Constraints on long-lived light scalars with flavor-changing couplingsand the KOTO anomaly

P. S. Bhupal Dev,1, ∗ Rabindra N. Mohapatra,2, † and Yongchao Zhang1, 3, ‡

1Department of Physics and McDonnell Center for the Space Sciences,Washington University, St. Louis, MO 63130, USA

2Maryland Center for Fundamental Physics, Department of Physics,University of Maryland, College Park, MD 20742, USA

3Department of Physics, Oklahoma State University,Stillwater, OK 74078, USA

Recently, the KOTO experiment at J-PARC has observed three anomalous events in the flavor-changing rare decay KL → π0νν̄, which indicates that the corresponding branching ratio is almosttwo orders of magnitude larger than the Standard Model (SM) prediction. Taking this intriguingresult at face value, we explore model implications of its viable explanation by a long-lived lightSM-singlet scalar (S) emission, i.e. KL → π0S, with S decaying outside the KOTO detector. Wederive constraints on the parameter space of such a light scalar in the context of three simplemodels: (i) a real singlet scalar extension of the SM; (ii) a B − L extension where neutrino massesarise via type-I seesaw mechanism from B − L breaking; and (iii) a TeV-scale left-right symmetricmodel. The flavor-changing couplings needed to explain the KOTO excess in models (i) and (ii)originate from tree-level mixing of the scalar with SM Higgs field (h), and in model (iii), from themixing of S and h with the neutral component of the heavy bidoublet Higgs field. After takinginto account the stringent constraints from high-precision searches for flavor-changing charged andneutral kaon decays at NA62, E949, KOTO and CHARM experiments, as well as the astrophysicaland cosmological constraints on a light scalar, such as those from supernova energy loss, big bangnucleosynthesis and relativistic degrees of freedom, we find that the light scalar interpretation of theKOTO excess is allowed in all these models. Parts of the parameter range can be tested in futureNA62 and DUNE experiments.

I. INTRODUCTION

In the Standard Model (SM) of particle physics, flavor-changing neutral currents (FCNCs) are absent at tree-level and are predicted to be small at loop level, sup-pressed by the Glashow-Iliopoulos-Maiani (GIM) mech-anism and the small off-diagonal Cabibbo-Kobayashi-Maskawa (CKM) matrix elements in the quark sector(or by the tiny neutrino masses in the lepton sector, ifwe allow nonzero neutrino masses to be part of the ‘new’SM) [1]. Any observation of FCNC above the SM pre-diction would therefore be a clear signature of beyondthe SM (BSM) physics. Very recently, the KOTO exper-iment at J-PARC has observed four candidate events inthe signal region of one such rare flavor-changing decayKL → π0νν̄ [2]. While one of the events is suspected tohave originated from SM activity upstream from the de-tector and can be vetoed away, the remaining three eventscannot be explained by currently known backgrounds,with the SM expectation of only 0.05±0.02 events. Thiscorresponds to a decay branching ratio (BR) of [2]

BR(KL → π0νν̄)KOTO19 = 2.1+2.0(+4.1)−1.1(−1.7) × 10−9 , (1)

at 68 (95)% confidence level (CL), where the uncertain-ties are primarily due to statistics. This result is consis-

∗ bdev@wustl.edu† rmohapat@umd.edu‡ yongchao.zhang@physics.wustl.edu

tent with their previously reported 90% CL upper boundof [3]

BR(KL → π0νν̄)KOTO18 < 3.0× 10−9 . (2)The central value in Eq. (1) is almost two orders of mag-nitude larger than the SM prediction of [4]

BR(KL → π0νν̄)SM =(3.4± 0.6

)× 10−11 . (3)

Needless to say, more experimental information on thesource of these intriguing events, as well as a carefulreevaluation of background estimations, is needed to con-firm whether the signal is indeed due to some BSMphysics. But given the far-reaching consequences, wetake the KOTO result (1) at face value and explore pos-sible implications for some simple BSM scenarios thatcan be independently tested in other ongoing or futureexperiments.

At the phenomenological level, the KOTO signal canbe interpreted as the emission of a new light, long-livedscalar particle S in the two-body kaon decay KL → π0S,which subsequently decays outside the KOTO detector,thus mimicking the invisible νν̄ final states in KL →π0νν̄ [5–7].1 In this paper, we consider possible ultra-violet (UV)-complete model frameworks for such a new

1 Other interpretations in terms of either a heavy mediator or anew light particle produced at fixed target and decaying off-axisto two photons (e.g. an axion-like particle) have also been dis-cussed [6]. Similarly, Ref. [8] has considered the possibility ofKL → π0QQ̄, where Q is a dark fermion of the dark sector.

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light scalar particle S with mS < mK − mπ0 and witha flavor-changing effective coupling of the form KπS sothat it can be emitted in kaon decay. Being light, thereare stringent constraints on this particle from laboratorysearches for FCNCs in the K, D and B meson decays.In particular, the scalar emission through the effectiveKπS coupling contributes to both neutral and chargedkaon decays, i.e. KL → π0S and K+ → π+S, whosebranching ratios (for an invisible S) are correlated bythe Grossman-Nir bound [9]

BR(KL → π0νν̄) ≤ 4.3 BR(K+ → π+νν̄) . (4)

No excess has been reported in the charged kaon decaymode K+ → π+νν̄, whose branching ratio is currentlyconstrained by the NA62 experiment [10] (and also byE949 experiment [11]) to be

BR(K+ → π+νν̄)NA62 < 2.44× 10−10 (5)

at 95% CL, which is consistent with the SM predictionof [4]

BR(K+ → π+νν̄)SM = (8.4± 1.0)× 10−11 . (6)

There also exist stringent constraints on a light, long-lived scalar decaying into charged lepton or photon pairsfrom the searches for `+`− and γγ in rare kaon decays atproton beam-dump experiments, such as CHARM [12].In addition, a light S particle will be constrained by as-trophysical and cosmological observations, such as thosefrom supernova energy loss, and effective relativistic de-grees of freedom (∆Neff) and/or additional energy injec-tion at the big bang nucleosynthesis (BBN) epoch.

To see whether the KOTO excess is consistent with allthese constraints, it is convenient to work within specificmodels so that the new scalar interactions with SM parti-cles have a definite profile. Due to the suppressed natureof these interactions, only models where the particle Shas no direct tree-level coupling to SM quarks need to beconsidered. We find the following three BSM scenarioswhich fall into this category:

(i) Scalar singlet model: Here the FCNC couplingsof S arise from its mixing with the SM Higgs fieldh, which has loop-induced FCNC couplings withSM quarks [13–15]. The new scalar S could belong-lived if the S − h mixing angle, θ is suitablysmall, while avoiding all existing laboratory con-straints [16–21]. This is a simple, two-parametermodel with only mS and θ as the unknown param-eters. We call this the SM+S model.

(ii) U(1)B−L model: A class of UV-complete modelswhere such a light scalar without tree-level couplingto SM fermions emerges naturally is based on thegauge group SU(2)L × U(1)I3R × U(1)B−L [22–25].In this case, three right-handed neutrinos (RHNs)Na(1, 1/2,−1) (with a = 1, 2, 3) are introduced forthe purpose of anomaly cancellation. The lightscalar S can be identified as the real part of a com-plex (B−L)-charged scalar ∆(1,−1, 2) that breaks

the U(1)I3R × U(1)B−L symmetry to U(1)Y of theSM and gives mass to the RHNs to implement thetype-I seesaw mechanism [26–30]. The long-livedproperty and the FCNC constraints of this modelon S have already been studied in great detail inRef. [31] (where S was denoted by H3), which willbe relied upon here. This model has some new, sup-pressed decay modes such as S → NN,Z ′Z ′ (whereZ ′ is the massive gauge boson associated with theU(1)B−L breaking) which are absent in the SM+Smodel. However, as long as all the three RHNsand the Z ′ boson are much heavier than the lightscalar S, they will not have any effect on the life-time of S, and the KOTO phenomenology of light Sin the U(1)B−L extension will be the same as in theSM+S model. In what follows, we assume this tobe the case and therefore do not separately discussthe U(1)B−L scenario for the KOTO explanation,except for the complementary collider signatures,which are different in the U(1)B−L case due to theadditional gauge-portal production.

(iii) Left-right symmetric model: The last classof models studied here is the left-right symmetricmodel (LRSM) based on the gauge group SU(2)L×SU(2)R×U(1)B−L [32–34]. Here the light scalar S(denoted by H3 in Refs. [31, 35]) can be identifiedas the real part of the neutral component of the(B − L)-charged, SU(2)R-triplet field ∆R(1,3, 2),which can be light and does not couple directly toSM quark fields prior to symmetry breaking [35–37]. It is therefore similar to the SM+S model inmany respects and can play a role in resolving theKOTO anomaly. The field ∆R is responsible forthe SU(2)R×U(1)B−L symmetry breaking and themodel, like the U(1)B−L model above, has the ex-tra motivation of being connected to neutrino massgeneration via type-I seesaw [26–30]. In contrastwith the previous two models, the FCNC couplingsof S in this case arise at tree-level, due to its mix-ing with the heavy scalar H1 from the bidoublet Φ(and the SM Higgs). Another special feature of thelight scalar S in the LRSM is that even for smallmixing angles , it can still decay into two photonsthrough the WR loop and the heavy charged scalarloops. This makes the FCNC limits, as well as thesupernova and BBN limits, on light S in the LRSMvery different from the other two models discussedabove.

As we show below, a limited parameter space that sat-isfies all the existing constraint allows an explanation ofthe KOTO excess in all the models. This allowed pa-rameter range can be tested in the future high-precisionintensity frontier experiments, such as NA62 and DUNE.

The rest of the paper is organized as follows: In Sec. II,we discuss the simplest real scalar extension of the SMin light of the KOTO excess vis-á-vis other laboratoryand astrophysical/cosmological constraints. Most of this

3

discussion is also applicable to the U(1)B−L case. InSec. III, we repeat the same exercise for the LRSM. Ourconclusions are given in Sec. IV.

II. SINGLET MODEL

The singlet scalar extension of the SM is one of thesimplest and well-motivated BSM scenarios [16]. Themost general renormalizable scalar potential of the SMHiggs doubletH and a real singlet scalar S can be writtenas

V = −µ21(H†H)− µ22S2

+λ1(H†H)2 + λ2H

†HS2 + λ3S4 , (7)

with µ21,2 > 0 being the mass parameters and λ1,2,3 be-ing the quartic couplings. We impose a Z2-symmetryunder which S → −S to prevent the S3 and SH†H tri-linear terms. After spontaneous symmetry breaking, theH and S fields obtain non-vanishing vacuum expectationvalues (VEVs), with 〈H〉 = (0, vEW)T with vEW ' 174GeV, the electroweak (EW) VEV and 〈S〉 = vS . Theh − S mixing (where h is the physical SM Higgs field,obtained by expanding the H-field around its VEV, i.e.H = (0, vEW + h)

T) is determined by the quartic cou-pling λ2. In the small mixing limit, the mass of the realcomponent of S is m2S ' 4λ3v2S to the leading order. Forsufficiently small λ3 and vS , the scalar S could be verylight, even down to a few MeV scale.2

In the U(1)I3R×U(1)B−L extension discussed in Sec. I,the S-field can be identified as the real part of a (B−L)-charged scalar, whose VEV breaks the U(1)I3R×U(1)B−Lgauge symmetry down to U(1)Y [31]. We assume theRHNs in this case are all heavier than S, so that thedecays of S are identical to those of the SM+S case, be-ing governed only by two parameters, namely, the scalarmass mS and the h− S mixing angle sin θ.

A. Fitting the KOTO anomaly

The couplings of S to the SM fermions arise fromits mixing with the SM Higgs h and are thus flavor-conserving at the tree level. However, FCNCs are gener-ated at one-loop level through Penguin diagrams involv-ing the W−top loop and CKM quark mixings. The effec-tive Lagrangian relevant for FCNC kaon decay is givenby [17]

Leff ⊃ ysd sin θSs̄LdR + H.c. ,

with ysd =3√

2GFm2tV∗tsVtd

16π2mS√2vEW

, (8)

2 When S is light, the SM Higgs might contribute radiatively tothe S mass, potentially making it heavier. However, this effectis highly suppressed by the h−S mixing angle sin θ, which needsto be small to make the S long-lived, as required for the KOTOexcess explanation.

where ysd is the effective loop-level coupling in the SM,GF is the Fermi constant, ms, t the strange and top quarkmasses, and Vtd, ts the CKM matrix elements. As a resultof the CP phase in the CKM matrix, the coupling ysdis complex. If kinematically allowed, this will induce theflavor-changing decays K± → π±S and KL → π0S, withthe partial widths

Γ(K± → π±S) ' mK± |ysd|2 sin2 θ

64π

m2K±

m2S×β2(mK± ,mπ± ,mS) , (9)

Γ(KL → π0S) 'mKL (Re ysd)

2sin2 θ

64π

m2K0

m2S×β2(mKL ,mπ0 ,mS) , (10)

with the kinematic function

β2(M, m1, m2) ≡[1− 2(m

21 +m

22)

M2+

(m21 −m22)2

M4

]1/2.

(11)

Note that the partial decay widths in Eqs. (9) and (10)are almost identical, except for the crucial difference thatthe decay KL → π0S depends only on the real part ofthe coupling ysd.

The same h−S mixing is also responsible for S decaysinto the SM quarks (u, d, s) and charged leptons at tree-level, and gluons and photons at one-loop level, just likethe SM Higgs boson decays. In the generic singlet modelall the decay modes of S are universally proportional tothe mixing angle sin θ, and therefore, the branching ratiosdepend only on the S mass but not on sin θ. As detailedin Ref. [31], if S is light, say below the GeV-scale, it tendsto be long-lived for a wide range of sin θ. If its averagedecay length is larger than the KOTO detector size, theprocess of interest will be

KL → π0S , S → invisible , (12)

with S decaying into anything outside the detector. Thishas the same final state as the decay KL → π0νν̄, i.e. twophotons from π0 → γγ and significant missing energy. Inthis case, the effective branching ratio is given by3

BReff(KL → π0S) = BR(KL → π0S) exp[−LΓS/b] ,(13)

where BR(KL → π0S) = Γ(KL → π0S)/ΓtotalKL , L = 3m for the KOTO detector, and b = ES/mS the Lorentzboost factor with energy ES ' 1.5 GeV. For the total

3 There is an O(1) correction factor to account for the kinematicaldifference between the 3-body SM decay KL → πνν̄ and the2-body decay KL → π0S in our scalar case, whose exact valuedepends on the scalar mass [3]. Here we have simply assumed itto be one, given the fact that there is no directional informationin the KOTO signal which only involves charge-neutral particlesand vetoes all charged particles.

4

decay width of KL, we use ΓtotalKL

= Γ(KL → π0S)+ΓSMKL ,where Γ(KL → π0S) is given by Eq. (10) and ΓSMKL =(1.29± 0.01)× 10−17 GeV [1].

Using Eq. (13), we calculate the preferred region inthe (mS , sin θ) parameter space that explains the KOTOexcess given by Eq. (1) at 95% CL. Our result is shown bythe green shaded region in Fig. 1, with the green dashedline corresponding to the KOTO central value in Eq. (1).The region with mS > 180 MeV is not included in thisfit, because it does not overlap with the KOTO signalregion [6].

B. Laboratory constraints

As shown by the Grossman-Nir bound [cf. Eq. (4)],the FCNC decays of charged and neutral kaons are cor-related. This relation has to do with general isospin sym-metry arguments, which relate the decay amplitudes ofK± to those of K0 and K0, and holds even for the 2-bodydecays KL → π0S and K+ → π+S [38]. For our singletscalar case, this can be explicitly seen from Eqs. (9) and(10). As a result, the current and future high-precisionmeasurements of the charged and neutral kaon rare de-cays can be used to set limits on the scalar mass mS andmixing angle sin θ, as discussed below.

In the charged kaon sector, the most stringent limitscome from searches of K+ → π+νν̄ at NA62 [10] andof K+ → π+X (with X being a long-lived particle) atE949 [11]. The NA62 experiment has put a 95% CLupper limit on BR(K+ → π+νν̄) [cf. Eq. (5)] which canbe translated into an exclusion region in the (mS , sin θ)plane, as shown by the blue line in Fig. 1. Here we haveconstructed an effective BR, similar to Eq. (13), replacingneutral mesons by charged mesons and modifying theexperimental parameters to L = 150 m and ES ∼ 37 GeVfor NA62. Again we have neglected the O(1) kinematicaldifference between the 3-body SM decay and the 2-bodydecay in our scalar case. We see from Fig. 1 that there isa gap in the NA62 excluded region around the pion mass.This is because of the fact that if the scalar mass mS isclose to π0 mass, we will have a large pion backgroundfrom the SM process K+ → π+π0 with π0 → νν̄. Withthe current limit of BR(π0 → νν̄) < 2.7 × 10−7 [1] andthe SM K+ → π+π0 branching ratio of 20.6% [1], theNA62 limit on K+ → π+S turns out to be two orders ofmagnitude weaker in this region, as shown by the gap inFig. 1.

The E949 experiment has reported 95% CL bounds onBR(K+ → π+X), where X is a long-lived particle, as afunction of the X mass [11]. The most stringent limiton the branching ratio BR(K+ → π+X) could reach upto 5.4 × 10−11 if the new particle X is stable. Usingthe same procedure as above for NA62, we evaluate theeffective branching ratio following Eq. (13), with decaylength L = 4 m and energy ES ' 710 MeV for E949. Thecorresponding exclusion region is shown by the magentaline in Fig. 1, which is up to a factor of few stronger

than the NA62 exclusion region in the low-mass range,but is weaker in the high-mass range and not applicablefor mS > 2mπ0 because in this case the S tends to decayquickly, compared to the E949 detector size of 4 m. Likethe NA62 limit, there is also a gap for the E949 constraintwhen mS ∼ mπ.

Similarly, a previous KOTO search has reported 90%CL upper limits on the 2-body decay BR(KL → π0X),where X is an invisible boson, as a function of the Xmass [3]. We can directly use this bound for our scalarcase, and following Eq. (13) with L = 3 m and ES ' 1.5GeV for KOTO, translate it into an exclusion region inthe (mS , sin θ) plane, as shown in Fig. 1 by the brownline. Note that there is no gap in the KOTO limit formS ∼ mπ, because the 2-body decay of K0L → π0π0is CP -violating and CKM-suppressed in the SM, with abranching ratio of 8.6× 10−4 [1].

Further limits on the light scalar can be derived fromthe e+e−, µ+µ− and γγ decay products of S produced inneutral and charged kaon decays at proton beam-dumpexperiment such as CHARM [12]. The production crosssection of S at CHARM is given by [18, 39, 40]

σS ' σppMpp[

1

2χsDK±BR(K

+ → π+S)

+1

4χsDKLBR(K

0 → π0S)], (14)

where σpp is the proton-proton cross section, Mpp = 11the average hadron multiplicity, and χs = 1/7 is the frac-tion of strange pair-production rate. In Eq. (14), the fac-tor DK ' `K/bKcτK (with K standing for both K± andKL) takes into account the re-absorption of Kaons beforedecaying [7], with `K = 15.3 cm the absorption length,bK = EK/mK the Lorentz boost factor with EK ' 25GeV, and τK the total Kaon decay width. It turns outthat the re-absorption factors are respectively 8.1×10−4and 2.0× 10−4 for K± and KL. Normalized to the neu-tral pion yield σπ0 ' σppMpp/3, we can predict the totalnumber of S particles produced: NS ' 2.9×1017σS/σπ0 .Then the number of events collected by the detectorwould be

Nevent = NS

( ∑χ=e,µ,γ

BR(S → χχ)

)

×[exp

(−LΓS

b

)− exp

(− (L+ ∆L)ΓS

b

)],

(15)

where L = 480 m is the CHARM beam dump base-line, ∆L = 35 m is the detector fiducial length, andb = ES/mS is the boost factor with ES ' EK/2 [12].Due to the huge number of events NS , the mixing an-gle sin θ is expected to be severely constrained, and themost stringent limits are from the ee and µµ channels,as the γγ channel is comparatively suppressed by theloop factor. Given that no signal event was found atCHARM, an upper limit of Nevent < 2.3 at the 90%

5

50 100 150 200 250

1.×10-4

5.×10-40.001

0.005

mS [MeV]

sinθ

generic singlet model

KOTO anomaly

KOTO

E949

NA62

CHARM

Figure 1. The parameter space favored by the KOTO anomaly [2] in the generic singlet scalar model (cf. Sec. II A) is shownby the green shaded region (95% CL), with the green dashed line corresponding to the central value quoted in Eq. (1). Forcomparison, we show the exclusion regions from a previous KOTO search for KL → π0νν̄ (brown) [3], NA62 search forK+ → π+νν̄ (blue) [10], E949 search for K+ → π+X (magenta) [11], and beam-dump experiment at CHARM (orange) [12];cf. Sec. II B. All the gray shaded regions are excluded.

CL on the contribution from BSM physics was set. Weuse this limit to derive the corresponding exclusion re-gion in the (mS , sin θ) plane, as shown by the orangeline in Fig. 1. For lighter S, the boost factor b becomeslarger, and fewer S decays happen inside the detector,thereby weakening the constraints. As can be seen fromFig. 1, even if all the laboratory constraints are taken intoconsideration, there is still a narrow parameter space inthe singlet model, i.e. 110 MeV

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by LHCb [54], BR(Bd → γγ) by BaBar [55] andBR(Bs → γγ) by Belle [56]. However, these limitsare much weaker and are not relevant to the KOTOanomaly.

If the light scalar S is long-lived, it can also be searchedfor at the LHC [57] and/or the dedicated long-lived par-ticle (LLP) detectors such as MATHUSLA [58]. At thehigh-energy colliders, S can be produced from the loop-level gluon fusion process gg → gS via mixing with theSM Higgs, and the cross section can go up to (25 pb)× sin2 θ [31]. The LLP searches at the HL-LHC and FCC-hh could probe a large parameter space of mS and sin θ(see Fig. 20 in Ref. [31]); however, they do not cover theKOTO parameter space of interest in Fig. 1, and hence,are not shown.

C. Astrophysical and cosmological constraints

For completeness, we consider in this subsection theastrophysical and cosmological constraints on the lightlong-lived scalar S. A sufficiently light S can be producedin significant amounts in the supernova core via the nu-clear bremsstrahlung process N+N → N+N+S, withN = p, n collectively standing for protons and neutrons.This process is induced by the mixing of S with the SMHiggs field and the effective couplings of the SM Higgsto nucleons. Through the couplings to quarks inside nu-cleons, the effective couplings ghNN of the SM Higgs tonucleons are of order ∼ 10−3 [59, 60]. Let us first makea ballpark estimate of the supernova limits. The totalenergy loss rate due to the emission of the light scalar Sis

Q ∼ Vcn2NσNN→NNS〈ES〉

∼ 3Vcn2Nα

2πg

2hNNT

7/2 sin2 θ

4π3/2m9/2N

, (16)

where Vc =4π3 R

3c is the supernova core volume with

Rc the core size, nN the nuclear density in the super-nova core, απ ' (2mN/mπ)2/4π the effective couplingof pion to nucleons with mN and mπ respectively themasses of nucleons and pion, T ' 30 MeV the tempera-ture in the supernova core, 〈ES〉 the averaged energy ofthe scalar S, and σNN→NNS the production cross sec-tion. Taking a typical supernova core size Rc = 10 km,nN = 1.2×1038cm−3, we get Q ' 6×1065 sin2 θ erg/sec.Comparing with the observed energy loss rate of 3×1053erg/sec [61], we get sin θ

7

the scalar sector:

Φ =

(φ01 φ

+2

φ−1 φ02

),

∆R =

(∆+R/√

2 ∆++R∆0R −∆

+R/√

2

). (19)

The SU(2)R×U(1)B−L symmetry is broken down to theSM U(1)Y gauge group, once the triplet develops a non-vanishing VEV 〈∆0R〉 = vR. The bidoublet Φ, with theVEVs 〈φ01〉 = κ and 〈φ02〉 = κ′ (where vEW =

√κ2 + κ′ 2),

is responsible for breaking the SM gauge group SU(2)L×U(1)Y down to U(1)em and for the generation of SMquark and charged lepton masses as well as the Diracmass matrix for the type-I seesaw.

In the bidoublet sector, the SM Higgs h is predomi-nantly from the real component of the neutral scalar φ01.There is a heavy CP-even scalar H1 from the real com-ponent of φ02, which couples to the SM quarks throughthe couplings

−LY ⊃ hqQLΦQR + h̃qQLΦ̃QR , (20)

with qL,R = (u, d)TL,R the left- and right-handed quark

doublets, Φ̃ = iσ2Φ∗ (with σ2 being the second Pauli ma-

trix), and hq and h̃q the quark coupling matrices. Aftersymmetry breaking, the tree-level couplings of H1 to theSM quarks are flavor-changing, which are governed bythe quark masses and the left- and right-handed quarkmixing matrices VL,R in the form of

−LY ⊃ H01 d̄idj[−√

2ξŶD +1√2

(V †L ŶUVR

)]ij

,(21)

with ξ = κ′/κ the VEV ratio in the bidoublet sector, i, j

the quark generation indices, and ŶU,D diagonal Yukawacoupling matrices for the SM up- and down-type quarks.The tree-level FCNC couplings of H1 contribute signif-icantly to the neutral K and B meson oscillations, andthus H1 is required to be superheavy, roughly above 15TeV [36, 68, 69].

The CP-even neutral component S from the triplet∆R couples predominantly to the BSM scalars, heavyWR and ZR gauge bosons and the heavy RHNs in theLRSM, and all the couplings of S to the SM particlesare from its mixings with the SM Higgs h and heavyH1 [36, 37]. Therefore in some region of the parameterspace, even if the radiative corrections to S mass aretaken into consideration, S can be very light, e.g. inthe sub-GeV-scale [31, 35]. Thus it might be a goodcandidate to explain the KOTO anomaly.

A. Fitting the KOTO anomaly

In the LRSM, the FCNC couplings of S are frommixing with the SM Higgs h and the heavy scalarH1 from the bidoublet. Denoting these mixing an-gles respectively by sin θ1, 2, the FCNC couplings of Sto s and d quarks will be proportional to the factor

of (ξ sin θ1 + sin θ2)(V †L ŶUVR

)12

, where the right-handedquark mixing matrix VR is almost the same as the CKMmatrix VL in the SM, up to some ambiguous signs [36, 70].For the sake of concreteness we set explicitly VR = VLthroughout this paper. Note that as a result of the CPphase in the VL,R matrices, this coupling is complex.The partial widths for the charged and neutral K mesondecays are given by [40, 71, 72]

Γ(K± → π±S) = GFmK± (ξ sin θ1 + sin θ2)

2

8√

2π

m2K±

m2S

∣∣∣(V †RM̂UVL)21

∣∣∣2(1− m2π±m2K±

)2β2(mK± ,mπ± ,mS) , (22)

Γ(KL → π0S) =GFmKL (ξ sin θ1 + sin θ2)

2

8√

2π

m2K0

m2S

∣∣∣Re(V †RM̂UVL)21

∣∣∣2(1− m2π0m2KL

)2β2(mKL ,mπ0 ,mS) , (23)

with the kinematic function β2(M,m1,m2) defined inEq. (11).

The mixing angles of S to h and H1 are strongly con-strained by the low-energy high-precision flavor data, de-pending on the S mass in the LRSM [31]. At the one-looplevel, S can decay into two photons, i.e. S → γγ, whichis induced by the WR boson and the singly- and doubly-charged scalars [31, 35]:

Γ(S → γγ) ' α2m3S

18π3v2R, (24)

where we have neglected the contributions from the SMfermion loops which are all highly suppressed by the mix-

ing angles sin θ1,2, and take the limit of light S (comparedto the BSM particles in the loop). Note that the partialwidth does not depend on the gauge coupling gR, as thedependence of WR couplings and WR mass on gR arecanceled out. Thus, the partial width of S to diphoton iseffectively suppressed only by the vR scale, independentof the mixing angles θ1,2.

As detailed in Refs. [31, 35], if S is below the GeV-scale, it tends to be long-lived. In the limit of sin θ1, 2 →0, the dominant decay mode of S is the diphoton chan-nel, and its lifetime only depends on its mass mS andthe vR scale [cf. Eq. (24)]. A long-lived S in the LRSMwith lifetime bcτS >∼ 3 m can be a good candidate for

8

the KOTO anomaly. Setting sin θ2 = 0, the preferredparameter space of mS and the S−h mixing angle sin θ1for the KOTO anomaly is shown by the shaded greenregion in Fig. 2.4 As in Fig. 1, the dashed green line cor-responds to the central value of the KOTO result, whilethe shaded green band is the 95% CL favored region fromEq. (1). The region with S mass mS > 180 MeV is notincluded here, because it does not have any overlap withthe KOTO signal region [6]. For concreteness, we set theVEV ratio ξ = κ′/κ = mb/mt which is natural for theknown hierarchy of bottom and top quark masses. Wehave evaluated the effective branching ratio in Eq. (13)for different vR values and found that it is almost inde-pendent of the vR value, as in the parameter space ofinterest the typical lifetime of S is much longer than theKOTO detector size of 3 m. As the FCNC couplings of Sare at tree-level in the LRSM, the mixing angles sin θ1 forthe KOTO anomaly (and the following constraints) areorders of magnitude smaller than in the generic singletmodel (cf. Fig. 1).

B. Laboratory constraints

As for the generic singlet model in Section II, the moststringent limits for the parameter space relevant for theKOTO anomaly are from the searches of K+ → π+νν̄at NA62 [10], K+ → π+X at E949 [11], KL → π0Xat KOTO [3], and the e+e−, µ+µ− and γγ decay prod-ucts from kaon decay at the CHARM beam-dump ex-periment [12]. Evaluations of these limits are quite simi-lar to those in the generic singlet model, as discussed inSec. II B and we do not repeat them here. As in Fig. 1,the current E949, NA62 and KOTO limits are shown re-spectively by the magenta, blue and brown lines in Fig. 2,and the future NA62 improvement is indicated by thedashed blue lines, which corresponds to a limit down to2.35×10−11 for the branching ratio BR(K+ → π+S) [73].

The limits from the CHARM beam-dump experimentare presented by the orange line in Fig. 2. Unlike thegeneric singlet model, the most stringent CHARM limitin the LRSM comes from the γγ channel, since this isthe dominant decay mode of S for small mixing angles.Therefore the event number depends on the vR scale, asillustrated with four benchmark values of vR = 10, 20,50 and 100 TeV. For a larger vR value, the lifetime ofS tends to be longer and as a result the CHARM limitsget weaker. With an improved proton-on-target number(PoT) of 5 × 1021, DUNE can collect 8 × 1021 kaons,with Mpp = 11 and χs = 1/7 [74]. With the energyES ' 12 GeV, the decay length parameters L = 500 mand ∆L = 7 m for the DUNE beam dump set up [74],

4 The other choice, namely, setting sin θ1 = 0 yields a very similarplot in the (mS , sin θ2) plane, with the mixing angle sin θ2 smallerthan sin θ1 in Fig. 2 by a factor of κ′/κ = mb/mt [31, 35], andis therefore not shown here.

and setting the Kaon absorption length at DUNE thesame as that for CHARM, the current CHARM limitson the mixing angle sin θ can be improved by two ordersof magnitude, as shown by the dashed purple curve inFig. 2. The decay K → πS can also be searched forin the SHiP experiment, but the PoT number 2× 1020 isalmost one order of magnitude lower than DUNE, and thelifetime that can be probed is also shorter [75]. Thus, weestimate that the prospect of S search at SHiP is weakerthan at CHARM and DUNE [31] and is not shown inFig. 2.

For all the calculations above in the LRSM, we haveset the VEV ratio ξ = κ′/κ = mb/mt. When the ξparameter is different, the KOTO region, the NA62 andCHARM limits for sin θ1 are all universally rescaled by ξ,and this does not help to enlarge the parameter space forthe KOTO anomaly. As for the generic singlet model, thelimits from the flavor-changing decays K → πχχ (withχχ = e+e−, µ+µ−, γγ) are not applicable to the long-lived S, and the limits from B meson decays, K and Bmeson oscillations are much weaker than those from theK mesons in the parameter space of interest.

As can be seen from Eqs. (22) and (23), the decayK → π+S for the KOTO anomaly and the KOTO, E949and NA62 limits are determined by the scalar mass mSand the mixing angle sin θ1, whereas the CHARM limitare mostly from the decay S → γγ which is dictatedby the scalar mass mS and the vR scale in the limit ofsmall mixing angles [cf. Eq. (24)]. Therefore the LRSMcould accommodate the KOTO anomaly while evadingthe stringent limits from CHARM (and the supernovalimits below), which is very different from the singletscalar model in Section II.

As in the generic singlet case in Section II, the long-lived scalar S in the LRSM can also be searched for asLLP in the high-energy colliders [31]. Unlike the singletscalar case, when the mixing angles sin θ1,2 are very small(cf. Fig. 2), the scalar S in the LRSM can be producedfrom the gauge interactions mediated by the heavy WR(and ZR) bosons, i.e. pp→WR(ZR)S. As a result of theMajorana nature of the heavy RHNs in the LRSM, thesmoking-gun signal of WR boson at hadron colliders issame-sign dilepton plus jets without significant missingenergy [76], and the current most stringent LHC same-sign dilepton limits requires that the WR mass mWR

>∼ 5TeV, depending on the RHN mass [77].5 For a 5 TeV WRboson, the production cross section of S at LHC 14 TeVis only σ(pp → WRS) ' 0.025 fb, and we cannot haveany LLP prospects for mS < 1 GeV at LHC or MATH-USLA if the SU(2)R gauge coupling is the same as thatfor SU(2)L (see the left panel of Fig. 17 in Ref. [31]). Itis easy to understand: in the limit of small mixing an-gles sin θ1,2, the decay width is proportional to m

3S/v

2R

5 Even if the RHNs are heavier than the WR boson, there are alsothe direct LHC searches of WR → tb̄, which exclude WR massbelow 3.25 TeV [78].

9

5 10 50 100 500

5.×10-101.×10-9

5.×10-91.×10-8

5.×10-81.×10-7

mS [MeV]

sinθ 1

LRSM

KOTO anomaly

KOTO

E949NA62

NA62 [future]

CHARMvR =10TeV

20TeV

50TeV

100TeV

DUNEΔN eff

supernova

→LLP →FCC-hh

Figure 2. The parameter space favored by the KOTO anomaly [2] in the LRSM (cf. Sec. III A) is shown by the green shadedregion (95% CL), with the green dashed line corresponding to the central value quoted in Eq. (1). For comparison, we showthe exclusion regions from a previous KOTO search for KL → π0νν̄ (brown) [3], NA62 search for K+ → π+νν̄ (blue) [10],E949 search for K+ → π+X (magenta) [11], and beam-dump experiment at CHARM (orange) [12]; cf. Sec. III B. All the grayshaded regions are excluded. The grey and pink shaded vertical regions are excluded by the ∆Neff and supernova constraintsrespectively (cf. Sec. III C). Also shown are the future prospects at NA62 (dashed blue) [73], DUNE (dashed purple) [74], thelong-lived particle searches at LHC (dashed red) and FCC-hh (dot-dashed red) [31]. For the CHARM limits we choose fourbenchmark values of vR = 10, 20, 50 and 100 TeV. Similarly, the solid and dashed lines for the supernova limits correspondrespectively to the luminosity of 2× 1053 erg and 3× 1053 erg for a fixed vR = 10 TeV (cf. Fig. 3).

(cf. Eq. (24)); so for a light S, the decay lifetime is solong that almost no S decays inside the LHC detector.At future 100 TeV colliders FCC-hh and SPPC, the pro-duction cross section of S can be almost four orders ofmagnitude larger than at LHC 14 TeV for mWR = 5 TeV,and we can have LLP prospects for below-GeV scale atFCC-hh and the dedicated LLP detectors therein [31].Setting mWR = 5 TeV, the LLP prospects at FCC-hhand the dedicated LLP detector is shown in Fig. 2 re-spectively by the dashed and dot-dashed red lines. Theregions to the right side of the red lines can be probed bythe LLP searches, which however do not have any allowedKOTO signal region.

C. Astrophysical and cosmological constraints

As in the generic singlet model case, if S is light, it canbe produced in the supernova core and get constrained bythe collapse luminosity. In the LRSM, S can be producedin two distinct channels:

(i) Nuclear bremsstrahlung processN+N → N+N+S, which originates from the mixing with the SMHiggs. In this case, the effective couplings of S tonucleons are highly suppressed by the mixing anglesin θ1 required for the KOTO explanation, thus the

5 10 50 100

0.05

0.10

0.50

1

5

10

mS [MeV]

luminosity

[1053erg] LRSMsin θ1 = sin θ2 = 0ℒν = 3×1053 erg

2×1053 erg

v R= 10 T

eV20TeV

50TeV

100 TeV

Figure 3. Supernova limits on the light scalar S in the LRSMas function of its mass for different values of vR = 10, 20,50 and 100 TeV. The horizontal solid and dashed grey linesindicate respectively the total energy loss of 2× 1053 erg and3 × 1053 erg due to neutrino emission. Here we have takenboth the S mixing angles θ1 (with the SM Higgs) and θ2 (withthe heavy bidoublet) to be zero, so only the gauge interactionsare relevant.

corresponding supernova limits are too weak and

10

not relevant to the KOTO anomaly. For the samereason, the reabsorption contribution to mean freepath of the S is also suppressed.

(ii) Photon fusion process, i.e. γγ → S, which is highlysuppressed by the ratio m2S/v

2R [cf. Eq. (24)]. As-

suming the photon momentum follows the Bose-Einstein distribution in the supernova core, we fol-low the calculations in Ref. [79] to estimate theproduction rate of S which turns out to be justat the order of ∼ 1053 erg for the benchmark val-ues of vR = (10 − 100) TeV, as shown in Fig. 3.For simplicity, we have set both the scalar mixingangles sin θ1,2 to be zero. The region of mS forwhich the luminosity exceeds the observed value of(2 − 3) × 1053 erg [61] can be excluded. For in-stance, the supernova limits for vR = 10 TeV areshown by the pink shaded region in Fig. 2, withthe solid and dashed lines corresponding to the lu-minosity of 2 × 1053 erg and 3 × 1053 erg respec-tively, which exclude respectively the mass ranges of15 MeV

11

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Constraints on long-lived light scalars with flavor-changing couplings and the KOTO anomalyAbstractI IntroductionII Singlet modelA Fitting the KOTO anomalyB Laboratory constraintsC Astrophysical and cosmological constraints

III Left-right symmetric modelA Fitting the KOTO anomalyB Laboratory constraintsC Astrophysical and cosmological constraints

IV Conclusion Acknowledgments References