We now solve the remaining integrals numerically. Before doing so, we gather
the analytical terms and find that they simplify dramatically to give

$\begin{split}F_{0}^{\text{a}}&:=F_{0}^{\text{SE}}+F_{0}^{XX}+F_{0}^{XH}+F_{0}^{\text{Spider}}+F_{0}^{XY,\text{a}}\\\
&\phantom{:}=-\frac{\lambda^{4}}{128\pi^{2}I_{13}}\biggl{\\{}H_{12,13}+H_{13,23}+2H_{12,23}\\\
&\phantom{:=}-\frac{Y_{123}}{32\pi^{2}}\left(\log\chi^{2}(1-\chi)^{2}-8\right)\biggr{\\}}\,.\end{split}$
(4.107)

We note in particular that all the divergent terms cancel each other. The
absence of $\log\tau_{4}^{2}$ is an encouraging sign that the final correlator
is conformal as expected.

##### Numerical integration

We perform a numerical integration of the sum of the remaining terms. An
efficient way to obtain the final result is to write an ansatz of
transcendental functions based on the form of the other terms. Such a method
was used successfully for instance to determine (4.20) in [350]. It is also
similar to the ansatz made in [10] to obtain (4.13) and (4.14).

A suitable ansatz is

$F_{0}^{XY,\text{n}}=\frac{\lambda^{4}}{\chi(1-\chi)}\left(\sum_{\vec{a}}\alpha_{\vec{a}}H_{\vec{a}}+\chi\sum_{\vec{a}}\beta_{\vec{a}}H_{\vec{a}}\right)\,,$
(4.108)

where we allow the sum to run up to transcendentality $3$. The different OPE
limits constrain $\vec{a}$ to contain only combinations of $0$ and $1$.

Figure 4.5: The orange dots in this plot show the function
$F^{XY,\text{n}}_{0}(\chi)$, defined as the sum of terms in the $XY$-diagrams
that could not be solved analytically. The blue line corresponds to the
analytical expression (4.109), obtained from comparing the ansatz (4.108) to
the numerical points. We observe a perfect agreement between the numerical and
analytical results.

It turns out that the coefficients $\alpha$ and $\beta$ are simple numbers. We
find the following solution:

$\displaystyle F_{0}^{XY,\text{n}}=\ $
$\displaystyle\frac{\lambda^{4}}{24576\pi^{8}}\frac{1}{\chi(1-\chi)}\bigl{(}2(6-\pi^{2})H_{1}+3(H_{0,1}+H_{1,0}-2H_{1,1})$
$\displaystyle+3(3H_{0,0,1}-H_{0,1,0}-2H_{1,0,0})+\chi\bigl{(}2(\pi^{2}-6)H_{0}+2(\pi^{2}-3)H_{1})$
$\displaystyle+6(H_{0,0}-H_{1,1})+3(3(H_{0,0,1}+H_{1,1,0})-(H_{0,1,0}+H_{1,0,1})$
$\displaystyle-2(H_{0,1,1}+H_{1,0,0}))-9\zeta_{3}\bigr{)}\bigr{)}\,.$ (4.109)

The comparison between numerical data and this analytical expression is
displayed in Figure 4.5. We observe a very convincing agreement between the
function (4.109) and the numerical integration of the $XY$-diagrams.

##### Final result

We now put the pieces together to write down the full correlator at NNLO.
Rewriting (4.107) in terms of HPLs and adding (4.109), we find the simple and
elegant result

$\displaystyle F_{0}^{(2)}=\ $
$\displaystyle\frac{1}{192\pi^{4}\chi(1-\chi)}\biggl{\\{}\pi^{2}H_{1}-3(H_{1,0,1}+H_{1,1,0}-2(H_{0,1,0}+H_{0,1,1}-H_{1,0,0}))$
$\displaystyle-3\chi\biggl{(}\frac{\pi^{2}}{3}H_{0}-(H_{0,0,1}+H_{1,1,0})+H_{0,1,0}+H_{1,0,1}+3\zeta_{3}\biggr{)}\biggr{\\}}\,.$
(4.110)

Note that here we have already unit-normalized the correlator. Solving (4.89)
with (4.110) yields the result (4.20), providing a check for the
bootstrability techniques developed in [350].

This concludes our exploration of multipoint correlators in the weak coupling
regime. In Section 4.5, we sketch possible future directions related to the
techniques used throughout this section.

###  4.3 Superconformal Ward identities

The superconformal Ward identity (4.2) plays a central role in the study of
four-point functions, both in the weak- and strong-coupling regimes. A natural
question arises: do multipoint correlators of half-BPS operators also satisfy
similar powerful identities? If so, what are their specific forms? In this
section, we put forth a conjecture that such identities exist, by proposing a
set of differential equations satisfied by our pool of perturbative results.
The resulting multipoint Ward identities take a similar form to (4.2) and pave
the way to a strong-coupling analysis of higher-point functions. As a
practical example, we show how these Ward identities can be solved in the case
of the five-point function
$\langle\,\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{2}\,\rangle$.

The results presented here are based on [2, 7].

####  4.3.1 Multipoint Ward identities

Using the recursion relations (4.47), (4.51) and (4.53), we have computed
correlators featuring up to eight external (half-BPS) operators. Concrete
examples of such correlators can be found in the supplementary Mathematica
notebook of [4]. They can be used here for postulating differential equations
satisfied by multipoint correlators.

##### The conjecture and checks

We found experimentally that all our correlators are annihilated by the
following family of differential operators:

$\sum_{k=1}^{n-3}\left(\frac{1}{2}\partial_{\chi_{k}}+\alpha_{k}\partial_{r_{k}}-(1-\alpha_{k})\partial_{s_{k}}\right)\,{\mathcal{F}}(\\{\chi;r,s,t\\})\raisebox{-8.61108pt}{$\Biggr{|}$}_{\raisebox{4.52083pt}{$\begin{subarray}{l}r_{i}\to\alpha_{i}\chi_{i}\\\
s_{i}\to(1-\alpha_{i})(1-\chi_{i})\\\
t_{ij}\to\alpha_{ij}\chi_{ij}\end{subarray}$}}=0\,,$ (4.111)

where $\alpha_{k}$ are arbitrary real numbers as in (4.2). These operators
represent a natural extension of the differential operator (4.15) that
captures the constraints of superconformal symmetry for four-point functions.

Even though we obtained these equations from weak-coupling correlation
functions, we expect these identities to also be satisfied in the strong-
coupling regime. As discussed in Section 2.2.1, this expansion is captured by
a well-understood AdS dual [103], which has been used to calculate planar
correlators in the $\lambda\gg 1$ limit. The leading order correlators are
given by simple Wick contractions of the fluctuations of the dual fundamental
string, which means that they correspond to the generalized free-field
expressions, see for instance (4.5) in [103]. It is easy to check that all
$n$-point correlators also satisfy (4.111) in the extreme strong-coupling
regime. Moreover, it was independently checked through Witten-diagram
computations in [368] that the next-to-leading order correlators also satisfy
our conjectured multipoint Ward identities.

We have therefore four non-trivial data points: the first two orders, both in
the weak- and strong-coupling regimes. It is then reasonable to assume that
the constraint (4.111) is non-perturbative and valid at all loop orders.

##### Comments and a word of caution

Since the superconformal Ward identities (4.111) are based on symmetry, they
are expected to have powerful consequences, some of which we discuss here. In
our study of planar correlators, their main application is to identify the
degrees of freedom of the correlator, as it was done in (4.14) for the four-
point function. But they are also handy for deriving superconformal blocks. In
[285], it was shown that one can bypass solving superconformal Casimir
equations by applying the Ward identity on an ansatz for the superblocks. In
Section 4.4, we apply this method for deriving the superblocks of the five-
point function
$\langle\,\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{2}\,\rangle$.

Moreover, note that these identities go beyond the planar limit.
Superconformal constraints are insensitive to gauge symmetry, which means that
they should also hold for non-planar corrections. This is indeed the case for
the first correction in the $N$ expansion at the leading order, and for
certain cases, we have checked that it also holds at the next-to-leading
order.

We conclude this section with a word of caution about (4.111). We should point
out that our conjecture cannot represent the full set of superconformal
constraints on the correlators. The reason is that our analysis of protected
operators only focuses on the highest-weight component while ignoring possible
fermionic descendants. Working in a suitable superspace, it is known that for
four-point functions, the full superconformal correlator can be reconstructed
from the highest weight. It is thus safe to set the fermions to zero in this
case. But starting with five-point and up, one expects nilpotent
invariants.131313We thank P. Heslop for discussions on this topic. In general,
for generic $n$-point functions, the Ward identities should be a collection of
partial differential equations relating the components associated with each
fermionic structure. In this light, the fact that we obtained a differential
operator that only acts on the highest weight and still annihilates the
correlator is unexpected. Moreover, the number of constraints seems to grow
unbounded with the number of external operators. This means that for a certain
$n$, the number of constraints would exceed the number of symmetries.141414We
are grateful to N. Gromov for insightful comments. At this point, the fate of
the Ward identities for a high number of external points is unclear, and it
would be desirable to conduct a proper superspace analysis (in the gist of
[285]) to prove that our experimental observation indeed corresponds to one of
the constraints imposed by superconformal invariance.

####  4.3.2 Solving the Ward identities

We now discuss how the solution (4.14) of the four-point Ward identity extends
to higher-point functions. We do not provide a general analysis, but instead
focus on highlighting the similarities between the four-point and higher-point
cases. As a practical example, we provide the solution of the Ward identities
(4.111) for the five-point function
$\langle\,\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{2}\,\rangle$.
This solution is then used for expressing the weak-coupling result compactly.
The results of this section will be crucial for the strong-coupling analysis
of Section 4.4.

##### An example:
$\langle\,\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{2}\,\rangle$

The five-point function
$\langle\,\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{2}\,\rangle$
is a correlator that we already considered at weak coupling, and we keep the
conventions defined in (4.79)-(4.80). The only difference is that here we use
new $R$-symmetry cross-ratios $\zeta$ and $\eta$ instead of $r$ and $s$, which
are the five-point pendants of (4.3):

$\begin{split}r_{1}=\zeta_{1}\zeta_{2}\,,\quad
s_{1}=(1-\zeta_{1})(1-\zeta_{2})\,,\\\ r_{2}=\eta_{1}\eta_{2}\,,\quad
s_{2}=(1-\eta_{1})(1-\eta_{2})\,.\end{split}$ (4.112)

The $R$-symmetry channel decomposition (4.80) becomes

$\begin{split}{\mathcal{F}}(\\{\chi;r,s,t\\})=\
&{\bar{F}}_{0}+\frac{\zeta_{1}\zeta_{2}}{\chi_{1}^{2}}{\bar{F}}_{1}+\frac{(1-\zeta_{1})(1-\zeta_{2})}{(1-\chi_{1})^{2}}{\bar{F}}_{2}\\\
&+\frac{\eta_{1}\eta_{2}}{\chi_{2}^{2}}{\bar{F}}_{3}+\frac{(1-\eta_{1})(1-\eta_{2})}{(1-\chi_{2})^{2}}{\bar{F}}_{4}+\frac{t}{\chi_{12}^{2}}{\bar{F}}_{5}\,.\end{split}$
(4.113)

This formulation is convenient, as the topological limit is reached for

$\displaystyle\zeta_{i}=\chi_{1}\,,\quad\eta_{i}=\chi_{2}\,,\quad
t=\chi_{12}^{2}\,.$ (4.114)

The Ward identities can be solved by considering an ansatz based on the
$R$-symmetry structure. It produces three constraints, which can then be used
for writing the general solution

${\mathcal{F}}(\\{\chi;r,s,t\\})={\mathds{F}}+\sum_{i=1}^{3}{\mathds{D}}_{i}f_{i}(\chi_{1},\chi_{2})\,,$
(4.115)

where we have defined the differential operators

$\begin{split}{\mathds{D}}_{1}&:=(v_{1}+v_{2})+v_{1}v_{2}\,(\partial_{\chi_{1}}+\partial_{\chi_{2}})\,,\\\
{\mathds{D}}_{2}&:=(w_{1}+w_{2})+w_{1}w_{2}\,(\partial_{\chi_{1}}+\partial_{\chi_{2}})\,,\\\
{\mathds{D}}_{3}&:=z+\chi_{12}(v_{1}v_{2}-w_{1}w_{2})\,(\partial_{\chi_{1}}+\partial_{\chi_{2}})\,,\end{split}$
(4.116)

with the shorthand variables

$v_{i}:=\chi_{1}-\zeta_{i}\,,\quad w_{i}:=\chi_{2}-\eta_{i}\,,\quad
z:=\chi_{12}^{2}-t\,.$ (4.117)

In terms of the $R$-symmetry channels given in (4.80) and (4.113), the
$f$-functions read

$\begin{split}f_{1}(\chi_{1},\chi_{2})&=-\frac{1}{\chi_{1}}F_{1}(\chi_{1},\chi_{2})+\frac{1}{1-\chi_{1}}F_{2}(\chi_{1},\chi_{2})\,,\\\
f_{2}(\chi_{1},\chi_{2})&=-\frac{1}{\chi_{2}}F_{3}(\chi_{1},\chi_{2})+\frac{1}{1-\chi_{2}}F_{4}(\chi_{1},\chi_{2})\,,\\\
f_{3}(\chi_{1},\chi_{2})&=-\frac{1}{\chi_{12}^{2}}F_{5}(\chi_{1},\chi_{2})\,.\end{split}$
(4.118)

Note that $F_{0}$ does not appear in this decomposition. This is because the
topological sector (4.84) fully eliminates one channel.

It may be possible to perform a more general analysis of the Ward identities
and their solutions. We reserve this for future work.

##### Revisiting the weak coupling

The solution (4.115) to the Ward identity reduces the number of functions from
six $R$-symmetry channels to just three functions $f_{i}$ and one constant
${\mathds{F}}$, which can be computed from localization.

Our weak-coupling results (4.81) and (4.82)-(4.83) can be expressed in terms
of these functions. The leading order gives

$\displaystyle f_{1}^{(0)}(\chi_{1},\chi_{2})$
$\displaystyle=-\frac{1}{\chi_{1}}\,,\qquad$ $\displaystyle
f_{2}^{(0)}(\chi_{1},\chi_{2})=\frac{1}{1-\chi_{2}}\,,$ (4.119) $\displaystyle
f_{3}^{(0)}(\chi_{1},\chi_{2})$
$\displaystyle=-\frac{1}{\chi_{12}^{2}}\,,\qquad$
$\displaystyle{\mathds{F}}^{(0)}=3\,.$

Notice the following relations:

$\begin{split}f_{1}^{(0)}(\chi_{1},\chi_{2})&=-f_{2}^{(0)}(1-\chi_{2},1-\chi_{1})\,,\\\
f_{3}^{(0)}(\chi_{1},\chi_{2})&=f_{3}^{(0)}(1-\chi_{2},1-\chi_{1})\,.\end{split}$
(4.120)

These equations hold non-perturbatively and follow from the crossing symmetry
of the correlator, given by

${\mathcal{F}}(\chi_{1},\chi_{2};r_{1},r_{2},s_{1},s_{2},t)={\mathcal{F}}(1-\chi_{2},1-\chi_{1};s_{2},s_{1},r_{2},r_{1},t)\,.$
(4.121)

At the next-to-leading order, the solution of the Ward identity reads

$\begin{split}f_{1}^{(1)}(\chi_{1},\chi_{2})=\
&-\frac{1}{8\pi^{2}}\biggl{(}\ell^{(2)}\left(\frac{\chi_{1}}{\chi_{2}}\right)+\ell^{(1)}\left(\frac{\chi_{1}}{\chi_{2}}\right)\\\
&+\frac{\chi_{1}}{\chi_{21}}\ell^{(1)}\left(\frac{\chi_{21}}{1-\chi_{1}}\right)+\frac{\pi^{2}}{6}\biggr{)}\,,\\\
f_{2}^{(1)}(\chi_{1},\chi_{2})=\ &-f_{2}^{(1)}(1-\chi_{2},1-\chi_{1})\,,\\\
f_{3}^{(1)}(\chi_{1},\chi_{2})=\
&-\frac{1}{8\pi^{2}\chi_{12}^{2}}\biggl{(}\ell^{(2)}\left(\frac{\chi_{21}}{1-\chi_{1}}\right)+\ell^{(1)}\left(\frac{\chi_{21}}{1-\chi_{1}}\right)\\\
&+(\chi_{1}\leftrightarrow 1-\chi_{2})+\frac{\pi^{2}}{6}\biggr{)}\,,\\\
{\mathds{F}}^{(1)}=\ &\frac{7}{48}\,.\end{split}$ (4.122)

The analysis done in this section will now be used for bootstrapping this
correlator in the strong-coupling regime.

###  4.4 Multipoint correlators at strong coupling

We have gone a long way in this chapter. We started with multipoint
correlation functions at weak coupling, from which we conjectured non-
perturbative Ward identities. This culminates now into a strong-coupling
computation of the five-point function
$\langle\,\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{2}\,\rangle$,
presenting preliminary results of [7].

####  4.4.1 Superconformal blocks

$\hat{\mathcal{O}}_{1}$$\hat{\mathcal{O}}_{1}$$\hat{\mathcal{O}}_{1}$$\hat{\mathcal{O}}_{1}$$\hat{\mathcal{O}}_{2}$

$\hat{\mathcal{O}}_{1}$$\hat{\mathcal{O}}_{1}$$\hat{\mathcal{O}}_{1}$$\hat{\mathcal{O}}_{1}$$\hat{\mathcal{O}}_{2}$

Figure 4.6: The two possible ways of performing OPEs in the correlator
$\langle\,\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{2}\,\rangle$.
The left figure shows the symmetric channel (4.124), while the asymmetric one
given in (4.127) is displayed on the right. The black bold lines refer to the
way the OPEs are being performed. Red lines refer to the product of the OPE
(4.123), while the orange line corresponds to (4.126).

Superconformal blocks play a crucial role in bootstrap algorithms. In the case
of the four-point functions [266], the superblocks were obtained with the help
of the Ward identities (4.2), bypassing the use of Casimir equations [285]. We
perform a similar analysis here and show that the five-point blocks can also
be entirely fixed with the Ward identities.

We consider here two ways of performing the OPEs, which are represented in
Figure 4.6. One can either consider two pairs of
$\hat{\mathcal{O}}_{1}\times\hat{\mathcal{O}}_{1}$ and leave
$\hat{\mathcal{O}}_{2}$ untouched, or first perform
$\hat{\mathcal{O}}_{1}\times\hat{\mathcal{O}}_{2}$. We call these channels
respectively symmetric and asymmetric.151515It might seem from Figure 4.6 that
these names should be inverted. They refer to the fact that in the symmetric
channel, the two OPEs are the same
($\hat{\mathcal{O}}_{1}\times\hat{\mathcal{O}}_{1}$), while in the asymmetric
one, they are distinct. They lead to two different block expansions, which we
discuss now.

##### The symmetric channel

The OPE between two half-BPS operators $\hat{\mathcal{O}}_{1}$ results in the
decomposition

${\mathcal{D}}_{1}\times{\mathcal{D}}_{1}\longrightarrow\mathds{1}+{\mathcal{D}}_{2}+\sum_{\hat{\Delta}>1}{\mathcal{L}}_{0,[0,0]}^{\hat{\Delta}}\,.$
(4.123)

Here we have used group-theoretical labels to designate the operators.
${\mathcal{D}}_{\hat{\Delta}}$ refers to half-BPS operators (not necessarily
single-trace), while the operators contained in
${\mathcal{L}}_{0,[0,0]}^{\hat{\Delta}}$ are unprotected longs with spin $s=0$
and $R$-symmetry charge $[a,b]=[0,0]$ (see [348] for more details about the
notation).

This leads to the expansion in superblocks

$\begin{split}{\mathcal{F}}(\\{\chi;r,s,t\\})=\
&\lambda_{112}^{\phantom{2}}({\mathcal{G}}_{\mathds{1},{\mathcal{D}}_{2}}+{\mathcal{G}}_{{\mathcal{D}}_{2},\mathds{1}})+\lambda_{112}^{2}\lambda_{222}^{\phantom{2}}{\mathcal{G}}_{{\mathcal{D}}_{2},{\mathcal{D}}_{2}}\\\\[8.00003pt]
&+\sum_{\hat{\Delta}}\lambda_{\vphantom{\hat{\Delta}}112}\lambda_{11\hat{\Delta}}\lambda_{22\hat{\Delta}}({\mathcal{G}}_{{\mathcal{D}}_{2},{\mathcal{L}}_{\hat{\Delta}}}+{\mathcal{G}}_{{\mathcal{L}}_{\hat{\Delta}},{\mathcal{D}}_{2}})\\\
&+\sum_{\hat{\Delta}_{1},\hat{\Delta}_{2}}\lambda_{11\hat{\Delta}_{1}}\lambda_{11\hat{\Delta}_{2}}\lambda_{2\hat{\Delta}_{1}\hat{\Delta}_{2}}{\mathcal{G}}_{{\mathcal{L}}_{\hat{\Delta}_{1}},{\mathcal{L}}_{\hat{\Delta}_{2}}}\,,\end{split}$
(4.124)

where we omitted the dependence on the cross-ratios on the right-hand side to
streamline the notation, and with
${\mathcal{L}}_{\hat{\Delta}}:={\mathcal{L}}_{0,[0,0]}^{\hat{\Delta}}$.

The superblocks themselves take the form (B.16) and are discussed in Appendix
B.2.1. Importantly, their form is fully fixed by (4.111), up to an overall
normalization constant. This illustrates the power of the superconformal Ward
identities.

Note that the expansion in superblocks largely truncates in the topological
sector. In particular, all the contributions from the longs drop out and we
are left with

${\mathds{F}}=2\lambda_{112}^{\phantom{2}}\lambda_{112}^{2}\lambda_{222}^{\phantom{2}}\,.$
(4.125)

##### The asymmetric channel

We now turn our attention to the asymmetric channel. In this case, we need in
addition to (4.123) the following OPE:

${\mathcal{D}}_{1}\times{\mathcal{D}}_{2}\longrightarrow{\mathcal{D}}_{1}+{\mathcal{D}}_{3}+\sum_{\hat{\Delta}}{\mathcal{L}}_{0,[0,1]}^{\hat{\Delta}}\,.$
(4.126)

This results in an expansion in superblocks of the following form:

$\begin{split}{\mathcal{F}}(\\{\chi;r,s,t\\})=\
&\lambda_{112}^{\phantom{2}}{\mathcal{G}}_{\mathds{1},{\mathcal{D}}_{1}}+\lambda_{112}^{3}{\mathcal{G}}_{{\mathcal{D}}_{2},{\mathcal{D}}_{1}}+\lambda_{112}^{2}\lambda_{123}^{\phantom{2}}{\mathcal{G}}_{{\mathcal{D}}_{2},{\mathcal{D}}_{1}}\\\\[8.00003pt]
&+\sum_{\hat{\Delta}_{1}}\left(\lambda_{11\hat{\Delta}_{1}}^{2}{\mathcal{G}}_{{\mathcal{L}}_{\hat{\Delta}_{1}},{\mathcal{D}}_{1}}+\lambda_{11\hat{\Delta}_{1}}\lambda_{13\hat{\Delta}_{1}}\lambda_{\vphantom{\hat{\Delta}}123}{\mathcal{G}}_{{\mathcal{L}}_{\hat{\Delta}_{1}},{\mathcal{D}}_{3}}\right)\\\
&+\sum_{\hat{\Delta}_{2}}\lambda_{\vphantom{\hat{\Delta}}112}^{\phantom{2}}\lambda_{12\hat{\Delta}_{2}}^{2}{\mathcal{G}}_{{\mathcal{D}}_{2},{\mathcal{L}}_{\hat{\Delta}_{2}}}\\\
&+\sum_{\hat{\Delta}_{1},\hat{\Delta}_{2}}\lambda_{11\hat{\Delta}_{1}}\lambda_{12\hat{\Delta}_{2}}\lambda_{1\hat{\Delta}_{1}\hat{\Delta}_{2}}{\mathcal{G}}_{{\mathcal{L}}_{\hat{\Delta}_{1}}{\mathcal{L}}_{\hat{\Delta}_{2}}}\,.\end{split}$
(4.127)

Here the subscripts in $\hat{\Delta}_{1}$ and $\hat{\Delta}_{2}$ are more than
a label; they are a shorthand notation to refer to the long operators
${\mathcal{L}}_{0,[0,0]}^{\hat{\Delta}_{1}}$ and
${\mathcal{L}}_{0,[0,1]}^{\hat{\Delta}_{2}}$, which have different
$R$-charges.

The blocks can be obtained using the same method as for the symmetric channel.
Meanwhile, the topological sector in this channel reads

${\mathds{F}}=\lambda_{112}^{\phantom{2}}+\lambda_{112}^{2}+\frac{3}{2}\lambda_{112}^{2}\lambda_{123}^{\phantom{2}}\,.$
(4.128)

####  4.4.2 Bootstrapping the correlator

We are now ready to initiate the bootstrap calculation of the correlator
$\langle\,\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{2}\,\rangle$
at strong coupling. In this section, we give the results for the leading and
next-to-leading orders, while the higher-loop orders will be presented in [7].

##### Generalized free-field theory

We need as an input for the higher orders the CFT data of the leading order,
which corresponds to generalized free-field (GFF) theory. It is easy to
compute the corresponding Witten diagrams, which are

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}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-22.76228pt}\pgfsys@moveto{2.13387pt}{-22.76228pt}\pgfsys@curveto{2.13387pt}{-21.58377pt}{1.17851pt}{-20.62842pt}{0.0pt}{-20.62842pt}\pgfsys@curveto{-1.17851pt}{-20.62842pt}{-2.13387pt}{-21.58377pt}{-2.13387pt}{-22.76228pt}\pgfsys@curveto{-2.13387pt}{-23.9408pt}{-1.17851pt}{-24.89615pt}{0.0pt}{-24.89615pt}\pgfsys@curveto{1.17851pt}{-24.89615pt}{2.13387pt}{-23.9408pt}{2.13387pt}{-22.76228pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-22.76228pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,\end{split}$
(4.129)

where the AdS propagators correspond to [103]

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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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(4.130)

This leads to the $R$-symmetry channel decomposition (see (4.80))

$F_{j}^{(0)}=\sqrt{2}\,,$ (4.131)

which in terms of the solution of the Ward identity (4.115) translates into

$\displaystyle
f_{1}^{(0)}=-\sqrt{2}\frac{1-2\chi_{1}}{\chi_{1}(1-\chi_{1})}\,,\qquad$
$\displaystyle
f_{2}^{(0)}=-\sqrt{2}\frac{1-2\chi_{2}}{\chi_{2}(1-\chi_{2})}\,,$ (4.132)
$\displaystyle f_{3}^{(0)}=-\frac{\sqrt{2}}{\chi_{12}^{2}}\,,$
$\displaystyle\mathds{F}=6\sqrt{2}\,.$

This result can be expanded in the superconformal blocks introduced in Section
4.4.1. For the symmetric channel, we find the following OPE coefficients:

$\begin{split}&\left.\lambda_{112}^{\phantom{2}}\right.=\sqrt{2}\,,\qquad\qquad\lambda_{112}^{2}\lambda_{222}^{\phantom{2}}=4\sqrt{2}\,,\\\\[8.00003pt]
&\left.\lambda_{112\vphantom{\hat{\Delta}}}\lambda_{11\hat{\Delta}}\lambda_{22\hat{\Delta}}\right|_{\hat{\Delta}\text{
even}}=\frac{\sqrt{\pi}(\hat{\Delta}-1)\Gamma(\hat{\Delta}+3)}{2^{\frac{4\hat{\Delta}+1}{2}}\Gamma\left(\hat{\Delta}+\frac{3}{2}\right)}\,,\\\
&\left.\lambda_{11\hat{\Delta}_{1}}\lambda_{11\hat{\Delta}_{2}}\lambda_{2\hat{\Delta}_{1}\hat{\Delta}_{2}}\right|_{\hat{\Delta}_{1,2}\text{
even}}=\frac{\pi(\hat{\Delta}_{1})_{3}(\hat{\Delta}_{2})_{3}\Gamma(\hat{\Delta}_{1}+\hat{\Delta}_{2})}{2^{\frac{4(\hat{\Delta}_{1}+\hat{\Delta}_{2})+7}{2}}\Gamma\left(\hat{\Delta}_{1}+\frac{3}{2}\right)\Gamma\left(\hat{\Delta}_{2}+\frac{3}{2}\right)}\,.\end{split}$
(4.133)

Note that the OPE coefficients with odd $\hat{\Delta}$ vanish.

From the asymmetric channel, we obtain

$\begin{split}&\left.\lambda_{112}^{\phantom{2}}\right.=\sqrt{2}\,,\qquad\lambda_{112}^{3}=2\sqrt{2}\,,\qquad\lambda_{112}^{2}\lambda_{123}^{\phantom{2}}=4\sqrt{2}\,,\\\
&\left.\lambda_{11\hat{\Delta}_{1}}^{2}\right|_{\hat{\Delta}_{1}\text{
even}}=\frac{\sqrt{\pi}(\hat{\Delta}_{1}-1)\Gamma(\hat{\Delta}_{1}+3)}{2^{\frac{4\hat{\Delta}_{1}+3}{2}}\Gamma\left(\hat{\Delta}_{1}+\frac{3}{2}\right)}\,,\\\
&\left.\lambda_{112\vphantom{\hat{\Delta}}}^{\phantom{2}}\lambda_{12\hat{\Delta}_{2}}^{2}\right|_{\hat{\Delta}_{2}\text{
even}}=\frac{\sqrt{\pi}(\hat{\Delta}_{2}-2)^{2}(\hat{\Delta}_{2})_{6}\Gamma(\hat{\Delta}_{2})}{2^{\frac{4\hat{\Delta}_{2}+7}{2}}3(\hat{\Delta}_{2}+1)(\hat{\Delta}_{2}+4)\Gamma\left(\hat{\Delta}_{2}+\frac{3}{2}\right)}\,,\\\
&\left.\lambda_{11\hat{\Delta}_{1}}\lambda_{12\hat{\Delta}_{2}}\lambda_{1\hat{\Delta}_{1}\hat{\Delta}_{2}}\right|_{\begin{subarray}{c}\hat{\Delta}_{1}\text{
even}\\\
\hat{\Delta}_{2}>\hat{\Delta}_{1}\end{subarray}}=2\sqrt{2}\hat{\Delta}_{21}(\hat{\Delta}_{1}-1)_{3}\\\
&\qquad\qquad\qquad\qquad\qquad\times\frac{\Gamma(\hat{\Delta}_{1}+3)\Gamma(\hat{\Delta}_{2}-1)\Gamma(\hat{\Delta}_{1}+\hat{\Delta}_{2}+4)}{\Gamma(2\hat{\Delta}_{1}+3)\Gamma(2\hat{\Delta}_{2}+3)}\,,\end{split}$
(4.134)

where all the other OPE coefficients are zero.

##### First correction

We now bootstrap the next order following the method sketched in Section
2.1.3. Our starting point is the crossing symmetry given in (4.121) that
relates the different functions $f_{j}(\chi_{1},\chi_{2})$. Additional
braiding constraints can be obtained by considering different orderings of the
operators in the correlator. In this case, the functions are equal up to a
phase.

We begin by writing an ansatz for $f_{1}$ and $f_{3}$161616Recall that $f_{2}$
is fixed by (4.121). based on the OPE limits. They take the form

$\begin{split}f_{1}(\chi_{1},\chi_{2})=\
&p_{1}(\chi_{1},\chi_{2})+r_{1}(\chi_{1},\chi_{2})\log\chi_{1}+r_{2}(\chi_{1},\chi_{2})\log\chi_{2}\\\
&+r_{3}(\chi_{1},\chi_{2})\log(1-\chi_{1})+r_{4}(\chi_{1},\chi_{2})\log(1-\chi_{2})\\\
&+r_{5}(\chi_{1},\chi_{2})\log\chi_{21}\,,\end{split}$ (4.135)

and

$\begin{split}f_{3}(\chi_{1},\chi_{2})=\
&p_{3}(\chi_{1},\chi_{2})+s_{1}(\chi_{1},\chi_{2})\log\chi_{1}+s_{2}(\chi_{1},\chi_{2})\log\chi_{2}\\\
&+s_{3}(\chi_{1},\chi_{2})\log(1-\chi_{1})+s_{4}(\chi_{1},\chi_{2})\log(1-\chi_{2})\\\
&+s_{5}(\chi_{1},\chi_{2})\log\chi_{21}\,,\end{split}$ (4.136)

where the functions $p_{i}$ and $r_{j}$ are rational functions.

The crossing and braiding symmetries result in the following constraints on
the rational functions present in (4.135) and (4.136):

$\begin{split}p_{1}(\chi_{1},\chi_{2})&=-p_{1}(1-\chi_{1},1-\chi_{2})\,,\\\
r_{1}(\chi_{1},\chi_{2})&=-r_{3}(1-\chi_{1},1-\chi_{2})\,,\\\
r_{2}(\chi_{1},\chi_{2})&=-r_{4}(1-\chi_{1},1-\chi_{2})\,,\\\
r_{5}(\chi_{1},\chi_{2})&=-r_{5}(1-\chi_{1},1-\chi_{2})\,,\end{split}$ (4.137)

and

$\begin{split}p_{3}(\chi_{1},\chi_{2})&=p_{3}(\chi_{2},\chi_{1})\,,\\\
p_{3}(\chi_{1},\chi_{2})&=p_{3}(1-\chi_{2},1-\chi_{1})\,,\\\
s_{2}(\chi_{1},\chi_{2})&=-s_{1}(\chi_{2},\chi_{1})\,,\\\
s_{3}(\chi_{1},\chi_{2})&=-s_{1}(1-\chi_{1},1-\chi_{2})\,,\\\
s_{4}(\chi_{1},\chi_{2})&=-s_{1}(1-\chi_{2},1-\chi_{1})\,,\\\
s_{5}(\chi_{1},\chi_{2})&=-s_{5}(\chi_{2},\chi_{1})\,,\\\
s_{5}(\chi_{1},\chi_{2})&=-s_{5}(1-\chi_{2},1-\chi_{1})\,.\end{split}$ (4.138)

This already eliminates five functions. Moreover, we also have relations that
relate $f_{1}$ to $f_{3}$:

$\displaystyle p_{1}(\chi_{1},\chi_{2})=\ $
$\displaystyle-\frac{1-\chi_{1}}{\chi_{2}^{2}}p_{3}\left(\frac{\chi_{1}}{\chi_{2}},\frac{1}{\chi_{2}}\right)+\frac{\chi_{1}}{(1-\chi_{2})^{2}}p_{3}\left(\frac{\chi_{12}}{1-\chi_{2}},-\frac{\chi_{2}}{1-\chi_{2}}\right)\,,$
$\displaystyle r_{1}(\chi_{1},\chi_{2})=\ $
$\displaystyle-\frac{1-\chi_{1}}{\chi_{2}^{2}}s_{1}\left(\frac{\chi_{1}}{\chi_{2}},\frac{1}{\chi_{2}}\right)+\frac{\chi_{1}}{(1-\chi_{2})^{2}}s_{5}\left(\frac{\chi_{12}}{1-\chi_{2}},-\frac{\chi_{2}}{1-\chi_{2}}\right)\,,$
$\displaystyle r_{2}(\chi_{1},\chi_{2})=\ $
$\displaystyle\frac{1-\chi_{1}}{\chi_{2}^{2}}\biggl{(}s_{1}\left(\frac{\chi_{21}}{\chi_{2}},-\frac{1-\chi_{2}}{\chi_{2}}\right)+s_{1}\left(-\frac{1-\chi_{2}}{\chi_{2}},\frac{\chi_{21}}{\chi_{2}}\right)$
$\displaystyle+s_{1}\left(\frac{\chi_{1}}{\chi_{2}},\frac{1}{\chi_{2}}\right)+s_{1}\left(\frac{1}{\chi_{2}},\frac{\chi_{1}}{\chi_{2}}\right)+s_{5}\left(\frac{\chi_{1}}{\chi_{2}},\frac{1}{\chi_{2}}\right)\biggr{)}$
$\displaystyle+\frac{\chi_{1}}{(1-\chi_{2})^{2}}s_{1}\left(-\frac{\chi_{2}}{1-\chi_{2}},\frac{\chi_{12}}{1-\chi_{2}}\right)\,,$
$\displaystyle r_{5}(\chi_{1},\chi_{2})=\ $
$\displaystyle-\frac{1-\chi_{1}}{\chi_{2}^{2}}s_{1}\left(\frac{\chi_{21}}{\chi_{2}},-\frac{1-\chi_{2}}{\chi_{2}}\right)$
$\displaystyle+\frac{\chi_{1}}{(1-\chi_{2})^{2}}s_{1}\left(\frac{\chi_{12}}{1-\chi_{2}},-\frac{\chi_{2}}{1-\chi_{2}}\right)\,,$
(4.139)

from which we can eliminate four more functions.

We are thus left with three rational functions to fix, which we choose to be
$p_{3}$, $s_{1}$, and $s_{5}$. Assuming a Regge behavior for the anomalous
dimensions, as well as the relations (4.138) and boundary conditions following
from the OPE expansion, we can fix these functions up to two constants, which
correspond to ${\mathds{F}}$ and $\lambda_{112}$. These are known from
localization, and we use them as input in our bootstrap routine.
${\mathds{F}}$ is given in (4.34), while $\lambda_{112}$ can be computed from
[104] and reads

$\lambda_{112}=\sqrt{2}-\frac{3}{2\sqrt{2}\sqrt{\lambda}}-\frac{9}{16\sqrt{2}\lambda}+{\mathcal{O}}(\lambda^{-3/2})\,.$
(4.140)

All in all, we find the remaining rational functions to be

$\begin{split}p_{3}(\chi_{1},\chi_{2})&=\frac{1}{2\sqrt{2}\sqrt{\lambda}}\left(\frac{4}{\chi_{1}\chi_{2}}+\frac{4}{(1-\chi_{1})(1-\chi_{2})}+\frac{19}{\chi_{12}^{2}}\right)\,,\\\
s_{1}(\chi_{1},\chi_{2})&=-\frac{\sqrt{2}}{\sqrt{\lambda}}\frac{\chi_{1}(\chi_{1}(1-3\chi_{2})+4\chi_{2}^{2})}{\chi_{2}^{2}\chi_{12}^{3}}\,,\\\
s_{5}(\chi_{1},\chi_{2})&=\frac{\sqrt{2}}{\sqrt{\lambda}}\left(\frac{1}{\chi_{1}^{2}}+\frac{1}{\chi_{2}^{2}}+\frac{1}{(1-\chi_{1})^{2}}+\frac{1}{(1-\chi_{2})^{2}}\right)\,,\end{split}$
(4.141)

from which the anomalous dimensions can be found to be degenerate and give

$\hat{\gamma}^{(1)}_{\hat{\Delta}}=\frac{\hat{\Delta}(\hat{\Delta}+2)}{2\sqrt{\lambda}}\,.$
(4.142)

This concludes our computation of the next-to-leading order using bootstrap
techniques. The full functions $f_{j}$ can be reconstructed from (4.141) and
the constraints listed above it. It is highly likely that the next order
should also be bootstrappable, and this computation is the object of current
work to appear in [7]. The first step in this direction consists of extracting
the CFT data from the correlator presented in this section, which then can be
used as input in the same way as it was done here.

###  4.5 Summary and outlook

In this chapter, we considered a family of multipoint correlation functions of
scalar fields in the $1d$ supersymmetric Wilson-line defect CFT. We developed
an efficient algorithm that takes the form of the recursion relations given in
(4.47), (4.49), (4.51) and (4.53), and which are valid up to the next-to-
leading order. This can be used to extract correlation functions of half-BPS
operators as well as of certain unprotected operators built of elementary
scalars.

Although these new perturbative results are interesting in their own right,
they also give access to a more general piece of information: multipoint Ward
identities. We conjectured a set of differential equations that annihilate all
our perturbative correlators. They take the form (see (4.111))

$\sum_{k=1}^{n-3}\left(\frac{1}{2}\partial_{\chi_{k}}+\alpha_{k}\partial_{r_{k}}-(1-\alpha_{k})\partial_{s_{k}}\right)\,{\mathcal{F}}(\\{\chi;r,s,t\\})\raisebox{-8.61108pt}{$\Biggr{|}$}_{\raisebox{4.52083pt}{$\begin{subarray}{l}r_{i}\to\alpha_{i}\chi_{i}\\\
s_{i}\to(1-\alpha_{i})(1-\chi_{i})\\\
t_{ij}\to\alpha_{ij}\chi_{ij}\end{subarray}$}}=0\,.$

These equations constitute a natural generalization of the Ward identities for
four-point functions, which are known to exist in several superconformal
setups. Ward identities play a crucial role in the superconformal bootstrap
program, and in Section 4.4 we used them for bootstrapping a five-point
function at strong coupling up to the next-to-leading order.

There are many interesting directions in which to further develop the
techniques presented here. It should be possible to push the bootstrap results
of Section 4.4 up to next-to-next-to-next-to-leading order, similarly to the
four-point function story (see (4.22) and [266]). Moreover, analogous
techniques can be used for computing the six-point function
$\langle\,\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\hat{\mathcal{O}}_{1}\,\rangle$,
whose structure was introduced in (4.70). In the following, we present in some
detail other selected follow-ups, which we feel deserve to be explored.

####  4.5.1 Multipoint correlators of fermions

A natural next step is to include more general operators in our recursion
formulas, such as fields that transform non-trivially under transverse
rotations. An example of such operators is fermions. We discuss here briefly
two ways of including fermions in our analysis.

##### Sketching a fermionic recursion relation

In Section 4.2.1, we have discussed the following key aspects of the recursion
relations:

  1. 1.

at leading order, the basic ingredient of the formulas (4.47) and (4.49) is
the scalar two-point function;

  2. 2.

at next-to-leading order, the basic ingredient of (4.51) and (4.53) is the
scalar four-point  function.

There is no inherent reason why these principles would not apply to fermionic
correlators. We provide here a sketch of how a recursion relation for fermions
would look like.

At the leading order, we expect the building block of the fermionic pendent to
(4.48) to simply be the two-point diagram

$\leavevmode\hbox to58.91pt{\vbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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} \pgfsys@invoke{\lxSVG@closescope
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}{}\pgfsys@moveto{48.3696pt}{0.0pt}\pgfsys@moveto{50.50346pt}{0.0pt}\pgfsys@curveto{50.50346pt}{1.17851pt}{49.54811pt}{2.13387pt}{48.3696pt}{2.13387pt}\pgfsys@curveto{47.19109pt}{2.13387pt}{46.23573pt}{1.17851pt}{46.23573pt}{0.0pt}\pgfsys@curveto{46.23573pt}{-1.17851pt}{47.19109pt}{-2.13387pt}{48.3696pt}{-2.13387pt}\pgfsys@curveto{49.54811pt}{-2.13387pt}{50.50346pt}{-1.17851pt}{50.50346pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{48.3696pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
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{}{}{}{}{}{}{{}}\pgfsys@moveto{28.16838pt}{14.22638pt}\pgfsys@lineto{29.92291pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
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{}{{}}{}{}{}{}{{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@lineto{56.90552pt}{28.45276pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{56.90552pt}{28.45276pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@moveto{10.66977pt}{0.0pt}\pgfsys@curveto{10.66977pt}{1.17851pt}{9.71442pt}{2.13387pt}{8.5359pt}{2.13387pt}\pgfsys@curveto{7.35739pt}{2.13387pt}{6.40204pt}{1.17851pt}{6.40204pt}{0.0pt}\pgfsys@curveto{6.40204pt}{-1.17851pt}{7.35739pt}{-2.13387pt}{8.5359pt}{-2.13387pt}\pgfsys@curveto{9.71442pt}{-2.13387pt}{10.66977pt}{-1.17851pt}{10.66977pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{48.3696pt}{0.0pt}\pgfsys@moveto{50.50346pt}{0.0pt}\pgfsys@curveto{50.50346pt}{1.17851pt}{49.54811pt}{2.13387pt}{48.3696pt}{2.13387pt}\pgfsys@curveto{47.19109pt}{2.13387pt}{46.23573pt}{1.17851pt}{46.23573pt}{0.0pt}\pgfsys@curveto{46.23573pt}{-1.17851pt}{47.19109pt}{-2.13387pt}{48.3696pt}{-2.13387pt}\pgfsys@curveto{49.54811pt}{-2.13387pt}{50.50346pt}{-1.17851pt}{50.50346pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{48.3696pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (4.143)

At the next-to-leading order, the four-point functions of fermions involve
both bulk and boundary diagrams, as already seen in (4.10)-(4.11) for the
scalar case. Specifically, the bulk diagrams in the channel $(12)(34)$ are

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