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arxiv: 2606.25433 · v1 · pith:LJLMKMURnew · submitted 2026-06-24 · ✦ hep-ph · hep-ex· quant-ph

Weak decay of the positronium ion

Nishat Ul Sani , M. Jamil Aslam , Ishtiaq Ahmed This is my paper

Pith reviewed 2026-06-25 21:10 UTC · model grok-4.3

classification ✦ hep-ph hep-exquant-ph
keywords positronium ionweak decaybranching ratioZ bosonneutrino pairortho-positroniumspin summationbound state decay
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The pith

The weak decay branching ratio of the positronium ion is comparable to that of ortho-positronium.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates the rare weak decay of the positronium ion Ps⁻ into an electron and a muon neutrino pair. It models the process as an effective two-body transition with a virtual Z boson replacing the photon. The decay rate is computed by evaluating all spin configurations of the initial and final states. The branching ratio is found to be comparable to the weak decay of ortho-positronium.

Core claim

By treating the three-body decay Ps⁻ → e⁻ ν_μ ν̄_μ as the effective two-body transition Ps⁻ → e⁻ Z*(→ ν_μ ν̄_μ), the decay rate is calculated explicitly for all spin configurations and agrees with the standard spin-summation formalism of quantum field theory, resulting in a branching ratio comparable to o-Ps → γ ν ν̄.

What carries the argument

The effective two-body transition Ps⁻ → e⁻ Z*(→ ν_μ ν̄_μ) obtained by replacing the photon with a virtual Z boson in the radiative decay channel.

Load-bearing premise

The three-body decay can be treated as an effective two-body transition Ps⁻ → e⁻ Z*(→ ν_μ ν̄_μ).

What would settle it

A precise measurement of the branching ratio for Ps⁻ → e⁻ ν_μ ν̄_μ that deviates substantially from the value obtained for the ortho-positronium weak decay would challenge the result.

Figures

Figures reproduced from arXiv: 2606.25433 by Ishtiaq Ahmed, M. Jamil Aslam, Nishat Ul Sani.

Figure 1
Figure 1. Figure 1: FIG. 1: The leading- order (LO) Feynman diagrams for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original 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 $\mathrm{Ps}^- \to e^- \nu_\mu \bar{\nu}_\mu$, obtained by replacing the photon in $\mathrm{Ps}^- \to e^- \gamma$ with a virtual $Z$ boson. Treating the three-body process as an effective two-body transition, $\mathrm{Ps}^- \to e^- Z^*\left(\to \nu_\mu\bar{\nu}_\mu\right)$, 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, $\mathrm{o\text{-}Ps} \to \gamma \nu \bar{\nu}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper investigates the rare weak decay Ps⁻ → e⁻ ν_μ ν̄_μ of the positronium ion by replacing the photon in the radiative decay with a virtual Z boson. It treats the three-body process as an effective two-body transition Ps⁻ → e⁻ Z*(→ ν_μ ν̄_μ), computes the rate by explicitly evaluating all spin configurations, reports agreement with the standard spin-summation formalism of QFT, and finds the branching ratio comparable to that of o-Ps → γ ν ν̄.

Significance. If the central approximation holds, the work supplies a new result for a rare weak channel in a three-body Coulomb bound state, extending studies of weak decays beyond two-body positronium systems. The explicit enumeration of spin states and cross-check against the standard formalism are strengths that would support the numerical claim if the reduction is justified.

major comments (1)
  1. [method of effective two-body transition (as described following the abstract)] The effective two-body reduction Ps⁻ → e⁻ Z*(→ ν_μ ν̄_μ) is load-bearing for the branching-ratio result. The manuscript must supply a detailed derivation showing how the spectator electron factors out of the matrix element, how the three-body Coulomb wave function is inserted into the two-body annihilation operator, and whether overlap corrections from indistinguishability or off-shell Z kinematics are included; without this, any systematic error propagates directly into the quoted rate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment. We appreciate the positive assessment of our explicit spin enumeration and its agreement with the standard QFT spin-summation formalism. We address the single major comment below.

read point-by-point responses

  1. Referee: [method of effective two-body transition (as described following the abstract)] The effective two-body reduction Ps⁻ → e⁻ Z*(→ ν_μ ν̄_μ) is load-bearing for the branching-ratio result. The manuscript must supply a detailed derivation showing how the spectator electron factors out of the matrix element, how the three-body Coulomb wave function is inserted into the two-body annihilation operator, and whether overlap corrections from indistinguishability or off-shell Z kinematics are included; without this, any systematic error propagates directly into the quoted rate.

    Authors: We agree that the effective two-body reduction is central to the result and that the current manuscript provides only a brief statement of the approximation without a full derivation. In the revised version we will add a dedicated subsection (or appendix) that derives the reduction explicitly. This will include: (i) the construction of the matrix element by projecting the three-body Coulomb wave function onto the positron-electron annihilation operator while treating the second electron as a spectator whose wave-function factor is integrated separately; (ii) the explicit insertion of the three-body bound-state wave function into the two-body operator; and (iii) a discussion of possible corrections arising from electron indistinguishability (via antisymmetrization of the initial state) and from the off-shell kinematics of the virtual Z. We will also quantify the expected size of these corrections under the kinematic conditions of the decay. These additions will be made without altering the numerical result already reported. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses explicit spin summation on effective two-body ansatz without reducing to fitted inputs or self-citations

full rationale

The provided abstract and description show a direct computation of the decay rate via explicit evaluation of all spin configurations for the effective two-body transition Ps⁻ → e⁻ Z*, with the result stated to agree with standard QFT spin-summation formalism. No parameters are fitted to data and then relabeled as predictions, no self-citations are invoked as load-bearing uniqueness theorems, and no ansatz is smuggled via prior work. The three-body to two-body reduction is presented as an explicit modeling choice whose validity can be assessed externally; it does not collapse by construction to the input assumptions. This is the common case of a self-contained calculation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; the calculation rests on standard QFT and bound-state assumptions whose details are not visible.

axioms (1)
  • standard math Standard quantum field theory formalism for weak decays and spin summation in bound states
    Invoked to evaluate the decay rate and compare with spin-summation method

pith-pipeline@v0.9.1-grok · 5742 in / 1106 out tokens · 19933 ms · 2026-06-25T21:10:15.285842+00:00 · methodology

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Reference graph

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    Decomposition of three-body phase space The three body phase space can be written as, ˆ dϕ3(P;k 1, k2, k3) = ˆ 3Y i=1 d3ki (2π)32Ei (2π)4δ4(P−k 1 −k 2 −k 3).(A.2) WhereP=p 1 +p 2 +p 3 is the total initial momentum. We can write the unity in terms of an integral as 1 = ˆ d3q (2π)32Eq dq2 2π (2π)4δ4(q−k 1 −k 2).(A.3) Hereq=k 1 +k 2 is the four momentum of t...

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