Strange partner of $T_{cc}^+$ from lattice QCD in $D^{(*)}D_s^{(*)}$ scattering

This lattice QCD study of $DD_s^*$ and $DD_s$ scattering near the $cc\bar{u}\bar{s}$ threshold reveals weak meson interactions and finds no evidence for pole structures that would indicate the existence of a strange partner to the $T_{cc}^+$ tetraquark.

The Gist

The Search for a "Strange" New Particle: A Lattice QCD Adventure

Imagine the universe is built out of tiny, fundamental Lego bricks called quarks. Usually, these bricks snap together in very specific, familiar patterns:

• Mesons: Two bricks (one quark, one anti-quark).
• Baryons: Three bricks (like protons and neutrons).

But for the last 20 years, physicists have been hunting for "exotic" Lego creations that don't fit these rules. They are looking for tetraquarks—structures made of four bricks.

Recently, scientists found a famous four-brick structure called $T_{cc}^+$ (two heavy charm bricks and two light up/down bricks). It was a huge discovery. Now, the big question is: Does a "strange" cousin of this particle exist?

This paper is the report of a team of scientists who tried to find that "strange cousin" (made of two charm bricks, one up brick, and one strange brick) using a supercomputer simulation called Lattice QCD.

Here is the story of their search, explained simply.

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1. The Detective Work: Building a Virtual Universe

You can't just build these particles in a lab; they are too unstable and fleeting. So, the scientists built a virtual universe inside a computer.

• The Grid: They created a 3D grid (a lattice) representing space and time. Think of it like a giant, invisible fishing net.
• The Simulation: They dropped "virtual" particles into this net and watched how they bounced off each other.
• The Goal: They wanted to see if the Charm-Charm-Strange combination would stick together tightly (like a bound state) or if they would just bounce off each other weakly.

2. The Experiment: Two Different Scenarios

The team looked at two different "dance floors" (scattering channels) where these particles could interact:

Scenario A: The Scalar Channel (The "S-Wave" Dance)

• The Players: A $D$ meson and a $D_s$ meson (two particles made of charm and strange quarks).
• The Observation: When the scientists watched these two dance, they noticed they seemed to push away from each other slightly.
• The Analogy: Imagine two magnets with the same pole facing each other. They don't snap together; they repel.
• The Result: The data showed no sign of a new particle forming here. The particles just bounced off each other weakly.

Scenario B: The Axialvector Channel (The "Coupled" Dance)

• The Players: This is more complex. It involves two different pairs of dancers ($D D_s^*$ and $D^* D_s$) that can switch partners mid-dance.
• The Observation: The scientists saw that the dancers were mixing a little bit, but not enough to form a tight, permanent group.
• The Analogy: Imagine two couples at a dance who occasionally swap partners for a spin, but they never decide to get married and stay together forever.
• The Result: Even in this complex mixing scenario, there was no evidence of a new, stable particle hiding near the energy threshold.

3. The Tools: How They "Saw" the Invisible

Since they couldn't see the particles directly, they used two clever mathematical tricks to interpret the data:

1. Lüscher's Formalism (The "Box" Method):

   • Imagine the particles are trapped in a small, finite box (the computer simulation).
   • If the particles interact, the size of the box changes the "notes" (energy levels) they can sing.
   • By listening to these notes, the scientists could work backward to figure out how the particles interact in the real, infinite world.

2. The Lippmann–Schwinger Equation (The "Blueprint" Method):

   • This is like building a theoretical blueprint of how the particles should interact based on physics laws, and then tweaking the blueprint until it matches the "notes" they heard in the box.

4. The Verdict: "No New Particle Found" (Yet)

After crunching the numbers and running thousands of simulations, the team concluded:

• No "Strange" Cousin: They did not find a stable, bound tetraquark (a new particle) in the energy range they studied.
• Weak Interactions: The particles interacted, but only very weakly. It was like a gentle nudge rather than a strong hug.
• Repulsion: In the simplest case, the particles actually pushed each other away.

5. Why This Matters (Even if the Answer is "No")

You might think, "If they didn't find it, why publish?"

In science, negative results are just as important as positive ones.

• Ruling Out Possibilities: Before this, theorists had many guesses about whether this "strange partner" existed. This paper says, "Based on our best computer models, it probably doesn't exist as a stable particle."
• Refining the Search: It tells experimentalists (the people building real particle colliders like the LHC) where not to look, or what kind of signals they should expect if the particle is actually a very short-lived ghost rather than a stable object.
• Future Work: The authors admit their simulation used "heavier" quarks than nature actually has (like using a slightly heavier version of the strange quark). They plan to run the simulation again with lighter, more realistic quarks to be 100% sure.

Summary Analogy

Imagine you are looking for a specific type of rare bird in a forest. You set up high-tech cameras and microphones (the Lattice QCD simulation) to listen for its call.

• You listen to the forest for hours.
• You hear the wind, the other birds, and the rustling leaves.
• You conclude: "We did not hear the call of this rare bird."

This doesn't mean the bird doesn't exist anywhere in the world, but it means it's not in this part of the forest, or it doesn't sing the way we thought it would. This paper is the report saying, "We checked this specific forest, and the bird isn't here."

Technical Summary

1. Problem Statement

The paper addresses the search for exotic hadrons, specifically the strange partner of the recently discovered doubly charmed tetraquark $T_{cc}^+$. While $T_{cc}^+$ has the quark content $cc\bar{u}\bar{d}$ and is stable under strong interactions, its strange counterpart with quark content $cc\bar{u}\bar{s}$ remains theoretically ambiguous.

• Theoretical Context: Previous model-dependent studies (quark models, QCD sum rules) suggest the $cc\bar{u}\bar{s}$ system is less likely to form a deeply bound state compared to $T_{cc}^+$. Instead, it might exist as a near-threshold resonance or a virtual state.
• The Gap: While lattice QCD studies have reported bound states for the bottom-strange analog ($bb\bar{u}\bar{s}$), the charm-strange system ($cc\bar{u}\bar{s}$) has not been explored in detail regarding scattering amplitudes. Distinguishing between bound, resonant, and virtual states requires a rigorous analysis of scattering amplitudes near the two-meson thresholds, which was previously missing for this channel.

2. Methodology

The study employs Lattice Quantum Chromodynamics (LQCD) to calculate scattering amplitudes from first principles.

• Ensembles: The calculation uses two CLS $N_f = 2+1$ Wilson-clover ensembles with:
  • Pion mass: $m_\pi \approx 280$ MeV (heavier than physical).
  • Lattice spacing: $a \approx 0.086$ fm.
  • Spatial volumes: $L/a = 24$ and $32$.
• Scattering Channels:
  • Scalar Channel ($J^P = 0^+$): Elastic $D D_s$ scattering.
  • Axialvector Channel ($J^P = 1^+$): Coupled-channel scattering between $D D_s^*$ and $D^* D_s$.
  • Note: The $D^* D_s^*$ channel was omitted as it opens at significantly higher energies.
• Operator Basis:
  • Constructed using two-meson interpolating operators ($D^{(*)} D_s^{(*)}$) projected onto cubic-group irreducible representations (irreps).
  • Local diquark-antidiquark operators were excluded based on previous findings that they have negligible impact on the low-lying spectrum.
• Correlation Analysis:
  • Utilized the distillation framework with $N_v = 75$ (large volume) and $45$ (small volume) Laplacian eigenvectors.
  • Applied the variational method (Generalized Eigenvalue Problem) to extract finite-volume energy spectra from correlation matrices.
• Amplitude Extraction:
  • Lüscher's Formalism: Used the generalized quantization condition to relate finite-volume energy levels to infinite-volume scattering matrices ($K$-matrix).
  • Lippmann–Schwinger Equation (LSE): Implemented a finite-volume version of the LSE to solve for scattering amplitudes using parametrized interaction potentials (contact interactions).
  • Moving Frames: Analyzed multiple inertial frames to access different partial waves ($S$-wave, $P$-wave) and constrain the amplitudes.

3. Key Contributions

• First-Principles Amplitude Analysis: This is the first lattice QCD study to extract infinite-volume scattering amplitudes and search for pole structures in the $cc\bar{u}\bar{s}$ system.
• Coupled-Channel Treatment: Successfully handled the near-degenerate thresholds of $D D_s^*$ and $D^* D_s$ in the $1^+$ channel, demonstrating the necessity of coupled-channel dynamics to explain the observed energy level shifts.
• Methodological Validation: Employed two complementary approaches (Lüscher and LSE) to cross-verify results, ensuring robustness against parametrization biases.
• Diagnostic Tests: Performed "pruned" operator basis tests and artificial rescaling of cross-correlations to confirm that observed energy shifts arise from inter-channel coupling rather than single-channel dynamics.

4. Results

Scalar Channel ($J^P = 0^+$, $D D_s$)

• Observation: The ground-state energies in $S$-wave irreps show positive shifts relative to non-interacting levels.
• Interpretation: This indicates a weakly repulsive interaction between $D$ and $D_s$ mesons.
• Amplitude Analysis:
  • The extracted $S$-wave amplitude ($k \cot \delta_0$) shows no zero crossings above threshold or intersections with bound-state constraints below threshold.
  • No poles (bound, virtual, or resonant) were found near the threshold.
  • The Argand trajectory shows limited phase motion, and $\rho^2|T|^2$ lacks a pronounced peak, ruling out resonant behavior.
• Conclusion: No evidence for a $cc\bar{u}\bar{s}$ bound state or resonance in the scalar channel.

Axialvector Channel ($J^P = 1^+$, $D D_s^* - D^* D_s$)

• Observation: In the $T_1^+(0)$ irrep, the two lowest energy levels exhibit opposite shifts: the ground state is shifted down (attractive), and the first excitation is shifted up (repulsive).
• Mechanism: Overlap factors and diagnostic tests confirm this pattern arises from significant channel mixing between $D D_s^*$ and $D^* D_s$.
• Amplitude Analysis:
  • Using an Effective Range Expansion (ERE), the scattering lengths were extracted: $a_{0,1} \approx 0.24$ fm and $a_{0,2} \approx -0.33$ fm.
  • Off-diagonal interaction terms are small, indicating weak channel coupling despite the mixing effects.
  • The amplitudes vary smoothly with energy, and the inelasticity remains near unity.
• Conclusion: No pole structures (bound or resonant) were found near the threshold. The system exhibits weak interactions and small channel mixing, inconsistent with a stable tetraquark state in this kinematic region.

5. Significance and Limitations

• Significance:
  • The study provides strong evidence that the strange partner of $T_{cc}^+$ ($cc\bar{u}\bar{s}$) does not exist as a bound state or narrow resonance near the $D^{(*)}D_s^{(*)}$ thresholds under the current kinematic conditions.
  • It clarifies the nature of interactions in the charm-strange sector, contrasting with the bound nature of the bottom-strange analog.
  • It establishes a robust framework for analyzing coupled-channel scattering in lattice QCD for heavy-light systems.
• Limitations & Future Work:
  • Unphysical Quark Masses: The calculation was performed at $m_\pi \approx 280$ MeV, which is heavier than the physical pion mass. This may affect binding energies.
  • Single Lattice Spacing: Results are from a single lattice spacing ($a \approx 0.086$ fm), leaving discretization errors (especially for charm quarks) unquantified.
  • Operator Basis: The exclusion of local diquark-antidiquark operators, while likely safe based on prior studies, means the search is not exhaustive for all possible tetraquark configurations.
  • Future Directions: The authors call for calculations at physical pion masses, multiple lattice spacings, and larger volumes to definitively rule out or confirm the existence of this state in the physical limit.

Final Verdict: Within the constraints of the current lattice data, the $cc\bar{u}\bar{s}$ system shows only weak meson-meson interactions with no evidence for a near-threshold tetraquark state.