The $\lambda$ mechanism of the $0\nu\beta\beta$-decay

The $\lambda$ mechanism ($W_L$-$W_R$ exchange) of the neutrinoless double beta decay ($0\nu\beta\beta$-decay), which has origin in left-right symmetric model with right-handed gauge boson at TeV scale, is investigated. The revisited formalism of the $0\nu\beta\beta$-decay, which includes higher order terms of nucleon current, is exploited. The corresponding nuclear matrix elements are calculated within quasiparticle random phase approximation with partial restoration of the isospin symmetry for nuclei of experimental interest. A possibility to distinguish between the conventional light neutrino mass ($W_L$-$W_L$ exchange) and $\lambda$ mechanisms by observation of the $0\nu\beta\beta$-decay in several nuclei is discussed. A qualitative comparison of effective lepton number violating couplings associated with these two mechanisms is performed. By making viable assumption about the seesaw type mixing of light and heavy neutrinos with the value of Dirac mass $m_D$ within the range $1~ \textrm{MeV}<m_D<1~ \textrm{GeV}$, it is concluded that there is a dominance of the conventional light neutrino mass mechanism in the decay rate.


I. INTRODUCTION
The Majorana nature of neutrinos, as favored by many theoretical models, is a key for understanding of tiny neutrino masses observed in neutrino oscillation experiments. A golden process for answering this open question of particle physics is the neutrinoless double beta decay (0νββ-decay) [1][2][3], in which an atomic nucleus with Z protons decays to another one with two more protons and the same mass number A, by emitting two electrons and nothing else. The observation of this process, which violates total lepton number conservation and is forbidden in the Standard Model, guaranties that neutrinos are Majorana particles, i.e., their own antiparticles [4]. The searches for the 0νββ-decay have not yielded any evidence for Majorana neutrinos yet. This could be because neutrinos are Dirac particles, i.e. not their own antiparticles. In this case we will never observe the decay. However, it is assumed that the reason for it is not sufficient sensitivity of previous and current 0νββ-decay experiments to the occurrence of this rare process.
Due to the evidence for neutrino oscillations and therefore for 3 neutrino mixing and masses the 0νββ-decay mechanism of primary interest is the exchange of 3 light Majorana neutrinos interacting through the left-handed V-A weak currents (m ββ mechanism). In this case, the inverse 0νββ-decay half-life is given by [1][2][3] T 0ν where G 01 , g A and M ν represent an exactly calculable phase space factor, the axial-vector coupling constant and the nuclear matrix element (whose calculation represents a severe challenge for nuclear theorists), respec-tively. m e is the mass of an electron. The effective neutrino mass, is a linear combination of the three neutrino masses m i , weighted with the square of the elements U ei of the first row of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix. The measured value of m ββ would be a source of important information about the neutrino mass spectrum (normal or inverted spectrum), absolute neutrino mass scale and the CP violation in the neutrino sector. However, that is not the only possibility. There are several different theoretical frameworks that provide various 0νββ-decay mechanisms, which generate masses of light Majorana neutrinos and violate the total lepton number conservation. One of those theories is the left-right symmetric model (LRSM) [5,6], in which corresponding to the left-handed neutrino, there is a parity symmetric right-handed neutrino. The parity between left and right is restored at high energies and neutrinos acquire mass through the see-saw mechanism, what requires presence of additional heavy neutrinos. In general one cannot predict the scale where the left-right symmetry is realized, which might be as low as a few TeVaccessible at Large Hadron Collider, or as large as GUT scale of 10 15 GeV.
The LRSM, one of the most elegant theories beyond the Standard Model, offers a number of new physics contributions to 0νββ-decay, either from right-handed neutrinos or Higgs triplets. The main question is whether these additional 0νββ-mechanisms can compete with the m ββ mechanism and affect the 0νββ-decay rate significantly. This issue is a subject of intense theoretical investigation within the TeV-scale left-right symmetry theories [7][8][9][10][11]. In analysis of heavy neutrino mass mechanisms of the 0νββ-decay an important role plays a study of related lepton number and lepton flavor violation processes in experiments at Large Hadron Collider [2,[12][13][14][15][16].
The goal of this article is to discuss in details the W L − W R exchange mechanism of the 0νββ-decay mediated by light neutrinos (λ mechanism) and its coexistence with the standard m ββ mechanism. For that purpose the corresponding nuclear matrix elements (NMEs) will be calculated within the quasiparticle random phase approximation with a partial restoration of the isospin symmetry [17] by taking the advantage of improved formalism for this mechanism of the 0νββ-decay of Ref. [18]. A possibility to distinguish m ββ and λ mechanisms in the case of observation of the 0νββ-decay on several isotopes will be analyzed. Further, the dominance of any of these two mechanisms in the 0νββ-decay rate will be studied within seesaw model with right-handed gauge boson at TeV scale. We note that a similar analysis was performed by exploiting a simplified 0νββ-decay rate formula and different viable particle physics scenarios in Refs. [7][8][9][10][11].

II. DECAY RATE FOR THE NEUTRINOLESS DOUBLE-BETA DECAY
Recently, the 0νββ-decay with the inclusion of righthanded leptonic and hadronic currents has been revisited by considering exact Dirac wave function with finite nuclear size and electron screening of emitted electrons and the induced pseudoscalar term of hadron current, resulting in additional nuclear matrix elements [18]. In this section we present the main elements of the revisited formalism of the λ mechanism of the 0νββ-decay briefly. Unlike in [18] the effect the weak-magnetism term of the hadron current on leading NMEs is taken into account.
If the mixing between left and right vector bosons is neglected, for the effective weak interaction hamiltonian density generated within the LRSM we obtain Here, G β = G F cos θ C , where G F and θ C are Fermi constant and Cabbibo angle, respectively. The coupling constant λ is defined as Here, M WL and M WR are masses of the Standard Model left-handed W L and right-handed W R gauge bosons, respectively. The left-and right-handed leptonic currents are given by The weak eigenstate electron neutrinos ν eL and ν eR are superpositions of the light and heavy mass eigenstate Majorana neutrinos ν j and N j , respectively. We have Here, U, S, T, and V are the 3 × 3 block matrices in flavor space, which constitute a generalization of the Pontecorvo-Maki-Nakagawa-Sakata matrix, namely the 6 × 6 unitary neutrino mixing matrix [19] The nuclear currents are, in the non-relativistic approximation, [20] Here, m N is the nucleon mass.
, q M ≡ q M (q 2 ) and q P ≡ q P (q 2 ) are, respectively, the vector, axial-vector, weak-magnetism and induced pseudoscalar form-factors. The nucleon recoil terms are given by where q n = p n − p ′ n is the momentum transfer between the nucleons. The initial neutron (final proton) possesses energy E ′ n (E n ) and momentum p ′ n (p n ). r n , τ + n and σ n , which act on the n-th nucleon, are the position operator, the isospin raising operator and the Pauli matrix, respectively.
By assuming standard approximations [18] for the 0νββ-decay half-life we get The effective lepton number violating parameters η ν (W L − W L exchange), η λ (W L − W R exchange) and their relative phase Ψ are given by The coefficients C I (I = mm, mλ and λλ) are linear combinations of products of nuclear matrix elements and phase-space factors: The explicit form and calculated values of phase-space factors G 0i (i=1, 2, 3, 4, 10 and 11) of the 0νββ-decaying nuclei of experimental interest are given in [18]. The NMES, which constitute the coefficients C I in Eq. (13), are defined as follows: The partial nuclear matrix elements M I , where I=GT, F, T, ωF, ωGT, ωT, qF, qGT, and qT are given by Here, O F,GT,T are the Fermi, Gamow-Teller and tensor operators 1, σ 1 · σ 2 and 3( σ 1 ·r 12 )( σ 2 ·r 12 ). The twonucleon exchange potentials h I (r) with =F, GT, T, ωF, ωGT, ωT, qF, qGT, and qT can be written as where and Here, E i , E f andĒ n are energies of the initial and final nucleus and averaged energy of intermediate nuclear states, respectively. r = (r r − r s ), r r,s is the coordinate of decaying nucleon and where q is the momentum exchange. The form factors g V (q 2 ), g A (q 2 ), g M (q 2 ) and g P (q 2 ) are defined in [21]. We note that factor 4 in definition of the two-nucleon exchange potentials h I (r) with I=ωF, ωGT, and ωT in Eq. (48) of Ref. [18] needs to be replaced by factor 2.  (11)) on partial contributions C 0k I associated with phase-space factors G 0k (k=2, 10 and 11) for nuclei of experimental interest. The partial contributions are identified by index k. The contributions from largest to the smallest are displayed in red, blue and black colors, respectively.

III. RESULTS AND DISCUSSION
The nuclear matrix elements are calculated in protonneutron quasiparticle random phase approximation with partial restoration of the isospin symmetry for 48 Ca, TABLE I. The nuclear matrix elements of the 0νββ-decay associated with m ββ and λ mechanisms and the coefficients Cmm, C mλ and C λλ (in 10 −14 yrs −1 ) of the decay rate formula (see Eq. (11)). The nuclear matrix elements are calculated within the quasiparticle random phase approximation with partial restoration of the isospin symmetry. The G-matrix elements of a realistic Argonne V18 nucleon-nucleon potential are considered [17]. The phase-space factors are taken from [18]. f λm = C λλ /Cmm, f G λm = G02/G01 and gA = 1.269 is assumed. Q ββ is the Q-value of the double beta decay in MeV. 48  Xe, which are of experimental interest. In the calculation the same set of nuclear structure parameters is used as in [17]. The pairing and residual interactions as well as the two-nucleon short-range correlations derived from the realistic nucleon-nucleon Argonne V18 potential are con-sidered [23]. The closure approximation for intermediate nuclear states is assumed with (Ē n − (E i + E f )/2) = 8 MeV. The free nucleon value of axial-vector coupling constant (g A = 1.25 − 1.27) is considered.
In Table I  those given in [17], which were obtained without consideration of the closure approximation. By glancing Table  I Table  I it follows that there is a significant difference between results of this work and the QRPA NMEs of Ref. [22], especially in the case of 100 Mo. This difference can be attributed to the progress achieved in the 0νββ-decay formalism due to inclusion of higher order terms of nucleon currents [18,21], the way of adjusting the parameters of nuclear Hamiltonian [24], description of short-range correlations [23] and restoration of the isospin symmetry [17]. Nuclear matrix elements M 2 − , M 1 + (λ mechanism) and M ν (m ββ mechanism) for 10 nuclei under consideration are given in Table I and displayed in Fig. 2. We note a rather good agreement between M 2 − and M ν for all calculated nuclear systems. It is because the contribution of M 1 + to M 2 − is suppressed by factor 9 and as a result M 2 − is governed by the M νω contribution (see Eq. (14)). Values of M 1 + exhibit similar systematic behavior in respect to considered nuclei as values of M ν and M 2 − , but they are suppressed by about factor 2-3 (with exception of 48 Ca).
The importance of the m ββ and λ mechanisms depends, respectively, not only on values of η ν and η λ parameters, which are unknown, but also on values of coefficients C I (I=mm, mλ, λλ), which are listed for all studied nuclei in Table I. They have been obtained by using improved values of phase-space factors G 0k (k=1, 2, 10 and 11) from [18]. We note that the squared value of M GT and fourth power of axial-vector coupling constant g A are included in the definition of coefficient C I unlike  Table I. in [18]. We see that C λλ is always larger when compared with C mm . The absolute value of C mλ is significantly smaller than C mm and C λλ . This fact points out on less important contribution to the 0νββ-decay rate from the interference of m ββ and λ mechanisms. For 10 nuclei of experimental interest the decomposition of coefficient C λλ (see Eq. (11)) on partial contributions C 0k I associated with phase-space factors G 0k (k=2, 10 and 11) is shown in Fig. 1. By glancing the plotted ratio C 0k I /C I we see that C λλ is dominated by a single contribution associated with the phase-space factor G 02 . From this and above analysis it follows that 0νββ-decay half-life to a good accuracy can be written as For a given isotope the factor f λm reflects relative sensitivity to the m ββ and λ mechanisms and f G λm is its approximation, which does not depend on NMEs. The values f λm and f G λm are tabulated in Table I and plotted as function of Q ββ in Fig. 3. We see that f λm depends only weakly on involved nuclear matrix elements (apart for the case of 48 Ca) what follows from a comparison of f λm with f G λm . The value of f λm is mainly determined by the Q-value of double beta decay process. From 10 analyzed nuclei the largest value of f mλ is found for 48 Ca and the smallest value for 76 Ge. A larger value of f λm means increased sensitivity to m ββ mechanism in comparison to λ mechanism and vice versa. and parameter η λ associated with right-handed currents mechanism imposed by current constraints on the 0νββ-decay half-life for nuclei of experimental interest. The calculation is performed with NMEs obtained within the QRPA with partial restoration of the isospin symmetry (see Table I). The upper limits on m ββ and η λ are deduced for a coexistence of the m ββ and λ mechanisms (Maximum) and for the case η λ = 0 or ην = 0 (On axis). gA = 1.269 and CP conservation (ψ = 0) are assumed.  Upper bounds on the effective neutrino mass m ββ and right-handed current coupling strength η λ are deduced from experimental half-lives of the 0νββ-decay by using the coefficients C mm , C mλ and C λλ of Table I. The maximum and the value on axis (m ββ = 0 or η λ = 0) are listed in Table II. The decays of 136 Xe and 76 Ge set the sharpest limit m ββ ≤ 0.13 eV and 0.18 eV, and η λ ≤ 1.7 10 −7 and 3.1 10 −7 , respectively. These are more stringent than those deduced from other experimental sources.
It is well known that by measuring different characteristics, namely energy and angular distributions of two emitted electrons, it is possible to identify which of m ββ and λ mechanisms is responsible for 0νββ-decay [18,20]. It might be achieved only by some of future 0νββ-decay experiments, e.g. the SuperNEMO [33] or NEXT [34]. A relevant question is whether the underlying m ββ or λ mechanism can be revealed by observation of the 0νββdecay in a series of different isotopes. In Fig. 4 this issue is addressed by an illustrative case of observation of the 0νββ-decay of 136 Xe with half-life T 0ν 1/2 = 6.86 10 26 yrs, which can be associated with m ββ = 50 meV or η λ = 9.8 10 −8 . The 0νββ-decay half-life predictions associated with a dominance of m ββ and λ mechanisms exhibit significant difference for some nuclear systems. We see that by observing, e.g., the 0νββ-decay of 100 Ge and 100 Mo with sufficient accuracy and having calculated relevant NMEs with uncertainty below 30%, it might be possible to conclude, whether the 0νββ-decay is due to m ββ or λ mechanism.
Currently, the uncertainty in calculated 0νββ-decay NMEs can be estimated up to factor of 2 or 3 depending on the considered isotope as it follows from a comparison of results of different nuclear structure approaches [3]. The improvement of the calculation of double beta decay NMEs is a very important and challenging problem. There is a hope that due to a recent progress in nuclear structure theory (e.g., ab initio methods) and increas- as follows [19] Here, 0 and 1 are the 3 × 3 zero and identity matrices, respectively. The parametrization of matrices A, B, R and S and corresponding orthogonality relations are given in [19]. If A = 1, B =1, R = 0 and S = 0, there would be a separate mixing of light and heavy neutrinos, which would participate only in left and right-handed currents, respectively. In this case we get η λ = 0, i.e., the λ mechanism is forbidden.
If masses of heavy neutrinos are above the TeV scale, the mixing angles responsible for mixing of light and heavy neutrinos are small. By neglecting the mixing between different generations of light and heavy neutrinos, the unitary mixing matrix U takes the form Here, m D represents energy scale of charged leptons and m LN V is the total lepton number violating scale, which corresponds to masses of heavy neutrinos. We see that U = U 0 can be identified to a good approximation with the PMNS matrix and V 0 is its analogue for heavy neutrino sector. Due to unitarity condition we find V 0 = U † 0 . Within this scenario of neutrino mixing the effective lepton number violating parameters η ν (m ββ mechanism) and η λ (λ mechanism) are given by with The importance of m ββ or λ-mechanism can be judged from the ratio It is naturally to assume that ζ m ≈ 1 and to consider the upper bound for the factor ζ λ , i.e., there is no anomaly cancellation among terms, which constitute these factors. Within this approximation η λ /η ν does not depend on scale of the lepton number violation m LN V and is plotted in Fig. 5. The Dirac mass m D is assumed to be within the range 1 MeV < m D < 1 GeV. The flavor and CP-violating processes of kaons and B-mesons make it possible to deduce lower bound on the mass of the heavy vector boson M W2 > 2.9 TeV [9]. From Fig. 5 it follows that within accepted assumptions the λ mechanism is practically excluded as the dominant mechanism of the 0νββ-decay.
In this section the light-heavy neutrino mixing of the strength m D /m LN V is considered. However, we note that there are models with heavy neutrinos mixings where strength of the mixing decouples from neutrino masses [35,36,[38][39][40]. This subject goes beyond the scope of this paper.

V. SUMMARY AND CONCLUSIONS
The left-right symmetric model of weak interaction is an attractive extension of the Standard Model, which may manifest itself in the TeV scale. In such case the Large Hadron Collider can determine the right-handed neutrino mixings and heavy neutrino masses of the seesaw model. The LRSM predicts new physics contributions to the 0νββ half-life due to exchange of light and heavy neutrinos, which can be sizable.
In this work the attention was paid to the λ mechanism of the 0νββ-decay, which involves left-right neutrino mixing through mediation of light neutrinos. The recently improved formalism of the 0νββ-decay concerning this mechanism was considered. For 10 nuclei of experimental interest NMEs were calculated within the QRPA with a partial restoration of the isospin symmetry. It was found that matrix elements governing the conventional m ββ and λ mechanisms are comparable and that the λ contribution to the decay rate can be associated with a single phase-space factor. A simplified formula for the 0νββ-decay half-life is presented (see Eq. 19)), which neglects the suppressed contribution from the interference of both mechanisms. In this expression the λ contribution to decay rate is weighted by the factor f λm , which reflects relative sensitivity to the m ββ and λ mechanisms for a given isotope and depends only weakly on nuclear physics input. It is manifested that measurements of 0νββ-decay half-life on multiple isotopes with largest deviation in the factor f λm might allow to distinguish both considered mechanisms, if involved NMEs are known with sufficient accuracy.
Further, upper bounds on effective lepton number violating parameters m ββ (η ν ) and η λ were deduced from current lower limits on experimental half-lives of the 0νββ-decay. The ratio η λ /η ν was studied as function of the mass of heavy vector boson M WR assuming that there is no mixing among different generations of light and heavy neutrinos. It was found that if the value of Dirac mass m D is within the range 1 MeV < m D < 1 GeV, the current constraint on M WR excludes the dominance of the λ mechanism in the 0νββ-decay rate for the assumed neutrino mixing scenario.