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S@M, a Mathematica Implementation of the Spinor-Helicity Formalism

D. Maitre, P. Mastrolia

arXiv:0710.5559v2hep-ph

TL;DR

On-shell methods use physical states and analytic structure to support scattering-amplitude calculations, but one-loop virtual corrections are a bottleneck for NLO results. S@M implements complex-spinor algebra and numerical evaluation in Mathematica; the paper presents it as a flexible tool for tree- and loop-level amplitude work, while noting ambiguities in some slashed-matrix conversions.

  • Problem

    One-loop virtual corrections are a bottleneck to making quantitative estimates for most processes at next-to-leading-order accuracy.

    Source: §1 · p. 2

  • Method

    S@M implements spinor-helicity algebra in Mathematica, with spinor manipulation, spinor shifts, and numerical evaluation at computational stages.

    Source: §1 · p. 4, §5 · p. 44

  • Results

    The package provides a flexible tool for basic spinor algebra and spinor shifts in analytic scattering-amplitude evaluation at tree and loop levels.

    Source: §5 · p. 44

  • Takeaways & Limitations

    The tool is oriented toward helicity-amplitude evaluation for Standard Model phenomenology and related applications.

    Source: §1 · p. 5

    When explicit slashed matrices are not embedded in a spinor chain, their conversion is ambiguous and To2DimSpinor converts them to Sm2 type.

    Source: §3.3.9 · p. 25

Abstract

from arXiv · show

In this paper we present the package S@M (Spinors@Mathematica) which implements the spinor-helicity formalism in Mathematica. The package allows the use of complex-spinor algebra along with the multi-purpose features of Mathematica. The package defines the spinor objects with their basic properties along with functions to manipulate them. It also offers the possibility of evaluating the spinorial objects numerically at every computational step. The package is therefore well suited to be used in the context of on-shell technology, in particular for the evaluation of scattering amplitudes at tree- and loop-level.

PROGRAM SUMMARY/NEW VERSION PROGRAM SUMMARY

S@M is a Mathematica implementation of the spinor-helicity formalism, designed for complex-spinor algebra. The program summary identifies Mathematica as its programming language and lists the implementation problem and solution method.

PROGRAM SUMMARY/NEW VERSION PROGRAM SUMMARY

1 Introduction

The introduction motivates on-shell amplitude methods and presents S@M as a Mathematica package for complex-spinor algebra. It outlines applications to tree- and loop-level scattering-amplitude calculations.

1 Introduction

  • On-shell methods use physical propagating states and exploit shared kinematic singularities to extract analytic information recursively or iteratively.The introduction describes poles at tree level and branch cuts at loop level as examples of shared structure.Source: §1 · p. 2
  • Complex on-shell momenta preserve momentum conservation while enabling factorization-based reconstruction of tree amplitudes through BCFW recurrence.The introduction describes this as a quadratic recursion and notes that it also works for massive particles [71].Source: §1 · p. 3
  • Generalized unitarity uses complex kinematics to impose multiple on-shell conditions and extract loop-amplitude coefficients from cuts.The introduction notes that four-dimensional cuts can leave rational terms undetermined, motivating complementary methods.Source: §1 · pp. 3–4
  • S@M implements the spinor-helicity formalism in Mathematica and supports complex-spinor algebra.The package provides spinor objects, their basic properties, manipulation functions, and numerical evaluation at computational steps.Source: §1 · p. 4
  • S@M is suitable for generating complex spinors associated with solutions of multi-particle factorization and generalized-cut conditions, implementing BCFW-like recurrences for high-multiplicity tree amplitudes and rational coefficients, and decomposing massive momenta onto massless ones.The cited examples include high-multiplicity tree amplitudes and rational coefficients.Source: §1 · p. 4
  • The package is oriented toward helicity-amplitude evaluation at leading order and beyond, including applications to Standard Model phenomenology.The paper describes its organization and says the final section presents three examples of S@M use.Source: §1 · pp. 4–5

2 Notation

Section 2 establishes the package’s spinor notation and conventions, covering two- and four-dimensional spinors, representation-independent labels, spinor products, and massive-spinor products.

2.1 Two-dimensional Spinors

  • The two-dimensional spinors for massless fermions are defined through the Dirac equation, with conventions for chirality, index raising, normalization, and complex momenta.The paper also gives an identity for representing spinors associated with real massless momenta.Source: §2.1 · pp. 5–6

2.2 Four-dimensional Spinors

  • The four-dimensional conventions use Dirac gamma matrices to represent massless spinors and evaluate spinor products with slashed-matrix insertions.The paper notes that positive- and negative-energy solutions of the massless Dirac equation agree up to normalization conventions.Source: §2.2 · pp. 6–7

2.2.1 Label Representation

  • The Label representation denotes spinors without committing to an explicit representation of their components.This notation is suited to algebraic manipulation of representation-independent spinor inner products.Source: §2.2.1 · p. 7

2.3 Spinor Products

  • Massless spinor inner products obey antisymmetry and vanishing same-label identities, and are related to momentum inner products as square roots up to a phase.The paper distinguishes these inner products from external spinor products used to decompose slashed matrices.Source: §2.3 · p. 8

2.4 Massive Spinors Products

  • Massive spinors are represented with two spin-polarization states relative to a fixed reference axis, rather than helicity states.Their products can be expressed using massless spinors through a light-cone decomposition with a massless reference momentum.Source: §2.4 · pp. 9–10

3 Mathematica Implementation

S@M provides notebook-friendly input and output and a symbolic framework for spinors, Lorentz vectors, slashed matrices, and spinor products. Its functions support multiple representations and automatic manipulations when objects are declared or otherwise recognized.

3 Mathematica Implementation

  • The package includes a Mathematica notebook containing the implementation examples, and the paper refers readers to an appendix for the full function list.The package is loaded at the beginning of a Mathematica session.Source: §3 · p. 10, §3.1 · p. 11, §3.3 · p. 12

3.1 Input and Output

  • Notebook input and output use a familiar bra-ket form, with palette and character-sequence entry options unavailable in the console version.Notebook output can be copied as input for a later evaluation; TraditionalForm can provide similar output in the console.Source: §3.1 · p. 10

3.2 General Structure

  • S@M supports label, two-dimensional Weyl, and four-dimensional Dirac representations, with a spinor symbol usable contextually as a spinor or its slashed matrix.The package’s functions act on defined spinor labels across representations.Source: §3.2 · p. 11, §3.3.1 · p. 12

3.3.1 Spinors Declaration

  • Spinors are declared with DeclareSpinor, while integer labels are automatically treated as spinors through Sp.Undeclared variables may lack automatic properties and prevent most functions from being used.Source: §3.3.1 · pp. 12–13

3.3.2 Spinor Representations

  • Spinor representations provide linear functions for two- and four-dimensional spinors, with contractions of two-dimensional spinors handled by Mathematica’s Dot operator.The package also specifies a standard contraction order for numerical evaluation.Source: §3.3.2 · pp. 13–14

3.3.3–3.3.5 Lorentz Vectors, Minkowski Products, and Invariants

  • Lorentz vectors can be declared explicitly or represented by the symbol of an associated declared spinor, and MP computes linear, symmetric Minkowski products.The invariant function s represents squared sums of momenta and accepts multiple arguments for multiparticle invariants.Source: §3.3.3 · p. 14, §3.3.4 · p. 15, §3.3.5 · p. 16

3.3.6–3.3.7 Slashed Matrices

  • Slashed matrices have label, two-dimensional, and four-dimensional representations, with declared spinors and vectors automatically supplying associated matrices.Other slashed matrices must be declared; the package provides tensor-product objects for matrices formed from two spinors.Source: §3.3.6 · pp. 16–18

3.3.8 Spinor Products

  • Spinor products use Spaa, Spab, Spba, and Spbb, with endpoint spinors and intermediate slashed matrices or objects interpretable as such.Spaa and Spbb take an even number of declared arguments, whereas Spab and Spba take an odd number.Source: §3.3.8 · pp. 20–21

-Standard order

S@M puts spinor products into a normal order by applying antisymmetry and spinor-chain identities, with the same rules in two- and four-dimensional representations.

-Standard order

  • Spinor products are reordered using antisymmetry and chain identities, including conversion between mixed angle- and square-bracket forms.The displayed identities also include the on-shell condition as a special case.Source: -Standard order · p. 21
  • Spaa and Spbb use opposite normal-ordering conventions, so ⟨a b⟩[b a] appears in the usual order.Source: -Standard order · p. 21
  • The same ordering rules apply to spinor products in the two- and four-dimensional representations.Source: -Standard order · p. 21

-Linearity

Spinor products automatically expand linear combinations of arguments that were declared as spinors.

-Linearity

  • Declared spinor arguments expand linearly; integer labels require explicit wrapping with Sp because arithmetic on integers occurs first.Source: -Linearity · p. 22

-Syntax correction

S@M automatically corrects spinor-product type mismatches when all arguments are declared package objects, and issues a warning.

-Syntax correction

  • When declared spinors, vectors, or slashed matrices do not match a product's expected argument count, S@M changes its type and warns.Source: -Syntax correction · p. 22

-Slashed matrices insertion

S@M provides symbolic rules and functions for manipulating spinor products, slashed-matrix chains, and numerical momenta, including conversions among representations.

-Slashed matrices insertion

3.5 Numerical Functions

  • Numerical functions expose momenta and spinor representations and evaluate products and invariants; unembedded slashed matrices convert to Sm2 because their conversion is ambiguous.The ambiguity applies to explicit slashed matrices not embedded in a spinor chain; products of them become Dot products starting with an Sm2 matrix.Source: §3.3.9 · p. 25, §3.5 · pp. 33–34

3.3.9 Spinor Manipulations

  • Spinor manipulations convert between label, two-dimensional, and four-dimensional forms, compactify or expand chains, apply Schouten identities, and shift or replace spinors.Applying a slashed matrix to a spinor yields an opposite-chirality spinor; SpOpen and SpClose open or close chains, while SpClose may depend on label ordering and need not be relabeling-invariant.Source: §3.3.9 · pp. 24–26

3.3.10 Constants

  • Gamma0 through Gamma5 provide the package's γ-matrices in representation (14), and these matrices can also be used as slashed matrices.Source: §3.3.10 · p. 31

3.3.11 Numerics

  • Numerical spinor-product evaluation is available after defining or generating the associated four-momenta.The package evaluates spinor products with or without inserted momentum slashed matrices, as well as invariants; declared momenta can be supplied as four-vectors or explicit λ and ˜λ spinors.Source: §3.3.11 · p. 32, §3.4 · p. 32, §3.5 · p. 34

3.4 Momentum Generation

  • GenMomenta generates arbitrary on-shell momenta whose sum is zero or a specified four-vector, with reproducibility available through SeedRandom.Source: §3.4 · pp. 32–33

4 Simple Examples

The section demonstrates S@M through three scattering-amplitude examples: a BCFW construction, a one-loop box coefficient, and spinor integration of a double-cut. The examples check the known MHV result and illustrate how cut coefficients and logarithmic terms are obtained.

4 Simple Examples

  • The section presents three package examples: BCFW reconstruction of a five-gluon MHV amplitude, a quadruple-cut box coefficient, and a spinor-integration double-cut.The examples are distributed with the package.Source: §4 · pp. 35–36

4.1 BCFW Construction

  • The BCFW construction numerically matches the known five-gluon MHV result and recovers its analytic formula.A Schouten identity puts the expression in a more familiar form, which is checked against the expected result.Source: §4.1 · pp. 37–38

4.2 A Box Coefficient

  • The box-coefficient calculation tests two internal-helicity configurations: one gives zero, and the second loop-momentum solution yields only vanishing contributions.The coefficient is computed by inserting the propagator spinors into the amplitudes at each box corner.Source: §4.2 · pp. 39–40

4.3 Spinor Integration

  • The double-cut integral is reduced by integration by parts in |λ] and evaluated through residues in |λ⟩; for the discussed term, only the residue at |λ⟩ = |2⟩ contributes.The pole at |λ⟩ = |4⟩ has zero residue because of a [4 λ]3 numerator factor; other cut terms can involve higher poles, while Cut[1,1] requires a Feynman parameter whose final integration produces logarithmic terms.Source: §4.3 · pp. 43–44

5 Conclusion and Outlook

S@M provides basic spinor algebra and spinor shifts for analytic scattering-amplitude evaluation at tree and loop levels, with numerical evaluation available at every computational stage.

5 Conclusion and Outlook

  • S@M combines basic spinor algebra and spinor shifts for analytic tree- and loop-level amplitude evaluation with numerical evaluation at every computational stage.Its flexible functions could be enriched with routines for more specific tasks.Source: §5 · p. 44

A Functions Index

The index lists package functions and their description pages, including DeclareLVectorMomentum on page 32, and stores the complete function list in $SpinorsFunctions.

A Functions Index

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