Position-space sampling for local multiquark operators in lattice QCD using distillation and the importance of tetraquark operators for $T_{cc}(3875)^+$

This paper introduces a position-space sampling method within the distillation framework to efficiently compute local multiquark operators, demonstrating that including local tetraquark operators significantly alters the extracted finite-volume spectrum and scattering phase shifts of the $T_{cc}(3875)^+$ state.

The Gist

Imagine you are trying to understand the structure of a complex building, like a skyscraper made of invisible Lego bricks. In the world of particle physics, these "bricks" are quarks, and the buildings they form are called hadrons. Most of the time, these buildings are simple: a pair of bricks (mesons) or a trio (baryons, like protons). But recently, scientists discovered some very strange, exotic buildings made of four or more bricks stuck together, like the $T_{cc}(3875)^+$, a tetraquark made of two charm quarks and two light quarks.

To study these exotic buildings, physicists use a super-computer simulation called Lattice QCD. Think of this simulation as a giant 3D grid (a lattice) where they try to calculate how these quarks interact.

Here is the problem: The more complex the building, the harder it is to calculate.

The Problem: The "All-Hands" Meeting

In this simulation, there's a clever trick called Distillation. Imagine you want to know how a message travels through a crowded city. Instead of asking every single person on every street corner (which takes forever), you only ask a select group of "key messengers" (the lowest energy modes) who know the main routes. This makes the calculation fast and efficient for simple buildings (like two-quark pairs).

However, when you try to study a four-quark building (a tetraquark) using this trick, the math explodes. It's like trying to organize a meeting where every key messenger has to talk to every other key messenger simultaneously. The number of conversations grows so fast that your computer runs out of memory and time. It's computationally impossible for large cities (large physical volumes).

The Solution: The "Spotlight" Strategy

The authors of this paper invented a new method called Position-Space Sampling.

Think of the city grid again. Instead of asking every key messenger on the entire grid, they decided to only ask messengers standing on a sparse grid—like a checkerboard where they only check the black squares, or a grid where they check every 8th street.

But here's the magic trick: To make sure they don't miss anything important (and to avoid bias), they randomly shift the checkerboard for every new simulation run.

• Run 1: Check every 8th street starting at block 1.
• Run 2: Check every 8th street starting at block 3.
• Run 3: Check every 8th street starting at block 5.

By averaging the results from all these random shifts, they get the exact same answer as if they had checked every single street, but they only did 1/64th of the work. It's like taking a high-resolution photo but only developing a few pixels at a time, then shifting the camera slightly and doing it again, until you have the full picture without ever needing to develop the whole film at once.

The Discovery: Why the "Local" View Matters

Once they solved the math problem, they applied this new method to study the $T_{cc}(3875)^+$ tetraquark. They wanted to know: Does the way these four quarks are glued together locally (right next to each other) matter, or is it enough to just look at them as two separate pairs drifting apart?

Previously, scientists mostly looked at the "drifting apart" view (bilocal operators). They thought the "glued together" view (local operators) might not be important.

The Result: They were wrong.
When the authors included the "glued together" (local) view in their calculations, the energy levels of the tetraquark shifted significantly.

• Analogy: Imagine trying to guess the weight of a suitcase. If you only look at the handle (the "drifting" view), you might guess it's light. But if you also look at the heavy metal frame inside (the "local" view), you realize it's actually quite heavy.
• The Impact: Ignoring the local view led to a systematic error. The "glued together" structure is crucial for understanding the true nature of this exotic particle.

The Big Picture

This paper is a breakthrough for two reasons:

1. The Tool: They built a "spotlight" method that makes it possible to study complex, multi-particle systems on supercomputers without crashing them.
2. The Insight: They proved that to understand these exotic particles, you can't just look at them from a distance; you have to zoom in and see how the quarks are tightly bound together.

In short, they found a way to look at the "microscopic glue" holding these exotic particles together, revealing that it plays a much bigger role in their identity than anyone previously realized.

Technical Summary

1. Problem Statement

In Lattice Quantum Chromodynamics (QCD), calculating hadronic two-point functions is essential for spectroscopy. The distillation method (using Laplacian Heaviside smearing) is the standard efficient framework for non-local operators (e.g., meson-meson scattering). However, it faces a severe computational bottleneck when applied to local multiquark operators (containing four or more quarks, such as tetraquarks).

• The Bottleneck: In standard distillation, computing two-point functions for local tetraquark operators requires constructing high-rank tensors (rank-4 for tetraquarks, rank-6 for hexaquarks) by summing over all spatial lattice points before contracting with perambulators.
• Cost Scaling: The computational cost for these contractions scales as $N_v^5$ for tetraquarks and $N_v^7$ for hexaquarks, where $N_v$ is the number of Laplacian eigenvectors. Since $N_v$ must scale with the physical volume to maintain a constant smearing radius, these calculations become prohibitively expensive for large volumes.
• The Gap: While local operators are crucial for capturing deeply bound states or specific internal structures (like diquark-antidiquark configurations), their high cost often forces researchers to rely solely on bilocal (scattering) operators, potentially missing critical spectral information.

2. Methodology

The authors propose a Position-Space Sampling method integrated within the distillation framework to mitigate the cost scaling of local multiquark operators.

• Core Concept: Instead of computing smeared propagators over the entire spatial lattice $\Lambda_3$, the method computes them only on sparse subspaces ($\tilde{\Lambda}_3 \subset \Lambda_3$).
• Unbiased Estimator:
  • Previous sparse-grid methods introduced bias due to incomplete momentum projection.
  • The authors implement a randomly shifted regular sparse grid. For each gauge configuration and source time, a random offset $\tilde{x}$ is chosen to define the sparse grid.
  • By averaging over these random offsets, the estimator remains unbiased, recovering the full lattice sum in the limit of infinite sampling.
• Cost Reduction:
  • The method decouples the contraction order: first, smeared propagators are constructed on the sparse grid, and then the momentum projection is performed.
  • This reduces the asymptotic cost scaling from $N_v^5$ (or higher) to $N_v^3$ (dominated by the computation of smeared propagators), which is independent of the number of quarks in the operator.
  • The variance is controlled by the point separation parameter $N_{sep}$. The authors demonstrate that for $N_{sep} \ge 8$, the statistical error is dominated by Monte Carlo noise rather than the sampling noise, making the method highly efficient.

3. Key Contributions

1. Algorithmic Innovation: Introduction of an unbiased position-space sampling technique that allows the inclusion of local multiquark operators in distillation-based variational analyses without prohibitive computational costs.
2. Efficiency Validation: Demonstration that the method works effectively for single-meson, single-baryon, and local tetraquark operators, identifying $N_{sep}=8$ as an optimal separation for the tested ensembles.
3. Phenomenological Application: A comprehensive study of the $T_{cc}(3875)^+$ tetraquark ($cc\bar{u}\bar{d}$), comparing spectra derived from:
   • A basis of purely bilocal scattering operators ($DD^*$, $D^*D^*$).
   • A mixed basis including local tetraquark operators (local $DD^*$, $D^*D^*$, and diquark-antidiquark).

4. Results

The study was performed using $N_f=2+1$ Wilson-clover fermions at the $SU(3)$-flavor-symmetric point on two CLS ensembles (B450 and N202).

• Spectral Shifts:
  • Ground State: The inclusion of local operators had a negligible effect on the ground state energy estimate (consistent within $1\sigma$).
  • Excited States: Significant shifts were observed for excited states. Specifically, the first excited state energy estimate shifted by approximately $3\sigma$ when local operators were included.
  • Convergence: The mixed basis (local + bilocal) showed fast convergence of energy levels with fewer operator groups. In contrast, the purely bilocal basis converged slowly, and even with many bilocal operators, it failed to reach the same energy estimates as the mixed basis.
• Operator Importance:
  • The local diquark-antidiquark operator alone was found to be sufficient to capture the spectral shifts, but combining local meson-meson operators with bilocal ones provided the most stable results.
  • On the coarser B450 ensemble, the effect was smaller ($\sim 0.5\sigma$) due to larger finite-volume spacing, but on the finer N202 ensemble, the shifts were substantial.
• Scattering Phase Shifts:
  • Using Lüscher's finite-volume quantization conditions, the authors extracted the $s$-wave $DD^*$ scattering phase shift.
  • The inclusion of local operators corrected the phase shift, reducing apparent discretization effects and improving agreement between the two lattice ensembles (B450 and N202).
  • The results suggest a virtual bound state (intersection of $q_{cm} \cot \delta_0$ at positive $\sqrt{-q_{cm}^2}$), though the proximity of the left-hand cut ($u$-channel) complicates the interpretation.

5. Significance

• Methodological Impact: This work resolves a major computational barrier in lattice QCD, making it feasible to include local multiquark operators in large-volume, high-precision spectroscopy studies. This is critical for systems where the internal structure may not be purely "molecular" (scattering) but involves compact multiquark configurations.
• Physics Impact on $T_{cc}$: The study proves that omitting local tetraquark operators leads to significant systematic errors in the extraction of the $T_{cc}(3875)^+$ spectrum, particularly for excited states. Relying solely on bilocal scattering operators may yield an incomplete or inaccurate picture of the hadron's nature.
• Future Directions: The methodology enables future studies of other exotic hadrons (e.g., pentaquarks, hexaquarks like the $d^*(2380)$) and allows for moving-frame analyses and studies at physical quark masses with controlled systematic uncertainties.

In conclusion, the paper establishes that a hybrid basis of local and bilocal operators is necessary for a reliable determination of the finite-volume spectrum of exotic multiquark states, and provides the computational tools to achieve this efficiently.