††institutetext: Mathematical Institute, University of Oxford, Woodstock Road,
Oxford, OX2 6GG, UK

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

Luis F. Alday, Tobias Hansen and Joao A. Silva<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

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 $n\ll\lambda^{1/4}$.

##  1 Introduction

It has been known since the work of Gubser, Klebanov and Polyakov that massive
string states in $AdS_{5}\times S^{5}$ couple to short single-trace operators
in $\mathcal{N}=4$ SYM Gubser:1998bc . These operators have conformal
dimensions which grow as $\Delta\sim 2\sqrt{\delta}\lambda^{1/4}$ at strong
t’Hooft coupling $\lambda$, where $\delta=1,2,\ldots$ is the string mass
level. In this paper we revisit these operators
$\mathcal{O}_{\delta,\ell,{\cal R}}$ for general spin $\ell$ and in the
$SO(6)$ representation ${\cal R}$.

One of the reasons we are interested in these operators is their appearance in
stringy correlators. In Alday:2022uxp ; Alday:2022xwz  we recently derived the
full $1/\sqrt{\lambda}$ correction to the Virasoro-Shapiro amplitude in
$AdS_{5}\times S^{5}$, i.e. the correlator
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$
at large $N$ and strong coupling including all stringy corrections, where
$\mathcal{O}_{2}$ 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
$\mathcal{O}_{\delta,\ell,[0,0,0]}$. A natural question is whether these
averages can be unmixed by considering more general correlators
$\langle\mathcal{O}_{p_{1}}\mathcal{O}_{p_{2}}\mathcal{O}_{p_{3}}\mathcal{O}_{p_{4}}\rangle$,
where $\mathcal{O}_{p_{i}}$ is a chiral primary of dimension $p_{i}$.111A
similar unmixing was carried out for light double trace operators in
supergravity in Alday:2017xua ; Aprile:2017bgs ; Aprile:2017xsp . 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
Gromov:2011de ; Basso:2011rs ; Gromov:2011bz , including the familiar Konishi
operator. These results agree with Alday:2022uxp ; Alday:2022xwz . One
advantage of our approach however, is that we can compute structure constants,
for which integrability techniques are still under development, see however
Basso:2022nny  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
Gubser:1998bc  and the representation theory for spin and R-symmetry is
similar to the weak coupling treatment of Bianchi:2003wx ; Beisert:2003te .
One result of this analysis is that operators on the leading Regge trajectory
$\ell=2\delta-2$ are non-degenerate while operators on the next Regge
trajectory $\ell=2\delta-3$ have degeneracy 2.

By combining the precise result for the supergravity correlator
$\langle\mathcal{O}_{p_{1}}\mathcal{O}_{p_{2}}\mathcal{O}_{p_{3}}\mathcal{O}_{p_{4}}\rangle$
from Rastelli:2016nze ; Aprile:2018efk  with the flat space limit formula of
Penedones:2010ue ; Fitzpatrick:2011hu  we derive the leading structure
constants for the coupling of two arbitrary chiral primaries with the massive
short operator $\mathcal{O}_{\delta,\ell,[n,0,0]}$. This result is valid for
generic R-charges of the three operators provided
$p_{i}\ll\lambda^{\frac{1}{4}}$. 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 $\mathbb{R}^{5}$ singlet. One could also
study the case $p_{i}\sim\lambda^{\frac{1}{4}}$ by considering a different
flat space limit that acts non-trivially on $S^{5}$, see Aprile:2020luw .

The structure constants of two chiral primaries and one short massive operator
on the leading Regge trajectory have previously been computed in
Minahan:2012fh ; Bargheer:2013faa ; Minahan:2014usa  and our computation
reproduces this result in this special case. One simplification of our
computation compared to Minahan:2012fh ; Bargheer:2013faa ; Minahan:2014usa
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 $\mathcal{N}=4$
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.

##  2 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 $\lambda$ is given by

$\Delta_{\cal K}=2\lambda^{1/4}-2+\frac{2}{\lambda^{1/4}}+\cdots.$ (1)

More generally, short operators at strong coupling are labelled by their
’level’ $\delta$, spin $\ell$ and $SO(6)$ representation ${\cal R}$. At strong
coupling and for small quantum numbers $\ell,{\cal R}$ (hence the name
short222Despite being called short, these operators are in long superconformal
multiplets.) compared to $\lambda^{1/4}$, their dimension grows as

$\Delta_{\delta,\ell,{\cal R}}=2\sqrt{\delta}\lambda^{1/4}+\cdots.$ (2)

The question we would like to answer is that of their degeneracy for a given
spin $\ell$ and $SO(6)$ representation. Following Gubser:1998bc  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
$\mathbb{R}^{1,4}\times S^{5}$ comes in representations of $SO(4)\times SO(6)$
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) under identification of the quadratic Casimirs of $SO(4,2)\times SO(6)$
and the 10d Poincare group in the limit where the AdS radius squared
$R^{2}=\alpha^{\prime}\sqrt{\lambda}$ is very large

$-\Delta^{2}=-R^{2}m^{2}+\ldots\,.$ (3)

The massive spectrum of type II string theory in flat space is given by

$m^{2}=\frac{4\delta}{\alpha^{\prime}}\,,\qquad\delta=1,2,3,\ldots\,.$ (4)

At each string level $\delta$ the $SO(9)$ (the massive little group for
$\mathbb{R}^{1,9}$) representations are encoded in the character polynomial

$T_{\delta}=T_{1}\otimes(\text{vac}_{\delta}\otimes\text{vac}_{\delta})\,,$
(5)

where $T_{1}$ is the long multiplet in flat space333We label irreducible
representations of $SO(d)$ (and their characters) by $[a_{1},\ldots,a_{r}]$,
where $r$ is the rank of $SO(d)$ and $a_{i}$ are non-negative integers which
give the coefficients in the decomposition of the highest weight vectors into
fundamental weights. Similarly irreducible representations of $SO(d_{1})\times
SO(d_{2})$ are labelled by $[a_{1},\ldots,a_{r_{1}};b_{1},\ldots,b_{r_{2}}]$.

$T_{1}=\left([2,0,0,0]+[0,0,1,0]+[1,0,0,1]\right)^{2}=\left({\bf
44+84+128}\right)^{2}\,,$ (6)

and $\text{vac}_{\delta}$ is given explicitly up to $\delta=7$ in
Bianchi:2003wx  and a generating function is given in Hanany:2010da . The
first few cases are

$\displaystyle{\rm vac}_{1}={}$ $\displaystyle[0,0,0,0]={\bf 1}\,,$ (7)
$\displaystyle{\rm vac}_{2}={}$ $\displaystyle[1,0,0,0]={\bf 9}\,,$
$\displaystyle{\rm vac}_{3}={}$ $\displaystyle[2,0,0,0]+[0,0,0,1]={\bf
44}+{\bf 16}\,,$ $\displaystyle{\rm vac}_{4}={}$
$\displaystyle[3,0,0,0]+[1,0,0,1]+[0,1,0,0]+[1,0,0,0]={\bf 156}+{\bf 128}+{\bf
36}+{\bf 9}\,.$

In order to extract from this the spectrum of massive string operators in
$\text{AdS}_{5}\times\text{S}^{5}$ we will mostly follow Bianchi:2003wx ;
Beisert:2003te .444A similar analysis was done for $SO(6)$ singlets in
Antunes:2020pof .

The first step is to replace the flat long multiplet $T_{1}$ in (5) by the
long superconformal multiplet $\mathcal{T}_{\text{sconf}}$. However, as we
really want to study the spectrum of superconformal primaries, we will not
multiply by $\mathcal{T}_{\text{sconf}}$. By considering only ${\rm
vac}_{\delta}^{2}$ we are directly counting superconformal primaries. We
have555We use SageMath sagemath  to compute tensor products and branchings.

$\displaystyle{\rm vac}_{1}^{2}={}$ $\displaystyle[0,0,0,0]\,,$ (8)
$\displaystyle{\rm vac}_{2}^{2}={}$
$\displaystyle[2,0,0,0]+[0,1,0,0]+[0,0,0,0]\,,$ $\displaystyle{\rm
vac}_{3}^{2}={}$
$\displaystyle[4,0,0,0]+[2,1,0,0]+2[2,0,0,1]+[0,2,0,0]+[2,0,0,0]+2[1,0,0,1]$
$\displaystyle+[0,0,0,2]+[0,0,1,0]+2[0,1,0,0]+[1,0,0,0]+2[0,0,0,0]\,.$

Next we branch the $SO(9)$ representations into irreducible representations of
$SO(4)\times SO(5)$, corresponding to the split into $AdS_{5}$ and $S^{5}$
directions. For the first mass levels this gives

$\displaystyle{\rm vac}_{1}^{2}={}$ $\displaystyle[0,0;0,0]\,,$ (9)
$\displaystyle{\rm vac}_{2}^{2}={}$
$\displaystyle[2,2;0,0]+[2,0;0,0]+[0,2;0,0]+2[0,0;0,0]+2[1,1;1,0]+[0,0;2,0]+[0,0;0,2]\,.$

The full expressions get very lengthy very quickly, but simplify if one looks
at specific representations.666For instance, we can focus on symmetric
traceless representations. These are the $[\ell,\ell]$ of $SO(4)$ and the
$[m,0]$ of $SO(5)$. Projecting onto these representations we get
$\displaystyle{\rm vac}_{1}^{2}|_{STT}={}$ $\displaystyle[0,0;0,0]\,,$ (10)
$\displaystyle{\rm vac}_{2}^{2}|_{STT}={}$
$\displaystyle[2,2;0,0]+2[0,0;0,0]+2[1,1;1,0]+[0,0;2,0]\,,$ $\displaystyle{\rm
vac}_{3}^{2}|_{STT}={}$
$\displaystyle[4,4;0,0]+4[2,2;0,0]+2[1,1;0,0]+6[0,0;0,0]+2[3,3;1,0]+8[1,1;1,0]$
$\displaystyle+2[0,0;1,0]+3[2,2;2,0]+4[0,0;2,0]+2[1,1;3,0]+[0,0;4,0]\,.$

To get the final degeneracy in terms of $SO(6)$ representations, we compactify
five dimensions into $\text{S}^{5}$ Salam:1981xd . This replaces each $SO(5)$
irrep $[m,n]$ by a Kaluza-Klein tower of $SO(6)$ representations
Bianchi:2003wx

${\rm
KK}_{[m,n]}=\sum_{r=0}^{m}\sum_{s=0}^{n}\sum_{p=m-r}^{\infty}\left[p,\,r+n-s,\,r+s\right]+\sum_{r=0}^{m-1}\sum_{s=0}^{n-1}\sum_{p=m-r-1}^{\infty}\left[p,\,r+n-s,\,r+s+1\right]\,.$
(11)

###  2.1 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
$1/\lambda$, 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 (3). A general operator
with dimension $\Delta$, spin $\ell$ and $SO(6)$ representation $[n,0,0]$ has
the $SO(4,2)$ Casimir

$c^{2}_{SO(4,2),\Delta,\ell}=\Delta(4-\Delta)+\ell(\ell+4-2)\,,$ (12)

and the $SO(6)$ Casimir

$c^{2}_{SO(6),[n,0,0]}=n(n+6-2)\,.$ (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 $(-+\ldots+)$
into

$k^{M}=(k^{\mu},k^{m})\,,\qquad\mu=0,\ldots 4\,,\quad m=5\ldots 9\,,$ (14)

and assign separately for the limit $R\to\infty$ (where $R$ is the radius of
both $AdS_{5}$ and $S^{5}$)

$\displaystyle c^{2}_{SO(4,2),\Delta,\ell}$
$\displaystyle=R^{2}k_{\mu}k^{\mu}+\ldots\,,$ (15) $\displaystyle
c^{2}_{SO(6),[n,0,0]}$ $\displaystyle=R^{2}k_{m}k^{m}+\ldots\,.$

For massive operators this agrees with the assignment Minahan:2012fh ;
Bargheer:2013faa .

Chiral primary operators $\mathcal{O}_{p}$ are the superconformal primaries of
$1/2$-BPS superconformal multiplets and their quantum numbers are fixed in
terms of a single integer $p\geq 2$, namely $\Delta=p$, $\ell=0$ and their
$SO(6)$ representation is $[p,0,0]$. (15) implies that in the flat space limit
the components $k^{m}$ of the momentum vanish

$k^{m}=O\left(\frac{1}{R}\right)\,.$ (16)

This is familiar for the flat space limit of correlators of chiral operators
in supersymmetric conformal field theories Chester:2018dga ; Alday:2020dtb .
As we will see in section 3, (16) implies that to leading order in $1/\lambda$
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.

###  2.2 Spectrum from the $SO(5)$ singlet

We show the spectrum that arises from the singlet of $SO(5)$ separately for
even and odd spins in tables 1 and 2, where we use the shorthand notation
$[\ell]\equiv[\ell,\ell]$ for traceless symmetric $SO(4)$ representations of
spin $\ell$. For each entry in these tables there is a tower of KK modes in
the $SO(6)$ representations

$\text{KK}_{[0,0]}=\sum\limits_{q=0}^{\infty}[q,0,0]\,.$ (17)

We observe that the multiplicity for superconformal primaries on the leading
even spin Regge trajectory is 1 and on the next even spin Regge trajectory it
is 6 for all but the first few operators.

$\delta$ |  $\text{vac}_{\delta}^{2}$ projected to $SO(4)|_{\text{even rank STT}}\times SO(5)|_{[0,0]}$
---|---
1 | [0]
2 | [2] + 2[0]
3 | [4] + 4[2] + 6[0]
4 | [6] + 6[4] + 24[2] + 22[0]
5 | [8] + 6[6] + 40[4] + 157[2] + 99[0]
6 | [10]+ 6[8] + 52[6] + 331[4] + 1104[2] + 547[0]
7 | [12]+ 6[10]+ 58[8] + 461[6] + 2570[4] + 7365[2] + 3112[0]
Table 1: The even spin spectrum arising from the $[0,0]$ of $SO(5)$. Each entry corresponds to the KK tower of $SO(6)$ representations (17). $\delta$ |  $\text{vac}_{\delta}^{2}$ projected to $SO(4)|_{\text{odd rank STT}}\times SO(5)|_{[0,0]}$
---|---
1 | 0
2 | 0
3 | 2[1]
4 | 8[3]+24[1]
5 | 12[5]+80[3]+172[1]
6 | 14[7]+128[5]+664[3]+1154[1]
7 | 14[9]+156[7]+1120[5]+4868[3]+7214[1]
Table 2: The odd spin spectrum arising from the $[0,0]$ of $SO(5)$. Each entry
corresponds to the KK tower of $SO(6)$ representations (17).

###  2.3 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 $1/\lambda$.
We show this part of the spectrum in tables 3 and 4, where each entry comes
with the KK tower

$\text{KK}_{[1,0]}=\sum\limits_{q=1}^{\infty}[q,0,0]+\sum\limits_{q=0}^{\infty}[q,1,1]\,.$
(18)

There are several important observations to make. The first one is that the
only $SO(6)$ representation that is entirely described by the singlet of
$SO(5)$ is the singlet $[0,0,0]$. Further, the leading Regge trajectory with
$\ell=2\delta-2$ (see table 1) receives only contributions from the singlet of
$SO(5)$. However, the leading odd spin Regge trajectory is $\ell=2\delta-3$
(see table 4) and arises entirely from the vector of $SO(5)$. The degeneracy
for operators on this trajectory is 2.

$\delta$ |  $\text{vac}_{\delta}^{2}$ projected to $SO(4)|_{\text{even rank STT}}\times SO(5)|_{[1,0]}$
---|---
1 | 0
2 | 0
3 | 2[0]
4 | 2[4] + 24[2] + 21[0]
5 | 2[6] + 46[4] + 244[2] + 157[0]
6 | 2[8] + 60[6] + 506[4] + 2054[2] + 1095[0]
7 | 2[10]+ 62[8] + 692[6] + 4689[4] + 15505[2] + 7055[0]
Table 3: The even spin spectrum arising from the $[1,0]$ of $SO(5)$. Each entry corresponds to the KK tower of $SO(6)$ representations (18). $\delta$ |  $\text{vac}_{\delta}^{2}$ projected to $SO(4)|_{\text{odd rank STT}}\times SO(5)|_{[1,0]}$
---|---
1 | 0
2 | 2[1]
3 | 2[3]+8[1]
4 | 2[5]+14[3]+46[1]
5 | 2[7]+16[5]+126[3]+330[1]
6 | 2[9]+18[7]+188[5]+1172[3]+2382[1]
7 | 2[11]+18[9]+228[7]+1938[5]+9714[3]+16210[1]
Table 4: The odd spin spectrum arising from the $[1,0]$ of $SO(5)$. Each entry
corresponds to the KK tower of $SO(6)$ representations (18).

##  3 Partial wave expansion in flat space

Next we discuss how the spectrum above contributes to the concrete example the
Virasoro-Shapiro amplitude

$\delta^{16}(Q)\frac{\Gamma\left(-S\right)\Gamma\left(-T\right)\Gamma\left(-U\right)}{\Gamma\left(S+1\right)\Gamma\left(T+1\right)\Gamma\left(U+1\right)}\,,$
(19)

where we use the dimensionless Mandelstam variables

$\begin{gathered}S=-\frac{\alpha^{\prime}}{4}(k_{1}+k_{2})^{2}\,,\qquad
T=-\frac{\alpha^{\prime}}{4}(k_{1}+k_{3})^{2}\,,\qquad
U=-\frac{\alpha^{\prime}}{4}(k_{1}+k_{4})^{2}\,.\end{gathered}$ (20)

The overall delta function $\delta^{16}(Q)$ contains the polarisation
dependence, written in an on-shell 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

$A(S,T)=\frac{\Gamma\left(-S\right)\Gamma\left(-T\right)\Gamma\left(-U\right)}{\Gamma\left(S+1\right)\Gamma\left(T+1\right)\Gamma\left(U+1\right)}\,.$
(21)

This amplitude has poles at $S=\delta$ corresponding to the exchange of states
of mass $m^{2}=4\delta/\alpha^{\prime}$. Let us briefly discuss how to
decompose the residues at these poles into irreducible representations of
$SO(4)\times SO(5)$ as listed in (9). We begin with the usual construction of
$SO(9)$ partial waves. The exchange diagram for a particle at mass level
$\delta$ and with $SO(9)$ spin $L$ has the form

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(22)

where the numerator is a sum over on-shell three-point amplitudes

$A_{3}^{I}(k_{1},k_{2})=\xi^{I}_{M_{1}\ldots
M_{L}}\prod\limits_{i=1}^{L}\sqrt{\frac{\alpha^{\prime}}{2}}(k_{1}-k_{2})^{M_{i}}\,,$
(23)

defined in terms of an orthonormal basis of polarisation tensors
$\xi^{I}_{M_{1}\ldots M_{L}}$ 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

$\sum\limits_{I=1}^{9}\xi_{M}^{I}\xi_{N}^{I}=\eta_{MN}-\frac{(k_{1}+k_{2})_{M}(k_{1}+k_{2})_{N}}{(k_{1}+k_{2})^{2}}\,,$
(24)

and the completeness relation for the spin $L$ representation can be written
in terms of the same vector polarisations (see Boels:2014dka )

$\sum\limits_{I}\xi_{M_{1}\ldots M_{L}}^{I}\xi_{N_{1}\ldots
N_{L}}^{I}=\pi^{(9)}_{I_{1}\ldots I_{L},J_{1},\ldots
J_{L}}\prod\limits_{i=1}^{L}\xi_{M_{i}}^{I_{i}}\xi_{N_{i}}^{J_{i}}\,.$ (25)

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
$x,\overline{x}\in\mathbb{R}^{d}$ (see Costa:2016hju  for details) in terms of
a Gegenbauer polynomial

$\pi^{(d)}_{\ell}(x,\overline{x})\equiv x^{I_{1}}\ldots
x^{I_{\ell}}\pi^{(d)}_{I_{1}\ldots I_{\ell},J_{1}\ldots
J_{\ell}}\overline{x}^{J_{1}}\ldots\overline{x}^{J_{\ell}}=\frac{\ell!}{2^{\ell}(\frac{d}{2}-1)_{\ell}}\left(x^{2}\overline{x}^{2}\right)^{\frac{\ell}{2}}C^{(\frac{d}{2}-1)}_{\ell}\left(\frac{x\cdot\overline{x}}{\sqrt{x^{2}\overline{x}^{2}}}\right)\,.$
(26)

By combining (22) and (25) one concludes that the residues of the amplitude
(21) have an expansion in terms of the partial waves

$\displaystyle P^{(d)}_{\ell}(S,T)={}$ $\displaystyle\pi^{(d)}_{I_{1}\ldots
I_{\ell},J_{1}\ldots
J_{\ell}}\prod\limits_{i=1}^{\ell}\frac{\alpha^{\prime}}{4S}\xi_{M_{i}}^{I_{i}}(k_{1}-k_{2})^{M_{i}}\xi_{N_{i}}^{J_{i}}(k_{4}-k_{3})^{N_{i}}\,,$
(27) $\displaystyle={}$
$\displaystyle\frac{\ell!}{2^{\ell}(\frac{d}{2}-1)_{\ell}}C^{(\frac{d}{2}-1)}_{\ell}\left(1+\frac{2T}{S}\right)\,,$

given by

$\mathop{\mathrm{Res}}\limits_{S=\delta}A(S,T)=\sum\limits_{L=0}^{2(\delta-1)}A_{\delta,L}P^{(9)}_{L}(\delta,T)\,,\qquad\delta=1,2,\ldots\,.$
(28)

In order to find the decomposition into $SO(4)\times SO(5)$ partial waves, we
have to implement branching rules as the ones used above in (9) in terms of
projectors. Labelling an irreducible representation of $SO(d)$ by $\rho_{d}$,
they generally take the form

$\rho_{9}=\sum\limits_{[\rho_{4};\rho_{5}]\in\rho_{9}}N_{[\rho_{4};\rho_{5}]}[\rho_{4};\rho_{5}]\,,$
(29)

where $N_{[\rho_{4};\rho_{5}]}$ are the multiplicities of each representation.
In terms of projectors, this relation takes the form777This relation also
exists for more general representations than traceless symmetric tensors.

$\pi^{\rho_{9}}_{\mathbf{I},\mathbf{J}}=\sum\limits_{[\rho_{4};\rho_{5}]\in\rho_{9}}\sum\limits_{k=1}^{N_{[\rho_{4};\rho_{5}]}}b^{\mathbf{i},\mathbf{m}}_{\mathbf{I},\rho_{9}\to[\rho_{4};\rho_{5}],k}\,\pi^{\rho_{4}}_{\mathbf{i},\mathbf{j}}\,\pi^{\rho_{5}}_{\mathbf{m},\mathbf{n}}\,b^{\mathbf{j},\mathbf{n}}_{\mathbf{J},\rho_{9}\to[\rho_{4};\rho_{5}],k}\,.$
(30)

Here bold indices encode multiple indices ($I=1,\ldots,9$, $i=1,\ldots,4$,
$m=5,\ldots,9$)

$\pi^{\rho_{9}}_{\mathbf{I},\mathbf{J}}\equiv\pi^{(9)}_{I_{1}\ldots,J_{1}\ldots}\,,\qquad\pi^{\rho_{4}}_{\mathbf{i},\mathbf{j}}\equiv\pi^{(4)}_{i_{1}\ldots,j_{1}\ldots}\,,\qquad\pi^{\rho_{5}}_{\mathbf{m},\mathbf{n}}\equiv\pi^{(5)}_{m_{1}\ldots,n_{1}\ldots}\,,$
(31)

and $b^{\mathbf{i},\mathbf{m}}_{\mathbf{I},\rho_{9}\to[\rho_{4};\rho_{5}],k}$
are fixed by the equation (30) itself. For example, the branching rule

$[1,0,0,0]=[1,1;0,0]+[0,0;1,0]\,,$ (32)

is written in terms of the projectors for vectors
$\pi^{(d)}_{I,J}=\delta_{I,J}$ and singlets $\pi^{(d)}=1$ as

$\pi^{[1,0,0,0]}_{I,J}=\delta_{I}^{i}\pi^{[1,1]}_{i,j}\pi^{[0,0]}\delta_{J}^{j}+\delta_{I}^{m}\pi^{[0,0]}\pi^{[1,0]}_{m,n}\delta_{J}^{n}=\delta_{I,J}\,.$
(33)

In order to find the partial waves for $SO(4)\times SO(5)$, we simply have to
insert (30) into (27). In the case where all momenta along the $SO(5)$
directions vanish $k_{1,2,3,4}^{m}=0$ 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

$P^{(9)}_{L}(S,T)\bigg{|}_{k_{1,2,3,4}^{m}=0}=\sum\limits_{k=0}^{\lfloor\frac{L}{2}\rfloor}B_{S,k}P^{(4)}_{L-2k}(S,T)\,,$
(34)

and (28) becomes a decomposition into 5D Lorentzian partial waves888The
coefficients $a_{\delta,\ell}$ are related to the coefficients
$f_{0}(\delta,\ell)$ introduced in Alday:2022uxp  by
$a_{\delta,\ell}=\frac{2^{\ell}}{\delta^{2}(\ell+1)}f_{0}(\delta,\ell)\,.$
(35)

$\mathop{\mathrm{Res}}\limits_{S=\delta}A(S,T)=\sum\limits_{\ell=0}^{2(\delta-1)}a_{\delta,\ell}P^{(4)}_{\ell}(\delta,T)\,,\qquad\delta=1,2,\ldots\,.$
(36)

Using the explicit form for the partial waves for $d=4$

$P^{(4)}_{\ell}(S,T)=\frac{\ell+1}{2^{\ell}}\,_{2}F_{1}\left(-\ell,\ell+2;\frac{3}{2};-\frac{T}{S}\right)\,,$
(37)

and the explicit form of the residues at $S=\delta$ we obtain

$\frac{(T+1)_{\delta-1}^{2}}{\Gamma(\delta+1)^{2}}=\sum_{\ell=0}^{2(\delta-1)}a_{\delta,\ell}\frac{\ell+1}{2^{\ell}}\,_{2}F_{1}\left(-\ell,\ell+2;\frac{3}{2};-\frac{T}{\delta}\right)\,,$
(38)

which allows to solve for the coefficients $a_{\delta,\ell}$ for
$\delta=1,2,\cdots$. These coefficients will serve as input to compute
${\mathcal{N}}=4$ SYM structure constants at leading order in $1/\lambda$.

##  4 Partial wave expansion in AdS

In this section we consider the four-point correlator of arbitrary chiral
primary operators in ${\mathcal{N}}=4$ 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 (19) in the flat space limit. More precisely, consider the four-
point function of four scalar superconformal primaries of $1/2$-BPS multiplets
$\mathcal{O}_{p_{i}}(x_{i},y_{i})$ with conformal dimension and R-charge
$p_{i}$ where $\sum_{i=1}^{4}p_{i}$ is even

$\langle\mathcal{O}_{p_{1}}(x_{1},y_{1})\cdots\mathcal{O}_{p_{4}}(x_{4},y_{4})\rangle\,.$
(39)

$y_{i}\in\mathbb{R}^{6}$ are polarisations satisfying $y_{i}^{2}=0$ 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
$\mathcal{T}(z,\overline{z},\alpha,{\overline{\alpha}})$. The reduced
correlator admits an expansion into superconformal long multiplets, labelled
by their superconformal primary with dimension $\Delta$, spin $\ell$ and
$SO(6)$ representation ${\cal R}$,999We suppress an overall factor $1/c$.

$\displaystyle\mathcal{T}(z,\overline{z},{\alpha},{\overline{\alpha}})={}$
$\displaystyle\sum\limits_{\Delta,\ell,{\cal
R}}C^{\\{p_{1}p_{2}\\}}_{\Delta,\ell,{\cal
R}}C^{\\{p_{3}p_{4}\\}}_{\Delta,\ell,{\cal
R}}G^{r,s}_{\Delta+4,\ell}(z,\overline{z})Z_{{\cal
R}}^{r,s}({\alpha},{\overline{\alpha}})\,.$ (40)

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

$\displaystyle
U=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}=z\overline{z},$
$\displaystyle
V=\frac{x_{23}^{2}x_{14}^{2}}{x_{13}^{2}x_{24}^{2}}=(1-z)(1-\overline{z}),$
(41)
$\displaystyle\frac{1}{\sigma}=\frac{y_{12}y_{34}}{y_{13}y_{24}}=\alpha\overline{\alpha},$
$\displaystyle\frac{\tau}{\sigma}=\frac{y_{23}y_{14}}{y_{13}y_{24}}=(1-\alpha)(1-\overline{\alpha})\,,$

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 (21). This relation is most
explicit in terms of the Mellin amplitude
$M(s,t,{\alpha},{\overline{\alpha}})$, defined through the Mellin transform

$\mathcal{T}(z,\overline{z},{\alpha},{\overline{\alpha}})=\int_{-i\infty}^{i\infty}\frac{dsdt}{(4\pi
i)^{2}}U^{s/2+2}V^{\frac{t-p_{2}-p_{3}}{2}}\Gamma_{p_{1}p_{2}p_{3}p_{4}}M(s,t,{\alpha},{\overline{\alpha}})\,,$
(42)

where we have introduced Mellin variables $s,t,u$ satisfying
$s+t+u=\sum_{i=1}^{4}p_{i}-4$ and

$\displaystyle\Gamma_{p_{1}p_{2}p_{3}p_{4}}={}$
$\displaystyle\Gamma\left(\frac{p_{1}+p_{2}-s}{2}\right)\Gamma\left(\frac{p_{3}+p_{4}-s}{2}\right)\Gamma\left(\frac{p_{1}+p_{4}-t}{2}\right)\Gamma\left(\frac{p_{2}+p_{3}-t}{2}\right)$
(43)
$\displaystyle\times\Gamma\left(\frac{p_{1}+p_{3}-u}{2}\right)\Gamma\left(\frac{p_{2}+p_{4}-u}{2}\right)\,.$

In the flat space limit we are considering, where $s,t$ become large while
$\alpha,\overline{\alpha}$ are kept fixed, we only access the singlet of
$SO(5)$. In this limit the whole Mellin amplitude must factorise into a
polynomial $P(\alpha,{\overline{\alpha}})$ which encodes the KK modes for the
singlet of $SO(5)$, times a function $\tilde{M}(s,t)$ of the Mellin variables
which encodes the $AdS_{5}$ directions

$\lim\limits_{s,t\to\infty}M(s,t,\alpha,{\overline{\alpha}})=\tilde{M}(s,t)P(\alpha,{\overline{\alpha}})\,.$
(44)

The AdS5 factor $\tilde{M}(s,t)$ is related to the flat space amplitude (21)
by the flat space limit

$A(S,T)=-2\lambda^{\frac{3}{2}}\lim\limits_{R\to\infty}\int_{\kappa-i\infty}^{\kappa+i\infty}\frac{d\alpha}{2\pi
i}\,e^{\alpha}\alpha^{-2-\frac{1}{2}\sum_{i}p_{i}}\tilde{M}\left(\frac{2\sqrt{\lambda}S}{\alpha},\frac{2\sqrt{\lambda}T}{\alpha}\right)\,.$
(45)

From this we immediately find

$\displaystyle A(S,T)$
$\displaystyle{}=-\frac{1}{STU}-2\sum\limits_{a,b=0}^{\infty}\hat{\sigma}_{2}^{a}\hat{\sigma}_{3}^{b}\alpha^{(0)}_{a,b}\,,$
(46) $\displaystyle\quad\Rightarrow\quad\tilde{M}(s,t)$
$\displaystyle{}=\frac{4\Gamma(\frac{1}{2}\sum_{i}p_{i}-1)}{(s-2)(t-2)(u-2)}+\sum\limits_{a,b=0}^{\infty}\frac{\Gamma(2a+3b+\frac{1}{2}\sum_{i}p_{i}+2)}{8^{a+b}\lambda^{\frac{3}{2}+a+\frac{3}{2}b}}\sigma_{2}^{a}\sigma_{3}^{b}\alpha^{(0)}_{a,b}\,,$

where we use the bases of crossing-symmetric polynomials

$\displaystyle\hat{\sigma}_{2}$
$\displaystyle=\frac{1}{2}(S^{2}+T^{2}+U^{2})\,,\qquad\hat{\sigma}_{3}=STU\,,\qquad\sigma_{2}$
$\displaystyle=s^{2}+t^{2}+u^{2}\,,\qquad\sigma_{3}=stu\,.$ (47)

To determine the factor $P(\alpha,{\overline{\alpha}})$ we can make use of the
known supergravity term in the Mellin amplitude. Taking the large $s,t$ limit
of the result from Rastelli:2016nze ; Aprile:2018efk  (assuming without loss
of generality $p_{21}\geq 0$, $p_{43}\geq 0$ and $p_{43}\geq p_{21}$) and
comparing to (44) we find that

$\lim\limits_{s,t\to\infty}M(s,t,\alpha,{\overline{\alpha}})=\frac{4\Gamma(\frac{1}{2}\sum_{i}p_{i}-1)}{stu}P(\alpha,{\overline{\alpha}})+O(\lambda^{-3/2})\,,$
(48)

with

$\displaystyle P(\alpha,{\overline{\alpha}})$
$\displaystyle=\frac{\sqrt{p_{1}p_{2}p_{3}p_{4}}}{2\Gamma(\frac{1}{2}\sum_{i}p_{i}-1)}\sum\limits_{\begin{subarray}{c}0\leq
i,j,k\leq\mathcal{E}-2\\\
i+j+k=\mathcal{E}-2\end{subarray}}\frac{\sigma^{\frac{p_{43}}{2}+i}\tau^{j}}{i!j!k!(i+\frac{p_{43}+p_{21}}{2})!(j+\frac{p_{43}-p_{21}}{2})!(k+|\frac{p_{1}+p_{2}-p_{3}-p_{4}}{2}|)!}\,,$
$\displaystyle\mathcal{E}$
$\displaystyle=p_{3}-\min(0,\tfrac{p_{1}+p_{2}-p_{3}-p_{4}}{2})\,,$ (49)

see also Alday:2020dtb . Next we want to extract from this the OPE
coefficients in the large $\lambda$ limit of the expansion (40). For the AdS
factor this can be done by taking the flat space limit of the conformal
partial wave decomposition

$\tilde{M}(s,t)=\sum_{\ell=0}^{\infty}\int_{-i\infty}^{i\infty}d\nu\,b_{\ell}(\nu^{2})M_{\nu,\ell}(s,t)\,,$
(50)

where $M_{\nu,\ell}(s,t)$ is the Mellin transform of a conformal partial wave
as defined in Costa:2012cb  and $b_{\ell}(\nu^{2})$ has poles at the conformal
dimensions of superconformal primaries with residues

$\displaystyle b_{\ell}(\nu^{2})\approx
b_{\Delta,\ell}\frac{2^{\ell}K^{\\{p_{i}+2\\},4}_{\Delta+4,\ell}}{\nu^{2}+(\Delta+4-2)^{2}}\,,$
(51)

where

$\displaystyle K^{\\{\Delta_{i}\\},d}_{\Delta,\ell}={}$
$\displaystyle\frac{\Gamma(\Delta+\ell)\,\Gamma(\Delta-\frac{d}{2}+1)\,(\Delta-1)_{\ell}}{4^{\ell-1}\Gamma\\!\left(\frac{\Delta+\ell+\Delta_{12}}{2}\right)\Gamma\\!\left(\frac{\Delta+\ell-\Delta_{12}}{2}\right)\Gamma\\!\left(\frac{\Delta+\ell+\Delta_{34}}{2}\right)\Gamma\\!\left(\frac{\Delta+\ell-\Delta_{34}}{2}\right)}$
(52)
$\displaystyle\frac{1}{\Gamma\\!\left(\frac{\Delta_{1}+\Delta_{2}-\Delta+\ell}{2}\right)\Gamma\\!\left(\frac{\Delta_{3}+\Delta_{4}-\Delta+\ell}{2}\right)\Gamma\\!\left(\frac{\Delta_{1}+\Delta_{2}+\Delta+\ell-d}{2}\right)\Gamma\\!\left(\frac{\Delta_{3}+\Delta_{4}+\Delta+\ell-d}{2}\right)}\,.$

As shown in Costa:2012cb ; Goncalves:2014ffa ; Alday:2022uxp , we can use

$K^{\\{p_{1}+2,p_{2}+2,p_{3}+2,p_{4}+2\\},4}_{\Delta+4,\ell}\underset{\Delta\gg
1}{\approx}\left(\frac{\Delta}{2}\right)^{8-\sum_{i}p_{i}}\frac{2^{2\Delta+2\ell+17}}{\pi^{3}\Delta^{2(\ell+5)}}\sin\left(\tfrac{\pi(\Delta-\ell-
p_{1}-p_{2})}{2}\right)\sin\left(\tfrac{\pi(\Delta-\ell-
p_{3}-p_{4})}{2}\right)\,,$ (53)

together with the fact that the flat space limit of $M_{\nu,\ell}(s,t)$ is the
5D flat space partial wave which appears in (36) to show that

$b_{\Delta,\ell}=\left(\frac{\Delta}{2}\right)^{\sum_{i}p_{i}-8}\frac{\pi^{3}\Delta^{10}}{\lambda
2^{2\Delta+\ell+16}\sin(\frac{\pi}{2}(\Delta-\ell-
p_{1}-p_{2}))\sin(\frac{\pi}{2}(\Delta-\ell-p_{3}-p_{4}))}a_{\delta,\ell}\,.$
(54)

The $S^{5}$ factor simply needs to be expanded into spherical harmonics

$P(\alpha,{\overline{\alpha}})=\sum\limits_{n=\max(p_{21},p_{43})}^{\min(p_{1}+p_{2},p_{3}+p_{4})-4}h_{n}Z_{[n,0,0]}(\alpha,{\overline{\alpha}})\,,$
(55)

where $n$ increases in steps of 2 and the coefficients can be written in terms
of sphere overlap integrals defined in (97) below

$h_{n}=\frac{S_{p_{1}-2,p_{2}-2,n}S_{p_{3}-2,p_{4}-2,n}}{\pi^{6}\prod_{i=1}^{4}\sqrt{\Gamma(p_{i})\Gamma(p_{i}-1)}}\,.$
(56)

In this way we find the leading OPE coefficients

$C^{\\{p_{1}p_{2}\\}}_{\Delta,\ell,[n,0,0]}C^{\\{p_{3}p_{4}\\}}_{\Delta,\ell,[n,0,0]}=b_{\Delta,\ell}h_{n}=\lambda_{p_{1},p_{2},n}\lambda_{p_{3},p_{4},n}\frac{\pi^{3}\Delta^{10}}{\lambda
2^{2\Delta+\ell+16}}a_{\delta,\ell}\,,$ (57)

where we defined

$\lambda_{p_{1},p_{2},n}=\left(\frac{\Delta}{2}\right)^{p_{1}+p_{2}-4}\frac{1}{\sin(\frac{\pi}{2}(\Delta-\ell-
p_{1}-p_{2}))}\frac{S_{p_{1}-2,p_{2}-2,n}}{\pi^{3}\sqrt{\Gamma(p_{1})\Gamma(p_{1}-1)\Gamma(p_{2})\Gamma(p_{2}-1)}}\,.$
(58)

Let us make some comments on this result. The full supergravity correlator of
Rastelli:2016nze  that we used to compute the limit (48) decomposes into all
$SO(6)$ representations of the form $[n-m,m,m]$, however (55) 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, (56) implies that $P(\alpha,{\overline{\alpha}})$ is the sum over
products of two sphere overlap integrals.101010Up to factors
$\sqrt{\Gamma(p_{i})\Gamma(p_{i}-1)}$ which can be attributed to the
normalisation of the external operators. We state this more explicitly in (99)
below.

###  4.1 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 (57) 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

$C^{\\{p_{1}p_{2}\\}}_{\Delta,\ell,[n,0,0]}C^{\\{p_{3}p_{4}\\}}_{\Delta,\ell,[n,0,0]}=\sum\limits_{I=1}^{N(\Delta,\ell,n)}C^{\\{p_{1}p_{2}\\}I}_{\Delta,\ell,[n,0,0]}C^{\\{p_{3}p_{4}\\}I}_{\Delta,\ell,[n,0,0]}\,,$
(59)

where $N(\Delta,\ell,n)$ is the number of superprimaries with the same quantum
numbers $(\Delta,\ell,n)$ at strong coupling. The quantity (59) is what we
actually computed. Let us explain what equation (57) implies for the
individual $C^{\\{p_{1}p_{2}\\}I}_{\Delta,\ell,[n,0,0]}$.

Let us introduce the following vectors

$A^{I}=C^{\\{p_{1}p_{2}\\}I}_{\Delta,\ell,[n,0,0]}\,,\qquad
B^{I}=C^{\\{p_{3}p_{4}\\}I}_{\Delta,\ell,[n,0,0]}\,.$ (60)

The triangle inequality for these two vectors reads $(A\cdot
B)^{2}\leq|A|^{2}|B|^{2}$. However, equation (57) implies that actually

$(A\cdot B)^{2}=|A|^{2}|B|^{2}\,.$ (61)

This implies that $A^{I}$ and $B^{I}$ are parallel vectors. Thus we conclude
that

$C^{\\{p_{1}p_{2}\\}I}_{\Delta,\ell,[n,0,0]}=\chi^{\\{p_{1}p_{2}\\}}_{\Delta,\ell,[n,0,0]}D^{I}\,,$
(62)

where

$\chi^{\\{p_{1}p_{2}\\}}_{\Delta,\ell,[n,0,0]}=\pm\lambda_{p_{1},p_{2},n}\sqrt{\frac{\pi^{3}\Delta^{10}}{\lambda
2^{2\Delta+\ell+16}}a_{\delta,\ell}}\,,$ (63)

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 Alday:2017xua ; Aprile:2017bgs ; Aprile:2017xsp  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.

##  5 Comparison with previous results

In this section we compare our results with those of Minahan:2014usa , 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 Minahan:2014usa  requires that the dimensions of all three
operators in the three-point function ($p_{1}$, $p_{2}$ and $\Delta$) are
large, of order $\lambda^{1/4}$. On first sight this limit seems to be
incompatible with our assumption that $p_{1},p_{2}$ are much smaller than
$\lambda^{1/4}$ in the large $\lambda$ limit. This assumption led to the
conclusion that to leading order in $1/\lambda$ 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 $p_{i}\ll\lambda^{1/4}$ when considering the
structure constants of these operators. In order to compare, it is thus enough
to take the additional limit $\Delta\gg p_{1},p_{2}$ of the results of
Minahan:2014usa .

The structure constants given in Minahan:2014usa  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

$\mathcal{L}_{p}=L^{+}_{p}+L^{-}_{p}=Q^{4}\mathcal{O}_{p}+\overline{Q}^{4}\mathcal{O}_{p}\,,$
(64)

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

$\displaystyle\left\langle\mathcal{L}_{p}(x_{1})\mathcal{L}_{p}(x_{2})\mathcal{L}_{p}(x_{3})\mathcal{L}_{p}(x_{4})\right\rangle=\frac{G(z,\overline{z},\alpha,{\overline{\alpha}})}{x_{12}^{4p}x_{34}^{4p}}=\frac{1}{x_{12}^{4p}x_{34}^{4p}}\sum_{\Delta,\ell,n}\widetilde{C}^{2}_{\Delta,\ell,n}G_{\Delta,\ell}(z,\overline{z})Z_{[n,0,0]}(\alpha,{\overline{\alpha}})\,.$
(65)

The OPE coefficients $\widetilde{C}_{\Delta,\ell,n}$ were computed in
Minahan:2014usa  for operators on the leading Regge trajectory, which are
descendants of the massive short superconformal primaries discussed above
$Q^{4}\overline{Q}^{4}\mathcal{O}_{\delta,\ell,[n,0,0]}$ and have dimension
$\Delta+4$, spin $\ell+4$ and R-charge $n$. In terms of the quantum numbers

$\Delta_{1,2}=p_{1,2}+2\,,\quad
n_{1,2}=p_{1,2}-2\,,\quad\Delta_{3}=2\sqrt{\frac{\ell+2}{2}}\lambda^{\frac{1}{4}}+2+O(\lambda^{\frac{1}{4}})\,,\quad
S=\ell+4\,,\quad n_{3}=n\,,$ (66)

their result is111111The factor $2^{\frac{3-S}{2}}$ accounts for our different
normalisation of conformal blocks and the whole correlator compared to
Minahan:2014usa ; Costa:2012cb ; Goncalves:2014ffa .

$\widetilde{C}_{\Delta+4,\ell+4,n}=2^{\frac{3-S}{2}}\frac{\sqrt{(\Delta_{1}-1)(\Delta_{2}-1)(\Delta_{3}-1)}}{2^{5/2}\pi}\,\frac{\Gamma({\alpha}_{1})\Gamma({\alpha}_{2})\Gamma({\alpha}_{3})\Gamma(\Sigma-2)}{\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma(\Delta_{3})}\,\mathcal{G}_{123}\,,$
(67)

with

$\Sigma=\frac{1}{2}(\Delta_{1}+\Delta_{2}+\Delta_{3})\,,\qquad\alpha_{i}=\Sigma-\Delta_{i}\,,$
(68)

and the coupling

$\mathcal{G}_{123}=\frac{8N^{2}\sqrt{\lambda}}{\pi^{2}}\,\langle
V_{\mathcal{L}}V_{\mathcal{L}}V_{S}\rangle S_{n_{1},n_{2},n_{3}}\,.$ (69)

The factor $\langle V_{\mathcal{L}}V_{\mathcal{L}}V_{S}\rangle$ was computed
using flat space vertex operators as

$\langle
V_{\mathcal{L}}V_{\mathcal{L}}V_{S}\rangle=\frac{(2\alpha^{\prime}\alpha_{1}\alpha_{2}\alpha_{3}\Sigma)^{S/2}}{N^{3}(\Delta_{3})^{S}\Gamma(S/2)}\,,$
(70)

and the factor $S_{n_{1},n_{2},n_{3}}$ in (69) comes from a sphere overlap
integral in $S^{5}$ and will be discussed in appendix B below.

The correlator (65) is related by a differential operator to the one of
superconformal primaries. In Alday:2022uxp  we used the differential operator
for the case $\langle 2222\rangle$ given in Drummond:2006by ;
Goncalves:2014ffa  to relate the OPE coefficients for the two correlators.
Here we will use the general differential operator given in Caron-Huot:2018kta
to generalise this computation to the case $\langle pppp\rangle$.

In Caron-Huot:2018kta  the authors provided the differential operator

$\Delta^{(8)}=\frac{z\overline{z}\alpha{\overline{\alpha}}}{(z-\overline{z})(\alpha-{\overline{\alpha}})}\left(\mathcal{D}_{z}-\mathcal{D}_{\alpha}\right)\left(\mathcal{D}_{z}-\mathcal{D}_{\overline{\alpha}}\right)\left(\mathcal{D}_{\overline{z}}-\mathcal{D}_{\alpha}\right)\left(\mathcal{D}_{\overline{z}}-\mathcal{D}_{{\overline{\alpha}}}\right)\frac{(z-\overline{z})(\alpha-{\overline{\alpha}})}{z\overline{z}\alpha{\overline{\alpha}}}\,,$
(71)

where

$\mathcal{D}_{x}\equiv
x^{2}\partial_{x}(1-x)\partial_{x}-\tfrac{1}{2}(r+s)x^{2}\partial_{x}-\tfrac{1}{4}rsx\,,$
(72)

which acts on the reduced correlator of $\mathcal{O}_{p}$ introduced in (40)
and computes

$\langle
L^{+}_{p}(x_{1})L^{+}_{p}(x_{2})L^{-}_{p}(x_{3})L^{-}_{p}(x_{4})\rangle\equiv\frac{\widetilde{G}(z,\overline{z},\alpha,{\overline{\alpha}})}{x_{12}^{4p}x_{34}^{4p}}=\frac{2}{x_{12}^{4p}x_{34}^{4p}}\frac{\Delta^{(8)}}{p^{4}(p^{2}-1)^{2}}\mathcal{T}(z,\overline{z},\alpha,{\overline{\alpha}})\,.$
(73)

We then use the relation (64) to write

$\displaystyle\langle\mathcal{L}_{p}\mathcal{L}_{p}\mathcal{L}_{p}\mathcal{L}_{p}\rangle=2\left(\langle
L^{+}_{p}L^{+}_{p}L^{-}_{p}L^{-}_{p}\rangle+\langle
L^{-}_{p}L^{+}_{p}L^{+}_{p}L^{-}_{p}\rangle+\langle
L^{+}_{p}L^{-}_{p}L^{+}_{p}L^{-}_{p}\rangle\right)$ $\displaystyle={}$
$\displaystyle\frac{2}{x_{12}^{4p}x_{34}^{4p}}\bigg{(}\widetilde{G}(z,\overline{z},\alpha,{\overline{\alpha}})+\left(\tfrac{z\overline{z}}{(1-z)(1-\overline{z})}\right)^{p+2}\left(\tfrac{(1-\alpha)(1-{\overline{\alpha}})}{\alpha{\overline{\alpha}}}\right)^{p-2}\widetilde{G}(1-z,1-\overline{z},1-\alpha,1-{\overline{\alpha}})$
$\displaystyle+\frac{(z\overline{z})^{p+2}}{(\alpha{\overline{\alpha}})^{p-2}}\widetilde{G}(1/z,1/\overline{z},1/\alpha,1/{\overline{\alpha}})\bigg{)}\,.$
(74)

Using also the crossing symmetry of
$\mathcal{T}(z,\overline{z},\alpha,{\overline{\alpha}})$, this relation
provides us with a differential operator $\widetilde{\Delta}^{(8)}$ relating
$\mathcal{T}(z,\overline{z},\alpha,{\overline{\alpha}})$ and
$G(z,\overline{z},\alpha,{\overline{\alpha}})$

$G(z,\overline{z},\alpha,{\overline{\alpha}})=\widetilde{\Delta}^{(8)}\,\mathcal{T}(z,\overline{z},\alpha,{\overline{\alpha}})\,.$
(75)

Following appendix F of Alday:2022uxp  we can act on a generic superconformal
block contributing to $\mathcal{T}(z,\overline{z},\alpha,{\overline{\alpha}})$
and obtain the precise contributions of the bosonic conformal blocks for each
component of this supermultiplet contributing to the correlator
$G(z,\overline{z},\alpha,{\overline{\alpha}})$. In order to compare to the
result (67) we can make two big simplifications. We only need the contribution
of the bottom component of the supermultiplet, which has spin $\ell+4$,
because this is the only component that contributes to the leading Regge
trajectory in the sense of the expansion (65). Furthermore we can take
$\Delta$ to be very large as before. The result is

$\displaystyle\mathcal{T}(z,\overline{z},\alpha,{\overline{\alpha}})$
$\displaystyle=G_{\Delta+4,\ell}(z,\overline{z})Z_{[n,0,0]}(\alpha,{\overline{\alpha}})\Rightarrow$
(76) $\displaystyle G(z,\overline{z},\alpha,{\overline{\alpha}})$
$\displaystyle=\frac{\Delta^{8}+O(\Delta^{7})}{2^{13}p^{4}(p^{2}-1)^{2}}G_{\Delta+4,\ell+4}(z,\overline{z})Z_{[n,0,0]}(\alpha,{\overline{\alpha}})+\text{subleading
Regge trajectories}\,.$ (77)

Using this relation and the result for the flat space partial wave
coefficients Alday:2022uxp

$a_{\frac{\ell}{2}+1,\ell}=\frac{(\ell+2)^{\ell-2}}{4^{\ell-1}\Gamma(\frac{\ell}{2}+1)^{2}}\,,$
(78)

we can now check that for operators on the leading Regge trajectory, our
result (57) agrees with (67) once we expand at large $\Delta$

$\frac{\Delta^{4}+O(\Delta^{\frac{7}{2}})}{2^{\frac{13}{2}}p_{1}p_{2}\sqrt{(p_{1}^{2}-1)(p_{2}^{2}-1)}}C^{\\{p_{1}p_{2}\\}}_{\Delta,\ell,[n,0,0]}=\widetilde{C}_{\Delta+4,\ell+4,n}\,.$
(79)

##  6 Conclusions

In this paper we studied the spectrum and structure constants of massive short
operators in planar ${\mathcal{N}}=4$ SYM theory at strong coupling by
considering strings on $AdS_{5}\times S^{5}$ 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 $\lambda$
become identical in the strict $\lambda=\infty$ 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 $\lambda^{1/4}$),
where integrability computations are expected to simplify. See Basso:2022nny
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 Alday:2022uxp ; Alday:2022xwz  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 $1/\sqrt{\lambda}$ expansion or to consider other correlators, for
example ones with external massive operators.

Of course it is possible to apply the methods of Alday:2022uxp ; Alday:2022xwz
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 $p_{i}\sim\lambda^{\frac{1}{4}}$ and
accesses the full spectrum of strings in 10 dimensions.

## Acknowledgements

The work of LFA and TH is supported by the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 787185). LFA is also supported in part by the STFC grant
ST/T000864/1. JS is supported by the STFC grant ST/T000864/1.

##  Appendix A Correlator Setup

In this appendix we state the definitions for the reduced correlator and long
superconformal block following the conventions of Caron-Huot:2018kta .

###  A.1 Reduced correlator

We normalise the two point function of $\frac{1}{2}$-BPS operators of
dimension $p$ as

$\displaystyle\langle\mathcal{O}_{p}(x_{1},y_{1})\mathcal{O}_{p}(x_{2},y_{2})\rangle=\frac{(y_{1}\cdot
y_{2})^{p}}{|x_{1}-x_{2}|^{2p}}\,,$ (80)

and write their four-point functions as

$\displaystyle\langle\mathcal{O}_{p_{1}}(x_{1},y_{1})\cdots\mathcal{O}_{p_{4}}(x_{4},y_{4})\rangle$
$\displaystyle\equiv
T_{p_{1}p_{2}p_{3}p_{4}}\mathcal{G}(z,\overline{z},{\alpha},{\overline{\alpha}})\,,$
(81) $\displaystyle T_{p_{1}p_{2}p_{3}p_{4}}$
$\displaystyle=g_{12}^{\frac{p_{1}+p_{2}}{2}}g_{34}^{\frac{p_{3}+p_{4}}{2}}\left(\frac{g_{24}}{g_{14}}\right)^{\frac{p_{2}-p_{1}}{2}}\left(\frac{g_{13}}{g_{14}}\right)^{\frac{p_{3}-p_{4}}{2}}\,,$

where $g_{ij}=\frac{y_{i}\cdot y_{j}}{x_{ij}^{2}}$. The superconformal Ward
identities imply that the correlator can generally be written in the form
Dolan:2004iy

$\displaystyle\mathcal{G}(z,\overline{z},{\alpha},{\overline{\alpha}})$
$\displaystyle=k\chi(z,{\alpha})\chi(\overline{z},{\overline{\alpha}})+\frac{(z-{\alpha})(z-{\overline{\alpha}})(\overline{z}-{\alpha})(\overline{z}-{\overline{\alpha}})}{({\alpha}-{\overline{\alpha}})(z-\overline{z})}$
$\displaystyle\times\left(-\frac{\chi(\overline{z},{\overline{\alpha}})f(z,{\alpha})}{{\alpha}z(\overline{z}-{\overline{\alpha}})}+\frac{\chi(\overline{z},{\alpha})f(z,{\overline{\alpha}})}{{\overline{\alpha}}z(\overline{z}-{\alpha})}+\frac{\chi(z,{\overline{\alpha}})f(\overline{z},{\alpha})}{{\alpha}\overline{z}(z-{\overline{\alpha}})}-\frac{\chi(z,{\alpha})f(\overline{z},{\overline{\alpha}})}{{\overline{\alpha}}\overline{z}(z-{\alpha})}\right)$
$\displaystyle+\frac{(z-{\alpha})(z-{\overline{\alpha}})(\overline{z}-{\alpha})(\overline{z}-{\overline{\alpha}})}{({\alpha}{\overline{\alpha}})^{2}(z\overline{z})^{2}}H(z,\overline{z},{\alpha},{\overline{\alpha}}),$
(82)

where $k$ is called the unit contribution, $f$ is the chiral correlator and
$H$ is the reduced correlator. These different parts can be extracted from
$\mathcal{G}$ as follows

$\displaystyle\begin{aligned}
k&=\mathcal{G}(z,\overline{z},z,\overline{z}),\\\
f(\overline{z},{\overline{\alpha}})&=\frac{{\overline{\alpha}}\overline{z}}{\overline{z}-{\overline{\alpha}}}\left(\mathcal{G}(z,\overline{z},z,{\overline{\alpha}})-k\chi(\overline{z},{\overline{\alpha}})\right),\end{aligned}$

and $H$ is obtained by subtracting the other contributions from (82). The
function $\chi$ is given by

$\chi(z,{\alpha})=\left(\frac{z}{\alpha}\right)^{\max(|p_{21}|,|p_{34}|)/2}\left(\frac{1-{\alpha}}{1-z}\right)^{\max(p_{21}+p_{34},0)/2}.$
(83)

We further define the interacting part of the reduced correlator by
subtracting the reduced correlator of the free theory

$\mathcal{T}(z,\overline{z},{\alpha},{\overline{\alpha}})=H(z,\overline{z},{\alpha},{\overline{\alpha}})-H^{\text{free}}(z,\overline{z},{\alpha},{\overline{\alpha}})\,.$
(84)

###  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
$\Delta$, spin $\ell$ 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

$\displaystyle\mathcal{A}_{\Delta,\ell,m,n}^{r,s}=\left\\{\begin{array}[]{l}\displaystyle
k=0,\qquad f(z,{\alpha})=0,\\\ \displaystyle
H(z,\overline{z},{\alpha},{\overline{\alpha}})=G^{r,s}_{\Delta+4,\ell}(z,\overline{z})Z_{[n-m,m,m]}^{r,s}({\alpha},{\overline{\alpha}})\end{array}\right..$
(85)

These definitions are in terms of the usual $SO(4,2)$ conformal blocks

$\displaystyle G_{\Delta,\ell}^{r,s}(z,\overline{z})$
$\displaystyle=\frac{z\overline{z}}{\overline{z}-z}\left[k_{\frac{\Delta-\ell-2}{2}}^{r,s}(z)k_{\frac{\Delta+\ell}{2}}^{r,s}(\overline{z})-k_{\frac{\Delta+\ell}{2}}^{r,s}(z)k_{\frac{\Delta-\ell-2}{2}}^{r,s}(\overline{z})\right],$
(86) $\displaystyle k_{h}^{r,s}(z)$
$\displaystyle=z^{h}\;_{2}F_{1}\left(h+\frac{r}{2},h+\frac{s}{2};2h,z\right),$
(87)

and the $S_{5}$ spherical harmonics, which are given in terms of the same
functions

$Z_{[n-m,m,m]}^{r,s}({\alpha},{\overline{\alpha}})=(-1)^{m}G_{-n,m}^{-r,-s}({\alpha},{\overline{\alpha}})\,.$
(88)

##  Appendix B Sphere integrals

Let us now relate the sphere overlap factor in the OPE coefficients to the
four-point partial wave (88). To encode the irreducible representation
$[n,0,0]$ of $SO(6)$ we introducing an orthonormal basis of traceless
symmetric $SO(6)$ tensors $C^{I}_{i_{1}\ldots i_{n}}$ labelled by
$I=1,\ldots,\dim([n,0,0])$ which satisfy

$C^{I_{1}}_{i_{1}\ldots i_{n}}C^{I_{2}}_{i_{1}\ldots
i_{n}}=\delta^{I_{1}I_{2}}\,,$ (89)

as well as the completeness relation

$C^{I}_{i_{1}\ldots i_{n}}C^{I}_{j_{1}\ldots j_{n}}=\pi^{(6)}_{i_{1}\ldots
i_{n},j_{1}\ldots j_{n}}\,,$ (90)

where $\pi^{(6)}_{i_{1}\ldots i_{n},j_{1}\ldots j_{n}}$ is the projector to
traceless symmetric $SO(6)$ representations (26). The scalar spherical
harmonics are defined by

$Y^{I}_{n}=z(n)C^{I}_{i_{1}\ldots i_{n}}\xi^{i_{1}}\ldots\xi^{i_{n}}\,,$ (91)

where $\xi\in S^{5}$, i.e. $\xi^{2}=1$ and the normalisation

$z(n)=\sqrt{2^{n-1}(n+1)(n+2)}\,,$ (92)

is chosen such that

$\int_{S^{5}}d\xi\,Y^{I_{1}}_{n}Y^{I_{2}}_{n}=\omega_{5}\delta^{I_{1}I_{2}}\,,$
(93)

where $\omega_{5}=\pi^{3}$ is the volume of $S^{5}$. To connect to our sphere
polarizations $y^{i}$ (satisfying $y^{2}=0$) from before we introduce the
object

$C^{I}_{n}(y)=C^{I}_{i_{1}\ldots i_{n}}y^{i_{1}}\ldots y^{i_{n}}\,.$ (94)

The structure constants were given in Minahan:2014usa  in terms of the sphere
overlap integrals

$\langle\psi_{n_{1}}(y_{1})\psi_{n_{2}}(y_{2})\psi_{n_{3}}(y_{3})\rangle=C^{I_{1}}_{n_{1}}(y_{1})C^{I_{2}}_{n_{2}}(y_{2})C^{I_{3}}_{n_{3}}(y_{3})\int_{S^{5}}d\xi\,Y^{I_{1}}_{n_{1}}Y^{I_{2}}_{n_{2}}Y^{I_{3}}_{n_{3}}\,,$
(95)

which were computed in Arutyunov:1999en

$\int_{S^{5}}d\xi\,Y^{I_{1}}_{n_{1}}Y^{I_{2}}_{n_{2}}Y^{I_{3}}_{n_{3}}=S_{n_{1},n_{2},n_{3}}\langle
C^{I_{1}}_{n_{1}}C^{I_{2}}_{n_{2}}C^{I_{3}}_{n_{3}}\rangle\,,$ (96)

where $\langle C^{I_{1}}_{n_{1}}C^{I_{2}}_{n_{2}}C^{I_{3}}_{n_{3}}\rangle$ is
the unique $SO(6)$ invariant that can be formed out of three copies of
$C^{I}_{i_{1}\ldots i_{n}}$ and

$\displaystyle S_{n_{1},n_{2},n_{3}}$
$\displaystyle=\frac{\omega_{5}}{2^{\widetilde{\Sigma}-1}}\frac{n_{1}!n_{2}!n_{3}!z(n_{1})z(n_{2})z(n_{3})}{(\widetilde{\Sigma}+2)!\widetilde{\alpha}_{123}!\widetilde{\alpha}_{231}!\widetilde{\alpha}_{312}!}\,,$
(97) $\displaystyle\widetilde{\Sigma}$
$\displaystyle=\tfrac{1}{2}(n_{1}+n_{2}+n_{3})\,,\qquad\widetilde{\alpha}_{ijk}=\tfrac{1}{2}(n_{i}+n_{j}-n_{k})\,.$

Furthermore, the completeness relation (90) implies the following relation to
the four-point partial waves (88)

$\displaystyle
C^{I_{1}}_{n_{1}}(y_{1})C^{I_{2}}_{n_{2}}(y_{2})C^{I_{3}}_{n_{3}}(y_{3})C^{I_{4}}_{n_{4}}(y_{4})\langle
C^{I_{1}}_{n_{1}}C^{I_{2}}_{n_{2}}C^{I}_{n_{5}}\rangle\langle
C^{I}_{n_{5}}C^{I_{3}}_{n_{3}}C^{I_{4}}_{n_{4}}\rangle$ $\displaystyle={}$
$\displaystyle\frac{1}{(n_{5}!)^{2}}(y_{1}\cdot
y_{2})^{\widetilde{\alpha}_{125}}(y_{1}\cdot\partial_{x})^{\widetilde{\alpha}_{152}}(y_{2}\cdot\partial_{x})^{\widetilde{\alpha}_{251}}(y_{3}\cdot
y_{4})^{\widetilde{\alpha}_{345}}(y_{3}\cdot\partial_{\overline{x}})^{\widetilde{\alpha}_{354}}(y_{4}\cdot\partial_{\overline{x}})^{\widetilde{\alpha}_{453}}\pi^{(6)}_{n_{5}}(x,\overline{x})$
$\displaystyle={}$ $\displaystyle(y_{1}\cdot
y_{2})^{\frac{n_{1}+n_{2}}{2}}(y_{3}\cdot
y_{4})^{\frac{n_{3}+n_{4}}{2}}\left(\frac{y_{2}\cdot y_{4}}{y_{1}\cdot
y_{4}}\right)^{\frac{n_{2}-n_{1}}{2}}\left(\frac{y_{1}\cdot y_{3}}{y_{1}\cdot
y_{4}}\right)^{\frac{n_{3}-n_{4}}{2}}Z_{0,n_{5}}^{r,s}({\alpha},{\overline{\alpha}})\,,$
(98)

which can be checked using (26). Using this relation, the observation (56)
that the expansion coefficients of the function
$P(\alpha,{\overline{\alpha}})$ can be written in terms of the coefficients
from (96) means that $P(\alpha,{\overline{\alpha}})$ is just the sum over
products of two sphere overlaps

$\displaystyle(y_{1}\cdot y_{2})^{\frac{p_{1}+p_{2}-4}{2}}(y_{3}\cdot
y_{4})^{\frac{p_{3}+p_{4}-4}{2}}\left(\frac{y_{2}\cdot y_{4}}{y_{1}\cdot
y_{4}}\right)^{\frac{p_{2}-p_{1}}{2}}\left(\frac{y_{1}\cdot y_{3}}{y_{1}\cdot
y_{4}}\right)^{\frac{p_{3}-p_{4}}{2}}P({\alpha},{\overline{\alpha}})$ (99)
$\displaystyle=\prod\limits_{i=1}^{4}\left(\frac{C^{I_{i}}_{p_{i}-2}(y_{i})}{\sqrt{\Gamma(p_{i})\Gamma(p_{i}-1)}}\right)\frac{1}{\omega_{5}^{2}}\sum\limits_{n}\int_{S^{5}}d\xi_{1}\,Y^{I_{1}}_{p_{1}-2}Y^{I_{2}}_{p_{2}-2}Y^{I}_{n}\int_{S^{5}}d\xi_{2}\,Y^{I_{3}}_{p_{3}-2}Y^{I_{4}}_{p_{4}-2}Y^{I}_{n}\,.$

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