Importance of local tetraquark operators for $T_{cc}(3875)^+$

This study demonstrates that incorporating local tetraquark operators into a variational analysis using Wilson-clover fermions and a novel position-space sampling method significantly alters the finite-volume energy spectrum and resulting $DD^*$ scattering phase shifts for the $T_{cc}(3875)^+$ tetraquark.

The Gist

Imagine you are trying to find a specific, very shy animal hidden deep inside a massive, foggy forest. You can't see the animal directly, so you have to listen for its footsteps and guess where it is based on the echoes bouncing off the trees.

This paper is about scientists trying to find a very rare, exotic particle called the $T_{cc}(3875)^+$. Think of this particle as a "double-charmed tetraquark." In the standard world of physics, particles are usually made of three quarks (like a proton) or a pair of quarks (like a meson). But this new particle is made of four quarks stuck together. It's like a molecular structure made of four distinct building blocks, which makes it an "exotic" creature that doesn't fit the old rulebook.

Here is the breakdown of what the scientists did, explained through simple analogies:

1. The Problem: How to "Hear" the Particle

To figure out the properties of this particle (like its mass and how stable it is), scientists use a supercomputer simulation called Lattice QCD. They create a tiny, digital universe (a grid) and try to calculate the energy levels of the particles living inside it.

To do this, they use a technique called the Variational Method. Imagine you are trying to tune a radio to find a specific station. You need to try different frequencies (operators) until you find the one that gives you the clearest signal.

• The "Bilocal" Operators: These are like listening to two separate radios talking to each other from a distance. They represent the particle as two separate mesons (like a $D$ and a $D^*$) floating near each other, loosely bound like a molecule.
• The "Local" Operators: These are like listening to a single radio where all four parts are glued together in one tight spot. They represent the particle as a compact, four-quark "blob."

The Challenge: For a long time, the "Local" operators were too expensive to compute. It was like trying to listen to a radio station that required a super-complex antenna that took too much power to build. So, scientists mostly relied on the "Bilocal" (loose) operators, hoping that would be enough.

2. The Innovation: A New Way to Listen

The authors of this paper developed a new trick called Position-Space Sampling.

• The Old Way: To calculate the "Local" signal, you had to check every single point in the digital forest. It was like trying to count every single leaf on every tree in a forest to find a bird. It was too slow and expensive.
• The New Way: Their new method is like sending out a few smart drones to randomly sample specific spots in the forest. Instead of checking every leaf, the drones check a cleverly chosen, sparse set of leaves. This gives them the same answer but uses a tiny fraction of the computing power.

This breakthrough made it possible to finally include the "Local" (compact) operators in their calculations without breaking the bank.

3. The Discovery: Why the "Local" Signal Matters

The team ran two simulations:

1. Simulation A: Used only the "Bilocal" (loose) operators.
2. Simulation B: Used a mix of "Bilocal" and "Local" operators.

The Result:
When they compared the two, they found that Simulation A was lying to them.

• The "Bilocal only" approach gave them energy estimates that were slightly off. It was like trying to tune your radio with a broken antenna; you get close to the station, but the signal is fuzzy, and you might miss the exact frequency.
• When they added the "Local" operators (Simulation B), the energy levels shifted significantly. The "Local" operators acted like a high-definition lens, sharpening the picture.
• Specifically, for the excited states (the "cousins" of the main particle), the difference was huge. Ignoring the local operators meant missing the true nature of the particle's energy.

4. The Conclusion: Don't Skip the Hard Part

The scientists concluded that if you want to understand these exotic particles accurately, you cannot just rely on the "loose" molecular operators. You must include the "compact" local ones, even though they are harder to calculate.

The Analogy:
Imagine you are trying to describe a house.

• If you only look at the driveway and the mailbox (the bilocal operators), you might think it's just a house with a car parked outside.
• But if you also look inside the walls and the foundation (the local operators), you realize the house has a secret basement and a unique structural design.
• The paper proves that if you ignore the "inside the walls" part, your description of the house is fundamentally flawed.

Why This Matters

This research helps us understand the "glue" that holds the universe together. By proving that these compact, four-quark structures are real and necessary to calculate correctly, the scientists are refining our understanding of how matter is built. They showed that with their new "drone sampling" method, we can finally get a clear, accurate picture of these exotic particles, rather than a blurry guess.

Technical Summary

1. Problem Statement

The doubly charmed tetraquark $T_{cc}(3875)^+$, observed by LHCb near the $D^*D$ threshold, represents a significant challenge for theoretical physics as it does not fit the traditional quark model. To determine its properties (mass, width, structure) from first principles, Lattice QCD (LQCD) is employed using Lüscher's finite-volume quantization conditions. This method relates the infinite-volume scattering amplitude to the discrete finite-volume energy spectrum.

The core problem addressed in this work is the operator basis selection in the variational method used to extract this spectrum.

• The Dilemma: Exotic hadrons like $T_{cc}$ may have a "molecular" structure (requiring non-local meson-meson scattering operators) or a compact "tetraquark" structure (requiring local four-quark operators).
• The Computational Bottleneck: While non-local bilocal operators (e.g., $D D^*$) are efficiently computed using the distillation framework, including local tetraquark operators in the same basis is computationally prohibitive. In traditional distillation, computing two-point functions for local operators involves rank-4 tensor contractions, leading to a computational cost scaling of $N_v^5$ (where $N_v$ is the number of Laplacian eigenvectors), compared to $N_v^3$ for bilocal operators. This makes it difficult to determine if local operators are necessary for an accurate spectrum.

2. Methodology

The authors employed a hybrid approach combining advanced LQCD techniques to overcome the computational barrier:

• Gauge Ensembles: Simulations were performed using $O(a)$-improved Wilson-clover fermions at the $SU(3)$-flavor symmetric point ($m_\pi \approx 420$ MeV). Two ensembles were used: B450 ($32^3 \times 64$) and N202 ($48^3 \times 128$). The charm quark mass was tuned to match the physical average of $D$ mesons.
• Operator Basis:
  • Bilocal Operators: A basis of 9 non-local $D^{(*)}D^*$ scattering operators with various momentum shells ($\mathbf{k}^2 = 0, 1, 2$).
  • Local Operators: 3 local operators were added: local $D D^*$, local $D^* D^*$, and a diquark-antidiquark operator.
• Position-Space Sampling in Distillation:
  • To make local operators affordable within distillation, the authors utilized their recently developed position-space sampling method.
  • Instead of computing full rank-4 tensors, they first construct smeared fermion propagators ($S_{f,sm}$) from the perambulators.
  • Momentum projections are then applied via stochastic sampling over randomly displaced sparse grids in position space.
  • This reduces the computational cost scaling back to $N_v^3$, making the inclusion of local operators feasible.
• Analysis Framework:
  • Variational Method: A Generalized Eigenvalue Problem (GEVP) was solved using a correlator matrix constructed from the mixed operator basis.
  • Energy Extraction: Ground and excited state energies were extracted by fitting effective energy differences ($\Delta E_{eff}$) to cancel statistical noise.
  • Lüscher Analysis: A single-channel $s$-wave Lüscher analysis was performed to convert the finite-volume energy levels into infinite-volume scattering phase shifts ($\delta_0$).

3. Key Contributions

1. Algorithmic Innovation: Successfully applied the position-space sampling method to local tetraquark operators within the distillation framework, reducing the computational complexity from $N_v^5$ to $N_v^3$.
2. Systematic Operator Importance Study: Conducted a rigorous comparison of finite-volume spectra derived from:
   • A purely bilocal basis.
   • A purely local basis.
   • A mixed basis (bilocal + local).
3. Quantification of Systematic Errors: Demonstrated that omitting local operators introduces significant systematic shifts in energy level estimates, which propagate to errors in scattering phase shifts.

4. Results

• Convergence Behavior:
  • Mixed Basis: The energy estimates for the ground state and first excited state converged rapidly when local operators were included in the basis.
  • Bilocal-Only Basis: The estimates converged slowly and showed a "step-like" decrease as more bilocal operators were added.
• Energy Shifts:
  • On the N202 ensemble, including local operators caused significant shifts in the energy levels. The purely bilocal basis resulted in a $1\sigma$ discrepancy for the ground state and a $2\sigma$ to $3\sigma$ discrepancy for the first excited state compared to the mixed basis.
  • On the B450 ensemble, shifts were smaller but still present, likely due to the less dense spectrum in the smaller volume.
• Scattering Phase Shifts:
  • The shifts in energy levels translated directly into shifts in the $D D^*$ scattering phase shifts ($\delta_0$).
  • The results from the mixed basis showed better agreement between the two different lattice volumes (N202 and B450), suggesting that the purely bilocal basis suffers from larger discretization or finite-volume artifacts.
  • The phase shifts indicate that the $T_{cc}$ is a virtual bound state at this pion mass (interacting with the positive $\sqrt{-q_{cm}^2}$ pole condition), though the proximity of the $u$-channel cut complicates the interpretation.

5. Significance

• Necessity of Local Operators: The study conclusively shows that relying solely on meson-meson scattering operators is insufficient for high-precision studies of tetraquarks like $T_{cc}$. Excluding local tetraquark operators leads to significant systematic errors in the extracted spectrum and scattering parameters.
• Efficiency: The work proves that local tetraquark operators can be included in large-scale distillation calculations without prohibitive computational costs, enabling more robust variational analyses for exotic hadrons.
• Physical Interpretation: By reducing systematic errors, the mixed basis provides a more reliable determination of the $T_{cc}$ nature, supporting the view of it as a virtual bound state and improving the precision of Lüscher analysis results.

In summary, this paper establishes that a diverse operator basis including both non-local molecular and local compact structures is essential for accurate Lattice QCD predictions of exotic hadron spectra, and provides the computational tools to achieve this efficiently.