[Paper Review] On the Question of the Point-Particle Nature of the Electron

Horace W. Crater, Cheuk-Yin Wong|arXiv (Cornell University)|Jun 27, 2014

Particle physics theoretical and experimental studies6 references3 citations

TL;DR

This paper proposes that if electrons are true point particles, the Two-Body Dirac Equations in QED predict a previously undiscovered, deeply bound $^{1}S_{0}$ positronium state with a binding energy of ~300 keV, distinct from the standard 6.8 eV state. This exotic state can be produced via two-photon decay of the conventional $^{1}S_{0}$ state, leading to a detectable four-photon decay signature with two photons at ~150 keV and two at ~360 keV, providing a direct experimental test of the electron's point-particle nature through the mixing probability $P_{up}$.

ABSTRACT

The electron and the positron treated as point particles in the Two Body Dirac equations of constraint dynamics for QED possess a new and as yet undiscovered peculiar ${}^1S_0$ bound-state which has a very large binding energy of about 300 keV, in addition to the usual ${}^1S_0$ positronium state with a binding energy of 6.8 eV. The production and detection of the peculiar ${}^1S_0$ state provide a test of the electron point-charge property. As the peculiar ${}^1S_0$ state lies lower than the usual ${}^1S_0$ state, the peculiar ${}^1S_0$ state can be produced by a two-photon decay of the usual $^{1}S_{0}$ state. We estimate the rate of the two-photon decay and show how it depends on the probability $P_{up}$ of the admixture of the peculiar component in the predominantly usual ${}^1S_0$ positronium. The produced peculiar ${}^1S_0$ state in turn annihilates into two photons with a total c.m. energy of about 723 keV. Thus the signature for this new peculiar ${}^1S_0$ positronium bound state would be the decay of the usual ${}^1S_0$ state into four photons, with two energies bunching around 150 and two around 360 keV. Such a four-photon decay of the usual ${}^1S_0$ state will not be present if the electron and positron are not point particles, or if the mixing probability $P_{up}$ is very small.

Motivation & Objective

• To investigate whether the electron's point-particle nature leads to observable deviations in $e^+e^-$ bound states within QED.
• To explore the existence of a new, deeply bound $^{1}S_0$ positronium state with ~300 keV binding energy, arising from point-like electron-positron interactions.
• To propose a detectable four-photon decay signature from the conventional $^{1}S_0$ state, mediated by the admixture of the exotic state.
• To quantify the decay rate and branching ratio as a function of the mixing probability $P_{up}$ between the standard and exotic $^{1}S_0$ states.

Proposed method

• Formulate the Two-Body Dirac Equations for $e^+e^-$ systems in constraint dynamics to describe bound states in QED.
• Identify a new $^{1}S_0$ bound state with large binding energy (~300 keV) due to point-particle nature, distinct from the standard 6.8 eV positronium state.
• Compute the two-photon decay amplitude of the conventional $^{1}S_0$ state into the exotic $^{1}S_0$ state using perturbation theory and matrix elements involving wavefunction overlap.
• Derive the decay rate $\Gamma$ as proportional to the mixing probability $P_{up}$, with the rate estimated at $\Gamma \approx 0.152 \times \Gamma_{1S_u \to 2\gamma} \times P_{up}$.
• Model the four-photon decay signature: two photons at ~150 keV and two at ~360 keV, arising from the exotic state's annihilation into two photons at ~723 keV c.m. energy.
• Use wavefunction overlaps and phase space integrals to compute the matrix element and decay width, incorporating relativistic corrections and angular dependencies.

Experimental results

Research questions

• RQ1 Does the point-particle assumption for electrons in QED lead to a new, deeply bound $^{1}S_0$ positronium state not present in standard QED?
• RQ2 Can the conventional $^{1}S_0$ positronium state decay into the exotic $^{1}S_0$ state via two-photon emission, and what is the rate of this process?
• RQ3 What is the four-photon decay signature of the conventional $^{1}S_0$ state if the exotic state is produced and subsequently annihilates into two photons?
• RQ4 How does the decay rate depend on the mixing probability $P_{up}$ between the standard and exotic $^{1}S_0$ states?
• RQ5 Can this decay process be experimentally distinguished from standard QED processes, providing a test of the electron's point-like nature?

Key findings

• The Two-Body Dirac Equations predict a new $^{1}S_0$ bound state with a binding energy of approximately 300 keV, significantly deeper than the standard 6.8 eV positronium state.
• The exotic $^{1}S_0$ state can be produced via two-photon decay of the conventional $^{1}S_0$ state, with the decay rate proportional to the mixing probability $P_{up}$.
• The total decay width for the four-photon process $1S_u \to 4\gamma$ is estimated as $\Gamma \approx 0.152 \times \Gamma_{1S_u \to 2\gamma} \times P_{up}$, leading to a branching ratio of $0.152 \times P_{up}$.
• The four-photon decay signature consists of two photons with energy ~150 keV and two with ~360 keV, arising from the exotic state's annihilation into two photons at ~723 keV center-of-mass energy.
• The decay rate is sensitive to the electron's point-particle nature, as the exotic state arises from the singular $\delta(\mathbf{r})$-like interaction at short distances inherent to point charges.
• The matrix element calculation shows that the dominant contribution to the decay amplitude comes from the wavefunction overlap $\langle 1S_p | 1S_u \rangle \sim 2^{5/4} \alpha^{3/2}$, with higher-order corrections suppressed by $\alpha^2$.