$\texttt{SMaSH}$ : Simplify Massive Spinor Helicity

Abstract: We present $\texttt{SMaSH}$, a $\texttt{Mathematica}$ package to do spinor helicity computations in four spacetime dimensions $\href{https://github.com/aakash-kmr/SMaSH}{\text{(github)}}$. It can handle massive spinor helicity computations with explicit little group indices which is a novel feature. It can also handle massless as well as off-shell spinor helicity variables. It is designed to compute perturbative computations; it comes with predefined three point amplitudes and propagators for any masses and spins (arXiv:1709.04891). It can implement the high energy limit over an expression, check the discrete $\tt{C,P,T}$ transformations, compute contact terms and impose gauge invariance for any scattering process. We have shown the usage of such functions for computing gauge invariant Weinberg minimal amplitudes (arXiv:2506:12431, arXiv:2504:06343). The package can also generate both real and complex numerical kinematics for any $n$-point scattering for arbitrary masses and energy scales by implementing the $\tt{RAMBO}$ algorithm. It is also rich with basic spinor helicity manipulations like Schouten simplification, Clifford algebra manipulation, conversion between spinor helicity and Lorentz vectors, derivative w.r.t. spinors and their scalars, helicity scaling etc.

• The paper introduces a Mathematica package that automates massive spinor-helicity computations with explicit little-group covariance.
• It integrates both analytic and numerical tools for managing mixed massive/massless kinematics and simplifying algebraic identities.
• The work enhances the reliability of amplitude computations in QFT, with applications ranging from collider phenomenology to gravitational scattering.

$\texttt{SMaSH}$: A Mathematica Package for Massive Spinor Helicity Computations

Motivation and Context

The development of the $\texttt{SMaSH}$ package addresses significant demands in contemporary scattering amplitude research, especially in organizing and computing massive and massless spinor-helicity objects in four-dimensional QFT. The spinor-helicity formalism, originally introduced for massless particles and now extended to include massive momenta with explicit little group structures, is essential for modern amplitude methods. These techniques are now ubiquitous in computations ranging from Standard Model processes to classical gravitational scattering phenomena, including recent advancements in binary black hole/higher-spin scattering and classical limits of quantum field amplitudes.

However, the manual manipulation of spinor-helicity expressions, in particular with consistent index contraction and the recursive use of Schouten identities, is both error-prone and inefficient for expressions beyond minimal complexity. While existing Mathematica packages cover various aspects of the massless formalism and, to a limited extent, the massive case, there is a gap in providing transparent, little-group-covariant manipulation and full numerical pipeline integration for both massive and massless legs.

Key Technical Innovations

1. Manifest Little Group Covariance

A central innovation in $\texttt{SMaSH}$ is its ability to perform computations with explicit SU(2) little group indices for massive legs. This feature maintains little group covariance throughout algebraic manipulations—an aspect not fully supported by existing packages which typically reduce massive spinors to pairs of massless ones early in the computation. The package supports explicit contraction and canonicalization of both spinor and little group indices, with deterministic rules to resolve ambiguities up to sign conventions and lexicographic ordering.

2. Integrated Massless, Massive, and Off-shell Kinematics

$\texttt{SMaSH}$ supports both massless and massive spinor variables, and seamlessly transitions between on-shell and off-shell kinematics. Variables can be declared as massive or massless, and external kinematic conditions (e.g., specific masses or momenta) can be declared to facilitate process- or frame-specific calculations. Off-shell variables are easily managed as first-class entities, broadening applicability beyond strictly on-shell S-matrix elements to effective field theory (EFT) operators and general correlators.

3. Automated Application of Algebraic Identities

The package automates several critical algebraic simplifications:

• Schouten Identities: Both manual and recursive, algorithmic application of Schouten relations for spinor contractions, extended with machine-learning-aided simplification strategies as motivated by recent literature [SchwartzSpinorHelicity].
• Clifford Algebra Manipulations: Automatic reduction of products of $\sigma$ matrices and Lorentz index contractions.
• Canonicalization: Consistent index naming and ordering, crucial in eliminating redundancy and ensuring term-by-term equivalence.

4. Amplitude Construction and Consistency Checks

$\texttt{SMaSH}$ computes three-point amplitudes with built-in, manifestly little-group-covariant expressions for arbitrary spin and masses, automatically incorporates propagator structures up to arbitrary spin, and provides direct support for constructing higher-point tree-level amplitudes including all contact terms. Gauge invariance can be checked or enforced algorithmically by reference-spinor shifting and contact-term completion.

5. Numerical Kinematics and Frame Generation

Via an implementation of the \texttt{RAMBO} algorithm, $\texttt{SMaSH}$ generates real and complex on-shell numerical kinematics for any $n$-point scattering, for fully general masses and energy scales. COM-frame configurations, specific mass choices, and reproducible seeds are supported. This enables direct cross-checking of analytic amplitude expressions numerically, as well as facilitating Monte Carlo approaches to phase space integration.

6. Comprehensive Manipulation and Conversion Tools

Additional functionality includes:

• Automated conversion between Lorentz vector and spinor-helicity representations.
• Discrete symmetry checks ($C$, $P$, $\texttt{SMaSH}$0, $\texttt{SMaSH}$1 transformations) for both massless and massive sectors, with conventions specified for practical amplitude calculations.
• Implementing the high energy limit and massless reduction systematically within the massive spinor-helicity framework.
• Basis transformation and symmetrization utilities to manage higher-spin indices and systematic operator enumeration.

Numerical and Symbolic Results

The documentation provides explicit worked examples, including the full construction and gauge-invariant completion of spin-1 Compton scattering amplitudes with arbitrary external helicities and flavor assignments. Notably, the contact-term completion strategy is implemented natively, and the package supports amplitude factorization checks by matching the analytic construction with the product of on-shell gauge-invariant three-point vertices. The package further demonstrates the ability to compute and numerically verify identities such as gauge independence and Schouten-equivalent forms via automated random kinematics.

Contrasts and Claims Relative to Prior Work

$\texttt{SMaSH}$2 is unique in combining:

• Complete support for explicit, unreduced massive little group structures in all amplitude manipulations.
• Transparent, documented conventions with user control over index placement and physical conventions (e.g., metric signature, raising/lowering conventions).
• Integration of both analytic and numerical tools from amplitude construction to observable calculation within a single environment, including discrete symmetry analysis.
• Support for both real and complex kinematic generation for analytic continuation and gauge invariance testing in nonphysical regions—a feature not generally accessible in other packages such as \texttt{S@M} [S@M], \texttt{SpinorHelicity4D} [spinorhelicity4d], or \texttt{SpinorsExtras} [spinorsextras].

Theoretical and Practical Implications

The release and adoption of $\texttt{SMaSH}$3 enables a new level of automation and reliability in both routine and state-of-the-art amplitude computations. It is particularly suited for:

• Four-point (and higher) amplitude computations with mixed massive/massless external states, especially in EFT matching and new-physics searches where mass effects are non-negligible.
• Classical limit analysis of gravitational and electromagnetic observables, including recent applications of amplitude methods to radiative observables in binary compact objects.
• Cross-checking and benchmarking modern analytic results with large kinematic or little group index complexity, which increasingly appear in higher-loop or higher-spin amplitude studies.

The implications extend to the systematic construction of operator bases in SMEFT and BSM analyses, the investigation of positivity bounds, and the ongoing development of numerical amplitude and event generation codes for collider and gravitational wave applications.

Future Directions

Several extensions are articulated:

• Optimization and acceleration of algebraic simplifications, possibly via more advanced machine learning or dedicated term-graph rewriting.
• Systematic enumeration/generation of independent amplitude bases for arbitrary $\texttt{SMaSH}$4-point, mixed-mass/spin processes (potential integration with on-shell EFT construction pipelines).
• Further integration with Standard Model and beyond Standard Model amplitude computations and improved support for classical limits, including classical observable extraction from QFT amplitudes.

Conclusion

$\texttt{SMaSH}$5 represents a significant progression in the automation and reliability of massive and massless spinor-helicity computations in four dimensions. By making little-group-covariant computations transparent, providing robust algebraic and numeric tools in a Mathematica environment, and supporting advanced tasks like discrete symmetry and gauge-invariance checks, it establishes a new computational standard for modern amplitude research. Its extensible platform is positioned to support both continued advances in amplitude methodology and direct applications to collider phenomenology and gravitational-wave astronomy.

Reference:

$\texttt{SMaSH}$6: Simplify Massive Spinor Helicity (2606.27928) Authors: Aakash Kumar, Arnab Rudra, Rahul Shaw