Multivariate elliptic kites and tetrahedral tadpoles
David Broadhurst, Open University, UK, December 4, 2022

Abstract: This work deals with two types of Feynman integrals in perturbative
quantum field theory: the 2-loop 2-point kite, with 5 arbitrary internal
masses, and its completion by a sixth propagator, to give a 3-loop tetrahedral
tadpole, with 6 arbitrary masses. The elliptic substructure of these integrals
should not be regarded as an obstruction. Rather, it is a bonus, thanks to the
arithmetic-geometric mean of Gauss. Compact formulæ are given, to handle all
cases. Zero-mass limits are carefully considered. Anomalous thresholds of
triangles in the kite pose no problem. The number theory of tadpoles is
investigated, with intriguing results. Instructive benchmarks are given at
60-digit precision, extendable to 200 digits in a few seconds.

In memoriam, Gabriel Barton (25 February 1934 to 11 October 2022) and Donald
Hill Perkins (15 October 1925 to 30 October 2022), trusted guides and mentors.

##  1 Introduction

A univariate elliptic kite in the two-loop electron propagator was tackled by
Afaf Sabry [1] sixty years ago, after work with Gunnar Källén [2] on the two-
loop photon propagator. The latter has an evaluation in terms of the classical
trilogarithm of Leonard Lewin’s influential books [3]. Thirty years later [4],
I extended both results to include all cases needed in unbroken gauge
theories. Dispersion relations, pioneered by Hans Kramers and Ralph Kronig [5]
and extended to particle physics by Murph Goldberger [6], were of the essence
in this work. I learnt about them from Gabriel Barton [7] who had keenly
attended Goldberger’s lectures in Princeton.

Work on univariate $L$-loop equal-mass sunrise integrals [8, 9], with $L+1$
edges, was a vital part of the preparation for Stefano Laporta’s result at 4
loops for the magnetic moment of the electron [10]. This subject was
revolutionized by Spencer Bloch and Pierre Vanhove [11], who gave a modular
parametrization of the 2-loop sunrise diagram in two spacetime dimensions.
With Matt Kerr, they achieved the same feat at 3 loops [12] where the
underlying differential equation is a symmetric square of the equation at 2
loops, as had been observed by Geoff Joyce [13], long before, in the context
of lattice Green functions for condensed matter problems. With David Bailey,
Jon Borwein and Larry Glasser [14] I exploited this in elliptic evaluations of
Bessel moments, which are related to Feynman integrals by the fact that the
Fourier transform of a massive propagator is a Bessel function. Many
conjectures [15, 16, 17, 18] on Bessel moments were later proved by Yajun Zhou
[19, 20, 21, 22].

Rich mathematical structure also resides in the 6-parameter tetrahedron made
by completing the kite with a sixth propagator, to obtain a 3-loop vacuum
integral which is called a tadpole. In [23, 24, 25], I considered binary
tetrahedral tadpoles, with unit or zero masses, discovering that all are
reducible to multiple polylogarithms of sixth roots of unity, despite elliptic
substructure in constituent kites. These results were used by Matthias
Steinhauser, whose computer-algebra package [26] reduces every binary 3-loop
tadpole, no matter what its subdivergences, to polylogarithmic constants
identified in [25].

Activity on elliptic substructure of massive Feynman integrals overlaps with
work in perturbative string theory [27, 28, 29], on massless Feynman
integrals, where elliptic substructure appears in a 2-loop 6-point function
with 4 off-shell legs [30, 31], and on developing [32, 33, 34, 35, 36] a
theory of iterated integrals with elliptic kernels, as generalizations of
better understood multiple polylogarithms [37, 38, 39, 40, 41].

Participation at recent workshops and conferences organized in Bonn,
Copenhagen, Ettal, Florence, Grenoble, Mainz, Potsdam, Uppsala, Zeuthen and
Zürich revealed the breadth and depth of activity in these related fields. As
yet, univariate cases have received more attention than multivariate cases,
since the latter are obviously more demanding. Every student of massive
Feynman integrals learns at an early stage that when three massive particles
meet at a vertex the square root of a Källén polynomial ensues [42]. Elliptic
substructure emerges when one integrates over a product of square roots. For
multiloop sunrise diagrams [43, 44, 45] there are no internal vertices and
multivariate analysis is not so hard. For the generic tetrahedral completion
of a kite, each of the four faces specifies a subintegration with square roots
associated to each of the three vertices that define the face. Yet the net
result may still be polylogarithmic, notwithstanding elliptic substructure.

The remainder of this article is organized as follows. After some short
preliminary definitions, Section 2 gives compact explicit formulæ that enable
one to evaluate any massive kite or tetrahedral tadpole numerically, to high
precision and at good speed, by one-dimensional quadrature of the derivative
$\sigma^{\prime}$ of the discontinuity of a kite, multiplied by a logarithm or
dilogarithm. The elliptic term in the integrand presents no problem, since it
is given by the extraordinarily fast process of the arithmetic-geometric mean
of Gauss [46, 47]. Nor are the anomalous thresholds of triangles problematic.
Section 3 gives zero-mass limits needed for gauge theories. As so often
occurs, the specific is more demanding than the general. With fewer
parameters, massless cases produce more complicated formulæ. Section 4 is
devoted to the number theory of tetrahedral tadpoles. Section 5 describes the
tests made of these procedures and gives benchmarks. Section 6 gives comments
and a summary.

###  1.1 Preliminaries

Define the two-loop scalar kite integral in 4-dimensional Minkowski space by

$\displaystyle
I(q^{2};m_{1}^{2},m_{2}^{2},m_{3}^{2},m_{4}^{2},m_{5}^{2})=-\frac{q^{2}}{\pi^{4}}\int{\rm
d}^{4}l\int{\rm d}^{4}k\,\prod_{j=1}^{5}\frac{1}{p_{j}^{2}-m_{j}^{2}-{\rm
i}\epsilon},$ (1)
$\displaystyle(p_{1},\,p_{2},\,p_{3},\,p_{4},\,p_{5})=(l,\,l-q,\,l-k,\,k,\,k-q)$
(2)

with a labelling chosen such that edge 1 is not adjacent to edge 5. Neither is
edge 2 adjacent to edge 4. Where appropriate, the arguments $m_{j}^{2}$ will
be omitted. Then $I(s)$ is analytic in the $s$-plane with a cut $s\in[s_{\rm
L},\infty]$ and a leftmost branch point $s_{\rm L}$ that is the lowest of the
thresholds $\\{s_{1,2},s_{4,5},s_{2,3,4},s_{1,3,5}\\}$, where
$s_{j,k}=(m_{j}+m_{k})^{2}$ and $s_{i,j,k}=(m_{i}+m_{j}+m_{k})^{2}$.

On the top lip of the cut, let $\Im I(s+{\rm i}\epsilon)=\pi\sigma(s)$. In the
general mass case, with $\sigma(s_{\rm L})=0$, it suffices to know the
derivative $\sigma^{\prime}(s)$, which gives the dispersion relation

$I(q^{2})=-\int_{s_{\rm L}}^{\infty}{\rm
d}s\,\sigma^{\prime}(s)\log\left(1-\frac{q^{2}}{s}\right).$ (3)

123456 213546 345261

Figure 1: Equivalent labellings of a tetrahedral tadpole

Consider the logarithmically divergent tetrahedral tadpole formed by joining
the external vertices of the kite with an inverse propagator
$q^{2}-m_{6}^{2}-{\rm i}\epsilon$. Regularization in $4-2\epsilon$ dimensions
gives a tadpole

$\displaystyle
T_{1,2,3}^{5,4,6}=\left(\frac{1}{3\epsilon}+1\right)6\zeta_{3}+3\zeta_{4}-F_{1,2,3}^{5,4,6}+O(\epsilon),$
(4) $\displaystyle F_{1,2,3}^{5,4,6}=\int_{s_{\rm L}}^{\infty}{\rm
d}s\,\sigma^{\prime}(s;m_{1}^{2},m_{2}^{2},m_{3}^{2},m_{4}^{2},m_{5}^{2})\left({\rm
Li}_{2}\left(1-\frac{m_{6}^{2}}{s}\right)+\frac{1}{2}\log^{2}\left(\frac{\overline{m}^{2}}{s}\right)\right)$
(5)

where $\overline{m}$ is the scale of dimensional regularization and the
dilogarithm is the analytic continuation of the sum ${\rm
Li}_{2}(z)=\sum_{n>0}z^{n}/n^{2}$, valid for $|z|<1$. The superscripts in (5)
are made redundant by the labelling convention: a superscript $j$ sits above a
subscript $k$ if and only if $j$ is congruent to $k$ modulo 3. Subscripts
identity triangles; superscripts identify vertices. Figure 1 shows that
$F_{1,2,3}=F_{2,1,3}=F_{3,4,5}$. The first identity is obvious from the point
of view of the kite in the integrand of (5). The second is highly non-trivial,
since now $m_{1}$ is external. The relation $F_{1,2,3}=F_{1,4,6}$ makes
$m_{3}$ external. Thus a single omnivorous tadople can digest 6 distinct
kites.

##  2 The generic kite

Consider the generic 5-parameter kite, with distinct non-zero masses and
thresholds. The derivative $\sigma^{\prime}(s)$ of its discontinuity receives
logarithmic contributions from two-particle cuts and elliptic contributions
from three-particle cuts. There may also be an algebraic contribution
associated with anomalous thresholds in the dispersion relations for triangles
that form the kite. I now give explicit compact formulæ for these three types
of contribution.

###  2.1 Non-elliptic non-anomalous contribution

In the absence of anomalous thresholds, the non-elliptic contribution to
$\sigma^{\prime}(s)$ may be obtained by diligent algebra and application of
Cauchy’s integral theorem. Here I give the 5-parameter result, written as
compactly as possible in (10,11).

Denote the square root of the symmetric Källén function by

$\Delta(a,b,c)=\sqrt{a^{2}+b^{2}+c^{2}-2(ab+bc+ca)}$ (6)

with convenient abbreviations $\Delta_{j,k}(s)=\Delta(s,m_{j}^{2},m_{k}^{2})$
and $\Delta_{i,j,k}=\Delta_{j,k}(m_{i}^{2})$. Then

$D_{j,k}(s)=\frac{r}{s-(m_{j}-m_{k})^{2}}\log\left(\frac{1+r}{1-r}\right),\quad
r=\left(\frac{s-(m_{j}-m_{k})^{2}}{s-(m_{j}+m_{k})^{2}}\right)^{1/2}$ (7)

is analytic in the $s$-plane with a cut $s\in[(m_{j}+m_{k})^{2},\infty]$,
where its discontinuity is $D_{j,k}(s+{\rm i}\epsilon)-D_{j,k}(s-{\rm
i}\epsilon)=-2\pi{\rm i}/\Delta_{j,k}(s)$. Next, define the real constants

$\alpha=\frac{(m_{1}^{2}-m_{4}^{2})(m_{2}^{2}-m_{5}^{2})}{m_{3}^{2}}-m_{3}^{2},\quad\beta=\frac{(m_{1}^{2}m_{5}^{2}-m_{2}^{2}m_{4}^{2})(m_{1}^{2}-m_{2}^{2}-m_{4}^{2}+m_{5}^{2})}{m_{3}^{2}}$
(8)

which help to condense the 5-parameter result. The (possibly complex)
constants

$s_{\pm}=\frac{m_{1}^{2}+m_{2}^{2}-2m_{3}^{2}+m_{4}^{2}+m_{5}^{2}-\alpha}{2}\,\pm\,\frac{\Delta_{1,3,4}\Delta_{2,3,5}}{2m_{3}^{2}}$
(9)

locate leading Landau singularities [48] of triangles that form the kite.

In the absence of anomalous thresholds, the non-elliptic contribution to
$\sigma^{\prime}(s)$ is

$\sigma^{\prime}_{\rm{N}}(s)=\Theta(s-s_{1,2})\sigma_{1,2}^{\prime}(s)+\Theta(s-s_{4,5})\sigma_{4,5}^{\prime}(s)$
(10)

with Heaviside steps denoted by $\Theta$. Putting edges 1 and 2 on-shell, I
obtain

$\displaystyle\Delta_{1,2}(s)\sigma_{1,2}^{\prime}(s)=\Re\left((s+\alpha)D_{4,5}(s)+L_{4,5}+\sum_{i=0,+,-}C_{i}\,\frac{D_{4,5}(s)-D_{4,5}(s_{i})}{s-s_{i}}\right),$
(11) $\displaystyle C_{\pm}=\alpha
s_{\pm}+\beta,\;C_{0}=-(m_{1}^{2}-m_{2}^{2})(m_{4}^{2}-m_{5}^{2}),\;s_{0}=0,\;L_{4,5}=\log\left(\frac{m_{4}m_{5}}{m_{3}^{2}}\right).$
(12)

To obtain $\sigma_{4,5}^{\prime}$, exchange $(m_{1},m_{2})$ and
$(m_{4},m_{5})$, noting that this does not alter the coefficients $C_{i}$ or
the arguments $s_{i}$.

The result in (10,11) holds if both of the conditions

$\displaystyle(m_{1}+m_{2})(m_{3}^{2}+m_{1}m_{2})$ $\displaystyle\geq$
$\displaystyle m_{1}m_{5}^{2}+m_{2}m_{4}^{2}$ (13)
$\displaystyle(m_{4}+m_{5})(m_{3}^{2}+m_{4}m_{5})$ $\displaystyle\geq$
$\displaystyle m_{5}m_{1}^{2}+m_{4}m_{2}^{2}$ (14)

are satisfied. However, these are not necessary conditions. The final decision
as to whether the non-elliptic contribution is in need of an anomalous
correction is arbitrated by the elliptic contribution, which is subject to no
uncertainty.

###  2.2 Elliptic contribution

The elliptic contribution, from intermediate states with three particles, has
the form

$\sigma^{\prime}_{\rm{E}}(s)=\Theta(s-s_{2,3,4})\sigma_{2,3,4}^{\prime}(s)+\Theta(s-s_{1,3,5})\sigma_{1,3,5}^{\prime}(s).$
(15)

It contains complete elliptic integrals of the third kind, which I shall
divide by complete integrals of the first kind. For real $k^{2}<1$, let

$P(n,k)=\frac{\Pi(n,k)}{\Pi(0,k)},\quad\Pi(n,k)=\int_{0}^{\pi/2}\frac{d\theta}{(1-n\sin^{2}\theta)\sqrt{1-k^{2}\sin^{2}\theta}}$
(16)

where $\Pi(0,k)=(\pi/2)/{\rm AGM}\,(1,\sqrt{1-k^{2}})$ is determined by the
arithmetic-geometric mean of Gauss. Then $P(n,k)$ is analytic in the $n$-plane
with a cut $n\in[1,\infty]$ on which its principal value is $1-P(k^{2}/n,k)$.

With $s=w^{2}$, an integration over the phase space of particles 2, 3 and 4
determines

$k^{2}=1-\frac{16m_{2}m_{3}m_{4}w}{W},\quad
W=(w_{+}^{2}-m_{+}^{2})(w_{-}^{2}-m_{-}^{2})$ (17)

with $w_{\pm}=w\pm m_{2}$ and $m_{\pm}=m_{3}\pm m_{4}$. Then I obtain

$\sigma^{\prime}_{2,3,4}(w^{2})=\frac{4\pi m_{3}m_{4}}{{\rm
AGM}\,(\sqrt{16m_{2}m_{3}m_{4}w},\sqrt{W})}\Re\left(\,\sum_{i=+,-}E_{i}\,\frac{P(n_{i},k)-P(n_{1},k)}{t_{i}-t_{1}}\right)$
(18)

with coefficients and arguments given, as compactly as possible, by

$\displaystyle
E_{\pm}=\frac{m_{2}^{2}-m_{3}^{2}+m_{5}^{2}}{2m_{5}^{2}}\pm\left(\frac{m_{4}^{2}-m_{5}^{2}-w^{2}}{2m_{5}^{2}}\right)\,\frac{\Delta_{2,3,5}}{\Delta_{4,5}(w^{2})},$
(19) $\displaystyle
t_{\pm}=\frac{\gamma\pm\Delta_{2,3,5}\Delta_{4,5}(w^{2})}{2m_{5}^{2}},\quad
t_{1}=m_{1}^{2},\quad
n_{i}=\frac{(w_{-}^{2}-m_{+}^{2})(t_{i}-m_{-}^{2})}{(w_{-}^{2}-m_{-}^{2})(t_{i}-m_{+}^{2})},$
(20)
$\displaystyle\gamma=(m_{2}^{2}+m_{3}^{2}+m_{4}^{2}-m_{5}^{2}+w^{2})m_{5}^{2}+(m_{2}^{2}-m_{3}^{2})(m_{4}^{2}-w^{2}).$
(21)

A strong check of (18) is its hidden invariance under the non-trivial
transformation
$(m_{1},m_{2},m_{3},m_{4},m_{5})\to(m_{5},m_{4},m_{3},m_{2},m_{1})$, despite
division by $m_{5}^{2}$ in (19,20). To obtain $\sigma^{\prime}_{1,3,5}$, take
the transform
$(m_{1},m_{2},m_{3},m_{4},m_{5})\to(m_{4},m_{5},m_{3},m_{1},m_{2})$.

An AGM procedure evaluates $P(n,k)=\Pi(n,k)/\Pi(0,k)$ to high precision, as
follows.

  1. 1.

Initialize $[a,\,b,\,p,\,q]=[1,\,\sqrt{1-k^{2}},\,\sqrt{1-n},\,n/(2-2n)]$.
Then set $f=1+q$.

  2. 2.

Set $m=ab$ and then $r=p^{2}+m$. Compute a vector of new values as follows:
$[(a+b)/2,\,\sqrt{m},\,r/(2p),\,(r-2m)q/(2r)]$. Then replace $[a,\,b,\,p,\,q]$
by those new values. Then add $q$ to $f$.

  3. 3.

If $|q/f|$ is sufficiently small, then return $P(n,k)=f$, else go to step 2.

This converges very quickly, for $n\notin[1,\infty]$. For real $n\geq 1$,
invoke the procedure with $n$ replaced by $k^{2}/n<1$, to obtain the principal
value $\Re P(n,k)=1-P(k^{2}/n,k)$.

###  2.3 Criterion for an anomalous contribution

If there is an anomalous contribution, it occurs above the higher of the two-
particle thresholds. Without loss of generality, suppose that $s_{4,5}\geq
s_{1,2}$. Then

$\sigma^{\prime}(s)=\sigma^{\prime}_{\rm N}(s)+\sigma^{\prime}_{\rm
E}(s)+C_{\rm
A}\frac{\Theta(s-s_{4,5})}{\Delta_{4,5}(s)}\,\Re\left(\frac{2\pi{\rm
i}\Delta_{4,5}(s_{-})}{s-s_{-}}\right)$ (22)

with $C_{\rm A}\neq 0$ if and only if
$(m_{1}+m_{2})(m_{3}^{2}+m_{1}m_{2})<m_{1}m_{5}^{2}+m_{2}m_{4}^{2}$ and at
least one of $\Delta_{1,3,4}$ and $\Delta_{2,3,5}$ is imaginary, in which case
$C_{\rm A}=\pm 1$ is the sign of the imaginary part of $\Delta_{4,5}(s_{-})$.
This value of $C_{\rm A}$ is required by the elliptic contribution at high
energy. With $L_{k}=m_{k}^{2}\log(s/m_{k}^{2})$, the large-$s$ behaviour

$s^{2}\sigma^{\prime}(s)=2L_{3}+\sum_{k=1,2,4,5}(L_{k}+m_{k}^{2})\,+\,O\left(\frac{\log(s)}{s}\right)$
(23)

invariably holds. The elliptic contribution $\sigma^{\prime}_{\rm E}$ in (22)
is oblivious to the anomalous threshold problem. Its high-energy behaviour
determines $C_{\rm A}$, ensuring (23). Thus the leading singularity of Lev
Landau in the anomalous non-elliptic sector neatly performs a task set by Carl
Friedrich Gauss in the elliptic sector. I tested this at 50-digit precision
with more than a million random assignments of masses. Each test took less
than a millisecond, thanks to the extraordinary efficiency of the AGM.

##  3 Zero-mass limits

Despite division by $m_{3}^{2}$ in (8,9), the two-particle cuts give finite
contributions as $m_{3}\to 0$, except in the special case that $m_{1}=m_{4}$
and $m_{2}=m_{5}$. As $m_{3}\to 0$ with $m_{1}\neq m_{4}$ and $m_{2}\neq
m_{5}$, diligent algebra shows that

$\displaystyle\Delta_{1,2}(s)\sigma_{1,2}^{\prime}(s)\to\Re\left((2s-s_{3})D_{4,5}(s)+\widehat{L}_{4,5}+\sum_{i=0,3}C_{i}\,\frac{D_{4,5}(s)-D_{4,5}(s_{i})}{s-s_{i}}\right),$
(24) $\displaystyle
s_{3}=-\frac{(m_{1}^{2}m_{5}^{2}-m_{2}^{2}m_{4}^{2})(m_{1}^{2}-m_{2}^{2}-m_{4}^{2}+m_{5}^{2})}{M},\quad
M=(m_{1}^{2}-m_{4}^{2})(m_{2}^{2}-m_{5}^{2}),$ (25)
$\displaystyle\widehat{L}_{4,5}=\log\left(\frac{m_{4}^{2}m_{5}^{2}}{M}\right),\quad
C_{3}=-\left(\frac{m_{1}^{2}}{u}-m_{2}^{2}u\right)\left(\frac{m_{4}^{2}}{u}-m_{5}^{2}u\right),\quad
u=\frac{m_{1}^{2}-m_{4}^{2}}{m_{2}^{2}-m_{5}^{2}}$ (26)

with $\sigma_{4,5}^{\prime}$ given by
$(m_{1},m_{2})\leftrightarrow(m_{4},m_{5})$. There is no anomalous term, since
both $\Delta_{1,3,4}$ and $\Delta_{2,3,5}$ are real for $m_{3}=0$. As
$m_{3}\to 0$ with $m_{1}=m_{4}$ and $m_{2}\neq m_{5}$

$\displaystyle\Delta_{2,4}(s)\sigma_{1,2}^{\prime}(s)$
$\displaystyle\to\Re\left((3s-m_{2}^{2}-2m_{4}^{2}-m_{5}^{2})D_{4,5}(s)+\log\left(\frac{m_{4}m_{5}^{3}}{(m_{2}^{2}-m_{5}^{2})^{2}}\right)\right.$
$\displaystyle+\left.(m_{2}^{2}-m_{4}^{2})(m_{4}^{2}-m_{5}^{2})\,\frac{D_{4,5}(s)-D_{4,5}(0)}{s}\right)$
(27)

with $\sigma_{4,5}^{\prime}$ given by $m_{2}\leftrightarrow m_{5}$. The
degenerate case with $m_{1}=m_{4}$ and $m_{2}=m_{5}$ will be considered after
adding contributions from three-particle cuts.

At $m_{3}=0$, the three-particle cuts yield logarithms. For real
$w>m_{j}+m_{k}$, let

$\widehat{P}_{j,k}(t,w)=\frac{(m_{k}^{2}-t)v(t)}{(w-m_{j})^{2}-t}\log\left(\frac{v(t)+v(m_{k}^{2})}{v(t)-v(m_{k}^{2})}\right),\;v(t)=\left(\frac{(w-m_{j})^{2}-t}{(w+m_{j})^{2}-t}\right)^{1/2}$
(28)

where $v(m_{k}^{2})$ is real and positive, while $v(t)$ may be complex. As
$m_{3}\to 0$

$\sigma_{2,3,4}^{\prime}(w^{2})\to\Re\left(\sum_{i=\pm}E_{i}\,\frac{\widehat{P}_{2,4}(t_{i},w)-\widehat{P}_{2,4}(m_{1}^{2},w)}{t_{i}-m_{1}^{2}}\right)$
(29)

with coefficients in (19) and arguments in (20) that are well behaved at
$m_{3}=0$.

With $m_{1}=m_{4}$ and $m_{2}=m_{5}$ all four thresholds collide as $m_{3}\to
0$, giving [4]

$\displaystyle\sigma^{\prime}(s)\to\Theta(s-s_{4,5})\,\frac{2\mu(y_{4})+2\mu(y_{5})-8\mu(y_{4}y_{5})}{\Delta_{4,5}(s)},$
(30) $\displaystyle\mu(y)=\log|1-y|+\frac{y\log|y|}{1-y},\quad
y_{k}=\frac{-2m_{k}^{2}}{s-m_{4}^{2}-m_{5}^{2}+\Delta_{4,5}(s)}.$ (31)

Next, consider cases with $m_{3}>0$ and one of the other masses vanishing.
Without loss of generality, take it to be $m_{4}$. As $m_{4}\to 0$, logarithms
from (28) appear in

$\sigma_{2,3,4}^{\prime}(w^{2})\to\Re\left(\sum_{i=\pm}E_{i}\,\frac{\widehat{P}_{2,3}(t_{i},w)-\widehat{P}_{2,3}(m_{1}^{2},w)}{t_{i}-m_{1}^{2}}\right).$
(32)

The logarithms in (11) are modified, to give

$\displaystyle\Delta_{1,2}(s)\sigma_{1,2}^{\prime}(s)\to\Re\left((s+\alpha)\widehat{D}_{5}(s)+\widehat{L}_{5}+\sum_{i=0,+,-}C_{i}\,\frac{\widehat{D}_{5}(s)-\widehat{D}_{5}(s_{i})}{s-s_{i}}\right),$
(33)
$\displaystyle\widehat{D}_{5}(s)=\frac{1}{s-m_{5}^{2}}\log\left(1-\frac{s}{m_{5}^{2}}\right),\quad\widehat{L}_{5}=\log\left(\frac{m_{5}^{2}}{m_{3}^{2}}\right).$
(34)

An elliptic contribution persists if two non-adjacent edges have vanishing
mass. Without loss of generality, take these edges to be 1 and 5. As $m_{1}\to
0$ and $m_{5}\to 0$

$(w^{2}-m_{4}^{2})\sigma^{\prime}_{2,3,4}(w^{2})\,\to\,-\frac{4\pi
m_{3}m_{4}\Re R(w^{2},m_{2}^{2},m_{3}^{2},m_{4}^{2})}{{\rm
AGM}\,(\sqrt{16m_{2}m_{3}m_{4}w},\sqrt{W})}$

where $R$ contains, in general, three elliptic terms. When $m_{2}\neq m_{3}$,
I obtain

$\displaystyle R(s,b,c,d)=P(\widehat{n},k)-\rho
P(n_{0},k)+(\rho-1)P(n_{3},k),$ (35)
$\displaystyle\widehat{n}=\frac{w_{-}^{2}-m_{+}^{2}}{w_{-}^{2}-m_{-}^{2}},\quad\frac{n_{0}}{\widehat{n}}=\frac{m_{-}^{2}}{m_{+}^{2}},\quad\frac{n_{3}}{\widehat{n}}=\frac{t_{3}-m_{-}^{2}}{t_{3}-m_{+}^{2}},\quad
t_{3}=\frac{(bd-cs)(b-c+d-s)}{(b-c)(d-s)},$ (36)
$\displaystyle\rho=\left(\frac{d-s}{b-c+d-s}\right)\left(\frac{(b+c)(d-s)+(b-c)(b+d)}{bd-
cs}\right)$ (37)

with $(s,b,c,d)=(w^{2},m_{2}^{2},m_{3}^{2},m_{4}^{2})$. In equal-mass cases
this simplifies greatly, giving

$R(s,c,c,d)=2P(\widehat{n},k)-2P(n_{0},k),\quad R(s,d,d,d)=\frac{s-9d}{6d}$
(38)

with a notable rational result for $m_{2}=m_{3}=m_{4}$, proven in [4].

Remaining cases with vanishing masses are logarithmic and hence somewhat
easier. Care is needed in cases like (30) when thresholds collide. In the
bivariate case where $m_{1}$, $m_{3}$ and $m_{4}$ vanish, with $m_{2}\neq
m_{5}$, I obtained the compact result

$\displaystyle
s\sigma^{\prime}(s)\to\Theta(s-m_{2}^{2})\,\tau(x_{2},x_{5})+\Theta(s-m_{5}^{2})\,\tau(x_{5},x_{2}),\quad
x_{k}=\frac{m_{k}^{2}}{s},$ (39)
$\displaystyle\quad\tau(x,y)=\nu(x,y)-\nu(y,x)-\mu(y),\quad\nu(x,y)=\frac{2}{1-x}\log\left|\frac{1-y}{x-y}\right|\quad$
(40)

after taking account of the threshold collisions $s_{2,3,4}\to s_{1,2}$ and
$s_{1,3,5}\to s_{4,5}$.

##  4 Tadpoles and number theory

Since $s\sigma^{\prime}(s)$ is dimensionless it depends only on the 5 ratios
$m_{k}^{2}/s$, with $k\in[1,5]$. Tetrahedral tadpoles have a simple behaviour
under the rescaling $m_{k}\to\kappa\,m_{k}$, with $k\in[1,6]$. This gives
$F\to F+12\zeta_{3}\log(\kappa)$ for the finite part $F$ in (5).

###  4.1 Perfect tadpoles

I define a tetrahedral tadpole to be perfect if and only if the Källén
function vanishes at each of its 4 vertices, thereby avoiding all resolutions
of square roots. Without loss of generality, suppose that
$m_{3}=\overline{m}=1$ and that $m_{k}\leq 1$ for $k\neq 3$. I promote the
subscripts and superscripts of $F$ to arguments that denote the 6 masses and
define the two-parameter family of perfect tadpoles

$\widehat{F}(x,y)=F_{(x,y,1)}^{(1-y,1-x,|x-y|)}=\widehat{F}(y,x)=\widehat{F}(1-x,1-y)$
(41)

with symmetries restricting distinct cases to $x\geq y\geq 1-x\geq 0$ and
hence $x\in[\frac{1}{2},1]$. In [25], I identified the tetralogarithms in two
perfect tadpoles, obtaining

$\displaystyle\widehat{F}(1,0)=F_{(1,1,0)}^{(1,1,0)}=17\zeta_{4}+16U_{3,1},\quad\widehat{F}(1,1)=F_{(1,1,1)}^{(0,0,0)}=12\zeta_{4},$
(42) $\displaystyle
U_{3,1}=\sum_{m>n>0}\frac{(-1)^{m+n}}{m^{3}n}=\tfrac{1}{2}\zeta_{4}+\tfrac{1}{2}\zeta_{2}\log^{2}(2)-\tfrac{1}{12}\log^{4}(2)-2\,{\rm
Li}_{4}(\tfrac{1}{2}).$ (43)

###  4.2 Fast elliptic determination of a perfect tadpole

Now consider the elliptic route to evaluating
$\widehat{F}(\frac{1}{2},\frac{1}{2})$. With $(m_{3},m_{6})=(1,0)$ and
$m_{1}=m_{2}=m_{4}=m_{5}=\frac{1}{2}$, I obtained

$\displaystyle\widehat{F}(\tfrac{1}{2},\tfrac{1}{2})=\tfrac{1}{2}\int_{1}^{\infty}{\rm
d}s\left(\widehat{\sigma}^{\prime}_{\rm N}(s)+\widehat{\sigma}^{\prime}_{\rm
E}(s)\right)\log^{2}(s),$ (44) $\displaystyle
w^{2}\widehat{\sigma}^{\prime}_{\rm
N}(w^{2})=\Theta(w-1)\left(2\log\left(\frac{r+1}{r-1}\right)-4r\log(2)\right),\;r=\frac{w}{\sqrt{w^{2}-1}},$
(45) $\displaystyle w^{2}\widehat{\sigma}^{\prime}_{\rm
E}(w^{2})=\frac{4\pi(1-P(n,k))\Theta(w-2)}{{\rm
AGM}(2\sqrt{w},(w-1)\sqrt{w^{2}+2w})},\;n=\frac{w^{2}-2w}{(w-1)^{2}},\;\frac{k^{2}}{n}=\frac{(w+1)^{2}}{w^{2}+2w}$
(46)

and readily discovered a new reduction of a perfect tadpole to tetralogarithms

$\widehat{F}(\tfrac{1}{2},\tfrac{1}{2})=30\zeta_{3}\log(2)-16\zeta_{4}-32U_{3,1}.$
(47)

###  4.3 Relations between tadpoles

$\frac{1}{2}$$\frac{1}{2}$1$\frac{1}{2}$$\frac{1}{2}$0
1$\frac{1}{2}$01$\frac{1}{2}$$\frac{1}{2}$
$\frac{1}{2}$$\frac{1}{2}$1$\frac{1}{2}$$\frac{1}{2}$1

Figure 2: The perfect tadpoles $\widehat{F}(\frac{1}{2},\frac{1}{2})$,
$\widehat{F}(1,\frac{1}{2})$ and $\widehat{G}(\frac{1}{2})$ in relation (49)

In addition to the two-parameter family $\widehat{F}(x,y)$ in (41) there is a
one-parameter family $\widehat{G}(x)=F_{(x,1-x,1)}^{(x,1-x,1)}$ of perfect
tadpoles, with $x\in[0,\frac{1}{2}]$ and
$\widehat{G}(0)=17\zeta_{4}+16U_{3,1}$.

I used the efficient AGM of Gauss to obtain 200 good digits of

$\widehat{G}(\tfrac{1}{2})=-\int_{1}^{\infty}{\rm
d}s\left(\widehat{\sigma}^{\prime}_{\rm N}(s)+\widehat{\sigma}^{\prime}_{\rm
E}(s)\right){\rm Li}_{2}(1-s)$ (48)

to which all routes are elliptic. This revealed the intriguing empirical
relation

$2\widehat{F}(\tfrac{1}{2},\tfrac{1}{2})+2\widehat{F}(1,\tfrac{1}{2})+\widehat{G}(\tfrac{1}{2})=42\zeta_{4}+24\zeta_{3}\log(2)$
(49)

with a non-elliptic route to $\widehat{F}(1,\tfrac{1}{2})$ that leads to
multiple polylogarithms in an alphabet of forms, ${\rm d}x/(x-a_{i})$, with
$a_{i}\in\\{0,1,-1,-2\\}$. Then (47) and (49) give

$\displaystyle\widehat{G}(\tfrac{1}{2})=47\zeta_{4}+40U_{3,1}+4\zeta_{3}\log(2)-12\overline{G}\,$
(50) $\displaystyle\overline{G}=2\zeta_{2}{\rm
Li}_{2}(\tfrac{1}{4})+3\zeta_{2}\log^{2}(2)+4\sum_{m>n>0}\frac{(-1)^{m}(-{\frac{1}{2}})^{n}}{m^{2}n^{2}}$
(51)

with 10000 digits of the perfect tadpoles in Figure 2 now obtainable in 2
seconds.

There are linear relations between binary tadpoles, with $m_{k}\in\\{0,1\\}$,
as here

$\displaystyle 3F_{(0,0,0)}^{(1,1,1)}$
$\displaystyle=F_{(1,1,1)}^{(0,0,0)}+2F_{(1,1,0)}^{(1,0,0)}$ (52)
$\displaystyle 3F_{(1,1,0)}^{(0,0,0)}$
$\displaystyle=F_{(1,0,0)}^{(0,0,0)}+2F_{(1,1,1)}^{(0,0,0)}$ (53)
$\displaystyle F_{(1,1,1)}^{(1,1,1)}+F_{(1,0,0)}^{(1,0,0)}$
$\displaystyle=F_{(1,1,0)}^{(1,1,0)}+F_{(0,0,0)}^{(1,1,1)}$ (54)

with tetralogarithms of second and sixth roots of unity neatly cancelling
[25].

Relation (54) is empirical. Like Figure 2, it relates a completely massive
elliptic tadpole to cases with zero mass that have non-elliptic routes. In the
course of the current work, I checked (54) at 200-digit precision.
011010 100101 110010

Figure 3: Binary tadpoles in relation (52)

###  4.4 Number fields of the alphabets of tadpoles

So far, one might guess that a tadpole with rational masses evaluates to
multiple tetralogarithms in an alphabet whose number field is no larger than
the compositum of the quadratic number fields associated by Gunnar Källén to
the vertices of the tetrahedron, namely the field
$Q(\Delta_{1,3,4},\Delta_{2,3,5},\Delta_{1,2,6},\Delta_{4,5,6})$. Yet that is
not the case.

Binary tadpoles evaluate to multiple polylogarithms in an alphabet containing
sixth roots of unity, with $\lambda=(1+\sqrt{-3})/2$ invariably appearing if
three massive edges meet at a vertex, where $\Delta_{i,j,k}=\sqrt{-3}$. For
example, with 5 unit edges

$F_{(1,1,1)}^{(1,1,0)}=\frac{109}{6}\left(\frac{\pi}{3}\right)^{4}+16\Re\left(\frac{{\rm
Li}_{2}^{2}(\lambda)}{6}+\sum_{m>n>0}\frac{\lambda^{3m+2n}}{m^{3}n}\right).$
(55)

So far, so good. Yet a counterexample to the idea that the Källén number field
is large enough to furnish the alphabet of a specific tadpole is provided by
relation (52) between the tadpoles of Figure 3. The first contains $\Re{\rm
Li}_{2}^{2}(\lambda)$. The second is perfect and evaluates to $12\zeta_{4}$.
So far, so good. The problem is with the third tadpole. While imperfect, it
has a Källén field that is rational, with $\Delta_{i,j,k}\in\\{0,1\\}$. Yet
its alphabet must involve $\lambda$, because of relation (52).

Faced with this rather limited, yet potent, evidence, I arrive at three
suggestions, each too weak to be dignified as a well-tested conjecture.

  1. 1.

Every tetrahedral tadpole with rational masses reduces to multiple or single
tetralogarithms whose alphabet lies in an algebraic number field.

  2. 2.

If the tadpole is perfect, the alphabet is rational.

  3. 3.

If the tadpole is imperfect, the alphabetic field may include the Källén
field.

As an experimentum crucis, I evaluated the totally massive imperfect tadpole
$F_{(\frac{1}{2},\frac{1}{2},\frac{1}{2})}^{(1,1,1)}$ with Källén field
$Q(\sqrt{-3})$. Seeking a reduction to multiple tetralogarithms in an alphabet
$\\{0,1,-1,-2,\lambda\\}$, with $\lambda=(1+\sqrt{-3})/2$, I achieved an
empirical evaluation

$F_{(\frac{1}{2},\frac{1}{2},\frac{1}{2})}^{(1,1,1)}=3\zeta_{3}\log(2)-4U_{3,1}+10\zeta_{4}+10{\rm
Cl}_{2}^{2}(\pi/3)-\tfrac{1}{2}\widehat{G}(\tfrac{1}{2})$ (56)

with a Clausen value ${\rm Cl}_{2}(\pi/3)=\Im{\rm Li}_{2}(\lambda)$, from the
Källén field, and a perfect tadpole $\widehat{G}(\frac{1}{2})$ given by
(50,51), in the rational alphabet $\\{0,1,-1,-2\\}$. Then, thanks to Gauss, it
took less than a minute to validate (56) at 600-digit precision.

##  5 Tests and benchmarks

By construction, the real derivative $\sigma^{\prime}(s)$ is regular at all
points on the cut apart from the thresholds
$\\{s_{1,2},s_{4,5},s_{2,3,4},s_{1,3,5}\\}$. This regularity is explicit in
(11) and (18). Any anomalous term in (22) is regular since, in such a case,
$s_{-}<s_{1,2}\leq s_{4,5}$ when $s_{-}$ is real. The three-particle
thresholds are benign, since $\sigma_{i,j,k}^{\prime}(s_{i,j,k})=0$. Near two-
particle thresholds, moderate a $1/\sqrt{x}$ singularity with $x=s-s_{j,k}$ by
using $y=\sqrt{x}$ as a local integration variable. It remains to consider
integration at large $s$.

I used lindep in Pari/GP to determine and record ornate rational polynomials
in 5 masses and 5 logs that give further terms in the $1/s$ expansion (23).
These are linear in logs with a mass dependence subject to symmetry
constraints. At order $1/s^{5}$, lindep found a relation with 170 non-zero
integers. By filing this exact rational data, I can obtain $5N$ good decimal
digits of dispersion integrals while integrating numerically only up to
$s=10^{N}s_{\rm R}$, where $s_{\rm R}$ is the rightmost threshold.

###  5.1 Tests

There are strong tests of the comprehensive formulæ that I have given.

  1. 1.

Elliptic terms must not depend on the order of phase-space integrations [49].

  2. 2.

The derivative of the discontinuity of a kite must satisfy the sum rule [4]

$\int_{s_{\rm L}}^{\infty}{\rm
d}s\,\sigma^{\prime}(s)\log\left(\frac{s}{s_{\rm L}}\right)=6\zeta_{3}.$ (57)

  3. 3.

The high energy behaviour (23) must hold irrespective of anomalous thresholds.

  4. 4.

The same tadpole must be obtained by integrating over 6 distinct kites.

These tests were invariably passed, at high precision, in a plethora of cases.

###  5.2 Benchmarks for kites

Benchmark 1: A useful benchmark was established by Stefan Bauberger and
Manfred Böhm [50], who gave 6 decimal digits of $B_{1}=I(50+{\rm
i}\epsilon;1,2,3,4,5)/50$, with all 4 cuts opened. This case is on the
borderline of being anomalous, with (13) satisfied as an equality and
$s_{-}=s_{1,2}$. The non-elliptic contributions come from

$\displaystyle\Delta_{1,2}(s)\sigma_{1,2}^{\prime}(s)$
$\displaystyle\to\left(s-\frac{1}{s}\right)\Re
D_{4,5}(s)+L_{4,5}+\frac{D_{4,5}(0)}{s}$ (58)
$\displaystyle\Delta_{4,5}(s)\sigma_{4,5}^{\prime}(s)$
$\displaystyle\to\left(s-\frac{1}{s}\right)\Re
D_{1,2}(s)+L_{1,2}+\frac{D_{1,2}(0)}{s}-\frac{2\pi(m_{1}+m_{2})m_{3}}{s-(m_{1}+m_{2})^{2}}$
(59)

with an arctangent responsible for $\pi$ in (59). For $B_{1}$, I obtain the
value
+0.173901219069555460362391997806756419040779085211744093645075
-0.118080028202009293890731446888246675922194086181504660940640*I
thereby extending the 6-digit result in [50] to 60 digits.

Benchmark 2: Stephen Martin [51] computed $B_{2}=-I(10+{\rm
i}\epsilon;1,3,5,2,4)/10$, in a non-anomalous case with only one open cut. For
$B_{2}$, I obtain the value
+0.718335353533534129653528554796276560425262176802655670356407
+0.390162199972762321424365961074218884677858368327292408622989*I
confirming the 8-digit result in [51].

###  5.3 Benchmarks for tadpoles

$\frac{5}{6}$$\frac{1}{3}$1$\frac{1}{6}$$\frac{2}{3}$$\frac{1}{2}$
$\frac{1}{2}$11$\frac{1}{2}$1$\frac{1}{2}$
1$\frac{1}{2}$$\frac{1}{2}$1$\frac{1}{2}$1

Figure 4: Tadpoles for benchmarks $B_{3}$, $B_{4}$ and $B_{5}$

I have chosen the benchmarks of Figure 4 as ambitious targets for adept users
of packages such as HyperInt [52, 53, 54]. Each is determined by a single
integral of a dilogarithm multiplying a derivative $\sigma^{\prime}(s)$ that
contains both elliptic and logarithmic terms. With efficient coding of the AGM
process, I obtain 200 good digits in a few seconds. Here I list only 60 of
those digits. In each case the scale of dimensional regularization in (5) is
$\overline{m}=1$, the largest mass.

Benchmark 3: The first example in Figure 4 is the simplest perfect tadpole
with 6 distinct non-zero rational masses. I suggest that its alphabet may be
rational. For its finite part $B_{3}=\widehat{F}(\frac{5}{6},\frac{1}{3})$, I
obtain
13.3861455348739022697615450327228552185248654855497464708212

Benchmark 4: The second example was conjecturally evaluated in (56). The
benchmark for $B_{4}=F_{(\frac{1}{2},\frac{1}{2},\frac{1}{2})}^{(1,1,1)}$ is
16.6059542811980228081648880073141697347243824321176643541089

Benchmark 5: The third example has two imperfections. I suggest that its
alphabetic field may include $Q(\sqrt{-3},\sqrt{5})$. The benchmark for
$B_{5}=F^{(\frac{1}{2},\frac{1}{2},\frac{1}{2})}_{(1,1,1)}$ is
16.5896999071871022548891317280131669711968061643643361121466

##  6 Comments and summary

My dominant impression in this work is of the contrast between relatively easy
generality and more demanding specificity in perturbative quantum field
theory.

Generic 5-mass kites and 6-mass tadpoles present few conceptual problems: 54
years ago, Don Perkins taught me that experimental data on the decay $\eta\to
3\pi$ is best displayed on a Dalitz plot [55] whose area is an elliptic
integral. Later that year, Gabriel Barton taught me that this elliptic
integral gives the discontinuity of a two-loop two-point function with three
internal massive particles. It was not hard to see how to extend that to the
kite with 5 masses. Diligent algebra and use of Cauchy’s integral theorem
leads to the generic formulæ of Section 2, which I strove to reduce to their
essentials, in a compact manner that leads to efficient coding, thanks to the
AGM of Gauss for complete elliptic integrals, whatever their kind.

There remained one general problem of principle. The kite has no anomalous
thresholds, yet the triangles that it contains may have dispersion relations
with anomalous thresholds [48] and complex branch points. Here I was both
challenged and encouraged by Spencer Bloch, who knew from algebraic geometry
that I was skating on thin ice, by taking principal values to obtain a
discontinuity as the real part of a combination of multi-sheeted complex
functions. Memories of Stanley Mandelstam’s ominous drawing in [56], depicting
contour deformation, came to haunt me. Yet I was protected by Gabriel Barton,
who had provided the key:

  1. 1.

If you know the discontinuity $\sigma$ of $f$, use a dispersion relation to
get $f$.

  2. 2.

If you know only $\sigma^{\prime}$, integrate it against a log.

  3. 3.

For the 4th-order photon propagator, $\sigma^{\prime}$ has logs, so $f$ has
Lewin’s tri-logs.

  4. 4.

For the 4th-order electron propagator, $\sigma^{\prime}$ is elliptic, so $f$
is harder to get.

  5. 5.

Källén did the easier case with Sabry and then left the harder case for Sabry.

Working with $\sigma^{\prime}$, I was confident that any anomalous term would
be algebraic, not an integral. Agreement with Gauss, at high energy, gave the
rule for the anomalous coefficient $C_{\rm A}\in\\{-1,0,1\\}$ in (22), without
having to deform any contour.

Turning to specifics, I remark how much more work one needs to do when masses
tend towards zero and thresholds collide. The algebra in Section 3 was
demanding and was enabled by efficient partial fractioning and multivariate
factorization in Tony Hearn’s open-source REDUCE, which is freely available
[57]. The rest of the work was done in Pari/GP, which is also freely available
[58]. There was no need to pay for expensive proprietary software. Nor was
hardware a problem. A modest laptop delivers 200 digits of a kite or tadpole
in a few seconds.

In summary:

  1. 1.

Elliptic substructure of 2-loop kites and 3-loop tadpoles is not a problem.
The time taken to evaluate a complete elliptic integral, of whatever kind, is
commensurate with the time for a logarithm and less than the time for a
dilogarithm. Thanks to Gauss, elliptic integrals should be embraced, not
feared.

  2. 2.

Anomalous terms are not problematic. They submit to Gauss, at high energy.

  3. 3.

The number theory of tadpoles is subtle. They may be polylogarithmic, even in
totally massive cases such as (50), to which every route is elliptic.

  4. 4.

Subsection 4.4 gives far-reaching suggestions on the number theory of
tadpoles.

  5. 5.

Subsection 5.3 gives demanding 60-digit benchmarks for ambitious users of
HyperInt to investigate those suggestions analytically.

### Acknowledgements

First and foremost, I thank the departed dead. The influences of Gabriel
Barton (1934-2022) and Don Perkins (1925-2022) have already been made clear. I
thank the brothers Borwein, Jonathan (1951-2016) and Peter (1953-2020), for
their fine monograph on the AGM, and Freeman Dyson (1924-2020) for encouraging
my efforts to relate quantum field theory to number theory. Turning to the
living, I thank Dirk Kreimer and his entire Arbeitsgruppe, who made my many
visits to Berlin so enjoyable and profitable. I thank mathematical advisors
for their patient instruction. In particular, the influence of Spencer Bloch,
Francis Brown, Pierre Deligne and Don Zagier extends deeply into the work
reported here, in both the polylogarithmic and elliptic sectors. I am grateful
to generous hosts for their invitations to the 10 recent workshops and
conferences mentioned in the introduction. I thank Yajun Zhou, for
communications on HyperInt, Kevin Acres and Michael St Clair Oakes, for close
scrutiny of a draft version, and William Edward Makin, for 60 years of
discussion of natural philosophy and its firm relation to other human
endeavours.

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