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

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 ([[??].sub.p]) 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 [[??].sub.p] 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 [R.sub.p] = [(-1).sup.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]

[mathematical expression not reproducible], (1)

where i, j, k are generation indices, [L.sub.i] and [Q.sub.i] are the lepton and quark left-handed [SU(2).sub.L] doublets, and [E.sup.c], [D.sup.c] are the charge conjugates of the right-handed leptons and quark singlets, respectively. Here [[lambda].sub.ijk], [[lambda]'.sub.ijk], and [[lambda]".sub.ijk] are the Yukawa couplings. The term proportional to [[lambda].sub.ijk] is antisymmetric in first two indices [i, j] and [[lambda]".sub.ijk] is antisymmetric in last two indices [j, k], implying 9([[lambda].sub.ijk]) + 27([[lambda]'.sub.ijk]) + 9([[lambda]".sub.ijk]) = 45 independent coupling constants among which 36 are related to the lepton flavor violation (9 from LL[E.sup.c] and 27 from LQ[D.sup.c]). 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., [M.sup.0] [right arrow] [[nu].sub.[alpha]][[bar.[nu]].sub.[beta]], [M.sup.[+ or -],0] [right arrow] [N.sup.[+ or -],0][[nu].sub.[alpha]][[bar.[nu]].sub.[beta]], and [M.sup.[+ or -],0] [right arrow] [N.sup.[+ or -],0][[nu].sub.[alpha]][[bar.[nu]].sub.[beta]], where M = K, B and N = [pi], K. At the quark level, all [M.sup.[+ or -],0] [right arrow] [N.sup.[+ or -],0] [[nu].sub.[alpha]][[bar.[nu]].sub.[beta]] decays are represented by s(b) [right arrow] d(d, s)q[[nu].sub.[alpha]][[nu].sub.[beta]] (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 ([alpha] = [beta]),

(2) lepton flavor violating ([alpha] [not equal to] [beta]) decays.

The first type of decays s(b) [right arrow] d(d, s)[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] ([alpha] = e, [mu], [tau]) 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].

The second type of decays s(b) [right arrow] d(d, s)[[nu].sub.[alpha]][[bar.[nu]].sub.[beta]] ([alpha] [not equal to] [beta]; [alpha], [beta] = e, [mu], [tau]) 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 in [[??].sub.p] 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 [right arrow] d[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]

In the SM, the effective Hamiltonian for the semileptonic (K [right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]], K [right arrow] [[pi].sup.0][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]) and pure leptonic [K.sub.L,S] [right arrow] [[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] processes is given by [62, 63]

[H.sub.eff] = [summation over (l)] [C.sub.SM] [([bar.s]d).sub.V-A] [([bar.[nu]]l[nu]l).sub.V-A]. (2)

In this case, all leptons couple universally with the electroweak gauge bosons, where

[C.sub.SM] = [G.sub.F][alpha]/[2[square root of (2)] [sin.sup.2][[theta].sub.w]] ([V.sup.*.sub.cs][V.sub.cd][X.sup.l.sub.NL] + [V.sup.*.sub.ts][V.sub.td]X([x.sub.t])), X([x.sub.t]) = [X.sub.0]([x.sub.t]) + [[alpha].sub.S]/4[pi] [X.sub.l] ([x.sub.t]), (3)

where [X.sub.0]([x.sub.t]) = ([x.sub.t]/8)(([x.sub.t] + 2)/([x.sub.t] - 1) + ((3[x.sub.t] - 6)/[([x.sub.t] - 1).sup.2]) log [x.sub.t]) and

[mathematical expression not reproducible], (4)

and [x.sub.t] = [[bar.m].sup.2.sub.t]([[mu].sub.t])/[M.sup.2.sub.W], [[mu].sub.t] = O([m.sub.t]), [C.sub.NL] and [B.sup.(1/2).sub.NL] are [Z.sup.0] penguin and box diagrams, respectively. [V.sup.*.sub.cs][V.sub.cd] are CKM matrix elements and [[alpha].sub.S] is the coupling strength of strong interactions.

In MSSM, the relevant effective Lagrangian for the decay process K [right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] is given by [57-61]

[mathematical expression not reproducible], (5)

where [alpha] = e, [mu]. The first term in (2) comes from the down squark exchange (where d and s are down type quarks). The dimensionless coupling constant [A.sup.sd.sub.[alpha][alpha]] is related to [[??].sub.p] Yukawa couplings [[lambda]'.sub.[alpha]k1][[lambda]'.sup.*.sub.[alpha]k2] by

[mathematical expression not reproducible]. (6)

The differential decay rate d[GAMMA]/d[q.sup.2] for the semileptonic decay processes is given by [62-64]

d[GAMMA]/d[q.sup.2] = [1/[2.sup.5][[pi].sup.5]] [[lambda].sup.3/2] (1, [r.sub.M], s) [m.sup.3.sub.K] [[absolute value of ([f.sup.+.sub.p]([q.sup.2]))].sup.2] [[absolute value of ([C.sub.l])].sup.2], (7)

where [lambda](1, [r.sub.M], s) = 1 + [r.sup.2.sub.M] + [s.sup.2] - 2[r.sub.M] - 2s - 2[r.sup.M]s with [r.sub.M] = [([m.sub.M]/[m.sub.B]).sup.2] and s = [q.sup.2]/[m.sup.2.sub.B], [C.sub.l] = [C.sub.SM] + z(1/8)[e.sup.i[theta]]; z is the general NP parameter. We have used the value for the form factor [f.sup.+.sub.p]([q.sup.2]) for the above decay processes of [K.sup.+] and [K.sup.0] as given in [65]. Since this work focuses on [[??].sub.p] MSSM, we will shift our focus to z (for the calculation of limits on couplings) and Yukawa couplings [mathematical expression not reproducible] (for the predictions of branching fraction). The decay rate for pure leptonic decay processes is given by

[GAMMA] (s [right arrow] d[nu]l[bar.[nu]]l) = [1/8[pi]] [m.sup.3.sub.K] [[absolute value of ([f.sub.p]([q.sup.2]))].sup.2] [[absolute value of (2[m.sub.l]/[m.sub.K] [C.sub.l])].sup.2]. (8)

The form factor [f.sub.p]([q.sup.2]) is given by [66]. [m.sub.K] represents the mass of strange meson and [m.sub.l] is the mass of lepton, where [C.sub.l] is same as that of semileptonic decays.

3. b [right arrow] d(s)[[nu].sub.[alpha]][[nu].sub.[alpha]]

In MSSM, the relevant effective Lagrangian for the decay process B [right arrow] [pi](K)[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] is given by [57-61]

[mathematical expression not reproducible], (9)

where [alpha] = e ,[mu]. The first term in (2) comes from the down squark exchange (where b and d(s) are down type quarks). The dimensionless coupling constant [A.sup.bd(s).sub.[alpha][alpha]] is given by

[mathematical expression not reproducible]. (10)

The differential decay rate for semileptonic decay processes is given by [62-64]

d[GAMMA]/d[q.sup.2] = [1/[2.sup.5][[pi].sup.5]] [[lambda].sup.3/2] (1, r, s) [m.sup.3.sub.B] [[absolute value of ([f.sup.+.sub.p] ([q.sup.2]))].sup.2] [[absolute value of ([C.sub.l])].sup.2], (11)

where

[C.sub.l] = [C.sub.SM] + [z/8] [e.sup.i[theta]], (12)

with

[C.sub.SM] = [G.sub.F][alpha]/2[square root of 2] [sin.sup.2][[theta].sub.w] ([V.sup.*.sub.cb][V.sub.cd(s)][X.sup.l.sub.NL] + [V.sup.*.sub.tb][V.sub.td(s)] X([x.sub.t])), (13)

with

X([x.sub.t]) = [X.sub.0]([x.sub.t]) + [[[alpha].sub.S]/4[pi]] [X.sub.l]([x.sub.t]), [x.sub.t] = [[bar.m].sup.2.sub.t]([[mu].sub.t])/[M.sup.2.sub.W], [[mu].sub.t] = O([m.sub.t]), (14)

and z as explained in the above section is the general NP parameter and [mathematical expression not reproducible]. We have used the form factor [f.sup.+.sub.p]([q.sup.2]) for the above decay processes of B- > [pi](K) as given in [67]. The decay rate for pure leptonic decay processes is given by [46, 50-52]

[GAMMA] (b [right arrow] d(s) [[nu].sub.l][[bar.[nu]].sub.l]) = [1/8[pi]] [m.sup.3.sub.B] [[absolute value of ([f.sup.p]([q.sup.2]))].sup.2] [[absolute value of (2[m.sub.l]/[m.sub.B] [C.sub.l])].sup.2]. (15)

The form factor [f.sub.p]([q.sup.2]) is given by [66], [m.sub.B] represents the mass of beauty meson and [m.sub.l] 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 [right arrow] X[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]; M [right arrow] [[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]), where M = K, B; [alpha] = e , [mu], [tau] and X = [pi], 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 [[??].sub.p] Yukawa couplings [[lambda]'.sub.[alpha]k[beta]][[lambda]'.sup.*.sub.[alpha]k[gamma]] 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 [C.sub.SM] 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., [absolute value of (z([[lambda]'.sub.[rho][omega][sigma]] [[lambda]'.sup.*.sub.[alpha][beta][gamma]]))]) 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 [+ or -]1[sigma] 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 ([[lambda]'.sub.[rho][omega][sigma]] [[lambda]'.sup.*.sub.[alpha][beta][gamma]]) involved are normalized to the square of [mathematical expression not reproducible]/100 GeV in all these tables and figures.

First, we will discuss the results related to semileptonic decay processes followed by pure leptonic decays. We have plotted graphs in Figures 2 and 3 for the study of process K [right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.a]. These plots relate the branching fraction of the said process with the magnitude and phase of NP parameters ([absolute value of (z([[lambda]'.sub.ik1][[lambda]'.sup.*.sub.ik2]))] and [theta]). 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 -1[sigma] level of measured branching fraction (0.7 x [10.sup.-7]). It clearly shows that only a narrow range of phase of NP parameter ([absolute value of ([theta])] [less than or equal to] [pi]/4) is allowed for given -1[sigma] level. The bounds on the magnitude of NP parameter are given in Table 2. Here the entry "[+ or -]......." means that these specific NP parameters cannot be defined for -1[sigma] as the corresponding [[??].sub.p] MSSM contribution exceeds the experimental limit on branching fraction in -1[sigma] 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 [right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]], [K.sup.0] [right arrow] [[pi].sup.0][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]], [K.sub.L,S] [right arrow] [[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]), the maximum limits for K [right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] 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 [theta].

Contour plots in Figure 4 represent the allowed region for NP parameters ([absolute value of (z([[lambda]'.sub.ik1][[lambda]'.sup.*.sub.ik2]))] and [theta]) for specific values of branching fraction of the process [K.sup.0] [right arrow] [[pi].sup.0][[nu].sub.[alpha]][[bar.[nu]].sub.a]. All four plots (comprising unpolarized (a) and polarized (b-d)) show that the maximum magnitude of NP parameter follows a catenary (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 [[??].sub.p] MSSM dominates over SM contribution by order of magnitude ~[10.sup.2], 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 [K.sup.0] [right arrow] [[pi].sup.0] [[nu].sub.[alpha]][[bar.[nu]].sub.a] with respect to the magnitude of NP parameter at several values of its phase [theta].

Contour graphs in Figure 6 illustrate the allowed region for NP parameters ([absolute value of (z([[lambda]'.sub.ik1][[lambda]'.sup.*.sub.ik3]))] and [theta]) of the process B [right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.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 ([absolute value of (z([[lambda]'.sub.ik1][[lambda]'.sup.*.sub.ik3]))]) at several values of its phase [theta]. 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 [theta]. The bounds on the magnitude of NP parameter are given in Table 4.

Contour graphs in Figure 8 depict the allowed region for NP parameters ([absolute value of (z([[lambda]'.sub.ik3][[lambda]'.sup.*.sub.ik2]))] and [theta]) of the process B [right arrow] K[[nu].sub.[alpha]][[bar.[nu]].sub.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 [right arrow] K[[nu].sub.[alpha]][[bar.[nu]].sub.a] with the magnitude of NP parameters at several values of its phase [theta]. 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 ([less than or equal to] 4 x [10.sup.-6]). This particular pattern shows that only a narrow range of phase [theta] of NP parameter is allowed in that case. The bounds on the magnitude of NP parameter are given in Table 5.

Plots in Figures 10 and 11 of the process [K.sub.L,S] [right arrow] [[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] display the variation of branching fraction (NP contribution only) with respect to the magnitude of NP parameter at several values of its phase [theta]. For pure leptonic decays of strange mesons involving neutrinos ([K.sub.L,S] [right arrow] [[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]), there is no experimental data available. Therefore, we use limits derived from K [right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] 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 [K.sub.L,S], respectively, which shows that [[??].sub.p] MSSM enhances SM contribution by order of 10 for [K.sub.L,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.

Plots in Figures 12 and 13 describe the variation of branching fraction (NP contribution only) of the process [B.sub.s,d] [right arrow] [[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] with the magnitude of NP parameters, ([absolute value of (z([[lambda]'.sub.ik1][[lambda]'.sup.*.sub.ik3))] at several values of [theta]. For pure leptonic decays of beauty involving neutrinos ([B.sub.s,d] [right arrow] [[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]), there is no experimental data available for these processes, and we use limits derived from B [right arrow] ([pi],K)[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] 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 [B.sub.s,d], which shows that [[??].sub.p] MSSM enhances SM contribution by order of magnitude 10 for [B.sub.s] and [10.sup.4] for [B.sub.d] and also the effect of destructive interference is not strong to affect SUSY enhancement.

5. Summary and Conclusion

Summarizing, we have carried out an analysis of semileptonic (M [right arrow] M'[[nu].sub.a][[bar.[nu]].sub.[alpha]]; M = K, B; M' = [pi], K; [alpha] = e , [mu], [tau]) and pure leptonic (M [right arrow][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]) decays of pseudoscalar mesons within the framework of [[??].sub.p] 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 [[??].sub.p] 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 ([absolute value of (z([[lambda]'.sub.[alpha]k[beta]] [[lambda]'.sup.*.sub.[alpha]k[gamma]]))] and [theta]). In general, SM contribution is dominated by [[??].sub.p] MSSM contribution, i.e., by a factor of 10 for the pure leptonic decays of [K.sub.L,S] and by [10.sup.2] and [10.sup.4] in case of [B.sub.s] and [B.sub.d] respectively. The interference term between [[??].sub.p] MSSM and SM is both constructive and destructive but except for the semileptonic decays of [K.sub.0], it does not affect the enhancement due to SUSY. This makes [[??].sub.p] 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.

https://doi.org/10.1155/2018/4176840

Conflicts of Interest

The authors declare that there are no conflicts of interest.

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Azeem Mir, (1) Farida Tahir, (1) Shakeel Mahmood (iD), (2) and Shi-Hai Dong (iD) (3)

(1) Physics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan

(2) Air University, PAF Complex, Service Road, E-9, Islamabad, Pakistan

(3) CIDETEC, Instituto Politecnico Nacional, Unidad Profesional, Adolfo Lopez Mateos, CDMX 07700, Mexico

Correspondence should be addressed to Shakeel Mahmood; shakeel mahmood@hotmail.com

Received 1 December 2017; Revised 15 March 2018; Accepted 11 April 2018; Published 1 August 2018

Academic Editor: Adrian Buzatu

Caption: Figure 1: Feynman diagrams of (a) b [right arrow] s[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]], (b) b [right arrow] d[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]], (c) s [right arrow] d[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] = 1,2,3.

Caption: Figure 2: Allowed region of general NP parameters (z, [theta]) for K [right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] at several values of branching fraction. [alpha] is (a) unpolarized, (b) e, (c) [mu], (d) [tau]. The three contours belong to branching fraction at [0.7, 1.7, 2.7] x [10.sup.-10].

Caption: Figure 3: Variations of branching fraction w.r.t NP parameter [absolute value of (z([[lambda]'.sub.ijk][[lambda]'.sub.lmn]))] at several values of [theta] for K [right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] is (a) unpolarized, (b) e, (c) [mu], (d) [tau]. The three bounds belong to branching fraction at [0.7 (-1[sigma]),1.7 (Mean), 2.7 (+1[sigma])] x [10.sup.-10] corresponding to an experimental measurement of (1.7 [+ or -] 0.7) x [10.sup.-10].

Caption: Figure 4: Allowed regions of general NP parameters (z, [theta]) for [K.sup.0] [right arrow] [[pi].sup.0][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] at specific values of branching fraction. [alpha] is (a) unpolarized, (b) e, (c) [mu], (d) [tau]. The three contours belong to branching fraction at [0.3, 1.3, 2.6] x [10.sup.-8].

Caption: Figure 5: Variations of branching fraction (NP contribution only) with respect to NP parameter [absolute value of ([[lambda]'.sub.ijk][[lambda]'.sub.lmn])] at several values of [theta] for [K.sup.0] [right arrow] [[pi].sup.0][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] is (a) unpolarized, (b) e, (c) [mu], (d) [tau].

Caption: Figure 6: Allowed regions of general NP parameters (z, [theta]) for B [right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] at specific values of branching fraction. [alpha] is (a) unpolarized, (b) e, (c) [mu], (d) [tau]. The three contours belong to branching fraction at [1.4-9.8] x [10.sup.-6].

Caption: Figure 7: Variation of branching fraction with respect to NP parameter [absolute value of (z([[lambda]'.sup.ijk][[lambda]'.sup.lmn]))] at several values of [theta] for B [right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] is (a) unpolarized, (b) e, (c) [mu], (d) [tau]. Experimental bound on the process is 9.8 x [10.sup.-5].

Caption: Figure 8: Allowed regions of general NP parameters (z, [theta]) for B [right arrow] K[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]] at specific values of branching fraction. [alpha] is (a) unpolarized, (b) e, (c) [mu], (d) [tau]. The three contours belong to branching fraction at [0.4-1.6] x [10.sup.-5].

Caption: Figure 9: Variations of branching fraction with respect to NP parameter [absolute value of (z([[lambda]'.sub.ijk] [[lambda]'.sub.lmn]))] at several values of [theta] for B [right arrow] K[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] is (a) unpolarized, (b) e, (c) [mu], (d) [tau]. Experimental bound on the process is 1.6 x [10.sup.-5].

Caption: Figure 10: Variations of branching fraction (NP contribution only) with respect to NP parameter [absolute value of ([[lambda]'.sub.ijk][[lambda]'.sub.lmn])] at several values of [theta] for [K.sub.S] [right arrow] [[nu].sub.[alpha]] [[bar.[nu]].sub.[alpha]]. [alpha] is (a) unpolarized, (b) e, (c) [mu], (d) [tau].

Caption: Figure 11: Variations of branching fraction (NP contribution only) with respect to NP parameter [absolute value of ([[lambda]'.sub.ijk][[lambda]'.sub.lmn])] at several values of [theta] for [K.sub.L] [right arrow] [[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] is (a) unpolarized, (b) e, (c) [mu], (d) [tau].

Caption: Figure 12: Variations of branching fraction (NP contribution only) with respect to NP parameter [absolute value of ([[lambda]'.sub.ijk][[lambda]'.sub.lmn])] at several values of [theta] for [B.sub.d] [right arrow] [[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] is (a) unpolarized, (b) e, (c) [mu], (d) [tau].

Caption: Figure 13: Variations of branching fraction (NP contribution only) with respect to NP parameter [absolute value of ([[lambda]'.sub.ijk][[lambda]'.sub.lmn])] at several values of [theta] for [B.sub.s] [right arrow] [[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] is (a) unpolarized, (b) e, (c) [mu], (d) [tau].
Table 1: Table listing the properties of processes under discussion
[68].  Here [absolute value of (z)] is the strength of the NP
parameter.

                          Experimental
Process                   Measurement              SM Prediction

B [right arrow]       < 9.8 x [10.sup.-5]        1.62 x [10.sup.-7]
[pi][nu][bar.[nu]]

[B.sup.0.sub.d]                                6.13 x [10.sup.-31](e)
[right arrow]                                4.57 x [10.sup.-27]([mu])
[nu][bar.[nu]]                               5.91 x [10.sup.-25]([tau])

B [right arrow]       < 1.6 x [10.sup.-5]        4.40 x [10.sup.-4]
K[nu][bar.[nu]]

[B.sup.0.sub.s]                                1.36 x [10.sup.-29](e)
[right arrow]                                1.02 x [10.sup.-25]([mu])
[nu][bar.[nu]]                               1.32 x [10.sup.-23]([tau])

[K.sup.[+ or -]]      < (1.7 [+ or -] 1.0)      8.23 x [10.sup.-11]
[right arrow]            x [10.sup.-5]
[pi][nu][bar.[nu]]

[K.sup.0.sub.s]                                7.24 x [10.sup.-33](e)
[right arrow]                                5.40 x [10.sup.-29]([mu])
[nu][bar.[nu]]                               6.98 x [10.sup.-27]([tau])

[K.sup.0.sub.L]
[right arrow]
[nu][bar.[nu]]

[K.sup.0] [right      < 2.6 x [10.sup.-5]       2.76 x [10.sup.-11]
arrow] [[pi].sup.0]
[nu][bar.[nu]]
(Bounds on New
Physics parameters
have been
calculated numeri-
cally for their
fit with SM
prediction)

Process                    Bounds on New Physics Parameter

                          Magnitude [absolute value of (z)]

B [right arrow]                  < 1.79 x [10.sup.-6]
[pi][nu][bar.[nu]]

[B.sup.0.sub.d]
[right arrow]
[nu][bar.[nu]]

B [right arrow]                  < 8.69 x [10.sup.-7]
K[nu][bar.[nu]]

[B.sup.0.sub.s]
[right arrow]
[nu][bar.[nu]]

[K.sup.[+ or -]]        (7.83 [+ or -] 2.01) x [10.sup.-9](e)
[right arrow]          (7.83 [+ or -] 2.01) x [10.sup.-9]([mu])
[pi][nu][bar.[nu]]    (7.81 [+ or -] 2.01) x [10.sup.-9]([tau])

[K.sup.0.sub.s]
[right arrow]
[nu][bar.[nu]]

[K.sup.0.sub.L]                 4.27 x [10.sup.-33](e)
[right arrow]                 3.19 x [10.sup.-29]([mu])
[nu][bar.[nu]]                4.06 x [10.sup.-27]([tau])

[K.sup.0] [right                 < 1.9 x [10.sup.-9]
arrow] [[pi].sup.0]             < 9.83 x [10.sup.-10]
[nu][bar.[nu]]                  < 6.96 x [10.sup.-10]
(Bounds on New                  < 5.68 x [10.sup.-10]
Physics parameters              < 5.09 x [10.sup.-10]
have been                       < 4.92 x [10.sup.-10]
calculated numeri-
cally for their
fit with SM
prediction)

Process               Bounds on New Physics Parameter

                             Phase ([theta])

B [right arrow]       [less than or equal to] 2[pi]
[pi][nu][bar.[nu]]

[B.sup.0.sub.d]       [less than or equal to] 2[pi]
[right arrow]
[nu][bar.[nu]]

B [right arrow]       [less than or equal to] 2[pi]
K[nu][bar.[nu]]

[B.sup.0.sub.s]       [less than or equal to] 2[pi]
[right arrow]
[nu][bar.[nu]]

[K.sup.[+ or -]]      [less than or equal to] 2[pi]
[right arrow]
[pi][nu][bar.[nu]]

[K.sup.0.sub.s]       [less than or equal to] 2[pi]
[right arrow]
[nu][bar.[nu]]

[K.sup.0.sub.L]       [less than or equal to] 2[pi]
[right arrow]
[nu][bar.[nu]]

[K.sup.0] [right               15[degrees]
arrow] [[pi].sup.0]            30[degrees]
[nu][bar.[nu]]                 45[degrees]
(Bounds on New                 60[degrees]
Physics parameters             75[degrees]
have been                      90[degrees]
calculated numeri-
cally for their
fit with SM
prediction)

Table 2: Bounds on NP parameters ([absolute value of
(z([[lambda]'.sub.ijk][[lambda]'.sub.lmn]))], [theta]) for K
[right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]].
[alpha] (B[r.sub.SM]) is (a) unpolarized (8.63x[10.sup.-11]), (b) e
(2.89x[10.sup.-11]), (c) [mu] (2.89x[10.sup.-11]), (d) [tau]
(2.85x[10.sup.-11]). Experimental limits are (1.7 [+ or -] 1.0)
x [10.sup.-10].

(a)

          Bounds on NP Parameter (unpolarized)

           [absolute value of (z([[SIGMA].sup.3.sub.i=1]
          [[lambda]'.sup.*.sub.ik2][[lambda]'.sub.ik1]))]
[theta]                    x [10.sup.-9]

0                       3.465 [+ or -] 3.209
30                      3.032 [+ or -] 2.653
60                      4.229 [+ or -] ....
90                      2.972 [+ or -] ....
120                     2.233 [+ or -] ....
150                     1.927 [+ or -] ....
180                      1.94 [+ or -] ....
210                     2.278 [+ or -] ....
240                     3.062 [+ or -] ....
270                     4.357 [+ or -] ....
300                     3.078 [+ or -] 2.721
330                     3.486 [+ or -] 3.233
360                     3.465 [+ or -] 3.209

(b)

          Bounds on NP Parameter (unpolarized)

           [absolute value of (z([[SIGMA].sup.3.sub.i=1]
          [[lambda]'.sup.*.sub.ik2][[lambda]'.sub.ik1]))]
[theta]                    x [10.sup.-9]

0                       5.575 [+ or -] 1.098
30                      4.356 [+ or -] 1.328
60                      4.229 [+ or -] ....
90                      2.972 [+ or -] ....
120                     2.233 [+ or -] ....
150                     1.927 [+ or -] ....
180                      1.94 [+ or -] ....
210                     2.278 [+ or -] ....
240                     3.062 [+ or -] ....
270                     4.357 [+ or -] ....
300                     4.504 [+ or -] 1.295
330                     5.629 [+ or -] 1.09
360                     5.575 [+ or -] 1.098

(c)

          Bounds on NP Parameter (unpolarized)

           [absolute value of (z[[lambda]'.sup.*.sub.1k2]
[theta]         [[lambda]'.sub.1k1]))] x [10.sup.-9]

0                       8.026 [+ or -] 1.87
30                      7.083 [+ or -] 1.967
60                      5.671 [+ or -] 2.079
90                      4.408 [+ or -] 2.072
120                     3.618 [+ or -] 1.956
150                     3.267 [+ or -] 1.865
180                     3.282 [+ or -] 1.87
210                     3.666 [+ or -] 1.967
240                     4.498 [+ or -] 2.079
270                     5.793 [+ or -] 2.072
300                     7.189 [+ or -] 1.956
330                     8.068 [+ or -] 1.865
360                     8.026 [+ or -] 1.87

(d)

          Bounds on NP Parameter (unpolarized)

           [absolute value of (z[[lambda]'.sup.*.sub.2k2]
[theta]         [[lambda]'.sub.2k1]))] x [10.sup.-9]

0                       8.026 [+ or -] 1.87
30                      7.083 [+ or -] 1.967
60                      5.671 [+ or -] 2.079
90                      4.409 [+ or -] 2.072
120                     3.618 [+ or -] 1.956
150                     3.267 [+ or -] 1.865
180                     3.282 [+ or -] 1.87
210                     3.666 [+ or -] 1.967
240                     4.499 [+ or -] 2.079
270                     5.793 [+ or -] 2.072
300                     7.189 [+ or -] 1.956
330                     8.068 [+ or -] 1.865
360                     8.026 [+ or -] 1.87

(e)

          Bounds on NP Parameter (unpolarized)

           [absolute value of (z[[lambda]'.sup.*.sub.3k2]
[theta]         [[lambda]'.sub.3k1]))] x [10.sup.-9]

0                       8.009 [+ or -] 1.87
30                      7.071 [+ or -] 1.966
60                      5.671 [+ or -] 2.076
90                      4.418 [+ or -] 2.069
120                     3.631 [+ or -] 1.954
150                     3.282 [+ or -] 1.865
180                     3.298 [+ or -] 1.87
210                     3.684 [+ or -] 1.966
240                     4.515 [+ or -] 2.076
270                     5.803 [+ or -] 2.069
300                     7.186 [+ or -] 1.954
330                     8.054 [+ or -] 1.865
360                     8.009 [+ or -] 1.87

Table 3: Bounds on NP parameters ([absolute value of
([[lambda]'.sub.ijk][[lambda]'.sub.lmn])], [theta])
derived from K [right arrow] [pi][[nu].sub.[alpha]]
[[bar.[nu]].sub.[alpha]] for [K.sub.0] [right arrow] [[pi].sup.0]
[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] (B[r.sub.SM])
is (a) unpolarized (2.94 x [10.sup.-11]), (b) e (9.78 x [10.sup.-12]),
(c) [mu], (d) [tau]. Experimental bound on the process is
2.6 x [10.sup.-8].

(a)

            Bounds on NP Parameter      Branching Fraction
                                         (x[10.sup.-10])
              [absolute value of
           ([[SIGMA].sup.3.sub.i=1]
          ([[lambda]'.sup.*.sub.ik2]
            [[lambda]'.sub.ik1]))]
[theta]         x [10.sup.-9]                   NP

0            3.465 [+ or -] 3.209         0. [+ or -] 0.
30           3.032 [+ or -] 2.653      2.484 [+ or -] 2.462
60           4.229 [+ or -] ....       8.212 [+ or -] ....
90           2.972 [+ or -] ....       5.407 [+ or -] ....
120          2.233 [+ or -] ....       2.289 [+ or -] ....
150          1.927 [+ or -] ....       0.568 [+ or -] ....
180           1.94 [+ or -] ....           0 [+ or -] ....
210          2.278 [+ or -] ....       0.794 [+ or -] ....
240          3.062 [+ or -] ....       4.304 [+ or -] ....
270          4.357 [+ or -] ....      11.621 [+ or -] ....
300          3.078 [+ or -] 2.721       7.75 [+ or -] 7.691
330          3.486 [+ or -] 3.233       3.46 [+ or -] 3.45
360          3.465 [+ or -] 3.209         0. [+ or -] 0.

                  Branching Fraction (x[10.sup.-10])

[theta]          Interference                Combined

0               0. [+ or -] 0.        0.003 [+ or -] 0.
30          -2.446 [+ or -] 2.426     0.041 [+ or -] 0.036
60          -8.099 [+ or -] ....      0.116 [+ or -] ....
90          -5.328 [+ or -] ....      0.082 [+ or -] ....
120          -2.25 [+ or -] ....      0.042 [+ or -] ....
150         -0.555 [+ or -] ....      0.017 [+ or -] ....
180              0 [+ or -] ....      0.003 [+ or -] ....
210         -0.796 [+ or -] ....      0.001 [+ or -] ....
240         -4.283 [+ or -] ....      0.023 [+ or -] ....
270        -11.541 [+ or -] ....      0.082 [+ or -] ....
300         -7.695 [+ or -] 7.634     0.058 [+ or -] 0.057
330          -3.44 [+ or -] 3.429     0.023 [+ or -] 0.021
360             0. [+ or -] 0.        0.003 [+ or -] 0.

(b)

            Bounds on NP Parameter      Branching Fraction
                                         (x[10.sup.-10])
              [absolute value of
          ([[lambda]'.sup.*.sub.1k2]
             [lambda]'.sub.1k1])]
[theta]         x [10.sup.-9]                   NP

0            8.026 [+ or -] 1.87          0. [+ or -] 0.
30           7.083 [+ or -] 1.967      2.757 [+ or -] 1.422
60           5.671 [+ or -] 2.079      5.585 [+ or -] 3.609
90           4.408 [+ or -] 2.072      4.843 [+ or -] 3.729
120          3.618 [+ or -] 1.956      2.589 [+ or -] 2.166
150          3.267 [+ or -] 1.865      0.722 [+ or -] 0.622
180          3.282 [+ or -] 1.87          0. [+ or -] 0.
210          3.666 [+ or -] 1.967      0.883 [+ or -] 0.736
240          4.498 [+ or -] 2.079      3.759 [+ or -] 2.863
270          5.793 [+ or -] 2.072      7.726 [+ or -] 4.9
300          7.189 [+ or -] 1.956      8.497 [+ or -] 4.306
330          8.068 [+ or -] 1.865      3.499 [+ or -] 1.536
360          8.026 [+ or -] 1.87          0. [+ or -] 0.

                  Branching Fraction (x[10.sup.-10])

[theta]          Interference                Combined

0               0. [+ or -] 0.         0.001 [+ or -] 0.
30          -2.719 [+ or -] 1.405      0.039 [+ or -] 0.017
60          -5.515 [+ or -] 3.568      0.071 [+ or -] 0.041
90          -4.782 [+ or -] 3.686      0.062 [+ or -] 0.043
120         -2.554 [+ or -] 2.14       0.036 [+ or -] 0.026
150          -0.71 [+ or -] 0.613      0.013 [+ or -] 0.009
180             0. [+ or -] 0.         0.001 [+ or -] 0.
210         -0.879 [+ or -] 0.731      0.005 [+ or -] 0.005
240         -3.732 [+ or -] 2.839      0.028 [+ or -] 0.024
270         -7.665 [+ or -] 4.857      0.062 [+ or -] 0.043
300         -8.429 [+ or -] 4.267      0.068 [+ or -] 0.038
330         -3.475 [+ or -] 1.523      0.025 [+ or -] 0.013
360             0. [+ or -] 0.         0.001 [+ or -] 0.

(c)

            Bounds on NP Parameter      Branching Fraction
                                         (x[10.sup.-10])
              [absolute value of
          ([[lambda]'.sup.*.sub.2k2]
            [[lambda]'.sub.2k1])]
[theta]         x [10.sup.-9]                   NP

0            8.026 [+ or -] 1.87          0. [+ or -] 0.
30           7.083 [+ or -] 1.967      2.757 [+ or -] 1.422
60           5.671 [+ or -] 2.079      5.585 [+ or -] 3.609
90           4.409 [+ or -] 2.072      4.843 [+ or -] 3.729
120          3.618 [+ or -] 1.956      2.589 [+ or -] 2.167
150          3.267 [+ or -] 1.865      0.722 [+ or -] 0.622
180          3.282 [+ or -] 1.87          0. [+ or -] 0.
210          3.666 [+ or -] 1.967      0.883 [+ or -] 0.736
240          4.499 [+ or -] 2.079      3.759 [+ or -] 2.863
270          5.793 [+ or -] 2.072      7.726 [+ or -] 4.9
300          7.189 [+ or -] 1.956      8.496 [+ or -] 4.305
330          8.068 [+ or -] 1.865      3.498 [+ or -] 1.536
360          8.026 [+ or -] 1.87          0. [+ or -] 0.

                  Branching Fraction (x[10.sup.-10])

[theta]          Interference                Combined

0               0. [+ or -] 0.         0.001 [+ or -] 0.
30          -2.719 [+ or -] 1.405      0.039 [+ or -] 0.017
60          -5.515 [+ or -] 3.568      0.071 [+ or -] 0.041
90          -4.782 [+ or -] 3.686      0.062 [+ or -] 0.043
120         -2.554 [+ or -] 2.14       0.036 [+ or -] 0.026
150          -0.71 [+ or -] 0.613      0.013 [+ or -] 0.009
180             0. [+ or -] 0.         0.001 [+ or -] 0.
210          -0.88 [+ or -] 0.731      0.005 [+ or -] 0.005
240         -3.732 [+ or -] 2.839      0.028 [+ or -] 0.024
270         -7.665 [+ or -] 4.857      0.062 [+ or -] 0.043
300         -8.429 [+ or -] 4.267      0.068 [+ or -] 0.038
330         -3.475 [+ or -] 1.523      0.025 [+ or -] 0.013
360             0. [+ or -] 0.         0.001 [+ or -] 0.

(d)

            Bounds on NP Parameter      Branching Fraction
                                         (x[10.sup.-10])
              [absolute value of
          ([[lambda]'.sup.*.sub.3k2]
            [[lambda]'.sub.3k1])]
[theta]         x [10.sup.-9]                   NP

0            8.009 [+ or -] 1.87          0. [+ or -] 0.
30           7.071 [+ or -] 1.966      2.748 [+ or -] 1.418
60           5.671 [+ or -] 2.076      5.582 [+ or -] 3.603
90           4.418 [+ or -] 2.069      4.856 [+ or -] 3.73
120          3.631 [+ or -] 1.954      2.603 [+ or -] 2.173
150          3.282 [+ or -] 1.865      0.727 [+ or -] 0.625
180          3.298 [+ or -] 1.87          0. [+ or -] 0.
210          3.684 [+ or -] 1.966       0.89 [+ or -] 0.739
240          4.515 [+ or -] 2.076       3.78 [+ or -] 2.869
270          5.803 [+ or -] 2.069      7.745 [+ or -] 4.899
300          7.186 [+ or -] 1.954      8.488 [+ or -] 4.299
330          8.054 [+ or -] 1.865      3.487 [+ or -] 1.533
360          8.009 [+ or -] 1.87          0. [+ or -] 0.

                   Branching Fraction (x[10.sup.-10])

[theta]          Interference                Combined

0               0. [+ or -] 0.         0.001 [+ or -] 0.
30          -2.711 [+ or -] 1.401      0.038 [+ or -] 0.017
60          -5.513 [+ or -] 3.562      0.071 [+ or -] 0.041
90          -4.795 [+ or -] 3.687      0.062 [+ or -] 0.043
120         -2.568 [+ or -] 2.146      0.036 [+ or -] 0.027
150         -0.715 [+ or -] 0.616      0.013 [+ or -] 0.009
180             0. [+ or -] 0.         0.001 [+ or -] 0.
210         -0.886 [+ or -] 0.734      0.005 [+ or -] 0.005
240         -3.753 [+ or -] 2.845      0.028 [+ or -] 0.024
270         -7.684 [+ or -] 4.856      0.062 [+ or -] 0.043
300         -8.421 [+ or -] 4.261      0.068 [+ or -] 0.038
330         -3.464 [+ or -] 1.52       0.024 [+ or -] 0.013
360             0. [+ or -] 0.         0.001 [+ or -] 0.

Table 4: Bounds on NP parameters ([absolute value of
(z([[lambda]'.sub.ijk][[lambda]'.sub.lmn]))], [theta]) for B
[right arrow] [pi][[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha]
(B[r.sub.SM]) is (a) unpolarized (1.73 x [10.sup.-7]) and 5.76 x
[10.sup.-8] for (b) e, (c) [mu] (d) [tau]. Experimental bound on the
process is 9.8 x [10.sup.-5].

(a)

            Bounds on NP Parameter (unpolarized)

            [absolute value of (z([[SIGMA].sup.3.sub.i=1]
           [[lambda]'.sup.*.sub.ik3][[lambda]'.sub.ik2]))]
[theta]                     x [10.sup.-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 (unpolarized)

            [absolute value of (z[[lambda]'.sup.*.sub.1k3]
[theta]          [[lambda]'.sub.1k2]))] x [10.sup.-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 (unpolarized)

            [absolute value of (z[[lambda]'.sup.*.sub.2k3]
[theta]          [[lambda]'.sub.2k2]))] x [10.sup.-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 (unpolarized)

            [absolute value of (z[[lambda]'.sup.*.sub.3k3]
[theta]          [[lambda]'.sub.1k1]))] x [10.sup.-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

Table 5: Bounds on NP parameters ([absolute value of
(z([[lambda]'.sub.ijk][[lambda]'.sub.lmn]))], [theta]) for B
[right arrow] K[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha]
(B[r.sub.SM]) is (a) unpolarized (4.69 x [10.sup.-6]) and (1.56 x
[10.sup.-6]) for (b) e, (c) [mu] (d) [tau]. Experimental bound on the
process is 1.6 x [10.sup.-5].

(a)

          Bounds on NP Parameter (unpolarized)

            [absolute value of (z([[SIGMA].sup.3.sub.i=1]
           [[lambda]'.sup.*.sub.ik3][[lambda]'.sub.ik2]))]
[theta]                     x [10.sup.-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 (unpolarized)

            [absolute value of (z[[lambda]'.sup.*.sub.1k3]
[theta]          [[lambda]'.sub.1k2]))] x [10.sup.-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 (unpolarized)

            [absolute value of (z[[lambda]'.sup.*.sub.2k3]
[theta]          [[lambda]'.sub.2k2]))] x [10.sup.-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 (unpolarized)

            [absolute value of (z[[lambda]'.sup.*.sub.3k3]
[theta]          [[lambda]'.sub.3k2]))] x [10.sup.-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                             8.7539

Table 6: Bounds on NP parameters ([absolute value of
([[lambda]'.sub.ijk][[lambda]'.sub.lmn])], [theta])
derived from K [right arrow] [pi][[nu].sub.[alpha]]
[[bar.[nu]].sub.[alpha]] for [K.sub.s] [right arrow]
[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] (B[r.sub.SM])
is (a) unpolarized (7.27 x [10.sup.-27]), (b) e (7.58 x [10.sup.-33]),
(c) [mu] (5.66 x [10.sup.-29]), (d) [tau] (7.27 x [10.sup.-27]).

(a)

            Bounds on NP Parameter      Branching Fraction
                                          (x[10.sup.-26])
              [absolute value of
           ([[SIGMA].sup.3.sub.i=1]
          ([[lambda]'.sup.*.sub.ik2]
            [[lambda]'.sub.ik1]))]
[theta]         x [10.sup.-9]                   NP

0            3.465 [+ or -] 3.209       2.69 [+ or -] 2.682
30           3.032 [+ or -] 2.653       1.958 [+ or -] 1.94
60           4.229 [+ or -] ....        2.157 [+ or -] ....
90           2.972 [+ or -] ....        1.065 [+ or -] ....
120          2.233 [+ or -] ....        0.601 [+ or -] ....
150          1.927 [+ or -] ....        0.448 [+ or -] ....
180           1.94 [+ or -] ....        0.454 [+ or -] ....
210          2.278 [+ or -] ....        0.626 [+ or -] ....
240          3.062 [+ or -] ....        1.131 [+ or -] ....
270          4.357 [+ or -] ....        2.289 [+ or -] ....
300          3.078 [+ or -] 2.721      2.035 [+ or -] 2.021
330          3.486 [+ or -] 3.233      2.726 [+ or -] 2.718
360          3.465 [+ or -] 3.209       2.69 [+ or -] 2.682

                  Branching Fraction (x[10.sup.-26])

[theta]          Interference                Combined

0           -1.952 [+ or -] 1.808       1.454 [+ or -] 0.873
30          -0.764 [+ or -] 0.669        1.91 [+ or -] 1.272
60           2.055 [+ or -] ....        4.928 [+ or -] ....
90           1.195 [+ or -] ....        2.976 [+ or -] ....
120         -0.808 [+ or -] ....        0.509 [+ or -] ....
150         -0.989 [+ or -] ....        0.175 [+ or -] ....
180          0.395 [+ or -] ....        1.565 [+ or -] ....
210          1.312 [+ or -] ....        2.654 [+ or -] ....
240         -0.079 [+ or -] ....        1.768 [+ or -] ....
270         -2.545 [+ or -] ....         0.46 [+ or -] ....
300         -0.476 [+ or -] 0.421       2.276 [+ or -] 1.6
330           1.87 [+ or -] 1.734       5.313 [+ or -] 4.453
360          1.109 [+ or -] 1.027       4.515 [+ or -] 3.709

(b)

            Bounds on NP Parameter      Branching Fraction
                                          (x[10.sup.-32])
              [absolute value of
          ([[lambda]'.sup.*.sub.1k2]
             [lambda]'.sub.1k1])]
[theta]         x [10.sup.-9]                   NP

0            8.026 [+ or -] 1.87       8.435 [+ or -] 3.725
30           7.083 [+ or -] 1.967       6.71 [+ or -] 3.46
60           5.671 [+ or -] 2.079       4.53 [+ or -] 2.93
90           4.408 [+ or -] 2.072       2.95 [+ or -] 2.27
120          3.618 [+ or -] 1.956        2.1 [+ or -] 1.76
150          3.267 [+ or -] 1.865      1.755 [+ or -] 1.515
180          3.282 [+ or -] 1.87        1.77 [+ or -] 1.52
210          3.666 [+ or -] 1.967       2.15 [+ or -] 1.79
240          4.498 [+ or -] 2.079       3.05 [+ or -] 2.32
270          5.793 [+ or -] 2.072        4.7 [+ or -] 2.98
300          7.189 [+ or -] 1.956      6.895 [+ or -] 3.495
330          8.068 [+ or -] 1.865      8.515 [+ or -] 3.735
360          8.026 [+ or -] 1.87       8.435 [+ or -] 3.725

                  Branching Fraction (x[10.sup.-32])

[theta]          Interference                Combined

0           -4.716 [+ or -] 1.097      4.472 [+ or -] 2.628
30          -1.846 [+ or -] 0.513      5.618 [+ or -] 2.947
60           2.876 [+ or -] 1.052       8.16 [+ or -] 3.982
90           1.834 [+ or -] 0.862      5.538 [+ or -] 3.132
120         -1.368 [+ or -] 0.742      1.486 [+ or -] 1.018
150         -1.741 [+ or -] 0.997      0.768 [+ or -] 0.518
180          0.703 [+ or -] 0.403      3.227 [+ or -] 1.923
210          2.198 [+ or -] 1.18       5.102 [+ or -] 2.97
240         -0.131 [+ or -] 0.058      3.673 [+ or -] 2.262
270         -3.524 [+ or -] 1.259       1.93 [+ or -] 1.72
300         -1.145 [+ or -] 0.314      6.504 [+ or -] 3.181
330          4.513 [+ or -] 1.045     13.782 [+ or -] 4.78
360           2.66 [+ or -] 0.622     11.848 [+ or -] 4.347

(c)

            Bounds on NP Parameter      Branching Fraction
                                          (x[10.sup.-28])
              [absolute value of
          ([[lambda]'.sup.*.sub.2k2]
            [[lambda]'.sub.2k1])]
[theta]         x [10.sup.-9]                   NP

0            8.026 [+ or -] 1.87        6.29 [+ or -] 2.78
30           7.083 [+ or -] 1.967      5.004 [+ or -] 2.58
60           5.671 [+ or -] 2.079      3.379 [+ or -] 2.184
90           4.409 [+ or -] 2.072      2.198 [+ or -] 1.692
120          3.618 [+ or -] 1.956      1.566 [+ or -] 1.311
150          3.267 [+ or -] 1.865       1.31 [+ or -] 1.128
180          3.282 [+ or -] 1.87       1.322 [+ or -] 1.136
210          3.666 [+ or -] 1.967      1.603 [+ or -] 1.336
240          4.499 [+ or -] 2.079      2.274 [+ or -] 1.732
270          5.793 [+ or -] 2.072      3.506 [+ or -] 2.224
300          7.189 [+ or -] 1.956      5.141 [+ or -] 2.605
330          8.068 [+ or -] 1.865       6.35 [+ or -] 2.787
360          8.026 [+ or -] 1.87        6.29 [+ or -] 2.78

                  Branching Fraction (x[10.sup.-28])

[theta]          Interference                Combined

0           -3.497 [+ or -] 0.815       3.35 [+ or -] 1.964
30          -1.374 [+ or -] 0.381      4.187 [+ or -] 2.199
60           2.132 [+ or -] 0.782      6.068 [+ or -] 2.965
90           1.366 [+ or -] 0.642       4.12 [+ or -] 2.334
120         -1.014 [+ or -] 0.548       1.11 [+ or -] 0.762
150         -1.295 [+ or -] 0.739      0.572 [+ or -] 0.389
180          0.518 [+ or -] 0.296      2.396 [+ or -] 1.432
210          1.632 [+ or -] 0.875      3.792 [+ or -] 2.211
240         -0.093 [+ or -] 0.043      2.738 [+ or -] 1.689
270         -2.616 [+ or -] 0.936      1.448 [+ or -] 1.289
300         -0.853 [+ or -] 0.232      4.845 [+ or -] 2.373
330          3.346 [+ or -] 0.773     10.254 [+ or -] 3.561
360          1.979 [+ or -] 0.461      8.826 [+ or -] 3.241

(d)

            Bounds on NP Parameter      Branching Fraction
                                          (x[10.sup.-26])
              [absolute value of
          ([[lambda]'.sup.*.sub.3k2]
            [[lambda]'.sub.3k1])]
[theta]         x [10.sup.-9]                   NP

0            8.009 [+ or -] 1.87       8.097 [+ or -] 3.584
30           7.071 [+ or -] 1.966      6.448 [+ or -] 3.328
60           5.671 [+ or -] 2.076      4.365 [+ or -] 2.818
90           4.418 [+ or -] 2.069      2.849 [+ or -] 2.188
120          3.631 [+ or -] 1.954      2.035 [+ or -] 1.698
150          3.282 [+ or -] 1.865      1.706 [+ or -] 1.466
180          3.298 [+ or -] 1.87        1.72 [+ or -] 1.477
210          3.684 [+ or -] 1.966      2.087 [+ or -] 1.734
240          4.515 [+ or -] 2.076      2.955 [+ or -] 2.244
270          5.803 [+ or -] 2.069      4.542 [+ or -] 2.874
300          7.186 [+ or -] 1.954      6.638 [+ or -] 3.362
330          8.054 [+ or -] 1.865      8.181 [+ or -] 3.596
360          8.009 [+ or -] 1.87       8.097 [+ or -] 3.584

                  Branching Fraction (x[10.sup.-26])

[theta]          Interference                Combined

0            -4.48 [+ or -] 1.046      4.328 [+ or -] 2.539
30          -1.768 [+ or -] 0.491       5.39 [+ or -] 2.836
60           2.734 [+ or -] 1.          7.81 [+ or -] 3.818
90           1.761 [+ or -] 0.825       5.32 [+ or -] 3.012
120         -1.304 [+ or -] 0.701      1.442 [+ or -] 0.997
150         -1.672 [+ or -] 0.951      0.744 [+ or -] 0.515
180          0.666 [+ or -] 0.378      3.097 [+ or -] 1.854
210          2.106 [+ or -] 1.124      4.904 [+ or -] 2.858
240         -0.115 [+ or -] 0.053      3.552 [+ or -] 2.19
270         -3.364 [+ or -] 1.2        1.889 [+ or -] 1.674
300         -1.102 [+ or -] 0.299      6.247 [+ or -] 3.063
330          4.288 [+ or -] 0.992      13.18 [+ or -] 4.589
360          2.543 [+ or -] 0.594      11.35 [+ or -] 4.178

Table 7: Bounds on NP parameters ([absolute value of
([[lambda]'.sub.ijk][[lambda]'.sub.lmn])], [theta])
derived from K [right arrow] [pi][[nu].sub.[alpha]]
[[bar.[nu]].sub.[alpha]] for [K.sub.L] [right arrow]
[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] (B[r.sub.SM])
is (a) unpolarized (4.29 x [10.sup.-27]), (b) e (4.48 x [10.sup.-33]),
(c) [mu] (3.34 x [10.sup.-29]), (d) [tau] (4.26 x [10.sup.-27]).

(a)

            Bounds on NP Parameter      Branching Fraction
                                         (x[10.sup.-26])
              [absolute value of
           ([[SIGMA].sup.3.sub.i=1]
          ([[lambda]'.sup.*.sub.ik2]
            [[lambda]'.sub.ik1]))]
[theta]         x [10.sup.-9]                   NP

0            3.465 [+ or -] 3.209      1.588 [+ or -] 1.583
30           3.032 [+ or -] 2.653      1.156 [+ or -] 1.146
60           4.229 [+ or -] ....       1.274 [+ or -] ....
90           2.972 [+ or -] ....       0.629 [+ or -] ....
120          2.233 [+ or -] ....       0.355 [+ or -] ....
150          1.927 [+ or -] ....       0.264 [+ or -] ....
180           1.94 [+ or -] ....       0.268 [+ or -] ....
210          2.278 [+ or -] ....       0.369 [+ or -] ....
240          3.062 [+ or -] ....       0.667 [+ or -] ....
270          4.357 [+ or -] ....       1.352 [+ or -] ....
300          3.078 [+ or -] 2.721      1.202 [+ or -] 1.193
330          3.486 [+ or -] 3.233       1.61 [+ or -] 1.605
360          3.465 [+ or -] 3.209      1.588 [+ or -] 1.583

              Branching Fraction  (x[10.sup.-26])

[theta]       Interference              Combined

0         -1.153 [+ or -] 1.067   0.858 [+ or -] 0.516
30        -0.451 [+ or -] 0.395   1.128 [+ or -] 0.75
60         1.213 [+ or -] ....     2.91 [+ or -] ....
90         0.705 [+ or -] ....    1.757 [+ or -] ....
120       -0.477 [+ or -] ....    0.301 [+ or -] ....
150       -0.584 [+ or -] ....    0.103 [+ or -] ....
180        0.233 [+ or -] ....    0.924 [+ or -] ....
210        0.775 [+ or -] ....    1.567 [+ or -] ....
240       -0.046 [+ or -] ....    1.044 [+ or -] ....
270       -1.503 [+ or -] ....    0.272 [+ or -] ....
300       -0.281 [+ or -] 0.248   1.344 [+ or -] 0.945
330        1.104 [+ or -] 1.024   3.136 [+ or -] 2.63
360        0.655 [+ or -] 0.606   2.666 [+ or -] 2.19

(b)

            Bounds on NP Parameter      Branching Fraction
                                         (x[10.sup.-32])
              [absolute value of
          ([[lambda]'.sup.*.sub.1k2]
             [lambda]'.sub.1k1])]
[theta]         x [10.sup.-9]                   NP

0            8.026 [+ or -] 1.87        4.98 [+ or -] 2.2
30           7.083 [+ or -] 1.967       3.96 [+ or -] 2.04
60           5.671 [+ or -] 2.079      2.675 [+ or -] 1.725
90           4.408 [+ or -] 2.072       1.74 [+ or -] 1.34
120          3.618 [+ or -] 1.956       1.24 [+ or -] 1.04
150          3.267 [+ or -] 1.865      1.035 [+ or -] 0.895
180          3.282 [+ or -] 1.87        1.05 [+ or -] 0.9
210          3.666 [+ or -] 1.967       1.27 [+ or -] 1.06
240          4.498 [+ or -] 2.079        1.8 [+ or -] 1.37
270          5.793 [+ or -] 2.072      2.775 [+ or -] 1.765
300          7.189 [+ or -] 1.956       4.07 [+ or -] 2.06
330          8.068 [+ or -] 1.865      5.025 [+ or -] 2.205
360          8.026 [+ or -] 1.87        4.98 [+ or -] 2.2

              Branching Fraction  (x[10.sup.-32])

[theta]       Interference              Combined

0         -2.784 [+ or -] 0.648   2.641 [+ or -] 1.552
30        -1.088 [+ or -] 0.3     3.317 [+ or -] 1.74
60         1.698 [+ or -] 0.626   4.818 [+ or -] 2.351
90         1.085 [+ or -] 0.51     3.27 [+ or -] 1.85
120       -0.808 [+ or -] 0.439   0.877 [+ or -] 0.601
150       -1.026 [+ or -] 0.589   0.454 [+ or -] 0.306
180         0.41 [+ or -] 0.235   1.906 [+ or -] 1.136
210        1.297 [+ or -] 0.694   3.012 [+ or -] 1.754
240       -0.076 [+ or -] 0.034   2.169 [+ or -] 1.336
270       -2.081 [+ or -] 0.749   1.139 [+ or -] 1.016
300       -0.674 [+ or -] 0.181    3.84 [+ or -] 1.879
330        2.668 [+ or -] 0.617   8.138 [+ or -] 2.822
360         1.57 [+ or -] 0.366   6.995 [+ or -] 2.566

(c)

            Bounds on NP Parameter      Branching Fraction
                                         (x[10.sup.-28])
              [absolute value of
          ([[lambda]'.sup.*.sub.2k2]
            [[lambda]'.sub.2k1])]
[theta]         x [10.sup.-9]                   NP

0            8.026 [+ or -] 1.87       3.713 [+ or -] 1.641
30           7.083 [+ or -] 1.967      2.954 [+ or -] 1.523
60           5.671 [+ or -] 2.079      1.995 [+ or -] 1.289
90           4.409 [+ or -] 2.072      1.298 [+ or -] 1.
120          3.618 [+ or -] 1.956      0.925 [+ or -] 0.774
150          3.267 [+ or -] 1.865      0.774 [+ or -] 0.666
180          3.282 [+ or -] 1.87        0.78 [+ or -] 0.671
210          3.666 [+ or -] 1.967      0.946 [+ or -] 0.789
240          4.499 [+ or -] 2.079      1.342 [+ or -] 1.023
270          5.793 [+ or -] 2.072       2.07 [+ or -] 1.313
300          7.189 [+ or -] 1.956      3.035 [+ or -] 1.538
330          8.068 [+ or -] 1.865       3.75 [+ or -] 1.646
360          8.026 [+ or -] 1.87       3.713 [+ or -] 1.641

              Branching Fraction (x[10.sup.-28])

[theta]       Interference              Combined

0         -2.064 [+ or -] 0.481   1.978 [+ or -] 1.16
30        -0.811 [+ or -] 0.225   2.472 [+ or -] 1.298
60         1.259 [+ or -] 0.461   3.582 [+ or -] 1.751
90         0.806 [+ or -] 0.378   2.433 [+ or -] 1.378
120       -0.599 [+ or -] 0.324   0.655 [+ or -] 0.45
150       -0.764 [+ or -] 0.436   0.338 [+ or -] 0.23
180        0.306 [+ or -] 0.175   1.414 [+ or -] 0.846
210        0.963 [+ or -] 0.517   2.238 [+ or -] 1.305
240       -0.055 [+ or -] 0.025   1.616 [+ or -] 0.997
270       -1.544 [+ or -] 0.552   0.854 [+ or -] 0.76
300       -0.503 [+ or -] 0.136    2.86 [+ or -] 1.402
330        1.976 [+ or -] 0.457   6.054 [+ or -] 2.102
360        1.169 [+ or -] 0.273   5.211 [+ or -] 1.914

(d)

            Bounds on NP Parameter      Branching Fraction
                                         (x[10.sup.-26])
              [absolute value of
          ([[lambda]'.sup.*.sub.3k2]
            [[lambda]'.sub.3k1])]
[theta]         x [10.sup.-9]                   NP

0            8.009 [+ or -] 1.87        4.78 [+ or -] 2.116
30           7.071 [+ or -] 1.966      3.806 [+ or -] 1.964
60           5.671 [+ or -] 2.076      2.577 [+ or -] 1.663
90           4.418 [+ or -] 2.069      1.682 [+ or -] 1.291
120          3.631 [+ or -] 1.954      1.202 [+ or -] 1.003
150          3.282 [+ or -] 1.865      1.007 [+ or -] 0.865
180          3.298 [+ or -] 1.87       1.016 [+ or -] 0.871
210          3.684 [+ or -] 1.966      1.233 [+ or -] 1.024
240          4.515 [+ or -] 2.076      1.746 [+ or -] 1.325
270          5.803 [+ or -] 2.069      2.682 [+ or -] 1.696
300          7.186 [+ or -] 1.954      3.919 [+ or -] 1.985
330          8.054 [+ or -] 1.865       4.83 [+ or -] 2.123
360          8.009 [+ or -] 1.87        4.78 [+ or -] 2.116

              Branching Fraction  (x[10.sup.-26])

[theta]       Interference              Combined

0         -2.646 [+ or -] 0.617   2.555 [+ or -] 1.499
30        -1.044 [+ or -] 0.29    3.182 [+ or -] 1.674
60         1.614 [+ or -] 0.591   4.611 [+ or -] 2.254
90          1.04 [+ or -] 0.487   3.141 [+ or -] 1.778
120        -0.77 [+ or -] 0.414   0.851 [+ or -] 0.588
150       -0.988 [+ or -] 0.561   0.439 [+ or -] 0.304
180        0.393 [+ or -] 0.223   1.828 [+ or -] 1.094
210        1.243 [+ or -] 0.664   2.896 [+ or -] 1.688
240       -0.069 [+ or -] 0.031   2.096 [+ or -] 1.294
270       -1.986 [+ or -] 0.708   1.116 [+ or -] 0.988
300       -0.651 [+ or -] 0.177   3.688 [+ or -] 1.808
330        2.531 [+ or -] 0.586   7.781 [+ or -] 2.709
360        1.501 [+ or -] 0.351   6.701 [+ or -] 2.467

Table 8: Bounds on NP parameters ([absolute value of
([[lambda]'.sub.ijk][[lambda]'.sub.lmn])], [theta])
derived from B [right arrow] [pi][[nu].sub.[alpha]]
[[bar.[nu]].sub.[alpha]] for [B.sub.d] [right arrow]
[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] (B[r.sub.SM])
is (a) unpolarized (6.35 x [10.sup.-25]), (b) e (6.53 x [10.sup.-31]),
(c) [mu] (4.87 x [10.sup.-27]), (d) [tau] (6.3 x [10.sup.-25]).

(a)

      Bounds on NP Parameter                 Branching Fraction
                                              (x[10.sup.-22])
              [absolute value of
           ([[SIGMA].sup.3.sub.i=1]
          ([[lambda]'.sup.*.sub.ik3]
            [[lambda]'.sub.ik1]))]
[theta]         x [10.sup.-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
                                                (x[10.sup.-27])
              [absolute value of
          ([[lambda]'.sup.*.sub.1k3]
             [lambda]'.sub.1k1])]
[theta]         x [10.sup.-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
                                                (x[10.sup.-23])
              [absolute value of
          ([[lambda]'.sup.*.sub.2k3]
            [[lambda]'.sub.2k1])]
[theta]         x [10.sup.-7]           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
                                               (x[10.sup.-21])
              [absolute value of
          ([[lambda]'.sup.*.sub.3k3]
            [[lambda]'.sub.3k1])]
[theta]         x [10.sup.-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       0.993

Table 9: Bounds on NP parameters ([absolute value of
([[lambda]'.sub.ijk][[lambda]'.sub.lmn])], [theta])
derived from B [right arrow] K[[nu].sub.[alpha]]
[[bar.[nu]].sub.[alpha]] for [B.sub.s] [right arrow]
[[nu].sub.[alpha]][[bar.[nu]].sub.[alpha]]. [alpha] (B[r.sub.SM])
is (a) unpolarized (1.41 x [10.sup.-23]), (b) e (1.45 x [10.sup.-29]),
(c) [mu] (1.08 x [10.sup.-25]), (d) [tau] (1.4 x [10.sup.-23]).

(a)

      Bounds on NP Parameter                 Branching Fraction
                                              (x[10.sup.-22])
              [absolute value of
           ([[SIGMA].sup.3.sub.i=1]
          ([[lambda]'.sup.*.sub.ik3]
            [[lambda]'.sub.ik2]))]
[theta]         x [10.sup.-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
                                                (x[10.sup.-28])
              [absolute value of
          ([[lambda]'.sup.*.sub.1k3]
             [lambda]'.sub.1k2])]
[theta]         x [10.sup.-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
                                                (x[10.sup.-24])
              [absolute value of
          ([[lambda]'.sup.*.sub.2k3]
            [[lambda]'.sub.2k2])]
[theta]         x [10.sup.-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
                                               (x[10.sup.-22])
              [absolute value of
          ([[lambda]'.sup.*.sub.3k3]
            [[lambda]'.sub.3k2])]
[theta]         x [10.sup.-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
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Title Annotation:Research Article
Author:Mir, Azeem; Tahir, Farida; Mahmood, Shakeel; Dong, Shi-Hai
Publication:Advances in High Energy Physics
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
Words:13525
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