PHYSICAL REVIEW LETTERS 126, 211602 (2021)

Crossing Symmetric Dispersion Relations for Mellin Amplitudes

Rajesh Gopakumar ,1 Aninda Sinha ,2 and Ahmadullah Zahed 2 1International Centre for Theoretical Sciences (ICTS-TIFR), Shivakote, Hesaraghatta Hobli, Bangalore North 560 089, India 2Centre for High Energy , Indian Institute of Science, C.V. Raman Avenue, Bangalore 560012, India (Received 23 February 2021; accepted 5 May 2021; published 27 May 2021)

We consider manifestly crossing symmetric dispersion relations for Mellin amplitudes of scalar four point correlators in conformal field theories. This allows us to set up the nonperturbative Polyakov bootstrap for the conformal field theories in Mellin space on a firm foundation, thereby fixing the contact term ambiguities in the crossing symmetric blocks. Our new approach employs certain “locality” constraints replacing the requirement of crossing symmetry in the usual fixed-t dispersion relation. Using these constraints, we show that the sum rules based on the two channel dispersion relations and the present dispersion relations are identical. Our framework allows us to connect with the conceptually rich picture of the Polyakov blocks being identified with Witten diagrams in anti-de Sitter space. We also give two sided bounds for Wilson coefficients for effective field theories in anti-de Sitter space.

DOI: 10.1103/PhysRevLett.126.211602

Introduction.—Conformal field theories (CFTs) are cen- remained obscure. Indeed, in [10], it was pointed out that tral to our modern understanding of strongly interacting contact terms need to be added to the basis; however, it was systems in high energy physics, condensed matter physics, not clear what would fix these terms. In the special case of and statistical mechanics. It is therefore crucial to develop 1D CFT, this issue was resolved in [11–13]; unfortunately, calculational tools to extract the dynamics of these theories. these methods do not carry over to higher dimensions. Four point correlation functions in CFTs are constrained by Recent works [14–17] have endeavored to clarify the crossing symmetry and unitarity in addition to the con- nonperturbative validity of the Polyakov bootstrap in d ≥ formal symmetries. The usual bootstrap consistency con- 2 by considering dispersion relations in Mellin space. ditions impose crossing symmetry on the conformal block However, these relations exhibit only two channel sym- expansion, and this places strong constraints on the scaling metry, with crossing symmetry additionally imposed, dimensions of operators as well as their three point thereby going against the spirit of Polyakov’s original functions. work. A nonperturbative version of the crossing symmetric An alternative approach is to make the crossing sym- bootstrap would no doubt shed light on the very successful metry manifest at the outset and instead impose other numerical developments arising from bootstrapping posi- consistency requirements. In 1974, Polyakov [1] proposed tion space correlators [18,19]. Furthermore, through the a version of the conformal bootstrap that used crossing connection to Witten diagrams, such an approach is the symmetric blocks. Consistency with the operator product natural one for CFTs with a large radius AdS dual. expansion (OPE) led to dynamical constraints. This idea In this Letter, we will consider a manifestly crossing lay dormant until recently [2,3], when its power was symmetric dispersion relation for Mellin amplitudes, focus- demonstrated in Mellin space [3,4]. The latter is the natural ing on identical scalars. This builds on certain relatively habitat for conformal correlators, playing the role that obscure investigations in the 1970s [20], whose utility was momentum space does for flat space scattering amplitudes recently demonstrated in the context of quantum field [5–9]. theories (QFTs) [21]. In this approach, we will employ a An attractive feature of this approach is that the basis of crossing symmetric parameterization of the Mellin varia- Polyakov is essentially that of exchange Witten diagrams bles ðs; tÞ (which are analogous to the usual Mandelstam used in the anti-de Sitter (AdS)/CFT correspondence [3,4]. invariants) and write the corresponding dispersion relation. Nevertheless, the nonperturbative validity of this approach This is to be contrasted with the more conventional fixed-t dispersion relation, which has symmetry manifest only in the ðs; uÞ channels [14,15]. The price we pay for making crossing symmetry mani- Published by the American Physical Society under the terms of fest is the potential presence of “nonlocal" singularities. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to More precisely, our amplitude will generally have unphys- the author(s) and the published article’s title, journal citation, ical poles in the Mellin variables (or rather, crossing and DOI. Funded by SCOAP3. symmetric combinations thereof). Thus, in our approach,

0031-9007=21=126(21)=211602(6) 211602-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 126, 211602 (2021) we need to impose “locality constraints” [Eq. (5)] for where zk are cube roots of unity. Note that a ¼ consistency. This then leads to our central claim: Once the ½s1s2s3=ðs1s2 þ s2s3 þ s3s1Þ is crossing symmetric. locality constraints are imposed, the crossing symmetric Mðs1;s2Þ is an analytic function of ðz; aÞ. Unlike the S- dispersion relations admit a Witten diagram expansion and, matrix amplitude, CFT Mellin amplitudes have poles in the moreover, all contact terms are fixed, as proposed in [10], twist of the exchanged operator (with accumulation points) up to a constant. rather than cuts. Instead of a dispersion relation in s for We should emphasize that the way the Witten diagrams fixed t, we now write (a twice subtracted) relation in z emerge from the crossing symmetric kernel is quite non- for fixed a,asin[20,21]. In our case, this becomes, trivial and only happens after imposition of the constraints. following steps similar to [20,21] (see supplementary Compared to the approach of [14,15], our locality con- material A of [21]), straints appear to be equivalent to the sum rules arising Z 1 ∞ ds0 from requiring crossing symmetry (whereas their expres- M s ;s α 1 A s0 ; s0 s0 ;a ð 1 2Þ¼ 0 þ 0 ½ 1 2ð 1 Þ sions do not have the unphysical poles above). This π τð0Þ s1 demonstrates that the expansion in Witten diagrams is 0 × Hðs ; s1;s2;s3Þ; ð2Þ robust and captures the requirements of both crossing 1 symmetry and locality, as might be expected. In our where α0 ¼ Mðs1 ¼ 0;s2 ¼ 0Þ is a subtraction constant, approach and that of the fixed-t dispersion relations, one Aðs1; s2Þ is the s-channel discontinuity, and then proceeds to impose the so-called Polyakov conditions. This is the requirement that there are no spurious double s s s H s; s ;s ;s 1 2 3 trace operators in the spectrum. It enables us to bootstrap ð 1 2 3Þ¼ s − s þ s − s þ s − s the dynamical conformal data of scaling dimensions and 1 2 3 OPE coefficients. s s þ 3a 1=2 s0 ðs; aÞ¼− 1 − ; ð3Þ We note that, in the context of QFTs, the analogous 2 2 s − a locality constraints [22] were also argued [21] to be equivalent to the crossing symmetry conditions for the which defines the crossing symmetric kernel H [20,21]. 0 fixed-t dispersion relation for effective field theories Here we have solved for s2 for fixed a in terms of the other 0 [23,24]. Analogously, we will also consider bounds on independent variable s1 from the defining relation below the Wilson coefficients of the Mellin amplitudes for [Eq. (1)]. Note that Eq. (2) is manifestly three channel 0 effective scalar field theories in AdS space. We give crossing symmetric since the dependence of s2 is only preliminary results for two sided bounds similar to those through a. In the Supplemental Material [26],wehave for QFTs [21,23,24]. numerically investigated how well Eq. (2) represents the Crossing symmetric dispersion relation.—We define the 2D Ising model Mellin amplitude. We have also examined Mellin amplitude Mðs1;s2Þ for a four point identical scalar there the conformal partial wave expansion from which we CFT correlator as can work out the domain of a where Eq. (2) converges. Let ð0Þ Z τ be the starting point of the chain of poles in s1. ds1 ds2 s1þ2Δϕ=3 s2−Δϕ=3 Following [15], the conformal partial wave expansion Gðu; vÞ¼ u v μðsiÞMðs1;s2Þ; 2πi 2πi converges when −τð0Þ=3 ≤ ReðaÞ < 2τð0Þ=3 for τð0Þ > 0 and τð0Þ < ReðaÞ < 2τð0Þ=3 for τð0Þ < 0. where s3 ¼ −s1 − s2, the measure factor Locality constraints.—The Mellin amplitude of identical 2 2 2 μðsiÞ¼Γ ðΔϕ=3 − s1ÞΓ ðΔϕ=3 − s2ÞΓ ðΔϕ=3 − s3Þ, and scalars has complete crossing symmetry and hence admits a Gðu; vÞ is the position space conformal correlator. Δϕ is manifestly symmetric expansion, the scaling dimension of the external scalars. M s1;s2 is ð Þ X∞ X∞ analogous to the usual flat space scattering amplitude, and p q pþq q Mðs1;s2Þ¼ Mp;qx y ¼ Mp;qx a ; ð4Þ s1, s2 are analogous to the Mandelstam invariants [related p;q¼0 p;q¼0 to the usual (s, t) by a shift [25] ]. Motivated by S-matrix amplitudes, where we can write a crossing symmetric with x ¼ −ðs1s2 þ s2s3 þ s3s1Þ and y ¼ −s1s2s3 (with dispersion relation [20,21], we will write a similar relation y ¼ ax). It is important that only positive powers of x, y for Mðs1;s2Þ, assuming its nonperturbative existence in a appear in this expansion around the crossing symmetric region of the complex plane [14]. We consider hyper- point ðx ¼ y ¼ 0Þ so as not to have unphysical singular- 3 surfaces ðs1 − aÞðs2 − aÞðs3 − aÞ¼−a , with a being a ities. However, as we will see, this will not be obviously real parameter. The si can then be parameterized as evident for our dispersion relation. Thus, we will need to impose the nontrivial constraints 3 aðz − zkÞ skðz; aÞ¼a − ;k¼ 1; 2; 3; ð1Þ z3 − 1 Mp;q ¼ 0;p<0: ð5Þ

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(Note: q ≥ 0 is built into our formalism.) In the fixed-t Witten diagram [Eq. (8)], the prefactor is the usual poly- dispersion relation, there are no such negative powers and nomial dependence on the Mellin variables in PΔ;lðs1;s2Þ, the corresponding sum rules turn out to follow from while the prefactor in Eq. (6) is the same Mack polynomial 0 requiring crossing invariance [27]. In our approach, on but with an argument s2,asshowninEq.(7). 0 the other hand, the constraints in Eq. (5) are the only Now, from Eq. (3),weseethats2ðτk;aÞ has singularities additional ones we need to impose, apart from the Polyakov when a ¼ y=x ¼ τk. These are unphysical since they indi- conditions to be discussed below. vidually give terms in the expansion that should not be Crossing symmetric block expansion.—The s-channel present in the full amplitude. Indeed, as we will see explicitly discontinuity in Eq. (2) comes from the series of poles in below, they will give terms with negative powers of x and the twist (including accumulation points): thus an expansion that is not of the form in Eq. (4). Note H τ ; s ;s ;s x 2τ − 3a = xa − xτ τ 3 X∞ that ð k 1 2 3Þ¼f½ ð k Þ ½ k þð kÞ g; ðkÞ therefore, we can expand Eq. (6) around a 0, x 0 Aðs1;s2Þ¼π cΔ;lPΔ;lðτk;s2Þδðτk − s1Þ: ¼ ¼ Δ;l;k to get only positive powers in the expansion [Eq. (4)]. Thus, what we need to do is to simultaneously expand ðkÞ Here cΔ;l is proportional to the square of the OPE Eq. (7) in powers of a and set the net negative powers of x to coefficient and is explicitly given in the Supplemental zero, as per Eq. (5). Since the meromorphic pieces of Eq. (6) Material. The PΔ;lðs1;s2Þ are the Mack polynomials that coincide with those of the exchange Witten diagrams, the net are the building blocks of the conformal partial wave result is to remove the spurious singularities coming 0 expansion and are also given for reference in the from s2ðτk;aÞ. This leaves a finite number of polynomial Supplemental Material. We have also defined the twist τk ¼ contactpieces,whichareallnowfixeduptoaconstant. ðΔ − lÞ=2 þ k − ð2Δϕ=3Þ with integer k ≥ 0. Using these, Let us illustrate the procedure. Consider the difference ð0Þ the dispersion relation [Eq. (2)] reads as [note: τk ≥ τ ] X 1 ðΔÞ ðiÞ Dl ¼ Ql;k ðaÞHðτk; s1;s2;s3Þ − MΔ;l;kðs1;s2Þ: ∞ ðkÞ τk X c i¼s;t;u M s ;s α Δ;l QðΔÞ a H τ ; s ;s ;s ; ð 1 2Þ¼ 0 þ τ l;k ð Þ ð k 1 2 3Þ ð6Þ Δ;l;k k Given that the Mack polynomial is of order l in the variables, ðΔÞ Ql;k ðaÞ and thus the difference Dl will have an unphysical where s3 ¼ −s1 − s2 and pole of order l=2 at a ¼ τk. As a concrete example, take the ðΔÞ 0 l ¼ 2 block. The Mack polynomial has a finite expansion Ql;k ðaÞ ≡ PΔ;l½τk;s2ðτk;aÞ: ð7Þ P P s ;s mþn≤l s m s nbðlÞ Δ;lð 1 2Þ¼ m¼0;n¼0ð 1Þ ð 2Þ m;n. Thus, in our case, The final answer [Eq. (6)] is fully crossing symmetric. Notice 2 2 2 P s ;s s2bð Þ s2bð Þ s s bð Þ s0 τ ;a s we have Δ;l¼2ð 1 2Þ¼ 2 0;2 þ 1 2;0 þ 1 2 1;1þ that since 2ð k Þs1¼τk ¼ 2,Eq.(6) gives the correct s bð2Þ s bð2Þ bð2Þ residues at the poles τk in each channel. In the 1 1;0 þ 2 0;1 þ 0;0, and therefore we find Supplemental Material, we perform several checks of the ð2Þ ð2Þ ð2Þ convergence of the representation [Eq. (6)]. xb0;2 xðb0;2 þ 2b2;0Þ — D : Witten diagram expansion. We can now relate the l¼2 ¼ a − τ þ τ ð9Þ block expansion in Eq. (6) to the Witten diagram expan- k k sion. Note that the s-channel Witten diagram can be written As expected, there is a single pole at a ¼ y=x ¼ τk that gives in terms of the meromorphic pieces [10]: rise to negative powers in x, y,aswellasadditional D 1 1 polynomial pieces. In other words, l¼2 takes the form MðsÞ s ;s P s ;s − : Δ;l;kð 1 2Þ¼ Δ;lð 1 2Þ τ − s τ ð8Þ k 1 k 2xbð2Þ ybð2Þ X∞ D 2;0 − 0;2 − xbð2Þ anτ−n−1: l¼2 ¼ 2 ½ 0;2 k ð10Þ Here, for convenience, we have subtracted an additional τk τ k n¼2 polynomial piece ∝ 1=τk compared to [10]. The t, ðtÞ u-channel Witten diagrams are related via MΔ;l;kðs1;s2Þ¼ The last term will not contribute once the requirement of ðsÞ ðuÞ ðsÞ Eq. (5) is imposed (on the full amplitude). The first two terms MΔ;l;kðs2;s3Þ, MΔ;l;kðs1;s2Þ¼MΔ;l;kðs3;s1Þ. are polynomials as expected and given by The key observation is the following. Since ½1=ðτk − s1Þ − 1=τk¼ð1=τkÞ½s1=ðτ − s1Þ, the crossing 2xbð2Þ ybð2Þ symmetric kernel Hðτk; s1;s2;s3Þ in Eq. (3) also has the MðcÞ s ;s 2;0 − 0;2 : Δ;l¼2;kð 1 2Þ¼ 2 ð11Þ same meromorphic pieces, in each channel, as the Witten τk τk exchange diagrams. However, the difference between the expansion [Eq. (6)] and the expansion in terms of the It should be clear that similar arguments hold for all l—the individual Witten diagrams arises in the prefactor. In the corresponding Dl will have spurious singularities that

211602-3 PHYSICAL REVIEW LETTERS 126, 211602 (2021) whenexpandedinpowersofa give negative powers of x X∞ ω s cðkÞ P τ ;s except for a finite number of pieces that are polynomials in x, p1;p2;p3 ð 2Þ¼ Δ;l Δ;lð k 2Þ Δ;l;k y. Once the locality constraints are imposed on the full amplitude, only these polynomial pieces survive. The 1 × Q l 4 2 Δϕ Δϕ ¼ case is also discussed explicitly in the Supplemental ð þ pi − τkÞð þ p3 þ τk þ s2Þ i¼1 3 3 Material. 1 We have therefore argued for the Witten diagram − Q : 2 Δϕ p s τ Δϕ p − τ expansion of the Mellin amplitude as i¼1 ð 3 þ i þ 2 þ kÞð 3 þ 3 kÞ X∞ X ð15Þ ðkÞ ðcÞ ðiÞ Mðs1;s2Þ¼ cΔ;l MΔ;l;kðs1;s2Þþ MΔ;l;kðs1;s2Þ : Δ;l;k i s;t;u ω s Ω s ¼ If we now Taylor expand p1;p2;p3 ð 2Þ and p1;p2;p3 ð 2Þ s 0 ð12Þ around 2 ¼ , we get, respectively,

∞ c X ð Þ r ðrÞ MΔ;l;kðs1;s2Þ are the polynomial pieces (for each l)inDl Ω s s Ω p1;p2;p3 ð 2Þ¼ ð 2Þ p1;p2;p3 after removing the spurious singularities at a ¼ τk. These r¼0 are precisely the contact terms that were not fixed in [10]. X∞ r ðrÞ ðcÞ ω s s ω l 0 M s ;s p1;p2;p3 ð 2Þ¼ ð 2Þ p1;p2;p3 ð16Þ Note that for the ¼ case, Δ;l¼0;kð 1 2Þ is a constant r¼0 that gives the subtraction constant α0.Ind ¼ 1, where there are no spins, this is the only contact term, as discussed in ωðiÞ ΩðiÞ i 1; …; 5 i and find that p1;p2;p3 ¼ p1;p2;p3 for ¼ . From ¼ detail in [13]. Our conclusion, then, is that the contact term 6 ambiguity anticipated in [10] is fixed for any spin, and the onward, there are nontrivial relations. To give a flavor of these relations, we exhibit, for r ¼ 6 and r ¼ 7 (where we Witten diagram expansion holds once the locality con- p straints are satisfied. Furthermore, the explicit form of the have put some of the i to zero), contact term shows why it did not contribute to the Oðϵ3Þ ωð6Þ − Ωð6Þ ∝ M results in [3,10], i.e., it can been seen to start contributing p1;p2;0 p1;p2;0 −1;2 ð17Þ only at Oðϵ4Þ. Polyakov conditions and sum rules.—The measure factor and μðsiÞ in the definition of the Mellin amplitude introduces Δ Δ 3p additional spurious poles in the full amplitude. We ωð7Þ − Ωð7Þ ∝ M − ϕð ϕ þ 1Þ M : p1;0;0 p1;0;0 −1;2 −2;3 ð18Þ must therefore impose the condition that this expansion 2Δϕ þ 3p1 not have the poles for double trace operators with Δ 2Δ l 2p M s ;s ¼ ϕ þ þ . Thus, ð 1 2Þ must vanish at The explicit expressions for the Mp;q appearing here are s1 ¼ Δϕ=3 þ p [3,10,14], which are the so-called given below in Eq. (19). What these relations indicate is Polyakov conditions. From Eq. (6), we get that once the locality constraints [Eq. (5)] are imposed, our Polyakov conditions are the same as those in [15].Wehave Δ ϕ checked that similar relations hold for any ðp1;p2;p3Þ for Fpðs2Þ ≡ M s1 ¼ þ p; s2 ¼ 0: ð13Þ 3 higher r. Positivity conditions for CFTs.—Positivity conditions on Derivation of sum rules in [15].—We define the combi- amplitudes are important for putting bounds on the Wilson ≥0 nations for pi ∈ Z coefficients of effective field theories (EFTs) in a dual AdS spacetime. We obtain them in our formalism following F s p1 ð 2Þ [21]. From Eq. (6) (see Supplemental Material), we find Ωp ;p ;p s2 ≡ − 1 2 3 ð Þ 2Δϕ ðp1 − p2Þðp1 þ p3 þ s2 þ Þ 3 X∞ k Δ;l;k Fp ðs2Þ M cð Þ Bð Þ;n≥ 1: − 2 n−m;m ¼ Δ;l n;m ð19Þ 2Δϕ Δ;l;k ðp2 − p1Þðp2 þ p3 þ s2 þ 3 Þ Δ;l;k Fp ðs2Þ ð Þ − 3 ; Here Bn;m is given by 2Δϕ 2Δϕ ðp1 þ p3 þ s2 þ Þðp2 þ p3 þ s2 þ Þ 3 3 Xm ðΔ;l;kÞ ðτkÞ mþq ð14Þ Bn;m ¼ Un;m;qð−1Þ PΔ;l;qðτk; 0Þ; ð20Þ q¼0 with Fp defined in (13). We can now compare Ωp ;p ;p ðs2Þ 1 2 3 q ω P τ ; 0 ∂s P τ ;s to the functional p1;p2;p3 in [15]. In our notation, where Δ;l;qð k Þ¼ 2 Δ;lð k 2Þjs2¼0 and

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n − q − 1 ! m 2n − 3q δ ðτkÞ ð Þ ð þ Þ where 0 is related to the EFT scale. Besides the fact that a Un;m;q ¼ mþ2n−qþ1 q!ðn − mÞ!ðm − qÞ!ðτkÞ two-sided bound in AdS space is novel, the derivation using q 1 ; q ;q− m; q 1 − 2nþm our approach is algebraically simpler compared to the 2 þ 2 2 þ 3 fixed-t dispersion. In the future, it will be desirable to × 4F3 ;4 : ð21Þ q 1;q 1 − n; q − 2nþm ðΔ;l;kÞ þ þ 3 understand the properties of Bm;n better. Discussion.—The new framework we have developed in τ ≥ 0 One can check numerically that (for k¼0 ) this Letter has allowed us to make contact with Polyakov’s P τ ; 0 ≥ 0 Δ;l;qð k Þ . A similar claim was made in [14]. original crossing symmetric bootstrap proposal and has ðτkÞ Also we find that Un;m;q ≥ 0.Asin[21], we can search shown the equivalence, in a nontrivial way, with the results ðr;mÞ of [14]. Furthermore, this approach clarifies the underlying for χn ðτkÞ such that reason for the existence of Witten diagram representations Xm for general CFT correlators. To this end, it will be good to ðr;mÞ ðΔ;l;kÞ ðτkÞ mþq χn ðτkÞBn;r ¼ Un;m;mð−1Þ PΔ;l;qðτk; 0Þ ≥ 0: broaden the preliminary discussion in the Supplemental r¼0 Material on the convergence of the Witten diagram expansion and apply it to new settings. It will also be ð0Þ Next, define M ðs1;s2Þ, the amplitude, after subtracting useful to compare the results developed in this Letter to the 0 off the twist zero contributions (if any). Let τð Þ denote the position space results in, e.g., [29,30]. Mellin space τ minimum nonzero twist k¼0. Using this in Eq. (19), we can expressions, in turn, permit ready comparison with flat ð0Þ write down positivity constraints on Mn−m;m similar to space results. Many of the positivity conditions for (com- those in [21] [with τð0Þ ≥ 0]: binations of) Gegenbauer polynomials appear to extend to Mack polynomials. Mathematically, it will be useful to Xm−1 develop proofs of these relations, which will enable a ð0Þ ðr;mÞ 0 ð0Þ Mn−m;m þ χn ½τð ÞMn−r;r ≥ 0: ð22Þ systematic study of corrections to EFTs in AdS space in the r¼0 large radius limit. Finally, we hope that this work will give new impetus to numerical bootstrap studies using χ A recursion relation for n was worked out in [21].For the constraints and functionals [such as in Eq. (14)] ð0;1Þ ð0;2Þ example [28], χn ðτkÞ¼ð2n þ 1Þ=2τk, χn ðτkÞ¼ defined here. 2n n 2 3 =4 τ 2 χð1;2Þ τ 2n 1 =2τ ½ ð þ Þþ ð kÞ , n ð kÞ¼ð þ Þ k. These We thank Parijat Dey, Kausik Ghosh, Apratim Kaviraj, ð0Þ ð0Þ ð0Þ imply, for instance, M0;1 > −½3=2τ M1;0. and Miguel Paulos for useful discussions. R. G.’s research Two-sided bounds.—If we assume that higher spin is supported by a J. C. Bose fellowship of the DST as well partial waves are suppressed, then we can argue for the as project RTI4001 of the Department of Atomic Energy, existence of two-sided bounds. We can check numerically Government of India. l < l that for , there always exists some positive β O l ¼ ð Þ such that [1] A. M. Polyakov, Nonhamiltonian approach to conformal Δ;l;k Δ;l;k α 3α Bð Þ < βBð Þ; ∀ Δ ∈ ; ; 66 1;1 1;0 ϕ 2 4 ð23Þ , Zh. Eksp. Teor. Fiz. , 23 (1974). [2] K. Sen and A. Sinha, On critical exponents without Feynman diagrams, J. Phys. A 49, 445401 (2016). where α ¼ d=2 − 1. Therefore, we get a two-sided bound [3] R. Gopakumar, A. Kaviraj, K. Sen, and A. Sinha, 3 valid for α < 2Δϕ < 2 α: Conformal Bootstrap in Mellin Space, Phys. Rev. Lett. 118, 081601 (2017).R. Gopakumar, A. Kaviraj, K. Sen, and 3 0 0 0 A. Sinha, A Mellin space approach to the conformal − Mð Þ < Mð Þ < βMð Þ: ð24Þ 2τð0Þ 1;0 0;1 1;0 bootstrap, J. High Energy Phys. 05 (2017) 027. [4] P. Dey, A. Kaviraj, and A. Sinha, Mellin space bootstrap for To derive a stronger upper bound analogous to the QFT global symmetry, J. High Energy Phys. 07 (2017) 019; P. bounds in [23,24], we will need to incorporate the locality Dey, K. Ghosh, and A. Sinha, Simplifying large spin constraints as well, a detailed study of which we leave for bootstrap in Mellin space, J. High Energy Phys. 01 (2018) 152. future work. In the case of AdS EFT for a scalar with mass m R [5] G. Mack, D-independent representation of conformal , with the AdS radius given by , we show in the field theories in D dimensions via transformation to mR ≫ 1 Supplemental Material that, for , auxiliary dual resonance models. Scalar amplitudes, arXiv:0907.2407. α 10α 11 MðAdSÞ < 1 ð þ Þ MðAdSÞ; [6] J. Penedones, Writing CFT correlation functions as AdS 0;1 þ 2 2 1;0 2ð2α þ 3Þm R ð2α þ 1Þδ0 scattering amplitudes, J. High Energy Phys. 03 (2011) 025. [7] M. F. Paulos, Towards Feynman rules for Mellin ampli- ð25Þ tudes, J. High Energy Phys. 10 (2011) 074.

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5 [8] L. Rastelli and X. Zhou, Mellin Amplitudes for AdS5 × S , [21] A. Sinha and A. Zahed, Crossing Symmetric Dispersion Phys. Rev. Lett. 118, 091602 (2017). Relations in QFTs, Phys. Rev. Lett. 126, 181601 (2021). [9] L. F. Alday, On Genus-one String Amplitudes on [22] In [23], these constraints were termed as “null constraints.” 5 AdS5 × S , J. High Energy Phys. 04 (2021) 005. [23] S. Caron-Huot and V. Van Duong, Extremal effective field [10] R. Gopakumar and A. Sinha, On the Polyakov-Mellin theories, arXiv:2011.02957. bootstrap, J. High Energy Phys. 12 (2018) 040. [24] A. J. Tolley, Z. Y. Wang, and S. Y. Zhou, New positivity [11] D. Mazac and M. F. Paulos, The analytic functional boot- bounds from full crossing symmetry, arXiv:2011.02400. strap. Part II. Natural bases for the crossing equation, J. [25] The notation in [10] was in terms of s, t, u, which are related High Energy Phys. 02 (2019) 163. to our si via s1 ¼ s − 2Δϕ=3, s2 ¼ t − 2Δϕ=3, [12] D. Mazáč, A crossing-symmetric OPE inversion formula, J. s3 ¼ u − 2Δϕ=3. High Energy Phys. 06 (2019) 082. [26] See Supplemental Material at http://link.aps.org/supplemental/ [13] P. Ferrero, K. Ghosh, A. Sinha, and A. Zahed, Crossing 10.1103/PhysRevLett.126.211602 for additional mathematical symmetry, transcendentality and the Regge behaviour of 1d details. CFTs, J. High Energy Phys. 07 (2020) 170. [27] For example, Eq. (141) in [14] is proportional M−1;2.In [14] J. Penedones, J. A. Silva, and A. Zhiboedov, Nonperturba- [16], these are called the “odd-spin” constraints. In fact, in tive Mellin amplitudes: Existence, properties, applications, the QFT context, the analogous conditions were termed as J. High Energy Phys. 08 (2020) 031. “null constraints” in [23].In[21], it was shown that these [15] D. Carmi, J. Penedones, J. A. Silva, and A. Zhiboedov, are, again, precisely what we term locality constraints. Our Applications of dispersive sum rules: ϵ-expansion and formalism allows us to write these constraints in complete holography, arXiv:2009.13506. generality. [16] S. Caron-Huot, D. Mazac, L. Rastelli, and D. Simmons- [28] As in [21], we find a recursion relation: Duffin, Dispersive CFT sum rules, arXiv:2008.04931. [17] C. Sleight and M. Taronna, The unique Polyakov blocks, J. High Energy Phys. 11 (2020) 075. Xm ðτkÞ Un;j;r ðr;mÞ jþrþ1 ðj;mÞ [18] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, χn ðτkÞ¼ ð−1Þ χn ðτkÞ ð26Þ UðτkÞ Bounding scalar operator dimensions in 4D CFT, J. High j¼rþ1 n;r;r Energy Phys. 12 (2008) 031. [19] D. Poland, S. Rychkov, and A. Vichi, The conformal bootstrap: Theory, numerical techniques, and applications, [29] A. Bissi, P. Dey, and T. Hansen, Dispersion relation for CFT Rev. Mod. Phys. 91, 015002 (2019). four-point functions, J. High Energy Phys. 04 (2020) 092. [20] G. Auberson and N. N. Khuri, Rigorous parametric [30] M. F. Paulos, Dispersion relations and exact bounds on CFT dispersion representation with three-channel symmetry, correlators, arXiv:2012.10454. Phys. Rev. D 6, 2953 (1972).

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