Wick rotations, Eichler integrals, and multi-loop Feynman diagrams

TL;DR

This paper employs contour deformations and modular forms to evaluate complex Feynman integrals in two-dimensional quantum field theory, connecting them to modular $L$-series and confirming recent conjectures.

Contribution

It introduces a novel approach combining contour deformations and modular forms to compute multi-loop Feynman diagrams explicitly.

Findings

Explicit evaluation of certain Bessel moments as constants or $L$-series values

Verification of recent conjectures by Broadhurst

Establishment of a link between Feynman integrals and modular forms

Abstract

Using contour deformations and integrations over modular forms, we compute certain Bessel moments arising from diagrammatic expansions in two-dimensional quantum field theory. We evaluate these Feynman integrals as either explicit constants or critical values of modular $L$-series, and verify several recent conjectures of Broadhurst.

Tables

Table 1. Table I. Values of X 6 , 3 ​ ( z ) subscript 𝑋 6 3 𝑧 X_{6,3}(z) , Z 6 , 3 ​ ( z ) subscript 𝑍 6 3 𝑧 Z_{6,3}(z) and their derivatives at z = 1 2 + i ​ 5 2 ​ 3 𝑧 1 2 𝑖 5 2 3 z=\frac{1}{2}+\frac{i\sqrt{5}}{2\sqrt{3}} [ with “rescaled Bologna constant” c = 5 ​ C = 1 240 ​ π 2 ​ Γ ​ ( 1 15 ) ​ Γ ​ ( 2 15 ) ​ Γ ​ ( 4 15 ) ​ Γ ​ ( 8 15 ) 𝑐 5 𝐶 1 240 superscript 𝜋 2 Γ 1 15 Γ 2 15 Γ 4 15 Γ 8 15 c=\sqrt{5}C=\frac{1}{240\pi^{2}}\Gamma\left(\frac{1}{15}\right)\Gamma\left(\frac{2}{15}\right)\Gamma\left(\frac{4}{15}\right)\Gamma\left(\frac{8}{15}\right) ]

Equations

$${\int }_0^{\infty }\frac{[\pi I_0(t)+i{K}_0(t){]}^m+[\pi I_0(t)-i{K}_0(t){]}^m}{i}[K_0(t){]}^m{t}^{n}dt=0$$

$${\int }_0^{\infty }\frac{[\pi I_0(t)+i{K}_0(t){]}^m-[\pi I_0(t)-i{K}_0(t){]}^m}{i}[K_0(t){]}^m{t}^{n}dt=0$$

$$\begin{array}{rl}& \frac{2^{1+2(n-1)[1-(-1{)}^m]}}{{\pi }^{m+1}}{\int }_0^{\infty }\frac{[\pi I_0(t)+i{K}_0(t){]}^m-[\pi I_0(t)-i{K}_0(t){]}^m}{i}\times \\ & \times [K_0(t){]}^m(2t{)}^{2n+m-3}dt\end{array}$$

$${\int }_0^{\infty }{I}_0(t)[K_0(t){]}^{4}tdt=\frac{\Gamma \left(\frac{1}{15}\right)\Gamma \left(\frac{2}{15}\right)\Gamma \left(\frac{4}{15}\right)\Gamma \left(\frac{8}{15}\right)}{240\sqrt{5}}$$

$$\left(\begin{array}{cc}\mathrm{IKM}(1,4;1)& \mathrm{IKM}(1,4;3)\\ \mathrm{IKM}(2,3;1)& \mathrm{IKM}(2,3;3)\end{array}\right)=\left(\begin{array}{cc}{\pi }^{2}C& {\pi }^2{\left(\frac{2}{15}\right)}^2\left(13C-\frac{1}{10C}\right)\\ \frac{\sqrt{15}\pi }{2}C& \frac{\sqrt{15}\pi }{2}{\left(\frac{2}{15}\right)}^2\left(13C+\frac{1}{10C}\right)\end{array}\right),$$

$$\frac{3}{{\pi }^2}\mathrm{IKM}(1,5;1)=\mathrm{IKM}(3,3;1)=$$

$$\mathrm{IKM}(2,4;1)=$$

$$=$$

$$f_{4,6}(z)=[\eta (z)\eta (2z)\eta (3z)\eta (6z){]}^2$$

$$\eta (z):=e^{\pi iz/12}\prod _{n=1}^{\infty }(1-e^{2\pi inz}),z\in \mathfrak{H}:=\{w\in \mathbb{C}|\mathrm{Im}w>0\}.$$

$$\mathrm{IKM}(4,4;1)=$$

$$\frac{1}{{\pi }^2}\mathrm{IKM}(1,7;1)=\mathrm{IKM}(3,5;1)=$$

$$\mathrm{IKM}(2,6;1)=$$

$$f_{6,6}(z)=\frac{[\eta (2z)\eta (3z){]}^9}{[\eta (z)\eta (6z){]}^3}+\frac{[\eta (z)\eta (6z){]}^9}{[\eta (2z)\eta (3z){]}^3}.$$

$$J_{\nu }(z):=$$

$$I_{\nu }(z):=$$

$$J_0(ix)=I_0(x)\text{ and}\frac{\pi i}{2}{H}_0^{(1)}(ix)=K_0(x)$$

$$H_0^{(1)}(x+i{0}^{+})=J_0(x)+i{Y}_0(x)\text{and}{H}_0^{(1)}(-x+i{0}^{+})=-J_0(x)+i{Y}_0(x)$$

$$H_0^{(1)}(z)=\sqrt{\frac{2}{\pi z}}{e}^{i\left(z-\frac{\pi }{4}\right)}\left[1+O\left(\frac{1}{|z|}\right)\right]\text{and}{H}_0^{(2)}(z)=\sqrt{\frac{2}{\pi z}}{e}^{-i\left(z-\frac{\pi }{4}\right)}\left[1+O\left(\frac{1}{|z|}\right)\right].$$

$${\int }_{i{0}^{+}-\infty }^{i{0}^{+}+\infty }[J_0^{}(z){]}^m[H_0^{(1)}(z){]}^n{z}^{\ell }dz:=\underset{\epsilon \to 0^{+}}{\lim }\underset{R\to \infty }{\lim }{\int }_{i\epsilon -R}^{i\epsilon +R}[J_0^{}(z){]}^m[H_0^{(1)}(z){]}^n{z}^{\ell }dz=0.$$

$${\int }_0^{\infty }[J_0(x){]}^m\left\{[J_0(x)+i{Y}_0(x){]}^n+(-1{)}^{\ell }[-J_0(x)+i{Y}_0(x){]}^n\right\}{x}^{\ell }dx=0,$$

$${\int }_0^{\infty }{I}_0(t)[K_0(t){]}^{4}tdt=$$

$${\left(\frac{2}{\pi }\right)}^4{\int }_0^{\infty }{I}_0(t)[K_0(t){]}^{4}tdt=-\mathrm{Re}{\int }_0^{i\infty }{J}_0^{}(z)[H_0^{(1)}(z){]}^{4}zdz,$$

$$-\mathrm{Re}{\int }_0^{\infty }{J}_0^{}(x)[H_0^{(1)}(x){]}^{4}xdx=-{\int }_0^{\infty }J(J^4-6J^2{Y}^2+Y^4)xdx,$$

$$J(J^4-6J^2{Y}^2+Y^4)-\frac{2J^2}{3}[(J+iY{)}^3-(-J+iY{)}^3]-\frac{(J+iY{)}^5-(-J+iY{)}^5}{10}=-\frac{8J^5}{15},$$

$$\mathrm{IKM}(1,4;1)=\frac{1}{8}{\int }_0^{\infty }\frac{dX}{X}{\int }_0^{\infty }\frac{dY}{Y}{\int }_0^{\infty }\frac{dZ}{Z}\frac{1}{(1+X+Y+Z)(1+X^{-1}+Y^{-1}+Z^{-1})-1},$$

$$\frac{4}{{\pi }^3}{\int }_0^{\infty }[K_0(t){]}^{3}dt={\int }_0^{\infty }[J_0(x){]}^{3}dx$$

$$\frac{4}{{\pi }^3}{\int }_0^{\infty }{I}_0(t)[K_0(t){]}^{3}dt={\int }_0^{\infty }[J_0(x){]}^{4}dx$$

$${\Gamma }_0(6):=\left\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)|a,b,c,d\in \mathbb{Z},ad-bc=1,c\equiv 0(\mathrm{mod}6)\right\}.$$

$$\displaystyle\left\{\begin{array}[]{r@{\;=\;}l@{ \quad\text{if}\quad}r@{\;\in\;}l}X_{6,3}\left(\tfrac{az+b}{cz+d}\right)&X_{6,3}(z),&\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)&\varGamma_{0}(6)_{+3};\\ \operatorname{Im}X_{6,3}(z)&0,&2\operatorname{Re}z&\mathbb{Z};\\ X_{6,3}\left(\tfrac{1}{2}+\tfrac{iy}{2\sqrt{3}}\right)&X_{6,3}\left(\tfrac{1}{2}+\tfrac{i}{2\sqrt{3}y}\right),&y&(0,\infty).\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{r@{\,:\,}r@{\;\longrightarrow\;}l}X_{6,3}\,:&\{z|\operatorname{Re}z=0,\operatorname{Im}z>0\}&(0,\infty)\\ X_{6,3}\,:&\left\{z\left|\operatorname{Re}z=\tfrac{1}{2},\operatorname{Im}z>\tfrac{1}{2\sqrt{3}}\right.\right\}&\left(-\tfrac{1}{16},0\right)\end{array}\right.$$

$${\hat{W}}_{3}z=\frac{3z-2}{6z-3}=\frac{1}{2}+\frac{i}{2\sqrt{3}y}\text{for }z=\frac{1}{2}+\frac{iy}{2\sqrt{3}}.$$

$${\int }_0^{\infty }{J}_0\left({\left[\frac{2\eta (2z)\eta (6z)}{\eta (z)\eta (3z)}\right]}^{3}t\right)I_0(t)[K_0(t){]}^{3}tdt=$$

Full text

*Keywords: Feynman integrals, Wick rotations, Bessel functions, Hankel transforms, random walks

MSC 2010: 11F03, 14E18 (Primary) 81T18, 81T40, 81Q30, 60G50 (Secondary)*

1. 1 Introduction

2. 1.1 Background and motivations

3. 1.2 Statement of results and plan of proof

4. 2 Bessel functions and their Wick rotations

5. 2.1 Some analytic properties of Bessel functions

6. 2.2 Contour deformations for Bessel moments

7. 3 Feynman diagrams with 5 Bessel factors

8. 3.1 A modular form associated with Bessel moments

9. 3.2 Symmetric squares and Eichler integrals

10. 3.3 Special values of Eichler integrals

11. 4 Feynman diagrams with 6 Bessel factors

12. 4.1 Modular parametrization for certain Hankel transforms

13. 4.2 Eichler integrals via Hankel fusions

14. 5 Feynman diagrams with 8 Bessel factors

15. 5.1 Hankel transforms and Wick rotations

16. 5.2 Critical $L$-values for Bessel moments

1. Introduction

1.1. Background and motivations

In quantum field theory (QFT), we encounter integrals over Bessel functions while performing diagrammatic expansions in the configuration space. For two-dimensional QFT, we need Bessel functions $J_{0}$ and $Y_{0}$, as well as modified Bessel functions $I_{0}$ and $K_{0}$, to define propagators and compute Feynman integrals [21, 1, 10, 12, 19].

We are interested in Bessel moments $\operatorname{\mathbf{JYM}}(\alpha,\beta;\nu):=\int_{0}^{\infty}[J_{0}(t)]^{\alpha}[Y_{0}(t)]^{\beta}t^{\nu}\operatorname{d}t$ and $\operatorname{\mathbf{IKM}}(a,b;n):=\int_{0}^{\infty}[I_{0}(t)]^{a}[K_{0}(t)]^{b}t^{n}\operatorname{d}t$, where the non-negative integers $\alpha,\beta,\nu,a,b,n$ are chosen to ensure convergence of the corresponding integrals. The Bessel moments $\operatorname{\mathbf{JYM}}$’s are useful auxiliary tools for computing $\operatorname{\mathbf{IKM}}$’s in two-dimensional QFT. Furthermore, the $\operatorname{\mathbf{IKM}}$’s also show up in the finite part for renormalized perturbative expansions of four-dimensional QFT: for example, $\operatorname{\mathbf{IKM}}(1,5;1)$ and $\operatorname{\mathbf{IKM}}(1,5;3)$ are part of the 4-loop contributions (from 891 Feynman diagrams) to electron’s magnetic moment [26, (19) and Fig. 3($a$)($a^{\prime}$)], according to the standard formulation of quantum electrodynamics (four-dimensional QFT).

The mathematical understanding of $\operatorname{\mathbf{JYM}}(\alpha,\beta;\nu)$ for $\alpha+\beta\geq 5$ and $\operatorname{\mathbf{IKM}}(a,b;n)$ for $a+b\geq 5$ is relatively scant. While numerical experiments have suggested a rich collection of identities relating various cases of $\operatorname{\mathbf{IKM}}(a,b;1)$ (each of which corresponding to a Feynman diagram containing $b-1$ loops) to special values of certain Hasse–Weil $L$-series for $a+b\in\{5,6,7,8\}$ [10, 12, 19], most of these conjectural evaluations are heretofore unproven.

In our recent work [43], we have shown that

[TABLE]

for $m\in\mathbb{Z}_{>1},n\in\mathbb{Z}_{\geq 0},\frac{m-n}{2}\in\mathbb{Z}_{>0}$, and

[TABLE]

for $m\in\mathbb{Z}_{>0},n\in\mathbb{Z}_{\geq 0},\frac{m-n-1}{2}\in\mathbb{Z}_{>0}$ (Bailey–Borwein–Broadhurst–Glasser sum rule [1, “final conjecture”, (220)], with generalizations). In addition, we have also confirmed that

[TABLE]

evaluates to a positive integer for all $m,n\in\mathbb{Z}_{>0}$ (Broadhurst–Mellit integer sequence [12, (149) in Conjecture 5] and Broadhurst–Roberts rational sequence [13, Conjecture 2]). While the aforementioned results resolve some longstanding conjectures, they barely scratch the surface of the algebraic and arithmetic nature of Bessel moments. For example, the determinant $\operatorname{\mathbf{IKM}}(1,4;1)\operatorname{\mathbf{IKM}}(2,$ $3;3)-\operatorname{\mathbf{IKM}}(2,3;1)\operatorname{\mathbf{IKM}}(1,4;3)=2\pi^{3}/\sqrt{3^{3}5^{5}}$ [conjectured in 12, (100)] and the sum rule $9\pi^{2}\operatorname{\mathbf{IKM}}(4,$ $4;1)-14\operatorname{\mathbf{IKM}}(2,6;1)=0$ [conjectured in 12, (147)] had not been covered by the real-analytic methods we employed in [43].

1.2. Statement of results and plan of proof

In this article, we supplement our previous work with complex analysis and modular forms, which are two powerful devices that not only produce new algebraic relations among different $\operatorname{\mathbf{IKM}}$ moments, but also connect Feynman diagrams to special $L$-values and Kluyver’s “random walk integrals” $\operatorname{\mathbf{JYM}}(n,0,1),n\in\mathbb{Z}_{\geq 5}$ [24, 8, 7].

The layout of this paper is described in the next four paragraphs.

Beginning with a brief survey of the analytic properties for (modified) Bessel functions in §2.1, we introduce Wick rotations, which are contour deformations that allow us to convert $\operatorname{\mathbf{IKM}}$ problems into $\operatorname{\mathbf{JYM}}$ problems, in §2.2. We demonstrate the usefulness of Wick rotations by a very short (yet self-contained) proof of the closed-form evaluation of a Bessel moment

[TABLE]

in terms of Euler’s gamma function $\Gamma(x):=\int_{0}^{\infty}u^{x-1}e^{-u}\operatorname{d}u$ for $x>0$. It is worth noting that nearly a decade had elapsed between the original proposal [1, 25] of (1.2.1) and its first rigorous (and highly technical) verification [4, 31]. Our simplified proof of (1.2.1) draws on its connection to a “random walk integral” $\operatorname{\mathbf{JYM}}(5,0;1)$.

In §3, we push the evaluation of (1.2.1) one step further, to give explicit verifications of all the entries in the following $2\times 2$ matrix:

[TABLE]

where $C=\frac{1}{240\sqrt{5}\pi^{2}}\Gamma\left(\frac{1}{15}\right)\Gamma\left(\frac{2}{15}\right)\Gamma\left(\frac{4}{15}\right)\Gamma\left(\frac{8}{15}\right)$ is the “Bologna constant” attributed to Broadhurst [9, 1] and Laporta [25]. (Here, the rigorous evaluation of the top-right entry $\operatorname{\mathbf{IKM}}(1,4;3)$ was previously unattested in the literature.) We accomplish this by using a modular function of level 6 (§3.1) that parametrizes a Picard–Fuchs differential equation of third order (§3.2) attached to a family of $K3$ surfaces formerly studied by Bloch–Kerr–Vanhove [4] and Samart [31]. In addition to proving (1.2.2) in §3.3, we work out the Eichler integral representations of $\operatorname{\mathbf{IKM}}(1,4;1)$, $\operatorname{\mathbf{IKM}}(1,4;3)$ and $\operatorname{\mathbf{IKM}}(1,4;5)$, which involve contour integrals over certain holomorphic modular forms.

We devote §4 to the verification of the following integral formulae [conjectured in 12, (109)–(111)]:

[TABLE]

where

[TABLE]

is a weight-4 modular form defined through the Dedekind eta function

[TABLE]

To prove these formulae relating Bessel moments to critical $L$-values (a special $L$-value $L(f,s)$ is said to be critical if $s$ is a positive integer less than the weight of the modular form $f$), we use modular parametrizations of Hankel transforms and the Parseval–Plancherel identity.

In §5, we fully exploit the techniques developed in the previous two sections, and confirm the following identities [cf. 12, (143)–(146)]:

[TABLE]

which involve a weight-6 modular form

[TABLE]

In addition, we also use explicit computations to verify the Eichler–Shimura–Manin relation $L(f_{6,6},5)/L(f_{6,6},3)=2\pi^{2}/21$ [cf. 12, (142)] and the sum rule $9\pi^{2}\operatorname{\mathbf{IKM}}(4,4;1)-14\operatorname{\mathbf{IKM}}(2,6;1)=0$ [cf. 12, (147)].

Broadhurst has recently proposed a vast set of conjectures [19, 12, 13, 14, 15, 16, 17, 18] connecting Feynman diagrams to special values of Hasse–Weil $L$-functions, whose local factors arise from Kloosterman sums [12, §§2–6]. Our current work only touches upon $\operatorname{\mathbf{IKM}}(a,b;1)$ for $a+b\in\{5,6,8\}$, where the corresponding $L$-series are modular. It is our hope that, by verifying a small subset of Broadhurst’s thought-inspiring conjectures about Bessel moments, we could make first steps towards an arithmetic understanding of these important mathematical constants deeply embedded in fundamental laws of nature, viz. quantum electrodynamics. On one hand, we have Feynman diagrams realized as motivic integrals, whose cohomology belongs to the realm of algebraic geometry; on the other hand, these Feynman integrals also evaluate to arithmetic objects, such as Eichler integrals and special $L$-values, whose symmetries embellish modern number theory.

2. Bessel functions and their Wick rotations

2.1. Some analytic properties of Bessel functions

For $\nu\in\mathbb{C},-\pi<\arg z<\pi$, the Bessel functions $J_{\nu}$ and $Y_{\nu}$ are defined by

[TABLE]

Hereafter, the fractional powers of complex numbers are defined through $w^{\beta}=\exp(\beta\log w)$ for $\log w=\log|w|+i\arg w$, where $|\arg w|<\pi$.

We will also need the cylindrical Hankel functions $H^{(1)}_{0}(z)=J_{0}(z)+iY_{0}(z)$ and $H^{(2)}_{0}(z)=J_{0}(z)-iY_{0}(z)$ of zeroth order, which are both well defined for $-\pi<\arg z<\pi$. In view of (2.1.1) and (2.1.2), we can verify

[TABLE]

as well as

[TABLE]

for $x>0$.

As $|z|\to\infty,-\pi<\arg z<\pi$, we have the following asymptotic behavior:

[TABLE]

The asymptotic behavior of $J_{0}(z)=[H_{0}^{(1)}(z)+H_{0}^{(2)}(z)]/2$ can be inferred accordingly.

2.2. Contour deformations for Bessel moments

In the next lemma, we present a mechanism that generates cancelation formulae for $\operatorname{\mathbf{JYM}}$. Special cases of this lemma (involving four Bessel factors) have already appeared in [42, §2].

Lemma 2.2.1** (Bessel–Hankel–Jordan).**

For $\ell,m,n\in\mathbb{Z}_{\geq 0}$ satisfying either $\ell-(m+n)/2<0;m<n$ or $\ell-m=\ell-n<-1$, we have

[TABLE]

Proof.

As the integrand goes asymptotically like $O(z^{\ell-(m+n)/2}e^{i(n-m)z})$ for $\operatorname{Im}z>0,|z|\to\infty$, we can close the contour in the upper half-plane with the help of Jordan’s lemma. $\blacksquare$

• Remark

Noting (2.1.4) and $J_{0}(-x)=J_{0}(x)$, we may reformulate (2.2.1) as

[TABLE]

which is a more convenient form to be used later. $\square$

In addition to closing the contour upwards (Lemma 2.2.1), sometimes we also need to turn the contour 90∘ clockwise, from the positive imaginary axis to the positive real axis. This trick is known as Wick rotation in QFT. Instead of stating and justifying the general procedures for Wick rotations, we illustrate with a concrete example that relates $\operatorname{\mathbf{IKM}}(1,4;1)$ to a well-studied integral in probability theory.

Theorem 2.2.2** (“Tiny nome of Bologna”).**

We have

[TABLE]

Proof.

Thanks to Jordan’s lemma, we can deform the contour in

[TABLE]

and identify it with its “Wick-rotated” counterpart:

[TABLE]

where $J$ (resp. $Y$) stands for $J_{0}(x)$ (resp. $Y_{0}(x)$) in the last expression. Now that

[TABLE]

we can verify the first equality in (1.2.1*′*), while referring back to (2.2.1*′*) in Lemma 2.2.1.

The “random walk integral” $\int_{0}^{\infty}[J_{0}(x)]^{5}x\operatorname{d}x$ has been thoroughly studied by Borwein and coworkers [8]. One can evaluate this integral through a special value of a modular form (to be elaborated later in §3.1). Here, we simply point out that the second equality in (1.2.1*′*) can be directly deduced from [8, (5.2)]. $\blacksquare$

• Remark

We pause to give a brief account for the history of the integral identity in (1.2.1). The closed-form evaluation in (1.2.1) was initially proposed by Broadhurst in the form of elliptic theta functions [1, (93)], and the current (equivalent) form involving products of gamma functions was suggested by Laporta [25, (7), (16), (17)]. Bloch–Kerr–Vanhove studied the momentum space reformulation of $\operatorname{\mathbf{IKM}}(1,4;1)$ as a triple integral of an algebraic function over the first octant:

[TABLE]

with a tour de force in motivic cohomology. They effectively verified (1.2.1) by casting $\operatorname{\mathbf{IKM}}(1,4;1)$ into $\frac{\pi^{3}}{8\sqrt{15}}\frac{[\eta(z)\eta(3z)]^{4}}{[\eta(2z)\eta(6z)]^{2}}$ for $z=\frac{3+i\sqrt{15}}{6}$ [4, (2.5.9)]. Drawing on a result of Rogers–Wan–Zucker [29, Theorem 5], Samart reanalyzed the triple integral formulation of $\operatorname{\mathbf{IKM}}(1,4;1)$, before finally expressing $\operatorname{\mathbf{IKM}}(1,4;1)$ as explicit gamma factors, and identifying it with a special $L$-value $\frac{3\pi}{2\sqrt{15}}L(f_{3,15},2)$ for the modular form $f_{3,15}(z)=[\eta(3z)\eta(5z)]^{3}+[\eta(z)\eta(15z)]^{3}$ [31, (35)]. $\square$

• Remark

In [8, §5], the authors remarked on the uncanny resemblance of the “random walk integral” $\int_{0}^{\infty}[J_{0}(x)]^{5}x\operatorname{d}x$ to the “tiny nome of Bologna”, without supplying a mechanistic interpretation later afterwards. Moreover, these authors recorded [8, Remark 7.3]

[TABLE]

and [8, between Theorems 7.6 and 7.7]

[TABLE]

after comparing explicit expressions of all the integrals in question, probably unaware that such equalities would follow easily from a Wick rotation and an application of Lemma 2.2.1 above. $\square$

3. Feynman diagrams with 5 Bessel factors

3.1. A modular form associated with Bessel moments

In this paper, we will mainly deal with modular forms of level 6, which respect the symmetries in the Hecke congruence group

[TABLE]

Furthermore, following the notation of Chan–Zudilin [20], we write $\widehat{W}_{3}=\frac{1}{\sqrt{3}}\left(\begin{smallmatrix}3&-2\\ 6&-3\end{smallmatrix}\right)$ and construct a group $\varGamma_{0}(6)_{+3}=\langle\varGamma_{0}(6),\widehat{W}_{3}\rangle$ by adjoining $\widehat{W}_{3}$ to $\varGamma_{0}(6)$. To set the stage for later developments in this article, we present some characteristics of a modular function on $\varGamma_{0}(6)_{+3}$.

Lemma 3.1.1** (A modular function of level 6).**

The function $X_{6,3}(z):=\left[\frac{\eta(2z)\eta(6z)}{\eta(z)\eta(3z)}\right]^{6},z\in\mathfrak{H}$ has the following properties:

[TABLE]

Moreover, the following mappings

[TABLE]

are bijective.

Proof.

The function $X_{6,3}$ is a Hauptmodul of $\varGamma_{0}(6)_{+3}$ with genus 0 [20, (2.2)], so it must satisfy the modular invariance relation, as displayed in the first line of (3.1.5). To prove the second line in (3.1.5), use the infinite product expansion for the Dedekind eta function in (1.2.6). To prove the last line in (3.1.5), note that

[TABLE]

The domains of the mappings in (3.1.8) are proper subsets of the fundamental domain for $\varGamma_{0}(6)_{+3}$, so these mappings are necessarily injective. Furthermore, by the second line in (3.1.5), these mappings are continuous real-valued functions defined on path-connected sets, so these injective mappings must also be monotone along the respective paths, and their continuous images are also path-connected. Consequently, the modular function $X_{6,3}$ induces bijective mappings from these two domains to their respective ranges, and the extent of the latter is inferred from the “boundary values” of the function $X_{6,3}$ at the extreme points of the domains of definition. $\blacksquare$

As a demonstration for the relevance of modularity in our studies of Bessel moments, we recall some known results from [30, 8], in slightly reorganized form. In particular, we will use the Chan–Zudilin notation $Z_{6,3}(z)=\frac{[\eta(z)\eta(3z)]^{4}}{[\eta(2z)\eta(6z)]^{2}}$ [20, (2.5)] for a modular form of weight 2 on $\varGamma_{0}(6)_{+3}$.

Proposition 3.1.2** (Bessel moments as modular forms).**

For $z/i>0$, we have

[TABLE]

which give modular parametrizations of $\int_{0}^{\infty}I_{0}(xt)I_{0}(t)[K_{0}(t)]^{3}t\operatorname{d}t$ and $\int_{0}^{\infty}J_{0}(xt)[J_{0}(t)]^{4}t\operatorname{d}t$ for $x\in(0,2)$.

Proof.

We recall from [1, (55) and (56)] the following formula

[TABLE]

where ${m\choose j}=\frac{m!}{j!(m-j)!}$ and $D_{n}$ is the $n$th Domb number. Meanwhile, we note that Rogers has shown in [30, Theorem 3.1] that

[TABLE]

holds for $|u|$ sufficiently small, where

[TABLE]

By termwise summation, we see that

[TABLE]

is valid for $x$ sufficiently small. Parametrizing the right-hand side of the equation above with modular forms (see [20, (2.8)] or [8, (4.13)]), we observe that (3.1.10) holds when $\operatorname{Im}z$ is sufficiently large and positive. By analytic continuation, the validity of (3.1.10) extends to the entire positive $\operatorname{Im}z$-axis, from which $x=\left[\frac{2\eta(2z)\eta(6z)}{\eta(z)\eta(3z)}\right]^{3}$ maps bijectively to $x\in(0,\infty)$.

Performing further analytic continuation on (3.1.10), we arrive at (3.1.11). Here, according to Lemma 3.1.1, we know that $x=\frac{1}{i}\left[\frac{2\eta(2z)\eta(6z)}{\eta(z)\eta(3z)}\right]^{3}$ maps $y\in\left(\frac{1}{2\sqrt{3}},\infty\right)$ bijectively to $x\in(0,2)$.

The integral identity in (3.1.12) paraphrases [8, (4.16)]. (A special case of this modular parametrization led to a closed-form evaluation of the “random walk integral” $\int_{0}^{\infty}[J_{0}(x)]^{5}x\operatorname{d}x$ in [8, (5.2)], which we quoted in our proof of Theorem 2.2.2. See also Table I.) $\blacksquare$

• Remark

For any CM point $z\in\mathfrak{H}$ (a complex number in the upper half-plane that solves a quadratic equation with integer coefficients), the absolute value $|\eta(z)|$ of the Dedekind eta function $\eta(z)$ can be explicitly written as the product of an algebraic number, a rational power of $\pi$, and rational powers of special values for Euler’s gamma function (see [32, §12] or [34, Theorem 9.3]). At any CM point $z$, the following expressions are computable algebraic numbers [39, (1.2.9) and Appendix 1]:

[TABLE]

where

[TABLE]

are Eisenstein series of weights 4 and 6. Higher order derivatives of the Dedekind eta function can be deduced from Ramanujan’s differential equations [28]:

[TABLE]

where $E_{2}^{*}(z)=E_{2}(z)+\frac{3}{\pi\operatorname{Im}z}$ is a holomorphic “weight-2 Eisenstein series”.

Samart has computed the values of $X_{6,3}(z)$ and $Z_{6,3}(z)$ at $z=\frac{1}{2}+\frac{i\sqrt{5}}{2\sqrt{3}}$ explicitly [31, Lemma 1]. We may combine his results with (3.1.28) to evaluate derivatives of $X_{6,3}(z)$ and $Z_{6,3}(z)$ at $z=\frac{1}{2}+\frac{i\sqrt{5}}{2\sqrt{3}}$, as summarized in Table I. $\square$

• Remark

As the Bessel differential equation leaves us [1, §1]

[TABLE]

we will have no difficulties in computing $\operatorname{\mathbf{IKM}}(2,3;1)=\frac{\sqrt{15}\pi}{2}C$, \operatorname{\mathbf{IKM}}(2,3;3)=\frac{\sqrt{15}\pi}{2}\left(\frac{2}{15}\right)^{2}\big{(}13C+\frac{1}{10C}\big{)} and \operatorname{\mathbf{IKM}}(2,3;5)=\frac{\sqrt{15}\pi}{2}\left(\frac{4}{15}\right)^{3}\big{(}43C+\frac{19}{40C}\big{)} from (3.1.11), with assistance from Table I. These Bessel moments were previously evaluated in [1, §5.10] with combinatorial techniques. $\square$

3.2. Symmetric squares and Eichler integrals

Central to the studies of Bloch–Kerr–Vanhove [4] and Samart [31] was the following motivic integral:

[TABLE]

and the geometry for the family of $K3$ surfaces that compactify the locus of $(1+X+Y+Z)(1+X^{-1}+Y^{-1}+Z^{-1})-u=0$ and resolve singularities. Inspired by their analysis, we give a modular parametrization of $\mathscr{I}(u)$ for $u\leq 16$. In [4] and [31], the authors parametrized the Feynman integral $\mathscr{I}(u)$ with the modular function $u(z)=-\left[\frac{\eta(z)\eta(3z)}{\eta(2z)\eta(6z)}\right]^{6}$, and needed sophisticated computations at the CM point $z_{*}=\frac{-3+i\sqrt{15}}{24}$ where $u(z_{*})=1$. In what follows, we will use a different modular parametrization (Lemma 3.2.1) to facilitate the representation of Bessel moments via Eichler integrals (Proposition 3.2.2).

Lemma 3.2.1** (Jacobian for a modular function).**

The modular parametrization

[TABLE]

satisfies

[TABLE]

With $q=e^{2\pi iz}$, we have the following asymptotic behavior

[TABLE]

near the infinite cusp ($z\to i\infty,q\to 0$).

Proof.

We can verify the following identity

[TABLE]

by showing that the ratio between both sides defines a bounded function on the compact Riemann surface $X_{0}(2)=\varGamma_{0}(2)\backslash(\mathfrak{H}\cup\mathbb{Q}\cup\{i\infty\})$, and that this ratio tends to $1$ as $z$ approaches the infinite cusp. Employing an identity due to Chan–Zudilin [20, (4.3)], we rewrite (3.2.5) as

[TABLE]

The two equations above add up to (3.2.3).

The expansion in (3.2.4) follows directly from (3.2.3) and $\eta(z)=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n})$. $\blacksquare$

Proposition 3.2.2** (Eichler integral representation of $\mathscr{I}(u)$).**

Let $\zeta(3)=\sum_{n=1}^{\infty}n^{-3}$ be Apéry’s constant. For $z/i>0$, we have

[TABLE]

which parametrizes $\int_{0}^{\infty}I_{0}(xt)[K_{0}(t)]^{4}t\operatorname{d}t$ for $x\in(0,2)$. Moreover, the equation above remains valid for z=\frac{1}{2}+iy,y\in\bigg{(}0,\frac{1}{2\sqrt{3}}\bigg{)}, corresponding to $x\in(0,2)$; and for $z=\frac{1}{2}+\frac{i}{2\sqrt{3}}e^{i\varphi},\varphi\in[0,\pi/3]$, corresponding to $x\in[2,4]$.

Proof.

Unlike the expressions $\int_{0}^{\infty}I_{0}(xt)I_{0}(t)[K_{0}(t)]^{3}xt\operatorname{d}t$ and $\int_{0}^{\infty}J_{0}(xt)[J_{0}(t)]^{4}xt\operatorname{d}t$ (covered in Proposition 3.1.2), which are annihilated by the Picard–Fuchs operator [8, (2.6) and (2.7)]

[TABLE]

the function $x\mathscr{I}(x^{2})=\int_{0}^{\infty}I_{0}(xt)[K_{0}(t)]^{4}xt\operatorname{d}t$ satisfies an inhomogeneous differential equation [cf. 4, Theorem 2.2.1]:

[TABLE]

For a solution to the homogeneous equation $\widehat{A}_{4}[f(x)]=0$, a modular parametrization [cf. 8, Remark 4.10] $x=\frac{1}{i}\left[\frac{2\eta(2z)\eta(6z)}{\eta(z)\eta(3z)}\right]^{3}$ leaves us general solutions in the form of

[TABLE]

where the constants $c_{0}$, $c_{1}$, $c_{2}$ can be determined by the behavior of $f(x)$ in specific contexts. We have the simple functional form in (3.2.10) because the operator $\widehat{A}_{4}$ is a symmetric square [8, Remark 4.6] and the corresponding family of $K3$ surfaces $(1+X+Y+Z)(1+X^{-1}+Y^{-1}+Z^{-1})=u$ admit Shioda–Inose structure (see [27, Corollary 7.1], [4, §3.2] and [31, §5]).

To construct a particular solution to the inhomogeneous equation in (3.2.9), we follow the Bloch–Kerr–Vanhove recipe [4, (2.3.9)], and derive the differential equation for the Wrońskian determinant $W(x)$ via

[TABLE]

Here, we determine the normalizing constant $\kappa=1024i/\pi^{3}$ for the Wrońskian

[TABLE]

by choosing a basis

[TABLE]

differentiating in $x$ with the help of (3.2.4) in Lemma 3.2.1 for small values of $q=e^{2\pi iz}\to 0$, and extracting the leading coefficient in the $q$-expansion $\frac{\kappa}{512}[1-30q+474q^{2}+O(q^{3})]=\frac{2i}{\pi^{3}}[1-30q+474q^{2}+O(q^{3})]$. Then, we simplify the integral representation of a particular solution [cf. 4, (2.3.8)]

[TABLE]

where

[TABLE]

using the cofactors

[TABLE]

With the parametrization $x=\frac{1}{i}\left[\frac{2\eta(2z)\eta(6z)}{\eta(z)\eta(3z)}\right]^{3},\mathscr{X}=\frac{1}{i}\left[\frac{2\eta(2\mathscr{Z})\eta(6\mathscr{Z})}{\eta(\mathscr{Z})\eta(3\mathscr{Z})}\right]^{3}$, we see that the general solution $f(\mathscr{X})$ to the inhomogeneous equation $\widehat{A}_{4}f(\mathscr{X})=-24\mathscr{X}^{3}$ is

[TABLE]

Since $Z_{6,3}(z)\to 1$ as $z\to i\infty$, we must have

[TABLE]

for our Eichler integral representations of Bessel moments.

When $z/i>0$ or $z=\frac{1}{2}+iy$ for $y>\frac{1}{2\sqrt{3}}$, according to Chan–Zudilin [20, (3.3) and (3.5)], we have

[TABLE]

where the two double sums appear in Ramanujan’s cubic theory for elliptic functions [3, Chap. 33]. Meanwhile, Borwein–Borwein–Garvan [5, Proposition 2.2(i)(ii) and Theorem 2.6(i)] identified the product of these two double sums with

[TABLE]

so we have a weight-4 modular form

[TABLE]

as given in the integrands of (3.2.6) and (3.2.7).

In addition to a routine analytic continuation, we need to check two more things for the extension of our modular parametrization to $x\in[2,4]$.

First, we show that the modular function $X_{6,3}(z)=\left[\frac{2\eta(2z)\eta(6z)}{\eta(z)\eta(3z)}\right]^{6}$ is real-valued along the geodesic segment $z=\frac{1}{2}+\frac{i}{2\sqrt{3}}e^{i\varphi},\varphi\in[0,\pi/3]$. From an analytic continuation of the last line in (3.1.5), it is clear that X_{6,3}\big{(}\frac{1}{2}+\frac{i}{2\sqrt{3}}e^{i\varphi}\big{)}=X_{6,3}\big{(}\frac{1}{2}+\frac{i}{2\sqrt{3}}e^{-i\varphi}\big{)}. By modular invariance with respect to $z\mapsto z-1$, we see that the same expression is also equal to X_{6,3}\big{(}-\frac{1}{2}+\frac{i}{2\sqrt{3}}e^{-i\varphi}\big{)}=\overline{X_{6,3}\big{(}\frac{1}{2}+\frac{i}{2\sqrt{3}}e^{i\varphi}\big{)}}, its own complex conjugate.

Then, by modifying our arguments in the second half of Lemma 3.1.1, we can check that X_{6,3}:\big{\{}\frac{1}{2}+\frac{i}{2\sqrt{3}}e^{i\varphi}\big{|}\varphi\in[0,\pi/3]\big{\}}\longrightarrow\big{[}-\frac{1}{4},-\frac{1}{16}\big{]} is bijective. $\blacksquare$

• Remark

In the proposition above, our modular parametrizations of the motivic integral $\mathscr{I}(u)$ differ from the Bloch–Kerr–Vanhove approach [4, (2.3.44)], but closely resemble certain Eichler integrals in our previous work [40, §4] that served as precursors to Epstein zeta functions. In fact, the only methodological innovation here is that we are now working with Eichler integrals on $\varGamma_{0}(6)_{+3}$, rather than on the simpler Hecke congruence group $\varGamma_{0}(4)$, as in [40, §4]. We refer our readers to [41, §2] for more arithmetic applications of inhomogeneous Picard–Fuchs equations. $\square$

3.3. Special values of Eichler integrals

If we want to compute the integral $\operatorname{\mathbf{IKM}}(1,4;2k+1)=\int_{0}^{\infty}I_{0}(t)[K_{0}(t)]^{4}t^{2k+1}\operatorname{d}t$ for $k\in\{1,2\}$, we may apply the differential identity in (3.1.29) to the Eichler integral representation in (3.2.7), at $z=\frac{1}{2}+\frac{i\sqrt{5}}{2\sqrt{3}}$. As we have closed-form evaluations of $X_{6,3}(z)$, $Z_{6,3}(z)$ and their derivatives at this specific CM point in Table I, the remaining challenge resides in the computation of the Eichler integral

[TABLE]

at $z=\frac{1}{2}+\frac{i\sqrt{5}}{2\sqrt{3}}$. Meanwhile, special values of higher-order derivatives, such as

[TABLE]

[with $c=\frac{1}{240\pi^{2}}\Gamma\left(\frac{1}{15}\right)\Gamma\left(\frac{2}{15}\right)\Gamma\left(\frac{4}{15}\right)\Gamma\left(\frac{8}{15}\right)$] are readily computable from the expression [see (3.2.24) and (3.3.3)]

[TABLE]

and entries in Table I.

Lemma 3.3.1** (Special values of $\mathscr{E}(z)$ and $\mathscr{E}^{\prime}(z)$).**

We have the following identities:

[TABLE]

Proof.

The evaluation in (3.3.6) comes from Theorem 2.2.2 and the special value for Z_{6,3}\bigg{(}\frac{1}{2}+\frac{i\sqrt{5}}{2\sqrt{3}}\bigg{)} in Table I.

Before computing $\mathscr{E}^{\prime}(z)$ at $z=\frac{1}{2}+\frac{i\sqrt{5}}{2\sqrt{3}}$, we need to consider

[TABLE]

Integrating by parts, we obtain

[TABLE]

Using the Wrońskian relation $I_{0}(t)K_{1}(t)+I_{1}(t)K_{0}(t)=1/t$, we get

[TABLE]

At the point $z=\frac{1}{2}+\frac{i\sqrt{5}}{2\sqrt{3}}$ where $X_{6,3}(z)=-\frac{1}{64}$, we differentiate both sides of

[TABLE]

in $z$, to deduce, respectively,

[TABLE]

and

[TABLE]

where $c=\sqrt{5}C=\frac{1}{240\pi^{2}}\Gamma\left(\frac{1}{15}\right)\Gamma\left(\frac{2}{15}\right)\Gamma\left(\frac{4}{15}\right)\Gamma\left(\frac{8}{15}\right)=\sqrt{5}\operatorname{\mathbf{IKM}}(1,4;1)/\pi^{2}$ is the “rescaled Bologna constant” introduced in Table I. Comparing the last two displayed equations, we arrive at the value of $\mathscr{E}^{\prime}\left(\frac{1}{2}+\frac{i\sqrt{5}}{2\sqrt{3}}\right)$ given in (3.3.7). $\blacksquare$

Lemma 3.3.2** (A special value of $\mathscr{E}^{\prime\prime}(z)$).**

We have the following identity:

[TABLE]

which entails

[TABLE]

Proof.

Upon comparison between (3.2.3) and (3.2.24), we see that

[TABLE]

Integrating (3.1.12), namely

[TABLE]

over the differential $\operatorname{d}X_{6,3}(z)$, we identify the left-hand side of (3.3.14) with

[TABLE]

Meanwhile, the integral representations in (3.3.2) and (3.3.3) tell us that the left-hand side of (3.3.14) is also equal to

[TABLE]

This verifies (3.3.15). $\blacksquare$

Theorem 3.3.3** ($\operatorname{\mathbf{IKM}}(1,4;3)$ and $\operatorname{\mathbf{IKM}}(1,4;5)$ via $\mathscr{E}(z)$, $\mathscr{E}^{\prime}(z)$ and $\mathscr{E}^{\prime\prime}(z)$).**

We have

[TABLE]

where $C=\frac{1}{240\sqrt{5}\pi^{2}}\Gamma\left(\frac{1}{15}\right)\Gamma\left(\frac{2}{15}\right)\Gamma\left(\frac{4}{15}\right)\Gamma\left(\frac{8}{15}\right)$ is the “Bologna constant”.

Proof.

As we twice differentiate (3.3.11) with respect to $z$, and set $X_{6,3}(z)=-\frac{1}{64}$ afterwards, we obtain a formula

[TABLE]

where the subscripts for $X_{6,3}$ and $Z_{6,3}$ are dropped, and the argument $z=\frac{1}{2}+\frac{i\sqrt{5}}{2\sqrt{3}}$ is suppressed throughout, to save space. Substituting known results from Table I and Lemmata 3.3.1–3.3.2, we may transcribe the last equality into

[TABLE]

which confirms the evaluation for $\operatorname{\mathbf{IKM}}(1,4;3)$.

Taking fourth-order derivatives on (3.3.11), we arrive at

[TABLE]

where

[TABLE]

can be derived in a similar vein as (3.3.10), and the relation $\operatorname{\mathbf{IKM}}(0,3;3)=2[2\operatorname{\mathbf{IKM}}(0,3;1)-1]/3$ has been proved in [1, §3.2]. Now that the left-hand side of (3.3.23) equals

[TABLE]

and its right-hand side amounts to

[TABLE]

we can simplify the relation above into

[TABLE]

This confirms the evaluation for $\operatorname{\mathbf{IKM}}(1,4;5)$. (Furthermore, based on the recursion for the rescaled moments $\operatorname{\mathbf{IKM}}(1,4;2n+1)/\pi^{2},n\in\mathbb{Z}_{\geq 0}$ [1, (11)], one can show that all of them are rational combinations of $C$ and $1/C$.) $\blacksquare$

• Remark

We have recently found [44, §2] that the closed-form evaluation of $\operatorname{\mathbf{IKM}}(1,4;3)$ can also be deduced from a result of Borwein–Straub–Vignat [7, Theorem 4.17], using Wick rotations. $\square$

• Remark

It is also possible to use factorizations of Wrońskians to compute the determinant of the matrix in (1.2.2), without evaluating the four individual Bessel moments. Such an algebraic approach is described in our recent manuscript [45, §2]. $\square$

4. Feynman diagrams with 6 Bessel factors

4.1. Modular parametrization for certain Hankel transforms

Instead of working directly on the modularity of Feynman integrals with 6 Bessel factors, we will first analyze a small building block with 4 Bessel factors. The latter problem can be solved using the classical elliptic integrals [cf. 36, §13.46, (9)], whose modular parametrization will be our major concern.

Lemma 4.1.1** (Some Wick rotations).**

1. (a)

The following identities hold:

[TABLE] 2. (b)

For $x\in[0,1)$, we have

[TABLE] 3. (c)

For $x\in[0,3)$, we have

[TABLE]

Proof.

1. (a)

As in the proof of Theorem 2.2.2, we compute

[TABLE]

where $J=J_{0}(x),Y=Y_{0}(x)$ in the last step. Applying Lemma 2.2.1 to

[TABLE]

we arrive at (4.1.1).

The proof of (4.1.2) is essentially similar. 2. (b)

By Jordan’s lemma, we can justify the following Wick rotation for $x\in[0,1)$:

[TABLE]

where $J=J_{0}(t),Y=Y_{0}(t)$ in the last expression. Meanwhile, by a variation on Lemma 2.2.1, we have

[TABLE]

so the claimed identity is proved. 3. (c)

To show (4.1.4), simply take a Wick rotation:

[TABLE]

where we use the abbreviation $J=J_{0}(t),Y=Y_{0}(t)$ as before.

For (4.1.5), Wick rotation alone brings us

[TABLE]

In the meantime, we extend the technique in Lemma 2.2.1 to

[TABLE]

which implies

[TABLE]

The equation above allows us to eliminate the term $\int_{0}^{\infty}Y_{0}(xt)JY^{2}t\operatorname{d}t$ from (4.1.11) and arrive at the right-hand side of (4.1.5). $\blacksquare$

Let $h(x)=\int_{0}^{\infty}J_{0}(xt)I_{0}(t)[K_{0}(t)]^{2}t\operatorname{d}t$ be the Hankel transform of the function $I_{0}(t)[K_{0}(t)]^{2}$, and $\widetilde{h}(x)=\int_{0}^{\infty}J_{0}(xt)[J_{0}(t)]^{3}t\operatorname{d}t$ be a “random walk integral” ($\widetilde{h}(x)=p_{3}(x)/x$, where $p_{3}(x)$ is the radial probability density of the distance travelled by a random walker in the plane, taking three consecutive steps of unit lengths). According to the Parseval–Plancherel theorem for Hankel transforms [cf. 1, (16)], we have

[TABLE]

In order to recast the left-hand sides of the equations above into Eichler integrals, we need to represent the Hankel transforms $h(x)$ and $\widetilde{h}(x)$ as modular forms.

Proposition 4.1.2** (Modular parametrizations of two Hankel transforms).**

1. (a)

For $x>0$, we have a hypergeometric evaluation

[TABLE]

which can be parametrized as

[TABLE]

where $\theta(z):=\sum_{n\in\mathbb{Z}}e^{\pi in^{2}z}$ is one of Jacobi’s elliptic theta functions (“Thetanullwerte”), and $w=-\frac{1}{2}+iy$ for $y>0$. 2. (b)

For $x\in(0,1)$, the function $p_{3}(x)/x=\int_{0}^{\infty}J_{0}(xt)[J_{0}(t)]^{3}t\operatorname{d}t$ admits a modular parametrization

[TABLE]

where $w=(1+e^{i\varphi})/6,\varphi\in(0,\pi)$; for $x>3$, we have $p_{3}(x)/x=0$.

Proof.

1. (a)

For sufficiently small $x$, we have

[TABLE]

by the Wick rotation in (4.1.3) and an analytic continuation of the hypergeometric representation for $\int_{0}^{\infty}J_{0}(xt)[J_{0}(t)]^{3}t\operatorname{d}t$ [8, (3.4)]. Setting $p=-\frac{2x^{2}}{x^{2}+3}$ in the following hypergeometric identity [3, Chap. 33, Theorem 6.1]:

[TABLE]

we recast (4.1.23) into (4.1.17). The validity of (4.1.17) extends to all $x>0$, by analytic continuation.

With a substitution x=i\big{[}\theta\big{(}-\frac{2}{3z}-1\big{)}\big{/}\theta\big{(}-\frac{2}{z}-3\big{)}\big{]}^{2}, one can verify

[TABLE]

by showing that the ratio between the two sides defines a bounded function on $X_{0}(3)=\varGamma_{0}(3)\backslash(\mathfrak{H}\cup\mathbb{Q}\cup\{i\infty\})$, and that the leading order $q$-expansions of both sides agree. One can also check that the geodesic $z=(5+e^{i\varphi})/12,\varphi\in(0,\pi)$ is mapped bijectively to $x\in(0,\infty)$, using a method similar to what was employed in the proof of Proposition 3.2.2.

Meanwhile, with the aforementioned relation between $x\in(0,\infty)$ and $z=(5+e^{i\varphi})/12$ for $\varphi\in(0,\pi)$, we paraphrase an identity [3, Chap. 33, Corollary 3.4] from Ramanujan’s notebook as follows:

[TABLE]

Multiplying both sides with

[TABLE]

we obtain

[TABLE]

Furthermore, by a theta function identity [2, Chap. 18, (24.31)] in Ramanujan’s notebook, we have

[TABLE]

and the last expression can be reduced by an identity

[TABLE]

also due to Ramanujan [2, Chap. 16, Entry 24(iii) and Chap. 20, Entry 1(ii)].

Finally, setting $\tau=1-\frac{1}{3z}$ and $\frac{2z-1}{3z-1}=1+2w\in i\mathbb{R}$ for $z=(5+e^{i\varphi})/12,\varphi\in(0,\pi)$, while simplifying eta functions with the modular transformation $\eta(-1/\tau^{\prime})=\sqrt{\tau^{\prime}/i}\eta(\tau^{\prime})$ where necessary, we arrive at the expression in (4.1.18). 2. (b)

The modular parametrization in (4.1.19) follows directly from analytic continuation of (4.1.18) and the Wick rotation relation in (4.1.3).

One notes that the smooth functions $p_{3}(x),x\in(0,1)$ and $p_{3}(x),x\in(1,3)$ satisfy the same ordinary differential equation of second order [8, Theorem 2.4], so $p_{3}(x)/x,x\in(1,3)$ must be a linear combination of

[TABLE]

for x=\left[\theta\left(1-\frac{1}{3w}\right)\big{/}\theta\left(3-\frac{1}{w}\right)\right]^{2}. Here, the linear combination must be proportional to $(1-3w)$, so as to guarantee finiteness of $p_{3}(x)/x$ in the $x\to 3-0^{+}$ regime. The precise prefactor can be determined by asymptotic analysis of $p_{3}(x)/x$ and $q$-expansion of the modular form. This proves (4.1.20).

For $x>3$, one can prove $\int_{0}^{\infty}J_{0}(xt)[J_{0}(t)]^{3}t\operatorname{d}t=0$ by extracting the real part from the following Wick rotation:

[TABLE]

Alternatively, one may invoke the probabilistic interpretation of $p_{3}(x)=\int_{0}^{\infty}J_{0}(xt)[J_{0}(t)]^{3}xt\operatorname{d}t$ to conclude that $p_{3}(x)/x=0$ for $x>3$. $\blacksquare$

• Remark

The modular parametrizations in the proposition above are foreshadowed by the following formula (see [36, §13.46, (9)] and [6, (3)]) for $x\in(0,1)\cup(1,3)$:

[TABLE]

and the fact that [3, Chap. 33, Lemma 5.5 and Theorem 5.6]

[TABLE]

Formally, we may regard (4.1.18) as an analytic continuation of the identities above, along with a modular transformation corresponding to (4.1.28). $\square$

• Remark

It is also possible to parametrize the aforementioned Hankel transforms without using Jacobi’s theta functions. For example, after comparing the Taylor expansion of $p_{3}(x),0\leq x<1$ due to Borwein–Straub–Wan–Zudilin [6, (3.2)] to Zagier’s Apéry-like recurrence (Case C) [38], we arrive at

[TABLE]

for $w/i>0$, which is an alternative formulation of (4.1.19). For yet another approach to this modular parametrization, see Broadhurst’s recent talks at Vienna ([16, §1.2] and [17, §1.2]), which refers to his earlier talk at Les Houches [11, §2.5]. $\square$

In addition to the usual Hankel transform $\int_{0}^{\infty}J_{0}(xt)f(t)t\operatorname{d}t$ of a function $f(t),t\in(0,\infty)$, we will also need the $Y$-transform $\int_{0}^{\infty}Y_{0}(xt)f(t)t\operatorname{d}t$ and the $K$-transform $\int_{0}^{\infty}K_{0}(xt)f(t)t\operatorname{d}t$ for certain Bessel moments.

Proposition 4.1.3** ($Y$- and $K$-transforms).**

1. (a)

We have

[TABLE]

where $w=-\frac{1}{2}+iy$ for $y>0$, and

[TABLE]

for $w/i>0$. 2. (b)

We have

[TABLE]

for $w/i>0$ and $w=(1+e^{i\varphi})/6,\varphi\in(0,\pi)$.

Proof.

1. (a)

We observe that the sequences $c_{3,k}:=\int_{0}^{\infty}[K_{0}(t)]^{3}t^{k}\operatorname{d}t$ and $s_{3,k}:=\int_{0}^{\infty}I_{0}(t)[K_{0}(t)]^{2}t^{k}\operatorname{d}t$ satisfy the same recursion [1, (8)], namely, $(k+1)^{4}c_{3,k}-2(5k^{2}+20k+21)c_{3,k+2}+9c_{3,k+4}=0$ and $(k+1)^{4}s_{3,k}-2(5k^{2}+20k+21)s_{3,k+2}+9s_{3,k+4}=0$ both hold for non-negative integers $k$. As a result, the function

[TABLE]

is annihilated by the differential operator

[TABLE]

and we have an inhomogeneous differential equation

[TABLE]

Meanwhile, differentiating under the integral sign and integrating by parts [cf. 35, §9], we can verify that

[TABLE]

In view of the analysis above, the left-hand side of (4.1.47) must be equal to

[TABLE]

where $k_{0}$ and $k_{1}$ are constants. Since $Y_{0}(xt)=\frac{2}{\pi}\log(xt)+O(1)$ as $x\to 0^{+}$, and $\int_{0}^{\infty}I_{0}(t)[K_{0}(t)]^{2}t\operatorname{d}t=\frac{\pi}{3\sqrt{3}}$ [1, (23)], we can determine $k_{1}=\frac{\pi^{2}}{2\sqrt{3}i}$ immediately. Superimposing with (4.1.18), we obtain

[TABLE]

which analytically continues to

[TABLE]

for $w/i>0$. Taking the $w\to i0^{+}$ limit, and recalling the evaluation $\int_{0}^{\infty}I_{0}(t)[K_{0}(t)]^{3}t\operatorname{d}t=\pi^{2}/16$ from [1, (54)], we find $k_{0}=0$.

Thus far, we have confirmed both (4.1.47) and (4.1.48). 2. (b)

We note that the expression $\int_{0}^{\infty}I_{0}(xt)[K_{0}(t)]^{3}t\operatorname{d}t+3\int_{0}^{\infty}K_{0}(xt)I_{0}(t)[K_{0}(t)]^{2}t\operatorname{d}t$ is continuous with respect to $x\in(0,3)$, and the right-hand side of (4.1.48) is smooth in a neighborhood of $i0^{+}$. Therefore, the validity of (4.1.48) extends to the geodesic $w=(1+e^{i\varphi})/6,\varphi\in(0,\pi)$, by analytic continuation.

Adding up (4.1.4) and (4.1.5), we derive (4.1.49) from (4.1.48). $\blacksquare$

4.2. Eichler integrals via Hankel fusions

We can now use the modular parametrizations in Proposition 4.1.2 to fuse Hankel transforms into Feynman integrals involving 6 Bessel factors, as planned in (4.1.14).

Proposition 4.2.1** (Eichler formulation of $\operatorname{\mathbf{IKM}}(2,4;1)$).**

We have

[TABLE]

Proof.

By the Parseval–Plancherel theorem for Hankel transforms, we have

[TABLE]

Here, for $\tau=2-\frac{1}{6w}$, the modular parameter [cf. (4.1.37) and (4.1.38)]

[TABLE]

satisfies [cf. (3.2.5*′*) and (3.2.5*′′*)]

[TABLE]

so (4.2.1) follows immediately. $\blacksquare$

Proposition 4.2.2** (Eichler formulation of $\operatorname{\mathbf{JYM}}(6,0;1)$).**

We have

[TABLE]

Proof.

Applying the arguments in the last proposition directly to (4.1.19) and (4.1.20), we obtain

[TABLE]

where the second integral runs along the semi-circular path $w=(1+e^{i\varphi})/6,\varphi\in(0,\pi)$.

Before arriving at the expression in (4.2.5), we need to perform modular transformations on the last integral.

Towards this end, we recall from Chan–Zudilin [20] that the group $\varGamma_{0}(6)_{+2}=\langle\varGamma_{0}(6),\widehat{W}_{2}\rangle$, constructed by adjoining $\widehat{W}_{2}=\frac{1}{\sqrt{2}}\left(\begin{smallmatrix}2&-1\\ 6&-2\end{smallmatrix}\right)$ to $\varGamma_{0}(6)$, enjoys a Hauptmodul

[TABLE]

and a weight-2 modular form

[TABLE]

With these notations, we see that $[\eta(z)\eta(2z)\eta(3z)\eta(6z)]^{2}=[Z_{6,2}(z)]^{2}X_{6,2}(z)$ is a modular form of weight 4 on $\varGamma_{0}(6)_{+2}$. In particular, we have

[TABLE]

Consequently, a variable substitution $w=\widehat{W}_{2}z$ brings us

[TABLE]

thereby completing the proof. $\blacksquare$

David Broadhurst considered the following modular form of weight 4 and level 6

[TABLE]

based on a suggestion from Francis Brown at Les Houches in 2010. Drawing on the work of Hulek et al. [22] that related the aforementioned modular form to a Kloosterman problem, Broadhurst conjectured that $\operatorname{\mathbf{IKM}}(2,4;1)$ is equal to $\frac{3}{2}L(f_{4,6},3)$ [12, (110)], where the special $L$-value can be written explicitly as [12, (108)]

[TABLE]

We now verify Broadhurst’s conjecture.

Theorem 4.2.3** ($\operatorname{\mathbf{IKM}}(2,4;1)$ as a critical $L$-value).**

We have

[TABLE]

Proof.

Judging from termwise integration of uniformly convergent series, we note that Broadhurst’s conjecture essentially says that

[TABLE]

What we will do is to show that this statement is consistent with our results in Propositions 4.2.1 and 4.2.2. Here, one can prove

[TABLE]

by a change of variable $w=-1/{(6z)}$ and the modular transformation $\eta(-1/\tau)=\sqrt{\tau/i}\eta(\tau)$, so the right-hand side of (4.2.14) is the same as $-\pi^{3}i\int_{0}^{i\infty}f_{4,6}(w)\operatorname{d}w.$ However, according to Propositions 4.2.1 and 4.2.2, we have

[TABLE]

Meanwhile, the Wick rotation in (4.1.1) tells us that this is precisely $\operatorname{\mathbf{IKM}}(2,4;1)$, as conjectured by Broadhurst. $\blacksquare$

Before applying Proposition 4.1.3 to the 4-loop sunrise diagram $\operatorname{\mathbf{IKM}}(1,5;1)$, we need a cancelation formula related to Hankel and $Y$-transforms.

Lemma 4.2.4** (Hilbert cancelation).**

Consider a continuous function $F(t),t>0$, whose Kramers–Kronig transform

[TABLE]

is well-defined, and has the following asymptotic behavior:

[TABLE]

Suppose that $\int_{0}^{\infty}J_{0}(xt)F(t)t\operatorname{d}t,x\in(0,\infty)$ and $\int_{0}^{\infty}Y_{0}(xt)F(t)t\operatorname{d}t,x\in(0,\infty)$ are both well-defined, then

[TABLE]

Proof.

According to the asymptotic behavior of $\widehat{\mathscr{K}}F$, we have a vanishing identity for all $x>0$:

[TABLE]

Here, the contour can be closed upwards, thanks to Jordan’s lemma. As $\operatorname{Im}\tau\to 0^{+}$, we have the following Plemelj jump relation for $\xi\in(-\infty,0)\cup(0,\infty)$:

[TABLE]

where $\mathscr{P}$ denotes Cauchy principal value. Here, the first term on the right-hand side of the equation above is the Hilbert transform of an even function $F(|t|)|t|,t\in(-\infty,0)\cup(0,\infty)$, so it must be an odd function in $\xi$ [23, §4.2]. Meanwhile, we know that

[TABLE]

so the vanishing identity in (4.2.20) brings us

[TABLE]

Now we compute

[TABLE]

The last contour integral is indeed vanishing, because the integrand remains bounded as $\tau\to i0^{+}$, and we can close the contour upwards, according to the asymptotic behavior of the Kramers–Kronig transform $\widehat{\mathscr{K}}F$. $\blacksquare$

Theorem 4.2.5** (Sunrise at 4 loops).**

We have

[TABLE]

as stated in (1.2.3).

Proof.

The first equality in (4.2.25) has been proved in [43, Lemma 3.1], as a special case ($m=3,n=1$) of (1.1.1). The last equality comes from the definition of $L$-functions via Mellin transforms of cusp forms. The rest of this proof will revolve around the second equality.

We combine (4.1.18) with (4.1.47), and carry out computations as in Proposition 4.2.1, to arrive at

[TABLE]

Here, we have used the Parseval–Plancherel identity

[TABLE]

and the Hilbert cancelation

[TABLE]

By an analog of Proposition 4.2.2, we fuse (4.1.19)–(4.1.20) and (4.1.49) together into the following formula:

[TABLE]

Again, a variable substitution $w=\widehat{W}_{2}z$ gives rise to

[TABLE]

Thus, we have

[TABLE]

by cancelation of Eichler integrals. We can rewrite the equation above as

[TABLE]

with the aid of (4.1.2). As we have [cf. 12, (107)]

[TABLE]

by termwise integration, this completes the proof. $\blacksquare$

Like the determinant of (1.2.2), Broadhurst–Mellit also proposed that [12, (113)]

[TABLE]

We have recently verified this conjecture in [45, §3], without explicitly computing individual matrix elements.

The Eichler integral representations for the first column in the determinant above have already been discussed. In a recent talk at the Erwin Schrödinger Institute [18, §7.3], Broadhurst has announced his discoveries of representations for the second column as integrals over modular forms. We now prove Broadhurst’s empirical formulae.

Theorem 4.2.6** (Broadhurst integrals for $\operatorname{\mathbf{IKM}}(1,5;3)$ and $\operatorname{\mathbf{IKM}}(2,4;3)$).**

Setting $v=3\left[\frac{\eta(3z)}{\eta(z)}\right]^{4}\left[\frac{\eta(2z)}{\eta(6z)}\right]^{2}$ and $G(z)=f_{4,6}(z)(v^{4}-6v^{2}+2-6v^{-2}+9v^{-4})$, we have

[TABLE]

Proof.

Writing $f(u):=\int_{0}^{\infty}J_{0}(\sqrt{u}t)I_{0}(t)[K_{0}(t)]^{2}t\operatorname{d}t$ for $u>0$, and using the Bessel differential equation along with $\widehat{B}_{3}f(u)=0$ [cf. (4.1.53)], one can show that

[TABLE]

By Hankel fusion and integration by parts, we have

[TABLE]

As we may recall from Proposition 4.2.1, the differential form $\frac{[f(u)]^{2}\operatorname{d}u}{2}$ translates into $\frac{\pi^{3}i}{3}[\eta(z)\eta(2z)$ $\eta(3z)\eta(6z)]^{2}\operatorname{d}z$ for $\operatorname{Re}z=-\frac{1}{2}$, and

[TABLE]

so $\operatorname{\mathbf{IKM}}(2,4;3)$ has an integral representation:

[TABLE]

Here, the path of integration can be shifted to $\operatorname{Re}z=\frac{1}{2}$, by periodicity of the integrand. To identify the integrand inside the braces of (4.2.40) with $G(z)/96$ in (4.2.35), simply compare their $q$-expansions up to sufficiently many terms [20, Remark 1]. This proves Broadhurst’s integral representation for $\operatorname{\mathbf{IKM}}(2,4;3)$ in (4.2.35).

To verify (4.2.36), we start by rewriting (4.1.47) as

[TABLE]

and noting that $\widehat{B}_{3}g(u)=0$. We can subsequently deduce Broadhurst’s integral representation for $\operatorname{\mathbf{IKM}}(1,5;3)$ from Hankel fusion and a vanishing identity for $F(t)=I_{0}(t)[K_{0}(t)]^{2}$:

[TABLE]

which is provable by a modest variation on Lemma 4.2.4. $\blacksquare$

5. Feynman diagrams with 8 Bessel factors

5.1. Hankel transforms and Wick rotations

We open this section by a confirmation of Broadhurst’s conjecture on $\operatorname{\mathbf{IKM}}(2,6;1)$.

Theorem 5.1.1** (Eichler integral formulation of $\operatorname{\mathbf{IKM}}(2,6;1)$).**

We have

[TABLE]

Proof.

By the Parseval–Plancherel theorem for Hankel transforms, we have

[TABLE]

With the modular parametrization in (3.1.10), and the Jacobian in (3.2.3), we transition from an integration over the variable $x\in(0,\infty)$ to its counterpart over the variable $z$ on the $\operatorname{Im}z$-axis. Accordingly, we see that

[TABLE]

descends from (5.1.2).

Meanwhile, one can establish the following identity

[TABLE]

by verifying that both sides are weight-6 modular forms on $\varGamma_{0}(6)$, and checking the $q$-expansions of both sides up to sufficiently many terms [20, Remark 1]. $\blacksquare$

• Remark

Encouraged by Yun’s recent contribution to Kloosterman sums [37], Broadhurst wrote [12, (135)]

[TABLE]

and conjectured that $\operatorname{\mathbf{IKM}}(2,4;1)=\frac{27}{2}L(f_{6,6},5)$ for [12, (141) and (145)]

[TABLE]

This said the same thing as

[TABLE]

which is also equivalent to (5.1.1) per a Fricke involution $z\mapsto-1/(6z)$ and a modular transformation $\eta(-1/\tau)=\sqrt{\tau/i}\eta(\tau)$. $\square$

• Remark

In an earlier version of his conjecture, Broadhurst formulated the modular form $f_{6,6}$ as [10, (90) and (91)]

[TABLE]

This is of course compatible with the left-hand side of (5.1.4), in view of an identity by Borwein–Borwein–Garvan [5, Proposition 2.2(i)(ii) and Theorem 2.6(i)]. $\square$

Before handling other Bessel moments $\operatorname{\mathbf{IKM}}(a,b;1)$ satisfying $a+b=8$, we need a modest generalization of Lemma 4.1.1 and modular parametrizations of some Hankel transforms not covered in §3.

Lemma 5.1.2** (Some identities for Bessel moments).**

1. (a)

The following formulae are true:

[TABLE] 2. (b)

For $x\in[0,2]$, we have

[TABLE] 3. (c)

For $x\in[0,2]$, we have

[TABLE]

Proof.

1. (a)

By Wick rotation, we have

[TABLE]

for $J=J_{0}(x),Y=Y_{0}(x)$. With

[TABLE]

we are able to reduce (5.1.14) into (5.1.9), by virtue of (2.2.1*′*) in Lemma 2.2.1.

One can prove (5.1.10) in a similar vein.

To prove (5.1.11), compute

[TABLE]

and invoke (2.2.1*′*). 2. (b)

By a variation on (4.1.9), we have the following vanishing identity when $x\in[0,2]$:

[TABLE]

with $J=J_{0}(t),Y=Y_{0}(t)$. 3. (c)

By Wick rotation, we can show that

[TABLE]

where $J=J_{0}(t),Y=Y_{0}(t)$. Meanwhile, when $x\in[0,4]$, we also have

[TABLE]

by an extension of Lemma 2.2.1. $\blacksquare$

Proposition 5.1.3** (Hankel transforms related to $\operatorname{\mathbf{JYM}}$).**

1. (a)

For $z=\frac{1}{2}+\frac{i}{2\sqrt{3}}e^{i\varphi},\varphi\in(0,\pi/3)$, we have

[TABLE]

where $x=i\left[\frac{2\eta(2z)\eta(6z)}{\eta(z)\eta(3z)}\right]^{3}$ maps $\varphi\in(0,\pi/3)$ bijectively to $x\in(2,4)$; for $x\geq 4$, we have

[TABLE]

Consequently, we have

[TABLE] 2. (b)

For $z=\frac{1}{2}+iy,y\in\left(\frac{1}{2\sqrt{3}},\infty\right)$ and $z=\frac{1}{2}+\frac{i}{2\sqrt{3}}e^{i\varphi},\varphi\in[0,\pi/3)$, the formula

[TABLE]

parametrizes $\int_{0}^{\infty}J_{0}(xt)[J_{0}(t)]^{2}[Y_{0}(t)]^{2}t\operatorname{d}t$ for $x\in(0,4)$, and brings us

[TABLE]

In addition, for $z=(1+e^{i\psi})/6,\psi\in[\pi/3,\pi)$, we have

[TABLE]

which parametrizes $\int_{0}^{\infty}J_{0}(xt)[J_{0}(t)]^{2}[Y_{0}(t)]^{2}t\operatorname{d}t$ for $x\in[4,\infty)$ and leads us to

[TABLE] 3. (c)

For $z=\frac{1}{2}+iy,y\in\left(\frac{1}{2\sqrt{3}},\infty\right)$, we have

[TABLE]

a formula that parametrizes $\int_{0}^{\infty}J_{0}(xt)[J_{0}(t)]^{3}Y_{0}(t)t\operatorname{d}t$ for $x\in[4,\infty)$. As a result, the following identity holds:

[TABLE]

Proof.

1. (a)

Judging from (3.2.10), we know that

[TABLE]

where the constants $c_{0}$, $c_{1}$ and $c_{2}$ can be determined by the continuity at $x=2$ and the asymptotic behavior as $x\to 4^{-}$ [8, Theorem 4.1]. This proves (5.1.20).

To show (5.1.21), read off the real part from the following Wick rotation:

[TABLE]

Applying the Parseval–Plancherel theorem for Hankel transforms to (3.1.12) and (5.1.20), we arrive at (5.1.22). 2. (b)

For $z=\frac{1}{2}+iy,y\in\left(\frac{1}{2\sqrt{3}},\infty\right)$, the Hankel transform formula in (5.1.23) follows from (3.1.12) and (5.1.12). The remaining arguments run parallel to those in (a). 3. (c)

To verify (5.1.27), simply combine (3.1.11) with (5.1.13). The rest founds on similar principles as the proof of (a). $\blacksquare$

• Remark

We note that Borwein et al. expressed $\int_{0}^{\infty}J_{0}(xt)[J_{0}(t)]^{4}t\operatorname{d}t,x\in(2,4)$ as generalized hypergeometric series [8, Theorem 4.7], but did not give a modular parametrization. $\square$

Proposition 5.1.4** ($Y$- and $K$-transforms).**

For $z=\frac{1}{2}+iy,y\in\left(\frac{1}{2\sqrt{3}},\infty\right)$, we have

[TABLE]

For $z/i>0$, we have

[TABLE]

Proof.

Let $\widehat{A}_{4}$ be the Picard–Fuchs operator given in (3.2.8), then one can verify

[TABLE]

by differentiation under the integral sign, and integration by parts [cf. 35, §9]. Comparing this to (3.2.9), we know that $\int_{0}^{\infty}I_{0}(xt)[K_{0}(t)]^{4}xt\operatorname{d}t+4\int_{0}^{\infty}K_{0}(xt)I_{0}(t)[K_{0}(t)]^{3}xt\operatorname{d}t$ is annihilated by $\widehat{A}_{4}$. Therefore, the left-hand side of (5.1.33) must assume the form

[TABLE]

for certain constants $k_{0}$, $k_{1}$, and $k_{2}$. Since $K_{0}(xt)=-\log(xt)+O(1)$ as $x\to 0^{+}$, and $\int_{0}^{\infty}I_{0}(t)[K_{0}(t)]^{3}t\operatorname{d}t=\pi^{2}/16$ [1, (54)], the left-hand side of (5.1.33) behaves like $\frac{\pi^{3}(2z-1+o(z))}{8i}Z_{6,3}(z)$ as $z\to\frac{1}{2}+i\infty$. This shows that $k_{1}=\frac{\pi^{3}}{8i}$ and $k_{2}=0$. To demonstrate that $k_{0}=0$, simply check the special value at $z=\frac{1}{2}+\frac{i\sqrt{5}}{2\sqrt{3}}$ against Theorem 2.2.2 and Table I.

As we perform analytic continuation on the left-hand side of (5.1.33) to the positive $\operatorname{Im}z$-axis, and extract the real part, we arrive at (5.1.34). $\blacksquare$

• Remark

From a Hilbert transform formula [cf. 43, (3.2)]

[TABLE]

we can deduce [cf. (4.2.23)]

[TABLE]

Thus, we may recast (5.1.34) into

[TABLE]

for $z/i>0$. $\square$

• Remark

From (4.1.48) and (5.1.33), we see that when $n$ is 3 or 4, and $u\in(0,1)$, the expression

[TABLE]

is annihilated by a differential operator (in $u$) of order $n-1$. The same pattern actually applies to all $n\in\mathbb{Z}_{\geq 2}$, and the corresponding differential operator has been constructed by Vanhove in [35, §9]. The steps of integrations by parts leading to these homogeneous differential equations are described in [45, Lemma 4.2]. Such homogeneous differential equations are crucial in our recent proofs [45, §4] of two determinant formulae proposed by Broadhurst–Mellit [12, Conjectures 4 and 7]. $\square$

5.2. Critical $L$-values for Bessel moments

A conjectural sum rule $9\pi^{2}\operatorname{\mathbf{IKM}}(4,4;1)-14\operatorname{\mathbf{IKM}}(2,$ $6;1)=0$ dated back to 2008 [1, at the end of §6.3, between (228) and (229)], and was restated as an open problem in 2016 [12, (147)]. It has also been conjectured that [12, (139) and (143)]

[TABLE]

With the preparations in §5.1, we can verify these claims.

Theorem 5.2.1** (Relation between $\operatorname{\mathbf{IKM}}(2,6;1)$ and $\operatorname{\mathbf{IKM}}(4,4;1)$).**

1. (a)

We have a vanishing identity

[TABLE] 2. (b)

We have a sum rule

[TABLE]

Proof.

1. (a)

We spell out both sides of (5.1.11) using Hankel fusions. The left-hand side becomes

[TABLE]

where we have transformed

[TABLE]

by a Fricke involution $z\mapsto-1/(6z)$ and a horizontal translation. The right-hand side becomes

[TABLE]

according to (5.1.9), (5.1.22), and (5.1.24). We bear in mind that $f_{6,6}(z)=[Z_{6,2}(z)]^{3}X_{6,2}(z)[1+9X_{6,2}(z)]$ is a modular form of weight 6 on $\varGamma_{0}(6)_{+2}$, which transforms under $\widehat{W}_{2}z=\frac{2z-1}{6z-2}$ as

[TABLE]

Thus, the identities

[TABLE]

allow us to rewrite (5.2.6) as

[TABLE]

Identifying (5.2.4) with (5.2.10), we arrive at (5.2.2), as claimed. 2. (b)

In the light of (5.1.9) and (5.1.10), we see that the proposed sum rule is equivalent to the following vanishing identity:

[TABLE]

We may compute

[TABLE]

where the first equality comes from (5.1.22) and (5.1.24), while the second and third equalities hinge on (5.2.7) and (5.2.2), respectively. $\blacksquare$

Theorem 5.2.2** (Relation between $L(f_{6,6},3)$ and $L(f_{6,6},5)$).**

1. (a)

We have

[TABLE] 2. (b)

We have

[TABLE] 3. (c)

The following integral identity holds:

[TABLE]

which implies

[TABLE]

where

[TABLE]

Proof.

1. (a)

According to (5.1.9), (5.1.22), (5.1.24) and (5.2.7), we have

[TABLE]

In the meantime, by complex conjugation, we have

[TABLE]

whereas $f_{6,6}(z)=f_{6,6}(z+1)$ brings us

[TABLE]

Therefore, we obtain

[TABLE]

after invoking $f_{6,6}(-1/(6z))=-216z^{6}f_{6,6}(z)$ in the last step.

All this allows us to rearrange (5.2.18) into (5.2.13). 2. (b)

In view of (5.1.9), (5.1.22), and (5.1.30), we have

[TABLE]

As before, we may reduce

[TABLE]

[TABLE]

and

[TABLE]

By virtue of the vanishing identity in (5.2.2), the right-hand side of (5.2.25) is also equal to

[TABLE]

Gathering the results above, we arrive at (5.2.14). 3. (c)

Eliminating

[TABLE]

from (5.2.13) and (5.2.14), we obtain

[TABLE]

which is equivalent to (5.2.15). [There is also an alternative way to arrive at the equation above, namely, by fusing (5.1.34*′*) with itself, and referring to (5.2.3).] Checking the definition of $L(f_{6,6},3)$ in (5.2.17) against termwise integration on the right-hand side of the following equation:

[TABLE]

we can verify (5.2.16). $\blacksquare$

• Remark

Previously, Broadhurst observed that $L(f_{6,6},5)/[\zeta(2)L(f_{6,6},3)]$ must be a rational number, according to Eichler–Shimura–Manin theory [cf. 33, Theorem 1], and found this rational number to be numerically $4/7$ [12, (142)]. $\square$

• Remark

As a by-product of the foregoing computations, one may eliminate $\operatorname{\mathbf{JYM}}(6,2;1)$ from (5.1.9) and (5.1.10), to deduce

[TABLE]

which gives $L$-series representations for a “random walk integral” $\operatorname{\mathbf{JYM}}(8,0;1)$.

Furthermore, we have recently shown [44, Theorem 5.1] that for each $j\in\mathbb{Z}_{>1}$, the function $\int_{0}^{\infty}J_{0}(xt)[J_{0}(t)]^{2j+1}t\operatorname{d}t,0\leq x\leq 1$ is a $\mathbb{Q}$-linear combination of

[TABLE]

This implies that, for all $n\in\mathbb{Z}_{>4}$, the “random walk integral” $\operatorname{\mathbf{JYM}}(n,0;1)$ is a $\mathbb{Q}$-linear combination of $\operatorname{\mathbf{IKM}}(a,b;1)/\pi^{b}$ for certain positive integers $a$ and $b$ satisfying $a+b=n$. $\square$

Finally, we verify Broadhurst’s conjectures regarding $\operatorname{\mathbf{IKM}}(1,7;1)$ and $\operatorname{\mathbf{IKM}}(3,5;1)$.

Theorem 5.2.3** (Sunrise at 6 loops).**

We have

[TABLE]

which is equivalent to (1.2.8).

Proof.

The first equality in (5.2.32), which says

[TABLE]

is a special case ($m=4,n=1$) of (1.1.2).

Fusing together (3.1.10) and (5.1.34), while noting that (see Lemma 4.2.4)

[TABLE]

we arrive at the last equality in (5.2.32), after some computations similar to those in Theorem 5.1.1. Alternatively, we can throw (3.1.10) and (5.1.34*′*) into the Parseval–Plancherel theorem for Hankel transforms, and invoke the first equality in (5.2.32).

It is clear that (5.2.32) is compatible with (1.2.8), up to a Fricke involution $z\mapsto-1/(6z)$ in the integrand. $\blacksquare$

Acknowledgments

This research was supported in part by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4).

This manuscript grew out of my research notes formerly intended for a project on automorphic representations [39, 40, 41] at Princeton in 2013, and was completed in 2017 during my stay in Beijing arranged by Prof. Weinan E (Princeton University and Peking University). I thank Dr. David Broadhurst for providing valuable background information in quantum field theory, as well as for sharing with me his slides for recent talks in Paris [13], Marseille [14], Edinburgh [15], Zeuthen [16] and Vienna [17, 18]. I also thank him for pointing out an error in an early draft.

I am deeply indebted to the anonymous referees for their careful examinations and detailed analyses of this work. I am grateful to them for their thoughtful comments and suggestions that helped me improve the presentation of this paper.