Excited $Σ$ states of the hydrogen-antihydrogen molecule

Imagine the universe as a giant dance floor. Usually, the dancers are made of "matter" (like us). But there's a secret side of the floor where the dancers are made of "antimatter." The paper you're asking about is a theoretical study of what happens when a "matter" dancer (a Hydrogen atom) meets an "antimatter" dancer (an Antihydrogen atom).

Here is the story of their dance, explained simply:

1. The Dance Floor Rules (The System)

When a Hydrogen atom and an Antihydrogen atom get close, they don't just bounce off each other. They form a temporary, wobbly molecule called H $\bar{\text{H}}$ .

Think of this molecule like a four-person dance troupe:

Two heavy leaders (the proton and the antiproton).
Two light followers (the electron and the positron).

The scientists wanted to map out the "music" (energy levels) this troupe can dance to. Specifically, they looked at the excited states—situations where the light followers are jumping around more wildly than usual.

2. The "Magic Mirror" (Q Symmetry)

The paper introduces a special rule called Q symmetry. Imagine a magic mirror placed exactly between the two heavy leaders.

If you reflect the light followers across this mirror and swap their positions, the dance looks exactly the same.
This rule splits all possible dances into two groups: "Even" dances and "Odd" dances.
The scientists calculated the energy for both groups, finding that the "Odd" dances are just as important as the "Even" ones, contrary to some previous guesses.

3. The Two Types of Dancers (Molecules vs. Free Floaters)

The biggest discovery in this paper is about the nature of the dancers.

The Molecular Dancers: Sometimes, the electron and positron stick to their respective leaders, forming a tight little molecule.
The Free Floaters (Positronium): Sometimes, the electron and positron decide to ignore the heavy leaders and dance with each other instead, forming a tiny, free-floating pair called Positronium.

The Analogy: Imagine a group of four people holding hands. Usually, they stay in a square. But sometimes, two of them let go of the group and start spinning in a circle by themselves, while the other two watch.

The paper shows that the "Free Floater" state (Positronium) isn't just a rare accident; it's a fundamental part of the system. The scientists found a way to see these "Free Floaters" appearing right alongside the "Molecular Dancers" in their calculations.

4. The "Trap" (Avoided Crossings)

Here is the most exciting part. The scientists found that the energy levels of the "Molecular Dancers" and the "Free Floaters" keep bumping into each other.

The Analogy: Imagine two roads running parallel. Suddenly, they get so close they almost crash, but instead of crashing, they swerve around each other. This is called an avoided crossing.
Because of these swerves, the "Free Floaters" and "Molecular Dancers" get mixed up.
The Result: This creates a massive number of "traps" or resonances. Think of these as energy pits where the atoms can get stuck for a tiny moment before breaking apart.

5. Why This Matters (The Collision)

The paper argues that if you shoot an Antihydrogen atom at a Hydrogen atom (even very slowly), they might not just bounce off.

Because there are so many of these "energy traps" (resonances) created by the excited states, the atoms are likely to get caught in one of them.
It's like throwing a ball into a forest with millions of hidden nets. Even if you throw it gently, it's very likely to get snagged.
Once caught, the atoms might rearrange themselves (turning into Protonium and Positronium) or annihilate (disappear in a flash of energy).

6. The "Crunch" Zone (The Critical Distance)

There is a specific point where the atoms get so close that the rules of the dance change completely. The paper admits that their math gets a bit shaky right at this "crunch" point (called the critical distance).

To get around this, they had to guess (extrapolate) what happens in that tiny, dangerous zone.
They checked their guess against a super-complex, full-scale simulation (a "four-body calculation") and found that, despite the guesswork, their map of the dance floor is surprisingly accurate.

The Bottom Line

This paper is a map. It tells us that when Hydrogen and Antihydrogen meet, they don't just have one or two ways to interact. They have a plethora (a huge abundance) of excited states and "traps" that can catch them.

If scientists want to understand exactly how these atoms collide, crash, or annihilate, they can no longer ignore these excited states. They have to account for the fact that the atoms can get stuck in these "energy nets" before they finally break apart or disappear. The paper provides the first detailed map of these hidden nets.

Technical Summary: Excited $\Sigma$ States of the Hydrogen-Antihydrogen Molecule

Problem Statement
The interaction between hydrogen (H) and antihydrogen ( $\bar{\text{H}}$ ) is a paradigmatic matter-antimatter system of interest for testing fundamental symmetries (CPT, WEP) and for sympathetic cooling experiments. While the ground-state interaction has been extensively studied, the behavior of the system at inter-hadronic separations $R$ below the critical distance $R_c$ (the Fermi-Teller radius) remains problematic. Below $R_c$ , the Born-Oppenheimer (BO) approximation breaks down as the energy of the rearrangement channel (protonium $p\bar{p}$ + positronium $e\bar{e}$ ) drops below that of the H $\bar{\text{H}}$ molecule, causing non-adiabatic corrections to diverge.

Existing scattering calculations for H– $\bar{\text{H}}$ collisions have largely relied on the ground-state BO potential curve. However, previous studies utilizing full four-body treatments have identified scattering resonances near the dissociation threshold associated with highly excited positronium states. This raises the question of whether excited leptonic states of the H $\bar{\text{H}}$ quasimolecule, which were previously neglected in adiabatic treatments, support ro-vibrational states that could lead to scattering resonances near the ground-state dissociation limit ( $\approx -1.0$ a.u.). Furthermore, a complete theoretical framework for $R < R_c$ is lacking, and the inclusion of rearrangement channels (protonium and positronium) in close-coupling calculations has been computationally demanding due to the need for distinct basis sets for molecular and free-particle states.

Methodology
The authors calculate the Born-Oppenheimer potential curves for excited $\Sigma$ states of the H $\bar{\text{H}}$ system, considering both even and odd $Q$ symmetries. The $Q$ symmetry operator, a combination of lepton mirroring and permutation, allows the separation of the Hilbert space into even and odd sectors, reducing computational effort.

Leptonic Problem: The authors employ explicitly correlated Kołos-Wolniewicz-type basis functions within the prolate spheroidal coordinate system. The wavefunction is expanded as a superposition of these basis functions.
- Dual-Base Approach: A key methodological innovation is the adoption of a "dual-base" basis set. Instead of a single set of exponential parameters, two distinct sets (bases) are used. One base is optimized for large inter-hadronic separations (describing the molecular/atomic character), and the second is optimized for small separations (describing the positronium character). This allows for a more efficient description of the continuous change in state character and, crucially, enables the representation of discretized free positronium states within the same diagonalization framework.
- Optimization: Base parameters are optimized using a gradient descent algorithm. For the first base, the ground state energy at $R=5.0$ $a_0$ is minimized. For the second base, parameters are optimized by minimizing the energy of specific excited states, starting from $R=5.0$ $a_0$ and iteratively moving to $R=0.8$ $a_0$ .
- Convergence: The convergence of the results is systematically tested by increasing the basis set size parameter $\Omega$ (where the sum of integer indices in the basis functions is $\le \Omega$ ).
Hadronic Problem: The resulting BO potential curves are used to solve the nuclear Schrödinger equation for ro-vibrational states. The radial wavefunction is expanded in a B-spline basis.
- Extrapolation below $R_c$ : Since the BO approximation breaks down near $R_c \approx 0.744$ $a_0$ , the potential curves are extrapolated to $R < 0.8$ $a_0$ . The authors assume a continuous change of state character rather than an avoided crossing with a free pair, extrapolating the leptonic energies to converge toward free positronium energies. A simple linear extrapolation is used, with sensitivity checks performed using higher-order polynomials.
- Singularity Handling: To handle the $-1/R$ divergence as $R \to 0$ , the potential is regularized using an exponential function in the immediate vicinity of the origin ( $R < 10^{-6}$ $a_0$ ).

Key Contributions and Results

Excited State Potentials: The study provides the first BO potential curves for excited $\Sigma$ states of H $\bar{\text{H}}$ with both even and odd $Q$ symmetries. The results confirm that for small $R$ , the attractive inter-hadronic Coulomb term ( $-1/R$ ) dominates, causing all potential curves to tend toward $-\infty$ .
Discretized Positronium States: By utilizing the dual-base approach, the authors successfully obtain discretized free positronium states within the molecular basis set. These states appear as energy levels that are nearly independent of $R$ (until they interact with molecular states). This validates the dual-base strategy and provides a pathway for close-coupling calculations that intrinsically include rearrangement channels without requiring separate basis functions for free positronium.
Avoided Crossings: The spectra reveal numerous avoided crossings between molecular states (which decrease in energy as $R$ decreases) and the discretized positronium states (which have nearly constant energy).
Ro-vibrational Spectrum: The calculation of ro-vibrational states based on these potentials shows a high density of states just above the ground-state dissociation threshold ( $\approx -1.0$ a.u.).
Comparison with H $_2$ : Unlike the H $_2$ molecule, where excited states are energetically inaccessible at low collision energies, the attractive nature of the H– $\bar{\text{H}}$ interaction ensures that all excited leptonic states become energetically degenerate with the ground-state scattering channel at some finite $R$ , regardless of the collision energy.
Validation: A full non-relativistic four-body calculation of the H $\bar{\text{H}}$ ground-state energy was performed for cross-checking. The result ($-459.219$ a.u.) agrees well with the BO result ($-460.347$ a.u., relative deviation $2.4 \times 10^{-3}$ ), suggesting the BO approach with extrapolation yields reasonably accurate energies despite the known breakdown of the approximation near $R_c$ .

Significance and Claims
The paper claims that the excited leptonic states of H $\bar{\text{H}}$ support a plethora of ro-vibrational states with energies close to the ground-state dissociation threshold. Consequently, these states must be considered in theoretical treatments of ground-state H– $\bar{\text{H}}$ collisions.

The authors argue that even at very low collision energies, scattering resonances can occur when the incident atoms become energetically degenerate with these excited molecular bound states. While the exact positions of these resonances are subject to uncertainties regarding the treatment of the system below the critical distance, the high density of states suggests that the resonance condition is likely fulfilled across a wide range of collision energies.

The work emphasizes that previous scattering calculations, which often neglected excited states or treated rearrangement channels via approximations like distorted waves, may be incomplete. The ability to describe both molecular and free positronium states within a single Kołos-Wolniewicz framework using a dual-base set offers a computationally efficient route for future close-coupling scattering calculations that explicitly include rearrangement channels. The paper concludes that refined scattering calculations including these excited states are necessary to obtain reliable cross-sections for H– $\bar{\text{H}}$ interactions.

Excited ΣΣΣ states of the hydrogen-antihydrogen molecule