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FlaSR: A Mathematica Package for the Automatic Generation of $U$ -spin Sum Rules

This paper introduces FlaSR, a Mathematica package that automatically generates higher-order $U$ -spin amplitude and amplitude-squared sum rules for systems of symmetry-related processes based on recent mathematical insights into flavor symmetries.

Original authors: Margarita Gavrilova, Livia Kong

Published 2026-08-28

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CC BY 4.0

Original authors: Margarita Gavrilova, Livia Kong

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe of subatomic particles is governed by a set of hidden patterns, much like the rules of a complex game that scientists are still learning to play. At the heart of this game are quarks, the tiny building blocks that make up protons, neutrons, and the particles that decay from them. Among these quarks, two specific types—the down quark and the strange quark—behave in a remarkably similar way, as if they were twins separated at birth. Physicists call this resemblance "U-spin," a symmetry that suggests these two particles should act identically if the universe were perfectly balanced. However, nature is rarely perfect. The strange quark is slightly heavier than its partner, and this small difference breaks the symmetry, causing the particles to behave differently. Understanding exactly how and why this symmetry breaks is crucial. It allows scientists to predict how particles should decay and to spot any deviations that might hint at new, unknown forces or particles lurking beyond our current understanding of physics.

For decades, researchers have tried to map out these relationships, but the task has been like trying to solve a massive jigsaw puzzle where the pieces keep changing shape. Every time a particle decays, it can split into many different combinations of other particles, and the mathematics required to track all these possibilities and find the hidden rules connecting them has been incredibly difficult. The calculations are so tedious and prone to human error that studying complex systems with many particles has often been out of reach. This is where a new tool, developed by Margarita Gavrilova and Livia Kong, changes the game. They have created a software package called FlaSR, which acts as an automated architect for these complex relationships. Instead of a physicist spending months manually crunching numbers to find a single rule, this software can instantly generate a complete list of every possible relationship between particle decays, even for systems involving many particles and subtle, higher-order effects.

The software works by taking a description of a particle system—what goes in, what comes out, and the forces involved—and applying a set of deep mathematical insights to find the connections. It does not rely on the old, slow method of breaking everything down into its smallest parts, which is like trying to understand a building by counting every single brick. Instead, it uses a smarter approach that sees the structure of the whole building at once. The program can handle systems where the symmetry is broken, which is the real-world scenario, and it can do so with a level of precision that was previously impossible to achieve by hand. It produces two main types of results: rules that connect the probabilities of different decays happening, and rules that connect the actual rates at which these decays occur. These rules are vital because they provide a strict test for the Standard Model of physics. If an experiment measures a decay rate that violates one of these computer-generated rules, it would be a clear signal that something new is happening in the subatomic world.

The researchers tested their tool on several real-world examples, including the decay of a particle called a D-meson into two other particles, and more complex scenarios involving three or more particles. In every case, the software successfully generated the full set of mathematical relationships that govern these processes. It was able to find rules that hold true even when the symmetry is broken to a high degree of precision, offering predictions that are much more stringent than those derived from simpler approximations. While the software is guaranteed to find every possible rule for the probabilities of decays, the researchers note that for the actual rates of decay, it finds a very strong set of rules, though they cannot yet mathematically prove it finds every single one. This is a minor limitation in a tool that otherwise solves a problem that was once considered too messy to tackle systematically.

The impact of this work extends beyond just making calculations faster. By automating the discovery of these relationships, the tool opens the door to studying much larger and more complex systems than ever before. It allows physicists to survey a vast landscape of particle decays and look for patterns that might have been missed. This is particularly important as new experiments produce more data on rare and complex decays. With FlaSR, scientists can now quickly check if their new data fits the established rules of the universe or if it points toward a revolution in our understanding of matter. The software does not just save time; it changes the scope of what is possible to investigate, turning a chaotic mess of possibilities into a clear, organized map of how the subatomic world behaves.

Technical Summary of FlaSR: A Mathematica Package for the Automatic Generation of U-spin Sum Rules

Problem Statement
Theoretical predictions for fully hadronic weak decays are hindered by the non-perturbative dynamics of Quantum Chromodynamics (QCD). While approximate flavor symmetries, specifically the SU(2) U-spin subgroup relating $d$ and $s$ quarks, offer model-independent relations (sum rules) among amplitudes and decay rates, deriving these relations beyond the exact symmetry limit is computationally challenging. The standard method, relying on Clebsch–Gordan decomposition, becomes a cumbersome bookkeeping task for large systems involving arbitrary external multiplets and higher-order symmetry-breaking spurion insertions. Furthermore, the resulting relations often appear as an arbitrary null-space basis of a coefficient matrix, obscuring their underlying symmetry structure and organization by the order of symmetry breaking.

Methodology
The paper presents FlaSR, a Mathematica package designed to automate the generation of higher-order U-spin sum rules. The package implements a systematic method derived from recent theoretical insights (Refs. [1, 2]) that avoids explicit Clebsch–Gordan decomposition. Instead, it relies on a series of theorems characterizing the mathematical structure of U-spin amplitude sum rules.

The core algorithmic workflow is as follows:

Input Definition: The user defines a system by specifying the irreducible representations (irreps) of the initial state particles, the final state particles, and the symmetry-limit effective Hamiltonian. The package supports both "abstract" input (pure SU(2) irreps) and "physical" input (particle multiplets with CKM factors).
Amplitude Sum Rules (ASRs): The package maps amplitude pairs to points on a $d$ -dimensional lattice, where $d = n/2 - 1$ ( $n$ being the number of "would-be" doublets). By forming weighted sums over subspaces of this lattice, the package derives the complete set of linearly independent amplitude sum rules at any order $b$ in the symmetry-breaking parameter $\epsilon$ . It utilizes an $a(s)$ -type basis (antisymmetric and symmetric under U-spin conjugation) to decouple sectors containing odd and even powers of $\epsilon$ .
Amplitude-Squared Sum Rules (A2SRs): To derive relations among physical observables (rates), the package employs a "squaring" procedure. It converts the linear amplitude sum rules into relations among squared amplitudes ( $|A|^2$ ) by manipulating the coefficient matrices. This involves separating dependent and residual amplitudes, squaring the relations, and eliminating interference terms via null-space analysis. The resulting relations are expressed in a $\Delta(\Sigma)$ -type basis. The package checks specific conditions (regarding the existence of lower-order sum rules) to ensure the validity of converting these into $\Delta$ or $\Sigma$ sum rules.
Implementation: The user interface is written in Wolfram Language (Mathematica), while computationally intensive routines are implemented in Python.

Key Contributions

Automation of Higher-Order Derivations: FlaSR provides the first automated tool capable of generating the full set of amplitude sum rules for arbitrary U-spin systems at arbitrary orders of symmetry breaking, a task previously limited by manual complexity.
Structural Clarity: By utilizing the lattice construction and $a(s)$ / $\Delta(\Sigma)$ bases, the package outputs sum rules organized by their order in symmetry breaking, revealing a universal structure shared by systems built from the same collection of SU(2) irreps.
Physical Applicability: The package handles physical particle multiplets, including cases with identical particles in the final state, by distinguishing between differential and integrated observables. It outputs sum rules in terms of physical processes, quantum numbers, or $n$ -tuples, and can restore CKM factors for physical interpretation.
Generalizability: While focused on U-spin in weak decays, the underlying group-theoretical framework applies to any SU(2) system (including isospin and V-spin) and processes beyond weak decays (e.g., strong interaction scattering, mass sum rules), provided the system consists of pure irreps and a complete set of channels.

Results and Scope
The paper demonstrates the package's capabilities through representative examples, including $D^0 \to P^+P^-$ , $D^-_q \to P^+P^-P^-$ , $B^0 \to D^0 \Omega^-_B \Omega^+_B$ , and mass sum rules for $\Omega^-_B$ .

Completeness: The package is guaranteed to generate the complete set of linearly independent amplitude sum rules at every order in symmetry breaking where such relations exist.
Limitations: For amplitude-squared sum rules (which translate to rate sum rules), the current version is not guaranteed to generate the complete set. The "squaring" procedure relies on specific conditions regarding the existence of lower-order relations; while these conditions are always satisfied for systems with at least one doublet, they require explicit checking for systems without doublets. Consequently, additional rate sum rules beyond those generated by the package may exist.

Significance
The authors position FlaSR as a tool to address the increasing need for comprehensive theoretical predictions in light of recent experimental advances in hadronic decay measurements. By enabling the systematic derivation of higher-order sum rules, the package facilitates:

More stringent theoretical predictions that account for U-spin breaking effects (proportional to $\Delta m = m_s - m_d$ ).
Precise tests of the flavor-symmetry framework against experimental data.
Probes into the pattern of flavor-symmetry breaking and non-perturbative QCD dynamics.
Potential sensitivity to physics beyond the Standard Model through deviations from predicted sum rules.

The paper concludes that while the current implementation of amplitude-squared sum rules has limitations regarding completeness, the framework provides a robust foundation for analyzing flavor symmetries, with extensions to higher orders in symmetry breaking for rate sum rules reserved for future work.

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