Three-body molecular states composed of $D^{(*)}$ and two nucleons

Using the Gaussian Expansion and Complex Scaling methods within a hadronic molecular framework, this study predicts the existence of robust, compact three-body bound states in the $DNN$ and $D^{*}NN$ systems driven by realistic nucleon-nucleon correlations and heavy-quark symmetry, while finding no three-body resonances.

The Gist

Imagine the subatomic world as a bustling, chaotic dance floor. Usually, we see particles dancing in pairs: two protons holding hands (like a deuteron), or a heavy charm meson dancing with a single proton. But what happens when you throw a third dancer onto the floor? Does the group just wobble, or do they lock into a tight, energetic huddle?

This paper, written by a team of physicists, explores exactly that scenario. They are investigating three-body "molecules" made of one heavy charm meson (either a D or a D-star) and two protons (nucleons). Think of it as a trio: a heavy, charismatic lead dancer (the meson) and two smaller, sturdy partners (the protons).

Here is the breakdown of their findings, translated into everyday language:

1. The Setup: The Dance Floor Rules

To understand how these particles interact, the scientists had to write down the "rules of the dance."

• The Protons: We already know how two protons interact very well. They have a specific "handshake" (the nuclear force) that lets them stick together to form a deuteron (the nucleus of heavy hydrogen).
• The Heavy Meson: The heavy charm meson is a bit of a mystery. It's heavy, and it follows special rules called "Heavy-Quark Symmetry." The scientists had to figure out how this heavy guest interacts with the protons. They used a theoretical framework (like a map) based on how other heavy particles behave to predict these interactions.

2. The Experiment: Solving the Puzzle

The team used powerful mathematical tools (like a super-advanced calculator) to solve the equations of motion for this trio. They asked: If we put these three particles together, will they stick together tightly, loosely, or fly apart?

They looked at two main scenarios:

• Scenario A (DNN): A standard charm meson (D) + two protons.
• Scenario B (D*NN): A "spin-1" charm meson (D*) + two protons. (Think of the D* as the same dancer but wearing a spinning top hat, which changes how they interact).

3. The Big Surprises

The "Super Glue" Effect (DNN)

In the first scenario, the scientists found something remarkable. Even if the heavy meson and one proton barely stick together (or don't stick at all), the moment you add the second proton, the whole group snaps together into a tight, compact ball.

• The Analogy: Imagine two people who can't quite hold hands with a third person. But if you bring in a fourth person who acts as a "glue," suddenly all three are huddled so tightly they look like a single, small marble.
• The Result: The trio forms a stable, deep-bound state that is much smaller and denser than a normal atomic nucleus. The heavy meson acts like a magnet, pulling the two protons close together.

The "Two-Track" System (D*NN)

The second scenario (with the spinning D* meson) was even more fascinating. Because the D* meson has a "spin" (like a spinning top), the forces change depending on how the trio is oriented.

• The Deep Track: In some orientations, the trio forms an incredibly tight, deep-bound state. They are so close they are practically touching.
• The Shallow Track: In other orientations, they form a looser, "halo-like" structure. It's like a fuzzy cloud where the particles are still connected but spread out further.
• The Analogy: Think of a couple dancing. Sometimes they dance a tight, fast tango (deep, compact state). Other times, they do a slow, wide Waltz where they hold hands but stay far apart (shallow, extended state). The paper found that the D*NN system can do both dances, depending on the spin.

4. What They Didn't Find

The scientists also looked for "resonances." In physics, a resonance is like a temporary, unstable huddle that forms for a split second and then immediately falls apart (like a group hug that ends in a laugh and a scatter).

• The Finding: They found no such unstable huddles in their calculations. Every state they found was a solid, permanent "molecule" that would stick around for a while.

5. Why Does This Matter?

This isn't just a math exercise. It tells us that heavy particles can act as super-glue for atomic nuclei.

• The Takeaway: Just because two particles don't stick well on their own doesn't mean a group of three can't form a super-tight bond. The presence of the heavy charm meson changes the rules, compressing the space between the protons.
• Future Hunt: The paper gives a "Wanted Poster" for experimentalists. It predicts exactly how heavy these new particles should be and what their "spin" should look like. Scientists at big facilities like the LHC (Large Hadron Collider) can now look for these specific "heavy-flavor molecules" in the debris of particle collisions.

Summary

In short, this paper shows that when you mix heavy charm mesons with protons, you don't just get a loose group. You get tight, compact, and surprisingly stable new forms of matter. The heavy meson acts like a cosmic magnet, pulling the protons into a dense cluster, creating a new kind of "nuclear molecule" that nature has hidden away, waiting to be discovered.

Technical Summary

1. Problem Statement

The paper investigates the existence and properties of three-body hadronic molecular states composed of a charmed meson ($D$ or $D^*$) and two nucleons ($N$), specifically the $DNN$ and $D^*NN$ systems.

• Context: While exotic hadrons (like $X, Y, Z$ states and pentaquarks) have been observed, systems involving open heavy-flavor mesons interacting with nucleons remain theoretically challenging.
• Gap: Previous studies often relied on simplified approximations (e.g., fixed-center approximations) or lacked a unified treatment of realistic nucleon-nucleon ($NN$) forces and heavy-meson–nucleon ($D^{(*)}N$) interactions constrained by Heavy-Quark Spin Symmetry (HQSS).
• Objective: To determine if these three-body systems support bound states or resonances, quantify their binding energies and spatial sizes, and understand the role of spin-dependent forces and three-body correlations in forming compact heavy-flavor states.

2. Methodology

The authors employ a unified potential-model framework combining realistic nuclear forces with HQSS-constrained meson-baryon interactions.

• Hamiltonian & Potentials:
  • $NN$ Interaction: Modeled using the high-precision CD-Bonn potential, based on a relativistic one-boson-exchange (OBE) framework involving $\pi, \sigma, \eta, \rho, \omega$ mesons.
  • $D^{(*)}N$ Interaction: Constructed within an OBE framework constrained by Heavy-Quark Spin Symmetry (HQSS) and chiral symmetry. The interaction includes scalar, vector, and pseudoscalar meson exchanges.
  • Regulator Dependence: The coupling constants for the $D^{(*)}N$ vertices are not fixed arbitrarily but are renormalized based on the pole positions of near-threshold exotic states ($X(3872)$, $T_{cc}(3875)$, and $Z_c(3900)$). This allows the study of how variations in the $Z_c(3900)$ virtual state (from $-5$ to $-35$ MeV) affect the three-body results.
• Numerical Techniques:
  • Gaussian Expansion Method (GEM): Used to solve the three-body Schrödinger equation. The wavefunction is expanded in Gaussian basis functions defined on Jacobi coordinates, efficiently capturing both short-range correlations and long-range asymptotics.
  • Complex Scaling Method (CSM): Applied to the Hamiltonian ($r \to r e^{i\theta}$) to investigate the analytic structure of the spectrum. This distinguishes genuine bound states (stable eigenvalues on the negative real axis) from resonances (poles with non-zero imaginary parts).

3. Key Contributions

• Unified Framework: Successfully integrates the realistic CD-Bonn $NN$ potential with HQSS-constrained $D^{(*)}N$ potentials to solve the full three-body problem without relying on the fixed-center approximation.
• Systematic Parameter Study: Quantifies the sensitivity of three-body binding to the short-range dynamics by varying the $Z_c(3900)$ pole position and the cutoff parameter $\Lambda$.
• Spin Hierarchy Analysis: Reveals a distinct spin-dependent structure in the $D^*NN$ system, driven by the spin-1 nature of the $D^*$ meson and tensor forces.
• Resonance Search: Explicitly searches for three-body resonances using CSM and finds none in the explored parameter space, confirming the bound-state nature of the predicted states.

4. Key Results

A. Two-Body Subsystems ($D^{(*)}N$)

• $DN$ System: The isoscalar channel ($I=0, J^P=1/2^-$) supports only a very shallow bound state or is unbound depending on the model parameters. The isovector channel ($I=1$) remains unbound.
• $D^*N$ System: The isoscalar channel ($I=0, J^P=3/2^-$) exhibits robust attraction and forms a compact bound state across various theoretical approaches. The isovector channel ($I=1, J^P=1/2^-$) is sensitive to short-range dynamics, appearing as a shallow bound state only for strong couplings.

B. Three-Body Systems ($DNN$ and $D^*NN$)

• $DNN$ System ($I=1/2, J^P=1^-$):

  • Supports a robust, compact bound state over a wide range of cutoffs ($\Lambda = 1.1 - 1.3$ GeV) and $Z_c$ pole variations.
  • Binding Energy: Ranges from $\sim 13$ MeV to $\sim 64$ MeV (depending on $\Lambda$ and $Z_c$).
  • Spatial Structure: The Root-Mean-Square (RMS) radius is $\sim 1.0 - 1.3$ fm, significantly smaller than the deuteron ($\sim 3.4$ fm). This indicates the $D$ meson acts as a strong attractor, compressing the two nucleons into a compact configuration rather than forming a loose "meson-deuteron" molecule.

• $D^*NN$ System:

  • $J^P = 0^-$ and $2^-$ Channels: Both support deeply bound and compact states.
    • Binding energies reach up to $\sim 100$ MeV (for $2^-$) and $\sim 87$ MeV (for $0^-$).
    • RMS radii are small ($\sim 1.0 - 1.3$ fm), indicating compact structures.
  • $J^P = 1^-$ Channel (Spin Hierarchy): Exhibits a unique two-branch pattern:
    1. Upper Branch (Deeply Bound): Highly compact states with binding energies $> 100$ MeV (up to $\sim 200$ MeV for large $\Lambda$) and RMS radii $< 1$ fm. These are dominated by short-range dynamics.
    2. Lower Branch (Shallow/Extended): Weakly bound states (few MeV to tens of MeV) with large RMS radii (up to $\sim 6.8$ fm), resembling halo-like or near-threshold configurations.
  • Pole Trajectories: CSM analysis confirms that all identified states are genuine bound states (poles on the negative real energy axis) with no imaginary parts, ruling out resonance interpretations.

5. Significance

• Theoretical Impact: The study demonstrates that three-body correlations can generate significantly stronger binding and spatial compression than suggested by two-body subsystems alone. Even when the $D^{(*)}N$ subsystem is weakly bound or unbound, the cooperative effect of realistic $NN$ correlations and heavy-quark symmetry can form compact heavy-flavor nuclei.
• Experimental Guidance: The paper provides quantitative benchmarks (binding energies, spin hierarchy, and spatial sizes) for future experimental searches at facilities like LHC, J-PARC, and GSI-FAIR.
  • The predicted compact nature of these states suggests they might be observable as narrow peaks in invariant mass spectra.
  • The specific spin hierarchy in $D^*NN$ (especially the two-branch structure in $1^-$) offers a clear signature for distinguishing molecular states from other exotic configurations.
• Methodological Validation: The successful application of GEM combined with CSM in the heavy-flavor sector validates these methods for studying complex few-body hadronic systems.

In conclusion, the paper establishes the existence of compact, deeply bound three-body molecular states in the charm-nuclear sector, driven by the interplay of heavy-quark symmetry and realistic nuclear forces, providing a new frontier for exploring non-perturbative QCD.