Helicity basis for three-dimensional conformal field theory
PPrepared for submission to JHEP
Helicity basis for three-dimensional conformal fieldtheory
Simon Caron-Huot, Yue-Zhou Li
E-mail: [email protected], [email protected]
Abstract:
Three-point correlators of spinning operators admit multiple tensor structurescompatible with conformal symmetry. For conserved currents in three dimensions, we pointout that helicity commutes with conformal transformations and we use this to construct three-point structures which diagonalize helicity. In this helicity basis, OPE data is found to bediagonal for mean-field correlators of conserved currents and stress tensor. Furthermore, weuse Lorentzian inversion formula to obtain anomalous dimensions for conserved currents atbulk tree-level order in holographic theories, which we compare with corresponding flat-spacegluon scattering amplitudes. a r X i v : . [ h e p - t h ] F e b ontents /CFT n limit from gluon scattering amplitudes 30 (cid:104) V V V (cid:105) from Witten-diagram 37B Simplifying Fourier transforms using spinors 40C More on conformal blocks 41
C.1 Series expansion of conformal blocks 41C.2 Inverting powers of cross-ratios times Gegenbauers 42C.3 Cross-channel expansion of blocks 43
D Four-dimensional gluon amplitudes in flat space 45 – 1 –
Introduction
It is an old proposition to use self-consistency conditions, such as unitarity, analyticity andcrossing symmetry, to “bootstrap” physical observables like the S-matrix of Lorentz invariantquantum field theories. Nonperturbatively, this philosophy has been successfully applied inrecent years to conformal field theories (CFT). This has allowed to nonperturbatively explorethe space of conformal theories, and to extract precision spectra for a number of specifictheories (for a review see [1]).A surprising feature of the bootstrap is that a small number of correlators often sufficeto obtain interesting constraints. Many studies therefore focus on four-point correlators ofscalar operators. Spinning correlators are technically more complicated but much progresshas been made and numerical studies involving them are now possible [2–6]. As nontrivialrepresentations of rotation groups, spinning operators are bound to involve fancier structures.Three-point functions, for example, can be constructed using the embedding formalism [7, 8],and four-point conformal blocks, key ingredient to the bootstrap, may then obtained byacting with corresponding spinning-up or weight-shifting operators on scalar seeds [8, 9].This heavy machinery comes at a cost. This is especially visible in analytic work, whichhas so far specialized to limits such as free theories, the Regge limit, or conformal colliderkinematics (see for example [10–16]).There are several motivations to pursue analytic work with spinning correlators. Amain one is the analogy with perturbative S-matrices, where massless spinning particles obeystringent self-consistency conditions. These include Weinberg’s derivation of perturbativegeneral relativity from soft limits [17], or to give just one more modern example, on-shellrecursion relations for gluon amplitudes [18, 19]. For strongly coupled conformal theorieswith a holographic AdS dual that includes weakly coupled gravity, stress-tensor correlatorsare thus expected to strongly constrain not only gravity, but its coupling to matter. Indeedany CFT has a stress tensor, which, like gravity, couples to every degree of freedom.A useful starting point for analytic approaches is good control of mean-field theory,around which one can start various approximations, be these in large spins, large N , small (cid:15) ,or other quantities [20–26]. When the mentioned technology is applied to spinning correlators,the OPE data become matrices in the space of tensor structures. But even making seeminglynatural choices, one finds dense, non-diagonal matrices already in mean field theory (MFT)[12]! It is difficult to bring oneself to study corrections to such a zeroth approximation.A possible way forward is the fascinating observation that the number of spinning struc-tures in CFT d is identical to the number of structures for scattering amplitudes in QFT d +1 [27]. While physically natural from the viewpoint of the bulk-point or flat space limits ofcorrelators, it is still unclear whether this counting extends to a useful map beyond thatlimit. Indeed, the non-diagonal nature of MFT correlators stands in sharp contrast with theQFT side, where diagonalizing trivial scattering S = 1 was never a big challenge! We shouldthen ask: can one find a basis of CFT three-point structures in which MFT correlators arediagonal? – 2 –n this paper we address this question in the special case of CFT , exploiting the fact thatin QFT massless particles come with two helicity states ± . We point out that the “helicity”of a conserved current is a meaningful (crossing-symmetric) concept also in CFT , whichformally implies that a helicity basis of three-point structures will automatically diagonalizecrossing symmetry. We will confirm this by computing explicit OPE data in MFT, as well asthe first correction to CFT current correlators dual to tree-level gluon scattering in AdS .This paper is organized as follows. In section 2, we construct the helicity basis for three-point functions and explain that it diagonalizes a well-defined operator h . We also intro-duce the group-theoretic concepts to be used in later sections, including three-point pairings,shadow transforms, Euclidean and Lorentzian inversion formula. In section 3, we use bothinversion formulas to independently obtain mean-field OPE data for conserved currents ofvarious spins. In section 4, we apply our scheme to study YM /CFT , using the Lorentzianinversion formula to extract the analytic-in-spin part of the leading-order double-twist anoma-lous dimensions of currents. In section 5, we explicitly check that the anomalous dimensionsof the double-twist states [ V V ] n,J at large- n agree with flat-space partial waves for tree-levelgluon scattering.This paper contains a number of technical appendices. In appendix A, we relate CFT three-point functions conserved currents to the bulk YM couplings, using the AdS embeddingformalism. In appendix B, we explain how to simplify certain calculations by representingpolarization vectors as spinors and give formulas for Fourier transforms. In appendix C, wereview the series expansion of scalar conformal blocks. Moreover, we show how to computeOPE data for correlators that are powers of cross-ratios multiplied with Gegenbauer polyno-mials, which may have applications to other problems; we also record simplified expansionsfor certain scalar, currents and stress-tensors exchanges. Finally, flat-space gluon amplitudes,including Yang-Mills and higher-derivative couplings, are reviewed in appendix D. The structure of conformal correlators for spinning external operators is by now well under-stood. Here we aim to concisely summarize key results so as to state our new three-pointstructures as early as possible (eq. (2.10) below). We eschew the use of embedding spaceand cross-ratios. Rather, we use conformal symmetry to place local operators at standardlocations such as (0 , x, ∞ ) as shown in figure 1, or (0 , x, y − , ∞ ) for four-points.In this frame, three point functions for scalar operators are determined by dimensionalanalysis up to a normalization: T ( x ) = (cid:104)O (0) O ( x ) O ( ∞ ) (cid:105) = 1 | x | ∆ +∆ − ∆ , | x | ≡ (cid:112) x µ x µ . (2.1)We define O i ( ∞ ) by taking the limit x − → − ∆ i (see eq. (114)– 3 – O (0) O ( x ) O ( ∞ ) Figure 1 : Conformal frame used for three-point functions: (cid:104)O (0) O ( x ) O ( ∞ ) (cid:105) .of [28]). We will often Fourier transform with respect to the second position x : T ( p ) = (cid:104)O (0) O ( p ) O ( ∞ ) (cid:105) = (cid:90) d d xe − ip · x (cid:104)O (0) O ( x ) O ( ∞ ) (cid:105) . (2.2)This was used in [12] to simplify calculations of shadow transforms and to compute conformalpairings, which all become simple algebraic operations.It is important to note that we do not Fourier transform all operators, as is sometimesconsidered in the literature, e.g. in [29]. The only Fourier integrals we will compute involvepowers of a single variable as in (2.1) which are rather straightforward. Physically, singlingout one operator is natural in conformal bootstrap applications, as we typically treat externaland internal states asymmetrically. We think of the third operator as the exchanged one O in the conformal block decomposition of a four-point correlator, as shown in figure 2. Multiple index contractions generally exist between spinning operators, and three-point struc-tures are correspondingly no longer unique. They are straightforward to classify in the aboveframe [30]. For pedagogical reasons, let us focus on the case where all operators are symmet-ric traceless tensors, O µ ...µ J , where J is the spin of the operator. In d = 3, this covers allbosonic operators. We work in index-free notation [7] and dot into the J ’th power of a nullpolarization vector (cid:15) µ . Our two-point functions follow the standard normalization: (cid:104)O (0) O ( ∞ ) (cid:105) = ( (cid:15) · (cid:15) ) J . (2.3)Any index contraction between the (cid:15) µi and x µ defines an allowed three-point function.For example, for two operators of spin-1 and a third of spin J (cid:104) V V O (cid:105) , a basis of fiveindependent (parity-even) monomials is easily enumerated: B V = (cid:26) (cid:15) · (cid:15) , (cid:15) · x (cid:15) · xx , (cid:15) · x (cid:15) · (cid:15) (cid:15) · x , (cid:15) · (cid:15) (cid:15) · x(cid:15) · x , (cid:15) · (cid:15) (cid:15) · (cid:15) ( (cid:15) · x ) x (cid:27) × ( (cid:15) · x ) J | x | ∆ +∆ − ∆ + J . (2.4)– 4 – O O O O (cid:105) = Σ O O O O O O Figure 2 : Four-point function factorized into three-point functions.Each monomial has homogeneity (1 , , J ) with respect to the three (cid:15) i . It will be usefulto treat structures analytically in the third spin J . The fact that 1 / ( (cid:15) · x ) appears in thedenominator implies that certain structures cease to exist at low spin. It will be possible touse a common labelling scheme for all values of J , but we will have to remember that certainstructures do not contribute at low J .Although our frame choice breaks permutation symmetry it is trivial to restore it. Forexample to exchange 1 and 2, we simply take translation by an amount − x and substitute x µ (cid:55)→ − x µ . Less trivially, to interchange operators 1 and 3, we use the inversion x µ (cid:55)→ x µ /x ≡ x − , acting with the inversion tensor on (cid:15) : T ( ∞ , x,
0) = x − T (0 , x − , ∞ ) (cid:12)(cid:12)(cid:12) (cid:15) µ (cid:55)→ I µν ( x ) (cid:15) ν , I µν ( x ) = δ µν − x µ x ν x . (2.5)There is no need to include inversions acting on (cid:15) , (cid:15) because inversion is included in thedefinition of O ( ∞ ). The structures in eq. (2.4) become (cid:26) (cid:15) · ˜ (cid:15) , − (cid:15) · x (cid:15) · xx , (cid:15) · x ˜ (cid:15) · (cid:15) (cid:15) · x , − (cid:15) · (cid:15) (cid:15) · x(cid:15) · x , (cid:15) · (cid:15) ˜ (cid:15) · (cid:15) ( (cid:15) · x ) x (cid:27) × ( (cid:15) · x ) J | x | ∆ +∆ − ∆ + J (2.6)where ˜ (cid:15) µ = (cid:15) µ − x µ (cid:15) · x/x .Let us now improve this in steps. Instead of just “listing all monomials”, a good ideais to use the SO( d −
1) symmetry which preserve the point x . An SO( d ) traceless symmetrictensor of rank J can be written as a direct sum of multiple SO( d −
1) tensors, with rank0 ≤ J (cid:48) ≤ J indices, roughly, how many indices are perpendicular to x . Three-point structuresare then in one-to-one correspondence with SO( d −
1) singlets in the tensor products of thethree representations from the three legs. Such a scheme was used for example in ref. [30].While effective for generic operators, this is not the scheme we shall use, since we are interestedin conserved currents. In x -space, conservation is a cumbersome differential constraint.The next improvement is to use instead SO( d −
1) tensors in momentum space, separatingindices that are parallel or perpendicular to p in the frame in eq. (2.2). For conserved currentsone simply has to drop all the structures that are not fully perpendicular to p . For example,for two conserved currents in d dimensions (which have scaling dimension ∆ = ∆ = d − This is really a substitution, not a symmetry transformation. It can be done whether or not the theory isparity symmetric. – 5 –here are just two allowed structures, proportional to: (cid:110) (cid:2) p ( (cid:15) · (cid:15) ) − ( p · (cid:15) )( p · (cid:15) ) (cid:3) (cid:2) p ( (cid:15) · (cid:15) ) − ( p · (cid:15) )( p · (cid:15) ) (cid:3) ( p · (cid:15) ) − p ( (cid:15) · (cid:15) ) − ( p · (cid:15) )( p · (cid:15) ) d − ,p ( (cid:15) · (cid:15) ) − ( p · (cid:15) )( p · (cid:15) ) (cid:111) × ( p · (cid:15) ) J | p | d − − ∆ − J . (2.7)These two structures are transverse with respect to (cid:15) and (cid:15) and are respectively SO( d − (cid:15) . The first structure is analyticfor spin J ≥
2, and the second for J ≥
0. In this example “transverse” simply means invariantunder (cid:15) i (cid:55)→ (cid:15) i + p i . For higher-rank conserved currents, the correct statement will involve anoperator D designed to preserve the constraint (cid:15) i = 0 [31]: p µ D (cid:15) µ T = p µ D (cid:15) µ T = 0 , D (cid:15)µ ≡ (cid:18) d − (cid:15) · ∂∂(cid:15) (cid:19) ∂∂(cid:15) µ − (cid:15) µ ∂∂(cid:15) · ∂∂(cid:15) . (2.8)Such a scheme could be used to label three-point structures in any dimension d , includingoperators O in mixed representations of SO( d ). We now specialize to d = 3, where furthersimplifications occur.In d = 3, SO( d −
1) irreps (transverse to p ) are one-dimensional and labelled by helicity ± J . For two conserved currents of any spin there are thus only four structures. A projectoronto the positive-helicity component of (cid:15) can be written by combining parity-even and oddstructures: (cid:15) µ Π µν ± p (cid:15) ν ≡ (cid:18) (cid:15) · (cid:15) − ( p · (cid:15) )( p · (cid:15) ) p ± i | p | ( (cid:15) , p, (cid:15) ) (cid:19) . (2.9)Here ( a, b, c ) = (cid:15) µνσ a µ b ν c σ denotes contraction with (cid:15) = +1 the antisymmetric tensor inEuclidean signature. The projector satisfies Π ± p = Π ± p and p · Π ± p = 0. For p along the zaxis, it can be written as (1 , i, µ (1 , − i, ν .Given two conserved currents of spin J and J in d = 3, we thus define a complete basisof four possible three-point couplings, including a convenient factor, as: T ± , ± ≡ (4 π ) ( − i √ J + J + J τ + τ − ∆ × ( (cid:15) Π ∓ p (cid:15) ) J ( (cid:15) Π ± p (cid:15) ) J × ( p · (cid:15) ) J − J − J | p | β − , (2.10)where β = (∆ + J ) + (∆ + J ) − (∆ + J ) and τ i = ∆ i − J i is the twist. The twosuperscripts represent the helicity of each operator. Note the reversal of the momentum inthe first projector, since the first operator has momentum − p , so that helicity retains itsphysical interpretation as spin along momentum axis. The transversality condition (2.8) isreadily verified for any J i .Eq. (2.10) defines the helicity basis we will use throughout. The opposite-helicity struc-tures T + − and T − +123 are only allowed for local operators (polynomial in (cid:15) ) when J ≥ J + J . There are momentum-space constructions for spinning operators in the literature, where all three positionsare Fourier transformed, see, e.g., [29, 32–34] and references therein, which enjoy potential applications toinflationary cosmology [35, 36]. – 6 –n the other hand, since SO(2) representations are one-dimensional, the projectors satisfythe identity: ( (cid:15) Π − (cid:15) )( (cid:15) Π + (cid:15) ) = ( (cid:15) Π − (cid:15) )( (cid:15) Π − (cid:15) ) = − ( p · (cid:15) ) p ( (cid:15) Π − (cid:15) ) , (2.11)which extends the range of same-helicity structures T ++123 and T −− to: J ≥ | J − J | . Theseranges coincide with the usual selection rules for the total angular momentum of two masslessparticles in flat four-dimensional space.Although eq. (2.10) is primarily meant to be used for conserved currents, where ∆ i = 1+ J i for i = 1 ,
2, we kept ∆ i free since the structures make sense for any ∆ i . In particular, we willuse the same expressions below for shadow-transformed operators. For spin-0 states, we keepthe same formula but drop superscripts.Once the helicity basis is defined in momentum space, it is often necessary to transformit to coordinate space. The Fourier-transform of a power-law is straightforward (cid:90) d d p (2 π ) d e ip · x p k = 4 k x k + d Γ( d + k ) π d Γ( − k ) . (2.12)Our strategy is to perform Fourier-transform for pure power-laws at first, and then replace p · (cid:15) → − i(cid:15) · ∂ . (2.13)Doing so, one finds that the parity-even and odd components produce disparate gamma-functions that don’t nicely combine. Many calculations are thus simplified by switching to anEven/Odd basis of parity eigenstates. Each parity sector contains two elements, representingstates with opposite or same helicity: (cid:110) T E, opp123 , T E, same123 (cid:111) ≡ Γ (cid:0) − τ − τ +∆ + J (cid:1) Γ (cid:0) τ + τ − τ (cid:1) × (cid:110) T + − + T − +123 √ , T ++123 + T −− √ (cid:111) , (cid:110) T O, opp123 , T O, same123 (cid:111) ≡ Γ (cid:0) − τ − τ +∆ + J (cid:1) Γ (cid:0) τ + τ − τ (cid:1) × (cid:110) T + − − T − +123 √ , T ++123 − T −− √ (cid:111) , (2.14)where we introduced gamma-factor normalizations for future convenience. These ensure thatthe transform produces polynomials in ∆ and J of the lowest possible degree, as the de-nominator cancels spurious double-twist poles from the Fourier transform.Fourier transforms may now be straightforwardly computed, by expanding the even/oddstructures into dot products of p with polarizations, up to a possible single odd factor ( p, (cid:15) i , (cid:15) j ).As a trivial example, in the scalar case J = J = 0, there is just a single structure T E O = 2 J | x | ∆ − J − ∆ − ∆ ( x · (cid:15) ) J . (2.15)As a more illustrative example, for two spin-1 currents (cid:104) V V O(cid:105) the two even structures turnout to be proportional to eq. (2.7) (in the same order). As it should, the transform takes the– 7 –orm of a matrix acting on the basic structures B V in eq. (2.4): (cid:18) T E, opp11 O T E, same11 O (cid:19) = 2 J n (cid:32) n − ˜ J ) 2( n −
1) (3 ˜ J − n +1) (3 ˜ J − n +1) ˜ J − (8 n +1) ˜ J +8 n n − n − J ) 2( n − J J J − J n (cid:33) · B V , (2.16)where n is defined through τ = τ + τ + 2 n , and ˜ J denotes the “spin shadow”: ˜ J = − − J in d = 3 [37]. The parity-odd structures can be similarly represented in terms of four oddmonomials: B (cid:48) V = (cid:26) ( (cid:15) , x, (cid:15) ) (cid:15) · xx , ( (cid:15) , x, (cid:15) ) (cid:15) · (cid:15) x · (cid:15) , ( (cid:15) , x, (cid:15) ) (cid:15) · xx , ( (cid:15) , x, (cid:15) ) (cid:15) · (cid:15) x · (cid:15) (cid:27) ( (cid:15) · x ) J − | x | ∆ +∆ − ∆ + J − , (2.17)in which (cid:32) T O, opp11 O T O, same11 O (cid:33) = 2 J (cid:32) (1 − n ) (1+ J +2 n ) (1 − n ) (1+ J +2 n )(1 − n ) ( − J +2 n ) ( − n ) (1 − J − n ) (cid:33) · B (cid:48) V . (2.18)Notice that so far n is simply a notation for the twist, but when n takes on (half-)integer valuesit will represent so-called double-twist operators. Parity-even double twists have integer n while parity-odd ones have half-integer n .A technical complication when dealing with higher-rank tensors and odd structures isthe presence of Gram determinant relations (antisymmetrizing any four vectors gives zero).In our calculations below, we circumvent this either by evaluating expressions on a symbolicthree-dimensional parametrization, or by using the spinor formulation in appendix B.The opposite-helicity structure in eq. (2.16) is physically allowed for J ≥
2, but thereis an important discrete exception: when O is a conserved current ( J = 1 and ∆ = 2).Then the complicated polynomial in the fifth column vanishes, shielding the problematicdenominator in eq. (2.4). The three structures: T E, opp , T E, same , T O, same then define valid(and independent) couplings between three currents. We verify in appendix A that thesemap, respectively , to bulk Yang-Mills couplings Tr F , and to parity even/odd parts of Tr F ! The reader may worry that our definition of helicity structures in eq. (2.10) is tied to thespecific frame (0 , x, ∞ ). However, it turns out to be independent of this! Here we constructa conformal integral transform, whose eigenvalue is helicity. Its existence will automaticallyimply that crossing is diagonal in the helicity basis.It is intuitively clear from holography that helicity should be frame-independent, sincemomentum-space currents with definite helicity source AdS gauge fields that are either self-dual or anti-self-dual near the boundary [35, 38]. (For a spinor-helicity formalism in AdS ,see also [39].) Since the self-dual decomposition is invariant under conformal isometries, weexpect it to agree between all channels.In momentum space, the operation which measures helicity is simply hJ µ ( p ) ≡ − i (cid:15) µνσ p σ | p | J ν ( p ) . (2.19)– 8 –ourier transforming this defines an integral transform: hJ µ ( x ) = (cid:90) d yH µν ( x − y ) J ν ( y ) , H µν ( x − y ) ≡ (cid:15) µνσ π ∂∂y σ x − y ) . (2.20)We now show that h commutes with conformal transformations. Normally, this would re-quire the kernel H to transform like a two-point function between a current and its shadow, (cid:104) J µ ( x ) ˜ J ν ( y ) (cid:105) . For a generic operator, this is impossible: conformal two-point functions be-tween operators of different dimension must vanish! (This follows easily from scale invariancein the frame ( x, y ) = (0 , ∞ ).) The loophole here is that since J ν ( y ) is conserved, the shadow˜ J ν is defined modulo a derivative: the kernel H only needs to be conformally invariant moduloa total derivative ∂ νy X µ ( x, y ).Let’s thus check invariance under inversion x µ (cid:55)→ x µ /x . Applying the standard trans-formation laws, a short calculation gives: I µµ (cid:48) ( x ) I νν (cid:48) ( y ) x y H µν ( x − − y − ) = 1 π (cid:20) (cid:15) µνσ ( y − x ) σ ( x − y ) + (cid:15) µνσ x σ ( x − y ) x + 2 ( x − y ) ν (cid:15) µρσ y ρ x σ x ( x − y ) (cid:21) (2.21)We have used the Schouten identity to eliminate terms with x µ or y µ . With a bit of inspection,we find that the sum of H and its transform is indeed a total derivative: H µν ( x − y ) + I µµ (cid:48) ( x ) I νν (cid:48) ( y ) x y H µν ( x − − y − ) = ∂∂y ν (cid:15) µρσ y ρ x σ π ( x − y ) . (2.22)This shows formally that h is invariant under inversion (up to an overall sign change):( hJ ) − = − h ( J − ) (2.23)where ( J − ) µ ( x ) = I µµ (cid:48) J µ (cid:48) ( x − ) /x denotes the inversion map. The sign change was expectedsince h is parity-odd. One could equivalently say that h is invariant under the combinationof inversion and parity.To illustrate the action of h , let us briefly consider two-point functions. A special featureof d = 3 CFTs is that two structures are allowed by conformal invariance [40]: (cid:104) J µ ( x ) J ν (0) (cid:105) = (cid:0) δ µν ∂ − ∂ µ ∂ ν (cid:1) τ π x + iκ π (cid:15) µνρ ∂ ν δ ( x ) , (2.24)where the coefficient κ of the contact term is defined modulo an integer. It is easy to see(for example using momentum space expressions from ref. [40]) that acting with h on J µ ( x )yields the same with τ and 8 κ/π interchanged. This confirm that h takes conformal two-pointfunctions to conformal two-point functions. Of course, just like the shadow transform, hJ isgenerally not a local operator.For higher-spin conserved currents, a similar transform can be defined hT µ ··· µ J ( x ) = (cid:90) d y ( H µ ν ( x − y ) T ν µ ...µ J ( y ) + ( J −
1) permutations of µ ) (2.25)– 9 –enerally, h = J , and one can easily verify that the structures in eq. (2.10) are eigenstates: h T ± , ± = ( ± J ) T ± , ± h T ± , ± = ( ± J ) T ± , ± . (2.26)Although we did not construct a total derivative akin to eq. (2.22) in the higher-spin case, webelieve h to be conformal as well, given the fact that all data computed in the next sectionswill turn out diagonal.In Lorentzian signature, there is a subtlety: h depends on operator ordering through thebranch choice | p | ≡ (cid:112) p ± i h and T , this means that taking discontinuities or commutators do notpreserve h eigenstates; one can explicitly see in eqs. (2.16)-(2.18) that even and odd struc-tures acquire different phases. This will be important below in our discussion of Lorentzianinversion. Since h is a conformal operation, three-point structures with definite helicity will be orthog-onal under all natural operations. Here we review two simple operations, which will formuseful building blocks later.The simplest may be the conformal pairing between three-point structures and shadowstructures: P a,b = (cid:16) T a , T b ˜1˜2˜3 (cid:17) ≡ (cid:90) d d x d d x d d x vol(SO( d + 1 , (cid:104)O O O (cid:105) a (cid:104) ˜ O ˜ O ˜ O (cid:105) b (2.27)= 12 d vol(SO( d − (cid:88) (cid:15) ,(cid:15) ,(cid:15) T a ( (cid:15) , (cid:15) , (cid:15) )(1) T b ˜1˜2˜3 ( (cid:15) ∗ , (cid:15) ∗ , (cid:15) ∗ )(1) , (2.28)where we have used the symmetry to put x = 1. The denominator is the volume form of the“little group” that keeps the frame (0 , , ∞ ) fixed [12].A good way to compute index contraction is to use the differential operator (2.8) (cid:88) (cid:15) f ( (cid:15) ∗ ) g ( (cid:15) ) = 1 J !( d − ) J f ( D (cid:15) ) g ( (cid:15) ) . (2.29)For example, for vector-vector-general (cid:104) V V O(cid:105) case, the pairings between Even or Odd struc-tures (2.16) and (2.18) is readily evaluated: P E O = 16 P s N E O (cid:32) ( J +1)( J +2)( J − J
00 1 (cid:33) , P O O = 16 P s N O O (cid:32) ( J +1)( J +2)( J − J
00 1 (cid:33) , (2.30)where P s is just the pairing of two scalars and one spinning operator [12] P s = 12 d vol(SO( d − d − J (cid:0) d − (cid:1) J (2.31) Our scalar structures are larger by a factor 2 J / than those of [12]: P here s = 2 J P there s . – 10 –nd for latter convenience we introduce the N factor, which is precisely the product of thegamma-functions in eq. (2.14) and its shadow: N EJ J O = (cid:0) τ + τ − τ (cid:1) J + J (cid:0) β − β − β (cid:1) J + J ,N OJ J O = (cid:0) τ + τ − τ (cid:1) J + J − (cid:0) β − β − β (cid:1) J + J − . (2.32)Many other examples can be straightforwardly worked out and it turns out that the three-point pairing is always orthogonal. In fact there is a rather mechanical explanation: the x -space pairing is also proportional to the momentum-space one [12] : P a,b = 1 / (2 π ) d d vol(SO)( d − (cid:88) (cid:15) ,(cid:15) ,(cid:15) T a ( (cid:15) , (cid:15) , (cid:15) )( p ) T b ˜1˜1˜1 ( (cid:15) ∗ , (cid:15) ∗ , (cid:15) ∗ )( − p ) . (2.34)Due to this, the diagonal pairing would be rather trivially diagonal in any d , using themomentum space basis discussed above eq. (2.7). Without derivation, we thus quote thediagonal 4 × d = 3 helicity basis (2.10): P ( h ,h ) , (¯ h (cid:48) , ¯ h (cid:48) )123 = δ h (cid:48) h δ h (cid:48) h × P s × | h | + | h | ( − | h − h | ( J + 1) | h − h | ( − J ) | h − h | . (2.35)Taking Even/Odd combinations (2.14) simply adds the N E/O factors, reproducing the J = J = 1 example quoted in eq. (2.30). The fact that the pairing is diagonal (with ¯ h = − h ) isa first hint that the structures are well chosen.A second natural and useful operation is the shadow transform S [ O ( x )] ≡ (cid:90) d d y (cid:104) ˜ O ( x ) ˜ O ( y ) (cid:105)O ( y ) , (2.36)which maps operators to their shadow operators nonlocally. Operating on three-point struc-tures this generally produces a shadow matrix S ([ O ] O O ) a b : (cid:104) S [ O ] O O (cid:105) a = S ([ O ] O O ) a b (cid:104) ˜ O O O (cid:105) b . (2.37)The shadow transform for conserved currents in d = 3 is simple: the two-point functionin momentum space can be diagonalized by helicity, which is always maximal for conservedcurrents. Using 2 ∆ − ˜∆ A j,j (cid:12)(cid:12) ∆= J +1 from eq. (E.11) of [12], we get simply S ([ ˜ O ] O O ) ( h (cid:48) ,h (cid:48) ) ( h ,h ) = δ h (cid:48) h δ h (cid:48) h ( − J π × C J , C J ≡ δ J, J )! . (2.38) This can be proven formally by moving gauge-fixing factors in the frame (0 , x, ∞ ): (cid:90) d d x vol(SO( d ) × SO(1 , T ( x ) ˜ T ( x ) = (cid:90) d d x (cid:90) d d p d d p (cid:48) e ix · ( p + p (cid:48) ) (2 π ) d vol(SO( d ) × SO(1 , T ( p ) ˜ T ( p (cid:48) ) . (2.33)The x integral simply gives a delta-function setting p (cid:48) = − p . – 11 –his holds when acting on the shadow of a conserved current ˜ O , or a scalar with the sametwist ∆ = 1. (We note that S is not invertible and S [ O ] = 0 acting on a conserved current.)The transform in the Even/Odd basis is of course also diagonal, but displays additional scalarfactors due to the gamma-functions in (2.14).The shadow transform with respect to O will be technically more difficult to compute;we will find below (see (3.15)) that it is also diagonal. A more interesting and nontrivial object is the correlator of four operators. The OperatorProduct Expansion distills those in terms of a given theory’s spectrum and OPE coefficients.Using conformal symmetry we can assume the four points are at (0 , x, y − , ∞ ) (where y − isthe point y µ /y ). Factoring out a conventional prefactor to trivialize the x → y → (cid:104)O (0) O ( x ) O ( y − ) O ( ∞ ) (cid:105) = | y | ∆ +∆ | x | ∆ +∆ G ( z, z ) . (2.39)Our notation O ( y − ) implies that we apply inversion tensors to the indices on the third(and fourth) operator. The complex variable z (which is complex conjugate to z in Euclideansignature) encodes the sizes and angles of the vectors x µ and y µ : zz = x y , z + z = 2 x · y . (2.40)Inserting a complete basis of states between O , O gives the operator product expansion G ( z, z ) = (cid:88) ∆ ,J,a,b λ O a λ O b G a,b ∆ ,J ( z, z ) (2.41)where the sum runs over the spectrum of the theory, and the λ ’s are OPE coefficients. Whenthe external operators have spin, there are generally multiple index contractions a , b tosum over representing the different three-point structures, each of which has an independentcoefficient. The special functions G a,bJ, ∆ ( z, z ) are the so-called conformal blocks, which wenormalize so they approach as x → O ):lim x → G a,b ∆ ,J ( z, z ) = x ∆ +∆ y ∆ +∆ (cid:88) (cid:15) O T a O ( x ) T b O ( y ) ≡ P a,b ∆ ,J ( x, y ) (2.42)For example, for scalar external operators in our normalization (2.15) one finds P ∆ ,J (ˆ x, ˆ y ) = ( | x || y | ) ∆ ( d − J (cid:0) d − (cid:1) J ˜ C J (cid:0) x · y | x || y | (cid:1) → z (cid:28) z (cid:28) ( zz ) ∆ / ( z/z ) J/ (2.43)where ˜ C j ( ξ ) = C J ( ξ ) /C J (1) = F ( − J, J + d − , d − , − ξ ) is a Gegenbauer normalized tounity at ξ = 1. In terms of cross-ratios, x · y | x || y | = z + z √ zz .– 12 –he conformal block G contains an infinite tower of terms suppressed by powers of x(or z, z ), arising from exchange of descendants ∂ k O ∆ . Series expansions for these termsare available from refs [41–43], as well as an efficient Zamolodchikov recursion algorithm, see[6, 44]. In practice we will use the spinning up/spinning down method. We write the spinningblock as a derivative of a scalar one, G a,b ∆ ,J = P a ( α ) P b ( β ) D ( α,β ) ↑ G ( α,β )∆ ,J . (2.44)Let us explain our notation here. The indices α, β, · · · span the space of spinning-up operators(see eq. (3.25) below), so that the P b ( β ) are constant matrices, that depend only on ∆ , J butnot on spacetime coordinates; ˜ G ( α,β ) is a scalar conformal blocks, where the superscriptsdenote the specific shift of conformal dimensions associated with the particular spinning-upoperator ( α, β ). Explicit operators will be written in section 3.3 below; a simple recursion forscalar conformal blocks is reviewed in appendix C.1. The OPE sum runs over the spectrum of the theory, which we generally don’t know exactly.For analytics it is often better to replace the sum by an integral, the “harmonic analysis”: G ( z, z ) = (cid:88) J,a,b (cid:90) d/ i ∞ d/ − i ∞ d ∆2 πi c a,b (∆ , J ) (cid:16) G a,b ∆ ,J ( z, z ) + shadow (cid:17) . (2.45)The “shadow term” is the same block with ∆ (cid:55)→ ˜∆ = d − ∆ and with a specific coefficient,see [45, 46]. This shadow term ensures that the parenthesis is Euclidean single-valued (i.e.does not have a branch cut) in the limits x → y − and x → ∞ . Explicitly, this term is S ( O O [ O ]) a c ( S ( O O [ ˜ O ]) − ) b d G c,d ˜∆ ,J . (2.46)To obtain the OPE (2.41) from the integral (2.45) one simply closes the contour to theright in the G term, and the formulas will match provided − Res ∆ (cid:48) → ∆ c a,b (∆ , J ) = λ O a λ O b . (2.47)The function c a,b (∆ , J ) will be useful below since it simultaneously encodes the spectrum(through the location of its poles) and OPE coefficients (through the residues); this enablesone to speak about OPE coefficients without having to first know the spectrum.As single-valued eigenfunctions of a Casimir differential operator, the harmonic functionssatisfy an orthogonality relation (cid:90) d d x · · · d d x vol(SO( d + 1 , (cid:104) (cid:105) a,b ∆ ,J (cid:104) ˜1˜2˜3˜4 (cid:105) c,d ˜∆ ,J = ( N (∆ , J ) ( a,b ) , ( c,d ) ) − [2 πδ ( ν − ν (cid:48) ) + shadow] , (2.48)where ∆ = d + iν and the tildes denote shadow operators; tensor indices are meant to becontracted between each operator and its shadow. Note that we abbreviate ( G ∆ ,J + shadow)– 13 –s (cid:104) (cid:105) ∆ ,J . The symmetry can be used to fix the points to (0 , x, , ∞ ) so the integral isreally just over x . The normalization N (∆ , J ) can be expressed in terms of the pairing P ( a,b ) of eq. (2.35), since the δ -function originates from the x → N (∆ , J ) ( a,b ) , ( c,d ) = µ (∆ , J ) ( P a,c O ) − ( P b (cid:48) ,d (cid:48)
34 ˜ O ) − S (34[ ˜ O ]) b (cid:48) b S (˜3˜4[ O ]) d (cid:48) d , (2.49)where the “Plancherel measure” is µ ( J, ∆) = ( d + 2 J − d + J − − d − ∆ − J − d − ∆ + J − d π d vol(SO( d ))Γ( d − J + 1)Γ( d − ∆)Γ(∆ − d ) . (2.50) Evaluated in terms of cross-ratios, this gives an integral over the complex- z plane c a,b (∆ , J ) = N (∆ , J ) ( a,b ) , ( c,d ) d − vol(SO( d − (cid:90) d zz z (cid:12)(cid:12)(cid:12)(cid:12) z − zzz (cid:12)(cid:12)(cid:12)(cid:12) d − (cid:16) ˜ G c,dd − ∆ ,J ( z, z ) + non-shadow (cid:17) G ( z, z )(2.52)where index contractions with G ( z, z ) is again implied. To extract the spectrum using thisformula one would have to know the exact correlator G ( z, z ), which of course is impracticalunless one already has solved the theory. The usefulness of this formula is that it providesanalytic estimates for the OPE data in certain limits. Specifically, following [12] we will usethis formula to extract OPE data in mean field theory in section 3. An effective method to go beyond MFT is to analytically continue the Euclidean inversionformula to Lorentzian signature, which gives the Lorentzian inversion formula [37, 47, 48]. Itexpresses OPE data as a sum of so-called t - and u -channel double-discontinuities.A practical advantage relevant for the present paper is that at tree-level in theories witha large- N expansion, the double-discontinuity is saturated by single-trace exchanges [49, 50],effectively giving AdS cutting rules (see also [51, 52]).The formula was generalized to the spinning case in ref. [37]. The t -channel contributionis given as: c ta,b (∆ , J ) = N L ( a,b ) , ( c,d ) (cid:90) dzdzz z (cid:12)(cid:12)(cid:12)(cid:12) z − zzz (cid:12)(cid:12)(cid:12)(cid:12) d − ˜ G c,dJ + d − , ∆ − d +1 ( z, ¯ z )dDisc[ G ( z, ¯ z )] , (2.53)where the tilde denotes that the external operators are shadow operators, and the tensorindexes are contracted between ˜ G and dDisc[ G ]. A key result of ref. [37] is an elegant way We used eq. (2.28) and the relation vol(SO( d − d − = vol S d − to write, for any conformal function ( · · · ): (cid:90) d d x · · · d d x vol(SO( d + 1 , · · · ) x d x d = 12 d − vol(SO( d − (cid:90) d zz z (cid:12)(cid:12)(cid:12)(cid:12) z − zzz (cid:12)(cid:12)(cid:12)(cid:12) d − ( · · · ) . (2.51) – 14 –o calculate the normalization factor N L , which is generally a matrix, in terms of “light-transforms”. The light-transform of a spinning operator is defined as L [ O ]( x, (cid:15) ) = (cid:90) ∞−∞ dα ( − α ) − ∆ − J O ( x − (cid:15)α , (cid:15) ) . (2.54)(Despite appearance, the integral has no branch point at α = 0 due to the behavior of O .We refer to [37] for further details on the precise branch choices, which we will ignore in thispresentation.) When the light-transform acts on the third operator of a three-point function,it simply induces a Weyl reflection for that operator (∆ (cid:55)→ − J, J (cid:55)→ − ∆) with an overalllight-transform matrix, i.e. (cid:104)O O L [ O ∆ ,J ] (cid:105) a = L ab ( O O [ O ]) (cid:104)O O O − ∆ , − J (cid:105) b . (2.55)We found that the integrand can be computed directly in our frame (0 , x, ∞ ), using a specialconformal transformation along the direction (cid:15) to keep x = ∞ and move x instead. Thisreduces to a simple substitution: x (cid:55)→ x + x α (cid:15) , (cid:15) (cid:55)→ (cid:15) + (cid:16) x · (cid:15) − x (cid:15) · (cid:15) α + 2 x · (cid:15) (cid:17) (cid:15) α − (cid:15) · (cid:15) α + 2 x · (cid:15) x (2.56)and we have to multiply three-point functions by (1 + 2 x · (cid:15) /α ) ∆ − ∆ . Using this rule, andintegrating over α following ref. [37], we find that the light-transform matrix in the Even/Oddbasis is actually independent of J and J : L E ( J J [ O ∆ ,J ]) = L (0) s (cid:32) (cid:33) , L O ( J J [ O ∆ ,J ]) = L (1) s (cid:32) (cid:33) , (2.57)where L (∆ ) s denotes the scalar light transform [37] L (∆ ) s = − i − β π Γ( β − (cid:0) β − ∆ (cid:1) Γ (cid:0) β +∆ (cid:1) . (2.58)We note that the light transform is not diagonal. Attempting to transform from the Even/Oddbasis to the helicity basis (via eq. (2.14)) would produce a matrix that is not only non-diagonal,but also dense. The reason the light transform does not commute with helicity is that itscalculation requires taking a discontinuity, which does not commute with h , as found at theend of subsection 2.2. We will therefore work in the Even/Odd basis, where the simple formof eq. (2.57) will enable us to write the Lorentzian inversion formula very explicitly below.The remaining ingredient is the inverse of a “Lorentzian” pairing between three-pointstructures [37], which reads in the x = 1 gauge: P a,b O ] ,L = ( − (cid:15) · d − d − vol(SO( d − (cid:88) (cid:15) ,(cid:15) T a ( (cid:15) , (cid:15) , (cid:15) ) T b ˜1˜23 S ( (cid:15) ∗ , (cid:15) ∗ , (cid:15) )(1) . (2.59)– 15 –he tilde denotes the shadow, and the superscript S denotes the full shadow where boththe scaling dimension and the spin are reflected (∆ (cid:55)→ d − ∆ , J (cid:55)→ − d − J ). Similarly to theEuclidean pairing discussed above, we find that it is nicely diagonal in the even/odd basis: P EJ J [ O L ] ,L = ( − J + J P s,L N EJ J O L × I , P OJ J O L ,L = − ( − J + J P s,L N OJ J O L × I , (2.60)where the factor N E/O is defined in eq. (2.32) and the subscript L denotes Weyl reflectionassociated with the light-transform (∆ (cid:55)→ − J, J (cid:55)→ − ∆). P s,L is simply the Lorentzian pairingof two scalars and one spinning operator [37] P s,L = ( − d − d vol(SO( d − . (2.61)The normalization N L ( a,b ) , ( c,d ) in the Lorentzian inversion formula (2.53) is then given as N L ( a,b ) , ( c,d ) = 12 ∆+ J (∆ + J −
1) ˆ L a,c ( O O [ O J, ∆ ]) ˆ L b,d ( O O [ O J, ∆ ]) , (2.62)where ˆ L is a sort of inverse of the light transform with respect to the pairing:ˆ L a,c ( O O [ O J, ∆ ]) L de ( O O [ O J, ∆ ]) P c,eL ( O O [ O − ∆ , − J ]) = − iδ da P s,L . (2.63)For scalars, it is straightforward to verify that the above expression reduces to N Ls = 14 κ (∆ , ∆ ) β , κ (∆ , ∆ ) β = Γ( β − ∆ )Γ( β +∆ )Γ( β − ∆ )Γ( β +∆ )2 π Γ( β − β ) . (2.64)More generally, we can write explicitly the normalization factor in the spinning Lorentzianinversion formula (2.53) in the Even/Odd basis: N L ( a,b ) , ( c,d ) = 14 κ (∆ , ∆ ) β ( − − (cid:80) i =1 J i N E/OJ J O L N E/OJ J O L (cid:32) (cid:33) a,c (cid:32) (cid:33) b,d (2.65)where a and c must have the same parity, as well as b and d . We set ∆ = 0 is a, b are evenand ∆ = 1 is they are both odd, and similarly for ∆ . (If operator 1 or 2 is a scalar, thereis only one structure a and we drop the corresponding matrix.)Performing (2.53) is a bit challenging because generally evaluating the spinning conformalblocks is a hard task. A nice idea, following ref. [9], is to “integrate-by-parts” the spin-upfrom eq. (2.44) acting on the block to get instead spinning-down operators acting on thecorrelator: c ta,b (∆ , J ) = N L ( a,b ) , ( c,d ) (cid:90) dzdzz z (cid:12)(cid:12)(cid:12)(cid:12) z − zzz (cid:12)(cid:12)(cid:12)(cid:12) d − ˜ G ( α,β ) J + d − , ∆ − d +1 ( z, ¯ z )dDisc[ P cα P dβ D ( α,β ) ↓ G ( z, ¯ z )] , (2.66)which will effectively reduce us to the scalar Lorentzian inversion formula. Eq. (3.31) belowgives a concrete expression in a specific basis of spin-down operators.– 16 – OPE data for spinning Generalized Free Fields
Using the shadow transform, the OPE data in GFF can be efficiently evaluated by the Eu-clidean inversion formula [12]. It is especially effective for three-point functions in momentum-space. To use this, it is best to write the Euclidean inversion formula (2.52) in a covariantway c a,b (∆ , J ) = ˆ N (∆ , J ) ( a,b ) , ( c,d ) (cid:90) d d x · · · d d x vol(SO( d + 1) , (cid:104) (cid:105) (Ψ ˜∆ i ˜∆ ,J ) c,d , (3.1)where the tildes denote shadow operators. The factor ˆ N is the same as N in eq. (2.49) butwith the factor S (˜3˜4[ O ]) dropped ( ie. replaced by identity). The harmonic function Ψ ∆ i ∆ ,J is a combination of block and shadow, which, importantly, can be written as integral of twothree-point functions (this is called the shadow representation):(Ψ ∆ i ∆ ,J ) a,b = S (34[ ˜ O ]) b c ( G a,cJ, ∆ + shadow) (3.2)= (cid:90) d d x (cid:104) O ( x ) (cid:105) a (cid:104) ˜ O ( x )34 (cid:105) b . (3.3)We now consider a Mean Field Theory four-point function: (cid:104) (cid:105) = (cid:104) (cid:105)(cid:104) (cid:105) + (cid:104) (cid:105)(cid:104) (cid:105) + (cid:104) (cid:105)(cid:104) (cid:105) . (3.4)We focus on the t -channel contribution (cid:104) (cid:105)(cid:104) (cid:105) to illustrate the algorithm for computing theOPE data of MFT. The u -channel contributions (cid:104) (cid:105)(cid:104) (cid:105) can be evaluated in the same way,while the s -channel (cid:104) (cid:105)(cid:104) (cid:105) is trivial to evaluate: it contributes to the identity exchange.Considering the term (cid:104) (cid:105)(cid:104) (cid:105) , the integrals over x , x in (3.1) boil down to the shadow-transform for ˜3 and ˜4, and the remaining integrals are all removed by the gauge-fixing,leaving a simple pairing [12]: c t, MFT a,b (∆ , J ) = µ (∆ , J )( P a,c
34 ˜ O ) − S ([˜1]˜2 ˜ O ) cd S (1[˜2] ˜ O ) de S (12[ ˜ O ]) eb . (3.5)This formula breaks the calculation of MFT coefficients into simple algebraic operations:three shadows and a pairing.The pairing and first two shadows were presented earlier in subsection 2.3. Before wecalculate the third shadow, let us revisit the shadow transform, defined in eq. (2.36). It canbe computed algebraically as multiplication in momentum space: (cid:88) (cid:15) (cid:48) K ˜1˜1 (cid:48) ( p ) T (cid:48) ( p ) a = S ([ O ] O O ) a b T ˜123 ( p ) b , (3.6)where K (cid:48) is the Fourier transform of the two-point function of O [53]: K (cid:48) ( p ) = J (cid:88) k =0 K k (∆ , J )( (cid:15) · p ) k ( p · (cid:15) (cid:48) ) k ( (cid:15) · (cid:15) (cid:48) ) J − k | p | − d − k , – 17 – k (∆ , J ) = π d/ Γ( J + 1)2 d − k Γ (cid:0) d + k − ∆ (cid:1) Γ( J − k + ∆ − − k + 1)Γ( J + ∆)Γ( J − k + 1) . (3.7)Applying this map to the helicity structures normalized as in eq. (2.10), we find the simpleresult K (cid:48) ( p ) T ± , ± ˜1 (cid:48) O ( p ) = π Γ( − ∆ )(∆ + J − − × T ± , ± O ( p ) , (3.8)which reproduces as ∆ → J + 1 the formula for conserved current in eq. (2.38).The third shadow transform is technically more difficult to evaluate since we defined ourstructures in a frame where x = ∞ . The trick is to use the rule eq. (2.5) to interchange O and O in position space, then Fourier transform back to the momentum space, wherewe can apply eq. (3.6). These steps are somewhat lengthy (we found the Fourier transform(B.4) helpful), but thankfully the last step turns out to simply multiply each structure by anoverall factor. This had to be the case since the shadow transform commutes with h and h .Trying a few cases we observe a simple pattern: S (12[ O ]) h ,h h (cid:48) ,h (cid:48) = δ h h (cid:48) δ h h (cid:48) − π Γ(2 + J − ∆)Γ(∆ − )Γ(∆ − J ) × (2 − ∆) | h + h | (∆ − | h + h | . (3.9)Combining the shadows (3.8) and (3.9) with the pairing (2.35) thus gives MFT coefficients(3.5): c t, MFT h ,h , ¯ h , ¯ h (∆ , J ) = δ h h δ h h − π Γ(∆ − − ) Γ(∆ + J )Γ( J − ∆ + 2) Γ( J + )Γ( J + 1) C J C J × ( − J ) | h − h | ( J + 1) | h − h | (∆ − | h + h | (2 − ∆) | h + h | , (3.10)where the constant C J is defined in eq. (2.38). The u-channel identity (if operators 1 and 3are identical) gives the same result times ( − J and with h and h swapped.Eq. (3.10) can be used in the harmonic decomposition (2.45). Where are the poles andcorresponding OPE data? To read off the local OPE data, we have to keep in mind thattensor structures in the helicity basis have poles at double-twist locations. To find OPE datafrom residues, it is best to convert to the Even/Odd basis defined in eq. (2.14), in which theposition-space structures do not have poles. Performing the rotation, we get extra gamma-functions which nicely combine to give scalar MFT coefficients, times the same matrix in theeven and odd cases: c t, MFT , E / O (∆ , J ) = c E / O (∆ , J ) s ( − J ) J J (∆ − | J − J | ( J +1) J J (2 − ∆) | J − J | ( − J ) | J − J | (∆ − J J ( J +1) | J − J | (2 − ∆) J J C J C J , (3.11)where we normalized by the OPE data for scalars of twist 1 or 2 in the even and odd cases,more precisely: c E (∆ , J ) s = c (1 ,
1; ∆ , J ) s , c O J (∆) s = c (1 ,
2; ∆ , J ) s . (3.12)– 18 –he scalar MFT data c (∆ , ∆ ; ∆ , J ) s can be found from earlier literature [54] and is recordedin eq. (C.9) (with p = ∆ + ∆ , a = b = ∆ − ∆ ).For future reference, let us summarize all the ingredients in the Even/Odd basis. Theproducts of “easy” shadows, S ([˜1]˜2 ˜ O ) S (1[˜2] ˜ O ), are given as S E = S Es N EJ J ˜ O ( − J + J × C J C J I , S O = S Os N (cid:48) J J ˜ O ( − J + J +1 × C J C J I , (3.13)where S E/Os are just the scalar factor for (∆ , ∆ ) = (1 ,
1) and (1 ,
2) respectively [37] S E = 4 π (1 − β )( τ − , S O = − π . (3.14)The third shadow (3.9) yields S E/O (12[ O ]) = S s (12[ O ]) (2 − ∆) | J − J | (∆ − | J − J | (2 − ∆) J J (∆ − J J , (3.15)with the same matrix for both even and odd, and where S s is just the shadow coefficients ofscalars [12, 55] S s (12[ O ]) = π d/ Γ(∆ − d )Γ( J + ∆ − (cid:0) ( J + ˜∆ + ∆ ) (cid:1) Γ (cid:0) ( J + ˜∆ − ∆ (cid:1) Γ(∆ − (cid:0) ( J + ∆ + ∆ ) (cid:1) Γ (cid:0) ( J + ∆ − ∆ ) (cid:1) Γ( J + ˜∆) (3.16)with ∆ = 0 for parity-even and ∆ = 1 for parity-odd cases. Finally, the pairing (2.35): P E/O O = δ h (cid:48) h δ h (cid:48) h × P s × N E/O O | h | + | h | ( − | h − h | ( J + 1) | h − h | ( − J ) | h − h | . (3.17)Multiplying these ingredients again according to (3.5) gives eq. (3.11). Let us now describe the OPE data which stems from eq. (3.11). When computing the integral(2.45) as a sum of poles, one finds two sorts of terms: double-twist poles at ∆ − J = 2 + 2 n from the gamma-function in eq. (3.11), and spurious poles from the block, at ∆ − J = 3 , , . . . .The position of the latter is set by their kinematical origin as zero-norm descendants (“nullstates”) of the exchanged primary.We are in the unfortunate situation that the physical and spurious poles overlap. Inprinciple, we should subtract the spurious poles using the results from ref. [44] for the polesof spinning 3d blocks. We pursue a simpler, heuristic method, to be justified in the next sub-section. For scalar mean-field-theory with ∆ = ∆ = 1, the poles are simpler and have beendiscussed in ref. [47]. Using eq. (3.9) there, we find that the spurious poles effectively double the OPE coefficient. On the other hand, the leading trajectory n = 0 has no correspondingspurious pole and so does not double. – 19 –uch a relative factor was also found in the spinning case [12], and so our tentativeguess is that the same happens in our basis and the spurious poles just double the non-leadingtrajectories, that is: λ E O λ E O (cid:12)(cid:12) n,J = − ∆=2+2 n + J c E, MFT (∆ , J )= 2 C J C J n +2 J ( J + 1) (2 n + J + ) ( n + ) ( n + J + 1) M (2 + 2 n + J, J ) ( n = 1 , , , . . . ) , (3.18a) λ O O λ O O (cid:12)(cid:12) n,J = − ∆=2+2 n + J c O, MFT (∆ , J )= 2 C J C J n +2 J ( J + 1) (2 n + J + ) ( n + 1) − ( n + J + ) − M (2 + 2 n + J, J ) ( n = , , , . . . ) , (3.18b)where M (∆ , J ) is the 2 × M (∆ , J ) = ( − J ) J J (∆ − | J − J | ( J +1) J J (2 − ∆) | J − J | ( − J ) | J − J | (∆ − J J ( J +1) | J − J | (2 − ∆) J J . (3.19)Some comments are in order. We recall that the first structure (opposite-helicity) existsonly for J ≥ J + J . This is reflected in an overall zero from ( − J ) J + J in the first entry.Even below this range, the denominator always have fewer zeros than the numerator, so thevanishing is never ambiguous. The range of the J -sums is built-in!The second structure (same-helicity) is more subtle. It generically exists only for J ≥| J − J | . But since 2 − ∆ = − n − J , it may look like the second entry of the matrix M divergesfor the lowest few trajectories. However, inspection of the structures T E O reveals that thesehave corresponding zero for precisely those cases (a special case is visible in eq. (2.18) with n = , J = 1). The conformal blocks thus has a double zero, which shields the singularityfrom the denominator. This means that mean-field-theory doesn’t have operators at theseplaces. For n = 0, we will find below that there is a single leading trajectory.The set of operators appearing in MFT can thus be characterized as: • Opposite-helicity: One operator for each n ≥ J ≥ J + J • Same-helicity: One operator for each n ≥ J ≥ max( | J − J | , J + J − n )This spectrum is depicted in fig. 3. (The helicity of the n = 0 double-twists is really unde-fined.)Let us discuss more the leading trajectory, n = 0. Since there are no spurious poles, onemight think that we should take half the above formula. This is correct but misleading. Thereason is that when n = 0 the same- and opposite- helicity structures become degenerate, asvisible from eq. (2.16). Helicity is simply not defined for n = 0. One can verify that this– 20 –appens whenever O , O are spinning operators, of any spin. The resolution is to rotate toa new basis near n = 0: (cid:32) T E, reg123 T E, sing123 (cid:33) = (cid:32) − ( − J ) J J ( J +1) | J − J | ( J +1) J J ( − J ) | J − J | n n (cid:33) (cid:32) T E, opp123 T E, same123 (cid:33) . (3.20)As the two structures degenerate, both combinations are smooth around n = 0. Since thesecond structure T E, sing123 has a non-vanishing double-discontinuity (in fact it has poles 1 /x ),its coefficient is guaranteed to vanish in MFT. The fact that the two structures become T reg effectively doubles the real n = 0 coefficient. In the rotated basis ( T E, reg123 , T E, sing123 ), theleading-trajectory data is thus given by λ E, rotated12 O λ E, rotated43 O (cid:12)(cid:12) ,J = 2Γ( J + 1) Γ(2 J + 1) × C J C J ( − J ) J + J ( J + 1) | J − J | ( J + 1) J + J ( − J ) | J − J | (cid:32) (cid:33) . (3.21)The above fully describes the OPE decomposition of t -channel exchange. To be fully explicit,let us write out the s -channel OPE decomposition of the full MFT correlator including identityin all three-channels, without any matrix, and including color indices in the case we haveseveral currents: G abcd, MFT = δ ab δ cd ++ (cid:88) n ≥ (cid:88) J ≥ J + J λ E, same12 O λ E, same43 O (cid:12)(cid:12) n,J (cid:16) δ bc δ ad + ( − J δ ac δ bd (cid:17) G E, same , E, same∆ ,J + (cid:88) n ≥ (cid:88) J ≥ J ( n ) λ E, opp12 O λ E, opp43 O (cid:12)(cid:12) n,J (cid:16) δ bc δ ad + ( − J δ ac δ bd (cid:17) G E, opp , E, opp∆ ,J + (cid:88) n ≥ (cid:88) J ≥ J + J λ O, same12 O λ O, same43 O (cid:12)(cid:12) n,J (cid:16) δ bc δ ad + ( − J δ ac δ bd (cid:17) G O, same , O, same∆ ,J + (cid:88) n ≥ (cid:88) J ≥ J ( n ) λ O, opp12 O λ O, opp43 O (cid:12)(cid:12) n,J (cid:16) δ bc δ ad − ( − J δ ac δ bd (cid:17) G O, opp , O, opp∆ ,J , (3.22)where J ( n ) = max( | J − J | , J + J − n ) and the λ ’s refer to elements of (3.18a). The lasttwo sums run over half-integer n .Typically, one would further decompose the global symmetry indices into s -channel irreps,and symmetrical versus antisymmetrical combinations. The t and u channels contributionsthen effectively remove half the spins (the double-twist operators with the wrong symmetry),and otherwise effectively double the coefficient.Let us cross-check the above MFT spectrum. MFT operators can be written as productsof two operators and their derivatives: ∂ O ∂ O ; the game is to enumerate linear com-binations that are primaries. An equivalent exercise is to enumerate three-point structuresof the form eq. (2.10) whose Fourier transform are polynomials in x . Although finding suchexplicit polynomials is somewhat cumbersome, it is straightforward to count them by makinga generating function. We now summarize this exercise.– 21 – ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ (a) two currents ( J = J = 1) ○ ○ ○ ○ ○○ ○ ○ ○ ○○ ○ ○ ○ ○○ ○ ○ ○ ○○ ○ ○ ○ ○ (b) one current and one stress tensor Figure 3 : Spectrum of double-twist operators of the form [
J J ] n,J and [ J T ] n,J . Double circlesindicate multiplicity: there is a single trajectory for n = 0 and two for each n ≥ q ∆ z J represents an SO(3) multiplet ofdimension ∆ and spin J (that is, 2 J + 1 states). Starting from a scalar operator φ ofdimension ∆, we could characterize its descendants in terms of symmetric-traceless tensors,times Laplacian: ( ∂ µ · · · ∂ µ J − traces)( ∂ ) n φ , which contributes a term q ∆+2 n + J z J . Summingover n and J gives a generating function q ∆ (1 − q )(1 − zq ) which enumerates descendants of a scalar.Omitting steps, we find similar generating functions for the descendants of conserved currentsand generic primaries: Z conserved J = q J +1 z J (1 − q )(1 − qz ) , Z generic∆ ,J = q ∆ z J + q (1 + z ) q J − z J q − z (1 − q )(1 − qz ) . (3.23)For conserved currents, the dimension-one generator responsible for − q is simply the curl (cid:126) ∇ × • , that is, the numerator of eq. (2.25). To find the primaries that enter the OPE productof two conserved currents, we have to match the generating functions: Z conserved J × Z conserved J = (cid:88) n,J c n,J Z generic2+ n + J,J (3.24)where the c ’s are multiplicities of the various representations appearing. Putting in themultiplicities from fig. 3 and comparing the series for various values of J , J , we find perfectagreement. Beyond MFT, the Euclidean inversion formula is less efficient as double-twist operators con-taminate the cross-channel OPE. We should thus seek another way to extract the relevantOPE data: using the Lorentzian inversion formula. As a warm-up, we demonstrate that wecan reproduce the above OPE data from the Lorentzian inversion formula, using spinning-down technology. As we will explain, within this framework it is straightforward to disentangle– 22 –hysical and spurious poles, so this calculation will also confirm the decomposition (3.22). Inthis subsection, we restrict attention to parity-even four currents (“VVVV”) as a concreteexample.In d = 3, all bosonic conformal blocks can be written as spin-ups of scalar conformalblocks. In embedding space, a convenient set of spinning-up differential operators is [8] D ijii = Z Ai (cid:32) ( X i · X j ) ∂∂X Aj + ( X i · Z j ) ∂∂Z Aj − X Aj ( X i · ∂∂X j ) − Z Aj ( X i · ∂∂Z j ) (cid:33) ,D ijij = Z Ai (cid:18) ( X i · X j ) ∂∂X Ai + X Aj ( Z i · ∂∂Z i ) − X Aj ( X i · ∂∂X i ) (cid:19) ,D ijiO = (cid:15) ABCDE Z Ai X Bi ∂∂X iC (cid:18) X Dj ∂∂X jE + Z Dj ∂∂Z jE (cid:19) . (3.25) D ijii increases the spin and decreases the conformal dimension of i th operator by one unitsimultaneously. On the other hand, D ijij increases the spin of i th operator by one unit anddecreases the conformal dimension of j th operator by one unit simultaneously, while the oddoperator D iO only changes the first spin but not the dimensions. Using these operators, (forexample) our two parity-even three-point structures (cid:104) V V O (cid:105) can be constructed by actingon scalar three-point functions (cid:104)O O O(cid:105) with five spin-up operators (cid:104) V V O (cid:105) a = P a ( α ) D ( α ) ↑ (cid:104)O O O(cid:105) ( α ) , D ( α ) ↑ = (cid:16) D D , H , D D , D D , D D (cid:17) , (3.26)where H is H = 2 (cid:0) ( X · Z )( Z · X ) − ( X · X )( Z · Z ) (cid:1) . (3.27)As mentioned previously, it is important to note that the operators act on different three-pointfunctions ( α ) as the dimensions ∆ and ∆ are shifted differently for different operators. Forexample, the first and the third structures are actually ( D D , D D ) (cid:104)O ∆ +1 O ∆ +1 O ∆ ,J (cid:105) ,and the fourth structure is D D (cid:104)O ∆ O ∆ +2 O ∆ ,J (cid:105) . Each of these can be written as acombination of the five basis monomials in eq. (2.4) and ultimately we are interested only inthe linear combinations which produce the two conserved structures in our basis (2.16). Wefind that these combinations, when acting on the “funny block” ˜ G ( c,d ) J + d − , ∆ − d +1 , with externalshadow operators are: P aα = − √ β +1)(4 − τ )(∆ − −
2) ( β +1)(∆ − − τ ) √ − √ J +3)( β +1)(4 − τ )( J +1)(∆ − − √ J +3)( β +1)(4 − τ )( J +1)(∆ − − √ J +5) − ( J +1) ( J +4))( J +1)(∆ − − √ β +1)(4 − τ )( J +1)( J +2) − J ( β +1)(4 − τ ) √ J +2) √ β +1)(4 − τ )( J +1)( J +2) √ β +1)(4 − τ )( J +1)( J +2) √ ( ∆ ˜∆ − J ( J +1) ) ( J +1)( J +2) , (3.28) where β = ∆ + J and τ = ∆ − J . (The coefficients are different if we want to get the currentsinstead of their shadows.)After integrating by parts, the spinning-up operators D ( α ) ↑ become spinning-down op-erators, in our case D ( α ) ↓ = (cid:16) ¯ D ¯ D , ¯ D H , ¯ D ¯ D , ¯ D ¯ D , ¯ D ¯ D (cid:17) . The spinning-down– 23 –perators can be constructed from weight-shifting operators in [9], and we find convenient todefine them so they are adjoints to the above. This is readily done using the operator D Z from eq. (2.8) :¯ D ijii = −D AZ i (cid:32) ( X i · X j ) ∂∂X Aj + ( X i · Z j ) ∂∂Z Aj − X Aj ( X i · ∂∂X j ) − Z Aj ( X i · ∂∂Z j ) (cid:33) , ¯ D ijij = −D AZ i (cid:18) ( X i · X j ) ∂∂X Ai − X jA (cid:18) d − X i · ∂∂X i ) + ( Z i · ∂∂Z i ) (cid:19)(cid:19) , ¯ D H ij = 2 (cid:0) ( X i · D Z j )( D Z i · X j ) − ( X i · X j )( D Z i · D Z j ) (cid:1) , ¯ D ijiO = − (cid:15) ABCDE D AZ i X Bi ∂∂X iC (cid:18) X Dj ∂∂X jE + Z Dj ∂∂Z jE (cid:19) (3.29)These are adjoint to the D ’s up to a spin-dependent factor which can be traced to eq. (2.29),namely: (cid:0) D T J J ... , T J +1 ,J ... (cid:1) = 1( J + d − )( J + 1) (cid:0) T J J ... , ¯ D T J +1 ,J ... (cid:1) . (3.30)This identity makes it trivial to integrate-by-parts. For ¯ D H ij there is an extra J + d − )( J +1) since both spins change.Interestingly, we find that ¯ D ijij vanishes identically on conserved currents, so the last threespin-down operators in our list vanish identically, reducing us to a two-dimensional basis. Itwould be interesting to understand these simplifications from the perspective of the bispinorformalism for AdS /CFT [56].To find the spinned-down Lorentzian inversion formula, we now have two options. Thefirst, as described so far, is to insert the matrix in eq. (3.28) inside eq. (2.65) and integrate-by-parts. Since the last three spin-down operators vanish, we can write eq. (2.66) in terms oftwo-by-two matrices. Generally, we have c ta,b (∆ , J ) = (cid:88) α,β κ ( α,β )∆+ J (cid:90) dzdzz z (cid:12)(cid:12)(cid:12)(cid:12) z − zzz (cid:12)(cid:12)(cid:12)(cid:12) d − ˜ G ( α,β ) J + d − , ∆ − d +1 ( z, ¯ z )dDisc[ˆ P a,α ˆ P b,β D ( α,β ) ↓ G ( z, ¯ z )] , (3.31)where, from eq. (2.65), ˆ P a,α = ( − J + J J ! J !( ) J ( ) J N E/OJ J O L (cid:32) (cid:33) ac P cα . (3.32)Explicitly, for J = J = 1, the parity-even matrix evaluates to:ˆ P Ea,α = 2 √ β − τ − × (cid:32) − J +1)( J +2) J ( J +2)2(∆ − − − (∆ − − (cid:33) , (3.33) While D Z now acts on an embedding-space 5-vector Z , the dimension-dependent factor d − remains the same as in eq. (2.8). See ref. [7]. For the odd operators, we only verified that D iO is the adjoint of ¯ D iO when acting on scalar operators,sufficient for our purposes. – 24 –here only ∆ = 0 appears in κ and the block. For odd structures, in the spin-down basis D O ↓ = ( ¯ D O ¯ D , ¯ D O ¯ D ),ˆ P Oa,α = −√ J + 1)(∆ − × (cid:32) J +2)(∆ − − J +2)(∆ − J (∆ −
3) 1 J (∆ − (cid:33) . (3.34)These matrices tell us how to convert the scalar inversion of the spinned-down correlators(given below in eq. (3.36)) to OPE data in opposite/same-helicity structures.There is a simple check: acting with the spin-down operators ˆ P a,α D α ↓ on the three-pointspinning structure T b O , we must get δ ba times a canonically normalized scalar three-pointstructure T O . In fact this gives a second method to directly find the matrix ˆ P a,α , by-passing the spinning Lorentzian inversion formula. We find precise agreement between thetwo methods. (The second one being admittedly more straightforward.)These operators can be applied to any correlator. We now consider t -channel identityexchange: G = H H ( − X · X ) ∆ +1 ( − X · X ) ∆ +1 , (3.35)which gives for example the even spinned-down correlator D ↓ G D (1 , ↓ G = − y (¯ y + 1)(24 y + 3 y (5 − y ) + 3 y (¯ y (4¯ y + 3) + 1) − y (¯ y + 1)(3¯ y (4¯ y + 3) + 1)+3(¯ y + 1) (¯ y (8¯ y + 7) + 1)) , D (2 , ↓ G = − y (¯ y + 1) (cid:0) y − y (¯ y + 1) + (¯ y + 1) (cid:1) , D (1 , ↓ G = D (2 , ↓ G = − y (¯ y + 1) (cid:0) y + y (1 − y ) + y (¯ y + 1)(5¯ y + 1) − y + 1) (3¯ y + 1) (cid:1) , (3.36)where we reparameterized the cross-ratios by ( z = y y , ¯ z = y ).Inserting in eq. (3.33) it remains to do the scalar inversion integrals of eqs. (3.36). Agood strategy is to expand in y → z twist-by-twist. Thisalso requires the lightcone expansion z → G J + d − , ∆ − d +1 ( z, ¯ z ) in the inversion formula(2.66), which can be done by noting (see, eq. (A.24) in [47]) κ ( β ) κ ( β + 2 p ) (1 − z ) a + b (1 − z ¯ z ) d − G J + d − , ∆ − d +1 (cid:12)(cid:12) q,p ∼ B q,p z J − ∆2 + n + d − k β +2 m (¯ z ) , (3.37)where B q,p can be recursively solved by the quadratic Casimir equation [47]. Moreover, wecan take use of the following integral formula to do the integral over ¯ z [47] I ˆ τ ( β ) = (cid:90) d ¯ z ¯ z (1 − ¯ z ) a + b κ a,bβ k a,bβ (¯ z ) dDisc[ (cid:0) − ¯ z ¯ z (cid:1) ˆ τ − b (¯ z ) − b ]= Γ( β − a )Γ( β + b )Γ( β − ˆ τ )Γ( − ˆ τ − a )Γ( − ˆ τ + b )Γ( β − β + ˆ τ + 1) . (3.38)– 25 –ith this strategy we can calculate the result analytically for any n >
0, and find a simplecommon formula given below.The case n = 0 is subtle as we discussed previously in subsection 3.2: the structuresbecome degenerate. In fact the whole matrix (3.33) blows up as τ →
2. The solution, asabove, is to apply a further rotation to the basis in eq. (3.20). In the ( T E, reg123 , T E, sing123 ) basis,the matrix (3.33) becomes:ˆ P E, rotated a,α = √ (cid:32) − J − J − J ( J +1)( J +2)(2 J +1) J ( J − J +1)( J +2)(2 J +1)12( J − J (2 J +1) − J +14( J − J +1) (cid:33) , (3.39)which is now nicely finite. The same rotation will also work in the computation of anomalousdimensions in the next section.For MFT correlators discussed here where D ↓ G is actually a finite sum of powers of cross-ratios times Gegenbauer polynomials, a more compact and comprehensive trick is availableto extract the OPE data, see appendix C.2. Our result, for n ≥
1, the coefficients of even(opposite/same) helicity structures are then: λ E O λ E O (cid:12)(cid:12) n,J = ( J + 1) (2 n + J + ) n +2 J +3 ( n + ) ( n + J + 1) (cid:32) J ( J − J +2)( J +1) (2 n + J +1)(2 n + J +2)(2 n + J )(2 n + J − (cid:33) , (3.40)which is precisely eq. (3.18a) with J = J = 1. For the leading trajectory, in the rotatedbasis we find λ E, rotated12 O λ E, rotated43 O (cid:12)(cid:12) ,J = 2Γ( J + 1) Γ(2 J + 1) × J ( J − J + 2)( J + 1) (cid:32) (cid:33) , (3.41)which again agrees with eq. (3.21) with J = J = 1. This confirm that spurious poles simplydouble the n > /CFT The simplicity and diagonal nature of the mean field OPE encourages us to look at theleading corrections. In this section, we study CFT current correlators that are dual tobulk YM gluon amplitudes at tree-level. The Lorentzian inversion formula will give usthe corresponding anomalous dimensions in terms t - and u - channel exchanges of conservedcurrents.These correlation functions have been previously discussed in momentum space. Resultsare remarkably tractable thanks to the fact that YM is conformally invariant (at tree-level)and AdS is conformally flat [35, 38, 57, 58]. Our goal is to obtain the corresponding OPEanomalous dimension, which we will then compare with the flat space limit in the next section.The flat space limit of AdS/CFT [59–61] ( R AdS → ∞ ) has not been much studied for spinningoperators (with a notable exception [62]) and we feel it is important to clarify it. Similarlyto the scalar case, one may expect (massless) amplitudes to be encoded in the z → ¯ z (cid:9) “bulk-point” limit [63, 64], or equivalently the large-twist limit of OPE data. This will be confirmedin the next section. – 26 –1 34 Figure 4 : Witten diagram for (cid:104)
V V V V (cid:105) with on-shell t -channel gluon exchange. Two evenand one odd coupling can be used in each vertex; u -channel is similar with 1 and 2 swapped. Our strategy is use spin-up/spin-down operators to reduce the calculation to scalar Lorentzianinversion formulas. The spin-down operators were described and validated in section 3.3,acting on identity exchange in the t - and u -channel. The exchanged operator is now a current,as shown in fig. 4. (Double-trace exchanges do not contribute to tree-level accuracy, thanksto the double-discontinuity.)From the CFT perspective, each current exchange involves two parity-even and one oddcoupling, described below eq. (4.1), which map one-to-one with bulk on-shell three-gluon cou-plings. These can be obtained from a bulk Lagrangian including higher-derivative corrections: L = − g F aµν F µνa + θ π F aµν ˜ F µνa − f abc g (cid:16) g H F µ νa F ν ρb F ρ µc + g (cid:48) H ˜ F µ νa ˜ F ν ρb ˜ F ρ µc (cid:17) + · · · , (4.1)where ˜ F µν = (cid:15) µνσρ F σρ . We show that in appendix A that the couplings satisfy: λ (e1) V V V = g YM √ , λ (e2) V V V = g H √ , λ (o2) V V V = g (cid:48) H √ π (4.2)where the structures refer to the even/odd basis in eq. (2.14). (We recall that the firststructure is the “opposite helicity” one which generically exists for spin J ≥ λ ( i ) V V V .We consider only the parity-even couplings. There are then four ways to dress the thegraph in fig. 4: G , Yang-Mills vertex to Yang-Mills vertex ,G , higher-derivative vertex to higher-derivative vertex ,G , Yang-Mills vertex to higher-derivative vertex ,G , higher-derivative vertex to Yang-Mills vertex . (4.3)– 27 –n each case the t -channel block can be written as the spin-up of a scalar block, so afterspinning down D ( c,d ) ↓ G ( z, ¯ z ) in eq. (2.66) gives a 8th order differential equation acting onscalar blocks. The cross-channel scalar blocks themselves are not known in closed form; inappendix C.3 we provide the series expansion of the log z term to any order in z , which issufficient to calculate anomalous dimensions exactly, in terms of ( y = z/ (1 − z ) , ¯ y = (1 − ¯ z ) / ¯ z ),i.e., eq. (C.16). For example, at the leading order in the lightcone expansion y →
0, we find D ↓ G = log yπ − y (¯ y +1) ( ¯ y +27¯ y +675¯ y +1225 ) y / y ( y − y − y − ) y / y ( y − y − y − ) y / − y ( y +26¯ y +9 ) ¯ y / + O ( y ) , (4.4)where we parameterize y = z/ (1 − z ) , ¯ y = (1 − ¯ z ) / ¯ z . At the leading order, D ↓ G has thesame expression as D ↓ G , but differs at the second and higher orders. Up to the leadingorder, D ↓ G = D ↓ G is D ↓ G = 3 log yπ − y (¯ y +5) ( ¯ y − y +171¯ y +245 ) y / y ( ¯ y +¯ y − y − ) y / y ( ¯ y +¯ y − y − ) y / − y ( y − y +3 ) ¯ y / + O ( y ) . (4.5)The above expansions eq. (4.4) and eq. (4.5) would then be used in principle to obtain theleading-twist anomalous dimensions by simply integrating over ¯ y using the formula (3.38).As discussed in subsection 3.2, the leading-twist analysis is a bit subtle due to a degeneracyin three-point structures, and is discussed below. As the rotation in eq. (3.20) removes alldivergences, the anomalous dimension can be computed using just the logarithmic term ineq. (4.4). Nontrivially, we find a result proportional to the leading order matrix (cid:18) (cid:19) , asrequired by the fact that there is a single leading-twist family (the number of operators can’tchange under small perturbations). The anomalous dimension is then γ E (cid:12)(cid:12) n =0 = − (cid:16) β − β +28 β − β +32( β − β − β − β ( β +2) (cid:0) ( λ e1 V V V ) + ( λ e2 V V V ) (cid:1) + − β λ e1 V V V λ e2 V V V (cid:17) ( T + ( − J U ) . (4.6)At subleading twists, the calculation uses analogous expressions together with the s -channel expansion (3.37) and (3.38). This yields the anomalous dimensions as analytic functions of β for fixed n ≥
1. Includingthe ˆ P matrix in eq. (3.33), we obtain (cid:104) cγ (cid:105) J, ∆ , which we then divide by the generalized freeOPE data (3.11) (with J = J = 1), to arrive at anomalous dimensions. It is importantto include both t - and u -channel identity in the denominator, which effectively doubles it as Since the second structure T E, sing11 O has a nonvanishing discontinuity, its ∼ ( λ (e1) V V V ) OPE coefficient willbe required to predict the one-loop dDisc, in addition to the given anomalous dimension. – 28 –iscussed below (3.22). In the pure Yang-Mills case we find: γ E = λ (e1) V V V ) π (cid:0) T + ( − J U (cid:1) diag (cid:32) ψ β − n − − ψ β + n − β − n )( β − n +2) + − n + β − ψ β − n − ψ β + n +2 + β +2 n − β +2 n − + n + β (cid:33) , (4.7)where diag represents the diagonal matrix, and we have factored out T - and U -channel colorstructures T = f bce f ade , U = f ace f bde . (4.8)These should be viewed as operators acting on the initial pair, for example both have theeigenvalue T, U (cid:55)→ C A when acting on a color-singlet state δ ab .Eq. (4.7) (for n ≥
1) gives the CFT analog of the four-point Parke-Taylor amplitude.We note that to all orders in the 1 /β , the two entries are related by the reciprocity relation β (cid:55)→ − β , which could have been anticipated from the off-diagonal nature of the lighttransform in eq. (2.57). The fact that it is diagonal will match with the vanishing of non-helicity-conserving flat space amplitudes at tree-level.The Yang-Mills self-interaction also gives diagonal anomalous dimension the odd double-twists (which have half-integer n ): γ O = λ (e1) V V V ) π diag (cid:16) ψ β − n − ψ β + n − β − n − β − n ) (cid:17) (cid:0) T − ( − J U (cid:1)(cid:16) ψ β − n − ψ β + n + β +2 n − β +2 n ) (cid:17) (cid:0) T + ( − J U (cid:1) . (4.9) Let us now record the pure higher-derivative corrections, which involve purely algebraic ex-pressions: γ E = λ (e2) V V V ) π diag ( n ( β − ( n +8( β − n +( β − β +4 ) (2 n − β − n − β )(2 n − β +2)(2 n − β +4) (cid:0) − T − ( − J U (cid:1) ( n ( β − − ( n +1) + β − n +1) β ) (2 n + β − n + β − n + β )(2 n + β +2) (cid:0) − T − ( − J U (cid:1) , (4.10) γ O = λ (e2) V V V ) π diag ( n ( β − ( n +8( β − n +( β − β +4 ) (2 n − β − n − β )(2 n − β +2)(2 n − β +4) (cid:0) T − ( − J U (cid:1) ( n ( β − − ( n +1) + β − n +1) β ) (2 n + β − n + β − n + β )(2 n + β +2) (cid:0) T + ( − J U (cid:1) . (4.11)The even and odd matrices are identical up to some signs, and again reciprocity β (cid:55)→ − β swaps the trajectories up to a minus sign.The G contributions (one Yang-Mills and one higher-derivative vertex) violate helicityconservation and give purely off-diagonal anomalous dimensions. Since the Lorentzian inver-sion formula gives us (cid:104) cγ (cid:105) J, ∆ , we divide the off-diagonal terms by the geometric mean of GFFcoefficients to define a symmetrical anomalous dimension matrix γ even12 = γ even21 : γ E = λ (e1) V V V λ (e2) V V V π − n ( β − √ ( β − n )( β − n +2)( β ( β − n − T +( − J U )( β − n +2)( β − n ) √ ( β − n − β − n − β +2 n − β +2 n − β +2 n )( β +2 n +2) (cid:32) (cid:33) . (4.12)– 29 –he odd γ is the same and γ is also identical up to an overall minus sign (such that thesum vanishes: γ O + γ O = 0, which will be in agreement with symmetries of the scatteringamplitude).We end this section by giving the large- n limit of above anomalous dimensions, whichwill be compared in the next section with flat-space 2-to-2 gluon scattering amplitudes: γ E | n →∞ = λ (e1) V V V ) π ( T + ( − J U )diag (cid:32) ψ J − − log(2 n ) + J − J +1)( J +2) ψ J +1 − log(2 n ) (cid:33) ,γ O | n →∞ = λ (e1) V V V ) π (cid:32) (cid:0) ψ J +1 − log(2 n ) − J ( J +1) (cid:1) ( T − ( − J U ) (cid:0) ψ J +1 − log(2 n ) (cid:1) ( T + ( − J U ) (cid:33) ,γ E/O | n →∞ = λ (e2) V V V ) π (cid:32) n ( ∓ T +( − J U )( J − J ( J +1)( J +2)
00 0 (cid:33) ,γ E/O | n →∞ = λ (e1) V V V λ (e2) V V V π − n ( T +( − J U ) √ ( J − J ( J +1)( J +2) (cid:32) (cid:33) ,γ E/O | n →∞ = λ (e1) V V V λ (e2) V V V π ∓ n ( T +( − J U ) √ ( J − J ( J +1)( J +2) (cid:32) (cid:33) . (4.13)We note that each higher-derivative correction λ ( e V V V comes accompanied with a power of n ∼ s , as expected from bulk dimensional analysis. Furthermore, we see that the differencebetween even- and odd- same-helicity anomalous dimensions vanishes at large- n : γ E, same11 − γ O, same11 = 128( λ (e1) V V V ) π T + ( − J U )( β + n − ∼ n . (4.14)This indicates that the same-helicity amplitude M ++++ vanishes in the flat-space limit (asexpected). However, we find it remarkable that it is not identically zero in AdS space. Thissuggests that, in a more precise treatment where the flat-space limit is defined as R → ∞ as opposed to s → ∞ , a distributional term near s = 0 may survive; such terms couldpotentially give a new perspective on four-dimensional unitarity and the rational one-loopamplitude M (1)++++ . We leave this to future work. n limit from gluon scattering amplitudes There is a close relation between the anomalous dimensions at large dimension in a CFT d and the scattering amplitude of a dual QFT d +1 in the flat space limit of AdS. This can beseen for example by considering kinematic configurations which focus particles — such as theanalytically continued z → z “bulk-point” limit, see for example [59–61, 63–65] . For masslessexternal particles (dual to our currents), since the past and future states are connected by This kinematic configuration is, however, modified if external particles are massive [66–69]. – 30 – X X X P Figure 5 : Bulk-point kinematics in Lorentzian cylinder of AdS. X and X are at Lorentziantime − π/ X and X are at Lorentzian time π/
2, where particles are focused on the bulk-point P .time π on the cylinder, the scattering phase is related to CFT anomalous dimensions by thesimple dictionnary γ n,J | n →∞ → − π a J , sR = 4 n , (5.1)where a J is the partial-wave amplitudes with angular momentum J , s is the Mandelstaminvariant of the bulk scattering process, and R is the AdS radius. We often take R = 1 belowfor simplicity and take s (cid:54) = 0, so the limit is equivalent to n → ∞ . (In general the amplitudemaps to a weighted average of anomalous dimensions. A one-loop example is provided in[49].) We expect this relation to work for spinning operators as well, for suitably definedpartial waves. Two-particle scattering states in QFT can be organized according to their SO(3) spin in therest frame of their total momentum, P = p + p . Since rotations commute with helicity, wecan choose a basis of states with definite helicity. For definiteness, we focus here on the caseof two massless spin 1 particles.We use the spinor-helicity formalism where each null momentum is factorized into aproduct of spinors, p/ i = | i ] (cid:104) i | , see [70]. Under little-group rotations of spinors | i ] and | i (cid:105) byopposite phases, a state of helicity h transforms like | i ] h . We treat two-particle states likea massive particle of momentum P and spin J , which in index-free notation is a polynomial ∼ | (cid:15) (cid:105) J in a left-handed spinor | (cid:15) (cid:105) . (There is no need to use right-handed spinors, since P can be used to convert one into the other, see [71].) Lorentz and little-group symmetries then– 31 –niquely fix the matrix elements of two-particle states Ψ J ± : (cid:104) − − | Ψ J −− (cid:105) = (cid:104) (cid:15) (cid:105) J (cid:104) (cid:15) (cid:105) J (cid:104) (cid:105) J − [12] , (cid:104) + − | Ψ J − + (cid:105) = (cid:104) (cid:15) (cid:105) J +2 (cid:104) (cid:15) (cid:105) J − (cid:104) (cid:105) J , (cid:104) + + | Ψ J ++ (cid:105) = (cid:104) (cid:15) (cid:105) J (cid:104) (cid:15) (cid:105) J (cid:104) (cid:105) J +1 / [12] , (cid:104) − + | Ψ J + − (cid:105) = (cid:104) (cid:15) (cid:105) J − (cid:104) (cid:15) (cid:105) J +2 (cid:104) (cid:105) J . (5.2)More precisely, symmetries fix the states up to a power of s = − P , which we chose so thatall states have the same dimension. We further define the state | Ψ Jh h (cid:105) to be orthogonal togluons of other helicity.In the above kinematic factors we treat the two particles as distinguishable. These arerelated to actual gluon states by adding color labels and accounting for Bose symmetry: fullydecorated states can be defined as (cid:104) h c h d | Ψ J,abh h (cid:105) = δ ad δ bc δ h h δ h h (cid:104) h h | Ψ Jh h (cid:105) + δ ac δ bd δ h h δ h h (cid:104) h h | Ψ Jh h (cid:105) . (5.3)Since interactions can change helicities, the action of the S-matrix on these states takes theform of a 4 × S| Ψ Jh a,h b (cid:105) = (cid:88) h ,h ,c,d S Jh a,h bh d,h c | Ψ Jh c,h d (cid:105) + multi-particles . (5.4)As is customary, we subtract the identity part: S = 1 + i A , where A is the scatteringamplitude. In the 2 → S J = ( δ δ + δ δ ) + ia J , where we use collectiveindices in δ = δ h h δ da . The partial wave a is then simply the amplitude in the | Ψ (cid:105) basis: a J = A ⊗ | Ψ J (cid:105) , (5.5)which can be computed as a phase-space integral. To be fully explicit with indices (see alsoeq. (2.16) of [72]): a Jh a,h bh d,h c = 12 (cid:88) h (cid:48) ,h (cid:48) ,a (cid:48) ,b (cid:48) (cid:90) d Ω64 π (cid:104) h c h d |A| a (cid:48) h (cid:48) b (cid:48) h (cid:48) (cid:105)(cid:104) h (cid:48) a (cid:48) h (cid:48) b (cid:48) | Ψ Jh a,h b (cid:105)(cid:104) h h | Ψ Jh ,h (cid:105) (5.6)= 116 π (cid:90) d Ω4 π (cid:104) h c h d |A| ah bh (cid:105) (cid:104) h h | Ψ Jh ,h (cid:105)(cid:104) h h | Ψ Jh ,h (cid:105) . (5.7)The second form will be particularly useful for calculations. Notice that the two terms ineq. (5.3) simply canceled the symmetry factor . In this integral, p and p are held fixedand d Ω represents the solid angle of (cid:126)p in the rest frame of P .The angular integral can be conveniently parametrized in terms of spinors via [73] | (cid:105) = cos θ | (cid:105) − sin θe iφ | (cid:105) , | (cid:105) = sin θe − iφ | (cid:105) + cos θ | (cid:105) , (5.8)– 32 –ith analogous expressions for the conjugate spinors |
1] and |
2] with the phase reversed φ (cid:55)→ − φ . In the rest frame of P , the variables θ and φ represent physically (half) theazimuthal and polar angle with respect to p . The measure is then (cid:90) d Ω4 π = (cid:90) π dφ π (cid:90) π sin(2 θ ) dθ . (5.9)It is important to note that both the numerator and denominator in eq. (5.7) depend on | (cid:15) (cid:105) , p and p , in addition to the integration variables θ, φ . However, since the result of theintegral is determined by symmetry, the ratio after doing the integral is guaranteed to be apure number independent of these variables.This method allows us to define partial waves without having to worry about the nor-malization of the states. The idea is that the eigenvalues of the matrix S J map to weightedaverages of CFT anomalous dimensions e − iπγ . To leading order in perturbation theory, thisrelation gives simply, as quoted: γ J ≈ − π a J . (5.10)Surprisingly, the exact same relation has an interpretation purely in the context of QFT: thephase of the S-matrix acting on form factors of local operators gives the dilatation operatorof the QFT: SF ∗ = e − iπD F ∗ [72]. This was used there to compute anomalous dimensionsof local operators of a QFT , as labelled by their two-particle form factors. (For example,the infrared-safe combination γ − γ − + − acting on a color-singlet state computes theQCD β -function.) Here γ J instead gives holographically a CFT anomalous dimension γ ( n ) where 4 n = sR is large. It is amusing that anomalous dimensions in the bulk QFT d +1 and boundary CFT d are computed by literally the same formula. On-shell amplitudes in YM are recorded in appendix D. We use these on-shell amplitudestogether with eq. (5.7) to extract the corresponding partial-wave amplitudes, from which wewill find perfect agreement with CFT eq. (4.13).We begin with the pure Yang-Mills theory, then add higher-derivative corrections. Using Yang-Mills amplitudes eq. (D.4), we can readily evaluate (5.7). For example, we obtain( a YM ) − + − + = g π (cid:104) (cid:15) (cid:105) J − (cid:104) (cid:15) (cid:105) J +2 (cid:90) π dφ (cid:90) π dθ ( (cid:104) (cid:15) (cid:105) cos θ − (cid:104) (cid:15) (cid:105) sin θe iφ ) J +2 × ( (cid:104) (cid:15) (cid:105) cos θ + (cid:104) (cid:15) (cid:105) sin θe − iφ ) J − cos θ × ( T cot θ + U tan θ ) , ( a YM ) − + + − = g π (cid:104) (cid:15) (cid:105) J − (cid:104) (cid:15) (cid:105) J +2 (cid:90) π dφ (cid:90) π dθe iφ ( (cid:104) (cid:15) (cid:105) cos θ − (cid:104) (cid:15) (cid:105) sin θe iφ ) J − × ( (cid:104) (cid:15) (cid:105) cos θ + (cid:104) (cid:15) (cid:105) sin θe − iφ ) J +2 sin θ × ( T cot θ + U tan θ ) , (5.11)– 33 –nd ( a YM ) + − − + = ( a YM ) − + + − when the integral is evaluated. Same-helicity partial-waveamplitudes give( a YM ) −− −− = ( a YM ) ++ ++ = g π (cid:104) (cid:15) (cid:105) J (cid:104) (cid:15) (cid:105) J (cid:90) π dφ (cid:90) π dθ ( (cid:104) (cid:15) (cid:105) cos θ − (cid:104) (cid:15) (cid:105) sin θe iφ ) J × ( (cid:104) (cid:15) (cid:105) cos θ + (cid:104) (cid:15) (cid:105) sin θe − iφ ) J × (cid:0) T cot θ + U tan θ (cid:1) , (5.12)and other helicity-violation terms identically vanish, e.g., a − + −− = a −− − + = 0. It is worthnoting that above integrals fail to converge due to IR divergence. In the context of computingUV anomalous dimensions in QFT, these could be subtracted using that the stress-tensor isprotected [72]. However, in our context these reflect physical divergences of bulk anomalousdimensions as R AdS → ∞ . We thus regularize the above equations by introducing a small-angle cut-off (cid:15) < θ < π − (cid:15) which we will then compare with the bulk cutoff n → ∞ . Theazimuthal integral can be readily evaluated, which gives( a YM ) − + − + = − g π (cid:0) γ E + log (cid:15) + ψ J − + j − j +2)( j +3) + j − (cid:1) T + g π ( J − ( − J U , ( a YM ) − + + − = − g π (cid:0) γ E + log (cid:15) + ψ J − + j − j +2)( j +3) + j − (cid:1) ( − J U + g π ( J − T , ( a YM ) −− −− = ( a YM ) ++ ++ = − g π (cid:0) γ E + log (cid:15) + ψ J +1 (cid:1) ( T + ( − J U ) . (5.13)As a simple check, acting on color-singlet states ( T, U (cid:55)→ C A ) and taking large spin, wereproduce the famous logarithmic scaling of gauge theories, γ = − aπ → + g π log J .To compare with anomalous dimensions evaluated in CFT, we should rotate to paritybasis ( a YM ) E = 12 diag (cid:32) ( a YM ) − + − + + ( a YM ) − + + − + (+ ↔ − )( a YM ) −− −− + ( a YM ) ++ ++ (cid:33) , ( a YM ) O = 12 diag (cid:32) ( a YM ) − + − + − ( a YM ) − + + − + (+ ↔ − )( a YM ) −− −− + ( a YM ) ++ ++ (cid:33) , (5.14)where (+ ↔ − ) denotes flipping all helicity. Imposing following simple identification (cid:15) = e − γ E n , (5.15)and using λ (e1) V V V = g YM / (16 √
2) from eq. (4.2), we then find a perfect match with the CFTanomalous dimension in eq. (4.13): γ E/O | n →∞ = − π ( a YM ) E/O . (5.16) Let us start with the pure higher-derivative interaction (e.g. at both vertices). Using the am-plitudes recorded in eq. (D.5), we can immediately conclude that ( a H ) −− −− = ( a H ) ++ ++ =– 34 –, because M H − − + + only have s -channel pole and thus is evaluated to be identicallyzero, which nicely agrees with predictions from CFT. On the other hand, ( a H ) − + − + and( a H ) − + + − contributes T and U separately, giving( a H ) − + − + = g π (cid:104) (cid:15) (cid:105) J − (cid:104) (cid:15) (cid:105) J +2 (cid:90) π dφ (cid:90) π dθ ( (cid:104) (cid:15) (cid:105) cos θ − (cid:104) (cid:15) (cid:105) sin θe iφ ) J +2 × ( (cid:104) (cid:15) (cid:105) cos θ + (cid:104) (cid:15) (cid:105) sin θe − iφ ) J − (cos θ ) sin θ (cos(2 θ ) − × U , ( a H ) − + + − = g π (cid:104) (cid:15) (cid:105) J − (cid:104) (cid:15) (cid:105) J +2 (cid:90) π dφ (cid:90) π dθe iφ ( (cid:104) (cid:15) (cid:105) cos θ − (cid:104) (cid:15) (cid:105) sin θe iφ ) J − × ( (cid:104) (cid:15) (cid:105) cos θ + (cid:104) (cid:15) (cid:105) sin θe − iφ ) J +2 (sin θ ) cos θ (cos(2 θ ) + 3) × T . (5.17)We can readily evaluate the integrals and find( a H ) − + − + = 3 g s π ( J − ( − J U , ( a H ) − + + − = 3 g s π ( J − T , (5.18)and simultaneously flipping helicity + ↔ − gives the same answer. Rotating to the Even/Oddparity basis readily gives( a H ) E/O = 3 g s π ( J − (cid:32) (cid:0) ∓ T + ( − J U (cid:1)
00 0 (cid:33) . (5.19)Using λ e2 V V V = g H / (8 √
2) from eq. (4.2) and s = 4 n from eq. (5.1), we achieve a perfectagreement with CFT anomalous dimensions from eq. (4.13). γ E/O | n →∞ = − π ( a H ) E/O . (5.20)The contact ambiguity that has the same scaling dimension as the a H interaction (seeeq. (D.5)) affects the J = 2 OPE data, making the preceding partial wave valid only for J >
2. We believe that all other results are valid for
J > M −− + − and M − +++ , which is not symmetricand thus give slightly different partial-wave amplitudes that form a non-symmetric and anti-diagonal matrix; eigenvalues of the resulting matrix should agree with CFT eigenvalues fromeq. (4.13) (that is, we only compare up to similarity transformation).For example, we find some of ( a mix ) = a (cid:12)(cid:12) g YM g H for M −− + − type mixing reads( a mix ) − + −− = − g YM g H s π (cid:104) (cid:15) (cid:105) J (cid:104) (cid:15) (cid:105) J (cid:90) π dφ (cid:90) π dθe − iφ ( (cid:104) (cid:15) (cid:105) cos θ − (cid:104) (cid:15) (cid:105) sin θe iφ ) J +2 × ( (cid:104) (cid:15) (cid:105) cos θ + (cid:104) (cid:15) (cid:105) sin θe − iφ ) J − sin(2 θ ) × ( T cot θ + U tan θ ) , ( a mix ) −− + − = − g YM g H s π (cid:104) (cid:15) (cid:105) J − (cid:104) (cid:15) (cid:105) J +2 (cid:90) π dφ (cid:90) π dθe iφ ( (cid:104) (cid:15) (cid:105) cos θ − (cid:104) (cid:15) (cid:105) sin θe iφ ) J × – 35 – (cid:104) (cid:15) (cid:105) cos θ + (cid:104) (cid:15) (cid:105) sin θe − iφ ) J sin θ cos θ × ( T cot θ + U tan θ ) . (5.21)( a mix ) − + ++ gives the same as ( a mix ) − + −− , and ( a mix ) −− − + is similar to ( a mix ) −− + − butflipping e iφ → e − iφ . Though the integrand looks a bit different when we flipping + ↔ − ,we find they give the same result( a mix ) − + −− = ( a mix ) − + ++ = − g YM g H s πJ ( J − (cid:0) T + ( − J U (cid:1) , ( a mix ) −− − + = ( a mix ) −− + − = − g YM g H s π ( J + 1)( J + 2) (cid:0) T + ( − J U (cid:1) , (5.22)and the same for flipping ± → ∓ . Now we can rotate to the parity basis. To compare withCFT calculation where we record γ and γ separately, we should be careful about clarifying a mix12 and a mix21 : γ corresponds to a mix with different helicity in ( h , h ) , and γ correspondsto a mix with same helicity in ( h , h ). We find( a mix12 ) E/O = 12 (cid:32) a mix ) − + −− + ( a mix ) + − ++ ( a mix ) −− + − + ( a mix ) ++ − + (cid:33) , ( a mix21 ) E/O = ± (cid:32) a mix ) − + ++ + ( a mix ) + − −− ( a mix ) −− − + + ( a mix ) ++ + − (cid:33) . (5.23)The signs work out so that, when we add the contributions from the two vertices, the parity-even part doubles and the odd part cancels out ( a O + a O = 0), as found in the precedingsection. Using the dictionary λ e2 V V V = g H / (8 √
2) and λ e2 V V V = g H / (16 √
2) from eq. (4.2) and s = 4 n , we find that the eigenvalues of a mix precisely coincide with γ and γ in eq. (4.13)up to − /π , i.e., γ E/O (cid:12)(cid:12) n →∞ ∼ − π ( a mix ) E/O , (5.24)and ∼ denotes the equivalence up to similarity transformation. In this paper, we introduced a helicity basis for conformal blocks of conserved currents ofany spins in three-dimensional CFTs. We observed that the concept of helicity is conformallyinvariant (see subsection 2.2) and can be defined without reference to any particular formalismsuch as momentum space. This ensures that the helicity basis plays nicely with crossingsymmetry. We found evidence of this in the OPE decomposition of mean-field correlators,which turns out nicely diagonal (see eq. (3.10), and we further computed the CFT OPEdata dual to tree-level gluon scattering scattering of Yang-Mills theory in AdS , includinghigher-derivative corrections.The YM calculation was done using the spinning Lorentzian inversion formula (seeeq. (4.7), (4.9) and following), which gives the OPE data for sufficiently large spin J > J ∗ ,where we expect J ∗ = 1 without including higher-derivative corrections and J ∗ = 2 with– 36 –hem. The anomalous dimensions follow a simple diagonal / off-diagonal pattern and preciselymatch, in the large-twist limit, with the partial waves in the flat space limit of the bulk theory,shown in eq. (4.13). We found a simple one-to-one dictionary between on-shell three-pointinteractions in bulk AdS and three-point helicity structures (see eq. (A.14)).We expect that a calculation of the 6 j symbol (also known as crossing kernel) in thehelicity basis could thus greatly help bootstrap calculations involving conserved currents andstress tensors in 3d CFTs. We expect the 6 j symbols to be diagonal in helicity basis. Whethera basis exists in higher dimensions which would diagonalize mean-field correlators remainsan open question. Better understanding the flat-space limit of massless-massless-massivethree-point functions could shed light on this question.In perturbation theory, our findings pave the way for a study of loop corrections in YM with a four-dimensional treatment of infrared effects. Compared with flat space, AdS physicscomes with a built-in infrared regulator, and an interesting fact is that leading double-twiststates (the n = 0 trajectory) do not have a definite helicity (see eq. (3.21)). The notion thatzero-energy gluons do not have helicity resonates with findings from the asymptotic symmetrycontext (see for example [74]), and it would be interesting to make this connection closer.Eq. (4.14) suggest that the tree-level amplitude for four same-helicity gluons is not identicallyzero even in flat space, but retains a sort of distributional component around zero energy,which could be important for unitarity calculations in flat space.Nonperturbatively, we expect the helicity basis to be particularly convenient for uncover-ing the implications of crossing symmetry on stress tensor correlators in CFT and the dualgravitational physics. Acknowledgments
We would like to thank for Nikhil Anand for useful conversations and initial collaboration onAdS calculations, and Petr Kravchuk for discussions. Work of S.C.-H. is supported by theNational Science and Engineering Council of Canada, the Canada Research Chair program,the Fonds de Recherche du Qu´ebec - Nature et Technologies, the Simons Collaboration onthe Nonperturbative Bootstrap, and the Sloan Foundation. Work of Y.-Z.L. is supported inparts by the Fonds de Recherche du Qu´ebec - Nature et Technologies. A (cid:104) V V V (cid:105) from Witten-diagram
In this appendix, we start from AdS Lagrangian in d = 3 to derive (cid:104) V V V (cid:105) three-pointfunctions. From helicity basis we constructed in the main text, it follows that (cid:104)
V V V (cid:105) hasthree independent structures, and it is expected the first structure corresponds to the Yang-Mills vertex and the higher-derivative coupling in AdS is captured by the second two (theodd and even “same-helicity” ones, which are analytic in spin for J ≥ L = − g F aµν F µνa + θ π F aµν ˜ F µνa − f abc g (cid:16) g H F µ νa F ν ρb F ρ µc + g (cid:48) H ˜ F µ νa ˜ F ν ρb ˜ F ρ µc (cid:17) + · · · , (A.1)where a, b, c are SU( N ) group indices, f abc is the structure constant, ˜ F µν = (cid:15) µνσρ F σρ and · · · is other terms that are not relevant to our purpose. After rescaling the fields by the couplingto make A canonically normalized, it follows that we have two three-point gluon verticesYang-Mills: − g YM f abc ∂ µ A aν A µb A νc , Higher-derivative: − g H f abc F µ νa F ν ρb F ρ µc + odd part , (A.2)where only the linearized part of F µν will contribute in the second case.It is most convenient to work with the AdS embedding formalism [75] where the bulk-to-boundary propagator with conformal dimension ∆ and spin J [75]Π ∆ ,J ( Y, W ; X, Z ) = C (∆ , J ) (cid:0) ( − X · Y )( W · Z ) + 2( W · X )( Z · Y ) (cid:1) J ( − X · Y ) ∆+ J , (A.3)where X and Z are embedding coordinate and auxiliary polarization respectively for boundaryCFT, similarly Y and W are ( d + 2)-dimensional embedding coordinate and polarization forthe bulk AdS d +1 , which are constrained by X = X · Z = Z = 0 , Y = − , Y · W = W = 0 , (A.4)and have the further redundancy Z (cid:39) Z + αX . The normalization factor reads C (∆ , J ) = ( J + ∆ − π d (∆ − − d ) . (A.5)Derivatives in AdS can be evaluated using the bulk covariant derivative operator [75] ∇ A = ∂∂Y A + Y A ( Y · ∂∂Y ) + W A ( Y · ∂∂W ) . (A.6)which commutes with the constraints. It is also convenient to introduce the differentialoperator K A [75] K WA = (cid:18) ∂∂W A + Y A ( Y · ∂∂W ) (cid:19) (cid:18) d −
32 + W · ∂∂W (cid:19) − W A (cid:18) ∂ ∂W · ∂W + ( Y · ∂∂W ) (cid:19) , (A.7)which helps do index contractions in AdS: (cid:88) W f ( W ∗ ) g ( W ) = 1 J !( d − ) J f ( K W ) g ( W ) . (A.8)– 38 –ith these ingredients, we are ready to compute (cid:104) V V V (cid:105) by performing the following integralsover (Euclidean) AdS Y = − (cid:104) V ( X ) V ( X ) V ( X ) (cid:105) YM = − g YM f abc C d − , (cid:90) EAdS dY (cid:88) W ,W ( W ∗ · ∇ Π d − ,J ( Y, W ∗ ; X , Z )) × Π d − ,J ( Y, W ; X , Z )Π d − ,J ( Y, W ; X , Z ) + (5 permutations) (cid:104) V ( X ) V ( X ) V ( X ) (cid:105) H = − g H f abc C d − , (cid:90) EAdS dY (cid:88) W ,W ,W × (cid:0) W ∗ · ∇ Π d − , ( Y, W ; X , Z ) − W · ∇ Π d − , ( Y, W ∗ ; X , Z ) (cid:1) × (cid:0) W ∗ · ∇ Π d − , ( Y, W ; X , Z ) − W · ∇ Π d − , ( Y, W ∗ ; X , Z ) (cid:1) × (cid:0) W ∗ · ∇ Π d − , ( Y, W ; X , Z ) − W · ∇ Π d − , ( Y, W ∗ ; X , Z ) (cid:1) , (A.9)where the factor C d − , ensures our V V two-point function follows the CFT normalization.The integrals can be done in elementary ways, for example using Feynman/Schwinger param-eters. We obtain (in d = 3): (cid:104) V V V (cid:105) YM = 3 g YM √ f abc H V + H V + H V + V V V ( − X · X ) ( − X · X ) ( − X · X ) , (cid:104) V V V (cid:105) H = − g H √ f abc H V + H V + H V + 5 V V V ( − X · X ) ( − X · X ) ( − X · X ) , (A.10)where H ij follows the definition in eq. (3.27) and V i is defined by (see [7] for more details) V i := V i,jk = ( X i · X k )( Z i · X j ) − ( X i · X j )( Z i · X k ) X j · X k . (A.11)To project onto the conformal frame (0 , x, ∞ ), we parameterize X i , Z i (in embedding lightconecoordinates) as X = (1 , , , Z = (0 , , (cid:15) ) , X = (1 , x , x ) , Z = (0 , (cid:15) · x, (cid:15) ) ,X = (0 , , , Z = (0 , , (cid:15) ) . (A.12)We thus end up with (cid:104) V V V (cid:105) YM = 3 g YM f abc √ | x | (cid:20) ( x · (cid:15) )( (cid:15) · (cid:15) ) + ( x · (cid:15) )( (cid:15) · (cid:15) ) − ( x · (cid:15) )( (cid:15) · (cid:15) ) + ( x · (cid:15) )( x · (cid:15) )( x · (cid:15) ) x (cid:21) , (cid:104) V V V (cid:105) H = − g H f abc √ | x | (cid:20) ( x · (cid:15) )( (cid:15) · (cid:15) ) + ( x · (cid:15) )( (cid:15) · (cid:15) ) − ( x · (cid:15) )( (cid:15) · (cid:15) ) − x · (cid:15) )( x · (cid:15) )( x · (cid:15) ) x (cid:21) . (A.13)Comparing the above results with M V B V (see eq. (2.4) and eq. (2.16)) for conserved currents,the agreement can be easily observed and the OPE coefficients can be readily read off λ (e1) V V V = g YM √ , λ (e2) V V V = g H √ , λ (o2) V V V = g (cid:48) H √ π , (A.14)– 39 –here we strip off color factors by defining (cid:104) V V V (cid:105) three-point functions as (cid:104)
V V V (cid:105) a = f abc × λ aV V V T a , (A.15)in which a runs through structures in eq. (2.14). B Simplifying Fourier transforms using spinors
We find that much of the calculations can be streamlined analytically by representing thepolarization vectors as a product of two spinors (see also [12]).Given a two-component spinor | (cid:15) (cid:105) , we define (cid:104) (cid:15) | ≡ | (cid:15) (cid:105) T · iσ , and parametrize the nullpolarizations as (cid:15) µi ≡ (cid:104) (cid:15) i | σ µ | (cid:15) i (cid:105) (B.1)where σ µ , µ = 1 , ,
3, are Pauli matrices. This vector is automatically null. Other usefulidentities include: (cid:104) a | σ µ | b (cid:105)(cid:104) c | σ µ | d (cid:105) = −(cid:104) ac (cid:105)(cid:104) bd (cid:105) − (cid:104) ad (cid:105)(cid:104) bc (cid:105) , ( (cid:15) , p, (cid:15) ) = i (cid:104) (cid:15) (cid:15) (cid:105)(cid:104) (cid:15) | p | (cid:15) (cid:105) . (B.2)The three-point helicity structures in eq. (2.10) are very simple in terms of spinors: T ± , ± ( p ) = (4 π ) τ + τ − ∆ (cid:18) − i (cid:104) (cid:15) | p | (cid:15) (cid:105)√ (cid:19) J − J − J (cid:104) (cid:15) (cid:15) (cid:105) J (cid:104) (cid:15) (cid:15) (cid:105) J | p | β − × (cid:18) − ξ ,p, (cid:19) J (cid:18) ξ ,p, (cid:19) J (B.3)where ξ i,p, ≡ (cid:104) (cid:15) i | p | (cid:15) (cid:105)| p |(cid:104) (cid:15) i (cid:15) (cid:105) is a measure of spin along the p axis.When we go to Fourier space using eq. (2.12) and its derivatives, we find remarkable sim-plifications thanks to the fact that the vector (cid:15) is orthogonal to all other vectors multiplying p . In fact the Fourier-transform involves only similar-looking objects and we were able toFourier-transform the generic term analytically: (cid:90) d p (2 π ) e ip · x p k ( − i (cid:104) (cid:15) | p | (cid:15) (cid:105) ) J ( ξ ,p, ) a ( ξ ,p, ) b = 2 k + J π (cid:104) (cid:15) | x | (cid:15) (cid:105) J x k +2 J +3 × (cid:88) a (cid:48) ,b (cid:48) f a,ba (cid:48) ,b (cid:48) Γ (cid:0) a (cid:48) + b (cid:48) +32 + k + J (cid:1) Γ (cid:0) a + b − k (cid:1) ( ξ ,x, ) a (cid:48) ( ξ ,x, ) b (cid:48) (B.4)where the sum runs over a (cid:48) ≤ a , b (cid:48) ≤ b such that ( a + b − a (cid:48) − b (cid:48) ) is even, and f is the followingcombinatorial factor f a,ba (cid:48) ,b (cid:48) = (2 i ) a (cid:48) + b (cid:48) a + b a ! a (cid:48) ( a − a (cid:48) )! b ! b (cid:48) !( b − b (cid:48) )! ( a + b − a (cid:48) − b (cid:48) )! (cid:0) a + b − a (cid:48) − b (cid:48) (cid:1) ! . (B.5)Using the integral (B.4) it is straightforward to convert the structures in eq. (B.3) back andforth between momentum and coordinate space. The other operations also have simple forms:– 40 – Conformal inversion: this takes ( ∞ , x, (cid:55)→ (0 , x µ /x , ∞ ) and | (cid:15) (cid:105) (cid:55)→ i x | (cid:15) (cid:105)| x | . The neteffect is simply: ξ ,x (cid:55)→ /ξ ,x and (cid:104) (cid:15) (cid:15) (cid:105) (cid:55)→ i (cid:104) (cid:15) (cid:15) (cid:105) ξ ,x . • Shadow transform: two-point functions in position and Fourier space are simply: (cid:104)O (0) O ( x ) (cid:105) = (cid:104) | x | (cid:105) J ( − J | x | ∆+ J , (cid:104)O (0) O ( p ) (cid:105) = (4 π ) Γ( − ∆)( ) J ( − J ∆ Γ(∆ + J ) | p | ∆ − (cid:104) (cid:15) (cid:15) (cid:105) J × F (cid:0) − J, − ∆ , , ξ ,p, (cid:1) . (B.6) • Index contractions: the sum over a basis of spin- J states (2.29) becomes: (cid:88) (cid:15) f ( (cid:15) ∗ ) g ( (cid:15) ) = ( − J (2 J )! f ( ∂ (cid:15) ) g ( (cid:15) ) . (B.7) C More on conformal blocks
C.1 Series expansion of conformal blocks
Here we review how to obtain a series expansion for conformal blocks using the conformalCasimir operator, following the work of ref. [41] for scalar blocks. The same recursion willcome in handy for doing certain inversion integrals in the next subsection. The conformalsymmetry generators act on a spinning primary O ( x, (cid:15) ) of dimension ∆ as D = x µ ∂ xµ + ∆ , J µν = x µ ∂ νx − x ν ∂ µx + (cid:15) µ ∂ ν(cid:15) − (cid:15) ν ∂ µ(cid:15) ,P µ = ∂ µx , K µ = x ∂ µx − x µ D + 2( x · (cid:15)∂ µ(cid:15) − (cid:15) µ x · ∂ (cid:15) ) , (C.1)where D , J , P and K generate respectively dilations, rotations, translations and specialconformal transformations. The Casimir operator is then C = D − J µν J µν + { P µ , K µ } ,which has eigenvalue C ∆ ,J = ∆(∆ − d ) + J ( J + d −
2) if O is a rank- J tensor.Four-point conformal blocks are (by definition) eigenfunctions of the Casimir acting onthe pair of operators 1 , C = D − J µν ( J ) µν + 12 { P µ , ( K ) µ } (C.2)where the subscript denote the fields on which the generators act: D ≡ D + D etc. Thisform of the Casimir operator however can’t be used for the correlator in the frame 0 , x, y − , ∞ .The problem is that P does not preserve the condition x = 0. Fortunately, there is a simplesolution: we can use conformal invariance of the four-point correlator to rewrite P (cid:55)→ − P .Accounting for a commutator, the C = (cid:20) D ( D − d ) − J µν ( J ) µν (cid:21) + K µx K yµ ≡ C (0) + C (1) . (C.3)– 41 –otice that C (0) is homogenous in x , while C (1) increases the weight in x and y by one unit.Furthermore, the former is diagonalized by the three-point structures P ab ∆ ,J in eq. (2.43). Thissuggests writing the block as a an infinite series in P ab ∆ ,J : G ( a,b ) J, ∆ ( z, z ) = ∞ (cid:88) m =0 m (cid:88) k = − m A ( aa (cid:48) )( bb (cid:48) ) m,k P a,b ∆+ m,J + k (ˆ x, ˆ y ) , (C.4)such that the Casimir (C.3) gives a recursion relation for the coefficients A . For example, forscalar operators, applying the Casimir to the Gegenbauer polynomials (2.43) gives C (0) P a,b ∆ ,J = C ∆ ,J P a,b ∆ ,J , C (1) P a,b ∆ ,J = γ a,b, − ∆ ,J P a,b ∆+1 ,J − + γ a,b, +∆ ,J P a,b ∆+1 ,J +1 , (C.5)with γ a,b, +∆ ,J = (∆ + J + 2 a )(∆ + J + 2 b ) ,γ a,b, − ∆ ,J = J ( d + J − − a + d − ∆ + J − − b + d − ∆ + J − d + 2 J − d + 2 J − , (C.6)from which one deduces the recursion [41] (cid:0) C ∆ ,J − C ∆+ m,J + k (cid:1) A m,k = γ − ∆+ m − ,J + k − A m − ,k +1 + γ +∆+ m − ,J + k +1 A m − ,k − . (C.7)Note a, b in γ a,b, ± ∆ ,J is not representing the structure index, they are simply a = 1 / − ∆ ) , b = 1 / − ∆ ). These coefficients eq. (C.6) will also play important role when we aredealing with GFF, see appendix C.2.This method allows to extend this result straightforwardly to spinning operators [76].We can use eq. (2.42) to construct P a,b ∆ ,J from three-point functions, and in general P a,b ∆ ,J canbe organized as Gegenbaur polynomials and their derivatives, which is consistent with grouptheoretical analysis for projectors [77]. C.2 Inverting powers of cross-ratios times Gegenbauers
In this appendix, we present a more compact approach to deal with the spinning GFF. Tobe more precise, there is a surprisingly concise and powerful trick that can be used performLorentzian inversion formula for a scalar GFF correlator extended with Gegenbauer polyno-mial, namely G = u p v p + a ˜ C J (cid:48) (cid:0) ξ (cid:48) (cid:1) , (C.8)where u = z ¯ z , v = (1 − z )(1 − ¯ z ) and ξ (cid:48) = (1 − u − v ) / (2 √ uv ). The punchline is that wefind a recursion relation for OPE data associated with above correlator, see eq. (C.11). Thisformula enjoys more general applications, since as just shown, conformal blocks admit seriesexpansion of precisely this form (after interchanging operators 3 and 4 operators). This wasused in [78] to estimate Lorentzian inversion integrals at large dimensions in the 3 d -Ising– 42 –odel. In this paper, we apply the formula to G = D ↓ G for spinning GFF, which is a finitesum of terms (C.8).The starting point of the recursion is the scalar case, J (cid:48) = 0. The relevant OPE data canbe found in literatures, at least for equal external operators a = b , e.g., [54, 79]. There is atrivial modification that also works for independent a, b, p : c a,b ,p,J (∆) = Γ (cid:0) d − p − a (cid:1) Γ (cid:0) d − p + b (cid:1) (cid:0) p + a (cid:1) Γ (cid:0) p − b (cid:1) Γ (cid:0) ∆+ J + a (cid:1) Γ (cid:0) ∆+ J − b (cid:1) Γ (cid:0) d − ∆+ J − a (cid:1) Γ (cid:0) d − ∆+ J + b (cid:1) × Γ(∆ − (cid:0) J + d (cid:1) Γ( d − ∆ + J )Γ( J + 1)Γ (cid:0) ∆ − d (cid:1) Γ(∆ − J ) Γ (cid:0) p − ∆+ J (cid:1) Γ (cid:0) p − d +∆+ J (cid:1) Γ (cid:0) − p + d +∆+ J (cid:1) Γ (cid:0) − p +2 d − ∆+ J (cid:1) . (C.9)This was tested by checking that the obtained OPE coefficients (obtained from the residuesat ∆ = p + J + 2 m ) reproduce the series expansion of the bracket in eq. (C.8) with J (cid:48) = 0to high order. To proceed on generalizing above OPE data to those with J (cid:48) (cid:54) = 0, we shallslightly modify P ∆ ,J in (C.4) by interchanging operator 3 and 4, for which eq. (C.5) becomes C (0) u p v p + a ˜ C J (cid:48) (cid:0) ξ (cid:48) (cid:1) = C p,J (cid:48) u p v p + a ˜ C J (cid:48) (cid:0) ξ (cid:48) (cid:1) , C (1) u p v p + a ˜ C J (cid:48) (cid:0) ξ (cid:48) (cid:1) = u p +12 v p +12 + a (cid:0) γ a, − b, − p,J (cid:48) ˜ C J (cid:48) − (cid:0) ξ (cid:48) (cid:1) + γ a, − b, + p,J (cid:48) ˜ C J (cid:48) +1 (cid:0) ξ (cid:48) (cid:1)(cid:1) , (C.10)with γ a,b, ± ∆ ,J already given in eq. (C.6). Since we can integrate-by-parts the Casimir operatorin the inversion integral, by eliminating ˜ C J (cid:48) +1 from this equation, we get a recursion relationin t -channel spin J (cid:48) : γ a, − b, + p − ,J (cid:48) − c a,bJ (cid:48) ,p,J (∆) = (cid:16) C ∆ ,J − C p − ,J (cid:48) − (cid:17) c a,bJ (cid:48) − ,p − ,J (∆) − γ a, − b, − p − ,J (cid:48) − c a,bJ (cid:48) − ,p,J (∆) . (C.11)Let’s end by explaining how do we extract OPE data in spinning GFF by using above formula.We first decompose D ↓ G , e.g., eq. (3.36) into a finite sum of (C.8), next we obtain OPE datafor each term by using eq. (C.11) and in the end we can sum them over to get a final answer. C.3 Cross-channel expansion of blocks
In this appendix, we expand (scalar) conformal blocks as z → z . To accomplish the computations of the anomalous dimensions in the main-text, we wouldneed t -channel conformal blocks with scalar-exchange, conserved-current-exchange and stress-tensor-exchange. In particular, since we are only concerned about the anomalous dimensions,the logarithmic part of t -channel conformal blocks are enough for our purpose. Our formulaecan be deduced from geodesic Witten-diagram [80] by doing a bit of guesswork as describedin [81], and is consistent with the most general t -channel conformal blocks in terms of ( u, v )rather than ( y, ¯ y ) provided recently in [82, 83] (see also [84]). Throughout this appendix, weuse the variables: y = z − z , ¯ y = 1 − zz . (C.12)– 43 –n the main text, we use these conformal blocks in the t -channel dDisc, where we take z (cid:55)→ − z (using y variables, it is y → ¯ y ). • Scalar exchangeFor scalar-exchange, we can provide a more general t -channel conformal blocks, beyondonly picking up logarithmic part. The explicit series is given by G , ∆ ( z, ¯ z ) = y ∆2 (1 + y ) b (1 + ¯ y ) a (cid:88) k (cid:16) ( − k ¯ y k + a + b Γ(∆)Γ( − a − b − k ) (cid:0) a + ∆2 (cid:1) k k !Γ( − b + ∆2 )Γ( − a − k + ∆2 ) s a,b,k ( y )+( a → − a, b → − b ) (cid:17) (C.13) where s a,b,k ( y ) = F (cid:16) ∆ − a , b +∆2 , − a − d +∆ − k +22 ; − d +2∆+22 , − a +∆ − k , − y (cid:17) . (C.14)In practice, what we use in the main text is the logarithmic part log ¯ y of above seriesfrom setting a = b = 0. Note the first line of eq. (C.13) does not have log ¯ y , and thesecond line gives us G , ∆ ( z, ¯ z ) = − (cid:88) k Γ(∆)Γ( k − ∆2 + 1) y ∆2 Γ( ∆2 ) Γ( k + 1) Γ( − k − ∆2 + 1) ¯ y k log ¯ y s , ,k ( y ) . (C.15) • Conserved-current exchangeThe log ¯ y part of t -channel conformal block with conserved-current-exchange is exhibitedas follows: G ,d − ( z, ¯ z ) = (cid:88) k N (1) k ¯ y k y d − log ¯ yy + 1 (cid:0) v d − ,k, − d − ky ( d − k )( d − k ) v d ,k, (cid:1) , (C.16)where N (1) k = − d − Γ (cid:0) d +12 (cid:1) Γ (cid:0) d + k (cid:1) √ π ( k !) Γ (cid:0) d (cid:1) Γ (cid:0) d − k (cid:1) , v p,k,m = F ( p, − k + m, p + 1 − k, − y ) . (C.17) • Stress-tensor exchangeThe log ¯ y part of t -channel conformal block with stress-tensor-exchange was also ob-tained in [81], it is given by G ,d ( z, ¯ z ) = (cid:88) k N (2) k ¯ y k y d − log ¯ yy + 1 (cid:0) ( d − d ( y + 1) + 2( ky + k − y − g d ,k ( y ) − (cid:0) d ( y + 1) + d ( k (4 y + 3) − y −
5) + 2( k − ky + k − y − (cid:1) g d − ,k ( y ) (cid:1) , – 44 –C.18)where N (2) k = 2 d +1 Γ (cid:0) d +32 (cid:1) Γ (cid:0) d + k + 1 (cid:1) √ π ( d + 2 k − d + 2 k )Γ (cid:0) d + 1 (cid:1) Γ( k + 1) Γ (cid:0) d − k + 1 (cid:1) ,g p,k ( y ) = F (cid:0) p, − k, ( d + 2 − k ) , − y (cid:1) . (C.19) D Four-dimensional gluon amplitudes in flat space
Here we record the bulk YM tree-level gluon amplitudes corresponding to the Lagrangian ineq. (A.1) used in the main text. We start with the three-point ones, from which the four-pointamplitudes are then determined by factorization (see [70], whose conventions we follow, fora pedagogical introduction), up to contact interactions with the mass dimension of g H . Theform of on-shell three-point amplitudes are fixed by Lorentz and little-group symmetries, upto coupling-dependent prefactors, which we find to be M YM1 − − + = i √ f abc g YM (cid:104) (cid:105) (cid:104) (cid:105)(cid:104) (cid:105) , M H1 − − − = i √ f abc ( g H − ig (cid:48) H ) (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) . (D.1)For M YM1 + + − and M H1 + + + , we simply replace angle-bracket by square-bracket and reversethe odd coupling g H (cid:48) . Tree-level four-point amplitudes can be cut into a product of on-shellthree-point amplitudes M (cid:12)(cid:12)(cid:12) p I → = M I M I p I . (D.2)We can use this factorization property to construct four-point amplitudes.Let’s first consider the pure Yang-Mills case. One might try to directly use (D.2) for allchannels and sum them over, however, this overcounts the pole structures, since the s -channelresidue has poles in t or u channel. The standard strategy (see [71]) is to make an ansatzwhich correctly counts helicity weight and number of derivatives without violating locality M − − + + = (cid:104) (cid:105) [34] ( Ast + Bsu + Ctu ) . (D.3)By demanding the factorization (D.2), one can readily obtain the Parke-Taylor form M YM = 2 g (cid:104) ij (cid:105) (cid:16) T (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) + U (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) (cid:17) , (D.4)where i, j are gluons that have negative helicity, T = f bce f ade is the t -channel color factorpreviously defined in eq. (4.8), and U is the same with a and b swapped.Note that the first term above actually contains s and t -channel poles, and the secondterm contains s and u poles.For pure higher-derivative coupling, the nonvanishing amplitudes again all have two glu-ons of each helicity: M H − − + + , M H − + + − and M H − + − + that arise from s -channel,– 45 – -channel and u -channel respectively. Using the factorization (D.2) and Bose symmetry, weobtain: M H − + + − = 2( g + g (cid:48) ) (cid:104) (cid:105) [23] T u − s t + c (cid:104) (cid:105) [23] , (D.5)and permutations thereof. The contact ambiguity c depends on higher-derivative terms in theLagrangian but doesn’t contribute to the analysis in the main text as it has finite support inspin. (The tree-level all-+ amplitude, also a pure contact term but controlled by a differentconstant, similarly does not contribute.)Finally, the mixed g YM g H amplitudes (including the higher-derivative correction on bothvertices) are quite similar to pure Yang-Mills amplitudes. For example, for M mix1 − + + + weconsider an ansatz suggested by its helicity scaling and derivative order: (cid:104) (cid:105)(cid:104) (cid:105) [23][34][24] times two-channel poles like 1 / ( st ). We then obtain: M YM − H1 − − − + = 2 g YM ( g H − ig (cid:48) H ) T (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) (cid:104) (cid:105)(cid:104) (cid:105) + (1 ↔ , M YM − H1 − + + + = 2 g YM ( g H + ig (cid:48) H ) T [23][34][24] [12][41] + (3 ↔ , (D.6)and permutations thereof. References [1] D. Poland, S. Rychkov and A. Vichi,
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