Anomalous magnetic moment of the positronium ion

We determine the gyromagnetic factor of the positronium ion, a three-body system consisting of two electrons and a positron, including first relativistic corrections. We find that the g-factor is modified by a term -0.51(1)alpha^2, exceeding 15 times the alpha^2 correction for a free electron. We compare this effect with analogous results found previously in atomic positronium and in hydrogen-like ions.


I. INTRODUCTION
The positronium ion Ps − is a bound state of two electrons and one positron. Discovered in 1981 [1], it is now being precisely studied with the goal of determining its lifetime [2,3], the binding energy, and the photodetachment cross section [4]. These observables have been precisely predicted [5][6][7][8][9][10][11]. The recent progress has occurred thanks to the prospect of intense positron sources on the experimental side [12][13][14] and by improved variational calculations of the three-body wave function and incorporation of relativistic and some radiative effects on the theory side.
In this paper we focus on the magnetic moment of this three-body system. In its ground state the two electrons are in a spatially symmetric wave function forming a spin singlet to make their total wave function antisymmetric. Thus the whole magnetic moment is due to the positron and, if we neglect the bound-state effects, it is given by g 2 e 2m where g is the gyromagnetic ratio of a free positron (or electron), g = 2 + α π + . . ., and α ≃ 1/137 is the fine structure constant. The free-particle g factor is known since recently to the astonishing five-loop order, O α π 5 [15].
The purpose of this paper is to determine to what extent the interaction of the positron with the two electrons modifies the magnetic moment of the ion. This effect is expected to be analogous to that in hydrogen-like atoms and ions, where the nuclear electric field modifies the g factor of an electron [16], and thus be a correction of order α 2 , enhanced relative to the free-particle effects in this order in the coupling constant. Effects of this origin have been studied with high precision in hydrogen-like ions [17][18][19]. Combined with measurements with a five-fold ionized carbon [20][21][22] they are the basis of the most precise determination of the electron mass.

II. HAMILTONIAN
We are interested in the lowest-order relativistic corrections, or effects O (1/c 2 ) (equivalently α 2 ). To this order, the Hamiltonian describing the two electrons (labels 1 and 2) and the positron (label 3) consists of the kinetic energy H 0 , the spin orbit interaction H 3 , the spin-other orbit term H 4 and the magnetic moment interaction H 5 . We number the terms in the Hamiltonian in a way consistent with previously published results [23]. The expressions are simplified since all particles have equal masses, m 1 = m 2 = m 3 ≡ m, where r ij ≡ r i −r j . We only retain the terms that can contribute to the magnetic moment in the desired order α 2 . The terms proportional to the electron spins s 1 and s 2 are symmetric in the particle indices 1 and 2. However, the Ps − wave function is antisymmetric in 1 and 2. Therefore the expectation values of these terms are zero, and they have been omitted. Note that in the expression for H 5 in [23], there is a factor mc 2 missing in the denominator of the term corresponding to the second term in the bracket of (II.4).

III. CENTER OF MASS COORDINATES
Expressions (II.1-II.4) refer to particle coordinates and momenta in the LAB frame. On the other hand, we determine the wave function in the center of mass (CM) system of the ion. In order to calculate the magnetic moment, we need the Hamiltonian expressed in the CM variables. This can be achieved by using the Krajcik-Foldy (KF) relations between the CM and LAB variables [24]. It turns out however that most of the terms of those relations do not contribute to the O (α 2 ) correction to the g factor and we only need where r i , p i , and s i are the LAB variables of the ith particle, and ρ i , π i , and σ i are the corresponding CM variables. M is the total mass of the system. We choose the center of mass as the origin, R = 0. None of the terms dependent on the total momentum of the system were found to contribute to the magnetic moment to order α 2 , so we also set P = 0.

IV. g FACTOR IN A TWO-BODY ATOMS
Before we consider the three-body ion, we show how the known corrections for simple one-electron atoms can be reproduced.

A. Positronium
Positronium is a two-body system with the symmetry due to equal masses, so the Hamiltonian simplifies. Among the parts of the Hamiltonian shown in eqs. (II.1-II.4), only H 0,3,4,5 contribute to the order α 2 . The Ps atom contains only the electron i = 1 and the positron i = 3, so all terms where the label i = 2 appears can be neglected. On the other hand, in H 3,4,5 , we have to account for the spin of the electron (not included in (II.2-II.4) in anticipation of cancellations in Ps − , due to the symmetry of its wave function). This is achieved by replacing s 3 → s 3 − s 1 .
We set e 3 = −e 1 = e and π 3 = −π 1 = π. Neglecting terms containing R and P , we find that in the transformation LAB→CM, eq. (III.1), the only term relevant for the Ps atom is while the momentum and spin transform trivially, p i → π i and s i → σ i . Since the transformation (IV.1) adds a term suppressed by 1/c 2 , we only need to apply it to the lowest order term H 0 , where it affects the vector potential in the kinetic term. The resulting contribution to the magnetic moment is (here and below we average over the directions of position and momentum, since we are interested in the S-wave ground state), The same effect arises from the kinetic energy of the positron. In total, The next corrections are expressed by position operators of e ± . We have, after the transformation to CM, r 1 → ρ 1 ≡ − r 2 , r 3 → ρ 3 ≡ + r 2 , and r 13 → −r. The sum of terms 3 and 4 in the Hamiltonian, eqs. (II.2-II. 3

), gives the magnetic interaction
Finally, H 5 gives (IV.5) The total magnetic moment interaction is the sum of (IV.3-IV.5). Its expectation value with the ground state spatial part of the wave function gives (IV.6) confirming the well known result [25][26][27]. The resulting interaction does not have diagonal elements neither in spin singlet nor triplet states of Ps. However, it mixes the m = 0 state of the triplet with the singlet. Measurements of the resulting splitting among the oPs states determine the hyperfine splitting of positronium.

B. Hydrogen
In hydrogen there are further simplifications, since the spin-other orbit term H 4 does not contribute in the leading order, due to the suppression by the proton mass. Also, there is no difference between the LAB and the CM frames, in the leading order in 1/M. Thus only H 5 and the spin-orbit term H 3 contribute (we replace s 3 → s and r 13 → r), and the total magnetic moment interaction in the ground state of H becomes in agreement with the classic result by Breit [16].

C. Hydrogen-like ions, including recoil effects
Now we consider an ion consisting of a nucleus with charge Ze and a single electron with −e. Among the systems, for which binding effects on the g factors have been evaluated, this is the closest one to the positronium ion, which is also charged and in which recoil effects are not suppressed, since there is no heavy nucleus.
Since we have already established which terms are relevant to the order we need, we set c = 1 from now on. The relevant terms of the KF transformation become, using r ≡ r e − r p , m for the mass of the electron and, only in this section, M for the mass of the nucleus, for easier comparison with ref. [25] r e → R + M M + m r + s e × p e 2m(M + m) , .
This introduces the spin interaction into the kinetic energy term H 0 , and in the ground state π 2 = Z 2 α 2 µ 2 where µ = M m M +m is the reduced mass. If the nuclear mass is taken as finite, the spin-orbit and spin-other orbit terms become (IV.10) Finally, the last correction comes from H 5 , The sum of (IV.9, IV.10, IV.11) gives the total magnetic moment interaction in the ion, es · B m 1 − Z 2 α 2 M 2 (3m + 2M) + Zm 2 (3M + 2m) 6(M + m) 3 , (IV. 12) in agreement with Eq. (43) in [25]. We note that the correction is symmetric with respect to the exchange of the electron and nucleus mass and charge, M ↔ m, Z ↔ 1; in the limit M ≫ m reproduces our non-recoil result (IV.7); and in the limit Z → 1, M → m agrees with the correction in the positronium atom (IV.6).

V. POSITRONIUM ION
For the positronium ion, the correction arises in a way similar to the Ps atom. Setting c = 1, we find where the first two terms arise from H 5 , the third from H 0 , and the last one from H 3 + H 4 . We use the notation ρ ij = ρ i − ρ j , π 2 ij = −∇ 2 ij . For the expectation value we use the wave function found using the variational calculation as described in [8] (see Appendix) and find g Ps − = g free + ∆g bound , Here g free = 2 1 + α 2π − 0.328 α π 2 + . . . is the g-factor of a free electron [15]. The error in (V.2) arises primarily from higher-order binding corrections, beyond the scope of this paper. Note that the binding correction (V.2) exceeds the same order effect, O (α 2 ), in g free , about 15 times. Our final prediction for the gyromagnetic factor of the positronium ion is We see that the correction (V.2) is smaller in magnitude than in hydrogen, Eq. (IV.7), where it is −0.67α 2 , but larger than in the positronium atom, Eq. (IV.7), −0.42α 2 . Indeed, this confirms the naive expectation that the value should be in between these two and closer to positronium. The entire magnetic moment of the three-body ion can be thought of as being due to the magnetic moment of the positron, whose gyromagnetic ratio g is modified by the binding to the two electrons. If the two electrons are considered as a kind of a nucleus in whose field the g factor of the positron is modified, it is heavier than in the positronium atom, but much lighter than in hydrogen. Can this quantity be measured? The main challenge is the very short lifetime of the ion, only four times longer than that of the atomic parapositronium, or about half a nanosecond. With an intense beam and a strong external magnetic field, a possible scenario of a measurement could be as follows. An ion with a known initial polarization could be subjected to the magnetic field, where its polarization (the direction of the positron spin) would precess. The annihilation process occurs predominantly within a spin-singlet electron-positron pair, so that the total spin direction of the ion is preserved by the surviving electron, and can be detected. Such a measurement, if precise enough to detect the binding effects obtained in this study, would provide a valuable insight into the inner structure of this exotic system.