On the spectrum and structure constants of short operators in N=4 SYM at strong coupling

We study short operators in planar $\mathcal{N}=4$ SYM at strong coupling, for general spin and $SO(6)$ symmetric traceless representations. At strong coupling their dimension grows like $\Delta \sim 2\sqrt{\delta} \lambda^{1/4}$ and their spectrum of degeneracies can be analysed by considering the massive spectrum of type II strings in flat space-time. We furthermore compute their structure constants with two arbitrary chiral primary operators. This is done by considering the four-point correlator of arbitrary chiral primary operators at strong coupling in planar $\mathcal{N}=4$ SYM, including the supergravity approximation plus the infinite tower of stringy corrections that contributes in the flat space limit. Our results are valid for generic rank $n$ symmetric traceless representations of $SO(6)$ and in particular for $n \gg 1$, as long as $n \ll \lambda^{1/4}$.


Introduction
It has been known since the work of Gubser, Klebanov and Polyakov that massive string states in AdS 5 × S 5 couple to short single-trace operators in N = 4 SYM [1].These operators have conformal dimensions which grow as ∆ ∼ 2 √ δλ 1/4 at strong t'Hooft coupling λ, where δ = 1, 2, . . . is the string mass level.In this paper we revisit these operators O δ,ℓ,R for general spin ℓ and in the SO(6) representation R.
One of the reasons we are interested in these operators is their appearance in stringy correlators.In [2,3] we recently derived the full 1/ √ λ correction to the Virasoro-Shapiro amplitude in AdS 5 ×S 5 , i.e. the correlator ⟨O 2 O 2 O 2 O 2 ⟩ at large N and strong coupling including all stringy corrections, where O 2 is the chiral primary operator in the stress-tensor multiplet.From this result one can extract averaged corrections to conformal dimensions and structure constants for all the massive short operators O δ,ℓ,[0,0,0] .A natural question is whether these averages can be unmixed by considering more general correlators ⟨O p 1 O p 2 O p 3 O p 4 ⟩, where O p i is a chiral primary of dimension p i . 1 As we will show in this paper, unfortunately this is not the case.
Another motivation to study short operators is to further bridge the gap between the conformal bootstrap and integrability.Explicit integrability results for dimensions of short operators at strong coupling are currently available for those operators who sit on the leading Regge trajectory [7][8][9], including the familiar Konishi operator.These results agree with [2,3].One advantage of our approach however, is that we can compute structure constants, for which integrability techniques are still under development, see however [10] for recent progress relevant for the case at hand.
To understand the spectrum of short massive operators at strong coupling, we relate it to the spectrum of type II strings in flat space, compactified on S 5 .The relation between conformal dimensions and masses is the one from [1] and the representation theory for spin and R-symmetry is similar to the weak coupling treatment of [11,12].One result of this analysis is that operators on the leading Regge trajectory ℓ = 2δ − 2 are non-degenerate while operators on the next Regge trajectory ℓ = 2δ − 3 have degeneracy 2.
By combining the precise result for the supergravity correlator ⟨O p 1 O p 2 O p 3 O p 4 ⟩ from [13,14] with the flat space limit formula of [15,16] we derive the leading structure constants for the coupling of two arbitrary chiral primaries with the massive short operator O δ,ℓ,[n,0,0] .This result is valid for generic R-charges of the three operators provided p i ≪ λ 1 4 .This restriction implies that in the flat space limit, all momenta in the S 5 directions vanish and all SO(6) representations are KK-modes of the R 5 singlet.One could also study the case p i ∼ λ 1 4 by considering a different flat space limit that acts non-trivially on S 5 , see [17].The structure constants of two chiral primaries and one short massive operator on the leading Regge trajectory have previously been computed in [18][19][20] and our computation reproduces this result in this special case.One simplification of our computation compared to [18][19][20] is that at each step we work at the level of long supersymmetry multiplets.Since the long superconformal multiplet must become the usual long multiplet in the flat space limit, we strip off the corresponding factor at each step: when analysing the spectrum and when considering partial wave decompositions in flat space and AdS.This paper is organised as follows.In section 2 we study the massive spectrum of type II superstrings compactified on S 5 , which we expect to coincide with the spectrum of the corresponding massive operators in N = 4 SYM at strong coupling.In section 3 we review the corresponding partial wave decomposition of the Virasoro-Shapiro amplitude in flat space and in section 4 we use the flat space limit of the conformal partial wave decomposition to compute the structure constants of two chiral primaries and one short massive operator.Section 4.1 discusses the question whether the general structure constants allow us to unmix OPE data from the AdS Virasoro-Shapiro amplitude.In section 5 we compare our structure constants to the previously known result for the special case of operators on the leading Regge trajectory.We conclude in section 6.

Spectrum of massive strings
In this section we study the spectrum of short operators at strong coupling.The simplest example is the Konishi operator whose conformal dimension at large λ is given by More generally, short operators at strong coupling are labelled by their 'level' δ, spin ℓ and SO(6) representation R. At strong coupling and for small quantum numbers ℓ, R (hence the name short2 ) compared to λ 1/4 , their dimension grows as 2) The question we would like to answer is that of their degeneracy for a given spin ℓ and SO(6) representation.Following [1] this maps to a computation of the massive spectrum of type II string theory in flat space.More precisely, the massive spectrum of type II strings on R 1,4 ×S 5 comes in representations of SO(4) × SO( 6) and is expected to match the spectrum of short operators at strong coupling.The masses in flat space m are directly related to the conformal dimensions (2.2) under identification of the quadratic Casimirs of SO(4, 2) × SO( 6) and the 10d Poincare group in the limit where the AdS radius squared The massive spectrum of type II string theory in flat space is given by At each string level δ the SO(9) (the massive little group for R 1,9 ) representations are encoded in the character polynomial where T 1 is the long multiplet in flat space3 and vac δ is given explicitly up to δ = 7 in [11] and a generating function is given in [21].The first few cases are (2.7) In order to extract from this the spectrum of massive string operators in AdS 5 × S5 we will mostly follow [11,12]. 4he first step is to replace the flat long multiplet T 1 in (2.5) by the long superconformal multiplet T sconf .However, as we really want to study the spectrum of superconformal primaries, we will not multiply by T sconf .By considering only vac 2 δ we are directly counting superconformal primaries.We have 5 Next we branch the SO(9) representations into irreducible representations of SO(4) × SO(5), corresponding to the split into AdS 5 and S 5 directions.For the first mass levels this gives vac 2 2 = [2, 2; 0, 0] + [2, 0; 0, 0] + [0, 2; 0, 0] + 2[0, 0; 0, 0] + 2[1, 1; 1, 0] + [0, 0; 2, 0] + [0, 0; 0, 2] .
The full expressions get very lengthy very quickly, but simplify if one looks at specific representations. 6o get the final degeneracy in terms of SO( 6) representations, we compactify five dimensions into S 5 [24].This replaces each SO(5) irrep [m, n] by a Kaluza-Klein tower of SO( 6) representations [11] (2.11)

Massive strings coupling to two chiral primaries
In the remainder of the paper we will study the structure constants of one short massive operator and two chiral primary operators to leading order in 1/λ, so it is important to understand which part of the massive spectrum admits such couplings.To this end we need to refine the matching of Casimirs which we did for short massive operators in (2.3).A [m, 0] of SO (5).Projecting onto these representations we get vac 2  1 |ST T = [0, 0; 0, 0] , vac 2  2 |ST T = [2, 2; 0, 0] + 2[0, 0; 0, 0] +  6),[n,0,0] = n(n + 6 − 2) . (2.13) In the flat space limit the SO(6) Casimir should equal the Casimir for the five spacelike dimensions that are compactified into S 5 , i.e. we can split the 10d momentum k M in flat spacetime with metric (− + . . .+) into and assign separately for the limit R → ∞ (where R is the radius of both AdS 5 and S 5 ) For massive operators this agrees with the assignment [18,19].Chiral primary operators O p are the superconformal primaries of 1/2-BPS superconformal multiplets and their quantum numbers are fixed in terms of a single integer p ≥ 2, namely ∆ = p, ℓ = 0 and their SO(6) representation is [p, 0, 0].(2.15) implies that in the flat space limit the components k m of the momentum vanish This is familiar for the flat space limit of correlators of chiral operators in supersymmetric conformal field theories [25,26].As we will see in section 3, (2.16) implies that to leading order in 1/λ only massive operators from the singlet of SO(5) couple to two chiral primaries.
For this reason we next analyse this part of the massive spectrum in some more detail.

Spectrum from the SO(5) vector
It is instructive to also have a closer look at the contributions of the vector [1, 0] of SO(5), even though they are not exchanged in the four-point functions of chiral primary operators at leading order in 1/λ.We show this part of the spectrum in tables 3 and 4, where each entry comes with the KK tower (

Partial wave expansion in flat space
Next we discuss how the spectrum above contributes to the concrete example the Virasoro-Shapiro amplitude where we use the dimensionless Mandelstam variables The overall delta function δ 16 (Q) contains the polarisation dependence, written in an onshell superspace formalism, and is fixed by supersymmetry.We can directly consider the contribution of exchanged superconformal primaries (as opposed to their full long multiplet) by stripping off this factor.We thus study the function This amplitude has poles at S = δ corresponding to the exchange of states of mass m 2 = 4δ/α ′ .Let us briefly discuss how to decompose the residues at these poles into irreducible representations of SO(4) × SO(5) as listed in (2.9).We begin with the usual construction of SO(9) partial waves.The exchange diagram for a particle at mass level δ and with SO( 9) spin L has the form where the numerator is a sum over on-shell three-point amplitudes defined in terms of an orthonormal basis of polarisation tensors ξ I M 1 ...M L in the representation [L, 0, 0, 0], which are transverse to the momentum of the particle k 1 + k 2 .For the vector representation these polarisations satisfy the completeness relation and the completeness relation for the spin L representation can be written in terms of the same vector polarisations (see [27]) Here we made use of the projector to traceless symmetric rank L tensors of SO(d), which can be most conveniently written by contracting it with vectors x, x ∈ R d (see [28] for details) in terms of a Gegenbauer polynomial By combining (3.4) and (3.7) one concludes that the residues of the amplitude (3.3) have an expansion in terms of the partial waves given by L (δ, T ) , δ = 1, 2, . . . .
In order to find the decomposition into SO(4)×SO( 5) partial waves, we have to implement branching rules as the ones used above in (2.9) in terms of projectors.Labelling an irreducible representation of SO(d) by ρ d , they generally take the form where N [ρ 4 ;ρ 5 ] are the multiplicities of each representation.In terms of projectors, this relation takes the form7 Here bold indices encode multiple indices (I = 1, . . ., 9, i = 1, . . ., 4, m = 5, . . ., 9) and b i,m I,ρ 9 →[ρ 4 ;ρ 5 ],k are fixed by the equation (3.12) itself.For example, the branching rule is written in terms of the projectors for vectors π I,J = δ I,J and singlets π (d) = 1 as In order to find the partial waves for SO(4) × SO(5), we simply have to insert (3.12) into (3.9).In the case where all momenta along the SO(5) directions vanish k m 1,2,3,4 = 0 we see that only the singlet of SO(5) can be exchanged.In this case there is a relation between SO(9) and SO(4) partial waves L−2k (S, T ) , Using the explicit form for the partial waves for d = 4 and the explicit form of the residues at S = δ we obtain which allows to solve for the coefficients a δ,ℓ for δ = 1, 2, • • • .These coefficients will serve as input to compute N = 4 SYM structure constants at leading order in 1/λ.

Partial wave expansion in AdS
In this section we consider the four-point correlator of arbitrary chiral primary operators in N = 4 SYM theory at leading non-trivial order in a 1/c expansion, where c = (N 2 − 1)/4 is the central charge.This is related to (3.1) in the flat space limit.More precisely, consider the four-point function of four scalar superconformal primaries of 1/2-BPS multiplets O p i (x i , y i ) with conformal dimension and R-charge p i where 4 i=1 p i is even y i ∈ R 6 are polarisations satisfying y 2 i = 0 that we contracted with the p i R-symmetry indices of each operator.Similar to the amplitude in flat space, we strip off the part of the correlator that is fixed by supersymmetry and focus on the reduced correlator T (z, z, α, α).The reduced correlator admits an expansion into superconformal long multiplets, labelled by their superconformal primary with dimension ∆, spin ℓ and SO(6) representation R,9 For the precise relation between the full and reduced correlator and the definition of the superconformal blocks, see appendix A. The cross-ratios are given by and r = p 21 = p 2 − p 1 , s = p 34 .In the flat space limit, the reduced correlator is related to the reduced amplitude (3.3).This relation is most explicit in terms of the Mellin amplitude M (s, t, α, α), defined through the Mellin transform where we have introduced Mellin variables s, t, u satisfying s + t + u = 4 i=1 p i − 4 and (4.5) In the flat space limit we are considering, where s, t become large while α, α are kept fixed, we only access the singlet of SO(5).In this limit the whole Mellin amplitude must factorise into a polynomial P (α, α) which encodes the KK modes for the singlet of SO( 5), times a function M (s, t) of the Mellin variables which encodes the AdS 5 directions lim s,t→∞ M (s, t, α, α) = M (s, t)P (α, α) . (4.6) The AdS 5 factor M (s, t) is related to the flat space amplitude (3.3) by the flat space limit From this we immediately find where we use the bases of crossing-symmetric polynomials To determine the factor P (α, α) we can make use of the known supergravity term in the Mellin amplitude.Taking the large s, t limit of the result from [13,14] (assuming without loss of generality p 21 ≥ 0, p 43 ≥ 0 and p 43 ≥ p 21 ) and comparing to (4.6) we find that lim s,t→∞ with see also [26].Next we want to extract from this the OPE coefficients in the large λ limit of the expansion (4.2).For the AdS factor this can be done by taking the flat space limit of the conformal partial wave decomposition where M ν,ℓ (s, t) is the Mellin transform of a conformal partial wave as defined in [29] and b ℓ (ν 2 ) has poles at the conformal dimensions of superconformal primaries with residues where As shown in [2,29,30], we can use 15) together with the fact that the flat space limit of M ν,ℓ (s, t) is the 5D flat space partial wave which appears in (3.18) The S 5 factor simply needs to be expanded into spherical harmonics where n increases in steps of 2 and the coefficients can be written in terms of sphere overlap integrals defined in (B.9) below

.18)
In this way we find the leading OPE coefficients where we defined Let us make some comments on this result.The full supergravity correlator of [13] that we used to compute the limit (4.10) decomposes into all SO(6) representations of the form [n − m, m, m], however (4.17) shows that only the symmetric tensors [n, 0, 0] survive the flat space limit.This is consistent with the fact that this limit explores only the singlet of SO (5).Moreover, (4.18) implies that P (α, α) is the sum over products of two sphere overlap integrals. 10We state this more explicitly in (B.11) below.

Operator mixing
We can now turn to the question on whether studying these general correlators allows us to unmix the OPE data for heavy operators extracted from these correlators.The result (4.19) should really be seen as a sum over products of OPE coefficients for multiple degenerate operators.Let us introduce a label I for these species and write more precisely where N (∆, ℓ, n) is the number of superprimaries with the same quantum numbers (∆, ℓ, n) at strong coupling.
The triangle inequality for these two vectors reads (A • B) 2 ≤ |A| 2 |B| 2 .However, equation (4.19) implies that actually This implies that A I and B I are parallel vectors.Thus we conclude that where and D I is a unit vector |D| 2 = 1.It is not possible to say more about the individual components D I , i.e. we cannot determine them at leading order in a way similar to [4][5][6] for double trace operators.The reason for this is ultimately that all the KK modes originate from the same amplitude in 10-dimensional flat space.We expect that it would be possible to unmix the OPE coefficients by considering correlators with different flat space analogues, for example by taking some of the external operators to be massive as well.

Comparison with previous results
In this section we compare our results with those of [20], where the three-point functions for two chiral primaries and one massive string operator on the leading Regge trajectory were computed from the flat space limit.The method of [20] requires that the dimensions of all three operators in the threepoint function (p 1 , p 2 and ∆) are large, of order λ 1/4 .On first sight this limit seems to be incompatible with our assumption that p 1 , p 2 are much smaller than λ 1/4 in the large λ limit.This assumption led to the conclusion that to leading order in 1/λ only the KK modes of the singlet of SO( 5) couple to two chiral primaries, see section 2.1.The reason that we can compare results anyway is that, as mentioned in section 2.3, the leading Regge trajectory arises only from the singlet of SO (5) in general, so we can lift the restriction p i ≪ λ 1/4 when considering the structure constants of these operators.In order to compare, it is thus enough to take the additional limit ∆ ≫ p 1 , p 2 of the results of [20].
The structure constants given in [20] are those for the maximal superconformal descendants of the operators we considered above.These operators are conformal, but not superconformal, primaries.For the external operators these are given by and they are scalars with conformal dimension p + 2 in the [p − 2, 0, 0] of SO (6).For p = 2 this operator is the Lagrangian of the theory.We consider the decomposition of the crossing symmetric correlator into bosonic conformal blocks ⟨L p (x 1 )L p (x 2 )L p (x 3 )L p (x 4 )⟩ = G(z, z, α, α) x 4p 12 x 4p 34 = 1 (5. 2) The OPE coefficients C ∆,ℓ,n were computed in [20] for operators on the leading Regge trajectory, which are descendants of the massive short superconformal primaries discussed above Q 4 Q 4 O δ,ℓ,[n,0,0] and have dimension ∆+4, spin ℓ+4 and R-charge n.In terms of the quantum numbers ) ) 11 The factor 2

3−S 2
accounts for our different normalisation of conformal blocks and the whole correlator compared to [20,29,30].
component that contributes to the leading Regge trajectory in the sense of the expansion (5.2).Furthermore we can take ∆ to be very large as before.The result is we can now check that for operators on the leading Regge trajectory, our result (4.19) agrees with (5.4) once we expand at large ∆ 2 ) 2 (5.16)

Conclusions
In this paper we studied the spectrum and structure constants of massive short operators in planar N = 4 SYM theory at strong coupling by considering strings on AdS 5 × S 5 in the flat space limit.It would be very interesting to compare the spectrum of degeneracies with explicit results from integrability.Where data from integrability is available, on the leading Regge trajectory, the degeneracies do match.More generally, we cannot rule out the possibility that two different operators at finite λ become identical in the strict λ = ∞ limit, and are seen as a single operator in flat space.It would be interesting to explore this.It would also be very interesting to reproduce our results for structure constants, from integrability.Note in particular that our results are valid for large R−charges (as long as they are smaller than λ 1/4 ), where integrability computations are expected to simplify.See [10] for relevant progress in this direction.
Our result for the structure constants of two chiral primaries and one massive short operator implies that the OPE data contained in the AdS Virasoro-Shapiro amplitude of [2,3] cannot be unmixed at leading order by considering generalisations to correlators of arbitrary chiral primary operators.Possible alternatives are to unmix the OPE data at a higher order in the 1/ √ λ expansion or to consider other correlators, for example ones with external massive operators.
Of course it is possible to apply the methods of [2,3] to more general correlators to compute corrections to the OPE data of massive operators with R-charge.We plan to report on this in a forthcoming paper.
Another possible avenue is to implement a flat space limit that probes large R-charges of the external operators p i ∼ λ 1 4 and accesses the full spectrum of strings in 10 dimensions.

A.2 Superconformal blocks
The superconformal multiplets that can be exchanged in the four-point functions we will consider can be labelled by the conformal dimension ∆, spin ℓ and SO( 6) representation [n − m, m, m] of the superconformal primary of the multiplet.Massive string states are in long multiplets, which have the following contributions to the unit contribution, chiral correlator and reduced correlator These definitions are in terms of the usual SO(4, 2) conformal blocks