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Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams

J. Ablinger, J. Blümlein, A. De Freitas, M. van Hoeij, E. Imamoglu, C. G. Raab, C. -S. Radu, C. Schneider

arXiv:1706.01299v1hep-thcs.SChep-phmath-phmath.AGmath.MP

TL;DR

Some master-integral equations are second order and do not factorize to first order in differential or Mellin-N space. The paper solves such cases analytically using hypergeometric functions, definite-integral letters, and elliptic q-series representations. The solutions also yield Mellin moments that map to rational numbers and multiple zeta values for fixed N.

  • Problem

    Some physical master-integral cases have second-order equations that do not fully factorize to first order in differential or difference-equation space.

    Source: §1 · p. 2

  • Method

    The paper solves the second-order equations analytically, using definite-integral letters for iterated integrals and modular-function and modular-form representations for elliptic cases.

    Source: §4 · p. 12, §6 · p. 23, §8 · p. 45

  • Results

    The master integrals for irreducible second-order equations with more than three singularities are solved fully analytically.

    Source: §8 · p. 45

  • Takeaways & Limitations

    For elliptic contributions, fixed Mellin moments map to rational numbers and multiple zeta values.

    Source: §8 · p. 46

    The q-series have many equivalent representations, and finding minimal representations remains an open aim.

    Source: §6.3.4 · p. 43

Abstract

from arXiv · show

We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the $\rho$-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-$N$ space either. The solution of the homogeneous equations is possible in terms of convergent close integer power series as $_2F_1$ Gau\ss{} hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using $q$-product and series representations implied by Jacobi's $\vartheta_i$ functions and Dedekind's $\eta$-function. The corresponding representations can be traced back to polynomials out of Lambert--Eisenstein series, having representations also as elliptic polylogarithms, a $q$-factorial $1/\eta^k(\tau)$, logarithms and polylogarithms of $q$ and their $q$-integrals. Due to the specific form of the physical variable $x(q)$ for different processes, different representations do usually appear. Numerical results are also presented.

1 Introduction

Many Feynman integrals can be expressed through iterated-integral classes, but some master-integral equations resist first-order factorization in both differential and Mellin-N space. This paper addresses such second-order cases with hypergeometric solutions, iterated integrals containing definite-integral letters, and elliptic q-series representations.

1 Introduction

  • Some physical cases lack full first-order factorization in differential equations in x or difference equations in N; the corresponding second-order differential equations can have more than three singularities.The paper addresses master integrals appearing in the 3-loop corrections of the ρ-parameter [39].Source: §1 · p. 2
  • The homogeneous equations are solved using Gauss 2F1 functions with rational parameters and rational-function arguments, with complete elliptic integrals K and E as special cases.The integrals considered are more general than sunrise and kite cases because the elliptic integral of the second kind E cannot be transformed away.Source: §1 · p. 2
  • The approach factors higher-order equations into first-order parts and second-order contributions, then iterates new letters whose definite-integral factors cannot be reduced to ordinary iterated integrals.The paper describes this as an algorithmic treatment of equations with second-order differential operators.Source: §1 · p. 3
  • For elliptic cases, the solutions are also represented using q-series and modular forms, including Lambert–Eisenstein series and q-factorial η denominators.The paper notes that different representations typically arise with the process-dependent form of x(q).Source: §1 · pp. 3–4
  • The inhomogeneous solutions use iterated integrals with letters defined by higher transcendental functions given as definite integrals, with overlapping series expansions around x = 0 and x = 1.The methods can accommodate letters given by multiple definite integrals, not only 2F1 functions or elliptic integrals.Source: §1 · p. 3

2 The Differential Equations

The master integrals are treated through second-order differential equations, whose homogeneous solutions are hypergeometric functions and whose inhomogeneous solutions are constructed by variation of constants. In the cases considered, the resulting nested integrals include elliptic functions.

2 The Differential Equations

  • Variation of constants gives a particular inhomogeneous solution using the two homogeneous solutions, while two constants are fixed by special physical requirements.The construction uses the equation’s Wronskian and coefficient functions.Source: §2 · p. 5
  • For the equations considered, the nested-integral solution involves higher transcendental homogeneous functions that can be expressed through complete elliptic integrals at rational argument.The cited construction also uses harmonic polylogarithms and rational functions.Source: §2 · p. 5
  • The section considers differential-equation systems for the O(ε^0) terms of the master integrals.The systems are decoupled into scalar equations in the section’s derivation.Source: §2 · pp. 5–6
  • The second-order differential equations have more than three singularities, and the equations considered have 2F1 solutions.The solution search uses Gauss hypergeometric functions with rational arguments.Source: §2 · pp. 6–7

3 Solution of the homogeneous equation

The homogeneous equations are solved using close-integer-series 2F1 functions, with some solutions expressible through complete elliptic integrals. The section also identifies an inhomogeneous integral that falls outside the usual iterative-integral class.

3 Solution of the homogeneous equation

  • The homogeneous solutions of Eq. (2.14) are obtained using 2F1 functions represented by close integer series.The solutions are derived using the algorithm of Ref. [56].Source: §3 · pp. 7–8
  • Equivalent solutions obtained through triangle-group relations retain the Wronskian up to a sign while differing from the original pair.A contiguous relation is applied in obtaining solutions (3.6, 3.7).Source: §3 · pp. 8–10
  • The 2F1 functions in solutions (3.6, 3.7) are represented by complete elliptic integrals.The text notes that 2F1 functions with 2a, 2b, c ∈ Z and c > 0 can be mapped to complete elliptic integrals.Source: §3 · pp. 9–10
  • Whether all 2F1 solutions from single-scale Feynman integrals reduce to complete elliptic integrals is unknown.The stated mapping algorithm applies when the 2F1 parameters match the required pattern.Source: §3 · p. 10
  • One homogeneous solution is expressed using E(z), K(z), E(1 − z), and K(1 − z), with a structure matching an earlier solution up to a prefactor.The corresponding ratios satisfy r2(x) r2(−x) = 1 − z(x).Source: §3 · p. 11
  • The inhomogeneous solution includes an integral that cannot be represented within the usual class of iterative integrals.The paper says this requires a generalization introduced in Section 4.Source: §3 · p. 11

4 Iterated Integrals over Definite Integrals

The section generalizes iterative integrals to accommodate letters containing definite integrals, including complete elliptic integrals. The broader construction allows multiple definite integrals and retains shuffle-type relations.

4 Iterated Integrals over Definite Integrals

  • The integrals arising here fall outside standard iterative-integral classes because their elliptic-integral arguments cannot be moved entirely to integration boundaries.The paper motivates a generalized class of integrals for this case.Source: §4 · p. 12
  • The new iterative integrals admit letters containing 2F1 functions, including the complete elliptic integrals E(z) and K(z).The integrals can include factors given by definite integrals.Source: §4 · p. 12
  • The construction allows multiple definite integrals, including those arising with Appell functions and higher functions, and the letters obey shuffle-type relations.The stated relations apply with respect to the letters containing the definite-integral factors.Source: §4 · p. 12
  • Widely overlapping series expansions provide accurate numerical representations using a finite number of analytic expansion coefficients.The paper applies these representations to the differential equations and also develops q-series representations using Dedekind η functions.Source: §4 · p. 13

5 The Solution of the Inhomogeneous Equation by Series Expansion

The inhomogeneous solutions are represented by series expanded around x = 0 and x = 1, with broad overlap that supports accurate evaluation across x ∈ [0, 1]. Their convergence behavior differs among cases, with branch points affecting the b-series expansions.

5 The Solution of the Inhomogeneous Equation by Series Expansion

  • The inhomogeneous solutions of Eqs. (2.18) and (2.19) are given as integrals containing the corresponding f8a, f9a, or f8b, f9b functions.The series expansions agree with those obtained by solving the differential equations through series Ansätze in [39].Source: §5 · pp. 15, 17, 21
  • Series around x = 0 and x = 1 overlap widely, giving accurate representations across x ∈ [0, 1] with expansions through O(x50) and O(y50).The physical constants make all solutions real in this interval.Source: §5 · pp. 13, 18, 21
  • The series for f10a(x) show larger endpoint deviations than those reported for f8a(x) and f9a(x).The f8a and f9a approximations show similar behavior, with small discrepancies near the opposite endpoints.Source: §5 · pp. 18, 20
  • For f8b(x) and f9b(x), the x = 0 series starts to diverge near x ∼ 0.4, while the x = 1 series remains valid below x ∼ 0.1.The f10b expansions have similar validity ranges to f8b and f9b.Source: §5 · pp. 20–21
  • The x = 1 expansions for f8a(x), f9a(x), and f10a(x) reflect the branch point at x = 1/3 in the homogeneous solution.The general expressions also allow expansions around other fixed values of x.Source: §5 · p. 21

6 Elliptic Solutions

The elliptic solutions are organized by the elliptic-integral structures and rational arguments involved. The paper relates these solutions to q-series built from modular forms and Lambert–Eisenstein series, with an additional η-factor when elliptic integrals of the second kind contribute.

6 Elliptic Solutions

  • The levels of complexity depend on the rational function r(z), whether elliptic integrals of the second kind contribute, and the other solution building blocks.The paper considers cases involving K(r(z)) and K′(r(z)), as well as cases with E(r(z)) and E′(r(z)).Source: §6 · pp. 22–23
  • When the elliptic integrals are K(z) or K(1−z), the Mellin-transformed difference equation factorizes to first order, allowing algorithmic solution and transition back to z-space.The z-space representation also permits analytic continuation to other kinematic regions.Source: §6 · p. 22
  • For cases involving E and E′, the q-series construction requires separating an additional η-factor from modular-form building blocks.The homogeneous and inhomogeneous solutions are constructed from polynomials of η-weighted Lambert–Eisenstein series; the remaining q-integral is expressible in elliptic polylogarithms.Source: §6 · pp. 22–23

6.1 From Elliptic Integrals to Lambert–Eisenstein Series

This section develops a modular-function framework for representing differential-equation building blocks, organizing Jacobi and Dedekind identities and modular-form relations that connect elliptic integrals with eta-ratios, elliptic polylogarithms, and Lambert–Eisenstein series. It also introduces functions for compact representations, including cases where elliptic-polylogarithm parameters depend on q.

6.1 From Elliptic Integrals to Lambert–Eisenstein Series

  • Relations among elliptic integrals, elliptic functions, and modular forms provide tools for representing complete elliptic integrals at rational arguments through elliptic polylogarithms.These relations support the later representation of the inhomogeneous solution based on homogeneous solutions ψ3 and ψ4.Source: §6.1 · p. 23

6.1.2 Representations of the Modulus and the Elliptic Integrals

  • The section represents the modulus and complete elliptic integrals of the first and second kinds using elliptic-curve parameters, Jacobi ϑ-functions, and Dedekind η-functions.It also relates the Weierstraß function’s differential equation to the elliptic curve.Source: §6.1 · p. 23, §6.1.2 · pp. 25–27

6.1.1 The Jacobi ϑi and Dedekind η Functions

  • Jacobi ϑ-functions and Dedekind η-functions supply q-series and product representations, while elliptic polylogarithms accommodate parameter patterns found in q-series from Feynman-integral calculations.The section collects these function relations as ingredients for subsequent representations.Source: §6.1.1 · pp. 23–25

6.1.3 Modular Forms and Modular Functions

  • All building blocks of the homogeneous solutions and inhomogeneous integrands can be expressed as η-ratios, which are modular functions or modular forms.An η-ratio can be multiplied by a suitable power of η to obtain a modular form, a form used to derive Lambert–Eisenstein series and elliptic-polylogarithm representations.Source: §6.1.3 · pp. 27–30

6.1.4 Elliptic Polylogarithms

  • Elliptic polylogarithms represent the Lambert–Eisenstein series spanning the η-ratios, and their multiplication, synchronization, and integration relations support later manipulations.The section also addresses synchronizing arguments such as q^m to q.Source: §6.1.4 · p. 30

6.1.5 Representations in terms of η-Weighted Lambert–Eisenstein Series

  • Modular forms are represented by Eisenstein series and products of matching total weight; in the cases considered, products of two suffice.E2(τ) − N E2(Nτ) is a weight-2 modular form for Γ0(N) with trivial character, and Lambert-series arguments q^m are synchronized to q, producing roots of unity in elliptic-polylogarithm parameters.Source: §6.1.5 · pp. 31–32
  • The functions Y, Z, and T allow more compact representations of a series of later building blocks.Their use is introduced for later representations.Source: §6.1.5 · p. 33
  • Some elliptic-polylogarithm parameters (x; y) can become q-dependent.Source: §6.1.5 · p. 33

6.2 The Emergence of E(r(z))

The analysis asks whether these solutions can use only elliptic integrals of the first kind. For the cases considered, differentiating a first-kind integral introduces the second-kind integral E, which cannot be transformed away.

6.2 The Emergence of E(r(z))

  • The associated equations fall outside the cited cases solvable as integrals over an algebraic curve, and mapping them back requires an additional differential operator.The equations considered are first transformed to Heun form with four singularities.Source: §6.2 · pp. 33–34
  • Differentiating an elliptic integral of the first kind introduces E, which cannot be transformed away in these equations.The corresponding solutions therefore retain both the first- and second-kind elliptic integrals.Source: §6.2 · p. 34
  • Unlike the equal-mass sunrise and kite cases, these master integrals require the second-kind elliptic integral E.The cited comparison attributes the first-kind-only solutions for sunrise and kite to their dispersive integral representations.Source: §6.2 · p. 34

6.3 The q-Series of the η-Ratio Representations of the Basic Building Blocks

This section develops q-series representations of elliptic-integral solution building blocks using modular forms, η-ratios, and Lambert–Eisenstein series, and describes transforming inhomogeneity terms into q-space to construct iterated integrals. It notes that these q-series admit many equivalent forms and raises the search for minimal representations without specifying a minimality criterion.

6.3.1 The kinematic variable x

  • The kinematic variable x is represented by an η-ratio whose q-series is expressed using Lambert–Eisenstein series.The section also describes synchronizing arguments to q and checking the resulting modular-form identity through expansion coefficients.Source: §6.3.1 · p. 36

6.3.2 How to find the complete q-series of the building blocks?

  • Known η-ratio q-series can be checked against modular forms by comparing enough non-vanishing expansion coefficients, avoiding a time-consuming Ansatz search.The authors recommend first consulting Sloan’s Online Encyclopedia of Integer Sequences, then applying Lemma 6.8 to verify equality.Source: §6.3.2 · p. 36

6.3.3 The Ingredients of the Homogeneous Solution

  • The homogeneous-solution ingredients, including elliptic integrals and a square-root factor, are given q-series representations using Lambert series and η-ratios.The section also assembles polynomial factors and lists a q-series for Jacobi’s η-function.Source: §6.3.3 · pp. 36–38

6.3.4 The Inhomogeneity

  • The authors derive valid q-series representations algorithmically, while noting that the large inhomogeneity expressions may admit further simplification.For the inhomogeneity representations, 38 Lambert series contribute in the chosen basis; the authors leave possible simplifications for later study.Source: §6.3.4 · p. 41
  • Using higher-weight symmetry relations and shuffle algebra, it is sufficient to calculate q-representations for H0,1, H1,−1, H0,0,1, H0,1,1, H0,1,−1, and H1,1,−1 up to weight w = 3.Source: §6.3.4 · p. 42
  • For higher inhomogeneity integrals, q-Pochhammer symbols arise, requiring a generalized q-integration relation; the partition function p(n) has no known finite rational closed form.The passage cites in connection with the q-Pochhammer symbols and regarding the partition function.Source: §6.3.4 · p. 43
  • Equivalent modular-form q-series abound; minimal representations might minimize elementary elliptic polylogarithms, with argument synchronization needed only before q-integration.Synchronizing the arguments is straightforward; the authors say it is needed only when the expression is to be integrated over q.Source: §6.3.4 · p. 43

7 Possible Extensions

The section describes extensions of the paper’s iterative-integral and elliptic representations, while noting that broader differential equations may require new function classes. It also identifies dispersion relations as a guide to finding relevant structures, with an exception for diagrams containing local operator insertions.

7 Possible Extensions

  • The new iterative integrals use letters that cannot be rewritten with the next integration variable appearing in only one integration boundary.In the examples studied, complete elliptic integrals form these letters, and the inhomogeneities allow η-weighted Lambert–Eisenstein-series integrands.Source: §7 · p. 44
  • Other second-order equations may yield 2F1 solutions that cannot be reduced to complete elliptic integrals, while more general equations may require higher-transcendental solutions and new letters.Such solutions may have definite-integral representations; in some cases, the Section 4 representation may be the final one.Source: §7 · p. 44
  • Elliptic-polylogarithm representations may admit further generalizations, including extensions related to generalized harmonic sums and binomially weighted sums.The passage connects generalized harmonic sums with Mellin transforms of Kummer-iterative integrals and the nome term q^jk.Source: §7 · p. 44
  • Dispersion relations can guide the search for relevant structures by exposing cuts related to Landau variables, but the technique does not directly apply to diagrams with local operator insertions.For those diagrams, the short-distance representation must be re-derived after cutting the corresponding usual Feynman diagram.Source: §7 · p. 44
  • Hyperelliptic and Abel integrals may arise at a higher level and would require new classes of functions for analytic representations.These integrals are described as multiperiodic, compared with the double periodicity of the elliptic case.Source: §7 · p. 45

8 Conclusions

The paper analytically solves irreducible second-order differential equations with more than three singularities arising in 3-loop QCD corrections to the ρ-parameter. It develops hypergeometric and elliptic representations, including iterative integrals with non-iterative letters.

8 Conclusions

  • The authors analytically solve irreducible second-order equations with more than three singularities for master integrals in the 3-loop QCD corrections to the ρ-parameter.These equations exemplify structures beyond singly factorizing integrals represented by iterated integrals over a general alphabet.Source: §8 · p. 45
  • The inhomogeneous solutions use iterative integrals with partly non-iterative letters; elliptic cases yield η-weighted Lambert–Eisenstein series and, generally, q-integrals outside the elliptic-polylogarithm class.The q-Pochhammer symbol in the denominator is one reason the q-integrals do not generally belong to that class.Source: §8 · pp. 45–46
  • Overlapping finite-region series expansions can achieve high accuracy, despite the potentially involved differential equations.The paper also presents the simple form of (8.1) as appealing for physics applications.Source: §8 · p. 45
  • A chosen modular-form basis enables an algorithmic representation of every η-ratio, while mapping to modular forms and comparing coefficients can establish equality of q-series.The authors identify finding these representations algorithmically as their main goal.Source: §8 · p. 46
  • For fixed Mellin N, moments with elliptic contributions map to rational numbers and multiple zeta values; large sets of moments can be computed algorithmically.The cited algorithm is described as a way to quantify the corresponding physical problem.Source: §8 · p. 46

A 2F1 Solutions of Second Order Differential Equations with more than three Singularities

  • The appendix describes mapping second-order equations to 2F1 solutions by changes of variable and exp-product transformations, or by first transforming to a simpler operator.The algorithm seeks hypergeometric solutions of the simplified equations and uses them to construct solutions of the original homogeneous parts.Source: A 2F1 Solutions of Second Order… · pp. 47, 50
  • In the examples, 2F1 solutions can be mapped to complete elliptic integrals of the first and second kinds using transformations including triangle-group relations.The second-kind integral cannot be avoided in the solutions considered here.Source: §8 · p. 46, A 2F1 Solutions of Second Order… · pp. 50–51

B The Equal Mass Sunrise: from Kinematics to Elliptic Polylogarithms

  • For the equal-mass sunrise, the choice of integration variable and kinematic module changes x(q), and consequently changes the q-representation of the inhomogeneity and contributing functions.Different processes also generally lead to different representations because of the specific form of x(q).Source: §8 · p. 45, B The Equal Mass Sunrise: from… · p. 51

B.1 The treatment by Bloch and Vanhove, [30].

This subsection reviews the Bloch–Vanhove treatment of the equal-mass sunrise and its q-series route to elliptic polylogarithms.

B.1 The treatment by Bloch and Vanhove, [30].

  • Bloch and Vanhove’s treatment is comparable to the approach in [35], but uses a different definition of t and therefore a somewhat different I(q).Both treatments ultimately lead to elliptic polylogarithms.Source: §B.1 · p. 51
  • The reviewed derivation transforms the expression using a relation by M.D. Rogers, then rewrites the result as a generalized Lambert series.The transformation is followed by the q-series and Lambert-series steps described in this treatment.Source: §B.1 · pp. 51–52
  • Further logarithmic q-integrals produce higher-weight elliptic polylogarithms.Source: §B.1 · p. 52

B.2 The treatment by Adams et al., [35].

This subsection reviews the Adams et al. treatment, in which the sunrise integral is expressed through a q-variable and modular-function representations.

B.2 The treatment by Adams et al., [35].

  • The treatment sets the integration variable t = m2y from the product of the squared modules.Source: §B.2 · p. 52
  • The variable y is inverted from a modular function for Γ0(4), while the kinematic variable is a modular function for Γ0(12).That kinematic variable also serves as the variable of the inhomogeneity and, generally, of harmonic polylogarithms and related functions.Source: §B.2 · p. 52
  • The q-dependent part is transformed into Lambert form and integrated twice to obtain a special solution.Source: §B.2 · p. 53

C A Series of Sums

This section gives polylogarithmic representations for single infinite sums from earlier sunrise-graph work, while noting that the remaining double sums are not fully reducible to iterative integrals.

C A Series of Sums

  • Single infinite sums from the earlier sunrise-graph representations are expressed using root-argument polylogarithms up to Li2(z).The authors state these can be calculated with techniques available in HarmonicSums.m.Source: C A Series of Sums · p. 53
  • The sums also involve harmonic sums and single cyclotomic harmonic sums.The excerpt defines harmonic sums and states that single cyclotomic harmonic sums contribute.Source: C A Series of Sums · pp. 53–54
  • The remaining double sums in [79] cannot be solved completely in terms of iterative integrals and therefore involve non-iterative integrals.Source: C A Series of Sums · p. 56

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