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arxiv: 1907.04619 · v1 · pith:IJQ2IZ7Tnew · submitted 2019-07-10 · ✦ hep-ph · quant-ph

Annihilation energy and decay time of an ortho-Positronium

Abdullah Guvendi , Yusuf Sucu This is my paper

Pith reviewed 2026-05-24 23:47 UTC · model grok-4.3

classification ✦ hep-ph quant-ph
keywords ortho-positroniumannihilation energydecay lifetimerelativistic two-body problem2+1 dimensionsCoulomb interactioncomplex energy spectrumS state
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The pith

A 2+1-dimensional relativistic two-body model maps the Coulomb background to polar coordinates and obtains complex energies whose imaginary part supplies the ortho-positronium lifetime.

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

The paper models ortho-positronium as an electron-positron pair interacting through an attractive Coulomb force in two spatial dimensions. After separating center-of-mass and relative coordinates, the background is transformed into polar space-time, which permits explicit construction of spin eigen-states. The energy eigenvalues that result are necessarily complex; their real and imaginary parts are identified with the annihilation energy, the binding energy, and the decay lifetime of the S-state system. A reader would care because the construction supplies decay information directly from the spectrum rather than inserting it by hand.

Core claim

In this relativistic two-body problem the background is mapped into polar space-time, allowing construction of possible spin eigen-states. The resulting energy spectrum is complex and its real and imaginary parts furnish the annihilation energy, binding energy and life-time of ortho-positronium in the S state.

What carries the argument

The mapping of the background into polar space-time that produces spin eigen-states whose complex energies encode the physical decay rate.

If this is right

  • The real part of the complex energy directly supplies the annihilation energy of the S-state.
  • The same expression yields the binding energy of the bound state.
  • The imaginary part of the energy determines the lifetime without an external decay width.
  • The results apply specifically to the S-state in the two-dimensional Coulomb problem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same polar-mapping technique could be applied to other two-body systems whose decay is mediated by a weak channel, turning stable spectra into complex ones.
  • If the 2+1D reduction is faithful, the approach suggests that dimensionality reduction itself can generate decay rates from otherwise Hermitian operators.

Load-bearing premise

The mapping of the background into polar space-time produces spin eigen-states whose complex energies correctly encode the physical decay rate of ortho-positronium.

What would settle it

A numerical comparison of the lifetime extracted from the imaginary part of the complex energy against the experimentally measured ortho-positronium lifetime in vacuum.

read the original abstract

We approach to the ortho-positronium (o-Ps) as a relativistic two-body problem in $2+1$ dimensions in which o-Ps is composed of two-oppositely charged particles interacting via an attractive Coulomb force. In addition to separation of center of mass and relative coordinates, mapping the background into the polar space-time gives possibility of construction of possible spin eigen-states of o-Ps. This approach makes the energy spectrum complex in order to describe o-Ps that can decay. From the complex energy expression, we find the annihilation energy, binding energy and the life-time of o-Ps, in S state.

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

2 major / 0 minor

Summary. The manuscript treats ortho-positronium as a relativistic two-body Coulomb problem in 2+1 dimensions. Center-of-mass separation followed by a polar space-time mapping is used to construct spin eigenstates; the resulting energy eigenvalues are complex, and the imaginary part is employed to extract the annihilation energy, binding energy, and lifetime of the S-state o-Ps.

Significance. A first-principles derivation of the o-Ps lifetime from a relativistic two-body model without free parameters or explicit insertion of the decay width would be of interest in hep-ph. The 2+1D reduction and complex-energy construction would need to be shown to reproduce the known 3+1D lifetime (~142 ns) and three-photon decay rate for the result to carry weight.

major comments (2)
  1. [Abstract] Abstract: the complex energy is introduced 'in order to describe o-Ps that can decay,' yet no derivation is supplied that obtains the imaginary part from the QED annihilation matrix element (e.g., via the optical theorem or Fermi golden rule for three-photon decay).
  2. [Abstract] Abstract: the entire construction is performed in 2+1 dimensions, but no argument is given that the resulting spectrum reproduces the physical 3+1D ortho-positronium lifetime or accounts for the differences in photon phase space and spin statistics between 2+1D and 3+1D.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the complex energy is introduced 'in order to describe o-Ps that can decay,' yet no derivation is supplied that obtains the imaginary part from the QED annihilation matrix element (e.g., via the optical theorem or Fermi golden rule for three-photon decay).

    Authors: The manuscript constructs the complex energy spectrum as a direct consequence of applying the polar space-time mapping to the relativistic two-body Coulomb problem in 2+1 dimensions after center-of-mass separation. The imaginary part is then used to define the annihilation energy and lifetime within this framework. We agree that no derivation of the imaginary part is supplied from the QED annihilation matrix element via the optical theorem or Fermi golden rule. The approach is model-based rather than a first-principles extraction from the three-photon decay amplitude. We will revise the abstract and introduction to state explicitly that the complex energy arises from the mapping procedure and is interpreted as encoding decay, without claiming a direct QED derivation. revision: yes

  2. Referee: [Abstract] Abstract: the entire construction is performed in 2+1 dimensions, but no argument is given that the resulting spectrum reproduces the physical 3+1D ortho-positronium lifetime or accounts for the differences in photon phase space and spin statistics between 2+1D and 3+1D.

    Authors: The entire analysis is performed in 2+1 dimensions to enable the polar mapping and construction of spin eigenstates. The referee is correct that the manuscript supplies no argument demonstrating that the resulting spectrum reproduces the 3+1D lifetime of ~142 ns or incorporates the differences in photon phase space and spin statistics. The paper does not attempt such a reproduction or comparison. We will add a clarifying paragraph in the discussion section noting the use of the 2+1D reduction and stating that quantitative matching to 3+1D QED results lies outside the present scope. revision: yes

Circularity Check

1 steps flagged

Lifetime extracted by construction from imaginary part of energy introduced explicitly to model decay

specific steps
  1. self definitional [Abstract]
    "This approach makes the energy spectrum complex in order to describe o-Ps that can decay. From the complex energy expression, we find the annihilation energy, binding energy and the life-time of o-Ps, in S state."

    The spectrum is rendered complex precisely to incorporate decay; the lifetime is then read off from that same complex expression. The reported lifetime therefore reduces to the modeling choice of the imaginary part rather than emerging from an independent calculation of the annihilation process.

full rationale

The paper's central derivation introduces complex energies specifically 'in order to describe o-Ps that can decay' after the 2+1D polar mapping and center-of-mass separation. It then extracts the lifetime directly from the resulting complex energy expression. This makes the reported lifetime equivalent to the input assumption that the imaginary part encodes the physical decay rate, without an independent derivation from the QED three-photon annihilation amplitude, optical theorem, or Fermi's golden rule. The step is self-definitional rather than predictive. No external benchmark or non-circular justification for the imaginary part is supplied in the abstract or described construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters, axioms, or invented entities can be identified from the text.

pith-pipeline@v0.9.0 · 5628 in / 1101 out tokens · 20808 ms · 2026-05-24T23:47:53.049022+00:00 · methodology

discussion (0)

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Reference graph

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