Eur. Phys. J. D (2015) THE EUROPEAN DOI: 10.1140/epjd/e2015-60427-6 PHYSICAL JOURNAL D Regular Article Theoretical study of true-muonium µ+µ− formationinmuon collision processes µ− + µ+e− and µ+ + pµ− True-muonium µ+µ− formation in µ− + µ+e− and µ+ + pµ− Kazuhiro Sakimotoa Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan Received 21 July 2015 / Received in final form 30 September 2015 Published online (Inserted Later) – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2015 Abstract. Formation processes of true muonium (μ+μ−) atoms are investigated theoretically 1) for negative-muon (μ−) collisions with a muonium (Mu = μ+e−) atom (i.e., μ− + μ+e− → μ+μ− + e−) and 2) for positive-muon (μ+) collisions with a muonic hydrogen (pμ−) atom (i.e., μ+ +pμ− → μ+μ− +p). The calculations are carried out by using two different types of approaches: one, applied to the μ− + e−μ+ system where electron emission plays an important role, is a semiclassical (SC) method, in which the ra- dial distance between the negative and positive muons is treated as a classical variable, and the remaining degrees of freedom are described by quantum mechanics. The other is a classical-trajectory Monte Carlo (CTMC) method, in which all the degrees of freedom are described by classical mechanics. The formation and ionization (or dissociation) cross sections and the state distributions of produced true-muonium atoms are presented. 1 Introduction Under these circumstances, as part of the Ultra Slow Muon project at J-PARC MUSE [19], it has been sug- + − Atomic bound states of a muon-antimuon pair μ μ are gested that the μ+μ− atoms may be efficiently produced called true muonium (or also known as di-muonium)[1–3]. by using atomic collision processes of a negative muon and Since a muon is 207 times heavier than an electron, the a Mu atom, i.e., μ+μ− s hydrogenic atom has a deep ground (1 ) state en- − + − + − − ergy of −1.41 keV, and a very small Bohr radius of mere μ + μ e → μ μ + e . (1) μ+μ− 512 fm. Due to this compactness, although the In planning such experiments, detailed information on the atom has not yet been observed experimentally, its spec- reaction (1) becomes important needless to say, though it troscopic study is expected to play an important role in pμ− cannot be obtained sufficiently at the present time [20]. aQEDtest[1–5]. A muonic hydrogen atom is also In the sense that the system is composed of an electron an interesting compact system, and recently experiments and two heavy particles with opposite charges, the reac- of measuring the Lamb shift and 2s hyperfine splitting pp p pμ− tion (1) is closely analogous to ¯ formation in ¯ +H(or of have become feasible [6,7]. However, in compar- pμ− formation in μ− + H), for which detailed theoretical ing between theory and experiment, one must consider a studies have been already progressed [21–23]. hadronic effect especially due to a finite proton size [6–9]. μ+μ− Although no experimental plan has been arranged so In this respect, the atom is a purely leptonic sys- far to use muon exchange in atomic collisions of a positive tem, and is ideally suited for the QED test. The pos- muon and a muonic hydrogen atom, i.e., sibility of producing μ+μ− atoms (via virtual photons on nuclei, positron-electron annihilation, etc.) in high- μ+ + pμ− → μ+μ− + p, (2) energy collider experiments has been discussed in sev- eral papers [3,10–13]. Unfortunately, it does not seem very this reaction may be also expected to be an efficient μ+μ− likely that the μ+μ− atoms manufactured by high-energy means of producing the atoms. As described be- pμ− colliders will be directly tractable for further performing fore, the atom is really utilized for testing QED [6,7]. μ+ high-precession spectroscopic measurements. Low-energy beams are available by inducing the ion- pμ− Recently, a source of muonium (Mu = μ+e−)atoms ization of the Mu atoms with lasers [15,24]. If the with a high vacuum yield becomes available in low atom is in the ground state (i.e., the principal quan- n temperature conditions [14,15]. Furthermore, many efforts tum number = 1), the reaction (2) is highly endoergic − have been made to obtain low-energy μ− beams [16–18]. ( 1.12 keV), and is of absolutely no use for producing the μ+μ− atoms. On the other hand, if n ≥ 2, the re- a e-mail: [email protected] action (2) becomes exoergic (≥0.774 keV), and can take Page 2 of 9 place even at low collision energies. The pμ− atoms are Schr¨odinger equation in the SC method is given by [21,23] produced after stopping and capture of negative muons ∂ in hydrogen media. It is noteworthy that highly-excited i Ψ JM(Rˆ , r,t)=HΨ˜ JM(Rˆ , r,t), (3) states (n 15) of the pμ− atoms are primarily produced ∂t − in elementary collision processes such as μ +H[22,23]. where (J, M) are the total angular momentum quantum The density of matter for stopping muons is usually so numbers of the μ− + μ+e− system, R =(R, Rˆ )isthe high that the highly-excited states rapidly cascade down position vector of μ− from μ+,andr =(r, ˆr)isthat to lower ones [25–27]. In the absence of collisions with sur- of e− from μ+. Here and in the following, atomic units roundings, however the highly-excited states could remain are used unless otherwise stated. The Hamiltonian H˜ unchanged for a long time. Furthermore, the availability of in equation (3)is: sufficiently long-lived pμ− atoms in the 2s state has been experimentally established [28]. Thus, it is quite conceiv- L˜2 p˜2 − H˜ V, able that the excited states of pμ atoms will be available = 2 + + (4) 2mRR 2mr as a target in the reaction (2). The present paper makes a theoretical study of the where L˜ is the μ+μ− angular momentum operator, p˜ is μ+μ− two types of the -formation processes (1)and(2) the electronic momentum operator, mR and mr are the re- by means of reasonably reliable methods. The ionization duced masses of μ− +μ+ and e− +μ+, respectively, and V → μ+ μ− e− → μ+ μ− p JM ( + + ) and dissociation ( + + )pro- is the interaction. The total wave function Ψ (Rˆ , r,t) cesses are also investigated. Electron emission dominates can be written as [21,23] the reaction dynamics in μ− + μ+e−, and accordingly the calculation for this system is carried out by using a semi- 1/2 JM 1 2J +1 J Jλ classical (SC) method, in which the radial distance be- Ψ (Rˆ , r,t)= D−M−λ(Rˆ )ψ (r,t), + − r 4π tween μ and μ is treated as a classical variable, and λ the other degrees of freedom are described by quantum (5) J mechanics. The SC method was applied top ¯ +H and where DMλ(Rˆ ) is the Wigner rotation matrix element. μ− +H [21,23], which are allied to the present μ− + μ+e− The parameter R is taken, as was done in previous system. The reliability of the SC method was demon- SC studies [21,23], to represent the classical radial mo- strated forp ¯ +H and μ− + H by comparing the results tion governed by the Born-Oppenheimer potential of the with accurate quantum mechanical (QM) results [21,23] μ− + μ+e− system [36]. In the μ− + μ+e− collisions, (for other theoretical methods, see Ref. [22]). One may the μ+e− atom is assumed to be initially in the 1s state. expect that the SC method is appropriate also for the in- The hydrogenic μ+μ− state is identified by (N,L), vestigation of the reaction (1). Although the accurate QM which are the principal and angular momentum quantum method could be applied to the reaction (1), its calcula- numbers. The distribution of the kinetic energy ε and an- tion encounters double-continuum difficulties at energies gular momentum l of emitted electrons for a specified L above the ionization threshold. On the other hand, the can be given by [23] SC method can be employed at energies both below and + − dP J dAJ above the ionization threshold. For the μ + pμ system, Ll 1 J ∗ Ll = Im (ALl) , (6) dε mr dr a classical-trajectory Monte Carlo (CTMC) method is in- r=r0 troduced. The application of the CTMC method to the Coulomb three-body collision problem was examined in where r0 is taken to be sufficiently large, and references [29–31]. The reaction (2) is similar to positron- + − + − J 1 Jl iεt Jλ ium (e e ) formation in e +H or e +H.¯ QM calculations A (r, ε)=√ U e Ylλ ψ (t) dt, (7) Ll π Lλ r have been carried out for this system [32]. However, since 2 λ muons are much heavier than electrons, the application of the QM treatment to the reaction (2)wouldnotbeso with the spherical harmonics Ylλ(ˆr)and e− ¯ easy. CTMC calculations were carried out for the + H 1/2 2L +1 system [33,34], and the results agree well with experimen- U Jl = (L0lλ|Jλ). (8) tal results of the e+ + H system [35]. Since muons can be Lλ 2J +1 regarded as a heavy particle, the CTMC method is ex- μ+ pμ− In the present study, the probability of the μ− capture to pected to work much better for the + system.