Physics Letters B 747 (2015) 551–555 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 7 Positronium energy levels at order mα : Vacuum polarization corrections in the two-photon-annihilation channel ∗ Gregory S. Adkins , Christian Parsons, M.D. Salinger, Ruihan Wang Franklin & Marshall College, Lancaster, PA 17604, United States a r t i c l e i n f o a b s t r a c t 7 Article history: We have calculated all contributions to the energy levels of parapositronium at order mα coming from Received 11 June 2015 vacuum polarization corrections to processes involving virtual annihilation to two photons. This work is Accepted 19 June 2015 motivated by ongoing efforts to improve the experimental determination of the positronium ground-state Available online 23 June 2015 hyperfine splitting. Editor: M. Cveticˇ © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 3 Keywords: (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . Quantum electrodynamics (QED) Positronium Hyperfine splitting (hfs) 1. Introduction with citations to the original literature. The most precise hfs mea- surements were performed by two groups in the 70s and early 80s Positronium, the electron–positron bound state, is a particularly [11–13]: simple and interesting system. The constituents of positronium = have no known internal structure. The properties of positronium E(Brandeis) 203 387.5(1.6) MHz, are governed almost completely by QED – strong and weak inter- E(Yale) = 203 389.10(74) MHz. (1) action corrections are below the level of current interest due to the small value of the electron mass. On the other hand, some features Both of these results are based on the observation of Zeeman mix- of positronium tend to complicate the analysis compared to, say, ing of ortho and para states in the presence of a static magnetic hydrogen or muonium. The no-recoil approximation is not relevant field. More recently, a great deal of work has been done both with for positronium – the mass ratio for positronium takes its max- the Zeeman approach and with other indirect and direct methods imum value of one. Also, because positronium is composed of a of measurement [14–21]. A new high-precision measurement uti- particle and its antiparticle, it exhibits real and virtual annihilation lizing the Zeeman method with improved control of systematics into photons. The states of positronium can be taken to be eigen- was recently reported [22] states of the discrete symmetries charge parity and spatial parity, = making positronium useful in searches for new, symmetry break- E(Tokyo) 203 394.2(1.6)stat(1.3)sys MHz. (2) ing interactions. Because of its unique properties and accessibility Theoretical work on the positronium hfs involves calculating to high-precision experiments, positronium is an ideal system for the energy splitting by use of bound-state methods in QED. The tests of the bound state formalism in quantum field theory and for principal modern approach involves the definition of an effective searches for new physics in the leptonic sector. non-relativistic theory through matching with full QED followed Since its discovery in 1951 [1], positronium has been the object by a bound-state perturbation calculation in the effective theory. of increasingly precise measurements of the ground state hyperfine The result has the form of a perturbation series in the fine struc- splitting (hfs), orthopositronium (spin-triplet) and parapositronium ture constant α augmented by powers of = ln(1/α). This series (spin-singlet) decay rates and branching ratios, n = 2fine structure, has the form and the 2S–1S interval. This progress is reviewed in Refs. [2–10] α α 2 E = mα4 C + C + C α2 + C 0 1 π 21 20 π * Corresponding author. α3 α3 α 3 + C 2 + C + C +··· (3) E-mail address: [email protected] (G.S. Adkins). 32 π 31 π 30 π http://dx.doi.org/10.1016/j.physletb.2015.06.050 0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 552 G.S. Adkins et al. / Physics Letters B 747 (2015) 551–555 where m is the electron mass. The coefficients C0–C31 are known analytically as reviewed in Ref. [23]. The most recent result, 17 217 C31 =− ln 2 + , (4) 3 90 was obtained by three groups in 2000 [24–26]. The numerical value of the theoretical prediction, including terms through C31, is E(th) = 203 391.69 MHz (5) with an uncertainty due to uncalculated terms that has been esti- mated as 0.16 MHz [25], 0.41 MHz [24], or 0.6 MHz [23]. Theory and the older experiments are separated by 2.6σ and 3.5σ in terms of the experimental uncertainties, but theory and the new experiment are consistent with one another. 7 4 3 The naive size of O (mα ) corrections is only mα (α/π) = 4.39 kHz, but contributions as large as several tenths of a MHz 7 Fig. 1. The three types of O (mα ) pure vacuum polarization corrections in the have been found coming from “ultrasoft” energy scales [27,28]. two-photon-annihilation channel. Graph (a) represents the contribution of the one- 7 Additional O (mα ) contributions have recently been obtained [23, photon-irreducible two-loop vacuum polarization function. This function is repre- sented by a single diagram, but there are two additional contributions that can be 29–31]. The present work is a contribution to a systematic calcula- 7 described as self-energy corrections inside the vacuum polarization loop that are tion of all corrections at O (mα ) begun in anticipation of yet more not shown. Graph (b) gives the one-photon-reducible contribution. Graph (c) shows precise measurements of the positronium hfs. a final two-loop vacuum polarization correction, one with a one-loop correction on 7 Contributions to the positronium hfs at O (mα ) can be classi- each virtual photon. Graphs with crossed photons are not displayed – the crossed fied as either annihilation or exchange depending on the presence photon graphs give energy contributions equal to those of the graphs that are dis- + − + − played. In addition, the contributions of graphs (a) and (b) must be doubled since or not of virtual annihilation e e → nγ → e e in the descrip- the correction could occur on either virtual photon. tion of the process. Among annihilation contributions, ones that in- volve virtual annihilation to an odd number of photons only affect The two-loop function has been given by Källén and Sabry [33], orthopositronium according to charge conjugation symmetry, while Schwinger [34], and in a form using “standard” notation for the ones involving an even number of photons only affect parapositro- dilogarithm function [35], by Eides, Grotch, and Shelyuto [36]. One nium. We consider here two-photon-annihilation processes affect- must note that the Källén–Sabry form is for the reducible vacuum ing parapositronium and focus specifically on processes containing polarization function (Figs. 1a plus 1b), not the irreducible function a vacuum polarization correction to one or both of the annihilation of Fig. 1a alone. The Schwinger and Eides–Grotch–Shelyuto forms photons. This set of contributions forms a gauge invariant set, and furthermore is insensitive to the particular bound-state formalism are for the irreducible function. used in its evaluation, and consequently it comprises a reasonable The energy shift due to the vacuum polarization graphs con- set of contributions to be evaluated in isolation from other types of sidered in this article is insensitive to the particular bound state terms. Most of the two-, three-, and four-photon annihilation con- formalism we employ because the energy and momentum values 2 tributions have non-vanishing imaginary parts, as can be seen from that contribute are purely “hard” – of order m, not mα or mα . We Cutkosky analysis [32]. However, the vacuum polarization function choose to use the formalism of Ref. [37], in which the energy shift vanishes for small k2 (where kμ is the photon momentum), so is an expectation value the vacuum polarization corrections discussed here vanish for an- E = i δ¯ K . (9) nihilation to on-shell virtual photons. These contributions to the energy shift E are purely real and that fact simplifies their eval- The two-body states , ¯ carry information about the positron- uation considerably. ium spin state χ : 0 χ 00 2. Pure vacuum polarization corrections → φ ¯ T → φ , (10) 0 00 0 χ † 0 The “pure vacuum polarization” corrections involving either the where√ χ is the two-by-two two-particle singlet spin matrix χ = two-loop vacuum polarization function or a product of two one- 1/ 2. The positronium states, in this approximation, have van- loop functions are shown in Fig. 1. The effect of a vacuum polar- ishing relative momentum, and φ = m3α3/(8πn3) is the wave ization (VP) correction is to modify a photon propagator according 0 function at spatial contact for a state of principal quantum num- to ber n and orbital angular momentum = 0. 1 1 1 1 1 → = − (p2) +··· (6) The explicit expression for the energy shift for the vacuum po- 2 2 2 2 R 2 p 1 + R (p ) p p p larization diagram of Fig. 1a is where the renormalized scalar vacuum polarization function can 4 − − 2 d p 2 i i be expressed in a spectral form as E1a = (−1)4iφ −R (p ) 0 (2π)4 p2 (P − p)2 1 i p2 × 00 − μ − ν 2 = tr † ( ieγ ) ( ieγ ) R (p ) dvg(v) .