Compositeness of near-threshold eigenstates with… — Plain-Language Explanation

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

In the world of subatomic physics, scientists are constantly discovering new, exotic particles. The big question is: Are these new particles fundamental building blocks, or are they just two existing particles loosely holding hands?

This paper by Tomona Kinugawa and Tetsuo Hyodo tackles this question, but with a specific twist: What happens when those two particles are electrically charged?

Here is the breakdown of their research in simple terms, using some everyday analogies.

1. The Setup: The "Velcro" and the "Magnets"

Usually, when physicists study how two particles stick together, they look at the "short-range" force. Think of this as Velcro. It's very strong when the particles are touching, but if you pull them apart even a tiny bit, the connection snaps.

However, many particles (like protons or atomic nuclei) also have an electric charge. This introduces the Coulomb force, which is like magnets.

If they have the same charge, they repel (like trying to push two North poles together).
If they have opposite charges, they attract (like a North and South pole).

The problem is that magnets have a "long reach." Even when the particles are far apart, the magnetic pull (or push) is still there. This long reach messes up the standard rules physicists use to determine if a particle is a "house of cards" (composite) or a "brick" (elementary).

2. The Goal: Measuring "Composite-ness"

The authors want to calculate a number called Compositeness ( $X$ ).

If $X = 1$ , the object is 100% a "house of cards" (two particles stuck together).
If $X = 0$ , it's a 100% "brick" (a fundamental particle).
If $X$ is somewhere in between, it's a mix.

In the past, scientists had a simple rule for the "Velcro-only" world: If a particle is barely holding together (very weakly bound), it is almost certainly a "house of cards" ( $X \approx 1$ ).

3. The Discovery: The "Magnet" Changes the Rules

The authors asked: Does this simple rule still work when we add the long-range "magnet" (Coulomb force)?

They used a sophisticated mathematical toolkit (Effective Field Theory) to simulate different scenarios. Here is what they found:

Scenario A: Weak Magnets (The "House of Cards" survives)

If the electric charge is small or the particles are heavy (making the magnetic effect weak compared to the Velcro), the old rule mostly holds.

The Analogy: Imagine two people holding hands with Velcro, but they are also wearing weak magnets that slightly pull them apart. If the Velcro is strong enough to keep them together, they are still clearly a "team" (composite).
Result: Near-threshold states (particles barely holding together) are still mostly "composite."

Scenario B: Strong Magnets (The "House of Cards" breaks)

If the electric repulsion is strong, the rules change completely.

The Analogy: Now imagine the two people are wearing powerful magnets that push them apart. Even if they are holding hands with Velcro, the magnetic push makes the "team" feel very unstable.
Result: The "house of cards" structure disappears. Even if the particles are barely bound, they look more like individual "bricks" that are just barely touching. The compositeness drops. The long-range repulsion hides the fact that they are a pair.

The "Ghost" Effect

One of the most interesting findings is about Resonances (particles that exist for a split second before flying apart).

In the "Velcro-only" world, a barely-bound particle turning into a resonance usually means it stops being a "house of cards."
In this "Velcro + Magnet" world, the authors found that even resonances can remain "composite dominant" if the magnet isn't too strong. The connection between the "bound" state and the "resonance" state is smoother than expected.

4. Real-World Applications

The authors didn't just do math; they applied their new formula to real things in the universe:

Protons ($pp$): Two protons repelling each other. They found this is a "virtual" state, mostly composite.
Beryllium-8 ( $^8$ Be): A nucleus made of two alpha particles. This is a famous "resonance." Their math confirms it is largely a "house of cards" (two alpha particles), which matches what nuclear physicists have suspected for decades.
Exotic Hadrons: They looked at hypothetical particles made of heavy quarks (like the $\Omega_{ccc}$ ). They found that even though these particles have strong electric charges, they are still likely "molecular" structures (composite) rather than new fundamental bricks.

The Big Takeaway

Think of the universe as a dance floor.

Short-range interactions are like dancers holding hands tightly. If they let go slightly, they are clearly a pair.
Coulomb interactions are like a strong wind blowing on the dance floor.
- If the wind is a breeze, the dancers are still clearly a pair, even if they sway a bit.
- If the wind is a hurricane, it becomes hard to tell if they are a pair or just two people standing near each other. The wind (Coulomb force) can make a "pair" look like two separate individuals.

In summary: This paper provides a new "ruler" to measure the internal structure of particles when electricity is involved. It tells us that while electricity can sometimes hide the fact that particles are stuck together, for many real-world systems (like nuclei and exotic hadrons), they are still mostly "houses of cards" built from smaller parts.

1. Problem Statement

The internal structure of exotic hadrons and nuclear clusters near scattering thresholds is a central topic in modern physics. In systems governed purely by short-range interactions, a "low-energy universality" exists: as the binding energy approaches zero, bound states become purely composite (molecular) with a compositeness $X \to 1$ . This is often quantified by the weak-binding relation involving the scattering length and effective range.

However, many realistic systems (e.g., nuclei like $^8$ Be, exotic hadrons like $T_{cc}$ , and $\alpha$ -clusters) involve electrically charged constituents, introducing a long-range Coulomb interaction. The presence of the Coulomb force:

Modifies the analytic structure of the scattering amplitude (introducing logarithmic branch cuts).
Qualitatively changes the threshold behavior of poles (e.g., bound states may turn directly into resonances rather than virtual states).
Breaks the standard short-range universality, making it unclear how to define and calculate the "compositeness" (the probability of finding a molecular component) in these systems.

The paper aims to establish a rigorous theoretical framework to quantify the compositeness of near-threshold $s$ -wave eigenstates in the presence of both Coulomb and short-range interactions.

2. Methodology

The authors employ a Nonrelativistic Effective Field Theory (EFT) tailored for systems with Coulomb plus short-range interactions.

Hamiltonian Construction: They define a Hamiltonian including a free part, a short-range interaction (modeled via a separable potential or a discrete bare state $\phi$ coupled to scattering states $\psi_1\psi_2$ ), and the non-separable Coulomb potential $V_C$ .
Scattering Amplitude: The scattering amplitude $f(k)$ is derived by separating the pure Coulomb contribution $f_C(k)$ from the Coulomb-distorted short-range contribution $f_{CS}(k)$ . The formalism utilizes the Coulomb Green function to solve the Lippmann-Schwinger equation analytically.
Pole Condition: The eigenstates (bound states, resonances, virtual states) are identified as poles in the complex momentum plane ( $k$ -plane). The authors derive the specific pole condition involving the Coulomb scattering length ( $a_s^C$ ), Coulomb effective range ( $r_e^C$ ), and the Bohr radius ( $a_B$ ).
Compositeness Definition:
- For bound states, compositeness $X$ is defined as the overlap of the bound state with the two-body scattering continuum.
- For resonances, the definition is extended using Gamow vectors to handle complex eigenenergies.
- Crucially, the authors derive an expression for $X$ in terms of the energy derivative of the self-energy $\Sigma(E)$ . This approach avoids the need for a separable potential, which is incompatible with the non-separable Coulomb interaction.
Universal Relation: They derive a generalized weak-binding relation for Coulomb systems:
$X = \left[ 1 - \frac{r_e^C}{R_C} \right]^{-1}$
where $R_C$ is an effective radius modified by the Coulomb interaction, dependent on the Bohr radius and the trigamma function.

3. Key Contributions

Generalized Formalism: The derivation of a model-independent expression for compositeness applicable to systems with non-separable Coulomb interactions, expressed solely through observables ( $a_s^C, r_e^C, a_B$ ).
Analytic Structure Analysis: A detailed classification of pole trajectories in the complex momentum plane, revealing how the logarithmic branch cut (unique to Coulomb systems) alters the transition between bound states, virtual states, and resonances.
Quantification of Coulomb Effects: A systematic numerical study showing how the ratio $|r_e^C|/a_B$ (the strength of Coulomb relative to short-range forces) dictates the internal structure.
Extension to Resonances: The application of various interpretation schemes (e.g., $\tilde{X}_{KH}, \bar{X}_C$ ) to extract physically meaningful compositeness values from complex resonance parameters.

4. Key Results

A. Pole Trajectories and Threshold Behavior

Repulsive Coulomb: In purely short-range systems, an $s$ -wave bound state crossing the threshold becomes a virtual state. In the presence of repulsive Coulomb, the bound state crosses the threshold and turns directly into a resonance (and an anti-resonance), bypassing the virtual state. This is attributed to the repulsive Coulomb barrier acting similarly to a centrifugal barrier.
Attractive Coulomb: Bound states originating from short-range interactions do not turn into virtual states or resonances; they remain bound but shift toward Coulomb levels. However, the trajectories of virtual states and resonances in the lower half-plane behave similarly to the repulsive case.

B. Compositeness and Universality

The behavior of compositeness $X$ depends critically on the parameter $\xi = |r_e^C|/a_B$ :

Weak Coulomb Regime ( $\xi \ll 1$ ): A "remnant" of short-range universality survives. Near-threshold bound states and even near-threshold resonances tend to be composite dominant ( $X \approx 1$ ). The Coulomb interaction is too weak to disrupt the molecular nature.
Strong Coulomb Regime ( $\xi \gg 1$ ): The enhancement of compositeness near the threshold is absent. Even near-threshold states can be elementary dominant (non-composite). The Coulomb interaction qualitatively modifies the structure, preventing the state from being purely molecular.
Threshold Limit: Unlike the short-range case where $X \to 1$ as binding energy $B \to 0$ , in the Coulomb case, $X$ at the threshold is determined by the interplay of $a_B$ and $r_e^C$ and does not necessarily approach unity.

C. Application to Realistic Systems

The authors applied the formalism to several physical systems (Table V & VIII):

$pp$ (proton-proton): Identified as a virtual state; found to be composite dominant.
$^8$ Be ( $\alpha\alpha$ ): A narrow resonance. Despite the large Coulomb repulsion, it is found to be composite dominant (consistent with its $\alpha$ -cluster structure), though interpretation schemes show some ambiguity due to the large real part of $X$ .
$\Omega^- \Omega^-$ and $\Omega_{ccc}^{++} \Omega_{ccc}^{++}$ : The $\Omega^- \Omega^-$ is a bound state; $\Omega_{ccc}^{++} \Omega_{ccc}^{++}$ is a resonance. Both are found to be composite dominant, suggesting the Coulomb interaction does not qualitatively alter their molecular nature in these specific weak-Coulomb regimes.
$\Xi^- \alpha$ and $\Omega^- p$ : Attractive Coulomb systems. Both are bound states and are interpreted as composite dominant.

5. Significance

Theoretical Foundation: The paper provides the first rigorous derivation of compositeness for Coulomb-plus-short-range systems, resolving the ambiguity of applying short-range universality rules to charged particles.
Reinterpretation of Threshold Rules: It clarifies that the "threshold rule" (states near thresholds are molecular) holds for charged systems only if the Coulomb interaction is weak relative to the short-range scale ( $|r_e^C|/a_B < 1$ ). For strong Coulomb interactions, this rule breaks down.
Impact on Hadron/Nuclear Physics: The results offer a theoretical basis for understanding the nature of exotic hadrons (like $X(3872)$ , $T_{cc}$ ) and nuclear clusters (like $^8$ Be). It suggests that even for resonances, if they are connected continuously to a composite bound state regime in the presence of weak Coulomb forces, they can retain a dominant molecular structure.
Methodological Advance: The use of the self-energy derivative to define compositeness provides a robust tool for analyzing unstable resonances in EFT, applicable beyond the specific models used here.

In summary, the work demonstrates that while the Coulomb interaction can qualitatively alter the pole structure and suppress compositeness in strong regimes, a "weak-Coulomb" universality persists, allowing near-threshold states in many realistic hadronic and nuclear systems to remain predominantly composite.

Compositeness of near-threshold eigenstates with Coulomb plus short-range interactions