††institutetext: Zhejiang Institute of Modern Physics, Department of Physics,
Zhejiang University,
866 Yuhangtang Road, Hangzhou, Zhejiang 310058, China††institutetext: Joint
Center for Quanta-to-Cosmos Physics, Zhejiang University,
866 Yuhangtang Road, Hangzhou, Zhejiang 310058, China

# Graviton Scattering in $\mathrm{AdS}_{5}\times\mathrm{S}^{5}$ at Two Loops

Zhongjie Huang  and Ellis Ye Yuan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We report a result for the third-order correction in $1/N$ expansion to the
four-point correlator of the stress tensor multiplet in $\mathcal{N}=4$ super
Yang–Mills theory at large ’t Hooft coupling, which corresponds to the two-
loop scattering of four gravitons in the dual
$\mathrm{AdS}_{5}\times\mathrm{S}^{5}$ supergravity. This is obtained by
bootstrapping an educated ansatz based on intuitions from the hidden
10-dimensional conformal symmetry.

##  1 Introduction

Correlators of chiral primary operators (CPO) are crucial for understanding
the correspondence between $\mathcal{N}=4$ super Yang–Mills with gauge group
$\mathrm{SU}(N)$ (SYM) and its AdS dual 111See Heslop:2022qgf  for a nice
review on these correlators.. In the supergravity limit they serve as useful
probe for particle dynamics in the bulk. The simplest cases at AdS tree level
were computed long ago using SUSY constraints and perturbation methods (e.g.,
Eden:2000bk ; Arutyunov:2002fh ; Arutyunov:2017dti ; Arutyunov:2018tvn ). With
the help of Mellin space techniques Mack:2009mi ; Mack:2009gy ;
Penedones:2010ue ; Fitzpatrick:2011ia  a unified formula was discovered for
arbitrary four CPOs Rastelli:2016nze ; Rastelli:2017udc . Because of the
essential complexity of bulk perturbation such computation barely continued
its success to loop levels.

Recently there arose a new approach from the boundary side, where analytic
conformal bootstrap methods are available (e.g., Komargodski:2012ek ;
Fitzpatrick:2012yx ; Alday:2016njk ; Caron-Huot:2017vep ; Carmi:2019cub ;
Mazac:2019shk ; Penedones:2019tng ; Caron-Huot:2020adz , a recent review is
available in Bissi:2022mrs ). In brief the idea is to rebuild the correlator
from its Lorentzian singularities. When applied to theories with a weakly
coupled bulk dual, this induces recursive relations among CFT data at
different orders of bulk perturbation Aharony:2016dwx . Specializing to
$\mathcal{N}=4$ SYM such analysis succeeded in generating new results at one
loop Alday:2017xua ; Aprile:2017bgs ; Alday:2017vkk ; Aprile:2017qoy ;
Aprile:2019rep ; Alday:2019nin .

In this paper we report the computation for the four-point correlator of the
simplest CPOs, the stress tensor multiplet, at bulk two loops.

##  2 Preliminaries

###  2.1 Structure of
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$

In $\mathcal{N}=4$ SYM with gauge group $\mathrm{SU}(N)$, the CPOs
$\mathcal{O}_{p}$ are half-BPS states with protected conformal dimension $p$
(for some integer $p\geq 2$) that transform as a Lorentz scalar and in the
$[0,p,0]$ (i.e., traceless symmetric) representation of the $\mathrm{SU}(4)$
R-symmetry. We normalize them such that

$\langle\mathcal{O}_{p}(x_{1},y_{1})\mathcal{O}_{p}(x_{2},y_{2})\rangle=\left(\frac{y_{12}^{2}}{x_{12}^{2}}\right)^{p},$
(1)

where $x_{ij}^{2}=(x_{i}-x_{j})^{2}$ and $y_{ij}^{2}=y_{i}\cdot y_{j}$.
$x^{\mu}$ is the spacetime coordinates. The 6d null vector $y^{a}$ is a
convenient way to take care of the R-symmetry representation. Among these
operators the simplest case $p=2$ is the stress tensor multiplet and is dual
to the graviton in $\mathrm{AdS}_{5}\times\mathrm{S}^{5}$.

As usual we define the crossratios

$\begin{split}u\equiv\,&\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}=z\bar{z},\quad
v\equiv\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}=(1-z)(1-\bar{z}),\\\
&\frac{y_{12}^{2}y_{34}^{2}}{y_{13}^{2}y_{24}^{2}}=\alpha\bar{\alpha},\quad\\!\qquad\frac{y_{14}^{2}y_{23}^{2}}{y_{13}^{2}y_{24}^{2}}=(1-\alpha)(1-\bar{\alpha}),\end{split}$
(2)

By superconformal Ward identities the correlator $\langle
2222\rangle\equiv\langle\mathcal{O}_{2}(x_{1},y_{1})\cdots\mathcal{O}_{2}(x_{4},y_{4})\rangle$
can be written as Eden:2000bk

$\begin{split}&\left(\frac{y_{12}^{2}y_{34}^{2}}{x_{12}^{2}x_{34}^{2}}\right)^{-2}\langle
2222\rangle\equiv\mathcal{G}=\mathcal{G}^{\rm
free}+\mathcal{I}\,\mathcal{H},\end{split}$ (3)

where $\mathcal{I}$ is a simple factor

$\mathcal{I}=\frac{(z-\alpha)(z-\bar{\alpha})(\bar{z}-\alpha)(\bar{z}-\bar{\alpha})}{(z\bar{z}\alpha\bar{\alpha})^{2}}.$
(4)

The free correlator $\mathcal{G}^{\rm free}$ is completely determined by Wick
contraction and is independent of the ’t Hooft coupling $\lambda$ (its
explicit expression can be found in, e.g., (2.11) of Aprile:2017bgs ). So non-
trivial dynamics of $\langle 2222\rangle$ is fully encoded in $\mathcal{H}$,
which is usually called the reduced correlator. Consider the supergravity
limit $N\to\infty$ and $\lambda\to\infty$, where we can take double expansion
of $\mathcal{H}$ with respect to small $1/\sqrt{\lambda}$ and $1/c$,
$c=\frac{N^{2}-1}{4}$ being the central charge. $1/\sqrt{\lambda}$ controls
the stringy corrections to the bulk effective action, which are beyond our
current scope of investigation, and so in this paper we only focus on the
supergravity sector, at order $\lambda^{0}$ 222Due to this ignorance of
stringy corrections our supergravity result is subject to counterterms whose
coefficients are not yet determined. This will be further discussed later in
the paper.. The $1/c$ expansion of this sector take the form

$\mathcal{H}|_{\rm
sugra}=c^{-1}\mathcal{H}^{(1)}+c^{-2}\mathcal{H}^{(2)}+c^{-3}\mathcal{H}^{(3)}+\cdots.$
(5)

This expansion has a clean bulk interpretation in terms of perturbation in
Newton’s constant. $\mathcal{H}^{(1)}$ corresponds to the tree-level
scattering, and was computed long ago in DHoker:1999mic ; Arutyunov:2000py .
$\mathcal{H}^{(2)}$ corresponds to the one-loop scattering, and its position
space result was recently obtained using analytic bootstrap on the CFT side
Aprile:2017bgs . Our target in this paper is to extend the position space
computation and solve $\mathcal{H}^{(3)}$, which describes the graviton
scattering at two loops in this theory.

Because our computation relies on the used of CFT data, it is helpful to
quickly review the conformal block expansion of $\langle 2222\rangle$. The
superconformal symmetry of SYM nicely organizes its spectrum into various
supermultiplets. Among these there are the protected multiplets (including the
half-BPS and the semi-short multiplets) and the long multiplets.
Correspondingly the correlator $\mathcal{G}$ can as well be decomposed as

$\mathcal{G}=\mathcal{G}_{\rm protected}+\mathcal{I}\mathcal{H}_{\rm long}.$
(6)

The contribution from the protected multiplets $\mathcal{G}_{\rm protected}$
are free from quantum corrections and its expression was determined in
Doobary:2015gia ; Aprile:2017bgs . The remaining part is then identical to a
summation over long multiplets only. Because the long multiplets appearing in
the OPE of $\mathcal{O}_{2}\times\mathcal{O}_{2}$ can only R-symmetry
representation $[0,0,0]$, whose corresponding superconformal blocks are
explicitly

$A_{\tau,\ell}^{[0,0,0]}(z,\bar{z},\alpha,\bar{\alpha})=\mathcal{I}g_{\tau+4,\ell}(z,\bar{z}),$
(7)

where $g_{\tau,\ell}$ denotes the ordinary conformal block

$\displaystyle g_{\tau,\ell}$
$\displaystyle=\frac{z\bar{z}}{\bar{z}-z}\left(k_{\frac{\tau-2}{2}}(z)k_{\frac{\tau+2\ell}{2}}(\bar{z})-(z\leftrightarrow\bar{z})\right),$
(8) $\displaystyle k_{h}(z)$ $\displaystyle=z^{h}{}_{2}F_{1}(h,h,2h,z).$ (9)

The shift of twist in $g$ relative to $A$ in (7) is consistent with the non-
trivial conformal weights carried by the factor $\mathcal{I}$. This indicates
that the quantity $\mathcal{H}_{\rm long}$ in (6) only depends on the position
space cross-ratios and it receives a decomposition onto the ordinary blocks as

$\displaystyle\mathcal{H}_{\rm long}(z,\bar{z})$
$\displaystyle=\sum_{i}a_{i}g_{\tau_{i}+4,\ell_{i}}(z,\bar{z}),$ (10)

where the index $i$ scans over the spectrum of the long multiplets in the
theory.

In the supergravity limit, the supergravity piece of $\mathcal{H}_{\rm long}$
can be expanded in analogy to (5)

$\mathcal{H}_{\rm long}|_{\rm sugra}=\mathcal{H}_{\rm
long}^{(0)}+c^{-1}\mathcal{H}_{\rm long}^{(1)}+c^{-2}\mathcal{H}_{\rm
long}^{(2)}+c^{-3}\mathcal{H}_{\rm long}^{(3)}+\cdots.$ (11)

Note that in the case of $\langle 2222\rangle$ both $\mathcal{G}_{\rm free}$
and $\mathcal{G}_{\rm protected}$ are non-vanishing only up to order $c^{-1}$.
This implies the identification

$\mathcal{H}^{(k)}=\mathcal{H}_{\rm long}^{(k)},\qquad k\geq 2.$ (12)

In other words, from the boundary point of view the graviton scattering at one
loop and higher only receives contributions from long multiplets. Because in
the supergravity limit all the multiplets in correspondence to stringy states
become infinitely heavy and get decoupled, these remaining long multiplets
have their twists at most finite at the leading order and are known to be in
general multi-trace operators.

To see the structure of $\mathcal{H}^{(k)}$ at loop levels more clearly,
consider the expansion of twists and OPE coefficients in (10) with respect to
small $1/c$ (we suppress stringy corrections)

$\displaystyle\tau_{i}$
$\displaystyle=\tau_{i}^{(0)}+c^{-1}\gamma_{i}^{(1)}+c^{-2}\gamma_{i}^{(2)}+\cdots,$
(13) $\displaystyle a_{i}$
$\displaystyle=a_{i}^{(0)}+c^{-1}a_{i}^{(1)}+c^{-2}a_{i}^{(2)}+\cdots.$ (14)

Operators with $a_{i}^{(0)}\neq 0$ already show up in the mean field theory,
and are the double-trace operators of the generic form
$[\mathcal{O}_{q_{1}}\mathcal{O}_{q_{2}}]_{n,\ell}\equiv\mathcal{O}_{q_{1}}\square^{n}\partial^{\ell}\mathcal{O}_{q_{2}}$,
with classical twist $\tau^{(0)}=q_{1}+q_{2}+2n$. So in the OPE
$\mathcal{O}_{2}\times\mathcal{O}_{2}$ all the classical twists are even, with
the smallest value at $\tau^{(0)}=4$. From (8) and (9) one observes that (in
s-channel) $\mathcal{H}^{(k)}$ contains terms with the factor $\log{u}$ up to
$\log^{k}{u}$. Terms with the maximal power $k$ are called the leading log
terms, which are

$\mathcal{H}^{(k)}\big{|}_{\log^{k}{u}}=\\!\sum_{\tau^{(0)},\ell}\\!\frac{\langle
a^{(0)}(\gamma^{(1)})^{k}\rangle_{\tau^{(0)},\ell}}{p!\,2^{p}}g_{\tau^{(0)}+4,\ell}(z,\bar{z}).$
(15)

The coefficients only depend on $a^{(0)}$ and $\gamma^{(1)}$, so this part
solely consists of double-trace operators, and can be recursively determined
once these data at the disconnected and the tree level are available. At
$\tau^{(0)}=4$ it is literally the product $a^{(0)}(\gamma^{(1)})^{k}$, as
there is a unique operator $\mathcal{O}_{2}\mathcal{O}_{2}$. At higher
$\tau^{(0)}$ there are multiple choices of $\\{q_{1},q_{2},n\\}$ for the same
classical twist, and the angle bracket indicates certain average due to
degeneracy among these operators. For more details of this degeneracy, see
e.g., Aprile:2017xsp ; Aprile:2018efk . More generally, at higher loops other
terms with $\log^{\geq 2}u$ can be recursively constructed from lower-loop
data in analogous ways. One example is shown in (33), which will be used as
one of the inputs to our computation. However, very often the determination of
these subleading log terms requires data of higher-trace operators, which are
not available by considering 4-point half-BPS correlators only.

The significance of $\mathcal{H}^{(k)}|_{\log^{k}{u}}$ can already be seen at
one loop, where it fully determines the behavior of $\langle 2222\rangle$
around its Lorentzian singularities, which further reconstructs the correlator
Alday:2017xua ; Aprile:2017bgs . Starting at two loops, while these leading
log terms still plays an important role Bissi:2020woe , the Lorentzian
singularities also relies on the subleading log terms as well. Due to the lack
of information about higher-trace operators as mentioned above, one cannot
expects to compute $\mathcal{H}^{(3)}$ as straightforwardly as at one loop.
Therefore we need to look for alternative sources of constraints for this
quantity.

###  2.2 Hidden Symmetries

A useful hint already resides in the structure of the leading log terms. It
was noted in Caron-Huot:2018kta  that the expansion (15) can in fact resum
into a very intuitive analytic expression

$\mathcal{H}^{(k)}\big{|}_{\log^{k}{u}}=\left[\Delta^{(8)}\right]^{k-1}\mathcal{F}^{(k)}(z,\bar{z}).$
(16)

$\Delta^{(8)}$ is an eighth-order differential operator

$\displaystyle\Delta^{(8)}=\frac{z\bar{z}}{\bar{z}-z}D_{z}(D_{z}-2)D_{\bar{z}}(D_{\bar{z}}-2)\frac{\bar{z}-z}{z\bar{z}}$
(17)

acting in s-channel, where $D_{z}=z^{2}\partial_{z}(1-z)\partial_{z}$. (16) is
a consequence of a conjectured 10d conformal symmetry observed from the
pattern of tree-level data, where the degeneracy in $\langle
a^{0}(\gamma^{(1)})^{k}\rangle$ is interpreted in terms of decomposition of
$\mathrm{SO}(10,2)$ blocks into those of its
$\mathrm{SO}(4,2)\times\mathrm{SO}(6)$ subgroup. The function
$\mathcal{F}^{(k)}(z,\bar{z})$ is built out of 10d blocks, which takes the
generic form

$\mathcal{F}^{(k)}(z,\bar{z})=\sum_{|\vec{a}|=0}^{k}\frac{p_{\vec{a}}(z,\bar{z})}{(\bar{z}-z)^{7}}G(\vec{a};z)+(z\leftrightarrow\bar{z}).$
(18)

$G(\vec{a};z)$ denotes the multiple polylogarithm (MPL), labeled by a weight
vector $\vec{a}$. In this case, components of $\vec{a}$ take values in
$\\{0,1\\}$. $p_{\vec{a}}(z,\bar{z})$ is some polynomial of $z$ and $\bar{z}$,
of degree $7$ in each. Finally, the summation over MPLs is carried up to
weight $k$. Detailed expressions were worked out in Caron-Huot:2018kta .

Interestingly, Aprile:2019rep  discovered that similar action of
$\Delta^{(8)}$ as in (16) simplifies the structure of $\langle 2222\rangle$ at
one-loop as well, with a slight modification 333The relation between our
$\mathcal{H}^{(2)}$ here and their $\mathcal{H}^{(2)}_{2222}$ is
$\mathcal{H}^{(2)}=u^{2}\mathcal{H}^{(2)}_{2222}/4$.

$\displaystyle\mathcal{H}^{(2)}=\Delta^{(8)}\mathcal{L}^{(2)}+\frac{1}{4}\mathcal{H}^{(1)},$
(19)

where

$\displaystyle\mathcal{L}^{(2)}=\sum_{|\vec{a}|+|\vec{a}^{\prime}|=0}^{4}\frac{p_{\vec{a},\vec{a}^{\prime}}(z,\bar{z})}{(\bar{z}-z)^{7}}G(\vec{a};z)G(\vec{a}^{\prime};\bar{z}).$
(20)

Like $p_{\vec{a}}$, the polynomial $p_{\vec{a},\vec{a}^{\prime}}(z,\bar{z})$
is again of degree $7$ in each variable. Now the summation is performed over
all possible combinations $G(\vec{a};z)G(\vec{a}^{\prime};\bar{z})$ of total
weight up to $4$. Note this maximal weight $4=2+2$ is identical to the weight
of the leading log terms including $\log^{2}{u}$.

This is a huge simplification as compared to the expression of
$\mathcal{H}^{(2)}$ itself. It turns out $\mathcal{L}^{(2)}$ can in fact be
fully bootstraped by generic consistency constratints on $\mathcal{L}^{(2)}$
and $\mathcal{H}^{(2)}$, without even knowing the leading log terms
Aprile:2019rep ! This strongly suggests a much wider use of $\Delta^{(8)}$.

##  3 Ansatz for $\mathcal{H}^{(3)}$

Based on the above discussions we draw the following ansatz for the two-loop
correlator of $\langle 2222\rangle$

$\displaystyle\mathcal{H}^{(3)}=\left[\Delta^{(8)}\right]^{2}\mathcal{L}^{(3)}+\kappa_{2}\mathcal{H}^{(2)}+\kappa_{1}\mathcal{H}^{(1)},$
(21)

where $\kappa_{1}$ and $\kappa_{2}$ are two parameters to be determined later
on. Similar to (20) the unknown function $\mathcal{L}^{(3)}$ admits a
decomposition onto MPLs, now up to weight $6$. Here we can set up the ansatz
in a more informed manner. The full correlator $\mathcal{H}^{(3)}$ is expected
to be single-valued on the Euclidean slice $\bar{z}=z^{*}$, with singularities
at $0,1,\infty$. When multiplied by $(z-\bar{z})^{4}/(z\bar{z})^{4}$ it
becomes invariant under an $\mathrm{S}_{3}\times\mathbb{Z}_{2}$ permutation
group. The $\mathbb{Z}_{2}$ associates to the exchange of variables
$z\leftrightarrow\bar{z}$. The $\mathrm{S}_{3}$ permutes $0,1,\infty$, and
consists of the following operations

$\begin{split}&\rho(123):\,z\mapsto
z,\quad\quad\quad\;\;\,\rho(132):\,z\mapsto\bar{z}/(\bar{z}-1),\\\
&\rho(213):\,z\mapsto 1-\bar{z},\quad\quad\rho(231):\,z\mapsto 1/(1-z),\\\
&\rho(312):\,z\mapsto 1-1/z,\quad\rho(321):\,z\mapsto 1/\bar{z},\end{split}$
(22)

(and $\bar{z}$ transforming in the conjugate way), which account for the full
crossing symmetries of the correlator. Therefore it is natural to choose a
basis for single-valued MPLs (SVMPLs) where each element falls into a specific
representation of $\mathrm{S}_{3}\times\mathbb{Z}_{2}$. We label them as

$G^{\rm SV}_{w,r,i}(z,\bar{z}),$ (23)

where $w$ denotes the uniform transcendental weight, and $r$ chooses the six
different $\mathrm{S}_{3}\times\mathbb{Z}_{2}$ representations
$\\{\mathbf{1}^{\pm},\bar{\mathbf{1}}^{\pm},\mathbf{2}^{\pm}\\}$ ($\pm$ stands
for symmetric/anti-symmetric representation of $\mathbb{Z}_{2}$). The third
index $i$ takes care of linearly independent SVMPLs with the same $w$ and $r$.
$\mathrm{S}_{3}$ acts on these functions as

$\displaystyle\rho(abc)\circ G^{\rm SV}_{w,r,i}$
$\displaystyle=\mathbf{R}_{(abc)}^{r}G^{\rm SV}_{w,r,i},$ (24)

where $\rho(abc)\circ$ means to change the variables according to (22), and
$\mathbf{R}_{(abc)}^{r}$ is the matrix for the element $(abc)$ in
representation $r$. Explicitly, we have
$\mathbf{R}_{(abc)}^{\mathbf{1}^{\pm}}\equiv 1$.
$\mathbf{R}_{(abc)}^{\bar{\mathbf{1}}^{\pm}}$ is $+1$ for even $(abc)$
permutations and $-1$ otherwise. $G^{\rm SV}_{w,\mathbf{2}^{\pm},i}$ is an
array of two basis functions, and the corresponding linear transformation can
be realized in various ways. The choice that we make is

$\begin{split}&\mathbf{R}_{(123)}^{\mathbf{2}^{\pm}}=\left(\begin{matrix}1&0\\\
0&1\end{matrix}\right),\quad\;\;\mathbf{R}_{(132)}^{\mathbf{2}^{\pm}}=\left(\begin{matrix}1&-1\\\
0&-1\end{matrix}\right),\\\
&\mathbf{R}_{(213)}^{\mathbf{2}^{\pm}}=\left(\begin{matrix}0&1\\\
1&0\end{matrix}\right),\quad\;\;\mathbf{R}_{(231)}^{\mathbf{2}^{\pm}}=\left(\begin{matrix}0&-1\\\
1&-1\end{matrix}\right),\\\
&\mathbf{R}_{(312)}^{\mathbf{2}^{\pm}}=\left(\begin{matrix}-1&1\\\
-1&0\end{matrix}\right),\quad\mathbf{R}_{(321)}^{\mathbf{2}^{\pm}}=\left(\begin{matrix}-1&0\\\
-1&1\end{matrix}\right).\end{split}$ (25)

MPLs are accompanied by a useful tool called _symbol_ (see appendix). Provided
a finite set of allowed symbol alphabets, one can construct such basis of
SVMPLs for any cutoff in the maximal transcendental weight. Up to one loop the
alphabets are observed to be $\\{z,\bar{z},1-z,1-\bar{z}\\}$, in accordance
with the singularities of the correlator at $0$, $1$ and $\infty$. We expect
this to still be valid at two loops, except that a fifth letter $z-\bar{z}$ is
also present (note this pattern already appear in the denominator of (20)).
The basis construction can be systematized following, e.g., the algorithm
introduced in Chavez:2012kn 444In that paper the authors constructed the SVMPL
basis functions by taking care of the $\mathbb{Z}_{2}$ group only. We follow
their strategy but consider $\mathrm{S}_{3}$ representations as well.. Without
the letter $z-\bar{z}$, the counting of independent representations for each
$w$ and $r$ are as follows

$w$ $r$ | $\;\mathbf{1}^{+}\;$ | $\;\mathbf{1}^{-}\;$ | $\;\bar{\mathbf{1}}^{+}\;$ | $\;\bar{\mathbf{1}}^{-}\;$ | $\;\mathbf{2}^{+}\;$ | $\;\mathbf{2}^{-}\;$
---|---|---|---|---|---|---
$0$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$
$1$ | $0$ | $0$ | $0$ | $0$ | $1$ | $0$
$2$ | $2$ | $1$ | $0$ | $0$ | $1$ | $0$
$3$ | $2$ | $0$ | $1$ | $0$ | $3$ | $1$
$4$ | $5$ | $3$ | $1$ | $0$ | $5$ | $2$
$5$ | $7$ | $3$ | $4$ | $2$ | $11$ | $5$
$6$ | $15$ | $10$ | $6$ | $3$ | $20$ | $12$

For instance, up to $w=2$ we have

$\begin{split}&G_{0,\mathbf{1}^{+}}^{\rm
SV}\\!=\\!1,\;\;G_{1,\mathbf{2}^{+}}^{\rm
SV}\\!=\\!(\log{u},\log{v}),\;\;G_{2,\mathbf{1}^{+},1}^{\rm
SV}\\!=\\!\zeta_{2},\\\ &G_{2,\mathbf{1}^{+},2}^{\rm
SV}=\log^{2}{u}-\log{u}\log{v}+\log^{2}{v},\\\ &G_{2,\mathbf{1}^{-}}^{\rm
SV}\\!=\\!2\mathrm{Li}_{2}(z)-2\mathrm{Li}_{2}(\bar{z})+\log{u}\log\frac{1-z}{1-\bar{z}},\\\
&G_{2,\mathbf{2}^{+}}^{\rm
SV}\\!\\!=\\!\left(\\!\frac{\log^{2}\\!{u}\\!-\\!2\log{u}\log{v}}{2},\frac{\log^{2}\\!{v}\\!-\\!2\log{u}\log{v}}{2}\\!\right)\\!.\end{split}$
(26)

Here we give an explicit example where the SVMPL of give $w$ and $r$ is not
unique, in which case there are multiple basis functions
$G_{2,\mathbf{1}^{+},i}^{\rm SV}$ (with $i=1,2$). In such situation one can of
course make arbitrary choice of basis functions (e.g.,
$G_{2,\mathbf{1}^{+},1}^{\rm SV}+G_{2,\mathbf{1}^{+},2}^{\rm SV}$ and
$G_{2,\mathbf{1}^{+},1}^{\rm SV}-G_{2,\mathbf{1}^{+},2}^{\rm SV}$ here), but
the dimension of the function space is invariant.

When the letter $z-\bar{z}$ is also included, starting at weight $3$ there are
additional independent functions. In particular at weight $3$ there is a
unique $\mathbf{1}^{-}$ function, which we denote as $Q_{3}$ 555The $Q_{3}$
function we construct here is to be distinguished from that defined in
Chavez:2012kn . We require this function to be in the $\mathbf{1}$
representation of $\mathrm{S}_{3}$.. There are even more functions at higher
weights but it turns out only $Q_{3}$ is needed (intuition about these extra
functions is discussed in the appendix). Hence in total there are 189
independent SVMPLs in our basis, and $\mathcal{L}^{(3)}$ has the expansion

$\displaystyle\mathcal{L}^{(3)}$
$\displaystyle=\sum_{w=0}^{6}\sum_{r,i}\frac{\omega^{w,r}_{i}(z,\bar{z})}{(\bar{z}-z)^{7}}G^{\rm
SV}_{w,r,i}(z,\bar{z}),$ (27) $\displaystyle\omega^{w,r}_{i}$
$\displaystyle=\sum_{j,k=0}^{7}c^{w,r}_{i,j,k}z^{j}\bar{z}^{k},\quad
c^{w,r^{\pm}}_{i,j,k}=\mp c^{w,r^{\pm}}_{i,k,j}.$ (28)

The undetermined coefficients $c^{w,r}_{i,j,k}$ are rational. The condition
(28) is to ensure that $\mathcal{L}^{(3)}$ is a $\mathbb{Z}_{2}$ invariant.
The power of denominator in (27), $7$, is exactly the same as that at one-loop
level (20) and that in the leading log terms at arbitrary loops (18). One may
be curious about taking a larger power (for instance $9$) to begin with, and
so there will be even more unknown variables. With the constraints to be
described later on, in our computation we find such enlarged ansatz leads to
the same result.

##  4 Bootstrap

We now set out to constrain the above ansatz. $\mathcal{L}^{(3)}$ is not a
full correlator, but is expected to admit a decomposition into s-channel
blocks. Therefore a set of generic properties can be directly imposed at the
level of $\mathcal{L}^{(3)}$, as listed below:

  1. 1.

In Euclidean region $\mathcal{L}^{(3)}$ should be finite at $z=\bar{z}$.

  2. 2.

As a sum of s-channel blocks with identical external operators, exchanging
operator 1 and 2 should leave $\mathcal{L}^{(3)}$ unchanged. Explicitly we
have

$\mathcal{L}^{(3)}(z,\bar{z})=\mathcal{L}^{(3)}\left(\frac{\bar{z}}{\bar{z}-1},\frac{z}{z-1}\right).$
(29)

The known leading log expression (16) provides an important constraint on
$\mathcal{H}^{(3)}$. In fact it can be imposed directly on
$\mathcal{L}^{(3)}$, as $\Delta^{(8)}$ acting on $\log^{k}{u}$ definitely
lowers its power. This leads to a theory-specific condition:

  1. A.

When expanding $\mathcal{L}^{(3)}$ in the s-channel, all the $\log^{k}{u}$
terms with $k>3$ have to vanish. And the $\log^{3}{u}$ terms of
$\mathcal{L}^{(3)}$ should match known data. By comparing (27) and (16) we
have

$\mathcal{L}^{(3)}(z,\bar{z})\big{|}_{\log^{3}{u}}=\mathcal{F}^{(3)}(z,\bar{z}),$
(30)

where the detailed expression of $\mathcal{F}^{(3)}$ can be found in Appendix
C of Caron-Huot:2018kta .

After solving the above constraints we pass on to the full correlator
$\mathcal{H}^{(3)}$ via the action of $[\Delta^{(8)}]^{2}$. Firstly there is
one last generic CFT constraint:

  1. 3.

$\mathcal{H}^{(3)}$ should respect the full crossing symmetries. With the help
of the $\mathrm{S}_{3}\times\mathbb{Z}_{2}$ SVMPL basis, this is equivalent to
requiring that, when decomposing the correlator as

$\frac{(z-\bar{z})^{4}}{z^{4}\bar{z}^{4}}\mathcal{H}^{(3)}=\\!\\!\\!\\!\sum_{w,r,i}\Omega^{w,r}_{i}(z,\bar{z})G^{\rm
SV}_{w,r,i}(z,\bar{z}),$ (31)

the rational coefficient functions $\Omega^{w,r}_{i}(z,\bar{z})$ transform in
a way such that each term on RHS is an $\mathrm{S}_{3}$ invariant, i.e.,

$\displaystyle\rho(abc)\circ\Omega^{w,r}_{i}$
$\displaystyle=\Omega^{w,r}_{i}(\mathbf{R}_{(abc)}^{r})^{-1}.$ (32)

Then there are more conditions specific to $\mathcal{N}=4$ SYM, mainly
regarding its spectrum:

  1. B.

The tree-level correlator $\mathcal{H}^{(1)}$ contains twist below the double-
twist threshold $4$ in its decomposition into ordinary conformal blocks, which
is manifested in that $\mathcal{H}^{(1)}$ has poles at $z=1$ and at
$\bar{z}=1$. As pointed out before, for $\langle 2222\rangle$
$\mathcal{H}^{(k\geq 2)}$ consists of only long super-multiplets (with
$\tau^{(0)}\geq 4$) and so is free of these poles. Therefore in our ansatz
(21) $\left[\Delta^{(8)}\right]^{2}\mathcal{L}^{(3)}$ expects to produce
exactly the same poles (which is indeed true) in order to cancel out the ones
in $\mathcal{H}^{(1)}$. This condition fixes $\kappa_{1}$ to $-\frac{1}{16}$.

  2. C.

Coefficients of the subleading $\log^{2}{u}$ terms in $\mathcal{H}^{(3)}$ take
the schematic form $\langle
a^{(1)}(\gamma^{(1)})^{2}+2a^{(0)}\gamma^{(1)}\gamma^{(2)}\rangle$. In general
they include contributions from long operators other than the double-trace
ones $[\mathcal{O}\mathcal{O}]$. Due to the lack of knowledge about these
extra operators these coefficients cannot be recursively determined in the
fashion of leading log coefficients (15). However, twist $4$ is a lucky
exception, where only $[\mathcal{O}_{2}\mathcal{O}_{2}]_{0,\ell}$ contribute.
There is even no need to solve any operator mixing and these coefficients are
easily determined as

$\begin{split}\langle
a^{(1)}(\gamma^{(1)})^{2}+2a^{(0)}\gamma^{(1)}\gamma^{(2)}\rangle_{4,\ell}=&\frac{2\langle
a^{(0)}\gamma^{(1)}\rangle_{4,\ell}\langle
a^{(1)}\gamma^{(1)}+a^{(0)}\gamma^{(2)}\rangle_{4,\ell}}{\langle
a^{(0)}\rangle_{4,\ell}}\\\ &-\frac{\langle
a^{(0)}\gamma^{(1)}\rangle_{4,\ell}^{2}\langle
a^{(1)}\rangle_{4,\ell}}{\langle a^{(0)}\rangle_{4,\ell}^{2}},\end{split}$
(33)

where the data shown on RHS are all available from $\langle 2222\rangle$ at
lower orders. This fixes $\kappa_{2}$ to $\frac{5}{4}$.

  3. D.

It is generally expected that in the bulk-point limit $\langle 2222\rangle$
reduces to the flat-space four-graviton scattering amplitude in 10d
Heemskerk:2009pn ; Gary:2009ae ; Maldacena:2015iua . Alday:2017vkk  proposed a
simpler connection between the two sides at loop level, by comparing the
discontinuities. Specifically, the leading divergence of
$\mathrm{dDisc}\,\mathcal{H}(z^{\circlearrowright},\bar{z})$ at the bulk point
limit ($z=\bar{z}+2\epsilon\bar{z}\sqrt{1-\bar{z}}$, $\epsilon\to 0$ after
continuing $z$ around $z=0$ clock-wisely) should match the discontinuity of
the scattering amplitude $\mathcal{A}$ across the t-channel cut. At two loops
we have

$\displaystyle(\bar{z}-z)^{23}\text{dDisc}\,\mathcal{H}^{(3)}(z^{\circlearrowright},\bar{z})\big{|}_{z\to\bar{z}}=\frac{\Gamma(22)(1-\bar{z})^{11}\bar{z}^{24}}{8\pi^{8}}\text{Disc}_{x>1}\mathcal{A}^{(2)}(x)\big{|}_{x=1/\bar{z}},$
(34)

where the two-loop supergravity amplitude $\mathcal{A}^{(2)}(x)$ has been
computed in Bissi:2020woe . This matching can also be done even before the
action of $[\Delta^{(8)}]^{2}$. See discussions in Section 2.4 of
Bissi:2020woe .

  4. E.

By construction it is not guaranteed that the ansatz produces OPE data with
sufficient analyticity in spin. At two loops it is expect that the OPE
coefficient are analytic functions of spin $\ell$ down to $\ell=2$ at
$\log^{2}u$, and to $\ell=6$ at $\log^{1}u$ and $\log^{0}u$, while the non-
analyticity at lower spins are closely tied to the allowed contourterm
ambiguities at two loops. To ensure sufficient analyticity we impose that the
ansatz is self-consistent under CFT dispersion relation, up to these
ambiguities. In our computation this constraint is implemented using the so-
called Lorentzian inversion formula Caron-Huot:2017vep ; Simmons-
Duffin:2017nub ; Alday:2017vkk ; Caron-Huot:2018kta , and the details are
presented in the appendix.

Note that after the action of $[\Delta^{(8)}]^{2}$ the number of unsolved
$c^{w,r}_{i,j,k}$ that actually show up in $\mathcal{H}^{(3)}$ is reduced.
This is because $[\Delta^{(8)}]^{2}$ has a non-trivial kernel, which generates
no physical effects at all. Apart from these disappearing variables there are
two types of remaining degrees of freedom at the end of the above computation.
The first type can be identified as ambiguities due to counterterms, which are
discussed in the next section.

The second type is genuinely tied to supergravity, which has support on
infinitely many spins. This type contains only one remaining variable (let us
call it $\mathcal{X}$). We are not aware of direct physical constraints to
determine this variable. However, there are considerations regarding the
mathematical structure of the result.

Note that at the end of the above computation the function space in which the
correlator lives greatly shrinks, by the fact that a large portion of basis
functions $G^{\rm SV}$ completely disappear 666In $(w,r)$ sectors with
degeneracy this observation relies on a further proper choice of basis
functions. For instance, in the $(2,\mathbf{1}^{+})$ sector, with the basis
shown in (26) we observe $G_{2,\mathbf{1}^{+},1}^{\rm SV}$ disappears at the
end.. By curiosity one can try further reducing the function space by tuning
the remaining variables. Remarkably, it turns out the only possibility is to
fix $\mathcal{X}$ to a unique value! To be explicit, this value kills one
$G_{5,\bar{\mathbf{1}}^{-}}^{\rm SV}$ and one $G_{4,\mathbf{2}^{+}}^{\rm SV}$
basis elements. Furthermore, while $\mathcal{L}^{(3)}$ does not need to
respect the full crossing symmetry in general, we find it is possible to
consistently do so, and this fixes $\mathcal{X}$ to exactly the same value as
above. These are strong hints that this choice should be special from the
physical perspective as well, hence we conjecture that it leads to the correct
supergravity result at two loops.

##  5 Results

Loop-level supergravity computations suffer from counterterms of the schematic
form $\partial^{2k}\mathcal{R}^{4}$. These vertices can build up tree-level
and one-loop diagrams, which we denote as $\mathcal{C}_{\text{tree},i}$ and
$\mathcal{C}_{\text{one-loop},q}$, $i$ and $q$ labeling different diagrams.
These diagrams are responsible for the cancellation of UV divergence in the
two-loop correlator of supergravity, at the cost of introducing ambiguities
unless the full string theory is considered.

The tree-level diagrams $\mathcal{C}_{\text{tree},i}$ are easily described in
Mellin space Mack:2009mi ; Mack:2009gy ; Penedones:2010ue

$\mathcal{C}_{\text{tree},i}=\int_{-i\infty}^{i\infty}\frac{\differential
s\,\differential t}{(2\pi
i)^{2}}u^{\frac{s}{2}+2}v^{\frac{t}{2}-2}\mathcal{M}_{\text{tree},i}\,\Gamma\left(2-\frac{s}{2}\right)^{2}\Gamma\left(2-\frac{t}{2}\right)^{2}\Gamma\left(2-\frac{\tilde{u}}{2}\right)^{2},$
(35)

with $s+t+\tilde{u}=4$. The corresponding Mellin amplitudes
$\mathcal{M}_{\text{tree},i}$ are just symmetric polynomials in
$s,t,\tilde{u}$ Alday:2014tsa ; Alday:2018pdi

$\displaystyle\mathcal{M}_{\text{tree},1}$ $\displaystyle=1,$ (36)
$\displaystyle\mathcal{M}_{\text{tree},2}$
$\displaystyle=s^{2}+t^{2}+\tilde{u}^{2},$ (37)
$\displaystyle\mathcal{M}_{\text{tree},3}$ $\displaystyle=st\tilde{u},$ (38)
$\displaystyle\mathcal{M}_{\text{tree},4}$
$\displaystyle=(s^{2}+t^{2}+\tilde{u}^{2})^{2},$ (39)
$\displaystyle\mathcal{M}_{\text{tree},5}$
$\displaystyle=st\tilde{u}(s^{2}+t^{2}+\tilde{u}^{2}),$ (40)
$\displaystyle\mathcal{M}_{\text{tree},6}$
$\displaystyle=s^{2}t^{2}\tilde{u}^{2}.$ (41)

For those who are not familiar with Mellin space method, we also offer the
results in position space

$\displaystyle\mathcal{C}_{\text{tree},1}$
$\displaystyle=4u^{4}(1+u\partial_{u})(2+u\partial_{u})(3+u\partial_{u})\partial_{u}^{3}\Phi(u,v),$
(42) $\displaystyle\mathcal{C}_{\text{tree},i}$
$\displaystyle=\widehat{\mathcal{M}}_{\text{tree},i}\,\mathcal{C}_{\text{tree},1},$
(43)

where

$\Phi(u,v)=\frac{1}{z-\bar{z}}\left(\text{Li}_{2}(z)-\text{Li}_{2}(\bar{z})+\log(z\bar{z})\log\frac{1-z}{1-\bar{z}}\right),$
(44)

and $\widehat{\mathcal{M}}_{\text{tree},i}$ are differential operators
obtained by replacing $s,t,\tilde{u}$ in $\mathcal{M}_{\text{tree},i}$ with

$\displaystyle\hat{s}$ $\displaystyle=-4+2u\partial_{u},$ (45)
$\displaystyle\hat{t}$ $\displaystyle=4+2v\partial_{v},$ (46)
$\displaystyle\hat{u}$ $\displaystyle=4-2u\partial_{u}-2v\partial_{v}.$ (47)

Among these tree-level ambiguities only $\mathcal{C}_{\rm tree,i\leq 5}$
appear in our computation, because $\mathcal{C}_{\rm tree,6}$ has a power of
$z-\bar{z}$ in the denominator that exceeds our ansatz, while the others can
all be put into the pattern as an action of $(\Delta^{(8)})^{2}$ or just come
from the leftover ambiguity in the one-loop reduced correlator
$\mathcal{H}^{(2)}$.

The one-loop diagrams $\mathcal{C}_{\text{one-loop},i}$ no longer take such a
simple form in Mellin space. The origin of these diagrams is further described
in Appendix B. Like the tree-level contact diagrams, they have the
characteristic feature that their leading log OPE coefficients have finite
support on spin Alday:2018pdi ; Drummond:2019hel . With this feature these
diagrams can be bootstrapped individually using similar ansatz. The allowed
solutions can be classified according to the power of $z-\bar{z}$ in the
denominator in its leading log ($\log^{2}u$) part, which is $7+2q$
($q=0,1,\ldots$) Alday:2018pdi . By this power counting the ambiguities that
can possibly show up in our ansatz corresponds to the solutions with $q$ up to
$8$. We list these nine solutions in the ancillary file. It turns out that in
order to fit the $(\Delta^{(8)})^{2}$ structure in (21) they are further
restricted to two possible linear combinations, which are responsible for the
remaining one-loop ambiguities left over in our computation.

We have checked that the remaining degrees of freedom from our computation
(except for $\mathcal{X}$) belong to the counterterm ambiguities described
above. Nevertheless, because we do not study string corrections in this paper,
all the above ambiguities can in principle enter the complete supergravity
result, including those that are not captured by our ansatz. We therefore
express the final result of this bootstrap as a special solution plus all such
counterterm ambiguities

$\boxed{\begin{split}\mathcal{H}^{(3)}=&\left[\Delta^{(8)}\right]^{2}\mathcal{L}^{(3)}+\frac{5}{4}\mathcal{H}^{(2)}-\frac{1}{16}\mathcal{H}^{(1)}\\\
&+\sum_{q=0}^{\infty}b_{1,q}\mathcal{C}_{\text{1-loop},q}+\sum_{i=1}^{6}b_{0,i}\mathcal{C}_{\text{tree},i}.\end{split}}$
(48)

Notice that $\mathcal{C}_{\text{1-loop},q}$ may potentially affect $\log^{1}$
and $\log^{0}$ data for all spin $\ell$. The coefficients $b_{1,q}$ and
$b_{0,i}$ can be determined only when the full string theory corrections to
the correlator are considered. We leave them for future work.

In $\mathcal{H}^{(3)}$ each coefficient function $\Omega^{w,r}_{i}(z,\bar{z})$
forms a representation of $\mathrm{S}_{3}\times\mathbb{Z}_{2}$ according to
(32). This motivate us to define a basic building block

$B_{d,k}(z,\bar{z})=\frac{(1+u+v)^{d-3k}(uv)^{k}}{(z-\bar{z})^{d}},\quad
d,k\in\mathbb{N},$ (49)

and then in each representation $r$ and given maximal power $d_{\rm max}$ of
$(z-\bar{z})$ we construct a basis $\mathcal{B}^{r}_{d_{\rm max}}$ for
$\Omega$

$\displaystyle\mathcal{B}^{\mathbf{1}^{+}}_{d_{\rm max}}$
$\displaystyle=\\{B_{d,k}\;|\;d\text{ even},d\leq d_{\rm max},k\leq\lfloor
d/3\rfloor\\},$ (50) $\displaystyle\mathcal{B}^{\mathbf{1}^{-}}_{d_{\rm max}}$
$\displaystyle=\\{B_{d,k}\;|\;d\text{ odd},d\leq d_{\rm max},k\leq\lfloor
d/3\rfloor\\},$ (51)
$\displaystyle\mathcal{B}^{\bar{\mathbf{1}}^{\pm}}_{d_{\rm max}}$
$\displaystyle=\frac{(1-u)(1-v)(u-v)}{(z-\bar{z})^{3}}\mathcal{B}^{\mathbf{1}^{\mp}}_{d_{\rm
max}-3},$ (52) $\displaystyle\mathcal{B}^{\mathbf{2}^{\pm}}_{d_{\rm max}}$
$\displaystyle=\frac{(1-2u+v,1+u-2v)}{(z-\bar{z})}\mathcal{B}^{\mathbf{1}^{\mp}}_{d_{\rm
max}-1}$
$\displaystyle\;\quad\bigcup\frac{(1-2u^{2}+v^{2},1+u^{2}-2v^{2})}{(z-\bar{z})^{2}}\mathcal{B}^{\mathbf{1}^{\pm}}_{d_{\rm
max}-2}.$ (53)

For our study of $\mathcal{H}^{(3)}$, $d_{\rm max}=19$. Then the correlator
can be expressed as

$\frac{(z-\bar{z})^{4}}{z^{4}\bar{z}^{4}}\mathcal{H}^{(3)}=\sum_{r}\mathcal{B}^{r}_{19}\mathcal{C}^{r}\mathcal{G}^{\rm
SV}_{r},$ (54)

and the dynamics of the theory is encoded in the coefficient matrices
$\mathcal{C}^{r}$. $\mathcal{G}^{\rm SV}_{r}$ denotes the set of SVMPL basis
for the representation $r$.

Comparing to the ansatz, we end up with a much smaller set of SVMPL basis.
Apart from $Q_{3}$, the counting of the basis in each sector reduces as

$w$ $r$ | $\;\mathbf{1}^{+}\;$ | $\;\mathbf{1}^{-}\;$ | $\;\bar{\mathbf{1}}^{+}\;$ | $\;\bar{\mathbf{1}}^{-}\;$ | $\;\mathbf{2}^{+}\;$ | $\;\mathbf{2}^{-}\;$
---|---|---|---|---|---|---
$0$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$
$1$ | $0$ | $0$ | $0$ | $0$ | $1$ | $0$
$2$ | ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| $1$ | $0$ | $0$ | $1$ | $0$
$3$ | ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| $1$ | $1$ | $0$ | ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| $1$
$4$ | ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| $1$ | $0$ | ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| $2$
$5$ | ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>
$6$ | ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>

The $\mathcal{C}^{r}$ matrices are still too huge to directly fit into this
paper, and we provide them in the ancillary file on arXiv. Because the
constraints on $\mathcal{X}$ is conjectural, in that file we also present the
coefficients of $\mathcal{X}$ before this last step for readers’ reference (we
have shifted the definition of this last variable such that $\mathcal{X}=0$
corresponds to our supergravity result).

##  6 Outlook

In this paper we bootstraped the simplest CPO correlator in $\mathcal{N}=4$
SYM at AdS two loops. Our computation is based on an ansatz that generalizes a
structure observed at one loop which was inspired by a conjectured 10d
conformal symmetry. Very interestingly, after exploiting all possible physical
constraints available to us, there arises a unique solution (up to contourterm
ambiguities). We expect that this result may inspire detailed studies of
graviton dynamics and scattering of other particles at two loops and at even
higher loops. Combined with string corrections that fix the counterterms this
will provide data for unprotected operators in SYM at strong coupling with
higher precision. Some of the future explorations are commented below.

Our supergravity result can be expressed in terms of repeated action of a
remarkable differential operator $\Delta^{(8)}$. In particular, if we plug in
the one-loop result (19) into (48) we obtain

$\displaystyle\mathcal{H}^{(3)}=\left[\Delta^{(8)}\right]^{2}\mathcal{L}^{(3)}+\frac{5}{4}\Delta^{(8)}\mathcal{L}^{(2)}+\frac{1}{4}\mathcal{L}^{(1)},$
(55)

plus additional counterterms, where $\mathcal{L}^{(1)}=\mathcal{H}^{(1)}$. The
coefficients above are both very simple rationals, which puts strong
confidence on the validity of this result as well as the structure of the
ansatz. It is then natural to write the correlator $\langle 2222\rangle$ at
arbitrary higher loops as

$\mathcal{H}^{(k)}=\sum_{j=1}^{k}\kappa^{(k)}_{j}\left[\Delta^{(8)}\right]^{j-1}\mathcal{L}^{(j)},$
(56)

where $\kappa^{(k)}_{k}=1$. Then at each order $k$ the problem reduces to
solving a single function $\mathcal{L}^{(k)}$ together with the
$\kappa^{(k)}_{j<k}$ coefficients. From the structure of the known leading log
terms $\mathcal{L}^{(j)}$ need to contain SVMPLs of weights up to $2k$, with
coefficients of the same structure as in (27)(28).

Moreover, the conjectured hidden symmetries in Caron-Huot:2018kta  also
implies that at tree-level other CPO correlators $\langle pqrs\rangle$ (after
subtracting a free part) can be extracted from $\langle 2222\rangle$ by
another differential operator $\mathcal{D}_{pqrs}$

$\displaystyle\mathcal{H}^{(1)}_{pqrs}-\mathcal{H}^{(1),\text{free}}_{pqrs}=\mathcal{D}_{pqrs}\mathcal{H}^{(1)}.$
(57)

The leading log terms at higher loops again enjoy the precise relation

$\displaystyle\mathcal{H}^{(k)}_{pqrs}\Big{|}_{\log^{k}{u}}=\left[\Delta^{(8)}\right]^{k-1}\mathcal{D}_{pqrs}\mathcal{F}^{(k)}(z,\bar{z}),$
(58)

where $\Delta^{(8)}$ now takes a more general form, which together with
$\mathcal{D}_{pqrs}$ can be found in Caron-Huot:2018kta . It was observed in
Aprile:2019rep  that one-loop correlators $\mathcal{H}^{(2)}_{22pp}$ admit
representations of the form (19) as well, although the non-derivative term is
no longer simply proportional to $\mathcal{H}^{(1)}_{22pp}$. It will be
interesting to investigate whether this may suggest similar method to
bootstrap generic $\mathcal{H}^{(3)}_{pqrs}$.

Finally, observe that the number of independent SVMPLs is greatly reduced in
the final result as compared to those in the ansatz. In many categories with
fixed $w$ and $r$ there is even a unique function remaining. It would be nice
to see whether these functions individually have any physical significance.

###### Acknowledgements.

The authors would like to thank Yunfeng Jiang, Song He, Paul Heslop, Lilin
Yang, Yang Zhang, Xinan Zhou, Huaxing Zhu for useful discussions. ZH and EYY
are supported by National Science Foundation of China under Grant No. 12175197
and Grand No. 12147103. EYY is also supported by National Science Foundation
of China under Grant No. 11935013, and by the Fundamental Research Funds for
the Chinese Central Universities under Grant No. 226-2022-00216.

##  Appendix A Multiple Polylogarithms

Multiple polylogarithms (MPLs) can be defined in terms of nested integrals
Duhr:2019tlz

$\displaystyle G(z)=$ $\displaystyle 1,$ $\displaystyle G(\vec{a};z)\equiv$
$\displaystyle G(a_{1},...,a_{n};z)=\int_{0}^{z}\frac{\differential
t}{t-a_{1}}G(a_{2},...,a_{n};t).$

Especially when $\vec{a}=(0,\dots,0)$, we define

$\displaystyle G(\underbrace{0,\dots,0}_{n};z)=\frac{1}{n!}\log{z}^{n}.$

The length $|\vec{a}|$ of vector $\vec{a}$ is called the weight of
$G(\vec{a};z)$. $G$ is naturally accompanied by a mathematical entity called
symbol, $\mathcal{S}[G]$, whose advantages include encoding information about
their singularities and simplifying identities among MPLs. $\mathcal{S}[G]$
can be obtained by applying differentiation iteratively. If

$\differential G=\sum_{i}G^{\prime}_{i}\ \differential\log R_{i},$

where $R_{i}$s are algebraic functions, then

$\mathcal{S}[G]=\sum_{i}S[G^{\prime}_{i}]\otimes R_{i}.$

As a simple example,
$\mathcal{S}[G(0,1;z)]\equiv\mathcal{S}[-\mathrm{Li}_{2}(z)]=(1-z)\otimes z$.
Expression in each entry of the $\otimes$ product is called a _symbol letter_
, and the collection of all letters the _alphabet_ of the symbol. Our
computation of both MPLs and their symbols utilizes the PolyLogTools
Mathematica package introduced in Duhr:2019tlz .

##  Appendix B About Extra MPLs with Letter $z-\bar{z}$

There are several intuitions about why $Q_{3}$ has to be included in the
computation at two loops. On the one hand, one may try to carry out the same
bootstrap program as above but without $Q_{3}$. Then in the comparison at the
bulk-point limit the ansatz fails to match the flat-space result. In fact, it
manages to cover the contribution from plannar diagrams in the supergravity
amplitude, but that from the non-plannar diagrams goes beyond.

On the other hand, when viewing in the full context of string theory with
$\lambda$ expansion, the one-loop supergravity correlator receives a string
correction from $\mathcal{R}^{4}$ residing at the $\lambda^{0}$ order (e.g.,
see Chester:2019pvm ; Chester:2020dja ). By unitarity it together with the
tree-level supergravity result recursively determines a one-loop counterterm
to the two-loop correlator, which is again at the same $\lambda^{0}$ order as
supergravity. Moreover, by origin this correction has to be proportional to
the $\lambda^{-3/2}$ correction to the one-loop correlator, and the latter was
computed in Drummond:2019hel  and was observed to contain $Q_{3}$ 777Again,
the $Q_{3}$ function in that paper differs from ours by MPLs without the
symbol letter $z-\bar{z}$.. Because in our setup the computation is subject to
ambiguities regarding counterterms, there is no reason that this function is
well separated from the supergravity.

One may also wonder about higher-weight single-valued MPLs with letter
$z-\bar{z}$. We found that including such MPLs with weight 4 leads to the same
result.

##  Appendix C Constraints from Lorentzian Inversion Formula

As mentioned previously the last step of our computation constrains the ansatz
using Lorentzian inversion formula Caron-Huot:2017vep ; Simmons-Duffin:2017nub
. As with other types of dispersion relations, this formula reconstructs (data
of) the entire correlator from its singularities in the Lorentzian region. The
core object here is the double discontinuity

$\mathrm{dDisc}\,\mathcal{H}(z,\bar{z})=\mathcal{H}(z,\bar{z})-\frac{1}{2}\mathcal{H}^{\circlearrowleft}(z,\bar{z})-\frac{1}{2}\mathcal{H}^{\circlearrowright}(z,\bar{z}),$
(59)

where $\circlearrowleft$ and $\circlearrowright$ means to analytically
continue $\bar{z}$ around $\bar{z}=1$ counter-clockwisely and clockwisely
respectively. Specific to the computation of $\langle 2222\rangle$ this
formula reads Alday:2017vkk ; Caron-Huot:2018kta

$C(\tau,\ell)\\!=\\!\kappa_{\bar{h}}\\!\int_{0}^{1}\frac{\mathrm{d}z}{z^{2}}\frac{\mathrm{d}\bar{z}}{\bar{z}^{2}}k_{1-h}(z)k_{\bar{h}}(\bar{z})\mathrm{dDisc}\left[\frac{\bar{z}-z}{\bar{z}z}\mathcal{H}\right],$
(60)

where
$\kappa_{\bar{h}}=\frac{(1+(-1)^{\ell})\Gamma(\bar{h})^{4}}{4\pi^{2}\Gamma(2\bar{h})\Gamma(2\bar{h}-1)}$,
$h=\frac{\tau+2}{2}$ and $\bar{h}=\frac{\tau+2\ell+4}{2}$. It encodes the
twists and the coefficients of blocks in the s-channel OPE in the form

$C(\tau,\ell)=\sum_{i}\frac{a_{i}}{\tau-\tau_{i}},$ (61)

where the summation is over different operators. Taking $1/c$ expansion
(13)(14) into consideration RHS of (61) further expands out, e.g., at order
$c^{-3}$

$\begin{split}C(\tau,\ell)\supset\frac{1}{c^{3}}\sum_{\tau^{(0)}}&\left(\frac{\langle
a^{(3)}\rangle}{\tau-\tau^{(0)}}+\frac{\langle
a^{(2)}\gamma^{(1)}+a^{(1)}\gamma^{(2)}+a^{(0)}\gamma^{(3)}\rangle}{(\tau-\tau^{(0)})^{2}}\right.\\\
&\left.\quad+\frac{\langle
a^{(1)}(\gamma^{(1)})^{2}+2a^{(0)}\gamma^{(1)}\gamma^{(2)}\rangle}{(\tau-\tau^{(0)})^{3}}+\frac{\langle
a^{(0)}(\gamma^{(1)})^{3}\rangle}{(\tau-\tau^{(0)})^{4}}\right).\end{split}$
(62)

These $\tau$ poles in $C(\tau,\ell)$ originate from $z$ integrals in (60). To
see this, we first expand the integrand of RHS at $z=0$. Since $k_{1-h}(z)\sim
z^{1-h}(1+\frac{1-h}{2}z+\cdots)$, the integrals to be done are

$\begin{split}\int_{0}^{1}\\!\frac{\differential
z}{z^{2}}z^{1-h+n}\log^{k}z=\frac{(-1)^{k+1}k!}{(h\\!-\\!n)^{k+1}}\sim\frac{1}{(\tau\\!+\\!2\\!-\\!2n)^{k+1}}.\end{split}$
(63)

Notice that only $\log^{k\geq k_{0}}z$ terms contribute to $(k_{0}+1)$-th
poles. For the computation of $\bar{z}$ integrals, see Appendix A in
Alday:2017zzv .

The data collected from the various coefficients in (62) directly show up in
the block expansion of $\mathcal{H}^{(3)}$ as well (as can be easily worked
out by taking $1/c$ expansion of (10))

$\begin{split}\mathcal{H}^{(3)}=&\sum_{\tau^{(0)},\ell}\left(\langle
a^{(3)}\rangle\ g_{\tau^{(0)},\ell}+\langle
a^{(2)}\gamma^{(1)}+a^{(1)}\gamma^{(2)}+a^{(0)}\gamma^{(3)}\rangle\,\partial_{\tau^{(0)}}g_{\tau^{(0)},\ell}\right.\\\
&\left.+\langle
a^{(1)}(\gamma^{(1)})^{2}+2a^{(0)}\gamma^{(1)}\gamma^{(2)}\rangle\frac{1}{2!}\partial^{2}_{\tau^{(0)}}\,g_{\tau^{(0)},\ell}+\langle
a^{(0)}(\gamma^{(1)})^{3}\rangle\frac{1}{3!}\partial^{3}_{\tau^{(0)}}\,g_{\tau^{(0)},\ell}\right).\end{split}$
(64)

Once we obtain these data from the Lorentzian inversion formula, we plug them
into the above expression (64) to reproduce $\mathcal{H}^{(3)}$ and compare
with the ansatz. This serves as a consistency constraint at each $\log u$
orders (note that in (64) $\log u$ can come out from
$\partial^{k}_{\tau^{(0)}}\,g_{\tau^{(0)},\ell}$, which tops up to
$\log^{k}u$). For example at the leading log order

$\sum_{\tau^{(0)},\ell}\frac{\langle
a^{(0)}(\gamma^{(1)})^{3}\rangle}{3!\,2^{3}}\,g_{\tau^{(0)}+4,\ell}(z,\bar{z})=\mathcal{H}^{(3)}\big{|}_{\log^{3}{u}}.$
(65)

However, (65) is a double infinite sum of two-variable functions, which is
hard to handle. A better way is to consider the series expansion of (65) up to
some order $\mathcal{O}(z^{h_{\text{max}}})$. Since
$g_{\tau^{(0)}+4,\ell}(z,\bar{z})\sim\mathcal{O}(z^{\frac{\tau^{(0)}+4}{2}})$,
the $\tau^{(0)}$ sum can thus be truncated to a finite sum

$\begin{split}\sum_{\tau^{(0)}=4}^{\scriptsize
2h_{\text{max}}-4}\sum_{\ell=0}^{\infty}\frac{\langle
a^{(0)}(\gamma^{(1)})^{3}\rangle}{3!\,2^{3}}\,g_{\tau^{(0)}+4,\ell}-\mathcal{H}^{(3)}\big{|}_{\log^{3}{u}}=\mathcal{O}(z^{h_{\text{max}}+1}).\end{split}$
(66)

For the remaining infinite sum over $\ell$, by (8) we only need to work out
for each $\tau^{(0)}$

$\sum_{\ell=0}^{\infty}\frac{\langle
a^{(0)}(\gamma^{(1)})^{3}\rangle}{3!\,2^{3}}\,k_{2+\frac{\tau^{(0)}}{2}+\ell}(\bar{z}).$
(67)

This infinite sum is supposed to be a linear combination of MPLs with alphabet
$\\{\bar{z},1-\bar{z}\\}$, where coefficients are rational functions with
denominator $\bar{z}^{\frac{\tau^{(0)}}{2}}$. The analytic result can be
easily worked out by setting up an ansatz and matching the Taylor series
around $\bar{z}=0$ with (67).

During the actual computation, since the leading log terms were already
determined by the matching (30) in previous steps, (66) itself only provides a
consistency check. However, we observe that non-trivial constraints enter in
the comparison at subsequent $\log u$ terms. We exhaust all possible
constraints of this type by gradually lifting the twist cutoff $h_{\rm max}$
till no more constraints further appear.

Subtlety comes when analyticity in spin can not go down to $\ell=0$ Caron-
Huot:2017vep . For example the $\log^{2}{u}$ data $\langle
a^{(1)}(\gamma^{(1)})^{2}+2a^{(0)}\gamma^{(1)}\gamma^{(2)}\rangle$ extracted
from $C(\tau,\ell)$ at $\ell=0$ may not be the true OPE data of
$\mathcal{H}^{(3)}$. In this case, we set $\ell=0$ data as unknown variables
when summing with conformal blocks. The same applies to $l=0,2,4$ data in
$\log^{1}u$ and $\log^{0}u$. This strategy takes care of possible counterterm
ambiguities appearing in the two-loop calculations.

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