Positronium Energy Levels at Order Mα7: Vacuum Polarization

Physics Letters B 747 (2015) 551–555

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Physics Letters B

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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

Article history: We have calculated all contributions to the energy levels of parapositronium at order mα7 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 measurements 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. The naive size of O (mα7) corrections is only mα4(α/π)3 = 4.39 kHz, but contributions as large as several tenths of a MHz Fig. 1. The three types of O (mα7) 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- Additional O (mα7) contributions have recently been obtained [23, photon-irreducible two-loop vacuum polarization function. This function is repre- 29–31]. The present work is a contribution to a systematic calcula- sented by a single diagram, but there are two additional contributions that can be described as self-energy corrections inside the vacuum polarization loop that are tion of all corrections at O (mα7) 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- + − + − or not of virtual annihilation e e → nγ → e e in the descrip- played. In addition, the contributions of graphs (a) and (b) must be doubled since the correction could occur on either virtual photon. tion of the process. Among annihilation contributions, ones that involve virtual annihilation to an odd number of photons only affect orthopositronium according to charge conjugation symmetry, while The two-loop function has been given by Källén and Sabry [33], ones involving an even number of photons only affect parapositro- Schwinger [34], and in a form using “standard” notation for the nium. We consider here two-photon-annihilation processes affect- dilogarithm function [35], by Eides, Grotch, and Shelyuto [36]. One ing parapositronium and focus specifically on processes containing must note that the Källén–Sabry form is for the reducible vacuum a vacuum polarization correction to one or both of the annihilation polarization function (Figs. 1a plus 1b), not the irreducible function photons. This set of contributions forms a gauge invariant set, and of Fig. 1a alone. The Schwinger and Eides–Grotch–Shelyuto forms 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) . (7) χ 0 γ (P /2 − p) − m p2 − 4m2/(1 − v2) 0 i 0 χ × tr (−ieγ ν) (−ieγ μ) (11) At one-loop order the spectral function g(v) takes the form γ (P /2 − p) − m 00 2 − 2 where P = (2m, 0 ) is the positronium 4-momentum in the center- =−v (1 v /3) α g1(v) . (8) of-mass frame, p is the 4-momentum of the vacuum polarization 1 − v2 π G.S. Adkins et al. / Physics Letters B 747 (2015) 551–555 553 corrected photon, the initial (−1) is a fermionic minus sign, and the factor of 4 accounts for the graph with crossed photons and for the fact that the vacuum polarization correction could act on either photon. After evaluating the traces and some simplifications, the energy shift above takes the form 5 4 2 2 mα d p 2p R (p ) E1a = (12) π iπ 2 p2(p − P )2(p2 − p · P )2 for the ground state (n = 1). The integral over p can be evaluated either using Feynman parameters or by the method of poles – clos- ing the p0 contour with a half-circle of infinite radius in the upper or lower half plane. In either case, the result for the energy shift becomes

1 g(v) 1 + v E = dv (2 − v)(1 + v)2 ln 2v(1 − v2) 2 0 Fig. 2. The four types of two-loop corrections to the two-photon-annihilation chan- 1 − v nel containing a one-loop vacuum polarization function. Graph (a) contains a self- − (2 + v)(1 − v)2 ln energy correction. Graph (b) displays a correction having a vertex correction on 2 the same photon that is modiﬁed by vacuum polarization (this is “Vertex Type A”). Graph (c) shows a correction where the vertex and vacuum polarization affect mα5 + 2v3 ln v + 2v(1 − v2) . (13) different virtual photons (this is “Vertex Type B”). Graph (d) contains a ladder cor- π rection. Each of these graphs must be multiplied by a factor of eight to account for equivalent graphs involving crossed annihilation photons, VP corrections on ei- For the one-loop vacuum polarization of (8) the energy shift is ther photon, and the self-energy, vertex, or ladder corrections appearing at various positions in the diagram. mα6 1 EVP;1 = IVP , IVP =− ζ(2) (14) 3. Terms involving one-loop vacuum polarization π 2 6 in accord with the result of [38]. The two-loop irreducible vacuum Additional vacuum polarization corrections are shown in Fig. 2. polarization contribution of Fig. 1a is They contain a one-loop vacuum polarization part combined with self-energy (SE), vertex, or ladder corrections to the bare two- 65 10 2 1 4 photon-annihilation process. These contributions form a set that is E1a = − ζ(4) + ζ(2) ln 2 − ln 2 24 3 18 best calculated together because, after renormalization of the self- 4 161 39 energy and vertex parts, each of these graphs contains an infrared − a4 + ζ(3) − ζ(2) ln 2 divergence that only cancels when all four are summed. 3 96 8 Our calculational method is to start with a form like (11) con- 475 43 mα7 taining the one-loop vacuum polarization correction of (7) and (8) + ζ(2) − (15) 192 96 π 3 and tack on a self-energy, vertex, or ladder correction. Convenient forms for the one-loop self-energy and vertex corrections were ≡ where ζ is the Riemann zeta function and a4 Li4(1/2). We didn’t given in [40]. The renormalized self-energy function corrects the actually “do” the integral for E1a. Rather, we obtained a nu- bare electron propagator according to merical result to high precision (100 digits) and used the PSLQ algorithm [39] to obtain (15). 1 α 1 → S1 + S2(p) (19) For the reducible contribution of Fig. 1b we were able to actu- γ p − m π γ p − m ally perform the exact integral. We used the spectral function where 2 − 2 = 2v (1 v /3) λ 1 g1b(v) S1 = ln + , (20a) 1 − v2 m 2 8 v2 v v2 1 + v α2 N(p) × − − 1 − ln (16) S2(p) = dxdufSE (20b) 9 3 2 3 1 − v π 2 D(p) =− extracted from the work of Källén and Sabry [33] in (13) to obtain with fSE 1/(2u) and = − − + 1 2 19 41 mα7 N(p) 2m (1 x)γ p (γ p m) (21a) E1b = ζ(3) − ζ(2) ln 2 + ζ(2) − . (17) 6 3 36 108 π 3 xm2 D(p) = p2 − m2 − . (21b) − Finally, for the term of Fig. 1c with a one-loop vacuum polar- (1 x)u ization correction on each annihilation photon, we were not able (All parametric integrals in the self-energy and vertex corrections to achieve an exact result. The numerical value of this contribution run from 0to1.) A photon mass λ was introduced as an infrared is regulator in the course of on-shell renormalization. The vertex correction has the form mα7 E =−0.045140511 . (18) α μ μ μ 1c π 3 γ μ → V + V (p , p) + V (p , p) (22) π 1 2 3 554 G.S. Adkins et al. / Physics Letters B 747 (2015) 551–555 d4q i Table 1 tr → 2 tr (−ieγ β ) Contributions to the p–Ps energy levels coming from one-loop vacuum polarization (2π)4 γ (−P /2 + q) − m (VP) corrections combined with self-energy (SE) (Fig. 2a), vertex (Figs. 2b and c), and ladder (Fig. 2d) corrections to the two-photon-annihilations graphs. For type A i × (−ieγ ν) (−ieγ μ) (type B) vertex corrections, the VP and vertex corrections act on the same pho- γ (P /2 + q − p) − m ton (different photons). The separate contributions to the self-energy, etc., parts are 7 3 − shown along with their totals, all in units of mα /π . The infrared divergence is × i − 0 χ i contained in L ≡ ln(λ/m) I . ( ieγβ ) (25) VP γ (P /2 + q) − m 00 q2 − λ2 Term Poles result Parameters result where the factor of 2 comes because the ladder correction could SE: 1 2L − 1 ζ(2) 2L − 1 ζ(2) 6 6 occur on either side of the diagram. Again an infrared divergence, SE: 2 −0.76281(16) −0.7629639(4) SE 2L − 1.03697(16) 2L − 1.0371196(4) here arising from the binding singularity, is regulated by including a photon mass λ. The contribution of (25) can be written as Vertex A: 1 −2L + 5 ζ(2) −2L + 5 ζ(2) 12 12 Vertex A: 2 −0.2272831(5) −0.2272846(14) d4q N(q) Vertex A: 3 0.1876296(12) 0.1876304(19) −2(4πα)2 (26) 4 Vertex A −2L + 0.6457357(13) −2L + 0.6457350(24) (2π) D(q)Z(q) − + 5 − + 5 = − 2 − 2 + 2 − 2 2 − 2 = Vertex B: 1 2L 12 ζ(2) 2L 12 ζ(2) where D(q) ((P/2 q) m )((P/2 q) m )(q λ ), Z(q) Vertex B: 2 0.0693357(4) 0.0693348(6) ((P/2 + q − p)2 − m2), and Vertex B: 3 0.2046073(11) 0.2046053(16) Vertex B −2L + 0.9593322(12) −2L + 0.9593293(18) N(q) = tr γ β (γ (−P /2 + q) + m) γ ν (γ (P /2 + q − p) + m) Ladder: 1 2L + 1 ζ(2) 2L + 1 ζ(2) 3 3 Ladder: 2 0.3124063(6) 0.3124057(21) μ 0 χ − − × γ (γ (P /2 + q) + m) γ . (27) Ladder: 3 0.5448084(11) 0.5448063(10) β 00 Ladder 2L + 0.3159093(13) 2L + 0.3159108(24) We isolate the small-q singularity by use of the decomposition: where p and p are the incoming and outgoing electron momenta N(q) N(0) N(0) 1 1 and = + − D(q)Z(q) D(q)Z(0) D(q) Z(q) Z(0) − μ = μ − λ − 5 N(q) N(0) V 1 γ ln , (23a) + (28) m 4 D(q)Z(q) μ μ −N V (p , p) = dxdu , (23b) consisting of terms 1, 2, and 3, which we evaluate in turn. The ﬁrst 2 4H term in (28) contains the infrared singular part of the q integral, 2 μ −x(H − xm ) but otherwise is simply proportional to IVP because V p p = μ dxdu dz (23c) 3 ( , ) γ ¯ . 2H N(0) 1 0 χ =−4m2tr γ ν γ μ (29) The parametric functions are Z(0) γ (P /2 − p) − m 00 is proportional to the right-hand trace of (11). The IR-singular = − 2 − 2 + − 2 − 2 H (1 x) u(m p ) (1 u)(m p ) binding integral is [41] − xu(1 − u)(p − p)2 + xm2, (24a) d4q −m2 mπ λ λ ¯ 2 2 = + ln − 1 + O . (30) H = xm + z(H − xm ), (24b) iπ 2 D(q) λ m m

μ λ μ N = γ γ (p + Q ) + m γ (γ (p + Q ) + m) γλ (24c) The mπ/λ term represents the part of the ladder correction that comes from exchange of a Coulomb photon and so is just part of =− + − where Q x(up (1 u)p). the binding that created the bound state in the first place, and so The self-energy contribution is pictured in Fig. 2a. The diagram must not be included again here [42]. Or, to put it another way, the shown must be multiplied by eight to account for the two places mπ/λ term is the piece that is removed in the matching calcula- where the SE correction could occur, the two photons that the VP tion if an effective field theory formalism had been used [43,41]. could correct, and the two types of graph (with uncrossed and 7 3 So term 1 contributes 2(ln(λ/m) − 1) IVP (mα /π ) to the energy crossed photons). All eight contribute equally. There are two parts shift. Terms 2 and 3 of the ladder correction were constructed to to the SE contribution: S1 and S2(p) of (19). The S1 contribution be IR safe and can be evaluated directly using λ → 0. (It is useful 7 3 6 is 2S1 IVP(mα /π ) where IVP is the O (mα ) VP correction of (14). to average over the directions of q when performing the d3q inte- 4 For the S2(p) contribution we could preform the d p integral as a gration as described in [44].) The results for the various parts of 0 whole using Feynman parameters, or do dp first by poles followed the ladder correction are reported in Table 1. by d3 p → 4π p2dp numerically. Both results are shown in Table 1, followed by their total. 4. Results and discussion There are two classes of vertex corrections: ones with the VP and vertex corrections affecting the same photon (type A, Fig. 2b) The results for all VP corrected two-photon-annihilation graphs or affecting different photons (type B, Fig. 2c). Each has three parts at O (mα7) are shown in Table 2. We see that all infrared di- as specified in (22) and (23). The various vertex correction inte- vergences cancel in the sum. The ground state parapositronium grals were performed both by poles and parameters, with results energy level correction due to vacuum polarization effects in the recorded in Table 1. two-photon-annihilation channel is The ladder correction shown in Fig. 2d gives an energy shift that looks like (11) except that the right-hand trace of (11) is re- mα7 placed by E =−0.153095(3) . (31) π 3 G.S. Adkins et al. / Physics Letters B 747 (2015) 551–555 555

Table 2 [14] V.G. Baryshevsky, O.N. Metelitsa, V.V. Tikhomirov, S.K. Andrukhovich, A.V. Positronium energy level corrections at O (mα7) coming from vacuum polarization Berestov, B.A. Martsinkevich, E.A. Rudak, Observation of time oscillation in corrections to two-photon-annihilations graphs. The tabulated results are contribu- 3γ -annihilation of positronium in a magnetic field, Phys. Lett. A 136 (1989) tions to I where E = (mα7/π 3)I. The poles and parameters results from Table 1 428–432. were combined to give the results shown here. The infrared divergence contained [15] S. Fan, C.D. Beling, S. Fung, Hyperfine splitting in positronium measured in L = ln(λ/m) IVP cancels in the sum. through quantum beats in the 3γ decay, Phys. Lett. A 216 (1996) 129–136. [16] P. Crivelli, C.L. Cesar, U. Gendotti, Advances towards a new measurement of the Term Diagram Energy shift 1S − 2S transition of positronium, Can. J. Phys. 89 (2011) 29–35. irreducible 2-loop VP 1a −0.9205630 [17] Y. Sasaki, A. Miyazaki, A. Ishida, T. Namba, S. Asai, T. Kobayashi, H. Saito, reducible 2-loop VP 1b −0.0712481 K. Tanaka, A. Yamamoto, Measurement of positronium hyperfine splitting with 1-loop VP on each 1c −0.0451405 quantum oscillation, Phys. Lett. B 697 (2011) 121–126. VP – SE 2a2L − 1.0371196(4) [18] A. Ishida, Y. Sasaki, G. Akimoto, T. Suehara, T. Namba, S. Asai, T. Kobayashi, VP – vertex A 2b −2L + 0.6457355(12) H. Saito, M. Yoshida, K. Tanaka, A. Yamamoto, Precise measurement of the VP – vertex B 2c −2L + 0.9593313(10) positronium hyperfine splitting using the Zeeman method, Hyperfine Interact. VP – ladder 2d2L + 0.3159096(12) 212 (2012) 133–140. [19] T. Yamazaki, A. Miyazaki, T. Suehara, T. Namba, S. Asai, T. Kobayashi, H. Saito, total −0.153095(3) I. Ogawa, T. Idehara, S. Sabchevski, Direct observation of the hyperfine transition of ground-state positronium, Phys. Rev. Lett. 108 (2012) 253401. [20] T. Namba, Precise measurement of positronium, Prog. Theor. Exp. Phys. 2012 The numerical value of this correction is small due to large cancel- (2012) 04D003. lations among the various terms. This parapositronium shift cor- [21] D.B. Cassidy, T.H. Hisakado, H.W.K. Tom, A.P. Mills Jr., Positronium hyperfine rects the hfs (spin-triplet minus spin-singlet) by the amount interval measured via saturated absorption spectroscopy, Phys. Rev. Lett. 109 (2012) 073401. [22] A. Ishida, T. Namba, S. Asai, T. Kobayashi, H. Saito, M. Yoshida, K. Tanaka, E = 0 67 kHz (32) hfs . . A. Yamamoto, New precision measurement of hyperfine splitting of positro- Higher S states with principal quantum number n have correc- nium, Phys. Lett. B 734 (2014) 338–344, http://dx.doi.org/10.1016/j.physletb. 3 2014.05.083. tions that are the same as (31) except divided by the n factor [23] G.S. Adkins, R.N. Fell, Positronium hyperfine splitting at order mα7: light-by- that comes from the square of the wave function at spatial con- light scattering in the two-photon-exchange channel, Phys. Rev. A 89 (2014) tact. States with = 0are not corrected by the terms considered 052518. 7 here at order O (mα7). Additional contributions in the two-photon- [24] B.A. Kniehl, A.A. Penin, Order α ln(1/α) contribution to positronium hyperfine splitting, Phys. Rev. Lett. 85 (2000) 5094–5097, http://dx.doi.org/ annihilation channel involve terms having a one-loop SE, vertex, or 10.1103/PhysRevLett.85.5094. ladder correction on each side of the diagram, and two-loop self- [25] K. Melnikov, A. Yelkhovsky, O (α3 ln α) corrections to muonium and positron- energy, vertex, etc., corrections on a single side of the diagram. ium hyperfine splitting, Phys. Rev. Lett. 86 (2001) 1498–1501, http://dx.doi.org/ Calculation of these terms is in progress. 10.1103/PhysRevLett.86.1498. [26] R.J. Hill, New value of mμ/me from muonium hyperfine splitting, Phys. Rev. Lett. 86 (2001) 3280–3283, http://dx.doi.org/10.1103/PhysRevLett.86.3280. Acknowledgements [27] S.R. Marcu, Ultrasoft contribution to the positronium hyperfine splitting, Mas- ter’s thesis, University of Alberta, 2011. We acknowledge the support of the National Science Founda- [28] M. Baker, P. Marquard, A.A. Penin, J. Piclum, M. Steinhauser, Hyperfine splitting 7 tion through Grant No. PHY-1404268 and of the Franklin & Mar- in positronium to O (α me ): one photon annihilation contribution, Phys. Rev. shall College Grants Committee through the Hackman Scholars Pro- Lett. 112 (2014) 120407. [29] M.I. Eides, V.A. Shelyuto, Hard nonlogarithmic corrections of order mα7 to hy- gram, Award Nos. 5789-91. perfine splitting in positronium, Phys. Rev. D 89 (2014) 111301(R). [30] G.S. Adkins, C. Parsons, M.D. Salinger, R. Wang, R.N. Fell, Positronium energy References levels at order mα7: light-by-light scattering in the two-photon-annihilation channel, Phys. Rev. A 90 (2014) 042502. [1] M. Deutsch, Evidence for the formation of positronium in gases, Phys. Rev. 82 [31] M.I. Eides, V.A. Shelyuto, Hard three-loop corrections to hyperfine splitting in (1951) 455–456, http://dx.doi.org/10.1103/PhysRev.82.455. positronium and muonium, arXiv:1506.00175 [hep-ph], 2015. [2] S. Berko, H.N. Pendleton, Positronium, Annu. Rev. Nucl. Part. Sci. 30 (1980) [32] R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. 543–581. Phys. 1 (1960) 429–433, http://dx.doi.org/10.1063/1.1703676. [3] A. Rich, Recent experimental advances in positronium research, Rev. Mod. Phys. [33] G. Källén, A. Sabry, Fourth order vacuum polarization, Dan. Mat. Fys. Medd. 53 (1981) 127–165. 29 (17) (1955) 1–10. [4] A.P. Mills Jr., S. Chu, Precision measurements in positronium, in: Advanced Se- [34] J. Schwinger, Particles, Sources, and Fields, vol. 2, Sec. 5-4, Addison–Wesley, ries on Directions in High Energy Physics, World Scientific, Singapore, 1990, Reading, MA, 1973. pp. 774–821. [35] L. Lewin, Polylogarithms and Associated Functions, Elsevier North Holland, [5] A. Rich, R.S. Conti, D.W. Gidley, J.S. Nico, M. Skalsey, J. Van House, P.W. Zitze- New York, 1981. witz, Tests of quantum electrodynamics and related symmetry principles using [36] M.I. Eides, H. Grotch, V.A. Shelyuto, Two-loop polarization contributions to positronium, in: A.O. Barut (Ed.), New Frontiers in Quantum Electrodynamics radiative-recoil corrections to hyperfine splitting in muonium, Phys. Rev. D 65 and Quantum Optics, Plenum Press, New York, 1990, pp. 257–274. (2001) 013003. [6] V.V. Dvoeglazov, R.N. Faustov, Y.N. Tyukhtyaev, Decay rate of a positronium. [37] G.S. Adkins, R.N. Fell, Bound-state formalism for positronium, Phys. Rev. A 60 Review of theory and experiment, Mod. Phys. Lett. A 8 (1993) 3263–3272. (1999) 4461–4475, http://dx.doi.org/10.1103/PhysRevA.60.4461. [7] M.I. Dobroliubov, S.N. Gninenko, A.Yu. Ignatiev, V.A. Matveev, Orthopositronium [38] G.S. Adkins, Y.M. Aksu, M.H.T. Bui, Calculation of the two-photon-annihilation lifetime problem, Int. J. Mod. Phys. A 8 (1993) 2859–2874. contribution to the positronium hyperfine interval at order mα6, Phys. Rev. A [8] S.G. Karshenboim, Precision study of positronium: testing bound state QED the- 47 (1993) 2640–2652. ory, Int. J. Mod. Phys. A 19 (2004) 3879–3896. [39] H.R.P. Ferguson, D.H. Bailey, S. Arno, Analysis of PSLQ, an integer rela- [9] A. Rubbia, Positronium as a probe for new physics beyond the standard model, tion finding algorithm, Math. Comput. 68 (1999) 351–369, http://dx.doi.org/ Int. J. Mod. Phys. A 19 (2004) 3961–3985. 10.1090/S0025-5718-99-00995-3. [10] S.N. Gninenko, N.V. Krasnikov, V.A. Matveev, A. Rubbia, Some aspects of [40] G.S. Adkins, R.N. Fell, J. Sapirstein, Two-loop renormalization of Feyn- positronium physics, Phys. Part. Nucl. 37 (2006) 321–346. man gauge QED, Phys. Rev. D 63 (2001) 125009, http://dx.doi.org/10.1103/ [11] A.P. Mills Jr., G.H. Bearman, New measurement of the positronium hyper- PhysRevD.63.125009. fine interval, Phys. Rev. Lett. 34 (1975) 246–250, http://dx.doi.org/10.1103/ [41] G.S. Adkins, R.N. Fell, J. Sapirstein, Two-loop correction to the orthopositronium PhysRevLett.34.246. decay rate, Ann. Phys. 295 (2002) 136–193. [12] A.P. Mills Jr., Line-shape effects in the measurement of the positronium hy- E.[42] W. Caswell, G.P. Lepage, J. Sapirstein, O (α) corrections to the decay rate of perfine interval, Phys. Rev. A 27 (1983) 262–267, http://dx.doi.org/10.1103/ orthopositronium, Phys. Rev. Lett. 38 (1977) 488–491. PhysRevA.27.262. E.[43] W. Caswell, G.P. Lepage, Effective lagrangians for bound state problems in [13] M.W. Ritter, P.O. Egan, V.W. Hughes, K.A. Woodle, Precision determination of QED, QCD, and other field theories, Phys. Lett. B 167 (1986) 437–442. − the hyperfine-structure interval in the ground state of positronium. V, Phys. [44]E. W. Caswell, G.P. Lepage, O (α2 ln(α 1)) corrections in positronium: hyperfine Rev. A 30 (1984) 1331–1338, http://dx.doi.org/10.1103/PhysRevA.30.1331. splitting and decay rate, Phys. Rev. A 20 (1979) 36–43.