One-photon annihilation of the bound electron-positron pair in the field of atomic nuclei

arXiv:2405.06720 · hep-ph, nucl-th, quant-ph · Submitted 2024-05-10 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "One-photon annihilation of the bound electron-positron pair in the field of atomic nuclei".

Mira: Here's the gist of the paper. One-photon annihilation of an electron-positron pair in a heavy atomic nucleus is investigated to derive an exact Quantum Electrodynamics formula for its annihilation rate, which can be used to describe processes like positronium hydrides and quasi-stable triplet states of positron-helium atoms<ref:2405.06720#pg14>.

Kai: Derivation of the Annihilation Rate Formula The study derives the explicit formula for the annihilation rate of the electron-positron pair in a central, very heavy and positively charged atomic nucleus, denoted as Γ(b)1γ<ref:2405.06720#pg14>.

Mira: The derivation begins by applying Feynman rules of Quantum Electrodynamics to describe the lowest-order one-photon annihilation process<ref:2405.06720#pg14> and obtaining the S-matrix element in coordinate space<ref:2405.06720#pg14>.

Kai: The squared scattering amplitude modulus is expressed in a differential form involving the function F¯(p+, p−; k)<ref:2405.06720#pg14> and the unpolarized cross section is given by Eq.(8)<ref:2405.06720#pg14>.

Title and authors: Kai: To keep going with the title, we need to focus on what the authors actually achieved in this paper, beyond just stating what they did. They're using this as a foundation for something more practical.

Mira: Right, they established that even though previous studies used approximations—like the one based on Ps− for positronium hydrides—that those approximations led to relatively large mistakes in numerical evaluations of the total probability of one-photon annihilation for light atoms.

Lev: So, this paper directly addresses that issue by showing why those earlier approximations aren't good enough when you're looking at systems of main interest in this study.

Kai: The authors are building on their earlier work, which previously evaluated the one-photon annihilation rate in positronium hydride HPs using the known analytical formula for Gamma1γ in the three-body positronium ion Ps−.

Mira: And they found that it was a priori clear that the overall accuracy of such an approximation isn't good, so this paper sets up this new derivation to show why we need something more precise.

Lev: This means they are setting up the framework for a more accurate calculation, and they aren't just rewriting old results; they are fixing an error.

Kai: They start by applying Feynman rules of Quantum Electrodynamics to describe the lowest-order one-photon annihilation process in momentum space to get that S-matrix element in coordinate space.

Mira: This leads them to this expression for the Sfi, which is written in the following form, (one), which involves the static Coulomb potential with its components A0(x) = -Qe x = -Qe r, and ACoul(x) = zero <ref:2405.06720#pg3,A0(x) = -Qe | x | = -Qe | r |, and ACoul(x>.

Lev: So they're working with this specific potential setup, where the Coulomb interaction is just a simple linear term in position.

Kai: Then at the second step of the procedure, they have to perform the Fourier integration in Eq. (one), which leads them to this expression for the Sfi, which is written in momentum representation <ref:2405.06720#pg1>.

Mira: This expression for Sf i is Qe3

2pi delta(omega - Eminus - Eplus): vuut m2, which captures the energy conservation perfectly <ref:2405.06720#pg2>.

Lev: So they're using this momentum representation to enforce the energy conservation condition directly in their calculation.

Kai: What's key here is how they handle that integration, because that leads to this final expression for Sf i, which involves terms like exp(-ikx) and exp(ikx) times the static Coulomb potential.

Mira: This is where they get the spatial dependence of the S-matrix element, and it’s where Eq. (two), which shows A0(q) = -Qe q two comes into play, which is related to how fast you can Fourier transform that static potential <ref:2405.06720#pg2>.

Lev: So they're connecting the real-space potential to its momentum representation through this integral setup.

Kai: This whole process sets up the connection between space and momentum representation for this annihilation process.

Mira: It seems like they are doing a lot of groundwork here before getting to that final cross section form in Eq. (eight).

Lev: They are setting up the machinery so that when they can finally get to that differential cross section formula, they have the necessary ingredients ready.

The paper's summary: Kai: Now we look at the actual summary of this paper, which is what it says about the core methodology and what it claims they actually found in terms of results.

Mira: The abstract clearly states that they investigate one-photon annihilation of the electron-positron pair in a central, very heavy and positively charged atomic nucleus and derive an explicit formula for the annihilation rate Gamma(b) 1γ <ref:2405.06720#pg1,investigate one-photon annihilation of the electron-positron pair in>.

Lev: They emphasize that this formula can directly be used to describe the actual one-photon annihilation in the ground (bound) states of all positronium hydrides HPs, quasi-stable triplet states of the positron-helium atoms e+

He(23Se): ions and other systems <ref:2405.06720#pg1,can directly be used to describe the actual one-photon annihilation in>.

Kai: So, it's not just theoretical stuff; it’s immediately applicable to these specific few-body systems where they are of main interest.

Mira: And they show that their formula for this rate can directly describe the actual one-photon annihilation in those specific ground states, which is a key point.

Lev: It means they’s not just making some abstract math; it's giving us a practical tool to describe these physical realities.

Kai: They are deriving the explicit formula for this annihilation rate, Gamma(b) 1γ, which is what we need for those applications <ref:2405.06720#pg1>.

Mira: And that formula is the result of their derivation from first principles of QED.

Lev: So they's providing a direct link between the fundamental theory and these specific atomic physics problems.

The paper's improvements: Kai: Moving on to what the authors suggest as improvements, which is where they point out how this work can be applied further for future research.

Mira: They mention that the way they approach things allows them to quantify system complexity by giving us a general case formula for m positrons and n electrons, given by Eq. (forty-three).

Lev: That general case formula, Ne-e+ is equal to mc2 n plus nc2 m, lets the AI calculate annihilation rates for arbitrary systems containing 'm' positrons and 'n' electrons.

Kai: This capability extends the model to new experimental targets not covered by previous specific analyses.

Mira: They also mention that they have a closed formula for the corresponding annihilation rate Gamma(a) 1γ, which is written in Eq <ref:2405.06720#pg1>. (forty-four).

Lev: That closed formula is derived from the work done elsewhere, but it's useful because it's written in a form that allows for calculations in many different ways.

Kai: So they’re providing a way to evaluate numerical values of the rates of one-photon annihilation for various atomic systems with positrons.

Mira: And this is basically giving us the ability to calculate these rates numerically for those systems where all electron's and positron's velocities always exceed some minimal value.

Lev: So they can calculate specific annihilation rates, such as those for HPs or quasi-stable ions, for use in astrophysical or solid-state physics applications.

Conclusion: Kai: To wrap up, what this means for the world is that we've derived an exact QED formula for one-photon annihilation of the bound electron-positron pair in the field of atomic nuclei.

Mira: This formula is a direct link between fundamental theory and these specific atomic physics problems.

Lev: The work provides a tool to calculate physical rates for those systems, and it gives us a closed analytical solution for Gamma(b) 1γ <ref:2405.06720#pg1>.

Kai: And the general case formula, Eq. (forty-three), where Ne-e+ is equal to mc2 n plus nc2 m, lets the AI calculate annihilation rates for arbitrary systems containing 'm' positrons and 'n electrons'.

Mira: It gives us a way to evaluate numerical values of the rates of one-photon annihilation for various atomic systems with positrons.

Lev: The study concludes that these formulas can be applied to evaluate numerical values of the rates of one-photon annihilation for various atomic systems with positrons, and they's provided an exact Quantum Electrodynamics formula for its annihilation rate.

Kai: So we're leaving this paper here, but this work gives us a solid foundation for calculating these physical rates in those specific contexts.

Mira: It’s a good piece of work that connects the fundamental theory to the specific physics of positronium hydrides and quasi-stable triplet states of positron-helium atoms.

Lev: And we're ready for whatever comes next in this research.

Alexei M. Frolov

University of Western Ontario

hep-ph, nucl-th, quant-ph

Submitted: 2024-05-10

Updated: 2026-10-16

Journal ref: Physics Letters A, {\bf 563}, article # 131073 (2025)

DOI: 10.1016/j.physleta.2025.131073

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 74/100

The gist: One-photon annihilation of an electron-positron pair in a heavy atomic nucleus is investigated to derive an exact Quantum Electrodynamics formula for its annihilation rate, which can be used to

Key concepts

One-photon annihilation: This refers to the process where an electron and a positron, which are bound together in an atom or nucleus, annihilate by emitting exactly one photon. The paper uses Quantum Electrodynamics (QED) rules to calculate the exact probability and rate of this specific interaction.
Differential Cross Section: This mathematical expression describes how the probability of annihilation changes depending on the angles at which the emitted photon is detected. It is used to map out the angular distribution of photons, similar to how it describes bremsstrahlung, allowing for detailed analysis of particle interactions.
Annihilation Rate ($\Gamma(b)_{1\gamma}$): This quantity represents how frequently a specific annihilation event occurs per unit time. The paper derives an explicit formula for this rate in atomic units, which is crucial because it allows researchers to predict the actual physical rates for systems like positronium hydrides.
General Few-Body System Formula: The study provides a general formula that accounts for any system containing 'm' positrons and 'n' electrons. This formula calculates the total number of possible triplets ($N_{e^-e^+}$) in such systems, enabling the calculation of annihilation rates for arbitrary atomic configurations.

Summary

One-photon annihilation of an electron-positron pair in a heavy atomic nucleus is investigated to derive an exact Quantum Electrodynamics formula for its annihilation rate, which can be used to describe processes like positronium hydrides and quasi-stable triplet states of positron-helium atoms<ref:2405.06720#pg14>.

Derivation of the Annihilation Rate Formula

The study derives the explicit formula for the annihilation rate of the electron-positron pair in a central, very heavy and positively charged atomic nucleus, denoted as Γ(b)1γ<ref:2405.06720#pg14>. The derivation begins by applying Feynman rules of Quantum Electrodynamics to describe the lowest-order one-photon annihilation process<ref:2405.06720#pg14> and obtaining the S-matrix element in coordinate space<ref:2405.06720#pg14>. The squared scattering amplitude modulus is expressed in a differential form involving the function F¯(p+, p−; k)<ref:2405.06720#pg14> and the unpolarized cross section is given by Eq.(8)<ref:2405.06720#pg14>.

Components of the Cross Section Formula

The final expression for the unpolarized cross section is given by Eq.(8)<ref:2405.06720#pg14>, which involves terms dependent on the positron and electron velocities in respect to the infinitely heavy nucleus, denoted as v+ and v−<ref:2405.06720#pg14>. The formula is written in a manifestly covariant form in four-dimensional space-time, which is a true relativistic invariant<ref:2405.06720#pg14>. The function F˜(p+, p−; k) is explicitly written as Eq.(9)<ref:2405.06720#pg14>, which incorporates spin sums for the positron and electron spinors<ref:2405.06720#pg14>.

Angular Distribution of Photons

The differential cross section formula, Eq.(8), can be used to describe the angular distribution of the emitted photons, which is presented in Eq.(38)<ref:2405.06720#pg14>. This formula is similar to the Bethe-Heitler formula [7], but for one-photon annihilation it contains only two angular variables, namely those of the positron and electron with respect to the photon direction<ref:2405.06720#pg14>. The angular problem is formulated using three 3D-vectors associated with the particles p+, p− and k<ref:2405.06720#pg14>.

Low-Energy Limit and Rate Calculation

The low-energy limit of the cross section is derived by using standard replacements in Eq.(38), such as E± ⇒ m0, ω ⇒ 2m0, and q ⇒ m0<ref:2405.06720#pg14>. In this approximation, only one term (of four) remains, resulting in the non-relativistic cross section dσnr<ref:2405.06720#pg14>. The corresponding annihilation rate Γ(b)1γ is given by Eq.(41)<ref:2405.06720#pg14>, which takes the form of a delta-function expectation value involving the three-particle delta-function<ref:2405.06720#pg14>. The final rate formula in atomic units is given by Eq.(42)<ref:2405.06720#pg14>.

General Case and Applications

The general case for a system with m positrons and n electrons is given by Eq.(43)<ref:2405.06720#pg14>, where Ne−e+ is the total number of triplets that can be formed from bound electrons and positrons in a given atomic system<ref:2405.06720#pg14>. The closed formula for the corresponding annihilation rate Γ(a)1γ has been derived in [2] and is written in the form of Eq.(44)<ref:2405.06720#pg14>. This study concludes that the formulas can be applied to evaluate numerical values of the rates of one-photon annihilation for various atomic systems with positrons<ref:2405.06720#pg14>. The researchers note that these tasks have no direct relation with quantum electrodynamics, and they stop their analysis here<ref:2405.06720#pg14>.

The gist

One-photon annihilation of the electron-positron pair in a heavy atomic nucleus is investigated to derive an exact Quantum Electrodynamics formula for its annihilation rate, which can be used to describe processes like positronium hydrides and quasi-stable triplet states of positron-helium atoms<ref:2405.06720#pg14>.

How it works

The derivation starts by applying Feynman rules of Quantum Electrodynamics to describe the lowest-order one-photon annihilation process in momentum space<ref:2405.06720#pg14>.
The squared scattering amplitude modulus is expressed in a differential form involving the function F¯(p+, p−; k)<ref:2405.06720#pg14> and the unpolarized cross section is given by Eq.(8)<ref:2405.06720#pg14>.
The final expression for the unpolarized cross section is given by Eq.(8)<ref:2405.06720#pg14>, which involves terms dependent on the positron and electron velocities in respect to the infinitely heavy nucleus, denoted as v+ and v−<ref:2405.06720#pg14>.
The angular distribution of photons is described by Eq.(38)<ref:2405.06720#pg14>, which is similar to the Bethe-Heitler formula [7], but for one-photon annihilation it contains only two angular variables<ref:2405.06720#pg14>.
The low-energy limit of the cross section is derived by using standard replacements in Eq.(38), such as E± ⇒ m0, ω ⇒ 2m0, and q ⇒ m0<ref:2405.06720#pg14>.
The corresponding annihilation rate Γ(b)1γ is given by Eq.(41)<ref:2405.06720#pg14>, which takes the form of a delta-function expectation value involving the three-particle delta-function<ref:2405.06720#pg14>.
The final rate formula in atomic units is given by Eq.(42)<ref:2405.06720#pg14>.

Key Formulas and Results

The final expression for the differential cross section is given by Eq.(35)<ref:2405.06720#pg14>, which takes a complex form involving the function F(p+, p−; k)<ref:2405.06720#pg14>. The low-energy limit of the cross section is described by Eq.(40)<ref:2405.06720#pg14>, which is written in the ‘velocity’ form<ref:2405.06720#pg14>. The final annihilation rate formula in atomic units is given by Eq.(42)<ref:2405.06720#pg14>.

General Case and Applications

Conclusion

The researchers have investigated the process of one-photon annihilation of the electron-positron pair at a very heavy, positively charged (Qe) atomic nucleus and found that their formulas can be applied to evaluate numerical values of the rates of one-photon annihilation for various atomic systems with positrons<ref:2405.06720#pg14>.

Acknowledgments

This manuscript is dedicated to the memory of Joachim Reinhardt who was always interested by similar QED problems<ref:2405.06720#pg14>.

Author Contributions

Alexei M. Frolov: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Writing - original draft (equal); Writing - review & editing (equal)<ref:2405.06720#pg14>.

Improvements for AI systems

textbfConcentration of QED Calculation Accuracy for Bound States: Improved Precision in Atomic Physics Simulations. The derived formula for one-photon annihilation rate can directly be used to describe the actual one-photon annihilation in the ground (bound) states of all positronium hydrides HPs, quasi-stable triplet states of the positron-helium atoms e+[He(23Se)] ions and other systems. This allows AI systems to predict experimental observables for these specific few-body systems with a level of accuracy previously unattainable by approximations in earlier studies [1].
textbfModeling Complex Three-Particle Angular Correlations: Enhanced Experimental Data Interpretation. The final formula for the differential cross section, Eq.(38), plays the same role, which the famous Bethe-Heitler formula [7] plays for description of angular distribution of bremsstrahlung. This enables AI to accurately predict and analyze angular correlations between incoming electron/positron momenta and outgoing photon directions in three-particle processes.
textbfBridging Theoretical Frameworks: Cross-Application Capability across QED Processes. The paper notes, one-photon annihilation of the (e−, e+)−pair at heavy nuclei is one of ten fundamental tasks that can generally be formulated for the processes with two-vertex Feynman diagrams. This suggests an AI system could use the derived framework to predict or simulate related inverse QED processes like bremsstrahlung by simply replacing the outgoing electron with an incoming positron.
textbfPredictive Capability for Bound State Annihilation Rates: Accurate Calculation of Physical Rates. The final expression for the annihilation rate, the explicit formula for the Γ(b)1γ rate (in sec−1) takes the form, provides a closed analytical solution applicable to systems where all electron’s and positron’s velocities always exceed some minimal value. This allows AI to calculate specific annihilation rates, such as those for HPs or quasi-stable ions, for use in astrophysical or solid-state physics applications.
textbfQuantifying System Complexity: Accurate Calculation of Annihilation Rates in General Few-Body Systems. The general case formula, Eq.(43), in general, for such a system with m positrons and n electrons one finds Ne−e+ = m C2 n + n C2 m, allows AI to calculate annihilation rates for arbitrary systems containing 'm' positrons and 'n' electrons. This capability extends the model to new experimental targets not covered by previous specific analyses.

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Key concepts

Summary

Improvements for AI systems

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