Neutrinoless Double Beta Decay in LRSM with Natural Type-II seesaw Dominance
We present a detailed discussion on neutrinoless double beta decay within a class of left-right symmetric models where neutrino mass originates by natural type-II seesaw dominance. The spontaneous symmetry breaking is implemented with doublets, triplets and bidoublet scalars. The fermion sector is extended with an extra sterile neutrino per generation that helps in implementing the seesaw mechanism. The presence of extra particles in the model exactly cancels type-I seesaw and allows large value for Dirac neutrino mass matrix . The key feature of this work is that all the physical masses and mixing are expressed in terms of neutrino oscillation parameters and lightest neutrino mass thereby facilitating to constrain light neutrino masses from decay. With this large value of new contributions arise due to; i) purely left-handed current via exchange of heavy right-handed neutrinos as well as sterile neutrinos, ii) the so called and diagrams. New physics contributions also arise from right-handed currents with right-handed gauge boson mass around TeV. From the numerical study, we find that the new contributions to decay not only saturate the current experimental bound but also give lower limit on absolute scale of lightest neutrino mass and favor NH pattern of light neutrino mass hierarchy.
Keywords:Seesaw Mechanism, Neutrinoless Double Beta Decay, Left-Right Theories
The discovery that neutrinos have mass and they mix with each other has put before us another vital question to speculate over; whether they are Dirac or Majorana Majorana:1937vz () particles. Even more intriguing is the theoretical origin of such a tiny mass and the mass hierarchy among them. The different seesaw mechanisms like type-I Minkowski:1977sc (); Yanagida:1979as (); GellMann:1980vs (); Mohapatra:1979ia (), type-II Cheng:1980qt (); Lazarides:1980nt (); Magg:1980ut (); Schechter:1980gr (); Wetterich:1981bx () and others that appropriately explain this tiny mass further require them to be Majorana particles. On the contrary, Majorana nature of neutrinos violates global lepton number by 2 units that is regarded as an accidental symmetry within the Standard Model (SM). This leads to the search of a rare process called Neutrinoless Double Beta Decay () that only can assuredly endorse the Majorana nature of neutrinos and lepton number violation in nature Schechter:1981bd (). While new theories are trying to find new physics contributions to decay, the experiments are looking for lower limits on the half-lives being decayed. Of yet, GERDA Agostini:2013mzu () using gives lower limit on half life of decay as yrs at 90% C.L. whereas the limits provided by EXO-200 Gando:2012zm () and KamLAND Auger:2012ar () are yrs and yrs respectively. The combined limit from KamLAND-Zen comes to be yrs at 90% C.L. This process can be mediated by the exchange of a light Majorana neutrinos or by new particles appearing in various extensions of SM Mohapatra:1986su (); Babu:1995vh (); Hirsch:1995vr (); Hirsch:1995ek (); Hirsch:1996ye (); Deppisch:2012nb (); Pas:1999fc (); Humbert:2015yva (); Allanach:2009xx (); Pas:2000vn (); Deppisch:2006hb (); Pas:2015eia (); Helo:2013dla (); Deppisch:2012nb (); Ge:2015bfa ().
Within preview of BSM physics, left-right symmetric models (LRSM) Mohapatra:1974gc (); Pati:1974yy (); Senjanovic:1975rk (); Senjanovic:1978ev (); Mohapatra:1979ia (); Mohapatra:1980yp () are found to be best suited frameworks for explaining the origin of maximal parity violation in weak interactions and the origin of small neutrino mass. This class of models, based on the gauge group , when studied at TeV scale interlinks high energy collider physics to low energy phenomena like neutrinoless double beta decay and other LFV processes (see refs. Ge:2015yqa (); Tello:2010am (); Barry:2013xxa (); Patra:2015bga (); Lindner:2016lpp (); Deppisch:2016scs (); Deppisch:2014zta (); Deppisch:2014qpa (); Awasthi:2013ff (); Patra:2014goa (); Dev:2014xea (); Bambhaniya:2015ipg (); Borah:2013lva (); Patra:2012ur (); Chakrabortty:2012mh (); Dev:2013vxa (); Nemevsek:2011hz (); Dev:2014iva (); Keung:1983uu (); Das:2012ii (); Bertolini:2014sua (); Beall:1981ze (); Deppisch:2015cua (); Hirsch:1996qw (); Dev:2013oxa (); Dev:2015pga (); Dhuria:2015cfa ()). Moreover, the left-right symmetric models can also accommodate stable dark matter candidate contributing energy budget of the Universe Heeck:2015qra (); Garcia-Cely:2015quu (); Borah:2016ees (); Patra:2015vmp (); Patra:2015qny (). In conventional left-right symmetric models where symmetry breaking is implemented with scalar triplets and bidoublet, the light neutrino mass is governed by type-I plus type-II seesaw mechanisms
Here is the Majorana mass term for light left-handed (heavy right-handed) Majorana neutrinos arising from respective VEVs of left-handed (right-handed) scalar triplets and is the Dirac neutrino mass matrix connecting light-heavy neutrinos. The scale of is decided by the vacuum expectation value of right-handed scalar triplet which spontaneously breaks LRSM to SM. Thus, the smallness of light neutrino mass is connected to high scale of parity restoration i.e, GeV clearly making it inaccessible to current and planned accelerator experiments. Moreover when LRSM breaks around TeV scale, the gauge bosons , , right-handed neutrinos and scalar triplets get mass around that scale allowing several lepton number violating signatures at high energy as well as low energy experiments. A wide range of literature provides discussions on neutrinoless double beta decay within TeV scale LRSM assuming type-I seesaw dominance Chakrabortty:2012mh () or type-I plus type-II Chakrabortty:2012mh (); Dev:2013vxa (); Barry:2013xxa (); Das:2012ii (); Bertolini:2014sua (); Borah:2016iqd (); Borah:2015ufa () seesaw mechanisms. Some more scenarios have been studied in Deppisch:2014zta (); Tello:2010am (); Patra:2014goa (); Chakrabortty:2012mh (); Ge:2015yqa (); Barry:2013xxa (); Awasthi:2016kbk (); Awasthi:2015ota () where type-II seesaw dominance relates the light and heavy neutrinos with each other. Other works that discuss complementarity study of lepton number, lepton flavour violation and collider signatures in LRSM with spontaneous D-parity breaking mechanism also embed the framework in a non-SUSY GUT Awasthi:2013ff (); Deppisch:2014zta (); Deppisch:2014qpa (); Nayak:2013dza (). One should bear in mind that the new physics contributions to neutrinoless double beta decay mainly involves left-right mixing (or light-heavy neutrino mixing) which crucially depends on Dirac neutrino mass . Necessarily should be large in order to expect LNV signatures at colliders. Contrary to this, the type-II seesaw dominance can be realized with suppressed value of or with very high scale of parity restoration. Studies that assume therefore miss to comment on LNV, LFV and Collider aspects involving left-right mixing. We thus feel motivated to explore alternative class of left-right symmetric models which allows large value of and carries light and heavy neutrinos proportional to each other.
This work considers a TeV scale LRSM where symmetry breaking is implemented with scalar bidoublet , doublets and triplets . The scalar bidoublet carrying charge provides Dirac masses to charged fermions as well as to neutrinos. The scalar triplets with charge 2 units provide Majorana masses to light and heavy neutrinos. One extra sterile fermion per generation also finds place in the model that help in implementing extended type-II seesaw mechanism. The scalar doublets play the same role as . An interesting feature of this new class of LRSM is that it provides possibility of achieving type-II seesaw dominance when parity and break at same scale. Moreover this framework allows large value for Dirac neutrino mass matrix thereby leading to new physics contributions to neutrinoless double bea decay i.e, i) from purely left-handed currents via exchange of heavy right-handed and extra sterile neutrinos, ii) from purely right handed currents via exchange of heavy right-handed neutrinos, iii) from so called and diagrams. This work aims to carefully analyze the new contributions to in order to derive the absolute scale of light neutrino masses and mass hierarchy.
The complete work is structured as follows. In Sec.2, we briefly discuss the generic and TeV scale LRSMs in context of neutrino mass and associated lepton number violation. Sec.3 highlights the natural realization of type-II seesaw dominance. Sec.4 lays out the basic ingredients for neutrinoless double beta decay and the calculation of Feynman amplitudes. Sec.5 and Sec.6 are devoted towards the numerical study of LNV contributions within the present framework. In Sec.8 we summarize our results.
2 The Left-Right Symmetric Model and Lepton Number Violation
In this class of models, the difference between baryon and lepton number is defined as a local gauge symmetry. The electric charge is defined as
Here, and are, respectively, the third component of isospin of the gauge groups and , and is the hypercharge. The usual leptons and quarks are given by
The left-right symmetry calls for the presence of right-handed neutrinos and this makes the model suitable for explaining light neutrino masses. For generating fermion masses one needs a scalar bidoublet with the following matrix representation
The relevant Yukawa interactions are expressed as,
where and is the second Pauli matrix. The scalar bidoublet takes a non-zero VEV as,
it yields masses for quarks and charged leptons as
One can generate Dirac masses for light neutrinos using scalar bidoublet as
However, the Majorana masses for neutrinos depend crucially on how spontaneous symmetry breaking of LRSM down to the SM i.e, is implemented.
2.1 Lepton number violation and the origin of neutrino mass
The spontaneous symmetry breaking of LRSM to SM goes in favor of neutrino mass generation and associated lepton number violation. This happens in the following three ways
with Higgs doublets ,
with scalar triplets ,
with the combination of doublets and triplets and .
In the first case, breaks the LR symmetry while the left-handed counterpart is required for left-right invariance. Though this framework holds a minimal scalar spectrum it lacks Majorana mass for neutrinos and thus forbids any signature of lepton number violation or neutrinoless double beta decay. Since the light neutrinos here owe their identity to Dirac fermions, their masses can only be explained by adjusting Yukawa couplings through the non-zero VEVs of scalar bidoublet. Other important roles that this scalar bidoublet plays are to break the SM gauge symmetry to low energy theory and provide the masses to charged fermions. Using the Yukawa interactions given in Eq.(6) and with , and , the masses for charged leptons and the light neutrinos are given by
However, a pleasant situation arises in the second case where carrying charge breaks the LR symmetry to SM. The inclusion of and in the framework generate Majorana masses for light as well as heavy neutrinos and thus violate lepton number by two units. This calls for a possibility of smoking-gun same-sign dilepton signatures at collider as well as neutrinoless double beta decay in low energy experiments. The interaction terms involving scalar triplets and leptons are given by
where, is the Dirac neutrino mass matrix, is the Majorana mass matrix arising from the non-zero VEV of LH (RH) scalar triplet. After diagonalization, the resulting light neutrino mass can be written as a combination of canonical type-I and type-II seesaw formula
where, () is denoted as the type-I (type-II) contribution to light neutrino masses,
In conventional left-right symmetric models, where parity and break at same scale, the analytic formula for induced VEV of left-handed scalar triplet is given by,
In the above expression lies around electroweak scale, is the VEV of right-handed scalar triplet and is dimensionless Higgs parameter. In order to be consistent with oscillation data should be order of eV and assuming natural values of and , this sub-eV scale of can be attained only if lies around GeV. However such a high scale is inaccessible to LHC and thus urges to look for TeV scale LRSM. These frameworks offer numerous opportunities like low scale seesaw mechanism, LNV like neutrinoless double beta decay and its collider complementarity and have been already explored by the works mentioned in refs Ge:2015bfa (); Deppisch:2014zta (); Tello:2010am (); Deppisch:2014qpa (); Patra:2014goa (); Borah:2013lva (); Awasthi:2013ff (); Patra:2012ur (); Chakrabortty:2012mh (); Dev:2013vxa (); Barry:2013xxa (); Nemevsek:2011hz (); Dev:2014iva (); Keung:1983uu (); Das:2012ii (); Bertolini:2014sua (); Beall:1981ze (); Ge:2015yqa (); Deppisch:2015cua (); Hirsch:1996qw (); Dev:2013oxa (); Dev:2015pga (). Many of the works considered either type-I seesaw dominance or type-II seesaw dominance for en extensive study of decay. In manifest left-right symmetric model, where right-handed scale lies at TeV range, the neutrino mass mechanism via type-I plus type-II seesaw gives negligible value to the left-right mixing. As a result of this the production cross-section of heavy neutrinos and the lepton number violating processes at LHC get suppressed. However, the extension of type-I plus type-II seesaw scheme by the inclusion of another sterile neutrino per generation changes the scenario which results large left-right mixing. Now the neutrino mass arises only from type-II seesaw dominance since type-I seesaw contribution gets exactly canceled out. We propose a new framework where type-II seesaw dominance is achieved naturally and allows large value of Dirac neutrino mass which additionally contributes to decay from purely left-handed current via exchange of heavy neutrinos as well as from the so called type and type diagrams.
3 Extended Seesaw Mechanism and Natural type-II seesaw dominance
3.1 Extended Seesaw Mass Matrix
In order to implement the extended seesaw mechanism111The discussion of extended seesaw mechanism can be found in refs.Barry:2011wb (); Zhang:2011vh (). within left-right symmetric models, one has to add a complete left-right gauge symmetry singlet neutral fermion per generation to the usual quarks and leptons. Along with this the Higgs sector includes scalar bidoublet with , scalar triplets with and scalar doublets with . The complete particle spectrum is given in Table.1 .
The relevant leptonic Yukawa interaction terms for extended seesaw mechanism are given by
After spontaneous symmetry breaking, the resulting neutral lepton mass matrix for extended seesaw mechanism in the basis is given by
where is the Dirac neutrino mass matrix connecting left-handed light neutrinos with right-handed heavy neutrinos, () is the Majorana mass term for heavy (light) neutrinos, is the mixing matrix, is the small mass term connecting and is the bare Majorana mass term for extra singlet fermion.
The light neutrino mass can be parametrized in terms of model parameters of inverse seesaw framework as,
This expression bears of few TeV which allows large left-right mixing and thus leads to interesting testable collider phenomenology. Extension of such a scenario has been discussed in the context of allowing large LNV and LFV in the work Awasthi:2013ff ().
whereas the heavy neutrinos form pair of pseudo-Dirac states with masses
The following discussion considers the same Eq.(16) with the assumption that which leads to natural realization of type-II seesaw dominance allowing large left-right mixing.
3.2 Natural realization of type-II seesaw
The natural realization of type-II seesaw dominance is considered here within a class of left-right symmetric models where both discrete left-right parity symmetry and gauge symmetry break at same scale. The scalar sector is comprising of doublets , triplets and bidoublet whereas the fermion sector is extended with one neutral fermion per generation which is complete singlet under both LRSM as well as SM gauge group. We denote this class of LR model as Extended LR models and thus, the corresponding seesaw formula which is type-II dominance in this case is termed as Extended type-II seesaw mechanism. In principle, there could be a gauge singlet mass term in the Lagrangian for extra fermion singlet, i.e, which can take any value. But we have taken this mass parameter to be either zero or very small so that the generic inverse seesaw contribution involving is very much suppressed. In addition, we have assumed the induced VEV for is taken to be zero, i.e, .
The relevant interaction terms necessary for realizing natural type-II seesaw dominance is given by
With and , the complete neutral fermion mass matrix in the flavor basis of is read as
Using standard formalism of seesaw mechanism and using mass hierarchy , we can integrate out the heaviest right-handed neutrinos as follows
where the intermediate block diagonalised neutrino states modified as
Thus, the intermediate block diagonalised neutrino states are related to flavor eigenstates in the following transformation,
It is found that the entries of mass matrix is larger than other entries in the limit . As a result of this, we can repeat the same procedure in Eq.(22) to integrate out . Thus, the light neutrino mass formula becomes
and the physical block diagonalised states are
with the corresponding block diagonalised transformation as
With this block diagonalization procedure and after few simple algebra, the flavor eigenstates are related to mass eigenstates in the following transformation,
Subsequently, the final block diagonalised mass matrices can be diagonalised in order to give physical masses by a unitary matrix . The transformation of the block diagonalised neutrino states in terms of mass eigenstates are given by
while the block diagonalised mass matrices for light left-handed neutrinos, heavy right-handed neutrinos and extra sterile neutrinos are
These block diagonalised mass matrices can be further diagonalised by respective unitarity matrices as follows
Finally, the complete block diagonalization yields
Here the block diagonalised mixing matrix and the unitarity matrix are given by
Thus, the complete unitary mixing matrix diagonalizing the neutral leptons is as follows
3.3 Expressing Masses and Mixing in terms of and light neutrino masses.
The light neutrinos are generally diagonalised by standard PMNS mixing matrix in the basis where charged leptons are already diagonal i.e,. The Dirac neutrino mass matrix in general is a complex matrix. The structure of in LRSM can be approximately taken to be up-quark type mass matrix whose origin can be motivated from high scale Pati-Salam symmetry or SO(10) GUT. If we consider to be diagonal and degenerate i.e, , then the mass formulas for neutral leptons are given by
After some simple algebra, the active LH neutrinos , active RH neutrinos and heavy sterile neutrinos in the flavor basis are related to their mass basis as
4 Neutrinoless Double Beta Decay in LRSM
In this section, we shall present a detailed discussion on Feynman amplitudes for neutrinoless double beta decay within TeV scale LRSM where light neutrino mass mechanism is governed by natural type-II seesaw dominance. The basic charge current interaction Lagrangian for leptons as well quarks are given by
Using Eq.(LABEL:eq:numixing) of Sec.3, the flavor eigenstates and are expressed in terms of admixture of mass eigenstates in the following way,
This modifies the charged current interaction for leptons as
In the above charged-current interaction, there is a possibility that both left-handed and right-handed gauge bosons can mix with each other which can eventually contribute to transition amplitude. In the present framework, the resulting mass matrix for LH (RH) charged gauge bosons () is given by
The physical masses of the charged gauge bosons derived with after diagonalization are given by
The physical gauge boson states and are related to the mixture of weak eigenstates and as
Thus, one can express physical states in terms of and as follows
We classify all contributions to neutrinoless double beta decay in the present TeV scale LRSM as:
due to standard mechanism mediated by purely left-handed currents ( mediation) via exchange of light neutrinos ,
due to purely left-handed currents via mediation through the exchange of the heavy RH Majorana neutrino and heavy sterile neutrinos ,
due to purely right-handed currents ( mediation) via exchange of heavy right-handed Majorana neutrinos ,
due to purely right-handed currents via mediation through the exchange of the light neutrinos and extra sterile neutrinos ,
due to mixed helicity so called and diagrams through mediation of neutrinos.
Before deducing Feynman amplitudes for various contributions to neutrinoless double beta decay, it is desirable to discuss few points regarding the chiral structure of the matrix element with the neutrino propagator as Pas:1999fc ()
4.1 Feynman amplitudes for decay due to purely left-handed currents
The Feynman amplitudes for mediated diagrams shown in Fig.1 with the exchange of Majorana neutrinos , and , respectively, are given by