Exotic $T^*_{csJ}$ and $T^*_{c\bar{s}J}$ states and coupled-channel scattering at the $SU(3)$ flavour symmetric point from lattice QCD

This lattice QCD study at the $SU(3)$ flavour symmetric point reveals attractive interactions and multiple pole singularities in the flavour-exotic $\mathbf{6}$ sector, identifying $J^P = 0^+, 1^+, 2^+$ partners of the experimentally observed $T^*_{cs0}(2870)^0$ and $T^*_{c\bar{s}0}(2900)$ states, while finding only weak interactions without poles in the $\overline{\mathbf{15}}$ sector.

The Gist

Imagine the universe is built from tiny, fundamental Lego bricks called quarks. Usually, these bricks snap together in very predictable ways: two bricks make a "meson" (like a proton's cousin), and three make a "baryon" (like a proton or neutron). For decades, physicists thought that was the only way nature played with these bricks.

But recently, experiments at the Large Hadron Collider (LHC) started finding strange new structures. These weren't just 2 or 3 bricks; they were 4-brick clusters (called tetraquarks) that didn't fit the old rules. They were "exotic" because their combination of flavors (types of quarks) was impossible for a simple 2- or 3-brick object. Two of these new discoveries were named $T^*_{cs0}(2870)^0$ and $T^*_{c\bar{s}0}(2900)$.

The big question was: What are these things? Are they four bricks glued tightly together in a tight knot (a "compact tetraquark")? Or are they two separate molecules (like a D-meson and a K-meson) loosely holding hands (a "molecule")? Or are they just a trick of the light?

This paper is a detective story written by a team called the Hadron Spectrum Collaboration. They didn't use a real LHC; they used a supercomputer to run a simulation of the universe from scratch. This is called Lattice QCD (Quantum Chromodynamics).

Here is how they solved the mystery, explained simply:

1. The Simulation: Building a Mini-Universe

To study these particles, the scientists created a digital grid (a lattice) representing space and time. They filled it with quarks and gluons (the "glue" holding them together).

• The Catch: Simulating the real world is incredibly hard because the math is messy. So, they made a few "cheats" to make the computer run faster:
  • They made the "light" quarks (up and down) heavier than they are in reality (about 700 MeV instead of the real 5 MeV). Think of this as studying a heavy, slow-motion version of a race car to understand its engine before worrying about aerodynamics.
  • They made the three light quarks (up, down, strange) have the exact same mass. This is called SU(3) flavor symmetry. It's like saying, "Let's pretend all three colors of Lego bricks are the same weight so the math is easier."

2. The Experiment: Listening to the Hum

In the real world, you can't just "see" these particles; you have to smash things together and look at the debris. In the computer simulation, they did something clever.

• They created a box of virtual space and asked: "What are the possible energy levels (notes) this box can hum at?"
• Just like a guitar string can only vibrate at specific frequencies, a box of quarks can only exist at specific energy levels.
• They used a massive library of different "operator" shapes (different ways to arrange the virtual quarks) to find these energy levels.

3. The Detective Work: The Lüscher Method

Once they had the list of energy levels (the "notes" the box hummed), they used a mathematical tool called the Lüscher formalism.

• The Analogy: Imagine you are in a dark room with a friend. You clap your hands, and you hear an echo. By listening to how the echo changes, you can figure out the size of the room and what the walls are made of, even though you can't see them.
• In this case, the "echo" is the shift in energy levels caused by the particles interacting. By analyzing these shifts, the scientists could calculate the scattering amplitude—a map of how these particles attract or repel each other.

4. The Findings: The "Hidden Sheet" Resonances

When they mapped out the interactions, they found something fascinating. In the "flavor 6" sector (one of the mathematical categories for these particles), they found six distinct poles.

• What is a pole? In physics, a "pole" in the math is like a spike. If you see a spike in the interaction map, it means a real particle exists there.
• The Twist: Most of these poles were found on what physicists call a "hidden sheet" of the complex energy plane.
  • The Analogy: Imagine a standard map of the world (the "physical sheet"). A normal resonance (like the famous $\rho$ meson) is a city right on the map. A "hidden sheet" pole is like a city that exists on a parallel dimension just underneath the map. You can't see it directly, but its gravity pulls on the map above, causing ripples.
  • This means these particles are virtual bound states or resonances that are very close to forming, but they are slightly "off" from being stable particles. They are like a magnet that is almost strong enough to hold two metal balls together, but they keep slipping apart.

5. Connecting the Dots

The paper connects these computer findings to the real-world particles LHCb found:

• The $T^*_{cs0}(2870)^0$ and $T^*_{c\bar{s}0}(2900)$ found in experiments are likely the same "family member" seen through the lens of this heavy-quark simulation.
• The simulation suggests these aren't just random bumps; they are real, distinct states.
• Crucially, the math predicts that if these exist, there should be partners with different spins (like a 1+ partner and a 2+ partner) that haven't been found yet. It's like finding a dog and predicting there must be a puppy and a bigger dog in the same litter, even if you haven't seen them yet.

6. The Conclusion

The study concludes that:

1. They are real: The exotic particles are likely molecular states (loosely bound pairs of mesons) rather than tight knots of four quarks.
2. They are hiding: They exist on those "hidden sheets," making them hard to spot but still very real.
3. More to find: The math predicts a whole family of these particles (partners with different spins) that experimentalists should go hunting for next.

In summary: This paper is like using a high-tech weather simulation to prove that a storm is coming. Even though they simulated the weather with slightly heavier clouds (heavier quarks), the patterns they found perfectly match the storms (exotic particles) we see in the real sky. They've confirmed the storm exists, explained how it forms, and told us exactly where to look for the next one.

Technical Summary

1. Problem Statement

Recent experimental observations by the LHCb collaboration have identified several exotic hadrons, specifically the $T^*_{cs0}(2870)^0$, $T^*_{c\bar{s}0}(2900)^0$, $T^*_{c\bar{s}0}(2900)^{++}$, and $T^*_{cs1}(2900)$. These states possess flavor quantum numbers that cannot be described by simple quark-antiquark ($q\bar{q}$) or three-quark ($qqq$) models, indicating they are likely tetraquarks or molecular states.

• The Challenge: Determining the nature of these states (compact tetraquark vs. hadronic molecule) requires a first-principles approach. However, at physical pion masses, these states lie above multi-hadron thresholds (e.g., $D\bar{K}\pi$), necessitating complex three-body (or more) scattering formalisms which are not yet fully mature for lattice QCD.
• The Goal: To perform the first lattice QCD investigation of these states using the Lüscher formalism for two-body scattering. To bypass the three-body threshold issue, the authors work at the SU(3) flavor symmetric point with unphysically heavy light quarks ($m_\pi \approx 700$ MeV). This pushes three-body thresholds higher in energy, allowing the exotic states to be studied within a two-body coupled-channel framework.

2. Methodology

The study employs Lattice QCD combined with the Lüscher finite-volume method to extract scattering amplitudes.

• Lattice Setup:
  • Ensembles: Five lattice volumes ($L/a_s = \{14, 16, 18, 20, 24\}$) with anisotropic lattices ($\xi = a_s/a_t \approx 3.47$).
  • Quarks: Three degenerate dynamical "light" quarks and one quenched charm quark.
  • Mass Scale: Pion mass $m_\pi \approx 700$ MeV.
  • Operators: Large bases of meson-meson interpolating operators (open-charm meson $\otimes$ light meson) projected into the flavor-exotic 6 and 15 irreducible representations (irreps) of SU(3)$_f$. Compact tetraquark operators were explicitly excluded to test the molecular hypothesis.
• Spectrum Extraction:
  • The finite-volume energy spectrum is extracted using the Variational Method (Generalized Eigenvalue Problem) on correlation function matrices.
  • Model averaging (Akaike Information Criterion) is used for energy levels with ambiguous fit ranges.
• Scattering Analysis:
  • Lüscher Formalism: The discrete finite-volume spectrum is related to infinite-volume scattering amplitudes ($t$-matrix) via the quantization condition.
  • Parameterization: The $t$-matrix is parameterized using K-matrix forms (polynomial, inverse polynomial, and ratio-of-polynomials) with both simple phase-space and Chew-Mandelstam prescriptions to ensure unitarity.
  • Channels: The analysis focuses on positive parity ($J^P = \{0, 1, 2, 3, 4\}^+$) sectors. The dominant channels involve open-charm mesons ($D, D^*$) and light mesons ($\eta_8, \omega_8$).
  • Partial Waves: The study considers S-waves, D-waves, and up to G-waves (though G-waves are largely found to be negligible in the energy region of interest).

3. Key Contributions

• First Lattice Study: This is the first lattice QCD calculation of coupled-channel scattering specifically targeting the flavor-exotic $T^*_{cs}$ and $T^*_{c\bar{s}}$ states.
• SU(3) Symmetry Exploration: By working at the flavor symmetric point, the authors map the scattering behavior of the 6 and 15 flavor multiplets, providing a unified view of the exotic spectrum before symmetry breaking.
• Pole Identification: The study rigorously identifies pole singularities in the complex energy plane, distinguishing between bound states, virtual bound states, and resonances on various Riemann sheets (physical, proximal, and hidden).

4. Key Results

A. Flavor 6 Sector (Exotic)

The flavor 6 sector exhibits strong attractive interactions in S-wave channels, leading to the discovery of six well-determined poles:

1. $J^P = 0^+$:
   • Virtual Bound State: Located below the $D\eta_8$ threshold, predominantly coupled to $D\eta_8$ ($1S_0$).
   • Resonance: Located below the $D^*\omega_8$ threshold, predominantly coupled to $D^*\omega_8$ ($1S_0$). This pole is identified as the SU(3) symmetric limit of the experimental $T^*_{cs0}(2870)^0$ and $T^*_{c\bar{s}0}(2900)$. The fact that they appear as a single pole in the symmetric limit suggests they are isospin partners ($I=0$ and $I=1$).
2. $J^P = 1^+$:
   • Virtual Bound State: Below $D^*\eta_8$ threshold, coupled to $D^*\eta_8$ ($3S_1$).
   • Virtual Bound State/Resonance: Below $D\omega_8$ threshold, coupled to $D\omega_8$ ($3S_1$).
   • Resonance: Below $D^*\omega_8$ threshold, coupled to $D^*\omega_8$ ($3S_1$). Identified as the $J^P=1^+$ partner of the $T^*_{cs0}(2870)^0/T^*_{c\bar{s}0}(2900)$.
3. $J^P = 2^+$:
   • Resonance: Below $D^*\omega_8$ threshold, coupled to $D^*\omega_8$ ($5S_2$). Identified as the $J^P=2^+$ partner of the $T^*_{cs0}(2870)^0/T^*_{c\bar{s}0}(2900)$.

• Nature of Poles: All identified poles reside on hidden Riemann sheets (not the proximal sheet), implying they may have less direct signatures in scattering cross-sections compared to standard resonances like the $\rho$ meson, but still manifest as rapid threshold enhancements.
• Higher Spins: No poles were found in the $J^P = \{3, 4\}^+$ sectors; interactions were mild.

B. Flavor 15 Sector (Exotic)

• The interactions in the flavor 15 sector are weak (slight attraction or repulsion).
• No robust poles were found in the energy region constrained for $J^P = \{0, 1, 2, 3, 4\}^+$. A single pole appeared in some $J^P=0^+$ parameterizations but was not consistently determined across all fits.

C. Physical Implications (Symmetry Breaking)

The authors extrapolate these findings to physical quark masses:

• The single $J^P=0^+$ pole in the flavor 6 sector splits into two distinct physical states:
  • An isospin-0 state ($T^*_{cs0}$) identified with $T^*_{cs0}(2870)^0$.
  • An isospin-1 state ($T^*_{c\bar{s}0}$) identified with $T^*_{c\bar{s}0}(2900)$.
• This supports the hypothesis that $T^*_{cs0}(2870)^0$ and $T^*_{c\bar{s}0}(2900)$ are members of the same flavor multiplet.
• The existence of the $J^P=1^+$ and $J^P=2^+$ partners implies the existence of unobserved states: a singly charged $T^*_{c\bar{s}0}(2900)^+$ (isospin partner) and higher-spin partners ($T^*_{cs1}, T^*_{cs2}$, etc.).

5. Significance

• Validation of Molecular/Tetraquark Models: The observation of strong attraction and poles in the flavor 6 sector, contrasted with weak interactions in the flavor 15 sector, supports the interpretation of these states as molecular or diquark-antidiquark configurations rather than conventional quark-model states. Specifically, the pattern aligns with heavy-quark spin symmetry expectations.
• Guidance for Experiment: The paper predicts the existence of specific isospin partners and higher-spin states ($J^P=1^+, 2^+$) that have not yet been observed. It suggests that the $T^*_{cs0}(2870)^0$ and $T^*_{c\bar{s}0}(2900)$ are indeed isospin partners, motivating searches for the missing charged component of the isospin-1 multiplet.
• Methodological Benchmark: The study demonstrates the feasibility of using the Lüscher formalism with heavy pion masses to disentangle complex multi-channel dynamics, providing a roadmap for future studies at physical pion masses once three-body formalisms are fully implemented.

In conclusion, this work provides the first ab initio evidence for a rich spectrum of exotic charmed-strange states, identifying the $T^*_{cs0}(2870)^0$ and $T^*_{c\bar{s}0}(2900)$ as a single flavor-symmetric entity that splits upon symmetry breaking, and predicts a tower of higher-spin partners.