# Wrońskian factorizations and Broadhurst–Mellit determinant formulae

@article{Zhou2017WroskianFA, title={Wrońskian factorizations and Broadhurst–Mellit determinant formulae}, author={Yajun Zhou}, journal={Communications in Number Theory and Physics}, year={2017}, volume={12}, pages={355-407} }

Drawing on Vanhove's contributions to mixed Hodge structures for Feynman integrals in two-di\-men\-sion\-al quantum field theory, we compute two families of determinants whose entries are Bessel moments. Via explicit factorizations of certain Wronskian determinants, we verify two recent conjectures proposed by Broadhurst and Mellit, concerning determinants of arbitrary sizes. With some extensions to our methods, we also relate two more determinants of Broadhurst--Mellit to the logarithmic…

## 16 Citations

Some Algebraic and Arithmetic Properties of Feynman Diagrams

- MathematicsTexts & Monographs in Symbolic Computation
- 2019

This article reports on some recent progresses in Bessel moments, which represent a class of Feynman diagrams in 2-dimensional quantum field theory. Many challenging mathematical problems on these…

Empirical Determinations of Feynman Integrals Using Integer Relation Algorithms

- MathematicsTexts & Monographs in Symbolic Computation
- 2021

Integer relation algorithms can convert numerical results for Feynman integrals to exact evaluations, when one has reason to suspect the existence of reductions to linear combinations of a basis,…

Analytic Structure of all Loop Banana Amplitudes

- Mathematics
- 2020

Using the Gelfand-Kapranov-Zelevinsk\u{\i} system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we…

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

- Mathematics
- 2017

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…

Eta Quotients and Rademacher Sums

- Mathematics
- 2019

Eta quotients on Γ0(6) yield evaluations of sunrise integrals at 2, 3, 4 and 6 loops. At 2 and 3 loops, they provide modular parametrizations of inhomogeneous differential equations whose solutions…

Q-LINEAR DEPENDENCE OF CERTAIN BESSEL MOMENTS

- Mathematics
- 2019

Let I0 and K0 be modified Bessel functions of the zeroth order. We use Vanhove’s differential operators for Feynman integrals to derive upper bounds for dimensions of the Q-vector space spanned by…

On Laporta’s 4-loop sunrise formulae

- MathematicsThe Ramanujan Journal
- 2019

We prove Laporta’s conjecture which relates the 4-loop sunrise diagram in 2-dimensional quantum field theory to Watson’s integral for 4-dimensional hypercubic lattice. We also establish several…

Differential equations, recurrence relations, and quadratic constraints for L-loop two-point massive tadpoles and propagators.

- MathematicsJournal of High Energy Physics
- 2019

Abstract
We consider L-loop two-point tadpole (watermelon) integral with arbitrary masses, regularized both dimensionally and analytically. We derive differential equation system and recurrence…

ON BORWEIN’S CONJECTURES FOR PLANAR UNIFORM RANDOM WALKS

- MathematicsJournal of the Australian Mathematical Society
- 2019

Let $p_{n}(x)=\int _{0}^{\infty }J_{0}(xt)[J_{0}(t)]^{n}xt\,dt$ be Kluyver’s probability density for $n$ -step uniform random walks in the Euclidean plane. Through connection to a similar problem in…

Wrońskian algebra and Broadhurst–Roberts quadratic relations

- MathematicsCommunications in Number Theory and Physics
- 2021

Through algebraic manipulations on Wronskian matrices whose entries are reducible to Bessel moments, we present a new analytic proof of the quadratic relations conjectured by Broadhurst and Roberts,…

## References

SHOWING 1-10 OF 38 REFERENCES

Bessel moments , random walks and Calabi – Yau equations

- Mathematics
- 2009

I prove a theorem that relates moments of Bessel functions to integrals recently considered in the context of random walks. Strong support is found, at 50 digit precision, for a conjecture that had…

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

- Mathematics
- 2017

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…

Hypergeometric Forms for Ising-Class Integrals

- MathematicsExp. Math.
- 2007

It is found that some Cn,k enjoy exact evaluations involving Dirichlet Lfunctions or the Riemann zeta function and almost certainly satisfy certain interindicial relations including discrete k-recurrences.

Feynman integrals, L-series and Kloosterman moments

- Mathematics
- 2016

This work lies at an intersection of three subjects: quantum field theory, algebraic geometry and number theory, in a situation where dialogue between practitioners has revealed rich structure. It…

The physics and the mixed Hodge structure of Feynman integrals

- Physics
- 2014

This expository text is an invitation to the relation between quantum field theory Feynman integrals and periods. We first describe the relation between the Feynman parametrization of loop amplitudes…

Feynman integrals and critical modular $L$-values

- Mathematics
- 2015

Broadhurst conjectured that the Feynman integral associated to the polynomial corresponding to $t=1$ in the one-parameter family $(1+x_1+x_2+x_3)(1+x_1^{-1}+x_2^{-1}+x_3^{-1})-t$ is expressible in…

A Feynman integral via higher normal functions

- MathematicsCompositio Mathematica
- 2015

We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two…

Hilbert transforms and sum rules of Bessel moments

- Mathematics
- 2017

Using Hilbert transforms, we establish two families of sum rules involving Bessel moments, which are integrals associated with Feynman diagrams in two-dimensional quantum field theory. With these…

Multiple Zeta Values and Modular Forms in Quantum Field Theory

- Mathematics
- 2013

This article introduces multiple zeta values and alternating Euler sums, exposing some of the rich mathematical structure of these objects and indicating situations where they arise in quantum field…

Short Walk Adventures

- MathematicsSpringer Proceedings in Mathematics & Statistics
- 2020

We review recent development of short uniform random walks, with a focus on its connection to (zeta) Mahler measures and modular parametrisation of the density functions. Furthermore, we extend…