Channel-coupling array theory for Coulomb three-body dynamics: Antiproton captureby hydrogen atoms

Nobuhiro Yamanaka1,* and Atsushi Ichimura2

1Department of Physics, University of Tokyo, Tokyo 113-0033, Japan2Institute of Space and Astronautical Science, JAXA, Sagamihara, Kanagawa 229-8510, Japan

�Received 1 March 2006; published 11 July 2006�

A theory of channel coupling array is developed for rearrangement reactions in Coulomb three-body sys-tems. Inhomogeneous coupled differential equations for the three Faddeev components are derived and directlysolved with the technique of complex rotation in the exterior region. This method is applied to slow antiprotoncollisions with hydrogen atoms. It is found for a collision energy of 6.8 eV that the antiproton is captureddominantly into protonium states with the principal quantum numbers around 40 and that the electron isreleased with energies around 2 eV. The electron velocity thereby obtained is much higher than the incidentvelocity of the antiproton. Time-dependent wave-packet propagation of the Faddeev components is also inves-tigated to illustrate the peculiar dynamics.

DOI: 10.1103/PhysRevA.74.012503 PACS number�s�: 36.10.�k, 34.90.�q, 02.70.Bf

I. INTRODUCTION

Antiproton �p̄�, the antiparticle of proton �p�, was discov-ered at the Bevatron in the Lawrence Berkeley Laboratory in1955 �1�. Recent progress in experimental techniques to pro-duce slow antiprotons opens up a new field of slow antipro-ton collisions with atoms �2–4�. In particular, much attentionis attracted to the formation process of antiprotonic atoms,p̄+A→ �p̄A+�NL+e.

In the theoretical treatment of antiproton capture by at-oms, difficulties arise from an extreme asymmetry of theantiproton mass Mp̄ to the electron mass me. For a collisionenergy of 1 atomic unit �a.u.�, the velocity is as slow as�0.05 a .u. Nevertheless, a large number of open channelsare involved in the collision. The antiproton is expected to becaptured into highly excited states of antiprotonic atoms withthe principal quantum number around N��Mp̄ /me�40 andthe angular momentum L�40. Hence, sophisticated ap-proaches need to be developed for describing such a colli-sion. Sakimoto �5� carried out a calculation with a wave-packet propagation method for the capture by hydrogenatoms and reported the total formation probabilities of pro-tonium atoms �p̄p�. Esry and Sadeghpour �6� and Hesse, Le,and Lin �7� presented hyperspherical close-coupling methodsto calculate partial probabilities for NL states but in a mass-scaled system. Zdánská, Sadeghpour, and Moiseyev �8� andOvchinnikov and Macek �9� employed adiabatic approachesto calculate the partial probabilities. A comprehensive reviewincluding former works is given in a recent article by Cohen�10�.

In the present study, we develop a new methodology forrearrangement reactions in a Coulomb three-body system, onthe basis of the following three principles. �a� According tothe Faddeev theory �11�, we decompose the total three-bodywave function in a unique way into three rearrangementcomponents, i.e., the Faddeev components. They satisfy theintegral equations in the momentum space, i.e., the Faddeevintegral equation �11�, in which boundary conditions are in-

corporated. �b� The Faddeev integral equation is hardly ap-plicable for a Coulomb three-body system because of atroublesome behavior associated with the integral kernel�12�. To avoid this difficulty, we convert the integral equationto coupled differential equations for the three components,which we call channel-coupling-array �CCA� equations. �c�The coupled differential equations are further rewritten in aninhomogeneous form to adapt the boundary conditions forrespective Faddeev components. The final form derived in�c� permits us to obtain the physical solution with the tech-nique of exterior complex scaling �ECS� method �13�. TheECS method has been originally developed for electron-impact ionization of hydrogen atoms �14,15�; it has an ad-vantage that explicit asymptotic forms of wave functions areunnecessary. We apply the ECS method for rearrangementreactions; this application enables us to carry out theFaddeev-CCA theory for a Coulomb three-body system.

In the present paper, we apply this ECS-CCA method tothe antiproton capture by a hydrogen atom, i.e., to a collisionsystem consisting of two heavy and one light particles withextreme mass asymmetry. In numerical calculations, thismethod is best applicable to such a system because the three-body relative coordinates can be conveniently taken owing tothe mass asymmetry �see Sec. III�. Furthermore, the modelspace can be approximately restricted to an S wave for thelight particle in respective Faddeev components. Calcula-tions are carried out at a collision energy of 6.8 eV forcapture probabilities into respective protonium NL states.We also examine a time-dependent wave-packet propagation�16� of the Faddeev components to illustrate a peculiardynamics in this rearrangement reaction. Atomic units�e=m= � =1� are used unless otherwise stated.

II. CHANNEL COUPLING ARRAY EQUATIONS

A. Faddeev equation

The Schrödinger equation of a Coulomb three-bodysystem is given by

�E − T − V1 − V2 − V3�� = 0 �1�

with the total kinetic energy operator T and the pairwiseCoulomb interactions V1, V2, and V3, where V1 denotes the*Electronic address: yam@nucl.phys.s.u-tokyo.ac.jp

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interaction between particles 2 and 3. The total wave func-tion � is given by a sum of three Faddeev components,

� = �1 + �2 + �3. �2�

They satisfy the Faddeev integral equation �11�,

�c = �0�c1 + lim�→0

Gc�E + i��Vc�b�c

�b, �3�

where the incident arrangement channel is taken to be 1. Theincident wave �0 is given by a product of a projectile planewave and a target bound state supported by V1 such that

�E − T − V1��0 = 0. �4�

Green’s function with the Coulomb interaction Vc is given by

Gc�E + i�� =1

E − T − Vc + i�. �5�

An infinitesimal positive number � plays a role of adiabaticswitch-on of the interaction Vc from time t=−�. From thisrole of �, the outgoing scattering boundary condition is in-corporated in the Faddeev equation �3�.

B. Inhomogeneous differential equation

The Faddeev components are written as a sum of the in-cident wave �0 and the outgoing scattering wave �c

+,

�c = �0�c1 + �c+, or ��1

�2

�3 = ��0

0

0 + ��1

+

�2+

�3+ . �6�

By substituting Eq. �6� into Eq. �3�, multiplying by Gc−1, and

taking a limit of �→ +0, we derive inhomogeneous coupleddifferential equations,

�E − T − Vc��c+ − Vc�

b�c

�b+ = Vc�0�̄c1, �7�

where �̄c1=1−�c1, or an array form as

�E − T − V1 − V1 − V1

− V2 E − T − V2 − V2

− V3 − V3 E − T − V3��1

+

�2+

�3+ = � 0

V2�0

V3�0 .

�8�

In these equations, the outgoing boundary condition shouldbe imposed again to solve the Faddeev components �c

+.The Faddeev component �c

+JM for total angular momen-tum J and its projection M are expanded over the products ofradial and angular parts as

�c+JM�Rc,rc� =

1

Rcrc�Lclc

cLclc+JM �Rc,rc�YLclc

JM �R̂c, r̂c� �9�

with

YLclcJM �R̂c, r̂c� = �

Mcmc

CLcMc,lcmc

JM YLcMc�R̂c�Ylcmc

�r̂c� , �10�

where three sets of Jacobi coordinates �Rc ,rc� shown in Fig.1 are used for the three Faddeev component c. In Eqs. �9�

and �10�, Lc and lc are the angular momenta associated with

R̂c and r̂c, Ylm the spherical harmonics, and CLcMc,lcmc

JM theClebsch-Gordan coefficients. The CCA equations for the ra-dial part cLclc

+JM are derived to be

�E − TcLclc− Vc�cLclc

+JM − Vc �b�c,Lblb

RcLclc,bLblbJM bLblb

+JM

= Vc�L0l0

RcLclc,0L0l0JM 0L0l0

JM �̄c1, �11�

where

TcLclc= −

1

2Mc

�2

�Rc2 +

Lc�Lc + 1�2McRc

2 −1

2�c

�2

�rc2 +

lc�lc + 1�2�crc

2

�12�

with the reduced masses Mc and �c associated with the rela-tive coordinates Rc and rc. The incident wave �0

JM is ex-panded as

�0JM�R1,r1� =

1

R1r1�L0l0

0L0l0JM �R1,r1�YL0l0

JM �R̂1, r̂1� , �13�

where

�E − T1L0l0− V1�0L0l0

JM = 0. �14�

In Eq. �11�, for convenience, we introduce a frame transfor-mation operator RcLclc,bLblb

JM �Rc ,rc ;Rb ,rb�, by which a radialfunction bLblb

+JM of the coordinates Rb and rb is expressed as afunction of Rc and rc.

We should mention that in the CCA equation �11�, theradial functions cLclc

+JM with different Lc and lc belonging to acommon Faddeev component c do not directly couple witheach other. Such a coupling is only caused via different Fad-deev components in the second or higher order processes.This property leads to rapid convergence of the total wavefunction with respect to the angular-momentum expansion�9�.

C. Time-dependent wave-packet equation

A time-dependent CCA equation �TD-CCA equation� isobtained in the following way. By multiplying by Gc

−1 andtaking a limit of �→ +0, Eq. �3� is converted to a homoge-neous differential equation for �c,

�E − T − Vc��c − Vc�b�c

�b = 0. �15�

By replacing E by i� /�t and �c�Rc ,rc� by �c�Rc ,rc , t�, wehave

FIG. 1. Three sets of Jacobi coordinates.

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i�

�t�c = T�c + Vc�

b

�b, �16�

or an array form

i�

�t��1

�2

�3 = �T + V1 V1 V1

V2 T + V2 V2

V3 V3 T + V3��1

�2

�3 , �17�

where the 3�3 matrix is called Hamiltonian array. The Fad-deev component �c

JM with J and M is expanded with Eq.�10� as

�cJM�Rc,rc,t� =

1

Rcrc�Lclc

cLclcJM �Rc,rc,t�YLclc

JM �R̂c, r̂c� . �18�

The TD-CCA equation for the radial part cLclcJM is derived to

be

i�

�tcLclc

JM = TcLclccLclc

JM + Vc �bLblb

RcLclc,bLblbJM bLblb

JM . �19�

III. NUMERICAL METHOD

We present numerical methods to directly solve the time-independent and dependent CCA equations �11� and �19�.For the antiproton and hydrogen system, the three sets ofcoordinates �Rc ,rc� are practically taken as those shown inFig. 2. They are convenient for a three-body system with twoheavy and one light particles, because the relative coordinatebetween the heavy particles is common in the three sets ofcoordinates �17�. Furthermore, the three Faddeev compo-nents are, respectively, expanded over single angular-momentum states with Lc=J and lc=0 in Eqs. �9� and �18�.

A. Time-independent CCA equation

We introduce the ECS method �13� to solve the time-independent CCA equation �11� for the Faddeev componentscLclc

+JM without using explicit asymptotic forms of the bound-ary conditions. The coordinate variables Rc and rc in theexterior regions are taken to be bended in the complex plane�see Fig. 3� as

Rc → Rc for Rc Rc0,

Rc0 + ei�c�Rc − Rc0� for Rc � Rc0,�20�

rc → rc for rc rc0,

rc0 + ei�c�rc − rc0� for rc � rc0�21�

with which the outgoing boundary condition is mimicked bytaking the argument as 0 �c ,�c � /4. The outgoing wave�eikR� decreases exponentially in the exterior region, whilethe incoming wave �e−ikR� diverges, hence excluded from thesolution. In this method, physical meanings are retained onlyin the interior region �Rc Rc0, rc rc0�.

The CCA equation �11� for the complex scaled coordi-nates �20� and �21� is discretized over grid points of Rci andrcj and reduced to a linear algebraic equation

�E − TcLclcij − Vc

jRcLclc,bLblbJM,ij,i�j� �bLblb

+JM,i�j� = VcjRcLclc,0L0l0

JM,ij,i�j� 0L0l0JM,i�j�,

�22�

where cLclc+JM,ij =cLclc

+JM �Rci ,rcj�, TcLclcij cLclc

+JM,ij

=TcLclccLclc

+JM �Rc ,rc��Rc=Rci,rc=rcj, Vc

j =Vc�rcj�, and RcLclc,bLblbJM,ij,i�j�

=RcLclc,bLblbJM �Rci ,rcj ;Rbi� ,rbj��. Note that in Eq. �22� a con-

vention albl=�lalbl is employed and Vcj =Vc�rcj� is replaced

by Vci =Vc�Rci�for c=2 because of a definition of the coordi-

nates �Rc ,rc� shown in Fig. 2. The incident wave is given as

0L0l0JM �R1,r1,t0� = jL0

�kR1�1sH �r1��L0J�l00 �23�

with the Ricatti-Bessel function jL and the ground-state wavefunction 1s

H of the hydrogen atom. To obtain accuratelywave functions, we introduce a nonuniform scale transfor-mation of the variables as described in Appendixes A and B.Respective radial functions cLclc

+JM are given over 240�240grid points with parameter values of Xmax=10, xmax=300,Xs=2, xs=60, �r=0.05 for the variable scaling functions�A9� and �A10�. The bending points in Eqs. �20� and �21� aretaken as Rc0=5.88 and rc0=176.40 for c=1, 2, and 3. Thenumerical accuracy obtained is better than 0.1% and 1 ppmfor energy eigenvalues of hydrogen and protonium atoms,respectively. The linear algebraic equations �22� are solvedin iteration with the conjugate gradient squared method �18�.

FIG. 2. Three sets of coordinates used in the present calculationto describe the three Faddeev components �c=1, 2, and 3� for anti-proton collisions with hydrogen atoms; Vc is the Coulomb interac-tion incorporated in the CCA equation for the Faddeev componentc.

FIG. 3. Two-dimensional radial variable �R ,r� used in the exte-rior complex scaling with Eqs. �20� and �21�.

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Total probabilities of antiproton capture in the partialwave J are obtained from the elastic wave function �kL0

,

PJ = �W�u,�kL0�

W�w,�kL0��2

, �24�

through the Wronskian

W�u,�kL0� = u�R1�

��kL0�R1�

��kR1�− �kL0

�R1��u�R1���kR1�

�25�

with the free incoming u and outgoing w wave functions.The �kL0

is given from a projection of the total wave func-tion �JM onto the initial target state,

1

R1�kL0

�R1�YL0M0�R̂1� = 1s

H �r1���JM�r1. �26�

Note that the total capture cross section is obtained as

� =�

k2�J

�2J + 1�PJ. �27�

Partial probabilities of antiproton capture into an NL state ofprotonium atoms are calculated from

PJ�NL� =� r22dr̂2�NL

+ �r2��r2=r0

2 , �28�

where the electronic outgoing wave NL+ is obtained from a

projection onto an NL-state �NL of protonium,

NL+ �r2� = �NL�R2���JM�R2

. �29�

B. Time-dependent CCA equation

The TD-CCA equation �19� is discretized over grid points�i , j�,

i�

�tcLclc

JM,ij = �TcLclcij + Vc

jRcLclc,bLblbJM,ij,i�j� �bLblb

JM,i�j�. �30�

The number of grid points is taken as Mmax�mmax=160�160, the variable scaling parameters Xmax=8, xmax=180,Xs=0, xs=20, and �R=�r=0.05. Equation �30� is solved bypropagating a wave packet in the time-evolution scheme de-veloped in Ref. �16� with minor improvements given inAppendix C.

The initial wave packet at t0=0 is constructed from asolution of the radial Schrödinger equation �14� in the inci-dent rearrangement channel as

0L0l0JM �R1,r1,t0� = gkL0

�R1�1sH �r1��L0J�l00 �31�

with the ground-state wave function 1sH of hydrogen atoms

and an incoming wave packet of antiprotons

gkL�R� =1

�w2��1/4 exp�−�R − R̄�2

2w2 �hL−�kR� , �32�

where R̄ and w are the center position and width of the initialwave packet, hL

−�kR�=exp�−ikR+ i�L /2� an asymptotic form

of the Ricatti-Hankel function, and k=�2M1E with a centerof mass energy E of the antiproton. The parameters in Eq.

�32� are taken as R̄=6.4, w=0.35. The width w ofwave packets causes a finite energy resolution of�E /E=2/kw� 0.27. The total probability PJ of antiprotoncapture �protonium formation� is calculated as the probabil-ity flow out of the incident channel,

PJ = 1 −� dR1� �JM�1sH �r1��r1

�2. �33�

IV. RESULTS AND DISCUSSION

A. Behaviors of three Faddeev components in the wavefunction

We calculate the wave-packet propagation at a collisionenergy of 6.8 eV and a partial wave of J=30 using the TD-CCA equation �19�. In Fig. 4, series �0� of panels show thetime evolution of probability densities for the total wavepacket and respective series �c=1, 2, and 3� for the Faddeevcomponent c. In the figure, the densities for the total packet�0� are plotted with the coordinates �R1 ,r1� and those for thecomponent c with �Rc ,rc�.

The total wave packet describes a full process of the an-tiproton capture. The panel �0a� in Fig. 4 shows the initialwave packet given by Eq. �31�. In the panels �0a�–�0c�, thepacket propagates to the left toward the origin. This behaviormeans that the antiproton propagates toward the target hy-drogen atom. The packet turns around at �R ,r���1.5,10� in�0c�–�0d� and propagates upward in the �0d�–�0f�. The up-ward propagation indicates a protonium formation in whichthe proton captures the antiproton and instead releases theelectron.

Behaviors peculiar to the CCA equation �16� are seen inrespective Faddeev components. In the Faddeev component1, as seen from the panels �1a�–�1c�, the antiproton ap-proaches the target up to a closest distance of R�1.5. In�1d�–�1f�, the wave packet decreases in amplitude and even-tually vanishes. The Faddeev component 3 gradually growswith time and propagates to the left in �3a�–�3c�, but dimin-ishes in �3d�–�3f� when the packet arrives at the closest dis-tance. The Faddeev component 2 gradually grows and propa-gates to the left in �2a�–�2c�. In �2c�–�2d�, this componentrapidly increases in amplitude, while the packets in the com-ponent 1 and 3 diminish. This growth is due to an inflowfrom the components 1 and 3. The packet in the component2 turns around at �R ,r���1.5,10� in �2c�–�2d� and propa-gates upward in �2d�–�2f�. In �2c�–�2d�, as the antiprotonapproaches the proton from a distance of R�1.5 to �0.7,binding of the electron becomes weaker and the electron iseventually released. The antiproton is consequently capturedby the proton.

We solve the time-independent CCA equation �11� withthe ECS method to calculate the total wave functions for acollision energy of 6.8 eV and partial waves of J=10, 20, 30,35, 40, and 50. The probability densities obtained are plottedin Fig. 5 with the coordinates �R1 ,r1� shown in Fig. 2�1�. It isseen in Fig. 5 that the density profiles have two components,

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one extending along the R axis and another along the r axis.This behavior is consistent to wave-packet behaviors in theTD-CCA wave function shown in Fig. 4. The first compo-nent seen in Fig. 5 represents the incoming propagation ofthe wave packet in Figs. 4�0a�–�0c�. Figure 6 shows theelastic part of the ECS-CCA wave functions obtained fromprojection, 1s

H �r1� �JM�r1. It is seen from Fig. 6�a� that for

J=30 the elastic part is incoming �e−ikR� with no outgoingcomponent �eikR�. It indicates that the incoming wave com-pletely attenuates, absorbed in inelastic channels. The secondcomponent seen in Fig. 5 corresponds to the upward propa-gation of the wave packet in Figs. 4�0d�–�0f� which repre-sents antiproton capture with electron release. In Fig. 5�f� forJ=50, this component does not appear. This means that theantiproton is not captured and only elastically scattered bythe hydrogen atom. Because of this, a standing wave appearsin the elastic part as shown in Fig. 6�b�.

The ECS-CCA wave functions exhibit fine stripes in Fig.5. The stripes parallel to the r axis come from nodes ofprotonium wave functions. For smaller partial waves J, thestripes are finer. This indicates that the antiproton is capturedinto high-N states of the protonium atom even for small J.Moreover, we see a ridge of the probability densities runningvertically and reflecting horizontally back and forth. Thisbehavior is related to a classical motion of the antiproton.The antiproton captured by the hydrogen atom moves be-tween two barriers, outer one by Coulomb potential by theproton and inner one by the centrifugal potential, while theelectron propagates away from the protonium formed.

B. Antiproton capture probabilities

We have calculated the total and partial capture probabili-ties by hydrogen atoms using the wave functions obtained inthe preceding section. Figure 7 shows partial-wave �J� de-

FIG. 4. Time evolution of wave-packet probability densities obtained with the TD-CCA method in antiproton collisions with hydrogenatoms for a partial wave of J=30 and an energy of 6.8 eV. The total packet is exhibited in the row �0�, while the three Faddeev components�c=1, 2, and 3� in the rows �1�, �2�, and �3�. Each Faddeev component �c� is plotted with the coordinates �Rc ,rc� shown in Fig. 2�c� and thetotal �0� with �R1 ,r1� in Fig. 2�1�.

FIG. 5. Probability densities of the total ECS-CCA wave function in antiproton collisions with hydrogen atoms for partial waves ofJ=10–50. The antiproton collision energy is 6.8 eV. The density profiles are plotted with the coordinates �R1 ,r1� shown in Fig. 2�1�. Thebending points in the ECS method are R0=5.88 and r0=176.40.

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pendence of the total probabilities �24� and �33� multipliedby the statistical weight �2J+1�. In the result of the ECS-CCA method, the capture probability almost reaches the uni-tarily limit �2J+1� for partial waves of J=0−35. Theprobability steeply decreases around J=38 and vanishes forJ�42. Here, a partial wave of J=38 corresponds to theimpact parameter of �1.8. This value is significantly largerthan the average radius of the hydrogen atom. This effectis caused by the polarization force induced by the antiproton.The result of the TD-CCA method is in good agreementwith the result of the ECS-CCA calculation at J=0, 10, 20,and 30, but slightly smaller at J=35 and larger at J=40 thanthe ECS-CCA result. These slight deviations come from afinite energy resolution of �1.8 eV of the wave packet. Incomparison with a previous result by Sakimoto �5�, we ob-

tain a good agreement for J=0–30, but a more steep de-crease around J=38; the capture cross section is in goodagreement. These results indicate the reliability of thepresent calculations.

The steep decrease obtained in the capture probabilityaround J=38 is related to a structure of the CCA equation�17�. The cth row of the Hamiltonian array in Eq. �17� onlyincludes Vc as a pairwise interaction. The coupling V2,for example, is responsible for a probability inflow into theFaddeev component 2 from the other components 1 and 3,but not for an outflow from the component 2 into 1 and 3,which is only caused by the couplings V1 and V3. As indi-cated in the panel of Fig. 4�0d�, the wave packet motionturns around in a region of �R ,r���1.5,10�, where the cou-pling is dominated by V2 ��V1, V3� because r�R. Hence,once the wave packet enters this region, the Faddeev com-ponent 2 hardly returns to the incident component 1. Thismeans that the protonium formation with electron release is aone-way process.

Figure 8 shows the partial capture probabilities multipliedby a statistical weight �2L+1� into protonium NL states. Fig-ure 8�a� shows the L-distributions for respective states withN=36–42. Note that the total energy, −6.8 eV, of the anti-proton and hydrogen system gives a limit of N=42 and thatthe energy of the ground-state hydrogen corresponds to pro-tonium states with N�30. In the wave function expansionemployed in the present calculation as L=J, the summationof partial probabilities over N gives the total capture prob-

FIG. 6. Elastic parts of the ECS-CCA wave function for anti-proton collisions with hydrogen atoms for partial waves of J=30and 50. The antiproton collision energy is 6.8 eV. The solid andbroken lines represent the real and imaginary parts of the functions.The bending point in the ECS method is R0=5.88.

FIG. 7. Partial-wave �J� dependence of the total capture prob-abilities times the statistical weight �2J+1� in antiproton collisionswith hydrogen atoms at a collision energy of 6.8 eV. Closed circlesand crosses represent the present results obtained from the ECS-CCA and the TD-CCA methods, respectively; broken line, a resultby Sakimoto �5�; dotted line, the unitarity limit of capture probabili-ties, i.e., 2J+1.

FIG. 8. Partial probabilities of antiproton capture into protoniumNL states in collisions with hydrogen atoms at a collision energy of6.8 eV. N and L are principal and angular-momentum quantumnumbers. This collision energy gives a limit of N=42 in the prob-ability distribution.

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abilities given in Fig. 7. For N=40–42, a sharp maximum isseen around L=36 in the distribution. For smaller N, theposition of its maximum shifts toward lower L.

Figure 8�b� shows N distributions for respectivestates with L=20–40. This distribution is equivalent tothe release energy distribution of the electron througha relation of Ee=−ENL−6.8 eV, where ENL is the internalenergy of protonium atoms. For L=38–40, the probabilityincreases with N. However, for L�37, the distributionhas a maximum around N=40 and its tail extends towardsmaller N. From these distributions, an average of theelectron release energy is estimated to be about 2 eV from�NL�2L+1�EePNL

J /�NL�2L+1�PNLJ . This energy is much

higher than the incident antiproton velocity.

V. SUMMARY

We have presented the CCA theory for rearrangement re-actions in a Coulomb three-body system and apply it to theantiproton capture by a hydrogen atom. The capture dynam-ics has been examined in terms of the three Faddeev compo-nents with the time-dependent and time-independent CCAequations. For a collision energy of 6.8 eV, the antiproton isfound to be captured into protonium states with the principalquantum numbers around 40; consequently, the electron isreleased with energies around 2 eV. The electron velocitythereby obtained is much higher than the incident velocity ofthe antiproton. The CCA theory developed in the presentpaper could be applicable also to other elementary atomicrearrangement processes; we leave it to future works.

ACKNOWLEDGMENTS

One of the authors �N.Y.� is grateful to Dr. R. S. Hayanoand Dr. Y. Yamazaki for helpful discussions and supports bythe Grant-in-Aid for Specially Promoted Research of MEXT�Japan�.

APPENDIX A: VARIABLE TRANSFORMATIONTECHNIQUE

In the CCA equations �11� and �19�, a variable transfor-mation technique is introduced to accurately describe wavefunctions. The radial coordinates Rc and rc are nonuniformlyscaled with appropriate functions Fc and fc of variables Xcand xc,

Rc = Fc�Xc�, rc = fc�xc� . �A1�

Accordingly, the radial wave functions are replaced as

cLclc+JM �Rc,rc� → cLclc

+JM �Xc,xc� = �Fc�fc�cLclc+JM �Rc,rc� ,

�A2�

cLclcJM �Rc,rc,t� → cLclc

JM �Xc,xc,t� = �Fc�fc�cLclcJM �Rc,rc,t�

�A3�

and the differential operators in Eq. �12� as

�2

�Rc2 →

1

Fc�

�2

�Xc2

1

Fc�+

1

2

Fc�

Fc�3 −

3

4

Fc�2

Fc�4 , �A4�

�2

�rc2 →

1

fc�

�2

�xc2

1

fc�+

1

2

fc�

fc�3 −

3

4

fc�2

fc�4 . �A5�

These operators would cause instabilities in numerical calcu-lations because of higher-order derivatives when the scalingfunctions F and f are not smooth. An alternative form isderived with integration by parts as

�2

�Rc2 → −

1

�Fc�

�

�Xc

�

1

Fc�

�

�Xc

�

1

�Fc�, �A6�

�2

�rc2 → −

1

�fc�

�

�xc

�

1

fc�

�

�xc

�

1

�xc�, �A7�

where differential operators with over-arrows are defined as

��x��−1

�fc�

�

�xc

�

1

fc�

�

�xc

�

1

�fc���x��

=� �

�xc

1

�fc��x��−

1

fc�� �

�xc

1

�fc��x�� . �A8�

The radial wave function is numerically described overMmax�mmax grid points in the two-dimensional �X ,x� plane.The grid points are uniformly taken up to Xmax and xmax withspacings of �X=Xmax/Mmax and �x=xmax/mmax. The scalingfunctions are taken as

Fc�X� = �X − Xs� + Xs exp�− AX� , �A9�

fc�x� = �x − xs� + xs exp�− ax� , �A10�

where A= �1/�X�ln�Xs / �Xs+�R−�X�� anda= �1/�x�ln�xs / �xs+�r−�x�� with the parameters Xs, xs,�R, and �r. The grid points in the �R ,r� plane are dense inan interacting region and sparse in an asymptotic region.

APPENDIX B: DISCRETIZED DIFFERENTIALOPERATOR

The differential operators in Eqs. �A4�–�A7� are dis-cretized with a discrete Fourier transformation technique.The second derivative of a one-dimensional function g�x� iswritten as

g� =d2g�x�

dx2 =� 2

��

0

�

dq�− q2�sin�qx�g̃�q� , �B1�

by using a Fourier transformation

g�x� =� 2

��

0

�

dq sin�qx�g̃�q� . �B2�

The derivative g� is written in an alternative form of

CHANNEL-COUPLING ARRAY THEORY FOR COULOMB¼ PHYSICAL REVIEW A 74, 012503 �2006�

012503-7

g� = �0

�

dx�� 2

��

0

�

dq�− q2�sin�qx�sin�qx���g�x�� ,

�B3�

by using an inverse transformation for g̃�q� in Eq. �B1�,

g̃�q� =� 2

��

0

�

dx� sin�qx��g�x�� . �B4�

Similarly, g and g� are written as

g = g�x� = �0

�

dx�� 2

��

0

�

dq sin�qx�sin�qx���g�x�� ,

�B5�

g� =dg�x�

dx= �

0

�

dx�� 2

��

0

�

dq�q�cos�qx�sin�qx���g�x�� .

�B6�

By performing a discrete integral with respect to q, we obtaina discrete form of the kth derivative gm

�k�=g�k��xm� at xm

=m�x as

gm�k� = �

m�=1

mmax−1

Dmm��k� gm�, �B7�

where the matrix Dmm��k� is given by

Dmm��0� = �mm�, �B8�

Dmm��1� = �− �− �m−m��q

2�tan���m + m��

2mmax�−1

− tan���m − m��2mmax

�−1� for m � m�,

−�q

2tan� �m

mmax�−1

for m = m�, �B9�

Dmm��2� = ��− �m−m��q2

2�sin���m + m��

2mmax�−2

− sin���m − m��2mmax

�−2� for m � m�,

�q2

2�sin� �m

mmax�−2

−2mmax

2 + 1

3� for m = m�, �B10�

where �q=� /xmax and g0=gmmax=0 from the boundary con-

dition. The matrix D̃mm��k� corresponding to the operators �A6�

and �A7� is given by

D̃ij�2� = − �

k=0

mmax � 1

�fk�Dki

�1� 1

�f i��� 1

�fk�Dkj

�1� 1

�f j�� . �B11�

APPENDIX C: TIME EVOLUTION SCHEME

A formal solution of Eq. �30� is derived to be

cLclcJM,ij�t� = ScLclc,bLblb

JM,ij,i�j� �t − t0�bLblbJM,i�j��t0� �C1�

with the time-evolution operator

ScLclc,bLblbJM,ij,i�j� �t − t0� = exp�− i�t − t0��TcLclc

ij + VcjRcLclc,bLblb

JM,ij,i�j� �� .

�C2�

For the discrete time variable tn=n�t with a short time pe-riod �t, a solution at tn is obtained through iteration of

cLclcJM,ij�t + �t� = ScLclc,bLblb

JM,ij,i�j� ��t�bLblbJM,i�j��t� . �C3�

By employing the split operator method, the time-evolutionoperator is written in a symmetric form of

ScLclc,bLblbJM,ij,i�j� ��t� = exp�− i

�t

2TcLclc

ij �exp�− i�tVcjRcLclc,bLblb

JM,ij,i�j� �

�exp�− i�t

2TcLclc

ij � + O��t3� . �C4�

The exponential operators are expanded into a Cayley’s frac-tional form as

exp�− i�Hll�� = �1 − i�

2Hll��−1�1 + i

�

2Hll�� + O��t3� .

�C5�

The inverse matrix �1− i�� /2�Hll��−1 is explicitly calculated,

because implicit schemes, e.g., the Crank-Nicholson scheme,are inefficient for the dense matrix Hll� due to the discrete

operators TcLclcij and RcLclc,bLblb

JM,ij,i�j� .

NOBUHIRO YAMANAKA AND ATSUSHI ICHIMURA PHYSICAL REVIEW A 74, 012503 �2006�

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