Study of the Rare Hyperon Decays in the Standard Model and New Physics*

Chinese Physics C Vol. 43, No. 9 (2019) 093104

Study of the s → dνν¯ rare hyperon decays in the Standard Model and new physics*

1) 2) Xiao-Hui Hu(胡晓会) Zhen-Xing Zhao(赵振兴) INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology, MOE Key Laboratory for Particle Physics, Astrophysics and Cosmology, School of Physics and Astronomy, Shanghai Jiao-Tong University, Shanghai 200240, China

Abstract: FCNC processes offer important tools to test the Standard Model (SM) and to search for possible new physics. In this work, we investigate the s → dνν¯ rare hyperon decays in SM and beyond. We find that in SM the branching ratios for these rare hyperon decays range from 10−14 to 10−11 . When all the errors in the form factors are included, we find that the final branching ratios for most decay modes have an uncertainty of about 5% to 10%. After taking into account the contribution from new physics, the generalized SUSY extension of SM and the minimal 331 model, the decay widths for these channels can be enhanced by a factor of 2 ∼ 7. Keywords: branching ratios, rare hyperon decays, form factors, light-front approach, new physics PACS: 12.39.-x, 12.60.-i DOI: 10.1088/1674-1137/43/9/093104

1 Introduction flight technique, and the corresponding observed upper limit is [8] : B + → π+νν < × −10, . The flavor changing neutral current (FCNC) trans- (K ¯)exp 14 10 at 95% CL (3) itions provide a critical test of the Cabibbo-Kobayashi- Similarly, the E391a collaboration reported the 90% C.L. Maskawa (CKM) mechanism in the Standard Model upper bound [9] (SM), and allow to search for possible new physics. In − B(K → π0νν¯) ⩽ 2.6 × 10 8. (4) SM, the FCNC transition s → dνν¯ proceeds through the Z- L exp penguin and electroweak box diagrams, and thus the de- The KOTO experiment, an upgrade of the E391a ex- 0 cay probabilities are strongly suppressed. In this case, a periment, aims at a first observation of the KL → π νν¯ de- precise study allows to perform very stringent tests of SM cay at J-PARC around 2020 [3, 10]. Given the goal of a and ensures large sensitivity to potential new degrees of 10% precision by NA62, the authors of Ref. [11] intend freedom. to carry out lattice QCD calculations to determine the A large number of studies have been performed of the long-distance contributions to the K+ → π+νν¯ amplitude. + + 0 + + 0 K → π νν¯ and KL → π νν¯ processes, and reviews of Analogous to K → π νν¯ and KL → π νν¯ , the rare these two decay modes can be found in [1–6]. On the the- hyperon decays Bi → B f νν¯ also proceed via s → dνν¯ at oretical side, using the most recent input parameters, the the quark level, and thus offer important tools to test SM SM predictions for the two branching ratios are [7] and to search for possible new physics. Compared to the + + + + −11 widely considered K → π νν¯ and K → π0νν¯ , there are B(K → π νν¯)SM = (8.4 1.0) × 10 , (1) L few studies devoted to rare hyperon decays B → B νν¯ . B → π0νν = . . × −11. i f (KL ¯)SM (3 4 0 6) 10 (2) This work aims to perform a preliminary theoretical re- The dominant uncertainty comes from the CKM matrix search of the rare hyperon decays both in and beyond elements and the charm contribution. On the experiment- SM. al side, the NA62 experiment at the CERN SPS has re- A study of the hyperon decays at the BESIII experi- ported the first search for K+ → π+νν¯ using the decay-in- ment is proposed using the hyperon parents of the J/ψ de-

Received 18 January 2019, Revised 4 May 2019, Published online 7 August 2019 * Supported in part by National Natural Science Foundation of China (11575110, 11735010, 11911530088), Natural Science Foundation of Shanghai (15DZ2272100) and Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education 1) E-mail: [email protected] 2) E-mail: [email protected] Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must main- tain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Pub- lishing Ltd

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/ (1 2) 0 cay. The electron-positron collider BEPCII provides a where CNL and BNL correspond to the Z -penguin and clean experimental environment. About 106-108 hyperons, the box-type contribution, respectively, given as [16] /ψ ψ [( )( Λ, Σ, Ξ and Ω, are produced in the J and (2S ) decays 24 x(mc) 48 24 696 4π C = K 25 K+ + K− − K with the proposed data samples at the BESIII experiment. NL 32 c 7 11 77 33 α (µ) Based on these samples, the sensitivity of the measure- ) ( ) s µ2 15212 −1 ment of the branching ratios of hyperon decays is in the + (1 − K ) + 1 − ln (16K+ − 8K−) − − c 2 range of 10 5-10 8. The author of Ref. [12] proposed that 1875 mc rare decays and decays with invisible final states may be 1176244 2302 3529184 − K+ − K− + K33 probed. 13125( 6875 48125 )] The paper is organized as follows. In Sec. 2, our com- 56248 81448 4563698 + K K+ − K− + K , puting framework is presented. Sec. 3 is devoted to per- 4375 6875 144375 33 [ ( ) forming the numerical calculations. The branching ratios 24 (1/2) x(mc) 4π 15212 − B = K 25 3(1 − K ) + (1 − K 1) of several rare hyperon decays are calculated in SM. The NL 4 c 2 α (µ) 1875 c new physics contribution, the Minimal Supersymmetric s ] µ2 r lnr 305 15212 15581 Standard Model (MSSM) and the minimal 331 model, are − ln − − + K + KK , 2 − 2 2 considered. We also discuss possible uncertainties from mc 1 r 12 625 7500 the form factors. The last section contains a short sum- (9) = 2/ 2 µ µ = O mary. where r ml mc( ), (mc) and αs(MW ) αs(µ) K = , Kc = , 2 Theoretical framework αs(µ) αs(mc) 6 − 12 − 1 K+ = K 25 , K− = K 25 , K33 = K2 = K 25 . (10) The next-to-leading order (NLO) effective Hamiltoni- In the following, we consider the transitions between → νν an for s d ¯ reads [13]: the baryon octet (Ξ, Σ, Σ and N) and the transitions from α ∑ − − GF ∗ l the baryon decuplet to the octet Ω → Ξ . H ﬀ = √ [V V X e π 2 θ cs cd NL 2 2 sin W l=e,µ,τ The transition matrix elements of the vector and axi- + ∗ ν ν + . ., al-vector currents between the baryon octets can be para- VtsVtd X(xt)](¯sd)V−A(¯l l)V−A h c (5) 2 metrized in terms of six form factors f1,2,3(q ) and l 2 where X(xt) and XNL are relevant for the top and the g1,2,3(q ): charm contribution, respectively. Their explicit expres- ′ ′ ′ ⟨B (P ,S )|d¯γµ(1 − γ )s|B (P,S )⟩ = = 2/ 2 8 z [ 5 8 z ] sions can be found in Ref. [13]. Here, xt mt mW. To ν ′ ′ 2 q 2 qµ 2 leading order in αs, the function X(xt) relevant for the top u¯(P ,S ) γµ f (q ) + iσµν f (q ) + f (q ) u(P,S ) z 1 M 2 M 3 z contribution reads [14, 15] [ ] ν α q qµ s − ′, ′ γ 2 + σ 2 + 2 γ , , X(x) =X0(x) + X1(x), u¯(P S z) µg1(q ) i µν g2(q ) g3(q ) 5u(P S z) [ 4π ] M M x 2 + x 3x − 6 (11) X (x) = − + ln x , ′ 0 8 1 − x (1 − x)2 where q = P − P , and M denotes the mass of the parent → ′ 23x + 5x2 − 4x3 x − 11x2 + x3 + x4 baryon octet B8. The form factors for the B8 B8 trans- X1(x) = − + ln x ition, f (q2) and g (q2) , can be expressed by the following 3(1 − x)2 (1 − x)3 i i formulas [17]: + 2 + 3 − 4 8x 4x x x 2 + ln x f =aF q2 + bD q2 , 2(1 − x)3 m m( ) m( ) 2 2 3 gm =aFm+3(q ) + bDm+3(q ), (m = 1,2,3), (12) 4x − x ∂X0(x) − L2(1 − x) + 8x ln xµ, (6) 2 2 (1 − x)2 ∂x where Fi(q ) and Di(q ) , with i = 1,2,··· ,6, are different functions of q2 for each of the six form factors. Some re- where xµ = µ2/M2 with µ = O(m ) and W ∫ t x marks are necessary [17]: − = lnt . ● The constants a and b in Eq. (12) are the SU(3) L2(1 x) dt − (7) 1 1 t Clebsch-Gordan coefficients that appear when an octet l operator is sandwiched between octet states. The function XNL corresponds to X(xt) in the charm sec- 2 = tor. It results from the renormalization group (RG) calcu- ● For q 0, the form factor f1(0) is equal to the elec- lation in next-to-leading-order logarithmic approxima- tric charge of the baryon, therefore F1(0) = 1 and D1(0) = 0. tion (NLLA) and is given as follows: ● The weak f2(0) form factor can be computed using l = − (1/2), the anomalous magnetic moments of proton and neutron XNL CNL 4B (8) NL (κp and κn) in the exact SU(3) symmetry. Here,

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1 3 F2(0) = κp + κn and D2(0) = − κn. parent (daughter) baryon mass in the initial (final) state. 2 2 The amplitudes for the negative helicity are obtained ● g1(0) is a linear combination of two parameters, F and D. from the relations, n→p 2 2 Ξ−→Ξ0 V V A A ● Since g = F (q ) + D (q ) = 0 and g = H−λ′,−λ = Hλ′,λ , H−λ′,−λ = −Hλ′,λ . (17) 2 5 5 2 V V V V 2 − 2 = 2 = 2 = D5(q ) F5(q ) 0, we get F5(q ) D5(q ) 0. Therefore, The complete helicity amplitudes are obtained by all pseudo-tensor form factors g2 vanish in all decays up V A Hλ′,λ = Hλ′,λ − Hλ′,λ . (18) to symmetry-breaking effects. V V V ● In the s → dνν¯ decay, the f3 and g3 terms are pro- Due to the lack of experimental data for the M1 and E2 portional to the neutrino mass and thus can be neglected transitions from the baryon decuplet to the octet, the vec- for the decays considered in this work. tor transition matrix element for Ω− → Ξ− can not be de- Since the invariant mass squared of lepton pairs in the termined. In this work we follow Ref. [18] , and consider only the axial-vector current matrix element [18, 20, 21]: hyperon decays is relatively small, it is expected that the { 2 − ′ ′ − ′ ′ A 2 q distribution in the form factors has small impact on the ⟨Ξ , | ¯γ γ |Ω , ⟩ = − , (P S z) d µ 5 s (P S z) u¯Ξ (P S z) C5 (q )gµν decay widths. We list the expressions for f , f and g at [ ] 1 2 1 A 2 A 2 α A 2 ′ 2 = +C (q )qµqν + C (q )γ +C (q )p q 0 in Table 1. 6 }3 4 Hence, Eq. (11) can be rewritten as: × − ν , . (qαgµν qνgαµ) uΩ− (P S z) (19) ⟨ ′ ′, ′ | ¯γ − γ | , ⟩ = B8(P S z) d µ(1 5)s B8(P S z) ν [ ] Here, u − (P,S ) represents the Rarita-Schwinger ν Ω z ′ ′ 2 q 2 2 − 3 u¯(P ,S ) γµ f (q ) + iσµν f (q ) − γµg (q )γ u(P,S z). spinor that describes the baryon decuplet Ω with spin . z 1 M 2 1 5 2 A 2 A 2 (13) In Ref. [22] it is shown that C3 (q ) and C4 (q ) are pro- portional to the mass difference of the initial and final ba- The helicity amplitudes of the hadronic contribution are ryons, and thus are suppressed. In the chiral limit, CA(q2) defined as 5 A 2 A 2 = 2 A 2 / 2 V ′ ′ ′ µ ∗ and C6 (q ) are related by C6 (q ) MNC5 (q ) q [20]. In Hλ′,λ ≡ ⟨B (P ,λ )|d¯γ s|B (P,λ)⟩ϵ µ(λV ), (14) V 8 8 V A = . . our calculations, we use C5 (0) 1 653 0 006 for − − − A ≡ ⟨ ′ ′,λ′ | ¯γµγ | ,λ ⟩ϵ∗ λ . Ω → Ξ , which is the same as Ω → Ξ0 in the SU(3) Hλ′,λ B (P ) d 5 s B8(P ) Vµ( V ) (15) V 8 limit [18]. The helicity amplitude can then be expressed Here, λ(′) denotes the helicity of the parent (daughter) ba- as: ryon in the initial (final) state, and λV is the helicity of the A − ′ ′ − ⋆µ H ′ = ⟨Ξ (P ,λ )|d¯γµγ s|Ω (P,λ)⟩ϵ (λ ) (20) λ ,λV 5 V V virtual intermediate vector particle. It can be shown that [ ] V,A ′ ′ A 2 A 2 ν ⋆µ the helicity amplitudes H ′ have the following simple =u¯Ξ− (P ,λ ) C (q )gµν+C (q )qµqν u − (P,λ)ϵ (λ ). λ ,λV 5 6 Ω V V forms [19]: (21) √ [ ] (′) Q− q2 Here, λ and λ have the same definition as in Eqs. (14)- V = − √ + ′ − , V H 1 ,0 i (M M ) f1 f2 2 q2 M (15). It can be shown that the helicity amplitudes √ A Hλ′,λ have the following simple forms [19]: Q+ V A = − √ − ′ , √ H 1 ,0 i (M M )g1 2 2Q+ E q2 A = A = √V A 2 , [ ] H 1 ,0 H− 1 ,0 i C5 (q ) √ + ′ 2 2 3 q2 V M M = − − + , √ H 1 ,1 i 2Q f1 f2 2 M A A Q+ A 2 √ H 1 =H 1 = i C (q ), A ,1 − ,−1 5 = − + . 2 2 3 H 1 ,1 i 2Q g1 (16) √ 2 A A A 2 = = + . ′ 2 2 ′ H 1 ,−1 H− 1 ,1 i Q C5 (q ) (22) In the above, Q = (M M ) − q , and M (M ) is the 2 2

′ Table 1. The form factors for the B → B transition, f1(0), f2(0) and g1(0) [17], where the experimental anomalous magnetic moments are κp = 1.793 0.087 and κn = −1.913 0.069 [18], with the two coupling constants F = 0.463 0.008 and D = 0.804 0.008 [18]. Here, g1/ f1 is posit- ive for the neutron decay, and all other signs are fixed using this sign convention. + B → B′ Λ → n Σ → p Ξ0 → Λ Ξ0 → Σ0 Ξ− → Σ− √ √ 3 3 f1(0) − −1 1 1 2 2 − √ √ √ 2 1 3 − κ + κ 3 − √ κ − κ κ − κ f2(0) − κp ( p 2 n) (κp + κn) ( p n) p n √ 2 √ 2 2 1 3 3 − √ + g1(0) − (F + D/3) −(F − D) (F − D/3) (F D) F + D 2 2 2

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′ The differential decay width for B → B νν¯ is given as: µc = 3 GeV,µt = 300 GeV , the NLO results are larger than dΓ dΓ dΓ the LO ones by about 10%. = L + T . (23) dq2 dq2 dq2 ● The LO results vary by about 50% from µ = 1 GeV, µ = 100 GeV to µ = 3 GeV,µ = 300 GeV, Here, dΓ /dq2 and dΓ /dq2 are the longitudinal and trans- c t c t L T while the NLO ones vary by about 30%. As expected, the verse parts of the decay width, and their explicit expres- NLO results depend less on the mass scales. sions are given by ● The branching ratio of Ω− → Ξ−νν¯ is the largest Γ 2 ′ d L q p 2 2 among the 6 channels. It is of the same order as for = N (|H 1 ,0| + |H− 1 ,0| ), (24) 2 π 3 2 2 2 + + 0 dq 12(2 ) M K → π νν¯ and KL → π νν¯ . Γ 2 ′ At present, there is a small number of experimental d T q p 2 2 2 2 = N (|H 1 ,1| + |H− 1 ,−1| + |H 1 ,−1| + |H− 1 ,1| ). studies, and thus most experimental constraints are less dq2 12(2π)3 M2 2 2 2 2 (25) severe. The prospects for rare and forbidden hyperon de- √ ′ cays at BESIII were analyzed in a recent publication Ref. In Eqs. (24) and (25), p = Q+Q−/2M is the mag- ′ [12]. We quote the experimental sensitivity for all decay nitude of the momentum of B in the rest frame of B, and modes in Table 3 . Unfortunately, one can see that the N = 2N (0) + N (mτ) with 1 1 current BESIII experiment will not be able to probe these ( ) 2 G α ∗ ∗ hyperon decays. We hope this may be improved at future N m = √F V V Xl m + V V X x . 1( l) 2 cd cs NL( l) td ts ( t) 2 2πsin ΘW experimental facilities like the Super Tau-Charm Factory. (26) 3.2 Uncertainties of the form factors Note that we have neglected the electron and muon masses. Note that due to the Ademollo-Gatto theorem [25], One can then obtain the decay width the form factor f1(0) does not receive any SU(3) sym- ∫ − ′ 2 metry breaking correction. However, f (0) can be com- (M M ) dΓ 2 Γ = dq2 . (27) puted using the anomalous magnetic moments of proton 2 0 dq and neutron (κp and κn) in the exact SU(3) symmetry. The experimental data for κp and κn already include the SU(3) symmetry breaking effects [18]: 3 Numerical results and discussion [ ] [ ] κ O 0 = . . , κ O 0 = − . . , 3.1 Calculations in SM p (ms ) 1 363 0 069 p (ms ) 1 416 0 049 (28) With the input parameters given in Table 2 and the [ ] κ O 0 + O 1 = . . , formulae from the last section, the LO and NLO results p (ms ) (ms ) 1 793 0 087 µ µ µ = , [ ] for c = 1 GeV, t = 100 GeV , and c 3 GeV κ O 0 + O 1 = − . . . p (ms ) (ms ) 1 913 0 069 (29) µt = 300 GeV , are listed Table 3. From the results in Table 3 one can see that: The uncertainties from κp and κn in the effect of SU(3) ● The branching ratios of the s → dνν¯ rare hyperon symmetry breaking is approximately 25%. We calculated −14 −11 + decays range from 10 to 10 . the effect of κp and κn on the branching ratio of Σ → pνν¯ ● For µc = 1 GeV,µt = 100 GeV , the NLO results are in the case of NLO with the energy scale µc = 1 GeV and smaller than the LO ones by about 30%, while for µt = 100 GeVsuch that:

Table 2. The input parameters used in this work.

The masses and lifetimes of baryons in the initial and final states [23]

mp = 938.2720813 MeV mΣ+ = 1189.37 MeV mΞ0 = 1314.86 MeV

mn= 939.5654133 MeV mΣ− = 1197.45 MeV mΞ− = 1321.71 MeV

mΛ = 1115.683 MeV mΣ0 = 1192.642 MeV mΩ− = 1672.45 MeV −10 −10 −10 τΞ0 = 2.90 × 10 s τΞ− = 1.639 × 10 s τΩ− = 0.821 × 10 s −10 −10 τΛ = 2.632 × 10 s τΣ+ = 0.8018 × 10 s

Physical constants and CKM parameters [23, 24]

−5 −2 2 GF = 1.16637387 × 10 GeV sin θW = 0.23122 αs(mZ )= 0.1182 α ≡ α(mZ )= 1/128

mτ=1776.86 MeV mc = 1.275 GeV mt = 173.0 GeV mW =80.379 GeV mZ = 91.1876 GeV A = 0.836 λ = 0.22453 ρ¯ = 0.122 η¯ = 0.355

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Table 3. The LO, NLO, NLO+SUSY and NLO+M331 results for the branching ratio of rare hyperon decays for µc = 1 GeV,µt = 100 GeV and µc = 3 GeV,µt = 300 GeV. + − − − − Branching ratio B(Λ → nνν¯) B(Σ → pνν¯) B(Ξ0 → Λνν¯) B(Ξ0 → Σ0νν¯) B(Ξ → Σ νν¯) B(Ω → Ξ νν¯) LO 2.85 × 10−12 6.88 × 10−13 1.06 × 10−12 1.77 × 10−13 2.17 × 10−13 1.78 × 10−11 µc = 1 GeV NLO 1.98 × 10−12 5.01 × 10−13 7.35 × 10−13 1.24 × 10−13 1.52 × 10−13 1.93 × 10−11 NLO+SUSY (Set.I) 8.14 × 10−12 2.06 × 10−12 3.02 × 10−12 5.08 × 10−13 6.23 × 10−13 7.94 × 10−11 −12 −13 −12 −13 −13 −11 µt = 100 GeV NLO+SUSY (Set.II) 3.78 × 10 9.55 × 10 1.40 × 10 2.36 × 10 2.89 × 10 3.69 × 10 NLO+M331 1.24 × 10−11 3.13 × 10−12 4.59 × 10−12 7.71 × 10−13 9.45 × 10−13 1.20 × 10−10

LO 1.10 × 10−12 2.65 × 10−13 4.10 × 10−13 6.83 × 10−14 8.37 × 10−14 1.07 × 10−11 µc = 3 GeV NLO 1.20 × 10−12 3.04 × 10−13 4.46 × 10−13 7.50 × 10−14 9.19 × 10−14 1.17 × 10−11 NLO+SUSY (Set.I) 5.85 × 10−12 1.48 × 10−12 2.17 × 10−12 3.65 × 10−13 4.47 × 10−13 5.71 × 10−11 −12 −13 −13 −13 −13 −11 µt = 300 GeV NLO+SUSY (Set.II) 2.35 × 10 5.94 × 10 8.72 × 10 1.47 × 10 1.80 × 10 2.29 × 10 NLO+M331 1.02 × 10−11 2.58 × 10−12 3.80 × 10−12 6.37 × 10−13 7.81 × 10−13 9.95 × 10−11

− − − − BESIII sensitivity [12] 3 × 10 7 4 × 10 7 8 × 10 7 9 × 10 7 − 2.6 × 10−5 [ ] B(Σ+ → pνν¯) O(m0) = (4.86 0.04) × 10−13, When all the above errors in the form factors are in- [ s ] + − cluded, we find that the final branching ratios for most B(Σ → pνν¯) O(m0) + O(m1) = (5.01 0.08) × 10 13. (30) s s decay modes have an uncertainty of about 5% to 10% . Next, we consider the uncertainty of the branching ratio of Σ+ → pνν¯ and Λ → nνν¯ in the case of NLO with the 3.3 Contribution from MSSM energy scale µc = 1 GeV and µt = 100 GeV. This uncer- The effective Hamiltonian for s → dνν¯ in the general- tainty comes from the parameters F = 0.463 0.008 and ized supersymmetry (SUSY) extension of SM is given in = . . D 0 804 0 008 [18] in the form factor g1(0): Eq. (5), with X(xt) replaced by [26] + −13 B(Σ → pνν¯) =(5.01 0.12) × 10 , Xnew = X(xt) + XH(xtH) +Cχ +CN. (35) − B(Λ → nνν¯) =(2.03 0.05) × 10 12. (31) = 2/ 2 Here, xtH mt mH , and XH(xtH) corresponds to the − − Ω → Ξ νν A = . . charged Higgs contribution. Cχ and CN denote the char- For the decay ¯ , C5 (0) 1 653 0 006 in the A = . . gino and neutralino contributions SU(3) symmetry, while C5 (0) 1 612 0 007 in the SU(3) 0 LL U LR U LR∗ U symmetry breaking [18]. In the case of NLO with the en- Cχ =X + X R + X R + X R , χ χ s d χ sLtR χ t d µ = µ = L L R L ergy scale c 1 GeV and t 100 GeV the branching ra- = D , − − CN XNRs d tio B(Ω → Ξ νν¯) is then calculated as: L L i B(Ω− → Ξ−νν¯)(sy) = (1.84 0.01) × 10−11, where Xχ and XN depend on the SUSY masses, and respectively on the chargino and neutralino mixing angles. B(Ω− → Ξ−νν¯)(br) = (1.93 0.01) × 10−11. (32) The explicit expressions for XH(x), Cχ and CN can be As an illustration of the effects of q2 distribution in found in Ref. [26]. The R parameters are defined in terms the form factors, we attempt to use the following para- of mass insertions, and their upper limits are listed in Ta- metrization for all form factors: ble 4 [26]. It should be mentioned that the phase ϕ of RU sLtR U 2 F(0) and R is a free parameter which ranges from 0 to 2π. F(q ) = , tRdL q2 (33) We set ϕ = 0 as a central result. 1 − m2 The parameters in Table 5 are adopted for detailed with m representing the initial hyperon mass. For ex- Table 4. Upper limits for the R parameters. Note that the phase of ample, for the NLO case of µc = 1 GeV and µt = 100 U U Rs t and R is unconstrained. GeV, we obtain: L R tRdL quantity upper limit B(Λ → nνν¯)(F(0)) = 1.98 × 10−12, D md˜ − R (−112 − 55i) L B(Λ → nνν¯)(F(q2)) = 2.03 × 10 12, sLdL 500 GeV mu˜ + −13 RU (−112 − 54i) L B(Σ → pνν¯)(F(0)) = 5.05 × 10 , sLdL ( ) 500 GeV + − m 3 B Σ → νν 2 = . × 13. RU { u˜L , } × iϕ, < ϕ < π ( p ¯)(F(q )) 5 16 10 (34) sLtR Min 231 43 e 0 2 ( 500 GeV) 2 We find that the differences between the two cases are U mu˜ ϕ R 37 L × ei ,0 < ϕ < 2π small, about a few percent. tRdL 500 GeV

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Table 5. Parameters and their ranges used in Ref. [27]. All mass parameters are in GeV.

parameters [27] the meaning of parameters [27] the range of parameters [27] Set.I [27] Set.II [27] β The angle of unitarity triangle −180◦ ⩽ β ⩽ 180◦ tanβ = 2 tanβ = 20

MA CP-odd Higgs boson mass 150 ⩽ MA ⩽ 400 333 260 ⩽ ⩽ M2 SU(2) gaugino mass; we use M1 GUT-related to M2 50 M2 800 181 750 μ Supersymmetric Higgs mixing parameter −400 ⩽ µ ⩽ 400 −375 −344

Msl Common flavour diagonal slepton mass parameter 95 ⩽ Msl ⩽ 1000 105 884

Msq Common mass parameter for the first two generations of squarks 240 ⩽ Msq ⩽ 1000 308 608 ⩽ ⩽ Mt˜L Squark mass parameter for the right stop 50 Mt˜R 1000 279 338 calculations [27]. The assumption M1 ≈ 0.5M2 was made the same as the SM inputs [23, 24]. The function X(xt) in SM [28]. With the above parameters, the branching ratios of Eq. (5) can be redefined as X(xt) = X (xt)+ ∆X with hyperon decays are listed in Table 3, and are signific- 2 θ 2 θ π ˜ ∗ ˜ ( ) sin W cos W 2 V32V31 MZ 2 antly enhanced compared with the SM results. Taking as ∆X = ∗ . (40) − − α 3 V V M ′ examples the decays Λ → nνν¯ and Ω → Ξ νν¯ with the ts td Z energy scale µc = 1 GeV and µt = 100 GeV , we obtain: With the modified function X(xt) and considering the − NLO contribution, the branching ratios of rare hyperon NLO : B(Λ → nνν¯) = 1.98 × 10 12, decays in the minimal 331 model can be calculated, as B Ω− → Ξ−νν = . × −11, ( ¯) 1 93 10 (36) shown in Table 3. The NLO+M331 predictions are much Set.I: B(Λ → nνν¯) = 8.14 × 10−12, larger than the NLO results in SM, and are two and four − − − times larger than the results of NLO+SUSY (Set. I) and B(Ω → Ξ νν¯) = 7.94 × 10 11, (37) NLO+SUSY (Set. II), respectively. Set.II : B(Λ → nνν¯) = 3.78 × 10−12, B(Ω− → Ξ−νν¯) = 3.69 × 10−11. (38) 4 Conclusions Comparing the results of NLO+SUSY (Set. I) and (Set. II) with the ones of NLO, we see that all branching ratios FCNC processes offer important tools to test SM and are roughly enhanced by a factor of 4 and 2, respectively. to search for possible new physics. The two decays + + 0 However, none of these results can be probed at the ongo- K → π νν¯ and KL → π νν¯ have been widely studied, ing experimental facilities, like the BESIII experiment while the corresponding baryon sector has not been ex- [12]. plored. In this work, we studied the s → dνν¯ rare hyperon decays. We adopted the leading order approximations for 3.4 Contribution from the minimal 331 model the form factors for small q2 , and derived expressions for The so-called minimal 331 model is an extension of the decay width. We applied the decay width formula to SM at the TeV scale, where the weak gauge group of SM both SM and new physics contributions. Different energy SU(2)L is extended to SU(3)L. In this model, a new neut- scales were considered. The branching ratios in SM range ral Z′ gauge boson can give very important additional from 10−14 to 10−11, and the largest is of the same order as + + 0 contributions, for it can transmit FCNC at the tree level. for the decays K → π νν¯ and KL → π νν¯ . When all the In Table 3, we denote this model as M331. More details errors in the form factors are included, we found that the of this model can be found in Ref. [29]. The minimal 331 final branching ratios for most decay modes have an un- model leads to a new term in the effective Hamiltonian certainty of about 5% to 10%. After taking into account [30]: the contribution from MSSM, the branching ratios are en- ∑ ∗ ( ) ′ ˜ ˜ hanced by a factor of ∼ . The branching ratios of hyp- Z GF V32V31 MZ 2 2 2 4 H ﬀ = √ cos θW (¯sd)V−A(¯νlνl)V−A +h.c., e ′ eron decays in the minimal 331 model are seven times 2 3 MZ l=e,µ,τ larger than the SM results. (39)

∗ − ′ = ˜ ˜ 2 = . × 6 with MZ 1 TeV, Re[(V32V31) ] 9 2 10 and The authors are grateful to Profs. Hai-Bo Li and Wei ˜ ∗ ˜ 2 = . × −8 Im[(V32V31) ] 4 8 10 [30]. The other parameters are Wang for useful discussions.

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