Advances in High Energy Physics

Volume 2018, Article ID 5093251, 10 pages

https://doi.org/10.1155/2018/5093251

## Effects of Leptonic Nonunitarity on Lepton Flavor Violation, Neutrino Oscillation, Leptogenesis, and Lightest Neutrino Mass

Department Of Physics, Gauhati University, Guwahati, Assam 781014, India

Correspondence should be addressed to Gayatri Ghosh; moc.liamg@hshgirtayag

Received 19 January 2018; Revised 20 March 2018; Accepted 2 April 2018; Published 25 June 2018

Academic Editor: Theocharis Kosmas

Copyright © 2018 Gayatri Ghosh and Kalpana Bora. 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}.

#### Abstract

Neutrino physics is a mature branch of science with all the three neutrino mixing angles and two mass squared differences determined with high precision. In spite of several experimental verifications of neutrino oscillations and precise measurements of two mass squared differences and the three mixing angles, the unitarity of the leptonic mixing matrix is not yet established, leaving room for the presence of small nonunitarity effects. Deriving the bounds on these nonunitarity parameters from existing experimental constraints, on cLFV decays such as , , and , we study their effects on the generation of baryon asymmetry through leptogenesis and neutrino oscillation probabilities. We consider a model where see-saw is extended by an additional singlet which is very light but can give rise to nonunitarity effects without affecting the form on see-saw formula. We do a parameter scan of a minimal see-saw model in a type I see-saw framework satisfying the Planck data on baryon to photon ratio of the Universe, which lies in the interval . We predict values of lightest neutrino mass and Dirac and Majorana CP-violating phases , , and , for normal hierarchy and inverted hierarchy for one-flavor leptogenesis. It is worth mentioning that all these four quantities are unknown yet, and future experiments will be measuring them.

#### 1. Introduction

Neutrinos have nonzero masses. There are 3 known flavors of neutrinos, , , and , each of which couples only to the charged lepton of the same flavor. , , and are superpositions of three mass eigenstates, , where and is the neutrino of definite mass . The cosmological constraints of the sum of the masses bound are from CMB, Planck 2015 data (CMB15+ LRG+ lensing + ) [1]. We note that the lepton mixing matrix has a big mixing and we know almost nothing about the phases. The discoveries of neutrino mass and leptonic mixing have come from the observation of neutrino flavor change, . CP violation interchanges every particle in a process by its antiparticle. This CP violation can be produced by the phase in . Neutrinos can have two types of mass term in the Lagrangian−Dirac and Majorana mass terms. To determine whether Majorana masses occur in nature, so that , the favorable approach to seek is Neutrinoless Double Beta Decay ().

In the conventional type I see-saw framework, there are Dirac and Majorana mass matrices and in the Lagrangian: The low-energy mass matrix is given by In the usual unitarity scenario, the three active neutrinos, the flavor eigenstates , , and are connected to the mass eigenstates , , and via , where . Here is the generalized neutrino mixing matrix which could be either unitary or nonunitary. In the diagonal charged lepton basis, is diagonalized by a unitary matrix as The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix is UP, where U is Here [3] (see [3] for recent global fit values) are the solar, atmospheric, and reactor angles for Normal Ordering (Inverted Ordering), respectively. The Majorana phases reside in P, where Cosmologists suggest that, just after the Big Bang, the Universe contained equal amounts of matter and antimatter. Today the Universe contains matter but almost no antimatter. This change needs that matter and antimatter act differently (CP violation). The CP-violating scenario to explain this change is leptogenesis. Leptogenesis is a natural outcome of the see-saw mechanism. In the see-saw picture, we assume that, just as there are 3 light neutrinos , there are 3 heavy right-handed neutrinos , where GeV, which were there in the Hot Big Bang. The decays modes arewhere are and are SM Higgs. CP violation effects in the decays may result from phases in the decay coupling constants. This leads to unequal numbers of leptons ( and ) and antileptons ( and ) in the Universe: In leptogenesis, CP-violating decays of heavy Majorana neutrinos create a lepton−antilepton asymmetry [4] and then B+L violating sphaleron processes [5] at and above the electroweak symmetry breaking scale convert part of this asymmetry into the observed baryon-antibaryon asymmetry. The heavy neutrinos are see-saw partners of the observed light ones.

Depending on mass of the lightest heavy RH Majorana neutrinos (whose decay causes leptogenesis) the leptogenesis can be of three types: unflavored (or one-flavor), two-flavored, and three-flavored leptogenesis. For unflavored leptogenesis, valid for GeV, we have taken here GeV where the flavor of the final state leptons plays no role. It can be shown that for lower values of it depends on the flavor of the final state leptons and hence is called flavored leptogenesis [6]. Here we consider unflavored leptogenesis. For unflavored leptogenesis, the decay asymmetry in the case of hierarchical heavy neutrinos is given by [7] where for 1, i.e., for hierarchical heavy neutrinos. The baryon asymmetry of the Universe is proportional to the decay asymmetry . The Dirac mass matrix in terms of a complex and orthogonal matrix is [8] The compatible quantity for leptogenesis is then If is real then there is no leptogenesis at all. Here, we have taken (see (4)). consists of the low-energy mixing elements and the CP phases. Since unitarity of neutrino mixing matrix has not been proved yet, if it is nonunitary, then, for the neutrino mixing matrix N to be nonunitary, we have connecting the flavor and mass states. The nonunitary matrix N can be written as where . If , which is diagonalized by a nonunitary mixing matrix, originates from the see-saw mechanism, we have And, thereupon, we have For nonunitary matrix, . Leptogenesis is no longer independent of the low-energy phases. It depends on the phases in as well as the phases in . Leptogenesis [9] is one of the exceedingly well-inspired frameworks which produces baryon asymmetry of the Universe through B + L violating electroweak sphaleron process [5, 10].

In the conventional type I see-saw mechanism, due to the mixing of the left-handed and right-handed neutrinos the PMNS matrix is nonunitary. Nevertheless this nonunitarity is too small to have any observable effects in lepton flavor violation (LFV) or neutrino oscillation (see [7, 11, 12]). If we want to connect the lepton mixing matrix to leptogenesis via the Casas-Ibarra parametrization, then there should not be any significant sizeable contribution to and leptogenesis other than the usual see-saw terms. Hence we must decouple the origin of unitary violation from these terms. Mixing of the light neutrinos with new physics creates nonunitarity in the low-energy mixing matrix. We consider here a model as used in [7] where the see-saw mechanism is enlarged by an additional singlet sector, which leads to a 9 9 mass matrix. Here where the upper left block is the one corresponding to usual type I see-saw mechanism. It can be diagonalized with a unitary matrix , such that The form of is as given in [7]. As discussed in [7] with proper choice of various elements in and of , it can be shown that which shows that is the lepton mixing matrix, as there is no other significant contribution to the mass term of the light neutrinos. Thus in this model the usual see-saw mechanism remains unaltered with unmodified leptogenesis. Here does not couple to the new singlets but a sizeable nonunitary lepton mixing matrix can be induced, thus providing us with a framework where we can apply (14).

We consider here that lepton asymmetry is generated by out-of-equilibrium decay of heavy right-handed Majorana neutrinos into Higgs and lepton within the framework of type I see-saw mechanism. In a hierarchical case of three right-handed heavy Majorana neutrinos , with the lepton asymmetry created by the decay of , the lightest one of three heavy right-handed neutrinos is [13]

Here = 174 GeV is the vacuum expectation value of the SM Higgs and , where

At temperatures, GeV, all the charged leptons are in equilibrium because the direct and inverse decays are very frequent and wash out any asymmetry. At moderate temperatures GeV ( GeV), some particles decouple and thus flavor effects play an important role in the calculation of lepton asymmetry [6, 14–18]. The regions of temperatures belonging to and are, respectively, denoted as two and three flavor regimes of leptogenesis [19].

The building blocks of matter are the quarks, the charged leptons, and the neutrinos. The discovery and study of the Higgs boson at the Large Hadron Collider (LHC) have provided strong evidence that the quarks and charged leptons derive their masses from a coupling to the Higgs field. Most theorists strongly believe that the origin of the neutrino masses is different from the origin of the quark and charged lepton masses. Neutrino oscillation has proved that neutrinos have nonzero masses. We and all matter may have descended from heavy neutrinos. We list the values of for one-flavor for different hierarchies and unitarity and nonunitarity of in Table 2 and check whether our values of are consistent with the constraints on the absolute scale of masses. The new results presented in this work are as follows:

(i) We have calculated new values of nonunitarity parameters of matrix from the bounds on rare cLFV decays.

(ii) Hence we predicted the absolute value of lightest mass in this regard. The values of lightest mass lie in the ranges of 0.0018 eV to 0.0023 eV, 0.048 eV to 0.056 eV, 0.05 eV to 0.054 eV, and 0.053 eV to 0.062 eV in one-flavor leptogenesis regime.

(iii) All these values satisfy the constraint .

(iv) The predicted values of CP-violating phases, , and Majorana phases and are , , , , , , , , , and . Here the calculated values of each of them, i.e., the Dirac CP-violating phase, , and the Majorana phases and , are found to be same.

The paper is organized as follows. In Section 2, we show the effect of low-energy phenomenology of nonunitarity on charged lepton flavor violating decays in type I see-saw theories and present the values of various parameters used in our analysis for the generation of baryon asymmetry of the Universe through the mechanism of leptogenesis. Section 3 contains our calculations and results. Section 4 contains analysis and discussions. Section 5 summarizes the work.

#### 2. Low-Energy Phenomenology of Nonunitarity and Leptogenesis

One interesting feature of nonunitarity of the PMNS matrix can be studied in rare charged lepton flavor violation (LFV) decay processes. In the light of unitarity violation in decays such as , the branching ratio is [7] AlsoUsing the latest updated constraint on [2], one can derive bounds on from (23). It can be shown that Now, we calculate the ratio Thus we find constraints on from (24), using the latest constraint on , where [2]. Again we have From our calculation, the ratio is And then we calculate the latest updated bounds on . The calculations are summarized in Table 1. The study of the effects of leptonic nonunitarity on two- and three-flavored leptogenesis is presented in [20]. We note that interesting results on cLFV in NUSM, NUHM, NUGM, and mSUGRA models are presented in [21] in which we have predicted some values of new SUSY particles that may be detected at next run of LHC.