aainstitutetext:
Department of Physics and Astronomy
Uppsala University, Box 516, SE-751 20 Uppsala, Sweden bbinstitutetext:
Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG,
UK

# AdS Virasoro-Shapiro amplitude with KK modes

Giulia Fardelli b Tobias Hansen b and Joao A. Silva
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We determine the first curvature correction for the string amplitude of two
supergravity states and two Kaluza-Klein modes on
$\text{AdS}_{5}\times\text{S}^{5}$, which is dual to the correlator
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{p}\mathcal{O}_{p}\rangle$
of half-BPS operators in $\mathcal{N}=4$ SYM theory. The result has the form
of an integral over the Riemann sphere as for the usual Virasoro-Shapiro
amplitude, with the insertion of single-valued multiple polylogarithms of
weight three. The result fixes OPE data of single-trace operators in
$\mathcal{N}=4$ SYM theory at strong coupling, including operators with non-
zero $R$-charge and odd spin. We successfully check our results by comparing
to data available from integrability, localisation and consistency with a
$10d$ effective action.

##  1 Introduction

Despite its prominent appearance in the AdS/CFT correspondence
Maldacena:1997re , the world-sheet theory of type IIB string theory on
$\text{AdS}_{5}\times\text{S}^{5}$ is still not fully known. While the dual
$\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory is very well
understood, making contact with the low energy effective action of strings in
AdS requires studying $\mathcal{N}=4$ SYM at strong ’t Hooft coupling
$\lambda$. The planar theory can be studied with integrability methods which
give precise values for the dimensions of unprotected operators at finite ’t
Hooft coupling $\lambda$, see Gromov:2023hzc  for the latest results. However,
at present, these methods are not developed enough to do the same for OPE
coefficients.111See Caron-Huot:2022sdy ; Cavaglia:2022qpg  for approaches that
combine integrability and bootstrap. In the series of papers Alday:2022uxp ;
Alday:2022xwz ; Alday:2023jdk ; Alday:2023mvu  a different approach was used
to study the AdS Virasoro-Shapiro amplitude as an expansion in
$\alpha^{\prime}/R^{2}$ by combining properties of the putative world-sheet
CFT and the dual CFT. The result is an expression for the world-sheet
integrand, an optimal target for future direct string theory computations.

In Alday:2022uxp ; Alday:2022xwz ; Alday:2023jdk ; Alday:2023mvu  the
correlator
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$
of four half-BPS operators of dimension two was studied. In this paper we
study
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{p}\mathcal{O}_{p}\rangle$
where $\mathcal{O}_{p}$ is a half-BPS operator of generic integer dimension
$p$.222We do require that $p\ll\lambda^{\frac{1}{4}}$. This enables us to
study the OPE data of single trace operators of nonzero $R$-charge and odd
spin, which appear in the OPE $\mathcal{O}_{2}\times\mathcal{O}_{p}$. Our main
result is a formula for the first Anti-de Sitter (AdS) curvature correction
$A^{(1)}(S,T)$ in
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{p}\mathcal{O}_{p}\rangle$
(see section 2.1 and in particular formula (10) for the precise definition).
It takes the form of a linear combination of terms, each of which is an
integral over the Riemann sphere

$\displaystyle\int d^{2}z|z|^{-2S-2}|1-z|^{-2T-2}G(S,T,z)\,,$ (1)

where the integrand $G(S,T,z)$ is a sum of single-valued multiple
polylogarithms of weight $3$, see section 3.2 for the precise formula. This
can be contrasted with the flat space Virasoro-Shapiro amplitude
$A^{(0)}(S,T)$ where the integrand is simply $1/(S+T)^{2}$

$\displaystyle A^{(0)}(S,T)=\int
d^{2}z|z|^{-2S-2}|1-z|^{-2T-2}\frac{1}{(S+T)^{2}}\,.$ (2)

While the flat space Virasoro-Shapiro amplitude determines OPE data at leading
order in the $\frac{1}{\sqrt{\lambda}}$ expansion, $A^{(1)}(S,T)$ determines
the OPE data at subleading order.

The paper is organised in the following way. In section 2 we review basic
facts about the correlator
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{p}\mathcal{O}_{p}\rangle$,
setting the stage for the computation of $A^{(1)}(S,T)$. Afterwards we compute
$A^{(1)}(S,T)$ using two different algorithms.

In section 3 we apply the approach of Alday:2023mvu , assuming that
$A^{(1)}(S,T)$ is given by a linear combination of integrals of the form (1).
We write an ansatz for the integrand in terms of single-valued multiple
polylogarithms of $z$ and rational functions of $S$ and $T$. This ansatz has a
finite number of parameters, which are fixed by consistency of the residues of
$A^{(1)}(S,T)$ with the OPE.

In section 4 we compute $A^{(1)}(S,T)$ by making use of dispersive sum rules,
following Alday:2022xwz . In this case we make an ansatz for the OPE data at
subleading order in $\frac{1}{\sqrt{\lambda}}$ (which involves an infinite
number of parameters). To fix these parameters we use locality (in the form of
null constraints Caron-Huot:2020cmc ; Caron-Huot:2021enk ). Besides this, we
assume that the Wilson coefficients that enter in the strong coupling
expansion are given by single-valued multiple zeta values. These two
conditions fully fix all of the parameters.

Both methods have numerous internal consistency checks and they both agree
with each other in the final answer. In section 5 we extract OPE data and
Wilson coefficients from $A^{(1)}(S,T)$ and compare them to the literature,
namely results from localisation, integrability and the putative existence of
an $AdS_{5}\times S_{5}$ effective action, finding full agreement. Section 6
contains our final conclusions. We have numerous appendices with auxiliary
results. In Appendix F the fully crossing symmetric dispersion relation of
Sinha:2020win ; Gopakumar:2021dvg  is adapted to the present case of
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{p}\mathcal{O}_{p}\rangle$
where only crossing symmetry between the $t$ and $u$ channels is present. This
might be useful in other contexts.

##  2 Basics

###  2.1 The correlator

In this paper we consider correlation functions involving half-BPS operators
$\mathcal{O}_{p}$ in planar $\mathcal{N}=4$ SYM, with SU$(N)$ gauge group, as
an expansion in inverse powers of the ’t Hooft coupling
$\lambda=g_{\mathrm{YM}}^{2}N$. These half-BPS operators are Lorentz scalars
transforming in the $[0,p,0]$ representation of the SU$(4)_{R}$ symmetry group
and they have protected conformal dimension $\Delta=p$. For $p=2$, this
corresponds to the superprimary of the stress-tensor multiplet, while for
$p>2$ they represent higher Kaluza-Klein supergravity excitations. In terms of
the $\mathcal{N}=4$ fundamental scalars, $\phi^{I}$, $I=1,\cdots,6$, they take
the schematic form

$\displaystyle\mathcal{O}_{p}=y_{I_{1}}\cdots
y_{I_{p}}\mathrm{tr}\left(\phi^{I_{1}}\cdots\phi^{I_{p}}\right)+\cdots$ (3)

where we have contracted the SO$(6)_{R}\simeq\mathrm{SU}(4)_{R}$ indices with
auxiliary null vectors $y_{I_{i}}$. The dots in the definition represent
contributions from multi-trace operators, which are suppressed by inverse
powers of the central charge $c=\frac{N^{2}-1}{4}$, and will not be relevant
in the following.333The presence of these additional contributions, on top of
the single-trace one, is necessary to make these operators dual to single-
particle states in AdS and they can be fixed by requiring that each operator
is orthogonal to any other multi-particle operator Alday:2019nin ;
Aprile:2019rep ; Aprile:2020uxk .

In this paper, we will focus on the four-point function involving two
$\mathcal{O}_{2}$’s and two other identical $\mathcal{O}_{p}$’s. It can be
written as Dolan:2001tt ; Dolan:2003hv ; Dolan:2004iy ; Caron-Huot:2018kta

$\displaystyle\,\,\langle\mathcal{O}_{2}(x_{1},y_{1})\mathcal{O}_{2}(x_{2},y_{2})\mathcal{O}_{p}(x_{3},y_{3})\mathcal{O}_{p}(x_{4},y_{4})\rangle=\frac{y_{12}^{2}y_{34}^{p}}{(x_{12}^{2})^{2}(x_{34}^{2})^{p}}\mathcal{G}_{\\{22pp\\}}(U,V;\alpha,\overline{\alpha})\,,$
(4)
$\displaystyle\mathcal{G}_{\\{22pp\\}}(U,V;\alpha,\overline{\alpha})=\mathcal{G}^{\mathrm{free}}_{\\{22pp\\}}(U,V;\alpha,\overline{\alpha})+\frac{(z-\alpha)(z-\overline{\alpha})(\overline{z}-\alpha)(\overline{z}-\overline{\alpha})}{(z\overline{z})^{2}(\alpha\overline{\alpha})^{2}}\mathcal{T}(U,V)\,,$

where definitions of cross-ratios and generalisations can be found in Appendix
A. The function $\mathcal{G}_{\\{22pp\\}}^{\mathrm{free}}$ represents the free
theory contributions and can be evaluated by means of Wick contractions — see
(74) for its explicit expression. The dynamical information is encoded in the
reduced correlator $\mathcal{T}(U,V)$, which can be further split as

$\displaystyle\mathcal{T}(U,V)=\mathcal{T}^{\mathrm{short}}(U,V)+\mathcal{T}^{\mathrm{long}}(U,V)\,.$
(5)

The first part receives contributions from protected short operators, while
the second part contains the contributions of long operators. See Appendix A
for more details. As we will soon see, the dimensions and OPE coefficients of
these long operators get corrections at large $\lambda$.

For our purposes, it is convenient to consider the Mellin transform
$M(s_{1},s_{2})$ of the reduced correlator

$\displaystyle\mathcal{T}(U,V)$
$\displaystyle=\int_{-i\infty}^{+i\infty}\frac{ds_{1}ds_{2}}{(4\pi
i)^{2}}\,U^{\frac{s_{1}}{2}+2+\frac{p}{3}}\,V^{\frac{s_{2}}{2}-1-\frac{p}{6}}\,\Gamma_{22pp}(s_{1},s_{2})\,M(s_{1},s_{2})\,,$
(6) $\displaystyle\Gamma_{22pp}(s_{1},s_{2})$
$\displaystyle=\Gamma\left(\frac{6-p}{3}-\frac{s_{1}}{2}\right)\Gamma\left(\frac{2p}{3}-\frac{s_{1}}{2}\right)\Gamma\left(\frac{p}{6}+1-\frac{s_{2}}{2}\right)^{2}\Gamma\left(\frac{p}{6}+1-\frac{s_{3}}{2}\right)^{2},$

where $s_{1}+s_{2}+s_{3}=0$.444These Mellin variables are related to the more
standard $s,t,u$ by $s_{1}=s-\frac{2p}{3}$, $s_{2}=t-\frac{2p}{3}$,
$s_{3}=u-\frac{2p}{3}$. Crossing symmetry of the correlator (4) under the
exchange of points $1\leftrightarrow 2$ and $3\leftrightarrow 4$, translates
for the Mellin amplitude to

$\displaystyle M(s_{1},s_{2})=M(s_{1},s_{3})\,.$ (7)

At leading order in $1/c$ and in an expansion around large $\lambda$, the
Mellin amplitude is given by the supergravity result ($\lambda\to\infty$) plus
a series of “stringy” corrections, described by polynomials in the Mellin
variables and representing contact Witten diagrams with higher derivative
vertices Alday:2018pdi ; Binder:2019jwn

$\displaystyle M(s_{1},s_{2})$
$\displaystyle=\frac{4p}{\Gamma(p-1)}\frac{1}{(s_{1}-\frac{6-2p}{3})(s_{2}-\frac{p}{3})(s_{3}-\frac{p}{3})}$
(8)
$\displaystyle+\sum_{b=0}^{\infty}\sum_{a=b}^{\infty}\frac{\Gamma(a+b+p+4)}{\Gamma(b+1)\Gamma(p)\Gamma(p-1)}s_{1}^{a-b}(s_{2}s_{3})^{b}\lambda^{-\frac{3}{2}-\frac{a}{2}-\frac{b}{2}}\left(\xi_{a,b}^{(0)}(p)+\frac{1}{\sqrt{\lambda}}\xi_{a,b}^{(1)}(p)+\ldots\right).$

The first term is the supergravity correlator and $\xi_{a,b}^{(k)}(p)$ are
usually referred to as Wilson coefficients.

Following Alday:2022xwz  we also consider the transform

$\displaystyle A(S,T)\equiv
2\Gamma(p-1)\Gamma(p)\int_{-i\infty+\kappa}^{+i\infty+\kappa}\frac{d\alpha}{2\pi
i}e^{\alpha}\alpha^{-p-4}M\left(\frac{2\sqrt{\lambda}S}{\alpha},\frac{2\sqrt{\lambda}T}{\alpha}\right)\,,$
(9)

which is like the flat space limit Penedones:2010ue , except that it applies
to each layer $\xi_{a,b}^{(k)}(p)$, whereas the flat space limit considers
only $\xi_{a,b}^{(0)}(p)$. Applied to the low energy expansion (8), it implies

$\displaystyle A(S,T)$
$\displaystyle=\frac{1}{\lambda^{\frac{3}{2}}}\sum\limits_{k=0}^{\infty}\frac{1}{\lambda^{k/2}}A^{(k)}(S,T)\,,$
(10) $\displaystyle A^{(k)}(S,T)$
$\displaystyle=\text{SUGRA}^{(k)}+\sum_{b=0}^{\infty}\sum_{a=b}^{\infty}\frac{2^{a+b+1}}{\Gamma(b+1)}S^{a-b}(TU)^{b}\xi_{a,b}^{(k)}(p)\,.$

The supergravity terms $\text{SUGRA}^{(k)}$ are given by

$\displaystyle\mathrm{SUGRA}^{(0)}=\frac{1}{STU}\,,\
\mathrm{SUGRA}^{(1)}=-\frac{p}{6}\,\frac{pS^{2}+2(p-3)TU}{(STU)^{2}}\,,\
\ldots\,,\ \mathrm{SUGRA}^{(k>p)}=0\,,$ (11)

and the general expression can be found in (82). The leading contribution is
the Virasoro-Shapiro amplitude in flat space

$A^{(0)}(S,T)=-\frac{\Gamma\left(-S\right)\Gamma\left(-T\right)\Gamma\left(-U\right)}{\Gamma\left(S+1\right)\Gamma\left(T+1\right)\Gamma\left(U+1\right)}\,,$
(12)

which fixes the first layer of Wilson coefficients $\xi_{a,b}^{(0)}(p)$.

###  2.2 Dispersion relation

The ingredients to write a dispersion relation for the Mellin amplitude are
its physical poles and residues and its behaviour at infinity. As briefly
discussed before, and more widely in Appendix A, the $(12)$ OPE of
$\mathcal{T}(U,V)$ contains superconformal primaries that are singlets under
SU$(4)_{R}$. We will indicate them by $\mathcal{O}_{s}$: $\tau_{s}$ will label
their twist and $C_{s}$ the product of OPE coefficients
$\langle{\mathcal{O}}_{2}{\mathcal{O}}_{2}{\mathcal{O}}_{s}\rangle\times\langle{\mathcal{O}}_{p}{\mathcal{O}}_{p}{\mathcal{O}}_{s}\rangle$.
In the Mellin amplitude, the exchange of these operators and their descendants
results in an infinite sequence of poles at $s_{1}=\tau_{s}+2m-\frac{2}{3}p$,
for $m\in\mathbb{N}_{0}$. Similarly, the Mellin amplitude has also poles in
$s_{2}$ and $s_{3}$, due to the exchange of superprimaries in the other OPE
channels. We will denote them by $\mathcal{O}_{t}$. They transform in the
$[0,p-2,0]$ representation of $R$-symmetry, have twist $\tau_{t}$ and we will
use $C_{t}$ to label the product of OPE coefficients
$\langle{\mathcal{O}}_{2}{\mathcal{O}}_{p}{\mathcal{O}}_{t}\rangle\times\langle{\mathcal{O}}_{2}{\mathcal{O}}_{p}{\mathcal{O}}_{t}\rangle$.
The exact position of their poles is at
$s_{2},s_{3}=\tau_{t}+2m-\frac{2}{3}p$, for $m\in\mathbb{N}_{0}$.

Due to the bound on chaos, Maldacena:2015waa  the planar Mellin amplitude is
bounded in the Regge limit Penedones:2019tng 555The condition
$\mathop{\mathrm{Re}}(s_{2})\leq\frac{p}{3}$ comes from the fact that in the
correlator
$\langle{\mathcal{O}}_{2}{\mathcal{O}}_{2}{\mathcal{O}}_{p}{\mathcal{O}}_{p}\rangle$
the lowest twist exchanged in the $t$ channel has twist equal to $p$.

$\lim_{s_{1}\rightarrow\infty}M(s_{1},s_{2})\leq
O(s_{1}^{-2})\,,\,\,\,\mathop{\mathrm{Re}}(s_{2})\leq\frac{p}{3}\,.$ (13)

This allows us to write a dispersion relation for the Mellin amplitude
$M(s_{1},s_{2})$. To do that, we keep the variable $s_{3}$ fixed and rewrite
the Mellin amplitude at a generic point as a contour integral in the $s_{1}$
plane. Afterwards, we deform the contour as illustrated in figure 1.

$s_{1}$$s_{1}^{\prime}$$\tau_{s}+2m-\frac{2}{3}p$$-s_{3}-\tau_{t}-2m+\frac{2}{3}p$
Figure 1: Contour deformation involved in the dispersion relation. The
integral around a generic point is equal to the sum over the physical poles
plus a contribution from the arc at infinity. The contribution coming from the
arc at infinity vanishes due to the bound on chaos.

The dispersion relation reads

$\displaystyle M(s_{1},s_{2})$
$\displaystyle=\oint_{s_{1}}\frac{ds^{\prime}}{2\pi
i}\frac{M(s^{\prime},-s_{3}-s^{\prime})}{s^{\prime}-s_{1}}$ (14)
$\displaystyle=\sum_{{\mathcal{O}}_{s}}\sum_{m=0}^{\infty}C_{s}\frac{\mathcal{Q}_{s}(s_{3};\tau_{s},\ell,m)}{s_{1}-\tau_{s}-2m+\frac{2}{3}p}+\sum_{{\mathcal{O}}_{t}}\sum_{m=0}^{\infty}C_{t}\frac{\mathcal{Q}_{t}(s_{3};\tau_{t},\ell,m)}{s_{2}-\tau_{t}-2m+\frac{2}{3}p}\,,$

where $\ell$ labels the spin of the exchanged operator. The kinematic
functions $\mathcal{Q}_{s}(s_{3};\tau_{s},\ell,m)$ and
$\mathcal{Q}_{t}(s_{3};\tau_{t},\ell,m)$ are related to Mack polynomials
Mack:2009mi  and their expressions can be found in Appendix C.

Similarly to (5), $M(s_{1},s_{2})$ receives contributions from short and long
multiplets. When plugged in into the dispersion relation (14), the information
of protected operators666See Appendix A for explicit OPE coefficients.
completely determines the supergravity amplitude. Away from the strict
$\lambda\to\infty$, single-trace long multiplets start gaining corrections to
their OPE data, thus producing the “stringy” $\frac{1}{\sqrt{\lambda}}$
corrections in (8). We will comment more on that in the next section.777Double
trace operators do not contribute to (14) at $\mathcal{O}(\frac{1}{c})$.

###  2.3 OPE data of stringy operators

At strong coupling, the superprimaries belonging to long multiplets gain large
anomalous dimensions. Holographically these are related to massive string
states. In fact, at leading order in $\frac{1}{\sqrt{\lambda}}$, their twist
$\tau$ is given by $mR$, where $m$ is the mass of the string state and $R$ is
the radius of AdS. In type IIB string theory, the string energy levels are
$m^{2}=4\frac{\delta}{\alpha^{\prime}}$, with $\delta\in\mathbb{N}$. As a
consequence

$\displaystyle\tau=2\sqrt{\delta}\,\lambda^{\frac{1}{4}},\,\,\,\delta\in\mathbb{N}\,,$
(15)

where recall $\lambda=\frac{R^{4}}{\alpha^{\prime 2}}$. Apart from $\delta$,
the OPE data of each stringy operator will depend on $\lambda$, $p$, the spin
$\ell$ and on other possible quantum numbers, which we will collectively
denote by $\hat{r}$. At leading order, multiple operators can share the same
$\delta,\ell$ and transform in the same $R$-symmetry representations. This
degeneracy (i.e. the range of $\hat{r}$) has been discussed in Alday:2023flc
and, unfortunately, can not be resolved by just looking at four-point
functions of half-BPS operators. As a consequence, in our setup, we will not
be able to obtain OPE data for the unmixed operators, but just to compute
averages over the quantum numbers $\hat{r}$

$\displaystyle\langle\ldots\rangle_{\delta,\ell}=\sum_{\hat{r}}\ldots\,.$ (16)

When the exchanged operators have very large dimensions, as in this case, we
can actually perform the sums $\sum_{m=0}^{\infty}$ in (14). It was shown in
Alday:2022uxp  that this sum is dominated by $m\sim\tau^{2}$, so we can set
$m=x\tau^{2}$ and replace the sum with the integral

$\sum_{m=0}^{\infty}\to\tau^{2}\int_{0}^{\infty}dx\,.$ (17)

Some details on this can be found in Appendix D. In this way we can completely
fix the functional dependence of the OPE data on $\lambda$. The twists go as

$\displaystyle\tau_{s}(\delta,\ell,\hat{r};\lambda,p)$
$\displaystyle=2\sqrt{\delta}\,\lambda^{\frac{1}{4}}+\tau_{s,1}(\delta,\ell,\hat{r};p)+\tau_{s,2}(\delta,\ell,\hat{r};p)\lambda^{-\frac{1}{4}}+\ldots\,,$
(18) $\displaystyle\tau_{t}(\delta,\ell,\hat{r};\lambda,p)$
$\displaystyle=2\sqrt{\delta}\,\lambda^{\frac{1}{4}}+\tau_{t,1}(\delta,\ell,\hat{r};p)+\tau_{t,2}(\delta,\ell,\hat{r};p)\lambda^{-\frac{1}{4}}+\ldots\,,$

and the OPE coefficients read

$\displaystyle C_{s}(\delta,\ell,\hat{r};\lambda,p)$
$\displaystyle=\frac{\pi^{3}(-1)^{p}\,\tau_{s}(\delta,\ell,\hat{r};\lambda,p)^{2p+2}\,2^{-2\ell-2p-2\tau_{s}(\delta,\ell,\hat{r};\lambda,p)-8}}{(\ell+1)\Gamma(p-1)\Gamma(p)\sin^{2}\left(\frac{\pi\tau_{s}(\delta,\ell,\hat{r};\lambda,p)}{2}\right)}f_{s}(\delta,\ell,\hat{r};\lambda,p)\,,$
(19) $\displaystyle C_{t}(\delta,\ell,\hat{r};\lambda,p)$
$\displaystyle=\frac{\pi^{3}(-1)^{\ell}\tau_{t}(\delta,\ell,\hat{r};\lambda,p)^{2p+2}2^{-2\ell-2p-2\tau_{t}(\delta,\ell,\hat{r};\lambda,p)-8}}{(\ell+1)\Gamma(p-1)\Gamma(p)\sin^{2}\left(\frac{\pi
p}{2}+\frac{\pi\tau(\delta,\ell,\hat{r};\lambda,p)}{2}\right)}f_{t}(\delta,\ell,\hat{r};\lambda,p)\,,$
$\displaystyle f_{s}(\delta,\ell,\hat{r};\lambda,p)$
$\displaystyle=f_{s,0}(\delta,\ell,\hat{r};p)+f_{s,1}(\delta,\ell,\hat{r};p)\lambda^{-\frac{1}{4}}+f_{s,2}(\delta,\ell,\hat{r};p)\lambda^{-\frac{1}{2}}+\ldots\,,$
$\displaystyle f_{t}(\delta,\ell,\hat{r};\lambda,p)$
$\displaystyle=f_{t,0}(\delta,\ell,\hat{r};p)+f_{t,1}(\delta,\ell,\hat{r};p)\lambda^{-\frac{1}{4}}+f_{t,2}(\delta,\ell,\hat{r};p)\lambda^{-\frac{1}{2}}+\ldots\,.$

##  3 Worldsheet method

In this section we determine $A^{(1)}(S,T)$ using the method developed in
Alday:2023jdk  for the correlator
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$.
We make an ansatz for the world-sheet integrand which has transcendentality
three and contains single-valued multiple polylogs and single-valued multiple
zeta values. This was motivated in Alday:2023jdk ; Alday:2023mvu  with a
simple toy model of strings in AdS expanded around flat space. The parameters
of the ansatz are fixed by matching the residues with those predicted by the
OPE. We determine these residues directly by applying the Borel transform (9)
to the OPE poles of the Mellin amplitude in (14).

###  3.1 Poles and residues

To discuss how the poles and residues of $A(S,T)$ in the $1/\sqrt{\lambda}$
expansion depend on OPE data, we follow Appendix C of Alday:2022xwz . For the
Mellin amplitude we know that non-perturbatively all its poles originate from
the OPE and have the form (we consider the poles in $s_{1}$ for concreteness)

$M(s_{1},s_{2})\sim\frac{C_{s}\mathcal{Q}_{s}(s_{3};\tau,\ell,m)}{s_{1}-\tau-2m+\frac{2}{3}p}\,,$
(20)

where $m$ labels conformal descendants. We will see that each conformal family
will become one particle in the flat space limit, so we have to consider the
sum over descendants888Considering this sum over poles is justified by the
dispersion relation (14).

$M_{\tau,\ell}(s_{1},s_{2})=\sum_{m=0}^{\infty}\frac{C_{s}\mathcal{Q}_{s}(s_{3};\tau,\ell,m)}{s_{1}-\tau-2m+\frac{2}{3}p}\,.$
(21)

The transform (9) of (21) is

$A_{\tau,\ell}(S,T)=2\Gamma(p-1)\Gamma(p)\int_{-i\infty+\kappa}^{+i\infty+\kappa}\frac{d\alpha}{2\pi
i}e^{\alpha}\alpha^{-p-4}\sum_{m=0}^{\infty}\frac{C_{s}\mathcal{Q}_{s}(\frac{2\sqrt{\lambda}U}{\alpha};\tau,\ell,m)}{\frac{2\sqrt{\lambda}S}{\alpha}-\tau-2m+\frac{2}{3}p}\,.$
(22)

We exchange the integral and summation and for each $m$ pick the pole at999At
this point we should explain why we do not pick up the pole at $\alpha=0$. The
reason is the absence of a bulk point UV singularity of planar $\mathcal{N}=4$
SYM at finite ’t Hooft coupling $\lambda$, as discussed in Maldacena:2015iua .
In the bulk point limit, we expect a divergence like
$\frac{1}{(z-\overline{z})^{d-3}}=\frac{1}{z-\overline{z}}$. Translating that
to Mellin space implies that the Mellin amplitude grows at most as $M(\xi
S,\xi T)\sim\xi^{-p-4}$ in the limit of very large $\xi$ Penedones:2019tng .

$\alpha=\alpha_{*}\equiv\frac{2\sqrt{\lambda}S}{\tau+2m-\frac{2}{3}p}\,.$ (23)

Next we have to sum over $m$ as explained in (17). We arrive at

$A_{\tau,\ell}(S,T)=-2\Gamma(p-1)\Gamma(p)\frac{\tau^{2}}{2\sqrt{\lambda}S}\int_{0}^{\infty}dx\
e^{\alpha_{*}}\alpha_{*}^{-p-2}C_{s}\mathcal{Q}_{s}\left(\tfrac{2\sqrt{\lambda}U}{\alpha_{*}};\tau,\ell,x\tau^{2}\right)\,.$
(24)

We can now expand the integrand at large $\lambda$, using that
$\tau\sim\lambda^{1/4}$. The answer has the form

$-2\Gamma(p-1)\Gamma(p)\frac{\tau^{2}}{2\sqrt{\lambda}S}e^{\alpha_{*}}\alpha_{*}^{-p-2}C_{s}\mathcal{Q}_{s}\left(\tfrac{2\sqrt{\lambda}U}{\alpha_{*}};\tau,\ell,x\tau^{2}\right)=\frac{1}{\lambda^{\frac{3}{2}}}\frac{e^{-\frac{1}{4x}+\frac{\sqrt{\lambda}S}{\tau^{2}x}}}{x^{2}}\sum_{i=0}^{\infty}\frac{1}{\lambda^{i/4}}P^{(i)}\left(\tfrac{1}{x}\right)\,,$
(25)

where $P^{(i)}\left(\tfrac{1}{x}\right)$ are polynomials in $\frac{1}{x}$ with
increasing degree, for instance degree $\frac{3}{2}i$ for even $i$. The
leading term is given by

$P^{(0)}\left(\tfrac{1}{x}\right)=\frac{\delta^{p}S^{-p-3}f_{s;0}(\delta,\ell)C_{\ell}^{(1)}\left(1+\frac{2T}{S}\right)}{4(\ell+1)}\,,$
(26)

where $C_{\ell}^{(1)}\left(x\right)$ is a Gegenbauer polynomial, as
appropriate for partial waves in flat space. We can now do the integrals in
$x$ using

$\int_{0}^{\infty}dx\,\frac{e^{-\frac{1}{4x}+\frac{\sqrt{\lambda}S}{\tau^{2}x}}}{x^{2}}=-\frac{\frac{\tau^{2}}{\sqrt{\lambda}}}{S-\frac{\tau^{2}}{4\sqrt{\lambda}}}=-\frac{4\delta}{S-\delta}+O\left(\lambda^{-\frac{1}{4}}\right)\,.$
(27)

The effect of the polynomials $P^{(i)}\left(\tfrac{1}{x}\right)$ is that each
additional power of $\frac{1}{x}$ increases the order of the pole by one, as
$\frac{1}{x}$ can be replaced by $\frac{\tau^{2}}{\sqrt{\lambda}}\partial_{S}$
acting on both sides of (27).

We first compute the poles and residues of $A_{\tau,\ell}(S,T)$ at order
$\lambda^{-\frac{1}{4}}$. Since $A_{\tau,\ell}(S,T)$ should have no
corrections at this order (10) we set the residues to zero, which fixes the
OPE data at this order to the values given in (47) below.

At the next order $P^{(2)}\left(\tfrac{1}{x}\right)$ is a polynomial of degree
3 so that there are poles up to fourth order. The residues in terms of OPE
data are $p$-dependent generalisations of equation (C.22) of Alday:2022xwz .
We include them in an ancillary Mathematica file. Crucially, the poles of
order four and three depend only on OPE data that was fixed at lower orders.

###  3.2 World-sheet correlator

Next we make an ansatz for the world-sheet integral representation of our
correlator that respects the crossing symmetry $A^{(1)}(S,T)=A^{(1)}(S,U)$

$A^{(1)}(S,T)=B^{(1)}_{1}(S,T)+B^{(1)}_{1}(S,U)+B^{(1)}_{1}(U,T)+B^{(1)}_{2}(S,T)+B^{(1)}_{2}(S,U)\,,$
(28)

where

$B^{(1)}_{i}(S,T)=\int
d^{2}z|z|^{-2S-2}|1-z|^{-2T-2}G^{(1)}_{i}(S,T,z)\,,\quad i=1,2\,,$ (29)

and crossing symmetry implies

$B^{(1)}_{1}(U,T)=B^{(1)}_{1}(T,U)\quad\Leftrightarrow\quad
G^{(1)}_{1}(S,T,z)=G^{(1)}_{1}(T,S,1-z)\,.$ (30)

Note that there is no symmetry constraint on $B^{(1)}_{2}(S,T)$ and the only
reason to include the terms $B^{(1)}_{1}(S,T)$ and $B^{(1)}_{1}(S,U)$ in the
ansatz is to make it easier to compare to the fully crossing symmetric case
$p=2$, since in this case it is easy to see that our ansatz implies

$B^{(1)}_{2}(S,T)=0\,,\quad p=2\,.$ (31)

We also impose symmetry under the exchange $z\leftrightarrow\overline{z}$ and
define the following combinations of single-valued multiple
polylogarithms101010See Brown:2004ugm ; Dixon:2012yy  and also Alday:2023jdk ;
Alday:2023mvu  for details on these functions in this context.

$\displaystyle\mathcal{L}^{s}_{w}(z)={}$
$\displaystyle\mathcal{L}_{w}(z)+\mathcal{L}_{w}(1-z)+\mathcal{L}_{w}(\overline{z})+\mathcal{L}_{w}(1-\overline{z})\,,$
(32) $\displaystyle\mathcal{L}^{a}_{w}(z)={}$
$\displaystyle\mathcal{L}_{w}(z)-\mathcal{L}_{w}(1-z)+\mathcal{L}_{w}(\overline{z})-\mathcal{L}_{w}(1-\overline{z})\,.$

We choose the following weight 3 basis Alday:2023mvu

$\displaystyle\mathcal{L}^{(1)s}={}$
$\displaystyle\Big{(}\mathcal{L}^{s}_{000}(z),\mathcal{L}^{s}_{001}(z),\mathcal{L}^{s}_{010}(z),\zeta(3)\Big{)}\,,$
(33) $\displaystyle\mathcal{L}^{(1)a}={}$
$\displaystyle\Big{(}\mathcal{L}^{a}_{000}(z),\mathcal{L}^{a}_{001}(z),\mathcal{L}^{a}_{010}(z)\Big{)}\,.$

Our ansatz for the worldsheet correlator is then

$\displaystyle G^{(1)}_{i}(S,T,z)$
$\displaystyle=\sum\limits_{u=1}^{4}r^{(1)s}_{i,u}(S,T)\mathcal{L}^{(1)s}_{u}+\sum\limits_{u=1}^{3}r^{(1)a}_{i,u}(S,T)\mathcal{L}^{(1)a}_{u}\,,$
(34)

where $r^{(1)s/a}_{i,u}(S,T)$ are rational functions of homogeneity 0 and we
assume the denominator to be $U^{2}$. Furthermore $r^{(1)s/a}_{1,u}(S,T)$ have
symmetry properties according to (30)

$r^{(1)s}_{1,u}(S,T)=\frac{f_{1,u}(p)(S+T)^{2}+f_{2,u}(p)ST}{(S+T)^{2}}\,,\qquad
r^{(1)a}_{1,u}(S,T)=f_{3,u}(p)\frac{S-T}{S+T}\,,$ (35)

whereas $r^{(1)s/a}_{2,u}(S,T)$ are of the form

$\frac{f_{4,u}(p)(S+T)^{2}+f_{5,u}(p)ST+f_{6,u}(p)(S^{2}-T^{2})}{(S+T)^{2}}\,.$
(36)

In total our ansatz depends on 32 functions of $p$.

We can now compute the residues of our ansatz at $S=1,2,\ldots$ and
$T=1,2,\ldots$ and match them with those computed in section 3.1 in terms of
OPE data. We further compute the residues at $S=0$ and $T=0$ and match them to
the supergravity term (11). As discussed in Alday:2023mvu  the functions
$G^{(1)}_{i}(S,T,z)$ have certain ambiguities, i.e. terms that we can add to
$G^{(1)}_{i}(S,T,z)$ without changing $A^{(1)}(S,T)$. With the ansatz
discussed here, there are 6 such ambiguities, so that our final answer has the
form

$G^{(1)}_{i}(S,T,z)=\sum\limits_{u=1}^{4}r^{(1)s}_{i,u}\mathcal{L}^{(1)s}_{u}+\sum\limits_{u=1}^{3}r^{(1)a}_{i,u}\mathcal{L}^{(1)a}_{u}+\sum_{j=1}^{6}a_{j}\left(\sum\limits_{u=1}^{4}\hat{r}^{(1)s}_{i,ju}\mathcal{L}^{(1)s}_{u}+\sum\limits_{u=1}^{3}\hat{r}^{(1)a}_{i,ju}\mathcal{L}^{(1)a}_{u}\right)\,,$
(37)

with

$\displaystyle r^{(1)s}_{1}$
$\displaystyle=\frac{1}{24}\left(-p^{2},2(p-2)p,p^{2}-2p-6,48\right)\,,$ (38)
$\displaystyle r^{(1)a}_{1}$
$\displaystyle=\frac{p^{2}(S-T)}{24(S+T)}\left(-1,2,1\right)\,,$
$\displaystyle r^{(1)s}_{2}$
$\displaystyle=\frac{p(p-2)}{24(S+T)}\left(3S,-2(2S+T),-2S-T,0\right)\,,$
$\displaystyle r^{(1)a}_{2}$
$\displaystyle=\frac{p(p-2)}{24(S+T)}\left(3S,-2(2S-T),-2S+T\right)\,.$

The coefficients $a_{j}$ are unfixed and multiply the ambiguities, which are
given in appendix E.

##  4 Dispersive sum rules method

In this section we compute $A^{(1)}(S,T)$ with a different method, following
Alday:2022xwz . We apply the low energy expansion (8) to a dispersion relation
to obtain dispersive sum rules, which relate Wilson coefficients to OPE data.
We then make an ansatz in terms of nested sums for certain combinations of OPE
data, and fix the parameters of the ansatz with locality and single-valuedness
constraints for the Wilson coefficients.

In (14) we have seen a way to write $M(s_{1},s_{2})$ dispersively in terms of
two channels. Unfortunately, written in this way, the symmetry under the
exchange of $s_{2}$ and $s_{3}$ is obscured. Therefore, to make it manifest,
in this section we implement a different dispersion relation, where, instead
of $s_{3}$, we keep fixed a new parameter $r=\frac{s_{2}s_{3}}{s_{1}}$. With
this change of coordinates, $s_{2}$ and $s_{3}$ are now treated on the same
footing, thus leading to a more symmetric dispersion relation and eventually
to simpler sum rules.111111In spirit, this is very similar to the crossing
symmetric dispersion relations of Sinha:2020win ; Gopakumar:2021dvg  for the
fully crossing symmetric case. The details on how this dispersion relation is
derived are deferred to Appendix F. In the following we simply report the
associated sum rules and the corresponding results for the Wilson coefficients
and OPE data.

###  4.1 Flat space OPE data

The first of the dispersive sum rules relating Wilson coefficients and OPE
data reads

$\displaystyle\xi_{a,b}^{(0)}(p)$
$\displaystyle=\sum_{\delta=1}^{\infty}\sum_{q=0}^{b}\frac{c_{s,a,b,q}^{(0)}F_{s;q}^{(0)}(\delta)+c_{t,a,b,q}^{(0)}F_{t;q}^{(0)}(\delta)}{\delta^{3+a+b}}\,,$
(39)

where

$\displaystyle F_{s;q}^{(0)}(\delta)$
$\displaystyle=\frac{4^{q}}{\Gamma(2q+2)}\sum_{\ell=0,2}^{2\delta-2}\langle
f_{s,0}\rangle_{\delta,\ell}(\ell-q+1)_{q}(2+\ell)_{q}\,,$ (40) $\displaystyle
F_{t;q}^{(0)}(\delta)$
$\displaystyle=\frac{4^{q}}{\Gamma(2q+2)}\sum_{\ell=0,1}^{2\delta-2}\langle
f_{t,0}\rangle_{\delta,\ell}(\ell-q+1)_{q}(2+\ell)_{q}\,,$

and

$\displaystyle c_{s,a,b,q}^{(0)}$
$\displaystyle=\frac{(-1)^{q}q2^{-a-b-1}\Gamma(2b-q)}{\Gamma(b-q+1)},~{}~{}~{}q>0\,,$
(41) $\displaystyle c_{s,a,b,0}^{(0)}$
$\displaystyle=2^{-a-1}\delta_{b,0},~{}~{}~{}q=0\,,$ $\displaystyle
c_{t,a,b,q}^{(0)}$
$\displaystyle=\frac{(-1)^{a+1+q}\Gamma(a)\Gamma(b+1)2^{-a-b-1}(-a-b+q)\,}{\Gamma(b-q+1)\Gamma(a-b+q+1)}$
$\displaystyle\times_{3}F_{2}(q,q-b,-a-b+q+1;-a-b+q,a-b+q+1;1),~{}~{}~{}q>0\,,$
$\displaystyle c_{t,a,b,0}^{(0)}$
$\displaystyle=\frac{(-2)^{-a-b-1}(a+b)\Gamma(b-a)}{\Gamma(1-a)},~{}~{}~{}q=0\,.$

The Wilson coefficients $\xi_{a,b}^{(0)}(p)$ can be read off from the flat
space Virasoro-Shapiro amplitude (see (105) and (106)). Thus we can
systematically analyse the equations (39), solving for the functions
$F_{s;q}^{(0)}(\delta)$ and $F_{t;q}^{(0)}(\delta)$. As an example, we work
out the $b=0$ case in appendix G.2. After studying many cases, the general
conclusion we extract is that

$\displaystyle
F_{s;q}^{(0)}(\delta)=F_{t;q}^{(0)}(\delta)=F_{q}^{(0)}(\delta),\ \forall
q\in\mathbb{N}_{0}\,,\quad\Leftrightarrow\quad\langle
f_{s,0}\rangle_{\delta,\ell}=\langle f_{t,0}\rangle_{\delta,\ell}=\langle
f_{0}\rangle_{\delta,\ell}\,,$ (42)

where $F_{q}^{(0)}(\delta)$ and $\langle f_{0}\rangle_{\delta,\ell}$ are the
same as in the
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$
case (Alday:2022xwz, ), i.e.

$\displaystyle
F_{q}^{(0)}(\delta)=\sum\limits_{d=\lfloor\frac{q+1}{2}\rfloor}^{q}\sum\limits_{\begin{subarray}{c}s_{1},\ldots,s_{d}\in\\{1,2\\}\\\
s_{1}+\ldots+s_{d}=q\end{subarray}}2^{\sum_{i}\delta_{s_{i},1}}\delta^{q}Z_{s_{1},\ldots,s_{d}}(\delta-1)\,,$
(43)

where $Z_{s_{1},\ldots,s_{d}}(\delta-1)$ is an Euler-Zagier sum defined by

$\displaystyle
Z_{s_{1},s_{2},s_{3},\ldots}(N)=\sum_{n=1}^{N}\frac{Z_{s_{2},s_{3},\ldots}(n-1)}{n^{s_{1}}}\,,\qquad
Z(N)=1\,,\qquad Z_{s_{1},s_{2},s_{3},\ldots}(0)=0\,.$ (44)

In particular we find that $\langle f_{t,0}\rangle_{\delta,\ell}=0$, if $\ell$
is odd. Also, we find that any $\langle f_{s,0}\rangle_{\delta,\ell}$ does not
depend on $p$, i.e. at leading order in $\lambda$ the full $p$-dependence of
the OPE coefficients is in the prefactor in (19). All this is in agreement
with the more general result of Alday:2023flc  for the product of leading OPE
coefficients
$\langle{\mathcal{O}}_{p_{1}}{\mathcal{O}}_{p_{2}}{\mathcal{O}}_{\Delta,\ell,[0,n,0]}\rangle\times\langle{\mathcal{O}}_{p_{3}}{\mathcal{O}}_{p_{4}}{\mathcal{O}}_{\Delta,\ell,[0,n,0]}\rangle$.

At the next order we impose that there are no $\lambda^{-\frac{1}{4}}$
corrections to the flat space Wilson coefficients. From the string theory
perspective, this would correspond to fractional powers of
$\alpha^{\prime}/R^{2}$, and we assume that such powers are absent. This leads
to the equation

$\displaystyle\ 0={}$
$\displaystyle\sum_{\delta=1}^{\infty}\sum_{q=0}^{b}\frac{c_{s,a,b,q}^{(0)}\big{(}F_{s;q}^{(1)}(\delta)-(3+a+b)T_{s;q}^{(1)}(\delta)\big{)}}{\delta^{\frac{7}{2}+a+b}}$
(45)
$\displaystyle+\sum_{\delta=1}^{\infty}\sum_{q=0}^{b}\frac{c_{t,a,b,q}^{(0)}\big{(}F_{t;q}^{(1)}(\delta)-(3+a+b)T_{t;q}^{(1)}(\delta)\big{)}}{\delta^{\frac{7}{2}+a+b}}\,,$

where

$\displaystyle F_{s;q}^{(1)}(\delta)$
$\displaystyle=\frac{4^{q}}{\Gamma(2q+2)}\sum_{\ell=0,2}^{2\delta-2}(\ell-q+1)_{q}(2+\ell)_{q}\left(\sqrt{\delta}\langle
f_{s,1}\rangle_{\delta,\ell}-\langle f_{s,0}\rangle_{\delta,\ell}\left(\ell
p+\ell+2p+\frac{7}{4}\right)\right),$ $\displaystyle T_{s;q}^{(1)}(\delta)$
$\displaystyle=\frac{4^{q}}{\Gamma(2q+2)}\sum_{\ell=0,2}^{2\delta-2}\langle
f_{s,0}\rangle_{\delta,\ell}(\ell-q+1)_{q}(2+\ell)_{q}(\tau_{s,1}(\delta,\ell)+\ell+2),$
$\displaystyle F_{t;q}^{(1)}(\delta)$
$\displaystyle=\frac{4^{q}}{\Gamma(2q+2)}\sum_{\ell=0,1}^{2\delta-2}(\ell-q+1)_{q}(2+\ell)_{q}\left(\sqrt{\delta}\langle
f_{t,1}\rangle_{\delta,\ell}-\langle f_{t,0}\rangle_{\delta,\ell}\left(\ell
p+\ell+\frac{p^{2}}{2}+\frac{15}{4}\right)\right),$ $\displaystyle
T_{t;q}^{(1)}(\delta)$
$\displaystyle=\frac{4^{q}}{\Gamma(2q+2)}\sum_{\ell=0,1}^{2\delta-2}\langle
f_{t,0}\rangle_{\delta,\ell}(\ell-q+1)_{q}(2+\ell)_{q}(\tau_{t,1}(\delta,\ell)+\ell+2).$
(46)

The solution to these equations is121212We assumed in this analysis that all
operators acquire the same finite shift, as argued in section $6.1.1$ of
Gromov:2023hzc .

$\displaystyle\tau_{s,1}(\delta,\ell)$
$\displaystyle=\tau_{t,1}(\delta,\ell)=-2-\ell\,,$ (47) $\displaystyle\langle
f_{s,1}\rangle_{\delta,\ell}$ $\displaystyle=\frac{\langle
f_{s,0}\rangle_{\delta,\ell}}{\sqrt{\delta}}\left(\ell
p+\ell+2p+\frac{7}{4}\right)\,,$ $\displaystyle\langle
f_{t,1}\rangle_{\delta,\ell}$ $\displaystyle=\frac{\langle
f_{t,0}\rangle_{\delta,\ell}}{\sqrt{\delta}}\left(\ell
p+\ell+\frac{p^{2}}{2}+\frac{15}{4}\right)\,.$

###  4.2 First curvature correction

At the next order we find the dispersive sum rule

$\displaystyle\xi^{(1)}_{a,b}(p)={}$
$\displaystyle\sum_{\delta=1}^{\infty}\sum_{q=0}^{b}\frac{c_{s,a,b,q}^{(0)}\big{(}F_{s;q}^{(2)}(\delta)-(3+a+b)T_{s;q}^{(2)}(\delta)\big{)}+c_{s,a,b,q}^{(2,0)}F_{s;q}^{(0)}(\delta)+c_{s,a,b,q}^{(2,1)}F_{s;q+1}^{(0)}(\delta)}{\delta^{4+a+b}}$
(48)
$\displaystyle+\sum_{\delta=1}^{\infty}\sum_{q=0}^{b}\frac{c_{t,a,b,q}^{(0)}\big{(}F_{t;q}^{(2)}(\delta)-(3+a+b)T_{t;q}^{(2)}(\delta)\big{)}+c_{t,a,b,q}^{(2,0)}F_{t;q}^{(0)}(\delta)+c_{t,a,b,q}^{(2,1)}F_{t;q+1}^{(0)}(\delta)}{\delta^{4+a+b}}\,,$

where

$\displaystyle F_{s;q}^{(2)}(\delta)$
$\displaystyle=\frac{4^{q}}{\Gamma(2q+2)}\sum_{\ell=0,1}^{2\delta-2}(\ell-q+1)_{q}(2+\ell)_{q}\left(\delta\langle
f_{s,2}\rangle_{\delta,\ell}-\frac{1}{4}(4p^{2}+9p+5)\ell\langle
f_{s,0}\rangle_{\delta,\ell}\right),$ $\displaystyle T_{s;q}^{(2)}(\delta)$
$\displaystyle=\frac{4^{q}}{\Gamma(2q+2)}\sum_{\ell=0,1}^{2\delta-2}\sqrt{\delta}(\ell-q+1)_{q}(2+\ell)_{q}\langle
f_{s,0}\tau_{s,2}\rangle_{\delta,\ell},$ $\displaystyle F_{t;q}^{(2)}(\delta)$
$\displaystyle=\frac{4^{q}}{\Gamma(2q+2)}\sum_{\ell=0,1}^{2\delta-2}(\ell-q+1)_{q}(2+\ell)_{q}\left(\delta\langle
f_{t,2}\rangle_{\delta,\ell}-\frac{1}{4}\left(2p^{3}-2p^{2}+9p+13\right)\ell\langle
f_{t,0}\rangle_{\delta,\ell}\right),$ $\displaystyle T_{t;q}^{(2)}(\delta)$
$\displaystyle=\frac{4^{q}}{\Gamma(2q+2)}\sum_{\ell=0,1}^{2\delta-2}\sqrt{\delta}(\ell-q+1)_{q}(2+\ell)_{q}\langle
f_{t,0}\tau_{t,2}\rangle_{\delta,\ell},$ (49)

and we wrote $c_{s,a,b,q}^{(2,0)}$, $c_{s,a,b,q}^{(2,1)}$,
$c_{t,a,b,q}^{(2,0)}$ and $c_{t,a,b,q}^{(2,1)}$ in (107). The OPE data (49) is
truly sensitive to AdS curvature corrections, since it depends on Wilson
coefficients $\xi^{(1)}_{a,b}(p)$ which are not fixed by the flat space
Virasoro-Shapiro amplitude. The Wilson coefficients $\xi^{(1)}_{a,b}(p)$ are
in general unknown.

The situation regarding the sum rules (48) is now different from the one we
encountered in section 4.1. In section 4.1 the LHS of the sum rules was known
and the Wilson coefficients were an input into the dispersive sum rules and
the OPE data was the output. To make progress we will proceed along the same
lines of Alday:2022xwz . Fixing $b$, we make a general ansatz for the OPE data
that enters (48), as a linear combination of Euler-Zagier sums involving a few
undetermined functions of $p$. Using (48) this determines the Wilson
coefficients in terms of those functions. Afterwards, we impose physical
conditions on the Wilson coefficients (namely single-valuedness) and these
will completely fix all of these functions.

Let us see how this comes about in practice. In complete analogy with the
$p=2$ case, we first assume that $T_{s;q}^{(2)}(\delta)$ and
$T_{t;q}^{(2)}(\delta)$ are linear combinations of Euler-Zagier sums with
maximal depth $q+1$ and maximal weight $q+2$ and that $F_{s;q}^{(2)}(\delta)$
and $F_{t;q}^{(2)}(\delta)$ are linear combinations of Euler-Zagier sums with
maximal depth $q+1$ and maximal weight $q+3$:

$\displaystyle T_{s;q}^{(2)}(\delta)={}$ $\displaystyle A^{1}_{q+2,q+1},$ (50)
$\displaystyle T_{t;q}^{(2)}(\delta)={}$ $\displaystyle A^{2}_{q+2,q+1},$
$\displaystyle F_{s;q}^{(2)}(\delta)={}$ $\displaystyle
A^{3}_{q+3,q+1}+\delta^{3}\zeta(3)A^{4}_{q,q},$ $\displaystyle
F_{t;q}^{(2)}(\delta)={}$ $\displaystyle
A^{5}_{q+3,q+1}+\delta^{3}\zeta(3)A^{6}_{q,q},$

for $q\geq 1$ where

$A^{i}_{w_{\text{max}},d_{\text{max}}}=\sum\limits_{w=1}^{w_{\text{max}}}\sum\limits_{d=1}^{d_{\text{max}}}\sum\limits_{\begin{subarray}{c}s_{1},\ldots,s_{d}\in\mathbb{N}\\\
s_{1}+\ldots+s_{d}=w\end{subarray}}c^{i}_{s_{1},\ldots,s_{d}}(p)\delta^{w}Z_{s_{1},\ldots,s_{d}}(\delta-1)\,.$
(51)

$c^{i}_{s_{1},\ldots,s_{d}}(p)$ are the functions of $p$ that are fixed by our
procedure. For $q=0$, see formula (112).

Second, we impose that the strong coupling expansion cannot contain negative
powers of the Mellin variables apart from the supergravity term. This simple
condition is nontrivial from the point of view of equations (48) and leads to

$\displaystyle\xi_{a,b}^{(1)}(p)=0,\,\,\text{for}\,\,a=0,\ldots,b-1,\forall\,b\in\mathbb{N}.$
(52)

Plugging (48) into (52) leads to constraints on the OPE data.

Third, we impose that $\xi_{a,b}^{(1)}(p)$ is in the ring of single-valued
multiple zeta values. For a practical introduction to this topic, see section
$3.1$ of Alday:2022xwz . We use the MAPLE program HyperlogProcedures
HyperlogProcedures  to do so. In practice, for any fixed $b$ it is enough to
impose single-valuedness for a few values of $a$ to fix all of the
undetermined functions. Afterwards, we check that the resulting expression for
$\xi_{a,b}^{(1)}(p)$ is single-valued for other values of $a$ that we did not
use before.131313For example, for $b=0$, we impose single-valuedness for
$a=0,\ldots,6$ and checked it for $a=7,\ldots,10$. For $b=1$ we imposed it for
$a=1,\ldots,8$ and checked it for $a=9,10,11$. For $b=2$ we imposed it for
$a=2,\ldots,8$ and checked it for $a=9,10,11$.

In this manner we solved (48) for $b=0,\dots,5$. In appendix G.3 we write all
the details on how to implement this for the $b=0$ case. From these cases, we
extract the following general conclusions

$\displaystyle T_{s;q}^{(2)}(\delta)$ $\displaystyle=T_{q}^{(2)}(\delta)\,,$
(53) $\displaystyle T_{t;q}^{(2)}(\delta)$
$\displaystyle=T_{q}^{(2)}(\delta)+\frac{p^{2}-4}{4}F_{q}^{(0)}(\delta)\,,$

where $T_{q}^{(2)}(\delta)$ is the value obtained in the
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$
case Alday:2022xwz  for $T_{s;q}^{(2)}(\delta)$.

As to the OPE coefficients, they are determined by

$\displaystyle F_{s;q}^{(2)}(\delta)$
$\displaystyle=F_{q}^{(2)}(\delta)+(p-2)\big{(}g_{1}(q,p)F_{q}^{(0)}(\delta)+g_{2}(q,p)F_{q+1}^{(0)}(\delta)\big{)}\,,$
(54) $\displaystyle F_{t;q}^{(2)}(\delta)$
$\displaystyle=F_{q}^{(2)}(\delta)+(p-2)\big{(}g_{3}(q,p)F_{q}^{(0)}(\delta)+g_{4}(q,p)F_{q+1}^{(0)}(\delta)\big{)}\,,$

where $F_{q}^{(2)}(\delta)$ is the value obtained in the
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$
case Alday:2022xwz  for $F_{s;q}^{(2)}(\delta)$ and

$\displaystyle g_{1}(q,p)$
$\displaystyle=-\frac{p^{2}}{3}+\frac{1}{6}p\left(3q^{2}+3q+5\right)+2q^{2}+3q+6,$
(55) $\displaystyle g_{2}(q,p)$ $\displaystyle=\frac{1}{2}(q+1)(pq+p+4q+5),$
$\displaystyle g_{3}(q,p)$
$\displaystyle=\frac{1}{24}\left(3p^{3}+4p^{2}+p\left(12q^{2}+18q+41\right)+48q^{2}+84q+78\right),$
$\displaystyle g_{4}(q,p)$ $\displaystyle=g_{2}(q,p).$

The solution for the OPE data completely fixes the Wilson coefficients
$\xi^{(1)}_{a,b}(p)$. At the next order in $\frac{1}{\sqrt{\lambda}}$ this
procedure does not allow us to fix all the undetermined functions.

##  5 Data and checks

###  5.1 OPE data

We can determine all of the averages of the OPE data for each $\delta$ and
$\ell$, either by computing the residues of the poles of the world-sheet
integral (28) and comparing with the residues computed in section 3.1, or by
equating (53) and (54) with (49). For example, for the leading Regge
trajectory we find

$\displaystyle\langle f_{s,0}\tau_{s,2}\rangle_{\delta,2\delta-2}={}$
$\displaystyle\frac{r_{0}(\delta)}{2\delta^{\frac{3}{2}}}(3\delta^{2}-\delta+2),$
(56) $\displaystyle\langle f_{t,0}\tau_{t,2}\rangle_{\delta,2\delta-2}={}$
$\displaystyle\frac{r_{0}(\delta)}{2\delta^{\frac{3}{2}}}(3\delta^{2}-\delta+\frac{p^{2}}{2}),$
$\displaystyle\langle f_{s,2}\rangle_{\delta,2\delta-2}={}$
$\displaystyle\frac{r_{0}(\delta)}{\delta^{2}}\Big{(}\frac{7\delta^{2}}{2}-\frac{7\delta}{12}-\frac{p^{3}}{3}+\left(2\delta^{2}-\delta+\frac{1}{2}\right)p^{2}$
$\displaystyle+\frac{1}{6}\left(24\delta^{2}+3\delta-1\right)p+\delta^{3}\left(2\zeta(3)-\frac{7}{6}\right)-\frac{35}{32}\Big{)},$
$\displaystyle\langle f_{t,2}\rangle_{\delta,2\delta-2}={}$
$\displaystyle\frac{r_{0}(\delta)}{\delta^{2}}\Big{(}\frac{7\delta^{2}}{2}+\frac{17\delta}{12}+\frac{p^{4}}{8}+\left(\delta-\frac{13}{12}\right)p^{3}+\left(2\delta^{2}-\frac{7\delta}{2}+\frac{23}{8}\right)p^{2}$
$\displaystyle+\frac{1}{6}\left(24\delta^{2}+3\delta-28\right)p+\delta^{3}\left(2\zeta(3)-\frac{7}{6}\right)+\frac{77}{32}\Big{)},$

where

$\displaystyle
r_{n}(\delta)=\frac{4^{2-2\delta}\delta^{2\delta-2n-1}(2\delta-2n-1)}{\Gamma(\delta)\Gamma\left(\delta-\left\lfloor\frac{n}{2}\right\rfloor\right)}\,.$
(57)

Using that the operators on the leading Regge trajectory are non-degenerate,
we can extract the twists for these operators with SU$(4)_{R}$ representation
$[0,p-2,0]$ from the second equation in (56)

$\displaystyle\tau^{[0,p-2,0]}\left(\tfrac{\ell}{2}+1,\ell\right)={}$
$\displaystyle\sqrt{2(\ell+2)}\lambda^{\frac{1}{4}}-\ell-2+\frac{3\ell^{2}+10\ell+2p^{2}+8}{4\sqrt{2(\ell+2)}}\lambda^{-\frac{1}{4}}+O(\lambda^{-\frac{3}{4}})\,.$
(58)

This is in agreement with Gromov:2011de .141414With the respect to the
conventions of (2.28) of Gromov:2011de , we need to identify $S=\ell+2$ and
remember the shift $J=(p-2)+2$. A few of the anomalous dimensions for $p>2$
have also been recomputed recently in Gromov:2023hzc  using the quantum
spectral curve, namely

$\displaystyle\tau_{2}^{[0,1,0]}(1,0)$ $\displaystyle=\frac{13}{4}\,,\quad$
$\displaystyle\tau_{2}^{[0,2,0]}(1,0)$
$\displaystyle=5\,,\qquad\tau_{2}^{[0,3,0]}(1,0)=\frac{29}{4}\,,$ (59)
$\displaystyle\tau_{2}^{[0,1,0]}(2,2)$
$\displaystyle=\frac{29}{4\sqrt{2}}\,,\quad$
$\displaystyle\tau_{2}^{[0,2,0]}(2,2)$ $\displaystyle=\frac{9}{\sqrt{2}}\,,$

also in perfect agreement with (58).

At the order we are considering, operators with odd spin only contribute to
the quantity $\langle f_{t,2}\rangle_{\delta,\ell}$, which is the leading
contribution to the OPE coefficients for these operators. For the first few
odd spin Regge trajectories our results for this quantity are

$\displaystyle\langle f_{t,2}\rangle_{\delta\geq 2,2\delta-3}={}$
$\displaystyle-\frac{r_{\frac{1}{2}}(\delta)}{\delta}(\delta-1)\left(p^{2}-4\right)\,,$
$\displaystyle\langle f_{t,2}\rangle_{\delta\geq 3,2\delta-5}={}$
$\displaystyle-\frac{2r_{\frac{3}{2}}(\delta)}{3}(\delta-2)(\delta-1)(\delta+4)\left(p^{2}-4\right)\,,$
(60) $\displaystyle\langle f_{t,2}\rangle_{\delta\geq 4,2\delta-7}={}$
$\displaystyle-\frac{r_{\frac{5}{2}}(\delta)}{45}(\delta-3)\left(10\delta^{4}+73\delta^{3}+128\delta^{2}-352\delta-192\right)\left(p^{2}-4\right)\,.$

Anomalous dimensions for operators with odd spin appear only at higher orders
in the $1/\sqrt{\lambda}$ expansion, i.e. in $A^{(k>1)}(S,T)$. In terms of
degeneracies, the leading odd spin Regge trajectory with $\ell=2\delta-3$ is
particularly simple: it was shown in Alday:2023flc  that all the operators on
this trajectory have degeneracy 2.

###  5.2 Wilson coefficients

The Wilson coefficients that appear in the low energy expansion (10) can be
computed either from the world-sheet correlator (37), as explained in
Alday:2023jdk  (which uses the methods of Zerbini:blah ), or by plugging the
solutions (53-54) into the dispersive sum rule (48) and performing the sums,
following Alday:2022xwz . The first few terms in the expansion are given by

$\displaystyle
A^{(1)}(S,T)=-\frac{p}{S^{2}T^{2}U^{2}}\left(\frac{p}{6}S^{2}+\frac{p-3}{3}TU\right)+p(p-2)S\zeta(5)$
(61)
$\displaystyle+\frac{\zeta(3)^{2}}{3}\left(\left(p^{2}-6p-36\right)S^{2}+2\left(p^{2}+18\right)TU\right)+\zeta(7)\left(2p(p-2)S^{3}-\left(2p^{2}+2p+\tfrac{441}{8}\right)STU\right)$
$\displaystyle+\frac{2\zeta(3)\zeta(5)}{3}\left(\left(p^{2}-10p-81\right)S^{4}+2\left(2p^{2}+4p+81\right)S^{2}TU-\left(2p^{2}+4p+81\right)T^{2}U^{2}\right)+\ldots\,.$

Some values of the Wilson coefficients $\xi_{a,b}^{(1)}(p)$ had been computed
before in the literature. Below we perform a detailed comparison of our
results with all available results in the literature and we find full
agreement. In Binder:2019jwn  the Mellin amplitude was computed to order
$\lambda^{-\frac{5}{2}}$ using localisation methods. We find full agreement
when comparing their results to our prediction for $\xi_{a,0}^{(1)}(p)$ for
$a=0,1$, which is

$\displaystyle\xi_{0,0}^{(1)}(p)$ $\displaystyle=0\,,$ (62)
$\displaystyle\xi_{1,0}^{(1)}(p)$ $\displaystyle=\frac{1}{4}(p-2)p\zeta(5)\,.$
(63)

In Abl:2020dbx  an algorithm to compute the $p$-dependence of Wilson
coefficients in a generic correlator
$\langle\mathcal{O}_{p_{1}}\mathcal{O}_{p_{2}}\mathcal{O}_{p_{3}}\mathcal{O}_{p_{4}}\rangle$
was proposed. This algorithm is based on considering an effective field theory
in full $AdS_{5}\times S_{5}$. With this algorithm it is possible to compute
the $p$-dependence of Wilson coefficients up to a finite number of
undetermined parameters. Take for example the Wilson coefficient
$\xi_{2,0}^{(1)}(p)$, which enters at $O(\lambda^{-3})$. Abl:2020dbx  predicts

$\displaystyle\xi_{2,0}^{(1)}(p)=\frac{1}{24}\zeta(3)^{2}(3\tilde{B}_{1}-24\tilde{E}_{0}+(p-6)p-20),$
(64)

where $\tilde{B}_{1}$ and $\tilde{E}_{0}$ are undetermined rational
numbers.151515More specifically, $B_{1}=\tilde{B}_{1}\zeta^{2}(3)$ and
$E_{0}=\tilde{E}_{0}\zeta^{2}(3)$ are defined in equation $(84)$ of
Abl:2020dbx . Our own computation gives

$\displaystyle\xi_{2,0}^{(1)}(p)=\frac{1}{24}((p-6)p-36)\zeta(3)^{2}\,.$ (65)

We see that (64) and (65) are fully compatible and together they imply
$3\tilde{B}_{1}-24\tilde{E}_{0}+16=0$. Combining this approach with our
results and extending the localisation constraints in Binder:2019jwn  at order
$\lambda^{-3}$ we were able to fix the Mellin amplitude up to an unknown
number $\beta$

$\displaystyle\begin{aligned}
&M(s_{1},s_{2})\Big{|}_{\lambda^{-3}}=\frac{\zeta_{3}^{2}(p)_{4}}{8\Gamma(p-1)}\Big{(}(p+4)_{3}\,s_{1}s_{2}s_{3}+\frac{(p+4)_{2}}{3}(p^{2}-6p-36)\,s_{1}^{2}\\\
&+\frac{2(p+4)_{2}}{3}(p^{2}+18)s_{2}s_{3}+\frac{(p+4)}{3}(p-2)((p^{3}-7p^{2}-24p+36)+\frac{3}{2}\beta(p+2))s_{1}\\\
&+\frac{2(p-12)}{27}(p^{5}-3p^{4}+2p^{3}+138p^{2}+288p+216)+\frac{\beta}{3}(p-2)^{2}p(p+2)\Big{)}\,.\end{aligned}$
(66)

##  6 Conclusions

In this paper we determined the first curvature correction to the AdS
Virasoro-Shapiro amplitude with external KK modes, for the correlator
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{p}\mathcal{O}_{p}\rangle$,
generalising the
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$
results of Alday:2022uxp ; Alday:2022xwz ; Alday:2023jdk . We expect that
higher curvature corrections can be obtained in an analogous way, as
demonstrated in Alday:2023mvu , where the second curvature correction for
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$
was computed.

We also used the opportunity to recapitulate and contrast the two methods that
were previously used to solve the problem at hand. The method introduced in
Alday:2023mvu  and used in section 3 assumes the existence of a world-sheet
integral with insertions of single-valued multiple polylogs, which is
suggested by the fact that world-sheet non-linear sigma models for AdS
expanded around flat space receive contributions from flat space string
amplitudes with insertions of additional soft gravitons Alday:2023jdk ;
Alday:2023mvu . The ansatz is then fixed essentially by consistency with the
OPE.

The method used in section 4 was introduced earlier, in Alday:2022xwz , and
involves an ansatz for certain combinations of OPE data in terms of Euler-
Zagier sums. This ansatz was inspired by the structure of the dispersive sum
rules and their leading solution which can be inferred from flat space. Of
course, both methods lead to the same results.

An obvious future direction in the same line as the present paper would be to
consider the correlator with generic half-BPS operators
$\langle\mathcal{O}_{p_{1}}\mathcal{O}_{p_{2}}\mathcal{O}_{p_{3}}\mathcal{O}_{p_{4}}\rangle$.
This would give access to operators in even more general $\text{SU}(4)_{R}$
representations, see (72).

Another generalisation would be to relax the assumption
$p\ll\lambda^{\frac{1}{4}}$. As discussed in Alday:2023flc  (see also
Aprile:2020luw ), in the regime $p\ll\lambda^{\frac{1}{4}}$ all operators that
are not KK modes of the $\text{SO}(5)$ singlet are suppressed by powers of
$\lambda^{-\frac{1}{4}}$. Studying correlators in the regime
$p\sim\lambda^{\frac{1}{4}}$ would make it easier to access those operators,
which can also be characterised as having momentum in the $S^{5}$ directions.

## Acknowledgements

We thank Fernando Alday, Joseph Minahan, Erik Panzer and Oliver Schnetz for
useful discussions. GF thanks the Oxford Mathematical Institute for
hospitality during the early stages of this work and Agnese Bissi, Parijat
Dey, Simon Ekhammar and Dima Volin for valuable discussions. The work of GF is
supported by Knut and Alice Wallenberg Foundation under grant KAW 2016.0129
and by VR grant 2018-04438. The work of TH is supported by the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No 787185). JS is supported by the STFC
grant ST/T000864/1.

##  Appendix A $\mathcal{N}=4$ stress tensor four point function

We normalise two point functions of half-BPS operators $\mathcal{O}_{p}(x,y)$
as

$\displaystyle\langle\mathcal{O}_{p}(x_{1},y_{1})\mathcal{O}_{p}(x_{2},y_{2})\rangle=g_{12}^{p}$
(67)

with $g_{ij}\equiv\frac{y_{ij}}{x_{ij}^{2}}$, where $y_{ij}=y_{i}\cdot y_{j}$
and $x_{ij}=x_{i}-x_{j}$. The correlator of four generic half-BPS operators
$\mathcal{O}_{p_{i}}$ can be written as Dolan:2003hv

$\displaystyle\langle\mathcal{O}_{p_{1}}(x_{1},y_{1})\mathcal{O}_{p_{2}}(x_{2},y_{2})\mathcal{O}_{p_{3}}(x_{3},y_{3})\mathcal{O}_{p_{4}}(x_{4},y_{4})\rangle$
$\displaystyle=\mathcal{K}_{\\{p_{i}\\}}(x_{i},y_{i})\mathcal{G}_{\\{p_{i}\\}}(z,\overline{z};\alpha,\overline{\alpha})\,,$
(68) $\displaystyle\mathcal{K}_{\\{p_{i}\\}}(x_{i},y_{i})$
$\displaystyle=g_{12}^{\frac{p_{1}+p_{2}}{2}}g_{34}^{\frac{p_{3}+p_{4}}{2}}\left(\frac{g_{24}}{g_{14}}\right)^{\frac{p_{21}}{2}}\left(\frac{g_{13}}{g_{14}}\right)^{\frac{p_{34}}{2}},$
(69)

where $p_{ij}=p_{i}-p_{j}$. We have introduced the cross-ratios

$\displaystyle U$
$\displaystyle=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}=z\overline{z}\,,\qquad\qquad
V=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}=(1-z)(1-\overline{z})\,,$
(70) $\displaystyle\frac{1}{\sigma}$
$\displaystyle=\frac{y_{12}y_{34}}{y_{13}y_{24}}=\alpha\overline{\alpha}\,,\qquad\qquad\frac{\tau}{\sigma}=\frac{y_{23}y_{14}}{y_{13}y_{24}}=(1-\alpha)(1-\overline{\alpha})\,.$
(71)

If we consider the $s$-channel superconformal block decomposition,
$\mathcal{G}_{\\{p_{i}\\}}$ contains all the SU$(4)_{R}$ representations in
$[0,p_{1}-2,0]\otimes[0,p_{2}-2,0]\cap[0,p_{3}-2,0]\otimes[0,p_{4}-2,0]$.
These can be labelled as

$\displaystyle[n-m,2m+s,n-m]\,,\ n=0,\cdots
P=\mathrm{min}\left(p_{i},\frac{p_{i}+p_{j}+p_{k}-p_{l}}{2}\right)-2\,,\
m=0,\cdots,n\,,$ (72)

with $s=\mathrm{max}(|p_{12}|,|p_{34}|)$.

We decompose $\mathcal{G}_{\\{p_{i}\\}}$ into its free-theory value plus the
reduced correlator $\mathcal{T}_{\\{p_{i}\\}}$

$\displaystyle\mathcal{G}_{\\{p_{i}\\}}(z,\overline{z};\alpha,\overline{\alpha})=\mathcal{G}_{\\{p_{i}\\}}^{\mathrm{free}}(z,\overline{z};\alpha,\overline{\alpha})+\frac{(z-\alpha)(z-\overline{\alpha})(\overline{z}-\alpha)(\overline{z}-\overline{\alpha})}{(z\overline{z})^{2}(\alpha\overline{\alpha})^{2}}\mathcal{T}_{\\{p_{i}\\}}(z,\overline{z};\alpha,\overline{\alpha})\,.$
(73)

In the case under analysis, namely the one with two $\mathcal{O}_{2}$’s and
two $\mathcal{O}_{p}$’s, the free correlator takes the form

$\displaystyle\mathcal{G}^{\mathrm{free}}_{\\{22pp\\}}=g_{12}^{2}g_{34}^{p}+\frac{p}{2c}\left(g_{12}g_{14}g_{23}g_{34}^{p-1}+g_{12}g_{13}g_{24}g_{34}^{p-1}\right)+\frac{p(p-1)}{2c}g_{13}g_{14}g_{23}g_{24}g_{34}^{p-2}\,.$
(74)

Only one representation is exchanged in the OPE decomposition and
$\mathcal{T}_{\\{p_{i}\\}}(z,\overline{z};\alpha,\overline{\alpha})$ is just a
known function of $\alpha,\overline{\alpha}$ times a function of the cross-
ratios

$\displaystyle\mathcal{T}_{\\{22pp\\}}(z,\overline{z};{\alpha},\overline{\alpha})$
$\displaystyle=\mathcal{T}_{\\{22pp\\}}(U,V)\,,$ (75)
$\displaystyle\mathcal{T}_{\\{p22p\\}/\\{2p2p\\}}(z,\overline{z};{\alpha},\overline{\alpha})$
$\displaystyle=({\alpha}\overline{\alpha})^{\frac{2-p}{2}}\mathcal{T}^{\prime}_{\\{p22p\\}/\\{2p2p\\}}(U,V)\,.$

By imposing superconformal Ward identities, it is possible to disentangle
contributions from short, protected operators and long, unprotected ones in
$\mathcal{T}_{\\{p_{i}\\}}$ — see Nirschl:2004pa ; Dolan:2004iy ; Dolan:2001tt
; Caron-Huot:2018kta ; Aprile:2017xsp  for details. So we can write

$\displaystyle\mathcal{T}_{\\{p_{i}\\}}=\mathcal{T}_{\\{p_{i}\\}}^{\mathrm{short}}+\mathcal{T}_{\\{p_{i}\\}}^{\mathrm{long}}\,,$
(76)

and each part can be expanded in superconformal blocks. For the correlators
under analysis, the protected contribution reads

$\displaystyle\mathcal{T}_{\\{22pp\\}}^{\text{short}}(z,\overline{z};{\alpha},\overline{\alpha})$
$\displaystyle=-\sum_{\lambda\geq
1}^{\infty}A_{2[\lambda+1]}^{(s)}G^{(0,0)}_{\lambda+5,\lambda-1}(z,\overline{z})\,,$
$\displaystyle\mathcal{T}_{\\{2p2p\\}}^{\text{short}}(z,\overline{z};{\alpha},\overline{\alpha})$
$\displaystyle=-(\alpha\overline{\alpha})^{\frac{2-p}{2}}\sum\limits_{\lambda\geq
1}^{\infty}A^{(t)}_{p[\lambda+1]}G^{(p-2,2-p)}_{\lambda+p+3,\lambda-1}(z,\overline{z})\,,$
(77)
$\displaystyle\mathcal{T}_{\\{p22p\\}}^{\text{short}}(z,\overline{z};{\alpha},\overline{\alpha})$
$\displaystyle=-(\alpha\overline{\alpha})^{\frac{2-p}{2}}\sum\limits_{\lambda\geq
1}^{\infty}(-1)^{\lambda}A^{(t)}_{p[\lambda+1]}G^{(2-p,2-p)}_{\lambda+p+3,\lambda-1}(z,\overline{z})\,,$

where we have introduced the conformal blocks

$\displaystyle G_{\Delta,\ell}^{(r,s)}(z,\overline{z})$
$\displaystyle=\frac{z\overline{z}}{\overline{z}-z}\left[k_{\frac{\Delta-\ell-2}{2}}^{r,s}(z)k_{\frac{\Delta+\ell}{2}}^{r,s}(\overline{z})-k_{\frac{\Delta+\ell}{2}}^{r,s}(z)k_{\frac{\Delta-\ell-2}{2}}^{r,s}(\overline{z})\right],$
(78) $\displaystyle k_{h}^{r,s}(z)$
$\displaystyle=z^{h}\;_{2}F_{1}\left(h+\frac{r}{2},h+\frac{s}{2};2h,z\right)\,.$
(79)

The OPE coefficients appearing in (77), necessary to reproduce the
supergravity correlator in the dispersion relation, are given by

$\displaystyle\begin{aligned}
A^{(s)}_{2[\lambda]}&=\frac{p}{2c}\frac{1+(-1)^{\lambda}}{2}\,\frac{2(\lambda!)^{2}}{(2\lambda)!}\,,\\\
A^{(t)}_{p[\lambda]}&=\frac{p}{2c}\frac{\Gamma(p+\lambda-1)\left((p-1)\Gamma(\lambda+1)+(-1)^{\lambda}(p-1)_{\lambda}\right)}{\Gamma(p+2\lambda-1)}\,.\end{aligned}$
(80)

Finally, the long contribution can be expanded in superconformal blocks as

$\displaystyle\mathcal{T}_{\\{22pp\\}}^{\text{long}}(z,\overline{z},{\alpha},\overline{\alpha})$
$\displaystyle=\sum\limits_{\Delta,\ell}C_{s}\,G^{(0,0)}_{\Delta+4,\ell}(z,\overline{z})\,,$
$\displaystyle\mathcal{T}_{\\{2p2p\\}}^{\text{long}}(z,\overline{z},{\alpha},\overline{\alpha})$
$\displaystyle=(\alpha\overline{\alpha})^{\frac{2-p}{2}}\sum\limits_{\Delta,\ell}C_{t}\,G^{(p-2,2-p)}_{\Delta+4,\ell}(z,\overline{z})\,,$
(81)
$\displaystyle\mathcal{T}_{\\{p22p\\}}^{\text{long}}(z,\overline{z},{\alpha},\overline{\alpha})$
$\displaystyle=(\alpha\overline{\alpha})^{\frac{2-p}{2}}\sum\limits_{\Delta,\ell}(-1)^{\ell}C_{t}\,G^{(2-p,2-p)}_{\Delta+4,\ell}(z,\overline{z})\,.$

##  Appendix B $\text{SUGRA}^{(k)}$

$\displaystyle\begin{aligned} &\mathrm{SUGRA}^{(1\leq k\leq
p)}=\frac{2^{-k}\Gamma(1+p)}{\Gamma(1+p-k)}\frac{1}{(STU)^{k+1}}\Bigg{\\{}\left(-\frac{p}{3}\right)^{k}S^{2k}+\left(-\frac{2}{3}(p-3)\right)^{k}(TU)^{k}\\\
&\,+\sum_{j=1}^{\lfloor\frac{k}{2}\rfloor}\left(-\frac{p}{3}\right)^{j}\left(-\frac{p-6}{3}\right)^{j}\left(-\frac{2}{3}(p-3)\right)^{k-2j}S^{2j}(TU)^{k-j}+\\!\\!\\!\\!\\!\sum_{j=\lfloor\frac{k}{2}\rfloor+1}^{k-1}\left(-\frac{p}{3}\right)^{j}S^{2j}(TU)^{k-j}\times\\\
&\,\times\sum_{i=0}^{k-j}\left(-\frac{p-6}{3}\right)^{k-j-i}\left(-\frac{2}{3}(p-3)\right)^{i}\frac{j(i+2j-k+1)_{-i-j+k-1}(-i-2j+k+1)_{i}}{i!(-i-j+k)!}\Bigg{\\}}\,,\end{aligned}$
(82) $\displaystyle\phantom{{}^{1\leq}}\mathrm{SUGRA}^{(k>p)}=0\,.$ (83)

##  Appendix C Mack polynomials

To reproduce the superconformal block expansion of $\mathcal{T}(U,V)$ in (81),
the Mellin amplitudes should have poles at precise locations, whose residues
can be shown to be given in terms of Mack polynomials
$Q_{\ell,m}^{\Delta_{12},\Delta_{34},\tau}(s)$ Mack:2009mi ; Dey:2016mcs

$\displaystyle\mathcal{Q}_{s}(s_{3};\tau,\ell,m)$
$\displaystyle=\kappa^{({4,4,p+2,p+2})}_{\ell,m,\tau+4}Q_{\ell,m}^{0,0,\tau+4}\left(s_{3}-2-\frac{p}{3}\right)\,,$
(84) $\displaystyle\mathcal{Q}_{t}(s_{3};\tau,\ell,m)$
$\displaystyle=\kappa^{({p+2,4,4,p+2})}_{\ell,m,\tau+4}(-1)^{\ell}Q_{\ell,m}^{p-2,2-p,\tau+4}\left(s_{3}-\frac{4}{3}p\right)\,.$

We obtain a closed expression for $Q_{\ell,m}$ starting from (3.47) of
Dolan:2011dv ,

$\displaystyle
P_{i(\Delta-d/2),\ell}(s,t)=\frac{\sqrt{\pi}2^{-d-\ell+3}\Gamma(d+\ell-2)}{\Gamma\left(\frac{d-1}{2}\right)\Gamma\left(\frac{d}{2}+\ell-1\right)}\alpha_{\ell}\,,$
(85)

where $P_{\nu,\ell}(s,t)$ is the Mellin transform for the conformal partial
wave in the convention of Costa:2012cb . From this data, it is straightforward
to obtain a compact expression for the Mack polynomials, by partially
resumming the expression of $\alpha_{\ell}$

$\displaystyle Q_{\ell,m}^{\Delta_{12},\Delta_{34},\tau}(s)$
$\displaystyle=\frac{2^{\ell}\ell!\Gamma(\ell+\tau-1)}{\Gamma(2\ell+\tau-1)}\sum_{k=0}^{\ell}\sum_{n=0}^{\ell-k}(-m)_{k}\left(m+\frac{s+\tau}{2}\right)_{n}\mu(\ell,k,n,\Delta_{12},\Delta_{34},\tau)\,,$
$\displaystyle\mu(\ell,k,n,\Delta_{12},\Delta_{34},\tau)$
$\displaystyle=\frac{(-1)^{k+n+1}(\ell+\tau-1)_{n}}{k!n!(-k-n+\ell)!}\left(n+1+\frac{\Delta_{34}-\Delta_{12}}{2}\right)_{k}$
$\displaystyle\times\quad\left(k+n-\frac{\Delta_{12}}{2}+\frac{\tau}{2}\right)_{-k-n+\ell}\left(k+n+\frac{\Delta_{34}}{2}+\frac{\tau}{2}\right)_{-k-n+\ell}$
(86)
$\displaystyle\times\,_{4}F_{3}\left(\begin{array}[]{cc}-k,-1-n-\ell,-1+\frac{\Delta_{12}}{2}+\frac{\tau}{2},-1-\frac{\Delta_{34}}{2}+\frac{\tau}{2}\\\
-\ell,\frac{\Delta_{12}}{2}-\frac{\Delta_{34}}{2}-k-n,-2+\tau\end{array};1\right)\,,$

and

$\displaystyle\kappa_{\ell,m,\tau}^{(p_{1},p_{2},p_{3},p_{4})}$
$\displaystyle=-\frac{2^{1-\ell}(\ell+\tau-1)_{\ell}\Gamma(2\ell+\tau)\Gamma\left(-\frac{p_{1}-p_{2}}{2}+\ell+\frac{\tau}{2}\right)}{m!\left(-1+\ell+\tau\right)_{m}\Gamma\left(-m+\frac{p_{1}}{2}+\frac{p_{2}}{2}-\frac{\tau}{2}\right)\Gamma\left(-m+\frac{p_{3}}{2}+\frac{p_{4}}{2}-\frac{\tau}{2}\right)}$
(88)
$\displaystyle\times\Gamma\left(\frac{p_{1}-p_{2}}{2}+\ell+\frac{\tau}{2}\right)\Gamma\left(-\frac{p_{3}-p_{4}}{2}+\ell+\frac{\tau}{2}\right)\Gamma\left(\frac{p_{3}-p_{4}}{2}+\ell+\frac{\tau}{2}\right).\,$

We checked that these polynomials satisfy the expected recursion relations
Costa:2012cb

$\displaystyle(\mathcal{D}_{s}-\lambda_{\ell})Q_{\ell,m}(s)={}$ $\displaystyle
4m\left(\frac{d}{2}-\tau-\ell-m\right)(2Q_{\ell,m}(s)-Q_{\ell,m-1}(s+2)-Q_{\ell,m-1}(s))\,,$
$\displaystyle\mathcal{D}_{s}Q_{\ell,m}(s)={}$
$\displaystyle((2m+s+\tau)Q_{\ell,m}(s+2)-2sQ_{\ell,m}(s))(\Delta_{12}-\Delta_{34}+2m+s+\tau)\,,$
$\displaystyle+(\Delta_{12}+s)(s-\Delta_{34})Q_{\ell,m}(s-2)\,,$ (89)
$\displaystyle\lambda_{\ell}={}$
$\displaystyle(\Delta_{12}+2m+\tau)(-\Delta_{34}+2m+\tau)+4\ell(2m+\tau-1)+4\ell^{2}\,.$

In particular the case $m=0$ takes a very simple form

$\displaystyle
Q_{\ell,0}^{(\Delta_{12},\Delta_{34})}(s)=-2^{\ell}\frac{\left(\frac{\tau}{2}-\frac{\Delta_{12}}{2}\right)_{\ell}\left(\frac{\Delta_{34}}{2}+\frac{\tau}{2}\right)_{\ell}}{(\ell+\tau-1)_{\ell}}\,_{3}F_{2}\left(\begin{array}[]{cc}-\ell,\frac{s}{2}+\frac{\tau}{2},\tau+\ell-1\\\
\frac{\tau}{2}-\frac{\Delta_{12}}{2},\frac{\Delta_{34}}{2}+\frac{\tau}{2}\end{array};1\right)\,.$

##  Appendix D Large twist sums

Given the dispersion relation

$\displaystyle
M(s_{1},s_{2})=\sum_{{\mathcal{O}}_{s}}\sum_{m=0}^{\infty}C_{s}\frac{\mathcal{Q}_{s}(s_{3};\tau_{s},\ell,m)}{s_{1}-\tau_{s}-2m+\frac{2}{3}p}+\sum_{{\mathcal{O}}_{t}}\sum_{m=0}^{\infty}C_{t}\frac{\mathcal{Q}_{t}(s_{3};\tau_{t},\ell,m)}{s_{2}-\tau_{t}-2m+\frac{2}{3}p}$
(91)

and given that the operators exchanged have twist $\tau\sim\lambda^{1/4}$, we
ask how do the OPE coefficients $C_{s}$, $C_{t}$ depend on $\lambda$ in such a
way as to reproduce the first stringy correction

$\displaystyle\frac{p(p+1)(p+2)(p+3)}{\lambda^{3/2}\Gamma(p-1)}\times\zeta(3).$
(92)

At large $\tau$ the sum in $m$ in (91) is dominated by terms of order
$\tau^{2}$. The large $\tau$ expansion of (91) can be obtained by setting
$m\rightarrow x\tau^{2}$ and replacing $\sum_{m=0}^{\infty}\rightarrow\int
dx\tau^{2}$. In this limit

$\displaystyle\lim_{\tau_{s}\rightarrow\infty}\int_{0}^{\infty}dx\tau_{s}^{2}\frac{\mathcal{Q}_{s}(s_{3};\tau_{s},\ell,x\tau_{s}^{2})}{s_{1}-\tau_{s}-2x\tau_{s}^{2}+\frac{2}{3}p}$
$\displaystyle=\frac{(\ell+1)(-1)^{-p}\Gamma(p+4)\sin^{2}\left(\frac{\pi\tau_{s}}{2}\right)2^{2\ell+2p+2\tau_{s}+13}}{\pi^{3}\tau_{s}^{2(p+4)}},$
(93)
$\displaystyle\lim_{\tau_{t}\rightarrow\infty}\int_{0}^{\infty}dx\tau_{t}^{2}\frac{\mathcal{Q}_{t}(s_{3};\tau_{t},\ell,x\tau_{t}^{2})}{s_{2}-\tau_{t}-2x\tau_{t}^{2}+\frac{2}{3}p}$
$\displaystyle=\frac{(-1)^{\ell}(\ell+1)\Gamma(p+4)2^{2\ell+2p+2\tau_{t}+13}\sin^{2}\left(\frac{1}{2}\pi(p-\tau_{t})\right)}{\pi^{3}\tau_{t}^{2(p+4)}}.$

Inserting (93) and (92) into (91) we derive (19).

##  Appendix E Ambiguities of the world-sheet integrand

The ambiguous terms in the world-sheet correlator (37) are given by

$\displaystyle\hat{r}^{(1)s}_{1,1}={}$
$\displaystyle\left(0,0,0,\tfrac{S^{2}+4ST+T^{2}}{(S+T)^{2}}\right)\,,\quad\hat{r}^{(1)a}_{1,1}=\left(0,0,0\right)\,,\quad\hat{r}^{(1)s}_{2,1}=\left(0,0,0,0\right)\,,\quad\hat{r}^{(1)a}_{2,1}=\left(0,0,0\right)\,,$
$\displaystyle\hat{r}^{(1)s}_{1,2}={}$
$\displaystyle\left(-\tfrac{5S^{2}+8ST+5T^{2}}{12(S+T)^{2}},\tfrac{S^{2}+ST+T^{2}}{3(S+T)^{2}},-\tfrac{5S^{2}+8ST+5T^{2}}{12(S+T)^{2}},4\right)\,,$
$\displaystyle\hat{r}^{(1)a}_{1,2}={}$
$\displaystyle\left(-\tfrac{5(S-T)}{12(S+T)},\tfrac{S-T}{3(S+T)},\tfrac{S-T}{4(S+T)}\right)\,,\quad\hat{r}^{(1)s}_{2,2}=\left(0,0,0,0\right)\,,\quad\hat{r}^{(1)a}_{2,2}=\left(0,0,0\right)\,,$
$\displaystyle\hat{r}^{(1)s}_{1,3}={}$
$\displaystyle\left(0,0,0,0\right)\,,\quad\hat{r}^{(1)a}_{1,3}=\left(0,0,0\right)\,,\quad\hat{r}^{(1)s}_{2,3}=\left(0,0,0,\tfrac{2\left(S^{2}+2ST\right)}{(S+T)^{2}}\right)\,,\quad\hat{r}^{(1)a}_{2,3}=\left(0,0,0\right)\,,$
$\displaystyle\hat{r}^{(1)s}_{1,4}={}$
$\displaystyle\left(0,0,0,1\right)\,,\quad\hat{r}^{(1)a}_{1,4}=\left(0,0,0\right)\,,\quad\hat{r}^{(1)s}_{2,4}=\left(0,0,0,-\tfrac{2S+T}{S+T}\right)\,,\quad\hat{r}^{(1)a}_{2,4}=\left(0,0,0\right)\,,$
$\displaystyle\hat{r}^{(1)s}_{1,5}={}$ $\displaystyle\left(1,0,1,0\right)\,,\
\hat{r}^{(1)a}_{1,5}=\tfrac{S-T}{S+T}\left(1,-2,-1\right)\,,\
\hat{r}^{(1)a}_{2,5}=\left(-\tfrac{T^{2}-2S^{2}}{4(S+T)^{2}},-\tfrac{2S^{2}+9ST+8T^{2}}{2(S+T)^{2}},-\tfrac{(S+2T)^{2}}{4(S+T)^{2}}\right)\,,$
$\displaystyle\hat{r}^{(1)s}_{2,5}={}$
$\displaystyle\left(\tfrac{2S^{2}+T^{2}}{4(S+T)^{2}},-\tfrac{2S^{2}+3ST+4T^{2}}{2(S+T)^{2}},-\tfrac{S^{2}-4T^{2}}{4(S+T)^{2}},\tfrac{4\left(4S^{2}+9ST-4T^{2}\right)}{(S+T)^{2}}\right)\,,$
$\displaystyle\hat{r}^{(1)s}_{1,6}={}$
$\displaystyle\left(0,1,0,0\right)\,,\quad\hat{r}^{(1)a}_{1,6}=\left(0,0,0\right)\,,\quad\hat{r}^{(1)a}_{2,6}=\left(\tfrac{6S^{2}+4ST+T^{2}}{8(S+T)^{2}},-\tfrac{6S^{2}+7ST+4T^{2}}{4(S+T)^{2}},-\tfrac{3S^{2}}{8(S+T)^{2}}\right)\,,$
$\displaystyle\hat{r}^{(1)s}_{2,6}={}$
$\displaystyle\left(\tfrac{6S^{2}+4ST-T^{2}}{8(S+T)^{2}},-\tfrac{6S^{2}+13ST+8T^{2}}{4(S+T)^{2}},\tfrac{-3S^{2}+4ST+8T^{2}}{8(S+T)^{2}},\tfrac{8S^{2}+6ST-8T^{2}}{(S+T)^{2}}\right)\,.$
(94)

##  Appendix F Crossing-symmetric dispersion relation

Our Mellin amplitude has the symmetry $M(s_{1},s_{2})=M(s_{1},s_{3})$.
Following Gopakumar:2021dvg ; Sinha:2020win , it is useful to construct a
dispersion relation where this symmetry is manifest, since that will lead to
simpler dispersive sum rules.
We will use the variables $s_{1}$ and $r$, where $r=\frac{s_{2}s_{3}}{s_{1}}$.
In terms of these variables we have

$s^{\prime}_{2}(s_{1},r)=\frac{1}{2}\left(-s_{1}+\sqrt{s_{1}(s_{1}-4r)}\right)\,,\qquad
s^{\prime}_{3}(s_{1},r)=\frac{1}{2}\left(-s_{1}-\sqrt{s_{1}(s_{1}-4r)}\right)\,,$
(95)

and we will consider

$\displaystyle\widetilde{M}(s_{1},r)=M(s_{1},s^{\prime}_{2}(s_{1},r)).$ (96)

Although $s^{\prime}_{2}(s_{1},r)$ has a branch cut from $s_{1}=0$ to
$s_{1}=4r$, $\widetilde{M}(s_{1},r)$ does not have a branch cut because of the
symmetry $M(s_{1},s^{\prime}_{2}(s_{1},r))=M(s_{1},s^{\prime}_{3}(s_{1},r))$.
In the new variables the bound on chaos (13) becomes

$\widetilde{M}(s_{1},r)=O(s_{1}^{-2})\,,\text{ for }|s_{1}|\to\infty\text{
with }\text{Re}(r)>-\frac{p}{3}\,.$ (97)

Thus, keeping $r$ fixed and with $\text{Re}(r)>-\frac{p}{3}$ we can write a
dispersion relation in the new variables

$\widetilde{M}(s_{1},r)=\oint_{s_{1}}\frac{ds_{1}^{\prime}}{2\pi
i}\frac{\widetilde{M}(s_{1}^{\prime},r)}{(s_{1}^{\prime}-s_{1})}\,.$ (98)

In the $s_{1}$ plane $\widetilde{M}(s_{1},r)$ has poles at
$s_{1}=\tau_{s}^{m}$ and $s_{1}=-\frac{(\tau_{t}^{m})^{2}}{\tau^{m}_{t}+r}$,
where $\tau^{m}_{s}=\tau_{s}+2m-\frac{2}{3}p$ and
$\tau_{t}^{m}=\tau_{t}+2m-\frac{2}{3}p$. This leads to

$\displaystyle\widetilde{M}(s_{1},r)=\sum_{{\mathcal{O}}_{s}}\sum_{m=0}^{\infty}C_{s}\frac{\tilde{\mathcal{Q}}_{s}(s^{\prime}_{2}(\tau_{s}^{m},r)-2-\frac{p}{3})}{s_{1}-\tau_{s}^{m}}-\sum_{{\mathcal{O}}_{t}}\sum_{m=0}^{\infty}C_{t}\frac{\tau_{t}^{m}(2r+\tau_{t}^{m})}{(r+\tau_{t}^{m})^{2}}\frac{\tilde{\mathcal{Q}}_{t}(-2-\frac{p}{3}-\frac{(\tau_{t}^{m})^{2}}{r+\tau_{t}^{m}})}{s_{1}+\frac{(\tau_{t}^{m})^{2}}{r+\tau_{t}^{m}}},$
(99)

where

$\displaystyle\tilde{\mathcal{Q}}_{s}(x)$
$\displaystyle\equiv\kappa^{({4,4,p+2,p+2})}_{\ell,m,\tau+4}Q_{\ell,m}^{0,0,\tau+4}(x)\,,$
(100) $\displaystyle\tilde{\mathcal{Q}}_{t}(x)$
$\displaystyle\equiv\kappa^{({p+2,4,4,p+2})}_{\ell,m,\tau+4}Q_{\ell,m}^{2-p,2-p,\tau+4}(x)\,.$

It is important to have formulas that connect OPE data with the low energy
expansion of $\widetilde{M}(s_{1},r)$. Expanding (99) as

$\displaystyle\widetilde{M}(s_{1},r)=\sum_{a,b=0}^{\infty}\xi_{a,b}s_{1}^{a}r^{b}\,,$
(101)

the coefficients read

$\displaystyle\xi_{a,b}=\sum_{{\mathcal{O}}_{s}}\sum_{m=0}^{\infty}C_{s}\sum_{q=0}^{b}\mathfrak{U}^{\tau_{s}^{m},\
s}_{a,b,q}\partial^{q}_{x}\mathfrak{Q}_{s}(x)|_{x=0}+\sum_{{\mathcal{O}}_{t}}\sum_{m=0}^{\infty}C_{t}\sum_{q=0}^{b}\mathfrak{U}^{\tau_{t}^{m},\
t}_{a,b,q}\partial^{q}_{x}\mathfrak{Q}_{t}(x)|_{x=0}\,,$ (102)

where

$\displaystyle\mathfrak{U}^{\tau,\,s}_{a,b,0}$
$\displaystyle=-\frac{\delta_{b,0}}{\tau^{a+1}}\,,$
$\displaystyle\mathfrak{U}^{\tau,\ s}_{a,b,q>0}$
$\displaystyle=\frac{(-1)^{q+1}\Gamma(2b-q)}{\Gamma(b+1)\Gamma(q)\Gamma(b-q+1)\tau^{a+b+1-q}}\,,$
$\displaystyle\mathfrak{U}^{\tau,\ t}_{a,b,0}$
$\displaystyle=\frac{(-1)^{-a-b}(a+b)(1-a)_{b-1}}{\Gamma(b+1)\tau^{a+b+1}}\,,$
(103) $\displaystyle\mathfrak{U}^{\tau,\ t}_{a,b,q>0}$
$\displaystyle=\frac{(-1)^{a-1}\Gamma(a)(a+b-q)\,_{3}F_{2}(q,q-b,-a-b+q+1;-a-b+q,a-b+q+1;1)}{\Gamma(q+1)\Gamma(b-q+1)\Gamma(a-b+q+1)\tau^{1+a+b-q}}\,,$

and

$\displaystyle\mathfrak{Q}_{s}(x)$
$\displaystyle\equiv\kappa^{({4,4,p+2,p+2})}_{\ell,m,\tau+4}Q_{\ell,m}^{0,0,\tau+4}(x-2-\frac{p}{3})\,,$
(104) $\displaystyle\mathfrak{Q}_{t}(x)$
$\displaystyle\equiv\kappa^{({p+2,4,4,p+2})}_{\ell,m,\tau+4}(-1)^{\ell}Q_{\ell,m}^{p-2,2-p,\tau+4}(-x-\frac{4}{3}p)\,.$

Using formula (102) and the large twist expansions of Appendix D we can derive
the expressions relating the Wilson coefficients and the OPE data that we
present in the main text. Finally, the Wilson coefficients
$\xi_{a,b}^{(0)}(p)$ can be read off from the flat space Virasoro-Shapiro
amplitude through Alday:2019nin

$\displaystyle\frac{4p}{\Gamma(p-1)}\frac{1}{s_{1}s_{2}s_{3}}+M_{\text{flat}}(s_{1},s_{2})=\frac{4}{\Gamma(p)\Gamma(p-1)s_{1}s_{2}s_{3}}\int_{0}^{\infty}d\beta\,e^{-\beta}\,\beta^{p}\,A_{\text{flat}}(2\beta
s_{1},2\beta s_{2})\,,$ (105)

where

$\displaystyle A_{\text{flat}}(s_{1},s_{2})$
$\displaystyle=\exp\left(\sum_{k=1}^{\infty}\frac{2\zeta(2k+1)}{2k+1}\left(\frac{1}{4\sqrt{\lambda}}\right)^{2k+1}\left(s_{1}^{2k+1}+s_{2}^{2k+1}+s_{3}^{2k+1}\right)\right)\,,$
(106) $\displaystyle M_{\text{flat}}(s_{1},s_{2})$
$\displaystyle=\sum_{a=0}^{\infty}\sum_{b=0}^{a}\frac{\Gamma(a+b+p+4)}{\Gamma(b+1)\Gamma(p)\Gamma(p-1)}s_{1}^{a-b}\big{(}s_{2}s_{3}\big{)}^{b}\lambda^{-\frac{3}{2}-\frac{a}{2}-\frac{b}{2}}\xi_{a,b}^{(0)}(p)\,.$

##  Appendix G Details about dispersive sum rules

###  G.1 Formulas

The functions appearing in (48) are given by

$\displaystyle c_{s,a,b,q}^{(2,0)}$
$\displaystyle=\frac{1}{96}\Big{(}-16a^{3}-16b^{3}-16b^{2}(p-3q+6)+32p^{3}-112p^{2}-480p+321$
$\displaystyle-16a^{2}(3b+p-3q+6)-80p^{2}q-80pq-168q^{2}+228q-48p^{2}q^{2}-144pq^{2}$
$\displaystyle+8b\left(4p^{2}+8pq-10p-3q^{2}+36q+2\right)$
$\displaystyle-8a\left(6b^{2}+4b(p-3q+6)-4p^{2}-8pq+10p+3q^{2}-36q-2\right)\Big{)}c_{s,a,b,q}^{(0)}\,,$
$\displaystyle c_{s,a,b,q}^{(2,1)}$
$\displaystyle=-\frac{1}{48}(q+1)\Big{(}-12a^{2}-12b^{2}-2b(8p-6q+33)+32p^{2}+56p-15$
$\displaystyle-2a(12b+8p-6q+33)+24p^{2}q+72pq+84q\Big{)}c_{s,a,b,q}^{(0)}\,,$
$\displaystyle c_{t,a,b,q}^{(2,0)}$
$\displaystyle=\frac{1}{96}\Big{(}-16a^{3}-16b^{3}-16b^{2}(p-3q+6)+32p^{3}-112p^{2}-480p+321$
$\displaystyle-16a^{2}(3b+p-3q+6)-80p^{2}q-80pq-168q^{2}+228q-48p^{2}q^{2}-144pq^{2}$
$\displaystyle+8b\left(4p^{2}+8pq-10p-3q^{2}+36q+2\right)-12(p-2)^{2}\left(2a+2b+p^{2}+6p-2q+17\right)$
$\displaystyle-8a\left(6b^{2}+4b(p-3q+6)-4p^{2}-8pq+10p+3q^{2}-36q-2\right)\Big{)}c_{t,a,b,q}^{(0)}\,,$
$\displaystyle c_{t,a,b,q}^{(2,1)}$
$\displaystyle=-\frac{1}{48}(q+1)\Big{(}-12a^{2}-12b^{2}-2b(8p-6q+33)+32p^{2}+56p-15$
$\displaystyle-2a(12b+8p-6q+33)+24p^{2}q+72pq+84q\Big{)}c_{t,a,b,q}^{(0)}\,.$
(107)

###  G.2 Dispersive sum rule for $b=0$

The simplest dispersive sum rule is the one for the flat space Wilson
coefficient $\xi_{a,0}^{(0)}(p)$. It leads to the equations

$\displaystyle\zeta(3+a)$
$\displaystyle=\frac{1}{2}\sum_{\delta=1}^{\infty}\frac{\sum_{\ell=0,2}^{2\delta-2}f_{s,0}(\delta,\ell)}{\delta^{3+a}}+\frac{1}{2}\sum_{\delta=1}^{\infty}\frac{\sum_{\ell=0,1}^{2\delta-2}f_{t,0}(\delta,\ell)}{\delta^{3+a}},~{}~{}~{}a\in
2\mathbb{N}_{0}\,,$ (108) $\displaystyle 0$
$\displaystyle=\sum_{\delta=1}^{\infty}\frac{\sum_{\ell=0,2}^{2\delta-2}f_{s,0}(\delta,\ell)}{\delta^{3+a}}-\sum_{\delta=1}^{\infty}\frac{\sum_{\ell=0,1}^{2\delta-2}f_{t,0}(\delta,\ell)}{\delta^{3+a}},~{}~{}~{}a\in(2\mathbb{N}_{0}+1)\,,$
(109)

i.e. the first equation holds for $a$ even and the second for $a$ odd. This
separation between $a$ even and odd has to do with the fact that in the flat
space limit the Mellin amplitude only contains terms of type
$(s_{1}^{2}+s_{2}^{2}+s_{3}^{2})^{n_{1}}(s_{1}s_{2}s_{3})^{n_{2}}$. The
equations for odd $a$ imply that

$\displaystyle\sum_{\ell=0,2}^{2\delta-2}f_{s,0}(\delta,\ell)=\sum_{\ell=0,1}^{2\delta-2}f_{t,0}(\delta,\ell).$
(110)

As to the equations with even $a$, since
$\zeta(3+a)=\sum_{\delta=1}^{\infty}\frac{1}{\delta^{3+a}}$ they imply that

$\displaystyle\sum_{\ell=0,2}^{2\delta-2}f_{s,0}(\delta,\ell)=1.$ (111)

###  G.3 Example

In this Appendix we demonstrate how we solve (48) for $b=0$. For $q=0$ the
ansatz for the OPE data is

$\displaystyle T_{s;q=0}^{(2)}(\delta)={}$ $\displaystyle
d_{1}(p)Z(\delta-1)+d_{2}(p)\delta
Z_{1}(\delta-1)+d_{3}(p)\delta^{2}Z_{2}(\delta-1)\,,$ (112) $\displaystyle
T_{t;q=0}^{(2)}(\delta)={}$ $\displaystyle d_{4}(p)Z(\delta-1)+d_{5}(p)\delta
Z_{1}(\delta-1)+d_{6}(p)\delta^{2}Z_{2}(\delta-1)\,,$ $\displaystyle
F_{s;q=0}^{(2)}(\delta)={}$ $\displaystyle d_{7}(p)Z(\delta-1)+d_{8}(p)\delta
Z_{1}(\delta-1)+d_{9}(p)\delta^{2}Z_{2}(\delta-1)$
$\displaystyle+d_{10}(p)\delta^{3}Z_{3}(\delta-1)+d_{11}(p)\delta^{3}\zeta(3)Z(\delta-1),$
$\displaystyle F_{t;q=0}^{(2)}(\delta)={}$ $\displaystyle
d_{12}(p)Z(\delta-1)+d_{13}(p)\delta
Z_{1}(\delta-1)+d_{14}(p)\delta^{2}Z_{2}(\delta-1)$
$\displaystyle+d_{15}(p)\delta^{3}Z_{3}(\delta-1)+d_{16}(p)\delta^{3}\zeta(3)Z(\delta-1).$

$d_{i}(p)$ are $16$ functions of $p$ that we will fix by our
procedure.161616The terms proportional to $Z(\delta-1)$ are required, by
contrast to the $q\geq 1$ case. The reason is that the functions in (49)
vanish for $(\delta=1,\,q\geq 1)$, but do not vanish when $(\delta=1,\,q=0)$.
We plug (112) into (48) for $b=0$ and use the formula

$\displaystyle\zeta(s,s_{1},s_{2},\ldots)=\sum_{\delta=1}^{\infty}\frac{Z_{s_{1},s_{2},\ldots}(\delta-1)}{\delta^{s}}\,,$
(113)

to express $\xi_{a,0}^{(1)}(p)$ in terms of multiple zeta values and as a
function of $d_{i}(p)$. Convergence of the $\delta$ sum implies that

$\displaystyle d_{10}(p)+d_{11}(p)+d_{15}(p)+d_{16}(p)=0.$ (114)

Finally, we impose that $\xi_{a,0}^{(1)}(p)$ is a linear combination of
single-valued multiple zeta values. We do so using the MAPLE program
HyperlogProcedures HyperlogProcedures . It was enough to impose this for
$a=0,\ldots,6$. In this manner we fix $13$ out of the $16$ functions
$d_{i}(p)$171717After fixing the functions $d_{i}(p)$ in this way we checked
that the resulting expression for $\xi_{a,0}^{(1)}(p)$ is single-valued for
$a=7,\ldots,10$.

$\displaystyle
d_{2}(p)=\frac{1}{4},\,d_{3}(p)=1,\,d_{5}(p)=\frac{1}{4},\,d_{6}(p)=1,d_{7}(p)=2d_{1}(p)+2d_{4}(p)-d_{12}(p)+\frac{p^{4}}{8}-\frac{5p^{3}}{12}+\frac{19p^{2}}{8}+\frac{25p}{6}$
(115)
$\displaystyle+\frac{13}{16},\,d_{8}(p)=2d_{1}(p)+2d_{4}(p)+\frac{p^{2}}{2}+3p-\frac{39}{8},\,d_{9}(p)=2,\,d_{10}(p)=-2,\,d_{13}(p)=2d_{1}+2d_{4}+\frac{p^{2}}{2}+3p$
$\displaystyle-\frac{39}{8},\,d_{14}(p)=2,\,d_{15}(p)=-2,\,d_{16}(p)=2.$

The $3$ functions $d_{1}(p)$, $d_{4}(p)$ and $d_{12}(p)$ are not fixed by this
procedure. However, they are fixed by repeating this exercise for
$b=0,1,\ldots$ .

## References

  * (1) J. M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231–252, [hep-th/9711200].
  * (2) N. Gromov, A. Hegedus, J. Julius and N. Sokolova, _Fast QSC Solver: tool for systematic study of N=4 Super-Yang-Mills spectrum_ , 2306.12379.
  * (3) S. Caron-Huot, F. Coronado, A.-K. Trinh and Z. Zahraee, _Bootstrapping $\mathcal{N}$ = 4 sYM correlators using integrability_, _JHEP_ 02 (2023) 083, [2207.01615].
  * (4) A. Cavaglià, N. Gromov, J. Julius and M. Preti, _Bootstrability in defect CFT: integrated correlators and sharper bounds_ , _JHEP_ 05 (2022) 164, [2203.09556].
  * (5) L. F. Alday, T. Hansen and J. A. Silva, _AdS Virasoro-Shapiro from dispersive sum rules_ , _JHEP_ 10 (2022) 036, [2204.07542].
  * (6) L. F. Alday, T. Hansen and J. A. Silva, _AdS Virasoro-Shapiro from single-valued periods_ , _JHEP_ 12 (2022) 010, [2209.06223].
  * (7) L. F. Alday, T. Hansen and J. A. Silva, _Emergent world-sheet for the AdS Virasoro-Shapiro amplitude_ , 2305.03593.
  * (8) L. F. Alday and T. Hansen, _The AdS Virasoro-Shapiro Amplitude_ , 2306.12786.
  * (9) S. Caron-Huot and V. Van Duong, _Extremal Effective Field Theories_ , _JHEP_ 05 (2021) 280, [2011.02957].
  * (10) S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, _AdS bulk locality from sharp CFT bounds_ , _JHEP_ 11 (2021) 164, [2106.10274].
  * (11) A. Sinha and A. Zahed, _Crossing Symmetric Dispersion Relations in Quantum Field Theories_ , _Phys. Rev. Lett._ 126 (2021) 181601, [2012.04877].
  * (12) R. Gopakumar, A. Sinha and A. Zahed, _Crossing Symmetric Dispersion Relations for Mellin Amplitudes_ , _Phys. Rev. Lett._ 126 (2021) 211602, [2101.09017].
  * (13) L. F. Alday and X. Zhou, _Simplicity of AdS Supergravity at One Loop_ , _JHEP_ 09 (2020) 008, [1912.02663].
  * (14) F. Aprile, J. Drummond, P. Heslop and H. Paul, _One-loop amplitudes in AdS ${}_{5}\times$S5 supergravity from $\mathcal{N}$ = 4 SYM at strong coupling_, _JHEP_ 03 (2020) 190, [1912.01047].
  * (15) F. Aprile, J. M. Drummond, P. Heslop, H. Paul, F. Sanfilippo, M. Santagata et al., _Single particle operators and their correlators in free $\mathcal{N}$ = 4 SYM_, _JHEP_ 11 (2020) 072, [2007.09395].
  * (16) F. A. Dolan and H. Osborn, _Superconformal symmetry, correlation functions and the operator product expansion_ , _Nucl. Phys. B_ 629 (2002) 3–73, [hep-th/0112251].
  * (17) F. A. Dolan and H. Osborn, _Conformal partial waves and the operator product expansion_ , _Nucl. Phys. B_ 678 (2004) 491–507, [hep-th/0309180].
  * (18) F. A. Dolan and H. Osborn, _Conformal partial wave expansions for N=4 chiral four point functions_ , _Annals Phys._ 321 (2006) 581–626, [hep-th/0412335].
  * (19) S. Caron-Huot and A.-K. Trinh, _All tree-level correlators in AdS ${}_{5}\times$S5 supergravity: hidden ten-dimensional conformal symmetry_, _JHEP_ 01 (2019) 196, [1809.09173].
  * (20) L. F. Alday, A. Bissi and E. Perlmutter, _Genus-One String Amplitudes from Conformal Field Theory_ , _JHEP_ 06 (2019) 010, [1809.10670].
  * (21) D. J. Binder, S. M. Chester, S. S. Pufu and Y. Wang, _$\mathcal{N}$ = 4 Super-Yang-Mills correlators at strong coupling from string theory and localization_, _JHEP_ 12 (2019) 119, [1902.06263].
  * (22) J. Penedones, _Writing CFT correlation functions as AdS scattering amplitudes_ , _JHEP_ 03 (2011) 025, [1011.1485].
  * (23) J. Maldacena, S. H. Shenker and D. Stanford, _A bound on chaos_ , _JHEP_ 08 (2016) 106, [1503.01409].
  * (24) J. Penedones, J. A. Silva and A. Zhiboedov, _Nonperturbative Mellin Amplitudes: Existence, Properties, Applications_ , _JHEP_ 08 (2020) 031, [1912.11100].
  * (25) G. Mack, _D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes_ , 0907.2407.
  * (26) L. F. Alday, T. Hansen and J. A. Silva, _On the spectrum and structure constants of short operators in N=4 SYM at strong coupling_ , 2303.08834.
  * (27) J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, _Looking for a bulk point_ , _JHEP_ 01 (2017) 013, [1509.03612].
  * (28) F. C. S. Brown, _Polylogarithmes multiples uniformes en une variable_ , _Compt. Rend. Math._ 338 (2004) 527–532.
  * (29) L. J. Dixon, C. Duhr and J. Pennington, _Single-valued harmonic polylogarithms and the multi-Regge limit_ , _JHEP_ 10 (2012) 074, [1207.0186].
  * (30) O. Schnetz, “ _HyperlogProcedures_ , https://www.math.fau.de/person/oliver-schnetz/.”
  * (31) N. Gromov, D. Serban, I. Shenderovich and D. Volin, _Quantum folded string and integrability: From finite size effects to Konishi dimension_ , _JHEP_ 08 (2011) 046, [1102.1040].
  * (32) P. Vanhove and F. Zerbini, _Single-valued hyperlogarithms, correlation functions and closed string amplitudes_ , _Adv. Theor. Math. Phys._ 26 (2022) , [1812.03018].
  * (33) T. Abl, P. Heslop and A. E. Lipstein, _Towards the Virasoro-Shapiro amplitude in AdS ${}_{5}\times S^{5}$_, _JHEP_ 04 (2021) 237, [2012.12091].
  * (34) F. Aprile and P. Vieira, _Large $p$ explorations. From SUGRA to big STRINGS in Mellin space_, _JHEP_ 12 (2020) 206, [2007.09176].
  * (35) M. Nirschl and H. Osborn, _Superconformal Ward identities and their solution_ , _Nucl. Phys. B_ 711 (2005) 409–479, [hep-th/0407060].
  * (36) F. Aprile, J. M. Drummond, P. Heslop and H. Paul, _Unmixing Supergravity_ , _JHEP_ 02 (2018) 133, [1706.08456].
  * (37) P. Dey, A. Kaviraj and A. Sinha, _Mellin space bootstrap for global symmetry_ , _JHEP_ 07 (2017) 019, [1612.05032].
  * (38) F. A. Dolan and H. Osborn, _Conformal Partial Waves: Further Mathematical Results_ , 1108.6194.
  * (39) M. S. Costa, V. Goncalves and J. Penedones, _Conformal Regge theory_ , _JHEP_ 12 (2012) 091, [1209.4355].