We study a $μ- τ$ reflection symmetry in neutrino sector realized at the GUT scale in the context of the seesaw model. In our scenario, the exact $μ- τ$ reflection symmetry realized in the basis where the charged lepton and heavy Majorana mass matrices are diagonal, leads to vanishing lepton asymmetries. We find that, in the minimal supersymmetry extension of the seesaw model with appropriate values of $\tanβ$, the renormalization group (RG) evolution from the GUT scale to seesaw scale can induce a successful leptogenesis. It is shown that the right amount of the baryon asymmetries $η_B$ can be achieved via so-called resonant leptogenesis, which can be realized at rather low seesaw scale in our scenario, so that the well-known gravitino problem is safely avoided. In this work, we consider both flavor dependent and flavor independent leptogenesis, and demonstrate how they lead to different amounts of baryon asymmetries in detail.
We study the possibility that the baryon asymmetry of the universe is generated in a minimal seesaw scenario where two right-handed Majorana neutrinos with degenerate masses are added to the standard model particle content. In the usual framework of thermal leptogenesis, a nonzero $CP$ asymmetry can be obtained through the mass splitting induced by the running of the heavy Majorana neutrino masses from their degeneracy scale down to the seesaw scale. Although, in the light of the present neutrino oscillation data, the produced baryon asymmetry turns out to be smaller than the experimental valu…
We consider a $μ-τ$ symmetry in neutrino sectors realized at GUT scale in the context of a seesaw model. In our scenario, the exact $μ-τ$ symmetry realized in the basis where the charged lepton and heavy Majorana neutrino mass matrices are diagonal leads to vanishing lepton asymmetries. We find that, in the minimal supersymmetric extension of the seesaw model with large $\tanβ$, the renormalization group (RG) evolution from GUT scale to seesaw scale can induce a successful leptogenesis even without introducing any symmetry breaking terms by hand, whereas such RG effects lead to tiny deviations…
We demonstrate how loop effects in gravitational backgrounds lead to a difference in the propagation of matter and antimatter, and show this is forbidden in flat space due to CPT and translation invariance. This mechanism, which is naturally present in beyond the standard model (BSM) theories exhibiting C and CP violation, generates a curvature-dependent chemical potential for leptons in the low-energy effective Lagrangian, allowing a matter-antimatter asymmetry to be generated in thermodynamic equilibrium, below the BSM scale.
We propose an extension of the standard model (SM) by including a dark sector comprising of three generations of heavy right-handed neutrinos, a singlet scalar and a singlet Dirac fermion, where the latter two particles are stable and are viable candidates of dark matter (DM). In the early Universe, the CP-violating out-of-equilibrium decay of heavy right-handed neutrinos to singlet Dirac fermion and scalar in the dark sector generates a net DM asymmetry. The latter is then transported to the visible sector via a dimension eight operator which conserves $B-L$ symmetry and is in thermal equilib…
We present and discuss constraints on supersymmetric type I seesaw models imposed by neutrino data, charged lepton flavor violation and thermal leptogenesis.
Minimal models exist in which thermal, resonant leptogenesis occurs at the electroweak scale. The effects of individual lepton flavours play a crucial role and allow successful leptogenesis with large couplings between some of the charged leptons and the heavy Majorana neutrinos. This leads to potentially observable signals for low energy experiments (such as MEG, MECO and PRIME) and future linear colliders.
In the framework of seesaw models with only two heavy Majorana neutrinos, nonzero leptonic asymmetries can be radiatively generated when exact heavy neutrino mass degeneracy ($M_1=M_2=M$) is imposed at a scale $Λ_D > M$. For a specific case, we show that an acceptable value for the baryon asymmetry of the Universe can be obtained considering thermal leptogenesis.
We study the role played in leptogenesis by the equilibration of lepton flavors, as could be induced in supersymmetric models by off diagonal soft breaking masses for the scalar lepton doublets $\tilde m_{αβ}$, or more generically by new sources of lepton flavor violation. We show that if $\tilde m_{αβ}\gsim 1 $GeV and leptogenesis occurs below $\sim 100 $TeV, dynamical flavor effects are irrelevant and leptogenesis is correctly described by a one-flavor Boltzmann equation. We also discuss spectator effects in low scale leptogenesis by taking into account various chemical equilibrium condition…
Flavour models may display a relation between the CP-violating asymmetry for leptogenesis and low-energy parameters. If the flavour symmetry produces an exact mass independent lepton mixing scheme at leading order (with type I see-saw) the CP-violating asymmetry would vanish in the absence of corrections. We present a model displaying the link between deviations from the mixing scheme and leptogenesis.
Thermal leptogenesis, that can be viewed as a consequence of the seesaw model, is a very natural mechanism to explain the matter anti-matter asymmetry of the Universe. Recently, lepton flavours have been included in the Boltzmann equations, modifying significantly the evaluation of the efficiency of leptogenesis to explain the observed baryon asymmetry.
Baryon and lepton numbers of the Universe in leptogenesis with Dirac neutrino and leptogenesis with Majorana neutrino scenarios are considered. It is shown that despite quite different features of Dirac and Majorana fermions both scenarios yield the same relation among the initial lepton and the final baryon asymmetries. Moreover right-handed neutrinos in the leptogenesis with Dirac neutrino scenario have very little impact on the effective number of relativistic degrees of freedom, constrained by BBN. Thus the two scenarios are similar from the cosmological point of view. It is also pointed o…
In models with flavour symmetries added to the gauge group of the Standard Model the CP-violating asymmetry necessary for leptogenesis may be related with low-energy parameters. A particular case of interest is when the flavour symmetry produces an exact mass independent lepton mixing scheme, leading to a vanishing CP-violating asymmetry. We present a model-independent discussion that confirms this always occurs for unflavoured leptogenesis in type I see-saw scenarios.
We investigate what would be the consequences for leptogenesis of the existence of a charged SU(2)_L singlet scalar delta^+. If such a scalar particle exists, it allows the right-handed neutrinos to couple not only to left-handed lepton and Higgs doublets as in ordinary leptogenesis, but also to a right-handed charged lepton and a delta^+. This provides a new source of leptogenesis which can be successful in a non-resonant way at scales as low as TeV. The incorporation of this scenario in left-right symmetric and unified models is discussed.
We discuss minimal non-supersymmetric models of resonant leptogenesis, based on an approximate flavour symmetries. As an illustrative example, we consider a resonant tau-leptogenesis model, compatible with universal right-handed neutrino masses at the GUT scale, where the required heavy-neutrino mass splittings are generated radiatively. In particular, we explicitly demonstrate, how a minimum number of three heavy Majorana neutrinos is needed, in order to obtain successful leptogenesis and experimentally testable rates for processes of lepton flavour violation, such as mu --> e gamma and mu --…
We propose a new class of leptogenesis bounds on the spontaneous symmetry breaking of global lepton number. These models have a generic feature of inducing new lepton number violating interactions, due to the presence of the Majorons. We analyzed the singlet Majoron model with right-handed neutrinos and find that the lepton number should be broken above 10^5 GeV to realize a successful leptogenesis because the annihilations of the right-handed neutrinos into the massless Majorons and into the standard model Higgs should go out of equilibrium before the sphaleron process is over. We then argue …
We discuss flavour dependent lepton asymmetries in the Exceptional Supersymmetric Standard Model (E$_6$SSM). In the E$_6$SSM, the right-handed neutrinos do not participate in gauge interactions, and they decay into leptons and leptoquarks. Their Majorana nature allows violation of lepton number. New particles and interactions can result in substantial lepton asymmetries, even for scales as low as $10^6 {\rm GeV}$.
We propose a radiatively induced quark and lepton mass model in the first and second generation with extra $U(1)$ gauge symmetry and vector-like fermions. Then we analyze the allowed regions which simultaneously satisfy the FCNCs for the quark sector, LFVs including $μ-e$ conversion, the quark mass and mixing, and the lepton mass and mixing. Also we estimate the typical value for the $(g-2)_μ$ in our model.
A discrepancy between the experimental and theoretical results was reported for the anomalous magnetic moment of the muon. In leptogenesis scenarios which are due to a decay of right-handed neutrinos, this anomaly leads to large violations of lepton flavour within the framework of supersymmetric see-saw models. It is shown that for a hierarchical right-handed neutrino mass spectrum, we generically expect to observe μ\to e γin the near future experiment.
We investigate how the baryon asymmetry of our universe via leptogenesis can be achieved within the framework of the seesaw model with Fritzsch type lepton mass matrices proposed by Fukugita $et. al$. We study the cases with CP-violating phases in charged lepton Yukawa matrix, however, with and without Dirac neutrino Yukawa phases. We consider both flavor independent and flavor dependent leptogenesis, and demonstrate how they lead to different amounts of lepton asymmetries in detail. In particular, it is shown that flavor dependent leptogenesis in this model can be worked out only when the CP …
It has been recently shown that the quantum Boltzmann equations may be relevant for the leptogenesis scenario. In particular, they lead to a time-dependent CP asymmetry which depends upon the previous dynamics of the system. This memory effect in the CP asymmetry is particularly important in resonant leptogenesis where the asymmetry is generated by the decays of nearly mass-degenerate right-handed neutrinos. We study the impact of the non-trivial time evolution of the CP asymmetry in the so-called Minimal Lepton Flavour Violation framework where the charged-lepton and the neutrino Yukawa coupl…
We study the constraints on a minimal supersymmetric seesaw model imposed by neutrino data, charged lepton flavor violation, thermal leptogenesis and perturbativity. We show that it is possible to constrain the three heavy Majorana neutrino masses as well as the complex Yukawa coupling matrix. Our results provide a first step towards a seesaw benchmark model for further phenomenological studies and model building.
Dirac leptogenesis (or Dirac neutrinogenesis), in which neutrinos are purely Dirac particles, is an interesting alternative to the standard leptogenesis scenario. In its supersymmetric version, the modified form of the superpotential required for successful baryogenesis contributes new, generically non-flavor-diagonal terms to the slepton and sneutrino mass matrices. In this work, we examine how current experimental bounds on flavor-changing effects in the lepton sector (and particularly the bound on Mu -> e Gamma) constrain Dirac leptogenesis and we find that it is capable of succeeding with …
A novel contribution to the leptonic CP asymmetries in type II seesaw leptogenesis scenarios is obtained for the cases in which flavor effects are relevant for the dynamics of leptogenesis. In the so-called flavoured leptogenesis regime, the interference between the tree-level amplitude of the scalar triplet decaying into two leptons and the one-loop wave-function correction with leptons in the loop, leads to a new nonvanishing CP asymmetry contribution. The latter conserves total lepton number but violates lepton flavour. Cases in which this novel contribution may be dominant in the generatio…
We propose a simple ansatz for neutrino phenomenology, in which the relevant lepton mass matrices take the universal Fritzsch-like form and the seesaw relation holds under a particular condition. There exist six textures of this nature, but their consequences on neutrino oscillations are exactly the same. We show that our scenario is viable to account for the cosmological baryon number asymmetry via thermal leptogenesis. Its predictions for the lepton-flavor-violating decays μ-> eγ, tau -> μγand τ-> eγare also presented. We find that the branching ratios of these rare processes depend strongly…
We consider a mirror world scenario, in which light Dirac neutrinos are generated from a seesaw mechanism and leptogenesis occurs at high scale without violating lepton number. After leptogenesis, the conservation laws of the theory imply the visible baryon-minus-lepton asymmetry to be equal to the mirror baryon-minus-lepton asymmetry. We extend previous work by presenting a detailed study of this Dirac leptogenesis mechanism by constructing the full set of Boltzmann Equations for both cases of unflavored and flavored regimes. We show that $Z_2$ breaking and lepton/mirror lepton flavor effects…
Several classes of neutrino mass models, which can naturally account for hierarchical masses and the bi-large pattern of neutrino mixing, are constructed from a bottom-up perspective based on the idea of decoupling of one right-chiral neutrino from the seesaw mechanism. The interplay between the predictions for the lepton-flavour violation in the MSSM with universal soft masses and for leptogenesis is studied. In particular, classes of neutrino mass models in which successful leptogenesis implies BR(mu->e gamma) potentially observable in upcoming experiments are identified.
The little Higgs scenario may provide an interesting framework to accommodate TeV scale leptogenesis because a TeV Majorana mass of the right-handed neutrino that we employ for the latter may find a natural place near the ultraviolet cutoff of the former. In this work we study how a light neutrino spectrum, generated radiatively, and TeV scale leptogenesis can be embedded in the simplest little Higgs framework. Alternatively, we highlight how the neutrino Yukawa textures of the latter are constrained.
We study the implications of a successful leptogenesis within the framework of Minimal Lepton Flavour Violation combined with radiative resonant leptogenesis and the PMNS matrix being the only source of CP violation, which can be obtained provided flavour effects are taken into account. We find that the right amount of the baryon asymmetry of the universe can be generated under the conditions of a normal hierarchy of the light neutrino masses, a non-vanishing Majorana phase, sin(theta_{13})>0.13 and m_{nu,lightest}<0.04 eV. If this is fulfilled, we find strong correlations among ratios of char…
We calculate the lepton electric dipole moments in a class of supersymmetric seesaw models and explore the possibility that they may provide a way to probe some of the CP violating phases responsible for the origin of matter via leptogenesis. We show that in models where the right handed neutrino masses, $M_R$ arise from the breaking of local B-L symmetry by a Higgs field with B-L=2, some of the leptogenesis phases can lead to enhancement of the lepton dipole moments compared to the prediction of models where $M_R$ is either directly put in by hand or is a consequence of a higher dimensional o…
We investigate the viability of thermal leptogenesis in type-I seesaw models with leptonic flavour symmetries that lead to tribimaximal neutrino mixing. We consider an effective theory with an A4 x Z3 x Z4 symmetry, which is spontaneously broken at a scale much higher than the electroweak scale. At the high scale, leptonic Yukawa interactions lead to exact tribimaximal mixing and the heavy Majorana neutrino mass spectrum is exactly degenerate. In this framework, leptogenesis becomes viable once this degeneracy is lifted either by renormalization group effects or by a soft breaking of the A4 sy…
We propose a model with an $U(1)_{B-L}$ gauge symmetry, in which small neutrino masses, dark matter and the matter-antimatter asymmetry in the Universe can be simultaneously explained. In particular, the neutrino masses are generated radiatively, while the matter-antimatter asymmetry is led by the leptogenesis mechanism, at TeV scale. We also explore allowed regions of the model parameters and discuss some phenomenological effects including lepton flavor violating processes.
We analyze lepton flavour violation (LFV) and the generation of the observed baryon-antibaryon asymmetry of the Universe (BAU) within a generalized minimal lepton flavour violation framework with three quasi-degenerate heavy Majorana neutrinos. The BAU which is obtained through radiative resonant leptogenesis can successfully be generated widely independent of the Majorana scale in this scenario and flavour effects are found to be relevant. Then we discuss the specific case in which CP violation is exclusively present at low-energies (a real R matrix) in the flavour sensitive temperature regim…
We investigate the viability of leptogenesis in models with three heavy right-handed neutrinos, where the charged-lepton and the neutrino Yukawa couplings are the only irreducible sources of lepton-flavour symmetry breaking (Minimal Lepton Flavour Violation hypothesis). We show that in this framework a specific type of resonant leptogenesis can be successfully accomplished. For natural values of the free parameters, this mechanism requires a high right-handed neutrino mass scale ($M_ν\gsim 10^{12}$ GeV). By means of a general effective field theory approach, we analyse the impact of the CP vio…
We demonstrate a common origin for high-scale leptogenesis and three-loop neutrino mass generation. Specifically we extend the standard model by two real singlet scalars, two singly charged scalars carrying different lepton numbers and two or more singlet fermions with Majorana masses. Our model respects a softly broken lepton number and an exactly conserved $Z_2^{}$ discrete symmetry. Through the lepton-number-violating decays of the real scalars and then the lepton-number-conserving decays of the charged scalars, we can obtain a lepton asymmetry stored in the standard model leptons. This lep…
We review the current status of the data on neutrino masses and lepton mixing and the prospects for measuring the CP-violating phases in the lepton sector. The possible connection between low energy CP violation encoded in the Dirac and Majorana phases of the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix and successful leptogenesis is emphasized in the context of seesaw extensions of the Standard Model with a flavor symmetry Gf (and CP symmetry).
We present a purely linear seesaw mechanism in a left-right symmetric framework and then realize a novel leptogenesis scenario for parametrizing the cosmic baryon asymmetty by the charged lepton masses and the light Majorana neutrino mass matrix up to an overall factor. Through the same Yukawa couplings, the lepton-number-conserving decays of the mirror charged leptons can generate three individual lepton asymmetries stored in the ordinary lepton flavors, while the lepton-number-violating processes for the Majorana neutrino mass generation can wash out part of these lepton asymmetries. The rem…
We reexamine the idea that bilinear R-parity violating couplings can be responsible for leptogenesis. We prove that, to have lepton number violation before the electroweak phase transition, misalignment between the lepton-Higgsino and slepton-Higgs sectors must be involved. The processes that generate a lepton asymmetry are bino or wino decays into a lepton-Higgs and slepton-Higgsino final states. Since these decays occur mainly while the gauginos are still in thermal equilibrium, they quantitatively fail to produce the observed baryon asymmetry of the universe.
We study how lepton flavor violation (LFV) is related to the reactor angle $θ_{13}$, LHC/dark matter (DM) signal with a successful leptogenesis in a radiative seesaw model through $A_4$ flavor symmetry breaking, in the light of the deviation of tri-bimaximal mixing (TBM) angles and the recent precision oscillation data. To do this, we consider the normal mass spectrum of light neutrino giving a successful leptogenesis at electroweak (EW) scale or more with a relatively large $\sinθ_{13}\simeq{\cal O}(0.1)$, where a successful leptogenesis requires LHC/DM signal $m_{-}\lesssim$ a few keV in our…
Dirac neutrinos with lepton-number-violating interactions can give rise to a new leptogenesis mechanism. In its simplest renormalizable realization, based on a gauged B-L symmetry spontaneously broken by four units, the decay of a new scalar creates an asymmetry in the right-handed neutrinos. A neutrinophilic two-Higgs-doublet model converts this asymmetry to the baryons, provides a natural explanation of the small neutrino masses, and can lead to an effective number of relativistic degrees of freedom of N(eff) = 3.29 due to the entropy-suppressed contribution of the right-handed neutrinos.
It is well-known that for unflavored leptogenesis there is in general no connection between low and high energy CP violation. We stress that for a non-unitary lepton mixing matrix this may not be the case. We give an illustrative example for this connection and show that the non-standard CP phases that are induced by non-unitarity can be responsible for observable effects in neutrino oscillation experiments as well as for the generation of the baryon asymmetry of the Universe. Lepton Flavor Violation in decays such as tau -> mu gamma can also be induced at an observable level. We also comment …
We consider a minimal extension of the Standard Model with one singlet neutrino per generation that can realize resonant leptogenesis at the electroweak scale. In particular, the baryon asymmetry in the Universe can be created by resonant lepton-to-baryon conversion of an individual lepton number, for example that of the $τ$-lepton. The solar and atmospheric neutrino data can be explained by a simple CP-violating Yukawa texture. The model has several testable phenomenological implications. Most interestingly, it predicts heavy Majorana neutrinos at the sub-TeV scale, which can be probed at $e^…
Seesaw models with a slightly broken lepton number symmetry can explain small neutrino masses, and allow for low-scale leptogenesis. We make a thorough analysis of leptogenesis within the simplest model with two right-handed (RH) neutrinos (or with N_3 decoupled). We obtain a semi-analytical formula for the final asymmetry in both supersymmetric and non-supersymmetric cases with a simple dependence on each parameter. The low-energy parameters factorize from the high-energy ones, and the high-energy phase must be non-zero. The role of the PMNS phases is carefully studied. Moreover, we find that…
In this Letter, we introduce leptogenesis via a varying Weinberg operator from a semi-classical perspective. This mechanism is motivated by the breaking of an underlying symmetry which triggers a phase transition that causes the coupling of the Weinberg operator to become dynamical. Consequently, a lepton anti-lepton asymmetry arises from the interference of the Weinberg operator at two different spacetime points. Using this semi-classical approach, we treat the Higgs as a background field and show a reflection asymmetry between the leptons and anti-leptons is generated in the vicinity of the …
Upcoming Lepton Flavour Violation experiments searching for $μ\rightarrow 3e$ and $μ$ to $ e $ conversion in nuclei processes will provide new opportunities to test the fundamental properties of the neutrino sector, and possibly the origin of matter. In recent work, it was shown that the Type II Seesaw mechanism alone can simultaneously explain the neutrino masses, Leptogenesis, and inflation. A key prediction of this model was the possibility of signals being produced in Lepton Flavour Violation decays. Searches at future experiments such as Mu3e and COMET will be integral to determining the …
We calculate flavour dependent lepton asymmetries within the $E_6$ inspired Supersymmetric Standard Model ($\rm E_6SSM$), which has an extra $U(1)_N$ symmetry. In this model, the right-handed neutrino doesn't participate in gauge interactions, allowing it to be used for both the see--saw mechanism and leptogenesis. Extra Higgs, leptons and leptoquarks predicted by the E$_6$SSM contribute to the ordinary lepton CP asymmetries induced by the decays of the lightest right--handed neutrino (and sneutrino) and give rise to a set of extra decay asymmetries. We find that the CP asymmetries can be rela…
A predictive Leptogenesis scenario is presented based on the Minimal Lepton Flavour Violation symmetry. In the realisation with three right-handed neutrinos transforming under the same flavour symmetry of the lepton electroweak doublets, lepton masses and PMNS mixing parameters can be described according to the current data, including a large Dirac CP phase. The observed matter-antimatter asymmetry of the Universe can be achieved through Leptogenesis, with the CP asymmetry parameter $\varepsilon$ described in terms of only lepton masses, mixings and phases, plus two real parameters of the low-…
The presence of flavor symmetries in the lepton sector may have several consequences for the generation of the baryon asymmetry of the Universe via leptogenesis. We review the mechanism in general type-I, type-II and type-III seesaw models. We then turn to the discussion of the cases when the asymmetry is generated in the context of seesaw models extended with flavor symmetries, before or after flavor symmetry breaking. Finally we explain how the interplay between type-I and type-II seesaws can (or not) lead to viable models for leptogenesis even when there is an exact mixing pattern enforced …
A supersymmetric see-saw model obeying the flavor symmetry L_mu - L_tau, which naturally predicts quasi-degenerate neutrinos, is investigated. Breaking of the symmetry is introduced in the Dirac mass matrix because it is the most economic choice in the sense that all interesting low and high energy phenomenology is made possible: we analyze the predictions for the low energy neutrino observables, for leptogenesis and for lepton flavor violating decays such as mu \to e gamma, where the SPS benchmark points for the SUSY parameters are used. It is outlined how these decays in connection with the …
In this paper, we examine how gauge interactions can dilute the lepton asymmetry in lepton induced baryogenesis. Constraints imposed on Majorana masses keep this dilution at an acceptable level.
Leptogenesis, i.e. the creation of a lepton asymmetry in the early Universe, may occur through the decay of heavy singlet (right-handed) neutrinos. If we require it not to be erased by physics beyond the Standard Model at the TeV energy scale, then only 2 candidates are possible if they are subgroups of $E_6$. These 2 solutions happen to be also the only ones within 1$σ$ of the atomic parity violation data and the invisible $Z$ width. Lepton flavor violation is predicted in one model, as well as in another unrelated model of neutrino masses where the observable decay of a doubly charged scalar…
We investigate the relation between non-unitarity of the leptonic mixing matrix and leptogenesis. We discuss how all parameters of the canonical type-I seesaw mechanism can, in principle, be reconstructed from the neutrino mass matrix and the deviation of the effective low-energy leptonic mixing matrix from unitary. When the mass M' of the lightest right-handed neutrino is much lighter than the masses of the others, we show that its decay asymmetries within flavour-dependent leptogenesis can be expressed in terms of two contributions, one depending on the unique dimension five (d=5) operator g…
We explore how the initial conditions affect the final lepton asymmetry in Soft Leptogenesis. It has been usually assumed that the initial state is a statistical mixture of sterile sneutrinos and anti-sneutrinos with equal abundances. We calculate the lepton asymmetry due to the most general initial mixture. The usually assumed equal mixture produces a small, but sufficient, lepton asymmetry which is proportional to the ratio of the supersymmetry breaking scale over the Majorana scale. A more generic mixture, still with equal contents of sneutrinos and anti sneutrinos, yields an unsuppressed l…
We consider, in the context of a 331 model with a single neutral right-handed singlet, the generation of lepton masses. At zeroth order two neutrinos and one charged lepton are massless, while the other leptons, two neutrinos and two charged leptons, are massive. However the charged ones are still mass degenerate. The massless fields get a mass through radiative corrections which also break the degeneracy in the charged leptons.
The origin of tiny neutrino masses and the baryon number asymmetry of the Universe are naturally interpreted by the canonical seesaw and leptogenesis mechanisms, in which there are the heavy Majorana neutrino mass matrix $M_{\rm R}$, the Dirac neutrino mass matrix $M_{\rm D}$, the charged-lepton mass matrix $M_\ell$ and the effective (light) neutrino mass matrix $M_ν$. We find that ${\rm Im}\left(\det\left[ M^\dagger_{\rm D} M_{\rm D}, M^\dagger_{\rm R} M_{\rm R} \right]\right)$, ${\rm Im}\left(\det\left[ M_\ell M^\dagger_\ell, M_νM^\dagger_ν\right]\right)$ and ${\rm Im}\left(\det\left[ M_\ell…
