Hello! e+ c s γ,Z νµ µ e− + Here is an equation with diagrams: = iλ, (1) − where the intersection is the vertex. Here is another diagram: q And another one: Many-body theory for positronium-atom interactions ∗ y z D. G. Green, andA. positron R. Swann, annihilationand diagram G. F. Gribakin Centre for Theoretical Atomic, Molecular, and Optical Physics,εν School of Mathematics ε and Physics, Queen’s University Belfast, Belfast BT7 1NN, Northern Ireland,µ United Kingdom And the last one: εn δ µν νµ V nε (Dated: April 26, 2018) = ⟨ | | ⟩⟨ | | ⟩ (3) ε ε ε + ε n νµ ν µ n A many-body-theory approach has been developed to study positronium-atom interactions.! As− first− applica- tions, we calculate the elastic scattering and momentum-transferAnd here is the diagram cross with sections a Γ block: and the pickoff annihilation rate 1 Γ K Zeff for Ps collisions with He and Ne. For He the cross section is in agreementε withν1 previous coupled-stateν2 ε′ calculations, while comparison with experiment for both atoms highlights discrepancies between various sets 1 (Γ) Γ of measured data. In contrast, the calculated Zeff (0.13 and 0.26ε for′ ΣE Heε and= Ne, respectively)µ1 areµ in2 excellent, (4) agreement with the measured values. ⟨ | | ⟩ Finally, my own attempt at the 2nd-order self-energy diagrams, n and its analog for the Z contribution Positronium (Ps) is a light “atom” consisting of an electron eff ε ε µ1 εε ν1 ε µ1 ν2 ε ε ε′ and its antiparticle, the positron. It is important for precision ′ µ2 m tests of QED [1] and for understanding galactic positron annihi- µ Γ m µ ε′ ΣE ε = 2 + n + . + , (5)(2) lation [2]. It also has numerous applications, from probing⟨ free| | ⟩ µ1 ε′ µ n µ space in condensed matter systems [3] to making antihydrogen n 2n [4] and studying free fall of antimatter [5]. These applicationsTotal positron self-energy: n ε′ ν ν require understanding of Ps interaction with normal matter, ενε′ ε 1 2 ε′ 1 which is far from complete. Recent experiments on Ps scatter- Γ ε′ ΣE ε = µ + µ µ , (6) ing on noble-gas atoms revealed some unexpected trends, e.g., ⟨ | | ⟩ 1 2 that the scattering cross section becomes very small at low Ps n energies [6]. Overall, there is a large uncertainty in the existing n Ps-atom scattering data [7], while calculations of theScreening rate of diagram pickoff annihilation in noble gases (where the positron from FIG. 1. The main contributions to the self-energy of the electron (top line) and positron (bottomν1 line) in theν field2 of the atom. Lines Ps annihilates with an atomic electron) [8–14] underestimate labeled " represent the electron or positronµ HF wave function. Lines the experimental data [15, 16] by as much as a factor of ten. labeled ν (µ) represent positron (excited electron) states, which are ν2µ2 δV µ1ν1 = (7) The theoretical description of Ps-atom interactions is chal- summed⟨ over.| Lines| labeled⟩ n and mn represent holes in the atomic lenging because of the composite nature of the collision part- ground state. Wavy lines represent Coulomb interactions. The shaded Γ µ1 µ2 ners and a significant cancellation between the short-rangeAnd now, the RPAEblock diagrammatic represents the equation: sum of the electron-positron ladder-diagram series [19, 31], which accounts forν′ virtual Ps formation. ω Ps-atom repulsion and van-der-Waals attraction. Accurate cal- ν culations must account for dynamical distortion of both ob- ν ν ν′ ω ω µ′ ω jects during the collision, which has only been achieved for electron-positron= + correlation correctionsν + to the annihilation simple targets, i.e., hydrogen and helium [17]. Calculations of µ 1 µ vertex, we obtain values of Zeff in excellent agreementµ′ with µ pickoff annihilation require account of important short-range experiment [15]. Atomic units (a.u.) are used throughout. µ electron-positron correlations, which provide corrections to MBT ofω electron- and positron-atomω interactions.—MBT the annihilation vertex [18–21], but have been neglected in all describesµ′ an electron or positron in the field of a many-electron previous calculations [22]. ν′ atom via theν′ Dyson equation for the (quasiparticle) wave func- Many-body theory (MBT) is a powerful and systematic + + ν (8) tion " [34]:ν µ method of accounting for virtual excitations of both objects ′ and the electron-positron correlation effects. It provided an Hˆ ± + Σˆ ± ±¹rº = " ±¹rº: (1) µ 0 " "µ " accurate description of low-energy electron-atom scattering ˆ ± [23–28] and positron interaction with atoms [19, 20, 29–33], Here H0 is the zeroth-order2 Hamiltonian, e.g., that of the with scattering cross sections, annihilation rates, and γ spectra electron (−) or positron ¹+º in the field of the Hartree-Fock ˆ ± all found to be in excellent agreement with experiment. (HF) ground-state atom, and Σ" is the nonlocal, energy- arXiv:1709.00394v3 [physics.atom-ph] 25 Apr 2018 In this Letter we show how to describe Ps interaction with dependent correlation potential [35], equal to the electron or a many-electron atom by combining the MBT description positron self-energy in the field of the atom. Equation (1) of electron-atom and positron-atom interactions, and includ- can be solved separately for each partial wave, with the wave ± −1 ˜± ing the important effect of screening of the electron-positron function in the form " ¹rº = r P"`¹rºY`m¹rˆº, where Y`m Coulomb interaction by the atom. As first applications of is a spherical harmonic. Rather than computing the self- ± 0 the theory, we calculate phase shifts, elastic-scattering and energy ΣE ¹r; r º in coordinate space, it is more convenient 0 ˆ ± momentum-transfer cross sections, and the pickoff annihila- to work with its matrix elements h" jΣE j"i in the HF basis 1 ± ± ± ± ± −1 ± tion rate Zeff for Ps on He and Ne. The cross sections are '" , where H0 '" = "'" , '" ¹rº = r P"`¹rºY`m¹rˆº, and 0 ˆ ± ¯ ± 0 ± 0 ± 0 ± found to be in agreement with previous coupled-state [17] h" jΣE j"i = P"0`¹r ºΣE`¹r; r ºP"l¹rº dr dr , with ΣE` the and model van der Waals [7] calculations. By accounting for self-energy for partial wave `. Using the completeness of the 2 r (a.u.) r (a.u.) 0 1 2 3 4 0 1 2 3 4 Ne 4 4 0.4 (a) (d) 3 3 ) . u . 2 2 0.4 0.2 (a 0 r 1 1 Strength parameter 0 -1 -0.5 0 0.5 0 0 0.2 4 4 Energy (a.u.) (b) (e) S± 3 3 FIG. 3. The dimensionless strength parameter E` of the correlation ) . potentials for the electron (blue) and positron (red) in the field of Ne, u . 0.0 (a 2 2 as a function of energy E, for ` = 0 (circles), ` = 1 (squares), ` = 2 0 r (triangles), and ` = 3 (diamonds). 1 1 0 0 0.2 4 4 − p, and d waves in Ne, calculated at E = 0. Their key feature is a “valley” along the diagonal r = r 0, whose width characterizes 3 (c) (f) 3 the degree to which Σˆ ± is nonlocal. The main contribution ) . u . 0.4 to electron- and positron-atom attraction comes from r & 1 (a 2 2 − + 0 0 a.u. (i.e., outside the atom). Here, Σ ¹r; r º is more negative r E` − 0 1 1 than ΣE`¹r; r º, meaning stronger attraction for the positron. As a consequence of the Pauli principle, the correlation poten- 0 0 0 1 2 3 4 0 1 2 3 4 tial for the electron is quite different for different partial waves. r (a.u.) r (a.u.) It is also significantly more nonlocal than that of the positron, with prominent repulsive areas for the s and p waves. These FIG. 2. Self-energy Σ± ¹r; r 0º for Ne for the electron (a) ` = 0, (b) features are due to the contribution of the second, exchange 0` − 0 ` = 1, and (c) ` = 2; and positron (d) ` = 0, (e) ` = 1, and (f) ` = 2. diagram to ΣE`¹r; r º. The energy dependence of the electron and positron correla- tion potentials can be analyzed by examining the dimensionless ± Í ± basis, it can be expressed as strength parameter SE` = − ">0h"jΣE` j"i/" [37]. Figure3 shows S± for Ne, as a function of energy for electron and ± 0 Õ ± 0 0 ± ± E` ΣE`¹r; r º = P"0`¹r ºh" jΣE` j"iP"`¹rº: (2) positron s, p, d, and (for the electron) f waves. It confirms ";"0 that the correlation potential is stronger for the positron. It also shows that its energy dependence is relatively weak on Figure1 shows the main contributions to the electron and the energy scale of Ps (0.25 a.u.). This is important for the de- positron self-energy. For the electron (top row), the first di- scription of Ps-atom interaction, as it allows us to use Σ± ¹r; r 0º agram accounts for the attractive long-range polarization po- E` 4 calculated for a fixed energy (E = 0). tential −αd/2r , where αd is the dipole polarizability of the atom. The other three diagrams contribute only at short range. MBT of Ps-atom interactions.— The wave function Ψ of These diagrams provide a good description of the electron in- Ps in the field of the atom satisfies the two-particle Dyson teraction with noble-gas atoms [24–27][36]. For the positron equation (also known as the Bethe-Salpeter equation [34]) (bottom row), the first diagram produces a long-range polar- ˆ − ˆ − ˆ + ˆ + ization potential similar to that for the electron.