Low-energy interactions between doubly charmed baryons and Goldstone bosons from lattice QCD

This paper presents a lattice QCD calculation of low-energy $S$-wave interactions between doubly charmed baryons and Goldstone bosons, revealing an attractive interaction with a virtual state pole in the $\Xi_{cc}K^{(1,0)}$ channel while the other studied channels are repulsive.

The Gist

Imagine the universe is built out of tiny, invisible LEGO bricks called quarks. Usually, these bricks snap together in groups of three to form particles we know, like protons and neutrons. Sometimes, they form pairs (like mesons). But there's a special, rare, and heavy family of LEGO structures called doubly charmed baryons.

Think of these as "heavy-duty" LEGO creations made of two super-heavy "charm" bricks and one light "up" or "down" brick. They are so heavy and exotic that they are like the "supercars" of the particle world—hard to build, hard to find, and full of mystery.

This paper is a report from a team of scientists who used a massive digital simulation (called Lattice QCD) to figure out how these heavy "supercars" interact with the "wind" that blows around them. In the particle world, this "wind" is made of Goldstone bosons (like pions and kaons), which are the lightest, most energetic particles zipping around.

Here is the story of what they found, broken down into simple concepts:

1. The Digital Laboratory

Since these particles are too small to see with a microscope and too heavy to catch in a bottle, the scientists built a giant 3D grid inside a supercomputer. This grid is like a digital chessboard where they placed their heavy "charm" bricks and the light "wind" bricks.

They ran this simulation four times, tweaking the "weight" of the light bricks (the pion mass) to see how the interactions changed. It's like testing how a heavy truck drives on a road when the road is made of soft mud versus hard concrete.

2. The Dance of Attraction and Repulsion

The main question was: When a heavy doubly-charmed baryon meets a light Goldstone boson, do they hug or do they push away?

In physics, we measure this by looking at the energy levels.

• Repulsion: If they push each other away, the energy goes up (like trying to squeeze two strong magnets together).
• Attraction: If they like each other, the energy goes down (like a magnet snapping onto a fridge).

The Results:

• The Pushers: In three out of four scenarios, the heavy baryon and the light boson pushed each other away. They didn't want to stick together.
• The Hugger: In one special scenario (called $\Xi_{cc}K$), they attracted each other. The energy dropped, suggesting they wanted to get close.

3. The "Ghost" State (The Virtual Pole)

This is the most exciting part. In the scenario where they attracted each other, the scientists found something strange. It wasn't a stable "bound state" (like a permanent hug where they stick together forever). Instead, it was a virtual state.

The Analogy: Imagine two people running toward each other to hug. They get very close, their arms reach out, and they almost touch, but they don't quite lock hands. They bounce off each other, but for a split second, they were so close that it felt like a hug.

In the quantum world, this "almost-hug" is called a virtual state. It's a fleeting moment of attraction that leaves a fingerprint on the data, even though no permanent new particle is formed. The scientists found a "ghost" of this interaction in their data.

4. Why Does This Matter?

You might ask, "So what? Who cares about heavy LEGO bricks?"

• Completing the Puzzle: We know about the "light" particles and the "heavy" ones, but the "double-heavy" ones are the missing piece of the puzzle. Understanding how they interact helps us finish the periodic table of particle physics.
• Testing the Rules: The universe follows strict rules (Quantum Chromodynamics). By simulating these interactions from scratch (without guessing), the scientists proved that our current theories are on the right track.
• Future Experiments: There are giant machines (like the Large Hadron Collider) trying to find these particles in real life. This paper gives the experimentalists a "map." It tells them, "Look here! If you see a signal in this specific channel, it might be this virtual state we found."

Summary

In short, these scientists built a digital universe to watch how heavy, double-charmed particles interact with light, fast-moving particles. They discovered that while most interactions are a "push," one specific combination results in a "pull," creating a fleeting, ghostly virtual state. This is a crucial step in understanding the fundamental forces that hold our universe together, moving us one step closer to decoding the secret language of the cosmos.

Technical Summary

1. Problem Statement and Motivation

Doubly charmed baryons (DCBs), consisting of two charm quarks and one light quark ($ccq$), are crucial for understanding non-perturbative Quantum Chromodynamics (QCD) and heavy quark symmetry. While the mass spectrum of DCBs (specifically $\Xi_{cc}^{++}$) has been experimentally observed and theoretically studied, their low-energy interactions with Goldstone bosons (pions and kaons) remain largely unexplored from first principles.

Previous theoretical efforts relied on Baryon Chiral Perturbation Theory (BChPT), which requires Low-Energy Constants (LECs). These LECs were previously estimated using Heavy Diquark-Antiquark (HDA) symmetry by fitting to $D\phi$ (charmed meson-Goldstone boson) interactions. However, a direct, ab initio lattice QCD calculation of DCB-Goldstone boson ($B_{cc}\phi$) scattering was missing. This study aims to fill that gap by calculating S-wave scattering lengths and phase shifts directly from lattice QCD simulations.

2. Methodology

Lattice Setup

• Ensembles: The authors utilized four $N_f = 2+1$ flavor gauge ensembles generated by the CLQCD collaboration.
• Parameters:
  • Lattice spacing: $a = 0.07746$ fm.
  • Pion masses: Two unphysical values, $M_\pi \approx 210$ MeV and $M_\pi \approx 300$ MeV.
  • Volumes: Two spatial volumes ($32^3$ and $48^3$) for each pion mass to provide multiple kinematic points.
• Actions: Tadpole-improved tree-level Symanzik gauge action and Clover fermion action for light/strange quarks. The Fermilab action was used for valence charm quarks to control discretization errors.
• Smearing: Distillation quark smearing with 100 eigenvectors was employed to improve signal-to-noise ratios.

Interpolating Operators and Channels

The study focused on four single-channel elastic scattering processes involving ground-state spin-1/2 DCBs ($\Xi_{cc}, \Omega_{cc}$) and Goldstone bosons ($\pi, K$). The channels are classified by Strangeness ($S$) and Isospin ($I$):

1. $\Omega_{cc} \bar{K}$ with $(S, I) = (-2, 1/2)$
2. $\Xi_{cc} K$ with $(S, I) = (1, 1)$
3. $\Xi_{cc} K$ with $(S, I) = (1, 0)$
4. $\Xi_{cc} \pi$ with $(S, I) = (0, 3/2)$

Two-hadron operators were constructed to transform under the $G_1^-$ irreducible representation of the cubic group $O_h^{(2)}$, corresponding to the S-wave ($J^P = 1/2^-$) in the continuum.

Analysis Techniques

• Energy Extraction: Correlation matrices were constructed and analyzed using the Variational Method (Generalized Eigenvalue Problem, GEVP) to extract finite-volume energy levels.
• Scattering Analysis:
  • Lüscher's Formula: Used to relate the discrete finite-volume energy levels to infinite-volume scattering phase shifts ($\delta_0$).
  • Effective Range Expansion (ERE): The scattering amplitude was parameterized as $p \cot \delta_0 = 1/a_0 + \frac{1}{2}r_0 p^2 + \dots$ to extract the scattering length ($a_0$) and effective range ($r_0$).
  • Dispersion Relation Correction (DRC): A correction was applied to energy levels to account for lattice artifacts in the dispersion relation ($c^2 \neq 1$).
  • Chiral Extrapolation: Results at unphysical pion masses were extrapolated to the physical point ($M_\pi \approx 135$ MeV) using a linear fit in $M_\pi^2$.
  • Pole Analysis: Scattering amplitudes were analytically continued to search for bound states or virtual states (poles on the real axis of the $s$-plane).

3. Key Results

Interaction Nature

• Attractive Channel: The $\Xi_{cc} K$ channel with $(S, I) = (1, 0)$ exhibits attractive interactions. The finite-volume energy levels are shifted downward relative to the non-interacting thresholds.
• Repulsive Channels: The other three channels ($\Omega_{cc} \bar{K}$, $\Xi_{cc} K$ with $I=1$, and $\Xi_{cc} \pi$) show repulsive interactions, indicated by upward energy shifts.

Scattering Parameters (Physical Point)

After extrapolation to the physical pion mass, the S-wave scattering lengths ($a_0$) are determined as:

• $a_0(\Xi_{cc} K_{(1,0)}) = +0.730(147)$ fm (Positive, indicating attraction).
• $a_0(\Xi_{cc} K_{(1,1)}) = -0.230(25)$ fm.
• $a_0(\Xi_{cc} \pi_{(0,3/2)}) = -0.144(39)$ fm.
• $a_0(\Omega_{cc} \bar{K}_{(-2,1/2)}) = -0.123(25)$ fm.

These results are consistent with previous BChPT predictions (both HB and EOMS schemes), validating the use of HDA symmetry in estimating LECs for these systems.

Virtual State Discovery

A significant finding is the identification of a virtual state pole in the $\Xi_{cc} K_{(1,0)}$ channel.

• The pole is located at $p^2 \approx -0.03$ GeV$^2$ (on the second Riemann sheet, Im $p < 0$).
• This suggests the interaction is attractive but not strong enough to form a bound state, yet it supports a virtual state. This mirrors findings in the corresponding $D\bar{K}$ system, reinforcing the HDA symmetry connection.

4. Significance and Impact

1. First-Principles Input: This work provides the first ab initio lattice QCD determination of DCB-Goldstone boson scattering parameters, moving beyond model-dependent estimates.
2. Validation of Symmetry: The agreement between lattice results and BChPT predictions confirms the reliability of using Heavy Diquark-Antiquark (HDA) symmetry to estimate Low-Energy Constants for doubly heavy baryons.
3. Spectroscopy Implications: The discovery of a virtual state in the $\Xi_{cc} K_{(1,0)}$ channel has profound implications for the spectroscopy of doubly heavy baryons. It suggests the potential existence of near-threshold structures that could be relevant for future experimental searches (e.g., at LHCb or BESIII).
4. Foundation for Future Work: The study establishes a robust framework for analyzing coupled-channel effects (mixing with $B_c D$ channels) and paves the way for high-precision studies of double-heavy baryon properties and dynamics.

In summary, this paper successfully bridges the gap between experimental observations of DCBs and theoretical QCD predictions, offering critical data on their low-energy interactions and revealing the existence of a virtual state that enriches our understanding of the heavy-quark sector.