Pionium as a source of false events in the K\toπν\barν decays
aa r X i v : . [ h e p - ph ] J un Pionium as a source of false events in the K → π ν ¯ ν decays Peter Lichard
Institute of Physics and Research Centre for Computational Physics and Data Processing,Silesian University in Opava, 746 01 Opava, Czech RepublicandInstitute of Experimental and Applied Physics,Czech Technical University in Prague,128 00 Prague, Czech Republic
We suggest that the unexpected rare K L decay events that appeared in the KOTO experimentcan be explained by accounting for pionia (A π ) that were produced in the K L → π A π decays andescaped (a very small portion of them) the decay volume before decaying into two π s. Agreementwith the observation is reached when one assumes that the participating pionia are those in a2p state. However, this explanation of the KOTO anomaly has to be abandoned because of thedisagreement with the NA62 upper limit. The false events rate caused by the 1s pionia is alsocalculated and shown to constitute only a small ( < Two important experiments investigating the rare kaondecays in flight are currently running. The main aimof both is to test the Standard Model and to constrainnew physics theories by precisely measuring the very rarekaon decays into a pion and two neutrinos. The NA62experiment at the CERN Super Proton Synchrotron [1–3] deals with positively charged kaons K + and aims tocollect enough K + → π + ν ¯ ν events to get a signal tobackground ratio of 10:1. The Standard Model predictsthe branching fraction [4] for this decay [5] B ( K + → π + ν ¯ ν ) = (8 . ± . × − . (1)The KOTO experiment [6] is being conducted at theHadron Experimental Facility (HEP) at the Japan Pro-ton Accelerator Research Complex (J-PARC). It was de-signed to observe the decay K L → π ν ¯ ν of long-livedneutral kaons. The theoretical branching fraction [5] is B ( K L → π ν ¯ ν ) = (3 . ± . × − . (2)Until recently, both rare kaon decay experiments pro-ceeded as expected, slowly pushing down the upperbounds of branching fractions. However, in September2019, Satoshi Shinohara (on behalf of the KOTO col-laboration) [7] announced the presence of four events inthe signal region in the situation where mere 0.10 ± e.g. , [8] and [9], aimed at finding new physicsinterpretations of this surprising result. The analysisperformed in Ref. [8] shows that if the four events inthe KOTO signal region were real, it would mean thebranching fraction of the underlying mechanism equal to B ( K L → π ν ¯ ν ) KOTO = 2 . +2 . . − . − . × − , (3)at the 68(95)% confidence level.In this Letter, we will pursue a conservative approach.We will mostly use the well-established experimentalfacts about the π + π − atom, pionium, supplemented withthe assumptions discussed below. Pionium (usually denoted as A π ) was discovered in1993 at the Institute of High Energy Physics at Ser-pukhov [10] and intensively studied in the Dimeson Rela-tivistic Atomic Complex (DIRAC) experiment [11] at theCERN Proton Synchrotron. Assuming pure coulombicinteraction, the binding energy of pionium can be calcu-lated from the hydrogen-atom-like formula, from whichone obtains b = 1 .
86 keV [12]. The decay to two neutralpions is dominant and the measured lifetime is [11] τ = 3 . +0 . − . × − s . (4)The NA48/2 Collaboration at the CERN SPS [13] stud-ied decays K ± → π ± π π and found an anomaly in the π π invariant mass distribution in the vicinity of 2 m π + that can be interpreted as the production of pionia inthe kaon decays and their subsequent two- π decay. Forour later considerations, it is important that the NA48/2Collaboration in their seminal paper [13] determined thebranching ratio R = Γ( K ± → π ± + A π )Γ( K ± → π ± π + π − ) = (1 . ± . × − . (5)The DIRAC collaboration also discovered [14] so-calledlong-lived π + π − atoms, which are the 2p atomic states(A ′ π ) with quantum numbers J P C = 1 −− . The coulom-bic binding energy of the 2p pionium is 0.464 keV, itslifetime was determined in Ref. [14] to be τ = 0 . +1 . − . × − s . (6)Such a long lifetime is caused by the fact that the decaymodes to the positive C-parity states π π and γγ arenow forbidden and the slow 2p →
1s transition dominates.After reaching the 1s state, a decay to two π s quicklyfollows: A ′ π → A π + γ → π π γ .We will investigate the possible role of pionium in pro-ducing the unexpected KOTO events. To this end, weneed at least a crude estimate of the branching fractionof decay K L → π + A π . We assume that the branchingratio ˜ R = Γ( K L → π + A π )Γ( K L → π π + π − )has the same value as that for charged kaons (5). Thenwe can write the branching-fraction estimate B ( K L → π + A π ) = R × B ( K L → π π + π − ) ≈ × − , (7)where we have also consulted Ref. [15].We can speculate that the pionium that appears inthe kaon decays is not the ground state (1s) pionium,but its excited (2p) partner and that the correspondingbranching fraction is the same B ( K L → π + A ′ π ) ≈ × − . (8)In what follows, we will pursue those two alternatives inparallel.To simplify the reasoning, we will ignore the momen-tum spread of the K L beam in the KOTO experiment andwill use its peak value P = 1 . l =3 m. Another quantitythat enters the game is the pionium mass. It is given by m a = 2 m π + − b , where b is the binding energy. We willtake its coulombic value.The laboratory energy of pionium with mass m a isuniformly distributed in an interval, the bounds of whichare given by formula E a ± = 1 M ( EE ∗ a ± P p ∗ a ) , where M , E , and P are the mass, energy, and momentumof the K L , respectively, E ∗ a = ( M + m a ) / (2 M ), p ∗ a =( M − m a ) / (2 M ). Numerically, E a − = 0 .
497 GeV and E a + = 1 .
456 GeV.To simplify the consideration farther [16], we will as-sume that all pionia have the same laboratory momen-tum, given as the mean value p a = 1 E a + − E a − Z E a + E a − p E − m a d E = 12( E a + − E a − ) × (cid:20) E a + p a + − E a − p a − − m a log E a + + p a + E a − + p a − (cid:21) , (9)where p a ± = q E a ± − m a . Numerically, p a =0 .
880 GeV/c.The probability that pionium travels the path s with-out decaying is given by S ( s ) = exp {− κs } , where κ = m a p a τ , (10) τ being the pionium mean lifetime. For the two types ofpionia we get κ = 3 . × m − and κ = 235 m − .If we denote the length of the decay volume as l , themean survival probability of pionium at the point whereit leaves the decay volume is¯ S = 1 l Z l S ( l − z )d z = 1 − exp {− κl } κl . (11) Numerical values for two kinds of pionium are¯ S = 1 . × − (12)¯ S = 1 . × − (13)Multiplying these numbers by the assumed branchingfractions (7) and (8), we obtain the branching fractions of K L → π + A π and K L → π + A ′ π events that look likethe K L → π ν ¯ ν events because pionia left the KOTOdecay volume undecayed: B (KOTO) = 2 . × − (14) B (KOTO) = 2 . × − (15)A comparison of (15) with (3) suggests that the anoma-lous events in the KOTO experiment could be explainedas undecayed 2p pionia. But every mechanism thatclaims success in explaining the KOTO anomaly mustalso be able to explain the absence of anomaly in theNA62 experiment. At first sight, the mechanism we pro-pose should not have a problem with that. The NA62decay volume is more than twenty times longer (l=65 m[1]) than that in the KOTO experiment, so pionia havemore time for decaying. But on the other hand, the par-ent kaon momentum (P=75 GeV/c [1]) is more than fiftytimes higher, so the pionium momentum p a will also bemuch higher. The mean survival probability, which de-pends on the product κl = m a /τ × l/p a , is thus similar tothat in the KOTO experiment. The detailed calculationalong the lines as above gives for the NA62 experimentresults B (NA62) = 2 . × − (16) B (NA62) = 3 . × − (17)The 2p branching fraction is in disagreement not onlywith the new (preliminary) upper limit of 2.44 × − [3],but also with the published upper limit of 1.4 × − [2],both at the 95% confidence level. Therefore, we mustabandon the alternative with 2p pionia and leave theKOTO anomaly unexplained.The false events caused by the undecayed 1s pioniaprovide a background to the K → π ν ¯ ν decays. Dividing(16) by (1) gives a background percentage in the NA62experiment of 2.8%. Similarly, the KOTO backgroundpercentage of 5.9% comes from Eqs. (14) and (2).By rejecting the false event mechanism based on the 2ppionia by the NA62 data, we, in fact, reject our assump-tion about the branching fraction (8). As this quantityis not known experimentally, room for speculations stillexists. If it were, e.g. , one hundred times smaller than as-sumed in (8), the background percentage would be stillten times higher than that coming from the 1s pioniashown above.To conclude: We have presented a mechanism thatcan produce false events in very rare kaon decays exper-iments. It cannot simultaneously explain the suspectedanomaly in the KOTO experiment and its absence in theNA62 experiment. However, it may provide backgroundsthat were not considered so far and that should be takeninto account when analyzing the existing experiments orplanning new ones [17]. ACKNOWLEDGMENTS
This work was partly supported by the Ministry of Ed-ucation, Youth and Sports of the Czech Republic Inter- Excellence projects No. LTT17018 and No. LTI17018. [1] E. Cortina Gil et al. , JINST , P05025 (2017); S.Martellotti on behalf of the NA62 Collaboration, KnEEnerg. Phys. , 372 (2018);[2] E. Cortina Gil et al. (NA62 Collaboration), Phys. Lett. B , 156 (2019);[3] G. Ruggiero (On behalf of the NA62 Collaboration), Newresult on K + → π + ν ¯ ν from the NA62 experiment,KAON2019, Perugia, Italy, 2019.[4] We use the terminology of [15]: Branching fraction B isthe ratio of a partial decay rate to the total decay rate;branching ratio is the ratio of two partial decay rates or,equivalently, of two branching fractions.[5] J. Brod, M. Gorbahn and E. Stamou, Phys. Rev. D ,034030 (2011); A.J. Buras, D. Buttazzo, J. Girrbach-Noe,and R. Knegjens, JHEP , 33 (2015).[6] E. Iwai et al. (KOTO Collaboration), Nucl. Phys. B(Proc. Suppl.) , 279 (2012); J.K. Ahn et al. (J-PARCKOTO Collaboration), Prog. Theor. Exp. Phys. ,021C01 (2017); J.K. Ahn et al. (KOTO Collaboration),Phys. Rev. Lett. , 021802 (2019).[7] S. Shinohara, Search for the rare decay K L → π ν ¯ ν at J-PARC KOTO experiment, KAON2019,Perugia, Italy, 2019; The KOTO Collabora-tion, Important Addendum dated 2020-02-22.http://indico.cern.ch/event/769729/contributions/3510939/. [8] T. Kitahara, T. Okui, G. Perez, Y. Soreq, and K. To-bioka, Phys. Rev. Lett. , 071801 (2020).[9] D. Egana-Ugrinovic, S. Homiller, and P. Meade,Phys. Rev. Lett. , 191801 (2020).[10] L.G. Afanasyev et al. , Phys. Lett. B , 200 (1993).[11] B. Adeva et al. , Phys. Lett. B , 50 (2005); , 24(2011).[12] The exact value of binding energy is not important in ourreasoning. We tried also the value of b =10 MeV, assumedby Uretsky and Palfrey [18], and got almost identicalresults.[13] J.R. Batley et al. (NA48/2 Collaboration), Phys. Lett. B , 173 (2006).[14] B. Adeva et al. , Phys. Lett. B , 12 (2015);Phys. Rev. Lett. , 082003 (2019).[15] M. Tanabashi et al. (Particle Data Group),Phys. Rev. D , 030001 (2018).[16] Alternatively, we also performed the evaluation by gener-ating the pionium energy randomly within given bounds.The final results differed from those with mean p a by lessthan 10%.[17] M. Moulson, KLEVER: An experiment to measureBR( K L → π ν ¯ ν ) at the CERN SPS, KAON2019, Peru-gia, Italy, 2019.[18] J. Uretsky and J. Palfrey, Phys. Rev.121