Moments in the CFT Landscape

Moments in the CFT Landscap e Li-Y uan Chiang, David P oland, Go rdon Rogelb erg Dep artment of Physics, Y ale University, 217 Pr osp e ct St, New Haven, CT 06520, USA E-mail: li-yuan.chiang@yale.edu , david.poland@yale.edu , gordon.rogelberg@yale.edu Abstract: W e dev elop a no v el n umerical b ootstrap for unitary , crossing-symmetric con- formal ﬁeld theories, fo cusing on moment observ ables deﬁned as weigh ted av erages ov er conformal data. Pro viding a global and coarse-grained prob e of the operator sp ectrum, this framework yields n umerically rigorous bounds on the op erator distribution using stan- dard semideﬁnite programming techniques. In the heavy correlator regime, these b ounds remain robust and conv erge rapidly tow ards analytically-derived p o wer la ws. A t ﬁnite ex- ternal dimensions, low-lying moments capture corrections to analytic heavy limit results, while repro ducing familiar b ootstrap solutions such as Ising-mo del kinks on the b ound- ary of moment space. Most imp ortan tly , the moment b ootstrap reveals new features in previously unexplored regions of the b o otstrap landscap e. The low er b ounds on momen t v ariables exhibit tw o contin uous families of kinks p ersisting across 2 < d < 6, reﬂecting non trivial sp ectral reorganizations connected to underlying op erator decoupling phenom- ena. These results demonstrate that momen t v ariables uncov er b o otstrap solutions and collectiv e structures that are diﬃcult to access within traditional numerical approaches. Con tents 1 In tro duction 1 2 The moment b ootstrap 5 2.1 OPE moments 5 2.2 Numerical b o otstrap setup 6 2.3 Summary of results 9 3 Momen ts in the hea vy correlator limit 11 3.1 Momen ts of heavy correlators 11 3.2 Comparison to gap maximization 13 3.3 Max-en trop y reconstruction 14 4 The Ising momen ts 16 4.1 Correlator b ounds 16 4.2 Bounds on the ∆-moments 16 4.3 Bounds on the ℓ -moments 17 4.4 Precise moments under a gap assumption 19 5 A tour of the moment landscap e 20 5.1 Ov erview of geometric features 21 5.2 Sp ectral reorganization b efore the “cliﬀ ” 23 5.3 The cliﬀ 24 5.4 First v alley 26 5.5 The hill 28 5.6 Second v alley 29 5.7 F ake-primary in terpretation of low er b ounds 31 5.8 The ∆ gap ≥ 2∆ ϕ lo w er b ound 36 6 Discussion 38 A Rational approximation of conformal blo c ks 40 B Momen ts from 2d correlators 41 C Ising momen ts across dimensions 43 D Maxim um en trop y reconstruction of coarse-grained sp ectra 43 E F ak e-primary remap of generalized free ﬁeld correlators 45 – ii – 1 In tro duction The conformal b ootstrap has emerged as one of the most p o werful nonp erturbativ e meth- o ds for studying conformal ﬁeld theories (CFT) in general spacetime dimensions. Lying at the core of these techniques are the crossing equations: an inﬁnite family of sum rules that enco de the consistency of the operator pro duct expansion (OPE) with conformal sym- metry , unitarity , and crossing symme tr y . Implemen ted as a semideﬁnite program (SDP), n umerically optimized functionals acting on these constraints can b e used to mak e remark- ably precise predictions ab out the low-lying sp ectra of lo cal op erators in a wide range of CFTs such as the 3d Ising CFT [ 1 – 5 ], O(N) vector mo dels [ 6 – 9 ], Gross-Neveu-Y uk a wa CFTs [ 10 , 11 ] and their sup ersymmetric extensions [ 12 – 17 ], 3d conformal gauge theories suc h as quan tum electro dynamics with fermionic and scalar matter [ 18 – 26 ], and more. The ma jority of developmen ts in the b ootstrap hav e b een aimed tow ards the goal of isolating kno wn models in theory space. By using knowledge of low-lying sp ectra, fusion algebras of lo cal op erators, and gaps inspired b y equations of motion from Lagrangian descriptions as inputs to the bo otstrap equations, w e can isolate islands in parameter space that reproduce CFT data, which we can then c heck against known p erturbativ e results, Mon te Carlo, and fuzzy sphere computations. These inv estigations hav e since b ecome synon ymous with the b ootstrap and are widely regarded as the b enc hmark of its success. In this pap er, we in tro duce the numeric al moment b o otstr ap , a framew ork in which crossing symmetry and unitarity are used to place n umerically rigorous b ounds on mo- men ts of the OPE data, rather than on individual low-lying op erator prop erties. This pro vides an alternativ e wa y to nonp erturbativ ely map out theory space. W e iden tify mo- men ts as natural b o otstrap observ ables that capture global prop erties of the sp ectral data app earing in conformal correlators and study these observ ables with semideﬁnite program- ming techniques. This allows us to identify new geometrically privileged p oin ts, or kinks , in the allo wed space of scalar four-p oin t correlators with identical external dimensions. W e then c haracterize these kink structures by identifying unique prop erties and op erator decoupling phenomena within their lo w-lying sp ectra. Additionally , w e use this new space of b ootstrap observ ables to re-characterize kno wn theories, suc h as the 3d Ising mo del. W e hop e that these results will serve as a starting p oin t for precision inv estigations of these new b o otstrap solutions. Related obse rv ations ha ve app eared in a num b er of contexts. In [ 27 ] the authors stud- ied b ounds on the v alue of the 4-p oin t function in the crossing-symmetric conﬁguration, observing kinks related to the Ising model. In [ 28 ], the authors used a basis of analytic func- tionals dual to extremal solutions for the generalized free ﬁeld to compute gap-maximizing functionals of a correlator of identical scalar op erators in a 2d CFT. In addition to yield- ing the kno wn kinks asso ciated with ⟨ σ σ σσ ⟩ and ⟨ ϵϵϵϵ ⟩ in the Ising mo del, the study also rev ealed tw o new kinks that hav e yet to b e fully understo od. Similarly , in [ 12 ], the authors unco v ered three families of kinks in the space of sup erconformal ﬁeld theories (SCFT) in dimensions tw o through four. One of these kinks was asso ciated with a known family of theories interpolating b et ween the d = 2 , N = (2 , 2) minimal mo del and the theory of a free c hiral m ultiplet in d = 4. The other tw o kinks are more mysterious, with one co- – 1 – inciding with a kinematic threshold at which certain op erators can app ear and the other in terp olating to a kink in the space of 4d N = 1 SCFTs, whic h had b een previously studied in [ 29 – 31 ]. Similar unexplained kink structures hav e also b een observed in the 4-fermion b ootstrap [ 32 – 34 ]. These inv estigations not only rev eal the uncanny ability of the b ootstrap program to isolate kno wn solutions, but also pic k out new privileged solutions to crossing that hav e previously unkno wn descriptions. The aforementioned studies mak e use of functionals that extremize a low-energy prop- ert y of the OPE sp ectrum, such as the scaling dimension of the low est lying non-iden tit y op erator (gap-maximization) or the OPE co eﬃcien t of a single kno wn op erator in the the- ory (OPE-maximization). These low-lying features of the CFT sp ectrum can b e used to accurately approximate correlation functions with similarly small external scaling dimen- sions. In [ 35 ], the authors show ed that the ability to do such an appro ximation comes from the fact that heavy states with ∆ > 2∆ ϕ are suppressed in the OPE by factors of size e − (∆ − 2∆ ϕ ) , where ∆ ϕ is the external scaling dimension. This fact makes gap-maximizing functionals a useful tec hnique for understanding the space of correlators that are dominated b y ligh t states close to the gap. This leads one to ask ho w one may understand correlators with large external di- mension, where the op erators dominating the OPE are no longer close to the gap. The functionals that we analyze in this work are aimed at addressing this problem: they cap- ture glob al prop erties of the OPE rather than characterizing a single low-lying op erator. These functionals, which we refer to as OPE momen ts, are deﬁned as av erages of p o wers of ∆ and ℓ o ver the OPE sum, weigh ted by the squared OPE co eﬃcien t times a conformal blo c k ev aluated at the self-dual p oin t z = ¯ z = 1 / 2. In [ 36 ], the authors prov ed that OPE momen ts in p o wers of ∆ enjo y asymptotic tw o-sided b ounds in the limit of large external dimension and are saturated by particular limits of correlators of identical scalars in a generalized free theory . In this w ork, w e develop and implement the numerical moment b ootstrap, using semideﬁnite programming tec hniques and the solv er SDPB [ 37 , 38 ] to rigorously extend these moment b ounds a wa y from the asymptotic regime to ﬁnite and small v alues of the external scaling dimension. This approac h not only repro duces the expected large-∆ ϕ b e- ha vior but also resolves the geometry of momen t space at ﬁnite ∆ ϕ , unco vering new families of kinks in the space of CFT correlators across spatial dimensions. Bounding momen ts using SDP metho ds is not new, and has a long history in opti- mization, probabilit y , and ﬁnance, see e.g. [ 39 – 43 ]. In ph ysics, similar spaces of moment v ariables hav e b een utilized recen tly in e.g. [ 44 – 50 ] to study the prop erties of eﬀectiv e ﬁeld theories (EFTs) in the context of the S-matrix b ootstrap. These works make use of the lo w energy s ∼ 0 expansion of amplitudes of the form F ( s ) = Z dM 2 p ( M 2 ) M 2 − s = X n a n s n , (1.1) where p ( M 2 ) is the sp ectral density and a n = Z dM 2 p ( M 2 ) M 2  1 M 2  n (1.2) – 2 – denote moments of the p ositiv e measure deﬁned by p ( M 2 ) / M 2 . These momen ts are lin- early related to Wilson co eﬃcien ts in the eﬀectiv e action of the theory , with the UV cutoﬀ t ypically set to the mass gap. By the p ositivit y of the sp ectral density , it follows that the (shifted) Hankel matrices of these moments are p ositiv e semideﬁnite and totally nonnega- tiv e: 1       a 0 a 1 a 2 . . . a 1 a 2 a 3 . . . a 2 a 3 a 4 . . . . . . . . . . . . . . .       ≥ 0 and       a 1 a 2 a 3 . . . a 2 a 3 a 4 . . . a 3 a 4 a 5 . . . . . . . . . . . . . . .       ≥ 0 . (1.3) These conditions restrict the moment sequences asso ciated with amplitudes in a unitary theory to liv e in a con vex “moment cone.” In addition to b eing studied geometrically using the well developed theory of conv ex p olytopes, these momen ts may b e extremized n umerically using SDP . These tec hniques allo w one to deriv e rigorous b ound s on all Wilson co eﬃcien ts in theories admitting a UV completion. With additional assumptions on the system of amplitudes and their relations, one can also isolate known theories such as sup erstring theory in the space of Wilson co eﬃcien ts [ 50 ]. Related constrain ts on sp ectra in the EFT b o otstrap ha ve also b een derived, leading to nontrivial relations among masses and spins in weakly-coupled theories [ 51 , 52 ]. A similar study of momen t v ariables was carried out in the con text of the 2d mo dular b ootstrap [ 53 ], which considered b ounds arising from the con v ex cone of momen ts of the torus partition function: Z p,q = X i n i ∆ p i J q i . (1.4) In this work, numerical bounds on the leading momen ts w ere obtained using SDP metho ds and were seen to hav e a signiﬁcant amoun t of structure at b oth small and large v alues of the central charge, particularly after imposing integralit y conditions [ 53 , 54 ]. In the setting of the conformal b ootstrap, p ositiv e geometry tec hniques ha ve been previously applied to moments deﬁned as the T aylor co eﬃcien ts of a 4-p oin t correlation function [ 55 – 57 ] expanded around the self-dual p oin t: G ( z ) = X n g n ( z − 1 / 2) n , (1.5) with g n = G ( n ) (1 / 2) n ! , (1.6) and G ( n ) (1 / 2) can then b e further decomp osed into con tributions from conformal blo c ks. Notable results of these studies include a geometric pro of of the existence of op erators in a window determined by the external dimension [ 56 ], and a geometric c haracterization of extremal functionals [ 57 ]. While this c hoice of momen ts is natural from the point of view of the T a ylor expansion, they are diﬃcult to interpret and are far remov ed from the prop erties of the underlying 1 This means that the determinants of all minors are nonnegative. – 3 – sp ectrum. In our work we will fo cus instead on “classical” moment v ariables, taking the form of integer p o wers of quan tum n umbers in tegrated against the positive measure deﬁned b y the conformal blo c k decomp osition (analogous to ( 1.4 )), which admits an in tuitive statistical interpretation and mak es the dep endence on the underlying op erator sp ectrum more direct. The T aylor co eﬃcien ts in ( 1.6 ) can then b e expressed as a linear combination of the momen t v ariables considered in our work, pro vided one extends the set to include in v erse momen ts that capture the p ole structure of the conformal blo c ks. That said, due to the fact that conformal blo c ks behav e as an approximate p o wer law ∼ r ∆ in the limit of large exchanged dimension, the geometric and classical moments coincide up to a simple prefactor at leading order in the “heavy” limit of large external dimension. This leads to a num b er of remark able prop erties ab out geometric moments in the heavy limit, suc h as their simple description as a Minko wski sum o ver con v ex p olytopes [ 56 ]. There are man y w ays to functionally decomp ose a CFT correlator in to a p ositiv e sum, eac h of which deﬁnes a determinate momen t problem. Known decomp ositions that can b e enco ded as positive measures include expansions in to t wist monomials [ 58 ], radial monomi- als [ 59 , 60 ], and conformal blo c ks. In this work, we study moments deﬁned o v er a p ositiv e measure which enco des the conformal blo c k decomp osition of the correlator. The purpose of this choice is that this decomp osition conv erges most rapidly , as inﬁnite families of lo cal op erators (whic h include descendants) are repac k aged into a conformal blo c k lab eled b y the low est weigh t primary op erator. This allo ws us to more directly compute moments of light correlators by summing ov er known data of the low-lying sp ectrum, weigh ted b y appropriate factors of scaling dimension and spin. In turn, momen ts of the conformal blo c k decomp osition are closely related to previously computed data, unlike those of decomp osi- tions that do not distinguish b et w een primary and descendant op erators. This w ork will b e organized as follows: we b egin with a review of the moment b o ot- strap and its numerical implementation in section 2 . In section 3 , w e n umerically compute momen t b ounds for hea vy correlators and see agreemen t with the b ounds derived analyt- ically in [ 36 ]. W e also study the con v ergence of the b ounds and show how one can use the maxim um en trop y reconstruction metho d to appro ximate the OPE densit y . Then, w e discuss momen ts of the ⟨ σ σ σ σ ⟩ correlator in the 3d Ising mo del in section 4 , comparing them against mom en ts computed using its kno wn sp ectral data. In section 5 , w e presen t newly discov ered families of kinks in moment space across spatial dimensions, discuss their relation to the lo cations of kno wn solutions to crossing, such as minimal mo dels, and fur- ther draw atten tion to solutions to crossing that sim ultaneously extremize a large n um b er of momen t v ariables. Lastly , we conclude with a discussion of our results in section 6 and include additional details in the app endices. – 4 – 2 The moment b o otstrap 2.1 OPE moments W e b egin b y deﬁning the momen t v ariables for the conformal blo c k expansion. Consider the four-p oin t correlation function of iden tical scalar primaries ⟨ ϕ ( x 1 ) ϕ ( x 2 ) ϕ ( x 3 ) ϕ ( x 4 ) ⟩ = 1 x 2∆ ϕ 12 x 2∆ ϕ 34 G ( z , ¯ z ) , (2.1) where we hav e the follo wing conformal blo c k expansion G ( z , ¯ z ) = X O λ 2 ϕϕ O g ∆ ,ℓ ( z , ¯ z ) . (2.2) W e deﬁne the moment v ariables as weigh ted sums ov er the conformal data, where the w eigh t is determined b y the squared op erator pro duct expansion (OPE) c oeﬃcients and the conformal blocks ev aluated at some ﬁxed kinematics. Sp eciﬁcally , the sp ectral densit y is deﬁned as p ℓ (∆) = X O λ 2 ϕϕ O g ∆ ,ℓ ( z , ¯ z ) δ (∆ − ∆ O ) δ ℓ,ℓ O . (2.3) Since unitarit y implies that λ 2 ϕϕ O ≥ 0, and the conformal blocks are positive at the self-dual p oin t, this deﬁnes a p ositiv e sp ectral measure o v er op erator dimensions and spins, leading to the following deﬁnition for the moments: ν k = X ℓ Z d ∆ p ℓ (∆) ∆ k = X O λ 2 ϕϕ O g ∆ O ,ℓ O ( z , ¯ z )∆ k O . (2.4) W e can further generalize this idea to moments inv olving b oth ∆ and ℓ : ν m,n = X ℓ Z d ∆ p ℓ (∆) ∆ m ℓ n = X O λ 2 ϕϕ O g ∆ O ,ℓ O ( z , ¯ z )∆ m O ℓ n O . (2.5) It is immediate from the deﬁnition that the momen t v ariables dep end on the cross ratios z and ¯ z through their dep endence on the conformal blo c k. In this pap er, we fo cus on momen ts ev aluated at the self-dual p oin t, deﬁned by z = ¯ z = 1 / 2, where the s - t crossing equation is trivially satisﬁed. All numerical b ounds in this pap er are computed at this p oin t. The zeroth moment is simply the correlator ev aluated at the chosen kinematic conﬁg- uration and ma y p ossibly b e unbounded from ab o v e [ 27 ] at ﬁnite external scaling dimen- sions, and hence so can the higher momen ts. Crucially , the r atios b et ween momen ts remain b ounded for any ﬁnite external dimension [ 36 ]. Here, we deﬁne the normalize d moments using a normalization with resp ect to the zeroth moment: ⟨ ∆ k ⟩ ≡ ν k ν 0 , ⟨ ∆ m ℓ n ⟩ ≡ ν m,n ν 0 , 0 . (2.6) These momen t v ariables admit a clear statistical interpretation, capturing coarse-grained information ab out the exc hanged conformal data including its mean, v ariance, skew, kur- tosis, etc. – 5 – Finally , w e may also consider momen ts deﬁned with the iden tit y excluded. This modi- ﬁcation only aﬀects the zeroth moment, shifting it as ν 0 → ν 0 − 1, allo wing us to focus more directly on the non trivial op erator con ten t at and b ey ond the gap state. F or conv enience, w e deﬁne ⟨ ∆ k ⟩ ∗ ≡ ν k ν 0 − 1 , ⟨ ∆ m ℓ n ⟩ ∗ ≡ ν m,n ν 0 , 0 − 1 , (2.7) whic h will denote the momen ts with the identit y op erator excluded from now on. On a related note, in a previous w ork [ 27 ], the authors sho wed that the gap-maximizing solution can be recov ered via a simpler optimization problem in volving the correlator itself. W e ﬁnd that their observ ation ﬁts naturally into a broader framework based on moment v ariables. The v alue of the correlator at the self-dual p oin t corresponds to the zeroth unnormalized momen t, and as we will show, the 3d Ising mo del not only extremizes the correlator but also exhibits extremal b eha vior in higher moments, suggesting that the momen t b o otstrap provides a natural and eﬃcient w ay to explore the structure of theory space. 2.2 Numerical b o otstrap setup In this section, w e show that the problem of b ounding the OPE moments can b e under- sto od as an inﬁnite-dimensional linear program, solv able with the conv entional extremal functional metho d via semideﬁnite programming [ 37 ]. Crossing equation A CFT correlator of identical scalar primaries enjoys the following crossing symmetry equation, which iden tiﬁes the s -channel and the t -channel conformal blo c k expansions: v − ∆ ϕ − u − ∆ ϕ = X O λ 2 ϕϕ O  u − ∆ ϕ g ∆ ,ℓ ( u, v ) − v − ∆ ϕ g ∆ ,ℓ ( v , u )  ≡ X O λ 2 ϕϕ O F ∆ ,ℓ ( u, v ) , (2.8) where the left-hand side of the equation represents the v acuum contribution. No w, let us rewrite the crossing equation b y pulling out the piece that contributes to the sp ectral densit y ( 2.3 ) v − ∆ ϕ − u − ∆ ϕ = X O λ 2 ϕϕ O g ∆ ,ℓ ( z ∗ , ¯ z ∗ ) F ∆ ,ℓ ( u, v ) g ∆ ,ℓ ( z ∗ , ¯ z ∗ ) , (2.9) where z ∗ and ¯ z ∗ deﬁne the p oin t of expansion and are set to 1 / 2 in this w ork. T o pro ceed, w e make use of the standard rational appro ximation of conformal blocks, whic h allows deriv ativ es of blo c ks ev aluated at the crossing-symmetric point to b e expressed as rational functions of the scaling dimension ∆ of the form ∂ m z ∂ n ¯ z g ∆ ,ℓ (1 / 2 , 1 / 2) ≈ (4 r ∗ ) ∆ Q A (∆ − ∆ ∗ A ) P m,n ℓ (∆) , (2.10) where P m,n ℓ (∆) are p olynomials in ∆. W e summarize the construction and conv entions of the rational appro ximation in app endix A . Here w e only emphasize that the appro ximation is controlled b y tw o parameters, r max and κ , whic h resp ectiv ely determine the truncation order of the radial expansion used to compute the conformal blo c ks and the subset of p oles retained in the rational function. 2 W e then act with the deriv ativ e operator ∂ m z ∂ n ¯ z on both 2 Appro ximations of this form can b e further optimized using the interpolation scheme developed in [ 61 ]. – 6 – sides to get b m,n = X O λ 2 ϕϕ O g ∆ ,ℓ ( z ∗ , ¯ z ∗ ) F m,n ℓ (∆) P 0 , 0 ℓ (∆) = X ℓ Z d ∆ p ℓ (∆) F m,n ℓ (∆) P 0 , 0 ℓ (∆) , (2.11) where b m,n comes from the v acuum contribution, and the null constraint p olynomial F m,n ℓ (∆) comes from expanding the crossing vector F ∆ ,ℓ ( u, v ), expressible in terms of a linear combination of P a,b ℓ (∆) with diﬀeren t a and b , whose coeﬃcients are themselv es p olynomials in ∆ ϕ . Constrain ts from unitarity Unitarit y implies that all OPE co eﬃcien ts λ ϕϕ O are real, and hence λ 2 ϕϕ O ≥ 0 . (2.12) In a self-dual kinematic conﬁguration, the conformal blo c ks are also non-negative, whic h guaran tees a p ositiv e measure. Moreov er, unitarity imposes the follo wing lo wer bounds on the scaling dimension ∆ of a primary op erator with spin ℓ : ∆ ≥      d − 2 2 , ℓ = 0 , ℓ + d − 2 , ℓ > 0 . (2.13) Bounding unnormalized momen ts W e now ask ho w small and how large the momen t v ariables can b e. Mathematically , this amounts to extremizing a momen t sub ject to the constrain ts of crossing symmetry and unitarit y . Let us write the rational functions in ( 2.11 ) as N m,n ℓ (∆) = F m,n ℓ (∆) /P 0 , 0 ℓ (∆), then the problem of maximizing an arbitrary linear combination of the unnormalized moments can b e written as Maximize ν = X ℓ Z ∞ ∆ min ( ℓ ) d ∆ p ℓ (∆) V ℓ (∆) sub ject to X ℓ Z ∞ ∆ min ( ℓ ) d ∆ p ℓ (∆) N ( α ) ℓ (∆) = b ( α ) , p ℓ (∆) ≥ 0 , (2.14) where the function V ℓ (∆) deﬁnes the ob jectiv e function, whic h is for example c hosen to b e ∆ m ℓ n in order to b ound ν m,n as deﬁned in ( 2.5 ). W e use the lab el α as a shorthand for those ( m, n ) pairs that pro duce indep enden t crossing constraints. One ma y regard the problem as an inﬁnite-dimensional linear pr ogram, with a v ariable p ℓ (∆) for eac h ph ysical op erator sub ject to the unitarity constraint. T o transform the problem into a form solv able by SDPB , we consider its dual program Minimize  b ·  z sub ject to  z ·  N ℓ (∆) − V ℓ (∆) ≥ 0 ∀ ∆ ≥ ∆ min ( ℓ ) and ℓ ∈ S , (2.15) where the vector dot product denotes a con traction of the index α , and S the included set of spins of the exc hanged op erators. F or identical external scalars, it is the collection of non-negativ e ev en num b ers (prior to an y truncation). One can easily chec k, by integrating – 7 – the inequality constraint ( 2.15 ) against the p ositiv e spectral density , that an y dual feasible solution ( y ,  z ) pro vides a rigorous upp er b ound on the momen t ν : ν = X ℓ ∈S Z ∞ ∆ min ( ℓ ) d ∆ p ℓ (∆) V ℓ (∆) ≤  z · X ℓ ∈S Z ∞ ∆ min ( ℓ ) d ∆ p ℓ (∆)  N ℓ (∆) =  b ·  z . Therefore, by minimizing  b ·  z one obtains the strongest upp er b ound on ν . In practice, one needs to truncate the set S to a ﬁnite size by imp osing a spin truncation, ℓ max , and consider the crossing constraints up to a ﬁnite deriv ativ e order m + n ≤ Λ. Imp osing a gap assumption corresp onds to adjusting ∆ min . Finally , we note that one can also b ound momen ts deﬁned as w eigh ted a v erages ov er op erators of a ﬁxed spin, ℓ ∗ . This amoun ts simply to mo difying the function V ℓ (∆) to V ℓ (∆) δ ℓ,ℓ ∗ , and the p olynomial program b ounding the momen t follows straightforw ardly: Minimize  b ·  z sub ject to  z ·  N ℓ (∆) − V ℓ (∆) δ ℓ,ℓ ∗ ≥ 0 ∀ ∆ ≥ ∆ min ( ℓ ) and ℓ ∈ S . (2.16) W e will denote these ﬁxed-spin momen ts by ν ( ℓ ∗ ) k = Z ∞ ∆ min ( ℓ ∗ ) d ∆ p ℓ ∗ (∆) V ℓ ∗ (∆) . (2.17) Bounding normalized moments Our goal is to b ound the ratios of the momen t v ari- ables by solving optimization problems of the form Minimize P ℓ R ∞ ∆ min( ℓ ) d ∆ p ℓ (∆) V (1) ℓ (∆) + β P ℓ R ∞ ∆ min ( ℓ ) d ∆ p ℓ (∆) V (0) ℓ (∆) + α sub ject to X ℓ Z ∞ ∆ min ( ℓ ) d ∆ p ℓ (∆) N ( α ) ℓ (∆) = b ( α ) . (2.18) Here we require that the denominator b e p ositiv e and include the constan ts α and β to accoun t for any p ossible v acuum contribution. F or example, setting α = β = 0 and V (0) ℓ (∆) = 1 repro duces the normalized moments of ( 2.6 ) without the iden tit y op erator. No w, w e introduce the auxiliary v ariables ˜ p ℓ (∆) ≡ p ℓ (∆) P ℓ R ∞ ∆ min ( ℓ ) d ∆ ′ p ℓ (∆ ′ ) V (0) ℓ (∆ ′ ) + α , t ≡ 1 P ℓ R ∞ ∆ min ( ℓ ) d ∆ ′ p ℓ (∆ ′ ) V (0) ℓ (∆ ′ ) + α , (2.19) so that the problem b ecomes a standard linear program: Minimize X ℓ ∈S Z ∞ ∆ min ( ℓ ) d ∆ ˜ p ℓ (∆) V (1) ℓ (∆) + β t sub ject to X ℓ ∈S Z ∞ ∆ min ( ℓ ) d ∆ ˜ p ℓ (∆)  N ℓ (∆) = t  b, X ℓ ∈S Z ∞ ∆ min ( ℓ ) d ∆ ˜ p ℓ (∆) V (0) ℓ (∆) + α t = 1 , ˜ p ℓ (∆) ≥ 0 , t ≥ 0 , (2.20) – 8 – whose dual is Maximize − y sub ject to V (1) ℓ (∆) + y V (0) ℓ (∆) +  z ·  N ℓ (∆) ≥ 0 ∀ ∆ ≥ ∆ min ( ℓ ) and ℓ ∈ S , β + αy −  b ·  z ≥ 0 , (2.21) whic h is a p olynomial program that can b e solved using SDPB . Sp ectrum extraction Having cast the problem of bounding the momen ts into an SDPB - solv able form, we no w turn to sp ectrum extraction. At the optimal solution, the extremal sp ectrum follo ws from the complementary slac kness conditions. This pro cedure is describ ed in detail in [ 3 ] and is implemen ted in the built-in spectrum program of SDPB . F or unnor- malized moments, spectrum outputs the scaling dimensions, spins, and the squared OPE co eﬃcien ts (up to the prefactor app earing in the rational approximation ( 2.10 )). F rom these, one directly reconstructs the sp ectral densit y deﬁned in ( 2.3 ). F or normalized mo- men ts, spectrum instead returns the normalized probabilities ˜ p ℓ (∆) introduced in ( 2.19 ). T o recov er the actual sp ectral density , one m ultiplies ˜ p ℓ (∆) b y the v alue of the correla- tor, t − 1 , which can b e ﬁxed by enforcing any one of the null constraints in ( 2.20 ). The normalized OPE co eﬃcien ts are deﬁned in a similar wa y: ˜ λ ϕϕ O = √ tλ ϕϕ O . (2.22) 2.3 Summary of results Figures 1a , 1b , and 1c presen t our main n umerical results, showing the allo wed regions for the ﬁrst normalized moment v ariable, ⟨ ∆ ⟩ ∗ = ν 1 ν 0 − 1 , (2.23) deﬁned excluding the identit y op erator, across v arious spatial dimensions. The geometry of the moment v ariable exhibits a remark ably rich structure, where b oth the upp er and lo w er b ounds displa y pronounced nonlinear features, sho wing that the space of consistent momen t v alues is far from trivial. The dashed curves in the plots indicate the moments computed from generalized free ﬁeld (GFF) correlators. As the external dimension ∆ ϕ increases, the numerical b ounds rapidly conv erge tow ards linear tra jectories, whic h coincide with the GFF correlators at leading order. Section 3 analyzes this b eha vior in the large-∆ ϕ limit and demonstrates agreemen t b et ween the numerical b ounds and analytical results derived in the heavy limit. In t wo and three dimensions, sharp kinks are clearly visible in the upp er b ounds, lo cated precisely at the external dimensions corresp onding to the critical Ising mo del. Section 4 provides a detailed study of these points, including the leading momen ts and the sp ectral in terpretation of these kinks that c haracterize the Ising CFTs. Finally , in the low er b ounds, we identify sharp turning p oin ts that p ersist across di- mensions, forming tw o new contin uous families of kinks! Section 5 is devoted to exploring the origin of these features in terms of the corresp onding extremal spectra, whic h includes the decoupling of op erators along the b oundaries and an analysis of p ossible connections to the fake-primary eﬀect [ 34 , 62 ]. – 9 – (a) d = 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2 4 6 8 Δ ϕ 〈Δ〉 * n = 1 n = 2 n = 10 Allowed regi on: Δ gap ≥ d - 2 2 ( or Δ gap ≥ Δ ϕ ) Allowed regi on: Δ gap ≥ 2 Δ ϕ 2d Isi ng mode l 〈σσ σσ〉 2d Isi ng mode l 〈ϵ ϵ ϵ ϵ 〉 GFF < ϕ n ϕ n ϕ n ϕ n > 2 Δ ϕ Free scala r theo ry Cosin e ver tex ope rator 〈 CCC C 〉 with C = V α + V - α 2 (b) d = 3 0.5 1.0 1.5 2.0 2.5 3.0 3.5 2 4 6 8 10 Δ ϕ 〈Δ〉 * n = 1 n = 2 n = 4 Allowed regi on: Δ gap ≥ d - 2 2 Allowed regi on: Δ gap ≥ Δ ϕ Allowed regi on: Δ gap ≥ 2 Δ ϕ 3d Isi ng mode l < σσσ σ > 3d Isi ng mode l < ϵ ϵ ϵ ϵ > GFF < ϕ n ϕ n ϕ n ϕ n > 2 Δ ϕ Free scala r theo ry (c) d = 4 1.0 1.5 2.0 2.5 3.0 3.5 4.0 2 4 6 8 10 Δ ϕ 〈Δ〉 * n = 1 n = 2 n = 3 Allowed regi on: Δ gap ≥ d - 2 2 Allowed regi on: Δ gap ≥ Δ ϕ Allowed regi on: Δ gap ≥ 2 Δ ϕ GFF < ϕ n ϕ n ϕ n ϕ n > 2 Δ ϕ Free scala r theo ry Figure 1 : Bounds on the ﬁrst normalized momen t (excluding the iden tity) as a function of the external scaling dimension, computed at Λ = 23 (see app endix A for other truncation parameters). The shaded regions show the allow ed moment space under diﬀeren t gap assumptions, with dark er shades indicating stronger gaps. The solid p oin ts corresp ond to moments extracted from ⟨ σ σ σ σ ⟩ and ⟨ ϵϵϵϵ ⟩ in the Ising mo del, while the black dashed lines indicate GFF solutions. The upp er b ounds are insensitive to the gap c hoice. See app endix B for details on the minimal mo del and vertex op erator moment computations. – 10 – 3 Momen ts in the hea vy correlator limit T o demonstrate the p o wer of the n umerical moment b o otstrap, in this section w e will study the hea vy correlator limit, a regime in whic h standard gap-maximizing functionals con verge extremely slo wly in the numerical b ootstrap. In contrast, our numerical momen t b ounds sho w excellent agreement with those analytically computed at leading order in large ∆ ϕ . W e then pro ceed with a sp ectral reconstruction of the op erator distribution obtained from the numerical moment b ounds. 3.1 Momen ts of hea vy correlators In the previous w ork [ 36 ], the authors deriv ed th e follo wing t wo sided b ounds on normalized momen ts in the limit of large external dimension ∆ ϕ : 2 k 2 ≤ lim ∆ ϕ →∞ ⟨ ∆ k ⟩ ∆ k ϕ ≤ 2 3 k 2 − 1 . (3.1) The correlators whose moments saturate these b ounds are giv en b y four-p oin t functions of normal ordered pro ducts of elemen tary ﬁelds φ n in a generalized free theory . The low er b ound is saturated b y the case where ∆ φ is held ﬁnite while n → ∞ , resulting in the total external dimension ∆ ϕ = ∆ φ n b ecoming inﬁnite, while the upp er b ound is saturated by the case of ∆ φ → ∞ with n = 1. In the ligh t correlator regime, the standard numerical b ootstrap provides rigorous b ounds through SDP , which con verge to known solutions suc h as the critical Ising mo del. A natural question is: c an the numeric al moment b o otstr ap eﬀe ctively pr ob e the he avy c orr elator r e gime? It turns out that the n umerical momen t bo otstrap performs strikingly w ell: the b ounds con v erge rapidly to ward the analytic pow er la ws, even deep in the hea vy correlator regime. The reason for this is that moment v ariables are not sensitiv e to the precise hea vy op erator sp ectrum and instead only prob e the collectiv e b eha vior of op erator con tributions. The fact that Λ deriv ativ es acting on the conformal blo c k pro duce an approximate rational function in ∆, which tends to w ards a p olynomial of degree Λ at large v alues of exchanged dimension, means that to accurately compute the asymptotic b eha vior of ⟨ ∆ n ⟩ moments for n ≤ Λ, one needs to only use deriv ative functionals up to order Λ, where the precise b ounds in the light correlator regime are corrected by including higher deriv atives in the functional basis. T o quan tify this b eha vior, w e examine in ﬁgure 2 the leading normalized momen ts obtained from the numerical b ootstrap in spatial dimension d = 3. The analytic p o wer-la w b ounds in ( 3.1 ) are sho wn as colored dashed lines for comparison. While w e n umerically b ound the moments in d = 3, similar b eha vior can also b e observ ed in other dimensions in the heavy correlator limit. F or the ﬁrst moment, the asymptotic upp er and low er bounds coincide at leading order, b oth given by √ 2 ∆ ϕ + O (1). In this case, one can derive an even stronger lo wer b ound b y assuming a scalar gap of ∆ gap ≥ d − 2 in d ≤ 4: √ 2 ∆ ϕ − d 8 (3 √ 2 − 8) ν 0 − 1 ν 0 ≤ ⟨ ∆ ⟩ ≤ √ 2 ∆ ϕ . (3.2) – 11 – 5 10 15 20 0.6 0.8 1.0 1.2 1.4 Δ ϕ 〈Δ〉 / Δ ϕ Allowed regi on: Δ gap ≥ d - 2 2 Allowed regi on: Δ gap ≥ d - 2 Asym ptoti c uppe r boun d: 2 k 2 = 2 Asym ptoti c lower boun d: 2 - 3 8 ( 3 2 - 4 ) 1 Δ ϕ Empi ric al ﬁ tted cur ve: 2 - 9 2 ( 3 2 - 4 ) 1 Δ ϕ 5 10 15 20 1 2 3 4 5 Δ ϕ 〈Δ 2 〉 / Δ ϕ 2 Allowed regi on: Δ gap ≥ d - 2 2 Asym ptoti c lower boun d: 2 k 2 = 2 Asym ptoti c uppe r boun d: 2 3 k 2 - 1 = 4 5 10 15 20 5 10 15 Δ ϕ 〈Δ 3 〉 / Δ ϕ 3 Allowed regi on: Δ gap ≥ d - 2 2 Asym ptoti c lower boun d: 2 k 2 = 2 3 / 2 Asym ptoti c uppe r boun d: 2 3 k 2 - 1 = 2 7 / 2 Figure 2 : Tw o-sided bounds on the ﬁrst three normalized moments including the iden tit y op erator as a function of the external scaling dimension ∆ ϕ in d = 3, computed at Λ = 23. The vertical axes are normalized by ∆ k ϕ , so that the asymptotic analytic b ounds in ( 3.1 ) app ear as straigh t horizontal lines. – 12 – 1.0 1.5 2.0 2.5 3.0 2 4 6 8 10 12 14 Δ ϕ 〈Δ〉 * or Δ gap Gap maxi miz ation: Λ = 5 Gap maxi miz ation: Λ = 9 Gap maxi miz ation: Λ = 13 〈Δ〉 * max imi zation : Λ = 5 〈Δ〉 * max imi zation : Λ = 9 〈Δ〉 * max imi zation : Λ = 13 2 2 Δ ϕ Figure 3 : Gap maximization compared to moment maximization (without the iden tit y con tribution) at the same deriv ativ e order. The momen t b ounds conv erge rapidly tow ard the linear tra jectory 2 √ 2 ∆ ϕ and repro duce the correct asymptotic scaling at v ery lo w deriv ativ e order. In contrast, gap maximization con verges muc h more slo wly and fails to pro duce an y b ound at large external scaling dimension. Although the tw o pro cedures optimize diﬀeren t observ ables, this comparison illustrates that, in the hea vy correlator regime, the moments provide more natural observ ables for the b o otstrap. Under this additional assumption, the n umerical low er b ounds shrink signiﬁcan tly , leaving a narrow region that approac hes the asymptotic curve ⟨ ∆ ⟩ → √ 2 ∆ ϕ − d 8 (3 √ 2 − 4) (3.3) at large ∆ ϕ , in excellent agreement with the analytic low er b ounds. When instead assuming the most conserv ative scalar gap from unitarity , i.e., ∆ gap ≥ ( d − 2) / 2, the deriv ation of ( 3.2 ) no longer applies. Indeed, the n umerical results lie b elo w that curv e, reﬂecting a signiﬁcan t contribution from the ligh test scalar op erator. In terestingly , the lo w er bound in d = 3 can still b e captured very w ell b y the empirical ﬁt ⟨ ∆ ⟩ ∆ ϕ ≈ √ 2 − 9 2 (3 √ 2 − 4) 1 ∆ ϕ . (3.4) It w ould b e interesting to derive this analytically . Examining the extremal sp ectra along the lo w er b oundary reveals an interesting phenomenon: b ey ond a certain ∆ ϕ , the ligh test scalar op erator actually b egins to saturate the unitarit y b ound. W e will return to this observ ation in section 5 , where we discuss its connection to the fak e-primary eﬀect. 3.2 Comparison to gap maximization The n umerical moment b ootstrap p erforms remark ably well in the hea vy correlator regime. Here w e mak e a brief comparison b et ween maximizing the ﬁrst normalized moment (with – 13 – the identit y remov ed), ⟨ ∆ ⟩ ∗ , and the conv entional gap-maximization approach. The opti- mal gaps and moments, obtained at the same deriv ativ e orders, are shown in ﬁgure 3 . As seen in the plot, the optimal gap b ounds at diﬀerent deriv ativ e orders conv erge reasonably w ell at small external scaling dimension, follo wing an appro ximately linear tra- jectory . How ever, at any ﬁxed deriv ativ e order Λ, the gap-maximization bounds deteriorate as the correlator b ecomes increasingly hea vy: b ey ond a suﬃcien tly large external weigh t, the metho d ceases to pro duce meaningful b ounds. In con trast, the momen t is alw ays b ounded ev en at an extremely low deriv ative order, suc h as Λ = 3, 3 and con v erges remark ably rapidly to ward the linear tra jectory 2 √ 2∆ ϕ . This illustrates a key qualitativ e diﬀerence: b oth conceptually and numerically , moment v ariables provide a natural and eﬀective language for the conformal b ootstrap in the heavy correlator regime. Rather than fo cusing on individual op erators, it is more informative to c haracterize the av erage prop erties of the full CFT sp ectrum. 3.3 Max-en trop y reconstruction Wh y do es the n umerical b o otstrap p erform so w ell in the heavy correlator regime? A natural ﬁrst attempt is to examine the extremal sp ectra directly . How ever, the extremal sp ectrum b y itself do es not provide a meaningful reconstruction of the OPE measure: it only sp eciﬁes the p ositions of ﬁnitely many op erators, whose n umber can grow at most linearly with Λ, while correlators at large external w eight are gov erned b y the collective con tribution of many states. Indeed, when we insp ect the extremal sp ectra extracted from the n umerical b ounds, the individual op erator lo cations app ear scattered, with no ob vious Gaussian proﬁle of the kind predicted analytically , which is not surprising. The hea vy limit concerns the coarse-grained structure of the OPE measure, and suc h structure is inheren tly in visible at the level of individual states. A key observ ation, ho wev er, comes to the rescue: at eac h external weigh t ∆ ϕ , the upp er b ounds of the leading momen ts are all saturated b y the same extremal spectrum. This is in sharp con trast to the low er b ounds on the leading momen ts, whose extremal sp ectra diﬀer almost ev erywhere. Because the upp er b ounds at a ﬁxed ∆ ϕ share the same sp ectrum, it b ecomes meaningful to attempt an inv ersion: given the ﬁrst few optimal moments of a single underlying me asur e, c an we r e c onstruct the me asur e itself ? This leads us to a version of the classical moment problem, which concerns the existence and uniqueness of a p ositiv e measure compatible with a given set of momen ts. In our setting, ho w ev er, w e only hav e access to a ﬁnite n umber of leading moments. As a result, the reconstruction problem is intrinsically underdetermined: man y distinct p ositiv e measures are compatible with the same ﬁnite set of moments, unless sp ecial degeneracies suc h as a ﬂat extension o ccur. This lack of uniqueness, nevertheless, do es not imply that a meaningful sp ectral re- construction is imp ossible, but instead reﬂects the inherently coarse-grained nature of the hea vy correlator regime. Motiv ated by the observ ation of [ 36 ] that, in the ∆ ϕ → ∞ limit, 3 While an upp er b ound on ⟨ ∆ ⟩ can b e obtained ev en at Λ = 1 [ 36 ], it requires a sligh tly larger Λ to obtain an upp er b ound on ⟨ ∆ ⟩ ∗ . – 14 – ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 Δ ϕ - 1 Δ ρ norm ( Δ ϕ - 1 Δ ) Max - entr opy rec onstr ucti on: Δ ϕ = 1.2 Max - entr opy rec onstr ucti on: Δ ϕ = 6 Max - entr opy rec onstr ucti on: Δ ϕ = 2 ● Gene rali zed fre e ﬁ eld : Δ ϕ = 1.2 ■ Gene rali zed fre e ﬁ eld : Δ ϕ = 6 ▲ Gene rali zed fre e ﬁ eld : Δ ϕ = 20 Asym ptoti c saddl e point locat ion: 2 2 Figure 4 : The maxim um-entrop y reconstruction of the OPE measure (smo oth solid curv es) compared to the exact discrete measure of the generalized free ﬁeld correlator ⟨ ϕϕϕϕ ⟩ . Diﬀerent colors represent diﬀeren t external scaling dimensions ∆ ϕ . The OPE measure is normalized to in tegrate to unity against ∆ − 1 ϕ ∆. double-t wist families in GFF correlators approach Gaussian (maximum-en tropy) distribu- tions determined b y their mean and v ariance, w e adopt the maxim um-entrop y principle as a ph ysically motiv ated prescription to select the simplest represen tative among all measures compatible with the known moments. This approach allows us to systematically study ﬁnite-∆ ϕ corrections to this universal structure. In app endix D w e describ e the details of ho w this is implemented. Figure 4 compares the resulting maximum-en tropy reconstruction based on the ﬁrst ﬁv e moments with the exact discrete OPE measure of the ⟨ ϕϕϕϕ ⟩ generalized free ﬁeld. The reconstructed density sho ws clear conv ergence tow ard the generalized free ﬁeld OPE distribution as w e approach the heavy limit. At the same time, the numerics rev eal sys- tematic deviations from a p erfect Gaussian, capturing next-to-leading-order corrections to the upp er b ounds b ey ond the leading analytic asymptotic appro ximation. F rom the numerical p ersp ectiv e, as established in section 2 , the moment v ariables are sp eciﬁc linear combinations of deriv ativ e op erators acting on the correlator that yield w eigh ted av erages of conformal data. Y et, as demonstrated here, they exhibit b eha vior fundamen tally diﬀeren t from standard functionals that extremize a single op erator’s prop- ert y . The moments eﬀectively “sense” where the dominant collective contribution from a large n umber of states is concentrated, and repro duce the correct coarse-grained behavior using only a ﬁnite num b er of Dirac delta functions in the extremal sp ectral density . This remark able feature makes moment v ariables not only an eﬃcient to ol for discov ering light correlator CFTs, but also a p o w erful prob e ev en in the heavy correlator regime. Finally , as is visible in the plots, the low er b ounds of the moments display extremely ric h structures, including sharp turning p oin ts at smaller v alues of ∆ ϕ . These patterns are particularly in triguing: instead of carving out theories by maximizing a gap, here we are minimizing the moments, essentially probing the opp osite side of the b o otstrap landscap e. – 15 – Understanding these minima requires taking into accoun t the dynamics of extremely light op erators, leading us into a region of the bo otstrap that, to our knowledge, remains largely unexplored. Section 5 is dev oted to analyzing these new bo otstrap structures through their extremal moments and sp ectra. 4 The Ising moments The 3d Ising mo del has play ed a central role in studies of the n umerical conformal b oot- strap. In the work of [ 1 ], the critical Ising mo del was iden tiﬁed as a sharp kink on the b oundary of the allo w ed theory space, corresp onding to gap maximization of the ϵ op er- ator in the σ × σ OPE. In this section, we revisit the b ootstrap for the 3d Ising mo del in the language of momen t v ariables. These quantities capture coarse-grained features of the CFT sp ectrum and pro vide an alternative geometric p ersp ectiv e of theory space. W e ﬁnd that, in the space of low-lying momen ts, the upp er b ounds on these observ ables also dev elop a sharp kink precisely at the 3d Ising v alue. W e present the numerical results, dra w connections to kno wn extremal prop erties of the Ising spectrum, and ﬁnally compare our b est determination of the momen ts with those computed from previous Ising data. 4.1 Correlator b ounds Correlator b ounds in d = 3 were previously examined in [ 27 ], whic h derived t wo-sided b ounds and analyzed their dep endence on the gap assumption. In particular, they iden tiﬁed a sharp kink in the low er bound of the correlator, which corresponds to the 3d Ising mo del, and related correlator extremization to the gap maximization. In our framework, the correlator is exactly the zeroth unnormalized momen t, and we repro duce their b ounds as sho wn in ﬁgure 5 . In the heavy correlator limit, the extremal solutions are saturated by the generalized free ﬁeld correlators. 4.2 Bounds on the ∆ -momen ts It is not only the zeroth momen t that places kno wn theories on the b oundary of the allo wed region. Bounds on the higher moments of scaling dimension reveal additional geometric features, including the sharp corners asso ciated with the Ising mo del. Here, w e consider the ﬁrst normalized ∆-moment without the inclusion of the identit y op erator, ⟨ ∆ ⟩ ∗ , with t w o-sided b ounds shown in ﬁgure 6 . A sharp kink is lo cated at ∆ ϕ ≈ 0 . 518. Similarly , the second moment ⟨ ∆ 2 ⟩ ∗ exhibits a kink at the same lo cation. The emergence of the kink in the leading ∆-moments is in tuitive. F or ligh t correlators, the extremal solution is dominated b y the gap state, whic h in this case corresp onds to the ϵ op erator in the σ × σ OPE. Since the 3d Ising mo del maximizes the gap and sits precisely at the kink of the gap-maximization plot, the ∆-moment b ounds naturally reﬂect this prop ert y . F or reference, ﬁgure 7 sho ws the gap-maximization results at the same deriv ativ e order, allowing for a direct comparison b et ween the tw o approaches. This pattern p ersists for higher moments, although the kink becomes less sharp, as hea vier op erators contribute more signiﬁcan tly to the w eighted av erages. Plots for momen ts across diﬀerent dimensions can b e found in app endix C . – 16 – 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1 2 3 4 5 Δ ϕ ν 0 Allowed regi on: Δ gap ≥ d - 2 2 = 1 2 Allowed regi on: Δ gap ≥ 5 3 Δ ϕ Allowed regi on: Δ gap ≥ 2 Δ ϕ 3d Isi ng GFF Figure 5 : Repro duction of the correlator b ounds in d = 3 as a function of the external scaling dimension ∆ ϕ , following [ 27 ]. The sharp leftmost edge, where all b ounds con verge, corresp onds to the free theory , while the red dot on the low er b ound marks the 3d Ising correlator, computed from the conformal data of [ 3 ]. The bounds w ere obtained at Λ = 11. 0.50 0.51 0.52 0.53 0.54 0.55 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Δ ϕ 〈Δ〉 * Allowed regi on: Λ = 11 Allowed regi on: Λ = 19 GFF 3d Isi ng Figure 6 : Upp er and lo wer bounds on the ﬁrst normalized moment ⟨ ∆ ⟩ ∗ = ν 1 / ( ν 0 − 1) at Λ = 11 , 19. The Ising mo del (red dot) sits at the kink. 4.3 Bounds on the ℓ -momen ts In complete analogy with the previous analysis, one can also b ound the moment v ariables asso ciated with spin. Figure 8 shows tw o-sided b ounds on the ﬁrst normalized ℓ -moment, ⟨ ℓ ⟩ ∗ = ν 0 , 1 / ( ν 0 , 0 − 1). Intriguingly , m uch like the gap-maximization and ∆-momen t bounds, the uppe r b ound dev elops a sharp kink precisely at the 3d Ising point. The strictly p ositiv e lo w er b ound also pro ves the necessary existence of spinning op erators in the OPE. The app earance of the kink in the ℓ -moment upp er bound is interesting, as it has no direct analogue in the conv entional gap-maximization analysis. T o gain further insight, let us instead examine the unnormalized moment ν 0 , 1 , whose tw o-sided b ounds are sho wn – 17 – 0.50 0.51 0.52 0.53 0.54 0.55 1.0 1.1 1.2 1.3 1.4 1.5 Δ σ Δ ϵ Allowed regi on: Λ = 11 Allowed regi on: Λ = 19 GFF 3d Isi ng Figure 7 : Gap maximization at Λ = 11 , 19. The Ising model (red dot) sits at the kink. 0.50 0.51 0.52 0.53 0.54 0.55 0.15 0.20 0.25 0.30 Δ ϕ 〈ℓ〉 * Allowed regi on: Λ = 11 Allowed regi on: Λ = 19 GFF 3d Isi ng Figure 8 : Two-sided b ounds on the ﬁrst normalized momen t of spin, ⟨ ℓ ⟩ = ν 0 , 1 / ( ν 0 , 0 − 1), excluding the identit y op erator con tribution. The upp er b ound features a sharp kink precisely at the 3d Ising p oin t. in ﬁgure 9 . The fact that the Ising mo del maximizes the spin moment is intuitiv e: since ν 0 , 1 receiv es contributions exclusiv ely from spinning op erators, the stress tensor provides the dominan t con tribution. The moment v ariable therefore attains its maxim um when the stress tensor couples most strongly to the system, i.e., when the cen tral charge reaches its minimal v alue. The formation of the kink, ho wev er, reﬂects a c haracteristic phenomenon of op erator decoupling, closely tied to the underlying equation of motion in the Wilson–Fisher theory . As observ ed in [ 2 ], the scaling dimensions of the spin-2 op erators in the extremal sp ectra undergo a sudden rearrangement up on crossing the Ising p oin t. The momen t- maximizing sp ectra exhibit the same b eha vior, giving rise to the kink observed in the b ounds. – 18 – 0.50 0.51 0.52 0.53 0.54 0.55 0.18 0.19 0.20 0.21 0.22 Δ ϕ ν 0,1 Allowed regi on: Λ = 19 Allowed regi on: Λ = 23 GFF 3d Isi ng Figure 9 : Allow ed region for the ﬁrst unnormalized moment of spin, ν 0 , 1 . Again, a kink can b e seen close to the 3d Ising p oin t. The lo wer bound is saturated b y the corresp onding momen t of the generalized free ﬁeld theory correlator ⟨ ϕϕϕϕ ⟩ . 4.4 Precise moments under a gap assumption Finally , we presen t our most precise n umerical b ounds on the unnormalized moments ob- tained under a ﬁxed external scaling dimension and scalar gap, ∆ σ = 0 . 5181488 , ∆ ϵ = 1 . 412625 , (4.1) in 3d, corresp onding to the current b est estimate of the scaling dimensions of the relev an t op erators in the 3d Ising CFT [ 5 ]. This aggressive gap assumption pushes the b o otstrap solution v ery close to the b oundary of the allow ed theory space, so that the resulting momen ts are exp ected to b e tightly constrained around their true v alues. T able 1 summarizes the tw o-sided b ounds on the low est unnormalized momen ts ν m,n . As can b e seen, these bounds are indeed v ery tigh t, in some cases ev en excluding the v alues computed from the stable op erators in [ 3 ], whic h are highlighted in red. This is not unexp ected, since the list of stable op er ators in [ 3 ] is not exp ected to include all op erators con tributing to the moments. In particular, higher-twist tra jectories and unstable op erators ma y giv e non-negligible con tributions. Quite remark ably , our numerical moment b ounds, obtained at a signiﬁcan tly low er deriv ative order, are already sensitiv e enough to detect the eﬀects of these missing op erators. As w e ha ve seen, the momen t b ounds presen ted here resem ble the familiar results from gap maximization and c -minimization, yet they encapsulate b oth within a single, uniﬁed framew ork. How ever, they diﬀer in tw o crucial asp ects. 1. The momen t v ariables are deﬁned as w eigh ted a verages o ver the en tire op erator spec- trum. The resulting b ounds therefore do not merely constrain a single op erator, suc h as the gap state, but instead imp ose rigorous conditions on the collectiv e b eha vior of all op erators ab o ve the gap. The moment b o otstrap carves out the theory space b y characterizing the coarse-grained b eha vior of an inﬁnite n umber of op erators, and – 19 – Momen t Bounds Computed from [ 3 ] ν 0 , 0 0.768530(25) 0.7685483(69) ν 1 , 0 1.247745(15) 1.247736(21) ν 0 , 1 0.2004013(17) 0.200381(14) ν 2 , 0 2.2725170(20) 2.272380(91) ν 1 , 1 0.6255126(41) 0.625408(68) ν 0 , 2 0.4246333(36) 0.424550(54) ν ( ℓ =0) 0 0.671220(26) 0.6712437(60) ν ( ℓ =0) 1 0.949868(19) 0.949883(12) ν ( ℓ =0) 2 1.3482213(37) 1.3482174(31) ν ( ℓ =2) 0 0.0945027(28) 0.09450447(26) ν ( ℓ =2) 1 0.2836123(20) 0.28361131(80) ν ( ℓ =2) 2 0.8513774(44) 0.8513735(25) T able 1 : Two-sided b ounds on the leading unnormalized moments ν m,n and ﬁxed-spin momen ts ν ( ℓ ) k , obtained with truncation parameters Λ = 27, ℓ max = 50, r max = 60, and κ = 20. Momen ts whose b ootstrap b ounds do not ov erlap with the v alues reconstructed from the “stable op erators” in the spectrum of [ 3 ] are highlighted in red. remark ably , this can b e achiev ed numerically , providing rigorous statemen ts ab out the full sp ectrum even at ﬁnite deriv ative order. As w e hav e seen in section 3 , even in the heavy correlator regime, where the sp ectrum b ecomes dense, the n umerical momen t bounds remain well b eha ved and con tinue to yield meaningful constraints. 2. As is evident from the plots, the momen t v ariables are b ounded not only from ab o v e, but also from b elo w. As the external scaling dimension increases, the low er b ounds deviate progressiv ely from the generalized free ﬁeld solutions and dev elop ric h, highly non trivial structures, including previously unobserved kinks. Section 5 is dev oted to a detailed analysis of these new structures and their physical interpretation. 5 A tour of the momen t landscap e In the previous sections, w e hav e sho wn that momen t v ariables are an eﬃcien t wa y of prob- ing the space of hea vy correlators numerically and can also pro vide a useful c haracterization of the ⟨ σ σσ σ ⟩ correlator in the critical Ising model. While these results are already encour- aging, the true strength of the moment b o otstrap lies in its ability to uncov er previously unexplored structures in the space of solutions to the crossing equations. In this section, we guide the reader through the rich set of geometric features that c haracterize our momen t b ounds. Such structures arise only along the lower b ounds of the momen ts for light correlators, as can b e seen in ﬁgure 2 , long b efore the b ounds conv erge to the smo oth p o wer la ws predicted analytically in the heavy correlator limit. This places us in a regime where the p ole structure of the conformal blo c ks b ecomes imp ortan t, and the extremal sp ectra reorganize in highly non trivial wa ys. – 20 – 0 . 5 1 . 0 1 . 5 2 . 0 0 1 2 3 4 0 . 5 1.0 1.5 0 . 0 1 0 2 3 4 3 4 2 5 6 Figure 10 : The lo wer b ound on the normalized ﬁrst moment ⟨ ∆ ⟩ ∗ as a function of ( d, ∆ ϕ − d − 2 2 ), with a cutaw ay of the allow ed region cross-section at d = 3, computed at Λ = 23. Tw o contin uous families of kinks are clearly visible on the surface, corresp onding to the sharp turning features seen in the contour plot, ﬁgure 11 . The ﬁrst family forms a smo oth curv e from ( d, ∆ ϕ ) = (2 + , 1 2 ) to ( d, ∆ ϕ ) = (6 , 2), where it merges in to the free scalar theory . The second family originates from ( d, ∆ ϕ ) = (2 + , 3 2 ) and p ersists to ward higher dimensions. The features are not presen t in exactly d = 2, whic h we discuss in section 5.7 . As we will demonstrate, the geometric structures in the moment b ounds are closely related to several distinct features of the extremal sp ectrum, including • op erator decoupling of low-lying op erators, • extremization of scaling dimensions, • saturation of unitarity b ounds, and • evidence for an extremal sp ectrum that sim ultaneously extremizes m ultiple lo w-lying momen t v ariables. In the following sections, we take a guided tour of the moment landscap e revealed by the n umerical b ootstrap, fo cusing on the range 2 < d < 6 and on the relationship b et ween these structures and the sp ectral prop erties listed ab o ve. 5.1 Ov erview of geometric features By examining the low er b ounds on ⟨ ∆ ⟩ ∗ , i.e., the ﬁrst normalized moment without the in- clusion of the identit y , across a wide range of spatial dimensions, a global momen t landscap e emerges: the low er b ound traces out a non-trivial surface in the ( d, ∆ ϕ ) plane (ﬁgures 10 and 11 ), and we identify tw o families of kinks. As we will see shortly , they are closely related to discontin uous or extremizing features of the extremal sp ectra, in particular to decoupling even ts inv olving lo w-lying op erators. – 21 – 0.0 0.5 1.0 1.5 2.0 2.5 3.0 2 3 4 5 6 7 8 Δ ϕ - d - 2 2 d 0.54 1.44 2.34 3.24 4.14 5.04 5.94 6.84 Figure 11 : Contour plot of the same moment landscap e sho wn in ﬁgure 10 . The ﬁrst family of kinks (the “cliﬀ ”) can b e traced contin uously from its endp oin t at d → 2 + to the free scalar theory at d = 6. Across all dimensions 2 < d < 6, the low er b ound at ﬁxed d follo ws the same qualitativ e progression of geometric landmarks as ∆ ϕ increases: cliﬀ − → ﬁrst v alley − → hill − → second v alley | {z } merge into a single “crater” in d ≳ 4 − → asymptotically linear ramp . (5.1) The tw o families of kinks remain robust throughout the entire range 2 < d < 6, as can b e clearly traced in ﬁgures 10 and 11 , o ccurring near the “cliﬀ ” and again near the onset of the asymptotically linear ramp. The in termediate features, by contrast, are more malleable: in d ≲ 4, they app ear as tw o well-separated lo cal minima with a p eak betw een them, while in d ≳ 4 they merge in to a broad, single “crater” that eliminates the W-shaped lo w er b ound, with the tw o families tracing roughly the t wo sides of the crater. As d → 2 + , the lo w er b ounds are disc ontinuous : in exactly d = 2, the low er b ound sho ws no kinks and closely tracks the linear tra jectory √ 2 ∆ ϕ (ﬁgure 1a ). W e will return to this phenomenon in section 5.7 . T o discuss the sp ectral origin of the b ootstrap landscap e, it is conv enient to zo om in on a representativ e slice. Figure 12 displa ys the low er b ound on ⟨ ∆ ⟩ ∗ in d = 3, where the W-shap ed proﬁle is stable under increases in the deriv ativ e order, Λ, and the spin truncation, ℓ max , indicating that the features are genuine consequences of the b o otstrap equations rather than numerical artifacts. Unless stated otherwise, the momen t b ounds – 22 – and extremal sp ectra in the follo wing subsections are obtained in d = 3; deformations of these structures with the spatial dimension will b e discussed explicitly where relev ant. Figure 12 : Low er b ound on the normalized ﬁrst momen t ⟨ ∆ ⟩ ∗ in d = 3. The W-shap ed b ound consists of a sharp “cliﬀ ” follow ed b y t wo “v alleys” separated by an in termediate “hill.” The qualitative structure and lo cations of these features remain stable under in- creases in the deriv ative order Λ and the spin truncation ℓ max . 5.2 Sp ectral reorganization b efore the “cliﬀ ” Although the momen t b ound presented earlier exhibits dramatic nonlinear b eha vior only after the “cliﬀ,” signiﬁcan t spectral reorganization driv en b y the lightest scalar has already o ccurred b eforehand. The extremal sp ectra in a representativ e slice, d = 3, are shown in ﬁgure 13 , where w e plot the scaling dimensions and (normalized) OPE co eﬃcien ts of the ligh test scalar and spin-2 op erators as functions of the external scaling dimension ∆ ϕ . As we mov e a wa y from the free scalar p oin t, a light scalar op erator b egins to contribute to the OPE with a v ery small co eﬃcien t. Let us denote this op erator b y φ . It remains in the sp ectrum until ∆ ϕ ≈ 0 . 514, where its OPE co eﬃcien t drops to zero and the op erator decouples. Bey ond this point, the gap state is replaced by a scalar whose scaling dimension closely follo ws the GFF tra jectory ∆ ≈ 2∆ ϕ , which we sc hematically denote by ϕ 2 . At ∆ ϕ ≈ 0 . 62, in terestingly , the ligh t scalar φ reapp ears in the sp ectrum. F rom there on w ard, its scaling dimension gradually approaches the unitarity b ound, and its con tribution to the OPE ev en tually b ecomes dominan t. Deformation across dimensions The decoupling phenomenon describ ed abov e can b e seen only within a ﬁnite window of spatial dimensions, 2 . 6 ≲ d ≲ 4 . 9. W e summarize the global structure in ﬁgure 14 , where regions with diﬀeren t colors corresp ond to extremal sp ectra with qualitativ ely diﬀerent scalar op erator conten t. – 23 – Within this window, there exists a sp ecial region that w e refer to as the “p ond”, where the scalar gap state is the ϕ 2 op erator instead of the ligh t scalar φ . Within our n umerical resolution, the “p ond” app ears to touc h the free scalar theory at d = 4, and is b ounded within the interv al 2 . 6 ≲ d ≲ 4 . 9. F or d ≥ 5, we do not observe any op erator decoupling until the “cliﬀ.” Instead, the lo w-lying conformal data exhibit qualitativ ely diﬀerent b eha vior, pro ducing another sharp corner in the moment b ound, as shown in ﬁgure 15 . W e will return to these structures in the next section. 5.3 The cliﬀ Op erator decoupling and “gap-minimization” A t the “cliﬀ ”, lo cated at ∆ ϕ ≈ 0 . 908 in d = 3, the second-lightest scalar suddenly decouples from the extremal sp ectrum, as sho wn in ﬁgure 13 . This decoupling is accompanied by a sharp drop in its OPE co eﬃ- cien t and thus a suppression of its contribution to the sp ectral density ( 2.3 ). Mean while, the lightest scalar has a scaling dimension very close to the unitarity b ound, d − 2 2 , which mak es the corresp onding conformal block extremely large, con tributing signiﬁcantly to the sp ectral densit y . T ogether, these eﬀects pro duce the sudden drop in the moment. An interesting feature of the extremal sp ectra is that the scaling dimension of the decoupled op erator lies extremely close to 2∆ ϕ , whic h is the v alue of the generalized free ﬁeld double-trace op erator [ ϕϕ ] 0 , 0 at the same external weigh t. F urthermore, as shown in ﬁgure 20 , the already small anomalous dimension of the lightest scalar reac hes its lo- cal minimum at precisely the same lo cation, essentially “minimizing” the gap, revealing solutions on the opp osite side of the usual b ootstrap landscap e! Deformation across dimensions Interestingly , the “cliﬀ ” discussed abov e b elongs to a con tin uous family of kinks across spatial dimensions, and a notable even t o ccurs at d = 6. Here, this family collides with the free scalar theory p oin t at ∆ ϕ = 6 − 2 2 = 2, terminating the family entirely , as can b e seen in ﬁgures 10 , 11 , and 14 . Near d = 6, b oth the anomalous dimensions and the OPE co eﬃcien ts of the lightest op erators exhibit sharp, discontin uous b eha vior as we approach the free scalar theory , as sho wn in ﬁgure 15 . Beyond d ≳ 5, we consisten tly see tw o distinguished corners along the lo w er b ound. The ﬁrst corner is not accompanied b y an op erator decoupling and coincides with a lo cal maxim um of the ligh test-scalar anomalous dimension and of the cen tral c harge. By con trast, the second corner belongs to the “cliﬀ ” family , which con tin uously connects to the op erator decoupling abov e at d = 3; there we ﬁnd a lo cal minimum of the lightest-scalar anomalous dimension and of the central charge. Finally , we comment on the op erator-decoupling even t in general dimensions, respon- sible for the “cliﬀ ” observ ed. As p oin ted out, the scaling dimension of the second-lightest scalar nearly sits at the generalized free ﬁeld v alue 2∆ ϕ at the cliﬀ, but it do es not coin- cide with this v alue uniformly across dimensions. W e observe a small p ositiv e anomalous dimension for d < 3 and a negative anomalous dimension for d > 3. – 24 – 0.6 0.7 0.8 0.9 1.0 1.1 1 2 3 4 5 6 Δ n ( ℓ = 0 ) vs. Δ ϕ 2 Δ ϕ 2 Δ ϕ + 2 2 Δ ϕ + 4 0.6 0.7 0.8 0.9 1.0 1.1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 λ n ( ℓ = 0 ) vs. Δ ϕ 0.6 0.7 0.8 0.9 1.0 1.1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 λ  n ( ℓ = 0 ) vs. Δ ϕ 0.6 0.7 0.8 0.9 1.0 1.1 3 4 5 6 7 8 Δ n ( ℓ = 2 ) vs. Δ ϕ 2 Δ ϕ + 2 2 Δ ϕ + 4 2 Δ ϕ + 6 0.6 0.7 0.8 0.9 1.0 1.1 0.0 0.5 1.0 1.5 2.0 λ n ( ℓ = 2 ) vs. Δ ϕ 0.6 0.7 0.8 0.9 1.0 1.1 0.0 0.2 0.4 0.6 0.8 1.0 λ  n ( ℓ = 2 ) vs. Δ ϕ Figure 13 : Extremal scaling dimensions and (normalized) OPE co eﬃcien ts (see section 2.2 for deﬁnition) of the leading scalar and spin-2 op erators in d = 3 as functions of ∆ ϕ , computed along the Λ = 23 lo w er b ound of the normalized ﬁrst moment. The blue, green, and red curves corresp ond to the lightest ( n = 1), second-lightest ( n = 2), and third- ligh test ( n = 3) op erators, resp ectiv ely . A t ∆ ϕ ≈ 0 . 514, a light scalar φ decouples. At ∆ ϕ ≈ 0 . 62, φ reappears and its scaling dimension subsequently drifts tow ard the unitarit y b ound. Near ∆ ϕ ≈ 0 . 91, corresp onding to the “cliﬀ ”, the second-lightest scalar decouples. A t ∆ ϕ ≈ 1 . 05, the ﬁrst “v alley” is reached, where the reconstructed correlator div erges, and so do the unnormalized OPE coeﬃcients. The normalized OPE co eﬃcien ts, ho wev er, remain p erfectly ﬁnite. – 25 – Figure 14 : A map of the “CFT park” illustrating the landscap e of the ligh test scalar op- erator spectrum prior to the ﬁrst kink family (the “cliﬀ ”). The colors indicate qualitativ ely distinct regions, with φ representing a ligh t scalar whose scaling dimension ev entually drifts to w ard the unitarity b ound, and ϕ 2 an op erator whose scaling dimension closely follo ws the 2∆ ϕ tra jectory . The intermediate “pond” region (in blue) is characterized b y the ab- sence of the ligh t scalar φ . The dashed curves mark the corresponding b oundaries at whic h op erator decoupling o ccurs. W e stress that the naming of these op erators is descriptiv e, as no underlying p erturbativ e explanation has b een disco vered. 5.4 First v alley Shortly after the “cliﬀ ”, the moment ⟨ ∆ ⟩ ∗ reac hes its ﬁrst lo cal minim um. This o ccurs where the contribution of the lightest scalar to the sp ectral densit y is dominant, and where the scaling dimension of the second-ligh test scalar is lo cally maximized. Notably , the anomalous dimension of the ligh test scalar also reaches a lo cal maxim um at the same lo cation. These b eha viors are sho wn in ﬁgures 13 and 20 . In d = 3, the ﬁrst v alley marks a sp ecial p oin t where the moment b ootstrap b egins ﬁnding qualitativ ely diﬀerent moment-minimizing solutions. Using the sp ectrum-extraction pro cedure describ ed in section 2.2 , we can explicitly reconstruct the correlation functions from the extremal sp ectra along the low er b ound. W e ﬁnd that upon passing the ﬁrst v alley , the reconstructed correlation function exhibits b eha vior consisten t with b ecoming – 26 – 0.00 0.05 0.10 0.15 0.20 3.3 3.4 3.5 3.6 3.7 3.8 3.9 〈Δ〉 * vs. δ ϕ "cli ﬀ " ( ﬁ rst kin k famil y ) 0.00 0.05 0.10 0.15 0.20 1.60 1.62 1.64 1.66 1.68 1.70 Δ O 1 vs. δ ϕ 0.00 0.05 0.10 0.15 0.20 0.0 0.1 0.2 0.3 0.4 0.5 λ ϕϕ O 1 2 vs. δ ϕ 0.00 0.05 0.10 0.15 0.20 0.15 0.20 0.25 0.30 0.35 0.40 λ ϕϕ T 2 vs. δ ϕ (a) d = 5 . 2 0.00 0.05 0.10 0.15 0.20 3.6 3.8 4.0 4.2 4.4 〈Δ〉 * vs. δ ϕ "cli ﬀ " ( ﬁ rst kin k famil y ) 0.00 0.05 0.10 0.15 0.20 1.800 1.805 1.810 1.815 1.820 1.825 Δ O 1 vs. δ ϕ 0.00 0.05 0.10 0.15 0.20 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 λ ϕϕ O 1 2 vs. δ ϕ 0.00 0.05 0.10 0.15 0.20 0.15 0.20 0.25 0.30 0.35 0.40 λ ϕϕ T 2 vs. δ ϕ (b) d = 5 . 6 Figure 15 : Moment low er b ounds (obtained at Λ = 23) and corresponding extremal sp ectra in representativ e spatial dimensions near d = 6, plotted as functions of δ ϕ ≡ ∆ ϕ − d − 2 2 . The red dashed line denotes the unitarity b ound. In addition to the “cliﬀ ” asso ciated with an op erator decoupling, we observ e a preceding sharp corner. The latter coincides with lo cal maxima of b oth the lightest-scalar scaling dimension and the cen tral c harge, while the cliﬀ corresp onds to lo cal minima of the same quan tities. – 27 – un b ounded ab o v e. This is reﬂected in the fact that when the primal-dual gap in SDPB is tigh tened, the reconstructed correlator contin ues to grow. The unboundedness of correla- tors persists un til the second v alley is reac hed, where the reconstructed correlation function returns to a ﬁnite v alue. The p ossibilit y that the correlation function may b e unbounded ab o ve once a certain threshold ϕ ≳ d − 2 is crossed has b een noted previously in [ 27 ]. This b eha vior admits a natural interpretation: b ey ond the threshold, the n umerical b o otstrap is able to construct crossing-symmetric and unitary solutions that contain no iden tity op erator. Such solutions can b e rescaled b y an arbitrarily large p ositiv e factor while preserving crossing symmetry and unitarity , rendering the correlation function unbounded ab o ve. Although at ﬁnite numerical precision, the reconstructed correlator is merely extremely large rather than strictly inﬁnite, its magnitude increases as the duality gap is reduced. This means that the n umerical bo otstrap is probing solutions in which the iden tity op erator is negligible. More precisely , the moment-minimizing solution is eﬀectiv ely obtained by the b ootstrap through increasing the o verall normalization of the non-identit y con tribution to the correlator while k eeping the ratio ν 1 / ( ν 0 − 1) minimized. In this limit, only the normalized sp ectral density ˜ p ℓ (∆) or the normalized OPE co eﬃcien t ˜ λ ϕϕ O , in tro duced in section 2.2 , remains meaningful, as it divides the conformal data by the correlation function (without the identit y con tribution). Deformation across dimensions So far, w e ha v e not observed any clear connection b e- t w een the un b ounded correlator and the geometric features of the low er b ound. Moreov er, for d ≥ 5, the reconstructed correlation function remains ﬁnite throughout the region we examine. The fact that moment-minimizing solutions in higher dimensions do not app ear to fav or identit y-suppressed directions is intriguing and merits further inv estigation. The ﬁrst v alley is also related to the presence of the stress tensor, which serves as a diagnostic for lo calit y . In d ≤ 3, the ﬁrst v alley marks the p oin t where the stress tensor decouples from the extremal sp ectrum. In d > 3, lo cal theories emerge after the ﬁrst v alley and occupy the low er b ounds more and more as d increases. Near d = 6, the stress tensor is present across the en tire lo wer b ound until the second v alley . 5.5 The hill Emergence of constant twist tra jectories Bet ween the tw o v alleys, an interesting phenomenon app ears in the extremal sp ectra: spinning op erators b egin to align along constan t-t wist tra jectories that closely follo w those of the GFF sp ectrum, up to small anomalous dimensions. In this regime, the leading double-twist family [ ϕϕ ] 0 ,ℓ t ypically acquires a small negativ e anomalous dimension, while the next family [ ϕϕ ] 1 ,ℓ sho ws a small p ositiv e anomalous dimension. These are illustrated in ﬁgure 16 . Deformation across dimensions The pattern abov e is consisten tly observed in the windo w 2 . 8 ≲ d ≲ 4, whic h coincides with the range of spatial dimensions where the “hill” in the moment landscap e is clearly seen. Outside this window, b oth the hill and the asso ciated twist tra jectories b ecome muc h less visible. A b etter understanding of this – 28 – 0 5 10 15 20 0 1 2 3 4 5 ℓ τ Δ ϕ = 1. 0 5 10 15 20 0 1 2 3 4 5 ℓ τ Δ ϕ = 1.4 0 5 10 15 20 0 1 2 3 4 5 ℓ τ Δ ϕ = 1.8 Figure 16 : F ormation and collapse of the constant-t wist tra jectories b et ween the t w o v alleys in d = 3, computed at Λ = 23. The red dashed lines indicate the leading GFF double-t wist tra jectories, 2∆ ϕ and 2∆ ϕ + 2, and the black dashed line the unitarit y b ound, d − 2 = 1. On the “hill” of the momen t lo wer b ound, higher-spin op erators align along near- GFF double-t wist tra jectories: the leading family [ ϕϕ ] 0 ,ℓ exhibits small negative anomalous dimensions, while the next family [ ϕϕ ] 1 ,ℓ sho ws small p ositiv e anomalous dimensions. These tra jectories are clearly visible in 2 . 8 ≲ d ≲ 4. connection b et ween the sp ectral and geometric prop erties of the momen t b ounds remains an op en and intriguing direction for future work. 5.6 Second v alley In the second v alley , where the reconstructed correlation functions return to ﬁnite v alues, sev eral notable even ts o ccur in close succession: (a) the lightest scalar saturates the unitarity b ound, and (b) the scaling dimension of the second-lightest scalar suddenly drops back to its previ- ously decoupled v alue, 2∆ ϕ , closely follow ed by (c) an extremal sp ectrum that minimizes all the leading momen ts sim ultaneously . The b eha vior of the leading moments around p oin t (c) is shown in ﬁgure 17 and ﬁgure 18 . Rather than comparing the extremal spectra directly , we use lo w-lying momen ts as eﬃcient prob es of sp ectral coincidence. Although the extremal sp ectra obtained b y minimizing diﬀerent moments app ear scattered at the level of individual op erators, their ﬁrst and second momen ts collapse to the same v alue at p oin t (c). Higher moments exhibit the same qualitative b eha vior, appro ximately coinciding at the same p oin t as w ell. This strongly suggests the existence of a single extremal solution that simultaneously minimizes sev eral low-lying momen ts. It would b e in teresting to study this p oin t further in future w ork, either by increasing the num b er of deriv ativ es or b y applying the mixed-correlator b ootstrap, whic h ma y help clarify the ph ysical prop erties of this distinguished solution. The scalar sector around p oin ts (a) and (b) is shown in ﬁgure 19 , where the scaling dimension of the second-lightest scalar op erator shows a sudden jump. Slightly b efore the second-ligh test scalar decouples at ∆ ϕ ≈ 1 . 95, the scaling dimension of the ligh test scalar, – 29 – 1.6 1.8 2.0 2.2 2.4 1.0 1.5 2.0 2.5 3.0 3.5 Δ ϕ 〈Δ〉 * 〈Δ〉 * ext rem al spe ctra 〈Δ 2 〉 * ext rem al spe ctra 〈Δ 3 〉 * ext rem al spe ctra 〈Δ 4 〉 * ext rem al spe ctra 〈Δ 5 〉 * ext rem al spe ctra Figure 17 : Shortly after the second v alley , there is a sp ecial p oin t where the moment- minimizing sp ectra coincide. This is illustrated by plotting the ﬁrst moment ev aluated using the extremal sp ectra data obtained by minimizing higher moments (here at Λ = 23), sho wn in diﬀeren t colors. Around ∆ ϕ ≈ 1 . 96, all extremal sp ectra share the same a verage scaling dimension. A similar analysis for the second momen t is shown in ﬁgure 18 . 1.6 1.8 2.0 2.2 2.4 2 3 4 5 6 7 8 9 Δ ϕ 〈Δ 2 〉 * 〈Δ〉 * ext rem al spe ctra 〈Δ 2 〉 * ext rem al spe ctra 〈Δ 3 〉 * ext rem al spe ctra 〈Δ 4 〉 * ext rem al spe ctra 〈Δ 5 〉 * ext rem al spe ctra Figure 18 : A t the same lo cation as in ﬁgure 17 , we plot the second moment, ev aluated using the same extremal spectra obtained b y minimizing diﬀerent low-lying moments as in ﬁgure 17 . Near ∆ ϕ ≈ 1 . 96, all extremal spectra yield to the same second moment. Similar b eha vior con tin ues for higher moments. – 30 – 1.80 1.85 1.90 1.95 2.00 0 5 10 15 Δ O n vs. Δ ϕ 2 Δ ϕ 2 Δ ϕ + 6 1.80 1.85 1.90 1.95 2.00 0.0 0.1 0.2 0.3 0.4 λ  ϕϕ O n vs. Δ ϕ Figure 19 : The scalar sector of the extremal sp ectra around the second v alley , computed at Λ = 27. Tw o notable phenomena occur in close pro ximity: the ligh test scalar approac hes and saturates the unitarit y bound, and the second-ligh test scalar returns to 2∆ ϕ . The plots sho w the scaling dimensions (left) and the normalized OPE co eﬃcien ts (right) of the four ligh test scalar op erators, lab eled in blue ( n = 1), green ( n = 2), red ( n = 3), and orange ( n = 4). sho wn in ﬁgure 20 , reac hes the unitarity b ound at ∆ ϕ ≈ 1 . 92. Numerically , we are unable to distinguish its dimension from the exact v alue ( d − 2) / 2. A t ﬁrst glance, it do es not make sense to hav e an op erator exactly at the unitarity b ound, as suc h an op erator would b ecome a free ﬁeld: its descendant ∂ 2 O b ecomes null, implying an equation of motion of a free ﬁeld, and the op erator is exp ected to decouple from the sp ectrum. Indeed, ﬁgure 19 shows that the OPE co eﬃcien t of the lightest scalar approac hes zero as it reaches the unitarity b ound. The subtlety , how ever, is that the momen ts are weigh ted by the pr o duct of the squared OPE co eﬃcien t and the conformal blo c k. While the OPE co eﬃcien t v anishes, the conformal blo c k diverges at the unitarit y b ound. Their pro duct can therefore remain ﬁnite and even dominan t in the sp ectral densit y . This is natural from the p oin t of view of n umerical b ootstrap, as the semideﬁnite program do es not directly “see” the divergence of the blo c k. The b ootstrap equations discussed in section 2.2 dep end only on the full spectral measure, and the extremal spectra indeed sho w that the ligh test scalar con tin ues to contribute signiﬁcantly even when its OPE co eﬃcien t tends to zero. This phenomenon is, in fact, not new in the b o otstrap literature and is known as the fake-primary eﬀe ct . In the next subsection we review this mechanism and explain ho w it pro vides a prop er physical interpretation of the unitarity-saturating extremal solutions. W e then apply this viewp oin t to study the structure of the moment low er b ounds. 5.7 F ak e-primary interpretation of low er b ounds Here w e give a brief review of the fake-primary eﬀect in tro duced in [ 34 , 62 ], presen ting the basic idea in the con text of the single-correlator b ootstrap. – 31 – 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.500 0.502 0.504 0.506 0.508 0.510 Δ ϕ Δ O 1 Cli ﬀ Fir st vall ey Secon d vall ey Figure 20 : The scaling dimension of the ligh test scalar op erator in d = 3 along the lo w er bound of the ﬁrst momen t, computed at Λ = 23. Its anomalous dimension is lo cally minimized at the cliﬀ, lo cally maximized at the ﬁrst v alley , and touc hes the unitarity b ound at ∆ ϕ ≈ 1 . 92, shortly b efore the decoupling of the second-lightest scalar. Its OPE co eﬃcien t v anishes at the same time it saturates the unitarity bound, as sho wn in ﬁgure 19 . W e will discuss this subtlety in section 5.7 . The residue of a conformal blo c k at any of its p oles is prop ortional to the conformal blo c k of its null descendant: g ∆ ,ℓ ( z , ¯ z ) → R A ∆ − ∆ ∗ A g ∆ A ,ℓ A ( z , ¯ z ) as ∆ → ∆ ∗ A . (5.2) P articularly , at the scalar unitarity b ound 4 , the residue is giv en by the scalar conformal blo c k of its shadow: g ∆ , 0 ( z , ¯ z ) → R II I n =1 ,ℓ =0 ∆ − d − 2 2 (4 r ) 2 h d − 2 2 +2 , 0 = R II I 1 , 0 ∆ − d − 2 2 g d +2 2 , 0 ( z , ¯ z ) , (5.3) where R II I n =1 ,ℓ =0 = 1 8 d  d − 2 2  3 . (5.4) Therefore, whenever a theory con tains an op erator ˜ O with dimension exactly ˜ ∆ = d + 2 2 , (5.5) it can pro duce tw o distinct v alues for a moment (unless the momen t is in v arian t under the shado w transformation). This ambiguit y arises b ecause the same contribution to the 4 Notice that the conformal blo c k of identical external scalars can only b e singular for ℓ = 0 and ∆ = ( d − 2) / 2 ab o ve or at the unitarity b o und. – 32 – 0.5 1.0 1.5 2.0 2.5 1 2 3 4 5 6 7 〈Δ〉 * vs. Δ ϕ Allowed regi on: Δ gap ≥ d - 2 2 Allowed regi on: Δ gap ≥ Δ ϕ Allowed regi on: Δ gap ≥ 2 Δ ϕ Fake pri mary rem ap of lower bound ( tol = 1 0 - 4 ) Fake pri mary rem ap of MFT 〈ϕϕϕϕ 〉 Fake pri mary rem ap of MFT 〈ϕ 2 ϕ 2 ϕ 2 ϕ 2 〉 Fake pri mary rem ap of MFT 〈ϕ 3 ϕ 3 ϕ 3 ϕ 3 〉 Fake pri mary rem ap of MFT 〈ϕ 4 ϕ 4 ϕ 4 ϕ 4 〉 Figure 21 : F ake-primary remap of the n umerical lo wer b ounds on the ﬁrst momen t in d = 3, computed at Λ = 23. The momen t b ounds shown here are the same as those in ﬁgure 1b . W e replace an op erator by its shadow only if its distance from the unitarity b ound is less than the tolerance. The solid green line represents the fake-primary remap of the lo w er b ound with a tolerance of 10 − 4 . See app endix E for details on the fake-primary remap of GFF moments. sp ectral density may b e interpreted either as coming from an op erator O saturating the unitarit y bound, or from its shado w ˜ O . Concrete examples of this eﬀect also app ear in the generalized free ﬁeld correlators at sp eciﬁc v alues of the external weigh t; see app endix E for details. F rom a n umerical p erspective, once an extremal solution contains a ligh test scalar whose scaling dimension lies within n umerical tolerance of the unitarity b ound, one can p erform a fak e-primary remap by simply replacing this op erator by its shadow while pre- serving its total sp ectral contribution. The outcome is shown in ﬁgure 21 . Remark ably , after the second v alley , for the region in which the extremal sp ectra consisten tly con tain an op erator exactly at or extremely close to the unitarity b ound, the remapp ed moments lie nearly on top of the low er b ounds obtained under the gap assumption ∆ gap ≥ ∆ ϕ . Curiously , the external scaling dimension ∆ ϕ = 1 . 875, where the generalized free ﬁeld OPE con tains an operator exactly at the shado w of the unitarit y bound, is very close to the second v alley . While it remains unclear whether this alignment is accidental or inherent, the observ ation is comp elling and deserv es further inv estigation. Finally , we emphasize that although the ﬁrst mome n t can b e fake-primary remapped to match the gapp ed low er b ound, essen tially follo wing the tra jectory of the ⟨ ϕ n ϕ n ϕ n ϕ n ⟩ GFF correlators at large n , the reconstructed higher moments do not match those of the GFF. The physical interpretation of the remapp ed low er b ound therefore remains to b e understo od. Ho w gap assumptions c hange the low er b ounds In [ 34 , 62 ], v arying the gap as- sumption was used as a diagnostic to determine whether the observed b o otstrap features – 33 – 0.5 1.0 1.5 2.0 1.0 1.5 2.0 Δ ϕ 〈Δ〉 * Allowed regi on: ϵ = 0 Allowed regi on: ϵ = 0.001 Allowed regi on: ϵ = 0.002 Allowed regi on: ϵ = 0.003 Allowed regi on: ϵ = 0.004 Allowed regi on: ϵ = 0.005 Figure 22 : Upp er and low er b ounds on the ﬁrst normalized moment ⟨ ∆ ⟩ ∗ with diﬀerent gap assumptions ∆ gap ≥ d − 2 2 + ϵ imp osed, computed at Λ = 19. As ϵ increases, the face of the “cliﬀ ” is gradually ruled out, consistent with the b eha vior sho wn in ﬁgure 20 . are gen uine or are instead artifacts of the fake-primary eﬀect. Motiv ated b y this strategy , w e p erform an analogous analysis for the moment lo wer bounds. Figure 22 shows the up- p er and lo wer b ounds on the ﬁrst normalized momen t ⟨ ∆ ⟩ ∗ in d = 3 as a function of the external scaling dimension while imp osing a scalar gap ∆ gap ≥ d − 2 2 + ϵ. (5.6) W e observ e that ev en a mild gap assumption, ϵ = 0 . 001, is suﬃcien t to rule out the second v alley . This b eha vior is exp ected, since the corresp onding solution relies on the presence of a scalar op erator saturating the unitarit y b ound at ∆ = ( d − 2) / 2, and is therefore incompatible with a strictly positive gap. In con trast, the cliﬀ, where the momen t v alue undergo es its ﬁrst sudden drop, p ersists as ϵ is increased and is not completely ruled out even for larger v alues of the gap. This indicates that the structures observ ed prior to the second v alley cannot b e fully attributed to the fak e-primary eﬀect. This b eha vior parallels observ ations in previous works, where lifting the scalar gap remov es fak e-primary features while leaving more robust b ootstrap b ounds only mildly aﬀected. A more reﬁned analysis, suc h as incorp orating mixed-correlator constraints, may help further clarify the origin of this feature, and we leav e this direction for future w ork. Deformation across dimensions As we ha v e seen throughout this work, the lo w er b ounds on the moments generically exhibit t wo families of kinks, b oth asso ciated with the decoupling of lo w-lying scalar op erators. The second v alley , in particular, marks the saturation of the scalar unitarit y b ound, b ey ond which the n umerical solution can b e consisten tly rein terpreted through a fake-primary remap inv olving a scalar at ∆ = d +2 2 . The detailed realization of this mechanism, how ever, dep ends on the spatial dimension. – 34 – 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2 4 6 8 Δ ϕ 〈Δ〉 * n = 1 n = 2 n = 10 Allowed regi on: Δ gap ≥ d - 2 2 Allowed regi on: Δ gap ≥ Δ ϕ Allowed regi on: Δ gap ≥ 2 Δ ϕ Mini mal mode l ( p , q ) = ( 4,3 ) Mini mal mode l ( p , q ) = ( 5,4 ) Mini mal mode l ( p , q ) = ( 6,5 ) GFF < ϕ n ϕ n ϕ n ϕ n > 2 Δ ϕ Figure 23 : Momen t b ounds in d = 2 + 10 − 6 , computed at Λ = 19. With a unitary gap assumption, the lo w er b ound exhibits t wo sharp kinks at the simple v alues ∆ ϕ = 1 2 and ∆ ϕ = 3 2 . After a fak e-primary remap (see section 5.7 ), the entire curv e is deformed upw ard to coincide with the lo wer b ound obtained b y imposing ∆ gap ≥ ∆ ϕ . Although this b eha vior is fake-primary driven in the limit d → 2 + , in general dimensions 2 < d < 5 the scalar op erator b efore the second v alley lies strictly abov e the unitarit y b ound, and the associated lo w er b ounds are not entirely go verned by fak e-primary eﬀects. App endix B describ es our metho d for computing the minimal mo del moments. In exactly d = 2, as shown in ﬁgure 1a , the low er bound exhibits no kinks of the t yp e observ ed in higher dimensions and instead closely follo ws the linear tra jectory √ 2 ∆ ϕ , coinciding with the b ound obtained by imp osing ∆ gap ≥ ∆ ϕ . This b eha vior changes disc ontinuously when mo ving sligh tly aw ay from tw o dimensions. In the sp ecial limit d → 2 + , distinct geometric landmarks that are well separated at generic dimensions merge into a single kink, where op erator decoupling coincides with a lo cal minim um of the moment. In d = 2 + 10 − 6 the low er b ound develops t w o sharp kinks at the half-integer v alues ∆ ϕ = 1 2 and 3 2 , as shown in ﬁgure 23 . Op erator decoupling occurs exactly at these p oin ts, and the entire lo wer-bound contains a scalar saturating the unitarity b ound. Consequently , the tra jectory is “fake” in the sense that, after an appropriate fake- primary remapping, it coincides numerically with the low er b ound obtained by imp osing ∆ gap ≥ ∆ ϕ . Imp osing a tiny gap immediately collapses the b ound on to the same curve. This tra jectory asymptotically matches that of generalized free ﬁeld correlators ⟨ ϕ n ϕ n ϕ n ϕ n ⟩ at large n and aligns with the b ounds observ ed in strictly d = 2. F or comparison, we also include the moments of known theories, including the lo w-lying minimal mo dels and the generalized free ﬁeld theory . The origin of the discontin uity can b e traced to the b eha vior of the scalar conformal blo c k at the unitarit y bound, d − 2 2 . In d = 2 + ϵ , the conformal blo c k div erges as the scalar dimension approac hes ϵ/ 2, so placing an op erator exactly at the unitarit y b ound, in terpreted as the fak e-primary image of another solution containing a scalar at the lo cation – 35 – of its shadow, leads to a signiﬁcant reduction of the momen t. In con trast, in exactly d = 2, the global scalar blo c k is regular at the unitarity b ound, preven ting an analogous mec hanism. Finally , w e em p hasize that in d = 2 + 10 − 6 , although the ﬁrst momen t after the second v alley can b e fak e-primary remapp ed to match that of the GFF, the extremal sp ectra themselv es do not coincide with those of the GFF. This discrepancy appears in the higher momen ts, indicating that the underlying solutions are not genuine GFFs. The ph ysical in terpretation of these extremal solutions therefore remains unclear and merits further in v estigation. 5.8 The ∆ gap ≥ 2∆ ϕ lo w er b ound W e end this section b y highlighting the structures of the lo wer b ound obtained under the gap assumption ∆ gap ≥ 2∆ ϕ . As a function of the external scaling dimension ∆ ϕ , the extremal solutions interpolate b et ween three qualitativ ely distinct regimes: (a) a family of solutions that trace the generalized free ﬁeld (GFF) ⟨ ϕϕϕϕ ⟩ correlators at small ∆ ϕ , (b) an in termediate regime where the moment bound reac hes a lo cal maxim um, accom- panied by the decoupling of the scalar at the gap 2∆ ϕ , (c) and a ﬁnal regime appro ximately saturated by the GFF correlators ⟨ ϕ n ϕ n ϕ n ϕ n ⟩ at large n . This interpolating b eha vior is observed consistently across diﬀerent spatial dimensions, as sho wn in ﬁgure 1 . F or ligh t correlators, the gap assumption signiﬁcan tly shrinks the allo wed theory space, and the low er b ound is tightly saturated b y the generalized free ﬁeld ⟨ ϕϕϕϕ ⟩ tra jectory . Indeed, the extremal sp ectra appro ximately repro duce the constan t-t wist double-trace op- erators. In particular, the ligh test scalar op erator exactly follows 2∆ ϕ , saturating the gap assumption. As ∆ ϕ increases, the anomalous dimensions of op erators near the constant-t wist tra jec- tories gradually dev elop, and at the lo cal maxim um of the low er b ound, the scalar op erator at 2∆ ϕ decouples from the extremal sp ectrum. This scalar decoupling o ccurs across v ari- ous spatial dimensions and remains v ery close to the lo cal maximum of the lo wer b ound. Figure 24 sho ws the extremal sp ectra along the gapp ed low er b ound in a representativ e slice, d = 3. Bey ond this p oin t, the lo wer b ound decreases and asymptotically approac hes the linear tra jectory √ 2 ∆ ϕ . This b eha vior coincides with the ∆ gap ≥ ∆ ϕ lo w er b ound, or with the ﬁrst moment of the GFF correlators ⟨ ϕ n ϕ n ϕ n ϕ n ⟩ at large n . Deformation across dimensions W e extend ∆ ϕ to larger v alues in this gapp ed setup to trac k the approach to the asymptotic linear tra jectory . F or d ≲ 5, there exists an in termediate windo w along the gapp ed low er b ound in which the extremal spectra contain a conserv ed stress tensor. Intriguingly , w e observ e that the stress tensor enters the extremal – 36 – 1.2 1.3 1.4 1.5 1.6 1.7 1.8 2 4 6 8 10 Δ n ( ℓ = 0 ) vs. Δ ϕ 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0.0 0.2 0.4 0.6 0.8 p  n ( ℓ = 0 ) vs. Δ ϕ 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0.0 0.5 1.0 1.5 λ  n ( ℓ = 0 ) vs. Δ ϕ 1.2 1.3 1.4 1.5 1.6 1.7 1.8 3 4 5 6 7 8 9 Δ n ( ℓ = 2 ) vs. Δ ϕ 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0.0 0.2 0.4 0.6 0.8 p  n ( ℓ = 2 ) vs. Δ ϕ 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0.0 0.5 1.0 1.5 2.0 λ  n ( ℓ = 2 ) vs. Δ ϕ Figure 24 : Extremal sp ectra of the leading scalar and spin-2 operators along the ∆ gap ≥ 2∆ ϕ lo w er b ound in d = 3, computed at Λ = 23. The red dashed lines represen t 2∆ ϕ + 2 n for n = 0 , 1 , 2 , 3. The stress tensor en ters the spectrum at ∆ ϕ ≈ 1 . 3. Its con tribution to the sp ectral density then increases steadily and even tually acquires an anomalous dimension at ∆ ϕ ≈ 3 . 5. A t ∆ ϕ ≈ 1 . 63, where the gapp ed low er b ound reaches its maximal v alue, the ligh test scalar op erator decouples. Prior to this p oin t, its scaling dimension remains ﬁxed at the assumed gap, 2∆ ϕ . – 37 – sp ectrum when the external scaling dimension is close to d +2 4 , where the fak e-primary remap of the GFF b ecomes ﬁrst p ossible, as argued in app endix E . As ∆ ϕ increases further, the stress tensor decouples as the solution merges in to the asymptotic linear regime. It w ould b e in teresting to b etter understand this in terp olating b eha vior analytically and how it ma y b e related to known physical theories. 6 Discussion The numerical moment b ootstrap dev elop ed in this work extends our understanding of OPE moment v ariables into the ligh t correlator regime. Through the use of semideﬁ- nite programming, it provides numerically rigorous b ounds that complement the analytic p o wer-la w b eha vior derived in the heavy correlator limit. Assuming maximal Shannon en- trop y of the OPE sp ectrum, the extremal solutions allow for accurate reconstruction of the asso ciated sp ectral density , conﬁrming conv ergence to w ard generalized free ﬁeld theories while yielding systematic corrections to the sp ectral densit y . In this wa y , the numerical momen t b ootstrap establishes a concrete bridge b et w een n umerical b o otstrap techniques and analytical heavy limit analysis. Conceptually , the momen t formulation oﬀers a complemen tary p erspective on the con- formal b o otstrap by probing the sp ectrum in a global and coarse-grained manner, com- pressing detailed op erator information in to w eighted av erages that remain w ell deﬁned ev en when the sp ectrum b ecomes dense. Perhaps surprisingly , such collective observ ables can nev ertheless b e studied in a n umerically rigorous wa y . Beyond the single heavy correlators studied in this work, the momen t form ulation naturally extends to mixed correlators in- v olving b oth hea vy and light external op erators. In particular, hea vy-hea vy-ligh t-light cor- relators provide a promising arena where moment v ariables may prob e ﬁnite-temp erature ph ysics and asp ects of thermalization in conformal ﬁeld theories. W e lea v e a systematic exploration of such mixed systems to future work. Another notable adv antage of this framework is its ability to unify a wide range of familiar bo otstrap phenomena. Lo w-lying momen ts of scaling dimension and spin naturally repro duce the Ising-mo del kinks and enco de gap and central-c harge extremization within a single optimization problem. Compared to traditional gap-exclusion methods, which require repeated scans o ver the parameter space, the momen t approac h iden tiﬁes ph ysically relev an t theories more eﬃciently while sim ultaneously yielding globally rigorous bounds on the op erator sp ectrum. This p oin ts to a natural future direction: by b ounding moment v ariables using mixed-correlator constraints ov er the b ootstrap island, one may establish rigorous statemen ts ab out conformal ﬁeld theories not only along the b oundary of the island but also in its interior [ 63 ]. Most in triguingly , the numerical moment b o otstrap reveals previously unexplored ge- ometric structures on the other side of the b ootstrap landscape. The lo wer b ounds of momen t v ariables exhibit robust features across spatial dimensions, closely tied to nontriv- ial sp ectral ev ents including operator decoupling, anomalous-dimension extremization, and the fak e-primary eﬀect. In particular, the emergence of tw o contin uous families of kinks – 38 – across 2 < d < 6 demonstrates that momen t v ariables are sensitive to sp ectral reorganiza- tions that are otherwise diﬃcult to access within traditional b o otstrap setups. These geometric features admit a simple organization in terms of lightest scalar op er- ators. The “p ond” region is con trolled by the decoupling and reapp earance of the lightest scalar op erator. As its OPE co eﬃcien t is n umerically small and its imprint on the moment geometry is therefore subtle, it will b e imp ortan t to understand whether this structure sharp ens or p ersists as the deriv ative order is increased. The tw o sharp kinks across dimensions are closely tied to the second-ligh test op er- ator, whose scaling dimension closely follows 2∆ ϕ . The ﬁrst kink (the “cliﬀ ”), which is con tin uously connected to the free scalar theory in d = 6, features the decoupling of this op erator. At the second kink, the op erator reapp ears, while the lightest scalar approaches the unitarit y b ound, and the extremal sp ectra sim ultaneously minimize m ultiple low-lying momen ts. Whether these structures arise from renormalization group ﬂows of known in- teracting scalar theories, or instead prob e an approximately unitary sector of more exotic solutions, remains an op en question. Clarifying the ph ysical origin of the “cliﬀ ” will likely require additional input, for instance from mixed-correlator b ootstrap analyses that may isolate these solutions into b ootstrap islands. Determining the ultimate fate of the second kink in higher spatial dimensions likewise remains an in triguing op en direction. Bet w een the tw o kinks lies the “hill” region, where the moment-minimizing solutions prob e qualitatively diﬀerent directions in the theory space, with the reconstructed corre- lators showing unbounded growth. In this regime, the OPE is dominated by only a few lo w-lying op erators, suggesting that analytic to ols such as the Loren tzian in version for- m ula [ 64 ] may provide further insigh t in to the observ ed sp ectra. Finally , the gapp ed low er b ound exhibits an in terp olation betw een asymptotic regimes with an intermediate window of lo cal theories, further motiv ating analytic inv estigations of these structures. In this w ork, we deﬁned moments at the self-dual p oin t z = ¯ z = 1 / 2, a kinematic con- ﬁguration where conformal blo c ks suppress contributions from op erators with large scaling dimensions. More generally , we can deﬁne momen ts of the OPE in diﬀerent kinematic regimes, whic h w ould b e sensitive to qualitatively diﬀerent sets of op erators. F or example, if we study moments of the OPE sum weigh ted by conformal blo c ks ev aluated near the double light cone limits, the resulting quantities w ould naturally b e most sensitive to the op erators app earing in the leading twist tra jectory . Alternatively , if w e study moments of the OPE deep in the Regge limit, these quantities may b e used to characterize Regge tra jectories, whic h play an imp ortan t role in the dynamics of this regime. It w ould b e in teresting to understand whic h moments are naturally adapted to diﬀerent kinematic lim- its, and ho w they may b e b ounded numerically or computed p erturbativ ely in expansions around these limits. In conclusion, the momen t b ootstrap reveals b oth subtle and unexp ected geometric structures in the space of CFT four-p oin t functions of identical scalars, while op ening up several promising directions for future work, including numerically probing the heavy correlator regime and reinforcing existing precision studies of conformal data. W e hop e that this work oﬀers a clear map of the emerging “moment landscap e” and encourages further exploration of its ph ysical origins. – 39 – Ac knowledgmen ts W e thank W ei Li for numerous discussions and collab oration on related CFT moment problems, Liam Fitzpatric k, Y u-tin Huang, Shao-Cheng Lee, T ony Liu, Matthew Mitchell, and Balt v an Rees for additional discussions, and Y uan Xin for v aluable guidance on computational asp ects in the early stages of this work. The authors were supp orted by DOE grant DE-SC0017660. Computations w ere p erformed on the Y ale Grace computing cluster, supp orted b y the facilities and staﬀ of the Y ale Univ ersity F aculty of Sciences High P erformance Computing Center. A Rational approximation of conformal blo c ks Our setup for the n umerical b ootstrap starts from the rational appro ximation of conformal blo c ks. Extensively used in n umerical b ootstrap studies, it expresses a conformal blo c k in terms of a rational function of the scaling dimension ∆ of the op erator exchanged. This appro ximation is based on an expansion in the radial co ordinate r , related to the cross ratio z via [ 59 ] ρ = r e iθ = 1 − √ 1 − z 1 + √ 1 − z , (A.1) and η = cos θ . The idea is that the conformal blo c k develops singularities at non-physical scaling dimensions ∆, whic h giv es a recursion relation of the conformal blo c k in terms of a sum o v er p oles in ∆. Such a relation was ﬁrst disco vered by Zamolo dc hiko v [ 65 , 66 ] in 2d and w as generalized to higher dimensions in [ 6 ] for identical external scalar primaries. Here, we follow the con ven tion in [ 67 ] and write the conformal blo c k as g ∆ ,ℓ ( r , η ) = (4 r ) ∆ h ∆ ,ℓ ( r , η ) , (A.2) where the regularized conformal blo c k h ∆ ,ℓ ( r , η ) admits the recursive representation h ∆ ,ℓ ( r , η ) = h ∞ ,ℓ ( r , η ) + X A R A ∆ − ∆ ∗ A (4 r ) n A h ∆ ∗ A + n A ,ℓ A ( r , η ) , (A.3) with the lo cations of the p oles ∆ ∗ A and the residues R A giv en in table 2 . Therefore, b y iterating the recursion relation up to order r max in the radial co ordinate r , w e arriv e at the following appro ximation for the conformal blo c k and its deriv atives, ∂ m z ∂ n ¯ z g ∆ ,ℓ (1 / 2 , 1 / 2) ≈ (4 r ∗ ) ∆ Q A (∆ − ∆ ∗ A ) P mn ℓ (∆) , (A.4) at the crossing-symmetric p oin t r ∗ = 3 − 2 √ 2 , η = 1. F or maximized computational eﬃciency , we w ould lik e the degree of the p olynomial P m,n ℓ (∆) to b e as small as p ossible. F ollowing the metho d ﬁrst prop osed in [ 6 ], a t w o-sided P ad´ e approximation is used to reduce the n um b er of p oles in the rational function, keeping those that app ear b elo w order κ in the radial expansion while matc hing the ﬁrst ⌊ | P κ | +1 2 ⌋ deriv atives at the unitarity bound and the remaining ⌊ | P κ | 2 ⌋ at ∆ → ∞ , where | P κ | denotes the n um b er of p oles app earing at radial expansion order κ . – 40 – A ∆ ∗ A n A ℓ A R A I n ( n ∈ 2 N ) 1 − ℓ − n n ℓ + n R I n = − n ( − 2) n ( n !) 2  1 − n 2  2 n I I n ( n ∈ 2 N ) ℓ + d − 1 − n n ℓ − n R II n = − n ℓ ! ( − 2) n ( n !) 2 ( ℓ − n )! × ( d + ℓ − n − 2) n  d 2 + ℓ − n  n  d 2 + ℓ − n − 1  n  1 − n 2  2 n I II n ( n ∈ N ) d 2 − n 2 n ℓ R II I n = − n ( − 1) n  d 2 − n − 1  2 n ( n !) 2  d 2 + ℓ − n − 1  2 n  d 2 + ℓ − n  2 n T able 2 : The lo cations of the p oles of the conformal blo c k and the corresp onding residue data, adapted from [ 67 ]. All conformal blo c k deriv ativ es used in this work w ere generated using a custom Julia library , blockDeriv.jl [ 68 ], developed for eﬃciently computing deriv ativ es of scalar con- formal blo c ks with identical external op erators, including kinematic conﬁgurations aw ay from the self-dual p oin t. In exactly d = 2, we instead generated the deriv ativ es using scalar blocks [ 69 ] for its sp ecialized treatmen t of the p ole structure in even dimensions. The blockDeriv.jl pack age is currently b eing prepared for public release. Unless stated otherwise, throughout this w ork w e adopt the truncation parameters sp eciﬁed in table 3 . The duality gap is alw a ys chosen to be 10 − 30 , except in regions where the extremal sp ectra require a tighter gap to stabilize, in which case it is set to 10 − 60 . W e ha v e v eriﬁed that the moment b ounds are insensitiv e to the truncation parameters. Λ κ ℓ max r max 5 8 30 60 9 8 30 60 11 8 30 60 13 8 30 60 19 14 50 60 23 18 50 60 27 20 50 60 T able 3 : T runcation parameters for bo otstrap computations in this w ork. B Momen ts from 2d correlators Minimal models play a central role in the conformal b ootstrap. They w ere among the ﬁrst theories studied using b ootstrap techniques and pro vide exactly solv able examples with signiﬁcan t implications. Computing moment v ariables in these CFTs is therefore a natural and imp ortan t task: not only is their OPE data explicitly known, but they also serv e as a sanit y c hec k for locating exactly solv able theories within the momen t landscap e charted in this work. – 41 – In this app endix, w e describ e how the minimal-mo del and v ertex op erator moments sho wn in ﬁgures 1a and 23 are computed. The metho d is conceptually straightforw ard. The unitary minimal mo dels are labeled b y ( p, q ) = ( p, p + 1) with p ≥ 3, and the conformal dimensions of Virasoro primary op erators are giv en by h r,s = ( ps − q r ) 2 − ( p − q ) 2 4 pq , (B.1) where the p ositiv e integers r and s lie in the ranges 1 ≤ r ≤ p − 1 , 1 ≤ s ≤ p. (B.2) The structure constants of these theories are known in closed form, following the classic w ork of Dotsenko and F ateev [ 70 – 72 ]. In principle, this makes the computation of correla- tion functions en tirely explicit once the Virasoro conformal blo c ks are av ailable. How ever, since our goal is to compute moments deﬁned as weigh ted av erages ov er global conformal primaries, it is necessary to decomp ose Virasoro blo c ks into sums of global blo c ks. F ortunately , as emphasized in [ 73 ], sev eral useful representations of Virasoro blo c ks can b e obtained from the recursion relations disco vered by Zamolo dc hiko v [ 65 , 66 ], including expansions in terms of an inﬁnite sum of global hypergeometric blo c ks. W e make use of this represen tation to express the Virasoro blocks as sums o v er the global conformal blocks, up to a truncation ∆ max in the scaling dimensions. Combined with the known structure constan ts, this determines the global conformal data, whic h enables us to compute the momen ts directly from their deﬁnition. W e present their v alues in table 4 , computed with ∆ max = 14. W e hav e chec ked that the resulting v alues are stable under increases in ∆ max and are quoted to three decimal places. ( p, q ) ( r , s ) h r,s ∆ ϕ ⟨ ∆ ⟩ ⋆ (3, 4) (1 , 2) 1 16 1 8 1 . 081 (1 , 3) 1 2 1 2 . 516 (4, 5) (1 , 2) 1 10 1 5 1 . 297 (1 , 3) 3 5 6 5 2 . 578 (2 , 1) 7 16 7 8 2 . 592 (2 , 2) 3 80 3 40 0 . 228 (5, 6) (1 , 2) 1 8 1 4 1 . 439 (1 , 3) 2 3 4 3 2 . 598 (2 , 1) 2 5 4 5 2 . 564 (2 , 2) 1 40 1 20 0 . 141 (2 , 3) 1 15 2 15 0 . 398 (2 , 4) 21 40 21 20 2 . 319 T able 4 : The ﬁrst normalized moment v ariable for identical external scalars, with the iden tit y con tribution remo v ed, in the low-lying minimal mo dels. Here we list the momen ts app earing in ﬁgure 23 , corresp onding to correlators of relev ant op erators. – 42 – The v ertex op erator moments can b e computed in a similar wa y , now by decomp osing the correlator in to a p ositiv e sum of global conformal blo c ks. F or the iden tical scalar correlator setup considered in this work, we take the cosine vertex op erator C α = V α + V − α √ 2 , (B.3) whic h giv es a non-v anishing four-p oin t function G ( z , ¯ z ) = 1 2  v ϕ + u 2 ϕ v − ϕ + v − ϕ  . (B.4) W e expand the correlation function in the s -channel OPE limit z = ¯ z = 0 up to order z max , so that only op erators with suﬃciently small ∆ contribute at this order. W e then matc h this expansion to a sum o v er global conformal blo c ks to extract the OPE data. The op erators app earing in the OPE consist of the global primaries in the V erma mo dules of the v acuum and of the vertex op erator C 2 α . Their quantum num b ers therefore organize in to t w o families, spaced b y in tegers ab o ve the corresp onding Virasoro primaries. With z max = 8, we compute the momen ts sho wn in ﬁgure 1 . C Ising moments across dimensions The seminal w ork of Wilson and Fisher [ 74 ] ﬁrst established that the critical point of the Ising mo del corresp onds to a nontrivial infrared ﬁxed p oin t of the ϕ 4 theory b elo w four dimensions, unifying the 2d and 3d Ising critical p oin ts with the Gaussian theory in d = 4. The conformal b o otstrap has pro vided precise nonp erturbativ e evidence for this picture. In particular, [ 75 ] extended the n umerical b ootstrap to fractional spatial dimensions, demonstrating the existence of a smooth family of CFTs in terp olating betw een the 2d and 3d Ising models and the free theory in 4d. More recen tly , reﬁned analyses hav e mapp ed this family with further accuracy , extracting the critical exp onen ts [ 76 ]. In our framework, the kinks asso ciated with the Ising ﬁxed p oin ts can b e eﬃcien tly trac k ed across spatial dimensions using the momen t v ariables discussed in this work. Fig- ure 25 shows the ﬁrst normalized moment (with the identit y contribution remov ed) as a function of the external dimension ∆ ϕ , after subtracting the unitarity b ound ∆ min = d − 2 2 . D Maxim um entrop y reconstruction of coarse-grained sp ectra The maxim um-entrop y principle provides a natural presc r iption for selecting a probability measure when only partial information is a v ailable. Originally formulated in the con text of statistical mec hanics b y Jaynes [ 77 ], it states that among all distributions consistent with a given set of constraints, the one maximizing the Shannon entrop y constitutes the least biased c hoice. Its ﬁrst systematic applications to quan tum-mec hanical systems w ere carried out by Mead and P apanicolaou [ 78 ], who emplo yed the metho d to reconstruct densities of states in harmonic solids and dynamical correlation functions in quantum spin systems. In quan tum ﬁeld theory , maximum-en tropy techniques hav e b ecome a standard to ol for reconstructing sp ectral functions from a ﬁnite set of Euclidean correlators, particularly in – 43 – 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Δ ϕ - Δ mi n 〈Δ〉 * d = 2.0 d = 2.2 d = 2.4 d = 2.6 d = 2.8 d = 3.0 d = 3.2 d = 3.4 d = 3.6 d = 3.8 d = 4.0 Figure 25 : Upp er b ounds on the ﬁrst normalized momen t (with the identit y op erator excluded), plotted as a function of the external scaling dimension ∆ ϕ after subtracting the unitarit y b ound ∆ min = d − 2 2 , in v arious spatial dimensions d . These bounds w ere computed at Λ = 19. One can see here the evolution of the Ising kinks across diﬀeren t dimensions. lattice QCD studies [ 79 ]. Related ideas hav e also app eared in other areas of high-energy ph ysics, for instance in the reconstruction of parton distribution functions from a ﬁnite set of moments [ 80 ]. Here w e outline the metho d of reconstructing a sp ectral density from a ﬁnite num b er of moments. Given the ﬁrst few normalized moments, ⟨ ∆ k ⟩ ∗ = Z d ∆ ˜ p (∆)∆ k , k = 1 , 2 , ..., N , (D.1) the task is to determine a p ositiv e distribution ˜ p (∆) that repro duces these v alues. F or general v alues of the moments, the solution to the momen t problem is not unique, as in- ﬁnitely many distributions can share the same ﬁrst N moments. The max-en tropy metho d pro vides a natural criterion for selecting among them: it c ho oses the distribution that maximizes the Shannon entrop y S [ ˜ p ] = − Z d ∆ ˜ p (∆) log ˜ p (∆) . (D.2) By the metho d of Lagrange multipliers, the maximizing distribution alwa ys takes the ex- p onen tial form ˜ p (∆) = 1 Z ( λ ) exp − N X n =1 λ n ∆ n ! , (D.3) where the co eﬃcien ts λ n are determined by enforcing the conditions that the sp ectral den- sit y repro duces the ﬁrst N given moments. In practice, these parameters can b e eﬃcien tly – 44 – solv ed numerically using existing implementations such as the Python pac k age PyMaxEnt [ 81 ]. In the context of this work, the max-entrop y reconstruction is used in section 3 to obtain a smo oth, positive approximation to the discrete OPE sp ectral measure, providing a weigh t-interpolating function that captures the collective b eha vior of hea vy op erators. E F ak e-primary remap of generalized free ﬁeld correlators As argued in section 5 , when a theory contains a scalar op erator exactly at ∆ = d +2 2 , the fake-primary eﬀect allows us to remap the operator to its shadow at the unitarit y b ound while maintaining crossing and unitarit y . A concrete example is provided by the generalized free ﬁeld theory: let us consider the correlation function ⟨ ϕϕϕϕ ⟩ , where the exc hanged operators in the ϕ × ϕ OPE hav e scaling dimensions ∆ [ ϕϕ ] n,ℓ = 2∆ ϕ + 2 n + ℓ. 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