# Study of Rare Mesonic Decays Involving Di-Neutrinos in Their Final State

Authors: Azeem Mir; Farida Tahir; Shakeel Mahmood; Shi- Hai Dong
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

We have studied phenomenological implication of R-parity violating (Rp) Minimal Supersymmetric Model (MSSM) via analyses of pure leptonic (M→νν¯) and semileptonic decays of pseudoscalar mesons (M→Xνν¯). These analyses involve comparison between theoretical predictions made by Rp MSSM and the Standard Model (SM) with the experimental results like branching fractions (Br) of the said process. We have found, in general, that Rp contribution dominates over the SM contribution, i.e., by a factor of 10 for the pure leptonic decays of KL,S and by 102 and 104 in case of Bs and Bd, respectively. Furthermore, the limits obtained on Rp Yukawa couplings (λαkβ′λαkγ′⁎) by using Br(M→Xνν¯) are used to calculate Br(M→νν¯). This demonstrates the role of Rp MSSM as a viable model for the study of new physics contribution in rare decays at places like Super B factories, KOTO (J-PARC) and NA62 at CERN.


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## Body

## 1. Introduction

Flavor Changing Neutral Currents (FCNC) that mediate different flavored fermions (quarks) of the same charge are one of the most important tools searching for physics beyond the Standard Model (SM). This is due to their rarity owing to the GIM mechanism [1]. FCNC processes involving leptons are strictly forbidden in SM due to lepton family number conservation contrary to established experimental facts [2–13], and such processes can only be accommodated through physics beyond the SM. However,lepton flavor conserving processes can proceed through both universal and non-universal weak neutral current interactions. Here, universal weak neutral current interactions correspond to the SM interactions, which are flavor as well as generation blind, and non-universal weak neutral current interactions represent new physics (NP) interactions which are flavor as well as generation sensitive. Analyses, involving the bounds on NP couplings, of such type of processes are good for comparative study of different models. In this paper, we have presented one class of such type of pure leptonic and semileptonic decays of pseudoscalar mesons involving di-neutrinos in their final state in the framework of SM and R-parity violating (Rp) supersymmetric (SUSY) model.Leptonic and semileptonic decays of beauty and strange mesons have played an important role in measuring parameters related to Cabibbo-Kobayashi-Maskawa (CKM), unitary angles, and also in probing CP-violation [14–16]. Many NP models like 2HDM [17] and Rp Minimal Supersymmetric Standard Model (MSSM) [18–20] have been explored in these processes [21–33] as well. Super B factories [34, 35] and experimental set-ups like KOTO at J-PARC and NA62 at CERN [36–39] hold a lot of potential in this regard. LHCb also holds a lot of promise for discovering prospects of NP in B decays [40, 41].MSSM [42–47] is the most economical version of SUSY. It is also the minimal extension of SM [42–47]. MSSM allows processes that violate baryon and lepton number. It also allows Lepton flavor violating (LFV) processes (that do violate lepton family number). R-parity, a discrete symmetry, is imposed to prevent baryon number, lepton number, and flavor violating processes. It is defined as Rp=(-1)3B+L+2S  [42, 48, 49]. R-parity conservation is phenomenologically motivated and if relaxed carefully allows one to analyze rare and forbidden decays while maintaining the stability of matter [46, 50–52]. The R-parity violating gauge invariant and renormalizable superpotential is [42, 48, 49](1)WRp=12λijkLiLjEkc+λijk′LiQjDkc+12λijk′′UiCDjCDkC+μiHuLi,where i,j,k are generation indices, Li and Qi are the lepton and quark left-handed SU(2)L doublets, and Ec, Dc are the charge conjugates of the right-handed leptons and quark singlets, respectively. Here λijk, λijk′, and λijk′′ are the Yukawa couplings. The term proportional to λijk is antisymmetric in first two indices [i,j] and λijk′′ is antisymmetric in last two indices [j,k], implying 9(λijk)+27(λijk′)+9λijk′′=45 independent coupling constants among which 36 are related to the lepton flavor violation (9 from LLEc and 27 from LQDc). We can rotate the last term away without affecting things of our interest.In this scenario for detailed illustration we will use the pure and semileptonic rare decays of pseudoscalar mesons with neutrinos in the final state, i.e.,M0→ναν¯β,M±,0→N±,0ναν¯β, and M±,0→N±,0ναν¯β, where M=K,B and N=π,K. At the quark level, all M±,0→N±,0ναν¯β decays are represented by s(b)→d(d,s)qναν¯β(q=d,s) and (all these processes can be) divided into two categories on the bases of lepton flavors, i.e.,(1)
lepton flavor conserving(α=β),(2)
lepton flavor violating(α≠β) decays.Thefirst type of decays s(b)→d(d,s)ναν¯α (α=e,μ,τ) is absent in the SM at tree level and is however induced by GIM mechanism [1] at the quantum loop level [53] which makes their effective strength very small, further suppression caused by the CKM matrix [54, 55]. These two suppressions make FCNC decays very rare. Furthermore, these processes will provide indirect test of high energy scales through a low energy process. Such type of processes has only short distance dominant contribution whereas long distance contribution is subleading [56], as we are taking pure and semileptonic decays, which can be accurately predicted in the SM due to the fact that the only relevant hadronic operators are just the current operators whose matrix elements can be extracted from their respective leading decays [57–61].Thesecond type of decays s(b)→d(d,s)ναν¯β(α≠β;α,β=e,μ,τ) is strictly forbidden to all orders in the SM due to lepton flavor violation, so their detection can clearly signal the presence of new interactions. Hence one can say that these are the “golden channels” for the study of NP.In this paper, we have analyzed the above-mentioned decays in the SM (first case) and then inRp violating MSSM. Our focus is to compare the NP contribution to the branching fraction of decay processes (under consideration) with the SM prediction and also with the experimental limits. In the forthcoming section, we will discuss these processes one by one.


## 2.s→dναν¯α

In the SM, the effective Hamiltonian for the semileptonic(K→πναν¯α,K→π0ναν¯α) and pure leptonic KL,S→ναν¯α processes is given by [62, 63](2)Heff=∑lCSMs¯dV-Aν¯lνlV-A.In this case, all leptons couple universally with the electroweak gauge bosons, where(3)CSM=GFα22sin2θwVcs∗VcdXNLl+Vts∗VtdXxt,Xxt=X0xt+αS4πXlxt,where X0(xt)=xt/8(xt+2/xt-1+3xt-6/(xt-1)2log⁡xt) and(4)Xlxt=-23xt+5xt2-4xt331-xt2+xt-11xt2+xt3+xt41-xt3log⁡xt+8xt+4xt2+xt3-xt421-xt3log⁡xt2-4xt-xt31-xt2∫1xtdtlog⁡t1-t+γm0xt∂X0xt∂xtlog⁡μt2MW2,XNLl=CNL-4BNL1/2,andxt=m¯t2(μt)/MW2, μt=O(mt),CNL and BNL(1/2) are Z0 penguin and box diagrams, respectively. Vcs∗Vcd are CKM matrix elements and αS is the coupling strength of strong interactions.In MSSM, the relevant effective Lagrangian for the decay processK→πναν¯α is given by [57–61](5)LRpeffs⟶d+να+ν¯α=4GF2Aαβsdν¯αγμPLναd¯γμPRs,where α=e,μ. The first term in (2) comes from the down squark exchange (where d and s are down type quarks). The dimensionless coupling constant Aααsd is related to Rp Yukawa couplings λαk1′λαk2′∗ by(6)Aααsd=24GF∑k=13λαk1′λαk2′∗2mdkc~2.The differential decay ratedΓ/dq2 for the semileptonic decay processes is given by [62–64](7)dΓdq2=125π5λ3/21,rM,smK3fp+q22Cl2,where λ(1,rM,s)=1+rM2+s2-2rM-2s-2rMs with rM=(mM/mB)2 and s=q2/mB2, Cl=CSM+z1/8eiθ; z is the general NP parameter. We have used the value for the form factor fp+(q2) for the above decay processes of K+ and K0 as given in [65]. Since this work focuses on Rp MSSM, we will shift our focus to z (for the calculation of limits on couplings) and Yukawa couplings λαk1′λαk2′∗/mdkc~2 (for the predictions of branching fraction). The decay rate for pure leptonic decay processes is given by(8)Γs⟶dνlν¯l=18πmK3fpq222mlmKCl2.The form factor fp(q2) is given by [66]. mK represents the mass of strange meson and ml is the mass of lepton, where Cl is same as that of semileptonic decays.


## 3.b→d(s)νανα

In MSSM, the relevant effective Lagrangian for the decay processB→π(K)ναν¯α is given by [57–61](9)LRpeffb⟶ds+να+ν¯α=4GF2Aααbdsν¯αγμPLναb¯γμPRds,where α=e,μ. The first term in (2) comes from the down squark exchange (where b and d(s) are down type quarks). The dimensionless coupling constant Aααbd(s) is given by(10)Aααbds=24GF∑k=13λαk12′λαk3′∗2mdkc~2.The differential decay rate for semileptonic decay processes is given by [62–64](11)dΓdq2=125π5λ3/21,r,smB3fp+q22Cl2,where(12)Cl=CSM+z8eiθ,with(13)CSM=GFα22sin2θwVcb∗VcdsXNLl+Vtb∗VtdsXxt,with(14)Xxt=X0xt+αS4πXlxt,xt=m¯t2μtMW2,μt=Omt,andz as explained in the above section is the general NP parameter and λαk1(2)′λαk3′∗/mdkc~2. We have used the form factor fp+(q2) for the above decay processes of B->π(K) as given in [67]. The decay rate for pure leptonic decay processes is given by [46, 50–52](15)Γb⟶dsνlν¯l=18πmB3fpq222mlmBCl2.The form factorfp(q2) is given by [66], mB represents the mass of beauty meson and ml is the mass of lepton.


## 4. Results and Discussions

We have carried out study of hypercharge changing two and three body decay processes of pseudoscalar mesons (M→Xναν¯α;M→ναν¯α), where M=K,B;α=e,μ,τ and X=π,K. This study considers two types of processes: polarized and unpolarized flavor of the lepton. The analysis carried out involves comparison of branching fraction of a certain decay process (mentioned above) calculated from both theoretical and experimental ground. This comparison not only helps to place bounds (listed in Tables 2, 4, and 5) on Rp Yukawa couplings λαkβ′λαkγ′∗ but also enables to predict (listed in Tables 3, 6–9) the enhancement of similar processes (having identical FCNC). The enhancement is given in three forms, namely, NP (contribution from Yukawa couplings only), Interference (product of SM contribution, coming from CSM and Yukawa couplings), and combined (NP+Interference). All the results are displayed in graphs plotted in Figures 2–13, which are composed of simple (variation of branching fraction with respect to the magnitude of NP parameter, i.e., z(λρωσ′λαβγ′∗)) and contour plot (region plot of magnitude and phase of NP parameter at different values of branching fraction within limits of experimental measurements). A visual error analysis for the experimental measurement of branching fraction is also presented in these graphs by constraining lines at mean and ±1σ level. Similar error analysis is repeated in tables. The Feynman diagrams and table listing experimental data [68] related to these processes are given in Figure 1 and Table 1, respectively. The Yukawa couplings (λρωσ′λαβγ′∗) involved are normalized to the square of mdc~/100GeV in all these tables and figures.Table 1
Table listing the properties of processes under discussion [68]. Here |z| is the strength of the NP parameter.
Process Experimental Measurement SM Prediction Bounds on New Physics Parameter Magnitude|z| Phase (θ)  B→πνν- < 9.8 × 10 - 5 1.62 × 10 - 7 < 1.79 × 10 - 6 ≤ 2 π  Bd0→νν-  6.13 × 10 - 31(e)  ≤ 2 π 4.57 × 10 - 27(μ) 5.91 × 10 - 25(τ)  B→Kνν- < 1.6 × 10 - 5 4.40 × 10 - 4 < 8.69 × 10 - 7 ≤ 2 π  Bs0→νν-  1.36 × 10 - 29(e)  ≤ 2 π 1.02 × 10 - 25(μ) 1.32 × 10 - 23(τ)  K±→πνν- < ( 1.7 ± 1.0 ) × 10 - 5 8.23 × 10 - 11 ( 7.83 ± 2.01 ) × 10 - 9(e) ≤ 2 π ( 7.83 ± 2.01 ) × 10 - 9(μ) ( 7.81 ± 2.01 ) × 10 - 9(τ)  Ks0→νν-  7.24 × 10 - 33(e)  ≤ 2 π 5.40 × 10 - 29(μ) 6.98 × 10 - 27(τ)  KL0→νν-  4.27 × 10 - 33(e)  ≤ 2 π 3.19 × 10 - 29(μ) 4.06 × 10 - 27(τ)  K 0 → π 0 ν ν -  (Bounds on New Physics parameters have been calculated numerically for their fit with SM prediction) < 2.6 × 10 - 5 2.76 × 10 - 11 < 1.9 × 10 - 9 15 ° < 9.83 × 10 - 10 30 ° < 6.96 × 10 - 10 45 ° < 5.68 × 10 - 10 60 ° < 5.09 × 10 - 10 75 ° < 4.92 × 10 - 10 90 °Table 2
Bounds on NP parameters (|z(λijk′λlmn′)|,θ) for K→πναν¯α. α(BrSM) is (a) unpolarized (8.63×10-11), (b) e (2.89×10-11),(c) μ (2.89×10-11), (d) τ (2.85×10-11). Experimental limits are (1.7±1.0)×10-10.
(a) Bounds on NP Parameter (unpolarized) θ | z ( Σ i = 1 3 λ ′ i k 2 ⋆ λ ′ i k 1 ) | × 1 0 - 9 0 3.465 ± 3.209 30 3.032 ± 2.653 60 4.229 ± . . . . 90 2.972 ± . . . . 120 2.233 ± . . . . 150 1.927 ± . . . . 180 1.94 ± . . . . 210 2.278 ± . . . . 240 3.062 ± . . . . 270 4.357 ± . . . . 300 3.078 ± 2.721 330 3.486 ± 3.233 360 3.465 ± 3.209  (b) Bounds on NP Parameter (unpolarized) θ | z ( Σ i = 1 3 λ ′ i k 2 ⋆ λ ′ i k 1 ) | × 1 0 - 9 0 5.575 ± 1.098 30 4.356 ± 1.328 60 4.229 ± . . . . 90 2.972 ± . . . . 120 2.233 ± . . . . 150 1.927 ± . . . . 180 1.94 ± . . . . 210 2.278 ± . . . . 240 3.062 ± . . . . 270 4.357 ± . . . . 300 4.504 ± 1.295 330 5.629 ± 1.09 360 5.575 ± 1.098  (c) Bounds on NP Parameter θ | z ( λ ′ 1 k 2 ⋆ λ ′ 1 k 1 ) | × 1 0 - 9 0 8.026 ± 1.87 30 7.083 ± 1.967 60 5.671 ± 2.079 90 4.408 ± 2.072 120 3.618 ± 1.956 150 3.267 ± 1.865 180 3.282 ± 1.87 210 3.666 ± 1.967 240 4.498 ± 2.079 270 5.793 ± 2.072 300 7.189 ± 1.956 330 8.068 ± 1.865 360 8.026 ± 1.87  (d) Bounds on NP Parameter θ | z ( λ ′ 2 k 2 ⋆ λ ′ 2 k 1 ) | × 1 0 - 9 0 8.026 ± 1.87 30 7.083 ± 1.967 60 5.671 ± 2.079 90 4.409 ± 2.072 120 3.618 ± 1.956 150 3.267 ± 1.865 180 3.282 ± 1.87 210 3.666 ± 1.967 240 4.499 ± 2.079 270 5.793 ± 2.072 300 7.189 ± 1.956 330 8.068 ± 1.865 360 8.026 ± 1.87  (e) Bounds on NP Parameter θ | z ( λ ′ 3 k 2 ⋆ λ ′ 3 k 1 ) | × 10 - 9 0 8.009 ± 1.87 30 7.071 ± 1.966 60 5.671 ± 2.076 90 4.418 ± 2.069 120 3.631 ± 1.954 150 3.282 ± 1.865 180 3.298 ± 1.87 210 3.684 ± 1.966 240 4.515 ± 2.076 270 5.803 ± 2.069 300 7.186 ± 1.954 330 8.054 ± 1.865 360 8.009 ± 1.87Figure 1
Feynman diagrams of (a)b→sναν¯α, (b) b→dναν¯α, (c) s→dναν¯α. α=1,2,3.
(a) (b) (c)Figure 2
Allowed region of general NP parameters(z,θ) for K→πναν¯α at several values of branching fraction. α is (a) unpolarized, (b) e, (c) μ, (d) τ. The three contours belong to branching fraction at [0.7,1.7,2.7]×10-10.
(a) (b) (c) (d)Figure 3
Variations of branching fraction w.r.t NP parameter|z(λijk′λlmn′)| at several values of θ for K→πναν¯α. α is (a) unpolarized, (b) e, (c) μ, (d) τ. The three bounds belong to branching fraction at [0.7 (-1σ),1.7 (Mean), 2.7 (+1σ)]×10-10 corresponding to an experimental measurement of (1.7±0.7)×10-10.
(a) (b) (c) (d)Figure 4
Allowed regions of general NP parameters(z,θ) for K0→π0ναν¯α at specific values of branching fraction. α is (a) unpolarized, (b) e, (c) μ, (d) τ. The three contours belong to branching fraction at [0.3,1.3,2.6]×10-8.
(a) (b) (c) (d)Figure 5
Variations of branching fraction (NP contribution only) with respect to NP parameterλijk′λlmn′ at several values of θ for K0→π0ναν¯α. α is (a) unpolarized, (b) e, (c) μ, (d) τ.
(a) (b) (c) (d)Figure 6
Allowed regions of general NP parameters(z,θ) for B→πναν¯α at specific values of branching fraction. α is (a) unpolarized, (b) e, (c) μ, (d) τ. The three contours belong to branching fraction at [1.4-9.8]×10-6.
(a) (b) (c) (d)Figure 7
Variation of branching fraction with respect to NP parameter|z(λijk′λlmn′)| at several values of θ for B→πναν¯α. α is (a) unpolarized, (b) e, (c) μ, (d) τ. Experimental bound on the process is 9.8 ×10-5.
(a) (b) (c) (d)Figure 8
Allowed regions of general NP parameters(z,θ) for B→Kναν¯α at specific values of branching fraction. α is (a) unpolarized, (b) e, (c) μ, (d) τ. The three contours belong to branching fraction at [0.4-1.6]×10-5.
(a) (b) (c) (d)Figure 9
Variations of branching fraction with respect to NP parameter|z(λijk′λlmn′)| at several values of θ for B→Kναν¯α. α is (a) unpolarized, (b) e, (c) μ, (d) τ. Experimental bound on the process is 1.6 ×10-5.
(a) (b) (c) (d)Figure 10
Variations of branching fraction (NP contribution only) with respect to NP parameterλijk′λlmn′ at several values of θ for KS→ναν¯α. α is (a) unpolarized, (b) e, (c) μ, (d) τ.
(a) (b) (c) (d)Figure 11
Variations of branching fraction (NP contribution only) with respect to NP parameterλijk′λlmn′ at several values of θ for KL→ναν¯α. α is (a) unpolarized, (b) e, (c) μ, (d) τ.
(a) (b) (c) (d)Figure 12
Variations of branching fraction (NP contribution only) with respect to NP parameterλijk′λlmn′ at several values of θ for Bd→ναν¯α. α is (a) unpolarized, (b) e, (c) μ, (d) τ.
(a) (b) (c) (d)Figure 13
Variations of branching fraction (NP contribution only) with respect to NP parameterλijk′λlmn′ at several values of θ for Bs→ναν¯α. α is (a) unpolarized, (b) e, (c) μ, (d) τ.
(a) (b) (c) (d)First, we will discuss the results related to semileptonic decay processes followed by pure leptonic decays. We have plotted graphs in Figures2 and 3 for the study of process K→πναν¯a. These plots relate the branching fraction of the said process with the magnitude and phase of NP parameters (z(λik1′λik2′∗) and θ). Contour plots in Figure 2 represent the allowed region for NP parameters (magnitude and phase) for specific values of branching fraction. All four plots (comprising unpolarized (a) and polarized (b-d)) show that the maximum magnitude of NP parameter oscillates with respect to its phase in general. The plot in Figure 2(a) shows a particular pattern at given error, i.e., at -1σ level of measured branching fraction (0.7×10-7). It clearly shows that only a narrow range of phase of NP parameter (θ≤π/4) is allowed for given -1σ level. The bounds on the magnitude of NP parameter are given in Table 2. Here the entry “±.......” means that these specific NP parameters cannot be defined for -1σ as the corresponding Rp MSSM contribution exceeds the experimental limit on branching fraction in -1σ error. Tables 2(a) and 2(b) show the same pattern as observed in Figures 2(a) and 3(a) numerically. Since Yukawa couplings for R-parity violation are identical for the processes (K→πναν¯α,K0→π0ναν¯α,KL,S→ναν¯α), the maximum limits for K→πναν¯α are used for calculating NP contribution to branching fraction of other processes. Figure 3 represent the variation of branching fraction with respect to the magnitude of NP parameter at several values of its phase θ.Contour plots in Figure4 represent the allowed region for NP parameters (z(λik1′λik2′∗) and θ) for specific values of branching fraction of the process K0→π0ναν¯a. All four plots (comprising unpolarized (a) and polarized (b-d)) show that the maximum magnitude of NP parameter follows acatenary(hanging chain) pattern, with the bottom level smoothening with decreasing error levels. The bounds on the magnitude of NP parameter and possible NP contribution are in Table 3, which shows that Rp MSSM dominates over SM contribution by order of magnitude 1~02, but the overall effect is to give a comparatively less contribution due to destructive interference. The plots in Figure 5 represent the variation of branching fraction (NP contribution only) for the process K0→π0ναν¯a with respect to the magnitude of NP parameter at several values of its phase θ.Table 3
Bounds on NP parameters (λijk′λlmn′,θ) derived from K→πναν¯α for K0→π0ναν¯α. α(BrSM) is (a) unpolarized (2.94.×10-11) and 9.78×10-12 for (b) e, (c) μ, (d) τ. Experimental bound on the process is 2.6×10-8.
(a) Bounds on NP Parameter Branching Fraction (×10-10) θ | Σ i = 1 3 ( λ ′ i k 2 ⋆ λ ′ i k 1 ) | × 10 - 9 NP Interference Combined 0 3.465 ± 3.209 0 . ± 0 . 0 . ± 0 . 0.003 ± 0 . 30 3.032 ± 2.653 2.484 ± 2.462 - 2.446 ± 2.426 0.041 ± 0.036 60 4.229 ± . . . . 8.212 ± . . . . - 8.099 ± . . . . 0.116 ± . . . . 90 2.972 ± . . . . 5.407 ± . . . . - 5.328 ± . . . . 0.082 ± . . . . 120 2.233 ± . . . . 2.289 ± . . . . - 2.25 ± . . . . 0.042 ± . . . . 150 1.927 ± . . . . 0.568 ± . . . . - 0.555 ± . . . . 0.017 ± . . . . 180 1.94 ± . . . . 0 ± . . . . 0 ± . . . . 0.003 ± . . . . 210 2.278 ± . . . . 0.794 ± . . . . - 0.796 ± . . . . 0.001 ± . . . . 240 3.062 ± . . . . 4.304 ± . . . . - 4.283 ± . . . . 0.023 ± . . . . 270 4.357 ± . . . . 11.621 ± . . . . - 11.541 ± . . . . 0.082 ± . . . . 300 3.078 ± 2.721 7.75 ± 7.691 - 7.695 ± 7.634 0.058 ± 0.057 330 3.486 ± 3.233 3.46 ± 3.45 - 3.44 ± 3.429 0.023 ± 0.021 360 3.465 ± 3.209 0 . ± 0 . 0 . ± 0 . 0.003 ± 0 .  (b) Bounds on NP Parameter Branching Fraction (×10-10) θ λ ′ 1 k 2 ⋆ λ ′ 1 k 1 × 10 - 9 NP Interference Combined 0 8.026 ± 1.87 0 . ± 0 . 0 . ± 0 . 0.001 ± 0 . 30 7.083 ± 1.967 2.757 ± 1.422 - 2.719 ± 1.405 0.039 ± 0.017 60 5.671 ± 2.079 5.585 ± 3.609 - 5.515 ± 3.568 0.071 ± 0.041 90 4.408 ± 2.072 4.843 ± 3.729 - 4.782 ± 3.686 0.062 ± 0.043 120 3.618 ± 1.956 2.589 ± 2.166 - 2.554 ± 2.14 0.036 ± 0.026 150 3.267 ± 1.865 0.722 ± 0.622 - 0.71 ± 0.613 0.013 ± 0.009 180 3.282 ± 1.87 0 . ± 0 . 0 . ± 0 . 0.001 ± 0 . 210 3.666 ± 1.967 0.883 ± 0.736 - 0.879 ± 0.731 0.005 ± 0.005 240 4.498 ± 2.079 3.759 ± 2.863 - 3.732 ± 2.839 0.028 ± 0.024 270 5.793 ± 2.072 7.726 ± 4.9 - 7.665 ± 4.857 0.062 ± 0.043 300 7.189 ± 1.956 8.497 ± 4.306 - 8.429 ± 4.267 0.068 ± 0.038 330 8.068 ± 1.865 3.499 ± 1.536 - 3.475 ± 1.523 0.025 ± 0.013 360 8.026 ± 1.87 0 . ± 0 . 0 . ± 0 . 0.001 ± 0 .  (c) Bounds on NP Parameter Branching Fraction (×10-10) θ | λ ′ 2 k 2 ⋆ λ ′ 2 k 1 | × 10 - 9 NP Interference Combined 0 8.026 ± 1.87 0 . ± 0 . 0 . ± 0 . 0.001 ± 0 . 30 7.083 ± 1.967 2.757 ± 1.422 - 2.719 ± 1.405 0.039 ± 0.017 60 5.671 ± 2.079 5.585 ± 3.609 - 5.515 ± 3.568 0.071 ± 0.041 90 4.409 ± 2.072 4.843 ± 3.729 - 4.782 ± 3.686 0.062 ± 0.043 120 3.618 ± 1.956 2.589 ± 2.167 - 2.554 ± 2.14 0.036 ± 0.026 150 3.267 ± 1.865 0.722 ± 0.622 - 0.71 ± 0.613 0.013 ± 0.009 180 3.282 ± 1.87 0 . ± 0 . 0 . ± 0 . 0.001 ± 0 . 210 3.666 ± 1.967 0.883 ± 0.736 - 0.88 ± 0.731 0.005 ± 0.005 240 4.499 ± 2.079 3.759 ± 2.863 - 3.732 ± 2.839 0.028 ± 0.024 270 5.793 ± 2.072 7.726 ± 4.9 - 7.665 ± 4.857 0.062 ± 0.043 300 7.189 ± 1.956 8.496 ± 4.305 - 8.429 ± 4.267 0.068 ± 0.038 330 8.068 ± 1.865 3.498 ± 1.536 - 3.475 ± 1.523 0.025 ± 0.013 360 8.026 ± 1.87 0 . ± 0 . 0 . ± 0 . 0.001 ± 0 .  (d) Bounds on NP Parameter Branching Fraction (×10-10) θ | λ ′ 3 k 2 ⋆ λ ′ 3 k 1 | × 10 - 9 NP Interference Combined 0 8.009 ± 1.87 0 . ± 0 . 0 . ± 0 . 0.001 ± 0 . 30 7.071 ± 1.966 2.748 ± 1.418 - 2.711 ± 1.401 0.038 ± 0.017 60 5.671 ± 2.076 5.582 ± 3.603 - 5.513 ± 3.562 0.071 ± 0.041 90 4.418 ± 2.069 4.856 ± 3.73 - 4.795 ± 3.687 0.062 ± 0.043 120 3.631 ± 1.954 2.603 ± 2.173 - 2.568 ± 2.146 0.036 ± 0.027 150 3.282 ± 1.865 0.727 ± 0.625 - 0.715 ± 0.616 0.013 ± 0.009 180 3.298 ± 1.87 0 . ± 0 . 0 . ± 0 . 0.001 ± 0 . 210 3.684 ± 1.966 0.89 ± 0.739 - 0.886 ± 0.734 0.005 ± 0.005 240 4.515 ± 2.076 3.78 ± 2.869 - 3.753 ± 2.845 0.028 ± 0.024 270 5.803 ± 2.069 7.745 ± 4.899 - 7.684 ± 4.856 0.062 ± 0.043 300 7.186 ± 1.954 8.488 ± 4.299 - 8.421 ± 4.261 0.068 ± 0.038 330 8.054 ± 1.865 3.487 ± 1.533 - 3.464 ± 1.52 0.024 ± 0.013 360 8.009 ± 1.87 0 . ± 0 . 0 . ± 0 . 0.001 ± 0 .Table 4
Bounds on NP parameters (|z(λijk′λlmn′)|,θ) for B→πναν¯α. α(BrSM) is (a) unpolarized (1.73×10-7) and 5.76×10-8 for (b) e, (c) μ, (d) τ. Experimental bound on the process is 9.8×10-5.
(a) Bounds on NP Parameter (unpolarized) θ | z ( Σ i = 1 3 λ ′ i k 3 ⋆ λ ′ i k 1 ) | × 1 0 - 6 0 1.0164 30 1.0298 60 1.0506 90 1.0735 120 1.0923 150 1.1016 180 1.0987 210 1.0844 240 1.0629 270 1.0402 300 1.0223 330 1.0137 360 1.0164  (b) Bounds on NP Parameter θ | z ( λ ′ 1 k 3 ⋆ λ ′ 1 k 1 ) | × 1 0 - 6 0 1.7907 30 1.8043 60 1.8252 90 1.8481 120 1.8667 150 1.8759 180 1.873 210 1.8589 240 1.8376 270 1.8148 300 1.7967 330 1.7879 360 1.7907  (c) Bounds on NP Parameter θ | z ( λ ′ 2 k 3 ⋆ λ ′ 2 k 1 ) | × 1 0 - 6 0 1.7907 30 1.8043 60 1.8252 90 1.8481 120 1.8667 150 1.8759 180 1.873 210 1.8589 240 1.8376 270 1.8148 300 1.7967 330 1.7879 360 1.7907  (d) Bounds on NP Parameter θ | z ( λ ′ 3 k 3 ⋆ λ ′ 1 k 1 ) | × 1 0 - 6 0 1.7907 30 1.8043 60 1.8252 90 1.8481 120 1.8667 150 1.8759 180 1.873 210 1.8589 240 1.8376 270 1.8148 300 1.7967 330 1.7879 360 1.7907Table 5
Bounds on NP parameters (|z(λijk′λlmn′)|,θ) for B→Kναν¯α. α(BrSM) is (a) unpolarized (4.69×10-6) and (1.56×10-6) for (b) e, (c) μ, (d) τ. Experimental bound on the process is 1.6×10-5.
(a) Bounds on NP Parameter (unpolarized) θ | z ( Σ i = 1 3 λ ′ i k 3 ⋆ λ ′ i k 2 ) | × 1 0 - 7 0 5.9349 30 5.512 60 4.4436 90 3.238 120 2.3595 150 1.9022 180 1.7667 210 1.9022 240 2.3595 270 3.238 300 4.4436 330 5.512 360 5.9349  (b) Bounds on NP Parameter θ | z ( λ ′ 1 k 3 ⋆ λ ′ 1 k 2 ) | × 1 0 - 7 0 8.7538 30 8.3927 60 7.4629 90 6.3358 120 5.3788 150 4.7829 180 4.5856 210 4.7829 240 5.3788 270 6.3358 300 7.4629 330 8.3927 360 8.7538  (c) Bounds on NP Parameter θ | z ( λ ′ 2 k 3 ⋆ λ ′ 2 k 2 ) | × 1 0 - 7 0 8.7538 30 8.3927 60 7.4629 90 6.3358 120 5.3788 150 4.7829 180 4.5856 210 4.7829 240 5.3788 270 6.3358 300 7.4629 330 8.3927 360 8.7538  (d) Bounds on NP Parameter θ | z ( λ ′ 3 k 3 ⋆ λ ′ 3 k 2 ) | × 1 0 - 7 0 8.7539 30 8.3927 60 7.4629 90 6.3357 120 5.3788 150 4.7829 180 4.5856 210 4.7829 240 5.3788 270 6.3357 300 7.4629 330 8.3927 360 8.7539Contour graphs in Figure6 illustrate the allowed region for NP parameters (z(λik1′λik3′∗) and θ) of the process B→πναν¯a for specific values of branching fraction. All four plots (comprising unpolarized (a) and (b-d)) show that the maximum magnitude of NP parameter oscillates gently with respect to its phase in general. Similarly, plots in Figure 7 represent the variation of branching fraction with respect to the magnitude of NP parameter (z(λik1′λik3′∗)) at several values of its phase θ. All four plots demonstrate the gentle oscillation behavior as observed in Figure 6 with sharply distinct curves for different values of phases of NP parameter θ. The bounds on the magnitude of NP parameter are given in Table 4.Contour graphs in Figure8 depict the allowed region for NP parameters (z(λik3′λik2′∗) and θ) of the process B→Kναν¯a for specific values of branching fraction bounded by the experimental limit, while plots in Figure 9 represent the variation of branching fraction of the process B→Kναν¯a with the magnitude of NP parameters at several values of its phase θ. All four plots (comprising unpolarized (a) and polarized (b-d)) show that the maximum magnitude of NP parameter oscillates with respect to its phase in general. The plot in Figure 8(a) shows a particular pattern below given limiting branching fraction (≤4×10-6). This particular pattern shows that only a narrow range of phase θ of NP parameter is allowed in that case. The bounds on the magnitude of NP parameter are given in Table 5.Plots in Figures10 and 11 of the process KL,S→ναν¯α display the variation of branching fraction (NP contribution only) with respect to the magnitude of NP parameter at several values of its phase θ. For pure leptonic decays of strange mesons involving neutrinos (KL,S→ναν¯α), there is no experimental data available. Therefore, we use limits derived from K→πναν¯α to calculate NP contribution to these processes. The bounds on the magnitude of NP parameter and possible NP contribution are given in Tables 6 and 7 for the decay of KL,S, respectively, which shows that Rp MSSM enhances SM contribution by order of 10 for KL,S. The interference term in this case is both constructive and destructive, but the destructive effect is not strong enough to affect the enhancement by SUSY.Table 6
Bounds on NP parameters (λijk′λlmn′,θ) derived from K→πναν¯α for Ks→ναν¯α. α(BrSM) is (a) unpolarized (7.27×10-27), (b) e (7.58×10-33), (c) μ (5.66×10-29), (d) τ (7.27×10-27).
(a) Bounds on NP Parameter Branching Fraction (×10-26) θ | Σ i = 1 3 ( λ ′ i k 2 ⋆ λ ′ i k 1 ) | × 1 0 - 9 NP Interference Combined 0 3.465 ± 3.209 2.69 ± 2.682 - 1.952 ± 1.808 1.454 ± 0.873 30 3.032 ± 2.653 1.958 ± 1.94 - 0.764 ± 0.669 1.91 ± 1.272 60 4.229 ± . . . . 2.157 ± . . . . 2.055 ± . . . . 4.928 ± . . . . 90 2.972 ± . . . . 1.065 ± . . . . 1.195 ± . . . . 2.976 ± . . . . 120 2.233 ± . . . . 0.601 ± . . . . - 0.808 ± . . . . 0.509 ± . . . . 150 1.927 ± . . . . 0.448 ± . . . . - 0.989 ± . . . . 0.175 ± . . . . 180 1.94 ± . . . . 0.454 ± . . . . 0.395 ± . . . . 1.565 ± . . . . 210 2.278 ± . . . . 0.626 ± . . . . 1.312 ± . . . . 2.654 ± . . . . 240 3.062 ± . . . . 1.131 ± . . . . - 0.079 ± . . . . 1.768 ± . . . . 270 4.357 ± . . . . 2.289 ± . . . . - 2.545 ± . . . . 0.46 ± . . . . 300 3.078 ± 2.721 2.035 ± 2.021 - 0.476 ± 0.421 2.276 ± 1.6 330 3.486 ± 3.233 2.726 ± 2.718 1.87 ± 1.734 5.313 ± 4.453 360 3.465 ± 3.209 2.69 ± 2.682 1.109 ± 1.027 4.515 ± 3.709  (b) Bounds on NP Parameter Branching Fraction (×10-32) θ | λ ′ 1 k 2 ⋆ λ ′ 1 k 1 | × 10 - 9 NP Interference Combined 0 8.026 ± 1.87 8.435 ± 3.725 - 4.716 ± 1.097 4.472 ± 2.628 30 7.083 ± 1.967 6.71 ± 3.46 - 1.846 ± 0.513 5.618 ± 2.947 60 5.671 ± 2.079 4.53 ± 2.93 2.876 ± 1.052 8.16 ± 3.982 90 4.408 ± 2.072 2.95 ± 2.27 1.834 ± 0.862 5.538 ± 3.132 120 3.618 ± 1.956 2.1 ± 1.76 - 1.368 ± 0.742 1.486 ± 1.018 150 3.267 ± 1.865 1.755 ± 1.515 - 1.741 ± 0.997 0.768 ± 0.518 180 3.282 ± 1.87 1.77 ± 1.52 0.703 ± 0.403 3.227 ± 1.923 210 3.666 ± 1.967 2.15 ± 1.79 2.198 ± 1.18 5.102 ± 2.97 240 4.498 ± 2.079 3.05 ± 2.32 - 0.131 ± 0.058 3.673 ± 2.262 270 5.793 ± 2.072 4.7 ± 2.98 - 3.524 ± 1.259 1.93 ± 1.72 300 7.189 ± 1.956 6.895 ± 3.495 - 1.145 ± 0.314 6.504 ± 3.181 330 8.068 ± 1.865 8.515 ± 3.735 4.513 ± 1.045 13.782 ± 4.78 360 8.026 ± 1.87 8.435 ± 3.725 2.66 ± 0.622 11.848 ± 4.347  (c) Bounds on NP Parameter Branching Fraction (×10-28) θ | λ ′ 2 k 2 ⋆ λ ′ 2 k 1 | × 10 - 9 NP Interference Combined 0 8.026 ± 1.87 6.29 ± 2.78 - 3.497 ± 0.815 3.35 ± 1.964 30 7.083 ± 1.967 5.004 ± 2.58 - 1.374 ± 0.381 4.187 ± 2.199 60 5.671 ± 2.079 3.379 ± 2.184 2.132 ± 0.782 6.068 ± 2.965 90 4.409 ± 2.072 2.198 ± 1.692 1.366 ± 0.642 4.12 ± 2.334 120 3.618 ± 1.956 1.566 ± 1.311 - 1.014 ± 0.548 1.11 ± 0.762 150 3.267 ± 1.865 1.31 ± 1.128 - 1.295 ± 0.739 0.572 ± 0.389 180 3.282 ± 1.87 1.322 ± 1.136 0.518 ± 0.296 2.396 ± 1.432 210 3.666 ± 1.967 1.603 ± 1.336 1.632 ± 0.875 3.792 ± 2.211 240 4.499 ± 2.079 2.274 ± 1.732 - 0.093 ± 0.043 2.738 ± 1.689 270 5.793 ± 2.072 3.506 ± 2.224 - 2.616 ± 0.936 1.448 ± 1.289 300 7.189 ± 1.956 5.141 ± 2.605 - 0.853 ± 0.232 4.845 ± 2.373 330 8.068 ± 1.865 6.35 ± 2.787 3.346 ± 0.773 10.254 ± 3.561 360 8.026 ± 1.87 6.29 ± 2.78 1.979 ± 0.461 8.826±3.241  (d) Bounds on NP Parameter Branching Fraction (×10-26) θ λ ′ 3 k 2 ⋆ λ ′ 3 k 1 × 10 - 9 NP Interference Combined 0 8.009 ± 1.87 8.097 ± 3.584 - 4.48 ± 1.046 4.328 ± 2.539 30 7.071 ± 1.966 6.448 ± 3.328 - 1.768 ± 0.491 5.39 ± 2.836 60 5.671 ± 2.076 4.365 ± 2.818 2.734 ± 1 . 7.81 ± 3.818 90 4.418 ± 2.069 2.849 ± 2.188 1.761 ± 0.825 5.32 ± 3.012 120 3.631 ± 1.954 2.035 ± 1.698 - 1.304 ± 0.701 1.442 ± 0.997 150 3.282 ± 1.865 1.706 ± 1.466 - 1.672 ± 0.951 0.744 ± 0.515 180 3.298 ± 1.87 1.72 ± 1.477 0.666 ± 0.378 3.097 ± 1.854 210 3.684 ± 1.966 2.087 ± 1.734 2.106 ± 1.124 4.904 ± 2.858 240 4.515 ± 2.076 2.955 ± 2.244 - 0.115 ± 0.053 3.552 ± 2.19 270 5.803 ± 2.069 4.542 ± 2.874 - 3.364 ± 1.2 1.889 ± 1.674 300 7.186 ± 1.954 6.638 ± 3.362 - 1.102 ± 0.299 6.247 ± 3.063 330 8.054 ± 1.865 8.181 ± 3.596 4.288 ± 0.992 13.18 ± 4.589 360 8.009 ± 1.87 8.097 ± 3.584 2.543 ± 0.594 11.35 ± 4.178Table 7
Bounds on NP parameters(λijk′λlmn′,θ) derived from K→πναν¯α for KL→ναν¯α. α(BrSM) is (a) unpolarized (4.29×10-27), (b) e (4.48×10-33), (c) μ(3.34×10-29), (d) τ(4.26×10-27).
(a) Bounds on NP Parameter Branching Fraction (×10-26) θ | Σ i = 1 3 ( λ ′ i k 2 ⋆ λ ′ i k 1 ) | × 1 0 - 9 NP Interference Combined 0 3.465 ± 3.209 1.588 ± 1.583 - 1.153 ± 1.067 0.858 ± 0.516 30 3.032 ± 2.653 1.156 ± 1.146 - 0.451 ± 0.395 1.128 ± 0.75 60 4.229 ± . . . . 1.274 ± . . . . 1.213 ± . . . . 2.91 ± . . . . 90 2.972 ± . . . . 0.629 ± . . . . 0.705 ± . . . . 1.757 ± . . . . 120 2.233 ± . . . . 0.355 ± . . . . - 0.477 ± . . . . 0.301 ± . . . . 150 1.927 ± . . . . 0.264 ± . . . . - 0.584 ± . . . . 0.103 ± . . . . 180 1.94 ± . . . . 0.268 ± . . . . 0.233 ± . . . . 0.924 ± . . . . 210 2.278 ± . . . . 0.369 ± . . . . 0.775 ± . . . . 1.567 ± . . . . 240 3.062 ± . . . . 0.667 ± . . . . - 0.046 ± . . . . 1.044 ± . . . . 270 4.357 ± . . . . 1.352 ± . . . . - 1.503 ± . . . . 0.272 ± . . . . 300 3.078 ± 2.721 1.202 ± 1.193 - 0.281 ± 0.248 1.344 ± 0.945 330 3.486 ± 3.233 1.61 ± 1.605 1.104 ± 1.024 3.136 ± 2.63 360 3.465 ± 3.209 1.588 ± 1.583 0.655 ± 0.606 2.666 ± 2.19  (b) Bounds on NP Parameter Branching Fraction (×10-32) θ | λ ′ 1 k 2 ⋆ λ ′ 1 k 1 | × 10 - 9 NP Interference Combined 0 8.026 ± 1.87 4.98 ± 2.2 - 2.784 ± 0.648 2.641 ± 1.552 30 7.083 ± 1.967 3.96 ± 2.04 - 1.088 ± 0.3 3.317 ± 1.74 60 5.671 ± 2.079 2.675 ± 1.725 1.698 ± 0.626 4.818 ± 2.351 90 4.408 ± 2.072 1.74 ± 1.34 1.085 ± 0.51 3.27 ± 1.85 120 3.618 ± 1.956 1.24 ± 1.04 - 0.808 ± 0.439 0.877 ± 0.601 150 3.267 ± 1.865 1.035 ± 0.895 - 1.026 ± 0.589 0.454 ± 0.306 180 3.282 ± 1.87 1.05 ± 0.9 0.41 ± 0.235 1.906 ± 1.136 210 3.666 ± 1.967 1.27 ± 1.06 1.297 ± 0.694 3.012 ± 1.754 240 4.498 ± 2.079 1.8 ± 1.37 - 0.076 ± 0.034 2.169 ± 1.336 270 5.793 ± 2.072 2.775 ± 1.765 - 2.081 ± 0.749 1.139 ± 1.016 300 7.189 ± 1.956 4.07 ± 2.06 - 0.674 ± 0.181 3.84 ± 1.879 330 8.068 ± 1.865 5.025 ± 2.205 2.668 ± 0.617 8.138 ± 2.822 360 8.026 ± 1.87 4.98 ± 2.2 1.57 ± 0.366 6.995 ± 2.566  (c) Bounds on NP Parameter Branching Fraction (×10-28) θ λ ′ 2 k 2 ⋆ λ ′ 2 k 1 × 10 - 9 NP Interference Combined 0 8.026 ± 1.87 3.713 ± 1.641 - 2.064 ± 0.481 1.978 ± 1.16 30 7.083 ± 1.967 2.954 ± 1.523 - 0.811 ± 0.225 2.472 ± 1.298 60 5.671 ± 2.079 1.995 ± 1.289 1.259 ± 0.461 3.582 ± 1.751 90 4.409 ± 2.072 1.298 ± 1 . 0.806 ± 0.378 2.433 ± 1.378 120 3.618 ± 1.956 0.925 ± 0.774 - 0.599 ± 0.324 0.655 ± 0.45 150 3.267 ± 1.865 0.774 ± 0.666 - 0.764 ± 0.436 0.338 ± 0.23 180 3.282 ± 1.87 0.78 ± 0.671 0.306 ± 0.175 1.414 ± 0.846 210 3.666 ± 1.967 0.946 ± 0.789 0.963 ± 0.517 2.238 ± 1.305 240 4.499 ± 2.079 1.342 ± 1.023 - 0.055 ± 0.025 1.616 ± 0.997 270 5.793 ± 2.072 2.07 ± 1.313 - 1.544 ± 0.552 0.854 ± 0.76 300 7.189 ± 1.956 3.035 ± 1.538 - 0.503 ± 0.136 2.86 ± 1.402 330 8.068 ± 1.865 3.75 ± 1.646 1.976 ± 0.457 6.054 ± 2.102 360 8.026 ± 1.87 3.713 ± 1.641 1.169 ± 0.273 5.211±1.914  (d) Bounds on NP Parameter Branching Fraction (×10-26) θ | λ ′ 3 k 2 ⋆ λ ′ 3 k 1 | × 1 0 - 9 NP Interference Combined 0 8.009 ± 1.87 4.78 ± 2.116 - 2.646 ± 0.617 2.555 ± 1.499 30 7.071 ± 1.966 3.806 ± 1.964 - 1.044 ± 0.29 3.182 ± 1.674 60 5.671 ± 2.076 2.577 ± 1.663 1.614 ± 0.591 4.611 ± 2.254 90 4.418 ± 2.069 1.682 ± 1.291 1.04 ± 0.487 3.141 ± 1.778 120 3.631 ± 1.954 1.202 ± 1.003 - 0.77 ± 0.414 0.851 ± 0.588 150 3.282 ± 1.865 1.007 ± 0.865 - 0.988 ± 0.561 0.439 ± 0.304 180 3.298 ± 1.87 1.016 ± 0.871 0.393 ± 0.223 1.828 ± 1.094 210 3.684 ± 1.966 1.233 ± 1.024 1.243 ± 0.664 2.896 ± 1.688 240 4.515 ± 2.076 1.746 ± 1.325 - 0.069 ± 0.031 2.096 ± 1.294 270 5.803 ± 2.069 2.682 ± 1.696 - 1.986 ± 0.708 1.116 ± 0.988 300 7.186 ± 1.954 3.919 ± 1.985 - 0.651 ± 0.177 3.688 ± 1.808 330 8.054 ± 1.865 4.83 ± 2.123 2.531 ± 0.586 7.781 ± 2.709 360 8.009 ± 1.87 4.78 ± 2.116 1.501 ± 0.351 6.701 ± 2.467Plots in Figures12 and 13 describe the variation of branching fraction (NP contribution only) of the process Bs,d→ναν¯α with the magnitude of NP parameters, (z(λik1′λik3′∗) at several values of θ. For pure leptonic decays of beauty involving neutrinos (Bs,d→ναν¯α), there is no experimental data available for these processes, and we use limits derived from B→(π,K)ναν¯α to calculate NP contributions to these processes. The bounds on the magnitude of NP parameter and possible NP contribution are given in Tables 8 and 9, respectively for the decay of Bs,d, which shows that Rp MSSM enhances SM contribution by order of magnitude 10 for Bs and 104 for Bd and also the effect of destructive interference is not strong to affect SUSY enhancement.Table 8
Bounds on NP parameters(λijk′λlmn′,θ) derived from B→πναν¯α for Bd→ναν¯α. α(BrSM) is (a) unpolarized (6.35×10-25), (b) e (6.53×10-31), (c) μ (4.87×10-27), (d) τ (6.3×10-25)
(a) Bounds on NP Parameter Branching Fraction (×10-22) θ Σ i = 1 3 λ ′ i k 3 ⋆ λ ′ i k 1 × 10 - 6 NP Interference Combined 0 1.0164 3.325 0.269 3.6 30 1.0298 3.413 0.151 3.57 60 1.0506 3.552 - 0.23 3.328 90 1.0735 3.709 - 0.23 3.485 120 1.0923 3.84 0.168 4.014 150 1.1016 3.905 0.289 4.2 180 1.0987 3.885 - 0.08 3.811 210 1.0844 3.784 - 0.308 3.482 240 1.0629 3.636 - 0.016 3.626 270 1.0402 3.482 0.291 3.779 300 1.0223 3.363 0.104 3.473 330 1.0137 3.307 - 0.252 3.061 360 1.0164 3.325 - 0.181 3.15  (b) Bounds on NP Parameter Branching Fraction (×10-27) θ | λ ′ 1 k 3 ⋆ λ ′ 1 k 1 | × 10 - 6 NP Interference Combined 0 1.7907 1.06 0.051 1.112 30 1.8043 1.08 0.025 1.106 60 1.8252 1.1 - 0.038 1.063 90 1.8481 1.13 - 0.04 1.091 120 1.8667 1.15 0.034 1.185 150 1.8759 1.17 0.046 1.217 180 1.873 1.16 - 0.012 1.149 210 1.8589 1.14 - 0.05 1.091 240 1.8376 1.12 - 0.005 1.116 270 1.8148 1.09 0.053 1.144 300 1.7967 1.07 0.018 1.089 330 1.7879 1.06 - 0.047 1.014 360 1.7907 1.06 - 0.031 1.03  (c) Bounds on NP Parameter Branching Fraction (×10-23) θ | λ ′ 2 k 3 ⋆ λ ′ 2 k 1 | × 1 0 - 6 NP Interference Combined 0 1.7907 0.792 0.037 0.829 30 1.8043 0.804 0.021 0.825 60 1.8252 0.823 - 0.03 0.793 90 1.8481 0.844 - 0.03 0.814 120 1.8667 0.861 0.022 0.883 150 1.8759 0.87 0.038 0.908 180 1.873 0.867 - 0.01 0.857 210 1.8589 0.854 - 0.04 0.814 240 1.8376 0.834 - 0.001 0.833 270 1.8148 0.814 0.039 0.853 300 1.7967 0.798 0.014 0.812 330 1.7879 0.79 - 0.034 0.756 360 1.7907 0.792 - 0.024 0.768  (d) Bounds on NP Parameter Branching Fraction (×10-21) θ | λ ′ 3 k 3 ⋆ λ ′ 3 k 1 | × 10 - 6 NP Interference Combined 0 1.7907 1.024 0.047 1.072 30 1.8043 1.04 0.026 1.067 60 1.8252 1.064 - 0.04 1.025 90 1.8481 1.091 - 0.04 1.052 120 1.8667 1.113 0.028 1.142 150 1.8759 1.124 0.048 1.173 180 1.873 1.12 - 0.014 1.107 210 1.8589 1.103 - 0.052 1.052 240 1.8376 1.078 - 0.003 1.076 270 1.8148 1.052 0.05 1.103 300 1.7967 1.031 0.018 1.05 330 1.7879 1.021 - 0.045 0.977 360 1.7907 1.024 - 0.032 0.993Table 9
Bounds on NP parameters(λijk′λlmn′,θ) derived from B→Kναν¯α for Bs→ναν¯α. α(BrSM) is (a) unpolarized (1.41×10-23), (b) e (1.45×10-29), (c) μ(1.08×10-25), (d) τ(1.4×10-23).
(a) Bounds on NP Parameter Branching Fraction (×10-22) θ |Σi=13(λ′ik3⋆λ′ik2)|×10-7 NP Interference Combined 0 5.9349 1.145 - 0.804 0.482 30 5.512 0.988 - 0.115 1.014 60 4.4436 0.642 0.574 1.357 90 3.238 0.341 0.197 0.679 120 2.3595 0.181 - 0.26 0.062 150 1.9022 0.118 - 0.18 0.079 180 1.7667 0.101 0.144 0.386 210 1.9022 0.118 0.228 0.487 240 2.3595 0.181 - 0.104 0.218 270 3.238 0.341 - 0.432 0.05 300 4.4436 0.642 0.014 0.797 330 5.512 0.988 0.741 1.87 360 5.9349 1.145 0.229 1.515  (b) Bounds on NP Parameter Branching Fraction (×10-28) θ |λ′1k3⋆λ′1k2|×10-7 NP Interference Combined 0 8.7538 2.56 - 1.216 1.489 30 8.3927 2.36 - 0.183 2.322 60 7.4629 1.86 0.996 3.001 90 6.3358 1.34 0.4 1.885 120 5.3788 0.97 - 0.612 0.503 150 4.7829 0.77 - 0.471 0.444 180 4.5856 0.7 0.387 1.232 210 4.7829 0.77 0.586 1.501 240 5.3788 0.97 - 0.246 0.869 270 6.3358 1.34 - 0.866 0.619 300 7.4629 1.86 0.028 2.033 330 8.3927 2.36 1.159 3.664 360 8.7538 2.56 0.352 3.057  (c) Bounds on NP Parameter Branching Fraction (×10-24) θ |λ′2k3⋆λ′2k2|×10-7 NP Interference Combined 0 8.7538 1.913 - 0.91 1.111 30 8.3927 1.758 - 0.134 1.732 60 7.4629 1.39 0.74 2.238 90 6.3358 1.002 0.296 1.406 120 5.3788 0.722 - 0.455 0.375 150 4.7829 0.571 - 0.347 0.332 180 4.5856 0.525 0.286 0.919 210 4.7829 0.571 0.44 1.119 240 5.3788 0.722 - 0.182 0.648 270 6.3358 1.002 - 0.648 0.462 300 7.4629 1.39 0.018 1.516 330 8.3927 1.758 0.867 2.733 360 8.7538 1.913 0.259 2.28  (d) Bounds on NP Parameter Branching Fraction (×10-22) θ |λ′3k3⋆λ′3k2|×10-7 NP Interference Combined 0 8.7539 2.472 - 1.177 1.435 30 8.3927 2.273 - 0.174 2.239 60 7.4629 1.797 0.956 2.893 90 6.3357 1.295 0.382 1.817 120 5.3788 0.933 - 0.588 0.485 150 4.7829 0.738 - 0.45 0.428 180 4.5856 0.678 0.37 1.188 210 4.7829 0.738 0.569 1.447 240 5.3788 0.933 - 0.235 0.838 270 6.3357 1.295 - 0.838 0.597 300 7.4629 1.797 0.022 1.959 330 8.3927 2.273 1.119 3.532 360 8.7539 2.472 0.335 2.947


## 5. Summary and Conclusion

Summarizing, we have carried out an analysis of semileptonic (M→M′νaν¯α;M=K, B;M′=π,K;α=e,μ,τ) and pure leptonic (M→ναν¯α) decays of pseudoscalar mesons within the framework of Rp MSSM. The analysis involves a detailed comparison of experimental results with respect to the theoretical prediction of the branching fraction of given processes. The comparison (listed in Tables 2–9) quantifies the effect of contribution from Rp MSSM to the branching fraction of processes under discussion. The analysis performed (listed in Tables 3, 6–9) also enables to compare the branching fraction of processes having identical FCNC. The plots in Figures 2–13 show the variations of branching fraction with respect to NP parameters (z(λαkβ′λαkγ′∗) and θ). In general, SM contribution is dominated by Rp MSSM contribution, i.e., by a factor of 10 for the pure leptonic decays of KL,S and by 102 and 104 in case of Bs and Bd respectively. The interference term between Rp MSSM and SM is both constructive and destructive but except for the semileptonic decays of K0, it does not affect the enhancement due to SUSY. This makes Rp MSSM a viable model for the comparison of NP contribution in rare decays at labs like Super B factories, KOTO (J-PARC) and NA62 at CERN.



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