Some Algebraic and Arithmetic Properties of Feynman Diagrams

Zhou, Yajun

doi:10.1007/978-3-030-04480-0_19

Cited by 9 publications

(9 citation statements)

References 55 publications

(108 reference statements)

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“…At L loops, with N = L + 2 Bessel functions, there is a simple result for a k × k determinant with k = (L + 1)/2 . The first non-trivial case is at L = 3 loops, where it was discovered [3] and is now proven [14] that…”

Section: Simple Determinants Up To L = 6 Loopsmentioning

confidence: 97%

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Quadratic relations between Feynman integrals

Broadhurst¹

2018

Proceedings of Loops and Legs in Quantum Field Theory — PoS(LL2018)

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Section: Simple Determinants Up To L = 6 Loopsmentioning

confidence: 97%

“…Until recently, relations between Feynman integrals and Eichler integrals of modular forms were conjectural [2,3,6,4], tested to many thousands of digits. For an account of how they were proved [10,11,12,13], see the lucid review by Yajun Zhou [14].…”

Section: Feynman Integrals From Modular Formsmentioning

confidence: 99%

Quadratic relations between Feynman integrals

Broadhurst¹

2018

Proceedings of Loops and Legs in Quantum Field Theory — PoS(LL2018)

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“…Such Bessel moments have also caught the interest of number theorists, one reason being that they, in some cases, evaluate to critical values of L-series of certain modular forms (as briefly reviewed in subsection 3.6). They satisfy many interesting relations [43][44][45][46] and are closely related to L-functions built from symmetric power moments of Kloosterman sums [47][48][49].…”

Section: Bessel Function Representation Of L-loop Banana Integralsmentioning

confidence: 99%

“…Furthermore, it is reasonable to have similar expectations for the period integrals of higher orders in the dimensional regularization parameter ε. 45 It would be interesting to collaborate with complementary expertise to provide actual program tools for this task.…”

Section: Jhep05(2021)066mentioning

confidence: 99%

Analytic structure of all loop banana integrals

Bönisch¹,

Fischbach²,

Klemm

et al. 2021

J. High Energ. Phys.

102

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Using the Gelfand-Kapranov-Zelevinskĭ system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana integrals with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel $$ \hat{\Gamma}\hbox{-} \mathrm{class} $$ Γ ̂ ‐ class evaluation in the ambient spaces of the mirror, while the imaginary part of the integral in this regime is determined by the $$ \hat{\Gamma}\hbox{-} \mathrm{class} $$ Γ ̂ ‐ class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius κ-constants, which determine the behaviour of the integrals when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogeneous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the banana integral in very short time to very high numerical precision for all values of the physical parameters. Using modular properties of the periods we determine the value of the maximal cut equal mass four-loop integral at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.

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“…1 The expressions involving I 0 and K 0 are included to elicit a connection to work by Bailey, Borwein, Broadhurst and Glasser (2008) on Feynman diagrams in two-dimensional Quantum Field Theory; Laporta (2008Laporta ( , 2018 on four-loop integrals in Quantum Electrodynamics; and Broadhurst (2013, Broadhurst and Mellit (2016), Broadhurst and Roberts (2018) and Zhou (2019b) on critical L-values attached to modular forms (as defined in Section 2). that provide another step towards finding their closed forms.…”

Section: Andmentioning

confidence: 99%

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

McCrorie

2020

Sankhya B

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This paper considers the representation of odd moments of the distribution of a four-step uniform random walk in even dimensions, which are based on both linear combinations of two constants representable as contiguous very well-poised generalized hypergeometric series and as even moments of the square of the complete elliptic integral of the first kind. Neither constants are currently available in closed form. New symmetries are found in the critical values of the L-series of two underlying cusp forms, providing a sense in which one of the constants has a formal counterpart. The significant roles this constant and its counterpart play in multidisciplinary contexts is described. The results unblock the problem of representing them in terms of lower-order generalized hypergeometric series, offering progress towards identifying their closed forms. The same approach facilitates a canonical characterization of the hypergeometry of the parbelos, adding to the characterizations outlined by Campbell, D'Aurozio and Sondow (2020, The American Mathematical Monthly127(1), 23-32). The paper also connects the econometric problem of characterizing the bias in the canonical autoregressive model under the unit root hypothesis to very well-poised generalized hypergeometric series. The confluence of ideas presented reflects a multidisciplinarity that accords with the approach and philosophy of Prasanta Chandra Mahalanobis.

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Some Algebraic and Arithmetic Properties of Feynman Diagrams

Cited by 9 publications

References 55 publications

Quadratic relations between Feynman integrals

Quadratic relations between Feynman integrals

Analytic structure of all loop banana integrals

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

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