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 eÀ, 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 eÀ [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. 0168-583X/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2003.08.023 N. Yamanaka, Y. Kino / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 40–43 41 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 À R0Þ À gkLðRÞ¼ exp À 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 À with the interaction and width of the wave packet at t0,andhL ðkRÞ the asymptotic Hankel function. The TDCC equation Z À Z þ Z Z À V ðR; rÞ¼ e e þ p e (4) is fast and stably solved with numerical tech- jrj jR þðm=meÀ 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 eÀ,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 antipro- ton. 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 À 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 À 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/EJÀ1=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 À16 2 cross section. sions in units of pa0 ¼ 0:880  10 cm . Closed circles rep- resent 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 À3 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  10À16 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 nÀ3 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].