Positivity of the effective range for finite range attractive potentials with a repulsive core

This paper rigorously proves that for finite-range potentials featuring an inner repulsive core and an outer attractive tail, the effective range remains strictly positive whenever the scattering length exceeds the potential's range, thereby providing a fundamental constraint on using the sign of the effective range to distinguish exotic hadron configurations.

The Gist

Imagine you are trying to figure out what's inside a mysterious, sealed box just by throwing small balls at it and watching how they bounce off. In the world of subatomic particles, physicists do something similar. They shoot particles at each other at very low speeds and analyze how they scatter to understand the invisible forces holding them together.

Two main numbers help them describe this "bounce":

1. The Scattering Length: Think of this as the "effective size" of the box. It tells you how far out the force reaches.
2. The Effective Range: This is a bit trickier. It measures how much the inside of the box "squishes" or "stretches" the path of the ball compared to if the box weren't there at all.

The Big Debate: What's Inside the Box?

Recently, scientists studying "exotic hadrons" (strange, heavy particles made of quarks) have been arguing about what these particles actually look like. There are two main theories:

• The "Loose Molecule" Theory: The particle is like a fluffy, loosely held-together cloud of smaller particles (like a molecule).
• The "Compact Multiquark" Theory: The particle is a tight, dense ball of quarks glued together (like a solid marble).

For a long time, physicists have used the sign (positive or negative) of that second number, the Effective Range, to guess which theory is right.

• Positive Effective Range: Suggests a loose, fluffy molecule.
• Negative Effective Range: Suggests a tight, compact ball.

The New Discovery: The "Repulsive Core" Rule

The author of this paper, Davide Germani, wanted to test a specific idea. He asked: "Can we create a 'tight ball' (negative effective range) just by adding a hard, repulsive wall inside a standard attractive force?"

Imagine a potential energy landscape as a valley.

• Standard Attractive Potential: A smooth valley where particles want to fall in.
• The Modification: What if we put a small, hard bump (a repulsive core) at the very bottom of that valley?

Many physicists thought, "If we put a hard bump in the middle, maybe we can force the effective range to become negative, proving the particle is compact."

The Paper's Verdict:
The author proved mathematically that this doesn't work.

He showed that as long as the "Scattering Length" (the effective size) is larger than the size of the box itself, adding a repulsive bump in the middle cannot make the effective range negative. It will always stay positive.

A Creative Analogy: The Trampoline and the Bouncy Castle

Imagine a trampoline (the attractive force) that pulls a ball down.

1. The Standard Case: You jump on a trampoline. The fabric stretches down. The "effective range" is positive because the fabric is pulling you in.
2. The "Compact" Attempt: Now, imagine you put a small, hard, bouncy castle in the very center of the trampoline. You try to jump on it.
   • The hard castle (the repulsive core) pushes you up a little bit in the center.
   • However, the author proved that if your jump is big enough (meaning the scattering length is large), the trampoline's overall pull is so strong that the hard castle in the middle doesn't change the overall nature of the bounce enough to flip the sign. The "stretchiness" of the whole system remains positive.

The math shows that the hard core actually makes the effective range larger (more positive), not negative. It's like the hard core forces the wave to "avoid" the center, making the interaction look even more spread out, not more compact.

What This Means for the "Compact" Theory

The paper concludes that if you want to explain a particle as a "compact multiquark" (which requires a negative effective range), you cannot just use a simple model with a hard repulsive core inside an attractive force.

If a particle has a negative effective range, it means the interaction is much more complex than just "attractive force with a hard bump." It likely requires:

• Multiple channels interacting at once (like several different doors opening and closing).
• Or forces that change depending on the energy of the collision.

In short: You can't fake a "compact" particle just by putting a hard wall inside a standard attractive force. If the math says the effective range is negative, the particle is doing something much more complicated than a simple repulsive core can explain.

Technical Summary

Technical Summary: Positivity of the Effective Range for Finite Range Attractive Potentials with a Repulsive Core

Problem Statement
In the phenomenological study of exotic hadrons, the sign of the effective range ($r_0$) in low-energy scattering is frequently invoked as a criterion to distinguish between compact multiquark configurations (associated with $r_0 < 0$) and loosely bound hadronic molecules (associated with $r_0 > 0$). While it is a well-established mathematical result that standard finite-range attractive potentials supporting a shallow bound state yield a strictly positive effective range, the specific conditions under which a negative $r_0$ can arise in single-channel local interactions remain a subject of investigation. Specifically, there is a need to determine whether the addition of an inner repulsive core to an attractive potential is sufficient to invert the sign of the effective range, a mechanism sometimes proposed to model compact exotic states without invoking coupled-channel dynamics.

Methodology
The paper employs rigorous quantum mechanical scattering theory to analyze single-channel, local, finite-range, spherically symmetric potentials. The analysis focuses on the Effective Range Expansion (ERE) in the low-energy regime ($kR \ll 1$), where the scattering amplitude is characterized by the scattering length ($a$) and the effective range ($r_0$).

The core of the methodology involves:

1. Integral Representation: Utilizing the integral representation of the effective range, $r_0 = 2 \int_0^R [\chi_0^2(r) - u_0^2(r)] dr$, where $u_0(r)$ is the physical zero-energy wave function and $\chi_0(r)$ is the corresponding asymptotic free solution normalized to unity at the origin.
2. Theoretical Proof: The authors construct a proof for potentials characterized by an inner repulsive core ($V(r) \geq 0$ for $r < r_1$) and an outer attractive tail ($V(r) \leq 0$ for $r_1 < r < R$). The proof relies on analyzing the convexity properties of the difference function $\Delta(r) = \chi_0(r) - u_0(r)$ under the condition that the scattering length exceeds the range of the potential ($a > R$).
3. Numerical and Analytical Verification: To illustrate the theorem, two specific phenomenological models are examined:
   • A piecewise spherical potential consisting of a repulsive barrier followed by an attractive well.
   • A superposition of repulsive and attractive Yukawa potentials, truncated to ensure finite range.
     In both cases, the authors compare the effective range of the core-plus-tail potential against a purely attractive potential of the same range and scattering length.

Key Contributions and Results
The primary contribution of the work is Theorem 1, which rigorously proves that for any real, local, finite-range potential with an inner repulsive core and an outer attractive tail, the effective range $r_0$ remains strictly positive, provided the scattering length satisfies $a > R$.

The proof demonstrates that under the condition $a > R$, the physical wave function $u_0(r)$ is strictly suppressed relative to the asymptotic solution $\chi_0(r)$ throughout the interaction region ($0 \leq r \leq R$). Specifically, the presence of the repulsive core forces $u_0(r)$ to be smaller than $\chi_0(r)$ in the inner region, while the attractive tail ensures the wave function matches the boundary conditions required for a positive scattering length. Consequently, the integrand $[\chi_0^2(r) - u_0^2(r)]$ remains non-negative, ensuring $r_0 > 0$.

Numerical results for the spherical barrier-well and Yukawa models confirm this theoretical bound. In both examples, the inclusion of a repulsive core resulted in an effective range ($r_0$) that was larger than that of a purely attractive potential with the same scattering length, rather than negative. For instance, in the Yukawa model, the effective range increased from 1.01 (purely attractive) to 1.23 (with repulsive core), while maintaining the same scattering length.

Significance and Claims
The paper claims that these findings impose a severe mathematical constraint on the modeling of exotic hadrons. The authors assert that adding an arbitrary inner repulsion to a single-channel local potential is fundamentally insufficient to produce a negative effective range.

Consequently, the paper argues that the observation of a negative effective range in phenomenological studies cannot be attributed solely to the shape of a single-channel local potential (i.e., a repulsive core plus attractive tail). Instead, if a negative $r_0$ is observed, it implies that the underlying physics likely involves mechanisms beyond simple single-channel local interactions, such as explicit coupled-channel dynamics or energy-dependent interactions. This result restricts the class of potential models that can legitimately be used to describe compact exotic hadrons without invoking these more complex dynamical features.