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

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Slow p physics: 1.ANTIMATTER: p -, Ps, Ħ, Pn, Ħ μ 2.Theory and experimental developments: ATRAP, ASACUSA, ALPHA … CERN 3.Anti-hydrogen, muonic antihydrogen, Ps atoms, slow antiproton capture, etc. 4.Theory: 3- and 4-body Coulomb and Coulomb + nuclear quantum systems: collisions, bound states, etc. 5.Few-Body, Faddeev type computations. 6.Test of different numerical methods and procedures: discretization & matrix methods. 7.One can (should) compare theoretical & available experimental results. 2 _ -

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LOW ENERGY ANTIPROTON ( pbar = p - = p ) and Ħ PHYSICS Collision experiments with antiprotons: H 2 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+ 4 He + and pbar+ 3 He + - ATOMCULE! at ASACUSA TEAM (RIKEN, JAPAN). 3 _- _-

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ASACUSA = ATOMIC SPECTROSCOPY AND COLLISIONS USING SLOW ANTIPROTONS ATRAP = ANTIHYDROGEN TRAP ALPHA = Antihydrogen Laser Physics Apparatus 4

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

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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? 6

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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 ….. Antimatter may even be related to the dark matter problem in astrophysics. 7

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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. 8 _- _- - _- _- _- _- _- _-

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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 - + H μ => (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! 9 _- _- _- _-

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MUON PHYSICS: H μ and Ħ μ 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 of Ħ μ atoms at low and very low energies from ~ 5eV down to ~ eV. This is done in this work. 10

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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 Ħ μ atom is ~207 times smaller than Ħ”. 11

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

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Some references (2). Hayano, R.S., Hori M., Horvath, D., “Antiprotonic helium and CPT invariance”, 2007, Rep. Prog. Phys., 70, 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,

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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 + p ̄ ) α + μ − is considered in this work, where μ − is a muon (m μ = 207m e ). 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. 16

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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). 17

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THREE-BODY SYSTEMS 18

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THREE-BODY SYSTEMS 19

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THREE-BODY SYSTEMS 20

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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! 21

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CONFIGURATION TRIANGLE (123) 22

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

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BUT WITH MUONS TOO: 24

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Energy levels in p - + H 25

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Energy levels in p - + H μ 26

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COULOMB THREE-BODY SYSTEMS 27

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TWO (NOT 3) SPATIAL CONFIGURATIONS 28

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

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

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

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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.

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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.

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ASYMPTOTICS for only two components: 34

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

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K-matrix formalism: K 12 = K 21 or K 12 / K 21 = 1.0 ! arXiv: v2 or J. Phys. B 46 (2013)

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Discretization Procedure: 42

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RESULTS FOR PROTONIUM FORMATION IN SLOW COLLISIONS BETWEEN p and H μ 51 _

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A. Igarashi, N. Toshima, “Application of hyperspherical close-coupling method to antiproton collisions with muonic hydrogen”, Eur. Phys. J. D 46, 425–430 (2008) 53

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RESULTS FOR ANTIHYDROGEN AND MUONIC ANTIHYDROGEN THREE-BODY FORMATION REACTIONS 56

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FEW-BODY REACTIONS for Ħ and Ħ μ FORMATION (1)-(5) 57

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TEST: MUON TRANSFER 3-BODY REACTIONS (6)-(8) of MUON CATALYZED FUSION CYCLE (μCF - cycle) 58

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RESULTS: arXiv: v2 or J. Phys. B 46 (2013) ). t + + (dμ) 1s → (tμ) 1s + d + 2). t + + (pμ) 1s → (tμ) 1s + p + 3). d + + (pμ) 1s → (dμ) 1s + p + 4). pbar + Ps → Ħ + e - 5). pbar + (Ps) μ → Ħ μ + μ - HERE: t + = 3 H + ; d + = 2 H + ; p + = 1 H + ; μ - pbar=p - ; Ps = (e + e - ); (Ps) μ =(μ + μ - ). 59

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t + + (dμ) 1s → (tμ) 1s + d + Rearrangement scattering at low energies; Only total angular momentum L=0, 1s+2s+2p in the expansion functions; Muon transfer reaction cross sections has been computed; AND… Low energy muon transfer rate: λ = σ v N = const ! 60

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Muon transfer: λ = σ v N λ = 2.6 ✖ s -1 This work, FH equations (1s+2s+2p and the total orbital momentum only L=0: at low energies this is good). λ = 2.8 ✖ s -1 (Kino, Kamimura, Hyper. Int. 82, 45 (1993)). λ = 2.8 ✖ s -1 & 3.5 ✖ s -1 (Breunlich et al. (Experiment) in PRL 58, 329 (1987)). 62

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t + + (pμ) 1s → (tμ) 1s + p + 63

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d + + (pμ) 1s → (dμ) 1s + p + 64

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pbar + Ps → Ħ + e - 66

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Ħ PRODUCTION RATE Now one can compute the rate of the Ħ - atom production: LEAR CERN : R = σ Ħ n d I Here: n - the density of Ps atoms, d - the linear dimension, of the interaction region, & I - intensity of the Ps atom beam. 69

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pbar + (μ + μ) - → Ħ μ + μ - Here are new results for the 3-particle reaction. Computation is done for low and very low energies: from 1 eV down to eV. Very stable results for the rearrangement scattering cross sections with good unitarity numbers, i.e. K 12 / K 21 ~ 1.0. Only 1s, 1s+2s and 1s+2s+2p modified CC (close-coupling) approximation has been applied. 70

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pbar + (Ps) μ → Ħ μ + μ - 71

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72 1.The Pn formation reaction was computed. We obtain a fairly good agreement between our results and with previous computations for the 1s Pn formation channel and for the elastic scattering channel. However, we obtain a not satisfactory agreement in the 2p Pn formation channel. 2. We plan to include the strong interaction between the hadrons now, i.e. p- and p+ in the 1s channel. 3. The antihydrogen (Ħ, Ħ μ ) formation cross sections have been computed from 3-charge-particle collisions at low and very low kinetic energies.

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73 4. Test 3-body systems (muon transfer) have also been computed with the use of the FH-type equations. 5. Normal and muonic antihydrogen formation thermal rates are computed at low and very low temperatures: T ~ K. This is important in view of possible near future, very-very low energy good antiproton beams at CERN! 6. Later, it may be useful to increase the expansion basis: 1s+2s+2p+3s+3p+3d+… AND 7. Consider higher values of the total 3-body angular momentum L, i.e. L > 0 (currently L=0 ).

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THIS IS IT ! THANK YOU FOR YOUR ATTENTION ! 74

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ADDITION: If time permits: Takayanagi’s modified wave number approximation (from 1960s) in order to take into account and compute contributions from higher values of the total three-body angular momentum L. 76

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The Takayanagi Method: Modified Wave Number Approximation: MWNA ( ’s)” In order to take into account an important contribution from the higher values of the total angular momentum L, i.e. L > 0. Important: below “J” and our “L” are the same 3-body angular momenta, i.e. J=L! This method (MWNA) was originally introduced in some molecular physics applications such as H 2 +H 2 : K. Takayanagi, Adv. At. Mol. Phys. pp (1965). Later this approach became a prototype for a very successful and widely used in computational quantum chemistry: The Famous J-shifting Approximation. 78

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