Near-threshold production of antihydrogen positive ion in positronium–antihydrogen collision

Near-threshold production of antihydrogen ion ( H̄+ ) in positronium–antihydrogen collisions is predicted by a rigorous four-body scattering calculation. The convergence of the cross sections for the rearrangement and for all competing reactions (elastic scattering, de/excitation and de/polarization of positronium) is carefully examined against partial waves. The multi-channel scattering solutions are composed of functions that diagonalize the four-body Hamiltonian, and scattering functions that satisfy the correct asymptotic boundary conditions. The production rates of H̄+ show large discrepancies compared to more approximate calculations.

TheH + production channel from the collision between Ps andH(1s), opens at 6.05 eV in center-of-mass energy for Ps(n = 1), 0.95 eV for Ps(n = 2) and 0.0017 eV for Ps(n = 3). Theoretical treatment of the reaction (1) involves considerable challenges; e.g., it is a rearrangement process where the particle configuration drastically changes from initial channel to final channel. Besides, the slow collision between Ps andH requires a simultaneous treatment of all constituent particles and would not allow the semi-classical approximations. It should be also stressed that the reaction (1) competes with several other inelastic processes. Below the threshold energy of e − +H + , there are fragmentation channels of Ps(nl) +H(1s) (n 3). The reaction (1) competes with Ps excitation and/or deexcitation. Due to the charge-conjugation symmetry, Ps +H collisions (1) are equivalent to Ps + H collisions. Low-energy scattering of Ps(1s) and H has been studied in a number of works before as a fundamental example of positronium-atom interaction [23][24][25][26]. So far, theoretical studies revealed the Ps(1s) + H(1s) total/differential elastic cross sections and scattering length [27][28][29][30][31][32][33][34][35][36], but the understanding of inelastic scattering still remains. The Born-based approximations and eikonal-based approximations have been adopted to predictH + production cross section; however, such approximations assume intermediate to high energy collisions, and may not be appropriate for low-energy scattering. Although the close-coupling method [32,37,38] and a coupled pseudo-state approach (CPA) [39] can be applied to low-energy scattering, these methods have so far not been applied to collisions with excited positronium. Comini and Hervieux have adopted a continuum distorted wave final state approximation (CDW-FS) [40][41][42] to calculate the cross sections forH + production in collisions ofH and Ps in excited states. However, the CDW-FS involves perturbative nature of the formulation. The suitability of this approach to the near-threshold production ofH + remains to be tested by more rigorous calculations.
A quantitative prediction ofH + production rate in Ps(nl) +H(1s) slow collision requires a rigorous theoretical treatment of the four-body non-adiabatic multi-branch scattering including a rearrangement reaction. In this communication we predict elastic/inelastic cross sections of these collisions in the vicinity of e − +H + threshold energy by a rigorous four-body scattering calculation. We describe the four-body correlation during the scattering by means of orthogonal finite space basis functions that diagonalize the full four-body Hamiltonian, H. We augment this description with the channel functions that satisfy the asymptotic boundary conditions, and calculate the complete multi-channel scattering states by solving the coupled channel equations. This procedure delivers the full scattering matrix S whose elements determine the elastic and inelastic cross sections.

Theory
We consider the Schrödinger equation for the non-relativistic time-independent scattering wavefunction [43], where the Hamiltonian H includes kinetic energy operators in center-of-mass system and all inter-particle Coulomb potential operators. We construct the total wavefunction Ψ as where Φ υ are square integrable four-body functions that describe the 'intermediate state' during the scattering with constants b υ , and ψ α are open channel functions that describe the asymptotically non-vanishing component. Construction of {Φ υ } plays a primary role in accurate determination of the cross section. In this work we construct {Φ υ } using a Gaussian expansion method (GEM) [44][45][46]. Φ υ is described in terms of finite range Gaussian functions written in the five sets of Jacobian coordinates {r c , R c , ρ c } in figure 1 and their positron-permutated sets. {Φ υ } are eigenfunctions of the four-body Hamiltonian H and corresponding eigenenergies E υ are given by The use of the basis functions written in several coordinate systems facilitates the description of the four-body interactions and multi-channel character of the scattering. It should be stressed that {Φ υ } provides the explicit description of virtualH excitation, virtual Ps excitation, virtual Ps + formation (mainly described by c = 4, 5 coordinate sets) and transient formation ofH + during the scattering. They describe also the mutual polarization of the atoms which is important in the description of their collisional interaction. Thep-e + and e + -e − correlations are expanded in terms of Gaussian functions whose maximum ranges are around 20 a 0 and 50 a 0 , respectively. In the present calculation, Φ υ are expanded in terms of 40 000-65 000 basis functions. The channel function ψ α of Ps(nl) +H(NL) is written in the c = 1 coordinate system and its positron-exchanged coordinate system as where [· · ·] denotes linear combination of the spherical harmonics with Clebsh-Gordan coefficients as defined in reference [47]. P is a permutation operator for two positrons. R Ps nl (r 1 ), RH NL (R 1 ) and R α (ρ 1 ) are radial functions of Ps(nl),H(NL), and the relative motion between them, respectively. In this communication we consider N = 1 and L = 0 for the ground stateH atom. Then the quantum number Λ of composite angular momentum Λ = L + l coincides with l.  Table 1. Definition of arrangement channels F and α. λ is angular momentum of relative motion of fragments. We list only the channels that can lead to rearrangement reaction.

F Fragments
The channel function of e − +H + is written as where φH + L=0 is a three-body wavefunction ofH + (it is numerically calculated using the same Gaussian basis functions and corresponds to binding energy E b = 0.027 718; which differs by 0.000 001 hartrees from the best variational result). R α is a radial function of the relative motion between e − andH + . The L = 0 denotes the zero total angular momentum ofH + . The radial functions R α (ρ c ) of the relative motion in equations (5) and (6) are determined by numerical integration during the process of solving the integro-differential equations that are part of the coupled-channel procedure.
The whole system can be characterized by its parity P, a total angular momentum J and its projection onto the z-axis M, where the z-axis can be chosen to be in an arbitrary direction. In collision between Ps(nl) andH(NL), the parity is defined by l, L and the angular momentum of relative motion λ, that is, in the initial channel, P = (−1) l+L+λ . Turning to the final channel of the rearrangement process, the ground state ofH + ( 1 S e ) possesses two positrons with singlet spin. As a consequence, the spatial part of its wavefunction has positive permutation parity. In the final channel, the parity of the whole system, e − +H + , is determined by the angular momentum of relative motion λ f . Therefore, noticing the vector sum of L and λ f configures J, the parity ofH Since we consider spin-independent interaction throughout this paper, the parity conservation between the initial and final channels induces constraints on the partial waves contributing toH + production. Considering for the example, the Ps(2p) +H(1s) →H + + e − reaction, J = 1 allows only s-wave and d-wave So far the channel α is not defined rigorously. The radial component of scattering function R α (ρ 1 ) in equation (5) depends on the fragments Ps(nl) andH(NL) and (λ, Λ, J, M). Similarly, R α (ρ 2 ) in equation (6) depends on J and M. Hereafter, we call α a 'detailed' channel that specifies the fragments and the angular momentum of its partial wave while a channel F specifies only the fragments. For example, F defines only a collision between Ps(1s) andH(1s) whereas α specifies also the partial waves and intermediate couplings. Table 1 displays α that access the final e − +H + channel together with the associated λ and J quantum numbers. We denote the initial detailed channel as α i and the angular momentum of its partial wave as λ i . Since we consider all energetically open channels above the e − +H + threshold energy, the scattering in J = 0 has 7 branches, J = 1 has 10 branches and J 2 has 11 branches. We denote the number of branches as α (J) max . For a given J, the dimension of the S-matrix is This representation of the scattering state was adopted to three-body muonic atom collisions [48,49] and recently used in four-body antihydrogen atom collisions [50,51]. The Schrödinger equation is converted to a set of coupled equations with the following conditions, and Here, · · · r c ,R c ,ρ c means the integration over the indicated coordinates leaving out integration over ρ c . Expressing the radial functions of the relative motion in the initial and final channels as R α (ρ c ) = χ α (ρ c )/ρ c , we get a set of coupled integro-differential equations for χ α . We solve it using a compact finite difference method [52] under proper boundary conditions at ρ c → ∞, where u (∓) λ is an incoming/outgoing spherical Hankel function for channels with fragmentation into Ps(nl) andH(1s), and a spherical Coulomb-Hankel function for the final channel with fragmentation into e − +H + . υ α denotes speed of relative motion between the fragments in channel α. χ α (ρ c ) is computed up to 800 a 0 for ρ 1 and 1200 a 0 for ρ 2 , and above these values S-matrix elements are well converged.
The S-matrix elements S (JM) αα i for each J and M give cross sections from an initial detailed channel α i to other detailed channel α as Since the initial wave involves all partial waves, the scattering cross section from F i to F can be expressed as whereσ (λ i ) FF i is a partial wave cross section of λ i angular momentum. In the present framework,σ (λ i ) FF i can be calculated by gathering the σ (JM) αα i . Since the scattering process does not depend on M, the partial wave cross sections are written asσ (λ i ) where the summation over α and α i runs over all detailed channels belonging to F and F i , respectively. Note that the allowed α i should be compatible with λ i . For example, Ps(3s) +H(1s) scattering (F i = 4) in p-wave (λ i = 1) is simply given by α i = 4 for J = 0. On the other hand, Ps(3p) +H(1s) scattering (F i = 5) in p-wave is given by the summation of α i = 5 of J = 0 and α i = 6 of J = 2 (these two detailed channels have the same λ i but different internal coupling). Our calculation does not assume the principle of detailed balance, and the S-matrix is not symmetric by the construction. Thus, we can use its symmetry as a check of the accuracy of our calculation. In this communication we evaluate the cross section with error σ αα i ± Δσ αα i by the difference between S αα i and S α i α . In addition to the symmetry of S-matrix, the unitarity condition can also be an indicator of calculation accuracy. If the solution of the Schrödinger equation were exact, the value U α i = α |S αα i | 2 would be unity due to the conservation of probability. The deviation, |1 − U α i |, can be used to estimate the error of scattering cross sections from a detailed channel α i . Throughout the presented calculations, the errors estimated from the unitarity condition are kept smaller than the accumulated errors from the symmetry condition of the S-matrix (U α i = 1.001 in average with the standard deviation 0.003 for all calculated energy values).

Partial wave convergence ofH + production cross section
We investigate theH + production as function of collision energy E i in the initial channel. Denoting the kinetic energy of relative motion in the final channel as E f , we concentrate on the near-thresholdH + production, 0 < E f 0.15 eV. Table 2 lists the partial wave cross sections ofH + production,σ (λ i ) 7F i , for E f = 0.01 and 0.1 eV. Each of the cross sections converge against the number and chosen forms of {Φ υ }, and the choice of radial grid for finite difference method.
In Ps(1s) +H(1s) scattering, the dominant contribution comes from d-wave collision (λ i = 2) with cross sectionσ (2) 7,1 ∼ 9a 2 0 . The total cross section ofH + production almost converges up to λ i = 4. The cross Table 2. Partial wave cross sections ofH + production,σ (λ i ) 7F i , at 0.01 eV and 0.1 eV in kinetic energy betweenH + and e − . The cross sections are in atomic unit (a 2 0 ). λ i is angular momentum of relative motion between Ps andH. Digits in parentheses denote the uncertainty in the last digit estimated by the difference of S αα i and S α i α . section at 0.01 eV is similar to that at 0.1 eV for all partial waves. This is because the change of collision energy, i.e. ΔE i /E i is small. Ps(2s) +H(1s) scattering also has the largest contribution of the d-wave collision inH + production. TheH + production cross section from Ps(2s) is much larger than that from Ps(1s). On the other hand, Ps(2p) +H(1s) scattering has the largest contribution of the f-wave collision (λ i = 3) inH + production while the g-wave contribution is the second dominant in the totalH + production cross section. The total cross section ofH + production from Ps(2p) tends to converge with increasing λ i . Comparing to the case of Ps(n = 1, 2), fewer partial waves are needed to converge the Ps(n = 3) cross sections which is understandable in view of lower collision energy. Each partial wave cross section changes substantially between E f = 0.01 and 0.1 eV, since in this case ΔE i /E i is large. Figure 2 shows theH + production (rearrangement) cross section σ rearr = σ 7F i against the collision energy E i together with elastic/inelastic scattering cross sections. For Ps(1s) +H(1s) collision (F i = 1), theH + production cross section is given by σ rearr = σ 71 , the elastic scattering cross section by σ ela = σ 11 , and Ps excitation cross section by σ ex = σ 21 + σ 31 + σ 41 + σ 51 + σ 61 . The zero-energy limit of σ ela is also confirmed to give the scattering length 4.329 a 0 , in agreement with the results in references [33,36,38]. Thē H + production whose channel opens at 6.0486 eV is the second dominant process after the elastic scattering. In the present energy region, these cross sections are nearly constant. σ ex is dominated by the excitation to n = 2 states followed by the n = 3 states whose channel opens at 6.0469 eV; the latter is smaller by one order of magnitude. The σ rearr starts at the finite value and is also constant in agreement with Wigner's threshold law [54] for the case when the final fragments are Coulomb attractive and when their relative velocity is much smaller than the velocity of the initial fragments. Figures 2(b) and(c) shows the results of Ps(n = 2) +H(1s) collisions (F i = 2, 3). Again, we find that σ rearr is finite at the threshold energy. For the case of Ps(2s) collision (F i = 2), theH + production cross section σ rearr = σ 72 should be compared with Ps excitation cross section σ ex = σ 42 + σ 52 + σ 62 and deexcitation cross section σ dex = σ 12 . Like the Ps(1s) collision, Ps(n = 2) +H(1s) collisions produceH + as the dominating inelastic process in scattering. One also finds that σ ex is zero at threshold and grows in agreement with Wigner's threshold law for neutral fragments (each partial wave growing as E λ f +1/2 f ). The behaviour of cross sections for Ps(2p) collision is similar to that for Ps(2s) collision, although the σ rearr from Ps(2p) is slightly larger than that from Ps(2s).

4.H + production and the elastic/inelastic branches
Turning to Ps(n = 3) the results for the l = 0, 1, 2 (F i = 4, 5, 6) are shown in figures 2(d), (e) and (f). σ rearr decreases rapidly just above the threshold energy, and is almost constant at higher energies. The deexcitation of Ps(3l) states is seen to increase with l. σ dex = σ 1F i + σ 2F i + σ 3F i obey Wigner's threshold law  [40], DWBA and FBA [53] and CPA [39]. The light-blue vertical arrows indicate the threshold energies ofH + production in our calculation. For comparison with our data, the results of references [39,53] have been multiplied by factor 4 as to account for the spin statistics of the positronic spin singlet scattering. The right-head arrows indicate theH + production cross section of CDW-FS(UC) at the lowest energy considered in reference [40].
for the case where the final fragments move fast and the initial fragments collide slowly, that is, the partial wave cross sections are proportional to E λ i −1/2 i , and thus σ dex are finite at the threshold ofH + formation. The major difference between the Ps(n = 3) collision and Ps(n = 2) collision is that the former results in smallerH + formation and larger Ps deexcitation. In the considered energy interval, theH + formation in Ps(3d) collisions is on average an order of magnitude smaller than in Ps(2p) collisions. At the same time, the cross section for Ps(n = 3) deexcitation is more than one order of magnitude larger than that forH + production [whereby the deexcitation to Ps(n = 2) occurs faster than to Ps(n = 1)]. The branches of rapid inelastic scattering may suppress the flux into the e − +H + branch due to the conservation of probability.
Our results are compared withH + production cross sections in previous works. CDW-FS calculations [40] were performed with several different wavefunctions ofH + and predicted monotonically decreasinḡ H + production cross sections. CDW-FS(UC) uses an uncorrelated Chandrasekhar wavefunction that reproducesH + three-body energy with error of binding energy, 0.38 eV. In the present work, we have used a more preciseH + wavefunction that reproducesH + three-body energy with more than 6 digits accuracy. That, together with very careful four-body description and coupling of all open channels, enabled us to predict the cross sections very close to theH + formation threshold. Comparing our results with previous works, CDW-FS overestimated the σ rearr for Ps(2s, 2p, 3p and 3d). In contrast, CPA underestimated σ rearr for Ps(1s, 2s). Our results forH + production in Ps(n = 3) +H(1s) scattering are much smaller than those calculated by FBA [53].
In relating the present results to the experiments one should bear in mind the particular experimental conditions. For instance, in the conditions of the GBAR experiment, with an almost stationary laser-excited Ps target, most of the center-of-mass energy comes from theH (orp colliding with the Ps target) impact energy, which means that theH + production requires 1 + mH/m Ps = 919.6 times larger kinetic energy ofH in the laboratory frame. In addition, since theH + production is allowed when the two positrons (i.e. one in the Ps and the other in theH) couple to the singlet state and is not allowed for coupling in the triplet state, the net cross section of σ 7F i in figure 2 should be reduced by the spin weight 1/4 7 .

Conclusion
In conclusion, the near-threshold production ofH + in the collision of Ps(nl) +H(1s) occurs as the most dominant inelastic process when n = 1, 2, and the second dominant one when n = 3. Close to the threshold energy, the largestH + production cross section is predicted for n = 2, followed by n = 1 and n = 3. We also find that the depletion of Ps(n = 2) population by the reactions competing withH + formation is smaller than the similar depletion of Ps(n = 3). While our results are limited to lower energies, we findH + production cross sections significantly smaller than previous calculations based on high-energy approximations. The presented results have been obtained by an ab initio calculation that does not contain any modeling approximations and rigorously deals with the non-adiabatic, multi-channel character of the composite-particle scattering. The numerical accuracy is reported in terms of the unitarity of the scattering matrix, and the scattering cross sections are seen to follow the trends predicted by the general scattering theory.