Advances in High Energy Physics
Volume 2018 (2018), Article ID 4176840, 20 pages
https://doi.org/10.1155/2018/4176840

Research Article

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

Azeem Mir,¹ Farida Tahir,¹ Shakeel Mahmood,² and Shi- Hai Dong³

¹Physics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan
²Air University, PAF Complex, Service Road, E-9, Islamabad, Pakistan
³CIDETEC, Instituto Politécnico Nacional, Unidad Profesional, Adolfo López 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

Copyright © 2018 Azeem Mir et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

We have studied phenomenological implication of R-parity violating () Minimal Supersymmetric Model (MSSM) via analyses of pure leptonic () and semileptonic decays of pseudoscalar mesons (). These analyses involve comparison between theoretical predictions made by MSSM and the Standard Model (SM) with the experimental results like branching fractions of the said process. We have found, in general, that contribution dominates over the SM contribution, i.e., by a factor of for the pure leptonic decays of and by and in case of and , respectively. Furthermore, the limits obtained on Yukawa couplings by using are used to calculate This demonstrates the role of 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.

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 () 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 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   [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]

where are generation indices, and are the lepton and quark left-handed doublets, and , are the charge conjugates of the right-handed leptons and quark singlets, respectively. Here , , and are the Yukawa couplings. The term proportional to is antisymmetric in first two indices and is antisymmetric in last two indices , implying independent coupling constants among which 36 are related to the lepton flavor violation (9 from and 27 from ). 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., , and , where and . At the quark level, all decays are represented by and (all these processes can be) divided into two categories on the bases of lepton flavors, i.e.,

(1)lepton flavor conserving ,(2)lepton flavor violating decays.

The first type of decays () 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 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 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.

In the SM, the effective Hamiltonian for the semileptonic and pure leptonic processes is given by [62, 63]

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

where and

and , and are penguin and box diagrams, respectively. are CKM matrix elements and is the coupling strength of strong interactions.

In MSSM, the relevant effective Lagrangian for the decay process is given by [57–61]

where . The first term in (2) comes from the down squark exchange (where and are down type quarks). The dimensionless coupling constant is related to Yukawa couplings by

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

where with and , ; is the general parameter. We have used the value for the form factor for the above decay processes of and as given in [65]. Since this work focuses on MSSM, we will shift our focus to (for the calculation of limits on couplings) and Yukawa couplings (for the predictions of branching fraction). The decay rate for pure leptonic decay processes is given by

The form factor is given by [66]. represents the mass of strange meson and is the mass of lepton, where is same as that of semileptonic decays.

3.

In MSSM, the relevant effective Lagrangian for the decay process is given by [57–61]

where . The first term in (2) comes from the down squark exchange (where and are down type quarks). The dimensionless coupling constant is given by

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

where

with

with

and as explained in the above section is the general NP parameter and . We have used the form factor for the above decay processes of as given in [67]. The decay rate for pure leptonic decay processes is given by [46, 50–52]

The form factor is given by [66], represents the mass of beauty meson and 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 (), where and . 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 Yukawa couplings 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 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., ) 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 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 involved are normalized to the square of in all these tables and figures.

Table 1: Table listing the properties of processes under discussion [68]. Here is the strength of the NP parameter.

Process
Experimental Measurement
SM Prediction
Bounds on New Physics Parameter

Magnitude
Phase ()

(e)

(μ)

()

(e)

(μ)

()

(e)

(μ)

()

(e)

(μ)

()

(e)

(μ)

()

(Bounds on New Physics parameters have been calculated numerically for their fit with SM prediction)

Table 2: Bounds on NP parameters (, for . is (a) unpolarized (8.63, (b) e (2.89,(c) (2.89, (d) (2.85. Experimental limits are

(a)

Bounds on NP Parameter (unpolarized)

0

30

60

90

120

150

180

210

240

270

300

330

360

(b)

Bounds on NP Parameter (unpolarized)

0

30

60

90

120

150

180

210

240

270

300

330

360

(c)

Bounds on NP Parameter

0

30

60

90

120

150

180

210

240

270

300

330

360

(d)

Bounds on NP Parameter

0

30

60

90

120

150

180

210

240

270

300

330

360

(e)

Bounds on NP Parameter

0

30

60

90

120

150

180

210

240

270

300

330

360

(a)

(b)

(c)

Figure 1: Feynman diagrams of (a) , (b) , (c) .

(a)

(b)

(c)

(d)

Figure 2: Allowed region of general NP parameters for at several values of branching fraction. is (a) unpolarized, (b) e, (c) , (d) . The three contours belong to branching fraction at .

(a)

(b)

(c)

(d)

Figure 3: Variations of branching fraction w.r.t NP parameter at several values of for . is (a) unpolarized, (b) e, (c) , (d) . The three bounds belong to branching fraction at 0.7 ,1.7 , 2.7 corresponding to an experimental measurement of

(a)

(b)

(c)

(d)

Figure 4: Allowed regions of general NP parameters for at specific values of branching fraction. is (a) unpolarized, (b) e, (c) , (d) . The three contours belong to branching fraction at

(a)

(b)

(c)

(d)

Figure 5: Variations of branching fraction (NP contribution only) with respect to NP parameter at several values of for . is (a) unpolarized, (b) e, (c) , (d) .

(a)

(b)

(c)

(d)

Figure 6: Allowed regions of general NP parameters for at specific values of branching fraction. is (a) unpolarized, (b) e, (c) , (d) . The three contours belong to branching fraction at .

(a)

(b)

(c)

(d)

Figure 7: Variation of branching fraction with respect to NP parameter at several values of for . is (a) unpolarized, (b) e, (c) , (d) . Experimental bound on the process is 9.8 .

(a)

(b)

(c)

(d)

Figure 8: Allowed regions of general NP parameters for at specific values of branching fraction. is (a) unpolarized, (b) e, (c) , (d) . The three contours belong to branching fraction at .

(a)

(b)

(c)

(d)

Figure 9: Variations of branching fraction with respect to NP parameter at several values of for . is (a) unpolarized, (b) e, (c) , (d) . Experimental bound on the process is 1.6 .

(a)

(b)

(c)

(d)

Figure 10: Variations of branching fraction (NP contribution only) with respect to NP parameter at several values of for . is (a) unpolarized, (b) e, (c) , (d) .

(a)

(b)

(c)

(d)

Figure 11: Variations of branching fraction (NP contribution only) with respect to NP parameter at several values of for . is (a) unpolarized, (b) e, (c) , (d) .

(a)

(b)

(c)

(d)

Figure 12: Variations of branching fraction (NP contribution only) with respect to NP parameter at several values of for . is (a) unpolarized, (b) e, (c) , (d) .

(a)

(b)

(c)

(d)

Figure 13: Variations of branching fraction (NP contribution only) with respect to NP parameter at several values of for . is (a) unpolarized, (b) e, (c) , (d) .

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 These plots relate the branching fraction of the said process with the magnitude and phase of NP parameters ( and ). 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 level of measured branching fraction (). It clearly shows that only a narrow range of phase of NP parameter () is allowed for given 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 as the corresponding MSSM contribution exceeds the experimental limit on branching fraction in 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 , the maximum limits for 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 .

Contour plots in Figure 4 represent the allowed region for NP parameters ( and ) for specific values of branching fraction of the process . 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 MSSM dominates over SM contribution by order of magnitude , 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 with respect to the magnitude of NP parameter at several values of its phase .

Table 3: Bounds on NP parameters (, derived from for . is (a) unpolarized ( and for (b) e, (c) , (d) . Experimental bound on the process is .

(a)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(b)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(c)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(d)

Bounds on NP Parameter
Branching Fraction ()

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

Table 4: Bounds on NP parameters (, for . is (a) unpolarized ( and for (b) e, (c) , (d) . Experimental bound on the process is .

(a)

Bounds on NP Parameter (unpolarized)

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

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

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

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 (, for . is (a) unpolarized () and () for (b) e, (c) , (d) . Experimental bound on the process is .

(a)

Bounds on NP Parameter (unpolarized)

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

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

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

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

Contour graphs in Figure 6 illustrate the allowed region for NP parameters ( and ) of the process 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 () at several values of its phase . 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 . 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 ( and ) of the process 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 with the magnitude of NP parameters at several values of its phase . 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 (). This particular pattern shows that only a narrow range of phase 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 display the variation of branching fraction (NP contribution only) with respect to the magnitude of NP parameter at several values of its phase . For pure leptonic decays of strange mesons involving neutrinos (), there is no experimental data available. Therefore, we use limits derived from 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 , respectively, which shows that MSSM enhances SM contribution by order of 10 for . 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 () derived from for . is (a) unpolarized (, (b) e (, (c) (, (d) ().

(a)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(b)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(c)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(d)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

Table 7: Bounds on NP parameters derived from for . is (a) unpolarized , (b) e , (c) , (d) .

(a)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(b)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(c)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(d)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

Plots in Figures 12 and 13 describe the variation of branching fraction (NP contribution only) of the process with the magnitude of NP parameters, ( at several values of . For pure leptonic decays of beauty involving neutrinos (), there is no experimental data available for these processes, and we use limits derived from 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 , which shows that MSSM enhances SM contribution by order of magnitude 10 for and for and also the effect of destructive interference is not strong to affect SUSY enhancement.

Table 8: Bounds on NP parameters , derived from for . is (a) unpolarized (6.35, (b) e (6.53, (c) (4.87), (d) (6.3)

(a)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(b)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(c)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0
1.7907
0.792

0.829

30
1.8043
0.804

0.825

60
1.8252
0.823

0.793

90
1.8481
0.844

0.814

120
1.8667
0.861

0.883

150
1.8759
0.87

0.908

180
1.873
0.867

0.857

210
1.8589
0.854

0.814

240
1.8376
0.834

0.833

270
1.8148
0.814

0.853

300
1.7967
0.798

0.812

330
1.7879
0.79

0.756

360
1.7907
0.792

0.768

(d)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

Table 9: Bounds on NP parameters derived from for . is (a) unpolarized , (b) e , (c) , (d) ).

(a)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(b)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(c)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

(d)

Bounds on NP Parameter
Branching Fraction (

NP
Interference
Combined

0

30

60

90

120

150

180

210

240

270

300

330

360

5. Summary and Conclusion

Summarizing, we have carried out an analysis of semileptonic (, ) and pure leptonic () decays of pseudoscalar mesons within the framework of 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 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 ( and ). In general, SM contribution is dominated by MSSM contribution, i.e., by a factor of 10 for the pure leptonic decays of and by and in case of and respectively. The interference term between MSSM and SM is both constructive and destructive but except for the semileptonic decays of , it does not affect the enhancement due to SUSY. This makes 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.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

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