Neutrinoless Double Beta Decay and $<\eta>$ Mechanism in the Left-Right Symmetric Model

The neutrinoless double beta ($0\nu\beta\beta$) decay is studied in the framework of left-right symmetric model. The coexistence of left and right handed currents induces rather complicated interactions in the mixing of lepton and hadrons, called $<\lambda>$ mechanism and $<\eta>$ mechanism in addition to the conventional effective neutrino mass $$ mechanism. In this letter we indicate the possible magnification of $<\eta>$ mechanism and importance to survey $0\nu\beta\beta$ decay of different nuclei for specifying New Physics beyond the Standard Model.


Introduction
Neutrinoless double beta (0νββ) decay is one of the key probes for the new physics beyond the Standard Model (BSM physics). In this letter, we consider this process in the framework of left-right (L − R) symmetric model [1,2], where the decay is concerned with the correlations between the L-handed light neutrinos and the R-handed heavy neutrinos. L − R symmetric model in the SO (10) grand unified theory appears in the intermediate stage [3,4] which includes and is related with the wide varieties of BSM physics besides 0νββ decay, like baryo-genesis via lepto-genesis and dark matters etc. There are two conditions to realize 0νββ decay in the context of this framework [5]. 1. ν e should be the same as its anti-particle and 2. the connecting neutrinos should have the same helicity. The latter condition is satisfied if neutrinos are massive or if the R-handed current coexists with the L-handed current. The first case of 2. is described as the well known effective neutrino mass, Here U αi (Greek (Latin) indicates flavour (mass) eigenstate) is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [6,7] in L-handed current. Substituting the observed values, Then, the inverted hierarchy (IH) case enhances < m ν > relative to the normal hierarchy (NH) case. Though the final answer to the hierarchy problem is given by observation, the theoretical predictions have been given by many models. One of the typical models is due to the predictive minimal SO(10) model [4]. Based on this model, we fitted the low energy spectra of all quark and lepton masses and the CKM and the PMNS mixing angles and their phases. Our results prefer the NH manifestly to the IH: That is, inputting the observed lepton masses and the PMNS angles into the model, we compared the outputs of quark masses and CKM matrices [8,9] in the model with the observations. We obtained χ 2 ≤ 1 for the NH case and χ 2 > 200 for the IH case [10]. In this model, the effective neutrino mass is also predicted including the Majorana phases as < m ν >≈ 1 meV.
On the other hand, the recent 0νββ experiment in the KamLAND-Zen [11] provides the most stringent upper limit on it. The half life T 1/2 of 0νββ decay in 126 Xe is Here G 0ν is the phase-space integral and M 0ν is the nuclear matrix element (NME), which leads to the < m ν >= 36 − 156 meV [11,12,13], already in the inverted hierarchy regions. Efforts to reduce the ambiguities of NME in different nuclear models are in progress [14,15,16,17]. We consider 0νββ decay in the L−R symmetric model. The model generates contact, heavy neutrino and light neutrino exchange quark-quark interactions The quark-quark interactions then mapped onto contact, pion exchange and light neutrino exchange interactions between two nucleons classified in a general from in effective field theory [18,19]. Here we focus on the light neutrino exchange mechanism in particular the < λ > and the < η > mechanisms in addition to the < m ν > mechanism [5,20]. The BSM physics appears in the leptonic currents, which restricts the structure of the hadronic current. Such interplay between leptonic and hadronic currents has been overlooked so far. Especially, 0νββ is very sensitive to the spatial momentum of neutrino propagator and interference between the vector and axial vector currents of nucleon, enhancing < η > mechanism [5,21,22]. This term is also very crucial to the heavy right-handed neutrino mass. In this letter, surveying into these entanglements of interplay and solving them, we narrow down the general forms of L-R symmetric models [18,19], leading to the mechanism of 0νββ to < m ν > and < η > mechanisms if experiments reveal the non-null result around the present upper bound.
This letter is organized as follows. In Sec.2, we discuss the structure of the leptonic current, assuring the low energy seesaw mechanism. Its hadronic counterparts are studied in Sec.3. We show a simple understanding of the mechanism to enhance a sensitivity to the < η > mechanism due to the V-A interference term. This mechanism has a potential to reveal R-handed current if the 0νββ decay is discovered above the NH region from the present and near future experiments. It is also discussed that the atom dependence (A-dependence) of 0νββ beta decay rate may clarify < m ν > and/or < η > mechanisms. Section 4 is devoted to discussions.

Right-handed weak current
We consider L − R symmetric model [2] in this section. The weak Hamiltonian is given by  Here j µ (J µ ) indicates leptonic (hadronic) current, and the L and R-handed leptonic currents, j Lµ and j Rµ , are given by where Also ν lL (N lR ) are L-handed (R-handed) weak eigenstates of the neutrinos, and Thus the system is mixed with rather simple leptonic world and composite hadronic world. There are many precedent works to have discussed 0νββ in the L − R symmetric models [23,24,25,26,27] etc. We are not concerned with the detailed new calculations of hadronic models but to try to give a firm foundation for low energy seesaw mechanism and to make clear the connection of neutrino potential with hadronic NMEs. The main diagrams of 0νββ decay in the L − R symmetric model are depicted in Fig.1 Here are the amplitude of (a) and (b) of Fig.2. Thus these diagrams give subdominant contributions. So we limit our arguments in Fig.1 hereafter. Its amplitude in closure approximation [20] is given as where the lepton tensor L νµ is Here e p i ,s ′ i (x) are electron wave functions with the energy e i , and the mixing matrices U, V * are omitted for simplicity. The energy denominator is given by A i = e i + < E n > −E i and E f + e 1 + e 2 = E i . Here E i/f and < E n > are energy of the initial/final nuclear state and the average energy of the intermediate nuclear state, respectively.
The nuclear tensor H νµ is given by the matrix element of the nuclear weak current as whereJ µ L,R are given in (11) and (12). The neutrino propagator becomes, In the presence of the R-handed current, we have (±ωγ 0 − k · γ)P β in addition to (3). The spatial momentum exchanged between nucleon by neutrino is significantly larger than neutrino mass term, |k| ≈ 100MeV ≫ E n − E i , m i , which gives a significant effect to the decay rate.
This mechanism gives interesting interplay between particle physics and nuclear physics, whose explanation is the main theme of this paper. The half life T 1/2 in this case [5] is given as Here ab includes NME and phase space integral. The other parts include BSM physics. The effective couplings < η > and < λ > are given as ψ is the relative phase between < m ν > and < λ > and < η >, where ′ indicates the summation over only the light neutrinos. However, U and V are independent and we set ψ = 0 hereafter. The details of λ and η are given by (30) and (31). We proceed to discuss the detailed structure of mixing matrices. Higgs sectors are composed of (3,1,2), (1,3,2) triplets (∆ L , ∆ R , respectively) and bi-doublet ( and The neutrino mass matrix is [29,30,31,32] Thus we have the extended Fermi couplings (7). In (8) and (9), ν lL (N lR ) are L-handed (Rhanded) weak eigenstates of the neutrinos. Using 3 × 3 blocks U, V, X, Y , the mass eigenstates ν ′ , N ′ are given as That is, where α (i) are the flavour (mass) eigenstates.
The constants λ and η in (7) are related to the mass eigenvalues of the weak bosons in the L and R− handed gauge sectors (W L , W R ) as follows: Here M W 1 and M W 2 are the masses of the mass eigenstates W 1 and W 2 , respectively, and ζ is the mixing angle which relates the mass eigenstates and the gauge eigenstates. We are considering L − R symmetric model. The gauge boson mass is generated from (22) and (23) and the mixing angle ζ is with and [34,35]. In the L − R symmetric model, we set g L = g R , which indicates further unification of at least rank five GUT, including SU (3) color. tan β is constrained from the Yukawa coupling is renormalizable up to the GUT scale, That is, the upper limit (lower limit) appears from the renormalizability of bottom (top) Yukawa coupling in GUT. Furthermore, in this case, large tan β induces too rapid proton decay since the proton life-time is proportional to 1/ tan β 2 and tan β is limited around 10 [10]. Reflecting these relatively low mass constraint, we will discuss on the low energy seesaw mechanism later. Corresponding to Figure 1, we will consider 0νββ decay in this scheme: Here and in the subsequent discussions in this section, we write the subdominant terms in addition to (20), illustrating the seesaw structure. In the latter sum, 3 and 6 indicate type I (25) and Inverse seesaw mechanisms (47), respectively.
with g L = g R , which was suppressed compared with the others due to M W L • W L − W R diagram; the neutrino mixing (λ) and W L − W R mixing (η) Here the first terms U ei V * ei dominate, and let us estimate the magnitude of V ei . The naive type I seesaw (24) gives tiny value for the above quantities. We are interested in TeV scale seesaw and consider the inverse seesaw mechanism [36] hereafter. Its 9 × 9 mass matrix is given by Here and Thus U of (25) in type I seesaw is modified in the inverse seesaw as where {U, V, W } and {X, Y, Z} are 3 × 3 and 3 × 6 matrices, respectively. All deviation from unitarity is determined by It goes from (44) and (47), Then and O m light M D is also valid for type I seesaw. Thus, O(V ) seems to be tiny. However, this is too naive estimation. This is because these estimations are due to the hierarchy assumption M ≫ M D ≫ µ and to the neglect of the generation. Alternative ideas may be free from the light neutrino mass constraint like (50). For type I seesaw, for example, if (M D ) i3 = 0 and (M −1 N ) 33 = 0, then, m ν = 0 and V is free from the light neutrino mass constraint 4 . For the case of inverse seesaw, we may consider another mass hierarchy M ≫ µ ≫ M D . Important is that there are windows of sizable V . The order of the magnitude of V should be estimated from the observations. As was shown in (37), < η > is not so suppressed and its contribution in (19) may be important when we take into account the large contribution from NME C

Nuclear matrix elements and role of < η > mechanism
The sensitivity of the 0νββ decay to the R-handed current can be roughly summarized as follows. The decay rate calculated from the neutrino mass terms with < m ν >∼ 0.1 eV corresponds to the R-handed current contribution of either < η >∼ 10 −9 or < λ >∼ 10 −7 [14]. The two order of magnitude difference of the sensitivity between the < η > mechanism and the < λ > mechanism comes from the interference between the nuclear vector and axial vector current. The combination of the V − A interference of nuclear current, which corresponds to NME χ R and χ P [5], and the phase space integral G i enhances sensitivity to the < η > mechanism. In the previous section, it is argued that < η > may not be as tiny as usually assumed, which opens an interesting possibility to reveal R-handed current through the < η > mechanism. We revisit this mechanism, showing a simplified derivation of the relevant nuclear operator on the 0 + − 0 + 0νββ decay. We then estimate the allowed region of < η > from the current upper limit of the 0νββ decay rare.
The transition amplitude R 0ν (15) can be written as where O(x, y) is the NME of the hadronic current. The interference terms of R-handed and L-handed current from the neutrino momentum (k) dependent term of the neutrino propagator The Dirac matrices are for the electron spinor. Using the hadronic currents of (11) and (12), we keep the interference terms between the SM hadronic L-handed current and the BSM Lhanded current ηJ µ L and the R-handed current λJ µ R , while the κJ µ R term can be neglected. Using J µ L/R = V µ ± A µ , we keep only interference term between the vector and the axial vector current in O(x, y), Here ∓ is for < λ > and < η > term, respectively. We keep the contribution of the spatial component of the axial vector current, i. e. Gamow-Teller operator, as Here ± is for < λ > and < η > term, respectively. The V − A interference terms remains only for the < η > mechanism. The interference terms between the time component of the vector current(V 0 ) and the space component of the axial vector current(A) including s-wave and p-wave electron contribute to the χ P term. The spatial components of the vector current(V ) and the axial vector current(A) with the s-wave electron contribute to the χ R term.
We examine further the V − A term. Either emitted electrons or nuclear currents have to compensate the p-wave nature of neutrino propagator. The main contribution comes from the s-wave electron and momentum dependent nuclear magnetization current. O(x, y), which become scalar effective nuclear operator for the 0 + − 0 + transition, is given as Here the magnetization current of the vector current is expressed as V (x) = ∇ × µ(x). In the non-relativistic and impulse approximation of nuclear current, µ(x) and A(x) are given by using the same spin-isospin flip operator ∼ τ + σ as [40], Here g A (0) = 1.27, g V (0)+g M (0) = 4.706 and M is mass of nucleon. Within this approximation, the < λ > term vanishes and only the < η > term remains. Therefore, the 0νββ decay for 0 + −0 + transition can be sensitive probe for the < η > mechanism of R-handed current. Assuming the s-wave electron wave function can be approximated by constant, the amplitude R 0ν is given by the NME of two-body operator M ij as with The neutrino-exchange two-body operator M ij , whose spin and momentum structure is similar to the ρ meson exchange nucleon-nucleon potential. After k integration, effective two-body nuclear operator consists of spin-spin and tensor interaction. It is noticed that even though the light neutrino exchange mechanism, the operator becomes contact two-nucleon interaction by approximating ω ∼ k and A i ∼ 0. The short distance nature of the operator is well recognized and has been examined in detail. It is essential to include both the finite size of weak nucleon current for the effective nuclear operator and the short range correlation for the nuclear wave function. The form factors of axial vector and weak magnetism of nucleon are usually parameterized in a dipole form as g A (k 2 ) = g A /(1− k 2 /1.14GeV 2 ) 2 and g V ( Improved form factors and their uncertainties are analyzed from the analyses of electron and neutrino scattering data on proton and deuteron [41,42]. The short range correlation, which is not taken into account in the model nuclear wave function, is taken into account by introducing short range correlation (SRC) function [43,44,45,46] F (r) = 1 − ce −ar 2 (1 − br 2 ) (60) 0 0.05 0.1 a Figure 3: Allowed region of < η > and < m ν > for 136 Xe. a,b,c are evaluated using C's of Refs. [48], [49] and [50] (model without p-n pairing), respectively.
with r = |r j − r i |. This correlation function vanishes at lim r = 0 when c = 1. The suppression rates relative to those without the SRC are not affirmative, depending on nuclear models and various SRCs, from 5% to 30 − 40%. See the most recent result, Fig.10 of [16].
The decay rate of a single nuclear species is not possible to reveal the mechanism of 0νββ decay including BSM physics. However, difference between the space and spin structure of the the effective nuclear operators for < m ν > and < η > mechanisms has a potential to generate A-dependence of the decay rate. A-dependence of the decay rate can be quantified by the normalized decay rate with respect to the reference 0νββ process for two extreme cases < m ν > alone (α = m ν ) and < η > alone (α = η) as and deviation of the ratio R A = R η A /R mν A from one indicates an ability of the process to find BSM physics. We estimated R A using C's from the Tables of [14]. In  Table 1: Ratio of decay rate R α A evaluated using C's of [50].
R A is appreciably small for 48 Ca, 82 Se, 96 Zr, 116 Cd. The result suggests that A-dependence of the decay rate can be a key to disentangle the mechanism of 0νββ decay and to discover the BSM signal of R-handed current. A precise A-dependence of R A may not yet established theoretically and it is expected to narrow down model dependence of NME, especially for interaction range comparable with the nucleon size.
As for the heavy neutrino exchange mechanism, there may be some enhancements [19,26,51] due to ππee vertex from the effective operator, which generate pion range interaction, We can obtain the NME due to this term using the master formula by [18,19] Here the notations and definitions are due to Appendix A.2 of [19]. Thus, this term enhances < λ > mechanism. However it may not change our order estimations because the original C ηη is much larger than C ij with i, j = m, λ by several orders.

Conclusion
We have studied 0νββ decay in the presence of R-handed current. We have tried to clarify the arguments of hadronic side and lepton's BSM physical one. As is well known, if neutrino masses obey IH, < m ν > mechanism works around 50 meV already marginal to the present and near future experiments. From NME, < λ > mechanism is suppressed and < η > mechanism can dominate the decay rate even around the present or near future experimental limits. This is the case if the neutrino masses belong to NH which is much more probable than IH from theoretical reasons. Even if we get the non-null result in 0νββ decay in a single species, though it is the great achievement, we can not limit BSM physics. It is very important to survey this process in different nuclei to specify BSM physics.