R. A. Sultanov 1,2) and S.K. Adhikari 2) 1) SCSU, St. Cloud, Minnesota, USA

1. Protonium Pn = (p+p-)α Formation in a Collision Between Slow Anti-Proton (p-) and Muonic Hydrogen: p- + Hμ => (p+ p-)α + μ- R. A. Sultanov1,2) and S.K. Adhikari2) 1)SCSU, St. Cloud, Minnesota, USA 2)IFT-UNESP, SAO PAULO, SP, BRAZIL Critical Stability Workshop , Santos, SP, Brazil October 12-17, 2014

2. _- Slow p physics: • ANTIMATTER: p-, Ps, Ħ, Pn, Ħμ • Theory and experimental developments:ATRAP, ASACUSA, ALPHA … CERN • Anti-hydrogen, muonic antihydrogen, Ps atoms, slow antiproton capture, etc. • Theory: 3- and 4-body Coulomb and Coulomb + nuclear quantum systems: collisions, bound states, etc. • Few-Body, Faddeev type computations. • Test of different numerical methods and procedures: discretization & matrix methods. • One can (should) compare theoretical & available experimental results.

3. LOW ENERGY ANTIPROTON( pbar = p- = p ) and Ħ PHYSICS _- • Collision experiments with antiprotons: H2 ionization with pbar: significant deviation between experimental results and theory. • Antiprotonic helium experiments: CPT & fundamental constants. Temp. T < 1 K! • pbar + He+ experiments and theory: pbar+4He+ and pbar+3He+ - ATOMCULE! at ASACUSATEAM(RIKEN, JAPAN).

4. ASACUSA= ATOMIC SPECTROSCOPY AND COLLISIONS USING SLOW ANTIPROTONSATRAP= ANTIHYDROGEN TRAPALPHA = Antihydrogen Laser PhysicsApparatus

5. ATHENA= ANTIHYDROGEN PRODUCTION AND PRECISION EXPERIMENTS========================In 2002, it was the first experiment to produce 50,000 low-energy Ħ.Nature 419, 456 (2002).

6. WHY ANTIMATTER? • GOALS: TO PRODUCE COLD Ħ; TO TRAP COLD Ħ; TO COMPARE H and Ħ WITH THE USE OF LASER SPECTR-OSCOPY. • CPT SYMMETRY: Particles and anti-particles have same masses, same mean life, same magnetic moments, and… opposite charges! • ATOM and ANTI-ATOM HAVE SAME STRUCTURE?

7. WHY ANTIHYDROGEN: Ħ ? • Spectroscopic measurement on Ħ in comparison with corresponding results for normal H. • This comparison may contribute to the verification OR falsification of the CPT conservation law. • Gravitational measurements of Ħ and H and ….. • Antimattermay even be related to the dark matterproblem in astrophysics.

8. WHY PROTONIUM: Pn ? _- • Pn= p-p+ = p p+ itself is a very interesting two-heavy particle system in view of the interplay between Coulomb and nuclear forces between the particles: p- & p+ • Laser spectroscopy of longer-living Pn’s in vacuum. • Pn formation & annihilation is related to charmonium production - a hydrogen-like atom: a bound state of a c quark and c anti-quark, i.e. cc • The pp+ annihilation can produce cc in the ground or excited states. - _- _- _-

9. _- Pn FORMATION & ANNIHILATION: cc • It would be interesting for future experiments with participation of antiprotons and muons to compute charmonium production (formation) rate from the reaction:p- + Hμ => (p+ p-)α + μ- • It would be interesting to use and check different NN interaction potentials in this reaction. • But first, the 3-charge-particle low energy scattering problem should be solved! • Particles have comparable masses: No adiabatic approximation! _-

10. MUON PHYSICS: Hμ and Ħμ even better ? • In book: “Introductory Muon Science”, 2003: K. Nagamine pointed out: muonic antihydrogen atom might be even better choice to check the CPT law than usual antihydrogen atom, i.e. Ħ. • Therefore, as a first step it would be interesting to compute the formation rates ofĦμ atoms at low and very low energies from ~ 5eV down to ~ 10-4 eV. This is done in this work.

11. WHY MUONIC ANTIHYDROGEN ? “If the CPT violating interaction is short range with an extremely massive exchange boson such an effect can be detected more easily in the case of muonic antihydrogen than in simple antihydrogen. This is because the size of the Ħμ atom is ~207 times smaller than Ħ”.

12. Some references (1), • Gabrielse, G. (ATRAP Collab.) et. al., “Adiabatic Cooling of Antiprotons”, 2011, Phys. Rev. Lett., 106, 073002. • Andresen, G. B. (ALPHA Collab.) et. al., “Evaporative Cooling of Antiprotons to Cryogenic Temperatures”, 2010, Phys. Rev. Lett., 105, 013003. • Pohl R. et. al. “The size of the proton”, 2010, Nature, 466, 213.

15. Some references (2). • Hayano, R.S., Hori M., Horvath, D., “Antiprotonic helium and CPT invariance”, 2007, Rep. Prog. Phys., 70, 1995. • Lauss, B. “Fundamental measurements with muons ”, 2009, N. P. A 827, 401c. • Klempt, E., Batty, C., Richard, J.-M. “The antinucleon-nucleon interaction at low energy: annihilation dynamics”, 2005, Phys. Rep. 413, 197. • Klempt, E., Bradamante,F.,Martin, A.,Richard,J.-M. “Antinucleon-nucleon interaction at low energy: Scattering and protonium.”, 2002, Phys. Rep. 368, 119.

16. In this work: • A bound state of a proton, p+, and its counterpart antiproton, p-, is a Protonium atom Pn = (p ̄ p). • The following three-charge-particle reaction: p ̄ + (p+μ−)1s → (p+p̄ )α + μ− is considered in this work, where μ− is a muon (mμ= 207me). • This is a 3-body Coulomb problem with heavy particles. • The cross sections and rates of the Pn formation reaction are computed in the framework of a few-body approach.

17. ALSO in this work: • Two 3-charge-particle collisions with participation of slow antiprotons and: 1) positronium atoms, Ps, & 2) muonic muonium atoms (“True Muonium”) is also considered: • Instead of 3 coupled Faddeev equations, just 2 coupled Faddeev-Hahn-type equations are used. • Solution: a modified close coupling expansion approach and K-matrix formalism (unitarity).

18. THREE-BODY SYSTEMS

19. THREE-BODY SYSTEMS

20. THREE-BODY SYSTEMS

21. THREE-BODY SYSTEMS COULOMB THREE-BODY SYSTEMS AT LOW ENERGIES, that is BEFORE THE 3-BODY BREAK-UP SRESHOLD. THERE ARE ONLY TWO SPATIAL CONFIGURATIONS, that is NOT THREE!

22. CONFIGURATION TRIANGLE (123)

23. “Usual” PROTONIUM Pn=(p-p+) FORMATION REACTIONS: n~30

24. BUT WITH MUONS TOO:

25. Energy levels in p- + H

26. Energy levels in p- + Hμ

27. COULOMB THREE-BODY SYSTEMS

28. TWO (NOT 3) SPATIAL CONFIGURATIONS

29. JACOBI COORDINATES, MASSES and EQUATIONS 3-body systemswith arbitrary masses: the 3-body W.F. is represented as:

30. Elastic – inelastic channels and rearrangement/transfer channels: p- + (μ+μ-)1s→ p- +(μ+μ-)1s (+2s,+2p,+3s…) → μ- + (Ħμ)1s (+2s,+2p,+3s…) i.e. CLOSED channels p ̄ +(p+μ−)1s → p ̄ + (p+μ−)1s(+2s,+2p,+3s…) → μ− + (p+p̄ )1s → μ− + (p+p̄ )2s → μ− + (p+p̄ )2p

31. FADDEEV-HAHN-type EQUATIONS A set of two coupled equations:

32. 1).Y. Hahn, Phys. Rev. 169, 794 (1968). 2). The constructed coupled equations satisfy the Schrödinger Eq. exactly. 3). The Faddeev decomposition avoids the overcompleteness problems. 4). Two-body subsystems are treated in an equivalent way.

33. 5). The correct asymptotics: guaranteed. 6). It simplifies the solution procedure and provides the correct asymptotic behavior of the solution below the 3-body break-up threshold. 7). FH-type equations have the same advantages as the Faddeev equations, because they are formulated for the w.f. components with correct physical asymptotes.

34. ASYMPTOTICS for only two components:

39. Vectors & Angles of the 3-Body System (123)

40. K-matrix formalism:K12 = K21or K12 / K21 = 1.0 !arXiv:1304.2434v2 or J. Phys. B 46 (2013) 215204

42. Discretization Procedure: