Anomalous magnetic moment of the positronium ion
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Anomalous magnetic moment of the positronium ion
Yi Liang, Paul L. McGrath, ∗ and Andrzej Czarnecki Department of Physics, University of Alberta,Edmonton, Alberta T6G 2E1, Canada
We determine the gyromagnetic factor of the positronium ion, a three-body systemconsisting of two electrons and a positron, including first relativistic corrections. Wefind that the g -factor is modified by a term − . α , exceeding 15 times the α correction for a free electron. We compare this effect with analogous results foundpreviously in atomic positronium and in hydrogen-like ions. I. INTRODUCTION
The positronium ion Ps − is a bound state of two electrons and one positron. Discoveredin 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 beenprecisely predicted [5–11]. The recent progress has occurred thanks to the prospect of intensepositron sources on the experimental side [12–14] and by improved variational calculationsof the three-body wave function and incorporation of relativistic and some radiative effectson the theory side.In this paper we focus on the magnetic moment of this three-body system. In its groundstate the two electrons are in a spatially symmetric wave function forming a spin singlet tomake their total wave function antisymmetric. Thus the whole magnetic moment is due tothe positron and, if we neglect the bound-state effects, it is given by g ~ e m where g is thegyromagnetic ratio of a free positron (or electron), g = 2 + απ + . . . , and α ≃ / is thefine structure constant. The free-particle g factor is known since recently to the astonishingfive-loop order, O (cid:16)(cid:0) απ (cid:1) (cid:17) [15].The purpose of this paper is to determine to what extent the interaction of the positronwith the two electrons modifies the magnetic moment of the ion. This effect is expectedto be analogous to that in hydrogen-like atoms and ions, where the nuclear electric fieldmodifies the g factor of an electron [16], and thus be a correction of order α , enhancedrelative to the free-particle effects in this order in the coupling constant. Effects of thisorigin have been studied with high precision in hydrogen-like ions [17–19]. Combined withmeasurements with a five-fold ionized carbon [20–22] they are the basis of the most precisedetermination of the electron mass. II. HAMILTONIAN
We are interested in the lowest-order relativistic corrections, or effects O (1 /c ) (equiva-lently α ). To this order, the Hamiltonian describing the two electrons (labels 1 and 2) and ∗ Present address: Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L3G1, Canada the positron (label 3) consists of the kinetic energy H , the spin orbit interaction H , thespin-other orbit term H and the magnetic moment interaction H . We number the terms inthe Hamiltonian in a way consistent with previously published results [23]. The expressionsare simplified since all particles have equal masses, m = m = m ≡ m , H = Π m + Π m + Π m + e r − e r − e r (II.1) H = − e m c s · r × Π r − e m c s · r × Π r (II.2) H = e m c s · r × Π r + e m c s · r × Π r (II.3) H = − emc s · B (cid:18) − Π m c (cid:19) (II.4)where r ij ≡ r i − r j . We only retain the terms that can contribute to the magnetic moment inthe desired order α . The terms proportional to the electron spins s and s are symmetricin the particle indices and . However, the Ps − wave function is antisymmetric in and . Therefore the expectation values of these terms are zero, and they have been omitted.Note that in the expression for H in [23], there is a factor mc missing in the denominatorof 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. Onthe other hand, we determine the wave function in the center of mass (CM) system of theion. In order to calculate the magnetic moment, we need the Hamiltonian expressed in theCM variables. This can be achieved by using the Krajcik-Foldy (KF) relations between theCM and LAB variables [24]. It turns out however that most of the terms of those relationsdo not contribute to the O ( α ) correction to the g factor and we only need r i = ρ i + X j σ j × π j mM c , p i = π i , s i = σ i , (III.1)where r i , p i , and s i are the LAB variables of the i th particle, and ρ i , π i , and σ i are thecorresponding CM variables. M is the total mass of the system. We choose the center ofmass as the origin, R = 0 . None of the terms dependent on the total momentum of thesystem were found to contribute to the magnetic moment to order α , 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 simpleone-electron atoms can be reproduced.
A. Positronium
Positronium is a two-body system with the symmetry due to equal masses, so the Hamil-tonian simplifies. Among the parts of the Hamiltonian shown in eqs. (II.1-II.4), only H , , , contribute to the order α . 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 , , , we have to account for the spin of the electron (not included in (II.2-II.4) in antici-pation of cancellations in Ps − , due to the symmetry of its wave function). This is achievedby replacing s → s − s .We set e = − e = e and π = − π = π . Neglecting terms containing R and P , we findthat in the transformation LAB → CM, eq. (III.1), the only term relevant for the Ps atom is r i → ρ i + X j σ j × π j mM c = ρ i + ( σ − σ ) × π m c , (IV.1)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 /c , we only need to applyit to the lowest order term H , where it affects the vector potential in the kinetic term.The resulting contribution to the magnetic moment is (here and below we average over thedirections of position and momentum, since we are interested in the S-wave ground state), Π = (cid:16) p − e c A (cid:17) → − (cid:26) π , e c B × ( σ − σ ) × π m c (cid:27) → e m c ( σ − σ ) · B π . (IV.2)The same effect arises from the kinetic energy of the positron. In total, Π + Π m → e m c ( σ − σ ) · B π . (IV.3)The next corrections are expressed by position operators of e ± . We have, after the transfor-mation to CM, r → ρ ≡ − r , r → ρ ≡ + r , and r → − r . The sum of terms 3 and 4in the Hamiltonian, eqs. (II.2-II.3), gives the magnetic interaction H + H → e m c ( σ − σ ) · r × (cid:0) e B × r (cid:1) r → e m c r ( σ − σ ) · B . (IV.4)Finally, H gives H → − emc ( σ − σ ) · B (cid:18) − π m c (cid:19) . (IV.5)The total magnetic moment interaction is the sum of (IV.3-IV.5). Its expectation value withthe ground state spatial part of the wave function gives − emc ( σ − σ ) · B (cid:28) − π m c − π m c − e mcr (cid:29) = − emc ( σ − σ ) · B (cid:18) − α (cid:19) , (IV.6)confirming the well known result [25–27]. The resulting interaction does not have diagonalelements neither in spin singlet nor triplet states of Ps. However, it mixes the m = 0 stateof the triplet with the singlet. Measurements of the resulting splitting among the oPs statesdetermine the hyperfine splitting of positronium. B. Hydrogen
In hydrogen there are further simplifications, since the spin-other orbit term H does notcontribute in the leading order, due to the suppression by the proton mass. Also, there isno difference between the LAB and the CM frames, in the leading order in /M . Thus only H and the spin-orbit term H contribute (we replace s → s and r → r ), H → e m c s · r × (cid:0) e c B × r (cid:1) r → e m c r s · B H → emc s · B (cid:18) − π m c (cid:19) and the total magnetic moment interaction in the ground state of H becomes emc s · B (cid:28) − π m c + e mc r (cid:29) = emc s · B (cid:18) − α (cid:19) , (IV.7)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, thisis the closest one to the positronium ion, which is also charged and in which recoil effectsare 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, foreasier comparison with ref. [25] r e → R + MM + m r + s e × p e m ( M + m ) , r p → R − mM + m r + s e × p e m ( M + m ) . (IV.8)This introduces the spin interaction into the kinetic energy term H , H = Π e m + Π p M → − e s · B M + m ) (cid:18) m + ZmM (cid:19) (cid:10) π (cid:11) , (IV.9)and in the ground state h π i = Z α µ where µ = MmM + m is the reduced mass.If the nuclear mass is taken as finite, the spin-orbit and spin-other orbit terms become H + H → α m r s · r × Π e − αmM r s · r × Π p → eZ α µ m M ( M + m ) s · B (cid:0) M − Zm (cid:1) . (IV.10)Finally, the last correction comes from H , H → e s · B m (cid:18) − Z α µ m (cid:19) . (IV.11)The sum of (IV.9, IV.10, IV.11) gives the total magnetic moment interaction in the ion, e s · B m (cid:18) − Z α M (3 m + 2 M ) + Zm (3 M + 2 m )6( M + m ) (cid:19) , (IV.12)in agreement with Eq. (43) in [25]. We note that the correction is symmetric with respectto the exchange of the electron and nucleus mass and charge, M ↔ m , Z ↔ ; in the limit M ≫ m reproduces our non-recoil result (IV.7); and in the limit Z → , M → m agreeswith 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 g = 2 (cid:20) − (cid:28) π m (cid:29) − (cid:28) π m (cid:29) − αm (cid:28) ρ · ρ ρ (cid:29)(cid:21) , (V.1)where the first two terms arise from H , the third from H , and the last one from H + H .We use the notation ρ ij = ρ i − ρ j , π ij = −∇ ij .For the expectation value we use the wave function found using the variational calculationas described in [8] (see Appendix) and find g Ps − = g free + ∆ g bound , ∆ g bound = − . α . (V.2)Here g free = 2 h α π − . (cid:0) απ (cid:1) + . . . i 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 ) , in g free , about15 times. Our final prediction for the gyromagnetic factor of the positronium ion is g Ps − = 2 . . (V.3)We see that the correction (V.2) is smaller in magnitude than in hydrogen, Eq. (IV.7), whereit is − . α , but larger than in the positronium atom, Eq. (IV.7), − . α . Indeed, thisconfirms the naive expectation that the value should be in between these two and closer topositronium. The entire magnetic moment of the three-body ion can be thought of as beingdue to the magnetic moment of the positron, whose gyromagnetic ratio g is modified by thebinding to the two electrons. If the two electrons are considered as a kind of a nucleus inwhose field the g factor of the positron is modified, it is heavier than in the positroniumatom, 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 mea-surement could be as follows. An ion with a known initial polarization could be subjected tothe 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 bedetected. Such a measurement, if precise enough to detect the binding effects obtained inthis study, would provide a valuable insight into the inner structure of this exotic system. ACKNOWLEDGMENTS
We thank Vladimir Shabaev for very helpful suggestions and Mariusz Puchalski for adviceon the numerical implementation of the variational method. This research was supportedby Science and Engineering Research Canada (NSERC).
Appendix A: Optimization and expectation values of operators
Here we briefly describe how the operators in Eq. (V.1) are evaluated using the variationalmethod. We expand the trial wave function in an explicitly correlated Gaussian basis,following the steps described in a study of the di-positronium molecule [28], φ = N X i =1 c i exp " − X a
A69 , 37 (1970).[28] M. Puchalski and A. Czarnecki, Phys. Rev. Lett.101