Two-loop supergravity on AdS${}_{5}\times$S5 from CFT

J. M. Drummond1 and H. Paul2
1 School of Physics and Astronomy and STAG Research Centre,
University of Southampton, Highfield, SO17 1BJ  2 Université Paris-Saclay,
CNRS, CEA, Institut de Physique Théorique,
91191, Gif-sur-Yvette, France

Abstract

We describe a construction of the two-loop amplitude of four graviton
supermultiplets in AdS${}_{5}\times$S5. We start from an ansatz for a
preamplitude from which we generate the full amplitude under the action of a
specific Casimir operator. The ansatz captures a recent ansatz of Huang and
Yuan and we confirm their result through similar constraints. The form of the
result suggests that all ambiguities are captured by the preamplitude which
determines the result up to tree-level ambiguities only. We identify a class
of four-dimensional ‘zigzag’ integrals which are perfectly adapted to
describing the leading logarithmic discontinuity to all orders. We also
observe that a bonus crossing symmetry of the preamplitude follows from the
transformation properties of the Casimir operator. Combined with the zigzag
integrals this allows us to construct a crossing symmetric function with the
correct leading logarithmic discontinuities in all channels.

From the two-loop result we extract an explicit expression for the two-loop
correction to the anomalous dimensions of twist-four operators of generic spin
which includes dependence on (alternating) nested harmonic sums up to weight
three. We also revisit the prescription of the bulk-point limit of AdS
amplitudes and show how it recovers the full flat-space amplitude, not just
its discontinuity. With this extended notion of the bulk-point limit we
reproduce the scale-dependent logarithmic threshold terms of type IIB string
theory in flat-space.

###### Contents

  1. 1 Introduction
    1. 1.1 Setup: the $\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$ correlator
    2. 1.2 OPE decomposition and double-trace spectrum
  2. 2 Zigzag integrals: a basis for the leading log
    1. 2.1 The operator $\Delta^{(8)}$ and the leading log
    2. 2.2 The Zigzag integrals
  3. 3 The bulk-point limit of AdS${}_{5}\times$S5 amplitudes
    1. 3.1 The perturbative expansion of the type IIB string amplitude
    2. 3.2 The contributions $\mathcal{H}^{(n,m)}$ in the bulk-point limit
  4. 4 Bootstrapping the two-loop correlator $\mathcal{H}^{(3)}$
    1. 4.1 Review of tree-level and one-loop
    2. 4.2 More on $\Delta^{(8)}$: symmetries under crossing
    3. 4.3 A minimal ansatz for the two-loop correlator
    4. 4.4 Bootstrapping $\mathcal{H}^{(3)}$: constraints
    5. 4.5 Bootstrapping $\mathcal{H}^{(3)}$: results
    6. 4.6 Comparison with the results of Huang and Yuan
    7. 4.7 Comments on triple-trace contributions to $\mathcal{H}^{(3)}$
  5. 5 Extracting the two-loop anomalous dimension
    1. 5.1 Comments on analyticity in spin of $\gamma^{(3)}$
    2. 5.2 $\gamma^{(3)}$ and the reciprocity principle
  6. 6 An exploration of wider ansätze
    1. 6.1 Including weight 4 functions with letter $x-\bar{x}$
    2. 6.2 Generalising the one-loop term
  7. 7 Outlook and open questions
  8. A Details on the basis of transcendental functions
  9. B The $\log^{2}(u)$ prediction at twist 4

##  1 Introduction

The AdS/CFT correspondence relates bulk AdS scattering amplitudes to
correlation functions in a boundary conformal field theory. The nature of bulk
dynamics can therefore be effectively explored by making use of tools from
conformal field theory. The archetypal example of the correspondence is that
of $\mathcal{N}=4$ super Yang-Mills theory as the boundary CFT, describing
type IIB superstrings in the AdS${}_{5}\times$S5 bulk. Many recent works have
explored this theory with much focus on tree-level four-point amplitudes in
the bulk and their one-loop corrections [1, 2, 3, 4]. Such bulk amplitudes
correspond to a large $N$ expansion of the correlation function of four half-
BPS operators in the conformal field theory.

Here we would like to explore the possibility of extending the analysis to
higher loop order, focussing on two loops as a first example. This may seem a
formidable problem, particularly from the bulk perspective, where already
tree-level and one-loop amplitudes pose significant computational problems [5,
6, 7, 8, 9, 10, 11, 12, 13, 14]. However, the end results for tree-level and
one-loop amplitudes, obtained mostly from boundary CFT considerations,
actually display surprising simplicity. In part, this is due to a hidden ten-
dimensional conformal symmetry of the tree-level amplitudes which in turn
leads to a significant simplification of the leading logarithmic discontinuity
at all loop orders.

We will focus on the four-point scattering of graviton supermultiplets, or
equivalently the four-point correlation function of four stress-energy
multiplets in $\mathcal{N}=4$ super Yang-Mills theory. In [15] it was observed
that the one-loop result, first computed in [4], can actually be simplified
significantly with the help of an eighth-order conformal Casimir operator.
This operator was known to simplify the form of the leading logarithmic
discontinuity to all loop orders [16]. It is related to the surprisingly
simple form of the tree-level contribution to the anomalous dimensions of the
double-trace operators exchanged in the OPE with two external half-BPS
operators [17, 16]. The simple form of the anomalous spectrum is in turn
related to the ten-dimensional conformal symmetry of the tree-level amplitudes
[18]. With the use of the eight-order Casimir the one-loop amplitude can be
replaced by a much simpler preamplitude, whose analytic structure is so
tightly constrained that in fact one does not even require the form of the
leading discontinuity to essentially fix it uniquely [15].

Here, in common with the recent work of Huang and Yuan [19], we explore the
extension of the use of this operator to higher loops. In particular we write
the amplitude in terms of a preamplitude from which we obtain the full
amplitude by two applications of the eighth-order Casimir. We observe that the
square of the Casimir actually exhibits a simple transformation under crossing
symmetry which allows us to restrict the preamplitude to be fully crossing
symmetric. We describe an anzatz for the preamplitude in terms of
polylogarithmic functions of two variables, including singularities which
appear in the one-loop string corrections to the supergravity amplitude [20].
Then we impose various OPE predictions arising from known tree-level and one-
loop data. Under the further assumption that the leading discontinuity is
manifested by the preamplitude, and imposing that the amplitude correctly
reproduces the two-loop flat-space supergravity amplitude upon taking the
bulk-point limit, we are led to confirm the result of [19]. In particular, we
fix some ambiguities left undetermined in their result and, as a consequence
of the interplay of crossing symmetry with the canonical form of the leading
discontinuity (as well as the conjectured absence of weight 4 functions
containing the letter $x-\bar{x}$), we provide a justification for the
vanishing of their last free parameter. We also explore certain wider ansätze
and analyse the corresponding freedoms they introduce into the final form of
the two-loop amplitude.

The result we obtain is very suggestive that in fact all ambiguities should
also be included in the form of the preamplitude ansatz, that is to say that
any finite contributions from tree-level and one-loop counterterms should also
respect the form of the ansatz given in terms of the preamplitude, as also
happens at one loop. This possibility restricts the final amplitude to just a
handful of ambiguities, which are exactly the expected tree-level ambiguities
consistent with the presence of genus-two contributions from the
$\partial^{4}\mathcal{R}^{4}$, $\partial^{6}\mathcal{R}^{4}$,
$\partial^{8}\mathcal{R}^{4}$ and $\partial^{10}\mathcal{R}^{4}$ corrections.
Notably, the $\mathcal{R}^{4}$ term does not appear as an independent
ambiguity, in analogy with its vanishing contribution at genus two.

Furthermore, from our result we extract the two-loop correction to the
anomalous dimensions of the twist-four double-trace operators
$\mathcal{O}_{2,\ell}\sim\mathcal{O}_{2}\partial^{\ell}\mathcal{O}_{2}$ for
generic spin $\ell$. In the supergravity limit (and neglecting $1/\lambda$
corrections), their dimensions admit a large $N$ expansion of the form

$\displaystyle\Delta_{2,\ell}=4+\ell+2\big{(}a\gamma^{(1)}+a^{2}\gamma^{(2)}+a^{3}\gamma^{(3)}+O(a^{4})\big{)},$
(1.1)

where we introduced $a\equiv 1/(N^{2}-1)$. To order $a^{3}$, the anomalous
dimensions read

$\displaystyle\begin{split}\gamma^{(1)}&=-\frac{48}{(\ell+1)(\ell+6)}\,,\\\\[3.0pt]
\gamma^{(2)}&=\frac{1344(\ell-7)(\ell+14)}{(\ell-1)(\ell+1)^{2}(\ell+6)^{2}(\ell+8)}-\frac{2304(2\ell+7)}{(\ell+1)^{3}(\ell+6)^{3}}-\frac{1080}{7}\,\delta_{\ell,0}\,,\\\\[3.0pt]
\gamma^{(3)}&=c_{3}\,\big{(}S_{-3}-S_{3}-2S_{1,-2}+3\zeta_{3}\big{)}+c_{2}\,S_{-2}+c_{1}\,S_{1}+c_{0}+c_{0}^{(a)}+\alpha\,\tilde{\gamma}^{(2,3)}_{2,\ell},\end{split}$
(1.2)

where the tree-level and one-loop contributions $\gamma^{(1)}$ and
$\gamma^{(2)}$ have been derived in [21] and [4], respectively, while the two-
loop contribution $\gamma^{(3)}$ constitutes a new result.111 Note that the
formula for $\gamma^{(3)}$ quoted above is strictly speaking only valid for
spins $\ell\geq 6$. This is because we are able to fix the two-loop amplitude
only up to certain tree-level ambiguities which contribute up to spin $\ell=4$
in the conformal block expansion.  The coefficients $c_{i}$ are rational
functions of spin and $S_{\vec{a}}\equiv S_{\vec{a}}(\ell+3)$ are nested
harmonic sums, whose precise definitions as well as an extended discussion of
the properties of $\gamma^{(3)}$ are given in Section 5.

The rest of the paper is organised as follows: the remainder of this section
is devoted to fixing our notation for the
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$
correlator and introducing its OPE decomposition.

In Section 2, we focus on the leading logarithmic divergence of supergravity
correlators to any loop-order. We notice that its transcendental structure is
described by a certain family of four-dimensional loop-integrals, the so-
called zigzag integrals, which are a particular class of the generalised
ladders discussed in [22]. This observation is powerful as it determines a
part of the supergravity correlator at any loop-order, greatly reducing the
number of transcendental functions with undetermined coefficients.

Next, in Section 3 we revisit the bulk-point limit of AdS amplitudes and we
show how known tree-level and one-loop AdS${}_{5}\times$S5 amplitudes match
their flat-space counterparts. Moreover, the extra $x-\bar{x}$ singularity of
the one-loop string corrections turns out to be a crucial ingredient which
reproduces the logarithmic threshold terms present in the perturbative
expansion of the type IIB string amplitude in flat-space. Extended to two-loop
order, the knowledge of the 10-dimensional two-loop supergravity amplitude
gives a non-trivial constraint for the corresponding AdS correlator.

The main result of our work is described in Section 4, where we spell out the
details on the minimal ansatz used as a starting point of our bootstrap
approach to the two-loop supergravity contribution. In particular, we discuss
in great detail the basis of transcendental functions and compare our result
to the one recently obtained in [19].

In Section 5, we make use of the obtained two-loop correlator to extract from
it new unprotected CFT data, recall also the previous comments around equation
(1.2). The result for the two-loop anomalous dimension extends the previously
known terms up to order $a^{3}$.

Lastly, in Section 6 we consider possible natural extensions of the minimal
ansatz and analyse the corresponding additional free parameters in the final
result. We then conclude with a number of open questions.

###  1.1 Setup: the
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$
correlator

We consider the four-point correlation function of the stress-tensor
superprimary $\mathcal{O}_{2}$, which is a half-BPS single-trace operator of
conformal dimension $\Delta=2$. It is given in terms of the fundamental scalar
fields $\Phi_{i}$ ($i=1,\ldots,6$) of the $\mathcal{N}=4$ supermultiplet by

$\displaystyle\mathcal{O}_{2}(x,y)=y^{i}y^{j}\leavevmode\nobreak\
\text{Tr}\big{(}\Phi_{i}(x)\Phi_{j}(x)\big{)},$ (1.3)

with $y^{i}$ being an auxiliary $so(6)$ null vector obeying $y\cdot y=0$, such
that $\mathcal{O}_{2}$ transforms in the traceless symmetric representation
$[0,2,0]$ of the R-symmetry group $su(4)$. In the context of the AdS/CFT
correspondence, this operator is dual to the scalar in the graviton multiplet
of type IIB supergravity on an AdS${}_{5}\times$S5 background, whereas its
higher charge cousins $\mathcal{O}_{p}$ with $p\geq 3$ are dual to Kaluza-
Klein modes arising from the compact S5 factor.222 To be more precise, the
exact form of ‘single-particle operators’ $\mathcal{O}_{p}$ which are dual to
single-particle states in AdS supergravity in general contains admixtures of
multi-trace operators, whose coefficients are $1/N$ suppressed with respect to
the usual single-trace term [16]. For an extensive study of the properties of
these single-particle operators and their free-theory correlators, see [23].

We are interested in the four-point correlator

$\displaystyle\langle\mathcal{O}_{2}(x_{1},y_{1})\mathcal{O}_{2}(x_{2},y_{2})\mathcal{O}_{2}(x_{3},y_{3})\mathcal{O}_{2}(x_{4},y_{4})\rangle=g_{12}^{2}g_{34}^{2}\,\mathcal{G}(u,v;\sigma,\tau),$
(1.4)

where $g_{ij}=y_{ij}^{2}/x_{ij}^{2}$ (with $y_{ij}^{2}=y_{i}\cdot y_{i}$) are
propagator factors which account for the conformal weight and the scaling
weights $y_{i}$ of the correlator, such that $\mathcal{G}$ depends only on the
conformal and $su(4)$ R-symmetry cross-ratios. These are defined by

$\displaystyle u=x\bar{x}$
$\displaystyle=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}},\qquad$
$\displaystyle\leavevmode\nobreak\
v=(1-x)(1-\bar{x})=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}},$ (1.5)
$\displaystyle\frac{1}{\sigma}=y\bar{y}$
$\displaystyle=\frac{y_{12}^{2}y_{34}^{2}}{y_{13}^{2}y_{24}^{2}},\qquad$
$\displaystyle\frac{\tau}{\sigma}=(1-y)(1-\bar{y})=\frac{y_{14}^{2}y_{23}^{2}}{y_{13}^{2}y_{24}^{2}}.$

As a consequence of superconformal symmetry the function $\mathcal{G}$ is
furthermore constrained to take the form [24, 25]

$\displaystyle\mathcal{G}(u,v;\sigma,\tau)=\mathcal{G}_{\text{free}}(u,v;\sigma,\tau)+\mathcal{I}\leavevmode\nobreak\
\mathcal{H}(u,v).$ (1.6)

For convenience, we normalise the correlator by a factor of $(N^{2}-1)^{2}$
such that the free theory contribution is given by

$\displaystyle\mathcal{G}_{\text{free}}(u,v;\sigma,\tau)=4\Big{(}1+u^{2}\sigma^{2}+\frac{u^{2}\tau^{2}}{v^{2}}\Big{)}+16a\Big{(}u\sigma+\frac{u\tau}{v}+\frac{u^{2}\sigma\tau}{v}\Big{)}.$
(1.7)

We call the second contribution to (1.6) the interacting or dynamical part, as
it is the only piece of the correlator which depends on the gauge coupling
$g_{\text{YM}}$ and hence contains all of the non-trivial dynamics of the
theory. It is of the factorised form shown above with the factor $\mathcal{I}$
being fixed by the superconformal Ward identities,

$\displaystyle\mathcal{I}=\frac{(x-y)(x-\bar{y})(\bar{x}-y)(\bar{x}-\bar{y})}{(y\bar{y})^{2}}\,.$
(1.8)

As a feature of the
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$
correlator, and more generally for next-to-next-to-extremal correlators, the
function $\mathcal{H}$ is then independent of the $su(4)$ variables.
Furthermore, it obeys the crossing symmetries

$\displaystyle\mathcal{H}(u,v)=\frac{1}{v^{2}}\mathcal{H}(u/v,1/v)=\frac{u^{2}}{v^{2}}\mathcal{H}(v,u),$
(1.9)

placing strong constraints on its functional form which we will exploit in our
bootstrap approach.

We will study the function $\mathcal{H}$ in the supergravity regime (reached
by first taking $N$ large with the ’t Hooft coupling
$\lambda=g_{\text{YM}}^{2}N$ being held fixed and then taking $\lambda$ to
infinity), where it describes the scattering of four supergravitons in type
IIB superstring theory on AdS${}_{5}\times$S5. Ignoring the
$\lambda$-dependence for a moment and focussing only on the supergravity
contributions, $\mathcal{H}$ admits the large $N$ expansion

$\displaystyle\mathcal{H}(u,v)=a\,\mathcal{H}^{(1)}(u,v)+a^{2}\,\mathcal{H}^{(2)}(u,v)+a^{3}\,\mathcal{H}^{(3)}(u,v)+O(a^{4})\,,$
(1.10)

where for convenience we choose to expand in small $a$, recall the definition
$a=1/(N^{2}-1)$.333 In terms of the central charge $c$, one has
$a=\frac{1}{4c}$. The first term in the expansion is given by the well known
tree-level supergravity correlator $\mathcal{H}^{(1)}$ [8, 26], while the
higher-order terms correspond to loop amplitudes in the bulk: the order
$a^{2}$ term $\mathcal{H}^{(2)}$ has been computed in [4]444See also
references [3] and [27]. and is dual to the one-loop AdS amplitude. Next comes
the two-loop correction $\mathcal{H}^{(3)}$ which is the main object of
interest here, and has recently also been considered in [19].

Let us now comment on the structure of $\lambda$-dependent terms, which we
have omitted in (1.10). These terms arise due to higher-derivative corrections
of the string theory effective action on AdS${}_{5}\times$S5, and in the flat-
space limit they are related to analogous terms in the low-energy expansion of
the ten-dimensional four-point scattering amplitude of massless string states
in type IIB string theory. To this end, we will denote a contribution at order
$a^{n}\lambda^{-\frac{m}{2}}$ by $\mathcal{H}^{(n,m)}$. To be consistent with
the expansion (1.10), we have $\mathcal{H}^{(n)}\equiv\mathcal{H}^{(n,0)}$.

At tree-level, such terms are given by contact Witten diagrams whose
interaction vertices are higher-derivative corrections to supergravity at
genus zero. These vertices are of the form $\partial^{2k}\mathcal{R}^{4}$ and
give rise to the tree-level terms $\mathcal{H}^{(1,k+3)}$.555 These string
corrections have been addressed in [28, 29, 30, 31, 32]. Note that in general
these terms receive higher-genus corrections which contribute to higher orders
in $a$, while still being ‘tree-level’ in the bulk.666 The existence of such
contributions ultimately follows from S-duality of type IIB string theory, see
references [33, 34, 35] for a study of modular invariance in this context and
also the recent work [36] for a new approach to $SL(2,\mathbb{Z})$ invariance
in the context of $\mathcal{N}=4$ SYM.

For instance, the $\mathcal{R}^{4}$ vertex receives a genus-one correction
$\mathcal{R}^{4}|_{\text{genus-1}}$ at order $a^{2}\lambda^{\frac{1}{2}}$.
This gives rise to a term $\mathcal{H}^{(2,-1)}$, which happens to be super-
leading with respect to the one-loop supergravity correction
$\mathcal{H}^{(2)}$. As argued for in [37], this super-leading term can be
thought of as a finite counter-term from string theory which regularises the
one-loop supergravity divergence in AdS${}_{5}\times$S5. The subsequent terms
are then suppressed by powers of $1/\lambda$, corresponding to either genus-
one corrections to the higher-derivative tree-level terms
$\mathcal{H}^{(1,k+3)}$ or genuine one-loop contributions induced by their
very presence at order $a$, which have been addressed in [38, 20, 39].

Similarly, at order $a^{3}$, there exist genus-two contributions from certain
tree-level contact terms which precede the two-loop supergravity correlator
$\mathcal{H}^{(3)}$. These are the genus-two corrections to the
$\partial^{4}\mathcal{R}^{4}$ and $\partial^{6}\mathcal{R}^{4}$ vertices,
corresponding to the terms $\mathcal{H}^{(3,-3)}$ and $\mathcal{H}^{(3,-2)}$
in our notation. Next, at order $a^{3}\lambda^{\frac{1}{2}}$, a new effect
takes place: there are two distinct contributions to $\mathcal{H}^{(3,-1)}$,
namely the genus-two correction to the tree-level
$\partial^{8}\mathcal{R}^{4}$ term and a one-loop contribution involving a
supergravity and a $\mathcal{R}^{4}|_{\text{genus-1}}$ vertex. Finally, the
genus-two correction to the $\partial^{10}\mathcal{R}^{4}$ term is at the same
order as two-loop supergravity, and thus will also contribute to
$\mathcal{H}^{(3)}$.

Lastly, let us remark that the coefficients of some of these super-leading
terms have been computed up to genus three using supersymmetric localisation
techniques, see [40, 41] for more details.

###  1.2 OPE decomposition and double-trace spectrum

Next, we introduce the OPE decomposition of the
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$
correlator. In particular, we will be interested into its unprotected (or
long) part, which receives contributions from both the free theory and the
interacting part and admits a decomposition into superconformal blocks

$\displaystyle\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle_{\text{long}}=g_{12}^{2}g_{34}^{2}\leavevmode\nobreak\
\mathcal{I}\leavevmode\nobreak\
\sum_{t,\ell}A_{t,\ell}\,G_{t,\ell}(x,\bar{x}).$ (1.11)

The sum above runs over all long superconformal primaries with half the twist
$t\equiv\frac{\Delta-\ell}{2}$ and even spin $\ell$ which are present in the
$su(4)$ singlet representation in the OPE of
$\mathcal{O}_{2}\times\mathcal{O}_{2}$. The $A_{t,\ell}$ are the squared OPE
coefficients and the functions $G_{t,\ell}(x,\bar{x})$ are related to the
usual four-dimensional conformal blocks by a shift in their dimensions, giving
[42, 43]

$\displaystyle
G_{t,\ell}(x,\bar{x})=(-1)^{\ell}(x\bar{x})^{t}\leavevmode\nobreak\
\frac{x^{\ell+1}F_{t+\ell+2}(x)F_{t+1}(\bar{x})-\bar{x}^{\ell+1}F_{t+\ell+2}(\bar{x})F_{t+1}(x)}{x-\bar{x}},$
(1.12)

with $F_{\rho}(x)={}_{2}F_{1}\left(\rho,\rho,2\rho;x\right)$ being the
standard hypergeometric function.

To leading order in the expansion around the supergravity limit introduced
above, the only unprotected operators exchanged in the OPE of
$\mathcal{O}_{2}\times\mathcal{O}_{2}$ are singlet channel double-trace
operators of even spin $\ell$. Such operators are constructed from products of
two half-BPS operators, and we denote their classical dimensions by
$\Delta^{(0)}=2t+\ell$. Note that generically there are many such operators
with the same quantum numbers, leading to operator mixing: for a given half-
twist $t$, there are $t-1$ such operators which we distinguish by their
degeneracy label $i=1,\ldots,t-1$. Schematically, the set of degenerate
double-trace operators is given by

$\displaystyle\mathcal{O}_{t,\ell,i}=\big{\\{}\mathcal{O}_{2}\square^{t-2}\partial^{\ell}\mathcal{O}_{2},\leavevmode\nobreak\
\mathcal{O}_{3}\square^{t-3}\partial^{\ell}\mathcal{O}_{3},\leavevmode\nobreak\
\ldots\leavevmode\nobreak\ ,\leavevmode\nobreak\
\mathcal{O}_{t}\square^{0}\partial^{\ell}\mathcal{O}_{t}\big{\\}}.$ (1.13)

In the supergravity limit, their dimensions $\Delta_{t,\ell,i}$ and OPE
coefficients
$A_{t,\ell,i}\equiv\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{t,\ell,i}\rangle^{2}$
admit the expansion

$\displaystyle\Delta_{t,\ell,i}$
$\displaystyle=\Delta^{(0)}+2\big{(}a\gamma^{(1)}_{i}+a^{2}\gamma^{(2)}_{i}+a^{3}\gamma^{(3)}_{i}+O(a^{4})\big{)}\,,$
(1.14) $\displaystyle A_{t,\ell,i}$
$\displaystyle=A^{(0)}_{t,\ell,i}+aA^{(1)}_{t,\ell,i}+a^{2}A^{(2)}_{t,\ell,i}+a^{3}A^{(3)}_{t,\ell,i}+O(a^{4})\,,$
(1.15)

where we have again suppressed the $\lambda$-dependence at it is understood
that the anomalous dimensions $\gamma^{(n)}_{i}$ themselves depend on $t$ and
spin $\ell$.

In the singlet channel and up to order $a$, the mixing of the operators
$\mathcal{O}_{t,\ell,i}$ has been fully resolved [4, 17], leading to a compact
expression for their tree-level anomalous dimensions777For a generalisation to
double-trace operators in any $su(4)$ channel, see [16].

$\displaystyle\gamma^{(1)}_{i}=-\frac{2(t-1)_{4}(t+\ell)_{4}}{(\ell+2i-1)_{6}}\,,$
(1.16)

and leading-order OPE coefficients

$\displaystyle
A^{(0)}_{t,\ell,i}=\frac{8(t+\ell+1)!^{2}t!^{2}(\ell+1)(2t+\ell+2)}{(2t)!(2t+2\ell+2)!}\leavevmode\nobreak\
R_{t,\ell,i}\,b_{t,i}\,,$ (1.17)

where

$\displaystyle\begin{split}R_{t,\ell,i}&=\frac{2^{1-t}(2\ell+3+4i)[(\ell+i+1)_{t-i-1}]^{{\rm
sign}(t-i-1)}(t+\ell+4)_{i-1}}{(\tfrac{5}{2}+\ell+i)_{t-1}}\,,\\\
b_{t,i}&=\frac{2^{(1-t)}(2+2i)!(t-2)!(2t-2i+2)!}{3(i-1)!(i+1)!(t+2)!(t-i-1)!(t-i+1)!}\,.\end{split}$
(1.18)

Note that the first factor in equation (1.17) can be thought of as an averaged
OPE coefficient $\langle A^{(0)}_{t,\ell}\rangle$ over the double-trace
degeneracies $i$, since $\sum_{i=1}^{t-1}R_{t,\ell,i}\,b_{t,i}=1$.

To conclude this section, let us indicate how the double-trace OPE data
introduced above is encoded within the supergravity correlators across
different orders in $a$. By plugging in the expansions (1.14) and (1.15) into
the superconformal block decomposition (1.11), one finds

$\displaystyle\begin{split}\mathcal{H}^{(1)}&=\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\log^{1}(u)\,\big{[}A^{(0)}\gamma^{(1)}\big{]}\,G_{t,\ell}(x,\bar{x})\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\log^{0}(u)\,\big{[}A^{(1)}+2A^{(0)}\gamma^{(1)}\partial_{\Delta}\big{]}\,G_{t,\ell}(x,\bar{x})\,,\\\
\mathcal{H}^{(2)}&=\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\log^{2}(u)\,\big{[}\tfrac{1}{2}A^{(0)}(\gamma^{(1)})^{2}\big{]}\,G_{t,\ell}(x,\bar{x})\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\log^{1}(u)\,\big{[}(A^{(1)}\gamma^{(1)}+A^{(0)}\gamma^{(2)})+2A^{(0)}(\gamma^{(1)})^{2}\partial_{\Delta}\big{]}\,G_{t,\ell}(x,\bar{x})\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\log^{0}(u)\,\big{[}A^{(2)}+2(A^{(1)}\gamma^{(1)}+A^{(0)}\gamma^{(2)})\,\partial_{\Delta}+2A^{(0)}(\gamma^{(1)})^{2}\partial_{\Delta}^{2}\big{]}\,G_{t,\ell}(x,\bar{x})\,,\\\
\mathcal{H}^{(3)}&=\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\log^{3}(u)\,\big{[}\tfrac{1}{6}A^{(0)}(\gamma^{(1)})^{3}\big{]}\,G_{t,\ell}(x,\bar{x})\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\log^{2}(u)\,\big{[}(\tfrac{1}{2}A^{(1)}(\gamma^{(1)})^{2}+A^{(0)}\gamma^{(1)}\gamma^{(2)})+A^{(0)}(\gamma^{(1)})^{3}\partial_{\Delta}\big{]}\,G_{t,\ell}(x,\bar{x})\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\log^{1}(u)\,\big{[}(A^{(2)}\gamma^{(1)}+A^{(1)}\gamma^{(2)}+A^{(0)}\gamma^{(3)})\\\
&\qquad\qquad\quad\leavevmode\nobreak\ \leavevmode\nobreak\
+2(A^{(1)}(\gamma^{(1)})^{2}+2A^{(0)}\gamma^{(1)}\gamma^{(2)})\partial_{\Delta}+2A^{(0)}(\gamma^{(1)})^{3}\partial_{\Delta}^{2}\big{]}\,G_{t,\ell}(x,\bar{x})\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\log^{0}(u)\,\big{[}A^{(3)}+2(A^{(2)}\gamma^{(1)}+A^{(1)}\gamma^{(2)}+A^{(0)}\gamma^{(3)})\partial_{\Delta}\\\
&\qquad\qquad\quad\leavevmode\nobreak\ \leavevmode\nobreak\
+2(A^{(1)}(\gamma^{(1)})^{2}+2A^{(0)}\gamma^{(1)}\gamma^{(2)})\partial_{\Delta}^{2}+\tfrac{4}{3}A^{(0)}(\gamma^{(1)})^{3}\partial_{\Delta}^{3}\big{]}\,G_{t,\ell}(x,\bar{x})\,.\\\
\end{split}$ (1.19)

To facilitate readability, all indices and three kinds of summation have been
suppressed: within each $\log(u)$-stratum, a finite sum over degeneracy labels
$i=1,\ldots,t-1$, as well as two infinite sums over half-twists $t$ and even
spins $\ell$.

Importantly, note that in the above we have only recorded the contributions
from double-trace operators $\mathcal{O}_{t,\ell,i}$. At twist 6 and above,
triple-trace (and more generally, higher-trace) operators are expected to
contribute and to mix with the double-trace spectrum. As we will show in
Section 4.7, consistency of the OPE indeed predicts the presence of operators
beyond just the double-trace spectrum in the $\log^{2}(u)$ part of
$\mathcal{H}^{(3)}$ starting from twist 6.

##  2 Zigzag integrals: a basis for the leading log

When combining the perturbative expansion in $a$ with the superconformal block
decomposition as shown for the first few orders in equation (1.19), one
notices that the leading logarithmic divergence (or leading log for short) is
entirely determined by the tree-level OPE data of long double-trace operators
$\mathcal{O}_{t,\ell,i}$. To be precise, at any order $a^{n}$, one has

$\displaystyle\mathcal{H}^{(n)}(u,v)|_{\log^{n}(u)}=\frac{1}{n!}\sum_{t,\ell}\sum_{i=1}^{t-1}A^{(0)}_{t,\ell,i}\big{(}\gamma^{(1)}_{i}\big{)}^{n}G_{t,\ell}(x,\bar{x}),$
(2.1)

which depends only on the leading-order OPE coefficients $A^{(0)}_{t,\ell,i}$
and tree-level anomalous dimensions $\gamma^{(1)}$ given in (1.17) and (1.16),
respectively. The simplicity of these suggest that a pattern will emerge in
the resummed form of the leading log. As we will discuss in the following,
this is indeed the case and moreover we can identify a family of four-
dimensional loop integrals, the so-called ‘zigzag’ integrals, which provide
the correct functional basis for the leading log. This observation will later
feed into our ansatz for the two-loop correlator $\mathcal{H}^{(3)}$, but here
we make the more general observation that this structure is present at any
loop-order.

###  2.1 The operator $\Delta^{(8)}$ and the leading log

To begin with, it is possible to directly perform the defining sums (2.1) case
by case and thus obtain explicit results for the leading log. These
expressions are of the general form

$\displaystyle\mathcal{H}^{(n)}(u,v)|_{\log^{n}(u)}=\frac{u^{2}\leavevmode\nobreak\
f_{\log}^{(n)}(x,\bar{x})}{(x-\bar{x})^{8n-1}},$ (2.2)

where $f_{\log}^{(n)}(x,\bar{x})$ is given by a linear combination of HPL’s of
transcendental weights up to $n$ with polynomial coefficients of (combined)
degree $12n-6$. However, as realised in [44, 16] and later explained in terms
of a hidden ten-dimensional conformal symmetry [18], one can greatly simplify
these expressions by making use of a certain eighth-order differential
operator denoted by $\Delta^{(8)}$. For the relevant case at hand, i.e. the
$su(4)$ singlet representation, this operator can be written in the fully
factorised form888 While here we focus on the
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{2}\rangle$
correlator which contributes only to the singlet-channel, a generalisation of
$\Delta^{(8)}$ to all $su(4)$ channels $[a,b,a]$ (relevant for four-point
correlators of arbitrary external charges) has been given in [16]. For a
beautiful reformulation into one compact formula see [18]. 999 The properties
of $\Delta^{(8)}$ under crossing transformations are discussed in Section 4.2.

$\displaystyle\Delta^{(8)}=\frac{u^{4}}{(x-\bar{x})}\partial_{\bar{x}}^{2}(1-\bar{x})^{2}\partial_{\bar{x}}^{2}\partial_{x}^{2}(1-x)^{2}\partial_{x}^{2}(x-\bar{x}).$
(2.3)

The key observation made in [44, 16] is that $\Delta^{(8)}$ satisfies an
eigenvalue equation when acting on long superconformal blocks
$u^{2}G_{t,\ell}$, with its eigenvalue being given (up to a factor of $-2$) by
the numerator of the tree-level anomalous dimensions $\gamma^{(1)}$ from
equation (1.16):

$\displaystyle\Delta^{(8)}\,u^{2}G_{t,\ell}(x,\bar{x})$
$\displaystyle=(t-1)_{4}(t+\ell)_{4}\,u^{2}G_{t,\ell}(x,\bar{x}).$ (2.4)

Since this numerator is independent of the degeneracy label $i$, one may pull
out $n-1$ factors of $\Delta^{(8)}$ from the sum (2.1) and thus remove $n-1$
powers of the numerator from $(\gamma^{(1)})^{n}$, leading to a considerably
simpler sum. Explicitly, this leads to

$\displaystyle\mathcal{H}^{(n)}(u,v)|_{\log^{n}(u)}=\frac{1}{u^{2}}\big{(}\Delta^{(8)}\big{)}^{n-1}g^{(n)}(x,\bar{x})\,,\qquad
g^{(n)}(x,\bar{x})=\frac{{u^{2}}\tilde{f}_{\log}^{(n)}(x,\bar{x})}{(x-\bar{x})^{7}},$
(2.5)

where the functions $\tilde{f}_{\log}^{(n)}(x,\bar{x})$ contain the same HPL’s
as $f_{\log}^{(n)}(x,\bar{x})$ but their coefficient polynomials have greatly
simplified now, with their degrees being reduced from $12n-6$ to only $10$.

A further simplification of the leading log can be achieved by making use of
the observation that the ten-dimensional conformal blocks diagonalise the
mixing of double-trace operators at order $a$ [18]. This is a consequence of
the hidden conformal symmetry of tree-level supergravity correlators,
effectively reducing the computation of the leading log to a single infinite
sum over spins. We refer to Section 5.5 of [18] for more details. Here we just
quote the form of the result, which reduces the information required to a
single-variable function $h^{(n)}(x)$,

$h^{(n)}(x)=\frac{(-2)^{n+2}}{n!}\sum_{l=0,2,4,\ldots}\frac{960\Gamma(l+1)\Gamma(l+4)x^{l+1}{}_{2}F_{1}(l+1,l+4;2l+8;x)}{\Gamma(2l+7)[(l+1)_{6}]^{n-1}}\,.$
(2.6)

The function $h^{(n)}(x)$ is related to the function $g^{(n)}(x,\bar{x})$
appearing in (2.5) via a third-order differential operator,

$g^{(n)}(x,\bar{x})=\mathcal{D}_{x}^{(3)}h^{(n)}(x)+(x\leftrightarrow\bar{x})\,,$
(2.7)

where $\mathcal{D}_{x}^{(3)}$ is given by

$\mathcal{D}_{x}^{(3)}=-\biggl{(}\frac{x\bar{x}}{x-\bar{x}}\biggr{)}^{7}+\frac{1}{2}\biggl{(}\frac{x\bar{x}}{x-\bar{x}}\biggr{)}^{6}x^{2}\partial_{x}-\frac{1}{10}\biggl{(}\frac{x\bar{x}}{x-\bar{x}}\biggr{)}^{5}x^{3}\partial_{x}^{2}x+\frac{1}{120}\biggl{(}\frac{x\bar{x}}{x-\bar{x}}\biggr{)}^{4}x^{4}\partial_{x}^{3}x^{2}\,.$
(2.8)

Note that under the crossing transformation,

$x\mapsto
x^{\prime}=\frac{x}{x-1}\,,\qquad\bar{x}\mapsto\bar{x}^{\prime}=\frac{\bar{x}}{\bar{x}-1}\,,$
(2.9)

the function $h^{(n)}$ and the operator $\mathcal{D}^{(3)}_{x}$ are both
antisymmetric,

$h^{(n)}(x^{\prime})=-h^{(n)}(x)\,\qquad\mathcal{D}^{(3)}_{x}\mapsto\mathcal{D}^{(3)}_{x^{\prime}}=-\mathcal{D}^{(3)}_{x}\,.$
(2.10)

Some contributions to $h^{(n)}(x)$ (or $g^{(n)}(x,\bar{x})$) turn out to be so
simple that closed form expressions have been found for them for any $n$. We
define the function $J_{k}(x)$ as a linear combination of weight $k$ harmonic
polylogarithms via

$J_{k}(x)=\sum_{a_{i}=0,1}H_{0a_{1}0a_{2}\ldots 1}(x)\,,\qquad
J_{k}(x^{\prime})=-\sum_{a_{i}=0,1}H_{a_{1}0a_{2}0\ldots 1}(x)\,.$ (2.11)

Note that $J_{1}(x)=-J_{1}(x^{\prime})=H_{1}(x)$. The function $h^{(n)}$ has
the form

$h^{(n)}(x)=a^{(n)}_{0}(x)+a^{(n)}_{1}(x)H_{1}(x)+\sum_{k=2}^{n}[a^{(n)}_{k}(x)J_{k}(x)-(x\leftrightarrow
x^{\prime})]\,,$ (2.12)

with $a^{(n)}_{k}(x)$ being polynomials in $1/x$ of degree at most six. The
leading terms obey $a^{(n)}_{0}(x^{\prime})=-a^{(n)}_{0}(x)$ and
$a^{(n)}_{1}(x^{\prime})=a^{(n)}_{1}(x)$ so that $h(x^{\prime})=-h(x)$. We see
that $h^{(n)}(x)$ is organised by terms of transcendental weight
$k=0,1,\ldots,n$. Since the functions $J_{n}$ for $n\geq 2$ obey

$x\partial_{x}J_{n}(x)=-J_{n-1}(x^{\prime})\,\qquad
x^{\prime}\partial_{x^{\prime}}J_{n}(x^{\prime})=x(1-x)\partial_{x}J_{n}(x^{\prime})=-J_{n-1}(x)\,,$
(2.13)

it follows that $g^{(n)}(x,\bar{x})$ is organised similarly,

$\displaystyle\begin{split}g^{(n)}(x,\bar{x})&=g^{(n)}_{0}(x,\bar{x})+\bigl{[}g^{(n)}_{1}(x,\bar{x})H_{1}(x)+(x\leftrightarrow\bar{x})\bigr{]}\\\
&+\sum_{k=2}^{n}\bigl{[}[g^{(n)}_{k}(x,\bar{x})J_{k}(x)+g^{(n)}_{k}(x^{\prime},\bar{x}^{\prime})J_{k}(x^{\prime})]+(x\leftrightarrow\bar{x})\bigr{]}\,.\end{split}$
(2.14)

Here $g^{(n)}_{0}$ obeys
$g^{(n)}_{0}(x^{\prime},\bar{x}^{\prime})=g^{(n)}_{0}(x,\bar{x})=g^{(n)}_{0}(\bar{x},x)$
and $g^{(n)}_{1}$ obeys
$g^{(n)}_{1}(x^{\prime},\bar{x}^{\prime})=-g^{(n)}_{1}(x,\bar{x})$.

The terms of leading transcendental weight can be written for all $n$ [45],

$a^{(n)}_{n}(x)=\frac{(-1)^{n}4^{4-n}15^{1-n}}{x^{5}n!}\bigl{(}(-6\cdot
5^{n}+10^{n}+50)x^{2}+3(5^{n}-25)x+30\bigr{)}\,.$ (2.15)

This expression reproduces101010up to an overall $n$-dependent prefactor due
to differences in conventions. the expression given in [16] for the highest
weight contribution (i.e. the $k=n$ term in (2.14)) to $g^{(n)}(x,\bar{x})$,

$g^{(n)}_{\rm
top}(x,\bar{x})=\bigl{[}J_{n}(x)\mathcal{D}_{x}^{(3)}a^{(n)}_{n}(x)-J_{n}(x^{\prime})\mathcal{D}_{x}^{(3)}a^{(n)}_{n}(x^{\prime})\bigr{]}+(x\leftrightarrow\bar{x})\,.$
(2.16)

Note that when the operator $\mathcal{D}^{(3)}_{x}$ acts on a polynomial in
$1/x$ of degree at most six, the result is always antisymmetric under
$(x\leftrightarrow\bar{x})$. For the leading weight term (2.16) it follows
that the explicit symmetrisation actually produces antisymmetric combinations
of harmonic polylogs of the form $J_{n}(x)-J_{n}(\bar{x})$ and
$J_{n}(x^{\prime})-J_{x}(\bar{x}^{\prime})$,

$g^{(n)}_{\rm
top}(x,\bar{x})=[J_{n}(x)-J_{n}(\bar{x})]\mathcal{D}_{x}^{(3)}a^{(n)}_{n}(x)-[J_{n}(x^{\prime})-J_{n}(\bar{x}^{\prime})]\mathcal{D}_{x}^{(3)}a^{(n)}_{n}(x^{\prime})\,.$
(2.17)

For the lower weight terms ($k<n$ in (2.14)) we generically have different
coefficients for $J_{k}(x)$, $J_{k}(\bar{x})$, $J_{k}(x^{\prime})$ and
$J_{k}(\bar{x}^{\prime})$. Note that a closed form has also been found for
$a^{(n)}_{n-1}$ in [45].

While these formulae are very explicit and a clear pattern is visible in the
HPL’s which do appear, the origins of these structures were left unexplained.
In the following, we describe a family of four-dimensional loop integrals
which give rise to these previously observed structures.

###  2.2 The Zigzag integrals

The relevant series are the zigzag series of integrals which obey

$x\bar{x}\partial_{x}\partial_{\bar{x}}Z^{(L)}(x,\bar{x})=Z^{(L-1)}(1-x,1-\bar{x})\,.$
(2.18)

where the series begins with the one-loop ladder function

$Z^{(1)}(x,\bar{x})=\phi^{(1)}(x,\bar{x})\,,$ (2.19)

and the solution for all higher $L$ is determined uniquely by imposing single-
valuedness. Because of the extra symmetry of $\phi^{(1)}$ we find
$Z^{(2)}=\phi^{(2)}$ but that thereafter the series is different from the
ladder series $\phi^{(L)}$ [46]. The functions $Z^{(n)}$ are antisymmetric
under $x\leftrightarrow\bar{x}$.

$x_{2}$$x_{0}$$x_{1}$ Figure 1: Example of a zigzag integral, with integration
vertices marked with dots.

Both the ladder series and the zigzag series arise from a more general class
of four-dimensional loop integrals [22]. Given a word $m=a_{1}...a_{L-1}$ with
letters $a_{i}\in\\{0,1\\}$, we define an $L$-loop, three-point integral as
follows,

$I_{m}(x_{0},x_{1},x_{2})=\frac{1}{\pi^{2L}}\int\frac{1}{x_{2b_{1}}^{2}}\prod_{i=1}^{L-1}\biggl{(}\frac{d^{4}x_{b_{i}}}{x_{b_{i}b_{i+1}}^{2}x_{b_{i}a_{i}}^{2}}\biggr{)}\frac{d^{4}x_{b_{L}}}{x_{b_{L}0}^{2}x_{b_{L}1}^{2}}\,.$
(2.20)

The integration vertices in the above definition are labelled by $x_{b_{i}}$
for $i=1,\ldots,L$ and we use the shorthand $x_{ij}^{2}=(x_{i}-x_{j})^{2}$. If
all the $a_{i}$ are zero then the above integral $I_{m}$ corresponds to the
three-point version of the dual of the ladder series.111111 I.e. the limit as
the fourth point is taken to infinity of the conformal four-point integral
dual to the $L$-loop ladder diagram. If we alternate the indices
$m=a_{1}a_{2}\ldots a_{L-1}=0101\ldots$ we get the zigzag series (with an
example illustrated in Fig. 1),

$I_{m}(x_{0},x_{1},x_{2})=\frac{1}{x_{01}^{2}}\frac{Z^{(L)}(z,\bar{z})}{z-\bar{z}}\,.$
(2.21)

Here we have used the variables,

$\frac{z\bar{z}}{(1-z)(1-\bar{z})}=\frac{x_{02}^{2}}{x_{12}^{2}}\,,\qquad\frac{1}{(1-z)(1-\bar{z})}=\frac{x_{01}^{2}}{x_{12}^{2}}\,.$
(2.22)

Note that the zigzag function is antisymmetric,
$Z^{(n)}(\bar{z},z)=-Z^{(n)}(z,\bar{z})$.

The differential equation (2.18) follows from considering the action of the
Laplace operator at the point $x_{2}$ on both sides of (2.21). Since the
Laplace operator acts on a single propagator in the integral, its action
produces a delta-function which localises one loop integral to produce an
$(L-1)$ loop integral, which is also a zigzag after swapping the points
$x_{0}$ and $x_{1}$. The differential equation together with the constraint
that the function should be single-valued (in Euclidean kinematics where
$\bar{z}$ is the complex conjugate of $z$) uniquely defines $Z^{(n)}$.

Here we give expressions for the first few zigzags $Z^{(n)}$ in terms of
single-valued polylogarithms [47],

$\displaystyle\begin{split}Z^{(1)}&=\mathcal{L}_{2}-\mathcal{L}_{10}\,,\\\
Z^{(2)}&=\mathcal{L}_{200}-\mathcal{L}_{30}\,,\\\
Z^{(3)}&=\mathcal{L}_{2210}-\mathcal{L}_{2120}-2\zeta_{3}(3\mathcal{L}_{20}+2\mathcal{L}_{21})\,,\\\
Z^{(4)}&=\mathcal{L}_{2230}-\mathcal{L}_{2320}-4\zeta_{3}(\mathcal{L}_{23}-\mathcal{L}_{220})-20\zeta_{5}\mathcal{L}_{20}\,,\\\
Z^{(5)}&=\mathcal{L}_{222120}-\mathcal{L}_{221220}+4\zeta_{3}(\mathcal{L}_{2221}-\mathcal{L}_{2212})+\zeta_{5}(4\mathcal{L}_{221}+15\mathcal{L}_{220})\\\
&\quad-12\zeta_{3}^{2}\mathcal{L}_{22}-\frac{441}{8}\zeta_{7}\mathcal{L}_{20}+18\zeta_{3}\zeta_{5}\mathcal{L}_{2}\,.\end{split}$
(2.23)

It is worth remarking that the detailed form of the zeta-values appearing
above is directly related to the values of certain vacuum integrals (also
called zigzags), whose name motivates our name for the $Z^{(n)}$, and whose
form was conjectured to all loops in [48] and proven in [49].121212 See also
the recent work [50].

Let us note that one may take many $\partial_{x}$ derivatives of the zigzag
functions and obtain other single-valued pure transcendental functions. In
fact one may take $m$ derivatives with $m\leq(n-1)$, alternating between
$(-x\partial_{x})$ and $(1-x)\partial_{x}$ and obtain another pure single-
valued function of weight $2n-m$,

$Z_{m}^{(n)}(x,\bar{x})=\underbrace{\ldots(1-x)\partial_{x}(-x\partial_{x})(1-x)\partial_{x}(-x\partial_{x})}_{m\text{
derivatives}}Z^{(n)}(x,\bar{x})\,.$ (2.24)

If we take a further derivative we obtain a sum of two pure functions of
weight $n$ with different rational prefactors. Obviously, the functions
$Z_{m}^{(n)}(x,\bar{x})$ do not exhibit any symmetry under
$x\leftrightarrow\bar{x}$ although they may be decomposed into symmetric and
antisymmetric parts.

None of the zeta value terms contribute to the leading discontinuity which is
therefore simple to calculate. Indeed the leading logarithmic singularities of
the zigzags obey

$\displaystyle\begin{split}Z^{(n)}\biggl{(}\frac{1}{x},\frac{1}{\bar{x}}\biggr{)}\bigg{|}_{\log^{n}(u)}&=\frac{(-1)^{n+1}}{n!}[J_{n}(x)-J_{n}(\bar{x})]\,,\\\
Z^{(n)}\biggl{(}\frac{1}{x^{\prime}},\frac{1}{\bar{x}^{\prime}}\biggr{)}\bigg{|}_{\log^{n}(u)}&=\frac{(-1)^{n+1}}{n!}[J_{n}(x^{\prime})-J_{n}(\bar{x}^{\prime})]\,.\end{split}$
(2.25)

These leading discontinuities are exactly what is required to match the
leading weight terms predicted by OPE, as given in (2.17). More precisely, to
match the top weight terms in the leading discontinuity we should include in
$\mathcal{H}^{(n)}(u,v)$ the contribution

$\frac{1}{u^{2}}\bigl{(}\Delta^{(8)}\bigr{)}^{n-1}\mathcal{G}^{(n)}_{\rm
top}(x,\bar{x})\,,$ (2.26)

where

$\mathcal{G}^{(n)}_{\rm
top}(x,\bar{x})=(-1)^{n+1}n!\bigl{[}\mathcal{D}_{x}^{(3)}a^{(n)}_{n}(x)\bigr{]}Z^{(n)}\biggl{(}\frac{1}{x},\frac{1}{\bar{x}}\biggr{)}+(x\leftrightarrow
x^{\prime})\,.$ (2.27)

Recall that $\mathcal{D}_{x}^{(3)}a^{(n)}_{n}(x)$ is actually an antisymmetric
function of $x$ and $\bar{x}$, as is $Z^{(n)}$, so that
$\mathcal{G}^{(n)}_{\rm top}$ is symmetric.

Moreover we can also see that taking $\partial_{x}$ derivatives of the
orientations of the zigzags given in (2.25) will give single-valued functions
whose leading discontinuities will give the contributions to
$g^{(n)}(x,\bar{x})$ of the form $J_{k}(x)$ and $J_{k}(x^{\prime})$ for $k<n$.
For $1\leq m\leq n-1$ we have

$\displaystyle\begin{split}Z_{m}^{(n)}\biggl{(}\frac{1}{x},\frac{1}{\bar{x}}\biggr{)}\bigg{|}_{\log^{n}(u)}&=\begin{cases}\frac{(-1)^{n-m+1}}{n!}J_{n-m}(x^{\prime})\,,\text{
$m$ odd}\\\ \frac{(-1)^{n-m+1}}{n!}J_{n-m}(x)\,,\text{ $m$ even}\end{cases}\\\
Z_{m}^{(n)}\biggl{(}\frac{1}{x^{\prime}},\frac{1}{\bar{x}^{\prime}}\biggr{)}\bigg{|}_{\log^{n}(u)}&=\begin{cases}\frac{(-1)^{n-m+1}}{n!}J_{n-m}(x)\,,\text{
$m$ odd}\\\ \frac{(-1)^{n-m+1}}{n!}J_{n-m}(x^{\prime})\,,\text{ $m$
even}\end{cases}\end{split}$ (2.28)

Since we have acted with $\partial_{x}$ derivatives the contributions of the
form $J_{n}(\bar{x})$ from (2.25) have disappeared. By replacing the
$J_{k}(x)$ and $J_{k}(x^{\prime})$ and conjugates in $g^{(n)}(x,\bar{x})$ by
the appropriate zigzag derivative, one may therefore construct a function
which matches the leading discontinuity to all weights from the zigzags
$Z^{(n)}$ and their derivatives. Let us consider the following combination,

$\displaystyle\mathcal{G}^{(n)}(x,\bar{x})=$
$\displaystyle\Biggl{[}\frac{n!}{4}\bigl{(}g^{(n)}_{0}(x,\bar{x})(1-x)\partial_{x}+2g^{(n)}_{1}(x,\bar{x})\bigr{)}\Bigl{(}Z^{(n)}_{n-1}\biggl{(}\frac{1}{x},\frac{1}{\bar{x}}\biggr{)}-(x\leftrightarrow
x^{\prime})\Bigr{)}+(x\leftrightarrow\bar{x})\Biggr{]}$
$\displaystyle-n!\sum_{k=2}^{n-1}\Bigl{[}\bigl{(}(-1)^{k}g^{(n)}_{k}(x,\bar{x})Z^{(n)}_{k-n}(z_{k})+(x\leftrightarrow
x^{\prime})\bigr{)}+(x\leftrightarrow\bar{x})\Bigr{]}$
$\displaystyle+\mathcal{G}^{(n)}_{\rm top}(x,\bar{x})\,.$ (2.29)

where $z_{k}=1/x$ for $(n+k)$ even and $z_{k}=1/x^{\prime}$ for $(n+k)$ odd.

If we take the leading discontinuity of this function we obtain
$g^{(n)}(x,\bar{x})$. We may then include a contribution to
$\mathcal{H}^{(n)}(u,v)$ as follows,

$\mathcal{H}^{(n)}(u,v)=\frac{1}{u^{2}}\bigl{(}\Delta^{(8)}\bigr{)}^{n-1}\mathcal{G}^{(n)}(x,\bar{x})+\text{
terms with no leading discontinuity}\,.$ (2.30)

For $n=1,2$ the zigzags are related to ladder integrals, as we have seen, and
hence for these values they have additional symmetries under crossing
transformations. For $n\geq 3$ the integrals do not exhibit any symmetries
under crossing transformations and the six possible crossing orientations
which we may list as,

$Z^{(n)}(x,\bar{x}),\,Z^{(n)}(1-x,1-\bar{x}),\,Z^{(n)}\biggl{(}\frac{1}{x^{\prime}},\frac{1}{\bar{x}^{\prime}}\biggr{)},\,Z^{(n)}(x^{\prime},\bar{x}^{\prime}),\,Z^{(n)}\biggl{(}\frac{1}{1-x},\frac{1}{1-\bar{x}}\biggr{)},\,Z^{(n)}\biggl{(}\frac{1}{x},\frac{1}{\bar{x}}\biggr{)},$

are linearly independent. Moreover, for $n\geq 3$ the only orientations which
contribute to the leading log term are those given in (2.25). This makes
matching the leading logarithmic discontinuities with a combination of all
orientations of zigzag integrals (which will be required for crossing
invariance of the final answer) straightforward at two loops and beyond.

In summary, we have seen how the leading discontinuity at any loop order
admits a compact expression with powers of the differential operator
$\Delta^{(8)}$ pulled out. Moreover, we have shown that the transcendental
structure of the leading log is described solely in terms of zigzag integrals
and derivatives thereof. As depicted schematically in equation (2.30), the
remaining difficulty in determining the full correlator $\mathcal{H}^{(n)}$
then lies in the construction of the remaining terms with no contribution to
the leading log, which in general will include transcendental functions beyond
the zigzag integrals discussed here. We will address this task for the two-
loop case in Section 4.

##  3 The bulk-point limit of AdS${}_{5}\times$S5 amplitudes

Let us next discuss the flat-space limit of supergraviton amplitudes on
AdS${}_{5}\times$S5 as described by the dual CFT correlator
$\mathcal{H}(x,\bar{x})$. In its position space representation considered
here131313A similar formulation of the flat-space limit for the Mellin space
representation of holographic correlators has been developed in [51, 52]. This
flat-space limit of Mellin amplitudes has proven to be a particularly useful
tool for constraining tree-level string corrections in AdS, see e.g.
references [53, 28, 29, 30, 31, 32]. A beautiful formalism adapted
specifically to the AdS${}_{5}\times$S5 case has been proposed in [54], and
for discussions of the flat-space limit for one-loop Mellin amplitudes see
references [38, 55, 56, 57]. this amounts to considering the so-called bulk-
point limit: By using sufficiently localised wave-packets in AdS, one can
focus onto a point in the bulk and thus effectively recover the corresponding
scattering process in flat-space. This manifests itself in a singularity of
the correlator $\mathcal{G}(x,\bar{x};y,\bar{y})$ which is of the form

$\displaystyle\mathcal{H}(x,\bar{x})\leavevmode\nobreak\
\xrightarrow[]{\leavevmode\nobreak\ \text{bpl}\leavevmode\nobreak\
}\leavevmode\nobreak\ \frac{\mathcal{F}(x)}{(x-\bar{x})^{p}}\,,$ (3.1)

for some positive integer power $p$ specified later. As explained in
references [58, 59, 60, 61], this is the expected behaviour of holographic
correlators with a local bulk dual. Note that in order to expose this bulk-
point singularity, one first needs to analytically continue the Euclidean
correlator to Lorentzian signature before taking the limit $\bar{x}\to x$. The
residue of the singularity is then directly related to the corresponding (in
this case ten-dimensional) flat-space scattering amplitude
$\mathcal{A}^{(10)}$ as a function of the scattering angle,

$\displaystyle\mathcal{A}^{(10)}\,\propto\,s^{k}\,\frac{\mathcal{F}(x)}{x^{l}(1-x)^{m}}\,,$
(3.2)

where $k,l,m$ are integers related to the dimension of the effective bulk
interaction vertex [59]. The dimensionless parameter $x$ is defined in terms
of the Mandelstam invariants $s$ and $t$, or equivalently in terms of the
scattering angle $\theta$, by

$\displaystyle x\equiv 1+\frac{t}{s}=\frac{1+\cos\theta}{2}.$ (3.3)

For our purposes, the relevant flat-space amplitude $\mathcal{A}^{(10)}$ is
given by the massless four-particle scattering amplitude of ten-dimensional
type IIB string theory, whose low-energy expansion we will review in the next
section.

The above relation (3.2) can be leveraged to put constraints on the AdS
correlator by using knowledge about the corresponding flat-space amplitude.
Let us first point out that in the context of the large $N$ expansion (1.10)
of the $\mathcal{N}=4$ SYM correlator, in [27] a simpler matching than given
by (3.2) has been proposed, namely a relation between the double-discontinuity
$\text{dDisc}\,\mathcal{H}(x,\bar{x})$ and the $t$-channel discontinuity
$\text{Disc}_{t}\,\mathcal{A}^{(10)}(x)$. Therein, the authors applied it to
the one-loop correlator $\mathcal{H}^{(2)}$, with later applications to one-
loop string corrections given in [28]. At two-loop order, this simpler
matching has been considered for the leading log of $\mathcal{H}^{(3)}$ in the
works [62, 45], and also the recently proposed two-loop bootstrap result of
[19] imposes this simpler matching.

In contrast, here we will describe how in the bulk-point limit the large $N$,
large $\lambda$ expansion of $\mathcal{H}(x,\bar{x})$ recovers the
corresponding low-energy expansion of the full type IIB string amplitude, not
just its discontinuity. In general, one would not expect this to lead to more
constraints for the AdS correlator than the simpler matching performed at the
level of the discontinuities, since the Lorentzian inversion formula of [63]
allows one to (in principle) reconstruct the full correlator from its double-
discontinuity.141414 In analogy with the reconstruction of flat-space
amplitudes from their discontinuity up to regular terms.  However, this
statement holds only modulo terms with vanishing double-discontinuity and
finite spin contributions in the conformal block decomposition. But this is
exactly the case for the tree-level string corrections $\mathcal{H}^{(1,m)}$
with $m>0$, which subsequently also appear at loop-order in the form of
finite-spin ambiguities.

Hence, besides providing an explicit test of the relation (3.2) across
different orders in the perturbative expansion, the extended matching proposed
here is sensitive to tree-level contributions and finite-spin ambiguities of
loop amplitudes in AdS. Unfortunately, as we will explain in the following,
some of the terms in the low-energy expansion of the genus-one string
amplitude depend on some constant scale $\mu$. This prevents us from fixing
the tree-level ambiguities in the one-loop string corrections
$\mathcal{H}^{(2,m)}$ which contribute to the bulk-point limit, since shifting
the scale alters the value of the non-analytic terms one would like to match.
Nevertheless, the bulk-point limit of the one-loop correlators
$\mathcal{H}^{(2,m)}$ sheds light on the role of functions containing the
letter $x-\bar{x}$. In particular, we will argue that the extra logarithmic
divergence due to the presence of the letter $x-\bar{x}$ is necessary to
recover the scale-dependent logarithmic terms of the form
$\log(\frac{\alpha^{\prime}s}{\mu})$, which do appear in the low-energy
expansion of the type IIB string amplitude starting from genus-one.

Lastly, moving to genus-two, we will describe a constraint for the bulk-point
limit of the two-loop correlator $\mathcal{H}^{(3)}$ which comes from matching
the two-loop supergravity amplitude in ten-dimensional flat-space.

###  3.1 The perturbative expansion of the type IIB string amplitude

Let us begin by assembling the necessary terms in the low-energy expansion of
the type IIB string theory scattering amplitude $\mathcal{A}^{(10)}$. Its
genus-expansion takes the general form

$\displaystyle\mathcal{A}^{(10)}$
$\displaystyle=\kappa_{10}^{2}\,g_{s}^{2}\,\frac{\widehat{K}}{64}\,\big{(}\mathcal{A}_{\text{genus-0}}+g_{s}^{2}\,\mathcal{A}_{\text{genus-1}}+g_{s}^{4}\,\mathcal{A}_{\text{genus-2}}+O(g_{s}^{6})\big{)},$
(3.4)

where $\kappa_{10}^{2}=2^{6}\pi^{7}\,(\alpha^{\prime})^{4}$ and $\widehat{K}$
is an overall kinematic factor depending on 10-dimensional momenta $k_{i}$ and
polarisation vectors $\xi_{i}$ of the external gravitons. In order to compare
with the bulk-point limit of holographic correlators on the product space
AdS${}_{5}\times$S5, we need to restrict $\mathcal{A}^{(10)}$ to transverse
kinematics by taking the momenta of the external gravitons to lie in AdS5 and
their polarisation vectors along S5, such that $k_{i}\cdot\xi_{j}=0$. As shown
in [64], with such kinematics the overall factor
$\widehat{K}\to\widehat{K}_{\perp}$ becomes proportional to the prefactor
$\mathcal{I}$ of the dynamical part dictated by supersymmetry, recall
equations (1.6) and (1.8), and hence these two factors will cancel in the
comparison (3.2). Moreover, we need to express the flat-space amplitude as a
function of the dimensionless parameter $x=1+\frac{t}{s}$, related to the
scattering angle via (3.3). This is achieved by pulling out the overall
scaling with the center-of-mass energy $s$ from the individual terms in the
expansion (3.4). We thus rewrite the low-energy expansion in the form

$\displaystyle\mathcal{A}^{(10)}_{\perp}=\kappa_{10}^{2}\,g_{s}^{2}\,\frac{\widehat{K}_{\perp}}{64}\,\Big{(}\frac{1}{(\alpha^{\prime}s)^{3}}\,\mathcal{A}^{(0,-3)}(x)\,+\,\sum_{g=0}\sum_{k=0}(2\pi^{2}g_{s}^{2})^{g}(\alpha^{\prime}s)^{k}\,\mathcal{A}^{(g,k)}(x)\Big{)},$
(3.5)

where $\mathcal{A}^{(10)}_{\perp}$ denotes the 10-dimensional amplitude in
transverse kinematics. The first term is the tree-level supergravity
contribution which scales as $1/s^{3}$, followed by an infinite tower of
string and higher-genus corrections, all scaling with positive powers of $s$.

The genus-zero contributions are given by the well-known Virasoro-Shapiro
amplitude, and for the first few orders in $\alpha^{\prime}$ one finds

$\displaystyle\begin{split}\mathcal{A}^{(0,-3)}&=\frac{64}{x(1-x)},\quad\mathcal{A}^{(0,0)}=2\zeta_{3},\quad\mathcal{A}^{(0,2)}=\frac{\zeta_{5}}{8}(x^{2}-x+1),\quad\mathcal{A}^{(0,3)}=\frac{\zeta_{3}^{2}}{2^{5}}x(1-x),\\\\[3.0pt]
\mathcal{A}^{(0,4)}&=\frac{\zeta_{7}}{2^{7}}(x^{2}-x+1)^{2},\quad\mathcal{A}^{(0,5)}=\frac{\zeta_{3}\zeta_{5}}{2^{8}}x(1-x)(x^{2}-x+1),\\\\[3.0pt]
\mathcal{A}^{(0,6)}&=\frac{\zeta_{3}^{3}}{2^{10}\,3}x^{2}(1-x)^{2}+\frac{\zeta_{9}}{2^{11}\,3}(3x^{6}-9x^{5}+19x^{4}-23x^{3}+19x^{2}-9x+3),\end{split}$
(3.6)

where the terms $\mathcal{A}^{(0,k)}$ with $k\geq 0$ correspond to tree-level
$\partial^{2k}\mathcal{R}^{4}$ higher-derivative corrections to the
supergravity effective action.

Next, the low-energy expansion of the genus-one contribution
$\mathcal{A}_{\text{genus-1}}$ has been worked out most systematically in
[65].151515See also the corrections/extensions given in references [66, 67].
Putting together the analytic and non-analytic contributions, we have

$\displaystyle\begin{split}\mathcal{A}^{(1,0)}&=\frac{1}{3}\,,\quad\mathcal{A}^{(1,1)}=\mathcal{A}^{\text{one-
loop}}_{\text{sugra}}(x)\,,\quad\mathcal{A}^{(1,2)}=0\,,\quad\mathcal{A}^{(1,3)}=\frac{\zeta_{3}}{2^{6}\,3}x(1-x)\,,\\\\[3.0pt]
\mathcal{A}^{(1,4)}&=-\frac{\zeta_{3}}{2^{6}\,45}\Big{[}x^{4}\log(x)+(1-x)^{4}\log(1-x)-i\pi+2(x^{2}-x+1)^{2}\log(\tfrac{\alpha^{\prime}\,s}{\mu_{4}})\Big{]}\,,\\\\[3.0pt]
\mathcal{A}^{(1,5)}&=\frac{29\,\zeta_{5}}{2^{11}\,45}x(1-x)(x^{2}-x+1)\,,\\\\[3.0pt]
\mathcal{A}^{(1,6)}&=\frac{\zeta_{3}^{2}}{2^{13}\,3}x^{2}(1-x)^{2}-\frac{\zeta_{5}}{2^{12}\,315}\Big{[}x^{4}(22x^{2}-x+1)\log(x)\\\
&\qquad\qquad\quad+(1-x)^{4}(22x^{2}-43x+22)\log(1-x)-i\pi(x^{2}-x+22)\\\
&\qquad\qquad\quad+(44x^{6}-132x^{5}+327x^{4}-434x^{3}+327x^{2}-132x+44)\log(\tfrac{\alpha^{\prime}\,s}{\mu_{6}})\Big{]}\,,\end{split}$
(3.7)

for some dimensionless scales $\mu_{4}$ and $\mu_{6}$, which are determined by
the perturbative string theory expansion. Note that the contribution to
$\mathcal{A}^{(1,1)}$ is given by the one-loop supergravity field theory
amplitude $\mathcal{A}^{\text{one-loop}}_{\text{sugra}}(x)$ which scales
linearly with $s$.

Lastly, we move to the genus-two term $\mathcal{A}_{\text{genus-2}}$ where
much less is known about its low-energy expansion. To our knowledge, so far
only the first two non-trivial analytic terms, corresponding to the genus-two
corrections of the $\partial^{4}\mathcal{R}^{4}$ and
$\partial^{6}\mathcal{R}^{4}$ terms, have been determined [68, 69, 70]:

$\displaystyle\mathcal{A}^{(2,0)}=\mathcal{A}^{(2,1)}=0\,,\quad\mathcal{A}^{(2,2)}=\frac{1}{2^{4}\,135}(x^{2}-x+1)\,,\quad\mathcal{A}^{(2,3)}=\frac{1}{2^{7}\,15}x(1-x)\,,$
(3.8)

The precise form of the terms beyond $\partial^{6}\mathcal{R}^{4}$ is not
known, and the only information about the structure of these super-leading
counter-terms to the two-loop supergravity amplitude comes from studying the
compactification of 11-dimensional two-loop supergravity [71]. From those
considerations, the next two terms in the expansion are expected to be of the
schematic form

$\displaystyle\mathcal{A}^{(2,4)}$
$\displaystyle\simeq\partial^{8}\mathcal{R}^{4}|_{\text{genus-2}}+\mathcal{A}_{\text{SG}|\mathcal{R}^{4}|_{\text{genus-1}}}^{\text{one-
loop}}\,,\quad\mathcal{A}^{(2,5)}\simeq\partial^{10}\mathcal{R}^{4}|_{\text{genus-2}}+\mathcal{A}_{\text{sugra}}^{\text{two-
loops}}\,,$ (3.9)

where $\mathcal{A}_{\text{SG}|\mathcal{R}^{4}|_{\text{genus-1}}}^{\text{one-
loop}}$ is a non-analytic contribution similar to the genus-one term
$\mathcal{A}^{(1,4)}$ from equation (3.7), arising due to a one-loop diagram
with one supergravity and one $\mathcal{R}^{4}$-vertex. Finally, the relevant
term for us is the contribution $\mathcal{A}^{(2,5)}$, which is given by the
sum of the genus-two correction to the tree-level
$\partial^{10}\mathcal{R}^{4}$ term (whose precise coefficient is not known)
and the ten-dimensional two-loop supergravity amplitude
$\mathcal{A}_{\text{sugra}}^{\text{two-loops}}$.

###  3.2 The contributions $\mathcal{H}^{(n,m)}$ in the bulk-point limit

As mentioned earlier, the bulk-point singularity of the CFT correlator can be
accessed only in the Lorentzian regime. The necessary analytic continuation
from the Euclidean correlator $\mathcal{H}(x,\bar{x})$ corresponds to a Wick
rotation of AdS global time, see e.g. [58]. In terms of cross-ratios, the
prescription is to take $x$ counter-clock wise around $0$ and $\bar{x}$ around
$1$. Then, the bulk-point singularity is exposed by taking the limit
$\bar{x}\to x$. We find that the terms $\mathcal{H}^{(n,m)}$ at order
$a^{n}\lambda^{-\frac{m}{2}}$ in the large $N$, large $\lambda$ expansion
around the supergravity limit diverge as

$\displaystyle\mathcal{H}^{(n,m)}(x,\bar{x})\leavevmode\nobreak\
\xrightarrow[]{\leavevmode\nobreak\ \text{bpl}\leavevmode\nobreak\
}\leavevmode\nobreak\ \frac{\mathcal{F}^{(n,m)}(x)}{(x-\bar{x})^{p}}\,,$
(3.10)

with the denominator power given by $p=8n+2m-1$.

For terms $\mathcal{H}^{(n,m)}$ given entirely in terms of SVHPL’s, which is
the case for all tree-level correlators $\mathcal{H}^{(1,m)}$ as well as the
one-loop supergravity correction $\mathcal{H}^{(2,0)}$, taking the bulk-point
limit as described above is relatively straightforward. For example,
performing the analytic continuation and taking the limit $\bar{x}\to x$ for
the first few zigzag integrals, one finds

$\displaystyle\begin{split}Z^{(1)}(x,\bar{x})\leavevmode\nobreak\
&\xrightarrow[]{\leavevmode\nobreak\ \text{bpl}\leavevmode\nobreak\
}\leavevmode\nobreak\ 4\pi^{2},\\\ Z^{(2)}(x,\bar{x})\leavevmode\nobreak\
&\xrightarrow[]{\leavevmode\nobreak\ \text{bpl}\leavevmode\nobreak\
}\leavevmode\nobreak\ -4\pi^{2}\big{(}H_{0,0}(x)+i\pi H_{0}(x)\big{)},\\\
Z^{(3)}(x,\bar{x})\leavevmode\nobreak\ &\xrightarrow[]{\leavevmode\nobreak\
\text{bpl}\leavevmode\nobreak\ }\leavevmode\nobreak\
8\pi^{2}\big{(}H_{3,1}(x)-i\pi(H_{3}(x)-\zeta_{3})+\tfrac{1}{6}\pi^{2}H_{2}(x)-\tfrac{11}{360}\pi^{4}\big{)}.\end{split}$
(3.11)

Then, the last step before comparing with the terms $\mathcal{A}^{(g,k)}$ from
the flat-space string amplitude is to convert the $\mathcal{N}=4$ SYM double-
expansion in $a$ and $1/\lambda$ to string theory quantities $g_{s}$ and
$\alpha^{\prime}$. Using the usual AdS/CFT relations, we have

$\displaystyle a=\frac{1}{N^{2}-1}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\
\frac{16\pi^{2}g_{s}^{2}\,(\alpha^{\prime})^{4}}{L^{8}}\,,\qquad\lambda^{-\frac{1}{2}}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\ \frac{\alpha^{\prime}}{L^{2}}\,,$ (3.12)

where $L$ is the AdS radius which we set to one. The first relation in (3.12)
gives that increasing the loop-order in AdS by one corresponds to four powers
of $\alpha^{\prime}$, and hence the residues $\mathcal{F}^{(n,m)}(x)$ from
(3.10) should be compared with the flat-space contributions
$\mathcal{A}^{(g,k)}$ with $g=n-1$ and $k=4n+m-7$.

#### Matching the genus-zero terms

We begin by comparing against the genus-zero terms $\mathcal{A}^{(0,k)}$ given
in (3.6), whose AdS counterparts $\mathcal{H}^{(1,m)}$ have been considered up
to order $(\alpha^{\prime})^{9}$ using the Mellin space representation.161616
Here, we use the results of [32], see also [72] for a manifestly
10-dimensional formulation.  Note that the comparison at tree-level is in some
sense tautological, since precisely the terms which contribute to the bulk-
point limit were fixed using the flat-space limit formulation in Mellin space
in the first place. Nevertheless, this serves as a consistency check and
enables us to fix some factors and the overall normalisation.

Converting the Mellin space results to their position space representation in
terms of linear combinations of $\overline{D}$-functions and taking the bulk-
point limit using the first line of (3.11), we find that the precise matching
reads171717 As explained below (3.4), the overall kinematic factor
$\widehat{K}$ in transverse kinematics cancels against the factor
$\mathcal{I}$. This cancellation is implicit in equation (3.13).

$\displaystyle\mathcal{A}^{(g=n-1,k)}(x)$
$\displaystyle=\frac{1}{2^{2k-3n+9}\,(2k+11)!}\frac{\mathcal{F}^{(n,m)}(x)}{x^{k+10}(x-1)^{k+6}}\,,$
(3.13)

where we recall that $k=4n+m-7$. The above relation between the bulk-point
limit of the correlators $\mathcal{H}^{(n,m)}$ and the corresponding terms in
the flat-space amplitude is in agreement with the general expectation
previously given in equation (3.2).

#### Matching the genus-one terms

At genus-one, the coefficient of the super-leading counter-term
$\mathcal{H}^{(2,-1)}$ is fixed from the flat-space limit alone [40] and hence
it matches the analytic genus-one contribution $\mathcal{A}^{(1,0)}$ by
construction, in analogy with the tree-level terms discussed before.

More interesting is the comparison with the non-analytic terms. The first such
contribution occurs at order $\alpha^{\prime}$ and is the one-loop
supergravity amplitude $\mathcal{A}^{\text{one-loop}}_{\text{sugra}}(x)$,
which can be determined from a field-theory computation as a sum over three
orientations of the massless one-loop box-integral in 10 dimensions, see e.g.
Appendix D of [27] for an explicit evaluation. Taking the bulk-point limit of
the supergravity correlator $\mathcal{H}^{(2)}$ from [4],181818 From the basis
of 15 independent SVHPL’s appearing in $\mathcal{H}^{(2)}$, only 6 elements
contribute to the bulk-point limit. These are the three orientations of the
two-loop zigzag $Z^{(2)}$, two orientations of $\log(u)Z^{(1)}$ and $Z^{(1)}$
itself.  we find that it precisely matches the full form of
$\mathcal{A}^{\text{one-loop}}_{\text{sugra}}(x)$, and not just its
discontinuity as reported in [27]. Note that in the matching one has to set
the quadratic divergence of $\mathcal{A}^{\text{one-loop}}_{\text{sugra}}(x)$
to zero since it is the super-leading term $\mathcal{A}^{(1,0)}$ which takes
the role of the one-loop counter-term. Furthermore, the one-loop ambiguity
$\alpha$ does not contribute to the bulk-point limit as its denominator
$(x-\bar{x})^{13}$ is sub-leading compared to $(x-\bar{x})^{15}$ of
$\mathcal{H}^{(2)}$, and we therefore do not get any constraint on $\alpha$
from the bulk-point limit.

The next two non-analytic genus-one contributions are the order
$(\alpha^{\prime})^{4}$ and $(\alpha^{\prime})^{6}$ terms
$\mathcal{A}^{(1,4)}$, $\mathcal{A}^{(1,6)}$ given in (3.7). The corresponding
AdS correlators $\mathcal{H}^{(2,3)}$ and $\mathcal{H}^{(2,5)}$ have been
constructed in [20], and it turns out that a function with a new type of
singularity is present. This weight-three function, $f^{(3)}(x,\bar{x})$,
contains the additional letter $x-\bar{x}$ and is therefore beyond the space
of SVHPL’s, making the computation of its bulk-point limit more involved. We
use the Mathematica package PolyLogTools [73] to perform the analytic
continuations and, introducing the parametrisation $\epsilon=x-\bar{x}$, to
then carefully take the limit $\epsilon\to 0$. We find that the function
$f^{(3)}$ contributes to the bulk-point limit with

$\displaystyle f^{(3)}(x,\bar{x})\leavevmode\nobreak\
\xrightarrow[]{\leavevmode\nobreak\ \text{bpl}\leavevmode\nobreak\
}\leavevmode\nobreak\
-8\pi^{2}\big{(}2\log(x)+2\log(1-x)+i\pi-3\log(\epsilon)\big{)}.$ (3.14)

Note the appearance of an extra logarithmic divergence parametrised by
$\log(\epsilon)$ due to the presence of the new letter $x-\bar{x}$.

Now, we can proceed to check the relation (3.13) between the bulk-point limit
of the correlators $\mathcal{H}^{(2,3)}$, $\mathcal{H}^{(2,5)}$,191919 Besides
$f^{(3)}$, also $Z^{(1)}$ and the two orientations of $\log(u)Z^{(1)}$
contribute to the bulk-point limit.  and the non-analytic genus-one
contributions $\mathcal{A}^{(1,4)}$, $\mathcal{A}^{(1,6)}$. We find perfect
agreement upon identifying the extra logarithmic divergence $\log(\epsilon)$
from the function $f^{(3)}$ with the scale-dependent terms
$\log(\frac{\alpha^{\prime}\,s}{\mu})$ according to

$\displaystyle\log(\epsilon)\simeq\frac{1}{2}\big{(}\log(x)+\log(1-x)-\log(\tfrac{\alpha^{\prime}s}{\mu})+i\pi\big{)},$
(3.15)

for a choice of scale $\mu$. Besides providing the expected matching, the
above identification can be further justified by checking that both sides of
the equation transform equally under all crossing transformations. Indeed,
after recasting (3.15) into the form
$\log(\epsilon^{2})=\log(x)+\log(1-x)-\log(\tfrac{\alpha^{\prime}s}{\mu})+i\pi$
and recalling $\epsilon=x-\bar{x}$, one can verify that the proposed
identification is consistent with crossing.

Lastly, note that the dependence of (3.15) on an à priori arbitrary scale
$\mu$ prevents us from fixing any tree-level ambiguities in the one-loop
correlators which might contribute to the bulk-point limit. Take for example
$\mathcal{H}^{(2,3)}$, which has been fully fixed up to four ambiguities
corresponding to the Mellin amplitudes
$\\{1,\sigma_{2},\sigma_{3},\sigma_{2}^{2}\\}$, where $\sigma_{n}\equiv
s^{n}+t^{n}+u^{n}$. In the bulk-point limit, or equivalently in the
formulation of the flat-space limit for Mellin amplitudes, only the overall
leading powers in the Mellin variables contribute, which in this case is given
by the term $\sigma_{2}^{2}$. Now, one can check that in the bulk-point limit
the coefficient of $f^{(3)}$ is proportional to
$s^{4}(x^{2}-x+1)^{2}\propto\sigma_{2}^{2}$, and thus a shift in the scale
$\mu$ in (3.15) simply corresponds to a redefinition of the coefficient of the
ambiguity.

#### Constraints for $\mathcal{H}^{(3)}$ from matching at genus-two

The leading term at genus two is of order $(\alpha^{\prime})^{2}$ and
corresponds to the genus-two correction of the $\partial^{4}\mathcal{R}^{4}$
term. In AdS, this contribution has been determined in [41] and one can check
that in the bulk-point limit it correctly reproduces the analytic contribution
$\mathcal{A}^{(2,2)}$ given in (3.8).

Let us skip forward and focus on the $(\alpha^{\prime})^{5}$ contribution,
which is of particular relevance to us as it provides a non-trivial constraint
for the two-loop correlator $\mathcal{H}^{(3)}$. As stated in (3.9), one
expects two distinct contributions to $\mathcal{A}^{(2,5)}$: an analytic
contribution from $\partial^{10}\mathcal{R}^{4}|_{\text{genus-2}}$ of the form
$\sigma_{2}\sigma_{3}\propto s^{5}\,x(1-x)(x^{2}-x+1)$ and a non-analytic
contribution due to the two-loop supergravity amplitude
$\mathcal{A}_{\text{sugra}}^{\text{two-loops}}$, which is given by the sum
over crossing orientations of the planar and non-planar double-box integral in
10 dimensions, see e.g. [74]. Its finite part has been evaluated in [45],
whose result we use to compute the function of angles
$\mathcal{A}_{\text{sugra}}^{\text{two-loops}}(x)$.202020 We record the
explicit expression for $\mathcal{A}_{\text{sugra}}^{\text{two-loops}}(x)$ in
an ancillary file. It is given in terms of HPL’s up to weight 4 with rational
coefficients. Notably, the $1/\epsilon^{2}$ pole in dimensional regularisation
cancels in the sum of the planar and non-planar integral such that the
remaining divergence is only of order $1/\epsilon$. This gives rise to a
scale-dependent term with a logarithmic contribution of the form
$s^{5}\log(\frac{\alpha^{\prime}s}{\mu})$. We find that its coefficient is
proportional to $x(1-x)(x^{2}-x+1)$, which is in agreement with the expected
presence of the analytic contribution since a change in the scale $\mu$ simply
amounts to a shift in the coefficient of
$\partial^{10}\mathcal{R}^{4}|_{\text{genus-2}}$.

Note that the scale-dependent contribution to $\mathcal{A}^{\text{two-
loops}}_{\text{sugra}}$ has a non-trivial consequence for the corresponding
AdS correlator. In the bulk-point limit, the two-loop supergravity correlator
$\mathcal{H}^{(3)}$ maps to the genus-two $(\alpha^{\prime})^{5}$ term
$\mathcal{A}^{(2,5)}$, and in order to match the scale-dependent logarithm we
need a non-zero contribution from $f^{(3)}(x,\bar{x})$ to the bulk-point limit
of $\mathcal{H}^{(3)}$. We will therefore necessarily need to include the
letter $x-\bar{x}$ in our ansatz for $\mathcal{H}^{(3)}$ at least to weight 3,
in contrast to the one-loop correlator $\mathcal{H}^{(2)}$ which is given in
terms of SVHPL’s alone. Furthermore, as in the case for the one-loop string
correlators discussed earlier, the dependence of the identification (3.15) on
an arbitrary scale $\mu$ (as well as our ignorance of the precise coefficient
of $\partial^{10}\mathcal{R}^{4}|_{\text{genus-2}}$) will prevent us from
fixing the expected tree-level ambiguity $\sigma_{2}\sigma_{3}$ in the two-
loop correlator $\mathcal{H}^{(3)}$.

##  4 Bootstrapping the two-loop correlator $\mathcal{H}^{(3)}$

Before explaining the details of our construction of the two-loop supergravity
contribution, it is useful to first step back and review the known results at
preceding orders. The structures appearing in the tree-level and one-loop
correlators, $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$, will then guide us
to make an educated ansatz for the two-loop correlator $\mathcal{H}^{(3)}$.

###  4.1 Review of tree-level and one-loop

The derivation of the tree-level correlator $\mathcal{H}^{(1)}$ dates back to
the early days of the AdS/CFT correspondence and belongs to the first explicit
results of an amplitude on AdS. In our conventions, it is given by [8, 26]

$\displaystyle\mathcal{H}^{(1)}=-16u^{2}\overline{D}_{2422}.$ (4.1)

In hindsight, this result can be bootstrapped from the following
considerations: the basis function for the top-weight part is given by the
first member of zigzag integrals, $Z^{(1)}$. The remaining lower-weight basis
elements are then simply given by $\log(u)$, $\log(v)$ and $1$. Each of these
four functions is multiplied by a rational coefficient function with
denominator power $(x-\bar{x})^{7}$. A subtlety at tree-level is that one also
has to allow for a (single) power of $v=(1-x)(1-\bar{x})$ in the denominator
in order to ensure cancellation of certain protected twist 2 contributions.
Finally, requiring crossing symmetry, no unphysical poles at $x=\bar{x}$ and
cancellation of the twist 2 sector fully fixes $\mathcal{H}^{(1)}$ and
precisely yields the result given above.

Considerably more involved is the one-loop correlator $\mathcal{H}^{(2)}$.
Explicitly given in its full form in position space in [4], it has later been
recast in a much simpler form by using the fact that its leading log can be
written as $\Delta^{(8)}$ acting on a simpler object, as explained in Section
2.1. Remarkably, one finds that this property of the leading log extends to
the full correlator, such that $\mathcal{H}^{(2)}$ can be written as
$\Delta^{(8)}$ acting on a ‘preamplitude’ $\mathcal{L}^{(2)}$ at the expense
of having to add some amount of the tree-level correlator $\mathcal{H}^{(1)}$
[15],

$\displaystyle\mathcal{H}^{(2)}=\frac{1}{u^{2}}\Delta^{(8)}\mathcal{L}^{(2)}+\mathcal{H}^{(1)}.$
(4.2)

Let us emphasise again that this achieves a remarkable simplification: the
coefficient functions of the transcendental basis elements in
$\mathcal{H}^{(2)}$ come with maximal denominator power $(x-\bar{x})^{15}$,
whereas the corresponding coefficient functions in $\mathcal{L}^{(2)}$ have at
most denominator power $(x-\bar{x})^{7}$, with the extra $8$ powers being
supplied through the action of the 8-th order differential operator
$\Delta^{(8)}$. In that regard, the coefficient functions at one-loop order
turn out to be of the same complexity as the tree-level ones.

The basis of transcendental functions appearing in $\mathcal{L}^{(2)}$ is
again consistent with our previous observations: at top-weight (weight 4 in
this case), it is given by the three independent orientations of the zigzag
integral $Z^{(2)}$, completed by all lower-weight functions with no
$\log^{3}(u)$ contribution to any channel. In contrast to the tree-level case
however, the bootstrap constraints described above are not sufficient to
entirely fix the full one-loop correlator. Instead, one is left with one
remaining free parameter, $\alpha$, which comes with a tree-level like
function given by $u^{2}\overline{D}_{4444}$, corresponding to a constant
Mellin amplitude. The presence of such a tree-level ambiguity is related to
the super-leading counter-term $\mathcal{R}^{4}|_{\text{genus-1}}$, and its
value is determined within the full type IIB string theory effective action on
AdS${}_{5}\times$S5. However, our bootstrap computation is not able to fix it
and one has to resort to other methods. For instance, as shown in [40], it is
possible to determine it by using supersymmetric localisation techniques,
yielding the value $\alpha=60$.

Interestingly, it is possible to write the one-loop ambiguity $\alpha$ as part
of the preamplitude $\mathcal{L}^{(2)}$, which is a remarkable property of the
corresponding $\overline{D}$-function. Note that this term contributes non-
analytically with a single spin $\ell=0$ conformal block to the one-loop
anomalous dimension $\gamma^{(2)}$, as recorded in equation (B.3) for the
twist 4 case. The presence of the $\alpha$-parameter at one-loop order will in
turn induce a non-analytic spin 0 contribution to the $\log^{2}(u)$-part of
the two-loop correlator.

###  4.2 More on $\Delta^{(8)}$: symmetries under crossing

As this will become important in what comes next, let us briefly comment on
the crossing properties of the $\Delta^{(8)}$ operator, which we introduced
earlier in Section 2. For convenience, we repeat its definition:

$\displaystyle\Delta^{(8)}=\frac{u^{4}}{(x-\bar{x})}\partial_{\bar{x}}^{2}(1-\bar{x})^{2}\partial_{\bar{x}}\partial_{x}^{2}(1-x)^{2}\partial_{x}^{2}(x-\bar{x}).$
(4.3)

Recall that $\Delta^{(8)}$ was useful in the context of the leading log of
$\mathcal{H}^{(n)}$, since pulling out $n-1$ powers of $\Delta^{(8)}$
drastically simplifies the leading log. In order for this to be consistent
with the $1\leftrightarrow 2$ exchange-symmetry of the OPE decomposition, this
operator itself needs to respect that symmetry. Indeed, one can check that
under the corresponding crossing transformation $x\mapsto
x^{\prime}=\frac{x}{x-1}$, the $\Delta^{(8)}$ operator (and consequently any
power of it) is left invariant

$\displaystyle\big{(}\Delta^{(8)}\big{)}^{k}\leavevmode\nobreak\
\xrightarrow[]{\leavevmode\nobreak\
x\,\rightarrow\,x^{\prime}\leavevmode\nobreak\ }\leavevmode\nobreak\
\leavevmode\nobreak\ \big{(}\Delta^{(8)}\big{)}^{k}.$ (4.4)

Remarkably, an accidental enhancement of crossing symmetry occurs for the case
when $k=2$. We find that

$\displaystyle\big{(}\Delta^{(8)}\big{)}^{2}\leavevmode\nobreak\
\xrightarrow[]{\leavevmode\nobreak\ x\,\rightarrow\,1-x\leavevmode\nobreak\
}\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{u^{4}}{v^{4}}\,\big{(}\Delta^{(8)}\big{)}^{2},$ (4.5)

while no other power of $\Delta^{(8)}$ obeys this extra symmetry. As a simple
consequence, the combination $\frac{1}{u^{2}}(\Delta^{(8)})^{2}$ has the same
crossing symmetries as the correlator $\mathcal{H}(u,v)$ itself, see equation
(1.9). This enhancement to full crossing symmetry is particularly useful in
the construction of the two-loop amplitude $\mathcal{H}^{(3)}$, as it allows
the preamplitude (introduced next) to be made fully crossing invariant from
the get go. In contrast, the preamplitude of the one-loop correlator obeys
only the $x\mapsto x^{\prime}$ crossing symmetry.

###  4.3 A minimal ansatz for the two-loop correlator

Based on the fact that $(\Delta^{(8)})^{2}$ can be pulled out from the leading
log, and further motivated by the simple structure of the one-loop correlator
as given in equation (4.2), we start with the following minimal ansatz for the
two-loop correlator,212121 We have found this structure to be the simplest
ansatz possible which satisfies all of the imposed bootstrap constraints (to
be discussed below), justifying the name ‘minimal ansatz’. For an exploration
of different modifications of the minimal ansatz, see Section 6.

$\displaystyle\mathcal{H}^{(3)}=\frac{1}{u^{2}}(\Delta^{(8)})^{2}\,\mathcal{P}^{(3)}+a_{2}\mathcal{H}^{(2)}+a_{1}\mathcal{H}^{(1)},$
(4.6)

with $\mathcal{P}^{(3)}$ denoting the two-loop preamplitude we would like to
compute. Let us point out that the structure of our minimal ansatz is of the
same form as proposed in the recent work by Huang and Yuan [19], albeit
different in some details. We will comment on the precise differences of the
final results in Section 4.6.

On general grounds, the two-loop preamplitude is of the form

$\displaystyle\mathcal{P}^{(3)}(x,\bar{x})=\sum_{i}\frac{p_{i}(x,\bar{x})}{(x-\bar{x})^{d_{i}}}\,\mathcal{Q}_{i}(x,\bar{x}),\quad\text{with
}p_{i}(x,\bar{x})=\sum_{n=0}^{d_{i}}\sum_{m=n}^{d_{i}}a_{n,m}^{(i)}\left(x^{n}\bar{x}^{m}+x^{m}\bar{x}^{n}\right),$
(4.7)

where the $\mathcal{Q}_{i}$ are pure transcendental functions (forming the
basis $\mathcal{Q}$ specified below) with coefficient polynomials
$p_{i}(x,\bar{x})$ containing free parameters $a_{n,m}^{(i)}$. The denominator
powers are given by $d_{i}=7$ ($d_{i}=6$) when the corresponding function
$\mathcal{Q}_{i}(x,\bar{x})$ is antisymmetric (symmetric) under the exchange
symmetry $x\leftrightarrow\bar{x}$. This ensures that overall the preamplitude
is a symmetric function of $(x,\bar{x})$.

Let us now proceed to specify our basis of transcendental functions. Our basis
contains SVHPL’s built from the alphabet $\\{x,\bar{x},1-x,1-\bar{x}\\}$ up to
transcendental weight 6, subject to the following conditions:

  * •

The functions at weight 6 are constrained by the observations on the leading
log discussed in Section 2. At two-loop order, the top-weight part is given by
the zizag-integral $Z^{(3)}$ which provides the correct leading-log
contribution to $\mathcal{H}^{(3)}$. In addition, we include all other weight
6 functions with no further $\log^{3}(u)$ contributions in any orientation and
which vanish at $x=\bar{x}$.222222 This is a necessary condition for ensuring
the cancellation of spurious poles at $x=\bar{x}$ discussed later. It turns
out that these two conditions constrain the additional weight 6 functions to
be antisymmetric.

  * •

For functions of weight $w=0,1,\ldots 5$, the only selection criterion we
impose is the fact that the OPE predicts the leading logarithmic divergence to
be of order $\log^{3}(u)$. Below weight 6, we will thus include all functions
(both symmetric and antisymmetric) with no $\log^{4}(u)$ contributions to any
channel. Recall that in Section 2.2 we argued that all functions contributing
to the leading log are in principle further constrained to be given by zigzags
or derivatives thereof. However, for simplicity we will refrain from imposing
this condition here, at the expense of possibly working with a slightly wider
transcendental basis than necessary. As a consequence, we will find that any
such extra functions have their coefficients set to zero in the final result.

  * •

Lastly, we need to consider the possibility of including functions with the
additional letter $x-\bar{x}$, which goes beyond the space of SVHPL’s. As
shown in [20], the one-loop $1/\lambda$ corrections $\mathcal{H}^{(2,m)}$
necessarily contain such a function at weight 3, denoted by
$f^{(3)}(x,\bar{x})$ therein.232323 In fact, $f^{(3)}(x,\bar{x})$ is the
unique such new function at weight 3. This is not the case at higher weights
any more, however. In particular, its presence is required because the one-
loop string corrections have finite spin contributions to the leading log, in
contrast to the infinite spin support of $\mathcal{H}^{(2)}|_{\log^{2}(u)}$.
Now, it is for the same reason that we need to include it as part of our
transcendental basis for $\mathcal{H}^{(3)}$: the one-loop ambiguity $\alpha$
from $\mathcal{H}^{(2)}$ induces a one-loop like contribution to the two-loop
supergravity correlator, which is exactly of the form of
$\mathcal{H}^{(2,3)}$.242424 Recall there is a similar term at order
$a^{3}\lambda^{\frac{1}{2}}$, which is a one-loop counter-term coming from a
diagram containing a supergravity and a $\mathcal{R}^{4}|_{\text{genus-1}}$
vertex. This term is in fact super-leading with respect to the two-loop
supergravity correlator, leading us to anticipate the presence of such a
contribution also in $\mathcal{H}^{(3)}$. We therefore expect $f^{(3)}$ to
contribute at two-loop supergravity level, in agreement with arguments from
the bulk-point limit. On the other hand, we exclude higher-weight functions
with letter $x-\bar{x}$ for now and revisit such a possibility later in
Section 6.1.

The above considerations leave us with a basis of 73 independent functions
$\mathcal{Q}_{i}(x,\bar{x})$. A schematic overview of the basis $\mathcal{Q}$
is presented in Table 1, while the precise definitions and symmetry properties
of all the functions are spelled out in great detail in Appendix A. Note that
we consider explicit $\zeta$-values as independent basis elements, such that
the free coefficients $a_{n,m}^{(i)}$ in the preamplitude (4.7) are simply
rational numbers. Before having imposed any constraints on
$\mathcal{P}^{(3)}$, our initial ansatz for the preamplitude contains 2308
free parameters.

$w$ | $x\leftrightarrow\bar{x}$ | $\mathcal{Q}_{i}(x,\bar{x})$ | total
---|---|---|---
6 | $-$ | $6\times Z^{(3)},\leavevmode\nobreak\ A^{(6)},\leavevmode\nobreak\ 3\times B^{(6)},\leavevmode\nobreak\ \zeta_{3}f^{(3)},\leavevmode\nobreak\ 2\times\zeta_{3}\log(u)Z^{(1)}$ | 13
| $+$ | - | 0
5 | $-$ | $6\times\widetilde{\Psi}^{(3)},\leavevmode\nobreak\ 3\times\widetilde{\Pi}^{(5)},\leavevmode\nobreak\ \zeta_{3}Z^{(1)}$ | 10
| $+$ | $6\times\Psi^{(3)},\leavevmode\nobreak\ 6\times\Pi^{(5)},\leavevmode\nobreak\ \Omega^{(5)},\leavevmode\nobreak\ 2\times\log(u)(Z^{(1)})^{2},\leavevmode\nobreak\ 3\times\zeta_{3}\log^{2}(u)$ | 18
4 | $-$ | $3\times\log^{2}(u)Z^{(1)},\leavevmode\nobreak\ 3\times Z^{(2)}$ | 6
| $+$ | $6\times\Upsilon^{(3)},\leavevmode\nobreak\ 2\times\log^{3}(u)\log(v),\leavevmode\nobreak\ (Z^{(1)})^{2}$ | 9
3 | $-$ | $f^{(3)},\leavevmode\nobreak\ 2\times\log(u)Z^{(1)}$ | 3
| $+$ | $4\times\log^{3}(u),\leavevmode\nobreak\ 3\times\Psi^{(2)}$ | 7
2 | $-$ | $Z^{(1)}$ | 1
| $+$ | $3\times\log^{2}(u)$ | 3
1 | $-$ | - | 0
| $+$ | $2\times\log(u)$ | 2
0 | $-$ | - | 0
| $+$ | $1$ | 1
Table 1: Overview of the basis of transcendental functions $\mathcal{Q}$,
ordered by their transcendental weight $w$. At each weight, we classify the
basis elements according to their symmetry under $x\leftrightarrow\bar{x}$
exchange, distinguishing antisymmetric ($-$) from symmetric ($+$) functions.
The third column explicitly lists the functions $\mathcal{Q}_{i}(x,\bar{x})$
together with their number of independent orientations under crossing (e.g. at
weight $w=1$, the entry $2\times\log(u)$ means there exist 2 orientations of
this basis element, given by $\log(u)$ and $\log(v)$ in this case). Finally,
in the last column we give the total number of functions in each category.

###  4.4 Bootstrapping $\mathcal{H}^{(3)}$: constraints

We now turn to the description of the constraints which the ansatz (4.6) for
$\mathcal{H}^{(3)}$ needs to obey. First of all, there are certain constraints
which can be imposed directly on the preamplitude $\mathcal{P}^{(3)}$:

  1. (1)

As a consequence of the simple transformation properties of
$(\Delta^{(8)})^{2}$ under crossing, we can take $\mathcal{P}^{(3)}$ to be
fully crossing symmetric:

$\displaystyle\mathcal{P}^{(3)}(x,\bar{x})=\mathcal{P}^{(3)}(x^{\prime},\bar{x}^{\prime})=\mathcal{P}^{(3)}(1-x,1-\bar{x}).$
(4.8)

  2. (2)

We can furthermore impose the matching of the predicted leading log directly
on the preamplitude $\mathcal{P}^{(3)}$, as discussed in Section 2. This
amounts to imposing:

$\displaystyle\begin{split}\mathcal{P}^{(3)}(x,\bar{x})=\,\,&\mathcal{G}^{(3)}(x,\bar{x})+\mathcal{G}^{(3)}(1-x,1-\bar{x})+\mathcal{G}^{(3)}(1-x^{\prime},1-\bar{x}^{\prime})\\\
&+\text{ terms with no $\log^{3}u$ in any channel,}\end{split}$ (4.9)

where $\mathcal{G}^{(3)}$ is the combination of zigzags and derivatives
introduced in (2.2). However, in practice we impose this only for the top-
weight part for simplicity and match the rest against our ansatz.

  3. (3)

The full correlator $\mathcal{H}^{(3)}(u,v)$ is expected to be non-singular at
$x=\bar{x}$, as such poles would be at unphysical locations. Note that since
the (repeated) application of $\Delta^{(8)}$ does not create any new poles at
$x=\bar{x}$, we can directly impose the cancellation of the 7 explicit poles
from the denominators in (4.7) within the preamplitude $\mathcal{P}^{(3)}$.

At this stage, the ansatz for $\mathcal{H}^{(3)}$ obeys the correct crossing
symmetries, has no poles at $x=\bar{x}$ and by construction matches the
correct leading log. However, we are still left with a total of 82 free
parameters (80 from $\mathcal{P}^{(3)}$ together with $a_{1}$ and $a_{2}$). On
the other hand, there are further constraints which $\mathcal{H}^{(3)}$ needs
to satisfy:

  1. (4)

Cancellation of all contributions with twists below 4. As in the one-loop
case, the presence of the tree-level expression $\mathcal{H}^{(1)}$ in our
ansatz is required in order to satisfy this constraint. This step gives 60
constraints on the free parameters, and in particular we find that $a_{1}$ is
fixed to take the value $a_{1}=-1$.

  2. (5)

Matching the predicted $\log^{2}(u)$ contribution at twist 4. According to the
OPE expansion (1.19), this term is given in terms of one-loop and tree-level
OPE data by the combination

$\displaystyle
x^{2}g(\bar{x})\equiv\sum_{\ell}\left(\frac{1}{2}\,A_{2,\ell}^{(1)}\,\big{(}\gamma_{2,\ell}^{(1)}\big{)}^{2}+A_{2,\ell}^{(0)}\,\gamma_{2,\ell}^{(1)}\,\gamma_{2,\ell}^{(2)}+A_{2,\ell}^{(0)}\big{(}\gamma_{2,\ell}^{(1)}\big{)}^{3}\partial_{\Delta}\right)G_{2,\ell}(x,\bar{x})\,,$
(4.10)

where we have kept only twist 4 contributions. Performing the sum over even
spins $\ell$ we obtain the explicit one-variable function $g(\bar{x})$
recorded in Appendix B. Notably, it contains a non-analytic spin 0
contribution from the one-loop ambiguity $\alpha$. We find 8 further
constraints from this matching, one of which determines $a_{2}=5$.

  3. (6)

Matching the flat-space contribution $\mathcal{A}^{(2,5)}$. As explained in
Section 3, the correlator $\mathcal{H}^{(3)}$ is mapped to the two-loop
supergravity amplitude in the bulk-point limit. We fix 6 more free parameters
by matching against the explicit result for
$\mathcal{A}_{\text{sugra}}^{\text{two-loops}}(x)$ from reference [45], whilst
keeping the ambiguities corresponding to the
$\partial^{10}\mathcal{R}^{4}|_{\text{genus-2}}$ correction term.252525 As
already emphasised in Section 3, we perform this matching at the level of the
full amplitude, not only its discontinuity. While this does not result in
giving more constraints, being able to match a complicated expression
involving HPL’s of up to weight 4 nevertheless constitutes a highly non-
trivial consistency check between the result for the flat-space amplitude from
[45] and our construction of the CFT correlator.

###  4.5 Bootstrapping $\mathcal{H}^{(3)}$: results

After having imposed the constraints (1) – (6) described above, we find that
the two-loop correlator $\mathcal{H}^{(3)}$ is completely fixed up to only 8
remaining free parameters, which all correspond to tree-level ambiguities. Our
final result takes the form

$\displaystyle\mathcal{H}^{(3)}=\frac{1}{u^{2}}(\Delta^{(8)})^{2}\,\mathcal{P}^{(3)}+5\,\mathcal{H}^{(2)}-\mathcal{H}^{(1)},$
(4.11)

with the tree-level and one-loop correlators given in (4.1) and (4.2),
respectively. Due to the complexity of the coefficient functions
$p_{i}(x,\bar{x})$, we attach our result for the preamplitude
$\mathcal{P}^{(3)}$ and also the full correlator $\mathcal{H}^{(3)}$ (for some
choice of ambiguities) in an ancillary file to the arXiv submission. We find
it remarkable that our highly constrained minimal ansatz is able to satisfy
all of the imposed constraints, and we emphasise that the only free parameters
left are of the form of tree-level contact diagrams.

We observe that not all of the transcendental functions
$\mathcal{Q}_{i}(x,\bar{x})$ which were included in the initial ansatz
actually contribute to the final result. In the notation of Table 1 and
organised by their transcendental weight $w$, the 10 basis elements with
vanishing coefficients (both in $\mathcal{P}^{(3)}$ and $\mathcal{H}^{(3)})$
read

  * •

$w=6$: $\quad 3\times B^{(6)},\leavevmode\nobreak\
2\times\zeta_{3}\log(u)Z^{(1)}$,

  * •

$w=5$: $\quad\Omega^{(5)},\leavevmode\nobreak\ 2\times\log(u)(Z^{(1)})^{2}$,

  * •

$w=4$: $\quad 2\times\log^{3}(u)\log(v)$.

Note that the functions of weight 4 and 5 listed above all have $\log^{3}(u)$
contributions and the vanishing of their coefficients is a consequence of the
observation made in Section 2.2 that the leading log is fully captured by the
zigzag functions $Z^{(n)}$ and derivatives thereof. Indeed, one can check that
the functions $\Omega^{(5)}$, $\log(u)(Z^{(1)})^{2}$ and $\log^{3}(u)\log(v)$
can not be expressed in terms of derivatives of zigzags, and hence they do not
appear in the two-loop correlator. On the other hand, the same argument does
not apply to the weight 6 functions listed above as they do not contribute to
the leading log. Currently, we do not have an explanation for the unexpected
vanishing of these particular functions and it would be very interesting to
understand the principle behind this observation.

Let us now turn to remaining tree-level ambiguities in $\mathcal{H}^{(3)}$.
Schematically, the 8 free parameters split into $4\times\overline{D}$ and
$4\times\zeta_{3}\overline{D}\,$,262626Recall that we do not allow for
explicit $\zeta$-values in the coefficient polynomials. Instead, all
$\zeta$-values are treated as separate basis elements of $\mathcal{Q}$,
explaining why $\overline{D}$ and $\zeta_{3}\overline{D}$ appear as
independent functions. whose coefficients we parametrise by $d_{j}$ and
$e_{j}$, with $j=1,2,3,4$. Such tree-level ambiguities are best described in
terms of their Mellin amplitudes, and we find they are given by the following
linear combinations of monomials
$\\{1,\sigma_{2},\sigma_{3},\sigma_{2}^{2},\sigma_{2}\sigma_{3}\\}$,

$\displaystyle d_{1}\simeq\sigma_{2}-\tfrac{16}{7},\qquad
d_{2}\simeq\sigma_{3}+\tfrac{32}{7},\qquad
d_{3}\simeq\sigma_{2}^{2}-\tfrac{128}{7},\qquad
d_{4}\simeq\sigma_{2}\sigma_{3}-\tfrac{256}{7},$ (4.12)

and the other 4 are simply related by $e_{j}=\zeta_{3}\,d_{j}$. In fact, these
are precisely the expected ambiguities at this order as they are the ones
which contribute to the $\partial^{10}\mathcal{R}^{4}|_{\text{genus-2}}$ term
in AdS. In the conformal block expansion, they contribute to the
$\log(u)$-part with finite spin support for spins $\ell=0,2,4$ only, and hence
there are no ambiguities with infinite spin support in our two-loop result
(4.11).

Interestingly, as in the one-loop case, these unfixed parameters are all
written as part of the preamplitude $\mathcal{P}^{(3)}$, which is a non-
trivial statement about the corresponding $\overline{D}$-functions. However,
note that the Mellin monomial ‘$1$’ does not appear as an independent
ambiguity in (4.12) since it can not be written as $(\Delta^{(8)})^{2}$ acting
on a preamplitude.272727 This can be easily seen in the positions space
representation: the $\overline{D}$-function corresponding to the Mellin
amplitude ‘$1$’ is $u^{2}\overline{D}_{4444}$ which has denominator power
$(x-\bar{x})^{13}$. This is too small as the minimal denominator power
generated by applying $(\Delta^{(8)})^{2}$ is $(x-\bar{x})^{17}$.  In light of
the vanishing of the $\mathcal{R}^{4}|_{\text{genus-2}}$ contribution (both in
AdS${}_{5}\times$S5 and in the flat-space string amplitude), there is no
super-leading two-loop $\mathcal{R}^{4}$ counterterm. This is consistent with
$u^{2}\overline{D}_{4444}$ not being an independent ambiguity of the two-loop
correlator $\mathcal{H}^{(3)}$ and we speculate that (4.12) indeed gives to
most general form of allowed ambiguities.

Lastly, let us mention how the one-loop ambiguity $\alpha$ enters our result
for $\mathcal{H}^{(3)}$. To this end, we have left it as a free coefficient in
our final expression even though its value is actually fixed.282828 Recall
that the localisation computation of [40] yields $\alpha=60$.  As anticipated,
we find that $\alpha$ is simply proportional to the one-loop string correction
$\mathcal{H}^{(2,3)}$ (with its tree-level ambiguities fixed to some values).
Furthermore, just like the ambiguities (4.12), it turns out that
$\mathcal{H}^{(2,3)}$ can in fact be written as part of the preamplitude, i.e.
$\mathcal{H}^{(2,3)}=\frac{1}{u^{2}}(\Delta^{(8)})^{2}\,\mathcal{P}^{(2,3)}$
for some $\mathcal{P}^{(2,3)}$. This is surprising as $\mathcal{H}^{(2,3)}$ is
really a one-loop amplitude and naively one would expect that only a single
power of $\Delta^{(8)}$ could be pulled out.

###  4.6 Comparison with the results of Huang and Yuan

Let us denote the result obtained in [19] by
$\mathcal{H}^{(3)}_{\text{HY}}$.292929 Here $\mathcal{H}^{(3)}_{\text{HY}}$
refers to the main part of their amplitude which we take from their ancillary
file. Our result for $\mathcal{H}^{(3)}$ agrees with the main part of their
correlator upon setting their free parameter $\mathcal{X}=0$. In particular,
after changing to our conventions, we find that both results agree on the
precise coefficients of the $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$
contributions to $\mathcal{H}^{(3)}$. The exact difference between our two
results reads

$\displaystyle\mathcal{H}^{(3)}_{\text{our}}-\mathcal{H}^{(3)}_{\text{HY}}|_{\mathcal{X}=0}=\tfrac{36}{7}\,\zeta_{3}\mathcal{H}^{(2,3)}-\big{(}\tfrac{5849}{1008}-\tfrac{\alpha}{240}\big{)}\,\mathcal{H}^{(2,3)}+\big{(}\overline{D}\text{-
and }\zeta_{3}\overline{D}\text{-ambiguities}\big{)},$ (4.13)

where the last term stands schematically for the tree-level ambiguities
expected from the $\partial^{10}\mathcal{R}^{4}|_{\text{genus-2}}$ correction,
i.e. terms with Mellin amplitudes
$\\{1,\sigma_{2},\sigma_{3},\sigma_{2}^{2},\sigma_{2}\sigma_{3}\\}$. The above
difference in our respective results is consistent with equation (37) of [19],
since (apart from the usual tree-level ambiguities mentioned before)
additional one-loop ambiguities have been left unfixed.

The condition that the free parameter $\mathcal{X}$ in their correlator
$\mathcal{H}^{(3)}_{\text{HY}}$ is required to vanish in the comparison (4.13)
is explained when considering the corresponding preamplitude
$\mathcal{P}^{(3)}_{HY}$, which we have reconstructed from their expression
for the full correlator. We find that the contribution of $\mathcal{X}$ is
sourced by the presence of new weight 4 functions containing the letter
$x-\bar{x}$ in the preamplitude $\mathcal{P}^{(3)}_{HY}$, which are however
annihilated once $\Delta^{(8)}$ is applied such that their contributions to
the full amplitude $\mathcal{H}^{(3)}_{HY}$ vanish. Since we have not included
such new functions at weight 4 in our preamplitude, we necessarily find that
$\mathcal{X}=0$ and we thereby fix their free parameter.

Note that the vanishing of $\mathcal{X}$ is conjectured in [19], but for a
different reason: the authors state that the condition $\mathcal{X}=0$ follows
from imposing full crossing invariance of the preamplitude
$\mathcal{P}^{(3)}_{HY}$, which at first sight seems to be inconsistent with
the enhancement of crossing symmetry for the particular case of
$(\Delta^{(8)})^{2}$, recall the discussion in Section 4.2. Indeed, we find
that $\mathcal{P}^{(3)}_{HY}$ can be made fully crossing symmetric regardless
of the value of $\mathcal{X}$ by adding terms which are in the kernel of
$(\Delta^{(8)})^{2}$. However, these extra terms come with $\log^{3}(u)$
contributions which would spoil the imposed matching of the leading log at the
level of the preamplitude. In other words, the presence of the free parameter
$\mathcal{X}$ is excluded by imposing full crossing symmetry together with
matching of the leading log on $\mathcal{P}^{(3)}$, i.e. constraints (1) and
(2) of Section 4.4. Of course, this relies on insisting on a ‘canonical’ form
for the leading-log preamplitude as defined by the OPE resummation or,
equivalently, the ten-dimensional conformal symmetry which diagonalises the
tree-level mixing problem.

In summary, as shown in equation (4.13), our correlator $\mathcal{H}^{(3)}$ is
in agreement with the results of [19], up to the following details:

  * •

Due to the absence of weight 4 functions with letter $x-\bar{x}$ in our
minimal ansatz, we determine their free parameter $\mathcal{X}=0$.

  * •

We fix the one-loop ambiguities present in equation (37) of [19] by carefully
tracking the contribution of the one-loop ambiguity $\alpha$ to order $a^{3}$.
This explains the presence of the $\mathcal{H}^{(2,3)}$ terms in the
comparison (4.13).

  * •

Some shifts in the tree-level ambiguities are required in the comparison,
hence the presence of the $\overline{D}$ terms in (4.13). However, note that
since we write the ambiguities as part of the preamplitude, we no longer
consider $u^{2}\overline{D}_{4444}$ as an independent ambiguity in
$\mathcal{H}^{(3)}$.

###  4.7 Comments on triple-trace contributions to $\mathcal{H}^{(3)}$

At some point in the perturbative large $N$ expansion new operators besides
the usual tower of double-trace operators $\mathcal{O}_{t,\ell,i}$ will
contribute. It is expected that the next relevant family of operators is given
by triple-trace operators, and arguments for their presence at two-loop order
have been given by considering unitarity cuts of the corresponding flat-space
amplitudes [62, 45] or also directly at the level of AdS diagrams, see, e.g.
[75].

Here, we present an argument based on consistency of the OPE decomposition. In
particular, we will show that there are contributions from new operators in
the $su(4)$ singlet representation to the $\log^{2}(u)$ term of
$\mathcal{H}^{(3)}$, starting from twist 6. The argument goes as follows:

Let us consider the twist 6 $\log^{2}(u)$ contribution at two-loop order,
which according to the expansion (1.19) is entirely determined in terms of
one-loop and tree-level OPE data. Now, assuming that only double-trace
operators contribute303030 Twist 6 corresponds to $t=3$ and there are two
degenerate double-trace operators $\mathcal{O}_{3,\ell,i}$ with $i=1,2$,
schematically given by (linear combinations of)
$\mathcal{O}_{2}\square\partial^{\ell}\mathcal{O}_{2}$ and
$\mathcal{O}_{3}\partial^{\ell}\mathcal{O}_{3}$, see the discussion above
(1.13). we proceed to resolve the order $a^{2}$ mixing problem, for which
besides $\mathcal{H}^{(2)}$ we also need to use information from the one-loop
$\langle\mathcal{O}_{2}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{3}\rangle$
and
$\langle\mathcal{O}_{3}\mathcal{O}_{3}\mathcal{O}_{3}\mathcal{O}_{3}\rangle$
correlators determined in [44, 15]. The solution of this one-loop mixing
problem yields explicit expressions for $\gamma_{3,\ell,i}^{(2)}$ and
$A^{(1)}_{3,\ell,i}$ for $i=1,2$, which we then use to compute the double-
trace prediction for the twist 6 $\log^{2}(u)$ part at order $a^{3}$. However,
one finds that this does not match the actual result extracted from our result
for $\mathcal{H}^{(3)}$ and we are led to conclude that more operators besides
the two double-trace operators enter the one-loop mixing problem at twist 6.

Based on intuition from large $N$ counting, we presume these additional
operators are given by triple-trace operator which will necessarily mix with
the double-trace ones.313131 Mixing between operators with different trace-
structures has already been observed in the case of so-called single-particle
operators $\mathcal{O}_{p}$ [16, 23], which are defined as the $\mathcal{N}=4$
half-BPS operators dual to single-particle states of AdS${}_{5}\times$S5
supergravity. The first case where half-BPS single-, double- and triple-trace
operators mix is $\mathcal{O}_{6}$, which in the $SU(N)$ theory is of the form
(see equation (15) of [23])
$\displaystyle\mathcal{O}_{6}=T_{6}+\beta_{1}\,T_{3,3}+\beta_{2}\,T_{4,2}+\beta_{3}\,T_{2,2,2},$
(4.14) where $T_{p_{1},p_{2},\ldots,p_{n}}(x)\equiv
T_{p_{1}}(x)\,T_{p_{2}}(x)\cdots\,T_{p_{n}}(x)$, with
$T_{p}(x)\equiv\text{Tr}(\Phi(x)^{p})$ being the usual dimension $p$ single-
trace operator. The coefficients $\beta_{i}$ are functions of $N$ with large
$N$ behaviours $\beta_{1,2}\sim\frac{1}{N}$ and
$\beta_{3}\sim\frac{1}{N^{2}}$, leading to the pattern that terms with $m$
traces are $1/N$ suppressed with respect to $(m-1)$-trace terms. In analogy,
we expect the triple-trace admixtures to the exchanged, unprotected double-
trace operators to be $1/N$ suppressed.  As a consequence, information about
their leading-order anomalous dimension is already contained within the four-
point correlators at one-loop order $a^{2}$.

Lastly, let us mention that the mismatch between the double-trace prediction
for the twist 6 $\log^{2}(u)$ part and the actual expression extracted from
$\mathcal{H}^{(3)}$ is already visible in the structure of the contributing
transcendental functions: the twist 6 contribution to
$\mathcal{H}^{(3)}|_{\log^{2}(u)}$ contains terms with up to $\log^{3}(v)$
divergences, whereas the twist 4 contribution due to only double-trace
exchanges is found to have at most $\log^{2}(v)$ terms, which can be verified
using the expression given in (B.4). This is consistent with the observation
that sums over the double-trace spectrum typically lead to at most
$\log^{2}(v)$ divergences (for example the leading log to all loop orders, see
also the discussion in [45]). We thus interpret the observed $\log^{3}(v)$
divergences in the two-loop $\log^{2}(u)$ part at twist 6 as a sign of triple-
trace contributions.

##  5 Extracting the two-loop anomalous dimension

With the explicit result for $\mathcal{H}^{(3)}$ at hand, we can proceed to
extract new CFT data from it. In particular, the quantity of physical interest
is the correction to the dimension of the twist 4 double-trace operators
$\mathcal{O}_{2,\ell}\sim\mathcal{O}_{2}\partial^{\ell}\mathcal{O}_{2}$.
Recall that at higher twists mixing between exchanged operators occurs, with
twist 4 being the only case where there exists only one unique operator for
each spin. This fact allows us to unambiguously determine its two-loop
anomalous dimension $\gamma^{(3)}$, which is encoded in the $\log(u)$-part of
the two-loop correlator.

In practice, extracting the two-loop anomalous dimension from
$\mathcal{H}^{(3)}|_{\log(u)}$ for low, finite spins $\ell$ is
straightforward: after subtracting the derivative terms specified by the OPE
expansion, see equation (1.19), and performing the block decomposition up to
some finite cut-off $\ell_{\max}$ (keeping twist 4 contributions only), one
ends up with the particular combination of CFT data
$(A^{(2)}_{2,\ell}\gamma^{(1)}_{2,\ell}+A^{(1)}_{2,\ell}\gamma^{(2)}_{2,\ell}+A^{(0)}_{2,\ell}\gamma^{(3)}_{2,\ell})$
for spins $\ell=0,2,\ldots,\ell_{\max}$. Since all quantities except
$\gamma^{(3)}$ in that combination are known from lower-order calculations,
one can thus solve for $\gamma^{(3)}$ spin by spin.

However, obtaining a closed-form expression directly from the above finite
spin data is more difficult. In order to find $\gamma^{(3)}$ as an analytic
function of spin, we employ the Lorentzian inversion formula of [63], whose
input is the so-called double-discontinuity $\text{dDisc}(\mathcal{H}^{(3)})$.
In this formalism, the exact same combination of CFT data as given above is
encoded in the double-poles of the inversion integral. Computing these
integrals323232 We made use of the method described in Appendix C of [27] to
recursively simplify the integrals. See also Appendix A of reference [76] for
a useful list of results for some inversion integrals. and solving for the
two-loop anomalous dimension, we obtain $\gamma^{(3)}_{2,\ell}$ as an analytic
function of $\ell$, which agrees with the finite spin data obtained from the
direct OPE expansion described above. For spins $\ell\geq 6$, the two-loop
anomalous dimension takes the form

$\displaystyle\gamma^{(3)}_{2,\ell}$
$\displaystyle=c_{3}\,\big{(}S_{-3}-S_{3}-2S_{1,-2}+3\zeta_{3}\big{)}+c_{2}\,S_{-2}+c_{1}\,S_{1}+c_{0}+c_{0}^{(a)}+\alpha\,\tilde{\gamma}^{(2,3)}_{2,\ell},$
(5.1)

where the $c_{i}$ are rational functions of spin and $S_{\vec{a}}\equiv
S_{\vec{a}}(\ell+3)$ are nested harmonic sums. For integer arguments and
weight-vectors $\vec{a}$ containing non-zero integer entries, they are defined
recursively by

$\displaystyle
S_{a_{1},a_{2},\ldots,a_{n}}(m)=\sum_{k=1}^{m}\frac{(\text{sgn}(a_{1}))^{k}}{k^{|a_{1}|}}S_{a_{2},\ldots,a_{n}}(k),\qquad
S_{\emptyset}(m)=1.$ (5.2)

Note that the coefficient functions $c_{i}$ have definite sign under the
symmetry $\ell\rightarrow-\ell-7$: apart from $c_{0}^{(a)}$ which is
antisymmetric, all other $c_{i}$ are symmetric and can therefore be expressed
in terms of even powers of the Casimir eigenvalue $J^{2}$, which at twist 4 is
given by $J^{2}=(\ell+3)(\ell+4)$. We will elaborate further on this so-called
reciprocity principle later on. The symmetric coefficient functions read

$\displaystyle\begin{split}c_{3}&=\frac{-221184J^{2}(J^{2}-2)(J^{8}-50J^{6}-653592J^{4}+30292416J^{2}+15169835520)}{5(J^{2}-6)^{2}(J^{2}-12)(J^{2}-20)(J^{2}-30)(J^{2}-42)(J^{2}-56)(J^{2}-72)},\\\\[3.0pt]
c_{2}&=\frac{-18432\,q_{2}(J^{2})}{(J^{2}-6)^{2}(J^{2}-12)^{2}(J^{2}-20)(J^{2}-30)^{2}(J^{2}-42)(J^{2}-56)^{2}(J^{2}-72)(J^{2}-90)(J^{2}-132)},\\\\[3.0pt]
c_{1}&=\frac{-27648J^{2}(J^{2}-2)(J^{8}+525J^{6}+1730258J^{4}-79817784J^{2}-39925126080)}{(J^{2}-6)^{2}(J^{2}-12)(J^{2}-20)(J^{2}-30)(J^{2}-42)(J^{2}-56)(J^{2}-72)},\\\\[3.0pt]
c_{0}&=\frac{384\,q_{0}(J^{2})}{5J^{2}(J^{2}-6)^{5}(J^{2}-12)^{2}(J^{2}-20)^{2}(J^{2}-30)^{2}(J^{2}-42)(J^{2}-56)^{2}(J^{2}-72)(J^{2}-90)(J^{2}-132)},\end{split}$
(5.3)

where $q_{2}(J^{2})$ and $q_{0}(J^{2})$ are non-factorisable polynomials given
by

$\displaystyle\begin{split}q_{2}(J^{2})&=J^{2}\,\big{(}23J^{20}-3252J^{18}-11511408J^{16}+3632264384J^{14}\\\
&\leavevmode\nobreak\
\qquad-187756129296J^{12}-46028945140416J^{10}+6505487171987328J^{8}\\\
&\leavevmode\nobreak\
\qquad-303456834615886848J^{6}+5448093169711196160J^{4}\\\
&\leavevmode\nobreak\
\qquad-28725812248908349440J^{2}+64442560728268800\big{)},\end{split}$ (5.4)

and

$\displaystyle\begin{split}q_{0}(J^{2})&=\big{(}3363J^{32}-1629859J^{30}+1808457782J^{28}-530500662732J^{26}\\\
&\quad+34917254916536J^{24}+5640280310229488J^{22}-1020874675751115744J^{20}\\\
&\quad+64205146187309426112J^{18}-1998754919048666890368J^{16}\\\
&\quad+32877932476802852450304J^{14}-274537156441056453513216J^{12}\\\
&\quad+863756354962716443394048J^{10}+2400097076662032032956416J^{8}\\\
&\quad-23573917734820546679930880J^{6}+46280241034580622311424000J^{4}\\\
&\quad-4409814873370414546944000J^{2}-66814046963069091840000\big{)}.\end{split}$
(5.5)

The contribution from the one-loop regulator $\alpha$ is, as expected,
proportional to the anomalous dimension induced by the one-loop string
correction $\mathcal{H}^{(2,3)}$ . We find
$\tilde{\gamma}^{(2,3)}_{2,\ell}=\frac{1}{480\zeta_{3}}\gamma^{(2,3)}_{2,\ell}$,
with $\gamma^{(2,3)}_{2,\ell}$ given by [29],

$\displaystyle\gamma^{(2,3)}_{2,\ell}=-\frac{1658880\zeta_{3}\,J^{2}(J^{2}-2)(J^{2}+4)(J^{2}+42)}{(J^{2}-6)(J^{2}-12)(J^{2}-20)(J^{2}-30)(J^{2}-42)(J^{2}-56)},\quad\text{for
}\ell\geq 6.$ (5.6)

Lastly, the antisymmetric contribution $c_{0}^{(a)}$ to the two-loop anomalous
dimension takes the simple form

$\displaystyle
c_{0}^{(a)}=\frac{193536(2\ell+7)(J^{4}-160J^{2}+2380)}{(J^{2}-6)^{4}(J^{2}-20)^{2}}.$
(5.7)

In the remainder of this section, we comment on some interesting features of
$\gamma^{(3)}$: we first point out some subtleties of our formula (5.1) at low
values of the spin, before discussing the reciprocity principle and the large
spin limit.

###  5.1 Comments on analyticity in spin of $\gamma^{(3)}$

While the two-loop anomalous dimension is manifestly finite for spins
$\ell\geq 10$, there are explicit poles for some low values of the spin
$\ell$. In particular, the coefficient functions $c_{2}$ and $c_{0}$ contain
the factors $(J^{2}-90)$ and $(J^{2}-132)$ in their denominators, giving rise
to (apparent) poles for spins $\ell=6$ and $\ell=8$, respectively. However, it
turns out that the numerator of the combination $c_{2}\,S_{-2}+c_{0}$ has a
zero at exactly those locations, thus cancelling the apparent divergences! Let
us emphasise that this cancellation is highly non-trivial, and it depends on
the precise spin dependence of the coefficients $c_{2}$ and $c_{0}$. Note that
there are no such spurious poles at higher values of the spin and the formula
(5.1) is manifestly finite for $\ell\geq 10$.

One can evaluate $\gamma^{(3)}$ for spins $\ell=6,8$ by using an analytic
continuation of the alternating harmonic sum $S_{-2}(\ell+3)$,333333 Taking
the analytic continuation of $S_{-2}(m)$ from odd arguments $m$, see e.g
reference [77], one has  $\displaystyle
S_{-2}(z)=\frac{1}{4}\Big{(}\psi^{(1)}\big{(}\tfrac{z+2}{2}\big{)}-\psi^{(1)}\big{(}\tfrac{z+1}{2}\big{)}\Big{)}-\frac{\zeta_{2}}{2}.$
(5.8) yielding the finite values

$\displaystyle\begin{split}\gamma^{(3)}_{2,\ell=6}&=-\frac{775569722924539}{79394167125}+\frac{9368130816\zeta(3)}{1028755}-\frac{409464\alpha}{54145},\\\\[3.0pt]
\gamma^{(3)}_{2,\ell=8}&=\frac{147901898176964147}{170666923100625}-\frac{1161059328\zeta(3)}{1820105}-\frac{16588\alpha}{13965}.\end{split}$
(5.9)

We find that these values, obtained through an analytic continuation of
harmonic sums, are in perfect agreement with the explicit low spin data which
we extracted from the OPE expansion of the $\log(u)$ part of
$\mathcal{H}^{(3)}$.

Another interesting feature of the formula for $\gamma^{(3)}$ is the presence
of the factor $(J^{2}-72)$ in the denominator of all 4 coefficient functions
$c_{i}$, see equation (5.3). This factor provides a pole at $\ell=5$ which is
not cancelled any more. Even though this pole is at an unphysical value of the
spin (only operators with even spins are exchanged in the OPE), it prevents us
from analytically continuing below spin 6.343434 Exactly the same phenomenon
occurs in the one-loop anomalous dimension given in equation (B.3):
$\gamma^{(2)}$ has a pole at $\ell=1$, signalling the presence of a non-
analytic spin 0 contribution from the one-loop ambiguity $\alpha$. This is
consistent with the presence of the tree-level ambiguities $d_{j},e_{j}$ given
in equation (4.12), which are left unfixed by our bootstrap program: being
related to the tree-level $\partial^{10}\mathcal{R}^{4}$ correction, they
contribute non-analytically to the two-loop anomalous dimension precisely for
finite spin values $\ell=0,2,4$, and $\gamma^{(3)}$ is expected to be analytic
only for $\ell\geq 6$.

###  5.2 $\gamma^{(3)}$ and the reciprocity principle

For conformal field theories, the reciprocity principle is the general
statement that anomalous dimensions of operators are functions of the
conformal spin [78]. This is equivalent to the statement that the large spin
expansion of operator dimensions contains only even inverse powers of the bare
Casimir $J^{2}$ [79]. After taking into account some shifts due to
$\mathcal{N}=4$ supersymmetry, the relevant expression for the Casimir
eigenvalue takes the form $J^{2}=(t+\ell+1)(t+\ell+2)$. When the large spin
expansion can be resummed, which is the case in our setting (up to some finite
spin ambiguities), this implies a discrete $\mathbb{Z}_{2}$ symmetry of the
operator spectrum under the map $\ell\mapsto-\ell-\tau(\ell,a)-3$, with $\tau$
being the full anomalous twist of the relevant operator.353535 As pointed out
in [44], this symmetry should be really thought of as a symmetry of the full
operator spectrum rather than individual operators, as in the generic case the
shift symmetry transforms many objects non-trivially. For example, the tree-
level anomalous dimensions $\gamma^{(1)}_{t,\ell,i}$ of the singlet channel
double-trace operators, recorded in equation (1.16), are mapped into each
other according to
$\displaystyle\gamma^{(1)}_{t,\ell,i}\mapsto\gamma^{(1)}_{t,-\ell-2t-3,i}=\gamma^{(1)}_{t,\ell,t-i}.$
(5.10) In this case, the shift symmetry maps data from the family of operators
with degeneracy label $i$ onto the family with label $i^{\prime}=t-i$.

At twist 4, there is a unique operator for each spin, and the above
transformation should leave the dimension $\Delta_{2,\ell}$ of the double-
trace operators
$\mathcal{O}_{2,\ell}\sim\mathcal{O}_{2}\partial^{\ell}\mathcal{O}_{2}$
invariant. Plugging-in the large $N$ expansion of their full anomalous twist
$\tau(\ell,a)=2(2+a\gamma^{(1)}+a^{2}\gamma^{(2)}+\ldots)$ and expanding to
order $a^{3}$, one finds the relations

$\displaystyle\begin{split}\tfrac{1}{2}\big{(}\gamma^{(1)}_{2,\ell}-\gamma^{(1)}_{2,-\ell-7}\big{)}&=0\,,\\\\[3.0pt]
\tfrac{1}{2}\big{(}\gamma^{(2)}_{2,\ell}-\gamma^{(2)}_{2,-\ell-7}\big{)}&=\gamma^{(1)}_{2,\ell}\partial_{\ell}\gamma^{(1)}_{2,-\ell-7}\,,\\\\[3.0pt]
\tfrac{1}{2}\big{(}\gamma^{(3)}_{2,\ell}-\gamma^{(3)}_{2,-\ell-7}\big{)}&=\gamma^{(1)}_{2,\ell}\,\partial_{\ell}\gamma^{(2)}_{2,-\ell-7}+\gamma^{(2)}_{2,\ell}\,\partial_{\ell}\gamma^{(1)}_{2,-\ell-7}+(\gamma^{(1)}_{2,\ell})^{2}\,\partial_{\ell}^{2}\gamma^{(1)}_{2,-\ell-7}\,.\end{split}$
(5.11)

In words, the antisymmetric contribution to the anomalous dimension at a given
order is entirely determined by lower-order data. One can easily verify that
the first two lines are satisfied by the tree-level and one-loop anomalous
dimensions $\gamma^{(1)}$ and $\gamma^{(2)}$, respectively.

Remarkably, our result for the two-loop anomalous dimension is in perfect
agreement with the prediction from the reciprocity symmetry: the only
antisymmetric contribution to $\gamma^{(3)}$ is given by
$c_{0}^{(a)}=\tfrac{1}{2}\big{(}\gamma^{(3)}_{2,\ell}-\gamma^{(3)}_{2,-\ell-7}\big{)}$
as recorded in equation (5.7), and indeed we find that $c_{0}^{(a)}$ satisfies
the last line of (5.11).

After subtracting the antisymmetric contribution $c_{0}^{(a)}$, the large spin
expansion of the two-loop anomalous dimension is of the form expected from
reciprocity symmetry [79],

$\displaystyle\gamma_{2,\ell}^{(3)}-c_{0}^{(a)}\approx\frac{\beta_{2}(\log(J))}{J^{4}}+\frac{\beta_{3}(\log(J))}{J^{6}}+\frac{\beta_{4}(\log(J))}{J^{8}}+\ldots,$
(5.12)

with only even inverse powers of $J$ appearing in the expansion. The leading
coefficient $\beta_{2}$ is given by

$\displaystyle\beta_{2}(\log(J))=\frac{384}{5}\big{(}3363-45\alpha+460\pi^{2}-864\zeta_{3}-24(15+4\pi^{2})(\log(J)+\gamma_{E})\big{)},$
(5.13)

with $\gamma_{E}$ being the Euler-Mascheroni constant.

##  6 An exploration of wider ansätze

There are several different possibilities of extending the minimal ansatz
(4.6), while still preserving the general structure with powers of
$\Delta^{(8)}$ acting on simpler preamplitudes which we think is well
motivated by properties of the leading log. In this section, we consider two
natural extensions of the minimal ansatz:

  1. 1.

We address the possibility of including new weight 4 functions with letter
$x-\bar{x}$ in the preamplitude $\mathcal{P}^{(3)}$. Previously, we argued
that the presence of such a function at weight 3 is required in order to
account for the spin 0 non-analyticity of the $\log^{2}(u)$ twist 4
prediction, and indeed we find the function $f^{(3)}$ to be an essential part
of our result for $\mathcal{H}^{(3)}$. Here, we explore the additional degrees
of freedom due to the presence of such new functions at weight 4, which à
priori are not necessary to match our bootstrap constraints.

  2. 2.

The minimal ansatz (4.6) is of particular simplicity in the sense that the
one-loop correction term is exactly given by the one-loop result
$\mathcal{H}^{(2)}$ itself. An alternative would be to generalise this term to
$\Delta^{(8)}$ acting on a more general one-loop like preamplitude different
from $\mathcal{L}^{(2)}$ in (4.2). We explore such an extension in Section
6.2.

As we describe in the following, these modifications of the minimal ansatz
lead to more free parameters in the final answer, which are no longer of the
form of tree-level ambiguities only. In general, such additional free
parameters will contribute to the two-loop anomalous dimension $\gamma^{(3)}$
with infinite spin support, although at most the coefficient functions $c_{2}$
and $c_{0}$ in (5.1) seem to be affected. In light of that it is all the more
remarkable that the minimal ansatz gets fully fixed up to the expected tree-
level ambiguities.

###  6.1 Including weight 4 functions with letter $x-\bar{x}$

At weight 4, there are 3 new functions when including the additional letter
$x-\bar{x}$. Two of them are simply given by $\log(u)f^{(3)}(x,\bar{x})$ and
$\log(v)f^{(3)}(x,\bar{x})$, while the third one, $f^{(4)}(x,\bar{x})$, is the
unique such function which is fully crossing antisymmetric, just like
$f^{(3)}$ at weight 3.

Including these 3 functions as additional basis elements, we now start with a
new preamplitude $\mathcal{P}^{\prime(3)}$ built from an extended
transcendental basis $\mathcal{Q}^{\prime}$ containing 76 elements. This new,
wider ansatz is again of the form

$\displaystyle\mathcal{H}^{\prime(3)}=\frac{1}{u^{2}}(\Delta^{(8)})^{2}\,\mathcal{P}^{\prime(3)}+a_{2}\mathcal{H}^{(2)}+a_{1}\mathcal{H}^{(1)}.$
(6.1)

We then proceed to impose the same bootstrap constraints described in Section
4.4. After imposing conditions (1) – (3), the new weight 4 functions give rise
to 15 additional free parameters. As before, in the process of imposing the
further constraints (4) – (6) the coefficients of the tree-level and one-loop
correction terms get fixed to $a_{1}=-1$ and $a_{2}=5$. However, we now end up
with 12 free parameters in the final result for $\mathcal{H}^{\prime(3)}$,
compared to only 8 ambiguities when starting from the minimal ansatz.

We find that the four additional free parameters contribute to all
antisymmetric weight 4 functions together with their lower weight
completions.363636 However, note that none of these parameters is related to
free coefficient $\mathcal{X}$ of reference [19]. As explained in Section 4.6,
starting from a fully crossing symmetric preamplitude which matches the
canonical form of the leading log as given by the OPE resummation excludes the
function $\mathcal{X}$.  In the OPE decomposition, they contribute with finite
spin support to the $\log^{2}(u)$-part for twists $\tau\geq 6$, whereas their
contributions to the $\log(u)$-part have infinite spin support starting from
twist 4. As a consequence, they contribute non-trivially to the two-loop
anomalous dimension, but we find their contributions are limited to the
coefficient function $c_{0}$, preserving analyticity in spin for $\ell\geq 6$.

###  6.2 Generalising the one-loop term

As another natural extension we consider a generalisation of the one-loop
correlator $\mathcal{H}^{(2)}$. The minimal ansatz is thus modified by
including a wider one-loop like correction term and takes the more general
form

$\displaystyle\mathcal{H}^{\prime\prime(3)}=\frac{1}{u^{2}}(\Delta^{(8)})^{2}\,\mathcal{P}^{(3)}+\frac{1}{u^{2}}\Delta^{(8)}\,\mathcal{P}^{(2)}+a_{1}\mathcal{H}^{(1)},$
(6.2)

where $\mathcal{P}^{(3)}$ is the original two-loop preamplitude from (4.7),
while the preamplitude $\mathcal{P}^{(2)}$ is analogous to (4.7) with
different coefficients $b^{(i)}_{n,m}$ which we set to zero for all basis
elements $\mathcal{Q}_{i}$ with transcendental weight greater than 4 or
contributions to $\log^{3}(u)$ in any orientation, in order not to spoil the
matching of the leading log by the first term.

We then impose the bootstrap constraints (1) – (3) on $\mathcal{P}^{(3)}$ and
$\mathcal{P}^{(2)}$ separately, with the modification that $\mathcal{P}^{(2)}$
is less constrained by crossing symmetry, obeying only

$\displaystyle\mathcal{P}^{(2)}(x,\bar{x})=\mathcal{P}^{(2)}(x^{\prime},\bar{x}^{\prime}),$
(6.3)

since this is the only symmetry respected by a single power of $\Delta^{(8)}$.
The other symmetry is then imposed in a second step on the full combination
$\frac{1}{u^{2}}\Delta^{(8)}\mathcal{P}^{(2)}$.

Proceeding to impose the constraints (4) – (6), we find that the coefficient
of the tree-level contribution is fixed to $a_{1}=4$ and we have

$\displaystyle\mathcal{H}^{\prime\prime(3)}=\frac{1}{u^{2}}(\Delta^{(8)})^{2}\,\mathcal{P}^{(3)}+\frac{1}{u^{2}}\Delta^{(8)}\,\mathcal{P}^{(2)}+4\mathcal{H}^{(1)},$
(6.4)

with $\mathcal{H}^{\prime\prime(3)}$ containing a total of 11 unfixed
parameters, 8 of which contribute only to $\mathcal{P}^{(3)}$ and these are
the expected tree-level ambiguities described in Section 4.5. One of the
additional parameters contributes only to $\mathcal{P}^{(2)}$ and it is
proportional to the tree-level ambiguity $u^{2}\overline{D}_{4444}$, which
contributes to the two-loop anomalous dimension only at spin $0$. The other 2
remaining parameters appear in both $\mathcal{P}^{(2)}$ and
$\mathcal{P}^{(3)}$: one of them, $\delta_{1}$, contributes non-trivially up
to weight 6 (with contributions to the functions $A^{(6)}$ and
$\zeta_{3}f^{(3)}$ at top-weight) and has infinite spin support in the
$\log^{2}(u)$-part for twists $\tau\geq 6$. The other parameter, $\delta_{2}$,
is proportional to a one-loop string correction with finite spin support in
the $\log^{2}(u)$-part. In the two-loop anomalous dimension they both
contribute with infinite spin support: $\delta_{1}$ contributes to $c_{2}$ and
$c_{0}$, while $\delta_{2}$ contributes only to $c_{0}$. Again, we find that
both contributions preserve analyticity in spin for $\ell\geq 6$.

Lastly, when combining the two types of generalisations discussed above, i.e.
starting from the extended ansatz (6.2) while at the same time allowing for
weight 4 functions with letter $x-\bar{x}$ (in both $\mathcal{P}^{(3)}$ and
$\mathcal{P}^{(2)}$), we find 9 additional undetermined coefficients compared
to the minimal ansatz. These comprise the 4 parameters described in Section
6.1, the 3 from the generalisation (6.2) as well as 2 more independent
parameters: one of them contributes up to weight 6 and has similar properties
as the parameter $\delta_{1}$ described above, while the second one is due to
the fact that the coefficient $a_{1}$ is no longer fixed.

##  7 Outlook and open questions

We conclude with mentioning some open questions and future directions:

  * •

Recall that the simple structure of the minimal ansatz (4.6) for the two-loop
correlator $\mathcal{H}^{(3)}$ was motivated by two facts, namely $(i)$ the
non-trivial property of the leading discontinuity allowing
$(\Delta^{(8)})^{2}$ to be pulled out and $(ii)$ the observation of reference
[15] that the one-loop correlator (originally constructed without assuming a
structure involving $\Delta^{(8)}$) can actually be written in the simple form
given in (4.2). However, beyond the remarkable fact that this very restrictive
minimal ansatz satisfies all of the bootstrap constraints and leads to a final
result with the expected tree-level ambiguities only, the origin of this
structural simplicity remains unclear to us. While the existence of the
operator $\Delta^{(8)}$ and its properties in relation to the leading log are
directly related to the hidden conformal symmetry, the structures we observe
at loop order seem to go beyond the original understanding of this symmetry.
It would be very interesting to investigate whether analogous features, such
as the appearance of lower order correlators for example, are present in
similar setups.373737 One natural candidate to consider would be the case of
the pure AdS5 background arising from gauged $\mathcal{N}=8$ supergravity.

  * •

We find it suggestive that a similar structure might persist at higher loops.
As already stated in [19], the generalisation of the minimal ansatz to any
loop order takes the simple form

$\displaystyle\mathcal{H}^{(n)}=\frac{1}{u^{2}}(\Delta^{(8)})^{n-1}\,\mathcal{P}^{(n)}+\sum_{i=1}^{n-1}a_{i}\,\mathcal{H}^{(i)}\,,$
(7.1)

such that at any order one is only left with determining the preamplitude
$\mathcal{P}^{(n)}$, which includes functions up to transcendental weight
$w=2n$, and the coefficients $a_{i}$ of the lower-loop results. Crucially, the
only growth of complexity in $\mathcal{P}^{(n)}$ at higher loop order is due
to the increasing size of the transcendental basis, while the number of free