Eur. Phys. J. D (2015) THE EUROPEAN DOI: 10.1140/epjd/e2015-60427-6 PHYSICAL JOURNAL D

Regular Article

Theoretical study of true- 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−) are investigated theoretically 1) for negative- (μ−) collisions with a muonium (Mu = μ+e−) (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 emission plays an important role, is a semiclassical (SC) method, in which the ra- dial distance between the negative and positive 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 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 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 on nuclei, -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 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 , 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. In + − the present paper, an additional CTMC calculation is car- form μ μ in the (N,L) state is defined by: ried out also for μ− + μ+e−, though an electron is not a    ε2 dP J  N+1/2 dP J dN heavy particle. P J Ll dε m Ll , NL = dε = R dε N3 (9) l ε1 l N−1/2

where ε1,2 are the electron energies corresponding to 2 Semiclassical method 1 N = N ± 2 through the conservation of energy [23] The SC method is applied to the μ+μ− formation and the mR mr Etot = ε − = E − , (10) ionization in μ− + μ+e− collisions. The time-dependent 2N2 2 Page 3 of 9 with E being the center-of-mass (CM) collision energy of atom having the principal quantum number n is given − + − 2 μ + μ e . The cross section for the formation of the by −mq/(2n ). In reference [37], the initial angular mo- (N,L) state is given by: mentum of the target was given as a fixed value. In the π  present study, no special value is specified for the initial SC J j σμ+μ− (N,L)= (2J +1)PNL. (11) angular momentum of BC, and a uniform distribution is mRE 2 2 J assumed for j [38], which has the range

2 2 2 The (N,L)-state distribution is defined as: 0 ≤ j ≤ jmax = n . (17)

SC SC SC + − F (N,L)=σμ+μ− (N,L) σμ+μ− , (12) The cross sections for μ μ formation and dissociation (or ionization) are calculated from where  SC SC σμ+μ− = σμ+μ− (N,L) (13) CTMC 2 Iμ+μ− σ + − = πb , (18) NL μ μ max I is the total formation cross section. If the electron energy Idis(ion) σCTMC = πb2 , (19) is 0 <ε

where  1/2 3 Classical-trajectory Monte Carlo method mR N = . (23) 2|Eμ+μ− | For the system A + BC = μ+ + pμ− or μ− + e−μ+,the CTMC method is employed to investigate the μ+μ− for- Then, the (N,L)-state distribution is calculated by: mation. Let (Q, q) denote the Jacobi coordinates associ- ated with the A + BC arrangement: Q being the position CTMC Iμ+μ− (N,L) q F (N,L)= , (24) vector of A from BC and being that of C from B. The Iμ+μ− classical equations of motion are

I + − N,L dQ P dP ∂U where μ μ ( ) is the number of trajectories resulting = , = − , (15) in the formation of the (N,L) state. dt mQ dt ∂Q The CTMC calculations with I = 5000−15 000 were dq p dp ∂U carried out. The values of bmax were chosen as bmax = = , = − , (16) − + − dt mq dt ∂q 3.0a0−6.64a0 for μ +μ e (n =1),bmax =0.09a0−0.2a0 + − + for μ + pμ (n =2),bmax =0.5a0 − 1.0a0 for μ + − + − where mQ and mq are the corresponding reduced masses, pμ (n =5),bmax =2.5a0 − 3.5a0 for μ + pμ (n = 10), + − and U is the interaction. A set of equations (15)and(16) bmax =4.0a0 − 10.0a0 for μ + pμ (n = 15), and + − were solved by a Runge-Kutta method. The numerical bmax =6.0a0 − 15.0a0 for μ + pμ (n = 20). To en- details of the CTMC calculation were described in ref- sure the numerical reliability, the CTMC calculations were erence [37]. The initial energy of the hydrogenic BC tested for pμ− formation in μ− + H and for pp¯ formation Page 4 of 9 inp ¯ + H with a somewhat smaller number of trajecto- 10 ries (I = 1000). The deviations in the cross section from )

− 2 the previous CTMC calculation of Cohen (μ +H)[30] 8

. E . . E . cm are 2 5% at =109eVand048% at =163eV,and Formation those from the CTMC calculation of Schultz (¯p +H)[31] -16 6 are 3.5% at E =8.16 eV and 1.8% at E =13.6eV.These SC deviations are within the statistical errors. CTMC 4

4Results 2 Cross section (10 4.1 µ− + µ+e− collisions 0 2 4 6 8 10 12 μ− μ+e− In this section, the results are presented for the + CM collision energy E (eV) collision system. Using the SC method, the reaction cross sections and the μ+μ−-state distributions were calculated 3.0 E − at collision energies =1 100 eV. Throughout this pa- ) 2 2.5 SC (formation) per, the collision energy E is represented in the CM frame. SC (ionization) The CTMC method was also applied to this system, and cm CTMC (formation) -16 2.0 - the comparison was made between the two methods. CTMC (formation in m + H) μ+μ− The -formation and ionization cross sections in 1.5 the low-, intermediate-, and high-energy regions are plot- E ted as a function of the CM energy together in Figure 1. 1.0 Since the μ+μ− formation is exoergic, its cross section in- creases with decreasing E even at low energies. The ion- 0.5 ization channel becomes open only at energies above the Cross section (10 threshold E = mr/2. For all the reaction channels, the 0.0 CTMC cross sections are overall much larger than the SC 10 15 20 25 30 ones. The same holds true with regard to the μ− +H CM collision energy E (eV) andp ¯ + H systems [21,40]. As was expected, the classical 2.5 treatment of the electron would be deficient. However, if one aims at observing only the qualitative behavior of the ) 2 cross sections, it can be said that the CTMC method is 2.0 cm

acceptable. -16 As seen in the middle part of Figure 1,theformation 1.5 and ionization channels are competitive, and their cross ΔE sections change places rapidly within a narrow range ( ) 1.0 of energies above the ionization threshold E = mr/2. The Ionization energy range ΔE is equal to the highest energy ε = ε0 that can be carried away by the emitted electrons [23]. The 0.5 SC present SC and CTMC calculations offer ΔE ∼ 15 eV. For Cross section (10 CTMC comparison, the middle part of Figure 1 also includes the 0.0 CTMC cross sections calculated by Cohen for pμ− forma- 20 40 60 80 100 tion in μ− +H[30]. The CTMC cross section of μ− +His CM collision energy E (eV) − smaller than the present CTMC result of μ +Mu, and the μ+μ− − Fig. 1. Cross sections for formation and ionization in ΔE μ − + − energy range seems to be slightly smaller for +H. μ + μ e collisions, calculated by the SC and CTMC meth- Another previous CTMC study [41] shows that the energy ods. CTMC cross sections for pμ− formation in μ− +H ob- range for pp¯ formation inp ¯ +HisΔE ∼ 5eV,whichis − tained by Cohen [30] are also plotted in the middle figure. The much smaller than that of μ + H. The reduced-mass ra- collision energy E is represented in the CM frame. tio ofp ¯ +H,μ− +Handμ− +Muisabout9:2:1.For the system of a negatively-charged particle and a hydro- gen isotope, it may be expected that the formation can with τ being the collision time. Since this relation implies take place at higher energies above the ionization thresh- that ε0 ∝ 1/τ, one can obtain (for the same collision en- 1/2 old (i.e., ΔE becomes larger) if the reduced mass of the ergy E) ΔE  ε0 ∝ (reduced mass) , which seems to be collision system is getting smaller. Since the ionization en- in accord with the present finding. ergy practically vanishes at small distances (R)inthese The μ+μ− formation in μ− +Muoffersa remark- systems [22,36,42], the effective transition energy in the able resemblance to associative electron detachment in the electron emission may be just equal to the electron ki- H− + H system [44], i.e., netic energy ε. Then, the so-called Massey criterion [43] − − suggests that the emitted electrons mostly have ε ∼ 1/τ, H +H→ H2 + e . (25) Page 5 of 9

Also for this reaction, a rapid decrease in the cross section 15 was observed at collision energies just above the detach- 14 E = 5 eV ment threshold [44]. In this case, the break-up detach- ment (→ H+H+e−) competes with the reaction (25). 13 An isotope effect was investigated by comparing the cross 12 sections between the H− +HandD− + D systems [45]: 0.08

Although no notable isotope effect was found, the en- N 11 ergy range corresponding to the present ΔE seemed to 0.06 − 10 be slightly (but non-negligibly) smaller for D +Dthan 0.04 − 9 for H + H. The origin of the only slight difference may 0.02 be the same as that for the μ− +Muand μ− + H systems: 8 The reduced-mass ratio between the isotope systems is ∼ 2 0.00 for the both cases of the associative detachment (D− +D 7 and H− + H) and the present type of formation reaction 0 5 10 15 (μ− +Handμ− +Mu). L Figure 2 shows the state distributions F (N,L)calcu- 22 lated by the SC method at CM collision energies E = E = 10 eV 5, 10, and 15 eV. The distribution is visualized as a bright- 20 ness scale image. At first glance, an image feature changes drastically with E. Generally, the distribution is shifted 18 to higher quantum numbers with increasing E. The pro-

N 16 duced states can have very high N up to ∞, but are lim- 0.020 ited to somewhat low L. Interestingly, the image feature 0.015 E 14 at = 10 eV is very similar to that of the QM calculation 0.010 for μ−+H at E =10eV[46] except for the related range of 12 0.005 quantum numbers: Since the reduced mass of pμ− is about twice as large as that of μ+μ−, it is natural that different 10 0.000 values of quantum numbers are involved in μ− +Muand μ− +H. 0 5 10 15 20 L Figure 3 shows the state distributions defined by: 70   E = 15 eV 0.004 F (N)= F (N,L),F(L)= F (N,L), (26) 60 0.003 L N 50 0.002 0.001 and compares the results between the SC and CTMC 40 0.000 methods. The CTMC method is in good agreement with N the SC method for the L distribution F (L), and offers 30 a much narrower distribution for F (N). In Figure 4,the average quantum numbers defined by: 20

  10 N¯ = NF(N), L¯ = LF(L) (27) N L 0 5 10 15 20 L are plotted as a function of E.ForbothL¯ and N¯, good Fig. 2. (N,L)-state distributions F (N,L)inμ− + μ+e− col- agreement can be obtained between the two methods. As lisions at the CM collision energies E =5, 10, and 15 eV, was sometimes assumed in previous studies, one might calculated by the SC method. expect that the most probable state of produced μ+μ− were estimated by the condition of energy matching, i.e., 4.2 µ+ + pµ− collisions N ∼ mR/mr  10. Figure 4 shows that this is the case only at very-low collision energies. The average quan- For the μ+ + pμ−(n) collision system, the CTMC method N¯ E + − tum number actually has significant dependence, and was used to investigate the μ μ formation and disso- E becomes very large at high . Instead, it is empirically ciation processes. The calculations were carried out for L¯ true that the average quantum number takes a value several initial states n between 2 and 20. Related energies of ∼ mR/mr at high collision energies. Thus, the simple at which the reaction processes prominently take place evaluation formula mR/mr should be used as the most differ depending on the initial state n. A rough indica- probable value of N only if E is very low and otherwise tion of such energies is given by the dissociation energy of − 2 3 2 as that of L. pμ (n), i.e., Dn = mq/(2n )=2.53 × 10 /n eV. Page 6 of 9

0.25 2.0 E = 10 eV ) 2 0.20 cm

F(N) 1.5 SC -19 n = 2 CTMC 0.15 F(L) Formation SC 1.0 Dissociation CTMC 0.10 Distribution 0.5 0.05 Cross section (10

0.00 0.0 0 1000 2000 3000 4000 0 5 10 15 20 25 E Quantum number CM collision energy (eV) 8 Fig. 3. N-andL-state distributions F (N)andF (L)in − + − μ μ e E )

+ collisions at the CM collision energy =10eV, 2 calculated by the SC and CTMC methods. cm

6

-18 n = 5 35 Formation 4 Dissociation 30 Average N SC 25 CTMC Average L 2 20 SC CTMC Cross section (10 15 0 0 100 200 300 400 500 600 10 CM collision energy E (eV) 5 Average quantum number 6

0 ) 2 2 4 6 8 10 12 14 5 cm CM collision energy E (eV) n = 15 -16 4 Fig. 4. Average quantum numbers N¯ and L¯ in μ− +μ+e− colli- Formation Dissociation sions, calculated by the SC and CTMC methods. The collision 3 energy E is represented in the CM frame. 2

1

Figure 5 shows the reaction cross sections as a function Cross section (10 of the CM collision energy E for the initial states n =2, 5, and 15. With increasing n, the cross sections become large, 0 and the related energies become low. Since the formation 0 20 40 60 80 reaction is exoergic for n ≥ 2, its cross section increases CM collision energy E (eV) E + − with decreasing . The energy dependence is overall sim- Fig. 5. Cross sections for μ μ formation and dissociation in ilar for all the cases of n ≥ 2, though a notable shoulder μ+ + pμ−(n)withn =2, 5, and 15, calculated by the CTMC structure appears at E ∼ Dn for n = 2. In contrast to the method. The collision energy E is represented in the CM frame. μ− + μ+e− system, it turns out that the formation can occur even at high energies much above the dissociation threshold (i.e., E Dn). Figure 6 shows the N-state distribution F (N)for The similarity in the E dependence observed in Fig- the initial states n =2, 5, and 15. It is seen that the ure 5 suggests that the cross sections for different n might μ+μ− states with higher N can be produced as E in- be scaled to each other. For hydrogenic atoms, the av- creases. However, the position N = N0 of the distri- erage radius and the binding energy both have a scale 2 bution peak remains mostly unchanged with E.Inthe factor n .InFigure7, accordingly the scaled cross sec- + − 4 4 μ + pμ system, the condition of energy matching offers tions σμ+μ− /n and σdis/n are plotted as a function 2 N ∼ n mQ/mq =0.746n, which is always close to N0 of the scaled energy n E for various initial states of regardless of the values of n and E. This result is quite n =2−20. It seems that the plotted data are mostly ap- different from that of the μ− + μ+e− system. proximated by a single smooth curve, except for the data Page 7 of 9

0.7 1.2 0.6 Formation ) 1.0

n = 2 2 0.5 cm 0.8 n = 2 E = 400 eV n = 5 ) 0.4 E = 1000 eV -20

N n = 10 ( E = 2000 eV 0.6

F n = 15

0.3 (10 4 n = 20 n 0.4 0.2 / s

0.1 0.2

0.0 0.0 0 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 2 4 N n E (10 eV) 0.6 3.5

0.5 3.0

n = 5 ) 0.4 2 2.5 cm

E = 100 eV ) 2.0 E = 200 eV -21 N 0.3 ( E = 300 eV Dissociation F 1.5 (10 0.2 4 n n = 2

/ 1.0 n = 5

s n 0.1 = 10 0.5 n = 15 n = 20 0.0 0.0 0 2 4 6 8 10 12 14 0.0 0.5 1.0 1.5 2.0 N 2 4 n E (10 eV)

0.20 4 Fig. 7. Scaled formation cross sections σµ+µ− /n and scaled 4 2 dissociation cross sections σdis/n plotted as a function of n E n = 15 μ+ pμ− n n , , , 0.15 in + ( )with =2 5 10 15, and 20, calculated by the E = 10 eV CTMC method. The collision energy E is represented in the E = 20 eV ) CM frame.

N E = 30 eV ( 0.10 F by A =1.09,B =1.79 × 10−4 if 200 ≤ x<4000, and A =1.90,B =2.92 × 10−4 if x ≥ 4000. 0.05

0.00 5 Summary and discussion 0 5 10 15 20 25 30 N The present paper has investigated the reaction processes N F N μ+ pμ− n n Fig. 6. -state distributions ( )in + ( )with = of true-muonium μ+μ− formationindifferenttwotypes , 2 5, and 15, calculated by the CTMC method. of atomic collisions: 1) between a negative muon μ− and a muonium atom μ+e− and 2) between a positive muon μ+ and a muonic hydrogen atom pμ−. The theoretical calcu- of the n = 2 formation cross sections just below n2E ∼ lations have been carried outwithuseoftheSCmethod 2 3 n Dn =2.53 × 10 eV. Analogous n-scaling laws can for describing the dynamics dominated by electron emis- − + − be established for the cross sections of collision processes sion in the μ + μ e system, and with use of the CTMC involving highly-excited Rydberg atoms [47–51]. By us- method for describing the muon exchange dynamics in the + − ing the scaling law found in the present study, the re- μ + pμ system. action cross sections of the μ+ + pμ− collisions can be It has been found that the μ+μ− formation cannot take easily estimated for any initial state n ≥ 2 and any en- place in the μ− + μ+e− system unless E<30 eV. Hence, ergy E. For reference, the scaled formation cross section when one tries to produce the μ+μ− atoms in collision ex- 4 −20 2 − + − − y = σμ+μ− /n in units of 10 cm can be roughly evalu- periments of μ +μ e , it is absolutely necessary that μ ated by a simple exponential fitting y = A exp(−Bx)with beams available in experiments [16–18] contain a sufficient x = n2E in units of eV, where the coefficients are given amount of <30 eV energy components (in the CM frame). Page 8 of 9

In contrast to the μ− + μ+e− system, the μ+μ− for- References mation has been found to be able to take place at much + − higher energies in the μ + pμ system. This finding may 1.D.Owen,W.W.Repko,Phys.Rev.A5, 1570 (1972) be an encouraging result for experimentalists. The essen- 2. J. Malenfant, Phys. Rev. D 36, 863 (1987) tial requirement for the practical use of the μ+ + pμ− − 3. S.J. Brodsky, R.F. Lebed, Phys. Rev. Lett. 102, 213401 collisions is that the pμ atoms in excited (n ≥ 2) states (2009) can be extracted and become available as a target. The − 4.U.D.Jentschura,G.Soff,V.G.Ivanov,S.G.Karshenboim, experimental evidence of the existence of long-lived pμ Phys. Rev. A 56, 4483 (1997) atoms in the 2s state [28] may be a favorable outlook for 5. H. Lamm, R.F. Lebed, J. Phys. G 41, 125003 (2014) + − manufacturing the μ μ atoms. It has been established 6. R. Pohl, A. Antognini, F. Nez, F.D. Amaro, F. Biraben, + − conveniently for the μ + pμ system that the formation J.M.R. 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