Study of the sdνˉν rare hyperon decays in the Standard Model and new physics

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Xiao-Hui Hu and Zhen-Xing Zhao. Study of the sdνˉν rare hyperon decays in the Standard Model and new physics[J]. Chinese Physics C, 2019, 43(9): 093104. doi: 10.1088/1674-1137/43/9/093104
Xiao-Hui Hu and Zhen-Xing Zhao. Study of the sdνˉν rare hyperon decays in the Standard Model and new physics[J]. Chinese Physics C, 2019, 43(9): 093104.  doi: 10.1088/1674-1137/43/9/093104 shu
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Study of the sdνˉν rare hyperon decays in the Standard Model and new physics

    Corresponding author: Xiao-Hui Hu, huxiaohui@sjtu.edu.cn
    Corresponding author: Zhen-Xing Zhao, star_0027@sjtu.edu.cn
  • 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 sdνˉν rare hyperon decays in SM and beyond. We find that in SM the branching ratios for these rare hyperon decays range from 1014 to 1011 . 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 27.

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    1.   Introduction
    • The flavor changing neutral current (FCNC) transitions provide a critical test of the Cabibbo-Kobayashi-Maskawa (CKM) mechanism in the Standard Model (SM), and allow to search for possible new physics. In SM, the FCNC transition sdνˉν proceeds through the Z-penguin and electroweak box diagrams, and thus the decay probabilities are strongly suppressed. In this case, a precise study allows to perform very stringent tests of SM and ensures large sensitivity to potential new degrees of freedom.

      A large number of studies have been performed of the K+π+νˉν and KLπ0νˉν processes, and reviews of these two decay modes can be found in [16]. On the theoretical side, using the most recent input parameters, the SM predictions for the two branching ratios are [7]

      B(K+π+νˉν)SM=(8.4±1.0)×1011,

      (1)

      B(KLπ0νˉν)SM=(3.4±0.6)×1011.

      (2)

      The dominant uncertainty comes from the CKM matrix elements and the charm contribution. On the experimental side, the NA62 experiment at the CERN SPS has reported the first search for K+π+νˉν using the decay-in-flight technique, and the corresponding observed upper limit is [8] :

      B(K+π+νˉν)exp<14×1010,at 95% CL.

      (3)

      Similarly, the E391a collaboration reported the 90% C.L. upper bound [9]

      B(KLπ0νˉν)exp2.6×108.

      (4)

      The KOTO experiment, an upgrade of the E391a experiment, aims at a first observation of the KLπ0νˉν decay at J-PARC around 2020 [3, 10]. Given the goal of a 10% precision by NA62, the authors of Ref. [11] intend to carry out lattice QCD calculations to determine the long-distance contributions to the K+π+νˉν amplitude.

      Analogous to K+π+νˉν and KLπ0νˉν, the rare hyperon decays BiBfνˉν also proceed via sdνˉν at the quark level, and thus offer important tools to test SM and to search for possible new physics. Compared to the widely considered K+π+νˉν and KLπ0νˉν, there are few studies devoted to rare hyperon decays BiBfνˉν. This work aims to perform a preliminary theoretical research of the rare hyperon decays both in and beyond SM.

      A study of the hyperon decays at the BESIII experiment is proposed using the hyperon parents of the J/ψ decay. The electron-positron collider BEPCII provides a clean experimental environment. About 106-108 hyperons, Λ, Σ, Ξ and Ω, are produced in the J/ψ and ψ(2S) decays with the proposed data samples at the BESIII experiment. Based on these samples, the sensitivity of the measurement of the branching ratios of hyperon decays is in the range of 105-108. The author of Ref. [12] proposed that rare decays and decays with invisible final states may be probed.

      The paper is organized as follows. In Sec. 2, our computing framework is presented. Sec. 3 is devoted to performing the numerical calculations. The branching ratios of several rare hyperon decays are calculated in SM. The new physics contribution, the Minimal Supersymmetric Standard Model (MSSM) and the minimal 331 model, are considered. We also discuss possible uncertainties from the form factors. The last section contains a short summary.

    2.   Theoretical framework
    • The next-to-leading order (NLO) effective Hamiltonian for sdνˉν reads [13]:

      Heff=GF2α2πsin2θWl=e,μ,τ[VcsVcdXlNL+VtsVtdX(xt)](ˉsd)VA(ˉνlνl)VA+h.c.,

      (5)

      where X(xt) and XlNL are relevant for the top and the charm contribution, respectively. Their explicit expressions can be found in Ref. [13]. Here, xt=m2t/m2W. To leading order in αs, the function X(xt) relevant for the top contribution reads [14, 15]

      X(x)=X0(x)+αs4πX1(x),X0(x)=x8[2+x1x+3x6(1x)2lnx],X1(x)=23x+5x24x33(1x)2+x11x2+x3+x4(1x)3lnx+8x+4x2+x3x42(1x)3ln2x4xx3(1x)2L2(1x)+8xX0(x)xlnxμ,

      (6)

      where xμ=μ2/M2W with μ=O(mt) and

      L2(1x)=x1dtlnt1t.

      (7)

      The function XlNL corresponds to X(xt) in the charm sector. It results from the renormalization group (RG) calculation in next-to-leading-order logarithmic approximation (NLLA) and is given as follows:

      XlNL=CNL4B(1/2)NL,

      (8)

      where CNL and B(1/2)NL correspond to the Z0-penguin and the box-type contribution, respectively, given as [16]

      CNL=x(mc)32K2425c[(487K++2411K69677K33)(4παs(μ)+152121875(1K1c))+(1lnμ2m2c)(16K+8K)117624413125K+23026875K+352918448125K33+K(562484375K+814486875K+4563698144375K33)],B(1/2)NL=x(mc)4K2425c[3(1K2)(4παs(μ)+152121875(1K1c))lnμ2m2crlnr1r30512+15212625K2+155817500KK2],

      (9)

      where r=m2l/m2c(μ), μ=O(mc) and

      K=αs(MW)αs(μ),Kc=αs(μ)αs(mc),K+=K625,K=K1225,K33=K2=K125.

      (10)

      In the following, we consider the transitions between the baryon octet (Ξ, Σ, Σ and N) and the transitions from the baryon decuplet to the octet ΩΞ.

      The transition matrix elements of the vector and axial-vector currents between the baryon octets can be parametrized in terms of six form factors f1,2,3(q2) and g1,2,3(q2):

      B8(P,Sz)|ˉdγμ(1γ5)s|B8(P,Sz)=ˉu(P,Sz)[γμf1(q2)+iσμνqνMf2(q2)+qμMf3(q2)]u(P,Sz)ˉu(P,Sz)[γμg1(q2)+iσμνqνMg2(q2)+qμMg3(q2)]γ5u(P,Sz),

      (11)

      where q=PP, and M denotes the mass of the parent baryon octet B8. The form factors for the B8B8 transition, fi(q2) and gi(q2) , can be expressed by the following formulas [17]:

      fm=aFm(q2)+bDm(q2),gm=aFm+3(q2)+bDm+3(q2),(m=1,2,3),

      (12)

      where Fi(q2) and Di(q2) , with i=1,2,,6, are different functions of q2 for each of the six form factors. Some remarks are necessary [17]:

      ● The constants a and b in Eq. (12) are the SU(3) Clebsch-Gordan coefficients that appear when an octet operator is sandwiched between octet states.

      ● For q2=0, the form factor f1(0) is equal to the electric charge of the baryon, therefore F1(0)=1 and D1(0)=0.

      ● The weak f2(0) form factor can be computed using the anomalous magnetic moments of proton and neutron (κp and κn) in the exact SU(3) symmetry. Here, F2(0)=κp+12κn and D2(0)=32κn.

      g1(0) is a linear combination of two parameters, F and D.

      ● Since gnp2=F5(q2)+D5(q2)=0 and gΞΞ02=D5(q2)F5(q2)=0, we get F5(q2)=D5(q2)=0. Therefore, all pseudo-tensor form factors g2 vanish in all decays up to symmetry-breaking effects.

      ● In the sdˉνν decay, the f3 and g3 terms are proportional to the neutrino mass and thus can be neglected for the decays considered in this work.

      Since the invariant mass squared of lepton pairs in the hyperon decays is relatively small, it is expected that the q2 distribution in the form factors has small impact on the decay widths. We list the expressions for f1, f2 and g1 at q2=0 in Table 1.

      BBΛnΣ+pΞ0ΛΞ0Σ0ΞΣ
      f1(0)32132121
      f2(0)32κp(κp+2κn)32(κp+κn)12(κpκn)κpκn
      g1(0)32(F+D/3)(FD)32(FD/3)12(F+D)F+D

      Table 1.  The form factors for the BB 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 positive for the neutron decay, and all other signs are fixed using this sign convention.

      Hence, Eq. (11) can be rewritten as:

      B8(P,Sz)|ˉdγμ(1γ5)s|B8(P,Sz)=ˉu(P,Sz)[γμf1(q2)+iσμνqνMf2(q2)γμg1(q2)γ5]u(P,Sz).

      (13)

      The helicity amplitudes of the hadronic contribution are defined as

      HVλ,λVB8(P,λ)|ˉdγμs|B8(P,λ)ϵVμ(λV),

      (14)

      HAλ,λVB8(P,λ)|ˉdγμγ5s|B8(P,λ)ϵVμ(λV).

      (15)

      Here, λ() denotes the helicity of the parent (daughter) baryon in the initial (final) state, and λV is the helicity of the virtual intermediate vector particle. It can be shown that the helicity amplitudes HV,Aλ,λVhave the following simple forms [19]:

      HV12,0=iQq2[(M+M)f1q2Mf2],HA12,0=iQ+q2(MM)g1,HV12,1=i2Q[f1+M+MMf2],HA12,1=i2Q+g1.

      (16)

      In the above, Q±=(M±M)2q2, and M (M) is the parent (daughter) baryon mass in the initial (final) state. The amplitudes for the negative helicity are obtained from the relations,

      HVλ,λV=HVλ,λV,HAλ,λV=HAλ,λV.

      (17)

      The complete helicity amplitudes are obtained by

      Hλ,λV=HVλ,λVHAλ,λV.

      (18)

      Due to the lack of experimental data for the M1 and E2 transitions from the baryon decuplet to the octet, the vector transition matrix element for ΩΞ can not be determined. In this work we follow Ref. [18] , and consider only the axial-vector current matrix element [18, 20, 21]:

      Ξ(P,Sz)|ˉdγμγ5s|Ω(P,Sz)=ˉuΞ(P,Sz){CA5(q2)gμν+CA6(q2)qμqν+[CA3(q2)γα+CA4(q2)p]×(qαgμνqνgαμ)}uνΩ(P,Sz).

      (19)

      Here, uνΩ(P,Sz) represents the Rarita-Schwinger spinor that describes the baryon decuplet Ω with spin 32. In Ref. [22] it is shown that CA3(q2) and CA4(q2) are proportional to the mass difference of the initial and final baryons, and thus are suppressed. In the chiral limit, CA5(q2) and CA6(q2) are related by CA6(q2)=M2NCA5(q2)/q2 [20]. In our calculations, we use CA5(0)=1.653±0.006 for ΩΞ , which is the same as ΩΞ0 in the SU(3) limit [18]. The helicity amplitude can then be expressed as:

      HAλ,λV=Ξ(P,λ)|ˉdγμγ5s|Ω(P,λ)ϵμV(λV)

      (20)

      =ˉuΞ(P,λ)[CA5(q2)gμν+CA6(q2)qμqν]uνΩ(P,λ)ϵμV(λV).

      (21)

      Here, λ() and λV have the same definition as in Eqs. (14)-(15). It can be shown that the helicity amplitudes HAλ,λVhave the following simple forms [19]:

      HA12,0=HA12,0=i2Q+3EVq2CA5(q2),HA12,1=HA12,1=iQ+3CA5(q2),HA12,1=HA12,1=iQ+CA5(q2).

      (22)

      The differential decay width for BBˉνν is given as:

      dΓdq2=dΓLdq2+dΓTdq2.

      (23)

      Here, dΓL/dq2 and dΓT/dq2 are the longitudinal and transverse parts of the decay width, and their explicit expressions are given by

      dΓLdq2=Nq2p12(2π)3M2(|H12,0|2+|H12,0|2),

      (24)

      dΓTdq2=Nq2p12(2π)3M2(|H12,1|2+|H12,1|2+|H12,1|2+|H12,1|2).

      (25)

      In Eqs. (24) and (25), p=Q+Q/2M is the magnitude of the momentum of B in the rest frame of B, and N=2N1(0)+N1(mτ) with

      N1(ml)=|GF2α2πsin2ΘW(VcdVcsXlNL(ml)+VtdVtsX(xt))|2.

      (26)

      Note that we have neglected the electron and muon masses.

      One can then obtain the decay width

      Γ=(MM)20dq2dΓdq2.

      (27)
    3.   Numerical results and discussion

      3.1.   Calculations in SM

    • With the input parameters given in Table 2 and the formulae from the last section, the LO and NLO results for μc = 1 GeV, μt = 100 GeV , and μc=3GeV,μt=300GeV , are listed Table 3.

      The masses and lifetimes of baryons in the initial and final states [23]
      mp=938.2720813 MeVmΣ+=1189.37 MeVmΞ0=1314.86 MeV
      mn=939.5654133 MeVmΣ=1197.45 MeVmΞ=1321.71 MeV
      mΛ=1115.683 MeVmΣ0=1192.642 MeVmΩ=1672.45 MeV
      τΞ0=2.90×1010sτΞ=1.639×1010sτΩ=0.821×1010s
      τΛ=2.632×1010sτΣ+=0.8018×1010s
      Physical constants and CKM parameters [23, 24]
      GF=1.16637387×105GeV2sin2θW=0.23122αs(mZ)=0.1182αα(mZ)=1/128
      mτ=1776.86 MeVmc=1.275 GeVmt=173.0 GeVmW=80.379 GeVmZ=91.1876 GeV
      A=0.836λ=0.22453ˉρ=0.122ˉη=0.355

      Table 2.  The input parameters used in this work.

      Branching ratioB(Λnνˉν)B(Σ+pνˉν)B(Ξ0Λνˉν)B(Ξ0Σ0νˉν)B(ΞΣνˉν)B(ΩΞνˉν)
      μc=1 GeVLO2.85×10126.88×10131.06×10121.77×10132.17×10131.78×1011
      NLO1.98×10125.01×10137.35×10131.24×10131.52×10131.93×1011
      μt=100 GeVNLO+SUSY (Set.I)8.14×10122.06×10123.02×10125.08×10136.23×10137.94×1011
      NLO+SUSY (Set.II)3.78×10129.55×10131.40×10122.36×10132.89×10133.69×1011
      NLO+M3311.24×10113.13×10124.59×10127.71×10139.45×10131.20×1010
      μc=3 GeVLO1.10×10122.65×10134.10×10136.83×10148.37×10141.07×1011
      NLO1.20×10123.04×10134.46×10137.50×10149.19×10141.17×1011
      μt=300 GeVNLO+SUSY (Set.I)5.85×10121.48×10122.17×10123.65×10134.47×10135.71×1011
      NLO+SUSY (Set.II)2.35×10125.94×10138.72×10131.47×10131.80×10132.29×1011
      NLO+M3311.02×10112.58×10123.80×10126.37×10137.81×10139.95×1011
      BESIII sensitivity [12]3×1074×1078×1079×1072.6×105

      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.

      From the results in Table 3 one can see that:

      ● The branching ratios of the sdνˉν rare hyperon decays range from 1014 to 1011 .

      ● For μc=1GeV,μt=100GeV , the NLO results are smaller than the LO ones by about 30%, while for μc=3GeV,μt=300GeV , the NLO results are larger than the LO ones by about 10%.

      ● The LO results vary by about 50% from μc=1 GeV, μt=100 GeV to μc=3GeV,μt=300GeV, while the NLO ones vary by about 30%. As expected, the NLO results depend less on the mass scales.

      ● The branching ratio of ΩΞνˉν is the largest among the 6 channels. It is of the same order as for K+π+νˉν and KLπ0νˉν.

      At present, there is a small number of experimental studies, and thus most experimental constraints are less severe. The prospects for rare and forbidden hyperon decays at BESIII were analyzed in a recent publication Ref. [12]. We quote the experimental sensitivity for all decay modes in Table 3 . Unfortunately, one can see that the current BESIII experiment will not be able to probe these hyperon decays. We hope this may be improved at future experimental facilities like the Super Tau-Charm Factory.

    • 3.2.   Uncertainties of the form factors

    • Note that due to the Ademollo-Gatto theorem [25], the form factor f1(0) does not receive any SU(3) symmetry breaking correction. However, f2(0) can be computed using the anomalous magnetic moments of proton 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]:

      κp[O(m0s)]=1.363±0.069,κp[O(m0s)]=1.416±0.049,

      (28)

      κp[O(m0s)+O(m1s)]=1.793±0.087,κp[O(m0s)+O(m1s)]=1.913±0.069.

      (29)

      The uncertainties from κp and κn in the effect of SU(3) symmetry breaking is approximately 25%. We calculated the effect of κp and κn on the branching ratio of Σ+pνˉν in the case of NLO with the energy scale μc=1GeV and μt=100GeVsuch that:

      B(Σ+pνˉν)[O(m0s)]=(4.86±0.04)×1013,B(Σ+pνˉν)[O(m0s)+O(m1s)]=(5.01±0.08)×1013.

      (30)

      Next, we consider the uncertainty of the branching ratio of Σ+pνˉν and Λnνˉν in the case of NLO with the energy scale μc=1GeV and μt=100GeV. This uncertainty comes from the parameters F=0.463±0.008 and D=0.804±0.008 [18] in the form factor g1(0):

      B(Σ+pνˉν)=(5.01±0.12)×1013,B(Λnνˉν)=(2.03±0.05)×1012.

      (31)

      For the decay ΩΞνˉν, CA5(0)=1.653±0.006 in the SU(3) symmetry, while CA5(0)=1.612±0.007 in the SU(3) symmetry breaking [18]. In the case of NLO with the energy scale μc=1 GeV and μt=100 GeV the branching ratio B(ΩΞνˉν) is then calculated as:

      B(ΩΞνˉν)(sy)=(1.84±0.01)×1011,B(ΩΞνˉν)(br)=(1.93±0.01)×1011.

      (32)

      As an illustration of the effects of q2 distribution in the form factors, we attempt to use the following parametrization for all form factors:

      F(q2)=F(0)1q2m2,

      (33)

      with m representing the initial hyperon mass. For example, for the NLO case of μc=1GeV and μt= 100 GeV, we obtain:

      B(Λnνˉν)(F(0))=1.98×1012,B(Λnνˉν)(F(q2))=2.03×1012,B(Σ+pνˉν)(F(0))=5.05×1013,B(Σ+pνˉν)(F(q2))=5.16×1013.

      (34)

      We find that the differences between the two cases are small, about a few percent.

      When all the above 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% .

    • 3.3.   Contribution from MSSM

    • The effective Hamiltonian for sdνˉν in the generalized supersymmetry (SUSY) extension of SM is given in Eq. (5), with X(xt) replaced by [26]

      Xnew=X(xt)+XH(xtH)+Cχ+CN.

      (35)

      Here, xtH=m2t/m2H± , and XH(xtH) corresponds to the charged Higgs contribution. Cχ and CN denote the chargino and neutralino contributions

      Cχ=X0χ+XLLχRUsLdL+XLRχRUsLtR+XLRχRUtRdL,CN=XNRDsLdL,

      where Xiχ and XN depend on the SUSY masses, and respectively on the chargino and neutralino mixing angles. The explicit expressions for XH(x), Cχ and CN can be found in Ref. [26]. The R parameters are defined in terms of mass insertions, and their upper limits are listed in Table 4 [26]. It should be mentioned that the phase ϕ of RUsLtR and RUtRdL is a free parameter which ranges from 0 to 2π. We set ϕ=0 as a central result.

      quantityupper limit
      RDsLdL(11255i)m˜dL500GeV
      RUsLdL(11254i)m˜uL500GeV
      RUsLtRMin{231(m˜uL500GeV)3,43}×eiϕ,0<ϕ<2π
      RUtRdL37(m˜uL500GeV)2×eiϕ,0<ϕ<2π

      Table 4.  Upper limits for the R parameters. Note that the phase of RUsLtR and RUtRdL is unconstrained.

      The parameters in Table 5 are adopted for detailed calculations [27]. The assumption M10.5M2 was made [28]. With the above parameters, the branching ratios of hyperon decays are listed in Table 3, and are significantly enhanced compared with the SM results. Taking as examples the decays Λnνˉν and ΩΞνˉν with the energy scale μc=1 GeV and μt=100 GeV , we obtain:

      parameters [27]the meaning of parameters [27]the range of parameters [27]Set.I [27]Set.II [27]
      βThe angle of unitarity triangle180β180tanβ=2tanβ=20
      MACP-odd Higgs boson mass150MA400333260
      M2SU(2) gaugino mass; we use M1 GUT-related to M250M2800181750
      μSupersymmetric Higgs mixing parameter400μ400375344
      MslCommon flavour diagonal slepton mass parameter95Msl1000105884
      MsqCommon mass parameter for the first two generations of squarks240Msq1000308608
      M˜tLSquark mass parameter for the right stop50M˜tR1000279338

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

      NLO: B(Λnνˉν)=1.98×1012,B(ΩΞνˉν)=1.93×1011,

      (36)

      Set.I: B(Λnνˉν)=8.14×1012,B(ΩΞνˉν)=7.94×1011,

      (37)

      Set.II: B(Λnνˉν)=3.78×1012,B(ΩΞνˉν)=3.69×1011.

      (38)

      Comparing the results of NLO+SUSY (Set. I) and (Set. II) with the ones of NLO, we see that all branching ratios are roughly enhanced by a factor of 4 and 2, respectively. However, none of these results can be probed at the ongoing experimental facilities, like the BESIII experiment [12].

    • 3.4.   Contribution from the minimal 331 model

    • The so-called minimal 331 model is an extension of SM at the TeV scale, where the weak gauge group of SM SU(2)L is extended to SU(3)L. In this model, a new neutral Z gauge boson can give very important additional contributions, for it can transmit FCNC at the tree level. In Table 3, we denote this model as M331. More details of this model can be found in Ref. [29]. The minimal 331 model leads to a new term in the effective Hamiltonian [30]:

      HZeff=l=e,μ,τGF2˜V32˜V313(MZMZ)2cos2θW(ˉsd)VA(ˉνlνl)VA+h.c.,

      (39)

      with MZ=1TeV, Re[(˜V32˜V31)2]=9.2×106 and Im[(˜V32˜V31)2]=4.8×108 [30]. The other parameters are the same as the SM inputs [23, 24]. The function X(xt) in Eq. (5) can be redefined as X(xt)=XSM(xt)+ ΔX with

      ΔX=sin2θWcos2θWα2π3˜V32˜V31VtsVtd(MZMZ)2.

      (40)

      With the modified function X(xt) and considering the NLO contribution, the branching ratios of rare hyperon decays in the minimal 331 model can be calculated, as shown in Table 3. The NLO+M331 predictions are much larger than the NLO results in SM, and are two and four times larger than the results of NLO+SUSY (Set. I) and NLO+SUSY (Set. II), respectively.

    4.   Conclusions
    • FCNC processes offer important tools to test SM and to search for possible new physics. The two decays K+π+νˉν and KLπ0νˉν have been widely studied, while the corresponding baryon sector has not been explored. In this work, we studied the sdνˉν rare hyperon decays. We adopted the leading order approximations for the form factors for small q2 , and derived expressions for the decay width. We applied the decay width formula to both SM and new physics contributions. Different energy scales were considered. The branching ratios in SM range from 1014 to 1011, and the largest is of the same order as for the decays K+π+νˉν and KLπ0νˉν. When all the errors in the form factors are included, we found that the final branching ratios for most decay modes have an uncertainty of about 5% to 10%. After taking into account the contribution from MSSM, the branching ratios are enhanced by a factor of 24. The branching ratios of hyperon decays in the minimal 331 model are seven times larger than the SM results.

      The authors are grateful to Profs. Hai-Bo Li and Wei Wang for useful discussions.

Reference (30)

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