(cid:122) (cid:125) G ( t )∆ (cid:48) ,J (cid:48) ( z, ¯ z ) (5.23)where ∆ (cid:48) = 2∆ σ + 2 n and the average spectral density P (0) (∆ (cid:48) , J (cid:48) ) ≡ P ∆ σ ∆ (cid:48) − J (cid:48) ,J (cid:48) is defined ineq. (3.3). The term with ( − J is regular in the z, ¯ z → G ( z, ¯ z ) = (cid:90) d ∆ (cid:48) dJ (cid:48) c (0) (∆ (cid:48) , J (cid:48) ) G ( t )∆ (cid:48) ,J (cid:48) ( z, ¯ z ) × (cid:104) C (∆ (cid:48) , J (cid:48) ) (cid:105) , (5.24)where the bracket is a sum over δ -function at each local operators, divided by the mean freespectral density: (cid:104) C (∆ (cid:48) , J (cid:48) ) (cid:105) ≡ (cid:88) ∆ (cid:48) f σσ O (cid:48) C (0) (∆ O , J (cid:48) ) δ (∆ (cid:48) − ∆ O ) . (5.25)Note that our normalization of the spectral density is slightly different from [46]. As discussedthere, the fact that the z, ¯ z → (cid:104) C (∆ (cid:48) , J (cid:48) ) (cid:105) , aftersuitably smearing out in ∆ (cid:48) and J (cid:48) , goes to 1 asymptotically, with computable corrections.Namely, exchange of a s -channel scalar operator of dimension ∆ produces a correction sup-pressed by a relative ∆ (cid:48)− :( z ¯ z ) ∆2 = (cid:90) d ∆ (cid:48) dJ (cid:48) c (0) (∆ (cid:48) , J (cid:48) ) G ( t )∆ (cid:48) ,J (cid:48) ( z, ¯ z ) × γ (0) γ ( d − γ (∆) γ (∆ + d −
2) ( h (cid:48) ¯ h (cid:48) ) − ∆ +subleading , (5.26)where γ ( x ) = Γ (cid:18) ∆ + ∆ − x (cid:19) Γ (cid:18) ∆ + ∆ − x (cid:19) (5.27)is a combination which will re-occur often, and h (cid:48) = ∆ (cid:48) + J (cid:48) − , ¯ h (cid:48) = ∆ (cid:48) − J (cid:48) − d + 1 (5.28)– 41 –re combinations which transform simply ( h (cid:48) (cid:55)→ ± h (cid:48) or ± ¯ h (cid:48) ) under all SO( d,
2) Weyl reflec-tions (∆ ↔ d − ∆, ∆ ↔ − J and j ↔ − d − J ). Although in eq. (5.26) we focus on theleading term at large-∆ (cid:48) and J (cid:48) , we find that using the Weyl-friendly form of h and ¯ h makessubleading terms smaller (suppressed by a relative 1 / ∆ (cid:48) ).What about s -channel operators with spin? Using the Casimir recursion in Dolan-Osborncoordinates (see section 2 of [33]) we could compute exactly the OPE coefficient dual to apower of z ¯ z (1 − z )(1 − ¯ z ) times a Gegenbauer polynomial. This will be detailed elsewhere [47] andhere we simply record a compelling formula that we observed for the leading behavior at large∆ (cid:48) and J (cid:48) :( z ¯ z ) ∆2 C J (cid:18) z + ¯ z √ z ¯ z (cid:19) = (cid:90) d ∆ (cid:48) dJ (cid:48) c (0) (∆ (cid:48) , J (cid:48) ) G ( t )∆ (cid:48) ,J (cid:48) ( z, ¯ z ) × γ (0) γ ( d − γ (∆ − J ) γ (∆ + J + d − × ( h (cid:48) ¯ h (cid:48) ) − ∆ C J (cid:18) h (cid:48) + ¯ h (cid:48) h (cid:48) ¯ h (cid:48) (cid:19) + subleading . (5.29)Notice the parallel between z and h − on the two sides of the formula, with Gegenbauers turn-ing onto Gegenbauers. This is the key observation made long ago in ref. [44] using an auxiliaryimpact parameter representation, which allowed them to generalize the statement that theFourier transform of a Gegenbauer is a Gegenbauer. Here we sidestepped the auxiliary spaceand we are simply making a statement about conformal blocks.Comparing eqs. (5.24) and (5.29) and summing over the s -channel OPE gives a formalseries expansion for the asymptotic spectral density: (cid:104) C (∆ (cid:48) , J (cid:48) ) (cid:105) = (cid:88) ∆ ,J f O f O γ (0) γ ( d − h (cid:48) ¯ h (cid:48) ) − ∆ γ (∆ − J ) γ (∆ + J + d − C J (cid:18) h (cid:48) + ¯ h (cid:48) h (cid:48) ¯ h (cid:48) (cid:19) + . . . (5.30)where the dots stand for the omitted subleading terms in eq. (5.29). The formula showsthat the presence of an operator (∆ , J ) on the s -channel OPE implies ( h (cid:48) ¯ h (cid:48) ) − ∆ correctionsto the large-dimension spectrum in the cross-channel with the same Gegenbauer angulardependence. The factors 1 /γ (∆ − J ) produce a double-zero when (∆ , J ) is a double-twistoperator: this was expected since such exponents can be generated by individual blocks anddo not affect the heavy spectrum. This factor grows at large ∆ and likely causes eq. (5.30)to be an asymptotics series in 1 /h . Eq. (5.30) represents a technical extension of ref. [46]to account for spinning s -channel operators. As explained there, a minimal but rigorous“smearing” can be provided via Cauchy moments.We now apply the same logic to the Regge limit (5.22) of the double discontinuity. Fromthe t -channel perspective, the double discontinuity simply multiplies the average in eq. (5.24)by two sines:dDisc G ( z, ¯ z ) = (cid:90) d ∆ (cid:48) dJ (cid:48) c (0) (∆ (cid:48) , J (cid:48) ) G ( t )∆ (cid:48) ,J (cid:48) ( z, ¯ z ) ×(cid:104) (cid:16) ∆ (cid:48) − J (cid:48) − ∆ − ∆ (cid:17) sin (cid:16) ∆ (cid:48) − J (cid:48) − ∆ − ∆ (cid:17) C (∆ (cid:48) , J (cid:48) ) (cid:105) . (5.31)– 42 –e need to find the average which reproduce the conformal Regge prediction (5.22). Thisrequires a generalization of eq. (5.29) where the Gegenbauer function is no longer a polyno-mial, and for which the Casimir recursion mentioned above eq. (5.29) does not terminate.However, we find the form of eq. (5.29) compelling enough to conjecture that it is valid ingeneral:( z ¯ z ) − J C − ∆ (cid:18) z + ¯ z √ z ¯ z (cid:19) ? = (cid:90) d ∆ (cid:48) dJ (cid:48) c (0) (∆ (cid:48) , J (cid:48) ) G ( t )∆ (cid:48) ,J (cid:48) ( z, ¯ z ) × γ (0) γ ( d − γ ( J − ∆) γ ( J − d + ∆) × ( h (cid:48) ¯ h (cid:48) ) J − C − ∆ (cid:18) h (cid:48) + ¯ h (cid:48) h (cid:48) ¯ h (cid:48) (cid:19) + subleading . (5.32)We do not have a proof of eq. (5.32), but we give indirect evidence for it below. Pluggingeq. (5.32) into eq. (5.22) and comparing with eq. (5.31), we obtain the following formula forthe asymptotics of the spectral density: (cid:104) C (∆ (cid:48) , J (cid:48) )2 sin ( · · · ) (cid:105) γ (0) γ ( d − (cid:39) d + i ∞ (cid:90) d − i ∞ d ∆2 πi Γ(∆ − (cid:0) ∆ − d (cid:1) C − ∆ ( η h ) Res J = j ∗ (∆) 12 b t (∆ , J )( h (cid:48) ¯ h (cid:48) ) J − (5.33)where η h = h (cid:48) +¯ h (cid:48) h (cid:48) ¯ h (cid:48) with h (cid:48) , ¯ h (cid:48) defined in eq. (5.28) and b t (∆ , J ) = c t (∆ , J ) K (∆ , J ) γ (∆ − J ) γ ( d − ∆ − J ) . (5.34)Eq. (5.33) is the main result of this subsection. It shows that the heavy spectrum must beregular in theories that are asymptotically transparent, at least in so far as probed by thefour-point function. That is, if j ∗ < (cid:48) j ∗ − implying that2 sin averages to zero. Operators whose dimensions differ appreciably from double-twists,∆ (cid:48) ≈ ∆ + ∆ + 2 n + J (cid:48) , if they exist, must thus have small coefficients.At large dimension with h (cid:48) ≈ ¯ h (cid:48) , we expect the integral (5.33) to be dominated near theintercept, which was studied in section 3.2. It would be interesting to confront the predictionof this formula with the heavy spectrum of the 3d Ising model.Conversely, for theories that are asymptotically opaque where dDisc →
1, we wouldnaively expect ∆ to be uniformly distributed modulo 2 so that 2 sin averages to 1, ie. thephase of e iπ ∆ must be random. The condition that dDisc → b t is power-behaved at large imaginary dimensions, since the γ -factors ineq. (5.34) imply that b t ∝ c t e π | Im∆ | , and all we know from the Lorentzian inversion formulais that c t is power-behaved. This means that the integral in eq. (5.33) may not convergepointwise for a given h (cid:48) , ¯ h (cid:48) . This is not a fundamental problem and simply means that weneed some smearing in ∆ (cid:48) : eq. (5.33) is an expression for the average spectral density. Justhow much smearing is needed is a question we leave to future work.– 43 – .3.1 Comments on conformal Regge theory As mentioned, a relation between Regge trajectories and the heavy spectrum is not new andwas discussed in the holographic context in [44]. To our knowledge, however, this was notdiscussed in the more general context. Let us thus make contact with the conventions andresults of the conformal Regge theory paper [7]. We begin by rewriting our Gegenbauer-likefunction C − ∆ in eq. (2.6) in terms of the harmonic function Ω defined there, which are thesame up to a proportionality factor:4 π d Ω iν ( η ) ≡ Γ(∆ − (cid:0) ∆ − d (cid:1) C − ∆ ( η ) (cid:12)(cid:12)(cid:12) ∆= d + iν . (5.35)We note that this is shadow-symmetric: Ω iν = Ω − iν . We may then rewrite our eq. (5.21) forthe Regge limit of the correlator aslim z, ¯ z → A ( z, ¯ z ) = (cid:90) + ∞−∞ dν π (cid:20) π d Ω iν (cid:18) z + ¯ z √ z ¯ z (cid:19)(cid:21) Res J = j ∗ (∆) (cid:20) γ (∆ − J ) γ ( d − ∆ − J ) b t + b u e − πiJ e − πiJ − z ¯ z ) − J (cid:21) (5.36)where A ≡ e iπ ( a + b ) G ( z, ¯ z ) (cid:9) −G ( z, ¯ z ), b u (∆ , J ) is defined similarly to eq. (5.34), and ∆ = d + iν where it appears. This form is in precise agreement with eq. (56) of [7]. For the doublediscontinuity we get the same formula with b t + b u e − πiJ e − πiJ − (cid:55)→ b t .We see that the only differences between the Lorentzian inversion and Mellin-space for-malisms is how the double-twist poles γγ are treated. What comes out of the Lorentzianinversion formula is c ∝ γγb and from this perspective it seems like an arbitrary choice toexplicit this factor in eq. (5.36). In holographic CFTs, however, the function b turns out tobe simple rational function (at tree-level) which makes the writing in eq. (5.36) natural. Inthe Mellin space approach of ref. [7], the double-trace poles are built-in.Comparing eqs. (5.36) and (5.33), we see that the principal change in going from ( z, ¯ z )space to ( h (cid:48) , ¯ h (cid:48) ) space is the disappearance of the γγ double-twist poles. This was alreadyobserved using the impact parameter representation in ref. [44], understood there to be directlyrelated to the ( h (cid:48) , ¯ h (cid:48) ) spectrum (called ( h, ¯ h ) in section 3.2 there); it was later explained thatgoing from the ( z, ¯ z ) Regge limit to the impact parameter representation simply cancelsfactors of γγ (see eq. (2.31) of [23]). Here our starting point was simply an observed identityregarding the asymptotics of blocks, eq. (5.32), generalized from integer spins, and we viewthe results of [23, 44] as further supporting that identity.To our knowledge, it is an open question whether the function b is power-behaved ornot, or equivalently, whether the integral (5.33) (or equivalently (2.31) of [23]) convergespointwise. This is not implied by the Lorentzian inversion formula, but it is known to betrue perturbatively in holographic theories. If this were to hold nonperturbatively, one could We used that K (∆ , J ) there = K (∆ ,J ) here (2∆ − d )4 − J π γ (∆ − J ) γ ( d − ∆ − J ) . Furthermore, comparing eqs. (28) and(43) of [7] with our eq. (2.8) we find b + J ( ν ) there = − J π ( b t + b u ) here , and, from eq. (54) there: β ( ν ) there = − π Res J = j ∗ (∆) b + J ( ν ) there . – 44 –magine a version of eq. (5.33) (including subleading corrections) that represents the exact spectral function, that is a sum over discrete δ -function, as opposed to just smeared averagesas considered here. We leave this to future investigation. In this paper we study the spectrum of the 3D Ising model at low spin, combining theLorentzian inversion formula developed in [6] with the numerical data from [5]. Two leadingtwist families are our main focus; [ σσ ] which is the leading Z -Even twist family and [ σ(cid:15) ] which is the leading Z -Odd twist family. Two compelling questions are studied in this work.First, can these trajectories which are proven to exist for J ≥ (cid:15) and shadow of σ ? Second, what is the intercept of the leading Regge trajectory,[ σσ ] ?We started by studying a benchmark case, the stress tensor in section 3.1. This is the spin2 operator in [ σσ ] family. To evaluate the inversion integral to high accuracy we used themethod of dimensional reduction to express 3 dimensional as sums of 2d ones with practicallyneglibigle error (see [18] and Appendix. A.1). We then summed the conformal blocks over theknown (truncated) spectrum determined in ref. [5], i.e., (cid:15) and operators belonging to [ σσ ] ,[ σσ ] and [ (cid:15)(cid:15) ] family up to spin 40 are included. We also added the high-spin tails to thesefamilies, which are under analytic control. In fig. 3 we compare the stress-tensor anomalousdimension obtained from different truncation in cross-channel expansion in terms of stabilityand overall error. We obtain a stable result for twist and OPE coefficient with a controllederror at 10 − levels (see table. 1).We then proceeded to continuously lower the value of ∆ (and spin) to reach the intercept.The intercept answers the question of whether high-energy scattering in the 2+1-dimensionalversion of the model is transparent or opaque. However, the method is very different fromstudying such scattering processes directly using the OPE, since the integrals computed insection 3.2 are dominated by a region of large impact parameter, where OPE convergenceis improved. Such shuffling around of information is familiar from dispersion relations. Inthe vicinity of the intercept we use accurate integral representations of the (cid:15) and T cross-channel blocks (see eq. 3.10 and eq. 3.11), as well as suitable approximations ( ρ -expansion)for the direct channel blocks. The truncation of spectrum is reflected in the z -dependence ofthe result as depicted in fig. 12, and we obtain the value of the intercept to be ∼ . (cid:15) operator and its shadow lie on a different branch of theleading trajectory, although we were not able to use this method to compute the properties of (cid:15) numerically stable way. In the Z -odd sector we find similar conclusions, with a compellingpicture of a odd-spin [ σ(cid:15) ] trajectory having a zero at the location of an (absent) spin-1current, and where the even-spin is compatible with passing through the shadow of σ . Thisqualitatively picture arises rigorously in the large- N O ( N ) model, as shown in section 5.2,– 45 –here the [ φ i φ j ] (discussed previously in [16, 38]) and [ φ i S ] trajectories corresponding to[ σσ ] and [ σ(cid:15) ] (see fig. 13).The finding that the intercept is below unity, j ∗ <
1, indicates transparency in the high-energy scattering of lumps or equivalently a negative Lyapunov exponent and absence ofchaos when the theory is placed in Rindler space. A specific prediction is regularity of theheavy spectrum in eq. (5.33). While not chaotic, the 3D Ising CFT is certainly not integrable,and a useful analogy may be the KAM theorem in classical mechanics, which states (veryroughly) that certain small enough deformations of an integrable system are not chaotic. .The approximately conserved quantities of the Ising CFT are likely higher-spin currents [4, 48]and transparency suggests that they become increasingly powerful for heavy operators withincreasing dimensions. Acknowledgments
We thank Anh-Khoi Trinh, David Simmons-Duffin, David Meltzer, Petr Kravchuk for discus-sions, and Alex Maloney for initial collaboration. Zahra Zahraee acknowledges support fromthe Schulich fellowship. Work of SCH is supported by the National Science and EngineeringCouncil of Canada, the Canada Research Chair program, the Fonds de Recherche du Qu´ebec- Nature et Technologies, and the Simons Collaboration on the Nonperturbative Bootstrap.
A Inversion Integrals
In this appendix we discuss various inversion integrals employed in the body of the paper.We first describe the dimensional reduction of 3-dimensional blocks over 2d ones, which reliesitself on a recursion for the series expansion of blocks; we present (for the first time) a closedformula for the latter. Then we give analytic formulas for the inversion integral of 2d block.Combined, these results provide an accurate way to compute the contribution of a singlecross-channel block. Lastly, we consider the collinear approximation to the exchange of asingle cross-channel block, which has been used to make comparison plots such as in section2.4.
A.1 Dimensional reduction for cross-channel blocks
In this section, we briefly review of the dimensional reduction method introduced in [18]and employed in the body of the paper. The idea is to break the (Euclidean) d -dimensionalconformal group, SO( d +1 , d ,1). This will help us to write differentrepresentations of the latter group in terms of the former. A primary operator in d -dimensionsis a sum of infinitely many primaries in ( d − K µ . Primaries of both of both groups are annihilated by K , .., K d − but only We thank Alex Maloney for this analogy. – 46 –O( d ) primaries are annihilated by K d . Loosely speaking, taking derivatives with P d gener-ates new SO( d −
1) primaries. The SO( d ) angular momentum multiplets also decompose intoSO( d −
1) multiplets. This consequently means that any d -dimensional conformal multiplet ofof spin J and dimension ∆ can be decomposed in terms of infinitely many d − ≤ (cid:96) ≤ J and dimensions ∆ + m with m ≥
0. This in turn means thatconformal blocks should also follow this decomposition rule so that a d -dimensional conformalblock can be written as follows: G ( a,b )∆ ,J ( z, ¯ z ; d ) = (cid:88) A ( a,b ) m,n (∆ , J ) G ( a,b )∆+ m,(cid:96) − n ( z, ¯ z ; d −
1) 0 ≤ n ≤ J, m = 0 , , ... (A.1)where again a = ∆ − ∆ and b = ∆ − ∆ .The coefficients with m = 0 describe the dimensional reduction of Gegenbauer polyno-mials and are given as A ( a,b )0 ,n (∆ , J ) = (cid:40) Z Jn/ , n even0 , otherwise (A.2)with, in our conventions, Z t ≡ ( − t ( ) t ( − J ) t t ! ( J − t + d − ) t ( − J − d − ) t (A.3)with ( a ) b ≡ Γ( a + b )Γ( a ) the Pochhammer symbol. Generally, the other coefficients vanish unless m ≡ n modulo 2. They can be obtained recursively by comparing the radial expansion of theblocks in the two dimensions. This recursion was given for identical operators in ref. [18] andwe state here the general case: A ( a,b ) m,n (∆ , J ) = m + n (cid:88) p =max( − m − n , (cid:16) Z J − n +2 pp a ( a,b ) m + n − p, m − n + p (∆ , J ; d ) (cid:17) − m (cid:88) m (cid:48) =1 (cid:88) n (cid:48) A ( a,b ) m − m (cid:48) ,n − n (cid:48) (∆ , J ) a ( a,b ) m (cid:48) + n (cid:48) , m (cid:48)− n (cid:48) (∆ + m − m (cid:48) , J − n + n (cid:48) ; d − . (A.4)where the sum over n (cid:48) ranges from max( − m (cid:48) , n − J + δ m (cid:48) + n − (cid:96), odd ) to min( m (cid:48) , n ) in steps of2. The coefficients a describe the radial expansion of blocks: G ( a,b )∆ ,J ( z, ¯ z ; d ) = (cid:88) r,s ≥ a ( a,b ) r,s (∆ , J ; d )( z ¯ z ) ∆+ r + s C J + s − r (cid:18) z + ¯ z √ z ¯ z ; d (cid:19) . (A.5)In the case of identical operators, a closed form solution to eq. (A.4) was given in ref. [18],which we reproduced. As shown in that reference, the expansion (A.1) converges very rapidly,always at least as fast as the ρ -series. Our subscripts differ from ref. [18] as: m here = 2 m there , as required for non-identical operators. – 47 – .2 New closed-form expression for radial expansion coefficients The coefficients in eq. (A.5) are to be determined using another recursion [33] (see for exampleappendix A of ref. [6] for nonidentical operators). We do not reproduce that recursion here,because, inspired by recent formulas by Li [49] and a bit of guesswork, we were able to finda closed formula! a ( a,b ) r,s (∆ , J ; d ) = (cid:0) ∆ − J +2 − d + a (cid:1) r (cid:0) ∆ − J +2 − d + b (cid:1) r r !(∆ − J +2 − d ) r ( − J − d − ) r ( J − r + d − ) r (cid:0) ∆+ J + a (cid:1) s (cid:0) ∆+ J + b (cid:1) s s !(∆ + J ) s × min( r,s ) (cid:88) p =0 (cid:32) ( − p F (cid:20) − r + p − s + p p ∆ − p − d − J + d − − r + p − J − s + p − d − ; 1 (cid:21) × ( − d ) p ( − d ) p ( − r ) p ( − s ) p ( d − − ∆+ J − r ) p ( − J ) r − p p !(∆ − d − ) p ( − J − s − d − ) p ( J + d − p − r (cid:33) . (A.6)Note from eq. (A.5) that r increases the twist of the descendants and s the conformal spin.The logic of this formula is that coefficients get progressively more complicated as one goesaway from the leading twist or leading conformal spin, where only the p = 0 term contributesin both cases; a pattern was guessed empirically by working away from these simple limits.The formula truncates in d = 2 and d = 4 due to the Pochhammers. In fact the limit to evenspacetime dimensions gives annoying 0 / d = 2 we usethe following simplified result: a ( a,b ) r,s (∆ , J ; d =2) = 11 + δ J, (cid:0) ∆ − J + a (cid:1) r (cid:0) ∆ − J + b (cid:1) r r !(∆ − J ) r (cid:0) ∆+ J + a (cid:1) s (cid:0) ∆+ J + b (cid:1) s s !(∆ + J ) s × (cid:18) − r ) J (∆ + s ) J (1 − ∆ − r ) J (1 + s ) J (cid:19) (A.7)which is valid for j + s − r ≥ A.3 Lorentzian inversion in 2d
The 3d to 2d series (A.1) is useful for this work because of exact results for inversion integralsthat exist in d = 2. Let us denote as c ∆ , ∆ , ∆ , ∆ ∆ (cid:48) ,J (cid:48) ( β, z ; d =2) the contribution eq. (2.21) comingfrom a 2d t -channel block of (∆ (cid:48) , J (cid:48) ), as defined in eq. (2.4). Using eq. (3.38) in [50] (see also[17, 51]), the result (integrated from ¯ z ≥ z ≥ z !) is written as: c ∆ , ∆ , ∆ , ∆ ∆ (cid:48) ,J (cid:48) ( β, z ; d =2) = z ∆1+∆22 (1 − z ) ∆2+∆32 I ∆12 ··· ∆42∆ (cid:48) + J (cid:48) (cid:16) β (cid:17) k ( a (cid:48) ,b (cid:48) )∆ (cid:48) − J (cid:48) (1 − z ) + ( J (cid:48) (cid:55)→ − J (cid:48) )1 + δ J (cid:48) , , (A.8)– 48 –here a (cid:48) = ∆ − ∆ , b (cid:48) = ∆ − ∆ and I h ··· h h (cid:48) ( h ) is the one-dimensional inversion given as: I h ··· h h (cid:48) ( h ) = Γ( h + h )Γ( h + h )Γ( h + h − h (cid:48) )Γ( h + h − h (cid:48) )Γ(2 h −
1) Γ( h − h (cid:48) + h + h − h + h (cid:48) − h − h + 1) × F (cid:20) h (cid:48) + h h (cid:48) + h h (cid:48) − h − h +1 h (cid:48) − h − h +12 h (cid:48) h + h (cid:48) − h − h h (cid:48) − h − h − h +2 ; 1 (cid:21) + 2 sin (cid:0) π ( h (cid:48) − h − h ) (cid:1) sin (cid:0) π ( h (cid:48) − h − h ) (cid:1) κ ( h ,h )2 h × F (cid:20) h + h h + h h + h + h − h + h + h − h h + h (cid:48) + h + h − h − h (cid:48) + h + h ; 1 (cid:21) (A.9)with h ij = h i − h j . This result, inserted in the 3d to 2d expansion (A.1), gives the formula(2.26) which is used repeatedly in this paper.We stress that the analytic result (A.9) is only valid when integrating over the completerange 0 ≤ ¯ z ≤ z ≤ ¯ z ≤
1. Since in practice we work at small z , we can correct for this discrepancyby subtracting from eq. (A.9) the integral of the first few terms in the ¯ z → z this is completely negligible, but for moderate values like z ∼ − this is important for our precision study.The hypergeometric series in eq. (A.9) terminates in special cases such as h (cid:48) = h . Thesecorrespond to power laws in the t -channel. Setting h + h = τ , h = a and h = b theformula reduces to the integral recorded in eq. (4.7) of [6] (using also a ↔ b symmetry): I ( a,b ) τ ( β ) ≡ (cid:90) dzz κ ( a,b ) β k ( − a, − b ) β ( z )dDisc (cid:34)(cid:18) − zz (cid:19) τ − b z − b (cid:35) = 1Γ (cid:0) − τ + b (cid:1) Γ (cid:0) − τ − a (cid:1) Γ (cid:16) β − a (cid:17) Γ (cid:16) β + b (cid:17) Γ( β −
1) Γ (cid:16) β − τ − (cid:17) Γ (cid:16) β + τ + 1 (cid:17) . (A.10)This integral allows to deal exactly with identity exchange and more generally large-spinperturbation theory. The first hypergeometric function in eq. (A.9) can be interpreted assumming up the (1 − ¯ z ) / ¯ z series according to this integral; this series is asymptotic, and thesecond hypergeometric can be interpreted as a nonperturbative correction at large spin [17]. A.4 Collinear expansion ¯ z → for cross-channel exchanged blocks Taking the limit z → G ∆ ,J ( z, ¯ z ) or ¯ z → G ∆ ,J (1 − ¯ z, − z ) is a straightforwardprocedure. In this limit the quadratic Casimir equation becomes a hypergeometric equationso the leading behaviour of the conformal blocks in this limit is G ∆ ,J → z ∆ − J k ∆+ J (¯ z ) , (A.11)where k β ( z ) is defined in (2.5). A nice way to organize the expansion, which was discussedfor example in [6], is in terms of these functions since they control the SL ( R ) part of the– 49 –onformal group that remains after taking the limit. With a convenient factor extracted andfocusing on d = 3, the expansion that we use is (cid:112) − z/ ¯ z G ∆ ,J ( z, ¯ z ) = ∞ (cid:88) m =0 z τ + m h ( m )∆ ,J (¯ z ) , (A.12)with h ( m )∆ ,J (¯ z ) = m (cid:88) n = − m h ( m,n )∆ ,J k β +2 n (¯ z ) . (A.13)The quadratic Casimir equation then gives the following recursion relation in m (see [6]): m (cid:88) n = − m ( n ( n + β −
1) + m ( m + τ − h ( m,n )∆ ,J k β +2 n (¯ z )= (cid:18) τ −
32 + m + a (cid:19) (cid:18) τ −
32 + m + b (cid:19) h ( m − ,J (¯ z ) − m (cid:88) m (cid:48) =1 (cid:18) m (cid:48) ¯ z m (cid:48) − m (cid:48) − z m (cid:48) − (cid:19) h ( m − m (cid:48) )∆ ,J (¯ z ) . (A.14)To isolate the coefficient of k β +2 n (¯ z ) on the right-hand-side we need to use the shift relation1¯ z k β (¯ z ) = k β − (¯ z ) + (cid:18) − abβ ( β − (cid:19) k β (¯ z ) + ( β − a )( β − b ) β ( β − k β +2 (¯ z ) , (A.15)to eliminate all explicit appearance of ¯ z , after which we can solve recursively for the coefficients h ( m,n )∆ ,J .The result of the recursion can finally be combined with the prefactor in eq. (A.12) toexpand the block in pure powers of z . The first two terms of this expansion are G ∆ ,J ( z, ¯ z ) ≈ z τ k β (¯ z ) + z τ +1 (cid:34) β − τ β − τ − k β − (¯ z ) + ( β − a )( β − b )( β + τ − β ( β − β + τ − k β +2 (¯ z )+ (cid:32) τ + 2 a + 2 b ab (cid:0) ( β − − τ + 1 (cid:1) β ( β − τ − (cid:33) k β (¯ z ) (cid:35) + O (¯ z τ +2 ) . (A.16)For blocks with J non-integer, the same formulas give the expansion of g pure∆ ,J ( z, ¯ z ). B Compact approximations from large-spin perturbation theory
In this appendix we record compact but surprisingly accurate approximations for the OPEdata based on large-spin perturbation theory. Our formulas are essentially simplified versionsof results from [5].Starting from the Lorentzian inversion formula, the idea is to truncate the t -channel sumto just identity and a small number of operators. For each operator, we keep only the leadingterm at large spin or β → ∞ , which comes from ¯ z →
1, and we extract anomalous dimensionsby looking at logarithmic terms as z →
0. – 50 –onsidering the exchange of an operator O of twist τ O = ∆ O − J O and conformal spin β O = ∆ O + J O , we take the double limit ( z, ¯ z ) → (0 ,
1) (for example starting from the ¯ z → z → , ¯ z → G (0 , O ,J O (1 − ¯ z, − z ) → − β O )Γ( β O (cid:1) (1 − ¯ z ) τ O (cid:16) log z + H ( β O − (cid:17) (B.1)where H ( x ) = ψ ( x + 1) − ψ (1) is the harmonic number. Plugging into the inversion integral(2.22) (replacing (1 − ¯ z ) by − ¯ z ¯ z ) and expanding eq. (A.10) at large β , we obtain the followingapproximation to the collinear generating function: C t ( z, β ) + C u ( z, β ) ≈ C (0)[ σσ ] ( β ) z ∆ σ (cid:34) − (cid:88) O f σσ O Γ( β O )Γ(∆ σ ) Γ( β O (cid:1) Γ (cid:0) σ − τ O (cid:1) log z + H ( β O − β − / τ O (cid:35) , (B.2)where we defined the mean-field theory coefficient on the leading trajectory: C (0)[ σσ ] ( β ) ≡ I (0 , − σ = 2Γ( β )Γ (cid:0) β (cid:1) σ ) Γ( β + ∆ σ − β − ∆ σ + 1) . (B.3)We stress that in eq. (B.2) only the lowest few operators should be included in the sum,which is not a convergent sum. We include only (cid:15) and T . As noted in the main text, weexpand in 1 / ( β −
1) because the series is even in that variable. Taking the coefficient of log z , and the constant, respectively, gives the “pocket-book” formula recorded for the twistin eq. (2.31), and a corresponding formula for the OPE coefficients (including the Jacobianfactor in eq. (2.20)): τ [ σσ ] ≈ σ − (cid:88) O = (cid:15),T λ σσ O Γ(∆ σ ) Γ (cid:0) ∆ σ − τ O (cid:1) Γ( β O )Γ (cid:0) β O (cid:1) (cid:18) β − (cid:19) τ O ,f σσ [ σσ ] ≈ C (0)[ σσ ] ( β )1 − dτ [ σσ ]0 dβ − (cid:88) O = (cid:15),T λ σσ O Γ(∆ σ ) Γ (cid:0) ∆ σ − τ O (cid:1) Γ( β O )Γ (cid:0) β O (cid:1) H (cid:0) β O − (cid:1) (cid:18) β − (cid:19) τ O . (B.4)Note that β = ∆+ J = τ +2 J enters the formula for τ . To compute the twist of an operator ofgiven spin J , we first evaluate the first line with β (cid:55)→ σ + 2 J to get a crude approximationto τ ; we then iterate using the improved value β (cid:55)→ τ + 2 J . The procedure converges rapidly.The resulting value of β is then inserted in both equations. In figure 24 we compare thisformula with the numerical data of ref. [5]. Both plots exhibit relative accuracy better than10 − for all twists and the stress tensor OPE coefficients, but the relative error is closer to10 − for the spin-4 OPE coefficient.Given the error budget discussed in section 3.1, we believe that the remarkable accuracyof the approximation at spin J = 2 is a lucky accident of that particular formula. Indeed, inthe absence of accident one would expect the discrepancy to be significantly larger for J = 2than for J = 4. – 51 – - (a) Twist, with difference (data)-(approx.) in inset. (b) OPE coefficients. Figure 24 : Comparison of the large-spin approximation (B.4) for the [ σσ ] family withthe numerical data of [5], for the twist and OPE coefficients (divided by mean field theory).Only numerical errors are shown: discrepancies on the left-hand side of the plots should beattributed to shortcomings in the approximation not data.For the subleading trajectories, [ σσ ] and [ (cid:15)(cid:15) ] are near-degenerate and mix substantially,as pointed out in [5]. We thus need to study a 2 × σ . We begin with identityexchange, first pretending that 2∆ (cid:15) ≈ σ + 2 so as to make the operators degenerate. Wethen expanding the Lorentzian inversion formula to second order in z where needed ( ie. tosubtract descendants of [ σσ ] ): M (0) ≡ (cid:32) C σσσσ C σσ(cid:15)(cid:15) C (cid:15)(cid:15)σσ C (cid:15)(cid:15)(cid:15)(cid:15) (cid:33) z ≈ σ +2 = Γ (cid:0) β (cid:1) Γ( β − ∆ σ − Γ(∆ σ ) (cid:16) β − (cid:17) − σ (cid:15) ) (cid:16) β − (cid:17) − (cid:15) . (B.5)To get anomalous dimensions we look for logarithmic terms log z . We keep two sources:identity exchange expanded to linear order in (2∆ (cid:15) − σ − σ -exchange. Multiplyingby ( M (0) ) − / on both sides to properly normalize the states, we find that the twists of the[ σσ ] and [ (cid:15)(cid:15) ] families are the eigenvalues of the following matrix (respectively the higherand lower eigenvalues): τ { [ σσ ] , [ (cid:15)(cid:15) ] } = (cid:32) σ +2 XX (cid:15) (cid:33) (B.6)with off-diagonal term X = 4 f σσ(cid:15) Γ(∆ (cid:15) )Γ(∆ σ − ∆ (cid:15) )Γ(∆ σ ) Γ (cid:0) ∆ (cid:15) (cid:1) Γ (cid:0) σ − ∆ (cid:15) (cid:1) ∆ (cid:15) − ∆ σ − √ σ − (cid:18) β − (cid:19) ∆ σ . (B.7)– 52 –
20 40 60 802.42.62.83.03.23.4 (a) Twist
10 20 30 40 50 600.50.60.70.80.9 (b) OPE coefficients divided by MFT (3.3).
Figure 25 : Same as fig. 24 but for the [ σσ ] and [ (cid:15)(cid:15) ] families (top and bottom, respectively).To find the OPE coefficients we compare a certain derivative of the matrix of generatingfunctions (2 z∂ z − τ [ (cid:15)(cid:15) ] ) C with the OPE:( τ [ σσ ] − τ [ (cid:15)(cid:15) ] ) (cid:32) f σσ [ σσ ] f σσ [ (cid:15)(cid:15) ] (cid:33) (cid:16) f σσ [ σσ ] f σσ [ (cid:15)(cid:15) ] (cid:17) = ( M (0) ) / (cid:2) τ { [ σσ ] , [ (cid:15)(cid:15) ] } − τ [ (cid:15)(cid:15) ] (cid:3) ( M (0) ) / . (B.8)The right-hand-side is an explicitly given matrix of rank 1 and so the equation allows tosolve for f σσ [ σσ ] and f (cid:15)(cid:15) [ σσ ] , up to overall sign conventions; couplings to the [ (cid:15)(cid:15) ] family areobtained similarly. These approximations are plotted in fig. 25. Note that the accuracy is lessthan for [ σσ ] since the approximation is more complicated due to the mixing yet far cruder(we did not even account for (cid:15) exchange). C Convexity of the leading trajectory
Here we give an elementary proof that the region of convergence of the Lorentzian inversionformula, in the real (∆ , J ) plane, is convex. We consider a correlator of identical operators,so dDisc G is a positive-definite distribution. Consider first the integration region where z (cid:28) ¯ z (cid:28)
1. In this region, the block in eq. (2.11) has an exponential-like dependence on J, ∆: G J + d − , ∆+1 − d ∝ z J − ∆2 + d − ¯ z ∆+ J . The basic point is that the exponential function is convex: e xA +(1 − x ) B ≤ xe A + (1 − x ) e B , ≤ x ≤ . (C.1)Therefore if the integral (2.11) converges at the two points (∆ A , J A ) and (∆ B , J B ), it auto-matically converges everywhere along the line segment joining them: the region of convergenceis convex. Increasing J can only improve convergence, and adding imaginary parts does notaffect convergence.The conformal blocks which enter the Lorentzian inversion formula are more complicatedfunctions than exponentials, but we can apply the same logic. A better model, which reflects– 53 –he shadow-symmetry of the curve, is a cosh function; indeed the z ∼ ¯ z (cid:28) c g ( z ) cosh (cid:0) (∆ − d ) log z ¯ z (cid:1) ≤ C ∆+1 − d (cid:18) z + ¯ z √ z ¯ z (cid:19) ≤ c g ( z ) cosh (cid:0) (∆ − d ) log z ¯ z (cid:1) (C.2)where g ( z ) = cosh (cid:0) (1 − d ) log z ¯ z (cid:1) and the constants c and c depend only on spacetimedimension but work uniformly for all ∆ , z, ¯ z . This shows that Lorentzian inversion convergesif and only if the cosh model converges, and since cosh is a sum of two exponentials wecan apply eq. (C.1). This takes care of the region z ∼ ¯ z (cid:28)
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