DDCPT-17/19
Unmixing Supergravity
F. Aprile , , J. M. Drummond , P. Heslop , H. Paul Mathematical Sciences and STAG Research Centre,University of Southampton, Highfield, SO17 1BJ, School of Physics and Astronomy and STAG Research Centre,University of Southampton, Highfield, SO17 1BJ, Mathematics Department, Durham University,Science Laboratories, South Rd, Durham DH1 3LE
Abstract
We examine the double-trace spectrum of N = 4 super Yang-Mills theory in the supergrav-ity limit. At large N double-trace operators exhibit degeneracy. By considering free-fieldand tree-level supergravity contributions to four-point functions of half-BPS operators weresolve the degeneracy for a large family of double-trace operators. The mixing problemreveals a surprisingly simple structure which allows us to obtain their three-point functionsat leading order in the large N expansion as well as their leading anomalous dimensions. a r X i v : . [ h e p - t h ] F e b Introduction
The behaviour of conformal field theories in the limit of large central charge has been asubject of great interest in recent years. One motivation for this interest is the AdS/CFTcorrespondence which relates gravitational theories in AdS to a CFT on the boundary [1–3].While much recent work has focussed on understanding general constraints on possible holo-graphic theories [4–12], it is also of interest to explore explicit examples to understand thedetails of the spectrum and interactions as these can sometimes reveal unexpected features.The archetypal holographic example is the correspondence between four-dimensional N = 4super Yang-Mills theory and type IIB superstring theory on AdS × S .The central quantities of interest under such a correspondence are the correlation functionsof gauge-invariant local operators. In the case of N = 4 super Yang-Mills theory, suchcorrelation functions are dependent on the gauge coupling g and the choice of gauge group,which we take to be SU ( N ). The limit of large central charge corresponds to the large N limit and, when taken with the ’t Hooft coupling λ = g N fixed and large, should leadto a regime of the theory where the massive string excitations decouple and which can bedescribed by IIB supergravity on AdS × S .The massless string modes include the graviton and its superpartners. These fields canpropagate in the AdS directions, while the S factor leads to a tower of Kaluza-Kleinmodes all carrying representations of SU (4). The graviton multiplet corresponds to theenergy-momentum multiplet of N = 4 super Yang-Mills theory and it is the simplestexample of a half-BPS multiplet. There is an infinite tower of related half-BPS operators,corresponding to the associated Kaluza-Klein modes. In terms of Yang-Mills fields thesuperconformal primary operators of these half-BPS multiplets take the form of a singletrace over scalar fields φ I which transform in the vector representation of SO (6), O p ( x, y ) = y I . . . y I p tr (cid:0) φ I ( x ) . . . φ I p ( x ) (cid:1) . (1)Here y I is an auxiliary null SO (6)-vector, y = 0. The label p denotes the fact that theprimary sits in the SU (4) representation [0 , p, p = 2 corresponding to theenergy-momentum multiplet. The fact that the operators O p are half-BPS means that theyalways possess their classical integer scaling dimensions. Their two-point and three-pointfunctions also receive no quantum corrections and take their free field theory forms.Here we will draw on general CFT techniques, in particular the operator product expansion(OPE), as well as explicit results for the tree-level supergravity contribution to correlationfunctions of half-BPS operators. A very compact solution for the most general half-BPSfour-point function (cid:104)O p O p O p O p (cid:105) at tree-level in the supergravity limit was presentedin [13]. The formula is given in Mellin space, and is deduced from general analytic principlesapplied to the Mellin representation, rather than a direct supergravity calculation. Theseproperties are based on the existence of the OPE and in particular the presence of exchangeddouble-trace operators as well as other properties such as crossing symmetry. The resultingformula agrees with the cases available in the literature obtained from representations interms of Witten diagrams and other techniques [14–23]. Further analysis examining theconsistency of the result of [13] with supergravity have been performed in [24].Here we systematically analyse the OPE of a particular class of four-point functions at large1 , using methods developed in many papers on the OPE of conformal and superconformaltheories [20, 25–29]. In the OPE of these correlators we expect both protected operatorsand unprotected ones. The only unprotected operators which we expect to be present inthe spectrum in the supergravity limit are multi-trace operators made from products ofderivatives of the operators O p . This is because we expect all other long operators tocorrespond to string excitations which have acquired large mass in the supergravity limit.Furthermore, of the long multi-trace operators, we expect only the double-trace operatorsof the schematic form O p (cid:3) n ∂ l O q , (2)to appear in the OPE at leading order in 1 /N . Higher multi-trace operators should alsoappear, but only at higher orders in 1 /N . Operators of the form (2) have classical dimen-sion p + q + 2 n + l and spin l . More often we will refer to the twist which is the differenceof the dimension and the spin (hence equal to p + q + 2 n in the above case) instead of thedimension itself. In the strict large N limit the dimension will be fixed to its classical value,regardless of the value of the Yang-Mills coupling. In a large N expansion the operators(2) will only acquire anomalous dimensions at order 1 /N .In the first instance we will consider the SU (4) singlet double-trace operators for which weneed p = q in (2). In that case the only quantum numbers which distinguish them are thetwist and the spin. It is therefore clear that the set of ( t −
1) operators {O (cid:3) t − ∂ l O , O (cid:3) t − ∂ l O , . . . , O t (cid:3) ∂ l O t } (3)are degenerate in the strict large N limit since they all have twist 2 t and spin l . Includingthe anomalous dimensions at order 1 /N will lift the degeneracy however.We label the ( t −
1) degenerate operators with fixed t and l by K t,l,i for i = 1 , . . . , t −
1. Inorder to resolve the degeneracy among the operators (3) we consider four-point correlatorsof the form (cid:104)O p O p O q O q (cid:105) for 2 ≤ p ≤ q ≤ t . To perform our OPE analysis we need twopieces of information about each correlator. Firstly we need the leading large N result whichcomes from disconnected contributions and can be obtained from free field theory. Secondlywe need the first 1 /N suppressed connected contribution, coming from the formula of [13].With these two pieces of information we find that we have enough information to resolve thedegeneracy of the sector of unprotected double trace operators. This yields the leading orderthree-point functions (cid:104)O p O p K t,l,i (cid:105) for each of the operators K t,l,i as well as the O (1 /N )contribution to their anomalous dimensions.Above we discussed the singlet channel but we are able to generalise the analysis to considerlong double-trace operators in the [ n, , n ] representation for any n . In this case we have( t − n −
1) degenerate operators of twist 2 t and spin l schematically given by {O n +2 (cid:3) t − n − ∂ l O n +2 , O n +3 (cid:3) t − n − ∂ l O n +3 , . . . , O t (cid:3) ∂ l O t } (4)Again the information required to resolve the degeneracy can be obtained just by consideringcorrelators of the form (cid:104)O p O p O q O q (cid:105) for 2 + n ≤ p ≤ q ≤ t .Even though the formula of [13] for the Mellin representation of the correlation functionsis very simple, there is no guarantee that solution of the mixing problem will be simple.2owever, we find a surprisingly simple structure appearing in both the anomalous dimen-sions and the OPE coefficients. To exhibit the simplicity here we quote the formula for theanomalous dimensions of the ( t − n −
1) double-trace operators in the [ n, , n ] representationwith classical twist 2 t and spin l . We write the full dimension as∆ [ n, ,n ] t,l,i = 2 t + l + 2 N η [ n, ,n ] t,l,i + . . . (5)where the ellipsis denotes terms of higher order in 1 /N . The quantity η [ n, ,n ] t,l,i is given by η [ n, ,n ] t,l,i = − t − − n ) t ( t + 1)( t + 2 + n )( t + l − n )( t + l + 1)( t + l + 2)( t + l + 3 + n )( l + 2 i + n − , (6)and i = 1 , . . . , t − n − N . In (6) we have used the Pochhammer symbol( x ) r = x ( x + 1) . . . ( x + r −
1) to compactify the expression.The three-point functions (cid:104)O p O p K t,l,i (cid:105) also exhibit a very nice structure with respect totheir spin dependence. For fixed t these naturally form a mixing matrix with l -dependententries. We find that the l dependence has a universal structure which can be precisely fixedby imposing orthogonality of the (normalised) matrix. Thus, having used the explicit datato identify this pattern, the three-point functions can then actually be determined with nomore reference to the explicit correlation functions. We find this universal structure quiteremarkable and suggests a further underlying structure yet to be identified.Very recently the two papers [33, 34] appeared using the idea of resolving the degeneracyamong the singlet double-trace operators to make statements about quantum correctionsto the classical supergravity results. In [33], the method of large spin perturbation theory(see [35]) was applied to derive formulas for the O (1 /N ) corrections to the anomalousdimensions of the singlet twist-four operators. In our paper [34], we used the resolved mix-ing for the singlet channel presented here in more detail to make a closed-form resummedprediction for the double discontinuity of the correlator (cid:104)O O O O (cid:105) at order 1 /N . Inparticular in [34] we already presented and used the result for the anomalous dimensions(6) in the singlet case n = 0. We were then able to use a polylogarithmic ansatz to constructa crossing symmetric amplitude, which was fixed almost uniquely by the double disconti-nuity. From this predicted amplitude we then extracted a closed-form all-spin formula forthe 1 /N correction to the singlet twist-four anomalous dimensions. The resulting formulaagrees with the dimensions quoted in [33]. N = 4 SYM
Half-BPS scalar operators in N = 4 SYM transform in the irrep [0 , p,
0] of the R -symmetrygroup SO (6) ⊂ SU (2 , |
4) and have protected dimension ∆ = p . At weak coupling, theseoperators can be described by the single-trace operators O Ip = C i ,...i p I Tr (cid:0) φ i . . . φ i p (cid:1) , I = 1 , . . . dim[0 , p,
0] (7) For details see the discussion in Section 6 of [34]. φ i =1 ,..., are elementary fields in the adjoint of SU ( N ), and the C i ...i k I provide a realbasis of traceless symmetric tensors for the irrep [0 , p, O Ip is dual to a scalar field ϕ Ip of type IIB supergravity compactified on AdS × S . Accordingto the AdS/CFT correspondence, the mass of ϕ Ip is related to the dimension of O Ip throughthe formula m L = p ( p − L is the AdS radius, and the corresponding irrep isobtained from Kaluza-Klein reduction on the five-sphere [36].Here we will be interested in four-point correlation functions. A generic four-point correlatorwill transform as a singlet inside the product ⊗ i =1 [0 , p i , SO (6) structurecan be conveniently done as follows, O p = y i . . . y i p Tr (cid:0) φ i . . . φ i p (cid:1) , (cid:126)y · (cid:126)y = 0 , (8)where y i is a complex null vector parametrizing the coset space SU (4) /S ( U (2) × U (2)).In the context of AdS/CFT bulk fields ϕ Ip are parametrized by harmonic variables on adifferent coset space, S ∼ SO (6) /SO (5), therefore the representation (8) is not directlyavailable. Four-point correlators obtained in this representation can be re-expressed interms of the other and reduced to the following general form (cid:104)O p ( (cid:126)x ) O p ( (cid:126)x ) O p ( (cid:126)x ) O p ( (cid:126)x ) (cid:105) = (cid:88) { d ij } (cid:32) (cid:89) ≤ i 0] = p (cid:88) k =0 p − k (cid:88) k =0 [ k , p − p + 2 k , k ] ( p ≤ p ) (51)Descendants are obtained by the action of the derivative operator L ( l ) ( x , ∂ x ) on theprimaries. A manifest N = 4 formulation of the OPE can be obtained by reorganizing thesum over operators into supermultiplets. Therefore, inserting the OPE of O p ( x ) O p ( x )and O p ( x ) O p ( x ) into the four-point correlator we obtain the representation (cid:104)O p O p O p O p (cid:105) = P (OPE) { p i } (cid:88) { τ, l, R } A { p i } R ( t | l ) S { p i } R ( t | l ) (52)where t = (∆ − l ) / S { p i } R ( t | l ) are superconformal blocks described below. Here thesum over representations runs over those which belong to ([0 , p , ⊗ [0 , p , ∩ ([0 , p , ⊗ [0 , p , A { p i } R ( t | l ) depend explicitly on the charges and are related to theOPE coefficients by A { p i } R ( t | l ) = (cid:88) O ∈ R C p p O C p p O , (53)where the sum is over all operators with spin l , leading order dimension ∆ and SU (4)representation R . The prefactor P (OPE) depends on the ordering of the charges. The blockdecomposition is invariant under swapping points 1 and 2, points 3 and 4 and swapping thepairs of points 1,2 and 3,4. Using this symmetry we can clearly always ensure that p ≥ p , p ≥ p and p − p ≤ p − p . Assuming such an ordering, then the following diagramexists in the free theory, p − dd p − d p p p p p d := p + p + p − p We then take the prefactor as represented by this diagram P (OPE) { p i } = g d g p − d g p − d g p with p ≥ p , p ≥ p , p − p ≤ p − p . (54)Comparing this with the prefactor (and corresponding diagram) taken out of the supergrav-ity correlator (13), we see that up to the appropriate permutation of points, the prefactorsare the same.Finally the superconformal blocks themselves, S { p i } R ( t | l ), can be derived using a variety ofapproaches and were first derived in [20, 26]. Here we explain them in a compact fashion interms of representations of GL (2 | u, v, σ, τ , it will be useful to use the variables x , x and y , y : u = x x , v = (1 − x )(1 − x ) , σ = 1 y y , τ = (cid:18) − y (cid:19) (cid:18) − y (cid:19) . (55)13n terms of these, the degree two polynomial (14), singled out from the “partial non-renormalization” theorem, becomes fully factorized: I ( u, v, σ, τ ) = v + σ uv + τ u + σv ( v − − u ) + τ (1 − u − v ) + στ u ( u − − v )= ( x − y ) ( x − y ) ( x − y ) ( x − y )( y y ) (56)Note that x i =1 , and y i =1 , are not to be confused with the space-time variables and internalharmonic variables that were introduced in previous sections. The above variables areconformally invariant. GL (2 | superconformal partial wave Conformal blocks and SU (4) harmonics are commonly introduced in the literature as [25,29] B t | l = ( − ) l x t + l +11 x t F t + l ( x ) F t − ( x ) − x t + l +12 x t F t − ( x ) F t + l ( x ) x − x (57) Y nm = − P n +1 ( y ) P m ( y ) − P m ( y ) P n +1 ( y ) y − y (58)where F t is related to F [ a, b ; c ] hypergeometrics and P n is related to Jacobi polynomialsJP ( a | b ) c through the definitions F t ( x ) = F (cid:104) t − p , t + p t (cid:105) ( x ) , P n ( y ) = y JP ( p − d | p − d ) n (cid:18) y − (cid:19) . (59)In particular, Y nm with m ≤ n is a polynomial of degree n in ( σ, t ), and B t | l is analytic in u and (1 − v ), i.e B t | l = (cid:88) n ≥ (cid:88) m ≥ max(0 ,l − n ) r nm [ t, l, p , p ] u t + n (1 − v ) m . (60)The series expansion of B t | l begins with leading term u t (1 − v ) l where t = (∆ − l ) / N = 4 representations and the corresponding superconformal blocks have been studiedextensively in the literature [27–29, 49–53]. They can be written as specific linear combina-tions of terms of the form B × Y corresponding to the component fields appearing in themultiplet. This way of writing it depends strongly on the type of multiplet and in particu-lar its shortening conditions. A more group theoretic approach was taken in [53] giving auniform description of all superconformal blocks via a determinantal formula associated toa GL (2 | 2) Young tableau which we review now.In this approach, an operator is defined on analytic superspace by specifying a GL (2 | λ , together with a charge γ , O γ,λ . The allowed Youngtableaux have the general shape λ = [ λ , λ , µ , µ ]. Here we specify the row lengths with the notation 2 µ denoting 2 , , . . . , 2, with µ entries in the list,that is µ rows of length 2. λ →← λ →↑ µ ↓↑ µ ↓ (61)The translation to standard quantum numbers depends on the precise shape and is sum-marised by the table below:Translation between N = 4 superconformal reps and superfields O γλ GL (2 | 2) rep λ (∆ − l ) / l R multiplet type[0] γ/ , γ, 0] half BPS[1 µ ] γ/ µ, γ − µ, µ ] quarter BPS[ λ, µ ] ( λ ≥ γ/ λ − µ, γ − µ − , µ ] semi-short[ λ , λ , µ , µ ] ( λ ≥ γ/ λ − λ − λ [ µ , γ − µ − µ − , µ ] long (62)Note that in the case of long multiplets the description of a superconformal representationin terms of O γλ is not unique. Indeed if µ > λ → λ + 1 , λ → λ + 1 , µ → µ − , γ → γ − . (63)The leading term in the long multiplet O γλ can be written schematically in the form ∂ λ − λ (cid:3) λ − φ γ | R . Then the above degeneracy in the description of long reps is a reflec-tion of the fact that this is the same representation as ∂ λ − λ (cid:3) λ − φ γ − | R .A further point is that Young tableaux only make sense for integer values of the row lengths.However one can analytically continue the long representations to non-integer values. This ispossible because all the long SL (2 | 2) representations have the same dimension. Specificallywe formally allow the first two row lengths λ , λ to be non-integer, with the difference λ − λ remaining integer. This then allows for anomalous dimensions. We even formallyallow the case λ → µ = 0. This corresponds to a representation approaching theunitary bound. In the limit when λ = 1 multiplet shortening occurs. So as representations,a long rep in this limit splits into two short reps . Specificallylim λ → O γ [ λ + l,λ , µ ] = O γ [ l +1 , µ ] ⊕ O γ − l +2 , µ ] . (64) We here only consider those representations the four-point function detects. S { p i } R = (cid:18) x x y y (cid:19) ( γ − p + p ) F αβγλ γ = p − p , p − p + 2 , . . . , min( p + p , p + p ) F αβγλ = ( − p +1 D − det (cid:18) F Xλ RK λ F Y (cid:19) , (65)where the matrix has dimension ( p + 2) × ( p + 2) with p = min { α, β } , α = ( γ − p + p ) , β = ( γ + p − p ) , (66)and for given α, β, γ , and Young tableaux λ , the matrix elements are defined as follows( F Xλ ) ij = (cid:16) [ x λ j − ji F ( λ j + 1 − j + α, λ j + 1 − j + β ; 2 λ j + 2 − j + γ ; x i )] (cid:17) ≤ i ≤ ≤ j ≤ p ( F Y ) ij = (cid:16) ( y j ) i − F ( i − α, i − β ; 2 i − γ ; y j ) (cid:17) ≤ i ≤ p ≤ j ≤ ( K λ ) ij = (cid:16) − δ i ; j − λ j (cid:17) ≤ i ≤ p ≤ j ≤ p R = (cid:32) x − y x − y x − y x − y (cid:33) D = ( x − x )( y − y )( x − y )( x − y )( x − y )( x − y ) (67)The square bracket around the components of F X indicate that only the regular part shouldbe taken. So if λ j < j one has to subtract off the first few terms in the Taylor expansion ofthe hypergeometrics.This formula as written deals with all cases. Note that the determinant yields a sum ofterms each of which contain at most two hypergeometrics in x i (from the first two rows) andtwo in y i (from the last two columns). When the determinant is expanded out, the formulayields different forms depending on whether the multiplet is 1/2 BPS, short or long, due tothe different nature of the matrix K λ in each case. All cases can be written in terms of atwo-variable or four-variable function. In this paper however we will not need the explicitforms in all cases. Instead we only need the superconformal blocks for long operators, whichwe use to perform the block expansion of the interacting piece of the correlator H . For thefree correlator we use an alternative approach outlined in the next section which turns outto be very efficient and much less complicated than the one outlined here. It is particularlyuseful for performing the free theory analysis which contains all the short multiplets. Thestudy of short multiplets is technically the most challenging from a superblock point ofview. N = 4 superconformal blocks. The long multiplets all have Young tableaux containing a two by two block. They thushave Young tableaux of the form λ = [ λ , λ , λ (cid:48) − , λ (cid:48) − λ (cid:48) ]16hat is the first and second rows have length λ , λ respectively and the first and secondcolumns have height λ (cid:48) , λ (cid:48) respectively, with λ , λ , λ (cid:48) , λ (cid:48) ≥ 2. In this case the determi-nantal formula factorises yielding F αβγλ long ( x | y ) = ( − λ (cid:48) + λ (cid:48) ( x − y )( x − y )( x − y )( x − y ) × F αβγλ ( x ) F αβγλ − ( x ) − x ↔ x x − x × G αβγλ (cid:48) ( y ) G αβγλ (cid:48) − ( y ) − y ↔ y y − y (68)where F αβγλ ( x ) := x λ − F ( λ + α, λ + β ; 2 λ + γ ; x ) G αβγλ (cid:48) ( y ) := y λ (cid:48) − F ( λ (cid:48) − α, λ (cid:48) − β ; 2 λ (cid:48) − γ ; y ) . (69)From table (62), this gives the superblock corresponding to a long multiplet of half twist t = γ/ λ − 2, spin l = λ − λ and SU (4) rep R = [ λ (cid:48) − λ (cid:48) , γ − λ (cid:48) , λ (cid:48) − λ (cid:48) ].This can be straightforwardly converted into a B × Y notation. From the defintion of B introduced in (57) we immediately recognize that F αβγλ ( x ) F αβγλ − ( x ) − x ↔ x x − x = 1( x x ) γ B t +2 | l . (70)Similarly, from the definition of α, β given in (66), and an hypergeometric identity , we find − G αβγλ (cid:48) − ( y ) G αβγλ (cid:48) ( y ) − y ↔ y y − y = ( y y ) γ + p − ( n + 1)! m !( n +2+ p ) n +1 ( m +1+ p ) m Y nm (72)where we used the definition of Y nm in (58) and the identification, m = p / γ/ − λ (cid:48) , n = p / γ/ − λ (cid:48) The SU (4) representation is then [ n − m, m + p , n − m ]. It is also convenient to definethe normalized SU (4) harmonics as follows,Υ nm = ( n + 1)! m !( n +2+ p ) n +1 ( m +1+ p ) m Y nm . (73)Including the prefactors from the definition of S , and relabelling S → L to highlight thatthis is a long operator, the long superblock then becomes L { p i } nm ( t | l ) = ( x − y )( x − y )( x − y )( x − y )( y y ) B t | l u p Υ nm . (74) The following identity might be useful F [ λ − α, λ − β, λ ] ( y ) = y − λ + β n !( n − β + 1) n JP ( − α − β | α − β ) n (cid:18) y − (cid:19) , n ≡ β − λ . (71) .2 Bosonised superblocks Here we outline a novel approach to performing a SCPW analysis, particularly useful forthe free theory in N = 4. It is based on the approach of [53], outlined above and based onanalytic superspace: The key observation there is that superconformal blocks in generalised analytic superspace with SU ( m, m | n ) symmetry exhibit a universal structure, thus onecan map the correlation functions into a generalised analytic superspace with SU ( m, m | n )symmetry for any m, n , perform the appropriate superblock expansion, and the block co-efficients thus obtained will be the same as the ones you would have obtained had youperformed the expansion in the original space. In particular it is convenient to map theproblem to the generalised conformal group SU ( m, m ) with n = 0.As we have seen the free theory 4-point function of any four 1/2 BPS operators is givenas a sum of products of powers of the superpropagators g ij (9). Now each term in the freetheory contains information about operators O γ λ for a specific value of γ , namely γ = d + d + d + d . (75)Note that graphically γ is simply the number of propagators going from the pair of points1,2 to the pair 3,4. Explicitly, every term in the free theory can be written as (cid:89) i 2) matrix Z .In the conventional approach one would then simply expand this in terms of SU (2 , | F αβγλ as described in the previous section, to obtain informationabout operators O γλ . However, the universal structure of SU ( m, m | n ) blocks alluded toabove suggests an alternative approach, namely, using blocks in a theory with SU ( m, m | n )for different values of m, n compared to the N = 4 SYM case. Any value of m, n will giveaccurate information on the block coefficients, but not necessarily complete information.This happens because now the block decomposition will only yield information on operators O γλ where λ is a valid non-zero GL ( m | n ) Young tableau. For example, choosing m =2 , n = 0, means we consider GL (2) Young tableaux, which will only give information aboutoperators with maximally two row tableaux. Whereas choosing m = 3 , n = 0 will give18nformation on operators whose Young tableaux have up to three rows etc. On the otherhand one could consider m = 0. Then we are considering SU (0 | n ) tableau which are just“transposed” SU ( n ) tableau, where columns and rows are swapped. Thus Young tableauxin the m = 0 case have maximally n columns.In N = 4 we have GL (2 | 2) Young tableau which have the hook structure given in (61),namely at most two rows have length greater than two and at most two columns havelength greater than two. On the other hand expanding the structure in (77) in terms ofsuper Schur polynomials using the Cauchy identity (see [53] for details in this context), onecan see that the corresponding Young tableaux have maximal height given by the power d .Furthermore the corresponding blocks must also then have corresponding Young tableauxof height d or less. This means that performing the expansion with m = d , n = 0 willgive complete information on all the conformal blocks.The advantage of using SU ( m, m ) blocks (with m = d ) instead of SU (2 , | 4) blocks, isthat they are much simpler (at least conceptually), and are given by the compact formula F αβγλ ( x ) = det (cid:16) x λ j + m − ji F ( λ j +1 − j + α, λ j +1 − j + β ; 2 λ j +2 − j + γ ; x i ) (cid:17) ≤ i,j ≤ m det (cid:16) x m − ji (cid:17) ≤ i,j ≤ m . (78)Note that the denominator here is the famous Vandermonde determinant and can be rewrit-ten (cid:81) i 4) ones. Thus one may suspect that, although the blocks areconceptually much simpler, computationally this approach would be slower. However, sincewe are only interested in Young tableau of specific shapes, and in particular below the thirdrow, they have at most 2 boxes, then we correspondingly only need to perform a very limitedexpansion in the variables x , . . . , x m . Also notice that it is convenient to multiply bothsides by the Vandermonde determinant. Then the blocks themselves are holomorphic.Finally, note that the procedure as outlined above gives information on free theory operators O γλ for fixed γ . This is fine for short muliplets as they are uniquely defined by this This is simply because generalised symmetrisation of odd indices for a supergroup corresponds toanti-symmetrisation. γ ’s consistent with that representation.Let us illustrate this procedure with a few simple examples. Twist two contribution to (cid:104)O O O O (cid:105) Firstly consider the twist 2 sector in the (cid:104)O O O O (cid:105) free theory. In the free theorythe twist two operators are semi-short (recall they combine with other short operators tobecome long in the interacting theory). They correspond to semi-short operators in (62)with γ = 2 , µ = 0. Now the full free correlator is given below (11). But only two of thesix terms (the fourth and fifth) have (75) γ = d + d + d + d = 2. Thus to extractall information about twist two operators from the free theory we perform the followingexpansion A conn ( g g g g + g g g g ) = P × A conn (1 + det − (1 − Z )) = P × (cid:88) λ A λ F λ ( x ) , (80)where here P := P (OPE) { p i } × (cid:18) g g g g (cid:19) ( γ − p + p ) = g g g g since γ = 2 , p i = 2. This formula can be understood in terms of an SU ( m, m | n ) theory forany values of m, n . The values of the CPW coefficients A λ will not depend on the group.Moreover the Cauchy identity implies that the left had side is a sum of Schur polynomials ofone row only, and so the case m = 1 , n = 0 will capture all the relevant information in thiscase. In this case there is only one variable x and the superconformal blocks involve just asingle Hypergeometric. In summary therefore the twist two operator CPW coefficients canbe deduced from the following decomposition A conn (cid:18) − x (cid:19) = (cid:88) λ A λ ] x λ F ( λ + 1 , λ + 1; 2 λ + 2; x ) (81)which has the well known solution for twist two operators [26] A λ ] = 2 A conn ( λ )! (2 λ )! . (82) Higher twist singlet contribution to (cid:104)O O O O (cid:105) Let us now consider the contribution of higher twist long singlet operators to (cid:104)O O O O (cid:105) .The maximal value γ can take for this correlator is 4 (see (65)). Comparing with (62) wesee that the only way we can achieve a long singlet multiplet is if γ = 4, µ = µ = 0.Three of the six terms in the free correlator (11) have γ = d + d + d + d = 4 (the20rst, third and sixth). Thus to extract all information about twist four singlet operatorswe consider the expansion A disc ( g g + g g ) + A conn g g g g = P × (cid:16) A disc (1 + det − (1 − Z )) + A conn det − (1 − Z ) (cid:17) = P × (cid:88) λ A λ F λ ( x ) , (83)where this time P := P (OPE) { p i } × (cid:18) g g g g (cid:19) ( γ − p + p ) = g g since γ = 4 , p i = 2. Here, since the maximal inverse power of det(1 − Z ) is 2, we can recovercomplete information using the m = 2 , n = 0 blocks. Completely explicitly, using (78,79),we expand (multiplying both sides by the Vandermonde determinant):( x − x ) (cid:18) A disc (cid:18) − x ) (1 − x ) (cid:19) + A conn − x )(1 − x ) (cid:19) = (cid:88) λ A λ det (cid:16) x λ j +2 − ji F ( λ j − j +3 , λ j − j +3; 2 λ j − j +6; x i ) (cid:17) ≤ i,j ≤ . (84)From (62) we see that the long, twist 2 t , spin l , singlet reps with γ = 4 have CPWcoefficients A ,λ =[ l + t,t ] in the above expansion. Leading large N , higher twist singlet contribution to (cid:104)O O O O (cid:105) In this paper we will be mostly concerned with the leading in 1 /N piece of the free theory,and we consider such a case for higher charge. Consider the leading large N free correlator (cid:104)O O O O (cid:105) = A disc (cid:0) g g + g g + g g (cid:1) . (85)These terms correspond to γ = 0 , , 10 respectively according to (75). For long singletreps we need γ = 10 (note that if we considered the full free theory, rather than the leadinglarge N part, we would have to consider different values of γ = 4 , , , 10 and sum over theresults.) Taking out the relevant prefactor from the second two terms we need to performthe following expansion A disc (1 + det − (1 − Z )) = (cid:88) λ A λ F 55 10 λ ( x ) . (86)Here we convert to the m = 5 , n = 0, SU (5 , 5) theory and so explicitly, using (78,79) weexpand (multiplying both sides by the Vandermonde determinant):( x − x )( x − x ) . . . ( x − x ) A disc (cid:18) − x ) . . . (1 − x ) (cid:19) = (cid:88) λ A λ det (cid:16) x λ j +5 − ji F ( λ j − j +6 , λ j − j +6; 2 λ j − j +12; x i ) (cid:17) ≤ i,j ≤ . (87) In fact for the connected piece one could even use the m = 1 , n = 0 blocks as for twist 2 above. t , spin l correspond to the Youngtableau λ = [ l + t − , t − , , , x , . . . x up to and including the term x l + t +11 x t x x x , thusonly a fairly limited expansion in the variables x , x , x is needed.Note that if we were to consider the full (cid:104)O O O O (cid:105) free correlator, this will have sectorswith different values of γ = 0 , , , .., 10. Thus, since the description of long multiplets interms of O γλ is not unique (see (62) and the discussion below) then the CPW coefficientof a long operator will be given by the sum of all coefficients A γλ consistent with that rep.For example the OPE coefficient of a long singlet operator of twist t , spin l will be givenby the sum of terms A l + t − ,t − , , , + A l + t − ,t − , , + A l + t − ,t − , + A l + t,t ] . (88) Twist two operators from (cid:104)O p O p O q O q (cid:105) Let us now consider the twist two contribution to any correlator of the form (cid:104)O p O p O q O q (cid:105) .The argument above for the (cid:104)O O O O (cid:105) correlator implies that we must have γ = 2 andthen (75) then implies that there are only two contributing diagrams Ag p − g n − ( g g + g g ) = A × P × (1 + det − (1 − Z )) = P × (cid:88) λ A λ F ( Z ) (89)Notice that once the prefactor has been divided out the computation is exactly the sameas the (cid:104)O O O O (cid:105) case described above with the solution (82) A λ ] = 2 A ( λ )! (2 λ )! . (90)Finally, at large N the value of A can be deduced by counting the number of inequivalentplanar graphs contributing times the number of colour loops in a double line notation as A = N p + q p q (91) In this section we obtain an expression for the normalization N p p p p relative to the setof correlators (cid:104)O p O p O q O q (cid:105) . Let us recall that N ppqq is automatically obtained from first-principle computations in supergravity. For example, in the cases (cid:104)O p O p O p O p (cid:105) with p =2 , , (cid:104)O O O q O q (cid:105) for any q . However, it does not follow from the solution of thebootstrap problem [13], and we will need to determine N ppqq from an independent analysis.The important observation will be the following: The OPE analysis of known supergravityfour-point correlators [29, 54] reveals that in the supergravity certain long operators areabsent from the spectrum. Therefore, in the decomposition G sugra p p p p = G free p p p p + G dyna p p p p (92)22 special cancellation takes place between the sector of H dyna given by D sing and free theory.Building on this observation, we first derive N qq and N qq and we then obtain a formulafor N ppqq which generalizes the result N pppp obtained in [20]. Twist-2 long cancellation in (cid:104)O O O q O q (cid:105) The propagator structure in (cid:104)O O O q O q (cid:105) is easily obtained from the case q = 2. In fact,the latter is maximally symmetric and contains only two crossing symmetric classes [18]:(2 , , 0) and (1 , , q > 2, these two classes breaks into four sub-classes. This isshown in the diagrammatic expansion below where the extra (red) thick line indicates theadditional q − g . (cid:104)O O O q O q (cid:105) = A A + + A exc . + A + A exc . + A The residual symmetry exchanges g ↔ g and g ↔ g . In particular, A ( u, v ) = A ( u/v, /v ) ,A ( u, v ) = A ( u/v, /v ) , A exc . ( u, v ) = A ( u/v, /v ) ,A exc . ( u, v ) = A ( u/v, /v ) . (93)As a result, in free theory, where the A i =1 , , , are constants, we shall find A exc . = A and A exc . = A . The remaining coefficients to determine are A free1 = 2 qN q , ( A free2 , A free3 , A free4 ) = (cid:0) q ( q − , q, q ( q − (cid:1) A free1 N . (94)Two exceptions to these formulas are, A free2 = 1 for q = 2, and A free2 = 0 for q = 3. TheOPE prefactor reduces to g g q , which corresponds to the diagram associated with A .The correlator can then be rewritten as, (cid:104)O O O q O q (cid:105) free = g g q A free1 (cid:20) qN (cid:16) uσ + uτv (cid:17) +2 qN (cid:18) ( q − (cid:18) u σ + u τ v (cid:19) + ( q − u στv (cid:19) (cid:21) . (95)23he dynamical part of the correlation function obtained from tree-level supergravity is [22], (cid:104)O O O q O q (cid:105) dyna = g g q N qq I ( u, v, σ, τ ) u q D q,q +2 , , , (96)which decomposes as follows, A dyna1 = v u q D q,q +2 , , , A dyna3 = − (cid:18) − vu + 1 (cid:19) A dyna1 , (97) A dyna2 = vu A dyna1 , A dyna4 = − (cid:18) vu − (cid:19) A dyna1 . (98)Symmetry properties of D q,q +2 , , , described in the Appendix, imply the relations (93). Thenumber of propagator structures equals the number of SU (4) channel in the correlator.These correspond to the intersection([0 , , ⊗ [0 , , ∩ ([0 , q, ⊗ [0 , q, , , ⊗ [0 , , , (99)which splits into the six channels,[0 , , ⊗ [0 , , 0] = [0 , , ⊕ [0 , , ⊕ [0 , , ⊕ [2 , , ⊕ [1 , , ⊕ [1 , , , (100)according to (51). In each SU (4) channel we shall find contributions from operators be-longing to different N = 4 representation. For example, in the singlet channel we expect acontribution from the stress energy tensor, which belongs to a short multiplet, and a contri-bution from a twist-2 scalar, which belongs to a long multiplet. Moreover, long multipletswhose lowest dimension operator belong to [0 , , 0] have precisely the same SU (4) contentof (100), thus will contribute to all six channels.In free theory, all operators have canonical dimensions and are present in the spectrum. Aproper study of the superconformal OPE is needed in order to recombine all such contri-butions into supermultiplets [29, 53]. Once this decomposition is achieved [32], it can beshown that twist-two long contributions cancel between G free22 qq and G dyna22 qq precisely for thesupergravity value N qq = − q ( q − A free1 N . (101)We can prove (101) using a simpler argument: In the [0 , , 0] channel of the correlator, theconformal block corresponding to the twist-two scalar in the corresponding long multiplethas a series expansion with leading power u (1 − v ) . As remarked in (60), conformal blockscorresponding to operators with twist 2 t and spin l > u t (1 − v ) l . Therefore twist-2 is the very first non trivial contribution at order 1 /N ,and the absence of a twist-2 long multiplet implies that of the corresponding leading power.In terms of propagator structure, the [0 , , 0] channel is proportional to, (cid:104)O O O q O q (cid:105) (cid:12)(cid:12)(cid:12) [0 , , ∼ A + u (cid:18) A v + A exc . (cid:19) + u (cid:18) A v + A exc . + 13 A v (cid:19) . (102)where A i = A free i + A dyna i . At order 1 /N , the twist-two long contribution comes fromthe second term proportional to A free3 , and from A dyna1 ∼ D q,q +2 , , . The expression for24 sing q,q +2 , , can be obtained from (33). The limit v → q A free1 N + N qq Γ[ q − 1] = 0 , (103)which then leads to the result (101). This simpler argument generalizes to (cid:104)O p O p O q O q (cid:105) for arbitrary p and q . In fact, it will always be the case that in the [0 , , 0] channel of freetheory the first and only contribution at order 1 /N comes from a twist-2 scalar belongingto the corresponding long multiplet. As we now show, minor modifications are needed inthe derivation of N ppqq when p ≥ 3. However, taking these into account we will be able toobtain N ppqq in general. Twist-2 long cancellation in (cid:104)O O O q O q (cid:105) Similarly to the previous discussion, the propagator structure in (cid:104)O O O q O q (cid:105) follows fromthat at q = 3. In this case there are three crossing symmetric classes [18]: (3 , , 0) containsthree disconnected diagrams; (2 , , 0) contains six connected diagrams; and (1 , , 1) containsa single connected diagram. The symmetry breaking pattern when q > (cid:104)O O O q O q (cid:105) = A A + + A exc . + A + A exc . + A + A exc . + A + A exc . + A 25n free theory we find A exc .i =2 , , , = A i =2 , , , and A free3 = A free4 , with all the other constantsgiven by A free1 = 3 qN q , ( A free2 , A free3 , A free5 , A free6 ) = 3 q ( q − , , q − , A free1 N . (104)The special cases are q = 3, A free2 = 1 and q = 4, A free2 = 0. The OPE prefactor is g g q and we can rewrite the correlator as (cid:104)O O O q O q (cid:105) free = g g q A free1 (cid:20) qN (cid:18) uσ + uτv + u σ + u τ v + 2 u στv (cid:19) +3 qN (cid:18) ( q − (cid:18) u σ + u τ v (cid:19) + ( q − (cid:18) u σ τv + u στ v (cid:19) (cid:19)(cid:21) . (105)From the Mellin integral (23) we find (cid:104)O O O q O q (cid:105) dyna = g g q N qq I ( u, v, σ, τ ) H dyna (106) H dyna33 qq = u q (cid:104) σD q − ,q +2 , , + τ D q − ,q +2 , , + 1 q − D q,q +2 , , + (cid:18) q − σ + τ (cid:19) D q,q +2 , , (cid:105) . (107)Results for q > (cid:104)O O O q O q (cid:105) thereare ten SU (4) channels corresponding to the intersection([0 , , ⊗ [0 , , ∩ ([0 , q, ⊗ [0 , q, , , ⊗ [0 , , . (108)These include contributions from long multiplets whose lowest dimension operators belongto [0 , , , , 1] and [0 , , H dyna of the form H dyna33 qq = u q q − (cid:20) D q,q +2 , , + q + 13 D q,q +2 , , + q − (cid:0) D q − ,q +2 , , + D q − ,q +2 , , (cid:1)(cid:21) Υ + u q (cid:20)(cid:20) D q − ,q +2 , , − D q − ,q +2 , , (cid:21) Υ + (cid:0) D q − ,q +2 , , + D q − ,q +2 , , + 2 D q,q +2 , , (cid:1) Υ (cid:21) (109)A new feature compared to (cid:104)O O O q O q (cid:105) is the presence of several D δ δ δ δ for each channel.This implies a more intricate recombination analysis of the superconformal OPE [29, 53].Nevertheless, since the very first contribution to the [0 , , 0] channel in free theory onlycomes from a twist-two scalar, the absence of a twist-two long multiplet in the spectrum26an be unambiguously detected from (cid:104)O O O q O q (cid:105) (cid:12)(cid:12)(cid:12) [0 , , ∼ A + u (cid:18) A v + A exc . (cid:19) + u (cid:18) A v + A exc . + 13 A v (cid:19) + u (cid:18) A v + A exc . + 16 A v + 16 A exc . (cid:19) . (110)where A i = A free i + A dyna i . Following a procedure similar to that outlined for (cid:104)O O O n O n (cid:105) ,we find that the the relevant terms are A free3 and A dyna1 = N qq q − u q v (cid:0) D q,q +2 , , + D q,q +2 , , (cid:1) , (111)where the precise form of D sing q,q +2 , , and D sing q,q +2 , , can be obtained from (33). Importantly,only D sing q,q +2 , , will provide the leading power u , and the other D q,q +2 , , can be discarded.The equation to be solved is then3 q A free1 N + N qq Γ[ q − q − N qq = − q ( q − A free1 N . (113) Normalization N ppnn The analysis of the singlet channel in (cid:104)O O O q O q (cid:105) captures the generic features of the fourpoint correlator (cid:104)O p O p O q O q (cid:105) . Two comments are in order: Firstly, the leading contribu-tions to the scalar channel of the correlator is (cid:104)O p O p O q O q (cid:105) (cid:12)(cid:12)(cid:12) [0 , , ∼ A + u (cid:18) A v + A exc . (cid:19) + O ( u ) , (114)where A free3 has been given in (91). Secondly, even though several D functions will contributeto A dyna1 only D q,q +2 , , is relevant for the twist-two cancellation. From the definition of I ( u, v, σ, τ ) and the Mellin formula (23) we obtain A dyna1 = ( p − q − − p u q v D q,q +2 , , + . . . (115)where the dots stands for those D δ δ δ δ which do not contribute to the argument. We haveassumed q ≥ p , and the non trivial coefficient ( p − q − − p can be checked explicitlyin the examples (48)-(49).It then follows from the twist-two long cancellation that pq A free1 N + N ppqq ( p − q − q − − p = 0 (116)with solution N ppqq = − p ( p − q ( q − p )! A free1 N . (117)27 Determining strong coupling data from the correlator Having described the structure of the free theory and tree-level supergravity results thatwe need, we now proceed to analyse the OPE. The knowledge of the OPE leads to anexact superconformal block representation of any four-point correlator, including both shortand long exchanged representations. If we restrict attention to the contribution of longmultiplets, which comes from the free theory as well as from H dyna p p p p , we find (cid:104)O p O p O p O p (cid:105) long = N Σ P (OPE) (cid:88) t, l, R A { p i } R ( t | l ) L { p i } R ( t | l ) , (118) A { p i } R ( t | l ) = (cid:88) O ∈ R C p p O C p p O . (119)Here the operators have been normalised as in (8) with Σ = ( p + p + p + p ) / 2. Theexplicit expression for L { p i } R ( t | l ) can be read from (74). Expanding both the dimensions andOPE coefficients up to leading order in 1 /N ,∆ O = ∆ (0) O + 2 N η O , C p p O = C (0) p p O + 1 N C (1) p p O , (120)we obtain the following refinement (cid:104)O p O p O p O p (cid:105) long = N Σ P (OPE) (cid:32) (cid:88) t (cid:88) l, R A { p i } R ( t | l ) L { p i } R ( t | l )+ 1 N log( u ) (cid:88) t (cid:88) l, R M { p i } R ( t | l ) L { p i } R ( t | l ) + . . . (cid:33) where at order 1 /N we omitted analytic terms in u , which will not be relevant for ourdiscussion. Here t = (∆ (0) O − l ) / A { p i } R ( t | l ) = (cid:88) O ∈ R C (0) p p O C (0) p p O , (121) M { p i } R ( t | l ) = (cid:88) O ∈ R η O C (0) p p O C (0) p p O . (122)The data on the l.h.s of these equations will be obtained from the explicit form of thecorrelators. In particular, disconnected free theory determines A R ( t | l ), whereas M R ( t | l ) isobtained from the leading log( u ) singularity of H dyna .A fundamental assumption we will make about the supergravity limit is that the onlyoperators surviving are in one-to-one correspondence with single-trace half-BPS operators O p and multi-trace operators O t,l built from products of the O p . In the large N expansionthree point functions of half-BPS operators are 1 /N suppressed, as the computation (20)shows, and in any case contribute to the protected sector in the OPE. We expect thedouble-trace operators to be the only long operators O t,l to have non-vanishing three-pointfunctions C (0) p p O . Triple-trace and higher multi-trace operators are expected to have their28hree-point functions suppressed by further powers of 1 /N , i.e. they will start contributingto C (1) p p O and higher.In the first instance we will focus on unprotected operators in the singlet representation of SU (4), since these are the operators whose data ultimately determine the loop correction( O (1 /N )) to (cid:104)O O O O (cid:105) [34]. The exchanged singlet operators in question have thefollowing description in the free theory: K free t,l,i = O i +1 (cid:3) t − i − ∂ l O i +1 + . . . (123)where the SU (4) indices are understood to be contracted to produce a singlet, and theellipsis denotes similar terms with the space-time derivatives distributed differently betweenthe two constituent operators, O i +1 . The precise combination will not be important here,but importantly there is a unique combination yielding a conformal primary operator. Theoperators given in (123) have spin l and dimension 2 t + l (i.e. twist 2 t ) while i = 1 . . . t − t − 1) different operators which have the same spin and dimension. As soon as thecoupling is turned on, these ( t − 1) operators will mix and develop anomalous dimensions.At strong coupling with large N , the operators again take their free theory dimensions, withanomalous dimensions developing at order 1 /N . Since the operators (123) are protectedat infinite N they all remain present in the spectrum even though they reside in longmultiplets. It no longer makes sense to write the operators explicitly as (123), but thenumber of operators is the same. Thus we denote by K t,l,i , with i = 1 , . . . , t − 1, thecorresponding operators at strong coupling. They are operators which have well-definedanomalous dimensions at O (1 /N ). This automatically means their two-point functions areorthogonal at O ( N ) and we can also normalise them, so we have (cid:104) K t,l,i K t,l,i (cid:48) (cid:105) = δ ii (cid:48) . (124)Since we only consider them at leading order in 1 /N , we will also drop the superscriptfrom the three-point functions C (0) p p K t,l,i and just write C p p K t,l,i instead.We wish to obtain the anomalous dimensions η t,l,i of the operators K t,l,i as well as theirlarge N three-point functions C ppK t,l,i . First note that at leading order in the large N limitthe OPE of O p O p contains the operators K t,l,i for all t ≥ p . Thus for fixed t , the four-point correlators (cid:104)O p O p O q O q (cid:105) with p ≤ q contain information about operators K t,l,i for all q ≤ t . Noting the p ↔ q symmetry we deduce that there are t ( t − / A { p i } R ( t | l ) coming from each correlatorin the free theory at leading order into the following symmetric matrix, (cid:98) A ( t | l ) (cid:12)(cid:12)(cid:12) [0 , , = A A . . . A tt A . . . A tt . . . . . . A tttt . (125)In fact, from the form of the large N free theory correlators one can see immediately that theabove matrix ˆ A is actually diagonal. Likewise we can organise the information M { p i } R ( t | l )29oming from the log u term at order 1 /N in each correlator into another symmetric matrix, (cid:98) M ( t | l ) (cid:12)(cid:12)(cid:12) [0 , , = M M . . . M tt M . . . M tt . . . . . . M tttt . Both in (cid:98) M ( t | l ) and (cid:98) A ( t | l ) we have just given the independent entries in the upper triangularpart explicitly.Consider now the ( t − 1) independent operators K t,l,i . They are associated to ( t − three-point functions C ppK t,l,i where i = 1 , . . . t − p = 2 , . . . , t , and ( t − 1) anomalousdimensions η t,l,i . In total therefore we have t ( t − 1) unknowns that need to be determined.Thus the matrices (150) and (126) contain the precise amount of data needed! The reasonfor this precise matching of degrees of freedom is that the operators K t,l,i are (in one-to-onecorrespondence with) bilinears of half-BPS single-trace operators. The matching is thus aremarkable feature of large ’t Hooft coupling and large N only, as in general there will bemany other types of operators contributing.Let us now examine the equations (121)-(122) in detail, beginning with low twist cases. Tosimplify notation a little, we redefine C ppK t,l,i in favor of c pi taking out a universal factorwhich we find is always present,( C ppK t,l,i ) = ( l + t + 1)! (2 l + 2 t + 2)! c pi , p = 2 , . . . , t, i = 1 , . . . , t − . (126)At fixed twist we expect c pi to depend non trivially on l . Here there is only one operator contributing and it only appears in the simplest correlator (cid:104)O O O O (cid:105) . Extracting the relevant superblock coefficient we obtain at leading order(from the disconnected free correlator)( C K t,l, ) = A ⇒ c = 43 ( l + 1)( l + 6) , (127) η ( C K t,l, ) = M ⇒ c η = − . (128)This clearly yields η = − l + 1)( l + 6) , c = (cid:114) l + 1)( l + 6)3 . (129)This result has been known for a long time [26]. Note the symmetry l → − − l .30 .2 Twist 6 The situation becomes more interesting when we move to twist 6. Here there are twooperators contributing, K ,l, and K ,l, . The free theory results give: c + c = 25 ( l + 1)( l + 8) ,c + c = 940 ( l + 1)( l + 2)( l + 7)( l + 8) ,c c + c c = 0 . (130)It is interesting at this point to compare briefly with the free gauge theory at large N .The relevant correlator (disconnected free correlator) is exactly the same as the one we arediscussing here at strong coupling. However, despite this one should not be tempted toassume the leading large N three-point functions are also the same at strong and weakcoupling. In the free theory at large N we recall that the two operators are explicitly givenas K ,l, = O ∂ l (cid:3) O + . . . and K ,l, = O ∂ l O + . . . . Although in general other operatorscontribute at weak coupling (single trace etc.), at large N only these two contribute (theOPE can easily be performed explicitly via Wick contractions to verify this). Further thethree point functions c weak22 and c weak31 are supressed at this order and thus the solution ofthe above equations reads simply: c weak22 = c weak31 = 0 , ( c weak21 ) = 25 ( l + 1)( l + 8) , ( c weak32 ) = 940 ( l + 1)( l + 2)( l + 7)( l + 8) , (131)and the three-point functions c weak pi are diagonal.The strong coupling interpretation of the equations turns out to be very different however,even though it arises from the same free disconnected correlator. The dynamical parts ofthe correlators give c η + c η = − ,c η + c η = − l + 9 l + 44) ,c c η + c c η = 432 , (132)and in particular the last equation means that here the three-point c pi functions cannot bediagonal. Instead we straightforwardly solve the above equations and obtain the solution η = − l + 1)( l + 2) , η = − l + 7)( l + 8) ,c = − (cid:115) l + 1)( l + 2)( l + 8)5(2 l + 9) , c = − (cid:115) l + 1)( l + 7)( l + 8)5(2 l + 9) ,c = (cid:115) l + 1)( l + 2)( l + 7) ( l + 8)40(2 l + 9) , c = − (cid:115) l + 1)( l + 2) ( l + 7)( l + 8)40(2 l + 9) . (133)31 .3 General twist The first task in attempting to understand the general structure is to generalise the equa-tions we obtain from the correlators via the superconformal block expansion. At leadingorder the situation is simpler, since off-diagonal correlators (cid:104)O p O p O q O q (cid:105) with p (cid:54) = q aresuppressed and therefore the matrix (cid:98) A ( t | l ) is diagonal. We have computed a number ofexplicit examples and spot the pattern that leads to the following general formula, A pppp (cid:12)(cid:12)(cid:12) [0 , , =24( l + 1)( t − t !) ( l + 2 t + 2)( l + t − l + t + 1)!) ( p + t )!( l + p + t + 1)!( p + 1)( p − p − (2 t )!( t + 2)!( l + t + 3)!(2 l + 2 t + 2)!( t − p )!( l − p + t + 1)! . (134)Let us notice that A pppp has completly factorized form. For fixed twist, we can define thematrix of three-point function coefficients C ( t | l ) = C K t,l, C K t,l, . . . C K t,l,t − C K t,l, C K t,l, . . .. . .C ttK t,l, (135)and rewrite the equations (121) in matrix form,˜ c ˜ c T = Id t − , C = (cid:98) A · ˜ c ( t | l ) (136)where the orthonormality property of the matrix ˜ c is manifest. Equations (122) become˜ c · diag ( η , . . . , η t − ) · ˜ c T = (cid:98) A − · (cid:98) M ( t | l ) · (cid:98) A − (137)The columns of ˜ c ( t | l ), are then eigenvectors of the matrix (cid:98) A − · (cid:98) M ( t | l ) · (cid:98) A − and theanomalous dimensions are the corresponding eigenvalues. Notice from the structure ofeq. (137) (recalling that ˆ A is diagonal) the remarkable property that det( (cid:98) M ) will factorise.From the explicit expressions for M ppqq obtained upon decomposing H dyna in superconformalblocks this property is completely obscure. In particular, M ppqq is found to be proportionalto a polynomial in l of degree 2( p − p ≤ q , which does not admit real roots. Theirexpressions are cumbersome and thus we will not display them explicitly.Let us rewrite in this new notation the solution at twists four and six from eqs. (129) and(133). The ˜ c matrix in these two cases is˜ c (2 | l ) = 1 , ˜ c (3 | l ) = (cid:113) l +22 l +9 (cid:113) l +72 l +9 − (cid:113) l +72 l +9 (cid:113) l +22 l +9 , (138) In more detail, we first computed the cases with p = t up to 6 and spotted a pattern for these whichwe then confirmed at p = 7. Next we considered cases for fixed p general t , some of which were alreadyavailable [26, 53]. We spotted a pattern for these up to a numerical p dependent coefficient using results upto p = 5. This final numerical factor we can then fix as a function of p uniquely by comparison with the p = t case. c (3 | l ) ˜ c (3 | l ) T = Id . We also repeat the formulae for theanomalous dimensions for later convenience, η ,l, = (cid:110) − l +1)( l +6) (cid:111) η ,l,i = (cid:110) − l +1)( l +2) , − l +7)( l +8) (cid:111) , (139)We now proceed by performing the superblock expansion to find (cid:98) M ( t | l ) up to higher valuesof t ≤ 12, and solve for anomalous dimensions and ˜ c ( t | l ). From the solution at twist eightwe obtain ˜ c (4 | l ) = (cid:113) l +2)( l +3)6(2 l +9)(2 l +11) (cid:113) l +3)( l +8)3(2 l +9)(2 l +13) (cid:113) l +8)( l +9)6(2 l +11)(2 l +13) − (cid:113) l +2)( l +8)(2 l +9)(2 l +11) − (cid:113) l +9)(2 l +13) (cid:113) l +3)( l +9)(2 l +11)(2 l +13) (cid:113) l +8)( l +9)6(2 l +9)(2 l +11) − (cid:113) l +2)( l +9)3(2 l +9)(2 l +13) (cid:113) l +2)( l +3)6(2 l +11)(2 l +13) , (140)and η ,l,i = (cid:110) − l +7)( l +1)( l +2)( l +3) , − l +3)( l +8) , − l +4)( l +8)( l +9)( l +10) (cid:111) . (141)For higher twists the solution becomes quite lengthy so we find it helpful to introduce amore compact notation for the square root factors. We define( n ) = √ l + n , [ n ] = √ l + n . (142)With this more compact notation the solution at twist ten takes the form,˜ c (5 | l ) = (cid:113) 32 (2)(3)(4)[9][11][13] (cid:113) 52 (3)(4)(9)[9][13][15] (cid:113) 52 (4)(9)(10)[11][13][17] (cid:113) 32 (9)(10)(11)[13][15][17] − (cid:113) 278 (2)(3)(9)[9][11][13] − (cid:113) 58 ( l +18)(3)[9][13][15] (cid:113) 58 ( l − (cid:113) 278 (4)(10)(11)[13][15][17] (cid:113) 52 (2)(9)(10)[9][11][13] − (cid:113) 32 ( l − − (cid:113) 32 ( l +16)(3)[11][13][17] (cid:113) 52 (3)(4)(11)[13][15][17] − (cid:113) 58 (9)(10)(11)[9][11][13] (cid:113) 278 (2)(10)(11)[9][13][15] − (cid:113) 278 (2)(3)(11)[11][13][17] (cid:113) 58 (2)(3)(4)[13][15][17] , (143)and η ,l,i = (cid:110) − l +7)( l +8)( l +1)( l +2)( l +3)( l +4) , − l +3)( l +4) , − l +9)( l +10) , − l +5)( l +6)( l +9)( l +10)( l +11)( l +12) (cid:111) . (144)We begin to see intriguing structure in the entries of the matrix as well as in the anomalousdimensions. Note the symmetry l → − t − − l which is an invariance of the set ofanomalous dimensions and an invariance up to signs of the ˜ c matrix under a flip aboutthe vertical axis. Note also that at twist ten we see for the first time the appearance ofpolynomials in l (without a square root) in the numerators of the central entries of (143).33t twist ten these polynomials are all linear, but their degrees increase as we increase thetwist further.Indeed, proceeding to compute the next few examples one gets a better idea of the struc-ture. The anomalous dimensions reveal a fairly simple structure that is consistent with theformula η [0 , , t,l,i = − t − ( t + l ) ( l + 2 i − , (145)where ( x ) n = x ( x + 1) . . . ( x + n − 1) is the Pochhammer symbol. Note that the anomalousdimensions are all negative for all physical values of l .The ˜ c ( t | l ) matrix is trickier to understand. Already from the results up to twist ten we notea pattern of square roots of linear factors of l . In addition we have seen that in the entriestowards the centre one finds fewer square root factors in the numerator, and polynomialsin l without a square root. Note that the entries of the matrix always have a finite (butpossibly vanishing) limit as l → ∞ . In fact, we can deduce the structure of ˜ c ( t | l ) for agiven twist in terms of an ansatz with some undetermined numbers,˜ c [0 , , pi = (cid:115) − t (2 l + 4 i + 3) (( l + i + 1) t − i − p +1 ) σ (( t + l + p + 2) i − p +1 ) σ (cid:0) l + i + (cid:1) t − × min( i − ,p − ,t − i − ,t − p ) (cid:88) k =0 l k a [0 , , p,i,k ) . (146)The powers of the Pochhammer factors inside the square root are signs given explicitly by σ = sgn( t − p − i + 1) , σ = sgn( i − p + 1) . (147)where p = 2 , . . . , t and i = 1 , . . . , t − 1. We notice that the square root structure in ˜ c pi follows from complicated combinatorics, which nevertheless can be captured by the two(non-analytic) sign functions σ and σ . Around the outer frame of the matrix, the unfixedpolynomial has degree 0, i.e. it is simply a constant. Its degree increases as we move towardsthe inside of the matrix. One can readily check (146) is consistent with the examples givenexplicitly above and we have tested the structure up to t = 12.Given the ansatz (146), we have reduced the problem to that of finding the constants a ( p, i, k ). Quite surprisingly, enforcing orthonormality of ˜ c ( t | l ) uniquely fixes the solution .In more detail, we first insist that the first row has unit norm, (cid:80) i ˜ c i = 1. This is a linearequation in a (2 , i, with a unique solution. In fact, the constraint is a rational functionof l and so this single equation can fix more than one constant. Then, orthogonality of therows (cid:80) ˜ c pi ˜ c qi = 0 for p (cid:54) = q gives a linear system in the remaining variables and uniquelyfixes them, up to an overall scale which is fixed by the unit norm condition.We find it remarkable both that there exist such orthonormal matrices with the structure(146) and that the matrix is uniquely fixed by orthonormality as a linear system. The fact We have checked this up to twist 48 ( t = 24). t = 24. This enables us to spot patterns and write down general formulae.We do not have a completely general formula for the full matrix ˜ c but we do have variouscases in closed form. In particular the top row of the matrix is given by the formula a [0 , , ,i, = 2 t − (2 i + 2)!( t − t − i + 2)!3( i − i + 1)!( t + 2)!( t − i − t − i + 1)! , i = 1 , . . . , t − . (148)This formula completely specifies all the three-point function of the form C O O K t,l,i whichwas an essential ingredient in the prediction of the one-loop supergravity correction to (cid:104)O O O O (cid:105) presented in [34]. [0 , , to [ n, , n ] representations Having given the general structure of the solution to the mixing problem for singlet double-trace operators, we may now proceed to analysing more general SU (4) representations.Specifically we can investigate operators in the series of representations [ n, , n ] which alsoarise in the OPE of correlation functions of the form (cid:104)O p O p O q O q (cid:105) . For each channel ofthe form [ n, , n ] the structure of this problem is analogous to that of singlet channel. Inparticular, at twist 2 t a basis of double trace operators in the [ n, , n ] representation willhave the schematic form {O n (cid:3) t − n − ∂ l O n , O n (cid:3) t − n − ∂ l O n , . . . , O t (cid:3) ∂ l O t } . (149)and we expect ( t − − n ) superconformal primary operators. As for the singlet doubletrace operators in (123), the precise form of these primary operators is a specific linearcombination of the element of the basis, with derivatives acting on the two constituentoperators. These operators again have integer classical dimensions for infinite N and receiveanomalous dimensions at order 1 /N .The analysis of the [ n, , n ] channel for fixed n follows a very similar logic to that presentedin the singlet case. Once again we conclude that the series of correlators (cid:104)O p O p O q O q (cid:105) for n + 2 ≤ p ≤ q ≤ t provides the right amount of information needed in order to solve foranomalous dimensions and three-point functions of the exchanged double trace operators.From the general form of the long superconformal blocks (74) it is straightforward to isolatethe appropriate channel, and organize the data from the superblock expansion into thesymmetric matrices (cid:98) M ( t | l ) (cid:12)(cid:12)(cid:12) [ n, ,n ] and (cid:98) A ( t | l ) (cid:12)(cid:12)(cid:12) [ n, ,n ] .Before presenting our general results we go through some specific examples. In this channel the matrices (cid:98) M ( t | l ) (cid:12)(cid:12)(cid:12) [1 , , and (cid:98) A ( t | l ) (cid:12)(cid:12)(cid:12) [1 , , have the form35 M ( t | l ) (cid:12)(cid:12)(cid:12) [1 , , = M M . . . M tt M . . . M tt . . . . . . M tttt , (150) (cid:98) A ( t | l ) (cid:12)(cid:12)(cid:12) [1 , , = A A . . . A tt A . . . A tt . . . . . . A tttt , (151)where (cid:98) A ( t | l ) is diagonal with entries A pppp (cid:12)(cid:12)(cid:12) [1 , , = 15( p − t − t + 2)( l + t )( l + t + 3)( p + 2)( t − t + 3)( l + t − t + l + 4) A pppp (cid:12)(cid:12)(cid:12) [0 , , (152)We can then introduce the orthonormal matrix ˜ c ( t | l ) and start solving explicitly the mixingproblem. For illustration, let us look at the first three cases:At twist six there is only one operator, therefore˜ c (3 | l ) = 1 η ,l, = − l )(6 + l ) (153)At twist eight there are two operators, and we find˜ c (4 | l ) = (cid:113) l +22 l +11 (cid:113) l +92 l +11 − (cid:113) l +92 l +11 (cid:113) l +22 l +11 (154)with anomalous dimensions η ,l,i = (cid:110) − l )(2+ l )(4+ l )(7+ l ) , − l )(4+ l )(7+ l )(9+ l ) (cid:111) (155)At twist ten it is becoming evident that the structure of eigenvectors and anomalous dimen-sion found in the singlet case generalises to [1 , , 1] with minor modification. In particular˜ c (5 | l ) = (cid:113) l +2)( l +3)8(2 l +11)(2 l +13) (cid:113) l +3)( l +10)4(2 l +11)(2 l +15) (cid:113) l +10)( l +11)8(2 l +13)(2 l +15) − (cid:113) l +2)( l +10)(2 l +11)(2 l +13) − √ √ (2 l +11)(2 l +15) (cid:113) l +3)( l +11)(2 l +13)(2 l +15) (cid:113) l +10)( l +11)8(2 l +11)(2 l +13) − (cid:113) l +2)( l +11)4(2 l +11)(2 l +15) (cid:113) l +2)( l +3)8(2 l +13)(2 l +15) (156)36ith anomalous dimensions η ,l,i = (cid:110) − l )(2+ l )(3+ l )(5+ l ) , − l )(8+ l ) , − l )(8+ l )(10+ l )(11+ l ) (cid:111) (157)The solution of the mixing problem up to t = 12 can be found straightforwardly and leadsto the expression η [1 , , t,l,i = − t − t ( t + 1)( t + 3)( t + l − t + l + 1)( t + l + 2)( t + l + 4)( l + 2 i ) (158)for the anomalous dimensions, and˜ c [1 , , pi = (cid:115) − t (2 l + 4 i + 5) (( l + i + 1) t − i − p +1 ) σ (( t + l + p + 2) i − p +2 ) σ (cid:0) l + i + (cid:1) t − × min( i − ,p − ,t − i − ,t − p ) (cid:88) k =0 l k a [1 , , p,i,k ) . (159)for the entries of the ˜ c ( t | l ) matrix, with σ = sgn( t − p − i + 1), and σ = sgn( i − p + 2),and p = 3 , . . . , t and i = 1 , . . . , t − 2. The orthogonality condition of the matrix againdetermines completely the value of these a ( p, i, k ) at any twist. [2 , , to [ n, , n ]In this section, we present general formulae for the matrices (cid:98) M ( t | l ) and (cid:98) A ( t | l ) given in termsof disconnected free theory data, anomalous dimensions and orthonormal ˜ c ( t | l ) matrices.Let us begin from free theory, where we have obtained the following result, A pppp ( t | l ) (cid:12)(cid:12)(cid:12) [ n, ,n ] = (160) p n ! p !( p − n + 2) n +3 ( p + 1 + n )!( p − − n )! × ( t !) (2 t )! ( l + 1)((1 + l + t )!) ( l + 2 t + 2)(2 l + 2 t + 2)! × ( l + t − p + 2) p − − n ( l + t + 4 + n ) p − − n ( l + 1 + t − n ) n ( l + 1 + t + 2) n × ( t − p + 1) p − − n ( t + 3 + n ) p − − n ( t − n ) n ( t + 2) n Introducing the ˜ c ( t | l ) [ n, ,n ] matrices and computing (cid:98) M ( t | l ) [ n, ,n ] for a large number of twistand several values of n we have been able to fit and test both the anomalous dimensionsand the entries of ˜ c ( t | l ) with the following formulae: For the anomalous dimensions we find, η [ n, ,n ] t,l,i = − t − − n ) t ( t + 1)( t + 2 + n )( t + l − n )( t + l + 1)( t + l + 2)( t + l + 3 + n )( l + 2 i + n − (161)37nd for the entries of the ˜ c ( t | l ) matrix,˜ c [ n, ,n ] pi = (cid:115) − t (2 l + 4 i + 3 + 2 n ) (( l + i + 1) t − i − p +1 ) σ (( t + l + p + 2) i − p + n +1 ) σ (cid:0) l + i + n + (cid:1) t − n − × min( i − ,p − n − ,t − n − i − ,t − p ) (cid:88) k =0 l k a [ n, ,n ]( p,i,k ) . (162)The signs are given explicitly by σ = sgn( t − i − p + 1) , σ = sgn( i − p + n + 1) . (163)All unspecified coefficients a ( p, i, k ) are again determined by imposing orthogonality of ˜ c . Let us now analyse some general behaviour of the spectrum of anomalous dimensions thatwe found. Here we follow some of the arguments discussed in [9]. Let us consider a verylarge, but finite value of N . From eq. (145) we find that our expression for the full twist ofthe operator K t,l,i in the singlet channel is∆ [0 , , t,l,i − l = 2 t − N η [0 , , t,l,i + . . . = 2 t − N ( t − ( t + l ) ( l + 2 i − + . . . . (164)We note that the numerator of the anomalous term behaves like t for large t and that thecoefficient is negative. Keeping the leading terms for large t we find∆ [0 , , t,l,i − l = 2 t − N (cid:18) t ( l + 2 i − + O ( t ) (cid:19) + . . . . (165)As argued in [9] these two facts imply that for some large classical twist t the correctionterm will dominate over the classical term. Indeed for t ∼ N we find the two terms are ofthe same order and so the anomalous dimension formula inevitably requires corrections toavoid violating the unitarity bound.In fact we can argue that one needs corrections even before t reaches values of order N .Since we have resolved the mixing of the ( t − 1) operators with the same classical twist 2 t we may consider the differences in their dimensions. As already observed, the anomalousdimensions are all negative and so as one increases 1 /N away from zero the dimensionsdecrease. One can see from the formula (145) that dimension of the operator with twist i = 1 decreases fastest and the dimension of the operator with i = t − K t,l,t − at level t and the fastestdescending one K t +1 ,l, at level ( t + 1). The difference in their dimensions is∆ [0 , , t +1 ,l, − ∆ [0 , , t,l,t − = 2 − N t ( l + 1) + O ( t ) (166)38 � - �� �� - �� �� - �� �� - �� �� - �� �� - � ��������������� � / � � Δ * Figure 1: Varying t we show on a log-log plot the value of the dimension ∆ (cid:63)l at the crossingpoint ∆ [0 , , t +1 ,l, − ∆ [0 , , t,l,t − = 0 as function of 1 /N for l = 0 (red), l = 2 (blue) and l = 4(green). The best fit given by the solid black line is ∆ (cid:63) ≈ u l /N / with u l =0 , , ≈ { , . , } .Hence we find for t ∼ N that the two operators will become degenerate and then crossover in the values of their dimensions. Such level crossing should not occur at generic pointsin moduli space, it should only be associated with points of increased symmetry, such asthe free theory limit. Thus we conclude that before we reach this point further correctionsto the anomalous dimensions become relevant. A plot of the value of the dimension at thecrossing point against 1 /N for l = 0 , , l [7, 8, 12, 59]. In particular, the anomalous dimensions are conjectured to be negative,monotonic and convex as a function of l , at least for large enough l .Our results for all anomalous dimensions are negative for any values of t and l . By examiningtheir precise form (145) (which are simply rational functions involving linear factors in l )one can straightforwardly see that for all values of i ≤ ( t + 1) / l > − i . This is simply because atlarge l , η t,l,i ∼ − t − t ( t +1)( t +2) l is monotonic and convex, and for decreasing values of l , thefirst zero or pole is the negative pole at l = 1 − i . For i > ( t + 1) / 2, as we reduce the value of l , the anomalous dimension hits a zero at l = − t before it reaches the pole at l = 1 − i . Thus monotonicity and convexity break down atsome point with convexity breaking down first. By considering the equation ∂ l η t,l,i = 0we can study for which value of l convexity breaks down as we reduce l . Assuming large t (so we can approximate the resulting large polynomial equation with its highest powers)the breakdown in convexity occurs at l ∼ √ i + 4 i − √ t − t . This is negative for i < √ ( √ ) t ∼ . t and so the anomalous dimension is still convex, and monotonic for The case i = ( t − / t odd is in fact a special case, but in fact has no zero’s and again the firstspecial point is a negative pole at l = − t − 4. It is thus negative, monotonic and convex for all l > − t − l in this range. For operators with i > . t on the other hand, the anomalousdimension ceases to be convex for some finite positive value of l . The worst offender isthe operator with the maximal value of i = t − 1. This ceases to be convex for l belowapproximately (1 + √ t ∼ . t . We have presented a detailed analysis of the double trace spectrum of N = 4 super Yang-Mills theory in the supergravity limit. We have shown that the known tree-level supergrav-ity results contain all the necessary information to resolve the degeneracy of the doubletrace operators in the large N limit. Here we have focussed on the correlation functions ofthe form (cid:104)O p O p O q O q (cid:105) since these are sufficient to resolve the degeneracy of the double-traceoperators in the [ n, , n ] representations of SU (4). Similar methods can be applied to themore general cases (cid:104)O p O p O p O p (cid:105) to resolve the mixing for more general representations.Our results for the leading order OPE coefficients and anomalous dimensions are surpris-ingly simple, even given the very compact Mellin space form of the tree-level supergravitycorrelators given in [13]. The fact that the anomalous dimensions admit such a simpleformula as (161) is remarkable. Even more remarkable perhaps is the universal structurewe find in the orthogonal ˜ c matrices. The fact that orthogonal matrices ˜ c of the form(159) exist at all is surprising. We should point out that modifications of the structureof the square root factors in (159) typically lead to no orthogonal solution at all. Indeedthe structure of the ˜ c -matrices in the [ n, , n ] case was first guessed based on this structurebefore being explicitly identified by analysing the relevant channels of the OPE. It wouldbe very interesting to understand whether the structure (159) arises due to some as yetunidentified simplicity which could suggest more about the higher order 1 /N corrections tothe quantities we have derived in this work.The results we have presented here for the singlet channel have already been used in [34] tocontruct a prediction for the one-loop correction to the (cid:104)O O O O (cid:105) correlator. Certainlysimilar analyses could be carried out to make one-loop predictions for more general corre-lators. This would rely on resolving the mixing for more general representations than wehave examined here.Finally, while we have focussed on N = 4 super Yang-Mills theory here, the phenomenon oflarge N degeneracy and the need for resolving mixing is presumably common to many holo-graphic theories. Essentially the phenomenon arises because of the presence of a compactfactor (here an S ) in the gravity background which leads to the presence of a Kaluza-Kleintower of modes related to the massless gravity modes. For fixed twist and spin one willthen typically have many double-trace operators one can consider and these will genericallymix. It would be interesting to consider both other models and the generic structure oflarge N CFTs further. 40 cknowledgements FA would like to thank Gleb Arutyunov, Jorge Russo, Kostas Skenderis, Arkady Tseytlin,and Konstantin Zarembo for discussions on related topics. JMD and HP are supportedby the ERC Grant 648630. PH acknowledges support from an STFC Consolidated GrantST/L000407/1 and also National Science Foundation under Grant No. NSF PHY-1125915.FA acknowledges support from STFC through Consolidated Grant ST/L000296/1. A D -functions The analytic part of a D -function is given by D analytic δ δ δ δ = ( − ) σ (cid:88) n,m ≥ u n n !( σ + n )! Λ δ δ δ + σδ + σ ( n ) ( δ + n ) m ( δ + σ + n ) m ( δ + δ + 2 n ) m f nm (1 − v ) m m ! (167)where f nm = (cid:104) + ψ ( n + 1) + ψ ( σ + 1 + n ) + 2 ψ ( δ + δ + 2 n + m ) − ψ ( δ + σ + n ) − ψ ( δ + n ) − ψ ( δ + σ + n + m ) − ψ ( δ + n + m ) (cid:105) and we recall the definitionΛ δ δ δ δ ( n ) ≡ Γ[ δ + n ]Γ[ δ + n ]Γ[ δ + n ]Γ[ δ + n ]Γ[ δ + δ + 2 n ] . (168)In general, the full D -functions can be recursively generated by the action of differen-tial operators on the four-dimensional scalar one-loop box integral Φ (1) ( u, v ), for whichthere is an explicit expression in terms of polylogarithms, see equation (36). Starting with D ( u, v ) := Φ (1) ( u, v ), when δ i , and Σ = ( δ + δ + δ + δ ) / D δ δ δ δ from the following recursion relations [18]: D δ +1 ,δ +1 ,δ ,δ = − ∂ u D δ δ δ δ ,D δ ,δ ,δ +1 ,δ +1 = ( δ + δ − Σ − u∂ u ) D δ δ δ δ ,D δ ,δ +1 ,δ +1 ,δ = − ∂ v D δ δ δ δ ,D δ +1 ,δ ,δ ,δ +1 = ( δ + δ − Σ − v∂ v ) D δ δ δ δ ,D δ ,δ +1 ,δ ,δ +1 = ( δ + u∂ u + v∂ v ) D δ δ δ δ ,D δ +1 ,δ ,δ +1 ,δ = (Σ − δ + u∂ u + v∂ v ) D δ δ δ δ , (169)The D -functions obey many transformation identities (stemming from the permutationsymmetries of the one-loop box integral), one of which is the permutation property D δ δ δ δ ( u, v ) = v δ + δ − Σ D δ δ δ δ ( u, v ) = u δ + δ − Σ D δ δ δ δ ( u, v ) , (170)41hich can be used to convert a D -function with negative σ into one with σ ≥ 0, as requiredfor the decomposition shown in equation (32).In some cases it is useful to use the reflection identity D δ δ δ δ ( u, v ) = D Σ − δ , Σ − δ , Σ − δ , Σ − δ ( u, v ) (171)to bring a D -function into a more convenient form.Finally, under crossing transformations of the cross-ratios ( u, v ) the D -functions behave as D δ δ δ δ ( u, v ) = D δ δ δ δ ( v, u ) , = u − δ D δ δ δ δ (cid:18) u , vu (cid:19) , = v δ − Σ D δ δ δ δ (cid:18) uv , v (cid:19) . 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