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arXiv:2606.25433

Weak decay of the positronium ion

Nishat Ul Sani, M. Jamil Aslam, Ishtiaq Ahmed

hep-phhep-exquant-phSubmitted Jun 24, 2026• Announced Jun 25, 2026
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Abstract

The positronium ion ($\mathrm{Ps}^-$), a coulombic three-body bound state of two electrons and a positron, predominantly decays via electron-positron annihilation into electromagnetic final states. While its radiative decay channels have been extensively studied, much less attention has been given to weak processes in this system. In this work, we investigate the rare decay $\mathrm{Ps}^- \to e^- ν_μ\barν_μ$, obtained by replacing the photon in $\mathrm{Ps}^- \to e^- γ$ with a virtual $Z$ boson. Treating the three-body process as an effective two-body transition, $\mathrm{Ps}^- \to e^- Z^*\left(\to ν_μ\barν_μ\right)$, we compute the decay rate by explicitly evaluating all spin configurations of the initial bound state and final particles. The result agrees with that obtained using the standard spin-summation formalism of quantum field theory. We find that the branching ratio is comparable to that of the weak decay of ortho-positronium, $\mathrm{o\text{-}Ps} \to γν\barν$.

Comments
13 pages, 1 figure, Accepted in Phys. Rev. A

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The positronium ion (Ps−\mathrm{Ps}^-Ps−), a coulombic three-body bound state of two electrons and a positron, predominantly decays via electron-positron annihilation into electromagnetic final states. While its radiative decay channels have been extensively studied, much less attention has been given to weak processes in this system. In this work, we investigate the rare decay Ps−→e−νμνˉμ\mathrm{Ps}^- \to e^- ν_μ\barν_μPs−→e−νμ​νˉμ​, obtained by replacing the photon in Ps−→e−γ\mathrm{Ps}^- \to e^- γPs−→e−γ with a virtual ZZZ boson. Treating the three-body process as an effective two-body transition, Ps−→e−Z∗(→νμνˉμ)\mathrm{Ps}^- \to e^- Z^*\left(\to ν_μ\barν_μ\right)Ps−→e−Z∗(→νμ​νˉμ​), we compute the decay rate by explicitly evaluating all spin configurations of the initial bound state and final particles. The result agrees with that obtained using the standard spin-summation formalism of quantum field theory. We find that the branching ratio is comparable to that of the weak decay of ortho-positronium, o-Ps→γννˉ\mathrm{o\text{-}Ps} \to γν\barνo-Ps→γννˉ.
Positronium hydride (PsH)(\text{PsH})(PsH) is a weakly bound (molecular) state of two electrons $({e}^{\ensuremath{-}})$, a positron (e+)({e}^{+})(e+), and a proton (p+)({p}^{+})(p+). Due to the ${e}^{\ensuremath{-}}\ensuremath{-}{e}^{+}$ annihilation, it decays to a number of processes, where the radiationless decay is one of the most important channels. In this decay process, the photons released from the ${e}^{\ensuremath{-}}\ensuremath{-}{e}^{+}$ annihilation are absorbed by the remaining ${e}^{\ensuremath{-}}$ and p+{p}^{+}p+ in PsH, giving us $\text{PsH}\ensuremath{\rightarrow}{e}^{\ensuremath{-}}{p}^{+}$. For a particular spin (singlet) configuration of the constituents of PsH, the radiationless decay rate of PsH was determined by Aslam et al. [Phys. Rev. A 104, 052803 (2021)]. In this work, we calculate the decay rate of $\text{PsH}\ensuremath{\rightarrow}{e}^{\ensuremath{-}}{p}^{+}$ by exercising all possible spin orientations of PsH and the final states ${e}^{\ensuremath{-}}$ and p+{p}^{+}p+. By summing all the contributions, we verify that the result is the same if we calculate it using the standard fermion spin-sum technique of quantum field theory.
The positronium molecule (Ps2)({\mathrm{Ps}}_{2})(Ps2​) has not been experimentally observed yet because its tiny (4.5 eV) binding energy cannot be detected when the molecule annihilates by emitting two photons with energy of 0.51 MeV each. It is shown in this paper that the electric dipole transition between the recently found L=1L=1L=1 excited state and the L=0L=0L=0 ground state with its characteristic photon energy of 4.94 eV is a clear signature of the existence of the positronium molecule and the possibility of its experimental observation is realistic. The probability of this transition is about 17% of the total decay rate. Another Coulomb four-body system containing positron HPs (the positronium hydride or hydrogen positride), is also included for comparison.
The positronium ion Ps−−{\mathrm{Ps}}^{--}Ps−− is often qualitative described as a positronium Ps with a loosely bound second electron, since the value of the annihilation rate for Ps−−{\mathrm{Ps}}^{--}Ps−− is nearly equal to that of the spin-averaged neutral positronium. This similarity however has never been fully explained. To study its internal structure, we decompose the electron-positron distribution function of Ps−−{\mathrm{Ps}}^{--}Ps−− into the sum of two terms: one for the electron closest to the positron and the other for the farthest. We show that the inner electron behaves almost as if the outer electron were not present, forming a Ps substructure inside Ps−−{\mathrm{Ps}}^{--}Ps−− and that the effect of electron correlation is such that the outer electron tries to stay as close as possible to the positron but without interfering too much with the distribution of the inner electron. We show that this phenomenon is closely related to the problem of the critical stability of the system (Z+,e−−,e−−)({Z}^{+},\phantom{\rule{0.28em}{0ex}}{e}^{--},\phantom{\rule{0.28em}{0ex}}{e}^{--})(Z+,e−−,e−−) with varying ZZZ charge.
Abstract The electron-positron annihilation into two photons is a standard technology in medicine to observe e.g. metabolic processes in human bodies. A new tomograph will provide the possibility to observe not only direct e+e− annihilations but also the 3 photons from the decay of ortho-positronium atoms formed in the body. We show in this contribution that the three-photon state with respect to polarisation degrees of freedom depends on the angles between the photons and exhibits various specific entanglement features. In particular genuine multipartite entanglement, a type of entanglement involving all degrees of freedom, is subsistent if the positronium was in a definite spin eigenstate. Remarkably, when all spin eigenstates are mixed equally, entanglement –and even stronger genuine multipartite entanglement– survives. Due to a “symmetrization” process, however, Dicke-type or W-type entanglement remains whereas GHZ-type entanglement vanishes. The survival of particular entanglement properties in the mixing scenario may make it possible to extract quantum information in the form of distinct entanglement features, e.g., from metabolic processes in human bodies.
Entanglement of three photons from the decay of ortho-positronium is re-analyzed. We use the full three body phase space to write down the entangled states classified according to the spin directions of the ortho-positronium. Even in the case when the spin is perpendicular to the decay plane, we find non-negligible phases entering the entangled state. This has not been noticed before. We advocate a fixed quantization axis of the spin for the sake of generality. A brief discussion of a three dimensional formalism for photons, including correlations, versus a two dimensional one is given.