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 with a and a pionium (πþπ− ) 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 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 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 . The NA62 experiment at the CERN Super 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 . . Þ × ð1Þ Dimeson Relativistic Atomic Complex (DIRAC) experi- ment [13] at the CERN . In these The KOTO experiment [7] is being conducted at the 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 is dominant 0 −11 and the measured lifetime is [13] BðKL → π νν¯Þ¼ð3.4 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, K → ππ0π0 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 . . 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

ΓðK → π A2πÞ R ¼ ¼ð1.61 0.66Þ × 10−5: ð5Þ Published by the American Physical Society under the terms of Γ K → ππþπ− 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 , which are the 2p atomic states E EE Pp ; 0 PC −− a ¼ ð a 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 E M2 m2 = 2M p 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 a ¼ a 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.

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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. The right quantity that should be compared with (3) is X K þ π K π þ T;max

B0 R B ; p 130 =c m 0 < 281 2p ¼ × 2p ð14Þ For T;max ¼ MeV , it implies X MeV. It is obvious that ϵπ0X0 should go rapidly to zero with mX0 where R is the ratio of efficiencies, approaching 281 MeV. Figure 2 in Ref. [9] confirms that. In order to get estimates of the efficiencies, we first 0 ϵπ0 0 randomly generate the momenta within the π νν¯ R A2π : ¼ ð15Þ system in the KL rest frame using the GENBOD program ϵπ0νν¯ from the old CERN Program Library [19]. Counting the π0 Recently, the authors of Ref. [9] considered a similar ratio, events with the transverse momentum greater than the KOTO cut of 130 MeV=c, we get an efficiency ϵπ0νν¯ ϵπ0X0 of 22%. RðmX0 Þ¼ ; ð16Þ ϵπ0νν¯ Next, we consider the 2p pionium with the Coulombic binding energy, the mass of which is ma ¼ 279.14 MeV. p 0 132 0 =c where X0 is an invisible . They performed a Using the formula (17) we get Tmax;π ¼ . MeV . Monte Carlo simulation taking into account the experi- Assuming isotropic decay in the KL rest frame, we can mental cuts, acceptances, and resolutions. The dependence estimate the portion of events with pT;π0 greater than 130 =c ϵ 0 132 − 130 =132 1 5% of ratio (16) on the boson mass mX0 can be obtained from MeV as π A0 ¼ð Þ ¼ . . It means the curve in the left pane of Fig. 2 [9]. In principle, it should an efficiency ratio (15) of be possible to use their results and get ratio (15) by setting m 0 m 0 m 0 RðCoul:Þ¼0.069 X to A2π . Unfortunately, the value for A2π , which is close to 280 MeV, cannot be read from the curve. We will therefore make our own crude estimate of the efficiency and an efficiency corrected branching ratio ratio (15). B0 ðCoul:Þ¼0.34þ0.83 × 10−9: The efficiencies are most influenced by the pT cut 2p −0.23 imposed in the KOTO experiment. To suppress the events þ − 0 0 A comparison with (3) shows that if the experimental cut on from the KL → π π π ,apT momentum of π greater than π0 130 MeV=c is required over the majority of the signal transverse momentum of is taken into account, the 2p region; at the downstream edge, the cut is even higher (up pioniums with the Coulombic binding energy cannot be a to 150 MeV=c). This cut also suppresses the events with source of the unexpected events in the KOTO experiment. 0 Yet, no direct measurement of the 2p pionium mass two-body final states containing, besides the π , a massive or binding energy exists. From its decay chain, we know particle X0. It may be an invisible light boson, as in Ref. [9], that its mass must be greater than 2m 0 . The binding or a pionium. In these events, the maximum transverse π b 2 m − m 0 ≈ π0 energy should thus be smaller than ð πþ π Þ momentum of is equal to the momentum of the outgoing 9.19 MeV. It is possible [20] that the binding energy decay products in the KL rest frame, 0 of A2π may be around 9 MeV [21]. In that case, we qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 270 14 p 0 139 0 =c ϵ 0 1 have a ¼ . MeV, Tmax;A ¼ . MeV , π A0 ¼ p λ m2 ;m2;m2 ; 139 − 130 =139 6 5% T;max ¼ ð K π X0 Þ ð17Þ ð Þ ¼ . , and 2mK Rðb ¼ 9 MeVÞ¼0.29: where we use the usual notation The efficiency corrected branching fraction now comes 2 2 2 λðx; y; zÞ¼x þ y þ z − 2xy − 2xz − 2yz: out as

π0 B0 b 9 1 5þ3.5 10−9; The maximum transverse momentum of in the 2pð ¼ MeVÞ¼ . −1.0 × 0 KL → π νν¯ events is higher because the lowest invariant mass of the two- system is zero (neutrinos’ masses which is, within the errors, compatible with the branching neglected), namely, fraction (3) characterizing the unexpected KOTO events.

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The KOTO Collaboration continues in analyzing its As an illustration of the idea, we have shown that, older data and taking new ones [22]. One of the four under some assumptions, the production of 2p pioniums 0 events reported in [8] has been shown to be due to a A2π in the KL decays and their subsequent escaping mistake, so only three good candidate events remain. from the signal region may explain the production of The new background estimate is 1.05 0.28 events. The unexpected events reported by the KOTO Collaboration number of expected Standard Model events is 0.04. [8]. Our key assumptions, which may or may not be It is interesting that the KOTO experiment sees more K correct, have been as follows: (i) main contribution than expected. It may be worth remaining that the KL → comes from pioniums that are in the 2p state, which ∓ Ke ν¯ðνÞ decays [23] have not been observed yet. lives much longer than the ground state; (ii) the 0 0 0 þ − Nevertheless, the KOTO Collaboration considered them branching ratio of KL → π A2π to KL → π π π is þ þ þ þ þ − [24] as a source of background and varied the branching the same as that of K → π A2π to K → π π π , fraction in reasonable bounds. A negligible contribution which was determined by the NA48=2 Collaboration was found. 0 [14]; (iii) the A2π binding energy is larger than the To conclude, we have suggested that events in which the Coulombic one, which helps pioniums to overcome the kaon decays into pionium, which then leaves the decay experimental pT cut. region without decaying, may contribute to the background in the K → πνν¯ experiments. In our opinion, this source of I thank Teppei Kitahara for pointing me toward the background should be taken into account when analyzing importance of efficiencies and Augusto Ceccucci, Matthew the existing experiments or planning new ones [25]. Moulson, and Taku Yamanaka for useful correspondence. However, to get reliable estimates of this background, This work was partly supported by the Ministry of the discovery and measurement of the following decays Education, Youth and Sports of the Czech Republic 0 will be very valuable: KL → π A2π (a neutral kaon partner Inter-Excellence Project No. LTI17018 and European of the charged kaon decay already observed [14]), Regional Development Fund Project No. CZ.02.1.01/0.0/ 0 0 0 KL → π A2π, and K → π A2π. 0.0/16_019/0000766.

[1] E. C. Gil et al., J. Instrum. 12, P05025 (2017); S. Martellotti [9] T. Kitahara, T. Okui, G. Perez, Y. Soreq, and K. Tobioka, (on behalf of the NA62 Collaboration), KnE Energy Phys. 3, Phys. Rev. Lett. 124, 071801 (2020). 372 (2018). [10] P. S. B. Dev, R. N. Mohapatra, and Y. Zhang, Phys. Rev. D [2] E. C. Gil et al. (NA62 Collaboration), Phys. Lett. B 791, 156 101, 075014 (2020); D. Egana-Ugrinovic, S. Homiller, (2019). and P. Meade, Phys. Rev. Lett. 124, 191801 (2020);M. [3] E. C. Gil et al. (NA62 Collaboration), J. High Energy Phys. Fabbrichesi and E. Gabrielli, Eur. Phys. J. C 80, 532 (2020); 11 (2020) 042. Y. Liao, H.-L. Wang, C.-Y. Yao, and J. Zhang, Phys. Rev. D [4] R. Marchevski (On behalf of the NA62 Collaboration), 102, 055005 (2020). Evidence for the decay Kþ → πþνν¯ from the NA62 experi- [11] B. Dutta, S. Ghosh, and T. Li, Phys. Rev. D 102, 055017 ment at CERN, Proceedings of the ICHEP 2020, Prague (2020); X. Liu, Y. Li, T. Li, and B. Zhu, J. High Energy (Virtual Conference), July 28, 2020, https://indico.cern.ch/ Phys. 10 (2020) 197. event/868940/contributions/3815641/. [12] L. G. Afanasyev et al., Phys. Lett. B 308, 200 (1993). [5] We use the terminology of [16]: branching fraction B is the [13] B. Adeva et al., Phys. Lett. B 619, 50 (2005); 704,24 ratio of a partial decay rate to the total decay rate; branching (2011). ratio is the ratio of two partial decay rates or, equivalently, of [14] J. R. Batley et al. (NA48/2 Collaboration), Phys. Lett. B two branching fractions. 633, 173 (2006). [6] J. Brod, M. Gorbahn, and E. Stamou, Phys. Rev. D 83, [15] B. Adeva et al., Phys. Lett. B 751, 12 (2015); Phys. Rev. 034030 (2011); A. J. Buras, D. Buttazzo, J. Girrbach-Noe, Lett. 122, 082003 (2019). and R. Knegjens, J. High Energy Phys. 11 (2015) 33. [16] P. A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. [7] E. Iwai et al. (KOTO Collaboration), Nucl. Phys. B, Proc. Phys. 2020, 083C01 (2020). Suppl. 233, 279 (2012); J. K. Ahn et al. (J-PARC KOTO [17] Alternatively, we also performed the evaluation by generating Collaboration), Prog. Theor. Exp. Phys. 2017, 021C01 the pionium energy randomly within given bounds. The final (2017); results differed from those with mean pa by less than 10%. J. K. Ahn et al. (KOTO Collaboration), Phys. Rev. Lett. 122, [18] We did not perform a serious study of error propagation in ¯ 021802 (2019). (11), but simply calculated the values of S in the boundary [8] S. Shinohara (on behalf of the KOTO collaboration), J. Phys. points of error intervals of decay lengths. Conf. Ser. 1526, 012002 (2020). [19] F. James, Report No. CERN 68-15, 1968.

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[20] P. Lichard, Phys. Rev. D 102, 073004 (2020). 2020, Prague (Virtual Conference), July 28, 2020, https:// [21] J. Uretsky and J. Palfrey, Phys. Rev. 121, 1798 (1961), indico..ch/event/868940/contributions/3815582/. considered the pionium binding energies even higher than [23] P. Lichard, Phys. Rev. D 55, 5385 (1997); P. Lichard and J. 10 MeV. Thompson, arXiv:hep-ph/0004224. 0 [22] N. Shimizu, Search for new physics via the KL → π νν¯ decay [24] T. Yamanaka (private communication). at the J-PARC KOTO experiment, Proceedings of the ICHEP [25] M. Moulson, J. Phys. Conf. Ser. 1526, 012028 (2020).

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