Compositness and wave function of shallow bound states in relation to scattering observables

This paper investigates the relationship between the compositeness of shallow bound states and scattering observables using a coupled-channel potential model that incorporates both quark and hadron degrees of freedom, with a specific focus on applying this framework to analyze the internal structure of exotic hadrons such as the $X(3872)$, $T_{cc}(3875)$, $D_{s0}(2317)$, and $D_{s1}(2460)$.

The Gist

Imagine you are trying to figure out what a mysterious, newly discovered object is made of. Is it a solid, single block of stone (an "elementary" particle), or is it a delicate house of cards built from smaller, separate pieces stuck together (a "composite" molecule)?

In the world of particle physics, scientists have found strange new particles called exotic hadrons (like the famous X(3872)). These are heavier than normal protons or neutrons, and physicists are debating: Are they fundamental building blocks, or are they "molecules" made of two smaller particles orbiting each other loosely?

This paper by Ibuki Terashima and Tetsuo Hyodo is like a detective's guidebook. It tries to answer that question by looking at how these particles scatter (bounce off) other things.

Here is the breakdown of their investigation using simple analogies:

1. The Two Ingredients: The "Bare" Stone and the "Cloud"

The authors use a model that mixes two types of ingredients:

• The "Bare" State (The Stone): Imagine a solid, compact core made of quarks (the fundamental particles). This is the "elementary" part.
• The Hadronic Cloud (The House of Cards): Imagine two other particles (hadrons) floating around and sticking together loosely. This is the "molecular" or "composite" part.

The real particle they observe is a mix of both. The paper defines Compositeness ($X$) as a percentage score:

• $X = 1$ (100%): The particle is purely a house of cards (a molecule).
• $X = 0$ (0%): The particle is purely a solid stone (elementary).
• $X = 0.5$: It's a 50/50 mix.

2. The Detective Work: Reading the "Footprints"

You can't see the inside of the particle directly. So, how do you know if it's a house of cards or a stone? You look at the footprints it leaves when it interacts with other particles.

In physics, these footprints are called scattering observables (specifically, scattering length and effective range).

• The Analogy: Imagine throwing a ball at a target.
  • If the target is a loose house of cards (a shallow bound state), the ball might bounce off in a very specific, sensitive way that tells you the structure is fragile and extended.
  • If the target is a solid stone, the ball bounces off differently.

The paper shows that for particles that are very weakly bound (like a house of cards barely holding together), the "footprints" (scattering data) are directly linked to the "Compositeness" score. If the particle is almost entirely a molecule ($X \approx 1$), the footprints look a certain way. If it's mostly a stone ($X \approx 0$), the footprints look very different.

3. The "Local Approximation" Trap

Sometimes, to make math easier, scientists use a shortcut called a "local approximation." They pretend the interaction happens at a single point, ignoring the complex, spread-out nature of the particle.

• The Analogy: Imagine trying to describe a fluffy cloud by pretending it's a single, hard marble.
• The Finding: The authors found that this shortcut works great if the particle is mostly a house of cards (molecular). The math stays accurate.
• The Warning: However, if the particle has a significant "stone" core (elementary component), this shortcut breaks down. It forces the math to say the particle is 100% a house of cards, even if it isn't. It's like the shortcut is "blind" to the solid stone inside.

4. The Case Files: Solving the Mysteries of X(3872) and Friends

The authors applied their detective methods to four real-world exotic particles:

• X(3872): The "Star" of the show.

  • Verdict: It is almost certainly a molecule ($X \approx 96\%$ to $100%$).
  • Why? The "footprints" (scattering data) match the profile of a loose house of cards perfectly. The idea that it's a solid stone is ruled out unless the universe is fine-tuned in a very unnatural way.

• $T_{cc}(3875)$: A double-charmed particle.

  • Verdict: Also likely a molecule, but with a bit more uncertainty. It's still mostly a house of cards, but the "stone" core might be slightly more present than in X(3872).

• $D_{s0}(2317)$ and $D_{s1}(2460)$:

  • Verdict: These are the "hybrids." They are a mix.
  • Why? Their "footprints" suggest they are not 100% molecules. They have a significant "stone" core mixed in. The $D_{s1}(2460)$ is even less of a molecule than the $D_{s0}(2317)$, meaning it has a bigger solid core.

The Big Takeaway

The paper concludes that for these exotic particles, nature prefers the "house of cards" (molecular) structure.

If a particle is very weakly bound (shallow), it is almost guaranteed to be a composite molecule. The only way for it to be a solid "stone" (elementary) is if the internal parameters of the universe are "fine-tuned" to a razor-thin precision, which is considered highly unlikely.

In short: By looking at how these particles bounce off each other, we can tell that the famous X(3872) is not a fundamental brick of the universe, but rather a delicate, beautiful molecule made of two smaller particles dancing together.

Technical Summary

Here is a detailed technical summary of the paper "Compositeness and wave function of shallow bound states in relation to scattering observables" by Ibuki Terashima and Tetsuo Hyodo.

1. Problem Statement

The internal structure of exotic hadrons (such as $X(3872)$, $T_{cc}$, and $D_{s0}(2317)$) remains a central question in hadron physics. These states are hypothesized to be either hadronic molecules (loosely bound states of hadrons) or compact multiquark states (elementary particles with quark degrees of freedom), or a mixture of both.

The concept of compositeness ($X$) quantifies the probability of finding a hadronic molecular component in the wave function of a bound state. While $X$ is not a direct observable and generally exhibits model dependence, it is known that for $s$-wave weakly bound states near threshold, $X$ can be related to scattering observables (scattering length $a_0$ and effective range $r_e$) via weak-binding relations.

Key Challenges Addressed:

• How does the compositeness depend on model parameters (binding energy, bare state energy, cutoff, interaction strength)?
• How do variations in $X$ manifest in physical observables like scattering phase shifts, wave functions, and effective ranges?
• What is the validity of using local potentials (common in few-body calculations) to describe systems where the underlying interaction is nonlocal due to quark degrees of freedom?
• Can a unified framework determine the compositeness of specific exotic hadrons using experimental and lattice QCD constraints?

2. Methodology

The authors employ a coupled-channel potential model within nonrelativistic quantum mechanics, coupling quark degrees of freedom (compact states) and hadronic degrees of freedom (scattering states).

A. Theoretical Framework

• Hamiltonian: The system is described by a Hamiltonian $H$ with a quark channel ($|q\rangle$) and a hadron channel ($|h\rangle$), coupled via a transition potential $V^t$.
• Feshbach Projection: Using the Feshbach method, the quark degrees of freedom are integrated out to derive an effective energy-dependent and nonlocal potential $V(r', r, E)$ for the hadron channel.
• Form Factors:
  • The transition matrix element uses a Yukawa-type form factor (equivalent to a monopole form factor in momentum space), introducing a momentum cutoff $\mu$.
  • The direct hadron-hadron interaction is modeled as a separable potential with the same form factor.
• Analytic Solutions: Due to the separable nature of the potential, the bound-state wave functions, $T$-matrix, and scattering observables are derived analytically.

B. Definition of Compositeness

The compositeness $X$ and elementarity $Z$ are defined as:
$$X = \int d^3p |\langle p | B \rangle|^2, \quad Z = |\langle \phi_0 | B \rangle|^2$$
where $|B\rangle$ is the bound state, $|p\rangle$ are scattering states, and $|\phi_0\rangle$ is the bare quark state. The normalization condition ensures $X + Z = 1$.
The authors derive $X$ using two consistent methods:

1. Wave Function Normalization: Using the normalization condition for energy-dependent potentials.
2. Lippmann-Schwinger Equation: Using the derivative of the inverse scattering amplitude.

C. Local Approximation (HAL QCD Method)

The paper investigates the local approximation of the nonlocal potential using the derivative expansion (leading order, $n=0$) as employed in the HAL QCD method.

• The local potential is constructed to reproduce the scattering wave function at a specific momentum $k_0$ (chosen as the threshold, $k_0=0$).
• A critical theoretical finding is that a local potential derived this way lacks explicit energy dependence ($\partial V / \partial E = 0$), which mathematically forces the calculated compositeness to be $X=1$ regardless of the true underlying structure.

3. Key Contributions and Results

A. Dependence of Compositeness on Parameters

Using the $X(3872)$ as a reference model, the authors analyzed how $X$ varies with different parameters:

1. Binding Energy ($B$): As $B$ increases (state becomes more deeply bound), $X$ decreases. However, for shallow bound states (small $B$), $X$ remains close to 1 over a wide range of parameters.
2. Bare Energy ($E_0$): This is the most sensitive parameter.
   • If $E_0 \gg -B$ (bare state far above threshold), $X \approx 1$ (molecular dominant).
   • If $E_0 \approx -B$ (bare state near the physical binding energy), $X$ drops significantly toward 0 (elementary dominant).
   • Achieving a small $X$ (elementary dominant) requires fine-tuning of $E_0$ to be extremely close to $-B$.
3. Cutoff ($\mu$): Increasing the cutoff generally decreases $X$, but the effect is modest for typical hadronic scales unless $E_0$ is also tuned.
4. Direct Interaction ($\omega_h$): Increasing repulsive direct interaction reduces the hadronic component, lowering $X$.

B. Relation to Scattering Observables

The study confirms the weak-binding relations for shallow states:

• $X \approx 1$: The scattering length $a_0$ is large ($a_0 \sim 1/\sqrt{2mB}$), and the effective range $r_e$ is small (of the order of the interaction range). The phase shift drops steeply near threshold.
• $X \approx 0$: The scattering length $a_0$ becomes small, and the effective range $r_e$ becomes large and negative. The phase shift behavior changes significantly, approaching $\pi$ near threshold.
• Wave Function: As $X$ decreases, the norm of the scattering component of the wave function decreases, while the amplitude near the origin increases to maintain normalization.

C. Validity of Local Approximation

• Composite-Dominant Case ($X \approx 0.99$): The local approximation (HAL QCD method) successfully reproduces the exact phase shifts, binding energy, and wave function. The error in $X$ (forced to 1) is negligible because the true value is already near 1.
• Elementary-Dominant Case ($X \approx 0.55$): The local approximation fails to capture the correct physics.
  • It forces $X=1$, leading to a large discrepancy in the effective range (predicting a small positive value instead of a large negative one).
  • Consequently, the phase shifts deviate significantly at higher energies, and the binding energy is mispredicted (approx. double the exact value).
  • Conclusion: The local approximation is valid only for weakly bound, composite-dominant states. It fails for states with significant elementary components.

D. Phenomenological Application

The framework was applied to four exotic hadrons using experimental scattering lengths and effective ranges (from PDG, LHCb, and Lattice QCD):

1. $X(3872)$: $X = 0.96^{+0.04}_{-0.00}$. Conclusion: Predominantly a hadronic molecule.
2. $T_{cc}(3875)$: $X = 1.00^{+0.00}_{-0.40}$. Conclusion: Predominantly molecular, though with larger uncertainty due to a smaller bare energy $E_0$.
3. $D_{s0}^*(2317)$: $X = 0.62^{+0.19}_{-0.35}$. Conclusion: Significant molecular component, but with a non-negligible elementary admixture.
4. $D_{s1}(2460)$: $X = 0.36^{+0.57}_{-0.20}$. Conclusion: The most elementary-dominant of the set, reflecting a smaller bare energy $E_0$ closer to the threshold.

4. Significance

• Quantitative Structure Determination: The paper provides a rigorous method to link the abstract concept of compositeness to measurable scattering observables ($a_0, r_e$), allowing for the determination of internal structure without relying solely on model assumptions.
• Validation of Approximations: It establishes clear boundaries for the validity of local potential models (like those used in HAL QCD). It demonstrates that while local potentials work well for molecular states, they are insufficient for states with significant compact quark core contributions.
• Fine-Tuning Requirement: The study highlights that realizing an elementary-dominant shallow bound state requires extreme fine-tuning of the bare energy $E_0$. This suggests that naturally occurring shallow exotic hadrons are likely to be predominantly molecular ($X \approx 1$).
• Unified Framework: By incorporating both quark and hadron degrees of freedom, the model naturally generates the "bare state" and explains how it evolves into a physical bound state, bridging the gap between constituent quark models and hadronic molecule pictures.

In summary, this work solidifies the theoretical foundation for interpreting exotic hadrons as molecular states when scattering data indicates large scattering lengths and small effective ranges, while cautioning against the use of simplified local potentials for states with significant compact core components.