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AMBRE - a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals
J. Gluza, K. Kajda, T. Riemann
TL;DR
AMBRE addresses the construction of Mellin-Barnes representations for Feynman integrals, where obtaining compact representations is useful for analytic and numerical evaluation. It implements a loop-by-loop Mathematica workflow and can produce scalar and tensor MB representations, with examples showing reduced representation complexity. Its effectiveness is scope-dependent: non-planar topologies and some massive or high-dimensional cases remain challenging.
Problem
Constructing useful, low-dimensional Mellin-Barnes representations for multi-loop and tensor Feynman integrals requires a systematic procedure, especially when analytic or numerical evaluation depends on representation dimension.
Method
AMBRE uses a Mathematica loop-by-loop procedure to construct Mellin-Barnes representations from Feynman integrals and optimize intermediate representations with Barnes lemmas.
Results
Constructing B5l2m2 from scratch leaves four integrals, all three-dimensional or simpler, compared with eleven integrals, one four-dimensional, from direct line contraction.
Takeaways & Limitations
AMBRE provides practical planar Feynman-integral representations and supports subsequent analytic expansion and evaluation through the package MB.
Takeaways & Limitations
The loop-by-loop approach is not efficient for some non-planar topologies, while massive tadpoles and large-dimensional MB integrals can also limit the approach.
Abstract
from arXiv · showhide
The Mathematica toolkit AMBRE derives Mellin-Barnes (MB) representations for Feynman integrals in d=4-2eps dimensions. It may be applied for tadpoles as well as for multi-leg multi-loop scalar and tensor integrals. AMBRE uses a loop-by-loop approach and aims at lowest dimensions of the final MB representations. The present version of AMBRE works fine for planar Feynman diagrams. The output may be further processed by the package MB for the determination of its singularity structure in eps. The AMBRE package contains various sample applications for Feynman integrals with up to six external particles and up to four loops.
1 Introduction
The paper presents AMBRE as a Mathematica package for constructing Mellin-Barnes representations of Feynman integrals, building on analytical and numerical MB methods. It introduces the package's organization and applications across the paper.
- AMBRE constructs Mellin-Barnes representations for Feynman integrals as a Mathematica-based tool.
- The paper situates AMBRE alongside analytical MB evaluations, differential-equation methods, and earlier systematic numerical approaches.
- The paper covers the MB formalism, AMBRE's features, one-loop and multi-loop examples, tadpoles, on-shell diagrams, non-planar topologies, and Mathematica functions.
2 Construction of Mellin-Barnes representations
AMBRE converts Feynman integrals into Mellin-Barnes representations by applying the MB decomposition to topology-dependent polynomials and integrating out Feynman parameters. A loop-by-loop strategy extends the construction to multi-loop scalar and tensor cases while exposing restrictions for higher-rank tensors.
- The Mellin-Barnes relation decomposes sums in the F-polynomial, after which contour-separated Γ-function poles define the integration contours and Feynman-parameter integrations are performed.
- The construction starts from an L-loop Feynman integral in d = 4 − 2ε dimensions with internal lines, masses, and external momenta.
- AMBRE replaces momentum integrations with Feynman-parameter integrations whose topology is characterized by the U and F functions.
- Tensor integrals require additional numerator structures; higher-rank tensors depend on a diagonalizing rotation and can become non-polynomial in Feynman parameters.
- The present version is restricted to scalar and vector integrals and/or one-loop integrals because higher-rank tensor dependence complicates the construction.
- The resulting representation is a single multidimensional MB integral for scalar integrals and a finite sum of MB integrals for L-loop tensor integrals, with loop-by-loop evaluation reducing the formalism to one-loop cases.
- The package MB can subsequently analytically expand the resulting integrals in ε and evaluate the resulting finite MB integrals.
3 Using AMBRE
AMBRE provides a semi-automatic Mathematica workflow for defining integrals, ordering loop integrations, constructing subloop F-polynomials, and generating MB representations. Its functions support scalar, one-loop tensor, and selected higher-rank numerator cases, with Barnes lemmas available to reduce representations.
- AMBRE is a semi-automatic Mathematica procedure for multi-loop calculations, with examples introducing its package-specific features.
- The workflow defines kinematic invariants, chooses an order for the L one-loop subloops, constructs each subloop, optimizes its F-polynomial, applies the MB formula, and integrates over Feynman parameters sequentially.
- Different choices of loop and MB integration order can produce different forms of the final MB representation.
- Version 1.0 constructs planar MB representations for scalar multi-loop multi-leg integrals, tensor one-loop integrals, and selected higher-rank numerator integrals.
- The Fullintegral function defines an integral, while the internal-momentum list controls the ordering of iterated integrations.
- SubLoop calculates a subloop F-polynomial, determines its MB representation, and integrates over Feynman parameters; intermediate F-polynomials can introduce additional propagators.
- AMBRE supports one-loop tensor numerators and represents their results as sums of component MB integrals, with momentum-flow definitions affecting the calculation.
- Barnes' first and second lemmas can be applied to suitable integration variables or paired invariant exponents to simplify MB representations.
4 One-loop integrals
AMBRE constructs Mellin–Barnes representations for one-loop scalar and tensor integrals, including massive, massless, multi-leg, and numerator cases. Its examples show that algebraic regrouping and Barnes’ lemmas can substantially reduce representation dimensions, especially in ε-expanded results.
- AMBRE applies its one-loop construction to massless gauge-theory and massive-QED integrals, including scalar and tensor cases.
- 4.1 Example: the pentagon diagram of massive QED: A naive massive-QED pentagon construction would yield a twelve-dimensional MB integral, but grouping F-polynomial terms reduces it to five dimensions.The intermediate seven-fold representation is reduced further by two applications of Barnes’ first lemma.
- 4.1 Example: the pentagon diagram of massive QED: Five-particle kinematics prevents further reduction of the pentagon representation because it depends on five variables plus a mass in Bhabha scattering.
- 4.2 Numerators: Tensor numerators generally produce sums of MB integrals, with the number of terms increasing with tensor rank, although some irreducible-numerator cases remain compact.Numerical checks covered two-, three-, and four-point functions with numerators containing up to eight scalar products.
- 4.3 More masses: A general one-loop scalar vertex gives a five-dimensional MB integral, whereas the massive-QED specialization yields a compact one-dimensional representation.
- 4.4 More legs: For higher-leg one-loop functions, the general MB dimension increases, but ε-expanded constant terms can require only up to three-dimensional integrals for the examples considered.The massive Bhabha five-point function has at most three-dimensional finite contributions, while massless and massive hexagons are generally eight-fold.
5 Multi-loop integrals: loop-by-loop integrations
AMBRE constructs multi-loop Mellin–Barnes representations by iteratively integrating subloops, with momentum flow and integration order determining representation complexity. For planar topologies, this approach preserves regular subtopologies and can produce low-dimensional representations, including applications to massive boxes, ladders, and massless multi-loop diagrams.
- Loop-by-loop method: The loop-by-loop method applies the one-loop Mellin–Barnes procedure successively to multi-loop integrals.It exploits the simplification that the one-loop delta-function sets U = 1, so the MB relation acts on F.
- Integration order: Choosing loops with fewer lines first generally minimizes the number of terms in intermediate F-polynomials.The resulting SubLoop[integral] function represents the first loop’s F-polynomial.
- Massive planar box: The massive two-loop planar box becomes a five-dimensional MB representation after removing redundant kinematic factors with Fauto[0].The modification also removes the factor 4^z6.
- Special numerators: After the first integration, the remaining propagators can have shifted indices, but no additional momentum structure appears.Selected irreducible numerators can therefore be handled with a single MB representation as in a scalar integral.
- Further examples: The three-loop planar example yields a 10-fold MB representation whose numerical result agrees with an earlier result.Massless examples include a six-dimensional four-loop self-energy and a two-loop five-point topology whose representation dimension depends on the derivation order.
- Further examples: Different integration choices for the same massless five-point kinematics produce different MB dimensions, including a 13-dimensional representation.The minimal reported dimension is seven when internal box momenta are integrated before pentagon momenta.
6 Tadpoles
AMBRE applies loop-by-loop integration to planar tadpoles, but massive four-loop cases can produce complicated MB representations and require careful control of integration order and intermediate F-polynomials. Numerical checks agree with an independent reference for the basic integral.
- Tadpole construction: The loop-by-loop procedure applies to planar tadpoles, but the final iteration may contain different massive and massless propagator configurations.AMBRE uses different formulas depending on whether the last integral contains one massive, mixed, or one massless propagator.
- Tadpole construction: A term (−m)^α can cause oscillatory numerical errors, so the preceding F-polynomial must be modified to produce equal-momentum massive and massless propagators.The same adjustment is required when the last integral contains a single massless propagator.
- Integration order: The order of integrations is critical because a different choice can produce two- or higher-dimensional representations.This makes integration ordering a direct determinant of the final MB dimension.
- Numerical check: The basic tadpole integral’s numerical MB result agrees with the cited independent result.The reported expansion includes coefficients through ǫ^4.
- Scope and limitations: Four-loop massive tadpoles can yield six-dimensional MB integrals, indicating natural limits for the MB approach in complicated massive multi-loop calculations.The paper contrasts this complexity with other approaches.
7 On-shell diagrams
AMBRE also constructs MB representations for on-shell self-energies, including a two-dimensional representation for a massive five-line example and a simpler analytically expandable case. The examples agree with results from the On-Shell2 package.
- On-shell self-energies: The on-shell self-energy SE5l3m2 is represented by a two-dimensional MB integral.The example corresponds to the F01101 diagram in the On-Shell2 notation.
- On-shell self-energies: The SE5l3m2 example agrees with the result obtained using On-Shell2.
- On-shell self-energies: SE3l1m is a simpler on-shell self-energy with one massive and two massless propagators.Its result can be expanded to any order in ǫ.
- On-shell self-energies: Barnes’ second lemma is used for SE3l1m, and its result also agrees with On-Shell2.
8 Non-planar topologies
AMBRE’s loop-by-loop procedure is inefficient for non-planar topologies and cannot reach the known minimal representation for the massless non-planar vertex. Automating equally effective non-planar representations remains open.
- The loop-by-loop procedure is not the most efficient approach for non-planar topologies.
- For the massless non-planar vertex, the known minimal representation is two-dimensional, whereas the loop-by-loop construction yields four dimensions.The hourglass subtopology gives a three-dimensional representation before the second part is added.
- Changing momentum-flow arrangements does not improve the loop-by-loop result, so another approach is needed to obtain the minimal integral.
- Whether non-planar representations can be automated as for planar cases remains an open question.
9 Summary
AMBRE constructs Mellin-Barnes representations for planar Feynman integrals and often achieves low-dimensional forms through an iterative loop-by-loop approach. Its effectiveness depends on topology, kinematics, and representation dimension, which also affect subsequent analytical or numerical evaluation.
- AMBRE constructs Mellin-Barnes representations for planar Feynman integrals and provides sample applications.
- The loop-by-loop approach typically produces minimal-dimension MB integrals, with Barnes’ lemmas helping reduce dimensions independently of momentum flow and iteration order.
- For kinematics with five or more legs, iteration order and momentum-flow choices affect the resulting representation.
- Some multidimensional MB integrals are difficult to evaluate stably and accurately, analytically or numerically.
- For many phenomenological and theoretical applications, AMBRE addresses the derivation stage for a large class of Feynman integrals.
A AMBRE functions list
The AMBRE appendix lists functions for entering Feynman integrals, preparing subintegrals, constructing MB representations, modifying polynomials, and applying Barnes-lemma simplifications.
- Fullintegral inputs the numerator, propagators, and internal momenta of a Feynman integral.
- invariants stores the list of kinematic invariants used in the integral.
- IntPart prepares a subintegral by collecting its numerator, propagators, and integration momentum.
- Subloop determines the U and F polynomials and an MB representation for a selected subintegral.
- Fauto[0] permits user-specified modifications of the F polynomial fupc.
- BarnesLemma applies Barnes’ first or second lemma and can attempt simplifying variable shifts.