PPrepared for submission to JHEP
OPE inversion in Mellin space
Carlos Cardona a a Niels Bohr International Academy and Discovery Center,University of Copenhagen, Niels Bohr InstituteBlegamsvej 17, DK-2100 Copenhagen , Denmark
E-mail: [email protected]
Abstract:
The fundamental ingredients that build the observables in conformal field theoryare the spectrum of operators and the OPE coefficients, or equivalently, the two and three-point functions of the theory. Recently an inversion formula solving the OPE coefficientsby a convolution over the light-cone double-discontinuities of the correlator has been foundby Simon Caron-Huot. Taking into account that the same OPE data determine the Mellinamplitude representation of the correlator, motivate us to look for an analogous inversionformula in Mellin space, which we develops partially on this paper. a r X i v : . [ h e p - t h ] A p r ontents The basic building blocks of correlation functions of local operators in conformal field theoriesare given by the spectrum and the OPE coefficients. Despite intense study of interactingconformal field theories over the last couple of decades, we still lack of a general frameworkthat allow us to compute those fundamental blocks from first principles and possibly atany coupling. The most promising candidate so far for such a framework is the conformalbootstrap program, which adheres to the ideal that physical observables, such as correlationfunction in conformal field theories, should be constrained or even solved by imposing onthem a minimal set of physical requirements, such as unitarity, locality, crossing, space-timesymmetries and perhaps a general set of inner symmetries.On the CFT department of this program, there has been a successful resurgence ofthis idea involving many new intriguing and exciting results. From the numerical approach,tremendous progress has been made mostly boosted by the techniques developed in [1], wherethe problem of bounding operator dimensions by crossing symmetry conditions was revisited.Among the most popular subsequent applications of this new techniques concerns to thefamous 3D Ising model [2–5]. The amount of work in this direction is so vast that we will noteven try to summarize it in this introduction, but instead we refer the reader to some niceupdated reviews on the topic [6, 7].On the analytic side of the story, the recent progress is as exciting and impressive. Bystudying the crossing equation of a four point correlation function of scalars as an expansion– 1 –n inverse powers of the spin for large spin exchanges [8–13], it has been developed a “pertur-bation theory on spin”[14] that can be applied equally at weak and strong coupling regimes.This has been used successfully on a number of examples, remarkably giving accurate resultseven for operators of spin as large as (or better, as small as) two [5, 13]. Another similar limitthat allows to constraint the OPE coefficients is the Regge limit at high energy scatteringwhere some interesting progress has also been made [15, 16]. In applications to AdS/CFTduality, more particularly to N = 4 SYM, the analytic boostrap have been used recently tocompute the OPE coefficients of operators in the stress-tensor multiplet [17–19] which latelyshould corresponds to loop corrections in AdS space, where not much is know so far, althoughsome modest progress have been done lately [20–24]In principle, boostrap techniques based on crossing symmetry are not exclusive for con-formal field theories and it should be possible to extend them to constraint S-matrix elementsof general field theories. Some important considerations on that regard were made lately aswell [25–27]Even more recently, Simon Caron-Huot developed the CFT analogous of the Froissart-Gribov formula for partial wave S-matrix expansion, which inverts the conformal block expan-sion in cross-ratios space and allows to write the OPE coefficients in terms of a convolutionover the double discontinuities of the four point correlation function, with a kernel corre-sponding to a particular conformal block (in practice, the collinear limit of it) [28]. Explicitlythe inversion formula looks like, c t ∆ ,J = κ β π (cid:90) dzd ¯ zµ ( z, ¯ z ) G ( J + d − − d ( z, ¯ z ) dDisc [ G ( z, ¯ z )] , (1.1)where dDisc [ G ( z, ¯ z )] denotes the double discontinuity on Lorentz signature arising by crossingthe branch cuts from the light-cone singularities, µ is a measure factor and the coordinates( z, ¯ z ) are esencially the cross-ratios. This formula was re-derived in coordinate space in[29] where instead of using the Froissart-Gribov trick, the authors used the orthogonalityrelation between partial waves and its shadows to invert the euclidean OPE to later Wickrotate to Lorentz signature in order to expose the light-cone branch cuts on which the doublediscontinuities arise .Equation (1.1) is the main motivation of this short note. Here we take some steps informulating an analogous inversion formula in Mellin space. As we will see in the main bodyof this work, it is possible to invert the conformal block expansion in Mellin space by usingortogonality relations among the Mellin residues. The Mellin amplitude representation of thecollinear limit of a four point function can be written in terms of almost-Hanh polynomials[15, 31, 32] which satisfy the desired orthogonality. It is then expected, than the full Mellinrepresentation of a given four-point function preserve this ortogonality property, since it canbe written in terms of Mack Polynomials [33] which likely should form an orthogonal basis . See also [30], where is also shown how the Lorentzian blocks can be derived directly by inspection of theboundary conditions of the conformal Casimirs To my knowledge, up today there is not a prove for the orthogonality of Mack polynomials. – 2 –ven in using a fomula such as (1.1) in cross-ratios space, one can see that the Mellin rep-resentation have the advantage that the cross-ratios dependence is through simple monomialsand hence it is expected that the computation of discontinuities would be easier to performin Mellin space than over more complicated functions of the cross-ratios. This fact has beenexploited before for example in [34]. The double discontinuity across the light-cone branchcuts in (1.1) is responsible for killing the double twist contributions to the OPE coefficient.In Mellin space it translates to introducing zeros on the Mellin variables at the right positionsto cancel the poles associated to the double twist operators.Even thought it might be possible that a Froissart-Gribov type of contour deformationwhich introduces the proper zeros exist purely in Mellin space, in this work we are not intendedto study those deformations, but rather are going to use the doble-discontinuity as an input.Finally, let us end this introduction by highlight some important applications of Mellinspace in conformal field theory. It has been remarkable useful and enlightening the applicationof it in taking the Regge limit on CFT correlation functions [15, 16], as well as in takingadvantadge of the S-matrix character of the Mellin amplitudes to extract information aboutcorrelation functions [35, 36], or even in some attempts to do the inverse trick and insteaduse the correlator knowledge to gain information on the S-matrix [37, 38]. Also very recentlya Mellin representation for half-BPS four-point functions in N = 4 SYM has been foundin [39, 40]. More aligned to the ideas of this paper, a very interesting boostrap approachin Mellin space has been developed in the works [32, 41–43] , where unlike here, crossingsymmetry is guaranteed by construction and the bootstrap equations are given by imposingthe conditions that kill double twist contributions instead. We will consider a conformal correlation function of four scalar primary operators. Conformalinvariance dictates that up to a fix prefactor, it is given only by a function of the cross ratios, u = z ¯ z = x x x x , v = (1 − z )(1 − ¯ z ) = x x x x , (2.1)as, (cid:42) (cid:89) i =1 φ i ( x i ) (cid:43) = 1( x ) ∆1+∆22 ( x ) ∆3+∆42 (cid:18) x x (cid:19) a (cid:18) x x (cid:19) b G ( u, v ) . (2.2)with a = ∆ − ∆ and b = ∆ − ∆ . The function G ( u, v ) can be expanded in terms of knownconformal blocks [45], G ( u, v ) = (cid:88) ∆ ,J c ∆ ,J G ( J )∆ ( u, v ) . (2.3)Which in even space-time dimension can be written as an expansion in terms of hypergeo-metric functions. For d = 4 we have, G ( J )∆ ( u, v ) = z ¯ z ¯ z − z (cid:16) k ∆ − J − ( z ) k ∆+ J (¯ z ) − k ∆+ J ( z ) k ∆ − J − (¯ z ) (cid:17) , (2.4) See also [44] where the Mellin bootstrap has been applied to study interacting theories in d > – 3 –ith k β ( z ) = z β/ F (cid:18) β a, β b ; β | z (cid:19) . (2.5)We want to utilize a Mellin representation for the conformal blocks [33], to do so we use theconventions in [46] to write, G ( u, v ) = (cid:90) ρ − i ∞ ρ + i ∞ ds dt (2 πi ) Γ( − t )Γ( − t − a − b )Γ( s + t + a )Γ( s + t + b )Γ (cid:0) τ − s (cid:1) Γ (cid:16) d − β − s (cid:17) M ( s, t ) u s v − ( s + t ) , (2.6)where M ( s, t ) is known as the Mellin amplitude and in the context of conformal field theorywas introduced by Mack in [33]. By comparing (2.6) to the conformal block expansion (2.3)we can conclude that it should exist an equivalent expansion in Mellin space as, M ( s, t ) = (cid:88) ∆ ,J c ∆ ,J m ( J ) ( s, t ) , (2.7)such as the transformation of the sub-amplitudes m ( J ) ( s, t ) correspond to the usual conformalblocks in cross-ratios g ( J )∆ ( u, v ), explicitly, g ( J )∆ ( u, v ) = (cid:90) ρ − i ∞ ρ + i ∞ ds dt (2 πi ) Γ( − t )Γ( − t − a − b )Γ( s + t + a )Γ( s + t + b )Γ (cid:0) τ − s (cid:1) Γ (cid:16) d − β − s (cid:17) m ( J ) ( s, t ) u s v − ( s + t ) , (2.8)where β = ∆+ J and τ = ∆ − J are the conformal spin and conformal twist respectively.Let us define for future reference some terms we are going to use along the main sections.We will refer to the subamplitudes m ( J ) ( s, t ) expanding (2.7) as Mellin blocks . We also define, γ λ,a = Γ( λ + a )Γ( λ − a ) , (2.9)In applications, we sometimes need the Mellin amplitude with the extra gamma functionsremoved, and consider the Mellin (inverse) transformation,˜ g ( u, v ) = (cid:90) ρ − i ∞ ρ + i ∞ ds dt (2 πi ) (cid:102) M ( s, t ) u s v − ( s + t ) , (2.10)we will call (cid:102) M the amputated Mellin amplitude. Finally, it is worth noticing that in therepresentation (2.8), crossing u ↔ v translates simply to m ( s, t, r ) = m ( r, t, s ) in Mellinspace, where the variables ( s, t, r ) satisfies the usual kinematical relation for four particles,namely, s + t + r = − (cid:88) i =1 ∆ i , (2.11)where ∆ i are the conformal weights of the scalar operators involved in the correlator.– 4 – .1 Primary “collinear” Mellin blocks The leading contribution to the collinear limit z → u → v → (1 − ¯ z )) in the coordinates-channel expansion of (2.4) is controlled by the collinear conformal blocks (2.5), u τ k β (1 − v ) = u τ (1 − v ) β F (cid:18) β a, β b , β | − v (cid:19) . (2.12)It can be seen that this block can be reproduced from the representation (2.8) by consideringthe leading residue at the pole s = τ from the Mellin amplitude block m ( J ) ( s, t ), which inturns can be expressed as [15, 32, 46], m ( β ) ( τ / , t ) = ( t + a ) τ/ ( t + b ) τ/ κ β P β/ ( a, b | t ) , (2.13)with κ β = γ β ,a γ τ ,b Γ( β )Γ( β − , (2.14)and P β/ ( a, b, τ | s ) = ( a ) β/ ( b ) β/ Γ (cid:16) β + 1 (cid:17) F (cid:20) − β , − t, β − a , b | (cid:21) , (2.15)where ( · · · ) n denotes the Pochhammer symbol . As it was previously noticed by G. Ko-rchemsky [31] and in [15, 32], the polynomials P β/ ( a, b | t ) are known in the literature ascontinuous Hahn polynomials and we have used the definition given at [47]. They satisfy theorthogonality relation [47] (cid:90) i ∞− i ∞ dt πi Γ( − t )Γ( − t − a − b )Γ (cid:16) t + τ a (cid:17) Γ (cid:16) t + τ b (cid:17) P J ( a, b, τ | t ) P K ( a, b, τ | t )= γ β ,a γ β ,b ( β −
1) Γ (∆ − δ J,K (2.16)However, the Mellin blocks (2.13) are not quite equal to Hanh polynomials due to the t − dependent factor in front of P β/ ( a, b, τ | s ) . Fortunately for us, the Mellin blocks stillsatisfy a similar orthogonality relation, which we will prove in the appendix and reads, (cid:90) i ∞− i ∞ dt πi Γ( − t )Γ( − t − a − b )Γ (cid:16) t + τ a (cid:17) Γ (cid:16) t + τ b (cid:17) m ( β ) ( τ / , t ) m ( β (cid:48) ) ( τ / , t )= Γ( β − κ β Γ (cid:16) β + 1 (cid:17) δ β,β (cid:48) (2.17) In this work, whenever we refer to s − channel or t − channel we mean in cross-ratios space, not in the Mellinvariables. In writing the Mellin block in this form, we have made the assumption that β/ β/ It is worth to mention that if we were defined the Mellin blocks in terms of the spin J instead of theconformal spin β , they would be equal to Hanh polynomials up to a normalization factor. – 5 –t the collinear limit, the four point function (2.3) can be expanded as, G ( u → , v ) ∼ (cid:88) β c τ,β u τ k β (1 − v ) . (2.18)Therefore we can use the Mellin block (2.13) to write down a corresponding Mellin blockdecomposition of the Mellin amplitude as, M ( τ / , t ) ∼ (cid:88) β c τ,β m ( β ) ( τ / , t ) . (2.19) By using the orthogonality relation (2.17) on (2.19) we get an inversion for the Mellin ampli-tude expansion in Mellin space. (cid:90) i ∞− i ∞ dt πi Γ( − t )Γ( − t − a − b )Γ (cid:16) t + τ a (cid:17) Γ (cid:16) t + τ b (cid:17) m ( β ) ( τ / , t )) M ( τ / , t )= c τ,β Γ( β − κ β Γ (cid:16) β + 1 (cid:17) . (2.20)Is worth remarking here that this formula is applicable on the region of validity of theexpansion (2.18), namely, for the collinear s − channel expansion. Obviously there is a similarformula valid for the analogous limit on a t − channel expansion due to cross symmetry inMellin space. Light-cone double-discontinuity
We are after an equation analogous to (1.1) in Mellin space, and hence we would like toconsider an inversion that does not contains contributions from double twist operators. Incross-ratios space, Caron-Huot realized that this is the role played by the light-cone disconti-nuities of the four point function, which can be isolated by using a clever Froissart-Grivot-likecontour deformation. Even thought it would be interesting to see whether or not such a con-tour deformation exist purely in Mellin space, in this work we will not intend to play a similargame and only are going to use it is as an input.Sticking at the collinear region, the double-discontinuity across the light-cone branch-cut u → G ( u, v )) = (cid:90) ρ − i ∞ ρ + i ∞ ds dt (2 πi ) Γ( − t )Γ( − t − a − b )Γ( s + t + a )Γ( s + t + b )Γ (cid:0) τ − s (cid:1) Γ (cid:16) d − β − s (cid:17) M dis ( s, t ) | u | s v − ( s + t ) , (2.21)where we have absorbed the phase from the discontinuity into a new object, M dis ( s, t ) ≡ M ( s, t ) sin ( π s ) π , (2.22)– 6 –uch that in practice, the actual inversion formula we are going to use is (cid:90) i ∞− i ∞ dt πi Γ( − t )Γ( − t − a − b )Γ (cid:16) t + τ a (cid:17) Γ (cid:16) t + τ b (cid:17) m ( β ) ( τ / , t ) M dis ( τ / , t )= c τ,β Γ( β − κ β Γ (cid:16) β + 1 (cid:17) . (2.23) In this section we would like to apply the inversion formula discussed in the section above tosome simple cases.
It would be convenient to use some examples that allow us to make a first comparison with theexpressions at [28]. Following [28], lets start with the vacuum contribution. At the collinearlimit we want to consider the t − channel block, f τ (cid:48) ( v, u ) = (cid:18) v − v (cid:19) τ (cid:48) + a (1 − v ) a . (3.1)The prime on τ (cid:48) indicates that this corresponds to the twist of the operators expandingthe crossed channel. Is easy to see that the corresponding amputated Mellin amplitudereproducing the function (3.1) is given by, (cid:102) M ( τ (cid:48) / a, t ) = Γ (cid:16) − τ (cid:48) (cid:17) Γ (cid:16) t + τ (cid:48) + a (cid:17) Γ ( t + a + 1) . (3.2)Therefore, the Mellin amplitude associated to the discontinuity is, (cid:102) M dis ( τ (cid:48) / a, t ) = Γ (cid:16) − τ (cid:48) (cid:17) Γ (cid:16) t + τ (cid:48) + a (cid:17) Γ ( t + a + 1) sin ( π ( τ (cid:48) / a )) π . (3.3)Inserting this expression into (2.23) we have, (cid:90) i ∞− i ∞ dt πi Γ( − t )Γ( − t − a − b )Γ (cid:18) t + τ (cid:48) a (cid:19) Γ (cid:18) t + τ (cid:48) b (cid:19) × m ( β ) ( τ (cid:48) / , t ) Γ (cid:16) − τ (cid:48) (cid:17) Γ (cid:16) t + τ (cid:48) + a (cid:17) Γ ( t + a + 1) sin ( π ( τ (cid:48) / a )) π = Γ( β − κ β Γ (cid:16) β + 1 (cid:17) c τ,β . (3.4)After using the series representation for the hypergeometric function F ( ... | z ) in the definitionof m ( β ) ( τ (cid:48) / , t ) and commuting the sum with the contour integral, it is straightforward toperform the integration to obtain, – 7 – τ (cid:48) ,β = Γ (cid:16) β − a (cid:17) Γ (cid:16) β + b (cid:17) Γ (cid:0) − τ (cid:48) − a (cid:1) Γ( β −
1) Γ (cid:16) β − τ (cid:48) − (cid:17) Γ (cid:16) β + τ (cid:48) + 1 (cid:17) ≡ I a,b τ (cid:48) , β , (3.5)where we have defined I a,b τ , β for future reference and to match the notation of [28].In the s-channel, the function f ( u, v ) (3.1) can be expanded in terms of collinear conformalblocks as [45] f ( u, v ) = ∞ (cid:88) k =0 c ∆ , k u ∆ (1 − v ) k F (cid:16) ∆ + k, ∆ + k,
2∆ + 2 k | − v (cid:17) . (3.6)where the coefficients c ∆ , k has been computed in [48] by solving the bootstrap equation forfree fields. Comparing this last expansion with (2.18) and (2.19) it can be also invertedstraightforwardly in Mellin space. We can use crossing on (3.1) to go to the s-channel, butin practice it is equivalent to use (3.5) by exchanging τ → − ∆ (as well as J → k and β/ → ∆ + k ). Taking also a = b = 0 corresponds to equal dimension operators . Aftermaking those replacements, we obtain the result from [48] c free∆+ k = I , − ∆ , ∆+ k = Γ(∆ + k ) Γ(∆) Γ(2∆ + k − k + 1)Γ(2∆ + 2 k + 1) . (3.7) We would like to considering here the OPE coefficients associated to double twist operatorswith large spin. In order to do so, we are going to use the same strategy followed by [8].As it has been argued in [8], large spin operators at low twist in the s-channel arecontrolled by the t-channel block expansion around u → v → u →
0, but since the series expansion ofthe hypergeometric function at (2.12) is around v = 1, in order to explore the region v → v = 0, which is given by (see for example [51]), F ( A, B, D | − v ) = v D − A − B Γ( D )Γ( A + B − D )Γ( B )Γ( A ) F ( D − A, D − B, − A − B + D | v )+ Γ( D )Γ( − A − B + D )Γ( D − B )Γ( D − A ) F ( A, B, A + B − D | v ) , (3.8)applying this formula on (2.12), we have at leading order around ( u, v ) → (0 , g ∆ ,τ = u τ (1 − v ) β (cid:18) Γ( β )Γ( − a − b )Γ( β/ − b )Γ( β/ − a ) + Γ( β )Γ( a + b )Γ( β/ a )Γ( β/ b ) v − a − b (cid:19) (3.9)one can see that when a + b = 0, the expansion around ( u, v ) → (0 ,
0) develops logarithmswhich are associated to the anomalous dimension of double twist operator. Extracting the I would like to thank Charlotte Sleight for kindly point it me out the distributional character of Mellinamplitudes for disconected correlators and as well as for call to my attention the related work [49, 50] – 8 –eading contribution to (2.12) at ( u, v ) → (0 ,
0) & a + b = 0, we obtain, g ( J )∆ ( u, v ) ∼ u τ (1 − v ) β Γ( β )Γ (cid:16) β + a (cid:17) Γ (cid:16) β − a (cid:17) ( − v ) . (3.10)To go to the crossed channel we need to exchange u ↔ v and multiply by a factor u ∆ v ∆ , (3.11)from here and on the remaining of this subsection we have taken equal dimension scalars forsimplicity in the expressions, i.e a = b = 0. At leading order in the crossed channel we have, g ( J )∆ ( v, u ) ∼ u ∆ v τ − ∆ Γ( β )Γ (cid:16) β (cid:17) ( − u ) . (3.12)As argue by [8], the leading contribution in the crossed channel expansion of G ( v, u ) is givenby the operators with the lowest twist. Ignoring the contribution from the unity we have, G ( v, u ) ∼ c min τ ∆ ,τ (cid:18) v − v (cid:19) τ − ∆ Γ( β )Γ (cid:16) β (cid:17) ( − u ) (3.13)where we have denote by c min τ ∆ ,τ as the OPE coefficient associated to the operator with thelowest twist that can show up on the t − channel expansion.On the direct channel the leading term (3.13) should admit a collinear expansion of theform (2.18). By using the fact that we are considering operators with low twist, (hencelow anomalous twist), we can expand at first order in the anomalous dimension u ∆ + δτ/ ∼ δτ u ∆ ln u . All in all at first order, the s − channel expansion in the collinear limit can bewritten as, G ( u → , v ) ∼ u ∆ ln u (cid:88) β c τ,β (1 − v ) β F (cid:18) β a, β b, β | − v (cid:19) . (3.14)such us the crossing equation takes the form, c min τ ∆ ,τ (cid:18) v − v (cid:19) τ − ∆ Γ( β )Γ (cid:16) β (cid:17) = (cid:88) β δτ c τ,β (1 − v ) β F (cid:18) β a, β b, β | − v (cid:19) . (3.15)Now it is trivial to invert the above expansion in Mellin space applying (2.23) and byusing essentially the same amputed Mellin amplitude (3.3) by making the replacement τ (cid:48) → τ − ∆ . (3.16)– 9 –nd adding the multiplicative constants, (cid:102) M dis ( τ (cid:48) / a, t ) = c min τ ∆ ,τ Γ( β )Γ (cid:16) β (cid:17) Γ (cid:16) − τ (cid:48) (cid:17) Γ (cid:16) t + τ (cid:48) + a (cid:17) Γ ( t + a + 1) sin ( π ( τ (cid:48) / a )) π . (3.17)Putting this back into (2.23) and going through the same computation as in the section above,we obtain c τ,β δτ c min τ ∆ ,τ Γ( β )Γ (cid:16) β (cid:17) I , τ − ∆ , β . (3.18)It was additionally argued in [8] that the coefficients c τ,β expanding the direct channel in(3.15) should correspond to those ones in the expansion of the vacuum c free∆ , k (3.7).Putting all those things together we can write, δτ c min τ ∆ ,τ Γ( β )Γ (cid:16) β (cid:17) I , τ − ∆ , β I , − ∆ , ∆+ k . (3.19)By taking the large − β on the right hand side of the above equation, we have the first correctionin 1 /β to the twist of large spin double twist operators in terms of the minimal twist operator,as previously done in [8, 28]. We are going to see this more generally in the next section. It should be possible in principle to compute leading corrections to the result above, as longas it is still possible to have an expansion in terms of collinear blocks. In this section wewould like to illustrate how it would works by using a simple example, namely a exchange ofa scalar block in the crossed channel. The collinear limit in the crossed channel for a scalarblock around v → g J, ∆ ( u → , v ) ∼ − ln u Γ(∆ (cid:48) )Γ (cid:0) ∆ (cid:48) (cid:1) F (cid:18) ∆ (cid:48) , ∆ (cid:48) (cid:48) − | v (cid:19) . (3.20)such as, taking a = b = 0 for simplicity, the corrections on higher order in powers of v for thecrossing equation looks like, c sub∆ (cid:48) ,τ (cid:48) (cid:16) v − v (cid:17) ∆ (cid:48) − ∆ Γ(∆ (cid:48) )Γ (cid:16) ∆ (cid:48) (cid:17) F (cid:16) ∆ (cid:48) , ∆ (cid:48) ; ∆ (cid:48) − | v (cid:17) = (cid:80) β δτ ( J )2 c free∆ , k (1 − v ) β F (cid:16) β , β ; β | − v (cid:17) . (3.21)Where δτ ( J ) is now explicitly understood as a function of J . A generalization for the am-putated Mellin (3.3), perhaps naive, but which reproduce the collinear expansion at the first– 10 –ine of (3.21) is as follows, M sub − dis (∆ (cid:48) / , t ) = c sub τ ∆ (cid:48) ,τ (cid:48) Γ (cid:16) − ∆ (cid:48) + ∆ (cid:17) Γ (cid:16) t + ∆ (cid:48) − ∆ (cid:17) Γ( t + 1) π sin (cid:18) π (cid:18) ∆ (cid:48) − ∆ (cid:19)(cid:19) Γ(∆ (cid:48) )Γ (cid:0) ∆ (cid:48) (cid:1) F (cid:20) ∆ (cid:48) / , ∆ (cid:48) / , t + ∆ (cid:48) / − ∆ ∆ (cid:48) − , ∆ (cid:48) − ∆ | (cid:21) (3.22)We can now invert the expansion on the second line of (3.21) as, (cid:90) i ∞− i ∞ dt πi Γ ( − t )Γ (cid:18) t + ∆ (cid:48) (cid:19) m ( β ) (∆ (cid:48) , t ) M sub − dis (∆ (cid:48) / , t )= δτ ( J )2 c free∆ , k Γ( β − κ β Γ (cid:16) β + 1 (cid:17) . (3.23)By expanding the hypergeometric function in the definition of M sub − dis (∆ (cid:48) / , t ) the com-putation follows essentially the same steps as in the previous sections term by term on theexpansion, so each individual terms give us a function I β,τ , more precisely, after performingthe contour integration we end up with, c sub − min τ ∆ ,τ e iπ (∆ − Γ(∆ (cid:48) )Γ (cid:0) ∆ (cid:48) (cid:1) ∞ (cid:88) k =0 (cid:16) ∆ (cid:48) (cid:17) k ( − k k ! (∆ (cid:48) − k I , ∆ (cid:48) − ∆ + k, β = δτ ( J )2 c free∆ , k . (3.24)This equation can be rewritten in a terms of a hypergeometric function but that form willnot be very illuminating either. We can however try a large β limit. By taking ∆ (cid:48) ∼
1, theleading term on a large β expansion on the right hand side is, δτ ( J )2 = c sub − min τ ∆ ,τ Γ(∆ (cid:48) )Γ (cid:0) ∆ (cid:48) (cid:1) Γ(∆ ) Γ (cid:0) ∆ − ∆ (cid:48) (cid:1) ∞ (cid:88) k =0 (cid:16) ∆ (cid:48) (cid:17) k ( − k k ! (∆ (cid:48) − k (cid:18) β (cid:19) ∆ (cid:48) +2 k = c sub − min τ ∆ ,τ Γ(∆ (cid:48) )Γ (cid:0) ∆ (cid:48) (cid:1) Γ(∆ ) Γ (cid:0) ∆ − ∆ (cid:48) (cid:1) (cid:18) β (cid:19) ∆ (cid:48) F (cid:32) ∆ (cid:48) , ∆ (cid:48) (cid:48) − | (cid:18) β (cid:19) (cid:33) . (3.25)or even better, taking the same limit but this time with ∆ β kept fixed lead us to, δτ ( J )2 = c sub − min τ ∆ ,τ Γ(∆ (cid:48) )Γ (cid:0) ∆ (cid:48) (cid:1) (cid:18) β (cid:19) ∆ (cid:48) F (cid:32) ∆ (cid:48) , ∆ (cid:48) (cid:48) − | (cid:18) β (cid:19) (cid:33) . (3.26)The last result agrees with the re-summation of the leading terms in the large J expansionfor the scalar block performed in [52]. – 11 – Conclusions and outlook
In this work we have considered the translation of Caron-Huot’s OPE inversion formula (1.1)from cross-ratios space to Mellin space in the collinear aproximation. In order to do so we havewritten the Mellin amplitude residue at s = τ / J = 0 and J = 1. Despite we have not used a contour deformation explicitly, we believe theinversion in Mellin space considered here might be still not valid for the lowest spins, sincein essence it is just a translation of (1.1). This and other important features of the inversion(1.1) remains to be better understood in Mellin space.Over the last decade we have gained some important understanding of the Mellin rep-resentation for correlation functions in conformal field theory, however, we feel it is still avery unexplored subject that might not only have something else to teach us but can be apotentially useful tool. Acknowledgments
I would like to thank to Juan Maldacena, Nima Arkani-Hamed, Simon Caron-Huot, Luis F.Alday, Hugh Osborn, Jake Bourjaily, David McGady and Charlotte Sleight for comments andenlightening discussions. This work is supported in part by the Danish National ResearchFoundation (DNRF91), ERC Starting Grant (No 757978) and Villum Fonden.
A Mellin blocks orthogonality
In the main body of this work we have defined the polynomial Mellin blocks as, m ( β ) ( τ / , t ) = ( t + a ) τ/ ( t + b ) τ/ κ β ( a ) β/ ( b ) β/ Γ (cid:16) β + 1 (cid:17) F (cid:20) − β , − t, β − a , b | (cid:21) , (A.1)– 12 –ith κ β = γ β ,a γ τ ,b Γ( β )Γ( β − , (A.2)In order to prove the orthogonality relation (2.17) we are going to follow the same route as forthe case of Hanh polynomials [53], as there, it is sufficient to show that m ( β ) ( t ) is orthogonalto one polynomial of each degree less than β and then check the constant when they are bothof the same degree. Lets consider the following polynomial of degree β (cid:48) , which includes thenormalization constants, q ( β (cid:48) ) ( τ / , t ) = ( − t − a − b ) β (cid:48) κ β (cid:48) Γ (cid:16) β (cid:48) + 1 (cid:17) (A.3) (cid:90) i ∞− i ∞ dt πi Γ( − t )Γ( − t − a − b )Γ (cid:16) t + τ a (cid:17) Γ (cid:16) t + τ b (cid:17) m ( β ) ( τ / , t ) q ( β (cid:48) ) ( τ / , t )= ( a ) β/ ( b ) β/ κ β Γ (cid:16) β + 1 (cid:17) κ β (cid:48) Γ (cid:16) β (cid:48) + 1 (cid:17) β/ (cid:88) n =0 (cid:16) − β (cid:17) n (cid:16) β − (cid:17) n n !( a ) n ( b ) n × (cid:90) i ∞− i ∞ dt πi Γ( − t + n )Γ (cid:18) − t − a − b + β (cid:48) (cid:19) Γ ( t + a ) Γ ( t + b ) , (A.4)we can perform the contour integral by means of Barnes first lemma, leading us to,Γ (cid:16) β + a (cid:17) Γ (cid:16) β + b (cid:17) Γ (cid:16) β (cid:48) − a (cid:17) Γ (cid:16) β (cid:48) − b (cid:17) κ β Γ( β (cid:48) )Γ (cid:16) β + 1 (cid:17) κ β (cid:48) Γ (cid:16) β (cid:48) + 1 (cid:17) F (cid:18) − β , β − | β (cid:48) (cid:19) , (A.5)here we can use Gauss summation formula (sometimes referred as Gauss hypergeometrictheorem), namely, F ( a, b ; c |
1) = ( c − b ) − a ( c ) − a , (A.6)to end up with,Γ (cid:16) β + a (cid:17) Γ (cid:16) β + b (cid:17) Γ (cid:16) β (cid:48) − a (cid:17) Γ (cid:16) β − b (cid:17) κ β Γ( β (cid:48) )Γ (cid:16) β + 1 (cid:17) ( − β (cid:16) − β (cid:48) (cid:17) β (cid:16) β (cid:48) (cid:17) β κ β (cid:48) Γ (cid:16) β (cid:48) + 1 (cid:17) , (A.7)which vanishes for β (cid:48) < β . The constant at β (cid:48) = β is equal toΓ( β − κ β Γ (cid:16) β + 1 (cid:17) . (A.8)This proves (2.17). – 13 – eferences [1] R. Rattazzi, V. S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4DCFT , JHEP (2008) 031, [ ].[2] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solvingthe 3D Ising Model with the Conformal Bootstrap , Phys. Rev.
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