# A Symbol and Coaction for Higher-Loop Sunrise Integrals

Andreas Forum  Matt von Hippel

###### Abstract

We construct a symbol and coaction for $l$-loop sunrise integrals, both for
the equal-mass and generic-mass cases. These constitute the first concrete
examples of symbols and coactions for integrals involving Calabi-Yau
threefolds and higher. In order to achieve a symbol of finite length, we
recast the differential equations satisfied by the master integrals of this
topology in the form of a unipotent differential equation. We augment the
basis of master integrals in a natural way by including ratios of maximal cuts
$\tau_{i}$. We discuss the relationship of this construction to constructions
of symbols and coactions for multiple polylogarithms and elliptic multiple
polylogarithms, in particular its connection to notions of transcendental
weight.

##  1 Introduction

Two mathematical structures, the symbol and coaction on multiple
polylogarithms, have revolutionized the way we compute loop-level Feynman
diagrams [1, 2, 3, 4, 5, 6]. Multiple polylogarithms are iterated integrals
over rational functions, and the symbol and coaction are operations that
disassemble these iterated integrals, revealing their branch cut and
derivative structure and trivializing identities between them. Since their
introduction to the physics community in ref. [2], the symbol and coaction
have been key tools, whether for simplifying results [5], clarifying new
structure [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24], reformulating differential equations [25], or in special cases
bootstrapping complete quantities from minimal constraints [26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38].

As the physics community considers more complicated processes with more loops
or more complicated kinematic dependence, we have begun to encounter Feynman
diagrams which cannot be expressed in terms of multiple polylogarithms. The
oldest such example is a two-loop propagator correction involving an exchange
of massive particles [39]. Now known as the two-loop sunrise, this diagram has
been the subject of intensive study [40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59]. The two-loop sunrise is part of a
broader class of diagrams that evaluate to integrals involving elliptic
curves. Recently, a formalism has been proposed for these integrals that
parallels that previously developed for multiple polylogarithms, involving
iterated integrals over these elliptic curves [60, 61]. In ref. [62], the
authors of the latter paper developed a symbol and coaction for these
functions. This construction made use of a general recipe for building symbols
and coactions for motivic periods described in ref. [63], specializing it to
the elliptic case.

Higher-loop propagator corrections can involve more complicated functions yet.
A particularly well-studied example is a higher-loop analogue of the sunrise
integral, also known as a banana integral [64, 65, 66]. While the equal-mass
case at three loops can be represented in terms of functions from the elliptic
toolbox, namely iterated integrals of modular forms, [67, 68], more general
three-loop sunrise integrals and sunrise integrals at higher loops require
more complicated functions. These integrals have been observed to have a
surprising connection to Calabi-Yau manifolds [69, 70, 71], with a Calabi-Yau
$l-1$-fold appearing at $l$ loops. This connection has recently been exploited
to construct differential equations and series expansions for these integrals
[72, 73, 74]. The latest paper in that series presented higher-loop sunrise
integrals in terms of a particularly suggestive iterated integral form,
involving elements of a particular basis for their maximal cuts called the
Frobenius basis.

Inspired by this result, we investigate the possibility of applying the
general constructions of ref. [63] to higher-loop sunrise integrals, making
use of the structure of their differential equations as laid out in ref. [74].
A key requirement for the construction of a symbol is the existence of
functions that satisfy a unipotent differential equation, that is a
homogeneous differential equation formulated with a nilpotent matrix. We find
that the formulation of ref. [74] does not immediately satisfy these
requirements, but it does once the basis is suitably enlarged. To enlarge the
basis, we make a natural choice of adding ratios of maximal cuts $\tau_{i}$,
analogous to the modulus $\tau$ of the torus associated with the elliptic
curve at two loops. With this enlarged basis and an appropriate choice of
normalization, we find that we can construct a symbol and coaction for the
higher-loop sunrise integrals. This represents the first specific examples of
a symbol and coaction constructed for integrals involving Calabi-Yau manifolds
with dimension greater than one. We find that our construction has an unusual
relationship to the notions of “transcendental weight” in the literature,
which are largely inspired by polylogarithmic Feynman integrals. We comment on
this relationship, related to different choices one can make in the symbol and
coaction construction, and highlight the importance of choices that match
particular practical goals.

The paper is organized as follows. We begin in section (2) with some
preliminaries: subsection (2.1) defines our integral of interest, subsection
(2.2) reviews the general symbol and coaction construction we will be using,
and subsection (2.3) describes the form of differential equations that we will
employ. Finally, subsection (2.4) presents a basis of integrals with useful
properties deriving from their relevance to Calabi-Yau manifolds. We then
present our symbol and coaction constructions, first for the equal-mass case
in section (3), then for the generic-mass case in section (4). We conclude in
section (5), discussing some implications of our work and directions for
further study. We also include one ancillary file,
SunriseSupplementaryMaterial.nb, with Mathematica functions that output the
symbol and coaction for the equal-mass sunrise integral through five loops.

##  2 Preliminaries

###  2.1 The $l$-loop Sunrise Integral

$p$$m_{1}$$m_{2}$$m_{l}$$m_{l+1}$ Figure 1: The $l$-loop sunrise integral.

The $l$-loop sunrise integral is depicted in Figure (1). It has $l+1$
propagators, with masses $m_{1}\ldots m_{l+1}$. In this paper we consider
integrals of this type with general propagator powers,

$I_{\underline{\nu}}(\underline{x},D)=\int\prod_{r=1}^{l}\frac{d^{D}k_{r}}{(i\pi)^{D/2}}\prod_{j=1}^{l+1}\frac{1}{(q_{j}^{2}-m_{j}^{2})^{\nu_{j}}}\,,$
(2.1)

with $q_{j}$ the momentum flowing through the relevant propagator.
$\underline{\nu}$ denotes the vector of integer $\nu_{j}$ propagator powers,
and $\underline{x}$ expresses the dependence of the integral on momentum
invariants:

$\underline{x}=\\{p^{2},m_{1}^{2},\ldots m_{l+1}^{2}\\}\,.$ (2.2)

In practice, it is always possible to factor out an overall power of one of
these momentum invariants, typically chosen to be $p^{2}$. The nontrivial
kinematic dependence of these integrals is then encoded in a vector of
dimensionless ratios,

$\underline{z}=\\{m_{1}^{2}/p^{2},\ldots m_{l+1}^{2}/p^{2}\\}\,.$ (2.3)

The set of $l$-loop sunrise integrals with general propagator powers are
related by integration-by-parts identities. They can thus be expressed in a
basis of master integrals. In $D=2-2\epsilon$ dimensions, there will be a
total of $2^{l+1}-1$ independent master integrals. We will represent these in
different bases as we continue, but we begin with the basis initially
discussed in ref. [74]. This basis consists of $l+2$ sectors, of which $l+1$
reduce to tadpole integrals,

$J_{l,i}(\underline{z},\epsilon)=-\frac{(-1)^{l+1}}{\Gamma(1+l\epsilon)}(p^{2})^{l\epsilon}\epsilon^{l}I_{1,\ldots,1,0,1,\ldots,1}(\underline{x},2-2\epsilon)=-\frac{\Gamma(1+\epsilon)}{\Gamma(1+l\epsilon)}\prod_{j=1,j\neq
i}^{l+1}z_{j}^{-\epsilon}\,,$ (2.4)

and one top sector containing the remaining integrals, indexed by
$\underline{k}\in\\{0,1\\}^{l+1}$,

$\displaystyle J_{l,\underline{0}}(\underline{z},\epsilon)$
$\displaystyle=\frac{(-1)^{l+1}}{\Gamma(1+l\epsilon)}(p^{2})^{1+l\epsilon}I_{1,\dots
1}(\underline{x},2-2\epsilon)$ (2.5) $\displaystyle
J_{l,\underline{k}}(\underline{z},\epsilon)$
$\displaystyle=(1+2\epsilon)\ldots(1+k\epsilon)\partial_{\underline{z}}^{\underline{k}}J_{l,\underline{0}}(\underline{z},\epsilon)\,,$
(2.6)

where $k=|k|$.

If all of the propagator masses are equal $m_{i}^{2}=m^{2}$, then there are
only $l+1$ distinct master integrals. We begin by representing them in the
following basis,

$\displaystyle J_{l,0}(z,\epsilon)$
$\displaystyle=\frac{(-1)^{l+1}}{\Gamma(1+l\epsilon)}(m^{2})^{l\epsilon}\epsilon^{l}I_{1,\ldots
1,0}(\underline{x},2-2\epsilon)=-\frac{\Gamma(1+\epsilon)}{\Gamma(1+l\epsilon)}$
(2.7) $\displaystyle J_{l,1}(z,\epsilon)$
$\displaystyle=\frac{(-1)^{l+1}}{\Gamma(1+l\epsilon)}(m^{2})^{1+l\epsilon}I_{1,\ldots
1}(\underline{x},2-2\epsilon)$ (2.8) $\displaystyle J_{l,k}(z,\epsilon)$
$\displaystyle=(1+2\epsilon)\ldots(1+k\epsilon)\partial_{z}^{k}J_{l,1}(z,\epsilon)\,.$
(2.9)

The number of independent master integrals also changes for special values of
$\epsilon$. In particular, the number of independent master integrals in the
top sector are reduced to $2^{l-1}-\left(\begin{subarray}{c}\\\ l-2\\\
\lfloor\frac{l+2}{2}\rfloor\end{subarray}\right)$ for $\epsilon=0$. This
reduction is due to the fact that the derivatives in eq. (2.6) generate the
cohomology groups in
$H_{\textnormal{hor}}^{l-1-k,k}(M_{l-1}^{CI}),k=0,...,l-1$. The cohomology
group $M_{l-1}^{CI}$ contains linear dependencies of these derivatives, which
means that the number of independent ones are instead lower than what is found
in eq. (2.6).

###  2.2 Symbols and Coactions for Motivic Periods

In this section we review the construction of symbols and coactions for
motivic periods of ref. [63], specifically the elements used in ref. [62].

The symbols we are familiar with from multiple polylogarithms take the
following form: for a function of transcendental weight $w$ whose derivative
satisfies

$dF_{w}=\sum_{i}F_{w-1}^{i}d\log\phi_{i}$ (2.10)

one obtains a symbol

$S(F_{w})=\sum_{i}S(F_{w-1}^{i})\otimes\phi_{i}\,.$ (2.11)

Symbols of this sort must have a few key features. In order for the definition
to be recursive in this way, the functions the symbol is defined for must
satisfy a homogeneous system of differential equations, so that derivatives do
not map outside the space of functions for which symbols are defined. In order
for symbols to be of finite length, the recursion must terminate.

Let us assume we have a basis of $n$ linearly independent functions
$\underline{I}$, which we can think of as integrations of a basis of forms
$\xi$ over a common contour,

$\underline{I}=\left(\int_{\gamma}\xi_{1},\ldots\int_{\gamma}\xi_{n}\right)\,,$
(2.12)

that satisfy a homogeneous system of differential equations,

$d\underline{I}=A\underline{I}\,,$ (2.13)

where $A$ is an $n\times n$ matrix of one-forms. In order for the symbol to be
of finite length, we require that $A$ is a nilpotent matrix, that is that
there is some positive integer $k$ such that $A^{k}=0$. A differential
equation with this property is called unipotent.

Once we have a basis of functions with this property, we can apply the recipe
of ref. [63] to define a symbol and coaction. This recipe will be unfamiliar
to readers used to symbols and coactions for polylogarithms, so before
reviewing the method it is worth clarifying a few points. For polylogarithms,
we are used to thinking of the symbol as the maximal iteration of a coaction:
a coaction applied repeatedly until we have broken down our functions into the
smallest possible logarithmic pieces. The entries of a symbol are not actually
logarithms, though: logarithms are periods, a pairing of a form and an
integration contour, while symbol entries do not have a contour. This is why
we can think of them as defined “up to $i\pi$”, or why “the symbol kills
constants”: a symbol entry does not specify the branch of the logarithm.

Such symbol entries are quite intuitive to manipulate when they resemble
logarithms, but when they are more general more care must be taken. This is
why the recipe of refs. [63, 62] will involve two ingredients. The first, a
motivic coaction, breaks up periods into pairs of a period and a pairing of
differential forms called a de Rham period. The second ingredient is the
symbol, and its role is now to make sense of these de Rham periods,
transforming them into formal descriptions of iterated integrals with no
specified contour (the so-called “bar construction”). We will now review how
to construct each of these ingredients.

The first ingredient, the motivic coaction, takes the following schematic
form:

$\Delta^{\mathfrak{m}}([\gamma,\omega])=\sum_{i}[\gamma,\omega_{i}]\otimes[\omega_{i},\omega]\,.$
(2.14)

The pair $[\gamma,\omega]$ is the notation of ref. [62] for a motivic period,
which pairs an integration cycle $\gamma$ with a form $\omega$ on which it can
be integrated. $[\omega_{i},\omega]$ is a pairing between differential forms
that represents a de Rham period.

To define our second ingredient, the symbol map, we begin with the matrix $A$
appearing in our differential equation. From this we define a matrix $T_{A}$,
with entries in the bar construction: formal specifications of iterated
integrals.

$T_{A}=1+[A]^{R}+[A|A]^{R}+[A|A|A]^{R}+\ldots$ (2.15)

Here objects $[A|A|\ldots]$ should be computed by multiplying the matrices and
concatenating the one-forms appearing in their entries into words within the
bar construction. The operation $R$ reverses words,

$[w_{1}|\ldots|w_{k}]^{R}=[w_{k}|\ldots|w_{1}]\,.$ (2.16)

Because the matrix $A$ is nilpotent, the series in eq. (2.15) is guaranteed to
terminate.

Using the matrix $T_{A}$, the symbol of a pair of forms $[\xi_{i},\xi_{j}]$ is
an entry of the matrix as follows:

$\mathcal{S}([\xi_{i},\xi_{j}])=\langle\xi_{i}|T_{A}|\xi_{j}\rangle=\left(T_{A}\right)_{ji}\,,$
(2.17)

Thus, each pair of forms in our basis is assigned an object in the bar
construction. These objects are tensor product of one-forms, which may be more
familiar: they are in essence our familiar notion of a symbol.

While one might worry that symbols of this sort are basis-dependent, in fact
changes of basis consistent with the overall form in eq. (2.13) will leave the
symbol invariant, provided we take into account relations on symbols implied
by integration by parts. For the case of closed forms $\omega_{i}$, these
relations read,

$\displaystyle[\omega_{1}|\ldots|\omega_{k}|df|\omega_{k+1}|\ldots|\omega_{n}]=[\omega_{1}|\ldots|\omega_{k}f|\omega_{k+1}|\ldots|\omega_{n}]-[\omega_{1}|\ldots|\omega_{k}|f\omega_{k+1}|\ldots|\omega_{n}]$
(2.18)
$\displaystyle[df|\omega_{1}|\ldots|\omega_{n}]=[f\omega_{1}|\ldots|\omega_{n}]-f[\omega_{1}|\ldots|\omega_{n}]$
(2.19)
$\displaystyle[\omega_{1}|\ldots|\omega_{n}|df]=f[\omega_{1}|\ldots|\omega_{n}]-[\omega_{1}|\ldots|\omega_{n}f]$
(2.20)

Combining our two ingredients, this symbol map and the motivic coaction,
naturally leads to a coaction of the following form:

$\Delta\equiv(\textrm{id}\otimes\mathcal{S})\Delta^{\mathfrak{m}}\,.$ (2.21)

Acting on the basis in eq. (2.12) with this coaction gives,

$\Delta(I_{a})=\sum_{c}I_{c}\otimes\mathcal{S}([\xi_{c},\xi_{a}])\,.$ (2.22)

As the symbol is only defined on unipotent quantities, this coaction as stated
is as well. There is however a simple way it can be extended to consider more
general quantities, at the cost of introducing an arbitrary choice.

In particular, a general motivic period $x$ can be decomposed into a bilinear
combination of unipotent periods $u_{i}$ and semi-simple periods $s_{i}$ as
follows:

$x=\sum_{i}u_{i}s_{i}\,.$ (2.23)

Semi-simple periods are defined to be periods for which the coaction acts
trivially,

$\Delta s=s\otimes 1\,,$ (2.24)

while coactions act on unipotent periods according to eq. (2.22). This
construction should be familiar from the polylogarithmic case, as it is the
way the coaction handles powers of $\pi$: these cannot appear in the final
entries as these are de Rham periods, but they can be consistently included in
the first entry of the coaction.

There are in principle many legitimate choices for this decomposition, with
different choices in general corresponding to inequivalent constructions of
the symbol and coaction111For example, it is always possible to include any
unipotent period in the semi-simple part instead, in the extreme case
trivializing the coaction completely.. Readers should keep this in mind in
what follows: while we make what we believe to be a reasonable choice in our
constructions, there were many other possible choices.

###  2.3 Gauss-Manin System

In general, a basis of master integrals for a set of Feynman diagrams will
satisfy a coupled system of differential equations [75, 76, 77, 78]. This can
be presented as a system of linear differential equations. Following ref.
[74], we refer to this presentation as a Gauss-Manin system.

$d\underline{I}=A\underline{I}$ (2.25)

If one instead focuses on the integrals in a particular sector, the system
becomes inhomogeneous, with the inhomogeneity coming from contributions in
lower sectors. Let us write the basis of master integrals in a single sector
as,

$\underline{J}=\\{J_{1},\ldots J_{n}\\}\,.$ (2.26)

We will have two cases of interest, the top sectors in the equal-mass and
general-mass cases for the sunrise. For the equal-mass case at $l$ loops,
$J_{1}\ldots J_{n}$ correspond to the $l$ master integrals
$J_{l,1},J_{l,2},\ldots J_{l,l}$ defined in eqs. (2.8) and (2.9). For the
general-mass case, they instead correspond to the top sector integrals
specified in eqs. (2.5) and (2.6). Denoting the relevant dimensionless
kinematic ratios by $\underline{z}$ in either case, we have,

$d\underline{J}(\underline{z},\epsilon)=\textbf{B}(\underline{z},\epsilon)\underline{J}(\underline{z},\epsilon)+\underline{N}(\underline{z},\epsilon)\,,$
(2.27)

where $\underline{N}(\underline{z},\epsilon)$ gives the inhomogeneous terms,
which come from lower sectors. These terms vanish for the maximal cut, as
such, the maximal cuts of top sector integrals satisfy the homogeneous form of
this differential equation. We are working with an $\epsilon$-regular basis of
integrals, so all expressions here are finite in the $\epsilon\rightarrow 0$
limit. In particular, we write
$\textbf{B}(\underline{z})=\textbf{B}(\underline{z},0)$.

One can write a general solution to the homogeneous equation in terms of a
matrix called the Wronskian,

$d\textbf{W}(\underline{z})=\textbf{B}(\underline{z})\textbf{W}(\underline{z})\,.$
(2.28)

Using the Wronskian, we can transform to a different basis, which will be a
useful starting point in the following:

$\underline{L}(\underline{z},\epsilon)=\textbf{W}(\underline{z})^{-1}\underline{J}(\underline{z},\epsilon)\,.$
(2.29)

In this basis, the differential equation becomes,

$d\underline{L}(\underline{z},\epsilon)=\textbf{W}(\underline{z})^{-1}\left[\textbf{B}(\underline{z},\epsilon)-\textbf{B}(\underline{z})\right]\textbf{W}(\underline{z})+\textbf{W}(\underline{z})^{-1}\underline{N}(\underline{z},\epsilon)\,,$
(2.30)

which becomes especially simple when restricted to integer dimension with
$\epsilon\rightarrow 0$, which we will do in the following:

$d\underline{L}(\underline{z})=\textbf{W}(\underline{z})^{-1}\underline{N}(\underline{z})\,.$
(2.31)

###  2.4 Frobenius Basis and Griffiths Transversality

In order to make the above differential equation explicit, one needs explicit
representations for the inverse of the Wronskian
$\textbf{W}(\underline{z})^{-1}$ and for the inhomogeneities
$\underline{N}(\underline{z})$. For the sunrise integrals the latter are
simple, coming from tadpole integrals. The inverse of the Wronskian is
potentially more complicated, and at higher loops one might expect it to
contain higher degree polynomials in the maximal cuts. However, ref. [74]
provides a few key simplifications in the specific case of the sunrise family,
owing to their Calabi-Yau structure.

An alternative to the Gauss-Manin presentation represents a coupled system of
differential equations in terms of a set of inhomogeneous higher-order
differential equations satisfied by each master integral, called Picard-Fuchs
differential equations

$\mathcal{L}_{k}I_{k}(\underline{z})=R_{k}(\underline{z}).$ (2.32)

The inhomogeneous term is related to master integrals from lower sectors.
Solutions of the homogeneous version (found by setting $R_{k}(z)=0$) close to
a given point $\underline{z}_{0}$ can be characterized by indicial equations,
polynomials whose roots give the leading exponents of $(z_{i}-z_{0,i})$ in a
power series around $\underline{z}_{0}$. If these roots are degenerate then
there exist logarithmic solutions, with higher powers of logarithms for higher
degeneracy, organized in what is called a Frobenius basis.

If the differential equations describe the moduli space of a Calabi-Yau
manifold then there is expected to be at least one point of maximal unipotent
monodromy, or MUM point [72]. At a MUM point, all of the indicial roots are
degenerate. There is then only one power series solution, accompanied by a
hierarchical set of logarithmic solutions of increasing power. As such, it
becomes quite straightforward to write down an ansatz for these solutions and
solve for the Frobenius basis to the desired order around a MUM point. This
Frobenius basis will serve as a particularly nice basis for the maximal cuts,
and thus the entries of $\textbf{W}(\underline{z})$

An additional property of Calabi-Yau manifolds of dimension $n$ is called
Griffiths transversality. This property can be used to derive quadratic
relations between elements of the Frobenius basis (and thus between maximal
cuts and entries in $\textbf{W}(\underline{z})$) of the following form:

$\underline{\Pi}(\underline{z})^{T}\Sigma_{l}\partial_{\underline{z}}^{\underline{k}}\underline{\Pi}(\underline{z})=\begin{cases}0&\textnormal{
for }0\leq|\underline{k}|<n\\\ C_{\underline{k}}(\underline{z})&\textnormal{
for }|\underline{k}|=n\end{cases}$ (2.33)

The functions $C_{\underline{k}}(\underline{z})$ are often referred to as
Yukawa couplings due to their role in Calabi-Yau compactifications. Ref. [74]
describes how to find explicit expressions for these functions from the
Picard-Fuchs differential equations and derives them for the equal-mass
sunrise. We will review that derivation in section (3) and perform the
analogous derivation for the generic-mass case in section (4). Once we have
explicit expressions for the $C_{\underline{k}}(\underline{z})$, we can use
the relations derived from Griffiths transversality to simplify the inverse of
the Wronskian, representing its entries linearly in the Frobenius basis.

##  3 The Equal-Mass Case

The goal of this section is to apply the equations of the previous section to
the concrete example of the equal-mass master integral basis defined in eq.
(2.6). We will begin with reviewing the maximal cuts of the equal-mass master
integrals, as these are used in the construction of the Wronskian. We will
also review the relations that can be found from Griffiths Transversality and
in particular use these relations to simplify the inverse Wronskian.
Finally we will provide a recipe for defining a unipotent differential
equation satisfied by these master integrals, which will in turn allow us to
construct a symbol and coaction.

###  3.1 Frobenius Basis

When all masses are equal the solution space of the Picard-Fuchs differential
equation is spanned by $l-1$ logarithmic solutions containing up to $l$ powers
of $\log(z)$. In the equal-mass case it is sufficient to consider only a
single Picard-Fuchs operator of the type

$\mathcal{L}_{l}=B_{l,l}(z)\partial_{z}^{l}+B_{l,l-1}(z)\partial_{z}^{l-1}+...+B_{l,0}(z),$
(3.1)

to generate the entire solution space. Such operators have been studied in
[73] and a list of the relevant operators at different loop orders is
provided. We look for solutions around the MUM-point $z_{0}=0$. At this MUM-
point the solution space is spanned by the Frobenius basis given by

$\varpi_{l,k}(z)=\sum_{j=0}^{k}\frac{1}{(k-j)!}\log^{k-j}(z)\Sigma_{l,j}(z),$
(3.2)

where $\Sigma_{l,k}(z)$ is a power series, normalized such that
$\Sigma_{l,k}(z)=\delta_{k,0}z+\mathcal{O}(z^{2})$. For $k=0$, we have

$\varpi_{l,0}(z)=\Sigma_{l,0}(z)=\sum_{k_{1},...,k_{l+1}\geq
0}\left(\frac{|k|!}{k_{1}!k_{2}!...k_{l+1}!}\right)^{2}z^{|k|+1},$ (3.3)

where $|k|=\sum_{i}^{l+1}k_{i}$. The remaining $\Sigma_{l,k}(z)$’s can be
derived by considering the fact that elements of the Frobenius basis satisfy
the homogenous version of the Picard-Fuchs equation, i.e all elements satisfy
$\mathcal{L}_{l}\varpi_{l,k}(z)=0$. We collect all the Frobenius elements into
a vector $\Pi_{l}(z)=(\varpi_{l,0}(z),\varpi_{l,1}(z),...,\varpi_{l,l-1}(z))$.
As mentioned earlier, Griffiths Transversality allows one to derive relations
between elements of the Frobenius basis. We will in the following derive these
relations, which we collectively refer to as Griffiths Relations. By using the
explicit expressions of the equal-mass Frobenius basis in the context of
Griffiths Transversality, it is possible to derive that the intersection
matrix $\Sigma_{l}$ has the form

$\Sigma_{l}=\left(\begin{matrix}0&\ldots&0&1\\\ 0&\ldots&-1&0\\\
0&\iddots&0&0\\\ (-1)^{l+1}&0&0&0\end{matrix}\right).$ (3.4)

We now consider Griffiths Transversality in the case where $|k|=n=l-1$. Our
goal is to derive a differential equation that is satisfied by the Yukawa
couplings $C_{l}(z)$.

We start out by using the product rule to derive relations between first
orders of derivatives with respect to the Frobenius basis:

$C_{l}(z)=\partial_{z}(\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{z}^{l-2}\underline{\Pi}(z))-\partial_{z}\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{l}^{l-2}\underline{\Pi}_{l}(z).$
(3.5)

The first term is zero, due to eq. (2.33). We can continue using the product
rule this way and find a more generalized version of eq. (2.33):

$C_{l}(z)=(-1)^{k}\partial_{z}^{k}\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{z}^{l-1-k}\underline{\Pi}_{l}(z).$
(3.6)

If one then differentiates the expression in eq. (2.33) and uses the product
rule successively, one finds,

$\begin{split}\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{z}^{l}\underline{\Pi}_{l}(z)&=\partial_{z}C_{l}(z)-\partial_{z}\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{z}^{l-1}\underline{\Pi}_{l}(z)\\\
&=l\partial_{z}C_{l}(z)+(-1)^{l}\partial_{z}^{l}\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\underline{\Pi}_{l}(z).\end{split}$
(3.7)

Due to the $(-1)^{l+1}$ symmetry of $\Sigma_{l}$, we can reduce this to

$\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{z}^{l}\underline{\Pi}_{l}(z)=\frac{l}{2}\partial_{z}C_{l}(z).$
(3.8)

So far we have only developed relations between different degrees of
differential orders of the Frobenius basis. To find the differential equation
whose solution is the Yukawa-coupling we make use of the fact that the Picard-
Fuchs operator is an $l$-order differential equation that annihilates the
Frobenius basis:

$\begin{split}0&=\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\mathcal{L}_{l}\underline{\Pi}_{l}(z)\\\
&=B_{l,l-1}(z)\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{z}^{l-1}\underline{\Pi}_{l}(z)+B_{l,l}(z)\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{z}^{l}\underline{\Pi}_{l}(z).\end{split}$
(3.9)

We can now use the relations we have found above, namely eq. (2.33) and eq.
(3.8), to find a differential equation that is satisfied by $C_{l}(z)$

$\partial_{z}C_{l}(z)+\frac{2}{l}\frac{B_{l,l-1}(z)}{B_{l,l}(z)}C_{l}(z)=0\,.$
(3.10)

This differential equation is solved by,

$C_{l}(z)=\frac{1}{z^{l-3}\prod_{k\in\Delta^{(l)}}(1-kz)},$ (3.11)

where

$\Delta^{(l)}=\bigcup_{j=0}^{\lceil\frac{l-1}{2}\rceil}\left\\{(l+1-2j)^{2}\right\\}.$
(3.12)

We can also use eq. (3.7) to generate a series of relations through
differentiation. To begin, we see that we can rewrite eq. (3.7), by using the
Picard Fuchs operator

$\begin{split}\partial_{z}C_{l}(z)&=\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{z}^{l}\underline{\Pi}_{l}(z)+\partial_{z}\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{z}^{l-1}\underline{\Pi}_{l}(z)\\\
&=-\sum_{j=1}^{l-1}\frac{B_{l,j}(z)}{B_{l,l}(z)}\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{z}^{j}\underline{\Pi}_{l}(z)+\partial_{z}\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{z}^{l-1}\underline{\Pi}_{l}(z).\end{split}$
(3.13)

Due to Griffiths relations, we realize that the only non-vanishing term in the
sum is $j=l-1$. We can then use the differential equation (3.10) to collect
terms. This gives us even more relations between elements of the Frobenius
basis

$\left(1-\frac{l}{2}\right)\partial_{z}C_{l}(z)=\partial_{z}\underline{\Pi}_{l}(z)^{T}\Sigma_{l}\partial_{z}^{l-1}\underline{\Pi}_{l}(z).$
(3.14)

We can proceed in this fashion, i.e taking the derivative of eq. (3.8) and
using the Picard-Fuchs operator to re-write the $l$-order differential.
Eventually we will have enough relations to calculate all entries in the
matrix

$\boldsymbol{\mathrm{Z}}_{l}(z)=\left(\begin{matrix}\underline{\Pi}(z)^{T}\Sigma_{l}\underline{\Pi}(z)&...&\underline{\Pi}(z)\Sigma_{l}\partial_{z}^{l-1}\underline{\Pi}(z)\\\
\vdots&\ddots&\vdots\\\
\partial_{z}^{l-1}\underline{\Pi}(z)\Sigma_{l}\underline{\Pi}(z)&...&\partial_{z}^{l-1}\underline{\Pi}(z)\Sigma_{l}\partial_{z}^{l-1}\underline{\Pi}(z)\end{matrix}\right).$
(3.15)

In terms of $\mathbf{Z}_{l}(z)$, it is possible to express the inverse
Wronskian as

$\mathbf{W}_{l}(z)^{-1}=\Sigma_{l}\mathbf{W}_{l}(z)^{T}\mathbf{Z}_{l}(z)^{-1}.$
(3.16)

It has a $(-1)^{l+1}\mathbf{Z}_{l}(z)=\mathbf{Z}_{l}(z)^{T}$ symmetry. We
explicitly show the result for the $l$’th column of the inverse Wronskian,
with $1\leq k\leq l$, as this will be of great use for us later on,

$W(z)_{k,l}^{-1}=\frac{(-1)^{l+k}\varpi_{l,l-k}(z)}{C_{l}(z)}.\\\ $ (3.17)

We once again emphasise the importance of the Griffiths relations. We see that
our results will be linear in elements of the Frobenius basis. Had we instead
inverted the Wronskian as if it was a generic matrix, we would be left with
degree $(l-1)$ polynomials of elements in the Wronskian.

###  3.2 Unipotent Differential Equation

As mentioned in the introduction to this section, the construction of a symbol
and coaction on our master integrals begins with eq. (2.31). We need only find
the inhomogeneous term $\underline{N}_{l}(z)$ to write the differential
equation explicitly. This term can be found by realising the connection
between the Picard-Fuchs equation and the Gauss-Manin equation. In particular,
acting with the Picard-Fuchs operator on the top sector yields

$\mathcal{L}_{l}\underline{J}_{l}(z)=\underline{N}_{l}(z).$ (3.18)

In the equal-mass case, this inhomogeneity has the form

$N_{l}(z)=\left(0,...,0,(-1)^{l+1}(l+1)!\frac{z}{z^{l}\prod_{k\in\Delta^{(l)}}(1-kz)}\right),$
(3.19)

where $\Delta^{(l)}$ is the same as eq. (3.12).

We can now combine eq. (3.17) and eq. (3.19) to explicitly write the
differential equation as

$dL_{l,k}(z)=(-1)^{k+1}\frac{(l+1)!}{z^{2}}\varpi_{l,k-1}(z)dz,\textnormal{
for }1\leq j\leq l$ (3.20)

We will now need to decompose this equation into a unipotent and semi-simple
part. As explained in section (2), there are many choices of doing so. We
choose to define the unipotent period

$\tau_{j}(z)=\frac{\varpi_{l,j}(z)}{\varpi_{l,l-1}(z)},\;\;\;\;0\leq j\leq
l-1.$ (3.21)

This particular choice of $\tau$ is chosen as it leads to fewer logarithmic
singularities in the high-momentum limit $z\rightarrow 0$, which is obvious
from inspection of eq. (3.2). Writing the differential equation in terms of
unipotent and semi-simple periods we obtain

$dL_{l,k}(z)=(-1)^{k+1}\frac{(l+1)!}{z^{2}}\varpi_{l,l-1}(z)\tau_{k-1}(z)dz.$
(3.22)

It has previously been mentioned that in order to build a coaction and symbol,
we need our master integrals to satisfy a unipotent differential equation

$d\underline{I}_{l}(z)=\mathbf{N}(z)\underline{I}_{l}(z),$ (3.23)

where $\mathbf{N}(z)$ is a nilpotent matrix. The master integrals in eq.
(3.22) do not satisfy a unipotent differential equation of this type. We can
instead define a larger basis, which satisfies a unipotent differential
equation as well as containing the master integrals. We define the expanded
$L$-basis as

$\underline{T}_{l}(z)=\left(L_{l,1}(z),...,L_{l,l}(z),\tau_{0}(z),...,\tau_{l-2}(z),1\right)$
(3.24)

This basis will satisfy a unipotent differential equation and enable us to
work out a coaction and symbol on the master integrals. For example, at $l=2$
we have the basis

$\underline{T}_{l}(z)=\left(L_{2,1}(z),L_{2,2}(z),\tau_{0}(z),1\right),$
(3.25)

which satisfies the unipotent differential equation

$d\left(\begin{matrix}L_{2,1}(z)\\\ L_{2,2}(z)\\\ \tau_{0}(z)\\\
1\end{matrix}\right)=\left(\begin{matrix}0&0&0&\frac{3!}{z^{2}}\varpi_{2,1}(z)dz\\\
0&0&-\frac{3!}{z^{2}}\varpi_{2,1}(z)dz&0\\\ 0&0&0&d\tau_{0}\\\ 0&0&0&0\\\
\end{matrix}\right)\left(\begin{matrix}L_{2,1}(z)\\\ L_{2,2}(z)\\\
\tau_{0}(z)\\\ 1\end{matrix}\right),$ (3.26)

where we have used eq. (3.22) to find the entries of the matrix. We can
perform this procedure for arbitrary loop order, finding a $2l\times 2l$
matrix with entries

$\mathbf{N}_{l}(z)_{i,j}=\begin{cases}(-1)^{j+1}\frac{(l+1)!}{z^{2}}\varpi_{l,l-1}(z)dz&\textnormal{
for }i+j=2l+1,\;\;\;1\leq j\leq l,\;\;\;l<i\leq 2l\\\
d\tau_{j-l-1}&\textnormal{ for }i=2l,\;\;\;l<j<2l\\\ 0&\textnormal{ otherwise
}\end{cases}.$ (3.27)

It can be shown that for any loop order $l$, $\mathbf{N}_{l}(z)$ has the
property $\mathbf{N}_{l}(z)^{3}=0$, and thus is indeed nilpotent.

We are particularly interested in the coaction and symbol for loop orders
$l\geq 4$, as this is the order at which the master integrals can no longer be
expressed in terms of elliptic functions. As an example, we present the
nilpotent matrix for $l=4$:

$\mathbf{N}_{4}(z)=\left(\begin{matrix}0&0&0&0&0&0&0&\frac{5!}{z^{2}}\varpi_{4,3}(z)dz\\\
0&0&0&0&0&0&-\frac{5!}{z^{2}}\varpi_{4,3}(z)dz&\\\
0&0&0&0&0&\frac{5!}{z^{2}}\varpi_{4,3}(z)dz&0&0\\\
0&0&0&0&-\frac{5!}{z^{2}}\varpi_{4,3}(z)dz&0&0&0\\\ 0&0&0&0&0&0&0&d\tau_{0}\\\
0&0&0&0&0&0&0&d\tau_{1}\\\ 0&0&0&0&0&0&0&d\tau_{2}\\\
0&0&0&0&0&0&0&0\end{matrix}\right).$ (3.28)

The structure is noticeably similar to that at two loops.

###  3.3 Symbol

We will now use our nilpotent matrix in eq. (3.27) to build a symbol operator.
The first step is to define the unipotent matrix $T_{N}$

$T_{N}=1+[\mathbf{N}_{l}]^{R}+[\mathbf{N}_{l}|\mathbf{N}_{l}]^{R}+[\mathbf{N}_{l}|\mathbf{N}_{l}|\mathbf{N}_{l}]^{R}+...,$
(3.29)

where we have suppressed $z$-dependence. Using eq. (3.27), we can derive a
general construction of $[\mathbf{N}_{l}|\mathbf{N}_{l}]^{R}$ and
$[\mathbf{N}_{l}|\mathbf{N}_{l}|\mathbf{N}_{l}]^{R}$

$\begin{split}[\mathbf{N}_{l}|\mathbf{N}_{l}]_{ij}^{R}&=\begin{cases}[d\tau_{j}|(-1)^{j+1}\frac{(l+1)!}{z^{2}}\varpi_{l,l-j}(z)dz]&\textnormal{
for }2\leq j\leq l,\;\;\;i=2l\\\ 0&\textnormal{otherwise}\end{cases}\\\
[\mathbf{N}_{l}|\mathbf{N}_{l}|\mathbf{N}_{l}]_{ij}^{R}&=0\end{split}.$ (3.30)

We now define the symbol on a pair of periods $[\xi_{i},\xi_{j}]$, where
$\xi_{i},\xi_{j}\in\underline{T}_{l}(z)$, as

$\mathcal{S}([\xi_{i},\xi_{j}])=(T_{N})_{ji}$ (3.31)

To see how this works in practice, consider the example of $l=2$. In this
case, the non-zero symbols are

$\begin{split}&\mathcal{S}([\xi_{i},\xi_{i}])=1\,,\\\
&\mathcal{S}([\tau_{0},L_{2,2}(z)])=\left[-\frac{3!}{z^{2}}\varpi_{2,1}(z)dz\right]\,,\\\
&\mathcal{S}([1,L_{2,1}(z)])=\left[-\frac{3!}{z^{2}}\varpi_{2,1}(z)dz\right]\,,\\\
&\mathcal{S}([1,L_{2,2}(z)])=\left[d\tau_{0}|\frac{3!}{z^{2}}\varpi_{2,1}(z)dz\right]\,,\\\
&\mathcal{S}([1,\tau_{0}(z)])=[d\tau_{0}]\,.\end{split}$ (3.32)

Symbols of the form $\mathcal{S}([1,\xi_{j}])$ are the type we would usually
refer to as $\mathcal{S}(\xi_{j})$ in the polylogarithmic case, the “symbol of
the function”. The other type of pair $\mathcal{S}([\tau_{0},L_{2,1}(z)])$ has
a different interpretation. It may be viewed as an integral of the form
defining $L_{2,1}(z)$ over a contour that encircles all of $\tau_{0}$’s
singularities.222Note that this is only an analogy, as de Rham pairs are not
actually integrals.

We delay a comparison of these expressions to the two-loop elliptic symbols in
the literature to the next subsection.

We can similarly work out the symbol in the more interesting case of $l=4$,
where the master integrals are no longer expressible in terms of elliptic
functions. In this case we find the non-zero symbols to be

$\begin{split}&\mathcal{S}([\xi_{i},\xi_{i}])=1\\\
&\mathcal{S}([\tau_{0}(z),L_{4,4}(z)])=\left[\frac{5!}{z^{2}}\varpi_{4,3}(z)\right],\\\
&\mathcal{S}([\tau_{1}(z),L_{4,3}(z)])=-\left[\frac{5!}{z^{2}}\varpi_{4,3}(z)\right],\\\
&\mathcal{S}([\tau_{2}(z),L_{4,2}(z)])=\left[\frac{5!}{z^{2}}\varpi_{4,3}(z)\right],\\\
&\mathcal{S}([1,L_{4,1}(z)])=-\left[\frac{5!}{z^{2}}\varpi_{4,3}(z)\right],\\\
&\mathcal{S}([1,L_{4,2}(z)])=\left[d\tau_{2}|\frac{5!}{z^{2}}\varpi_{4,3}(z)\right],\\\
&\mathcal{S}([1,L_{4,3}(z)])=-\left[d\tau_{1}|\frac{5!}{z^{2}}\varpi_{4,3}(z)\right],\\\
&\mathcal{S}([1,L_{4,4}(z)])=\left[d\tau_{0}|\frac{5!}{z^{2}}\varpi_{4,3}(z)\right],\\\
&\mathcal{S}([1,\tau_{0}(z)])=\left[d\tau_{0}\right],\\\
&\mathcal{S}([1,\tau_{1}(z)])=\left[d\tau_{1}\right],\\\
&\mathcal{S}([1,\tau_{2}(z)])=\left[d\tau_{2}\right].\\\ \end{split}$ (3.33)

By first inspection these all appear to be of length two. This may be
surprising, as naively one might expect higher-length functions to appear at
higher loop orders, as polylogarithmic functions at higher orders are often of
higher transcendental weight. We will comment on this surprise in the next
subsection, and offer a partial explanation.

###  3.4 Coaction

Now that we have defined symbols on all pairs of elements in
$\underline{T}_{l}(z)$, we can use

$\Delta(T_{l,k})=\sum_{i=1}^{2l}\left(T_{l,i}(z)\otimes\mathcal{S}([\xi_{i},\xi_{k}])\right),$
(3.34)

to calculate the coaction on elements in this basis.
We again start with $l=2$ example. In this case we find

$\begin{split}&\Delta(L_{2,1}(z))=L_{2,1}(z)\otimes
1-1\otimes\left[\frac{3!}{z^{2}}\varpi_{2,1}(z)dz\right]\,,\\\
&\Delta(L_{2,2}(z))=L_{2,2}(z)\otimes
1+1\otimes\left[d\tau_{0}|\frac{3!}{z^{2}}\varpi_{2,1}(z)dz\right]+\tau_{0}(z)\otimes\left[\frac{3!}{z^{2}}\varpi_{2,1}(z)dz\right]\,,\\\
&\Delta(\tau_{0}(z))=\tau_{0}\otimes 1+1\otimes[d\tau_{0}]\,.\end{split}$
(3.35)

As we know how the coaction acts on every element of $\underline{T}_{l}(z)$,
it is possible to calculate the coaction of our original master integrals in
$\underline{J}_{l}(z)$. Remember that the two are related by
$\underline{J}_{l}(z)=\mathbf{W}_{l}(z)\underline{L}_{l}(z)$, meaning that we
can express elements of $\underline{J}_{l}(z)$ in terms of elements in
$\underline{L}_{l}(z)$ and $\mathbf{W}_{l}(z)$. For instance, in the two-loop
example, we have that

$\begin{split}J_{2,1}(z)&=L_{2,1}(z)\varpi_{2,1}(z)\tau_{0}(z)+L_{2,2}(z)\varpi_{2,1}(z)\,,\\\
J_{2,2}(z)&=L_{2,1}(z)\left[\varpi_{1}(z)\partial_{z}\tau_{0}(z)+\tau_{0}(z)\partial_{z}\varpi_{2,1}(z)\right]+L_{2,2}\partial_{z}\varpi_{2,1}(z)\,.\end{split}$
(3.36)

To find the coaction on these quantities, we make use of the compatibility
requirement between multiplication and coaction:

$\Delta(T_{l,k}(z)T_{l,j}(z))=\Delta(T_{l,k}(z))\cdot\Delta(T_{l,j}(z))\,,$
(3.37)

as well as the coaction’s action on semi-simple quantities such as
$\varpi_{2,1}(z)$ here, described generically in eq. (2.24). Doing this, we
obtain,

$\begin{split}\Delta(J_{2,1}(z))&=J_{2,1}(z)\otimes
1+\varpi_{2,1}(z)\left(L_{2,1}(z)\otimes[d\tau_{0}]+1\otimes\left[d\tau_{0}|\frac{3!}{z^{2}}\varpi_{2,1}(z)dz\right]\right)\\\
\Delta(J_{2,2}(z))&=J_{2,2}(z)\otimes
1+\partial_{z}\varpi_{2,1}(z)\left(L_{2,1}(z)\otimes[d\tau_{0}]+1\otimes\left[d\tau_{0}|\frac{3!}{z^{2}}\varpi_{2,1}(z)dz\right]\right),\end{split}$
(3.38)

where we have used the relations in eq. (2.19) and (2.20) as well as tensor
identities for multiplication to simplify the results. The results we obtain
are similar to those of Broedel et. al [62], at least in their broad
structure.

We now move on to calculate the coaction at higher loop orders. We consider
the case $l=4$ and obtain the following results:

$\begin{split}&\Delta(\tau_{k}(z))=\tau_{k}\otimes
1+1\otimes\left[d\tau_{k}\right],\\\ &\Delta(L_{4,1}(z))=L_{4,1}(z)\otimes
1-1\otimes\left[\frac{5!}{z^{2}}\varpi_{4,3}(z)dz\right]\\\
&\Delta(L_{4,2}(z))=L_{4,2}(z)\otimes
1+1\otimes\left[d\tau_{2}|\frac{5!}{z^{2}}\varpi_{4,3}(z)dz\right]+\tau_{2}(z)\otimes\left[\frac{5!}{z^{2}}\varpi_{4,3}dz\right],\\\
&\Delta(L_{4,3}(z))=L_{4,3}(z)\otimes
1-1\otimes\left[d\tau_{1}|\frac{5!}{z^{2}}\varpi_{4,3}(z)dz\right]-\tau_{1}(z)\otimes\left[\frac{5!}{z^{2}}\varpi_{4,3}dz\right],\\\
&\Delta(L_{4,4}(z))=L_{4,4}(z)\otimes
1+1\otimes\left[d\tau_{0}|\frac{5!}{z^{2}}\varpi_{4,3}(z)dz\right]+\tau_{0}(z)\otimes\left[\frac{5!}{z^{2}}\varpi_{4,3}dz\right].\end{split}$
(3.39)

We include Mathematica code to generate these coactions through five loops in
an ancillary file, SunriseSupplementaryMaterial.nb, along with code to
generate the symbols from the previous section.

The structure is similar to that of the $l=2$ case. Again it is possible to
calculate the coaction for the $\underline{J}_{4}(z)$ basis using the same
procedure as for the $l=2$ example. For instance the coaction of the first
element in $\underline{J}_{4}(z)$ is

$\begin{split}\Delta(J_{4,1}(z))=&J_{4,1}(z)\otimes
1+\varpi_{4,3}(z)\bigg{(}L_{4,1}(z)\otimes[d\tau_{0}]\\\
&+\tau_{1}\otimes\left[d\tau_{2}|\frac{5!}{z^{2}}\varpi_{4,3}(z)dz\right]+L_{4,2}(z)\otimes[d\tau_{1}]+1\otimes\left[d\tau_{2}|d\tau_{1}|\frac{5!}{z^{2}}\varpi_{4,3}(z)dz\right]\\\
&-\tau_{2}\otimes\left[d\tau_{1}|\frac{5!}{z^{2}}\varpi_{4,3}(z)dz\right]+L_{4,3}(z)\otimes[d\tau_{2}]-1\otimes\left[d\tau_{1}|d\tau_{2}|\frac{5!}{z^{2}}\varpi_{4,3}(z)dz\right]\\\
&+1\otimes\left[d\tau_{0}|\frac{5!}{z^{2}}\varpi_{4,3}(z)dz\right]\bigg{)}.\end{split}$
(3.40)

This expression should partially reassure us. If we think of the symbol of the
function $J_{4,1}(z)$ as encoded in terms of the coaction of the form
$1\otimes X$, then the symbols present in these terms have length three, not
two. This property is also present at three loops, where it agrees with the
length of the elliptic multiple polylogarithm functions used to represent the
equal-mass banana graph at this order in ref. [68]. While this agreement with
the literature is reassuring, it may be strange that the length of our symbols
continues to be three at four loops. In fact, the symbols obtained from our
formalism for the master integrals $J_{l\geq 3,i}$ will always be of length
three, at any loop order, contrary to the expectation that the length of
functions increases at higher orders in perturbation theory along with their
transcendental weight.

One perspective on the situation is as follows: It was explained earlier in
section (2.2) that there is considerable freedom in choosing semi-simple and
unipotent periods when defining a symbol. Indeed it was not needed to expand
the $\underline{L}_{l}(z)$-basis with $l$ elements of $\tau_{k}(z)$ to define
a basis that satisfies a unipotent differential equation. We could have simply
expanded the $\underline{L}_{l}(z)$ basis with a simple “1”. Had we done so,
all symbols would be of length one and no new information would be obtained.
Due to the arbitrariness of decomposing functions into semi-simple and
unipotent parts, it is possible that there exists another choice, in which the
symbol length is even longer than that which we found. It is therefore
important to note that even though we have found symbol lengths of three for
$l>2$, this may be an artifact of choosing our particular basis and its
decomposition of semi-simple and unipotent periods.

##  4 The Generic-Mass Case

We now move to the generic-mass case. We will in this section attempt to
generalize the results of the previous section. We will therefore consider the
$\epsilon=0$ reduced basis of master integrals expressed in eq. (2.6). Many of
the expressions we had in the equal-mass case are in the generic-mass case
replaced by very long and complicated polynomials in all $l+1$ scales. As we
are mostly interested in the structure of the symbol and coaction, we will in
general not derive these expressions explicitly, but instead describe how they
may be derived.

###  4.1 Frobenius Basis

It is possible to extend the one-dimensional Frobenius basis in eq. (3.2) to
the multivariate case. In this case we consider a set of differential
operators $\mathcal{D}=\\{\mathcal{L}_{1},...,\mathcal{L}_{s}\\}$. The
solution space is spanned by the set of linearly independent functions
$f(\underline{z})$ that are simultaneously annihilated by all of the operators
in $\mathcal{D}$. This set of differential operators is by no means unique, in
the sense that two different sets of operators may generate the exact same
solution space [74]. We are looking for operators of the form

$\mathcal{L}_{l,k}=\sum_{\underline{p}}^{|\underline{p}|=l}\beta_{k,\underline{p}}(\underline{z})\partial^{\underline{p}}_{\underline{z}},$
(4.1)

where $p$ is an $l+1$ vector to generate our solution space. This particular
form of operators is important as it allows us to relate an $l$-order
differential to lower order differentials, which is needed for the derivation
of generic-mass Griffiths relations.

It is again possible to find a solution around a MUM-point, located at
$\underline{z}_{0}=0$. Around this point there exists one power series type
solution

$\varpi_{0}(\underline{z})=\sum_{j_{1},...,j_{l+1}\geq
0}\left(\frac{|j|!}{j_{1}!...j_{l+1}!}\right)^{2}\prod_{n=1}^{l+1}z_{n}^{j_{n}}\,.$
(4.2)

The rest of the solutions are graded by increasing orders of logarithms. In
the generic-mass case however, there are typically multiple distinct solutions
at each order. For instance, solutions for $l>2$ have $l+1$ single-logarithmic
solutions, normalized as

$\varpi_{1}^{k}(\underline{z})=\log(z_{k})+\mathcal{O}(\underline{z}),$ (4.3)

In total one finds that there are
$\lambda=2^{l+1}-\left(\begin{subarray}{c}l+2\\\
\lfloor\frac{l+2}{2}\rfloor\end{subarray}\right)$ elements in the Frobenius
basis. For all loop orders there exists only one solution with total
logarithmic order $l-1$, which we denote as $\varpi_{\lambda}(\underline{z})$
[73].

The elements in the generic-mass Frobenius basis are also related through
Griffiths Transversality (eq. (2.33)). Just as in the equal-mass case, this
can be used to simplify the inverse Wronskian. We are therefore motivated to
find generic-mass version of the Griffiths relations derived in section (2.4).

The intersection matrix $\Sigma_{l}$ in the generic-mass case has the same
$(-1)^{l+1}$ symmetry as in the equal-mass case. The construction of the
intersection matrix is however not the same. Using the expansions for the
generic-mass Frobenius basis one can derive that the three and four loop
intersection matrices have the forms:

$\Sigma_{3}=\left(\begin{matrix}0&0&0&0&0&1\\\ 0&0&-1&-1&-1&0\\\
0&-1&0&-1&-1&0\\\ 0&-1&0&0&-1&0\\\ 0&-1&-1&-1&0&0\\\ 1&0&0&0&0&0\\\
\end{matrix}\right)\\\
,\;\;\;\;\Sigma_{4}=\left(\setcounter{MaxMatrixCols}{12}\begin{matrix}0&0&0&0&0&0&0&0&0&0&0&1\\\
0&0&0&0&0&0&-1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&-1&0&0&0&0\\\
0&0&0&0&0&0&0&0&-1&0&0&0\\\ 0&0&0&0&0&0&0&0&0&-1&0&0\\\
0&0&0&0&0&0&0&0&0&0&-1&0\\\ 0&1&0&0&0&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0&0&0\\\
0&0&0&0&1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0&0&0&0\\\
-1&0&0&0&0&0&0&0&0&0&0&0\\\ \end{matrix}\right).$ (4.4)

From now on we will assume that $n=|\underline{k}|=l-1$ in eq. (2.33) and
derive relations from that. Just as in the equal-mass case, we can gain more
relations by using the product rule. However, since $\underline{k}$ is now a
vector, we have a choice of which derivative to “pull out” of
$\partial_{z_{i}}^{\underline{k}}$. We can keep our notation general by
pulling out $\partial_{z_{i}}$, as long as the restriction to cases where
$\underline{k}_{i}\geq 1$ is understood. By pulling out one $\partial_{z_{i}}$
we get the equation

$C_{\underline{k}}(\underline{z})=\partial_{z_{i}}\left(\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}-1_{i}}\underline{\Pi}(\underline{z})\right)-\partial_{z_{i}}\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}-1_{i}}\underline{\Pi}(\underline{z}),$
(4.5)

where $\underline{k}-1_{i}$, means that we are subtracting one from the $i$’th
entry in $\underline{k}$. Equivalently one can think of
$1_{i}=(0,...,0,1,0,...,0)$ as an $l+1$ vector, where the $i$ indicates the
entry that has the “$1$”. The first term on the right hand side is zero due to
the fact that $|k-1_{i}|<l-1$. We can once again pull out a derivative
$\partial_{z_{j}}$ from the second term in eq. (4.5), where $j$ need not be
different from $i$

$C_{\underline{k}}(\underline{z})=-\partial_{z_{j}}\partial_{z_{i}}\left(\partial_{z_{i}}\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}-1_{i}-1_{j}}\underline{\Pi}(\underline{z})\right)+\partial_{z_{j}}\partial_{z_{i}}\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}-1_{i}-1_{j}}\underline{\Pi}(\underline{z}).$
(4.6)

It is once again possible to show that the first term on the right hand side
vanishes due to eq. (2.33). We can keep going in this way, and in general we
obtain

$C_{\underline{k}}(\underline{z})=\partial_{z}^{\underline{s}}\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}-\underline{s}}\underline{\Pi}(\underline{z}),$
(4.7)

where $|s|\leq|k|$ and $k_{i}-s_{i}\geq 0$ for all $1,\leq i\leq l+1$.
We will now derive the next set of Griffiths relations, by taking the $i$’th
derivative of eq. (2.33)

$\begin{split}\partial_{z_{i}}\left(\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}}\underline{\Pi}(\underline{z})\right)&=\partial_{z_{i}}C_{\underline{k}}(\underline{z})\\\
\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}+1_{i}}\underline{\Pi}(\underline{z})&=\partial_{z_{i}}C_{\underline{k}}(\underline{z})-\partial_{z_{i}}\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}}\underline{\Pi}(\underline{z}).\end{split}$
(4.8)

If we now pull out an $\partial_{z_{j}}$ from
$\partial_{\underline{z}}^{\underline{k}}$ in the second term, we find

$\begin{split}\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}+1_{i}}\underline{\Pi}(\underline{z})&=\partial_{z_{i}}C_{\underline{k}}(\underline{z})-\partial_{z_{j}}\left(\partial_{z_{i}}\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}-1_{j}}\underline{\Pi}(\underline{z})\right)+\partial_{z_{j}}\partial_{z_{i}}\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}-1_{j}}\underline{\Pi}(\underline{z})\\\
&=\partial_{z_{i}}C_{\underline{k}}(\underline{z})-\partial_{z_{j}}C_{\underline{k}+1_{i}-1_{j}}(\underline{z})+\partial_{z_{j}}\partial_{z_{i}}\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}-1_{j}}\underline{\Pi}(\underline{z}).\end{split}$
(4.9)

(We must still be careful to apply this only in cases where
$\underline{k}_{j}-1_{j}\geq 0$.) Performing this procedure successively, we
find

$\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}+1_{i}}\underline{\Pi}(\underline{z})=\partial_{z_{i}}C_{\underline{k}}(\underline{z})+\sum_{n=1}^{l+1}(-1)^{|\underline{k}_{n}+1|}\underline{k}_{n}\partial_{z_{n}}C_{\underline{k}+1_{i}-1_{n}}(\underline{z})+(-1)^{|\underline{k}+1_{i}|}\partial_{z}^{\underline{k}}\underline{\Pi}(\underline{z})\Sigma_{l}\underline{\Pi}(\underline{z}).$
(4.10)

Due to the $(-1)^{l+1}$ symmetry of $\Sigma_{l}$ we can write (4.10) as

$\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}+1_{i}}\underline{\Pi}(\underline{z})=\frac{1}{2}\partial_{z_{i}}C_{\underline{k}}(\underline{z})-\sum_{n=1}^{l+1}(-1)^{\underline{k}_{n}}\frac{\underline{k}_{n}}{2}\partial_{z_{n}}C_{\underline{k}+1_{i}-1_{n}}(\underline{z}).$
(4.11)

We can insert eq. (4.11) back into eq. (4.8) to find yet more relations

$\partial_{z_{i}}\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}}\underline{\Pi}(\underline{z})=\frac{1}{2}\partial_{z_{i}}C_{\underline{k}}(\underline{z})+\sum_{n=1}^{l+1}(-1)^{\underline{k}_{n}}\frac{\underline{k}_{n}}{2}\partial_{z_{n}}C_{\underline{k}+1_{i}-1_{n}}(\underline{z}).$
(4.12)

We will now derive a generic-mass version of the differential equation in eq.
(3.10). To do this we again use the fact that the Picard-Fuchs operators
annihilate the Frobenius basis

$\begin{split}0&=\underline{\Pi}(\underline{z})\Sigma_{l}\mathcal{L}_{l,k}\underline{\Pi}(\underline{z}),\\\
&=\underline{\Pi}(\underline{z})\Sigma_{l}\beta_{l,\underline{k}+1_{i}}^{(l)}\partial_{\underline{z}}^{\underline{k}+1_{i}}\underline{\Pi}(\underline{z})+\underline{\Pi}(\underline{z})\Sigma_{l}\beta_{l,\underline{k}}^{(l)}\partial_{\underline{z}}^{\underline{k}}\underline{\Pi}(\underline{z}).\end{split}$
(4.13)

Combining this with eq. (2.33) and (4.11) we get

$\sum_{n=1}^{l+1}(-1)^{k_{n}}\frac{k_{n}}{2}\partial_{z_{n}}C_{\underline{k}+1_{i}-1_{n}}(\underline{z})=\frac{1}{2}\partial_{z_{i}}C_{\underline{k}}(\underline{z})+\frac{\beta_{l,k}^{(l)}(\underline{z})}{\beta_{l,\underline{k}+1_{i}}^{(l)}}C_{\underline{k}}.$
(4.14)

This differential equation may be solved iteratively, but would be somewhat of
a distraction from the main text, so we explain it in Appendix (A).
Finally we can obtain more relations by using the fact that the Picard-Fuchs
operators $\mathcal{L}_{l,i}$ annihilate the maximal cuts,

$\begin{split}\partial_{z_{i}}\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}}\underline{\Pi}(\underline{z})&=\partial_{z_{i}}C_{\underline{k}}(\underline{z})-\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{k}+1_{i}}\underline{\Pi}(\underline{z})\\\
&=\partial_{z_{i}}C_{\underline{k}}(\underline{z})+\underline{\Pi}(\underline{z})\Sigma_{l}\sum_{\underline{p}/(\underline{k}+1_{i})}^{|p|=l-1}\frac{\beta_{i,\underline{p}}^{(l)}(\underline{z})}{\beta_{i,\underline{k}+1_{i}}^{(l)}(\underline{z})}\partial_{\underline{z}}^{\underline{p}}\underline{\Pi}(\underline{z}).\end{split}$
(4.15)

We can keep taking derivatives of eq. (4.8) and using the Picard-Fuchs
operator to gain more and more relations. Eventually we will have enough
relations to construct a generic-mass version of the matrix

$\mathbf{Z}_{l}(\underline{z})=\left(\begin{matrix}\underline{\Pi}(\underline{z})\Sigma_{l}\underline{\Pi}(\underline{z})&...&\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{s}_{\lambda}}\underline{\Pi}(\underline{z})\\\
\vdots&\ddots&\vdots\\\
\partial_{\underline{z}}^{\underline{s}_{\lambda}}\underline{\Pi}(\underline{z})\Sigma_{l}\underline{\Pi}(\underline{z})&...&\partial_{\underline{z}}^{\underline{s}_{\lambda}}\underline{\Pi}(\underline{z})\Sigma_{l}\partial_{\underline{z}}^{\underline{s}_{\lambda}}\underline{\Pi}(\underline{z})\end{matrix}\right).$
(4.16)

Note that the differentials that go into $\mathbf{Z}_{l}(\underline{z})$ are
the same as the ones that are used to construct the Wronskian. The matrix has
the same symmetry as in the equal-mass case;
$\mathbf{Z}_{l}(\underline{z})=(-1)^{l+1}\mathbf{Z}(\underline{z})^{T}$. One
can show that in the generic-mass case the inverse Wronskian is still
expressible as

$\mathbf{W}(\underline{z})^{-1}=\Sigma_{l}\mathbf{W}(\underline{z})^{T}\mathbf{Z}(\underline{z})^{-1}.$
(4.17)

In the generic-mass case, we are then once again able to describe the inverse
Wronskian linearly in terms of entries in $\mathbf{W}_{l}(\underline{z})$.

###  4.2 Unipotent Differential Equation

We proceed, following essentially the same analysis as in the equal-mass case.
First we need to find the form of the inhomogeneous term in eq. (2.30). We may
once again act with the Picard-Fuchs operators on our basis of master
integrals $\underline{J}_{l}(\underline{z})$. Doing so, we find that only the
very first element is trivial. The non-trivial elements are complicated
polynomials in all scales. To concentrate on the relevant structure, we will
here simply express the inhomogeneous part as

$\mathbf{N}_{l}(\underline{z})=\left(\begin{matrix}0\\\
\sum_{i}^{l+1}S_{1,i}(\underline{z})dz_{i}\\\ \vdots\\\
\sum_{i}^{l+1}S_{l-1,i}(\underline{z})dz_{i}\end{matrix}\right)\,,$ (4.18)

where the $S_{i,j}$’s are determined by the inhomogeneities obtained from eq.
(2.32).

We now define the ratios
$\tau_{k}=\varpi_{k}(\underline{z})/\varpi_{\lambda}(\underline{z})$, which we
will later take as unipotent periods in our basis. We then use eq. (2.31) to
express the differential equation our master integrals satisfy, in the
$L$-basis. As opposed to the equal-mass case, the entries in the inverse
Wronksian are not necessarily proportional to just one element (c.f. eq.
(3.17)), but can in general be linear combinations of all elements of the
Frobenius basis. This is for instance the case at $l=3$. With this in mind we
can write the differential equation eq. (2.31) as

$dL_{k}(\underline{z})=\sum_{i=0}^{l-1}c_{k,i}\big{(}X_{i}(\underline{z})\tau_{i}(\underline{z})+D_{i}(\underline{z})\big{)}\,,$
(4.19)

where $c_{k,i}\in\\{-1,0,1\\}$ and,

$\begin{split}X_{k}(\underline{z})=\alpha_{k,\underline{0}}(\underline{z})\varpi_{\lambda}(\underline{z})+\sum_{\underline{s}\in
T}\alpha_{k,\underline{s}}(\underline{z})\partial_{\underline{z}}^{\underline{s}}\varpi_{\lambda}(\underline{z}),\\\
D_{k}(\underline{z})=\sum_{\underline{s}\in
T}\sum_{\begin{subarray}{c}\underline{m},\underline{s}\\\
n\neq\underline{0}\\\
\underline{m}+\underline{n}=\underline{s}\end{subarray}}\alpha_{k,\underline{s}}(\underline{z})\partial_{\underline{z}}^{\underline{m}}\varpi_{\lambda}\partial_{\underline{z}}^{\underline{n}}\tau_{k}(\underline{z}).\end{split}$
(4.20)

The set $T$ contains all vectors such that for $\underline{s}\in T$ the
derivative $\partial_{\underline{z}}^{\underline{s}}J_{l,0}(\underline{z})$ is
a master integral. The $\alpha$ coefficients are one-forms of the form
$\alpha(\underline{z})=\sum_{i}^{l+1}\omega_{i}(\underline{z})dz_{i}$, where
the $\omega_{i}$ are rational functions.

We now define a new basis, which will allow us to build a unipotent
differential equation, by expanding the $L_{k}(\underline{z})$ basis with all
$\tau_{k}(\underline{z})$ elements:

$T_{l}(\underline{z})=\left(L_{l,1}(\underline{z}),...,L_{l,\lambda}(\underline{z}),\tau_{0}(\underline{z}),...,\tau_{\lambda-1}(\underline{z}),1\right).$
(4.21)

This basis satisfies the unipotent differential equation

$d\underline{T}(\underline{z})=\mathbf{N}_{l}(\underline{z})\underline{T}_{l}(\underline{z}).$
(4.22)

The nilpotent matricies for $l=3,4$ have the following structure:

$\mathbf{N}_{3}(\underline{z})=\left(\begin{matrix}0&\overbrace{...}^{\times
5}&0&0&0&0&0&X_{5}+D_{5}\\\
0&...&0&0&-X_{2}&-X_{3}&-X_{4}&-D_{2}-D_{3}-D_{4}\\\
0&...&0&-X_{1}&0&-X_{3}&-X_{4}&-D_{1}-D_{3}-D_{4}\\\
0&...&0&-X_{1}&-X_{2}&0&-X_{4}&-D_{1}-D_{2}-D_{4}\\\
0&...&0&-X_{1}&-X_{2}&-X_{3}&0&-D_{1}-D_{2}-D_{3}\\\
0&...&X_{0}&0&0&0&0&D_{0}\\\ 0&...&0&0&0&0&0&d\tau_{0}\\\
0&...&0&0&0&0&0&\vdots\\\ 0&...&0&0&0&0&0&d\tau_{\lambda-1}\\\
0&...&0&0&0&0&0&0\\\ \end{matrix}\right),$ (4.23)
$\mathbf{N}_{4}(\underline{z})=\left(\setcounter{MaxMatrixCols}{14}\begin{matrix}0&\overbrace{...}^{\times
11}&0&0&0&0&0&0&0&0&0&0&0&X_{0}+D_{0}\\\
0&...&0&0&0&0&0&0&-X_{6}&0&0&0&0&-D_{6}\\\
0&...&0&0&0&0&0&0&0&-X_{7}&0&0&0&-D_{7}\\\
0&...&0&0&0&0&0&0&0&0&-X_{8}&0&0&-D_{8}\\\
0&...&0&0&0&0&0&0&0&0&0&-X_{9}&0&-D_{9}\\\
0&...&0&0&0&0&0&0&0&0&0&0&-X_{10}&-D_{10}\\\
0&...&0&X_{1}&0&0&0&0&0&0&0&0&0&D_{1}\\\
0&...&0&0&X_{2}&0&0&0&0&0&0&0&0&D_{2}\\\
0&...&0&0&0&X_{3}&0&0&0&0&0&0&0&D_{3}\\\
0&...&0&0&0&0&X_{4}&0&0&0&0&0&0&D_{4}\\\
0&...&0&0&0&0&0&X_{5}&0&0&0&0&0&D_{5}\\\
0&...&-X_{11}&0&0&0&0&0&0&0&0&0&0&-D_{11}\\\
0&...&0&0&0&0&0&0&0&0&0&0&0&d\tau_{0}\\\ 0&...&0&0&0&0&0&0&0&0&0&0&0&\vdots\\\
0&...&0&0&0&0&0&0&0&0&0&0&0&d\tau_{\lambda-1}\\\
0&...&0&0&0&0&0&0&0&0&0&0&0&0\\\ \end{matrix}\right),$ (4.24)

where we have suppressed the explicit $\underline{z}$ dependence.

We note that the nilpotent matrices are very similar in structure to the
intersection matrices for these cases. In this sense, it is relatively simple
to construct the nilpotent matrix once one knows the intersection matrix.

###  4.3 Symbol

We are now ready to construct a symbol on our master integrals. We once again
define a unipotent matrix of the form in eq. (3.29) using the nilpotent
matrices found in the previous section. Similarly to the equal-mass case the
symbols are all of length two. We list all relevant results for the three- and
four-loop cases.

$l=3$:

$\begin{split}\mathcal{S}([\xi_{i},\xi_{i}])&=1\\\
\mathcal{S}([1,L_{1}])&=[D_{5}]+[X_{5}]\\\
\mathcal{S}([1,L_{2}])&=-[d\tau_{2}|X_{2}]-[d\tau_{3}|X_{3}]-[d\tau_{4}|X_{4}]-[D_{2}]-[D_{3}]-[D_{4}]\\\
\mathcal{S}([1,L_{3}])&=-[d\tau_{1}|X_{1}]-[d\tau_{3}|X_{3}]-[d\tau_{4}|X_{4}]-[D_{1}]-[D_{3}]-[D_{4}]\\\
\mathcal{S}([1,L_{4}])&=-[d\tau_{1}|X_{1}]-[d\tau_{2}|X_{2}]-[d\tau_{4}|X_{4}]-[D_{1}]-[D_{2}]-[D_{4}]\\\
\mathcal{S}([1,L_{5}])&=-[d\tau_{1}|X_{1}]-[d\tau_{2}|X_{2}]-[d\tau_{3}|X_{3}]-[D_{1}]-[D_{2}]-[D_{3}]\\\
\mathcal{S}([1,L_{6}])&=[d\tau_{0}|X_{0}]+[D_{0}]\\\
\mathcal{S}([1,\tau_{k}])&=[d\tau_{k}]\\\ \end{split}$ (4.25)

$l=4$:

$\begin{split}\mathcal{S}([\xi_{i},\xi_{i}])&=1\\\
\mathcal{S}([1,L_{1}])&=[D_{0}]+[X_{0}]\\\
\mathcal{S}([1,L_{2}])&=-[d\tau_{6}|X_{6}]-[D_{6}]\\\
\mathcal{S}([1,L_{3}])&=-[d\tau_{7}|X_{7}]-[D_{7}]\\\
\mathcal{S}([1,L_{4}])&=-[d\tau_{8}|X_{8}]-[D_{8}]\\\
\mathcal{S}([1,L_{5}])&=-[d\tau_{9}|X_{9}]-[D_{9}]\\\
\mathcal{S}([1,L_{6}])&=-[d\tau_{10}|X_{10}]-[D_{10}]\\\
\mathcal{S}([1,L_{7}])&=[d\tau_{1}|X_{1}]+[D_{1}]\\\
\mathcal{S}([1,L_{8}])&=[d\tau_{2}|X_{2}]+[D_{2}]\\\
\mathcal{S}([1,L_{9}])&=[d\tau_{3}|X_{3}]+[D_{3}]\\\
\mathcal{S}([1,L_{10}])&=[d\tau_{4}|X_{4}]+[D_{4}]\\\
\mathcal{S}([1,L_{11}])&=[d\tau_{5}|X_{5}]+[D_{5}]\\\
\mathcal{S}([1,L_{12}])&=-[d\tau_{11}|X_{11}]-[D_{11}]\\\
\mathcal{S}([1,\tau_{k}])&=[d\tau_{k}]\end{split}$ (4.26)

###  4.4 Coaction

Now that we have the symbols defined, we can calculate the coaction on our
master integrals as discussed for the equal-mass case. We find the following:

$l=3$:

$\begin{split}\Delta(L_{3,1}(\underline{z})=&1\otimes D_{5}+1\otimes
X_{5}+L_{3,1}(\underline{z})\otimes 1\\\
\Delta(L_{3,2}(\underline{z})=&L_{3,2}(\underline{z})\otimes
1-1\otimes[d\tau_{2}|X_{2}]-1\otimes[d\tau_{3}|X_{3}]-1\otimes[d\tau_{4}|X_{4}]-1\otimes(D_{2}+D_{3}+D_{4})\\\
&-\tau_{2}\otimes X_{2}-\tau_{3}\otimes X_{3}-\tau_{4}\otimes X_{4}\\\
\Delta(L_{3,3}(\underline{z})=&L_{3,3}(\underline{z})\otimes
1-1\otimes[d\tau_{1}|X_{1}]-1\otimes[d\tau_{3}|X_{3}]-1\otimes[d\tau_{4}|X_{4}]-1\otimes(D_{1}+D_{3}+D_{4})\\\
&-\tau_{1}\otimes X_{1}-\tau_{3}\otimes X_{3}-\tau_{4}\otimes X_{4}\\\
\Delta(L_{3,4}(\underline{z})=&L_{3,4}(\underline{z})\otimes
1-1\otimes[d\tau_{1}|X_{1}]-1\otimes[d\tau_{2}|X_{2}]-1\otimes[d\tau_{4}|X_{3}]-1\otimes(D_{1}+D_{2}+D_{4})\\\
&-\tau_{1}\otimes X_{1}-\tau_{2}\otimes X_{2}-\tau_{4}\otimes X_{4}\\\
\Delta(L_{3,5}(\underline{z})=&L_{3,5}(\underline{z})\otimes
1-1\otimes[d\tau_{1}|X_{1}]-1\otimes[d\tau_{2}|X_{2}]-1\otimes[d\tau_{3}|X_{3}]-1\otimes(D_{1}+D_{2}+D_{3})\\\
&-\tau_{1}\otimes X_{1}-\tau_{2}\otimes X_{2}-\tau_{3}\otimes X_{3}\\\
\Delta(L_{3,6}(\underline{z})=&L_{3,6}(\underline{z})\otimes
1+1\otimes[d\tau_{0}|X_{0}]+1\otimes D_{0}+\tau_{0}\otimes X_{0}\\\
\Delta(\tau_{k})=&\tau_{k}\otimes 1+1\otimes[d\tau_{k}]\end{split}$ (4.27)

$l=4$:

$\begin{split}\Delta(L_{4,1}(\underline{z})=&L_{4,1}(\underline{z})\otimes
1+1\otimes X_{0}+1\otimes D_{0}\\\
\Delta(L_{4,2}(\underline{z})=&L_{4,2}(\underline{z})\otimes
1-1\otimes[d\tau_{6}|X_{1}]-1\otimes D_{1}-\tau_{6}\otimes X_{1}\\\
\Delta(L_{4,3}(\underline{z})=&L_{4,3}(\underline{z})\otimes
1-1\otimes[d\tau_{7}|X_{2}]-1\otimes D_{2}-\tau_{7}\otimes X_{2}\\\
\Delta(L_{4,4}(\underline{z})=&L_{4,4}(\underline{z})\otimes
1-1\otimes[d\tau_{8}|X_{3}]-1\otimes D_{3}-\tau_{8}\otimes X_{3}\\\
\Delta(L_{4,5}(\underline{z})=&L_{4,5}(\underline{z})\otimes
1-1\otimes[d\tau_{9}|X_{4}]-1\otimes D_{4}-\tau_{9}\otimes X_{4}\\\
\Delta(L_{4,6}(\underline{z})=&L_{4,6}(\underline{z})\otimes
1-1\otimes[d\tau_{10}|X_{5}]-1\otimes D_{5}-\tau_{10}\otimes X_{5}\\\
\Delta(L_{4,7}(\underline{z})=&L_{4,7}(\underline{z})\otimes
1+1\otimes[d\tau_{1}|X_{6}]+1\otimes D_{6}+\tau_{1}\otimes X_{6}\\\
\Delta(L_{4,8}(\underline{z})=&L_{4,8}(\underline{z})\otimes
1+1\otimes[d\tau_{2}|X_{7}]+1\otimes D_{7}+\tau_{2}\otimes X_{7}\\\
\Delta(L_{4,9}(\underline{z})=&L_{4,9}(\underline{z})\otimes
1+1\otimes[d\tau_{3}|X_{8}]+1\otimes D_{8}+\tau_{3}\otimes X_{8}\\\
\Delta(L_{4,10}(\underline{z})=&L_{4,10}(\underline{z})\otimes
1+1\otimes[d\tau_{4}|X_{9}]+1\otimes D_{9}+\tau_{4}\otimes X_{9}\\\
\Delta(L_{4,11}(\underline{z})=&L_{4,11}(\underline{z})\otimes
1+1\otimes[d\tau_{5}|X_{10}]+1\otimes D_{10}+\tau_{5}\otimes X_{10}\\\
\Delta(L_{4,12}(\underline{z})=&L_{4,12}(\underline{z})\otimes
1-1\otimes[d\tau_{0}|X_{11}]-1\otimes D_{11}-\tau_{0}\otimes X_{11}\\\
\Delta(\tau_{k})=&\tau_{k}\otimes 1+1\otimes[d\tau_{k}]\end{split}$ (4.28)

Unlike in the $l=3$ equal-mass case, here there can be no direct comparison
with existing expressions. We only remark that the structure we observe
appears to be quite consistent across loop orders, up to the complexity of the
coefficients $X_{k}$ and $D_{k}$. It is again possible to transform back to
the original $\underline{J}_{l}(\underline{z})$ basis, with the transformation
$\underline{J}_{l}(\underline{z})=\mathbf{W}_{l}(\underline{z})\underline{L}_{l}(\underline{z})$.
Doing so we again find that for $l\geq 3$ the symbol length is three.

##  5 Conclusions

We have constructed a coaction and symbol for the sunrise integral at
arbitrary loop order, both for the equal-mass case and for the generic-mass
case. We built these coactions and symbols by re-arranging the differential
equations satisfied by the sunrise master integrals into a unipotent form,
augmenting them with ratios of maximal cuts $\tau_{i}$. We find that, counter
to our experience in the case of polylogarithms, the lengths of these symbols
do not increase with loop order: instead, they saturate at length three to all
loops. This suggests that our symbol entries are increasing in complexity with
loop order, which may mean they satisfy even more complicated relations along
the lines of the symbol-prime of ref. [79]. At two and three loops, our
symbols and coactions should in principle be comparable with those derived for
the equal-mass sunrise at two loops in ref. [62], for the equal-mass sunrise
at three loops in ref. [68], and for the unequal-mass sunrise at two loops in
ref. [79]. While the structures we observe are in general similar, we have not
yet investigated whether we can show direct equivalence.

We highlight two choices in our construction, which if varied could yield
substantially different results. First, we made a choice to expand our basis
with the ratios $\tau_{i}$, and not with additional functions. In general, one
could imagine introducing more intermediate functions related by differential
equations, leading to longer symbols. Such functions would not be Feynman
integrals, at least not in the same dimension as the sunrise integrals we
consider, but this should not be especially unfamiliar: when constructing
polylogarithmic functions to bootstrap scattering amplitudes as in refs. [26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], one must in general consider
integrals that are not themselves Feynman integrals.

Second, we made a choice in our separation of our integrals into semi-simple
and unipotent factors. Different choices can lead to different lengths, and to
substantially different symbols more generally, by factoring out different
transcendental functions from the functions considered by the symbol. While we
believe our choice is reasonable (motivated in part by the symbol and coaction
on elliptic multiple polylogarithms), it is not the only possible choice, and
a different choice may be more informative in other circumstances.

It is important to note that the letters of our symbol contain higher powers
of logarithms around the MUM point. As such, taking a limit that reduces the
sunrise to a polylogarithm should have a nontrivial effect on our symbol,
almost certainly not directly yielding a polylogarithmic symbol. In future
work we would like to investigate the approach to pseudothresholds, the most
straightforward such limit to study.

In general, we expect a symbol for Calabi-Yau integrals that matches the
length of polylogarithmic symbols to require a somewhat different
construction. It would need to involve a substantially larger basis, likely as
part of a basis of functions more general than the sunrise master integrals
alone. It would be extremely interesting to construct such a basis, which may
demand investigation of more general diagrams involving Calabi-Yau manifolds
along the lines of [80, 81, 82, 83].

Acknowledgements

We thank Kilian Bönisch, Matthias Wilhelm, and Cristian Vergu for helpful
discussions. This work was supported by the Danish National Research
Foundation (DNRF91), the research grant 00015369 from Villum Fonden, and a
Starting Grant (No. 757978) from the European Research Council.

##  Appendix A Appendix: Solution to differential equations satisfied by
generic-mass Yukawa-couplings

Solving the differential equations in eq. (4.14) can be done in an iterative
fashion. We will start by considering the explicit example of solving the
equations for $l=3$. We will in the end comment on how the $l=3$ case
generalizes quite easily for higher loop orders.

In the case of $l=3$, the Griffiths Transversality condition (eq. (2.33)) only
has non-trivial solutions for $|k|=2$. We start with the explicit example of
$\underline{k}=(2,0,0)$. In this case, we have three equations (one for each
$i$):

$\displaystyle\partial_{z_{1}}C_{2,0,0}(\underline{z})$
$\displaystyle=2\frac{\beta_{l,(2,0,0)}^{(l)}(\underline{z})}{\beta_{l,(3,0,0)}^{(l)}(\underline{z})}C_{2,0,0}(\underline{z}),$
(A.1) $\displaystyle\partial_{z_{1}}C_{1,1,0}(\underline{z})$
$\displaystyle=\frac{1}{2}\partial_{z_{2}}C_{2,0,0}(\underline{z})+\frac{\beta_{l,2,0,0}^{(l)}(\underline{z})}{\beta_{l,(2,1,0)}^{(l)}(\underline{z})}C_{2,0,0}(\underline{z}),$
(A.2) $\displaystyle\partial_{z_{1}}C_{1,0,1}(\underline{z})$
$\displaystyle=\frac{1}{2}\partial_{z_{3}}C_{2,0,0}(\underline{z})+\frac{\beta_{l,(2,0,0)}^{(l)}(\underline{z})}{\beta_{l,(2,0,1)}^{(l)}(\underline{z})}C_{2,0,0}(\underline{z}).$
(A.3)

The first of these equations, namely eq. (A.1), is a differential equation
only in the function $C_{2,0,0}(\underline{z})$. This differential equation
has the general solution

$C_{2,0,0}(\underline{z})=c_{1}(z_{2},z_{3})\exp\left(2\int_{1}^{z_{1}}dt\frac{\beta_{l,k}^{(l)}(t,z_{2},z_{3})}{\beta_{l,(3,0,0)}^{(l)}(t,z_{2},z_{3})}\right),$
(A.4)

where $c(z_{2},z_{3})$ is an integration constant. Note that due to the
symmetry of $z_{1},z_{2},z_{3}$ in eq. (4.14), the solutions for $C_{0,2,0}$
and $C_{0,0,2}$ are the same as in eq. (A.4), only with the role of $z_{1}$
exchanged with $z_{2}$ and $z_{3}$ respectively. The solution for
$C_{2,0,0},C_{0,2,0}$ and $C_{0,0,2}$ can be used to solve the differential
equations involving $C_{1,1,0},C_{1,0,1},C_{0,1,1}$ by simple integration.

This procedure generalizes to higher loop order. One starts at
$\underline{k_{1}}=(0,...,l-1,...,0)$, to find the solutions of the type in
eq. (A.4). One then moves on to $\underline{k_{2}}=(0,...,l-2,,0,...,1,...,0)$
and solves the equations using the functions $C_{\underline{k_{1}}}$ just
obtained. It is possible to continue in this fashion until one reaches the
differential equations corresponding to
$\underline{k}_{l}=(1,1,...,1,0,1,...,1)$. Due to the way the differentials
are present in eq. (2.6), we note that the $\mathbf{Z}_{l}(\underline{z})$
will only depend on Yukawa couplings of the type
$C_{1,...,1,0,1,...,1}(\underline{z})$.

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