Source-linked AI summary
Expansion by regions: revealing potential and Glauber regions automatically
Bernd Jantzen, Alexander V. Smirnov, Vladimir A. Smirnov
TL;DR
Finding all regions in expansion-by-regions calculations is difficult because the earlier asy.m code missed potential and Glauber regions. The paper introduces an algorithm and asy2.m update that automates the required decompositions and transformations, and reports that the updated procedure reveals these regions or issues a warning when it cannot ensure completeness.
Problem
Identifying all relevant regions remains difficult, and asy.m does not detect potential regions in threshold expansions or Glauber regions.
Method
The paper develops an algorithm implemented in asy2.m that automates parameter decompositions and variable transformations to reveal potential and Glauber regions.
Results
The updated code outputs relevant regions as scalings of integration parameters and is reported to find potential and Glauber regions.
Takeaways & Limitations
With PreResolve enabled, asy2.m is reported either to reveal all relevant regions or to issue a warning when opposite-sign or unknown-sign terms prevent successful resolution.
Takeaways & Limitations
The completeness assurance is qualified: unresolved opposite-sign monomials or symbols with unknown signs can prevent all regions from being revealed.
Abstract
from arXiv · showhide
When performing asymptotic expansions using the strategy of expansion by regions, it is a non-trivial task to find the relevant regions. The recently published Mathematica code asy.m automates this task, but it has not been able to detect potential regions in threshold expansions or Glauber regions. In this work we present an algorithm and its implementation in the update asy2.m which also reveals potential and Glauber regions automatically.
1 Introduction
Expansion by regions provides a general way to obtain asymptotic expansions, but identifying all relevant regions remains difficult, especially for threshold and Glauber limits. The paper addresses this gap by extending asy.m with an algorithm and asy2.m implementation that automatically reveals these regions.
- 1 Introduction: Expansion by regions divides loop-momentum space into regions, expands the integrand in each region, integrates over the full domain, and discards scaleless integrals.The same strategy can be formulated for parametric representations through scaling relations among alpha or Feynman parameters.
- 1 Introduction: Identifying the typical regions for a given limit remains a non-trivial task that is often approached through progressively more complicated examples and checks against known results.Only a limited number of regions contribute, because other regions produce scaleless integrals.
- 1 Introduction: The earlier asy.m algorithm uses a convex-hull construction to find parameter scalings that yield non-scaleless integrals.It can expand parametric integrals or translate the scalings into loop-momentum regions.
- 1 Introduction: asy.m works for many limits, including Sudakov-type limits, but does not reveal potential regions in threshold expansions or Glauber regions.Its completeness result applies when the parametric polynomial F contains terms with the same sign.
- 1 Introduction: The paper presents an algorithm and asy2.m update that automatically identify relevant regions, including potential and Glauber regions.The paper also explains automated decompositions, variable transformations, and the matching of loop-momentum and parametric-region descriptions.
2 Expansion by regions in parametric representations
In parametric representations, expansion regions are specified by parameter scalings, after which the integrand and measure are expanded in the small parameter. Homogeneity, delta-function treatment, and scaleless integrals determine the resulting contributions and the limited set of relevant regions.
- 2 Expansion by regions in parametric representations: The alpha representation uses homogeneous polynomials U and F, with degrees h and h+1; F depends linearly on kinematic invariants and squared masses, while U does not.For standard propagators, U and F are the Symanzik polynomials and can be generated with UF.m.
- 2 Expansion by regions in parametric representations: A region is defined by scaling each alpha or Feynman parameter as a power of the small parameter, then expanding the transformed integrand in that parameter.The measure contributes an additional power given by the sum of the parameter-scaling exponents.
- 2 Expansion by regions in parametric representations: The contribution of a region includes the parametric prefactor, the measure factor ρ^∑l rl, and the integral with its integrand expanded in powers of ρ.The delta-function argument is expanded consistently, so parameters may become integrated from 0 to ∞.
- 2 Expansion by regions in parametric representations: Region scalings are defined only up to adding the same constant to every exponent, because homogeneity leaves the contribution unchanged.This freedom also permits the delta function to be rescaled back to standard form.
- 2 Expansion by regions in parametric representations: After retaining sufficiently many terms and setting ρ = 1, the integral is represented by those terms plus a remainder that vanishes sufficiently fast in the limit.Only a limited number of regions contribute; the others yield scaleless integrals and are set to zero.
3 Revealing potential contributions
The threshold expansion exposes a potential region that the earlier asy.m code missed, while asy2.m finds both hard and potential regions automatically after suitable parametric transformations.
- Threshold example: The earlier asy.m code reported only the hard region, whereas asy2.m finds both hard and potential regions in the threshold expansion.The potential region is associated with x1 − x2 ∼ y1/2.
- Threshold example: The missing potential contribution arises near x1 ≈ x2, where opposite-sign terms in the parametric polynomial can cancel.The relevant scaling is more precisely x1 − x2 ∼ y1/2.
- Threshold example: Decomposing the integration domain into x1 ≤ x2 and x2 ≤ x1 maps the problematic line x1 = x2 to a boundary suitable for region analysis.The two resulting integrals are equal in this example, and the variable transformation is applied separately to the subdomains.
- Threshold example: WilsonExpand[] analyzes parametric integrals using the polynomials F and U, integration parameters, kinematic scalings, and an optional delta-function restriction.The Delta option applies when the sum over an arbitrary non-empty subset of parameters is restricted to 1.
- Threshold example: The output encodes regions through parameter scalings and associated variable transformations, Jacobians, and new-variable scalings.A common shift of all scaling exponents leaves a region unchanged, so {0,−1/2} is equivalent to {1/2,0}.
- Threshold example: For the potential region, the resulting contribution is non-vanishing, expressible through gamma functions at general ε, and begins at order y1/2−ε.The result agrees with the cited prior analysis.
- Algorithm: asy2.m’s preresolution recursively tests variable combinations that reduce opposite-sign monomials, selecting decompositions with fewer such terms or fewer sectors.The method requires the relative signs of all terms in F, so kinematic substitutions must replace quantities by numbers or powers of the small parameter.
4 Revealing Glauber contributions
For the five-point integral, asy.m finds the hard and two collinear regions but misses the leading Glauber contribution. After decomposing the parametric integral and changing variables, asy2.m reveals the relevant regions and reproduces the expansion through NNLO.
- The Glauber region provides the leading contribution, scaling as (m2)^−2−ε, while collinear terms begin at (m2)^−1−ε and the hard term at (m2)^0.
- asy.m returns three regions—one hard and two collinear—but none supplies the leading (m2)^−2−ε contribution.
- The missing Glauber contribution arises when either (x2−x3) or (x4−x5) scales as (m2)^1, requiring a twofold variable transformation after decomposing their sign domains.
- With preresolution and variable changes, asy2.m finds the region patterns {0,0,0,0,0}, {0,1,0,0,0}, and {0,0,0,1,0}, up to parameter permutations.
- The transformed calculation agrees with the momentum-space Glauber contribution, while the previously found collinear regions yield scaleless integrals and vanish under analytic regularization.
- Contributions from the three asy2.m regions reproduce all five-point-integral terms through NNLO, including the hard, NLO collinear, and leading Glauber residues.
- For general diagrams, AlphaRepExpand[] attempts automatic sign-domain decomposition and warns when opposite-sign monomials or unknown-sign symbols may prevent all regions from being revealed.
5 Disentangling regions via propagator powers
Generic propagator powers separate region contributions and align parametric-space expansions with loop-momentum expansions, clarifying the roles of Glauber and collinear regions.
- Region matching: Distinct propagator-power scalings separate hard, collinear, and Glauber contributions and allow matching between loop-momentum and parametric expansions.The hard, two collinear, and Glauber regions receive distinct leading scalings in m^2, while generic powers disentangle their contributions.
- Glauber contributions: The leading contribution comes from the sum of the two LO Glauber contributions, whose individual singularities cancel after combining them.Each Glauber contribution is singular as the analytic regulators vanish, but their sum is finite.
- NLO structure: NLO Glauber terms vanish under λ3 = λ2 and λ5 = λ4, leaving the NLO result entirely from LO collinear contributions.The cancellation occurs both in parametric space and in loop-momentum space.
- NNLO structure: At NNLO, the NLO collinear contributions vanish under the same relations, leaving NNLO Glauber contributions and a LO hard contribution.These remaining contributions agree between loop-momentum and parametric expansions and reproduce the results when all λi are set to zero.
- Mellin–Barnes check: A Mellin–Barnes representation confirms the region assignments through pole families for hard, collinear, and Glauber contributions.The pole structure also shows that Glauber terms do not contribute at NLO and collinear terms are absent at NNLO.
6 Summary of asy2.m
asy2.m provides commands for identifying asymptotic regions from loop and parametric representations, with preprocessing and options for generalized integrals and delta constraints.
- Region identification: AlphaRepExpand[] identifies regions for loop integrals, while its output records Feynman-parameter scalings in powers of the expansion parameter.The loop-integral interface accepts loop momenta, propagator denominators, invariant substitutions, and kinematic scalings.
- Preprocessing: The updated code automates decompositions and variable transformations, and PreResolve attempts to eliminate cancellations before region finding.Its output includes transformations, Jacobians, and the scalings of the new variables.
- Options and scope: The code supports additional options for verbose output, scalar-product notation, and general integrals not necessarily originating from Feynman diagrams.The implementation also documents restrictions of PreResolve for non-integer propagator powers.
- Generalized integrals: WilsonExpand[] reveals regions for general parameter integrals by taking the relevant polynomials, integration variables, and expansion rules as inputs.It can also handle generalized delta-function constraints when the constraint choice is irrelevant for the integral.
- Delta constraints: The Delta option is valid when the delta-function linear combination has nonnegative coefficients with at least one positive coefficient and the integrand has the required homogeneity.Under these conditions, the specific coefficients in the delta constraint do not affect the integral.
7 Conclusion
The paper presents asy2.m as an automated parametric-space algorithm that finds potential and Glauber regions, and illustrates expansion by regions on a non-Feynman one-dimensional integral.
- Conclusion: asy2.m extends asy.m by automatically finding potential and Glauber regions and automating the required decompositions and variable transformations.The algorithm operates at the level of parametric representations and uses AlphaRepExpand[] with preprocessing support.
- Illustration: For a one-dimensional integral with t → 0, WilsonExpand[] finds two regions and their leading terms can be evaluated analytically for general λ.The integral is initially assumed absolutely convergent for Reλ < −1/2 and then analytically continued.
- Illustration: A onefold Mellin–Barnes representation confirms the asymptotic expansion by shifting the contour and taking residues at gamma-function poles.This example demonstrates successful expansion by regions beyond Feynman integrals.