Abstract
We have developed a coupled-rearrangement-channel method allowing the rigorous non-adiabatic treatment of the multi-channel scattering problem for four particles. We present the study of the binding, resonant and collisional properties of the \(\bar {H}-Ps\) system with the total angular momentum J = 0+ (singlet positronic configuration). The binding energy, the life-times of the resonant states and the collisional cross sections are calculated and discussed. We present the preliminary cross sections for the elastic and inelastic \(\bar {H}-Ps\) scattering, notably for the excitation of Ps and for the rearrangement reaction producing the \(\bar {H}^{+}\) ions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Perez, P., Rosowsky, A.: A new path toward gravity experiments with antihydrogen. Nucl. Instr. Meth. Phys. Res. A 545, 20 (2005)
Chardin, G., et al.: GBAR, Proposal to measure the Gravitational Behaviour of Antihydrogen at Rest, CERN-SPSC-P-342, 30/09/2011. Technical report (2011)
Indelicato, P., et al.: The GBAR project, or how does antimatter fall?. Hyp. Int. 228, 141 (2014)
Perez, P., et al.: The GBAR antimatter gravity experiment. Hyp. Int. 233, 21 (2015)
Yan, Z.-C., Ho, Y.K.: Ground state and S-wave autodissociating resonant states of positronium hydride. Phys. Rev. A 59, 2697 (1999)
Ivanov, I. A., Mitroy, J., Varga, K.: Positronium-hydrogen scattering using the stochastic variational method. Phys. Rev. A 65, 032703 (2002)
Mitroy, J.: Energy and expectation values of the PsH system. Phys. Rev. A 73, 054502 (2006)
Bubin, S., Varga, K.: Ground-state energy and relativistic corrections for positronium hydride. Phys. Rev. A 84, 012509 (2011)
Biswas, P.K.: Effect of H− ion formation on positronium-hydrogen elastic scattering. J. Phys. B 34, 4831 (2001)
Blackwood, J.E., McAlinden, M.T., Walters, H.R.J.: Positronium scattering by atomic hydrogen with inclusion of target excitation channels. Phys. Rev. A 65, 032517 (2002)
Comini, P., et al.: \(\bar {\mathrm {H}}^+\) production from collisions between positronium and keV antiprotons for GBAR. Hyp. Int. 228, 159 (2014)
Woods, D., Ward, S.J., Van Reeth, P.: Detailed investigation of low-energy positronium-hydrogen scattering. Phys. Rev. A 92, 022713 (2015)
Hiyama, E., Kino, Y., Kamimura, M.: Gaussian expansion method for few-body systems. Prog. Theor. Exp. Phys. 51, 223 (2003)
Hiyama, E.: Gaussian expansion method for few-body systems and its applications to atomic and nuclear physics. Progr. Theor. Exp. Phys. 01A204, 1–36 (2012)
Kamimura, M.: Nonadiabatic coupled-rearrangement-channel approach to muonic molecules. Phys. Rev. A 38, 621 (1988)
Kino, Y., Kamimura, M.: Non-adiabatic calculation of muonic atom-nucleus collisions. Hyp. Int. 82, 45 (1993)
Taylor, J.R.: Scattering Theory. Wiley, Hoboken (1972)
Messiah, A.: Quantum Mechanics. North-Holland (1970)
Piszczatowski, K., Voronin, A., Froelich, P.: Nonadiabatic treatment of hydrogen-antihydrogen collisions. Phys. Rev. A 89, 062703 (2014)
Drachman, R.J., Houston, K.: Positronium-hydrogen elastic scattering. Phys. Rev. A 12, 885 (1975)
Zhao, J., Corless, R.M.: Compact finite difference method for integro-differential equations. App. Math. Comput. 177, 271 (2006)
Acknowledgments
We gratefully acknowledge the financial support from the Japan Society for the Promotion of Science (JSPS) and from the Swedish Research Council. T. Y. was financially supported by Grant-in-Aid for JSPS Research Fellow Grant Number JP16J02658 and JSPS Overseas Challenge Program for Young Researchers. Y. K. was supported by JSPS KAKENHI Grant Number JP17K05592.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the Topical Collection on Proceedings of the 13th International Conference on Low Energy Antiproton Physics (LEAP 2018) Paris, France, 12-16 March 2018
Edited by Paul Indelicato, Dirk van der Werf and Yves Sacquin
Appendix
Appendix
We display below the generic form of the coupled integro-differential equations for channel functions, to show the appearance of the non-local potentials and coupling of the rearrangement channels arising from the permutation of the two positrons. For simplicity and brevity we show the case of the elastic \(\bar {H}-Ps\) scattering. The case involving inelastic and rearrangement channels is a non straightforward generalization of the equations below.
The channel functions are solutions of the following equations:
where \(k_{\alpha }^{2} = 2\mu _{\alpha } \left (E - \epsilon _{\overline {\mathrm {H}}(1s)} - \epsilon _{\text {Ps}(1s)}\right )\), α specifies a coordinate set (see Fig. 1) and \(\bar \alpha \) specifies a coordinate set generated by permutation of two positrons; η = 1 is used for the space-symmetric and η = − 1 for the space-antisymmetric case, respectivly. In the following formulation, we use η = 1. In the above we have introduced the following non-local potentials:
where \(\mathcal {J}_{\alpha \gamma }\) is the Jacobian of the coordinate transformation from the set {rα,qα,Rα} to the set {qα,Rγ,Rα}, defined through the relationship \(\mathrm {d}\mathbf {r}_{\alpha } \mathrm {d}\mathbf {q}_{\alpha } \mathrm {d}\hat {\mathbf {R}}_{\alpha } \) \(=\mathcal {J}_{\alpha \gamma } \mathrm {d}\mathbf {q}_{\alpha } \mathrm {d}\mathbf {R}_{\gamma } \mathrm {d}\hat {\mathbf {R}}_{\alpha }\).
Furthermore,
where Vαυ(Rα) is defined as
There is also a local potential
with the interaction \( V_{\text {int}}^{(\alpha )}\) defined by the relations
where T(Rα) is the operator for the kinetic energy of relative motion along Rα, \(h_{\bar {H}}\) the Hamiltonian for \(\bar {H}\), hPs the Hamiltonian for Ps, so that Vint is the Coulomb interaction between \(\bar {H}\) and Ps.
The appearance of non-local potentials in the master equation (10) is of uttermost importance, since e.g. for the elastic scattering the local potential \(\tilde V_{\text {int}}^{(\alpha )} (R_{\alpha })\) (15) describing the effective \(\bar {H}-Ps\) interaction vanishes, due to the particular symmetry of Ps, whose center of mass coincides with its geometrical center.
The above master equation (10) is solved using the compact finite difference method (CFDM) [19, 21]. The solution \(\chi _{J}^{(JM)}\) is calculated as a vector on the grid
where r1, r2, ⋯, rN are the non-uniformly distributed grid points extending from the short range to the long range that in our calculations was typically outstretched to more than 100 a.u..
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Froelich, P., Yamashita, T., Kino, Y. et al. Four-body treatment of the antihydrogen-positronium system: binding, structure, resonant states and collisions. Hyperfine Interact 240, 46 (2019). https://doi.org/10.1007/s10751-019-1572-0
Published:
DOI: https://doi.org/10.1007/s10751-019-1572-0