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Physical Review A

( e− , e+ )-pair annihilation in the positronium molecule Ps2

Alexei M. Frolov, Sergei I. Kryuchkov, Vedene H. Smith

Published Jun 1, 1995• 38 citations
Publisher

Abstract

The (${\mathit{e}}^{\mathrm{\ensuremath{-}}}$,${\mathit{e}}^{+}$)-pair annihilation in the bound $^{1}$${\mathit{S}}_{0}$ state of the ${\mathrm{Ps}}_{2}$ molecule (${\mathit{e}}^{\mathrm{\ensuremath{-}}}$${\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$${\mathit{e}}^{+}$) is studied. It is shown that the expectation values of the two-, three-, and four-body Dirac \ensuremath{\delta} functions 〈${\mathrm{\ensuremath{\delta}}}_{+\mathrm{\ensuremath{-}}}$〉, 〈${\mathrm{\ensuremath{\delta}}}_{++\mathrm{\ensuremath{-}}}$〉 (\ensuremath{\equiv}〈${\mathrm{\ensuremath{\delta}}}_{+\mathrm{\ensuremath{-}}\mathrm{\ensuremath{-}}}$〉), and 〈${\mathrm{\ensuremath{\delta}}}_{++\mathrm{\ensuremath{-}}\mathrm{\ensuremath{-}}}$〉 determine the photonless, one- and many-photon annihilation rates, i.e., the values of ${\mathrm{\ensuremath{\Gamma}}}_{0\ensuremath{\gamma}}$, ${\mathrm{\ensuremath{\Gamma}}}_{1\ensuremath{\gamma}}$, and ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}\ensuremath{\gamma}}$, where n\ensuremath{\ge}2. The appropriate analytical expressions are presented together with numerical estimates. As expected, the two-photon annihilation process dominates.

DOI
10.1103/physreva.51.4514

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Physical Review A•Mar 21, 2005•15 citations
The (e\ensuremath−{\mathit{e}}^{\mathrm{\ensuremath{-}}}e\ensuremath−,e+{\mathit{e}}^{+}e+)-pair annihilation in the bound ^{1}$${\mathit{S}}_{0} state of the Ps2{\mathrm{Ps}}_{2}Ps2​ molecule ({\mathit{e}}^{\mathrm{\ensuremath{-}}}$${\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$${\mathit{e}}^{+}) is studied. It is shown that the expectation values of the two-, three-, and four-body Dirac \ensuremath{\delta} functions 〈\ensuremathδ+\ensuremath−{\mathrm{\ensuremath{\delta}}}_{+\mathrm{\ensuremath{-}}}\ensuremathδ+\ensuremath−​〉, 〈\ensuremathδ++\ensuremath−{\mathrm{\ensuremath{\delta}}}_{++\mathrm{\ensuremath{-}}}\ensuremathδ++\ensuremath−​〉 (\ensuremath{\equiv}〈\ensuremathδ+\ensuremath−\ensuremath−{\mathrm{\ensuremath{\delta}}}_{+\mathrm{\ensuremath{-}}\mathrm{\ensuremath{-}}}\ensuremathδ+\ensuremath−\ensuremath−​〉), and 〈\ensuremathδ++\ensuremath−\ensuremath−{\mathrm{\ensuremath{\delta}}}_{++\mathrm{\ensuremath{-}}\mathrm{\ensuremath{-}}}\ensuremathδ++\ensuremath−\ensuremath−​〉 determine the photonless, one- and many-photon annihilation rates, i.e., the values of \ensuremathΓ0\ensuremathγ{\mathrm{\ensuremath{\Gamma}}}_{0\ensuremath{\gamma}}\ensuremathΓ0\ensuremathγ​, \ensuremathΓ1\ensuremathγ{\mathrm{\ensuremath{\Gamma}}}_{1\ensuremath{\gamma}}\ensuremathΓ1\ensuremathγ​, and \ensuremathΓn\ensuremathγ{\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}\ensuremath{\gamma}}\ensuremathΓn\ensuremathγ​, where n\ensuremath{\ge}2. The appropriate analytical expressions are presented together with numerical estimates. As expected, the two-photon annihilation process dominates.
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.
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.
Close to converged energies and expectation values for e+Li{e}^{+}\mathrm{Li}e+Li are computed using a ground state wave function consisting of 1200 explicitly correlated Gaussians. The best estimate of the e+Li{e}^{+}\mathrm{Li}e+Li energy was $\ensuremath{-}7.532 895 5$ hartree which has a binding energy of 0.002 482 hartree against dissociation into Ps+Li+\mathrm{Ps}+{\mathrm{Li}}^{+}Ps+Li+. The $2\ensuremath{\gamma}$ annihilation rate for the spin singlet state was $6.996\ifmmode\times\else\texttimes\fi{}{10}^{9}\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$. The annihilation rate for the triplet state, taking into account core annihilation and the $3\ensuremath{\gamma}$ decay, was $9.36\ifmmode\times\else\texttimes\fi{}{10}^{6}\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$.
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.
The $({e}^{\ensuremath{-}},{e}^{+})$-pair annihilation is considered in the positron-helium e+[He(23S)]{e}^{+}[\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)]e+[He(23S)] four-body ion, in which the positron forms a bound state with the helium atom in its lowest triplet state [i.e., in the 23S(L=0)2\phantom{\rule{0.2em}{0ex}}^{3}S\phantom{\rule{0.3em}{0ex}}(L=0)23S(L=0) state]. A number of bound-state properties of this system have been determined for the e+[He(23S)]{e}^{+}[\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)]e+[He(23S)] ions, where the $^{\ensuremath{\infty}}\mathrm{He}$, 4He^{4}\mathrm{He}4He, and 3He^{3}\mathrm{He}3He helium nuclei are considered. Some applications related to the positron annihilation in the e+[He(23S)]{e}^{+}[\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)]e+[He(23S)] ion are also discussed.