An extension of the two Higgs doublet model including inverse seesaw neutrinos and neutral Higgs bosons was constructed based on the $A_4$ symmetry in order to explain the recent neutrino oscillation data. This model can distinguish two well-known normal and inverted order schemes of neutrino data once both the effective masses $m_β$ in tritium beta decays and $\langle m\rangle$ in the neutrinoless double beta decay are observed. The lepton flavor violating decays of the charged leptons $e_b\rightarrow e_aγ$, $μ\rightarrow3e$, the Standard model-like Higgs boson decays $h\rightarrow e_be_a$, and the $μ$-e conversions in some nuclei are generated from loop corrections. The experimental data of the branching ratio Br$(μ\rightarrow eγ, 3e)$ predict that the upper bounds of Br$(τ\rightarrow μγ,eγ)$ and Br$(h\rightarrow e_{a}e_b)$ are much smaller than the planned experimental sensitivities. In contrast, the $μ$-e conversions are the promising signals for experiments.
We discuss the impact of fermion mass matrices on some lepton number violating processes, namely baryogenesis via leptogenesis and neutrinoless double beta decay, and on some lepton flavor violating processes, namely radiative lepton decays in supersymmetric seesaw models.
Now the bounds on the lepton-flavor violating (LFV) decay modes of tau lepton are being improved by the Belle experiment in the KEK B factory. In this paper the LFV decay of tau lepton is discussed in the supersymmetric seesaw model, using the current neutrino oscillation data.
We propose a left-right symmetric framework with universal seesaw mechanism for the generation of masses of the Standard Model quarks and leptons. Heavy vector-like singlet quarks and leptons are required for generation of Standard Model-like quark and lepton masses through seesaw mechanism. A softly broken $Z_2$ symmetry distinguishes the lepton sector and the quark sector of the model. This leads to the presence of some lepton-specific interactions that can produce unique collider signatures which can be explored at the current Large Hadron Collider run and also future colliders.
The inverse seesaw mechanism of neutrino mass, i.e. m_nu = (m_D^2/m_N^2)epsilon_L where epsilon_L is small, is discussed in the context of the U(1)_Sigma model. This is a gauge extension of the Standard Model of particle interactions with lepton triplets (Sigma^+,Sigma^),Sigma^-) as (Type III) seesaw anchors for obtaining small Majorana neutrino masses.
We discuss a supersymmetric inverse seesaw model in which lepton flavour violating decays can be enhanced either by flavour violating slepton contributions or by the non-unitarity of the charged current mixing matrix. As an example we calculate Br(mu -> e gamma) taking into account both heavy lepton exchange as well as supersymmetric diagrams in a minimal supergravity framework. We find that the for the same parameters the rate can be enhanced with respect to seesaw model expectations, with or without supersymmetry.
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.
We study the implications of the global $U(1)_R$ symmetry present in minimal lepton flavor violating implementations of the seesaw mechanism for neutrino masses. Our discussion is done in the context of explicit minimal type-I seesaw scenarios, where depending on the R-charge assignment different models can be constructed. We study the charged lepton flavor violating phenomenology of a concrete realization paying special attention to $μ\to eγ$ and $μ\to 3e$.
In a version of the 3-3-1 model proposed by Duong and Ma the introduction of the scalar sextet for giving mass to the charged leptons is avoided by adding a singlet charged lepton. We show that in this case the $τ$ lepton gains mass through a seesaw--like mechanism. Besides we show how to generate neutrino masses at the tree and at the 1-loop level with the respective Maki-Nakagawa-Sakata leptonic mixing matrices.
We investigate a seesaw type mass matrix $M_f\simeq m_L M_F^{-1} m_R$ for quarks and leptons, $f$, under the assumptions that the matrices $m_L$ and $m_R$ have common structures for the quarks and leptons, and that the matrix $M_F$ characterizing the heavy fermion sector has the form [(unit matrix)+ (democratic-type matrix)]. We obtain well-satisfied relations for quark masses and mixings related to the charged lepton masses.
Within low-scale seesaw mechanisms, such as the inverse and linear seesaw, one expects (i) potentially large lepton flavor violation (LFV) and (ii) sizeable non-standard neutrino interactions (NSI). We consider the interplay between the magnitude of non-unitarity effects in the lepton mixing matrix, and the constraints that follow from LFV searches in the laboratory. We find that NSI parameters can be sizeable, up to percent level in some cases, while LFV rates, such as that for μ-> e γ, lie within current limits, including the recent one set by the MEG collaboration. As a result the upcoming …
In minimal supersymmetric seesaw models, the contribution to lepton flavour violation from Z-penguins is usually negligible. In this study, we consider the supersymmetric inverse seesaw and show that, in this case, the Z-penguin contribution dominates in several lepton flavour violating observables due to the low scale of the inverse seesaw mechanism. Among the observables considered, we find that the most constraining one is the muon to electron conversion rate which is already restricting the otherwise allowed parameter space of the model. Moreover, in this framework, the Z-penguins exhibit …
In leptogenesis scenario, the decays of heavy Majorana neutrinos generate lepton family asymmetries, $Y_e, Y_μ$ and $Y_τ$. They are sensitive to CP violating phases in seesaw models. The time evolution of the lepton family asymmetries are derived by solving Boltzmann equations. By taking a minimal seesaw model, we show how each family asymmetry varies with a CP violating phase. For instance, we find the case that the lepton asymmetry is dominated by $Y_μ$ or $Y_τ$ depending on the choice of the CP violating phase. We also find the case that the signs of lepton family asymmetries $Y_μ$ and $Y_τ…
We study the lepton electric dipole moments in the framework of the supersymmetric type II seesaw model where the exchange of heavy SU(2)_W triplets generates small neutrino masses. We show that the CP violating phase of the bilinear soft supersymmetry breaking term associated with the SU(2)_W triplets contributes to lepton electric dipole moments mainly through threshold corrections to the gaugino masses at the seesaw scale. As a consequence, the ratio of the electric dipole moments of the muon and the electron is the same as the ratio of their masses in a wide region of parameter space.
We study the possibility of connecting leptonic Dirac CP phase $δ$, lightest neutrino mass and baryon asymmetry of the Universe within the framework of a model where both type I and type II seesaw mechanisms contribute to neutrino mass. Type I seesaw gives rise to Tri-Bimaximal (TBM) type neutrino mixing whereas type II seesaw acts as a correction in order to generate non-zero $θ_{13}$. We derive the most general form of type II seesaw mass matrix which can not only give rise to correct neutrino mixing angles but also can generate non-trivial value of $δ$. Considering both the cases where type…
We calculate the contribution to the anomalous magnetic moment of leptons in the type-I and type-III seesaw models. We show that, even if the scale of the new physics is pushed down to the electroweak scale, this contribution is not large enough to explain the measured discrepancy of the muon anomalous magnetic moment.
We present a novel framework within the conformal inverse seesaw scheme allowing large lepton number violation while the neutrino mass formula is still governed by the low-scale inverse seesaw mechanism. This model includes new contributions to rare low-energy lepton number violating processes like neutrinoless double beta decay. We find that the lifetime for this rare process due to heavy sterile neutrinos can saturate current experimental limits. The characteristic collider signature of the present conformal inverse seesaw scheme includes, same-sign dilepton plus two jets and same-sign dilep…
We study the supersymmetric version of the type-II seesaw mechanism assuming minimal supergravity boundary conditions. We calculate branching ratios for lepton flavour violating (LFV) scalar tau decays, potentially observable at the LHC, as well as LFV decays at low energy, such as $l_i \to l_j + γ$ and compare their sensitivity to the unknown seesaw parameters. In the minimal case of only one triplet coupling to the standard model lepton doublets, ratios of LFV branching ratios can be related unambigously to neutrino oscillation parameters. We also discuss how measurements of soft SUSY breaki…
We present a general framework for models in which the lepton mixing matrix is the product of the maximal mixing matrix U_ω times a matrix constrained by a well-defined Z_2 symmetry. Our framework relies on neither supersymmetry nor non-renormalizable Lagrangians nor higher dimensions; it relies instead on the double seesaw mechanism and on the soft breaking of symmetries. The framework may be used to construct models for virtually all the lepton mixing matrices of the type mentioned above which have been proposed in the literature.
We propose a model for the vector-like lepton to explain the small muon mass by a seesaw mechanism, based on lepton-specific two Higgs doublet models with a local $U(1)'$ symmetry. There is no bare muon mass for a nonzero $U(1)'$ charge of the leptophilic Higgs doublet, so the physical muon mass is generated due to the mixing between the vector-like lepton and the muon after the leptophilic Higgs doublet and the dark Higgs get VEVs. In this scenario, the non-decoupling effects of the vector-like lepton give rise to leading contributions to the muon $g-2$, thanks to the light $Z'$ and the light…
In the context of a typical model for fermion mass matrices, possibly based on the horizontal U(2) symmetry, we explore the effect of the type II seesaw mechanism on lepton mixings. We find that the combined contribution of type I and type II terms is able to explain the large but not maximal 1-2 mixing and the near maximal 2-3 mixing, while the 1-3 mixing angle is predicted to be small.
Lepton flavor violating Higgs boson decays are studied within the context of seesaw models with Majorana massive neutrinos. Two models are considered: The SM-seesaw, with the Standard Model Particle content plus three right handed neutrinos, and the MSSM-seesaw, with the Minimal Supersymmetric Standard Model particle content plus three right handed neutrinos and their supersymmetric partners. The widths for these decays are derived from a full one-loop diagrammatic computation in both models, and they are analyzed numerically in terms of the seesaw parameters, namely, the Dirac and Majorana ma…
We show that models with exotic leptons transforming as E ~ (1,3,-1) under the standard model gauge symmetry are well suited for generating neutrino mass via a radiative inverse seesaw. This approach realizes natural neutrino masses and allows multiple new states to appear at the TeV scale. The exotic leptons are therefore good candidates for new physics that can be probed at the LHC. Furthermore, remnant low-energy symmetries ensure a stable dark matter candidate, providing a link between dark matter and the origins of neutrino mass.
A previous short analysis of the seesaw mechanism, based on quark-lepton symmetry, experimental data and hierarchical neutrino spectrum, is enlarged to include small but not zero U_{e3}, inverted mass hierarchy, and the qualitative effect of Majorana phases. The structure of the heavy neutrino mass matrix obtained in several cases is discussed. We find two leading forms for this matrix. One is diagonal and stands at the unification scale or above. The other is off-diagonal and stands at the intermediate scale.
We provide an $A_4$ based flavor symmetric scenario to accommodate the inverse seesaw mechanism for explaining light neutrino masses and mixing. We find that the lepton mixing, in particular the tri-bimaximal mixing pattern and its deviation through nonzero $θ_{13}$, is originated solely from the flavor structure of the lepton number violating contribution of the neutral lepton mass matrix. Here we discuss in detail how a nonzero value of $θ_{13}$ is correlated with the other parameters in the framework and its impact on the Dirac CP phase $δ$. We also analyze the non-unitarity effects on lept…
We investigate the LHC discovery potential for electroweak scale heavy neutrino singlets (seesaw I), scalar triplets (seesaw II) and fermion triplets (seesaw III). For seesaw I we consider a heavy Majorana neutrino coupling to the electron or muon. For seesaw II we concentrate on the likely scenario where the new scalars decay to two leptons. For seesaw III we restrict ourselves to heavy Majorana fermion triplets decaying to light leptons plus gauge or Higgs bosons, which are dominant except for unnaturally small mixings. The possible signals are classified in terms of the charged lepton multi…
We study the possibility of generating non-zero reactor mixing angle $θ_{13}$ by perturbing the $μ-τ$ symmetric neutrino mass matrix. The leading order $μ-τ$ symmetric neutrino mass matrix originates from type I seesaw mechanism whereas the perturbations to $μ-τ$ symmetry originate from type II seesaw term. We consider four different realizations of $μ-τ$ symmetry: Bimaximal Mixing(BM), Tri-bimaximal Mixing (TBM), Hexagonal Mixing (HM) and Golden Ratio Mixing (GRM) all giving rise to $θ_{13} = 0, θ_{23} = \fracπ{4}$ but different non-zero values of solar mixing angle $θ_{12}$. We assume a mini…
In minimal supergravity (mSugra), the neutrino sector is related to the slepton sector by means of the renormalization group equations. This opens a door to indirectly test the neutrino sector via measurements at the LHC. Concretely, for the simplest seesaw type-I, we present the correlations between seesaw parameters and ratio of stau lepton flavour violating (LFV) branching ratios. We find some simple, extreme scenarios for the unknown right-handed parameters, where ratios of LFV rates correlate with neutrino oscillation parameters. On the other hand, we scan the mSugra parameter space, for …
We study the impact of the inverse seesaw mechanism on several low-energy flavour violating observables such as tau decaying to three muons in the context of the Minimal Supersymmetric Standard Model. As a consequence of the inverse seesaw, the contributions of the right-handed sneutrinos significantly enhance the Higgs-mediated penguin diagrams. We find that different flavour violating branching ratios can be enhanced by as much as two orders of magnitude. We also comment on the impact of the Higgs-mediated processes on the leptonic B-meson decays and on the Higgs flavour violating decays.
We study the impact of the inverse seesaw mechanism on several leptonic and hadronic low-energy flavour-violating observables in the context of the Minimal Supersymmetric Standard Model. Indeed, the contributions of the light right-handed sneutrinos from the inverse seesaw significantly enhance the Higgs-mediated penguin diagrams. We find that this can increase the different branching ratios by as much as two orders of magnitude.
In light of recent analyses that have shown that nontrivial hidden-sector dynamics in models of supersymmetry breaking can lead to a significant impact on the predicted low-energy supersymmetric spectrum, we extend these studies to consider hidden-sector effects in extensions of the MSSM to include a seesaw model for neutrino masses. A dynamical hidden sector in an interval of mass scales below the seesaw scale would yield renormalization-group running involving both the anomalous dimension from the hidden sector and the seesaw-extended MSSM renormalization group equations (RGEs). These effect…
We discuss a GUT realization of the supersymmetric triplet seesaw mechanism(recently proposed by us in hep-ph/0604083 and further analyzed in hep-ph/0607298) where the exchange of the heavy triplet states generates both neutrino masses and soft SUSY breaking terms.
We review Lepton Flavour Violation (LFV) in the supersymmetric version of the seesaw mechanism (type I, II, III) and in Left-Right models. The LFV needed to explain neutrino masses and mixings is the only source of LFV and has experimental implications both in low-energy experiments where we search for the radiative decays of leptons, and at the LHC where we look at its imprint on the LFV decays of the sparticles and on slepton mass splittings. We discuss how this confrontation between high- and low-energy LFV observables may provide information about the underlying mechanism of LFV.
We examine the lepton-flavor violation caused by a Yukawa coupling matrix $y_{ν,ij}$ for right-handed neutrinos in the supersymmetric standard model. We stress that decay rates for $τ\rightarrowμγ$ and $μ\rightarrow eγ$ may reach the range to be accessible to near future experiments if left-right mixing terms in the slepton mass matrix are substantially large.
We present an update of previous work on charged lepton flavor violation (LFV) in the seesaw model. The most recent neutrino data fits and post WMAP mSUGRA benchmark scenarios are used as input. In this framework we compare the sensitivity of rare radiative decays on the right-handed Majorana mass scale M_R with the reach in slepton-pair production at a future linear collider.
We suggest a low-scale model based on $A_4\times Z_4 \times Z_2$ symmetry and a global lepton number $U(1)_L$ symmetry capable of generating the current neutrino data. The neutrino mass smallness is reproduced by the linear seesaw mechanism. The model can explain the current observed pattern of lepton mixing in which the reactor and atmospheric angles get the best-fit values, and the solar angle and Dirac phase lie within $3σ$ limits. The obtained values of the sum of neutrino mass and the effective neutrino mass are below the present experimental limits.
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…
Lepton flavor violating Higgs boson decays (LFVHD) are studied in the context of the Minimal Supersymmetric Standard Model (MSSM) enlarged with three right handed neutrinos and their supersymmetric partners, and with the neutrino masses being generated by the seesaw mechanism. We compute the partial widths for these decays to one-loop order and analyze numerically the corresponding branching ratios in terms of the MSSM and seesaw parameters. We analyze in parallel the lepton flavor changing $ł_j\to l_i γ$ decays and explore the maximum predicted rates for LFVHD, mainly for $H^0,A^0 \to τ{\bar …
We present a type-III version of inverse seesaw or, equivalently an inverse version of type-III seesaw. Naturally small neutrino masses arise at low-scale from the exchange of neutral fermions transforming as hyperchargeless SU(2) triplets. In order to implement tri-bimaximal lepton mixing we supplement the minimal SU(3)xSU(2)xU(1) gauge symmetry with an A4-based flavor symmetry. Our scenario induces lepton flavour violating (LFV) three body decays that can proceed at the tree level, while radiative li to lj gamma decays and mu-e conversion in nuclei are also expected to be sizeable. LFV decay…
We show that a TeV scale inverse seesaw model for neutrino masses can be realized within the framework of a supersymmetric SO(10) model consistent with gauge coupling unification and observed neutrino masses and mixing. We present our expectations for non-unitarity effects in the leptonic mixing matrix some of which are observable at future neutrino factories as well as the next generation searches for lepton flavor violating processes such as μ--> e + γ. The model has TeV scale W_R and Z' bosons which are accessible at the Large Hadron Collider.
Analytic expressions of lepton-flavour- and lepton-number-violating decays of charged leptons are derived in the context of general $SU(2)_L\otimes U(1)_Y$ seesaw scenarios that are motivated by grand unified theories (GUT's) or superstring models, in which left-handed and/or right-handed neutral singlets are present. Possible constraints imposed by cosmology and low-energy data are briefly discussed. The violation of the decoupling theorem in flavour-dependent graphs due to the presence of heavy neutral leptons of Dirac or Majorana nature is emphasized. Numerical estimates reveal that the dec…
We discuss the leptonic flavor structure generated by a brane shifted extra dimensional seesaw model with a single right handed neutrino in the bulk. In contrast to previous works, no unitarity approximation for the $3 \times 3$ submatrix has been employed. This allows to study phenomenological signatures such as lepton flavor violating decays. A strong prediction of the model, assuming CP conservation, are the ratios of flavor violating charged lepton decay and Z decay branching ratios which are correlated with the neutrino mixing angles and the neutrino mass hierarchy. Furthermore, it is pos…
Diverse mass and mixing patterns between the quarks and leptons makes it challenging to construct a simple grand unified theory of flavor. We show that SO(10) SUSY GUTs with type II seesaw mechanism giving neutrino masses provide a natural framework for addressing this issue. A simple ansatz that the dominant Yukawa matrix (the 10-Higgs coupling to matter) has rank one, appears to simultaneously explain both the large lepton mixings as well as the observed quark flavor hierarchy in these models. A testable prediction of this ansatz is the neutrino mixing, U_{e3}, which should be observable in …
The tau lepton plays important role in distinguishing neutrino mass patterns and determining the chirality nature in heavy scalar mediated neutrino mass models, in the light of the neutrino oscillation experiments and its polarization measurement. We investigate the lepton flavor signatures with tau lepton at LHC upgrades, i.e. HL-LHC, HE-LHC and FCC-hh, through leptonic processes from doubly charged Higgs in the Type II Seesaw. We find that for the channel with one tau lepton in final states, the accessible doubly charged Higgs mass at HL-LHC can reach 655 GeV and 695 GeV for the neutrino mas…
In minimal supersymmetric models the $Z$-penguin usually provides sub-dominant contributions to charged lepton flavour violating observables. In this study, we consider the supersymmetric inverse seesaw in which the non-minimal particle content allows for dominant contributions of the $Z$-penguin to several lepton flavour violating observables. In particular, and due to the low-scale (TeV) seesaw, the penguin contribution to, for instance, $\Br(μ\to 3e)$ and $μ-e$ conversion in nuclei, allows to render some of these observables within future sensitivity reach. Moreover, we show that in this fr…
We review and update the current phenomenological constraints on minimal type I seesaw extensions of the Standard Model in which New Physics content can be probed at the electroweak scale. In this class of models, the flavour structure of the neutrino Yukawa couplings is determined by the requirement of reproducing neutrino oscillation data. The strongest constraints on the seesaw parameter space are imposed by the very recent upper limit on μ --> e γ decay rate. Searches of non-standard Higgs boson decays into a light and a heavy neutrino may also provide and independent test of these seesaw …
We propose a scheme in that the masses of the heavier leptons obey seesaw type relations. The light lepton masses, except the electron and the electron neutrino ones are generated by one loop level radiative corrections. We work in a version of the 3-3-1 electroweak model that predicts singlets (charged and neutral) of heavy leptons beyond the known ones. An extra U(1)_Omega symmetry is introduced in order to avoid the light leptons get masses at the tree level. The electron mass induces an explicit symmetry breaking at U(1). We discuss also the mixing matrix among four neutrinos. The new ener…
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 discuss the possible enhancement of the tau lepton events at LHC when the left-handed stau doublet becomes light (which can be even lighter than the right-handed stau). This is illustrated in the constrained supersymmetric seesaw model where the slepton doublet mass is suppressed by the effects of a large neutrino Yukawa coupling. We study a few representative parameter sets in the sneutrino coannihilation regions where the tau sneutrino is NLSP and the stau coannihilation regions where the stau is NLSP both of which yield the thermal neutralino LSP abundance determined by WMAP.
We summarize the predictions for the radiative decays l_j->l_i γwithin the context of the supersymmetric type II seesaw mechanism considering universal boundary conditions for the soft SUSY breaking terms. The dependence on the low-energy neutrino parameters is discussed and the deviations from the analytical results for large $\tanβ$ analyzed.
In a seesaw mass matrix model M_f = m_L M_F^{-1} m_R^\dagger with a universal structure of m_L \propto m_R, as the origin of m_L (m_R) for quarks and eptons, flavor-triplet Higgs scalars whose vacuum expectation values v_i are proportional to the square roots of the charged lepton masses m_{ei}, i.e. v_i \propto \sqrt{m_{ei}}, are assumed. Then, it is investigated whether such a model can explain the observed neutrino masses and mixings (and also quark masses and mixings) or not.
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 …
In the type I + II seesaw formula the mass matrix of light neutrinos m_nu receives contributions from the exchanges of both heavy Majorana neutrinos and SU(2)_L-triplet Higgs bosons. We propose a new parameterization for the Dirac-type Yukawa coupling matrix of neutrinos in this case, which generalizes the well known Casas-Ibarra parameterization to type I + II seesaw and is useful when the triplet term in m_nu is known. Neutrino masses and mixing, lepton flavor violation in decays like mu -> e γwithin mSUGRA models and leptogenesis can then be studied within this framework. We illustrate the …
In type II seesaw model of neutrino mass generation, we study a remarkable signal of same-sign tetra-lepton (SS4L) signal at the Large hadron collider. When doubly charged Higgs is lightest, the heavier singly charged or neutral Higgs boson produces a doubly charged Higgs boson through its fast gauge decay. This leads to a novel signature of same-sign tetra-leptons resulting from a pair production of same-sign doubly charged Higgs bosons. We study production cross section for the SS4L signal in parameter space of the mass splitting among triplet components and the triplet vacuum expectation va…
