Imprint of a new light particle at KOTO?

Yi Liao,1, 2, ∗ Hao-Lin Wang,1, † Chang-Yuan Yao,1, ‡ and Jian Zhang1, §

1School of Physics, Nankai University, Tianjin 300071, China2Center for High Energy Physics, Peking University, Beijing 100871, China

Recently, the KOTO experiment reported their new preliminary result of searching for the decayKL → π0νν̄. 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 prediction inthe charged meson decay K+ → π+νν̄. Furthermore, the two decays are bound by a well-motivatedrelation from an analysis of isospin symmetry that is hard to break by the new physics of heavyparticles. In this work, we study the issue by a systematic effective field theory approach withthree of the simplest scenarios, in which the KL may decay into a new light neutral particle X, i.e.,KL → π0X, KL → γγX, or KL → π0XX. We assess the feasibility of the scenarios by simulationsand by incorporating constraints coming from NA62 and other relevant experiments. Our mainconclusion is that the scenario K → πXX for a long lived scalar X seems more credible than theother two when combining distributions and other experimental constraints while the region belowthe KOTO’s blind box provides a good detection environment to search for all three scenarios for arelatively heavy X.

I. INTRODUCTION

The flavor changing neutral current decays of the neu-tral and charged kaons K → πνν̄ 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 KL → π0νν̄ using data collected during years 2016−2018 [1]. With the KOTO single event singularity (SES),SESπ0νν̄ = 6.9× 10−10, which is larger than the SM pre-diction B

(KL → π0νν̄

)SM

= (0.34± 0.06) × 10−10 [2],a total of 0.10 ± 0.02 events would be expected. Butsurprisingly, three candidate events apparently fell intothe signal region. As the experimental analysis is still inprogress, any theoretical attempt to interpreting the po-tential discrepancy has to be taken with great caution.On the other hand, if the three candidate events wereconfirmed in the future, it would imply a decay branch-ing ratio [3]:

B(KL → π0νν̄

)KOTO

= 2.1+2.0(+4.1)−1.1(−1.7) × 10−9, (1)

at 68% (95%) C.L., and would pose a strong challenge tothe SM.

In the charged sector, the NA62 experiment recentlyreported their result [4]:

B(K+ → π+νν̄

)NA62

=(0.47+0.72

−0.47

)× 10−10, (2)

which is consistent with a previous measurement bythe BNL-E949 Collaboration, B (K+ → π+νν̄)E949 =(1.73+1.15

−1.05

)× 10−10 [5]. Combining the two one obtains

B(K+ → π+νν̄

)Exp

=(0.86+0.62

−0.58

)× 10−10, (3)

∗ liaoy@nankai.edu.cn† whaolin@mail.nankai.edu.cn‡ yaocy@nankai.edu.cn§ zhangjianphy@nankai.edu.cn

or an upper bound at 95% C.L.,

B(K+ → π+νν̄

)Exp≤ 2.05× 10−10. (4)

This confirms the SM prediction B (K+ → π+νν̄)SM =(0.84± 0.10)× 10−10 [2].

This potential tension between the neutral and chargedsectors is even exacerbated due to a theoretical relationbetween the two. Based on the well-motivated assump-tion that the decays are dominated by the interactionswith isospin change ∆I = 1/2, they are related by theso-called Grossman-Nir (GN) bound [6],

B(KL → π0νν̄) . 4.3B(K+ → π+νν̄). (5)

Together with the bound in the charged sector of equa-tion (4), this implies B

(KL → π0νν̄

)< 8.8 × 10−10 at

95% C.L., which would lead to at most 1.3 instead of 3candidate events at KOTO. This bound is hard to breakas it is immune to low energy effects at leading orderof new heavy particles, and may only be violated by∆I = 3/2 interactions of higher dimensional operatorsbetween quarks and neutrinos; for recent discussions andattempts, see, for instance, Refs. [7–10].

The challenge here is to accommodate the KOTO re-sult while respecting the measurements at NA62 and oth-ers. In this work we propose to relax the tension by as-suming a light neutral particle that can appear in thefinal states of kaon decays. We will consider three sce-narios, one of which has been suggested in the literature,and will assess by simulations whether they are practi-cally feasible. Since π0 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, KL → π0X, KL → γγX,and KL → π0XX, in which X appears as missing energy.

We will first examine in Sec. II the most popular sce-nario K → πX [3, 11–17]. We will assess whether it isfeasible by simulating its event distribution and compar-

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ing it with that measured at KOTO. We will also incor-porate constraints coming from other measurements onKL and K+ decays. In Sec. III we consider the loop-induced process of KL → γγX which could disguise it-self as a signal event for the decay searched for at KOTObut has no counterpart at NA62. If the new particle cou-ples only in pairs to quarks for one reason or another,it will also appear in pairs in the kaon decays. We willtherefore make an analysis on the decays K → πXX inSec. IV. As we are conscious of the preliminary statusof the KOTO result, we will show what constraint couldbe set on the KL decay branching ratio if any of thethree candidate events were confirmed or completely fal-sified in the future. In the course of our analysis we willmake some suggestions that may deserve further study inthe KOTO and NA62 experiments. We summarize ourmain results in the last section V. Our analysis in thefollowing sections is based on the effective field theoryapproach. The effective interactions at leading order andthe amplitudes and branching ratios for various decaysare detailed in Appendixes A, B, and C, and the simula-tion setup and validation is discussed in Appendixes D.We mention for completeness that if there are new lightparticles other than the X particle that mediate the de-cays considered here, they could break the GN bound in adifferent manner or change the decay kinematical distri-butions to avoid the GN bound. We refer the interestedreader to Refs. [10, 18] on this approach.

II. KL → π0X

The signal region for the KL → π0νν̄ search at KOTOis also suitable for the two-body decay KL → π0X, whereX 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 Zvtx−PT plane of the twodecays are different. Here Zvtx is the π0 decay vertex po-sition projected onto theKL beam direction, and PT is itstransverse momentum with respect to the beam. We haveperformed the simulations by applying the kinematicalcuts proposed in KOTO’s 2015 data analysis [19] and as-suming the signal region chosen from its 2016−2018 dataanalysis [1]. While the decay KL → π0νν̄ has a relativelyuniform distribution in the signal region, KL → π0Xconcentrates in a narrow interval of PT . Furthermore,the heavier the X is, the lower the maximal PT is. For aheavy X, it is hard to accommodate the candidate eventswith a high PT at KOTO.

The branching ratio for KL → π0X corresponding toa specific number of candidate events, Nsignal, that canbe accommodated is estimated by the relation

B(KL → π0X

)KOTO

= Nsignal · SESπ0X (6)

= Nsignal · SESπ0νν̄ ·επ0νν̄

επ0X(mX). (7)

Here ε is the detection efficiency in the signal region atKOTO [1] upon imposing various kinematical cuts [19].We have estimated the ratio of the two efficiencies by sim-ulations and then used the above quoted SESπ0νν̄ (withan assumed relative error of 10%) to obtain SESπ0X . Thetotal background is 0.10±0.02 in the signal region (aboutone half from the decay KL → π0νν̄ and another halffrom its SM background) and 0.08±0.06 in the region be-low the blind box [1]. The branching ratio for KL → π0Xwith two-sided 68% C.L. limits is shown in Fig. 2 as afunction of mX in various mass intervals where a specificnumber Nsignal of candidate events are accommodated.Note that the choice of the interval delimiters is not sharpbut only a rough estimate based on simulations. WhenX is so heavy (roughly mX > 190 MeV) that none of thecandidate events can be accommodated or even all of itssignals drop below the blind box (mX > 270 MeV), wethen use the above background estimates to set a 90%C.L. upper bound on the decay branching ratio.

Now we examine whether the decay KL → π0X 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 mX ∈(100, 165) ∪ (260, 354) MeV. These intervals were nottaken into account in the NA62 experiment since its sen-sitivity to the decay K+ → π+X is considerably de-graded by the large backgrounds K+ → π+π0(γ) andK+ → π+π0π0, 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 onecannot provide a perfect solution either. For X in thefirst interval, i.e., mX ∼ mπ0 , the decay KL → π0X canonly accommodate two candidate events but not the onewith a high PT ∼ 238 MeV. If X is stable and invisibleand if we leave aside the last event, we may obtain thebranching ratio with two-sided limits at 68% (95%) C.L.at, e.g., mX = 135 MeV,

B(KL → π0X

)KOTO

= 1.12+1.21(+2.70)−0.74(−0.97) × 10−9. (8)

It might be appropriate to recall that a stable and invis-ible X has been studied as a light dark matter particlethat appears singly or in a pair in rare kaon to pion de-cays; see, for instance, Refs. [20–22]. But such a stableparticle would hardly relax the tension between the NA62and preliminary KOTO results due to the GN bound.Our strategy is to utilize their different experimental set-tings so that X escapes the KOTO detector but decaysin the NA62 detector without breaking the GN bound.The only potential loophole may appear in the regionaround mX ∼ mπ which has been cut in the NA62 ex-periment. But as analyzed above, even this fortuitoussituation does not work out perfectly.

For X of the other mass, it must be so long lived to beinvisible at KOTO while short lived to decay into SM par-ticles to be vetoed at NA62, thus avoiding the constraint

3

FIG. 1. Reconstructed events (blue scatters) in the Zvtx − PT plane after all the cuts for KL → π0νν̄ (top left) and forKL → π0XS with mXS = 10 MeV (top right), 135 MeV (bottom left), and 220 MeV (bottom right). The results for a vectorXV 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.

0 50 100 150 200 250 3000

2

4

6

8

10

3 Events 2 Events 1 Event 0 Event

(68% CL)

(68% CL)

(68% CL)Allowed Region (90% CL)

FIG. 2. Branching ratio for KL → π0XS(V ) with 68%C.L. limits in various mass intervals that can accommo-date a specific number of candidate events, and its 90%C.L. upper limit as a function of mass in the KOTO sig-nal region (solid line) and in the region below the blind box(dashed line). The dots correspond to the points mXS(V )

=1, 10, 20, . . . , 130, 135, 140, . . . , 290, 300 MeV.

from the GN bound. In order to incorporate all three can-didate events, X must be light enough, mX . 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 mX . 60 MeV, the signals of K+ → π+X at NA62would be predominantly located in its signal region 1 forthe K+ → π+νν̄ search [4] with missing mass squaredm2

miss ∈ (0, 0.01) GeV2, where no events were observed.Since we did not simulate the decays K+ → π+X, π+νν̄at NA62, we attempted an estimate on some simplifyingassumptions and by employing its combined 2016 and2017 data. We assumed an equal SES = (0.346±0.017)×10−10 [4] for both decays and an mX -independent effi-ciency. Utilizing the expected signals (0.81 ± 0.10) andbackground (0.60 ± 0.06) for K+ → π+νν̄ correspond-ing to their total expectations in signal regions 1 and2 at NA62 [4] and their distributions [23], we obtainedB (K+ → π+X)NA62 < 0.4×10−10 at 90% C.L.. Very re-cently, the NA62 group presented their preliminary resultthrough systematic simulations based on the 2017 data,and set a limit B (K+ → π+X)NA62 < (0.5−2.0)×10−10

at 90% C.L. for the low mX region [24]. By applying itsbound shape (which is related to the K+ → π+X effi-ciency) to the combined 2016 and 2017 data, we arriveat a slightly better bound,

B(K+ → π+X

)NA62

< (0.4− 1.6)× 10−10, (9)

for mX ∈ (0, 100) MeV. We will make our fur-ther analysis based on this bound. A similar limit,

4

B (K+ → π+X)E949 < (0.5− 1.2)× 10−10, was obtainedby E787/E949 [25].

Now we can employ the above results to obtain a strongconstraint on the branching ratio of KL → π0X. Wefirst recall that the real, physical branching ratio maydiffer its measured value in a specific detector (det) in anexperiment such as KOTO or NA62 if X decays [3]:

B (K → πX)real = B (K → πX)det eLp

mXcτX , (10)

where τX is the lifetime of X, and p/mX and L are theeffective boost factor and detector size, respectively. Weapply the above relation to both KL → π0X at KOTOand K+ → π+X at NA62, and employ equation (5) forreal branching ratios to link the two, so that we have

(r − 1) lnB(KL)real

& r lnB(KL)KOTO − ln[4.3B(K+)NA62

], (11)

where B with a subscript indicates its measured or realvalue, and r = (L/p)NA62/(L/p)KOTO. We use the val-ues (L/p)KOTO = (3 m)/(1.5 GeV) and (L/p)NA62 =(150 m)/(47 GeV). One word of explanation concerningthe choice p ≈ 47 GeV at NA62 follows. The X par-ticle is emitted spherically symmetrically from the K+

decay at rest. When the latter is boosted to an energyof 75 GeV, we analyzed that the momentum of the Xis evenly distributed in the range pX ∈ (40, 60) GeV forthe momentum of the π+ that falls in the signal regionat NA62, pπ+ ∈ (15, 35) GeV [24]. Upon comparing withRef. [16] we found that the choice pX ≈ 47 GeV fits besttheir simulation result and is also consistent with the re-sult in Ref. [3]. We note in passing that the stated choiceEX ≈ 37 GeV in Ref. [3], though followed in many laterpapers, actually neither fits its own numerical result norfalls in the observed pπ+ range at NA62. Plugging in thecentral value of B(KL)KOTO from Fig. 2 and the upperbound on B(K+)NA62 in equation (9), we arrive at thelower bound:

B(KL → π0X

)real

> (0.5− 3.1)× 10−8, (12)

at 90% C.L. for mX ∈ (0, 100) MeV. Applying thisbound to equation (10) for the KOTO detector, we ob-tain an upper bound on the lifetime τX of X as a functionof mX shown in Fig. 3. A similar analysis is done for theE949 experiment which searched for the two-body decayK+ → π+X of a stopping K+ and has the parametersLE949 = 1.5 m and pX =

√λ(m2

K ,m2π,m

2X)/(2mK) with

the usual triangular function λ(a, b, c) = a2 + b2 + c2 −2ab − 2bc − 2ca. The Kµ2

experiment [26] set an up-per bound on B(K+ → π+X) independently of whetherthe X decays or not. We translate it via the GN boundinto an upper bound on B(KL), and then apply it to theKOTO detector again through equation (10). This thenyields a lower bound on τX which is also shown in Fig. 3.

This lower bound of order 10−8 has been strictly re-stricted or excluded by other measurements. Being light,X can only decay to e+e− and/or γγ. For X → e+e−, wehave used the light scalar search result at Orsay eN →

0 20 40 60 80 10010-12

10-11

10-10

10-9

10-8

FIG. 3. Excluded regions in the mX − τX plane for the decayKL → π0X. The purple (blue) region is excluded by the GNbound using measurements of KL at KOTO [1] and of K+ atNA62 [4, 24] (E949 [25]). The orange region corresponds tothe exclusion of K+ → π+X from Kµ2 [26]. The gray region isexcluded by the untagged KL decay branching ratio [27]. Thecyan region bounded by the solid (dashed) curve is excludedby Orsay [28] ([29]). The black dashed curves correspond toB(KL → π0X) = 10−4, 10−6 and 10−8, and the green curveis the lower bound on τX from NA48 [30] assuming X → γγ.

eNφ with φ → e+e− [28, 29] to obtain the constraintsshown in Fig. 3. (For other potential constraints, see dis-cussions in Ref. [16].) We see that there is a small survivalspace for mX < 60 GeV, i.e., with mX ∈ (45, 60) MeVand τX ∈ (0.04, 0.11) ns. For X → γγ, the limits mainlycome from the measurements of B(KL → π0γγ) andits spectrum at KTeV [31] and NA48 [30]. The resultsB(KL → π0γγ

)= (1.29 ± 0.03 ± 0.05) × 10−6 at KTeV

and B(KL → π0γγ

)= (1.36±0.03±0.03)×10−6 at NA48

are consistent with the SM prediction ∼ 1 × 10−6 [32–35]. Furthermore, both theoretical calculations and ex-perimental observations give a consistent distributionwhich is dominated by the invariant mass interval mγγ ∈(160, 360) MeV. In the following we take the NA48 ex-periment to describe how to employ its result to set alower bound on τX for a light X as shown in Fig. 3.

The NA48 experiment detects four photons in the de-cay KL → π0γ3γ4 with π0 → γ1γ2, assuming that allphotons originate from the same KL decay vertex. Fora branching ratio satisfying the bound equation (12),the X particle from the decay KL → π0X will mostlydecay in the KOTO detector and thus definitely de-cay in the NA48 detector considering its parametersLNA48/pNA48 = (100 m)/(15 GeV). This means thatthe decay KL → π0X with X → γγ could mimic theKL → π0γγ signal at NA48. Since the X genericallydecays at a displaced vertex, the momenta of its daugh-ter photons (γ3γ4) will have a larger opening angle (θ34)than they would have if they had originated from thesame vertex as the other photon pair (γ1γ2) due to mo-

5

mentum conservation. Using the kinematical relationcos θ34 = 1 −m2

34/(2E3E4), where E3,4 and m234 are re-

spectively the energies and invariant mass squared of thephotons γ3γ4, we see that a given mX will appear as asmaller m34 < mX in the NA48 analysis; in other words,the true value of mX looks as though it is being shifteddownwards in the invariant mass analysis of NA48. Ifthe lower limit equation (12) held true for a light X,one would expect a peak or at least enhancement aroundm34 < mX < 60 GeV in the data. However, no suchenhancement was observed in the experiment. Actually,NA48 reported no signal events for m34 ∈ [0, 40] MeV,and set the limit B

(KL → π0γγ

)< 0.6 × 10−8 (90%

C.L.) for m34 ∈ [30, 110] MeV [30]. Using the data pro-vided in [30], we have estimated the upper bound form34 ∈ [0, 40] MeV, and found it to be about the sameas for m34 ∈ [30, 110] MeV. Taking into account theconstraint y = |pK · (k3 − k4)|/m2

K < 0.2 at NA48, wefound by a simple simulation that the acceptance forthe two-body decay chain KL → π0X with X → γγis about half of that for the three-body decay, which im-plies B

(KL → π0X with X → γγ

). 1.2 × 10−8. We

then use it to obtain a lower bound for τX as shown inFig. 3. We can see that there is no survival space forKL → π0X with X → γγ. These are great obstacles tothe interpretation of the KOTO result in terms of thedecay KL → π0X, especially when X is a light particle.

III. KL → γγX

In the KOTO experiment the directions of the photonpair were not recorded, and the reconstruction of thedecay KL → π0νν̄ was based on the assumption that thephoton pair arises from the π0 decay which in turn is aproduct of the KL decay in the beam line. This motivatesus to consider the decay KL → γγX in which the photonpair would be misidentified as coming from the π0 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. Hencethe X here could, in principle, act as dark matter, butthis scenario is not supported by the preliminary KOTOresult of distributions discussed below.

The decay KL → γγX can appear at the one-looporder. The effective field theory calculation of the de-cay rate is delegated to Appendixes B. Assuming paritysymmetry, X could be a pseudoscalar or an axial vector.In this section, we analyze its phenomenological aspectsby simulations. The distribution in the Zvtx − PT planeof the reconstructed events is depicted in Fig. 4 at threetypical masses of the pseudoscalar XP . A similar dis-tribution was found for the axial vector case and thusnot shown separately. As we can see from the figure, theKOTO’s three candidate events could be covered over arange of mX , but the decay events more tend to be lo-cated below the signal region. If we naively insist thatthe KOTO’s three events are faked by KL → γγX, more

events should be found below the signal region, whichhowever is not the case. Therefore, we will not pursuethis idea further, but employ KOTO’s results to work outan experimental upper limit on the decay KL → γγX in-stead.

We choose as our signal region for KL → γγX speci-fied by Zvtx ∈ [2.9, 5.1] m and PT ∈ [0, 120] MeV, whichexcludes KOTO’s three events but includes most of ourdecay events with a background of 0.08 ± 0.06. Accord-ing to the observed zero event in the region [1], we obtainan upper limit on B(KL → γγXP ) and relevant Wilsoncoefficients as a function of mXP in the top and bot-tom panels of Fig. 5, respectively. Also included is theprojected future detection capability of KOTO (dashedcurves) when an SESπ0νν̄ = 3.0×10−11 is reached. In thebottom panel, we also display limits coming from otherexperiments: K+ → π+π0νν̄ at E787 [36], KL → π0π0νν̄at E391a [37], and the K0 − K̄0 mixing [27]. For a lightXP , the limit is dominated by KL → π0π0XP , while fora heavy XP the strongest limit comes from the K0− K̄0

mixing; in between (roughly for mXP ∈ (220, 350) MeV)the KOTO can yield the best upper bound in the future.Similar limits for an axial vector XA are shown in Fig. 6.

IV. KL → π0XX

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. [18]. We consider here a bosonic Xwhich, for one reason or another, only couples in pairsto the quarks and may be a scalar or vector of eitherparity. Our simulation results for the signal distributionin the Zvtx − PT plane are shown in Fig. 7 at three dif-ferent masses. We find that the events for a light Xare almost evenly distributed in the signal region as inthe case for KL → π0νν̄ but differ significantly from thedecay KL → π0X as shown in Fig. 1. As X becomesheavier, the maximal PT also drops. But the truncationof the distribution around the maximal PT is not as sharpas in the two-body decay, and this leaves some space toexplain the KOTO signal with a high PT . For even largermasses, the distribution will shift below the blind box.

The branching ratio for KL → π0XX from the abovesimulation is shown in Fig. 8. The KOTO candidatesignals prefer a light scalar XS with mXS . 30 MeV,which must be unstable and have an appropriate lifetimeto avoid the GN bound. The requirement is that with ahigh probability the two XSs should be both invisible toKOTO while at least one of them decays into SM par-ticles (γγ or e+e−) which can be vetoed at NA62. Thisimplies in passing that the X particle would unlikely playthe role of stable dark matter. Qualitatively speaking, interms of kinematics, the experimental constraints on theπXSXS mode are looser than those on the πXS mode.But to assess the feasibility of the scenario, we have to

6

FIG. 4. Reconstructed events (blue scatters) in the Zvtx − PT plane after all the cuts for KL → γγXP with mXP = 10 MeV(left), 320 MeV (middle), and 430 MeV (right). With the same KL sample the difference in the total number of scatters forvarious mXP arises from different cut efficiencies.

100 200 300 400 50010-10

10-9

10-8

10-7

10-6

10-5

10-4

Excluded by current KOTO (90% CL)

Excluded by future KOTO (90% CL)

100 200 300 400 50010-10

10-9

10-8

10-7

10-6

FIG. 5. Top: upper limit on B(KL → γγXP ) from KOTO’scurrent data (solid) and projected future capability (dashed).Bottom: upper limits on Wilson coefficients: |CPsd| fromK+ → π+π0 + invisible by E787 experiment (red solid),|Re(CPsd)| from K0− K̄0 mixing (green solid), K+ → π+π0 +invisible by E391a experiment (blue solid), KL → γγXP byKOTO’s current result (purple solid) and future expectation(orange dashed).

determine the appropriate lifetime and mass of XS . Thephysical branching ratio for K → πXSXS at KOTO orNA62 can be expressed as

B(K → πXSXS)real

= B(K → πXSXS)det

∫e

(L1p1

+L2p2

)mXScτXS ×

f(L1, L2, p1, p2) dL1dL2dp1dp2, (13)

where f represents the probability that the two XS ’spropagate respectively at the momentum p1, p2 for adistance L1, L2 to exit the detector without decay. The

100 200 300 400 50010-10

10-9

10-8

10-7

10-6

10-5

10-4

Excluded by current KOTO (90% CL)

Excluded by future KOTO (90% CL)

100 200 300 400 50010-10

10-9

10-8

10-7

10-6

FIG. 6. Same as in Fig. 5 except for an axial vector XA.

integration over the signal region ensures that all situa-tions of the K decay have been taken into account. How-ever, it is difficult for us to manage a systematic simula-tion to cover all information due to the complexity of thedisplaced-vertex simulation. Considering that the twoXSs have a large boost at both KOTO and NA62, theyshould have a high probability of flying in the beam di-rection with a small angle to the veto plates. For furtherestimation, we found that p1 = p2 ≡ E and L1 = L2 ≡ Lserve as a good approximation, which leads to the sim-plification:

B(K → πXSXS)real

≈ B(K → πXSXS)det

∫dLdE e

2LE

mXScτXS F (L,E).(14)

The two-dimensional probability function F (L,E) ismuch simpler compared to the four-dimensional one. ForKL → π0XSXS at KOTO, it can be extracted from our

7

FIG. 7. Reconstructed events (blue scatters) in the Zvtx − PT plane after all the cuts for KL → π0XX with mX = 10 MeV(left), 70 MeV (middle), and 100 MeV (right) for a scalar (top) or vector (bottom) X.

0 50 100 1500

5

10

15

20

3 Events 2 Events 1 Event 0 Event

(68% CL)(68% CL)

(68% CL) Allowed Region (90% CL)

0 50 100 1500

1

2

3

4

2 Events 1 Event 0 Event

(68% CL)

(68% CL)

Allowed Region (90% CL)

FIG. 8. Same as Fig. 2 but for KL → π0XX with a scalar(top) or vector (bottom) X.

current simulation; for K+ → π+XSXS at NA62, wehave adopted a rough simulation in which only some cutson the signal region are considered.

As an example of simulation we consider the scenario

that a scalar XS of mass mXS = 10 MeV decays into aphoton pair. The branching ratio measured at KOTOcan be read off in Fig. 8, with the central value beingB(KL → π0XSXS)KOTO = 2.7 × 10−9. For NA62, as-suming that K+ → π+XSXS has the same acceptanceas K+ → π+νν̄, we obtain B(K+ → π+XSXS)NA62 <1.60 × 10−10 at 95% C.L.. By incorporating all theseinto equation (14), we obtain B(KL → π0XSXS)real asa function of the lifetime τXS shown in Fig. 9 as thegreen curve, and an upper limit on it (red curve) fromNA62 with the aid of the GN bound. This yields theallowed region with τXS . 2.2 × 10−7 s, within whichthe veto information can also be gained from the fig-ure: at least 74.6% of the K+ → π+XSXS signals arevetoed at NA62, while for τXS & 10−8 s the two XSsat KOTO nearly do not decay before exiting the detec-tor. On the other hand, the untagged KL branching ra-tio [27] constrains B(KL → π0XSXS) < 1%, which leadsto τXS & 1.7 × 10−10 s. But we emphasize once againthat this is only a rough estimate, and an appropriate de-termination of the lifetime and mass of XS 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 π0

decays into a pair of photons and neutrinos appear asmissing energy, we have considered three scenarios, i.e.,KL → π0X, γγX, π0XX. Our results can be summa-rized as follows. The simplest scenario KL → π0X is

8

0 1 2 3 40

1

2

3

4

5

6

FIG. 9. B(KL → π0XSXS

)real

as a function of τXS atmXS =10 MeV (green curve) together with the upper limit fromNA62 upon using the GN bound (red).

difficult to accommodate all three candidate events atKOTO, especially the one with a high PT . 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 tends to favor a lightX, our com-prehensive analysis on the NA62 and other experimentssets a strong constraint on the scenario KL → π0X: thedecay X → γγ is essentially excluded while the decayX → e+e− leaves open a very small region in the mX -τXplane. Since the scenario KL → γγX has no constraintat NA62, it is free of the GN bound. While the three can-didate 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 KL → π0XSXS ,where XS is a scalar with mXS . 30 MeV. To accom-modate the measurements at NA62, XS should be un-stable but long lived, whose lifetime is estimated to be1.7 × 10−10 s . τXS . 2.2 × 10−7 s at mXS = 10 MeV.But a more precise result would necessitate sophisticatedsimulations which we hope the KOTO and NA62 Collab-orations will perform in their future experimental analy-sis.

ACKNOWLEDGMENTS

We would like to thank Hai-Bo Li for helpful discus-sions, K. Tobioka for electronic communications, and J.Liu for explanations concerning Ref. [16]. This work wassupported in part by Grants No. NSFC-11975130, No.NSFC-11575089, the China Postdoctoral Science Foun-dation Grant No. 2018M641621, by the National Key Re-search and Development Program of China under GrantNo. 2017YFA0402200, and by the CAS Center for Excel-

lence in Particle Physics (CCEPP).We thank the anonymous referee for informing us of

the NA62 group’s preliminary result based on their 2017data after we submitted the first version of this work.

Appendix A: EFFECTIVE FIELD THEORYFRAMEWORK

For our purpose of accounting for the KOTO anomalywe assume a new real neutral particle X of mass below afew hundred MeV. It may be a scalar (XS), pseudoscalar(XP ), vector (XV ) or axial vector (XA) particle as appro-priate to the scenario under consideration. We start withthe low energy effective field theory that contains the Xparticle 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:

LX = CSpr(dpdr)XS + CPpr(d

piγ5dr)XP

+CVpr(dpγµd

r)XµV + CApr(d

pγµγ5dr)Xµ

A, (A1)

where p, r refer to the d, s quarks and the Wilson coef-ficients are 2 × 2 Hermitian matrices in the d, s space.When studying the third scenario K → πXX, we switchoff the above single-X couplings but switch on the fol-lowing double-X couplings:

LXX = (s̄d)(gSXXX + gVXX

µXµ

)+ H.c., (A2)

where X and Xµ may have any parity without affectingour discussions in the work.

Below the chiral symmetry breaking scale the Nambu-Goldstone bosons become the dynamical degrees of free-dom, whose low energy physics is determined at leadingorder by [38, 39]

Lχ =F 2

0

4Tr(DµU(DµU)†

)+F 2

0

4Tr(χU† + Uχ†

), (A3)

where U = exp(i√

2Φ/F0

)exponentiates the octet

Nambu-Goldstone bosons

Φ =

π0√

2+ η√

6π+ K+

π− − π0√

2+ η√

6K0

K− K̄0 −√

23η

, (A4)

and

DµU = ∂µU − ilµU + iUrµ, (A5)

χ = 2B(s− ip). (A6)

As usual, F0 is the decay constant in the chiral limit andB parametrizes the quark condensate 〈q̄q〉 = −3BF 2

0 .The new field X gets involved in the form of additional

9

FIG. 10. Feynman diagrams for KL → γγX at first nonvan-ishing order. The large (small) dot stands for the effective X(standard QED) interactions in equation (A3).

terms in the external sources to QCD, which are Hermi-tian matrices in the light flavor space:

lµ = CVXµV − C

AXµA, (A7)

rµ = CVXµV + CAXµ

A, (A8)

s = CSXS , (A9)

p = −CPXP , (A10)

in the case of single-X couplings in equation (A1), and

spr = δ3pδ

2r

(gSXXX + gVXXµX

µ)

+δ2pδ

3r

(gS∗X XX + gV ∗X XµX

µ), (A11)

in the case of double-X couplings in equation (A2). Notethat the QED interaction is contained as usual in thesources lµ, rµ, which we do not write explicitly. Theeffective interactions contained in equation (A3) will beapplied to calculate the quantities in the following ap-pendices.

Appendix B: DECAY AMPLITUDES ANDDISTRIBUTIONS

The decay width for KL → π0X is, for a scalar XS ,

dΓπ0XS

dpπ,z=

B2

16πm2KL

[Re CSsd

]2, (B1)

where pπ,z is the π0 momentum component in the KL

beam direction, or, for a vector XV ,

dΓπ0XV

dpπ,z=λ(m2

KL,m2

π0 ,m2X)

16πm2KLm2X

[Im CVsd

]2, (B2)

with λ(x, y, z) = x2 + y2 + z2 − 2xy − 2yz − 2zx.The decay KL(k)→ γ(q1)γ(q2)X(p) involves only neu-

tral particles, and can only take place at the one-loop or-der whose Feynman diagrams are shown in Fig. 10. Theresult is finite, and the (spin-summed) squared matrixelement is, for a pseudoscalar XP ,

|MP |2 =

(αBRe CPsd6√

2πF0s

)2

|f(rπ) + f(rK)|2 , (B3)

where rπ,K = s/(4m2π±,K±), s = (q1 + q2)2, and α ≈

1/137 is the fine structure constant, or for an axial vectorXA,

|MA|2 =

(αRe CAsd

8√

2πF0mXAs

)2

|f(rπ) + f(rK)|2

×λ(m2KL ,m

2XA , s). (B4)

The one-loop function is

f(r) = 4+

−4

rarcsin2(

√r), r ≤ 1

1

r

[2 ln

(√r −√r − 1

)+ iπ

]2, r > 1

(B5)

For simulations, we use the distribution

dΓγγXP (A)

ds dt=

1

512π3m3KL

|MP (A)|2, (B6)

where t = (k − q1)2.The decay width for KL(k)→ π0(p)X(q1)X(q2) is eas-

ily computed to be, for a scalar XS or vector XV respec-tively,

dΓπ0XSXS

ds dt=B2(Re gSX)2

128π3m3KL

, (B7)

dΓπ0XVXV

ds dt=B2(Re gVX)2

128π3m3KL

[2 + (2rX − 1)2

], (B8)

where rX = s/(4m2X), s = (q1 + q2)2, and t = (k − q1)2.

Appendix C: OTHER EXPERIMENTALCONSTRAINTS

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 [36, 37]

B(K+ → π0π+νν̄) < 4.3× 10−15, (C1)

B(KL → π0π0νν̄) < 8.1× 10−7. (C2)

The corresponding amplitudes for the decay K(k) →π(p1)π(p2)X(q) read, for a pseudoscalar XP ,

M(K+ → π+π0XP ) =B

2F0CPsd

u− tm2K −m2

X

, (C3)

M(K̄0 → π0π0XP ) =BCPds

2√

2F0

s+m2X −m2

K

m2K −m2

X

, (C4)

and for an axial vector XA,

M(K+ → π+π0XA) = iCAsdF0

(p2 − p1) · ε∗, (C5)

M(K̄0 → π0π0XA) = iCAds√2F0

k · ε∗, (C6)

where s = (p1 + p2)2, t = (k − p1)2, u = (k − p2)2, and εis the polarization vector of XA.

Moreover, the K0 − K̄0 mixing can give limits on thecouplings. The experimental measured quantities are theKL − KS mass difference ∆MK and the CP violationparameter εK [40–42], whose current experimental valuesare [27],

∆MK = (3.484± 0.006)× 10−12 MeV, (C7)

|εK | = (2.228± 0.011)× 10−3. (C8)

10

Considering the theoretical uncertainties from long-distance contributions in ∆MK , we require the new con-tribution do not exceed the experimental value. Rela-tively, the calculation of εK is more credible; hence we,require the new contribution to be less than 30% of itsexperimental value. The limits on the Wilson coefficientsthen read, for a pseudoscalar XP ,∣∣(Re CPsd)

2 − (Im CPsd)2∣∣ < 1

2RA, (C9)∣∣Re CPsd Im CPsd

∣∣ < 0.3√2|εK |RA, (C10)

with RP = ∆MKmK(m2K −m2

XP)/(B2F 2

K), and for anaxial vector XA,∣∣(Re CAsd)

2 − (Im CAsd)2∣∣ < 1

2RA, (C11)∣∣Re CAsd Im CAsd

∣∣ < 0.3√2|εK |RA, (C12)

with RA = ∆MKm2XA/(F 2

KmK).

Appendix D: SIMULATION OF KL DECAY

In this appendix we will briefly describe how we do thesimulations for various KL 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 KL → π0X with KOTO’s to verify our simulation.

We first generate initial KL particles according tothe momentum distribution of KL measured experimen-tally [43], which will have a certain probability of decayin the detector. All of the KL decay modes in this paperproduce two photons, and the distributions of the ener-gies and positions of the photons are largely dependenton the probability distribution functions. For the two-body decay KL → π0X, we use a uniform distributionfunction as in Eqs. (B1) and (B2). For the three-bodydecay KL → π0νν̄, we adopt the same distribution func-tion as in equation (S1) in the supplemental materialof Ref. [3]. The distribution functions of KL → Xγγand KL → π0XX are determined by equation (B6), andEqs. (B7) and (B8), respectively. It is worth mentioningthat for the decay modes KL → π0νν̄, KL → π0X, andKL → π0XX, the two photons are originated from theπ0, while for KL → Xγγ, the two photons are directly

generated by KL; only the former decays will lead to aninvariant mass of the two photons mγγ ' mπ0 . Thenthe photons are captured by the CsI calorimeter in thedetector, and we record their energies and positions inour simulation. In order to better simulate the detec-tor’s response to the photons, we include the energy andposition resolution of the CsI calorimeter, which can befound in Ref. [44].

0 50 100 150 200 2500

2.×10-9

4.×10-9

6.×10-9

8.×10-9

1.×10-8

FIG. 11. Validation of B(KL → π0X) upper bound at90% C.L. by comparing our result (red dots) with KOTO’s[19] (black dots). The horizontal line is the upper limit forB(KL → π0νν̄) < 3.0× 10−9 at 90% C.L..

By assuming that the two photons produced on thebeam axis are from π0 decay (in the interesting case ofKL → Xγγ the two photons are also required to havean invariant mass mγγ ' mπ0 in order to fake the sig-nals), we can reconstruct the decay location Zvtx and π0’stransverse momentum PT by combining the informationof the photons’ energies and positions. Then we use thesame selection criteria as KOTO [19] to analyze the re-constructed events. We use the signal region in KOTO’sanalysis of 2015 data [19] for validation, whose result isshown in Fig. 11, but employ the new signal region [1]to analyze our decay modes. It should be noted that wedo not consider shape-related cuts and veto cuts, whoseefficiencies are considered the same for different processesand can be estimated as r = εKOTO

π0νν̄ /επ0νν̄ , where επ0νν̄ is

from our rough simulation and εKOTOπ0νν̄ is the real accep-

tance from KOTO’s systematic simulation. We calculatethe real acceptance of each process by multiplying ourresult with r. After applying the reconstruction and var-ious kinematical cuts, we finally obtain the distributionsof signal events in the Zvtx−PT plane, and the branchingratios and Wilson coefficients of some processes can beevaluated accordingly.

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