WICK ROTATIONS, EICHLER INTEGRALS, AND

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WICK ROTATIONS, EICHLER INTEGRALS, AND MULTI-LOOP FEYNMAN DIAGRAMS YAJUN ZHOU

A BSTRACT. 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 Lseries, and verify several recent conjectures of Broadhurst.

C ONTENTS 1. Introduction 1.1. Background and motivations 1.2. Statement of results and plan of proof 2. Bessel functions and their Wick rotations 2.1. Some analytic properties of Bessel functions 2.2. Contour deformations for Bessel moments 3. Feynman diagrams with 5 Bessel factors 3.1. A modular form associated with Bessel moments 3.2. Symmetric squares and Eichler integrals 3.3. Special values of Eichler integrals 4. Feynman diagrams with 6 Bessel factors 4.1. Modular parametrization for certain Hankel transforms 4.2. Eichler integrals via Hankel fusions 5. Feynman diagrams with 8 Bessel factors 5.1. Hankel transforms and Wick rotations 5.2. Critical L-values for Bessel moments Acknowledgments References

1 1 1 3 3 3 5 5 7 11 15 15 20 24 24 28 33 33

Date: July 5, 2017. Keywords: Feynman integrals, Wick rotations, Bessel functions, Hankel transforms, random walks MSC 2010: 11F03, 14E18 (Primary) 81T18, 81T40, 81Q30, 60G50 (Secondary) * 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). i

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

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1. I NTRODUCTION 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 J0 and Y0 , as well as modified Bessel functions I 0 and K 0 , to define propagators and compute FeynmanR integrals [15, 1, 9, 10, 13]. ∞ We are interested in Bessel moments JYM(α, β; γ) := 0 [ J0 ( t)]α [Y0 ( t)]β tγ d t and IKM(a, b; c) := R∞ a b c 0 [ I 0 ( t)] [K 0 ( t)] t d t, where the non-negative integers α, β, γ, a, b, c are chosen to ensure convergence of the corresponding integrals. The Bessel moments JYM’s are useful auxiliary tools for computing IKM’s in two-dimensional QFT. Furthermore, the IKM’s also show up in the finite part for renormalization group expansions of four-dimensional QFT: for example, IKM(1, 5; 1) and IKM(1, 5; 3) are part of the 4-loop contributions (amongst 891 Feynman diagrams in total) to electron’s magnetic moment [20, (19) and Fig. 3(a)(a0 )], according to the standard formulation of quantum electrodynamics (four-dimensional QFT). The mathematical understanding of JYM(α, β; γ) for α + β ≥ 5 and IKM(a, b; c) for a + b ≥ 5 is relatively scant. While numerical experiments have suggested a rich collection of identities relating various cases of 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 ∈ {5, 6, 7, 8} [9, 10, 13], most of these conjectural evaluations are heretofore unproven. In our recent work [33], we have effectively shown that the following Bessel moment Z 2 c+1 ∞ [π I 0 ( t) + iK 0 ( t)]m − [π I 0 ( t) − iK 0 ( t)]m [K 0 ( t)]m t c d t (1.1.1) m + 1 i π 0 is a non-negative rational number whenever a positive integer m and a non-negative integer c add up to an odd number m + c. Furthermore, such a rational number vanishes if and only if m > c + 1 (a variation on the Bailey–Borwein–Broadhurst–Glasser sum rule [1, “final conjecture”, (220)]); such a rational number reduces to an integer if m is even (Broadhurst–Mellit integer sequence [10, (149) in Conjecture 5]). While the aforementioned results resolve some longstanding conjectures, they barely scratch the surface of the algebraic and arithmetic nature of Bessel pmoments. 3 For example, the determinant IKM(1, 4; 1) IKM(2, 3; 3) − IKM(2, 3; 1) IKM(1, 4; 3) = 2π / 33 55 [conjectured in 10, (100)] and the sum rule 9π2 IKM(4, 4; 1) − 14 IKM(2, 6; 1) = 0 [conjectured in 10, (147)] had not been covered by the real-analytic methods we employed in [33]. 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 IKM moments, but also connect Feynman diagrams to special L-values and Kluyver’s “random walk integrals” JYM( n, 0, 1), n ∈ Z≥3 [18, 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 IKM problems into 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 ¡1¢ ¡2¢ ¡4¢ ¡8¢ Z ∞ Γ 15 Γ 15 Γ 15 Γ 15 4 I 0 ( t)[K 0 ( t)] t d t = (1.2.1) p 0 240 5 R∞ in terms of Euler’s gamma function Γ( x) := 0 u x−1 e−u d u for x > 0. It is worth noting that nearly a decade had elapsed between the original proposal [1, 19] of (1.2.1) and its first rigorous (and highly technical) verification [4, 25]. Our simplified proof of (1.2.1) draws on its connection to a “random walk integral” JYM(5, 0; 1).

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YAJUN ZHOU

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 × 2 matrix: ¡ ¢ ¡ ¢ ! µ ¶ Ã 2 1 2 2 2 π C π 13 C − IKM(1, 4; 1) IKM(1, 4; 3) 15 10C ¢ , p = p15π (1.2.2) ¡ ¢ ¡ 15π 2 2 1 IKM(2, 3; 1) IKM(2, 3; 3) C 13 C + 2 2 15 10C ¡1¢ ¡2¢ ¡4¢ ¡8¢ 1 p where C = Γ 15 Γ 15 Γ 15 Γ 15 is the “Bologna constant” attributed to Broadhurst [1] 240 5π2 and Laporta [19]. (Here, the rigorous evaluation of the top-right entry 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 K 3 surfaces formerly studied by Bloch–Kerr–Vanhove [4] and Samart [25]. In addition to proving (1.2.2) in §3.3, we work out the Eichler integral representations of IKM(1, 4; 1), IKM(1, 4; 3) and 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 10, (109)– (111)]: Z i∞ 3 3 2 IKM(1, 5; 1) = IKM(3, 3; 1) = − 6π f 4,6 ( z) z d z = L( f 4,6 , 2), (1.2.3) 2 2 π 0 Z π2 π3 i ∞ f 4,6 ( z) d z = L( f 4,6 , 1) IKM(2, 4; 1) = i 0 2 Z i∞ 3 3 (1.2.4) = 6π i f 4,6 ( z) z2 d z = L( f 4,6 , 3), 2 0 where

f 4,6 ( z) = [η( z)η(2 z)η(3 z)η(6 z)]2 is a weight-4 modular form defined through the Dedekind eta function ∞ Y η( z) := eπ iz/12 (1 − e2π inz ), z ∈ H := {w ∈ C| Im w > 0}.

(1.2.5)

(1.2.6)

n=1

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. 10, (143)–(146)]: Z i∞ 3 IKM(4, 4; 1) = 4π i f 6,6 ( z) z2 d z = L( f 6,6 , 3), (1.2.7) 0 Z i∞ 1 9 4 IKM(1, 7; 1) = IKM(3, 5; 1) = 6π f 6,6 ( z) z3 d z = L( f 6,6 , 4), (1.2.8) 2 4 π 0 Z 27 9π 5 i ∞ f 6,6 ( z) z4 d z = L( f 6,6 , 5), (1.2.9) IKM(2, 6; 1) = i 0 4 which involve a weight-6 modular form

f 6,6 ( z) =

[η(2 z)η(3 z)]9 [η( z)η(6 z)]9 + . [η( z)η(6 z)]3 [η(2 z)η(3 z)]3

(1.2.10)

In addition, we also use explicit computations to verify the Eichler–Shimura–Manin relation L( f 6,6 , 5)/L( f 6,6 , 3) = 2π2 /21 [cf. 10, (142)] and the sum rule 9π2 IKM(4, 4; 1) − 14 IKM(2, 6; 1) = 0 [cf. 10, (147)]. Broadhurst has recently proposed a vast set of conjectures [13, 10, 12, 11] connecting Feynman diagrams to special values of Hasse–Weil L-functions, whose local factors arise from Kloosterman sums [10, §§2–6]. Our current work only touches upon IKM(a, b; 1) for a + b ∈ {5, 6, 8}, where the

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

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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. B ESSEL FUNCTIONS AND THEIR W ICK ROTATIONS 2.1. Some analytic properties of Bessel functions. For ν ∈ C, −π < arg z < π, the Bessel functions Jν and Yν are defined by ³ z ´2k+ν ∞ X Jµ ( z) cos(µπ) − J−µ ( z) (−1)k Jν ( z) := , (2.1.1) , Yν ( z) := lim µ→ν sin(µπ) k=0 k!Γ( k + ν + 1) 2 which may be compared to the modified Bessel functions I ν and K ν : ³ z ´2k+ν ∞ X I −µ ( z) − I µ ( z) π 1 I ν ( z) := , K ν ( z) := lim . 2 µ→ν sin(µπ) k=0 k!Γ( k + ν + 1) 2

(2.1.2)

Hereafter, the fractional powers of complex numbers are defined through wβ = exp(β log w) for log w = log |w| + i arg w, where | arg w| < π. We will also need the cylindrical Hankel functions H0(1) ( z) = J0 ( z) + iY0 ( z) and H0(2) ( z) = J0 ( z) − iY0 ( z) of zeroth order, which are both well defined for −π < arg z < π. In view of (2.1.1) and (2.1.2), we can verify π i (1) J0 ( ix) = I 0 ( x) and H ( ix) = K 0 ( x) (2.1.3) 2 0 as well as

H0(1) ( x + i 0+ ) = J0 ( x) + iY0 ( x) and

H0(1) (− x + i 0+ ) = − J0 ( x) + iY0 ( x)

for x > 0. As | z| → ∞, −π < arg z < π, we have the following asymptotic behavior: s s · µ ¶¸ · µ ¶¸ 2 i ( z− π ) 2 − i ( z− π ) 1 1 (1) (2) 4 4 H0 ( z) = e e 1+O and H0 ( z) = 1+O . πz | z| πz | z|

(2.1.4)

(2.1.5)

The asymptotic behavior of J0 ( z) = [ H0(1) ( z) + H0(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 JYM. Special cases of this lemma (involving four Bessel factors) have already appeared in [35, §2]. Lemma 2.2.1 (Bessel–Hankel–Jordan). For `, m, n ∈ Z≥0 satisfying either ` − ( m + n)/2 < 0; m < n or ` − m = ` − n < −1, we have Z i0+ +∞ [ J0 ( z)]m [ H0(1) ( z)]n z` d z = 0. (2.2.1) i0+ −∞

Proof. As the integrand goes asymptotically like O ( z`−(m+n)/2 e i(n−m)z ) for Im z > 0, | z| → ∞, we can close the contour in the upper half-plane with the help of Jordan’s lemma.  Remark Noting (2.1.4) and J0 (− x) = J0 ( x), we may reformulate (2.2.1) as Z ∞ n o [ J0 ( x)]m [ J0 ( x) + iY0 ( x)]n + (−1)` [− J0 ( x) + iY0 ( x)]n x` d x = 0,

(2.2.10 )

0

which is a more convenient form to be used later.



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YAJUN ZHOU

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 IKM(1, 4; 1) to a well-studied integral in probability theory. Theorem 2.2.2 (“Tiny nome of Bologna”). We have ¡1¢ ¡2¢ ¡4¢ ¡8¢ Z ∞ 4Z ∞ Γ π 15 Γ 15 Γ 15 Γ 15 . I 0 ( t)[K 0 ( t)]4 t d t = [ J0 ( x)]5 x d x = p 30 0 0 240 5

(1.2.10 )

Proof. Thanks to Jordan’s lemma, we can deform the contour in µ ¶4 Z ∞ Z i∞ 2 4 I 0 ( t)[K 0 ( t)] t d t = − Re J0 ( z)[ H0(1) ( z)]4 z d z, π 0 0 and identify it with its “Wick-rotated” counterpart: Z ∞ Z ∞ (1) 4 − Re J0 ( x)[ H0 ( x)] x d x = − J ( J 4 − 6 J 2 Y 2 + Y 4 ) x d x, 0

(2.2.2)

(2.2.3)

0

where J (resp. Y ) stands for J0 ( x) (resp. Y0 ( x)) in the last expression. Now that 2J2 ( J + iY )5 − (− J + iY )5 8J5 [( J + iY )3 − (− J + iY )3 ] − =− , (2.2.4) 3 10 15 we can verify the first equalityRin (1.2.10 ), while referring back to (2.2.10 ) in Lemma 2.2.1. ∞ The “random walk integral” 0 [ J0 ( x)]5 x 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 (1.2.10 ) can be directly deduced from [8, (5.2)]. 

J ( J 4 − 6 J 2Y 2 + Y 4) −

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 [19, (7), (16), (17)]. Bloch–Kerr–Vanhove studied the momentum space reformulation of IKM(1, 4; 1) as a triple integral of an algebraic function over the first octant: Z Z Z 1 ∞ d X ∞ dY ∞ d Z 1 IKM(1, 4; 1) = , (2.2.5) 8 0 X 0 Y 0 Z (1 + X + Y + Z )(1 + X −1 + Y −1 + Z −1 ) − 1 with a tour de force in motivicpcohomology. They effectively verified (1.2.1) by casting IKM(1, 4; 1) 3

[η(z)η(3z)]4

π into p for z = 3+ i6 15 [4, (2.5.9)]. Drawing on a result of Rogers–Wan–Zucker [23, The8 15 [η(2z)η(6z)]2 orem 5], Samart reanalyzed the triple integral formulation of IKM(1, 4; 1), before finally express3π ing IKM(1, 4; 1) as explicit gamma factors, and identifying it with a special L-value p L( f 3,15 , 2)

2 15

for the modular form f 3,15 ( z) = [η(3 z)η(5 z)]3 + [η( z)η(15 z)]3 [25, (35)].  Remark RIn [8, §5], the authors remarked on the uncanny resemblance of the “random walk ∞ integral” 0 [ J0 ( x)]5 x d x to the “tiny nome of Bologna”, without supplying a mechanistic interpretation later afterwards. Moreover, these authors recorded [8, Remark 7.3] Z Z ∞ 4 ∞ 3 [K 0 ( t)] d t = [ J0 ( x)]3 d x (2.2.6) π3 0 0 and [8, between Theorems 7.6 and 7.7] Z Z ∞ 4 ∞ 3 I ( t )[ K ( t )] d t = [ J0 ( x)]4 d x (2.2.7) 0 0 π3 0 0 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. 

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3. F EYNMAN DIAGRAMS WITH 5 B ESSEL 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 ¶¯ ¾ ½µ a b ¯¯ a, b, c, d ∈ Z, ad − bc = 1, c ≡ 0 (mod 6) . (3.1.1) Γ0 (6) := c d ¯ ¡ ¢ c3 = p1 3 −2 and construct Furthermore, following the notation of Chan–Zudilin [14], we write W 6 − 3 3 c3 〉 by adjoining W c3 to Γ0 (6). To set the stage for later developments in a group Γ0 (6)+3 = 〈Γ0 (6), W this article, we present some characteristics of a modular function on Γ0 (6)+3 . i h η(2z)η(6z) 6 Lemma 3.1.1 (A modular function of level 6). The function X 6,3 ( z) := η(z)η(3z) , z ∈ H has the following properties: ¡ az+b ¢ ¡a b¢  X = X ( z ) , if  6,3 6,3 c d ∈ Γ0 (6)+3 ;  cz+ d Im X ( z ) = 0 , if 2 Re z ∈ Z; (3.1.2) ³ 6,3 ´ ³ ´  i y i 1 1  X p p = X 6,3 2 + , if y ∈ (0, ∞). 6,3 2 + 2 3

2 3y

Moreover, the following mappings ( X 6,3 : n ¯ { z| Re z = 0, Im z > 0o} −→ (0, ∞) ¡ 1 ¢ ¯ 1 −→ − 16 X 6,3 : z ¯Re z = 21 , Im z > p ,0

(3.1.3)

2 3

are bijective. Proof. The function X 6,3 is a Hauptmodul of Γ0 (6)+3 with genus 0 [14, (2.2)], so it must satisfy the modular invariance relation, as displayed in the first line of (3.1.2). To prove the second line in (3.1.2), use the infinite product expansion for the Dedekind eta function in (1.2.6). To prove the last line in (3.1.2), note that 3z − 2 1 i c3 z = W = + p 6z − 3 2 2 3 y

for z =

1 iy + p . 2 2 3

(3.1.4)

The domains of the mappings in (3.1.3) are proper subsets of the fundamental domain for Γ0 (6)+3 , so these mappings are necessarily injective. Furthermore, by the second line in (3.1.2), 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.  As a demonstration for the relevance of modularity in our studies of Bessel moments, we recall some known results from [24, 8], in slightly reorganized form. In particular, we will use the [η(z)η(3z)]4

Chan–Zudilin notation Z6,3 ( z) = [η(2z)η(6z)]2 [14, (2.5)] for a modular form of weight 2 on Γ0 (6)+3 . Proposition 3.1.2 (Bessel moments as modular forms). For z/ i > 0, we have ¸ ¶ Z ∞ µ· 2η(2 z)η(6 z) 3 π2 J0 t I 0 ( t)[K 0 ( t)]3 t d t = Z6,3 ( z), η( z)η(3 z) 16 0 R∞ which gives a modular parametrization of 0 J0 ( xt) I 0 ( t)[K 0 ( t)]3 t d t for x > 0. For z = 1 p , we have 2 3 ¸ ¶ Z ∞ µ · 1 2η(2 z)η(6 z) 3 π2 I0 t I 0 ( t)[K 0 ( t)]3 t d t = Z6,3 ( z) i η( z)η(3 z) 16 0

(3.1.5) 1 2

+ i y, y >

(3.1.6)

6

YAJUN ZHOU

T ABLE I. Values of X 6,3 (z), Z6,3 (z) and their derivatives at z = p ¡ 1 ¢ ¡ 2 ¢ ¡ 4 ¢ ¡ 8 ¢¤ Bologna constant” c = 5C = 2401π2 Γ 15 Γ 15 Γ 15 Γ 15

1 2

+

p i p5 2 3

Z6,3 (z)

p 8 3c π

X 60 ,3 (z)

1 − 64 p 3 15c 32i

Z60 ,3 (z)

48c(3c − 1) p 5π i

X 600,3 (z)

9c(9c + 1) 16

Z600,3 (z)

X 6,3 (z)

X 6000,3 (z)

p 27 15c(18c2 − 18c − 1) 80i

Z6000,3 (z)

X 60000 ,3 (z)

81c(753c3 + 54c2 − 27c − 1) 20

Z60000 ,3 (z)



£ with “rescaled

p 48 3c(62c2 − 18c + 3) 5π

1728ic(57c3 − 62c2 + 9c − 1) p 5 5π

p 1728 3c(266c4 − 228c3 + 124c2 − 12c + 1) 5π

and µ · ¸ ¶ 1 2η(2 z)η(6 z) 3 3(2 z − 1) J0 t [ J0 ( t)]4 t d t = Z6,3 ( z), (3.1.7) i η( z)η(3 z) 4π i 0 R∞ R∞ which give modular parametrizations of 0 I 0 ( xt) I 0 ( t)[K 0 ( t)]3 t d t and 0 J0 ( xt)[ J0 ( t)]4 t d t for x ∈ (0, 2). Z



Proof. We recall from [1, (55) and (56)] the following formula !à ! µ ¶ n à !2 à µ ¶ Z ∞ π2 n ! 2 X n 2( n − k) 2 k π2 n ! 2 3 2n+1 I 0 ( t)[K 0 ( t)] t dt = = Dn, (3.1.8) 16 4n k=0 k 16 4n n−k k 0 ¡ ¢ m! where mj = j!(m and D n is the nth Domb number. Meanwhile, we note that Rogers has shown − j)! in [24, Theorem 3.1] that µ 1 1 2 ¯ ¶ ∞ D X , 2 , 3 ¯¯ 27 u2 n n 3 = (1 − u ) u (3.1.9) 3 F2 ¯ n 3 1, 1 4(1 − u) n=0 4 holds for | u| sufficiently small, where à a1 , . . . , a p p Fq b1 , . . . , b q

Q p Γ(a j +n) ¯ ! ¯ n ∞ X j =1 Γ(a j ) x ¯ . ¯ x := 1 + Q q Γ(b k + n) n! ¯ n=1

By termwise summation, we see that Z ∞ J0 ( xt) I 0 ( t)[K 0 ( t)]3 t d t = 0

k=1

π2

16 + x2

µ 3 F2

(3.1.10)

Γ(b k )

1 1 2 3, 2, 3

1, 1

¯ ¶ ¯ 108 x4 ¯ ¯ (16 + x2 )3

(3.1.11)

is valid for x sufficiently small. Parametrizing the right-hand side of the equation above with modular forms (see [14, (2.8)] or [8, (4.13)]), we observe that (3.1.5) holds when Im z is sufficiently large and positive. By analytic continuation, the validity of (3.1.5) extends to the entire positive h i 2η(2z)η(6z) 3 maps bijectively to x ∈ (0, ∞). Im z-axis, from which x = η(z)η(3z) Performing further analytic continuation on (3.1.5), we arrive at (3.1.6). Here, according to h i ³ ´ 2η(2z)η(6z) 3 1 Lemma 3.1.1, we know that x = 1i η(z)η(3z) maps y ∈ p , ∞ bijectively to x ∈ (0, 2). 2 3 The integral identity in (3.1.7) paraphrases [8, (4.16)]. (A special case parameR ∞of this modular 5 trization led to a closed-form evaluation of the “random walk integral” 0 [ J0 ( x)] x d x in [8, (5.2)], which we quoted in our proof of Theorem 2.2.2. See also Table I.) 

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Remark For any CM point z ∈ H (a complex number in the upper half-plane that solves a quadratic equation with integer coefficients), the absolute value |η( z)| of the Dedekind eta function η( z) can be explicitly written as the product of an algebraic number, a rational power of π, and rational powers of special values for Euler’s gamma function (see [26, §12] or [28, Theorem 9.3]). At any CM point z, the following expressions are computable algebraic numbers [31, (1.2.9) and Appendix 1]: · ¸ E 2 ( z) 12 d log η( z) i E 4 ( z) E 6 ( z) = − , , , (3.1.12) 4 4 8 dz 4 Im z [η( z)] π i [η( z)] [η( z)] [η( z)]12 where

E 4 ( z) = 1 + 240

∞ n3 e2π inz X

n=1 1 − e

, 2π inz

E 6 ( z) = 1 − 504

∞ n5 e2π inz X

n=1 1 − e

2π inz

(3.1.13)

are Eisenstein series of weights 4 and 6. Higher order derivatives of the Dedekind eta function can be deduced from Ramanujan’s differential equations [22]:  d E ∗2 ( z) [E ∗2 ( z)]2 − E 4 ( z)    z = ,   dz 12    d E 4 ( z) E ∗2 ( z)E 4 ( z) − E 6 ( z) (3.1.14) z = ,  dz 3     d E ( z) E ∗ ( z)E ( z) − [E ( z)]2  6 4  6  z = 2 , dz 2 3 where E ∗2 ( z) = E 2 ( z) + π Im z is a holomorphic “weight-2 Eisenstein series”. p

Samart has computed the values of X 6,3 ( z) and Z6,3 ( z) at z = 21 + i p5 explicitly [25, Lemma 2 3 1]. We p may combine his results with (3.1.14) to evaluate derivatives of X 6,3 ( z) and Z6,3 ( z) at

z = 21 +

i p5 , 2 3

as summarized in Table I.

Remark As the Bessel differential equation leaves us [1, §1] µ 2 ¶k ∂ 1 ∂ + I 0 ( xt) = t2k I 0 ( xt), ∀ k ∈ Z>0 , 2 x ∂x ∂x



(3.1.15)

p p ¡ ¢ ¡ ¢ π 15π 2 2 1 C , IKM(2 , 3; 3) = 13C + 10C we will have no difficulties in computing IKM(2, 3; 1) = 15 2 2 15 p ¡ ¢ ¡ ¢ π 4 3 19 43C + 40C from (3.1.6), with assistance from Table I. These Bessel and IKM(2, 3; 5) = 15 2 15 moments were previously evaluated in [1, §5.10] with combinatorial techniques. 

3.2. Symmetric squares and Eichler integrals. Central to the studies of Bloch–Kerr–Vanhove [4] and Samart [25] was the following motivic integral: Z ∞ p I ( u ) := I 0 ( ut)[K 0 ( t)]4 t d t 0 Z Z Z 1 ∞ d X ∞ dY ∞ d Z 1 = , (3.2.1) 8 0 X 0 Y 0 Z (1 + X + Y + Z )(1 + X −1 + Y −1 + Z −1 ) − u and the geometry for the family of K 3 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 I ( u) for u ≤ 16. In [4] and [25], the authors parametrized the Feynman inteh i η(z)η(3z) 6 gral I ( u) with the modular function u( z) = − η(2z)η(6z) , and needed sophisticated computations p

i 15 at the CM point z∗ = −3+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).

8

YAJUN ZHOU

Lemma 3.2.1 (Jacobian for a modular function). The modular parametrization · ¸ 1 2η(2 z)η(6 z) 3 x= i η( z)η(3 z)

(3.2.2)

satisfies ½ ¾ [η( z)η(2 z)]3 1 dx [η(3 z)η(6 z)]3 = πi +9 . x dz η(3 z)η(6 z) η( z)η(2 z)

(3.2.3)

With q = e2π iz , we have the following asymptotic behavior p 1 dx 4 q dx = = [1 + 9 q + 30 q2 + 112 q3 + 297 q4 + O ( q5 )] q d q 2π i d z i

(3.2.4)

near the infinite cusp ( z → i ∞, q → 0). Proof. We can verify the following identity d η(2 z) π i [η( z)]8 + 32[η(4 z)]8 log = , dz η( z) 12 [η(2 z)]4

∀z ∈ H

(3.2.5)

by showing that the ratio between both sides defines a bounded function on the compact Riemann surface X 0 (2) = Γ0 (2)\(H ∪ Q ∪ { i ∞}), and that this ratio tends to 1 as z approaches the infinite cusp. Employing an identity due to Chan–Zudilin [14, (4.3)], we rewrite (3.2.5) as ½ ¾ η(2 z) π i [η( z)η(2 z)]3 [η(3 z)η(6 z)]3 d log = + 27 , ∀ z ∈ H. (3.2.50 ) dz η( z) 12 η(3 z)η(6 z) η( z)η(2 z) Meanwhile, a cubic transformation brings us [14, second equation below (4.5)] ½ ¾ η(6 z) π i [η( z)η(2 z)]3 [η(3 z)η(6 z)]3 d log = +3 , ∀ z ∈ H. dz η(3 z) 4 η(3 z)η(6 z) η( z)η(2 z)

(3.2.500 )

The two equations above add up to (3.2.3). Q n The expansion in (3.2.4) follows directly from (3.2.3) and η( z) = q1/24 ∞  n=1 (1 − q ). P∞ Proposition 3.2.2 (Eichler integral representation of I ( u)). Let ζ(3) = n=1 n−3 be Apéry’s constant. For z/ i > 0, we have ¸ ¶ Z ∞ µ· 2η(2 z)η(6 z) 3 t [K 0 ( t)]4 t d t J0 η ( z ) η (3 z ) 0 # " Z i∞ · 0 0 ¸3 © ª η (2 z 7ζ(3) ) η (6 z ) = Z6,3 ( z) + 12π3 i [η( z0 )η(2 z0 )]4 + 9[η(3 z0 )η(6 z0 )]4 ( z − z0 )2 d z0 , (3.2.6) 8 η( z0 )η(3 z0 ) z ³ ´ R∞ 1 which parametrizes 0 J0 ( xt)[K 0 ( t)]4 t d t for x > 0. For z = 21 + i y, y ∈ p , ∞ , we have 2 3

¸ ¶ µ · ∞ 1 2η(2 z)η(6 z) 3 I0 t [K 0 ( t)]4 t d t i η( z)η(3 z) 0 " # ¸3 Z i∞ · ª 7ζ(3) η(2 z0 )η(6 z0 ) © 3 0 0 4 0 0 4 0 2 0 = Z6,3 ( z) + 12π i [η( z )η(2 z )] + 9[η(3 z )η(6 z )] ( z − z ) d z , (3.2.7) 8 η( z0 )η(3 z0 ) z R∞ which parametrizes 0 I 0 ( xt)[K 0 ( t)]4 t d t for x ∈ (0, 2). Moreover, the equation above remains valid ³ ´ 1 1 p for z = 2 + i y, y ∈ 0, , corresponding to x ∈ (0, 2); and for z = 21 + pi e iϕ , ϕ ∈ [0, π/3], correspondZ

2 3

ing to x ∈ [2, 4].

2 3

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

R∞

9

R∞

Proof. Unlike the expressions 0 I 0 ( xt) I 0 ( t)[K 0 ( t)]3 xt d t and 0 J0 ( xt)[ J0 ( t)]4 xt d t (covered in Proposition 3.1.2), which are annihilated by the Picard–Fuchs operator [8, (2.6) and (2.7)] ¸ µ µ ¶3 · µ ¶ ¶3 d d 2 d 3 d 4 b + 3 + 64 x A 4 := x x + 1 − 4x 5 x −1 dx dx dx dx = ( x − 4)( x − 2) x3 ( x + 2)( x + 4) + x(7 x4 − 32 x2 + 64)

the function xI ( x2 ) = 4, Theorem 2.2.1]:

R∞ 0

d3 d2 4 2 + 6 x ( x − 10) d x3 d x2

d + ( x2 − 8)( x2 + 8), dx

(3.2.8)

I 0 ( xt)[K 0 ( t)]4 xt d t satisfies an inhomogeneous differential equation [cf. b4 [ xI ( x2 )] = −24 x3 . A

(3.2.9)

b4 [ f ( x)] = 0, a modular parametrization [cf. 8, For a solution to the homogeneous equation A h i3 2η(2z)η(6z) leaves us general solutions in the form of Remark 4.10] x = 1i η(z)η(3z)

f ( x) = Z6,3 ( z)( c 0 + c 1 z + c 2 z2 ), (3.2.10) x where the constants c 0 , c 1 , c 2 can be determined by the behavior of f ( x) in specific contexts. We b4 is a symmetric square [8, have the simple functional form in (3.2.10) because the operator A Remark 4.6] and the corresponding family of K 3 surfaces (1 + X + Y + Z )(1 + X −1 + Y −1 + Z −1 ) = u admit Shioda–Inose structure (see [21, Corollary 7.1], [4, §3.2] and [25, §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 Wronskian determinant W ( x) via d 6 x4 ( x2 − 10) log W ( x) = − dx ( x − 4)( x − 2) x3 ( x + 2)( x + 4) 3 d =− log[(16 − x2 )(4 − x2 )]. 2 dx ´ Here, we determine the normalizing constant κ = 1024 i /π3 for the Wronskian   y0 ( x) y1 ( x) y2 ( x) κ W ( x) = = det  y00 ( x) y10 ( x) y20 ( x)  2 [(16 − x )(4 − x2 )]3/2 y000 ( x) y100 ( x) y200 ( x)

(3.2.11)

(3.2.12)

by choosing a basis

y j ( x) x

= Z6,3 ( z) z j ,

j ∈ {0, 1, 2},

(3.2.13)

differentiating in x with the help of (3.2.4) in Lemma 3.2.1 for small values of q = e2π iz → 0, κ and extracting the leading coefficient in the q-expansion 512 [1 − 30 q + 474 q2 + O ( q3 )] = π2i3 [1 − 30 q + 474 q2 + O ( q3 )]. Then, we simplify the integral representation of a particular solution [cf. 4, (2.3.8)] Z X f(X , x) Aˆ 4 [ xI ( x2 )] d x W y∗ (X ) = (3.2.14) W ( x)( x − 4)( x − 2) x3 ( x + 2)( x + 4) 0 where   y0 ( x) y1 ( x) y2 ( x) f(X , x) = det  y0 ( x) y10 ( x) y20 ( x)  , W (3.2.15) 0 y0 (X ) y1 (X ) y2 (X )

10

YAJUN ZHOU

using the cofactors ¶ µ  dz y0 ( x) y1 ( x)   = x2 [ Z6,3 ( z)]2 det 0 ,  0  y ( x ) y ( x ) dx  0 1  ¶ µ  dz y ( x) y2 ( x) = x2 [ Z6,3 ( z)]2 det 00 (2 z), (3.2.16) 0 y ( x ) y ( x )  dx  0 2 µ ¶   dz 2  y ( x) y2 ( x)   det 10 = x2 [ Z6,3 ( z)]2 z . y1 ( x) y20 ( x) dx i h i h 2η(2z)η(6z) 3 2η(2Z )η(6Z ) 3 With the parametrization x = 1i η(z)η(3z) , X = 1i η(Z )η(3Z ) , we see that the general solub4 f (X ) = −24X 3 is tion f (X ) to the inhomogeneous equation A

X Z6,3 (Z )( c 0 + c 1 Z + c 2 Z 2 )+ + 12π i X Z6,3 (Z ) 3

Z

i∞ p

Z

p 1 + 4 X 6,3 ( z) 1 + 16 X 6,3 ( z)[ Z6,3 ( z)]2 X 6,3 ( z)(Z − z)2 d z.

Since Z6,3 ( z) → 1 as z → i ∞, we must have Z ∞ 7ζ(3) , c 1 = 0, c 2 = 0 c 0 = IKM(0, 4; 1) = [K 0 ( t)]4 t d t = 8 0

(3.2.17)

(3.2.18)

for our Eichler integral representations of Bessel moments. 1 When z/ i > 0 or z = 12 + i y for y > p , according to Chan–Zudilin [14, (3.3) and (3.5)], we have 2 3

p p 1 + 4 X 6,3 ( z) 1 + 16 X 6,3 ( z) Ã !Ã ! X 4π i(m2 +mn+n2 )z [η(2 z)η(6 z)]2 X 2π i(m2 +mn+n2 )z = e e , [η( z)η(3 z)]4 m,n∈Z m,n∈Z

(3.2.19)

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 [η(3 z)η(6 z)]3 [η( z)η(2 z)]3 +9 , η(3 z)η(6 z) η( z)η(2 z)

(3.2.20)

so we have a weight-4 modular form p p [ Z6,3 ( z)]2 X 6,3 ( z) 1 + 4 X 6,3 ( z) 1 + 16 X 6,3 ( z) · ¸ ª η(2 z)η(6 z) 3 © [η( z)η(2 z)]4 + 9[η(3 z)η(6 z)]4 , = η( z)η(3 z)

(3.2.21)

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 ∈ [2, 4]. h i 2η(2z)η(6z) 6 First, we show that the modular function X 6,3 ( z) = η(z)η(3z) is real-valued along the geodesic

segment z = 12 + pi e iϕ , ϕ ∈ [0, π/3]. From an analytic continuation of the last line in (3.1.2), it is ¡ 12 3 i iϕ ¢ ¡ ¢ clear that X 6,3 2 + p e = X 6,3 12 + pi e− iϕ . By modular invariance with respect to z 7→ z − 1, 2 3 2 3 ¡ ¢ ¡ ¢ we see that the same expression is also equal to X 6,3 − 12 + pi e− iϕ = X 6,3 12 + pi e iϕ , its own 2 3 2 3 complex conjugate. Then, by¯modifying our arguments in the second half of Lemma 3.1.1, we can check that X 6,3 : ©1 ª £ ¤ i 1 p e iϕ ¯ϕ ∈ [0, π/3] −→ − 41 , − 16 is bijective.  2+ 2 3

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

11

Remark In the proposition above, our modular parametrizations of the motivic integral I ( u) differ from the Bloch–Kerr–Vanhove approach [4, (2.3.44)], but closely resemble certain Eichler integrals in our previous work [32, §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 Γ0 (6)+3 , rather than on the simpler Hecke congruence group Γ0 (4), as in [32, §4]. We refer our readers to [34, §2] for more arithmetic applications of inhomogeneous Picard–Fuchs equations.  3.3. RSpecial values of Eichler integrals. If we want to compute the integral IKM(1, 4; 2 k + ∞ 1) = 0 I 0 ( t)[K 0 ( t)]4 t2k+1 d t for k ∈ {1, 2}, we may apply the differential identity in (3.1.15) to the p

Eichler integral representation in (3.2.7), at z = 12 + i p5 . As we have closed-form evaluations of 2 3 X 6,3 ( z), Z6,3 ( z) and their derivatives at this specific CM point in Table I, the remaining challenge resides in the computation of the Eichler integral ¸3 Z i∞ · ª η(2 z0 )η(6 z0 ) © 7ζ(3) 3 0 0 4 0 0 4 E ( z) := 12π i [ η ( z ) η (2 z )] + 9[ η (3 z ) η (6 z )] ( z − z0 )2 d z0 + 0 0 η( z )η(3 z ) 8 z Z ∞ ¢ ¡ p 1 (3.3.1) = I 0 8 − X 6,3 ( z) t [K 0 ( t)]4 t d t, Z6,3 ( z) 0 along with its derivatives ¸3 Z i∞ · ª η(2 z0 )η(6 z0 ) © 0 3 E ( z) := 24π i [η( z0 )η(2 z0 )]4 + 9[η(3 z0 )η(6 z0 )]4 ( z − z0 ) d z0 , 0 0 η( z )η(3 z ) z ¸3 Z i∞ · ª 0 η(2 z0 )η(6 z0 ) © 00 3 0 0 4 0 0 4 E ( z) := 24π i [ η ( z ) η (2 z )] + 9[ η (3 z ) η (6 z )] dz , η( z0 )η(3 z0 ) z at z = 21 +

(3.3.2) (3.3.3)

p i p5 . 2 3

Meanwhile, special values of higher-order derivatives, such as à p ! p ! p p i 1 i 1 5 5 E 0000 + p = − 108 3π c2 (3 c + 1), (3.3.4) E 000 + p = 27 i 5π c2 , 2 2 3 2 2 3 £ ¡ 1 ¢ ¡ 2 ¢ ¡ 4 ¢ ¡ 8 ¢¤ with c = 2401π2 Γ 15 Γ 15 Γ 15 Γ 15 are readily computable from the expression [see (3.2.21) and (3.3.3)] p p E 000 ( z) = −24π3 i [ Z6,3 ( z)]2 X 6,3 ( z) 1 + 4 X 6,3 ( z) 1 + 16 X 6,3 ( z), (3.3.5) Ã

and entries in Table I. Lemma 3.3.1 (Special values of E ( z) and E 0 ( z)). We have the following identities: à p ! 1 i 5 π3 E + p = p , 2 2 3 8 15 à p ! 1 i 5 π3 3π IKM(0, 3; 1) E0 + p = − . p 2 2 3 20 i 2 5i Proof. The evaluation in (3.3.6) comes from Theorem 2.2.2 and the special value for Z6,3 in Table I. p Before computing E 0 ( z) at z = 21 + i p5 , we need to consider 2 3 ¯ Z ∞ Z ∞ ∂ ¯¯ 4 I 0 ( xt)[K 0 ( t)] t d t = I 1 ( t)[K 0 ( t)]4 t2 d t. ∂ x ¯ x=1 0 0 Integrating by parts, we obtain Z ∞ Z ∞ Z ∞ I 1 ( t)[K 0 ( t)]4 t2 d t = −2 I 0 ( t)[K 0 ( t)]4 t d t + 4 I 0 ( t)K 1 ( t)[K 0 ( t)]3 t2 d t. 0

0

0

(3.3.6) (3.3.7) ³

1 2

+

p ´ i p5 2 3

(3.3.8)

(3.3.9)

12

YAJUN ZHOU

´ Using the Wronskian relation I 0 ( t)K 1 ( t) + I 1 ( t)K 0 ( t) = 1/ t, we obtain Z ∞ Z Z 4 ∞ 2 ∞ 4 2 3 I 1 ( t)[K 0 ( t)] t d t = [K 0 ( t)] t d t − I 0 ( t)[K 0 ( t)]4 t d t 5 5 0 0 0 2 = [2 IKM(0, 3; 1) − IKM(1, 4; 1)]. 5 At the point z = 12 +

p i p5 2 3

(3.3.10)

1 where X 6,3 ( z) = − 64 , we differentiate both sides of Z ∞ ¡ p ¢ I 0 8 − X 6,3 ( z) t [K 0 ( t)]4 t d t = Z6,3 ( z)E ( z)

(3.3.11)

0

with respect to z, to obtain, respectively, Z ∞ 0 − 32 X 6,3 ( z) I 1 ( t)[K 0 ( t)]4 t2 d t 0 p Z ∞ p 6 15 ic 4 2 [2 IKM(0, 3; 1) − IKM(1, 4; 1)] I 1 ( t)[K 0 ( t)] t d t = = 3 15 ic 5 0

(3.3.12)

and à p 2 p p ! 2 3 π ic (3 c − 1) 8 3 c 1 i 5 0 Z6,3 ( z)E ( z) + Z6,3 ( z)E 0 ( z) = − + E0 + p , (3.3.13) 5 π 2 2 3 p ¡1¢ ¡2¢ ¡4¢ ¡8¢ p Γ 15 Γ 15 Γ 15 = 5 IKM(1, 4; 1)/π2 is the “rescaled Bologna conwhere c = 5C = 2401π2 Γ 15 stant” in Table I. Comparing the last two displayed equations, we arrive at the value ³ introduced p ´ i p5 0 1 of E 2 + given in (3.3.7).  2 3

Lemma 3.3.2 (Bessel moments associated with E 00 ( z)). For z =

1 2

+ i y, y ∈

³

´ , we have the

1 p ,∞ 2 3

following modular parametrization of Bessel moments: · ¸ Z ¸ ¶ µ · 6 2η(2 z)η(6 z) 3 ∞ 1 2η(2 z)η(6 z) 3 00 E ( z) = t I 0 ( t)[K 0 ( t)]3 d t, I1 i η( z)η(3 z) i η( z)η(3 z) 0

(3.3.14)

In particular, we have Ã

E 00

p ! p 3 3π 1 i 5 + p = IKM(0, 3; 1). 2 2 3 5

Proof. Upon comparison between (3.2.3) and (3.2.21), we see that · ¸ ª Z6,3 ( z) d X 6,3 ( z) η(2 z)η(6 z) 3 © [η( z)η(2 z)]4 + 9[η(3 z)η(6 z)]4 = . η( z)η(3 z) 2π i dz Thus, we can prove (3.3.14) by integrating (3.1.6), namely Z ∞ ¡ p ¢ π2 Z6,3 ( z) I 0 8 − X 6,3 ( z) t I 0 ( t)[K 0 ( t)]3 t d t = 16 0

(3.3.15)

(3.3.16)

(3.3.17)

over the differential d X 6,3 ( z), while keeping in mind that ∂ xI 1 ( xt) ∂x

t

= xI 0 ( xt).

We compute à p ! Z 1 Z 1 i 5 00 1 3 E + p =6 I 1 ( t) I 0 ( t)[K 0 ( t)] d t = 9 {[ I 0 ( t)]2 − 1}[K 0 ( t)]2 K 1 ( t) d t, 2 2 3 0 0

(3.3.18)

(3.3.19)

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

13

´ starting from integration by parts. Once again, we use the Wronskian relation I 0 ( t)K 1 ( t) + I 1 ( t)K 0 ( t) = 1/ t to further deduce à p ! ¸ Z · 18 ∞ I 0 ( t) i 5 00 1 E + p = − K 1 ( t) [K 0 ( t)]2 d t, (3.3.20) 2 2 3 5 0 t before splitting the remaining integral into two parts ¸ ¸ Z ∞ Z ∞· Z ∞· I 0 ( t) − 1 1 I 0 ( t) 2 2 − K 1 ( t) [K 0 ( t)] d t = [K 0 ( t)] d t + − K 1 ( t) [K 0 ( t)]2 d t, t t t 0 0 0

(3.3.21)

which will be treated in the next two paragraphs. The first integral can pbe computed with the Taylor expansion for [ I 0 ( t) − 1]/ t, the moment formula IKM(0, 2; 2 n − 1) = ∞

Z 0

π[(n−1)!]3 ¡ ¢ 4Γ n+ 12

for n ∈ Z>0 [1, (7)], and a subsequent hypergeometric reduction:

¡ ¢ £ ¡ ¢¤ ∞ X π 3ψ(1) 31 − ψ(1) 65 ( n!)2 ζ(3) I 0 ( t) − 1 2 [K 0 ( t)] d t = = − , p 3 t 3 96 3 n=1 4 n (2 n)!

(3.3.22)

where ψ(1) ( x) = d2 log Γ( x)/ d x2 . According to [31, Remark 2.2.3.1] and [1, §3.2], we also have £ ¡ ¢ ¡ ¢¤ ¡ ¢¤ p £ ¡ ¢ ¸ ∞ · π 3ψ(1) 13 − ψ(1) 56 π ψ(1) 13 − ψ(1) 32 1 1 3π X − = = p p 8 n=0 (3 n + 1)2 (3 n + 2)2 96 3 24 3 p 3π IKM(0, 3; 1). (3.3.23) = 6 ´ The second portion will be treated with the Wronskian relation I 0 ( t)K 1 ( t) + I 1 ( t)K 0 ( t) = 1/ t, and integration by parts, as follows: ¸ Z ∞· Z ∞ Z ∞ 1 2 2 − K 1 ( t) [K 0 ( t)] d t = [ I 0 ( t) − 1]K 1 ( t)[K 0 ( t)] d t + I 1 ( t)[K 0 ( t)]3 d t t 0 0 0 Z 4 ∞ = I 1 ( t)[K 0 ( t)]3 d t. (3.3.24) 3 0 We evaluate the last integral via the following representation of a Feynman diagram in momentum space Z ∞ Z Z p 1 ∞ d X ∞ dY 1 3 I 0 ( ut)[K 0 ( t)] t d t = , (3.3.25) 4 0 X 0 Y (1 + X + Y )(1 + X −1 + Y −1 ) − u 0 which is a slight variation on [4, §2.1]. Integrating over u ∈ (0, 1), we arrive at Z ∞ Z Z 1 ∞ d X ∞ dY ( X + Y + 1)( X Y + X + Y ) I 1 ( t)[K 0 ( t)]3 d t = log . 8 0 X 0 Y ( X + 1)(Y + 1)( X + Y ) 0

(3.3.26)

We complete the integration over X ∈ (0, ∞) using dilogarithms, and subsequently obtain µ ¶ Z Z ∞ 1 ∞ dY 1 ζ(3) 3 = I 1 ( t)[K 0 ( t)] d t = log(1 + Y ) log 1 + . (3.3.27) 8 0 Y Y 4 0 Combining the analysis in the last two paragraphs, we reach the evaluation in (3.3.15).



Remark One may eliminate IKM(0, 3; 1) from the two integral evaluations in (3.3.7) and (3.3.15), before combining them into the following curious identity: ¸ Z i∞ · ª η(2 z)η(6 z) 3 © 240 [η( z)η(2 z)]4 + 9[η(3 z)η(6 z)]4 (2 z − 1) d z = 1. (3.3.28) p 1 i p5 η( z)η(3 z) 2+ 2 3

Interested readers may convert the formula above into new identities for elliptic polylogarithms and lattice sums, following the approaches by Bloch–Kerr–Vanhove [4] and Samart [25]. 

14

YAJUN ZHOU

Theorem 3.3.3 (IKM(1, 4; 3) and IKM(1, 4; 5) via E ( z), E 0 ( z) and E 00 ( z)). We have µ ¶2 µ ¶ µ ¶3 µ ¶ 1 19 2 2 2 4 IKM(1, 4; 3) = π 13C − , IKM(1, 4; 5) = π 43C − , 15 10C 15 40C ¡1¢ ¡2¢ ¡4¢ ¡8¢ 1 p where C = Γ 15 Γ 15 Γ 15 Γ 15 is the “Bologna constant”. 2

(3.3.29)

240 5π

1 Proof. As we twice differentiate (3.3.11) with respect to z, and set X 6,3 ( z) = − 64 afterwards, we obtain a formula Z ∞ 02 00 − 32(64 X + X ) I 1 ( t)[K 0 ( t)]4 t2 d t + 1024 X 02 IKM(1, 4; 3) 0 00

= Z E + 2Z E + ZE , 00

0

0

(3.3.30) p i p5 2 3

is suppressed where the subscripts for X 6,3 and Z6,3 are dropped, and the argument z = 12 + throughout, to save space. Substituting known results from Table I and Lemma 3.3.1, we may transcribe the last equality into p 6 5π2 c(26 c2 − 1) 2 135 c IKM(1, 4; 3) = , (3.3.31) 25 which confirms the evaluation for IKM(1, 4; 3). Taking fourth-order derivatives on (3.3.11), we arrive at Z £ 0000 ¤ ∞ 002 04 000 0 02 00 − 32 X + 64(3 X + 24576 X + 4 X X + 768 X X ) I 1 ( t)[K 0 ( t)]4 t2 d t 0

+ 1024[3 X 002 + 24576 X 04 + 4 X 000 X 0 + 768 X 02 X 00 ] IKM(1, 4; 3) Z ∞ 02 00 02 − 65536 X (3 X + 128 X ) I 1 ( t)[K 0 ( t)]4 t4 d t + 1048576 X 04 IKM(1, 4; 5) 0

= Z 0000 E + 4 Z 000 E 0 + 6 Z 00 E 00 + 4 Z 0 E 000 + Z E 0000 ,

(3.3.32)

where ∞

4 I 1 ( t)[K 0 ( t)]4 t4 d t = [IKM(0, 3; 3) − IKM(1, 4; 3)] (3.3.33) 5 0 can be derived in a similar vein as (3.3.10), and the relation IKM(0, 3; 3) = 2[2 IKM(0, 3; 1) − 1]/3 has been proved in [1, §3.2]. Now that the left-hand side of (3.3.32) equals ¸ · 648 c(78 c3 − 36 c2 + 18 c − 1) 2 π2 c − × 2 IKM(0, 3; 1) − p 5 5 5 2 − 2916 c (3 c + 1)(5 c − 1) IKM(1, 4; 3) Z

4 [4 IKM(0, 3; 1) − 3 IKM(1, 4; 3) − 2] + 18225 c4 IKM(1, 4; 5) 15 and its right-hand side amounts to − 14580( c − 1) c3 ×

216π2 c(1330 c4 − 684 c3 + 124 c2 + 12 c − 3) + 7776( c − 1) c3 p 25 5 2592 c(228 c3 − 186 c2 + 18 c − 1) IKM(0, 3; 1), − 25 we can simplify the relation above into

(3.3.34)

(3.3.35)

− 729 c2 [4(11 c2 + 6 c − 1) IKM(1, 4; 3) − 25 c2 IKM(1, 4; 5)] =

216π2 c(862 c4 − 468 c3 + 16 c2 + 18 c − 3) . p 25 5

(3.3.36)

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

15

This confirms the evaluation for IKM(1, 4; 5). (Furthermore, based on the recursion for the rescaled moments IKM(1, 4; 2 n + 1)/π2 , n ∈ Z≥0 [1, (11)], one can show that all of them are rational combinations of C and 1/C .)  4. F EYNMAN DIAGRAMS WITH 6 B ESSEL 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. 29, §13.46, (9)], whose modular parametrization will be our major concern. Lemma 4.1.1 (Some Wick rotations). (a) The following identities hold: Z Z ∞ π4 ∞ 2 4 [ J0 ( x)]6 x d x, [ I 0 ( t)] [K 0 ( t)] t d t = 30 0 0 Z ∞ Z π5 ∞ 5 I 0 ( t)[K 0 ( t)] t d t = − [ J0 ( x)]5 Y0 ( x) x d x. 12 0 0 (b) For x ∈ [0, 1), we have Z Z ∞ π2 ∞ J0 ( xt)[ J0 ( t)]3 t d t. I 0 ( xt) I 0 ( t)[K 0 ( t)]2 t d t = 6 0 0 (c) For x ∈ [0, 3), we have Z Z ∞ π3 ∞ 3 J0 ( xt)Y0 ( t){3[ J0 ( t)]2 − [Y0 ( t)]2 } t d t, I 0 ( xt)[K 0 ( t)] t d t = − 8 0 0 Z Z ∞ π3 ∞ 2 3 J0 ( xt)Y0 ( t){3[ J0 ( t)]2 + [Y0 ( t)]2 } t d t K 0 ( xt) I 0 ( t)[K 0 ( t)] t d t = − 8 0 0 Z π3 ∞ Y0 ( xt)[ J0 ( t)]3 t d t. − 4 0 Proof. (a) As in the proof of Theorem 2.2.2, we compute µ ¶4 Z ∞ Z i∞ 2 2 4 [ I 0 ( t)] [K 0 ( t)] t d t = − [ J0 ( z)]2 [ H0(1) ( z)]4 z d z π 0 0 Z ∞ = − Re [ J0 ( x)]2 [ H0(1) ( x)]4 x d x Z ∞0 =− J 2 ( J 4 − 6 J 2 Y 2 + Y 4 ) x d x,

(4.1.1) (4.1.2)

(4.1.3)

(4.1.4)

(4.1.5)

(4.1.6)

0

where J = J0 ( x), Y = Y0 ( x) in the last step. Applying Lemma 2.2.1 to − J 2( J 4 − 6 J 2Y 2 + Y 4)

2J3 8J6 J [( J + iY )5 − (− J + iY )5 ] + [( J + iY )3 − (− J + iY )3 ] = , 10 3 15 we arrive at (4.1.1). The proof of (4.1.2) is essentially similar. (b) By Jordan’s lemma, we can justify the following Wick rotation for x ∈ [0, 1): µ ¶2 Z ∞ Z i∞ 2 2 I 0 ( xt) I 0 ( t)[K 0 ( t)] t d t = J0 ( xz) J0 ( z)[ H0(1) ( z)]2 z d z π 0 0 Z ∞ = Re J0 ( xt) J0 ( t)[ H0(1) ( t)]2 t d t 0 Z ∞ = J0 ( xt) J ( J 2 − Y 2 ) t d t, +

0

(4.1.7)

(4.1.8)

16

YAJUN ZHOU

where J = J0 ( t), Y = Y0 ( t) in the last expression. Meanwhile, by a variation on Lemma 2.2.1, we have Z ∞ Z ∞ ( J + iY )3 − (− J + iY )3 J0 ( xt) tdt = J0 ( xt) J ( J 2 − 3Y 2 ) t d t = 0, (4.1.9) 2 0 0 so the claimed identity is proved. (c) To show (4.1.4), simply take a Wick rotation: µ ¶3 Z ∞ Z i∞ 2 3 I 0 ( xt)[K 0 ( t)] t d t = − Im J0 ( xz)[ H0(1) ( z)]3 z d z π 0 Z0∞ = − Im J0 ( xt)[ H0(1) ( t)]3 t d t Z ∞0 =− J0 ( xt)Y (3 J 2 − Y 2 ) t d t, (4.1.10) 0

where we use the abbreviation J = J0 ( t), Y = Y0 ( t) as before. For (4.1.5), Wick rotation alone brings us µ ¶3 Z ∞ 2 K 0 ( xt) I 0 ( t)[K 0 ( t)]2 t d t π 0 Z ∞ Z ∞ 2 = −2 J0 ( xt) J Y t d t − Y0 ( xt) J ( J 2 − Y 2 ) t d t. 0

(4.1.11)

0

In the meantime, we extend the technique in Lemma 2.2.1 to Z ∞ [ J0 ( xt) + iY0 ( xt)]( J + iY )3 − [− J0 ( xt) + iY0 ( xt)](− J + iY )3 t d t = 0, 2i 0 which implies Z ∞ Z ∞ 2 2 J0 ( xt)Y (3 J − Y ) t d t + Y0 ( xt) J ( J 2 − 3Y 2 ) t d t = 0. 0

(4.1.12)

(4.1.13)

0

R∞ The equation above allows us to eliminate the term 0 Y0 ( xt) JY 2 t d t from (4.1.11) and arrive at the right-hand side of (4.1.5).  R∞ Let h( x) = 0 J0 ( xt) I 0 ( t)[K 0 ( t)]2 t d t be the Hankel transform of the function I 0 ( t)[K 0 ( t)]2 , and R e( x) = ∞ J0 ( xt)[ J0 ( t)]3 t d t be a “random walk integral” ( h e( x) = p 3 ( x)/ x, where p 3 ( x) is the radial h 0 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 Z ∞ Z ∞ Z ∞ Z ∞ 2 4 2 6 e( x)]2 x d x. [ I 0 ( t)] [K 0 ( t)] t d t = [ h( x)] x d x, [ J0 ( t)] t d t = [h (4.1.14) 0

0

0

0

In order to compute the left-hand side of the equations above as Eichler integrals, we need to e( x) as modular forms. represent the Hankel transforms h( x) and h Proposition 4.1.2 (Modular parametrizations of two Hankel transforms). (a) For x > 0, we have a hypergeometric evaluation µ 1 2 ¯ 4 ¶ Z ∞ π 1 , 3 ¯¯ x (9 + x2 ) 2 3 J0 ( xt) I 0 ( t)[K 0 ( t)] t d t = p , (4.1.15) 2 F1 1 ¯ (3 + x2 )3 0 3 3 + x2 which can be parametrized as  " ¡ ¢ #2  Z ∞ 1 θ 1 − 3w π η(3w)[η(2w)]6  I 0 ( t)[K 0 ( t)]2 t d t = p J0  i t , (4.1.16) ¡ ¢ 1 0 θ 3− w 3 3 [η(w)]3 [η(6w)]2

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

17

P 2 where θ ( z) := n∈Z eπ in z is one of Jacobi’s elliptic theta functions (“Thetanullwerte”), and w = − 12 + i y for y > 0. R∞ (b) For x ∈ (0, 1), the function p 3 ( x)/ x = 0 J0 ( xt)[ J0 ( t)]3 t d t admits a modular parametrization " ¡ ¢ #2  Z ∞ 1 θ 1 − 3w 2 η(3w)[η(2w)]6 3   t [ J0 ( t)] t d t = p , (4.1.17) J0 ¢ ¡ 1 0 θ 3− w 3π [η(w)]3 [η(6w)]2

where w/ i > 0; for x ∈ (1, 3), the function p 3 ( x)/ x can be parametrized as " ¡ ¢ #2  Z ∞ 1 6 θ 1 − 3w  [ J0 ( t)]3 t d t = 2(1p− 3w) η(3w)[η(2w)] , J0  t ¡ ¢ 1 0 θ 3− w 3π [η(w)]3 [η(6w)]2

(4.1.18)

where w = (1 + e iϕ )/6, ϕ ∈ (0, π); for x > 3, we have p 3 ( x)/ x = 0. Proof. (a) For sufficiently small x, we have µ Z ∞ 1 π 2 J0 ( xt) I 0 ( t)[K 0 ( t)] t d t = p 2 F1 0 3 3 − x2

1 2 3, 3

1

¯ ¶ ¯ x2 (9 + x2 )2 ¯− , ¯ (3 − x2 )3

(4.1.19)

by the Wick rotation in (4.1.3) and an analytic continuation of the hypergeometric represenR∞ 2 tation for 0 J0 ( xt)[ J0 ( t)]3 t d t [8, (3.4)]. Setting p = − x2x 2 +3 in the following hypergeometric identity [3, Chap. 33, Theorem 6.1]: µ 1 2 ¯ ¶ µ 1 2 ¯ 2 ¶ ¯ p(3 + p)2 ¯ p (3 + p) , , 3 3 ¯ = (1 + p)2 F1 3 3 ¯¯ , (4.1.20) 2 F1 1 ¯ 2(1 + p)3 1 4 we recast (4.1.19) into (4.1.15). The validity of (4.1.15) extends to all x > 0, by analytic continuation. £ ¡ 2 ¢± ¡ ¢¤2 With a substitution x = i θ − 3z − 1 θ − 2z − 3 , one can verify ½ · ¸ ¾−1 x4 (9 + x2 ) 1 η( z) 12 = 1+ 27 η(3 z) (3 + x2 )3

(4.1.21)

by showing that the ratio between the two sides defines a bounded function on X 0 (3) = Γ0 (3)\(H ∪ Q ∪ { i ∞}). One can also show that the geodesic z = (5 + e iϕ )/12, ϕ ∈ (0, π) is mapped bijectively to x ∈ (0, ∞), using a method similar to what was employed in the proof of Proposition 3.2.2. Meanwhile, with the aforementioned relation between x ∈ (0, ∞) and z = (5 + e iϕ )/12 for ϕ ∈ (0, π), we paraphrase an identity [3, Chap. 33, Corollary 3.4] from Ramanujan’s notebook as follows: p p p p µ 1 2 ¯ 4 ¶ · µ ¶¸2 12 4 3 ¯ x (9 + x2 ) 3 x 1 + x2 9 + x2 2 z − 1 , 3 3 ¯ = η . (4.1.22) 2 F1 1 ¯ (3 + x2 )3 3z − 1 3 + x2 Multiplying both sides with ¡ −1 ¢ p p 12 η 2z 3 1 + x2 η( z) −1 = = ¡ 3z ¢, p p 12 6z 3 2 η(3 z) η −3 x 9+ x 3z−1

where

¡ 2 −1 ¢ 3 −1

∈ Γ0 (3),

(4.1.23)

we obtain p µ 3 3 1 + x2 2 F1 3 + x2

1 2 3, 3

1

¯ 4 ¶ £ ¡ 2z−1 ¢¤3 ¯ x (9 + x2 ) η 3z−1 ¯ ¡ 6z−3 ¢ . ¯ (3 + x2 )3 = η 3z −1

(4.1.24)

18

YAJUN ZHOU

Furthermore, by a theta function identity [2, Chap. 18, (24.31)] in Ramanujan’s notebook, we have v " ¡ 2 ¢ #4 ¡ 2 1¢ u u p θ − 3z − 1 θ − 9z − 3 3 t 2 1+ x = 1− = 1 − ¡ 2 3¢ , (4.1.25) ¡ 2 ¢ θ −z −3 θ −z −3 and the last expression can be reduced by an identity ¡ ¢ · ¸ θ 23τ − 1 η(τ) η(6τ) 3 , 1− =2 θ (6τ − 9) η(2τ) η(3τ)

∀τ ∈ H,

(4.1.26)

also due to Ramanujan [2, Chap. 16, Entry 24(iii) and Chap. 20, Entry 1(ii)]. 1 2z−1 Finally, setting τ = 1 − 3z and 3z = 1 + 2w ∈ i R for z = (5 + e iϕ )/12, ϕ ∈ (0, π), while simplify−1 p ing eta functions with the modular transformation η(−1/τ0 ) = τ0 / i η(τ0 ) where necessary, we arrive at the expression in (4.1.16). (b) The modular parametrization in (4.1.17) follows directly from analytic continuation of (4.1.16) and the Wick rotation relation in (4.1.3). One notes that the smooth functions p 3 ( x), x ∈ (0, 1) and p 3 ( x), x ∈ (1, 3) satisfy the same ordinary differential equation of second order [8, Theorem 2.4], so p 3 ( x)/ x, x ∈ (1, 3) must be a linear combination of η(3w)[η(2w)]6

wη(3w)[η(2w)]6 [η(w)]3 [η(6w)]2

and (4.1.27) [η(w)]3 [η(6w)]2 £ ¡ ¢± ¡ ¢¤ 1 1 2 for x = θ 1 − 3w θ 3− w . Here, the linear combination must be proportional to (1 − 3w), so as to guarantee finiteness of p 3 ( x)/ x in the x → 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 (4.1.18). R∞ For x > 3, one can prove 0 J0 ( xt)[ J0 ( t)]3 t d t = 0 by extracting the real part from the following Wick rotation: Z ∞ Z 2i ∞ (1) 3 H0 ( xt)[ J0 ( t)] t d t = [ I 0 ( t)]3 K 0 ( xt) t d t, ∀ x > 3. (4.1.28) π 0 0 R∞ Alternatively, one may invoke the probabilistic interpretation of p 3 ( x) = 0 J0 ( xt)[ J0 ( t)]3 xt d t to conclude that p 3 ( x)/ x = 0 for x > 3.  Remark The modular parametrizations in the proposition above are foreshadowed by the following formula (see [29, §13.46, (9)] and [6, (3)]) for x ∈ (0, 1) ∪ (1, 3): µ 1 1 ¯ ¶ Z ∞ ¯ (3 − x)(1 + x)3 1 , 3 ¯ , (4.1.29) J0 ( xt)[ J0 ( t)] t d t = 2 p Re 2 F1 2 2 ¯ 1 16 x π x 0 and the fact that [3, Chap. 33, Lemma 5.5 and Theorem 5.6] µ 1 1 ¯ ¶ ¯ (3 + t2 )(1 − t2 )3 θ ( z) , 2 2 2 ¯− = [ θ (3 z )] , for t = , z / i > 0. 2 F1 ¯ 1 θ (3 z) 16 t2

(4.1.30)

Formally, we may regard (4.1.16) as an analytic continuation of the identities above, along with a modular transformation corresponding to (4.1.20).  R∞ In addition to the usual Hankel f ( t), t ∈ (0, ∞), we will R ∞ transform 0 J0 ( xt) f ( t) t d t of a function R∞ also need the Y -transform 0 Y0 ( xt) f ( t) t d t and the K -transform 0 K 0 ( xt) f ( t) t d t for certain Bessel moments.

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

19

Proposition 4.1.3 (Y - and K -transforms). (a) We have  " ¡  " ¡ ¢ #2  ¢ #2  Z ∞ Z ∞ 1 1 θ 1 − θ 1 − 3w 3π 3w Y0  i J0  i t [K 0 ( t)]3 t d t − t I 0 ( t)[K 0 ( t)]2 t d t ¡ ¢ ¡ ¢ 1 1 2 0 0 θ 3− w θ 3− w =

π2 (2w + 1) η(3w)[η(2w)]6 , p 2 3 i [η(w)]3 [η(6w)]2

(4.1.31)

where w = − 21 + i y for y > 0, and " ¡ " ¡ ¢ #2  ¢ #2  Z ∞ Z ∞ 1 1 θ 1 − 3w θ 1 − 3w t [K 0 ( t)]3 t d t + 3 t I 0 ( t)[K 0 ( t)]2 t d t I0  K0  ¡ ¢ ¡ ¢ 1 1 0 0 θ 3− w θ 3− w π2 w η(3w)[η(2w)]6 =p 3 i [η(w)]3 [η(6w)]2

(4.1.32)

for w/ i > 0. (b) We have ∞

Z

3 0

" ¡ " ¡ ¢ #2  ¢ #2  Z ∞ 1 1 θ 1 − 3w θ 1 − 3w J0  t [ J0 ( t)]2 Y0 ( t) t d t + Y0  t [ J0 ( t)]3 t d t ¢ ¢ ¡ ¡ 1 1 0 θ 3− w θ 3− w

4w η(3w)[η(2w)]6 = −p 3π i [η(w)]3 [η(6w)]2

(4.1.33)

for w/ i > 0 and w = (1 + e iϕ )/6, ϕ ∈ (0, π). R∞ R∞ Proof. (a) We observe that the sequences c 3,k := 0 [K 0 ( t)]3 t k d t and s 3,k := 0 I 0 ( t)[K 0 ( t)]2 t k d t satisfy the same recursion [1, (8)], namely, ( k + 1)4 c 3,k − 2(5 k2 + 20 k + 21) c 3,k+2 + 9 c 3,k+4 = 0 and ( k + 1)4 s 3,k − 2(5 k2 + 20 k + 21) s 3,k+2 + 9 s 3,k+4 = 0 both hold for non-negative integers k. As a result, the function µ 1 2 ¯ 2 ¶ Z ∞ p π 1 , 3 ¯¯ u (9 + u) 2 3 J0 ( ut) I 0 ( t)[K 0 ( t)] t d t = p (4.1.34) 2 F1 1 ¯ (3 + u)3 0 3 3+u

is annihilated by the differential operator d d2 b3 := u( u + 1)( u + 9) + (3 u2 + 20 u + 9) + ( u + 3), B 2 du du and we have an inhomogeneous differential equation ½Z ∞ ¾ p 3 3 b B3 J0 ( ut)[K 0 ( t)] t d t = . 2 0

(4.1.35)

(4.1.36)

Meanwhile, differentiating under the integral sign and integrating by parts, we can verify that ½Z ∞ ¾ p 2 b3 B (4.1.37) Y0 ( ut)π I 0 ( t)[K 0 ( t)] t d t = 1. 0

In view of the analysis above, the left-hand side of (4.1.31) must be equal to η(3w)[η(2w)]6

[η(w)]3 [η(6w)]2

[ k 0 + k 1 (2w + 1)]

(4.1.38)

20

YAJUN ZHOU

where k 0 and k 1 are constants. Since Y0 ( xt) = π2 log( xt)+O (1) as x → 0+ , and 2

R∞ 0

I 0 ( t)[K 0 ( t)]2 t d t =

π [1, (23)], we can determine k 1 = p immediately. Superimposing with (4.1.16), we obtain 2 3i  " ¡  " ¡ ¢ #2  ¢ #2  Z ∞ Z ∞ 1 1 θ 1 − 3w θ 1 − 3w 3π J0  i t [K 0 ( t)]3 t d t − H0(1)  i t I 0 ( t)[K 0 ( t)]2 t d t ¡ ¢ ¡ ¢ 1 1 2i 0 0 θ 3− w θ 3− w µ ¶ η(3w)[η(2w)]6 π2 w = k0 + p , (4.1.39) [η(w)]3 [η(6w)]2 3i

π p 3 3

which analytically continues to " ¡ " ¡ ¢ #2  ¢ #2  Z ∞ Z ∞ 1 1 θ 1 − θ 1 − 3w 3w t [K 0 ( t)]3 t d t + 3 K0  t I 0 ( t)[K 0 ( t)]2 t d t I0  ¡ ¢ ¡ ¢ 1 1 0 0 θ 3− w θ 3− w µ ¶ η(3w)[η(2w)]6 π2 w = k + (4.1.40) p 0 [η(w)]3 [η(6w)]2 3i R∞ for w/ i > 0. Taking the w → i 0+ limit, and recalling the evaluation 0 I 0 ( t)[K 0 ( t)]3 t d t = π2 /16 from [1, (54)], we find k 0 = 0. Thus far, we have confirmedR both (4.1.31) and (4.1.32). R∞ ∞ (b) We note that the expression 0 I 0 ( xt)[K 0 ( t)]3 t d t + 3 0 K 0 ( xt) I 0 ( t)[K 0 ( t)]2 t d t is continuous with respect to x ∈ (0, 3), and the right-hand side of (4.1.32) is smooth in a neighborhood of i 0+ . Therefore, the validity of (4.1.32) extends to the geodesic w = (1 + e iϕ )/6, ϕ ∈ (0, π), by analytic continuation. Adding up (4.1.4) and (4.1.5), we derive (4.1.33) from (4.1.32).  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 IKM(2, 4; 1)). We have Z ∞ Z 1 π3 i − 2 + i ∞ 2 4 [ I 0 ( t)] [K 0 ( t)] t d t = [η(w)η(2w)η(3w)η(6w)]2 d w. 1 3 −2 0 Proof. By the Parseval–Plancherel theorem for Hankel transforms, we have ¯2 Z ¯Z Z ∞ ¯ p 1 ∞ ¯¯ ∞ 2 2 4 ¯ d u. ut ) I ( t )[ K ( t )] t d t [ I 0 ( t)] [K 0 ( t)] t d t = J ( 0 0 0 ¯ 2 0 ¯ 0 0 1 Here, for τ = 2 − 6w , the modular parameter [cf. (4.1.25) and (4.1.26)] " ¡ ¢ #4 " ¡1 ¢ #3 1 1 θ θ 1 − − 3w u = x2 = − = 1 − ¡3 9w −1 ¢ ¢ ¡ 1 1 θ 3− w θ 3− w · ¸ · ¸ · ¸ · ¸ η(τ) 3 η(6τ) 9 η(6w) 3 η(w) 9 =8 −1 = −1 η(2τ) η(3τ) η(3w) η(2w)

satisfies [cf. (3.2.50 ) and (3.2.500 )] · ¸ · ¸ du η(6w) 3 η(w) 9 [η(3w)η(6w)]3 [η(w)]8 [η(6w)]6 = −18π i = −18π i , dw η(3w) η(2w) η(w)η(2w) [η(2w)]10 so (4.2.1) follows immediately.

(4.2.1)

(4.2.2)

(4.2.3)

(4.2.4)



WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

Proposition 4.2.2 (Eichler formulation of JYM(6, 0; 1)). We have Z ∞ Z 12 i∞ 6 [ J0 ( t)] t d t = [η(w)η(2w)η(3w)η(6w)]2 d w π i 0 0 Z 1 6 2 + i∞ [η(w)η(2w)η(3w)η(6w)]2 d w. − π i 21

21

(4.2.5)

Proof. Applying the arguments in the last proposition directly to (4.1.17) and (4.1.18), we obtain Z ∞ Z 12 i∞ 6 [ J0 ( t)] t d t = [η(w)η(2w)η(3w)η(6w)]2 d w πi 0 0 Z + 12 0+ i0 [η(w)η(2w)η(3w)η(6w)]2 (3w − 1)2 d w, (4.2.6) + π i 13 + i0+ where the second integral runs along the semi-circular path w = (1 + e iϕ )/6, ϕ ∈ (0, π). Before arriving at the expression in (4.2.5), we need to perform modular transformations on the last integral. c2 〉, conTowards this end, we recall¡ from¢ Chan–Zudilin [14] that the group Γ0 (6)+2 = 〈Γ0 (6), W 1 2 − 1 c structed by adjoining W2 = p 6 −2 to Γ0 (6), enjoys a Hauptmodul 2 ¸ · η(3 z)η(6 z) 4 X 6,2 ( z) = (4.2.7) η( z)η(2 z) and a weight-2 modular form

Z6,2 ( z) =

[η( z)η(2 z)]3 . η(3 z)η(6 z)

(4.2.8)

With these notations, we see that [η( z)η(2 z)η(3 z)η(6 z)]2 = [ Z6,2 ( z)]2 X 6,2 ( z) is a modular form of weight 4 on Γ0 (6)+2 . In particular, we have c2 z)η(2W c2 z)η(3W c2 z)η(6W c2 z)]2 = 4(3 z − 1)4 [η( z)η(2 z)η(3 z)η(6 z)]2 . [η(W (4.2.9) c2 z brings us Consequently, a variable substitution w = W Z 0+ i0+ 12 [η(w)η(2w)η(3w)η(6w)]2 (3w − 1)2 d w π i 13 + i0+ Z 1 6 2 + i∞ =− [η( z)η(2 z)η(3 z)η(6 z)]2 d z, π i 12

(4.2.10)



thereby completing the proof. David Broadhurst considered the following modular form of weight 4 and level 6 ∞ X f 4,6 ( z) = [η( z)η(2 z)η(3 z)η(6 z)]2 = a 4,6 ( n) e2π inz ,

(4.2.11)

n=1

based on a suggestion from Francis Brown at Les Houches in 2010. Drawing on the work of Hulek et al. [16] that related the aforementioned modular form to a Kloosterman problem, Broadhurst conjectured that IKM(2, 4; 1) is equal to 23 L( f 4,6 , 3) [10, (110)], where the special L-value can be written explicitly as [10, (108)] µ ¶ ∞ a ∞ a X X 2π n 2π2 n2 −2πn/p6 4,6 ( n) 4,6 ( n) L( f 4,6 , 3) := = 1+ p + e . (4.2.12) 3 n3 n3 6 n=1 n=1 (Here, the first equality comes from the original definition of L-series, and the second equality results from the reflection formula [10, (106)] for the L-function in question, or equivalently, a Fricke involution of the corresponding modular form. The second infinite series representation,

22

YAJUN ZHOU

which converges faster, was convenient in numerical experiments [9, 10].) We now verify Broadhurst’s conjecture. Theorem 4.2.3 (IKM(2, 4; 1) as a special L-value). We have Z ∞ 3 IKM(2, 4; 1) = [ I 0 ( t)]2 [K 0 ( t)]4 t d t = L( f 4,6 , 3). 2 0

(4.2.13)

Proof. Judging from termwise integration of uniformly convergent series, we note that Broadhurst’s conjecture essentially says that ¶ µ Z ∞ Z i∞ 1 2 4 3 2 (4.2.14) [ I 0 ( t)] [K 0 ( t)] t d t = 6π i p f 4,6 (w) w − d w. 6 0 i/ 6 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 Z i∞ Z i/p6 3 2 3 6π i p f 4,6 (w)w d w = −π i f 4,6 ( z) d z (4.2.15) i/ 6

0

p by a change of variable w = −1/(6 z) and the modular transformation η(−1/τ) = τ/ i η(τ), so the R i∞ right-hand side of (4.2.14) is the same as −π3 i 0 f 4,6 (w) d w. However, according to Propositions 4.2.1 and 4.2.2, we have Z i∞ Z Z π4 ∞ 3 ∞ −π3 i f 4,6 (w) d w = [ J0 ( t)]6 t d t − [ I 0 ( t)]2 [K 0 ( t)]4 t d t. (4.2.16) 12 2 0 0 0 Meanwhile, the Wick rotation in (4.1.1) tells us that this is precisely IKM(2, 4; 1), as conjectured by Broadhurst. 

Before applying Proposition 4.1.3 to the 4-loop sunrise diagram 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 Z ∞ F (| t|)| t| d t c , τ∈H (4.2.17) (K F )(τ) := −∞ π(τ − t) is well-defined, and has the following asymptotic behavior:   lim (KcF )(τ) = 0,   τ→ i0+ µ ¶ (4.2.18) 1   c  (K F )(τ) = O , as |τ| → ∞. |τ| R∞ R∞ Suppose that 0 J0 ( xt)F ( t) t d t, x ∈ (0, ∞) and 0 Y0 ( xt)F ( t) t d t, x ∈ (0, ∞) are both well-defined, then ¸ ·Z ∞ ¸ Z ∞ ·Z ∞ J0 ( xt)F ( t) t d t Y0 ( xτ)F (τ)τ d τ x d x = 0. (4.2.19) 0

0

0

Proof. According to the asymptotic behavior of KcF , we have a vanishing identity for all x > 0: Z i0+ +∞ H0(1) ( xτ)(KcF )(τ) d τ = 0. (4.2.20) i0+ −∞

Here, the contour can be closed upwards, thanks to Jordan’s lemma. As Im τ → 0+ , we have the following Plemelj jump relation for ξ ∈ (−∞, 0) ∪ (0, ∞): Z ∞ F (| t|)| t| d t − iF (|ξ|)|ξ|, (4.2.21) (KcF )(ξ + i 0+ ) = P −∞ π(ξ − t)

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

23

where 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 ∈ (−∞, 0) ∪ (0, ∞), so it must be an odd function in ξ [17, §4.2]. Meanwhile, we know that ( J0 ( xξ) + iY0 ( xξ), ξ > 0, H0(1) ( xξ + i 0+ ) = (4.2.22) − J0 ( x|ξ|) + iY0 ( x|ξ|), ξ < 0, so the vanishing identity in (4.2.20) brings us · Z Z ∞ Z ∞ Y0 ( xt)F ( t) t d t = − J0 ( xξ) P 0

Now we compute Z ∞ ·Z

¸ F (| t|)| t| d t d ξ. −∞ π(ξ − t)

0





¸ ·Z



¸

Y0 ( xτ)F (τ)τ d τ x d x ¸ ¾ ¸ ½Z ∞ · Z ∞ Z ∞ ·Z ∞ F (| t|)| t| d t dτ xd x =− J0 ( xt)F ( t) t d t J0 ( xτ) P 0 0 0 −∞ π(τ − t) · Z ∞ ¸ Z ∞ Z i0+ +∞ c 1 F (| t|)| t| d t [(K F )(τ)]2 d τ =− F (τ) P d τ = Im . 4 τ 0 −∞ π(τ − t) i0+ −∞ 0

0

J0 ( xt)F ( t) t d t

(4.2.23)

0

(4.2.24)

The last contour integral is indeed vanishing, because the integrand remains analytic as τ → i 0+ , and we can close the contour upwards, according to the asymptotic behavior of the Kramers– Kronig transform KcF .  Theorem 4.2.5 (Sunrise at 4 loops). We have Z Z ∞ Z i∞ 3 ∞ 3 5 3 3 2 I 0 ( t)[K 0 ( t)] t d t = [ I 0 ( t)] [K 0 ( t)] t d t = − 6π f 4,6 ( z) z d z = L( f 4,6 , 2), 2 2 π 0 0 0 as stated in (1.2.3).

(4.2.25)

Proof. The first equality in (4.2.25) has been proved in [33, Lemma 3.1]. The last equality comes from termwise integration. The rest of this proof will revolve around the second equality. We combine (4.1.16) with (4.1.31), and carry out computations as in Proposition 4.2.1, to arrive at Z ∞ Z 1 π4 − 2 + i ∞ 5 I 0 ( t)[K 0 ( t)] t d t = f 4,6 (w)(1 + 2w) d w. (4.2.26) 2 − 21 0 Here, we have used the Parseval–Plancherel identity ¾ ½Z ∞ ¾ Z ∞ ½Z ∞ Z 2 3 J0 ( xt) I 0 ( t)[K 0 ( t)] t d t J0 ( xτ)[K 0 (τ)] τ d τ x d x = 0

0

0

0

and the Hilbert cancelation ¾ ½Z Z ∞ ½Z ∞ 2 J0 ( xt) I 0 ( t)[K 0 ( t)] t d t 0

0

0



2



I 0 ( t)[K 0 ( t)]5 t d t

(4.2.27)

¾

Y0 ( xτ) I 0 (τ)[K 0 (τ)] τ d τ x d x = 0.

(4.2.28)

By an analog of Proposition 4.2.2, we fuse (4.1.17)–(4.1.18) and (4.1.33) together into the following formula: Z Z Z ∞ + 8 0+ i0 8 i∞ 5 f 4,6 (w)w d w + f 4,6 (w)w(1 − 3w) d w. (4.2.29) [ J0 ( t)] Y0 ( t) t d t = π 0 π 13 + i0+ 0 c2 z gives rise to Again, a variable substitution w = W Z Z 1 + 8 0+ i0 4 2 + i∞ f 4,6 (w)w(1 − 3w) d w = f 4,6 ( z)(1 − 2 z) d z. π 13 + i0+ π 21

(4.2.30)

24

YAJUN ZHOU

Thus, we have Z Z Z ∞ 8 i∞ 8 ∞ 5 f 4,6 (w)w d w = I 0 ( t)[K 0 ( t)]5 t d t [ J0 ( t)] Y0 ( t) t d t + 5 π 0 π 0 0 by cancelation of Eichler integrals. We can rewrite the equation above as Z Z 4 ∞ 8 i∞ f 4,6 (w)w d w = − 5 I 0 ( t)[K 0 ( t)]5 t d t, π 0 π 0 with the aid of (4.1.2). This completes the proof.

(4.2.31)

(4.2.32)



5. F EYNMAN DIAGRAMS WITH 8 B ESSEL FACTORS 5.1. Hankel transforms and Wick rotations. We open this section by a confirmation of Broadhurst’s conjecture on IKM(2, 6; 1). Theorem 5.1.1 (Eichler integral formulation of IKM(2, 6; 1)). We have ¾ ½ Z ∞ Z [η( z)η(6 z)]9 π5 i∞ [η(2 z)η(3 z)]9 2 6 + d z. [ I 0 ( t)] [K 0 ( t)] t d t = 4i 0 [η( z)η(6 z)]3 [η(2 z)η(3 z)]3 0 Proof. By the Parseval–Plancherel theorem for Hankel transforms, we have ¯2 Z ∞ Z ∞ ¯Z ∞ ¯ ¯ 2 6 3 ¯ [ I 0 ( t)] [K 0 ( t)] t d t = J0 ( xt) I 0 ( t)[K 0 ( t)] t d t¯¯ x d x. ¯ 0

0

(5.1.1)

(5.1.2)

0

With the modular parametrization in (3.1.5), and the Jacobian in (3.2.3), we transition from an integration over the variable x ∈ (0, ∞) to its counterpart over the variable z on the Im z-axis. Accordingly, we see that Z ∞ [ I 0 ( t)]2 [K 0 ( t)]6 t d t 0 ½ ¾ Z 3 π5 i ∞ [η(3 z)η(6 z)]3 2 [η( z)η(2 z)] = [η( z)η(2 z)η(3 z)η(6 z)] +9 dz (5.1.3) 4i 0 η(3 z)η(6 z) η( z)η(2 z) descends from (5.1.2). Meanwhile, one can establish the following identity ½ ¾ 3 [η(3 z)η(6 z)]3 2 [η( z)η(2 z)] +9 [η( z)η(2 z)η(3 z)η(6 z)] η(3 z)η(6 z) η( z)η(2 z) [η(2 z)η(3 z)]9 [η( z)η(6 z)]9 + (5.1.4) [η( z)η(6 z)]3 [η(2 z)η(3 z)]3 by verifying modular invariance (with respect to Γ0 (6)+2 ) and checking the q-expansions of both sides up to sufficiently many terms [14, Remark 1].  =

Remark Encouraged by Yun’s recent contribution to Kloosterman sums [30], Broadhurst wrote [10, (135)] ∞ X [η( z)η(6 z)]9 [η(2 z)η(3 z)]9 f 6,6 ( z) = + = a 6,6 ( n) e2π inz (5.1.5) 3 3 [η( z)η(6 z)] [η(2 z)η(3 z)] n=1 and conjectured that IKM(2, 4; 1) = 27 2 L( f 6,6 , 5) for [10, (141) and (145)] µ ¶ ∞ a ∞ a X X 2π n 2π2 n2 2π3 π4 n4 −2πn/p6 6,6 ( n) 6,6 ( n) L( f 6,6 , 5) := = 1+ p + + p + e . 3 27 n5 n5 6 9 6 n=1 n=1

(5.1.6)

This said the same thing as 9π5 IKM(2, 4; 1) = i

Z

i ∞ ½ [η(2 z)η(3 z)]9 p i/ 6

[η( z)η(6 z)]9 + [η( z)η(6 z)]3 [η(2 z)η(3 z)]3

¾µ

¶ 1 z + d z, 36 4

(5.1.7)

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

25

which is also equivalent to (5.1.1) per a Fricke involution z 7→ −1/(6 z) and a modular transformap tion η(−1/τ) = τ/ i η(τ).  Remark In an earlier version of his conjecture, Broadhurst formulated the modular form f 6,6 as [9, (90) and (91)] Ã !Ã ! X X 2 2 2 2 f 6,6 ( z) = [η( z)η(2 z)η(3 z)η(6 z)]2 e2π i(m +mn+n )z e4π i(m +mn+n )z . (5.1.8) m,n∈Z

m,n∈Z

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)].  Before handling other Bessel moments 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). (a) The following formulae are true: µ ¶6 Z ∞ Z 2 8 ∞ 2 6 [ I 0 ( t)] [K 0 ( t)] t d t = − [ J0 ( x)]6 {[ J0 ( x)]2 − 7[Y0 ( x)]2 } x d x, (5.1.9) π 7 0 0 µ ¶4 Z ∞ Z 4 ∞ 2 4 4 [ I 0 ( t)] [K 0 ( t)] t d t = − [ J0 ( x)]6 {[ J0 ( x)]2 − 5[Y0 ( x)]2 } x d x. (5.1.10) π 5 0 0 Z Z ∞ 1 ∞ 4 4 [ J0 ( x)]6 {[ J0 ( x)]2 − 10[Y0 ( x)]2 } x d x. (5.1.11) [ J0 ( x)] [Y0 ( x)] x d x = − 5 0 0 (b) For x ∈ [0, 2], we have Z ∞ Z ∞ 4 J0 ( xt)[ J0 ( t)] t d t = 3 J0 ( xt)[ J0 ( t)]2 [Y0 ( t)]2 t d t. (5.1.12) 0

0

(c) For x ∈ [0, 2], we have µ ¶3 Z ∞ Z ∞ 2 3 I 0 ( xt) I 0 ( t)[K 0 ( t)] t d t = −2 J0 ( xt)[ J0 ( t)]3 Y0 ( t) t d t. π 0 0 Proof. (a) By Wick rotation, we have µ ¶6 Z ∞ Z i∞ 2 2 6 [ I 0 ( t)] [K 0 ( t)] t d t = Re [ J0 ( z)]2 [ H0(1) ( z)]6 z d z π 0 0 Z ∞ = Re [ J0 ( x)]2 [ H0(1) ( x)]6 x d x Z ∞0 = J 2 ( J 6 − 15 J 4 Y 2 + 15 J 2 Y 4 − Y 6 ) x d x,

(5.1.13)

(5.1.14)

0

for J = J0 ( x), Y = Y0 ( x). With

J 2 ( J 6 − 15 J 4 Y 2 + 15 J 2 Y 4 − Y 6 ) J [( J + iY )7 − (− J + iY )7 ] − J 3 [( J + iY )5 − (− J + iY )5 ] 14 8 = − J 6 ( J 2 − 7Y 2 ), 7 we are able to reduce (5.1.14) into (5.1.9), by virtue of (2.2.10 ) in Lemma 2.2.1. One can prove (5.1.10) in a similar vein. To prove (5.1.11), compute −

J3 [( J + iY )5 − (− J + iY )5 ] = J 4 ( J 4 − 10 J 2 Y 2 + 5Y 4 ) 2 and invoke (2.2.10 ).

(5.1.15)

(5.1.16)

26

YAJUN ZHOU

(b) By a variation on (4.1.9), we have the following vanishing identity when x ∈ [0, 2]: Z ∞ Z ∞ ( J + iY )3 − (− J + iY )3 J0 ( xt) J tdt = J0 ( xt) J 2 ( J 2 − 3Y 2 ) t d t = 0, 2 0 0 with J = J0 ( t), Y = Y0 ( t). (c) By Wick rotation, we can show that µ ¶3 Z ∞ Z ∞ 2 3 I 0 ( xt) I 0 ( t)[K 0 ( t)] t d t = − J0 ( xt)(3 J 3 Y − JY 3 ) t d t, π 0 0 where J = J0 ( t), Y = Y0 ( t). Meanwhile, when x ∈ [0, 4], we also have Z ∞ Z ∞ ( J + iY )4 − (− J + iY )4 tdt = J0 ( xt)( J 3 Y − JY 3 ) t d t = 0, J0 ( xt) 8 i 0 0 by an extension of Lemma 2.2.1.

(5.1.17)

(5.1.18)

(5.1.19)



Proposition 5.1.3 (Hankel transforms related to JYM). (a) For z = 12 + pi e iϕ , ϕ ∈ (0, π/3), we have 2 3 ¸3 ¶ Z ∞ µ · 2η(2 z)η(6 z) 1 − 6 z + 12 z2 J0 i t [ J0 ( t)]4 t d t = Z6,3 ( z) (5.1.20) η( z)η(3 z) 4π i 0 i h 2η(2z)η(6z) 3 where x = i η(z)η(3z) maps ϕ ∈ (0, π/3) bijectively to x ∈ (2, 4); for x ≥ 4, we have Z ∞ J0 ( xt)[ J0 ( t)]4 t d t = 0. (5.1.21) 0

Consequently, we have Z ∞ [ J0 ( x)]8 x d x 0

=

36 πi

1 2 + i∞

Z

i 1 p 2+2 3

³

f 6,6 ( z)(1 − 2 z)2 d z +

4 πi

1 i p 2+2 3

Z

1 i p 4+4 3

f 6,6 ( z)(1 − 6 z + 12 z2 )2 d z.

´

and z = 12 + pi e iϕ , ϕ ∈ [0, π/3), the formula 2 3 ¸3 ¶ Z ∞ µ · 2η(2 z)η(6 z) 2z − 1 J0 i t [ J0 ( t)]2 [Y0 ( t)]2 t d t = Z6,3 ( z), η( z)η(3 z) 4π i 0

(b) For z = 21 + i y, y ∈

(5.1.22)

1 p ,∞ 2 3

(5.1.23)

R∞ parametrizes 0 J0 ( xt)[ J0 ( t)]2 [Y0 ( t)]2 t d t for x ∈ (0, 4), and brings us Z ∞ [ J0 ( x)]6 [Y0 ( x)]2 x d x 0

12 = πi

Z

1 2 + i∞ 1 i p 2+2 3

4 f 6,6 ( z)(1 − 2 z) d z − πi 2

Z

1 i p 2+2 3 1 i p 4+4 3

f 6,6 ( z)(1 − 2 z)(1 − 6 z + 12 z2 ) d z.

In addition, for z = (1 + e iψ )/6, ψ ∈ [π/3, π), we have ¸ ¶ Z ∞ µ · z(1 − 3 z) 2η(2 z)η(6 z) 3 t [ J0 ( t)]2 [Y0 ( t)]2 t d t = − J0 i Z6,3 ( z), η( z)η(3 z) πi 0 R∞ which parametrizes 0 J0 ( xt)[ J0 ( t)]2 [Y0 ( t)]2 t d t for x ∈ [4, ∞) and leads us to Z ∞ [ J0 ( x)]4 [Y0 ( x)]4 x d x

(5.1.24)

(5.1.25)

0

4 = πi

Z

1 2 + i∞ 1 i p 4+4 3

64 f 6,6 ( z)(1 − 2 z) d z + πi 2

1 i p 4+4 3

Z 0

f 6,6 ( z) z2 (1 − 3 z)2 d z.

(5.1.26)

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

(c) For z = 12 + i y, y ∈

³

1 p 2 3

27

´ , ∞ , we have

¸ ¶ µ · 1 2η(2 z)η(6 z) 3 t [ J0 ( t)]3 Y0 ( t) t d t = − Z6,3 ( z), (5.1.27) J0 i η( z)η(3 z) 4π 0 R∞ which parametrizes 0 J0 ( xt)[ J0 ( t)]3 Y0 ( t) t d t for x ∈ (0, 2); for z = 21 + pi e iϕ , ϕ ∈ [0, π/3), the 2 3 identity ¸ ¶ Z ∞ µ · 1 − 6 z + 6 z2 2η(2 z)η(6 z) 3 t [ J0 ( t)]3 Y0 ( t) t d t = Z6,3 ( z) (5.1.28) J0 i η( z)η(3 z) 4π 0 R∞ parametrizes 0 J0 ( xt)[ J0 ( t)]3 Y0 ( t) t d t for x ∈ [2, 4); for z = (1 + e iψ )/6, ψ ∈ [π/3, π), we have ¸ ¶ Z ∞ µ · 2η(2 z)η(6 z) 3 3 z2 3 J0 i t [ J0 ( t)] Y0 ( t) t d t = − Z6,3 ( z), (5.1.29) η( z)η(3 z) 2π 0 R∞ a formula that parametrizes 0 J0 ( xt)[ J0 ( t)]3 Y0 ( t) t d t for x ∈ [4, ∞). As a result, the following identity holds: Z ∞ [ J0 ( x)]6 [Y0 ( x)]2 x d x Z



0

=−

4 πi

Z

144 − πi

1 2 + i∞ 1 i p 2+2 3

Z 0

f 6,6 ( z) d z −

1 i p 4+4 3

4 πi

Z

i 1 p 2+2 3 1 i p 4+4 3

f 6,6 ( z)(1 − 6 z + 6 z2 )2 d z

f 6,6 ( z) z4 d z.

(5.1.30)

Proof. (a) Judging from (3.2.10), we know that Z ∞ J0 ( xt)[ J0 ( t)]4 t d t = Z6,3 ( z)( c 0 + c 1 z + c 2 z2 ), 0

x ∈ (2, 4),

(5.1.31)

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 → 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: Z ∞ Z 2i ∞ (1) 4 H0 ( xt)[ J0 ( t)] t d t = [ I 0 ( t)]4 K 0 ( xt) t d t, ∀ x ≥ 4. (5.1.32) π 0 0 Applying the Parseval–Plancherel theorem for Hankel transforms to (3.1.7) and (5.1.20), we arrive at (5.1.22). ´ ³ 1 , ∞ , the Hankel transform formula in (5.1.23) follows from (3.1.7) and (b) For z = 12 + i y, y ∈ p 2 3 (5.1.12). The remaining arguments run parallel to those in (a). (c) To verify (5.1.27), simply combine (3.1.6) with (5.1.13). The rest founds on similar principles as the proof of (a).  R∞ Remark We note that Borwein et al. expressed 0 J0 ( xt)[ J0 ( t)]4 t d t, x ∈ (2, 4) as generalized hypergeometric series [8, Theorem 4.7], but did not give a modular parametrization.  ³ ´ 1 , ∞ , we have Proposition 5.1.4 (Y - and K -transforms). For z = 12 + i y, y ∈ p 2 3

¸ ¶ µ · ¸ ¶ Z ∞ µ · Z ∞ 2η(2 z)η(6 z) 3 2η(2 z)η(6 z) 3 4 I0 i t [K 0 ( t)] t d t + 4 K0 i t I 0 ( t)[K 0 ( t)]3 t d t η ( z ) η (3 z ) η ( z ) η (3 z ) 0 0 =

π3 (2 z − 1)

8i

Z6,3 ( z).

(5.1.33)

28

YAJUN ZHOU

For z/ i > 0, we have ¸ ¶ ¸ ¶ Z ∞ µ· Z ∞ µ· 2η(2 z)η(6 z) 3 2η(2 z)η(6 z) 3 4 t [K 0 ( t)] t d t − 2π Y0 t I 0 ( t)[K 0 ( t)]3 t d t J0 η ( z ) η (3 z ) η ( z ) η (3 z ) 0 0 π3 z

Z6,3 ( z). 4i b4 be the Picard–Fuchs operator given in (3.2.8), then one can verify Proof. Let A ½Z ∞ ¾ 3 b A4 K 0 ( xt) I 0 ( t)[K 0 ( t)] xt d t = 6 x3 =

(5.1.34)

(5.1.35)

0

by differentiation under the integralRsign, and integration by parts. Comparing this to (3.2.9), we R∞ ∞ b4 . Therefore, know that 0 I 0 ( xt)[K 0 ( t)]4 xt d t + 4 0 K 0 ( xt) I 0 ( t)[K 0 ( t)]3 xt d t is annihilated by A the left-hand side of (5.1.33) must assume the form £ ¤ Z6,3 ( z) k 0 + k 1 (2 z − 1) + k 2 (2 z − 1)2 , (5.1.36) R ∞ for certain constants k 0 , k 1 , and k 2 . Since K 0 ( xt) = − log( xt)+O (1) as x → 0+ , and 0 I 0 ( t)[K 0 ( t)]3 t d t = π2 /16 [1, (54)], the left-hand side of (5.1.33) behaves like

shows that k 1 = 1 2

p i p5 2 3

3

π 8i

π3 (2z−1+ o(z)) Z6,3 ( z) 8i

as z →

1 2

+ i ∞. This

and k 2 = 0. To demonstrate that k 0 = 0, simply check the special value at

z= + 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 Im z-axis, and extract the real part, we arrive at (5.1.34).  Remark From a Hilbert transform formula [cf. 33, (3.2)] Z ∞ 2π I 0 ( t)[K 0 (| t|)]3 | t| d t = {[π I 0 (τ)]2 − [K 0 (|τ|)]2 }[K 0 (|τ|)]2 τ, ∀τ ∈ R r {0}, P π(τ − t) −∞ we can deduce [cf. (4.2.23)] Z ∞ Z ∞ 2 2 2 J0 ( xt){[π I 0 ( t)] − [K 0 ( t)] }[K 0 ( t)] t d t = −2π Y0 ( xt) I 0 ( t)[K 0 ( t)]3 t d t, ∀ x > 0. 0

(5.1.37)

(5.1.38)

0

Thus, we may recast (5.1.34) into ¸ ¶ Z ∞ µ· 2η(2 z)η(6 z) 3 πz J0 Z6,3 ( z) t [ I 0 ( t)]2 [K 0 ( t)]2 t d t = η( z)η(3 z) 4i 0

(5.1.340 )



for z/ i > 0.

5.2. Critical L-values for Bessel moments. A conjectural sum rule 9π2 IKM(4, 4; 1)−14 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 [10, (147)]. It has also been conjectured that [10, (143)] Z ∞ ∞ a X 6,6 ( n) [ I 0 ( t)]4 [K 0 ( t)]4 t d t = L( f 6,6 , 3) := . (5.2.1) n3 0 n=1 With the preparations in §5.1, we can verify these claims. Theorem 5.2.1 (Relation between IKM(2, 6; 1) and IKM(4, 4; 1)). (a) We have a vanishing identity Z 1 + i∞ Z 1 + pi 2 2 2 3 2 f 6,6 ( z)(1 − 2 z) d z + f 6,6 ( z)(1 − 4 z + 8 z2 ) d z = 0. (5.2.2) 1 i p 2+2 3

1 i p 4+4 3

(b) We have a sum rule 9π 2

Z 0



[ I 0 ( t)]4 [K 0 ( t)]4 t d t − 14

Z 0



[ I 0 ( t)]2 [K 0 ( t)]6 t d t = 0.

(5.2.3)

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

29

Proof. (a) We spell out both sides of (5.1.11) using Hankel fusions. The left-hand side becomes Z 0



20 [ J0 ( x)] [Y0 ( x)] x d x = πi 4

4

Z

1 2 + i∞

i 1 p 2+2 3

4 f 6,6 ( z)(1 − 2 z) d z + πi 2

1 i p 2+2 3

Z

1 i p 4+4 3

f 6,6 ( z)(1 − 2 z)2 d z,

(5.2.4)

where we have transformed i 1 p 4+4 3

Z 0

1 f 6,6 ( z) z (1 − 3 z) d z = 4

Z

1 = 4

Z

2

2

− 21 + i ∞ − 21 +

i p 2 3

1 2 + i∞ 1 i p 2+2 3

f 6,6 ( z)(1 + 2 z)2 d z

f 6,6 ( z)(1 − 2 z)2 d z,

(5.2.5)

by a Fricke involution z 7→ −1/(6 z) and a horizontal translation. The right-hand side becomes −

1 5

Z 0



[ J0 ( x)]6 {[ J0 ( x)]2 − 10[Y0 ( x)]2 } x d x

· µ ¶ Z 1 + pi 2 2 3 4 11 4 1 + f 6,6 ( z)(1 − 2 z) d z − f 6,6 ( z) z− 1 i 5π i 14 + pi 9 3 3 p 2+2 3 4 3 µ ¶2 µ ¶3 µ ¶4 ¸ 1 1 1 +12 z − − 192 z − + 144 z − d z, 3 3 3

84 = 5π i

Z

1 2 + i∞

2

(5.2.6)

according to (5.1.9), (5.1.22), and (5.1.24). We bear in mind that f 6,6 ( z) = [ Z6,2 ( z)]3 X 6,2 ( z)[1 + c2 z = 2z−1 as 9 X 6,2 ( z)] is a modular form of weight 6 on Γ0 (6)+2 , which transforms under W 6z−2 c2 z) = −8(3 z − 1)6 f 6,6 ( z). f 6,6 (W

(5.2.7)

Thus, the identities Z

144

1 i p 2+2 3

i 1 p 4+4 3

Z

192

1 i p 2+2 3

i 1 p 4+4 3

¶ Z 1 i 1 4 4 2 + 2p3 f 6,6 ( z) z − f 6,6 ( z) d z, dz= 3 9 14 + pi 4 3 µ ¶3 µ ¶ Z 1 + pi 1 32 2 2 3 1 f 6,6 ( z) z − dz= − f 6,6 ( z) z − d z 3 3 14 + pi 3 µ

(5.2.8)

(5.2.9)

4 3

allow us to rewrite (5.2.6) as Z 1 ∞ − [ J0 ( x)]6 {[ J0 ( x)]2 − 10[Y0 ( x)]2 } x d x 5 0 Z 1 Z 1 + pi 2 2 3 4 84 2 + i∞ 2 = f 6,6 ( z)(1 − 2 z) d z + f 6,6 ( z)(1 − 4 z − 12 z2 ) d z. 1 i 1 5π i 2 + p 5π i 4 + pi 2 3

(5.2.10)

4 3

Identifying (5.2.4) with (5.2.10), we arrive at (5.2.2), as claimed. (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: Z 0



[ J0 ( x)]6 {2[ J0 ( x)]2 − 5[Y0 ( x)]2 } x d x = 0.

(5.2.11)

30

YAJUN ZHOU

We may compute Z ∞ [ J0 ( x)]6 {2[ J0 ( x)]2 − 5[Y0 ( x)]2 } x d x 0

µ ¶ · Z 1 i 28 32 1 1 2 + 2p3 + z− f 6,6 ( z)(1 − 2 z) d z + f 6,6 ( z) 1 i π i 41 + pi 9 3 3 p 2+2 3 4 3 µ ¶ µ ¶ µ ¶ ¸ 1 2 1 3 1 4 +96 z − − 96 z − + 1152 z − dz 3 3 3 Z 1 Z 1 i 12 2 + i∞ 12 2 + 2p3 2 = f 6,6 ( z)(1 − 2 z) d z + f 6,6 ( z)(1 − 4 z + 8 z2 ) d z = 0, π i 12 + pi π i 14 + pi

12 = πi

1 2 + i∞

Z

2

2 3

(5.2.12)

4 3

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.  Theorem 5.2.2 (Relation between L( f 6,6 , 3) and L( f 6,6 , 5)). (a) We have Z ∞ 7 [ I 0 ( t)]2 [K 0 ( t)]6 t d t 6π 5 i 0 Z 1 + i∞ Z 1 + pi Z 1 + pi 2 2 2 3 4 4 3 2 2 = f 6,6 ( z) z2 d z. f 6,6 ( z)(1 − 2 z ) d z + f 6,6 ( z) z d z − 2 i 1 p 2+2 3

1 i p 4+4 3

(5.2.13)

0

(b) We have Z ∞ 21 [ I 0 ( t)]2 [K 0 ( t)]6 t d t 2π 5 i 0 Z 1 + i∞ Z 2 2 = f 6,6 ( z)(2 + 17 z ) d z + 23 1 i p 2+2 3

i 1 p 2+2 3 1 i p 4+4 3

2

f 6,6 ( z) z d z + 17

i 1 p 4+4 3

Z 0

f 6,6 ( z) z2 d z.

(c) The following integral identity holds: Z Z i∞ 2 i∞ 4 f 6,6 ( z) z2 d z = 0, f 6,6 ( z) z d z + 7 0 0

(5.2.14)

(5.2.15)

which implies

L( f 6,6 , 5)

4 = , ζ(2)L( f 6,6 , 3) 7

(5.2.16)

where

L( f 6,6 , 3) :=

∞ a X 6,6 ( n)

n=1

n3

=

∞ a X 6,6 ( n)

n=1

n3

µ ¶ 4π n 2π2 n2 −2πn/p6 2+ p + e . 3 6

Proof. (a) According to (5.1.9), (5.1.22), (5.1.24) and (5.2.7), we have µ ¶ Z Z 1 7 2 6 ∞ 48 2 + i∞ 2 6 − [ I 0 ( t)] [K 0 ( t)] t d t + f 6,6 ( z)(1 − 2 z)2 d z i 1 8 π πi 2 + p 0 2 3 µ ¶ µ ¶ µ ¶ µ ¶ ¸ · 1 i Z + p 4 2 2 3 8 4 1 1 2 1 3 1 4 = f 6,6 ( z) + z − + 12 z − − 120 z − + 144 z − dz π i 41 + pi 9 3 3 3 3 3 4 3 Z 1 + pi 48 2 2 3 = f 6,6 ( z) z2 d z. π i 14 + pi 4 3

(5.2.17)

(5.2.18)

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

31

In the meantime, by complex conjugation, we have 1 2 + i∞

Z

1 i p 2+2 3

Z

2

f 6,6 ( z)(1 − 2 z) d z =

− 12 + i ∞

− 21 +

i p 2 3

f 6,6 ( z)(1 + 2 z)2 d z,

(5.2.19)

where as f 6,6 ( z) = f 6,6 ( z + 1) brings us Z

1 2 + i∞

i 1 p 2+2 3

Z

f 6,6 ( z)(1 − 4 z) d z +

Therefore, we obtain Z 1 + i∞ 2

1 i p 2+2 3

=−

1 i p 2+2 3 1 2 + i∞

Z =−

− 12 +

i p 2 3

Z

f 6,6 ( z)(1 + 4 z) d z = −2

1 2 + i∞

i 1 p 2+2 3

f 6,6 ( z) d z.

(5.2.20)

f 6,6 ( z)(1 − 2 z)2 d z

1 2 + i∞

Z

− 21 + i ∞

i 1 p 2+2 3

1 2 + i∞

Z

f 6,6 ( z) d z + 2 f 6,6 ( z) d z + 2

f 6,6 ( z) z d z + 2

1 i p 2+2 3

1 i p 4+4 3

ÃZ

Z

2

0

1 2 + i∞

Z +

− 21 + i ∞

− 12 + pi 2 3

f 6,6 ( z) z2 d z

!

f 6,6 ( z) z2 d z,

i 1 p 2+2 3

(5.2.21)

after invoking f 6,6 (−1/(6 z)) = −216 z6 f 6,6 ( z) in the last step. All this allows us to rearrange (5.2.18) into (5.2.13). (b) In view of (5.1.9), (5.1.22), and (5.1.30), we have µ ¶ Z Z 1 7 2 6 ∞ 4 2 + i∞ 2 6 − f 6,6 ( z)[9(1 − 2 z)2 + 7] d z [ I 0 ( t)] [K 0 ( t)] t d t − 1 i 8 π πi 2 + p 0 2 3 · µ ¶ µ ¶ µ ¶ µ ¶ ¸ Z 1 + pi 4 2 2 3 8 32 1 1 2 1 3 1 4 = f 6,6 ( z) + z − + 12 z − − 120 z − + 396 z − dz π i 14 + pi 9 3 3 3 3 3 4 3 Z 1 i 1008 4 + 4p3 + f 6,6 ( z) z4 d z. (5.2.22) πi 0 As before, we may reduce Z

1 2 + i∞ 1 i p 2+2 3

1 2 + i∞

Z = −2

f 6,6 ( z)[9(1 − 2 z)2 + 7] d z

1 i p 2+2 3

f 6,6 ( z) d z + 18

1 i p 4+4 3

Z

36 0

1 i p 4+4 3

ÃZ

4

0

f 6,6 ( z) z d z =

Z

1 2 + i∞

Z +

1 i p 2+2 3

1 2 + i∞ 1 i p 2+2 3

!

f 6,6 ( z) z2 d z,

f 6,6 ( z) d z,

(5.2.23)

(5.2.24)

and Z

1 i p 2+2 3 1 i p 4+4 3

1 = 3

Z

µ ¶ µ ¶ µ ¶ µ ¶ ¸ 8 32 1 1 2 1 3 1 4 f 6,6 ( z) + z − + 12 z − − 120 z − + 396 z − dz 9 3 3 3 3 3

1 i p 2+2 3 1 i p 4+4 3

·

f 6,6 ( z)(−7 + 28 z + 36 z2 ) d z.

(5.2.25)

32

YAJUN ZHOU

By virtue of the vanishing identity in (5.2.2), the right-hand side of (5.2.25) is also equal to Z 1 i 92 2 + 2p3 f 6,6 ( z)(1 − 2 z) d z + f 6,6 ( z) z2 d z i 1 i 1 3 p p + + 2 2 3 4 4 3 ÃZ 1 i Z 1 + i∞ Z 1 i Z 1 + i∞ ! p 4+4 3 2 14 92 2 + 2p3 7 2 2 f 6,6 ( z) d z + f 6,6 ( z) z d z + f 6,6 ( z) z2 d z. =− + 1 1 i i 3 12 + pi 3 0 3 p p + + 2 4

7 3

Z

1 2 + i∞

2

2 3

2 3

(5.2.26)

4 3

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

1 2 + i∞ 1 i p 2+2 3

from (5.2.13) and (5.2.14), we obtain Z i∞ f 6,6 ( z) z2 d z = 0

f 6,6 ( z) d z

7 18π5 i

Z 0



(5.2.27)

[ I 0 ( t)]2 [K 0 ( t)]6 t d t,

which is equivalent to (5.2.15). With termwise integration, we can verify (5.2.16).

(5.2.28)



Remark Previously, Broadhurst observed that L( f 6,6 , 5)/[ζ(2)L( f 6,6 , 3)] must be a rational number, according to Eichler–Shimura–Manin theory [cf. 27, Theorem 1], and found this rational number to be numerically 4/7 [10, (142)].  Remark As a by-product of the foregoing computations, one may eliminate JYM(6, 2; 1) from (5.1.9) and (5.1.10), to deduce Z Z Z Z ∞ 80 i∞ 280 i∞ 70 i∞ 2 8 f 6,6 ( z) d z = − f 6,6 ( z) z d z = f 6,6 ( z) z4 d z, (5.2.29) [ J0 ( x)] x d x = 9π i 0 πi 0 πi 0 0 which gives L-series representations for a “random walk integral” JYM(8, 0; 1).



Finally, we verify Broadhurst’s conjectures regarding IKM(1, 7; 1) and IKM(3, 5; 1). Theorem 5.2.3 (Sunrise at 6 loops). We have Z ∞ Z ∞ Z 2 3 5 7 6 π [ I 0 ( t)] [K 0 ( t)] t d t = I 0 ( t)[K 0 ( t)] t d t = −π 0

0

0

i∞

f 6,6 ( z) z d z,

Proof. The first equality in (5.2.30), which says Z ∞ [π I 0 ( t) + iK 0 ( t)]4 − [π I 0 ( t) − iK 0 ( t)]4 [K 0 ( t)]4 t d t = 0, i 0

(5.2.30)

(5.2.31)

is a special case of the generalized Bailey–Borwein–Broadhurst–Glasser sum rule [cf. (1.1.1)]. Fusing together (3.1.5) and (5.1.34), while noting that (see Lemma 4.2.4) ¾ ½Z ∞ ¾ Z ∞ ½Z ∞ 3 3 J0 ( xt) I 0 ( t)[K 0 ( t)] t d t Y0 ( xτ) I 0 (τ)[K 0 (τ)] τ d τ x d x = 0, (5.2.32) 0

0

0

we arrive at the last equality in (5.2.30), after some computations similar to those in Theorem 5.1.1. Alternatively, we can throw (3.1.5) and (5.1.340 ) into the Parseval–Plancherel theorem for Hankel transforms, and invoke the first equality in (5.2.30). It is clear that (5.2.30) is compatible with (1.2.8), up to a Fricke involution z 7→ −1/(6 z) in the integrand. 

WICK ROTATIONS, EICHLER INTEGRALS AND MULTI-LOOP FEYNMAN DIAGRAMS

33

Acknowledgments. This manuscript grew out of my research notes formerly intended for a project on automorphic representations [31, 32, 34] 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 Prof. 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 [12] and Marseille [11].

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