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2019 | Buch

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Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

herausgegeben von: Prof. Dr. Johannes Blümlein, Prof. Dr. Carsten Schneider, Prof. Dr. Peter Paule

Verlag: Springer International Publishing

Buchreihe : Texts & Monographs in Symbolic Computation

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book includes review articles in the field of elliptic integrals, elliptic functions and modular forms intending to foster the discussion between theoretical physicists working on higher loop calculations and mathematicians working in the field of modular forms and functions and analytic solutions of higher order differential and difference equations.

Inhaltsverzeichnis

Frontmatter

Eta QuotientsEta quotients and Rademacher SumsRademacher sums

Abstract

Eta quotients on \(\varGamma _0(6)\) 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 are readily obtained by expanding in the nome q. Atkin–Lehner transformations that permute cusps ensure fast convergence for all external momenta. At 4 and 6 loops, on-shell integrals are periods of modular forms of weights 4 and 6 given by Eichler integrals of eta quotients. Weakly holomorphic eta quotients determine quasi-periods. A Rademacher sum formula is given for Fourier coefficients of an eta quotient that is a Hauptmodul for \(\varGamma _0(6)\) and its generalization is found for all levels with genus 0, namely for \(N = 1,2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25\). There are elliptic obstructions at \(N = 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49,\) with genus 1. We surmount these, finding explicit formulas for Fourier coefficients of eta quotients in thousands of cases. We show how to handle the levels \(N=22, 23, 26, 28, 29, 31, 37, 50\), with genus 2, and the levels \(N=30,33,34,35,39,40,41,43,45,48,64\), with genus 3. We also solve examples with genera 4, 5, 6, 7, 8, 13.

Kevin Acres, David Broadhurst

On a Class of Feynman Integrals Evaluating to Iterated Integrals of Modular Forms

Abstract

In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic curves and modular forms. Feynman integrals, which evaluate to iterated integrals of modular forms go beyond the class of multiple polylogarithms. Nevertheless, we may bring for all examples considered the associated system of differential equations by a non-algebraic transformation to an \(\varepsilon \)-form, which makes a solution in terms of iterated integrals immediate.

Luise Adams, Stefan Weinzierl

Iterative Non-iterative Integrals in Quantum Field Theory

Abstract

Single scale Feynman integrals in quantum field theories obey difference or differential equations with respect to their discrete parameter N or continuous parameter x. The analysis of these equations reveals to which order they factorize, which can be different in both cases. The simplest systems decouple to linear differential equations which factorize to first-order. For them complete solution algorithms exist. The next interesting level is formed by those cases that decouple to linear differential equations in which also irreducible second-order factors emerge. We give a survey on the latter case. The solutions can be obtained as general \(_2F_1\) solutions. The corresponding solutions of the associated inhomogeneous differential equations form so-called iterative non-iterative integrals. There are known conditions under which one may represent the solutions by complete elliptic integrals. In this case one may find representations in terms of meromorphic modular forms, out of which special cases allow representations in the framework of elliptic polylogarithms with generalized parameters. These are in general weighted by a power of \(1/\eta (\tau )\), where \(\eta (\tau )\) is Dedekind’s \(\eta \)-function. Single scale elliptic solutions emerge in the \(\rho \)-parameter, which we use as an illustrative example. They also occur in the 3-loop QCD corrections to massive operator matrix elements and the massive 3-loop form factors.

Johannes Blümlein

Analytic Continuation of the Kite Family

Abstract

We consider results for the master integrals of the kite family, given in terms of ELi-functions which are power series in the nome q of an elliptic curve. The analytic continuation of these results beyond the Euclidean region is reduced to the analytic continuation of the two period integrals which define q. We discuss the solution to the latter problem from the perspective of the Picard–Lefschetz formula.

Christian Bogner, Armin Schweitzer, Stefan Weinzierl

A Four-Point Function for the Planar QCD Massive Corrections to Top-Antitop Production in the Gluon-Fusion Channel

Abstract

In these proceedings we present the study of a four-point function that is involved in the evaluation of the Master Integrals necessary to compute the two-loop massive QCD planar corrections to \(t\bar{t}\) production in the gluon fusuin channel, at hadron colliders. The solution involves complete elliptic integrals of the first and second kind and one- or two-fold integrations of such elliptic integrals multiplied by ratios of polynomials, inverse square roots and logarithms or dilogarithms.

Roberto Bonciani, Matteo Capozi, Paul Caucal

From Modular Forms to Differential Equations for Feynman Integrals

Abstract

In these proceedings we discuss a representation for modular forms that is more suitable for their application to the calculation of Feynman integrals in the context of iterated integrals and the differential equation method. In particular, we show that for every modular form we can find a representation in terms of powers of complete elliptic integrals of the first kind multiplied by algebraic functions. We illustrate this result on several examples. In particular, we show how to explicitly rewrite elliptic multiple zeta values as iterated integrals over powers of complete elliptic integrals and rational functions, and we discuss how to use our results in the context of the system of differential equations satisfied by the sunrise and kite integrals.

Johannes Broedel, Claude Duhr, Falko Dulat, Brenda Penante, Lorenzo Tancredi

One-Loop StringString ScatteringScattering AmplitudesAmplitudes as Iterated EisensteinEisenstein Integrals

Abstract

In these proceedings we review and expand on the recent appearance of iterated integrals on an elliptic curve in string perturbation theory. We represent the low-energy expansion of one-loop open-string amplitudes at multiplicity four and five as iterated integrals over holomorphic Eisenstein series. The framework of elliptic multiple zeta values serves as a link between the punctured Riemann surfaces encoding string interactions and the iterated Eisenstein integrals in the final results. In the five-point setup, the treatment of kinematic poles is discussed explicitly.

Johannes Broedel, Oliver Schlotterer

Expansions at Cusps and Petersson Products in Pari/GP

Abstract

We begin by explaining how to compute Fourier expansions at all cusps of any modular form of integral or half-integral weight thanks to a theorem of Borisov–Gunnells and explicit expansions of Eisenstein series at all cusps. Using this, we then give a number of methods for computing arbitrary Petersson products. All this is available in the current release of the Pari/GP package.

Henri Cohen

CM Evaluations of the Goswami-Sun Series

Abstract

In recent work, Sun constructed two q-series, and he showed that their limits as \(q\rightarrow 1\) give new derivations of the Riemann-zeta values \(\zeta (2)=\pi ^2/6\) and \(\zeta (4)=\pi ^4/90\). Goswami extended these series to an infinite family of q-series, which he analogously used to obtain new derivations of the evaluations of \(\zeta (2k)\in \mathbb {Q}\cdot \pi ^{2k}\) for every positive integer k. Since it is well known that \(\varGamma \left( \frac{1}{2}\right) =\sqrt{\pi }\), it is natural to seek further specializations of these series which involve special values of the \(\varGamma \)-function. Thanks to the theory of complex multiplication, we show that the values of these series at all CM points \(\tau \), where \(q:=e^{2\pi i\tau }\), are algebraic multiples of specific ratios of \(\varGamma \)-values. In particular, classical formulas of Ramanujan allow us to explicitly evaluate these series as algebraic multiples of powers of \(\varGamma \left( \frac{1}{4}\right) ^4/\pi ^3\) when \(q=e^{-\pi }\), \(e^{-2\pi }\).

Madeline Locus Dawsey, Ken Ono

Automatic Proof of Theta-Function Identities

Abstract

This is a tutorial for using two new MAPLE packages, thetaids and ramarobinsids. The thetaids package is designed for proving generalized eta-product identities using the valence formula for modular functions. We show how this package can be used to find theta-function identities as well as prove them. As an application, we show how to find and prove Ramanujan’s 40 identities for his so called Rogers–Ramanujan functions G(q) and H(q). In his thesis Robins found similar identities for higher level generalized eta-products. Our ramarobinsids package is for finding and proving identities for generalizations of Ramanujan’s G(q) and H(q) and Robin’s extensions. These generalizations are associated with certain real Dirichlet characters. We find a total of over 150 identities.

Jie Frye, Frank Garvan

The Generators of all Polynomial Relations Among Jacobi Theta Functions

Abstract

In this article, we consider the classical Jacobi theta functions \(\theta _i(z)\), \(i=1,2,3,4\) and show that the ideal of all polynomial relations among them with coefficients in \(K :=\mathbb {Q}(\theta _2(0|\tau ),\theta _3(0|\tau ),\theta _4(0|\tau ))\) is generated by just two polynomials, that correspond to well known identities among Jacobi theta functions.

Ralf Hemmecke, Cristian-Silviu Radu, Liangjie Ye

Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms

Abstract

We describe algorithms to compute elliptic functions and their relatives (Jacobi theta functions, modular forms, elliptic integrals, and the arithmetic-geometric mean) numerically to arbitrary precision with rigorous error bounds for arbitrary complex variables. Implementations in ball arithmetic are available in the open source Arb library. We discuss the algorithms from a concrete implementation point of view, with focus on performance at tens to thousands of digits of precision.

Fredrik Johansson

Multi-valued Feynman Graphs and Scattering Theory

Abstract

We outline ideas to connect the analytic structure of Feynman amplitudes to the structure of Karen Vogtmann’s and Marc Culler’s Outer Space. We focus on the role of cubical chain complexes in this context, and also investigate the bordification problem in the example of the 3-edge banana graph.

Dirk Kreimer

Interpolated Sequences and Critical L-Values of Modular Forms

Abstract

Recently, Zagier expressed an interpolated version of the Apéry numbers for \(\zeta (3)\) in terms of a critical L-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier’s six sporadic sequences are essentially critical L-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown’s cellular integrals and critical L-values of modular forms of odd weight.

Robert Osburn, Armin Straub

Towards a Symbolic Summation Theory for Unspecified Sequences

Abstract

The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide such a simplification without exceptions. If it fails, it may suggest a more advanced simplification introducing in addition a single nested sum where the summand has to satisfy a particular constraint. More precisely, an explicitly given parameterized telescoping equation must hold. Restricting to the case that the arising unspecified sequences are specialized to the class of indefinite nested sums defined over hypergeometric, multi-basic or mixed hypergeometric products, it can be shown that this constraint is not only sufficient but also necessary.

Peter Paule, Carsten Schneider

Differential EquationsDifferential Equations and Dispersion RelationsDispersion Relations for Feynman Amplitudes

Abstract

The derivation of the Cutkosky’s cutting rule by means of the Veltman’s Largest Time Equation is described in detail, and the use of cut graphs, imaginary parts and dispersive representations within the Differential Equation approach to the evaluation of Feynman graph amplitudes is discussed.

Ettore Remiddi

Feynman Integrals, Toric Geometry and Mirror Symmetry

Abstract

This expository text is about using toric geometry and mirror symmetry for evaluating Feynman integrals. We show that the maximal cut of a Feynman integral is a GKZ hypergeometric series. We explain how this allows to determine the minimal differential operator acting on the Feynman integrals. We illustrate the method on sunset integrals in two dimensions at various loop orders. The graph polynomials of the multi-loop sunset Feynman graphs lead to reflexive polytopes containing the origin and the associated variety are ambient spaces for Calabi-Yau hypersurfaces. Therefore the sunset family is a natural home for mirror symmetry techniques. We review the evaluation of the two-loop sunset integral as an elliptic dilogarithm and as a trilogarithm. The equivalence between these two expressions is a consequence of (1) the local mirror symmetry for the non-compact Calabi-Yau three-fold obtained as the anti-canonical hypersurface of the del Pezzo surface of degree 6 defined by the sunset graph polynomial and (2) that the sunset Feynman integral is expressed in terms of the local Gromov-Witten prepotential of this del Pezzo surface.

Pierre Vanhove

Modular and Holomorphic Graph Functions from Superstring Amplitudes

Abstract

We compare two classes of functions arising from genus-one superstring amplitudes: modular and holomorphic graph functions. We focus on their analytic properties, we recall the known asymptotic behaviour of modular graph functions and we refine the formula for the asymptotic behaviour of holomorphic graph functions. Moreover, we give new evidence of a conjecture appeared in [4] which relates these two asymptotic expansions.

Federico Zerbini

Some Algebraic and Arithmetic Properties of Feynman Diagrams

Abstract

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 Bessel moments have been formulated as a vast set of conjectures, by David Broadhurst and collaborators, who work at the intersection of high energy physics, number theory and algebraic geometry. We present the main ideas behind our verifications of several such conjectures, which revolve around linear and non-linear sum rules of Bessel moments, as well as relations between individual Feynman diagrams and critical values of modular L-functions.

Yajun Zhou

Titel: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory
herausgegeben von: Prof. Dr. Johannes Blümlein
Prof. Dr. Carsten Schneider
Prof. Dr. Peter Paule
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-04480-0
Print ISBN: 978-3-030-04479-4
DOI: https://doi.org/10.1007/978-3-030-04480-0