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HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters

T. Huber, D. Maître

arXiv:0708.2443v2hep-ph

TL;DR

Hypergeometric-function expansions are needed in dimensional-regularization calculations, but existing implementations were restricted to integer parameters and half-integer methods were not available in a user-friendly widely used computer-algebra package. The paper introduces a new operator-based algorithm, implements it for certain classes in Mathematica's HypExp, and demonstrates it on Feynman diagrams up to four loops, reproducing known results and obtaining new results for A6,2.

  • Problem

    Existing implementations were restricted to integer-valued parameters, while half-integer methods lacked a user-friendly package in a widely used computer-algebra program.

  • Method

    The algorithm expresses a hypergeometric-function expansion through integration and differentiation operators acting on a basis function of the same type.

  • Results

    Feynman-diagram examples up to four loops reproduce known results and provide new results, including for the massless three-loop master integral A6,2.

  • Takeaways & Limitations

    The extended HypExp package demonstrates that half-integer hypergeometric expansions can be applied to multiloop Feynman-diagram calculations.

  • Takeaways & Limitations

    The method currently covers only specified classes of hypergeometric functions with half-integer parameters, although it might extend to broader classes.

Abstract

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In this article, we describe a new algorithm for the expansion of hypergeometric functions about half-integer parameters. The implementation of this algorithm for certain classes of hypergeometric functions in the already existing Mathematica package HypExp is described. Examples of applications in Feynman diagrams with up to four loops are given.

NEW VERSION PROGRAM SUMMARY

HypExp 2 extends the package to expand hypergeometric functions about integer and half-integer parameters, adding half-integer support through a new Mathematica algorithm.

  • NEW VERSION PROGRAM SUMMARY: The package uses the HPL package included in its distribution.
  • NEW VERSION PROGRAM SUMMARY: HypExp 2 supersedes the previous version published in Comput. Phys. Commun. 175(2006)122.
  • NEW VERSION PROGRAM SUMMARY: HypExp 2 adds the ability to expand hypergeometric functions about half-integer-valued parameters.The expandable classes with half-integer parameters are listed in the long write-up.
  • NEW VERSION PROGRAM SUMMARY: The program addresses expansions about parameters that are integer and/or half-integer valued.
  • NEW VERSION PROGRAM SUMMARY: The new algorithm is implemented in Mathematica as an extension of the existing HypExp package.

1 Introduction

Hypergeometric functions require parameter expansions in dimensional-regularization calculations, while existing implementations were restricted to integer parameters and half-integer methods lacked a user-friendly package. This work introduces and implements a method extending HypExp to address that gap.

  • 1 Introduction: Dimensional regularization makes expansion coefficients beyond the constant term relevant because divergences appear as poles in 1/ε^n.
  • 1 Introduction: Existing systematic implementations expanded hypergeometric functions about integer-valued parameters, not half-integer-valued parameters.
  • 1 Introduction: Half-integer parameters arise in hypergeometric functions from computations involving massive particles.
  • 1 Introduction: The paper introduces a new half-integer expansion method and implements it in an extended public Mathematica package, HypExp.
  • 1 Introduction: The paper presents the algorithm, its HypExp implementation, Feynman-diagram examples, analytic continuation, and additional appendices.

2 Expansion of hypergeometric functions about half-integer parameters

The paper develops an algorithm that reduces hypergeometric-function expansions with half-integer parameters to basis-function expansions and operator manipulations, then derives all-order basis expansions using HPLs. The method is generalized to arbitrary PFP−1 classes and implemented for selected classes in HypExp.

  • 2.2 Algorithm: The algorithm expresses a hypergeometric function’s expansion through differentiation and integration operators acting on a basis function of the same type.This reduces the general expansion problem to computing one basis hypergeometric function and operating on its expansion.
  • 2.1 Definitions and notation: Hypergeometric parameters are organized by integer finite parts and ǫ-dependent coefficients, with HF type P_i^j indicating which numerator and denominator parameters are half-integers.The notation separates numerator parameters A, a, α from denominator parameters B, b, β.
  • 2.2 Algorithm: Fractions and polynomials arising in the hypergeometric series are rewritten as integration and differentiation operators, enabling reduction to the chosen basis function.Partial fractions produce denominators of the forms 1/(i + j + γǫ)^n and 1/(i + j + 1/2 + γǫ)^n.
  • 2.2.2 General method: For arbitrary PFP−1 functions, the resulting formula relates any function of type P_r^s to a same-type basis function, provided its expansion and operator actions are available.The basis expansion and a procedure for integrating and differentiating it are the two required ingredients.
  • 2.3.2 Simple example: The first terms of the derived expansion are presented explicitly, and the approach is extended through implementation in the Mathematica package HypExp.The package adds expansion capabilities for selected hypergeometric functions with half-integer parameters.
  • 2.3 All-order expansion of the basis functions: The basis-function strategy yields explicit all-order expansions for some hypergeometric-function types, with achievable depth determined by the available depth of the corresponding basis expansion.The construction uses HPL products, division, and derivative factorization to manipulate expansions.

3 Extension of the Mathematica package HypExp

HypExp 2 extends the Mathematica package HypExp to expand supported hypergeometric functions with half-integer parameters, while adding related expression-handling features and complex-argument output support.

  • HypExp 2 extends the existing Mathematica package HypExp to expand JFJ−1-functions of specified types with half-integer parameters.
  • Outputs use rational functions, logarithms, polylogarithms, Nielsen polylogarithms, and HPLs, with complex-argument HPL support required for half-integer expansions.
  • HypExp returns the ǫ-expansion of a supported Hypergeometric2F1 or HypergeometricPFQ through a requested order O(ǫn).
  • The package leaves non-hypergeometric inputs unchanged and avoids returning SeriesData so the surrounding expression is not automatically expanded.
  • Series was updated to expand compound expressions containing hypergeometric functions, including their other ǫ-dependent factors.
  • HypExp is preferable for high expansion orders or large expressions, while simplification is not applied systematically because it can take longer than expansion.

4 Examples and applications

The package is applied to Feynman-diagram integrals through four loops, reproducing known results and providing new results such as the massless three-loop master integral A6,2.

  • Four examples, including Feynman diagrams with up to four loops, demonstrate applications of the package.
  • 4.1 One-loop vertex correction: The one-loop vertex correction is evaluated from a one-dimensional Mellin–Barnes representation with unit propagator powers.
  • 4.1 One-loop vertex correction: The expansion agrees with a recent reference result, and HPL output can be converted into logarithms, polylogarithms, and Nielsen polylogarithms.
  • 4.2 Massless three-loop master integral: The massless three-loop master integral A6,2 is expanded using HPLs, argument transformations, product expansion, and HPL integration, in principle to all orders.
  • 4.3 Two-loop vacuum diagram: For the two-loop equal-mass vacuum diagram, 4F3 functions reduce to linear combinations of 2F1 functions that HypExp can expand, reproducing a known result for n1 = n2 = n3 = 1.
  • 4.4 Four-loop tadpole: The four-loop tadpole is treated through Mellin–Barnes residues, conversion to a Meijer-G function, and re-expansion in hypergeometric functions, agreeing with prior findings.

5 Conclusion and outlook

The paper presents an algorithm for half-integer hypergeometric expansions, implements it in HypExp, and demonstrates applications through four-loop Feynman-diagram examples.

  • The method expresses hypergeometric functions through integration and differentiation operators acting on a same-type basis function.The basis-function choice is not unique and can be selected for convenience.
  • The approach requires an expansion of one same-type hypergeometric function and integration and differentiation routines for the functions involved.
  • The method may extend to hypergeometric functions of higher P or to other types of hypergeometric functions.
  • The expansions can be obtained through appropriate limits for related hypergeometric functions.
  • The algorithm has been implemented for specified classes of hypergeometric functions in the existing Mathematica package HypExp.
  • Examples involving Feynman diagrams with up to four loops reproduce known results and provide new results for the massless three-loop master integral A6,2.

A Analytic continuation of the expansion

The appendix describes how the expansion is handled outside the package’s primary interval, including branch cuts, complex arguments, and square-root sign conventions.

  • The package results are given for arguments in the interval (0, 1), while evaluation elsewhere in the complex plane requires care.
  • For x < 0, the HPL argument and reality properties depend on the argument transformation and the number of + weights.
  • The factor of √x removes the square-root sign ambiguity when the same convention is used in the HPL argument and prefactor.
  • The discussed expansion remains real for x < 0 under consistent square-root conventions.
  • For x > 1, the HPL branch cut corresponds to the hypergeometric function’s branch cut and generates an imaginary part.

B Computation of the basis functions

The basis-function computation uses the operator strategy to derive explicit expansions for several hypergeometric-function types, including cases requiring new weights.

  • Integration and differentiation operators with appropriate coefficients connect hypergeometric functions of the same type, making the basis-function choice arbitrary.
  • The strategy is applied to functions of types 21, 30, 31, and 32.
  • The expansion of hypergeometric functions of type 31 requires introducing new weights.
  • Some expansions could also be computed by other methods, but this method produces them directly in a form suitable for the algorithm.

B.1 HFs of type 21

For hypergeometric functions of type 21, the paper computes a basis-function expansion and specifies recursive rules for its coefficient functions.

  • The section considers the expansion of the basis function B1 0({0, 0; 1}, {α1, α2; β1}).
  • The coefficient functions c are determined recursively by a set of stated rules.
  • The rules set c(0) and c(1) to zero and define c(−1) as 2α2.
  • Further rules recursively relate indexed coefficients, including cases beginning with −1 or 0 and continuing with specified index patterns.

B.2 HFs of type 31

The section develops parameterized expansions for HFs of type 31 using recursively determined coefficient functions and basis functions. In the integer-parameter limit, the expansion contains only HPLs of weights 0 and −.

  • Parameterization: The expansion is expressed as a function of the parameters s, d1, d2, and d3, with definitions specialized to this HF class.The supplied formulas define these combinations from the α and β parameters.
  • Recursive construction: Coefficient functions are determined recursively by applying parity-dependent rules to the weights and parameters.The rules distinguish even and odd numbers of + weights and conditions on the first weight.
  • Basis functions: The basis function is obtained by subtracting unity from the hypergeometric function, after which separate expansions for B1 and B0 are used.The section also gives parameter substitutions for obtaining related basis-function expressions.
  • Integer-parameter limit: In the integer-parameter limit, all HPLs with weight + disappear, leaving only HPLs with weights 0 and −.This matches the expected representation in terms of HPLs with argument x² for integer parameters.
  • Consistency check: The results of Eq. (43) are recovered by setting either α1 or α2 equal to one of the β parameters.

B.5 HFs of type 31

The section expands a type-31 hypergeometric function through basis-function expressions containing HPLs with the new weight +. It identifies a current implementation boundary arising from unsupported HPL weights.

  • Expansion: The expansion of the basis function B1 for the specified type-31 HF is obtained from the expansion of the corresponding hypergeometric function.
  • HPL structure: The resulting expression contains HPL terms with weight sequences including (0,0,+), (+,−,+), (0,−,+), and (+,0,+).Their coefficients depend on combinations of the α and β parameters.
  • New weights: The construction requires defining new HPL weights because weights +, −, and 0 alone do not suffice for all half-integer-parameter expansions.
  • Implementation boundary: These new weights do not limit the algorithm, but they are unsupported by the current HPL package implementation.
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