Exploring Leptoquark Effects in Hyperon and Kaon Decays with Missing Energy

PHYSICAL REVIEW D 102, 075032 (2020)

Exploring leptoquark effects in hyperon and kaon decays with missing energy

Jhih-Ying Su1 and Jusak Tandean 1,2 1Department of Physics, National Taiwan University, Taipei 106, Taiwan 2Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan

(Received 21 February 2020; accepted 10 September 2020; published 27 October 2020)

We entertain the possibility that scalar leptoquarks (LQs) generate consequential effects on the strangeness-changing decays of hyperons and kaons involving missing energy carried away by a pair of invisible fermions. Although such processes have suppressed rates in the standard model (SM), they could get significant enhancement in the presence of the LQs. In order to respect the available data on the kaon modes K → πνν¯ and increase the rates of the hyperon decays substantially at the same time, two different scalar LQs are needed. If the LQs have Yukawa couplings solely to SM fermions, we find that the hyperon rates cannot attain values within the reach of ongoing or near-future experiments because of the combined constraints from the measurements on kaon mixing and lepton-flavor-violating processes. However, if we include light right-handed neutrinos in the LQ interactions, their contributions can evade the leading restrictions and translate into hyperon rates which may be big enough to be probed by upcoming searches. Thus, these hyperon modes could provide a new avenue for seeking sterile neutrinos.

DOI: 10.1103/PhysRevD.102.075032

I. INTRODUCTION close to the SM expectations [5]. This implies that the room → π The flavor-changing neutral current (FCNC) decays of for NP in K E can no longer be considerable. strange hadrons involving missing energy are of major Nevertheless, as pointed out in Refs. [6,7], the impact of importance because they are known to be dominated by NP could still be large through operators having exclusively parity-odd ds quark bilinears. The reason is that short-distance physics [1]. Therefore these decays offer → π beneficial access not only to the underlying weak dynamics these operators do not participate directly in K E, of the standard model (SM) but also to manifestations of hence not restricted by their data, and are instead subject to possible new physics (NP) beyond it. Such processes arise restraints from the channels with no or two pions, namely → → ππ0 mainly from the quark transition s → dE with the missing K E and K E, which at the moment have com- – energy (E) being carried away by one or more invisible paratively loose empirical bounds [5,8 10]. Future efforts particles. In the SM this proceeds from loop diagrams [1] to improve upon the data on the latter two sets of channels and a pair of undetected neutrinos (νν¯) act as the invisibles. would then be highly desirable. The effective s → dE operators governing all of the This entails that the SM branching fractions of the aforementioned kaon modes affect their baryon counter- corresponding hadron decays are very small, which makes parts as well and consequently may also be testable in the them good places to look for indications of NP. If it exists, FCNC decays of hyperons with missing energy [6,7],on there could be additional ingredients which modify the SM which there has been no measurement yet [5]. It is therefore amplitudes and/or give rise to extra channels with invisible of keen interest that there has recently been a proposal to nonstandard particles. These changes may lead to signs that pursue them in the BESIII experiment [11], which is are sizable enough to be discovered by running or forth- currently in operation and has acquired an ample hyperon coming searches. dataset [12]. As demonstrated in Refs. [6,7], the constraints To date the most intensive quests for s → dE have from K → E and K → ππ0E are sufficiently loose to allow focused on the kaon modes K → πνν¯ and produced upper these hyperon modes to have branching fractions greatly limits on their branching fractions [2–5] which are fairly exceeding their SM predictions and within the proposed sensitivity reach of BESIII for these processes [11]. Its search for them may then be expected to yield valuable Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. information regarding NP which could be lurking in the Further distribution of this work must maintain attribution to underlying quark transition. the author(s) and the published article’s title, journal citation, The foregoing motivates us in this paper to examine and DOI. Funded by SCOAP3. these strangeness-changing (jΔSj¼1) hyperon decays

2470-0010=2020=102(7)=075032(10) 075032-1 Published by the American Physical Society JHIH-YING SU and JUSAK TANDEAN PHYS. REV. D 102, 075032 (2020) with missing energy in the contexts of relatively simple quarks. In each of these instances, it is necessary to have at NP models. It also serves to complement earlier model- least two different scalar LQs in order to respect the independent analyses in which the missing energy was existing K → πνν¯ data and raise the hyperon rates signifi- assumed to be carried away by an invisible pair of spin-1=2 cantly at the same time.1 We show that these two scenarios particles [6] or spinless ones [7]. Within a specific model can furnish very dissimilar sets of predictions for the the parameters determining the strength of the s → dE hyperon rates which may be experimentally tested. Part operators often enter other observables, which may be well of the reason for the dissimilarity is that due to gauge constrained by their respective data. In exploring this, one invariance the SM neutrinos belong to weak doublets along could thus learn how the different restrictions could probe with the charged leptons, which causes the first scenario to various aspects of a model and what modifications to it, if be subject to more restraints than the second. We will also still feasible, may need to be made to comply with the comment briefly on how our findings may have to be constraints. Moreover, this exercise could provide rough altered in order to accommodate the B anomalies if guidance about what kind of NP might be responsible if they turn out to proceed from NP. In Sec. V we draw certain clear signatures beyond the SM appear from the our conclusions. hunts for these hyperon modes. Here we entertain in particular the possibility that heavy II. LEPTOQUARK INTERACTIONS leptoquarks (LQs) with spin 0 mediate the NP contributions to the FCNCs. Over the last few years LQs have gained Among LQs which can have renormalizable couplings plenty of attention in the literature (e.g., [13–22]) because to SM fermions without violating baryon- and lepton- proposed models containing them are among those that number conservations and SM gauge symmetries, there are could offer the preferred explanations for the so-called three which have spin 0 and can at tree level induce B-physics anomalies. Thus, while extra data are awaited to FCNC transitions with missing energy among down-type establish whether some or all of these anomalies originate quarks [19]. Following the nomenclature of Ref. [19], from NP, it is germane to investigate if LQs can bring about we denote these scalar LQs, with their assignments under 3 2 1 appreciable manifestations in the hyperon sector as well. the SM gauge groups SUð Þcolor ×SUð ÞL ×Uð ÞY,by ¯ ˜ ¯ Furthermore, such a hyperon study has never been con- S1ð3; 1; 1=3Þ, R2ð3; 2; 1=6Þ, and S3ð3; 3; 1=3Þ. From this ducted before as far as we know, and it is timely, now that point on, we concentrate on the contributions of the first we anticipate the aforesaid hyperon measurements by two. In terms of their components, BESIII. In contrast, potential LQ effects on K → πE and ! ˜ 2=3 several other kaon transitions have been extensively treated 1=3 ˜ R2 – S1 ¼ S1 ; R2 ¼ −1 3 ; ð1Þ in the past [16 36]. R˜ = Besides the scalar LQs, we will include light right- 2 handed (RH) neutrinos in the theory. These are singlets where the superscripts indicate their electric charges. under the SM gauge groups and, via the LQs, can have We incorporate into the theory three RH neutrinos, renormalizable links to SM quarks. Our introduction of RH which are singlets under the SM gauge groups and called neutrinos is well motivated on a couple of grounds. First, N1, N2, and N3. We assume that they and the SM neutrinos their existence will be required if empirical research in the are all of Dirac nature. In addition, we suppose that N1;2;3 future proves that neutrinos are Dirac in nature. Second, as have masses small enough to be neglected in the hyperon will be seen later on, compared to those with only SM and kaon processes of concern and are sufficiently long- fermions the FCNCs involving the RH neutrinos and SM lived that they do not decay inside detectors. quarks might have substantially amplified influence on the We express the Lagrangian for the renormalizable hyperon decays. ˜ interactions of S1 and R2 with SM fermions plus N1 2 3 as The organization of the rest of this article is as follows. In ; ; Sec. II we describe how the interactions of the new LL c ˜ RR c RL ˜ T L ¼ðY1 q iτ2l þ Y1 d N ÞS1 þ Y2 d R2iτ2l particles, namely the scalar LQs and the light RH neutrinos, LQ ;jy j y ;jy j y ;jy j y ˜ LR ˜ N with SM fermions can affect the s → dE operators. In þ Y2;jyqjR2 y þ H:c:; ð2Þ Sec. III we derive the amplitudes for the hyperon and kaon ˜ decay modes of interest and formulate their rates. In Sec. IV where the Yjy and Yjy are generally complex elements of we deal with the pertinent limitations on the Yukawa the LQ Yukawa matrices, summation over family indices j, couplings of the LQs and present our numerical results. y ¼ 1, 2, 3 is implicit, qj ðlyÞ and dj symbolize a left- To illustrate the potential impact of the scalar LQs on these handed quark (lepton) doublet and right-handed down-type hyperon and kaon processes, we discuss two distinct cases. In the first one, only SM quarks and leptons take part in the 1Invoking two scalar LQs to suppress certain processes and Yukawa interactions of the LQs. In the second case, we let enhance others has previously been employed to address the B the RH neutrinos couple directly to the LQs as well as the anomalies, such as in [14,15].

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RR RR LR LR quark singlet, respectively, and τ2 stands for the second −Y˜ Y˜ Y˜ Y˜ CV CA 1;1x 1;2y 2;2x 2;1y 2 L NN0 ¼ NN0 ¼ 2 þ 2 ; Pauli matrix. To obtain decay amplitudes from LQ,we 8 8 mS1 m ˜ need to write it in terms of mass eigenstates. For the R2 ˜ RR ˜ RR ˜ LR ˜ LR −Y1 1 Y1 2 Y2 2 Y2 1 processes of interest, it is convenient to do so in the mass c˜ V c˜ A ; x ; y − ; x ; y NN0 ¼ NN0 ¼ 2 2 ; ð6bÞ basis of the down-type fermions, in which case 8m 8m ˜ S1 R2 V U U ν kj kL jk k q ¼ ;l¼ ;d¼ D ; ð3Þ N j D j l j jR where the indices x and y label the lepton or species. jL jL In the remainder of this article, we explore how these LQ-induced operators may affect especially the aforemen- where k ¼ 1, 2, 3 is implicitly summed over, V ≡ V ckm tioned hyperon decays with missing energy as well as U ≡ U is the Cabibbo-Kobayashi-Maskawa quark ð PMNSÞ → ππ0 (Pontecorvo-Maki-Nakagawa-Sakata neutrino) mixing K E. However, we will also touch on how our 1 treatment might need to be adjusted in order to accom- matrix, f ¼ ð1 ∓ γ5Þf, and U1 2 3 ð¼u; c; tÞ, D1 2 3 L;R 2 ; ; ; ; modate the B-physics anomalies should future data estab- d; s; b , ν1 2 3, and l1 2 3 e; μ; τ represent mass ð¼ Þ ; ; ; ; ð¼ Þ lish them to be caused by NP. eigenstates. Since the left-handed neutrinos’ masses are ν U ν ν tiny, we can work with lj ¼ jk k instead of j. Thus, Eq. (2) becomes III. AMPLITUDES AND RATES

In the hyperon sector Ldsff0 in Eq. (5) brings about the L YLL V U cl − D cν 0 ¯0 LQ ¼f 1;jy½ kjð kLÞ yL ð jLÞ ly decay modes B → B f f involving the pairs of baryons 0 þ 0 0 0 − − ˜ RR c BB ¼ Λn; Σ p; Ξ Λ; Ξ Σ ; Ξ Σ , all having spin 1=2, þ Y1 ðD Þ N gS1 ;jy jR y Ω− → Ξ−f f¯0 Ω− 3 2 2 3 −1 3 and , where has spin = . To deal with the YRL D l ˜ = − ν ˜ = þ 2;jy jRð yLR2 ly R2 Þ amplitudes for these decays, we need the baryonic matrix

LR ˜ 2=3 ˜ −1=3 elements of the quark bilinears in Eq. (5) which can be Y˜ V U R D R N H:c: 4 þ 2;jyð kj kL 2 þ jL 2 Þ y þ ð Þ estimated with the aid of chiral perturbation theory at B → B0f f¯0 ˜ 0 leading order [6,39].For the results are [6] These couplings allow S1 and R2 to mediate dsff interactions at tree level, the ff0 pair being unobserved, f ν N 0 ¯ η η with ¼ or . Assuming that both of the LQs are heavy, hB jdγ sjBi¼VB0Bu¯ B0 γ uB; L we find that LQ gives rise to the low-energy effective 0 ¯ η η mB0 þ mB η Lagrangian B dγ γ5s B AB0Bu¯ B0 γ − Q γ5uB; h j j i¼ 2 −Q2 ð7Þ mK ¯ η ¯ V A 0 −L ff0 ¼ dγ sf γηðC 0 þ γ5C 0 Þf ds ff ff pffiffiffi pffiffiffi pffiffiffi η ¯γ γ f¯γ c˜ V γ c˜ A f 0 where VB0B ¼ −3= 6; −1; 3= 6; −1= 2; 1 and AB0B ¼ þ d 5s ηð ff0 þ 5 ff0 Þ þ H:c:; ð5Þ pffiffiffi pffiffiffi pffiffiffi −ðDþ3FÞ= 6;D−F;ð3F−DÞ= 6, −ðDþFÞ= 2;DþF 0 þ 0 0 0 − − the pertinent coefficients being given by for B B ¼ nΛ;pΣ ; ΛΞ ; Σ Ξ ; Σ Ξ , respectively, u¯ B0 and uB represent the Dirac spinors of the baryons, and −YLL YLL YRL YRL Q − CV −CA 1;1x 1;2y 2;2x 2;1y ¼ pB pB0 , with pB and pB0 being their momenta. For νν0 ¼ νν0 ¼ 2 þ 2 ; − − ¯0 8m 8m ˜ Ω → Ξ f f we have [6] S1 R2 LL LL RL RL Y1 1 Y1 2 Y2 2 Y2 1 ˜ V ˜ A ; x ; y ; x ; y c 0 ¼ −c 0 ¼ þ ; ð6aÞ ˜ η ˜ νν νν 2 2 η κ 8m 8m ˜ − ¯ η − Q Q κ S1 R2 hΞ jdγ γ5sjΩ i¼Cu¯ Ξ u þ u ; ð8Þ Ω 2 − ˜ 2 Ω mK Q

η ˜ − − − 2 where Q ¼ pΩ pΞ and uΩ is a Rarita-Schwinger spinor. In Eq. (2) we include only the minimal ingredients which C serve our purposes pertaining to the s → dE reactions of interest. The constants D, F, and above occur in the lowest-order c chiral Lagrangian and can be fixed from baryon data. In In principle, one could have more Yukawa terms, such as ujeyS1, with uj (ey) designating a right-handed up-type quark (charged numerical work, we will adopt the same values of these and † ˜ lepton) field, and/or introduce scalar couplings, such as H S1R2 other input parameters as those given in Ref. [6]. involving the SM Higgs doublet H leading to mixing between the With Eqs. (7) and (8), we derive the amplitudes for ¯ − − ¯ LQ components of the same electric charge [37,38]. Such B → B0f f 0 and Ω → Ξ f f 0. Since in the scenarios under extensions would be accompanied by new free parameters and f f 0 contribute to processes beyond the scope of this paper. Therefore, consideration the and masses are assumed to be small we refrain from considering LQ interactions not contained in compared to the pion mass, we then arrive at the differential Eq. (2). rates [6]

075032-3 JHIH-YING SU and JUSAK TANDEAN PHYS. REV. D 102, 075032 (2020) pffiffiffi Γ λ1=2 λ 2 d B→B0f f¯0 BB0 BB0 2 2 i V A − ˆ − ˆ M − 0 − ¯ ¯ − c˜ γ c˜ ¯ ¼ 3 3 þðmB mB0 Þ s s K →π π f f 0 ¼ uf ðp0 p−Þð ff0 þ 5 ff0 Þvf 0 ; dsˆ 64π mB 3 fK CV 2 CA 2 V2 i × ðj ff0 j þj ff0 j Þ B0B M →π0π0f f¯0 ¼ pffiffiffi u¯ f ðp1 þ p2Þ KL 2fK λBB0 2 2 þ þðmB þ mB0 Þ sˆ − sˆ c˜ V c˜ V γ c˜ A c˜ A 3 × ½ f 0f þ ff0 þ 5ð f 0f þ ff0 Þvf¯ ð13Þ 2 2 2 c˜ V c˜ A A in the mf f 0 0 limit. The expressions in Eq. (12) lead to × ðj ff0 j þj ff0 j Þ B0B ; ð9Þ ; ¼ the differential decay rates

1 2 3=2 = 2 Γ − − ¯ λ dΓΩ−→Ξ−f f¯0 λΩ−Ξ− C λΩ−Ξ− d K →π f f 0 Kþπþ V 2 A 2 4ˆ ¼ ðjC 0 j þjC 0 j Þ; ¼ 3 3 2 þ s ˆ 192π3 3 ff ff dsˆ 768π mΩ− 3mΩ− ds mKþ 2 2 2 3=2 − − − ˆ c˜ V c˜ A Γ 0 λ × ½ðmΩ þ mΞ Þ sðj 0 j þj ff0 j Þ; ð10Þ d →π f f¯0 0 0 ff KL K π CV −CV 2 CA −CA 2 ¼ 3 3 ðj f 0f ff0 j þj f 0f ff0 j Þ: ð14Þ dsˆ 768π 0 mK ˆ 2 λ 4 − 2 2 ˆ 2 where s ¼ðpf þ pf 0 Þ and XY ¼ mX ðmY þ sÞmX þ 2 − ˆ 2 ðmY sÞ . As in Ref. [6], in our numerical evaluation of the From Eq. (13), we arrive at the double-differential decay hyperon rates we include form-factor effects not yet taken rates into account in Eqs. (7) and (8). Particularly, in Eq. (9) we 2 2 V → 1 2ˆ V A → Γ − 0 − ¯ apply the changes B0B ð þ s=MVÞ B0B and B0B d K →π π f f 0 1 2ˆ 2 A 0 97 4 ˆ ˆ ð þ s=MAÞ B0B with MV ¼ . ð Þ GeV and MA ¼ dsdς 1 25 15 . ð Þ GeV, in line with the parametrization commonly β3λ˜1=2 ςˆ K− ˜ 2 2 employed in experimental analyses of the charged-current λ − 12sˆ ςˆ c˜ V c˜ A ; ¼ 18 4π 5 2 3 ð K þ Þðj ff0 j þj ff0 j Þ semileptonic decays of hyperons [40–44]. Moreover, in ð Þ fKmK− C C 1 − ˆ 2 2Γ Eq. (10) we modify to =ð s=MAÞ. d K →π0π0f f¯0 L L In the kaon sector, dsff0 in Eq. (5) is consequential for dsdˆ ςˆ → πf f¯0 → ππ0f f¯0 → f f¯0 K and K but not for K which is ˜3=2 βςˆλ 0 helicity suppressed because in this study we focus on the K ˜ V ˜ V 2 ˜ A ˜ A 2 ¼ ðjc 0 þ c 0 j þjc 0 þ c 0 j Þ; ð15Þ ≃ 0 48 4π 5 2 3 f f ff f f ff mf ;f 0 case. It follows that the relevant mesonic matrix ð Þ fKmK0 elements are where π− ¯γη − η η h ðpπÞjd sjK ðpKÞi ¼ pK þ pπ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 − ¯ η − 4 2 hπ ðp0Þπ ðp−Þjdγ γ5sjK i mπ 2 2 βςˆ 1 − ; ςˆ p0 p− p1 p2 ; pffiffiffi ¼ ςˆ ¼ð þ Þ ¼ð þ Þ μ − μ q˜ q˜ η i 2 η η ðp0 p−Þ μ p − p− ; 2 2 ¼ ð 0 Þþ 2 − q˜ 2 λ˜ ¼ðm − sˆ − ςˆÞ − 4sˆ ςˆ : ð16Þ fK mK K K 0 0 ¯ η ¯ 0 hπ ðp1Þπ ðp2Þjdγ γ5sjK i 2 ˆ ςˆ 0≤ ˆ ≤ − 0 −2 μ μ The s and integration ranges are s ðmK ;K mπÞ q˜ q˜ η 2 1 2 − i η η ðp1 þ p2Þ μ 4 ≤ ςˆ ≤ − 0 − ˆ = p p ; and mπ ðmK ;K s Þ for the K and KL chan- ¼ ð 1 þ 2Þþ 2 − q˜ 2 ð11Þ fK mK nels, respectively. 0 0 0 0 ¯ If f ≠ f , the extra channel KL → π ðπ Þf f also occurs, where f 155.6 4 MeV [5] is the kaon decay constant, whose rate formula is the same as Γ 0 0 ¯0 but with f and K ¼ ð Þ KL→π ðπ Þf f V A q˜ ¼p − −p0 −p− ¼p ¯ 0 −p1 −p2, and we have ignored f 0 f 0 ≠ f C ; c˜ V;A K K interchanged. For ,if f 0f and f 0f are not zero, they form-factor effects. Assuming isospin symmetry and mak- CV;A c˜ V;A are generally independent from ff0 and ff0 and also induce ing use of charge conjugation, we also have the relations − − 0 ¯ ¯ − − ¯ pffiffiffi K → π π f 0f as well as B → B0f 0f and Ω → Ξ f 0f . π0 ¯γη ¯ 0 − π− ¯γη − 2 − π0 ¯γη 0 ð Þ h jd sjK i¼ h jd sjK i= ¼ h js djK i and In the instances we discuss in the next section, the π0π0 ¯γηγ 0 π0π0 ¯γηγ ¯ 0 h js 5djK i¼h jd 5sjK i. Hence the ampli- emitted fermions are either SM neutrinos with different − → π− π0 f f¯0 → π0 π0 f f¯0 tudes for K ffiffiffi ð Þ and KL ð Þ with the flavors or the RH neutrinos, implying that the SM and NP p 0 ¯ 0 approximation 2KL ¼ K þ K are [6,35] amplitudes do not interfere in the decay rates. Furthermore, the SM contributions to the hyperon modes and K → ππ0E M 2¯ CV γ CA are significantly smaller than the currently allowed room K−→π−f f¯0 ¼ uf pKð ff0 þ 5 ff0 Þvf¯0 ; for potential NP in these processes [6] and hence can be V V A A M →π0f f¯0 ¼ u¯ f p ½C 0 − C 0 þ γ5ðC 0 − C 0 Þvf¯0 ; ð12Þ KL K f f ff f f ff neglected.

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IV. CONSTRAINTS AND NUMERICAL RESULTS opposite relative sign, and since all of these couplings participate in the hyperon rates in Eqs. (9) and (10), one For definiteness and simplicity, we look at a couple of way to amplify the hyperon rates and satisfy Eq. (20) distinct situations as examples. In the first one (A) the RH CV;A 0 simultaneously is if νν0 ¼ , which implies neutrinos, N1;2;3, are absent. In the second example (B) LL LL RL RL they are present and couple to the LQs and quarks. Y1 1 Y1 2 Y2 2 Y2 1 c˜ V −c˜ A ; x ; y ; x ; y Numerically, we will treat the nonzero Yukawa couplings νν0 ¼ νν0 ¼ 2 ¼ 2 : ð21Þ 4m 4m ˜ phenomenologically, letting one or more of them vanish or S1 R2 differ substantially from the others, in order to comply with Consequently, as can be deduced from Eqs. (17)–(19), the various constraints. K → πE decays are now due to the SM alone, whereas Before proceeding, we integrate the differential rates of K → ππ0E and the hyperon modes can still undergo LQ- the baryon and kaon decays from the previous section to 3 induced modifications. In the rest of this subsection, we obtain the branching fractions investigate the extent to which the hyperon rates may be

¯0 V 2 ˜ v 2 7 4 enhanced with the choice in Eq. (21) subject to additional B Λ → nf f 8.89 C 0 14.1 c 0 10 GeV ; ð Þ¼ð j ff j þ j ff j Þ relevant requirements. However, we shall also briefly look B Σþ → f f¯0 10 2 CV 2 3 67 c˜ v 2 107 4 CV;A ≠ 0 ð n Þ¼ð . j ff0 j þ . j ff0 j Þ GeV ; at the νν0 case. 0 ¯0 V 2 ˜ v 2 7 4 The interactions in Eq. (4) can impact the mixing of BðΞ → Λf f Þ¼ð18.5jC 0 j þ 1.99jc 0 j Þ10 GeV ; ff ff neutral kaons K0 and K¯ 0 via four-quark operators which B Ξ0 → Σ0f f¯0 0 57 CV 2 2 75 c˜ v 2 107 4 ð Þ¼ð . j ff0 j þ . j ff0 j Þ GeV ; arise from box diagrams, with the LQs and SM leptons − − ¯0 V 2 ˜ v 2 7 4 running around the loops, and are given in the effective BðΞ → Σ f f Þ¼ð0.70jC 0 j þ 3.38jc 0 j Þ10 GeV ; ff ff Hamiltonian [19,33,36] − − ¯0 V 2 9 4 P BðΩ → Ξ f f Þ¼1.60jc˜ 0 j × 10 GeV ; ð17Þ ff ð YLL YLL Þ2 HLQ x 1;1x 1;2x ¯ γη ¯ γ Δ 2 ¼ 2 2 sL dLsL ηdL − − ¯0 V 2 10 4 j Sj¼ 128π B K → π f f 2 59 C 10 ; mS1 ð Þ¼ . j ff0 j × GeV P 0 2 10 4 RL RL 2 B → π f f¯0 2 91 CV − CV 10 ð Y2 2 Y2 1 Þ ðKL Þ¼ . j f 0f ff0 j × GeV ; ð18Þ x ; x ; x ¯ γη ¯ γ þ 2 2 sR dRsR ηdR þ H:c: ð22Þ 64π m ˜ R2 B − → π−π0f f¯0 1 26 c˜ v 2 106 4 ðK Þ¼ . j ff0 j × GeV ; Its matrix element between the K0 and K¯ 0 states contributes B → π0π0f f¯0 4 24 c˜ v c˜ v 2 106 4 ðKL Þ¼ . j f 0f þ ff0 j × GeV . ð19Þ Δ ∝ ϵ ∝ to the kaon-mixing parameters mK ReMKK¯ and K 0 0 ¯ H 2 0 CA CV ImMKK¯ , where MKK¯ ¼hK j jΔSj¼2jK i=ð mK Þ. Given These results already incorporate the relations j ff0 j¼j ff0 j Δ ϵ ˜ A ˜ V that mK and K have been well measured, the constraints and jc 0 j¼jc 0 j from Eq. (6). ff ff from them on the LQ effects are stringent [36]. Nevertheless, we see from Eq. (22) that these restrictions A. SM fermions only can be evaded by assigning the contributing elements of the YLL YRL Among FCNC kaon decays with missing energy, first and second rows of 1 and 2 to separate columns. the most precise data available to date are the branching The interactions in Eq. (4) also give rise to lepton-flavor- B þ → πþνν 1 7 1 1 10−10 violating dsll0 operators in fractions [5] ðK Þexp ¼ . ð . Þ × and BðK → π0νν¯Þ < 3.0 × 10−9 at 90% confidence level L exp −L ¯γη l¯γ V γ A l0 (CL). Evidently, they are not very far from what the SM dsll0 ¼ d s ηð ll0 þ 5 ll0 Þ þ1.0 −11 ¯ η ¯ 0 B þ → πþνν¯ 8 5 10 γ γ lγ ˜ 0 γ ˜ 0 l predicts [33]: ðK Þsm ¼ð . −1.2 Þ × and þ d 5s ηðVll þ 5All Þ þ H:c:; ð23Þ B →π0νν¯ 3 2þ1.1 10−11 ðKL ÞSM ¼ð . −0.7 Þ× . The extent to which these experimental numbers deviate from their SM counter- where from tree-level LQ-exchange diagrams [19,35,36] parts reflects how much possible NP is permitted to YRL YRL influence these modes. We may then adopt the upper ends V −A ˜ −˜ 2;2x 2;1y ll0 ¼ ll0 ¼ Vll0 ¼ All0 ¼ 2 : ð24Þ of the 90%-CL ranges of the deviations, 8m ˜ R2 ΔB K− → π−E < 2 7 10−10; ð Þ . × Accordingly, if the pair of the lepton-family indices of ΔB → π0 3 0 10−9 YRL YRL 12 ðKL EÞ < . × ; ð20Þ 2;2x 2;1y has the values xy ¼ , 21, corresponding to 0 ¯ η ¯ η ll eμ; μe, the dγ s and dγ γ5s terms in Eq. (23), plus as caps on the NP contributions. In view of Eq. (14), this ¼ V;A applies to C 0 in Eq. (6a). In the absence of N1 2 3,we νν ; ; 3 V;A ˜ V;A Studies focusing on the hyperon modes affected by NP set C 0 ¼ c 0 ¼ 0. NN NN manifesting the same chiral structure as that of the SM operator, V A V A In general, the two parts of C 0 ¼ −C 0 in Eq. (6) do not C −C −c˜ V c˜ A νν νν namely with νν0 ¼ νν0 ¼ νν0 ¼ νν0 in Eq. (5), can be found c˜ V;A cancel each other. Since they also enter νν0 but with the in [45,46].

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∓ ∓ their H.c., will bring about K → πe μ and KL → e μ , their empirical bounds are unavoidable by the selections of respectively, both of which have severe experimental the Yukawa couplings in the previous paragraph but way restraints. They can be eluded by ensuring that xy is milder than the aforesaid restrictions on e-μ violation. neither 12 nor 21. These choices moreover do not lead Numerically, among the τ decay constraints, the one on to the one-loop diagrams causing the μ → eγ; 3e decays τ− → μ−K¯ 0 turns out to be the least demanding on the and μ → e transitions in nuclei and hence can escape the coefficients, as can be inferred from Ref. [47], and bears limits from their quests. on the combinations already displayed in Eq. (27). The conditions described in the preceding two para- This translates into graphs dictate that some of the elements of the Yukawa LL RL matrices Y1 and Y2 be substantially smaller than the jy2;23y2;12j 0.036 others. To avoid further constraints from processes involv- 2 < 2 ð28aÞ m ˜ TeV ing the b quark, we assume that the third-row elements R2 of these matrices are sufficiently tiny. For the sake of simplicity, we can then approximate the matrices by setting and in view of Eq. (26) their relatively suppressed elements to zero.4 Thus, we can write down a sample set of the desired jy1;13y1;22j 0.036 matrices as 2 < 2 : ð28bÞ 0 1 mS1 TeV 00y1;13 B C YLL 0 0 1 ¼ @ y1;22 A; Before moving on to the resulting hyperon predictions, 00 0 we mention here additional potentially relevant observables 0 1 on which empirical information is available, but which at 0 y2;12 0 B C the moment entail much weaker restraints than Eq. (28). YRL 00 2 ¼ @ y2;23 A: ð25Þ The S1 parameters also participate in the decay τ → μγ 00 0 induced by one-loop diagrams [19], but the rate turns out to be far below its current measured bound because of loop Incorporating these into Eqs. (21) and (24), we obtain suppression and the fact that the LQ interactions with the

charged leptons are chiral [36], exclusively left-handed as y1 13y1 22 y2 23y2 12 c˜ V −c˜ A ; ; ; ; Eq. (4) specifies. Although the couplings of the Z boson to ντνμ ¼ ντνμ ¼ 2 ¼ 2 ; ð26Þ 4m 4m ˜ S1 R2 a pair of SM leptons receive one-loop modifications from the LQ interactions [15], the restrictions inferred from the y y2 12 V −A ˜ −˜ 2;23 ; existing Z data are not yet competitive to Eq. (28). Direct τμ ¼ τμ ¼ Vτμ ¼ Aτμ ¼ 2 ; ð27Þ 8m ˜ R2 LQ searches at colliders have turned up negative to date [5,48–50], ruling out the lighter LQs, and most recently with the other coefficients in Ldsff0 and Ldsll0 in Eqs. (5) ATLAS [48,49] (CMS [50]) has excluded scalar LQs and (23) being zero. Alternatively, we could pick one of decaying fully into a quark and charged lepton (neutrino) three other sets, from which the analogous products of at 95% CL for masses up to 1.8 (1.1) TeV, but this can be nonzero matrix elements are given by ðy1;12y1;23;y2;22y2;13Þ, easily satisfied by the parameters already fulfilling Eq. (28). ðy1 11y1 23;y2 21y2 13Þ, and ðy1 13y1 21;y2 23y2 11Þ, respecti- We can now put Eqs. (26) and (28) together with the ; ; ; ; ; ; ; ; ¯ vely. branching fractions of the kaon decays K → ππ0f f 0 and the ¯ − − ¯ If xy ¼ 13, 31, 23, 32 in Eq. (24), the operators in hyperon ones, B → B0f f 0 and Ω → Ξ f f 0. Thus, employ- Eq. (23) generate the lepton-flavor-violating jΔSj¼1 tau ing the formulas in Eqs. (17) and (19), we predict the upper decays τ → lK¯ ðÞ; lKðÞ; lπK∓ with l ¼ e, μ, and so limits

− 0 − −10 0 0 −10 BðK → π π ντν¯μÞ < 1.0 × 10 ; BðKL → π π ντνμÞ < 6.9 × 10 ; ð29Þ

−8 þ −9 BðΛ → nντν¯μÞ < 1.1 × 10 ; BðΣ → pντν¯μÞ < 3.0 × 10 ; 0 −9 0 0 −9 BðΞ → Λντν¯μÞ < 1.6 × 10 ; BðΞ → Σ ντν¯μÞ < 2.2 × 10 ; 0 0 −9 − − −7 BðΞ → Σ ντν¯μÞ < 2.7 × 10 ; BðΩ → Ξ ντν¯μÞ < 1.3 × 10 ; ð30Þ

4This kind of treatment of LQ Yukawa matrices has been applied to other contexts, such as in [19–21].

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0 0 0 0 where BðKL → π π ντνμÞ is due to both KL → π π νμν¯τ [13–22]. To exhibit one specific sample set of parameters 0 0 ≃ and KL → π π ντν¯μ which have the same rate, as the which can achieve this, we have ðy1;22;y1;23;y1;32;y1;33Þ −0 2 −0 1 3 3 −0 05 ≃3 3 second line of Eq. (19) indicates. ð . ; . ; . ; . Þ and mS1 . TeV from Ref. [20], CV;A 0 Instead of choosing ντνμ ¼ which led to Eq. (26),we along with y1;13 values which comply with Eq. (28b) and also alternatively try to maximize the hyperon rates by perturbativity. For S1 to be responsible also for the b → c CV c˜ V allowing the two different terms in ντνμ and ντνμ as defined anomalies would however require the inclusion of its in Eq. (6a) to have any values compatible with the couplings to right-handed SM fermions [20]. conditions in Eqs. (20) and (28), but again with the matrices in Eq. (25). The branching fractions we find are just B. Including right-handed neutrinos marginally larger than those in Eqs. (29) and (30) and ˜ RR ˜ LR In the presence of N , we turn on the Y1 and Y2 terms CV ≃ 1 0 10−10 −2 c˜ v ≃ y correspond to j ντνμ j . × GeV and j ντνμ j specified in Eq. (4). To fulfill Eq. (20) once more, we can 9 1 10−9 −2 CV;A 0 . × GeV . first set NN0 ¼ in Eq. (6b), implying that The numbers in Eq. (29) are substantially bigger, by a ˜ RR ˜ RR ˜ LR ˜ LR few orders of magnitude, than their SM counterparts, −Y1 1 Y1 2 −Y2 2 Y2 1 − 0 − −14 0 0 ˜ V ˜ A ; x ; y ; x ; y B → π π νν¯ ∼ 10 B → π π νν¯ ∼ c 0 ¼ c 0 ¼ ¼ ; ð32Þ ðK ÞSM and ðKL ÞSM NN NN 2 2 −13 4m 4m ˜ 10 [51–53], but still lie significantly under the existing S1 R2 B K− → π0π−νν¯ < 4 3 10−5 measured bounds ð Þexp . × [8] CV;A ≠ 0 B →π0π0νν¯ 8 1 10−7 but subsequently we shall also touch on the NN0 case. and ðKL Þexp < . × [9] both at 90% CL. ˜ RR ˜ LR As we will see shortly, Y1 and Y2 can bring about much Likewise, the hyperon results in Eq. (30) greatly exceed larger effects than YLL and YRL discussed in the previous 10−13 − 10−11 1 2 their SM expectations, which fall within the example, and so we can ignore the contributions of the range [6], but the former are not yet close to the sensitivity latter to the s → dE reactions. levels of BESIII estimated in Ref. [11] for the branching Similarly to the first case (A), the Yukawa couplings in Λ → νν¯ Σþ → νν¯ Ξ0 → Λνν¯ Ξ0 → Σ0νν¯ fractions of n , p , , , Eq. (32) affect K0 − K¯ 0 mixing through box diagrams, with Ω− → Ξ−νν¯ 3 10−7 4 10−7 8 10−7 and , which are × , × , × , the LQs and RH neutrinos going around the loops, giving 9 10−7 2 6 10−5 × , and . × , respectively. rise to the effective Hamiltonian One could attempt to get around instead the τ decay P constraints by arranging the nonvanishing elements of the 2 ð Y˜ LR Y˜ LR Þ Yukawa matrices to be on, say, the third columns. It follows H˜ LQ x 2;2x 2;1x ¯ γη ¯ γ jΔSj¼2 ¼ 2 2 sL dLsL ηdL that the kaon-mixing restrictions apply to these elements. 128π m ˜ P R2 This option turns out to be more restraining than Eq. (28) 2 ð Y˜ RR Y˜ RR Þ and hence leads to comparatively smaller kaon and hyperon x 1;1x 1;2x ¯ γη ¯ γ þ 128π2 2 sR dRsR ηdR þ H:c: ð33Þ rates. Another possibility would be to enlist the third scalar mS1 ¯ LQ referred to earlier, S3ð3; 3; 1=3Þ, but we have found that it would not improve on the situation described in the last Evidently, the limitations from kaon-mixing data can again paragraph. be eluded by assigning the nonzero elements of the first and The above exercise then suggests that with only SM second rows of the Y˜ matrices to separate columns. As for fermions participating in the interactions of the scalar LQs processes which do not conserve charged-lepton flavor, it would be unlikely for the jΔSj¼1 hyperon decays with their data no longer offer pertinent constraints on the Y˜ missing energy to have rates sizable enough to be within the matrix elements because they are not associated with SM reach of BESIII. Therefore, future facilities such as super leptons. charm-tau factories may be needed to test this particular LQ The main restrictions would then come from the empiri- scenario. cal bounds on K → ππ0E. To illustrate the implication, we Here we discuss briefly what we might have to do consider this sample set of matrices that satisfy the above differently in order to repeat the preceding steps and requisites: simultaneously explain, at least partially, the B anomalies. 0 1 0 1 It turns out that, among various options, changing only the 0 y˜1;12 0 00y˜2;13 B C B C first of the matrices in Eq. (25) into RR LR Y˜ ¼ @00y˜1 23 A; Y˜ ¼ @0 y˜2 22 0 A: ð34Þ 0 1 1 ; 2 ; 00 0 00 0 00y1;13 B C LL Y1 ¼ @ 0 y1;22 y1;23 A ð31Þ These yield 0 y1;32 y1;33 −y˜1 12y˜1 23 −y˜2 22y˜2 13 ˜ V ˜ A ; ; ; ; may suffice to account for the recent anomalous measure- c ¼ cN N ¼ ¼ ; ð35Þ N2N3 2 3 4 2 4 2 þ − m m ˜ ments of b → sμ μ transitions within their 2σ ranges S1 R2

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c˜ V;A 5 B →π0π0νν¯ the other ff0 vanishing. Given that the ðKL Þexp As mentioned earlier, the latest search by CMS [50] for bound cited in the last subsection is stronger than scalar LQs decaying fully into a quark and neutrino has B − → π0π−νν¯ B → ruled out masses up to 1.1 TeV at 95% CL. This is the ðK Þexp one, we can impose ðKL π0π0 8 10−7 B → π0π0 applicable to the case in which the neutrino is a right- EÞLQ < × , where ðKL EÞLQ ¼ N 0 0 ¯ 0 0 ¯ handed one, y, and can accommodate Eq. (37) because BðK → π π N2N3ÞþBðK → π π N3N2Þ¼2BðK → L L L jy˜1 12y˜1 23j and jy˜2 22y˜2 13j can each have a size up to the π0π0N N¯ ; ; ; ; 2 3Þ. This translates into perturbativity limit of 4π. These parameters also enter loop ˜ V 2 −14 −4 diagrams containing the LQs and SM quarks and contrib- jcN N j < 9.4 × 10 GeV ð36Þ 2 3 uting to the invisible partial width of the Z boson, but we and consequently have checked that the impact is negligible. ˜ ˜ ˜ ˜ Incorporating Eq. (37) into the hyperon decay rates from jy1;12y1;23j jy2;22y2;13j 1.2 2 ¼ 2 < 2 : ð37Þ Sec. III, we obtain the maximal branching fractions m m ˜ TeV S1 R2

¯ −5 þ ¯ −6 BðΛ → nN2N3Þ < 1.3 × 10 ; BðΣ → pN2N3Þ < 3.5 × 10 ; 0 ¯ −6 0 0 ¯ −6 BðΞ → ΛN2N3Þ < 1.9 × 10 ; BðΞ → Σ N2N3Þ < 2.6 × 10 ; − − ¯ −6 − − ¯ −4 BðΞ → Σ N2N3Þ < 3.2 × 10 ; BðΩ → Ξ N2N3Þ < 1.5 × 10 : ð38Þ

The higher ends of these predictions well exceed their information on potential NP contributions which is counterparts in Eq. (30) and also the corresponding complementary to that gained from the kaon sector, BESIII sensitivity levels [11]: BðΛ → nνν¯Þ < 3 × 10−7, including restraints on them which may be stricter than BðΣþ → pνν¯Þ < 4 × 10−7, BðΞ0 → Λνν¯Þ < 8 × 10−7, those provided by kaon measurements. Lastly, another BðΞ0 →Σ0νν¯Þ<9×10−7, and BðΩ− →Ξ−νν¯Þ<2.6×10−5. lesson is that, as the Yukawa matrices in Eq. (34) is It follows that BESIII might soon uncover NP clues in these generally independent from that in Eq. (31), it might processes or, if not, come up with improved restrictions on happen that distinct segments of the same LQ scenario the interactions of the scalar LQs with the RH neutrinos. could lead to noticeable NP signs in both hyperon and b- We remark that the numbers in Eq. (38) are consistent hadron data. with the indirect upper bounds on the branching fractions of yet-unobserved decay modes of Λ, Σþ, Ξ0, Ξ−, and Ω−, V. CONCLUSIONS 1 4 10−2 8 0 10−3 3 4 10−4 8 3 10−4 namely . × , . × , . × , . × , and We have explored the effects of scalar leptoquarks on 1 6 10−2 . × , respectively, which were inferred in Ref. [54] dsff0 interactions that involve light invisible spin-1=2 from the data (at 2 sigmas) on the observed channels [5]. f f 0 V;A fermions, and , and induce the FCNC decays of strange Instead of fixing C ¯ ¼ 0 which led to Eq. (35), we also N2N3 hadrons with missing energy, E, carried away by the try alternatively to maximize the hyperon rates by allowing V V fermions. We concentrate on the case in which the LQ the two terms in C ¯ and c˜ ¯ as defined in Eq. (6b) to N2N3 N2N3 influence may be insignificant on the kaon mode K → πE vary freely and at the same time applying the conditions in but can be substantial on the hyperon decays B → B0E and Eqs. (20) and (36), but still employing the matrices in Ω− → Ξ−E, as well as on K → ππ0E. This can occur Eq. (34). In this instance, the results we get turn out 0 V because K → πE are sensitive exclusively to dsff operators to be barely bigger than those in Eq. (38),withjC ¯ j ≃ N2N3 with parity-even quark bilinears, whereas the hyperon −10 −2 ˜V −7 −2 1.0 × 10 GeV and jc ¯ j ≃ 3.1 × 10 GeV . modes are affected not only by this kind of operators but N2N3 The examples presented in this subsection and the also by those with solely parity-odd quark bilinears. We previous one indicate that experiments on these hyperon show that this possibility can be realized in a model transitions can serve as a valuable tool to discriminate NP containing scalar LQs by using two of them which have models. Furthermore, the acquired data could supply different chiral couplings to the d and s quarks. If SM fermions alone participate in the interactions of the LQs, we find that due to the combined restrictions from kaon mixing 5As before, there are alternatives, such as those with and lepton-flavor-violating processes the hyperon rates the corresponding products of nonzero matrix elements cannot reach values likely to be detectable in the near future. ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ being given by ðy1;13y1;22; y2;23y2;12Þ, ðy1;11y1;23; y2;21y2;13Þ, and In contrast, with the inclusion of light SM-gauge-singlet ˜ ˜ ˜ ˜ ðy1;13y1;21; y2;23y2;11Þ. right-handed neutrinos which also couple directly to the

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LQs, extra s → dE channels can arise, with the RH neutrinos addition suggests that the hyperon sector could serve as being emitted invisibly, and moreover the constraints from another environment in which to seek sterile neutrinos. To lepton-flavor-violation data can be evaded. As a conse- conclude, our analysis based on simple NP scenarios quence, the resulting hyperon rates are permitted to increase illustrates the importance of experimental quests for these to levels which may be discoverable by the ongoing BESIII hyperon modes, as the outcomes would complement those or future efforts such as at super charm-tau factories.6 This in of measurements on their kaon counterparts and could help distinguish NP models.

6It is worth noting that considerable branching fractions of the ACKNOWLEDGMENTS B → B0 Ω− → Ξ− hyperon modes E and E are also attainable if This research was supported in part by the MOST (Grant the missing energy is carried away by a single massless dark photon [54]. No. MOST 106-2112-M-002-003-MY3).

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