Antihydrogen Formation in Antiproton–Positronium Collisions

Nuclear Instruments and Methods in Physics Research B 214 (2004) 40–43 www.elsevier.com/locate/nimb

Antihydrogen formation in antiproton–positronium collisions

Nobuhiro Yamanaka a,*, Yasushi Kino b a RIKEN (Institute of Physical and Chemical Research), Wako, Saitama 351-0198, Japan b Department of Chemistry, Tohoku University, Sendai 980-8578, Japan

Abstract

Rearrangement in antiproton (pp) and positronium (Ps) collisions, p þ Ps ! H þ e, is a promising process to produce large amount of antihydrogen atom (H). The formation cross section is calculated by using a time-dependent coupled channel (TDCC) method. Numerical accuracy of the TDCC method is demonstrated in a calculation of Ps- formation cross sections in positron and hydrogen collisions. The present result shows a dominant peak of the cross section around a center of mass collision energy of 10 eV. Ó 2003 Elsevier B.V. All rights reserved.

PACS: 34.70.+e; 36.10.Dr Keywords: Antihydrogen; Antiproton; Positronium; Time-dependent coupled channel method

1. Introduction p þ Ps ! H þ e [3]. Antihydrogen-formation cross sections were reported in many theoretical Cold antihydrogen production at the CERN works with, e.g., the close-coupling (CC) method antiproton decelerator (AD) has been recently re- [4,5], and the hyperspherical close-coupling ported [1,2]. The production of the atomic anti- (HSCC) method [6], and the classical trajectory matter stimulates spectroscopic studies for the test Monte Carlo (CTMC) simulation [7]. However, of fundamental principles of physics, e.g. the CPT those predictions are inconsistent with each other. invariance and the weak equivalence principle, and In the present work, we calculate antihydrogen- collision studies to reveal interaction between an- formation cross sections using a time-dependent timatter and matter (see, for review, [3]). Produc- coupled channel (TDCC) method [8,9] in which tion of large amount of antihydrogen atom is coupled-channel equations for scattering wave essential to these experiments. functions are time-dependently solved with a wave A promising process of large amount antihy- packet. Numerical accuracy of the TDCC method drogen (H) production is rearrangement in anti- is demonstrated in a calculation of Ps-formation proton (pp) and positronium (Ps) collisions, i.e. cross sections in positron–hydrogen collisions, eþ þ H ! Ps þ p. The present result is compared

* with previous theoretical results and the experi- Corresponding author. E-mail addresses: [email protected] (N. Yama- ment [10] for the charge-conjugate reaction naka), [email protected] (Y. Kino). p þ Ps ! H þ eþ.

2. TDCC equation and numerical method and an incoming wave packet of the antiproton g , wJMJ ðR; r; t Þ¼g ðRÞ/PsðrÞd d , where kL Ll 0 kL" 1s LJ #l0 The total Hamiltonian for a three-body colli- 2 sion system which consists of an electron, a posi- 1 ðR R0Þ gkLðRÞ¼ exp hL ðkRÞ; ð7Þ tron and an antiproton is given by ðw2pÞ1=4 2w2 pffiffiffiffiffiffiffiffiffiffi 1 b 2 1 2 H ¼ P þ p^ þ V ðR; rÞð1Þ where k ¼ 2ME is the wave number with COM 2M 2m collision energy E, R0 and w the localization radius with the interaction and width of the wave packet at t0,andhL ðkRÞ the asymptotic Hankel function. The TDCC equation Z Z þ Z Z V ðR; rÞ¼ e e þ p e (4) is fast and stably solved with numerical tech- jrj jR þðm=me Þrj nique developed in [8].

Z Z þ Antihydrogen-formation cross sections are ob- þ p e ; ð2Þ jR ðm=meþ Þrj tained by projecting bound-state wave functions H /nlm of antihydrogen atoms on to time-evolved where R denotes the position vector of the anti- wave functions WJMJ ðR; r; 1Þ, proton from the center of mass (COM) of the p X r ¼ ð2J þ 1ÞqJ ð8Þ electron–positron pair, r the relative position vec- H k2 tor between the electron and the positron, Pb and p^ J the conjugate momentum operators of R and r, Zi with and mi the atomic number and mass of i-particle Z DE X 2 (i ¼ e,eþ,orpp), and M and m the reduced masses J 0 JMJ H 0 q ¼ dR W ðR; r; 1Þj/nlmðr Þ ; ð9Þ r0 associated with R and r. nlm The total wave function is expanded into a se- 0 0 JMJ where R and r are the position vectors of the ries of the angular momentum eigenfunction YLl in a form of electron and the positron relative to the antiproton. Ps-formation cross section in positron–hy- X JMJ 1 JMJ JMJ b drogen collisions is also calculated in the similar W ðR; r; tÞ¼ wLl ðR; r; tÞYLl ðR;^rÞ; ð3Þ Rr Ll treatment. where J is the total angular momentum and L and b l the orbital angular momenta associated with R 3. Result and discussion and ^r. From these relations and the time-depen- o JMJ JMJ dent Schroodinger€ equation: i tW ¼ HW ,we Fig. 1 shows Ps-formation cross sections. The have TDCC equations present result is in very good agreement with the o experiment [11] from the Ps-formation threshold JMJ i wLl ðR; r; tÞ (6.8 eV) to 50 eV, and also with recent CC [12,13] ot h i X and HSCC [14] calculations. From this agreement, JMJ JMJ JMJ T 0 0 V R r t ¼ Ll dLL dll þ LlL0l0 wL0l0 ð ; ; Þ; ð4Þ the usefulness of the TDCC method has been 0 0 L l demonstrated. where Fig. 2 shows antihydrogen-formation cross o2 o2 sections. The cross section exhibits a dominant JM 1 LðL þ 1Þ 1 lðl þ 1Þ T J ¼ þ þ ; peak around a COM collision energy of 10 eV. This Ll 2M oR2 2MR2 2m or2 2mr2 peak structure comes from threshold behavior in ð5Þ partial-formation cross sections of excited antihy-

JMJ JMJ JMJ drogen atoms. The formation channel into the V 0 0 ¼hY jV jY 0 0 i: ð6Þ LlL l Ll L l ground state opens for any energy; the threshold H Ps The initial radial function is constructed by the energy Eth ¼ Enl E1s is negative for nl ¼ 1s. The Ps product of the ground-state Ps wave function /1s formation channels into the 2s and 2p states open 42 N. Yamanaka, Y. Kino / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 40–43

4.0 result indicates that the classical mechanical cal-

0 culation is not appropriate in low energies. 2 a

π We should mention that the antihydrogen-for- 3.0 mation cross section should rise in the low energy limit through an S-wave contribution according to WignerÕs threshold law [15], rJ ðEÞ/EJ1=2, 2.0 H;1s where rJ is the partial-wave J contribution to H;1s the ground-state (1s) formation cross section. 1.0 However, this contribution is negligibly small for E 0:01 eV as conﬁrmed from the partial cross Cross section (units of ) J 0 sections rPs;1s calculated in [12–14] for Ps forma- 0 10 20 30 40 50 tion, through the detailed balance, rJ ðEÞ¼ðE þ H;1s Energy (eV) J 1=4Þ=ð2EÞrPs;1sðE þ 6:8eVÞ. Therefore, the previ- Fig. 1. Ps-formation cross section in positron–hydrogen colli- ous results [4–6] indicate no appreciable rise of the 2 16 2 cross section. sions in units of pa0 ¼ 0:880 10 cm . Closed circles represent the present result of a TDCC calculation; dash-dotted In comparison with the present result, the line, CC(28, 3) [12]; broken line, CC(30, 3) [13]; dotted line, CC(3, 3) [5] underestimates the cross section. This HSCC [14]; crosses, the experiment of Zhou et al. [11]. is caused by non-incorporation of break-up channels into wave functions. This defect has been improved in the CC(28, 3) and CC(13, 8) calcula- 20 tions [4], but somewhat overestimates the cross 0 2

a section. In the CC(28, 3), only six bound states up π 15 to n ¼ 3 of antihydrogen atom were incorporated into wave functions, where n is the principle quantum number. Contributions of formation 10 channels into higher excited states have been ad- ded with an empirical correction in which partial 3 5 cross sections are proportional to n . However, this treatment would be overestimation, because it

Cross section (units of ) does not satisfy the flux conservation. In fact, the 0 01020304050 result without the correction is in excellent agree- Energy (eV) ment with the present result (see Fig. 2). The other result of the HSCC calculation [6] is in good Fig. 2. Antihydrogen-formation cross section in antiproton–Ps agreement for low energies, but significantly devi- collisions in units of pa2 ¼ 0:880 1016 cm2. Closed circles 0 ates from the present result as the energy increases. represent the present calculation; solid and broken lines, CC(28, 3)–CC(13, 8) with and without the n3 correction [4]; Finally, the agreement with the experiment [10] is dash-dotted line, CC(3, 3) [5]; dash-double dotted line, CTMC good. However, the precision of the experiment is [7]; dotted line, HSCC [6]; crosses, the experiment of Merrison insufficient to discuss validity of the TDCC et al. [10]. The collision energy is given in the center-of-mass method. Hence, we desire more precise experi- frame. ment. at Eth ¼ 3:4 eV. The threshold energy Eth shifts toward higher energy for higher excited states. The 4. Conclusion emergence of the formation channels causes the increase of the cross section from 3.4 eV and the We have calculated antihydrogen-formation peak around 10 eV. The CTMC simulation [7] fails cross sections using the TDCC method. In a calcu- in reproducing the peak structure. The cross sec- lation of Ps-formation, its usefulness has been dem- tion obtained rises as the energy decreases. This onstrated. For antihydrogen formation, discrepancy N. Yamanaka, Y. Kino / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 40–43 43 with previous results was found. The cross section (2002) 213401; has a dominant peak around 10 eV. The present Phys. Rev. Lett. 89 (2002) 233401. [3] M. Charlton, J. Eades, D. Horvaath, R.J. Hughes, C. result is in excellent agreement with the CC calcu- Zimmermann, Phys. Rep. 241 (1994) 65; 3 lations [4] without the n correction. M.H. Holzscheiter, M. Charlton, Rep. Prog. Phys. 62 (1999) 1; J. Eades, F.J. Hartmann, Rev. Mod. Phys. 71 (1999) 373. Acknowledgements [4] J. Mitroy, G. Ryzhikh, J. Phys. B 30 (1997) L371; J. Mitroy, Aust. J. Phys. 48 (1995) 893. We are grateful to Dr. P. Froerich and Dr. A. [5] J. Mitroy, A.T. Stelbovics, Phys. Rev. Lett. 72 (1994) 3495; J. Mitroy, A.T. Stelbovics, J. Phys. B 27 (1994) 3257; Ichimura for helpful discussions. N.Y. appreciates J. Mitroy, K. Ratnavelu, J. Phys. B 28 (1995) 287; support by Special Postdoctoral Researchers Pro- K. Ratnavelu, J. Mitroy, A.T. Stelbovics, J. Phys. B 29 gram of RIKEN. Y.K. wishes to acknowledge the (1996) 2775. Grant-in-Aid for Scientific Research, Ministry of [6] A. Igarashi, N. Toshima, T. Shirai, J. Phys. B 27 (1994) Education and Culture and Inoue Foundation for L497. [7] R.J. Whitehead, J.F. McCann, I. Shimamura, Phys. Rev. A Science for their generous financial support. 64 (2001) 023401. [8] N. Yamanaka, Y. Kino, Phys. Rev. A 64 (2001) 042715; N. Yamanaka, Y. Kino, Phys. Rev. A 65 (2002) 062709; References Hyperfine Interact. 138 (2001) 85; N. Yamanaka, Y. Kino, Y. Takano, H. Kudo, A. [1] M. Amoretti, C. Amsler, G. Bonomi, A. Bouchta, P. Bowe, Ichimura, Phys. Rev. A 67 (2003) 052712. C. Carraro, C.L. Cesar, M. Charlton, M.J.T. Collier, M. [9] http://nucl.phys.s.u-tokyo.ac.jp/yam/tdcc/. Doser, V. Filippini, K.S. Fine, A. Fontana, M.C. Fujiwara, [10] J.P. Merrison, H. Bluhme, J. Chevallier, B.I. Deutch, P. R. Funakoshi, P. Genova, J.S. Hangst, R.S. Hayano, M.H. Hvelplund, L.V. Jørgensen, H. Knudsen, M.R. Poulsen, Holzscheiter, L.V. Jørgensen, V. Lagomarsino, R. Landua, M. Charlton, Phys. Rev. Lett. 78 (1997) 2728. D. Lindeloo,€ E. Lodi Rizzini, M. Macri, N. Madsen, G. [11] S. Zhou, W.E. Kauppila, C.K. Kwan, T.S. Stein, Phys. Manuzio, M. Marchesotti, P. Montagna, H. Pruys, C. Rev. Lett. 72 (1994) 1443; Regenfus, P. Riedler, J. Rochet, A. Rotondi, G. Rouleau, S. Zhou, H. Li, W.E. Kauppila, C.K. Kwan, T.S. Stein, G. Testera, A. Variola, T.L. Watson, D.P. van der Werf, Phys. Rev. A 55 (1997) 361. Nature 419 (2002) 456. [12] J. Mitroy, J. Phys. B 29 (1996) L263. [2] G. Gabrielse, N.S. Bowden, P. Oxley, A. Speck, C.H. [13] A.A. Kernoghan, D.J.R. Robinson, M.T. McAlinden, Storry, J.N. Tan, M. Wessels, D. Grzonka, W. Oelert, G. H.R.J. Walters, J. Phys. B 29 (1996) 2089. Schepers, T. Sefzick, J. Walz, H. Pittner, T.W. Haansch,€ [14] A. Igarashi, N. Toshima, Phys. Rev. A 50 (1994) 232. E.A. Hessels (ATRAP Collaboration), Phys. Rev. Lett. 89 [15] E.P. Wigner, Phys. Rev. 73 (1948) 1002.