}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{32.14322pt}{-11.11368pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\footnotesize$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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=\frac{1}{|x_{1}^{\perp}|}\left\\{\tan^{-1}\left(\frac{\tau_{12}}{|x_{1}^{\perp}|}\right)-\tan^{-1}\left(\frac{\tau_{13}}{|x_{1}^{\perp}|}\right)\right\\}\,.$
(C.30)

This result can be used for instance to derive the value of all the vertices
listed in (2.66). These integrals are particularly useful to derive the
expression given in (4.104).

####  C.3.2 One-loop integrals

We now consider the defect integrals encountered in the one-loop diagrams of
Section 4.2. We first show how to solve the $T$-integrals, before listing the
results for $U$-integrals, which are relevant for correlators of unprotected
operators.

#####  $T$-integrals

The $T$-integrals are defined via222This class of integrals also appears in
[349], where they are defined slightly differently and labelled as
$B_{ij;kl}$.

$T_{ij,kl}:=\partial_{ij}\int_{\tau_{k}}^{\tau_{l}}d\tau_{m}\,\varepsilon(\tau_{i}\,\tau_{j}\,\tau_{m})\,Y_{ijm}\,.$
(C.31)

It is the integral along the line of a $Y$-vertex of the type scalar-scalar-
gluon. Here $\varepsilon(\tau_{i}\,\tau_{j}\,\tau_{k})$ encodes the change of
sign due to the path ordering and the color trace. It is formally defined as

$\varepsilon(\tau_{i}\,\tau_{j}\,\tau_{k}):=\operatorname{sgn}\tau_{ij}\,\operatorname{sgn}\tau_{ik}\,\operatorname{sgn}\tau_{jk}\,.$
(C.32)

An important identity satisfied by $\varepsilon(\tau_{i}\,\tau_{j}\,\tau_{k})$
is

$\partial_{i}\,\varepsilon(\tau_{i}\,\tau_{j}\,\tau_{k})=2(\delta(\tau_{ij})-\delta(\tau_{ik}))\,,$
(C.33)

which allows to extract the divergences from the integrals encountered for
instance in the diagrams

$\displaystyle=\int_{\tau_{k}}^{\tau_{l}}d\tau_{m}\,\varepsilon(\tau_{i}\,\tau_{j}\,\tau_{m})\,\partial_{ij}Y_{ijm}=T_{ij,kl}-4Y_{iij}\,.$
(C.34)

Note that this identity is valid only for
$\tau_{k}<\tau_{i}<\tau_{j}<\tau_{l}$ and must be adapted for other cases.

The $T$-integrals are finite and can be evaluated. When the range of
integration is the entire line, the integral is easy to perform and results in

$T_{ij,(-\infty)\infty}=-\frac{I_{ij}}{12}\,.$ (C.35)

This follows from

$T_{ij,(-\infty)i}=-T_{ij,ij}=T_{ij,(-\infty)i}=-\frac{I_{ij}}{12}\,.$ (C.36)

Another direct consequence of (C.36) is

$\left.T_{ij,kl}\right|_{\tau_{k}<\tau_{i}<\tau_{j}<\tau_{l}}=-\frac{I_{ij}}{12}-T_{ij,(-\infty)k}-T_{ij,l\infty}\,.$
(C.37)

There also exists a beautiful relation between $T$\- and $Y$-integrals:

$I_{ik}T_{jk,ki}+I_{jk}T_{ik,jk}=-\frac{I_{ik}I_{jk}}{12}+I_{ik}I_{jk}\left(\frac{1}{I_{ik}}+\frac{1}{I_{jk}}-\frac{2}{I_{ij}}\right)Y_{ijk}\,.$
(C.38)

The $T$-integrals can be performed explicitly for different orderings of the
$\tau$’s, and here we give the results assuming
$\tau_{1}<\tau_{2}<\tau_{3}<\tau_{4}$ [349]:

$\begin{split}T_{12,34}&=\frac{I_{12}}{8\pi^{2}}\left(4L_{R}\left(\frac{\tau_{12}}{\tau_{14}}\right)-4L_{R}\left(\frac{\tau_{12}}{\tau_{13}}\right)-C_{123}+C_{124}\right)\,,\\\
T_{34,12}&=\frac{I_{34}}{8\pi^{2}}\left(4L_{R}\left(\frac{\tau_{34}}{\tau_{14}}\right)-4L_{R}\left(\frac{\tau_{34}}{\tau_{24}}\right)-C_{341}+C_{342}\right)\,,\\\
T_{14,23}&=\frac{I_{14}}{8\pi^{2}}\left(4L_{R}\left(\frac{\tau_{24}}{\tau_{14}}\right)-4L_{R}\left(\frac{\tau_{34}}{\tau_{14}}\right)-C_{412}-C_{143}\right)\,,\\\
T_{23,41}&=\frac{I_{23}}{8\pi^{2}}\left(-4L_{R}\left(\frac{\tau_{23}}{\tau_{13}}\right)-4L_{R}\left(\frac{\tau_{23}}{\tau_{24}}\right)-C_{234}-C_{123}\right)\,,\end{split}$
(C.39)

where we defined the help function

$C_{ijk}:=-32\pi^{4}\,\tau_{ij}(\tau_{ik}+\tau_{jk})\,Y_{ijk}\,.$ (C.40)

The Rogers dilogarithm $L_{R}(\chi)$ is defined as [452]

$L_{R}(\chi):=-\frac{1}{2}(H_{1,0}-H_{0,1})\,,$ (C.41)

with $H_{\vec{a}}:=H_{\vec{a}}(\chi)$.

It is easy to take pinching limits of the integrals given above. For example,
we have

$T_{12,23}=\frac{I_{12}}{8\pi^{2}}\left(4L_{R}\left(\frac{\tau_{12}}{\tau_{13}}\right)-\frac{2\pi^{2}}{3}+C_{123}\right)+Y_{112}\,,$
(C.42)

using the fact that $L_{R}(1)=\frac{\pi^{2}}{6}$. All the other pinching
limits can be obtained in the same way.

#####  $U$-integrals

Correlation functions involving the scalar coupling to the line defect
generate another type of boundary integral. We denote them by $U_{i,jk}$, and
they are defined as

$U_{i,jk}=\ \scalebox{0.85}{ \leavevmode\hbox to58.91pt{\vbox
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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:=\int_{\tau_{j}}^{\tau_{k}}d\tau_{n}\,I_{in}\,,$ (C.43)

where the interrupted scalar line should be connected to the orange dots on
the line. These integrals are easy to perform explicitly and give

$U_{i,jk}=-\frac{1}{4\pi^{2}}\left(\frac{1}{\tau_{ij}}-\frac{1}{\tau_{ik}}\right)\,,$
(C.44)

which is valid for both the orderings $\tau_{a}<\tau_{i}<\tau_{j}$ and
$\tau_{i}<\tau_{j}<\tau_{a}$. Special cases, such as $\tau_{a}=\tau_{i}$, can
be obtained by regularizing the divergences in the integral using point-
splitting.

At the next-to-leading order, we also encounter the integrals

$U^{(2)}_{ij,kl}:=\int_{\tau_{k}}^{\tau_{l}}d\tau_{m}\,I_{im}U_{j,nl}\,.$
(C.45)

We compute them explicitly for three different configurations, from which the
other relevant ones can be obtained by taking the appropriate limits. Assuming
the ordering $\tau_{1}<\tau_{2}<\tau_{3}<\tau_{4}$, we get

$\begin{split}U^{(2)}_{12,34}=\
&\frac{I_{12}}{4\pi^{2}}\left\\{-\frac{\tau_{12}^{2}\tau_{34}}{\tau_{23}\tau_{24}\tau_{14}}+\frac{1}{\tau_{23}\tau_{24}}\left((\tau_{2}^{2}+\tau_{3}\tau_{4})\log\frac{\tau_{23}\tau_{14}}{\tau_{13}\tau_{24}}\right.\right.\\\
&\left.\left.\phantom{\frac{I_{12}}{4\pi^{2}}\
}+\tau_{2}(\tau_{3}+\tau_{4})\log\frac{\tau_{13}\tau_{24}}{\tau_{23}\tau_{14}}+\tau_{12}\tau_{34}\right)\right\\}\,,\\\
U^{(2)}_{34,12}=\
&\frac{I_{34}}{4\pi^{2}}\left\\{-\frac{\tau_{34}^{2}\tau_{12}}{\tau_{13}\tau_{14}\tau_{23}}+\frac{1}{\tau_{13}\tau_{23}}\left((\tau_{3}^{2}+\tau_{1}\tau_{2})\log\frac{\tau_{14}\tau_{23}}{\tau_{13}\tau_{24}}\right.\right.\\\
&\left.\left.\phantom{\frac{I_{12}}{4\pi^{2}}\
}+\tau_{3}(\tau_{1}+\tau_{2})\log\frac{\tau_{13}\tau_{24}}{\tau_{14}\tau_{23}}+\tau_{34}\tau_{12}\right)\right\\}\,,\\\
U^{(2)}_{14,23}=\
&\frac{I_{14}}{4\pi^{2}}\left\\{\frac{\tau_{14}^{2}\tau_{23}}{\tau_{12}\tau_{13}\tau_{34}}+\frac{1}{\tau_{12}\tau_{13}}\left((\tau_{1}^{2}+\tau_{2}\tau_{3})\log\frac{\tau_{12}\tau_{43}}{\tau_{42}\tau_{13}}\right.\right.\\\
&\left.\left.\phantom{\frac{I_{12}}{4\pi^{2}}\
}+\tau_{1}(\tau_{2}+\tau_{3})\log\frac{\tau_{42}\tau_{13}}{\tau_{12}\tau_{43}}+\tau_{41}\tau_{23}\right)\right\\}\,.\end{split}$
(C.46)

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