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Weak decay of the positronium ion

This paper investigates the rare weak decay of the positronium ion ( $\mathrm{Ps}^-$ ) into an electron and a muon neutrino-antineutrino pair via a virtual $Z$ boson, calculating its decay rate and finding a branching ratio comparable to that of the weak decay of ortho-positronium.

Original authors: Nishat Ul Sani, M. Jamil Aslam, Ishtiaq Ahmed

Published 2026-06-25

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CC BY 4.0

Original authors: Nishat Ul Sani, M. Jamil Aslam, Ishtiaq Ahmed

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a tiny, unstable family of three particles living together in a very small, invisible house. This family is called the Positronium Ion (Ps⁻). It consists of two electrons (negative charge) and one positron (positive charge, the electron's antimatter twin).

Usually, this family falls apart in a very predictable way: the positron and one electron hug each other, vanish, and turn into pure light (a photon). This is like a couple deciding to leave a party by turning into a flash of fireworks, while the third guest (the remaining electron) just watches and has to move out of the way to keep the energy balance. This is the "electromagnetic" decay, and it happens very often.

The Big Question
The scientists in this paper asked: "What if, instead of turning into light, they tried to turn into something else entirely?" Specifically, they looked at a very rare, "weak" process where the positron and electron vanish and turn into a ghostly pair of particles called neutrinos (which barely interact with anything) and a leftover electron.

In the language of particle physics, this is like the couple deciding to turn into invisible smoke (neutrinos) instead of fireworks, while the third guest still has to move.

The Two Ways to Solve the Puzzle
The authors used two different mathematical "lenses" to calculate how likely this rare event is. They wanted to make sure their answer was rock-solid, so they checked their work twice.

The "Step-by-Step" Lens: They imagined the process happening in two stages. First, the family turns into an electron and a "virtual" heavy particle (a Z boson). Then, that heavy particle instantly splits into the two neutrinos. They calculated the odds for every possible way the spins (tiny internal magnets) of the particles could be arranged.
The "Big Picture" Lens: They treated the whole event as one giant, complex calculation, adding up every possible way the particles could interact at once, using standard rules of quantum mechanics.

The Surprising Discovery
When they did the math, they found something interesting about the "spins" of the particles.

Think of the two electrons in the family as a pair of dancers. They can dance in sync (spins aligned) or in opposition (spins opposite).
The math showed that the "weak" decay (turning into neutrinos) only happens if the two electrons are dancing in opposition (a spin-singlet state).
If they try to dance in sync, the process simply cannot happen; the probability is zero. It's as if the universe has a strict rule: "Only the opposite-spin dancers are allowed to turn into neutrinos."

How Rare is This?
The result is that this weak decay is incredibly rare.

The "fireworks" decay (turning into a photon) happens almost all the time.
The "neutrino" decay happens so rarely that for every one time it occurs, the fireworks decay happens about one quintillion (10¹⁹) times.

To put this in perspective, the paper compares this to another rare event involving a simpler version of the family (just the electron and positron, without the third guest). The odds are roughly the same: both are incredibly unlikely compared to their normal, light-producing decay.

Why Does This Matter?
The paper doesn't claim this will lead to new medical treatments or energy sources. Instead, it's a "clean" test of our understanding of the universe's rules. By calculating this rare event so precisely using two different methods and getting the exact same answer, the scientists have confirmed that our current theories (the Standard Model) correctly predict how these tiny, three-particle families behave when they interact with the "weak" force. It's like checking a very complex math problem with two different calculators to ensure the answer is right.

In Summary
The paper is a detailed calculation showing that a three-particle atom (two electrons, one positron) can very rarely decay into an electron and two invisible neutrinos. This only happens if the two electrons have opposite spins, and it is so rare that it's essentially a "needle in a haystack" event compared to the atom's normal way of decaying. The authors proved this using two different mathematical approaches, and both agreed perfectly.

Technical Summary: Weak Decay of the Positronium Ion

Problem Statement
The positronium ion ( $Ps^-$ ), a Coulombic three-body bound state consisting of two electrons and a positron, is predominantly known for its electromagnetic decay via electron-positron annihilation into photons ( $Ps^- \to e^-\gamma$ ). While the radiative decay channels of positronium systems have been extensively studied within Quantum Electrodynamics (QED), their weak-interaction decay channels have received comparatively little attention. This paper investigates the rare electroweak decay process $Ps^- \to e^- \nu_\mu \bar{\nu}_\mu$ . This process is the weak analog of the electromagnetic decay, obtained by replacing the final-state photon with a virtual $Z$ boson that subsequently decays into a muon neutrino-antineutrino pair. The study aims to compute the decay rate and branching ratio, addressing the unique kinematic and spin-configuration features of this three-body bound system compared to standard positronium ($Ps$) decays.

Methodology
The authors employ two complementary theoretical approaches to calculate the decay rate, demonstrating their consistency:

Factorized Two-Body Transition Approach:
- The process is treated as a factorized transition: $Ps^- \to e^- Z^*$ followed by $Z^* \to \nu_\mu \bar{\nu}_\mu$ .
- The authors explicitly evaluate the scattering amplitudes for all possible spin configurations of the initial bound state and the final particles.
- The three-body phase space is decomposed into a product of two two-body phase spaces. The decay rate is expressed in terms of the effective two-body rate ( $Ps^- \to e^- Z^*$ ) integrated over the virtual $Z$ boson's invariant mass squared ( $q^2$ ).
- The calculation utilizes the leading-order (LO) Feynman diagrams (eight in total), separating contributions from spin-triplet (ortho-like) and spin-singlet (para-like) annihilations. The bound-state effects are incorporated via the wave function at the origin, $\Psi(0,0,0)$ .
Direct Three-Body QFT Calculation:
- A direct calculation is performed using standard Quantum Field Theory (QFT) techniques.
- The squared amplitude is evaluated using the "Casimir trick" (trace methods over Dirac matrices) after summing over final-state spins and averaging over initial-state spins.
- This approach utilizes the FeynArts package for diagram generation and FeynCalc for algebraic manipulation and trace evaluation.
- The result is compared directly with the factorized approach to ensure theoretical consistency.

Key Contributions and Results

Spin Configuration Analysis: A primary finding is that the decay rate is non-vanishing only when the annihilating electron-positron pair is in a spin-singlet state ( $S=0$ ). Despite the initial $Ps^-$ state allowing for both singlet and triplet configurations of the constituent pair, the amplitudes for all other spin configurations (including the triplet state) vanish. This contrasts with the electromagnetic decay where both configurations contribute.
Decay Rate Calculation: Both theoretical methods yield identical results for the total decay rate:
$\Gamma(Ps^- \to e^- \nu_\mu \bar{\nu}_\mu) \approx 1 \times 10^{-20} \, \text{s}^{-1}$
The calculation confirms that the result is consistent with the standard spin-summation formalism of QFT.
Branching Ratio: The branching ratio for this weak decay relative to the dominant electromagnetic decay ( $Ps^- \to e^- \gamma$ ) is calculated to be:
$\text{Br}(Ps^- \to e^- \nu_\mu \bar{\nu}_\mu) \approx 10^{-19}$
This extreme suppression is parametrically understood as scaling with $G_F^2 m^4 / \alpha$ , where $G_F$ is the Fermi constant and $\alpha$ is the fine-structure constant.
Comparison with Positronium: The paper notes that this branching ratio is comparable to that of the weak decay of ortho-positronium ( $o\text{-}Ps \to \gamma \nu \bar{\nu}$ ), which is also estimated at $\approx 10^{-19}$ . The similarity arises because the bound-state factors (related to the wave function at the origin) cancel out in the ratio of weak to electromagnetic decay widths, leaving the result determined by the relative strengths of the interactions.

Significance and Claims
The paper claims to provide the first detailed investigation of the weak decay channel $Ps^- \to e^- \nu_\mu \bar{\nu}_\mu$ . Its significance lies in:

Extending NRQED Factorization: It demonstrates that the factorization properties of non-relativistic QED (NRQED), which separate short-distance coefficients from bound-state probabilities, hold for this electroweak process, similar to previous findings in dipositronium ( $Ps_2$ ) and positronium hydride ($PsH$).
Three-Body Weak Interaction Probe: The decay offers a "clean probe" of purely three-body leptonic weak interactions within a bound-state environment.
Methodological Consistency: The work establishes the equivalence between a factorized two-body approach and a direct three-body QFT calculation for this specific bound-state system, validating the use of effective two-body transitions for such rare decays.

The authors conclude that while the decay is theoretically significant for understanding weak interactions in bound states, the branching ratio is highly suppressed ( $\sim 10^{-19}$ ), making experimental observation extremely challenging with current technology.

Technical Summary: Weak Decay of the Positronium Ion

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