aainstitutetext: II. Institut für Theoretische Physik, Universität Hamburg,
22761 Hamburg, Germanybbinstitutetext: Bogoliubov Laboratory of Theoretical
Physics, Joint Institute for Nuclear Research, Dubna, Russiaccinstitutetext:
Budker Institute of Nuclear Physics, Novosibirsk, Russiaddinstitutetext:
Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow,
Russia,

# Massive Two-Loop Heavy Particle Diagrams

B. Assi a B.A. Kniehl b,c,d A.I. Onishchenko

###### Abstract

We determine the master integrals for vertex and propagator diagrams that
appear in effective field theories containing heavy fields. The integrals
involve at least one heavy line, and the standard lines include an arbitrary
mass scale. The evaluation is done analytically with modern techniques. We
employ the methods of differential equations and dimensional recurrence
relations to evaluate said integrals up to two-loop order.

##  1 Introduction

Heavy Particle Effective Theories (HPET) have a wide range of applicability,
and its use cases have expanded dramatically in recent years. They are
apparent when a field of arbitrary spin in a given theory is taken to have a
large mass compared to other propagating massive degrees of freedom. HPETs
were originally conceived in the context of Quantum Electrodynamics (QED) and
Quantum Chromodynamics (QCD), such as in Heavy Quark Effective Theory (HQET),
Non-Relativistic (NR) QCD and QED and variations therein pineda1997effective ;
brambilla2000potential ; georgi1990effective . More recently, it has also been
applied in the Electroweak (EW) regime jantzen2005two ; jantzen2006two ;
chiu2008electroweak ; chiu2008electroweak0 ; chiu2008electroweak2 ; assi1 ;
assi3 , as well as beyond the Standard Model (SM) such as in the context of
heavy dark matter ovanesyan2015heavy ; beneke2019wino , $Z^{\prime}$ bosons
bauer2020low ; mecaj2020effective and black hole interactions
damgaard2019heavy ; aoude2020shell ; bern2021gravitational .

When dealing with such theories beyond leading perturbative order, one is
faced with loop diagrams containing eikonal lines. In this work, we determine
these at two-loop order employing a set of modern techniques, in particular
differential equations and dimensional recurrence relations, which have been
successful in similar contexts smirnov2010three ; lee2016evaluating . We
further include a non-zero mass-scale in the standard lines for theoretical
models with massive propagating degrees of freedom. The mass scale bounds the
infrared (IR) regime for the two- and three-point diagrams studied here. Even
in theories with exclusively massless propagating degrees of freedom such as
QED/QCD and gravity, the IR structure needs to be correctly understood
ablinger2018heavy ; bern2021gravitational . The diagrams considered here are
especially useful in the evaluation of form factors of a given model. The form
factor is most well-known for its uses in perturbative analyses of scattering
processes occurring at the LHC and future colliders chiu2009factorization ;
chiesa2013electroweak . Form factors are of primary consideration instead of
specific processes as they form the fundamental building blocks for a vast
array of processes. For instance, they have been employed to study di-jet,
$\bar{t}t$, squark pair, and dark matter (DM) production in various models
chiu2008electroweak2 ; beenakker2010supersymmetric ; ovanesyan2015heavy ;
ciafaloni2011weak . It is also the simplest amplitude that can be used to
study the IR behaviour of a theory of interest. For further reference in the
context of the SM, the QCD form factors of quarks have been evaluated to
three-loop order gehrmann2010calculation ; von2017quark ; bernreuther2005two ;
blumlein2019heavy ; ablinger2018heavy , and the EW corrections using both EFT
and IR evolution equations are currently being studied to two-loop order
ciafaloni2000electroweak ; fadin2000resummation ; kuhn2000summing ;
feucht2004two ; jantzen2005two ; denner2001one ; hori2000electroweak ;
jantzen2006two ; assi3 .

On the other hand, there has also been significant progress in the realm of
Feynman diagram evaluation. When previously, certain classes of multi-loop
diagrams were intractable, they have now become determinable with the help of
novel techniques. Most notably, diagrams with masses are now attainable with
the differential equations method diffeqn1 ; diffeqn2 ; diffeqn3 ; diffeqn4 ;
diffeqn5 . The basis of which is set upon differentiating the master integrals
(MIs) of interest, forming a system of differential equations and reducing
said system to so-called $\varepsilon$-form epform1 ; epform2 ; epform-
criterium . Given that such a reduction is achievable epform-criterium and the
obtained differential system is rational, the MIs are expressible in terms of
multiple polylogarithms (MPLs) goncharov2011multiple ; remiddi2000harmonic .
At present, we have also good progress in understanding equations not
reducible to $\varepsilon$-form111See for example epform-elliptics for non-
algebraic transformation to $\varepsilon$-form in the elliptic case and ep-
regular-basis for a notion of regular basis for non-polylogarithmic integrals.
and functions beyond multiple polylogarithms such as for example elliptical
polylogarithms (EPLs) levin1994elliptic ; brown2013multiple ; Adams:2014vja ;
bloch2015elliptic ; adams2016kite ; remiddi2017elliptic ; broedel2018elliptic
; Broedel:2018iwv ; Broedel:2018qkq ; Broedel:2019tlz ; weinzierl2021modular ;
bezuglov2021massive , or entirely novel functions bloch2015feynman ;
Primo:2017ipr ; adams2018planar ; adams2018analytic ; Bourjaily:2017bsb ;
Bourjaily:2018ycu ; Bourjaily:2018yfy ; bezuglov2021massive ; Bonisch:2021yfw
. In our case, the diagrams we encounter are reducible to $\varepsilon$-form
with rational kernels and thus, can be written in terms of MPLs. However, when
we take external lines off-shell, the integrals are only reducible to
$A+B\varepsilon$ form, as we will see. One must thus resort to EPLs to solve
such MIs. This work provides results for the massive heavy-heavy, heavy-light
and propagator diagrams at two-loop order. The results are explicitly given up
to $\mathcal{O}(\varepsilon^{2})$ working in $D=4-2\varepsilon$ dimensions,
which is the appropriate order for SM-like theories chiu2008electroweak ;
chiu2008electroweak0 ; chiu2008electroweak2 ; assi3 . However, this order is
arbitrary, as the results are simply attainable for any order in
$\varepsilon$.

Outlining this paper, we begin in Section 2 by presenting our two-loop
integral families and their associated differential equations. We then reduce
each set of differential equations to either $\varepsilon$-form or
$A+B\varepsilon$ form, illustrating the sequence of balance transformations
required. In Section 3 we proceed to solve the differential equations and
explicitly present results for each integral expanded up to an appropriate
order in $\varepsilon$. In Appendix A we further calculate the corresponding
one-loop integrals for completeness.

##  2 Integral families and differential equations

Figure 1: Prototype topologies of one-loop vertex and self-energy diagrams:
$(a)$ is heavy-heavy vertex, $(b)$ is heavy-light and $c$ is selfnergy. Solid
lines represent massive particles, double lines represent heavy particles and
dashed lines correspond to massless propagators. Arrows represent direction of
momenta. Figure 2: Prototype topologies of two-loop vertex diagrams. Solid
lines represent massive particles, double lines represent heavy particles,
dashed lines correspond to massless propagators. Arrows represent direction of
momenta. $(a_{i})$ and $(b_{i})$ correspond to heavy-heavy and heavy-light
topologies. We also include the case of light self-energy insertions as is
apparent in $(a_{3})$.

The master integrals for one loop HPET vertices and self-energy shown in Fig.
1 are easy to calculate. Nevertheless, for completeness, we present their
calculation in Appendix A. The massive HPET vertices and self-energy at two-
loop level have MIs with topologies represented by Figs. 2 $(a,b)$ and 3,
respectively. We begin by considering the prototype topologies for the heavy-
heavy vertex in Figs. 2 $(a)$, the master integrals of which can be expressed
in terms of a single integral family

$J^{HH}_{\nu_{1},\ldots,\nu_{10}}=\int\frac{d^{d}l_{1}d^{d}l_{2}}{(i\pi^{d/2})^{2}}\prod_{i=1}^{10}\frac{1}{(D_{i}+i0)^{\nu_{i}}}\,,$
(1)

where

$\displaystyle\quad D_{1}=l_{2}\cdot v_{1},\quad D_{2}=l_{1}\cdot v_{1},\quad
D_{3}=(l_{1}-l_{2})^{2}-M^{2},\quad D_{4}=l_{2}^{2}-M^{2},\quad
D_{5}=l_{1}\cdot v_{2},$ $\displaystyle\quad D_{6}=l_{2}\cdot v_{2},\quad
D_{7}=(l_{2}-l_{1})\cdot v_{2},\quad D_{8}=l_{1}^{2}-M^{2},\quad
D_{9}=l_{2}^{2},\quad D_{10}=(l_{1}-l_{2})^{2}.$ (2)

Here, $v_{1,2}$ are the heavy particle velocities, and $M$ is the mass of
exchanged bosons. It is convenient to rescale integration momenta with respect
to $M$ and factor out the overall $M$ dependence of the above integrals. So,
in what follows, we will imply $M=1$. After that, the integrals depend only on
$w\equiv v_{1}\cdot v_{2}$ scalar product.

Similarly, in the case of heavy-light vertex, the required prototype
topologies in Figs. 2 $(b)$ can be assembled into the following single
integral family

$J^{HL}_{\nu_{1},\ldots,\nu_{11}}=\int\frac{d^{d}l_{1}d^{d}l_{2}}{(i\pi^{d/2})^{2}}\prod_{i=1}^{11}\frac{1}{(D_{i}+i0)^{\nu_{i}}}\,,$
(3)

where

$\displaystyle\quad D_{1}=(l_{2}+p_{2})^{2}-m^{2},\quad
D_{2}=(l_{1}+p_{2})^{2}-m^{2},\quad D_{3}=l_{1}\cdot v_{1},\quad
D_{4}=(l_{1}-l_{2})\cdot v_{1},$ $\displaystyle\quad D_{5}=l_{2}^{2},\quad
D_{6}=(l_{1}-l_{2})^{2},\quad D_{7}=l_{2}\cdot v_{1},\quad
D_{8}=l_{1}^{2},\quad D_{9}=(l_{2}+p_{2})^{2},$ $\displaystyle\quad
D_{10}=l_{2}^{2}-m^{2},\quad D_{11}=(l_{1}-l_{2})^{2}-m^{2}.$ (4)

Here $v_{1}$ is the heavy field velocity, $p_{2}$ and $m$ are the full theory
field momentum and mass. Again, upon rescaling of integration momenta the
overall dependence on $m$ can be factored out and the rescaled integrals
depend only on the scalar product $w\equiv v_{1}\cdot p_{2}/m$ and we will
again imply that in the above integral family definition $m=1$.

Figure 3: Heavy field self-energy topologies. The MIs associated to other
topologies are subsets of the MIs required for topologies illustrated.

Lastly, the self-energy diagrams which contribute to heavy field
renormalization and residual mass term are examined manohar2007heavy . The
prototype topologies are shown in Fig. 3, and the corresponding single
integral family is defined as

$J^{SE}_{\nu_{1},\ldots,\nu_{8}}=\int\frac{d^{d}l_{1}d^{d}l_{2}}{(i\pi^{d/2})^{2}}\prod_{i=1}^{8}\frac{1}{(D_{i}+i0)^{\nu_{i}}}\,,$
(5)

where

$\displaystyle\quad D_{1}=(p+l_{1})\cdot v,\quad D_{2}=l_{2}\cdot v,\quad
D_{3}=l_{1}^{2}-M^{2},\quad D_{4}=l_{2}^{2}-M^{2},$ $\displaystyle\quad
D_{5}=(l_{1}-l_{2})^{2}-M^{2},\quad D_{6}=(p+l_{1}+l_{2})\cdot v,\quad
D_{7}=l_{2}^{2},\quad D_{8}=(l_{1}-l_{2})^{2}.$ (6)

Here $p$ and $v$ are the heavy particle residual momentum and velocity and $M$
is the mass of exchanged field. The overall $M$-dependence can be again
factored out, so in what follows we will assume $M=1$. The left integrals
depend only on the scalar product $w\equiv v\cdot p/M$. Note, that the
relation between the heavy field self-energy, $\Sigma(p)$, the bare field
counter-term $\delta Z_{h}$ and the residual heavy field mass, $\delta m_{h}$,
are given by,

$\displaystyle\delta Z_{h}={}$ $\displaystyle i\partial_{v\cdot
p}\tilde{\Sigma}|_{v\cdot p=0}$ (7) $\displaystyle\delta m_{h}={}$
$\displaystyle-i\tilde{\Sigma}|_{v\cdot p=0}.$ (8)

To determine these quantities, one only requires the MIs on-shell at $v\cdot
p=0$, thus eliminating the momentum, $p$, from the propagators. The resulting
MIs are, therefore, simple enough to evaluate with standard techniques.
Maintaining $v\cdot p\neq 0$ is interesting in the case of off-shell studies.
However, as we will see, in this case, we encounter integrals with elliptic
structure.

With the use of IBP relations IBP1 ; IBP2 all integrals in the described
integral families can be reduced to the set of so called IBP master integrals.
To evaluate the latter it is convenient to use the method of differential
equations diffeqn1 ; diffeqn2 ; diffeqn3 ; diffeqn4 ; diffeqn5 . As all master
integrals we consider are dependent on a particular scalar product $w$, it is
natural to consider differential equations for our master integrals with
respect to $w$. To take the derivative with respect to an arbitrary scalar
product $p\cdot q$, for two arbitrary vectors $p$ and $q$, one can follow one
of two equivalent ways,

$\frac{\partial}{\partial(p\cdot q)}=\frac{(p\cdot q)p-p^{2}q}{(p\cdot
q)^{2}-p^{2}q^{2}}\cdot\frac{\partial}{\partial p}=\frac{(p\cdot
q)q-q^{2}p}{(p\cdot q)^{2}-p^{2}q^{2}}\cdot\frac{\partial}{\partial q}$ (9)

In our study, we take derivatives with respect to the parameter $w$, defined
in each integral family considered. Upon re-reducing the differentiated
results with IBP identities, we obtain a linear combination of MIs, leading to
a set of coupled differential equations. More precisely, the derivative of a
given MI will inevitably lie in the same sector or sub-sector, meaning they
contain the same set of non-zero $\nu_{i}$, or a smaller set, compared to the
original MI. Thus, one can combine all MIs and their derivatives into a linear
system of differential equations. As we will see below, these systems can be
further reduced either to $\varepsilon$ or $A+B\varepsilon$ forms in the cases
of vertex and self-energy integral families, respectively. From there, we
solve each system iteratively at each order in a Laurent expansion about small
$\varepsilon$.

###  2.1 Heavy-heavy vertex

Figure 4: Master integrals for heavy-heavy vertex (#: 1-8). Green lines denote
propagators for heavy particles, solid - propagators for massive particles and
dashed - massless propagators. Figure 5: Master integrals for heavy-heavy
vertex (#: 9-16). Green lines denote propagators for heavy particles, solid -
propagators for massive particles and dashed - massless propagators. Figure 6:
Master integrals for heavy-heavy vertex (#: 17-26). Green lines denote
propagators for heavy particles, solid - propagators for massive particles and
dashed - massless propagators.

In the heavy-heavy vertex case we have 26 master integrals shown in Figs. 4-6.
The latter could be conveniently represented as a column vector
$\boldsymbol{A}(w)$ with component $A_{i}(w)$ corresponding to
$J_{\nu_{1},\ldots,\nu_{10}}^{HH}(w)$ master $\\#i$. Upon differentiation of
$\boldsymbol{A}(w)$ with respect to $w$ and reduction by IBP identities, we
have a differential system,

$\partial_{w}\boldsymbol{A}(w)=\mathbb{M}(w,\varepsilon)\boldsymbol{A}(w).$
(10)

with a $26\times 26$ matrix, $\mathbb{M}(w,\varepsilon)$ which is neither
Fuchsian nor in $\varepsilon$-form. Using balance transformations and the
algorithm outlined in epform2 this system can be however transformed into
$\varepsilon$-form. First, we transform to $\varepsilon$-form diagonal blocks.
The largest diagonal block (for masters $\\#17,\\#19$ and $\\#20$) is given by
$3\times 3$ matrix

$M_{3}=\left(\begin{array}[]{ccc}-\frac{2w^{2}-\varepsilon}{2(w-1)w(w+1)}&\frac{3}{4(w-1)w(w+1)}&\frac{(2w-1)(2w+1)}{4(w-1)w(w+1)\varepsilon}\\\
\frac{\varepsilon^{2}}{(w-1)w(w+1)}&-\frac{2w^{2}-3\varepsilon}{2(w-1)w(w+1)}&-\frac{1}{2(w-1)w(w+1)}\\\
\frac{2\varepsilon^{2}\left(2w^{2}\varepsilon-2w^{2}-2\varepsilon-1\right)}{(w-1)w(w+1)(2w-1)(2w+1)}&\frac{3\varepsilon\left(2w^{2}\varepsilon+2w^{2}-2\varepsilon-1\right)}{(w-1)w(w+1)(2w-1)(2w+1)}&-\frac{4w^{4}-4w^{2}\varepsilon-3w^{2}-2\varepsilon-1}{(w-1)w(w+1)(2w-1)(2w+1)}\\\
\end{array}\right).$ (11)

Using Libra package Libra we can easily find the sequence of balance
transformations to transform eigenvalues of $M_{3}$ matrix residues at all
poles except at $w=\pm 1$ to $n\varepsilon$ form, where $n$ is some integer.
The required overall transformation matrix is given by

$T_{3}=\left(\begin{array}[]{ccc}\frac{1}{w+1}&0&0\\\ 0&\frac{1}{w-1}&0\\\
\frac{2\varepsilon^{2}}{w+1}&-\frac{3\varepsilon}{(w-1)(4w^{2}-1)}&\frac{4w}{4w^{2}-1}\\\
\end{array}\right),$ (12)

. The eigenvalues at $w=\pm 1$ have the form $\pm 1/2+n\varepsilon$ and to
normalize them further we require variable change. It is clear222See also
epform-criterium . that the appropriate variable employ is

$w=\frac{1-\beta^{2}}{1+\beta^{2}}\,,\quad\beta=\sqrt{\frac{1-w}{1+w}}$ (13)

Now, we again use Libra to find the required sequence of balance
transformations to normalize eigenvalues at $\beta=0,\infty$ to $n\varepsilon$
form and find $\beta$-independent transformation to factor out the overall
$\varepsilon$-dependence. The required transformation matrix was found to be

$T_{3}^{\prime}=\left(\begin{array}[]{ccc}\frac{1}{\beta}&0&0\\\
0&\frac{\beta\varepsilon}{2}&0\\\ 0&0&\frac{\varepsilon^{2}}{4}\\\
\end{array}\right),$ (14)

which together with previously found matrix $T_{3}$ reduces $M_{3}$ to
$\varepsilon$-form,

$S_{3}=\varepsilon\left(\begin{array}[]{ccc}\frac{2w}{w^{2}-1}&0&\frac{\beta}{4(w-1)}\\\
0&\frac{6w}{(w^{2}-1)(4w^{2}-1)}&\frac{\beta}{(w-1)(4w^{2}-1)}\\\
\frac{16\beta}{w-1}&\frac{12\beta}{(w-1)(4w^{2}-1)}&-\frac{8w}{4w^{2}-1}\\\
\end{array}\right).$ (15)

Next, we repeat these steps for other diagonal blocks. Finally, fuchsifying
off-diagonal blocks and factoring out overall $\varepsilon$-dependence333See
epform2 for detail description of these steps. the original system of
differential equations is reduced to canonical or $\varepsilon$-form:

$\partial_{w}\tilde{\boldsymbol{A}}(w)=\varepsilon\mathbb{S}(w)\tilde{\boldsymbol{A}}(w)\,,$
(16)

where $\boldsymbol{A}=\mathbb{T}\cdot\tilde{\boldsymbol{A}}$. In what follows
we will refer to the vector of master integrals $\tilde{\boldsymbol{A}}$ as
the vector of canonical master integrals. It turns out that in this case the
only new variable appearing in the process of reduction is $\beta$ and thus it
is more convenient to consider differential system with respect to $\beta$

$\partial_{\beta}\tilde{\boldsymbol{A}}(\beta)=\varepsilon\tilde{\mathbb{M}}(\beta)\tilde{\boldsymbol{A}}(\beta)\,,$
(17)

This way, as we will see in the next section, the corresponding solution of
the differential system can be written in terms of MPLs. The corresponding
expressions for the canonical $\tilde{\mathbb{M}}$ and transformation matrices
$\mathbb{T}$ can be found in the accompanying Mathematica notebook.

###  2.2 Heavy-light vertex

Figure 7: Master integrals for heavy-light vertex (#: 1-8). Green lines denote
propagators for heavy particles, solid - propagators for massive particles and
dashed - massless propagators. Figure 8: Master integrals for heavy-light
vertex (#: 9-18). Green lines denote propagators for heavy particles, solid -
propagators for massive particles and dashed - massless propagators.

Here, we have 18 master integrals shown in Figs. 7 and 8. Writing the latter
as a column vector $\boldsymbol{B}(w)$ the corresponding differential
equations system can be written as

$\partial_{w}\boldsymbol{B}(w)=\mathbb{M}(w,\varepsilon)\boldsymbol{B}(w).$
(18)

with a $18\times 18$ matrix, $\mathbb{M}(w,\varepsilon)$. The reduction to
$\varepsilon$-form in the present case proceeds similar to the case of heavy-
heavy vertex. That is, we first reduce to $\varepsilon$-form diagonal blocks.
Then, we fuchsify off-diagonal blocks and finally factor out the overall
$\varepsilon$-dependence. What is important is that in this case similar to
the case of heavy-heavy vertex the only new variable introduced is $\beta$
(13). For example, take $2\times 2$ diagonal block (for masters $\\#13$ and
$\\#14$)

$M_{2}=\left(\begin{array}[]{cc}-\frac{2w^{2}\varepsilon+w^{2}-4\varepsilon}{(w-1)w(w+1)}&-\frac{4}{w}\\\
-\frac{\varepsilon(4\varepsilon+1)}{(w-1)w(w+1)}&-\frac{w^{2}+4\varepsilon+1}{(w-1)w(w+1)}\\\
\end{array}\right).$ (19)

Using balance transformations we again transform eigenvalues of $M_{2}$ matrix
residues at all poles except at $w=\pm 1$ to $n\varepsilon$ form, where $n$ is
some integer. This can be done for example with transformation matrix

$T_{2}=\left(\begin{array}[]{cc}\frac{1}{w+1}&0\\\
-\frac{(w-2)\varepsilon}{2\left(w^{2}-1\right)}&\frac{w}{w^{2}-1}\\\
\end{array}\right).$ (20)

To reduce eigenvalues at $w=\pm 1$ we require the same variable change to
$\beta$-variable (13) as in the case of heavy-heavy vertex. Next, we again use
Libra to find a sequence of balance transformations to normalize eigenvalues
at $\beta=0,\infty$ to $n\varepsilon$ form and find $\beta$-independent
transformation to factor out the overall $\varepsilon$-dependence. The
required transformation matrix was found to be

$T_{2}^{\prime}=\left(\begin{array}[]{cc}\frac{1}{\beta}&0\\\
-\frac{\varepsilon}{2\beta}&\frac{\varepsilon}{2}\\\ \end{array}\right),$ (21)

which together with previously found matrix $T_{2}$ transforms $M_{2}$ to
$\varepsilon$-form,

$S_{2}=\varepsilon\left(\begin{array}[]{cc}\frac{2w}{w^{2}-1}&-\frac{2\beta}{w-1}\\\
-\frac{4\beta}{w-1}&-\frac{4w}{w^{2}-1}\\\ \end{array}\right).$ (22)

Altogether, in this case we can also write down the transformed differential
system with respect to $\beta$ in $\varepsilon$-form
($\boldsymbol{B}=\mathbb{T}\cdot\tilde{\boldsymbol{B}}$)

$\partial_{\beta}\tilde{\boldsymbol{B}}(\beta)=\varepsilon\tilde{\mathbb{M}}(\beta)\tilde{\boldsymbol{B}}(\beta)\,,$
(23)

with the corresponding expressions for canonical $\tilde{\mathbb{M}}$ and
transformation $\mathbb{T}$ matrices located in the accompanying Mathematica
notebook.

###  2.3 Heavy propagator

Figure 9: Master integrals for heavy propagator (#: 1-12). Green lines denote
propagators for heavy particles, solid - propagators for massive particles and
dashed - massless propagators.

In the case of the heavy propagator, we have 12 master integrals as depicted
in Fig. 9. The reduction of the corresponding differential system for a vector
of master integrals $\boldsymbol{C}$ is achieved as in the cases of heavy-
heavy and heavy-light vertices. In particular, we first reduce diagonal blocks
where possible to $\varepsilon$-form and then fuchsify off-diagonal blocks.
Some of the diagonal blocks will also require a variable change to the $\beta$
variable, as defined in (13). However, contrary to the vertex cases considered
previously, we have elliptic sub-graphs represented by elliptic sunsets
(master integrals $\\#4$ and $\\#5$). This particular diagonal block can not
be reduced to $\varepsilon$-form444See epform-criterium for the criterion of
such reducibility.. Still, one can reduce the latter to $A+B\varepsilon$ form.
Also, instead of factoring out the overall $\varepsilon$-dependence in the
final step, we find a ($\beta(w)$-independent) diagonal transformation matrix,
which reduces the entire system to $A+B\varepsilon$ form. Altogether, in this
case we write down the transformed differential system with respect to $w$ in
$A+B\varepsilon$ form
($\boldsymbol{C}=\mathbb{T}\cdot\tilde{\boldsymbol{C}}$):

$\partial_{w}\tilde{\boldsymbol{C}}(w)=\tilde{\mathbb{M}}(w)\tilde{\boldsymbol{C}}(w)\,,$
(24)

with corresponding expressions for canonical $\tilde{\mathbb{M}}$ and
transformation $\mathbb{T}$ matrices located in accompanying Mathematica
notebook. Here, the matrix $\tilde{\mathbb{M}}$ does contain square roots in
$w$ induced by the variable $\beta$. As we already noted, for on-shell studies
we need to know master integrals only for $w=0$. Still, as will be shown in
next section, we will also able to supply all the ingredients necessary to
have Frobenius solution in terms of a generalized power series in $w$ for
$w\neq 0$.

Figure 10: Boundary master integrals for heavy-heavy vertex (#: 1-12). Green
lines denote propagators for heavy particles, solid - propagators for massive
particles and dashed - massless propagators.

##  3 Solution of differential equations and results

Figure 11: Boundary master integrals for heavy-light vertex (#: 1-5). Green
lines denote propagators for heavy particles, solid - propagators for massive
particles and dashed - massless propagators. Figure 12: Boundary master
integrals for heavy propagator (#: 1-10). Green lines denote propagators for
heavy particles, solid - propagators for massive particles and dashed -
massless propagators.

Given our reduced systems of differential equations to either $\varepsilon$ or
$A+B\varepsilon$ form in the previous section, we are now ready to solve them.
In the case of heavy-heavy and heavy-light vertices, where we managed to
reduce the corresponding differential systems to $\varepsilon$-form, the
solution can be readily written in terms of $\mathcal{P}$-exponents as

$\boldsymbol{J}(\beta)=\mathbb{T}(\beta)\tilde{\boldsymbol{J}}(\beta)=\mathbb{T}(\beta)\mathcal{P}\exp{\left[\varepsilon\int_{0}^{\beta}\tilde{\mathbb{M}}(t)dt\right]\mathbb{L}\cdot\boldsymbol{c}}\,,$
(25)

where $\boldsymbol{J}$ is either vector of $\boldsymbol{A}$ or
$\boldsymbol{B}$ masters, $\mathbb{T}$ is the transformation matrix to
canonical basis and $\tilde{\mathbb{M}}$ is corresponding differential
equations matrix in canonical basis. $\mathbb{L}$ is the adapter matrix
converting the vector of constants $\boldsymbol{c}$ into boundary conditions
for canonical master integrals. The expressions for said boundary conditions
can be found in the accompanying Mathematica notebook. The components of the
vector of constants $\boldsymbol{c}$ have the form $c_{i}(\beta^{j})$, where
$c_{i}(\beta^{j})$ is the coefficient of $\beta^{j}$ in the $\beta$-expansion
of the $i$-th master. In the case of the heavy-heavy and heavy-light vertices,
$\boldsymbol{c}$-vectors are given by

$\boldsymbol{c}_{HH}=\Big{(}c_{1}(\beta^{0}),c_{2}(\beta^{0}),c_{3}(\beta^{0}),c_{6}(\beta^{-1+2\varepsilon}),c_{5}(\beta^{0}),c_{5}(\beta^{-1+2\varepsilon}),c_{8}(\beta^{-1}),c_{9}(\beta^{-1}),c_{10}(\beta^{-1}),\\\
c_{11}(\beta^{-1}),c_{12}(\beta^{-2}),c_{13}(\beta^{0}),c_{14}(\beta^{0}),c_{15}(\beta^{0}),c_{16}(\beta^{0}),c_{18}(\beta^{-1+2\varepsilon}),c_{17}(\beta^{0}),c_{17}(\beta^{-1+2\varepsilon}),\\\
c_{17}(\beta^{1+2\varepsilon}),c_{21}(\beta^{-1}),c_{22}(\beta^{-1}),c_{23}(\beta^{-1}),c_{24}(\beta^{0}),c_{4}(\beta^{0}),c_{25}(\beta^{0}),c_{26}(\beta^{-1})\Big{)}^{\top}$
(26)

and

$\boldsymbol{c}_{HL}=\Big{(}c_{1}(\beta^{0}),c_{2}(\beta^{0}),c_{3}(\beta^{0}),c_{3}(\beta^{5-6\varepsilon}),c_{5}(\beta^{0}),c_{6}(\beta^{1-2\varepsilon}),c_{7}(\beta^{-1+2\varepsilon}),c_{8}(\beta^{1-4\varepsilon}),\\\
c_{9}(\beta^{-1+2\varepsilon}),c_{10}(\beta^{-4\varepsilon}),c_{11}(\beta^{2-4\varepsilon}),c_{12}(\beta^{-4\varepsilon}),c_{13}(\beta^{-4\varepsilon}),c_{13}(\beta^{-1+2\varepsilon}),\\\
c_{15}(\beta^{2-4\varepsilon}),c_{16}(\beta^{0}),c_{17}(\beta^{-2\varepsilon}),c_{17}(\beta^{-1+2\varepsilon})\Big{)}^{\top}\,.$
(27)

It is easy to see, that all $c_{i}(\beta^{j})$ constants other than
$c_{i}(\beta^{0})$ are zero either due to the regularity of master integrals
at $\beta=0$ or due to the absence of sub-graphs in the large mass expansion
(corresponding to the expansion at $\beta=0$) of these vertices. The remaining
constants will be calculated in the next sub-section with the use of
dimensional recursion relations.

In the heavy propagator case, due to the presence of elliptic sub-graphs in
its corresponding differential system, a $\mathcal{P}$-exponent solution is
not attainable. However, we can still obtain a Frobenius solution in terms of
a generalised power series in the $w$-variable. The solution can be written as

$\boldsymbol{C}(w)=\mathbb{T}(w)\tilde{\boldsymbol{C}}(w)=\mathbb{T}(w)\mathcal{F}(w)\cdot\mathbb{L}\cdot\boldsymbol{c},$
(28)

where $\mathcal{F}(w)$ is the Frobenius solution for canonical master
integrals. $\mathbb{L}$ is again the adapter matrix converting the vector of
constants $\boldsymbol{c}$ into boundary conditions for canonical master
integrals. the vector of constants $\boldsymbol{c}$ in this case is given by

$\boldsymbol{c}_{SE}=\Big{(}c_{1}(\beta^{0}),c_{2}(\beta^{0}),c_{3}(\beta^{0}),c_{4}(\beta^{0}),c_{4}(\beta^{2-2\varepsilon}),c_{6}(\beta^{0}),c_{7}(\beta^{0}),c_{8}(\beta^{0}),c_{9}(\beta^{0}),\\\
c_{10}(\beta^{0}),c_{11}(\beta^{3-4\varepsilon}),c_{12}(\beta^{0})\Big{)}^{\top}\,.$
(29)

Using Feynman parameters it is easy to see that
$c_{4}(\beta^{2-2\varepsilon})=0$ and

$c_{11}(\beta^{3-4\varepsilon})=(-1)^{4\varepsilon}2^{4-4\varepsilon}\Gamma(1-\varepsilon)^{2}\Gamma(4\varepsilon-3)\,.$
(30)

Like the case of vertices, the remaining constants are more convenient to
determine using dimensional recurrence relations. It is also possible to
determine Frobenius solutions for canonical master integrals $\mathcal{F}(w)$,
see for example frobenius1 ; frobenius2 ; frobenius3 ; frobenius4 ; frobenius5
; frobenius6 ; frobenius7 . Moreover, the Libra package Libra itself contains
built-in tools for obtaining such a solution. The reader is advised to consult
the accompanying Mathematica notebook.

###  3.1 Boundary constants from dimensional recurrences

We need to calculate the remaining $c_{i}(\beta^{0})$ constants in each case.
First, we note that for $\beta=0$ in the case of vertices, or $w=0$ in the
case of heavy propagators, the master integrals, as illustrated in Figs. 4-9,
are no longer masters and additional reduction using partial fractioning and
IBP identities is possible. In the case of the heavy-heavy vertex, additional
boundary master integrals can be written in terms of a single integral family

$K^{HH}_{\nu_{1},\ldots,\nu_{8}}=\int\frac{d^{d}l_{1}d^{d}l_{2}}{(i\pi^{d/2})^{2}}\prod_{i=1}^{8}\frac{1}{(D_{i}+i0)^{\nu_{i}}}\,,$
(31)

where

$\displaystyle\quad D_{1}=(l_{1}-l_{2})^{2}-1,\quad D_{2}=l_{2}^{2}-1,\quad
D_{3}=l_{1}\cdot v,\quad D_{4}=l_{2}\cdot v,$ $\displaystyle\quad
D_{5}=(l_{2}-l_{1})\cdot v,\quad D_{6}=l_{1}^{2}-1,\quad D_{7}=l_{2}^{2},\quad
D_{8}=(l_{1}-l_{2})^{2}$ (32)

and $v$ is the heavy field velocity. Altogether in this case we have 12 master
integrals shown in Fig. 10. Similarly, in the case of heavy-light vertex the
single integral family is given by

$K^{HL}_{\nu_{1},\ldots,\nu_{8}}=\int\frac{d^{d}l_{1}d^{d}l_{2}}{(i\pi^{d/2})^{2}}\prod_{i=1}^{8}\frac{1}{(D_{i}+i0)^{\nu_{i}}}\,,$
(33)

where

$\displaystyle\quad D_{1}=(l_{1}+v)^{2}-1,\quad D_{2}=(l_{1}-l_{2})\cdot
v,\quad D_{3}=l_{2}^{2},\quad D_{4}=(l_{1}-l_{2})^{2},$ $\displaystyle\quad
D_{5}=(l_{2}+v)^{2}-1,\quad D_{6}=l_{1}^{2},\quad D_{7}=l_{2}^{2}-1,\quad
D_{8}=(l_{1}-l_{2})^{2}-1$ (34)

and the corresponding 5 boundary master integrals are shown in Fig. 11.
Finally, for the heavy propagator case the single integral family takes the
form

$K^{SE}_{\nu_{1},\ldots,\nu_{7}}=\int\frac{d^{d}l_{1}d^{d}l_{2}}{(i\pi^{d/2})^{2}}\prod_{i=1}^{7}\frac{1}{(D_{i}+i0)^{\nu_{i}}}\,,$
(35)

where

$\displaystyle\quad D_{1}=v\cdot l_{1},\quad D_{2}=v\cdot l_{2},\quad
D_{3}=l_{1}^{2}-1,\quad D_{4}=l_{2}^{2}-1,$ $\displaystyle\quad
D_{5}=(l_{1}-l_{2})^{2}-1,\quad D_{6}=l_{2}^{2},\quad D_{7}=(l_{1}-l_{2})^{2}$
(36)

with 10 boundary master integrals shown in Fig. 12.

The simplest way to calculate the exact expressions in $\varepsilon$ for these
boundary master integrals is to use dimensional recurrence relations
tarasov1996connection ; dimrecSunrise ; lee2010space ; lee2010calculating . In
all cases we present here, the recurrence relations are all first-order.
Consider, for instance, the evaluation of the $K^{SE}_{0011100}(d)$ master
integral. In this case, we have the following dimensional recurrence relation
($\nu=d/2$)

$K^{SE}_{0011100}(\nu+1)=\frac{3}{2\nu(2\nu-1)}K^{SE}_{0011100}(\nu)+\frac{3}{2\nu(2\nu-1)}K^{SE}_{0001100}(\nu)\,,$
(37)

where $K^{SE}_{0001100}(\nu)$ is the product of two one-loop tadpoles

$K^{SE}_{0001100}(\nu)=\Gamma(1-\nu)^{2}\,.$ (38)

It is sufficient to consider the solution of this recurrence relation in a
basic strip $\nu\in[1,2)$. First, we determine the solution of the homogeneous
difference equation,

$J_{h}(\nu)=\frac{2\cdot 3^{\nu-1}\pi\csc(2\pi\nu)}{\Gamma(2\nu-1)}\,,$ (39)

where an extra periodic factor $\csc(2\pi\nu)$ was chosen to be well-behaved
at imaginary infinities and singularities close to those of the original
master integral $K^{SE}_{0011100}(\nu)$ (see lee2010space ; lee2010calculating
for further details). The easiest way to see them is through numerical
evaluation in a chosen strip with sector decomposition Binoth:2000ps ;
Binoth:2003ak ; Binoth:2004jv ; Heinrich:2008si ; Bogner:2007cr ;
Bogner:2008ry ; Kaneko:2009qx . In particular, we used the function SDAnalyse
from Fiesta Fiesta4 . Knowing the homogeneous solution, the particular
solution can be found with substitution into original recurrence relation the
ansatz $K^{SE}_{0011100}(\nu)=r(\nu)J_{h}(\nu)$ and obtaining the recurrence
relation for $r(\nu)$

$r(\nu+1)=r(\nu)+J_{h}^{-1}(\nu)\Gamma(1-\nu)^{2}\,.$ (40)

This first order recurrence relation is easy to solve and we have

$r(\nu)=-\sum_{i=0}^{\infty}\frac{3^{2+i-\nu}}{2}\frac{\Gamma(2+i-\nu)^{2}}{\Gamma(4+2i-2\nu)}=-\frac{3^{2-\nu}\Gamma(2-\nu)^{2}}{2\Gamma(4-2\nu)}~{}_{2}F_{1}\left(1,2-\nu;\frac{5}{2}-\nu;\frac{3}{4}\right)\,.$
(41)

The general solution is then given by

$K^{SE}_{0011100}(\nu)=J_{h}(\nu)r(\nu)+w(\nu)J_{h}(\nu)\,,$ (42)

where the periodic constant $w(\nu)$ is fixed from the leading term of the
expansion at $\nu=3/2$. Altogether, we have the exact solution,

$K^{SE}_{0011100}(\nu)=\frac{4\cdot
3^{\nu-3/2}\pi^{2}\csc(2\pi\nu)}{\Gamma(2\nu-1)}\\\
-\frac{3\pi\csc(2\pi\nu)\Gamma^{2}(2-\nu)}{\Gamma(4-2\nu)\Gamma(2\nu-1)}~{}_{2}F_{1}\left(1,2-\nu;\frac{5}{2}-\nu;\frac{3}{4}\right)\,.$
(43)

Employing the same technique, one can then calculate all other boundary master
integrals. The corresponding results can be found in the accompanying
Mathematica notebook. Expanding the exact results at $d=4-2\varepsilon$ and
accounting for obvious $\exp(2\gamma_{E}\varepsilon)$ factors, we get

#### Heavy-heavy:

$\displaystyle a_{1}={}$
$\displaystyle\frac{1}{\varepsilon^{2}}+\frac{2}{\varepsilon}+\left(3+\frac{\pi^{2}}{6}\right)+\left(-\frac{2\zeta_{3}}{3}+4+\frac{\pi^{2}}{3}\right)\varepsilon+\left(-\frac{4\zeta_{3}}{3}+5+\frac{\pi^{2}}{2}+\frac{7\pi^{4}}{360}\right)\varepsilon^{2}$
$\displaystyle+\left(-2\zeta_{3}-\frac{\pi^{2}\zeta_{3}}{9}-\frac{2\zeta_{5}}{5}+6+\frac{2\pi^{2}}{3}+\frac{7\pi^{4}}{180}\right)\varepsilon^{3}+\left(-\frac{8\zeta_{3}}{3}-\frac{2\pi^{2}\zeta_{3}}{9}+\frac{2\zeta_{3}^{2}}{9}-\frac{4\zeta_{5}}{5}\right.$
$\displaystyle\left.+7+\frac{5\pi^{2}}{6}+\frac{7\pi^{4}}{120}+\frac{31\pi^{6}}{15120}\right)\varepsilon^{4}+\mathcal{O}(\varepsilon^{5}),$
(44) $\displaystyle a_{2}={}$
$\displaystyle-\frac{2\pi}{\varepsilon}+\pi(4l_{2}-6)+\pi\varepsilon\left(-14-\frac{2\pi^{2}}{3}-4l_{2}^{2}+12l_{2}\right)+\pi\varepsilon^{2}\left(\frac{16\zeta_{3}}{3}-30-2\pi^{2}\right.$
$\displaystyle\left.+\frac{8l_{2}^{3}}{3}-12l_{2}^{2}+28l_{2}+\frac{4}{3}\pi^{2}l_{2}\right)+\pi\varepsilon^{3}\left(16\zeta_{3}-\frac{32}{3}\zeta_{3}l_{2}-62-\frac{14\pi^{2}}{3}-\frac{\pi^{4}}{5}-\frac{4l_{2}^{4}}{3}\right.$
$\displaystyle\left.+8l_{2}^{3}-28l_{2}^{2}-\frac{4}{3}\pi^{2}l_{2}^{2}+60l_{2}+4\pi^{2}l_{2}\right)+\pi\varepsilon^{4}\left(\frac{112\zeta_{3}}{3}+\frac{16\pi^{2}\zeta_{3}}{9}+\frac{64\zeta_{5}}{5}+\frac{32}{3}\zeta_{3}l_{2}^{2}\right.$
$\displaystyle\left.-32\zeta_{3}l_{2}-126-10\pi^{2}-\frac{3\pi^{4}}{5}+\frac{8l_{2}^{5}}{15}-4l_{2}^{4}+\frac{56l_{2}^{3}}{3}+\frac{8}{9}\pi^{2}l_{2}^{3}-60l_{2}^{2}-4\pi^{2}l_{2}^{2}+124l_{2}\right.$
$\displaystyle\left.+\frac{28}{3}\pi^{2}l_{2}+\frac{2}{5}\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}),$
(45) $\displaystyle a_{3}={}$
$\displaystyle\frac{32\pi^{2}\varepsilon}{3}+\varepsilon^{2}\left(\frac{704\pi^{2}}{9}-\frac{256}{3}\pi^{2}l_{2}\right)+\varepsilon^{3}\left(\frac{10880\pi^{2}}{27}+\frac{112\pi^{4}}{9}+\frac{1024}{3}\pi^{2}l_{2}^{2}-\frac{5632}{9}\pi^{2}l_{2}\right)$
$\displaystyle+\varepsilon^{4}\left(-\frac{1984\pi^{2}\zeta_{3}}{9}+\frac{147200\pi^{2}}{81}+\frac{2464\pi^{4}}{27}-\frac{8192}{9}\pi^{2}l_{2}^{3}+\frac{22528}{9}\pi^{2}l_{2}^{2}-\frac{87040}{27}\pi^{2}l_{2}\right.$
$\displaystyle\left.-\frac{896}{9}\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}),$
(46) $\displaystyle a_{4}={}$ $\displaystyle
2\pi^{2}+\varepsilon\left(8\pi^{2}-8\pi^{2}l_{2}\right)+\varepsilon^{2}\left(24\pi^{2}+\pi^{4}+16\pi^{2}l_{2}^{2}-32\pi^{2}l_{2}\right)+\varepsilon^{3}\left(-\frac{28\pi^{2}\zeta_{3}}{3}+64\pi^{2}\right.$
$\displaystyle\left.+4\pi^{4}-\frac{64}{3}\pi^{2}l_{2}^{3}+64\pi^{2}l_{2}^{2}-96\pi^{2}l_{2}-4\pi^{4}l_{2}\right)+\varepsilon^{4}\left(-\frac{112\pi^{2}\zeta_{3}}{3}+\frac{112}{3}\pi^{2}\zeta_{3}l_{2}+160\pi^{2}\right.$
$\displaystyle\left.+12\pi^{4}+\frac{5\pi^{6}}{12}+\frac{64}{3}\pi^{2}l_{2}^{4}-\frac{256}{3}\pi^{2}l_{2}^{3}+192\pi^{2}l_{2}^{2}+8\pi^{4}l_{2}^{2}-256\pi^{2}l_{2}-16\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}),$
(47) $\displaystyle a_{5}={}$
$\displaystyle\frac{3}{2\varepsilon^{2}}+\frac{9}{2\varepsilon}+\frac{1}{4}\left(8\sqrt{3}\Im\left(\text{Li}_{2}\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)\right)+42+\pi^{2}\right)+\sum_{n=1}^{4}a_{1,n}\varepsilon^{n}+\mathcal{O}(\varepsilon^{5}),$
(48) $\displaystyle a_{6}={}$
$\displaystyle-\frac{2\pi}{\varepsilon}+\left(-8\pi-\frac{4\pi^{2}}{\sqrt{3}}+4\pi
l_{2}\right)+\sum_{n=1}^{4}a_{2,n}\varepsilon^{n}+\mathcal{O}(\varepsilon^{5}),$
(49) $\displaystyle a_{7}={}$ $\displaystyle
6\pi^{2}+\varepsilon\left(24\pi^{2}-24\pi^{2}l_{2}\right)+\varepsilon^{2}\left(72\pi^{2}+3\pi^{4}+48\pi^{2}l_{2}^{2}-96\pi^{2}l_{2}\right)+\varepsilon^{3}\left(-28\pi^{2}\zeta_{3}+192\pi^{2}\right.$
$\displaystyle\left.+12\pi^{4}-64\pi^{2}l_{2}^{3}+192\pi^{2}l_{2}^{2}-288\pi^{2}l_{2}-12\pi^{4}l_{2}\right)+\varepsilon^{4}\left(-112\pi^{2}\zeta_{3}+112\pi^{2}\zeta_{3}l_{2}+480\pi^{2}\right.$
$\displaystyle\left.+36\pi^{4}+\frac{5\pi^{6}}{4}+64\pi^{2}l_{2}^{4}-256\pi^{2}l_{2}^{3}+576\pi^{2}l_{2}^{2}+24\pi^{4}l_{2}^{2}-768\pi^{2}l_{2}-48\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}),$
(50) $\displaystyle a_{8}={}$ $\displaystyle
4\pi^{2}+\varepsilon\left(16\pi^{2}-16\pi^{2}l_{2}\right)+\varepsilon^{2}\left(48\pi^{2}+2\pi^{4}+32\pi^{2}l_{2}^{2}-64\pi^{2}l_{2}\right)+\varepsilon^{3}\left(-\frac{56\pi^{2}\zeta_{3}}{3}+128\pi^{2}\right.$
$\displaystyle\left.+8\pi^{4}-\frac{128}{3}\pi^{2}l_{2}^{3}+128\pi^{2}l_{2}^{2}-192\pi^{2}l_{2}-8\pi^{4}l_{2}\right)+\varepsilon^{4}\left(-\frac{224\pi^{2}\zeta_{3}}{3}+\frac{224}{3}\pi^{2}\zeta_{3}l_{2}+320\pi^{2}\right.$
$\displaystyle\left.+24\pi^{4}+\frac{5\pi^{6}}{6}+\frac{128}{3}\pi^{2}l_{2}^{4}-\frac{512}{3}\pi^{2}l_{2}^{3}+384\pi^{2}l_{2}^{2}+16\pi^{4}l_{2}^{2}-512\pi^{2}l_{2}-32\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}),$
(51) $\displaystyle a_{9}={}$
$\displaystyle-\frac{4\pi^{2}}{3\varepsilon}+\left(\frac{16}{3}\pi^{2}l_{3}-\frac{8\pi^{2}}{3}\right)+\sum_{n=1}^{4}a_{3,n}\varepsilon^{n}+\mathcal{O}(\varepsilon^{5}),$
(52) $\displaystyle a_{10}={}$ $\displaystyle\frac{1}{2}a_{9},$ (53)
$\displaystyle a_{11}={}$
$\displaystyle\frac{1}{2\varepsilon^{2}}+\frac{3}{2\varepsilon}+\left(\frac{7}{2}+\frac{\pi^{2}}{4}\right)+\left(-\frac{4\zeta_{3}}{3}+\frac{15}{2}+\frac{3\pi^{2}}{4}\right)\varepsilon+\left(-4\zeta_{3}+\frac{31}{2}+\frac{7\pi^{2}}{4}+\frac{7\pi^{4}}{80}\right)\varepsilon^{2}$
$\displaystyle+\left(-\frac{28\zeta_{3}}{3}-\frac{2\pi^{2}\zeta_{3}}{3}-\frac{16\zeta_{5}}{5}+\frac{63}{2}+\frac{15\pi^{2}}{4}+\frac{21\pi^{4}}{80}\right)\varepsilon^{3}+\left(-20\zeta_{3}-2\pi^{2}\zeta_{3}+\frac{16\zeta_{3}^{2}}{9}\right.$
$\displaystyle\left.-\frac{48\zeta_{5}}{5}+\frac{127}{2}+\frac{31\pi^{2}}{4}+\frac{49\pi^{4}}{80}+\frac{869\pi^{6}}{30240}\right)\varepsilon^{4}+\mathcal{O}(\varepsilon^{5}),$
(54) $\displaystyle a_{12}={}$
$\displaystyle-\frac{2\pi}{\varepsilon}+\pi(4l_{2}-8)+\pi\varepsilon\left(-24-\frac{10\pi^{2}}{3}-4l_{2}^{2}+16l_{2}\right)+\pi\varepsilon^{2}\left(\frac{28\zeta_{3}}{3}-64-\frac{40\pi^{2}}{3}+\frac{8l_{2}^{3}}{3}\right.$
$\displaystyle\left.-16l_{2}^{2}+48l_{2}+\frac{20}{3}\pi^{2}l_{2}\right)+\pi\varepsilon^{3}\left(\frac{112\zeta_{3}}{3}-\frac{56}{3}\zeta_{3}l_{2}-160-40\pi^{2}-\frac{238\pi^{4}}{45}-\frac{4l_{2}^{4}}{3}+\frac{32l_{2}^{3}}{3}\right.$
$\displaystyle\left.-48l_{2}^{2}-\frac{20}{3}\pi^{2}l_{2}^{2}+128l_{2}+\frac{80}{3}\pi^{2}l_{2}\right)+\pi\varepsilon^{4}\left(112\zeta_{3}+\frac{140\pi^{2}\zeta_{3}}{9}+\frac{124\zeta_{5}}{5}+\frac{56}{3}\zeta_{3}l_{2}^{2}\right.$
$\displaystyle\left.-\frac{224}{3}\zeta_{3}l_{2}-384-\frac{320\pi^{2}}{3}-\frac{952\pi^{4}}{45}+\frac{8l_{2}^{5}}{15}-\frac{16l_{2}^{4}}{3}+32l_{2}^{3}+\frac{40}{9}\pi^{2}l_{2}^{3}-128l_{2}^{2}\right.$
$\displaystyle\left.-\frac{80}{3}\pi^{2}l_{2}^{2}+320l_{2}+80\pi^{2}l_{2}+\frac{476}{45}\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}),$
(55)

#### Heavy-light:

$\displaystyle b_{1}={}$
$\displaystyle\frac{1}{2\varepsilon^{2}}+\frac{5}{4\varepsilon}+\left(\frac{11}{8}+\frac{5\pi^{2}}{12}\right)+\left(\frac{11\zeta_{3}}{3}-\frac{55}{16}+\frac{25\pi^{2}}{24}\right)\varepsilon+\left(\frac{55\zeta_{3}}{6}-\frac{949}{32}+\frac{55\pi^{2}}{48}+\frac{101\pi^{4}}{240}\right)\varepsilon^{2}$
$\displaystyle+\left(\frac{121\zeta_{3}}{12}+\frac{55\pi^{2}\zeta_{3}}{18}+\frac{359\zeta_{5}}{5}-\frac{8575}{64}-\frac{275\pi^{2}}{96}+\frac{101\pi^{4}}{96}\right)\varepsilon^{3}+\left(-\frac{605\zeta_{3}}{24}+\frac{275\pi^{2}\zeta_{3}}{36}\right.$
$\displaystyle\left.+\frac{121\zeta_{3}^{2}}{9}+\frac{359\zeta_{5}}{2}-\frac{64189}{128}-\frac{4745\pi^{2}}{192}+\frac{1111\pi^{4}}{960}+\frac{3335\pi^{6}}{6048}\right)\varepsilon^{4}+\mathcal{O}(\varepsilon^{5}),$
(56) $\displaystyle b_{2}={}$
$\displaystyle-\frac{1}{\varepsilon^{2}}-\frac{4}{3\varepsilon}+\frac{\pi^{2}}{2}+\frac{56}{9}+\left(-\frac{118\zeta_{3}}{3}+\frac{1520}{27}+\frac{2\pi^{2}}{3}\right)\varepsilon+\left(-\frac{472\zeta_{3}}{9}+\frac{24224}{81}-\frac{28\pi^{2}}{9}\right.$
$\displaystyle\left.+\frac{299\pi^{4}}{120}\right)\varepsilon^{2}+\left(\frac{6608\zeta_{3}}{27}+\frac{59\pi^{2}\zeta_{3}}{3}-\frac{6478\zeta_{5}}{5}+\frac{330176}{243}-\frac{760\pi^{2}}{27}+\frac{299\pi^{4}}{90}\right)\varepsilon^{3}$
$\displaystyle+\left(\frac{179360\zeta_{3}}{81}+\frac{236\pi^{2}\zeta_{3}}{9}-\frac{6962\zeta_{3}^{2}}{9}-\frac{25912\zeta_{5}}{15}+\frac{4213376}{729}-\frac{12112\pi^{2}}{81}-\frac{2093\pi^{4}}{135}\right.$
$\displaystyle\left.+\frac{91909\pi^{6}}{15120}\right)\varepsilon^{4}+\mathcal{O}(\varepsilon^{5}),$
(57) $\displaystyle b_{3}={}$
$\displaystyle\frac{1}{\varepsilon^{2}}+\frac{2}{\varepsilon}+\left(3+\frac{\pi^{2}}{6}\right)+\left(-\frac{2\zeta_{3}}{3}+4+\frac{\pi^{2}}{3}\right)\varepsilon+\left(-\frac{4\zeta_{3}}{3}+5+\frac{\pi^{2}}{2}+\frac{7\pi^{4}}{360}\right)\varepsilon^{2}+\left(-2\zeta_{3}\right.$
$\displaystyle\left.-\frac{\pi^{2}\zeta_{3}}{9}-\frac{2\zeta_{5}}{5}+6+\frac{2\pi^{2}}{3}+\frac{7\pi^{4}}{180}\right)\varepsilon^{3}+\left(-\frac{8\zeta_{3}}{3}-\frac{2\pi^{2}\zeta_{3}}{9}+\frac{2\zeta_{3}^{2}}{9}-\frac{4\zeta_{5}}{5}+7+\frac{5\pi^{2}}{6}\right.$
$\displaystyle\left.+\frac{7\pi^{4}}{120}+\frac{31\pi^{6}}{15120}\right)\varepsilon^{4}+\mathcal{O}(\varepsilon^{5}),$
(58) $\displaystyle b_{4}={}$
$\displaystyle\frac{32\pi^{2}\varepsilon}{3}+\varepsilon^{2}\left(\frac{704\pi^{2}}{9}-\frac{256}{3}\pi^{2}l_{2}\right)+\varepsilon^{3}\left(\frac{10880\pi^{2}}{27}+\frac{112\pi^{4}}{9}+\frac{1024}{3}\pi^{2}l_{2}^{2}-\frac{5632}{9}\pi^{2}l_{2}\right)$
$\displaystyle+\varepsilon^{4}\left(-\frac{1984\pi^{2}\zeta_{3}}{9}+\frac{147200\pi^{2}}{81}+\frac{2464\pi^{4}}{27}-\frac{8192}{9}\pi^{2}l_{2}^{3}+\frac{22528}{9}\pi^{2}l_{2}^{2}-\frac{87040}{27}\pi^{2}l_{2}\right.$
$\displaystyle\left.-\frac{896}{9}\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}),$
(59) $\displaystyle b_{5}={}$
$\displaystyle\frac{3}{2\varepsilon^{2}}+\frac{17}{4\varepsilon}+\left(\frac{59}{8}+\frac{\pi^{2}}{4}\right)+\left(-\zeta_{3}+\frac{65}{16}+\frac{49\pi^{2}}{24}\right)\varepsilon+\varepsilon^{2}\left(\frac{151\zeta_{3}}{6}-\frac{1117}{32}+\frac{475\pi^{2}}{48}+\frac{7\pi^{4}}{240}\right.$
$\displaystyle\left.-8\pi^{2}l_{2}\right)+\varepsilon^{3}\left(192\text{Li}_{4}\left(\frac{1}{2}\right)+\frac{2125\zeta_{3}}{12}-\frac{\pi^{2}\zeta_{3}}{6}-\frac{3\zeta_{5}}{5}-\frac{13783}{64}+\frac{3745\pi^{2}}{96}-\frac{103\pi^{4}}{96}\right.$
$\displaystyle\left.+8l_{2}^{4}+16\pi^{2}l_{2}^{2}-52\pi^{2}l_{2}\right)+\varepsilon^{4}\left(1248\text{Li}_{4}\left(\frac{1}{2}\right)+1152\text{Li}_{5}\left(\frac{1}{2}\right)+\frac{19255\zeta_{3}}{24}-\frac{361\pi^{2}\zeta_{3}}{36}\right.$
$\displaystyle\left.+\frac{\zeta_{3}^{2}}{3}-\frac{9317\zeta_{5}}{10}-\frac{114181}{128}+\frac{26563\pi^{2}}{192}-\frac{7073\pi^{4}}{960}+\frac{31\pi^{6}}{10080}-\frac{48l_{2}^{5}}{5}+52l_{2}^{4}-32\pi^{2}l_{2}^{3}\right.$
$\displaystyle\left.+104\pi^{2}l_{2}^{2}-230\pi^{2}l_{2}+\frac{104}{15}\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}).$
(60)

#### Heavy propagator:

$\displaystyle c_{2}={}$ $\displaystyle a_{5},\quad c_{5}=a_{6},\quad
c_{8}=\frac{1}{2}a_{9}$ (61) $\displaystyle c_{1}={}$
$\displaystyle\frac{1}{\varepsilon^{2}}+\frac{2}{\varepsilon}+\left(3+\frac{\pi^{2}}{6}\right)+\left(-\frac{2\zeta_{3}}{3}+4+\frac{\pi^{2}}{3}\right)\varepsilon+\left(-\frac{4\zeta_{3}}{3}+5+\frac{\pi^{2}}{2}+\frac{7\pi^{4}}{360}\right)\varepsilon^{2}+\left(-2\zeta_{3}\right.$
$\displaystyle{}\left.-\frac{\pi^{2}\zeta_{3}}{9}-\frac{2\zeta_{5}}{5}+6+\frac{2\pi^{2}}{3}+\frac{7\pi^{4}}{180}\right)\varepsilon^{3}+\left(-\frac{8\zeta_{3}}{3}-\frac{2\pi^{2}\zeta_{3}}{9}+\frac{2\zeta_{3}^{2}}{9}-\frac{4\zeta_{5}}{5}+7+\frac{5\pi^{2}}{6}+\frac{7\pi^{4}}{120}\right.$
$\displaystyle{}\left.+\frac{31\pi^{6}}{15120}\right)\varepsilon^{4}+\mathcal{O}(\varepsilon^{5}),$
(62) $\displaystyle c_{3}={}$
$\displaystyle-\frac{2\pi}{\varepsilon}+\pi(4l_{2}-6)-\frac{2}{3}\varepsilon\left(\pi\left(21+\pi^{2}+6l_{2}^{2}-18l_{2}\right)\right)+\frac{2}{3}\pi\varepsilon^{2}\left(8\zeta_{3}-45-3\pi^{2}+4l_{2}^{3}\right.$
$\displaystyle{}\left.-18l_{2}^{2}+42l_{2}+2\pi^{2}l_{2}\right)-\frac{1}{15}\varepsilon^{3}\left(\pi\left(-240\zeta_{3}+160\zeta_{3}l_{2}+930+70\pi^{2}+3\pi^{4}+20l_{2}^{4}\right.\right.$
$\displaystyle{}\left.\left.-120l_{2}^{3}+420l_{2}^{2}+20\pi^{2}l_{2}^{2}-900l_{2}-60\pi^{2}l_{2}\right)\right)+\frac{1}{45}\pi\varepsilon^{4}\left(1680\zeta_{3}+80\pi^{2}\zeta_{3}+576\zeta_{5}\right.$
$\displaystyle{}\left.+480\zeta_{3}l_{2}^{2}-1440\zeta_{3}l_{2}-5670-450\pi^{2}-27\pi^{4}+24l_{2}^{5}-180l_{2}^{4}+840l_{2}^{3}+40\pi^{2}l_{2}^{3}\right.$
$\displaystyle{}\left.-2700l_{2}^{2}-180\pi^{2}l_{2}^{2}+5580l_{2}+420\pi^{2}l_{2}+18\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}),$
(63) $\displaystyle c_{4}={}$
$\displaystyle\frac{32\pi^{2}\varepsilon}{3}+\varepsilon^{2}\left(\frac{704\pi^{2}}{9}-\frac{256}{3}\pi^{2}l_{2}\right)+\frac{16}{27}\varepsilon^{3}\left(680\pi^{2}+21\pi^{4}+576\pi^{2}l_{2}^{2}-1056\pi^{2}l_{2}\right)$
$\displaystyle{}-\frac{32}{81}\varepsilon^{4}\left(558\pi^{2}\zeta_{3}-4600\pi^{2}-231\pi^{4}+2304\pi^{2}l_{2}^{3}-6336\pi^{2}l_{2}^{2}+8160\pi^{2}l_{2}\right.$
$\displaystyle{}\left.+252\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}),$
(64) $\displaystyle c_{6}={}$ $\displaystyle
2\pi^{2}+\varepsilon\left(8\pi^{2}-8\pi^{2}l_{2}\right)+\varepsilon^{2}\left(24\pi^{2}+\pi^{4}+16\pi^{2}l_{2}^{2}-32\pi^{2}l_{2}\right)+\varepsilon^{3}\left(-\frac{28\pi^{2}\zeta_{3}}{3}+64\pi^{2}+4\pi^{4}\right.$
$\displaystyle{}\left.-\frac{64}{3}\pi^{2}l_{2}^{3}+64\pi^{2}l_{2}^{2}-96\pi^{2}l_{2}-4\pi^{4}l_{2}\right)+\frac{1}{12}\varepsilon^{4}\left(-448\pi^{2}\zeta_{3}+448\pi^{2}\zeta_{3}l_{2}+1920\pi^{2}+144\pi^{4}\right.$
$\displaystyle{}\left.+5\pi^{6}+256\pi^{2}l_{2}^{4}-1024\pi^{2}l_{2}^{3}+2304\pi^{2}l_{2}^{2}+96\pi^{4}l_{2}^{2}-3072\pi^{2}l_{2}-192\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}),$
(65) $\displaystyle c_{7}={}$ $\displaystyle
4\pi^{2}+\varepsilon\left(16\pi^{2}-16\pi^{2}l_{2}\right)+\varepsilon^{2}\left(48\pi^{2}+2\pi^{4}+32\pi^{2}l_{2}^{2}-64\pi^{2}l_{2}\right)+\varepsilon^{3}\left(-\frac{56\pi^{2}\zeta_{3}}{3}+128\pi^{2}\right.$
$\displaystyle{}\left.+8\pi^{4}-\frac{128}{3}\pi^{2}l_{2}^{3}+128\pi^{2}l_{2}^{2}-192\pi^{2}l_{2}-8\pi^{4}l_{2}\right)+\frac{1}{6}\varepsilon^{4}\left(-448\pi^{2}\zeta_{3}+448\pi^{2}\zeta_{3}l_{2}+1920\pi^{2}\right.$
$\displaystyle{}\left.+144\pi^{4}+5\pi^{6}+256\pi^{2}l_{2}^{4}-1024\pi^{2}l_{2}^{3}+2304\pi^{2}l_{2}^{2}+96\pi^{4}l_{2}^{2}-3072\pi^{2}l_{2}-192\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}),$
(66) $\displaystyle c_{9}={}$
$\displaystyle\frac{1}{2\varepsilon^{2}}+\frac{3}{2\varepsilon}+\frac{1}{4}\left(14+\pi^{2}\right)+\frac{1}{12}\left(-16\zeta_{3}+90+9\pi^{2}\right)\varepsilon+\left(-4\zeta_{3}+\frac{31}{2}+\frac{7\pi^{2}}{4}+\frac{7\pi^{4}}{80}\right)\varepsilon^{2}$
$\displaystyle{}+\left(\pi^{2}\left(\frac{15}{4}-\frac{2\zeta_{3}}{3}\right)-\frac{28\zeta_{3}}{3}-\frac{16\zeta_{5}}{5}+\frac{63}{2}+\frac{21\pi^{4}}{80}\right)\varepsilon^{3}+\left(\pi^{2}\left(\frac{31}{4}-2\zeta_{3}\right)-20\zeta_{3}\right.$
$\displaystyle{}\left.+\frac{16\zeta_{3}^{2}}{9}-\frac{48\zeta_{5}}{5}+\frac{127}{2}+\frac{49\pi^{4}}{80}+\frac{869\pi^{6}}{30240}\right)\varepsilon^{4}+\mathcal{O}(\varepsilon^{5}),$
(67) $\displaystyle c_{10}={}$
$\displaystyle-\frac{2\pi}{\varepsilon}+4\pi(l_{2}-2)+\pi\varepsilon\left(-24-\frac{10\pi^{2}}{3}-4l_{2}^{2}+16l_{2}\right)+\frac{4}{3}\pi\varepsilon^{2}\left(7\zeta_{3}-48-10\pi^{2}+2l_{2}^{3}\right.$
$\displaystyle{}\left.-12l_{2}^{2}+36l_{2}+5\pi^{2}l_{2}\right)-\frac{2}{45}\varepsilon^{3}\left(\pi\left(-840\zeta_{3}+420\zeta_{3}l_{2}+3600+900\pi^{2}+119\pi^{4}+30l_{2}^{4}\right.\right.$
$\displaystyle{}\left.\left.-240l_{2}^{3}+1080l_{2}^{2}+150\pi^{2}l_{2}^{2}-2880l_{2}-600\pi^{2}l_{2}\right)\right)+\frac{4}{45}\pi\varepsilon^{4}\left(1260\zeta_{3}+175\pi^{2}\zeta_{3}+279\zeta_{5}\right.$
$\displaystyle{}\left.+210\zeta_{3}l_{2}^{2}-840\zeta_{3}l_{2}-4320-1200\pi^{2}-238\pi^{4}+6l_{2}^{5}-60l_{2}^{4}+360l_{2}^{3}+50\pi^{2}l_{2}^{3}-1440l_{2}^{2}\right.$
$\displaystyle{}\left.-300\pi^{2}l_{2}^{2}+3600l_{2}+900\pi^{2}l_{2}+119\pi^{4}l_{2}\right)+\mathcal{O}(\varepsilon^{5}).$
(68)

where $\zeta_{n}=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$ is the Riemann zeta
function, $l_{k}=\ln{k}$. Here $a_{i}$ constants correspond to $\\#i$ boundary
masters in Fig. 10, $b_{i}$ constants to $\\#i$ masters in Fig. 11 and $c_{i}$
constants to $\\#i$ masters in Fig. 12. The constants $a_{i,j}$ are given in
an accompanying Mathematica notebook. Note, that some of their analytical
expressions contain derivatives of hypergeometric functions. The latter can be
considered as new elliptic constants555The differential equation system for
heavy-heavy vertex was reduced to $\varepsilon$-form before. Still elliptics
is present in its boundary conditions., which can be evaluated with very high
precision using their triangle sum representation and SummerTime package
SummerTime , see accompanying Mathematica notebook for details.

###  3.2 Results

The complete set of results for the canonical MIs up to
$\mathcal{O}(\varepsilon^{3})$ for the heavy-heavy vertex and
$\mathcal{O}(\varepsilon^{2})$ for the heavy-light vertex are given in an
accompanying Mathematica notebook. The same notebook also contains results for
IBP MIs up to $\mathcal{O}(\varepsilon^{2})$ in the case of heavy-heavy and
heavy-light vertexes and up to $\mathcal{O}(\varepsilon^{2},w)$ in the case of
the heavy propagator. Moreover, we provide exact results for the boundary
integrals and associated asymptotic coefficient vectors, $\boldsymbol{c}$, as
well as the set of matrices, $\\{\mathbb{M},\mathbb{T},\mathbb{L}\\}$ for all
cases. The interested reader can employ these to reproduce our results. Due to
the size of the expressions for the canonical masters, we only present terms
up to an appropriate order. Also, in the case of the heavy propagator, we only
present results for IBP masters. We omit overall pre-factors of mass scale as
these can be determined by inspection.

#### Heavy-heavy:

$\displaystyle\tilde{A}_{1}={}$
$\displaystyle\frac{1}{\varepsilon^{2}}+\frac{2}{\varepsilon}+\left(3+\frac{\pi^{2}}{6}\right)+\frac{1}{3}\left(-2\zeta_{3}+12+\pi^{2}\right)\varepsilon+\frac{1}{360}\left(-480\zeta_{3}+1800+180\pi^{2}\right.$
$\displaystyle{}\left.+7\pi^{4}\right)\varepsilon^{2}+\left(-\frac{1}{9}\pi^{2}(\zeta_{3}-6)-2\zeta_{3}-\frac{2\zeta_{5}}{5}+6+\frac{7\pi^{4}}{180}\right)\varepsilon^{3}+\mathcal{O}\left(\varepsilon^{4}\right),$
(69) $\displaystyle\tilde{A}_{2}={}$
$\displaystyle\frac{35\pi}{3}+\frac{35}{18}\pi\varepsilon(44-12l_{2})+\frac{35}{54}\pi\varepsilon^{2}\left(680+6\pi^{2}+36l_{2}^{2}-264l_{2}\right)$
$\displaystyle{}-\frac{35}{81}\varepsilon^{3}\left(4\pi\left(18\zeta_{3}-1150+9l_{2}^{3}-99l_{2}^{2}+510l_{2}\right)+\pi^{3}(18l_{2}-66)\right)+\mathcal{O}\left(\varepsilon^{4}\right),$
(70) $\displaystyle\tilde{A}_{3}={}$
$\displaystyle\frac{32\pi^{2}\varepsilon}{3}+\frac{64}{9}\pi^{2}\varepsilon^{2}(11-12l_{2})+\frac{16}{27}\varepsilon^{3}\left(21\pi^{4}+8\pi^{2}\left(85+72l_{2}^{2}-132l_{2}\right)\right)+\mathcal{O}\left(\varepsilon^{4}\right),$
(71) $\displaystyle\tilde{A}_{4}={}$
$\displaystyle-\frac{2\pi}{\varepsilon}+4\pi(l_{2}-2)-\frac{2}{3}\varepsilon\left(\pi\left(36+5\pi^{2}+6l_{2}^{2}-24l_{2})\right)\right)+\frac{4}{3}\pi\varepsilon^{2}\left(7\zeta_{3}-48+2l_{2}^{3}\right.$
$\displaystyle{}\left.-12l_{2}^{2}+5\pi^{2}(l_{2}-2)+36l_{2}\right)-\frac{2}{45}\varepsilon^{3}\left(\pi\left(30\left(-28\zeta_{3}+2(7\zeta_{3}-48)l_{2}+120\right.\right.\right.$
$\displaystyle{}\left.\left.\left.+l_{2}^{4}-8l_{2}^{3}+36l_{2}^{2}\right)+119\pi^{4}+150\pi^{2}\left(6+l_{2}^{2}-4l_{2}\right)\right)\right)+\mathcal{O}\left(\varepsilon^{4}\right),$
(72) $\displaystyle\tilde{A}_{5}={}$
$\displaystyle-\frac{i(G(-i;\beta)-G(i;\beta))}{18\varepsilon}-\frac{1}{9}i(G(-i;\beta)-G(i;\beta)+G(0,-i;\beta)$
$\displaystyle{}-G(0,i;\beta)-G(-i,-i;\beta)+G(-i,i;\beta)-G(i,-i;\beta)$
$\displaystyle{}+G(i,i;\beta))+\mathcal{O}\left(\varepsilon^{1}\right),$ (73)
$\displaystyle\tilde{A}_{6}={}$ $\displaystyle-20i\varepsilon(\pi
G(-i;\beta)-\pi
G(i;\beta))+\varepsilon^{2}\left(40\pi^{2}G(-1;\beta)-40\pi^{2}G(1;\beta)\right.$
$\displaystyle{}\left.+80i\pi G(-1,-i;\beta)-80i\pi G(-1,i;\beta)-40i\pi
G(0,-i;\beta)+40i\pi G(0,i;\beta)\right.$ $\displaystyle{}\left.-40i\pi
G(-i,-i;\beta)+40i\pi G(-i,i;\beta)-40i\pi G(i,-i;\beta)+40i\pi
G(i,i;\beta)\right.$ $\displaystyle{}\left.+80i\pi G(1,-i;\beta)-80i\pi
G(1,i;\beta)+\frac{10}{3}i\pi(12l_{2}-44)G(-i;\beta)\right.$
$\displaystyle{}\left.-\frac{10}{3}i\pi(12l_{2}-44)G(i;\beta)\right)+\mathcal{O}\left(\varepsilon^{3}\right),$
(74) $\displaystyle\tilde{A}_{7}={}$
$\displaystyle-\frac{1}{9\varepsilon^{2}}-\frac{2}{9\varepsilon}+\frac{1}{54}\left(-48G(-i,-i;\beta)+48G(-i,i;\beta)+48G(i,-i;\beta)\right.$
$\displaystyle{}\left.-48G(i,i;\beta)-7\pi^{2}-18\right)+\mathcal{O}\left(\varepsilon^{1}\right),$
(75) $\displaystyle\tilde{A}_{8}={}$
$\displaystyle\frac{i(G(-i;\beta)-G(i;\beta))}{3\varepsilon}+\frac{2}{3}i(G(-i;\beta)-G(i;\beta))+\varepsilon\frac{1}{18}i\left(18+\pi^{2}\right)\left(G(-i;\beta)\right.$
$\displaystyle{}\left.-G(i;\beta)\right)+\mathcal{O}\left(\varepsilon^{2}\right),$
(76) $\displaystyle\tilde{A}_{9}={}$
$\displaystyle\frac{i(G(-i;\beta)-G(i;\beta))}{\varepsilon}+2i(G(-i;\beta)-G(i;\beta))+\varepsilon\frac{1}{6}i\left(18+\pi^{2}\right)\left(G(-i;\beta)\right.$
$\displaystyle{}\left.-G(i;\beta)\right)+\mathcal{O}\left(\varepsilon^{2}\right),$
(77) $\displaystyle\tilde{A}_{10}={}$ $\displaystyle\varepsilon\left(5i\pi
G(-i;\beta)-5i\pi
G(i;\beta)+\frac{5\pi}{2}\right)+\varepsilon^{2}\left(-10i\pi
G(0,-i;\beta)+10i\pi G(0,i;\beta)\right.$ $\displaystyle{}\left.+10i\pi
G(-i,-i;\beta)-10i\pi G(-i,i;\beta)+10i\pi G(i,-i;\beta)-10i\pi
G(i,i;\beta)\right.$
$\displaystyle{}\left.-\frac{5}{6}i\pi(12l_{2}-44)G(-i;\beta)+\frac{5}{6}i\pi(12l_{2}-44)G(i;\beta)+\pi\left(\frac{95}{6}-5l_{2}\right)\right)+\mathcal{O}\left(\varepsilon^{3}\right),$
(78) $\displaystyle\tilde{A}_{11}={}$
$\displaystyle\frac{35\pi\varepsilon}{6}+\frac{35}{18}\varepsilon^{2}(12i\pi
G(0,-i;\beta)-12i\pi G(0,i;\beta)+25\pi-6\pi l_{2})$
$\displaystyle{}+\frac{35}{54}\varepsilon^{3}\left(-72\pi^{2}G(0,-1;\beta)+264i\pi
G(0,-i;\beta)-264i\pi G(0,i;\beta)\right.$
$\displaystyle{}\left.+72\pi^{2}G(0,1;\beta)-144i\pi G(0,-1,-i;\beta)+144i\pi
G(0,-1,i;\beta)\right.$ $\displaystyle{}\left.+72i\pi G(0,0,-i;\beta)-72i\pi
G(0,0,i;\beta)+72i\pi G(0,-i,-i;\beta)\right.$ $\displaystyle{}\left.-72i\pi
G(0,-i,i;\beta)+72i\pi G(0,i,-i;\beta)-72i\pi G(0,i,i;\beta)\right.$
$\displaystyle{}\left.-144i\pi G(0,1,-i;\beta)+144i\pi G(0,1,i;\beta)-72i\pi
l_{2}G(0,-i;\beta)\right.$ $\displaystyle{}\left.+72i\pi
l_{2}G(0,i;\beta)+3\pi^{3}+415\pi+18\pi l_{2}^{2}-150\pi
l_{2}\right)+\mathcal{O}\left(\varepsilon^{4}\right),$ (79)
$\displaystyle\tilde{A}_{12}={}$
$\displaystyle(G(-i,-i;\beta)-G(-i,i;\beta)-G(i,-i;\beta)+G(i,i;\beta))+2\varepsilon(G(-i,-i;\beta)$
$\displaystyle{}-G(-i,i;\beta)-G(i,-i;\beta)+G(i,i;\beta)-G(-i,0,-i;\beta)+G(-i,0,i;\beta)$
$\displaystyle{}+G(-i,-i,-i;\beta)-G(-i,-i,i;\beta)+G(-i,i,-i;\beta)-G(-i,i,i;\beta)$
$\displaystyle{}+G(i,0,-i;\beta)-G(i,0,i;\beta)-G(i,-i,-i;\beta)+G(i,-i,i;\beta)$
$\displaystyle{}-G(i,i,-i;\beta)+G(i,i,i;\beta))+\mathcal{O}\left(\varepsilon^{2}\right),$
(80) $\displaystyle\tilde{A}_{13}={}$
$\displaystyle\frac{1}{2\varepsilon^{2}}+\frac{1}{\varepsilon}+\frac{1}{12}\left(8\sqrt{3}\Im\left(\text{Li}_{2}\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)\right)+18+\pi^{2}\right)+\frac{1}{12}\varepsilon\left(4a_{1,1}\right.$
$\displaystyle{}\left.-8\sqrt{3}\Im\left(\text{Li}_{2}\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)\right)-\pi^{2}-66\right)+\mathcal{O}\left(\varepsilon^{2}\right),$
(81) $\displaystyle\tilde{A}_{14}={}$
$\displaystyle\frac{35\pi}{9}+\frac{10}{27}\pi\varepsilon\left(77+\left(7\sqrt{3}-12\right)\pi-21l_{2}\right)+\frac{5}{162}\varepsilon^{2}\left(-63a_{2,1}+8\pi^{2}\left(-195+35\sqrt{3}\right.\right.$
$\displaystyle{}\left.\left.+144l_{2}\right)-56\pi(15l_{2}-58)\right)+\mathcal{O}\left(\varepsilon^{3}\right),$
(82) $\displaystyle\tilde{A}_{15}={}$
$\displaystyle\frac{4\pi^{2}}{9}-\frac{8}{9}\varepsilon\left(\pi^{2}(2l_{2}-1)\right)+\frac{2}{9}\pi^{2}\varepsilon^{2}\left(6+\pi^{2}+16l_{2}^{2}-16l_{2}\right)-\frac{4}{27}\varepsilon^{3}\left(2\pi^{2}\left(7\zeta_{3}-6+16l_{2}^{3}\right.\right.$
$\displaystyle{}\left.\left.-24l_{2}^{2}+18l_{2}\right)+\pi^{4}(6l_{2}-3)\right)+\mathcal{O}\left(\varepsilon^{4}\right),$
(83) $\displaystyle\tilde{A}_{16}={}$
$\displaystyle-\frac{2\pi^{2}}{3\varepsilon}+\frac{4}{3}\pi^{2}(2l_{3}-1)+\frac{1}{2}\varepsilon
a_{3,1}+\frac{1}{2}\varepsilon^{2}a_{3,2}+\frac{1}{2}\varepsilon^{3}a_{3,3}+\mathcal{O}\left(\varepsilon^{4}\right),$
(84) $\displaystyle\tilde{A}_{17}={}$
$\displaystyle-\frac{1}{6}i\varepsilon\left(\pi^{2}(G(-i;\beta)-G(i;\beta))+6(G(-i,-i,-i;\beta)-G(-i,-i,i;\beta)\right.$
$\displaystyle{}\left.-G(-i,i,-i;\beta)+G(-i,i,i;\beta)-G(i,-i,-i;\beta)+G(i,-i,i;\beta)+G(i,i,-i;\beta)\right.$
$\displaystyle{}\left.-G(i,i,i;\beta))\right)+\mathcal{O}(\varepsilon^{2}),$
(85) $\displaystyle\tilde{A}_{18}={}$
$\displaystyle-\frac{35}{3}\pi\varepsilon^{2}\left(2\pi
G\left(-\sqrt{3};\beta\right)-2\pi
G\left(\sqrt{3};\beta\right)-3i\left(G(-i,-i;\beta)-G(-i,i;\beta)\right.\right.$
$\displaystyle{}\left.\left.+G(i,-i;\beta)-G(i,i;\beta)-G\left(-\sqrt{3},-i;\beta\right)+G\left(-\sqrt{3},i;\beta\right)-G\left(\sqrt{3},-i;\beta\right)\right.\right.$
$\displaystyle{}\left.\left.+G\left(\sqrt{3},i;\beta\right)\right)\right)+\mathcal{O}(\varepsilon^{3}),$
(86) $\displaystyle\tilde{A}_{19}={}$
$\displaystyle\frac{\varepsilon}{3\sqrt{3}}\left(\pi^{2}\left(-G\left(-\frac{1}{\sqrt{3}};\beta\right)+G\left(\frac{1}{\sqrt{3}};\beta\right)+4G\left(-\sqrt{3};\beta\right)-4G\left(\sqrt{3};\beta\right)\right)\right.$
$\displaystyle{}\left.+6\left(-3G\left(-\frac{1}{\sqrt{3}},-i,-i;\beta\right)+3\left(G\left(-\frac{1}{\sqrt{3}},-i,i;\beta\right)+G\left(-\frac{1}{\sqrt{3}},i,-i;\beta\right)\right.\right.\right.$
$\displaystyle{}\left.\left.\left.-G\left(-\frac{1}{\sqrt{3}},i,i;\beta\right)+G\left(\frac{1}{\sqrt{3}},-i,-i;\beta\right)-G\left(\frac{1}{\sqrt{3}},-i,i;\beta\right)-G\left(\frac{1}{\sqrt{3}},i,-i;\beta\right)\right.\right.\right.$
$\displaystyle{}\left.\left.\left.+G\left(\frac{1}{\sqrt{3}},i,i;\beta\right)+G\left(-\sqrt{3},-i,-i;\beta\right)-G\left(-\sqrt{3},-i,i;\beta\right)-G\left(-\sqrt{3},i,-i;\beta\right)\right.\right.\right.$
$\displaystyle{}\left.\left.\left.+G\left(-\sqrt{3},i,i;\beta\right)-G\left(\sqrt{3},-i,-i;\beta\right)+G\left(\sqrt{3},-i,i;\beta\right)+G\left(\sqrt{3},i,-i;\beta\right)\right.\right.\right.$
$\displaystyle{}\left.\left.\left.-G\left(\sqrt{3},i,i;\beta\right)\right)+2i\Im\left(\text{Li}_{2}\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)\right)(G(-i;\beta)-G(i;\beta))\right)\right)+\mathcal{O}(\varepsilon^{2}),$
(87) $\displaystyle\tilde{A}_{20}={}$
$\displaystyle-6G(-i,-i;\beta)+6G(-i,i;\beta)+6G(i,-i;\beta)-6G(i,i;\beta)-\frac{17\pi^{2}}{12}+\mathcal{O}(\varepsilon),$
(88) $\displaystyle\tilde{A}_{21}={}$
$\displaystyle\frac{i(G(-i;\beta)-G(i;\beta))}{6\varepsilon}+\frac{1}{3}i(G(-i;\beta)-G(i;\beta))$
$\displaystyle{}+\frac{1}{36}i\varepsilon\left(24\sqrt{3}\Im\left(\text{Li}_{2}\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)\right)+18+\pi^{2}\right)(G(-i;\beta)-G(i;\beta))$
$\displaystyle{}-\frac{1}{36}i\varepsilon^{2}(G(-i;\beta)-G(i;\beta))\left(-12a_{1,1}+24\sqrt{3}\Im\left(\text{Li}_{2}\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)\right)-8\zeta_{3}\right.$
$\displaystyle{}\left.+7\pi^{2}+246\right)+\mathcal{O}\left(\varepsilon^{3}\right),$
(89) $\displaystyle\tilde{A}_{22}={}$
$\displaystyle\frac{35}{3}i\varepsilon(\pi G(-i;\beta)-\pi
G(i;\beta))+\varepsilon^{2}\left(\frac{35}{18}i\pi(12l_{2}-44)G(i;\beta)\right.$
$\displaystyle{}\left.-\frac{35}{18}i\pi(12l_{2}-44)G(-i;\beta)\right)+\varepsilon^{3}\left(\frac{35}{54}i\pi\left(680+6\pi^{2}+36l_{2}^{2}-264l_{2}\right)G(-i;\beta)\right.$
$\displaystyle{}\left.-\frac{35}{54}i\pi\left(680+6\pi^{2}+36l_{2}^{2}-264l_{2}\right)G(i;\beta)\right)+\mathcal{O}\left(\varepsilon^{4}\right),$
(90) $\displaystyle\tilde{A}_{23}={}$
$\displaystyle-\frac{35}{6}\pi\varepsilon^{3}\left(2\pi
G\left(0,-\sqrt{3};\beta\right)-2\pi
G\left(0,\sqrt{3};\beta\right)-3i\left(G(0,-i,-i;\beta)-G(0,-i,i;\beta)\right.\right.$
$\displaystyle{}\left.\left.+G(0,i,-i;\beta)-G(0,i,i;\beta)-G\left(0,-\sqrt{3},-i;\beta\right)+G\left(0,-\sqrt{3},i;\beta\right)\right.\right.$
$\displaystyle{}\left.\left.-G\left(0,\sqrt{3},-i;\beta\right)+G\left(0,\sqrt{3},i;\beta\right)\right)\right)+\mathcal{O}(\varepsilon^{4}),$
(91) $\displaystyle\tilde{A}_{24}={}$
$\displaystyle\frac{1}{2\varepsilon^{2}}+\frac{3}{2\varepsilon}+\frac{1}{4}\left(14+\pi^{2}\right)+\frac{1}{12}\left(-16\zeta_{3}+90+9\pi^{2}\right)\varepsilon+\frac{1}{80}\left(-320\zeta_{3}+1240+140\pi^{2}\right.$
$\displaystyle{}\left.+7\pi^{4}\right)\varepsilon^{2}+\frac{1}{240}\left(-2240\zeta_{3}-160\pi^{2}\zeta_{3}-768\zeta_{5}+7560+900\pi^{2}+63\pi^{4}\right)\varepsilon^{3}+\mathcal{O}\left(\varepsilon^{4}\right),$
(92) $\displaystyle\tilde{A}_{25}={}$ $\displaystyle
2\pi^{2}-8\varepsilon\left(\pi^{2}(l_{2}-1)\right)+\varepsilon^{2}\left(\pi^{4}+8\pi^{2}\left(3+2l_{2}^{2}-4l_{2}\right)\right)-\frac{4}{3}\varepsilon^{3}\left(\pi^{2}\left(7\zeta_{3}-48+16l_{2}^{3}\right.\right.$
$\displaystyle{}\left.\left.-48l_{2}^{2}+72l_{2}+\pi^{2}(3l_{2}-3)\right)\right)+\mathcal{O}\left(\varepsilon^{4}\right),$
(93) $\displaystyle\tilde{A}_{26}={}$
$\displaystyle\frac{i(G(-i;\beta)-G(i;\beta))}{2\varepsilon}+\frac{3}{2}i(G(-i;\beta)-G(i;\beta))+\varepsilon\frac{1}{4}i\left(14+\pi^{2}\right)\left(G(-i;\beta)\right.$
$\displaystyle{}\left.-G(i;\beta)\right)+\varepsilon^{2}\frac{1}{12}i\left(-16\zeta_{3}+90+9\pi^{2}\right)\left(G(-i;\beta)-G(i;\beta)\right)+\mathcal{O}\left(\varepsilon^{3}\right).$
(94)

#### Heavy-light:

$\displaystyle\tilde{B}_{1}={}$
$\displaystyle\frac{1}{\varepsilon^{2}}+\frac{2}{\varepsilon}+\left(3+\frac{\pi^{2}}{6}\right)+\frac{1}{3}\left(-2\zeta_{3}+12+\pi^{2}\right)\varepsilon+\frac{1}{360}\left(-480\zeta_{3}+1800\right.$
$\displaystyle{}\left.+180\pi^{2}+7\pi^{4}\right)\varepsilon^{2}+\mathcal{O}\left(\varepsilon^{3}\right),$
(95) $\displaystyle\tilde{B}_{2}={}$
$\displaystyle\frac{7}{60\varepsilon^{2}}+\frac{7}{30\varepsilon}+\left(\frac{7}{20}+\frac{7\pi^{2}}{72}\right)+\frac{7}{180}\left(22\zeta_{3}+12+5\pi^{2}\right)\varepsilon$
$\displaystyle{}+\frac{7\left(1760\zeta_{3}+600+300\pi^{2}+101\pi^{4}\right)\varepsilon^{2}}{7200}+\mathcal{O}\left(\varepsilon^{3}\right),$
(96) $\displaystyle\tilde{B}_{3}={}$
$\displaystyle\frac{2}{105\varepsilon^{2}}+\frac{4}{105\varepsilon}+\frac{1}{105}\left(-16G(-i,-i;\beta)+16G(-i,i;\beta)+16G(i,-i;\beta)\right.$
$\displaystyle{}\left.-16G(i,i;\beta)-\pi^{2}+6\right)-\frac{2}{315}\varepsilon\left(48G(-i,-i;\beta)-48G(-i,i;\beta)\right.$
$\displaystyle{}\left.-48G(i,-i;\beta)+48G(i,i;\beta)-144G(-i,0,-i;\beta)+144G(-i,0,i;\beta)\right.$
$\displaystyle{}\left.+144G(-i,-i,-i;\beta)-144G(-i,-i,i;\beta)+144G(-i,i,-i;\beta)\right.$
$\displaystyle{}\left.-144G(-i,i,i;\beta)+144G(i,0,-i;\beta)-144G(i,0,i;\beta)-144G(i,-i,-i;\beta)\right.$
$\displaystyle{}\left.+144G(i,-i,i;\beta)-144G(i,i,-i;\beta)+144G(i,i,i;\beta)-118\zeta_{3}\right.$
$\displaystyle{}\left.+3\pi^{2}-12\right)+\mathcal{O}\left(\varepsilon^{2}\right),$
(97) $\displaystyle\tilde{B}_{4}={}$
$\displaystyle-\frac{8i(G(-i;\beta)-G(i;\beta))}{105\varepsilon}-\frac{16}{105}i(G(-i;\beta)-G(i;\beta)-3G(0,-i;\beta)$
$\displaystyle{}+3G(0,i;\beta)+3G(-i,-i;\beta)-3G(-i,i;\beta)+3G(i,-i;\beta)$
$\displaystyle{}-3G(i,i;\beta))+\mathcal{O}\left(\varepsilon^{1}\right),$ (98)
$\displaystyle\tilde{B}_{5}={}$
$\displaystyle-\frac{64\pi^{2}}{105}+\frac{128}{105}\pi^{2}\varepsilon(4l_{2}-1)-\frac{32}{315}\varepsilon^{2}\left(\pi^{2}\left(18+7\pi^{2}+192l_{2}^{2}-96l_{2}\right)\right)+\mathcal{O}\left(\varepsilon^{3}\right),$
(99) $\displaystyle\tilde{B}_{6}={}$
$\displaystyle\frac{4i(G(-i;\beta)-G(i;\beta))}{3\varepsilon}+\frac{8}{3}i(G(-i;\beta)-G(i;\beta)-G(0,-i;\beta)+G(0,i;\beta)$
$\displaystyle{}+G(-i,-i;\beta)-G(-i,i;\beta)+G(i,-i;\beta)-G(i,i;\beta))+\mathcal{O}\left(\varepsilon^{1}\right),$
(100) $\displaystyle\tilde{B}_{7}={}$
$\displaystyle\frac{14}{45}i\varepsilon\left(\pi^{2}(G(-i;\beta)-G(i;\beta))+6(G(-i,-i,-i;\beta)-G(-i,-i,i;\beta)\right.$
$\displaystyle{}\left.-G(-i,i,-i;\beta).+G(-i,i,i;\beta)-G(i,-i,-i;\beta)+G(i,-i,i;\beta)+G(i,i,-i;\beta)\right.$
$\displaystyle{}\left.-G(i,i,i;\beta))\right)+\mathcal{O}(\varepsilon^{2}),$
(101) $\displaystyle\tilde{B}_{8}={}$
$\displaystyle-\frac{4i(G(-i;\beta)-G(i;\beta))}{15\varepsilon}-\frac{8}{15}i(G(-i;\beta)-G(i;\beta)-2G(0,-i;\beta)$
$\displaystyle{}+2G(0,i;\beta)+2G(-i,-i;\beta)-2G(-i,i;\beta)+2G(i,-i;\beta)$
$\displaystyle{}-2G(i,i;\beta))+\mathcal{O}\left(\varepsilon^{1}\right),$
(102) $\displaystyle\tilde{B}_{9}={}$
$\displaystyle\frac{1}{189}i\varepsilon\left(\pi^{2}(G(-i;\beta)-G(i;\beta))+6(G(-i,-i,-i;\beta)-G(-i,-i,i;\beta)\right.$
$\displaystyle{}\left.-G(-i,i,-i;\beta)+G(-i,i,i;\beta)-G(i,-i,-i;\beta)+G(i,-i,i;\beta)+G(i,i,-i;\beta)\right.$
$\displaystyle{}\left.-G(i,i,i;\beta))\right)+\mathcal{O}(\varepsilon^{0}),$
(103) $\displaystyle\tilde{B}_{10}={}$
$\displaystyle\frac{1}{441}\left(2G(-i,-i;\beta)-2G(-i,i;\beta)-2G(i,-i;\beta)+2G(i,i;\beta)+\pi^{2}\right)$
$\displaystyle{}+\frac{2}{441}\varepsilon\left(\pi^{2}G(-i;\beta)+\pi^{2}G(i;\beta)+2G(-i,-i;\beta)-2G(-i,i;\beta)\right.$
$\displaystyle{}\left.-2G(i,-i;\beta)+2G(i,i;\beta)-4G(0,-i,-i;\beta)+4G(0,-i,i;\beta)\right.$
$\displaystyle{}\left.+4G(0,i,-i;\beta)-4G(0,i,i;\beta)-6G(-i,0,-i;\beta)+6G(-i,0,i;\beta)\right.$
$\displaystyle{}\left.+10G(-i,-i,-i;\beta)-10G(-i,-i,i;\beta)+2G(-i,i,-i;\beta)-2G(-i,i,i;\beta)\right.$
$\displaystyle{}\left.+6G(i,0,-i;\beta)-6G(i,0,i;\beta)-2G(i,-i,-i;\beta)+2G(i,-i,i;\beta)\right.$
$\displaystyle{}\left.-10G(i,i,-i;\beta)+10G(i,i,i;\beta)+\pi^{2}-4\pi^{2}l_{2}\right)+\mathcal{O}\left(\varepsilon^{2}\right),$
(104) $\displaystyle\tilde{B}_{11}={}$
$\displaystyle-\frac{8}{9}\left(2G(-i,-i;\beta)-2G(-i,i;\beta)-2G(i,-i;\beta)+2G(i,i;\beta)+\pi^{2}\right)$
$\displaystyle{}-\frac{16}{9}\varepsilon\left(\pi^{2}G(-i;\beta)+\pi^{2}G(i;\beta)+2G(-i,-i;\beta)-2G(-i,i;\beta)\right.$
$\displaystyle{}\left.-2G(i,-i;\beta)+2G(i,i;\beta)-4G(0,-i,-i;\beta)+4G(0,-i,i;\beta)\right.$
$\displaystyle{}\left.+4G(0,i,-i;\beta)-4G(0,i,i;\beta)-2G(-i,0,-i;\beta)+2G(-i,0,i;\beta)\right.$
$\displaystyle{}\left.+6G(-i,-i,-i;\beta)-6G(-i,-i,i;\beta)-2G(-i,i,-i;\beta)+2G(-i,i,i;\beta)\right.$
$\displaystyle{}\left.+2G(i,0,-i;\beta)-2G(i,0,i;\beta)+2G(i,-i,-i;\beta)-2G(i,-i,i;\beta)\right.$
$\displaystyle{}\left.-6G(i,i,-i;\beta)+6G(i,i,i;\beta)+\pi^{2}-4\pi^{2}l_{2}\right)+\mathcal{O}\left(\varepsilon^{2}\right),$
(105) $\displaystyle\tilde{B}_{12}={}$
$\displaystyle-\frac{2}{11}(G(-i,-i;\beta)-G(-i,i;\beta)-G(i,-i;\beta)+G(i,i;\beta))$
$\displaystyle{}-\frac{4}{11}\varepsilon(G(-i,-i;\beta)-G(-i,i;\beta)-G(i,-i;\beta)+G(i,i;\beta)$
$\displaystyle{}-2G(0,-i,-i;\beta)+2G(0,-i,i;\beta)+2G(0,i,-i;\beta)-2G(0,i,i;\beta)$
$\displaystyle{}-3G(-i,0,-i;\beta)+3G(-i,0,i;\beta)+5G(-i,-i,-i;\beta)-5G(-i,-i,i;\beta)$
$\displaystyle{}+G(-i,i,-i;\beta)-G(-i,i,i;\beta)+3G(i,0,-i;\beta)-3G(i,0,i;\beta)$
$\displaystyle{}-G(i,-i,-i;\beta)+G(i,-i,i;\beta)-5G(i,i,-i;\beta)+5G(i,i,i;\beta))+\mathcal{O}\left(\varepsilon^{2}\right),$
(106) $\displaystyle\tilde{B}_{13}={}$ $\displaystyle
i\varepsilon^{2}(2G(-i,0,-i,-i;\beta)-2(G(-i,0,-i,i;\beta)+G(-i,0,i,-i;\beta)-G(-i,0,i,i;\beta)$
$\displaystyle{}+G(-i,-i,0,-i;\beta)-G(-i,-i,0,i;\beta)-2G(-i,-i,i,-i;\beta)+2G(-i,-i,i,i;\beta)$
$\displaystyle{}-G(-i,i,0,-i;\beta)+G(-i,i,0,i;\beta)+2G(-i,i,-i,-i;\beta)-2G(-i,i,-i,i;\beta)$
$\displaystyle{}+G(i,0,-i,-i;\beta)-G(i,0,-i,i;\beta)-G(i,0,i,-i;\beta)+G(i,0,i,i;\beta)$
$\displaystyle{}-G(i,-i,0,-i;\beta)+G(i,-i,0,i;\beta)+2G(i,-i,i,-i;\beta)-2G(i,-i,i,i;\beta)$
$\displaystyle{}+G(i,i,0,-i;\beta)-G(i,i,0,i;\beta)-2G(i,i,-i,-i;\beta)+2G(i,i,-i,i;\beta))$
$\displaystyle{}+3\zeta_{3}(G(i;\beta)-G(-i;\beta)))+\mathcal{O}(\varepsilon^{3}),$
(107) $\displaystyle\tilde{B}_{14}={}$
$\displaystyle\frac{1}{4}(G(-i,-i;\beta)-G(-i,i;\beta)-G(i,-i;\beta)+G(i,i;\beta))$
$\displaystyle{}+\frac{1}{2}\varepsilon(G(-i,-i;\beta)-G(-i,i;\beta)-G(i,-i;\beta)+G(i,i;\beta)$
$\displaystyle{}-2G(0,-i,-i;\beta)+2G(0,-i,i;\beta)+2G(0,i,-i;\beta)-2G(0,i,i;\beta)$
$\displaystyle{}-G(-i,0,-i;\beta)+G(-i,0,i;\beta)+3G(-i,-i,-i;\beta)-3G(-i,-i,i;\beta)$
$\displaystyle{}-G(-i,i,-i;\beta)+G(-i,i,i;\beta)+G(i,0,-i;\beta)-G(i,0,i;\beta)$
$\displaystyle{}+G(i,-i,-i;\beta)-G(i,-i,i;\beta)-3G(i,i,-i;\beta)+3G(i,i,i;\beta))+\mathcal{O}\left(\varepsilon^{2}\right),$
(108) $\displaystyle\tilde{B}_{15}={}$
$\displaystyle\frac{32}{9}(G(-i,-i;\beta)-G(-i,i;\beta)-G(i,-i;\beta)+G(i,i;\beta))$
$\displaystyle{}+\frac{64}{9}\varepsilon(G(-i,-i;\beta)-G(-i,i;\beta)-G(i,-i;\beta)+G(i,i;\beta)$
$\displaystyle{}-2G(0,-i,-i;\beta)+2G(0,-i,i;\beta)+2G(0,i,-i;\beta)-2G(0,i,i;\beta)$
$\displaystyle{}-G(-i,0,-i;\beta)+G(-i,0,i;\beta)+3G(-i,-i,-i;\beta)-3G(-i,-i,i;\beta)$
$\displaystyle{}-G(-i,i,-i;\beta)+G(-i,i,i;\beta)+G(i,0,-i;\beta)-G(i,0,i;\beta)$
$\displaystyle{}+G(i,-i,-i;\beta)-G(i,-i,i;\beta)-3G(i,i,-i;\beta)+3G(i,i,i;\beta))+\mathcal{O}\left(\varepsilon^{2}\right),$
(109) $\displaystyle\tilde{B}_{16}={}$
$\displaystyle\frac{19}{60\varepsilon^{2}}+\frac{19}{30\varepsilon}+\frac{1}{360}\left(342-13\pi^{2}\right)+\frac{1}{180}\varepsilon\left(-374\zeta_{3}+228+\pi^{2}(96l_{2}-13)\right)+\mathcal{O}\left(\varepsilon^{2}\right),$
(110) $\displaystyle\tilde{B}_{17}={}$
$\displaystyle\frac{1}{15}\left(-8G(-i,-i;\beta)+8G(-i,i;\beta)+8G(i,-i;\beta)-8G(i,i;\beta)-\pi^{2}\right)$
$\displaystyle{}+\frac{1}{15}\varepsilon\left(4\pi^{2}G(-1;\beta)-8\pi^{2}G(-i;\beta)-8\pi^{2}G(i;\beta)+4\pi^{2}G(1;\beta)\right.$
$\displaystyle{}\left.-16G(-i,-i;\beta)+16G(-i,i;\beta)+16G(i,-i;\beta)-16G(i,i;\beta)\right.$
$\displaystyle{}\left.+32G(-1,-i,-i;\beta)-32G(-1,-i,i;\beta)-32G(-1,i,-i;\beta)+32G(-1,i,i;\beta)\right.$
$\displaystyle{}\left.+16G(0,-i,-i;\beta)-16G(0,-i,i;\beta)-16G(0,i,-i;\beta)+16G(0,i,i;\beta)\right.$
$\displaystyle{}\left.+16G(-i,0,-i;\beta)-16G(-i,0,i;\beta)-64G(-i,-i,-i;\beta)+64G(-i,-i,i;\beta)\right.$
$\displaystyle{}\left.+32G(-i,i,-i;\beta)-32G(-i,i,i;\beta)-16G(i,0,-i;\beta)+16G(i,0,i;\beta)\right.$
$\displaystyle{}\left.-32G(i,-i,-i;\beta)+32G(i,-i,i;\beta)+64G(i,i,-i;\beta)-64G(i,i,i;\beta)\right.$
$\displaystyle{}\left.+32G(1,-i,-i;\beta)-32G(1,-i,i;\beta)-32G(1,i,-i;\beta)+32G(1,i,i;\beta)\right.$
$\displaystyle{}\left.+21\zeta_{3}-2\pi^{2}+10\pi^{2}l_{2}\right)+\mathcal{O}\left(\varepsilon^{2}\right),$
(111) $\displaystyle\tilde{B}_{18}={}$
$\displaystyle\frac{2}{3}i\varepsilon(\pi^{2}(G(-i;\beta)-G(i;\beta))+6(G(-i,-i,-i;\beta)-G(-i,-i,i;\beta)$
$\displaystyle{}-G(-i,i,-i;\beta)+G(-i,i,i;\beta)-G(i,-i,-i;\beta)+G(i,-i,i;\beta)$
$\displaystyle{}+G(i,i,-i;\beta)-G(i,i,i;\beta)))+\mathcal{O}(\varepsilon^{2}).$
(112)

The integrals above are expressed in terms of Goncharov polylogarithms, which
are generalisations of HPLs and defined recursively as poincare1884groupes ;
goncharov2010classical ; goncharov2011multiple

$G(b,a_{1}\cdots a_{n};x)=\int_{0}^{x}\frac{dy}{y-b}G(a_{1}\cdots
a_{n};y),\quad G(;x)=1,$ (113)

with

$G(0\cdots 0;x)=\frac{1}{n!}\ln^{n}{x}$ (114)

and the integration path being a straight line from 0 to $x\in\mathbb{C}$.

#### Heavy propagator:

$\displaystyle C_{1}={}$
$\displaystyle\frac{1}{\varepsilon^{2}}+\frac{2}{\varepsilon}+\left(3+\frac{\pi^{2}}{6}\right)+\frac{1}{3}\left(-2\zeta_{3}+12+\pi^{2}\right)\varepsilon+\frac{1}{360}\left(-480\zeta_{3}+1800+180\pi^{2}\right.$
$\displaystyle{}\left.+7\pi^{4}\right)\varepsilon^{2}+\mathcal{O}(\varepsilon^{3},w^{2}),$
(115) $\displaystyle C_{2}={}$
$\displaystyle\frac{3}{2\varepsilon^{2}}+\frac{9}{2\varepsilon}+\frac{1}{4}\left(8\sqrt{3}\Im\left(\text{Li}_{2}\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)\right)+42+\pi^{2}\right)+\varepsilon
a_{1,1}+\varepsilon^{2}a_{1,2}+\mathcal{O}(\varepsilon^{3},w^{2}),$ (116)
$\displaystyle C_{3}={}$
$\displaystyle\frac{2w}{\varepsilon^{2}}+\frac{2w-2\pi}{\varepsilon}+\left(\frac{1}{3}\left(6+\pi^{2}\right)w+\pi(4l_{2}-6)\right)+\varepsilon\left(\frac{1}{3}w\left(-4\zeta_{3}+6+\pi^{2}\right)\right.$
$\displaystyle{}\left.-\frac{2}{3}\pi\left(21+\pi^{2}+6l_{2}^{2}-18l_{2}\right)\right)+\varepsilon^{2}\left(\frac{1}{180}w\left(-240\zeta_{3}+360+60\pi^{2}+7\pi^{4}\right)\right.$
$\displaystyle{}\left.+\frac{2}{3}\pi\left(8\zeta_{3}-45+4l_{2}^{3}-18l_{2}^{2}+42l_{2}+\pi^{2}(2l_{2}-3)\right)\right)+\mathcal{O}(\varepsilon^{3},w^{2}),$
(117) $\displaystyle C_{4}={}$
$\displaystyle\frac{2w}{\varepsilon^{2}}+\frac{4w}{\varepsilon}+\frac{1}{3}\left(24+\pi^{2}\right)w+\varepsilon\left(\frac{2}{3}w\left(-2\zeta_{3}+24+\pi^{2}\right)+\frac{32\pi^{2}}{3}\right)+\varepsilon^{2}\left(\frac{1}{180}w\left(-480\zeta_{3}\right.\right.$
$\displaystyle{}\left.\left.+5760+240\pi^{2}+7\pi^{4}\right)+\frac{64}{9}\pi^{2}(11-12l_{2})\right)+\mathcal{O}(\varepsilon^{3},w^{2}),$
(118) $\displaystyle C_{5}={}$
$\displaystyle-\frac{2}{\varepsilon^{2}}-\frac{4}{\varepsilon}+\left(-4\pi^{2}w-\frac{\pi^{2}}{3}-8\right)+\varepsilon\left(16\pi^{2}w(2l_{2}-1)-\frac{2}{3}\left(-2\zeta_{3}+24+\pi^{2}\right)\right)$
$\displaystyle{}+\varepsilon^{2}\left(\frac{1}{180}\left(480\zeta_{3}-5760-240\pi^{2}-7\pi^{4}\right)-\frac{2}{3}w\left(7\pi^{4}+96\pi^{2}\left(1+2l_{2}^{2}-2l_{2}\right)\right)\right)$
$\displaystyle{}+\mathcal{O}(\varepsilon^{3},w^{2}),$ (119) $\displaystyle
C_{6}={}$
$\displaystyle\frac{w}{\varepsilon^{2}}+\frac{2w-2\pi}{\varepsilon}+\left(-\frac{4}{3}\pi\left(6+\sqrt{3}\pi-3l_{2}\right)+\frac{1}{6}w\left(24\sqrt{3}\Im\left(\text{Li}_{2}\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)\right)+24\right.\right.$
$\displaystyle{}\left.\left.+\pi^{2}\right)\right)+\varepsilon\left(a_{2,1}-\frac{1}{6}w\left(-12a_{1,1}+24\sqrt{3}\Im\left(\text{Li}_{2}\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)\right)-8\zeta_{3}+7\pi^{2}+222\right)\right)$
$\displaystyle{}+\varepsilon^{2}\left(\frac{1}{180}w\left(-360a_{1,1}+360a_{1,2}+480\zeta_{3}-7\pi^{4}-240\pi^{2}-5760\right)+a_{2,2}\right)+\mathcal{O}(\varepsilon^{3},w^{2}),$
(120) $\displaystyle C_{7}={}$ $\displaystyle
2\pi^{2}+\varepsilon\left(16\pi^{2}w-8\pi^{2}(l_{2}-1)\right)+\varepsilon^{2}\left(32\pi^{2}w(3-4l_{2})+\pi^{4}+8\pi^{2}\left(3+2l_{2}^{2}-4l_{2}\right)\right)$
$\displaystyle{}+\mathcal{O}(\varepsilon^{3},w^{2}),$ (121) $\displaystyle
C_{8}={}$ $\displaystyle-\frac{8(\pi w)}{\varepsilon}+\left(4\pi
w(4l_{2}-4)+4\pi^{2}\right)+\varepsilon\left(-\frac{4}{3}\pi
w\left(24+2\pi^{2}+12l_{2}^{2}-24l_{2}\right)\right.$
$\displaystyle{}\left.-16\pi^{2}(l_{2}-1)\right)+\varepsilon^{2}\left(\frac{8}{3}w\left(4\pi\left(2\zeta_{3}-6+l_{2}^{3}-3l_{2}^{2}+6l_{2}\right)+\pi^{3}(2l_{2}-2)\right)+2\left(\pi^{4}\right.\right.$
$\displaystyle{}\left.\left.+8\pi^{2}\left(3+2l_{2}^{2}-4l_{2}\right)\right)\right)+\mathcal{O}(\varepsilon^{3},w^{2}),$
(122) $\displaystyle C_{9}={}$
$\displaystyle-\frac{2\pi^{2}}{3\varepsilon}+\left(\frac{4}{3}\pi^{2}(2l_{3}-1)-\frac{16\pi^{2}w}{3\sqrt{3}}\right)+\varepsilon\left(\frac{4}{9}w\left(3a_{2,1}+2\pi^{3}+8\left(3+\sqrt{3}\right)\pi^{2}\right.\right.$
$\displaystyle{}\left.\left.+\pi\left(72+12l_{2}^{2}-48l_{2}\right)\right)+\frac{a_{3,1}}{2}\right)+\varepsilon^{2}\left(\frac{a_{3,2}}{2}-\frac{4}{9}w\left(6a_{2,1}-3a_{2,2}+8\pi\left(2\zeta_{3}-6+l_{2}^{3}\right.\right.\right.$
$\displaystyle{}\left.\left.\left.-3l_{2}^{2}+6l_{2}\right)+48\pi^{2}(4l_{2}-3)+2\pi^{3}(2l_{2}-2)\right)\right)+\mathcal{O}(\varepsilon^{3},w^{2}),$
(123) $\displaystyle C_{10}={}$
$\displaystyle\frac{1}{2\varepsilon^{2}}+\frac{3}{2\varepsilon}+\frac{7}{2}+\frac{\pi^{2}}{4}+\varepsilon\left(\frac{15}{2}+\frac{3\pi^{2}}{4}-\frac{4\zeta_{3}}{3}\right)+\varepsilon^{2}\left(\frac{31}{2}+\frac{7\pi^{2}}{4}+\frac{7\pi^{4}}{80}-4\zeta_{3}\right)$
$\displaystyle{}+\mathcal{O}(\varepsilon^{3},w^{2}),$ (124) $\displaystyle
C_{11}={}$ $\displaystyle 0+\mathcal{O}(\varepsilon^{3},w^{2}),$ (125)
$\displaystyle C_{12}={}$
$\displaystyle\frac{w}{\varepsilon^{2}}+\frac{2w-2\pi}{\varepsilon}+\left(\frac{1}{2}\left(8+\pi^{2}\right)w+4\pi(l_{2}-2)\right)+\varepsilon\left(w\left(-\frac{8\zeta_{3}}{3}+8+\pi^{2}\right)-\frac{2}{3}\left(5\pi^{3}\right.\right.$
$\displaystyle{}\left.\left.+6\pi\left(6+l_{2}^{2}-4l_{2}\right)\right)\right)+\varepsilon^{2}\left(\frac{1}{120}w\left(-640\zeta_{3}+1920+240\pi^{2}+21\pi^{4}\right)+\frac{4}{3}\pi\left(7\zeta_{3}\right.\right.$
$\displaystyle{}\left.\left.-48+2l_{2}^{3}-12l_{2}^{2}+5\pi^{2}(l_{2}-2)+36l_{2})\right)\right)+\mathcal{O}(\varepsilon^{3},w^{2}).$
(126)

##  4 Conclusion

We have employed the differential equations method to treat diagrams with
heavy field insertions up to three-points two-loop order. Our analysis focused
on vertex or form factor diagrams as these can be combined and used in studies
beyond three-point order as they form the building blocks for a broad class of
processes. By including a mass scale, our results are applicable for a wider
range of theories and provide an IR structure for massless models studied. The
treatment of the heavy-heavy and heavy-light vertex diagrams was achieved
explicitly by reducing integrals to $\varepsilon$-form and expressing their
results in terms of MPLs. On the other hand, the self-energy contributions
needed for heavy field and residual mass renormalization were determined by a
series expansion in the off-shell heavy field energy. Exact off-shell self-
energies were shown to require treatment with elliptic polylogarithms. We
instead employed the Frobenius method to obtain solutions as an expansion in
small off-shell energies. Thus, we provide further proof-positive of the power
of the differential equations and dimensional recurrence approaches,
advocating for their use even when more exotic propagators are present.

Acknowledgements: We are indebted to R. N. Lee, O. L. Veretin and V. A.
Smirnov for stimulating discussions. This work was supported in part by the
German Research Foundation DFG through Grant No. KN 365/12-1. The work of
A.I.O was supported in part by the Foundation for the Advancement of
Theoretical Physics and Mathematics "BASIS" and Russian Science Foundation,
grant 20-12-00205.

##  Appendix A One-loop master integrals

Here, for completeness, we present a calculation of one-loop master integrals.

###  A.1 Heavy-heavy vertex

In the case of heavy-heavy vertices, the single integral family for prototype
diagram shown in Fig. 1 (a) is given by

$I^{HH}_{\nu_{1},\nu_{2},\nu_{3}}=\int\frac{d^{d}l}{i\pi^{d/2}}\frac{1}{(l\cdot
v_{1})^{\nu_{1}}(l\cdot v_{2})^{\nu_{2}}(l^{2}-M^{2})^{\nu_{3}}}.$ (127)

Upon IBP reduction, all integrals in this family reduce to three master
integrals $I^{HH}_{001},I^{HH}_{011},I^{HH}_{111}$. The first two master
integrals are easy to calculate with Feynman parameters, and we have

$I^{HH}_{001}=-\Gamma(\varepsilon-1)M^{2-2\varepsilon}\,,\quad
I^{HH}_{011}=\sqrt{\pi}\Gamma\left(-\frac{1}{2}+\varepsilon\right)M^{1-2\varepsilon}$
(128)

The third master integral can be conveniently calculated with the use of
differential equation with respect to $w=v_{1}\cdot v_{2}$

$\frac{d}{dw}I^{HH}_{111}(w)=\frac{w}{1-w^{2}}I^{HH}_{111}(w)-\frac{2(\varepsilon-1)}{1-w^{2}}\frac{1}{M^{2}}I^{HH}_{001}\,.$
(129)

One can then easily solve Eq. (129) by variation of parameters and boundary
condition, $I^{HH}_{111}(1)=-2M^{-2\varepsilon}\Gamma(\varepsilon)$, to
obtain,

$I^{HH}_{111}=-\frac{2\Gamma(\varepsilon)M^{-2\varepsilon}}{\sqrt{1-w^{2}}}\arccos(w)\,,$
(130)

which mimics the well-known result from Feynman parametrization and its
modification for HQET-like propagators manohar2007heavy .

###  A.2 Heavy-light vertex

Here, the single integral family for prototype diagram shown in Fig. 1 (b) is
given by

$I^{HL}_{\nu_{1},\nu_{2},\nu_{3}}=\int\frac{d^{d}l}{i\pi^{d/2}}\frac{1}{(l\cdot
v_{1})^{\nu_{1}}((l+p_{2})^{2}-m^{2})^{\nu_{2}}(l^{2})^{\nu_{3}}}.$ (131)

Upon IBP reduction all integrals in this family reduce to two master integrals
$I^{HL}_{010},I^{HL}_{110}$. The first master we already have encountered in
previous sub-section

$I^{HL}_{010}=-\Gamma(\varepsilon-1)m^{2-2\varepsilon}\,$ (132)

The second master integral can be again calculated with the use of
differential equation, this time with respect to $w=v_{1}\cdot p_{2}/m$

$\frac{d}{dw}I^{HL}_{111}(w)=\frac{w(1-2\varepsilon)}{1-w^{2}}I^{HL}_{110}(w)+\frac{2(1-\varepsilon)}{1-w^{2}}\frac{1}{m}I^{HL}_{010}$
(133)

The Eq. (133) is easily solved by variation of parameters and boundary
condition,
$I^{HL}_{110}(0)=\sqrt{\pi}\Gamma(-1/2+\varepsilon)m^{1-2\varepsilon}$, to
obtain,

$I^{HL}_{110}=-2m^{1-2\varepsilon}\Gamma(\varepsilon)w~{}_{2}F_{1}\left(1,\varepsilon;\frac{3}{2};w^{2}\right)+\sqrt{\pi}m^{1-2\varepsilon}\Gamma\left(-\frac{1}{2}+\varepsilon\right)(1-w^{2})^{1/2-\varepsilon}\,.$
(134)

###  A.3 Heavy propagator

In this case, the single integral family for prototype diagram shown in Fig. 1
(c) is given by

$I^{SE}_{\nu_{1},\nu_{2}}=M^{d-\nu_{1}-2\nu_{2}}\int\frac{d^{d}l}{i\pi^{d/2}}\frac{1}{(l\cdot
v-w)^{\nu_{1}}(l^{2}-1)^{\nu_{2}}}$ (135)

Here we have the same master integrals as in previous sub-section. These
integrals are given by

$\displaystyle I^{SE}_{01}=-\Gamma(\varepsilon-1)M^{2-2\varepsilon}$ (136)

and

$I^{SE}_{11}=-2M^{1-2\varepsilon}\Gamma(\varepsilon)w~{}_{2}F_{1}\left(1,\varepsilon;\frac{3}{2};w^{2}\right)+\sqrt{\pi}M^{1-2\varepsilon}\Gamma\left(-\frac{1}{2}+\varepsilon\right)(1-w^{2})^{1/2-\varepsilon}\,.$
(137)

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