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Possible Manifestation of the 2P Pionium in Particle Physics Processes

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Possible Manifestation of the 2P Pionium in Particle Physics Processes PHYSICAL REVIEW D 102, 073004 (2020) Possible manifestation of the 2p pionium in particle physics processes Peter Lichard Institute of Physics and Research Centre for Computational Physics and Data Processing, Silesian University in Opava, 746 01 Opava, Czech Republic and Institute of Experimental and Applied Physics, Czech Technical University in Prague, 128 00 Prague, Czech Republic (Received 11 September 2020; accepted 29 September 2020; published 16 October 2020) 0 We suggest a few particle physics processes in which excited 2p pionium A2π may be observed. They include the eþe− → πþπ− annihilation, the V0 → π0lþl− and KÆ → πÆlþl− (l ¼ e, μ) decays, and the photoproduction of two neutral pions from nucleons. We analyze available experimental data and find that they, in some cases, indicate the presence of 2p pionium, but do not provide definite proof. DOI: 10.1103/PhysRevD.102.073004 I. INTRODUCTION The DIRAC collaboration recently discovered [8] so- πþπ− The first thoughts about an atom composed of a positive called long-lived atoms. These objects are apparently excited 2p states of the ground-state pionium A2π. The pion and a negative pion (pionium, or A2π in the present- day notation) appeared almost sixty years ago. Uretsky discovery was enabled by modifying the original DIRAC and Palfrey [1] assumed its existence and analyzed the setup by adding a Pt foil downstream of the production Be possibilities of detecting it in the photoproduction off target. The breakup of the long-lived states happened in that hydrogen target. Up to this time, such a process has not foil, placed at a distance of 96 mm behind the target. The magnetic field between the target and the foil does not been observed. They also hypothesized about the possibil- þ − þ þ influence the path of neutral atoms, but the π π pairs ity of decay K → π A2π, which has recently been observed in the experiment we mention below. coming from various sources are made more divergent. 2 0 Pionium was discovered in 1993 at the 70 GeV proton The longevity of p pionium (A2π) is caused by the PC ¼ 1−− synchrotron at Serpukhov, Russia [2]. The A2π atoms were fact that its quantum numbers J prevent it from 0 0 produced in a Ta target and in the same target they broke-up decaying into the positive C-parity π π and γγ states. It into their constituents with approximately equal energies must first undergo the 2p → 1s transition to the ground and small relative momenta, which distinguished them state. The mean lifetime of 2p pionium “ ” πþπ− from the free pairs. þ1.08 −11 τ2 ¼ 0 45 10 : ð Þ Using a similar method, the properties of ground-state p . −0.30 × s 1 pionium were intensively studied in the Dimeson Rela- is close to the value which comes for the πþπ− atom tivistic Atomic Complex (DIRAC) experiment at the CERN assuming a pure Coulomb interaction [8]. After reaching Proton Synchrotron [3]. A careful analysis showed that the the 1s state, a decay to two π0s quickly follows: þ0.28 −15 mean pionium lifetime is τ ¼ 3.15−0 26 × 10 s. It decays 0 0 0 . A2π → A2π þ γ → π π γ. into two neutral pions [1,4] and, to a much lesser extent, 0 The quantum numbers of A2π allow its coupling to the into two photons [5]. ρ0 48 2 electromagnetic field. Therefore, it can mediate, like the The NA = [6] experiment at CERN Super Proton meson, the interaction of neutral hadronic systems with that Synchrotron (SPS) observed a cusplike structure in the field or with a C ¼ −1 system of charged leptons and π0π0 Æ → πÆπ0π0 invariant mass distribution from K decay. photons. However, in contrast to the ρ0 meson, the coupling The enhancement can be interpreted as the contribution Æ Æ to the photon of which is fixed by the hypothesis of vector- from the decays K → π A2π (considered in [1]) followed 0 0 0 meson dominance (VMD) [9], the coupling of A2π to the by the decay A2π → π π [7]. 0 photon is unknown [10]. In addition, the width of A2π is extremely narrow. It comes out as 1.46 × 10−10 MeV if we take the central value of the measured lifetime (1). Published by the American Physical Society under the terms of In this paper, we will elaborate on some consequences of the Creative Commons Attribution 4.0 International license. 0 the fact that A2π interacts with the electromagnetic field. To Further distribution of this work must maintain attribution to 0 the author(s) and the published article’s title, journal citation, this end, we need an estimate of the A2π mass, which is 3 and DOI. Funded by SCOAP . related to the binding energy b by M ¼ 2mπþ − b. 2470-0010=2020=102(7)=073004(11) 073004-1 Published by the American Physical Society PETER LICHARD PHYS. REV. D 102, 073004 (2020) Assuming pure Coulombic interaction, the binding energy TABLE I. Parameters of the fits to BABAR πþπ− data [13] over of pionium can be calculated from the hydrogen-atom the full energy range assuming no pole below the threshold formula (second column) and the 2p pionium with the Coulombic binding energy b (third column) or with b ¼ 9 MeV (last column). m α2 b ¼ r ; ð2Þ No pionium b ¼ 0.464 keV b ¼ 9 MeV n 2n2 2 R1 (GeV ) 0.7086(31) 0.7114(28) 0.7060(30) where mr is the reduced mass in energy units (used M1 (GeV) 0.75629(18) 0.75645(21) 0.75646(21) throughout this paper), α ≈ 1=137.036 is the fine-structure Γ1 (GeV) 0.14361(33) 0.14374(36) 0.14377(36) 3 2 constant, and n is the principal quantum number. Putting R2 × 10 (GeV ) 7.65(28) 7.73(29) 7.74(28) n ¼ 2 for the first excited state and mr ¼ mπþ =2, we get M2 (GeV) 0.78203(18) 0.78204(18) 0.78204(18) b ¼ 0.4645 keV. The strong interactions may shift the Γ2 (MeV) 8.16(34) 8.23(35) 8.24(34) δ2 −2.02ð36Þ −2.02ð37Þ −2.02ð36Þ energies given by Eq. (2) [11]. As far as we know, there is 2 no experimental or theoretical indication that the binding R3 (GeV ) 0.477(17) 0.483(38) 0.476(38) energies of the ground (n ¼ 1) or first excited state (n ¼ 2) M3 (GeV) 1.426(12) 1.414(15) 1.413(15) Γ3 (GeV) 0.465(22) 0.456(24) 0.455(23) differ significantly from their Coulombic values (2). δ3 2.63(13) 2.71(13) 2.74(13) Nevertheless, Uretsky and Palfrey considered binding 2 R4 (GeV ) 2.63(13) 0.333(40) 0.328(39) energies even higher than 10 MeV in their analysis [1]. M4 (GeV) 1.817(18) 1.810(19) 1.809(18) Today we know that pionium decays into two neutral pions, Γ4 (GeV) 0.353(25) 0.333(30) 0.330(29) so the binding energy b must be smaller than 2ðmπþ − δ4 −1.58ð20Þ −1.31ð25Þ −1.26ð25Þ 2 mπ0 Þ ≈ 9.19 MeV. R5 (GeV ) 0.045(24) 0.033(18) 0.032(17) In this paper we will consider, besides the Coulombic M5 (GeV) 2.239(28) 2.239(23) 2.238(22) value b ¼ 0.4645 keV, the binding energy of b ¼ 9 MeV. Γ5 (GeV) 0.166(90) 0.126(85) 0.122(78) δ −0 94ð41Þ −0 71ð48Þ −0 66ð46Þ We hope that the phenomena we are going to study will be 5 . 2 able to decide between these two extreme values. R6 (GeV ) 0.0099(33) 0.0125(43) M6 (GeV) 2mπþ − b 2mπþ − b Γ6 (GeV) 0 0 II. POSSIBLE MANIFESTATION OF 2p PIONIUM δ6 −1.566ð64Þ 1.141(68) IN THE e + e − → π + π − PROCESS χ2=NDF 236.7=318 233.5=316 233.4=316 Confidence level 100% 100% 100% Recently, we have succeeded [12] in locating 2p kaonium as a bound-state pole in the amplitude of the eþe− → KþK− process. The pole corresponding to 2p Γ kaonium lies on the real axis in the complex s-plane below Here, Mi and i determine the position and width of the þ − the K K threshold. Our aim here is to find a pole in ith resonance, respectively. The residuum Ri includes the eþe− → πþπ− amplitude that would correspond to 2p product of two constants. One characterizes the coupling pionium by fitting the data on the eþe− → πþπ− cross of the ith resonance to the photon (up to the elementary section. Our experience with 2p kaonium shows that the charge e, which is taken off to form, after squaring, an α in crucial role in discovering the bound-state pole, which lies the prefactor) and the other is the coupling of the below the reaction threshold, is played by the cross section resonance to the pseudoscalar meson pair. The phases δ data at low energies, as close to the threshold as possible i regulate the interference between the resonances. We δ ¼ 0 [12]. Unfortunately, almost all eþe− → πþπ− experiments put 1 . have concentrated on the ρ=ω region or on energies above We first perform our fit over the full BABAR energy “ ” ρ ω ρ0 ρ00 ϕð1020Þ. The only exception is the BABAR experiment range assuming five standard resonances ( , , , , ρ000 [13], which in 2012 covered a wide energy range from 0.3 ) [15]. Similarly as it was done in Ref. [13] when fitting to 3.0 GeV by exploring the initial-state radiation method the data on the pion form factor, we fit the cross-section [14].
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