PHYSICAL REVIEW D 102, 113005 (2020) Pionium as a source of false events in the K → πνν¯ decays 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 1 June 2020; revised 3 November 2020; accepted 20 November 2020; published 18 December 2020) We suggest that the decay modes of kaons with a pion and a pionium (πþπ− atom) in the final state can constitute a not yet considered background to the very rare decay K → πνν¯. In fact, a part of pioniums may escape the decay region before decaying into two π0s (or to π0π0γ in the case of excited pionium). To illustrate the importance of this background, we show that it may even explain, under some assumptions, the unexpected KL decay events that appeared in the KOTO experiment. DOI: 10.1103/PhysRevD.102.113005 Two important experiments investigating the rare kaon this surprising result. The analysis performed in Ref. [9] decays in flight are currently running. The main aim of both shows that if the four events in the KOTO signal region were is to test the Standard Model and to constrain new physics real, it would mean a branching fraction of the underlying theories by precisely measuring the very rare kaon decays mechanism equal to into a pion and two neutrinos. The NA62 experiment at the CERN Super Proton Synchrotron [1–4] deals with pos- B K → π0νν¯ 2 1þ2.0ðþ4.1Þ 10−9 ð L ÞKOTO ¼ . −1.1 −1.7 × ð3Þ itively charged kaons Kþ and aims to collect enough ð Þ Kþ → πþνν¯ events to get a signal to background ratio at the 68(95)% confidence level. 10∶1 þ − of . The Standard Model predicts the branching The π π atom, pionium (usually denoted as A2π), was fraction [5] for this decay [6], discovered in 1993 at the Institute of High Energy Physics at Serpukhov, Russia [12] and intensively studied in the B Kþ → πþνν¯ 8 4 1 0 10−11: ð Þ¼ð . Æ . Þ × ð1Þ Dimeson Relativistic Atomic Complex (DIRAC) experi- ment [13] at the CERN Proton Synchrotron. In these The KOTO experiment [7] is being conducted at the Hadron experiments, the pioniums were produced by the proton Experimental Facility at the Japan Proton Accelerator beam impinging on a target. In the same target, the Research Complex. It was designed to observe the decay pioniums broke-up into their constituents with approxi- K → π0νν¯ L of long-lived neutral kaons. The theoretical mately equal energies and small relative momenta. branching fraction [6] is The decay of pionium to two neutral pions is dominant 0 −11 and the measured lifetime is [13] BðKL → π νν¯Þ¼ð3.4 Æ 0.6Þ × 10 : ð2Þ þ0.28 −15 τ1 ¼ 3.15−0 26 × 10 s: ð4Þ Until recently, both rare kaon decay experiments proceeded s . as expected, slowly pushing down the upper bounds The NA48=2 Collaboration at the CERN SPS [14] studied of branching fractions. However, in September 2019, KÆ → πÆπ0π0 decays and found an anomaly in the π0π0 Satoshi Shinohara (on behalf of the KOTO Collaboration) invariant mass distribution in the vicinity of 2mπþ that can be [8] announced the presence of four events in the signal interpreted as the production of pioniums in the kaon decays 0 10 0 02 region in the situation where mere . Æ . events and their subsequent two-π0 decay. For our later consid- were expected. Very soon, several papers appeared, e.g., 48=2 – erations, it is important that the NA Collaboration in Refs. [9 11], aimed at finding new physics interpretations of their seminal paper [14] determined the branching ratio Æ Æ ΓðK → π A2πÞ R ¼ ¼ð1.61 Æ 0.66Þ × 10−5: ð5Þ Published by the American Physical Society under the terms of Γ KÆ → πÆπþπ− the Creative Commons Attribution 4.0 International license. ð Þ Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, Another important discovery concerns the exited pio- and DOI. Funded by SCOAP3. nium. The DIRAC Collaboration observed [15] so-called 2470-0010=2020=102(11)=113005(5) 113005-1 Published by the American Physical Society PETER LICHARD PHYS. REV. D 102, 113005 (2020) πþπ− 1 long-lived atoms, which are the 2p atomic states E EEà Ppà ; 0 PC −− aÆ ¼ ð a Æ aÞ (A2π) with quantum numbers J ¼ 1 . Its lifetime was M measured in Ref. [15] with the result where M, E, and P are the mass, energy, and momentum K Eà M2 m2 = 2M pà τ 0 45þ1.08 10−11 : of the L, respectively, a ¼ð þ aÞ ð Þ, a ¼ 2p ¼ . −0.30 × s ð6Þ 2 2 ðM − maÞ=ð2MÞ. Numerically, Ea− ¼ 0.497 GeV and Ea ¼ 1.456 GeV. Such a long lifetime is caused by the fact that the decay þ To simplify the consideration further [17], we will modes to the positive C-parity states π0π0 and γγ are now assume that all pioniums have the same laboratory momen- forbidden and the slow 2p → 1s transition dominates. After tum, given as the mean value, reaching the 1s state, a decay to two π0s quickly follows: 0 0 0 Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi → 2πγ → π π γ Ea A2π A . 1 þ 2 2 1 The decay of the charged kaons into excited pionium A0 pa ¼ E − madE ¼ 2π Ea − Ea− E 2ðEa − Ea−Þ has not been reported yet. Neither has been the neutral kaon þ a− þ E p decay into any pionium. E p − E p − m2 aþ þ aþ ; × aþ aþ a− a− a log ð9Þ In order to show that the pioniums may contribute to the Ea− þ pa− background in the K → πνν¯ experiments by important pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p E2 −m2 p 0 880 = amount, we will estimate it in the KOTO experiment. To where aÆ ¼ aÆ a. Numerically, a ¼ . GeV c. this end, we need at least a crude estimate of the branching The probability that pionium travels the path s without 0 fraction of decay KL → π A2π. We assume that the branch- decaying is given by ing ratio SðsÞ¼expf−s=lg; 0 ΓðKL → π A2πÞ R˜ l ¼ 0 þ − where is the mean decay length of pionium, ΓðKL → π π π Þ paτ l ¼ ; ð10Þ has the same value as that for charged kaons (5). Then we ma can write the branching-fraction estimate τ being the pionium mean lifetime. For the two types of B K → π0 R B K → π0πþπ− l 2 98þ0.27 μ l 4 2þ10.2 ð L A2πÞ¼ × ð L Þ pioniums, we get 1s ¼ . −0.25 m and 2p ¼ . −2.8 mm. L ≈ 2 × 10−6; ð7Þ If we denote the length of the signal region as , the mean survival probability of pionium at the point where it leaves the signal region is where we have also consulted Ref. [16]. Z We will also consider possibility that the pionium that 1 L l S¯ S L − z z 1 − −L=l : appears in the kaon decays is not the ground state (1s) ¼ L ð Þd ¼ L ½ expf g ð11Þ pionium, but its excited (2p) partner and that the corre- 0 sponding branching fraction are the same, Numerical values for two kinds of pionium are [18] 0 0 −6 BðKL → π A2πÞ ≈ 2 × 10 : ð8Þ S¯ 1 75þ0.16 10−6; 1s ¼ . −0.15 × S¯ 2 5þ6.0 10−3: In what follows, we will pursue those two alternatives in 2p ¼ . −1.7 × parallel. To simplify the reasoning, we will ignore the momentum Multiplying these numbers by the assumed branching fractions (7) and (8), we obtain the branching fractions of spread of the KL beam in the KOTO experiment and will K → π0 K → π0 0 use its peak value P 1.4 GeV=c [7]. The length of the L A2π and L A2π events that look like the ¼ 0 KOTO signal region is L ¼ 1.7 m. Another quantity that KL → π νν¯ events because pioniums left the KOTO decay enters the game is the pionium mass. It is given by volume undecayed, ma ¼ 2mπþ − b, where b is the binding energy. We will B 3 50þ0.32 10−12; take its coulombic value, which can be calculated from the 1s ¼ . −0.30 × ð12Þ hydrogen-atom-like formula. One obtains b ¼ 1.86 keV b 0 464 B 5 0þ12.0 10−9: for the pionium ground state and ¼ . keV for the 2p ¼ . −3.4 × ð13Þ excited 2p state. The laboratory energy of pionium with mass ma is Branching fraction (12) is by 3 orders of magnitude smaller uniformly distributed in an interval, the bounds of which than branching fraction (3), which characterizes the pres- are given by the formula ence of unexpected events in the KOTO signal region. 113005-2 PIONIUM AS A SOURCE OF FALSE EVENTS IN THE … PHYS. REV. D 102, 113005 (2020) Therefore, we will not follow the 1s pionium option any m2 − m2 p K π 230 5 =c: Tmax;νν ¼ ¼ . MeV longer. 2mK A comparison of (13) with (3) suggests that the anoma- 0 lous events in the KOTO experiment could be explained as Given a pT cut, we can restrict the masses of X s that can undecayed 2p pioniums. But to make a realistic compari- be detected as son, we must take into account that the experimental rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K → π0νν¯ K → π0 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi efficiencies for the L and L A2π are differ- 2 2 2 2 m 0 < m m − 2m m p : ent.