Imprint of a new light particle at KOTO?
AAn imprint of a new light particle at KOTO?
Yi Liao,
1, 2, ∗ Hao-Lin Wang, † Chang-Yuan Yao, ‡ and Jian Zhang § School of Physics, Nankai University, Tianjin 300071, China Center for High Energy Physics, Peking University, Beijing 100871, China
Recently, the KOTO experiment reported their new preliminary result of searching for the decay K L → π ν ¯ ν . Three candidate events were observed in the signal region, which exceed significantlythe expectation based on the standard model. On the other hand, the new NA62 and previousBNL-E949 experiments yielded a consistent result and confirmed the standard model predictionin the charged meson decay K + → π + ν ¯ ν . Furthermore, the two decays are bound by a well-motivated relation from analysis of isospin symmetry which is hard to break by new physics ofheavy particles. In this work we study the issue by a systematic effective field theory approachwith three simplest scenarios, in which the K L may decay into a new light neutral particle X , i.e., K L → π X , K L → γγX , or K L → π XX . We assess the feasibility of the scenarios by simulationsand by incorporating constraints coming from NA62 and other relevant experiments. Our mainconclusion is that only the scenario K → πXX for a long-lived light scalar X has the potential toaccommodate the three candidate events at KOTO and the NA62 result simultaneously while theregion below the KOTO’s blind box provides a good detection environment to search for all threescenarios for a relatively heavy X . I. INTRODUCTION
The flavor changing neutral current decays of the neu-tral and charged kaons K → πν ¯ ν provide a clean venueto examine precisely the standard model (SM) and tosearch for new physics beyond it. Recently, the KOTOexperiment reported their preliminary result for the de-cay K L → π ν ¯ ν using data collected during years 2016-2018 [1]. If the three candidate events observed in thesignal region are confirmed in the future, it would implya decay branching ratio [2]: B (cid:0) K L → π ν ¯ ν (cid:1) KOTO = 2 . +2 . . − . − . × − , (1)at 68% (95%) CL, corresponding to the KOTO sin-gle event sensitivity, SES π ν ¯ ν = 6 . × − . Thisvalue is significantly larger than the SM prediction B (cid:0) K L → π ν ¯ ν (cid:1) SM = (0 . ± . × − [3], whichwould pose a strong challenge to the SM.In the charged sector, the NA62 experiment recentlyreported their result [4]: B (cid:0) K + → π + ν ¯ ν (cid:1) NA62 = (cid:0) . +0 . − . (cid:1) × − , (2)which is consistent with a previous measurement bythe BNL-E949 Collaboration, B ( K + → π + ν ¯ ν ) E949 = (cid:0) . +1 . − . (cid:1) × − [5]. Combining the two one obtains B (cid:0) K + → π + ν ¯ ν (cid:1) Exp = (cid:0) . +0 . − . (cid:1) × − , (3)or an upper bound at 95% CL, B (cid:0) K + → π + ν ¯ ν (cid:1) Exp ≤ . × − . (4) ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] This confirms the SM prediction B ( K + → π + ν ¯ ν ) SM =(0 . ± . × − [3].This tension between the neutral and charged sectorsis even exacerbated due to a theoretical relation betweenthe two. Based on the well-motivated assumption thatthe decays are dominated by the interactions with isospinchange ∆ I = 1 /
2, they are related by the so-calledGrossman-Nir (GN) bound [6], B ( K L → π ν ¯ ν ) (cid:46) . B ( K + → π + ν ¯ ν ) . (5)Together with the bound in the charged sector of Eq. (4)this implies B (cid:0) K L → π ν ¯ ν (cid:1) < . × − at 95% CL,which would lead to at most 1.3 instead of 3 candidateevents at KOTO. This bound is hard to break as it is im-mune to low energy effects at leading order of new heavyparticles, and may only be violated by ∆ I = 3 / π decays into a pair of photons andthe neutrino pair appears as missing energy, there arethree simplest scenarios that could mimic the SM decaysearched for at KOTO, namely, K L → π X , K L → γγX ,and K L → π XX , in which X appears as missing energy.We will first examine in Sec. II the most popular sce-nario K → πX [2, 11–17]. We will assess whether it isfeasible by simulating its event distribution and compar-ing it with that measured at KOTO. We will also in-corporate constraints coming from other measurementson K L and K + decays. In Sec. III we consider the loop-induced process of K L → γγX which could disguise itself a r X i v : . [ h e p - ph ] M a y as a signal event for the decay searched for at KOTO buthas no counterpart at NA62. If the new particle couplesonly in pairs to quarks for one reason or another, it willalso appear in pairs in the kaon decays. We will thereforemake an analysis on the decays K → πXX in Sec. IV.In the course of our analysis we will make some sugges-tions that may deserve further study in the KOTO andNA62 experiments. We summarize our main results inthe last section V. Our analysis in the following sectionsis based on the effective field theory approach. The ef-fective interactions at leading order and the amplitudesand branching ratios for various decays are detailed inApp. A, B, and C, and the simulation setup and valida-tion is discussed in App. D. II. K L → π X The signal region for the K L → π ν ¯ ν search at KOTOis also suitable for the two-body decay K L → π X , where X acts as missing energy. Assuming parity and workingto leading order in low energy expansion, X can be ascalar or vector particle. However, as shown in Fig. 1,the signal distributions in the Z vtx − P T plane of the twodecays are different. Here Z vtx is the π decay vertex po-sition projected onto the K L beam direction, and P T isits transverse momentum with respect to the beam. Wehave performed the simulations by applying the kinemat-ical cuts proposed in KOTO’s 2015 data analysis [18] andassuming the signal region chosen in its 2016-2018 dataanalysis [1]. While the decay K L → π ν ¯ ν has a relativelyuniform distribution in the signal region, K L → π X concentrates in a narrow interval of P T . Furthermore,the heavier the X is, the lower the maximal P T is. For aheavy X , it is hard to accommodate the candidate eventswith a high P T at KOTO.The branching ratio for K L → π X corresponding toa specific number of candidate events, N signal , that canbe accommodated is estimated by the relation B (cid:0) K L → π X (cid:1) KOTO = N signal · SES π X (6)= N signal · SES π ν ¯ ν · (cid:15) π ν ¯ ν (cid:15) π X ( m X ) . (7)Here (cid:15) is the detection efficiency in the signal region atKOTO [1] upon imposing various kinematical cuts [18].We have estimated the ratio of the two efficiencies by sim-ulations and then used the above quoted SES π ν ¯ ν (withan assumed relative error of 10%) to obtain SES π X . Thetotal background is 0 . ± .
02 in the signal region (aboutone half from the decay K L → π ν ¯ ν and another halffrom its SM background) and 0 . ± .
06 in the region be-low the blind box [1]. The branching ratio for K L → π X with two-sided 68% CL limits is shown in Fig. 2 as afunction of m X in various mass intervals where a specificnumber N signal of candidate events are accommodated.Note that the choice of the interval delimiters is not sharpbut only a rough estimate based on simulations. When X is so heavy (roughly m X >
190 MeV) that none of thecandidate events can be accommodated or even all of itssignals drop below the blind box ( m X >
270 MeV), wethen use the above background estimates to set a 90%CL upper bound on the decay branching ratio.Now we examine whether the decay K L → π X offersa feasible interpretation to the candidate events observedin the KOTO signal region by a comprehensive analysisof the decay distribution and the limits set by other ex-periments.We consider first the X mass intervals m X ∈ (100 , ∪ (260 , K + → π + X is considerably de-graded by the large backgrounds K + → π + π ( γ ) and K + → π + π π , π + π + π − , respectively. Thus the restric-tive GN bound is practically not in action. Neverthe-less, the second interval is obviously not supported bythe KOTO signals as seen in Fig. 1 and the first one can-not provide a perfect solution either. For X in the firstinterval, i.e., m X ∼ m π , the decay K L → π X can onlyaccommodate two candidate events but not the one witha high P T ∼
238 MeV. If X is stable and invisible and ifwe leave aside the last event, we may obtain the branch-ing ratio with two-sided limits at 68% (95%) CL at, e.g., m X = 135 MeV, B (cid:0) K L → π X (cid:1) KOTO = 1 . +1 . . − . − . × − . (8)For X of other mass, it must be so long-lived to be in-visible at KOTO while short-lived to decay into SM par-ticles to be vetoed at NA62, thus avoiding the constraintfrom the GN bound. In order to incorporate all three can-didate events, X must be light enough, m X (cid:46)
60 MeV,but still it is difficult to offer a feasible solution. Firstof all, not all observed events are gracefully in the maindistribution region as can be seen in Fig. 1. More impor-tantly, as we will detail below, this light mass region istightly constrained by other experiments.For m X (cid:46)
60 MeV, the signals of K + → π + X at NA62would be predominantly located in its signal region 1 forthe K + → π + ν ¯ ν search [4] with missing mass squared m ∈ (0 , .
01) GeV , where no events were observed,implying B (cid:0) K + → π + X (cid:1) NA62 < .
38 (0 . × − , (9)at 90% (95%) CL. As we did not simulate the decays K + → π + X, π + ν ¯ ν at NA62, we obtained the upper limitby assuming an equal SES = (0 . ± . × − for both decays and by utilizing that the expected sig-nals and background for K + → π + ν ¯ ν are respectively0 . ± .
15 and 1 . ± .
14 corresponding to their totalexpectations in signal regions 1 and 2 at NA62 [4]. Asimilar limit, B ( K + → π + X ) E787 < . × − (90%CL) for m X (cid:46)
115 MeV, was obtained by E787 [19].Now we can employ the above results to obtain a strongconstraint on the branching ratio of K L → π X . Wefirst recall that the real, physical branching ratio may FIG. 1. Reconstructed events (blue scatters) in the Z vtx − P T plane after all the cuts for K L → π ν ¯ ν (top left) and for K L → π X S with m X S = 10 MeV (top right), 135 MeV (bottom left), and 220 MeV (bottom right). The results for a vector X V are the same, as shown in Eqs. (B1) and (B2). The region surrounded by the red solid (black dashed) lines is the signalregion (blind box) implemented in the KOTO analysis [1] which yielded the three candidate events in red squares. ( % CL ) ( % CL ) ( % CL ) Allowed Region ( % CL ) FIG. 2. Branching ratio for K L → π X S ( V ) with 68%CL limits in various mass intervals that can accommodatea specific number of candidate events, and its 90% CL up-per limit as a function of mass in the KOTO signal re-gion (solid line) and in the region below the blind box(dashed line). The dots correspond to the points m X S ( V ) =1 , , , . . . , , , , . . . , ,
300 MeV. differ its measured value in a specific detector (det) in anexperiment such as KOTO or NA62 if X decays [2]: B ( K → πX ) real = B ( K → πX ) det e Lp mXcτX , (10)where τ X is the lifetime of X , and p/m X and L are the effective boost factor and detector size, respectively. Weapply the above relation to both K L → π X at KOTOand K + → π + X at NA62, and employ Eq. (5) for realbranching ratios to link the two, so that we have( r −
1) ln B ( K L ) real (cid:38) r ln B ( K L ) KOTO − ln (cid:2) . B ( K + ) NA62 (cid:3) , (11)where r = ( L/p ) NA62 / ( L/p ) KOTO ≈ .
49, and for brevity B with a subscript indicates its measured or real value.Plugging in the central value B ( K L ) KOTO ≈ . × − from Fig. 1 and the upper bound on B ( K + ) NA62 inEq. (9), we arrive at the lower bound at 90% (95%) CL: B (cid:0) K L → π X (cid:1) real > . . × − . (12)This lower bound of order 10 − has been strictly ex-cluded by other measurements. Being light, X canonly decay to e + e − and/or γγ . For X → e + e − , B (cid:0) K L → π X (cid:1) (suppressing the subscript) is tightlyconstrained by the measurements of B ( K L → π e + e − ),which is expected to be less than 2 . × − at 90% CLat the KTeV experiment [20]. For X → γγ , the limitsmainly come from the measurements of B ( K L → π γγ )and its spectrum at KTeV [21] and NA48 [22]. The re-sults B (cid:0) K L → π γγ (cid:1) = (1 . ± . ± . × − atKTeV and B (cid:0) K L → π γγ (cid:1) = (1 . ± . ± . × − at NA48 are consistent with the SM prediction ∼ × − [23–26]. Furthermore, both theoretical calculationsand experimental observations give a consistent distribu-tion which is dominated by the invariant mass interval m γγ ∈ (160 , X , one would expect a peak or atleast enhancement around m γγ = m X ∈ (0 ,
60) MeVin the experiments. But no such enhancement was ob-served at either experiment. Actually, NA48 reportedno signal events for m X ∈ [0 ,
40] MeV, and set thelimit B (cid:0) K L → π X (cid:1) < . × − (90% CL) for m X ∈ [30 , K L → π X , especially when X is a light particle. III. K L → γγX In the KOTO experiment the directions of the photonpair were not recorded, and the reconstruction of thedecay K L → π ν ¯ ν was based on the assumption that thephoton pair arises from the π decay which in turn is aproduct of the K L decay in the beamline. This motivatesus to consider the decay K L → γγX in which the photonpair would be misidentified as coming from the π decayand X appears as missing energy. It is worth mentioningthat this process is not constrained by the GN bound asits counterpart in the charged sector is absent.The decay K L → γγX can appear at the one-loop or-der. The effective field theory calculation of the decayrate is delegated to App. B. Assuming parity symmetry, X could be a pseudoscalar or an axial-vector. In thissection we analyze its phenomenological aspects by sim-ulations. The distribution in the Z vtx − P T plane of thereconstructed events is depicted in Fig. 3 at three typicalmasses of the pseudoscalar X P . A similar distributionwas found for the axial-vector case and thus not shownseparately. As we can see from the figure, the KOTO’sthree candidate events could be covered over a range of m X , but the decay events more tend to be located belowthe signal region. If we naively insist that the KOTO’sthree events are faked by K L → γγX , more events shouldbe found below the signal region, which however is notthe case. Therefore, we will not pursue this idea further,but employ KOTO’s results to work out an experimentalupper limit on the decay K L → γγX instead.We choose as our signal region for K L → γγX speci-fied by Z vtx ∈ [2 . , .
1] m and P T ∈ [0 , . ± .
06. Accord-ing to the observed zero event in the region [1], we obtainan upper limit on B ( K L → γγX P ) and relevant Wilsoncoefficients as a function of m X P in the top and bot-tom panels of Fig. 4 respectively. Also included is theprojected future detection capability of KOTO (dashedcurves) when an SES π ν ¯ ν = 3 . × − is reached. In thebottom panel, we also display limits coming from otherexperiments: K + → π + π ν ¯ ν at E787 [27], K L → π π ν ¯ ν at E391a [28], and the K − ¯ K mixing [29]. For a light X P , the limit is dominated by K L → π π X P , while fora heavy X P the strongest limit comes from the K − ¯ K mixing; in between (roughly for m X P ∈ (220 , X A are shown in Fig. 5. IV. K L → π XX Finally, we discuss the case in which a pair of X par-ticles appear in the K meson decays for which the kine-matics will be very different from that of K → πX thatwe studied in Sec. II. For a fermionic X this has beensuggested in Ref. [30]. We consider here a bosonic X which for one reason or another only couples in pairs tothe quarks and may be a scalar or vector of either par-ity. Our simulation results for the signal distribution inthe Z vtx − P T plane are shown in Fig. 6 at three differ-ent masses. We find that the events for a light X arealmost evenly distributed in the signal region as in thecase for K L → π ν ¯ ν but differ significantly from the de-cay K L → π X as shown in Fig. 1. As X becomes heav-ier, the maximal P T also drops. But the truncation ofthe distribution around the maximal P T is not as sharpas in the two-body decay, and this leaves some space toexplain the KOTO signal with a high P T . For even largermasses, the distribution will shift below the blind box.The branching ratio for K L → π XX from above sim-ulation is shown in Fig. 7. The KOTO candidate signalsprefer a light scalar X S with m X S (cid:46)
30 MeV, which mustbe unstable and have an appropriate lifetime to avoid theGN bound. The requirement is that with a high prob-ability the two X S s should be both invisible to KOTOwhile at least one of them decays into SM particles ( γγ or e + e − ) which can be vetoed at NA62. Qualitativelyspeaking, in terms of kinematics, the experimental con-straints on the πX S X S mode are looser than those on the πX S mode. But to assess the feasibility of the scenario,we have to determine the appropriate lifetime and massof X S . The physical branching ratio for K → πX S X S atKOTO or NA62 can be expressed as B ( K → πX S X S ) real = B ( K → πX S X S ) det (cid:90) e (cid:16) L p + L p (cid:17) mXScτXS × f ( L , L , p , p ) dL dL dp dp , (13)where f represents the probability that the two X S spropagate respectively at the momentum p , p for a dis-tance L , L to exit the detector without decay. The in-tegration over the signal region ensures that all situationsof the K decay have been taken into account. However,it is difficult for us to manage a systematic simulation tocover all information due to the complexity of displaced-vertex simulation. Considering that the two X S s have alarge boost at both KOTO and NA62, they should havea high probability of flying in the beam direction witha small angle to the veto plates. For further estimation, FIG. 3. Reconstructed events (blue scatters) in the Z vtx − P T plane after all the cuts for K L → γγX P with m X P = 10 MeV(left), 320 MeV (middle), and 430 MeV (right). With the same K L sample the difference in the total number of scatters forvarious m X P arises from different cut efficiencies.
100 200 300 400 50010 - - - - - - - Excluded by current KOTO ( % CL ) Excluded by future KOTO ( % CL )
100 200 300 400 50010 - - - - - FIG. 4. Top: upper limit on B ( K L → γγX P ) from KOTO’scurrent data (solid) and projected future capability (dashed).Bottom: upper limits on Wilson coefficients: | C Psd | from K + → π + π + invisible by E787 experiment (red solid), | Re( C Psd ) | from K − ¯ K mixing (green solid), K + → π + π +invisible by E391a experiment (blue solid), K L → γγX P byKOTO’s current result (purple solid) and future expectation(orange dashed). we found that p = p ≡ E and L = L ≡ L serve as agood approximation, which leads to the simplification: B ( K → πX S X S ) real ≈ B ( K → πX S X S ) det (cid:90) dLdE e LE mXScτXS F ( L, E ) . (14)The two-dimensional probability function F ( L, E ) ismuch simpler compared to the four-dimensional one. For K L → π X S X S at KOTO, it can be extracted from ourcurrent simulation; for K + → π + X S X S at NA62, wehave adopted a rough simulation in which only some cuts
100 200 300 400 50010 - - - - - - - Excluded by current KOTO ( % CL ) Excluded by future KOTO ( % CL )
100 200 300 400 50010 - - - - - FIG. 5. Same as in Fig. 4 except for an axial-vector X A . on the signal region are considered.As an example of simulation we consider the scenariothat a scalar X S of mass m X S = 10 MeV decays into aphoton pair. The branching ratio measured at KOTOcan be read off in Fig. 7, with the central value being B ( K L → π X S X S ) KOTO = 2 . × − . For NA62, as-suming that K + → π + X S X S has the same acceptanceas K + → π + ν ¯ ν , we obtain B ( K + → π + X S X S ) NA62 < . × − at 95% CL. By incorporating all these intoEq. (14), we obtain B ( K L → π X S X S ) real as a func-tion of the lifetime τ X S shown in Fig. 8 as the greencurve, and an upper limit on it (red curve) from NA62with the aid of the GN bound. This yields the allowedregion with τ X S (cid:46) . × − s, within which the veto in-formation can also be gained from the figure: at least74.6% of the K + → π + X S X S signals are vetoed atNA62, while for τ X S (cid:38) − s the two X S s at KOTOnearly do not decay before exiting the detector. On FIG. 6. Reconstructed events (blue scatters) in the Z vtx − P T plane after all the cuts for K L → π XX with m X = 10 MeV(left), 70 MeV (middle), and 100 MeV (right) for a scalar (top) or vector (bottom) X . ( % CL ) ( % CL ) ( % CL ) Allowed Region ( % CL ) ( % CL ) ( % CL ) Allowed Region ( % CL ) FIG. 7. Same as Fig. 2 but for K L → π XX with a scalar(top) or vector (bottom) X . the other hand, the untagged K L branching ratio [29]constrains B ( K L → π X S X S ) < τ X S (cid:38) . × − s. But we emphasize once again thatthis is only a rough estimate, and an appropriate deter- FIG. 8. B (cid:0) K L → π X S X S (cid:1) real as a function of τ X S at m X S =10 MeV (green curve) together with the upper limit fromNA62 upon using the GN bound (red). mination of the lifetime and mass of X S could only beachieved by a systematic detector simulation. We advo-cate that the KOTO and NA62 collaborations will takethis endeavor in their future experimental analysis. V. CONCLUSION
We have investigated by simulations the feasibility tointerpret the recent KOTO result in terms of a new neu-tral particle that appears in the kaon decays. Since π decays into a pair of photons and neutrinos appear asmissing energy, we have considered three scenarios, i.e., K L → π X, γγX, π XX . Our results can be summa-rized as follows. The simplest scenario K L → π X isdifficult to accommodate all three candidate events atKOTO, especially the one with a high P T . The signalevents for a relatively heavy X are mainly distributedbelow the KOTO’s signal region, which have been em-ployed to work out a bound on the decay branching ratio.While the KOTO result favors a light X , our compre-hensive analysis on the NA62 and other experiments setsa strong constraint that essentially excludes this poten-tial. Since the scenario K L → γγX has no constraint atNA62, it is free of the GN bound. While the three candi-date events can be accommodated, the distributions donot fit: more events would be expected below the blindbox. We have used the latter to set a constraint on thebranching ratio, and compared it with those from othermeasurements and the expected KOTO’s future capabil-ity. In terms of the signal distribution, the KOTO can-didate events favor the third scenario K L → π X S X S ,where X S is a scalar with m X S (cid:46)
30 MeV. To accom-modate the measurements at NA62, X S should be un-stable but long-lived, whose lifetime is estimated to be1 . × − s (cid:46) τ X S (cid:46) . × − s at m X S = 10 MeV.But a more precise result would necessitate sophisticatedsimulations which we hope the KOTO and NA62 collabo-rations will perform in their future experimental analysis. ACKNOWLEDGMENTS
We would like to thank Hai-Bo Li for helpful dis-cussions and K. Tobioka for electronic communications.This work was supported in part by the grants No.NSFC-11975130, No. NSFC-11575089, the China Post-doctoral Science Foundation grant No. 2018M641621, bythe National Key Research and Development Programof China under grant No. 2017YFA0402200, and by theCAS Center for Excellence in Particle Physics (CCEPP).
Appendix A: Effective field theory framework
For our purpose of accounting for the KOTO anomalywe assume a new real neutral particle X of mass below afew hundreds MeV. It may be a scalar ( X S ), pseudoscalar( X P ), vector ( X V ) or axial-vector ( X A ) particle as appro-priate to the scenario under consideration. We start withthe low energy effective field theory that contains the X particle in addition to the light quarks and leptons andhas the QCD and QED gauge symmetries. As we aremainly concerned with the transitions between the downand strange quarks, we only consider their couplings tothe X field: L X = C Spr ( d p d r ) X S + C Ppr ( d p iγ d r ) X P + C Vpr ( d p γ µ d r ) X µV + C Apr ( d p γ µ γ d r ) X µA , (A1)where p, r refer to the d, s quarks and the Wilson coef-ficients are 2 × d, s space.When studying the third scenario K → πXX , we switch off the above single- X couplings but switch on the fol-lowing double- X couplings: L XX = (¯ sd ) (cid:0) g SX XX + g VX X µ X µ (cid:1) + H.c. , (A2)where X and X µ may have any parity without affectingour discussions in the work.Below the chiral symmetry breaking scale the Nambu-Goldstone bosons (NGBs) become the dynamical degreesof freedom, whose low energy physics is determined atleading order by [31, 32] L χ = F (cid:0) D µ U ( D µ U ) † (cid:1) + F (cid:0) χU † + U χ † (cid:1) , (A3)where U = exp (cid:0) i √ /F (cid:1) exponentiates the octet NGBsΦ = π √ + η √ π + K + π − − π √ + η √ K K − ¯ K − (cid:113) η , (A4)and D µ U = ∂ µ U − il µ U + iU r µ , (A5) χ = 2 B ( s − ip ) . (A6)As usual, F is the decay constant in the chiral limit and B parameterizes the quark condensate (cid:104) ¯ qq (cid:105) = − BF .The new field X gets involved in the form of additionalterms in the external sources to QCD, which are Hermi-tian matrices in the light flavor space: l µ = C V X µV − C A X µA , (A7) r µ = C V X µV + C A X µA , (A8) s = C S X S , (A9) p = − C P X P , (A10)in the case of single- X couplings in Eq. (A1), and s pr = δ p δ r (cid:0) g SX XX + g VX X µ X µ (cid:1) + δ p δ r (cid:0) g S ∗ X XX + g V ∗ X X µ X µ (cid:1) , (A11)in the case of double- X couplings in Eq. (A2). Note thatthe QED interaction is contained as usual in the sources l µ , r µ , which we do not write explicitly. The effectiveinteractions contained in Eq. (A3) will be applied to cal-culate the quantities in the following appendices. Appendix B: Decay amplitudes and distributions
The decay width for K L → π X is, for a scalar X S , d Γ π X S dp π,z = B πm K L (cid:2) Re C Ssd (cid:3) , (B1) FIG. 9. Feynman diagrams for K L → γγX at first nonvan-ishing order. The large (small) dot stands for the effective X (standard QED) interactions in Eq. (A3). where p π,z is the π momentum component in the K L beam direction, or, for a vector X V , d Γ π X V dp π,z = λ ( m K L , m π , m X )16 πm K L m X (cid:2) Im C Vsd (cid:3) , (B2)with λ ( x, y, z ) = x + y + z − xy − yz − zx .The decay K L ( k ) → γ ( q ) γ ( q ) X ( p ) involves only neu-tral particles, and can only take place at the one-looporder whose Feynman diagrams are shown in Fig. 9. Theresult is finite, and the (spin-summed) squared matrixelement is, for a pseudoscalar X P , |M P | = (cid:18) αB Re C Psd √ πF s (cid:19) | f ( r π ) + f ( r K ) | , (B3)where r π,K = s/ (4 m π ± ,K ± ), s = ( q + q ) , and α ≈ /
137 is the fine structure constant, or for an axial-vector X A , |M A | = (cid:18) α Re C Asd √ πF m X A s (cid:19) | f ( r π ) + f ( r K ) | × λ ( m K L , m X A , s ) . (B4)The one-loop function is f ( r ) = 4+ − r arcsin ( √ r ) , r ≤ r (cid:2) (cid:0) √ r − √ r − (cid:1) + iπ (cid:3) , r > d Γ γγX P ( A ) ds dt = 1512 π m K L |M P ( A ) | , (B6)where t = ( k − q ) .The decay width for K L ( k ) → π ( p ) X ( q ) X ( q ) is eas-ily computed to be, for a scalar X S or vector X V respec-tively, d Γ π X S X S ds dt = B (Re g SX ) π m K L , (B7) d Γ π X V X V ds dt = B (Re g VX ) π m K L (cid:2) r X − (cid:3) , (B8)where r X = s/ (4 m X ), s = ( q + q ) , and t = ( k − q ) . Appendix C: Other experimental constraints
In this appendix we list other experimental constraintsthat have been used in Sec. III for a comprehensive anal-ysis. The experimental upper limits on the four-bodykaon decays are [27, 28]: B ( K + → π π + ν ¯ ν ) < . × − , (C1) B ( K L → π π ν ¯ ν ) < . × − . (C2)The corresponding amplitudes for the decay K ( k ) → π ( p ) π ( p ) X ( q ) read, for a pseudoscalar X P , M ( K + → π + π X P ) = B F C Psd u − tm K − m X , (C3) M ( ¯ K → π π X P ) = BC Pds √ F s + m X − m K m K − m X , (C4)and for an axial vector X A , M ( K + → π + π X A ) = i C Asd F ( p − p ) · (cid:15) ∗ , (C5) M ( ¯ K → π π X A ) = i C Ads √ F k · (cid:15) ∗ , (C6)where s = ( p + p ) , t = ( k − p ) , u = ( k − p ) , and (cid:15) is the polarization vector of X A .Moreover, the K − ¯ K mixing can give limits on thecouplings. The experimental measured quantities are the K L − K S mass difference ∆ M K and the CP violationparameter (cid:15) K [33–35], whose current experimental valuesare [29], ∆ M K = (3 . ± . × − MeV , (C7) | (cid:15) K | = (2 . ± . × − . (C8)Considering the theoretical uncertainties from long-distance contributions in ∆ M K , we require the new con-tribution do not exceed the experimental value. Rela-tively, the calculation of (cid:15) K is more credible, hence werequire the new contribution to be less than 30% of itsexperimental value. The limits on the Wilson coefficientsthen read, for a pseudoscalar X P , (cid:12)(cid:12) (Re C Psd ) − (Im C Psd ) (cid:12)(cid:12) < R A , (C9) (cid:12)(cid:12) Re C Psd Im C Psd (cid:12)(cid:12) < . √ | (cid:15) K | R A , (C10)with R P = ∆ M K m K ( m K − m X P ) / ( B F K ), and for anaxial-vector X A , (cid:12)(cid:12) (Re C Asd ) − (Im C Asd ) (cid:12)(cid:12) < R A , (C11) (cid:12)(cid:12) Re C Asd Im C Asd (cid:12)(cid:12) < . √ | (cid:15) K | R A , (C12)with R A = ∆ M K m X A / ( F K m K ). × - × - × - × - × - FIG. 10. Validation of B ( K L → π X ) upper bound at90% CL by comparing our result (red dots) with KOTO’s[18] (black dots). The horizontal line is the upper limit for B ( K L → π ν ¯ ν ) < . × − at 90% CL. Appendix D: Simulation of K L decay In this appendix we will briefly describe how we do thesimulations for various K L decays. A systematic simula-tion is complicated and time consuming, so we contentourselves with a simplified version of it in this work. Wefind that even in this simple framework we can get accu-rate results as in a systematic simulation. In the follow-ing, we will explain our procedure and compare our resulton K L → π X with KOTO’s to verify our simulation.We first generate initial K L particles according tothe momentum distribution of K L measured experimen-tally [36], which will have a certain probability of decayin the detector. All of the K L decay modes in this paperproduce two photons, and the distributions of the ener-gies and positions of the photons are largely dependent onthe probability distribution functions. For the two-bodydecay K L → π X , we use a uniform distribution func-tion as in Eqs. (B1) and (B2). For the three-body decay K L → π ν ¯ ν , we adopt the same distribution function asin Eq. (S1) in the supplemental material of Ref. [2]. Thedistribution functions of K L → Xγγ and K L → π XX are determined by Eq. (B6), and Eqs. (B7) and (B8)respectively. It is worth mentioning that for the decaymodes K L → π ν ¯ ν , K L → π X , and K L → π XX ,the two photons are originated from the π , while for K L → Xγγ , the two photons are directly generated by K L ; only the former decays will lead to an invariant massof the two photons m γγ (cid:39) m π . Then the photons arecaptured by the CsI calorimeter in the detector, and werecord their energies and positions in our simulation. Inorder to better simulate the detector’s response to thephotons, we include the energy and position resolutionof the CsI calorimeter, which can be found in Ref. [37].By assuming that the two photons produced on thebeam axis are from π decay (in the interesting case of K L → Xγγ the two photons are also required to havean invariant mass m γγ (cid:39) m π in order to fake the sig-nals), we can reconstruct the decay location Z vtx and π ’s transverse momentum P T by combining the informa-tion of photons’ energies and positions. 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