Review Article  Open Access
Berin Belma Şirvanlı, "Rare Decay in Family Nonuniversal Model", Advances in High Energy Physics, vol. 2015, Article ID 298604, 5 pages, 2015. https://doi.org/10.1155/2015/298604
Rare Decay in Family Nonuniversal Model
Abstract
The rare decay with polarized photon is studied in the framework of a family nonuniversal model. The branching ratio and photon polarization asymmetry to the model parameters are calculated and compared with the Standard Model. Deviations from the Standard Model will indicate the presence of new physics.
1. Introduction
One of the main aims of the LHC is investigating meson decays. Two distinct aspects may indicate why we should investigate Bmesons.
From theoretical aspects, the rare decays occur at loop level in the SM. Also the flavor changing neutral current (FCNC) transitions are forbidden at the tree level in Standard Model (SM). Besides, these processes only occur at the loop level in the weak interactions. Any finding about the FCNC transitions could be quite beneficial. These are plenty of areas in which they could be useful for calculation such as CabibboKobayashiMaskawa (CKM) matrix elements and leptonic decay constant. Moreover, these rare decays are able to prove new physics beyond the SM. Moreover, when the photon is emitted from external charged leptons, it is proportional to lepton mass which gives small contribution. They are helicity suppressed by a factor of . Despite this suppression factor, this decay, that is, , has been observed at LHC [1]. If we inspect channel, there is no such suppression.
From experimental point of view, their observation can be difficult because of the low efficiency. In case of massless neutrino, the decay is forbidden in the SM because of the helicity conservation. In the rare decay, since the helicity suppression is removed, we expect the larger branching ratio (BR). For the rare decay, branching ratio is of the order of ~, so that these decays might have very clear experimental signature.
If the rare decay has the same order branching ratio as that of , it can be measurable in the near future. In [2], this process was studied also within the framework of a family nonuniversal model.
The purpose of this study was to examine the rare decay in family nonuniversal model by taking into account the polarization of the photon. decay has been investigated in the SM by using the constituent quark model and pole models [3] for the determination of the leptonic decay constants .
In case of massless neutrino, this decay was studied within the framework of the lightcone QCD sum rules method in [4]. In [4], the Hamiltonian formed of a single term representing the four vector interactions of the left handed neutrinos. But, according to the results obtained from the Super Kamiokande experiment [5, 6], the neutrinos have mass which could have right handed components. Similar FCNC decays in detail are studied in [7, 8]. We note that the rare decay has been previously studied in [9, 10] in a model independent way. The final state photon can be revealed by examining the polarization in a radiative decay mode like . Investigating the effects of polarized photon may provide another kinematical variable, like to the differential and total branching ratios for radiative decays [11].
In this work we will investigate sensitivity of such “photon polarization asymmetry” in decay to the new Wilson coefficients in the family nonuniversal model.
The paper is organized as follows: in Section 2, we present the family nonuniversal model and form of the effective Hamiltonian and the parameterization of the hadronic matrix elements in terms of appropriate form factors. We then calculate the differential decay width and the photon polarization asymmetry for the decay. Section 3 is devoted to the numerical analysis and discussion of our results.
2. Matrix Element for the Decay
For the exclusive transition, the decay is described at quark level. In the SM, the effective Hamiltonian for the transition is [12–14] where and are the elements of the CabibboKobayashiMaskawa (CKM) matrix. Consider Here, and is a term, which gives very small contribution. The explicit expression of can be found in [15].
If the mixing between and is neglected, the contribution coming from can be described by just modifying the Wilson coefficient without introducing any new operator structure. The expression of the effective Hamiltonian in this case can be written as follows [16–18]: where and correspond to the interaction vertex of with quark and leptons. It follows from (1) and (2) that in order to take into account the contributions coming from the boson it is enough to modify the Wilson coefficients in the following way: where is the strong coupling constant:
Using the effective Hamiltonian, the matrix elements for the decay at hadronic level can be calculated. Although the rare decay is forbidden because of helicity conservation in case of massless neutrino, when a photon is emitted from initial quark lines, this process replaces it with the corresponding radiative one. Thus, we have eliminated the helicity suppression. After this operation, the rare decay has the following properties.(1)If a photon is emitted from internal charged particles, then one gets a suppression factor . Therefore the contributions of such diagrams can be safely neglected.(2)The Wilson coefficient is the same for the and as a consequence of Law’s low energy theorem [19].
As the last step for calculation, the matrix elements that we need for the decay are defined as follows (see [20–25]): where and are the fourvector polarization and four momenta of the photon, respectively, is the momentum transfer, and is the momentum of the meson. Using (1), the matrix element for decay can be written as follows: where and .
The differential decay rate of the rare decay was calculated as a function of dimensionless parameter , where is the photon energy.
We have to examine the polarization of photons to obtain the final state photon in such a radiative decay. In fact, the main question is how sensitive is the branching ratio to the model parameters when the photon is in the positive or negative helicity states.
In the center of mass frame of for the rare decay, we can prove and where fourmomentum and polarization vectors, and , are as follows: where , , and . We get the angle in (8) as . is the angle between the momentum of the meson and that of in the center of mass frame of .
With this information, we can obtain where with for , respectively.
In order to observe the effects of polarized photon, we have to calculate a variable “photon polarization asymmetry” [11]: where
3. Results and Discussion
In this part, we will indicate our numerical analysis about the branching ratio (BR) and the photon polarization asymmetry for the rare decay. To make some numerical estimates, the explicit forms of the form factors , , , and are necessary in (9) and (12). In the framework of lightcone QCD sum rules, the form factors were calculated in [20, 21], in terms of two parameters and as where the values and for the transition are listed in Table 1.

We have performed the numerical analysis about the branching ratio (BR) and the photon polarization asymmetry for decay in the family nonuniversal model. In this study, we have been used to the input parameters as follows:
Besides the remaining input parameters of the family nonuniversal model, using the latest improvement measurements on meson decays in [26, 27] for the coupling parameters , and are obtained under two circumstances:
Using the bound on the mass of the boson with what follows from analysis of decay (see, e.g., [26, 27] and LHC data [1]) for the and parameters we get what is presented in Table 2.

In Figure 1, we indicate the dependence of the and for decay in the family nonuniversal model. The superscripts and represent the positive and negative helicity states of photon, respectively. We reach the information from these figures that the branching ratio in both cases is very sensitive to the model parameters. For S1 scenario, is larger about times and for S2 scenario is larger about times compared to that of the SM prediction. For S1 scenario, gets even larger enhancement, which is about times, and for S2 scenario is larger about times compared to the SM.
In Figure 2, we show the dependence of the integrated photon polarization asymmetry as a function of for decay in the family nonuniversal model. From this figure, we see that, in scenario S2, at low photon energies for predicts are larger four times than that one in SM.
As a conclusion, the branching ratio of the rare decay was examined when photon has positive and negative helicities. By the same token, the photon polarization asymmetry of this decay was calculated by using the family nonuniversal model parameters. One can conclude that measurement of at low photon energies can give beneficial information about new physics. It would be possible to detect this rare process in the LHC. At the LHC and TeV hadronic machines pair per year [26, 27] will be produced. Therefore, the number of expected events is which detect this rare decay. The signature of this rare decay will be single photon and missing energy.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author would like to thank T. M. Aliev and N. K. Pak for invaluable comments and useful discussions.
References
 R. Aaij, C. A. Beteta, A. Adametz et al., “First evidence for the decay ${B}_{s}^{0}\to {\mu}^{+}{\mu}^{}$,” Physical Review Letters, vol. 110, no. 2, Article ID 021801, 9 pages, 2013. View at: Publisher Site  Google Scholar
 B. B. Sirvanli, “Lepton asymmetries for the ${B}_{s}\to \gamma {l}^{+}{l}^{}$ decay in a family nonuniversal Z′ model,” Physical Review D, vol. 89, Article ID 095016, 2014. View at: Publisher Site  Google Scholar
 C. D. Lü and D. X. Zhang, “${B}_{s}\left({B}_{d}\right)\to \gamma v\stackrel{}{v}$,” Physics Letters B, vol. 381, no. 1–3, pp. 348–352, 1996. View at: Publisher Site  Google Scholar
 T. M. Aliev, A. Özpineci, and M. Savc, “Rare $B\to v\stackrel{}{v}\gamma $ decay in light cone QCD sum rule,” Physics Letters B, vol. 393, no. 12, pp. 143–148, 1997. View at: Publisher Site  Google Scholar
 Y. Fukuda, T. Hayakawa, E. Ichihara et al., “Evidence for oscillation of atmospheric neutrinos,” Physical Review Letters, vol. 81, no. 8, pp. 1562–1567, 1998. View at: Publisher Site  Google Scholar
 M. S. Alam, J. Kim, Z. Ling et al., “First measurement of the rate for the inclusive radiative penguin decay $b\to s\gamma $,” Physical Review Letters, vol. 74, p. 2885, 1995. View at: Publisher Site  Google Scholar
 D. Melikhov and N. Nikitin, “Rare radiative leptonic decays ${B}_{d,s}\to {l}^{+}{l}^{}\gamma $,” Physical Review D, vol. 70, Article ID 114028, 2004. View at: Publisher Site  Google Scholar
 Y. G. Aditya, K. J. Healey, and A. A. Petrov, “Faking ${B}_{s}\to {\mu}^{+}{\mu}^{}$,” Physical Review D, vol. 87, Article ID 074028, 2013. View at: Publisher Site  Google Scholar
 O. Cakir and B. Sirvanl, “Rare ${B}_{s}\to \nu $ overline $\nu $γ decay beyond the standard model,” Acta Physica Polonica B, vol. 34, pp. 2643–2650, 2003, http://arxiv.org/abs/hepph/0210019. View at: Google Scholar
 B. Sirvanl and G. Turan, “Rare ${B}_{S}\to \gamma v\stackrel{\u0305}{v}$ decay with polarized photon and new physics effects,” Modern Physics Letters A, vol. 18, pp. 47–56, 2003. View at: Publisher Site  Google Scholar
 S. Rai Choudhury and N. Gaur, “Supersymmetric effects in ${B}_{s}\to {\mathcal{l}}^{+}{\mathcal{l}}^{}\gamma $ decays,” http://xxx.lanl.gov/abs/hepph/0205076. View at: Google Scholar
 G. Buchalla, A. J. Buras, and M. E. Lautenbacher, “Weak decays beyond leading logarithms,” Reviews of Modern Physics, vol. 68, no. 4, pp. 1125–1244, 1996. View at: Publisher Site  Google Scholar
 M. Misiak, “The $\text{b}\to {\text{se}}^{+}{\text{e}}^{}$ and $\text{b}\to \text{s}\gamma $ decays with nexttoleading logarithmic QCDcorrections,” Nuclear Physics B, vol. 393, no. 12, pp. 23–45, 1993. View at: Publisher Site  Google Scholar
 M. Misiak, “Erratum,” Nuclear Physics B, vol. 439, no. 12, pp. 461–465, 1995. View at: Google Scholar
 G. Buchalla and A. J. Buras, “QCD corrections to rare K and Bdecays for arbitrary top quark mass,” Nuclear Physics B, vol. 400, no. 1–3, pp. 225–239, 1993. View at: Publisher Site  Google Scholar
 C. H. Chen and H. Hatanaka, “Nonuniversal Z′ couplings in B decays,” Physical Review D, vol. 73, Article ID 075003, 2006. View at: Publisher Site  Google Scholar
 V. Barger, L. Everett, J. Jiang, P. Langacker, T. Liu, and C. E. M. Wagner, “Family nonuniversal $U{\left(1\right)}^{\prime}$ gauge symmetries and $b\to s$ transitions,” Physics Letters D, vol. 80, Article ID 055008, 2009. View at: Publisher Site  Google Scholar
 T. M. Aliev and M. Savci, “Lepton polarization effects in ${\Lambda}_{b}\to \Lambda {\mathcal{l}}^{+}{\mathcal{l}}^{}$ decay in family nonuniversal ${Z}^{\text{'}}$ model,” Physics Letters B, vol. 718, no. 2, pp. 566–572, 2012. View at: Publisher Site  Google Scholar
 G. L. Lin and Y. P. Yao, “Top quark mass dependence of the decay ${B}_{s}\to \gamma \gamma $ in the standard electroweak model,” Physical Review D, vol. 42, pp. 2319–2323, 1990. View at: Publisher Site  Google Scholar
 G. Eilam, I. Halperin, and R. R. Mendel, “Radiative decay $B\to l\nu \gamma $ in the light cone QCD approach,” Physics Letters B, vol. 361, no. 1–4, pp. 137–145, 1995. View at: Publisher Site  Google Scholar
 T. M. Aliev, A. Özpineci, and M. Savc, “Rare $B\to {\mathcal{l}}^{+}{\mathcal{l}}^{}\gamma $ decay and new physics effects,” Physics Letters B, vol. 520, pp. 69–77, 2001. View at: Publisher Site  Google Scholar
 A. Buras and M. Münz, “Effective Hamiltonian for $B\to {X}_{s}{e}^{+}{e}^{}$ beyond leading logarithms in the naive dimensional regularization and ’t Hooft–Veltman schemes,” Physical Review D, vol. 52, article 186, 1995. View at: Google Scholar
 C. H. Chen and C. Q. Geng, “Study of ${\mathrm{\Lambda}}_{b}\to \mathrm{\Lambda}v\stackrel{\u0305}{v}$ with polarized baryons,” Physical Review D, vol. 63, Article ID 054005, 2001. View at: Publisher Site  Google Scholar
 G. Eilam, I. Halperin, and R. R. Mendel, “Radiative decay $B\to lv\gamma $ in the light cone QCD approach,” Physics Letters, Section B: Nuclear, Elementary Particle and HighEnergy Physics, vol. 361, no. 1–4, pp. 137–145, 1995. View at: Google Scholar
 T. M. Aliev, A. Ozpineci, and M. Savci, “${B}_{q}\to {l}^{+}{l}^{}\gamma $ decays in light cone QCD,” Physical Review D, vol. 55, p. 7059, 1997. View at: Google Scholar
 Q. Chang, X. Q. Li, and Y. D. Yang, “A comprehensive analysis of hadronic $b\to s$ transitions in a family nonuniversal ${Z}^{\prime}$ model,” Journal of Physics G, vol. 41, no. 10, Article ID 105002, 2014. View at: Publisher Site  Google Scholar
 A. J. Buras and J. Girrbach, “Lefthanded Z′ and Z FCNC quark couplings facing new $b\to s{\mu}^{+}{\mu}^{}$ data,” Journal of High Energy Physics, vol. 121, article 009, 2013. View at: Google Scholar
Copyright
Copyright © 2015 Berin Belma Şirvanlı. 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 SCOAP^{3}.