${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}\blacksquare}$-blocks
in (57)), then we consider its pullback on
$\mathcal{M}_{1,5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
or
$\mathcal{M}_{1,5}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$
respectively. In contrast, if the entry falls outside this classification
(i.e., if it belongs to a
${\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{1}{.2}{1}{0.2}\pgfsys@color<EMAIL_ADDRESS>or any of the remaining diagonal entries in (57)), it is a dlog form, and
either torus is a valid choice. Note that this choice may break the overall
${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$
symmetry of the differential equation pulled back on the tori. This may also
have a nontrivial effect on whether or not the quantities defined on
$\mathcal{M}_{1,5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
and
$\mathcal{M}_{1,5}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$
couple in the differential equation solution at a given order in
$\varepsilon$. In section 6, we will briefly discuss a naive organization
strategy to use in order to prevent this from happening.

Once we determine on which torus a given entry is defined, we can pull back
the corresponding kinematic 1-form onto it. The target space of differential
forms is universally spanned by modular forms and Kronecker-Eisenstein forms,
which we review below (see section 5.1).

As we shall see, once we determine into which of the above categories an entry
of $\mathbf{B}$ falls, information about the location of its poles and modular
properties is enough to fix its pullback from a natural class of ansatzes:
linear combinations of modular and Kronecker-Eisenstein forms over
$\mathbbm{Q}+i\mathbbm{Q}$. In particular, the pole structure will be used to
determine the $z$-dependence of Kronecker-Eisenstein forms (see section 5.2),
while the modular properties will be used to establish the modular weight of
the differential forms that are allowed to appear (the modular behavior of
each matrix element is summarized in (144); the matrix contains dlog forms of
modular weight $0$ and (quasi-)modular forms of weights $\\{0,1,2,3,4\\}$).
The coefficients in $\mathbbm{Q}+i\mathbbm{Q}$ are always fixed numerically
from a PSLQ fit _a posteriori_.666This can be done, e.g., using Mathematica’s
FindIntegerNullVector[].

For the PSLQ reconstruction to work, it is important that a branch is chosen
consistently throughout the whole calculation. To be more precise, the root
ordering in (12) will pick a branch for
$\tau^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}$,
$z_{1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}$,
$z_{2}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}$
and
$\tau^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$,
$z_{4}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$,
$z_{5}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$.
Additionally, we enforce (51) to fix the branch for
$z_{4}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}$
and do similarly for
$z_{5}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}$
by requiring

$X_{5}>(\sqrt{X_{0}}+\sqrt{X_{1}})^{2}\text{ and
}X_{5}>(1+\sqrt{X_{2}})^{2}\,.$ (100)

Finally, we pick a branch for
$z_{1}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$
and
$z_{2}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$
by adding the constraints

$\begin{split}&X_{1}<(\sqrt{X_{0}}-\sqrt{X_{5}})^{2}\text{ and
}X_{1}<(1-\sqrt{X_{4}})^{2}\,,\\\ &X_{2}<(\sqrt{X_{0}}-\sqrt{X_{4}})^{2}\text{
and }X_{2}<(1-\sqrt{X_{5}})^{2}\,.\end{split}$ (101)

We checked the stability of the PSLQ reconstruction and the integrability
condition of the resulting matrix over a random sample of
$\mathcal{O}(10^{6})$ numerical points in $\mathcal{K}$ satisfying these
conditions.

In what follows, we discuss the pullback procedure in detail and provide the
complete matrix $\mathbf{B}$ in terms of differential forms on the tori (for
the eyeball and kite in sections 5.3 and 5.4, respectively). The result of the
full pullback is available in a Mathematica format in the ancillary file
attached to this article .

###  5.1 Review on differential forms on the torus

In this section, we outline the definitions and additional conventions
essential for the discussion that follows. We start with the aforementioned
_Kronecker-Eisenstein form_ elliptic_calc ; weinzierl2022feynman

$\omega_{k}(z,\tau)=(2\pi)^{2-k}\Big{(}g^{(k-1)}(z,\tau)\text{d}z+(k-1)g^{(k)}(z,\tau)\frac{\text{d}\tau}{2\pi
i}\Big{)}\,,$ (102)

where the $g$-kernels are defined from the generating _Kronecker-Eisenstein
series_ weil1976elliptic ; Brown:2011wfj ; Broedel:2014vla ; Broedel:2018qkq
(following the $\pi$-prescription of (weinzierl2022feynman, , eq. (13.190)))

$F(z,\eta,q)=\pi\dfrac{\theta_{1}^{\prime}(0,\tau)\theta_{1}(\pi(z+\eta),\tau)}{\theta_{1}(\pi
z,\tau)\theta_{1}(\pi\eta,\tau)}=\sum_{\alpha=0}^{\infty}\eta^{\alpha-1}g^{(\alpha)}(z,\tau)\,,$
(103)

where $\theta_{1}^{\prime}(z,\tau)=\partial_{z}\theta_{1}(z,\tau)$ and
$\theta_{1}(z,\tau)$ is the odd Jacobi theta function. The $g^{(k)}(z,\tau)$’s
are the genus-1 analogous of the rational function $1/z$. In particular, the
$g$-kernels have a simple pole at each lattice point $z=a+b\tau$ where
$a,b\in\mathbbm{Z}$ and $b\neq 0$. For $k=1$, $g^{(1)}(z,\tau)$ has a simple
pole on all lattice points, including those with $b=0$.

Importantly, the $g$-kernels transform under $\text{SL}(2,\mathbbm{Z})$ as
_quasi-modular forms_ (c.f., (26))

$g^{(k)}\Big{(}\dfrac{z}{c\tau+d},\dfrac{a\tau+b}{c\tau+d}\Big{)}=(c\tau+d)^{k}\sum^{k}_{j=0}\dfrac{(2\pi
i~{}c~{}z)^{j}}{j!(c\tau+d)^{j}}g^{(k-j)}(z,\tau)\,.$ (104)

It is not hard to see that this implies that $\omega_{k}(z,\tau)$ is a quasi-
modular form of weight $k$ (according to our definition of modular weight in
(25)). For $k=0,...,4$ we have the following useful expansions around
$q=e^{i\pi\tau}=0$

$\begin{split}g^{(0)}(z,\tau)&=1\\\ g^{(1)}(z,\tau)&=\pi\cot(\pi
z)+4\pi\sin(2\pi z)q^{2}+\mathcal{O}(q^{4})\,,\\\
g^{(2)}(z,\tau)&=-\dfrac{\pi^{2}}{3}+8\pi^{2}\cos(2\pi
z)q^{2}+\mathcal{O}(q^{4})\,,\\\ g^{(3)}(z,\tau)&=-8\pi^{3}\sin(2\pi
z)q^{2}+\mathcal{O}(q^{4})\,,\\\
g^{(4)}(z,\tau)&=-\frac{\pi^{4}}{45}-\frac{16\pi^{4}}{3}\cos(2\pi
z)q^{2}+\mathcal{O}(q^{4})\,.\end{split}$ (105)

Clearly, for $-\infty<\text{Im}z<\infty$, only $g^{(1)}(z,\tau)$ has a pole
(at the origin), while $g^{(2,3,4)}(z,\tau)$ do not. In fact, one can show
that $g^{(k>1)}(z,\tau)$ are regular at the origin bogner_unequal_2020 .

Finally, for some entries of $\mathbf{B}$, we also need _modular forms_ ,
which are puncture independent. Such forms appear whenever the $\text{d}z$
dependence of the ansatz is fixed, but the $\text{d}\tau$ dependence is not.
In that regard, we introduce the modular forms

$\displaystyle\eta_{2}(\tau)$
$\displaystyle=[e_{2}(\tau)-2e_{2}(2\tau)]\frac{\textnormal{d}\tau}{2\pi
i}\,,$ (106a) $\displaystyle\eta_{4}(\tau)$
$\displaystyle=e_{4}(\tau)\frac{\textnormal{d}\tau}{(2\pi i)^{3}}\,,$ (106b)

where the $e_{\bullet}$’s are _Eisenstein series_ (as defined in
(weinzierl2022feynman, , eq. (13.126))). In terms of more tractable elliptic
functions, these are given by

$e_{2}(\tau)=\frac{2\pi^{2}}{6}\frac{\theta_{1}^{{}^{\prime\prime\prime}}(0,\tau)}{\theta_{1}^{{}^{\prime}}(0,\tau)}\quad\textnormal{and}\quad
e_{4}(\tau)=\frac{\pi^{4}}{90}\big{(}{\theta_{2}^{8}\\!\big{(}0,\tau\big{)}+\theta_{3}^{8}\\!\big{(}0,\tau\big{)}+\theta_{4}^{8}\\!\big{(}0,\tau\big{)}}\big{)}\,.$
(107)

It is not difficult to show that
$e_{2}(\tau)-2e_{2}(2\tau)\in\mathcal{M}_{2}(\Gamma_{0}(2))$ and
$e_{4}(\tau)\in\mathcal{M}_{4}(\textnormal{SL}(2,\mathbbm{Z}))$, where
$\mathcal{M}_{k}(\Gamma)$ represents the set of modular forms of weight $k$ in
the congruence subgroup $\Gamma$ of $\textnormal{SL}(2,\mathbbm{Z})$. For more
on modular forms and congruence subgroups, see Adams:2017ejb .

###  5.2 All poles from the diagonal dlog terms and connection to Landau

In order to pull $\mathbf{B}$ back to the torus, we need to construct an
ansatz of differential 1-forms and know where the entries of $\mathbf{B}$ can
diverge on the torus. Our strategy is to identify a (optimally _minimal_) set
of matrix elements $B_{ij}$ that contains all possible poles in a way that is
easily mapped to the torus. In our case, we find that this is a property
satisfied by the diagonal dlog terms of $\mathbf{B}$.777This observation is
motivated by the naive expectation that the diagonal should “know” about all
off-diagonal singularities, which could, in principle, be made more precise
using the language of spectral sequences.

The dlog terms are modular invariant objects that often pull back to specific
linear combinations of $\omega_{2}(z,\tau)$. An important example is

$\Omega_{2}(z,\tau)=\omega_{2}(z,\tau)-2\omega_{2}(z,2\tau)\,.$ (108)

Note that the $g$-kernels with the argument $2\tau$ (hidden in
$\omega_{2}(z,2\tau)$) are required to make the above expressions modular
invariant and do not change the location of any singular points on the torus.
Although
$g^{(k)}(z,2\tau)=\frac{1}{2}\left[g^{(k)}(\frac{z}{2},\tau)+g^{(k)}(\frac{z+1}{2},\tau)\right]$
(see (weinzierl2022feynman, , §13); alternatively, this can be extracted from
the formula described in appendix C) these $g$-kernels are singular only on a
subset of the points where $g$-kernels with argument $\tau$ are singular.

To determine what $z$ is, given a dlog entry of $\mathbf{B}$, it is
instructive to compare the $q$-expansion of $\omega_{2}(z,\tau)$ with that of
the entry. On one hand, we have

$\omega_{2}(z,\tau)=\text{d}\log\dfrac{\theta_{1}(\pi
z,\tau)}{\eta(\tau)}=\text{d}\log\sin(\pi z)+\mathcal{O}(q^{2})\,,$ (109)

where $\eta(\tau)$ is the Dedekind eta function. This allows us to rewrite any
linear combination of $\omega_{2}$’s at leading order in $q$ as

$\sum_{j}c_{j}\,\omega_{2}(f_{j}(\boldsymbol{z}),\tau)=\text{d}\log\Big{(}\prod_{j}\sin^{c_{j}}\pi
f_{j}(\boldsymbol{z})\Big{)}+\mathcal{O}(q^{2})\,,$ (110)

where $c_{j}\in\mathbbm{Q}+i\mathbbm{Q}$. On the other hand, the kinematic
dlog entries we wish to pull back factorize as

$\text{d}\log\Big{(}\dfrac{A_{i_{1}}\ldots A_{i_{n}}}{A_{j_{1}}\ldots
A_{j_{n}}}\Big{)}=\text{d}\log A_{i_{1}}+\ldots+\text{d}\log
A_{i_{n}}-\text{d}\log A_{j_{1}}-\ldots-\text{d}\log A_{j_{n}}\,.$ (111)

Here, each algebraic function $A_{i}$ can be pulled back separately onto the
torus (i.e., expressed as 0-forms in terms of $\tau$ and the $z$ variables)
before being expanded in $q$. Then, expressing the leading-order terms of
these expansions as the dlog of a product of sines gives an efficient way of
reading off the $Z$-arguments $f_{j}(\boldsymbol{z})$ appearing in (110).

Crucially, we observe that applying this procedure to all diagonal dlog terms
is _sufficient_ to find all the arguments needed to express the matrix
$\mathbf{B}$ completely in terms of modular and Kronecker-Eisenstein forms.

##### Landau analysis of the
${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$-sunrise
subtopology

For an integral family such as the sunrise, all of the elliptic structure
appears in the top sector. When the kinematics is localized to the Landau
variety of the top sector, the elliptic geometry necessarily degenerates,
meaning that $\tau^{\alpha}\to 0$ or $i\infty$ (with
$\alpha={{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$).
In fact, it is fairly easy to show that the elliptic modulus $k^{2}_{\alpha}$
and its complement $1-k^{2}_{\alpha}$ are rational functions of the
polynomials that define the Landau variety for the corresponding sunrise
diagram; when the kinematics is restricted to a subvariety in $\mathcal{K}$
defined by any of these polynomials, $k^{2}_{\alpha}\to 0$ or $1$ and
$\tau^{\alpha}\to i\infty$ or $0$. In the following, we focus (without loss of
generality) on
$\alpha={{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
and suppress this label on the variables.

Near any
(${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$-sunrise)
Landau singularity where $\tau\to i\infty$, the polynomials in the original
kinematic variables $X_{i}$ that describe this Landau singularity are given by
the leading term in the corresponding $q$-series expansion. For example, we
have

$\displaystyle X_{0}$ $\displaystyle=16q^{2}\sin^{2}(\pi z_{1})\sin^{2}(\pi
z_{2})\to 0\,,$ (112a) $\displaystyle X_{1}$ $\displaystyle=\sin^{2}(\pi
z_{1})\sin^{-2}\big{(}\pi(z_{1}+z_{2})\big{)}\,,$ (112b) $\displaystyle X_{2}$
$\displaystyle=\sin^{2}(\pi z_{2})\sin^{-2}\big{(}\pi(z_{1}+z_{2})\big{)}\,,$
(112c)

when $\tau\to i\infty$ (or equivalently $q\to 0$). Recalling that
$z_{3}=1-z_{1}-z_{2}\simeq z_{1}+z_{2}\text{ mod }\Lambda_{(1,\tau)}$, we see
that the subleading Landau equations $X_{i=1,2}=0,\infty$ are satisfied
whenever the
${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$-punctures
vanish in the $\tau\to i\infty$ limit. In summary, we find that the subleading
Landau singularities $X_{i=1,2}=0,\infty$ of the sunrise integral are
associated to the points $(\tau,z_{i=1,2})\to(i\infty,0)$ in
$\mathcal{M}_{1,3}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$.

The above analysis of the $\tau\to i\infty$ limit is the first step towards
constructing an ansatz for the pullback of a kinematic quantity to
$\mathcal{M}_{1,3}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$.
For example, from (112b), we know that the form $\text{d}\log(X_{1})$ should
have simple poles whenever $z_{1}$ and $z_{1}+z_{2}$ vanish. Therefore, we
conclude that $z_{1}$ and $z_{1}+z_{2}$ should appear in the arguments of the
$g$-kernels in our ansatz. Indeed, one finds

$\displaystyle\text{d}\log(X_{1})=2\Omega_{2}(z_{1}+z_{2},\tau)-2\Omega_{2}(z_{1},\tau)\,,$
(113)

where each $\Omega_{2}$ is modular invariant.

In contrast to the subleading Landau singularities, the (pseudo-)normal
threshold singularities
$(\pm\pm\pm)=1\pm\sqrt{X_{0}}\pm\sqrt{X_{1}}\pm\sqrt{X_{2}}=0$ (c.f., (91))
arise in the limit $\tau\to 0$ (or equivalently $q\to 1$) giroux_loop-by-
loop_2022 . Since this limit defines the boundary of the analyticity domain of
$\theta_{1}(z,q)$, studying the (pseudo-)normal thresholds on
$\mathcal{M}_{1,3}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
is, strictly speaking, only possible after analytic continuation of
$\theta_{1}(z,q)$ via a modular $S$-transformation
($\tau\mapsto-1/\tau$).888Since the (psudo-)normal thresholds correspond to
$k^{2}\to\infty$, the punctures must also be analytically continued to the
region with $k^{2}>1$ before taking the (psudo-)normal threshold limit.

However, even if we cannot access these singularities without doing a
$S$-transformation, the $\tau\to i\infty$ limit can still be used to find the
corresponding singular locus on
$\mathcal{M}_{1,3}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
since $\text{d}\log(\text{theshold singularity})$ is modular invariant. For
example, consider the complementary elliptic modulus $1-k^{2}$. It is a
rational function of all the polynomials defining the normal and pseudo-normal
thresholds:

$\displaystyle 1-k^{2}=\frac{(--+)(-+-)(+--)(+++)}{(---)(-++)(+-+)(++-)}\,.$
(114)

Multiplying the numerator and the denominator of (114) yields the polynomial

$T=(--+)(-+-)(+--)(+++)(---)(-++)(+-+)(++-)\,,$ (115)

which vanishes at all normal and pseudo-normal thresholds of the
${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$-sunrise.
By pulling back $\text{d}\log T$ and looking at its poles, we learn how the
zeroes of $T$ are embedded into
$\mathcal{M}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}_{1,3}$.
As outlined earlier, to pull back this differential form, we start by
analyzing the $\tau\to i\infty$ limit:

$\displaystyle\lim_{\tau\to i\infty}T=\left[2\sin(\pi z_{1})\sin(\pi
z_{2})\sin^{-1}\big{(}\pi(z_{1}+z_{2})\big{)}\right]^{4}.$ (116)

This suggests that we should use the $Z$-arguments $z_{1}$, $z_{2}$ and
$z_{1}+z_{2}$ in our ansatz. Fixing the coefficients using the PSLQ algorithm,
we find

$\displaystyle\text{d}\log
T=4\Big{[}\Omega_{2}(z_{1}+z_{2},\tau)-\Omega_{2}(z_{1},\tau)-\Omega_{2}(z_{2},\tau)-3\eta_{2}(\tau)\Big{]}\,,$
(117)

where $\eta_{2}$ is a form that only depends on $\tau$ (c.f., (106a)) and is
added _after_ fixing the $\text{d}z$-part. It is interesting that the
$Z$-arguments for the $T=0$ Landau singularities are the same as the combined
$Z$-arguments of the $X_{1}=0$ and $X_{2}=0$ Landau singularities (c.f., eqs.
112b and 112c). In the end, the $Z$-arguments associated with the full Landau
variety in
$\mathcal{M}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}_{1,3}$
are fully determined by the location of the subleading Landau singularities!
The major difference between $\text{d}\log T$ and $\text{d}\log X_{1}$ is the
pure $\text{d}\tau$ term in $\text{d}\log T$. Indeed, the subleading Landau
singularities occur at $\tau\to i\infty$ while the (pseudo-)normal thresholds
occur at $\tau\to 0^{+}$ in our parameterization.

##### Landau analysis of the (1234)-eyeball (and kite)

For integral families such as the eyeball and kite diagrams, the elliptic
structure is inherited from a subtopology. Thus, it is no longer obvious that
the pullback of the entire Landau variety corresponds to the degenerate limits
$\tau^{\alpha}\to i\infty$ or $\tau^{\alpha}\to 0$; components not associated
with the elliptic subtopology could be missing. This opens up the interesting
possibility that part of the Landau variety pulls back to _finite_ (nonzero)
$\tau^{\alpha}$, which we explore further below.

While we will focus on the Landau singularities of the (1234)-eyeball here,
the discussion carries over straightforwardly to the kite. Once again, we
suppress the label
${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
for readability.

Even if there are some components of the Landau variety that are not
associated with the $\tau^{\alpha}\to i\infty$ limit, the $q^{\alpha}$-series
is still extremely useful for determining the location of Landau singularities
as illustrated in (116). For example, using the $\tau\to i\infty$ limit of
$X_{4}$ in (124), we find that the subleading Landau equation $X_{4}=0$ is
associated with the vanishing of $z_{1}$, $z_{1}+z_{2}\pm z_{4}$ and $z_{1}\pm
z_{2}$

$\displaystyle\text{d}\log X_{4}$
$\displaystyle=2\omega_{2}\left(z_{1},2\tau\right){+}2\omega_{2}\left(z_{2},2\tau\right){-}2\omega_{2}\left(z_{1},\tau\right){+}\omega_{2}\big{(}z_{1}{+}\tfrac{z_{2}+z_{4}}{2},\tau\big{)}{+}\omega_{2}\left(z_{1}{+}\tfrac{z_{2}-z_{4}}{2},\tau\right)$
$\displaystyle\quad-\omega_{2}\left(\tfrac{z_{2}-z_{4}}{2},\tau\right)-\omega_{2}\left(\tfrac{z_{2}+z_{4}}{2},\tau\right)-2\omega_{2}\left(z_{1}+z_{2},2\tau\right)-2\eta_{2}\left(\tau\right)\,.$
(118)

Note that the expression above is manifestly modular invariant on the left-
hand side, even though it is not expressed in terms of $\Omega_{2}$’s.

At this stage, one may wonder why we pulled back the complicated polynomial
$T$ in (115) instead of one of its simpler components $(\pm\pm\pm)$. Although
we can use PSLQ to match the $\text{d}z$-part of our ansatz with the
$\text{d}z$-part of $\text{d}\log(\pm\pm\pm)$, it seems that a more robust
ansatz beyond $\eta_{2}$ is needed to match the $\text{d}\tau$-parts. We
report that naively adding Kronecker-Eisenstein forms evaluated at rational
points $\omega_{2}(\mathbbm{Q}+i\mathbbm{Q},N\tau)$ to the PSLQ ansatz does
not seem to solve this issue. In fact, it might be impossible to pull back the
$\text{d}\tau$-part of $\text{d}\log(\pm\pm\pm)$ using only rational
coefficients in the ansatz. Therefore, it currently appears that only the dlog
of _special_ polynomials can be easily pulled back to
$\mathcal{M}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}_{1,3}$.

Interestingly, we can also pull back functions or 0-forms. For example, using
(5.2), we find that

$\displaystyle\frac{1}{X_{4}}$ $\displaystyle=\frac{\partial z_{4}}{\partial
X_{4}}\bigg{[}g^{(1)}(z_{2}{+}z_{4},\tau){-}g^{(1)}(z_{2}{-}z_{4},\tau){+}g^{(1)}(z_{1}{+}\tfrac{z_{2}{+}z_{4}}{2},\tau){-}g^{(1)}(z_{1}{+}\tfrac{1}{2}(z_{2}{-}z_{4}),\tau)\bigg{]}\,.$
(119)

Without knowing how to pull back a dlog-form, the coefficient $\partial
z_{4}/\partial X_{4}$ could be predicted simply by requiring modular
invariance; one would find that the coefficients of a $g$-kernel ansatz needs
to transform like the derivative of a puncture. Furthermore, note that while
$z_{1}$ and $z_{2}$ appear as $Z$-arguments in (5.2), they do not appear as
$Z$-arguments in (119). This suggests that either much of the singularity
structure of the $X_{4}^{-1}$ pullback is hidden inside the partial derivative
$\partial z_{4}/\partial X_{4}$ or that there are some nontrivial
cancellations in the pullback of $\text{d}\log X_{4}$.

Moreover, the pullback of 0-forms in terms of $g$-kernels (e.g., (119)) seems
to be new. That is, we are unaware of other explicit results that convert
rational functions of kinematics $X_{i}$ into linear combinations of
$g$-kernels. It would be interesting if such expressions could help sharpen
statements about the pullback of the Landau variety.

In principle, the list of $Z$-arguments associated with the Landau variety
could be constructed without knowledge of the differential equation; only the
punctures and the Landau variety in $\mathcal{K}$ are needed. This is
essentially what we have done since the singularities appearing in the
pullback of $\mathbf{B}$ are just the Landau singularities in disguise.
However, there is an important advantage to the way the Landau singularities
are organized in the dlog-forms on the diagonal of $\mathbf{B}$: their
arguments are correctly factorized. Identifying and factorizing the square-
root singularities of a Feynman integral is a well-known and difficult problem
that is the subject of ongoing research (see Drummond:2019cxm ; Mago:2020kmp ;
Mago:2021luw ; Ren:2021ztg ; Yang:2022gko ; Henke:2021ity  for some recent
references).

For example, the singularity $\sqrt{\lambda_{134}}=0$ appears in a very
specific way. One instance of this comes from the matrix element $B_{25,25}$
(${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$-eyeball—${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$-eyeball
component of $\mathbf{B}$):

$\displaystyle\text{d}\log\frac{\sqrt{\lambda_{134}}+X_{1}-X_{4}+1}{\sqrt{\lambda_{134}}-X_{1}+X_{4}-1}=\frac{\left(X_{1}+X_{4}-1\right)\text{d}X_{1}-2X_{1}\text{d}X_{4}}{X_{1}\sqrt{\lambda_{134}}}\,.$
(120)

The above dlog form is easy to map to the torus and consequently, the
coefficients of $\text{d}X_{i}$ (rational functions of $X_{i}$) are also easy
to pull back. For example, from the $\text{d}X_{4}$-component, we find

$\displaystyle\frac{-2}{\sqrt{\lambda_{134}}}=\frac{\partial z_{4}}{\partial
X_{4}}\bigg{[}\mathcal{G}^{(1)}_{2}\Big{(}\tfrac{1}{2}(z_{2}+z_{4})\Big{)}+\mathcal{G}^{(1)}_{2}\Big{(}\tfrac{1}{2}(z_{2}-z_{4})\Big{)}\bigg{]}\,,$
(121)

where $\mathcal{G}^{(k)}_{2}(z)=g^{(k)}(z,\tau)-2k\ g^{(k)}(z,2\tau)$ are
simply the linear combinations of $g$-kernels that appear in the components of
the modular invariant differential form $\Omega_{2}(z,\tau)$ in (108).
Interestingly, we find that it is much harder to derive the pullback of
$1/\lambda_{134}$ in terms of $g$-kernels. Of course, we can simply square
both sides of eq. 121 but this would correspond to something with double poles
on the torus, which is unexpected because $1/\lambda_{134}$ has only simple
poles in the roots of $\lambda_{134}$.

On the other hand, it is not too difficult to construct the pullback of
$\text{d}\log\lambda_{134}$:

$\displaystyle\text{d}\log(\lambda_{134})$
$\displaystyle=2\Omega_{2}(z_{1},\tau)+\Omega_{2}(z_{2},\tau)-2\Omega_{2}(z_{1}{+}z_{2},\tau)+\Omega_{2}(z_{4},\tau)-\omega_{2}(z_{2},\tau)$
$\displaystyle\qquad-\omega_{2}(z_{4},\tau)+2\omega_{2}(\tfrac{z_{2}{+}z_{4}}{2},\tau)+2\omega_{2}(\tfrac{z_{2}{-}z_{4}}{2},\tau)-4\eta_{2}(\tau)\,.$
(122)

Once again we note the absence of $Z$-arguments in (121) when compared to
(5.2) and ask whether there are singularities secretly hiding in the partial
derivative $\partial z_{4}/\partial X_{4}$ or if there are nontrivial
cancellations in (5.2).

###  5.3 A warm up: the eyeball pullback

As a warm-up, let’s first exemplify this approach from the perspective of the
$I_{1,1,1,1,0}$ eyeball integral. This integral is naturally expressed on
$\mathcal{M}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}_{1,4}$.
To avoid clutter, within this section we will suppress the
${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$-label
on the variables, since the label
${{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$
is never involved.

##### Building an ansatz

We begin by listing all the distinct, linearly independent combinations of
dlog forms appearing on the diagonal of the eyeball’s row. We find the
following

$\text{d}\log\\{X_{0},\,X_{4},\,\lambda_{024},\,\lambda_{314}\\}\,,$ (123)

where $\lambda_{ijk}=\lambda(X_{i},X_{j},X_{k})$ is defined below (41c). Next,
we change variables from $X_{0}$, $X_{1}$, $X_{2}$, and $X_{4}$ to $\tau$,
$z_{1}$, $z_{2}$, and $z_{4}$ (as previously defined in sections sections 2.2
and 4). Then, we retain only the leading order term of their $q$-expansion and
factorize the $\mathrm{d}\log$ argument into $\sin$ functions. This yields

$\displaystyle\text{d}\log\Big{(}X_{0}|_{q^{2}}\Big{)}$
$\displaystyle=\text{d}\log\Big{(}\sin^{2}\Big{(}\pi
z_{1}\Big{)}\sin^{2}\Big{(}\pi z_{2}\Big{)}\Big{)}\,,$ (124a)
$\displaystyle\text{d}\log\Big{(}X_{4}|_{q^{0}}\Big{)}$
$\displaystyle=\text{d}\log\dfrac{\sin\Big{(}\dfrac{\pi}{2}(2z_{1}{+}z_{2}{+}z_{4})\Big{)}\sin\Big{(}\dfrac{\pi}{2}(2z_{1}{+}z_{2}{-}z_{4})\Big{)}\sin^{2}\Big{(}\pi
z_{2}\Big{)}}{\sin^{2}\Big{(}\pi(z_{1}{+}z_{2})\Big{)}\sin\Big{(}\dfrac{\pi}{2}(z_{2}{+}z_{4})\Big{)}\sin\Big{(}\dfrac{\pi}{2}(z_{2}{-}z_{4})\Big{)}}\,,$
(124b) $\displaystyle\text{d}\log\Big{(}\lambda_{024}|_{q^{0}}\Big{)}$
$\displaystyle=\text{d}\log\dfrac{\sin^{2}\Big{(}\pi
z_{1}\Big{)}\sin^{4}\Big{(}\pi
z_{2}\Big{)}}{\sin^{2}\Big{(}\pi(z_{1}{+}z_{2})\Big{)}\sin^{2}\Big{(}\dfrac{\pi}{2}(z_{2}{+}z_{4})\Big{)}\sin^{2}\Big{(}\dfrac{\pi}{2}(z_{2}{-}z_{4})\Big{)}}\,,$
(124c) $\displaystyle\text{d}\log\Big{(}\lambda_{134}|_{q^{0}}\Big{)}$
$\displaystyle=\text{d}\log\dfrac{\sin^{2}\Big{(}\pi
z_{1}\Big{)}\sin^{2}\Big{(}\pi z_{2}\Big{)}\sin^{2}\Big{(}\pi
z_{4}\Big{)}}{\sin^{2}\Big{(}\pi(z_{1}{+}z_{2})\Big{)}\sin^{2}\Big{(}\dfrac{\pi}{2}(z_{2}{+}z_{4})\Big{)}\sin^{2}\Big{(}\dfrac{\pi}{2}(z_{2}{-}z_{4})\Big{)}}\,.$
(124d)

From (124), we can deduce all the building blocks necessary to fix the ansatz
(110) for all terms in the eyeball row of $\mathbf{B}$. In particular, the
spanning set of forms contains only
$\omega_{k}(\mathcal{L}_{i}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}(z_{1},z_{2},z_{4}),N\tau)$
and $\eta_{k}(\tau)$ where $N=1,2$, $k=1,2,3$ is the modular weight and

$\big{\\{}\mathcal{L}_{i}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}\big{\\}}_{\begin{subarray}{c}i=1\\\
i\neq
4\end{subarray}}^{9}=\Big{\\{}z_{1},z_{2},z_{4},z_{1}+z_{2},\frac{z_{2}+z_{4}}{2},\frac{z_{2}-z_{4}}{2},z_{1}+\frac{z_{2}+z_{4}}{2},z_{1}+\frac{z_{2}-z_{4}}{2}\Big{\\}}\,,$
(125)

are the $Z$-arguments of the Kronecker-Eisenstein forms. Note that the case
$i=4$ is skipped since it is reserved for
$\mathcal{L}_{4}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}=z_{5}$,
which will only appear in the full kite; this way, all single term
$Z$-arguments are indexed first. Furthermore, recall that $\eta_{k}(\tau)$ is,
_a priori_ , included in this basis as a means to account for _pure_
$\mathrm{d}\tau$ corrections in the Kronecker-Eisenstein form ansatz (110).

##### The eyeball pullback

Below, we present the restults for the pullback of all entries that are part
of the (1234)-eyeball line of $\mathbf{B}$. The presentation is organized by
modular weight.

##### _Modular weight 1_

$\begin{split}{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{25,12}}&=2\omega_{1}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{3},\tau\Big{)}\,.\end{split}$
(126)

##### _Modular weight 2_

$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{25,10}}$
$\displaystyle=\dfrac{1}{2}\left[\omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{6},\tau\Big{)}-\omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{7},\tau\Big{)}+\omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{8},\tau\Big{)}-\omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{9},\tau\Big{)}\right]\,,$
(127a)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{25,11}}$
$\displaystyle=\frac{1}{2}\left[3\omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{7},\tau\Big{)}-3\omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{6},\tau\Big{)}+\omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{8},\tau\Big{)}-\omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{9},\tau\Big{)}\right]\,,$
(127b)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{25,17}}$
$\displaystyle=\Omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{6},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{7},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{8},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{9},\tau\Big{)}\,,$
(127c)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{25,18}}$
$\displaystyle=\Omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{7},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{6},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{8},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{9},\tau\Big{)}\,,$
(127d)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{25,24}}$
$\displaystyle=-\Omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{6},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{7},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{8},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{9},\tau\Big{)}\,,$
(127e)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{25,25}}$
$\displaystyle=-2\omega_{2}(\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{1},2\tau)-2\omega_{2}(\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{2},2\tau)-2\omega_{2}(\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{3},\tau)-4\omega_{2}(\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{5},\tau)$
$\displaystyle\quad+6\omega_{2}(\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{5},2\tau)+3\omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{6},\tau\Big{)}+3\omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{7},\tau\Big{)}+\omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{8},\tau\Big{)}$
(127f)
$\displaystyle\quad+\omega_{2}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{9},\tau\Big{)}+6\eta_{2}(\tau)\,.$

The mass-permutation symmetry of the eyeball graph requires
$B_{25,18}=B_{25,24}\bigr{|}_{m_{1}\leftrightarrow m_{3}}$. As discussed in
appendix A, we find that $z_{2}$ and $z_{4}$ are left invariant under the
exchange of $m_{1}$ and $m_{3}$, while $z_{1}\mapsto 1-z_{1}-z_{2}$ under the
same permutation.

##### _Modular weight 3_

$\begin{split}\frac{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{25,9}}}{4}&=\omega_{3}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{3},\tau\Big{)}{-}3\omega_{3}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{6},\tau\Big{)}{+}3\omega_{3}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{7},\tau\Big{)}{-}\omega_{3}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{8},\tau\Big{)}{+}\omega_{3}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{9},\tau\Big{)}\,.\end{split}$
(128)

###  5.4 The kite pullback

Next, we turn to the full kite integral, which is naturally expressed on
either
$\mathcal{M}_{1,5}^{\alpha={{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}$.

##### Building an ansatz

Just as with the eyeball, we now want to determine which $Z$-arguments to feed
into the pullback ansatz for the kite row. Once again, we should list the
minimal set of linearly independent dlog forms, denoted
$\mathfrak{S}_{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}\cup\mathfrak{S}_{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}$.
Here, the entries of
$\mathfrak{S}_{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}$
can naturally be expressed on the
$\mathcal{M}_{1,5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$,
while the entries in
$\mathfrak{S}_{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}$
are naturally expressed on
$\mathcal{M}_{1,5}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$.
The list is as follows

$\mathfrak{S}_{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}\cup\mathfrak{S}_{{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}}=\text{d}\log\\{X_{0},\,X_{1},\,X_{2},\,X_{4},\,X_{5},\,\lambda_{024},\,\lambda_{314},\lambda_{315},\,\lambda_{325},\,\lambda_{01245}\\}\cup\\{\begin{subarray}{c}1\leftrightarrow
4\\\ 2\leftrightarrow 5\end{subarray}\\}\,,$ (129)

where $\lambda_{ijk}$ and $\lambda_{01245}$ are defined below (41c) and in
(62), respectively. Besides the function $\lambda_{01245}$, which we shall
discuss soon, all the above dlog forms may be expressed (up to mass
permutations; see appendix A) as Kronecker-Eisenstein and modular forms using
the results we found for the eyeball (i.e., (124)).

Following the procedure detailed in section 5.2, we find that
$\text{d}\log\lambda_{01245}$ introduces _four_ new $Z$-arguments:
$\mathcal{L}_{10}$, $\mathcal{L}_{11}$, $\mathcal{L}_{12}$, and
$\mathcal{L}_{13}$ (the analogues of equation (124) in this case are too
lengthy to be quoted here). These $Z$-arguments can be naturally expressed on
the moduli space of either torus. Depending on which torus we wish to work on,
we use $\omega_{k}(\mathcal{L}_{i}(z_{1},z_{2},z_{4},z_{5}),n\tau)$ with $n$,
$k$ as before and either

$\displaystyle\big{\\{}\mathcal{L}_{i}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}\big{\\}}_{i=1}^{17}$
$\displaystyle{=}\Big{\\{}z_{1},z_{2},z_{4},z_{5},z_{1}{+}z_{2},\frac{z_{2}{+}z_{4}}{2},\frac{z_{2}{-}z_{4}}{2},z_{1}{+}\frac{z_{2}{+}z_{4}}{2},z_{1}{+}\frac{z_{2}{-}z_{4}}{2},\frac{z_{1}{+}z_{5}}{2},\frac{z_{1}{-}z_{5}}{2},$
(130a) $\displaystyle
z_{2}{+}\frac{z_{1}{+}z_{5}}{2},z_{2}{+}\frac{z_{1}{-}z_{5}}{2},\underbracket{{\frac{z_{1}{{+}}z_{2}{{+}}z_{4}{{+}}z_{5}}{2}},{\frac{z_{1}{{+}}z_{2}{{+}}z_{4}{-}z_{5}}{2}},{\frac{z_{1}{{+}}z_{2}{-}z_{4}{{+}}z_{5}}{2}},{\frac{z_{1}{{+}}z_{2}{-}z_{4}{-}z_{5}}{2}}}_{\text{new
$Z$-arguments on
$\mathcal{M}_{1,5}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}$}}\Big{\\}}\,,$
$\displaystyle\big{\\{}\mathcal{L}_{i}^{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}\big{\\}}_{i=1}^{17}$
$\displaystyle{=}\Big{\\{}z_{4},z_{5},z_{1},z_{2},z_{4}{+}z_{5},\frac{z_{5}{+}z_{1}}{2},\frac{z_{5}{-}z_{1}}{2},z_{4}{+}\frac{z_{5}{+}z_{1}}{2},z_{4}{+}\frac{z_{5}{-}z_{1}}{2},\frac{z_{4}{+}z_{2}}{2},\frac{z_{4}{-}z_{2}}{2},$
(130b) $\displaystyle
z_{2}+\frac{z_{4}{+}z_{2}}{2},z_{2}+\frac{z_{4}{-}z_{2}}{2},\underbracket{{\frac{z_{4}{+}z_{5}{+}z_{1}{+}z_{2}}{2}},{\frac{z_{4}{+}z_{5}{+}z_{1}{-}z_{2}}{2}},{\frac{z_{4}{+}z_{5}{-}z_{1}{+}z_{2}}{2}},{\frac{z_{4}{+}z_{5}{-}z_{1}{-}z_{2}}{2}}}_{\text{new
$Z$-arguments on
$\mathcal{M}_{1,5}^{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}$}}\Big{\\}}.$

with
$(\tau,z_{i})=(\tau^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},z_{i}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}})$
and
$(\tau,z_{i})=(\tau^{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}},z_{i}^{{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}})$,
respectively.

##### The kite on the torus

The entries in the kite row exhibit the following modular structure

$\Big{(}\begin{array}[]{cccccccccccccccccccccccccccccc}-&-&-&-&-&-&-&-&\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}{3}&\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}{2}&\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}{2}&\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}{1}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}{3}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}{2}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}{2}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}{1}&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\text{d}\mathrm{l}}&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\text{d}\mathrm{l}}&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\text{d}\mathrm{l}}&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\text{d}\mathrm{l}}&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\text{d}\mathrm{l}}&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\text{d}\mathrm{l}}&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\text{d}\mathrm{l}}&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\text{d}\mathrm{l}}&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\text{d}\mathrm{l}}&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\text{d}\mathrm{l}}&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\text{d}\mathrm{l}}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}\text{d}\mathrm{l}}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}\text{d}\mathrm{l}}&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\text{d}\mathrm{l}}\end{array}\Big{)}\,.$
(131)

Here, entries marked with “$-$” vanish, those labeled as “dl” represent dlog
forms, and finally, entries denoted with $1,2,3$ are quasi-modular forms of
the corresponding modular weight. Additionally, the color coding used here is
as before: the entries in red are pulled back onto
$\mathcal{M}_{1,5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$,
while those in blue are pulled back onto
$\mathcal{M}_{1,5}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$.
The dlog terms could be pulled back onto either
$\mathcal{M}_{1,5}^{\alpha={{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}$;
the predominance of red for most dlog entries is simply _a choice_.999When
integrating the differential equation in terms of iterated integrals, this
choice may have important effects as it can lead to mixing of the elliptic
curves; we will get back to this issue later in section 6. The few blue dlog
entries are specifically associated with the eyeballs that include the
${{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$-sunrise
as a subtopology.

All entries in this row are provided below explicitly. Note that we use the
labels
${{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}$
or
${{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}$
on the $Z$-arguments $\mathcal{L}_{i}^{(ijk)}$ to indicate which torus the
entry is pulled back onto, and we simply write $\tau$ for the respective
moduli, omitting the label. As before, we organize the presentation by modular
weight and use the shorthand
$\tilde{B}_{ij}=B_{ij}|_{\mathcal{L}_{i}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}\rightarrow\mathcal{L}_{i}^{{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}},\tau^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}\rightarrow\tau^{{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}}}$.

##### _Modular weight 1_

$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,12}}$
$\displaystyle=\omega_{1}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{1},\tau\Big{)}+\omega_{1}\Big{(}\mathcal{L}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}}_{2},\tau\Big{)}\,,$
(132a)
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}B_{30,16}}$
$\displaystyle=\tilde{B}_{30,12}=\omega_{1}\Big{(}\mathcal{L}^{{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}}_{1},\tau\Big{)}+\omega_{1}\Big{(}\mathcal{L}^{{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}}_{2},\tau\Big{)}\,.$
(132b)

##### _Modular weight 2_

$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,10}}$
$\displaystyle=\frac{1}{4}\Big{[}\sum_{k={8,9,12,13}}\omega_{2}\Big{(}\mathcal{L}_{k}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\sum_{\ell={6,7,10,11}}\omega_{2}\Big{(}\mathcal{L}_{\ell}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\Big{]}\,,$
(133a)
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}B_{30,14}}$
$\displaystyle=\tilde{B}_{30,10}\,,$ (133b)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,11}}$
$\displaystyle=\frac{1}{4}\Big{[}\sum_{k={8,9,10,11}}\omega_{2}\Big{(}\mathcal{L}_{k}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\sum_{\ell={6,7,12,13}}\omega_{2}\Big{(}\mathcal{L}_{\ell}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\Big{]}\,,$
(133c)
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}B_{30,15}}$
$\displaystyle=\tilde{B}_{30,11}\,,$ (133d)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,17}}$
$\displaystyle=\frac{1}{2}\Big{[}\Omega_{2}\Big{(}\mathcal{L}_{6}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{7}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{8}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{9}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\Big{]}\,,$
(133e)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,18}}$
$\displaystyle=-\frac{1}{2}\Big{[}\Omega_{2}\Big{(}\mathcal{L}_{6}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{7}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{8}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{9}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\Big{]}\,,$
(133f)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,19}}$
$\displaystyle=\Omega_{2}\Big{(}\mathcal{L}_{10}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{11}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\,,$
(133g)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,20}}$
$\displaystyle=\Omega_{2}\Big{(}\mathcal{L}_{6}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{7}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\,,$
(133h)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,21}}$
$\displaystyle=\frac{1}{2}\Big{[}\Omega_{2}\Big{(}\mathcal{L}_{10}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{11}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{12}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{13}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\Big{]}\,,$
(133i)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,22}}$
$\displaystyle=-\frac{1}{2}\Big{[}\Omega_{2}\Big{(}\mathcal{L}_{10}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{11}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{12}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{13}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\Big{]}\,,$
(133j)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,23}}$
$\displaystyle=\frac{1}{2}\Big{[}\Omega_{2}\Big{(}\mathcal{L}_{12}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{13}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{10}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{11}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\Big{]}\,,$
(133k)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,24}}$
$\displaystyle=\frac{1}{2}\Big{[}\Omega_{2}\Big{(}\mathcal{L}_{8}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{9}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{6}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{7}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\Big{]}\,,$
(133l)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,25}}$
$\displaystyle=\frac{1}{2}\Big{[}\sum_{k={6,8,16,17}}\omega_{2}\Big{(}\mathcal{L}_{k}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\sum_{\ell={7,9,14,15}}\omega_{2}\Big{(}\mathcal{L}_{\ell}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\Big{]}\,,$
(133m)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,26}}$
$\displaystyle=\frac{1}{2}\Big{[}\sum_{k={10,12,15,17}}\omega_{2}\Big{(}\mathcal{L}_{k}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\sum_{\ell={11,13,14,16}}\omega_{2}\Big{(}\mathcal{L}_{\ell}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\Big{]}\,,$
(133n)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,27}}$
$\displaystyle=\frac{1}{2}\Big{[}\Omega_{2}\Big{(}\mathcal{L}_{14}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{15}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{16}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{17}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\tau\Big{)}\Big{]}\,,$
(133o)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,28}}$
$\displaystyle=\frac{1}{2}\Big{[}\Omega_{2}\Big{(}\mathcal{L}_{14}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\tau\Big{)}+\Omega_{2}\Big{(}\mathcal{L}_{15}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{16}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{17}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\tau\Big{)}\Big{]}\,,$
(133p)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,29}}$
$\displaystyle=-\frac{1}{2}\Big{[}\Omega_{2}\Big{(}\mathcal{L}_{14}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{15}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{16}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\Omega_{2}\Big{(}\mathcal{L}_{17}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\Big{]}\,,$
(133q)
$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,30}}$
$\displaystyle=-2\omega_{2}\Big{(}\mathcal{L}_{1}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},2\tau\Big{)}-2\omega_{2}\Big{(}\mathcal{L}_{2}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},2\tau\Big{)}+6\omega_{2}\Big{(}\mathcal{L}_{5}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},2\tau\Big{)}-2\omega_{2}\Big{(}\mathcal{L}_{5}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}$
$\displaystyle+2\omega_{2}\Big{(}\mathcal{L}_{6}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+2\omega_{2}\Big{(}\mathcal{L}_{7}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+2\omega_{2}\Big{(}\mathcal{L}_{10}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}+2\omega_{2}\Big{(}\mathcal{L}_{11}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}$
(133r)
$\displaystyle{-}\omega_{2}\Big{(}\mathcal{L}_{14}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}{-}\omega_{2}\Big{(}\mathcal{L}_{15}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}{-}\omega_{2}\Big{(}\mathcal{L}_{16}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}{-}\omega_{2}\Big{(}\mathcal{L}_{17}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}{+}6\eta_{2}(\tau)\,.$

##### _Modular weight 3_

$\displaystyle{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}B_{30,9}}$
$\displaystyle=2\Big{[}\sum_{\ell=14}^{17}\omega_{3}\Big{(}\mathcal{L}_{\ell}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}-\sum_{\ell=5}^{13}\omega_{3}\Big{(}\mathcal{L}_{\ell}^{{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}},\tau\Big{)}\Big{]}\,,$
(134a)
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}B_{30,13}}$
$\displaystyle=\tilde{B}_{30,9}\,.$ (134b)

##  6 Boundary value and integration

In this section, we provide a boundary value for the kite integral basis in
(79). We also detail the (numerical) integration technique, highlighting
certain technical challenges. Subsequently, we elaborate on the stress tests
conducted on the integrated results within a simplified region of the
kinematic space and conclude with an outlook on numerical integration
improvements.

###  6.1 Boundary value

We compute the boundary value in a region of the kinematic space where all the
master integrals are finite and simple. A suitable choice for this is the
kinematic point $\mathbf{X}_{0}\in\mathcal{K}$, located in the neighborhood

$m_{i}=m>0\quad\text{and}\quad p^{2}\to 0^{+}\qquad\forall~{}i\,.$ (135)

At $\mathbf{X}_{0}$, the master basis in (79) becomes significantly simpler,
primarily due to the identities

$\frac{\partial X_{i}}{\partial\tau^{\alpha}}=\left.\frac{\partial
X_{0}}{\partial\tau^{\alpha}}\right|_{\mathbf{X}_{0}}=0\qquad\text{for}\quad
i\neq 0\quad\text{and}\
\alpha={{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}\,,$
(136)

which is easily checked numerically. Analytically, these identities follow
from the observation that, in the neighborhood of $\mathbf{X}_{0}$, we have
$X_{i}\to 1$ for all $i\neq 0$ and approach

$\displaystyle\tau^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}$
$\displaystyle=\tau^{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}=i\infty\,,$
$\displaystyle
z_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
$\displaystyle=z_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{4}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=z_{5}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=\frac{1}{3}\,,$
(137) $\displaystyle
z_{4}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
$\displaystyle=z_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{1}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=z_{2}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=\frac{1}{2}-i\infty\,.$

Thus, the only nonzero components of the basis at $\mathbf{X}_{0}$ are

$\begin{split}\mathbf{J}_{0}=\varepsilon^{2}\Big{(}\underbracket{J_{1}}_{\times
8},\frac{\sqrt{3}J_{2}}{2},\underbracket{0}_{\times
2},-\frac{\sqrt{3}J_{2}}{4},\frac{\sqrt{3}J_{2}}{2},\underbracket{0}_{\times
2},-\frac{\sqrt{3}J_{2}}{4},i\sqrt{3}J_{2},\underbracket{0}_{\times
3},i\sqrt{3}J_{2},\underbracket{0}_{\times 9}\Big{)}\,,\end{split}$ (138)

where $J_{1}=I_{1,1,0,0,0}$ and $J_{2}=I_{1,1,1,0,0}$. In this case, we see
that there are only two101010This counting takes into account IBP relations
between the integrals at $\mathbf{X}_{0}$. It also accounts for the fact that
$I_{0,1,1,0,1}$ converges to the sunrise $I_{1,1,1,0,0}$ in a sufficiently
small neighborhood of $\mathbf{X}_{0}$. This can be directly observed through
an analysis by regions and by noting that both integrals satisfy the same
Symanzik polynomials in the contributing region. out of the 30 integrals in
(1) to evaluate explicitly in the neighborhood of $\mathbf{X}_{0}$.

The double-tadpole $I_{1,1,0,0,0}$ is independent of $p^{2}$ and is easily
evaluated in $d=2-2\varepsilon$ dimensions using Feynman parameters. The
result is

$J_{1}=I_{1,1,0,0,0}(2-2\varepsilon;\mathbf{X}_{0})=e^{2\gamma_{\text{EM}}\varepsilon}\Gamma^{2}(\varepsilon)\,.$
(139)

Next, we evaluate the sunrise integral around $\mathbf{X}_{0}$ using the
method of regions Beneke:1997zp ; Jantzen:2011nz ; Jantzen:2012mw . We find
that there is only one region contributing to the integral. Consequently, the
parametric representation of the integrand can be expanded at leading order in
$p^{2}$ before loop integration. Moreover, performing the change of variables
in (Adams:2013nia, , eq. (74)) on the expanded integrand yields a fully off-
shell 1-loop triangle integrand with massless internal states in
$4+2\varepsilon$ dimensions, which is given in (Bern:1993kr, , App. V).
Putting everything together, we obtain the exact result

$J_{2}=I_{1,1,1,0,0}(2{-}2\varepsilon;\mathbf{X}_{0})=\mathcal{N}\Big{[}\left(-e^{\frac{2i\pi}{3}}\right)^{-\varepsilon}F_{\varepsilon}\left({\tfrac{2i\pi}{3}}\right){-}\left(-e^{-\frac{2i\pi}{3}}\right)^{-\varepsilon}F_{\varepsilon}\left({-\tfrac{2i\pi}{3}}\right){+}\frac{\pi}{\varepsilon}\Big{]}\,,$
(140)

where
$\mathcal{N}=\tfrac{e^{2\gamma_{\text{EM}}\varepsilon}\Gamma(1{+}2\varepsilon)}{(-1)^{1+2\varepsilon}3^{\frac{1}{2}+\varepsilon}}$
and
$F_{\varepsilon}(z)=\frac{3i\Gamma^{2}(\varepsilon+1)}{2\varepsilon^{2}\Gamma(2\varepsilon+1)}\,_{2}F_{1}(-2\varepsilon,-\varepsilon,1-\varepsilon,e^{z})$.
The $\varepsilon$-expansion of both (139) and (140) can be easily performed
using the identity Adams:2015ydq

$\displaystyle{}_{2}F_{1}\left(-2\varepsilon,-\varepsilon;1-\varepsilon;x\right)$
$\displaystyle=1+2\varepsilon^{2}\mathrm{Li}_{2}\left(x\right)+\varepsilon^{3}\left[2\mathrm{Li}_{3}\left(x\right)-4\mathrm{Li}_{2,1}\left(x,1\right)\right]$
$\displaystyle+\varepsilon^{4}\left[2\mathrm{Li}_{4}\left(x\right)-4\mathrm{Li}_{3,1}\left(x,1\right)+8\mathrm{Li}_{2,1,1}\left(x,1,1\right)\right]+{\mathcal{O}}\left(\varepsilon^{5}\right)\,,$
(141)

where $\mathrm{Li}_{n_{1},n_{2},...,n_{k}}$ denotes the classical multiple
polylogarithm. Explicit expressions for the expansion can be found in the
ancillary file  and were verified numerically up to weight three using
PySecDec Borowka:2017idc ; Heinrich:2023til .

###  6.2 Discussion: numerical integration, checks and outlook

An important motivation for deriving the differential equation in (79) and the
boundary condition in (138) is to systematize the numerical evaluation of the
master integrals to which they correspond. The starting point is the formal
solution in (10).

For elliptic Feynman integrals, one strategy is to express the iterated
integrals that show up in the solution in terms of _elliptic multiple
polylogarithms_ Broedel:2014vla ; Broedel:2017kkb ; Walden:2020odh ;
weinzierl2022feynman

$\tilde{\Gamma}\left(\left.\begin{subarray}{c}n_{1}\\\
w_{1}\end{subarray}\begin{subarray}{c}n_{2}\\\
w_{2}\end{subarray}\begin{subarray}{c}...\\\
...\end{subarray}\begin{subarray}{c}n_{k}\\\
w_{k}\end{subarray};w\right|\tau\right)=\int_{0}^{w}\textnormal{d}w^{\prime}\,g^{(n_{1})}(w^{\prime}{-}w_{1},\tau)\tilde{\Gamma}\left(\left.\begin{subarray}{c}n_{2}\\\
w_{2}\end{subarray}\begin{subarray}{c}...\\\
...\end{subarray}\begin{subarray}{c}n_{k}\\\
w_{k}\end{subarray};w^{\prime}\right|\tau\right)\,,$ (142)

if we integrate along a path in $z$-space or in terms of _iterated integrals
over modular forms_ $\\{f_{i}\\}$ and ELi _-series_ Adams:2014vja ;
Adams:2015gva ; Adams:2015ydq ; bogner_unequal_2020 ; Weinzierl:2020fyx ;
Walden:2020odh ; weinzierl2022feynman  if we integrate along a path in
$\tau$-space

$\displaystyle\mathfrak{M}(f_{1},f_{2},...,f_{k}|\tau)$
$\displaystyle=\displaystyle\int\limits_{i\infty}^{\tau}f_{1}\left(t_{1}\right)\displaystyle\int\limits_{i\infty}^{t_{1}}f_{2}\left(t_{2}\right)...\displaystyle\int\limits_{i\infty}^{t_{k-1}}f_{k}\left(t_{k}\right)\,,$
(143a)
$\displaystyle\mathrm{ELi}_{n_{1},...,n_{\ell};m_{1},...,m_{\ell};\sigma_{1},...,\sigma_{\ell-1}}^{x_{1},...,x_{\ell};y_{1},...,y_{\ell};\bar{q}}$
$\displaystyle=\prod_{\alpha=1}^{\ell}\sum_{\begin{subarray}{c}j_{\alpha}=1\\\
k_{\alpha}=1\end{subarray}}^{\infty}\frac{x_{\alpha}^{j_{\alpha}}}{j_{\alpha}^{n_{\alpha}}}\frac{y_{\alpha}^{k_{\alpha}}}{k_{\alpha}^{m_{\alpha}}}\bar{q}^{j_{\alpha}k_{\alpha}}\prod\limits_{i=1}^{\ell-1}\left(j_{i}k_{i}{+}...{+}j_{\ell}k_{\ell}\right)^{-\sigma_{i}/2}\,,$
(143b)

where $\bar{q}=e^{2\pi i\tau}$. Their convergence properties are summarized in
Walden:2020odh . Of course, the high-precision numerical evaluation of (142)
and (143) comes with few obvious challenges, which we enumerate below.

##### Technical challenge I: _elliptic curves mixing_

In cases where many elliptic curves are involved, the function space spanned
by (142) and (143) is sufficient _only_ if the quantities defined on
$\mathcal{M}_{1,5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
and
$\mathcal{M}_{1,5}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$
fully decouple.

As for the kite family, the two sources of elliptic functions in the
differential equation, namely the sunrises _rectangular_ (shaded) blocks in

$\mathbf{B}=\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[scale={1}]{bmat.pdf}}}}}}}$}\,,$
(144)

completely decouple. Here, as before, “$\text{d}\mathrm{l}$” indicates that
the entry is a dlog form in kinematic variables, while the “$-$” entries
vanish. Otherwise, the number gives the (quasi-)modular weight of the entry.
Below, it is assumed that these non-dlog entries have _already_ been pulled
back to the moduli space of their respective torus:
$\mathcal{M}_{1,5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
for entries overlapping with the red-shaded region and
$\mathcal{M}_{1,5}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$
for those overlapping with the blue-shaded region.

Now, from (144), one can show to arbitrarily high order in the solution’s
Laurent expansion that iterated integrals, which can potentially depend
simultaneously on both
${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
and
${{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$,
appear _only_ through the interaction of dlog entries with any entries in
either the blue or red shaded rectangular blocks. From this perspective, a
simple strategy to prevent mixing of
${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
and
${{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$
becomes quite clear: we always have the freedom to pullback the dlog form on
the torus so that no mixing occurs.

Although conceptually clear, this way of organizing the calculation requires
keeping track of _local_ choices for the pullback of the dlog forms at the
level of each iterated integral order by order in the $\varepsilon$-expansion.
Of course, from a technical point of view, this quickly becomes inefficient.
To us, this serves as a first indication that the approach of performing the
numerical evaluation of the master basis through that of (142) and (143) might
not be optimal when _multiple_ tori are involved. A potential alternative
direction is mentioned towards the end of this section.

##### Technical challenge II: _convergence_

Once we have ensured that
${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
and
${{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$
decouple in the iterated integrals, there is yet another technical challenge
to face when numerically evaluating the master integrals from (142) and (143).
Similarly to Feynman integrals that evaluate to MPLs, we often want to
evaluate these integrals at points near or beyond the boundary of their region
of convergence (for methods addressing these issues more formally, see, for
example, Passarino:2016zcd ; Duhr:2019rrs ).

For the kite master, this occurs, for example, when the differential equation
is solved to finite $\tau$’s starting from (6.1). Because
$z_{4}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{1}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=z_{2}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=\frac{1}{2}-i\infty$
remain fixed, the leading-order solution

$\displaystyle J_{30}^{(2)}$
$\displaystyle{=}\sum_{j=6}^{9}{\Delta_{6,7}^{j}}\Big{(}\left[\Omega_{2,j,1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\Omega_{2,1,1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}\right]{-}2\left[\Omega_{2,j,1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\Omega_{2,1,2}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}\right]{-}\left[\Omega_{2,j,1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\Omega_{2,5,1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}\right]{+}2\left[\Omega_{2,j,1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\Omega_{2,5,2}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}\right]\Big{)}$
(145)
$\displaystyle{+}\sum_{j=10}^{13}{\Delta_{10,11}^{j}}\Big{(}\left[\Omega_{2,j,1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\Omega_{2,2,1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}\right]{-}2\left[\Omega_{2,j,1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\Omega_{2,2,2}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}\right]{-}\left[\Omega_{2,j,1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\Omega_{2,5,1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}\right]{+}2\left[\Omega_{2,j,1}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\Omega_{2,5,2}^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}\right]\Big{)}$
$\displaystyle+({{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}})\,,$

where $\Delta_{n,m}^{j}=(-1)^{\delta_{j,n}{+}\delta_{j,m}}$,
$\Omega_{a,b,c}^{d}=\omega_{a}(\mathcal{L}_{b}^{d},c~{}\tau_{d})$
and111111Note that, in practice, the lower bound of the integration domain
($\tau=i\infty$ or $q=0$) may introduce (spurious) poles. These singularities
are typically dealt with through the use of tangential base-point
regularization scheme deligne1989groupe ; Brown:2014pnb ; Panzer:2015ida ;
Bonisch:2021yfw .

$[w_{1},w_{2},...,w_{k}]=\displaystyle\int\limits_{i\infty}^{\tau}w_{1}\left(t_{1}\right)\displaystyle\int\limits_{i\infty}^{t_{1}}w_{2}\left(t_{2}\right)...\displaystyle\int\limits_{i\infty}^{t_{k-1}}w_{k}\left(t_{k}\right)\,,$
(146)

once expressed, for example, through ELi-series (c.f., (143b)), does not
converge _numerically_. Naturally, this divergence arises because the solution
involves a series representation that extends beyond its radius of
convergence; in this particular case, the use of the $g$-kernels
quasiperiodicity in $\tau$ resolves the issue (see (weinzierl2022feynman, ,
eq. (13.200))). Furthermore, such divergences are absent from subtopologies
involving three or fewer propagators, corresponding to master integrals
independent of the “extra” punctures
$z_{4}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{1}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=z_{2}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=\frac{1}{2}-i\infty$.
In particular, we find that the sunrise components after the $\tau$
integrations match (bogner_unequal_2020, , eq. (106)), as expected.

##### Numerical checks in the soft ($p^{2}=0^{+}$) regime

For the kite family, a simple alternative to avoid the need of continuing the
integration path beyond the region of convergence of (142) and (143) is to
_first_ integrate the differential equation from (6.1) to finite values of
$z_{4}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},z_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},z_{1}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$
and
$z_{2}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$,
for example, setting
$z_{4}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{1}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=z_{2}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=\frac{1}{2}$,
while keeping everything else constant.

A technical benefit of performing the integration at fixed
$\tau^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=\tau^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=i\infty$
is the significant simplification that occurs at the level of the pullback
differential equation (144). In fact, along the path $\gamma_{0}$
parameterized by the straight line
$z_{4}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{1}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=z_{2}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=t$,
where $t$ varies from $\frac{1}{2}-i\infty$ to $\frac{1}{2}$, the matrix
elements are purely trigonometric and the resulting iterated integrals combine
_nontrivially_ into finite quantities. The result can then be efficiently
computed using Mathematica’s NIntegrate[] function, as exemplified in the
ancillary file .

At this point, let us emphasize that the map121212To be more precise, the
relevant map is obtained by solving simultaneously the equations
$\text{sn}^{2}\left(\theta_{3}^{2}\left(0,e^{i\pi\tau_{\alpha}}\right)\frac{\pi
z_{i}^{\alpha}}{2},\frac{\theta_{2}^{4}\big{(}0,e^{i\pi\tau_{\alpha}}\big{)}}{\theta_{3}^{4}\left(0,e^{i\pi\tau_{\alpha}}\right)}\right)=u_{i}^{\alpha}(\boldsymbol{x})\qquad\forall~{}i=1,2,4,5~{}\text{and}~{}\alpha={{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}\,,$
for $\boldsymbol{x}=\\{X_{0},X_{1},X_{2},X_{4},X_{5}\\}$. Here,
$\text{sn}(x,k)$ denotes the Jacobi sn-function, and the $u$’s are given in
eqs. 29, 50, 95 and 96. The above equation is nothing but the formal inverse
of, e.g., (28) and is easily checked numerically. between the space
$\mathfrak{M}=\mathcal{M}_{1,5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\times\mathcal{M}_{1,5}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$
(parametrized by the moduli spaces of the two tori) and the kinematic space
$\mathcal{K}$ is _not_ invertible everywhere. For example, the map fails to be
surjective because, while $\mathfrak{M}$ is 10-dimensional, $\mathcal{K}$ is
only 5-dimensional. (This phenomenon is intrinsically related to the fact that
we are considering a multiscale problem with _multiple_ tori. In particular,
it does not occur for the unequal mass sunrise discussed in
bogner_unequal_2020  because the number of kinematic degrees of freedom
happens to match that on the torus, i.e., the map is an isomorphism.) For this
reason, it turns out that the point

$\begin{split}\tau^{{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}}&=\tau^{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}}=i\infty\,,\\\
z_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}&=z_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{4}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=z_{5}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=\frac{1}{3}\,,\\\
z_{4}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}&=z_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{1}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=z_{2}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=\frac{1}{2}\,,\end{split}$
(147)

we land on after integrating along $\gamma_{0}$ is _not_ the image of any
(complex) point in $\mathcal{K}$.131313This can be verified using, e.g.,
Mathematica’s FindInstances[]. This implies that the numerical result does not
coincide with the master integrals evaluated over $\mathcal{K}$, but instead
corresponds to their extensions to $\mathfrak{M}$. Practically, this makes
direct comparisons with software like PySecDec Heinrich:2023til  unfeasible
for some coordinate values in $\mathfrak{M}$ such as (147).

Nonetheless, we can validate our results by moving from (147) to points in the
$\mathfrak{M}$ space that have support in $\mathcal{K}$. By restricting
ourselves to the submanifold of $\mathfrak{M}$ where
$\tau^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=\tau^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=i\infty$,
we can efficiently obtain numbers using the same method as outlined earlier
(integrating successively along straight lines in $\mathfrak{M}$ with
NIntegrate[]).141414In doing so, one must ensure that the differential
equation on $\mathfrak{M}$ (depending on 10 variables) remains integrable
along each integration path. For instance, the master integrals computed at
the particular point

$\begin{split}\tau^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}&=\tau^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=i\infty\,,\quad
z_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{4}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=1/4\,,\quad
z_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{5}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=1/2\,,\\\
z_{4}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=&z_{1}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=-\frac{2i\log(i\sqrt{2+\sqrt{3}})}{\pi}\,,\quad
z_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=z_{2}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=\frac{i\log(8+3\sqrt{7})}{2\pi}\,,\end{split}$
(148)

(corresponding to
$\mathbf{X}=\\{X_{0},X_{1},X_{2},X_{4},X_{5}\\}=\\{0^{+},1,2,1,2\\}$ in
$\mathcal{K}$) perfectly agrees with the prediction of PySecDec at leading and
subleading order in $\varepsilon$. Similar checks were carried out
successfully at various other points within the soft region.

##### Outlook: _improving numerics_

We conclude this section by noting that, outside the _simplifying_ soft limit
$\tau^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}=\tau^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}=i\infty$,
we found it impractical to apply the NIntegrate[]-based method (used above)
for high-precision numerical evaluation of iterated integrals. The primary
hindrance being the slow convergence caused by the integration kernels being
elliptic ($g$-kernels) rather than trigonometric.

Although numerical values can, in principle, be obtained by directly
evaluating iterated integrals in the form of (142) and (143), we report that
this approach converges too slowly using the available tools (e.g.,
Walden:2020odh ) to make _satisfactory_ comparisons of our results with those
found in the literature Broadhurst:2022bkw  or through PySecDec within
_generic_ kinematic regions.

This is a situation that we wish to improve in the near future. In particular,
we believe that there are more efficient and generalizable approaches to the
problem of numerical integration of differential equations defined on moduli
spaces. Prime candidates are generalizations of the series methods employed,
e.g., in Caron-Huot:2020vlo ; Abreu:2020jxa ; Hidding:2020ytt ;
Armadillo:2022ugh  for the high-precision evaluation of polylogarithmic
Feynman integrals. For all of these approaches to work, the only inputs
required are a differential equation, a boundary value, and a path of
integration, thus bypassing the need for an explicit evaluation of MPLs. In
this regard, potential generalizations to _elliptic_ Feynman integrals are
exciting because they are meant to be agnostic of elliptic curve mixing and of
explicit evaluation of eMPLs in generic kinematic regions.

##  7 Conclusion

In this paper, we present the first analytic results for the 5-mass kite
family of Feynman integrals near two space-time dimensions using the method of
differential equations.

Our approach involved investigating the relationship between two elliptic
curves associated with the sunrise subsectors of the 5-mass kite family of
Feynman integrals. In particular, we introduced a method to construct an
$\varepsilon$-form basis, leveraging the underlying $\text{SL}(2,\mathbbm{Z})$
covariance of these integrals. Moreover, we identified a relevant
parameterization of the kinematic space in terms of the moduli and location of
punctures on a torus. The puncture locations were physically motivated and
take the form of integrals over maximal cuts. From there, we described how to
systematically pull back the $\varepsilon$-form differential equation from
kinematic space to a much more compact form involving both modular and KE
forms on the moduli space of the tori. Specifically, we pointed out how to
identify the relevant set of $Z$-arguments (combinations of punctures to feed
the ansatz of KE forms) through a simple analysis of the $q$-expansions of the
_diagonal_ entries of the differential equation. The output of this procedure
is one of the main results of this paper: a schematic representation of the
differential equation is quoted in (144) and two of its rows are explicitly
recorded in eqs. 126, 127, 128, 132, 133 and 134. The full matrix in can be
found in the ancillary files: . We further leveraged this analysis to sharpen
the connection between Landau singularities in the kinematic space
$\mathcal{K}$ with those on the moduli space. This was examplified by deriving
explicit expressions for various polynomials that define the Landau variety
and their associated dlog-forms (c.f., eqs. 117, 5.2, 119, 121 and 5.2) in
terms of $g$-kernels and modular forms. To our knowledge, this is the first
time that such relations between 0-forms in $\mathcal{K}$ and $g$-kernels have
been explicitly written down. Lastly, we derived a boundary value for the kite
family of master integrals (see (138)) that facilitates its numeric evaluation
and a cross-check of our results in a particular simplifying limit (unequal
mass soft limit). Our hope is that these developments lead to a number of new
directions, some of which are briefly discussed below.

We anticipate that the methods introduced in this paper will also be
applicable to more general multiscale 2-loop examples involving one or more
elliptic curves. Now, since the kite is the most general 2-point function (in
cubic theories), natural candidates for future work are the 3- and 4-point
analogues: the 4-mass parachute (see (Fevola:2023fzn, , fig. 1 (h))) and the
5-mass crossed-box/acnode graph (see (Mizera:2021ujs, , fig. 3)),
respectively. In particular, the latter still involves two elliptic curves and
five masses, but also depends on an additional variable: the momentum transfer
$t$.

Furthermore, it would be interesting to explore how the ideas introduced in
this paper could be applied to examples more directly related to Standard
Model processes. A prime example is the 3-parameter double box, which was
recently discussed in G_rges_2023 . As highlighted in this reference, this
graph is associated with an elliptic curve and contributes to the planar
corrections of H+jet production through a loop of massive (top) quarks “cut in
half” vertically by a gluon. At LHC energies ($\sqrt{s}\approx 13$ TeV), the
cross section for H+jet production is about half of the inclusive Higgs
production rate. Although amplitudes for this process have been obtained at
NNLO Harlander:2002wh ; Anastasiou:2002yz ; Ravindran:2003um ;
Bonciani:2019jyb , most of these calculations have been performed in the large
top-quark mass effective theory, where the top quark is integrated out and the
Higgs boson directly couples to gluons through higher-dimensional operators
(for lower order results in the full theory see, e.g., Ellis:1987xu ;
Neumann:2018bsx ; Jones:2018hbb ; Bonciani:2022jmb ). However, these effective
calculations are expected to break down for jet energy _greater_ than $\sim
1.6$ times the Higgs mass $m_{H}=125~{}\text{GeV}$ Grazzini:2013mca ;
Neumann:2014nha ; Dawson:2014ora . In view of the forthcoming high-luminosity
phase of the LHC and higher-energy muon colliders, it seems sensible to direct
future efforts towards lower-order perturbative calculations within the _full_
theory. At present, a complete NNLO calculation is only conceivable
numerically, although an $\varepsilon$-form differential equation is known
G_rges_2023 .

While systematic, the $\varepsilon$-form algorithm of G_rges_2023  introduces
integrals over a rational function times the period of the elliptic curve.
However, the relationship between these integrals and the class of functions
discussed earlier in (142) and (143) is rather obscure. It is natural to ask
to what extent these functions are related to $g$-kernels and modular forms.

It would also be interesting to further solidify our understanding of the
interplay between the $Z$-arguments on the tori’s moduli space and Landau
singularities in kinematic space. Although a partial analysis for the kite
integral was initiated in section 5, investigating whether $Z$-arguments can,
in some way, be derived more directly from Landau analysis is an exciting
avenue for future research. This is particularly relevant for the Symbol Prime
bootstrap Wilhelm:2022wow ; Morales:2022csr ; McLeod:2023qdf ; Cao:2023tpx ;
He:2023umf . Since it is possible to extract the symbol of the physical non
$\varepsilon$-form kite integral $I_{1,1,1,1,1}$ from the boundary condition
and differential equation provided in this work (the gauge transformation to
$\varepsilon$-form must be undone), it would be interesting to compare our
symbol with the result from a bootstrap.

As discussed in section 6, verifying our results involved a challenging high-
precision numerical evaluation of the basis of master integrals by solving the
pulled back differential equation in _generic_ kinematic regions. In
particular, using publicly available tools (see Walden:2020odh ), we found
that the numerical evaluation of the eMPLs entering the evaluation of the kite
master integral converged very slowly. This made the comparison with the
literature Broadhurst:2022bkw  and PySecDec (public numeric evaluation
software) in generic kinematic regions particularly difficult and, in our
view, unsatisfactory. A pressing problem is therefore to clarify how to
efficiently extract high-precision numerical values for multiscale elliptic
Feynman integrals given a differential equation written in KE and modular
forms. A promising direction involves the development of a series method that
extends those established in Caron-Huot:2020vlo ; Abreu:2020jxa ;
Hidding:2020ytt  to the context of elliptic Feynman integrals.

Lastly, looking beyond the elliptic world, it would also be interesting to see
if the method introduced in this paper can be generalized to more complicated
higher-genus geometries. For example, given a Feynman diagram involving a
hyperelliptic curve (see Marzucca:2023gto  for examples), is there a
systematic way to define punctures when needed? (Note that, while the
generalization of modular and KE forms to higher-genus surfaces is still under
extensive investigation DHoker:2023vax ; DHoker:2023khh , we find it
legitimate to ask this question even at this early stage). Of course, once an
appropriate set of differential forms is established, extending our method to
higher-dimensional geometries, such as Calabi-Yau spaces, also presents a
natural avenue for future research.

## Acknowledgements

We thank Claude Duhr for the suggestion of this project, as well as for many
fruitful discussions. We also thank Babis Anastasiou, Simon Caron-Huot, Miguel
Correia, Christoph Nega, Oliver Schlotterer, Marcus Spradlin, Lorenzo
Tancredi, Anastasia Volovich, Fabian Wagner, Stefan Weinzierl and Matthias
Wilhelm for helpful discussions. A special thanks goes to Sebastian Mizera for
sharing results on the Landau singularities of the kite integral before
publication of the PLD.jl package. We also thank Simon Caron-Huot, Claude
Duhr, Oliver Schlotterer and Matthias Wilhelm for comments on the manuscript.
FP and AP thank Uppsala University for hospitality during parts of this work.
The authors acknowledge the hospitality of ETH Zürich and the University of
Zürich during the workshop and associated school _Elliptics 2023_. MG’s work
is supported by the National Science and Engineering Council of Canada (NSERC)
and the Canada Research Chair program. FP’s work was co-funded by the European
Union (ERC Consolidator Grant LoCoMotive 101043686). Views and opinions
expressed are those of the author(s) only and do not necessarily reflect those
of the European Union or the European Research Council. Neither the European
Union nor the granting authority can be held responsible for them. AP is
supported by the US Department of Energy under contract DESC0010010 Task F. YS
acknowledges support from the Centre for Interdisciplinary Mathematics at
Uppsala University.

##  Appendix A Punctures and $Z$-arguments under mass permutations

In this appendix, we provide some examples of how the discrete symmetries of
the sunrise, eyeball, and kite topologies act on the punctures and
$Z$-arguments.

The
${{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$\-
and
${{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}}$-sunrises
are invariant under the action of the respective symmetric groups
$S_{3}(\\{m_{1},m_{2},m_{3}\\})$ and $S_{3}(\\{m_{3},m_{4},m_{5}\\})$, while
the (1234)-, (1235)-, (2345)-, and (1345)-eyeball diagrams are only invariant
under the respective exchanges $m_{1}\leftrightarrow m_{3}$,
$m_{2}\leftrightarrow m_{3}$, $m_{2}\leftrightarrow m_{5}$ and
$m_{3}\leftrightarrow m_{4}$. The kite has two discrete symmetries, namely
$m_{1}\leftrightarrow m_{4}$, $m_{2}\leftrightarrow m_{5}$ and
$m_{1}\leftrightarrow m_{5}$, $m_{2}\leftrightarrow m_{4}$. In table 1, we
provide a partial list of how these symmetries affect punctures and
$Z$-arguments.

Mass permutations | Puncture permutations |  $Z$-argument permutations
---|---|---
$m_{1}\leftrightarrow m_{2}$ | $z_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow z_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$ | $\mathcal{L}_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow\mathcal{L}_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
$m_{1}\leftrightarrow m_{3}$ | $z_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow 1-z_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}-z_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$ | $\mathcal{L}_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow-\mathcal{L}_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
$m_{2}\leftrightarrow m_{3}$ | $z_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow 1-z_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}-z_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$ | $\mathcal{L}_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow-\mathcal{L}_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$

Mass permutations | Puncture permutations |  $Z$-argument permutations
---|---|---
$m_{1}\leftrightarrow m_{3}$ | $z_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow 1-z_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}-z_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$ | $\mathcal{L}_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow-\mathcal{L}_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},\,\mathcal{L}_{8}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow-\mathcal{L}_{9}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$

Mass permutations | Puncture permutations |  $Z$-argument permutations
---|---|---
$m_{1}\leftrightarrow m_{2},\,m_{4}\leftrightarrow m_{5}$ | $z_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow z_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}},\,z_{4}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow z_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$ |  | $\mathcal{L}_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow\mathcal{L}_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},\,\mathcal{L}_{4}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow\mathcal{L}_{5}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},$
---
$\mathcal{L}_{6}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow\mathcal{L}_{10}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},\,\mathcal{L}_{7}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow\mathcal{L}_{11}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},$
$\mathcal{L}_{8}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow\mathcal{L}_{12}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},\,\mathcal{L}_{9}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow\mathcal{L}_{13}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},$
$\mathcal{L}_{15}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow\mathcal{L}_{16}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}},\,\mathcal{L}_{17}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow-\mathcal{L}_{17}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}$
$m_{1}\leftrightarrow m_{4},\,m_{2}\leftrightarrow m_{5}$ |  | $\tau^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow\tau^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}},$
---
$z_{1}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow
z_{4}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}},\,z_{2}^{{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}(123)}}\leftrightarrow
z_{5}^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}(345)}},$