Empirical determinations of Feynman integrals
using integer relation algorithms
Kevin Acres111Cryptography Research Centre, Technology Innovation Institute,
Abu Dhabi, UAE,
<EMAIL_ADDRESS>and David Broadhurst222School of Physical Sciences, Open
University, Milton Keynes MK7 6AA, UK,
<EMAIL_ADDRESS>10 March 2021

Abstract: Integer relation algorithms can convert numerical results for
Feynman integrals to exact evaluations, when one has reason to suspect the
existence of reductions to linear combinations of a basis, with rational or
algebraic coefficients. Once a tentative reduction is obtained, confidence in
its validity is greatly increased by computing more decimal digits of the
terms and verifying the stability of the result. Here we give examples of how
the PSLQ and LLL algorithms have yielded remarkable reductions of Feynman
integrals to multiple polylogarithms and to the periods and quasi-periods of
modular forms. Moreover, these algorithms have revealed quadratic relations
between Feynman integrals. A recent application concerning black holes
involves quadratic relations between combinations of Feynman integrals with
algebraic coefficients.

##  1 Introduction

The mathematical problem at hand is easy to state: given numerical
approximations to $n>2$ real numbers, $x_{k}$, is there at least one probable
relation $\sum_{k=1}^{n}z_{k}x_{k}=0$ with integer coefficients $z_{k}$, at
least two of which are non-zero? If so, produce such a relation.

By way of example, in 1985 Broadhurst studied periods coming from 6-loop
counterterms [7] in $\phi^{4}$ theory and found, with good confidence, the
relations

$P_{6,1}=168\zeta_{9},\quad
P_{6,2}=\frac{1063}{9}\zeta_{9}+8\zeta_{3}^{3},\quad
16P_{6,3}+P_{6,4}=1440\zeta_{5}\zeta_{3}$ (1)

between the periods $P_{6,k}$, as labeled in the later census by Schnetz [25],
and Riemann zeta values $\zeta_{s}=\sum_{n>0}1/n^{s}$. There was a strong
intuition that $P_{6,3}$ and $P_{6,4}$ might involve the multiple zeta value
(MZV)

$\zeta_{5,3}=\sum_{m>n>0}\frac{1}{m^{5}n^{3}}=0.03770767298484754401130478\ldots$
(2)

along with $\zeta_{8}$ and the product $\zeta_{5}\zeta_{3}$. Yet such
relations were not discovered, at the level of accuracy then attainable.

A decade later, Broadhurst and Kreimer [8] solved this problem, by improving
accuracy for the periods and using the PSLQ algorithm developed by Ferguson
and Bailey [18], which identified

$P_{6,3}=\mbox{$\frac{36}{5}$}\left(12\zeta_{5,3}-29\zeta_{8}\right)+252\zeta_{5}\zeta_{3}.$
(3)

Moreover, they found $\zeta_{3,5,3}$, with weight 11 and depth 3, in some
7-loop periods.

Much experimenting with PSLQ led to the Broadhurst-Kreimer conjecture [9] that
the number $N(w,d)$ of primitive MZVs of weight $w$ and depth $d$ is generated
by

$\prod_{w>2}\prod_{d>0}(1-x^{w}y^{d})^{N(w,d)}=1-\frac{x^{3}y}{1-x^{2}}+\frac{x^{12}y^{2}(1-y^{2})}{(1-x^{4})(1-x^{6})}$
(4)

with a final term inferred by relating MZVs to alternating sums.

##  2 PSLQ and LLL

PSLQ came from work by Helaman Ferguson and Rodney Forcade [17] in 1977,
implemented in multiple-precision fortran by David Bailey in 1992, improved
and parallelized by Bailey and Broadhurst [2] in 1999, and used extensively in
the study of Feynman integrals since then.

The algorithm proceeds as follows. First we initialize:

  1. 1.

For $j:=1$ to $n$: for $i:=1$ to $n$: if $i=j$ then set $A_{ij}:=1$ and
$B_{ij}:=1$ else set $A_{ij}:=0$ and $B_{ij}:=0$; endfor; endfor.

  2. 2.

For $k:=1$ to $n$: set $s_{k}:={\bf
sqrt}\left(\sum_{j=k}^{n}x_{j}^{2}\right)$; endfor. Set $t=1/s_{1}$.
For $k:=1$ to $n$: set $y_{k}:=tx_{k};\;s_{k}:=ts_{k}$; endfor.

  3. 3.

For $j:=1$ to $n-1$: for $i:=1$ to $j-1$: set $H_{ij}:=0$; endfor;
set $H_{jj}:=s_{j+1}/s_{j}$; for $i:=j+1$ to $n$: set
$H_{ij}:=-y_{i}y_{j}/(s_{j}s_{j+1})$; endfor; endfor.

  4. 4.

For $i:=2$ to $n$: for $j:=i-1$ to $1$ step $-1$: set $t:={\bf
round}(H_{ij}/H_{jj})$; $y_{j}:=y_{j}+ty_{i}$; for $k:=1$ to $j$: set
$H_{ik}:=H_{ik}-tH_{jk}$; endfor;
for $k:=1$ to $n$: set $A_{ik}:=A_{ik}-tA_{jk}$, $B_{kj}:=B_{kj}+tB_{ki}$;
endfor; endfor; endfor.

Thus the numerical data, in the vector $x$, is converted to a vector $y$, by
taking square roots of partial sums of squares. Then the matrices $H$, $A$ and
$B$ are created, with a crucial rounding in Step 4.

Then we proceed by iteration:

  1. 1.

Select $m$ such that $({\bf 4/3})^{i/2}|H_{ii}|$ is maximal when $i=m$. Swap
the entries of $y$ indexed $m$ and $m+1$, the corresponding rows of $A$ and
$H$, and the corresponding columns of $B$.

  2. 2.

If $m\leq n-2$ then set $t_{0}:={\bf sqrt}(H_{mm}^{2}+H_{m,m+1}^{2})$,
$t_{1}:=H_{mm}/t_{0}$ and $t_{2}:=H_{m,m+1}/t_{0}$; for $i:=m$ to $n$: set
$t_{3}:=H_{im}$, $t_{4}:=H_{i,m+1}$, $H_{im}:=t_{1}t_{3}+t_{2}t_{4}$ and
$H_{i,m+1}:=-t_{2}t_{3}+t_{1}t_{4}$; endfor; endif.

  3. 3.

For $i:=m+1$ to $n$: for $j:=\min(i-1,m+1)$ to $1$ step $-1$: set $t:={\bf
round}(H_{ij}/H_{jj})$ and $y_{j}:=y_{j}+ty_{i}$; for $k:=1$ to $j$: set
$H_{ik}:=H_{ik}-tH_{jk}$; endfor; for $k:=1$ to $n$: set
$A_{ik}:=A_{ik}-tA_{jk}$ and $B_{kj}:=B_{kj}+tB_{ki}$; endfor; endfor; endfor.

  4. 4.

If the largest entry of $A$ exceeds the precision, then fail, else if a
component of the $y$ vector is very small, then output the relation from the
corresponding column of $B$, else go back to Step 1.

The constant $4/3$ in the swap of Step 1 of the iteration ensures that the
algorithm will find a relation, provided that one exists and the data have
been specified to sufficient accuracy. In practice, reliable results may be
obtained with a smaller constant.

For big problems, parallelized PSLQ [2] has been vital, especially for the
magnetic moment of the electron [21]. For smaller problems, there is an
alternative.

###  2.1 LLL

In 1982, Arjen Lenstra, Hendrik Lenstra and László Lovász gave the LLL
algorithm [22] for lattice reduction to a basis with short and almost
orthogonal components. An extension of this underlies the lindep procedure in
Pari-GP [24], which we here apply to the problem of determining the period
$P_{6,3}$.

    P63=107.71102484102;\\ only 14 digits specified
    V=[P63,zetamult([5,3]),zeta(8),zeta(5)*zeta(3)];
    for(d=11,16,U=lindep(V,d);U*=sign(U[1]);print([d,U~]));
    [11, [4, -827, 173, -460]]
    [12, [4, -827, 173, -460]]
    [13, [4, -827, 173, -460]]
    [14, [5, -432, 1044, -1260]]
    [15, [5, -432, 1044, -1260]]
    [16, [196, 1652, -9045, -9701]]

In this case, using 14 good digits of $P_{6,3}$, we happen to obtain the
correct result (3). In practice, one would need several more digits, for
better confidence. In what follows, all claimed relations have been checked
using at least 100 more digits than were needed for the discoveries, making
the probability of a mistake less than $1/10^{100}$.

###  2.2 Improvement and parallelization of PSLQ

Multi-level improvement: perform most operations at 64-bit precision, some at
intermediate precision (Bailey and Broadhurst [2] chose 125 digits) and only
the bare minimum of the most delicate operations at full precision (more than
10000 digits, for some big problems).

Multi-pair improvement: swap up to $0.4n$ disjoint pairs of the $n$ indices at
each iteration. In this case, it is not proven that the algorithm will
succeed, but it has not yet been found to fail.

Parallelization: distribute the disjoint-pair jobs; for each pair, distribute
the full-precision matrix multiplication in the outermost loop.

###  2.3 Examples

Bailey and Broadhurst [2], working at 10000 digits, found that the constant
associated with the fourth bifurcation of the logistic map is the root of a
polynomial of degree 240.

They tested a conjecture on alternating sums of the form

$\zeta\left(\begin{array}[]{cccc}\sigma_{1},&\sigma_{2}&\cdots&\sigma_{d}\\\
s_{1},&s_{2}&\cdots&s_{d}\end{array}\right)=\sum_{k_{1}>k_{2}>\cdots>k_{d}>0}\frac{\sigma_{1}^{k_{1}}}{k_{1}^{s_{1}}}\,\frac{\sigma_{2}^{k_{2}}}{k_{2}^{s_{2}}}\,\cdots\,\frac{\sigma_{d}^{k_{d}}}{k_{d}^{s_{d}}}$
(5)

where $\sigma_{j}=\pm 1$ are signs and $s_{j}>0$ are integers, namely that at
weight $w=\sum_{j}s_{j}$ every convergent alternating sum is a rational linear
combination of elements of a basis of size $F_{w+1}=F_{w}+F_{w-1}$, i.e. the
Fibonacci number with index $w+1$. At $w=11$, many integer relations of size
$F_{12}+1=145$ were found, at 5000-digit precision.

For weights $w\leq 20$, inverse binomial sums [6] of the form

$S(w)=\sum_{n=1}^{\infty}\frac{1}{n^{w}{2n\choose n}}$ (6)

were reduced to multiple polylogarithms of the sixth root of unity [10], with
$S(20)$ given by 106 terms.

###  2.4 Relations in the multiple zeta value data mine

The Broadhurst-Kreimer conjecture (4) came from the PSLQ discovery that

$\displaystyle 2^{5}\cdot 3^{3}\zeta_{4,4,2,2}$ $\displaystyle=$
$\displaystyle 2^{14}\sum_{m>n>0}\frac{(-1)^{m+n}}{(m^{3}n)^{3}}+2^{5}\cdot
3^{2}\,\zeta_{3}^{4}+2^{6}\cdot 3^{3}\cdot 5\cdot 13\,\zeta_{9}\,\zeta_{3}$
(7) $\displaystyle{}+2^{6}\cdot 3^{3}\cdot 7\cdot
13\,\zeta_{7}\,\zeta_{5}+2^{7}\cdot
3^{5}\,\zeta_{7}\,\zeta_{3}\,\zeta_{2}+2^{6}\cdot
3^{5}\,\zeta_{5}^{2}\,\zeta_{2}$ $\displaystyle{}-2^{6}\cdot 3^{3}\cdot 5\cdot
7\,\zeta_{5}\,\zeta_{4}\,\zeta_{3}-2^{8}\cdot
3^{2}\,\zeta_{6}\,\zeta_{3}^{2}-\mbox{$\frac{13177\times
15991}{691}$}\,\zeta_{12}$ $\displaystyle{}+2^{4}\cdot 3^{3}\cdot 5\cdot
7\,\zeta_{6,2}\,\zeta_{4}-2^{7}\cdot 3^{3}\,\zeta_{8,2}\,\zeta_{2}-2^{6}\cdot
3^{2}\cdot 11^{2}\zeta_{10,2}$

which shows that, at weight 12, a depth 4 MZV is reducible to terms of depth
$d\leq 2$, and their products, if one allows an alternating double sum in the
basis.

When constructing the MZV data mine, Blümlein, Broadhurst and Vermaseren [4]
proved this, by massive use of computer algebra. It would be much harder to
prove an LLL discovery at weight 21 and depth 7, where

$81\zeta_{6,2,3,3,5,1,1}+326\sum_{j>k>l>m>n>0}\frac{(-1)^{k+m}}{(jk^{2}lm^{2}n)^{3}}$
(8)

was reduced to 150 terms containing MZVs of depths $d\leq 5$.

##  3 Counterterms at 7 loops

Broadhurst found reductions to MZVs for a pair of 7-loop periods [11]

$\displaystyle P_{7,8}$ $\displaystyle=$
$\displaystyle\frac{22383}{20}\zeta_{11}+\frac{4572}{5}\left(\zeta_{3,5,3}-\zeta_{3}\zeta_{5,3}\right)-700\zeta_{3}^{2}\zeta_{5}$
(9)
$\displaystyle\quad+\,1792\zeta_{3}\left(\frac{9}{320}\left(12\zeta_{5,3}-29\zeta_{8}\right)+\frac{45}{64}\zeta_{5}\zeta_{3}\right)$
$\displaystyle P_{7,9}$ $\displaystyle=$
$\displaystyle\frac{92943}{160}\zeta_{11}+\frac{3381}{20}\left(\zeta_{3,5,3}-\zeta_{3}\zeta_{5,3}\right)-\frac{1155}{4}\zeta_{3}^{2}\zeta_{5}$
(10)
$\displaystyle\quad+\,896\zeta_{3}\left(\frac{9}{320}\left(12\zeta_{5,3}-29\zeta_{8}\right)+\frac{45}{64}\zeta_{5}\zeta_{3}\right)$

that had been expected to involve alternating sums.

These results were later proven, one by the methods of Erik Panzer [23] and
the other by the methods of Oliver Schnetz [26]. Their methods yielded
complicated combinations of alternating sums, which were then reduced to the
MZV formulas (9,10) by use of proven results in the MZV data mine [4].

The period $P_{7,11}$ in the census of Schnetz [25] is much more demanding.
All other periods up to 7 loops reduce to MZVs; only $P_{7,11}$ requires
multiple polylogarithms of sixth roots of unity, of the form (5) with
$\sigma_{j}^{6}=1$.

Panzer evaluated $\sqrt{3}P_{7,11}$ in terms of 4589 such sums, each of which
he evaluated to 5000 digits. Then he found an empirical reduction to a
72-dimensional basis. The rational coefficient of $\pi^{11}$ in his result was
[23]

$C_{11}=-\frac{964259961464176555529722140887}{2733669078108291387021448260000}$
(11)

whose denominator contains 8 primes greater than 11, namely 19, 31, 37, 43,
71, 73, 50909 and 121577.

Using LLL, it was possible to find a much better basis, with no prime greater
3 in the denominator of any coefficient. Let $A={\rm d}\log(x)$, $B=-{\rm
d}\log(1-x)$ and $D=-{\rm d}\log(1-\exp(2\pi{\rm i}/6)x)$ be letters, forming
words $W$ that define iterated integrals $Z(W)$. Let

$W_{m,n}=\sum_{k=0}^{n-1}\frac{\zeta_{3}^{k}}{k!}A^{m-2k}D^{n-k},$ (12)

$P_{n}=(\pi/3)^{n}/n!$, $I_{n}={\rm Cl}_{n}(2\pi/3)$ and $I_{a,b}=\Im
Z(A^{b-a-1}DA^{2a-1}B)$. Using

$\displaystyle I_{2,9}$ $\displaystyle=$ $\displaystyle
91(11T_{2,9})-898T_{3,8}+11I_{4,7}-292P_{11}$ (13) $\displaystyle I_{3,8}$
$\displaystyle=$ $\displaystyle 24(11T_{2,9})+841T_{3,8}-190I_{4,7}-255P_{11}$
(14)

to transform to $T_{2,9}$ and $T_{3,8}$, the result becomes

$\displaystyle\sqrt{3}P_{7,11}$ $\displaystyle=$ $\displaystyle-10080\Im
Z(W_{7,4}+W_{7,2}P_{2})+50400\zeta_{3}\zeta_{5}P_{3}$ (15)
$\displaystyle{}+\left(35280\Re
Z(W_{8,2})+\frac{46130}{9}\zeta_{3}\zeta_{7}+17640\zeta_{5}^{2}\right)P_{1}$
$\displaystyle{}-13277952T_{2,9}-7799049T_{3,8}+\frac{6765337}{2}I_{4,7}-\frac{583765}{6}I_{5,6}$
$\displaystyle{}-\frac{121905}{4}\zeta_{3}I_{8}-93555\zeta_{5}I_{6}-102060\zeta_{7}I_{4}-141120\zeta_{9}I_{2}$
$\displaystyle{}+\frac{42452687872649}{6}P_{11}.$

##  4 Periods and quasi-periods in electrodynamics

The magnetic moment of the electron, in Bohr magnetons, has quantum
electrodynamic contributions $\sum_{L=0}^{4}a_{L}(\alpha/\pi)^{L}$ given up to
$L=4$ loops by [21]

$\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\phantom{-}1\phantom{.5}\quad[{\rm Dirac},\,1928]$ (16)
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle\phantom{-}0.5\quad[{\rm
Schwinger},\,1947]$ (17) $\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle-0.32847896557919378458217281696489239241111929867962\ldots$
(18) $\displaystyle a_{3}$ $\displaystyle=$
$\displaystyle\phantom{-}1.18124145658720000627475398221287785336878939093213\ldots$
(19) $\displaystyle a_{4}$ $\displaystyle=$
$\displaystyle-1.91224576492644557415264716743983005406087339065872\ldots$
(20)

In 1957, corrections by Petermann and by Sommerfield resulted in

$a_{2}=\frac{197}{144}+\frac{\zeta_{2}}{2}+\frac{3\zeta_{3}-2\pi^{2}\log
2}{4}\,.$ (21)

In 1996, Laporta and Remiddi obtained

$\displaystyle a_{3}$ $\displaystyle=$
$\displaystyle\frac{28259}{5184}+\frac{17101\zeta_{2}}{135}+\frac{139\zeta_{3}-596\pi^{2}\log
2}{18}$ (22)
$\displaystyle{}-\frac{39\zeta_{4}+400U_{3,1}}{24}-\frac{215\zeta_{5}-166\zeta_{3}\zeta_{2}}{24}\,.$

The 3-loop contribution contains a weight-4 depth-2 alternating sum

$U_{3,1}=\sum_{m>n>0}\frac{(-1)^{m+n}}{m^{3}n}=\frac{\zeta_{4}}{2}+\frac{(\pi^{2}-\log^{2}2)\log^{2}2}{12}-2\sum_{n>0}\frac{1}{2^{n}n^{4}}.$
(23)

Equally fascinating is the Bessel moment [14]

$B=-\int_{0}^{\infty}\frac{27550138t+35725423t^{3}}{48600}I_{0}(t)K_{0}^{5}(t){\rm
d}t$ (24)

in the evaluation by Laporta [21], at 4800 digits, of

$a_{4}=P+B+E+U\approx 2650.565-1483.685-1036.765-132.027\approx-1.912$ (25)

where $P$ comprises multiple polylogs, $E$ comprises integrals whose
integrands contain logs and products of elliptic integrals and $U$ comes from
6 light-by-light integrals, still under investigation.

###  4.1 Bessel moments and modular forms

Gauss noted on 30 May 1799 that the lemniscate constant

$\int_{0}^{1}\frac{{\rm
d}x}{\sqrt{1-x^{4}}}=\frac{(\Gamma(1/4))^{2}}{4\sqrt{2\pi}}=\frac{\pi/2}{{\rm
agm}(1,\sqrt{2})}$ (26)

is given by the reciprocal of an arithmetic-geometric mean. This is an example
of the Chowla-Selberg formula at the first singular value [5]. In 1939, Watson
[27] encountered the sixth singular value, in work on integrals from condensed
matter physics. Here, $\left(\sum_{n\in{\bf Z}}\exp(-\sqrt{6}\pi
n^{2})\right)^{4}$ gives the product of $\Gamma(k/{24})$ with $k=1,5,7,11$, as
observed by Glasser and Zucker [20] in 1977. In 2007, Broadhurst and Laporta
identified a Feynman period at the fifteenth singular value [3], where
$\left(\sum_{n\in{\bf Z}}\exp(-\sqrt{15}\pi n^{2})\right)^{4}$ gives the
product of $\Gamma(k/{15})$ with $k=1,2,4,8$.

With $N=a+b$ Bessel functions and $c\geq 0$, we define moments

$M(a,b,c)=\int_{0}^{\infty}I_{0}^{a}(t)K_{0}^{b}(t)t^{c}{\rm d}t$ (27)

that converge for $b>a\geq 0$. Then the 5-Bessel matrix

$\left[\begin{array}[]{lr}M(1,4,1)&M(1,4,3)\\\
M(2,3,1)&M(2,3,3)\end{array}\right]\,=\,\left[\begin{array}[]{lr}\pi^{2}C&\pi^{2}\left(\frac{2}{15}\right)^{2}\left(13C-\frac{1}{10C}\right)\\\
\frac{\sqrt{15}\pi}{2}C&\frac{\sqrt{15}\pi}{2}\left(\frac{2}{15}\right)^{2}\left(13C+\frac{1}{10C}\right)\end{array}\right]$
(28)

involves a single new constant

$C=\frac{\pi}{16}\left(1-\frac{1}{\sqrt{5}}\right)\left(\sum_{n=-\infty}^{\infty}\exp(-\sqrt{15}\pi
n^{2})\right)^{4}=\frac{1}{240\sqrt{5}\pi^{2}}\prod_{k=0}^{3}\Gamma\left(\frac{2^{k}}{15}\right)$
(29)

and its reciprocal. The determinant $2\pi^{3}/\sqrt{3^{3}5^{5}}$ of matrix
(28) is an algebraic multiple of a power of $\pi$. This is an example of an
all-loop result discovered by Broadhurst and Mellit [12] and proven by Yajun
Zhou [28].

The L-series for $N=5$ Bessel functions comes from a modular form of weight 3
and level 15:

$\displaystyle\eta_{n}$ $\displaystyle=$ $\displaystyle
q^{n/24}\prod_{k>0}(1-q^{nk}),\quad q=\exp(2\pi{\rm i}\tau),$ (30)
$\displaystyle f_{3,15}(\tau)$ $\displaystyle=$
$\displaystyle(\eta_{3}\eta_{5})^{3}+(\eta_{1}\eta_{15})^{3}=\sum_{n>0}A_{5}(n)q^{n}$
(31) $\displaystyle L_{5}(s)$ $\displaystyle=$
$\displaystyle\sum_{n>0}\frac{A_{5}(n)}{n^{s}}\quad{\rm for}~{}s>2$ (32)
$\displaystyle L_{5}(1)$ $\displaystyle=$
$\displaystyle\sum_{n>0}\frac{A_{5}(n)}{n}\left(2+\frac{\sqrt{15}}{2\pi
n}\right)\exp\left(-\frac{2\pi n}{\sqrt{15}}\right)$ (33) $\displaystyle=$
$\displaystyle
5C\,=\,\frac{5}{\pi^{2}}\int_{0}^{\infty}I_{0}(t)K_{0}^{4}(t)t{\rm d}t\,.$
(34)

###  4.2 Periods and quasi-periods for the Laporta problem

Laporta’s 4-loop work [21] engages the first row of the 6-Bessel determinant

${\rm det}\left[\begin{array}[]{lr}M(1,5,1)&M(1,5,3)\\\
M(2,4,1)&M(2,4,3)\end{array}\right]=\frac{5\zeta_{4}}{32}$ (35)

associated to a modular form
$f_{4,6}(\tau)=(\eta_{1}\eta_{2}\eta_{3}\eta_{6})^{2}$ with weight 4 and level
6 [11, 29]. At top left we have $M(1,5,1)$, from the on-shell 4-loop sunrise
diagram, in two spacetime dimensions. Below it, $M(2,4,1)$ comes from cutting
an internal line. The second column comes from differentiating the first, with
respect to the external momentum, to produce quasi-periods associated with a
weakly holomorphic modular form

$\widehat{f}_{4,6}(\tau)=\mu{f}_{4,6}(\tau),\quad\mu=\frac{1}{32}\left(w+\frac{3}{w}\right)^{4}-\frac{9}{16}\left(w+\frac{3}{w}\right)^{2},\quad
w=\frac{3\eta_{3}^{4}\eta_{2}^{2}}{\eta_{1}^{4}\eta_{6}^{2}}.$ (36)

With $s=1,2$, we computed compute 10,000 digits of the Eichler integrals

$\frac{\Omega_{s}}{(2\pi)^{s}}=\int_{1/\sqrt{3}}^{\infty}{f}_{4,6}\left(\frac{1+{\rm
i}y}{2}\right)y^{s-1}dy,\quad\frac{\widehat{\Omega}_{s}}{(2\pi)^{s}}=\int_{1/\sqrt{3}}^{\infty}\widehat{f}_{4,6}\left(\frac{1+{\rm
i}y}{2}\right)y^{s-1}dy.$ (37)

###  4.3 Laporta’s intersection number

LLL readily gave 4 linear relations

$\frac{2}{\pi^{2}}\left[\begin{array}[]{rr}4M(1,5,1)&\mbox{$\frac{36}{5}$}\left(M(1,5,1)+M(1,5,3)\right)\\\\[5.0pt]
\mbox{$\frac{5}{3}$}M(2,4,1)&3\left(M(2,4,1)+M(2,4,3)\right)\end{array}\right]=\left[\begin{array}[]{rr}-\Omega_{2}&\widehat{\Omega}_{2}\\\
-\Omega_{1}&\widehat{\Omega}_{1}\end{array}\right]$ (38)

between Feynman integrals, the periods $\Omega_{1,2}$ and the quasi-periods
$\widehat{\Omega}_{1,2}$.

The intersection number is the determinant of this matrix, namely $1/12$.
Broadhurst and Roberts [14] converted this into a quadratic relation between 4
hypergeometric series:

$\begin{array}[]{lclrrrrrrrrr}F_{a}&=&{}_{4}F_{3}(&1/2,&2/3,&2/3,&5/6;&7/6,&7/6,&4/3;&1)\\\
F_{b}&=&{}_{4}F_{3}(&-1/2,&1/6,&1/3,&4/3;&-1/6,&5/6,&5/3;&1)\\\
F_{c}&=&{}_{4}F_{3}(&1/6,&1/3,&1/3,&1/2;&2/3,&5/6,&5/6;&1)\\\
F_{d}&=&{}_{4}F_{3}(&-7/6,&-1/2,&-1/3,&2/3;&-5/6,&1/6,&1/3;&1)\end{array}$
(39)

namely

$7F_{a}F_{b}+10F_{c}F_{d}=40,$ (40)

which was later proven by Yajun Zhou [30].

##  5 Quadratic relations

In this section, we give quadratic relations between Feynman integrals,
recently discovered by using the LLL algorithm. If one has $n$ integrals,
there are $n(n+1)/2$ products to consider, in linear combinations with
rational or algebraic coefficients, which may give a rational or algebraic
multiple of a power of $\pi$. This problem soon explodes. We begin with a
conjecture obtained after intensive use of LLL and tested with up to $n=100$
Feynman integrals.

Conjecture: (Broadhurst and Roberts [14]) With the Feynman, de Rham and Betti
matrices below,

$F_{N}D_{N}F_{N}^{\tt tr}=B_{N}.$ (41)

The elements of the Feynman matrices $F_{N}$ are the Bessel moments

$\displaystyle F_{2k+1}(u,a)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{a-1}}{\pi^{u}}M(k+1-u,k+u,2a-1)$ (42) $\displaystyle
F_{2k+2}(u,a)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{a-1}}{\pi^{u+1/2}}M(k+1-u,k+1+u,2a-1)$ (43)

with $u$ and $a$, as well as later indices $v$ and $b$, running from 1 to $k$.
$F_{N}^{\tt tr}$ is the transpose of $F_{N}$.

The Betti matrices $B_{N}$ have rational elements given by

$\displaystyle B_{2k+1}(u,v)$ $\displaystyle=$
$\displaystyle(-1)^{u+k}2^{-2k-2}(k+u)!(k+v)!Z(u+v)$ (44) $\displaystyle
B_{2k+2}(u,v)$ $\displaystyle=$
$\displaystyle(-1)^{u+k}2^{-2k-3}(k+u+1)!(k+v+1)!Z(u+v+1)$ (45) $\displaystyle
Z(m)$ $\displaystyle=$ $\displaystyle\frac{1+(-1)^{m}}{(2\pi)^{m}}\zeta_{m}.$
(46)

For the de Rham matrices $D_{N}$, let $v_{k}$ and $w_{k}$ be the rationals
generated by

$\displaystyle\frac{J_{0}^{2}(t)}{C(t)}$ $\displaystyle=$
$\displaystyle\sum_{k\geq
0}\frac{v_{k}}{k!}\left(\frac{t}{2}\right)^{2k}=1-\frac{17t^{2}}{54}+\frac{3781t^{4}}{186624}+\ldots$
(47) $\displaystyle\frac{2J_{0}(t)J_{1}(t)}{tC(t)}$ $\displaystyle=$
$\displaystyle\sum_{k\geq
0}\frac{w_{k}}{k!}\left(\frac{t}{2}\right)^{2k}=1-\frac{41t^{2}}{216}+\frac{325t^{4}}{186624}+\ldots$
(48)

where $J_{0}(t)=I_{0}({\rm i}t)$, $J_{1}(t)=-J_{0}^{\prime}(t)$ and

$C(t)=\frac{32(1-J_{0}^{2}(t)-tJ_{0}(t)J_{1}(t))}{3t^{4}}=1-\frac{5t^{2}}{27}+\frac{35t^{4}}{2304}-\frac{7t^{6}}{9600}+\ldots$
(49)

Construct rational bivariate polynomials $H_{s}=H_{s}(y,z)$ by the recursion

$H_{s}=zH_{s-1}-(s-1)yH_{s-2}-\sum_{k=1}^{s-1}{s-1\choose
k}\left(v_{k}H_{s-k}-w_{k}zH_{s-k-1}\right)$ (50)

for $s>0$, with $H_{0}=1$ and $H_{-1}=0$. Use these to define

$d_{s}(N,c)=\frac{H_{s}(3c/2,N+2-2c)}{4^{s}s!}.$ (51)

Finally, construct de Rham matrices with the rational elements

$D_{N}(a,b)=\sum_{c=-b}^{a}d_{a-c}(N,-c)d_{b+c}(N,c)c^{N+1}.$ (52)

The discovery of formula (52) for the coefficients of these quadratic
relations involved intensive use of LLL, at high numerical precision. At 20
loops, there are 100 Feynman integrals to consider, with 5050 products. Javier
Fresán, Claude Sabbah and Jeng-Daw Yu [19] have verified that our formulas
hold up to 20 loops, after which they ran out of computing power. They
encountered subtleties when $N$ is divisible by 4. These are entirely avoided
by our uniform formula (52). Yajun Zhou [31] has given an illuminating
classical proof, with generalizations.

###  5.1 Quadratic relations at weight 6 and level 24

Here we establish relations between Bessel moments (27) with $a+b=6$,
$c=0,2,4$, and Eichler integrals of modular forms of weight 6 and level 24,
constructed from eta quotients. This connection was suggested by the discovery
of the linear and quadratic relations

$\displaystyle\frac{M(0,6,0)}{M(2,4,0)}=\frac{3M(0,6,2)-8M(0,6,4)}{3M(2,4,2)-8M(2,4,4)}$
$\displaystyle=$ $\displaystyle 3\pi^{2},$ (53) $\displaystyle{\rm
det}\left[\begin{array}[]{lr}M(0,6,0)&3M(0,6,2)-8M(0,6,4)\\\
M(1,5,0)&3M(1,5,2)-8M(1,5,4)\end{array}\right]$ $\displaystyle=$
$\displaystyle\frac{5\pi^{6}}{16}.$ (56)

We begin by defining three eta quotients, subject to two algebraic relations:

$\displaystyle r$ $\displaystyle=$
$\displaystyle\left(\frac{\eta_{2}\eta_{12}}{\eta_{4}\eta_{6}}\right)^{6}=\frac{s-t}{st}=9t-8s=q-6q^{3}+15q^{5}+O(q^{7}),$
(57) $\displaystyle s$ $\displaystyle=$
$\displaystyle\left(\frac{\eta_{4}\eta_{12}}{\eta_{2}\eta_{6}}\right)^{3}=q+3q^{3}+6q^{5}+O(q^{7}),$
(58) $\displaystyle t$ $\displaystyle=$
$\displaystyle\left(\frac{\eta_{6}\eta_{12}}{\eta_{2}\eta_{4}}\right)^{2}=q+2q^{3}+7q^{5}+O(q^{7}).$
(59)

The moments $M(0,6,0)$ and $M(1,5,0)$ are related to periods of the modular
form

$f_{1}(\tau)=\left(\eta_{2}\eta_{4}\eta_{6}\eta_{12}\right)^{3}\left(\frac{1}{s^{2}}-64s^{2}\right)=q-9q^{3}-34q^{5}-240q^{7}+81q^{9}+O(q^{11}).$
(60)

To find its quasi-periods, we form a column vector of 5 cusp forms

$\displaystyle{\bf f}(\tau)$ $\displaystyle=$
$\displaystyle\left(\eta_{2}\eta_{4}\eta_{6}\eta_{12}\right)^{3}\left[\begin{array}[]{lllll}1/s^{2}-64s^{2}\\\
1/s^{2}+64s^{2}\\\ 1/r^{2}-r^{2}\\\ 1/t^{2}-81t^{2}\\\
1/t^{2}+81t^{2}+54\end{array}\right]={\bf T}\left[\begin{array}[]{lllll}q\\\
q^{3}\\\ q^{5}\\\ q^{7}\\\ q^{9}\end{array}\right]+O(q^{11}),$ (71)
$\displaystyle{\bf T}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{rrrrr}1&-9&-34&-240&81\\\
1&-9&94&144&81\\\ 1&9&38&120&81\\\ 1&-7&-74&-24&-383\\\
1&47&-74&-24&697\end{array}\right],$ (77)

with the Hecke matrix ${\bf T}$ recording the first 5 non-vanishing Fourier
coefficients of the 5 modular forms. The first three components of ${\bf f}$
are new forms, while the remaining two are old forms.

Since there are 16 cusp forms of weight 6 and level 24, we began our
investigation with a far more fearsome 16-dimensional problem. After intensive
study of the relationship between Rademacher sums [1] and the determinants and
permanents [15] of matrices of periods and quasi-periods of modular forms, we
were able to reduce the Bessel-moment problem to the 5-dimensional problem
presented here.

For each of the 5 cusp forms $f_{k}$, we seek a weakly holomorphic form
$\widehat{f}_{k}$, such that the periods of $f_{k}$ and quasi-periods of
$\widehat{f}_{k}$ yield a determinant that is a rational multiple of a power
of $\pi$. For $k=1$, this will solve the Bessel-moment problem.

To construct 5 weakly homomorphic modular forms, we define a column vector

${\bf
g(\tau)}=\left(\frac{\eta_{12}^{5}}{\eta_{4}\eta_{6}^{2}}\right)^{6}\left[\begin{array}[]{l}1\\\
a+35\\\ a^{2}+40a+646\\\ a^{3}+45a^{2}+840a+8352\\\
a^{4}+50a^{3}+1059a^{2}+12308a+84817\end{array}\right],\,\quad
a=72\frac{\eta_{4}\eta_{12}^{5}}{\eta_{2}^{5}\eta_{6}},$ (78)

with monic polynomials in $a=O(q^{2})$, determined by the requirement that

$\frac{-27}{\tau^{6}}g_{k}\left(-\frac{1}{24\tau}\right)=\frac{1}{q^{2k-1}}+O(q),$
(79)

which records the singular behaviour near the cusp of $g_{k}(\tau)$ at
$\tau=0$. We avoid this singularity by taking Eichler integrals from
$\tau=\frac{1}{4}$ to $\tau={\rm i}\infty$, with extremely good behaviour of
$g_{k}(\tau)$ at the end-points.

The Fourier expansion of $g_{k}$ begins at $q^{11}$. Thus we may add, to any
combination of the weakly holomorphic forms in ${\bf g}$, a combination of the
cusp forms in ${\bf f}$, since the latter are determined by their expansions
up to $q^{9}$, recorded in the Hecke matrix ${\bf T}$. Our Ansatz for the
weakly homomorphic partners in $\widehat{\bf f}$ has the form

$\widehat{\bf f}=\widetilde{\bf T}^{-1}({\bf Ug+VT^{\rm-1}f}),\quad{\bf
U}=18^{2}\left[\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&3^{5}&0&0&0\\\
0&0&5^{5}&0&0\\\ 0&0&0&7^{5}&0\\\ 0&0&0&0&9^{5}\end{array}\right],$ (80)

which uses inverses of ${\bf T}$ and its transpose $\widetilde{\bf T}$. The
diagonal matrix ${\bf U}$ reflects the singular behaviour (79) of the weakly
holomorphic modular forms $g_{k}$ near $\tau=0$.

Our final challenge is to determine the matrix ${\bf V}$, which we expect,
from previous work, to be symmetric, with a vanishing first row and column.
Here we give our eventual result

${\bf V}=6^{2}\left[\begin{array}[]{rrrrr}0&0&0&0&0\\\ 0&-4&54&648&5995\\\
0&54&2916&77508&1150848\\\ 0&648&77508&3039444&64431936\\\
0&5995&1150848&64431936&1865595908\end{array}\right],$ (81)

with explanation of how it was obtained, empirically, by integer-relation
searches.

We define periods, quasi-periods and determinants as follows:

$\left[\begin{array}[]{c}P_{k}(s)\\\
\widehat{P}_{k}(s)\end{array}\right]=-{\rm
i}\int_{0}^{\infty}\left[\begin{array}[]{c}f_{k}((1+{\rm i}y)/4)\\\
\widehat{f}_{k}((1+{\rm i}y)/4)\end{array}\right]y^{s-1}{\rm d}y,\quad
D_{k}(s,t)=P_{k}(s)\widehat{P}_{k}(t)-P_{k}(t)\widehat{P}_{k}(s),$ (82)

with $k=1,2,3,4,5$ and $s=1,2,3,4,5$. The Eichler integrals are from
$\tau=\frac{1}{4}$ to $\tau={\rm i}\infty$, along the vertical line where the
real part of $\tau$ is $\frac{1}{4}$ and hence $q=\exp(2\pi{\rm i}\tau)$ is
pure imaginary. Since $f_{k}$ and $\widehat{f}_{k}$ have Fourier expansions in
odd powers of $q$, they too are pure imaginary. Hence the periods and quasi-
periods are real.

Our criteria for the elements of ${\bf V}$ are the determinant conditions

$D_{k}(1,3)=D_{k}(1,5)=D_{k}(2,4)=0,\quad D_{k}(1,2)=\frac{d_{k}}{\pi^{5}},$
(83)

where $d_{k}$ is a rational number. The first three conditions ensure the
matching of the period polynomial of $f_{k}$ with the quasi-period polynomial
of $\widehat{f}_{k}$. The fourth condition encodes a non-trivial quadratic
relation between periods and quasi-periods. Moreover, we were able to use
Rademacher sums [1] and Petersson inner products [24] to show that
$d_{k}=\frac{21}{2},-\frac{135}{14},-\frac{15}{2},\frac{15}{2},-\frac{6}{7}$,
for $k=1,2,3,4,5$.

Thus we have $5\times 4=20$ integer-relation conditions with which to
determine $4+3+2+1=10$ independent elements of ${\bf V}$, giving us great
confidence in our results.

Here we record the relations between periods, quasi-periods and Bessel moments
at $k=1$, which have been checked at 1000-digit precision:

$\displaystyle P_{1}(1)=21P_{1}(3)=9P_{1}(5)$ $\displaystyle=$ $\displaystyle
28\,\frac{M(0,6,0)}{\pi^{6}},$ (84) $\displaystyle P_{1}(2)=3P_{1}(4)$
$\displaystyle=$ $\displaystyle 8\,\frac{M(1,5,0)}{\pi^{5}},$ (85)
$\displaystyle\widehat{P}_{1}(1)=21\widehat{P}_{1}(3)=9\widehat{P}_{1}(5)$
$\displaystyle=$ $\displaystyle
7\,\frac{17M(0,6,0)+48(3M(0,6,2)-8M(0,6,4))}{80\pi^{6}},$ (86)
$\displaystyle\widehat{P}_{1}(2)=3\widehat{P}_{1}(4)$ $\displaystyle=$
$\displaystyle 2\,\frac{17M(1,5,0)+48(3M(1,5,2)-8M(1,5,4))}{80\pi^{5}}.$ (87)

###  5.2 Quadratic relations at levels 14 and 34

In 2019, Philip Candelas, Xenia de la Ossa, Mohamed Elmi and Duco van Straten
announced a discovery of a family of Calabi-Yau manifolds with rank-2
attractor points [16].

They compactified a 10-dimensional supergravity theory on a Calabi-Yau three-
fold with complex structure, to obtain 4-dimensional black holes, with event
horizons whose areas are determined by their electric and magnetic charges and
by ratios of periods of modular forms of weight 4 and levels 14 or 34.

Hearing of this on a visit to Oxford in November 2019, Broadhurst observed
that their Calabi-Yau periods come from solutions to a homogeneous
differential equation associated with 4-loop sunrise integrals, namely

$\displaystyle M_{m,n}(z)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}I_{0}(xz)[I_{0}(x)]^{m}[K_{0}(x)]^{5-m}x^{2n+1}{\rm
d}x$ (88) $\displaystyle N_{m,n}(z)$ $\displaystyle=$ $\displaystyle
z\int_{0}^{\infty}I_{1}(xz)[I_{0}(x)]^{m}[K_{0}(x)]^{5-m}x^{2n+2}{\rm d}x$
(89)

with $m\in\\{0,1,2\\}$, integers $n\geq 0$ and real $z^{2}<(5-2m)^{2}$. The
uncut diagram $M_{0,0}(z)$ satisfies an inhomogeneous differential equation.

The external mass is $z$. At $z=1$ we obtain Laporta’s on-shell periods, for
the magnetic moment of the electron at 4 loops, coming from the modular form
$f_{4,6}(\tau)=(\eta_{1}\eta_{2}\eta_{3}\eta_{6})^{2}$ with level 6. With mass
$z=\sqrt{17}-4$, we obtain level-34 periods. At the space-like point
$z=\sqrt{-7}$, we obtain level-14 periods.

At each of the levels 14 and 34, Candelas et al. considered 16 Calabi-Yau
periods, coming from 4 solutions to a homogeneous fourth-order differential
equation, together with the first 3 derivatives of each solution. They were
unable to identify all of these 16 periods.

Using LLL we found that 8 Feynman integrals, at each level, suffice to solve
their problem, completely. These 8 integrals determine a pair of periods and a
pair of quasi-periods, at each of the weights 2 and 4. Hence they satisfy two
quadratic relations. At level 34, the coefficients in these relations are
algebraic numbers in ${\bf Q}(\sqrt{17})$.

###  5.3 Level 14, at space-like momentum

At level 14, with $z=\sqrt{-7}$, we identified

$f_{4,14}(\tau)=\frac{(\eta_{2}\eta_{7})^{6}}{(\eta_{1}\eta_{14})^{2}}-4(\eta_{1}\eta_{2}\eta_{7}\eta_{14})^{2}+\frac{(\eta_{1}\eta_{14})^{6}}{(\eta_{2}\eta_{7})^{2}}$
(90)

as the relevant modular form of weight 4. Its periods are critical values of
the L-function
$L(f_{4,14},s)=((2\pi)^{s}/\Gamma(s))\int_{0}^{\infty}f_{4,14}({\rm
i}y)y^{s-1}{\rm d}y$, with

$\displaystyle L(f_{4,14},3)$ $\displaystyle=$ $\displaystyle
M_{1,0}(\sqrt{-7})=\int_{0}^{\infty}J_{0}(\sqrt{7}x)I_{0}(x)K_{0}^{4}(x)x{\rm
d}x=\frac{\pi^{2}}{7}L(f_{4,14},1)$ (91)
$\displaystyle\mbox{$\frac{1}{2}$}L(f_{4,14},2)$ $\displaystyle=$
$\displaystyle
M_{2,0}(\sqrt{-7})=\int_{0}^{\infty}J_{0}(\sqrt{7}x)I_{0}^{2}(x)K_{0}^{3}(x)x{\rm
d}x.$ (92)

There is also a modular form of weight 2 to consider,
$f_{2,14}(\tau)=\eta_{1}\eta_{2}\eta_{7}\eta_{14}$. This provides a modular
parametrization of a quartic elliptic curve, namely

$\displaystyle d^{2}$ $\displaystyle=$
$\displaystyle(1+h)(1+8h)(1+5h+8h^{2}),$ (93) $\displaystyle h$
$\displaystyle=$
$\displaystyle\left(\frac{\eta_{2}\eta_{14}}{\eta_{1}\eta_{7}}\right)^{3}=q+3q^{2}+6q^{3}+13q^{4}+O(q^{5}),$
(94) $\displaystyle d$ $\displaystyle=$
$\displaystyle\frac{q}{f_{2,14}}\frac{{\rm d}h}{{\rm
d}q}=1+7q+27q^{2}+92q^{3}+259q^{4}+O(q^{5}),$ (95)

yielding an L-value and a j-invariant:

$L(f_{2,14},1)=\frac{\omega_{+}}{3},\quad j\left(\frac{\omega_{+}+{\rm
i}\omega_{-}}{2\omega_{+}}\right)=\left(\frac{5\times 43}{28}\right)^{3},$
(96)

with periods determined by arithmetic-geometric means

$\omega_{\pm}=\frac{2\pi}{{\rm agm}\left(\sqrt{2^{9/2}\pm
13},\,2^{11/4}\right)}$ (97)

and also by Feynman integrals:

$\displaystyle\frac{\omega_{+}}{2}$ $\displaystyle=$ $\displaystyle
3M_{2,0}(\sqrt{-7})+4N_{2,0}(\sqrt{-7}),$ (98)
$\displaystyle\frac{\pi\omega_{-}}{2}$ $\displaystyle=$ $\displaystyle
3M_{1,0}(\sqrt{-7})+4N_{1,0}(\sqrt{-7}).$ (99)

The quasi-periods at weight 2 are $\widehat{\omega}_{\pm}$, with

$\displaystyle\frac{3\widehat{\omega}_{+}}{16}$ $\displaystyle=$
$\displaystyle 7M_{2,0}(\sqrt{-7})+8N_{2,0}(\sqrt{-7})+28M_{2,1}(\sqrt{-7}),$
(100) $\displaystyle\frac{3\pi\widehat{\omega}_{-}}{16}$ $\displaystyle=$
$\displaystyle 7M_{1,0}(\sqrt{-7})+8N_{1,0}(\sqrt{-7})+28M_{1,1}(\sqrt{-7}).$
(101)

Suppressing the argument $z=\sqrt{-7}$, we obtain the quadratic relation

${\rm det}\left[\begin{array}[]{lr}3M_{2,0}+4N_{2,0}&M_{2,0}+28M_{2,1}\\\
3M_{1,0}+4N_{1,0}&M_{1,0}+28M_{1,1}\end{array}\right]=-\frac{3\pi^{2}}{32}$
(102)

from Legendre’s relation for complete elliptic integrals.

At weight 4 we found

${\rm det}\left[\begin{array}[]{lr}M_{2,0}&39N_{2,0}-427M_{2,1}-112N_{2,1}\\\
M_{1,0}&39N_{1,0}-427M_{1,1}-112N_{1,1}\end{array}\right]=\frac{3\pi^{2}}{32}$
(103)

as the quadratic relation between the periods and quasi-periods of $f_{4,14}$.

We define the weight-4 periods and quasi-periods as

${\cal G}_{m}=M_{m,0},\quad\widehat{\cal
G}_{m}=7(35M_{m,0}-122M_{m,1})+2(39N_{m,0}-112N_{m,1}),$ (104)

for $m=1,2$ and $z=\sqrt{-7}$. The quasi-periods come from Eichler integrals
of a weakly holomorphic form obtained by multiplying $f_{2,14}^{2}$ by a
polynomial that is quartic in $h$ and linear in $d/h$. For one of the quasi-
periods, the dependence on $d/h$ is irrelevant. We used LLL to determine that

$\displaystyle\widehat{\cal G}_{2}$ $\displaystyle=$ $\displaystyle
5\pi^{2}\int_{1/\sqrt{7}}^{\infty}g\left(\frac{1+{\rm i}y}{2}\right)y{\rm d}y$
(105) $\displaystyle\quad g(\tau)$ $\displaystyle=$
$\displaystyle(253+645h+1446h^{2}+2064h^{3}+1024h^{4})f_{2,14}^{2}.$ (106)

Then the other quasi-period comes from the determinant ${\cal
G}_{2}\widehat{\cal G}_{1}-{\cal G}_{1}\widehat{\cal G}_{2}=3(\pi/4)^{2}$.

###  5.4 Level 34, with mass $\sqrt{17}-4$

At level 34, with $z=u=\sqrt{17}-4$, we used Pari-GP to identify the modular
form of weight 4. Let $\chi(n)$ be the Dirichlet character defined for prime
$p$ by $\chi(17)=0$ and otherwise by $\chi(p)=\pm 1$ according as whether $p$
is or is not a square modulo 17. Pari-GP declares that there are 12 cusp forms
of level 34 and weight 4 with this character. Feynman integrals choose a pair
of new forms whose Fourier coefficients, $A_{4}(n)$ and $\overline{A}_{4}(n)$,
are Gaussian integers, related by complex conjugation.

Let $L_{4}(s)$ be the analytic continuation of

$L_{4}(s)=\sum_{n>0}\frac{A_{4}(n)}{n^{s}}=\frac{1}{1+2^{1-s}}\prod_{p>2}\frac{1}{1-A_{4}(p)p^{-s}+\chi(p)p^{3-2s}}$
(107)

with the choice of sign $A_{4}(3)=2{\rm i}$. For prime $p$, $A_{4}(p)$ is real
if $\chi(p)=+1$ and imaginary if $\chi(p)=-1$, while $A_{4}(17)/17=1-4{\rm i}$
is truly complex.

Feynman integrals determine the critical L-values at weight 4:

$\displaystyle L_{4}(3)$ $\displaystyle=$
$\displaystyle\left(\frac{13-u+(1+13u){\rm i}}{17}\right)M_{1,0}(u),$ (108)
$\displaystyle L_{4}(2)$ $\displaystyle=$ $\displaystyle
4\left(\frac{5-3u+(3+5u){\rm i}}{17}\right)M_{2,0}(u),$ (109) $\displaystyle
L_{4}(1)$ $\displaystyle=$ $\displaystyle\left(\frac{7-11u+(11+7u){\rm
i}}{\pi^{2}}\right)M_{1,0}(u).$ (110)

At weight 2 they determine the periods and quasi-periods of the elliptic curve

$y^{2}=\left(x+\frac{5-u}{8}\right)\left(x+\frac{5+u}{8}\right)\left(x+\frac{3+u}{2}\right)$
(111)

whose real and imaginary periods are

$\omega_{1}=\frac{4\pi}{{\rm
agm}\left(\sqrt{4u},\sqrt{14+10u}\right)},\quad\omega_{2}=\frac{-4\pi{\rm
i}}{{\rm agm}\left(\sqrt{14+6u},\sqrt{14+10u}\right)}.$ (112)

The elliptic periods $\omega_{1,2}$ and quasi-periods $\widehat{\omega}_{1,2}$
are determined by

$\displaystyle\frac{\omega_{1}}{4}$ $\displaystyle=$ $\displaystyle{\cal
P}_{2}=(2+3u)M_{2,0}(u)+4(4+u)N_{2,0}(u)$ (113) $\displaystyle\frac{\pi{\rm
i}\omega_{2}}{4}$ $\displaystyle=$ $\displaystyle{\cal
P}_{1}=(2+3u)M_{1,0}(u)+4(4+u)N_{1,0}(u)$ (114)
$\displaystyle\frac{3\widehat{\omega}_{1}}{8(1+u)}$ $\displaystyle=$
$\displaystyle\widehat{\cal
P}_{2}=M_{2,0}(u)+2(5+u)N_{2,0}(u)+2u(3+u)(4+u)M_{2,1}(u)$ (115)
$\displaystyle\frac{3\pi{\rm i}\widehat{\omega}_{2}}{8(1+u)}$ $\displaystyle=$
$\displaystyle\widehat{\cal
P}_{1}=M_{1,0}(u)+2(5+u)N_{1,0}(u)+2u(3+u)(4+u)M_{1,1}(u)$ (116)

with Legendre’s condition giving ${\cal P}_{1}\widehat{\cal P}_{2}-{\cal
P}_{2}\widehat{\cal P}_{1}=3(\pi/4)^{2}/(1+u).$

At weight 4, the periods ${\cal H}_{m}=M_{m,0}(u)$ and quasi-periods

$\displaystyle\widehat{\cal H}_{m}$ $\displaystyle=$ $\displaystyle
81M_{m,0}(u)+3(2+u)(u-6)N_{m,0}(u)$ (117) $\displaystyle+$
$\displaystyle{}u^{2}(2+u)(4+u)(96+11u)M_{m,1}(u)+136(1-u)N_{m,1}(u)$

yield the intersection number ${\cal H}_{1}\widehat{\cal H}_{2}-{\cal
H}_{2}\widehat{\cal H}_{1}=3(\pi/8)^{2}/u.$

The numbers which remained unidentified in [16] are now easy to determine.
They involve the permanents [15] of the matrices of Feynman integrals whose
determinants yield intersection numbers that are algebraic multiples of powers
of $\pi$.

##  6 Summary

PSLQ and LLL have enlivened quests for analytical results, provided strong
tests on conjectures and condensed huge expressions. Parallel PSLQ was of the
essence in Laporta’s work in electrodynamics. LLL led to a conjecture on
quadratic relations for all loops, to determinations of quasi-periods at
weight 6 and level 24 and to exact results for black-hole problems that
involve modular forms of levels 14 and 34. Our new results at levels 14, 24
and 34 were obtained from extending methods developed in our work on eta
quotients [1]. The permanents which were lacking in [16] yield Rademacher sums
[1] that are sums of products of Bessel functions and Kloosterman sums [13] .

## References

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