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$D \to (K \pi)_{\mathbf{27}}$ at the SU(3)-flavour-symmetric point I: Methodology and strong phase determination

This paper presents the first part of an SU(3)-flavour-symmetric lattice QCD study that determines the strong scattering phase shift for the $(K\pi)_{\mathbf{27}}$ channel and infers the corresponding strong phase of the $D \to K\pi$ weak decay, while outlining the methodology for computing the full decay amplitude in a forthcoming publication.

Original authors: Matthew Black, Felix Erben, Maxwell T. Hansen, Fabian Joswig, Nelson Pitanga Lachini, Rajnandini Mukherjee, Srijit Paul, Antonin Portelli

Published 2026-09-01

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CC BY 4.0

Original authors: Matthew Black, Felix Erben, Maxwell T. Hansen, Fabian Joswig, Nelson Pitanga Lachini, Rajnandini Mukherjee, Srijit Paul, Antonin Portelli

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

In the subatomic world, particles called mesons are unstable travelers that decay, or break apart, into other particles. Among these, the charm meson is unique because it is the only particle containing a heavy charm quark that decays into other hadrons, which are particles made of quarks. Scientists study these decays to understand why the universe contains more matter than antimatter, a fundamental mystery known as charge-parity violation. To solve this, researchers must calculate how these particles interact with extreme precision. However, the forces that bind quarks together are governed by a theory called quantum chromodynamics, which is notoriously difficult to solve with standard math when particles are moving slowly or interacting strongly. This is where supercomputers come in, simulating the universe on a grid to calculate these interactions from first principles.

A team of physicists has taken a significant step forward in this effort by performing a complex simulation of a specific charm meson decay. They focused on a scenario where a charm meson transforms into a kaon and a pion, two other types of mesons. In the real world, the masses of the quarks that make up these particles are different, making the math incredibly messy. To simplify the problem, the researchers created a virtual world where the up, down, and strange quarks all have the exact same mass. In this perfectly balanced environment, the laws of physics possess a special symmetry that makes the calculations much more manageable. By working in this simplified setting, the team could isolate the specific forces at play without the noise of real-world mass differences, providing a clean test of their methods.

The researchers used a powerful technique called distillation, which acts like a filter to focus the computer's attention on the most important parts of the particle interactions. They built a digital model of the universe using three different grid sizes, allowing them to see how their results changed as the grid became finer and more detailed. By watching how the kaon and pion scattered off each other in this virtual box, they were able to measure the strength of the force between them. They found that the interaction was weakly repulsive, meaning the particles pushed each other away slightly. From this scattering data, they calculated a specific value called the scattering length, which describes how much the particles deflect one another. Their final result for this value is 0.926 with a small margin of error, a number that serves as a crucial benchmark for future, more complex calculations.

Perhaps the most important outcome of this work is the determination of a "strong phase," a specific angle that describes how the particles' wave-like nature shifts during the decay. The team calculated this angle to be minus 38.4 degrees. This number is vital because it represents a piece of the puzzle needed to predict the full decay rate of the charm meson in the real world. Without knowing this angle, scientists cannot accurately predict whether the decay happens more often for matter or antimatter, which is essential for understanding the universe's imbalance. The researchers confirmed their findings by using two different mathematical strategies to handle the grid sizes, and both methods led to the same consistent answer, giving them high confidence in the result.

This study is part one of a larger project. While the team has successfully mapped out the scattering behavior and determined the strong phase, the final piece of the puzzle—the actual calculation of the decay rate itself—will be presented in a follow-up paper. The current work establishes the necessary foundation, proving that their computational tools can handle the intricate dance of quarks and gluons required to study these rare events. By mastering the simplified, symmetric version of the problem, the team has paved the way for a complete prediction of the charm meson decay, bringing us closer to understanding the deep secrets of matter and antimatter.

Technical Summary: $D \to (K\pi)_{27}$ at the SU(3)-flavour-symmetric point: I: Methodology and strong phase determination

Problem Statement
Nonleptonic $D$ -meson decays are critical for testing the Standard Model's flavour structure and searching for new sources of CP violation. However, interpreting experimental measurements, such as the time-integrated CP asymmetry difference $\Delta A_{CP}$ , requires quantitative control over non-perturbative hadronic amplitudes. While lattice QCD has successfully addressed $K \to \pi\pi$ and semileptonic $B$ decays, calculations for charm decays involving multi-hadron final states remain challenging due to the complexity of finite-volume formalisms, the presence of multiple open channels, and the need for precise renormalization of weak operators.

This paper addresses the first step in a long-term program to provide reliable predictions for $D$ -meson decay amplitudes with fully controlled physical masses. Specifically, it focuses on the $D \to (K\pi)_{27}$ decay amplitude within an exact $SU(3)_F$ -flavour-symmetric limit (where $m_u = m_d = m_s$ ). In this limit, the decay amplitude is governed by a single reduced matrix element, $A_{27}$ , corresponding to the 27-plet irreducible representation of the flavour symmetry group. The primary goal of this manuscript is to determine the strong phase of this weak decay, which is equivalent to the elastic $K\pi$ scattering phase shift in the 27-plet channel, and to outline the methodology for extracting the full decay amplitude in a subsequent work.

Methodology
The calculation is performed using the distillation framework on three $SU(3)_F$ -symmetric gauge ensembles generated by the OpenLat collaboration. The ensembles utilize exponentiated Wilson-clover fermions and the Lüscher-Weisz gauge action, tuned such that the pion and kaon masses are degenerate ( $M_\pi = M_K \approx 410$ MeV). The study spans three lattice spacings ( $a \approx 0.12, 0.094, 0.064$ fm) to facilitate continuum extrapolation.

The technical workflow proceeds as follows:

Operator Construction: A matrix of Euclidean correlation functions is constructed using single-hadron operators projected to definite spatial momenta. These operators are organized into irreducible representations of the finite-volume little group. The distillation framework is employed to efficiently compute the necessary "all-to-all" propagators, utilizing Laplacian eigenvectors to project quark fields into a subspace.
Spectrum Extraction: A Generalized Eigenvalue Problem (GEVP) is solved to extract the finite-volume energy spectrum of the $K\pi$ system. This is performed across multiple total momentum frames ( $P^2 = 0, 1, 2, 3, 4$ ) to access a wide range of center-of-mass energies, extending from threshold up to $\approx 4M_\pi$ (below the $D$ -meson mass).
Phase Shift Determination: The finite-volume energies are related to infinite-volume scattering parameters via Lüscher's quantization condition. Given the $SU(3)_F$ symmetry and the specific channel, the analysis is restricted to the $s$ -wave ( $l=0$ ) in the 27-plet representation. The scattering phase shift $\delta_{27}(E)$ is parametrized using scattering length ( $a_0$ ) and effective range ( $r_0$ ) expansions.
Continuum Limit Strategies: The authors employ two distinct strategies to take the continuum limit ( $a \to 0$ $a \to 0$ ):
- Strategy (i): Extrapolating the extracted scattering parameters (specifically the scattering length $a_0$ ) directly from the finite-lattice-spacing results.
- Strategy (ii): Taking the continuum limit of the finite-volume energy levels themselves at fixed physical volume, and then extracting the scattering parameters from these continuum energies.
Decay Amplitude Framework: The paper details the setup for computing the three-point correlation functions required to extract the weak matrix element $\langle (K\pi)_{27} | H_W | D \rangle$ . It describes how to construct local four-quark operators within the distillation framework without storing prohibitively large four-quark fields, utilizing generalised perambulators. It also outlines the application of the Lellouch–Lüscher formalism to relate the finite-volume matrix elements to the physical infinite-volume decay amplitude.

Key Results

Scattering Length: The study determines the scattering length for the $I=3/2$ $SU(3)_F$ $K\pi$ system. Using the scattering-length-only parametrisation and the continuum extrapolation of finite-volume energies, the central result is:
$a_0 = 0.926(59)_{\text{stat}}(53)_{\text{sys}} \, \text{GeV}^{-1} = 0.926(79) \, \text{GeV}^{-1}.$
The results from both continuum extrapolation strategies are found to be consistent.
Strong Phase: Using the determined scattering length and the $D$ -meson mass in the $SU(3)_F$ limit ( $M_D \approx 1900$ MeV), the strong phase of the weak decay amplitude is inferred:
$\delta_{27}(M_D) = -38.4(2.4)^\circ.$
Lellouch–Lüscher Factors: The paper provides the necessary Lellouch–Lüscher factors for converting finite-volume matrix elements to infinite-volume amplitudes for all extracted energy levels across the three ensembles. These factors depend on the derivative of the scattering phase shift, which is evaluated using the determined parametrisation.
Methodological Validation: The consistency between the two continuum extrapolation strategies and the agreement between different fit analyses (Analysis 1 and Analysis 2) validate the robustness of the extracted spectrum and phase shifts.

Significance and Claims
The authors position this work as a necessary prerequisite for a full calculation of $D$ -meson decay amplitudes. The significance lies in:

First Step in a Controlled Program: This is the first step in a series of controlled calculations moving toward physical-mass predictions. By working in the $SU(3)_F$ limit, the authors isolate the $A_{27}$ reduced matrix element, avoiding the complications of mixing multiple irreducible representations and the power-divergent subtractions associated with singlet contractions.
Dual Continuum Strategy: A novel aspect of this work is the application of two different continuum limit strategies (extrapolating parameters vs. extrapolating energies). The agreement between these methods provides a strong check on the consistency of the dataset and the handling of discretization effects.
Strong Phase Determination: The determination of $\delta_{27}(M_D)$ is a critical input for predicting CP violation in charm decays. While the overall phase of a decay amplitude is convention-dependent, the relative strong phases between different contributions are observable. This result provides the first ingredient for such predictions once the remaining amplitudes are determined.
Methodological Blueprint: The paper serves as a technical guide for computing three-point functions involving local four-quark operators in the distillation framework, specifically addressing storage and computational cost challenges by factorizing intermediate objects.

The authors explicitly state that the complete determination of the weak matrix elements and the resulting decay amplitude will be presented in a forthcoming manuscript (Part II). This paper establishes the scattering input and the methodological framework required for that final calculation.

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